Genetic improvement of Small Millets [1st ed. 2024] 9819972310, 9789819972319

Table of contents :
Contents
Study on the Dynamic Performance of X-shaped Vibration Isolator with Friction Damping Based on Incremental Harmonic Balance Method
1 Introduction
2 Mathematical Modeling
2.1 Constitutive Equation of Plate-Type Friction Damping
2.2 Modeling of X-shape Vibration Isolator with Plate-Type Friction Damping Element
3 Frequency Response and Displacement Transmissibility
3.1 IHBM Procedure
3.2 Frequency Response
3.3 Displacement Transmissibility
4 Conclusions
Appendix I
Appendix II
Appendix III
Appendix IV
References
Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating for Rotating Engine in Experiment
1 Introduction
2 Experimental Design
3 Results and Discussion
3.1 Results of Rotor Vibration
3.2 Results of Stator Vibration
4 Conclusions
References
Semi-analytical Expression of Force and Stiffness of Perpendicular Polarized Ring Magnets for Nonlinear Dynamic Analysis
1 Introduction
2 Semi-analytical Calculations of Force and Stiffness
2.1 Notion and Geometry
2.2 Semi-analytical Expression of Magnetic Force
2.3 Semi-analytical Expression of Magnetic Stiffness
3 Simulation Verification
3.1 Size Parameters
3.2 Magnetic Force
3.3 Magnetic Stiffness
4 Parameters Analysis
4.1 Air Gap
4.2 Width of Axial Magnetized Magnet
4.3 Height of Axial Magnetized Magnet
5 Conclusion
References
On-Orbit Reconfiguration Dynamics and Control of Heterogeneous Intelligent Spacecraft
1 Introduction
2 Problem Formulation
2.1 On-Orbit Reconfiguration Mission
2.2 System Dynamics
3 Controller Design
3.1 Separation Phase
3.2 Unit Reconfiguration Phase
3.3 Reassembly Phase
4 Numerical Simulations
5 Conclusions
References
Study on the Effect of Angular Misalignment on the Contact Load and Stiffness of Cylindrical Roller Bearings
1 Introduction
2 Quasi-Static Model of CRB
2.1 Calculation of Contact Load Between Roller and Raceway
2.2 Calculation of Stiffness
2.3 Equilibrium Equation of CRB
3 The Effect of Angular Misalignment
4 Model Validation
5 Results and Discussion
6 Conclusion
References
Dynamic Modeling and Features of GTF Engine Rotor System
1 Introduction
2 Rotor Dynamics Model of GTF Engine
2.1 Low-Pressure Rotor Units
2.2 Membrane Disk Coupling
2.3 Helical Gear Meshing Unit
2.4 GTF Engine Rotor System Motion Equation
3 Simulation Analysis of GTF Engine Rotor System
3.1 Node Division
3.2 Modal Analysis
3.3 Unbalance Responses
4 Conclusions and Discussions
References
Nonlinear Dynamic Analysis of Rub-Impact Rod-Fastening Combined Rotor Systems with Internal Damping
1 Introduction
2 Modeling of RFCR System
2.1 Oil-Film Forces Model
2.2 Rub-Impact Force Model
2.3 Contact Model between Rotor Discs
2.4 System Governing Equations of RFCR
3 Model Validation
4 Results of the Numerical Simulation
4.1 Bifurcation and Nonlinear Response Analysis
4.2 Effect of Internal Damping Force
4.3 Effect of Friction Coefficients
5 Conclusion
References
A Multiscale Fracture Model to Reveal the Toughening Mechanism in the Bioinspired Bouligand Structure
1 Introduction
2 The Multiscale Fracture Model in Bioinspired Bouligand Structures
2.1 Material Descriptions
2.2 Constitutive Relation
2.3 The Fracture Model of the Twisted Crack
2.4 The Definition of the Energy Release Rate
3 Results and Discussions
4 Conclusions
A Appendix
A.1 The Stiffness Constants of the Lamina
A.2 The Transforming Matrix of the Tensor
A.3 -8pt The Macro-constitutive Relation in the Global Coordinate System
References
Decoupled Multi-mode Controllable Electrically Interconnected Suspension for Improved Vehicle Damping Performance
1 Introduction
2 The Model of Multi-mode Controllable EIS System
2.1 Half-Car Model
2.2 Circuit of Multi-mode Controllable EIS
3 Controller Designer for Multi-mode Controllable EIS
3.1 H∞ Controller Design
3.2 Multi-mode Control Logic
4 Numerical Simulation for Multi-mode Controllable EIS
4.1 Sinusoidal Excitation
4.2 Bump Excitation
4.3 Random Excitation
5 Conclusion
References
Adaptive Robust Sliding-Mode Control of a Semi-active Seat Suspension Featuring a Variable Inertance-Variable Damping Device
1 Introduction
2 VIVD Seat Suspension
2.1 VIVD Seat Suspension Model
2.2 VIVD Device Prototype
2.3 Mechanical Admittance Analysis
2.4 Parameter Identification
3 Controller Design
3.1 Adaptive Robust Sliding-Mode Controller
3.2 Force Tracking Controller
4 Experimental System and Results Analysis
4.1 Experimental Setup
4.2 Vibration Control Test
5 Conclusion
References
An Adaptive Controller for Payload Swing Suppression of Ship-Mounted Boom Cranes
1 Introduction
2 Ship-Mounted Boom Crane Model
2.1 Dynamics for Ship-Mounted Cranes
2.2 Math Model Transformation
3 Controller Design and Stability Analysis
3.1 Controller Design
3.2 Stability Analysis
4 Numerical Simulations
4.1 Simulation 1
4.2 Simulation 2
5 Conclusion
References
Study on Dynamic Modeling and Vibration Noise Suppression Method of AUV
1 Introduction
2 FE Model of AVU
3 Working Principle and Parameter Design of RC
3.1 Principle Derivation of RC
3.2 Simplification and Parameter Design of RC
4 Result and Discussion
4.1 Dynamic Response Analysis of AUV
4.2 Response Analysis of AUV with RC
5 Conclusions
References
Simultaneous Vibration Absorbing and Energy Harvesting Mechanism of the Tri-Magnet Bistable Levitation Structure: Modeling and Simulation
1 Introduction
2 Mathematical Modelling
2.1 Equivalent Modeling of the Cantilever Beam with BEVA
2.2 Design of the Symmetric Bistable Potential Wells
2.3 Calculation of Electromagnetic Damping
3 Results and Discussions
3.1 Parameter identification and verification
4 Conclusions
References
Design and Research of Triboelectric Energy Harvester for Low Frequency Nonlinear Vibration
1 Introduction
2 Prototype Design
2.1 Quasi-Static Analysis
3 Structural Dynamics Model
3.1 Approximate Analytical Solution
3.2 Non-dimensionalization
3.3 System Response Near Resonance
4 Electric Model of Triboelectric Energy Harvester
5 Numerical Simulations and Results
6 Conclusions
References
Energy Harvesting Study of Piezoelectric Vibration Harvester with Double Parallel Slender Structure
1 Modeling of Piezoelectric Energy Harvester with Cantilevered Double Straight Beams
1.1 Mechanical Equilibrium Control Equations with Electrically Coupled Effects
1.2 Control Equations for Circuits with Mechanical Coupling Effects
2 Green's Function Solutions for Piezoelectric Vibrations of Double Straight Beams
2.1 Vibration and Piezoelectric Equations Solving
2.2 Decoupling of Electromechanical Eulerian Double Beams Systems
3 Eddy-Current Induced Vibration of Piezoelectric Energy Harvester
4 Numerical Analysis
4.1 Effect of External Excitation Distribution on Voltage
4.2 Influence of the Elasticity Coefficient of the Middle Layer on the Voltage of a Piezoelectric Energy Harvester
5 Conclusions
Appendix A
References
A Broadband Energy Harvester with Three-to-One Internal Resonance
1 Introduction
2 Mathematical Model
3 Dynamical Analysis
4 Comparison of the Proposed Scheme and the Traditional Cases
5 Numerical Validations
6 Conclusions
References
Optimization Design of High-Pressure Simulated Rotor
1 Introduction
2 Modeling of High-Pressure Simulated Rotor
3 Optimization of the Simulated Rotor System
3.1 Damping Characteristic Calculation Mode
3.2 Calculation Model for Squirrel Cage
3.3 Newton Method
3.4 Dynamic Analysis of the Original Rotor System
3.5 The Impact of Axial Width of SFD on Damping Characteristics
3.6 Optimization of the Simulated Rotor System
4 Conclusion
References
Energy Transfer of Particle Impact Damper Systems
1 Introduction
2 The Integrated System and Computational Method
3 Analysis on Energy Transfer and Granular Motions
4 Conclusions
References
An Empirical Control Research on the Lexical Approach to Business Correspondence Writing in Vocational Colleges
1 Instruction
2 Research Methods
2.1 Research Questions
2.2 Participants
2.3 Instruments
2.4 Statistic Methods
3 Research Analyses
3.1 Statistics and Analyses of Questionnaire Survey
3.2 Analyses on Students' Use of Chunks
3.3 Analyses on SPSS Statistic Data
4 Major Findings and Discussions
4.1 Answers to Research Questions
4.2 Pedagogical Implication
References
Analytical Analysis of Nonlinear Vortex-Induced Vibration of Pipes Conveying Fluid
1 Introduction
2 Modelling and Formulations
3 Method of Multiple Scales
3.1 Primary Resonance of First Mode
3.2 Primary Resonance of Second Mode
4 Results and Discussions
4.1 Primary Resonance of First Two Modes
4.2 Frequency Lock-In Domain
4.3 Effect of the Other Parameters on Structure
5 Conclusions
References
RBF Neural Network for Feature Selection Using Sparsity Method
1 Introduction
2 Structure of RBF for Feature Selection
2.1 Parameter Initialization Using Clustering Algorithm
2.2 Interpretable Integrated Feature Selection Process
3 Simulation Result
3.1 SINC
3.2 Chem
4 Conclusions and Future Works
References
Periodic Motions and Bifurcations of a Spring-Driven Joint System with Periodic Excitation
1 Introduction
2 Methodology
2.1 Modeling
2.2 Mapping and Periodic Motions
3 Bifurcation and System Analysis
4 Conclusions
References
Enhanced Vibration Characteristics of Honeycomb Plates Composed of Metamaterials with NTE
1 Introduction
2 Mechanical Metamaterials Composed of Trapezoid Unit
3 Enhanced Vibration Characteristics
3.1 Vibration Characteristics
3.2 Geometric Parameters’ Effect on Frequencies
4 Comparison
5 Concluding Remarks
References
Vibration Reduction of Limited Series Nonlinear Energy Sink
1 Introduction
2 Mechanical Model
2.1 Limited Amplitude Series NES Coupling System
2.2 Series NES Coupling System
2.3 Evaluation of Vibration Reduction Effect
3 Approximate Analytical Method
4 Vibration Suppression Performance of the Limited Series NES
4.1 Parameter Selection
4.2 Parameter Analysis of Limited Series NES
5 Conclusions
References
Understanding Neural Rhythmic Mechanisms Through Self-oscillations of Complex Neural Networks and Their Adaptation
1 Introduction
2 Neuron Model and Network Topology
3 Rhythm and Synchronization of Scale-Free Networks
4 Neural Network Rhythms with STDP
5 Conclusions and Prospects
References
Multi-attention Based Multi-scale Temporal Convolution Network for Remaining Useful Life Prediction of Rolling Bearings
1 Introduction
2 Theoretical Background
2.1 Temporal Convolutional Network
2.2 Attention Mechanism
3 Proposed Method
3.1 Multi-scale Method
3.2 Channel Attention Mechanism
3.3 Feature Attention Mechanism
4 Verification
4.1 Case Introduction
4.2 Evaluation Metrics
4.3 Experimental Results and Analysis
5 Conclusions
References
Dynamics Analysis of the Cooperative Dual Marine Lifting Systems Subject to Sea Wave Disturbances
1 Introduction
2 Dynamics Model and Analysis
2.1 Kinematic Analysis
2.2 Dynamics analysis
3 Simulation Results and Analysis
4 Conclusions
References
Nonlinear Hierarchical Control for Unmanned Quadrotor Transportation Systems with Saturated Inputs
1 Introduction
2 Dynamics Analysis and Error Definition
3 Controller Development
3.1 Outer Loop Control
3.2 Inner Loop Control
4 Stability and Convergence Analysis
5 Simulation Results
6 Conclusion
References
Predicting the Endless Stop-Band Behaviour of the NS-MRE Isolator
1 Instruction
2 Structure and Analyses of the NS-MRE Isolator
2.1 Structure and Working Principle of the NS-MRE Isolator
2.2 The Simulation of Magnetic Field Distribution by COMSOL
3 The Effect of the Negative Stiffness on the Stop-Band with Infinite Period Structure
4 Transmission Spectrum Analyses with Finite Period Structures
5 Conclusions
References
Human-Mechanical Biomechanical Analysis of a Novel Knee Exoskeleton Robot for Rehabilitation Training
1 Introduction
2 Design of the Exoskeleton
3 Development of Human-Exoskeleton Model
3.1 Human Model
3.2 Exoskeleton Model
3.3 Human-Exoskeleton Interaction Model
4 Results and Analysis
4.1 Kinetic Parameters
4.2 Muscle Parameters
5 Conclusions
References
Adaptive Sliding Mode Control for Active Suspensions of IWMD Electric Vehicles Subject to Time Delay and Cyber Attacks
1 Introduction
2 Problem Formulation
2.1 Suspension Model of IWMD Vehicles
2.2 Adaptive Sliding Mode Controller
3 Main Results
3.1 Stability Analysis of the Sliding Motion
3.2 Reachability Analysis
4 Simulation Results
4.1 Bump Response
4.2 Random Response
5 Conclusion
References
Generative Adversary Network Based on Cross-Modal Transformer for CT to MR Images Transformation
1 Introduction
2 Related Works
3 Methodology
3.1 Generator
3.2 Discriminator
3.3 Loss Function
4 Experiment and Analysis
4.1 Paired Dataset and Experimental Environment
4.2 Result
5 Conclusion
References
A Novel Robust Finite-Time Control for Active Suspension Systems with Naturally Bounded Inputs
1 Introduction
2 Problem Formulation
2.1 System Description
2.2 Control Objects
3 Main Results
4 Simulation Results and Analysis
5 Conclusions
References
Quasi-Zero Stiffness Magnetic Vibration Absorber
1 Introduction
2 Mechanical Model
2.1 Analyzing Magnetic Force
2.2 Negative Stiffness Mechanism
2.3 Dynamic Equation of LO with the Novel NES
3 Analyzing Dynamic Characteristics of the Novel NES
3.1 Analyzing Parameters of NES
3.2 Harmonic Balance Analysis
3.3 Transient Response of LO with the Novel NES
3.4 Steady State Response of LO with the Novel NES
4 Conclusions
References
New Software Bionic Haptic Actuator Design Based on Barometric Array
1 Introduction
2 Structure Design and Production
2.1 Physical Design
2.2 Manufacture Method
3 Tactile Information Acquisition System
3.1 System Work Plan Design
3.2 Tactile Data Processing Method Based on Kalman Filter
4 Experiment and Data Analysis
5 Conclusions
References
Adaptive Robust Tracking Control for Aerial Work Platform Vehicle with Guaranteed Prescribed Performance
1 Introduction
2 Problem Formulation
2.1 Dynamic Model of AWPV
2.2 Desired Trajectory
2.3 Trajectory Tracking with Prescribed Performance
3 Adaptive Robust Tracking Control Design with PTSSP
3.1 Diffeomorphism Approach and State Transformation
3.2 Tracking Control for Nominal Transformed AWPV
3.3 Deal with Initial State Deviation
3.4 Adaptive Robust Design for Transformed AWPV Accounting for Uncertainties
4 Performance Validation
4.1 CFC for Nominal AWPV
4.2 AWPV with Initial State Deviation and Uncertainties
5 Conclusions
References
A Novel Model Predictive Control Strategy for Continuum Robot: Optimization and Application
1 Introduction
2 Dynamic Model
3 Model Predictive Control Method Analysis
3.1 Linearization and Discretization of the System
3.2 Stability Analysis and Optimal Control
4 Simulation Result
5 Conclusion
References
Finite Element Analysis and Error Compensation for Wrinkled Bellow-Like Soft Robotic Manipulator Kinematics Modeling
1 Introduction
2 Design of Wrinkled Bellow-Like Soft Robotic Manipulator Model
3 Kinematic Modeling of WBSRM
3.1 General Unit Kinematics
3.2 Kinematics of WBSRM
3.3 Specific Mapping of Virtual Units
4 Finite Element Analysis of WBSRM
4.1 Morphological Analysis of WBSRM
5 Comparison and Analysis of Kinematics Modeling and Simulation Results
6 Conclusion
References
Suppression of Galloping Oscillations Using Perforated Bluff Bodies
1 Introduction
2 Design and Modelling
2.1 System Design
2.2 Mathematical Modelling
3 Numerical Simulations and Experimental Validation
3.1 Experimental Set-Up
3.2 Dynamic Responses and Galloping Suppression Ratio
3.3 Aerodynamic Transverse Force Coefficient (CFy)
4 Conclusions
References
Motion Planning for Wave-Like-Actuated Manta-Inspired Amphibious Robots
1 Introduction
2 Design of the Manta-Inspired Amphibious Robot
2.1 Analysis of Manta Ray Motion Mode
2.2 Design of Mechanical Structure
3 Establishment of the Manta-Inspired Amphibious Robot’s System Model
4 Motion Planning Algorithms, Simulations and Experiments
4.1 Path Planning
4.2 Velocity Planning
4.3 Experiments
5 Conclusion
References
Time-Optimal Anti-swing Trajectory Planning of Double Pendulum Crane Based on Chebyshev Pseudo-spectrum Method
1 Introduction
2 Problem Statement
3 Trajectory Planning
3.1 Transformation of System Models
3.2 Trajectory Planning Based on Chebyshev Pseudo-spectrum Method
4 Simulation Results
5 Conclusion
References
Nonlinear Inertia and Its Effect Within an X-shaped Mechanism
1 Introduction
2 Modeling of the Novel Nonlinear Inertia Unit Coupling with an X-mechanism
3 Effect of Model Parameters on Nonlinear Characteristics
3.1 Effect of Length of Rods
3.2 Effect of the Mass Ratio η
3.3 Effect of the Angle α
3.4 Effect of the Layers n
3.5 Unique Adjustable Vibration Suppression Properties
3.6 A Summary of Parametric Influence
4 Understanding of Nonlinear Inertia-Related Forces
4.1 Influence on Transmissibility with Different Nonlinear Types
4.2 Influence on Interactive Force Around Equilibrium
5 Prototyping and Experiments
6 Conclusion
References
Metamaterial Beam with Bistable and Monostable-Hardening Attachments for Broad-Band Vibration Attenuation and Energy Harvesting
1 Introduction
2 Proposed Metastructure with Alternately Arranged Bistable and Monostable Hardening Oscillators
3 Effect of Load Resistance and Electromechanical Coupling
4 Conclusion
References
An Improved Incremental Classifier and Representation Learning Method for Elderly Escort Robots
1 Introduction
2 Incremental Classifier and Representation Learning Method
3 The Improved iCaRL Method
3.1 Overview of the Proposed Method
3.2 Density-Peaks-Based Prototype Selection
3.3 Memory Network for Replay
4 Experimental Results
5 Conclusion
References
Exact Dynamic Analysis of Viscoelastic Double-Beam System Using Dynamic Stiffness Method
1 Introduction
2 Motion Equation and General Solution of the Double-Beam System
2.1 Governing Differential Equation of the Double-Beam System
2.2 Solution of the Differential Equations of Vibration
2.3 Vibration Function Characterized by the Dynamic Boundary Displacements
3 Dynamic Stiffness Matrix
4 Analysis of the Free-Vibration Characteristics
4.1 Solution of Modal Frequencies
4.2 Solution of the Vibration Modes
5 Dynamic Response Analysis
5.1 Solution of the Dynamic Response
6 Numerical Examples
7 Conclusion
Appendix:
References
Nonlinear Control Strategy for Tower Cranes with Variable Cable Lengths and Multivariable State Constraints
1 Introduction
2 Problem Formulation
2.1 Tower Crane System Dynamics
2.2 Control Objective
3 Control System Design
4 Simulink Results
4.1 Simulink Group1
4.2 Simulink Group2
5 Conclusion
References
Design and Dynamic Analysis of a Flexible Inertia Device for Vehicle Suspensions
1 Introduction
2 Inertia Device and System Model
2.1 Configuration and Working Principle
2.2 System Model
3 Suspension with the Inertia Device Vibration Control
3.1 Equivalent Change of Inertia
3.2 1/4 Suspension with the FI Device Model
3.3 Key Inertia Parametric Analysis
4 Simulation Study
5 Conclusion
References
A Quasi-Zero Stiffness Nonlinear Absorber Based on Centrifugal Force
1 Introduction
2 Mathematical Model
2.1 Structural Model of the Rotor System with a QZS Nonlinear Absorber
2.2 Structural Model of the Rotor System with a QZS Nonlinear Absorber
3 Vibration Suppression Analysis of the Rotor System
4 Conclusion
References
A Cellular Strategy for Eliminating the Failure of Nonlinear Energy Sinks Under Strong Excitation
1 Introduction
2 Dynamics Equations and Approximate Analytical Solutions
3 Effects of Cellular Strategy
3.1 Effect of Cellular Strategy on SN Bifurcation
3.2 Effect of Cellular Strategy on Damping Efficiency
4 Conclusion
References
A Stable Adjustable Nonlinear Energy Sink
1 Introduction
2 Mechanical Model
3 Static Equilibrium Position Bifurcation
4 Dynamics and Vibration Reduction Performance Analysis
4.1 Approximate Analytical Solutions
4.2 Vibration Reduction Analysis
5 Experimental Verification
5.1 Experimental Device
5.2 Experimental Results
6 Conclusion
References
Gait-Planning-Based Path Planning for Crocodile-Inspired Pneumatic Soft Robots
1 Introduction
2 Design of Robots and Gaits
2.1 Robots Introduction
2.2 Gaits Analysis of Crocodile-Inspired Pneumatic Soft Robots
2.3 Gait-Planning-Based Path Planning
2.4 Experimental Results
3 Conclusion
References
Research on Nonlinear Energy Sink Vibration Reduction of Floating Raft System
1 Introduction
2 Mechanical Model and Modal Analysis of the Floating Raft System
3 NESs in the Floating Raft System
4 Comparison of Two NES Distribution Modes
4.1 (Nes-M-1)
4.2 (Nes-M-2)
5 Conclusions
References
X-mechanism Guided Elastic QZS Vibration Isolator Design for Beneficial Nonlinear Stiffness
1 Introduction
2 Design Principle
3 Loading Simulations
4 Experimental Loading Performance
5 Conclusion
References
Design and Vibration Control of Secondary Suspension for Maglev Train Based on Magnetorheological Fluid Damper
1 Introduction
2 Structure of the Secondary Suspension of High-Speed Maglev Train
3 Vibration Characteristics Analysis of Secondary Suspension
3.1 Analysis of Vibration Transmissibility
3.2 Selection of Working Mode and Structural Form of MR Damper
4 Bouc-Wen Model of Magnetorheological Damper
5 Fuzzy Control Algorithm of Magnetorheological Damper
5.1 Analysis of Fuzzy Control Algorithm Simulation Results
5.2 Comparison of Simulation Results
6 Conclusion
References
Nonlinear Dynamic Analysis of Flywheel Rotor Systems with Multiple Fit Clearances
1 Introduction
2 Theoretical Modeling and Analysis
2.1 Nonlinear Support Stiffness of Rolling Bearings
2.2 Transversal Stiffness of Clearance Fit
2.3 Dynamic Model of the Rotor-Bearing-Pedestal System
3 Analysis and Discussions
3.1 Effect of Unbalanced Mass
3.2 Effect of Fit Clearances
4 Conclusion
References
Comparison Studies of Dynamic Characteristics for Coupled Bearing-Rotor Systems with Fixed and Pivot-Supported Pads
1 Introduction
2 Mixed-Lubrication Dynamic Modeling of Coupled Bearing-Rotor Systems
2.1 Mixed-Lubrication Model
2.2 Dynamic Model
2.3 Calculation Procedure
3 Results and Discussions
4 Conclusions
References
Nonlinear Dynamic Performance and Analysis Model of Pump Valve System of Diaphragm Pump Hydraulic End
1 Introduction
2 Cone Valve Nonlinear Motion Model
2.1 Simplified Model of the Cone Valve
2.2 Movement Law of the Cone Valve in the Steady State
2.3 Movement of Pump Valve in Unstable State
3 Results and Analysis and Discussion
3.1 Influence of Impact Speed and Spring Stiffness
3.2 Effect on the Overall Performance of the Hydraulic End of the Diaphragm Pump
4 Conclusion
References
Bird-Inspired Nonlinear Oscillator with Triboelectric Nanogenerator for Vibration Control and Energy Harvesting
1 Introduction
2 Static Analysis
3 Excellent Performance of BINO-TENGD for Vibration Control
4 BINO-TENG for Low-Frequency Vibration Energy Harvesting Under Bistable Conditions
5 Application Value in Bridge Vibration Control and Detection as Well as Energy Capture
6 Conclusion
References
Improvement of Small Target Detection Algorithm Based on YOLOV5
1 Introduction
2 Basic Algorithm Framework and Introduction
3 Improved YOLOv5
3.1 Improved C4 Module
3.2 Feature Fusion Improvement
3.3 Adding Detection Heads
4 Experimental Results
5 Summarize
References
Aerodynamic Analysis of a Double Elastic Panel-Cavity with One Side Exposed to Supersonic Flow
1 Introduction
2 Numerical Methodology
2.1 Model Description
2.2 Structural Modeling of Composite Laminated Panel
2.3 Acoustic Modeling of the Compressible Fluid in Cavity
2.4 Piston Aerodynamic Theory of Supersonic Flow
3 Theoretical Monolithic Model of the Panel-Cavity Aeroelastic System
4 Verification
5 Results and Discussion
5.1 Effects of Dynamic Pressure on Flutter Dynamics
5.2 Effects of Stiffness Ratio on Flutter Dynamics
6 Conclusions
References
Observer-Based Robust Control for Active Suspension Systems by Employing Beneficial Disturbances and Coupling Effects
1 Introduction
2 Mathematical Model for Active Suspension Systems
3 Main Results
3.1 Nonlinear Disturbance Observer Design
3.2 Coupling and Disturbance Effects Indicators
3.3 Observer-Based Robust Control Method
4 Experimental Results and Analysis
5 Conclusions
References
Parametric Study on Performance of Parallel Asymmetric Nonlinear Energy Sinks
1 Introduction
2 Mechanical Model
3 Characterization Study of the System
3.1 Band Interval of Higher Branches of Response
3.2 Frequency Band of Strongly Modulated Response
4 Performance under Shock Excitation
5 Conclusions
References
Harnessing LSTM for Nonlinear Ship Deck Motion Prediction in UAV Autonomous Landing Amidst High Sea States
1 Introduction
2 Data Preparation
3 Model Training
4 Model Validation
5 Summary
References
Aircraft Anti-Skid Braking Nonlinear Control Based on ADRC
1 Introduction
2 System Description
2.1 Tire-Road Contact Friction Model
2.2 Overall Aircraft Landing Systems
3 System Description
3.1 Problem Formulation
3.2 Active Disturbance Rejection Controller for Aircraft Anti-Skid Braking System
4 Simulation Results on Semi-physical Simulation Platform
5 Discussion
References
Design of Synchronous Charge Extraction Multi-input Piezoelectric Energy Harvesting Circuit
1 Introduction
2 Circuit Simulation and Analysis
2.1 Circuit Principle
2.2 Circuit Analysis
2.3 Circuit Simulation
3 Experimental Verification
3.1 Experimental Setup
3.2 Experiment
4 Conclusion
References
Vibration Absorption of Nonlinear Energy Sink for Non-resonant Frequency Band of Rectangular Plate
1 Introduction
2 Experimental Device
3 Vibration Absorption Experiment
4 Conclusions
References
Uniform Load Assembly Method of PEMFC Under Vibration Based on the Wave Spring Suspension Support
1 Introduction
2 Experimental Test and Validation
2.1 Pressure Sensitive Film Test Experiment
2.2 Flexible Pressure Sensor Test Experiment
2.3 Validation of Electrochemical Performance
3 Conclusions
References
Origami-Inspired Vibration Isolation with Inherent Nonlinear Inerter
1 Introduction
2 Mathematic Design and Dynamic Modelling
3 POC Design, Fabrication and Experiment Setup
4 Test Result Discussion
5 Conclusion
References
Analytical Optimization Analysis of Inerter-Based Vibration Absorbers with Negative Stiffness
1 Introduction
2 Traditional Optimization Methods for Dynamic Excitations
2.1 Governing Equation
2.2 Optimization Based on Fixed Point Approach
3 Optimization for Excitations Combining Static and Dynamic Components
3.1 Static and Dynamic Performance Analysis
3.2 Optimization Procedure
4 Empirical Formulas for Practical Design
5 Example
6 Conclusions
References
Author Index

Citation preview

Lecture Notes in Electrical Engineering 1152

Xingjian Jing Hu Ding Jinchen Ji Daniil Yurchenko   Editors

Advances in Applied Nonlinear Dynamics, Vibration, and Control – 2023 The Proceedings of 2023 International Conference on Applied Nonlinear Dynamics, Vibration, and Control (ICANDVC2023)

Lecture Notes in Electrical Engineering

1152

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Napoli, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, München, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, University of Karlsruhe (TH) IAIM, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Dipartimento di Ingegneria dell’Informazione, Sede Scientifica Università degli Studi di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Intelligent Systems Laboratory, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, Department of Mechatronics Engineering, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Intrinsic Innovation, Mountain View, CA, USA Yong Li, College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Subhas Mukhopadhyay, School of Engineering, Macquarie University, Sydney, NSW, Australia Cun-Zheng Ning, Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Department of Intelligence Science and Technology, Kyoto University, Kyoto, Japan Luca Oneto, Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Genova, Genova, Italy Bijaya Ketan Panigrahi, Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Federica Pascucci, Department di Ingegneria, Università degli Studi Roma Tre, Roma, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany Germano Veiga, FEUP Campus, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Haidian District Beijing, China Walter Zamboni, Department of Computer Engineering, Electrical Engineering and Applied Mathematics, DIEM—Università degli studi di Salerno, Fisciano, Salerno, Italy Junjie James Zhang, Charlotte, NC, USA Kay Chen Tan, Department of Computing, Hong Kong Polytechnic University, Kowloon Tong, Hong Kong

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Xingjian Jing · Hu Ding · Jinchen Ji · Daniil Yurchenko Editors

Advances in Applied Nonlinear Dynamics, Vibration, and Control – 2023 The Proceedings of 2023 International Conference on Applied Nonlinear Dynamics, Vibration, and Control (ICANDVC2023)

Editors Xingjian Jing City University of Hong Kong, Kowloon Tong Hong Kong, China Jinchen Ji University of Technology Sydney Sydney, NSW, Australia

Hu Ding Shanghai University Shanghai, China Daniil Yurchenko University of Southampton Southampton, UK

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-981-97-0553-5 ISBN 978-981-97-0554-2 (eBook) https://doi.org/10.1007/978-981-97-0554-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Contents

Study on the Dynamic Performance of X-shaped Vibration Isolator with Friction Damping Based on Incremental Harmonic Balance Method . . . . . . Zhongren Yang, Haiping Liu, Hongbo Li, and Tian Wang Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating for Rotating Engine in Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ruixian Ma, Yaqing Wei, Quankun Li, Rui Wang, Mingfu Liao, Kaiming Wang, and Pin Lv

1

15

Semi-analytical Expression of Force and Stiffness of Perpendicular Polarized Ring Magnets for Nonlinear Dynamic Analysis . . . . . . . . . . . . . . . . . . . Ying Zhang, Wei Wang, and Junyi Cao

25

On-Orbit Reconfiguration Dynamics and Control of Heterogeneous Intelligent Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dengliang Liao, Xingyi Pan, Xilin Zhong, Zhengtao Wei, and Ti Chen

39

Study on the Effect of Angular Misalignment on the Contact Load and Stiffness of Cylindrical Roller Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zihang Li, Xilong Hu, Choangyang Wang, Haoze Wang, and Lihua Yang

51

Dynamic Modeling and Features of GTF Engine Rotor System . . . . . . . . . . . . . . . Heyu Hu, Bin Shi, Tianxiang Wang, Quankun Li, Mingfu Liao, Kang Zhang, and Fali Yang

64

Nonlinear Dynamic Analysis of Rub-Impact Rod-Fastening Combined Rotor Systems with Internal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chongyang Wang, Zihang Li, Haoze Wang, Xilong Hu, and Lihua Yang

77

A Multiscale Fracture Model to Reveal the Toughening Mechanism in the Bioinspired Bouligand Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yunqing Nie, Dongxu Li, and Luojing Zhou

91

Decoupled Multi-mode Controllable Electrically Interconnected Suspension for Improved Vehicle Damping Performance . . . . . . . . . . . . . . . . . . . . 105 Pengfei Liu, Donghong Ning, Guijie liu, and Haiping Du Adaptive Robust Sliding-Mode Control of a Semi-active Seat Suspension Featuring a Variable Inertance-Variable Damping Device . . . . . . . . . . . . . . . . . . . . 120 Guangrui Luan, Pengfei Liu, Donghong Ning, and Guijie Liu

vi

Contents

An Adaptive Controller for Payload Swing Suppression of Ship-Mounted Boom Cranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Bincheng Li, Peng Liao, Menghua Zhang, Donghong Ning, and Guijie Liu Study on Dynamic Modeling and Vibration Noise Suppression Method of AUV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Kangyu Zhang, Chao Fu, Kuan Lu, Kaifu Zhang, Hui Cheng, and Dong Guo Simultaneous Vibration Absorbing and Energy Harvesting Mechanism of the Tri-Magnet Bistable Levitation Structure: Modeling and Simulation . . . . . 165 Junjie Xu and Yonggang Leng Design and Research of Triboelectric Energy Harvester for Low Frequency Nonlinear Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Yinqiang Huang, Huajiang Ouyang, and Zihao Liu Energy Harvesting Study of Piezoelectric Vibration Harvester with Double Parallel Slender Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Xiang Zhao and Haotian Jiang A Broadband Energy Harvester with Three-to-One Internal Resonance . . . . . . . . 209 Le Yang, Wenan Jiang, Xingjian Jing, and Liqun Chen Optimization Design of High-Pressure Simulated Rotor . . . . . . . . . . . . . . . . . . . . . 221 Zhongyu Yang, Jiali Chen, and Yinli Feng Energy Transfer of Particle Impact Damper Systems . . . . . . . . . . . . . . . . . . . . . . . . 234 Xiang Li, Li-Qun Chen, Lawrence A. Bergman, and Alexander F. Vakakis An Empirical Control Research on the Lexical Approach to Business Correspondence Writing in Vocational Colleges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Jin Zhang Analytical Analysis of Nonlinear Vortex-Induced Vibration of Pipes Conveying Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Jian Liu and Yu Wang RBF Neural Network for Feature Selection Using Sparsity Method . . . . . . . . . . . 273 Tao Gao, Jun Yang, Yongyong Xu, Baosheng Qian, Bin Wang, and Ruoxi Yu

Contents

vii

Periodic Motions and Bifurcations of a Spring-Driven Joint System with Periodic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Yufan Zhou, Zhongliang Jing, Jianzhe Huang, Xiangming Dun, and Hailei Wu Enhanced Vibration Characteristics of Honeycomb Plates Composed of Metamaterials with NTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Qiao Zhang and Yuxin Sun Vibration Reduction of Limited Series Nonlinear Energy Sink . . . . . . . . . . . . . . . 310 Ting-Kai Du and Hu Ding Understanding Neural Rhythmic Mechanisms Through Self-oscillations of Complex Neural Networks and Their Adaptation . . . . . . . . . . . . . . . . . . . . . . . . 323 Peihua Feng, Luoqi Ye, Xinaer Adilihazi, Zhilong Liu, and Ying Wu Multi-attention Based Multi-scale Temporal Convolution Network for Remaining Useful Life Prediction of Rolling Bearings . . . . . . . . . . . . . . . . . . . 334 Yiwen Cong Dynamics Analysis of the Cooperative Dual Marine Lifting Systems Subject to Sea Wave Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Gang Li, Xin Ma, and Yibin Li Nonlinear Hierarchical Control for Unmanned Quadrotor Transportation Systems with Saturated Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Lincong Han and Menghua Zhang Predicting the Endless Stop-Band Behaviour of the NS-MRE Isolator . . . . . . . . . 377 Qun Wang, Zexin Chen, Jian Yang, and Shuaishuai Sun Human-Mechanical Biomechanical Analysis of a Novel Knee Exoskeleton Robot for Rehabilitation Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Mengmeng Yan, Guanbin Gao, Xin Chen, Yashan Xing, and Sheng Lu Adaptive Sliding Mode Control for Active Suspensions of IWMD Electric Vehicles Subject to Time Delay and Cyber Attacks . . . . . . . . . . . . . . . . . . . . . . . . . 403 Wenfeng Li, Jing Zhao, Mengqi Deng, Zhijiang Gao, and Pak Kin Wong Generative Adversary Network Based on Cross-Modal Transformer for CT to MR Images Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Zhenzhen Wu, Weijie Huang, Xingong Cheng, and Hui Wang

viii

Contents

A Novel Robust Finite-Time Control for Active Suspension Systems with Naturally Bounded Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Zengcheng Zhou, Xingjian Jing, and Menghua Zhang Quasi-Zero Stiffness Magnetic Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . . 444 Xuan-Chen Liu and Hu Ding New Software Bionic Haptic Actuator Design Based on Barometric Array . . . . . 457 Zige Yu, Sai Li, Mengying Lin, Hang Hu, Yingying Li, Qian Lei, and Zixin Huang Adaptive Robust Tracking Control for Aerial Work Platform Vehicle with Guaranteed Prescribed Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Jiawen Dai, Zheshuo Zhang, Bangji Zhang, Jie Bai, and Hui Yin A Novel Model Predictive Control Strategy for Continuum Robot: Optimization and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Yakang Wang, Yuzhe Qian, and Weipeng Liu Finite Element Analysis and Error Compensation for Wrinkled Bellow-Like Soft Robotic Manipulator Kinematics Modeling . . . . . . . . . . . . . . . . 498 Haibin Huang, Yingjie Li, Yingbo Huang, and Jing Na Suppression of Galloping Oscillations Using Perforated Bluff Bodies . . . . . . . . . 514 Juntong Xing, Masoud Rezaei, Huliang Dai, and Wei-Hsin Liao Motion Planning for Wave-Like-Actuated Manta-Inspired Amphibious Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Yixuan Wang, Qingxiang Wu, Xuebing Wang, and Ning Sun Time-Optimal Anti-swing Trajectory Planning of Double Pendulum Crane Based on Chebyshev Pseudo-spectrum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Ken Zhong and Yuzhe Qian Nonlinear Inertia and Its Effect Within an X-shaped Mechanism . . . . . . . . . . . . . 554 Zhenghan Zhu and Xingjian Jing Metamaterial Beam with Bistable and Monostable-Hardening Attachments for Broad-Band Vibration Attenuation and Energy Harvesting . . . . . . . . . . . . . . . 570 Che Xu and Liya Zhao An Improved Incremental Classifier and Representation Learning Method for Elderly Escort Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Ke Huang, Mingyang Li, Yiran Wang, Weijie Huang, and Menghua Zhang

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Exact Dynamic Analysis of Viscoelastic Double-Beam System Using Dynamic Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 Fei Han, Nianfeng Zhong, and Tao Yang Nonlinear Control Strategy for Tower Cranes with Variable Cable Lengths and Multivariable State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Hui Guo, Wei Peng, Menghua Zhang, Chengdong Li, and Fei Jiao Design and Dynamic Analysis of a Flexible Inertia Device for Vehicle Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 Bohuan Tan, Xingui Tan, Jingang Liu, Hai Li, and Yilong Xie A Quasi-Zero Stiffness Nonlinear Absorber Based on Centrifugal Force . . . . . . . 638 Hulun Guo and Zhiwei Cao A Cellular Strategy for Eliminating the Failure of Nonlinear Energy Sinks Under Strong Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 Sun-Biao Li and Hu Ding A Stable Adjustable Nonlinear Energy Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 You-cheng Zeng, Hu Ding, and Jinchen Ji Gait-Planning-Based Path Planning for Crocodile-Inspired Pneumatic Soft Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Yize Ma, Qingxiang Wu, Zehao Qiu, and Ning Sun Research on Nonlinear Energy Sink Vibration Reduction of Floating Raft System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 Hong-Li Wang and Hu Ding X-mechanism Guided Elastic QZS Vibration Isolator Design for Beneficial Nonlinear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 Chuanping Liu and Xingjian Jing Design and Vibration Control of Secondary Suspension for Maglev Train Based on Magnetorheological Fluid Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Yougang Sun, Dandan Zhang, Hongyu Ou, Guobin Lin, and Haiyan Qiang Nonlinear Dynamic Analysis of Flywheel Rotor Systems with Multiple Fit Clearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 Qinkai Han, Zhaoye Qin, and Fulei Chu Comparison Studies of Dynamic Characteristics for Coupled Bearing-Rotor Systems with Fixed and Pivot-Supported Pads . . . . . . . . . . . . . . . . 733 Wennian Yu, Chaodong Zhang, and Lu Zhang

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Contents

Nonlinear Dynamic Performance and Analysis Model of Pump Valve System of Diaphragm Pump Hydraulic End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 Jiameng Zhang, Wensheng Ma, Chunchuan Liu, and Zicheng Zhao Bird-Inspired Nonlinear Oscillator with Triboelectric Nanogenerator for Vibration Control and Energy Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Jiayi Liu, Yingxuan Cui, Tao Yang, and Xingjian Jing Improvement of Small Target Detection Algorithm Based on YOLOV5 . . . . . . . 775 Shoujun Lin, Lixia Deng, Huanyu Chen, Lingyun Bi, and Haiying Liu Aerodynamic Analysis of a Double Elastic Panel-Cavity with One Side Exposed to Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 Hao Liu, Yegao Qu, and Guang Meng Observer-Based Robust Control for Active Suspension Systems by Employing Beneficial Disturbances and Coupling Effects . . . . . . . . . . . . . . . . . 797 Menghua Zhang, Zengcheng Zhou, Qiang Liu, and Xingjian Jing Parametric Study on Performance of Parallel Asymmetric Nonlinear Energy Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 Huiyang Li and Jianen Chen Harnessing LSTM for Nonlinear Ship Deck Motion Prediction in UAV Autonomous Landing Amidst High Sea States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820 Feifan Yu, Wenyuan Cong, Xinmin Chen, Yue Lin, and Jiqiang Wang Aircraft Anti-Skid Braking Nonlinear Control Based on ADRC . . . . . . . . . . . . . . 831 FengRui Yu, Jingting Zou, Xinmin Chen, Yue Lin, and Feifan Yu Design of Synchronous Charge Extraction Multi-input Piezoelectric Energy Harvesting Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 Bin Zhang, Hao Sun, Ruibo Chai, Shizhou Lu, and Shengxi Zhou Vibration Absorption of Nonlinear Energy Sink for Non-resonant Frequency Band of Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858 Jie Wu and Jianen Chen Uniform Load Assembly Method of PEMFC Under Vibration Based on the Wave Spring Suspension Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 Biyu Pan, Dong Guan, Rui Wang, Zhen Chen, and Ting Chen Origami-Inspired Vibration Isolation with Inherent Nonlinear Inerter . . . . . . . . . 876 Kan Ye, J. C. Ji, Jianchun Li, and Keisuke Yamada

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Analytical Optimization Analysis of Inerter-Based Vibration Absorbers with Negative Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 Jing Bian and Ning Su Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

Study on the Dynamic Performance of X-shaped Vibration Isolator with Friction Damping Based on Incremental Harmonic Balance Method Zhongren Yang1,2 , Haiping Liu1,2(B) , Hongbo Li1 , and Tian Wang1 1 School of Mechanical Engineering, University of Science and Technology Beijing,

Beijing 100083, China [email protected] 2 Shunde Innovation School, University of Science and Technology Beijing, Foshan 528300, Guangdong, China

Abstract. In order to improve the vibration isolating performance of the existing X-shaped vibration isolator, a novel X-shaped vibration isolator is constructed by adding plate-type friction damping element which does not decrease the overall stiffness of the isolating system. First of all, the geometrical relationships of the Xshaped isolator with friction damping are presented, then the mathematical model of the isolating system is established. Furthermore, to validate effectiveness of the proposed dynamic model, the incremental harmonic balance method is utilized to derive steady-state solutions of the isolating system under base excitation, and the accuracy of the theoretical solutions is verified through numerical simulation method. At last, the vibration suppression performance of the isolating systems with and without the plate-type friction damping element are evaluated in terms of displacement transmissibility. The calculation results reveal that the developed X-shaped vibration isolator with plate-type friction damping has a better vibration reduction performance near the resonance frequency. Keywords: X-shaped vibration isolator · Incremental harmonic balance method · Friction damping

1 Introduction Vibration control poses a formidable challenge in numerous practical engineering domains. Increasing research on improving vibration control performance by introducing nonlinear elements into the vibration controlling system. Wherein, X-shaped structures, which exhibit exceptional nonlinear characteristics for vibration isolation, have been extensively documented and scrutinized [1–3, 13–15]. Incremental harmonic balance method (IHBM), as a semi-analytical method which combines incremental method and harmonic balance method, is firstly proposed by Lau et al. [4]. Previous studies have shown the merits of the IHBM in dealing with nonlinear systems: (1) can be applied to analysis the strong nonlinear vibration problems, and obtain high-precision solutions; (2) is a calculation method for solving nonlinear © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 1–14, 2024. https://doi.org/10.1007/978-981-97-0554-2_1

2

Z. Yang et al.

differential equations, which is suitable for the quantitative analysis of a large range of parameter changes [5–7]. Damping is a main energy-consuming element in a vibration isolating system. Generally, based on the generating mechanisms, damping can be divided into: electromagnetic damping [8], fluid damping [9], frictional damping [10] and so on. Among them, frictional damping elements, as a high-energy-consuming and cost-effective solution, are widely used in civil buildings. In our previous research [11, 12, 16] dynamic characteristics of a X-shaped vibration isolator with amplifying mechanism without considering the friction damping have been analyzed. However, the friction damping in the hinged position may influence the vibration isolation effect and should not be ignored. Therefore, in this paper, the platetype friction damping structure is added at the vertical hinge of the developed X-shaped vibration isolator. Then, the amplitude-frequency response and displacement teansmissiblity of the X-shape structure under basic excitaonare derived by IHBM. Finally, the vibration isolation performance of X-shaped vibration isolator with plate-type friction damping is compared and evaluated. The organization of this paper is as follows. In Sect. 2, the mathematical model of the X-shaped vibration isolator with the plate-type friction damping is established through the constitutive equation of friction damping and the geometrical relation of the isolating system. In Sect. 3, the IHBM is used to solve the dynamic equation, and the frequency response curves and displacement transmissibility curves are illustrated. Finally, conclusions are drawn in the final section.

2 Mathematical Modeling In this section, the mathematical model of the X-shaped vibration isolator with friction damping is presented. In this model, the amplifying function is implemented by a primary supporting spring and plate-type friction dampers, which is arranged in the hinged joints along the vertical direction, as shown in Fig. 1(a) and (b). From Fig. 1(a), the X-shaped vibration isolator is composed of a main supporting spring k v along vertical direction and a four-end articulated rod with amplifying mechanism including damping c and spring k h . At equilibrium position, y1 and x 2 represent the horizontal distance and vertical distance of hinged points, respectively. And the angles of the rigid rods with respect to horizontal lines is θ. It can be observed that red dotted line indicates the initial position, and horizontal distance is y2 . By comparison of initial and equilibrium positions, x 1 and ϕ are downward distance and angle displacement with mass M, respectively, and the horizontal displacement of the movable hinged points is denoted by y1 -y2 .

Study on the Dynamic Performance of X-shaped Vibration Isolator

3

Fig. 1. (a) The X-shaped vibration isolator with friction damping (blue circles mark the arranged positions of the plate-type friction damping element); (b) structural schematic of the single platetype friction damping structure; (c) force body diagram of the frictional plates.

2.1 Constitutive Equation of Plate-Type Friction Damping Schematic diagram of the friction plates in a free status is depicted, as shown in Fig. 1(c). F ks and F s are the forces on the left and right sides of the friction plate by preloading bolts, N 1 and N 2 are the interaction forces generated by the contact between the friction elements respectively Fks = −N2 , Fs = −N1 , Fks = Fs

(1)

f 1 and f 2 are the friction force generated on the left and right sides of the friction damping plate f1 = N1 · μ , f2 = N2 · μ

(2)

The friction plate is circular ring, as shown in Fig. 1(b), R is the outer diameter, r is the inner diameter, and the friction force per unit area is f =

N1 · μ f1 = S π(R2 − r 2 )

(3)

By integrating, the friction force can be transformed into a torque by the expression  R 2 N1 μ(R3 − r 3 ) Mf = f · x · 2π xdx = (4) 3 R2 − r 2 r 2.2 Modeling of X-shape Vibration Isolator with Plate-Type Friction Damping Element According to Fig. 1(a) Fx xi = 8Mf θi

(5)

where x i is the vertical displacement under force F x , and θ i is the angles of the rigid rods with respect to horizontal lines xi = 2(l sin(θ + ϕ) − l sin(θ + ϕ − θ )) ≈ 2l sin(θ + ϕ)θi

(6)

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According to Eqs. (4) and (6), output force F x of the friction damper can be expressed as Fx =

4Mf 8 Fks μ(R3 − r 3 ) = l cos(θ + ϕ) 3 (R2 − r 2 )l cos(θ + ϕ)

(7)

The geometrical relationships of the variables in the structure can be derived as   l 2 − (l sin θ + x/2)2 l 2 − (x1 + x2 )2 cos(θ + ϕ) = = (8) l l Substituting Eq. (8) into Eq. (7), one can obtain Fx =

Fks μ(R3 − r 3 ) 8  3 (R2 − r 2 ) l 2 − (l sin θ + x/2)2

(9)

In the vertical direction, the friction force by plate-type friction damping element is related to the motion direction of the X-shaped vibration isolator, the sign function is introduced to indicate the direction of friction Fx =

Fks μ(R3 − r 3 )sign(˙x) 8  3 (R2 − r 2 ) l 2 − (l sin θ + x/2)2

(10)

According to the mechanical model as shown in Fig. 1(a), the motion equation of the X-shape vibration isolating system along the vertical direction can be given by using Newton’s second law of motion dy1 d 2u ) tan(θ + ϕ) + kv u + (2kh y1 + 2c 2 dt dt Fks μ(R3 − r 3 )sign( du 8 dt )  + = −M ω2 Z0 cos ωt 2 2 2 3 (R − r ) l − (l sin θ + u/2)2 M

(11)

where u = x − z is the relative displacement between the inertial mass and the base, and x is the vertical displacement of the inertial mass. The geometrical relationships of the variables in the structure can be derived as tan(θ + ϕ) = y1 = l cos θ − y˙ 1 =



l sin θ + x/2 l cos θ − y1

l 2 − (l sin θ + x/2)2

l sin θ + x/2 1  x˙ 2 l 2 − (l sin θ + x/2)2

(12) (13) (14)

Study on the Dynamic Performance of X-shaped Vibration Isolator

5

Substituting Eqs. (12)–(14) into Eq. (11), one can obtain d 2u M 2 + kv u+ ⎡ dt  ⎤  2 2 ⎢ 2kh l cos θ − l − (l sin θ + x/2) ⎥ l sin θ + u/2 ⎢ ⎥ ⎢ ⎥ du l sin θ + u/2 ⎣ ⎦ l 2 − (l sin θ + u/2)2 ) tan(θ + ϕ) +c  l 2 − (l sin θ + x/2)2 dt +

(15)

Fks μ(R3 − r 3 )sign( du 8 dt )  = −M ω2 Z0 cos(ωt) 3 (R2 − r 2 ) l 2 − (l sin θ + u/2)2

To facilitate calculation, the following functions are defined in Eq. (15)   2 2 f1 (σ ) = 2kh l cos θ − l − (l sin θ + x/2) +c 

l sin θ + x/2

(16)

l 2 − (l sin θ + x/2)2 f2 (σ ) =

f3 (σ ) =

l2

(l sin θ + x/2)2 − (l sin θ + x/2)2

Fks μ(R3 − r 3 )sign(σ˙ ) 8  3 (R2 − r 2 ) l 2 − (l sin θ + x/2)2

(17) (18)

The f 1 (σ ) and f 2 (σ ) can be expanded at σ = 0 by using the fourth-order Taylor series, as follows f1 (σ ) = λ0 + λ1 σ + λ2 σ 2 + λ3 σ 3

(19)

f2 (σ ) = λ4 + λ5 σ + λ6 σ 2 + λ7 σ 3

(20)

Where, the coefficients λ0 , λ1 , λ2 , λ3 , λ4 , λ5 , λ6 and λ7 refer to the Taylor series expansion terms. And, the detailed expressions are listed in Appendix I.

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Due to the existence of symbolic functions in function f 3 (σ ), the Taylor series cannot be directly used to define the following functions in Eq. (18). Therefore, some fuctions are defined as g1 (σ ) =

Fks μ(R3 − r 3 ) 8  3 (R2 − r 2 ) l 2 − (l sin θ + x/2)2

(21)

g2 (σ ) = sign(σ˙ )

(22)

f3 (σ ) = g1 (σ ) · g2 (σ )

(23)

Eq. (18) can be expressed as

where g1 (σ ) is continuous and can be expanded using the Taylor series g1 (σ ) = q0 + q1 σ + q2 σ 2 + q3 σ 3

(24)

where, the specific expression of coefficients q0 , q1 , q2 , q3 are listed in Appendix II. So function f 3 (σ ) can be expressed as ⎡ ⎤ 8 Fks μ(R3 − r 3 ) 4 Fks μ(R3 − r 3 ) sin θ + (σ ) ⎢ 3 (R2 − r 2 )l cos θ ⎥ 3 (R2 − r 2 )l 2 cos3 θ ⎢ ⎥ ⎢ ⎥ 3 3 ⎢ 1 Fks μ(R − r )(cos 2θ − 2) 2 ⎥ (25) f3 (σ ) = ⎢ − ⎥ · sign(σ˙ ) (σ ) 2 − r 2 )l 3 cos5 θ ⎢ 3 ⎥ (R ⎢ ⎥ ⎣ 1 F μ(R3 − r 3 )(cos 2θ − 2) sin θ ⎦ ks (σ 2 ) − 2 2 4 7 6 (R − r )l cos θ For simplifying, some nondimensional parameters are introduced

c kv ω x ωn = , τ = ωn t, = , x0 = , ζ = √ M ωn l 2 Mkh f0 =

F0 u Z0 kh μ(R3 − r 3 ) Fks , u , z , γ , fks = = = = , d = 0 0 1 c 2 2 2 M ωn l l l kv (R − r )l M ωn2 l

where τ is the nondimensional time, is the frequency ratio, γ 1 is the stiffness ratio between the spring elements k h and k v , u0 and z0 are the displacement ratios of u and Z 0 with respect to the rod length l, ζ is the damping ratio. The Eq. (15) can be converted into the following  ⎡  ⎤ sin θ 1+sin2 θ 2 sin θ+sin3 θ 3 du0 2ζ tan2 θ + cos u + u + u 0 4θ 6θ 0 8θ 0 dτ 4 cos 2 cos ⎢ ⎥ ⎢ +(1 + γ1 tan2 θ )u0 + 3 sin θ γ1 u2 + 1+sin2 θ γ1 u3 ⎥ 2 4 8 ⎢ ⎥ 0 0 4 cos θ 8 cos θ d u0 ⎢ ⎛ ⎞ ⎥ 4 sin θ 8 + ⎢ ⎥ 2 dτ ⎢ ⎜ 3 cos θ dc fks + 3 cos3 θ dc fks u0 ⎥ ⎟ ⎣ +⎝ cos(2θ − 2) sign(˙ u ) ⎠ 0 ⎦ cos(2θ − 2) sin θ 2 3 dc fks u0 − dc fks u0 − 3 cos5 θ 3 cos5 θ = 2 z0 cos( t) (26)

Study on the Dynamic Performance of X-shaped Vibration Isolator

7

3 Frequency Response and Displacement Transmissibility 3.1 IHBM Procedure Dynamics equation of the X-shaped vibration isolating system is solved by IHBM. Firstly, defining a new variable j = τ . 1 + 3 sin2 θ 2 sin θ + sin3 θ 3 sin θ σ+ σ + σ )σ˙ 4 cos θ 4 cos6 θ 2 cos8 θ 3 sin θ 1 + sin2 θ 2 γ γ1 σ 3 σ + + (1 + γ1 tan2 θ )σ + 1 4 cos4 θ 8 cos8 θ ⎞ ⎛ 8 4 sin θ cos(2θ − 2) 2 d f σ − f σ dc fks + d c ks c ks ⎟ ⎜ cos3 θ 3 cos5 θ + ⎝ 3 cos θ ⎠sign(σ˙ ) cos(2θ − 2) sin θ 3 d − f σ c ks 3 cos5 θ = ρa0 cos ς

2 σ¨ + 2 ζ (tan2 θ +

(27)

The Newton-Raphson Incremental process σ = σ0 + σ, = 0 + 

(28)

For sign functions, it is necessary to establish a first-order Taylor expansion formula sign(u) = −1 + 2H (u)

(29)

⎧ ⎨ 0, u < 0 H (u) = 21 , u = 0 ⎩ 1, u > 0

(30)

The derivative of H(u) is the Dirac function δ(u), which is defined as follows  +∞ δ(x − a)g(x)dx = g(a) (31) −∞

Strictly speaking, δ and H should be defined as generalized functions, whose derivatives can be obtained through distribution theory d d sign(u) = 2 H (u) = 2δ(u) du du

(32)

Using a first-order Taylor expansion sign(u + u) = sign(u) + 2δ(u)u

(33)

Substituting the nonlinear term in Eq. (33) into the incremental equation sign(σ˙ + σ˙ ) = sign(σ˙ ) + 2δ(σ˙ )σ˙

(34)

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Substituting Eqs. (34) and (28) into Eq. (27), and omitting the higher order term 20 σ¨ + 2 0 (ξl0 + ξl1 + ξl2 + ξl3 )σ˙ + (Kl1 + Kl2 + Kl3 + Sl1 + Sl2 + Sl3 )σ +Sl0 + (Dl0 + Dl1 + Dl2 + Dl3 )σ˙ + 0 (ξl1  + ξl2  + ξl3 ) = r − [2 0 σ¨ 0 + 2σ˙ (ξl0 + ξl1 + ξl2 + ξl3 )] (35) where, r is the error value, and its value is zero when δ 0 and 0 are the exact solutions   20 δ¨0 + 2 0 (ξr0 + ξr1 + ξl2 + ξl3 )σ˙ 0 + Sr0 r = ρa0 cos ς − (36) +(Sr1 + Sr2 + Sr3 + Kr0 + Kr1 + Kr2 + Kr3 ) where, each parameter refers to the Appendix III. Harmonic balance process can be expressed as follows ⎧ N1 N1   ⎪ ⎪ ⎪ σ0 = ak cos((k − 1)ζ ) + bk cos(kζ ) = Cs A0 ⎨ k=1

k=1

N1 N1   ⎪ ⎪ ⎪ ak cos((k − 1)ζ ) + bk cos(kζ ) = Cs A0 ⎩ σ0 = k=1

(37)

k=1

Cs = [1, cos ζ, cos 2ζ, . . . . . . , cos(N1 − 1)ζ, sin ζ, sin 2ζ, . . . . . . , sin N2 ζ ]

(38)

A0 = [a1 , a2 , . . . . . . , aN1 , b1 , b2 , . . . . . . , bN2 ]

(39)

A0 = [a1 , a2 , . . . . . . , aN1 , b1 , b2 , . . . . . . , bN2 ]

(40)

S = Cs , A = A0 , A = A0

(41)

σ0 = S · A, σ = S · A

(42)

⎤ 20 σ¨ + 2 0 (ξl0 + ξl1 + ξl2 + ξl3 )σ˙ + (Kl1 + Kl2 ⎦d ς δ(σ )T ⎣ +Kl3 + Sl1 + Sl2 + Sl3 )σ +Sl0 + (Dl0 + Dl1 + Dl2 + Dl3 )σ˙ + 0 (ξl1  + ξl2  + ξl3 ) ⎡



2π 0



2π

=

δ(σ )T [r − (2 0 σ¨ 0 + 2σ˙ (ξl0 + ξl1 + ξl2 + ξl3 )) ]d ς

(43)

0

Kmc A = R − Rmc

(44)

Kmc = 20 M + 2 0 (ξl0 + ξl1 + ξl2 + ξl3 ) + KL1 + KL2 + Kl3 + SL0 + SL1 + SL2 + SL3 + DL0 + DL1 + DL2 + DL3 + 0 (ξl1  + ξl2  + ξl3 )

(45)

Study on the Dynamic Performance of X-shaped Vibration Isolator

 R=P−

 20 M + 2 0 (ξR0 + ξR1 + ξR2 + ξR3 ) · A0 − SR0 +(SR1 + SR2 + SR3 + KR1 + KR2 + KR3 ) Rmc = 2 0 M + 2(ξL0 + ξL1 + ξL2 + ξL3 )

9

(46) (47)

The above-mentioned individual parameter matrix refers to the Appendix IV. Additionally, the differential equations of motion can be written as time domain equations ⎧ u˙ 1 = u2 ⎪    ⎪ ⎪  ⎪ ⎪ l sin θ + x/2 1 ⎪ ⎪ u2 2kh l cos θ − l 2 − (l sin θ + u1 /2)2 + c  ⎨ u˙ 2 = − M l 2 − (l sin θ + x/2)2 ⎪ 8 Fks μ(R3 − r 3 )sign(u2 ) l sin θ + x1 /2 kv x1 ⎪ ⎪  − · − ⎪ ⎪ 2 − (l sin θ + x /2)2 ⎪ M 3M (R2 − r 2 ) l 2 − (l sin θ + u1 /2)2 l ⎪ 1 ⎩ + M ω2 Z0 cos ωt (48)

3.2 Frequency Response The frequency response curve of the X-shaped vibration isolating system is shown as Fig. 2 with system parameters: c = 0.1 Ns/m, θ = 60°, M = 1 kg, F = 10 N, F ks = 10 N, Z 0 = 0.01 m, k h = 1000 N/m, l 1 = 0.1 m, R = 0.0175, r = 0.005, μ = 0.2. The IHBM is used to calculate the dynamic responses in frequency domain, and taking the third harmonic with N 1 = 4 and N 2 = 3 in order to verify the vaildation of the steady-state solution with the lHBM, the numerical solutions are obtained by using the Runge-Kutta method. For the purpose of convenience, the numerical solutions obtained by using direct numerical integration is also demonstrated in the same graph. From Fig. 2, it can be found that the analytical results are in good agreement with the numerical solution. Thus, it is feasible to utilize the analytical results for further studies.

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Fig. 2. Frequency response curves under the base excitation when c = 0.1 Ns/m, θ = 60°, M = 1 kg, F = 10 N, F ks = 10 N, Z 0 = 0.01 m, k h = 1000 N/m, l 1 = 0.1 m, R = 0.0175, r = 0.005, μ = 0.2

3.3 Displacement Transmissibility The displacement transmissibility, which includes relative and absolute displacement transmissibility, is a widely used evaluation index for assessing the dynamic performance of the isolating system in engineering fields. To evaluate the performance of the vibration isolator with plate-type friction damping, the relative displacement transmissibility is selected as an evaluation index under base excitation conditions. For the sake of comparison, the relative displacement transmissibility curves of three types vibration isolators are plotted in Fig. 3. It can be seen from Fig. 3 that the amplitude of the relative displacement transmissibility of the X-shaped vibration isolator with plate-type friction damping is significantly attenuated compared with that of the conventional linear isolator and X-shaped vibration isolator without friction damping. Therefore, the X-shaped vibration isolator with platetype friction damping can achieve greater isolation performance and wider isolation frequency band. One of the most factors contributing to these improvements is the increased damping within the isolation system. Additional friction damping elements increase the overall damping capacity, which improves efficiency of the isolation system. As a result, compared to its original counterpart, this X-shaped vibration isolator with plate-type friction damping exhibits smaller magnitudes for its resonance peaks.

Study on the Dynamic Performance of X-shaped Vibration Isolator

11

Fig. 3. Relative displacement transmissibility with different vibration isolators under the base excitation

Referring to Fig. 3, it can be observed that the dynamic performance of the X-shaped vibration isolator with plate-type friction damping elements has been enhanced.

4 Conclusions In this article, a developed X-shaped vibration isolator with friction damping characteristic, by adding the plate-type friction damping elements at the vertical hinges, has been studied through theoretical method. The frequency response curves under the base excitation are obtained by using the incremental harmonic balance method (IHBM) and verified by the direct numerical integration method. Furthermore, the relative displacement transmissibility of the X-shaped vibration isolator with plate-type friction damping elements is compared with that of the linear counterpart and the X-shaped vibration isolator without the plate-type friction damping elements. The results demonstrate that the developed X-shaped vibration isolator with plate-type friction damping achieves better isolation performance around the resonant frequency. Acknowledgments. This research is supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2021B1515120049), and the Foshan Science and Technology Innovation Funds for Industry University Cooperation Project (No. BK22BE021).

Appendix I λ0 = 0, λ1 = kh tan2 θ, λ2 = λ4 = tan2 θ, λ5 =

3 kh sin θ 3 kh sin θ , λ3 = 4 l cos4 θ 4 l cos4 θ

sin θ 1 + sin2 θ sin θ + sin3 θ , λ6 = 2 , λ7 = 4 6 l cos θ 4l cos θ 2l 3 cos8 θ

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Appendix II q0 = q1 = q2 = − q3 = −

8 Fks μ(R3 − r 3 ) 3 (R2 − r 2 )l cos θ

4 Fks μ(R3 − r 3 ) sin θ 3 (R2 − r 2 )l 2 cos3 θ

1 Fks μ(R3 − r 3 )(cos 2θ − 2) 3 (R2 − r 2 )l 3 cos5 θ

1 Fks μ(R3 − r 3 )(cos 2θ − 2) sin θ 6 (R2 − r 2 )l 4 cos7 θ

Appendix III ξl0 = ζ tan2 θ, ξl1 = ζ ξl1  = 2ζ

sin θ 1 + sin2 θ 2 sin θ + sin3 θ 3 σ σ σ0 , ξ = ζ , ξ = ζ 0 l2 l3 0 cos4 θ 4 cos6 θ 2 cos8 θ

sin θ 1 + 3 sin2 θ sin θ + sin3 θ σ˙ 0 , ξl2  = ζ σ˙ 0 σ0 , ξl3  = 3ζ σ˙ 0 σ02 4 6 cos θ 4 cos θ 2 cos8 θ

3 sin θ sin θ + sin3 θ γ γ1 σ02 σ , K = 3ζ 1 0 l3 2 cos4 θ 2 cos8 θ 4 2 sin θ Sl0 = dc fks sign(σ˙ 0 ), Sl1 = dc fks sign(σ˙ 0 ) 3 cos θ 3 cos3 θ cos(2θ − 2) dc fks sign(σ˙ 0 )σ0 Sl2 = − 3 cos5 θ cos(2θ − 2) sin θ dc fks sign(σ˙ 0 )σ02 Sl3 = − 3 cos5 θ 8 8 dc fks δ(σ˙ 0 )σ0 , Dl1 = dc fks δ(σ˙ 0 )σ0 Dl0 = 3 cos θ 3 cos θ 2 cos(2θ − 2) cos(2θ − 2) sin θ dc fks δ(σ˙ 0 )σ03 dc fks δ(σ˙ 0 )σ02 , Dl3 = − Dl2 = − 2 cos7 θ 3 cos5 θ sin θ σ0 ξr0 = ζ tan2 θ, ξr1 = ζ cos4 θ 1 + sin2 θ 2 sin θ + sin3 θ 3 σ0 , ξr3 = ζ σ0 ξr2 = ζ 6 4 cos θ 2 cos8 θ 4 dc fks δ(σ˙ 0 ), Kr1 = 1 + γ1 tan2 θ Kr0 = 3 cos θ 3 sin θ 1 + 4 sin2 θ γ1 σ0 , Kr3 = γ1 σ02 Kr2 = 4 2 cos θ 8 cos8 θ 4 2 sin θ dc fks sign(σ˙ 0 ), Sr1 = dc fks sign(σ˙ 0 ) Sr0 = 3 cos θ 3 cos3 θ cos(2θ − 2) cos(2θ − 2) sin θ dc fks sign(σ˙ 0 )σ0 , Sr3 = − dc fks sign(σ˙ 0 )σ02 Sr2 = − 5 6 cos θ 3 cos5 θ Kl1 = 1 + γ1 tan2 θ, Kl2 =

Study on the Dynamic Performance of X-shaped Vibration Isolator

13

Appendix IV 

2π

M = 0



2π

ξL0 = 

ξL1  =

2π

˙ ς, ξL3 = S T · ξl2 · Sd

0 2π

0

˙ ς, ξL2  = S · ξl1  · Sd  

2π

2π

2π

2π

2π

˙ ς, KL2 = S T · Kl1 · Sd ˙ ς, KR2 = S T · Kr1 · Sd

0

2π

SL0 =  

2π

2π

2π

2π

2π

˙ ς S T · ξl3  · Sd

˙ ς, S T · ξr1 · Sd

0 2π

˙ ς S T · ξr3 · Sd

0



2π

˙ ς S T · Kl3 · Sd

0

2π

˙ ς, KR3 = S T · Kr2 · Sd

˙ ς, SL1 = S T · Sl0 · Sd ˙ ς, SL3 = S · Sl2 · Sd





2π

˙ ς S T · Kr3 · Sd

0



2π

˙ ς S T · Sl1 · Sd

0 2π

T

˙ ς, SR1 = S T · Sr0 · Sd ˙ ς, SR3 = S T · Sr2 · Sd

˙ ς S T · Sl3 · Sd

2π

 

2π

˙ ς S T · Sr1 · Sd

0 2π

˙ ς S T · Sr3 · Sd

0

˙ ς, DR1 = S T · Dr0 · Sd

0

0

2π

0

˙ ς, KL3 = S T · Kl2 · Sd

0

DR2 =



0

0

DR0 =

˙ ς S T · ξl3 · Sd

0

2π

SR2 =





0

SR0 = 



2π



0

SL2 =



0



˙ ς, S T · ξl1 · Sd

0

˙ ς, ξ3  = S · ξl2  · Sd

0



2π

T

˙ ς, ξR3 = S T · ξr2 · Sd

0

KR1 =



˙ ς, ξR1 = S T · ξr0 · Sd

0





0

ξR2 = KL1 =



0

T

ξR0 =



˙ ς, ξL1 = S · ξl0 · Sd T

0

ξL2 = 

¨ ς S T · 1·Sd

˙ ς, DR3 = S T · Dr2 · Sd

 

2π

˙ ς S T · Dr1 · Sd

0 2π

˙ ς S T · Dr3 · Sd

0

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Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating for Rotating Engine in Experiment Ruixian Ma1(B) , Yaqing Wei1 , Quankun Li1 , Rui Wang1 , Mingfu Liao1 , Kaiming Wang2 , and Pin Lv3 1 Northwestern Polytechnical University, Xi’an 710072, China

[email protected]

2 AECC Commercial Aircraft Engine Co., Ltd, Shanghai 200241, China 3 Goldwind Science and Technology Co., Ltd, Urumqi 830026, China

Abstract. Rotor-stator rubbing is common in rotating engines, and the induced vibrational characteristics of the rotor and stator are affected by the rubbing materials. The soft coating made by vulcanized rubber is considered for rubbing case, followed by the experimental investigations on the kind of progressive rubbing between the seal-labyrinth and the seal-case of a rotor. The vibrational waveforms and the spectral features of the rotor and stator, together with the shaft orbits during the rubbing process are analyzed. The center of the rotor is forced to shift from the balance point of zero displacement, in the meanwhile, the vibrational acceleration of the stator is increased radically and finally nonlinear instability is excited during the rubbing. The shaft orbit varies from stable to unstable, and recovers to be stable eventually. Rotor-stator coupling resonance is captured both in the rotor and stator vibrational signals. High-order super-harmonics (corresponding to the resonance of the rotor and stator), the modulations of the fundamental rotating frequency are excited by the soft rubbing. Keywords: Rotor-Stator Rubbing · Coupling Resonance · Rotating Power Machines

1 Introduction In modern rotating engines, the clearance between the rotor and stator is generally designed to be quite small to reduce the power loss. As a result, the possibility and the severity of rubbing between the rotor and stator is also increased, which may induced by excessive vibration of the rotor. The resulting rubbing generates genitive effects on the engines safety. In general, two methods can be employed to reduce the effects of rotor-stator rubbing. One is identifying the rubbing characteristics, and then removing the rubbing faults in advance. The other one is attaching abradable coating on the surface where the rubbing ICANDVC2023 best presentation paper. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 15–24, 2024. https://doi.org/10.1007/978-981-97-0554-2_2

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occurs to relieve its impacts. Many researchers have contributed to the analysis of rotor vibration under the action of rubbing. Chu et al. [1] and Ma et al. [2] analyzed the vibration of a rotor with rubbing at various rotating speeds. Different features in terms of the sub-harmonics, high-order harmonics and the vibration amplitudes were observed for the rubbing occurring before and after the rotor critical speed. Sun et al. [3] carried out steady-state response analysis for a dual-rotor system rubbing. Additionally, prevenient studies show the vibration features of rotor-stator rubbing are affected by the abradable coating. The results obtained by Batailly et al. [4] suggested that the casing vibration acceleration during rubbing was effectively reduced by the seal coating. Yang et al. [5] simulated the fixed-point rubbing phenomenon with different bump materials on the casing, and the results suggested that softer bump material generated less impacts on the rotor vibration. The thermal effects and the wear of abradable coating for the blade-casing rubbing were considered in Refs. [6–8]. Their results showed that the blade vibration might strongly depends on the contacting status. Stringer et al. [9] studied the effects of immerging velocity on the wear mechanisms. In the calculations of rubbing between blade and coated casing performed by Berthoul et al. [10], the results proved the wear appearance along the circumferential direction and radial direction is related to the blade modes. In addition, the wear severity of the coating is strongly relevant to the coating material parameters. In general, published investigations of the rotor-stator coupling vibration for rubbing mainly focused on the type of blade-casing rubbing (especially for the blade resonance). On the other hand, the coatings were basically made of metal-based abradable material. In special novel seal structures, rubber has been utilized for the abradable coating of the casing, in conjunction with labyrinth-teeth manufactured on thin-shell drumlike rotor. This combination of rotor structure and the coating material is significantly different from the traditional design, may resulting in completely vibration features when rotor and stator rub However, to the authors acknowledgement, relevant studies are rare. In sight of the limitation mentioned above, an experimental rig is designed for the rotor-stator rubbing test with drum-like rotor and rubber-coated stator in this study. The vibration displacement of the rotor and the acceleration of the stator are measured during the rubbing process, and the vibration signals are analyzed to identify the rubbing induced features.

2 Experimental Design The schematic diagram of the test rig and the locations of vibration sensors are depicted in Fig. 1. The system is mainly consisted of rotor, stator and driven motor. The rotor is supported by three rolling bearings, two of which are located at the free end and the other is at the driven end. In special, the squeeze film damper (SFD) and elastic support are installed for the No.1 bearing at the free end and No.5 bearing at the driven end. As for the rubbing components, the rotor is a kind of thin-shell drum, wherein two labyrinth teethes are manufactured along the circumferential direction as is shown in the enlarged part in Fig. 1. The rubbing stator is a rubber-coated casing. The maximum power of the driven motor is 220 kW, and the maximum speed is 20000 r/min.

Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating

17

Fig. 1. Schematic diagram of test rig.

As can be seen in Fig. 1, the eddy current displacement sensor is adopted to measure the rotor drum vibration both along the horizontal direction and vertical direction. Accelerometer is used to collect the stator casing vibration, which is also installed both along the two directions. The parameters of these two kinds of vibrational sensors are presented in Table 1. Table 1. Parameters of sensors. Type

Sensitivity

Measurement range

Frequency Range

Location

Eddy current sensor

8 mV/µm

1.5 mm

0–2600 Hz

Rotor (drum)

Accelerometer

100 mV/g

±50 g

0–10000 Hz

Stator (casing)

3 Results and Discussion 3.1 Results of Rotor Vibration The variation of rotating speed along with data collecting time is shown in Fig. 2, wherein the speed is increased gradually in stepped form. However, the speed can be noticed to change abruptly around from 426 s to 432 s (as is presented in the enlarged figure of Fig. 2), which indicates the drum and the casing is rubbing. The fluctuation of the rotating speed is induced by the driven torque of the motor and opposing torque generated by rubbing. The rubbing is so serious that the opposing torque overcomes the driven torque, resulting into the deceleration of rotating speed. With the rubbing coating being worn off rapidly, the rotating speed is forced to increase again by the motor driven torque until it reaching the target speed of 2400 r/min.

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Fig. 2. Rotating speed versus time.

The time waveforms of drum displacement during rubbing period are presented in Fig. 3(a) and Fig. 3(b) for the horizontal direction and vertical direction respectively. It can be seen that during the period of 427 s to 429 s (428 s particularly), the displacement waveform in the vertical direction is changed dramatically. This period corresponds to the fluctuation cycle of rotating speed which is observed when sever rubbing occurs. Due to the rotor-stator rubbing, the vibration amplitude is decreased to some extent together with the symmetry breaking around the zero point of the rotor displacement. Conversely, the vibration displacement of drum is recovered to be stale again after 429 s, which indicates the rubbing becoming reduced.

(a) Horizontal direction

(a) Vertical direction

Fig. 3. Time waveforms of drum displacement during rubbing.

Figures 4(a)–4(f) demonstrate the rotor orbit of the drum during the rubbing cycle at different rotating speeds with time advancing. As is seen, the rotor orbit is stable at 2121 r/min before rubbing, showing a typical elliptical pattern. When rotor and stator start to rub, the rotor orbit is forced to be unstable relatively with a decrease of rotating speed to 2098 r/min. As the rubbing being more serious, the rotor orbit shows a chaos status, in agreement with the displacement waveforms at 428 s with a speed of 2031 r/min. After the rapid wear of rubber coating of the casing by the labyrinth teeth, the clearance between

Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating

19

the coating and labyrinth teeth is increased gradually, contributing to the recovery of rotor orbit stability, as illustrated from Figs. 4(d) to Figs. 4(f).

(a) 2121 r/min

(b) 2098 r/min

(c) 2031 r/min

(d) 2228 r/min

(e) 2305 r/min

(f) 2329 r/min

Fig. 4. Rotor orbit of the drum during rubbing.

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For comparison, the rotor orbits of the drum without rubbing at corresponding typical speed of Fig. 4 are depicted in Fig. 5, wherein stable vibration of the drum can be observed at all speeds distinctly. The comparisons with respect to Fig. 4 further suggest that the serious unsteady movements of the rotor at Figs. 4(b) and Figs. 4(c) are excited by the rotor-stator rubbing.

(a) 2038 r/min

(b) 2225 r/min

(c) 2326 r/min

Fig. 5. Rotor orbit of the drum without rubbing.

For the purpose of investigating the spectral features of rubbing induced vibration, Figs. 6(a)–Figs. 6(f) show the spectrums of drum displacement during rubbing in consistent with the rotating speeds in Fig. 4. As can be seen in Fig. 6(a), at the initial stage of rubbing the effects of rotor-stator contacting and friction are completely small that the dominated frequencies are the fundamental rotating frequency 35.38 Hz as well as the 2X component. With rubbing being serious (Figs. 6(b) and 6(c)), several higher order harmonics vibration are excited, including the 6X and 8X components. As for the vibration at 482.8 Hz in Fig. 6(c)(as well as 490 Hz in Figs. 6(d)–6(e)) and at 750 Hz in Figs. 6(b)–6(c), these components are contributed to the rotor-stator coupling vibration. In particular, the vibration at 750 Hz corresponds to the natural frequency of the casing, which will be shown in Sect. 3.2. The component around 490 Hz is attributed to the first order resonance of the drum as can be seen in Fig. 7. From Figs. 6(d)–6(f), the vibration

Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating

21

frequency of the drum gradually changed to be dominated by the fundamental frequency again, indicating the negligible effect of rubbing.

(a) 2121 r/min

(b) 2098 r/min

(c) 2031 r/min

(d) 2228 r/min

(e) 2305 r/min

(f) 2329 r/min

Fig. 6. Spectrums for drum displacement during rubbing.

(a) 499.95 Hz

(b) 860.69 Hz

Fig. 7. Natural frequencies and modes of rotor drum.

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3.2 Results of Stator Vibration The time waveforms of vibration acceleration of casing during rubbing period are presented in Fig. 8, wherein the horizontal and vertical vibrations are shown in Fig. 8(a) and 8(b) respectively. It can be seen that the initial phase of rubbing between 426s and 426.5 s, the vibration acceleration of casing increases gradually, nevertheless a sharp drop of acceleration amplitude can be noticed. After the sharp drop, the vibration amplitude shows a repetitive increase as before. However, at about 426.8 s, the acceleration amplitude abruptly increases to over 50 g, indicating the nonlinear instability phases of the casing under the action of rubbing.

(a) Horizontal direction

(b) Vertical direction

Fig. 8. Time waveforms of acceleration for rubbing casing.

Figure 9 presents the Fourier transform results for the data of the first graduallyincrease phase in Fig. 8, and the corresponding rotating speed is 2123 r/min. The characteristic frequency around 745 Hz can be noticed easily. This frequency is in agreement with the frequency of 750 Hz observed in rotor vibration. On the contrary, the vibration at the rotating frequency is negligible. In order to confirm the vibration source of the dominant frequency 745 Hz, the calculated first four orders of natural frequency and modes of casing are shown in Fig. 10. The fourth order frequency of 774 Hz can be noticed, which is identically in coincident with the characteristic frequency captured both for rotor and stator at around 750 Hz. Therefore, the rubbing excited resonance of the casing may contribute significantly to both rotor and stator vibration. In addition, modulated side-band can be observed at 709.9 Hz and 780.7 Hz, which are modulated by the resonance frequency 745 Hz of the stator and the rotating frequency of the rotor.

Vibrations Induced by Rubbing Between Labyrinth and Rubber-Coating

(a) Horizontal direction

(b) Vertical direction

Fig.9. Spectrums of acceleration for rubbing casing.

(a) 527.8 Hz

(b) 594.2 Hz

(c) 602.64 Hz

(d) 774.25 Hz

Fig. 10. Natural frequencies and modes of casing.

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4 Conclusions Vibration induced by the rubbing between a seal-labyrinth of drum-like rotor and a rubber-coated casing is investigated experimentally and the results are discussed both for the rotor displacement and stator acceleration. When the rubbing is serious, the center of the rotor is forced to shift from the balance point of zero displacement, along with the decrease of rotating speed. Both the higher order harmonics of 6X and 8X and the natural frequencies of the drum and the casing are excited by the rubbing. Strong coupling between the rotor and stator vibration can be identified. The natural frequencies of the drum and the casing all can be captured in the spectrums of the rotor vibration displacement. The vibration of the casing shows a character of nonlinear instability under the action of serious rubbing. A modulation between the rotating frequency and the stator natural frequency arises. Acknowledgements. The financial support from the National Key Research and Development Program (Grant 2022YFB4201400), the Innovation Capability Support Program of Shaanxi (Grant 2023-CX-TD-30), China Scholarship Council (202306290109) and the Fundamental Research Funds for the Central Universities (Grant D5000210481, 3102020OQD705) is gratefully acknowledged.

References 1. Chu, F.L., Lu, W.: Experimental observation of nonlinear vibrations in a rub-impact rotor system. J. Sound Vib. 283(3–5), 621–643 (2005) 2. Ma, H., Shi, C., Han, Q., et al.: Fixed-point rubbing fault characteristic analysis of a rotor system based on con-tact theory. Mech. Syst. Sig. Process. 38, 137–153 (2013) 3. Sun, C., Chen, Y., Hou, L.: Steady-state response characteristics of a dual-rotor system induced by rub-impact. Nonlinear Dyn. 86, 91–105 (2016) 4. Batailly, A., Legrand, M.: Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for flexible casings with friction and abradable coating. J. Sound Vib. 348, 344–364 (2015) 5. Yang, Y., Cao, D.Q., Xu, Y.Q.: Rubbing analysis of a nonlinear rotor system with surface coatings. Int. J. Non-Linear Mech. 84, 105–115 (2016) 6. Millecamps, A., Brunel, J.F., Dufrenoy, P., et al.: Influence of thermal effects during bladecasing con-tact experiments. In: Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2009-86842, San Diego (2009) 7. Nyssen, F., Batailly, A.: Sensitivity analysis of rotor/stator interactions accounting for wear and thermal effects within low-and high-pressure compressor stages. Coatings 10(1), 74.1– 74.23 (2020) 8. Agrapart, Q., Nyssen, F., Lavazec, D., et al.: Multi-physics numerical simulation of an experimentally predicted rubbing event in aircraft engines. J. Sound Vibr. 460, 114869.1–114869.25 (2019) 9. Stringer, J., Marshall, M.B.: High speed wear testing of an abradable coating. Wear 294–295, 257–263 (2012) 10. Berthoul, B., Batailly, A., Stainier, et al.: Phenomenological modeling of abradable wear in turbomachines. Mech. Syst. Sig. Process. 98, 770–785 (2018)

Semi-analytical Expression of Force and Stiffness of Perpendicular Polarized Ring Magnets for Nonlinear Dynamic Analysis Ying Zhang1 , Wei Wang2 , and Junyi Cao1(B) 1 Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, School

of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China [email protected] 2 School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou 450001, China

Abstract. Magnetic coupling arrays composed of ring permanent magnets have been widely applied in industrial occasions for achieving nonlinearity, such as passive magnetic bearings, quasi-zero stiffness isolators and multi-stable energy harvesters. These magnetic couplings can be divided into basic configurations including axial magnetization, radial magnetization, and perpendicular magnetization. For the purpose of structure design and parameter optimization, the semianalytical expressions of first two configurations have been analyzed to obtain high accuracy and low computational cost in previous literatures, while the semianalytical calculation of perpendicular magnetization has not still been investigated. Therefore, the semi-analytical expressions of magnetic force and stiffness for perpendicular polarized ring magnets are proposed. Then, the magnetic forces calculated by the proposed method, numerical simulation, and COMSOL software under different parameters are obtained. The results show that the proposed semi-analytical calculation has higher accuracy and less computational time than numerical simulation. Moreover, the influence of structural parameters on magnetic stiffness is analyzed. It can be demonstrated that with the increase of air gap, the decrease of the width of axial magnetized magnet, and the decrease of the height of axial magnetized magnet, the magnetic force and magnetic stiffness are both reduced. In general, the proposed semi-expression model can be applied for the design and optimization in the practical applications of ring permanent magnets. Keywords: Energy harvesting · Nonlinear dynamics · Vibration control · Magnetic modeling

1 Introduction The permanent magnets have been widely used in many significant occasions. For example, passive magnetic bearings are composed of several permanent magnets polarized in axial or radial directions [1–3]. In these occasions, many basic magnetic coupling © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 25–38, 2024. https://doi.org/10.1007/978-981-97-0554-2_3

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configurations are used to provide magnetic force and stiffness to maintain the working performance of devices, such as in quasi-zero stiffness (QZS) isolator [4, 5]. In these engineering applications, it is of great significance to design, analyze, and optimize the structural parameters of ring permanent magnets due to the limited space. Therefore, to obtain the magnetic force or magnetic stiffness produced by ring permanent magnets for nonlinear dynamic analysis is becoming an open issue. The magnetic dipole method can be used to calculate the magnetic field or the magnetic force [6], but regarding the permanent magnet as the dipole may yield low accuracy if two magnets have close distance. Besides, the magnetic charge theory has been widely applied to calculate the magnetic force of permanent magnets [7]. This method considers the magnetization of magnets as positive and negative magnetic charges distributed in the surfaces of magnets. Then, according to the Coulomb’s law, the magnetic force can be calculated by integrating the magnetic force between magnetic charges. The fully analytical expressions of perpendicular or parallel cubic magnets [8], and rotational cubic magnets [9] were investigated. However, it is difficult to obtain the accurate fully analytical expressions of magnetic force produced by ring permanent magnets. According to the Coulomb’s law, the numerical method can be used to calculate the magnetic force by dispersing magnetic charge [10], but the high computational cost is not suitable for structural design and parametric analysis. Some semi-analytical or analytical expressions have been proposed to calculate magnetic field produced by ring permanent magnets [11–13]. Ravaud et al. [14] presented analytical formulations of magnetic field of ring magnets according to Colombian approach. Babic et al. [15] proposed an improved Colombian-based analytical calculation of magnetic fields created by ring magnet. Besides, since ring permanent magnets can be approximately composed of tile magnets, the magnetic field of ring permanent magnets can be calculated by superposing the magnetic fields produced by tile magnets [16–18]. However, the analytical formulations of magnetic force exerted between two ring permanent magnets are difficult to be obtained up till now. To address this issue, Ravaud et al. [19, 20] deduced the semi-analytical expressions of magnetic force and stiffness of passive magnetic bearings using permanent magnets with axial magnetizations and radial magnetizations for parametric studies. Then, Ravaud et al. [21] presented simplified analytical expressions of magnetic force and stiffness for ring permanent magnets with perpendicular polarizations where the inner ring polarization is perpendicular to the outer ring polarization. Therefore, much effort has been devoted to calculating magnetic force and magnetic stiffness of ring permanent magnets by presenting semi-analytical or simplified analytical expressions. However, the semi-analytical expression of perpendicular polarization of ring magnet remains uninvestigated. In addition, the simplified calculation of perpendicular polarization does not take the magnetic charge volume density of radial-magnetized ring magnet into consideration. It would be unavailable for thick radial-magnetized ring magnet. Therefore, the semi-analytical expressions of the magnetic force and magnetic stiffness of perpendicular polarized ring permanent magnets are presented by taking both surface and volume charge densities into consideration. Then, the accuracy and the time

Semi-analytical Expression of Force and Stiffness

27

cost of the proposed semi-analytical expressions are compared with numerical simulation and COMSOL software. Finally, the influence of structural parameters of permanent magnets on magnetic force and magnetic stiffness is analyzed.

2 Semi-analytical Calculations of Force and Stiffness The circular Halbach array composed by ring magnets can be widely applied for QZS isolators [22] and multi-stable energy harvesters, as shown in Fig. 1. The spring and Halbach array can be combined to provide nonlinear stiffness. The QZS isolator can be applied to isolate the low-frequency vibration, while the multi-stable energy harvester can be beneficial for energy extraction from broadband excitation. In order to investigate the nonlinear characteristics, it is necessary to model the magnetic force of Halbach array. Therefore, this section will present semi-analytical expressions of the magnetic force and stiffness exerted by perpendicular polarized ring permanent magnets for calculating the magnetic stiffness of Halbach array. The perpendicular polarization is composed of two ring magnets whose polarizations are axial and radial respectively. The perpendicular polarization can be seen in many magnetic arrays, such as Halbach array. Moreover, the semi-analytical expressions proposed in this section can improve the calculating accuracy and reduce the computational time.

Fig. 1. QZS isolator and multi-stable energy harvester by using Halbach array

2.1 Notion and Geometry Figure 2 shows the basic configurations of ring permanent magnets which composes the Halbach array, including axial magnetization, radial magnetization, and perpendicular magnetization. These three basic configurations can be applied to achieve a variety of magnetic couplings. In this paper, the concern is the configuration of perpendicular magnetization. Therefore, the semi-analytical expression of magnetic force and stiffness for perpendicular magnetization will be derived. The representation of perpendicular magnetization with two ring magnets is shown in Fig. 3. For the ring magnet polarized in axial direction, r 1 and r 2 are the inner and

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Fig. 2. Basic configurations of ring permanent magnets: (a) axial magnetization, (b) radial magnetization, and (b) perpendicular magnetization.

outer radiuses, z1 and z2 are the bottom and top heights, θ 1 is the position angle from 0 to 2π, and the magnetization is J. For the ring magnet polarized in radial direction, R1 and R2 are inner and outer radiuses, Z 1 and Z 2 are bottom and top heights, θ 2 is the position angle from 0 to 2π, and the magnetization is J.

Fig. 3. Representation of perpendicular magnetization with two ring magnets: (a) space diagram, (b) sectional diagram.

2.2 Semi-analytical Expression of Magnetic Force The axial polarized and radial polarized ring magnets can be equivalent to magnetic charges. For uniform axial polarized magnet, there is only surface charge σ 1 = J. For non-uniform radial polarized magnet, there are surface charge σ 2 = J and volume charge ρ 2 = J/R, where R is the radius where the volume charge locates. In this research, the axial magnetic force F z is the concern. Therefore, the axial magnetic force F z can be calculated by Fz = FSS + FSV

(1)

where F SS is the axial magnetic force exerted by surface charges between axial and radial polarized magnets, F SV is the axial magnetic force exerted by the surface charge of axial polarized magnet and the volume charge of radial polarized magnet. For F SS , it can be expressed as FSS = fss (z1 , R1 ) + fss (z1 , R2 ) + fss (z2 , R1 ) + fss (z2 , R2 )

(2)

Semi-analytical Expression of Force and Stiffness

29

where  r  Z  2π  2π 2 2 σ σ (z − Z)rR fss (z, R) = − 1 2 dθ dθ dZ dr  3 1 2 4π μ0 r1 Z1 0 0 r 2 + R2 − 2rR cos(θ1 − θ2 ) + (z − Z)2 2

(3)

where μ0 is the permeability of the vacuum. According to the symmetry of axial magnetic force, f ss (z, R) can be rewritten as σ1 σ2 fss (z, R) = − 2μ0



r2 r1



Z2



Z1

2π 0

(z − Z)rR   3 dθ dZdr (4) r 2 + R2 − 2rR cos(θ ) + (z − Z)2 2

After the integration with respect to r and Z, Eq. (4) can be rewritten as  2π σ1 σ2 f1 (r2 , z, R, Z2 , θ ) − f1 (r1 , z, R, Z2 , θ ) − f1 (r2 , z, R, Z1 , θ ) 4π μ0 0 (5) +f1 (r1 , z, R, Z1 , θ )

fss (z, R) = −

where

 

2r − 2R cos(θ ) f1 (r, z, R, Z, θ ) = R RArcTanh cos(θ ) + α 2α  α = r 2 + R2 + (z − Z)2 − 2rR cos(θ )

(6) (7)

For F SV , it can be expressed as FSV = fsv (z1 ) + fsv (z2 )

(8)

 R  r  Z  2π  2π 2 2 2 σ σ (z − Z)r fsv (z) = − 1 2 dθ dθ dZ dr dR  3 4π μ0 R1 r1 Z1 0 0 2 2 r + R − 2rR cos(θ1 − θ2 ) + (z − Z)2 2

(9)

where

According to the symmetry of axial magnetic force, f sv (z) can be rewritten as  R  r  Z  2π 2 2 2 σ σ (z − Z)r fsv (z) = − 1 2 dθ dZ dr dR 3 2μ0 R1 r1 Z1 0  r 2 + R2 − 2rR cos(θ1 − θ2 ) + (z − Z)2 2

(10)

After the integration with respect to r, R and Z, Eq. (10) can be rewritten as  σ1 σ2 2π (f2 (r2 , z, R2 , Z2 , θ ) − f2 (r1 , z, R2 , Z2 , θ ) 2μ0 0 −f2 (r2 , z, R1 , Z2 , θ )+f2 (r1 , z, R1 , Z2 , θ ) + f2 (r2 , z, R2 , Z1 , θ ) −f2 (r1 , z, R2 , Z1 , θ ) − f2 (r2 , z, R1 , Z1 , θ )+f2 (r1 , z, R1 , Z1 , θ ))dθ fsv (z) = −

where

(11)

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f2 = −r 2 + 2Rα + 2r 2 ln(R − r cos(θ) + α)   2 − R − 2(z − Z)2 + R2 cos(2θ ) cot(θ) csc(θ) ln(r − R cos(θ) + α)   +(z − Z)2 csc2 (θ) ln −r 2 − 2(z − Z)2 + r 2 cos(2θ ) 

⎞  ⎛   4 sin3 (θ)σ β + γ − 2 R2 + (z − Z)2 sin(θ) + 2α (δ + φ)2 2 ⎟ i(z − Z) (iR + (z − Z) cot(θ)) csc(θ) ⎜ ⎟ + ln⎜ √ ⎝−  3 ⎠ δ+φ 2 2 2 (z − Z) (z − Z + ir sin(θ)) −((z − Z) cos(θ) + iR sin(θ)) 

⎞  ⎛   4i sin3 (θ) −β − γ − 2 R2 + (z − Z)2 sin(θ) + 2α (−δ + φ)2 2 ⎟ ⎜ (z − Z) (R + (z − Z) cot(θ)) csc(θ) ⎜ ⎟ ln⎝− − √  3 ⎠ −δ + φ 2 2 2 (z − Z) (z − Z + ir sin(θ)) −(i(z − Z) cos(θ) + R sin(θ))

(12)

 α=

r 2 + R2 + (z − Z)2 − 2rR cos(θ )

(13)

β = −2irz + 2irZ

(14)

γ = 2iR(z − Z) cos(θ )

(15)

δ = −i(z − Z) cos(θ )

(16)

φ = R sin(θ )

(17)

σ = (z − Z) cos(θ ) + iR sin(θ )

(18)

2.3 Semi-analytical Expression of Magnetic Stiffness The axial magnetic stiffness K z can be expressed as Kz = KSS + KSV

(19)

where K SS is the axial magnetic force produced by surface charges between axial and radial polarized magnets, K SV is the axial magnetic force produced by the surface charge of axial polarized magnet and the volume charge of radial polarized magnet. For K SS , it can be expressed as KSS = kss (z1 , R1 ) + kss (z1 , R2 ) + kss (z2 , R1 ) + kss (z2 , R2 )

(20)

where kss (z, R) = −

∂ σ1 σ2 ∂z 2μ0



r2

r1



Z2 Z1



2π 0

(z − Z)rR   3 dθ dZdr r 2 + R2 − 2rR cos(θ ) + (z − Z)2 2 (21)

Semi-analytical Expression of Force and Stiffness

31

After calculating the derivative with respect to z and the integration with respect to R and z, Eq. (21) can be expressed as  σ1 σ2 2π k1 (r2 , z, R, Z2 , θ ) − k1 (r1 , z, R, Z2 , θ ) kss (z, R) = − 2μ0 0 − k1 (r2 , z, R, Z1 , θ )+k1 (r1 , z, R, Z1 , θ )dθ (22) where

  R(z−Z) R2 + (z − Z)2 − rR cos(θ ) k1 (z, R) =    r 2 + R2 + z 2 − 2zZ + Z 2 − 2rR cos(θ ) R2 +(z − Z)2 − R2 cos2 (θ ) (23)

For K SV , it can be expressed as KSV = ksv (z1 ) + ksv (z2 )

(24)

where ksv (z) = −

    ∂ σ1 σ2 R2 r2 Z2 2π (z − Z)r dθ dZdrdR 3 ∂z 2μ0 R1 r1 Z1 0  2 r + R2 − 2rR cos(θ ) + (z − Z)2 2

(25)

After calculating the derivative with respect to z and the integration with respect to r and Z, Eq. (25) can be rewritten as  σ1 σ2 2π ksv (z) = − (k2 (r2 , z, R2 , Z2 , θ ) − k2 (r1 , z, R2 , Z2 , θ ) 2μ0 0 − k2 (r2 , z, R1 , Z2 , θ )+k2 (r1 , z, R1 , Z2 , θ ) + k2 (r2 , z, R2 , Z1 , θ ) −k2 (r1 , z, R2 , Z1 , θ ) − k2 (r2 , z, R1 , Z1 , θ )+k2 (r1 , z, R1 , Z1 , θ ))dθ (26) where  

1 R − r cos(θ) k2 = (z − Z) 4a tanh csc2 (θ) 4 α

   √ −2Rβ+2γ sin(θ)+2r √ √cos(θ)(β−R sin(θ)) cot(θ) −rβ + (z − Z)2 cot(θ) 2 2a tanh 2α δ+φ − √ β δ+φ



√  √ √  2(Rβ+γ sin(θ)−r √ cos(θ)(β+R sin(θ))) cot(θ) rβ + (z − Z)2 cot(θ) δ−φ 2 2a tanh α δ−φ − √ β −δ + φ

 α = r 2 + R2 + (z − Z)2 − 2rR cos(θ )  β = −(z − Z)2

γ = r 2 + (z − Z)2   δ = r 2 − (z − Z)2 − r 2 + (z − Z)2 cos(2θ )  φ = 2r sin(2θ ) −(z − Z)2

(27) (28) (29) (30) (31) (32)

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3 Simulation Verification This section is aimed to verify the effectiveness of the proposed semi-analytical model for the calculations of magnetic force and magnetic stiffness. The results from the semianalytical model, numerical simulation, and COMSOL software are compared, including the accuracy and the computational time cost. The semi-analytical model and the numerical simulation are both finished by MATLAB. The numerical simulation is to discrete the integrating variables directly. 3.1 Size Parameters The sizes of ring permanent magnets for simulation verification are listed in Table 1. There are three sizes used to compare the magnetic force and stiffness. Table 1. Sizes of ring permanent magnets for simulation verification r 1 [mm] r 2 [mm] z1 [mm] z2 [mm] R1 [mm] R2 [mm] Z 1 [mm] Z 2 [mm] Size 1 10

15

0

10

25

30

0

10

Size 2 10

20

0

10

30

40

0

10

Size 3 10

25

0

10

35

50

0

10

3.2 Magnetic Force The magnetic forces of perpendicular magnetization are compared among COMSOL, numerical simulation, and semi-analytical calculation, as shown in Fig. 4, with the magnetization J of 1.4 T. The displacement is changing from -20 mm to 20 mm, with the step of 1 mm. For numerical simulation, the steps of Z, r, R, and θ are 0.1 mm, 0.1 mm, 0.1 mm and 0.1 rad respectively. For the semi-analytical calculation, the step of θ is 0.1 rad. It can be seen from Fig. 4 that the varying trends of magnetic forces for size 1, size 2, and size 3 are all the same. The maximal magnetic force can be seen when the displacement is 0. With the increase of displacement, the magnetic force finally reaches the peak value in +y direction. In addition, the magnetic forces calculated by numerical simulation and semi-analytical expression are both close to the result of COMSOL. For size 1 in Fig. 4(a), the maximal magnetic forces of COMSOL, semi-analytical calculation, and numerical simulation are −36.794 N, −36.472 N and −36.207 N respectively when the displacement is 0. The errors of semi-analytical calculation and numerical simulation are 0.88% and 1.60% respectively. For size 2 in Fig. 4(b), the maximal magnetic forces from COMSOL, semi-analytical calculation, and numerical simulation are −100.54 N, −98.995 N and −98.338 N respectively when the displacement is 0. The errors of semi-analytical calculation and numerical simulation are 1.54% and 2.19% respectively. For size 3 in Fig. 4(c), the maximal magnetic forces of COMSOL,

Semi-analytical Expression of Force and Stiffness

33

Fig. 4. Magnetic forces under different sizes: (a) size 1, (b) size 2, (c) size 3.

semi-analytical calculation, and numerical simulation are −169.846 N, −166.024 N and −165.025 N when the displacement is 0. The errors of semi-analytical calculation and numerical simulation are 2.25% and 2.84% respectively. Therefore, it indicates that both semi-analytical calculation and numerical simulation can have high accuracy, and the semi-analytical calculation is more precise than numerical simulation. In addition, the comparisons of error and time for magnetic force calculation are listed in Table 2. The computational times of numerical simulation for size 1, size 2, and size 3 are 88.922 s, 345.870 s, and 766.836 s respectively, while the computational times of semi-analytical calculation are only 0.505 s, 0.525 s, and 0.526 s. It can be concluded that the accuracy and the computational time of semi-analytical calculation are both better than numerical simulation. Table 2. Comparisons of error and time for magnetic force calculation. Methods

Size 1

Size 2

Size 3

Error

Time

Error

Time

Error

Time

COMSOL

\

190 s

\

236 s

\

310 s

Numerical simulation

1.60%

88.922 s

2.19%

345.870 s

2.84%

766.836 s

Semi-analytical

0.88%

0.505 s

1.54%

0.525 s

2.25%

0.526 s

3.3 Magnetic Stiffness Figure 5 illustrates the magnetic stiffness under three sizes of magnets, including COMSOL, numerical simulation, and semi-analytical calculation, with the magnetization J of 1.4 T. The displacement is changing from -20 mm to 20 mm, with the step of 1 mm. For numerical simulation, the steps of Z, r, R, and θ are 0.1 mm, 0.1 mm, 0.1 mm and 0.1 rad respectively. For the semi-analytical calculation, the step of θ is 0.1 rad. It can be seen from Fig. 5 that when the displacement is 0, the magnetic stiffness is 0. With the increase of displacement, the magnetic stiffness can firstly increase to the peak value and then decrease to 0 for all sizes magnets. For size 1 in Fig. 5(a),

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the maximal magnetic stiffnesses from COMSOL, numerical simulation, and semianalytical calculation are 4.619 × 103 N/m, 4.486 × 103 N/m, and 4.567 × 103 N/m respectively. The errors of semi-analytical calculation and numerical simulation are 1.13% and 2.88% respectively. For size 2 in Fig. 5(b), the maximal magnetic stiffnesses from COMSOL, numerical simulation, and semi-analytical calculation are 1.0968 × 104 N/m, 1.0544 × 104 N/m, and 1.0755 × 104 N/m respectively. The errors of semianalytical calculation and numerical simulation are 1.94% and 3.87% respectively. For size 3 in Fig. 5(c), the maximal magnetic stiffnesses of COMSOL, numerical simulation, and semi-analytical calculation are 1.6581 × 104 N/m, 1.5791 × 104 N/m, and 1.6013 × 104 N/m respectively. The errors of semi-analytical calculation and numerical simulation are 3.43% and 4.76% respectively. Therefore, the results show that the magnetic stiffness of semi-analytical calculation can have higher accuracy than numerical simulation.

Fig. 5. Magnetic stiffnesses under different sizes: (a) size 1, (b) size 2, (c) size 3.

The comparisons of error and time for magnetic stiffness calculation is summarized in Table 3. The computational times of numerical simulation for size 1, size 2, and size 3 are 89.052 s, 345.953 s, and 766.890 s respectively, while the computational times of semi-analytical calculation are only 0.136 s, 0.136 s, and 0.138 s. Therefore, it can be concluded that the accuracy and the computational time of semi-analytical calculation are both better than numerical simulation. Table 3. Comparisons of error and time for magnetic stiffness calculation. Methods

Size 1

Size 2

Size 3

Error

Time

Error

Time

Error

Time

COMSOL

\

190.053 s

\

236.056 s

\

310.079 s

Numerical simulation

2.88%

89.052 s

3.87%

345.953 s

4.76%

766.890 s

Semi-analytical

1.13%

0.136 s

1.94%

0.136 s

3.43%

0.138 s

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35

4 Parameters Analysis It is well known that structural parameters have an important impact on magnetic force and magnetic stiffness in the perpendicular magnetization. Therefore, it is necessary to analyze the influence of structural parameters including air gap, width of axial magnetized magnet, and height of axial magnetized magnet using the proposed semi-analytical expressions. 4.1 Air Gap To analyze the influence of air gap, the widths w1 , w2 and heights h1 , h2 of axial polarized and radial polarized ring magnets are 10 mm and 10 mm, 10 mm and 10 mm respectively. Then, the different air gaps g can be obtained by changing R1 and R2 . Figure 6 shows that the magnetic forces and stiffnesses under different air gaps g, ranging from 2 mm to 6 mm, with the step of 0.1 mm. It can be seen from Fig. 6 that with the increase of g, the peak value of magnetic force decreases from 335.367 N to 170.397 N. The number of extremums is 3 when g is 2 mm, but it decreases to 1 when g increases to 6 mm. In addition, with the increase of g, the peak value of magnetic stiffness decreases from 50907.7 N/m to 22356.6 N/m.

Fig. 6. Magnetic force and stiffness under different air gaps: (a) magnetic force, (b) magnetic stiffness.

4.2 Width of Axial Magnetized Magnet To investigate the influence of the width of axial magnetized magnet, the air gap g is 10 mm, the width w2 of radial magnetized magnet is 10 mm, and the heights h1 , h2 of axial polarized and radial polarized ring magnets are 10 mm and 10 mm respectively. Figure 7 shows the magnetic forces and stiffnesses under different widths of axial magnetized magnet w1 , ranging from 2 mm to 6 mm, with the step of 0.1 mm. The result shows that with the increase of w1 , the peak value of magnetic force rises from 24.601 N to 65.272 N, while the peak value of magnetic stiffness increases from 3034.14 N/m to 7462.87 N/m.

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Fig. 7. Magnetic forces and stiffnesses under different widths of axial magnetized magnet: (a) magnetic force, (b) magnetic stiffness.

4.3 Height of Axial Magnetized Magnet For the analysis of the height of axial magnetized magnet, the air gap g is 10 mm, the widths w1 , w2 of axial and radial magnetized magnet are 10 mm and 10 mm respectively, and the height h2 of radial polarized ring magnets is 10 mm. Figure 8 illustrates the magnetic forces and stiffnesses under different heights of axial magnetized magnet h1 , ranging from 8 mm to 12 mm. With the increase of h1 , the peak value of magnetic force increases from 82.562 N to 112.800 N, the peak value of magnetic stiffness rises from 9143.22 N/m to 11879.2 N/m.

Fig. 8. Magnetic forces and stiffnesses under different heights of axial magnetized magnet: (a) magnetic force, (b) magnetic stiffness.

Generally, the quantitative analysis of magnetic force and stiffness under different parameters can be very beneficial for the design and optimization of ring permanent magnets.

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5 Conclusion The semi-analytical expressions of magnetic force and stiffness induced by perpendicular polarized ring permanent magnets are presented. By comparing with the results from COMSOL software, the accuracy of the semi-analytical expression is higher than numerical simulation, and the computational time of the semi-analytical expression is much shorter than numerical simulation for magnetic force and stiffness calculation. Based on the semi-analytical expression, the parameters quantitative analysis of magnetic force and magnetic stiffness is carried out. With the increase of air gap, the decrease of the width of axial magnetized magnet, and the decrease of the height of axial magnetized magnet, the magnetic force and magnetic stiffness are both reduced. Moreover, the proposed semi-analytical expressions can be beneficial to obtaining the expected magnetic force and stiffness in the design and the optimization of ring permanent magnets.

References 1. Yonnet, J.-P.: Passive magnetic bearings with permanent magnets. IEEE Trans. Magn. 14, 803–805 (1978) 2. Yonnet, J.-P.: Permanent magnet bearings and couplings. IEEE Trans. Magn. 17, 1169–1173 (1981) 3. Samanta, P., Hirani, H.: Magnetic bearing configurations: theoretical and experimental studies. IEEE Trans. Magn. 44, 292–300 (2008) 4. Zhou, Z., Chen, S., Xia, D., He, J., Zhang, P.: The design of negative stiffness spring for precision vibration isolation using axially magnetized permanent magnet rings. J. Vib. Control 25, 2667–2677 (2019) 5. Yan, B., Ma, H., Jian, B., Wang, K., Wu, C.: Nonlinear dynamics analysis of a bi-state nonlinear vibration isolator with symmetric permanent magnets. Nonlinear Dyn. 97, 2499–2519 (2019) 6. Yung, K.W., Landecker, P.B., Villani, D.D.: An analytic solution for the force between two magnetic dipoles. Magn. Electr. Separat. 9 (1970) 7. Furlani, E.P.: Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications. Academic Press, Cambridge (2001) 8. Akoun, G., Yonnet, J.-P.: 3D analytical calculation of the forces exerted between two cuboidal magnets. IEEE Trans. Magn. 20, 1962–1964 (1984) 9. Charpentier, J.-F., Lemarquand, G.: Optimal design of cylindrical air-gap synchronous permanent magnet couplings. IEEE Trans. Magn. 35, 1037–1046 (1999) 10. Furlani, E.P.: A formula for the levitation force between magnetic disks. IEEE Trans. Magn. 29, 4165–4169 (1993) 11. Selvaggi, J.P., Salon, S., Kwon, O.-M., Chari, M.V.K.: Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders by employing toroidal harmonics. IEEE Trans. Magn. 43, 3833–3839 (2007) 12. Conway, J.T.: Inductance calculations for noncoaxial coils using Bessel functions. IEEE Trans. Magn. 43, 1023–1034 (2007) 13. Conway, J.T.: Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils. IEEE Trans. Magn. 44, 453–462 (2008) 14. Ravaud, R., Lemarquand, G., Lemarquand, V., Depollier, C.: Analytical calculation of the magnetic field created by permanent-magnet rings. IEEE Trans. Magn. 44, 1982–1989 (2008) 15. Babic, S., Akyel, C.: Improvement in the analytical calculation of the magnetic field produced by permanent magnet rings. Progr. Electromagn. Res. C 5, 71–82 (2008)

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16. Ravaud, R., Lemarquand, G., Lemarquand, V., Depollier, C.: The three exact components of the magnetic field created by a radially magnetized tile permanent magnet. PIER 88, 307–319 (2008) 17. Ravaud, R., Lemarquand, G., Lemarquand, V.: Magnetic field created by tile permanent magnets. IEEE Trans. Magn. 45, 2920–2926 (2009) 18. Ravaud, R., Lemarquand, G.: Analytical expression of the magnetic field created by tile permanent magnets tangentially magnetized and radial currents in massive disks. Progr. Electromagn. Res. B 13, 20 (2009) 19. Ravaud, R., Lemarquand, G., Lemarquand, V.: Force and stiffness of passive magnetic bearings using permanent magnets. Part 1: axial magnetization. IEEE Trans. Magn. 45, 2996–3002 (2009) 20. Ravaud, R., Lemarquand, G., Lemarquand, V.: Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: radial magnetization. IEEE Trans. Magn. 45, 3334–3342 (2009) 21. Ravaud, R., Lemarquand, G., Lemarquand, V.: Halbach structures for permanent magnets bearings. PIER M 14, 263–277 (2010) 22. Zhang, Y., Liu, Q., Lei, Y., Cao, J., Liao, W.H.: Halbach high negative stiffness isolator: modeling and experiments. Mech. Syst. Sig. Process. 188, 110014 (2023)

On-Orbit Reconfiguration Dynamics and Control of Heterogeneous Intelligent Spacecraft Dengliang Liao, Xingyi Pan, Xilin Zhong, Zhengtao Wei, and Ti Chen(B) State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao Street, Nanjing 210016, China [email protected]

Abstract. An ultra-large space structure constructed by modular intelligent spacecraft can meet the mission requirements of variable environment and multiple working conditions through reconfiguration. This paper focuses on the dynamics and control problems in the on-obit reconfiguration of heterogeneous intelligent spacecraft, which consists of 125 rigid spacecraft modules and 2 flexible spacecraft modules. The relative position motion of the spacecraft is described by the Clohessy-Wiltshire (C-W) equations, and the attitude dynamics is expressed on SO(3). The reconfiguration mission is decomposed into three distinct phases: separation, unit reconfiguration and reassembly, where the unit reconfiguration phase is further divided into three steps: separation, pre-assembly and docking. The reassembly phase can also be divided into two steps: pre-assembly and docking. To ensure the safety of the reconfiguration mission, a compound controller which combines a collision avoidance controller and a PD controller is designed for the pre-assembly steps, while only PD control is used for the docking steps. Some numerical results are shown to verify the effectiveness of the proposed controller. Keywords: On-orbit reconfiguration · C-W equations · PD control · Heterogeneous spacecraft

1 Introduction Ultra-large space structures are usually constructed through on-orbit assembly, however, the assembled configurations are often fixed and designed in advance, which limits their applications. In addition, conventional spacecraft have long design, construction and deployment cycles and high maintenance costs, making it difficult to meet the needs of human exploration of the universe. A prospective way to solve these problems is adopting on-obit reconfiguration technology. On-obit reconfiguration enables spacecraft to adapt to specific mission requirements through configuration transformation. Moreover, modular reconfigurable spacecraft (MRS) can be troubleshoot and upgraded by module replacement, which makes the spacecraft highly redundant and robust, reduces the difficulty of on-orbit maintenance, increases the spacecraft’s expected lifetime, and also © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 39–50, 2024. https://doi.org/10.1007/978-981-97-0554-2_4

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helps increase the spacecraft’s resistance to destruction. For example, in the event of a space debris collision or enemy spacecraft attack, by configuration transformation, or separating into modules to evade attacks and reassembling after the threat is removed so as to achieve self-defense. Therefore, the dynamics and control problems in the on-orbit reconfiguration of heterogeneous intelligent spacecraft are worth studying. The on-obit reconfiguration technology has drawn significant attention among the aerospace powers. Since the beginning of the 21st century, countries such as Japan, Germany and the USA have conducted research on MRS, proposed concepts such as CellSat [1], iBOSS [2], and Phoenix project [3], respectively, and further developed typical applications such as on-orbit services and satellite-based exploration. As for recent years, Northwestern Polytechnic University proposed the concept of heterogeneous cellular satellite, and studied its reconfiguration planning strategy and distributed control algorithm [4]. The American Aerospace Corporation announced the Hive project, using pivoting cube modular satellite components for large-space structures, which can not only roll and crawl between modules but can self-separate and restore on orbit [5]. The European Commission funded the MOSAR project, aiming to complete the technical demonstration of MRS on orbit [6]. However, the research on the configuration scheme and reconfiguration strategy of ultra-large structures is still scarce, and most of them have not been tested on orbit and are still at the stage of prototype development and technology verification. Moreover, most of the MRS concepts proposed usually rely on space robotics or specially designed mechanical structure to realize configuration transformation. This paper mainly focuses on another way, that is separating into modules, then reassembling to the target configuration via rendezvous and docking. In this kind of MRS system, the Autonomous Multiple Spacecraft Assembly (AMSA) technology plays an important role, which has received considerable attention over the years. For example, Badawy and McInnes presented the autonomous on-orbit assembly of a large space structure using superquadratic surfaces to describe the assembly elements and defining the repulsive potential energy field based on the radial Euclidean distance [7]. Zou and Meng considered the absence of absolute and relative angular velocity information and proposed a distributed control algorithm to achieve the leader–follower cooperative attitude tracking of multiple rigid bodies on SO(3) [8]. Considering the flexible appendages additionally, Chen and his colleagues completed the autonomous assembly of a team of flexible spacecraft without inter-member collision using potential field based method [9]. The objectives of this paper are as follows: (1) to design control strategy to achieve reconfiguration of an ultra-large space structure; (2) to numerically validate the control strategy and the controller performance. In the remaining part of this paper, Sect. 2 describes the reconfiguration mission and gives the dynamic equations of the concerned system, and Sect. 3 presents the designed controller. Afterwards, Sect. 4 presents the numerical results. Finally, some conclusions are drawn in Sect. 5.

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2 Problem Formulation 2.1 On-Orbit Reconfiguration Mission As shown in Fig. 1, the on-orbit reconfiguration mission of concern in this study is to realize the autonomous reconfiguration of the heterogeneous intelligent spacecraft. The central primary structure of the spacecraft consists of 125 rigid spacecraft modules, while 2 flexible spacecraft are attached to its two ends. As illustrated in Figs. 1, 2, 3, 4, 5 and 6, the entire mission process includes three distinct phases, i.e., (a) the separation phase, (b) the unit reconfiguration phase, and (c) the reassembly phase. Firstly, the two flexible modules separate from the primary structure, afterwards, the primary structure separates into five secondary structures, then each of the secondary structures separates into five tertiary structures, independently. After phase (a) is completed, two flexible modules and 25 rigid tertiary structures are waiting at their desired position. During phase (b), each of the tertiary structures reconfigures from the cross-shape to the line-shape, which can be completed in three steps, namely, separation, pre-assembly and docking. Once the relative distance between the separated modules and the remaining structure is great enough, the separated modules start moving to the pre-assembly position then slowly approach to the assembly configuration and dock. Finally, the 25 tertiary structures reassemble to the primary structure in line-shape and the two flexible modules assemble to its two ends, which can be considered as a reverse process of phase (a). The assembly can also be further divided into two steps, i.e., the pre-assembly and the docking, similar to that in phase (b). In particular, the collision avoidance issues are taken into account for the pre-assembly steps in both phase (b) and phase (c).

Fig. 1. Two configurations of the heterogeneous intelligent spacecraft

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Fig. 2. Three Separations in phase (a)

Fig. 3. Three steps in phase (b)

Fig. 4. 1st Assembly in phase (c)

On-Orbit Reconfiguration Dynamics and Control

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Fig. 5. 2nd Assembly in phase (c)

Fig. 6. 3rd Assembly in phase (c)

2.2 System Dynamics Under the assumption that a Target Spacecraft (TS) is moving in a circular orbit and the distance between the TS and a Chaser Spacecraft (CS) is much smaller than the orbital radius, the relative translational motion of the CS with respect to the TS can be described by C-W equations [10]. As shown in Fig. 2, the system is described by three sets of coordinates, i.e., the earth-center inertial frame Fe  Oxe ye ze , the body frame on the TS Ft  Txyz, and the body frame on the ith CS Fci  Ci xci yci zci , where points T and Ci are the centers of mass of the TS and the ith CS, respectively. Note that for frame Txyz, x lies along the tangent to the orbit and positive x is in the direction counter to the TS’s direction of motion, y is in the direction of the radius from earth’s center to the TS’ s, and z is determined according to the right-handed system. The coordinate vector of point Ci in the frame Txyz is ri , and the coordinate vectors of points T and Ci in the frame Oxe ye ze is ρ t and ρ ci , respectively.

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Fig. 7. Relative motion coordinates

Based on the coordinate system above, the C-W Equations have the following form fxi /mi = x¨ i − 2ω˙yi fyi /mi = y¨ i + 2ω˙xi − 3ω2 yi

(1)

fzi /mi = z¨i + ω zi 2

where mi is the mass of the ith CS, ω is the angular velocity of the TS, fxi , fyi , fzi denote the components of the control force vector f of the ith CS represented in frame Txyz, and xi , yi , zi denote the components of the relative position vector ri shown in Fig. 7. The relative orientation between two frames can be represented by a rotation matrix R ∈ SO(3), , where SO(3) is the  group of 3 × 3 orthogonal matrices with the determinant of 1, i.e., SO(3) = {R ∈ R3×3 det(R) = 1, RT R = RRT = I 3 }. The attitude kinematics and dynamics of the ith rigid spacecraft can be described based on the rotation matrix as R˙ i = Ri s(ωi ) J i ω˙ i + s(ωi )J i ωi = ui

(2)

where Ri ∈ R3×3 is the rotation matrix from frame Fci to frame Fe , J i ∈ R3×3 is the inertia matrix with respect to frame Fci , ωi ∈ R3 is the angular velocity with respect to Fe and expressed in frame Fci , and ui ∈ R3 is the control moment expressed in frame Fci . For a vector a = [a1 , a2 , a3 ]T , s(a) is a skew-symmetric matrix defined as ⎡

⎤ 0 −a3 a2 s(a) = ⎣ α3 0 −a1 ⎦ −a2 a1 0

(3)

and the inverse operation of s is denoted by S, i.e., S(s(a)) =a. As for flexible spacecraft, the vibration of flexible appendages is mainly coupled to the rotation [11], therefore the C-W equations can still be used to describe their relative

On-Orbit Reconfiguration Dynamics and Control

45

translational motion, while the attitude kinematics and dynamics equations of the ith flexible spacecraft can be expressed as [12] R˙ i = Ri s(ωi ) J i ω˙ i + δ Ti η¨ i = −s(ωi )(J i ωi + δ Ti η˙ i ) + ui η¨ i + C ni η˙ i + K ni ηi = −δ i ωi

(4)

where ηi ∈ R3 is the modal coordinate vector, δ i ∈ Rn×3 is the coupling matrix between the attitude motion and the flexible vibration, C ni = diag{ 2ζj ωnj } ∈ Rn×n is the 2 } ∈ Rn×n is the stiffness matrix in which ζ and damping matrix and K ni = diag{ ωnj j ωnj are the jth order natural frequency and damping ratio, respectively. Note that only the first n elastic modes are considered in this paper.

3 Controller Design The module in the center of the primary structure is selected as the only TS for the entire reconfiguration mission to establish C-W equations with all the other modules or structures. In addition, the Leader Spacecraft (LS) for each phase is marked in yellow in Figs. 1, 2, 3, 4, 5 and 6. For efficient control, different spacecraft will act as the Leader Spacecraft (LS) at different progresses of the mission to be chased by their neighboring spacecraft, which are named Follower Spacecraft (FS). 3.1 Separation Phase To simulate the separation function of a spacecraft docking mechanism, controller with a spring force form on the ith FS is designed as f i = k(rLio + l − rLi )

(5)

where k and l are the stiffness coefficient and the original length vector of the spring, respectively, rLi = ri −rL is the relative position vector between the ith FS and its LS, and rLio is the initial vector of rLi . Once the spring force ends its action, i.e., ri  ≥ rio + l, a PD controller is designed to drive the separated FS to the desired position, which has following form f i = −kp1 (rLi − rLides ) − kd 1 r˙Li

(6)

where kp1 and kd 1 are the control gains, r˙Li is the vector of time derivative of rLi , rLides is the vector of desired relative position. 3.2 Unit Reconfiguration Phase The purpose of this phase is to reconfigure the 25 tertiary structures from cross-shape into line-shape, each of which can be completed in the following three steps.

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Step A. Separation. In this step, the controller on the separated spacecraft takes the same form as Eq. (5). Since the modules are maneuvering in a relatively small range in this phase, PD controller is not added for separation, i.e., the separated FS moves freely with an initial velocity obtained from the spring force. Step B. Pre-assembly. Once the relative distance between the separated FS and the LS is greater than a certain value, i.e., ri  ≥ rsep , the system switches into Step B., where the attitude of spacecraft needs to be taken into account. In this paper, the attitude error and angular velocity error between ith spacecraft and jth spacecraft are defined as eωij = ωj − Ri Rj ωi 1 eRij = S(RTi Rj − RTj Ri ) 2

(7)

Based on this, the PD controller used to synchronize the attitude of ith FS with its LS is designed as ui = −kp2 eRiL − kd 2 eωiL

(8)

Moreover, it is necessary to consider the collision avoidance issue in this step to avoid the undesired collisions. The collision avoidance force for the ith spacecraft comes from the following avoidance potential  2 N  kavj (rij  − δij2 )2 Vavi = (9)   2 2  2 − d 2 )2 j=1 (δij − dij )( rij ij where N is the number of the collision avoidance objects, rij = ri − rj denotes the vector of relative position between the ith spacecraft and the jth object, δij and dij represent the radius of the danger and avoidance zones, respectively. In this paper, dij is chosen as the summation of the radius of the involved envelope circles. kavj is defined as   0 rij  ≥ δij   kavj = (10) k0 rij  ≤ δij where k0 is a positive constant. Then, the collision avoidance force acting on the ith FS reads   N k (r 2 − δ 2 )r  avj ij ∂Vavi ij ij =− (11) f avi = −  2 2 ∂rij   ( rij − d )3 j=1

ij

Hence, the compound controller used to drive the separated FS to the pre-assembly position reads f i = −kp1 (rLi − rLides ) − kd 1 r˙Li + f avi

(12)

For the reason that the collision avoidance force may prevent the controlled spacecraft from converging to the desired position, the switching conditions for entering Step C

On-Orbit Reconfiguration Dynamics and Control

47

are the relative position and relative velocity perpendicular to the direction of docking, the attitude error and the angular velocity error simultaneously satisfying the prescribed thresholds, i.e., eωiL  ≤ δω , eRiL  ≤ δR , rLi (n) ≤ δr and r˙Li (n) ≤ δv . In this paper, x(n) denotes the nth component of vector x. For example, if docking is processing along rLi (1), then the corresponding switching conditions are rLi (2) ≤ δr , rLi (3) ≤ δv , r˙Li (2) ≤ δr and r˙Li (2) ≤ δv . Step C. Docking. Only PD control is used in this step, the controllers take the same form as Eqs. (6) and (8). If the relative position between the two spacecraft is bounded within a given small value δd , the docking is considered complete.

3.3 Reassembly Phase The assemblies in this phase are completed by two steps, i.e., pre-assembly and docking, where the controllers are identical to those in 3.2.

4 Numerical Simulations In this section, numerical simulations are conducted to demonstrate the effectiveness of the proposed controller and reconfiguration strategy. In this paper, rigid module is considered as a uniform cube, with a side length of 0.5 m, a mass of 62.5 kg, and an inertia matrix of diag{2.6, 2.6, 2.6}kg · m2 . The flexible module has a mass of 300 kg, an inertia matrix of [350 3 4; 3 270 10; 4 10 190] kg · m2 , and its rigid central body is also considered as a uniform cube with a side length of 0.5 m, while the span of its flexible appendage takes 10 m. The other parameters of flexible module are taken as ⎡ ⎤ 6.4565 −1.2564 1.1169 δ i = ⎣ 1.2781 0.9176 2.4890 ⎦ kg1/2 · m/s2 , 2.1563 −1.6726 −0.8367 ωn1 = 0.7681 rad/s, ωn2 = 1.1038 rad/s, ωn3 = 1.8733 rad/s, ζ1 = 0.0056, ζ2 = 0.0086, ζ3 = 0.013. At the beginning of the mission, the primary structure is considered in a circular orbit with an angular velocity of 7.2722 × 10−5 rad/s, whose initial attitude is Rin = I 3 . Some of the controller gains and the desired relative positions rLides for each phase are represented in Table 1. Note that rLides are listed in the order marked in Figs. 2, 3, 4, 5 and 6. The attitude controller gains are always chosen as kp2 = 1, kd 2 = 20 for rigid spacecraft, and kp2 = 15, kd 2 = 25 for flexible modules. The parameters of spring are chosen as k = 1000(N/m), l = 0.05(m). The collision avoidance parameters are shown in Table 2 and k0 in Eq. (10) is always chosen as 1. In addition, parameters of switching condition are always chosen as rsep = 1.5 (m),δω = 0.001 (m), δR = 0.001 (m), δr = 0.01 (m) and δv = 0.001 (m). Figure 8 shows the response of the relative positions of all spacecraft to the TS with time during the whole process of reconfiguration. Note that the reconfiguration mission is conducted in the orbital plane, so the relative positions along z-axis are always zero. Three separations are realized at around 60s, 150s and 240s, respectively. From 300s to 1000s, the unit reconfiguration is completed, and all 25 cross-shaped tertiary structures are reconfigured into line-shaped tertiary structures. From 1000s to 2000s, 5 groups of

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D. Liao et al. Table 1. Controller gains and rLides used in each phase

Table 2. Avoidance Parameters used in the simulations

tertiary structures were assembled into 5 secondary structures. From 2000s to 3000s, 5 secondary structures were assembled into the primary structure in line-shape. In the time from 3000s to 4500s, two flexible spacecraft are assembled to the two ends of the primary structure, and the whole reconfiguration mission is completed. As shown in Figs. 9 and 10, the attitude error and modal oscillations of the flexible spacecraft gradually decay within a very small range from 0s to 3000s. At 3000s, the flexible spacecraft starts to carry out attitude adjustment, at which time the peak value of modal oscillation increases, and as the attitude error gradually decreases, the modal oscillation also gradually decays, and the peak value of modal oscillation decays from 0.018 to 0.0005 during the time from 3000s to 4000s.

On-Orbit Reconfiguration Dynamics and Control

Fig. 8. Relative position of all spacecraft and the TS in the whole mission

Fig. 9. Attitude error of flexible spacecraft and the TS

Fig. 10. Modal coordinates of flexible spacecraft in the whole mission

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5 Conclusions In this paper, we solve the on-orbit reconfiguration of heterogeneous intelligent spacecraft and describe the dynamics involved based on the C-W equations and SO(3). The reconfiguration mission is carried out in stages, corresponding controllers are designed, and collision avoidance force is incorporated in the pre-assembly. Numerical simulations are performed for the whole process of the mission. The experiments will be performed to demonstrate the reconfiguration concept and the controller performance in future studies. Acknowledgments. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12102174 and 11832005, and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant No. MCMS-I-0122K01).

References 1. Tanaka, H., Yamamoto, N., Yairi, T., Machida, K.: Reconfigurable cellular satellites maintained by space robots. J. Robot. Mechatron. 18(3), 356–364 (2006) 2. Michael, G., et al.: Modular robots for on-orbit satellite servicing. In: 2012 IEEE International Conference on Robotics and Biomimetics (ROBIO). IEEE (2012) 3. David, B., et al.: Phoenix program status-2013. In: AIAA SPACE 2013 Conference and Exposition (2013) 4. Haitao, C., et al.: Cellular space robot and its interactive model identification for spacecraft takeover control. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE (2016) 5. Taylor, A.B., Helvajian, H.: HIVE: a space architecture concept (2020) 6. Pierre, L., et al.: MOSAR: modular spacecraft assembly and reconfiguration demonstrator. In: 15th Symposium on Advanced Space Technologies in Robotics and Automation (2019) 7. Badawy, A., McInnes, C.R.: On-orbit assembly using superquadric potential fields. J. Guid. Control. Dyn. 31(1), 30–43 (2008) 8. Zou, Y., Meng, Z.: Velocity-free leader–follower cooperative attitude tracking of multiple rigid bodies on SO(3). IEEE Trans. Cybern. 49(12), 4078–4089 (2018) 9. Chen, T., Wen, H., Haiyan, H., Jin, D.: On-orbit assembly of a team of flexible spacecraft using potential field based method. Acta Astronaut. 133, 221–232 (2017) 10. Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (1960) 11. Mazzini, L.: Flexible Spacecraft Dynamics, Control and Guidance. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-25540-8 12. Hu, Q., Xiao, B.: Fault-tolerant sliding mode attitude control for flexible spacecraft under loss of actuator effectiveness. Nonlinear Dyn. 64, 13–23 (2011)

Study on the Effect of Angular Misalignment on the Contact Load and Stiffness of Cylindrical Roller Bearings Zihang Li1,2 , Xilong Hu1,2 , Choangyang Wang1,2 , Haoze Wang1,2 , and Lihua Yang1,2(B) 1 State Key Laboratory for Strength and Vibration of Mechanical Structures,

Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China [email protected] 2 School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China

Abstract. Cylindrical Roller Bearings (CRB) find extensive application in rotating machinery. However, the issue of bearing angular misalignment has received limited research attention. Therefore, a four-degree-of-freedom quasi-static model of CRB is introduced, specifically considering angular misalignment in two directions. The model enables an investigation into the impact of angular misalignment on load distribution, contact moment, and bearing stiffness. The obtained results reveal that angular misalignment in two radial directions only marginally affects load distribution but significantly influences the contact moment. Furthermore, it is observed that misalignment angles in different directions have varying effects on rollers positioned at distinct azimuths. Moreover, the influence of angular misalignment on angular stiffness is considerably more substantial than its impact on radial stiffness. These findings carry significant implications for addressing misalignment concerns in CRB. Keywords: Cylindrical Roller Bearings · Angular Misalignment · Quasi-static mode · Stiffness

1 Introduction Rolling element bearings find extensive application in high-speed precision machine tools, high-speed trains, aero engines, and other high-speed power systems. They possess the unique capability to withstand both radial and axial loads, exhibiting remarkable attributes such as high speed, stability, and reliability. Among various types of rolling element bearings, Cylindrical Roller Bearing (CRB) stands out as a representative example. Due to their ability to endure substantial radial loads, they are commonly employed in situations involving heavy loads or when higher stiffness and fatigue resistance characteristics are required. At the beginning of the last century, Stribeck [1] first studied the load distribution of rolling element bearings and proposed that the maximum load-carrying rolling elements can be calculated by a constant, which is 4.37 for ball bearings. Later Palmgren [2] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 51–63, 2024. https://doi.org/10.1007/978-981-97-0554-2_5

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refined the method and proposed that for roller bearings, the constant is 5. Sjovall [3] considered the bearing radial clearance and proposed an integral method to calculate the bearing load distribution. Jones [4] introduced centrifugal force and gyroscopic moment into the bearing mechanics model, breaking the limitation that the previous model could not be used for high-speed applications. Harris [5] presented to use the slicing method to calculate the load distribution of roller bearings, which is more accurate and widely used in later studies. DeMul et al. [6, 7] developed a general mathematical model with highly modular characteristics for ball and roller bearings, which can obtain the overall bearing stiffness matrix. Tong and Hong [8–12] extended DeMul’s model and could consider the effect of misalignment based on the original model. Numerous scholars have conducted extensive and in-depth research on the mechanical model and mechanical characteristics of CRB. Some researchers have even explored the effects of angular misalignment. However, several shortcomings still persist. The current research primarily focuses on angular misalignment in ball bearings and tapered roller bearings, while comparatively less attention has been devoted to misalignment in CRB. Furthermore, existing studies concerning angular misalignment in CRB primarily examine its impact on fatigue life, while the investigation into bearing load distribution, contact moment, and overall stiffness remain limited. To address these research gaps, it is essential to establish a four-degree-of-freedom quasi-static model for CRB to comprehensively investigate the influence of misalignment on bearing load and stiffness. Such findings would significantly contribute to assessing the performance of bearing load and stiffness due to misalignment in CRB.

2 Quasi-Static Model of CRB 2.1 Calculation of Contact Load Between Roller and Raceway Figure 1 depicts a schematic diagram of a four-degree-of-freedom quasi-static model for CRB. In Fig. 1a, the global coordinate system is illustrated, with the center O representing the fixed  outer ring of the bearing. The inner ring, subjected to a load  position of the FT = Fy , Fz , My , Mz in two radial directions, generates corresponding displacements   expressed as δT = δy , δz , ζy , ζz . The azimuth angle of each roller is denoted by φj . Figure 1b displays the local coordinate system for each roller, with the center Oj located at the roller’s center. The displacement of the inner ring under load can be expressed as uT = (ur , θ ) in the local coordinate system. Additionally, due to the deformation of inner ring, the rollers undergo corresponding displacements, represented as vT = (vr , ϕ) in the local coordinate system. The relationship between the global displacement δ of the inner ring in the global coordinate system and the displacement u in the local coordinate system can be converted using the transfer matrix [Rφ ]. Thus, the relationship between the two can be expressed as follows   cos φ sin φ 0 0 [Rφ ] = (1) 0 0 − sin φ cos φ

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Global coordinate system

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Local coordinate system

Fig. 1. Four-degree-of-freedom quasi-static model of CRB

The force state of each roller is represented by the red line in Fig. 2, and the equilibrium equation of the roller considering centrifugal force is     Qi − Qe + Fc 0 = (2) 0 Mi − Me where, Qi , Qe , M i and M e are the contact load and moment between the roller and raceway respectively, which are calculated by the slice method. This method can simply simulate the non-Hertzian line contact, analyze the contact state between the roller and raceway, and give the loading condition of each slice. The width of each slice is lk which can be expressed as lk = Lw /ns Assuming that the shaded part of the figure is the k th slice of the roller in contact with the raceway, the contact force can be expressed as 10

qk = ck9 lk

(3)

where c is the material-related coefficient, for steel bearings, it can be calculated by the following formula c=

1 1

(4)

1.24 × 10−5 Lw9

k is the contact compression deformation between the k th slice of the roller and the raceway, which can be expressed as k = 0 + ξ lk − hk

(5)

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Fig. 2. Scematic diagram of roller force state and slicing method

hk is the gap between the roller and the raceway resulting from the convexity of the roller for logarithmic profile modification hk = 0.00035Da ln(

1 1 − (2 Llkw )2

For the contact between roller and inner raceway ⎧ ⎨  = u − v − Pd 0 r r 4 ⎩ ξ =θ −ϕ For the contact between roller and outer raceway ⎧ ⎨  = v − Pd 0 r 4 ⎩ ξ =ϕ

)

(6)

(7)

(8)

Thus the forces and moments between each roller and the raceway can be expressed as

⎧ ns ⎪ ⎪ ⎪ Q = qk ⎪ ⎪ ⎨ k=1

ns ⎪ ⎪ ⎪ ⎪ M = qk lk ⎪ ⎩

(9)

k=1

2.2 Calculation of Stiffness Following the definition of the stiffness matrix for a multi-degree-of-freedom nonlinear system, the dynamic stiffness matrix of CRB can be mathematically expressed as the

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partial derivative of the external load F with respect to the bearing displacement δ, denoted as K = ∂FT /∂δ T . The derivation process for obtaining the overall stiffness matrix is as follows:     Fc 0 B = Bi + Be + = (10) 0 0 Therefore, the forces and moments of the inner and outer rings are expressed as

T T Bi = Qi , Mi , Be = Qe , Me , respectively. According to the definition of stiffness, the equilibrium equation is derived from the local displacement vector v to obtain the Jacobi matrix, which is the local contact stiffness matrix of the bearing ∂B ∂Bi ∂Be = T + T = JBi + JBe = JB ∂vT ∂v ∂v Where

⎡

JBi

∂Qi ⎢ ∂vr =⎢ ⎣ ∂Mi ∂vr

⎡ ⎤ ⎤ ∂Qi ∂Qe ∂Qe − − ⎢ ∂ϕ ⎥ ∂ϕ ⎥ ⎥, JBe = ⎢ ∂vr ⎥ ⎣ ⎦ ∂Me ∂Mi ∂Me ⎦ − − ∂ϕ ∂vr ∂ϕ

(11)

(12)

According to Eq. (12), a partial derivative of the equilibrium equation with respect to the displacement component u of the inner ring yields: ∂Bi ∂Be + T =0 ∂uT ∂u

(13)

For the convenience of subsequent calculations, w = u − v is defined here, so the following relations can be obtained: Bi = Bi (w), Be = Be (v), and v is considered as a function about u, i.e., v = v(u) Observing that for constant u, d w = −d v    10 ∂v ∂Bi ∂Be ∂v · − + T · T =0 (14) ∂vT ∂uT ∂v ∂u 01 Thus the implicit relation can be obtained ∂v = JB−1 JBi ∂uT

(15)

It is known that Y = −Bi , so it can be converted to ∂Be ∂Y = = JBe JB−1 JBi ∂uT ∂uT

(16)

According to the chain derivative rule, the overall equilibrium equation of CRB to the overall displacement vector δ for Jacobi matrix can get the bearing stiffness matrix K, that is K=

∂F ∂δT

=−

Z j=1

[[Rφ ]T Y [Rφ ]]j

(17)

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where the local contact stiffness matrix of the rolling body Y = ∂Y/∂uT The obtained displacement variables are brought into the derived stiffness matrix expression to obtain the stiffness matrix of the CRB ⎤ ⎡ kyy kyz kyζy kyζz ⎥ ⎢ k ⎥ ⎢k k k (18) K = ⎢ zy zz zζy zζz ⎥ ⎣ kζy y kζy z kζy ζy kζy ζz ⎦ kζz y kζz z kζz ζy kζz ζz

2.3 Equilibrium Equation of CRB The overall solution flow is shown in Fig. 3.

Fig. 3. Quasi-static model solving process of CRB

Substitute Eqs. (3) to (9) into Eq. (2) to obtain the static equilibrium equations of each roller. The equilibrium equations of each roller contain only two unknown quantities vr and ϕ, under the premise that the global displacement δ of the inner ring of the bearing is known. The Newton-Raphson method is used to solve the nonlinear equation system, and the contact load between the roller and raceway can be calculated after the roller displacement is obtained. When the roller local equations are solved, the contact load between the inner race and the roller can be converted to the global coordinate system of the inner ring of the bearing through the transfer matrix, and all the loads acting on

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the inner ring of the roller can be summed up to obtain the global equilibrium equations of the bearing, expressed as: F+

Z

[Rφ ]Tj Yj = 0

(19)

j=1

where Z indicates the number of rollers of the bearing, the subscript j refers to the number jth roller. The unknown quantity required to be solved is the overall displacement of the inner ring, and the same as the roller local balance equation system, the overall balance equation system of the inner ring is also solved by the Newton-Raphson method. It is worth noting that, with this method, the solution process needs to calculate the inner ring equilibrium equations of the Jacobi matrix, which is equivalent to the global bearing stiffness matrix Eq. (17), only in the value of the difference between a negative sign.

3 The Effect of Angular Misalignment Misalignment is a critical factor in rolling element bearing applications and often cannot be effectively avoided. Xu [13] classifies misalignment into three types: parallel misalignment, angular misalignment and combined misalignment. Parallel misalignment is often caused by axial preload, while angular misalignment can be caused by a number of factors. Because CRB generally can not bear axial load, so here ignore its axial misalignment, only to study the impact of angular misalignment. The misalignment state is shown in Fig. 4. When misalignment occurs in the bearing, the inner (outer) ring is equivalent to an equivalent pre-displacement load being applied, which can be expressed as δTmis = (ζy , ζz , γy , γz )

(20)

The angular misalignment displacement is denoted as δTmis = (0, 0, γy , γz )

(21)

And it is kept constant during the solution of the model.

4 Model Validation The results in Ref. [14] were used for verification, and the bearing parameters are shown in Table 1. The verification results are shown in Fig. 5, where the working parameters are: radial clearance 0.03494 mm, inner ring speed of 15000 rpm, as well as radial load of 350 N, 1500 N and 4000 N respectively. Since the results in the reference do not consider misalignments, the validation is also performed in that case. It can be seen from Fig. 5 that the model agrees well with the literature results, and there is only a small deviation at the radial load of 4000 N. The overall is within the error range, which can verify the correctness of the model in this paper.

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(a) Ideal alignment

(b) Angular misalignment

Fig. 4. Angular misalignment of CRB

Table 1. The parameters of CRB Parameters

Value

Parameters

Value

Outer diameter

75 mm

Radial clearance

0.03494 mm

Inner diameter

55 mm

Roller length

9 mm

Pitch diameter

65 mm

Roller number

14

Fig. 5. Quasi-static model validation

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5 Results and Discussion As shown in Fig. 6, the contact load and moment between roller and raceway change with angular misalignment, the operating conditions are radial clearance 0.01 mm, inner ring speed 15000 rpm, radial load 10000 N and remaining unchanged.

Fig. 6. Contact loads and moments vary with angular misalignment

It can be seen that the angular misalignment of the two radial directions has little effect on the contact load distribution, but has a significant effect on the contact moment. The contact moment of the roller located at 0° under the action of γy is equal to 0, while the contact moment of the roller at this position under the action of γz is the largest. Such a difference is due to the external load in the y direction on the inner ring of the bearing. In Fig. 7, the variation of roller load density with angular misalignment is presented. The rollers are numbered from the bottom, ranging from 0 to Z-1 in counterclockwise order. Both Fig. 7(a) and 7(b) display the load density of roller 0, which represents the contact load between each slice and the inner ring.

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(a) The qi of the roller 0 varies with γy

(c)

The qi of the roller 1 varies with γy

(b)

The qi of the roller 0 varies with γz

(d) The qi of the roller 1 varies with γz

Fig. 7. Roller load density varies with angular misalignment

Observing the results, it can be noted that γy has a minor effect on the load density of roller 0, whereas the impact of γz is considerably more significant. As the misalignment value increases, the load density on the side with a smaller effective length of the roller decreases, while the load density on the side with a larger effective length increases. Conversely, the effects of γy and γz in Fig. 7(c) and 7(d) exhibit an opposite trend, with the impact of γz being generally more pronounced. Figure 8 illustrates the variation of bearing stiffness with angular misalignment. The black line represents the radial stiffness, while the red line represents the angular stiffness. The solid line corresponds to the y-directional stiffness, and the dashed line represents the z-directional stiffness. Upon observation, it becomes evident that K yy is relatively insensitive to changes in angular misalignment in both directions. In Fig. 8(a), K zz is initially small when the bearing is aligned, experiences a sudden increase with γy , and subsequently fluctuates within a small range. In Fig. 8(b), K zz exhibits a slow initial increase with γz , followed by a nonlinear upward trend as γz becomes larger. The

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(a) Stiffness changes with γy

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(b) Stiffness changes with γz

Fig. 8. Stiffness varies with angular misalignment

trends of the angular stiffness in both directions are similar, displaying a rapid decrease initially, followed by a leveling off as the angular misalignment increases. Furthermore, the stiffness Kθyθy decreases by 30.77% and 52.31% under the influence of γy and γz , respectively. The effect on Kθzθz is relatively comparable, showing a similar trend. Figure 9 demonstrates the trend of stiffness with misalignment angle at different radial clearances. Observing the results, it can be noted that the K yy is generally larger, while Kθzθz is greater under the influence of angular misalignment. The impact of different radial clearances on K yy is minimal, showing little variation. On the other hand, K zz undergoes significant changes only when e = 0.01, and the magnitude of change decreases as the radial clearance increases. Additionally, Kθzθz experiences significant changes under the influence of angular misalignment, with the effect of γz being noticeably greater than that of γy . The angular stiffness both display a non-linear decrease as the misalignment angle increases. This non-linear trend becomes more apparent at smaller misalignment angles and tends to flatten out as the misalignment angle becomes larger.

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(a) Radial stiffness changes with γy

(c) Angular stiffness changes with γy

(b) Radial stiffness changes with γz

(d) Angular stiffness changes with γz

Fig. 9. Variation of stiffness with angular misalignment at different radial clearances

6 Conclusion In this paper, the angular misalignment effect in two directions is considered to establish a four-degree-of-freedom quasi-static model of CRB, and the roller and raceway contact load and stiffness are studied in depth, and the conclusions are as follows: (1) In the radial force F y under the action of two directions of angular misalignment has little effect on the contact load distribution between rollers and raceways, but the contact moment has a significant impact. The larger the γy , the larger the contact moment, but the lowermost roller is not subject to the moment load. And under the action of γz , all bearing area rollers are subject to moment load, and the smaller the azimuth angle, the greater the load.

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(2) For the roller at position 0, the roller load density under the action of γy is almost constant, while the effect of γz is very significant. The misalignment angle in both directions has an effect on the load density of the roller at position 1, and the trend is opposite. (3) The effect of angular misalignment on the angular stiffness is much larger than that on the radial stiffness. The radial stiffness changes very little, while the angular stiffness tends to decrease nonlinearly when the misalignment angle is small, and tends to level off when the misalignment angle is large. A similar pattern is observed for the stiffness at different radial clearances. In general, the larger the radial clearance, the smaller K yy , K zz and Kθyθy are, while the larger Kθzθz .

References 1. Stribeck, R.: Ball bearings for various loads. Trans. ASME 29, 420–463 (1907) 2. Palmgren, A.: Ball and Roller Bearing Engineering. SKF Industries Inc, Philadelphia (1959) 3. Sjoväll, H.: The load distribution within ball and roller bearings under given external radial and axial load. Teknisk Tidskrift 9, 895–921 (1933) 4. Jones, A.B.: A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions. J. Basic Eng. 82(2), 309–320 (1960) 5. Harris, T.A.: Rolling bearing analysis. Wiley 4, 1–481 (1966) 6. DeMul, J.M., Vree, J.M., Maas, D.A.: Equilibrium and associated load distribution in ball and roller bearings loaded in five degrees of freedom while neglecting friction—Part I: general theory and application to ball bearings. J. Tribol. 111(1), 142–148 (1989) 7. De Mul, J.M., Vree, J.M., Maas, D.A.: Equilibrium and associated load distribution in ball and roller bearings loaded in five degrees of freedom while neglecting friction—Part I: general theory and application to ball bearings. ASME J. Tribol. 111(1), 142–148 (1989) 8. Tong, V.-C., Hong, S.-W.: The effect of angular misalignment on the stiffness characteristics of tapered roller bearings. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 231(4), 712–727 (2017) 9. Tong, V.C., Hong, S.W.: The effect of angular misalignment on the running torques of tapered roller bearings. Tribol. Int. 95, 76–85 (2016) 10. Tong, V.-C., Hong, S.-W.: Fatigue life of tapered roller bearing subject to angular misalignment. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 230(2), 147–158 (2016) 11. Tong, V.-C., Hong, S.-W.: Study on the running torque of angular contact ball bearings subjected to angular misalignment. Proc. Inst. Mech. Eng. Part J 232(7), 890–909 (2018) 12. Tong, V.C., Hong, S.W.: Modeling and analysis of double-row cylindrical roller bearings. J. Mech. Sci. Technol. 31, 3379–3388 (2017) 13. Xu, T.F.: Study of the Effect of Misalignment on the Mechanical Characteristics of Paired Angular Contact Ball Bearings and Their Rotor Dynamics. Xi’an Jiaotong University (2023) 14. Zhang, Z.L., et al.: Dynamic characteristic analysis of high speed cylindrical roller bearing. J. Aerosp. Power 26(02), 397–403 (2011)

Dynamic Modeling and Features of GTF Engine Rotor System Heyu Hu1 , Bin Shi2 , Tianxiang Wang1 , Quankun Li1(B) , Mingfu Liao1 , Kang Zhang3 , and Fali Yang4 1 School of Power and Energy, Northwestern Polytechnical University, Xi’an, China

[email protected]

2 Aero Engine Corporation of China, Chengdu, China 3 Aviation Military Representative Office of Army Armament, Department in Chengdu,

Chengdu, China 4 Aero Engine Corporation of China, Shenyang, China

Abstract. To consider the problem that coupling, support position, bearing form and connection mode of the rotor system caused by adding a gear system between the fan and the compressor in the Geared Turbo Fan (GTF) engine, dynamic modeling and features of the GTF engine rotor system are studied in this paper. For the modelling, a star gear meshing model including herringbone teeth and five-way shunt is designed for star gear train; the inner gear ring connected to the fan rotor and the sun gear connected to the low-pressure turbine rotor are meshed through the gear meshing unit, and then the rotor dynamic model including disk, shaft, support structure and membrane disk coupling is established by finite element method. After that, some dynamic features like modal properties and unbalanced responses are analyzed through the proposed dynamic model of the GTF engine rotor system. The combined model of low-pressure unit and star gear unit is established, which can provide an efficient method for the dynamic analysis and design of GTF engine. Keywords: GTF Engine · Dynamic Model · Dynamic Feature · Finite Element Method

1 Introduction With the rapid development of civil aviation engine technology, low fuel consumption, low pollution emissions and high safety have become the main objectives of high bypass ratio turbofan engines. Improving thermal efficiency of engine and improving propulsive efficiency of engine are the main methods of reducing fuel consumption rate. Improving the propulsive efficiency of engine has become the main means now, GTF engine is a typical representative [1–3]. The GTF engine adds a star gear system behind the fan so that the fan is no longer directly driven by the low-pressure turbine shaft, and the fan and the low-pressure turbine do not need to work at the same speed. Therefore, when the fan speed is 3000 ~ 4000 r · min−1 , the compressor and low-pressure turbine system can work at the speed of 8000 ~ 10000 r · min−1 [4]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 64–76, 2024. https://doi.org/10.1007/978-981-97-0554-2_6

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Researchers at home and abroad have carried out a number of studies on gearbox modeling and gear transmission rotor dynamics of GTF engines. In order to make the fan reach an ideal power level, Buyukataman et al. [5] gave examples in the whole structural, gearbox design and bearing selection of GTF engine. Combined with the dynamic characteristics of gearbox, cooling system design, lubricating oil layout, material selection and other factors, the guiding opinions on the relatively high performance of herringbone planetary gear reducer supported by sliding bearing are given. Dent [6] studied the feasible powertrains of gearboxes such as offset shafting, star gear train and inner gear ring output. For the whole structural, Zhang Dezhi et al. [7] analyzed the unique characteristics of GTF engine from the aspects of whole structural and fuel supply mode. The technical problems of gearbox matching design, rotor system support structure adjustment and optimal working speed design of each rotor are emphasized. Wang et al. [8] compared the support modes of low-pressure rotor and fan rotor of GTF engine, studied the influence of bearing on the dynamic characteristics of rotor and gearbox, and analyzed the load sharing characteristics of star gearbox. For the design of gears, Feng et al. [9] analyzed the related technical problems of gear transmission structure and the related design of five-way shunt star gear reducer. Starting from the flexible design of the input shaft of GTF engine, Liu et al. [10] optimized the response surface of the input shaft of GTF engine by finite element analysis. Based on the concept of gear transmission system of GTF engine, Hou et al. [11] analyzed the structure characteristics of star gearbox of GTF engine. Gear structure design, gear fatigue strength and planetary gear structure optimization design are studied. Through CFD simulation, Bao et al. [12] studied the windage loss of the tooth surface, end face and inner hole characteristic surface of herringbone planetary gear. Ma et al. [13–15] considered that the transmission error and geometric eccentricity have great influence on gear dynamics. Based on the beam model, the full-degree-of-freedom motion model of the planetary gearbox is established. The response characteristics of the gear rotor under different coupling relationships are analyzed. When there are multiple planetary gear shafts in the gear rotor system, the motion of the planetary gear shaft cannot be simplified as a node. Therefore, the influence of planetary gear shaft motion should be fully considered when establishing the motion model of GTF engine rotor system.

2 Rotor Dynamics Model of GTF Engine The finite element method has great advantages in establishing the motion equation of the rotor system [16–18]. Therefore, this section will use the finite element method to establish the motion equation of the GTF rotor system. The model of rotor can be divided into discrete rigid disks, shafts with distributed mass and elasticity, and discrete bearings with stiffness and damping.

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2.1 Low-Pressure Rotor Units The low-pressure rotor units include disk unit, shaft unit and bearing unit, related motion of equations are derived as follows. Disk kinetic energy equation can be obtained, as: ⎡ ⎤T ⎡ ⎡ ˙ ⎤T ⎡ ⎤⎡ x˙ ⎤ ⎤⎡ θ˙ ⎤ θx x˙ x md 0 0 Id 0 0 1 ⎢ ⎥ ⎥ 1 ⎥ ⎢ ⎢ d 1⎢ ⎥ ⎣ T = ⎣ y˙ ⎦ 0 md 0 ⎦⎣ y˙ ⎦+ ⎣ θ˙y ⎦ ⎣ 0 Id 0 ⎦⎣ θ˙y ⎦+ 2d Ip − d θx θ˙y Ip 2 2 2 0 0 md 0 0 Id z˙ z˙ θ˙z θ˙z (1) where, md is the quality of the disk; Id is the diameter moment of inertia of the disk equivalent to the node; Ip is the polar moment of inertia of the disk equivalent to the node. According to the Lagrange equation,  d ∂T d ∂T d = Qd (2) − dt ∂ q˙ ∂q and substituting (1) into (2), the motion equation of the disk in the fixed coordinate system is constructed, as M d q¨ d − G d q˙ d =Qd where mass matrix and gyroscopic effect matrix are ⎤ ⎡ ⎡ md 0 0 0 0 0 0000 ⎢0 m 0 0 0 0 ⎥ ⎢0 0 0 0 d ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ d ⎢ 0 0 md 0 0 0 ⎥ d ⎢ 0 0 0 0 M =⎢ ⎥G =⎢ ⎢ 0 0 0 Id 0 0 ⎥ ⎢0 0 0 0 ⎥ ⎢ ⎢ ⎣ 0 0 0 0 Id 0 ⎦ ⎣ 0 0 0 −Ip 0 0 0 0 0 Id 0000

(3)

0 0 0 Ip 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0

The derivation process of the motion equation of the shaft element is the same as that of the disk element. Motion equation of the shaft element is shown, as M e q¨ e − G e q˙ e +K e qe =Qe

(4)

where, M e is the shaft mass matrix; G e is the gyroscopic effect matrix; K e is the shaft stiffness matrix. Considering linear stiffness and damping, the bearing motion equation is constructed, as C b q˙ b + K b qb =Qb

(5)

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where damping matrix and stiffness matrix are ⎡ ⎡ ⎤ cxx kxx ··· cθz x ··· ⎢ ⎢ ⎥ cyy kyy ⎢ ⎢ ⎥ ⎢. ⎢. ⎥ .. ⎢ ⎢ ⎥ . . . czz . kzz ⎥K b =⎢ . C b =⎢ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ cθx θx kθx θx ⎢ ⎢ ⎥ ⎣ ⎣ ⎦ cθy θy kθy θy cxθz ... cθz θz kxθz ...

kθz x .. .

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⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

kθz θz

When the cross stiffness and damping are ignored, the elements on the non-diagonal line are all zero. 2.2 Membrane Disk Coupling The membrane disk coupling is used to connect the low-pressure turbine shaft and the gearbox. The concentricity can be compensated. The structure of the membrane disk coupling is designed as shown in Fig. 1.

shaft1

membrane disk coupling

shaft2

Fig. 1. Structure diagram of membrane disk coupling.

The main part of the membrane disk coupling is equivalent to the shaft according to the actual structure size. The connection part between the membrane disk coupling and the two ends of the shaft is equivalent to spring-damping. The coupling relationship is established. Damping matrix and stiffness matrix are shown [19]:



  kc −kc cc −cc Kcoup = Ccoup = (6) −kc kc −cc cc where kc = diag[kxx , kyy , kzz , kθx θx , kθy θy , kθz θz ]; cc = diag[cxx , cyy , czz , cθx θx , cθy θy , cθz θz ], The stiffness and damping are determined by the parameters of the actual structure. 2.3 Helical Gear Meshing Unit The meshing relationship between the sun gear and the planetary gear is derived by the concentrated mass method [20]. The coupling motion relationship between star gear train and rotor system is established. The vibration of the planetary gear is described in the coordinate system as shown in Fig. 2.

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planetary gear fan shaft inner gear ring

y

turbine shaft

x

Op

O

X

sun gear

Fig. 2. Coordinate description diagram of rotor system.

In the coordinate system, each gear has three translational degrees of freedom and three rotational degrees of freedom. The displacement vector of the gear pair element is described as follows: q = [xi , yi , zi , θxi , θyi , θzi , xj , yj , zj , θxj , θyj , θzj ] The sun gear is denoted by the subscript s. The planetary gear is denoted by the subscript p. The inner gear ring is denoted by the subscript c. The dynamic model of planetary gear is constructed as shown in Fig. 3. yj

planetary gear

θ yj

θ xj x j

sun gear

yi θ yi

β

θ zj

β

zj

θxi xi

θ zi

inner gear ring

zi

Fig. 3. Pairwise meshing model diagram.

According to the meshing theory, the value of the meshing force is related to the meshing stiffness and damping, which can be expressed as F = cm [δ˙1 (t)−δ˙2 (t)]+km [δ1 (t)−δ2 (t)]

(7)

where cm is the meshing damping; km is the meshing stiffness; δ1 (t) and δ2 (t) are the ·

·

1

2

vibration displacement of two gears along meshing line; δ (t) and δ (t) are the vibration velocity of two gears along meshing line. The meshing relationship of the sun gear as the driving wheel is deduced. The base circle and meshing line of the sun gear and planetary gear are projected onto the XOY

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plane. The meshing model of sun gear and planetary gear is established as shown in Fig. 4.

planetary gearP

sun gearS

ys

O

meshing α ps point ϕp

yp

Op

xp

xs

Fig. 4. Sun gear and planetary gear meshing diagram.

As shown in Fig. 4, αps is the pressure angle; ϕpn is the planetary gear position angle. The three translational degrees of freedom of the planetary gear are projected in the direction of the meshing line, as: ⎧ ⎪ ⎨ Wxp = xp sin ψspn cos β Wyp = yp cos ψspn cos β (8) ⎪ ⎩ Wzp = zp sin β where xp , yp , zp are the displacement of translational freedom of planetary gear; Wxp , Wyp , Wzp are the projection of translational displacement of planetary gear on meshing line. The three rotational degrees of freedom of the planetary gear are projected in the direction of the meshing line, as ⎧ ⎪ ⎨ Wθxp = −rprp θxp sin ϕpn sin β ⎪ ⎩

Wθyp = rprp θyp cos ϕpn sin β Wθzp = −rprp θzp cos β cos αps

(9)

where θxp , θyp , θzp are the planetary gear angular displacement; Wθxp , Wθyp , Wθzp are the projection of rotational displacement of planetary gear on meshing line. The three rotational degrees of freedom of the sun gear are projected in the direction of the meshing line, as ⎧ ⎪ ⎨ Wθxs = rprs θxs sin ϕpn sin β Wθys = −rprs θys cos ϕpn sin β (10) ⎪ ⎩ Wθzs = rprs θzs cos β cos αps The three translational degrees of freedom of the sun gear are projected in the direction of the meshing line, as ⎧ ⎪ ⎨ Wxp = xp sin ψspn cos β Wyp = yp cos ψspn cos β (11) ⎪ ⎩ Wzp = zp sin β

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The total relative displacement of the two gears can be expressed as δsp = (Wxs − Wxp ) + (Wys − Wyp ) + (Wzs − Wzp ) +(Wθxs − Wθxp ) + (Wθys − Wθyp ) + (Wθzs − Wθzp )

(12)

The gear meshing force can be expressed as Fn = Cm δ˙sp +Km δsp

(13)

where Cm is the gear meshing damping; Km is the gear meshing stiffness. 2.4 GTF Engine Rotor System Motion Equation The mass matrix, gyroscopic effect matrix, stiffness matrix, damping matrix, coupling matrix and meshing matrix are assembled. The system is divided into nodes. When meshing occurs at nodes i and j, the meshing matrix is as follows:



 Mss Msp kii kij = Kmesh = kmesh Mps Mpp kji kjj The four block matrices in the meshing matrix are assembled to the corresponding node positions of the subscript according to the meshing nodes. The unit matrix assembly diagram is shown in Fig. 5. node node node

nodeL

nodeM

node 1

node1

shaft unit

node

disk unit nodeL

bearing unit meshing unit

nodeM

membrane disk coupling

node 1

node1

Fig. 5. Meshing matrix assembly diagram.

The motion equation of the rotor system is established as M s q¨ s + (C s − s G s )˙qs + K s qs = 0

(14)

where C s is the total damping matrix of the system; K s is the total stiffness matrix of the system; G s is the total gyroscopic effect matrix of the system; s is the total speed matrix of the system.

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The state vector is used to solve Eq. (14). The critical speed and vibration mode of the rotor system are obtained. The state vector is defined, as   h = q˙ s ; qs (15) Assuming h = h0 eλt , the characteristic equation of the rotor system is obtained, as   −(Ms )−1 (Cs − Gs ) −(Ms )−1 Ks h = λh (16) [I] [0] The problem of solving the modal of the system is transformed into the eigenvalue and eigenvector of the characteristic equation.

3 Simulation Analysis of GTF Engine Rotor System 3.1 Node Division Based on the real engine, the GTF engine rotor model is established as shown in Fig. 6. The main characteristics of the GTF engine include fan disk, fan shaft, star gearbox, compressor disk, turbine disk, low-pressure turbine shaft, bearing, planetary carrier and membrane disk coupling, with a total length of about 2300 mm. The rotor structure diagram is shown in Fig. 6.

fan disk

rear fulcrum front fulcrum front fulcrum of compressor of the fan of the gearbox the LP turbine disk

turbine disk

inter-shaft fulcrum of the LP turbine

rear fulcrum of the LP turbine

Fig. 6. Rotor structure diagram.

The GTF engine rotor is a 0–2-G-1–0-2 support structure. Rear fulcrum of the fan and front fulcrum of the gearbox are double row thrust bearings. Front fulcrum of the low-pressure turbine is deep groove ball bearing. Inter-shaft fulcrum of the low-pressure turbine and rear fulcrum of the low-pressure turbine are cylindrical roller bearings. All planetary bearings are cylindrical roller bearings. Each component is divided into nodes as shown in Fig. 7.

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disk node 











 

























shaft node meshing node





bearing node

 

















 





















Fig. 7. Model node Division.

3.2 Modal Analysis The GTF engine rotor system is established, which includes fan rotor, low-pressure turbine rotor and planetary gear rotor. The resonance of the whole rotor system may be excited by each rotor. Therefore, considering different rotors as the main excitation rotor, the critical speed of each order of the system is obtained. Under the main excitation of the fan rotor, the critical speed and modal shape are calculated as shown in Fig. 8. The solid line is the calculated result of unbalanced response amplitude of the measuring point.

Relative displacement

fan disk LP turbine disk fulcrum gearbox

Axial position / m

(a) Fan rotor excited cambell diagram

(b) Fan rotor excited modal shape

Fig. 8. Fan rotor excited cambell diagram and modal shape

The first critical speed of the system is 2320 r · min−1 . The modal shape excited by the fan rotor is the turbine second bending mode. Under the main excitation of the low-pressure rotor, the critical speed and modal shape are calculated as shown in Fig. 9.

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Relative displacement

fan disk LP turbine disk fulcrum gearbox

Axial position / m

(a) LP rotor excited cambell diagram (b)1st LP rotor excited modal shape

Axial position / m

(c)2nd LP rotor excited modal shape

fan disk LP turbine disk fulcrum gearbox

Relative displacement

Relative displacement

fan disk LP turbine disk fulcrum gearbox

Axial position / m

(d)3rd LP rotor excited modal shape

Fig. 9. LP rotor excited cambell diagram and modal shape

The first critical speed of the system is 3046 r · min−1 . The first modal shape excited by the low-pressure rotor is the turbine second bending mode. The second critical speed of the system is 6293 r · min−1 . The second modal shape excited by the low-pressure rotor is the first-order pitch vibration mode of fan. The third critical speed of the system is 6971 r·min−1 . The third modal shape excited by the low-pressure rotor is second-order bending vibration mode of turbine. 3.3 Unbalance Responses Under the fan disk 0° and 160 g.cm unbalance, the steady-state unbalance response amplitude changes of the fan disk, the compressor disk and the turbine disk are shown in Fig. 10. Each disk is insensitive to the unbalanced excitation of the fan disk. The vibration state of the low-pressure turbine shaft is less affected by the fan rotor, and the vibration amplitude does not exceed 1µm in the working speed.

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(a) fan disk

(b)compressor disk and turbine disk

Fig. 10. The steady-state unbalance response of the fan disk 0° and 160 g.cm unbalance

Under the turbine disk 180° and 160 g.cm unbalance, the steady-state unbalance response amplitude changes of the fan disk, the compressor disk and the turbine disk are shown in Fig. 11.

(a) fan disk

(b)compressor disk and turbine disk

Fig. 11. The steady-state unbalance response of the turbine disk 180° and 160 g.cm unbalance

The vibration amplitude of the turbine disk is close to 100 µm. The low-pressure turbine shaft excites the first critical speed of the fan rotor. The vibration amplitude of the fan disk is close to 10 µm. The unbalanced excitation of the turbine rotor has little effect on the fan rotor.

4 Conclusions and Discussions Aiming at the rotor system of GTF engine, a modeling design method is proposed. The star gear-rotor coupling model is established. Theoretical research are carried out. The conclusions are as follows: (1) The low-pressure unit model of GTF engine rotor including disk unit, shaft unit and support unit is established. Based on the meshing theory, the low-pressure unit model and star gearbox unit are combined. The rotor dynamics model of GTF engine is constructed.

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(2) The modal analysis of the rotor dynamics model of GTF engine is carried out. The calculation results show that fan rotor excited cambell diagram exists one critical speed. LP rotor excited cambell diagram exists three critical speeds. The modal shapes of each order are reasonable. (3) The unbalanced response of the rotor dynamics model of GTF engine is analyzed. The calculation result shows that each disk is insensitive to the unbalanced excitation of the fan disk. The vibration state of the low-pressure turbine shaft is less affected by the fan rotor. The unbalanced excitation of the turbine rotor has little effect on the fan rotor. Acknowledgements. This work is supported by the Fundamental Research Funds for the Central Universities (3102020OQD705), China Scholarship Council (202306290109), National Key Research and Development Program, Grant (2022YFB4201400) and the Innovation Capability Support Program of Shaanxi (Grant 2023-CX-TD-30).

References 1. Chen, C.H., Xin, Q.: Conceptual design of fan gear drive system. Aeronaut. Sci. Technol. 131(04), 8–11 (2011) 2. Xue, B.J., Zhou, Y.: Analysis of design technical characteristics of gear-driven turbofan engine with large culvert ratio. Sci. Technol. Innovation 40(13), 40 (2017) 3. Cao, S.Q., Hou, L.L.: Summary of research on design and dynamics of GTF engine. J. Mech. Eng. 55(13), 53–63 (2019) 4. Ma, Y.H., Cao, C., Hao, Y., Zhang, B., Hong, J.: Structure and dynamics analysis of rotor system of gear drive fan engine. J. Aeronaut. Power 30(11), 2753–2761 (2015) 5. Buyukataman, K., Wilton, S.: Gearbox system design for ultra-high bypass engines. In: Joint Propulsion Conference (2013) 6. Dent, E., Hirsch, R.A.: Design of aircraft turbine fan drive gear transmission system. In: Advances in Aeronautical Science and Engineering, pp. 1–88 (1970) 7. Zhang, D.Z., Zhang, J.X., Wang, F.: Analysis of structural design characteristics of gear-driven turbofan engine. Aeronaut. Engine. Mach. 37(04), 1–4 (2011) 8. Wang, S., Zhu, R., Jin, F.: Study on load sharing behavior of coupling gear-rotor-bearing system of GTF aero-engine based on multi-support of rotors. Mech. Mach. Theory 147, 103764 (2020) 9. Feng, J., Qin, K., Wang, D.: Discussion on conceptual design of star fan drive gearbox. Prog. Aeronaut. Eng. 6(4), 490–495 (2015) 10. Liu, C., Fang, Z.D.: Response surface design of flexible transmission shaft of GTF reducer. Mech. Transm. 42(6), 53–57 (2018) 11. Hou, M.X., Li, J.H., Zhang, M.Q.: Study on design technology of planetary gear transmission system of GTF engine. Aero-engine 40(2), 61–64 (2014) 12. Bao, H.Y., Wang, C.L., Lu, F.X.: Analysis of wind resistance loss of planetary gear in fandriven gearbox of GTF engine based on CFD. J. Central South Univ. (Natural Science Edition) (4), 971–978 (2020) 13. Ma, H., Wang, Q.B., Huang, J., Zhang, Y.M.: Vibration characteristics of coupled helical gear rotor system with different degrees of freedom. Vibr. Test. Diagn. 34(04), 650–657 (2014) 14. Ma, H., Wang, Q.B., Huang, J., Zhang, Y.M.: Vibration response analysis of helical gear coupled rotor system considering geometric eccentricity. J. Aeronaut. Power 28(01), 16–24 (2013)

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15. Ma, H., Zhu, L.S., Wang, Q.B., Zhang, Y.M.: Analysis of modal coupling characteristics of helical gear-parallel shaft rotor system. Chin. Proc. Electr. Eng. 32(29), 131–136 (2012) 16. Doirflein, T.M., Wilton, S.A., Allmon, B.L.: Method and apparatus for supporting rotor assemblies during unbalances, US (2004) 17. Branagan, M.: Rotor dynamic Analyses Using Finite Element Method. University of Virginia (2014) 18. Liao, M.F.: Aeroengine Rotor Dynamics. Northwest University of Technology Press (2015) 19. Michael, B.: Rotor dynamic Analyses Using Finite Element Method. University of Virginia (2014) 20. Liu, C., Qin, D., Lim, T.C., et al.: Dynamic characteristics of the herringbone planetary gear set during the variable speed process. J. Sound Vib. 333(24), 6498–6515 (2014)

Nonlinear Dynamic Analysis of Rub-Impact Rod-Fastening Combined Rotor Systems with Internal Damping Chongyang Wang1,2 , Zihang Li1,2 , Haoze Wang1,2 , Xilong Hu1,2 , and Lihua Yang1,2(B) 1 State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong

University, Xi’an 710049, Shaanxi, China [email protected] 2 School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China

Abstract. The objective of this study is to examine the nonlinear dynamic behavior of the rod-fastening combined rotor-bearing (RFCR) system with rub-impact, with a specific focus on the influence of internal damping. A dynamic model for the RFCR system is formulated using the finite element method based on the Lagrange equation. This model takes into account the system’s nonlinear characteristics, including rub-impact forces and oil-film forces. Through the analysis of bifurcation diagrams, Poincaré maps, time-domain plots, and frequency spectra, the effects of internal damping and stator stiffness on the system’s instability and nonlinear response are investigated. The findings reveal that both internal damping and stator stiffness have a significant impact on the vibration and instability of the RFCR system at varying speeds. Particularly, systems with rubbing faults exhibit pronounced nonlinearity and instability in high-speed regions. Furthermore, the presence of internal damping disrupts the stability of the P3 motion in the system, and the effect of divergence becomes more pronounced as the friction coefficient increases. In conclusion, considering internal damping is crucial when undertaking dynamic modeling and analysis of such complex rotors, as it plays a vital role in fault diagnosis and vibration control for practical RFCR systems. Keywords: RFCR · Internal damping · Rub-impact · Nonlinear dynamics

1 Introduction The rod-fastening combination rotor (RFCR), as a mechanical component of gas turbines, significantly influences the overall performance and safety of the system. Within the rotor system, the existence of nonlinear effects such as rub-impact and oil film leads to more complex and highly nonlinear vibration behavior. The discontinuity in the structure of RFCR introduces nonlinear stiffness and frictional damping, which critically affect the dynamic response of the rotor, especially under extreme conditions like unbalance and rotor rub-impact. Therefore, accurate modeling and dynamic analysis of RFCR assume utmost significance. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 77–90, 2024. https://doi.org/10.1007/978-981-97-0554-2_7

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Many scholars have researched the interfacial contact characteristics of RFCR. Currently, the calculation models for contact stiffness mainly include the equivalent spring and virtual material layer models. The equivalent spring model treats the contact characteristics as springs with stiffness and damping. In the works of Yuan [1], Gao [2], Li [3], Zhang [4], the equivalent spring model has been developed by means of theory, finite element experiments, and so on. The virtual material layer model could be found in works of in the works of Sun [5], Zhao [6, 7]. The RFCR is susceptible to various types of failures due to the high-speed and high-temperature operating conditions. Li et al. [8] investigated the dynamic response of RFCR under loose bolt faults using the mode superposition method. Wang et al. [10] revealed the stability and bifurcation characteristics of RFCR with transverse cracks. Hu et al. [11–13] conducted a comprehensive investigation of the nonlinear dynamic properties of the RFCR, incorporating several important factors such as the nonlinear rubbing force, nonlinear oil film force, and unbalanced mass. Hei et al. [14–16] analyzed the nonlinear dynamic behavior of RFCR by a combined bearing system consisting of fixed and tilting pads. Zhang et al. [17, 18] comprehensively analyzed RFCR, considering various influential factors such as rotor cracks and rubbing. Currently, research on the dynamic response of RFCR considering damping is still insufficient, and most studies employ constant damping coefficients. To address these limitations, this study establishes a dynamic model of the RFCR with considering damping effects and nonlinear oil film forces under rubbing faults, based on the Lagrangian equations. The Coulomb friction model is employed to introduce the damping effect of the contact interface. Finally, numerical methods are utilized to obtain the vibration response of the RFCR. Bifurcation diagrams, vibration waveforms, spectra, shaft orbits, and Poincaré maps are employed to illustrate the influence of internal damping on the RFCR system.

2 Modeling of RFCR System In this study, to effectively investigate the influence of internal damping on the nonlinear dynamic characteristics and stability of RFCR systems, a single-contact interface RFCR model is established using the Lagrangian equations, as shown in Fig. 1. The model considers the effects of oil film support, rub-impacts, and the contact between the two discs. This modeling approach not only ensures the representation of key structural features exhibited by heavy-duty gas turbine rotors but also simplifies the computational process, facilitating efficient analysis.

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Fig. 1. Schematic diagram of RFCR

2.1 Oil-Film Forces Model Figure 2 depicts a schematic diagram of the oil-film force acting on the journal. By using half Sommerfeld boundary condition, the dimensionless expressions for the nonlinear oil-film force in the x and y directions are      (X − 2Y˙ )2 + (Y + 2X˙ )2 fbx X × V − sin α × G − 2 cos α × S =− × fby 3Y × V + cos α × G − 2 sin α × S 1 − X2 − Y2 (1) V (X , Y , α) =

2 + (Y cos α − X sin α) × G(X , Y , α) 1 − X2 − Y2

(2)

X cos α + Y sin α 1 − (X cos α + Y sin α)2

(3)

S(X , Y , α) =

2 Y cos α − X sin α π G(X , Y , α) = √ × ( + arctan √ ) 2 2 2 1−X −Y 1 − X2 − Y2 α = arctan

Y + 2X˙ Y + 2X˙ π π − sign − sign(Y + 2X˙ ) ˙ ˙ 2 2 X − 2Y X − 2Y     Fbx f = sW bx Fby fby

(4) (5) (6)

R 2 L 2 where S = ηωRL W ( c ) ( 2R ) represents the Sommerfeld coefficient, η represents the lubricant oil viscosity, and W represents the external load undertaken by the bearing.

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Fig. 2. Schematic diagram of journal bearing oil-film force

2.2 Rub-Impact Force Model The local rub-impact model of the rotor can be simplified as shown in Fig. 3. F n is the normal rub-impact force, F t is the tangential friction force, and θ is the angle between the normal direction of the rub-impact point and the x-axis.

Fig. 3. Schematic diagram of Rub-impact force model

Assuming there is no radial friction and rub-impact force when the rotor is at rest and the rotor center coincides with the center of the stator, the gap between the rotor and stator is denoted as δ. When the radial displacement u of the rotor axis exceeds the gap δ, radial friction occurs. The normal rub-impact force F n and tangential nonlinear friction force F t can be expressed as follows  k (u − δ), u > δ (7) Fn = c 0, u≤δ The force components in the x and y directions, when rub-impact occurs, can be decomposed on the coordinate system as follows         kc (u − δ) 1 −μ x − cos θ sin θ Px Fn = =− (8) Py − sin θ − cos θ Ft μ 1 y u

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2.3 Contact Model between Rotor Discs Stiffness Model. Several researchers have analyzed the influence of preload on the contact between the discs as a bent beam with nonlinear stiffness [12], as shown in Eq. (9), where k1 represents the linear contact stiffness and k1 represents the nonlinear contact stiffness.       −Fcx2 k1 (x2 − x3 ) + k  (x2 − x3 )3 Fcx1 = = (9) Fcy1 −Fcy2 k1 (y2 − y3 ) + k1 (y2 − y3 )3 Damping Model. For a system subjected to frictional forces, its equation of motion can be expressed as follows: mδ¨ + k δ˙ + Ff = 0

(10)

The magnitude of the frictional force F f is constant and always in the opposite direction of the velocity. For a Coulomb damping system, the energy dissipated per cycle E c is given by the equation: EC = 4Ff δ0

(11)

The maximum amplitude of the system is represented by δ 0 . For a viscous damping system, the energy dissipated per cycle, E v , is given by the equation: Ev = π Cωδ 2

(12)

In the equation, C represents the damping coefficient, and ω represents the angular frequency. To obtain a damping coefficient that is convenient for dynamic calculations, the solution of the coefficient is based on the equivalent energy dissipation between the Coulomb damping system and the viscous damping system. By formulating Eq. And Eq. And solving for the equivalent viscous damping coefficient, C eq : Ceq =

Ec π ωδ02

(13)

The internal damping force, proportional to the velocity of the disk relative to the rotating reference frame (ζ-η-z), can be expressed as:

     Fdxi x˙ i ci 0 (14) =  Fdyi 0 ci y˙ i The coordinate transformation of the transverse displacement between the fixed coordinate system (Fig. 4) and the rotating coordinate system is given by: q = R˜q q=

      x cos ωt − sin ωt x ,R= , q˜ =  y y sin ωt cos ωt

(15) (16)

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Fig. 4. Schematic diagram of the fixed coordinate system and rotating coordinate system

The first derivative of Eq. (2) is: ˙q q˙ = Rq˙˜ + R˜

(17)

By utilizing the coordinate transformation formula, the expression for the damping force in the fixed reference frame can be derived as follows:         Fdxi ci 0 x˙ 0 ci x = + (18) Fdyi −ci 0 0 ci y˙ y 

The equation of motion for the disc in the fixed reference frame is given by:           x˙ cos t m 0 x¨ k ci x c + ci 0 + . (19) = me 2 + e 0 ce + ci y˙ −ci k sin t 0 m y¨ y

2.4 System Governing Equations of RFCR The overall solution flow is shown in Fig. 5. The global motion equation of the RFCR system with 8 DOFs can be derived as: M q¨ + C q˙ + Kq = Fu + Fb − Fg − Fcx − Fdx + Px

(20)

To analyze and solve the problem, the introduction of dimensionless displacement parameter and dimensionless time parameter t is employed. Substituting the dimensionless transformation into Eq., the dimensionless mathematical model can be derived as follows: ⎧ m1 cω2 X¨ 1 + m1 cωcb X˙ 1 + k(X1 − X2 ) = fx (X1 , Y1 , X˙ 1 , Y˙ 1 ) ⎪ ⎪ ⎪ ⎪ ⎪ m1 cω2 Y¨ 1 + m1 cωcb Y˙ 1 + k(Y1 − Y2 ) = fy (X1 , Y1 , X˙ 1 , Y˙ 1 ) − m1 g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m2 cω2 X¨ 2 + m2 cωcs X˙ 2 − k(X1 − X2 ) = Fux2 − Fcx + Fdx + PX ⎪ ⎪ ⎪ ⎪ ⎨ m2 cω2 Y¨ 2 + m2 cωcs Y˙ 2 − k(Y1 − Y2 ) = Fuy2 − Fcy − Fdy + PY − m2 g (21) ⎪ m3 cω2 X¨ 3 + m3 cωcs X˙ 3 + k(X3 − X4 ) = Fux3 + Fcx − Fdx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m3 cω2 Y¨ 3 + m3 cωcs Y˙ 3 + k(Y3 − Y4 ) = Fuy3 + Fcy + Fdy − m3 g ⎪ ⎪ ⎪ ⎪ ⎪ m4 cω2 X¨ 4 + m4 cωcb X˙ 4 − k(X3 − X4 ) = fx (X4 , Y4 , X˙ 4 , Y˙ 4 ) ⎪ ⎪ ⎪ ⎩ m4 cω2 Y¨ 4 + m4 cωcb Y˙ 4 − k(Y3 − Y4 ) = fy (X4 , Y4 , X˙ 4 , Y˙ 4 ) − m4 g

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where F u represents the unbalance force, which can be expressed as ⎧ 2 ⎪ ⎪ Fux2 = m2 e2 ω cos(ωt + ϕ1 ) ⎪ ⎪ ⎨ Fuy2 = m2 e2 ω2 sin(ωt + ϕ1 ) ⎪ Fux3 = m2 e2 ω2 cos(ωt + ϕ1 ) ⎪ ⎪ ⎪ ⎩ Fuy3 = m2 e2 ω2 sin(ωt + ϕ1 )

Fig. 5. Flow chart for calculating dynamic responses of the RFCR.

3 Model Validation The results in [12] were used for verification, and the RFCR parameters are shown in Table 1. The verification results are presented in Fig. 6, illustrating the bifurcation diagram of the system within the range of 300–2500 rad/s. It can be observed that the periodic motion characteristics obtained in this study exhibit a strong agreement with the findings reported in [12].

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Parameter

Value

Lumped mass, m1 , m2 , m3 , m4 (kg)

4,32.1,32.1,4

Damp coefficient, cb , cs (N·s/m)

1050,2100

Stiffness of the shaft, k (N/m)

2.5×107

Stiffness coefficient of contact interface, k r (N/m)

2.5×107

Radial stiffness of stator, k c (N/m)

1×107

Unbalance eccentricity, e1 , e2 (mm)

0.05,0.05

Gravitational acceleration, g

9.81

Initial clearance, δ (mm)

0.18

Bearing clearance, c (mm)

0.11

Bearing diameter, DB (mm)

50

Bearing length, L B (mm)

12

Lubricating viscosity, η (Pa·s)

0.018

Vector angle, ϕ

0

Fig. 6. Comparison of the bifurcation diagrams of the rotor system in this paper and [12]:(a) the results of this paper; (b) the results of [12].

4 Results of the Numerical Simulation 4.1 Bifurcation and Nonlinear Response Analysis This section investigates the effects of rub-impact on the RFCR. Bifurcation diagrams, the Largest Lyapunov Exponent (LLE), and the three-dimensional spectrum are effective tools for studying the nonlinear dynamical behavior of nonlinear dynamical systems. The bifurcation diagram and three-dimensional spectrum of disk 1 in the x-direction are shown in Fig. 7, with the rotor speed ranging from 300 to 2500 rad/s. The bifurcation diagram reveals that the rotor speed directly influences the dynamic characteristics of the

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rotor system. Furthermore, to delve into the evolution process of the dynamic behavior, time domain waveform, motion trajectory, frequency spectrum and Poincaré map at different speeds are obtained, as shown in Fig. 8–10.

(a) Bifurcation diagram

(c) Three-dimensional spectrum

(b)

(d)

LLE

Colormap

Fig. 7. Bifurcation diagram, LLE, three-dimensional spectrum, and colormap of RFCR

The results indicate that when the speed is below 508 rad/s, the motion exhibits synchronous vibration with a period-1 and only one isolated point. As the speed increases, the rotor enters period-4 motion at a speed of ω = 612 rad/s. As shown in Fig. 8 (b), the Poincaré map exhibits four isolated points, and the spectrogram reveals the appearance of a sub-synchronous whirl besides the fundamental frequency. As the speed further increases to ω = 1108 rad/s, the system transitions into quasi-periodic motion, and the Poincaré map forms a closed elliptical ring. At a speed of 1680 rad/s, the system enters period-3 motion. Finally, it returns to quasi-periodic motion.

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Fig. 8. Numerical analysis results at ω = 700 rad/s

Fig. 9. Numerical analysis results at ω = 1200 rad/s

Fig. 10. Numerical analysis results at ω = 1800 rad/s

4.2 Effect of Internal Damping Force To investigate the influence of contact surface damping on rotor dynamics, the bifurcation diagram and three-dimensional spectrum of the center of disk 1 in the x-direction were obtained considering the damping effect (μ = 0.05), as shown in Fig. 11(a,b). Considering or not considering the damping effect, the dynamic behavior evolution and frequency components are largely consistent. However, including damping causes more divergent vibration at high speeds. Furthermore, from the comparison in Fig. 11(c,d), it can be noted that the frequency shifts towards the right side when considering damping at high speeds. To better understand the impact of damping on the dynamic response, this study investigated the time-domain and frequency-domain responses at 1800 rad/s (Fig. 12). It can be observed that at 1800 rad/s, the damping effect becomes more prominent, resulting in a more divergent behavior compared to the original periodic motion. This can be attributed to the fact that the damping effect is related to both the rotational speed and the relative amplitude. At low speeds, where the vibration amplitude between the disks is small, the damping effect is relatively small compared to the effects of oil film and contact-induced effects. However, as the speed increases and the system’s amplitude grows, the damping effect becomes more significant and cannot be ignored.

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(a) with internal damping

(b) without internal damping

(c) with internal damping

(d) without internal damping

Fig. 11. Bifurcation diagram and colormap of RFCR with or without internal damping

Fig. 12. Numerical analysis results at ω = 1800 rad/s

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4.3 Effect of Friction Coefficients The friction coefficient is a key influencing parameter for the damping effect. To investigate the impact of the friction coefficient on the dynamic characteristics of the RFCR, an analysis was conducted on the responses for varying friction coefficients. From Fig. 13 and Fig. 14, it can be observed that as the friction coefficient increases, the damping effect leads to a significant divergence of the P3 motion after 1680 rad/s. Figure 14 shows that as the friction coefficient increases, the frequency shift becomes more pronounced at high speeds (ω > 2200rad/s).

Fig. 13. Bifurcation diagram of RFCR for Different Friction Coefficients

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Fig. 14. Colormap of RFCR for Different Friction Coefficients

5 Conclusion This study considers the damping effect and establishes a dynamic model of the RFCR system with both rubbing fault and nonlinear oil film forces. In-depth investigations are conducted on the periodic characteristics and response properties of the system, leading to the following conclusions: (1) Under rub-impact conditions, the system exhibits various motion types, including P1-P2-P4-(quasi-periodic)-P3. (2) Damping has a minor influence on the system’s periodic characteristics but can alter its response properties. In high-speed vibrations considering internal damping, the amplitude significantly increases. (3) Increasing the friction coefficient further induces system divergence and increases its instability.

References 1. Yuan, Q., Gao, R.: Analysis of dynamic characteristics of gas turbine rotor considering contact effects and pre-tightening force. In: Proceedings of the ASME Turbo Expo, pp. 983–988 (2008) 2. Gao, J., Yuan, Q.: Effects of bending moments and pretightening forces on the flexural stiffness of contact interfaces in rod-fastened rotors. ASME. J. Eng. Gas Turbines Power 134(10), 102503 (2012)

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3. Li, H., Liu, H., Yu, L.: Contact stiffness of rough mechanical joint surface. J. Xi’an Jiaotong Univ. 45(6), 69–74 (2011) 4. Zhang, Y., Du, Z.: Determination of contact stiffness of rod-fastened rotors based on modal test and finite element analysis. J. Eng. Gas Turbines Power 132(9), 275–282 (2010) 5. Sun, Y., Xiao, H., Xu, J.: Investigation into the interfacial stiffness ratio of stationary contacts between rough surfaces using an equivalent thin layer. Int. J. Mech. Sci. 163, 105147 (2019) 6. Zhao, R., Jiao, Y., Chen, Z., Li, Z., Qu, X.: Nonlinear analysis of a dual-disk rotor system considering elastoplastic contact. Int. J. Non-Linear Mech. 141, 103925 (2022) 7. Zhao, R., et al.: Multi-scale contact induced period-doubling vibrations in rotor systems: numerical and experimental studies. Mech. Syst. Signal Process. 195, 110251 (2023) 8. Li, P., Yuan, Q., Zhao, B., Gao, J.: Dynamics of a Rod-Fastened Rotor Considering the Bolt Loosening Effect. In: Volume 7A: Structures and Dynamics. p. V07AT33A007. American Society of Mechanical Engineers, Phoenix, Arizona, USA (2019) 9. Chen, L., Qian, Z.W., Chen, W.: Influence of structural parameters on the bistable response of a disk-rod-fastening rotor. J. Vib. Meas. Diagn. 32(05), 767–772+862–863 (2012) 10. Wang, N.S., Liu, H., Liu, Y., et al.: Stability and bifurcation of a flexible rod-fastening rotor bearing system with a transverse open crack. J. Vibroeng. 20(8), 3026–3039 (2018) 11. Hu, L., Liu, Y.B., Teng, W., et al.: Nonlinear coupled dynamics of a rod fastening rotor under rub-impact and initial permanent deflection. Energies 9(11), 883 (2016) 12. Hu, L., Liu, Y.B., Zhao, L., et al.: Nonlinear dynamic response of a rub-impact rod fastening rotor considering nonlinear contact characteristic. Arch. Appl. Mech. 86, 1869–1886 (2016) 13. Hu, L., Liu, Y.B., Zhao, L., et al.: Nonlinear dynamic behaviors of circumferential rod fastening rotor under unbalanced pre-tightening force. Arch. Appl. Mech. 86, 1621–1631 (2016) 14. Hei, D., Lu, Y.J., Zhang, Y.F., et al.: Nonlinear dynamic behaviors of a rod fastening rotor supported by fixed–tilting pad journal bearings. Chaos Soliton Fract. 69, 129–150 (2014) 15. Hei, D., Lu, Y.J., Zhang, Y.F., et al.: Nonlinear dynamic behaviors of rod fastening rotor– hydrodynamic journal bearing system. Arch. Appl. Mech. 85, 855–875 (2015) 16. Hei, D., Zheng, M.R.: Investigation on the dynamic behaviors of a rod fastening rotor based on an analytical solution of the oil film force of the supporting bearing. J. Low Freq. Noise V. A. 40(2), 707–739 (2020) 17. Zhang, Y., Xiang, L., Su, H., Hu, A., Yang, X.: Dynamic analysis of composite rod fastening rotor system considering multiple parameter influence. In: Applied Mathematical Modelling, vol. 105, pp. 615–630 (2022) 18. Zhang, Y., Xiang, L., Su, H., Hu, A., Chen, K.: Nonlinear dynamic response on multi-fault rod fastening rotor with variable parameters. In: Applied Mathematical Modelling, vol. 114, pp. 147–161 (2022)

A Multiscale Fracture Model to Reveal the Toughening Mechanism in the Bioinspired Bouligand Structure Yunqing Nie(B) , Dongxu Li, and Luojing Zhou College of Aerospace Science and Engineering, National University of Defense Technology, 109 Deya Road, Changsha 410073, Hunan, People’s Republic of China [email protected]

Abstract. The Bouligand structure has been observed in a variety of biological materials, such as lamellar bone and arthropod cuticles. It is a hierarchical architecture that exhibits excellent damage-resistant performance, which arouse many interests for the mechanists and structure engineers. However, there still lacks a deep understanding of the toughening mechanisms in the Bouligand structure. For the purpose of revealing the toughening effect of twisting cracks, this paper developed a multiscale fracture mechanics model with considering the non-homogeneity and anisotropic properties. Firstly, the macro and micro constitutive properties of the Bouligand structure are analyzed. Then, a multiscale fracture model is established to characterize the energy release rates and the local stress intensity factors at the crack front of twisting cracks which are formed within the Bouligand structure. Based on the model, serious of digital calculations are carried out. The digital results demonstrate that the decrease of the local energy release rate can be attributed to two mechanisms. One is that the multiscale structure causes the stress release of the crack tip nearby. The other is that the twisting crack leads to the loading mode transformation from the single-mode to the mixedmode, which is the main reason of the fracture toughness increasing. The research results shown in this paper can provide structure engineers some suggestive guidelines for the design of high-performance composites.

Keywords: Bouligand structure Toughening mechanisms

1

· Fracture toughness · Crack twist ·

Introduction

Almost all engineering structural materials are required to be both strong and tough. However, the properties of strength and toughness in most synthetic materials are often mutually exclusive [6]. Nature has developed novel ways to build lightweight, strong, and tough materials over millions of years of evolution [2,19]. Biological materials provide inspiration to revolutionize the techniques for producing novel materials with improved properties and functions. Nevertheless, c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 91–104, 2024. https://doi.org/10.1007/978-981-97-0554-2_8

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the mechanical performance of current bioinspired composites falls short of that of biological materials [19]. Significant efforts need to be made to elucidate the secrets of microstructures and mechanisms that enable these excellent mechanical performances. Bouligand structures, which have been observed in the lamellar bone [5,20] and the exoskeleton of crustaceans [4,8], are typical examples that demonstrate how microstructure can combine the behaviors of stiffness, strength, and toughness. Extensive mechanical analyses have been conducted to reveal the composition-structure-property relationship of Bouligand composites, especially their toughening mechanisms. Suksangpanya et al. developed a theoretical model to provide additional insights into the local stress intensity factors at the crack front of twisting cracks formed within the Bouligand structure [12]. A numerical method based on the crack driving force was used to depict the effect of inhomogeneous and anisotropic material behavior on fracture properties [1]. Crack twisting, driven by the fiber architecture, is the main fracture mechanisms, which has been confirmed by a experimental approach [13]. Song et al. proposed that a combination of crack tilting and crack bridging determines the effective fracture toughness of the fiber-reinforced composite with the plywood structure [10]. A discrete element method was developed to capture the main fracture mechanisms in fibrous laminates. Yang et al. also established a theoretical modelling approach to study the fracture properties and toughening mechanism of the helicoidal structures [21]. While Bouligand structures show anisotropic material properties within each layer and they possess inhomogeneity, e.g. multilayers with varying Young’s modulus. It is not clear which role the material inhomogeneity and anisotropy plays for the Bouligand structure. This paper aims to provide a multiscale fracture model to investigate the toughening mechanisms of the twisting crack in the Bouligand structure. This paper is organized as follows. Firstly, the macro and micro constitutive properties of the Bouligand structure are derived. Then, a multiscale fracture model, which considering the non-homogeneity and anisotropic properties, is developed to characterize the energy release rates and the local stress intensity factors at the crack front of twisting cracks formed within the Bouligand structure. Thirdly, the toughening mechanism of the twisting crack is discussed. Finally, the main conclusions drawn from this study are summarized.

2 2.1

The Multiscale Fracture Model in Bioinspired Bouligand Structures Material Descriptions

The bouligand structure has been found in a variety of biological materials. However, there still lacks a deep understanding about the toughening mechanisms of these composites under mode I loading. In this section, a ideal bouligand structure is extracted from the lamellar bone (see Fig. 1(a)). The Bouligand structure can be characterized by two main parameters: The pitch angle γ and the interlayer spacing d. As Fig. 1(b) shows, γ is the helicoidal angle between orientations

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of adjacent layers, and d is the distance between adjacent fiber layers. It should be noted that in biological materials, the pitch angle and the interlayer spacing have a gradient variation in the interlayer spacing moving normal to the surface. While we ignore the variation of γ and d, which is not the focus of this paper.

Fig. 1. The Bouligand structure. (a) SEM micrograph of the lamellar bone in goat tibia [7]. (b) 3D Schematic representation.

In this work, we focus on the toughening mechanism of the Bouligand structure. According to the experiments of biological materials [4,8], the Bouligand structure mainly bears loads along the in-plane direction of the interlayer, and the crack mostly extends along the thickness of the interlayers. A crack which is a semi-finite flat plane perpendicular to the layers, is defined as Fig. 2. The crack grows along the alignment of the fibers, and we assume the crack front is always a straight line and parallel to the fibers. With the crack extension, a continuous twisting crack surface is generated, as shown in Fig. 2. A global coordinate system (X, Y, Z) is defined, whose origin locates at the middle of the initial flat crack front. The initial crack face is perpendicular to Y axis, the initial crack edge is along with Z axis. The laminas are stacked along X axis. The twisting crack can be written as [12]    γ dφ (1) = −Z · tan X Y = −Z · tan X dX d where φ is the twisted angle, and it can be expressed as φ = Xγ/d. In order to properly describe the anisotropic properties and the fracture toughness at the crack front, we also define three coordinate systems: the transforming global coordinate system (X  , Y  , Z  ), the local coordinate system at the twisting crack front (x, y, z), and the local material coordinate system (xk , yk , zk ). The transforming global coordinate system (X  , Y  , Z  ) can be get by transforming the global coordinate system (X, Y, Z) to the point P (X0 , Y0 , Z0 ) in the twisted crack surface. The local coordinate system (x, y, z) can be defined by using the level set method. z axis is perpendicular to the twisted crack face.

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It can be get by rotating (X  , Y  , Z  ) around Z  axis with an angle α∗ , and then rotating around x axis with an angle φ. The kinked angle α∗ is given by [12] ⎧ ⎫

⎨ ⎬ 2 φ cos φ 1 + tan α∗ = cos−1  (2) ⎩ ⎭ 2 [(Zγ/d)sec2 φ] + 1 + tan2 φ The Bouligand structure shows anisotropic material properties within each layer. It’s necessary to define a local material coordinate system (xk , yk , zk ) in P . The laminas is stacked along xk axis, and the fiber in the lamina is along zk axis.

Fig. 2. Schematic illustration of a twisting crack in Bouligand structure.

2.2

Constitutive Relation

In the bioinspired Bouligand structure, there exist several different scales. For example for the smashing mantis shrimp in Odontodactylus Scyllarus, the thickness of the single lamina is often in the sub-micro scale, and the pitch distance (the thickness of the laminate in a period) is in the micro-scale [4,11]. The effect of multi-scale must be taken into consideration. Here we define two scales in this paper: the macro scale and the micro scale. The micro property is defined in the scale of the single lamina, and the macro-property is defined in the scale of the laminate or the length of the crack. Then we give the constitutive relations of different scales. (1) The micro-constitutive relation in the local material coordinate system According to the theory of composite mechanics, the macromechanical behavior of a lamina is written as [3] {σ} = [C] {ε}

(3)

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where the stress tensor {σ} = [σ11 σ22 σ33 σ23 σ13 σ12 ]T , the strain tensor {ε} = [ε11 ε22 ε33 γ23 γ13 γ12 ]T , and the stiffness matrix has the form ⎤ ⎡ C11 C12 C13 0 0 0 ⎢ C12 C22 C23 0 0 0 ⎥ ⎥ ⎢ ⎢ C13 C23 C33 0 0 0 ⎥ ⎥ ⎢ (4) [C] = ⎢ ⎥ ⎢ 0 0 0 C44 0 0 ⎥ ⎣ 0 0 0 0 C55 0 ⎦ 0 0 0 0 0 C66 The stiffness constants in Eq. (4) are shown in Appendix. (2) The micro-constitutive relation in the local twisted coordinate system The micro stress-strain relationship in the local twisted coordinate system is expressed as {σ (l) } = [c] {ε(l) } (5) The local twisted coordinate system can be get by rotating the system (xk , yk , zk ) around zk axis with a angle α∗ . According to the transforming law of tensor, the stiffness matrix can be given by [16] ⎤ ⎡ c11 c12 c13 0 0 c16 ⎢ c12 c22 c23 0 0 c26 ⎥ ⎥ ⎢ ⎢ c13 c23 c33 0 0 c36 ⎥ T ⎥ ⎢ (6) [c] = [Tz ] [C] [Tz ] = ⎢ ⎥ ⎢ 0 0 0 c44 c45 0 ⎥ ⎣ 0 0 0 c45 c55 0 ⎦ c16 c26 c36 0 0 c66 where the transforming matrix [Tz ] is shown in Appendix. (3) The micro-constitutive relation in the global coordinate system The Bouligand structure can be considered that is formed by a repeating sublaminate. The sublaminate (the typical cell) contains N orthotropic fiber composite laminas of arbitrary fiber orientations. The micro-constitutive relation in the system (X  , Y  , Z  ) reflects the stress-strain relationship in the kth lamina, and it is given by   (7) {σ (k) } = c(k) {ε(k) } The system (X  , Y  , Z  ) can be get by rotating

the system (xk , yk , zk ) around xk axis with a angle φ. The stiffness matrix c(k) of the kth lamina in the global coordinate system can be calculated by ⎤ ⎡ (k) c11 c12 (k) c13 (k) c14 (k) 0 0 ⎢ c12 (k) c22 (k) c23 (k) c24 (k) 0 0 ⎥ ⎥ ⎢ (k)   (k) (k) (k) ⎢ c c c 0 0 ⎥ c T 13 23 33 34 (k) ⎥ ⎢ (8) = [Tx ] [C] [Tx ] = ⎢ (k) c (k) (k) (k) 0 0 ⎥ ⎥ ⎢ c14 c24 c34 c44 (k) (k) ⎣ 0 0 0 0 c55 c56 ⎦ 0

0

0

0

c56 (k) c66 (k)

where the transforming matrix [Tx ] is depicted in Appendix.

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(4) The macro-constitutive relation in the global coordinate system The mechanical response in the micro-scale has been given in the above. In this paragraph, we establish the relation between the micro-property and the macro-constitutive relation. Using the theory of thick laminates, the macroconstitutive relation of the Bouligand structure in the system (X  , Y  , Z  ) can be written as c] {ε(g) } (9) {σ (g) } = [¯ where the macro-stiffness matrix [¯ c] is [14] ⎡ c¯11 c¯12 c¯13 c¯14 ⎢ c¯12 c¯22 c¯23 c¯24 ⎢ ⎢ c¯13 c¯23 c¯33 c¯34 [¯ c] = ⎢ ⎢ c¯14 c¯24 c¯34 c¯44 ⎢ ⎣ 0 0 0 0 0 0 0 0

0 0 0 0 c¯55 c¯56

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ c¯56 ⎦ c¯66

(10)

The effective elastic constants can be calculated by the micro properties of N laminas (Details is shown in Appendix). In this section, the constitutive relations of the Bouligand structure in micro and macro scales have been established. Next, these properties will be used to establish a multiscale fracture model. 2.3

The Fracture Model of the Twisted Crack

Considering a given point P (X0 , Y0 , Z0 ) on a straight crack front of the twisted crack, the auxiliary stress field at this point can be expressed in terms of the local system (r, θ, z) and the global system (R, Θ, Z) in the micro-scale, as σij (l) (i, j = r, θ, z) and σij (k) (i, j = R, Θ, Z). (r, θ, z) is the related cylindrical coordinate of (x, y, z) and (R, Θ, Z) is the related cylindrical coordinate of (X  , Y  , Z  ). σij (l) (i, j = r, θ, z) is the local stress tensor. It can be expressed by the local stress intensity factor, according to Sih’s theory [9] I II Re [μ1 μ2 ] − √k2πr Re [μ1 + μ2 ] σrr (l) (r, 0) = − √k2πr k (l) I σθθ (r, 0) = √2πr II σrθ (l) (r, 0) = √k2πr (l) σrz (r, 0) = − √kIII Re [μ3 ] 2πr kIII (l) √ σθz (r, 0) = 2πr I II σzz (l) (r, 0) = v √k2πr (1 − Re [μ1 μ2 ]) − v √k2πr Re [μ1 + μ2 ]

(11)

where kI , kII , kIII are the local stress intensity factors in mode I, II, and III. σij (k) (i, j = R, Θ, Z) is the micro-stress tensor in the global system (R, Θ, Z). It can be obtained through a complex mathematical transformation to the macrostress field σij (g) (i, j = X  , Y  , Z  ) in the global system (X  , Y  , Z  ). The global stress field σij (g) can be expressed in terms of the global coordinate system (X  , Y  , Z  ) [9,15] 3  1 m σij (g) (R, Θ) = √ Km gij (s¯ij , Θ) (i, j = X  , Y  , Z  ) 2πR m=1

(12)

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where KI , KII , KIII are the global stress intensity factors in mode I, II, and m (s¯ij , Θ) are defined by the elastic properties of orthotropic III. The functions gij materials, which are also given in Appendix. And [¯ s] = [¯ c]−1 is the macrocompliance matrix. Using Eq. (9), we could derive the global strain field [ε(g) ] = [¯ s][σ (g) ]. According to Sun’s theory of thick laminates [14], the micro stress σij (k) (i, j = X  , Y  , Z  ) and strain εij (k) (i, j = X  , Y  , Z  ) can be calculated using the macro-stress εY  Y  (k) = εY  Y  (g) ; εZ  Z  (k) = εZ  Z  (g) ; γY  Z  (k) = γY  Z  (g) σX  X  (k) = σX  X  (g) ; σX  Z  (k) = σX  Z  (g) ; σX  Y  (k) = σX  Y  (g)

σX  X  (k) −c12 (k) εY  Y  (k) −c13 (k) εZ  Z  (k) −c14 (k) γY  Z  (k) c11 (k) c55 (k) σX  Y  (k) −c56 (k) σX  Z  (k) = c55 (k) c66 (k) −c56 (k) 2 c66 (k) σX  Z  (k) −c56 (k) σX  Y  (k) = c55 (k) c66 (k) −c56 (k) 2 (k) = c12 εX  X  (k) + c22 (k) εY  Y  (k) + c23 (k) εZ  Z  (k) + c24 (k) γY  Z  (k) = c13 (k) εX  X  (k) + c23 (k) εY  Y  (k) + c33 (k) εZ  Z  (k) + c34 (k) γY  Z  (k) = c14 (k) εX  X  (k) + c24 (k) εY  Y  (k) + c34 (k) εZ  Z  (k) + c44 (k) γY  Z  (k)

εX  X  (k) = γX  Y  (k) γX  Z  (k) σY  Y  (k) σZ  Z  (k) σY  Z  (k)

(13) The stress tensor in the cylindrical coordinate system (R, Θ, Z) and cartesian coordinate system (X  , Y  , Z  ) has the following relation     σij(i,j=R,Θ,Z) (k) = [R ] σij(i,j=X  ,Y  ,Z  ) (k) [R ]T (14) where the transforming matrix is ⎡

⎤ cos α∗ sin α∗ 0 [R ] = ⎣ − sin α∗ cos α∗ 0 ⎦ 0 0 1

(15)

Now we get the micro-stress tensor σij (l) (i, j = r, θ, z) and σij (k) (i, j = R, Θ, Z) in point P . The former is expressed by the local stress intensity factor kI , kII , kIII , and the latter can be obtained by the function of the global stress intensity factor KI , KII , KIII . These two forms of micro-stress has the following relationship       T σij (l)  = [R] σij (k)  [R] (16) ∗ θ=0

Θ=α

where [R] is the transforming matrix of the stress tensor. It can be expressed as [16] ⎡ ⎤ 1 0 0 [R] = ⎣ 0 cos φ sin φ ⎦ (17) 0 − sin φ cos φ Substituting Eqs. (12–15) into Eq. (16), the local stress intensity factor kI , kII , kIII can be calculated when the initial global stress intensity factor KI , KII , KIII and the geometric parameters are defined. Now, the multi-scale fracture model to reveal the toughening mechanism of the twisted crack has been established.

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The Definition of the Energy Release Rate

In this section, we extract three related energy release rate using the fracture mechanics for anisotropic materials. The overall energy release rate of the initial crack G0 can be calculated by [17,18]      μ1 +μ2 s¯22 1 2 + K μ G0 = 12 s¯11 Im −KI2 ss¯¯22 (μ + μ ) + K K μ − 1 2 I II 1 2 II s¯11 μ1 μ2 11 μ1 μ2 + 21

(s¯44 s¯55 −¯s245 )

3/2

2 KIII

s¯44 s¯55

(18) where s¯ij is the constants of the macro-compliance matrix. μ1 , μ2 , μ3 is the characteristic factors, and can be calculated by s¯ij (see Appendix). We define another the local energy release rate of the non-twisted crack Gm as      2 μ1 +μ2 S22 1 2   + k μ (μ + μ ) + k k μ − Gm = 12 S11 Im −k  I SS22 1 2 1 2 II I II S11 μ1 μ2 11 μ1 μ2 + 12

(S44 S55 −S45 2 )

3/2

S44 S55

k  III 2

(19)  , k  III are the local stress intensity factor in the initial crack tip, where kI , kII [Sij ] = [C]−1 is the constants of the micro-compliance matrix of the lamina in this zone. Similarly μ1 , μ2 , μ3 can be calculated by Sij (see Appendix). The local energy release rate of the twisted crack Gt is defined as      μ1 +μ2 s22 1 2 + k μ Gt = 12 s11 Im −kI2 ss22 (μ + μ ) + k k μ − 1 2 I II 1 2 II s11 μ1 μ2 11 μ1 μ2 3/2

+ 12

(s44 s55 −s45 2 ) s44 s55

2 kIII

(20) where [sij ] = [s]−1 is the constants of the micro-compliance matrix of the material near the twisted crack tip. μ1 , μ2 , μ3 can be calculated by sij (see Appendix). Here we make some description about the physics significance of the above three energy release rates. The overall energy release rate G0 represents the overall crack driving force, which is directly related to the far field load. The local energy release rate Gm depicts the realistic crack driving force in the initial crack tip. As we know, the Bouligand structure is multi-scale and non-homogeneous. The principal direction of this structure is different along the stacking direction. There is a great difference in the mechanical property of the macro- and microscales. This non-homogeneity lead to a significant decrease of the micro-crack driving force, compared to the macro energy release rate G0 . Therefore, Gm /G0 characterize the toughening effect of the non-homogeneity structure. The local energy release rate Gt demonstrates the micro-crack driving force in the twisted crack. The twisted crack surface results in a relax of the auxiliary stress field near the crack tip. Gt /Gm represents the toughening effect of the twisted crack.

3

Results and Discussions

This paper aims to provide an analytical approach for consulting the improved fracture performance of the bioinspired Bouligand structure. To this end, fibre

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bundles and matrix are not considered separately, and the elastic properties of the lamellar bone are summarized in Table 1, as obtained by Ref. [1,21]. The geometric parameters are given by: the pitch angle γ = 10◦ , the interlayer number of the sub-laminate N = 18, and the interlayer spacing d = 10/N µm [1,12,21]. Table 1. Anisotropic elastic properties of lamellar bone [1, 21] Parameter E1 = E2 (GPa) E3 (GPa) G12 = G31 = G23 (GPa) ν32 = ν31 ν23 = ν13 ν21 = ν12 Value

5

10

2

0.25

0.125

0.33

Firstly, we analyze the effect of the stiffness ratio E3 /E1 to Gm /G0 . As Fig. 3 shows, with the increase of E3 /E1 , Gm /G0 gradually decreases. And when E3 /E1 < 1, Gm /G0 is also larger than 1. When E3 /E1 > 1, Gm /G0 is also smaller than 1. Gm /G0 < 1 depicts that the material non-homogeneity has a positive effect to the fracture toughness. Therefore, a larger E3 /E1 leads to a significant toughness contribution due to the non-homogeneity. While, it should be note that only when the material along the direction of the crack edge is softener than that along the direction parallel to the crack face, the nonhomogeneity reduces the crack driving force. This conclusion that why the crack edge in biological materials is always along the fiber direction [4,8]. Other geometric parameters, such as γ and d cause ignorable influence to the toughening effect of the non-homogeneity.

Fig. 3. The effect of the stiffness ratio E3 /E1 to the energy release rate ratio Gm /G0 .

Next, the toughening effect of the twisted crack is examined. Figure 4 shows the nondimensional energy release rate Gt /G0 , the nondimensional stress intensity factor kI /KI , kII /KI , kIII /KI as a function of X/d and Z/d in the twisted crack surface. As previously mentioned, Gt /G0 of the initial flat crack front at Z/d = 0 has a deep decrease, which results from the effect of the multi-scale structure. The nondimensional crack driving force Gt /G0 decreases with the increase of X/d and Z/d (see Fig. 4(a)). As Fig. 4(b) shows, kI /KI has a similar variation with Gt /G0 . Figure 4(c) shows that kII /KI is 0 when Z/d = 0,

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and rises with increasing Z/d. kII /KI is mainly influenced by the kinked angle α∗ . Figure 4(d) depicts that kIII /KI increases with increasing X/d. For constant X/d, kIII /KI almost remains constant. kIII /KI is mainly effected by the twisted angle φ. It can be concluded that the twisted crack can enhance the fracture performance by transforming single mode load (KI ) to mixed-mode loading (kI , kII , kIII ). And the mixed-mode behavior is mainly dominated by two parameters: the twisted angle φ and the kinked angle α∗ .

(a)

(b)

(c)

(d)

Fig. 4. Countor of Gt /G0 , kI /KI , kII /KI , kIII /KI with respect to X/d and Z/d.

To properly characterize the fracture resistance amplification, we also introduce three toughening ratios C GC 0 /Gm = G0 /Gm C Gm /GC t = Gm /Gt C GC /G t = G0 /Gt 0

(21)

C where GC 0 /Gm denotes the toughening ratio of the materials non-homogeneity. C C Gm /Gt reflects the toughening effect of the twisted crack, and the effect of

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C materials non-homogeneity is not considered. GC 0 /Gt is the overall toughening ratio. The contour of the toughening ratio with respect X/d and Z/d is shown in Fig. 5. It demonstrates that when the crack begins to grow, the toughening effect of the materials non-homogeneity plays a dominant role to the overall fracture toughness. With the crack twist, the effect of crack twist gradually is the main source of the fracture toughness.

Fig. 5. The contour of the toughening ratio with respect to X/d and Z/d

Finally, the effect of the twisted angle φ and the kinked angle α∗ to the C fracture toughness is depicted in Fig. 6. For small φ and α∗ , GC 0 /Gm is greater C C than Gm /Gt , which demonstrates that the toughening effect of the materials non-homogeneity play a more important role. With the increase of φ and α∗ , the dominant toughening mechanism is changed to the crack twist.

Fig. 6. The contour of the toughening ratio with respect to φ and α∗ .

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4

Conclusions

In this study, we first derived the macro and micro constitutive properties of the Bouligand structure. Then a multiscale fracture model considering the nonhomogeneity and anisotropic properties was established. Finally, the toughening mechanism of the twisting crack in the Bouligand structure was discussed. The digital results demonstrate that the decrease of the local energy release rate can be attributed to two mechanisms. One is that the multiscale structure causes the stress release of the crack tip nearby. The other is that the twisting crack leads to the loading mode transformation from the single-mode to the mixed-mode, which is the main reason of the fracture toughness increasing. The research results shown in this paper can provide structure engineers some suggestive guidelines for the design of high-performance composites.

A

Appendix

A.1

The Stiffness Constants of the Lamina

The stiffness constants in Eq. (4) are shown in Appendix. 23 v32 13 v31 31 v23 C22 = 1−v C12 = v21E+v C11 = 1−v E2 E3 Δ E1 E3 Δ 2 E3 Δ 12 v31 21 v32 12 v21 C23 = v32E+v C13 = v31E+v C33 = 1−v E1 E2 Δ 1 E3 Δ 2 E3 Δ C44 = G12 ; C55 = G31 ; C66 = G23

where Δ= A.2

1 − v12 v21 − v23 v32 − v31 v13 − 2v21 v32 v13 E1 E2 E3

(22)

(23)

The Transforming Matrix of the Tensor

The transforming matrix [Tz ] is ⎡ cos2 α∗ sin2 α∗ 2 ∗ ⎢ sin α cos2 α∗ ⎢ ⎢ 0 0 [Tz ] = ⎢ ⎢ 0 0 ⎢ ⎣ 0 0 − sin α∗ cos α∗ sin α∗ cos α∗

⎤ 0 0 0 2 sin α∗ cos α∗ 0 0 0 −2 sin α∗ cos α∗ ⎥ ⎥ ⎥ 1 0 0 0 ⎥ (24) ∗ ∗ ⎥ 0 0 cos α − sin α ⎥ ∗ ∗ ⎦ 0 0 sin α cos α 0 0 0 cos2 α∗ − sin2 α∗

The transforming matrix [Tx ] expressed as ⎤ ⎡ 1 0 0 0 0 0 ⎢ 0 cos2 φ sin2 φ −2 sin φ cos φ 0 0 ⎥ ⎥ ⎢ 2 ⎢ 0 sin2 φ cos φ 2 sin φ cos φ 0 0 ⎥ ⎥ [Tx ] = ⎢ ⎢ 0 sin φ cos φ − sin φ cos φ cos2 φ − sin2 φ 0 0 ⎥ ⎥ ⎢ ⎣0 0 0 0 cos φ sin φ ⎦ 0 0 0 0 − sin φ cos φ

(25)

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The Macro-constitutive Relation in the Global Coordinate System

The effective elastic constants are calculated by The terms can be calculated by the micro properties of N laminas [14] N N   (k) (k) (k) vk c12 c¯13 = vk c13 c¯14 = vk c14 k=1 k=1 k=1    N N    (k) (k) (1) (k) (k) c12 − c¯12 vk c12 − c12 /c11 = vk c22 + k=1 k=2    N N    (k) (k) (1) (k) (k) c12 − c¯12 vk c13 − c13 /c11 = vk c23 + k=1 k=2    N N    (k) (k) (1) (k) (k) c12 − c¯12 vk c14 − c14 /c11 = vk c24 + k=1 k=2    N N    (k) (k) (1) (k) (k) c13 − c¯13 vk c13 − c13 /c11 = vk c33 + k=1 k=2    N N    (k) (k) (1) (k) (k) c13 − c¯13 vk c14 − c14 /c11 = vk c34 + k=1 k=2    N N    (k) (k) (1) (k) (k) c14 − c¯14 vk c14 − c14 /c11 = vk c44 + k=1 k=2  N  (k) = vk c55 /Δk /Δ k=1  N  (k) = vk c56 /Δk /Δ k=1  N  (k) = vk c66 /Δk /Δ (k)

c¯11 = c11 c¯12 = c¯22 c¯23 c¯24 c¯33 c¯34 c¯44 c¯55 c¯56 c¯66

N 

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k=1

where  N  N 2   (k) (k) (k) vk c55 /Δk vk c66 /Δk − vk c56 /Δk k=1 k=1 2 k=1  (k) (k) (k) Δk = c55 c66 − c56 

Δ=

N 

(27)

References 1. Fischer, F.D., Kolednik, O., Predan, J., Razi, H., Fratzl, P.: Crack driving force in twisted plywood structures. Acta Biomater. 55, 349–359 (2017). https://doi.org/ 10.1016/j.actbio.2017.04.007 2. Huang, W., et al.: Multiscale toughening mechanisms in biological materials and bioinspired designs. Adv. Mater. 31(43), 1901561 (2019). https://doi.org/10.1002/ adma.201901561 3. Jones, R.M.: Mechanics of Composite Materials. CRC Press, Boca Raton (2018) 4. Okumura, K.: Simple model for the toughness of a helical structure inspired by the exoskeleton of lobsters. J. Phys. Soc. Jpn. 82(12), 124802 (2013). https://doi. org/10.7566/JPSJ.82.124802

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5. Reznikov, N., Shahar, R., Weiner, S.: Three-dimensional structure of human lamellar bone: The presence of two different materials and new insights into the hierarchical organization. Bone 59, 93–104 (2014). https://doi.org/10.1016/j.bone.2013. 10.023 6. Ritchie, R.O.: The conflicts between strength and toughness. Nat. Mater. 10(11), 817–822 (2011). https://doi.org/10.1038/nmat3115 7. Rui, H., Dongxu, L.: An inspiration from the microstructure of the cortical bone in goat tibia. Bioinspired Biomimetic Nanobiomater. 9(1), 53–63 (2020). https:// doi.org/10.1680/jbibn.19.00021 8. Shang, J.S., Ngern, N.H.H., Tan, V.B.C.: Crustacean-inspired helicoidal laminates. Compos. Sci. Technol. 128, 222–232 (2016). https://doi.org/10.1016/j. compscitech.2016.04.007 9. Sih, G.C., Paris, P.C., Irwin, G.R.: On cracks in rectilinearly anisotropic bodies. Int. J. Fract. Mech. 1(3), 189–203 (1965). https://doi.org/10.1007/BF00186854 10. Song, Z., Ni, Y., Cai, S.: Fracture modes and hybrid toughening mechanisms in oscillated/twisted plywood structure. Acta Biomater. 91, 284–293 (2019). https:// doi.org/10.1016/j.actbio.2019.04.047 11. Suksangpanya, N.: Fracture analysis in biomimetic Bouligand architectures. Ph.D. thesis, Purdue University (2016) 12. Suksangpanya, N., Yaraghi, N.A., Kisailus, D., Zavattieri, P.: Twisting cracks in Bouligand structures. J. Mech. Behav. Biomed. Mater. 76, 38–57 (2017). https:// doi.org/10.1016/j.jmbbm.2017.06.010 13. Suksangpanya, N., Yaraghi, N.A., Pipes, R.B., Kisailus, D., Zavattieri, P.: Crack twisting and toughening strategies in Bouligand architectures. Int. J. Solids Struct. 150, 83–106 (2018). https://doi.org/10.1016/j.ijsolstr.2018.06.004 14. Sun, C.T., Li, S.: Three-dimensional effective elastic constants for thick laminates. J. Compos. Mater. 22(7), 629–639 (1988). https://doi.org/10.1177/ 002199838802200703 15. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. ASME Press, New York (2000) 16. Ting, T.C.t.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, Oxford (1996), 45 17. Walters, M.C.: Domain-integral methods for computation of fracture-mechanics parameters in three-dimensional functionally-graded solids. Ph.D. thesis, University of Illinois at Urbana-Champaign (2005) 18. Walters, M.C., Paulino, G.H., Dodds, R.H.: Computation of mixed-mode stress intensity factors for cracks in three-dimensional functionally graded solids. J. Eng. Mech. 132(1), 1–15 (2006). https://doi.org/10.1061/(ASCE)0733-9399(2006)132: 1(1) 19. Wegst, U.G.K., Bai, H., Saiz, E., Tomsia, A.P., Ritchie, R.O.: Bioinspired structural materials. Nat. Mater. 14(1), 23–36 (2015). https://doi.org/10.1038/ nmat4089 20. Weiner, S., Arad, T., Sabanay, I., Traub, W.: Rotated plywood structure of primary lamellar bone in the rat: Orientations of the collagen fibril arrays. Bone 20(6), 509– 514 (1997). https://doi.org/10.1016/S8756-3282(97)00053-7 21. Yang, F., Xie, W., Meng, S.: Crack-driving force and toughening mechanism in crustacean-inspired helicoidal structures. Int. J. Solids Struct. 208–209, 107–118 (2021). https://doi.org/10.1016/j.ijsolstr.2020.10.016

Decoupled Multi-mode Controllable Electrically Interconnected Suspension for Improved Vehicle Damping Performance Pengfei Liu1 , Donghong Ning1(B) , Guijie liu1 , and Haiping Du2 1 College of Engineering, Ocean University of China, Qingdao 266110, China

[email protected] 2 School of Electrical, Computer and Telecommunications Engineering, University of

Wollongong, Wollongong, NSW 2522, Australia

Abstract. This paper proposes a new multi-mode controllable electrically interconnected suspension (EIS) system to further improve the driving smoothness and stability of passive suspension vehicles. The multi-mode controllable EIS utilizes the bi-directional conversion of mechanical and electrical energy of the electromagnetic damper and adds an active energy dissipation control circuit module to the semi-active EIS, which can realize active and semi-active multi-mode switching control of the EIS. Due to the decoupling characteristics of interconnected suspension, the switch in the designed active control circuit can achieve the effect of active segment decoupling control in conjunction with the adjustable resistor in the semi-active electrical network, which will greatly reduce the control difficulty of the EIS system. To deeply study the performance of each part of the multi-mode controllable EIS, the dynamics model, multi-mode controllable circuit model, and state equation of the semi-vehicle EIS are established. The target force required to improve the performance of the suspension is solved by an H∞ state feedback controller, and the control logic is designed for the coordination relationship between the bridge switch and the adjustable resistor module of the multi-mode controllable EIS so that the force generated by the EIS can better track the target force and achieve the purpose of semi-vehicle vibration control. The acceleration, body attitude, and power consumption information of the semi-active EIS only, active EIS only, and multi-mode controllable EIS are compared and analyzed by numerical simulation. Keywords: Interconnected suspension · Decoupling control · Active and semi-active · Vibration performance

1 Introduction The vehicle is a multi-degree-of-freedom vibration coupling mechanical system. The stiffness and damping of conventional suspensions are tightly coupled, and a compromise must be made between conflicting optimization objectives, which cannot simultaneously take into account the ride comfort and handling stability of the vehicle [1]. Suspension © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 105–119, 2024. https://doi.org/10.1007/978-981-97-0554-2_9

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interconnection technologies have been recognized by many scholars in solving the problem of conflicting vehicle smoothness and handling stability [2], such as anti-roll bars [3], pneumatically interconnected suspension [4], and hydraulically interconnected suspensions [5]. Electrically Interconnected Suspension (EIS) is a multi-degree-of-freedom vibration control system formed by connecting each electromagnetic damper (EMD) of the vehicle through an electrical network based on energy recovery of electromagnetic suspension (EMS) and with the technology of decoupling control of multiple suspensions interconnection [6]. As well, by constructing a suitable electrical network and adjusting the circuit component parameters, the decoupling control of multiple vibration modes and electromagnetic force of the body can be realized. In terms of interconnection structure, EIS replaces other interconnected suspension’s mechanical, oil, and gas pipes with circuits, and replaces other mechanical damping or hydraulic valves with controllable circuit components, which has the advantages of high energy conversion efficiency, fast control response, and simple control system [7]. In 2008, Hayashi et al. proposed a half-car coupled electromagnetic device based on EMDs, which is formed by connecting two EMD’s DC motors through a circuit containing multiple resistors [8]. Subsequently, in 2016, they proposed a circuit that couples four EMDs using a resistive element, allowing for independent tuning of the damping characteristics of the vehicle’s bounce, roll, and pitch motions [9]. However, due to the inability to adjust the resistance components in real-time, the vibration control performance of this coupling electromagnetic device is only equivalent to the effect of passive interconnected suspension. In 2020, Ning et al. [7] introduced the concept of circuit port network to describe the relationship between voltage and current in a semi-vehicle EIS using the impedance parameters of a two-port network to illustrate the characteristics of EIS decoupling control. The controller of the semi-vehicle EIS was designed using the H ∞ state feedback method, and the simulation results showed that the vibration control performance of the proposed system in the vertical and rolling directions was significantly improved. In 2021, Liu et al. [6] used the equivalent circuit method to obtain the time-domain model of the complex port circuit of EIS. They designed vertical and rolling vibration controllers using the features of EIS decoupled control, which simplified the difficulty of designing vibration controllers for multi-degree-of-freedom vehicles. Based on mechanical hardware-in-the-loop tests, the decoupled control characteristics and time-domain response of the semi-vehicle EIS were demonstrated. In 2022, Xia et al. proposed an optimization method for EN in EIS, adopted an innovative approach to determine the optimization scheme of EN structure in EIS to achieve satisfactory ride comfort and handling stability, and obtained the optimization parameters of EN using a genetic algorithm [10, 11]. Liao et al. developed a multi-functional EIS system with variable inertance and damping in the vertical direction and variable stiffness in the roll direction to improve the ride comfort of the vehicle [12]. In summary, the current research on EIS mainly utilizes the damping control realized by the vibration energy of the suspension itself [13], but the semi-active adjustable damping suspension only produces the force opposite to the direction of the suspension movement, and cannot produce active control force according to the demand for

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suspension damping [14], resulting in limited enhancement of the vehicle’s vibration damping performance. The study shows that the electromagnetic suspension with electric motor as the energy conversion medium can not only recycle the vibration energy of the suspension to realize the low-energy consumption of semi-active suspension but also consume electric energy to generate active control force according to the demand of control force to realize the high-performance characteristics of active suspension [15]. In order to further explore the vibration control performance of EIS, this paper proposes a multi-mode controllable electric interconnected suspension based on the semi-vehicle suspension model, so that it can give full play to the instinctive advantages of the electromagnetic suspension’s main/semi-active integration. The main contributions of this paper are: (1) An active and semi-active integrated multi-mode controllable EIS system is proposed; (2) A reasonable control logic is designed based on the characteristics of decoupled control of the multi-mode controllable EIS; (3) The performance of the multi-mode controllable EIS is analyzed from several aspects of suspension comfortable and handling stability. The rest of the paper is organized as follows: we will introduce the multi-mode controllable EIS half-car model and analyze its working principle in Sect. 2; Sect. 3 demonstrates the active and semi-active control logic of the multi-mode controllable EIS; then, in Sect. 4, we will use numerical simulation to verify the force tracking control effect of the multi-mode EIS and its vibration control performance under different road excitation conditions; Finally, Sect. 5 presents the conclusions of this research.

2 The Model of Multi-mode Controllable EIS System 2.1 Half-Car Model The half-car model with multi-mode controllable EIS is shown in which includes the vertical and rolling vibrations of the vehicle body and the vertical vibrations of the two wheels. Where ms and Is represent the vehicle body mass and moment of inertia, respectively, and mt is the wheel mass. ls Represents the distance from the suspension to the center of the body mass. zs is the vertical displacement and θs is the roll angle of the body mass. ztl and ztr are the vertical displacements of the wheels, and zgl and zgr are the road surface inputs (Fig. 1). When the body vibrates around the center of mass vertically and with small rotational amplitude, the vertical displacements zsl , zsr of the left and right sides of the vehicle body at the connection point with the suspension are respectively:  zsl = zs + θs ls (1) zsr = zs − θs ls The EMD in a multi-mode controllable EIS consists of a ball screw and a DC motor that replaces the damper in a conventional vehicle. The electromagnetic force generated by the EMD is related to the current flowing through the coil of the DC motor, and if its

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Fig. 1. The half-car model with multi-mode controllable EIS

intrinsic inertial force and friction are considered, the force fdl generated by the EMD on the left side can be described by Eq. (2). ⎧ fdl = fel + fil + ffl ⎪ ⎪ ⎨ fel = −rg ki il (2) ⎪ f = rg2 I0 (¨zsl − z¨tl ) il ⎪ ⎩ ffl = cf (˙zsl − z˙tl ) + f0 sign(˙zsl − z˙tl ) where fel , fbl , ffl are the electromagnetic force, inherent inertia force, and mechanical friction force generated by the EMD, respectively; ki , I0 are the torque constant and the rotor’s rotational inertia of the DC motor, respectively; rg is the transmission ratio of the ball crew; cf , f0 are the coefficient of kinetic friction and static friction of the EMD, respectively. Similarly, the force generated by the EMD on the right-hand side can be obtained. Thus, the kinetic equations for the four degrees of freedom vibration of the half-car suspension system are established as: ⎧ ms z¨s + ks (zsl − ztl ) + ks (zsr − ztr ) = Fdl + Fdr ⎪ ⎪ ⎨ Is θ¨s + (k  s (zsl − ztl ) − ks (zsr − ztr ))ls = (Fdl − Fdr )ls (3) ⎪ m z ¨ + k t ztl − zgl + ct z˙tl − z˙gl  − ks (zsl − ztl ) = −F dl ⎪ ⎩ t tl mt z¨tr + kt ztr − zgr + ct z˙tr − z˙gr − ks (zsr − ztr ) = −F dr where ks and kt are the stiffnesses of the left and right suspensions, respectively, and ct is the equivalent damping of the tires. 2.2 Circuit of Multi-mode Controllable EIS The multi-mode controllable EIS consists of two EMDs, and the circuit of the designed multi-mode controllable EIS is shown in Fig. 2, which includes semi-active and active control circuits, respectively.

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R1 M

R2

S2

S3

S1

S4

M

R2 S6

S7

S5

S8

R1 Active control circuit

Active control circuit

Semi-active control circuit

Fig. 2. The multi-mode controllable EIS circuit

All switches in the circuit are on by default, when the switches s1 (s5 ) and s1 (s5 ) are closed, the voltage of the battery will not be able to act in the circuit, and the current in the circuit is controlled by adjusting the two pairs of adjustable resistor modules (R1 , R2 ) to realize the semi-active control of the EIS; when the switches s1 (s5 ) and s3 (s7 ) or s2 (s6 ) and s4 (s8 ) are closed, the electric energy in the circuit simultaneously comes from the kinetic energy of the suspension and the battery at the same time, and the direction and magnitude of the current acting on the DC motor through the PWM signal realizes the active control of the EIS. The relationship between the modes and switching states of the system is shown in Table 1. Table 1. The relationship between the EIS modes and switching states Status

s1 (s5 ), s3 (s7 )

s2 (s6 ),s4 (s8 )

s1 (s5 ),s2 (s6 )

Active control

close

–

–

–

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–

–

–

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R1 R2 il ri

ri

i1

 el 

 

R1

rs ul

i3

ir

er i2

rs ur

R2

Fig. 3. Mesh analysis for multi-mode controllable EIS circuit

Therefore, we can describe the multimode controllable EIS circuit as shown in Fig. 3. The DC motor is simplified into a series connection of a voltage source and coil resistance, represented as el and er in series with ri for the left and right sides of the

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EIS, respectively. The active control circuit is simplified into a series connection of an adjustable voltage source and a switchable resistor, denoted as ul and ur in series with rs for the left and right sides of the EIS, respectively. The mesh currents i1 , i2 and i3 are assigned to the three meshes. Kirchhoff’s voltage laws (KVL) is applied to the meshes, then using Ohm’s law to express the voltages in terms of the mesh currents yields: ⎧ ⎪ ⎪ −el − ul + (r i +rs )i1 + R1 (i1 − i2 ) + R2 (i2 − i3 ) = 0 ⎪ ⎪ ⎪ ⎨ er + ur + (ri + rs )(i2 − i3 ) + R1 (i2 − i1 ) + R2 i2 = 0 (4) −er − ur + +(ri + rs )(i3 − i2 ) + R1 i3 + R2 (i3 − i1 ) = 0 ⎪ ⎪ ⎪ il = i1 ⎪ ⎪ ⎩ ir = i3 − i2 Collating Eq. (4), we can obtain:  il + ir = (el + er + ul + ur )/(r i + rs + R1 ) il − ir = ( el − er + ul − ur ) /(r i + rs + R2 )

(5)

Generally, the excitation voltage of a DC motor is directly proportional to its rotational speed, thus  el + er = rg ki (˙zsl − z˙tl + z˙sr − z˙tr ) (6) el − er = rg ki (˙zsl − z˙tl − z˙sr + z˙tr ) Combined with Eq. (2), the vertical electromagnetic force Fe and rotational electromagnetic torque Te of the multi-mode controllable EIS act on the vehicle body can be obtained as: ⎧

⎨ Fe = − r 2 k 2 (˙zsl − z˙tl + z˙sr − z˙tr ) + rg ki (ul + ur ) /(r i + rs + R1 )

g i (7) ⎩ Te = − r 2 k 2 (˙zsl − z˙tl − z˙sr + z˙tr )ls + rg ki (ul − ur )ls /(r i + rs + R2 ) g i It can be seen that the multi-mode controllable EIS has multiple decoupled control modes for the vertical electromagnetic forces and rotational electromagnetic moments of the vehicle body: (1) When ul = ur = 0, the vertical electromagnetic force of the multi-mode controllable EIS acting on the vehicle body is in the opposite direction to the suspension’s two sides relative vibration velocities sum, and its value increases with the decrease of the resistance R1 ; the rotational electromagnetic moment of the multi-mode controllable EIS acting on the vehicle body is in the opposite direction to the suspension’s two sides relative vibration velocities difference, and its value increases with the decrease of the resistance R2 . (2) When ul = ur = 0, the vertical electromagnetic force of the multi-mode controllable EIS acting on the vehicle body increases in absolute value as the sum of ul and ur increases; the rotational electromagnetic torque of the multi-mode controllable EIS acting on the vehicle body increases in absolute value as the difference between ul and ur increases. At the same time, to ensure that Fe and Te are large enough, it is necessary to adjust the values of the resistors R1 and R2 in the circuit to the minimum resistance.

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3 Controller Designer for Multi-mode Controllable EIS 3.1 H∞ Controller Design In this section, we design the H∞ controller by assuming that the left and right EMDs are ideal actuators that can generate ideal forces ul and ur . Considering the balance between system performance and energy consumption, we prioritize controlling the EMD to generate electromagnetic force to track the ideal force by adjusting the resistors R1 and R2 in the form of self-supplied energy. When the semi-active control is difficult to ensure a better tracking effect, the voltages Ul and Ur are then adjusted to input external energy into the EMD, thus realizing the multi-mode controllable EIS to control the vertical and rolling vibration of the half-car. Ignoring the nonlinear part of friction in the EMD, the vector is defined as X = T [ zs θs ztl ztr ] , the dynamic equation of half-car multi-mode controllable EIS can be transformed into matrix form (8) (M + B)X¨ + C X˙ + KX + LU = Fg ⎤ ⎡ ms 0 0 0     ⎢ 0 Is 0 0 ⎥ Ls K s LTs −Ls K s Ls C f LTs −Ls C f ⎥ ⎢ ,K = ,C = , where M = ⎣ −K s LTs K s + K t −C f LTs C f + C t 0 0 mt 0 ⎦ 0 0 0 mt   T      T Ls Bs Ls −Ls Bs 1 ls ks 0 kt 0 B = , L , K , Cf = = , K = = s s t −Bs LTs Bs 1 −ls 0 ks 0 kt      T     cf 0 1 ls −1 0 ct 0 bs 0 Ul , Ct = , Bs = , L = , , U = Ur 0 cf 0 ct 0 bs 1 −ls 0 −1 T  Fg = 0 0 kt zgl kt zgr , bs = rg2 I0 . For the convenience of expression, the state quantity of the EIS is selected as T  X1 = zsl − ztl zsr − ztr z˙sl z˙sr z˙tl z˙tr ztl − zgl ztr − zgr , the road incentives W1 =  T z˙gl z˙gr as interference input to the system. Then the state equation of the system can be written as (9) X˙ 1 = A1 X1 + B1 W1 + B2 U     06×2 I 4×4 04×4 where A1 = M 0 , B2 = M −1 0 , B1 = −(M + B)−1 K (M + B)−1 C −I 2×2 ⎤ ⎡ T Ls −I 2×2 02×2 02×2   ⎢ 02×2 02×2 LT 02×2 ⎥ 04×2 s ⎥ M0 , M0 = ⎢ ⎣ 02×2 02×2 02×2 I 2×2 ⎦. −(M + B)−1 Lse 02×2 I 2×2 02×2 02×2 According to the comprehensive evaluation performance of the suspension, the out z −z −ztr kt (ztl −zgl ) kt (ztr −zgr ) sl put of the system is defined as Z1 = τ z¨s θ¨s zfmaxtl zsrzfmax , we can Ff Ff get Z1 = τ (C 1 X1 + D2 U)

(10)

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where τ = diag( τ1 τ2 τ3 τ4 τ5 τ6 ) is the weight matrix of suspension perfor⎤ ⎡ −ks mt aLs −cf mt aLs cf mt aLs −bs kt aLs ⎢ 1 02×2 02×2 02×2 ⎥ mance, C 1 = ⎣ zfmax I 2×2 ⎦, D2 = mt aLs , a = kt 02×2 02×2 02×2 Ff I 2×2 1

1

diag( ms bs +2mt bs +ms mt Is bs +2mt bs l 2 +Is mt ), zfmax is the maximum travel of the suspension, s Ff = (ms + 2mt )g/2 is the static load of the tire. For the EIS system, combined with (10), according to the H∞ state feedback control law, we can get U = K X , where K is the feedback gain matrix. By solving the linear 





matrix inequality (LMI), we can get K . ⎡ ⎤ ∗ + P(A1 + B2 K ) ∗ ∗ ⎢ ⎥ H=⎣ −γ 2 I ∗ ⎦ < 0 BT1 P D1 −I C + D2 K 



(11)

where, P = P T > 0, γ is the desired level of disturbance attenuation. Pre- and post multiplying (11) by diag P −1 I I and its transposition, respectively, and defining Q = P −1 , Y = K Q, we have ⎤ ⎡ ∗ + AQ + B2 Y B1 ∗ (12) ∃=⎣ ∗ −γ 2 I ∗ ⎦ < 0 D1 −I CQ + D2 Y 

Use the parameters of the half-car EIS in Table 1 to solve and  optimize through the LMI toolbox in MATLAB, when τ = diag 1 1.4 1 1 1.2 1 .2 , and γ = 11.8, we can   7559 −4148 −5361 −495 1365 −258 87031 −10784 . get K = −4148 7559 −495 −5361 −258 1365 −10784 87031 

3.2 Multi-mode Control Logic Combining the ideal forces of EMD on both sides obtained from the H∞ controller, according to the decoupling characteristics of the multi-mode controllable EIS, it can be obtained that  I1d = −(Ul + Ur )/ki rg (13) I2d = −(Ul − Ur )/ki rg where I1d and I2d are the target currents required by the DC motor when the EIS controls the vertical and rotational vibration of the body, respectively. Since the values of both resistors R1 and R2 need to be greater than zero, the multimode EIS operates in the self-supply mode only when (el + er )I1d > 0 or (el − er )I2d > 0. At this time, we can close the switches s1 (s5 ), s2 (s6 ), that is ul = ur = 0, and prioritize the adjustment of the resistance in the circuit, which we can get according to Eq. (5)  R1d = (el + er )/I1d − ri − rs ((el + er )I1d > 0) (14) R2d = (el − er )/I2d − ri − rs ((el − er )I2d > 0)

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Since the adjustment range of the resistor is bounded, it is necessary to determine whether the obtained R1d and R2d are in the interval [ Rmin Rmax ], and if the value of R1d and R2d is greater than Rmax , the value of R1d and R2d is taken as Rmax . If the value of R1d and R2d is less than Rmi , it means that the self-supply of the EIS system alone cannot guarantee the electromagnetic force to track the target force better. In this case, take the value of R1d and R2d as Rmin , and adjust the switches s1 (s5 ), s3 (s7 ) or s2 (s6 ), s4 (s8 ) according to the direction of the excitation voltage of the electromagnetic damper motors, which is also known according to Eq. (5)  uld + urd = I1d (ri + rs + rmin ) − el − er (15) uld − urd = I2d (ri + rs + rmin ) − el + er when (el + er )I1d > 0 or (el − er )I2d > 0, the semi-active control mode of the EIS still cannot meet the demand, but Eq. (9) can still be used to calculate the uld and urd . Since the regulation range of the voltage is also bounded, it is still necessary to ensure that the obtained voltages uld and urd are in the interval [ −umax umax ]. By synthesizing the above control logic, the control flow chart of multi-mode controllable EIS for vertical vibration of the body can be obtained, as shown in Fig. 4, and similarly, the control flow of rotational vibration of the body can be obtained. W1

Vehicle model

X1

HĞ controller ul , ur I1d = (Ul +Ur)/rgki

fdl , fdr el + er

el + er I1d >0

No

uld+urd = I1d rirsRmin - el - er

Yes R1d =el+er I1d - ri

Multi-mode controllable EIS

Semi-active control RminİR1dİRmax Yes ul, ur, R1

R1 = R1d ul=ur=0

R1d = Rmin Yes

No RmaxİR1d Yes R1d = Rmax

No

RminıR1d Active control uld , urd ę[-Umax Umax]

ul, ur, R1

Fig. 4. The EIS vertical vibration control flow

4 Numerical Simulation for Multi-mode Controllable EIS To compare and analyze the damping performance of the multi-mode controllable EIS in different control modes, this paper constructs the half-car passive suspension model, semi-active and active modes of the multi-mode controllable EIS, respectively.

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The parameters of the vehicle and multi-mode controllable EIS in the simulation model are shown in Table 2. Table 2. The parameters of the vehicle and multi-mode EIS Parameters

Value

Parameters

Value

ki

0.254 Nm/A

ms

700 kg

ke

0.254 V · s/rad

mt

45 kg

ri

1.61 

ks

22000 N/m

p

0.020 m

cs

1500 N · s/m

I0

1590 g · cm2

kt

200000 N/m

f0

40 N

ls

0.65 m

cf

150 N · s/m

Is

350 kg · m2

rs

0.05 

he

0.4 m

Umax

48 V

zfmax

0.08 m

4.1 Sinusoidal Excitation To verify the control effect of force generated by the EIS on tracking ideal force. As shown in Eq. (21), the same and reverse direction road excitations are input to the simulation model, respectively.  zgl = zgr = 0.02sin(3π t) (Same direction input) (16) zgl = −zgr = 0.02sin(3π t) (Reverse direction input) Under the same road input, the simulation results of the passive suspension with damping of 1500 Nm/s and the multi-mode controllable EIS are obtained. The vehicles’ sprung mass acceleration and rotational angular acceleration are shown in Fig. 5 (a) and (b), respectively.

(a)

(b)

Fig. 5. The sprung acceleration under sinusoidal input: (a) Same direction; (b) Reverse direction.

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From Fig. 5, it can be seen that the unsprung mass acceleration and rotational angular acceleration of the multi-mode controllable EIS vehicle are smaller than those of the passive suspension, indicating that the designed H∞ controller and control logic is applicable to the vehicle suspension, and that the multi-mode controllable EIS with controllable damping can effectively reduce the vibration of the vehicle and improve the driving performance. Fig. 6 shows the EIS vertical electromagnetic force tracking the ideal force and the rotational electromagnetic torque tracking the ideal torque, respectively. From the force tracking trend, it can be seen that when the multi-mode EIS is working in semi-active mode, the electromagnetic force changes are discontinuous due to the lack of external energy input, so the electromagnetic force cannot completely track the ideal force. Its effect on suspension vibration control is worse than that of the active mode with the addition of external energy input.

Fig. 6. Force tracking effects of multi-mode controllable EIS under sinusoidal excitation: (a) vertical force; (b) rotational torque.

4.2 Bump Excitation Bump road is a commonly used excitation for suspension vibration testing. For the halfcar suspension system, the road will cause both vertical and roll vibrations of the vehicle. Therefore, the road profile is specially designed to verify the effectiveness of the EIS vibration control. Assume that the vertical vibration profile is zl1 = zr1 = 0.02(1 − cos(8π t)) (0 ≤ t ≤ 1)

(17)

and a roll vibration profile is zl2 = −zr2 = 0.02(1 − cos(4π t)) (0 ≤ t ≤ 1)

(18)

The two road profiles are combined as the mHIL system input, where zgl = zl1 + zl2 and zgr = zr1 + zr2 .

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Fig. 7. The sprung acceleration under bump road input

Figure 7 shows the vertical acceleration and rotational angular acceleration of the vehicle’s unsprung mass under bump excitation. Compared with the passive suspension, the peak acceleration of the multi-mode controllable EIS is significantly reduced and the vehicle vibration is stabilized more quickly.

(a)

(b)

Fig. 8. The circuit control of multi-mode controllable EIS.

Figure 8 (a) and (b) show the variation curves of resistance and voltage in the multimode controllable EIS circuit, respectively. It can be seen that in the initial stage of the bump excitation, the vertical vibration of the body is dominant, resistor R1 is frequently adjusted, and Ul and Ur are mostly in the same direction; in the later stage of the excitation, the lateral tilt vibration of the body is dominant, resistor R2 is frequently adjusted, and Ul and Ur are mostly in the opposite direction, which is consistent with the design concept of the controller. 4.3 Random Excitation To comprehensively evaluate the time-domain performance of the EIS, a random excitation test is implemented. According to the ISO 8608–2016 classification of road roughness level, the C class random road profile is obtained.

Decoupled Multi-mode Controllable Electrically Interconnected Suspension

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

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Fig. 9. Vibration control performance of multi-mode controllable EIS under the random road

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Figure 9 (a) to (h) show the vibration control performance of the multi-mode controllable EIS under random road excitations, including the full-time vertical acceleration, full-time rotational angular acceleration, part-time vertical acceleration, part-time rotational angular acceleration, dynamic travel of the left suspension, dynamic travel of the right suspension, dynamic load of the left tire, and dynamic load of the right tire, respectively. It can be seen that the overall performance of the multi-mode controllable EIS semi-active mode is slightly worse than that of the active control mode, but they both outperform the passive suspension by significantly attenuating the peaks of the vehicle vibration acceleration and maintaining them in a low range.

5 Conclusion In this paper, the circuit topology, control method and vibration control performance of a novel multi-mode controllable EIS are investigated to obtain the following conclusions: (1) The multi-mode controllable EIS utilizes the bidirectional conversion of mechanical and electrical energy of the electromagnetic damper, which can realize active and semi-active multi-mode switching control and still has the decoupling characteristics. (2) The designed control logic of the H∞ state feedback controller, resistance, and voltage is suitable for multi-mode controllable EIS to achieve a better vibration control level. (3) The multi-mode controllable EIS is characterized by a self-supply of energy in the semi-active mode, but its performance is significantly weaker relative to the active control mode under more intense excitation. Acknowledgment. The research is supported by the Taishan Scholars Program of Shandong Province (NO. Tsqn202211062), the China Postdoctoral Science Foundation (NO. 2023M733339), and the Nature Science Foundation of Shandong Province (NO. ZR2023QE113).

References 1. Smith, W.A., Zhang, N.: Recent developments in passive interconnected vehicle suspension. Front. Mech. Eng. China 5, 1–18 (2010). https://doi.org/10.1007/s11465-009-0092-z 2. Zhang, N., Smith, W.A., Jeyakumaran, J.: Hydraulically interconnected vehicle suspension: background and modelling. Veh. Syst. Dyn. 48, 17–40 (2010). https://doi.org/10.1080/004 23110903243182 3. Gao, J., Wu, F., Li, Z.: Study on the effect of stiffness matching of anti-roll bar in front and rear of vehicle on the handling stability. Int. J. Automot. Technol. 22, 185–199 (2021). https:// doi.org/10.1007/s12239-021-0019-1 4. Zhu, H., Yang, J., Zhang, Y.: Dual-chamber pneumatically interconnected suspension: modeling and theoretical analysis. Mech. Syst. Signal Process. 147, 107125 (2021). https://doi. org/10.1016/j.ymssp.2020.107125 5. Qi, H., Zhang, B., Zhang, N., Zheng, M., Chen, Y.: Enhanced lateral and roll stability study for a two-axle bus via hydraulically interconnected suspension tuning. SAE Int J. Veh. Dyn. Stab. NVH. 3, 5–18 (2018). https://doi.org/10.4271/10-03-01-0001

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6. Liu, P., Zheng, M., Ning, D., Zhang, N., Du, H.: Decoupling vibration control of a semi-active electrically interconnected suspension based on mechanical hardware-in-the-loop. Mech. Syst. Signal Process. 166, 108455 (2022). https://doi.org/10.1016/j.ymssp.2021.108455 7. Ning, D., Du, H., Zhang, N., Sun, S., Li, W.: Controllable electrically interconnected suspension system for improving vehicle vibration performance. IEEE/ASME Trans. Mechatron. 25, 859–871 (2020). https://doi.org/10.1109/TMECH.2020.2965573 8. Hayashi, R., Suda, Y., Nakano, K.: Anti-rolling suspension for an automobile by coupled electromagnetic devices. J. Mech. Syst. Transp. Logist. 1, 43–54 (2008). https://doi.org/10. 1299/jmtl.1.43 9. Fukumori, Y., Hayashi, R., Okano, H., Suda, Y., Nakano, K.: Study on coupled shock absorber system using four electromagnetic dampers. J. Phys. Conf. Ser. 744, 012217 (2016). https:// doi.org/10.1088/1742-6596/744/1/012217 10. Xia, X., Ning, D., Liu, P., Du, H., Zhang, N.: Electrical network optimization for electrically interconnected suspension system. Mech. Syst. Signal Process. 187, 109902 (2023). https:// doi.org/10.1016/j.ymssp.2022.109902 11. Gao, Z., Xia, X., Liao, Y., Ning, D., Liu, P., Du, H.: Electrical network optimization based electrically interconnected suspension control for vehicle cabin. In: SAE International, Warrendale, PA (2023). https://www.sae.org/publications/technical-papers/content/2023-01-0172/. Accessed 5 Mar 2023 12. Liao, Y., Ning, D., Liu, P., Du, H., Li, W.: A versatile semi-active electrically interconnected suspension with heave-roll vibration decoupling control. IEEE Trans. Ind. Electron. 1–10 (2022). https://doi.org/10.1109/TIE.2022.3196359 13. Taghavifar, H.: A novel energy harvesting approach for hybrid electromagnetic-based suspension system of off-road vehicles considering terrain deformability. Mech. Syst. Signal Process. 146, 106988 (2021). https://doi.org/10.1016/j.ymssp.2020.106988 14. Ning, D., Sun, S., Li, H., Du, H., Li, W.: Active control of an innovative seat suspension system with acceleration measurement based friction estimation. J. Sound Vib. 384, 28–44 (2016). https://doi.org/10.1016/j.jsv.2016.08.010 15. Li, Y., Zheng, L., Liang, Y., Yu, Y.: Adaptive compensation control of an electromagnetic active suspension system based on nonlinear characteristics of the linear motor. J. Vib. Control 26, 1873–1885 (2020). https://doi.org/10.1177/1077546320909985

Adaptive Robust Sliding-Mode Control of a Semi-active Seat Suspension Featuring a Variable Inertance-Variable Damping Device Guangrui Luan, Pengfei Liu(B) , Donghong Ning(B) , and Guijie Liu Ocean University of China, Qingdao 266000, China {liupengfei,ningdonghong}@ouc.edu.cn

Abstract. This article introduces a novel mechanical variable inertance-variable damping (VIVD) seat suspension based on an adaptive robust sliding-mode (ARSM) controller, including its characteristics validation and performance experiment. In this paper, a variable damping (VD) device and a flywheel are connected in series to form a variable inertance (VI) device with real-time controllable inertance, which is connected in parallel with another VD device to form a VIVD device. A two-layer control scheme is proposed where an upper desired controller is designed based on adaptive robust sliding-mode control and the desired control force is calculated; then a force tracking control strategy with energy priority storage (EPS) is designed as the lower layer controller. Under road random excitation, the VIVD seat suspension exhibits 21.89% and 9.56% lower RMS acceleration values compared to the passive seat suspension and a semi-active traditional sliding-mode control seat suspension. The new system demonstrates advantages in controllability and energy efficiency, with energy consumption falling within the milliwatt range. The proposed semi-active VIVD device shows potential in vehicle vibration control. Keywords: Variable inertance-variable damping · Seat suspension · Adaptive robust sliding-mode · Two-layer control scheme · Energy priority storage

1 Introduction Commercial heavy-duty vehicles play an essential role in the logistics and transport industry. When working the driver of a heavy vehicle is subjected to severe vibrations from rough road and the vehicle’s powertrain, suffering from the injuries causing low back pain [1]. The seat suspension is in direct contact with the human body, its vibration isolation performance impacts on human comfort, thus, research into various types of seat suspension is conducted in the field of vehicle vibration control [2]. Generally, the seat suspensions are classified as passive [3], semi-active [4] and active suspensions [5]. Compared to passive seat suspensions, semi-active seat suspensions are controllable with a fast response for different vibration conditions. Moreover, they possess the advantages of a simpler mechanical structure, less power consumption and fail-safe characteristics compared with active suspensions. The semi-active system © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 120–135, 2024. https://doi.org/10.1007/978-981-97-0554-2_10

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includes the variable damping (VD) device [6],variable stiffness (VS) device [7], and variable inertance (VI) device [8]. The inertance is characteristic of the inerter, which is a two terminals mechanical element corresponding to the spring and damper [13]. The force produced by an inerter is the inertance multiplied by the relative acceleration of the two ends [14], similar to the force produced by a spring is the stiffness multiplied by the relative displacement of the two ends. Ning et al. verified that a VD device and passive inerter can form mechanical networks with specific topologies, the magnetorheological fluid damper [15] and electromagnetic damper (EMD) [16] are utilized to design the novel VI device. After that, the semi-active variable equivalent stiffness and inertance (VESI) device implemented by an electrical network is proposed to improve the vibration reduction performance at both vertical and roll directions [17]. This paper proposed a VI device structure with a passive inerter and a electromagnetic variable damping device in serial, and validated its mechanical properties that are different from those of VD devices. Besides the innovation in mechanical structure, there are also considerable researches on control strategies for semi-active systems. The adaptive fuzzy sliding-mode controller, H ∞ controller [18], direct voltage controller based on TS fuzzy model [19], Lyapunov type robust controller [20] etc. are used for the control of semi-active magnetorheological (MR) seat suspension. Hu et al. utilized a suboptimal control law, called steepest gradient control in the semi-active damper for the control, meanwhile, low-order admittance networks are employed in the passive parts to optimize the low-order positive real admittance functions [21]. By utilizing the mechanical properties variability with advanced control strategies, the vibration of the suspension is attenuated to a large extent. This paper proposes a novel semi-active VIVD device structure and a two-layer control scheme. The upper controller is an adaptive robust sliding-mode controller that calculates the desired values. The lower controller ensures that vibration energy can be stored in the VI device, prioritising the use of this energy for vibration suppression. The key advantages of the proposed device and its controller are listed as: 1. The two mechanical properties, damping and inertance, can be independently validated and controlled. 2. A new ARSM controller and semi-active tracking control strategy are designed for VIVD seat suspension control. The remaining sections of the paper are structured as follows: Sect. 2 presents a model and prototype of the VIVD seat suspension, which is then the subject of analysis and validation. The ARSM controller and force tracking performance of the VIVD device is discussed in Sect. 3; Sect. 4 shows the vibration control performance of the VIVD seat suspension. Finally, Sect. 5 presents the conclusions of this research.

2 VIVD Seat Suspension 2.1 VIVD Seat Suspension Model A VI device can be created by serially connecting a VD device and a mechanical inerter. In [11], a flywheel is utilized as a mechanical inerter, connected in series with the VD unit to constitute a mechanical VI device. Moreover, the VD device exhibits efficient vibration

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suppression at higher excitation frequencies. Theoretically, by parallelly connecting VD and VI devices and implementing a suitable control strategy, vibration reduction can be achieved across a broader excitation range.

Fig. 1. The model of the VIVD seat suspension

This paper proposes a mechanical VIVD device that replaces the passive dampers in the seat suspension for vibration control. When VIVD devices are used in seat suspensions, the inherent inertance of the device and the friction between the components must be considered. Therefore, a simplified model of the VIVD seat suspension is illustrated in Fig. 1 mZ¨s = −K(Zs − Zv ) − Fout

(1)

Fout = Fr + Fj + FVD + FVI

(2)

where Zs and Zv are the displacement of the seat and cab chassis, K is the stiffness of seat suspension, Fr and Fj are the friction force and inherent inertance force of the VIVD device. FVD is the output force of VD device, The damping and inertance of the VIVD can be controlled by the controller. 2.2 VIVD Device Prototype In this article a prototype of the VIVD device is designed. Figure 2 Depicts the prototype of VIVD device. The orange section represents the motions of the VIVD device, since both the VD and the VI device are mounted between the seat and the cab chassis, the relative displacement is denoted as Zs − Zv , The ball screws are employed to transform the reciprocating motion of the seat suspension into a rotary motion represented by α, α = r(Zs − Zv )

(3)

r = 2π/d

(4)

where r is the transmission ratio of the ball screw, d is the lead of the ball screw, the d = 0.032 m in this design, and α is ration angle of the ball screw.

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The VI and VD device both employ the permanent magnet synchronous motors (PMSM) as the damping adjustment device, the Cvd and Cvi are the damping generated by the PMSM. The shaft of the PMSM is connected with the ball screw, thus, the α is the ration angle of shaft. In the VD device, the PMSM directly is connected with the shell, which is mounted on the bottom of the seat. The force Fvd generated by VD device is: Fvd = rCvd α˙ + Fjvd + Frvd

(5)

where α˙ is the rotational speed of the ball screw. Fjvd and Frvd are the inherent inertance and the friction force generated by the VD device.

Fig. 2. The prototype of the VIVD device (a). VD device (b). VI device

However, in the VI device, the PMSM is not connected with “ground”, the PMSM is connected to the flywheel and the two are rotated together, β is the ration angle of the flywheel. The relative rotational speed of ball screw and flywheel generates the output force FVI of the VI device. ˙ Fvi = rCvi (α˙ − β)+F jvi + Frvi

(6)

where β˙ is the rotational speed of the flywheel. Fjvi and Frvi are the inherent inertance and the friction force generated by the VI device. By adjusting the damping of the PMSM, the equivalent inertance of the VI device becomes controllable in real-time. In the ideal scenario, with the PMSM damping set to zero, the flywheel freely rotates and remains disconnected from the ball screw. Conversely, when the PMSM damping is set to infinity, the flywheel behaves as if it is rigidly connected to the ball screw. Consequently, the output force of the VI device is dependent on the equivalent inertance of the flywheel and can be expressed as follows: ¨ Fvi = rbβ+F jvi + Frvi

(7)

Thus, the dynamic model of the VI device can be built with the consideration of the flywheel:   (8) Cvi α˙ − β˙ = bβ¨

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The damping generated by the PMSM can be controlled through varying the external resistance of the circuit. The controllable rotary damping of the PMSM is: CT =

Ki Ke (Re + Ri )

(9)

where CT is the real-time damping of the PMSM; Ki , Ri Ke are the voltage constant and current constant of PMSM. Ri , Re are the internal resistance of PMSM, external resistance, respectively. 2.3 Mechanical Admittance Analysis Admittance is the reciprocal of impedance. In this section, we establish the mechanical admittance in the frequency domain of the VIVD device. The analysis of mechanical admittance allows us to observe the variations in damping and inertance characteristics of the VIVD device and verify the validity of its topology. This section will focus on the controllability of the VIVD device with regard to damping and inertance. To facilitate controller design, we simplify uncontrollable forces, such as friction force and inertial friction of the shaft. The mechanical admittance of VIVD device is:   r 2 bCvi 2 b2 Cvi ω2 2 + jω (10) Yi = r Cvd + b2 ω2 + Cvi 2 b2 ω2 + Cvi 2 Expressed in terms of resistance paraments as:   k2 r 2 bk 4 (Ri + Rvi )2 b2 k 2 ω2 (Ri + Rvi ) + Yi = r 2 + jω Ri + Rvd b2 ω2 (Ri + Rvi )2 + k 4 (Ri + Rvi )2 b2 ω2 + k 4

(11)

where k = Ki Ke The real part of the admittance represents the mechanical conductance and provides information about the equivalent damping. Besides, the imaginary part of the admittance corresponds to the mechanical susceptance, which carries information about the equivalent inertance. The equivalent damping and equivalent inertance are denoted as: ce = Yi(real) = be =

k2 b2 k 2 ω2 (Ri + Rvi ) + Ri + Rvd b2 ω2 (Ri + Rvi )2 + k 4

Yi(imag) r 2 bk 4 (Ri + Rvi )2 = ω (Ri + Rvi )2 b2 ω2 + k 4

(12) (13)

Figure 3 illustrates that as the resistance decreases, both the mechanical conductance and the mechanical susceptance increase. With increasing frequency, the mechanical conductance continues to rise, whereas the mechanical susceptance initially increases, reaches a certain value, and then gradually decreases. In Fig. 4, it can be observed that the equivalent damping and equivalent inertance of the VIVD device increase as the resistance of the external resistor decreases. Perform a detailed analysis of the frequency variations, Fig. 4 (a) reveals that the maximum

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inertance of the VIVD device is 250 kg, and the equivalent inertance decreases with increasing frequency. On the other hand, Fig. 4 (b) demonstrates that the VIVD device exhibits different minimum damping values with varying external resistors, and the equivalent damping gradually increases with increasing frequency. 1200 Rvd=Rvi= 0.3 Ohm Rvd=Rvi= 3.3 Ohm

Mechanical Susceptance (S)

1000

Rvd=Rvi= 8.3 Ohm Rvd=Rvi= 15.3Ohm

800

Rvd=Rvi=50.3Ohm

Fre q

uen c

y

600

400

200

0 1000

2000 3000 Mechanical Conductance (S)

4000

Fig. 3. The admittance of the VIVD device

(a). Equivalent damping

(b). Equivalent inertance

Fig. 4. Equivalent Mechanical properties of VIVD device

2.4 Parameter Identification Figure 5 depicts a force-displacement test rig utilized for parameter identification. The designed VIVD device includes frictional forces that vary with velocity and inherent inertial forces that change with acceleration. Additionally, the parameters of the PMSM, such as current constants, voltage constants, and internal resistance, need to be identified. Friction and inertia forces are modeled as follows:   (14) Fr = fr sat Z˙s − Z˙v Fj = r 2 J (Z¨s − Z¨v )

(15)

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 ⎧  ˙s − Z˙v > τ 1 Z ⎨     sat Z˙s − Z˙v = (1/τ )(Z˙s − Z˙v ) − τ ≤ Z˙s − Z˙v ≤ τ  ⎩ ˙ −1 Zs − Z˙v < −τ

(16)

Firstly, the excitation motion is x = Asin(2π ft), where A = 0.01 m, f = 1.5 Hz. The impact of external resistors on VIVD devices is verified by connecting resistors of varying values (50, 15, 8, 3, 0 ) to the circuit. Test results are shown in the. During VI device testing, the force generated by the VI device and the motion displacement exhibit negative stiffness characteristics, which increase as the resistance connected to the circuit decreases. The results indicate that adjusting the external resistance value of Re can modify the inertance of the device. By analyzing the characteristic curves of VD devices, it is observed that the area of the enclosed curve increases as the external resistor Re decreases, thereby indicating that the damping coefficient of both VD devices can be controlled by adjusting the external resistance Re . The results demonstrate that the designed VIVD device possesses controllable inertance and damping. The controller can select suitable external resistors Re to finely tune the damping and inertance of the seat suspension, thereby achieving effective seat control Figs. 5 and 6.

Fig. 5. Test platform

In the Figs. 7 and 8, different frequencies and amplitudes were set for the sinusoidal excitation to test the VI and VD devices respectively. When the frequency and amplitude increase, the area of the curved enclosure increases, which implies an increase in the output energy of the VIVD device. At the same time, the relative accuracy of the dynamic modelling of the VIVD device is verified by matching the experimental results with the simulation results under different operating conditions. Table 1 shows the results of the parameter identification.

Adaptive Robust Sliding-Mode Control exp_50 sim_50 exp_15 sim_15 exp_8 sim_8 exp_3 sim_3 exp_0 sim_0 External resistors(ohm)

200

100

sim_0 exp_0 sim_3 exp_3 sim_8 exp_8 sim_15 exp_15 sim_50 exp_50 External resistors(ohm)

300 200 100

Force (N)

0

Force (N)

127

-100

0 -100

-200 -200 -300 -0.010

-0.005

0.000

0.005

-300 -0.010

0.010

-0.005

0.000

0.005

Displacement (m)

Displacement (m)

(a). VI device

(b). VD device

0.010

Fig. 6. Characteristic curves of VIVD devices connected to different resistors Frequency (Hz)

400

exp_1.0 exp_1.5 exp_2.0 exp_2.5

300

exp_1.0 exp_1.5 exp_2.0 exp_2.5

400

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(b). VD device

Fig. 7. Characteristic curves of VIVD devices under different frequency 600

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Fig. 8. Characteristic curves of VIVD devices under different amplitude

0.03

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Parameter

Symbol

Value

Coefficient of friction of VI device

frvi

35 N

Coefficient of friction of VD device

frvd

200 × 10−7 kgm2

Inherent inertance of VI device

Jvi

30 N

Inherent inertance of VD device

Jvd

150 × 10−7 kgm2

Moment of inertia of flywheel

b

160 × 10−5 kgm2

Current constants (voltage constants) of PMSM

Ki (Ke )

0.41 Nm/s (Vs/rad)

Internal resistance of PMSM

Ri

10.5 Ohm

3 Controller Design 3.1 Adaptive Robust Sliding-Mode Controller Ensuring the absolute accuracy of various parameters, such as friction and inertial forces, during the seat suspension design process is challenging. In this paper, we adopt the approach of attributing all uncertain characteristics to changes in the mass on the spring. This enables the controller to modify the seat suspension characteristics and adapt to changes in the dynamic characteristics of objects and perturbations. The uncertain dynamic characteristics model of the seat suspension can be described as: M Z¨s = − − fdes

(17)

 = K(Zs − Zv ) + Fr + Fj

(18)

where  is the disturbances and total uncertainty in the uncertainty component of the model, u(t) is the desired dynamic output force of the VIVD device, M is the actual measurement of the mass on the spring. The sliding surface is defined as: s = e˙ + ce = Z˙s − Z˙d + ce

(19)

where Zd is the desired displacement, which is assumed as zero. e is tracking error of the controller. M s˙ = M (Z¨s − Z¨d + c˙e)

(20)

Considering the Lyapunov functions as: V =

1 2 1 ˜2 Ms + M 2 2γ

(21)



˜ =M −M M

(22)

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where M is the estimated value of M . So, its derivative is:  1  1 ˜ ˙˜ ˙˜ ˜M V˙ = Ms˙s + M M = s M Z¨s − M Z¨d + Mc˙e + M γ γ

(23)

where γ > 0. Substituting the formula (20):   1 ˜ ˙˜ V˙ = s(− − fdes − M Z¨d − Mc˙e ) + M M γ Thus, the controller can be designed as:   fdes = −M Z¨d − Mc˙e + ks + δsign(s)

(24)



(25)

where δ should be greater than the upper limit of the perturbation, δ ≥ max where max is the maximum values of  in the vehicle travelling. Then,

  ˙˜ ˜ Z¨d − Mc˙e + 1 M ˜M V˙ =s − − ks − sign(s) + M γ   ˙˜ ˜M ˜ s Z¨d − Mc˙e + 1 M = − ks − δ | s | −s + M γ

(26)

(27)

Adaptive law is designed to:   ˙ M = −γ s Z¨d − Mc˙e

(28)

V˙ = −ks2 − δ | s | −s ≤ −ks2 ≤ 0

(29)



And,

If and only if s= 0, the V˙ = 0, the system is asymptotically stable. But when V˙ = 0, ˜ → 0. At this point V is no longer decreasing, therefore, there is no guarantee that M Thus, the design of the adaptive law should ensure that the value of The design of the adaptive law should ensure that the value of M lies between M lies between [Mmin , Mmax ], Mmin = min /Zsmax + M

(30)

Mmax = max /Zsmin + M

(31)

The adaptive low is redesigned: ˙ M = Proj



˙



M

   −γ s Z¨d − Mc˙e

(32)

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⎧ ⎪ ⎨ 0 if M ≥ Mmax and · > 0 Proj ˙ (·) = 0 if M ≤ Mmin and · < 0 ⎪ M ⎩ · otherwise





(33)

Besides the sign(s) is replaced by sat(s) in the implementation of the controller to suppress the chattering phenomenon in the sliding mode [16]. 3.2 Force Tracking Controller Due to the lack of external energy input, the VIVD device is unable to produce the desired force fdes . To address this, a force tracking controller based on energy priority storage (EPS) is designed in this section. The control scheme is display in the Fig. 9. The sensors of seat suspension captured system status. Then, the upper layer ASMC controller calculated the fdes , the lower layer force tracking controller based on EPS calculated the output damping (Cvi , Cvd ) of VIVD device. Eventually, the external resistance is selected to modify the mechanical characteristics of the seat suspension. In the force tracking controller with EPS, the VI device is given the highest control priority to gather more energy for vibration attenuation. Meanwhile, the VD device is maintained at minimal damping, and the damping of the VI device changes in real-time while the flywheel collects the vibration energy. Cvd = Cmin

(34)

  Cvi = (fdes − fvdmin )/ r 2 α˙ − β˙

(35)

fvdmin = rCmin α˙

(36)

where Cmin and fvdmin are the minimum value of damping and force of PMSM, respectively. The Cvd and Cvi are the damping of the VD and VI device, respectively.

Fig. 9. Control scheme

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When the Cvi > Cmax , VD device enters the working state, energy stored in VI devices is released. Cvi = Cmax

(37)

 Cvd = (f des − fvimax )/ r 2 α˙

(38)

  fvimax = Cmax r 2 α˙ − β˙

(39)

where Cmax and fvimax are the maximum value of the damping and output force of the PMSM, respectively. Finally, the right resistor is picked so that the PMSM produces the correct damping. Rj = Ki Ke /Cj − Ri k

(40)

where Rj and Cj are the resistance value and damping of the VI and VD device, the j is the symbol of the VD or VD device.

4 Experimental System and Results Analysis 4.1 Experimental Setup The experimental setup is shown in Fig. 10. The VIVD device is mounted on a quarter car suspension testing platform which is under the control of an NI myRIO; the vibration platform can generate desired vibration based on the commands from Computer 2. The relative displacement of the seat suspension is measured by a pulse type wire displacement sensor; and another wire sensor is applied to measure the displacement of the output displacement of the seat suspension. A force sensor is mounted under the VIVD seat suspension to measure the force of the VIVD seat suspension. A acceleration sensors are used to obtain the top accelerations of the seat suspension. An NI CompactRIO 9038 is used to control two controllable resistor based on sensors’ feedback. 4.2 Vibration Control Test In the random excitation, the characteristics of controller and vibration control performance of VIVD seat suspension are verified. A SM controller was designed, and a resistance of 8  was connected to the circuit to make the VIVD seat suspension a passive suspension with fixed damping and inertance. These two types of suspension were used as comparison objects to verify the vibration control ability of ARSM control.

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Fig. 10. Experimental setup. 100

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-0.3

-0.2

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0.0

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ARSM SM Passive

Acceleration (m/s2

2 1 0 -1 -2 -3 0

10

20

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40

50

Fig. 12. Acceleration in the time domain

Figure 11 illustrates the energy output range of the VIVD seat suspension. It reveals that the VIVD seat suspension generates a positive output without any external energy input. This indicates that the flywheel can harvest vibration energy and utilize it to suppress the vibration. Figure 12 show the acceleration in the time domain, In the Fig. 12, the semi-active suspensions with ARSM controllers demonstrated reductions of 9.56%

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1000

ARSM SM Passive

800 600 400 200 0 2

6

4 Frequency (Hz)

8

Fig. 13. Acceleration in the frequency domain

ARSM SM Passive

1.6 1.4 1.2

Rate

1.0 0.8 0.6 0.4 0.2 0.0

RMS

FW-RMS

VDV

Fig. 14. Parameters of vibration

and 21.89% in root-mean-square (RMS) acceleration compared with semi-active suspensions with SM controllers and passive seat suspension, respectively. Figure 13 shows the vibration absorption effect of the three seat suspensions in the frequency domain. When greater than 4 Hz, there is almost no difference in the vibration control ability of the three seat suspensions, but in the frequency range of 0.Hz-3 Hz, which is most perceptible to the human body, the semi-active seat suspension equipped with the ARSM controller exhibits the most significant mitigation effect. Figure 14 visualizes the vibration parameters, and the relative rates of the other two suspensions compared to the passive seat were calculated separately using the parameters of the passive seat suspension as a baseline.

5 Conclusion In this study, the dynamic model of the VIVD seat suspension is determined firstly, and the topology of the VIVD device is established using the analysis method of mechanical admittance. Next, the prototype of the VIVD device is designed and manufactured, and its parameters are identified, the relevant parameters of the VIVD device are determined preliminarily. Following this, a two-layer control scheme is designed. An adaptive robust sliding mode controller is designed based on the traditional sliding-mode controller,

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which corrects the error between the modelling process and the actual physical model through the adaptive adjustment of the structural parameters, Simultaneously, it enhances controller robustness and reduces the chattering phenomenon of sliding-mode control. The lower layer force tracking controller prioritizes the VI device for execution, ensuring efficient energy harvesting for vibration suppression by the VIVD device. Finally, the vibration control performance of both the conventional sliding mode control and the proposed ARSM controller is evaluated through traditional under random excitation. The ARSM control improves the vibration mitigation performance of the semi-active seat suspension by 9.56% using the same device, thus validating the effectiveness of the proposed control approach. Funding. This research was funded by the Taishan Scholars Program of Shandong Province (NO.tsqn202211062) and the Nature Science Foundation of Shandong (NO. 2022HWYO-067).

References 1. Kumar, S.: Vibration in operating heavy haul trucks in overburden mining. Appl. Ergonomics 35(6), 509–520 (2004). https://doi.org/10.1016/j.apergo.2004.06.009 2. Liu, P., Ning, D., Luo, L., Zhang, N., Haiping, Du.: An electromagnetic variable inertance and damping seat suspension with controllable circuits. IEEE Trans. Ind. Electron. 69(3), 2811–2821 (2022). https://doi.org/10.1109/TIE.2021.3066926 3. Le, T.D., Ahn, K.K.: A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat. J. Sound Vib. 330(26), 6311–6335 (2011). https:// doi.org/10.1016/j.jsv.2011.07.039 4. Yao, G.Z., Yap, F.F., Chen, G., Li, W.H., Yeo, S.H.: MR damper and its application for semiactive control of vehicle suspension system. Mechatronics 12(7), 963–973 (2002). https:// doi.org/10.1016/S0957-4158(01)00032-0 5. Ning, D., Sun, S., Zhang, J., Haiping, Du., Weihua, Xu., Wang, Li.: An active seat suspension design for vibration control of heavy-duty vehicles. J. Low Freq. Noise, Vib. Act. Control 35(4), 264–278 (2016). https://doi.org/10.1177/0263092316676389 6. Sun, S.S., et al.: Horizontal vibration reduction of a seat suspension using negative changing stiffness magnetorheological elastomer isolators. Int. J. Veh. Des. 68(1/2/3), 104 (2015). https://doi.org/10.1504/IJVD.2015.071076 7. Deng, L., et al.: Investigation of a seat suspension installed with compact variable stiffness and damping rotary magnetorheological dampers. Mech. Syst. Signal Process. 171, 108802 (2022). https://doi.org/10.1016/j.ymssp.2022.108802 8. Ning, D., et al.: An electromagnetic variable inertance device for seat suspension vibration control. Mech. Syst. Signal Process. 133, 106259 (2019). https://doi.org/10.1016/j.ymssp. 2019.106259 9. Smith, M. C.: Synthesis of mechanical networks: the inerter, In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002. 2002, 10. Smith, M.C., Wang, F.-C.: Performance benefits in passive vehicle suspensions employing inerters. Veh. Syst. Dyn. 42(4), 235–257 (2004). https://doi.org/10.1080/004231104123312 89871 11. Ning, D., et al.: A rotary variable admittance device and its application in vehicle seat suspension vibration control. J. Franklin Inst. 356(14), 7873–7895 (2019). https://doi.org/10.1016/ j.jfranklin.2019.04.015

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12. Ning, D., Haiping, Du., Zhang, N., Jia, Z., Li, W., Wang, Y.: A semi-active variable equivalent stiffness and inertance device implemented by an electrical network. Mech. Syst. Signal Process. 156, 107676 (2021). https://doi.org/10.1016/j.ymssp.2021.107676 13. Xiu-Mei, Du., Miao, Yu., Jie, Fu., Peng, Y.-X., Shi, H.-F., Zhang, H.: H∞ control for a semiactive scissors linkage seat suspension with magnetorheological damper. J. Intell. Mater. Syst. Struct. 30(5), 708–721 (2018). https://doi.org/10.1177/1045389X18778340 14. Haiping, Du., James Lam, K.C., Cheung, W.L., Zhang, N.: Smart Mater. Struct. 22(10), 105016 (2013). https://doi.org/10.1088/0964-1726/22/10/105016 15. Park, C., Jeon, D.: Semiactive vibration control of a smart seat with an MR fluid damper considering its time delay. J. Intell. Mater. Syst. Struct. 13(7–8), 521–524 (2002). https://doi. org/10.1106/104538902030343 16. Hu, Y., Wang, K., Chen, M. Z. Q.: Semi-active suspensions with low-order mechanical admittances incorporating inerters, In: The 27th Chinese Control and Decision Conference (2015 CCDC), 2015, pp. 79–84 17. Li, Z., Zuo, L., Luhrs, G., Lin, L., Qin, Y.: Electromagnetic energy-harvesting shock absorbers: design, modeling, and road tests. IEEE Trans. Veh. Technol. 62(3), 1065–1074 (2013). https:// doi.org/10.1109/TVT.2012.2229308

An Adaptive Controller for Payload Swing Suppression of Ship-Mounted Boom Cranes Bincheng Li1 , Peng Liao1 , Menghua Zhang2 , Donghong Ning1(B) , and Guijie Liu1 1 Engineering College, Ocean University of China, Qingdao 266500, China

[email protected], {liaopeng,ningdonghong, liuguijie}@ouc.edu.cn 2 School of Electrical Engineering, University of Jinan, Jinan 250022, China [email protected]

Abstract. The ship-mounted crane is a typical system for completing marine projects, which often need to work in complex sea conditions. Besides, it has complex nonlinearity and is also a typical underdrive system. These factors increase the difficulty of the controller design. Traditional control strategies based on gravity compensation are mostly used in full-drive systems, which can effectively reduce the influence of gravity uncertainty on the system positioning error and save driving energy. However, the actuator oscillation problem of underdrive systems, such as the swing of the payload of ship-mounted crane systems, has not been fully solved. This paper proposes an adaptive swing suppression controller based on the gravity compensation of a (3-DOF) ship-mounted boom crane. Specifically, the influence of gravity on system stability is reduced by estimating unknow gravity parameters through adaptive functions. Combine the swing angle information of the payload with the control output to eliminate the remaining swing of the payload. The auxiliary term in the controller reduces the difficulty of subsequent stability analysis, and also has a certain resistance effect to steady-state oscillation. Next, with the help of the Lyapunov stability theory and LaSalle’s invariance theorem, the proof of the asymptotic stability of the closed-loop system is completed. Finally, this paper verifies the effectiveness of the proposed control method successively through numerical simulations, and the simulation data has the potential to improve the robustness of the proposed controller in different marine environments. Keywords: Ship-mounted boom crane · underdrive nonlinear system · payload swing suppression · gravity compensation · asymptotic stability

1 Introduction The ship-mounted crane is mainly used for ocean transportation, ocean fishing, ocean mining, etc., which often work in complex marine environments and are severely restricted by ocean conditions, making them unable to operate under harsh sea conditions. When influenced by waves and currents, ships may experience unexpected motion, such as rolling. However, ship motion can cause irregular swing of the payload of the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 136–150, 2024. https://doi.org/10.1007/978-981-97-0554-2_11

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ship-mounted crane system, which limits its working performance and efficiency, and poses safety hazards. In addition, the ship-mounted crane system is a typical underactuated system with more degrees of freedom than control inputs, which undoubtedly increases the difficulty of the ship-mounted crane system controller design. To fill this scope, researchers have proposed different controllers based on the characteristics of ship-mounted crane systems, the exiting controllers can be roughly divided into linear control, sliding mode control, adaptive control, feedback control based on energy control or intelligent control. The most classic linear controller is the Proportional Integral Derivative (PID) controller, but due to its low precision, it is usually used in combination with other control techniques. To reduces the remaining swing of the payload to a certain extent, a controller composed of a PI controller, a low-pass filter and a notch filter is proposed [1]. Besides, some PID controllers also combine adaptive sliding mode controllers [2], robust control methods, neural networks methods [3], etc. However, most PID controllers are designed based on the linearized system model, and the neglected nonlinear part often causes large steady-state oscillations, which cannot show good performance in practical applications. Consequently, most existing controllers are based on nonlinear models, avoiding the problems of linearization. Due to the excellent resistance of the sliding mode control method to interference, more researchers have applied it to the control strategy of shipmounted cranes. Zhang et al. proposed a model-independent PD-SMC method [4], to overcome the influence of underdrive characteristics on the stability of the system, a special control method to suppress the swing of the payload is designed. A sliding mode control method for ship-mounted container cranes is proposed, which effectively suppresses the lateral swing of the payload [5]. Kim proposed a continuous integral sliding mode control with input saturation [6], which solves positioning errors and large fluctuations caused by unknown nonlinear disturbances and parameter uncertainties. A second-order sliding mode control law for trajectory tracking and anti-swing control is proposed, which significantly reduces the influence of nonlinear disturbances [7]. Adaptive control can solve the problem of unknown parameters in the form of online prediction. Yang et al. [8] proposed an adaptive control method for online gravity compensation to compensate for positioning errors caused by the uncertainty of gravity parameters. An adaptive control method based on trajectory planning [9] is applied to tower cranes to keep acceleration and speed within a realistic range to keep the motion of the payload as close as possible to the desired origin. Zhang et al. [10] proposed an adaptive tracking control method to accurately estimate gravity-related parameters online, which solves the problem of unknown gravity. Ramli et al. [11] proposed a predictive unit amplitude shaper and adaptive feedback controller based on which control parameters can be updated online in real-time to gradually suppress the swing of the payload. Wu et al. [12] propose an adaptive dynamic output feedback controller, which designs an adaptive update rate for unknown gravity parameters while ensuring the limit of input saturation. Huang et al. [13] designed a filter-based adaptive control, which maintains steady-state oscillation through the filter, and lifts the system to reach steady-state speed. Zhai et al. [14] designed an adaptive fuzzy controller based on a nonlinear observer to stabilize cargo relative to the deck, solving problems with uncertainty, input dead zones, and external disturbances.

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The system energy of ship-mounted cranes and the corresponding feedback controllers are considered when designing the Lyapunov candidate functions. Qian et al. [15] proposed an energy-based nonlinear coupled control method. Lu et al. [16] proposed an output feedback control method by designing a class energy function. Based on the energy function, Sun et al. [17] designed a feedback controller with an integral term to reduce the steady-state error caused by inaccurate friction compensation and ensure the performance of pendulum suppression and positioning. Lu et al. proposed a nonlinear feedback controller [18] that effectively responds to uncertainty/disturbance, in particular, updates the control gain in real-time according to the dynamic response of the system, thereby further improving the efficiency of the designed controller. Chen et al. [19] constructed an energy storage function composed of kinetic energy and potential energy, and proposed a nonlinear output feedback inverse pendulum controller. Cao et al. have proposed a nonlinear model predictive control method [20] and predictive controller based on Lyapunov [21]. In order to ensure the recursive feasibility and stability of the established framework, a shrinkage constraint based on an energy controller was further introduced. With the development of intelligent control, some researchers [22–25] have applied it to the control of ship-mounted crane systems. This control method has good performance for parameter optimization and unknown quantity prediction. Some researchers have designed control methods to meet different control needs by studying the dynamic characteristics of ship-mounted cranes, such as fuzzy robust fault-tolerant control method with event triggering mechanism [26], and nonlinear controller with disturbance observers [27], etc. In this paper, the dynamic characteristics and control objectives of the ship-mounted boom cranes are comprehensively considered, and an adaptive controller based on gravity compensation is designed. The gravity term of the system is predicted online through the control strategy of gravity compensation, so as to avoid the swing angle of the amplified payload caused by the influence of uncertain gravity parameters. Secondly, the swing angle of the payload is combined with the control output to improve the control performance of the controller on the underdrive state parameters. We also carefully design the auxiliary control term to ensure the effective performance of the controller. Finally, through stability analysis, it can be concluded that the ship-mounted boom crane system can achieve progressive stability at its equilibrium point and meet engineering requirements. The rest of the paper is organized as follows. In Sect. 2, the dynamic model and brief analysis of ship-mounted cranes are given. In Sect. 3, an adaptive controller is proposed and its stability analysis is carried out. The simulation results are presented in Sect. 4. Part of the summary and experience is arranged in Sect. 5.

2 Ship-Mounted Boom Crane Model In this section, the dynamic model of the 3-DOF ship-mounted boom crane will be described, and the parameters will be processed to clarify the control objectives and facilitate the design of the controller. Before giving the dynamic model of the shipborne crane, the following description is required.

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1) The rolling motion of the ship is only considered under the influence of sea waves, while the influence of other nonlinear disturbances is ignored. 2) The rope is regarded as rigid, and its elastic deformation and quality are ignored. 3) The payload is considered a point of uniform mass.

Fig. 1. Schematic model of the ship-mounted boom crane

2.1 Dynamics for Ship-Mounted Cranes The adopted-model of 3- DOF ship-mounted boom crane is shown in Fig. 1, where xw and yw are the axis of the world coordinate system. xs and ys are the axis of the coordinate system fixed on the ship’s deck, which are parallel and perpendicular to the deck, respectively. The fixed parameters include the boom length PL and the length d from its center of gravity to o, the moment of inertia of the boom I , the payload mass mp , the boom mass m, and the gravitational constant g. The state variables include the angle of cargo swing θ relative to the ys -axis, the boom swing angle φ relative to the xs -axis, and the time-varying rope length L. This paper uses the kinematics analysis method based on the Jacobian matrix and the Lagrange equation to model the dynamics of the ship-mounted boom crane system. The dynamics of the ship-mounted boom crane system can be deduced as follows: M (q)¨q + C(q, q˙ )˙q + G(q) = u

(1)

where q = [φ(t)L(t)θ (t)]T denotes the configuration vector M (q), C(q, q˙ ) ∈ R3×3 , and G(q), u ∈ R3×1 represent the inertia matrix, Coriolis matrix, gravity vector and control torque/force vector, respectively. The explicit expressions of M (q), C(q, q˙ ), G(q) and u are given as follows: ⎡ ⎤ I + mp PL2 −mp PL Cφ−θ −mp PL LSφ−θ ⎦ M (q) = ⎣ −mp PL Cφ−θ mp 0 ⎡

−mp PL LSφ−θ

0

mp L2

⎤ ˙ c13 −mp PL Sφ−θ (θ˙ − α) C(q, q˙ ) = ⎣ mp PL Sφ−θ (φ˙ − α) ˙ 0 −mp L(θ˙ − α) ˙ ⎦ ˙ ˙ ˙ ˙ mp L(θ − α) ˙ mp LL −mp PL LCφ−θ (φ − α) 0

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G(q) =

where c13

T   mp PL + md gCφ−α −mp gCθ−α mp Lg Sθ−α

T  u = Mc FL 0    = −mp PL Sφ−θ L˙ − LCφ−θ θ˙ − α˙ .

(2)

2.2 Math Model Transformation In order to facilitate state analysis and controller design, the current state variables are redefined as follows ζ1 = φ − α, ζ2 = L, ζ3 = θ − α  T ζ = ζ1 ζ2 ζ3

(3)

Thus, ζ1 represents the angle between boom and xw , ζ2 represents the length of rope, and ζ3 represents the angle between rope and yw ,therefore, the original model (1) can be rewritten in the following form   M (ζ )ζ¨ + C ζ , ζ˙ ζ˙ + G(ζ ) = u (4) and the error vector is defined as follows T

e = e1 e2 e 3 , e1 = ζ1 − ζ1d , e2 = ζ2 − ζ2d , e3 = ζ3 − ζ3d

(5)

where ζ1d , ζ2d , ζ3d are the desired values of the state variables respectively. In addition, we can get the following properties. Property 1: M (ζ ) is a symmetric positive definite matrix and satisfies [28]

  1 ˙ ˙ ˙ζ T M (ζ ) − C ζ , ζ ζ˙ = 0 2

(6)

Property 2: The gravity vector satisfies the condition of linear parameterization, such that [29, 30] G(ζ ) = g(ζ )p

(7)

where g(ζ ) is a known matrix composed of state variables, and p is an unknown parameter vector. Property 3: There is a positive constant R, so that the partial derivative of gravity vector G(ζ ) satisfies ∂G(ζ ) ≤ R ⇒ G(ζ1 ) − G(ζ2 ) ≤ Rζ1 − ζ2 , ∀ζ1 , ζ2 ∈ Rn ∂ζ

(8)

To facilitate the design and stability proof of the controller, we make the following reasonable assumption.

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Assumption 1. Considering the actual situation, the rotation range of the boom should be limited π π (9) − < ζ1 (t) = φ(t) − α(t) < 2 2 The initial and desired rope length is always positive, and that is ζ2 (0) = L2 (0) > 0, ζ2d = L2d > 0

(10)

The payload swing angle with respect to the vertical direction, i.e., ζ3 is always bounded in the sense of π π (11) − < ζ3 (t) = θ (t) − α(t) < 2 2 Assumption 2. In practical applications, in order to ensure the stability and safety of the ship-mounted cranes during operation, it is required ζ¨1 , ζ¨2 < g, in this case, the swing angle of the payload is usually kept at (− π2 , π2 ),both angular velocity and angular acceleration are kept within a reasonable range, so we have reason to make the following assumption:   ζ˙3  < σ (12) kd > kθ σ

(13)

where σ and kθ is a positive constant. Remark 1: Due to the nature of Lyapunov-based controller design techniques, condition (12) and (13) is conservative, which is merely sufficient condition to guarantee the stability. Thus, when implementing the designed controller, we can moderately relax these gain conditions. Remark 2: For the convenience of description, the following abbreviations are used in this paper: Sφ−θ = sin(φ − θ ),Cφ−θ = sin(φ − θ ), Cφ−α = sin(φ − α), Sθ−α = sin(θ − α), Cθ−α = cos(θ − α), C1 = cos ζ1 , S3 = sin ζ3 , C3 = cos ζ3 , S1−3 = sin(ζ1 − ζ3 ), C1−3 = cos(ζ1 − ζ3 ).

3 Controller Design and Stability Analysis In this section, an adaptive controller for payload swing suppression of ship-mounted boom cranes will be developed, this paper also carefully designs a Lyapunov candidate function to verify its stability.

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3.1 Controller Design Based on the dynamic model of the system in (4) and the error vector in (5), the proposed controller is designed as follows: Mc = pˆ 1 C1 − kd 1 ζ˙1 − kp1 e1 − kau1 κ1 z1 − ksw1 ζ˙3 e˙ 1

(14)

FL = −ˆp2 − kd 2 ζ˙2 − kp2 e2 − kau2 κ2 z2 − ksw2 ζ˙3 e˙ 2

(15)

where Kd 1 , Kd 2 , Kp1 , Kp2 , Kau1 , Kau2 , Ksw1 , Ksw2 are positive control gains. κi =

ei2 ζi2

+

ei (ζi − ei )zi , i = 1, 2 ζi3

(16)

and zi is the auxiliary state variables, which is defined as z˙i = e˙ i − γi zi , i = 1, 2

(17)

where γi is a positive control parameter. Meanwhile, from (4) and (7) the unknown gravitational parameters can be defined as p1 = (mp PL + md )g

(18)

p2 = mp g

(19)

The update law for pˆ 1 , pˆ 2 is designed as pˆ˙ 1 = −Ψ1 ζ˙1 C1

(20)

pˆ˙ 2 = Ψ2 ζ˙2

(21)

where Ψ1 , Ψ2 are positive constants. Remark 3: The proportional derivative control method can provide sufficient driving force and is a simple and efficient control method. The design of the adaptive function ensures that the unknown gravity parameters can converge to the true value. Particularly, by feeding back the information of the underdrive state variables, the control method specially designed for the underdrive term ensures that the remaining swing of the payload disappears quickly. In addition, the design of the auxiliary term facilitates the subsequent stability analysis and has a certain resistance to steady-state oscillations. 3.2 Stability Analysis Theorem 1: The proposed adaptive control method (14), (15) and update law (20), (21) make the equilibrium point of the system in (4) asymptotically stable (ζ1 → ζ1d , ζ2 →ζ2d , ζ3 → 0,ζ˙1 → 0,ζ˙2 → 0, ζ˙3 → 0), which can identify the unknown gravity parameters online (ˆp1 → p1 , pˆ 2 → p2 ) and quickly eliminate the residual swing of the payload.

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Proof: We first construct the following Lyapunov function candidate: V1 =

1 T 1 1 ζ˙ M (ζ )ζ˙ + mp gζ2 (1 − C3 ) + kp1 e21 + kp2 e22 2 2 2 1 1 + Ψ1−1 p˜ 12 + Ψ2−1 p˜ 22 + V0 2 2

(22)

where p˜ i = pˆ i − pi , i = 1, 2. is the estimate error vector and V0 is defined as V0 =

2 



zi

kaui

i=1

0

ei2 τ ζi2

dτ

(23)

The derivative of (22) is: 1 T ˙ T V˙ 1 = ζ˙ M (ζ )ζ˙ + ζ˙ M (ζ )ζ¨ + mp g ζ˙2 (1 − C3 ) + mp gζ2 S3 ζ˙3 2 + kp1 e1 e˙ 1 + kp1 e2 e˙ 2 + Ψ1−1 p˜ 1 p˜˙ 1 + Ψ2−1 p˜ 2 p˜˙ 2 + V˙ 0

(24)

Substituting (4) into (24) yields 1 T ˙ T V˙ 1 = ζ˙ M (ζ )ζ˙ + ζ˙ (u − C(ζ , ζ )ζ˙ − G(ζ )) + mp g ζ˙2 (1 − C3 ) + mp gζ2 S3 ζ˙3 2 + kp1 e1 e˙ 1 + kp1 e2 e˙ 2 + Ψ1−1 p˜ 1 pˆ˙ 1 + Ψ2−1 p˜ 2 pˆ˙ 2 + V˙ 0 (25) From Property 1, it can be concluded that T V˙ 1 = ζ˙ (u − G(ζ )) + mp g ζ˙2 (1 − C3 ) + mp gζ2 S3 ζ˙3 + kp1 e1 e˙ 1 + kp1 e2 e˙ 2 + Ψ −1 p˜ 1 pˆ˙ 1 + Ψ −1 p˜ 2 pˆ˙ 2 + V˙ 0 1

2

(26)

With some further arrangements and by substituting update law (20) (21) into (26) yields V˙ 1 = ζ˙1 (Mc − (mp PL + md )gC1 ) + ζ˙2 (FL + mp gC3 ) − ζ˙3 mp gζ2 S3 +mp g ζ˙2 (1 − C3 ) + mp gζ2 S3 ζ˙3 + kp1 e1 e˙ 1 + kp1 e2 e˙ 2 + Ψ1−1 p˜ 1 (Ψ1 ζ˙1 C1 ) + Ψ2−1 p˜ 2 (Ψ2 ζ˙2 ) + V˙ 0

(27)

Substituting (14), (15) into (27), it can be rewritten as V˙ 1 = ζ˙1 (ˆp1 C1 − kd 1 ζ˙1 − kp1 e1 − kau1 κ1 z1 − ksw1 ζ˙3 e˙ 1 ) − ζ˙1 p1 C1 + ζ˙2 (−ˆp2 − kd 2 ζ˙2 − kp2 e2 − kau2 κ2 z2 − ksw2 ζ˙3 e˙ 2 ) + ζ˙2 mp gC3 − ζ˙3 mp gζ2 S3 +p2 ζ˙2 − mp g ζ˙2 C3 + mp gζ2 S3 ζ˙3 + kp1 e1 e˙ 1 + kp1 e2 e˙ 2 + p˜ 1 ζ˙1 C1 + p˜ 2 ζ˙2 +

2  ∂V0 ∂ζi i=1

∂ζi ∂t

+

2  ∂V0 ∂zi i=1

(28)

∂zi ∂t

where 2  ∂V0 ∂ζi i=1

∂ζi ∂t

= kau1

e1 (ζ1 − e1 )z12 ζ13

ζ˙1 + kau2

e2 (ζ2 − e2 )z22 ζ23

ζ˙2

(29)

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∂zi ∂t

= kau1

e12 z1 ζ12

(˙e1 − γ1 z1 ) + kau2

e22 z2 ζ22

(˙e2 − γ2 z2 )

(30)

With some further arrangements V˙ 1 = −kd 1 ζ˙12 − kθ1 ζ˙3 ζ˙12 − kd 2 ζ˙22 − kθ2 ζ˙3 ζ˙22 − kau1

γ1 e12 z12 ζ12

− kau2

γ2 e22 z22

(31)

ζ22

Then, by inserting (12) into (31), it is not difficult to find that V˙ 1 ≤ − kd 1 ζ˙12 + kθ1 σ ζ˙12 − kd 2 ζ˙22 − kθ2 σ ζ˙22 − kau1

γ1 e12 z12 ζ12

= −(kd 1 − kθ1 σ )ζ˙12 − (kd 2 ζ˙22 − kθ2 σ )ζ˙22 − kau1

− kau2

γ1 e12 z12 ζ12

γ2 e22 z22 ζ22

− kau2

γ2 e22 z22 ζ22

(32)

According to the previous assumption about the limit of the swing frequency in (13), V˙ 1 ≤ 0, ∀t ≥ 0 is always valid during the entire control process, which yields that V1 (t) ≤ V1 (0) ∈ L∞ ⇒ ζ˙ , e, e˙ , z, p˜ ∈ L∞ ⇒ pˆ , pˆ˙ ∈ L∞

(33)

Meanwhile, it implies that  the update law for pˆ is bounded. Further, we define a set to complete the proof of Theorem 1 as follows:     = ζ1 , ζ2 , ζ3 , ζ˙1 , ζ˙2 , ζ˙3 |V˙ = 0 Based on that, we further define as the largest invariant set included in follows from (34) that in :



(34) . It then

ζ˙1 = ζ˙2 = 0 ⇒ e˙ 1 = e˙ 2 = 0 ⇒ e¨ 1 , e¨ 2 , ζ¨1 , ζ¨2 = 0

(35)

we also did the following works when t → ∞, for further analysis: e1 = λ1 , e2 = λ2 , ζ1 = λ3 , ζ2 = λ4 , p˜ 1 = λ5 , p˜ 2 = λ6

(36)

where λ1 to λ6 are constants. Further, if z 1 , z2 = 0 in (32), one has that z˙1 , z˙2 = 0, and which yields  (37) z1 = λ1 − γ1 z1 (t)dt = 0  z 2 = λ 2 − γ2

z2 (t)dt = 0

(38)

Obviously, one can conclude that λ1 = e1 = 0

(39)

An Adaptive Controller for Payload Swing Suppression

λ2 = e2 = 0

145

(40)

Substituting (35), (36) into (4) yields −mp PL λ4 (S1−3 ζ¨3 − C1−3 ζ˙32 ) + (mp PL + md )gC1 = Mc

(41)

−mp λ4 ζ˙32 − mp gC3 = FL

(42)

mp λ24 ζ¨3 + mp λ4 gS3 = 0

(43)

Meanwhile, substituting (14), (20), (36) into (41) results in −mp PL λ4 S1−3 ζ¨3 + mp PL λ4 C1−3 ζ˙32 = λ5 C1

(44)

With some further arrangements, (44) can be rewritten as mp PL λ4 S1−3 (ζ¨1 − ζ¨3 − ζ¨1 ) + mp PL λ4 C1−3 (ζ˙1 − ζ˙3 − ζ˙1 )2 = λ5 C1

(45)

which, by utilizing ζ˙1 , ζ¨1 = 0, (45) can be reduced as mp PL λ4 S1−3 (ζ¨1 − ζ¨3 ) + mp PL λ4 C1−3 (ζ˙1 − ζ˙3 )2 = λ5 C1

(46)

Integrating both sides of (46) with regard to time yields mp PL λ4 S1−3 (ζ˙1 − ζ˙3 ) = λ5 C1 t + β1

(47)

wherein C1 is a constant (ζ1 = e1 + ζ1d = ζ1d ), and β1 denotes a constant to be determined. If λ5 = 0, then mp PL λ4 S1−3 (ζ˙1 − ζ˙3 ) → ∞ as t → ∞. However, it can be concluded from (33) that (ζ˙1 − ζ˙3 ) ∈ L∞ ⇒ mp PL λ4 S1−3 (ζ˙1 − ζ˙3 ) ∈ L∞ , which leads to an apparent contradiction. Consequently, the following conclusion can be derived: p˜ 1 = λ5 = 0 ⇒ mp PL λ4 S1−3 (ζ˙1 − ζ˙3 ) = β1

(48)

Integrating both sides of (48) results in −mp PL λ4 C1−3 = β1 t + β2

(49)

where β2 is also a constant. We can also get β1 = 0 ⇒ mp PL λ4 S1−3 (ζ˙1 − ζ˙3 ) = 0

(50)

Then, it can be concluded that S1−3 (ζ˙1 − ζ˙3 ) = 0

(51)

S1−3 = 0 or ζ˙1 − ζ˙3 = 0

(52)

which implies that

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Both can get the following conclusion: ζ˙1 − ζ˙3 = 0 ⇒ ζ˙1 = ζ˙3

(53)

From (35) and (53), which implies ζ˙3 = 0 ⇒ ζ¨3 = 0

(54)

Substituting (54) into (43), we are led to mp λ4 gS3 = 0 ⇒ S3 = 0 ⇒ ζ3 = e3 = 0

(55)

Substituting (15), (21), (35), (36), (54) and (55) into (42) and making some arrangements, one derives −mp λ4 ζ˙32 − mp gC3 = g2 (ζ )ˆp2 − kp2 e2 ⇒ −mp gC3 = g2 (ζ )ˆp2 ⇒ 0 = λ6 = p˜ 2

(56)

Gathering the results in (35), (39), (40), (48), (54), (55) and (56), one derives that the largest invariant set contains only the desired equilibrium point. According to the LaSalle’s invariance theorem, one can conclude that the desired equilibrium point [ζ1 ζ2 ζ3 ζ˙1 ζ˙2 ζ˙3 ]T = [ζ1d ζ2d 0 0 0 0]T is asymptotically stable, and the unknown gravity parameters are correctly estimated. Thus, the proof of Theorem 1 is completed.

4 Numerical Simulations In order to further verify the effectiveness of the proposed control method, data simulations are carried out using the MATLAB/Simulink software system. The 3-DOF shipmounted boom crane system mainly consists of a boom, a rope, and a suspended payload, with m = 4.54 kg, mp = 5 kg, PL = 1.5 m, d = 0.75 m, I = 5.29 kg · m2 , g = 9.8 m/s2 . In addition, the numerical simulation results are compared with the existing full-state feedback controller, which shows excellent performance in the control problem of ship-mounted boom cranes. 4.1 Simulation 1 In this group, the motion of the ship is regarded as the disturbance input of the system, which is set as α = 4 sin t ◦ . Then, the initial value of the state variables are given as ζ1 = 20◦ , ζ2 = 0.5 m, ζ3 = 0◦ , and the expected value is ζ1d = 50◦ , ζ2d = 0.7 m, ζ3d = 0◦ . With slight modifications, the full state feedback controller is specified as Mc = (mp PL + md )gC1 − kp01 e1 − kd 01 e˙ 1 FL = −mp g − kp02 e2 −

2λ0 e2 (ζ2 − e2 ) − kd 02 e˙ 2 ζ23

(57) (58)

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where the control gains are set as kp01 = 100, kd 01 = 250, kp02 = 150, kd 02 = 250, λ0 = 1. Then, for the proposed controller given in (14) and (15), the partial control gain and parameter values are chosen as kp1 = 200, kd 1 = 500, kau1 = 50, ksw1 = 3000, γ1 = 0.1, Ψ1 = 31, kp2 = 250, kd 2 = 500, kau2 = 10, Ψ2 = 95, γ2 = 0.1, ksw2 = 250. Figure 2 depict the simulation results of the proposed controller and the full-state feedback controller. One can see that: 1) The proposed controller has high positioning accuracy, the state variables ζ1 , ζ2 more quickly tend to the desired value after 8 s and 5 s respectively, and the driving motor consumes less energy, which may be because the control method of gravity compensation reduces the influence of gravity on the system. 2) It is seen that the proposed controller suppresses and eliminates the payload swing within a smaller range (ζ3 max = 0.18◦ , and almost no residual payload swing angle) than the full-state feedback controller (ζ3 max = 0.36◦ ) after 5 s. ζ3 max after 5 s of the proposed controller account for about 50.0% corresponding to the full-state feedback controller. 3) The proposed controller can realize online estimation of uncertain gravity parameters, and the effectiveness of the gravity compensation control method is verified. 4.2 Simulation 2 In this group, to further verify the robustness of the proposed controller, the disturbance input is changed to multiple “Bump” disturbances. It can be obtained from Fig. 3, under

Fig. 2. Results of Simulation 1 (red solid line: the proposed controller; blue dotted line: the fullstate feedback controller; black dashed-dotted line: the desired values)

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the action of two controllers, the ship-mounted crane system can quickly maintain stability after being subjected to intermittent “Bump” disturbances. The control performance of the controller proposed in this paper is slightly better than that of the full-state feedback controller. Specifically, the controller proposed in this article still achieves faster precise positioning, faster suppression of payload swing and accurate estimation of unknown gravity parameters, with satisfactory robustness to external disturbances.

Fig. 3. Results of Simulation 2 (red solid line: the proposed controller; blue dotted line: the fullstate feedback controller; black dashed-dotted line: the desired values)

5 Conclusion This paper has proposed an adaptive controller based on gravity compensation for payload swing suppression of ship-mounted boom cranes. For controller design, the information of the underdrive state variable has been combined with the control output, to ensure that the remaining swing of the payload disappear quickly. Also, an adaptive law has been presented to identify the unknown gravitational parameters. Moreover, an auxiliary term has been used to ensure the stability and reduce the steady-state oscillation of the system. The simulations have been implemented to verify the effectiveness of the proposed controller, where the state variables ζ1 , ζ2 has been quickly tended to the desired value after 8 s and 5 s respectively, and the angle of payload swing after 5 s has been limited in a smaller range that ζ3 max = 0.18◦ , which only account for 50% corresponded to the full-state feedback controller. In addition, the solution of some practical problems, such as unmeasurable nonlinear disturbances, control saturation, etc., will also be the focus of our future research.

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Acknowledgment. We are grateful for the anonymous referees’ valuable comments and suggestions. The study was financially supported by the National Natural Science Foundation of China (Grant No. 52088102).

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Study on Dynamic Modeling and Vibration Noise Suppression Method of AUV Kangyu Zhang1,2 , Chao Fu1,2 , Kuan Lu1,2(B) , Kaifu Zhang3(B) , Hui Cheng3 , and Dong Guo1,2 1

School of Mechanics Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, China [email protected] 2 Intelligent Aircraft Structural Strength and Design Institute, Northwestern Polytechnical University, Xi’an 710072, China 3 School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China [email protected] Abstract. This paper focuses on the key technical problems of the autonomous underwater vehicle (AUV) vibration and noise reduction. The dynamic model of AUV with propeller-shaft-motor by the second Lagrange method is established. The nonlinear bearing and other connecting parts are fully considered, which is solved by Runge-Kutta method, the dynamic behavior mechanism of AUV is revealed. Taking the vibration amplitude of the shell as the cost function and according to the amplitude-frequency response characteristics of the system, the resonance changer (RC) device by parameter design is designed. The results show that the resonance response amplitude can be greatly reduced. The theoretical model of this paper reveals the dynamic response characteristics of AUV. The results can provide new improvement ideas for the optimization design of vibration and noise reduction of AUV, and have theoretical guiding significance. Keywords: Autonomous underwater vehicle · Rotor dynamics · Nonlinear vibration reduction and isolation · Dual-beam model · Resonance changer

1

Introduction

Since the First World War, the autonomous underwater vehicle (AUV) represented by torpedo has attracted wide attention[1,18,19,23]. Especially in recent years, with the continuous rise of international maritime disputes, maritime military has been paid more attention, and the demand for defending national K. Zhang and C. Fu—Co-first author. Supported by the National Natural Science Foundation of China (U2241243, 12072263) and the Practice and Innovation Funds for Graduate Students of Northwestern Polytechnical University (PF2023007). c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 151–164, 2024. https://doi.org/10.1007/978-981-97-0554-2_12

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interests in offshore and even ocean has increased sharply. AUV has played an indispensable role in national defense by virtue of its strong concealment and sudden attack ability [8,9]. Among them, underwater torpedoes become remarkable in maritime warfare with unique advantages of good concealment, great destructive power and high hit probability [6,20]. Its high stealth technology is one of the important indicators to evaluate the advancement of modern AUV. Researches show that for every 5 dB increase in the radiated noise of such systems, the enemy’s alarm distance will increase by 50%, and the effective hit rate will be reduced by 25% [16]. Underwater torpedo is launched into the water by platforms such as land, ships and submarines carrying combat after discovering the target. The good concealment is not only the guarantee that AUV can launch a sudden attack, but also the barrier to maintain its vitality. The noise is an important index related to the combat effectiveness and the vitality of the launch platform. The vibration of the internal component structure and the external fluid environment is one of the important sources of noise. With the development of technology, various anti-submarine weapons have brought great challenges to the survival of AUV. How to effectively reduce the vibration of component devices and overall structure has become an urgent problem to be solved, which is of great strategic significance for improving the overall combat performance. With the purpose of solving the above problems, many researchers have done massive research work [14,15,17,25]. Resonance changer (RC) is a dynamic shock absorber using fluid medium, which was first proposed by Goodwin in the 1950s and developed in the 21st century [5]. It is widely used to suppress the axial force transmission of ships [2]. It is composed of a cylindrical piston and an oil cavity, which is connected by a slender pipe. The force is generated at the piston through the compression or expansion of the hydraulic oil inside the oil cavity, the value can be reflected in the form of the vibration equation. The optimal equivalent mass, equivalent damping and equivalent stiffness are obtained through the structural parameter design of RC. Hence, its natural frequency is equal to the resonance frequency of the system to achieve the purpose of anti-resonance. Dylejko used the transmission matrix approach to characterise the dynamic response of the propeller-shaft system, and developed a hydraulic dynamic vibration absorber to reduce the vibration transmission and avoiding excitation of hull axial resonances [3]. Sascha investigated the low frequency structural and acoustic responses of a simplified axisymmetric submarine model to fluctuating propeller forces along the submarine axis, the structure is modelled using the finite element method, so that more complex features such as ring-stiffeners, bulkheads and the propulsion system can be taken into account, and minimise excitation of the hull via the propeller shaft is by optimising the parameters of RC [11–13]. This paper focuses on the reduction of AUV vibration and noise. Based on the double-beam system, a refined dynamic model with propeller-shaft-shell is established, and the nonlinear bearing and other connecting parts are fully considered. The dynamic response of the double-beam system supported by

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nonlinear bearings is analyzed, and the dynamic behavior mechanism of AUV is revealed. Taking the vibration amplitude of the shell surface as the cost function and according to the amplitude-frequency response characteristics of the system, the RC by parameter design is added. The results show that the resonance response amplitude can be greatly reduced.

2

FE Model of AVU

The radiated noise of AUV is mainly caused by its power motor, which includes the propeller in a spatial non-uniform wake and the vibration of shell outer surface with contacting fluid. Hence, this paper ignores the axial excitation and focuses on the radial vibration of the shell with motor rotation. Considering the rotation characteristics under actual working conditions, the axisymmetric model was adopted. Here, the power cabin structure of the AUV, depicted in Fig. 1, was modeled by the finite element method (FEM) as a dual-beam (Timoshenko beam) [4]. The internal propulsion shaft system is simplified as a solid beam, which mainly consists of propeller, motor, shaft and ball bearings. The shell system is simplified as a hollow beam.

Fig. 1. Physical model of the AUV power cabin.

The global coordinate frame of AUV system, which is established on the center of the propeller. The torsional and axial displacement components are tiny and can be neglected, so that each element node has four DOFs with xi , yi translation along x and y direction and rotation around x-axis and y-axis, the bearing element node have two DOFs with xj , yj translation along x and y direction. The dynamic differential equation of AUV double-beam system is derived by Lagrange method. The second Lagrange equation of the vibration system can be expressed as   d ∂(T − V ) ∂D ∂(T − V ) − = Fk , k = 1, 2, · · · , n (1) − ˙ ∂t ∂Uk ∂ Uk ∂ U˙ k Hence, the generalized displacement vector of the system is expressed as T  U = xi yi θxi θyi · · · xj yj · · · (2) The total kinetic energy of the system T is composed of the translational and rotational kinetic energy of the mass point. The total potential energy of the

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system V includes the deformation energy of the propulsion shaft and the shell, and the energy stored in the deformation of the bearing outer ring support spring and the shell support spring. The external excitation comes from the eccentric excitation of propeller and motor. Only the Rayleigh dissipation energy of the propulsion shaft D is considered. The total kinetic energy, potential energy, dissipation energy and excitation force are substituted into Eq. (1), the dynamic equations of AUV system can be obtained as [10,21] ¨ + (C + ΩG)U(t) ˙ MU(t) + KU(t) = Fg + Fe + Fb (U, t) + Frc (U, t)

(3)

where M, K, C, G are the mass, stiffness, damping and gyroscopic matrices of the total system, representively; Fg is the gravity vector; Fe represent eccentric excitation; Fb represent Hertz contact force of each supporting bearings in the radial surface; Frc is the exciting force induced by RC, Ω is the rotational speed of the shaft. As for the ball bearing, the inner ring is fixed to the shaft and the outer ring is connected with the shell by a linear spring-mass-damper system. There is the point contact between the inner/outer rings and the ball, based on the Hertz contact theory, the nonlinear restoring force produced by the deformation induced by contact between the ball and the raceway in the x and y directions can be written as [7,24] 

θij =

Fbxi Fbyi

 = Cbi +

2π(j − 1) + Ωi t, N bi

Nbi  j=1

n H(δij )δij



cosθij sinθij



δij = xi cosθij + yi sinθij − δi0

(4)

(5)

where x, y are the center displacements of the inner ring, θij , δij are the corner position of the j th ball and the normal contact deformation between the j th ball and the raceway, Cbi is the Hertz contact stiffness related to material and shape of the bearing, n = 32 is the nonlinear Hertz contact coefficient, H(δij ) is the Heaviside function, Nbi , δi0 , Ωi are the number of balls, initial radial clearance and rotational speed of the cage (Equal to outer ring) of the ith bearing, respectively.

3 3.1

Working Principle and Parameter Design of RC Principle Derivation of RC

In this paper, RC simplified to a parallel spring-mass-damper system is attached separately to the radial cross-perpendicular directions of shaft, which consists of cylinder piston, oil cavity, hydraulic oil and pipe as shown in Fig. 2. The derivation of detailed principle of RC requires the following main assumptions: 1. The cylinder piston, oil cavity, and pipe are all rigid, which means no elastic deformation occurs with being subjected to force;

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2. The flow of hydraulic oil in the pipe is assumed to laminar flow; 3. Since most of the hydraulic oil is concentrated in oil cavity, compression or expansion of hydraulic oil occurs only in oil cavity, and in cylinder piston can be neglected.

Fig. 2. Principle model of resonance changer.

The force acting on hydraulic oil in pipe can be written as F1 = πr2 P (t)

(6)

where r is the cross-section radius of pipe, P is the pressure acting on cylinder piston, the value of which is real-time changing. According to the third assumption in this section, the pressure on hydraulic oil in oil cavity caused by compression can be written as P1 =

πrp2 B (xi+1 − xi ) V0

(7)

where rp is the cross-section radius of piston, V0 is the volume of oil cavity, B is the bulk modulus of the hydraulic oil, (xi+1 − xi ) is the relative displacement of cylinder piston. Thus, the force of hydraulic oil in oil cavity to the hydraulic oil in pipe is expressed as (8) F2 = πr2 P1 Viscous damping characteristics in hydraulic oil leads to losing pressure in laminar flow in pipe, which can be obtained by the second assumption in this section. 32μLrp2 8μLν Δp = = (x˙ i+1 − x˙ i ) (9) r4 r4 where ν is the average flow velocity of hydraulic oil in pipe, μ is the dynamic viscosity of hydraulic oil, L is the length of pipe, (x˙ i+1 − x˙ i ) is the relative velocity of cylinder piston. Therefore, the viscous damping force generated by hydraulic oil in pipe can be written as F3 = πr2 Δp

(10)

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The hydraulic oil in cavity is taken as the object of force analysis, and the resultant force is expressed as xi+1 − x ¨ i ) = F1 − F 2 − F 3 ρπLrp2 (¨

(11)

where ρ is the density of hydraulic oil, (¨ xi+1 − x ¨i ) is the relative acceleration of cylinder piston. The exciting force induced by RC is obtained by simplifying Eq. (11) and can be expressed as xi+1 − x ¨i ) + cr (x˙ i+1 − x˙ i ) + kr (xi+1 − xi ) Frc = mr (¨

(12)

where mr , cr , kr are equivalent mass, damping and stiffness, respectively, which are defined as mr = πρL 3.2

rp4 , r2

cr = 8πμL

rp4 , r4

kr = Bπ 2

rp4 V0

(13)

Simplification and Parameter Design of RC

The application process of RC includes: Firstly, the dynamic differential equation Eq. (3) of the double-beam system is solved, and the resonance frequency of the system is obtained by frequency sweepping. The RC is simplified as a singleDOF system attached to the propulsion shaft. The natural frequency ωdr is a function of the structural parameters of RC. The natural frequency by parameter design and optimization is the same as the resonance frequency of the doublebeam system, so as to achieve the purpose of anti-resonance. The equivalent mass, damping and stiffness of RC are shown in Eq. (13). According to the characteristics of single-DOF system, its natural frequency and damping ratio can be expressed as Bπr2 4μ LV0 (14) , ζ= 3 ωnr = ρLV0 r ρBπ Due to the existence of damping term, the natural frequency with damping can be written as

kr c2r ωdr = 1 − ζ 2 ωnr = − (15) mr 4kr mr

4 4.1

Result and Discussion Dynamic Response Analysis of AUV

Based on the Runge-Kutta method [22], the vibration differential equation of the system Eq. (3) is solved, the structural parameters of the motor and propeller are listed in Table 1, the parameters of shell and bearing are listed in Table 2 and Table 3.

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Table 1. Model properties of the propulsion shaft system Physical parameter

Value

Length of shaft ls (m)

1

Length of shaft element lse (m)

0.1

Radius of shaft rs (m)

0.01

Mass of motor mm (kg)

47.35

Mass of propeller mp (kg)

1.85

Diameter inertia of motor Jdm (kg · m2 )

0.4309

Diameter inertia of propeller Jdp (kg · m2 ) 0.0013 Polar inertia of motor Jm (kg · m2 )

0.1515

Polar inertia of propeller Jp (kg · m2 )

0.0023

Eccentricity of motor em (mm)

0.5

Eccentricity of propeller es (mm)

1

Elastic support stiffness kb (N/m)

5 × 108

Density of shaft ρs (kg/m )

7850

3

Poisson ratio of shaft μs

0.3

Young’s modulus of shaft Es (G · Pa)

210

Shear’s modulus of shaft Gs (G · Pa)

80

Table 2. Model properties of the shell system Physical parameter

Value

Length of shell lc (m)

2

Length of shell element lce (m)

0.1

Radius of shell rc (m)

0.245

Thickness of shell tc (mm)

5

Elastic support stiffness kc (N/m)

4.53 × 107

Density of shell ρs (kg/m3 )

7850

Density of fluid medium ρ0 (kg/m3 )

1000

Poisson ratio of shell μc

0.3

Young’s modulus of shell Ec (G · Pa)

210

Shear’s modulus of shell Gc (G · Pa) Sound velocity in fluid medium c0 (km · s

80 −1

) 1.5

Rayleigh damping coefficient α

1.3

Rayleigh damping coefficient β

1 × 10−6

157

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K. Zhang et al. Table 3. Model properties of the bearing Physical parameter

Value

Number of balls Nbi

15

Initial radial clearance δ0 (m)

5 × 10−6

Mass of bearing outer ring mb (kg)

0.3

Hertz contact stiffness Cb (N/m−1.5 ) 13.34 × 109

Figure 3 show the bifurcation diagram of the shell with nonlinear bearing in the y direction under the mult-speed of motor, the response curve is distributed in the negative area in 0–35 Hz, which is caused by the displacement generated by gravity is greater than the vibration response displacement. 16.3–27.7 Hz is unstable frequency band. As shown in Fig. 4, amplitude-frequency response of shell with nonlinear bearings supporting is higher with the increase of revolution speed in the y direction. There is a peak value of the absolute value of the response at 32.6 Hz (resonance frequency of the system), which reaches 36.58 µm. Hence, 32.6 Hz is also the natural frequency in the design of RC structural parameters. The response law of x direction is similar to that of y, which is not analyzed in detail here. In order to explore the dynamic response characteristics of the AUV doublebeam system at the resonance zone, the stable frequency point 22 Hz and the frequency point of 32.6 Hz at the maximum resonance peak are selected as the comparison. The time history, frequency spectrum, axis orbit and phase portrait are shown in Fig. 5 and Fig. 6. Since the response results of the double-beam system in the x and y direction are similar, the letter is just selected.

Fig. 3. Bifurcation diagram of shell with nonlinear bearings supporting.

Figure 5(a) shows multiple amplitudes of the steady-state time history cased by the nonlinear factors of the bearing, which is in an unstable state. Figure 5(b) gives the spectrum diagram of the steady-state process of 7–8 s. The main frequency is 22 Hz, which is the eccentric excitation frequency of the shaft, and it is the dominant component. Furthermore, 2X, 3X and 4X components can also

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Fig. 4. Amplitude-frequency response of shell with nonlinear bearings supporting.

be observed. Among them, 2X accounts for the smallest component and can be ignored. The dynamic behavior of the double-beam system at other resonance regions is similar of 22 Hz, which is not repeated here. As a comparative analysis, an resonance frequency point of 32.6 Hz is selected for dynamic analysis, and the results are shown in Fig. 6, the maximum vibration peak in the stable state is 26.37 µm, which is 3.2 times that at 22 Hz in Fig. 6(a). Besides, the steadystate time history curve of the system has a few amplitudes, which is periodic or quasi-periodic motion. As shown in Fig. 6(c) and Fig. 6(d), the axis orbit is more regular and the motion trajectory is relatively stable in comparison to Fig. 5.

Fig. 5. The typical dynamic behaviors of shell at 22 Hz with supporting by nonlinear bearings. (a) time history; (b) frequency spectrum; (c) axis orbit; (d) phase portrait.

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Fig. 6. The typical dynamic behaviors of shell at 32.6 Hz with supporting by nonlinear bearings. (a) time history; (b) frequency spectrum; (c) axis orbit; (d) phase portrait.

4.2

Response Analysis of AUV with RC

Essentially, RC is an anti-resonance device, the literature shows that the vibration reduction effect is relatively optimal when it is installed near the vibration source. Therefore, in this paper, RC is installed between the 7th and 8th element nodes on the propulsion shaft (near the power motor), and a pair of forces with equal size and opposite direction are generated at the two nodes. From Eq. (14) and Eq. (15), the following conclusions about the RC singleDOF system can be obtained, the natural frequency of RC is independent of the cross-sectional diameter of the cylindrical piston, and depends on the structural parameters ωdr (r, rp , L, V0 ). However, these parameters of RC limited by weight and space size of AUV cannot be arbitrarily selected. Furthermore, Eq. (13) indicates that the equivalent stiffness depends on the cross-sectional radius of the piston and is inversely proportional to the volume of the oil chamber. Figure 7 shows the influence of oil cavity volume (0.5–3.0 dm3 ) and crosssection radius (20–50 mm) of piston on equivalent stiffness, the equivalent stiffness is sensitive to the cross-sectional radius of the piston. Figure 8 shows the influence of length of pipe (0–1 m) and cross-section radius of pipe (0–15 mm) on natural frequency of RC with V0 = 2.5 dm3 . Hence, the resonance frequency of 32.6 Hz at the maximum response of the system in Fig. 4 is selected as the natural frequency of RC, and a set of structural parameters of RC are obtained, as shown in Table 4. Figure 9 and Fig. 10 show the amplitude-frequency response of shell under supporting by nonlinear bearings with RC and no in x and y. The response amplitude at the resonance frequency design point 32.6 Hz is reduced from 19.9 µm to 12.83 µm, which is reduced by 35.53%, and reduced by 23.51% at 42.2 Hz, which

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Table 4. Physical parameters of RC Physical parameter

Value

Length of pipe L (m)

1.5

Cross-section radius of pipe r (mm)

5.87

Cross-section radius of piston rp (mm)

35

Volume of oil cavity V0 (m3 )

2.5 × 10−3

Density of hydraulic oil ρ (kg/m3 )

950

Bulk modulus of hydraulic oil B (G · Pa)

1.38 × 109

Dynamic viscosity of hydraulic oil μ (Pa · s) 0.23

Fig. 7. The influence of oil cavity volume and cross-section radius of piston on equivalent stiffness.

show that RC can greatly reduce the resonance amplitude value. Meanwhile, the maximum resonance frequency is shifted from 32.6 Hz and 42.2 Hz to 36.2 Hz, which can provide some theoretical experience for AUV vibration isolation. It has similar response rules in the x and y, which are not introduced in detail.

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Fig. 8. The influence of length of pipe and cross-section radius of pipe on natural frequency of RC (V0 = 2.5 dm3 ).

Fig. 9. Amplitude-frequency response of shell under supporting by nonlinear bearings with RC and no RC in x direction.

Fig. 10. Amplitude-frequency response of shell under supporting by nonlinear bearings with RC and no RC in y direction.

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5

163

Conclusions

This paper focuses on the key technical problems of AUV vibration and noise reduction. The dynamic model of AUV with propeller-shaft-motor by the second Lagrange method is established and the nonlinear bearing and other connecting parts are fully considered, which is solved by Runge-Kutta method, the dynamic behavior mechanism of AUV is revealed. Taking the vibration amplitude of the shell as the cost function and according to the amplitude-frequency response characteristics of the system, the RC device by parameter design is added. The results show that the resonance response amplitude can be greatly reduced. The theoretical model of this paper reveals the dynamic response characteristics of AUV. The results can provide new improvement ideas for the optimization design of vibration and noise reduction of AUV, and have certain theoretical guiding significance.

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12. Merz, S., Kessissoglou, N., Kinns, R., Marburg, S.: Passive and active control of the radiated sound power from a submarine excited by propeller forces. J. Ship Res. 57(01), 59–71 (2013) 13. Merz, S., Kinns, R., Kessissoglou, N.: Structural and acoustic responses of a submarine hull due to propeller forces. J. Sound Vib. 325(1–2), 266–286 (2009) 14. Pan, Z., Li, X., Ma, J.: A study on free vibration of a ring-stiffened thin circular cylindrical shell with arbitrary boundary conditions. J. Sound Vib. 314(1–2), 330– 342 (2008) 15. Peters, H., Kinns, R., Kessissoglou, N.: Effects of apparent mass on the radiated sound power from fluid-loaded structures. Ocean Eng. 105, 83–91 (2015) 16. Qian, Z.: Overview of torpedo noise control technology. In: The 11th Symposium on Ship Underwater Noise, pp. 9–14. Xi’an, Shaanxi, China (2007). (in Chinese) 17. Rogacheva, N.N.: The Theory of Piezoelectric Shells and Plates. CRC Press, Boca Raton (1994) 18. Shaochun, D., Shijian, Z., Jingjun, L.: Statistical energy analysis on structure vibration and sound radiation from torpedo. J. Zhejiang Univ. Eng. Sci. 43(7), 1222–1224 (2009) 19. Wei, J., Chen, M., Hou, G., Xie, K., Deng, N.: Wave based method for free vibration analysis of cylindrical shells with nonuniform stiffener distribution. J. Vib. Acoust. 135(6), 061011–061011 (2013) 20. Yin, S., Wang, H., Aijun, G.: Torpeto Vibration and Noise Reduction Technology. National Defence Industry Press, Beijing (2011) 21. Zhang, H., Lu, K., Zhang, W., Fu, C.: Investigation on dynamic behaviors of rotor system with looseness and nonlinear supporting. Mech. Syst. Sig. Process. 166, 108400 (2022) 22. Zhang, K., Lu, K., Gu, X., Fu, C., Zhao, S.: Dynamic behavior analysis and stability control of tethered satellite formation deployment. Sensors 22(1), 62–62 (2022) 23. Zhang, Y., Han, J., Huang, B., Zhang, D., Wu, D.: Excitation force on a pump-jet propeller: the effect of the blade number. Ocean Eng. 281, 114727 (2023) 24. Zhang, Z., Chen, Y., Cao, Q.: Bifurcations and hysteresis of varying compliance vibrations in the primary parametric resonance for a ball bearing. J. Sound Vib. 350, 171–184 (2015) 25. Zimmerman, R., D’Spain, G., Chadwell, C.: Decreasing the radiated acoustic and vibration noise of a mid-size AUV. IEEE J. Ocean Eng. 30(1), 179–187 (2005)

Simultaneous Vibration Absorbing and Energy Harvesting Mechanism of the Tri-Magnet Bistable Levitation Structure: Modeling and Simulation Junjie Xu and Yonggang Leng(B) School of Mechanical Engineering, Tianjin University, Tianjin 300350, China [email protected]

Abstract. This paper investigates efficient simultaneous vibration absorbing and energy harvesting utilizing a bistable electromagnetic vibration absorber (BEVA). The vibration absorber employs a tri-magnet levitation structure, where the magnetic mechanism between the cylindrical and ring magnets achieves the symmetric bistable characteristic. Furthermore, it is attached to the free end of a cantilever beam subject to transient external excitation. The equivalent model of the cantilever beam with the BEVA system is obtained by exploiting the extended Hamilton’ s principle, while the magnetic forces are derived based on the equivalent magnetic charge method. The dry friction of the BEVA and the mechanical damping of the cantilever beam are obtained by parametric identification of the experimental data for accurate numerical simulations. From the vibration attenuation and energy dissipation viewpoints, the performance of the absorber under transient excitation is assessed. It is observed that inter-well oscillations occur at certain excitation levels, while the energy dissipation efficiency of the BEVA is improved substantially compared to the intra-well oscillations. Keywords: Vibration absorbing · Energy harvesting · Bistable system

1 Introduction Extensive harmful vibration in mechanical, aerospace and civil engineering fields requires effective solutions for elimination. The linear dynamic vibration absorber (DVA), also called the tuned mass damper (TMD), was proposed and used to mitigate vibration at the fundamental frequency of the main structure [1]. To improve the robustness of the DVA and broaden the frequency band of the vibration attenuation, researchers have developed the nonlinear dynamic vibration absorber (NDVA) [2–5]. Vakakis [6] first named a kind of NDVA with strong nonlinear stiffness and the targeted energy transfer mechanism as the nonlinear energy sink (NES). Specifically, the nonlinear stiffness indicates that the NES has non-constant natural frequencies. Therefore, ICANDVC2023 best presentation paper © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 165–180, 2024. https://doi.org/10.1007/978-981-97-0554-2_13

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transient resonance capture can be generated in the 1:1 resonant manifold of the system, resulting in an irreversible energy transfer phenomenon [7, 8]. Ding and Chen [9] reviewed NES research comprehensively and summarized several important designs for optimal damping effects and NES shortcoming offset. The significant advantage of NES is reflected in the increased energy absorption efficiency of the attached elements to the primary system [10]. Similarly, in the field of energy harvesting, there is a concern about how to improve the efficiency of energy conversion. Energy harvesting technology is now extensively employed in mechanical vibration devices for converting vibration energy to electrical energy. Conversion can be achieved by piezoelectric, electromagnetic and triboelectric components. Similar to the development of DVA, research into vibration energy harvesters have also evolved from linear to nonlinear structures. In order to introduce nonlinear characteristics, structural design combined with permanent magnets has been of widespread interest of scholars [11–13]. Yang et al. [14] proposed a harvester with bistable and tri-stable nonlinear enhancement mechanisms by adding four external magnets to a conventional tri-magnet electromagnetic energy harvester. Gao et al. [15] presented a multi-stable electromagnetic energy harvester by symmetrically arranging the eight external magnets on two planes. The results indicated that multi-stable configurations could increase the output current. A comprehensive investigation of the nonlinear dynamics of the bistable harvester by Jung et al. [16] revealed that promoting inter-well oscillations and exploiting high amplitude characteristics are critical points in favor of energy harvesting. In recent years there have been exploratory investigations on simultaneous vibration absorption and energy harvesting [17–22]. Pennisi et al. [23] utilized magnetic forces to counteract the linear term of the elastic force and consequently achieved a pure cubic NES. In parallel, an electromagnetic transducer was employed to convert the energy absorbed by the NES into electrical energy. Huang and Yang [24] investigated the energy trapping and harvesting performance of a multi-stable dynamic absorber from the perspective of energy shunting. Rezaei et al. [25] investigated simultaneous energy harvesting and vibration suppression utilizing a tunable bi-stable magneto-piezoelastic absorber (BPMA). The results indicated that the performance of the BMPA in energy harvesting and vibration mitigation was significantly improved due to the chaotic, strongly modulated response. Following this research, the complex dynamic behavior of the tristable magneto-piezoelastic absorber (TPMA) was further explored [26]. Although the feasibility of simultaneous vibration absorption and energy harvesting has been theoretically demonstrated, there is still a research gap in such dual-function absorber for cantilever beam vertical vibration attenuation. This paper proposes a novel tri-magnet bistable electromagnetic vibration absorber (BEVA) for simultaneous vibration absorbing and energy harvesting. The symmetrical bistable potential well in the direction of gravity is achieved by the magnetic interaction of the ring magnet and the cylindrical magnet. This paper is structured as follows. Section 2 describes the equivalent model of the cantilever beam with BEVA attached at the free end and the design of a symmetrical bistable potential well. In Sect. 3, the energy dissipation mechanism of the system is analyzed based on simulation results. Section 4 draws conclusions.

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2 Mathematical Modelling 2.1 Equivalent Modeling of the Cantilever Beam with BEVA The equivalent model of the cantilever beam with BEVA structure is obtained in this section by exploiting the extended Hamilton’s principle. As shown in Fig. 1(a), the considered system is composed of a cantilever beam with the BEVA rigidly attached at the end. Figure 1(b) shows a cross-sectional diagram of the BEVA. This structure consists of a cylindrical levitating magnet, two ring magnets, a Teflon tube, and two coils. The levitating magnet can move axially inside the tube. When the cantilever beam is excited, part of the energy is transferred to the levitating magnet and dissipated by electromagnetic damping and dry friction. The extended Hamilton’s principle can be stated as:  t2  t2 δWnc dt = 0 (1) δ (T − U )dt + t1

t1

here, δ is the variational operator, T is the system total kinetic energy, U is the total potential energy, W nc represents the work done by non-conservative forces, and t1 -t2 is an arbitrary time span.

Fig. 1. (a) Schematic diagram of the cantilever beam with BEVA, (b) Cross-sectional diagram of the BEVA, (c) Illustration of the parameter configuration of the BEVA.

The total kinetic energy of the understudied system can be expressed as: T =T1 + T2 + T3  2 1   2 1 lc 1  = ρA(w˙ + y˙ b )2 ds + Mt w˙ |lc + y˙ b + It w˙ |lc 2 0 2 2     2   1  2 + m w˙ |lc + y˙ b + y˙ m + y˙ m w |lc 2

(2)

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In Eq. (2), T1 , T2 and T3 represent the kinetic energy of the cantilever beam, the kinetic energy of the BEVA external structure, and the kinetic energy of the levitating magnet, respectively. ρ, A and lc are the density, cross-sectional area and length of the cantilever beam. s is the arclength of the beam. w, yb and ym represent the lateral deflection of the beam relative to the base, the displacement of the base and the displacement of the levitating magnet in the local coordinate system (x, y), respectively. The local system is related to the rotation of the free end of the beam, as shown in Fig. 1(a). Moreover, Mt , It denotes the mass and rotational inertia of the external structure of the absorber, respectively. m is the mass of the levitating magnet. The over-dot denotes partial differentiation with respect to time, and the over-prime shows differentiation to the undeformed length. The total potential energy of the coupled system can also be represented as: U =U1 + U2 + U3 lc 1 lc  = EI ∫ w 2 ds + ∫ ρAg(w + yb )ds + Mt g(w|lc + yb ) 2 0 0 + mg(w|lc + yb + ym ) + ∫ Fmag (ym )dym

(3)

In Eq. (3), U1 , U2 and U3 represent the elastic potential energy of the cantilever beam, total gravitational potential energy of the system, and magnetic potential energy of the levitating magnet. Here, g is the gravitational acceleration, and Fmag (ym ) indicates the magnetic force of the levitating magnet subjected to the ring magnets at both ends. Next, the virtual work done by non-conservative forces consists of three parts, namely, the virtual work done by the mechanical viscous damping c1 of cantilever beam, the electromagnetic damping ce of the BEVA, and the dry friction Ffric of the BEVA. Thus, the virtual work can be expressed as:  δWnc = −

lc

˙ − ce (ym )˙ym δym − Ffric c1 wδwds

0

y˙ m δym |˙ym |

(4)

Substituting Eqs. (2), (3), and (5) into Eq. (1), the coupled nonlinear governing equations of the system are obtained as follows: 2  ρAw¨ + δ(s − lc )mw¨ |lc + δ(s − lc )m¨ym + δ  (s − lc )m˙ym w |lc + c1 w˙ + EIw”” +ρAg + δ(s − lc )(Mt + m)g = −[ρA + δ(s − lc )(Mt + m)]¨yb

 2

m 1 + w |lc y¨ m + mw¨ |lc + 2m w |lc w˙ ’ |lc y˙ m + ce (ym )˙ym

+Ffric

y˙ m + Fmag (ym ) + mg = −m¨yb |˙ym |

(5)

(6)

The reduced-order model of partial differential equations (PDEs) are obtained by utilizing the Galerkin discretization. The following ordinary differential equations (ODEs)

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169

governing the system are obtained as: ∞ ∞ q¨ n + 2ζn ωn q˙ n + ωn2 qn + Ani q¨ i + Bni qi x˙ 2 + Cn x¨ + Dn g = −Dn y¨ b |m¨ym + m

∞ 

i=1

Eij qi qj y¨ m + 2m

i,j

+Ffric

∞ 

i=1

Eij qi q˙ j y˙ m + m

i,j

∞ 

ϕi (lc )¨qi + ce (ym )˙ym (7)

i=1

y˙ m + Fmag (ym ) + mg = −m¨yb |˙ym |

where the coefficients are defined as: Ani = mϕn (lc )ϕi (lc ) Bni = −mϕn (lc )ϕi (lc ) Cn = mϕn (lc ) lc

Dn = ∫ ρAϕn (s)ds + (Mt + m)ϕn (lc ) 0

Eij = ϕi (lc )ϕj (lc )

2.2 Design of the Symmetric Bistable Potential Wells The interaction force between magnets can be calculated by the magnetic dipole model [27]. In this paper, the magnetization direction of each magnet is axial, and the equivalent magnetic charge is uniformly distributed on the end faces of each magnet. As shown in Fig. 1(c), the local coordinate system is established. The magnetic force on the levitating magnet can be expressed as the superposition of the interactions between all equivalent magnetic charge surfaces [28]:



hlev hlev d d Fmag (ym ) = F13 − + hbot + ym − F14 + + hbot + ym 2 2 2 2



d d hlev hlev − + ym + F24 + + ym −F23 2 2 2 2



(8) d d hlev hlev + − ym + F54 − − ym −F53 2 2 2 2



d d hlev hlev + + htop − ym − F64 − + htop − ym +F63 2 2 2 2 where Fij , Fkj (i = 1, 2, j = 3, 4, k = 5, 6) denotes the interaction force between different equivalent magnetic charge surfaces. i, j and k denote the end surfaces of the bottom magnet, the levitating magnet and the top magnet, respectively. Here, the magnetic force between surface-i and the surface-j can be expressed as:

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  Fij dij =

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2π Rbot 2π Rlev 0 rbot 0 0

μ0 Mbot Mlev  4π

dij pq (pcosα − qcosβ) + (psinα − qsinβ)2 + dij 2 2

 23 dq d β dp d α

(9) where dij is the distance between the two surfaces, μ0 is the permeability of vacuum. As shown in Fig. 1(c), for the BEVA in the local coordinate system, the relative potential energy of the levitating magnet is defined as follows:   (10) Ur = mgdym − Fmag (ym )dym here, the first integral describes part of the gravitational potential energy of the levitating magnet. The bistable nonlinear mechanism has been investigated using the single-sided bipolarity of the ring magnet in the previous study [28, 29]. To avoid undesirable effects of the levitating magnet on the primary structure, a smaller-sized levitating magnet has been chosen in this paper. The magnets and other relevant parameters of the BEVA are listed in Table 1. As depicted in Fig. 2, the relative potential energy for different ring magnet spacings (d ) are plotted. When the spacing decreases, the potential wells on both sides gradually move closer to each other, and the potential barrier between them gets lower, which means that the levitating magnet is more susceptible to inter-well oscillations. As the spacing decreases, the bistable system will degrade to a monostable system. A fixed spacing (d = 39 mm) was selected for the BEVA because the depths of the potential wells on both sides are equal at this point.

Fig. 2. The relative potential energy for different ring magnet spacings.

2.3 Calculation of Electromagnetic Damping As shown in Fig. 1(c), two coils are attached axially symmetrically to the top (y = dcoil ) and bottom (y = −dcoil ) of the tube. This electromagnetic structure is utilized to convert vibration energy into electricity while providing a damping force to impede the

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Table 1. Properties of the BEVA. Parameter

Value

Bottom ring magnet (N52) Internal radius rbot

8.5 mm

External radius Rbot

12 mm

Height hbot

1.5 mm

Magnetization Mbot

7.2 × 105 A/m

Levitating magnet (N52) Radius Rlev

4 mm

Height hlev

10 mm

Magnetization Mlev

9.8 × 105 A/m

Density

7500 kg/m3

Top ring magnet (N52) Internal radius rtop

8.55 mm

External radius Rtop

11.5 mm

Height htop

2.5 mm

Magnetization Mtop

6.6 × 105 A/m

Top (bottom) ring-shaped Coil (38AWG) Internal radius rcoil

7.5 mm

External radius Rcoil

12.3 mm

Height hcoil

10 mm

Turns N

3000

Position dcoil

7 mm

Load resistor Rl

790

Total coils resistance Rc

786

movement of the levitating magnet. From the Faraday law of electromagnetic induction, the voltage induced in the top coil can be written as: Uv−top = −

d ϕtop dym d ϕtop =− dt dym d t

(11)

where ϕtop is the total magnetic flux through the top coil. According to the charge model, the magnetic field due to the levitating magnet is given as B [27], which is varied with ym . In reference to Fig. 1 (c), the magnetic flux emanating from the levitating magnet through all turns of the top coil is expressed as: N ¨ ϕtop (ym ) = (12) B(ym )dxdy 1

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where B is projection of the B onto the y axis. Similarly, for the bottom coil, we obtain ϕbot (ym ) and Uv−bot . When the device is excited, the magnetic fluxes of the upper and lower coils normally change. Assuming the top and bottom coils are connected and winded in the same directions. The total voltage generated in coils is given as: Uv = Uv−top + Uv−bot

(13)

According to the principle of energy conversion, the generated electric power is equal to the mechanical power dissipated by the electromagnetic damping force, i.e. 2 ce (ym )˙ym =

Uv2 Rl + Rc

(14)

The above equations give the electromagnetic damping coefficient as:  ce (ym ) =

d ϕtop dym

+

d ϕbot dym

Rl + Rc

2 (15)

And it must be mentioned that the coil inductance has a negligible effect at low frequency vibrations.

3 Results and Discussions 3.1 Parameter identification and verification In this section, the electromechanical equations of motion derived in the previous section are validated. For this purpose, the structural parameters of the system are obtained experimentally and compared with the simulation results. First, the dry friction of the BEVA is obtained through parameter identification methods based on the experimental frequency response. Then the mechanical damping ratio of the cantilever beam is identified from the transient response. Finally, the transient electromechanical response of the system, which is simulated using the identified parameters, is compared with the experimental results.

Fig. 3. (a) Experimental system for BEVA parameter identification, (b) Output voltage at 0.8 g.

First, the BEVA system is analyzed for frequency response, as shown in Fig. 3(a). In doing so, the waveform generator (Agilent 33500B) outputs a harmonic signal which is passed through a power amplifier (YMC LA-200) to the shaker (YMC VT-200).

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The excitation signal, monitored by the accelerometer (LC0103TA), is simultaneously acquired with the load voltage using the acquisition device at a sampling rate of 10 kHz. The Runge–Kutta method is used to solve ODEs. The step size is 0.0001 s, corresponding to the experimental sampling rate. Figure 3(b) shows the BEVA output voltage simulation results at 0.8 g, which agrees very well with the experiment. The BEVA can produce a relatively high-level voltage in the frequency range of 10.5–15 Hz because of the periodic inter-well oscillations. In the frequency range of 15.5–18.5 Hz, the BEVA performs chaotic oscillations [28], at which point fluctuations in output voltage are acceptable. The dry friction of the BEVA is identified through the genetic algorithm as 0.0042N.

Fig. 4. Experimental system for simultaneous vibration absorption and energy harvesting.

Fig. 5. The experimental measured excitation acceleration.

Second, the primary structure damping is experimentally obtained. The experimental system is shown in Fig. 4. Here, the waveform generator generates a triangular pulse signal. The laser (KEYENCE LK-H050) acquires the displacement response of the cantilever beam. Figure 5 shows the experimentally measured acceleration signal with a maximum amplitude of 2.7 g and a minimum amplitude of −3.6 g. This acceleration data was substituted into Eq. (14) for simulation. It should be mentioned that only the firstorder mode (n = 1) is considered in Galerkin’s series. Accordingly, only the damping ratio of the first-order mode needs to be identified. For simplicity, the cantilever beam attached to the BEVA external body (without a levitating magnet) at the free-end is called the ‘linear oscillator (LO) without BEVA’. When the levitating magnet is placed in BEVA external body, the coupling system is named the ‘linear oscillator (LO) with BEVA’. Figure 6(a) shows the displacement response of the LO without BEVA. The damping ratio of the first dominant mode can be easily find by log-decrement and the value is 0.004. The other relevant parameters of the system are listed in Table 2. As shown in Fig. 6, the simulation and experimental time-domain and frequency-domain responses are consistent.

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Fig. 6. The transient response of the LO without BEVA. (a) Displacement, (b) Amplitude Spectrum.

Finally, the developed electromechanical equations of motion for the coupled system are experimentally validated. Figure 7 shows the transient response of the LO with BEVA. In Fig. 7(a), the simulation and experimental displacements coincide almost precisely in the 0–2 s. After 2 s, the displacements appear to be inconsistent, which is acceptable. In addition, the voltage response shows a high agreement between simulation and experimental results within 0–0.5 s. After a while, the agreement between experimental and simulation results decreases, possibly due to manufacturing errors in the device. This discrepancy is acceptable, and there is no doubt that the equivalent model of the cantilever beam attached absorber proposed in this paper is correct.

Fig. 7. The transient response of the LO with BEVA. (a) Displacement, (b) Load voltage.

Transient vibration absorption and energy harvesting mechanism. In this section, the mechanism of the BEVA in vibration absorption and energy harvesting under an initial excitation is investigated. The initial excitation can be in the form of initial displacement (q1 (0) = 0) or initial velocity ( q˙ 1 (0) = 0). In this paper, the initial velocity (such as 0.05 m/s, 0.1 m/s) is chosen as the initial excitation. The energy dissipation of the system is considered as follows: System I: LO without BEVA: EI −in

1 = (˙q1 (0))2 , EI −out = 2



t

2ζ ω(˙q1 (t))2 dt

(16)

0

where the subscript ‘in’ indicates energy input and the subscript ‘out’ indicates energy dissipation.

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Table 2. Properties of the cantilever beam with BEVA. Parameter

Value

Cantilever beam (65Mn) Young’s modulus E

198 GPa

Mass density ρ

7850 kg/m3

Length l c

229.5 mm

Width wc

20 mm

Height hc

1.6 mm

BEVA external body (ABS) Mass Mt

68 g

Moment of inertiai It

310.25 g · cm2

Other Mass of the levitating magnet m

3.77 g

Dry friction of BEVAii Ffric

0.0042 N

Damping ratioii ζ

0.004

i The BEVA external body is approximately considered as a homogeneous cylinder. ii These values were identified from the experimental data.

System II: LO with BEVA: EII −in =

t 1 (1 + A11 )(˙q1 (0))2 , EII −out1 = ∫ 2ζ ω(˙q1 (t))2 dt 2 0 t

EII −out2 = ∫ 0

t (Uv (t))2 dt, EII −out3 = ∫ Ffric |˙ym (t)|dt Rl + Rc 0

(17)

here, EII −out1 is the mechanical damping dissipation energy, EII −out2 is the BEVA electromagnetic damping dissipation energy and EII −out3 is the BEVA dry friction dissipation energy. The total energy of the system in real-time can be obtained as Etotal (t) = Ein − Eout . (1) q˙ 1 (0) = 0.05 m/s Firstly, the initial velocity of the cantilever beam is set as q˙ 1 (0) = 0.05 m/s. The responses of the system are shown in Fig. 8. It can be seen from Fig. 8(a) that the vibration of the cantilever beam with BEVA is suppressed. In this case, the levitating magnet oscillates in a single potential well (Fig. 8(b)), corresponding to a small output voltage (Fig. 8(d)). The attachment of BEVA accelerates the energy dissipation of the system (Fig. 8(c)). For system I, the time required to dissipate 97.5% of the initial energy is 10.4 s. With the BEVA attached, this time is reduced to 4.9 s. Figure 9 shows the different forms of energy dissipation in System II. The result indicates that the BEVA

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Fig. 8. Transient responses of the system I and II when q˙ 1 (0) = 0.05 m/s. (a) Displacement (cantilever beam), (b) Displacement (the levitating magnet), (c) Total energy, (d) Load voltage.

relies mainly on dry friction for energy dissipation under low excitation. (2) q˙ 1 (0) = 0.1 m/s

Fig. 9. Energy dissipation in the system II (LO with BEVA) when q˙ 1 (0) = 0.05 m/s.

When the initial condition of q˙ 1 (0) = 0.1 m/s, the time responses of the system I and II are shown in Fig. 10. Figure 10(a) shows that the displacement of the cantilever beam in system II decays rapidly within 0–2 s, which is caused by the levitating magnet performing inter-well oscillation (Fig. 10(b)). For system I, the cantilever beam performs free-vibration, so the time required to dissipate 97.5% of the initial energy is still 10.4s (Fig. 10(c)). However, for System II, the inter-well oscillation significantly increases the energy dissipation speed, taking only 2.2 s to achieve 97.5% energy dissipation. The large-amplitude vibration of the levitating magnet also corresponds to an increase in output voltage, as shown in Fig. 10(d), with a peak load voltage of 2.5 V. An attractive phenomenon emerges in Fig. 11, where the energy dissipated by the friction is the primary part of the energy dissipated in System II. At the same time, the electromagnetic damping energy dissipation also shows an increase.

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Fig. 10. Transient responses of the system I and II when q˙ 1 (0) = 0.1 m/s. (a) Displacement (cantilever beam), (b) Displacement (the levitating magnet), (c) Total energy, (d) Load voltage.

Fig. 11. Energy dissipation in the system II (LO with BEVA) when q˙ 1 (0) = 0.1 m/s.

4 Conclusions In this research, a bi-stable electromagnetic vibration absorber (BEVA) is utilized for simultaneous vibration absorbing and energy harvesting. The vibration absorber employs a tri-magnet levitation structure, where the bistable characteristic is achieved by the magnetic mechanism between the cylindrical and ring magnets. The proposed absorber

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is connected to a cantilever beam subjected to transient excitation to suppress vibrations in the vertical direction and harvest electrical energy. Firstly, the continuous magnetomechanical control equations are derived using Hamilton’s principle, while the magnetic forces are derived based on the equivalent magnetic charge method. The parameters of the coupled system are identified using experimental data. Next, numerical simulations are carried out to reveal the vibration absorption and energy harvesting mechanism of the BEVA based on the validated system model. The results show that large-amplitude interwell oscillation contributes to faster energy dissipation while generating high output load voltage. The analysis of the energy dissipation ratio of each component shows that at low excitation, the BEVA dissipates energy mainly through dry friction. As the energy input increases, the proportion of energy dissipated by electromagnetic damping increases. Acknowledgements. This work was supported by the Natural Science Foundation of China (Grant No. 52275122 and 12132010).

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Design and Research of Triboelectric Energy Harvester for Low Frequency Nonlinear Vibration Yinqiang Huang1 , Huajiang Ouyang2 , and Zihao Liu1(B) 1 The Mechanical and Electrical Engineering College, Hainan University, Haikou, China

[email protected] 2 School of Engineering, University of Liverpool, Liverpool L69 3GH, UK

Abstract. Although great progress has been made in the study of triboelectric energy harvesting, most of the efforts are aimed at the manufacturing and experimental demonstration of the harvesters. For simultaneous energy harvesting and vibration control, there is still a strong need of structural design and in-depth theoretical research of structural dynamics of triboelectric harvesters. In this paper, a harvester in the form of a cantilever beam and two curved surfaces as constraints is proposed. When the cantilever beam vibrates under a lowfrequency excitation, the contact between the cantilever beam and one of the curved surfaces is gentle and gradual. Compared with a triboelectric harvester working in the traditional contact-separation mode, this contact can achieve energy harvesting while avoiding the introduction of vibro-impact of the structure, but introduces complex nonlinear vibration. Through the planar rigid body kinematics and a quasi-static analysis, the differential equation of motion for the cantilever beam including the ninth-order geometric nonlinearity for the contact is established. The mathematical model for combining the structural dynamics and electrical dynamics is established. Finally, the approximate analytical solution of the model is obtained by using the harmonic balance method, and the stability of the model under different structural parameters is analyzed by Floquet theory. Numerical simulation results show that when the frequency excitation is 5.72 Hz, the peak output voltage is 6.9 V and the average power is 1.9 μW. When the frequency is between 5.72 Hz and 5.93 Hz, the response exhibits bifurcation. Compared with the traditional cantilever beam absorber, the frequency response curve of this structure is deflected due to the nonlinear factors brought about by the curved surface, and the frequency band of vibration hysteresis is narrow. The broadband capacity of the nonlinear spring is proven in the frequency domain for two chosen surface curvature orders, with one low and one high amplitude of excitation. Therefore, the structure proposed in this paper can maintain a larger amplitude in the broadband, thus playing a role in broadening the working frequency band of vibration absorption. In summary, the structure can not only realize vibration energy harvesting without hard impact, but also work as a vibration absorber with nonlinear characteristics. Keywords: Triboelectric Energy Harvester · Nonlinear cantilever beam · Nonlinear vibration absorber · Nonlinear dynamics · Nonlinear modeling

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 181–194, 2024. https://doi.org/10.1007/978-981-97-0554-2_14

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1 Introduction Wireless sensor networks have been widely used in various fields, including aerospace engineering, intelligent transportation, smart factories, and other distributed environments, to perform sensing tasks. However, the energy supply for these sensors presents some challenges. These challenges include the use of complex wired power supply systems, limitations of battery power, high maintenance costs, and environmental pollution problems. Additionally, the ambient often contains significant vibrations, such as those produced by mechanical equipment and engineering structures that are commonly used in the industrial and transportation industries. These vibrations can cause significant harm, leading to economic losses and even casualties. Hence, there is an increasing need for vibration control solutions. Moreover, such environment often exhibits lowfrequency and broadband vibrations. Therefore, it is imperative to investigate how to effectively harness this mechanical vibration energy to simultaneously power the sensors and control vibration responses. However, there is limited research that focuses on both functions. Vibration-based energy harvesters offer a promising solution to recycle and utilize the wasted mechanical energy in the environment by converting it into electrical energy. In recent years, piezoelectric [1–4] and electromagnetic [5–7] energy harvesters have been extensively studied. Since the introduction of this concept in 2012 [8], triboelectric energy harvesters (TEHs) have also gained widespread attention due to their unique characteristics. TEHs are self-powered systems that collect energy from environmental vibrations. They offer advantages such as cost efficiency, high energy conversion rates, lightweight, small size, and wide availability of materials [9]. These characteristics make TEHs particularly suitable for collecting low-frequency vibration energy. With their structural design flexibility, TEHs have an immense potential as a replacement for traditional energy supplies in engineering applications. Electrical charges in TEHs are generated by the friction or contact between triboelectric materials. When two thin organic/inorganic films with distinct surface electron affinities come into contact or undergo separation in the normal direction or sliding in the tangential direction, triboelectrification and electrostatic induction occur. This leads to the generation of an electric potential difference due to the relative motion caused by a mechanical force. As a result, electrons on the surfaces of the two materials are driven to flow between the two electrodes [10]. TEHs typically operate in four working modes: vertical contact-separation mode [11], lateral sliding mode [12], single-electrode mode [13], and free-standing triboelectric-layer mode [14]. TEHs can be widely used for collecting various types of energy in various environments, such as wind energy, water energy, and mechanical energy. Here, some designs will be reviewed briefly. Fu et al. [15] collected vibration energy through triboelectricity generated by the vibration and impact between three parallel cantilever beams. Zhao et al. [16] proposed a novel cantilever type TEH for low-frequency vibration energy collection. In this design, the free end of the cantilever beam is attached to a contact plate with a polytetrafluoroethylene film. When the cantilever beam vibrates upwards, the contact plate is pushed to separate from the top plate, resulting in the generation of triboelectric electricity. Typically, the working mode of a triboelectric energy harvester is separate. However, with further research, mixed working mode THEs and hybrid TEHs

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have also been proposed. Zhao and Ouyang [17] introduced an integrated horizontal sliding and contact separation triboelectric energy harvester. This design includes a slider embedded inside a capsule-structure THE. Under external excitation, the slider produces sliding friction and vibro-impact with the inner wall of the capsule, thereby generating electricity. Currently, most research on THEs has focused on the preparation, experimentation, and surface treatment of triboelectric materials. Research has demonstrated that patterned triboelectric surfaces have higher energy generation efficiency than patternless ones [22]. Various surface patterns have been fabricated and studied, including pyramid and cube patterns [23], nanowire arrays [24], and nanopores [25]. The exploration of triboelectric materials has greatly improved the power generation efficiency and provided new directions for blue energy harvesting. The use of nonlinearity to expand the resonant bandwidth of a vibration energy harvester has received widespread attention. In addition to the above research, Wang et al. [26] investigated a Piezoelectric energy harvester that utilized a cantilever beam and two symmetric surfaces with given geometry. Similarly, Kluger et al. [7] used a similar structure to an electromagnetic vibration energy harvester to capture vibration energy generated during walking. Nonlinearity has proven to be influential not only in the field of vibration energy harvesting but also in vibration control. Silva et al. [27] applied geometrically nonlinear passive dampers to the main vibration system, and their results showed that the proposed device had a broad frequency response, indicating its ability to effectively engage in targeted energy transfer suitable for applications like vibration attenuation. Despite the extensive study of the vibration energy harvesters with a geometrically nonlinear clamping cantilever structure, this particular structure has not yet been exploited for triboelectric energy harvesting. The properties and contact mode of the contact surfaces play a crucial role in triboelectric harvesting, thus the characteristics of geometrically nonlinear surfaces hold considerable significance and warrant further investigation. This paper proposes a nonlinear surface contact cantilever triboelectric energy harvester (NSCCTEH) and offers the following scientific contributions for triboelectric energy harvesting. Firstly, the NSCCTEH is theoretically investigated for the first time in triboelectric energy harvesting encompassing structural dynamics and electrodynamic coupling. In contrast to the traditional “separation and closing” mode, this structure exhibits a gradual separation and gradual contact process. Secondly, unlike traditional TEH that typically has the function of only energy harvesting, the proposed NSCCTEH structure can simultaneously perform energy harvesting and vibration control under excitation. Thirdly, this paper expands the resonant bandwidth by incorporating geometric nonlinearity, resulting in a deflection of the frequency response curve. Consequently, a larger amplitude is maintained within a wider frequency band, achieving the goal of expanding the working frequency band. To better demonstrate these characteristics, the paper provides the derivation of the structural dynamics model and stability analysis of the structure, followed by the analysis of the equivalent electrical model. Finally, the paper discusses the dynamic response and the power output, and the influencing parameters.

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2 Prototype Design The proposed TEH in this paper consists of two parts: one is a nonlinear spring resulting from a cantilever beam constrained between two curved surfaces; the other part consists of a generator set which includes a polytetrafluoroethylene film (PTFE) and a metal film (Cu) covering the two curved surfaces and a cantilever beam made of copper. The unique structure improves the dynamic performance by the variable-curvature curved surfaces, such as preventing secondary vibration caused by impact-contact of the absorber, minimizing friction by reducing the relative sliding between the cantilever beam and the curved surface, and inhibiting excessive deflection of the cantilever beam by the curved surface. In a word, the proposed structure is a kind of vibration energy harvester and also is a nonlinear vibration absorber.

Fig. 1. Simplified model. where LC is the cantilever length, hg is the gap between the surface end and the undeflected cantilever, and LS is the length of the surface in the x direction, xd is the demarcation point, LF is the length of the un-contact part of the cantilever in the x direction; The tip displacement y of the beam depends on the vertical force F and the shape of the contact surface.

2.1 Quasi-Static Analysis The structural configuration of the TEH is described in Fig. 1, where two variablecurvature surfaces grip a cantilever beam with a mass at another end of the beam. The unique variable-curvature surface design of the TEH ensures that the curvature radius on the surface is maximum at the root and decreases along its length. In contrast, the curvature radius of the free cantilever beam at the base is minimum and increases along its length. This opposite trend in the variation of curvature radii along the beam length allows for immediate contact between the beam and the surface even with minimal force

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applied. The contact area between the cantilever beam and the surface gradually increases as the beam deforms. The improved surface shape can be represented as follows:  n (1) S = hg Lx S

where L S is the length of the surface in the x direction; hg is the distance between the undeflected cantilever beam and the curved surface at x = LS ; n is an integer number greater than 2 to ensure that the curvature radius at the root of the surface is 0. Any one of the two curved surfaces act as a nonlinear spring. To determine the characteristics of the nonlinear springs, it is necessary to obtain the force-displacement relationship. For this purpose, a vertical force F is applied at the end of the cantilever beam. The tip displacement y of the beam depends on the vertical force F and the shape of the contact surface. It can be divided into three components: y = y1 + y2 + y3 . The first component y1 is the displacement caused by the bending of the non-contact part of the cantilever beam. The second component y2 is due to the rotation of the beam segment (from xd to LS ) at the contact boundary point xd , The third component y3 is the deflection of the beam at the contact boundary point xd , Therefore, the displacement y of the cantilever beam tip under the action of external force F is:  FL3  y = y1 + y2 + y3 = 3EIF + dS .LF + S(xd ) (2) dx  x=xd

The relationship between the length of the contact boundary point xd and the restoring force FR : FR =

xdn−2 EIhg LnS n(n − 1) LC −xd

(3)

For a detailed derivation of tip displacement y and restoring force FR , please refer to [26]. The change of nonlinear spring force FR with displacement y of the tip of the cantilever beam is shown in Fig. 2(a). When the cantilever is gradually bent along the curved surface, the relationship between the contact length of the beam and the displacement of the beam tip under different curvature orders is shown in Fig. 2(b). In all cases, the maximum contact length between the cantilever occurs at the maximum displacement of the beam tip and its value is bounded. Near the the clamped boundary, with the increase of parameter, n, the contact length suddenly increases, and then, with the increase of displacement, the contact length gradually increases until the maximum contact length is reached. In this process, compared with different parameter values of n, The contact length for parameter n = 5 is greater than the contact lengths for the other two parameters, n, under small deformation conditions of the cantilever beam. As the tip displacement, y, increases to around 0.06 m, the contact lengths for all three parameters gradually intersect until they reach the same maximum contact length. This suggests that, in the structures discussed in this chapter, surfaces with higher values of n are more suitable for use as TEH. This is because even with small tip displacements, larger contact lengths (contact area) can be achieved. Additionally, the maximum curvature bending on the cantilever beam can be controlled, and the maximum normal stress on the beam can be kept within a certain value under any external force. This effectively prevents structural damage under extreme excitation.

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(a)

(b)

Fig. 2. The relationship between restoring force and tip displacement (a) and the relationship between contact length and tip displacement (b) under different n values

3 Structural Dynamics Model In this section, a structural dynamics model is established to comprehensively study the dynamic response of several nonlinear springs composed of different curvature surfaces and cantilever beams under simple harmonic base excitation. Numerical simulation is utilized to explore the influence of various parameters on vibration characteristics and conduct stability analysis of periodic solutions. 3.1 Approximate Analytical Solution Assuming that the system is driven by external excitation, the excitation function ub = Acos(ωb t), where A and ωb represent the excitation amplitude and the excitation frequency, respectively. The dynamic equations of the system are numerically solved using the Matlab differential equation solver ode45, which uses a variable step size and implements the Runge-Kutta algorithm, suitable for solving nonstiff differential equations. The equation governing the lateral motion of the beam is: m¨y(t) + c˙y(t) + FR = −m¨ub

(4)

where m is the equivalent mass of the tip mass and the cantilever beam mass, and c is the mechanical damping coefficient. The next step is to determine the approximate expression of the nonlinear resilience FR of Eq. (2), which is found to be through curve-fitting: FR = kL y + kNL y9

(5)

where kL and kNL are linear and nonlinear stiffness coefficients, respectively. 3.2 Non-dimensionalization Based on the resulting nonlinear restoring force FR , substituting Eq. (5) into Eq. (4) can represent a new differential equation expression: m¨y(t) + c˙y(t) + kL y + kNL y9 = −m¨ub

(6)

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The relative contribution of constant terms in Eq. (6) must be evaluated through nondimensionalization. By scaling the time variable by a characteristic time, Tc , let: t Tc ,

τ= where Tc =



m kL ,

μ = Tc y˙ , μ = T2c y¨

μ = y,

(7)

substitute Eq. (7) into (6), and simplify to get: μ + λμ + μ + εμ9 = 2r cosr τ .

where: λ=

√c , mkL

ε=

kNL kL ,

r = ωb



(8)

m kL

(9)

3.3 System Response Near Resonance Because there are higher-order terms in Eq. (8), it is difficult to obtain a solution in a closed form, so an approximate solution is sought by the harmonic balance method, which is suitable for systems with polynomial nonlinear terms. A solution to Eq. (8) is assumed to have a single frequency of the form: μ(t) ≈ μh = A cos(r τ ) + B sin(r τ )

(10)

where A and B are Fourier coefficients. The interest is in the low-frequency behaviour. The detailed derivation process is mathematically extensive and produces very long expressions. Therefore, this article briefly describes some of the steps of the process. Substituting Eq. (10) into Eq. (8) gives the power expressions of sine and cosine, and then through a series of trigonometric manipulations (e.g., sin(r τ )3 = 14 sin(3r τ ) + 3 4 sin(r τ )), an expression with the highest harmonic (9r τ ) is obtained. After ignoring all higher-order harmonic terms greater than (3r τ ) due to their tiny coefficients, the following solutions can be derived [27]: A − 2r A + B+

63 9 128 εA

63 9 128 B ε

+

189 4 5 64 εB A

− B2r +

+

63 2 7 32 εB A

189 4 5 64 A B ε

+

+ 2r +

63 6 3 32 A B ε

+

63 63 8 6 3 128 εB A + 32 εB A

+ Br λ = 0

− Ar λ +

=0

63 8 128 A Bε

63 2 7 32 A B ε

(11) (12)

Substituting A = a cos(θ ), B = bsin(θ ) into Eq. (11) and Eq. (12) for a polar substitution, and then eliminating the trigonometric term through some techniques, and performing a series of combined simplifications, the relationship between frequency and amplitude is finally obtained: 3969 18 2 16384 a ε

−

63 10 2 64 a εr

+

63 10 64 a ε

+ a2 4r − 2a2 2r + a2 + λ2 a2 2r = 2 4r (13)

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4 Electric Model of Triboelectric Energy Harvester As shown in Fig. 3, the design proposed in this paper consists of two independent triboelectric energy harvesters, that is, the upper and lower surfaces of the entire structure with the beam form two TEHs in contact-separation mode, represented by TEH1 and TEH2, respectively. For TEHi, the equivalent model includes an open-circuit voltage Voci and variable capacitance Ci , which are assumed to be only functions of the relative displacement between the beam and the surface, independent of other motion parameters such as velocity and acceleration [28]. Because of the symmetrical surface structure used in this design, TEH2 and TEH1 have identical dynamic behaviour. And the relationship between contact length xd and displacement y has been obtained in quasi-static analysis. So, the equivalent model of TEH1 and TEH2 includes the open circuit voltage Voci and variable capacitance Ci combined into one expression, that is, the equivalent capacitance C and the open circuit voltage Voc (Fig. 4).

Fig. 3. Simplify electrical models

Fig. 4. Working cycle of triboelectric pair.

In order to determine the capacitance C and open circuit voltage Voc = Va +Vd of the system, considering a capacitor of infinitesimal length dLh , one can get the infinitesimal

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capacitance of the TEH for a given distance. Upon integration over the length, the total capacitance between two electrodes can be obtained, and the electromechanical model can be established accordingly [31]. Reviewing the quasi-static analysis in Sect. 2.1, it can be seen that the relative displacement function of the infinitesimal length dLh and the corresponding surface on the cantilever beam can to be rewritten in Eq. (1) as:  n (14) Sx = hg LLh S

where Sx is the distance from any point on the untouched surface to the x-axis, Lh = xd → xdMax is the distance range from the contact point of the beam and the surface to the longest contact point, and then rewrite equation Eq. (2) as: yx (t) =

hg xdn−2  2 3xd 3Lns

+ 3nxd (Lh − xd ) + n(n − 1)(Lh − xd )2



(15)

where yx (t) expresses the deflection of any different elements on the cantilever beam of the lower separation part at a certain time. The vertical distance between the corresponding different infinitesimal length dLh of the canti-lever and the surface separation part can be expressed as: δ(t) = Sx − yx =

  3hg Lnh −hg xdn−2 3xd2 +3nxd (Lh −xd )+n(n−1)(Lh −xd )2 3Lns

(16)

For TEHi, the equivalent capacitance C can be divided into two capacitance components, where the free cantilever part and the surface non-contact part are non-parallel plate capacitors Ci.1 , the cantilever and surface contact parts are equivalent to the parallel plate capacitance Ci.2 , and the potential difference Vd inside the dielectric layer, which are expressed as: C = Ci.1 + Ci.2 xdMax

Ci.1 = ∫ xd

ε0 b ds +δ dLh , Ci.2

Vd = − σεs 0ds

(17) =

ε0 Ac ds

(18) (19)

where ds = εdrs1 is the effective thickness of the PTFE film, d1 is the thickness of the PTFE film, εrs is the relative permittivity, Ac = xd b, represents the area of the contact part, Ab = (xdMax − xd )b, represents the area of the untouched part, b is the beam width, σs is the surface density of the frictional charge, ε0 is the vacuum permittivity, and xdMax is the maximum contact length. Since the free cantilever plates are not parallel to the PTFE on the curved surface, the potential difference Va between the two plates cannot be calculated a simple shunt capacitor model. In this model, the Va is estimated by considering the electric field energy of the region between the free cantilever plate and the PTFE. It should be noted that the potential difference between the free cantilever plate and the PTFE at each position along the length direction is equal. In other words, the free cantilever plate and the dielectric layer are actually two equipotential planes to which the electric field lines

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should be perpendicular. After finding the arc length h corresponding to each differential infinitesimal length dLh , the total electric field energy W can be expressed as: xdMax

W = ∫ xd

1 2 2 ε0 Ea dv

xdMax

= ∫ xd



2 Va

1 2 ε0



h



xdMax

b hdLh = 21 ε0 Va2 b ∫ xd

1



dLh

(20)

h

where Ea is the electric field strength between the two plates, v is the volume of space gap, combined with the electric field energy formula W = simplifications, Va expression is shown: Va =

σs (xdMax −xd ) x 1 ε0 ∫xdMax dLh  d

Ci.1 Va2 2 ,

after a series of

(21)

h

The electricity generation equation of the triboelectric harvesters can be written as: V = −Q C + Voc

(22)

Applying Ohm’s law on Eq. (22), one can get the electric differential equation as: R dQ dt +

Q C

− Voc = 0

(23)

For continuous motion, the electric output signal from the harvesters is also timedependent. In such a case, the average output power is used to assess the performance of vibration-based energy harvesters [10]. The corresponding average output power P for TEHi and the total average power P can be obtained by: P=

∫t0 V 2 (t)dt Rt

(24)

5 Numerical Simulations and Results Figure 5 shows the amplitude-frequency response curves of the harmonic balance solution and the numerical solution, and the amplitude of the system can be determined at this frequency when the displacement reaches a steady state under the sweeping excitation of each frequency step and equal amplitude. As shown in Fig. 5(a), there is a stable solution to the steady-state response in the range of frequencies less than 5.72 Hz; when the frequency is in the range of 5.72 Hz–5.93 Hz, the response bifurcates, and the system has two stable solutions and one unstable solution. In Fig. 5, it is clear that the main formant is shifted towards the high frequency, and the jump occurs to the right of the formant. As can be seen from Fig. 6(a), as the excitation amplitude increases, the jump frequency of the system increases, the unstable frequency range and the main resonance frequency range also increase, and the degree of bending of the frequency response curve increases. It can also be seen from the figure that the frequency response curve is bent due to nonlinearity, and there are multi-value regions in the above figure, which is also the cause of the jumping phenomenon. The stability of the system is also verified by the evolution of the Floquet multiplier in Fig. 6(b), where the Floquet multiplier evolves as the excitation frequency increases. The frequencies at which the Floquet multiplier

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leaves and re-enters from the unit circle are 5.72 Hz and 5.93 Hz, respectively, which are also consistent with the numerical simulation and approximate analytical solution in Fig. 5. Figure 7 shows the simulation results of the contact length, displacement, and output voltage of the acquirer with a surface order n = 3 under the excitation of an external resistor of 10 M under the excitation of the resonance frequency of 5.72 Hz and the excitation amplitude  = 1.2 mm. The simulated output voltage is 6.9 V peak and the average output power is 1.9 μW. Future work will involve parameter optimization, including the surface width, surface parameter n, cantilever length; and nano-treatment of contact surfaces. In addition to the current geometrically nonlinear structure, other nonlinear factors such as magnetic nonlinearity will also be introduced.

Fig. 5. Harmonic balance method and numerical solution of amplitude-frequency response under different conditions are applied

(a)

(b)

Fig. 6. (a) The relationship between the response of the system and the amplitude of the excitation under the same curvature order. (b) Evolution of stability

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Fig. 7. (a) The amplitude of the beam tip and the contact length between the beam and the surface over time (b) THE output voltage waveform

6 Conclusions This paper proposes a NSCCTEH which avoids the vibro-impact working characteristics of the design and greatly increases the service life due to low friction. It can be vibrated under low-frequency excitation force, and the contact between the cantilever beam and the curved surface is a relatively gentle gradual contact. This method achieves energy harvesting while avoiding vibro-impact. However, it also introduces relatively complex nonlinear vibration. In this paper, the first step is to establish a dynamic differential equation containing a ninth-order geometric nonlinearity through planar rigid body kinematics combined with quasi-static analysis. The structural coefficients are then designed using parameter identification. The next step is to obtain the approximate analytical solution of the model using the harmonic balance method and analyze the stability of the model under different structural parameters using Floquet theory. Finally, a coupled mathematical model of structural dynamics and electrodynamics is established. This paper presents the first theoretical study of the nonlinear surface contact cantilever triboelectric energy harvester. It includes investigations into structural dynamics and electrical analysis. This study is essential for the efficient design and manufacture of energy harvesters. Through numerical simulation, the dynamic behavior and electrical output performance of the triboelectric energy harvester (TEH) were studied. It was discovered that the introduction of a geometric nonlinear contact surface benefits the TEH by enabling it to maintain a large steady-state amplitude in a wide frequency band. This expansion of the working frequency band allows for greater vibration absorption. Furthermore, even without the traditional contact separation typically used to increase electrical output, the TEH still demonstrates a relatively good electrical output. Unlike other vibrationbased energy harvesters, this unique structure of the TEH avoids the need for new shock excitation while simultaneously maintaining the electrical output characteristics found in both traditional contact separation and horizontal sliding TEHs. In summary, this structure not only facilitates vibration energy harvesting without the need for shock in

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low-frequency vibration environments, but it can also serve as a shock absorber with nonlinear characteristics. These findings highlight the potential of the TEH in low-frequency environments for energy harvesting and vibration absorption.

References 1. Madinei, H., Khodaparast, H.H., Adhikari, S., Friswell, M.I.: Design of MEMS piezoelectric harvesters with electrostatically adjustable resonance frequency. Mech. Syst. Sig. Process. 81, 360–374 (2016) 2. Abdelkefi, A., Najar, F., Nayfeh, A.H., Ayed, S.B.: An energy harvester using piezoelectric cantilever beams undergoing coupled bending-torsion vibrations. Smart Mater. Struct. 20, 115007 (2011) 3. Fu, H., Yeatman, E.M.: Rotational energy harvesting using bi-stability and frequency upconversion for low-power sensing applications: theoretical modelling and experimental validation. Mech. Syst. Sig. Process. 125, 229–244 (2019). https://doi.org/10.1016/j.ymssp.2018. 04.043 4. Vocca, H., Neri, I., Travasso, F., Gammaitoni, L.: Kinetic energy harvesting with bistable oscillators. Appl. Energy 97, 771–776 (2012) 5. Arroyo, E., Badel, A., Formosa, F., Wu, Y., Qiu, J.: Comparison of electromagnetic and piezoelectric vibration energy harvesters: model and experiments. Sens. Actuators 183, 148– 156 (2012) 6. Wang, X., Liang, X., Shu, G., Watkins, S.: Coupling analysis of linear vibration energy harvesting systems. Freq. Anal. Vib. Energy Harvest. Syst. 1(1), 203–229 (2016) 7. Kluger, J.M., Sapsis, T.P., Slocum, A.H.: Robust energy harvesting from walking vibrations by means of nonlinear cantilever beams. J. Sound Vib. 341, 174–194 (2015) 8. Wang, Z.L., Lin, L., Chen, J., Niu, S., Zi, Y.: Triboelectric Nanogenerators. Springer, Cham (2016) 9. Jin, C., Kia, D.S., Jones, M., Towfighian, S.: On the contact behavior of micro-/nano-structured interface used in vertical-contact-mode triboelectric nanogenerators. Nano Energy 27, 68–77 (2016) 10. Zhang, C., Wang, Z.L.: Triboelectric Nanogenerators. Springer, (2016) 11. Niu, S., et al.: Theoretical study of contactmode triboelectric nanogenerators as an effective power source. Energy Environ. Sci. 6, 3576–3583 (2013) 12. Wang, S., Lin, L., Xie, Y., Jing, Q., Niu, S., Wang, Z.L.: Sliding-triboelectric nanogenerators based on in-plane charge-separation mechanism. Nano Lett. 13, 2226–2233 (2013) 13. Yang, Y., et al.: Single-electrode-based sliding triboelectric nanogenerator for self-powered displacement vector sensor system. ACS Nano 7(8), 7342–7351 (2013) 14. Fu, Y., Ouyang, H., Davis, R.B.: Nonlinear dynamics and triboelectric energy harvesting from a three-degree-of-freedom vibro-impact oscillator. Nonlinear Dyn. 92, 1985–2004 (2018) 15. Fu, Y., Ouyang, H., Davis, R.B.: Triboelectric energy harvesting from the vibro-impact of three cantilevered beams. Mech. Syst. Sig. Process. 121, 509–531 (2019) 16. Zhao, C., Yang, Y., Upadrashta, D., Zhao, L., Lund, H., Kaiser, M.J.: Design, modeling and experimental validation of a low-frequency cantilever triboelectric energy harvester. Energy 214, 118885 (2021) 17. Zhao, H., Ouyang, H.: A capsule-structured triboelectric energy harvester with stick-slip vibration and vibro-impact. Energy 235, 121393 (2021) 18. Yang, J., et al.: Broadband vibrational energy harvesting based on a triboelectric nanogenerator. Adv. Energy Mater. 4(6), 1–9 (2014)

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19. Saadatnia, Z., Asadi, E., Askari, H., Esmailzadeh, E., Naguib, H.E.: A heaving point absorberbased triboelectric-electromagnetic wave energy harvester: an efficient approach toward blue energy. Int. J. Energy Res. 42., 2431–2447 (2018) 20. Liu, G., Guo, H., Xu, S., Hu, C., Wang, Z.L.: Oblate spheroidal triboelectric nanogenerator for all-weather blue energy harvesting. Adv. Energy Mater. 9, 1900801 (2019) 21. Jiang, T., et al.: Robust swing-structured triboelectric nanogenerator for efficient blue energy harvesting. Adv. EnergyMater. 10(23), 2000064 (2020) 22. Jin, C., Kia, D.S., Jones, M., Towfighian, S.: On the contact behaviour of micro-/nanostructured interface used in vertical-contact-mode triboelectricnanogenerators. Nano Energy 27, 68–77 (2016) 23. Wang, S.H., Lin, L., Wang, Z.L.: Nanoscale triboelectric-effect-enabled energy conversion for sustainably powering portable electronics. Nano Lett. 12(12), 6339–6346 (2012) 24. Wang, S., Lin, L., Xie, Y., Jing, Q., Niu, S., Wang, Z.L.: Sliding-triboelectric nanogenerators based on in-plane charge-separation mechanism. Nano Lett. 13(5), 2226–2233 (2013) 25. Yang, J., et al.: Broadband vibrational energy harvesting based on a triboelectric nanogenerator. Adv. Energy Mater. 4(6), 1301322 (2014) 26. Wang, C., Zhang, Q., Wang, W., Feng, J.: A low-frequency, wideband quad-stable energy harvester using combined nonlinearity and frequency up-conversion by cantilever-surface contact. Mech. Syst. Sig. Process. 112, 305–318 (2018) 27. Silva, C.E., Gibert, J.M., Maghareh, A., Dyke, S.J.: Dynamic study of a bounded cantilevered nonlinear spring for vibration reduction applications: a comparative study. Nonlinear Dyn. 101(2), 893–909 (2020) 28. Niu, Y.S., et al.: Simulation method for optimizing the performance of an integrated triboelectric nanogenerator energy harvesting system. Nano Energy 8, 150–156 (2014) 29. Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-14023-6 30. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics, 2nd edn. Wiley-VCH, Weinheim (2004) 31. Boisseau, S., Despesse, G., Ricart, T., Defay, E., Sylvestre, A.: Cantilever-based electretenergy harvesters. Smart Mater. Struct. 20(10), 105013 (2011)

Energy Harvesting Study of Piezoelectric Vibration Harvester with Double Parallel Slender Structure Xiang Zhao(B) and Haotian Jiang School of Civil Engineering and Surveying, Southwest Petroleum University, Chengdu 610500, People’s Republic of China [email protected]

Abstract. In this paper, we use a cantilevered double parallel slender structure single deformation piezoelectric energy harvester as a model and combine it with the Lamb-Oseen vortex model, where the effect from fluid vortices is used as an external load, and a metal sheet is attached to the free end of the energy harvester for capturing the shear force generated by wind-generated vortices on the double beams model. The closed-form solution of the bending forced vibration of the piezoelectric energy harvester is solved by establishing the relevant model and deriving the equations. Euler- Bernoulli beam assumptions are used to develop a coupled electromechanical model for the harvester with an intermediate spring layer and a transverse damping is considered, and Green’s functions and Laplace transform techniques are used to solve the vibration equations for the coupled piezoelectric vibration system. By solving for the voltage as a function of Green’s functions and using Matlab software, we can obtain the functional relationship between the voltage of the harvester and the elastic coefficient of the interlayer and the position of the metal plate setting. Keywords: Piezoelectric energy harvester · Euler- Bernoulli beam model · Green’s function · Laplace transform · Vortex-induced vibration

1 Modeling of Piezoelectric Energy Harvester with Cantilevered Double Straight Beams 1.1 Mechanical Equilibrium Control Equations with Electrically Coupled Effects This paper investigates a cantilevered double-beam single-deformation piezoelectric energy harvester, which is subjected to external irregularly distributed loads P1 , P2 . As shown in Fig. 1, the energy harvester is based on a homogeneous Eulerian beam model, which consists of a combination of an upper piezoelectric layer and a lower structural layer of the beam system, and the length of the piezoelectric beams are assumed to be L.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 195–208, 2024. https://doi.org/10.1007/978-981-97-0554-2_15

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Fig. 1. Cantilevered double beams single deformation piezoelectric energy harvester

Fig. 2. Cross-section of piezoelectric energy harvester

Figure 2 shows a cross-section of a single-deformation piezoelectric energy harvester, where: bs is the width of the beam structural layer; bp is the width of the piezoelectric layer; hs is the width of the beam structural layer; hp is the thickness of the piezoelectric layer; ha is the distance from the neutral axis (NA) to the lowermost surface of the beam structural layer; hb is the distance from the neutral axis (NA) to the bottom of the piezoelectric layer, and hc is the distance from the neutral axis (NA) to the top surface of the piezoelectric layer. When we consider an air damping coefficient ca , the vibration control equations for a double beams system can be written as [1–3]: ∂ 2 wrel1 (x, t) ∂wrel1 (x, t) ∂ 2 M1 (x, t) +m + ca + K(w1 − w2 ) = P1 2 x ∂t ∂t 2

(1)

∂ 2 wrel2 (x, t) ∂wrel2 (x, t) ∂ 2 M2 (x, t) + m + c + K(w2 − w1 ) = P2 a x2 ∂t ∂t 2

(2)

where ∂wrel (x, t) is the transverse deflection of the beam, M (x, t) is the internal bending moment of the beam, ca is the viscous air damping coefficient, and m is the mass per unit length of the beam.

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In order to obtain an expression for the internal bending moment M (x, t) of the beams, we can utilize the intrinsic relationship between the piezoelectric layer and the structural layer of the beams [4], and brought into the Eqs. (1)(2): ∂ 4 wrel1 (x, t) ∂wrel1 (x, t) ∂ 2 wrel1 (x, t) + ca + K(w1 − w2 ) +m 4 ∂t ∂t 2   ∂x d δ(x − x1 ) d δ(x − x2 ) = P1 +ϑv1 (t) − dx dx

(3)

∂ 2 wrel2 (x, t) ∂ 4 wrel2 (x, t) ∂wrel2 (x, t) + m + c + K(w2 − w1 ) a 4 ∂t ∂t 2   ∂x d δ(x − x1 ) d δ(x − x2 ) − = P2 +ϑv2 (t) dx dx

(4)

(EI )eff

(EI )eff

where: δ(x) is the Dirac function, (EI )eff is the bending stiffness of the composite cross-section, ϑ is the coupling coefficient, they can be expressed as: (EI )eff =

Es bs (h3b − h3a ) + Ep bp (h3c − h3b ) 3

ϑ =−

Ep bp d31 2 (hc − h2b ) 2hp

(5) (6)

1.2 Control Equations for Circuits with Mechanical Coupling Effects Equations (3)(4) are the vibration control equations for the double beams system under electrical coupling, we utilize the following piezoelectric intrinsic relations [4]: s Dy (x, t) = d31 Ep εxx (x, t) − ε33

v(t) hp

(7)

s is the where Dy (x, t) is the potential shift parallel to the beam thickness direction, ε33 dielectric constant, and εxx (x, t) is the average bending strain. The bending deformation of the structure will generate a potential shift in the piezoelectric layer, which will be collected by the electrodes, and the charge can be obtained by integrating the potential shift over the electrode region q(t).Since the current ii (t) is related to the capacitance, the voltage across the resistor can be expressed as [4]:  x2  s b (x − x ) ε33 ∂ 2 wreli (x, t) p 2 1 dvi (t) dx − (8) d31 Ep hpc bp vi (t) = Rli ii (t) = −Rli ∂x2 ∂t hp dt x1

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2 Green’s Function Solutions for Piezoelectric Vibrations of Double Straight Beams 2.1 Vibration and Piezoelectric Equations Solving Assuming that the external transverse loads are time-harmonic loads, we can let the deflections and voltages take a similar form to the following, separating the time parameters from the displacements: pi (x, t) = Pi (x)eit , w(x, t) = W (x)eit , v(t) = Veit

(9)

Substitution of Eq. (9) into Eqs. (3) (4) and (8): W1 + a2 W1 + a4 W2 = b2 P1 (x) − b3 V1 [δ  (x − x1 ) − δ  (x − x2 )]

(10)

W2 + c2 W2 + c4 W1 = d2 P2 (x) − d3 V2 [δ  (x − x1 ) − δ  (x − x2 )]

(11)

iCp Rl + 1 Vi = −iβ Rl



x2 x1

Wi (x)dx

(12)

where: ica − μ1 2 + K K 1 , a4 = − , b2 = , b3 = E1 I1 E1 I1 E1 I1 ica − μ2 2 + K K 1 c2 = , c4 = − , d2 = , d3 = E2 I2 E2 I2 E2 I2 a2 =

ϑ E1 I1 ϑ E2 I2

(13)

For a linear system, the principle of superposition should be satisfied. Therefore, the dynamic response of a double-beam system subjected to loads P1 (x) and P2 (x) is the sum of the responses of the systems subjected to P1 (x) and P2 (x) respectively. This shows that the solution of Eqs. (10) and (11) is the sum of the solutions of the following four cases. Case 1: W1 + a2 W1 + a4 W2 = b2 δ(x − x0 )

(14)

W2 + c2 W2 + c4 W1 = 0

(15)

where δ(·) is the Dirac delta function [5, 6] and x0 denotes the location where the unit harmonic load acts. Laplace transformations of Eqs. (14), (15): Q(s)W 1 (s) = (s4 + c2 )(s3 W1 (0) + s2 W1 (0) + sW1 (0) + W1 (0) + b2 e−sx0 ) −a4 (s3 W2 (0) + s2 W2 (0) + sW2 (0) + W2 (0)) Q(s)W 2 (s) = (s4 + a2 )(s3 W2 (0) + s2 W2 (0) + sW2 (0) + W2 (0)) −c4 (s3 W1 (0) + s2 W1 (0) + sW1 (0) + W1 (0) + b2 e−sx0 )

(16)

(17)

Energy Harvesting Study of Piezoelectric Vibration Harvester

Q(s) = (s4 + a2 )(s4 + c2 ) − a4 c4

199

(18)

Divide Q(s) to the right end of the equations and perform an inverse transformation of W 1 (s) and W 2 (s): G11 (x, x0 ) = H (x − x0 )φ11 (x − x0 ) + φ21 (x)W1 (0) + φ31 (x)W1 (0) + φ41 (x)W1 (0) +φ51 (x)W1 (0) + φ61 (x)W2 (0) + φ71 (x)W2 (0) + φ81 (x)W2 (0) + φ91 (x)W2 (0) (19) G12 (x, x0 ) = H (x − x0 )φ12 (x − x0 ) + φ22 (x)W1 (0) + φ32 (x)W1 (0) + φ42 (x)W1 (0) +φ52 (x)W1 (0) + φ62 (x)W2 (0) + φ72 (x)W2 (0) + φ82 (x)W2 (0) + φ92 (x)W2 (0) (20) The expression of ϕ are displayed in Appendix A. Using the boundary conditions at x = 0, The terms in the Green’s functions with a coefficient of 0 can be removed: G11 (x, x0 ) = H (x − x0 )φ11 (x − x0 ) + φ41 (x)W1 (0)

+φ51 (x)W1 (0) + φ81 (x)W2 (0) + φ91 (x)W2 (0)

G12 (x, x0 ) = H (x − x0 )φ12 (x − x0 ) + φ42 (x)W1 (0) +φ52 (x)W1 (0) + φ82 (x)W2 (0) + φ92 (x)W2 (0)

(21) (22)

Using the boundary conditions at the end of x = L, we can derive the second and third order derivatives of the above equations to obtain the following matrix equation: ⎡  ⎡  (L) φ  (L) φ  (L) ⎤ ⎡ W (0) ⎤  (L − x ) ⎤ φ41 (L) φ51 −φ11 1 0 81 91 ⎢ φ  (L) φ  (L) φ  (L) φ  (L) ⎥ ⎢ W  (0) ⎥ ⎢ −φ  (L − x0 ) ⎥ 81 91 51 ⎢ 41 ⎥·⎢ 1 ⎥ = ⎢ 11 ⎥ (23) ⎣ φ  (L) φ  (L) φ  (L) φ  (L) ⎦ ⎣ W  (0) ⎦ ⎣ −φ  (L − x0 ) ⎦ 42

52

82

92

 (L) φ  (L) φ  (L) φ  (L) φ42 82 92 51

2

W2 (0)

12

 (L − x ) −φ12 0

Case 2: The Green’s functions for case 2 can be obtained by solving the following equations: W1 + a2 W1 + a4 W2 = 0

(24)

W2 + c2 W2 + c4 W1 = d2 δ(x − x0 )

(25)

The solution process is similar to case 1, its Green’s functions are expressed as: G21 (x, x0 ) = H (x − x0 )φ 11 (x − x0 ) + φ41 (x)W1 (0) +φ51 (x)W1 (0) + φ81 (x)W2 (0) + φ91 (x)W2 (0) G22 (x, x0 ) = H (x − x0 )φ 12 (x − x0 ) + φ42 (x)W1 (0)

+φ52 (x)W1 (0) + φ82 (x)W2 (0) + φ92 (x)W2 (0)

(26) (27)

The expression of ϕ are displayed in Appendix A. According to the superposition principle, the displacement solution can be expressed as:     L G (x; ξ ) G21 (x; ξ ) W11 (x) 11 (28) = P1 (ξ ) + P2 (ξ )d ξ W21 (x) G12 (x; ξ ) G22 (x; ξ ) 0

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Case 3: The Green’s functions for case 3 can be obtained by solving the following equations: W1 + a2 W1 + a4 W2 = b3 V1 δ  (x − x0 )

(29)

W2 + c2 W2 + c4 W1 = 0

(30)

The Green’s functions can be solved in a similar way: G31 (x, x0 ) = V1 H (x − x0 )φ 11 (x − x0 ) + φ41 (x)W1 (0) +φ51 (x)W1 (0) + φ81 (x)W2 (0) + φ91 (x)W2 (0) G32 (x, x0 ) = V1 H (x − x0 )φ 12 (x − x0 ) + φ42 (x)W1 (0)

+φ52 (x)W1 (0) + φ82 (x)W2 (0) + φ92 (x)W2 (0)

(31)

(32)

The expression of ϕ are displayed in Appendix A. Case 4: The Green’s functions for case 4 can be obtained by solving the following equations: W1 + a2 W1 + a4 W2 = 0

(33)

W2 + c2 W2 + c4 W1 = d3 V2 δ  (x − x0 )

(34)

The Green’s functions can be solved in a similar way: G41 (x, x0 ) = V2 H (x − x0 )φˆ 11 (x − x0 ) + φ41 (x)W1 (0) +φ51 (x)W1 (0) + φ81 (x)W2 (0) + φ91 (x)W2 (0)

(35)

G42 (x, x0 ) = V2 H (x − x0 )φˆ 12 (x − x0 ) + φ42 (x)W1 (0) +φ52 (x)W1 (0) + φ82 (x)W2 (0) + φ92 (x)W2 (0)

(36)

The expressions of ϕˆ are displayed in Appendix A. According to the superposition principle, the displacements W12 and W22 can be expressed by the Green’s functions G31 , G32 , G41 , G42 and can be expressed as the following volume integrals.    L G (x; ξ ) W12 (x) 31 = [δ(ξ − x2 ) − δ(ξ − x1 )] W22 (x) G32 (x; ξ ) 0  (37) G41 (x; ξ ) + [δ(ξ − x2 ) − δ(ξ − x1 )]d ξ G42 (x; ξ )

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2.2 Decoupling of Electromechanical Eulerian Double Beams Systems According to the principle of superposition of linear systems, the steady state displacement W1 can be divided into W11 , W12 two parts; similarly W2 can be divided into W21 , W22 two parts. So, the expressions for W1 and W2 are:  L W1 = W11 + W12 = G11 (x; ξ )P1 (ξ ) + G21 (x; ξ )P2 (ξ )d ξ 0 (38)  L +

G31 (x; ξ )[δ(ξ − x2 ) − δ(ξ − x1 )] + G41 (x; ξ )[δ(ξ − x2 ) − δ(ξ − x1 )]d ξ  L W2 = W21 + W22 = G12 (x; ξ )P1 (ξ ) + G22 (x; ξ )P2 (ξ )d ξ 0

0



L

+

(39)

G32 (x; ξ )[δ(ξ − x2 ) − δ(ξ − x1 )] + G42 (x; ξ )[δ(ξ − x2 ) − δ(ξ − x1 )]d ξ

0

Substituting Eqs. (38) (39) into Eq. (8) yields a functional relationship between the voltage and the Green’s functions for the four cases: m1 V1 = m2 + m3 V2 n1 V2 = n2 + n3 V1 

x2 iCp Rl + 1   + iβ G 31 (x; x2 ) − G 31 (x; x1 )dx m1 = Rl

(40)

x1

x2 

L

m2 = −iβ

G11 (x; ξ )P1 (ξ ) + G21 (x; ξ )P2 (ξ )d ξ

dx

0

x1 x2

m3 = −iβ







G 41 (x; x2 ) − G 41 (x; x1 )dx x1

n1 =

iCp Rl + 1 Rl x2 

n2 = −iβ

x2 + iβ

(41) 



G 42 (x; x2 ) − G 42 (x; x1 )dx x1

 G12 (x; ξ )P1 (ξ ) + G22 (x; ξ )P2 (ξ )d ξ

dx

0

x1 x2

n3 = −iβ

L







G 32 (x; x2 ) − G 32 (x; x1 )dx x1

The expressions for voltage we can easily derive from the algebraic Eq. (40) through the linear expressions for voltage V1 and voltage V2 : V1 =

m2 n1 + m3 n2 m1 n2 + m2 n3 , V2 = m1 n1 − m3 n3 m1 n1 − m3 n3

(42)

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3 Eddy-Current Induced Vibration of Piezoelectric Energy Harvester

Fig. 3. Schematic diagram of a piezoelectric energy harvester subjected to vortex shedding aerodynamic loads

As shown in Fig. 3, vortex shedding will generate aerodynamic loads. The LambOseen vortex model [7], which is used in this system, produces a vertical load that can be expressed as [8]: 1 Fv = − Wf ρa 2

 0

Lf

(vc r −

2 r 2 )dx 4

(43)

where ρa is the air density; Wf and Lf are the width and length of the sheet, respectively; D is the diameter of the solid cylinder, which is an immovable obstacle; U0 is the mean fluid velocity; r = ((x − xc )2 + yc2 )0.5 is the distance from the point A(x, 0) on the sheet to the center C(xc , yc ) of the vortex, where (xc , yc ) is the position and vc is the velocity of the center of the vortex; and the vortex strength is: = ( 0 /4π ve t)exp(−r02 /4ve t)

(44)

where 0 = (U0 D)/2St is the initial velocity cycle, St is the Strouhal number related to the Reynolds number, r0 is the radius of the rigid vortex core [8], ve is the equivalent dissipation factor of the vortex, t = dr /U0 , dr is the relative distance between the vortex center and the plate. In this study, the Reynolds number of St = 0.21 ranges from 103 to 105 , corresponds to a flow rate of 0.05 to 5 m/s. In addition, U0 can be expressed by the velocity vc [8].  When the vortex center is located at yc = 1.3 × D 2, the vortex coming off the cylinder is stable. When the vortex moves to the middle of the  thin lobe, the equivalent harmonic force Fv reaches its maximum value, i.e. xc = Lf 2, dr = d . Therefore, the expression of Fmax [8]:   Lf Lf Lf 1 1.3D 2 2 1.3D 2 Fmax = − Wf ρa ) − [(x − )2 + ( ) ]}dx { vc (x − )2 + ( 2 2 2 4 2 2 0 (45)

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Substituting into Eq. (42) gives the expression for the voltage: iβ V1 = −

x2     (x; L)(1 − λ)F G11 (x; L)λFmax + G21 max dx × n1

x1

m1 n1 − m3 n3 x2     (x; L)(1 − λ)F m3 × iβ G12 (x; L)λFmax + G22 max dx x1

−

m1 n1 − m3 n3 m1 × iβ

V2 = −

x2     (x; L)(1 − λ)F G12 (x; L)λFmax + G22 max dx

x1

m1 n1 − m3 n3 x2     (x; L)(1 − λ)F iβ G11 (x; L)λFmax + G21 max dx × n3

−

(46)

(47)

x1

m1 n1 − m3 n3 V = V1 + V2

(48)

4 Numerical Analysis Through the computational derivation of the theory, we can get the expression of the voltage generated by the base model under vortex-induced vibration. In the following, we will explore the effects of different external excitation distributions and elasticity coefficients of springs on the voltage by setting up two sub-models without changing the boundary conditions and Green’s functions, respectively. 4.1 Effect of External Excitation Distribution on Voltage Since the position of the metal sheet in this model can be moved up and down, thus the magnitude of the external excitation force received by the two beams in the vibration model of the cantilevered double beams system can be changed, as shown in Fig. 4. We modeled the data by Matlab software, and after numerical analysis, we can get the magnitude of the voltage collected by the piezoelectric trap in different cases. As shown in Figs. 5, 6, 7. In this model λ is the external excitation distribution coefficient. Therefore the external excitation forces on the two beams are Fv1 = λFv and Fv2 = (1 − λ)Fv respectively. Figures 5 and 6 show us the relationship between the voltage of the piezoelectric energy harvester and the frequency of the external excitation force, respectively. Since the physical properties of the two beams are set up the same way, the two beams produce peak voltages at the same frequency of the external excitation force, which is 3388 Hz as well as 9317 Hz. When we change the Lambda (i.e., we change the magnitude of the external excitation force applied to the two beams), the performance of the piezoelectric energy

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Fig. 4. Schematic diagram of piezoelectric energy harvester subjected to vortex shedding aerodynamic loads (without considering the spring layer)

Fig. 5. Variation of V1 with aerodynamic load excitation frequency for different lambda

Fig. 6. Variation of V2 with aerodynamic load excitation frequency for different lambda

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Fig. 7. Variation of piezoelectric energy harvester Voltage with Lambda when  = 2661

harvester’s power generation changed, but it did not affect the frequency magnitude of the external excitation force corresponding to the peak voltage. From Fig. 7, it can be seen that the voltage of the piezoelectric energy harvester varies linearly with the magnitude of the external excitation force for a certain frequency of the external excitation force, and the sum of the voltages generated by the two beams is a constant value. 4.2 Influence of the Elasticity Coefficient of the Middle Layer on the Voltage of a Piezoelectric Energy Harvester In this model, as shown in Fig. 8. The two piezoelectric energy harvesters have a spring layer with an elasticity coefficient of K between them. An external excitation force of magnitude both 0.5Fv is transmitted to the piezoelectric energy harvesters by setting two identical metal sheets mounted on the cantilever end of each of the two beams. Thus, the relationship between the elasticity coefficient of the middle layer and the voltage of the piezoelectric energy harvester is explored, as shown in Fig. 9.

Fig. 8. Schematic of a piezoelectric energy harvester with an intermediate layer subject to vortex shedding aerodynamic loading

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Fig. 9. Variation of piezoelectric energy harvester voltage with elasticity coefficient of the middle layer,  = 2661.

Figure 9 shows the change rule of piezoelectric energy harvester voltage with the elasticity coefficient of the intermediate layer K when  = 2661. From this figure, it can be seen that: the power generation is highest when the intermediate layer is not added; when the elasticity coefficient is in the range of 0–0.7, the power generation efficiency of the double-beam system is fluctuating; when the elasticity coefficient is greater than 0.7, the constraint capacity of the double-beam system is strong, and the voltage fluctuates in a small range but the overall are smaller than the voltage of not adding the spring layer The voltage of the double-cantilever beams system fluctuates in a small range, but is generally smaller than that without the spring layer. Thus, the addition of an intermediate spring layer to the energy harvester of the double cantilever beams has an inhibiting effect on the power generation efficiency, and when the elasticity coefficient is too large, the generated voltage will gradually converge to a constant value.

5 Conclusions In this paper, we build a double parallel cantilever energy harvesting system by using the vortex excitation force generated by vortex shedding as the external load. We discuss the effects of different sizes of external loads on the power generation efficiency of the dual-beam system; we also explore the effects on the voltage magnitude with the addition of an intermediate spring layer. The following are the main conclusions of this paper. (1) The voltage of each beam of the energy harvester of the double-parallel structure varies linearly with the magnitude of the external excitation force, but does not affect the frequency of the external excitation force corresponding to the peak voltage. (2) The addition of an intermediate layer will reduce the power generation efficiency of the double-parallel cantilever energy harvester to a certain extent, and the voltage will be bullied to fluctuate; when the elasticity coefficient of the intermediate layer is large, the voltage is suppressed and tends to be constant.

Energy Harvesting Study of Piezoelectric Vibration Harvester

Appendix A (k)

φ 11 (x) = −

8 

8 

(k)

sik Ai (x)a4 d2 ; φ 12 (x) =

i=1

φ 11 (x) =

8 

sik Ai (x)(si4 + a2 )d2

i=1

Ai (x)(si4 + c2 )b3 si ; φ 12 (x) = −

i=1

φˆ 11 (x) = −

8 

φ31 (x) =

8 

8 

Ai (x)a4 d3 si ; φˆ 12 (x) =

i=1

Ai (x)(si4 + c2 )b2 , φ21 (x) =

8 

i=1

i=1 8 

Ai (x)(si4 + c2 )si2 , φ41 (x) =

i=1

φ51 (x) =

Ai (x)(si4 + a2 )d3 si

8 

8 

Ai (x)(si4 + c2 ), φ61 (x) = −

i=1

φ71 (x) = − φ91 (x) = −

Ai (x)c4 b3 si

i=1

i=1

φ11 (x) =

8 

i=1 8 

Ai (x)(si4 + c2 )si3 Ai (x)(si4 + c2 )si Ai (x)a4 si3

i=1

8  i=1 8 

Ai (x)a4 si2 , φ81 (x) = −

8 

Ai (x)a4 si

i=1

Ai (x)a4

i=1

φ12 (x) = − φ32 (x) = −

8 

Ai (x)c4 b2 , φ22 (x) = −

i=1

i=1

8 

8 

Ai (x)c4 si2 , φ42 (x) = −

i=1

φ52 (x) = −

8 

8 

Ai (x)c4 , φ62 (x) =

φ92 (x) =

8 

Ai (x)c4 si

Ai (x)(si4 + a2 )si3

i=1

Ai (x)(si4 + a2 )si2 , φ82 (x) =

i=1 8 

Ai (x)c4 si3

i=1

i=1

φ72 (x) =

8 

8 

Ai (x)(si4 + a2 )si

i=1

Ai (x)(si4 + a2 )

i=1

Ai (x) =

esi x (i = 1 ∼ 8) (si − s1 )..(si − si−1 )(si − si+1 )..(si − s8 )

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References 1. Stojanovi´c, V., Kozi´c, P., Pavlovi´c, R., et al.: Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading. Arch. Appl. Mech. 81(12), 1993–2005 (2011) 2. Xiaobin, L.I., Shuangxi, X.U., Weiguo, W.U., et al.: An exact dynamic stiffness matrix for axially loaded double-beam systems. Sadhana 39(3), 607–623 (2014) 3. Zhao, X., Chen, B., Li, Y.H., et al.: Forced vibration analysis of Timoshenko double-beam system under compressive axial load by means of Green’s functions. J. Sound Vib. 464, 115001 (2020) 4. Erturk, A., Inman, D.J.: A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J. Vib. Acoust. 130(4) (2008) 5. Abu-Hilal, M.: Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions. J. Sound Vib. 267(2), 191–207 (2003) 6. Li, X.Y., Zhao, X., Li, Y.H.: Green’s functions of the forced vibration of Timoshenko beams with damping effect. J. Sound Vib. 333(6), 1781–1795 (2014) 7. Erratum [M]. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics, pp. 777–780. Springer, Heidelberg (2006). https://doi.org/10.1007/978-3-540-29028-5_13 8. Hu, Y., Yang, B., Chen, X., et al.: Modeling and experimental study of a piezoelectric energy harvester from vortex shedding-induced vibration. Energy Convers. Manage. 162, 145–158 (2018)

A Broadband Energy Harvester with Three-to-One Internal Resonance Le Yang1 , Wenan Jiang1(B) , Xingjian Jing2 , and Liqun Chen3 1

2

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China [email protected] Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China 3 Department of Mechanics, Shanghai University, Shanghai 200072, China

Abstract. In this study, we present a novel two-degree-of-freedom (TDOF) nonlinear energy harvester with internal resonance. To show the performance, a TDOF nonlinear electromagnetic harvester is designed and the mathematical model is derived. The electromechanical coupling system is solved by adopting the harmonic balance method, and the first-order harmonic solutions of the system are provided. The displacement and current frequency response curves are created along with the modulation equations. The advantage of the proposed harvester is that compared to the conventional single-degree-of-freedom (SDOF) nonlinear model and the corresponding TDOF linear system, the results achieve that the proposed scheme can enhance the bandwidth of the harvesting energy. The influences of different excitation amplitudes f1 and f2 on the response are discussed. The accuracy of the analytical first-order harmonic results is demonstrated by numerical simulations, and the existence of multiple periodic solutions for time history at the same external force frequencies is quantitatively proved.

Keywords: Vibration energy harvesting resonance · Nonlinearity

1

· Broadband · Internal

Introduction

Vibration energy harvesting has emerged as a promising technology for powering wireless sensors and other low-power electronics. Conventional approaches to energy harvesting typically focus on narrowband systems, which are limited in their ability to capture energy across a wide range of frequencies. While a number of nonlinear techniques have proven effective in resolving the bandwidth limitation problem faced by linear harvesters. In light of this, numerous review papers on nonlinear energy harvesting exist, including Tang et al. [1], Pellegrini et al. [2], Harne and Wang [3], Daqaq et al. [4], Wei and Jing [5], Yang et al. [6] and Yang et al. [7]. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 209–220, 2024. https://doi.org/10.1007/978-981-97-0554-2_16

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When nonlinearities are presented in a multi-modal harvester, internal resonance is a possibility as long as the linear natural frequencies of the system are nearly commensurable. Internal resonance causes energy transfer between the modes, enhancing energy transmission and expanding the frequency range where energy can be harvested. Since then, there are numerous studies on nonlinear multiple modes, such as two-to-one internal resonance [8–20], three-to-one internal resonance [21,22], and one-to-one internal resonance [23]. Despite having many peaks, the linear multi-degree-of-freedom system only performs efficiently when harvesting near the natural frequency. To date, how to achieve the harvester to realize broadband oscillations remains an important problem in harvester design. Wu et al. [24] designed a two-degree-of-freedom nonlinear harvester and demonstrated the proposed scheme can achieve wider bandwidth between two natural frequencies. To emphasize this scheme, this paper implements the TDOF nonlinear technique to enhance the bandwidth issue of the energy harvester. Structures with geometric nonlinearities, such as beams and plates, are typically modeled as Duffing oscillators with symmetric potential wells, and it has been reported that three-to-one resonance results in the effective energy transfer between the modes [25]. It is reported that modulated motions can happen at excitation frequencies close to the lower mode frequency if the natural frequencies are almost commensurate, which is the main characteristic of three-to-one internal resonance. Chin and Nayfeh [26] investigated the first and second modes of hinged-clamped beams, and employed the method of multiple scales to obtain the response of the system under the three-to-one internal resonances. Nayfeh et al. [27] applied the perturbation method and determined the response of the integral partial differential equations for the three-to-one and one-to-one internal resonances. Additionally, Emam and Nayfeh [28] studied the nonlinear response of a buckled beam under a three-to-one internal resonance between the first and the third modes. Riedel and Tan [29] reported the forced response of a nonlinear axially moving strip with the coupled transverse motion and created the threeto-one internal resonance between the first two transverse modes. Pakdemirli and Ozkaya [30] investigated the three-to-one internal resonances case for a continuous system with cubic nonlinearity. Oz and Ozkaya [31] analyzed the transverse vibrations of a simply supported curved beam resting on a non-linear elastic foundation. In light of three-to-one internal resonance, Wang et al. [32] investigated the influence of the elastic foundation on the response of the beam. Hou et al. [33] established a two-degree-of-freedom model consisting of an NES and a coupled linear oscillator to form the three-to-one internal resonance system. Wei et al. [34] considered the three-to-one internal resonance in a two-beam structure connected with nonlinear joints. Xu et al. [35] explored the three-to-one internal resonance analysis for a suspension bridge with spatial cable through a continuum model. However, three-to-one internal resonance is rarely reported in energy harvesting techniques, so it is necessary to study the mechanism of its vibration.

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211

The primary motivation of this paper is to present a novel TDOF nonlinear energy harvester with three-to-one internal resonance that exhibits a wide frequency range and double peaks characteristics as a result of its cubic nonlinearity configuration. The second motivation is to reveal the nature of TDOF with cubic nonlinearity under periodic motion using analytical investigation and numerical verification. Harmonic balancing analysis demonstrates the characteristic that the amplitude-frequency response curve contains wide frequency double peaks. An electromagnetic energy harvester is designed, as shown in Fig. 1, to demonstrate this capability. After selecting the appropriate physical parameters, the amplitude-frequency curve has a broader bandwidth than conventional harvesters.

2

Mathematical Model

The physical model of a TDOF nonlinear electromagnetic energy harvester is exhibited in Fig. 1, the system comprises of two nonlinear mass-spring oscillators with linear stiffness coupling. In the meanwhile, a magnet is fixed on the frame, and linked to the mass Mx via a spring. Under the external force F1,2 cos ωt, the coil follows the mass Mx to a motion relative to the magnet. Based on Faraday’s law of electromagnetic induction, this system generates an electric current. The dynamical equations of the coupling system can be written as ¨ + Dx x˙ + Kx x + K3x x3 + Dxy (x˙ − y) ˙ + Kxy (x − y) + BILcoil = f1 cos ωt (1) Mx x My y¨ + Dy y˙ + Ky y + K3y y 3 + Dxy (y˙ − x) ˙ + Kxy (y − x) = f2 cos ωt

(2)

Lind I˙ + RI − BLcoil x˙ = 0

(3)

where x, y are the output displacements, Mx , My are the structural masses, Dx , Dy , Dxy are the damping coefficients, Kx , Ky , Kxy are the linear stiffness coefficients, K3x , K3y are the cubic nonlinearity coefficients, I is the current, R is the resistive load, B is the magnetic strength, Lcoil is the coil length, Lind is the inductance, F1,2 cos ωt is the external forces. Further, if the electric coupling and damping are disregarded, the equations of motion can be simplified to ¨ + (Kx + Kxy )x − Kxy y = 0 Mx x

(4)

My y¨ − Kxy x + (Ky + Kxy )y = 0

(5)

x = u1 f (t), y = u2 f (t)

(6)

we set

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Taking Eq. (6) into Eq. (4) and (5), the equations can be derived as Mx u1 f¨(t) + (Kx + Kxy )u1 f (t) − Kxy u2 f (t) = 0

(7)

My u2 f¨(t) − Kxy u1 f (t) + (Ky + Kxy )u2 f (t) = 0

(8)

then

−

f  (t) −kxy u2 + (kx + kxy )u1 −kxy u1 + (kx + kxy )u2 = = =λ f (t) Mx u 1 My u 2

(9)

We assume λ = ω 2 and take it into the Eq. (9) (Kx + Kxy − Mx ω 2 )u1 − Kxy u2 = 0

(10)

(Ky + Kxy − My ω 2 )u2 − Kxy u1 = 0

(11)

One can solve the natural frequencies 2 ω1,2 =

1 [Mx (Ky + Kxy ) + My (Kx + Kxy ) 2Mx My   2 2 ∓ (Mx (Ky + Kxy ) − My (Kx + Kxy )) + 4Mx My Kxy

(12)

To achieve three-to-one resonance between the first two modes, the parameters of the proposed harvester adopted in the present work are chosen as Mx = My = 1kg, Kx = Ky = 1N/m, Kxy = 4N/m. For subsequent calculations, the system (1–3) can be dimensionless as

where

x ¨ + ξ11 x˙ − ξ12 y˙ + k11 x − k12 y + αx3 + θI = f1 cos ωt

(13)

y¨ + ξ21 x˙ − ξ22 y˙ + k22 y − k21 x + βy 3 = f2 cos ωt

(14)

I˙ + λI − χx˙ = 0

(15)

D +D

D

K +K

K

3x ξ11 = xMx xy , ξ12 = Mxy , k11 = xMx xy , k12 = Mxy ,α = K Mx , x x Dy +Dxy Dxy Ky +Kxy Kxy K3y ξ21 = My , ξ22 = My , k21 = My , k22 = My , β = My , BLcoil F1 F2 R coil , f2 = M , θ = BL f1 = M Mx , λ = Lind , χ = Lind . x y

A Broadband Energy Harvester with Three-to-One Internal Resonance F1cosωt

213

F2cosωt

KxθK3x

KyθK3y

Coil

S

N

R

S

N

My

Mx

Magent Circuit

Dy

Dxy

Dx

x

y

Fig. 1. Schematics of a nonlinear TDOF energy harvester

3

Dynamical Analysis

This section introduces the dynamical response of the system, and the first-order harmonic response is assumed as x = a1 cos ωt + b1 sin ωt, y = a2 cos ωt + b2 sin ωt

(16)

where a1,2 and b1,2 are the magnitudes of the sine and cosine functions, respectively. Taking Eq. (16) into Eq. (15), the steady current can be derived as I=

χω [(a1 ω + λb1 ) cos ωt + (b1 ω − λa1 ) sin ωt] ω 2 + λ2

(17)

Inserting Eqs. (16) and (17) into Eqs. (13) and (14), yields the first-order harmonic responses − a1 ω 2 + ξ11 b1 ω + k11 a1 + 0.75αa31 + 0.75αa1 b21 − k12 a2 − ξ12 b2 ω +

θχω(a1 ω + λb1 ) = f1 ω 2 + λ2

(18)

− b1 ω 2 − ξ11 a1 ω + k11 b1 + 0.75αb31 + 0.75αa21 b1 − k12 b2 + ξ12 a2 ω +

θχω(b1 ω − λa1 ) =0 ω 2 + λ2

(19)

− a2 ω 2 + ξ21 b2 ω + k21 a2 + 0.75βa32 + 0.75βa2 b22 − k22 a1 − ξ22 b1 ω = f2 − b2 ω − ξ21 a2 ω + k21 b2 + 2

0.75βb32

+

0.75βa22 b2

+ k22 b1 + ξ22 a1 ω = 0

and the relation between current and displacement amplitude is  λχω a21 + b21 I= ω 2 + λ2

(20) (21)

(22)

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In the meantime, the output power can be constructed as P = I 2R =

λ3 χ2 ω 2 (a21 + b21 ) (ω 2 + λ2 )

2

(23)

To show the broadband characteristics of the proposed scheme, we refer to the results of amplitude-frequency response curve in the literature [36] and choose the physical parameters as Mx = My = 1kg, Dx = Dy = Dxy = 0.01N · s/m, Kx = 1.21N/m, Ky = 3.24N/m, Kxy = 1N/m, K3x = 1N/m, K3y = 0.5N/m, Lind = 1H, Lcoil = 1m, B = 1T , R = 10Ω, f1 = 0.02N , f2 = 0.4N . Consequently, the two natural frequencies of the system are derived as 1 and 3 s−1 , and the displacement and current curves are plotted in Fig. 2 for two sets of different excitation amplitudes f1 and f2 , respectively. As demonstrated in Fig. 2(a)-(b), the peak of resonance near the first natural frequency gradually increases as the growth of the excitation amplitude f1 , while the peak of resonance near the second natural frequency is a slight decrease. Moreover, the response of the proposed scheme with two peaks that bend to the right, results in a hardening spring characteristic. In addition, more than one solution can exist for some excitation frequency marked by solid and dotted lines, where the solid line denotes the steady solution, and the dotted line describes the unsteady solution. The amplitude and bandwidth enlarge as the excitation amplitude f2

Fig. 2. The displacement and the current versus the frequency for different f1 and f2 .

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increases, as shown in Fig. 2(c)-(d). Meanwhile, it is found that for small excitation amplitudes f2 = 0.1 and f2 = 0.2 there is only one solution while for large value f2 = 0.4 there are two resonance ranges with three solutions. Thus, the frequency curves of the system with internal resonance have a broadband characteristic of bending to the right near both natural frequencies.

4

Comparison of the Proposed Scheme and the Traditional Cases

In this section, we compare the performance of the proposed scheme with that of two conventional cases, the SDOF nonlinear system and the TDOF linear case. Figure 3 compares the displacement and current for the proposed scheme and the traditional SDOF nonlinear system as a function of frequency. This figure demonstrates that the response of the proposed scheme has two peaks and is characteristic of a hardening spring, causing the responses to bend to the right. Additionally, the system’s amplitude and resonance band are significantly widened as a result of the coupling and nonlinear interaction of My . As a result, the responses of the proposed scheme have both larger amplitude and frequency bandwidth than the traditional SDOF nonlinear case. Figure 4 further details the comparison of periodic solutions between the proposed scheme and the TDOF linear results. It is discovered that the linear system only has two peaks near two natural frequencies, while the nonlinear system has a response that curves to the right. The proposed scheme’s solution, in contrast to the linear system’s response, can take on multiple values. Consequently, we draw the conclusion that the TDOF nonlinear energy harvester with internal resonance has a much wider bandwidth than that of the traditional resonance cases.

Fig. 3. Comparisons of the TDOF nonlinear harvester and the SDOF nonlinear case.

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Numerical Validations

To evaluate the accuracy of analytical solutions, Eqs. (13)–(15) are numerically implemented by adopting the fourth-order Runge-Kutta method. Obtained results from Eqs. (13)–(15) are described in Fig. 5 marked by red and green solid

Fig. 4. Comparisons of the TDOF nonlinear harvester and the TDOF linear case.

Fig. 5. Comparisons of analytical solutions and numerical results.

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lines, and the analytical solutions are marked with the black solid line. It is discovered that a degree of a good consistency for the frequency response curve of periodic solutions. Moreover, the jump phenomenon of bending to the right is qualitatively verified, and the existence of two peaks is quantitatively proven. In the meantime, the displacement and current time history curves for various external force frequencies are shown in Figs. 6 and 7. As can be seen, multiple periodic solutions can exist for displacement and current time history in the same external force frequencies that are getting close to the resonance frequency.

Fig. 6. Time-history response of frequency Ω = 1.4 with initial conditions (a) x(0) = 0, x(0) ˙ = 0, (b) x(0) = 1, x(0) ˙ = 0.

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Fig. 7. Time-history response of frequency Ω = 1.5 with initial conditions (a) x(0) = 0, x(0) ˙ = 0, (b) x(0) = 1, x(0) ˙ = 0.

6

Conclusions

The paper demonstrated that a two-degree-of-freedom nonlinear system with three-to-one internal resonance can significantly increase the bandwidth of the energy harvester. We discover that the response of the system has two peaks that bend to the right, producing a hardening spring characteristic, through suitable parameter selection. Additionally, more than one solution can exist for some excitation frequency, there are two resonance ranges with three solutions. The frequency curves of the system with internal resonance exhibit a broadband characteristic of bending to the right near both natural frequencies, in contrast to traditional resonance cases.

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4. Daqaq, M.F., Masana, R., Erturk, A., et al.: On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl. Mech. Rev. 66, 040801 (2014) 5. Wei, C.F., Jing, X.J.: A comprehensive review on vibration energy harvesting: modelling and realization. Renew. Sust. Energ. Rev. 74, 1–18 (2017) 6. Yang, Z.B., Zhou, S.X., Zu, J., et al.: High-performance piezoelectric energy harvesters and their applications. Joule 2, 642697 (2018) 7. Shahruz, S.M.: Design of mechanical band-pass filters for energy scavenging. J. Sound Vib. 292, 987998 (2006) 8. Chen, L.Q., Jiang, W.A.: Internal resonance energy harvesting. J. Appl. Mech. 82, 031004 (2015) 9. Cao, D.X., Leadenham, S., Erturk, A.: Internal resonance for nonlinear vibration energy harvesting. Eur. Phys. J. Spec. Top. 224, 2867–2880 (2015) 10. Chen, L.Q., Jiang, W.A., Panyam, M., et al.: A broadband internally-resonant vibratory energy harvester. J. Acoust. Vib. 138, 061007 (2016) 11. Jiang, W.A., Chen, L.Q., Ding, H.: Internal resonance in axially loaded beam energy harvesters with an oscillator to enhance the bandwidth. Nonlinear Dynam. 85, 2507–2520 (2016) 12. Wu, Y.P., Ji, H.L., Qiu, J.H., et al.: A 2-degree-of-freedom cubic nonlinear piezoelectric harvester intended for practical low-frequency vibration. Sensor Actuat. A Phys. 264, 1–10 (2017) 13. Wu, Y., Ji, H.L., Qiu, J.H., et al.: An internal resonance based frequency upconverting energy harvester. J. Intel. Mat. Syst. Str. 29, 2766–2781 (2018) 14. Yang, W., Towfifighian, S.: Internal resonance and low frequency vibration energy harvesting. Smart Mater. Struct. 26, 095008 (2017) 15. Rocha, R.T., Balthazar, J.M., Tusset, A.M., et al.: Nonlinear piezoelectric vibration energy harvesting from a portal frame with two-to-one internal resonance. Meccanica 52, 2583 (2017) 16. Yan, Z.M., Hajj, M.: Nonlinear performances of an autoparametric vibration-based piezoelastic energy harvester. J. Intel. Mat. Syst. Str. 28, 254271 (2017) 17. Xiong, L.Y., Tang, L.T., Mace, B.R.: A comprehensive study of 2:1 internalresonance-based piezoelectric vibration energy harvesting. Nonlinear Dynam. 91, 1817–1834 (2018) 18. Liu, H.J., Gao, X.M.: Vibration energy harvesting under concurrent base and flow excitations with internal resonance. Nonlinear Dynam. 96, 10671081 (2019) 19. Karimpour, H., Eftekhari, M.: Exploiting double jumping phenomenon for broadening bandwidth of an energy harvesting device. Mech. Syst. Signal Pr. 139, 106614 (2020) 20. Fan, Y.M., Ghayesh, M.H., et al.: High-efficient internal resonance energy harvesting: modelling and experimental study. Mech. Syst. Signal Pr. 180, 109402 (2022) 21. Garg, A., Dwivedy, S.K.: Nonlinear dynamics of parametrically excited piezoelectric energy harvester with 1:3 internal resonance. Int. J. Nonlin. Mech. 111, 8294 (2019) 22. Shi, Y.X., Wu, Z., Liu, W., et al.: Three-to-one internal resonance of L-shaped multi-beam structure with nonlinear joints. J. Mar. Sci. Eng. 10, 1461 (2022) 23. Jiang, W.A., Ma, X.D., Han, X.J., et al.: Broadband energy harvesting based on one-to-one internal resonance. Chin. Phys. B 29, 100503 (2020) 24. Wu, H., Tang, L.H., Yang, Y.W., et al.: Development of a broadband nonlinear two-degree-of-freedom piezoelectric energy harvester. J. Intel. Mat. Syst. Str. 25, 1875–1889 (2014)

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25. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. John Wiley and Sons (2008) 26. Chin, C., Nayfeh, A.H.: Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dynam. 12(2), 129–154 (1997) 27. Nayfeh, A.H., Lacarbonara, W., Chin, C.: Non-linear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear Dynam. 18(3), 253–273 (1999) 28. Samir, A.E., Ali, H.N.: Non-linear response of buckled beams to 1:1 and 3:1 internal resonances. Int. J. Nonlin. Mech. 52, 12–25 (2013) 29. Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Nonlin. Mech. 37(1), 101–116 (2002) 30. Pakdemirli, M., Ozkaya, E.: Three-to-one internal resonances in a general cubic non-linear continuous system. J. Sound Vib. 268(3), 543–553 (2003) 31. Oz, H.R., Ozkaya, E.: Three-to-one internal resonances in a curved beam resting on an elastic foundation. Int. J. Appl. Mech. Eng. 10(4), 667–678 (2005) 32. Wang, L.H., Ma, J.J., Li, L.F., Peng, J.: Three-to-one resonant responses of inextensional beams on the elastic foundation. J. Vib. Acoust. 135, 11–15 (2013) 33. Shuai, H., et al.: Enhanced energy harvesting of a nonlinear energy sink by internal resonance. Int. J. Appl. Mech. 11(10), 1950100 (2019) 34. Wei, J., Yu, T., Jin, D.P., Liu, M., Tian, Y.S., Cao, D.Q.: Three-to-one internal resonance in a two-beam structure connected with nonlinear joints. Arch. Appl. Mech. 91, 3835 (2021) 35. Xu, L., Hui, Y., Zhu, W.D., Hua, X.G.: Three-to-one internal resonance analysis for a suspension bridge with spatial cable through a continuum model. Eur. J. Mech. A Solid 90, 104354 (2021) 36. Lenci, S.: Exact solutions for coupled duffing oscillators. Mech. Syst. Signal Pr. 165, 108299 (2022)

Optimization Design of High-Pressure Simulated Rotor Zhongyu Yang1,2,3(B) , Jiali Chen1,2,3 , and Yinli Feng1,3,4 1 Key Laboratory of Light Duty Gas Turbine, Institute of Engineering Thermophysics, Chinese

Academy of Sciences, Beijing, China [email protected] 2 School of Aeronautics and Astronautics, University of Chinese Academy of Sciences, Beijing, China 3 Innovation Academy for Light-Duty Gas Turbine, Chinese Academy of Science, Beijing, China 4 School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing, China

Abstract. Combined elastic support which include squirrel cage and squeeze film damper (SFD) are widely used in aero-engines, gas turbine, and steam turbine. Squirrel cage can vary the critical speed and strain energy distribution of the rotor system by changing the stiffness. SFD can effectively suppress rotor vibration and reduce transmitted forces. Given the inherently nonlinear behavior of SFD, a poorly designed damper has the potential to exacerbate rotor vibrations, posing a significant safety risk to engine operation. The influence of axial width of the SFD on dynamics behavior of the rotor-SFD system, such as critical speed, mode shape, vibration response has been developed in this paper. The Newton method was employed to optimize the critical speed and vibration response. The results of optimization reduced the 2nd critical speed by 24%, reduced the vibration amplitude of acceleration by 92%, and reduced the vibration amplitude of displacement by 88.3%. The contents and methods of this paper can provide guidance for the vibration optimization of rotor-squeeze film damper system. Keywords: Width · Squirrel Cage · Critical Speed · Vibration · Newton method

1 Introduction Combined elastic support which include squirrel cage and squeeze film damper (SFD) are widely used in rotating machines for its advantages of low cost, simple structure and outstanding vibration attenuation effect. However, it would cause complex motions of the rotor system under specific operating conditions. The dynamic characteristics of combined elastic support have been extensively investigated by both theoretical and experimental approaches. Han et al. [1] established a finite element model of a rotor-squirrel cage-SFD system, and analyzed the effect of bearing bias on the squirrel cage. Luo et al. [2] established a finite element model considering the coupling effect between composite bearings and rotors, analyzed the nonlinear contact and oil film forces of composite bearings, and derived the high-dimensional partial © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 221–233, 2024. https://doi.org/10.1007/978-981-97-0554-2_17

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differential equations with local nonlinearity. Ma et al. [3] combined the finite element method with the free interface modal synthesis method to solve the dynamic equations of a dual-rotor-composite bearing system, analyzed the influence of unbalance on the nonlinear dynamic characteristics of the system and verified the results through experiments. Jaroslav Zapomˇel et al. [4] investigated the influence of energy dissipation in the damping process of a rigid rotor-composite bearing system, and analyzed methods to minimize energy loss in rotor systems. Gao et al. [5] developed a finite element model for a flexible asymmetric rotor-SFD system and explored the influence of the coupling effect between combined elastic support and the rotor on the nonlinear characteristics of the rotor system. Chen et al. [6] derived the equations of motion for a rotor-SFD system considering inertial forces and static eccentricity, investigated the effects of structural parameters, unbalance, and stiffness of elastic support on the transient response of the rotor. Shao et al. [7] examined the impact of nonlinear oil film forces on the dynamic characteristics of a rotor system, and investigated the effects of structural parameters of the SFD on the vibration response. Luis San Andrés et al. [8] studied the impact of different end seals on the rotor dynamic characteristics, compared various methods for calculating damping coefficients and substantiated the result through experimental results. Kaihua Lu et al. [9] conducted experiments to revealed the influence of the viscosity of oil on the vibration characteristics of a rotor-combined bearing system, and analyzed the vibration reduction characteristics of the SFD. The effect of axial width of SFD on damping characteristics of SFD has been analyzed in this paper based on the principles of rotor dynamics and fluid mechanics. Newton method was employed for the optimization of rotor dynamics to address the issue of excessive critical speed and vibration amplitude. Structural optimization is reasonable based on the calculation results of finite element method, which can provide a reference for the design of rotor.

2 Modeling of High-Pressure Simulated Rotor Based on the situation of the experimental setup, the main design requirements for simulated rotor system are as follows: (1) The maximum axial dimension of simulated rotor system is 1500 (mm) and the maximum radial dimension is 200 (mm). (2) The maximum first two critical speeds of the rotor is 30000 (rpm). (3) The maximum value of D/N of the rotor bearing is 2.5 × 106 (mmr/min). (4) The bearing seat needs to provide enough space to arrange the lubrication oil path, ensuring the sealing and lubrication of the rotor system. Based on the above requirements, the simulated rotor referenced the structure of a high-pressure rotor system in aircraft engines which mainly includes the disc, lubrication oil channel, locking nuts, gasket, and others. This simulated rotor adopted a 1-0-1 bearing configuration, the length of shaft in this model is 440 (mm), the maximum diameter is 50 (mm), and the weight is 17 (kg). The specific structure is shown in Fig. 1. In this model, the diameter of journal of the SFD is 80 (mm), the outer diameter is 82 (mm), and the width is 15 (mm). The specific structure of the SFD is shown in Fig. 2.

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Fig. 1. Schematic diagram of the simulated rotor system structure

Fig. 2. Schematic diagram of the SFD structure

In this model, the number of bars in the squirrel cage is 8, the length is 30 (mm), the width is 4.1 (mm), the thickness is 2 (mm). The specific structure of the squirrel cage is shown in Fig. 3.

Fig. 3. Schematic diagram of the squirrel cage structure

A computational model for the damping characteristics has been established based on the theories of fluid mechanics and rotor dynamics. The results of the finite element method indicate that the rotor has superior overall rigidity and high strength. The strain energy distribution in this rotor system is reasonable, which ensures normal operation of the rotor. However, the 2nd critical speed exceeds 30000 (rpm), and the vibration amplitude of the rotor system exceeds the expected level, which severely affects the stability of the system during operation. Regarding the aforementioned issue, an optimized design for the rotor system had been presented in this paper. Firstly, the influence of the squirrel cage structure on

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stiffness have been determined, and the impact of the axial width of SFD on the stiffness and damping coefficient was investigated. Accordingly, the finite element method was employed to determine the stiffness and damping coefficient which meet the engineering requirements. Finally, the appropriate parameters of squirrel cage structure and the axial width of SFD were selected using the Newton method. The detailed explanation has been provided in Chapter 3.

3 Optimization of the Simulated Rotor System This design optimization prioritizes varying the critical speed by changing the structure of the squirrel cage, and reducing the vibration amplitude of the rotor system by selecting an appropriate axial width of the SFD. The calculation model for the damping characteristics of SFD and the Newton method have been used to optimize the rotor system. This optimization method can serve as a reference for engineering practice. 3.1 Damping Characteristic Calculation Mode For the rotor-SFD system, the damping properties of the oil film are crucial. The bearing 1 utilizes a combination elastic supporting structure consisting of SFD, squirrel cage, and rolling bearing, while bearing 2 adopts a rigid supporting structure with ball bearing. The schematic diagram of the SFD is shown in Fig. 4.

Fig. 4. Schematic diagram of the SFD

The basic equation for calculating the oil film force of the SFD is the generalized Reynolds equation for the hydrodynamic bearings [10]. The equation in Cartesian coordinates can be expressed as:     ∂ ρh3 ∂p ∂ ρh3 ∂p . + . ∂x μ ∂x ∂z μ ∂z (3.1)   ∂(ρh) ∂(ρh) ∂ρ + 12 + 12h = 6 ωb + ωj − 2 ∂θ ∂t ∂θ where, ρ is the density of the lubricant, h is the oil film thickness, μ is the viscosity of the lubricant, p is the oil film pressure, ωb , ωj are the angular velocities of the bearing and journal respectively, and Ω is the angular velocity of the journal orbiting.

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The Reynolds equation for compressible flow in polar coordinates as follows:     ∂ ρh3 ∂p ∂ ρh3 ∂p . + R2 . ∂θ μ ∂θ ∂z μ ∂z (3.2)   ∂(ρh) ∂ρ 2 2 ∂(ρh) + 12R + 12h = 6R ωb + ωj − 2 ∂θ ∂t ∂θ where, R is the radius of the journal, e is the radial velocity. When ωb = ωj = 0, ρ = 0, μ = 0. The transient Reynolds equation for the SFD can be simplified to:     ∂ ∂p ∂ ∂p + R2 (1 + ε cos θ )3 (1 + ε cos θ )3 ∂θ ∂θ ∂z ∂z (3.3) 2 12μR = (ε sin θ + ε˙ cos θ ) c2 where, c is the average gap of the oil film, ε is the eccentricity of the SFD, and ε = e/c, h = c + ecosθ. In the prototype SFD, there are no hermetic seals at both ends, and the pressure at either end is similar to the external pressure. Due to the squeezing effect, the pressure in the central region of the damper is considerably higher, while the circumferential pressure gradient is low, as in (3.4): ∂p ∂p >> ∂z ∂θ

(3.4)

Equation (3.3) can be simplified as:   ∂ 12μ ∂p = 2 (ε sin θ + ε˙ cos θ ) (1 + ε cos θ )3 ∂z ∂z c

(3.5)

The hypothesis of short bearing theory is available when L/D < 0.25. Taking the central section of the damper as the coordinate axis and using atmospheric pressure as the reference point, the boundary conditions at z = -L/2 are p = p1 = 0, and at z = L/2 are p = p2 = 0, as shown in Fig. 5. Integrating (3.5) twice with respect to z yields the pressure distribution of the oil film. p(θ, z) = −

6μ c2 (1 + ε cos θ )3

(ε sin θ + ε˙ cos θ )(

L2 − z2 ) 4

(3.6)

where, L is the width of the SFD, D is the diameter of the SFD. The radial and tangential forces on the journal caused by the pressure of the lubricant film in the damper can be calculated with the following formula: 

Fr Ft



L/2

= −L/2

θ2

θ1



cos θ −p(θ, z) Rd θ dz sin θ

(3.7)

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Fig. 5. Stress analysis diagram of SFD

Substituting (3.6) into (3.7), when the system is in the half-film state with θ 1 = and θ 2 = 2 , the radial and tangential forces on the journal can be determined as follows:   ⎧ 3 2 2) μRL 2ε π ε ˙ (1 + 2ε ⎪ ⎪ ⎪ F = . + ⎪ ⎨ r c2 (1 − ε2 )2 (1 − ε2 )5/2   (3.8) ⎪ ⎪ π ε μRL3 2ε ε˙ ⎪ ⎪ Ft = . + + (p1 + p2 )LR ⎩ c2 (1 − ε2 )3/2 (1 − ε2 )2 When the system is in the full-film state with θ 1 = 0 and θ 2 = 2 , the radial and tangential forces on the journal can be determined as follows: ⎧ ⎪ μRL3 π ε˙ (1 + 2ε2 ) ⎪ ⎪ Fr = . ⎨ c2 (1 − ε2 )5/2 (3.9) ⎪ μRL3 π ε ⎪ ⎪ ⎩ Ft = . c2 (1 − ε2 )3/2 The stiffness coefficient K and damping coefficient C of the SFD are: ⎧ Fr ⎪ ⎨K = − e F ⎪ ⎩C = − t e

(3.10)

The negative sign represents wherein the distribution of radial force and tangential force is opposite to the direction of the eccentricity and precession speed. 3.2 Calculation Model for Squirrel Cage The advantages of using a squirrel cage are that it can control the critical speed and the distribution of strain energy of the rotor system by changing the stiffness of the squirrel cage. The cross-section of the squirrel cage closes to rectangular, and the radial stiffness and axial stiffness of the bars are not equal. Therefore, under the action of the radial force F transmitted by the bearing, the load distributed to each bar is different. The boundary conditions at both ends of the cage bars are depicted in Fig. 6. It is assumed that one end is fixed and the other end is constrained to undergo only radial displacement without any rotational degrees of freedom.

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Fig. 6. The force model of squirrel cage

When one end of the cage bar is subjected to a force F ϕ resulting from the F, it is equivalent to a free end with an additional bending moment, the value of which is: Mϕ =

Fϕ l 2

(3.11)

where, l is the length of the cage bar. The displacement at the end of the cage bar can be calculated as follows: Y =

Fϕ l 3 12EJ

(3.12)

Since the cross-section of the cage bar is rectangular and the direction of the force is not aligned with the principal axis of the cross-section, the displacement should be calculated along the principal axis. The radial and circumferential displacements of the cage bar can be calculated as follows: Yr = Y cos ϕ, Fr = Fϕ cos ϕ Yt = Y sin ϕ, Ft = Fϕ sin ϕ

(3.13)

where, Y r is the radial displacement, F r is the radial load, Y θ is the circumferential displacement, and F θ is the circumferential load. The moments of inertia about the radial and circumferential axes are given respectively by: bh3 12 hb3 Jt = 12

Jr =

(3.14)

where, J r is the moment of inertia of the radial axes, J t is the moment of inertia of the circumferential axes. The total displacement of the cage bar in the direction of force is: Y = Yt sin ϕ + Yr cos ϕ =

Fϕ l 3 cos2 ϕ sin2 ϕ [ 3 + ] E h b hb3

(3.15)

If the force acting on the differential segment of the cage bar is df , then: df = Fϕ

rd ϕ B

(3.16)

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where, r is the radius, and B is the arc length between two cage bars. Substituting (3.15) into (3.16): df = E

0

π

dϕ b3 h3 rY 2 l3B b cos2 ϕ + h2 sin2 ϕ

2 dϕ = (π + i2π ) b2 cos 2ϕ + h2 sin 2ϕ bh

(3.17) (3.18)

where, i is 0 and a positive integer. When i = 0.

π 0

2π dϕ = b2 cos 2ϕ + h2 sin 2ϕ bh

(3.19)

The total force F acting on the entire squirrel cage is:

F=

2π

2π rb2 h2 YE l3B

df =

0

(3.20)

The stiffness coefficient of the entire squirrel cage is: k=

F 2π rb2 h2 E = Y l3b

(3.21)

3.3 Newton Method The Newton method is characterized by its rapid convergence and simple implementation, which makes it widely utilized for solving nonlinear optimization problems. The Hessian matrix of a multivariate function takes on the following form: ⎡ 2 ⎤ ∂ f ∂ 2f ⎢ 2 ∂x1 ∂x2 ⎥ ⎢ ∂x ⎥ Hf = ⎢ 1 (3.22) ⎥ ⎣ ∂ 2f ⎦ ∂ 2f 2 ∂x1 ∂x2 ∂x2 The gradient of the function is as follows: ∇f (x1 , x2 ) = (

∂f ∂f T , ) ∂x1 ∂x2

(3.23)

Let the independent variable be x = (x 1 , x 2 )T then the iterative formula can be expressed as: xk+1 = xk − Hf−1 (x)∇f (x) The computation process of the Newton method is shown in Fig. 7.

(3.24)

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Fig. 7. The computation process of the Newton method

3.4 Dynamic Analysis of the Original Rotor System The stiffness of bearing 1 of the original rotor system is 5 × 106 (N/m), the stiffness of bearing 2 is 1 × 108 (N/m), the journal diameter of SFD is 80 (mm), the width of SFD is 15 (mm), the eccentricity of SFD is 0.1, and the clearance of SFD is 0.2 (mm). The critical speeds of the rotor’s first three orders were calculated using rotor dynamic analysis software, as shown in Table 1. The transient response of the bearing 1 during the acceleration process is shown in Fig. 8. And Fig. 9. As can be seen, The 2nd critical speed is higher than the design requirement, and the vibration amplitude of the rotor system is relatively large during the acceleration process, which seriously endangers the stable operation of the rotor system. Table 1. The critical speeds of the original system Critical Speed (rpm) 1st order 2nd order

20567

3rd order

103155

32849

Fig. 8. Transient response of acceleration of the bearing 1 in the original rotor system

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Fig. 9. Transient response of displacement of the bearing 1 in the original rotor system

3.5 The Impact of Axial Width of SFD on Damping Characteristics The impact of axial width of SFD on the damping characteristics of the SFD is shown in Fig. 10. And Fig. 11. An increase in axial width of SFD leads to an increase in the effective area of oil film extrusion, and resulting in an increasing trend of oil film damping and a decrease in oil film stiffness. The damping coefficient also exhibits an increase in nonlinearity while the stiffness coefficient consistently displays nonlinear characteristics.

Fig. 10. The variation law of the damping coefficient with the axial width of SFD

Fig. 11. The variation law of the stiffness coefficient with the axial width of SFD

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3.6 Optimization of the Simulated Rotor System According to the analysis above, the original rotor system can be concluded to have issues of high 2nd critical speed and excessive vibration amplitude. This will seriously endanger the stable operation of the rotor system during acceleration. Therefore, measures need to be taken to reduce the 2nd critical speed, decrease the vibration amplitude, and improve stability and reliability. Firstly, regarding the issue of higher 2nd critical speed, as analyzed above, it is known that the stiffness of the squirrel cage has a significant impact on the critical speed of the rotor system. The initial parameters of the original squirrel cage were chosen as the starting point for the optimization algorithm, with the stiffness of squirrel cage being 4.6 × 106 (N/m). The Newton method was used to screen the data along the reverse gradient direction. Finally, it was determined, when the stiffness of squirrel cage was 5.7 × 105 (N/m), the 2nd critical speed met the design requirements. At this point, the number of bars in the squirrel cage is 6, the length is 40 (mm), the width is 3 (mm), the thickness is 1.8 (mm). The optimized rotor system has the first three critical speeds as shown in Table 2, with the 1st critical speed at 14651 (rpm) and the 2nd critical speed at 25263 (rpm), which meet the design requirements (2). The optimized squirrel cage was subjected to strength verification, and the results showed that it has good mechanical properties, which can ensure the normal operation of the rotor system. Table 2. The critical speeds of the optimized system Critical Speed (rpm) 1st order 2nd order

14651

3rd order

57979

25263

To address the problem of excessive vibration amplitude in the rotor system, this paper, considering the design constraints of the rotor system, such as critical speed and the DN value of the bearings, proposes the solution of increasing the damping effect of SFD by changing the size of axial width of SFD. The initial parameters of the original rotor system were chosen as the starting point for the optimization algorithm, with the axial width of the SFD is 15 (mm). The Newton method was used to screen the data along the reverse gradient direction. Finally, it was determined, when the axial width of the SFD was increased to 40 (mm), the vibration of the rotor system met the design requirements. The transient response of the bearing 1 during the acceleration process were shown in Fig. 12. And Fig. 13. The optimized rotor was subjected to strength verification, and the results showed that the overall rigidity of the rotor is relatively high, the strain energy distribution in the rotor system is reasonable, and meets the design requirements, which can ensure the normal operation of the rotor.

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Fig. 12. Transient response of acceleration of the bearing 1 in the optimized rotor system

Fig. 13. Transient response of displacement of the bearing 1 in the optimized rotor system

4 Conclusion Based on the finite element method, this paper analyzed the problems in the finite element model of the high-pressure simulated rotor and proposed optimization measures to address these shortcomings. The following conclusions were drawn: • The axial width of the SFD directly affects the stiffness and damping of the squeeze film, and changing the axial width of the SFD within a certain range can effectively reduce the vibration amplitude of the rotor system. • To reduce the 2nd critical speed, this paper used Newton method to obtain the optimal solution for the stiffness of the squirrel cage, and adjusting the structural parameters of the squirrel cage. • To reduce the vibration amplitude of the rotor system, this paper used Newton method to obtain the optimal solution for the axial width of the SFD, which effectively reduced the vibration amplitude and improved the stability and reliability of the rotor system. • Through the rotor dynamic analysis of the optimized simulated rotor, the results showed the optimized rotor system possesses favorable rotor dynamic characteristics and can meet all design requirements, which can provide support for subsequent research.

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References 1. Han, Q., Chen, Y., Zhang, H., et al.: Vibrations of rigid rotor systems with misalignment on squirrel cage supports. J. Vibroeng. 18(7), 4329–4339 (2016) 2. Luo, Z., Wang, J., Tang, R., et al.: Research on vibration performance of the nonlinear combined support-flexible rotor system. Nonlinear Dyn. 98, 113–128 (2019) 3. Xinxing, M.A., Hui, M.A., Haiqin, Q.I.N., et al.: Nonlinear vibration response characteristics of a dual-rotor-bearing system with squeeze film damper. Chin. J. Aeronaut. 34(10), 128–147 (2021) 4. Zapomˇe, J., Ferfecki, P., Kozánek, J.: Application of the controllable magnetorheological squeeze film dampers for minimizing energy losses and driving moment of rotating machines. In: Cavalca, K.L., Weber, H.I. (eds.) 10th International Conference on Rotor Dynamics, vol. 60, pp. 132–143. Mechanisms and Machine Science (2018) 5. Tian, G.A.O., Shuqian, C.A.O., Yongtao, S.U.N.: Nonlinear dynamic behavior of a flexible asymmetric aero-engine rotor system in maneuvering flight. Chin. J. Aeronaut. 33(10), 2633– 2648 (2020) 6. Chen, X., Gan, X., Ren, G.: Dynamic modeling and nonlinear analysis of a rotor system supported by squeeze film damper with variable static eccentricity under aircraft turning maneuver. J. Sound Vib. 485, 1879–1928 (2020) 7. Shao, J., Jigang, W., Cheng, Y.: Nonlinear dynamic characteristics of a power-turbine rotor system with branching structure. Int. J. Non-Linear Mech. 148, 21–37 (2022) 8. Andrés, L.S., Koo, B., Jeung, S.-H.: Experimental force coefficients for two sealed ends squeeze film dampers (Piston Rings and O-Rings): an assessment of their similarities and differences. J. Eng. Gas Turb. Power-Trans. ASME 141(2), 1–13 (2019) 9. Kaihua, L., He, L., Yan, W.: Experimental study of Iersfds for vibration reduction of gear transmissions. J. Vibroeng. 21(2), 409–419 (2019) 10. Jialiu, G.: Dynamic Characteristics of Rotors with Squeeze Film Dampers, 1st edn. National Defence Industry Press, Beijing (1985)

Energy Transfer of Particle Impact Damper Systems Xiang Li1(B) , Li-Qun Chen2 , Lawrence A. Bergman3 , and Alexander F. Vakakis4 1 Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China

[email protected]

2 Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and

Engineering Science, Shanghai University, Shanghai 200444, China 3 Aerospace Engineering, University of Illinois, Urbana, IL 61801, USA 4 Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801, USA

Abstract. In this work, investigations on energy transfers from a linear oscillator (called primary structure) forced by a shock to a strongly nonlinear attachment, namely a particle impact damper (PID) are carried out. The granules inside the cavity of the PID are arranged in initial topology with clearances. The granular interactions are realistically modeled by combining a Hertzian dissipative contact model and a Coulomb-tanh friction model, which is useful to accurately explore the strong and highly discontinuous nonlinear characteristics of this system subject to shocks of varying intensities. The discrete element method is employed for these simulations, considering granular translations and rotations. Generally, by optimizing the size of the cavity one can improve the shock mitigation performance. The capacity of energy transfer and energy dissipation in the granular medium is enhanced and maintained by a collect-and-collide regime. Intense nonlinear contacts via granule-to-granule and granule-to-wall interactions result in intense and irreversible energy transfer of shock energy from the directly forced primary structure to the PID. Once this energy is transferred, it gets efficiently scattered to high frequencies and dissipated by the inelastic granular collisions and frictional effects due to relative granular rotations. The energy transfer reported herein provides strong motivation for developing the next generation of PID technology. Keywords: Particle Impact Damper · Energy Transfer · Granular Motion

1 Introduction Particle impact dampers (PIDs) can be employed under the harsh environment conditions for efficient passive vibration suppression. The PID is got by putting some particles into a container or a hole which is fixed to the system, making them conceptual simplicity and low costs [1]. Furthermore, based on reasonable designs, they can work effectively over a wide frequency range. Due to these strengths, the PIDs are promising and effective damping devices, and have huge potential applications in many fields, such as high-rise buildings, circuit boards, machinery, rails, etc. [2]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 234–244, 2024. https://doi.org/10.1007/978-981-97-0554-2_18

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In recent years, energy exchange or energy transfer in the particle damping system attracts people’s attention. Lu et al. [1] concluded that particle damping technology is part of the category of nonlinear energy sink (NES), i.e. nonlinear targeted energy transfer (TET). The concept of the NES is about irreversible energy transfer from a primary structure to the NESs via strongly nonlinear interactions [3–5]. Several works related to energy exchange are briefly introduced as follows. Mao et al. [6, 7] reported that these particles absorb kinetic energy from the primary structure via particle-wall and particleparticle interactions. They provided an important insight that momentum is transferred to the particles via particle-wall collisions over a short time; most significant energy dissipation occurs during middle time; and little energy dissipation happens at a much longer time. Bai et al. [8] utilized particle dynamics simulations to further investigate energy transfer and energy dissipation. They found that collisions between the particles and the structure mainly lead to energy transfer, but frictions play little role in energy transaction. Besides, they concluded that the capacity of both energy transfer and energy dissipation governs particle damping performance. However, irreversible energy transfer phenomena have not been found and given in published papers. This one-way energy transfer regime still needs to be investigated. Some researchers investigated granular dynamics behaviors to gain further insights into the damping mechanism of the PIDs. Salueña et al. [9] reported three regimes of horizontally vibrating granules, i.e. solid, convective and gas-like regimes, in which the convective state indicates less energy dissipation rate. Eshuis et al. [10] widely studied granular motion phenomena in the vertical direction which includes bouncing bed, undulations, granular Leidenfrost effect, convection rolls, and granular gas. Bannerman et al. [11] and Sack et al. [12] found two different modes of granular dynamics in the absence of weights, i.e. the collect-and-collide regime and the gas regime, which both depend on excitation intensities. They reported that the collect-and-collide state at large intensities indicates efficient energy dissipation, whereas energy dissipation rate of the particle damper for the gas-like state at small intensities is much smaller than in the collect-and-collide state. Zhang et al. [13] discussed the damping performance of the particle dampers and the motion modes of all particles under various gap clearances and excitation intensities. They found that particles are in the Leidenfrost state when the particle dampers have optimal energy dissipation. Besides, they concluded that optimal damping performance is owing to directly energy dissipation and partly energy conversion. Meyer and Seifried [14] presented five different motion modes via velocity graphs, including solid-like, local fluidization, global fluidization, bouncing bed, and decoupled states. They showed that damping performance is enhanced when particles are in bouncing bed states and in the transition from local to global fluidization. These studies above revealed some relationships between the damping performance and the states of the granular matter. But the relations between granular motion regimes and one-way energy transfer in the particle damping system have not been discussed and figured out. The explanation about their inherent relations between them will contribute to TET regimes in the particle damping system as well as particle damping mechanism. This study focuses on energy exchange of some particle damping systems and demonstrates irreversible energy transfer. The PID can be designed through placing the same number of granules in different equilibrium positions or organizing granules in various

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topologies. The strongly nonlinear and highly non-smooth systems are simulated via the discrete element method (DEM). Dynamic responses, and energy components are computed based on convergent solutions. This work is organized as follows. Section 2 introduces the studied system and computational method. Section 3 explores the dynamics and energy transfer of the PIDs in which the granules are placed in topology, in which energy transfer and the granular motions are demonstrated as well as shock mitigation performance. Section 4 concludes this work.

2 The Integrated System and Computational Method A studied particle damping system is introduced in this section. Figure 1 depicts a schematic drawing about a single-degree-of-freedom primary structure coupled with a PID. In the horizontal direction, this primary  structure is forced by a shock excitation which can be expressed as F(t) = F0 sin[ tπ0 · min(t, t0 )], where F 0 is the amplitude of the shock force, t 0 is the duration of the shock force and t is the time variable.

Fig. 1. The schematic drawing of an integrated system: a single-degree-of-freedom primary structure coupled with a particle impact damper (PID).

For purpose of mitigating shock responses, the PID is employed to transfer input energy from the primary structure to the particle(s). As Fig. 1 shows, a rectangular cavity is inserted into this structure. The length and the height of this cavity are denoted by d 1 and d 2 , respectively. Since the two-dimensional motions in the horizontal plane are considered, the width of this cavity just needs to be larger than the radius of the granule. The size of this cavity, i.e. the length and the height, can be designed to get better vibration control performance. The PID is composed of some particles which are assumed as identical, spherical and linearly elastic granules. The particle(s) can be placed in topology together with introduction of clearances. Besides, the enclosure of the container is rigid, and no compressions exist between particles and the walls of the container or between particles before oscillation. The total mass of the whole system including the primary structure and the PID is equal to M. In this work, the total mass of the coupled system is the same in order to remove the added mass effects [15]. The mass, stiffness, and damping of this primary structure are represented by mps , K and C, respectively. In the beginning, the primary structure and the particles are all at rest with

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zero initial displacements and zero initial velocities. The equations of motion for the full system are listed below, mPS z¨ + C z˙ + Kz = F + Fd     Nij + fij + mi u¨ i = (Nik + fik ) j k     Ii θ¨i = Ri nij × fij + Ri (nik × fik ) j

(1)

k

where overdot indicates differentiation with respect to time t; the subscripts i and j denote the i-th and j-th granule, respectively; the subscript k represents the k-th point on the wall of the PID container; ui and θi denote the displacement vector and the angular displacement vector of particle i, respectively; mi , Ii , and Ri represent the mass, the radius and the moment of inertia of particle i, respectively; Nij and Nik are the normal contact forces between granules i and j, and particle i and the point on the wall k, respectively; fij or fik represent the corresponding tangential forces; nij and nik are the unit vectors; and lastly, F d is summation of contact forces acting on the primary structure in x direction, due to particle-wall interactions. Based on the contact law of Hertz, nonlinear dissipative contact model and Coulomb-tanh friction model are accepted to explore strong nonlinearity and highly discontinuous characteristics. So, the contact force Nij and tangential contact force fij are   3 2 ˙ Nij = − Aij δn,ij + γij δn,ij nij (2)   fij = −μ Nij tanh ks δ˙t,ij tij Here, δn,ij denotes the relative compression deformation along the radical direction  between particles i and j, and its expression is δn,ij = max Ri + Ri − uj − ui , 0 . The

√ 1/4 contact coefficient Aij = 43 Eeff Reff and the damping coefficient γij = αn meff Aij δn,ij . The normal force Nik and the tangential force fik can also be computed using Eq. (2) (see Ref. [16]). It is vital for energy analysis to get dynamic responses of the particle damping systems. The discrete element method (DEM) [16] is adopted to simulate the granule – structure system. The DEM is based on a principle that the simulated time step is so small that one particle is not disturbed by any particle except its immediate neighbors during this time step. As a consequence, sufficient dynamic information of the complicated and nonlinear systems can be accessed via DEM simulations, which is of great significance to estimate energy transfer. DEM simulations are based on the contact law of Hertz [17] and an assumption about small elastic deformations for granules [18]. Equations (1) are solved simultaneously based on 4th order Runge-Kutta algorithm in the MATLAB software. According to the DEM rule, the time step which is selected to integrate the equations should be small enough so that the disturbance does not affect any other granule except its immediate neighbors in a single step.

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3 Analysis on Energy Transfer and Granular Motions In this section, energy transfer and granular dynamics in this integrate system are demonstrated. The parameters of the full system are listed in Table 1. The spherical granules are made from the steel. Some parameters related to the PID or the granule can also be found in Table 1. During simulations, the convergence of the DEM simulation is checked through comparing all energies in the coupled system when various time steps are employed. Based on simulation results, the smoothing parameter k s = 200 s/m and time step Δt = 1.2 × 10–8 s are accepted to compute the transient responses. Table 1. System parameters for the structure with a PID. Parameter

Value

Unit

Mass, M

20

kg

Stiffness coefficient, K

8 × 104

N/m

Damping coefficient, C

25.30

Ns/m

Mass ratio of the PID, ε

6%

—

Young’s modulus, E

2 × 109

Pa

Density of a granule, ρ

7850

kg/m3

Passion ratio, υ

0.3

—

Restitution coefficient, α n

6.313 × 10–3

—

Friction coefficient, μ

0.099

—

The size of the cavity which holds five granules in the topology is optimized at F 0 = 5 × 103 N. The criterion of optimization is the normalized cumulative dissipated energies up to final time. Much larger cumulative dissipated energy in the PID or weaker cumulative dissipated energy in the primary structure indicate much better performance of energy exchange and energy dissipation. These normalized cumulative energies up to final time are computed for purpose of seeking some better sizes of the cavity, i.e. the length d 1 and the height d 2 . Three-dimensional figures about normalized dissipative energies are depicted in Fig. 2 in which x-, y- and z-axis refer to the length of the cavity d 1 , the height of the cavity d 2 and the normalized dissipative energy. Better energy dissipations via PIDs are achieved through changing the sizes of the containers. When the mass of these granules and the topology are fixed, the size of the cavity or the clearance has effects on energy dissipation. For better shock mitigation, a designed container of the PID should not be too small or much larger. Largest normalized energy dissipations in the PIDs are found, and their values are 0.68, 0.32 for the cavity size, d 1 = 0.13 m and d 2 = 0.10 m. Energy exchange or transfer phenomena and shock mitigation are discussed below. The optimum sizes of the cavity in the PID at F 0 = 5 × 103 N, namely d 1 = 0.13 m and d 2 = 0.10 m, are selected together with other system parameters listed in Tables 1. Figure 3 shows some results about contact forces, energy transaction and vibration reduction for

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Fig. 2. Normalized cumulative dissipative energy for various sizes of the cavity which holds five granules in asymmetric topology at F 0 = 5 × 103 N: a PID, b primary structure.

the PID at F 0 = 5 × 103 N. Summation of contact forces acting on the primary structure in x direction due to particle-wall interactions, i.e. F d , in time series is shown in Fig. 3a. Figure 3b depicts instantaneous energies which are normed by the total input energy in the system as well as the enlarged view from 0 to 0.6 s. Besides, the instantaneous energy distributions are described in Fig. 3c through dividing the mechanical energy of the primary structure and the PID by the total instantaneous energy. Figure 3d shows both displacements of the primary structure and their amplitude spectra for the system with or without the PID. Furthermore, velocities of the primary structure and particle 1 and their wavelet spectra are presented in Figs. 3e,f. The blue dashed lines in Figs. 3e,f mark the natural frequency, i.e. about 10 Hz. Significant energy transfer phenomena occur between the PID and the primary structure (see Fig. 3b). Especially, irreversible energy exchange regime is found in the optimum PID with five granules. It should be noted that this irreversible energy transfer is governed by strongly nonlinear and highly discontinuous characteristics. As Figs. 3e,f depict, these energies have high-frequency components. Significant energy owning high-frequency components is localized in these granules, cf. Figures 3f. These transferred energies are dissipated through inelastic impacts or frictions owing to granular rotations. Therefore, the optimized PID with five particles dissipates much more energy. Besides, as compared to natural frequency of the primary structure, Fig. 3d indicates that free vibration frequency of the primary structure coupled with a PID does not have obvious change. The optimized PID are also discussed at different intensities of the shock force. For the PID with the optimum size of the cavity, the dissipative energies are estimated at various shock force amplitudes which are F 0 = 100 N, 500 N, 1 × 103 N, 2.5 × 103 N, 5 × 103 N, 7.5 × 103 N and 1 × 104 N. Figure 4 shows the dissipative energies in the PID and the primary structure at various total input energies. At low input energy levels, less energy is transferred into and dissipated by the PIDs for all cases. As input energy or shock amplitude increases, much more normalized energies are transferred into the PIDs and then dissipated by themselves. As the number of the particles increases, the performance of energy dissipation for the PID is improved. To figure out differences on energy dissipation between at low energies and at high energies, contact forces and normalized instantaneous energies in time series are analyzed for the case of five particles at F 0 = 500 N (i.e. a low-energy level) and F 0 = 1 ×

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Fig. 3. At F 0 = 5 × 103 N: a contact forces F d , b instantaneous energies normed by total input energy, c instantaneous energies normalized by total instantaneous energy, d displacements of primary structure and their amplitude spectra, e velocities of primary structure and their wavelet spectra, f velocities of particle 1 and their wavelet spectra.

Fig. 4. Normalized cumulative dissipative energy in the primary structure and PID at various shock force amplitudes. The left dashed lines and the right dot-dashed line refer to the total input energies at F 0 = 500 N and F 0 = 1 × 104 N, respectively.

104 N (i.e. a high-energy level). Figure 5 depicts contact forces F d and normalized instantaneous energies for two kinds of input energy levels. Obviously, according to Fig. 5, the system at the low-energy level spends much more time to dissipate the input energy than that at the high-energy. Due to large and intensive contact forces (cf. Figure 5a), much energy is absorbed by the PID (cf. Figure 5b) and better energy localization (cf. Figure 5c) when the shock force amplitude is 1 × 104 N. However, the contact force is small and sparse in time series, which leads to less energy transaction (cf. Figure 5e) and bad energy localization (cf. Figure 5f) to some extent. The motions of granules in the PID are discussed for purpose of trying to explain differences on energy transfer and energy localization at low and high energies. Snapshots

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Fig. 5. At F 0 = 1 × 104 N: a contact forces F d , b instantaneous energies normalized by total input energy, c instantaneous energies normalized by the total instantaneous energy; and at F 0 = 500 N: d contact forces F d , e instantaneous energies normalized by total input energy, f instantaneous energies normalized by the total instantaneous energy.

about granular motions at some moments for the system at F 0 = 1 × 104 N and at F 0 = 500 N are presented in Figs. 6 and 7, respectively. At F 0 = 1 × 104 N or the high-energy level, the motions of granules including translational and rotational motions are quite quick inside the cavity together with state of aggregation. This phenomenon is called the collect-and-collide regime which is supposed to be most efficient damping in a driven granular matter system [12]. In this work, this collect-and-collide regime is further explained in the PID system. The phenomena of granular aggregation and separation in a short duration create and increase the chances of strong nonlinear interactions, i.e. inelastic collisions and frictions. Much input energy is transferred into the granules and dissipated among these granules, which leads to efficient energy transaction (cf. Fig. 5b) and energy localization (cf. Fig. 5c). Afterwards, the motions for the case of five particles at F 0 = 500 N (the low-energy level) are analyzed. According to snapshots in Fig. 7, a gaseous state is observed where both translational and rotational motions of granules are very slow. This gas-like behavior leads to small nonlinear interactions and less energy exchange between the particle and the structure. As depicted in Figs. 5e,f, energy transfer and energy localization are not very good. Therefore, energy dissipation in a gas regime is less effective than in the collect-and-collide regime. It has been reported in the driven granular matter [12] that the energy dissipation rate in the gas regime is smaller than in the collect-and-collide regime. Here, it is proved again from the view of energy transfer.

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Fig. 6. Snapshots from the animation for the case of five particles at F 0 = 1 × 104 N: a t = 0.06 s, b t = 0.15 s, c t = 0.23 s, d t = 0.43 s, e t = 0.78 s, f t = 1.25 s.

Fig. 7. Snapshots from animation for the case of five particles at F 0 = 500 N: a t = 0.06 s, b t = 0.62 s, c t = 1.5 s, d t = 2.2 s, e t = 3.01 s and f t = 3.58 s.

4 Conclusions Energy transfer phenomena are demonstrated through investigating a shock-excited primary structure coupled with a particle impact damper (PID). The PID is designed with five granules in asymmetric topology with clearances. Nonlinear dissipative contact model and Coulomb-tanh friction model are employed to explore strong nonlinearity and highly discontinuous characteristics in particle-based damping systems. Owing to

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simulation convergence and energy conversion check, discrete element method is a promising approach to simulate strongly nonlinear and highly non-smooth equations of motion. Energy exchange and energy dissipation, especially one-way energy transfer, are discussed. Some conclusions are given below: Strong and intense nonlinear contact interactions between particles and the structure may indicate occurrence of strong and ‘irreversible’ energy transfer regime as well as energy localization, which enhances energy dissipation. Strongly nonlinear and highly discontinuous characteristics govern irreversible energy transfer in particle-based dynamics systems. Granular motions in the collect-and-collide regime contribute to particle-wall interactions and thus energy transfer, whereas particles in gas state lead to less effective energy transfer and energy dissipation. Acknowledgements. This research was supported in part by the National Natural Science Foundation of China (No. 12202160), and the China Scholarship Council (XL).

References 1. Lu, Z., Wang, Z., Masri, S.F., Lu, X.: Particle impact dampers: Past, present, and future. Struct. Control Heal. Monit. 25, 1–25 (2018) 2. Gagnon, L., Morandini, M., Ghiringhelli, G.L.: A review of particle damping modeling and testing. J. Sound Vib. 459, 114865 (2019) 3. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., Michael, M.D., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer Science & Business Media, Berlin (2008). https://doi.org/10.1007/s11071-022-07216-w 4. Vakakis, A.F., Gendelman, O. V., Bergman, L.A., Mojahed, A., Gzal, M.: Nonlinear targeted energy transfer: state of the art and new perspectives. Nonlinear Dyn. 1–31 (2022) 5. Ding, H., Chen, L.Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100, 3061–3107 (2020) 6. Mao, K., Wang, M.Y., Xu, Z., Chen, T.: DEM simulation of particle damping. Powder Technol. 142, 154–165 (2004) 7. Mao, K., Wang, M.Y., Xu, Z.Z., Chen, T.: Simulation and characterization of particle damping in transient vibrations. J. Vib. Acoust. 126, 202–211 (2004) 8. Bai, X.M., Keer, L.M., Wang, Q.J., Snurr, R.Q.: Investigation of particle damping mechanism via particle dynamics simulations. Granul. Matter 11, 417–429 (2009) 9. Salueña, C., Esipov, S.E., Pöschel, T., Simonian, S.S.: Dissipative properties of granular ensembles. In: Proceedings SPIE 3327, Smart Structures and Materials 1998: Passive Damping and Isolation, pp. 23–29 (1998) 10. Eshuis, P., van der Weele, K., van der Meer, D., Bos, R., Lohse, D.: Phase diagram of vertically shaken granular matter. Phys. Fluids 19, 123301 (2007) 11. Bannerman, M.N., Kollmer, J.E., Sack, A., Heckel, M., Mueller, P., Pöschel, T.: Movers and shakers: Granular damping in microgravity. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 84, 1–9 (2011) 12. Sack, A., Heckel, M., Kollmer, J.E., Zimber, F., Pöschel, T.: Energy dissipation in driven granular matter in the absence of gravity. Phys. Rev. Lett. 111, 1–5 (2013) 13. Zhang, K., Chen, T., Wang, X., Fang, J.: Rheology behavior and optimal damping effect of granular particles in a non-obstructive particle damper. J. Sound Vib. 364, 30–43 (2016)

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14. Meyer, N., Seifried, R.: Toward a design methodology for particle dampers by analyzing their energy dissipation. Comput. Part. Mech. 8, 681–699 (2021) 15. Fang, X., Tang, J.: Granular damping in forced vibration: Qualitative and quantitative analyses. J. Vib. Acoust. 128, 489–500 (2006) 16. Wang, C., Zhang, Q., Vakakis, A.F.: Wave transmission in 2D nonlinear granular-solid interfaces, including rotational and frictional effects. Granul. Matter 23, 1–24 (2021) 17. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985) 18. Starosvetsky, Y., Jayaprakash, K., Hasan, M.A., Vakakis, A.F.: Topics on the Nonlinear Dynamics and Acoustics of Ordered Granular Media. World Scientific Press, Singapore (2017)

An Empirical Control Research on the Lexical Approach to Business Correspondence Writing in Vocational Colleges Jin Zhang(B) Shandong Youth University of Political Science, Jinan 250103, China [email protected]

Abstract. With the trend of economic globalization, business correspondence plays an increasingly important role in transmitting information, dealing with business activities and better cooperation. However, the current business correspondence teaching approach hasn’t come to a better effect to meet the above requirements. With the development of philosophy, psychology and corpus linguistics, more and more linguists have been aware of the limitation of the traditional approach. Lewis (1997) conducted a series of studies on lexical chunks and considered language is composed of chunks producing continuous coherent texts when they are combined. Chinese scholars have done some research on the appliance of lexical chunks, but how to implement this approach to business correspondence writing for vocational college students, is still at the margins. Accordingly, based on the lexical approach proposed by Lewis, this paper combines characteristics of languages used to compose business correspondence with the current learning status of vocational college students and processes the analyses of significant differences and the relationship between kinds of lexical chunks and the writing competence of vocational college students with the assistant of SPSS. The results indicate that lexical approach could effectively enhance students’ of business correspondence writing competence.

Keywords: Lexical approach Vocational colleges

· Business correspondence writing ·

Supported by Shandong Provincial Education Science “14th Five-Year” Plan Project 2023 Research on the “Trinity” Practical Approach of Integrating Excellent Traditional Chinese Culture into International Education; Teaching Achievement Award Incubation Programme of Shandong Youth University of Political Science 2022 Study on the Construction of a “Trinity” Civic Education System for the Chinese Culture Course Cluster in the Context of New Liberal Arts (CGPY202213); Teaching Reform Research Projects of Shandong Youth University of Political Science 2022 Practice of Classroom Teaching Reform Based on the PAD Model in the Context of the Epidemic (JGYB202214). c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 245–258, 2024. https://doi.org/10.1007/978-981-97-0554-2_19

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Instruction

With the acceleration of economic globalization, international economic activities have become increasingly frequent. As a way of communication, business correspondence plays a more and more significant role in transmitting information as well as dealing with business activities. Hence, those who acquire both business expertise and proficiency in English are urgently needed. In order to meet the demand for those talents, more and more vocational colleges begin to offer business correspondence courses to students in preliminary level. On the other side, as for vocational students, knowing how to write business correspondence will obviously help them catch more opportunities in job-hunting. In traditional correspondence writing, syntax is the major concern and grammar-based approach is often used. In this kind of teaching method, students succeed in applying grammatical rules into accurate use of language. But sometimes their compositions are awkward and inaccurate although they are correct in grammar. Later, influenced by the impact of task-based approach, teachers pay more attention to communicative language teaching. Students take topics as the thread and regard tasks as the centre. This teaching method believes language should be mastered in terms of its actual use in reality as much as possible. On the other hand, it focuses too much on language performance and often leads to inaccurate use of language. Furthermore, students of vocational colleges are relatively limited in English proficiency whose language production is apparently weak. In correspondence writing, they always encounter the following issues: Due to less acquisition of vocabulary, they can’t choose the appropriate words according to the context. Sentence patterns that they used are not standardized with many syntax errors. They know little about writing skills and the logical relation of letters is not reasonable. Therefore, it is obvious that grammar-based approach or task-based approach can’t help vocational college students write effectively or fluently. Moreover, compared with university’s four-year schooling system, time for English courses in vocational colleges is limited since the students only have two years or less time for English courses. Hence, a new teaching mode, under which students know how to write ordinary business correspondence in a short time, needs to be explored.

2 2.1

Research Methods Research Questions

The paper aims to illustrate the impact of lexical approach on business correspondence writing by answering the questions as follows: (1) Is it appropriate to put lexical approach into practice to business correspondence writing? (2) Can lexical approach contribute to enhancing vocational college students’ correspondence writing competence?

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(3) If it really works, what is the relationship between the lexical chunks used by students and business correspondence writing competence? 2.2

Participants

The participants in the research are 60 sophomores from two natural classes in school of economics and management of one vocational college. By random designation, we separate the two classes into experimental class (EC) and control class (CC). The average scores of their business English tests are mostly lower than 90 (full score 150). This experiment was carried out in the second academic year. Prior to the experiment, they have received over eight year’s English learning in school. In addition, they have been offered related business English courses after enrolling in vocational college so that they have some foundation in business English. Within this experiment, we instructed the two classes in business correspondence writing on the same teaching conditions. 2.3

Instruments

Questionnaire. The questionnaire (Appendix I) composed of 10 questions intends to explore whether the students enhance their awareness of lexical chunks after the whole semester’s study under the guidance of lexical approach. It not only attempts to reveal their impression and cognition of using lexical chunks in business correspondence writing but also intends to find out the students’ competence of using lexical chunks independently and consciously so that the chunk-based teaching could be revised according to students’ responses. Pretest. In order to test the students’ language output competence and provide data for the later comparison, students from two classes are required to write a business correspondence concerning self-introduction of a company at the beginning of the teaching experiment. The requirements and details are as follows. You are an importer of electronic goods in US. Recently you got a piece of information about a Rainbow Electronic Products Co. Ltd in China and you are very interested in the electronic fan they produced. Please write a letter to introduce yourself and show that you want to establish business relationship with them. Students are required to finish the business correspondence within 45 min without any help of dictionary or other assistants in order to obtain the natural composition which can reflect their writing proficiency. Students’ writings of pretest are carefully collected, saved and graded by the same experienced teacher. A revised scoring criterion (Appendix III) with reference to the grading standards of BEC preliminary writing will be used to evaluate students’ writings.

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Post-test. When the experiment is finished, students from both classes are needed to complete another business correspondence in 45 minutes. The topic and requirements are as follows. Please write a business correspondence sending a contract. The letter should include the following contents: A. To confirm order (No.237) for bed sheet and pillow case. B. To send contract (No. BP 103) asking for counter-signature. C. To express your hope for opening of the L/C. D. To ensure promote shipment. E. Expectation for further order. The writings of students are graded by the same teacher using the same scoring criteria to provide further research data. Interviews. After the post-test, a follow-up interview with 5 questions (Appendix II) is implemented to know about the learning status of students. 20 students (10 from either class) are selected to receive this interview, which aims to explore the students’ learning status in the following aspects: students’ confidence and intrinsic learning motivation; students’ self-evaluation on their academic performance; difficulties students encountered and how they solved them. By this interview, teaching effectiveness concerning lexical approach could be revealed. Data Collection. Data of this experiment come from the questionnaire, two writing tests and the interviews. The statistics obtained from the questionnaire are carefully analyzed in order to reveal students’ learning status, perceived ability to lexical chunks and difficulties in business correspondence writing. The lexical chunks that students used in their writing tests are valued, which are identified, counted and classified following the criteria given by Nattinger and DeCario (2000). In the process of counting chunks, repeated chunks in one letter are deemed as one chunk. The answers to the interviews will supply further information about the relationship between lexical approach and improvement of their writing competence of business correspondence. 2.4

Statistic Methods

The SPSS (Statistic Package of Social Science) software is applied to make an analysis of the data. The used statistic methods are as follows: (1) T-test, which is applied to test the correlation between learners’ scores in pretest and post-test so as to explore the significant difference among them. (2) Descriptive analysis, which is used to contrast the usage of lexical chunks between top-ranked students and bottom-ranked students. (3) Pearson Correlation Coefficient is used to explore the usage of different lexical chunks by all participants.

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Research Analyses Statistics and Analyses of Questionnaire Survey

All 60 participants from both experimental class and control class received the questionnaire. The first 4 questions are mainly related to their learning status. Questions 5 concerns students’ attitude towards learning strategy. Question 6– 8 mainly involve students’ perception of lexical chunks and ways of learning lexicalized language items. The last two questions concern their difficulties in business correspondence writing. The detailed data are displayed in Table 1. Table 1. Data of Questionnaire Survey 1

2

3

4

5

6

7

A 0%

53% 87% 45% 0%

B 3%

30% 63% 88% 25% 52% 5%

8

28% 13% 0%

9

10

30% 45%

53% 38% 55%

C 45% 42% 15% 90% 75% 12% 48% 28% 20% 72% D 37% 75% 18% 62% –

8%

32% 10% 12% 78%

E 15% 93% 25% 12% –

–

2%

8%

0%

–

Results of the first question show that most participants lack confidence in their English study. We can see the reasons for it from the results of the second and third question indicating that most students fail to master language output competence which requires lots of linguistic as well as grammatical knowledge. About 90% of the participants have realized the differences between business English and general English, but few of them think about business English learning from the perceptive of lexical chunks. Analyses of the results from the first 4 questions help us find out that the main reason for lack of confidence is the students’ ineffective learning strategy. Results of question 5 indicate that 75% of the students have recognized the importance of learning strategy in language learning, which lays a psychological foundation for the acceptance of a new learning method. Results of question 6–8 indicate that most of the participants have been aware of the existence of lexical chunks and have paid attention to the memory of them. But their vocabulary scope is limited as they only memorize the lexical chunks stressed by teachers. They have not realized the role that lexical chunks play in language learning, not to mention to be good at using lexical chunks in language output. Results of the last 2 questions reflect that in correspondence writing, students are unable to organize language naturally and they do not acquire the written skills contributing to coherence, idiom and fluency of writing. Statistic and analyses of the results obtaining from the questionnaire give the author a lot of inspiration in instruction design and provide objective data for this teaching experiment.

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Analyses on Students’ Use of Chunks

So as to figure out students’ use of chunks, the following data are extracted from the students’ writing: the number of total words, the figure of each classified lexical chunks, the figure of different lexical chunks, the figure of words in lexical chunks and the figure of recurring lexical chunks. Students’ Use of Chunks in Pretest. Detailed data of students’ use of chunks in pretest are displayed in Table 2, in which the repeated lexical chunks are calculated only once. Table 2. Comparison of Data in Pretest Students in CC

Students in EC

Total words

2135 Total words

2168

Average words

71

72

Average words

Poly words

175

Poly words

186

Institutional expressions

2

Institutional expressions

1

Phrasal constraints

83

Phrasal constraints

77

Sentence builders

80

Sentence builders

84

Total figure of lexical chunks

340

Total figure of lexical chunks

348

No. of words in lexical chunks 1029 No. of words in lexical chunks 1158 No. of different lexical chunks 177

No. of different lexical chunks 189

No. of recurring chunks

No. of recurring chunks

CC(Control Class)

163

159

EC(Experimental Class)

In Table 2, the number of total words, different categories of chunks and lexical chunks indicate that chunk knowledge mastered by students of both classes is almost at the same level, though the data in EC are slightly higher than those in CC. The number of words in lexical chunks shows that the lexical chunks students used occupy approximately 50%. With reference to the actual contents of students’ writing and the comparison between numbers of different kinds of chunks, the information depicts that students have mastered a certain number of lexical chunks, but these chunks are mostly poly words which consist of two or three words. Hence, as for participants, poly words are the lexical chunks which are easier to be acquired. The number of phrasal constraints and sentence builders show that these two types of chunks are less valued by participants. Comparison between the number of lexical chunks and that of recurring number illustrates that approximately 47% of all the used chunks are repeated by students. That is to say, students tend to use the same chunks to express certain meaning. Additionally, there is one point worthy of our attention, which is the number of institutional expressions. According to Nattinger and DeCario (2000)

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institutional expressions are pragmatically identifiable utterances which are often used to convey communicative meaning in oral social interaction. This type of lexical chunks is seldom used in business correspondence owing to the characters of written language used in correspondence. Therefore, students should try to avoid the use of such lexical chunks. Students’ Use of Chunks in Post-test. Table 3 shows the detailed information of students’ application of lexical chunks in post-test with the same statistic method. Table 3. Comparison of Data in Post-test Students in CC

Students in EC

Total words

2531 Total words

3096

Average words

84

Average words

103

Poly words

187

Poly words

193

Institutional expressions

1

Institutional expressions

0

Phrasal constraints

118

Phrasal constraints

169

Sentence builders

93

Sentence builders

143

Total figure of lexical chunks

399

Total figure of lexical chunks

505

No. of words in lexical chunks 1359 No. of words in lexical chunks 2216 No. of different lexical chunks 178

No. of different lexical chunks 286

No. of recurring chunks

No. of recurring chunks

CC(Control Class)

210

219

EC(Experimental Class)

Compared with CC, data in EC reflect that students in EC acquired more knowledge about lexical chunks than those in CC. The number of total words and different types of chunks shows that vocabulary scope of students in EC is larger than that of students in CC. In the writing of CC, lexical chunks occupy 53% of the words while the occupation rate is 72% in the writing of EC. From the comparison between the number of words in lexical chunks and that of total lexical chunks, we can see that lexical chunks mastered by students in EC are more complex than those mastered by students in CC. Besides poly words, they acquired many phrasal constrains and sentence builders. In contrast with the number of total chunks, the number of recurring chunks reflects that 40% of the used chunks are repeated by EC students while in CC students’ writing, the repetition rate is 54%. The data show that when expressing the same utterance, EC students has more options to choose than CC students.

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Analyses on SPSS Statistic Data

In this section the statistic data are collected and analyzed by SPSS to demonstrate two aspects respectively: the general disparities between scores of EC and CC in pretest and post-test as well as the relationship between lexical chunks and students’ writing competence of business correspondence. Significant Difference Analysis by Paired-Samples T Test. (1) T Test of Pretest Scores. With the assist of paired-sample T Test, a comparison between scores of EC and CC in pretest is listed in the following table. Table 4. Paired Samples Statistics Mean

N Std. Deviation Std. Error Mean

Pair 1 CC 17.2667 30 3.40318 EC 17.4333 30 2.90877

.62133 .53107

Table 5. Paired Samples Test Paired Differences

Pair 1

CC -EC

Mean

Std. Deviation

Std. Error Mean

95% Confidence Interval of the Difference Lower

Upper

−.16667

4.41067

.80527

−1.81364

1.48031

t

df

Sig. (2-tailed)

−.207

29

.936

In Table 4, basic information is reflected by the key index “mean”. By subtraction, it is obvious to notice that there is no apparent differences between the mean score of CC and EC. The index “Sig. (2-tailed) in Table 5 also illustrates this point as its figure is 0.936, a lot higher than the critical point 0.05. Hence, a conclusion can be drawn that before the teaching experiment the students’ writing competence of business correspondence in both classes is at the same level. (2) T Test of Post-test Scores. Tables 6 and 7 display the comparison of post-test scores of students in control class and experimental class. In Table 6, the mean score of students in experimental class is 1.7, a lot higher than that of students in control class. “Sig.” figure in Table 7 is 0.016, which is even lower than the critical point 0.05. Both statistic results demonstrate that a significant difference exist between the scores of these two classes in post-test. The “Std. Deviation” (standard deviation) of experimental class in Table 6 is 2.29667, which is not only lower than that of control class in Table 6 but also slightly lower than the figure of experimental class in pretest. This variation indicates that the gap between students’ writing competence is gradually narrowed.

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Table 6. Paired Samples Statistics Mean

N Std. Deviation Std. Error Mean

Pair 1 CC 18.3333 30 3.09987 EC 20.0333 30 2.29667

.56596 .41931

Table 7. Paired Samples Test Paired Differences

Pair 1

CC -EC

Mean

Std. Deviation

Std. Error Mean

95% Confidence Interval of the Difference Lower

Upper

−1.70000

3.62130

.66116

−3.05221

−.34779

t

df

Sig. (2-tailed)

−2.571

29

.016

(3) A Longitudinal Comparison. In order to test and verify that the academic performances of students in experimental class have indeed improved significantly, a longitudinal comparison is performed with the same data analysis method-T Test. Table 8. Paired Samples Statistics Mean

N Std. Deviation Std. Error Mean

Pair 1 EC Post-test 20.0333 30 2.29667 EC Pretest 17.4333 30 2.90877

.41931 .53107

From Table 8, we can see that there is a big gap between the mean of pretest and post-test. The index “Sig.” in Table 9 is 0.000, a lot lower than 0.05, revealing that the students’ writing competence has significantly improved. All the statistic data showed by the tables above certify the effectiveness of lexical approach. Cultivation of chunking ability is of great significance to the improvement of writing competence. Table 9. Paired Samples Test Paired Differences Mean

Std. Deviation

t Std. Error Mean

95% Confidence Interval of the Difference

.48090

1.61645 3.58355

Lower Pair 1 EC Post-test 2.60000 2.63400 - EC Pretest

df Sig. (2-tailed)

Upper 5.407 29 .000

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Function of Lexical Chunks in Improving Students’ Competence of Writing Business Correspondence. In order to explore the role that lexical chunks play in the cultivation of writing competence and find out which type of chunks could promote students’ writing ability to a large extent, two kinds of statistic methods are used here to make an analysis. (1) Descriptive Statistics about Students on Different Ranking. In this section, writings of 10 top-ranked students and of 10 bottom-ranked students from all the participants in post-test are picked out as samples for analysis. Two descriptive statistics about their use of lexical chunks are made to give a bird’s eye view on significance of acquiring plenty of lexical chunks (Tables 10 and 11). Table 10. Descriptive Statistics (top-ranked students) Mean Std. Deviation N scores

24.3

1.337

10

poly words

8.00

.667

10

phrasal constraints

6.80

.789

10

institutionalized expressions .00

.000

10

sentence builders

.632

10

6.20

From the mean figures of both tables, we can find a huge gap between the scores of two groups. Comparing the mean of poly words, phrasal constraints, sentence builders, it could be found out that top-ranked students master even more lexical chunks than bottom-ranked students. Among these numerical disparities, the D-value of sentence builders is most notable, the second is that of phrasal constraints while the last is that of poly words. The institutionalized expressions seem have no connection with writing scores. So an indication could be displayed that sentence builders and phrasal constraints have certain significance to students’ correspondence writing. Table 11. Descriptive Statistics (bottom-ranked students) Mean Std. Deviation N scores

15.00 1.333

10

poly words

5.4

.669

10

phrasal constraints

2.60

.679

10

institutionalized expressions .20

.422

10

sentence builders

.767

10

1.80

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Major Findings and Discussions Answers to Research Questions

Based on the above analyses, we have come to the answers to the research questions we have mentioned before. As for the first question whether it is appropriate to apply lexical approach to business correspondence writing, with the help of theoretical support and analysis of lexicalized character of business correspondence we mentioned, it is certain that lexical approach is ideal for business correspondence writing. Moreover, owing to the features of being easily memorized and used, the lexical approach is also suitable in improving students’ writing competence with low proficiency. In terms of the second question whether lexical approach can contribute to improving vocational college students’ correspondence writing competence, the data of this teaching experiment proved that lexical approach can effectively improve vocational students’ correspondence writing competence. Because lexical chunks are not only syntactically correct but are also used to convey certain contextual meaning, pragmatic meaning and pragmatic meaning. Mastering plenty of lexical chunks, students don’t feel difficult to write an accurate, coherent, idiomatic business correspondence fluently. If it really works, what is the relationship between the lexical chunks used by students and business correspondence writing competence? The answer to the third question could be reflected from two aspects. On the one hand, from the analysis of both lexical chunks as well as the relationship with business correspondence and the statistic data obtained from the experiment, a result can be found that the more lexical chunks students acquire, the better they write. On the other hand, due to different characteristics of each type of lexical chunks, not all lexical chunks can help to improve writing competence. Sentence builders, phrasal constraints and poly word enjoy advantages in correspondence writing owing to their unique functional meaning. 4.2

Pedagogical Implication

The previous research and experiments have provided sufficient evidence to illustrate the importance of lexical chunks to business correspondence writing. Thus, there is no denying that it has become the main teaching objective to enhance students’ acquisition of adequate lexical chunks in order to foster their correspondence writing competence. In the following, some tips for implementing lexical approach to correspondence writing will be displayed. Tips for Syllabus Design. According to the feedback obtained in the process of this experiment and with the reference to Lewis’ ideas on syllabus design (1993). Some tips are given on syllabus design of correspondence writing. In the first place, the materials used to provide lexical chunks should be contextualized rather than sentence based so that the meaning of the whole

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lexical items is clear in relation to the discourse in which it occurs. Only in this way could students conform to adequate pictures of language in use. In the second place, large high-frequency words and lexical phrases without any requirement for analysis should be given to the vocational students with low English proficiency. Therefore, with the enlarged size of mental lexicon, the condition under which students will form certain learning ability and communicative power could be created so that the competence in English might be developed. In the third place, a balance will be maintained between high-frequency words as well as poly words carrying considerable meaning and phrasal sentence pattern with low meaning content. The well-balanced range of lexical chunks ensures the development of communicative power. Also, the subsequent series of activities reflecting different types of lexical chunks will guarantee this development. Tips for Classroom Teaching. In this experiment, results of the questionnaire indicate that before stepping into the teaching of lexical chunks, we need to help students develop the basic concept of lexical chunks. Since sentence structures have been in the dominant position in language teaching for a long time, students have been accustomed to a learning mode in which grammatical rules are explored by analyzing sentences structure, which is the most important issue that should be highlighted in classroom teaching. So changing the wrong perception of the grammar as well as vocabulary dichotomy and shifting the core of language learning to the lexical chunks are the tasks we need to complete in the first place. An introduction of lexical chunks and their categories will give students a general awareness, which could be completed before class or in class. However, it is required to elaborate the idea by identifying lexical categories from the text or classifying them into different categories. Gradually, the attention of students could be transferred into lexical chunks with the following tips in the process of teaching. In the first place, chunking, the technical term created by Nattinger (1997), could be considered as a means by which lexical terms are memorized to some extent. This is the most important learning ability students need to cultivate in the stage of language input. In business correspondence writing, students’ recognition of bits that coherent written texts and discourses are made up of is essential to the competence of correspondence writing. Therefore, it is essential in the authentic teaching activities to explain the detailed chunking information to students and offer them enough materials to be sample letters or paragraphs so that the awareness of chunks could be enhanced by them. In the second place, learning lexical chunks involves a great deal more than simple memorization. Besides conscious learning of lexical item, the learning and acquisition of lexical chunks could be assisted by the following classroom procedures. (1) Help students identify lexical items correctly. (2) Encourage students to analyze the function of lexical chunks.

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(3) Help students to gradually transfer from short-term to long-term memory. (4) Encourage students to make full use of the dictionary as a learning resource. In the third place, different kinds of lexical chunks require different treatment in classroom teaching. Treating all the lexical chunks similarly will lead to chaos and inefficient learning. Different lexical chunks should be exploited in different ways as some types of lexical chunks deserve more time than others. Particularly, the lexical chunks represent precisely the frame of discourse as a whole and require special attention to make texts cohesive. In the fourth place, chunks note books are encouraged in class teaching. For lexical chunks, taking notes is an effective way of learning. Students need to be encouraged to organize the learned chunks in a way that is easy to be retrieved efficiently. In the fifth place, correctly response to the errors in correspondence writing. Teachers should avoid taking a relatively careless view on the errors in students’ written work. The errors should be analyzed and marked with different categories. Generally, the errors in students’ written work are reflected in two aspects, namely, surface error and content error. Surface error refers to the literal error, the mistaken use of words or time, etc while the content error refers to the mistakes in the process of drafting and editing. Different tags for different mistakes allow students to clearly recognize where their deficiencies are.

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13. Schmidt, R.: Awareness and second language acquisition. Annu. Rev. Appl. Linguist. 13(1), 206–226 (2013) 14. Hutchinson, T., Waters, A.: English For Specific Purposes. Shanghai Foreign Language Education Press (2012) 15. Littlewood, W.: Foreign and Second Language Learning. Foreign Language Teaching and Research Press (2020) 16. Weinert, R.: The role of formulaic language in second language acquisition: a review. Appl. Linguist. 16(2), 180–205 (1995) 17. Widdowson, H.G.: Knowledge of language and ability for use. Appl. Linguist. 19(10), 128–137 (2019) 18. Yule, G.: Pragmatics. Shanghai Foreign Language Education Press (2020) 19. Siying, C.: Lexical chunk learning and the improvement of language output ability: an empirical study of vocabulary teaching. J. Ningbo Univ. (5), 85–90 (2006) 20. Pingwen, C.: Lexical chunk theory and its application in English language teaching. Basic Engl. Stud. (6), 55–61 (2007) 21. Yingchun, C.: An investigation of the characteristics and learning methods of business English vocabulary. Modern Mall (10), 397–398 (2007) 22. Shiping, D.: A review of domestic research on second language lexical chunk teaching. Chin. Foreign Lang. 7, 63–75 (2008) 23. Huang, Y., Wang, H.: The “Chinese picture” of second language lexical chunk research. Foreign Lang. 3, 76–81 (2011) 24. Taizhi, L.: The advantages of lexical chunks in teaching English writing for foreign trade and the output training method. Foreign Lang. 1, 34–40 (2016) 25. Aijuan, L.: The frequency and categories of lexical chunks used by Chinese university students and. J. Changchun Normal Univ. 7, 137–140 (2010) 26. Chengyi, M.: Lexical chunks and their usage characteristics in English learners’ conversation. J. PLA Foreign Lang. Inst. 2, 58–62 (2008) 27. Caiying, W.: The application of vocabulary teaching method in the teaching of foreign trade correspondence. Examination Weekly (16), 51–52 (2017) 28. Wen, W.: The use of lexical chunk theory in business English teaching. J. Anhui Radio Telev. Univ. 2, 70–78 (2019) 29. Qian, Z.: The application of lexical chunk teaching method in teaching business correspondence writing. Youthful Years 8, 81–82 (2011) 30. Zhengzhong, Z.: Lexical chunks in business correspondence. J. Yangzhou Univ. 6, 219–221 (2007) 31. Xia, Z.: A Corpus-based Study on Chinese Advanced English Learners’ Use of Lexical Chunks. Science Press (2012)

Analytical Analysis of Nonlinear Vortex-Induced Vibration of Pipes Conveying Fluid Jian Liu and Yu Wang(B) College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China [email protected]

Abstract. In this paper, the vortex-induced vibrations of a long flexible pipe conveying steady fluid are investigated via a two-mode discretization of the governing differential equations. The governing nonlinear partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by applying the Galerkin’s method, which are then studied numerically for the pipe with primary resonances during lock-in for each of the first two modes. The method of multiple scales is utilized to obtain the steady-state responses of the coupled equations. It is found that the frequency-amplitude relationships present typical nonlinear phenomena, including jumping and multi-value. Numerical integrations are directly implemented in the vibration equations to verify the aforementioned analytical results. Furthermore, an analytical expression that predicts the lockin phenomenon range of external fluid velocity is derived. The influences of the velocity of internal and external fluids on the dynamical characteristics are discussed in detail. It is shown that the values of the external fluid velocity triggering the start and stop of the lock-in phenomenon will change with the value of the internal fluid velocity of the pipe. Keywords: Pipe conveying steady fluid · Vortex-induced vibration · Primary resonance · Lock-in domain · Method of multiple scale

1 Introduction Vortex-induced vibration (VIV) is a common phenomenon that could be observed in many potential areas, such as the chemical industry, heat exchanger tubes, bridges, power transmission lines, and marine structures such as cables and risers placed within ocean currents [1]. This kind of vibration would produce vortex-induced forces, which cause the cylindrical structures to transversely vibrate. The lock-in effects will happen when the shedding frequency is close to the natural frequency of the structure, and it is easy to excite the body to produce transverse resonance, which generates large vibrations and leads to fatigue failure of the cylindrical structure. So, it is thus imperative for structural engineers to analyze and simulate the effects of VIV on the dynamic responses of cylindrical pipes with or without internal fluid flow [2–6]. ICANDVC2023 best presentation paper. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 259–272, 2024. https://doi.org/10.1007/978-981-97-0554-2_20

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By coupling the structure and wake oscillators, Facchinetti et al. [7] analyzed the VIV of the coupled models and developed the most appropriate coupling terms and values of model parameters. Guo et al. [8] designed the first experimental study of vortex-excited vibrations in a flow-conducting tube. Blevins and Coughran [9] measured the vortex-induced vibration of an elastically supported circular cylinder in water with reduced velocity in an experiment. Païdoussis [10] pointed out that the critical internal fluid velocity means that the pipe begins to happen bucking. So, it is vital to study the VIV of pipes conveying fluid in the case of internal fluid velocity beyond the critical value. Keber et al. [11] investigated the weakly nonlinear dynamic properties of marine risers conveying internal flow and proposed a wake vibrator model that can be used to analyze the vortex-excited vibrations of the structures. More recently, Dai et al. [12] studied the nonlinear dynamic responses of pipes subjected to VIV by considering either subcritical or supercritical internal fluid velocities. Wang et al. [13] developed a new three-dimensional nonlinear model, considering the coupling between the structure and the fluid, for studying VIV of flexible pipes conveying internal fluid flow. Yang et al. [14] considered both the geometric and hydrodynamic nonlinearities to establish a threedimensional nonlinear dynamic model that can be presented to characterize the behavior of a flexible pipe conveying fluid under vortex-induced vibration. Xie et al. [15] analyzed the nonlinear dynamic behaviors of a flexible pipe conveying variable-density fluid and simultaneously subjected to vortex-induced vibrations (VIV) by the method of numerical simulations. Scholars also pay more attention to the pipe conveying pulsating fluid. OzÖz [16] point out the effect of internal pulsating flows on the dynamics of pipes could not be ignored due to the possible existence of parametric resonances. Panda et al. [17] investigated the nonlinear dynamics of a hinged-hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance. As a foundation, Dai et al. [18] further analyzed the steady-state responses of 2-D pipes conveying pulsating fluid flow under the action of external cross-flow. Ni et al. [19] investigated the in-plane and out-of-plane dynamics of a curved pipe conveying fluid by employing the method of multiple scales, and the method of multiple scales is employed in the instability regions of combination parametric resonance and principal parametric resonance. Zhang et al. [20] studied the dynamical modeling, multipulse orbits, and chaotic dynamics of a cantilevered pipe conveying pulsating fluid with harmonic external force by using the energy-phase method for the first time. Tan et al. [21] concentrated on presenting the characteristics of parametric resonances of the Timoshenko pipe with the pulsation of supercritical high-speed fluids. Hence, in the current work, a flexible pipe being not only subjected to axial steady internal flows but also placed in cross-flows can be considered, and the new phenomena might be found in the solution process of the method of multiple scales. The remainder of the paper is organized as follows: In Sect. 2, the physical model of fluid-structure interactions for this problem is constructed first. The method of multiple scales is adopted to analyze the nonlinear governing equations of the system with consideration of the primary resonances of the first two modes in Sect. 3. Frequency-response curves for the pipe in the case of the first two modes of primary resonances on the premise of lock-in

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and the results predicting lock-in domain are displayed in Sect. 4. Finally, the main findings of the present work are drawn out in Sect. 5.

2 Modelling and Formulations The analytical system can be made of a slender pipe conveying internal axial fluid with velocity U i in a uniform external cross-flow fluid with velocity U e , as shown in Fig. 1.

y L x z

D

Ui

Ue Fig. 1. Schematic of a pipe conveying fluid subjected to VIV.

For simplification, therefore, it is assumed that the motions of the flexible pipe would be restricted in the plane perpendicular to the external flow direction. This assumption has been demonstrated to be quite reasonable when its in-line displacement is relatively small [22]. In addition, the pipe can be simplified to a Euler-Bernoulli beam because of its exceedingly large aspect ratio of the pipe. And the internal and external fluids are thought to be incompressible. Based on these assumptions, if gravity, buoyancy, internal damping, externally imposed tension, and pressurization effects are either absent or neglected, the equation of motion for the fluid-conveying pipe subjected to vortex-induced forces can be expressed as m

2 ∂ 2 y(x, t) ∂ 4 y(x, t) ∂ 2 y(x, t) 2 ∂ y(x, t) + m + EI + 2m U U i f f i ∂t 2  ∂x4 ∂t∂x ∂x2 2  2  L EAp ∂ y(x, t) ∂y(x, t) − dx = f (x, t), 2L 0 ∂x ∂x2

(1)

where y(x,t) is the lateral displacement of the pipe, and x and t are the axial coordinate and time, respectively. mf , mp , and md respectively are defined as the mass per unit length of the internal fluid, the pipe and the additional fluid and m = mf + mp + md . E is Young’s modulus of pipe’s material. EI is the flexural stiffness of the pipe, Ap is the cross-section area of the pipe, L is the pipe length, and f (x,t) represents the effect of vortex-induced force acting on the pipe, which may be given as f (x, t) = fD (x, t) + fL (x, t),

(2)

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where f D (x,t) is the hydrodynamic damping force acting in the lateral direction, and f L (x,t) is the lift force. These can be respectively written as ∂y(x, t) 1 , fD (x, t) = − CD ρ0 DUe 2 ∂t fL (x, t) =

1 CL0 ρ0 DUe2 q(x, t), 4

(3) (4)

where D is the external diameter of the pipe, ρ 0 is the fluid density, C D and C L0 are the damping and lift coefficients in the transverse direction, which can be respectively taken as 1.2 and 0.3 [7]. In Eq. (4), q(x,t) is defined as the reduced lift coefficient, which presents the behavior of the near wake and can be modeled according to the following van der Pol equation.   ∂q(x, t) P ∂ 2 y(x, t) ∂ 2 q(x, t) 2 2 + ω , q(x, t) + λω − 1 q = s s ∂t 2 ∂t D ∂t

(5)

where ωs is the vortex-shedding frequency, and the values found for λ and P are 0.3 and 12 respectively [7]. Otherwise, Eq. (1) and Eq. (5) can be dimensionless by introducing the following non-dimensional quantities, which are written as mf EI y x ,v = η = ,ξ = ,τ = t Ui L, D L mL4 EI mf Ap D2 ρ0 DUe L2 ,γ = , c = CD √ , m 2I 2 mEI m CL ρ0 Ue2 L4 , Ωs = ωs L2 , α= 0 4EI EI

β=

then the nonlinear coupled equations for VIV of the fluid-conveying pipe can be written in dimensionless forms as follows:      1 ∂η 2 2

∂ 2η ∂ 4η ∂η ∂ 2η ∂ 2η 2∂ η + 2 + v + + c βv − γ d ξ − αq = 0, ∂τ 2 ∂ξ 4 ∂τ ∂ξ ∂τ ∂ξ 2 ∂ξ ∂ξ 2 0 (6)   ∂q ∂ 2q ∂ 2η 2 2 + Ω q + λΩ − 1 q − P = 0. s s ∂τ 2 ∂τ ∂τ 2

(7)

In this study, what we primarily focus on is to explore the primary resonances and lock-in conditions on the dynamics of pipes involving the first two modes, which can be convenient for comparing numerical integrations (NI) with MMS. Therefore, the Galerkin’s method is used to discretize the equation of motion. η(ξ, τ ) =

2  i=1

φi (ξ )ηi (τ ),

(8)

Analytical Analysis of nonlinear Vortex-Induced Vibration of Pipes Conveying Fluid

q(ξ, τ ) =

2 

φi (ξ )qi (τ ),

263

(9)

i=1

where φ i (ξ ) = sin(iπ ξ ) are the eigenfunctions of a simply supported beam, ηi and qi are the corresponding generalized coordinates. Substituting Eq. (8), Eq. (9) into Eq. (6), Eq. (7), depending on the orthogonality of the displacement function and multiplying φ i (ξ ) by on both sides and integrating from 0 to 1, four coupled equations of ηi and qi can be expressed as (we substitute ηi and qi for ηi and qi )  

η¨ 1 + 2 βvB12 η˙ 2 + ω1 η1 − γ R11 η1 H11 η12 + H22 η22 − αq1 + cη˙ 1 = 0, (10)  

η¨ 2 + 2 βvB21 η˙ 1 + ω2 η2 − γ R22 η2 H11 η12 + H22 η22 − αq2 + cη˙ 2 = 0,

(11)

  q¨ 1 + Ωs2 q1 − λΩs q˙ 1 + λΩs q22 q˙ 1 + 3 2q12 q˙ 1 + 2q1 q2 q˙ 1 − P η¨ 1 = 0,

(12)

  q¨ 2 + Ωs2 q2 − λΩs q˙ 2 + λΩs q12 q˙ 2 + 3 2q22 q˙ 2 + 2q2 q1 q˙ 1 − P η¨ 2 = 0,

(13)

where the other coefficients in the Eqs. (10–13) can be computed from the integrals of the eigenfunctions φ i (ξ ) = sin(iπ ξ ). These coefficients are given by  1  1 αn = Φn Φn(4) d ξ + v2 Φn Φn d ξ, (14) 0



1

Hnn = 0

Φn Φn d ξ, Rnn =

0



1 0

Φn Φn d ξ, Bnm =

 0

1

Φn Φn d ξ.

(15)

Taking the first order mode for instance, in order to obtain the frequency of structure and wake oscillator and the stability of corresponding solution, Eqs. (10) and (12) can be simplified by dropping the damping and non-linear terms, which may be written as η¨ 1 + α1 η1 − αq1 = 0,

(16)

q¨ 1 + Ωs2 q1 − P η¨ 1 = 0.

(17) 

Now, considering the response of Eqs. (16) and (17) as η1 (τ ) = η 1 eikτ , q1 (τ ) =  ikτ q 1 e and substituting them into Eqs. (16) and (17)     −k 2 + α1 η 1 − α q 1 = 0, (18)     pk 2 η 1 + −k 2 + Ωs2 q 1 = 0. For nonzero solution the determinant of coefficients must be equal to zero, so   k 4 + Pα − Ωs2 − α1 k 2 + Ωs2 α1 = 0.

(19)

(20)

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However, when the interaction between fluid and structure is considered, the Eqs. (10) and (12) can be written as η¨ 1 + α1 η1 − αq1 + cη˙ 1 = 0,

(21)

q¨ 1 + Ωs2 q1 − λΩs q˙ 1 − P η¨ 1 = 0.

(22)

And with the similar processes, one can be obtained the following equation:     k 4 + (Ωs λi − ci)k 3 + cΩs λ + Pα − Ωs2 − α1 k 2 + ciΩs2 − Ωs α1 λi k + Ωs2 α1 = 0.

(23)

The vibration frequency of the wake oscillator can be calculated by solving the equation with secant method.

3 Method of Multiple Scales In this section, the method of multiple scales is employed to analyze the coupling system. Firstly, introducing the small dimensionless bookkeeping parameter ε to signify different orders of amplitude, one can rescale v = εv, α = εα, c = εc, λ = ελ, P = εP. In order to solve Eqs. (10–13), the uniform expansion of the solution can be assumed as η1 = u11 (T0 , T1 ) + εu12 (T0 , T1 ), η2 = u21 (T0 , T1 ) + εu22 (T0 , T1 ) q1 = u31 (T0 , T1 ) + εu32 (T0 , T1 )

(24)

or η1 = u11 (T0 , T1 ) + εu12 (T0 , T1 ), η2 = u21 (T0 , T1 ) + εu22 (T0 , T1 ) q2 = u41 (T0 , T1 ) + εu42 (T0 , T1 )

(25)

where T 0 = t and T 1 = εt denote the fast and slow time scales. The first and second order time derivatives are d2 d = D0 + εD1 , 2 = (D0 + εD1 )2 , dt dt

(26)

where D0 = ∂ / ∂T 0 , D1 = ∂ / ∂T 1 . Order ε0 D02 u11 + ω12 u11 = 0, D02 u21 + ω22 u21 = 0, D02 u31 + Ωs2 u31 = 0, D02 u41 + Ωs2 u41 = 0. (27) Order ε1

D02 u12 + ω12 u12 = −2D0 D1 u11 − 2B12 v βD0 u21 − cD0 u11   3 2 , +αu31 + γ R11 H11 u11 + H22 u11 u21

D02 u22 + ω22 u22 = −2D0 D1 u21 − 2B21 v βD0 u11 − cD0 u22   2 3 , +αu41 + γ R22 H11 u11 u21 + H22 u21

(28)

(29)

Analytical Analysis of nonlinear Vortex-Induced Vibration of Pipes Conveying Fluid

D02 u32

+ Ωs2 u32

  3 2 2 = λΩs u41 D0 u31 + 2u31 u41 D0 u41 + D0 u31 − u31 D0 u31 2

265

(30)

+PD02 u11 − 2D0 D1 u31 ,

D02 u42

+ Ωs2 u42

  3 2 2 = λΩs 2u31 u41 D0 u41 + u31 D0 u41 − u41 D0 u41 − D0 u41 2

(31)

+PD02 u22 − 2D0 D1 u41 ,

3.1 Primary Resonance of First Mode In this subsection, the reduced-order equations for the primary resonance of the first mode are constructed to analyze the dynamical behaviors of the pipe. The nearness of Ω s to ω1 is described by using the detuning parameter σ. Therefore, the relationship between frequencies can be respectively written as Ωs = ω1 + εσ.

(32)

The solution of Eq. (27) can be expressed as u11 = A1 (T1 )eiω1 T0 + c.c, u21 = A2 (T1 )eiω1 T0 + c.c, u31 = A3 (T1 )eiΩs T0 + c.c. (33) Substituting Eq. (32) and (33) into Eqs. (10–12) and eliminating the secular terms yield the following equations. −2A1 jω1 − cA1 jω1 + 3γ H11 R11 A21 A1 + 8H11 R11 A1 A2 A2 + αA3 ejσ T1 = 0 −2A3 jΩs −

3λΩs2 jA23 A3 + λΩs2 jA3 + PA1 j 2 ω12 e−jσ T1 = 0 2

(34)

(35)

where j2 = −1. Then the polar forms of Aj are expressed as Aj =

1 aj (T1 )eiθj (T1 ) , (j = 1, 2, 3) 2

(36)

where aj and θ j are the amplitude and phase angle, respectively. Substituting Eq. (36) into Eqs. (34) and (35) and separating the real and imaginary parts, one can obtain the modulation equations as αa3 sin(θ3 − θ1 + σ T1 ) ca1 + , 2 2ω1

(37)

3λΩs a33 Pa1 ω12 sin(θ1 − θ3 − σ T1 ) λΩs a3 + + , 16 2 2

(38)

a˙ 1 = − a˙ 3 = −

a1 θ˙1 = −

3γ H11 R11 a13 αa3 cos(θ3 − θ1 + σ T1 ) − , 8 2

(39)

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a3 θ˙3 =

Pa1 ω12 cos(θ1 − θ3 − σ T1 ) . 2Ωs

(40)

Now assuming ϕ 1 = θ 3 – θ 1 + σ T 1 , αa3 sin(ϕ1 ) ca1 + , 2 2ω1

(41)

Pa1 ω12 sin(ϕ1 ) 3λΩs a33 λΩs a3 + + , 16 2 2Ωs

(42)

a˙ 1 = − a˙ 3 = − ϕ˙1 =

Pa1 cos(ϕ1 )ω12 3γ π 4 a12 αa3 cos(ϕ1 ) − + + σ. 2Ωs a3 8ω1 2a1 ω1

(43)

By putting derivative of a, b and ϕ 1 equal to zero in Eqs. (41–43), the equation governing equilibrium solution amplitude will be obtained. 3.2 Primary Resonance of Second Mode The case for this primary resonance will appear when the external excitation frequency Ω s ≈ ω2 . Again, let Ωs = ω2 + εσ.

(44)

The solution of Eq. (27) can be expressed as u11 = A1 (T1 )eiω1 T0 + c.c, u21 = A2 (T1 )eiω1 T0 + c.c, u41 = A4 (T1 )eiΩs T0 + c.c, (45) Aj =

1 aj (T1 )eiθj (T1 ) , (j = 1, 2, 4). 2

(46)

By applying the similar procedures as above, the equation about primary resonance of second mode can be expressed as αa4 sin(ϕ2 ) ca2 + , 2 2ω2

(47)

Pa2 ω22 sin(ϕ2 ) 3λΩs a43 λΩs a4 + + , 16 2 2Ωs

(48)

a˙ 2 = − a˙ 4 = − ϕ˙ 2 =

Pa2 cos(ϕ2 )ω22 6γ π 4 a22 αa4 cos(ϕ2 ) − + + σ, 2Ωs a4 ω2 2a2 ω2

(49)

where ϕ 2 = θ 4 – θ 2 + σ T 1 . The frequency response relationship equation for the second-order primary resonance can be obtained from Eqs. (47–49).

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4 Results and Discussions In this part, numerical results of the nonlinear responses obtained by the analytical solutions are presented in the presence of the first two modes of primary resonance, respectively. And the validations of the abovementioned multiple scale analysis are carried out. Meanwhile, the dynamical responses of the pipe for the coupled system with different system parameters under lock-in conditions are displayed, and the relationship between the unstable domain of the linear system and the lock-in domain is discussed. In the following calculations, Young’s modulus E = 210 GPa, the length L = 150 m, the outer diameter D = 0.25 m, the inner diameter Di = 0.125 m, the density of the pipe ρ p = 7850 kg/m3 , the density of the external fluid ρ o = 1020 kg/m3 , and the density of the internal fluid ρ i = 870 kg/m3 . And the values of the internal flow velocity considered in this study are all less than the critical fluid velocity [12] that causes the buckling failure of the pipe. The shedding frequency and external fluid reduced velocity are defined as Ω s = U r SΩ 1 and U r = 2π U e / (Ω 1 D), respectively, where S is the Strouhal number, which is constant with a value of 0.2 for a large range of Reynolds numbers [18]. 4.1 Primary Resonance of First Two Modes The Numerical results of the nonlinear responses obtained by the analytical solutions are showed in the presence of primary resonance of first two modes. And the abovementioned multiple scale analysis is verified by applying numerical integrations.

F1

D1

G1

C1 A1

stable unstable NI

H1

B1

σ a) Ωs≈ω1

I1

stable unstable

E2 Amplitude a2

Amplitude a1

E1

F2

D2

NI G2

C2 A2

H2

I2

B2

σ b) Ωs≈ω2

Fig. 2. Frequency-response curves of the first two orders.

The responses are described in terms of the maximum vibration amplitude A/D (multiple of pipe’s diameter D) at x = 0.5L. Figure 2 a) presents the response curves of a1 when Ω s ≈ ω1 . And the red stars are the numerical results obtained by directly integrating the dynamical Eq. (13). It can be seen that the analytical results agree well with those of the numerical integration. The jumping and multiple-value phenomena can be observed. Meanwhile, the stability of the responses is listed in Table 1. It also can be observed from the Fig. 2 a) that the overall trend of pipeline maximum vibration amplitude is to increase first and then decrease with the tuning parameters. Unlike the normal amplitude-frequency curve, the curve is smoother rather than sharp near

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zero in the Fig. 2 a), which can be illustrated well in the following lock-in frequency phenomenon. Otherwise, the stability of the steady-state periodic responses of the system can be judged by inspecting the eigenvalues of the Jacobi matrix. The method is: a point can be taken on each branch and the eigenvalues of the Jacobi matrix of these points are obtained, which are used to determine the stability of the corresponding branch. Table 1. The eigenvalues of the Jacobian matrix with frequencies in the multi-value regimes in Fig. 2a. A1

B1

C1

D1

F1

G1

H1

I1

σ = -4

σ = −2.4

σ=− 2.4

σ=− σ=1 2.4

σ = 2.7

σ = 2.7

σ= 2.7

σ=4

−32.2

−0.1

0.8

−11.59

−8.9

−7.9

−0.3

−4.0

−1.6

−0.1

−0.3

−0.6

−3.4

−31.5 18.9

2.0

−2.1

−2.2

−0.6 ± 19.8i −0.4 ± 3.2i −11.7 −7.49 −8.1 −0.3 1.2 stable

stable

unstable stable

E1

stable

unstable unstable stable stable

In Fig. 2 b), the responses are described in terms of the maximum vibration amplitude A/D of pipe at x = 0.25 L for the primary resonance of the second mode. Obviously, compared to the primary resonance of the first mode, the maximum amplitude does not show a considerable change. The jumping and multiple-value phenomena can also be observed. Meanwhile, the stability of the responses is listed in Table 2. The analytical results agree well with those of the numerical integration. Table 2. The eigenvalues of the Jacobian matrix with frequencies in the multi-value regimes in Fig. 2b. A2

B2

C2

D2

F2

G2

H2

I2

σ = −12

σ = −9

σ = −9

σ = −9 σ = 5

σ= 10.7

σ= 10.7

σ= 10.7

σ= 20

−129.1 −0.02

3.8

−46.9

−36.5 −31.2

−1.1

−14.7

−5.7

−2.5

−1.3

−0.01

−13.7

−117.5 −71.4

7.9

−8.3

9.4

−1.8 ± 20.6i −1.9 ± 8.5i −51.9 −31.4 −34.2 −0.5 6.2 stable

stable

unstable stable

E2

stable

unstable unstable stable

stable

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4.2 Frequency Lock-In Domain In Fig. 4, the main reason for increasing structure oscillations may be found in the variations in frequency of the wake oscillations due to the interaction between fluid and structure shown. At the same time, Fig. 3 a) shows that the vortex shedding frequency is locked on the natural frequency of the structure for a special domain of the fluid speed. It demonstrates that outside of this domain, the frequency of vortex shedding is in good agreement with the predicted value by the Strouhal number. The domain where the vortex synchronizes itself to the natural frequency of the pipe is called the lock-in domain [7], which causes the structure to oscillate with a large amplitude. 20

20

considering the interaction between structure and wake without considering the interaction

16

 structure

16

unstable

 wake

domain

[10]

of linear system

12

k

k

12

8 8

4 4 2

3

4

5

6

7

8

9

10

0

0

1

2

3

4

5

6

7

8

9

10

Ur Ur a) Relationship between the frequency of the b) Eigenvalue variations of linear system wake oscillator and external cross-flow reduced with respect to the variations of external velocity with considering damping and nonlinear cross-flow reduced velocity. terms.

Fig. 3. The prediction of frequency lock-in domain

The relationship between the frequency lock-in domain and the instability domain of an undamped linear system is considered. The variations of the eigenvalue of the undamped linear system with respect to the external cross-flow reduced velocity are shown in Fig. 3 a) using Eq. (17). The lock-in domain of Fig. 3 b) corresponds to the unstable domain of Fig. 3 a), as can be seen by comparing Figures Fig. 3 b) and Fig. 3 a). It demonstrates that while a linear system may extract a good forecast of the lock-in domain, it is not possible to anticipate the magnitude of structure and wake oscillation. According to the findings, the analytical multiple scale perturbation method can accurately estimate the lock-in domain and amplitude. 4.3 Effect of the Other Parameters on Structure In the above calculation, the internal fluid velocity of the pipe is zero. The response of the pipe should be focused on when the velocity of both the internal fluid and the external

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flow field changes. The responses are described in terms of the maximum vibration amplitude A/D at x = 0.5 L with respect to the external fluid reduced velocity U r in Fig. 4, which presents a tendency is consistent with the result of Literature.18. Figure 4 also indicates that the internal fluid velocity of the pipe has a relevant effect on the values of the lock-in domain. As expected, it shows that by increasing the internal fluid velocity, the lock-in domain begins at a higher external fluid speed. Moreover, even though the range of the lock-in phenomenon increases, which can be explained by the solution of Eqs. (18) and (19) well. 0.4

 v=0  v=1  v=2

0.3

A/D

v=3 0.2

0.1

0.0 2

4

6

8

10

12

14

16

Ur Fig. 4. Relationship between the peak amplitude and the external cross-flow reduced velocity for the first mode with different steady internal fluid velocities. 0.3

1.2

E=200GPa E=210GPa E=220GPa

0.8

A/D

A/D

0.2

 0.5c  2c  c

0.1

0.4

0.0 2

4

6

8

10

Ur Fig. 5. The influence of Young’s modulus on maximum amplitude of pipe

0.0

2

4

6

8

10

Ur Fig. 6. The influence of damping coefficient on VIV

In Fig. 5, the influence of Young’s modulus on the maximum amplitude of the pipe is discussed, and it is easy to observe, with the increase of Young’s modulus, the maximum amplitude would decrease because Young’s modulus directly affects the flexural stiffness of the pipe. Finally, based on the exploration of multi-valued phenomena, damping coefficients can be adapted to observe the vibrations of the pipe. As expected, the maximum vibration amplitude of the pipe will become larger when the damping coefficient

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is reduced, and the multi-value phenomenon that occurs above will disappear when the damping coefficient becomes large in Fig. 6. So it may be explained that other special amplitudes are not excited because of the increase in damping coefficient.

5 Conclusions In this paper, the primary objective is to explore the dynamic behavior of the pipe under the combination effect of internal and external flow by using the method of multiple scales and verify the accuracy of the analytical method by numerical methods. Based on the formulations and results, several significant conclusions can be drawn. 1. For the case of parametric resonance of the first two modes of the piping system for a given lock-in condition, the range of variation of the peak pipe amplitude increases and then decreases with the increase of the detuning parameter σ, with jumping phenomena and multi-value regions. 2. By determining the stability of the solution of the linear system, it is found that the unstable domain of the solution is in good agreement with the frequency lock-in domain. Hence, the linear system can well predict the frequency lock-in domain of VIV. 3. When the internal fluid velocity of the pipe remains constant, with the increase of the velocity of external cross-flow, the maximum amplitude of the pipe increases first and then decreases. And when the internal fluid velocity of the pipe increases, it happens that the lock-in domain enlarges on both sides, and the frequency lock-in phenomenon will be delayed.

References 1. Brika, D., Laneville, A.: Vortex-induced vibrations of a long flexible circular cylinder. J. Fluid Mech. 250, 481–508 (1993) 2. Zdravkovich, M.M.: Different modes of vortex shedding: an overview. J. Fluids Struct. 10(5), 427–437 (1996) 3. Sarpkaya, T.: A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct. 19(4), 389–447 (2004) 4. Gabbai, R.D., Benaroya, H.: An overview of modeling and experiments of vortex-induced vibration of circular cylinders. J. Sound Vib. 282(3–5), 575–616 (2005) 5. Huang, X., Zhang, H., Wang, X.: An overview on the study of vortex-induced vibration of marine riser. J. Marine Sci. 27(04), 95–101 (2009) 6. Païdoussis, M.P., Price, S.J., De Langre, E.: Fluid-Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University, London (2010) 7. Facchinetti, M.L., De Langre, E., Biolley, F.: Coupling of structure and wake oscillators in vortex-induced vibrations. J. Fluids Struct. 19(2), 123–140 (2004) 8. Guo, H.Y., Lou, M.: Effect of internal flow on vortex-induced vibration of risers. J. Fluids Struct. 24(4), 496–504 (2008) 9. Blevins, R.D., Coughran, C.S.: Experimental investigation of vortex-induced vibration in one and two dimensions with variable mass, damping, and reynolds number. J. Fluids Eng. 131(10), 101202 (2009)

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10. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow. Academic press, London (1998) 11. Keber, M., Wiercigroch, M.: Dynamics of a vertical riser with weak structural nonlinearity excited by wakes. J. Sound Vib. 315(3), 685–699 (2008) 12. Dai, H.L., Wang, L., Qian, Q., Ni, Q.: Vortex-induced vibrations of pipes conveying fluid in the subcritical and supercritical regimes. J. Fluids Struct. 39(5), 322–334 (2013) 13. Wang, L., Jiang, T.L., Dai, H.L., Ni, Q.: Three-dimensional vortex-induced vibrations of supported pipes conveying fluid based on wake oscillator models. J. Sound Vib. 422, 590–612 (2018) 14. Yang, W., Ai, Z., Zhang, X., Chang, X., Gou, R.: Nonlinear dynamics of three-dimensional vortex-induced vibration prediction model for a flexible fluid-conveying pipe. Int. J. Mech. Sci. 138, 99–109 (2018) 15. Xie, W., Gao, X., Wang, E., Xu, W., Bai, Y.: An investigation of the nonlinear dynamic response of a flexible pipe undergoing vortex-induced vibrations and conveying internal fluid with variable-density. Ocean Eng. 183, 453–468 (2019) 16. OzÖz, H.R.: Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity. Int. J. Non-Linear Mech. 36(7), 1031–1039 (2001) 17. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vib. 309(3), 375–406 (2008) 18. Dai, H.L., Wang, L., Qian, Q., Ni, Q.: Vortex-induced vibrations of pipes conveying pulsating fluid. Ocean Eng. 77, 12–22 (2014) 19. Ni, Q., Tang, M., Wang, Y., Wang, L.: In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid. Nonlinear Dyn. 75, 603–619 (2014) 20. Zhang, Y.F., Yao, M.H., Zhang, W., Wen, B.C.: Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance. Aerosp. Sci. Technol. 68, 441–453 (2017) 21. Tan, X., Ding, H.: Parametric resonances of Timoshenko pipes conveying pulsating high-speed fluids. J. Sound Vib. 485, 115594 (2020)

RBF Neural Network for Feature Selection Using Sparsity Method Tao Gao1(B) , Jun Yang1 , Yongyong Xu1 , Baosheng Qian1 , Bin Wang1 , and Ruoxi Yu2 1

The 15th Research Institute of China Electronics Technology Group Corporation, Beijing 100191, China gaotao [email protected], wbin [email protected] 2 School of Information Science and Technology, Tsinghua University, Beijing 100191, China [email protected] Abstract. The simplicity of RBF neural network (RBFNN) mainly depends on the input (feature) and hidden nodes. To compact the structure of RBFNN, in this paper, a weight decay regularizer based integrated feature selection (FS) strategy is proposed to prune the input nodes. The training procedure of our FS method is explainable: firstly using clustering algorithm, the initialization process of RBFNN is detailedly described; then a novel memory based gradient method is used to promote the process of feature selection and parameter optimization; finally, the weight decay terms for FS will tend to different values. Using two regression problems, the validation of our model to realize feature selection and predict the real outputs is proved. Keywords: RBF · Neural network · Gradient memory selection · Sparsity · Interpretability

1

· Feature

Introduction

The RBFNN has been proved to be a universal approximator and it can build any nonlinear mappings between the inputs and outputs. To improve the application ability, researchers mainly focus on designing the network structure, devising optimization methods for parameters (i.e., RBF centers, widths and output weights), determining the network size (including feature selection and hidden node pruning) [1] and the improvement of model robustness [2]. Considering the noise/uncertainty existed in the data, the interval/general type-2 RBFNN is constructed [3,4]. Differ from the type-1 RBFNN, type-2 models are with four layers and they need the type-reducing layer to deal with the hidden node information. Although the initial structure of RBFNNs can be randomly generated, usually they are initialized using clustering algorithms, for random initialization procedure is unexplainable. Clustering algorithms, such as Supported by the Major Project of National Natural Science Foundation of China (No. U19B2019). c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 273–284, 2024. https://doi.org/10.1007/978-981-97-0554-2_21

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k-means, FCM and their variants, can be used to structure initialization [5]. In addition, the sensitivity based method is another alternative to generate the RBFNN. In [6], using sensitivity a novel BRFNN construction method is proposed, where the sensitivity is defined as the expectation of the square of output change caused by the noise of the Gaussian centers, and the number of hidden nodes and the initial Gaussian centers are determined by the maximization of the output’s sensitivity to the training data. In [7], stochastic sensitivity measure is used to construct a localized generalization error bound which is applied to RBFNN structure selection. To optimize the initial network, optimization methods including gradient and evolutionary methods and so on are widely used to adjust the RBFNN structure parameters [8]. In this paper, we use clustering algorithm to initialize Gaussian centers and output weights of RBFNN simultaneously. In addition, the memory-based gradient method is designed to tune all the parameters. To obtain the compact structure, various feature selection and hidden node pruning methods are proposed to delete the bad or redundant nodes existed in the RBFNNs. To realize feature selection, a RBFNN-based sensitivity measure (SM) is presented in [9], which perturbs the input feature one by one and then computes the SM. By comparing all the values of SM of all features, the feature with low values will be eliminated. In [10], an integrated feature selection method for RBFNNs is proposed, and this method uses the weighted Euclidean distance to select the important features. To compute the feature selection weights, a logarithmic barrier function based error function is designed. In [11], to rank the importance of each feature, a novel feature selection method for RBFNNs using the partial derivative is proposed, and for the most important features, the derivatives vary most through the range of feature values. In [12], to delete the bad features, a feature modulator is introduced, which can change the value of Gaussian function automatically during training. In addition, considering the redundant features, the Pearson correlation coefficient is designed to be the regularizer to control the redundancy between the selected important features. To compress the structure of RBFNN, an adaptive PSO method is designed to select the important hidden nodes, in the meanwhile, it can optimize the structure parameters [1]. In [13], using sensitivity measure, a novel rule (hidden node) generation approach for fuzzy neural network, which is based on radial basis function neurons, is proposed to realize on-line hidden node growing and pruning. On the basis of [11], Jarkko proposes simultaneous feature selection and hidden node pruning method for RBF networks. Except for the generalization ability and simplicity, the robustness improvement of different RBFNNs is another important task. In [2], using the KullbackLeibler divergence, an objective function is defined to improve the fault tolerance of RBFNNs. To improve the robustness of the RBFNNs, the sensitivity definition is designed as the regularization term in the error function, and experiments show that the sensitivity of the optimal weight of the new model to additive noise and multiplicative noise is lower than that of the traditional model [15].

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Inspired by the sparsity method [16], in this paper, we design a regularization term based feature selection method for RBFNN, and the sparsity term is designed using the wight decay. It is an integrated feature selection method and its feature selection procedure is interpretable. For convenience, we call our model FS-RBFNN. The organization of the remaining part of this paper is as below. In Sect. 2, the RBF used to realize feature selection is introduced. Section 3 shows two supporting simulation results. Some conclusions and future works are presented in Sect. 4.

2

Structure of RBF for Feature Selection

Fig. 1. Topological structure of RBF for feature selection.

Just like Fig. 1 shows, the RBF is a model with three layers, and each layer possesses different number of nodes. Feature Layer: Feature layer is an input layer to map the original/normalized sample features directly. For one dataset, supposing its sample owns t input features, i.e., x = (x1 , x2 , · · · , xt )T ⊂ Rt , and corresponding to the number of features, there should be t nodes in the first layer. In this paper, we simply consider that the features include good features and bad features, and if the bad features are deleted, the structure of RBF will be simplified.

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Feature Selection Layer: This layer in Fig. 1 has n nodes, and usually each node realizes the Gaussian mapping of the input vector xj using the following expression. ||xj − mi ||2 ), (1) fji = exp(− b2i where xj = (xj1 , xj2 , · · · , xjt )T , mi = (mi1 , mi2 , · · · , mit )T . Actually, expression (1) can be rewritten as below: fji =

t 

μip (xjp ),

(2)

p=1

where

 (xjp − mip )2  . μip (xjp ) = exp − b2i

(3)

For improving the approximation ability of RBFNN, different supervised or unsupervised methods are designed to optimize the parameters mip and bi . Differ from the traditional methods, we do not update the Gaussian parameters and let bi = 1. In addition, to realize feature selection, the weight wp is employed to evaluate the importance of the pth feature xp . Hence, the Gaussian function we use is as below:   (4) μip (xjp ) = exp − (xjp − mip )2 wp2 , we call expression (4 Gate F unction), and wp works like the Gate W idth to control the open and close of the Gate. When wp = 0, μip (xjp ) = 1, the feature xp has no influence on the value of fji , hence it can be considered as a bad feature; otherwise, it is a good feature. ⎧ ⎨ xp is unimportant, wp2 < γ, (5) ⎩ xp is important, wp2 ≥ γ, where γ is the threshold for feature selection. Output Layer: The output node uses the following expression to compute yj : yj =

n 

fji ui ,

(6)

i=1

where ui is the output weight to adjust the value of fji . 2.1

Parameter Initialization Using Clustering Algorithm

We focus on dealing with the single output regression problems, and supposing xj = (xj1 , xj2 , · · · , xjt )T ∈ Rt is the jth input sample, oj ∈ R is the jth corresponding desired output. Let XT r = (x1 , x2 , · · · , xJT r ) be the

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original or normalized training input matrix, and OT r = (o1 , o2 , · · · , oJT r ) be the corresponding training output matrix. Constructing the augmented matrix XT∗ r = (x∗1 , x∗2 , · · · , x∗JT r ), where x∗j = (xTj , oj )T , and JT r denotes the total number of training samples. Using a certain clustering algorithm, XT∗ r is clustered into n groups, and each group has one clustering center vi∗ = ((vix )T , vio )T , where i = 1, 2, · · · , n. For an unfamiliar dataset, we are blind to its features, so we would rather take them as bad features by setting the initial value of wpk=0 be almost 0 at the = (mi1 , mi2 , · · · , mit )T = vix , and start of training. For other parameters, mk=0 i k=0 we fix the value of mip during training; ui = vio , which will be updated using the supervised learning algorithm as wp , where k denotes the iteration. 2.2

Interpretable Integrated Feature Selection Process

For the RBF, to simultaneously select the necessary features and realize parameter identification, the error function is constructed by the sum of the squared error and a weight decay (L2 norm) based regularization term, E (w) =

JT r t   (yj − oj )2 +λ wp2 , 2 p=1 j=1

(7)

where w = (w1 , w2 , · · · , wt , u1 , u2 , · · · , un ) ∈ Rt+n is the vector (argument of the error function) consisting of Gate W idths and output weighs, oj is the ideal output, JT r is the total number of training samples. To find an optimal option w∗ satisfying E (w∗ ) = min E (w), the following gradient memory-based update rule is used:

dk =

wk+1 = wk + ηdk , ⎧   −Ew wk , k = 0, ⎨

(8)

  −Ew wk + βdk−1 , k ≥ 1,

(9)

⎩

where k is the iterations, β ∈ [0, 1], when β = 0, expression (9) becomes the traditional gradient descent method and the value of β denotes the magnitude of past gradient information. The gradient of E(w) with respect to wp , i.e., Ewp , is computed as follows: Ewp =

JT r 

(yj − oj )

j=1

where

∂yj ∂wp

∂yj + 2λwp , ∂wp

(10)

is deduced as below:

∂yj = ∂wp

n  fji ui ) ∂( i=1

∂wp

=

∂(fj1 u1 + fj2 u2 + · · · + fjn un ) , ∂wp

(11)

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where

∂fji ui ∂wp

is computed by the following expression: ∂

∂fji ui , = ∂wp

t 

μip (xjp )ui

p=1

=

∂wp =

t 

∂μi1 (xj1 )μi2 (xj2 ) · · · μit (xjt ) ui ∂wp μip (xjp )(−2(xjp − mip )2 wp )ui .

(12)

p=1

Using (10), (11) and (12), we can get Ewp = −2n

JT r 

(yj − oj )

j=1

t 

μip (xjp )(−2(xjp − mip )2 wp )ui + 2λwp .

(13)

p=1

It is easy to get Eui as below: Eui =

JT r  j=1

Tr  ∂yj = (yj − oj )fji . ∂ui j=1

J

(yj − oj )

(14)

Generally speaking, the integrated learning algorithm can be divided into three steps: (1) when k = 0, the learning process starts, and all the features are considered as bad by setting the value of wp to be almost 0. Can we initialize the wp to high values so that at the start of training every feature is important and then discard the bad features through training? The answer is no. Our goal is to select good features and eliminate bad and indifferent features through gradient-based learning algorithm. Hence, we must guarantee that all features are bad features at the beginning of training. If we keep all the doors open (high width values) at the beginning, then we can not discard indifferent features because an indifferent feature neither will cause any problem nor will help for the error function, and hence they will be selected by the system. For the redundant features, if they have discriminating power, the doors will remain open. Hence, we initialize the width with values almost 0 so that every door is nearly closed at the beginning of the training. (2) when 0 < k < kmax , for the good feature, the value of wp gradually changes, and finally tends to a valid nonzero value, but wp s for bad features are compressed and converge to zero; (3) when k = kmax , the learning procedure ends, we can use the optimal value wp∗ to choose the important features based on (5).

3

Simulation Result

To validate the effectiveness of our proposed method, two datasets named SINC and Chem are used to conduct the simulation. All the simulations are carried

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out in MATLAB R2018b running an Intel(R) Core(TM) i7-9700 CPU. As a prediction performance criterion, we use the Root Mean Squared Error (RMSE):

J

1  RM SE =  (yj − oj )2 , (15) J j=1 where J is the total number of training/test samples. 3.1

SINC

SINC is a dataset constructed using the following difference equation: yd (t + 1) =

yd (t) + u3 (t), 1 + yd2 (t)

(16)

where t denotes the time, u(t) is the activation signal at time t, and yd (t) denotes the output. To generate the samples, the activation signal is set to be u(t) = sin(2πt/100), and tmax = 400. Finally, we obtain 400 samples and each sample has two inputs (i.e., yd (t) and ut ) and one output yd (t + 1). To test the feature selection merit of our model FS-RBFNN, two bad features satisfying normal distribution are injected for all the samples, and the first noised 200 samples are used as the training dataset and the remaining for test. The hyper-parameters are set as: γ = 0.01, η = 0.0005, β = 0.9, kmax = 1000, λ = 0.5, n = 6, which are obtained using several trial-and-error experiments.

Fig. 2. The feature selection process of FS-RBFNN for SINC.

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Fig. 3. The training regression results of RBFNN and FS-RBFNN for SINC.

After one trial, the feature selection procedure is plotted in Fig. 2. From this figure, we see that at the start, the Gate W idths for feature selection are with low values, and all the features are considered unimportant. With the training begins, the Gate W idths are changing gradually using the memory-based gradient method, and at the end w12 and w22 converge to nonzero values, but the values of w32 and w42 are almost 0, which is lower than the threshold 0.01. Using the expression (5), we conclude that the first two features are good features, and features 3 and 4 are not important. Figure 2 validates that with the help of t  wp2 can make the Gate F unction work like λ = 0.50 the regularization term p=1

a Gate to select the important features x1 and x2 from all the original features.

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Fig. 4. The average training error of FS-RBFNN for SINC.

Test actual outputs with all features and selected features are shown in Fig. 3. Visually, the two models can approximate the real outputs. Actually, the RMSE of our proposed model (0.0530) is lower than that of the traditional RBFNN (0.0593). More importantly, the structure of our model is compact. The training error plotted in Fig. 4 verifies the convergence of our feature selection model in real applications. 3.2

Chem

Chem data is a real data describing the production process in the chemical plant. Using 5 inputs named monomer concentration (x1 ), change of monomer concentration (x2 ), monomer flow rate (x3 ), and two local temperatures inside the plant (x4 and x5 ), the operator has controlled the device to get 70 actual outputs (the set point of the monomer flow rate). For this real data, several domain experts have confirmed that the last two features are bad [17]. Similar to the SINC problem, for the unfamiliar dataset, the feature selection model first uses the Gate F unctions to open a small crack to judge each feature with great care. If FS-RBFNN considers one feature is important, the gate for feature selection will gradually open until the feature enters the gate. But for the bad feature, our proposed model opens the gate with a small crack, and even shuts down the gate. For one trial with random initialization, the feature selection process of FSRBFNN for Chem data is depicted in Fig. 5. From this figure, we infer that the feature selection model shows a big welcome with a wide opened gate for features x1 , x2 and x3 , for the weight decay term values corresponding to the three features are bigger than w42 and w52 . Due to the lower values of w42 and w52 ,

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features x4 and x5 are regarded as bad features by our model. The approximation results in Fig. 6 illustrate that after feature selection, RS-RBFNN can describe the underlying structure of chemical plant, and its regression error is 77.1868, which is lower than its competitor.

Fig. 5. The feature selection process of FS-RBFNN for Chem.

Fig. 6. The prediction result of RBFNN and FS-RBFNN for Chem.

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283

Conclusions and Future Works

In this paper, using weight decay regularizer, an integrated feature selection algorithm for RBFNN is proposed. Using simulation results, the validation of our method FS-RBFNN is proved, i.e., FS-RBFNN not only can approximate the real output with less RMSE, but can simplify the structure of RBFNN. More importantly, the feature selection procedure of our model is interpretable. To improve the whole interpretability of RRFNN, in the future work, we will use the fuzzy rule to improve the interpretability of the RBF forward procedure and design the hidden node generation method using output weights.

References 1. Han, H.-G., Lu, W., Hou, Y., Qiao, J.-F.: An adaptive-PSO-based self-organizing RBF neural network. IEEE Trans. Neural Netw. Learn. Syst. 29(1), 104–117 (2018) 2. Leung, C.S., Sum, J.P.F.: A fault-tolerant regularizer for RBF networks. IEEE Trans. Neural Netw. 19(3), 493–507 (2008) 3. Solis, A.R., Panoutsos, G.: Interval type-2 radial basis function neural network: a modeling framework. IEEE Trans. Fuzzy Syst. 23(2), 457–473 (2015) 4. Solis, A.R., Melin, P., Hernandez, U.M., Panoutsos, G.: General type-2 radial basis function neural network: a data-driven fuzzy model. IEEE Trans. Fuzzy Syst. 27(2), 333–347 (2019) 5. Dash, C.S.K., Behera, A.K., Dehuri, S., Cho, S.-B.: Radial basis function neural networks: a topical state-of-the-art survey. Open Comp. Sci. 6(1), 33–63 (2016) 6. Shi, D., Yeung, D.S., Gao, J.: Sensitivity analysis applied to the construction of radial basis function networks. Neural Netw. 18(7), 951–957 (2005) 7. Yeung, D.S., Ng, W.W.Y., Wang, D.F., Tsang, E.C.C., Wang, X.-Z.: Localized generalization error model and its application to architecture selection for radial basis function neural network. IEEE Trans. Neural Netw. 18(5), 1294–1305 (2007) 8. Gutierrez, P.A., Martinez, C.H., Estudillo, F.J.M.: Logistic regression by means of evolutionary radial basis function neural networks. IEEE Trans. Neural Netw. 22(2), 246–263 (2011) 9. Ng, W.W.Y., Yeung, D.S.: Input dimensionality reduction for radial basis neural network classification problems using sensitivity measure. Int. Conf. Mach. Learn. Cybern. 4, 2214–2219. IEEE (2002) 10. Tikka, J.: Input selection for radial basis function networks by constrained optimization. In: de S´ a, J.M., Alexandre, L.A., Duch, W., Mandic, D. (eds.) ICANN 2007. LNCS, vol. 4668, pp. 239–248. Springer, Heidelberg (2007). https://doi.org/ 10.1007/978-3-540-74690-4 25 11. Tikka, J., Hollmen, J.: Selection of important input variables for RBF network using partial derivatives. ESANN, pp. 167–172 (2008) 12. Pal, N.R., Malpani, M.: Redundancy-constrained feature selection with radial basis function networks. In: The 2012 International Joint Conference on Neural Networks (IJCNN), pp. 1–8. IEEE (2012) 13. Han, H.G., Qiao, J.F.: A self-organizing fuzzy neural network based on a growingand-pruning algorithm. IEEE Trans. Neural Netw. 18(6), 1129–1143 (2010) 14. Tikka, J.: Simultaneous input variable and basis function selection for RBF networks. Neurocomputing 72, 2649–2658 (2009)

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15. Wang, J., Chang, Q., Gao, T., Zhang, K., Pal, N.R.: Sensitivity analysis of TakagiSugeno fuzzy neural network. Inf. Sci. 582, 725–749 (2022) 16. Li, X., Wang, Y., Ruiz, R.: A survey on sparse learning models for feature selection. IEEE Trans. Cybern. 52(3), 1642–1660 (2022) 17. Lin, C.-T., Pal, N.R., Wu, S.-L., Liu, Y.-T., Lin, Y.-Y.: An interval type-2 neural fuzzy system for online system identification and feature elimination. IEEE Trans. Fuzzy Syst. 26(7), 1442–1455 (2015)

Periodic Motions and Bifurcations of a Spring-Driven Joint System with Periodic Excitation Yufan Zhou1 , Zhongliang Jing1(B) , Jianzhe Huang1 , Xiangming Dun1 , and Hailei Wu2 1

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China {z1299663705,zljing}@sjtu.edu.cn 2 Shanghai Aerospace Control Technology, Shanghai, China

Abstract. Spring-driven joint system is a nonlinear dynamic system that is applied in flexible robot arms. Similar to the forced pendulum system, it has state variable in the sinusoidal function, but are more complicated with driven spring length coupling with the joint angle. With traditional analytical approaches, it is difficult to analyze such a nonlinear system since both the system state and the excitation are correlated with the nonlinear sinusoidal term. In this paper, a spring-driven joint system with periodic excitation will be discussed. The implicit discrete maps approach will be applied to solve the periodic motions for such a joint system, and the stability condition will be discussed. The analytical expressions for periodic motions for such a spring-driven system can be recovered with a series of Fourier functions. The bifurcation diagram for such a system will be given to show the complexity of the motions when the frequency of the excitation varies. From analytical bifurcation for period-1 and period-2 motions, the jump phenomenon and the continuity of periodic motion can be analytically explained. Keywords: Periodic motions

1

· Forced pendulum · Bifurcation trees

Introduction

Spring-driven joint system is a nonlinear dynamic system that is applied in flexible robot arms, which is shown in Fig. 1. For a flexible arm with numerous links, it becomes burdensome to set traditional motor-driven joints for each link. Scarifying some maneuverability, the spring-driven joint can actuate the arm with only a few springs and the whole arm gets much lighter and more flexible. Similar joint drive system can be found in many flexible arms, such as the tendon-driven robot finger in [1,2], and the cable-driven snake robot in [3,4]. The spring-driven joint is then extracted from the flexible arm and modeled as a more common joint system as shown in Fig. 1. Similar to the forced pendulum system, it has a pendulum-like part and an outside driven part. However, the c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 285–296, 2024. https://doi.org/10.1007/978-981-97-0554-2_22

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excitation on spring introduces uncertainty to the joint motion, which has not been thoroughly studied before, and the coupling between driven spring length and joint angle also makes the system more complicated. The research of the forced pendulum can be found from last century, since it gives a better understanding of complex motions in nonlinear dynamical systems, and works can be found in [5–7]. In 2000, Luo and Han [8] predict the resonant and stochastic layers for the periodically driven pendulum. To find the approximate analytical solutions for periodic motions in nonlinear dynamical systems, Luo [9] developed an analytical method based on the generalized harmonic balance method. Such a method is used by Luo and Huang [10] to investigate the periodic motions and chaos in the Duffing oscillator. In 2015, the aforementioned analytical method is further developed by Luo [11]. This semi-analytic approach based on implicit discrete maps method gives analytical solution of periodic motions for most of the nonlinear system with ordinary differential equations. This semi-analytical approach has been applied in many nonlinear systems, such as mobile piston system in high pressure gas cylinder [12], brushless motor with unsteady external torque [13], and power system including power disturbance [14], etc. It is also used to analysis the periodic motions in pendulum systems under periodic excitations, such as periodically forced pendulum [15] and periodically forced spring pendulum [16]. In this paper, a study of the spring-driven joint system will be conducted through excitation frequencies. The dynamic equations of the spring-driven joint system will be formulated and modeled. The periodic motions are discretized into multiple continuous mappings and solved via Newton-Raphson iteration. The motion stability will also be discussed and the analytic conditions for the saddlenode and period-doubling bifurcation will be determined. From the bifurcation analysis, the system characteristics will be comprehensively discussed and the intrinsic mechanism of motion switch and motion jump with parameter varies will be systematically explained.

2

Methodology

In this section, a semi-analytical method for periodic motions in the spring-driven joint system will be presented. The corresponding implicit discrete mapping structures of periodic motions will be employed, and the stability and bifurcation are also discussed. 2.1

Modeling

From the introduction, a spring-driven system is extracted from the flexible arm, and is shown in Fig. 1. The system consists of a movable joint, which rotates around point O, and an arm body connected to a spring at Q. The spring passes through a smooth fixed hole P and is driven by a lead screw. As the lead screw moves, both the length of the spring inside the joint (lj ) and outside the joint (ldr ) change simultaneously, leading to a corresponding alteration in the arm’s

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Fig. 1. Illustration of the spring-driven joint system.

attitude θ. With a linear spring equation, the spring drive moment M can be calculated as M = −T h = k (lcr − l) h = kh (lcr − ldr − lj )

(1)

where T is the spring force, h is the arm of spring force; l = ldr + lj is the whole length of spring, lcr is the spring’s critical length; k is the equivalent stiffness coefficient of spring and k = EA lcr . With the geometry relationship lj h = r1 r2 sin(θ+α) √ r1 r2 sin(θ+α) and h = , the system equation can be derived r12 +r22 −2r1 r2 cos(θ+α) as r1 r2 sin(θ + α) I θ¨ + Dθ˙ + K (θ − θ0 ) = −kr1 r2 sin(θ + α) + k (lcr − ldr )  2 r1 + r22 − 2r1 r2 cos(θ + α)

(2)

where I, K and D are system’s rotational inertia, stiffness coefficient and damping coefficient; r1 , r2 , α are the system parameters shown in Fig. 1; θ0 is the initial stiffness setting angle; ldr represents the excitation from driven length and ldr (t) = ldr0 + Acos(ωt). Define a state vector as x ≡ (x, y) ≡ (θ, ω).

(3)

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In state space, such a system can be expressed in a simplified term as 

x˙ = y y˙ = Q + [Q0 − Q1 cos(Ωt)] √

sin(θ+α)

λ2 −2λ cos(θ+α)+1

− A1 x − A2 x˙ − A3 sin(x + α)

(4) where A1 and A2 are parameters correlated with rotational stiffness and damping coefficient, respectively. A3 is a stiffness-alike coefficient correlated with joint parameters. Q is introduced by initial stiffness setting angle θ0 , Q0 is introduced by the initial driving length setting ldr0 , λ = rr21 is the radius ration. Q1 and Ω are excitation amplitude and frequency, respectively. Using a midpoint scheme for the time interval t ∈ [tk , tk+1 ], the foregoing system can be discretized to form an implicit map Pk (k = 0, 1, 2, ...) as Pk : (xk−1 , yk−1 ) → (xk , yk ) ⇒ (xk , yk ) = Pk (xk−1 , yk−1 ) . 2.2

(5)

Mapping and Periodic Motions

For a periodic motion of such a spring-driven joint system which the period equals m times the period of excitation (2mπ = Ω), the motion can be discretized into mN partitions. For each of the segments, Pk maps from one state (xk−1 ,yk−1 ) at t = tk−1 to the next state point (xk , yk ) at t = tk , where k = 1, 2 . . . mN . Thus, the period-m periodic motion can be represented by a discrete mapping structure as P = PmN ◦ PmN −1 ◦ · · · ◦ P2 ◦ P1 : (x0 , y0 ) → (xmN , ymN )   

(6)

mN − actions

with Pk : (xk−1 , yk−1 ) → (xk , yk ) ⇒ (xk , yk ) = Pk (xk−1 , yk−1 ) (k = 1, 2, . . . , mN ). (7) The mathematical expressions for mapping Pk can be obtained based on the implicit midpoint scheme as ⎧ h ⎪ ⎪ xk = xk−1 + (yk−1 + yk ) ⎪ ⎪ 2⎧ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ x +x ⎪ ⎪

  ⎪ ⎨ sin k−12 k + α ⎨ h yk = yk−1 + h (Q0 − Q1 cos Ω tk−1 + )  ⎪ 2 x +x ⎪ ⎪ ⎪ ⎩ λ2 − 2λ cos k−12 k + α + 1 ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ A2 (yk−1 + yk ) xk−1 + xk A1 (xk−1 + xk ) ⎪ ⎪ − − A3 sin +α ⎩ +Q − 2 2 2

where h is the time difference between two mappings, and h = tk − tk−1 .

(8)

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Since the motion is periodic, an additional constraint for periodic motion is given by xmN = x0 + 2lπ, ymN = y0 , l = 0, ±1, ±2, ...;

(9)

For a period-m motion with mN maps, we get 2mN equations with (7), and additional two more equations with the periodic constraint (9). Therefore, the 2mN + 2 node points for such a period-m motion can be solved analytically through Newton-Raphson method with 2mN + 2 equations. Once the node points x∗k (k = 0, 1, 2, ...mN ) of the period-m motion are obtained, the stability and bifurcation can be discussed by the corresponding Jacobian matrix. The small neighborhood of the node point x∗k can be written as xk = x∗k + Δxk (k = 0, 1, 2, ...mN ). Then linearize equation at the equilibrium and the first order of Taylor’s expansion is kept, it gives ΔxmN = DP Δx0 = DPmN · DPmN −1 · · · DP2 · DP1 Δx0   

(10)

mN -multiplication

where

∂x  k ∂xk DPk = = ∂x∂yk−1 k ∂xk−1 (x∗ ,x∗ ) ∂xk−1 k k−1 

∂xk ∂yk−1 ∂yk ∂yk−1

(x∗k ,x∗k−1 )

for k = 1, 2, . . . , mN. Based on the chain rule and mathematical expressions in Eq. (8), for the mapping Pk , the Jacobian matrix DP can be calculated using (12). The corresponding eigenvalues are computed by

where DP =

|DP − λI| = 0

(11)

  1  ∂xk ∂xk (x∗ ,x∗ ) k=mN k k−1

(12)

With the Jacobian matrix DP , we get two eigenvalues λ1 and λ2 . According to the continuous system theory, the periodic motion is stable if all |λi | < 1 for (i = 1, 2). Otherwise, the motion is unstable. The bifurcation happens at the boundary between the stable and unstable motions, where the magnitude of one of the eigenvalues is 1. And the three types of bifurcation can be expressed as (1) If one of λi = −1 and |λj | < 1 for (i, j ∈ 1, 2 and j = i), the period-doubling bifurcation of periodic motion occurs. (2) If one of λi = 1 and |λj | < 1 for (i, j ∈ 1, 2 and j = i), the saddle-node bifurcation of the periodic motion occurs. (3) If |λ1,2 | = 1 is a pair of complex eigenvalues, the Neimark bifurcation of the periodic motion occurs.

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With the node points set of the period-m motions, the analytic solution for such a period-m motion can be recovered as x

(m)

=

(m) a0

+

∞

bk/m cos

k=1

 

kΩ kΩ t + ck/m sin t m m

(13)

where (m)

a0

=

bk/m = ck/m =

3

 mN −1 1 (m) 2jπ xj − a1 mN j=0 NΩ



 mN −1 kjπ 2 (m) 2jπ xj − a1 cos mN j=0 NΩ mN

(14)



 mN −1 kjπ 2 (m) 2jπ xj − a1 sin . mN j=0 NΩ mN

Bifurcation and System Analysis

In this section, bifurcation diagrams are presented to show the routes of periodic motions in the spring-driven joint system. To simulate the spring-driven joint system, the parameters tabulated in Table 1 are adopted. For the excitation magnitude, the resulting moment is chosen as Q1 = 5.4, and the numerical bifurcation diagram with frequency varies from 1 to 9 is illustrated in Fig. 2. A series of periodic motions and chaos can be found in such a frequency range. In the frequency region which Ω < 1.60, the system has a simple period-1 motion. As the frequency increases, the period-1 motion “jumps” 2 times at Ω = 1.60 and 2.79 and evolves into a period-2 motion at Ω = 3.78, which exists in the range of Ω = (3.78, 5.60). Another branch of period-2 motion can be observed in the frequency range of Ω = (5.67, 6.55). With numerous period doubling bifurcation between Ω = (5.60 : 5.67), the period-2 motion develops into period-m motion and finally turns into chaos. When frequency is higher than 6.55, the system’s motion returns into a period-1 motion. Examples of period-2 and chaotic motions in phase plane are plotted in Fig. 3. Table 1. System parameters for spring-driven joint system. A1

A2

A3

Q

Q0 λ

3.000 0.250 9.203 5.564 0

α

1.342 0.180

As shown from the numerical bifurcation diagram, the spring-driven joint system has a simple priod-1 motion when the excitation frequency is relatively low or high. However, complicated periodic motions and chaos can be observed

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mod(k, N)

mod(k, N)

6.28

4.71

4 3 2

3.14

1

1.57

-1

0

-2 -3

0.00 1

2

3

4

5

6

7

8

9

1

2

3

4

(a)

5

6

7

8

9

(b)

Fig. 2. Numerical bifurcation for spring-driven joint system varying excitation frequency Ω = [1, 9] for Q1 = 5.4.

when the frequency varies between 3.78 and 6.55. Additionally, the “jump” phenomenon occurs several times at Ω = 1.60, 2.79, 5.59 and 6.56, which introduce discontinuity to the numerical bifurcation tree. Thus, with the semi-analytical method, the analytical bifurcation will be plotted to analysis the development of periodic motions, and the continuity between stable and unstable periodic motions will be shown.

3

6

2

4

1

2

0

0

-1

-2

-2

-4

-3 -0.5

0

0.5

(a)

1

-6 -1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

(b)

Fig. 3. Period-2 motion and chaotic motion for spring-driven joint system with Q1 = 5.4: (a) Ω = 4.01, (b)Ω = 5.61

Based on the aforementioned implicit discrete maps approach with N = 2048 points per excitation period, the m = 1 periodic motion can be solved with initial guess by Newton-Raphson method. The magnitude of the driven excitation is chosen to be Q1 = 5.4. The node points for every period of excitation are plotted by varying excitation frequency Ω, which are demonstrated in Fig. 4. The black solid curve is the stable period-1 motion, and the red dashed curve denotes the

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unstable period-1 motion. When the “jump” phenomenon occurs, the analytical bifurcation shows that the stable period-1 motions are continued with unstable period-1 motions. There are two routes of period-1 motion in the given frequency range, and there are several frequency bands where two kinds of stable period-1 motion can exist simultaneously. To clearly show the development for period-1 motion evolving to period-2 motion, the bifurcation tree has been divided in to three partitions, and will be discussed separately.

6.28

mod(k, N)

mod(k, N)

4

4.71

3 2 1 0 -1

3.14

-2 -3 1.57

-4 -5 -6

0.00 1

2

3

4

5

(a)

6

7

8

9

1

2

3

4

5

6

7

8

9

(b)

Fig. 4. Periodic nodes for (θmod(k,N ) , ωmod(k,N ) ) of period-1 motion varying excitation frequency Ω = [1, 10] for Q1 = 12.

The part 1 bifurcation tree of periodic motions in the range of Ω = (0.950, 3) is illustrated in Fig. 5. The acronyms “P-1” and “P-2” denote the period-1 and period-2 motion, respectively. From Fig. 5, it can be seen that for Ω = (1, 1.606) and (1.456, 2.794), it has stable period-1 motion. The stable period-2 motion only exists in a narrow frequency range where Ω = (2.448, 2.497). For stable period-1 motion, it becomes unstable at the saddle-node bifurcation at Ω = 1.456, 1.606 and 2.794, where the original stable period-1 motion “jumps” to another branch of stable period-1 motion. And these two routes of stable period-1 motion are connected through unstable motions. When periodic doubling bifurcation appears for period-1 motion at Ω = 2.497, the period-1 motion evolves to a stable period-2 motion which exists between (2.448, 2.497). For stable period-2 undergoes saddle-node bifurcation at Ω = 2.497, the period2 motion disappears and it continues to the next part of period-1 motion. It can also be observed that the period-doubling bifurcation for period-1 motion is exactly the saddle-node bifurcation for the period-2 motion. The critical frequencies of part 1 for period-1 and period-2 motions are summarized in Table 2. For another part of periodic motions in the range of Ω = (2.497, 9), the analytic bifurcation tree of period-1 and period-2 motions are shown in Fig. 6. For the period-1 motion, it undergoes four times of period-doubling bifurcation, so that there exist two routes of period-2 motion. The first route of stable period-2 motion starts with the period doubling bifurcation of period-1 motion at

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mod(k, N)

mod(k, N)

6.28

4.71

4 3 2 1

P-1

0 3.14

-1

P-2

-2

P-1

P-2

P-1

-3

1.57

-4 -5

0.00

-6

P-1 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

1

1.2

(a)

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

(b)

Fig. 5. Analytical bifurcation for period-1 motion to period-2 motion of part 1 with Q1 = 5.4 Table 2. Bifurcation critical frequencies for period-1 and period-2 motions of part 1. Motion type Critical frequency Bifurcation type P-1

1.456 1.606 2.497 2.794

SN SN PD SN

P-2

2.448 2.497

PD SN

Ω = 3.778, and ends with another period doubling bifurcation of period-1 motion at Ω = 6.412. The second route of period-2 motion is unstable and exists only in the frequency bands of Ω = (5.200, 5.708), where the period-1 motion becomes stable again. For the stable period-2 motion develops from period-1 motion at Ω = 4.778, it becomes unstable at the saddle-node bifurcation where Ω = 5.604. After another saddle-node bifurcation at Ω = 4.870, the period-2 motion becomes stable again, and this period-2 motion maintains stable between Ω = (4.870, 5.142) and (5.342, 6.552). Note that two period-doubling bifurcation happens at Ω = 5.142 and 5.342, which means that the period-2 motion becomes unstable for Ω = (5.142, 5.342), and develops into another period-4 motion which finally develops into chaos. Then with another saddle-node bifurcation at Ω = 6.552, this period-2 motion becomes unstable and disappears into the stable period-1 motion at Ω = 6.412 (Table 3). The third part of periodic motion exists between Ω = (2, 9), which is shown in Fig. 7. The stable period-1 motion undergoes period-doubling bifurcation at Ω = 5.128 and developing into another branch of stable period-2 motion. The stable period-2 motion becomes unstable after a period-doubling bifurcation at

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Ω = 5.128, and join the unstable period-1 motion which continues when the frequency increases (Table 4).

6 mod(k, N)

mod(k, N)

6.28

4.71

5 4 3 2

3.14

P-2

1

P-2

P-1

P-1

0 -1 1.57

P-1

-2

P-2

P-2

P-1

-3 -4

0.00 3

4

5

6

7

8

9

3

4

(a)

5

6

7

8

9

(b)

Fig. 6. Analytical bifurcation for period-1 motion to period-2 motion of part 2 with Q1 = 5.4

Table 3. Bifurcation critical frequencies for period-1 and period-2 motions of part 2. Motion type Critical frequency Bifurcation type P-1

3.778 5.201 5.708 6.412

PD PD PD PD

P-2

3.778 4.870 5.142 5.342 5.604 6.412 6.552

SN SN PD PD SN SN SN

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1.2

P-1 4.71

P-2

mod(k, N)

mod(k, N)

6.28

295

1 0.8 0.6 0.4

P-2

0.2

P-1

3.14

P-1

0

P-1

-0.2 -0.4

1.57

-0.6 -0.8 0.00 1

2

3

4

5

6

7

8

9

1

2

(a)

3

4

5

6

7

8

9

(b)

Fig. 7. Analytical bifurcation for period-1 motion to period-2 motion of part 2 with Q1 = 5.4 Table 4. Bifurcation critical frequencies for period-1 and period-2 motions of part 3. Motion type Critical frequency Bifurcation type

4

P-1

5.128

PD

P-2

4.658 5.128

PD SN

Conclusions

In this paper, the periodic motions and bifurcation in spring-driven joint system has been analyzed. The periodic excitation has been considered, and bifurcation diagrams have been obtained by varying the frequency of the periodic excitation. With the implicit discrete maps approach, the bifurcation routes for analytical solution of periodic motion have been calculated. The numerical bifurcation trees are not continuous but are connected with the analytical solutions. With the analytical bifurcation routes, the development of periodic motions can be better understood. The period-1 motion becomes period-2 at the period-doubling bifurcation, and the chaotic motion is formed due to the catastrophic effect of period doubling bifurcation. Additionally, there are few frequency bands that several different stable periodic motions can exist simultaneously. Acknowledgement. This work is partially supported by the National Natural Science Foundation of China under Grant 61673262, National GF Basic Research Program under JCKY2021110B134, Science and Technology Commission of Shanghai Municipality under Grant 16JC1401100, National Science Foundation of Chongqing under Grant No. cstc2021jcyj-msxmX0089 and the Fundamental Research Funds for the Central Universities.

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References 1. Ding, S., Lu, H., Huang, X.: Feedback control for underactuated trussed robot finger. In: Proceedings of the 33rd Chinese Control Conference, pp. 1862–1867 (2014). https://doi.org/10.1109/ChiCC.2014.6896913 2. Ozawa, R., Hashirii, K., Yoshimura, Y., Moriya, M., Kobayashi, H.: Design and control of a three-fingered tendon-driven robotic hand with active and passive tendons. Auton. Robot. 36(1–2), 67–78 (2014) 3. Peng, J., Xu, W., Liu, T., Yuan, H., Liang, B.: End-effector pose and arm-shape synchronous planning methods of a hyper-redundant manipulator for spacecraft repairing. Mech. Mach. Theory 155, 104062 (2021) 4. Liu, T., Xu, W., Yang, T., Li, Y.: A hybrid active and passive cable-driven segmented redundant manipulator design, kinematics and planning. IEEE/ASME Trans. Mechatron. PP(99), 1–1 (2020) 5. Zaslavskii, G.M.: Stochastic instability of nonlinear oscillations. J. Appl. Mech. Tech. Phys. 8(2), 8–11 (1967) 6. D’Humieres, D., Beasley, M.R., Huberman, B.A., Libchaber, A.: Chaotic states and routes to chaos in the forced pendulum. Phys. Rev. A 26(6), 3483–3496 (1982) 7. Mawhin, J., Willem, M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ. 52(2), 264–287 (1984) 8. Luo, A.C.J., Han, R.P.S.: The dynamics of stochastic and resonant layers in a periodically driven pendulum. Chaos Solitons Fractals 11(14), 2349–2359 (2000) 9. Luo, A.C.J., Huang, J.Z.: Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. J. Vib. Control 18(14), 1661–1674 (2011) 10. Luo, A.C.J., Huang, J.: Analytical dynamics of period-m flows and chaos in nonlinear systems. Int. J. Bifurcat. Chaos 22(04), 1250093-1-1250093-29 (2012) 11. Luo, Albert, C.J.: Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems. Int. J. Bifurcat. Chaos 25(03), 1550044 (2015) 12. Donghua, W., Jianzhe, H.: Periodic motions and chaos for a damped mobile piston system in a high pressure gas cylinder with p control. Chaos, Solitons Fractals 95, 168–178 (2017) 13. Jianzhe, H., Zhongliang, J.: Feedback control of unstable periodic motion for brushless motor with unsteady external torque. Eur. Phys. J. Spec. Top. 228, 1809–1822 (2019) 14. Jianzhe, H.: Periodic motions and chaos in power system including power disturbance. Eur. Phys. J. Spec. Top. 228, 1793–1808 (2019) 15. Guo, Y., Luo, A.C.J.: Routes of periodic motions to chaos in a periodically forced pendulum. Int. J. Dyn. Control 5, 551–569 (2017) 16. Guo, Y., Luo, A.C.J.: Periodic motions and bifurcations of a periodically forced spring pendulum varying with excitation amplitude. In: ASME 2020 International Mechanical Engineering Congress and Exposition (2020)

Enhanced Vibration Characteristics of Honeycomb Plates Composed of Metamaterials with NTE Qiao Zhang and Yuxin Sun(B) National Key Laboratory of Strength and Structural Integrity, School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China [email protected]

Abstract. Mechanical metamaterials with tailorable negative thermal expansion (NTE) are significant and potential to be applied in stability control, precise instrument, space equipment, etc. This paper established the honeycomb plates which are composed of metamaterial lattices consisting of trapezoid units with NTE. The vibration characteristics of these honeycomb plates were analyzed by Euler beam elements in finite element analysis. The frequency characteristics and harmonic response analysis of these honeycomb plates were investigated under uniform temperature increments. In addition, the effects of geometric parameters on the fundamental frequency and thermal stress were discussed. Finally, the fundamental frequency and thermal stress were compared between the present metamaterials with trapezoid units and the corresponding metamaterials with triangular units. These results indicate that the NTE effect in metamaterials can enhance the fundamental frequency and reduce the deflection dynamic amplification factor of structures under uniformly raised temperatures. The present metamaterials provide a thought for designing and developing heat-resistant structures. Keywords: metamaterial · NTE · honeycomb · vibration

1 Introduction Mechanical metamaterials have been focused on because of their unique negative parameters, such as negative Poisson’s ratio (NPR), negative thermal expansion (NTE), negative compressibility (NC), negative stiffness (NS), negative moisture expansion (NME), etc., [1]. NTE indicates the materials’ contraction behavior with temperature increments, which can ameliorate problems in conventional materials and structures with positive thermal expansion. The thermal contraction behavior is observed and evaluated by the occupied volume reduction of metamaterials in thermal environments via decreasing the apparent distance. According to the primary deformation mode of the NTE mechanism, there are two categories of mechanical metamaterials with NTE: bending-dominated and

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 297–309, 2024. https://doi.org/10.1007/978-981-97-0554-2_23

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stretching-dominated types [2]. In bending-dominated metamaterials, the thermal contraction behavior is realized by bi-material beams’ bending deformation under temperature increments. By contrast with bending-dominated metamaterials, stretchingdominated metamaterials directly relies on the difference of thermal stretching deformations, leading to the thermal shrinkage of whole structures. Generally, stretchingdominated metamaterials are characterized by higher stiffness due to the stretching or compressing deformation under various loads. A two-dimensional triangular unit was designed by Grima et al. [3] by connecting rods with different materials together through hinges, which can compose stretchingdominated mechanical metamaterials with a wide range of CTE from negative value to positive value. UCSB lattice was proposed and developed in [4–6]. A combination of stretching rods and chiral beams was performed to construct the microstructure with isotropic near-zero CTE [7]. A triangular structure consisting of corner-hinged beam-like elements with different positive CTE, can be assembled into more complex systems with tailorable thermal expansion [8]. By employing the triangular microstructure, Wei and co-authors [2, 9–13] did plenty of work on planar stretching-dominated metamaterials with NTR. Later, more planar mechanical metamaterials were proposed using the bimaterial triangular unit and auxetic structures [14–18]. Especially, a planar bi-metallic lattice with tailorable CTE was proposed and manufactured using Al, Ti, and Invar alloy by Xu et al. [19]. Besides, mechanical metamaterials with NPR and NTE were proposed based on the interconnected array of rings and sliding rods [20, 21]. Considering the advantage of the bi-material triangular microstructure, the hierarchical design was adopted to propose metamaterials with higher negative CTE [22–24]. To date, quite a few three-dimensional extensions of the above planar mechanical metamaterials have been established [25–33]. Notably, the prominent stress concentration exists at the vertex joint of stretchingdominated mechanical metamaterials based on bi-material triangular units, resulting in premature material failure probably. Besides, there may be a discrepancy between the ideal model and the practical sample of the connection joint. Therefore, inspired by the triangular metamaterial unit, the trapezoid unit was proposed by Zhang and Sun [34] to alleviate the problem. Herein, mechanical metamaterials with programmable CTE based on the trapezoid unit are established, and their vibration characteristics are investigated. These metamaterials have potential for stability controlling [35], precision instruments [36], satellite supports [37], etc., in which the tailorable CTE can improve structures’ mechanical performance.

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2 Mechanical Metamaterials Composed of Trapezoid Unit As shown in Fig. 1, the trapezoid unit cell composed of beams with different CTE is adopted to compose the mechanical metamaterial honeycomb plate. The trapezoid unit presents the thermal contraction behavior in its height direction and the thermal expansion in its width direction under temperature increments. Herein, the length of the waist beam is kept at 10 mm by default. Two geometric parameters can be tailorable to  modulate the structure co, including the length ratio λ = L3 2L2 and the characteristic angle θ . 3

2

Fig. 1. The basic representative unit with NTE in the height direction

Based on the trapezoid unit in Fig. 1, metamaterial lattice structures with six configurations are developed in Fig. 2 by sharing the long beam and sharing the vertex, referred to as Edge-shared Triangle (ET), Vertex-shared Triangle (VT), Edge-shared Quadrangle (EQ), Vertex-shared Quadrangle (VQ), Edge-shared Hexagon (EH), and Vertex-shared Hexagon (VH), respectively. The ET, EQ, EH, and VT metamaterials can be characterized by NTE and PTE via adjusting geometric parameters, while only the PTE effect exists in VQ and VH metamaterials. As exhibited in Fig. 3 with the specific geometric parameters λ = 1, θ = 25◦ , the metamaterial lattices ET, EQ, and EH show the shrinkage behavior of volume (or area) under uniform increments of temperature, where the dimensional size of the red deformed structure is smaller than that of the black undeformed structure. By contrast, VT, VQ, and VH metamaterial lattices present a thermal expansion response, similar to conventional materials under a positive temperature increment.

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Fig. 2. Metamaterial lattice of (a) ET, (b) VT, (c) EQ, (d) VQ, (e) EH and (f) VH composed of the trapezoid unit. The letters ‘E’ and ‘V’ denote the edge-shared and vertex-shared types. The letters ‘T’, ‘Q’, and ‘H’ represent the Triangle, Quadrangle, and Hexagon shapes, respectively.

3 Enhanced Vibration Characteristics 3.1 Vibration Characteristics The honeycomb plate composed of these metamaterial lattices is considered herein to investigate the effect of NTE on the vibration characteristic of structures. The thickness of the nearly square porous plate shown in Fig. 3 is taken as 10 mm.

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Fig. 3. NTE and PTE behaviors of metamaterial lattices.

As described above, the metamaterial lattices with NTE will contract under temperature increments. Due to boundary conditions, the constrained contraction behavior leads to internal stretching stress, enhancing the bending stiffness of the honeycomb plate. As shown in Fig. 4(a), (c), and (e), for the ET, EQ, and EH metamaterials comprising three different materials, their fundamental frequencies increase with temperature rise expectedly, which can benefit the stability control of structures under thermal environments. By comparison, the VT, VQ, and VH metamaterial plates’ frequencies decrease with temperature increments due to compressive thermal stress from PTE in Fig. 4(b), (d), and (f). Moreover, when only one material is adopted to fabricate these honeycomb plates, their temperature-frequency relations are similar to that of conventional materials because of compressive thermal stress resulting from the positive thermal expansion.

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Fig. 4. Effects of temperature on the fundamental frequency of the metamaterial honeycomb plates (a) ET, (b) VT, (c) EQ, (d) VQ, (e) EH, and (f) VH. The red curve denotes that the lattice comprises three materials with different CTE, while the black curve represents the corresponding lattice made of a single material.

Vibration of Honeycomb Plates made of Metamaterials with NTE

(b)

12

0 10 20 30 40 50

D

8

4

16

0 10 20 30 40 50

12 8

D

(a)

303

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As exhibited in Fig. 5, under different temperature increments, the steady-state harmonic response analysis is performed on these metamaterial honeycomb plates subjected to uniformly distributed load. The central nondimensional deflection dynamic amplification factor D is the ratio of the dynamic response deflection to the static deflection without temperature increment. The abscissa β is defined as the ratio of the applied loading frequency to the free-vibration fundamental frequency of the honeycomb plate without temperature increment. Apparently, in metamaterial lattices ET, EQ, and EH

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from Fig. 5(a), (c), and (e), the NTE effect not only increases the resonant frequency of structures under temperature increments, but also decreases the dynamic amplification factor D, which is conducive to stiffness design in thermal environments. For VT, VQ, and VH with PTE in Fig. 5(b), (d), and (f), PTE leads to the larger dynamic amplification factor D. 3.2 Geometric Parameters’ Effect on Frequencies To adopt the same periodic number to compose the honeycomb plate, the length and width of the plate have to vary with the geometric parameters. Hence, to eliminate the effect of the dimensional size on the plate’s natural frequency, the following nondimensional fundamental frequency is adopted to capture the influence of geometric parameters on free vibration fundamental frequency.  ω = ω ω∗ , (1) in which ω, ω∗ denote the fundamental frequencies of the honeycomb plate and corresponding homogeneous plate with the same dimensional size as the honeycomb plate. It is apparent from Fig. 6 that the larger characteristic angle can reduce the fundamental frequency of the edge-shared type metamaterial lattices (ET, EQ, and EH), while the fundamental frequency of VQ and VH metamaterial lattices can be enhanced by increasing the angle. For VT metamaterial, there exists a particular angle around 30° at which the maximum fundamental frequency can be achieved. That is because the VT metamaterial is characterized by the PTE and NTE before and after around 30 degree, respectively. Moreover, the extremely large or small length ratio can increase the frequency for all these metamaterial honeycomb plates.

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4 Comparison All  nondimensional thermal stress σ T = −Eα and fundamental frequency ω = E ρ(1 − v2 ) of the present metamaterials with 0 ≤ θ < θmax and 0.1 ≤ λ ≤ 10 are plotted in Fig. 7, by comparing with the  corresponding  triangular metamaterials obtained by assuming λ = 0.001. E = E E2 , ρ = ρ ρ2 , α = α α2 , v are the nondimensional effective modulus, relative density, nondimensional coefficient of thermal expansion, and Poisson’s ratio of these metamaterials. The subscript 2 denotes the waist beam. The top right location with high natural frequency and high stretching thermal stress in these figures of Fig. 7 is the designing aim of metamaterials to stabilize structures in thermal environments. On the contrary, the bottom left corner must be avoided when designing metamaterials. In comparison with triangular metamaterials, the present metamaterials with trapezoid units have less or zero compressive thermal stress under the same specific stiffness (fundamental frequency), which can be seen from the arrow identifications in Fig. 7(b), (c), and (e) of VT, EQ, and EH especially. For VQ and VH configurations in Fig. 7(d) and (f), the performance of the present metamaterials is inferior to that of metamaterials composed of triangular units. Hence, the geometric configurations EQ, EH, and VT are adoptable in constructing heat-resistant metamaterials with high stiffness and low thermal stress.

5 Concluding Remarks The paper employs metamaterial lattices consisting of trapezoid units with NTE to compose honeycomb plates. The vibration characteristics of the honeycomb plates are analyzed by the finite element method with Euler beam elements. The following points can be drawn from the above discussion: The negative thermal expansion can increase the natural frequencies and decrease the deflection dynamic magnification factor of metamaterial structures under a thermal environment, which is conducive to stability controlling and load capacity. The enhancing effect of temperature on the fundamental frequency and bending stiffness of honeycomb plates can be tailorable by modulating the length ratio and characteristic angle in ET, EQ, EH, and VT metamaterials. EQ, EH, and VH configurations are suitable for building heat-resistant metamaterials with decent stiffness and low or zero thermal stress.

References 1. Lim, T.-C.: Mechanics of Metamaterials with Negative Parameters. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-6446-8 2. Chen, J., Xu, W., Wei, Z., Wei, K., Yang, X.: Stiffness characteristics for a series of lightweight mechanical metamaterials with programmable thermal expansion. Int. J. Mech. Sci. 202–203, 106527 (2021) 3. Grima, J.N., Farrugia, P.S., Gatt, R., Zammit, V.: A system with adjustable positive or negative thermal expansion. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 463(2082), 1585–1596 (2007)

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4. Steeves, C.A., dos Santos e Lucato, S.L., He, M., Antinucci, E., Hutchinson, J.W., Evans, A.G.: Concepts for structurally robust materials that combine low thermal expansion with high stiffness. J. Mech. Phys. Solids 55(9), 1803–1822 (2007) 5. Berger, J., Mercer, C., McMeeking, R.M., Evans, A.G.: The design of bonded bimaterial lattices that combine low thermal expansion with high stiffness. J. Am. Ceram. Soc. 94(S1), 42–54 (2011) 6. Toropova, M., Steeves, C.: Bimaterial lattices with anisotropic thermal expansion. J. Mech. Mater. Struct. 9(2), 227–244 (2014) 7. Jefferson, G., Parthasarathy, T.A., Kerans, R.J.: Tailorable thermal expansion hybrid structures. Int. J. Solids Struct. 46(11), 2372–2387 (2009) 8. Miller, W., Mackenzie, D.S., Smith, C.W., Evans, K.E.: A generalised scale-independent mechanism for tailoring of thermal expansivity: positive and negative. Mech. Mater. 40, 351–361 (2008) 9. Wei, K., Chen, H., Pei, Y., Fang, D.: Planar lattices with tailorable coefficient of thermal expansion and high stiffness based on dual-material triangle unit. J. Mech. Phys. Solids 86, 173–191 (2016) 10. Wei, K., Peng, Y., Qu, Z., Zhou, H., Pei, Y., Fang, D.: Lightweight composite lattice cylindrical shells with novel character of tailorable thermal expansion. Int. J. Mech. Sci. 137, 77–85 (2018) 11. Wei, K., Yang, Q., Ling, B., Qu, Z., Pei, Y., Fang, D.: Design and analysis of lattice cylindrical shells with tailorable axial and radial thermal expansion. Extreme Mech. Lett. 20, 51–58 (2018) 12. Wei, K., Xiao, X., Chen, J., Wu, Y., Li, M., Wang, Z.: Additively manufactured bi-material metamaterial to program a wide range of thermal expansion. Mater. Des. 198, 109343 (2021) 13. Wei, K., Peng, Y., Qu, Z., Pei, Y., Fang, D.: A cellular metastructure incorporating coupled negative thermal expansion and negative Poisson’s ratio. Int. J. Solids Struct. 150, 255–267 (2018) 14. Liu, K.-J., Liu, H.-T., Li, J.: Thermal expansion and bandgap properties of bi-material triangle re-entrant honeycomb with adjustable Poisson’s ratio. Int. J. Mech. Sci. 242, 108015 (2023) 15. Li, Y., Chen, Y., Li, T., Cao, S., Wang, L.: Hoberman-sphere-inspired lattice metamaterials with tunable negative thermal expansion. Compos. Struct. 189, 586–597 (2018) 16. Lim, T.-C.: An auxetic metamaterial with tunable positive to negative hygrothermal expansion by means of counter-rotating crosses. Physica Status Solidi (b) 258(8), 2100137 (2021) 17. Ai, L., Gao, X.L.: Metamaterials with negative Poisson’s ratio and non-positive thermal expansion. Compos. Struct. 162, 70–84 (2017) 18. Raminhos, J.S., Borges, J.P., Velhinho, A.: Development of polymeric anepectic meshes: auxetic metamaterials with negative thermal expansion. Smart Mater. Struct. 28(4), 045010 (2019) 19. Xu, M., et al.: Planar bi-metallic lattice with tailorable coefficient of thermal expansion. Acta. Mech. Sin. 38(7), 421546 (2022) 20. Lim, T.-C.: Auxetic and negative thermal expansion structure based on interconnected array of rings and sliding rods. Physica Status Solidi (b) 254(12), 1600775 (2017) 21. Lim, T.-C.: Negative environmental expansion for interconnected array of rings and sliding rods. Physica Status Solidi (b) 256(1), 1800032 (2019) 22. Wei, K., Peng, Y., Wen, W., Pei, Y., Fang, D.: Tailorable thermal expansion of lightweight and robust dual-constituent triangular lattice material. J. Appl. Mech. 84(10), 101006 (2017) 23. Xu, H., Farag, A., Pasini, D.: Multilevel hierarchy in bi-material lattices with high specific stiffness and unbounded thermal expansion. Acta Mater. 134, 155–166 (2017) 24. Yu, H., et al.: Building block design for composite metamaterial with an ultra-low thermal expansion and high-level specific modulus. Compos. Struct. 300, 116131 (2022)

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Vibration Reduction of Limited Series Nonlinear Energy Sink Ting-Kai Du1 and Hu Ding1,2,3(B) 1

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China [email protected] 2 Shanghai Institute of Aircraft Mechanics and Control, Zhangwu Road, Shanghai 200092, China 3 Shaoxing Institute of Technology, Shanghai University, Shaoxing 312074, China

Abstract. In engineering, vibration is ubiquitous. In order to avoid damage caused by vibration to some devices, passive vibration absorbers are usually used for vibration reduction in engineering. The concept of nonlinear energy sink (NES) has been proposed for a long time. However, most previous studies on NES have typically conducted vibration reduction research by coupling a single-degree-of-freedom (1DOF) NES with a single-degree-of-freedom primary system. This paper proposes a vibration control strategy using a limited amplitude series NES. A dynamic model using a limited series NES was established for the two-degree-offreedom (2DOF) primary system. The influence trend of the parameters of the limited series NES parameters on the vibration control effect was studied through dynamic analysis. The results show that the amplitudelimiting series NES has a better vibration absorption effect and stability than the ordinary series NES under the same mass of the total vibration absorber. This study provides a new and efficient vibration control strategy. Keywords: Nonlinear Energy Sink · Passive Vibration Control Vibration Reduction · Amplitude-Limiting

1

·

Introduction

In engineering, vibration is ubiquitous and inevitable. For vibration, existing research has many control strategies. These research studies include active control [1] and passive control, and passive control is further divided into vibration reduction [2], vibration isolation [3–5], etc. In previous passive vibration reduction applications, tuned mass damper (TMD) is widely used in life [6–9]. However, traditional TMD is effective within a specific narrow frequency band. The concept of a nonlinear energy sink (NES) was first proposed by Vakakis in 2001 [10]. Compared to the TMD, NES has the advantages of a broad vibration c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 310–322, 2024. https://doi.org/10.1007/978-981-97-0554-2_24

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suppression band [11], small additional mass [12], and not changing the natural frequency of the primary system. NES has been widely used for the vibration reduction of beams [13–15], plates [16–18], and shells [19]. Nowadays, NES has many creative and excellent designs. Zang et al. designed a lever-type nonlinear energy sink, which is through amplifying the working displacement to obtain better performance [20]. A track-type NES that achieves nonlinear restoring force through linear spring compression is a more effective design for engineering applications [21–23]. Zeng et al. constructed a bistable nonlinear energy sink (BNES) using a bucking beam and found that this BNES can effectively reduce micro amplitude vibration [24]. Moreover, they also constructed a tristable nonlinear energy sink (TNES) using a bucking beam and magnet that this TNES can effectively reduce large amplitude vibration [25]. The BNES [24,26,27] and the TNES [25] constructed using multiple energy potentials have the characteristic of making it easier for the system to enter the strongly modulated response (SMR). In addition, Geng et al. designed a NES with amplitude-limiting device, and found that NES coupled with amplitudelimiting device has better vibration reduction performance [28,29]. Most of the above studies mainly focus on the use of 1DOF NES for vibration reduction of a linear oscillator (LO). For the study of multi-DOF systems, Gendelman et al. conducted research on the vibration reduction and targeted energy transfer of a 2DOF NES for the transient response of a 1DOF system, and found that the asymmetric stiffness design between the two NES can enhance the targeted energy transfer effect of the 2DOF NES [30]. Wierschem et al. researched the use of 2DOF NES for vibration reduction of the transient response of a 2DOF system [31]. Their research found that 2DOF NES has good vibration reduction performance. Grinberg et al. researched the transient response of a two degree of freedom NES to a single degree of freedom system, and found that the additional DOF of the NES considerably broadens the range of amplitudes where efficient mitigation is possible [32]. Dang et al. conducted a 2DOF NES enhanced by inerters, and find this 2DOF inerter-enhanced NES can easily to achieve SMR [33]. In the subsequent content of this article, the 2DOF NES will be called series NES. By utilizing the wider frequency band vibration reduction effect of series NES, this paper applies it to reduce the vibration of a 2DOF linear system. In this paper, the discussion of the system used in this article and the evaluation method of vibration reduction effect are presented in Sect. 2; the explanation of approximate analytical method (harmonic balance method, HBM) and RungeKutta (RK) method is presented in Sect. 3. Section 4 will conduct research on series NES and limited series NES. All of the conclusions will be summed up in Sect. 5.

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Mechanical Model Limited Amplitude Series NES Coupling System

The schematic diagram of a 2DOF primary system coupling a nonlinear vibration absorber is shown in Fig. 1. The nonlinear vibration absorber is composed with two NES, and it can be called series NES. m1 is the mass of the first oscillator of the series NES, and m2 is the mass of the second oscillator of the series NES; c3 is the damping of the first oscillator of the series NES, and c4 is the damping of the second oscillator of the series NES; k3 is the cubic stiffness of the first oscillator of the series NES, and k4 is the cubic stiffness of the second oscillator of the series NES; k5 is the piecewise linear stiffness of the first oscillator of the series NES, and k6 is the piecewise linear stiffness of the second oscillator of the series NES. And L is the unlimited range. M1 , M2 is the mass of the 2DOF primary system; c1 , c2 is the damping of the 2DOF primary system; k1 , k2 is the linear stiffness of the 2DOF primary system. x1 is the displacement of the first oscillator of the primary system, x2 is the displacement of the second oscillator of the primary system. u1 is the displacement of the first oscillator of the series NES, u2 is the displacement of the second oscillator of the series NES. The displacement excitation x0 = A cos(ωt), where A is the excitation amplitude, and ω is the excitation frequency.

Fig. 1. The schematic of coupling system.

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The governing equations of the coupling system can be derived as ¨1 + c1 (x˙ 1 − x˙ 0 ) + c2 (x˙ 1 − x˙ 2 ) + k1 (x1 − x0 ) + k2 (x1 − x2 ) = 0, M1 x ¨2 + c2 (x˙ 2 − x˙ 1 ) + c3 (x˙ 2 − u˙ 1 ) + k2 (x2 − x1 ) + k3 (x2 − u1 )3 + h1 (x2 − u1 ) M2 x = 0, ¨1 + c3 (u˙ 1 − x˙ 2 ) + c4 (u˙ 1 − u˙ 2 ) + k3 (u1 − x2 )3 + k4 (u1 − u2 )3 + h2 (u1 − u2 ) m1 u = h1 (x2 − u1 ), ¨2 + c4 (u˙ 2 − u˙ 1 ) + k4 (u2 − u1 )3 = h2 (u1 − u2 ) m2 u (1) where the h1 (x2 − u1 ) and h2 (u1 − u2 ) are the piecewise stiffness restoring force. They can be derived as ⎧ ⎪ ⎨k5 (x2 − u1 + L), x2 − u1 < −L, h1 (x2 − u1 ) = 0, (2) −L ≤ x2 − u1 ≤ L, ⎪ ⎩ k5 (x2 − u1 − L), x2 − u1 > L ⎧ ⎪ ⎨k6 (u1 − u2 + L), u1 − u2 < −L, h2 (u1 − u2 ) = 0, (3) −L ≤ u1 − u2 ≤ L, ⎪ ⎩ k6 (u1 − u2 − L), u1 − u2 > L For the piecewise curves represented by Eq. (2) and Eq. (3), the hyperbolic tangent function can be used to fit the stiffness into a continuous function. They can be derived as   1 [1 + tanh (ε (x2 − u1 − L))] H1 (x2 − u1 ) = k5 (x2 − u1 − L) 2   (4) 1 + k5 (x2 − u1 + L) [1 − tanh (ε (x2 − u1 + L))] 2   1 H2 (u1 − u2 ) = k5 (u1 − u2 − L) [1 + tanh (ε (u1 − u2 − L))] 2   (5) 1 + k5 (u1 − u2 + L) [1 − tanh (ε (u1 − u2 + L))] 2 2.2

Series NES Coupling System

If the limiting device is removed from the system shown in Fig. 1, it will become a series NES coupling system as shown in Fig. 2. The governing equations of the new coupling system can be derived as M1 x ¨1 + c1 (x˙ 1 − x˙ 0 ) + c2 (x˙ 1 − x˙ 2 ) + k1 (x1 − x0 ) + k2 (x1 − x2 ) = 0, ¨2 + c2 (x˙ 2 − x˙ 1 ) + c3 (x˙ 2 − u˙ 1 ) + k2 (x2 − x1 ) + k3 (x2 − u1 )3 = 0, M2 x ¨1 + c3 (u˙ 1 − x˙ 2 ) + c4 (u˙ 1 − u˙ 2 ) + k3 (u1 − x2 )3 + k4 (u1 − u2 )3 = 0, m1 u ¨2 + c4 (u˙ 2 − u˙ 1 ) + k4 (u2 − u1 )3 = 0 m2 u

(6)

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Fig. 2. The schematic of new coupling system.

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Evaluation of Vibration Reduction Effect

In this article, the energy of the primary system is used as the evaluation of vibration reduction effect. The transient energy of the primary system at a certain time can be derived as T (t) =

1 1 1 1 M1 x˙ 21 + M2 x˙ 22 + k1 (x1 − x0 )2 + k2 (x2 − x1 )2 2 2 2 2

(7)

The displacement and velocity at each moment can be obtained using the 4th order Runge Kutta method. To eliminate the influence of Initial condition, the time step is chosen to be 1/100 of the present frequency period, and 1000 periods are computed for each frequency to determine the energy at each instance. The RMS root mean square is then calculated by selecting the energy at each instance within the preceding 50 periods, denoted as E = RM S(T (t)). The evaluation of vibration reduction effect can be derived as ηi =

Ei−LO − Ei−N ES × 100%, i = 1, 2 Ei−LO

(8)

where, ηi is the vibration reduction efficiency of the i-th natural frequency. Ei−LO is the max energy of the primary system at the i-th natural frequency when it is not coupled with NES, Ei−N ES is the max energy of the primary system at the i-th natural frequency when it is coupled with NES.

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Approximate Analytical Method

In this article, the harmonic balance method (HBM) is used to analyze the coupled system. Pseudo arc length method (PALM) is used to solve nonlinear equations constructed by HBM. In this article, the displacement of the primary system and NES can be written as the superposition of harmonics xi = uj =

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(aik cos(kωt) + bik sin(kωt)) , i = 1, 2, (9) (aik cos(kωt) + bik sin(kωt)) , j = 1, 2, i = j + 2

k=1

By substituting Eq. (9) into either Eq. (1) or Eq. (6), the harmonic coefficient is extracted, resulting in the formation of nonlinear equations that are composed of harmonic coefficient. The Jacobian matrix of the harmonic coefficients can be yielded by the Galerkin process. Then, the values of the harmonic coefficient are solved by the PALM. Incorporate harmonic coefficient into the solution results, calculate 100 steps per period, calculate displacement within two periods, and use RMS root mean square to obtain the amplitude under present displacement excitation. For the stability analysis of the present obtained solution, the Lyapunov first method can be used to solve it. The displacement of the primary system and NES can be written as xi = uj =

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(aik (t) cos(kωt) + bik (t) sin(kωt)) , i = 1, 2, (10) (aik (t) cos(kωt) + bik (t) sin(kωt)) , j = 1, 2, i = j + 2

k=1

By substituting Eq. (10) into either Eq. (1) or Eq. (6). The order of the ordinary differential equation at the zero solution is reduced, and the Jacobian matrix is computed. The eigenvalues of the Jacobian matrix are used for determining the system’s behavior. The system is stable if all eigenvalues are negative; otherwise, it is unstable.

4 4.1

Vibration Suppression Performance of the Limited Series NES Parameter Selection

The parameter selection of the primary system is shown in Table 1, and the parameter selection of the series NES is shown in Table 2. κ is the asymmetric stiffness coefficient of series NES, its value is equal to k4 /k3 and normally κ = 0.1.

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Based on the parameters selected for the series NES, the limiting parameters for the limited series NES are selected as k5 = k6 = 800 Nm−1 , L = 0.02 m. The displacement excitation amplitude A = 0.001 m. By incorporating the data from Table 1 and Table 2, and using the HBM described in Sect. 3 in conjunction with the 4th order Runge Kutta (R-K) method, the amplitude frequency response of the system can be obtained as shown in Fig. 3. From Fig. 3, it can be seen that the solution of the harmonic balance method corresponds to the solution of the 4th order R-K method. The forward and backward scanning results of the R-K method correspond well, proving that the peak NES of the coupled system exhibits periodic response under this parameter as shown in Fig. 4. Table 1. The parameters of LO. Item

Notation Value

Mass of LO1

M1

3 kg

Mass of LO2

M2

3 kg

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k1

8000 Nm−1

Stiffness of LO2

k2

8000 Nm−1

Damping of LO1 c1

0.3 Nsm−1

Damping of LO2 c2

0.3 Nsm−1

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Value

Proportion of series NES total mass

–

3.33%

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μ = m2 /m1 1

Nonlinear cubic stiffness of series NES headend k3

4.2

1.6 × 106 Nm−3

Nonlinear cubic stiffness of series NES end

k4

κ × k3

Damping of series NES headend

c3

3 Nsm−1

Damping of series NES end

c4

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Parameter Analysis of Limited Series NES

As shown in Fig. 3, the series NES has a good vibration reduction effect on the primary system under the current parameters. The comparison between the limited series NES using the default limiting device parameters and the series NES is shown in Fig. 5. Through comparison, it can be found that the amplitude-limiting device can not only increase the reliability of NES engineering

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applications, but also further improve the vibration reduction effect of NES. At the first natural frequency, the vibration reduction effect of the limited series NES is better than that of the series NES. At the second natural frequency, the limiting device has no effect. The comparative phase diagrams of the two at twoorder natural frequencies are shown in Fig. 6 and Fig. 7. It can be clearly seen that the amplitude-limiting device effectively limits the amplitude of the NES at the first-order natural frequency. But it does not take effect at the second-order natural frequency. 10 0 Without NES With Series NES(HBM) R-K(forward) R-K(backward)

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The amplitude frequency response of the primary system under different piecewise stiffness is shown in Fig. 8. According to Fig. 8, it can be seen that under the same limiting distance, the greater the piecewise stiffness and the stronger the limiting amplitude, the stronger the vibration reduction effect of NES gradually strengthens. The influence of the limiting width L on the vibration reduction effect is shown in Fig. 9. When L = 0.04 m, the limiting device was not triggered, and the vibration reduction effect of NES remained almost unchanged. However, when L = 0.02 m, the limiting device is triggered, and the vibration reduction effect of NES is further strengthened. When L = 0 m, the system is equivalent to coupling a linear oscillator, and the vibration reduction effect becomes worse. The vibration reduction efficiency curves of different limiting widths L calculated by R-K are shown in Fig. 10. As the limiting width L gradually increases, the vibration damping effect of the limited series NES at the first natural frequency changes from poor to good, and then becomes close to that of the ordinary series NES. However, the vibration damping effect of the limited series NES at the second-order natural frequency changes from good to 0.015 k5=k6=0 N/m

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Conclusions

This article investigates the vibration reduction performance and dynamic behavior of the limited series NES. Regarding the vibration reduction effect, in addition to the limited series NES, the original series NES was compared with an uncontrolled system for necessary comparison. Research has found that the limiting device can further improve the vibration reduction effect of the series NES, which has significant benefits in the practical engineering field. On the other hand, by analyzing the parameters of the limited amplitude series NES, some parameters with better vibration reduction effect were obtained. In summary, the conclusion of this article can be summarized as follows: (1) Series NES can effectively reduce multi-mode resonance. And the limited series NES has better vibration reduction effect. By coupling series NES on the primary system, the vibration of the system can be effectively reduced. (2) When the amplitude of the series NES is too large, amplitude-limiting device can effectively reduce the amplitude of the series NES and increase the safety performance of the series NES. (3) The selection of NES parameters for limited series NES has a significant impact on the vibration reduction effect of NES. Moderate amplitudelimiting can enhance the vibration reduction effect of series NES. Acknowledgments. The authors gratefully acknowledge the support of the China National Funds for Distinguished Young Scholars (No. 12025204) and the Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018).

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Understanding Neural Rhythmic Mechanisms Through Self-oscillations of Complex Neural Networks and Their Adaptation Peihua Feng1(B) , Luoqi Ye1 , Xinaer Adilihazi2 , Zhilong Liu3 , and Ying Wu1 1

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State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China {fphd2017,wying36}@xjtu.edu.cn, ylq xjtu [email protected] 2 Aplos Machines Manufacturing (Shanghai) Co., Ltd., Shanghai, China [email protected] State Key Laboratory of Compressor Technology, Hefei General Machinery Research Institute, Hefei, China Abstract. This study delves into the self-oscillation properties of complex neural networks to elucidate the intrinsic mechanisms driving biological rhythm generation and adaptation to external periodic signals. We scrutinize the influence of electrical coupling and Spike-TimingDependent Plasticity (STDP) on network synchronization. Employing a neural network characterized by scale-free topology, we observe that neurons with higher node degrees necessitate activation by an increased number of fellow neurons. Central neurons emerge as pivotal in facilitating swift excitation propagation. In contrast, low-degree circuits sustain activity during burst intervals, with circuit length inherently dictating the rhythmic period. Notably, despite their inherent complexity and diverse rhythm generation, neural networks can adaptively select a lowdegree loop congruent with external input rhythms via Hebbian learning principles. These insights offer profound implications for comprehending the variances in human biological rhythms across different environments and hold significant value for planning extended space expeditions. Keywords: Neural rhythms

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Introduction

Maintaining alertness and vigilance is crucial for astronauts during space missions, especially in the harsh space environment. However, studies have shown that factors such as microgravity and changes in the light-dark cycle can disrupt biological rhythms, leading to a series of physiological changes such as reduced sleep [1]. By identifying and classifying the factors that affect biological rhythms, we can predict the effects of space environment on biological rhythms and prepare astronauts to adapt to these changes. Specifically, we aim to investigate whether c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 323–333, 2024. https://doi.org/10.1007/978-981-97-0554-2_25

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changes in synaptic connections between neurons can help the nervous system readapt to new external rhythms in microgravity. By constructing a neural network with synaptic plasticity and using variable rhythmic signals to stimulate the network, we aim to reveal the mechanisms of spontaneous rhythm generation and adaptive regulation, and establish the relationship between network topology and rhythmic adaptation [2]. The human brain, with its intricate architecture and multifaceted functionalities, has often been likened to a complex network. Leveraging the principles of complex network theory provides a systematic and quantitative approach to uncovering the structural and functional intricacies of the brain. In this context, the small-world network model, proposed by Watts and Strogatz, has emerged as a prominent framework for mimicking the balance between local specialization and global integration observed in brain networks [3]. Graph theoretical approaches, for instance, have become increasingly pivotal in exploring connectivity patterns in both structural and functional brain networks, elucidating phenomena such as network resilience, modularity, and the role of hub regions [4]. The potency of complex networks to replicate the brain’s topology not only aids in advancing our comprehension of neurological processes but also offers potential avenues for studying maladaptive and pathological conditions. Scale-free networks, characterized by a skewed degree distribution where few nodes (hubs) possess a disproportionately high number of connections, have been identified as an essential structure in representing biological nervous systems. Their unique topology mimics the hierarchical and modular organization seen in neural systems, emphasizing the role of hubs in efficient communication and resilience to random failures [5]. Regarding the neural rhythm problem studied in this paper, in fact, complex network dynamics offer a holistic understanding of neural rhythms by capturing the collective behavior of interacting neurons. In recent years, studies have highlighted the importance of network topology in influencing neural oscillations and synchrony [6]. The interplay between structural connectivity and functional patterns, for instance, provides deep insights into how the brain self-organizes and responds to stimuli [7]. Furthermore, the modularity and hierarchy inherent in neural networks, often modelled through complex networks, have shown to be vital for diverse cognitive functions and rhythmic variations [8]. Investigations into phase synchronization, metastable states, and network motifs have further enriched our understanding of the intricate dynamics and stability underlying neural rhythms [9]. Synaptic plasticity, the dynamic modulation of synaptic strength in response to neuronal activity, plays a pivotal role in learning, memory, and the adaptive capabilities of the nervous system. Research has established that synaptic modifications, both long-term potentiation (LTP) and long-term depression (LTD), are not mere by-products of activity but crucial determinants in shaping neural network dynamics and consequently, neural rhythms [10]. Recent studies have further illuminated the complex interplay between synaptic plasticity and oscillatory patterns in various frequency bands, revealing the mechanisms by which synaptic changes can either stabilize or disrupt rhythmic activity [11]. More-

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over, the behavior or properties of the entire neural network can be influenced by events happening at the microscale (individual synapses) [12]. Synaptic plasticity can also induce chimera states in neuronal network [13,14]. In addition, Zhou et al. used supervised learning to change the connection strength of neurons to realize the artificial neural network “remembering” the time information of external signals [15]. Mi et al. used the spontaneous collective oscillation of the neural network to explain the rhythmic firing phenomenon of the zebrafish optic nerve and found a ring structure composed of low-node degree neurons in a complex network [16]. Then Liao et al. searched for oscillatory backbones to explain the periodic firing behavior of neural networks with multiple frequencies [17]. The aforementioned study integrates the network topology and the periodic collective firing of the neural network, providing valuable insights for future research in this area.

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In our study, we employ the FitzHugh-Nagumo (FHN) model (Eq. 1), a simplified but powerful representation capturing the essential dynamics of neuronal firing [18]. The FHN model, a two-dimensional system, mimics the action potential generation and the subsequent refractory period observed in real neurons through the dynamics of its excitable and recovery variables. Its qualitative behavior, combined with computational tractability, renders the FHN model invaluable for large-scale neural network simulations. Here f (ui ) a piecewise function with a specific expression of taking 0 when u is less than 1/3 and taking 1 when u is greater than 1. Otherwise, it takes 1 − 6.75ui (ui − 1)2 . N

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τ v˙ i =f (ui ) − vi Drawing from the inherent heterogeneity in biological systems, our constructed network adopts a scale-free topology, reflecting the hierarchically organized and non-uniform connectivity patterns observed in real neural networks2. Within this structure, a unique feature is implemented: neurons with high node degrees (hub neurons) require simultaneous firing from multiple neurons to become active, underscoring their role as information integration points. In contrast, neurons with lower node degrees can be activated by a single firing neuron, emphasizing their role in the rapid propagation of information. Chemical synapses, predominant in biological systems, are utilized as the coupling mechanism between neurons, enabling dynamic, bidirectional interactions and serving as a foundation for synaptic plasticity studies within our model3. The chemical coupling is denoted by Fij and Fij = C0 (uj − ui ). In our study, the parameters are set as follows: a = 0.84, b = 0.07, ε = 0.04 and C0 = 0.174 such that only neurons with a node degree greater than or equal to 6 can be classified as hub neurons. The fourth-order Runge-Kutta method is used to solve the governing equation with a time step of 0.01.

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Rhythm and Synchronization of Scale-Free Networks

Scale-free networks have attracted considerable attention in the realm of neuroscience due to their inherent ability to mirror certain aspects of real-world neural systems. To delve deeper into their rhythmic and synchronous properties, we embarked on an exploration by generating an arbitrary scale-free network. For our study, specific parameters were set: the power law exponent, denoted as γ, was fixed at 3. This network comprised a total of N = 210 neurons. In tandem with this, the average node degree of the network < k > was established at 4. The choice of these parameters was instrumental in ensuring a balance between complexity and computational tractability, providing an optimal foundation for our investigation into the network’s dynamic behaviors. In our study’s simulation, each neuron’s membrane potential within the network was initialized with a random number uniformly distributed between 0 and 1. These initial conditions served as the baseline for subsequent investigations. Over a span of 150 s, numerical simulations were conducted a total of 100 times. We observe that the network transitioned into a self-sustaining periodic oscillatory steady state in approximately 10% of these simulations. Within this subset, 4% exhibited low-frequency self-oscillations, while 6% revealed highfrequency self-oscillations. Drawing upon prior research, it is well established that the oscillation period is intrinsically determined by the length of the loops formed by neurons with low node degrees; specifically, the number of neurons constituting such loops [16]. An extended loop corresponds to a larger self-excited oscillation period, and conversely, a shorter loop from low-degree nodes results in a diminished period which is distinctly illustrated in Fig. 1(c) and (d). However, not all simulations led to sustained oscillations. In some instances, the network demonstrated only two oscillations before regressing to a quiescent state, depicted in Fig. 1(a). In other scenarios, despite multiple oscillations manifesting initially, the network still invariably settled into a resting state, as evident in Fig. 1(b). On the occasions where the network did evolve into self-sustaining periodic oscillations, the average activity of all neurons portrayed a distinctive pattern: oscillatory attractors characterized by intense bursts of synchronized neuronal firing interspersed with extended intervals where only a sparse number of neurons were active. Such dynamics underscore the network’s capability to uphold a rhythmic and synchronized output. To elucidate the underlying mechanism governing this network behavior, we scrutinized the average firing activity of individual neurons during the rhythmic firing process, with a keen focus on Fig. 1(c). Three distinct neuronal firing patterns emerged from our analysis. Firstly, as exemplified in Fig. 2(a), certain neurons are triggered only once during a singular oscillatory cycle. Intriguingly, the number of spikes from neuron 1 in this figure aligns perfectly with those in Fig. 1(c), both recording six spikes. Such neurons that display this specific firing behavior correspond directly to the hub neurons within the network. Secondly, another firing pattern is characterized by neurons that are triggered twice

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(a) Non-sustained self-oscillation with two peaks

(b) Non-sustained self-oscillation with multiple peaks

(c) Sustained self-oscillation with low frequency

(d) Sustained self-oscillation with high frequency

Fig. 1. The average membrane potential of all neurons in the neural network. (a) non-sustained self-excited oscillation with two peaks, (b) non-sustained self-excited oscillation with multiple peaks ultimately returning to the resting state, (c) sustained self-excited oscillation with low-frequency, (d) sustained high-frequency self-excited oscillation.

within one oscillatory cycle. This is depicted in Fig. 2(b). An examination of neuron number 5 reveals that it produces twice the number of spikes compared to Fig. 1(c), tallying up to 12 spikes. Neurons that exhibit this dual-trigger behavior typically possess lower connectivity, and they play a crucial role in forming the loop structure within the network. The third and final firing pattern is demonstrated in Fig. 2(c). These neurons, in comparison to those in Fig. 1(c), display an intriguing firing behavior, averaging 1.5 spikes per oscillatory cycle. Notably, such neurons are not positioned on the periphery but are relatively proximate, in terms of connection paths (see Fig. 3), to the loop formed by the neurons with the shortest low-node degree. A deep dive into these varied firing patterns has offered significant insights, unearthing the internal dynamics and mechanisms that drive network behavior.

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(a) Firing of Neuron 1

(b) Firing of Neuron 5

(c) Firing of Neuron 31

Fig. 2. Evolution of firing activity of a single neuron over time.

Fig. 3. The loop formed by nodes with low node degree in the network which is the oscillation skeleton.

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In the orchestration of rhythmic synchronous firing, distinct roles emerge for different components within the network. Hub neurons act as primary conduits, rapidly disseminating excitation throughout the network. Conversely, lowdegree loops maintain a persistent level of activity during the interburst intervals. Intriguingly, the length of these loops appears to be a determining factor in the rhythm’s duration. Within this framework, neurons situated within the low-degree loops are observed to fire twice within a singular cycle: once during sporadic bursts and subsequently during the synchronous bursts instigated by central activity. In contrast, certain neurons, as illustrated in Fig. 4, initiate firing solely during the synchronous firing phase. The presence of electrical synapses between neurons emerges as pivotal, ensuring that, as excitation propagates through the lower-degree circuitry, only a solitary central neuron or a very limited number are activated. Additionally, these low-degree loops take on a secondary yet critical role: they preserve the seeds of excitation, thwarting synchronous firing incited by central neurons, thus underpinning the rhythmic cadence of network activity.

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Neural Network Rhythms with STDP

Hebb’s learning rule, often encapsulated by the maxim “cells that fire together, wire together,” postulates that synaptic strength is augmented between two neurons if they are activated concurrently. This enhancement underpins associative learning, where recurrent and simultaneous activations reinforce synaptic connections. In contrast, if one neuron fires without the simultaneous activation of its partner, the synaptic strength may remain unchanged or even decrease. SpikeTiming-Dependent Plasticity (STDP) refines this concept with precise temporal specificity. In STDP, the relative timing between the pre- and post-synaptic spikes determines the direction and magnitude of synaptic modification. If a presynaptic neuron fires just before its postsynaptic counterpart, the synapse is strengthened (potentiation); conversely, if the postsynaptic neuron fires before the presynaptic one, the synapse is weakened (depression). In essence, while Hebb’s rule broadly establishes a connection between co-active neurons, STDP delves into the intricate temporal dynamics of this relationship. Thus, STDP can be seen as an extended, more temporally-detailed version of Hebbian plasticity. The mechanism of STDP is as follows: the synaptic connection strength of two neurons A and B is gij , and the change of neuron connection strength is a function of the STDP learning rule F is determined by ⎧ (−|Δt|)/τ+ , Δt > 1 ⎪ ⎨ A+ e Δt = 0 F (Δx) = c, (2) ⎪ ⎩ (−|Δt|)/τ− − A− e , Δt < 0 where Δt = tj − ti , ti and tj are the firing times of neurons i and j. In our study, A+ = 0.07, A− = 0.016 and τ+ = τ− = 0.02. We subjected the neural network to a stimulus at intervals of 80ms over the initial 1500ms duration. This was implemented by randomly selecting 4% of the neurons during each stimulus and elevating their membrane potential to 1.

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Fig. 4. The comparison between the firing of single neuron and the firing of the network as a whole.

Post the 1500 ms timeframe, external stimulation to the neural network ceased. Throughout this secondary phase, we documented the precise firing times of neurons and adjusted the synaptic strengths between them utilizing the STDP rule. Simultaneously, once the external stimulus was halted, the application of STDP was also discontinued. The outcomes of these simulations are depicted in Fig. 5 and Fig. 6. From Fig. 5, it’s evident that there’s a probability where the

Fig. 5. Under the action of STDP, the state evolution diagram of the neural network under the stimulation of external rhythm signals (the “memory” of the external signal rhythm has not been formed). External stimuli are applied on the left side of the red dotted line, and external stimuli are removed on the right side.

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network fails to manifest periodic synchronous firing patterns aligning with the rhythm of the external stimuli. Conversely, Fig. 6 elucidates that, following 20 consecutive external stimuli, the network begins to exhibit a synchronous firing pattern in harmony with the external rhythm upon removal of the stimuli. This data provides numerical evidence that, under the influence of STDP, the neural network is capable of “retaining” the rhythmic pattern of an external stimulus.

Fig. 6. Under the action of STDP, the network state evolution diagram of the neural network under the stimulation of the external rhythm signal (under the action of the two, the neural network forms a “memory” of the rhythm of the external signal). External stimuli are applied on the left side of the red dotted line, and external stimuli are removed on the right side.

5

Conclusions and Prospects

In our comprehensive exploration of neural rhythm generation, we delved into the realm of complex network dynamics. Building on existing research, our studies elucidated that such complex networks inherently showcase intricate internal structures. This complexity subsequently manifests as multifaceted self-excited oscillations in the network dynamics. Through our findings, we can assertively corroborate the perspective that the network’s topological structure is intrinsically tied to its temporal characteristics. As we further explored, loop structures, primarily formed of neurons with low node degrees, are quintessential in determining the overall oscillatory period of the network. On the other hand, hub neurons, or neurons with high node degrees, effectively drive the collective synchrony across the network. An increase in complexity within the loop structures, as evidenced by multiple intertwined loops, transitions the network from a singular oscillatory pattern to more intricate behaviors, with multiple peaks evident within a single oscillatory cycle. Our numerical experimentation with Spike-Timing-Dependent Plasticity (STDP) for the dynamic adjustment of synaptic connections presented noteworthy results. When subjected to periodic external stimuli, the network predominantly exhibited epilepsy-like burst behaviors. However, on specific instances, the network maintained oscillations resonating with the external stimulus rhythm

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even after its withdrawal. This behavior suggests an inherent capacity within the neural networks to adapt and potentially “retain” certain periodic inputs, offering avenues for deeper exploration into neural memory mechanisms at a network level. Our research offers pivotal insights into the dynamics of complex neural networks, emphasizing the integral relationship between their topological structure and temporal behavior. These findings not only enhance our comprehension of biological neural systems but also pave the way for advancements in artificial neural network design. The insights derived from this study pave the way for a deeper comprehension of how neural networks, particularly with a scale-free topology, generate and adapt to rhythmic stimuli. Understanding the intricate dance of hub neurons, low-degree loops, and synaptic plasticity via STDP offers a framework to predict and possibly manipulate neural responses. As space missions continue to extend in duration and distance, discerning the adaptability of biological rhythms becomes paramount. This research could inform astronaut training regimens, helping them prepare and adapt to the vast expanse of space. Moreover, the findings might also have terrestrial implications: for patients with neurological disorders, especially rhythmic disturbances like epilepsy. By harnessing the adaptability mechanisms revealed, it’s conceivable to develop advanced therapeutic interventions that fine-tune neural synchrony and rhythms, paving a new frontier in neurology. Acknowledgements. We would like to thank Prof. Yuanyuan Mi for the very useful discussion. This work was supported by the key National Natural Science Foundation of China (Grant Nos.12132012), and Youth program of National Natural Science Foundation of China under Grant Nos. 12002252, and Opening project of State Key Laboratory (Grant No. SKL-YSJ201913).

References 1. Mistlberger, R.E., Skene, D.J.: Social influences on mammalian circadian rhythms: animal and human studies. Biol. Rev. 79(3), 533–556 (2004) 2. Buzsaki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science 304(5679), 1926–1929 (2004) 3. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998) 4. Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10(3), 186–198 (2009) 5. Barab´ asi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999) 6. Buzs´ aki, G.: Rhythms of the Brain. Oxford University Press (2006) 7. Sporns, O.: Networks of the Brain. MIT Press (2016) 8. Bassett, D.S., Bullmore, E.D.: Small-world brain networks. Neuroscientist 12(6), 512–523 (2006) 9. Breakspear, M.: Dynamic models of large-scale brain activity. Nat. Neurosci. 20(3), 340–352 (2017)

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10. Bliss, T.V.P., Collingridge, G.L.: A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361(6407), 31–39 (1993) 11. Buzs´ aki, G., Wang, X.J.: Mechanisms of gamma oscillations. Annu. Rev. Neurosci. 35, 203–225 (2012) 12. Turrigiano, G.G., Nelson, S.B.: Homeostatic plasticity in the developing nervous system. Nat. Rev. Neurosci. 5(2), 97–107 (2004) 13. Huo, S., Tian, C., Kang, L., et al.: Chimera states of neuron networks with adaptive coupling. Nonlinear Dyn. 96, 75–86 (2019) 14. Feng, P., Wu, Y.: Pattern selection in multilayer network with adaptive coupling. Int. J. Bifurcation Chaos 33(05), 2330012 (2023) 15. Bi, Z., Zhou, C.: Understanding the computation of time using neural network models. Proc. Natl. Acad. Sci. 117(19), 10530–10540 (2020) 16. Mi, Y., Liao, X., Huang, X., et al.: Long-period rhythmic synchronous firing in a scale-free network. Proc. Natl. Acad. Sci. 110(50), E4931–E4936 (2013) 17. Liao, X., Xia, Q., Qian, Y., et al.: Pattern formation in oscillatory complex networks consisting of excitable nodes. Phys. Rev. E 83(5), 056204 (2011) 18. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J . 1(6), 445–466 (1961)

Multi-attention Based Multi-scale Temporal Convolution Network for Remaining Useful Life Prediction of Rolling Bearings Yiwen Cong(B) China Nanhu Academy of Electronics and Information Technology, Jiaxing 314000, China [email protected]

Abstract. Remaining useful life (RUL) prediction is the key to prognostic health management (PHM) of rolling bearings. Multiple sensors are usually used to monitor the comprehensive data of the rolling bearing operation. To the best of our knowledge, most of the existing research work does not consider the effects of different sensors and different features to RUL prediction results, so they cannot accurately understand more comprehensive information about rolling bearings. Therefore, it is necessary to weigh the information of different sensors and assign weights to the features extracted by the neural network. This paper proposes a RUL prediction method of rolling bearings based on multi-attention and multi-scale temporal convolution network (MA-MTCN). Specifically, the channel attention mechanism is used to weigh different sensors and the feature attention mechanism is used to weigh different features which extract by multi-scale temporal convolutional networks. The temporal convolutional networks are used to extract the complete features and time series features between data. Finally, the feasibility of this method is verified by PHM2012 dataset. Experimental results prove that the MA-MTCN method can reduce the error of RUL prediction to less than 5%. Keywords: Remaining Useful Life Prediction · Channel Attention Mechanism · Feature Attention Mechanism · Multi-scale Method · Temporal Convolutional Network

1 Introduction Prognosis and Health Management (PHM) is a kind of health management technology for large equipment maintenance, which is put forward by domestic and foreign experts and scholars in big data environment in recent years [1]. With the development of the intelligent equipment and industrial internet, PHM of rotating machinery has received extensive attention in recent years [2]. In particular, rolling bearings are the most important key components in rotating machinery. As one of the core components of rotating machinery, the main role of rolling bearing in rotating machinery is to support the rotating axis and other parts on the axis to ensure the position accuracy and rotation accuracy of the shaft in operation. The running state of the rolling bearing directly affects the running state of the rotating machinery. Once the rolling © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 334–347, 2024. https://doi.org/10.1007/978-981-97-0554-2_26

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bearing fails, it affects the operation of the whole large equipment, so its remaining useful life should be predicted to select the best maintenance and replacement time [3]. The RUL prediction of the bearing not only helps to grasp the health status of the system in real time, but also provides a theoretical basis for the scientific formulation of health management strategies [4]. During the past decades, various research works have been proposed to predict RUL of rolling bearings. These RUL methods can be divided into two categories: modelbased methods [5] and data-driven methods [6]. With the development of industry, the RUL prediction method based on mechanism modeling is gradually hard to meet the needs of complex operation of modern machinery [7]. With the arrival of big data, the amount of sensor data becomes larger and a large number of sensor data promote the development of RUL prediction based on data. More researchers begin to pay attention to data-driven RUL prediction methods and this method can be further divided into statistical data-driven methods and deep learning methods [8]. The RUL prediction methods of complex degenerate system based on deep learning mainly include the method based on depth neural network (DNN), depth confidence network (DBN), convolution neural network (CNN), recurrent neural network (RNN). The following is a detailed overview of these deep learning RUL prediction methods. DNN is the simplest and most typical structure in deep learning [9]. Li proposed a method combining the traditional regression model and DNN [10]. The method can reduce the influence of noise and other factors on the RUL prediction results. DBN can automatically extract typical features of input data by stacking RBM, thus avoiding the uncertainty caused by manual feature extraction [11]. A model combining DBN and SVM method is proposed to predict RUL [12], which combines the feature extraction ability of DBN with the nonlinear fitting ability of SVM.CNN is proposed to predict the RUL of bearings [13]. The parameter sharing and spatial pooling of CNN solves the problem of too many parameters of DBN network. This method achieves better RUL prediction results than other networks on C-MAPSS dataset and PHM2008 dataset. RNN has a special internal connection structure, which adds a recursive connection on the basis of the feedforward connection [14]. It not only has the ability to pass information backwards, but also can process data with time relations. RNN is proposed to deal with the 2008 PHM challenge dataset directly [15]. The results of the RUL prediction show that the high-precision of RNN. Long-short term memory network (LSTM) is proposed to strengthen the memory ability of RNN to long time series data by introducing the concept of gate [16]. LSTM is proposed to predict the RUL of aero-engine [17]. It combined Dropout in the network to improve the generalization ability of the model, and finally proved that the effect of LSTM network is obviously better than RNN in RUL prediction. But there are still too many parameters in LSTM because there are three gates. GRU network is proposed on the basis of LSTM, and it proposed to predict RUL of the rolling bearing [18]. Verified by the dataset, it can get the same prediction results as LSTM but with less time. On the basis of the comprehensive analysis of existing RUL prediction methods of rolling bearings, there are still some issues in existence: (1) In previous methods, there is only one-scale network to extract features, the features are not comprehensive. Incomplete features will reduce the accuracy of prediction.

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(2) In practice, the contribution of data from multiple sensors and features to RUL prediction is different. However, most of the existing method defaults the weight of each feature is same. It also reduces the RUL prediction performance. (3) In the previous degenerate feature extraction methods, the number of network parameters used to extract features are large. This results the low network training efficiency, which is also reduce the prediction performance of RUL. In order to deal with these problems, this paper propose a RUL prediction method of rolling bearing based on multi-attention and multi-scale temporal convolution network. Specially, the weights of different sensors are assigned by the channel attention mechanism. Then, a multi-scale temporal convolutional network is used to extract features and time series between data in different scales. After that, the feature attention mechanism is used to assign weights of different features. Finally, input the features into the fully connected layers and estimate the RUL. The main innovations of this paper are as follows: (1) This method uses multi-scale TCN to extract features from different time scales. It can extract time series and complete features and increase the accuracy of RUL prediction. (2) This method solves the problem of the influence of different sensors and features on RUL prediction results. It improves the accuracy of RUL prediction. (3) This method uses TCN to extract features, and TCN has less parameters than other networks. It solves the problem of low efficiency of network training and increase the accuracy of RUL prediction.

2 Theoretical Background 2.1 Temporal Convolutional Network Temporal convolutional network (TCN) is an improved CNN. In the past, people generally used RNN to extract the features of time series signals, because the recurrent structure of RNN can express time series well, but RNN always has too many parameters. Due to the fixed size of convolution kernel, CNN cannot capture long-term dependent information well. But TCN can solve this problem, it is a kind of one-dimensional CNN, so it has the global parameters sharing features of it. It means TCN can deal with multi-thread parallel processing in the same time, which improves the speed of extracting time series features of the network. For input x0…xT, the output of TCN is expressed as follow: 



y0 , ..., yT = f (x0 , ..., xT )

(1)

The above formula shows that the input and output dimensions of TCN are the same, and the reason is TCN use one-dimension fully convolution neural network. The casual convolution ensures the complete extraction of time series features. The reason why TCN can extract time series features well comes from the dilated convolution and residual connections in its structure, which are described in more detail below.

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2.1.1 Dilated Convolution Dilated convolution is to add space to the convolution content of standard convolution in order to increase the receptive field. Compared to traditional convolution, dilated convolution has one more parameter, that is dilatation rate. And dilated convolution allows interval sampling of input during convolution, the sampling rate is controlled by dilatation rate. Dilated convolution is defined as follows: (F ∗d X )(xt ) =

K 

fk xt−(K−k)d

(2)

k=1

Among them, X represent the input and X = (x1,x2,…,xT), F represent filter and F = (f1,f2,…,fK), k represent the number of layers, d represents the dilatation rate. The structure of dilated convolution is shown in Fig. 1.

Fig. 1. Dilated convolution layer.

It can be seen from Fig. 1 that the value of each layer t time only depends on the value of the t-1 time of the upper layer. It reflects the features of causal convolution. The extraction of information from the upper layer in each layer is leaping, and the dilated rate increases exponentially by two layer by layer. It reflects the features of dilated convolution. Due to the use of dilated convolution, each layer has to do padding, and the size of padding is (k-1)d. 2.1.2 Residual Connections Residual block is instead of the convolution layer in traditional convolution network, and that is also the reason why the input and output of TCN have the same dimension. The output of the residual block is defined as follows: o = Activation(X + F(X ))

(3)

where X represents the input of the residual block, F represents dilated convolution operation, and the structure of the residual block is shown in Fig. 2. It can be seen in Fig. 2 that every residual block has two dilated convolution layers, normalization of weights, ReLU activation function and Dropout.

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z

( i 1)

(i )

(i )

(i )

( z1 ,..., z T )

( i 1)

( z1

( i 1)

,..., z T )

Fig. 2. Residual block structure.

2.2 Attention Mechanism The attention mechanism imitates the attention module of the human’s vision system, which is similar to the distribution of attention when dealing with certain things. Generally, people’s attention is focus on the more important areas, other unimportant areas are ignored, and the attention mechanism is similar to this. Since the all-round development of deep learning, the attention mechanism has been gradually introduced into the deep learning network and become an important part of it. According to the type of attention mechanism, attention mechanism can be divided into hard attention and soft attention. When the hard attention is applied to CNN, there only has two weights, 0 and 1, that means one region noticed and the other is completely ignored. But the soft attention mechanism is different, the range of it is between 0 and 1, each region gets different weights of attention. In region where more attention needs to be allocated, larger weights are assigned, while less important region are assigned smaller weights. Therefore, the existence of attention mechanism can well magnify the important parts and eliminate the unimportant areas, thus solving the problem of LSTM and GRU network processing long time series.

x1

1

o1

x2

2

o2

xk

k

ok

Fig. 3. Attention mechanism.

Figure 3 shows the basic structure of the soft attention mechanism, where x represents the input of the attention mechanism, α represents the different weights assigned by the

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attention mechanism to the input, and o represents the weighted output of the attention mechanism.

3 Proposed Method This section describes in detail the proposed MA-MTCN and shows the architecture in Fig. 4, which contains multi-scale feature extraction module and RUL estimation module. In MA-MTCN, the input is the online monitoring data from different sensors, then the channel attention mechanism assigns weights to different sensors. Use multiscale method to extract complete features of the data itself and the complete timing series between the data, then the feature attention mechanism module assigns weights to different features. Finally, RUL estimation module predicts the remaining useful life of the rolling bearing.

11

12

1k 2 1k 1 1k

21 22 1

2

2k 2

3

2k 1 2k

31 32

3k 2

3k 1 3k

Fig. 4. Attention mechanism used in MA-MTCN.

3.1 Multi-scale Method Convolution kernels of different sizes are used to extract different features, and they have different abilities for different feature extraction. Large convolution kernels are more suitable for global features, while for some local features, smaller convolution kernels are more suitable. In the case of multi sensors monitoring at the same time, not only the scales of signals from different sensors are different, but also the degradation information contained in different time periods are also different. In order to extract the comprehensive features of the whole monitoring signal, this paper proposes a multi-scale method.

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In Fig. 4, there are three scales in the input sensor data. There are three kinds of kernel sizes, which are F × 1, 2F × 1, 4F × 1. The three scales are independent of each other, and signal features are extracted respectively. The features extracted from different scales complete each other and the three different scales have the same parameter settings except for the size of the convolution kernel. The signal features extracted from these three scales are assigned their respective weights and finally input into the RUL estimation layer. 3.2 Channel Attention Mechanism The channel attention mechanism in this paper is used to assign weight of different sensors. The sensor data is {x1, x2, … xk-2, xk-1, xk}. It can be seen in Fig. 4 that the input of the channel attention mechanism are the sensor data which from different sensors in the same time, and the following equations calculate their weights: exp(ωk xk ) αk =  exp(ωk xk )

(4)

where k means the number of sensors and the parameter in the formula is ωk. The output is calculated as follow: ok = xk · αk

(5)

where ok is the output of the channel attention mechanism 3.3 Feature Attention Mechanism After assigning weights to the monitoring data of different sensors, the features are extracted by TCN of different scales. We need to use the feature attention mechanism to distribute the weights of these features again. The features extracted by different scales for the final RUL is different, so it uses the feature attention mechanism to allocate weights. The equations are as follow: exp(ωf ff ) βf =  exp(ωf ff )

(6)

where f means features and the parameter in the formula is ωf. The output is calculated as follow: of = ff · βf

(7)

where of is the output of the feature attention mechanism.

4 Verification The experimental analysis of the vibration signal data monitored by multiple sensors is carried out to prove the effectiveness of the MA-MTCN method in this section. In addition, the method proposed in this paper is compared with several other methods which have been proved to be effective.

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4.1 Case Introduction The life prediction data come from IEEE PHM 2012 Data Challenge [19]. Under normal circumstances, the life of the bearing can be transported for tens of thousands of hours. Such a long running time is impossible to actually run in the environment of the test room, it is necessary to use accelerated life test. So the purpose of accelerating failure can be achieved by applying additional load or increasing rotational speed to the bearing. The data acquisition platform of rolling bearings is shown in Fig. 5 and the details are shown in [20].

Fig. 5. PHM2012 data acquisition test bench.

Table 1. PHM2012 dataset. Test condition

1

2

3

Rotating speed

1800rpm

1650rpm

1500rpm

Load

4000N

4200N

5000N

H

Bearing1_1

Bearing2_1

Bearing3_1

Bearing1_2

Bearing2_2

Bearing3_2

m

Bearing1_3

Bearing2_3

Bearing3_3

Bearing1_4

Bearing2_4

Bearing1_5

Bearing2_5

Bearing1_6

Bearing2_6

Bearing1_7

Bearing2_7

The vibration signals in the horizontal and vertical direction are measured by two high-frequency accelerometers 3035B DYTRAN in this platform, and the maximum collection range is 50 g. If it is found that the amplitude of the collected acceleration

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signal is more than 20 g continuously in the process of measuring the acceleration signal. However, the bearing has not stopped working because of its failure. In order to prevent the damage of the test platform caused by the damage of the bearing, it can be considered that the bearing has failed. The operation data of bearing speed 1800rpm and radial load 4000N are selected to verify the method in this paper. The PHM 2012 dataset contains seven different bearings and each bearing run in three different operating conditions, which is shown in Table 1 in details. In the experiment, we choose one operating condition which is rotation at the constant speeds of 1800rpm and with the constant payload weights of 4000N. The training sets in this case are Bearing1 and Bearing2, and the test sets are Bearing3, Bearing4, Bearing5 and Bearing6. 4.2 Evaluation Metrics In the RUL prediction of rolling bearings, the prediction error is a general evaluation index, which is defined as follows: Er = ActRUL − PreRUL

(8)

where ActRUL indicates the true remaining life value of the rolling bearing and PreRUL indicates the predicted remaining life value of the rolling bearing. The calculation formula of percentage error is shown as follows: Eri =

ActRULi − PreRULi × 100% ActRULi

(9)

where ActRULi represents the real residual life value of the i-th predicted object, and PreRULi represents the remaining life predicted value of the i-th predicted object. In the actual situation, the risk of lead prediction (Eri > 0) is lower than that of delay prediction (Eri < 0), so the score value of the prediction object is defined as follows: ⎧ Eri ⎪ )), Eri ≤ 0 ⎨ exp(− ln(0.5)( 5 Ai = (10) ⎪ ⎩ exp(ln(0.5)( Eri )), Er > 0 i 20

Fig. 6. Trend graph of score value versus percentage error.

It can be seen from Fig. 6 that when the absolute values of positive and negative percentage errors are equal, the score of lead prediction is higher, that means the lead

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prediction is better than the delay prediction. The final remaining life prediction results are scored as follows: Score =

N 1  Ai N

(11)

i=1

where N is the number of test objects, the higher the score, the more accurate the remaining life prediction results. 4.3 Experimental Results and Analysis This part is the experiment results of MA-MTCN method on PHM2012 dataset. The specific content is to compare the prediction accuracy of MA-MTCN method and other methods in RUL prediction. Firstly, use the ordinary least squares method to fit the real RUL into a straight line and normalize. It makes the comparison between the prediction results and the actual situation more intuitively. The y-axis range is between 0 and 1. The blue line in the figure represents the true value, the orange line represents the predicted average value, and the green area represents the 95% confidence interval. Figure 7(a) is the RUL prediction result of Bearing 3, Fig. 7(b) is the RUL prediction result of Bearing 4, Fig. 7(c) is the RUL prediction result of Bearing 5, and Fig. 7(d) is the RUL prediction result of Bearing 6.

Fig. 7. RUL prediction result of test dataset: (a) bearing 3, (b) bearing 4, (b) bearing 5, (d) bearing 6.

It can be seen from Fig. 7 that the MA-MTCN method can well predict the RUL of rolling bearings. The RUL curve predicted by the model can basically fit the actual RUL

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curve. At the same time, it can be seen that the predicted RUL value of the MA-MTCN model is below the actual RUL curve in most cases. That means the lead prediction is in most cases. Compared with the delay prediction, lead predictive maintenance can better ensure the normal operation of the equipment and reduce the probability of damage. In order to verify the necessary of attention mechanism and the multi-scale method, four networks are used for comparative analysis with MA-MTCN. These four networks are single TCN, MTCN, MTCN with channel attention mechanism (CA-MTCN), and MTCN with feature attention mechanism (FA-MTCN). The hyper parameter settings of the four networks are exactly the same as MA-MTCN, and the comparison of the prediction results for Bearing 6 are shown in Fig. 8, and the scores and variances of the five methods are shown in Table 2. Table 2. Comparison of the Prediction Results of the Five Methods Methods

Evaluation indicators

Mean value

RMSE

TCN

Score

225.61

43.23

MTCN

Score

241.37

32.23

CA-MTCN

Score

250.69

30.59

FA-MTCN

Score

256.91

65.43

MA-MTCN

Score

264.82

6.98

It can be seen from the results in the Fig. 8 and Table 2 that the MA-MTCN method has achieved higher scores and smaller variance than other methods. The score of MTCN is higher than that of TCN, which proves the necessary of multi-channel method. The score of MTCN with attention mechanism is higher than that without attention mechanism, which proves the necessary of attention mechanism. The score of MA-MTCN is higher than that of MTCN with single attention, which proves the necessary of multi-attention mechanism. At the same time, the variance of the predicted results of MA-MTCN on four bearings is lower than that of the other four methods, which proves that this method has higher stability and more accurate predicted results. In order to further verify the effectiveness of MA-MTCN, three other methods are used to compare with MA-MTCN, PF-based method [21], CLSTM-based method [22] and AGRU-based method [23]. The comparison results are shown in Figs. 9 and Table 3. It can be seen from Fig. 9 that the prediction results of the MA-MTCN method proposed in this chapter are more stable and accurate than those of other methods. The specific comparison between the actual residual life and the predicted residual life is shown in Table 3. It is not difficult to see that the prediction error value of the MAMTCN method proposed in this chapter is far less than that of other methods. Compared with other life prediction methods, the prediction results of this method are more accurate and stable, this further shows that the MA-MTCN method proposed in this chapter is an efficient RUL prediction method.

Multi-attention Based Multi-scale Temporal Convolution Network

(a)The comparison of TCN and MTCN.

345

(b)The comparison of MTCN with CA-MTCN and FA-MTCN.

(C) The comparison of MA-MTCN with CA-MTCN and FA-MTCN. Fig. 8. The comparison of five methods.

Fig. 9. The comparison results between this method and the other three methods.

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Y. Cong Table 3. Performance estimation results of four different network

Testset

ActRUL

PreRUL

MA-MTCN

PF

Bearing3

5730

5725.14

0.08

−0.35

54.73

7.62

Bearing4

339

352.87

−4.09

5.6

38.69

−157.71

Bearing5

1610

1615.89

−0.37

100

Bearing6

1460

1670.38

−14.41

28.08

CLSTM

−99.4 −120.07

AGRU

−72.57 0.93

5 Conclusions This paper proposes a RUL prediction method of rolling bearing based on MA-MTCN, which is consist of multi-scale feature extraction module and RUL estimation module. Firstly, the input of the network is the monitoring data from different sensors. Data of these sensors can fully show the operation of rolling bearings. Then the channel attention mechanism assigns weights to different sensors. The multi-scale feature extraction module extracts the complete features of the data itself and the complete timing series between the data. The feature attention mechanism assigns weights to different features. Finally, RUL estimation module makes a prediction of the rolling bearings. The PHM2012 dataset is used to verify the reliability of the MA-MTCN method and some other methods. The experiments results show that the proposed method is more accurate than existing state-of-the art methods and reduce the error of RUL prediction to less than 5%. Moreover, the prediction performance is more stable. It is worth mentioning that the parameters settings in all network models mentioned in this paper are based on experience. In the future, we can try to use some parameter optimization setting algorithms to automatically set the most appropriate parameters in the model.

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Dynamics Analysis of the Cooperative Dual Marine Lifting Systems Subject to Sea Wave Disturbances Gang Li, Xin Ma(B) , and Yibin Li Center for Robotics, School of Control Science and Engineering, Shandong University, Jinan 250061, China [email protected]

Abstract. Strong coupling and complex nonlinearities increase the difficult of the dynamic modeling of the cooperative dual marine lifting system (DMLS). Moreover, it is a great challenge for DMLS model analysis while considering complex sea wave disturbances. To solve these issues, this paper establishes an accurate model of the DMLS by Lagrangian method without any simplifications. The dynamic relationship among the dual horizontal mechanism displacement, the dual lifting boom heights and the payload attitudes is analyzed accurately. The dynamic model considers ship rolling, heave and sway motion induced to sea wave disturbances. Finally, the simulation results are verified the effectiveness of the proposed dynamic model. Keywords: Dual marine lifting systems · Dynamic modeling · Cooperative control · Sea wave disturbances

1 Introduction With the increase in a huge demand for the marine resources, the dual marine lifting system (DMLS) is used to hoist the offshore platform or the large-scale payload in the marine environment (See Fig. 1). Compared with a single marine lifting system, the DMLS has more state variables, stronger coupling dynamics and geometric constraints. Moreover, the ship motion makes it difficult to control the DMLS since complex sea wave disturbances. In order to research the cooperative control method for the DMLS subject to sea wave disturbances, it is urgent to establish the dynamic model and analysis for the DMLS. The dynamic modeling and controller design for the ship-mounted lifting system (especially the offshore crane) have attracted attention of the scholars around the world. Compared with the land-fixed cranes (such as rotary cranes [5, 6], tower cranes [3, 4] and overhead cranes [1, 2],), the dynamic model of the offshore crane are more complex because the external disturbance is caused by ship sway, pitch, surge, roll, heavy and yaw motion [7].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 348–361, 2024. https://doi.org/10.1007/978-981-97-0554-2_27

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Numerous research works is focused on dynamic modeling and controller design for single pendulum offshore crane with ship roll and heave motion in the two dimensional (2-D) planer [8–10]. In order to accomplish the complex transportation task, the jib can luff and rotate simultaneously in the three dimensional (3-D) workspace. The aforementioned modeling and controllers based on 2-D offshore cranes fail to work for the 3-D offshore cranes in real practical situations. Not only have more control inputs and degrees-of-freedom, but also may occur complex spherical pendulum phenomenon. Specifically, [11, 12] establish the modeling of spherical pendulum dynamics in 3-D offshore crane with jib luffing and rotary motion. A previous work proposes an output feedback regulation controller to suppress the payload spherical swing for 3-D offshore crane with varying cable lengths [13]. Some recent works research complex double pendulum dynamics in offshore crane by utilizing Lagrangian method [14–16]. Offshore platform

Horizontal mechanism 1

Lifting boom 1

Lifting boom 2

Horizontal mechanism 2

Ship 1

Ship 2

Fig. 1. Operation configuration using DMLS.

The aforementioned dynamic modeling and controller design is based on single offshore crane system. Compared with the single offshore crane, the dual offshore crane has more state variables, stronger coupling dynamics and geometric constraints. A dynamic model of dual offshore crane with ship roll motion is established by using Lagrangian method [17]. An optimal learning sliding mode controller is designed for dual offshore crane with external disturbances caused by ships roll motion [18]. An adaptive sliding mode controller based on neural network can ensure that full states of the dual offshore crane converge to zero within a finite time [19]. Based on the in-depth analysis of the existing research works of ship-mounted lifting system, there are many problems to be solved as follows: (1) The existing ship-mounted lifting system is usually the offshore crane, which is a kind of the underactuated system [20–22]. Due to underactuated characteristic, the offshore crane has a limited ability to adjust the payload under the complex sea wave disturbances.

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(2) In practices, the dual offshore crane is used to lift large-scale payload in the marine environment. The existing dual offshore crane model only considers the ship roll motion and ignores the effect of the ship heave and sway motion induced to complex sea wave disturbance. To solve the abovementioned problems, we design an DMLS model that simultaneously considers ship roll, heave and sway motion. The proposed DMLS is a full actuated system, which includes dual horizontal mechanism and dual lifting boom. The main contribution is a novel dynamics model of DMLS with ship roll, heave and sway motion. This paper provides the exact dynamic model equations of DMLS without any linearization. This model can be used by other researchers. Numerical simulation results demonstrate that the accuracy of the dynamic model.

2 Dynamics Model and Analysis 2.1 Kinematic Analysis The model of DMLS in the O-XY coordinates is shown in Fig. 2. The parameters and variables are presented in Table 1.

Y

d

Payload P

α1 ( t )

B2

Lifting boom 1

θ (t )

B1

Horizontal G mechanism 1 2 G1

Lifting boom 2

G4

lb

Horizontal mechanism 2 G3

b2 (t )

lb

O2

b1 (t )

O1

α2 (t )

h

lc la

Ship 1 a1 (t )

lc

la

c2 ( t )

c1 ( t ) s1 ( t )

Ship 2

a2 (t )

O

s2 ( t )

X

Fig. 2. Model of DMLS.

The barycenter of the horizontal mechanism 1, the lifting boom 1, the horizontal mechanism 2 and the lifting boom 2 are defined as G1 (x1 , y1 ), G2 (x2 , y2 ), G3 (x3 , y3 ), G4 (x4 , y4 ), respectively. In the O-XY coordinate, x1 , y1 , x2 , y2 , x3 , y3 , x4 , y4 are expressed as follows:    x1 = −s1 + a1 + 21 la cos α1 + lc sin α1 (1) y1 = c1 + lc cos α1 − a1 + 21 la sin α1

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  x2 = −s1 + (a1 + la ) cos α1 + lc + 21 b1 + 21 lb sin α1 y2 = c1 − (a1 + la ) sin α1 + lc + 21 b1 + 21 lb cos α1    x3 = s2 − a2 + 21 la cos  α2 − lc sin α2 y3 = c2 + lc cos α2 − a2 + 21 la sin α2    x4 = s2 − (a2 + la ) cos α2 −  lc + 21 b1 + 21 lb sin α2 y4 = c2 − (a1 + la ) sin α2 + lc + 21 b1 + 21 lb cos α2



(2) (3) (4)

Table 1. Parameters and variables of the DMLS. Para

Parameter definition

Units

m1 , m2 , m3

Horizontal mechanism, lifting boom and payload masses

kg

la , lb

Horizontal mechanism and lifting boom lengths

m

lc

The distance between the center of ship and the horizontal mechanism

m

ai (i = 1, 2)

Horizontal mechanism i displacement

m

bi (i = 1, 2)

Lifting boom i displacement

m

d

Payload width

m

h

Half of the payload height

m

θ

Payload swing angle

deg

αi (i = 1, 2)

Ship i roll angle

deg

si (i = 1, 2)

Ship i sway displacement

m

ci (i = 1, 2)

Ship i heave displacement

m

Fai (i = 1, 2)

Horizontal mechanism i actuating force

N

Fbi (i = 1, 2)

Lifting boom i actuating force

N

  The barycenter of the payload P xp , yp in the O-XY coordinate are expressed as 

xp = yp =

xB1 +xB2 2 yB1 +yB2 2

− h sin θ + h cos θ

(5)

where B1 (xB1 , yB1 ) and B2 (xB2 , yB2 ) are the position of the top the lifting boom 1 and lifting boom 2, respectively. In the O-XY coordinate, xB1 , yB1 , xB2 , yB2 are expressed as follows:  xB1 = x2 + 21 (b1 + lb ) sin α1 (6) yB1 = y2 + 21 (b1 + lb ) cos α1  xB2 = x4 − 21 (b2 + lb ) sin α2 (7) yB2 = y4 + 21 (b2 + lb ) cos α2

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The geometric constraints can be derived from the X and Y directions as follows cos θ =

xB2 − xB1 d

(8)

sin θ =

yB2 − yB1 d

(9)

2.2 Dynamics analysis Differentiating (1)–(5) with respect to time, we can get velocity signals x˙ 1 , y˙ 1 , x˙ 2 , y˙ 2 , x˙ 3 , y˙ 3 , x˙ 4 , y˙ 4 , x˙ p , y˙ p . The entire kinetic energy of the DMLS K is calculated by using the velocity signals x˙ 1 , y˙ 1 , x˙ 2 , y˙ 2 , x˙ 3 , y˙ 3 , x˙ 4 , y˙ 4 , x˙ p , y˙ p as follows  1   1   1   1  2 m1 x˙ 1 + y˙ 12 + m2 x˙ 22 + y˙ 22 + m1 x˙ 32 + y˙ 32 + m2 x˙ 42 + y˙ 42 2 2 2 2  1  2 2 (10) + mp x˙ p + y˙ p 2

K=

The entire potential energy P of DMLS is obtained as follows: P = m1 gy1 + m2 gy2 + m1 gy3 + m2 gy4 + mp gyp

(11)

We defined a Lagrange function L  K − P and calculate the following equations: ⎧ d ∂L ∂L ⎪ ⎪ dt ∂ a˙ 1 − ∂a1 = F1 ⎪ ⎪ ⎨ d ∂L − ∂L = F2 dt ∂ b˙ 1 ∂b1 (12) d ∂L ∂L ⎪ − ⎪ dt ∂ a ˙ ∂a2 = F3 2 ⎪ ⎪ d ∂L ∂L ⎩ dt ˙ − ∂b2 = F4 ∂ b2

The dynamic model equations of DMLS can be obtained from (12) as follows. (1) The dynamic model equation of the horizontal mechanism 1 is given as 

h h2 1 − mp cos(α1 + α2 ) + mp sin(α1 + α2 ) + mp sin(α1 + α2 ) 2 a¨ 2 4 d d  h h2 1 + − mp sin(α1 + α2 ) + mp cos(α1 + α2 ) − mp sin(α1 + α2 ) 2 b¨ 2 4 d d  2 2  1  h  h 1 1 1  + mp 2 lb + lc + m2 b1 + mp b1 + mp b1 2 + m2 lb + (m1 + m2 )lc + mp lb + lc α¨ 1 2 4 2 4 d d   h2 h h2 1 1 − (m1 + m2 ) + mp + mp 2 sin α1 c¨ 1 − mp sin α1 + mp cos α1 + mp 2 sin α1 c¨ 2 4 4 d d d   h2 h h2 1 1 − (m1 + m2 ) + mp + mp 2 cos α1 s¨1 − − mp cos α1 + mp sin α1 + mp 2 cos α1 s¨2 4 4 d d d

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⎛

⎞   1  1 h2  ⎜ mp d 2 lb + lc cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + 4 mp la sin(α1 + α2 )⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟ h h 1 ⎜− ⎟ m l sin(α + α cos(α + α + m cos(α + α m − ) )b )b p a p p 1 2 1 2 2 1 2 2 ⎜ ⎟ 2 2 4 d d ⎟α¨ 2 +⎜ ⎜ ⎟ 2 ⎜ ⎟ h 1 h ⎜ + mp sin(α + α )a − ⎟ m sin(α + α + m cos(α + α l )a ) p p a 1 2 2 1 2 2 1 2 ⎜ ⎟ 2 4 d d ⎜ ⎟ ⎝ ⎠   h h h + mp lb + lc sin(α1 + α2 ) + mp cos(α1 + α2 )a2 + mp sin(α1 + α2 )b2 d d d  h h2 1 + mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) a˙ 2 α˙ 2 2 d d  h h2 1 + − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 cos(α1 + α2 ) b˙ 2 α˙ 2 2 d d  2 2 h h 1 1 1 − mp 2 la + (m1 + m2 )a1 + mp a1 + mp 2 a1 + m1 la + m2 la + mp la α˙ 12 4 2 4 d d ⎛ ⎞ 2  1 h 1  ⎜ 4 mp la cos(α1 + α2 ) − mp la d 2 cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟  h h 1 ⎜ − mp ⎟ l sin(α + l + α cos(α + α − m cos(α + α m + ) )a )a c p p 1 2 1 2 2 1 2 2⎟ b ⎜ 2 2 4 d d ⎟α˙ 2 +⎜ ⎜ ⎟ 2 2 ⎜ ⎟   h h 1 ⎜ + mp sin(α + α )b − mp sin(α1 + α2 )b2 + mp lb + lc cos(α1 + α2 ) ⎟ 1 2 2 ⎜ ⎟ 2 4 d d ⎜ ⎟ ⎝ ⎠ h h h − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b2 − mp sin(α1 + α2 )a2 d d d   h2 1 1 h2 + m2 + mp + 2mp 2 b˙ 1 α˙ 1 + m1 + m2 + mp + 2 mp a¨ 1 2 4 d d   h 1 − m1 + m2 + mp g sin α1 − mp g cos α1 = F1 2 d

(2) The dynamic model equation of the lifting boom 1 is expressed as h h2 1 mp sin(α1 + α2 ) + mp cos(α1 + α2 ) − mp sin(α1 + α2 ) 2 a¨ 2 4 d d  h h2 1 − − mp cos(α1 + α2 ) + mp sin(α1 + α2 ) + mp sin(α1 + α2 ) 2 b¨ 2 4 d d  2 2 h h 1 1 1 1 − mp 2 la + m2 a1 + mp a1 + mp a1 2 + m2 la + mp la α¨ 1 2 4 2 4 d d   h2 h h2 1 1 1 + m2 + mp + mp 2 cos α1 c¨ 1 − − mp cos α1 + mp sin α1 + mp 2 cos α1 c¨ 2 2 4 4 d d d   1 1 h2 h h2 1 − m2 + mp + mp 2 sin α1 s¨1 + mp sin α1 + mp cos α1 − mp 2 sin α1 s¨2 2 4 4 d d d  h h2 1 + − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 cos(α1 + α2 ) a˙ 2 α˙ 2 2 d d ⎛ ⎞  1 h2 1  ⎜ 4 mp la cos(α1 + α2 ) − mp d 2 la cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟   h 1 h ⎜− mp lb + lc sin(α1 + α2 ) + mp cos(α1 + α2 )a2 − 2 mp cos(α1 + α2 )a2 ⎟ ⎜ ⎟ 2 4 d d ⎜ ⎟α¨ 2 −⎜ ⎟ 2 ⎜ ⎟   ⎜ + 1 mp sin(α + α )b − h mp sin(α + α )b + mp h l + lc cos(α + α ) ⎟ 1 2 2 1 2 2 1 2 b ⎜ ⎟ 4 d d2 ⎜ ⎟ ⎝ ⎠ h h h − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b2 − mp sin(α1 + α2 )a2 d d d  h h2 1 + − mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) b˙ 2 α˙ 2 2 d d 

−

(13)

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G. Li et al.   1   1   h2  h2 1  − mp 2 lb + lc + mp + m2 b1 + mp 2 b1 + m2 lb + 2lc + mp lb + lc α˙ 12 4 4 4 d d ⎞ ⎛  h2 h2 1  ⎜ mp lb d 2 cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + mp d 2 lc cos(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ h2 1 1 ⎟ ⎜ + mp la sin(α1 + α2 ) − mp la sin(α1 + α2 ) − mp cos(α1 + α2 )b2 ⎟ ⎜ 2 4 4 d ⎟ ⎜ ⎟ 2 ⎜ 2 ⎟α˙ ⎜ h h 1 +⎜ ⎟ 2 ⎟ ⎜ + mp d 2 cos(α1 + α2 )b2 + 4 mp sin(α1 + α2 )a2 − mp d sin(α1 + α2 )a2 ⎟ ⎜ ⎟ ⎜  h h ⎟ ⎜ + m + m cos(α + α + l + α l l sin(α ) ) ⎟ ⎜ p a p c 1 2 1 2 b ⎟ ⎜ d d ⎠ ⎝ h h + mp cos(α1 + α2 )a2 + mp sin(α1 + α2 )b2 d d    h2 1 1 1 − m2 + mp + 2mp 2 a˙ 1 α˙ 1 + m + mp + h2 mp b¨ 1 2 4 2 4 d   h 1 1 + m2 + mp g cos α1 − mp g sin α1 = F2 2 2 d

(3) The dynamic model equation of the horizontal mechanism 2 is described as 

h h2 1 − mp cos(α1 + α2 ) + mp sin(α1 + α2 ) + mp cos(α1 + α2 ) 2 a¨ 1 4 d d ⎛

⎞  h2 1  h2 ⎜ mp d 2 lb cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + d 2 mp lc cos(α1 + α2 )⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 h2 1 ⎜ + mp la sin(α1 + α2 ) − ⎟ m l sin(α + α cos(α + α − m ) )b p a p 1 2 1 2 1 ⎜ ⎟ 2 4 4 d ⎜ ⎟ ⎜ ⎟ 2 2 ⎟α¨ 1 h 1 h +⎜ ⎜+ m cos(α + α + sin(α + α − m sin(α + α m )b )a )a p p p 1 2 1 1 2 1 1 2 1 ⎟ ⎜ ⎟ 2 2 4 d d ⎜ ⎟ ⎜ ⎟  h h ⎜ ⎟ ⎜ + mp lb + lc sin(α1 + α2 ) + mp la cos(α1 + α2 ) ⎟ ⎜ ⎟ d d ⎝ ⎠ h h + mp cos(α1 + α2 )a1 + mp sin(α1 + α2 )b1 d d  h h2 1 + − mp sin(α1 + α2 ) + mp cos(α1 + α2 ) − mp sin(α1 + α2 ) 2 b¨ 1 4 d d   2 2 h 1 h 1 + m1 + m2 + mp + 2 mp a¨ 2 + m2 + mp + 2mp 2 b˙ 2 α˙ 2 4 2 d d   1  h2  h2 1 1 1  + mp 2 lb + lc + m2 b2 + mp b2 + mp b2 2 + m2 lb + (m1 + m2 )lc + mp lb + lc α¨ 2 2 4 2 4 d d   2 h h h2 1 1 − mp sin α2 + mp cos α2 + mp 2 sin α2 c¨ 1 − (m1 + m2 ) + mp + mp 2 sin α2 c¨ 2 4 d 4 d d   2 h h h2 1 1 − − mp cos α2 + mp sin α2 + mp 2 cos α2 s¨1 − (m1 + m2 ) + mp + mp 2 cos α2 s¨2 4 d 4 d d  h h2 1 + mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) a˙ 1 α˙ 1 2 d d  h h2 1 + − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 cos(α1 + α2 ) b˙ 1 α˙ 1 2 d d ⎛ ⎞  h2 1  1 ⎜ 4 mp la cos(α1 + α2 ) − mp la d 2 cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟  h h 1 ⎜ − mp ⎟ l sin(α + l + α cos(α + α − m cos(α + α + m ) )a )a c p p 2 1 2 1 2 1 1 2 1⎟ b ⎜ 2 4 d d ⎟α˙ 2 +⎜ ⎜ ⎟ 1 2 ⎜ ⎟ ⎜ + 1 mp sin(α + α )b − mp h sin(α + α )b + mp h l + lc  cos(α + α ) ⎟ 1 2 1 1 2 1 1 2 ⎟ b ⎜ 2 4 d d ⎜ ⎟ ⎝ ⎠ h h h − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b1 − mp sin(α1 + α2 )a1 d d d

(14)

Dynamics Analysis of the Cooperative Dual Marine Lifting Systems  h2 h2 1 1 1 − mp 2 la + (m1 + m2 )a2 + mp a2 + mp 2 a2 + m1 la + m2 la + mp la α˙ 22 4 2 4 d d   h 1 − m1 + m2 + mp g sin α2 + mp g cos α2 = F3 2 d

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(15)

(4) The dynamic model equation of the lifting boom 2 is provided as 

1 h h2 mp sin(α1 + α2 ) + mp cos(α1 + α2 ) − mp sin(α1 + α2 ) 2 a¨ 1 4 d d  h h2 1 − − mp cos(α1 + α2 ) + mp sin(α1 + α2 ) + mp cos(α1 + α2 ) 2 b¨ 1 4 d d ⎛ ⎞  h2 1  1 ⎜ 4 mp la cos(α1 + α2 )mp − d 2 la cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟   h h 1 ⎜ + mp cos(α + α )b − ⎟ l sin(α m + l + α cos(α + α m + ) )a p c p 1 2 1 1 2 1 2 1 b ⎜ ⎟ 2 d 4 d ⎜ ⎟α¨ 1 −⎜ ⎟ 2 2 ⎜ ⎟ ⎜ − h mp cos(α + α )a + 1 mp sin(α + α )b − h mp la sin(α + α )b ⎟ 1 2 1 1 2 1 1 2 1 ⎜ ⎟ 4 d2 d2 ⎜ ⎟ ⎝ ⎠  h h h + mp lb + lc cos(α1 + α2 ) − mp la sin(α1 + α2 ) − mp sin(α1 + α2 )a1 d d d  h2 h2 1 1 1 1 − mp 2 la + m2 a2 + mp a2 + mp a2 2 + m2 la + mp la α¨ 2 2 4 2 4 d d   2 1 h h h2 1 1 − − mp cos α2 + mp sin α2 + mp 2 cos α2 c¨ 1 + m2 + mp + mp 2 cos α2 c¨ 2 4 d 2 4 d d   2 2 1 1 h h h 1 + mp sin α2 + mp cos α2 − mp 2 sin α2 s¨1 − m2 + mp + mp 2 sin α2 s¨2 4 d 2 4 d d  2 h h 1 + − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 cos(α1 + α2 ) a˙ 1 α˙ 1 2 d d  h h2 1 + − mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) b˙ 1 α˙ 1 2 d d ⎞ ⎛ 2  h h2 1  ⎜ mp lb d 2 cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + mp d 2 lc cos(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 h2 1 ⎟ ⎜ + mp la sin(α1 + α2 ) − mp sin(α + α cos(α + α m − ) )b p 1 2 1 2 1 ⎟ ⎜ 2 4 4 d ⎟ ⎜ ⎟ 2 ⎜ ⎟α˙ h2 h h 1 +⎜ ⎜ + mp 1 cos(α + α + + α − m + α m sin(α sin(α )b )a )a p p 1 2 1 1 2 1 1 2 1 ⎟ ⎟ ⎜ 2 4 d d d ⎟ ⎜ ⎟ ⎜  h h ⎟ ⎜ ⎟ ⎜ + mp la cos(α1 + α2 ) + mp lb + lc sin(α1 + α2 ) ⎟ ⎜ d d ⎠ ⎝ h h + mp cos(α1 + α2 )a1 + mp sin(α1 + α2 )b1 d d   1  h2  h2 1 1 1  − mp 2 lb + lc + m2 b2 + mp 2 b2 + m2 lb + m2 lc + mp lb + lc α˙ 22 4 4 2 4 d d   2 2   1 h h 1 m + mp + 2 mp b¨ 2 − m2 + mp + 2mp 2 a˙ 2 α˙ 2 + 2 4 2 d d   h 1 1 + m2 + mp g cos α2 − mp g sin α2 = F4 2 2 d −

(16)

where F1  F1a − F1f , F2  F2a − F2f , F3  F3a − F3f , F4  F4a − F4f . F1a , F2a , F3a and F4a are the actuating forces controlling the horizontal mechanism 1, the lifting boom 1, the horizontal mechanism 2 and the lifting boom 2. F1f , F2f , F3f and F4f represent the friction force of the horizontal mechanism 1, the

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lifting boom 1, the horizontal mechanism 2 and the lifting boom 2 as [23,24]:    ⎧ F1f = f1 tanh a˙ 1 ε − kr1|˙a1 |˙ ⎪ a1 ⎪ ⎨ F2f = f2 tanh b˙ 1  ε  − kr2 ˙b1 ˙b1 (17) ⎪ a2 F3f = f3 tanh a˙ 2 ε − kr3|˙a2 |˙ ⎪  ⎩ F4f = f4 tanh b˙ 2 ε − kr4 ˙b2 ˙b2 where f1 , f2 , f3 , f3 , ε, kr1 , kr2 , kr3 , kr4 are friction coefficients. The complex the dynamic model (13)–(16) can be further rewritten as follows: M (q)¨q + C(q, q˙ )˙q + G(q) = U + D where ⎡

⎤ ⎤ ⎡ m11 0 m13 m14 0 c12 c13 c14 ⎢ ⎥ ⎢ ⎥ 0 c23 c24 ⎥ ⎢ 0 m22 m23 m24 ⎥ ⎢c M (q) = ⎢ ⎥ ⎥, C(q, q˙ ) = ⎢ 21 ⎣ m13 m23 m33 0 ⎦ ⎣ c31 c32 0 c34 ⎦ m14 m24 0 m44 c41 c42 c43 0     T T T T G(q) = g1 g2 g3 g4 , U = F1 F2 F3 F4 , q = a1 b1 a2 b2 , D = d1 d2 d3 d4 h h2 1 h2 1 m11 = m1 + m2 + mp + 2 mp , m13 = − mp cos(α1 + α2 ) + mp sin(α1 + α2 ) + mp 2 cos(α1 + α2 ), 4 4 d d d  h h2 h2 1 1 m + mp + mp 2 , m14 = − mp sin(α1 + α2 ) − mp cos(α1 + α2 ) + mp 2 sin(α1 + α2 ), m22 = 4 d 4 2 d d h h2 1 1 h2 m23 = − mp sin(α1 + α2 ) − mp cos(α1 + α2 ) + mp 2 sin(α1 + α2 ), m33 = m1 + m2 + mp + 2 mp , 4 d 4 d d  h h2 h2 1 1 m + mp + mp 2 , m24 = mp cos(α1 + α2 ) − mp sin(α1 + α2 ) − mp 2 cos(α1 + α2 ), m44 = 4 d 4 2 d d   h2 h2 1 1 c12 = m2 + mp + 2mp 2 α˙ 1 , c21 = − m2 + mp + 2mp 2 α˙ 1 , 2 2 d d  2 1 h h mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) α˙ 2 , c13 = 2 d d  h h2 1 c14 = − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 sin(α1 + α2 ) α˙ 2 , 2 d d  h h2 1 c23 = − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 sin(α1 + α2 ) α˙ 2 , 2 d d  1 h h2 c24 = − mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) α˙ 2 , 2 d d  1 h h2 mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) α˙ 1 , c31 = 2 d d  h h2 1 c32 = − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 sin(α1 + α2 ) α˙ 1 , 2 d d   h2 h2 1 1 c34 = m2 + mp + 2mp 2 α˙ 2 , c43 = − m2 + mp + 2mp 2 α˙ 2 , 2 2 d d  h h2 1 c41 = − mp cos(α1 + α2 ) + 2mp sin(α1 + α2 ) + 2mp 2 sin(α1 + α2 ) α˙ 1 , 2 d d  h h2 1 c42 = − mp sin(α1 + α2 ) + 2mp cos(α1 + α2 ) − 2mp 2 sin(α1 + α2 ) α˙ 1 , 2 d d

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Dynamics Analysis of the Cooperative Dual Marine Lifting Systems   h 1 g1 = − m1 + m2 + mp g sin α1 − mp g cos α1 , 2 d   h 1 1 m2 + mp g cos α1 − mp g sin α1 , g2 = 2 2 d   1 h g3 = − m1 + m2 + mp g sin α2 + mp g cos α2 , 2 d   h 1 1 g4 = m2 + mp g cos α2 − mp g sin α2 , 2 2 d

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  1  h2  h2 1 1 1  d1 = mp 2 lb + lc + m2 b1 + mp b1 + mp b1 2 + m2 lb + (m1 + m2 )lc + mp lb + lc α¨ 1 2 4 2 4 d d ⎞ ⎛ 2     h 1 1 h2 ⎜ mp d 2 lb + lc cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + 4 mp la sin(α1 + α2 ) − d 2 mp la sin(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ + ⎜ − 1 mp cos(α + α )b + h mp cos(α + α )b + 1 mp sin(α + α )a − h mp sin(α + α )a ⎟α¨ 2 1 2 2 1 2 2 1 2 2 1 2 2 ⎟ ⎜ 4 4 d2 d2 ⎟ ⎜ ⎠ ⎝  h h h h + mp la cos(α1 + α2 ) + mp lb + lc sin(α1 + α2 ) + mp cos(α1 + α2 )a2 + mp sin(α1 + α2 )b2 d d d d   1 h2 h h2 1 − (m1 + m2 ) + mp + mp 2 sin α1 c¨ 1 − mp sin α1 + mp cos α1 + mp 2 sin α1 c¨ 2 4 4 d d d   2 1 h h h2 1 − (m1 + m2 ) + mp + mp 2 cos α1 s¨1 − − mp cos α1 + mp sin α1 + mp 2 cos α1 s¨2 4 4 d d d  2 2 h h 1 1 1 − mp 2 la + (m1 + m2 )a1 + mp a1 + mp 2 a1 + m1 la + m2 la + mp la α˙ 12 4 2 4 d d ⎞ ⎛   h2 h2  1 1  ⎜ 4 mp la cos(α1 + α2 ) − mp la d 2 cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) − mp d 2 lb + lc sin(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ 2 2 ⎟ 2 ⎜ + ⎜ 1 mp cos(α + α )a − mp h cos(α + α )a + 1 mp sin(α + α )b − mp h sin(α + α )b ⎟α˙ 2 1 2 2 1 2 2 1 2 2 1 2 2 ⎟ ⎜4 2 2 4 d d ⎟ ⎜ ⎠ ⎝  h h h h mp lb + lc cos(α1 + α2 ) − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b2 − mp sin(α1 + α2 )a2 d d d d h2 h2 1 1 1 1 d2 = − mp 2 la + m2 a1 + mp a1 + mp a1 2 + m2 la + mp la α¨ 1 2 4 2 4 d d ⎞ ⎛    h2 1 1  h2 ⎜ 4 mp la cos(α1 + α2 ) − mp d 2 la cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) − d 2 mp lb + lc sin(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ − ⎜ + 1 mp cos(α + α )a − h mp cos(α + α )a + 1 mp sin(α + α )b − h mp sin(α + α )b ⎟α¨ 2 1 2 2 1 2 2 1 2 2 1 2 2 ⎟ ⎜ 2 2 4 4 d d ⎟ ⎜ ⎠ ⎝   h h h h + mp lb + lc cos(α1 + α2 ) − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b2 − mp sin(α1 + α2 )a2 d d d d   h2 h h2 1 1 1 + m2 + mp + mp 2 cos α1 c¨ 1 − − mp cos α1 + mp sin α1 + mp 2 cos α1 c¨ 2 2 4 4 d d d   2 2 h h h 1 1 1 − m2 + mp + mp 2 sin α1 s¨1 + mp sin α1 + mp cos α1 − mp 2 sin α1 s¨2 2 4 4 d d d  2 2  1   1   h  h 1  − mp 2 lb + lc + mp + m2 b1 + mp 2 b1 + m2 lb + 2lc + mp lb + lc α˙ 12 4 4 4 d d ⎛ ⎞ 2 2  h h 1  ⎜ mp lb d 2 cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + mp d 2 lc cos(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ h2 h2 1 1 ⎜ + mp la sin(α1 + α2 ) − mp la ⎟ sin(α + α cos(α + α + m cos(α + α − m ) )b )b p p 1 2 1 2 2 1 2 2 ⎜ ⎟ 2 2 2 4 4 d d +⎜ ⎟α˙ 2 ⎜ ⎟   h h h 1 ⎜ ⎟ ⎜ + mp sin(α1 + α2 )a2 − mp sin(α1 + α2 )a2 + mp la cos(α1 + α2 ) + mp lb + lc sin(α1 + α2 )⎟ ⎜ ⎟ 4 d d d ⎝ ⎠ h h + mp cos(α1 + α2 )a2 + mp sin(α1 + α2 )b2 d d  1  h2  h2 1 1 1  d3 = mp 2 lb + lc + m2 b2 + mp b2 + mp b2 2 + m2 lb + (m1 + m2 )lc + mp lb + lc α¨ 2 2 4 2 4 d d

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 1 h h2 h2 1 mp sin α2 + mp cos α2 + mp 2 sin α2 c¨ 1 − (m1 + m2 ) + mp + mp 2 sin α2 c¨ 2 4 d 4 d d   h h2 h2 1 1 − − mp cos α2 + mp sin α2 + mp 2 cos α2 s¨1 − (m1 + m2 ) + mp + mp 2 cos α2 s¨2 4 d 4 d d ⎛ ⎞  h2 1  h2 ⎜ mp d 2 lb cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + d 2 mp lc cos(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟ ⎜ + 1 mp la sin(α + α ) − h mp la sin(α + α ) − 1 mp cos(α + α )b + h mp cos(α + α )b ⎟ 1 2 1 2 1 2 1 1 2 1 ⎜ ⎟ 2 2 4 4 d d ⎜ ⎟α¨ 1 +⎜ ⎟ 2 ⎜ ⎟ ⎜ + 1 mp sin(α + α )a − h mp sin(α + α )a + mp h l + lc  sin(α + α ) ⎟ 1 2 1 1 2 1 1 2 b ⎜ ⎟ 4 d d2 ⎜ ⎟ ⎝ ⎠ h h h + mp la cos(α1 + α2 ) + mp cos(α1 + α2 )a1 + mp sin(α1 + α2 )b1 d d d ⎛ ⎞  h2 1  1 ⎜ 4 mp la cos(α1 + α2 ) − mp la d 2 cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 2 ⎜ ⎟   ⎜ − mp h l + lc sin(α + α ) + 1 mp cos(α + α )a − mp h cos(α + α )a ⎟ 1 2 1 2 1 1 2 1 b ⎜ ⎟ 2 2 2 4 d d ⎜ ⎟α˙ +⎜ ⎟ 1 2 ⎜ ⎟   ⎜ + 1 mp sin(α + α )b − mp h sin(α + α )b + mp h l + lc cos(α + α ) ⎟ 1 2 1 1 2 1 1 2 b ⎜ ⎟ 4 d d2 ⎜ ⎟ ⎝ ⎠ h h h − mp la sin(α1 + α2 ) + mp cos(α1 + α2 )b1 − mp sin(α1 + α2 )a1 d d d  h2 h2 1 1 1 − mp 2 la + (m1 + m2 )a2 + mp a2 + mp 2 a2 + m1 la + m2 la + mp la α˙ 22 4 2 4 d d  h2 h2 1 1 1 1 d4 = − mp 2 la + m2 a2 + mp a2 + mp a2 2 + m2 la + mp la α¨ 2 2 4 2 4 d d ⎛ ⎞ 2   1 h 1 ⎜ 4 mp la cos(α1 + α2 )mp − d 2 la cos(α1 + α2 ) + 4 mp lb + lc sin(α1 + α2 ) ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟   ⎜ + mp h cos(α + α )b − h mp l + lc sin(α + α ) + 1 mp cos(α + α )a ⎟ 1 2 1 1 2 1 2 1 b ⎜ ⎟ 2 d 4 d ⎜ ⎟α¨ 1 −⎜ ⎟ 2 2 ⎜ ⎟ ⎜ − h mp cos(α + α )a + 1 mp sin(α + α )b − h mp la sin(α + α )b ⎟ 1 2 1 1 2 1 1 2 1 ⎜ ⎟ 2 2 4 d d ⎜ ⎟ ⎝ ⎠  h h h + mp lb + lc cos(α1 + α2 ) − mp la sin(α1 + α2 ) − mp sin(α1 + α2 )a1 d d d   1 1 1 h h2 h2 − − mp cos α2 + mp sin α2 + mp 2 cos α2 c¨ 1 + m2 + mp + mp 2 cos α2 c¨ 2 4 d 2 4 d d   2 2 h h h 1 1 1 + mp sin α2 + mp cos α2 − mp 2 sin α2 s¨1 − m2 + mp + mp 2 sin α2 s¨2 4 d 2 4 d d ⎞ ⎛ 2 2   h h 1 ⎜ mp lb d 2 cos(α1 + α2 ) − 4 mp lb + lc cos(α1 + α2 ) + mp d 2 lc cos(α1 + α2 )⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 h2 1 ⎟ ⎜ + mp la sin(α1 + α2 ) − mp sin(α + α cos(α + α − m ) )b p 1 2 1 2 1 ⎟ ⎜ 2 4 4 d ⎟ ⎜ ⎟ 2 ⎜ 2 ⎟α˙ h h h 1 +⎜ ⎟ 1 ⎜ + mp cos(α + α + + α − m + α m sin(α sin(α )b )a )a p p 1 2 1 1 2 1 1 2 1 ⎟ ⎜ 2 4 d d d ⎟ ⎜ ⎟ ⎜   h h ⎟ ⎜ ⎟ ⎜ + mp la cos(α1 + α2 ) + mp lb + lc sin(α1 + α2 ) ⎟ ⎜ d d ⎠ ⎝ h h + mp cos(α1 + α2 )a1 + mp sin(α1 + α2 )b1 d d   1  h2  h2 1 1 1  − mp 2 lb + lc + m2 b2 + mp 2 b2 + m2 lb + m2 lc + mp lb + lc α˙ 22 4 4 2 4 d d −

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3 Simulation Results and Analysis This section provides the simulation verification for the proposed dynamic model of DMLS in Matlab. The parameters of DMLS are set as follows: m1 = 6 kg, m2 = 2 kg, m3 = 1.5 kg, la = 0.2 m, lb = 0.1 m, lc = 0.1 m, h = 0.05 m, d = 0.3 m.

The ships rolling, heave and sway motion induced to sea wave disturbances are chosen as ⎧ ⎪ ⎪ s1 = −0.05*sin(t + 0.5) − 0.4 ⎪ ⎪ ⎪ c1 = 0.05*sin(t − 0.5) + 0.15 ⎪ ⎪ ⎨ s2 = 0.05*sin(t + 0.5) + 0.4 (19) ⎪ c2 = 0.05*sin(t + 0.5) + 0.15 ⎪  ⎪ ⎪ ⎪ α1 = π 18sin(t) ⎪ ⎪ ⎩ α2 = −π 18sin(t)

Fig. 3. Ships motion.

Figure 3 shows the ship heave, sway and rolling motion induced by the sea wave disturbances. Figure 4 shows the horizontal mechanisms motion displacements and lifting booms motion displacement. It can be seen from Fig. 3 and Fig. 4 that the horizontal mechanisms and lifting booms produce the periodic motion due to the ships periodic motion.

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Fig. 4. Horizontal mechanisms and lifting booms motion.

4 Conclusions In this article, a DMLS with ship roll, heave and sway motion is established by Lagrangian method. To ensure the accuracy of the dynamic model, approximation and linearization are not used in the modeling process. As far as we know, it is the first DMLS dynamic model, which includes dual horizontal mechanism moving, the dual lifting boom hoisting, payload swing, ship roll, heave and sway motion. In the future, we will design a novel cooperative controller for the DMLS to achieve the control objectives of positioning and anti-swing with complex sea wave disturbances. Acknowledgement. This work is supported by the Key Research and Development Project of Shandong Province under Grant 2021CXGC010701 and the Joint Fund of the National Nature Science Foundation of China and Shandong Province under Grant U1706228.

References 1. Yang, T., Sun, N., Chen, H., Fang, Y.: Motion trajectory-based transportation control for 3-D boom cranes: analysis, design, and experiments. IEEE Trans. Industr. Electron. 66(5), 3636–3646 (2019) 2. Li, G., Ma, X., Li, Z., Li, Y.: Kinematic coupling-based trajectory planning for rotary crane system with double-pendulum effects and output constraints. J. Field Robot. 40(2), 289–305 (2023) 3. Zhang, M., Jing, X.: Adaptive neural network tracking control for double-pendulum tower crane systems with nonideal inputs. IEEE Trans. Syst. Man Cybern. Syst. 52(4), 2514–2530 (2022) 4. Li, G., Ma, X., Li, Z., Li, Y.: Time-polynomial-based optimal trajectory planning for doublependulum tower crane with full-state constraints and obstacle avoidance. IEEE/ASME Trans. Mechatron. 28(2), 919–932 (2023)

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5. Jaafar, H.I., Mohamed, Z., Ahmad, M.A.: Control of an underactuated double-pendulum overhead crane using improved model reference command shaping: design, simulation and experiment. Mech. Syst. Sig. Process. 151, 107444 (2021) 6. Li, G., Ma, X., Li, Z., Li, Y.: Optimal trajectory planning strategy for underactuated overhead crane with pendulum-sloshing dynamics and full-state constraints. Nonlinear Dyn. 109, 815– 835 (2022) 7. Cao, Y., Li, T.: Review of antiswing control of shipboard cranes. IEEE/CAA J. Automatica Sinica 7(2), 346–354 (2020) 8. Lu, B., Fang, Y., Sun, N., Wang, X.: Antiswing control of offshore boom cranes with ship roll disturbances. IEEE Trans. Control Syst. Technol. 26(2), 740–747 (2018) 9. Tuan, L.A., Cuong, H.M., Trieu, P.V., Nho, L.C., Thuanb, V.D., Anh, L.V.: Adaptive neural network sliding mode control of shipboard container cranes considering actuator backlash. Mech. Syst. Sig. Process. 112, 233–250 (2018) 10. Cao, Y., Li, T., Hao, L.: Lyapunov-based model predictive control for shipboard boom cranes under input saturation. IEEE Trans. Autom. Sci. Eng. 20(3), 2011–2021 (2023) 11. Kim, G.H., Hong, K.S.: Adaptive sliding-mode control of an offshore container crane with unknown disturbances. IEEE/ASME Trans. Mechatron. 24(6), 2850–2861 (2019) 12. Wu, Q., Ouyang, H., Xi, H.: Adaptive nonlinear control for 4-DOF ship-mounted rotary cranes. Int. J. Robust Nonlinear Control 33(3), 1957–1972 (2023) 13. Chen, H., Sun, N.: An output feedback approach for regulation of 5-DOF offshore cranes with ship yaw and roll perturbations. IEEE Trans. Industr. Electron. 69(2), 1705–1716 (2022) 14. Sun, N., Wu, Y., Liang, X., Fang, Y.: Nonlinear stable transportation control for doublependulum shipboard cranes with ship-motion-induced disturbances. IEEE Trans. Industr. Electron. 66(12), 9467–9479 (2019) 15. Wu, Y., Sun, N., Chen, H., Fang, Y.: New adaptive dynamic output feedback control of doublependulum ship-mounted cranes with accurate gravitational compensation and constrained inputs. IEEE Trans. Industr. Electron. 69(9), 9196–9205 (2022) 16. Li, Z., Ma, X., Li, Y.: Nonlinear partially saturated control of a double pendulum offshore crane based on fractional-order disturbance observer. Autom. Constr. 137, 104212 (2022) 17. Hu, D., Qian, Y., Fang, Y., Chen, Y.: Modeling and nonlinear energy-based anti-swing control of underactuated dual ship-mounted crane systems. Nonlinear Dyn. 106, 323–338 (2021) 18. Qian, Y., Hu, D., Chen, Y., Fang, Y.: Programming-based optimal learning sliding mode control for cooperative dual ship-mounted cranes against unmatched external disturbances. IEEE Trans. Autom. Sci. Eng. 20(2), 969–980 (2023) 19. Qian, Y., Zhang, H., Hu, D.: Finite-time neural network-based hierarchical sliding mode antiswing control for underactuated dual ship-mounted cranes with unmatched sea wave disturbances suppression. IEEE Trans. Neural Netw. Learn. Syst. (2023). https://doi.org/10. 1109/TNNLS.2023.3257508. (in press) 20. Lee, S., Son, H.: Antisway control of a multirotor with cable-suspended payload. IEEE Trans. Control Syst. Technol. 29(6), 2630–2638 (2021) 21. Roy, S., Baldi, S., Ioannou, P.A.: An adaptive control framework for underactuated switched Euler–Lagrange systems. IEEE Trans. Autom. Control 67(8), 4202–4209 (2022) 22. Shi, H., et al.: A review for control theory and condition monitoring on construction robots. J. Field Robot. 40(4), 934–954 (2023)

Nonlinear Hierarchical Control for Unmanned Quadrotor Transportation Systems with Saturated Inputs Lincong Han and Menghua Zhang(B) School of Electrical Engineering, University of Jinan, Jinan 250022, China [email protected]

Abstract. An unmanned quadrotor presents excellent mobility to fly freely in complex environments, which makes it an ideal choice for aerial transferring tasks. During the transferring process, it is very challenging to eliminate the swing, since there is no direct control on the payload. The quadrotor transportation system presents the great challenge of the cascaded underactuated–underactuated property, which makes it extremely difficult to simultaneously implement accurate quadrotor positioning and efficient payload swing suppression. In this paper, a nonlinear hierarchical control scheme is proposed for a quadrotor transportation system, which takes full advantage of the cascade property of the system and separates the controller design for the inner loop and the outer loop, respectively, to facilitate the design procedure. More specifically, for the outer loop subsystem, based on the proposed energy storage function, a virtual control vector is designed, which introduces a saturation function to make the desired attitude free of any singularities. For the inner loop, a coordinate-free geometric attitude tracking controller is designed on the Lie group to drive the quadrotor to its desired attitude. Based on that, an energy control law is proposed by taking the practical input constraints into account, which achieves both precise quadrotor positioning and efficient payload swing elimination, and so on. Keywords: Underactuated Systems · Quadrotor · Energy Analysis · Saturation

1 Introduction Recently, unmanned aerial vehicles (UAVs) have attracted much attention from mechatronics community. Among numerous applications, cargo transportation by air is a vital one, which has received considerable interests recently. The conventional method is grasping a payload by the grippers equipped on a quadrotor [1], yet it meets the problem of slow response for attitude variation due to the additional inertia of the load acting on the quadrotor [2]. Another more interesting transportation method is suspending a payload beneath the quadrotor by a cable. By contrast, this method retains the good agility of a quadrotor itself, and it is especially advantageous when transporting toxic substances or huge-volume objects [3]. Therefore, the study for the cable-suspended payload-quadrotor system is of both practical engineering significance and theoretical importance. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 362–376, 2024. https://doi.org/10.1007/978-981-97-0554-2_28

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So far, the quadrotors control problem is still a fairly open topic. Almost all previous schemes, except the open-loop methods, require full state feedback [4]. However, equipping additional velocity sensors not only increases the cost but also makes the system cumbersome. Hence, it is a practical requirement to design high-performance control laws without velocity feedback. This paper studies the control problem for unmanned quadrotor transportation systems and proposes a hierarchical control scheme, which considers the full dynamics of the system with eight DOFs instead of the simplified planar case. And the pseudo-velocity signal and input saturation are introduced. Compared with the control of the quadrotor without a suspended payload, the work in this paper is much more challenging because of the increased system DOFs [5] and the newly introduced objective of payload swing suppression, still with the same control inputs. The designed controller successfully drives the quadrotor to the desired position accurately and suppresses the payload swing simultaneously, as shown by both theoretical analysis and experimental results, which thus enhances the safety and transportation efficiency of the system. Specifically, after analyzing the dynamics by the Lagrange method, we know that the cascaded inner–outer loops are coupled by a nonlinear interconnection term [6]. For the outer loop, a novel energy storage function is shaped and then analyzed to construct a virtual control vector in a concise form. Subsequently, the thrust input and the desired attitude for the quadrotor are derived, and regarding the inner loop, a coordinate-free geometric tracking controller on the Lie group [7] is adopted to drive the quadrotor to its desired attitude. The stability of the integral closed-loop system is guaranteed by Lyapunov techniques and LaSalle’s invariance theorem, together with the lemma about cascade systems. Convincing experimental verifications for the control of a full dynamics quadrotor transportation system are obtained by the proposed closed-loop control scheme, which presents the following merits. 1) Different from the existing results, owing to the expression of the attitude dynamics on the configuration manifold of the Lie group and the devised saturated virtual control input, the designed control algorithm is free of any singularities. 2) Due to the careful mathematical manipulation, the coupling term between the inner loop and the outer one satisfies the growth restriction condition, which brings us great flexibility for the controller design in the sense that control laws for the outer loop and the inner one can be constructed separately, and as long as the two subsystems are both asymptotically stable, the asymptotic stability of the overall system can be theoretically ensured. 3) As clearly illustrated by experimental results, the proposed control scheme achieves satisfactory control performance even if the rope length is changed, which is of key importance for practical applications. The rest of this paper is organized as follows. In Section II, we set up the model for the unmanned quadrotor transportation system and then introduce tracking errors for the outer/inner loop subsystems. Subsequently, controller development is detailed in Section III. The corresponding stability analysis is provided in Section IV. Section V presents the experimental results. Finally, Section VI provides summaries and discusses the future work of this paper.

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2 Dynamics Analysis and Error Definition

Fig. 1. Schematic for the unmanned quadrotor transportation system.

Consider the unmanned quadrotor transportation system with full dynamics depicted by Fig. 1. Based on the Lagrange’s modeling technique, we can represent the system with the following dynamic equations: Mc (q)¨q + Vc (q, q˙ )˙q + G(q) = u

(1)

R˙ = Rsk()

(2)

˙ + sk()J  = τ J

(3)

where (1) depicts the system outer loop dynamics, including the quadrotor translation and the payload swing; (2) and (3) depict the system inner loop dynamics on the quadrotor rotation, with sk(·) : R3 → so(3) transforming a vector into a skew-symmetric matrix, such that x × y = sk(x)y for any x, y ∈ R3 . For the outer loop dynamics, q(t) ∈ R5 denotes the outer loop state vector, and Mc (q), Vc (q, q˙ ) ∈ R5×5 , G(q), and u ∈ R5 represent the inertia matrix, the centripetal-Coriolis matrix, the gravity vector, and the resultant force applied to the outer loop subsystem, respectively. The detailed expressions for these signals are provided as follows: q = [ξ T , θx , θy ]

(4)

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⎡

mc11 0 0 mc14 ⎢ 0 m 0 ⎢ c22 0 ⎢ Mc = ⎢ 0 0 mc33 mc34 ⎢ ⎣ mc41 0 mc43 mc44 mc51 mc52 mc53 0

⎤ mc15 mc25 ⎥ ⎥ ⎥ mc35 ⎥ ⎥ 0 ⎦ mc55

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mc11 = mc22 = mc33 = M + m, mc44 = ml 2 Cy2 , mc55 = ml 2 , mc41 = mc14 = mlCx Cy , mc43 = mc34 = mlSx Cy , mc51 = mc15 = −mlSx Sy , mc52 = mc25 = mlCy , mc53 = mc35 = mlCx Sy ⎤ ⎡ 0 0 0 Vc14 Vc15 ⎢0 0 0 0 V ⎥ ⎢ c25 ⎥ ⎥ ⎢ (6) Vc = ⎢ 0 0 0 Vc34 Vc35 ⎥ ⎥ ⎢ ⎣ 0 0 0 Vc44 Vc45 ⎦ 0 0 0 Vc54 0 Vc14 = −ml θ˙x Sx Cy − ml θ˙y Cx Sy , Vc44 = −ml 2 θ˙y Cy Sy , Vc34 = ml θ˙x Cx Cy − ml θ˙y Sx Sy , Vc54 = ml 2 θ˙x Cy Sy , Vc15 = −ml θ˙x Cx Sy − ml θ˙y Sx Cy , Vc25 = −ml θ˙y Sy , V35 = −ml θ˙x Sx Sy + ml θ˙y Cx Cy , Vc45 = −ml 2 θ˙x Cy Sy G = [0, 0, 0, mglSx Cy , mglCx Sy ]T

(7) (8)

where ξ (t) = [x(t), y(t), z(t)]T ∈ R3 denotes the position of the quadrotor’s mass center in the inertial reference frame; θx (t) ∈ R and θy (t) ∈ R are the projected swing signals; R(t) ∈ SO(3) is the rotation matrix of the quadrotor from the body-fixed frame to the inertial frame; (t) ∈ R3 represents the angular velocity of the quadrotor in the bodyfixed frame; M , m, l, g ∈ R, and J ∈ R3×3 stand for the quadrotor mass, the payload mass, the rope length, the gravitational acceleration, and the moment of inertia for the quadrotor with respect to the body-fixed frame, respectively; f (t) ∈ R is the applied thrust and τ (t) ∈ R3 denotes the moment vector of the quadrotor in the body-fixed frame, and they are the control inputs for the overall system; and Cx , Cy , Sx , and Sy are abbreviations of cos(θx ), cos(θy ), sin(θx ), and sin(θy ), respectively. As widely done in the literature on suspended systems [8, 9] we make the following reasonable assumption. Assumption 1: The payload swing angles satisfy θx (t), θy (t) ∈ (−π/2, π/2), implying that the payload will never go above the quadrotor due to the practical physical constraints. Property 1: Based on the structure of (5), it can be shown that Mc is a positive-definite symmetric matrix.

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Property 2: According to the equations given by (5) and (6), it is straightforward to see that the centripetal-Coriolis matrix Vc (q, q˙ ) and the time derivative of Mc (q) satisfy the following skew-symmetric relationship: 1 ˙ 3 xT ( M c − Vc )x = 0 ∀x ∈ R 2

(9)

Let ξd = [xd , yd , zd ]T ∈ R3 be the desired position; then, the control objective is to drive the quadrotor from its initial position to the desired one, while eliminating the payload swing in the sense that q → qd = [ξ Td , 0, 0]T

(10)

To facilitate subsequent controller development and analysis, define the outer loop tracking errors eq (t), eq˙ (t) ∈ R5 as eq (t) = q(t) − qd = [eξ (t)T , θx (t), θy (t)]T

(11)

eq˙ (t) = e˙ q (t) = [˙eξ (t)T , θ˙x (t), θ˙y (t)]T

(12)

where eξ (t) = ξ (t) − ξ d = [ex (t), ey (t), ez (t)]T ∈ R3 . Then, the outer loop dynamics (1) can be reorganized as e˙ q = eq˙

(13)

e˙ q˙ = Mc−1 (u − Vc eq˙ − G)

(14)

For the inner loop subsystem, define the attitude tracking error eR ∈ R3 and the angular velocity tracking error e ∈ R3 as [7] eR =

1 vex(RTd R − RT Rd ) 2

e =  − RT Rd d

(15) (16)

where vex(·) : so(3) → R3 is the inverse operation of sk(·); Rd ∈ R3×3 , d ∈ R3 are the desired attitude and desired angular velocity of the quadrotor, respectively, which will be determined subsequently. The time derivatives for eR and e are calculated as e˙ R =

1 vex(RTd Rsk(e ) + sk()) 2

˙ d) J e˙  = τ − sk()J  + J (sk()RT Rd d − RT Rd 

(17) (18)

Subsequently, for the quadrotor transportation system with the cascade structure of (13), (14) and (17), (18), a nonlinear control scheme will be designed to achieve the control objective. It is worthwhile to point out that the inner loop subsystem and the outer loop one are coupled by a nonlinear interconnection term, which is caused by the rotation matrix R appearing in the resultant force as shown in (8).

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Remark 1: As can be seen from (1)–(3), the full dynamics of the quadrotor transportation system presents eight DOFs. Remark 2: With four control inputs composed of f ∈ R and τ ∈ R3 , the quadrotor itself is underactuated and provides the overall transportation system’s driving force, which presents a “double” underactuated property. This is completely different from other underactuated systems whose driving part is fully actuated, because the outer loop control input u(t) is not only related to the thrust force f (t), but also directly depends on the quadrotor’s attitude R(t) determined by the moment τ (t).

3 Controller Development This section provides details for the design of the nonlinear hierarchical flight controller, including the controller construction for both the outer loop and the inner one. 3.1 Outer Loop Control The total energy of the outer loop subsystem consists of the kinetic part and the potential part as 1 T q˙ M c q˙ + mgl(1 − Cx Cy ) (19) 2 which is locally positive definite q˙ (t), θx (t) and θy (t). The derivative of (19) along the system dynamics can be derived as follows: E=

1 ˙ ˙ ) + mgl(θ˙x Sx Cy + θ˙y Cx Sy ) E˙ = q˙ T (Mc q¨ + M cq 2 = e˙ Tξ [fRe3 − (M + m)ge3 ]

(20)

where Property 2 has been utilized. Furthermore, to construct the “shaped” energy storage function, define the following auxiliary function: Es = [kpx , kpy , kpz ]A where kpx , kpy , and kpz are positive control gains, and vector A ∈

(21) R3

takes the form of

A = [ln cosh(ex ), ln cosh(ey ), ln cosh(ez )]T

(22)

According to (19) and (21), the total energy storage function is constructed as V = E + Es

(23)

Taking the time derivative of (23) and utilizing (20), we are led to the following results: V˙ = E˙ + [kpx , kpy , kpz ]A˙ = e˙ Tξ [fRe3 − (M + m)ge3 + kp B]

(24)

wherein kp = diag([kpx , kpy , kpz ]) ∈ R3×3 is a positive definite diagonal matrix, and B ∈ R3 is defined as follows: B = [tanh(ex ), tanh(ey ), tanh(ez )]T

(25)

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To deal with the coupling term, fRe3 can be divided into the following two parts: fRe3 =

f eT3 RTd Re3 +

[(eT3 RTd Re3 )Re3 − Rd e3 ]

f eT3 RTd Re3

Rd e3

(26)

where the signal  ∈ R3 and the to-be-constructed virtual control vector fd = [fd 1 , fd 2 , fd 3 ]T ∈ R3 are defined as =

f [(eT3 RTd Re3 )Re3 − Rd e3 ] eT3 RTd Re3 fd =

f eT3 RTd Re3

Rd e3

(27) (28)

respectively. Furthermore, (28) implies that vectors fd and Rd e3 have the same direction; then, the desired unit direction vector b3d = Rd e3 ∈ R3 can be obtained as b3d = Rd e3 =

fd fd 

(29)

Substituting the results of (29) into (28) yields   f f d  f d f fd  = fd = T T   f f d Re3 d f d Re3 indicating that f = f Td Re3

(30)

It is seen that as far as the virtual controller f d is designed, the real control f can be directly obtained by (30). Therefore, the kernel step is the design of f d . To this end, noting from (24) and (26) that V˙ = e˙ Tξ [f d +  − (M + m)ge3 + Kp B]

(31)

construct f d in the following saturated way: f d = −Kp B + (M + m)ge3 − Kd C

(32)

where Kd = diag([kdx , kdy , kdz ]) ∈ R3×3 denotes a positive definite diagonal matrix, and C ∈ R3 is defined as follows: C = [tanh(εx ), tanh(εy ), tanh(εz )]T where εx (t), εy (t), εz (t) are dynamically generated as follows: εx (t) = μx + kdx ex , μ˙ x = −kdx (μx + kdx ex )

(33)

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εy (t) = μy + kdy ey , μ˙ y = −kdy (μy + kdy ey ) εz (t) = μz + kdz ez , μ˙ z = −kdz (μz + kdz ez )

369

(34)

It is easily indicated from (35) that ε˙ x = −kdx εx + kdx e˙ x ε˙ y = −kdy εy + kdy e˙ y ε˙ z = −kdz εz + kdz e˙ z

(35)

Then, substituting (32) into (31) produces V˙ = e˙ Tξ [−K d C + ]

(36)

Hereto, the thrust force f (t) and the desired direction vector b3d can be calculated from (29), (30), and (32) as f = f Td Re3 = ( − Kp B + (M + m)ge3 − Kd C)T Re3 −Kp B + (M + m)ge3 − Kd C f  b3d =  d  =  f d  −Kp B + (M + m)ge3 − Kd C 

(37) (38)

As per practical requirements, it is demanded to design a controller to stabilize the system state using only output feedback under saturated control inputs as |fd 1 (t)| ≤ fd 1 max

(39)

|fd 2 (t)| ≤ fd 2 max

(40)

|fd 3 (t)| ≤ fd 3 max

(41)

where fd 1 max , fd 2 max , fd 3 max denote the largest permitted control inputs. And considering the characteristic tanh(·) ≤ 1, we choose the control gains in (32) satisfying kpx + kdx < fd 1 max

(42)

kpy + kdy < fd 2 max

(43)

kpz + kdz < fd 3 max

(44)   to make the last term of f d nonzero, implying that f d  = 0; therefore, the desired direction vector b3d is free of any singularity problems on the basis of (38). To identify the desired attitude Rd , choose an arbitrary vector b1c ∈ R3 not parallel to b3d ; then, we can obtain the desired attitude from Rd = [b2d × b3d ; b2d ; b3d ]

(45)

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where b2d ∈ R3 represents the direction of the body-fixed axes calculated as b2d =

b3d × b1c b3d × b1c 

(46)

Subsequently, we aim to rewrite the outer loop subsystem dynamics to separate out the coupling term. According to (8), (27), and (28), the resultant force u for the outer loop subsystem can be divided into the following three parts: u = ud − m0 + u

(47)

where ud ∈ R5 is derived from vector f d , m0 ∈ R5 denotes the total mass of the quadrotor and the payload, and u ∈ R5 represents the interconnection between the inner loop and the outer one, with their explicit expressions as ud = [fdT , 0, 0]T

(48)

m0 = [0, 0, (M + m)g, 0, 0]T

(49)

u = [T , 0, 0]T

(50)

Substituting (47) into (14), one has e˙ q = Mc−1 (ud − m0 − Vc eq˙ − G) + Mc−1 u

(51)

By defining e(t) = [eTq (t), eTq˙ (t)]T ∈ R10 , and then utilizing (13) and (51), it is derived that e˙ = α(e, f d , ξ d ) + β

(52)

where α(·) ∈ R10 can be expressed as eq˙ + Mc−1 (ud − m0 − Vc eq˙ − G) α=

(53)

and β ∈ R10 is the coupling term defined as follows: Mc−1 u β=

(54)

10

where matrices



10 ,

01

01

01

∈ R10×5 are constants explicitly expressed as   I5×5 05×5 , = = 10 01 05×5 I5×5

(55)

where I5×5 stands for a 5 × 5 identity matrix, and 05×5 stands for a 5 × 5 zero matrix. When temporarily neglect the coupling term β, the outer loop subsystem (52) can be transformed into e˙ = α(e, f d , ξ d ) for which the following conclusion can be made.

(56)

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Theorem 1: The designed saturated virtual controller f d given by (32) guarantees that the system state of (56) converges to the equilibrium point in the following sense: T T lim [eTξ , e˙ Tξ , θx , θy , θ˙x , θ˙y ]T = [03×1 , 03×1 , 0, 0, 0, 0]T

t→∞

(57)

Proof: Taking (23) as the Lyapunov function candidate, according to (36) and the fact that  = 03×1 for system (56), one can conclude that V˙ = −εx kdx tanh(εx ) − εy kdy tanh(εy ) − εz kdz tanh(εz ) ≤ 0

(58)

which indicates that the closed-loop system (56) is Lyapunov stable at the origin. It is implied from (19), (21), and (23) that ex , ey , ez , e˙ x , e˙ y , e˙ z , εx , εy , εz θx , θy , θ˙x , θ˙y ∈ L∞

(59)

To accomplish the proof, we define φ as

φ = {(ex , ey , ez , e˙ x , e˙ y , e˙ z , θx , θy , θ˙x , θ˙y ) V˙ (t) = 0 }

(60)

and let be the largest invariant set in φ. In , based on (58), it is straightforward to make the following conclusions: e˙ x = e˙ y = e˙ z = 0 ⇒ e¨ x = e¨ y = e¨ z = 0, ex = βx , ey = βy , ez = βz

(61)

with βx , βy , and βz being undetermined constants. Based on a similar argument proposed in [10] we can show that βx = βy = βz = 0, and further, we know that the largest invariant set includes only the equilibrium point. By invoking LaSalle’s invariance theorem, it can be concluded that for (56), the system states converge to the desired ones asymptotically, which completes the proof.

3.2 Inner Loop Control In order to avoid possible singularity problems appeared in quadrotor attitude control, for inner loop subsystem (17) and (18), we adopt the following coordinate-free control approach [7] τ = − kR eR − k e + sk()J  ˙ d) − J (sk()RT Rd d − RT Rd 

(62)

where kR , k ∈ R are positive constants, and the desired angular velocity of the quadrotor d can be calculated from d = vex(RTd R˙ d )

(63)

As shown in [7] the moment input (62) guarantees that the zero equilibrium of eR and e is exponentially stable.

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4 Stability and Convergence Analysis Due to the existence of the coupling term β in (52), Theorem 1 and the results in Section Inner Loop Control usually cannot sufficiently guarantee the stability of the closed-loop system. Inspired by the theory on cascade systems exploited in [11, 12], we tend to show that the coupling term satisfies a growth restriction condition, which can then be utilized to show the stability of the overall system. Next, the growth restriction condition will be verified, and before proving the boundness of the coupling term β, we first provide the following lemma and its proof. Lemma 1: There exist positive constants cf and kf such that the virtual control vector f d presents the following property:    f d (e) ≤ kf e, for e ≥ cf (64) kf cf , for e ≥ cf Proof: Combining (32) with (25), (33), and exploiting the trigonometric relation that |tanh(x)| < |x|, ∀x ∈ R, the following result is derived:     f d  = −Kp B + (M + m)ge3 − Kd C    ≤ Kp B + (M + m)g + Kd C     ≤ (M + m)g + max(λp , λd )(eq  + eq˙ ) √ ≤ (M + m)g + 2 max(λp , λd )e √ (M + m)g + e] ≤ 2 max(λp , λd )[ √ 2 max(λp , λd ) where λp and λd denote the maximum eigenvalue of Kp and Kd , respectively. By setting √ kf = 2 2 max(λp , λd ), (M + m)g cf = √ 2 max(λp , λd ) the results in (64) can be obtained. Lemma 2: There exist a positive constant c and a class - κ function γ (·) differentiable at [eTR , eT ]T = 06×1 , such that the coupling term β satisfies the following growth restriction condition: β ≤ γ (eR )e, fore ≥ c

(65)

Proof: Substituting (29) and (30) into (27), one derives       = f d eT3 RTd Re3 − Rd e3  (66)  T T  After some geometry transformations, we can see that e3 Rd Re3 − Rd e3  represents the sine of the angle between Re3 and Rd e3 , and eR  represents the sine of the eigenaxis rotation angle between Rd and R [7] thus    ≤ f d eR 

Nonlinear Hierarchical Control

≤ kf eeR , fore ≥ cf

373

(67)

where (64) has been utilized. In accordance with (50), (54), and (67), we can conclude that     β ≤  Mc−1  u  01   F  −1  = M   01 c F    −1  ≤ kf  Mc  eR e, fore ≥ cf (68) 01

F

·F where stands the Frobenius 

 for norm. By choosing γ (eR ) = kf  01 Mc−1 F eR , c = cf , results for this lemma are proven. Theorem 2: The proposed thrust control law (37) and the moment control law in (62) guarantee that the quadrotor is driven to the desired position, while the payload swing is damped out in the sense that T lim [ξ T , ξ˙ , θx , θy , θ˙x , θ˙y , eTR , eT ]T   T T T = ξ Td , 03×1 , 0, 0, 0, 0, 03×1 , 03×1

t→∞

(69)

Proof: From Lemma 2, Theorem 1, and the results in Section Inner Loop Control, by invoking the theorem on the stability of cascade systems (which is validated through Lemma 2), the asymptotic stability of the equilibrium point is guaranteed. Remark 5: The attitude representation way on the Lie group not only avoids singularity problems, but also facilitates the controller design process and stability analysis. Specifically, on the nonlinear configuration Lie group SO(3), we extract the term (t) in (27), which is the basis of the coupling term β(t). By proving the growth restriction condition in Lemma 2, the stability of the overall system can be guaranteed from the stability of the two subsystems presented in Sect. 3.

5 Simulation Results In this section, a simulation is conducted to verify the performance of the proposed approach. The system parameters are determined as M = 0.625 kg, m = 0.075 kg, g = 9.8 kg · m/s2 , l = 0.535 m μx = 0, μy = 0,

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μz = 0, fd 1 max = 8N , fd 2 max = 8N , fd 3 max = 15N , J = diag([0.005, 0.005, 0.013])kg · m2 . The control gains are set as K p = diag([3.4, 3.4, 10]), K d = diag([1.83, 1.83, 3.5]) K R = diag([6.6, 6.6, 6.3]), K  = diag([0.6, 0.6, 1]) where K R ∈ R3×3 and K  ∈ R3×3 are extended forms of the previously defined control gains kR ∈ R and k ∈ . Vector b1c is chosen as [1, 0, 0]T [1, 0, 0]T in this paper. We compare the performance of the proposed controller with PD controller without any external disturbances. The quadrotor initial position and the desired one are set as ξ(0) = [−0.5, 1.3, 1.0]T and ξd = [0.5, 0.3, 1.5]T . The obtained results are illustrated in Figs. 2 and 3, wherein the record of f d for the proposed method is also provided. The quantitative results, including the quadrotor final position error eξ , the maximum payload swing angles max , the maximum thrust force fmax . By comparing the experimental results of eξ and max , one can see that the proposed method achieves higher positioning accuracy and smaller payload swing with almost the higher transportation efficiency.

Fig. 2. Results for PD controller method in simulation. (a) Quadrotor position and payload swing angles. (b) Quadrotor attitude. (c) Control inputs.

In summary, the results of simulations clearly show the proposed method’s superior performance in terms of quadrotor positioning accuracy and payload swing suppression.

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Fig. 3. Results for proposed method in simulation. (a) Quadrotor position and payload swing angles. (b) Quadrotor attitude. (c) Control inputs.

6 Conclusion A hierarchical control scheme was proposed for the unmanned quadrotor transportation system, which can effectively suppress and eliminate the payload swing and guarantee accurate quadrotor positioning. The kernel contribution of this paper was the construction of cascade system formulation and its stability analysis based on the theory of systems in cascade. This allowed us to design controllers for the inner loop and the outer one separately and thus brought much convenience for controller design and analysis. In particular, Lyapunov based analysis was employed to theoretically ensure the performance of the hierarchical control scheme. Results of simulation were included to show the proposed method’s superior control performance. Acknowledgements. This work was supported in part by the National Natural Science Foundation of China under Grant No. 62273163, the Outstanding Youth Foundation of Shandong Province Under Grant No. ZR2023YQ056, and the Key R&D Project of Shandong Province under Grant No. 2022CXGC010503.

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6. Cao, N., Lynch, A.F.: Inner–outer loop control for quadrotor UAVs with input and state constraints. IEEE Trans. Control Syst. Technol. 24(5), 1797–1804 (2015) 7. Lee, T., Leok, M., McClamroch, N.H.: Geometric tracking control of a quadrotor UAV on SE (3). In: 49th IEEE Conference on Decision and Control (CDC), pp. 5420–5425. IEEE (2010) 8. Sun, N., Wu, Y., Fang, Y., et al.: Nonlinear antiswing control for crane systems with doublependulum swing effects and uncertain parameters: Design and experiments. IEEE Trans. Autom. Sci. Eng. 15(3), 1413–1422 (2017) 9. Ngo, Q.H., Hong, K.S.: Adaptive sliding mode control of container cranes. IET Control Theory Appl. 6(5), 662–668 (2012) 10. Gioioso, G., Franchi, A., Salvietti, G., et al.: The flying hand: a formation of UAVs for cooperative aerial tele-manipulation. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 4335–4341. IEEE (2014) 11. Zhao, B., Xian, B., Zhang, Y., et al.: Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology. IEEE Trans. Industr. Electron. 62(5), 2891– 2902 (2014) 12. Chen, F., Jiang, R., Zhang, K., et al.: Robust backstepping sliding-mode control and observerbased fault estimation for a quadrotor UAV. IEEE Trans. Industr. Electron. 63(8), 5044–5056 (2016)

Predicting the Endless Stop-Band Behaviour of the NS-MRE Isolator Qun Wang1 , Zexin Chen2 , Jian Yang3 , and Shuaishuai Sun1,4(B) 1 CAS Key Laboratory of Mechanical Behaviour and Design of Materials, School of Engineering Science, University of Science and Technology of China, Hefei 230026, Anhui, China [email protected] 2 School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong, NSW 2522, Australia 3 School of Electrical Engineering and Automation, Anhui University, Hefei 230039, Anhui, China 4 Institute of Deep Space Sciences, Deep Space Exploration Laboratory, Hefei 230026, China

Abstract. The periodic structure of the MRE isolator could generate special frequency range in which the waves cannot be propagated. This paper proposed a Negative Stiffness Magnetorheological Elastomer (NS-MRE) isolator with acoustic metamaterial characteristics. This particular structure generates an endless stop-band to restrict the transmission of the vibration. Furthermore, the permanent magnet arrays in the periodic structures of the NS-MRE isolator add dynamic negative stiffness to reduce the overall stiffness in the horizontal vibration direction without sacrificing vertical load-bearing stiffness. That can solve contradictory requirements for large load-bearing capacity and excellent vibration isolation performance at low frequencies as the former requires for large stiffness and the later requires for small stiffness. Using the mass-spring model, the generation mechanism of the stop-band was theoretically analysed. This paper deeply analyses the difference between the NS-MRS isolator and MRE isolator, in particular the stop-band of the single-mass periodic structure from the perspectives of finite period and infinite period, and its potential application for attenuating vibration amplitudes. Keywords: MRE · negative stiffness · stop-band · isolator

1 Instruction Vibration isolation is essential in various engineering fields including precision instruments, vehicle engineering, industrial equipment and civil buildings [1, 2]. The unwanted vibration may destruct the accuracy of instruments [3, 4], break the fixation of electronic devices [5], harm the safety of people [6], produce harsh noise [7] and result in economic losses or catastrophic accidents. Traditional passive isolation methods [8, 9], which use Q. Wang and Z. Chen—These authors contributed equally to this work. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 377–389, 2024. https://doi.org/10.1007/978-981-97-0554-2_29

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springs and dampers, are simple and convenient to apply but have negative effect on vibration isolation at the resonant frequency, which has restricted their applications in different working environments. By contrast, active isolation can adapt actively to different vibration environments by using motors to introduce active control in vibration isolation [10]. However, its complex structure and extra energy requirement limit its wide applications. To provide a control strategy which is better than the passive control at reducing vibrations while cheaper and simpler than active control, semi-active control systems were put forward. Generally, they can perform changeable stiffness or damping, which means they acquires advantages of low power consumption and high controllability [11]. The semi-active system can be realized by utilizing intelligent materials and structures [12, 13]. As one member of the smart material family, MRE is widely used in the implementation of semi-active systems due to that MRE can behave controllable modulus and damping upon the application of external magnetic field. In addition, MRE can avoid some problems of its counterpart, i.e. Magnetorheological fluid [14] (MRF), such as iron particle deposition, sealing difficulty and environmental pollution. Therefore, MRE has been extensively used for the implementation of isolators. Leng et al. designed a variable stiffness isolator by embedding an MRE-based device into a X-shaped structure [15]. Li Y and Li J proposed an MRE isolator for seismic base isolation [16, 17], which consists of multi-layered rubber material with metal sheets embedded. As to the acoustic metamaterials, the periodic structures could generate specific frequency range in which the transmission of vibration is restrained [18, 19]. The reason is that the dispersion curves of such structures do not exist in some frequency range. There have been considerable researches on bandgaps in the middle of two dispersion curve [18–20]. The main mechanisms of bandgaps generation are Bragg scattering and local resonance [21, 22]. Yao et al. proposed the multi-resonator structure, which effectively spliced bandgaps of each resonator to enlarge frequency range of the flexural wave mitigation [19]. Wang et.al designed a digital metamaterial to produce tuneable bandgap for broadband vibration isolation at low frequency [23]. With all these advantages of MRE and acoustic metamaterial, some MRE isolators with acoustic metamaterial structures attracted much attention of researchers. Chen et.al innovatively designed and prototyped a metamaterial MRE isolator with controllable local resonance bandgaps for vibration isolation [24]. Wang et al. proposed a novel metamaterial MRE sandwich beam to generate real-time tuneable local resonance bandgaps [25]. However, the bandgaps resulting from these two mechanisms have a limited width. The frequency range within these bandgaps is determined by the lower and upper cutoff frequencies. Interestingly, it is barely noticed that the frequency range above all dispersion curves bears similarities to the bandgaps [26, 27], in particular the single mass period structure. In other words, it should be highlighted that there is an endless stop-band without dispersion curves for a single mass period structure, which can stop the transmission of the vibration waves after cut-off frequency. But stop-band also have some drawbacks similar to that of bandgaps, such as high cut-off frequency because of high stiffness [28] and limited application scenarios [29]. The direct approach to reduce the cut-off frequency is to decrease the stiffness of the materials. The lower stiffness not only advances the occurrence of the stop-band but

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also improves the performance of low-frequency vibration isolation. It should be noted, though, that isolators require high stiffness materials to ensure sufficient load-bearing capability. Therefore, there is a direct conflict between the load-bearing capability and the low-frequency vibration isolation [30]. Luckily, the negative stiffness concept provides the potential to meet these requirements. Negative-stiffness components, such as oblique pre-stressed springs [31, 32], pre-stressed rods [33, 34], buckling beams [35, 36], and magnetic springs [37–41], produce no force at rest but provide force in the same direction of movement as the vibration occurs. The negative stiffness produced in vibration can reduce the overall stiffness in the vibration direction but the original stiffness in the loading direction which is perpendicular to the vibration direction remains, which balances the stiffness requirement contradiction. As a result, the array of magnets is particularly suitable to add negative stiffness to compact acoustic metamaterial structures. Based on these, a novel NS-MRE isolator with negative stiffness is proposed. The total structure is based on the period structure and embeds an array of magnets in each copper ring of the period structure. There are repulsive forces generated between the adjacent layers in horizontal vibration, which is equivalent to the elastic force from the magnetic springs. The ratio between the repulsive force and the displacement is called negative stiffness. The sum of negative stiffness and positive stiffness is lower than positive stiffness, so the existence of magnetic springs reduces the overall stiffness of the NS-MRE isolator in the horizontal direction, while that is unable to affect the stiffness in the vertical direction. The paper is structured as follows. Section 2 shows the acoustic metamaterial characteristic of the NS-MRE isolator and the simulation of magnetic field distribution by COMSOL. In Sect. 3, based on the mass-spring model theory, analysis of the cut-off frequency of the infinite period are illustrated. In Sect. 4, the transmissibility spectrum of the finite period is presented and the particularity of the periodic structure including springs and mass is explained. Finally, the conclusion of the NS-MRE isolator is shown in Sect. 5.

2 Structure and Analyses of the NS-MRE Isolator 2.1 Structure and Working Principle of the NS-MRE Isolator The structure of the NS-MRE isolator is shown in Fig. 1(a). The external structure of the isolator consists of the top steel plate, the base steel plate, the periodic steel rings and the periodic MRE slices. For each steel ring, a copper ring installed with cylindrical magnets are fixed on the inside diameter of the steel ring. The iron core locates at the centre of the isolator and the copper coil is wounded around the iron core. The end cap of the iron core is used to limit the movement of the copper coil and increase the contact area between the iron core and the base steel plate. There is a 2 mm gap between the iron core and the top steel plate to prevent the transmission of the vibration. Figure 1(b) shows a unit cell of the periodic structure, which mainly includes MRE slices, a steel ring, a copper ring, and a magnet array. 18 cylindrical magnets are spaced placed in the copper ring. The MRE is a mixture of carbonyl iron particles (with the diameter of 38 μm), silicone rubber (Selleys Pty Ltd, Australia) and silicone oil (XIAMETER

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PMX-200, 50CS, United States) with a mass ratio of 7:2:1. Four MRE slices with the radian of π6 are evenly distributed on the steel ring. The relative geometrical parameters of the unit cell are as follows: d = 6 mm, D1 = 84 mm, D2 = 104 mm, D3 = 124 mm, h1 = 2 mm, h2 = 7 mm The positive stiffness of the MRE slice is 78835 N/m, and the negative stiffness between two layers of magnet arrays is - 8000 N/m. The mass of a single periodic cell is 0.35 kg.

Fig. 1. (a) The cutaway view of the NS-MRE isolator. (b) A unit cell of the periodic structure. (c) The fabricated NS-MRE isolator without top steel plate.

The description of the above structure reveals two key features of the NS-MRE isolator: its acoustic metamaterial structure that produces a stop-band, and the negative stiffness that is provided by the permanent magnet arrays. According to the understanding of the dispersion curve, the structure generates an infinite stop-band when it is periodic and its masses only have one kind. And the stop-band range theoretically spans from the lower cut-off frequency to the unlimited upper cut-off frequency. For convenience, the lower cut-off frequency will be directly expressed by the cut-off frequency. The vibration amplitude of the top steel plate decreases rapidly near the cut-off frequency. The cutoff frequency depends on the mass and the stiffness. When it comes to the NS-MRE isolator, the cut-off frequency is determined by the mass of the steel and copper rings, and the sum of the stiffness of the MRE and the negative stiffness provided by magnets. The extra-magnetic environment is regulated by the currents, while the magnetic field influence the stiffness of the MRE slices. As a result, various cut-off frequencies can be achieved with different currents. Moreover, the negative stiffness reduces the total stiffness in the horizontal direction, leading to lower cut-off frequency. 2.2 The Simulation of Magnetic Field Distribution by COMSOL The section view of the magnetic field of the MRE isolator simulated in COMSOL Multiphysics is presented in Fig. 2(a). As indicated by the red arrow, a closed magnetic circuit is established from the iron core through the top steel plate, periodic steel rings and periodic MRE slices, the base steel plate, and back to the iron core. The magnetic

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flux density is positively correlated with the coil current. As shown in Fig. 2(b), the magnetic flux density of the MRE layer indicated by the red circle in Fig. 2(a) increases with the increasing coil current. Moreover, the magnet arrays are verified to have hardly effects on the magnetic field of the MRE slices. Considering that excessive current may cause the MRE overheat which may influence its stiffness, the coil current of the isolator is chosen as 0.3 A, 0.6 A and 0.9 A. The maximum magnetic flux density of the MRE slice is 115 mT when the current is 0.9 A.

(a)

(b)

Fig. 2. (a) Magnetic field distribution of the NS-MRE isolator (0.6 A). (b) The magnetic flux density of MRE in the red circle.

3 The Effect of the Negative Stiffness on the Stop-Band with Infinite Period Structure As shown in Fig. 3 (a), the structure of the isolator can be theoretically modelled by an infinite periodic structure with one mass and one spring in a single period. The green balls in the figure represent the concentrated masses which include the mass of the periodic steel rings and the copper rings. The black spring represents the stiffness of the MRE layer. By using the mass-spring model, the isolator with infinite periods can be represented and the corresponding oscillation equation can be easily obtained as follows: m¨xn = k(xn−1 − xn ) + k(xn+1 − xn )

(1)

where m represents the mass of a metal layer, k represents the stiffness of one MRE layer. xn is the displacement of the mass. When the system is excited with a harmonic excitation, the acceleration of each vibrator can be written as follows: x¨ n = −ω2 xn

(2)

According to the Bloch theorem, the displacement can be rewritten as: xn−1 = xn e−iKa = An ei(−Ka−2πf t) , xn+1 = xn eiKa = An ei(Ka−2πf t)

(3)

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where An is the amplitude, K is the wave vector, a is the cell constant (9 mm), f is the vibration frequency. Finally, the Eq. (4) can be obtained by substituting Eq. (2) and Eq. (3) into Eq. (1). It illustrates that each wave vector K has the unique vibration solution called lattice wave and the corresponding vibration frequency is f.  1 2k(1 − cos(Ka)) (4) f (K) = 2π m The dispersion relation of the lattice wave is shown in Fig. 3(b). At the boundary of the Brillouin zone, the group velocity of the lattice wave is zero, which is equal to that of the standing wave produced by Bragg reflections. Among all the solutions, the vibration frequency has a maximum value f1 . The vibration transmission through the periodic structure is allowed when the vibration frequency is lower than the maximum value f1 . In other words, there is an endless stop-band above the maximum frequency f1 and this maximum frequency f1 is called cut-off frequency.

(a)

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Fig. 3. (a) The model of infinite periodic springs oscillator. (b) One-dimensional dispersion diagram

With practical parameters, the cut-off frequency f1 can be computed as:   1 78835 1 k = = 151.1Hz (5) f1 = π m π 0.35 When permanent magnet arrays are added to the periodic structure, the cut-off frequency is affected by the magnetic force. The linear magnetic force can be expressed with a linear term formula as follow: Fm (u) = qu

(6)

where u is the relative displacement of two steel layer, q (N/m) is the stiffness coefficient of the magnetic spring. As shown in Fig. 4(a), there are two different types of springs: the black springs represent the MRE layer and the red one represents the magnetic springs. The vibration equation with magnetic springs is written as follow: m¨xn = (k + q)(xn−1 − xn ) + (k + q)(xn+1 − xn )

(7)

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Similar to the calculation in the case without permanent magnets, the dispersion relation of the period structures with magnets is as follow:  1 2(k + q)[1 − cos(Ka)] (8) f (K) = 2π m The dispersion diagram with magnetic force is shown in Fig. 4(b). The cut-off frequency f2 with magnetic springs can be calculated as:  (k + q) (9) f2 = 2 m The negative stiffness coefficient q is 800 N/m. And the other parameters are the same value with Eq. (5):   1 (78835 − 8000) 1 (k + q) f2 = = = 143.2Hz (10) π m π 0.35

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Fig. 4. (a) The model of the infinite periodic springs oscillator with magnetic springs. (b) Onedimensional dispersion diagram with magnetic force

4 Transmission Spectrum Analyses with Finite Period Structures In this part, the NS-MRE isolator, which has 8 periodic units, is analysed with the massspring model. As shown in Fig. 5, k represents the stiffness of MRE (black springs), q is the stiffness of the magnetic spring (red springs) between the two layers of mass, M represents the mass of the top layer, m represents the mass of each metal layer. The displacement of metal layer is expressed as xn . The bottom and the top displacements of the NS-MRE isolator is x0 and x9 , respectively. For the periodic unit cells, both of the terminal unit cells only have the magnetic spring at one side and the inner unit cells have the magnetic spring at the both sides. The dashed box represents a single periodic cell.

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Fig. 5. The model of finite periodic spring oscillator with magnetic force

For the unit cells in the inner part of the periodic structure, the vibration equation is: m¨xn = (k + q)(xn−1 − xn ) + (k + q)(xn+1 − xn )n = 2, ..., 7

(11)

Kinetic equations of the terminal unit cells are: m¨x1 = k(x0 − x1 ) + (k + q)(x2 − x1 )

(12)

m¨x8 = k(x9 − x8 ) + (k + p)(x7 − x8 )

(13)

Besides, kinetic equation of the top steel plate is: M¨x9 = k(x8 − x9 )

(14)

xn is the imaginary number to contain phase and amplitude information, which could be written as |xn |eiωt . Acceleration x¨ n can be replaced by −ω2 xn . The simple harmonic term eiωt belongs to the two sides of equations from (11) to (14). That all can be neglected because of the linear relationship.   k−Mω2

x9 . With Eq. (14) and x9 , the displacement x8 can be represented as k When x9 and x8 are known, the value of x7 can be obtained by solving the Eq. (13).

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Similarly, the value of xn−1 can be obtained by solving the Eq. (11) when xn+1 and xn are known. Finally, x0 can be obtained from Eq. (12). With the excitation amplitude and the top layer amplitude, the expression of vibrational transmissibility is as follow:   |x9 | (15) T = 20 log |x0 | The theoretical transmissibility of the NS-MRE isolator was calculated using MATLAB and the results are shown in Fig. 6. The amplitude of the top steel plate is set to be 1. It can be observed that the transmissibility of both isolators drastically decreases after reaching their respective cut-off frequencies, with the transmissibility curve of the NS-MRE isolator dropping earlier. In short, the cut-off frequency of the NS-MRE isolator is shifted significantly to the left comparing to the MRE isolator without negative stiffness. 200

MRE isolator NS-MRE isolator

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151.1 Hz 143.2Hz

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Fig. 6. Theoretical transmissibility of the two isolators

The period structure, consisting of masses and springs, is responsible for the sudden decrease in transmissibility after the cut-off frequency, resembling a cliff-like profile. To investigate the reason why the stop-band is generated, two kinds of single freedom structures are designed to compare with the NS-MRE isolator. One of the single freedom structures is a set of springs (a black spring and a red spring), and the other is interconnected series springs (two black springs and seven sets of springs). A set of springs can be used to reveal the difference between the periodic structure and the single structure. The series springs can be used to verify the function of the masses in the generation of stop-band. So, the transmissibility of an inner cell comprising a set of springs or series springs alone is calculated. For instance, as shown in Fig. 7, if assuming that only a set of springs or only series springs are considered without the presence of the masses between the bottom plate and the top plate, their equivalent stiffness values are 70835 N/m and 8052 N/m, respectively. The calculated resonance frequencies are 48.6 Hz and 16.4 Hz,

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respectively. The calculation formula is as follows:   1 70835 1 k+q  = = 48.6 Hz f = 2π M 2π 0.76 k  =

(16)

78835 × 70835 k(k + q) = = 8052 N/m 9k + 2q 9 × 78835 − 2 × 8000    1 k 1 8052  f = = = 16.4 Hz 2π M 2π 0.76

(a)

(17)

(18)

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Fig. 7. (a) A set of spring (b) Interconnected series springs without masses

It can be observed that single freedom structures exhibit a peak at the resonance frequency, followed by a steady decrease in transmissibility as the frequency increases, as shown in Fig. 8. If the structure between the bottom and top plate is simplified to include only interconnected series springs, its equivalent stiffness is lower than that of a set of springs. As a result, the resonance peak and the subsequent decrease in 200

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Fig. 8. Transmissibility of the NS-MRE isolator and two single freedom systems

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transmissibility occur earlier. The descent rate of transmissibility in single freedom structures is relatively gradual. In contrast, the transmissibility curve of the NS-MRE isolator experiences a sharp drop at the cut-off frequency, indicating that the waves are trapped in the stop-band. The grey area represents the stop-band of the NS-MRE isolator, which cannot occur even if the stiffness of the single freedom structures is significantly reduced. This confirms that the generation of the stop-band has no relation with the vibration frequency which is far away from the resonance frequency.

5 Conclusions This paper proposed a novel NS-MRE isolator with the acoustic metamaterial characteristics and the negative stiffness characteristics. By utilizing the Bloch theorem, predicting the appearance of the endless low frequency stop-band. And the transmission spectrum of vibration amplitude of the finite period structures also proves the existence of the stop-band. This attenuation due to the stop-band is much greater than the attenuation caused by being away from resonance frequency. Furthermore, the magnetic springs in the isolator create negative stiffness, which reduces its vibration stiffness and lowers the cut-off frequency, leading to a wider attenuation frequency range. Through simulations of the magnetic field, the feasibility of controlling the stiffness of the NS-MRE isolator was foreseeable, which. The theoretical analyses results of the infinite period structure were consistent with the theoretical analyses of the finite periodic structures, indicating that the correctness of generation mechanism of the stop-band. The lower cut-off frequency of the NS-MRE isolator, compared to MRE isolator, enables this isolator cater to much more application scenarios in the future.

References 1. Gripp, J.A.B., Rade, D.A.: Vibration and noise control using shunted piezoelectric transducers: a review. Mech. Syst. Sig. Process. 112 (2018) 2. Yan, G., Zou, H.-X., Wang, S., Zhao, L.-C., et al.: Bio-inspired vibration isolation: methodology and design. Appl. Mech. Rev. 73 (2021) 3. Lee, D.-O., Park, G., Han, J.-H.: Hybrid isolation of micro vibrations induced by reaction wheels. J. Sound Vib. 363 (2016) 4. Liu, C., Jing, X., Daley, S., Li, F.: Recent advances in micro-vibration isolation. Mech. Syst. Sig. Process. 56–57 (2015) 5. Park, T.-Y., Shin, S.-J., Park, S.-W., Kang, S.-J., et al.: High-damping PCB implemented by multi-layered viscoelastic acrylic tapes for use of wedge lock applications. Eng. Fract. Mech. 241 (2021) 6. Sun, Y., Gong, D., Zhou, J., Sun, W., et al.: Low frequency vibration control of railway vehicles based on a high static low dynamic stiffness dynamic vibration absorber. Sci. China Technol. Sci. 62 (2018) 7. Kalel, N., Darpe, A., Bijwe, J.: Propensity to noise and vibration emission of copper-free brake-pads. Tribol. Int. 153 (2021) 8. Rivin, E.: Passive vibration isolation. Appl. Mech. Rev. 57 (2005) 9. Jun, L., Caizhang, W.: Ultra-low frequency active vertical vibration isolation system. Sci. China 40 (1997)

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Human-Mechanical Biomechanical Analysis of a Novel Knee Exoskeleton Robot for Rehabilitation Training Mengmeng Yan1 , Guanbin Gao1,2(B) , Xin Chen1 , Yashan Xing1,2 , and Sheng Lu3,4 1 Faculty of Mechanical and Electrical Engineering, Kunming University of Science and

Technology, Kunming 650500, China [email protected] 2 Yunnan International Joint Laboratory of Intelligent Control and Application of Advanced Equipment, Kunming 650500, China 3 The First People’s Hospital of Yunnan Province, Kunming 650034, China 4 Yunnan Key Laboratory of Digital Orthopedics, Kunming 650034, China

Abstract. Exoskeleton robots are a new kind of wearable assistive and rehabilitation training equipment, with user adaptability and assistive effectiveness being important performance indicators. To improve the comfort of wearers and the effectiveness of rehabilitation training, this paper designs a knee rehabilitation training exoskeleton robot and analyzes its performance. Firstly, the exoskeleton robot is designed to reduce the lower limb movement load by placing the power source and motors at the wearer’s waist, and transmitting power to the knee through Bowden cables. A three-dimensional model of the exoskeleton robot is established. Secondly, a human-exoskeleton robot biomechanical simulation environment is constructed, including scenarios of the human body alone, the human body wearing a passive exoskeleton, and the human body wearing an active exoskeleton. Biomechanical simulations of the human-exoskeleton interactions are conducted in these three scenarios. The results show that the designed exoskeleton does not significantly impede human movement. When wearing the exoskeleton, the force and torque of the knees during squatting exercises are significantly reduced, and the muscle force and muscle activation of the quadriceps and gluteus maximus are significantly decreased, while the impact on the calf muscles is minimal. This verifies the good user adaptability and assistive effectiveness of the designed exoskeleton. Keywords: Adaptability · Cable-Driven · Exoskeleton · Rehabilitation training

1 Introduction The exoskeleton robot is a new kind of equipment integrating multi-disciplinary knowledge of mechanics, electronics, biomechanics and human-computer interaction, etc. Its structure is designed from the human body’s point of view, and based on providing protection and body support for the wearer, it can also perform certain functions and tasks © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 390–402, 2024. https://doi.org/10.1007/978-981-97-0554-2_30

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under the wearer’s control, which has now begun to be applied in the field of rehabilitation training. Rehabilitation exoskeleton robots are mainly used for upper limbs [1], lower limbs [2], fingers [3], hip [4], knee [5], ankle [6], and other parts of the booster and rehabilitation training, of which the knee exoskeleton has the most applications and is mainly used for the rehabilitation of the knee joints and the surrounding muscle tissues. The knee joint is one of the largest joints in human movement, which has a wide range of motion and bears high force, has always been a key part of sports injury and rehabilitation, and the elderly and people with limited mobility usually face the problem of decreased knee strength and diminished athletic ability. It has been found that cumulative knee disorders account for 65% of musculoskeletal disorders of the lower extremity, with deep squatting being one of the main contributors to knee disorders [7]. The design and study of knee exoskeletons is essential to restore, strengthen, and assist the function of the human knee. Excessive mass is still one of the main problems of wearable exoskeletons, in order to solve this problem, researchers have begun to choose a more compact and lightweight drive mode while optimizing the structure, and the flexible drive mode represented by cable drive has gradually entered into the field of vision [8–10]. Cable-driven exoskeleton robots can realize more degrees of freedom control compared with the traditional rigid drive mechanism, which makes it more flexible. Moreover, the cable is light in mass and small, which helps to reduce the overall weight of the exoskeleton robot, streamline the structure and volume of the exoskeleton, and improve the wearing comfort [11]. In order to reduce the wearer’s load, improve the wearing comfort, and enhance the assisting effect, this paper designs a rigid-flexible coupled exoskeleton robot for knee rehabilitation training, which adopts the Bowden cable to drive the knee joint for flexion/extension, to assist the knee injury patients in rehabilitation training. To evaluate the effect of the exoskeleton on the knee joint, a human-exoskeleton interaction simulation model was built in biomechanics software to simulate human movement, analyze and compare the changes in the muscle parameters of the lower limbs before and after wearing, and validate the rationality of the exoskeleton robot mechanism and the assisting effect of the exoskeleton robot designed in this paper.

2 Design of the Exoskeleton To reduce the load on the knee joint during human activities, the exoskeleton robot designed in this paper adopts motor drive and Bowden cable transmission to control the knee joint for flexion and extension to provide assistance to the human lower limbs for knee rehabilitation training. The structure of the exoskeleton robot is as shown in Fig. 1, which mainly consists of two symmetrical mechanical legs on the left and the right, the energy power part of the waist-integrated motors, batteries, and hardware circuitry, the and a Bowden cable connecting the mechanical legs to the integrated structure. The heavier motor-integrated part is placed at the back waist and is worn on the human body through a belt. The exoskeleton robot controls the rotation of the motors so that the fixed winding pulleys on the motors drive the two Bowden cables to rotate in two directions. The Bowden cables act as flexible cables that transmit forces directly to the knee winding pulleys,

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which drive the calves and thighs to form a two-direction relative motion, simulating the flexion and extension movements at the knee joint. The knee exoskeleton is designed with one joint degree of freedom, i.e., flexion/extension degree of freedom of the knee joint. Considering that the human calf has different degrees of valgus, the calf swing member is designed to adapt to the different degrees of valgus of the human body, to make the exoskeleton fit the human body more closely. The knee joint is equipped with a mechanical limit structure with a range of 0–130° to prevent the knee from being damaged by over-extension when the rotation angle exceeds the safety value due to malfunction. Considering the lightweight design, some of the main stress parts are made of aluminum alloy, while the rest of the parts are made of carbon fiber.

Fig. 1. The structure of the exoskeleton robot

3 Development of Human-Exoskeleton Model In this paper, three simulation models are built in AnyBody by combining the human body model with the exoskeleton model, the first one is a separate human squatting simulation model, the second one is a simulation model in which the human body wears an unpowered exoskeleton for squatting, and the third one is a simulation model in which the human body wears a powered exoskeleton and the exoskeleton drives the human body to perform squatting. The reasonableness of the structural design of the knee exoskeleton robot is examined by comparing the simulation data of the first model with the second model, and the assistive effect of the knee exoskeleton robot is examined by comparing the simulation data of the first model with the third model.

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3.1 Human Model To evaluate the assistive effect of the knee exoskeleton conveniently, the movements with large knee joint force in daily life were selected for simulation, and squatting is one of the common and important movements in daily life, which requires high strength and stability of the human lower limbs, and involves the coordinated movement of multiple muscles. Moreover, most of the existing knee exoskeletons are designed to assist walking, and due to the interference of their structure, the knee bending angle is limited, and they usually cannot do movements with excessive knee flexion such as deep squatting [12]. Simulation evaluation of human wearing exoskeleton for squatting exercise can more realistically simulate the actual use scenario and verify the effectiveness and adaptability of exoskeleton in daily life. The AnyBody Modeling System (AMS) modeling calculates human muscle and joint forces through inverse dynamics analysis, but since muscles are unilateral element that can only pull and not push, and muscle systems tends to have more muscles than strictly necessary to balance the external force, i.e., the phenomenon of muscle redundancy, and there is a static indeterminacy problem. The AMS, through the muscle recruitment optimization approach to solve this problem, and its objective function is shown in Eq. 1:   fi p (1) G= Ni where, G is the function to minimize; f is the muscle force; N is the normalization factor, here expressed as muscle strength; i is the muscle identifier; p is the power exponent, indicating muscle synergy, with the default of p = 3 being the better muscle recruitment scheme. The human body was modelled using AMS, and its lower limb model used the new Twente Lower Extremity Model (TLEM 2.0) model, the TLEM 2.0 lower limb model includes 169 muscles and 6 degrees of freedom of joints [13]. These include 3 degrees of freedom in the hip joint (flexion/extension, adduction/abduction, inversion/eversion), 1 degree of freedom in the knee joint (flexion/extension), and 2 degrees of freedom in the ankle joint (plantar/dorsiflexion, inversion/eversion). The AnyBody Managed Model Repository (AMMR) is a unique open collection of human body parts represented as mechanical elements, and examples of how to model activities of daily living [13]. The human body model is derived from the Squat model in AMMR, which is 1.75 m and 75 kg. In order to improve the computing speed, the muscles of the upper limbs of the human body are removed, and only all the muscles of the lower limbs are retained. The initial state of the human body model was set to keep the initial posture of squatting with the arms crossed in front of the chest. Add the drive for the human joints, so that the human body to complete the squatting action. From the knee flexion angle in Fig. 2, the knee angle of the human body in the initial position is 10°, and the squatting process is completed in 1.5 s, at which time the knee flexion angle is 90°, and the human body starts to stand up after 1.5s and completes the whole set of actions in 3 s, and then restores the initial position.

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3.2 Exoskeleton Model The designed knee exoskeleton robot is exported from the 3D design software SolidWorks to AnyBody through the data exchange interface, and the mass attributes of its parts, such as mass, center of mass, and moment of inertia, can be computed in SolidWorks, and the relevant mass attributes are defined in AnyBody and the fit and connection of the exoskeleton parts are performed. Due to the negative correlation between the complexity of the mechanism and the computational efficiency, the exoskeleton model can be simplified into three parts: Energy-Drive part, Thigh-Knee part, Knee-Shank part, and since the Bowden cable is flexible, it can be ignored, and its transmission process can be equated to adding a drive at the knee joint. The exoskeleton totals 6.5 kg, is symmetrical for the left and right legs, weighs on one leg, and contains one joint degree of freedom, i.e., knee flexion/extension freedom in the sagittal plane. 3.3 Human-Exoskeleton Interaction Model When implementing the Human-Exoskeleton Interaction Model, the exoskeleton model is first connected to the human model. Corresponding reference points are set on the exoskeleton model and the human model to prepare for the next connection. The energydriven component is worn on the waist of the human body through a belt, and is relatively static with the human body during movement, so this component is fixedly connected to the human lumbar spine. The bi-directional cable-driven knee exoskeleton robot works on the principle that the power output from the motor end drives the Bowden cable, which

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in turn drives the knee wheel to rotate with respect to the thigh, which in turn causes the exoskeleton calf to rotate the human calf. Therefore, the thigh-knee component is fixedly connected to the human thigh, and it is ensured that the exoskeleton knee joint is aligned with the human knee joint in the axis perpendicular to the sagittal plane. Adding a connection in AnyBody constrains the three degrees of freedom, and after the above connection, the remaining one degree of freedom has not yet been constrained, and over-constraint will occur if the exoskeleton calf is connected to a reference point of the human calf, and therefore, it is impossible to Use a method similar to the above corresponding reference point connection, and just add a constraint on the calf part. Setting the angle between the exoskeleton leg and the human leg in the sagittal plane to 0 ensures that the exoskeleton leg is connected to the human leg with a constraint added. The connected Human-Exoskeleton Model is shown in Fig. 3. There are two scenarios in which the human body wears an exoskeleton for movement, one is when the exoskeleton is worn as a load on the human body for movement (Human-unpowered Exoskeleton), and the other is when the exoskeleton is used as an actuator to drive the human body for movement (Human-powered Exoskeleton). When the exoskeleton is used as a load, the exoskeleton is simply connected to the human body and follows the human body to perform the movement, while the driver is still on the human body. When the exoskeleton is used as a driver, the human knee drive is switched off, the driver is added to the knee joint of the exoskeleton, and the exoskeleton drives the human body to move. Kinematic analysis and inverse dynamics analysis were performed on the three models to obtain the dynamics information and muscle parameters of the human model. Comparing the knee flexion angles of the three simulation models, the error is reduced to 0.0002°, proving that the modelling is correct.

Fig. 3. Human-Exoskeleton model

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4 Results and Analysis Once the Human-Exoskeleton Model is completed, it can be simulated and analyzed to derive the human kinetic parameters (joint reaction force, joint moment) and muscle parameters (muscle metabolism, muscle force, muscle activity). By comparing the model information of the three simulation models, namely, Human Model (H), Human-unpowered Exoskeleton Model (H-uE), and Human-powered Exoskeleton Model (H-pE), the effects of exoskeleton usage on human movement parameters can be analyzed. 4.1 Kinetic Parameters Knee Reaction Force There are three main directions of forces that play an important role at the knee joint, Medial-Lateral (M-L), Anterior-Posterior (A-P), and Proximal-Distal (P-D). The ML Force refers to the forces that act on the medial and lateral sides of the knee joint, where there are ligamentous and cartilaginous structures. These structures help maintain stability of the knee joint and prevent excessive lateral movement of the knee in both the internal and external directions. A-P Force refers to the forces acting anteriorly and posteriorly on the knee joint. The A-P Force helps to control the flexion and extension movements of the knee joint while maintaining stability of the knee joint in both the anterior and posterior directions. The M-L Force is the force acting on the anterior and posterior sides of the knee joint. It controls knee deceleration and stabilizes the speed of the squat during squatting, and produces a corresponding reaction force during squatting to return the knee to the upright position. The P-D Force refers to the forces acting in the proximal (lateral thigh bone) and distal (lateral calf bone) directions of the knee, which is oriented along the axis of the knee joint during human movement. The P-D Force is created by the action of the muscles and ligaments that help to stabilize the knee joint during flexion and extension. As shown in Fig. 4, the reaction forces of the three simulation models, H Model, H-uE Model, and H-pE Model, are shown in the three directions of the knee joint, respectively. Compared with the human squatting process alone, when the human body is wearing an unpowered exoskeleton, i.e., the exoskeleton is used as a load, the knee joint reaction force increases by 34.4 N and 170.9 N in the M-L and A-P directions, respectively, and the P-D Force in the squatting process has an insignificant difference and decreases by 87.6 N at the lowest point of the squatting process. The trend of the knee joint reaction force in the three directions is roughly the same, and the effect on human movement is small when the exoskeleton is used as a load. Were roughly the same, and the exoskeleton had less effect on human movement when used as a load. When the human body wears a powered exoskeleton, i.e., when the exoskeleton drives the human body to perform squatting, the knee joint reaction force changes in the three directions in a trend like that of the human body movement alone, but the difference in the values is obvious, with reductions of about 271.9 N, 1226.6 N, and 522.7 N. This

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indicates that there is a significant reduction in the joint force of the human body during the movement of the human body driven by the exoskeleton. Knee Flexion Moment In Fig. 4, the three models have basically the same trend of change in knee flexion moment during movement, the overall moment increases when the exoskeleton is unpowered, and the maximum bending moment increases by 11.3 N·m, and the overall moment decreases when the exoskeleton is powered, and the maximum bending moment decreases by 24.2 N·m. 600

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4.2 Muscle Parameters Muscle Metabolism Metabolism during activity is the process by which the body produces and expends energy during various physical activities and sports. The higher the intensity of the activity, the higher the accompanying muscle metabolism. Metabolic reduction has been considered an important metric for evaluating exoskeleton devices, and its feasibility has been verified in a variety of exercise scenarios [14–16]. As shown in Fig. 5,

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the metabolism of the body muscles is not the same when completing both squatting and rising movements, and the metabolism produced during the rising movement is significantly higher than that of squatting, which is due to the fact that not only does the movement need to be completed during the rising movement, but it also needs to overcome the force of gravity to do the work, which requires an additional supply of energy than that of the squatting movement. Comparing the exercise of the three models, the muscle metabolism when wearing the unpowered exoskeleton was very similar to that of the human body alone, and the maximal metabolism of the squatting and standing up maneuvers increased by 13.2 J and 58.8 J respectively, which was not a significant change. When wearing a powered exoskeleton, although the trend of metabolic change was still increasing and then decreasing, the time of generating maximal metabolism for both squatting and standing up was advanced, and the maximal metabolism decreased by 102 J and 911.3 J respectively, which was a significant difference, indicating that the exoskeleton has an obvious role in providing energy when driving the human body. 1200 H H-uE H-pE

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Muscle Force There are four muscle groups in the lower extremity that are involved in the synergistic action of the human body when completing the squat: the quadriceps, gluteus maximus, hamstrings, and calf muscle groups. The quadriceps group is located in the anterior thigh and consists of four muscles, rectus femoris, vastus lateralis, vastus medialis, and vastus intermedius, which are the primary power muscles in the squat, and which, when

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contracted, propel the knee into flexion for the purpose of squatting and standing up. The gluteus maximus is in the gluteal area and is also an important power muscle group in the squat. The contraction of the gluteus maximus helps push the hip joint into extension, which is essential for standing up and returning to an upright position. The posterior thigh muscle group is located at the back of the thigh and includes biceps femoris, semitendinosus and semimembranosus. These muscles also play an important role in the squat, assisting in knee flexion and helping to control knee extension when rising. The calf muscles include gastrocnemius and soleus. They are located at the back of the calf and play an important role in stabilizing the knee and ankle during squatting. 0

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In this paper, knee flexor muscle force and one of each of the four muscle groups mentioned above are analyzed: flexor muscle force, femoral muscle force, gluteus maximus muscle force and gastrocnemius muscle force. By looking at Fig. 6, the muscle force change curves for each group of muscles can be seen for the human body alone as well as for the two exoskeleton wearing modes, and by comparing the maximum muscle values, it can be concluded that exoskeleton-driven human body movement reduces the human body’s muscle force. In this case, the four muscle forces do not change much when the exoskeleton is used as a load, and the overall trend is similar to that when the human body is moving. When the exoskeleton actuates human movement, knee flexor

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muscle force, femoral muscle force, and gluteus maximus muscle force were all significantly reduced, and in the deepest part of the squat, they were reduced by 266.4 N, 1528.2 N, and 760.4 N, respectively, and femoral and gluteus maximus muscles were especially obvious, and the gluteus maximus muscle force was almost reduced to the minimum with the help of the exoskeleton, and the femoral muscle force, which plays an important role in the knee flexion and extension, was also reduced by nearly 2/3. Gastrocnemius muscle showed a similar trend with a slight increase in muscle force, which showed that the knee exoskeleton had little effect on the gastrocnemius muscle. From the above analysis, the knee exoskeleton robot has a greater effect on the anterior and posterior thigh muscles and the gluteus maximus in the squatting exercise, which can achieve the effect of assisting the human body to move and reduce the muscle force of the thighs and the gluteal muscles, but has a smaller effect on the calf muscles. The reason for this phenomenon is that in the squatting movement, the feet are not moving relative to the ground, the thighs and hips are in a state with a large range of motion relative to the calves, and the knee exoskeleton mainly provides assistance to the thighs and hips, and the calves mainly play a role in bearing the weight. 0.26

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Muscle Activation Muscle activation is the degree to which a particular muscle or group of muscles is engaged and produces force to the maximum extent possible during the performance of a particular movement or activity. In simple terms, it describes how active and engaged a muscle is during a specific movement, and can also be expressed as a ratio of muscle activity to muscle strength. By analyzing the activation level of the major muscles of the lower limbs before and after wearing the exoskeleton, we can get a detailed understanding of the activation level and fatigue state of the muscles during squatting, and when the value is greater than 1, the muscles are in a fatigue or damage state. As shown in Fig. 7, the maximum muscle activation and the activation of the main active muscles (biceps femoris, semitendinosus and gastrocnemius) during squatting were observed and analyzed. The muscle activation levels were found to have the same trend and exceeded 1 for all three states, indicating that there was no muscle damage or muscle fatigue throughout the exercise. When the exoskeleton is unpowered, the maximum muscle activation level increases by about 0.018, which is due to the increased load on the human body, and the muscles of the lower limbs need to provide more energy. Maximum muscle activation was significantly lower when the body was powered by the exoskeleton, and there was a lag and advance at the minimum level of activation. The biceps femoris and semitendinosus muscles showed a significant decrease in muscle activation after wearing the exoskeleton, but the effect on the gastrocnemius muscle was not significant and tended to increase in the deepest part of the squat. The knee exoskeleton robot has muscle compensation for the thigh muscles during squatting, but has little effect on the calf muscles.

5 Conclusions In this paper, an exoskeleton robot for knee rehabilitation training is designed. The bidirectional cable drive can not only help the knee joint in both flexion and extension directions, but also has the advantages of streamlined structure, freedom and flexibility. To evaluate its effect on human activities, the squatting model, which has a greater impact on the human knee joint, was selected for simulation experiments. The human-exoskeleton interaction platform was built by a biomechanical software, and the squatting simulation was conducted before and after the human body wore the unpowered exoskeleton and the powered exoskeleton. The results showed that the knee exoskeleton had no obstruction to the human body during movement when it was used as a load, and when the knee exoskeleton drove the human body, the exoskeleton provided a booster effect on the knee joint. The activation level of all muscles in the lower limbs is less than 1, which indicates that the knee exoskeleton does not cause muscle fatigue and muscle damage phenomenon to human movement, and has muscle compensation effect on the anterior and posterior thigh muscle groups and gluteus maximus, but has little effect on the calf muscle groups. The knee exoskeleton robot structure is reasonably designed, and has no obvious obstruction to human lower limb activities, so it can assist human movement and help patients to carry out rehabilitation training. Acknowledgement. This work was supported by Yunnan Major Scientific and Technological Projects under grants (202001AS070028, 202102AA310042, 202202AG050002) and Kunming

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University of Science and Technology & the First People’s Hospital of Yunnan Province Joint Special Project on Medical Research (KUST-KH2022003Y).

References 1. Zhou, L., Li, Y., Bai, S.: A human-centered design optimization approach for robotic exoskeletons through biomechanical simulation. Robot. Auton. Syst. 91, 337–347 (2017) 2. Weiqun, W., Guang, H.Z., Lina, T., Feng, Z., Yixiong, C., Min, T.: A novel leg orthosis for lower limb rehabilitation robots of the sitting/lying type. Mech. Mach. Theory 74, 337–353 (2013) 3. Duojin, W., Qingyun, M., Qiaoling, M., Xinwei, L., Hongliu, Y.: Design and development of a portable exoskeleton for hand rehabilitation. IEEE Trans. Neural Syst. Rehabil. Eng. Public. IEEE Eng. Med. Biol. Soc. 26(12), 2376–2386 (2018) 4. Pan, Y.T., et al.: Effects of bilateral assistance for hemiparetic gait post-stroke using a powered hip exoskeleton. Ann. Biomed. Eng. 51(2), 410–421 (2022) 5. Chen, J., et al.: A pediatric knee exoskeleton with real-time adaptive control for overground walking in ambulatory individuals with cerebral palsy. Front. Robot. AI 8, 2296–9144 (2021) 6. Ying, F., Lerner, Z.F.: Bilateral vs. paretic-limb-only ankle exoskeleton assistance for improving hemiparetic gait: a case series. IEEE Robot. Autom. Lett. 7(2), 1246–1253 (2022) 7. Reid, C.R., Bush, P.M., Cummings, N.H., McMullin, D.L., Durrani, S.K.: A review of occupational knee disorders (review). J. Occup. Rehabil. 20(4), 489–501 (2010) 8. Aguirre-Ollinger, G., Yu, H.: Lower-limb exoskeleton with variable-structure series elastic actuators: phase-synchronized force control for gait asymmetry correction. IEEE Trans. Robot. 37(3), 763–779 (2021) 9. Yu, S., Huang, T.-H., Wang, D., Lynn, B., Sayd, D., Silivanov, V., et al.: Design and control of a high-torque and highly backdrivable hybrid soft exoskeleton for knee injury prevention during squatting. IEEE Robot. Autom. Lett. 4(4), 4579–4586 (2019) 10. Junlin, W., Xiao, L., Hao, H.T., Shuangyue, Y., Yanjun, L., Tianyao, C., et al.: Comfortcentered design of a lightweight and backdrivable knee exoskeleton. IEEE Robot. Autom. Lett. 3(4), 4265–4272 (2018) 11. Wang, K.C.H., Xing, M.: Application progress of rehabilitation exoskeleton robot in limb rehabilitation. J. Mech. Transm. 46(4), 10–21 (2022) 12. Pratt, J., Krupp, B., Morse, C., Collins, S.: The roboknee - an exoskeleton for enhancing strength and endurance during walking. In: IEEE International Conference on Robotics and Automation, New Orleans, LA, USA (2004) 13. The Anybody Modeling System. https://www.anybodytech.com/. Accessed 19 July 2023 14. Collins, S.H., Wiggin, M.B., Sawicki, G.S.: Reducing the energy cost of human walking using an unpowered exoskeleton. Nature 522(7555), 212–215 (2015) 15. Mooney, L.M., Rouse, E.J., Herr, H.M.: Autonomous exoskeleton reduces metabolic cost of human walking. J. Neuroeng. Rehabil. 11(1), 151 (2014) 16. Kim, J., Panizzolo, F.A., Zhou, Y.M., Baker, L.M., Galiana, I., Malcolm, P., et al.: Reducing the metabolic cost of running with a tethered soft exosuit. Sci. Robot. 2(6), 6708 (2017)

Adaptive Sliding Mode Control for Active Suspensions of IWMD Electric Vehicles Subject to Time Delay and Cyber Attacks Wenfeng Li1 , Jing Zhao1 , Mengqi Deng1 , Zhijiang Gao1 , and Pak Kin Wong1,2(B) 1 Department of Electromechanical Engineering, University of Macau, Taipa, Macao

[email protected] 2 Zhuhai UM Science and Technology Research Institute, Zhuhai 519031, China

Abstract. This paper addresses the control problem for active suspensions of the in-wheel motor driven electric vehicle with consideration of time delay and cyber attacks. The main purpose is to develop an adaptive sliding mode control (SMC) method to improve the suspension performances by handling the issues of time delay and cyber attacks. Firstly, by considering a dynamic vibration absorber to mitigate vibrations, an active suspension model is constructed, in which both the spring dynamic nonlinearity and the damper dynamic segmentation are approximated by the Takagi-Sugeno fuzzy model. Secondly, by introducing an integraltype sliding surface, sufficient conditions are developed to ensure the sliding motion satisfies the asymptotical stability and desired performance requirements despite the occurrence of time delay. Based on the reachability to the sliding surface, a SMC approach is developed such that the closed-loop suspension system can achieve the desired performances of the sliding surface. For a better calculation of the controller gains, the controller design condition is converted to an optimization problem. Finally, various simulation tests are implemented to verify the merits of the proposed adaptive control method. Keywords: Active Suspension · In-wheel motor driven · Cyber attacks · Time delay · Sliding mode control

1 Introduction The electric vehicle has been devoted a great deal of efforts in last decades because it has the advantages of energy-saving and environmental friendliness [1–5]. Specifically, the in-wheel motor-driven (IWMD) vehicle has the prominent advantages of independently and precisely controllable torque, high transmission efficiency, large torque, and fast response. As a result, the IWMD electric vehicle is the main development direction of electric vehicles [6]. However, in the IWMD electric vehicles, installing the motors in the drive wheels indicates that the excitation is directly transmitted to the vehicle body. At the same time, in un-sprung modes, the frequency of vertical excitation increases and ICANDVC2023 best presentation paper © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 403–417, 2024. https://doi.org/10.1007/978-981-97-0554-2_31

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the ride comfort and contact stability performance decreases. Hence, some treatments are needed to reduce the impact on ride comfort [7]. Bridgestone company has designed a suspension structure with a wheel motor, which is effective to improve the suspension performances as a dynamic vibration absorber (DVA). It can greatly counteract the road vibration inputs and improve the road holding performance compared to the conventional electric vehicles, but this structure requires the use of active suspension control [8, 9]. Hence, the advanced control technologies of the suspension systems are important to promote the development of IWMD electric vehicles. To improve suspension responses, many control methods were presented in the field of active suspension control including the adaptive control [10], the robust control [11], the neural network control [12], the sliding model control (SMC) [13] and so on [14– 17]. Among the above control methods, the SMC possesses the advantages of strong robustness and ability to maintain stability despite the system uncertainties and external disturbances. However, traditional SMC usually faces an obvious chattering phenomenon. In line of this consideration, various methods including the terminal SMC, the higher-order SMC and the adaptive sliding model control have been presented to avoid the above problem and to improve the control performances [18]. However, the former two approaches have the disadvantages of needing accurate mathematical models and complicated controller design. Based on the above-mentioned issues, the adaptive SMC approach has been widely investigated [19, 20]. The authors in [21] investigated the adaptive SMC problem for the stabilization of Markovian jump systems. The authors in [22] proposed an adaptive SMC method for the uncertain active suspensions. The wide application of network brings many benefits, such as improving the system operation rate and communication efficiency. On the other hand, the network may also generate some problems such as time delay and cyber attacks [23]. For electric vehicles, due to the interaction of information among sensors, controllers, and actuators, the active suspension systems are easily susceptible to the cyber attacks. The cyber attacks consume or take up reasonable resources of the system by using malicious data, which may degrade the control performances of active suspensions [24]. To deal with the cyber attacks, the SMC has been widely investigated due to its obvious advantages in resisting external disturbance [25]. However, the parameter uncertainties and time delay were not fully addressed in the above adaptive SMC methods. In the control strategy of the vehicle, the signal transmission inevitably causes time delay. It should be noted that the time delay can decrease the stability and responses of the vehicle suspension systems [5, 26]. In recent years, there are many control strategies for the time-delay active suspensions, but most of them ignored the cyber attacks [27– 29]. Although the adaptive SMC has an advantage in handling the cyber attacks, it is still necessary to consider the time delay in the design process of adaptive sliding mode controller. This paper aims to develop an adaptive SMC method for the IWMD vehicle active suspensions subject to time delay and cyber attacks. This paper has the following features as: (1) An active suspension model is established by considering the spring and damper dynamic nonlinearities, dynamic vibration absorber, time delay and cyber attacks. (2) An integral-type sliding surface is established with consideration of time delay. (3) An

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adaptive SMC method is proposed to improve the vehicle suspension performance and meet the suspension constraint requirements.

2 Problem Formulation 2.1 Suspension Model of IWMD Vehicles To simplify the complexity of the suspension dynamics analysis and controller design, a quarter-vehicle suspension model is considered. It can be seen from Fig. 1 that the suspension dynamics can be described as: ⎧ ms z¨s = −ks (t)(zs − zu ) − cs (t)(˙zs − z˙u ) + u(t) ⎪ ⎪ ⎪ ⎨ m z¨ = k (t)(z − z ) + c (t)(˙z − z˙ ) + k (z − z ) + c (˙z − z˙ ) u u s s u s s u w w d d d d (1) ⎪ −kt (zu − zw ) − ct (˙zu − z˙w ) − u(t) ⎪ ⎪ ⎩ md z¨d = −kd (zd − zu ) − cd (˙zd − z˙u ) where zd , zu and zs stand for the vertical displacements of the motor mass md , unsprung mass mu and sprung mass ms . zw and u(t) denote the road disturbances and the actuator force. The suspension damping and stiffness coefficients are ks and cs . The motor damping and stiffness coefficients are kd and cd . The tire damping and stiffness coefficients are kt and ct , respectively. And the nonlinear stiffness and damping coefficients are described as [30, 31]:   ks (t) = kl 1 + n(zs − zu )2  (2) cs1 , z˙ s − z˙ u ≥ 0 cs (t) = cs2 , z˙ s − z˙ u < 0 where n is the ratio of nonlinear to linear segment kl of the stiffness coefficient. cs1 and cs2 represent damping coefficient of extension and compression. Defining two premise variables as δ1 (t) = (zs − zu )2 and δ2 (t) = z˙s − z˙u , Eq. (2) can be morphed into ks (t) = kl + nkl δ1 (t), cs (t) = cs1 υ(δ2 (t)) + cs2 (1 − υ(δ2 (t))

(3)

where υ(t) is a step function. Regarding to the controller design, the following four key suspension performances are considered as [32]: (1) Ride comfort: It is usually characterized by the body acceleration, thus the following condition is considered: min z¨s (t)

(4)

(2) Handling stability: It can be characterized by the suspension travel, which needs to be smaller than its maximum value, namely, |zs (t) − zu (t)| ≤ zmax

(5)

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zs

ms u (t )

ks

cs

zd

md cd

kd

zu

mu kt

ct

zw

Fig. 1. Quarter-vehicle active suspension model with a DVA

(3) Road holding ability: The tire should be connected to the road closely and firmly, requiring the dynamic load less than or equal to static load, namely, kt (zu (t) − zw (t)) ≤ (ms + mu + md )g

(6)



T Defining the state vector as x(t) = zs − zu z˙s zd − zu z˙d zu − zw z˙u the road perturbate input as ω(t) = z˙w , and the control output as z(t) =

T z¨s a1 (zs − zu ) a2 (zd − zu ) a3 (zu − zw ) , the suspension model can be converted to 

x˙ (t) = Ax(t) + B1 ω(t) + B2 u(t)

(7)

z(t) = Cx(t) + D1 ω(t) + D2 u(t) where ⎡

0

1

ks mu

cs (t) mu

⎢ − ks (t) − cs (t) ms ⎢ ms ⎢ 0 0 ⎢ A=⎢ 0 ⎢ 0 ⎢ ⎣ 0 0 ⎡

0

⎤

0 0 0 0 0 1 kd − md − mcdd 0 0 kd mu

⎡ ⎢ 1 ⎥ − ks (t) ⎢ ms ⎥ ⎢ ms ⎥ ⎢ ⎢a ⎢ 0 ⎥ B2 = ⎢ ⎥, C = ⎢ 1 ⎢ 0 ⎥ ⎣0 ⎥ ⎢ ⎣ 0 ⎦ 0 − m1u

cd mu s (t) − cm s 0 0 0

0 −1 cs (t) 0 ms 0 −1 cd 0 md 0 1 d +ct − mktu − cs (t)+c mu 0 0 0 a3

0 0 a2 0

⎤

⎤ 0 ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎥, B1 = ⎢ ⎥, ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ ⎦ ⎣ −1 ⎦ ⎡

ct mu

⎡ ⎤ ⎡ ⎤ 1 s (t) 0 cm 0 ms s ⎢ ⎥ ⎢0⎥ ⎢ 0 00 ⎥ ⎥ ⎥, D1 = ⎢ ⎣ 0 ⎦, D2 = ⎢ ⎣ 0 00 ⎦ 0 0 00

⎤ ⎥ ⎥ ⎥. ⎦

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Moreover, since the opening-up property of the IWMD vehicle, the cyber attacks are considered when building the suspension system model, which can be shown as  x˙ (t) = Ax(t) + B1 ω(t) + B2 [u(t) + h(x(t))] (8) z(t) = Cx(t) + D1 ω(t) + D2 [u(t) + h(x(t))] where h(x(t)) represents the cyber attacks on suspension system, which meets the following condition and ε > 0 is a constant: h(x(t)) ≤ εx(t)

(9)

Based on (2)-(3), the membership functions are 2 2 2 P1 (δ1 (t)) = (zmax − δ1 (t))/(zmax − zmin ), Q1 (δ2 (t)) = υ(δ2 (t)) 2 2 2 )/(zmax − zmin ), Q2 (δ2 (t)) = 1 − υ(δ2 (t)) P2 (δ1 (t)) = (δ1 (t) − zmin

(10)

2 2 Therefore, the condition δ1 (t) = P1 (δ1 (t))zmin + P2 (δ1 (t))zmax can be obtained with P1 (δ1 (t)) + P2 (δ1 (t)) = 1. Then, the following Takagi-Sugeno (T-S) fuzzy model is obtained as: Fuzzy Rule i: IF δ1 (t) is Pv (δ1 (t)) and δ2 (t) is Qv (δ2 (t)), THEN  x˙ (t) = Ai x(t) + B1i ω(t) + B2i [u(t) + h(x(t))] (11) z(t) = Ci x(t) + D1i ω(t) + D2i [u(t) + h(x(t))]

where v = 1, 2 and i = 1, 2, 3, 4. And the matrices Ai , B1i , B2i , Ci , D1i and D2i of system can be obtain by corresponding δ1 (t) and δ2 (t), respectively. Furthermore, one has ⎧ 4  ⎪ ⎪ ⎪ x ˙ (t) = ηi [Ai x(t) + B1i ω(t) + B2i (u(t) + h(x(t)))] ⎪ ⎪ ⎨ i=1 (12) 4 ⎪  ⎪ ⎪ ⎪ ηi [Ci x(t) + D1i ω(t) + D2i (u(t) + h(x(t)))] ⎪ ⎩ z(t) = i=1

 where the grades of membership ηi satisfies ηi ≥ 0 and 4i=1 ηi = 1, and η1 = P1 (δ1 (t)) × Q1 (δ2 (t)), η2 = P1 (δ1 (t)) × Q2 (δ2 (t)), η3 = P2 (δ1 (t)) × Q1 (δ2 (t)), η4 = P2 (δ1 (t)) × Q2 (δ2 (t)). 2.2 Adaptive Sliding Mode Controller Firstly, the following sliding surface is constructed as: s(t) = Hx(t) −

 t 4 4  0 i=1 j=1

ηi ηj HAi x(α)d α −

 t 4 4 

ηi ηj HB2i Kj x(β − τ (β))d β

0 i=1 j=1

(13)

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where the constant matrix H ∈ 1×4 will be design to guarantee that HB2i is nonsingular and HB1i = 0. Based on the parallel distribution compensation, the following control law can be obtained: Control Rule j: IF δ1 (t) is Pv (δ1 (t)) and δ2 (t) is Qv (δ2 (t)), THEN u(t) = ηj Kj x(t − τ (t)) − h(x(t)) where the local control gain matrix Kj will be obtained by subsequent calculations. Hence, we have the global fuzzy controller as follows: u(t) = ηj Kj x(t − τ (t)) − h(x(t))

(14)

By combining Eq. (14) and Eq. (12), the closed-loop model of IWMD vehicle active suspension system is ⎧ 4 4  ⎪  ⎪ ⎪ ⎪ x˙ (t) = ηi ηj [Ai x(t) + B1i ω(t) + B2i Kj x(t − τ (t))] ⎪ ⎪ ⎨ i=1 j=1 (15) 4 4  ⎪  ⎪ ⎪ ⎪ z(t) = ηi ηj [Ci x(t) + D1i ω(t) + D2i Kj x(t − τ (t))] ⎪ ⎪ ⎩ i=1 j=1

As shown in Fig. 2, the control flow of the IWMD vehicles nonlinear suspension is presented. The active suspension system is subject to both road interference and cyber attacks. However, the closed-loop system (15) will be asymptotically stable and achieve the desired performance after being processed by the designed sliding mode controller. In a word, a robust adaptive SMC method is proposed for the IWMD vehicle active suspension such that the asymptotical stability and the following condition are satisfied: z(t)2 ≤ γ ω(t)2 , ω(t) ∈ L2 [0, ∞)

h

u

Fig. 2. Control workflow of the IWMD vehicles suspension system

(16)

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3 Main Results 3.1 Stability Analysis of the Sliding Motion Theorem 1. For the given positive scalars τ1 , τ2 , γ , κ1 , κ2 , and control gains K j . The closed-loop system (15) ensure the asymptotical stability and guarantee the H∞ performance in (16), if there exist  positive definite matrices E 1 , E 2 , F 1 , F 2 , X , general F2 W > 0, such that matrix W that satisfies ∗ F2 ii < 0

(17)

ij + ji < 0 (i < j)

(18)

  where i, j = 1, 2, 3, 4 ij =

11

F2 W ∗ F2

12

ij ij

22

∗ ij

 >0

(19)

 , and

⎤

F1 B2i K j 0 B1i ⎢ ∗ −F − F − E F − W W 0 ⎥ ⎥ ⎢ 1 2 1 2 ⎥ ⎢ =⎢∗ 0 ⎥, ∗ [W − F 2 ]s F 2 − W ⎥ ⎢ ⎣∗ ∗ ∗ −F 2 − E 2 0 ⎦ ∗ ∗ ∗ ∗ −γ 2 I ⎤ ⎡ τ1 X ATi (τ2 − τ1 )X ATi PCiT T T T T T T ⎥ ⎢ (τ2 − τ1 )K j B2i K j D2i ⎥ ⎢ τ1 K j B2i ⎥ ⎢ =⎢ ⎥, 0 0 0 ⎥ ⎢ ⎣ 0 0 0 ⎦ T T T (τ2 − τ1 )B1i D1i τ1 B1i ⎡

11

ij

12

ij

22 =diag{2κX − κ 2 F 1 , 2κX − κ 2 F 2 , −I }, = [X Ai ]s + E 1 + E 2 − F 1 . −1 Moreover, the control gain is calculated as Kj = K j X . Proof. Constructing a Lyapunov function as:  t  T T V (t) = x (t)Xx(t) + x (θ )E1 x(θ )d θ +  +1

0

−τ1



t−τ1 t

t+θ

x˙ T (α)F1 x˙ (α)d αd θ + (τ1 − τ2 )



t

t−τ2 −τ1  t −τ2

t+θ

xT (θ )E1 x(θ )d θ (20) x˙ T (α)F2 x˙ (α)d αd θ

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The derivative of (20) is obtained as V˙ (t) = x˙ T (t)Xx(t) + xT (t)X x˙ (t) + xT (t)(E1 + E2 )x(t) −xT (t − τ1 )E1 x(t − τ1 ) − xT (t − τ2 )E2 x(t − τ2 ) + τ12 x˙ T (t)F1 x˙ (t)  t−τ1  t x˙ T (θ )F1 x˙ (θ )d θ − (τ2 − τ1 ) x˙ T (θ )F2 x˙ (θ )d θ +(τ2 − τ1 )2 x˙ T (t)F2 x˙ (t) − τ1 t−τ1

t−τ2

(21) Based on Jensen’s inequality [33], the following condition holds 



t

−τ

x˙ T (θ )F1 x˙ (θ )d θ ≤

t−τ1

x(t) x(t − τ1 )

T 

−F1 F1 ∗ −F1



x(t) x(t − τ1 )

 (22)

According to Reciprocal convex inequality [34], it follows that  t−τ1 −(τ2 − τ1 ) x˙ T (s)R2 x˙ (s)ds ≤ t−τ2 ⎤T ⎡

⎡

⎤⎡ ⎤ x(t − τ1 ) W −F2 F2 − W x(t − τ1 ) ⎣ x(t − τ (t)) ⎦ ⎣ ∗ [W − F2 ]s F2 − W ⎦⎣ x(t − τ (t)) ⎦ ∗ ∗ −F2 x(t − τ2 ) x(t − τ2 )

(23)



 F2 W And the matrix W meets the conditions > 0. ∗ F2 Combining (21)–(23), the following condition can be obtained: J = V˙ (t) + z T (t)z(t) − γ 2 ωT (t)ω(t) ≤ ς T (t)ij ς (t)

(24)

where ς T (t) = col{x(t), x˙ (t), x(t − τ1 ), x(t − τ (t)), x(t − τ2 ), ω(t)}, and ij =

4 4   i=1 j=1

ηi ηj ij =

4 

ηi2 ii +

i=1

4 3  

ηi ηj ( ij + ji )

(25)

i=1 j>i

12 −1 12 T where ij = 11 ij − ij 22 ( ij ) . In other words, when the following condition is true, J < 0 holds:

ii < 0

(26)

ij + ji < 0 (i < j)

(27)

Moreover, when ω(t) = 0, the derivative V˙ (t) < 0 of Lyapunov function can be easily obtained, thus the IWMD vehicle active suspension system (15) is asymptotically stable. Under V (0) = 0 and V (∞) ≥ 0, one has  ∞  ∞ z T (θ )z(θ )d θ − γ 2 ωT (θ )ω(θ )d θ < 0 (28) 0

0

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Thus, the H∞ performance can be guaranteed. Define  = diag{X −1 , X −1 , X −1 , X −1 , I , I , I , I }, and pre- and post-multiply  −1 and its transposition to (26) and (27). Define Kj = K j X , E 1 = X −1 E1 X −1 , E 2 = X −1 E2 X −1 , F 1 = X −1 F1 X −1 , F 2 = X −1 E2 X −1 , and W = X −1 W3 X −1 . Based on −1 Schur complement and X F α X ≤ 2κα X −κα2 F α , the Eqs. (17) and (18) can be obtained. Thus, the asymptotic stability and the performance (16) of the suspension system (15) can be guaranteed.

3.2 Reachability Analysis Theorem 2. The active suspension system (15), can approach a specified sliding mode surface s(t) = 0 in limited time with the following SMC strategy as u(t) =

4 

ηj Kj x(t − τ (t)) − ϕ(t)sgn(s(t))

(29)

j=1

where ϕ(t) = μ + ε(t)x(t), and the updating rule of ε(t) can be expressed as ε˙ (t) = ρs(t)x(t)

(30)

where μ > 0 and ρ > 0 the known constants. Proof: Considering the Lyapunov function Vs (t) shown below: ⎛ ⎞−1 4 1 T ⎝ 1 2 ε (t) ηj HB2i ⎠ s(t) + Vs (t) = s (t) 2 2ρ

(31)

j=1

Combining (13) and (29), the derivative of sliding-surface function s(t) can be derived as s˙ (t) = H

4 

ηi B1i (−ϕ(t)sgn(s(t)) + h(x(t)))

(32)

i=1

In addition, the derivative of Vs (t) can be expressed as ⎛ ⎞−1 4  1 ηj HB2i ⎠ s˙ (t) + ε(t)˙ε (t) V˙ s (t) = sT (t)⎝ ρ j=1

(33)

≤ −μs(t) < 0 where s(t) = 0, which indicates the trajectories of the suspension system can all arrive at the specified sliding-mode surface s(t) = 0 in a finite time.

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4 Simulation Results Various simulations are carried out to verify the performance of the adaptive SMC for active suspensions of IWMD vehicles under cyber attacks. The parameters of suspension model are shown in Table 1. Given scalars τ1 = 0.015 s,τ2 = 0.020 s,a1 = 5, a2 = 25, a3 = 10, κ1 = 0.1, κ2 = 0.1 and μ = 0.5, ρ = 0.1, and the matrix H = [ 100 100 0 0 100 100 ] such nonsingular and HB1i = 0. And the function of cyber attacks is set as that HB2i is  h(x(t)) = − 4j=1 0.2Kj x(t). To highlight the effectiveness of the proposed method, the robust H ∞ control method is selected as a comparison, which is labeled ‘Comparison’ in subsequent results. In addition, the H ∞ performance index γmin = 17.2039 and controller gains can be obtained by solving Theorem 1. Table 1. Parameters of active suspension model. Parameter

Value

Unit

Parameter

Value

Unit

ms

340

kg

n

10

-

mu

40

kg

kt

32000

N/m

md

30

kg

ct

0

Ns/m

kl

32000

N/m

kd

41000

N/m

cs1

1350

Ns/m

cd

1000

Ns/m

cs2

1650

Ns/m

zmax

0.1

m

4.1 Bump Response The following bumpy road is selected as the disturbance input: ⎧A 2π V L ⎪ t)), 0 ≤ t ≤ ⎨ (1 − cos( 2 L V zw = ⎪ ⎩ 0, t > L V

(34)

where A = 0.05 m and l = 6 m, respectively, and v = 10 m/s. Figure 3 plots the bump response results for IWMD vehicle active suspension systems. Figure 3(a) shows that the body acceleration of the proposed control method has lower peaks than the other two cases, which means the ride comfort is better guaranteed. Figure 3(b)-(c) illustrates that the fluctuation ranges of the suspension travel and tire deflection are smaller than that of the comparison and the passive suspension, which means the handling stability and road holding ability are ensured by the proposed control method. Figure 3(d) illustrates the actuator force under two control strategies. In other words, Fig. 3 shows the adaptive SMC method has an excellent performance. Figure 4 presents the variation trends of s(t) and the adaptive parameter ε(t). It can be seen from Fig. 4(a) that the sliding variable changes with bump road interference.

Adaptive Sliding Mode Control for Active Suspensions of IWMD 0.1 Proposed Controller Comparison Passive

1

Suspension travel (m)

Body acceleration (m/s2 )

2

413

0

-1

-2 0

0.5

1

1.5 Time (s)

2

2.5

0.05

0 Proposed Controller Comparison Passive

-0.05

-0.1 0

3

(a) Body acceleration

0.5

1

1.5 Time (s)

2

2.5

3

(b) Suspension travel

0.016

300

Active force (N)

Tire deflection (m)

200 0.008

0 Proposed Controller Comparison Passive

-0.008

-0.016 0

0.5

1

1.5 Time (s)

2

2.5

100 0 -100

Proposed Controller Comparison

-200 -300 0

3

(c) Tire deflection

0.5

1

1.5 Time (s)

2

2.5

3

(d) Actuator force

Fig. 3. Bump response of IWMD vehicle suspension system 20

120

Adaptive parameter

Sliding variable

100 10

0

-10

80 60 40 20

-20 0

0.5

1

1.5 Time(s)

2

2.5

3

(a) Sliding-mode surface

0 0

0.5

1

1.5 Time(s)

2

2.5

3

(b) Adaptive parameter

Fig. 4. The trajectory of sliding mode

Figure 4(b) shows that the parameter ε(t) varies from the system states to a constant. In addition, Fig. 4 shows that the sliding motion is achieved in finite time. 4.2 Random Response The following random road excitation is considered:  z˙w (t) = 2π n0 Gq (n0 )(v0 + at)n(t) − (v0 + at)2π nc zr (t)

(35)

where v0 = 5 m/s, a = 0.5 m/s2 , n0 =0.1 m−1 and nc =0.01 m−1 , n(t) denotes a white noise and Gq (n0 ) = 64 × 10−6 m3 is selected in this simulation of random road. The dynamic responses of the IWMD vehicle suspension system under random response are plotted in Fig. 5. It is observed from Fig. 5(a) that the body acceleration is greatly reduced by the proposed controller compared to the other two cases. Figure 5(b)-(c) presents the suspension travel constraint and the tire deflection constraint are guaranteed, which also shows that the performance of the design SMC method is

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Suspension travel (m)

Body acceleration (m/s2 )

0.4

0.2

0

-0.2

Proposed Controller Comparison Passive

-0.4 0

0.5

1

1.5 Time (s)

2

2.5

3

0

-0.008

-0.016 0

(a) Body acceleration 10 -3

-2

1

1.5 Time (s)

2

2.5

1.5 Time (s)

2

2.5

3

Proposed Controller Comparison

0

0.5

1

(b) Suspension travel

2

-4 0

0.5

70

Proposed Controller Comparison Passive

Active force (N)

Tire deflection (m)

4

Proposed Controller Comparison Passive

0.008

3

35

0

-35

-70 0

(c) Tire deflection

0.5

1

1.5 Time (s)

2

2.5

3

(d) Actuator force

4

8

2

6

Adaptive parameter

Sliding variable

Fig. 5. Random response of IWMD vehicle suspension system

0

-2

-4 0

0.5

1

1.5 Time(s)

2

2.5

(a) Sliding-mode surface

3

4

2

0 0

0.5

1

1.5 Time(s)

2

2.5

3

(b) Adaptive parameter

Fig. 6. The trajectory of sliding mode under random road

more outstanding than the comparison and passive suspension. The actuator forces of the proposed control scheme and comparison are revealed in Fig. 5(d). The changing trends of the sliding surface and adaptive parameter under random road interference are illustrated in Fig. 6. From Fig. 6(a) we can see that the sliding surface s(t) is changing along with the system state, which means the sliding mode motion can be achieved in a finite time. Figure 6(b) shows that the parameter ε(t) increases along with the time, which is in line with the trend of the definition of ε(t).

5 Conclusion An adaptive SMC problem for active suspensions of IWMD electric vehicles under time delay and cyber attacks was investigated in this paper. First of all, a T-S fuzzy model has been constructed to capture the nonlinearity of the IWMD vehicle suspension dynamics, and it provided a powerful foundation for the controller design. Then, a set of sufficient conditions were developed to ensure the asymptotical stability and constraint performances of the sliding motion despite the occurrence of time delay and cyber

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attacks. Moreover, an adaptive SMC strategy was proposed to guarantee the closedloop active suspension system can achieve the desired performances guaranteed by the specific sliding surface. Finally, the simulation results illustrated the superiority of the proposed adaptive SMC scheme. In the future, the energy-saving control issue will be investigated for the suspension systems [35, 36]. Acknowledgements. This work was supported in part by the National Natural Science Foundation of China under Grant 52175127, in part by the Guangdong Basic and Applied Basic Research Foundation under Grant 2022A1515011495, Grant 2022A1515110301 and Grant 2023A1515012327, in part by the research grant of the University of Macau under Grant MYRG2022–00099-FST and Grant MYRG-GRG2023–00235-FST-UMDF, in part by the research grant of the University of Macau under Grant UMMTP-2022-PD01.

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Generative Adversary Network Based on Cross-Modal Transformer for CT to MR Images Transformation Zhenzhen Wu1 , Weijie Huang1(B) , Xingong Cheng1(B) , and Hui Wang2 1 School of Electrical Engineering, University of Jinan, Jinan 250000, Shandong, China

[email protected], [email protected] 2 High School Attached To Shandong Normal University, Jinan 250000, Shandong, China

Abstract. Acquiring Magnetic Resonance (MR) images in the current medical imaging tasks is expensive and time-consuming. We need technology to acquire multi-contrast MR images. Nowadays, studying the synthesis of MR images through deep learning algorithms to improve diagnostic efficiency is a hot topic. However, cross-modal translations are very challenging. This paper proposes an efficient and effective generative adversary network based on Cross-Modal Transformer (C-M Transformer) to address the issues of blurred synthetic images and unstable training, in order to achieve the conversion from Computed Tomography (CT) images to MR images. Firstly, the original input CT image is filtered to generate high-frequency detail images, and then the high-frequency detail images and the original images are respectively fed into the U-shaped network structure for feature extraction. After four downsamplings, the features are sent to the C-M Transformer for feature fusion. In C-M Transformer, the detail feature stream is Q, and the original feature stream is K and V. We added a Masked Attention to provide an average feature representation of two streams. The fused feature image is fed to the upsampling portion of the U-shaped structure to generate the MR image. It can be shown through the experimental results that the method outperforms mainstream algorithms in terms of Mean square error (MAE), Peak signal-to-noise ratio (PSNR), and Structural similarity (SSIM). This method generates MR images that show bone marrow signals in the vertebral body more clearly and accurately than other methods. It can clearly show the position of the lumbar vertebral plate and so on. The results of this method can be used to assist in orthopedic diagnosis after approval by the physician. Keywords: CT to MR · Transformer · Generative Adversary Network

1 Introduction In paramedical diagnostics, MR is significant because of its excellent imaging ability of soft tissues and its non-invasive nature, but MR scans also have unavoidable drawbacks. First, MR scans take longer than CT. Secondly, accompanied by excessive noise, MR scan is operated in an isolated and narrow space for a long time, and patients need to wear © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 418–430, 2024. https://doi.org/10.1007/978-981-97-0554-2_32

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special earmuffs to cooperate with the examination, which makes it difficult for patients with claustrophobia and children to complete the examination. In addition, due to the strong magnetic field generated by the MR machine accompanied by the magnetic field effect, the placement of metal monitoring and resuscitation equipment in the machine room is prohibited. MR scans cannot be performed on emergency and critically ill patients, as well as patients with metal implants such as standard pacemakers, metal coronary stents, and artificial joints in their bodies. Because MRI images have good softtissue contrast and provide images in planes perpendicular to each other, information about disease in several spinal regions can be assessed.MR images provide excellent visualization of the intervertebral discs, dural sacs, epidural space, nerve components, paraspinal soft tissues, and the spinal cord. Sometimes MRI images are essential but may not be available due to a number of limitations. If CT can be converted to MR through medical imaging modality conversion, patients have the opportunity to receive CT-based MR diagnostic information at the same time as the CT image is taken. Transformer is a deep neural network based on a self-attention mechanism and parallelization of processing data. Deep learning-based modal transformation methods for medical images are mainly with the help of Generative Adversarial Networks (GAN) [1, 2]. GAN comprises a generator and a discriminator whose parameters are alternately updated for better performance in unique adversarial training [3]. However, the generation threshold for obtaining both modal images at the same time is high, and the training of the GAN is highly unstable and difficult to optimize, which may also lead to a loss of feature resolution and fine details. To address this challenge, based on the inspiration of Transformer [4–6], in this paper, we design an adversarial neural network based on C-M Transformer to realize CT to MR image transformation, which is used to learn to communicate accurately across modalities.

2 Related Works Chartsias et al. used the CycleGAN model and unpaired cardiac CT data to generate cardiac MR images [7]. However, since the heart remains beating during the collection of cardiac images, paired CT and MR data from the same patient cannot be obtained. The author only used an indirect evaluation method, which uses segmentation methods to verify that the generated data can be used to expand the data volume and improve segmentation accuracy. Jin et al. used paired and unpaired brain CT and MR image data to generate corresponding MR images to enhance tumor target identification for CT image-based treatment planning [8]. Their deep learning network is based on CycleGAN improvement, an unsupervised learning neural network. However, this article does not compare the difference in effectiveness between traditional supervised learning methods and unsupervised learning methods in MR images generation tasks. Yang et al. used prostate CBCT images to generate corresponding MR images to improve the soft tissue contrast of CBCT-guided radiotherapy-based CBCT images, thereby improving adaptive radiotherapy based on CBCT guidance [9]. Liming Xu et al. proposed BPGAN, a bidirectional prediction model between MR and CT images, which learns two nonlinear mappings by projecting the same pathological features from one domain to the other via a cyclic consistency strategy [10]. Jiayuan Wang et al. proposed a bi-directional learning model, DC-cycleGAN, to realize the transformation of CT images to MRI images

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by introducing a double contrast loss in the discriminator, which indirectly establishes constraints between the actual source image and the synthesized image [11]. Unlike the above methods, the C-M Transformer uses a two-stream network with a high-frequency filtered edge information branch, which enhances the generator encoder’s grasp of image edge details. Added a Masked Attention to provide an average feature representation of two streams. We have demonstrated the effectiveness of the C-M Transformer in improving the quality of generated MR.

3 Methodology This paper proposes a generative adversary network based on C-M Transformer for CT to MR images transformation. The generative adversarial network is a U-net structured network [12, 13]. The input raw CT image is filtered with high frequency to form a detailed image, and the high-frequency detail image and the raw image are fed into the feature extraction network to extract features and feature fusion via the cross-modal transformer. C-M Transformer architecture follows the typical Transformer architecture, and the difference is that Masked Attention is used in the encoding layer instead of Multi-Head Attention. Since two streams are joined, one for edge information and one for raw image features, Positional Encoding is added to XA and XB for embedding spatial correlation to form a kind of interaction between the two features. After feature fusion, it is fed to the up-sampled portion of the U-net structure, which ultimately generates the MR image. An overview of the C-M Transformer framework is shown in Fig. 1. In this section, a generator consisting of a decoder and an encoder is described in detail, and the improved Transformer structure (Sect. 3.1), discriminator (Sect. 3.2), and loss function (Sect. 3.3).

Fig. 1. Overview of framework for C-M Transformer.

3.1 Generator In this paper, we propose a new generator consisting of an encoder and a decoder with the network structure shown in Fig. 2.

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Feature Extraction. In the encoder section, the original input CT image is first filtered to generate its high-frequency detail image. The high-frequency detail image and the original image are respectively fed into the feature extraction network to extract the edge feature map XA and original feature map XB . C-M Transformer. After four downsamplings, the C-M Transformer is added to the lowest output of the encoder. Figure 3 shows the network structure of the C-M Transformer. We add positional encoding to XA and XB for embedding spatial correlations. Q represent the detailed feature maps filtered out by high-frequency filtering, while K and V represent the original feature maps. Stacking position with Q, K, V and pass through Masked Attention, which provides an average feature representation of two streams. Figure 4 shows the internal structure of Masked Attention. A is the initial correspondence between Q and K. After learning from Q and K, the masked dependency map became: Amask = ReLU (A)

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After normalizing A, the result is: A˜ mask = softmax(α · Amask )

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Ultimately, it can be concluded Masked Attention: Masked Attention = Xcor = A˜ mask V

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Fig. 3. The network structure of C-M Transformer.

Decoder. In the decoder section, the synthetic MR with the same size as the Groundtruth MR is output through the Convolutional Block Attention Module (CBAM) [30], Convolutional Layer, BatchNorm Layer, ReLU Activation Function, ConTransfer Layer, Tanh Activation Function.

3.2 Discriminator The discriminator is a tool that aids in the training of the generator, and the discriminator uses PatchGAN [15], the structure of which is shown in Fig. 5. This discriminator has two sets of inputs. The first set of inputs are spliced images of CT and synthetic MR, while the second set is spliced images of CT and corresponding Ground-truth MR. The PatchGAN discriminator measures whether the synthesized MR image is real or not for each of the 70 × 70 matrices in the image.

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Fig. 4. The internal structure of Masked Attention.

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3.3 Loss Function Structural Consistency Loss. We want the structure of the generated synthetic MR image S to be consistent with the structure of the input CT image LA . But the organization of CT images may produce some changes in the process of cross-modal translation. To ameliorate this problem, this paper proposes Structural Consistency Loss (Lscl ), using spatial correlation maps [16] to enhance the structural coherence between the original CT image and the CT high-frequency detail image. First, the element K is randomly selected from the feature tensor QL and the spatial correlation map PL is computed separately. Perform the same operation on CT image XB , CT high-frequency information image XA , and synthetic MR image S for each identical K to obtain PXA ,PXB and PS . Then, calculate the cosine distance between PXA and PS , PXB and PS . Finally, take their average value of:    LsclA = 1 − cos PX , PS  (4) A

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LsclA + LsclB (6) Lscl = 2  1 2  Which, PL = PL , PL , . . . , PLK , ∈ RK×H ×W can be executed at the feature tensor level, A spatial correlation map P m maps the correlation between an element m and other elements n in a feature tensor of size H × W = 256 × 256. The correlation map P m maps the correlation between an element m and other elements n in the feature tensor. The mathematical expression for the mapping of LA onto a pixel m is given by: T    PLmA = QLKA QLnA (7) Which, QLKA ∈ RC×1 and QLnA ∈ RC×H ×W is the feature tensor of LA on channel C, which is denoted PLmA ∈ R1×H ×W . Adversarial Loss. Generally speaking, both generators and discriminators can be simultaneously bounded by Adversarial Loss (Lal ), This is the key to realizing adversarial training. The L2 loss function [18] was modified to make it more robust. During training, the output probability of discriminator is close to 0, and the Loss function of discriminator is:     ELA (D(G(LA )))2 + ELB (1 − D(LB ))2 (8) Ldisc = 2 Which, LA is the input-CT, LB is the ground-truth MR. meanwhile, G(LA ) also needs to make the generator output probability close to 0, which can lead to an adversarial loss: 

 (9) Lal = E 1 − (D(G(LA )))2 Total Loss.

LG = λscl Lscl + λal Lal

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Which λscl , λal are the weight of the above three losses, these are 21 , 21 . The loss function of the discriminator is:LD =Ldisc .

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4 Experiment and Analysis 4.1 Paired Dataset and Experimental Environment CT and MR data collected from spine patients with actual clinical visits were registered by 3D Slicer software [19] under the guidance of specialized physicians. The aligned CT slices all have their corresponding MR slices, while the network is trained by supervised learning. The dataset is shown in Fig. 6. This experiment is trained in Python framework. Input CT images were populated to 256 × 256 sizes after training, and the size of the synthesized MR is restored to the original size of the input CT. The experiment was conducted for a total of 200 epochs.

Fig. 6. The partial paired data of lumbar spine median sagittal position.

4.2 Result This experiment uses 740 pairs of CT and MR data to train the C-M Transformer, as well as the CycleGAN [20] and Pix2Pix [21] modal transformation algorithms. Qualitative and quantitative comparisons between C-M Transformer and CycleGAN and Pix2Pix modal shift algorithms are reflected in the Mean Absolute Error (MAE), Peak Signalto-Noise Ratio (PSNR), and Structural Similarity (SSIM). The three evaluation metrics of the algorithm are shown in Table 1. In general terms, C-M Transformer achieved the best results in all three evaluations metrics. Of these, the evaluation results of MAE

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also reflect that C-M Transformer cross-modal translation has the highest accuracy. In the evaluation of PSNR, the experimental results of the C-M Transformer are more significant than those of CycleGAN and Pix2Pix, indicating that the Synthetic MR generated by C-M Transformer is closer to Ground-truth MR. The results of the SSIM evaluation demonstrate the effective maintenance of structural consistency by the C-M Transformer. Table 1. Comparison of different methods in three evaluations indicators.

CycleGAN

MAE↓

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28.24 ± 15.07

27.79 ± 1.98

0.57 ± 0.13

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32.53 ± 2.07

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The results of CycleGAN, Pix2Pix, and C-M Transformer cross-modal transformations are compared, as shown in Fig. 7. The images show, from top to bottom, the Input CT image, the experimental results of CycleGAN, the experimental results of Pix2Pix and the experimental results of C-M Transformer, and the Ground-truth MR image. The sagittal image of the lumbar vertebral plate is in the circle identified in Example 1. The CycleGAN experimental results did not show the lumbar vertebral plate configuration, and the C-M Transformer experimental results showed a more coherent and clearer shape of the lumbar vertebral plate compared to the Pix2Pix experimental results. The circle identified in Example 2 is the bone marrow portion of the lumbar vertebral body, and it can be clearly seen that the results of the C-M Transformer experiment have a more transparent and more accurate bone marrow signal than the results of the Pix2Pix experiment. A common cauda equina structure in the lumbar spine, easily compressed by a herniated disc, is shown in the circle marked by Example 3. The C-M Transformer experimental images are better handled in terms of detail and closer to Ground-truth MR than the Pix2Pix experimental images. The lumbar intervertebral disc portion is in the circle marked by Example 4. As can be seen from the C-M Transformer experiment results, the herniated intervertebral disc has a clear and perfect morphology, which is closer to the Ground-truth MR image. The above examples can fully prove that C-M Transformer has a relatively significant advantage in image detail processing.

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Fig. 7. Comparison between the results of CycleGAN、Pix2Pix、C-M Transformer.

The loss functions for the three sets of experiments are shown in Fig. 8. The loss function curve of the Pix2Pix model is shown in Fig. 8(b), the overall value decreases significantly, but there are some local jumps. The loss function curve of the C-M Transformer model is shown in Fig. 8(c), the overall value shows a smooth decreasing trend, and the jumps are reduced compared with the Pix2Pix model. From the above analysis, it can be concluded that the proposed C-M Transformer can obtain Synthetic MR that is closer to Ground-truth MR.

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Fig. 8. The loss functions for the three sets of experiments.

5 Conclusion The results in this paper show that a generative adversary network based on CrossModal Transformer can better accomplish the transformation of the spine from CT to MR. A two-stream structure is proposed, and a cross-modal transformer is designed for feature fusion at the end of encoding. Both global and high-frequency features can be obtained, which significantly improves the realism and reliability of cross-modal transitions. Experiments have shown that MR images converted by C-M Transformer can show clearer and more accurate bone marrow signals in the vertebral body and can clearly show the position of the lumbar vertebral plate and other locations. It can play a vital reference role in the judgment of surgical conditions and the selection of surgical plans. For doctors, it can solve the problem that the time and individual differences in clinical diagnosis limit the timely acquisition of MR Medical imaging information that can be used for diagnosis. Providing diagnostic references to reduce the risk of missing the optimal treatment time has particular clinical auxiliary diagnostic value. It can reduce financial and time costs for patients, is friendly to patients with metal implants in their bodies, those with claustrophobia, and younger patients, and can improve the experience for those types of patients.

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Acknowledgments. This work was supported in part by the Youth Foundations of Shandong Province under Grant No. ZR2021QF100, the National Natural Science Foundation of China under Grant No.62273163, the Outstanding Youth Foundation of Shandong Province under Grant No.ZR2023YQ056, the Key R&D Project of Shandong Province under Grant No.2022CXGC010503.

References 1. Yi, X., Walia, E., Babyn, P.: Generative adversarial network in medical imaging: a review. Med. Image Anal. 58, 101552 (2019) 2. Choudhary, A., et al.: Advancing medical imaging informatics by deep learning-based domain adaptation. Yearbook Med. Inform. 29(01), 129–138 (2020) 3. Lim, S., Shin, M., Paik, J.: Point cloud generation using deep adversarial local features for augmented and mixed reality contents. IEEE Trans. Consum. Electron. 68(1), 69–76 (2022) 4. Dosovitskiy, A., et al.: Transformers for image recognition at scale. arXiv preprint arXiv: 2010.11929 (2020) 5. Heo, B., et al.: Rethinking spatial dimensions of vision transformers. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (2021) 6. Vaswani, A., et al.: Attention is all you need. In: Advances in Neural Information Processing Systems, vol. 30 (2017) 7. Chartsias, A., Joyce, T., Dharmakumar, R., Tsaftaris, S.A.: Adversarial image synthesis for unpaired multi-modal cardiac data. In: Tsaftaris, S.A., Gooya, A., Frangi, A.F., Prince, J.L. (eds.) SASHIMI 2017. LNCS, vol. 10557, pp. 3–13. Springer, Cham (2017). https://doi.org/ 10.1007/978-3-319-68127-6_1 8. Jin, C.-B., et al.: Deep CT to MR synthesis using paired and unpaired data. Sensors 19(10), 2361 (2019). https://doi.org/10.3390/s19102361 9. Lei, Y., et al.: Male pelvic multi-organ segmentation aided by CBCT-based synthetic MRI. Phys. Med. Biol. 65(3), 035013 (2020) 10. Xu, L., et al.: BPGAN: Bidirectional CT-to-MRI prediction using multi-generative multiadversarial nets with spectral normalization and localization. Neural Netw. 128, 82–96 (2020) 11. Wang, J., Wu, Q.M.J., Pourpanah, F.: DC-cycleGAN: bidirectional CT-to-MR synthesis from unpaired data. Comput. Med. Imaging Graph. 102249 (2023) 12. Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9351, pp. 234–241. Springer, Cham (2015). https://doi.org/10.1007/978-3-31924574-4_28 13. Qin, X., et al.: U2-Net: going deeper with nested U-structure for salient object detection. Pattern Recogn. 106, 107404 (2020) 14. Xu, R., et al.: Face transfer with generative adversarial network. arXiv preprint arXiv:1710. 06090 (2017) 15. Hou, X., et al.: Deep feature consistent variational autoencoder. In: 2017 IEEE Winter Conference on Applications of Computer Vision (WACV). IEEE (2017) 16. Gao, X., Fang, Y.: A note on the generalized degrees of freedom under the L1 loss function. J. Statist. Plann. Inference 141(2), 677–686 (2011) 17. Mao, X., et al.: Least squares generative adversarial networks. In: Proceedings of the IEEE International Conference on Computer Vision (2017) 18. Fedorov, A., et al.: 3D Slicer as an image computing platform for the quantitative imaging network. Magn. Reson. Imaging 30(9), 1323–1341 (2012)

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A Novel Robust Finite-Time Control for Active Suspension Systems with Naturally Bounded Inputs Zengcheng Zhou1,2 , Xingjian Jing2(B) , and Menghua Zhang3 1 Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong,

People’s Republic of China 2 Department of Mechanical Engineering, City University of Hong Kong, Hong Kong,

People’s Republic of China [email protected] 3 School of Electrical Engineering, University of Jinan, Jinan 250022, People’s Republic of China

Abstract. This paper focuses on a novel robust finite-time control for active suspension systems with external disturbance where the control inputs are naturally bounded by a prior known range to avoid input saturation. To achieve finite-time stability, a novel nonsingular terminal sliding mode variable with an integral term is designed. More importantly, the control inputs are naturally bounded all the time due to the characteristics of hyperbolic tangent functions such that extra saturation compensation methods can be avoided. A disturbance compensation technique is deliberately designed in the proposed control to enhance the robustness of the system while the bounded property can be ensured simultaneously. The whole control structure is relatively simple yet effective compared with existing control techniques which is much easier to implement in practical active suspension systems. Additionally, the overall stability of the closed-loop system is verified by the Lyapunov theorem. Various simulation results are provided to demonstrate the robustness and effectiveness of the proposed control design. Keywords: Active Suspension Systems · Robustness · Finite-Time Control · Bounded Inputs

1 Introduction Vehicle suspension systems, which play an important role in vehicle ride comfort, driving safety, and maneuverability, arouse great attention in recent years [1–3]. Besides, vehicle suspension systems are used to support the weight of the vehicle and passengers, deal with the vibration between the vehicle body and wheels, and maintain contact between tires and the ground [4]. Normally, vehicle suspension systems can be divided into three categories including passive suspensions, semi-active suspensions, and active suspensions [5]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 431–443, 2024. https://doi.org/10.1007/978-981-97-0554-2_33

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The passive suspension systems consist of dampers and springs whose coefficients are fixed during the whole time, which usually leads to the unsatisfactory anti-vibration performance of ride comfort and road handling abilities [6]. To address the disadvantages of passive suspensions, variable dampers are considered a great improvement in semiactive suspension systems by adjusting the rate of energy dissipation [7]. Furthermore, compared with passive and semi-active suspensions, active suspension systems (ASSs) are able to achieve a better suspension response due to the extra actuator assembled between the vehicle body and the wheel axle which can effectively add and dissipate energy from the system [8]. Hence, various advanced control techniques are adopted for ASSs including the H-infinity control [9], adaptive control [10], back-stepping control [11], etc. Specifically, the sampled-data method was integrated with the robust H-infinity control for ASSs where the whole controller was converted to a convex optimization problem [9]. An adaptive trajectory tracking control was designed for ASSs while the actuators suffer from unknown dead zones and hysteresis nonlinearities [10]. Furthermore, a back-stepping tracking control was constructed for uncertain ASSs while the safety constraints can be ensured simultaneously [11]. Despite the satisfactory performance that can be obtained by these controllers, more efforts should be devoted to the control design of ASSs to improve the anti-vibration ability and ride comfort. Considering the physical limitations of the actuators in ASSs, only limited forces can be provided by the actuators which leads to the input saturation problem. Excessive control signals may cause damage to the actuators or the instability of the systems. Therefore, the control input signals are constrained within the predefined upper and lower bounds. However, the resulting inputs are non-smooth when saturation occurs, and extra saturation compensation techniques should be integrated to enhance the system stability. Some naturally bounded functions, such as the inverse tangent function arctan, signum function sign and the hyperbolic tangent function tanh, are widely used in recent controllers to deal with the input saturation problems. For instance, a robust proportionalderivative sliding mode control was proposed for ASSs where the control signals were bounded utilizing the arctan and sign functions [12]. An adaptive neural network control was developed for uncertain robots where the control inputs are asymmetrically bounded using the tanh functions [13]. Besides the results above, the robust controllers for ASSs with bounded inputs are rarely considered and need more effort. One should also note that the aforementioned controllers with naturally bounded inputs can only obtain asymptotic stability with a slow convergence rate. The finite-time stability of the closed-loop system can provide a higher convergence speed while the robustness and disturbance rejection ability can be improved. Some efforts have been made in the control theory community to achieve finite-time stability with naturally bounded inputs for various mechanical systems. An adaptive finite-time control was designed for Euler-Lagrange systems with prior known bounded inputs generated by tanh and sign functions [14]. Tian et al. proposed a continuous finite-time bounded control for double integrator systems using the extension of super-twisting algorithms [15]. Furthermore, the finite-time trajectory tracking control was proposed for rigid robots with bounded inputs and terminal sliding mode techniques [16]. A novel cluster formation control was designed for networked marine surface vehicles with a predefined time estimator and bounded input constraints [17]. However, most of these controllers

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are complex and difficult to be implemented in practical applications. Therefore, it is meaningful to propose an easy yet effective finite-time control for ASSs with prior known bounded inputs. To effectively address the issues above, a novel robust finite-time control is proposed for ASSs with bounded control inputs. The characteristics of hyperbolic tangent functions and the adaptive technique with bounded outputs are investigated such that the whole control inputs are constrained within the prior known bounds. Furthermore, the newly designed nonsingular integral terminal sliding mode variable is constructed for ASSs to achieve finite-time stability while the singular phenomenon can be effectively avoided. This paper can provide a simple yet effective controller design for practical ASS applications. The rest of this paper is organized as follows. In Sect. 2, the problem formulation including the system description and the control objects is stated. The main results of the robust finite-time control design with bounded inputs are presented in Sect. 3. After that, some comparative simulations are provided in Sect. 4 to illustrate the effectiveness and robustness of the closed-loop system. Finally, Sect. 5 concludes the main results of this paper.

2 Problem Formulation 2.1 System Description Without loss of generality, a quarter-vehicle active suspension system is considered in this paper shown in Fig. 1 borrowed from [18]. The main components of the system are the masses, springs, dampers, and an active actuator. The dynamic equations of the ASS can be written as [19, 20] ˙ s, D ˙ u) + F +  ¨ s = −Rs (Ds , Du ) − Rd (D ms D ˙ s, D ˙ u ) − Rt (Du , Dr ) − Rb (D ˙ u, D ˙ r) − F ¨ u = Rs (Ds , Du ) + Rd (D mu D

(1)

with Rs (Ds , Du ) = cs1 (Ds − Du ) + cs2 (Ds − Du )3 ˙ s, D ˙ u ) = cd 1 (D ˙s −D ˙ u ) + cd 2 (D ˙s −D ˙ u )2 Rd (D Rt (Du , Dr ) = ct (Du − Dr ) ˙ u, D ˙ r ) = cb (D ˙u −D ˙ r) Rb (D

(2)

where the definitions of system parameters and variables for the ASS are provided in Table 1.  T The following state variables are defined x = x1 x2 x3 x4 where x1 = Ds , x2 = ˙ s denote the vertical displacement and velocity of the sprung mass, respectively; x3 = D ˙ u represent the vertical displacement and velocity of the unsprung mass, Du , x4 = D respectively. Then the system dynamic equations of the ASSs can be rewritten as follows. ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎨ m x˙ = P(x) + F +  s 2 (3) ⎪ x˙ 3 = x4 ⎪ ⎪ ⎩ mu x˙ 3 = Q(x) − F

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with P(x) = −cs1 (x1 − x3 ) − cs2 (x1 − x3 )3 − cd 1 (x2 − x4 ) − cd 2 (x2 − x4 )2

(4)

˙ r ). Q(x) = ct (x3 − Dr ) + cb (x4 − D

(5)

In practice, the system parameters are difficult to obtain due to system nonlinearities and external disturbance. Besides, an unstable environment may affect the accuracy of the values of parameters, which should be carefully considered. Therefore, control robustness is required for the uncertain ASSs to achieve satisfactory anti-vibration performance. Table 1. The definitions of system parameters and variables for the ASS Parameter/variable

Definition

ms

Sprung mass

mu

Unsprung mass

Ds

Vertical displacement of the sprung mass

Du

Vertical displacement of the unsprung mass

Dr

Road input profile

cs1 , cs2

Stiffness coefficients of the nonlinear spring

cd 1 , cd 2

Damping coefficients of the damper

ct

Stiffness coefficient of the tire

cb

Damping coefficient of the tire

F

Control signal



Unknown external disturbance

Some useful assumptions are made for the ASSs to facilitate the controller design. ˙ are bounded. Assumption 1: The unknown disturbance  and its first-time derivative  ˙ r are bounded Assumption 2: The input profile Dr and its first-time derivative D  road    ˙ by |Dr | ≤ δr1 and Dr ≤ δr2 with δr1 , δr2 > 0. Remark 1: Assumptions 1 and 2 are common assumptions in the control design for nonlinear ASSs shown in [21, 22] which are reasonable under normal working conditions.

2.2 Control Objects In this paper, the main control objects are to design a robust finite-time control with bounded control inputs such that the vibration of the vehicle body can be effectively

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Fig. 1. Quarter-vehicle ASS model

suppressed, and the tire load Rdyn = Rt + Rb and suspension space Dp = Ds − Du are bounded and remain within the permitted ranges for driving safety given by   Rdyn  ≤ (ms + mu )g = Rsat (6)   Dp  ≤ Dmax

(7)

where g is the gravitational constant, Rsat means the static tire load, Dmax denotes the maximum suspension space.

3 Main Results In this section, a robust finite-time control design with naturally bounded inputs is proposed for the ASSs. In order to achieve finite-time stability of the closed-loop system, a novel nonsingular terminal sliding mode variable with the integral term is described as t μ1 [x1 ]p + μ2 [x2 ]q d τ (8) s = x2 + 0

where 0 < p < 1, q = 2p/(1 + p), μ1 , μ2 > 0 are positive design constants, [x]∗ denotes the abbreviation of |x|∗ sgn(x). An auxiliary velocity signal is defined as t ϑ =− μ1 [x1 ]p + μ2 [x2 ]q d τ − λ1 ζ (9) 0

where λ1 > 0 is a design constant, and ζ is an auxiliary state variable generated by ζ˙ = λ2 s + tanh(s), ζ (0) = 0.

(10)

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Next, a new state variable can be constructed as χ = x2 − ϑ.

(11)

Then the state variable becomes χ =x2 +

t

μ1 [x1 ]p + μ2 [x2 ]q d τ + λ1 ζ

0

=s + λ1 ζ

(12)

Based on the previous definitions, the robust finite-time control is given by

F = −λ3 tanh( χ

1

)+Ω

(13)

where λ3 > 0 is a design parameter, is the adaptive variable compensating for the external disturbance denoted by

Ω = −λ4 tanh( χ

2

with λ4 , λ5 are positive design constants, rate.

) − λ5 Ω , Ω(0) ≤ λ4 λ5 1

and

2

(14)

> 0 are used to adjust the convergence

Theorem 1. For the nonlinear suspension system (1), the proposed controller (13) with the nonsingular integral terminal sliding mode surface (8) and the adaptive variable (14) can guarantee that the signals are bounded all the time, Ds can converge to zero in a finite time, the control input of the closed-loop system can be naturally bounded by

|F| ≤ λ3 + λ4 λ5 . (15) and the tire load Rdyn = Rt + Rb and suspension space Dp can keep within the following ranges   Rdyn  ≤ (ms + mu )g = Rsat (16)   Dp  ≤ Dmax

(17)

Proof: The detailed proof of Theorem 1 is omitted which is available upon request.

4 Simulation Results and Analysis In this section, some comparative simulation results on a quarter-vehicle suspension system are presented to demonstrate the effectiveness and robustness of the proposed control. The values of the system parameters for the ASS are given in Table 2.

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Three different suspension systems are evaluated in this paper. • Passive: Passive suspension without force inputs; • SPDSM: Active suspension system using the saturated PD sliding mode control [12]; • Proposed: Active suspension system utilizing the proposed robust finite-time control with bounded inputs. The control gains for the SPDSM control are tuned as λ = 40, kp = 300, kd = 300 and ks = 10. Besides, the parameter values of the proposed control are designed as λ1 = λ2 = 10, λ3 = 50, λ4 = 6, λ5 = 2, μ1 = 10, μ2 = 1, p = 0.9 and 1 = 2 = 0.1. The external disturbance for three suspensions is selected as  = 0.2 sin(0.1t). In this paper, two different cases are considered for the suspension systems including the sinusoidal and random road inputs presented in Fig. 2. In Case 1, the sinusoidal road input is designed as Dr = 0.005 sin(6π t). In Case 2, a random road profile is provided. The simulation results are depicted in the following Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Table 2. Parameters of the active suspension system Symbols

Values

Symbols

Values

ms cs1

2.45 kg

mu

1 kg

900 N/m

cs2

10 N/m3

cd

8 Ns/m

ct

1250 N/m

cb

5 Ns/m

Fig. 2. Road inputs in two Cases

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Fig. 3. Vertical displacement Ds in Case 1

¨ s in Case 1 Fig. 4. Vehicle body acceleration D

Fig. 5. Control force input F in Case 1

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Fig. 6. Tire deflection Du − Dr in Case 1

Fig. 7. Suspension space Dp in Case 1

Fig. 8. Vertical displacement Ds in Case 2

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¨ s in Case 2 Fig. 9. Vehicle body acceleration D

Fig. 10. Control force F in Case 2

Fig. 11. Tire deflection Du − Dr in Case 2

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Fig. 12. Suspension space Dp in Case 2

The vertical displacement results are presented in Fig. 3 and Fig. 8, which present the satisfactory control performance of the SPDSM control and the proposed control compared with the passive suspension. However, the proposed control method shows superior control performance with smaller vertical displacement of the sprung mass, thus it can provide better anti-vibration results compared with the existing SPDSM control. Furthermore, these figures also indicate that the proposed control can achieve finite-time stability, that is to say, the convergence rate is higher than the SPDSM method. The vehicle body acceleration results are recorded in Fig. 4 and Fig. 9, which are important evaluation indicators of ride comfort. As is obviously shown, the amplitudes of the vehicle body acceleration of the proposed control under two cases are smaller than those of the passive suspension system and the SPDSM method. Thus, the higher ability of the proposed control in improving ride comfort can be demonstrated effectively. Furthermore, the control forces given in Fig. 5 and Fig. 10 depict that the control inputs in both active suspensions under two different cases are always bounded while the proposed control has a smaller prior known upper bound of the control force. For system stability and driving safety, the tire load and suspension space should be restricted in permitted ranges. According to the simulation results shown in Fig. 6 and Fig. 11, the tire deflections Ds − Du for three different suspensions are always bounded and kept in a reasonable set. Besides, as observed from Fig. 7 and Fig. 12, the suspension space values behave within the allowable ranges while the maximum suspension space is designed to be 3.8cm according to [23]. All these results demonstrate the robustness and effectiveness of the proposed control method.

5 Conclusions In this paper, a novel robust finite-time control with prior known bounded inputs is proposed for ASSs in the presence of unknown external disturbance. First, a nonsingular integral terminal sliding mode surface is defined in the control to achieve finite-time stability. Then the control inputs are constructed based on the hyperbolic tangent functions and the newly designed adaptive technique which are constrained in the prior known

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range. One should note that the input saturation problem can be avoided without any extra saturation compensation designs. The robustness and effectiveness of the proposed control are verified by various simulation results. Acknowledgments. This work is supported by the Hong Kong RGC General Research Fund (11202323), the startup fund for Laboratory of Nonlinear Dynamics, Vibration, and Control, City University of Hong Kong (9380140), the National Natural Science Foundation of China under Grant No. 62273163, the Outstanding Youth Foundation of Shandong Province Under Grant No. ZR2023YQ056, the Key R&D Project of Shandong Province under Grant No. 2022CXGC010503.

References 1. Pan, H., Sun, W., Gao, H., Jing, X.: Disturbance observer-based adaptive tracking control with actuator saturation and its application. IEEE Trans. Autom. Sci. Eng. 13(2), 868–875 (2016) 2. Liu, B., Saif, M., Fan, H.: Adaptive fault tolerant control of a half-car active suspension systems subject to random actuator failures. IEEE/ASME Trans. Mechatron. 21(6), 2847– 2857 (2016) 3. Guo, X., Zhang, J., Sun, W.: Robust saturated fault-tolerant control for active suspension system via partial measurement information. Mech. Syst. Signal Process. 191, 110116 (2023) 4. Huang, Y., Na, J., Wu, X., Gao, G.: Approximation-free control for vehicle active suspensions with hydraulic actuator. IEEE Trans. Ind. Electron. 65(9), 7258–7267 (2018) 5. Hua, C., Chen, J., Li, Y., Li, L.: Adaptive prescribed performance control of half-car active suspension system with unknown dead-zone input. Mech. Syst. Signal Process. 111, 135–148 (2018) 6. Sun, W., Gao, H., Kaynak, O.: Adaptive backstepping control for active suspension systems with hard constraints. IEEE/ASME Trans. Mechatron. 18(3), 1072–1079 (2013) 7. Yin, X., Zhang, L., Zhu, Y., Wang, C., Li, Z.: Robust control of networked systems with variable communication capabilities and application to a semi-active suspension system. IEEE/ASME Trans. Mechatron. 21(4), 2097–2107 (2016) 8. Zhao, F., Ge, S.S., Tu, F., Qin, Y., Dong, M.: Adaptive neural network control for active suspension system with actuator saturation. IET Control Theory Appl. 10(14), 1696–1705 (2016) 9. Gao, H., Sun, W., Shi, P.: Robust sampled-data $H_{\infty}$ control for vehicle active suspension systems. IEEE Trans. Control Syst. Technol. 18(1), 238–245 (2010) 10. Pan, H., Sun, W., Jing, X., Gao, H., Yao, J.: Adaptive tracking control for active suspension systems with non-ideal actuators. J. Sound Vib. 399, 2–20 (2017) 11. Pang, H., Zhang, X., Xu, Z.: Adaptive backstepping-based tracking control design for nonlinear active suspension system with parameter uncertainties and safety constraints. ISA Trans. 88, 23–36 (2019) 12. Zhang, M., Jing, X., Huang, W., Li, P.: Saturated PD-SMC method for suspension systems by exploiting beneficial nonlinearities for improved vibration reduction and energy-saving performance. Mech. Syst. Signal Process. 179, 109376 (2022) 13. Kong, L., He, W., Dong, Y., Cheng, L., Yang, C., Li, Z.: Asymmetric bounded neural control for an uncertain robot by state feedback and output feedback. IEEE Trans. Syst. Man Cybern: Syst. 51(3), 1735–1746 (2021) 14. Huang, B., Zhang, S., He, Y., Wang, B., Deng, Z.: Finite-time anti-saturation control for Euler-Lagrange systems with actuator failures. ISA Trans. 124, 468–477 (2022)

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15. Tian, B., Li, Z., Zong, Q.: A continuous multivariable finite-time control scheme for double integrator systems with bounded control input. IEEE Trans. Autom. Control 67(11), 6068– 6073 (2022) 16. Pliego-Jiménez, J., Arteaga-Pérez, M.A., López-Rodríguez, M.: Finite-time control for rigid robots with bounded input torques. Control. Eng. Pract. 102, 104556 (2020) 17. Liang, C.D., Ge, M.F., Liu, Z.W., Wang, L., Park, J.H.: Model-free cluster formation control of NMSVs with bounded inputs: a predefined-time estimator-based approach. IEEE Trans. Intell. Veh.,1–11 (2022) 18. Pan, H., Jing, X., Sun, W.: Robust finite-time tracking control for nonlinear suspension systems via disturbance compensation. Mech. Syst. Signal Process. 88, 49–61 (2017) 19. Pan, H., Sun, W., Gao, H., Yu, J.: Finite-time stabilization for vehicle active suspension systems with hard constraints. IEEE Trans. Intell. Transp. Syst. 16(5), 2663–2672 (2015) 20. Na, J., Huang, Y., Wu, X., Su, S.F., Li, G.: Adaptive finite-time fuzzy control of nonlinear active suspension systems with input delay. IEEE Trans Cybern. 50(6), 2639–2650 (2020) 21. Liu, Y.J., Zhang, Y.Q., Liu, L., Tong, S., Chen, C.L.P.: Adaptive finite-time control for halfvehicle active suspension systems with uncertain dynamics. IEEE/ASME Trans. Mechatron. 26(1), 168–178 (2021) 22. Ho, C.M., Tran, D.T., Ahn, K.K.: Adaptive sliding mode control based nonlinear disturbance observer for active suspension with pneumatic spring. J. Sound Vib.Vib. 509, 116241 (2021) 23. Pan, H., Sun, W.: Nonlinear output feedback finite-time control for vehicle active suspension systems. IEEE Trans. Ind. Inf. 15(4), 2073–2082 (2019)

Quasi-Zero Stiffness Magnetic Vibration Absorber Xuan-Chen Liu1 and Hu Ding1,2(B) 1 Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied

Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China [email protected] 2 Shaoxing Institute of Technology, Shanghai University, Shaoxing 312074, China

Abstract. A magnetic-enhanced nonlinear energy sink with quasi-zero stiffness (QZS-ME-NES) is proposed, which mainly focuses on the influence of linear stiffness on vibration reduction. Quasi-zero stiffness is generally used in the field of vibration isolation. In terms of vibration absorption, it is inevitable that there is linear stiffness in nonlinear stiffness. Based on the magnetic force expression of permanent magnet, the dynamic equations of the linear oscillator (LO) equipped with the novel NES are established. Its dynamic characteristics are analyzed by numerical and approximate analytical solutions. The transient response and steady-state response of the LO equipped with the NES are studied. Moreover, the vibration attenuation performance of the novel NES is compared with that of the triplemagnet magnetic suspension dynamic vibration absorber (TMSDVA). The results show that the novel NES not only has the ability of adaptive broadband vibration suppression, but also significantly enhances the vibration attenuation of LO. In summary, the proposed NES is a reliable and effective vibration attenuation strategy. Keywords: Nonlinear energy sink · Magnetic force · Quasi-zero stiffness

1 Introduction Vibration suppression of engineering structures is a long-standing problem, which has received continuous attention due to its importance in engineering applications. Therefore, scholars have proposed a large number of vibration control technologies [1–3]. Among them, passive control technology has been widely studied and applied due to its simple and reliable structure and no need to provide energy [4, 5]. The most famous vibration absorber is the linear tuned mass damper (TMD) [6–8]. Linear tuned mass damper has been proved to be an efficient passive linear vibration suppression technique. The shortcomings of TMD as a vibration absorber are well known and well documented [9]. The main related problem is related to the narrow bandwidth of the optimal control, which requires accurate tuning. In the past few decades, in order to overcome these limitations, a lot of research has been carried out. One of the main ideas is to use nonlinearity to improve the design of vibration absorbers [10]. Researchers have begun © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 444–456, 2024. https://doi.org/10.1007/978-981-97-0554-2_34

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to pay attention to nonlinear energy sinks (NESs), which can effectively respond to the amplitude characteristics of external forcing in a wide frequency range [11, 12]. The concept of NES depends on the vanishing linear stiffness, resulting in essentially nonlinear restoring force. Since NES has no natural frequency, it can adapt to the frequency of the primary system. Subsequently, targeted energy transfer (TET) can occur in an irreversible manner [13–15]. Since nonlinear energy sinks have shown good vibration reduction performance, various nonlinear energy sinks have been proposed. Such as inertial NES [16], track NES [17], bi-stable NES [18], NES with piecewise spring [19], vibro-impact NESs (VI-NESs) [20], and other types of NES [21–23]. Among all kinds of NESs, the influence of magnetic force on vibration reduction has been widely studied and applied in recent years due to the excellent characteristics of non-contact force generated by magnetic force. Geng et al. [24] proposed magneticenhanced NES (ME-NES) using non-contact magnetic force to limit the large amplitude vibration of NES, which can prevent the damage of collision to the structure and improve the reliability of NES engineering application. Chen et al. [25] proposed a new type of triple-magnet magnetic levitation dynamic vibration absorber (TMSDVA), which improves the practical application ability of nonlinear dynamic vibration absorber. Yao et al. [26] found that Magnetic NES effectively attenuates the vibration of unbalanced rotor system, and the occurrence range of SMR is broadened. Feudo et al. [27] used magnetic vibration absorber to reduce vibration of multi-layer structures and other types of NES. In addition to the application of vibration suppression, magnetic force can be applied to energy harvesting to improve the efficiency of energy harvesting [28, 29]. All the above studies reflect the high efficiency of non-contact magnetic force for vibration reduction. However, most of the research on magnetic force is based on its nonlinear restoring force and ignores the influence of its linear stiffness on the vibration reduction of the primary system. The nonlinear magnetic force with quasi-zero stiffness is mainly used in the field of vibration isolation [30, 31]. Therefore, there is almost no research on the influence of linear stiffness. In this paper, vibration reduction is based on nonlinear restoring force generated by magnetic force. A quasi-zero stiffness magnetically enhanced NES is formed by a mechanism that provides negative linear stiffness to offset the linear stiffness generated by the magnetic force. According to the mechanical mechanism of magnetic force, the expression of nonlinear restoring force of magnetic force is derived. The theoretical model of the main system equipped with QZS-ME-NES is established. The dynamic response characteristics are analyzed by approximate analytical solutions and numerical methods. The vibration suppression of quasi-zero stiffness NES and TMSDV in transient response and steady-state response is compared.

2 Mechanical Model 2.1 Analyzing Magnetic Force The interaction force of permanent magnets is affected by many parameters, so it is difficult to express the interaction force of permanent magnets. In the early stage of magnetic force research, some relevant scholars derived a relatively complex magnetic force calculation formula, and the calculated results were in good agreement with the

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actual force [32]. Some scholars have theoretically and experimentally tested the accuracy of this approximate expression [33], which proved that the result was correct. The magnetic force expression in the above literature is adopted, and the model magnetic force expression can be approximated as   B2 A2 (h + R)2 1 1 2 G0 = r + − , (1) π μ0 h2 d2 (d + 2h)2 (d + h)2 where Br is the residual magnetic flux density, μ0 is the vacuum permeability, h is the thickness of the magnet, R is the radius of the circular magnet, A is the area of the magnet and d is the distance between two permanent magnets. For the selection of permanent magnet types, this paper selects a permanent magnet with N35, Br is 1.2 T, h is 0.003 m, and μ0 is the constant of 4π × 10–7 H/m. When the central mass is moved to the right by a distance b. The magnetic forces of both sides of the central mass are different, so the magnetic forces of the left and right permanent magnets can be expressed as   1 Br2 A2 (h + R)2 1 2 + − GL = , (2) π μ0 h2 (d + b)2 (d + b + 2h)2 (d + b + h)2   1 Br2 A2 (h + R)2 1 2 , (3) + − GR = π μ0 h2 (d − b)2 (d − b + 2h)2 (d − b + h)2 where GL and GR are the forces of the left and right permanent magnets respectively. The expression of the nonlinear restoring force of the magnetic force can be obtained as  B2 A2 (h + R)2 1 1 G= r − 2 2 π μ0 h (d − b) (d + b)2 1 1 (4) + − 2 (d − b + 2h) (d + b + 2h)2  2 2 . + − 2 (d + b + h) (d − b + h)2

2.2 Negative Stiffness Mechanism In order to eliminate linear stiffness, a negative linear stiffness mechanism is introduced to offset the positive linear stiffness, and a novel NES is proposed to offset the linear stiffness. Considering applying a pair of linear stiffness springs to the attached mass, the mechanism can produce a negative linear stiffness under suitable conditions. It can be used to eliminate the linear stiffness. It is assumed that the displacement of the attached mass is a. The nonlinear restoring force expression of the mechanism can be obtained   a Q(a) = k2 l 2 + a2 − l0 √ , (5) 2 l + a2

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where l 0 is the physical original length and l is the actual length. Due to its significant strong nonlinearity, the Taylor series of the above formula is expanded to the third order, then Eq. (5) is rewritten as Q(a) =

l0 l − l0 k2 a + 3 k2 a3 . l 2l

(6)

When l < l0 , they are in a compressed state, and the primary linear term is negative, then the device has both negative linear stiffness and cubic nonlinear stiffness. Therefore, in order to eliminate the influence of linear stiffness, it is feasible to add a pair of compressed linear springs on the attached mass to generate negative linear stiffness. Moreover, by adjusting l and l0 parameters, the linear stiffness of the device can be adjusted to meet the demand for eliminating different linear stiffness caused by magnetic parameter changes. 2.3 Dynamic Equation of LO with the Novel NES The mechanical model of the system is described in Fig. 1. The designed LO equipped with the NES is composed of a central attached mass, four permanent magnets and a pair of compression spring mechanisms that provide negative stiffness. The NES is embedded in the primary mass. The attached mass is in the unbalanced position, whereas the attached mass is in the balanced position when the negative stiffness mechanism spring is compressed. Based on the design of the NES, the differential equation of motion of the LO equipped with the NES can be expressed as M x¨ 1 + k1 x1 + c1 x˙ 1 = cN (˙x2 − x˙ 1 ) + I (x2 − x1 ) + f sin(ωt),

(7)

m¨x2 + cN (˙x2 − x˙ 1 ) + I (x2 − x1 ) = 0

(8)

where M represents the mass of LO, k 1 and c1 are the linear stiffness coefficient and damping coefficient of the LO respectively. m, I and cN respectively represent the attached mass, nonlinear stiffness coefficient and damping coefficient. x 1 and x 2 are the displacement of the LO and the attached mass, respectively. f is the harmonic excitation amplitude. ω is the external excitation frequency. “.” denotes the derivative of time t. The restoring force expression I(x 2 -x 1 ) of the novel NES can be expressed as   Br A2 (h + R)2 4 4 8 I (x2 − x1 ) = (x2 − x1 ) + − π μ0 h2 d3 (d + 2h)2 (d + 2h)3   Br A2 (h + R)2 8 8 16 (x2 − x1 )3 + + − π μ0 h2 d5 (d + 2h)5 (d + h)5 (9)   Br A2 (h + R)2 12 12 24 5 (x2 − x1 ) + + − π μ0 h2 d7 (d + 2h)7 (d + h)7 l − l0 l0 k2 (x2 − x1 ) + 3 k2 (x2 − x1 )3 . + l 2l

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k

x Fig. 1. Structure diagram.

Table 1. Damping system setting parameters of the novel NES. Item

Value

Item

Value

M

2 kg

Br

1.2 T

k1

5000 N/ m

h

0.003 m

c1

1 Ns/m

R

0.015 m

m

0.1 kg

μ0

4π × 10–7 H/m

cN

2 Ns/m

d

0.01 ~ 0.2 m

k2

1000 N/m

l0

%1.%2 m

3 Analyzing Dynamic Characteristics of the Novel NES 3.1 Analyzing Parameters of NES Based on the calculation and analysis, the parameters of the LO with the NES are selected as presented in Table 1. The primary mass, stiffness, and damping of the LO are assumed to be M = 2 kg, k 1 = 5000 N/m, and c1 = 1 N·s/m, respectively. This means that the natural frequency of the primary system ω0 is 50 rad/s. For the traditional NES, when the attached mass is 2% of the LO, the TET mechanism can be triggered. As the attached mass increases, the damping effect of the NES is improved. However, when the mass of the NES is greater than 10%, the improvement of the damping effect is limited. To improve the damping performance of the NES and minimize the attached mass, the attached mass is selected as 5%, that is, m = 0.1 kg.

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3.2 Harmonic Balance Analysis Harmonic balance method is used to solve the coupled system response equation. The solution of the equation is assumed to be xi =

n 

ai,j cos(jωt) +

j=1

n 

bi,j sin(jωt),

(10)

j=1

where ai,j and bi,j represent harmonic coefficients, i is the number of assumed solutions, and i = 1 and 2, harmonic order is j, and j = 1, 2, 3,…n. Substitute Eq. (10) into Eqs. (7)-(8). The Jacobian matrix is derived by using the Galerkin method, and then the displacement of the system can be assumed to be   n  n Ai =

(ai,j )2 + (bi,j )2 . (11) j=1

j=1

3.3 Transient Response of LO with the Novel NES According to the design of the novel NES. Magnetic nonlinearity is affected by magnetic parameters. In addition, the damping performance affected by the change of the initial energy in the transient response. In this section, the initial energy, magnetic gap and magnetic radius are studied. Generally, the vibration absorption performance of NES can be measured by the energy dissipation rate, which is also applicable to the novel NES. The energy dissipation rate η can be expressed as t cN [˙x2 (t) − x˙ 1 (t)]2 dt × 100%, (12) η= 0 1 2 2 M x˙ 1 (0) where t represents the integration time. The LO converges in 400 cycles. Each period of LO is about 0.1 s. To unify the time calculation measurement, the integration time t is taken as 40 s for all cases. The magnetic force is mainly related to the magnetic gap d, magnetic radius R and thickness h. For the same magnetic force, the greater the R and h, the smaller the distance d between the two magnets. It is shown that if R and h of permanent magnet are determined, different magnetic restoring forces can be obtained by adjusting d. Therefore, it is assumed that the radius of the permanent magnet R = 0.02 m and the thickness h = 0.003 m. For a negative stiffness mechanism, when the linear stiffness of the magnetic force is eliminated, an additional nonlinear restoring force is applied to the system. The additional nonlinear force is mainly affected by the linear spring stiffness k 2 and the initial original length l0 of the spring. For better application and research convenience, k 2 = 1000 N/m and l 0 = 0.05 m are temporarily calculated. The effect of magnetic gap d in different magnetic regions on the damping efficiency of the novel NES is discussed. In Fig. 2, when magnetic gap d = 0.04 m, 0.08 m and 0.12 m, the damping performance of the novel NES is considered. When magnetic gap d = 0.04 m, there is a

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low damping effect for the novel NES with various initial velocities. When the magnetic gap d is appropriate, the vibration suppression of the novel NES is very effective. By comparing the energy dissipation rate of d = 0.08 m and d = 0.08 m, the minimum initial acceleration corresponding to the maximum energy dissipation rate is x˙ 1 (0) = 0.6 m/s and x˙ 1 (0) = 1 m/s. Therefore, when d = 0.08 m, its damping performance is stronger than that of d = 0.12 m.

Fig. 2. Energy dissipation rate of LO with different magnetic gaps.

The velocity x˙ 1 (0) = 0.6 m/s is selected. The energy dissipation rate of TMSDVA and the novel NES with the change of d is compared, as plotted in Fig. 3(a). Compared with TMSDVA, the energy dissipation rate of the novel NES is higher than that of TMSDVA, so the novel NES has better adaptability to magnetic gap d. With the increase of d, the energy dissipation rate of TMSDVA decreases. The novel NES can enhance the damping performance compared with TMSDVA. In Fig. 3(b) for d = 0.08m, the energy dissipation rate of the novel NES is higher than that of TMSDVA. Therefore, the vibration reduction performance of the novel NES is stronger than that of TMSDVA.

Fig. 3. (a) Energy dissipation rate of LO with different magnetic gaps, (b) Energy dissipation rate of LO.

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In Fig. 4, when initial velocity x˙ 1 (0) = 0.6 m/s and magnetic gap d = 0.08 m, the time-history response of the LO with the novel NES converges faster than that of the LO with the TMSDVA.

Fig. 4. Comparison of time-history response.

3.4 Steady State Response of LO with the Novel NES According to the expression of magnetic force G, the approximate analytical solution is obtained to solve the problem directly. Therefore, the Taylor series expansion of Eq. (4) is performed. The relative displacement of the main mass and the attached mass is z = x 2 − x 1 . The nonlinear restoring force G is expanded by Taylor series. If the number of expansion items is too small, the calculation accuracy is not high, which will cause errors to the research data and lead to inaccurate research results. Too many expanded terms will result in low efficiency of approximate solution. Therefore, the curves with different expansion terms in Taylor series expansion are compared and the accuracy is guaranteed while the fewer expansion terms are selected to improve the computational efficiency. Parameters R = 0.01 m, h = 0.003 m and d = 0.08 m are selected. In order to improve the computational efficiency and satisfy the requirement of solving accuracy, the 5th order Taylor series expansion is selected. The nonlinear restoring force after Taylor series expansion is expressed as   4 8 Br A2 (h + R)2 4 (z) + − G(z) = π μ0 h2 d3 (d + 2h)2 (d + 2h)3   Br A2 (h + R)2 8 8 16 (13) + (z)3 + − π μ0 h2 d5 (d + 2h)5 (d + h)5   Br A2 (h + R)2 12 12 24 (z)5 + O(z 7 ). + + − π μ0 h2 d7 (d + 2h)7 (d + h)7 The effect of different magnetic gap d of the novel NES on the amplitude of LO is considered under the excitation of different harmonic amplitude f . In order to measure

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the damping effect of the novel NES under steady-state excitation, the resonance peak ratio of LO controlled by the novel NES and without control is used. The expression can be expressed as p=

AUN − ANovel NES × 100%, AUN

(14)

where ANovel NES represents the resonance peak value of LO with the novel NES, and AUN represents the resonance peak value of LO without NES. As plotted in Fig. 5, in the case of small amplitude vibration, the magnetic linear restoring force is greater than the nonlinear restoring force and is dominated by the linear restoring force. In this case, there is a resonance peak with frequency advance and large peak. At this time, the damping effect is not ideal. It is also found that the linear stiffness of the magnetic force results in a large resonance peak, which is moved forward. By comparing the amplitude-frequency response curve of the novel NES, it is found that after eliminating the linear stiffness, the vibration reduction can be achieved in the resonance frequency range. The vibration reduction effect is also excellent. In Fig. 6, the approximate analytical result of the LO with the TMSDV is periodic response. The time-history curve and the fast Fourier transform (FFT) are analyzed under the resonance frequency. It can be seen from Fig. 7(a) that the vibration suppression of the novel NES presents a modulated response within the resonance frequency and has a good vibration suppression. In Fig. 7(b), chaotic phenomenon occurs in the resonance frequency of the novel NES. In Fig. 7(c) a quasi-periodic response further verifies the chaotic phenomenon and modulation response.

Fig. 5. Amplitude-frequency curves.

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Fig. 6. Dynamic responses of LO with NES in resonant region: (a) time-history curve, (b) phase diagram, (c) FFT spectrum.

Fig. 7. Dynamic responses of LO with NES in resonant region: (a) time-history curve, (b) phase diagram, (c) FFT spectrum.

As shown in Fig. 8. It is found that compared with TMSDVA, the novel NES also has a wider effective magnetic gap interval after linear stiffness elimination.

p (%)

100

80

0.04 m Vibration reduction efficiency

60

40

20

Extended interval Novel NES TMSDVA

0 0.05

0.1

0.15

Magnetic gap

0.2

0.25

d (m)

Fig. 8. Comparison of vibration reduction efficiency of LO with radius change with novel NES and TMSDVA

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4 Conclusions In this paper, a magnetic enhanced NES with quasi-zero stiffness by combining negative stiffness mechanism with permanent magnets. By analyzing the force mechanism of negative stiffness mechanism and permanent magnets, the expression of magnetic force with quasi-zero stiffness characteristics is derived. Through rational design, the magnetic force is appropriate between the primary mass and the attached mass. The dynamic model of the primary system coupled with the novel NES is established. The analytical and numerical analysis of the novel NES is investigated. The conclusion is as follows: (1) Compared with TMSDVA, the novel NES has no linear stiffness, which can achieve broadband vibration reduction and improve the vibration reduction effect. (2) Appropriate magnetic force can improve the vibration reduction performance of the novel NES and can better control NES. (3) Compared with TMSDVA, the novel NES has better adaptability to parameters and enhanced vibration reduction. Acknowledgments. The authors would like to gratefully acknowledge the support of the National Science Fund for Distinguished Young Scholars (Grant No. 12025204) and the Program of Shanghai Municipal Education Commission (No. 2019–01-07–00-09-E00018).

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11. Xia, Y.W., Ruzzene, M., Erturk, A.: Bistable attachments for wideband nonlinear vibration attenuation in a metamaterial beam. Nonlinear Dyn. 102(3), 1285–1296 (2020) 12. Wang, T., Tang, Y., Yang, T.Z., Ma, Z.S., Ding, Q.: Bistable enhanced passive absorber based on integration of nonlinear energy sink with acoustic black hole beam. J. Sound Vibr. 544 (2023) 13. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., Mojahed, A., Gzal, M.: Nonlinear targeted energy transfer: state of the art and new perspectives. Nonlinear Dyn. 108(2), 711–741 (2022) 14. Hubbard, S.A., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Targeted energy transfer between a model flexible wing and nonlinear energy sink. J. Aircr. 47(6), 1918–1931 (2010) 15. Jiang, X., Michael Mcfarland, D., Bergman, L.A., Vakakis, A.F.: Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dyn. 33(1), 87–102 (2003) 16. Zhang, Z., Ding, H., Zhang, Y.W., Chen, L.Q.: Vibration suppression of an elastic beam with boundary inerter-enhanced nonlinear energy sinks. Acta Mech. Sin. 37(3), 387–401 (2021) 17. Lu, X.L., Liu, Z.P., Lu, Z.: Optimization design and experimental verification of track nonlinear energy sink for vibration control under seismic excitation. Struct. Control Health Monit. 24(12) (2017) 18. Li, M., Li, Y.Q., Liu, X.H., Dai, F.H.: A bi-stable nonlinear energy sink using the cantilever bi-stable hybrid symmetric laminate. Mech. Syst. Signal Process. 186 (2023) 19. Geng, X.F., Ding, H., Mao, X.Y., Chen, L.Q.: Nonlinear energy sink with limited vibration amplitude. Mech. Syst. Signal Process. 156 (2021) 20. Li, T., Gourc, E., Seguy, S., Berlioz, A.: Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations. Int. J. Non Linear Mech. 90, 100–110 (2017) 21. Zang, J., Yuan, T.C., Lu, Z.Q., Zhang, Y.W., Ding, H., Chen, L.Q.: A lever-type nonlinear energy sink. J. Sound Vibr. 437, 119–134 (2018) 22. Mao, X.Y., Ding, H., Chen, L.Q.: Nonlinear torsional vibration absorber for flexible structures. J. Appl. Mech.-Trans. ASME 86(2) (2019).https://doi.org/10.1115/1.4042045 23. Lu, Z., Wang, Z.X., Masri, S.F., Lu, X.L.: Particle impact dampers: past, present, and future. Struct. Control Health Monit. 25(1) (2018) 24. Geng, X.F., Ding, H., Jing, X.J., Mao, X.Y., Wei, K.X., Chen, L.Q.: Dynamic design of a magnetic-enhanced nonlinear energy sink. Mech. Syst. Signal Process. 185, 109813 (2023). https://doi.org/10.1016/j.ymssp.2022.109813 25. Chen, X.Y., Leng, Y.G., Fan, S.B., Su, X.K., Sun, S.L., Xu, J.J., et al.: Research on dynamic characteristics of a novel triple-magnet magnetic suspension dynamic vibration absorber. J. Vibr. Control (2023) 26. Yao, H.L., Wang, Y.W., Xie, L.Q., Wen, B.C.: Bi-stable buckled beam nonlinear energy sink applied to rotor system. Mech. Syst. Signal Process. 138 (2020) 27. Lo Feudo, S., Touze, C., Boisson, J., Cumunel, G.: Nonlinear magnetic vibration absorber for passive control of a multi-storey structure. J. Sound Vibr. 438, 33–53 (2019) 28. Challa, V.R., Prasad, M.G., Shi, Y., Fisher, F.T.: A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater. Struct. 17(1), 015035 (2008). https:// doi.org/10.1088/0964-1726/17/01/015035 29. Fakeih, E., Almansouri, A.S., Kosel, J., Younis, M.I., Salama, K.N.: A wideband magnetic frequency up-converter energy harvester. Adv. Eng. Mater. 23(6) (2021)

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New Software Bionic Haptic Actuator Design Based on Barometric Array Zige Yu1 , Sai Li1 , Mengying Lin1 , Hang Hu1 , Yingying Li1 , Qian Lei1(B) , and Zixin Huang1,2,3 1

School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan 430205, People’s Republic of China [email protected] 2 Hubei Key Laboratory of Digital Textile Equipment, Wuhan Textile University, Wuhan 430200, People’s Republic of China 3 Institute of Robotics and Automatic Information Systems, College of Artificial Intelligence, Nankai University, Tianjin 300350, China

Abstract. To address the limitations of single-point tactile perception in soft biomimetic actuators, such as fixed positioning, limited coverage area, low resolution at long distances from the geometric center, and challenges in maintaining high sensitivity and stability over a broad range, a novel soft biomimetic tactile actuator based on a pneumatic array is designed and manufactured. Using liquid silicone rubber and BMP280 pressure sensor, the actuator was crafted by an injection molding process. Silicone gel was injected into the pressure sensor array, and vacuum extraction created a sealed space within the sensor cavity. The pressure excitation applied on the surface of the actuator could be converted into electrical signals, enabling tactile pressure detection. An STM32 microprocessor is utilized for building haptic information acquisition and processing system, with sensitivity assessed using the Kalman filter algorithm. Experimental results show that the soft biomimetic tactile actuator has high sensitivity and minimal repetition error, which can effectively detects changes in tactile force output, mitigates noise interference, and enhances the precision and stability of physiological pressure data feedback in physiotherapy applications.

Keywords: Pressure sensor array Soft bionic actuator

· Kalman filtering · Haptic sensing ·

This work was supported by Hubei Province Nature Science Foundation (2023AFB380), and the Scientific Research Foundation of Wuhan Institute of Technology (K2021027), and the Hubei Key Laboratory of Intelligent Robot (Wuhan Institute of Technology) (HBIRL202105), and the Hubei Key Laboratory of Digital Textile Equipment (Wuhan Textile University) (KDTL2022003), and the Graduate Innovative Fund of Wuhan Institute of Technology (CX2022123, CX2022149). c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 457–468, 2024. https://doi.org/10.1007/978-981-97-0554-2_35

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Introduction

Due to its high flexibility, good compliance, excellent adaptability and natural safety interactivity [1–3], soft bionic robots have broad application prospects in intelligent medical [4], health monitoring [5] and human-computer interaction [6]. With the rapid development of machine vision [7] and artificial intelligence [8], solving the accurate and rapid perception of the external environment of soft bionic robots has become the core bottleneck of realizing intelligence [9–11]. Different from visual perception information, tactile perception technology has unique advantages in dynamic characterization of target object features and description of physical characteristics such as stiffness and surface texture of object in non-structural environment [12–14]. Many researchers have explored and studied the haptics of soft biomimetic robots. Matteo Bianchi [15] proposed a simple touch-based method based on Pisa/IIT SoftHand, in which the finger tip is equipped with an inertial measurement unit sensor as a tactile sensing device. By detecting the acceleration resulting from external object contact, object manipulation is achieved, with limited system accuracy. In order to improve the accuracy of tactile recognition, researchers often increase the number of tactile sensor array points or improve the sensor resolution. Zhang [16] designed a magnetostrictive tactile sensor array for measuring pressure, which breaks through the limitation of collecting information from a single contact of the magnetostrictive sensor and increases the detection area of the tactile sensor. The magnetostrictive sensor array is large in size due to structural limitations, and the freedom and mobility of soft robot applications are limited. Based on this, this paper selects liquid silicone rubber and BMP280 barometer chip as materials, and uses vacuum pumping method to pour barometer chip [17–19] with silica gel to make a pressure array software bionic tactile actuator. The actuator has high sensitivity at a low cost, and by arranging and combining multiple sensing units, it creates an effective sensing region of a specific area, mitigating the limitations of single point contact. In addition, a haptic information acquisition system is designed with discrete Kalman filter algorithm to filter and preprocess the tactile information of pneumatic sensing to obtain smooth and stable tactile data information.

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Structure Design and Production

In order to enable the robot to accurately obtain tactile information data during work through the actuator and make correct behavioral feedback after accurate analysis of the tactile information data, a soft bionic tactile actuator based on the air pressure array is designed. It is jointly made by cheap air pressure sensor array and silicone software, with better structural flexibility and high safety. The manufacture and maintenance of the actuator is also very simple and the cost is low.

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Physical Design

Pressure Sensor Array. The air pressure sensor array consists of four BMP280 air pressure sensors, and the number of sensors can then be changed according to the application scenario to achieve effective tactile information acquisition in different areas. The BMP280 consists of a MEMS diaphragm with Wheatstone bridge, an instrument amplifier, a temperature sensor, a multiplexed circuit, a digital-to-analog converter and an I2C bus, and its components are very compact and small in package to meet the needs of space-constrained portable electronic devices. In addition, the BMP280 adopts a piezoresistive sensor technology with high precision, high linearity and long-term stability, and supports two interface types I2C and SPI, with a maximum rate of 3.4MHz in I2C mode. In this design, I2C interface mode is used. In the I2C interface mode, the SCK pin is connected to the clock signal line SCL of the master chip, and the SDI is connected to the data line SDA. When multiple BMP280 devices are used at the same time, the SDO pin can distinguish different slave devices under different level states. The technical parameters of the BMP280 pressure sensor are shown in Table 1. Table 1. Technical parameters of BMP280 air pressure sensor Parameter name

Parameter values

Working range Operating temperature range Relative precision absolute precision Working supply current Response time

300-1100HPa -40 ◦ C-85 ◦ C ±0.12HPa ±1HPa 2.8µA 5.5 ms

Fig. 1. BMP280 barometer

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The size of the packaged barometer is only 2×2.5×0.95 mm, and the surface is distributed with air holes for collecting pressure values. The array obtained after assembling four pressure sensors is shown in Fig. 1. Design Method of Soft Biomimetic Tactile Actuator. According to the structural principle of the sensor, a kind of soft biomimetic tactile actuator modeled on human skin is designed. The actuator consists of two parts: a pressure sensor array and a silicone software. The silicone software wraps the pressure sensor array in it, and the silicone software can ensure the actuator’s safe and flexible contact with the outside world. The actuator can not only realize more accurate tactile information perception in structured environment, but also realize more sensitive tactile information acquisition in unstructured environment, as shown in Fig. 2.

Fig. 2. Software bionic tactile actuator model

The working principle of the soft bionic tactile actuator is based on the gap between the MEMS barometer chip of the air pressure sensor BMP280 and the metal housing, and the air pressure value of the outside air can be sensed through the vent, but the pressure in the form of touch cannot be sensed. To this end, the method of silicone sensor array is adopted, and the air left in the cavity after pouring is extracted by vacuum pumping, so that the sensor cavity forms a closed space, and when the tactile pressure is applied on the surface of the silicone software, it can be transmitted to the MEMS chip, and finally the tactile pressure detection is completed. The sensor array and the silicone software as a whole constitute a tactile actuator. When pressure is applied to the surface of the silicone software, the pressure will be transmitted to the air pressure sensor of the tactile actuator in real time, and the tactile sensing function can be realized, as shown in Fig. 3.

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(b) After the silicone is poured

Fig. 3. Section view of BMP280 before and after pouring silicone

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Preparation of Materials and Equipment. The preparation of the soft bionic tactile actuator requires materials such as air pressure sensor and silica gel. Vacuum pumping is completed by vacuum pump and heat gun is used to accelerate the forming. The specific materials and equipment for preparation are shown in Table 2. Table 2. Materials and equipment for the preparation of pneumatic sensors Name

Type

Air pressure sensor BMP280

Silica gel

Food grade silicone

Vacuum pump

RS-4

Heat gun

Deer Fairy 858D+ series

Sketch map

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Preparation Method. Considering the size of the air pressure sensor array, four BMP280 air pressure sensors are tightly designed on the front of the integrated board, and then the air pressure sensor array on the integrated board is poured silica gel. The method of injection molding is adopted for silicone pouring, which has the advantages of convenient operation and high efficiency. It is widely used in the preparation technology of soft robot. Medical A and B silica gel with strength of 5 were selected as the main raw materials. The a and b silicone is first mixed in a 1:1 ratio and stirred for 3 to 5 min to fully fuse, then the vacuum pump is used to eliminate the bubbles for the first time, and the defoamed ab silicone is poured on the pressure sensor array. Then the whole thing is put into the vacuum pump again for secondary bubble elimination. Finally, heat with a heat gun to solidify, as shown in Fig. 4.

Fig. 4. Manufacturing and preparation process

After the silicone is cured, a layer of silicone software is formed right above the pressure sensor array. By pressing the surface silicone software, the pressure can be transmitted to the barometer, thus completing the detection of tactile information. After trimming it and installing the terminal interface, the soft bionic tactile actuator can be obtained, as shown in Fig. 5.

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Fig. 5. Soft biomimetic tactile actuator

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This section mainly describes the haptic information acquisition system based on Kalman filter to obtain the initial haptic information by using the haptic actuator, and to process the collected data through the Kalman filter algorithm to improve the accuracy and stability of haptic information. 3.1

System Work Plan Design

The haptic information acquisition system adopts a distributed architecture design scheme, and the overall structure is shown in Fig. 6. The system is mainly composed of tactile actuator module, controller module and data display module. The tactile actuator module communicates with the controller module through I2C bus, and synchronously transmits the initial tactile information collected to STM32 MCU. The host computer communicates with STM32 microcontroller through communication converter, and uses Kalman filter method to reduce the noise of the received original sensor data.

Fig. 6. System overall structure diagram

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Tactile Data Processing Method Based on Kalman Filter

The initial tactile information collected by the pneumatic tactile actuator is affected by the interference caused by environmental noise and the instability in the sensor, resulting in a certain degree of abnormal data, which then affects the accuracy and accuracy of the pneumatic sensor. At present, the widely used filtering algorithms are Kalman filter, particle filter and Gaussian filter. Among them, the Kalman filter algorithm can predict the current state value of the system according to the observed value of the system at the previous time by introducing the equation of state, which can effectively reduce the influence of unstable factors such as noise in the sensing system on the output accuracy of the sensor, and obtain smooth sensor data information [20]. Since the process of collecting tactile data by pneumatic tactile actuator can be regarded as a discrete state system, the discrete Kalman filtering algorithm to filter the collected tactile data information to improve the collecting accuracy. ˆ k−1 from The discrete Kalman filter can calculate the current time estimate X ˆ − , where A and B are matrix coefficients, U (k) the previous time state estimate X k is the control quantity of the measurement model, and its basic state prediction equation is ˆ − = AX ˆ k−1 + BUk X k

(1)

Pk− = APk−1 AT + Q Pk−

(2) T

where is the system state covariance matrix of priori estimate A, A is transposed matrix of A, Q is the system noise covariance matrix, the Pk− is Xk− covariance, Pk−1 is Xk−1 covariance. The measurement equation obtained by modifying the prior estimates is Kk =

Pk− C T CPk− C T + R

(3)

ˆk = X ˆ − + Kk (yk − C X ˆ −) X k k

(4)

Pk = (I − Kk C)Pk−

(5)

where Pk is the covariance matrix of the system; R is the covariance matrix of measurement noise; Kk is the Kalman gain coefficient matrix; C is the observation model matrix; yk is the measured value at time k. For the software bionic tactile actuator designed in this paper, its function is to collect and process tactile information through the actuator, without considering other state variables such as speed or position, and there is only a single model measurement of the pressure sensor. Then Eqs. (1) and (2) can be simplified as follows: ˆ− = X ˆ k−1 X (6) k

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(7)

Substituting Eqs. (6) and (7) into Eqs. (3), (4) and (5) for simplification can be obtained P− (8) Kk = −k Pk +R ˆk = X ˆ − + Kk (yk − X ˆ −) X k k

(9)

Pk = (I − Kk )Pk−

(10)

According to formula (8), (9) and (10), the noise filtering of the data information collected by the tactile actuator can be completed.

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Experiment and Data Analysis

The sensitivity of tactile information feedback plays an important role in the application of soft biomimetic tactile actuators, which determines the ability and accuracy of the actuators to perceive external stimuli. Tactile actuators with high sensitivity can more accurately simulate human tactile perception, thus playing a key role in several fields such as robot operation, medical device use and virtual reality technology improvement. In order to evaluate the tactile information feedback sensitivity of the soft biomimetic tactile actuator after silicone pouring, we conducted a tactile sensitivity detection experiment. Based on the tactile perception principle of the soft bionic tactile actuator, the pressure feedback data measured before and after silicone pouring were compared, and the tactile feedback sensitivity after silicone pouring was tested, select the same position to press the tactile actuator point by point, and use the Kalman filter algorithm to reduce the noise of the data. The pressure information data before the silicone pouring was recorded in real time, and the corresponding curve was drawn according to the data. The sensitivity and response of the tactile actuator to the pressing force before the silicone pouring could be observed, as shown in Fig. 7. Similarly, the tactile sensitivity test after silicone pouring was carried out, and the tactile information data was drawn as a graph, as shown in Fig. 8. According to the analysis of the tactile information data curve, it can be seen that the tactile information data measured by the actuator under the same sampling number is similar to the pressure information data before the pouring, and the soft biomimetic tactile actuator after the silicone pouring still has a good tactile sensitivity. After Kalman filtering, with the increase of filtering times, the influence of the set initial filter value on the filtering model gradually decreases until it disappears. Finally, the filtered tactile information data of the actuator gradually tends to be smooth and stable, which reduces the interference of noise on the real data collected and ensures the effectiveness of the tactile information collected by the actuator during operation. The experiment verifies

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Fig. 7. Sensor output data before silicone pouring

Fig. 8. Tactile information data of actuators after silicone pouring

the sensitivity and data validity of the soft bionic tactile actuator, which can better reflect the ability of simulating human tactile perception and meet the needs of the robot for external tactile perception.

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Conclusions

In this paper, a soft biomimetic tactile actuator based on air pressure array is designed and made. The actuator is made by using silica gel to cast sensor array, which can mimic human skin to realize tactile sensing function. Experiments show that the tactile information changes can be accurately detected when the pressure is applied at different positions within the effective sensing range of the actuator after effective Kalman filtering. The software bionic tactile actuator designed in this paper can change the shape of the actuator according to the application function requirements, optimize the number and volume of sensor

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arrays, and provide a new idea for using software actuators for fine planning operations and human-computer interaction tasks in different situations in the future.

References 1. Lin, D., Yang, F., Gong, D., et al.: Bio-inspired magnetic-driven folded diaphragm for biomimetic robot. Nat. Commun. 14(1), 163 (2023) 2. Yin, C., Wei, F., Fu, S., et al.: Visible light-driven jellyfish-like miniature swimming soft robot. ACS Appl. Mater. Interfaces 13(39), 147–154 (2021) 3. Wang, F., Wang, L., Shen, F., et al.: High-performance nano-biocomposite ionic soft actuators based on microfibrillated cellulose/ionic liquid electrolyte membrane. Polym. Compos. 44(2), 963–970 (2023) 4. Lee, H., Kim, S., Yeo, K., et al.: Fully portable continuous real-time auscultation with a soft wearable stethoscope designed for automated disease diagnosis. Sci. Adv. 8(21), eabo5867 (2022) 5. Wang, C., Wu, Y., Dong, X., et al.: In situ sensing physiological properties of biological tissues using wireless miniature soft robots. Sci. Adv. 9(23), eadg3988 (2023) 6. Yeon, H., Lee, H., Kim, Y., et al.: Long-term reliable physical health monitoring by sweat pore-inspired perforated electronic skins. Sci. Adv. 7(27), eabg8459 (2021) 7. Wang, S., Sun, Z.: Hydrogel and machine learning for soft robots’ sensing and signal processing: a review. J. Bionic Eng. 20(3), 845–857 (2023) 8. Wan, X., Huang, Z., Li, Y., et al.: Distributed pneumatic physiotherapy robot for human acupoints. In: 2023 IEEE 12th Data Driven Control and Learning Systems Conference, pp. 429–433. Xiangtan, China (2023) 9. Sun, Z., Zhu, M., Zhang, Z., et al.: Smart soft robotic manipulator for artificial intelligence of things (AIOT) based unmanned shop applications. In: 2021 IEEE 34th International Conference on Micro Electro Mechanical Systems, Gainesville, FL, USA, pp. 591-594 (2021) 10. Bai, H., Kim, Y., Shepherd, R.: Autonomous self-healing optical sensors for damage intelligent soft-bodied systems. Sci. Adv. 8(49), eabq2104 (2022) 11. Hu, D., Giorgio-Serchi, F., Zhang, S., et al.: Stretchable e-skin and transformer enable high-resolution morphological reconstruction for soft robots. Nat. Mach. Intell. 5(3), 261–272 (2023) 12. Yao, K., Zhou, J., Huang, Q., et al.: Encoding of tactile information in hand via skin-integrated wireless haptic interface. Nat. Mach. Intell. 4(10), 893–903 (2022) 13. Siles, E.L.: Soft law for unbiased and nondiscriminatory artificial intelligence. IEEE Technol. Soc. Mag. 40(4), 77–86 (2021) 14. Wang, X., Chen, C., Zhu, L., et al.: Vertically integrated spiking cone photoreceptor arrays for color perception. Nat. Commun. 14(1), 34–44 (2023) 15. Bianchi, M., Averta, G., Battaglia, E., et al.: Touch-based grasp primitives for soft hands: applications to human-to-robot handover tasks and beyond. In: 2018 IEEE International Conference on Robotics and Automation , Brisbane, QLD, Australia, pp. 7794-7801 (2018) 16. Zhang, B., Wang, B., Li, Y., et al.: Magnetostrictive tactile sensor array for object recognition. IEEE Trans. Magn. 55(7), 1–7 (2019) 17. Huang, Z., Li, X., Wang, J., et al.: Human pulse detection by a soft tactile actuator. Sensors 22, 5047 (2022)

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18. Li, S., Wang, J., Li, X., et al.: Information express of soft bionic finger based on visual recognition. In: 2022 IEEE International Conference on Cyborg and Bionic Systems (CBS), Wuhan, China, pp. 240-244 (2023) 19. Wang, S., Dong, X., Liu, G., et al.: Low-cost single-frequency DGNSS/DBA combined positioning research and performance evaluation. Remote Sens. 14(3), 586 (2022) 20. Liu, H., Hu, F., Su, J., et al.: Comparisons on Kalman-filter based dynamic state estimation algorithms of power systems. IEEE Access 8, 51035–51043 (2020)

Adaptive Robust Tracking Control for Aerial Work Platform Vehicle with Guaranteed Prescribed Performance Jiawen Dai1 , Zheshuo Zhang2(B) , Bangji Zhang1,2 , Jie Bai2 , and Hui Yin1 1

2

College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, People’s Republic of China Intelligent Transportation System Research Center, Hangzhou City University, Hangzhou 310015, People’s Republic of China [email protected]

Abstract. An aerial work platform vehicle (AWPV) plays a crucial role in conducting industrial activities at elevated heights, but its tracking control presents significant challenges due to complex nonlinearities and uncertainties. This study introduces a constraint-based adaptive robust control approach for AWPV that reliably achieves trajectory tracking while ensuring prescribed transient and steady-state performance (PTSSP). The control strategy takes into account time-variant uncertainties with unknown bounds, which may change rapidly and irregularly. The desired trajectories and PTSSP are respectively formulated as equality and inequality servo constraints. To incorporate the inequality servo constraints into the equality ones, a diffeomorphism approach is employed, leading to the formulation of new equality servo constraints. Consequently, the control task transforms into guiding the AWPV to adhere to the new equality servo constraints. Accordingly, an adaptive robust control method is designed to follow these constraints, incorporating an adaptive law for estimating online uncertainty bounds and compensating for uncertainties. Notably, no approximations or linearizations are utilized. The effectiveness and robustness of the proposed AWPV tracking control approach are rigorously validated through proofs and simulation results. This represents a pioneering effort in uncertain AWPV tracking control while ensuring PTSSP. Keywords: Aerial work platform vehicle · Uncertainty · Adaptive robust control · Prescribed performance · Constraint-following · Tracking control

1

Introduction

Working at height happens from time to time in many industrial activities. The time-consuming and laborious traditional scaffold mode in high-altitude Supported by the National Natural Science Foundation of China [Grant Nos. 52102434, 52105096] and the open project of State Key Laboratory of Traction Power [Grant No. TPL2208]. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 469–483, 2024. https://doi.org/10.1007/978-981-97-0554-2_36

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operation is rapidly replaced by Aerial work platform Vehicle (AWPV), which has an elevating platform, providing temporary access in the air and making aerial working much easier [1]. The workers on the platform operate it moving up or down to reach the required height. With a given goal, a smooth trajectory can be automatically planned and then tracked by the control system of AWPV. Due to working at heights, the performance of control system is crucial to the safety of workers [2], and consequently the desired trajectory should be robustly and accurately tracked [3]. In addition, it is strictly required that the platform is kept upright under any operating conditions, as shown in Fig. 1. Researchers have designed the control of AWPV considering the nonlinearity of motion [4]. Although their study proved the effectiveness of their control, they have not taken uncertainty into consideration.

Fig. 1. Keeping platform upright during motion of AWPV.

Uncertainty is an inevitable issue in the practical AWPV system, such as the change of platform mass for different workers or the wind disturbance [5]. These uncertainties would greatly degrade control system performance if they are not properly handled [6,7]. AWPV working standards ANSI/SAIA A92 and OSHA 1926.453 provide limitations for load and working environment, and there are certain ranges of other uncertainties associated with different properties, like boom mass or length. Therefore, the characteristic of AWPV uncertainties is (possibly fast and irregularly) time-variant and bounded with unknown bounds [8]. In order to robustly track desired trajectories of AWPV under uncertainties, Jia et al. [9] proposed a control scheme based on artificial neural network (ANN). Hu et al. [10] designed a sliding mode control for uncertainties with ANN to reduce chatting. Another study by Hu et al. [3] suggested the backstepping control for the complex nonlinearity of AWPV system and ANN for the uncertainties. The control methods by ANN are adequate for time-variant uncertainties with a learnable pattern, but their ability to deal with irregularly time-variant uncertainties with unknown bounds (possibly non-learnable) has never been verified.

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By formulating tracking tasks as equality servo constraints, the adaptive robust constraint-following control (CFC), originated by Chen et al. [11] to deal with holonomic and nonholonomic equality servo constraints of mechanical systems, has shown its ability of achieving trajectory-tracking for the constrained nonlinear mechanical systems subject to (possibly fast and irregularly) timevariant uncertainties with unknown bounds [11–13]. Compared to control by ANN, adaptive robust CFC saves many efforts of training model. Furthermore, control by ANN is approximation-based, whose approximation errors are difficult to be controlled thus affecting the system performance [14]. By contrast, CFC is constraint-based control [15] without requirement of the Lagrangian multiplier, linearizations and/or nonlinear cancellation [16,17], thus approximationfree [18,19]. In addition, satisfying the Gauss’s minimum principle and the Lagrange’s form of D’Alembert’s principle, the CFC exhibits a modest control magnitude [20]. Therefore, adaptive robust CFC is promising for the tracking control of the nonlinear AWPV system under complex uncertainties. Even though aforementioned studies have proved the convergence of the controlled AWPV, there was no guarantee of converging speed or finally converging accuracy. For practical application of AWPV, which has high requirements for safety, it is meaningful that both prescribed transient and steady state performance (PTSSP) are guaranteed simultaneously to limit tracking errors as well as converging speed [21]. Particularly, it is required that the converging speed is more than a certain value, and the tracking error eventually converges to a predefined area near zero [22]. Nevertheless, it is difficult to control AWPV to robustly achieve trajectory-tracking while guaranteeing the PTSSP in the presence of (possibly fast and irregularly) time-variant uncertainties with unknown bounds [23], which has not been reported in any articles. Although the adaptive robust CFC may be a promising tool to fulfill this task, how to guarantee the PTSSP in adaptive robust CFC is still challenging, even existing extensive studies on constrained motion of mechanical systems [24,25], since the PTSSP generally cannot be directly treated as equality servo constraints. In this study, a constraint-based adaptive robust control is designed for AWPV that robustly achieves trajectory-tracking and satisfies PTSSP in the presence of (possibly fast and irregularly) time-variant uncertainties with unknown bounds. The trajectory-tracking task is translated as a series of equality servo constraints for the states of AWPV, while the PTSSP is translated as a series of inequality servo constraints for the trajectory-tracking errors. In order to incorporate inequality servo constraints into equality servo constraints, a “diffeomorphism”, denoting a continuously differentiable map with a continuously differentiable inverse [20], is employed to transform the states of AWPV into new ones which are only with equality servo constraints and free from inequality servo constraints in mathematical form [26]. The control task is thus transformed into designing a control rendering the transformed AWPV to follow the new equality servo constraints. Furthermore, we design a class of adaptive robust CFC for the transformed AWPV, in which an adaptive law is established for online uncertainty bounds estimation to compensate uncertainties. By solving the equation

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established based on the adaptive robust CFC and diffeomorphism approach, the control inputs can render AWPV to track the desired trajectories and guarantee the PTSSP at the same time. The contributions of this paper are threefold. First, it is of interest to design a AWPV tracking control by the constraint-based control method for the possibly first time. Second, it is of practical value to consider the complex uncertainties and PTSSP at the same time. This study proposes an adaptive robust CFC to deal with both of them. Third, it is of innovation to introduce diffeomorphism into AWPV control design to deal with the inequality servo constraints brought by PTSSP.

2 2.1

Problem Formulation Dynamic Model of AWPV

Fig. 2. Dynamic model of AWPV.

The AWPV model, depicted in Figure. 2, exhibits a 2-Dof structure encompassing the luffing angle(θ1 ) of the main boom and the corresponding auxillary boom angle(θ2 ). The main boom possesses a length(L1 ), while the auxiliary boom features a length(L2 ). An unchanging angle(ϑ) is established between the auxiliary boom and the platform. Within this setup, the masses of the main boom(mb1 ), auxiliary boom(mb2 ), and the platform itself(m) are significant factors. Notably, the mass of platform(mp ) may vary when workers of unspecified mass occupy the platform, introducing uncertainties into the system. Additionally, potential errors in measuring the masses(mb1 and mb2 ), boom lengths(L1 and L2 ), and the fixed angle(ϑ) further contribute to the presence of uncertainties. The dynamic model of the AWPV, encompassing uncertainties, can be described as follows: M (q(t), υ(t))¨ q (t) + C(q(t), ˙ q(t), υ(t))q(t) ˙ + G(q(t), υ(t)) = u(t).

(1)

where denote q ∈ R2 , q˙ ∈ R2 and q¨ ∈ R2 as the state, velocity, and acceleration vectors, respectively. The independent variable t ∈ R represents time,

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while υ ⊂ Rm denotes the vector of uncertainties. It is important to note that the uncertainties are (potentially rapidly and irregularly) time-variant, bounded within unknown limits bounds Υ , and may encompass both system parameters and external disturbances. Furthermore, the matrices M ∈ R2×2 , C ∈ R2×2 and ˙ q(t) and uncertainties υ(t), while u ∈ R2 G ∈ R2 are intricately linked to q(t), represents the control input. Explicit expressions for these vectors and matrices are provided below:     m11 m12 C11 C12 T T T q = [θ1 ,θ2 ] , G=[g1 , g2 ] , u = [u1 , u2 ] , M = , C= . m21 m22 C21 C22 Complete expressions for the elements pertaining to these vectors and matrices can be found in Appendix. 2.2

Desired Trajectory

To ensure the stability of the platform throughout the process (as depicted in Fig. 1), it is imperative that the luffing angle of the main boom θ1 reaches the target position θ1d , while the luffing angle of the auxiliary boom θ2 adjusts in accordance with θ1 to maintain the platform in an upright position, meanwhile, the luffing velocities of two booms should be zero after arrival to guarantee the stability of AWPV, that is, ⎧ ⎨ lim θ1 (t) = θ1d , θ2 (t) =ϑ − θ1 (t) , t→∞ T   T (2) ⎩ lim θ˙1 (t) θ˙2 (t) = 0 0 t→td

where td is the reaching time. The cycloid curve emerges as a viable trajectory for AWPV states, effectively fulfilling all the aforementioned requirements [27]. Consequently, the ensuing application of the cycloid curve is adopted to delineate the desired trajectories of the states. 

d) + θ10 , t < td (θ1d − θ10 ) ttd − sin(2πt/t 2π Θ1 (t) = (3) θ1d , t ≥ td Θ2 (t) = ϑ − Θ1 (t). Remark 1. It is easy to notice that the trajectories in (3) are second-order continuously differentiable, a property that is crucial for the diffeomorphism approach proposed in this paper. 2.3

Trajectory Tracking with Prescribed Performance

In order to achieve trajectory-tracking control, where the states q ∈ R2 follow the desired trajectories Θ ∈ R2 , it is preferable for system (1) to adhere to a set of equality servo constraints: |qi (t) − Θi (t)| = 0 i = 1, 2.

(4)

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where Δi (q, t) := qi (t) − Θi (t) are the trajectory-tracking errors, which can indicate the tracking performance of the AWPV system. For the practical application of AWPV, in addition to realizing trajectory tracking, it is particularly important to ensure the convergence speed of the tracking and the final convergence error, which is qualified by the PTSSP. In this study, the PTSSP is described by the following function: βi (t) = (βi0 − βi∞ ) e−hi t + βi∞ ,

(5)

where hi determining the convergence speed, which are positive constants. Problem Formulation. The problem is formulated as: designing a control u ∈ R2 , that takes initial state deviation and uncertainties υ ∈ Rm with unknown bounds Υ into account, to drive the states of AWPV q ∈ R2 tracking the desired trajectories Θ ∈ R2 and satisfying the PTSSP indicated by βi (t), i.e., rendering |qi (t) − Θi (t)| < βi (t) , i = 1, 2.

(6)

Consequently, the guaranteed PTSSP of AWPV is equivalent to its tracking error satisfying a series of inequality servo constraints.

3 3.1

Adaptive Robust Tracking Control Design with PTSSP Diffeomorphism Approach and State Transformation

The concept of diffeomorphism can be described as follows: If both an invertible map F : f1 → f2 and its inverse F −1 : f2 → f1 are continuously differentiable, the map F is referred to as a diffeomorphism. Utilizing the transformative power of diffeomorphism, the states can be seamlessly converted from one coordinate space to another. Further, multiple constraints can be amalgamated, and if an appropriate mapping is judiciously selected to transition the states to the unrestrained descriptions, then the inequalities can be relaxed. The control should drive the motion of AWPV following the desired trajectories Θi (t) in (4) and satisfying the PTSSP βi (t) in (6) at the same time. In order to simplify the control design, a diffeomorphism approach is proposed, transforming the constrained states to an unconstrained one, and is expressed as: π·(qi (t)−Θi (t)) i (q,t) , i = 1, 2. zi (q, t) := tan π·Δ (7) 2βi (t) = tan 2βi (t) T

where z = [z1 , z2 ] . It can be found that when Δi ∈ (−βi , βi ), the z ∈ (−∞, ∞), meaning that the transformed states zi are in unrestricted range. Using the diffeomorphism approach depicted in Fig. 3, the motion of AWPV will consistently adhere to the PTSSP description (5) and the servo constraint (6), if the designed control can guarantee the boundedness of zi (t). Substituting Δi (q, t) := qi (t) − Θi (t) into (7) yields qi =

2βi arctan zi + Θi , π

(8)

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Fig. 3. The diffeomorphism approach.

Bringing the expressions for qi and its first and second order derivatives into (1), there is the transformed AWPV as Ξ(z(t), υ(t), t)¨ z (t) + Ψ (z(t), ˙ z(t), υ(t), t)z(t) ˙ + Λ(z(t), υ(t), t) = u(t) ,

where Ξ=

2β2 2β1 m m π(1+z12 ) 11 π(1+z22 ) 12 2β2 2β1 m m π(1+z12 ) 21 π(1+z22 ) 22

(9)

,

(10)

Remark 2. The reversibility of Ξ is an important requirement in the CFC  app  roach. There are |Ξ| = 4 (m11 m22 − m12 m21 ) β1 β2 / π 2 1 + z12 1 + z22 and βi (t) > 0, it is easy to know that m11 m22 − m12 m21 > 0, so Ξ is reversible. The control problem of original AWPV system, which stated in Sect. 2.3, can be rephrased as rendering the transformed AWPV system to adhere the following servo constraints z˙i (t) + ζi zi (t) = 0, i = 1, 2,

(11)

The first and second order forms of the constraints can be separately written in the following matrix form z˙ = c (z) , (12) z¨ = b (z) ˙ , T

(13) T

where c = [−ζ1 x1 , −ζ2 x2 ] , b = [−ζ1 x˙ 1 , −ζ2 x˙ 2 ] , and constants ζ1,2 ∈ (0, ∞). 3.2

Tracking Control for Nominal Transformed AWPV

The Ξ, Ψ and Λ in system (9) can be decomposed as: ⎧ N Δ ⎪ ⎨ Ξ(z, δ, t) = Ξ (z, t) + Ξ (z, δ, t) ˙ z, t) + Ψ Δ (z, ˙ z, δ, t) Ψ (z, ˙ z, δ, t) = Ψ N (z, ⎪ ⎩ N Δ Λ(z, δ, t) = Λ (z, t) + Λ (z, δ, t)

(14)

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where superscripts N and Δ stand for nominal portions and uncertain portions, respectively. Since the parameters of the nominal system are determined by the designer, so Ξ N > 0 is always feasible. The nominal system of transformed AWPV (9) can be expressed as Ξ N (z, t)¨ z (t) + Ψ N (z, ˙ z, t)z(t) ˙ + ΛN (z, t) = u(t).

(15)

Theorem 1. The system (15) is servo constraint controllable [18] with respect to constraint (13) for all (z, ˙ z, t) ∈ R2 × R2 × R and the control is given by    (16) ˙ z, t) = D−1 (z, t) b(z) ˙ + D(z, t) Ψ N (z, ˙ z, t)z(t) ˙ + ΛN (z, t) , τ1 (z,  −1 where D(z, t) := Ξ N (z, t) . 3.3

Deal with Initial State Deviation

Based on the first-order form of constraints (12), let δ (z, ˙ z) := z˙ − c (z) and the control for system (15) to deal with initial state deviation can be designed as: τ2 (z, ˙ z, t) = −κΞ N (z, t)Q−1 δ(z, ˙ z),

(17)

where κ ∈ (0, ∞) is constant; Q ∈ R2×2 is a given matrix and Q > 0. The tracking control for the nominal transformed AWPV (15) containing initial state deviation is proposed as (18) uCT 2 = τ1 + τ2 .

Theorem 2. The control in (18) renders system performance δ globally exponentially stable [28, Definition 3.3]. 3.4

Adaptive Robust Design for Transformed AWPV Accounting for Uncertainties

Next, the uncertainty of the transformed AWPV system will be dealt with by designing adaptive robust control. Define −1 (19) E(z, υ, t) := Ξ N (z, t) × (Ξ(z, υ, t)) − I,   −1 −1 ΔD(z, υ, t) := (Ξ(z, υ, t)) − Ξ N (z, t) = D(z, t)E(z, υ, t). (20) Assumption 1. Let W :=

T   1 min λmin QΔDΞ N Q−1 + QΔDΞ N Q−1 , 2 υ∈Υ

(21)

where λmin denotes the minimum eighenvalue of the matrix. There is a constant ˙ z, t) ∈ R2 × R2 × R, there is: δW > −1 , which maybe unknown, then for all (z, W ≥ δW ,

(22)

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Remark 3. The constant δW is typically elusive as the uncertainty bound Υ remains unknown. In the exceptional circumstance where Ξ = Ξ N , which means there no uncertainty for system, ΔD = 0 and W = 0, permitting the choice of δW = 0. Consequently, by virtue of continuity, the assumption 1 can indicate the effect of uncertainties on the possible deviation of Ξ from Ξ N to be within a certain threshold, which is unidirectional. It is noted that the terms in Ξ, Ψ z˙ + Λ are either constant, states and velocik ties, or their quadratic. As [12], there is an unknown constant vector α ∈ (0, ∞) , k = 3 and a known function( which may be interpreted as the uncertainty bound) ˙ + 1)2 + α2 (z + 1)2 + α3 H(α, z, ˙ z, t) =α1 (z   T ˙ + 1)2 (z + 1)2 1 = α1 α2 α3 (z ˜ z, ˙ z, t) =αT H( such that for all (z, ˙ z, t) ∈ R2 × R2 × R,    −1 (1 + δW ) max QΔD (−Ψ z˙ − Λ + τ1 + τ2 ) + QD −Ψ Δ z˙ − ΛΔ  υ∈Υ

≤ H (α, z, ˙ z, t) .

(23)

(24)

Now, the control can be updated as uCT 3 = τ1 + τ2 + τ3 .

(25)

and the τ3 , which is designed for uncertainties, is given by: τ3 (˜ α, z, ˙ z, t) = −Ξ N (z, t)Q−1 ε (˜ α, z, ˙ z, t) μ (˜ α, z, ˙ z, t) H (˜ α, z, ˙ z, t) ,

(26)

where μ (˜ α, z, ˙ z, t) =δ (z, ˙ z) H (˜ α, z, ˙ z, t) ,  1/ μ (˜ α, z, ˙ z, t) if μ > ψ ε (α, ˜ z, ˙ z, t) = , 1/ψ if μ ≤ ψ

(27) (28)

The ψ is a constant defined as positive, and the α, ˜ which has a positive initial value(˜ αi (t0 ) > 0, i = 1, 2, 3), is the online estimate of α. The α is related to the uncertainty bounds, and the α ˜ is obtained by the following adaptive law: ˜ δ − k2 α α ˜˙ = k1 H ˜.

(29)

where k1 > 0 and k2 > 0. Remark 4. The adaptive law (29) is designed to estimate the unknown uncertainty bounds online, thus designed control is adaptive robust. In scenarios where the trajectory tracking error is large, α will increase to compensate the uncertainties. Conversely, the α will decreases to prevent the control effort from being excessively large when the tracking error is small enough.

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T  T Theorem 3. Define φ := δ T , (˜ α − α) ∈ R2+3 . With the control in (25), the system (9) obtains the following stability properties:(i) uniform boundedness(UB) and (ii) uniform ultimate boundedness (UUB): (i) For any r > 0, there is a d (r) < ∞ such that if φ (t0 ) ≤ r, then φ (t) ≤ d (r) for all t ≥ t0 ; (ii) For any r > 0 with φ (t0 ) ≤ r, there ¯ r , where is a d > 0 such that φ (t) ≤ d¯ for any d¯ > d as t ≥ t0 + T d, ¯ r < ∞. 0 ≤ T d, Remark 5. Theorem 3 provides assurance that the proposed control ensures the boundedness of δ. This signifies that the proposed control will facilitate the convergence of the system motion to (11), thereby satisfying (4) and (5) consistently. Remark 6. It can be found that the control design presented in this paper is free of any linearization and linear approximation. Furthermore, the system (1) also can converge to servo constraint (4) with the control proposed in Sects. 3.2, 3.3 and 3.4. However, without the help of diffeomorphism approach, which is proposed in Sect. 3.1, the PTSSP will not be guaranteed. This assertion will be demonstrated in the subsequent performance validation. The detailed control design process is illustrated in Fig. 4. The desired trajectory is formulated as equality constraints, and the PTSSP is formulated as inequality constraints, which is transformed into equality one by utilizing diffeomorphism approach.

Fig. 4. Control design flowchart.

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Performance Validation

In this section, performance of AWPV by various CFC method is studied. The control objective is to accomplish the task of moving the platform from an initial angle of θ10 =5◦ to a desired angle of θ1d = 75◦ within a duration of td = 60s, while maintaining the platform in an upright position with θ20 =115◦ and θ2d =45◦ . The desired PTSSP for tracking errors, which can be obtained using (5) with the following parameters: β10 = 10◦ , β1∞ = 2◦ , β20 = 8◦ and β2∞ = 1◦ . The remaining parameters are as follows: mp = 200 kg, mb1 = 1000 kg, mb2 = 400 kg, L1 = 15 m, L2 = 4 m and ϑ = 120◦ . 4.1

CFC for Nominal AWPV

The initial speed is θ˙10 =θ˙20 =0◦ /s. A CFC by τ1 in (16) alone is applied for the nominal original AWPV system in (1), without initial state deviation or uncertainties. The states θ1 and θ2 under CFC is illustrated in Fig. 5 (a), which also shows the desired trajectories Θ1 and Θ2 obtained by (3). It can be seen that by CFC the states follow the desired trajectories perfectly throughout luffing process. The results by LQR are also drawn in Fig. 5, as a comparison. LQR cannot drive states to follow the desired trajectories because of the complex nonlinearity of AWPV. The performance index of original AWPV system is represented by the tracking errors Δ1 and Δ2 , drawn in Fig. 5 (b). The PTSSP β1 and β2 are the limitation for the tracking errors. It can be seen from Fig. 5 (b) that the CFC can guarantee the nominal system following PTSSP, but the LQR generates unacceptable tracking errors. In total, the CFC alone can drive the motion of nominal AWPV tracking the desired trajectories and satisfying the PTSSP at the same time, without initial state deviation or uncertainties.

(a)

(b)

Fig. 5. (a) Luffing angles and (b) tracking errors of nominal AWPV system without initial state deviation or uncertainties.

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AWPV with Initial State Deviation and Uncertainties

In this section, both initial state deviations and uncertainties are taken into accounting. Meanwhile, the same tracking task as before is considered. The parameters with uncertain values include: (i)the uncertainties for masses, mΔ b = Δ N Δ N · sin t, m = 0.02m and m = 0.01m ; (ii)for structure length, 0.02mN b b1 b1 b2 b2 N Δ N LΔ 1 = 0.02L1 sin (t + π/4) and L2 = 0.01L2 sin (t + π/3); and (iii)for fixed angel, ϑΔ = 0.01ϑN .

(a)

(b)

Fig. 6. (a) Tracking error and (b) control force with initial deviation and uncertainties.

Three different control methods are applied to drive the AWPV system finishing the task: CT3 means the proposed control (25) in this study for the transformed AWPV (9) with diffeomorphism approach; C3 means a similar tracking control, however, for the original AWPV (1) without diffeomorphism approach; C2 is a tracking control by (18) for the original AWPV (1) without diffeomorphism approach. The control parameters are chosen as κ = 0.2, k1 = 0.2 and k2 = 1. The tracking errors Δ1 and Δ2 are drawn in Fig. 6(a), which also shows the PTSSP β1 and β2 , limitation for the tracking errors. It can be seen that the proposed CT3 can guarantee the AWPV following the desired trajectories as well as satisfying the PTSSP, and the interference of initial state deviation and uncertainties can be suppressed. C3 can also reduce the disturbance of initial state deviation and uncertainties, but the PTSSP cannot be satisfied. Meanwhile, the Fig. 6(a) show that C2 causes large tracking error under uncertainties. Figure 6(b) shows the control forces by these methods. It can be seen that CT3 and C3 generate modest forces compared to CT2. In other words, without the help of τ3 in (26), more control forces result in worse effects. In addition, CT3 and C3 generate similar forces even though only CT3 can satisfy the PTSSP.

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In summary, the proposed control method in this study can successfully guarantee the PTSSP when tracking the desired trajectories in the presence of initial state deviation and uncertainties.

5

Conclusions

The trajectory-tracking control for uncertain AWPV with PTSSP requirement is investigated. A novel constraint-based adaptive robust control is proposed to robustly achieve trajectory-tracking as well as guarantee PTSSP, even in the presence of (possibly fast and irregularly) time-variant uncertainties with unknown bounds. The desired trajectories and the PTSSP are formulated as equality and inequality servo constraints, respectively. By deliberately introducing a diffeomorphism approach, the inequality servo constraints “disappears” in the control design and the adaptive robust CFC is only based on the new yielded equality servo constraints. It is emphasized that this could be for the first time to study the AWPV tracking control with PTSSP guaranteed, associated with uncertainties of mass, length and angles, which may be (possibly fast and irregularly) time-variant and their bounds are unknown. The Lyapunov-based analysis proves that the proposed adaptive robust control is able to achieve trajectorytracking and satisfy PTSSP for the AWPV with uncertainties. The comparisons of results for different control methods show the effectiveness of the proposed adaptive robust tracking control for AWPV. Future work on the optimization of designed tunable control parameters (e.g. κ, k1 and k2 ) is worth-pursuing.

Appendix   m11 =mp (L2 − L1 )2 + 2L1 (L1 − L2 Cθ2 )   + L21 mb1 /3 + mb2 2L22 + 6L1 (L1 − L2 Cθ2 ) /6 m12 = L2 (2L2 (3mp + mb2 ) − 3L1 Cθ2 (2mp + mb2 )) /6 m21 = L2 (2L2 (3mp + mb2 ) − 3L1 Cθ2 (2mp + mb2 )) /6 m22 = L22 (3mp + mb2 ) /3 where Cθ2 = cos θ2 , Cθ1 = cos θ1 , Sθ2 = sin θ2 , Sθ1 = sin θ1 . C11 = L1 L2 (2mp + mb2 ) Sθ2 θ2 1 L1 L2 (2mp + mb2 ) Sθ2 θ2 2 1 = − L1 L2 (2mp + mb2 ) Sθ2 θ2 2 C22 = 0;

C12 = C21

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1 g1 =gL1 mp Cθ1 + gL1 Cθ1 mb1 + gL1 Cθ1 mb2 2 1 + gL2 (2mp + mb2 ) (Sθ1 Sθ2 − Cθ1 Cθ2 ) 2 1 g2 = lg (2mp + mb2 ) (Sθ1 Sθ2 − Cθ1 Cθ2 ) 2

References 1. Krasucki, J., Rostkowski, A., Gozdek, L  , Barty´s, M.: Control strategy of the hybrid drive for vehicle mounted aerial work platform. Autom. Constr. 18, 130–138 (2009) 2. Deng, M., Ji, A., Zhang, L., Wang, H., Zhao, Z.: Lateral vibration behaviors of the straight boom of the aerial work platform in the horizontal linear motion. Autom. Constr. 134, 104095 (2022) 3. Hu, H., Song, Y., Fan, P., Diao, C., Cai, N.: A backstepping controller with the RBF neural network for folding-boom aerial work platform. Complexity 2022, 4289111 (2022) 4. Miao, M., Yuan, H., Song, X., Shen, Z., Zhao, W.: Folding-boom aerial working vehicle tracking and control. Chin. J. Constr. Mach. 4, 319–323 (2013) 5. Zhang, Z., Dhanasekar, M., Ling, L., Thambiratnam, D.P.: Effectiveness of a raised road: rail crossing for the safety of road vehicle occupants. Eng. Fail. Anal. 97, 258–273 (2019) 6. Zhang, Z., Qin, A., Zhang, J., Zhang, B., Fan, Q., Zhang, N.: Fuzzy sampled-data H∞ sliding-mode control for active hysteretic suspension of commercial vehicles with unknown actuator-disturbance. Control. Eng. Pract. 117, 104940 (2021) 7. Sun, S., et al.: A new generation of magnetorheological vehicle suspension system with tunable stiffness and damping characteristics. IEEE Trans. Industr. Inf. 15, 4696–4708 (2019) 8. Wang, C., Wang, Z., Zhang, L., Cao, D., Dorrell, D.G.: A vehicle rollover evaluation system based on enabling state and parameter estimation. IEEE Trans. Industr. Inf. 17, 4003–4013 (2020) 9. Jia, P., Li, E., Liang, Z., Qiang, Y.: Adaptive neural network control of an aerial work platform’s arm. In: Proceedings of the 10th World Congress on Intelligent Control and Automation, pp. 3567–3570. IEEE (2012) 10. Hu, H., Cai, N., Cui, L., Ren, Y., Yu, W.: A neural network-based sliding mode controller of folding-boom aerial work platform. Adv. Mech. Eng. 9, 1687814017720876 (2017) 11. Chen, Y.-H., Zhang, X.: Adaptive robust approximate constraint-following control for mechanical systems. J. Franklin Inst. 347, 69–86 (2010) 12. Yin, H., Chen, Y.-H., Yu, D., L¨ u, H., Shangguan, W.: Adaptive robust control for a soft robotic snake: a smooth-zone approach. Appl. Math. Model. 80, 454–471 (2020) 13. Sun, H., Yang, L., Chen, Y., Zhao, H.: Constraint-based control design for uncertain underactuated mechanical system: leakage-type adaptation mechanism. IEEE Trans. Syst. Man Cybern. Syst. 51, 7663–7674 (2020) 14. Mehrasa, M., Babaie, M., Zafari, A., Al-Haddad, K.: Passivity ANFIS-based control for an intelligent compact multilevel converter. IEEE Trans. Industr. Inf. 17, 5141–5151 (2021)

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15. Chen, Y.-H.: Constraint-following servo control design for mechanical systems. J. Vib. Control 15, 369–389 (2009) 16. Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. Roy. Soc. Math. Phys. Eng. Sci. 439, 407–410 (1992) 17. Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach (1996) 18. Yin, H., Chen, Y.-H., Huang, J., L¨ u, H.: Tackling mismatched uncertainty in robust constraint-following control of underactuated systems. Inf. Sci. 520, 337–352 (2020) 19. Yin, H., Chen, Y.-H., Yu, D.: Fuzzy dynamical system approach for a dualparameter hybrid-order robust control design. Fuzzy Sets Syst. 392, 136–153 (2020) 20. Yin, H., Chen, Y.-H., Yu, D.: Vehicle motion control under equality and inequality constraints: a diffeomorphism approach. Nonlinear Dyn. 95, 175–194 (2019) 21. Zhang, Z., Zhang, J., Dai, J., Zhang, B., Qi, H.: A Fusion algorithm for estimating time-independent/-dependent parameters and states. Sensors 21, 4068 (2021) 22. Pan, Y., Yang, C., Pan, L., Yu, H.: Integral sliding mode control: performance, modification, and improvement. IEEE Trans. Industr. Inf. 14, 3087–3096 (2017) 23. Yuan, Q., Lew, J., Piyabongkarn, D.: Motion control of an aerial work platform. In: 2009 American Control Conference, pp. 2873–2878. IEEE (2009) 24. Yin, H., Chen, Y.-H., Yu, D., L¨ u, H.: Nash-game-oriented optimal design in controlling fuzzy dynamical systems. IEEE Trans. Fuzzy Syst. 27, 1659–1673 (2019) 25. Qin, W., Liu, F., Yin, H., Huang, J.: Constraint-based adaptive robust control for active suspension systems under sky-hook model. IEEE Trans. Ind. Electron. 69, 5152–5164 (2021) ´ D` 26. Acosta, J.A., oria-Cerezo, A., Fossas, E.: Diffeomorphism-based control of nonlinear systems subject to state constraints with actual applications. In: 2014 IEEE Conference on Control Applications (CCA), pp. 923–928. IEEE (2014) 27. Ouyang, H., Tian, Z., Yu, L., Zhang, G.: Motion planning approach for payload swing reduction in tower cranes with double-pendulum effect. J. Franklin Inst. 357, 8299–8320 (2020)

A Novel Model Predictive Control Strategy for Continuum Robot: Optimization and Application Yakang Wang, Yuzhe Qian(B) , and Weipeng Liu Hebei University of Technology, College of Artificial Intelligence, Tianjin 300130, China [email protected] Abstract. It is very important for continuous robots to achieve accurate and rapid control. However, the current continuum robot control faces many challenges. First, they often have complex nonlinear dynamics, including kinematics and dynamics equations, which makes it difficult to build accurate models, and conventional control methods do not work well on these complex systems. Secondly, the motion of a continuous robot system is continuous and coherent, requiring real-time control strategies to maintain stability and accuracy. These problems bring great challenges to the control of continuum robots. In this paper, the nonlinear system of continuum robot is modeled, and then the real-time control of continuum robot is realized by the model predictive control method. The real-time control problem of continuum robot is effectively solved and satisfactory control effect is obtained. In other words, for the continuous robot system, nonlinear modeling is first carried out, and then a new linear model of the system is obtained by linearizing and discretizing the nonlinear system model with feedback linearization method. On this basis, model prediction method is applied to the linearized model to achieve effective control of the target Angle. By solving the constraints of the model predictive control method, the control problem of the target Angle is successfully realized. The simulation also verifies the feasibility of the model predictive control method, which can realize the target Angle approaching the target value quickly. Compared with the traditional PD control method, the superiority of the model predictive control method is also proved. Keywords: Continuum robot · Model predictive control · Nonlinear model · Optimal control · System linearization · Dynamic modeling

1

Introduction

In recent years, with the development of science and technology, how to make high-risk activities safer has become the research goal of many researchers. More This work was supported by the National Natural Science Foundation of China under Grant 62103128. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 484–497, 2024. https://doi.org/10.1007/978-981-97-0554-2_37

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and more bionic robots begin to appear and assist humans in high-risk work to improve work efficiency and protect workers’ safety [1–4]. One kind of bionic robot is invented and applied based on octopus tentacles [5–7]. This kind of robot has a high degree of flexibility and can flexibly realize various control objectives at the end of the robot. This kind of robot is often called “continuum robot” [8–10]. Because of the high flexibility of the continuum robot, it can realize the functions of detection, search and rescue in the disaster site. In addition, continuum robots are also used. in underwater operations, pipeline operations, and minimally invasive surgery in medicine [11]. The control object of this paper is the pulmonary nodule surgical robot, and the pictures of the surgical robot are shown in Fig. 1. As a continuum robot, how to accurately and quickly achieve the control goal has become a focus of many researchers. Compared with traditional rigid link robots, continuous robots have jointless continuous skeleton, redundant degrees of freedom and significant compliance [12–14], which provide excellent operational capabilities in environmental interaction and object manipulation, but these characteristics also make system control challenging. The control system studied in this paper has nonlinear dynamics characteristics, and there are uncertainties or perturbations, so the optimal control needs to be achieved through prediction and optimization, and the traditional control methods can’t meet the performance requirements of the system. By calculating kinetic energy and elastic potential energy of a continuum robot, An et al. proposed a dynamic model of a continuum robot [16–18]. In [15,26], Chen et al. applied model predictive control to crane and other control models, and achieved satisfactory control effects for these models that also required precise control. Liu et al. introduced the relevant content of nonlinear model predictive control in [19–21]. In [22–25], feedback linearization is used to linearize nonlinear systems, and the linearization model is not much different from the original model by calculation.

Fig. 1. Surgical robot pictures.

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Based on the above problems, this paper uses Euler-Bernoulli beam equation and Lagrange equation to model the continuum robot, and obtains a nonlinear model with universality. Then the nonlinear model is linearly discretized to obtain the linearized model of the system, and finally it is applied to the model predictive control to achieve the optimal control action in the nonlinear system. The structure of the article is as follows. In Sect. 1, the lung nodule robot is briefly described, and the lung nodule surgical robot is classified as a continuum robot. In Sect. 2, the dynamic model of the continuum robot is established. In Sect. 3, the system model is linearized and controlled by model prediction method. In Sect. 4, the simulation results of the system show that the model predictive control is better than the traditional PD control method. The fifth chapter is the conclusion.

2

Dynamic Model

In this section, we will model and analyze the continuum robot. The continuum robot studied in this paper consists of a main skeleton, a secondary skeleton and a disk. We first drew the geometric model of the continuum robot, as shown in Fig. 2. In order to more clearly explain the structure of the continuum robot, we omitted the secondary skeleton of the robot in the geometric model, because the kinetic energy of the secondary skeleton is the same as that of the main skeleton.

Fig. 2. Geometric model of continuum robot

The meanings of symbols and representatives in Fig. 3 are shown in Table 1. The structure of a continuum robot is composed of a main skeleton, a subskeleton and a disk. Therefore, the kinetic energy of a continuum robot consists of these three components. Suppose there is a point R on the base disk, and the coordinates of the point R is:

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Table 1. Parameters of geometric model of continuum robot Symbol Representative meaning arc-length parameter of the segment OR (s=0 at the base disk and s=l at the end disk) the distance from the primary backbone to each secondary backbone on the disk curvature radius of the primary backbone, defined in the bending plane the bending angle of the primary backbone tangent in the X1 Z1 plane at the point R. β is the bending angle at the end disk the rotation angle of the bending plane division angle (θ = 2π , is the number of secondary n backbones) the total mass of the primary backbone the total mass of the secondary backbone distance between adjacent disks

s r ro βR

γ θ m1 m2 h

P = [x

y

z]T = [

s (1 − cos βp ) cos γ βp

s (1 − cos βp ) sin γ βp

s sin β]T (1) βp

First, the kinetic energy of the main skeleton is calculated, and the derivative of Eq. (1) is obtained. The velocity of point R is: ⎧ sβ sβ dβ sβ dγ dx 1 l l ⎪ ⎪ dt = β [s sin l cos γ − β (1 − cos l ) cos γ] dt − β (1 − cos l ) sin γ] dt ⎪ ⎪ ⎪ ⎨ dy sβ sβ dβ sβ dγ 1 l l dt = β [s sin l sin γ − β (1 − cos l ) sin γ] dt + β (1 − cos l ) cos γ] dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dz = 1 (s cos sβ − l sin sβ ) dβ dt β l β l dt

(2)

From the velocity obtained by Eq. (2), the kinetic energy of the main skeleton can be obtained: Ek1

1 = 2



l

[( 0

dx 2 dy dz ) + ( )2 + ( )2 ]ρAdx dt dt dt

(3)

where ρ is the density of the primary backbone and A is the cross-section area of the primary backbone. Substitute Eqs. (2) into (3) to get: Ek1 =

1 dβ 1 dy m1 l2 ( )2 K1 + m1 l2 ( )2 K2 6 dt 8 dt

(4)

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K1 = (β 3 + 6β − 12 sin β + 6β cos β)/β 5

(5)

K2 = (6β − 8 sin β + sin 2β)/β 3

(6)

where m1 is the mass of the primary backbone and K1 and K2 are the kinetic energy equivalent factor. The kinetic energy of the secondary skeleton is composed of two parts, and the kinetic energy of the first part is the same as that of the main skeleton. The second part of the kinetic energy comes from the driving kinetic energy: Ek2 =

dβ dγ 1 dβ 1 dγ 1 m1 (l2 K1 + K3 )( )2 + m1 (3l2 K2 + 4K5 )( )2 + m1 K4 (7) 2 dt 8 dt 2 dt dt K3 = r2 [cos2 (γ) + cos2 (−γ + θ) + cos2 (γ + θ)]

(8)

K4 = r2 β[− sin 2γ + sin 2(−γ + θ) − sin 2(γ + θ)]

(9)

K5 = r2 β 2 [sin2 (γ) + sin2 (−γ + θ) + sin2 (γ + θ)]

(10)

where m1 is the mass of the secondary backbone and K3 , K4 and K5 are kinetic energy equivalent factors. Finally, the kinetic energy of the gaskets and end disks is required, and since all disks are attached to the main trunk, the speed of the disks can be derived from Eq. (1). The speed of each disk can be expressed as: ⎧ khβ khβ dβ khβ dγ dx 1 l l ⎪ ⎪ dt = β [kh sin l cos γ − β (1 − cos l ) cos γ] dt − β (1 − cos l ) sin γ] dt ⎪ ⎪ ⎪ ⎨ dy khβ khβ dβ khβ dγ 1 l l dt = β [kh sin l sin γ − β (1 − cos l ) sin γ] dt + β (1 − cos l ) cos γ] dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dz = 1 (kh cos khβ − l sin khβ ) dβ dt β l β l dt (11) The total kinetic energy of all gaskets and end plates can be expressed as: Ek3 =

n  1 k=1

2

m2 [(

dx 2 dy dz ) + ( )2 + ( )2 ] dt dt dt

(12)

Substitute Eqs. (11) into (12) to get: Ek3 = K6 =

1 dβ 1 dγ m2 ( )2 K6 + m2 ( )2 K7 2 dt 2 dt

h2 n(n+1)(2n+1) 6β 2

− βhl3

2

+ (2n + 1) βl 4 − (n+1) sin β−n sin β 2 sin2 2n

(13) (n+1)β n

l2 sin β+sin β4 sin β

(n+1)β n

n

(14)

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K7 =

(n+1)β sin 2β + sin 2(n+1)β l2 3n 3 sin β + sin n n + − [ + +] β2 2 4 sin nβ 4 sin 2β n

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(15)

where, m2 is the mass of a disk and K6 and K7 are kinetic energy equivalent factors. By combining formulas (4) (7) (13), the total kinetic energy is obtained: 2 Ek = 16 (4m1 l2 K1 + 3m1 K3 + 3m2 K6 )( dβ dt ) dy 2 dγ 1 2 + 2 (m1 l K2 + m1 K5 + m2 K7 )( dt ) + 12 m1 K4 dβ dt dt

(16)

After calculating the sum of kinetic energy of the three parts, it is necessary to continue to calculate the elastic potential energy of the continuum robot, which is given by Eq. (14): 

l

EP = 4 0

EI dβp 2 2EI 2 ( ) ds = β 2 ds l

(17)

where EI is the stiffness of the material,βp ,=sβ/l as mentioned above. The Lagrange equation is expressed as: ∂Ek ∂Ep d ∂EK − + = Qj , (j = 1, 2) dt ∂ p˙j ∂pj ∂pj

(18)

By substituting Eqs. (16) and (17) into (18), the dynamic model of the continuum robot is finally obtained:    V V g M11 M12 q¨ + 11 12 q˙ + 1 = u (19) M21 M22 V21 V22 g2

T q= βγ

(20)

where M is the inertia matrix of the system, V is the matrix related to the Coriolis centripetal force of the system, and G is the matrix related to the elasticity of the system.

3 3.1

Model Predictive Control Method Analysis Linearization and Discretization of the System

At present, model prediction methods (MPC) are mostly applied to linear systems, therefore, it is necessary to transform the nonlinear system model of continuous robot into a linear system model. Specifically, the discrete system model is obtained by linearizing and discretizing the dynamic model of the continuum robot obtained in Sect. 2. On this basis, the model predictive control analysis is carried out to ensure the satisfactory performance of the system. Since the G matrix is full of linear determinations, we move G to the right side of the equation, Eq. (19) can be expressed as:

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M q¨ + V q˙ = U

(21)

In this paper, the nonlinear state feedback method and inverse dynamic control are used to linearize the system dynamics. The following control rate can be obtained by nonlinear compensation: ˆ v + Vˆ q˙ U =M

(22)

ˆ and Vˆ represent the estimators of M and V , respectively, which are where M calculated based on measurements of the system state, while v is the new control input to be designed. Usually due to the inevitable need to make some approximations during modeling, or intentional simplification of compensation operations, there are: ˆ = M + ΔM M

(23)

Vˆ q˙ = V q˙ + ΔV q˙

(24)

By substituting Eqs. (22) into (21) parallel Eqs. (23) and (24), we can get: q¨ = M −1 (M + ΔM )v + M −1 ΔV q˙ = v + η

(25)

where η is a nonlinear function of the system state: η = M −1 ΔM v + M −1 ΔV q˙

(26)

If you want the target angle is (qd , q˙d , q¨d ), the angle error can be defined as ˙ the second derivative can be obtained e = qd −q, the first derivative is e˙ = q˙d − q, from equation (26). e¨ = q¨d − v − η

(27)

The derived error dynamic equation is used to study the convergence from the actual state to the expected state. For this purpose, the design system state is: ξ = [e, e] ˙T

(28)

In this paper, in order to simplify the system and facilitate processing, the input and output of the design are usually linearized expressions, that is, the design input v is a linear combination of tracking Angle and feedback. v = q¨d + w(ξ)

(29)

The error state equation is obtained by combining the Eq. (27): ξ˙ = F ξ − Gw(ξ) − Gη

(30)

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where F and G need to obey formula (29), having the following form:   0I 0 F = G= 00 I

491

(31)

For complete nonlinear compensation (η(q, q) ˙ = 0), the control method is linear control as follows: 

w(ξ) = −KP e − KD e˙ = −KP −KD ξ

(32)

where KP and KD are diagonal matrices. By simplifying, we get: e¨ − KD e˙ − KP e = 0

(33)

Through calculation, the linearized equation is finally obtained: x˙m = Axm + Bu

(34)

y = xm

(35)

where  A=

 0m Im 0m×1 B = −M −1 KP −M −1 KD −M −1 e

(36)

Now select the sampling period T, and Eqs. (35) and (36) can be discretized as follows: xm (k + 1) = AP xm (k) + BP u(k) y(k) = CP xm (k), CP = I 3.2

(37)

Stability Analysis and Optimal Control

In Sect. 2, we obtain the dynamic model of the continuum robot, and in the first section of the third chapter, we linearize and discretize the continuum robot system, and get the discrete model of the system. In this section, the system is optimized to reach the target position quickly, and its stability is proved. By performing the difference operation on Eq. (38) of the equation, we can get: Δxm (k + 1) = AP Δxm (k) + BP Δu(k)

(38)

y(k + 1) = CP Δxm (k + 1) = CP AP Δxm (k) + CP BP Δu(k)

(39)

According to the superposition principle of linear systems, the output of the system is: y(k) = a1 u(k − 1) + a2 u(k − 2) + · · · · · · + am u(k − m)

(40)

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The incremental value of the system is: Δy(k) = a1 Δu(k − 1) + a2 Δu(k − 2) + · · · · · · + am Δu(k − m)

(41)

The cost function of the model is: J0∗ (x0 ) = minU0 p(xN ) +

N −1 i=0

q(xi , ui )

(42)

p(xN ) = xTN P xN

(43)

q(xi , ui ) = xTi Qxi + uTi Rui

(44)

The first term of the cost function is the terminal cost, and the second term is the stage cost. In this paper, we assume that the terminal constraint of the continuum robot is 0, Xn is 0. The goal of the proof of stability is to prove that the cost function is a Lyapunov function: J0∗ (x1 ) − J0∗ (x0 ) < 0, x0 = 0

(45)

Derivation process: J0∗ (x1 ) = minU0

N

i=1 q(xi , ui )

(46)

Substitute Eqs. (42) into (46) to get: J0∗ (x1 ) = J0∗ (x0 ) − q(x0 , u∗o ) + q(0, 0)

(47)

where q(0, 0) = 0, the stability is proved.

4

Simulation Result

In this section, we conduct simulation tests of model predictive control and PD control in MATLAB/Simulink environment, and compare the two methods. During the simulation of MPC, the sampling time was selected as 0.1 s,T = 0.1s, and the exact values of relevant parameters were shown in Table 2. The simulation results are shown in the figure. From Figs. 3, 4 and 5, the solid blue line represents the simulation value, the red dashed line represents the target value, the value of the upper subgraph is the Angle, the value of the lower subgraph is the angular velocity. We can find that the proposed model predictive control method successfully achieves the control goal of accurately and quickly reaching the required Angle. As is shown in Fig. 3, using the model predictive control method, the simulation results require control angles from 0 to 1. As is shown in Fig. 4, using the model predictive control method, the simulation results require control angles from 1 to 0.

A Novel Model Predictive Control Strategy Table 2. Parameters of geometric model of continuum robot Symbol

Data size

l

150

r

3

a

2π/3

I

0.102

m1

1.086

m2

0.848

E

65

Fig. 3. The velocity and angular velocity of the system

Fig. 4. The velocity and angular velocity of the system

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Fig. 5. The velocity and angular velocity of the system

Fig. 6. The velocity and angular velocity of the system after adding interference

As is shown in Fig. 5, using the model predictive control method, the simulation results require control angles from −1 to 1. Using the model predictive control method, the simulation results require control angles from 0 to 1. At 5 s, a disturbance was input to the system, and the system successfully recovered to a stable state in a relatively fast time, as shown in Fig. 6. In addition, the model predictive control method is compared with the common PD control method. It can be seen from the simulation effect diagram

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Fig. 7. Comparative simulation of model predictive control and PD control

that the model predictive control method is closer to the target value than the traditional PD control method, and the simulation results are shown in Fig. 7. The simulation results require control angles from 0 to 1. The blue solid line represents the model predictive control simulation value, the green dotted line represents the PD control simulation value, the red dotted line represents the target value, the value of the upper subgraph is the Angle, and the value of the subgraph is the angular velocity.

5

Conclusion

In this paper, the MPC method is applied to the continuous robot system, which has a good performance in tracking the Angle of the robot arm. When the lung nodule surgical robot is working, the doctor gives the command, the robot can respond quickly. The simulation results show that the performance of the proposed model predictive control method is better than that of the traditional PD control method. In the future, our further work is to study the anti-interference performance of the lung nodule surgery machine, mainly to prevent the doctor from mistouching when manipulating the robot. In addition, the stability analysis of model predictive control methods with constraints is a challenging problem, and we will continue to study this problem in future work and hope to achieve results.

References 1. Briot, S., Boyer, F.: A geometrically exact assumed strain modes approach for the geometrico- and kinemato-static modelings of continuum parallel robots. IEEE Trans. Rob. 39(2), 1527–1543 (2023)

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Finite Element Analysis and Error Compensation for Wrinkled Bellow-Like Soft Robotic Manipulator Kinematics Modeling Haibin Huang1,2 , Yingjie Li1,2 , Yingbo Huang1,2(B) , and Jing Na1,2 1 Faculty of Mechanical and Electrical Engineering, Kunming University of Science and

Technology, Kunming 650500, People’s Republic of China [email protected] 2 Yunnan International Joint Laboratory of Intelligent Control and Application of Advanced Equipment, Kunming 650500, People’s Republic of China

Abstract. In this paper, a finite element analysis (FEA) and error compensation method are conducted for the wrinkled bellow-like soft robotic manipulator (WBSRM) kinematics modeling. Firstly, WBSRM is designed, which has good axial expansibility and radial bending deformation. The link and restriction of the interference plate make the WBSRM have better motion effect and can be better used implemented for FEA. The wrinkled bellow actuator has good axial scalability. Then, the model of WBSRM based on the constant small curvature (CSC) methodology is established. To verify the accuracy of the established kinematics model, the FEA approach is employed, in which the local coordinate system is established to obtain the local coordinate value and the curve length is calculated by curve fitting and integration, such that the idealized constraint limit of the common actuator length can be avoided. Finally, the FEA simulation result and the established kinematics model are compared and analyzed. Concerning on the modeling error between the established mathematical model and the simulation result, the back propagation neural network (BPNN) technique is introduced to realize the error compensation. The obtained method results show that the compensation method can effectively eliminate the modeling error and then dedicates to characterize the detailed kinematics model of the WBSRM. Keywords: Wrinkled bellow-like soft robotic manipulator (WBSRM) · Kinematics modeling · Finite element analysis (FEA) · Back propagation neural network (BPNN)

1 Introduction In recent years, with the advancements of material science and biomimetic technology, the development of soft robotics has further progressed, and the role of soft robotics in the field of robotics is becoming increasingly significant [1]. Inspired by creatures in nature, particularly the invertebrates such as jellyfish and octopuses, which can achieve flexible movements through their flexibility, the design of soft robotics has inspired a © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 498–513, 2024. https://doi.org/10.1007/978-981-97-0554-2_38

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growing number of researchers to work on the research and development of soft robotics. Different from traditional rigid robotics, soft robotics possess superior compliance and infinite degrees of freedom. They can adapt to a variety of forms and can be changed according to the demands of the usage environment, making their applications more extensive in human-robot interactive environments, such as in healthcare, post-disaster rescue, and other areas [2, 3]. However, the development and application of soft robotics still face many challenges concerning on the modeling and control compared with the traditional rigid-bodies robotics [4]. Walker et al. [5] first proposed that the OctArm continuum manipulator to the OctArm VI [6] are constantly being optimized. The OctArm continuum manipulator moves by charging and discharging, and has good bearing capacity. However, there is a certain friction in the fiber network, which affects the motion performance. In addition, Festo’s researchers [7] have developed and designed bionic handling assistant, which uses laser sintering to make bellows actuators with light weight. This model is printed directly and cannot be assembled flexibly. Gong et al. [8] designed a soft robotic manipulator r made of silicone rubber, and used the core to constrain the axial elongation of the manipulator. By coordinating the pressure in the gas path, the manipulator can bend in any direction, but cannot achieve elongation. In summary, the structure of soft robotic manipulator can be divided into two categories: combination structure design and multi-cavity structure design. In order to improve the flexibility and elongation of soft robotic manipulator, a wrinkled bellows soft manipulator is designed. The fiber-constrained actuator (McKibben) designed by Schulte Jr et al. [9] produces radial expansion under the constraints of the inflatable state and the external fiber braided sleeve, and shrinks in the axial direction, but it cannot elongate, and there is friction. The embedded fiber-reinforced soft actuator studied by Connoll et al. [10] makes it produce axial contraction and limits the movement of radial expansion, but its scalability effect is general and it is not easy to simulate and analyze. In order to avoid the complexity of the manufacturing of traditional knitted fabric bending actuators, Roche et al. [11] proposed a method of making pneumatic bending soft actuators by laminating a laser-cut fabric shell onto thermoplastic polyurethane (TPU), but this method requires higher equipment and process. Mosadegh et al. [12] designed a multi-chamber soft actuator, but it can only achieve bending and cannot be elongated. In order to achieve better elongation and reduce the energy loss caused by the radial expansion effect, Elsayed et al. [13] studied the bellows actuator. The actuators can be divided into three categories: fiber-constrained soft actuators [9] embedded fiber-reinforced soft actuators [14] and multi-chamber soft actuators [12]. In order to improve the flexibility and elongation of the soft manipulator, the wrinkled bellows soft manipulator is designed and its actuator structure is designed. Building a more accurate kinematic model is important for the soft robotics control. The commonly used kinematics modeling method is the constant curvature (CC) [15] modeling method, which regards the soft robotic manipulator as a circular arc with CC. However, due to the diversity of soft robotic manipulator, their curvatures are usually inconsistent. Therefore, Walker et al. [7] improved the CC modeling method and proposed the piecewise constant curvature (PCC) method. The trunk of the soft robotic manipulator is divided into multiple parts with constant length and curvature, and the

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model is simplified. However, the PCC is used for cylindrical soft manipulators without external forces. Rucker et al. [16] proposed a variable curvature (VC) kinematics based on the Cosserat-rod model, which is highly accurate and suitable for design and optimization, but it is computationally intensive and not intuitive enough for describing complex shapes of geometric robots. Regarding at the above problems, Wang et al. [17] proposed a geometric design method for continuum robots based on free-form piecewise approximation, which is more intuitive to describe continuum robots, but does not analyze soft robotics with multi-actuator coupling. In addition, Drotman et al. [18] carried out static simulation analysis on the soft robotics and obtained good simulation results, but did not simulate and verify the mapping relationship of its kinematic model. The common verification methods are experimental verification method and visual experiment verification method, but its intuitive and idealized characteristics make it impossible to accurately verify the kinematics model, and it is impossible to optimize the soft robotics better [19]. Rad C et al. [20] proposed a data-driven approximate kinematics (DAK) model to estimate the shape and opening degree of soft pneumatic actuators under corresponding pressures. This method is suitable for the control algorithm of pneumatic actuator without sensor. Wang et al. [21] proposed a kinematics modeling method for soft robotics ontology perception and used a recurrent neural network deep learning method. However, this method is purely based on machine learning, and there is no kinematic description of the machine model. Based on the above discussions, this paper proposes a kinematic model error compensation method based on back propagation neural network (BPNN) approach to accommodate the kinematic model error, so as to improve the accuracy of the wrinkled bellowlike soft robotic manipulator (WBSRM) model. Firstly, a parallel WBSRM with good flexibility is constructed. Then, a constant small curvature (CSC) kinematics model is proposed based on the piecewise constant curvature theory. Then, the finite element simulation of WBSRM is carried out and its morphological analysis is carried out. Finally, for the error of the mathematical model, BPNN is used for error compensation analysis to improve the accuracy of the theoretical model.

2 Design of Wrinkled Bellow-Like Soft Robotic Manipulator Model In order to analyze the theoretical model and then simulate and verify the WBSRM, this section introduces the structural design of a WBSRM with good bending and elongation characteristics. The structural design of WBSRM mainly includes two aspects: the overall structural design and the structural design of the actuators. Next, we first introduce the overall structural design of WBSRM, and then design the structure of the wrinkled bellows actuators. The WBSRM model is assembled in parallel by wrinkled bellows actuators, and better bending and elongation effects are achieved through the coupling of the motion of the actuator. In this paper, the wrinkled bellows are fabricated from hyperelastic material, and the extension and contraction movements of the wrinkled bellows actuators are realized by inflating the inflatable pump or vacuum negative pressure pumping. However, when inflating the wrinkled bellows actuator, not only does the actuators elongate axially,

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but it also expands radially. In order to optimize the better actuators performance of the soft body starter and to reduce radial expansion to improve the mechanical performance of the WBSRM, a structure design was developed for the WBSRM. The structure design of the parallel WBSRM is shown in Fig. 1.

Fig. 1. (a) Structure schematic diagram of wrinkled bellow-like soft robotic manipulator, (b) The schematic diagram of the wrinkled bellow-like soft robotic manipulator top view.

The structure of the WBSRM is shown in Fig. 1(a), where the WBSRM consists of three wrinkled bellows actuators and link and interference plates. As shown in Fig. 1(b), the actuators are arranged in a 120◦ array with three wrinkled bellows actuators, with the diameter of the array’s circular arc being D. In order to reduce the unwanted radial expansion effect of the wrinkled bellows actuators, the interference plate is installed to restrain its radial expansion effect, and the installation of the interference plate strengthens the stiffness of the WBSRM. In addition, the link plate plays a fixed and sealing role in the wrinkled bellows actuators, and the size of the circular arc diameter D of the actuator array directly affects the performance of the WBSRM. Therefore, in the actual structure design process, the diameter D of the WBSRM can be designed according to the specific use environment. As shown in Fig. 1(a), the working principle of the WBSRM is simple, each actuator is driven separately, and it is easy to carry out simulation analysis. The internal pressure of the actuator is adjusted by air pump inflation or vacuum pumping, so as to achieve elongation or shortening motion. Due to the fixation of the link plate and the limitation of the interference plate, each actuator is coupled in different motion states, so as to realize the elongation and bending of the WBSRM. The motion of WBSRM mainly depends on the coupling effect of actuator motion. In order to improve the flexibility of WBSRM, the structure of the actuators are designed. The wrinkled bellow structure has a good axial expansion effect compared with the ordinary bellow structure [22, 23]. Therefore, the actuator is designed as a wrinkled bellow structure, as shown in Fig. 2. In Fig. 2, L indicates the total length of the actuator, L1 indicates the axial distance between two arcs of the wrinkled bellow actuator, D1 and D2 represent the radial distance from the center of the inner and outer corrugated arcs, respectively. R1 , R2 , R3 , R4 represent the radius of the wrinkled arcs of the wrinkled bellow actuator, C represents the angle of wrinkle of wrinkled bellow actuator, t refers to the wall thickness of the wrinkled bellow actuator. These parameters will directly affect the driving performance of the actuator.

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Fig. 2. Schematic diagram of the geometry of a wrinkled bellows actuator.

3 Kinematic Modeling of WBSRM In this section, the CSC kinematics modeling method of WBSRM is proposed. Firstly, the kinematics of the general unit is derived. Then, the general section kinematics is obtained by connecting the base unit and the head unit in series with the general unit. Finally, the actuator of the WBSRM is mapped to the arc of the CSC kinematics. The CSC kinematic modeling proposed in this paper is based on the theory of PCC [15]. The CC kinematic model divides the forward kinematic model into two mappings, as illustrated in Fig. 3. The first mapping is from the actuator space of the soft robotics to the configuration space, and the second mapping is from the configuration space of the soft robotics to the task space of the WBSRM. The specific mapping relationship from the actuator space to the configuration space describes the relationship from the actuator length to the arc parameters of the soft robotics. The independent mapping relationship from the configuration space of the soft robotics to the task space describes the independent mapping relationship from the arc parameters of the soft robotics to the position and attitude, which is independent of the shape of the soft robotics, so it is suitable for most soft robotics. The CSC kinematics modeling of the WBSRM is based on the idea of the VS modeling method [24], that is, in the first mapping of the PCC forward kinematics modeling, the cross-section model is a finite number of imaginary planes, and the PCC modeling method is used for this cross-section model.

Fig. 3. Kinematics space and its mapping relationship of soft robotics.

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3.1 General Unit Kinematics According to the independent mapping relationship of the main curve of the manipulator in the CC theory [15], the homogeneous transformation matrix of the virtual unit can be written as follows to describe the arc curve of the WBSRM:   i−1 i−1 R(φ , κ , l O(φ , κ , l ) ) i−1 i i i i i i   i i (1) i U = 0 1 

where i−1 R(φi , κi , li ) is the rotation matrix of the backbone curve with respect to the i 

coordinate system, and i−1 O(φi , κi , li ) is the translation vector of coordinate i relai 



tive to i−1 , The specific descriptions i−1 R(φi , κi , li ) and i−1 O(φi , κi , li ) are given as i i follows: ⎤ ⎡ 2 c φi (cκi li − 1) + 1 sφi cφi (cκi li − 1) cφi sκi li i−1 ⎣ sφi cφi (cκi li − 1) c2 φi (1 − cκi li ) + cκi li sφi sκi li ⎦ (2) i R(φi , κi , li ) = −cφi sκi li −sφi sκi li cκi li T  i−1 cφi (1−cκi li ) sφi (1−cκi li ) sκi li (3) i O(φi , κi , li ) = κi κi κi where c is the abbreviation for cosine and s is the abbreviation for sine. 3.2 Kinematics of WBSRM The kinematics model of the WBSRM is composed of the bottom unit, virtual units of the CC part and the top unit. Both the base unit and the top unit are rigid bodies, their  mi corresponding transformation matrices b0 S and  h S are as follows: ⎤ 100 0 ⎢0 1 0 0 ⎥ b ⎥ ⎢ 0 S = ⎣ 0 0 1 l ⎦, b 000 1 ⎡

⎤ 100 0 ⎢0 1 0 0 ⎥ mi ⎥ ⎢ h S = ⎣ 0 0 1 l ⎦ h 000 1 ⎡

(4)

where lb and lh denote the thickness of the rigid base unit and the rigid top unit, respectively.

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Since the bottom unit, the virtual unit and the top unit are connected in series with each other, the homogeneous transformation matrix of the kinematic equation of the WBSRM is given as follows: ⎤ 100 0  mi ⎢0 1 0 0 ⎥  b i−1 i−1 ⎥ S=⎢ T =0 S i U i U ⎣ 0 0 1 lb ⎦ i=1 i=1 h 000 1 ⎤ ⎡ ⎤ ⎡ r11 r12 r13 px 100 0 ⎢ 0 1 0 0 ⎥ ⎢ r21 r22 r21 py ⎥ ⎥ ⎢ ⎥ ×⎢ ⎣ 0 0 1 lh ⎦ = ⎣ r31 r32 r33 pz ⎦ 000 1 0 0 0 1 m

i 

mi

⎡

(5)

where px , py and pz are the tool center (TC) coordinates of the WBSRM. r11 to r33 are the attitude parameters of the WBSRM. 3.3 Specific Mapping of Virtual Units In order to illustrate the mapping from actuator space to configuration space of the WBSRM, and concurrently mapping the actuator space to the arc of CSC kinematics, a setup for the WBSRM is demonstrated as shown in Fig. 4. Figure 4(a) shows the schematic diagram of the complete structure of the WBSRM, and Fig. 4(b) shows the horizontal section view of the xi yi plane of the coordinate system of the virtual unit of the WBSRM. As shown in Fig. 4(a), the green part is the bottom and head of the WBSRM, and the middle gray part is the virtual unit layer. The three actuators are independent of each other and are distributed in a 120◦ array along the center of the circle with d as the radius. The length of the actuator is lk and k is the number of the actuator. milik is the length of the virtual actuator. It can be seen from Fig. 4(a) that lk = i=1 lik , where the virtual actuator is assumed to be uniformly elongated, so there is lik = lk /mi . The right-hand coordinate system has been applied to define the coordinate system of the WBSRM, where the z-axis passes through the center of the robot’s main body and the x-axis passes through actuator 1. In addition, the TC coordinate system is defined, which is located at the center of the upper surface of the top unit of the WBSRM. According to the specific mapping relationship of the three-actuator continuum robot in the CC kinematics theory [15], the specific mapping of the virtual units in the actuators can be identified as follows: √  3(li3 − li2 ) −1 (6) φi = tan li2 + li3 − 2li1  2 + l2 + l2 − l l − l l − l l 2 li1 i1 i2 i1 i3 i2 i3 i2 i3 (7) κi = di (li1 + li2 + li3 )

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Fig. 4. The complete structure diagram of the soft robotics.

di (li1 + li2 + li3 ) li =  2 + l2 + l2 − l l − l l − l l li1 i1 i2 i1 i3 i2 i3 i2 i3 ⎛ ⎞ 2 + l2 + l2 − l l − l l − l l li1 i1 i2 i1 i3 i2 i3 i2 i3 ⎠ × sin−1 ⎝ 3di

(8)

From (6) to (8), it can seen that the specific mapping of the virtual unit actuator: φi , κi and li depend on the length of the virtual unit actuator li1 , li2 , li3 and the radius di of the position of the virtual unit actuator array. However, due to the uniform elongation of the virtual actuator, the φi , κi and li in the specific mapping of the virtual unit actuator ultimately depend on the length of the actuator li , the radius of the virtual unit actuator array position di , and the number of virtual unit actuator mi .

4 Finite Element Analysis of WBSRM The structure of the WBSRM has been given in Fig. 1 to validate the kinematics model of the WBSRM. Firstly, the structure of the WBSRM is parameterized, then the material properties of WBSRM are given and the boundary conditions are defined. Finally, the finite element analysis (FEA) is conducted for the designed WBSRM. The structure design of the wrinkled bellow actuator is shown in Fig. 2. In order to carry out the detailed parameters set in FEA is given as L = 195.32 mm, L1 = 19.48 mm, D1 = 22 mm, D2 = 38.3 mm, R1 = 1.5 mm, R2 = 4.5 mm, R3 = 3 mm, R4 = 6 mm, C = 50.3◦ , t = 3 mm. The structure of the WBSRM is shown in Fig. 1(a). As shown in Fig. 1(b), the radial distance of the wrinkled bellows actuator installed along the fixed plate array is D = 32 mm. To verify the accuracy of the kinematics model of the WBSRM, the positive and negative pressures are imposed on the three actuators of the WBSRM in FEA procedure. The bottom link plate of the WBSRM is fixed, and the positive and negative pressures are directly imposed on the inner surface of the soft actuator. Assuming that the sealing effect of the WBSRM is good, the whole WBSRM is fixedly linked as a whole in the FEA environment. In the design of FEA boundary conditions, the material of the link plate and the interference plate is Polylactic Acid (PLA), and the material parameters used are shown

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in Table 1. The material of the actuator is the hyperelastic material Dragon skin30. In order to better carry out the FEA, the Yeoh third-order model [1] is used. The strain energy density function is described as follows: W =

n    2 λ − λ−2 .i.Ci (Ii − 3)i−1

(9)

i−1

where Ci is the material parameter, λ is the volume ratio of the material after deformation and before deformation. Note that the Dragon skin30 is approximately incompressible material, such that we take λ = 1.I1 is the deviatoric stress invariant. The detailed material parameters are shown in Table 2. In the simulation environment of FEA software, the material properties of the parts of the WBSRM are first defined as shown in Tables 1 and 2. Then the boundary conditions are set as follows: the tetrahedral element automatic division method is used to mesh the parts, and the mesh size is set to 6. The bottom link plate is fixed, and the positive and negative pressure is imposed on the inner surface of the actuator as the power source and the influence of gravity is ignored. Because the material of the actuator of the WBSRM is hyperelastic material, the large deflection is opened. Furthermore, to ensure the simulation process converges, appropriate sub-steps should be set. Apply pressure P1 , P2 , P3 to the actuators of the WBSRM, where P2 = P3 . A total of 22 sets of pressures act on the actuators of the WBSRM for FEA. The first set of pressure values are: P1 = −400pa, P2 = P3 = 200pa, arranged according to the arithmetic sequence, until the last set of pressure values is: P1 = −8800pa, P2 = P3 = 4400pa. As shown in Fig. 5, the FEA results are shown when P1 = −8800pa and P2 = P3 = 4400pa. Table 1. Material properties of polylactic acid Material properties

Tensile Strength (MPa) ISO 527

Young’s Modulus (MPa) ISO 527

Elongation at Break, (%) ISO527

Flexural Strength (MPa) ISO 178

Bending Modulus (MPa) ISO 178

Value

45.6

2641

2.4

87.7

1900

Material properties

Notched Impact Strength (KJ/m2) ISO 178

Heat Distortion Temperature (°C) ISO 75 1.8MPa

Glass Transition Temperature (°C)

Melting Point (°C)

Value

2.7

58.1

61

150

4.1 Morphological Analysis of WBSRM In the process of FEA, to obtain the simulation results of the deformation of the soft robotic manipulator, it is necessary to establish a local coordinate system in the structure

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Table 2. Material parameters for dragon skin30 in Mpa Material parameter

C1

C2

C3

Value

1.00 × 10–1

1.19 × 10–1

6.04 × 10–4

Fig. 5. Cloud diagram exemplifying Finite element analysis.

of the WBSRM. Firstly, the global coordinate system is established at the center of the bottom surface of the bottom link plate of the WBSRM, as shown in Fig. 4. Then, a local coordinate system is established at the center of the top surface of the top link plate of the WBSRM, and a local coordinate system is established on the body of the wrinkled bellow actuator. Finally, the deformation value of the local coordinate position of WBSRM is obtained by the pointer method. The pointer method is a method to obtain the local coordinate position deformation by using the relationship between the position information and the coordinate system between the nodes. It is realized by setting the pointer point on the model. In order to obtain the coordinate values of the TC and the drive of the WBSRM in the FEA environment, the formula established is as follows: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ xin xde xac ⎣ yac ⎦ = ⎣ yin ⎦ + ⎣ yde ⎦ (10) zac zin zde where xac ,yac and zac are the actual coordinate values of the WBSRM after simulation relative to the global coordinate system, xin , yin and zin are the initial coordinate values of the WBSRM relative to the global coordinate system, xde , yde and zde are the deformation of the local coordinate values of the WBSRM after simulation. According to the established (10), the actual length of the actuator can be obtained, and the kinematic model of the WBSRM is verified. However, the length of the actuator cannot be obtained in the static simulation environment. The widely used method for the length capture of the actuator is the visual method in the experiment, which approximates the shape curve to the center curve of the actuator, so there is an error. Based on the simulation results, this paper proposes a plane curve polynomial fitting method and a three-dimensional space curve fitting method to obtain the length of the actuator.

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Fig. 6. Spatial curve fitting diagram of the actuator.

As shown in Fig. 6, the projection of the actuator space curve in the xz plane is shown in Fig. 6(a), the projection of the actuator space curve in the yz plane is depicted in Fig. 6(b) and the fitting diagram of the actuator space curve in the three-dimensional space is given in Fig. 6(c). For the case where the fitting effect is not ideal, the idea of smooth curve approximation method can be used to write code for spatial curve fitting, such as Smoothing Spline function fitting method. The length of the actuators are obtained by using the plane curve polynomial fitting method and the three-dimensional space curve fitting method to obtain the length of the actuators shape curve of the WBSRM in the aforementioned 22 groups of pressure simulation environments, as shown in Fig. 7(a). In addition, the TC trajectory of the corresponding WBSRM in the simulation environment can be obtained as shown in Fig. 7(b).

Fig. 7. (a) Simulated data pertaining to the length of the actuator, (b) Coordinate values at the tool center point of the soft robotic entity.

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5 Comparison and Analysis of Kinematics Modeling and Simulation Results FEA of the WBSRM indicates that under the same boundary conditions, different TC coordinate values can be obtained by applying different pressure values. To verify the accuracy of the kinematics theoretical model, the length value of the actuator is substituted into the formula (5) for calculation, and the theoretical TC coordinates px , py and pz of the kinematics model of the WBSRM can be obtained. The TC coordinate value in the FEA environment and the theoretical TC coordinate value are shown in Fig. 8. It can be seen from Fig. 9(a), (b), (c) and (d) that when the deformation of WBSRM is relatively small, the error between the TC coordinate value in the simulation environment and the theoretical TC coordinate value is small and approximately coincides. However, as the deformation of the WBSRM increases, the two coordinate points begin to separate, and the difference between the coordinate values gradually increases, making the theoretical model inaccurate for solid model analysis. To analyze the error of the WBSRM, the formula is established as follows: ⎡ ⎤ ⎡  ⎤ ⎡ ⎤ px xac x ⎣ y ⎦ = ⎣ y ⎦ − ⎣ py ⎦ (11) ac   z zac pz where x , y and z represent the error of the TC coordinate value between the FEA and the theoretical model.

Fig. 8. Fitting curve of tool center coordinate value for finite element simulation and theoretical modeling: (a) 3D-tool center coordinate value fitting curve, (b) x coordinate value fitting effect diagram, (c) y coordinate value fitting effect diagram, (d) z value coordinate fitting effect diagram.

To enhance the accuracy of the theoretical model, the error compensation approach is conducted for the inaccuracy of the theoretical model of the WBSRM. Firstly, it is assumed that the value obtained by the WBSRM in the FEA environment is the standard

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value, and then the error compensation is performed on the TC coordinate value of the theoretical model of the WBSRM. The error compensation formula can be described as follows: ⎤ ⎡ r11 r12 r13 px + x ⎢ r21 r22 r23 py + y ⎥ ⎥ (12) T =⎢ ⎣ r31 r32 r33 pz + z ⎦ 0 0 0 1 To realize the error compensation, the BPNN method is employed to compensate the error of TC coordinate value in the theoretical model of WBSRM. Firstly, the length of the actuator obtained by the WBSRM in the FEA environment is taken as the input value of the BPNN. Secondly, the error of the TC coordinate value between the FEA and the theoretical model is taken as the output value of BPNN, and then the BPNN is trained. In the BPNN training, the input layer includes three elements, namely the length of the soft actuator l1 , l2 , l2 . The output layer has three elements: x , y and z . In this model, the hidden layer and weight are adjusted according to the results of the training performance graph and the linear regression graph to optimize the training effect. As shown in Fig. 9, the performance of the BPNN is shown. It can be seen that as the number of training times increases, the mean square error (MSE) gradually decreases until the best performance is achieved. As shown in the green line annotation in Fig. 9, MSE = 4.1252e−6. Figure 10 shows the linear regression results. The abscissa in the linear regression result graph is the actual output value, and the ordinate is the corresponding predicted output value. The correlation of the training set is equal to 1 as shown in Fig. 10(a), the correlation of the validation set is equal to 1 is depicted in Fig. 10(b) and the correlation of the test set is equal to 1 is given in Fig. 10(c). It can be seen from Fig. 10 that the linear regression effect is better at this time. Finally, the proportion of training, testing and validation is set to 75%, 10% and 15%, and the number of hidden neurons is set to 15. The training algorithm is Levenberg-Marquardt.

Fig. 9. The performance of the back propagation neural network.

It can be seen from Fig. 11(a), (b), (c) and (d) that the TC coordinate fitting curve of the WBSRM in the FEA environment almost coincides with the theoretical TC coordinate fitting curve with the suggest BPNN-based error compensation method. One can find from Fig. 11 that the proposed method can effectively compensate the error of the WBSRM model and improve the accuracy of the model.

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Fig. 10. Linear regression diagram: (a) Training set linear regression diagram, (b) Verification set linear regression diagram, (c) Test set linear regression diagram, (d) Overall linear regression diagram.

Fig. 11. The tool center coordinate value of finite element analysis and theoretical modeling by back propagation neural network error compensation: (a) 3D-tool center coordinate value fitting curve, (b) x coordinate value fitting effect diagram, (c) y coordinate value fitting effect diagram, (d) z value coordinate fitting effect diagram.

6 Conclusion In this paper, a FEA and error compensation analysis method based on WBSRM kinematics modeling is proposed, which can accurately describe the theoretical model of the TC position of WBSRM. This method includes the structural design of WBSRM. Firstly, the structure of the WBSRM assembly is designed, and then the structure of the wrinkled bellows actuator is designed. The structural design of WBSRM provides a solid model with good flexibility. Aiming at the kinematics modeling of WBSRM, a CSC kinematics modeling method for WBSRM is proposed, and the relationship between the actuator and the center position of the tool is established. Through the parameterization of the structure of the WBSRM, the material properties of the WBSRM are given and the boundary conditions are defined. Finally, the FEA of the designed WBSRM is carried out, and the local coordinate system is established and integrated to obtain the actuator

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length and the TC coordinate value. According to the error between the calculated theoretical TC coordinate value and the simulated coordinate value, BPNN is used for error compensation analysis. After BPNN error compensation analysis, the kinematic model established for WBSRM is more accurate. Acknowledgment. This work was supported by National Natural Science Foundation of China under Grants 62273169, 62003153, 62373174 and partially supported by Yunnan Fundamental Research Projects under Grants 202201AW070005, 202001AU070001, 202101BE070001–060.

References 1. Marechal, L., et al.: Toward a common framework and database of materials for soft robotics. Soft Robot. 8, 284–297 (2021) 2. Kwok, K.W., et al.: Soft robot-assisted minimally invasive surgery and interventions: advances and outlook. Proc. IEEE. 110, 871–892 (2022) 3. Rus, D., Tolley, M.T.: Design, fabrication and control of soft robots. Nature 521, 467–475 (2015) 4. Armanini, C., Boyer, F., Mathew, A.T., Duriez, C., Renda, F.: Soft robots modeling: a structured overview. IEEE Trans. Robot. 39(3), 1728–1748 (2023). https://doi.org/10.1109/TRO. 2022.3231360 5. Trivedi, D., Lotfi, A., Rahn, C.D.: Geometrically exact models for soft robotic manipulators. IEEE Trans. Robot. 24, 773–780 (2008) 6. Grissom, M.D., et al.: Design and experimental testing of the OctArm soft robot manipulator. In: Defense and Security Symposium, Orlando (Kissimmee), FL (2006) 7. Hannan, M.W., Walker, I.D.: Kinematics and the implementation of an elephant’s trunk manipulator and other continuum style robots: hannan and walker: an elephant’s trunk manipulator and other continuum style robots. J. Robotic Syst. 20, 45–63 (2003) 8. Gong, Z., et al.: An opposite-bending-and-extension soft robotic manipulator for delicate grasping in shallow water. Front. Robot. AI. 6, 26 (2019) 9. Schulte Jr, H.: The characteristics of the McKibben artificial muscle. The Application of External Power in Prosthetics and Orthotics, pp. 94–115, (1961) 10. Connolly, F., Polygerinos, P., Walsh, C.J., Bertoldi, K.: Mechanical programming of soft actuators by varying fiber angle. Soft Rob. 2, 26–32 (2015) 11. Quevedo-Moreno, D., Roche, E.T.: Design and modeling of fabric-shelled pneumatic bending soft actuators. IEEE Robot. Autom. Lett. 8, 3110–3117 (2023) 12. Mosadegh, B., et al.: Pneumatic networks for soft robotics that actuate rapidly. Adv. Funct. Mater. 24, 2163–2170 (2014) 13. Elsayed, Y., et al.: Finite element analysis and design optimization of a pneumatically actuating silicone module for robotic surgery applications. Soft Rob. 1, 255–262 (2014) 14. Suzumori, K., et al.: Development of flexible microactuator and its applications to robotic mechanisms. In: Proceedings of the 1991 IEEE International Conference on Robotics and Automation, pp. 1622–1627. IEEE Comput. Soc. Press, Sacramento, CA, USA (1991) 15. Iii, R.J.W., Jones, B.A.: Design and kinematic modeling of constant curvature continuum robots: a review. Int. J. Robot. Res. 29, 1661–1683 (2010) 16. Caleb Rucker, D., Jones, B.A., Webster III, R.J.: A geometrically exact model for externally loaded concentric-tube continuum robots. IEEE Trans. Robot. 26(5), 769–780 (2010). https:// doi.org/10.1109/TRO.2010.2062570

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17. Wang, S., Blumenschein, L.H.: A geometric design approach for continuum robots by piecewise approximation of freeform shapes. In: 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5416–5423 (2022) 18. Drotman, D., Jadhav, S., Karimi, M., de Zonia, P., Tolley, M.T.: 3D printed soft actuators for a legged robot capable of navigating unstructured terrain. In: 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 5532–5538 (2017) 19. Luo, R., et al.: Design and kinematic analysis of an elephant-trunk-like robot with shape memory alloy actuators. In: 2017 IEEE 2nd Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), pp. 157–161. IEEE, Chongqing, China (2017) 20. Rad, C., Hancu, O., Lapusan, C.: Data-driven kinematic model of pneunets bending actuators for soft grasping tasks. Actuators. 11, 58 (2022) 21. Wang, L., et al.: Soft robot proprioception using unified soft body encoding and recurrent neural network. Soft Rob. 10(4), 825–837 (2023). https://doi.org/10.1089/soro.2021.0056 22. Du Pasquier, C., Chen, T., Tibbits, S., Shea, K.: Design and computational modeling of a 3D printed pneumatic toolkit for soft robotics. Soft Rob. 6, 657–663 (2019) 23. MacCurdy, R., et al.: Printable hydraulics: a method for fabricating robots by 3D co-printing solids and liquids. In: 2016 IEEE International Conference on Robotics and Automation (ICRA), pp. 3878–3885. IEEE, Stockholm, Sweden (2016) 24. Mahl, T., et al.: A variable curvature modeling approach for kinematic control of continuum manipulators. In: 2013 American Control Conference, pp. 4945–4950 (2013)

Suppression of Galloping Oscillations Using Perforated Bluff Bodies Juntong Xing1 , Masoud Rezaei1 , Huliang Dai2(B) , and Wei-Hsin Liao1 1

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China [email protected] 2 Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China [email protected] http://www.mae.cuhk.edu.hk/~whliao

Abstract. This study explores the potential effect of utilizing perforated bluff bodies on suppressing galloping oscillations. Six types of perforated bluff bodies with vertically parallel two holes, four holes, and six holes are proposed, and three kinds of surface treatments including allsided and non-all-sided designs are implemented to vary the perforated shape. A comprehensive aero-electromechanical mathematical modelling is established for better analysing the proposed passive vibration suppression system. Wind tunnel experiments and numerical simulations reveal that increasing the number of holes on perforated bluff bodies can effectively reduce the vibration amplitude and increase the galloping cutin wind speed. Furthermore, only the all-sided six-hole perforated bluff body has the greatest potential to suppress galloping oscillations. Further wind tunnel computational fluid dynamics (CFD) simulations are also performed to obtain the transverse force coefficients so as to investigate the underlying aerodynamic reasons behind the observed phenomena. Compared with most of the existing galloping oscillations suppression methods, this technique has the advantages of no external energy, lightweight, and simple design, etc., which makes it promising for possible practical applications. Keywords: Galloping oscillations suppression · Passive control · Perforated bluff bodies · Wind tunnel experiments · Transverse force coefficient

1

Introduction

In the past decades, researchers have been fascinated by different fluid-structure interaction phenomena, because of the widespread emergence such as skyscrapers, chimneys, bridges, towers, etc.[9,12] When flow passes these objects, flowinduced vibrations (FIVs) can occur, which may lead to catastrophic incidents if c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 514–528, 2024. https://doi.org/10.1007/978-981-97-0554-2_39

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the vibration amplitude is too large. Therefore, how to prevent structures from destructive fluctuations has always been an important topic for both industry and academia. From the perspective of aeroelastic instability, FIV can be categorized into four types, namely, galloping, flutter, vortex-induced vibrations (VIVs), and wake galloping [2,16]. In particular, galloping has a large oscillation amplitude which is usually one to ten times the across-wind dimension of the bluff body [22]. Therefore, galloping oscillations are more detrimental and need to be mitigated for the sake of structural safety. Researchers have proposed different approaches to suppress galloping oscillations [4]. Two most important approaches are active and passive control. Regarding active control methods [25], although some techniques such as linear, nonlinear, and time-delayed algorithms have been proven to be effective, all of them require additional energy sources, which are not easy to access in harsh application scenarios. Therefore, different passive control methods which do not require energy sources can be used as alternatives. However, classical passive control techniques introduce different energy sinks to the system, such as tuned mass dampers and nutation dampers [11]. These methods bring additional mass to the host structure, which makes the system awkward and unwieldy. Hence, emerging methods by adjusting surface characteristics of the bluff body, for example, adding protrusions on its surface [27,28], have been proposed recently to tackle this problem. Similarly, making the bluff body perforated has also been introduced to effectively suppress VIVs in recent studies [23,24], however, there is a lack of investigations focusing on galloping oscillations. Therefore, in this study, various perforated bluff bodies are designed for the purpose of comparing their galloping suppression efficiency. Besides, wind tunnel experiments and numerical simulations are conducted to demonstrate and validate the proposed design. This paper is organized as follows. In Sect. 2, the design of different types of perforated bluff bodies is illustrated and a comprehensive aero-electromechanical mathematical model is established based on the extended Hamilton’s principle, Euler-Bernoulli beam theory, and the quasi-steady assumption. In Sect. 3, wind tunnel experiments are conducted to validate the proposed design and numerical simulations are also performed to validate the established model. Besides, galloping suppression ratios and underlying aerodynamic reasons on the basis of transverse force coefficients are also analysed. Finally, conclusions are summarized in Sect. 4.

2

Design and Modelling

In this section, first, the designs for the galloping suppression system and six perforated bluff bodies are described, respectively. After that, a comprehensive mathematical model for the overall system is established for further numerical studies.

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2.1

System Design

The schematic of the proposed galloping suppression system is shown in Fig. 1(a). A slender and hollow bluff body is attached to the free end of a cantilever beam perpendicularly and the system is placed perpendicularly to the wind. Under such arrangements, the initial angle of attack (AoA) of the bluff body with respect to the windward direction can be considered to be zero. A Macro Fiber Composite (MFC) sensing unit is bounded to the fixed end of the cantilever beam, serving as a voltage signal generator for studying the galloping suppression ratios. The bluff body is fabricated by 3D printing using resin material, and the cantilever beam is made of 304 stainless steel and fabricated by laser cutting method. In this study, in order to compare the galloping mitigation effectiveness of different perforated bluff bodies, six types of perforated bluff bodies are proposed and depicted in Fig. 1(b). Based on the type of surface treatment, they can be further divided into all-sided and non-all-sided perforated bluff bodies. Furthermore, in order to better distinguish these bluff bodies, they are named ‘Two holes’, ‘Four holes’, ‘Six holes/Six-All’, ‘Six-Front’, ‘Six-Side’, and ‘SixBack’, respectively, in the following discussions. The geometrical and material properties of the galloping suppression system are listed in Table 1. Two holes

(b)

(a) ± w(x,t)

Four holes

Six holes

All-sided perforated bluff bodies

Wind MFC sensing unit

Lbb

Original Bluff body

R z

Dbb

x

y

Bluff body

Different surface treatments

Front

Side

Back

Fig. 1. (a) Schematic of the proposed galloping suppression system with an original bluff body attached to the beam end; (b) Six perforated bluff bodies with different surface treatments for comparative studies.

2.2

Mathematical Modelling

For the sake of mathematical modelling, a schematic of the proposed cantilever beam-type galloping oscillations suppression system is depicted as Fig. 2. The coupled aero-electromechanical governing equations can be derived based on the extended Hamilton’s principle [1,26] and Euler-Bernoulli beam theory [19,20]. In this principle, the variational indicator (V.I.) can be expressed as Eq. (1):

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Table 1. Geometrical and material properties of the galloping oscillations suppression system. Bluff body (3D printing by Resin): Mass (Mt ) Width (Dbb ) Length (Lbb )

12 g 30 mm 70 mm

Cantilever beam (304 Stainless Steel): Young’s modulus (Eb ) Mass density (ρb ) Length (Lb ) Width (Wb ) Thickness (tb )

193 GPa 7930 kg/m3 75 mm 12 mm 0.3 mm

MFC sensing unit (MFC): Young’s modulus (Ep ) Mass density (ρp ) Length (Lp ) Width (Wp ) Thickness (tp ) Piezoelectric constant (d31 ) Permittivity (e33 )

15.86 GPa 5440 kg/m3 28 mm 14 mm 0.33 mm −170 pm/V 19.36 nF/m

Others: Air density (ρa ) Damping ratio (ζ)

1.24 kg/m3 0.001

t2 V.I. =

(δT − δU + δWnc )dt = 0

(1)

t1

where T , U , and δWnc denote the total kinetic energy, potential energy, and virtual work of nonconservative forces, respectively. The total kinetic energy of the system is composed of the kinetic energy of the beam and the tip perforated bluff body, which can be expressed explicitly as Eq. (2): 1 T = 2

Lb 0

 2 2 1 Dbb  1  w˙ Lb m (x) w˙ 2 dx + Mt w˙ Lb + + It w˙ Lb 2 2 2

(2)

where m(x) denotes the beam mass per unit length, and It denotes the mass moment of inertia of the bluff body with respect to its centre. These two parameters can be expressed as Eq. (3). In Eq. 2, w stands for the displacement of

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Dbb

Uf α

Lb

. wt R

FD FL

α

Fg

Fig. 2. Mathematical modelling schematic of the cantilever beam-type galloping oscillations suppression system (Top view).

a specific position on the beam, and Lb stands for the cantilever beam length. Besides, the overdot represents the differentiation with respect to time, and the prime represents the differentiation with respect to position.  m (x) = ρb Wb tb + ρp Wp tp [H(x) − H(x − Lp )] (3) It = 16 Mt Dbb 2 In Eq. 3, ρ, W , and t denote the mass density, width, and thickness, respectively; subscripts b and p refer to the beam and MFC patch, respectively; Mt and Dbb stand for the mass and width of the bluff body, respectively. Furthermore, H(x) is the Heaviside function and defined as Eq. (4):  1, x ≥ 0 (4) H (x) = 0, x < 0 The total potential energy includes three parts, namely, the mechanical potential energy of the beam (Ub ) and MFC patch (Upm ) as well as the electrical potential energy of the MFC patch (Upe ). It can be expressed as Eq. (5): U = Ub + Upm + Upe =

1 2

L b 0

Y I(x)κ2 dx +

1 2

L b 0

˙ − 1 Cp R2 Q˙ 2 2θκRQdx 2

(5)

where Y I(x), θ, Cp , κ, R, and Q respectively denote the effective bending stiffness, piezoelectric coupling coefficient, the capacitance of the MFC patch, the curvature of the beam [10], electrical open-circuit load resistance, and the current passing through the load. The explicit expressions of the first four abbreviations are given in Eq. (6):  ⎧ 3 1 Y I(x) H(x − Lb ) ⎪  = 12 W b Eb tb  H(x − Lp ) − ⎪

t +t 3 

t −t 3  ⎪  ⎪ tb −tp 3 tb +tp 3 p p ⎪ 1 b b ⎪ + Wp Ep H(x) − H(x − Lp ) + − ⎪ 2 2 2 2 ⎪ + 3 Wb Eb ⎪ 

⎨  

  2 2 Ep d31 tb +tp tb −tp H(x) − H(x − Lp ) Wp − θ = − 2t 2 2 ⎪ p ⎪ ⎪ ⎪ e33 Wp Lp ⎪ ⎪ C = ⎪ ⎪ tp ⎪ p ⎩ κ = w + 1 w w2 2

(6)

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where E, d31 , and e33 denote Young’s modulus, piezoelectric voltage constant, and permittivity, respectively. Here the double prime represents the second differentiation with respect to position. The total virtual work of nonconservative forces comprises mechanical viscous dissipation, electrical load resistance dissipation, and aerodynamic excitation. It can be expressed as Eq. (7): Lb δWnc = −

˙ cwδwdx ˙ − RQδQ + Fg

0

Lb  0

 Dbb  δ (x − Lb ) δwdx δ (x − Lb ) − 2

(7) where c denotes the dimensional damping of the system, and Fg stands for the galloping aeroelastic force [5,13], which can be expressed by Eq. (8): Fg =

1 ρa Uf 2 Lbb Dbb CF y 2

(8)

where ρa denotes the air density, Uf denotes the wind speed, Lbb denotes the length of bluff bodies, and CF y represents the transverse force coefficient. Note that the quasi-steady assumption [13] is adopted to model the galloping aeroelastic force. In this assumption, the lift and drag coefficients in the course of oscillation are assumed to be the same as the values measured at the corresponding steady AoA in wind tunnel experiments or equivalent CFD simulations. Therefore, as depicted in Fig. 2, the transverse force coefficient is related to the lift (CL ) and drag (CD ) coefficients, which can be expressed as Eq. (9). Moreover, using the curve fitting method, CF y can be further expressed as a seventh-order polynomial function [17]. CF y = − (CL + CD tan α) sec α ≈ A1 α − A3 α3 + A5 α5 − A7 α7

(9)

where α can be expressed as Eq. (10): α=

w˙ Lb + D2bb w˙ Lb w˙ t = Uf Uf

(10)

where wt denotes the deflection of the bluff body centre. By substituting Eqs. (2), (5), and (7) into Eq. (1), the governing equations of the proposed galloping suppression system can be derived as Eq. (11): ⎧        ⎪     1 2 ⎪ ⎪ m w ¨ + c w ˙ + Y Iw w w + Y I w w + θ 1 + v ⎪ ⎪ ⎪ 2 ⎪ ⎪   ⎪ ⎪ Dbb  ⎨ δ (x − Lb ) = Fg δ (x − Lb ) − (11) 2 ⎪ ⎪ ⎫ ⎧ ⎪ ⎪   ⎬ Lb ⎪ ⎪  ∂ ⎨ 1 1 2 ⎪ ⎪ C v˙ + v = θw 1 + w dx ⎪ ⎪ ⎩ p ⎭ R ∂t ⎩ 2 0

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In order to numerically solve the above governing equation, Galerkin’s method [14,18] is utilized to discretize it into the reduced-order model. In this method, the mechanical response of the system can be expressed in a series form as Eq. (12): M  w(x, t) = φi (x)qi (t) (12) i=1

where spatial functions φi (x) are chosen to be the i-th mode shape of the cantilever beam with a tip bluff body, temporal functions qi (t) are the generalized time-dependent modal coordinates, and M is the number of modes under consideration. The mode shapes are provided in Appendix A. Using the orthogonality condition of the chosen mode shapes, the reduced-order model can be finally obtained as Eq. (13): ⎧ ⎪ 2 ⎪ ⎪ ⎨ q¨n + 2ζn ωn q˙n + ωn qn + ⎪ ⎪ ⎪ ˙ + ⎩ Cp v(t)

1 R v(t)

=

M 

M 

Anijk qi qj qk + BnI v(t) +

BnII q˙n +

n=1

M 

CnijI qi qj v(t) = Dn Fg

i,j=1

i,j,k=1 M 

(13)

CnijII (q˙n qi qj + qn q˙i qj + qn qi q˙j )

i,j=1

where ζn , ωn , and Anijk are the modal damping coefficient, natural frequency, and geometric nonlinearity of the unimorph, respectively; BnI and BnII denote the modal linear electromechanical coupling coefficients; CnijI and CnijII denote the modal nonlinear electromechanical coupling coefficients; Dn denotes the modal coefficient of the aerodynamic forces. The aforementioned parameters are defined in Appendix B.

3

Numerical Simulations and Experimental Validation

In this section, experimental and numerical studies are conducted to investigate the galloping dynamic responses of the proposed suppression system with different perforated bluff bodies. Numerical studies are performed in MATLAB by solving Eq. (13) using the Runge-Kutta method. Experiments are conducted in an open-circuit wind tunnel to validate the mathematical modelling. 3.1

Experimental Set-Up

As shown in Fig. 3(a), the prototype of the proposed galloping suppression system was placed in the test section of an open-circuit wind tunnel. The real-time wind speed was measured and recorded by an anemometer (testo 405i). The details of the galloping suppression system are shown in Fig. 3(b). An MFC sensing unit (M2814-P2, Smart Material Corporation) was bounded to the fixed end of the cantilever beam for studying the energy mitigation rate and a laser sensor (HG-C1100, Panasonic) was mounted parallel to the beam structure to detect its tip deflection. The output signals were recorded by an oscilloscope (STO1104C, Micsig) for further processing. A series of wind sweep experiments were conducted under the speed range of 0 to 5.1 m/s.

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(a)

(b)

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Laser sensor

Anemometer MFC sensing unit Perforated bluff bodies

Wind tunnel Oscilloscope

Fig. 3. (a) Experimental set-up of the proposed galloping suppression system in an open-circuit wind tunnel; (b) Details of the galloping suppression system.

3.2

Dynamic Responses and Galloping Suppression Ratio

In order to investigate the performance of galloping oscillations suppression using all-sided perforated bluff bodies, forward wind sweep experiments are conducted in the electrical open-circuit condition. Experimental and numerical results of the beam tip deflection and RMS voltage versus wind speed are plotted in Fig. 4. In this study, the dynamic response of the original bluff body is regarded as a comparison benchmark. It can be observed that increasing the holes of perforated bluff bodies can effectively reduce the vibration amplitude and increase the galloping cut-in speed. Moreover, the perforated bluff body with a series of vertically parallel six holes has a cut-in wind speed of 1.9 m/s and a beam tip amplitude of 4.2 mm at 5.1 m/s, which are 533% higher and 45% lower compared with those of the original bluff body, respectively. By comparing the experimental results with the simulation results, it can be concluded that the mathematical model can well predict the galloping oscillations of each perforated bluff body. Furthermore, both mechanical and electrical responses of the bluff bodies have an identical developing trend, which originates from the fact that the vibration energy is proportional to the square of the vibration amplitude.

(b)15

8 Original-Exp Original-Num Two holes-Exp Two holes-Num Four holes-Exp Four holes-Num Six holes-Exp Six holes-Num

6 4

RMS Voltage (V)

Beam Tip Deflection (mm)

(a)

2 0

0

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2

3

Wind Speed (m/s)

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Original-Exp Original-Num Two holes-Exp Two holes-Num Four holes-Exp Four holes-Num Six holes-Exp Six holes-Num

10

5

0

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2

3

4

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Fig. 4. Experimental and numerical dynamic responses of different all-sided perforated bluff bodies: (a) Beam tip deflection versus wind speed; (b) RMS voltage versus wind speed.

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To evaluate the galloping suppression performance quantitatively, two indexes of amplitude suppression ratio (Ras ) and energy suppression ratio (Res ) are introduced, which can be calculated by Eq. (14):   ⎧ ⎨ Ras = wpw−wo  × 100%  2 o 2  x=Lb (14) ⎩ R = Vp −Vo × 100% es V2 o

where wp and wo denote the beam tip deflection of the perforated and original bluff bodies, respectively; Vp and Vo denote the respective output voltage of the MFC sensing unit of the perforated and original bluff bodies. The respective suppression ratios for each perforated bluff body are plotted in Fig. 5. From these results, it can also be concluded that the perforated bluff body, ‘Six holes’, outperforms the others in terms of galloping oscillations suppression. Besides, for each suppression ratio curve, there exists an obvious ‘valley’ region, and after that, the curves gradually converge to a constant value. The reason for this phenomenon is that the perforated design delays the occurrence of galloping oscillations to a higher cut-in wind speed, but cannot effectively suppress the amplitude growth rate after that. This indicates that the proposed method is more effective at low wind speeds. (b)

0

Energy Suppression Ratio (%)

Amplitude Suppression Ratio (%)

(a)

-20 -40 -60 Two holes Four holes Six holes

-80 -100

0

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3

Wind Speed (m/s)

4

5

0 Two holes Four holes Six holes

-20 -40 -60 -80 -100

0

1

2

3

4

5

Wind Speed (m/s)

Fig. 5. Suppression rate analyses of (a) Amplitude suppression ratio and (b) Energy suppression ratio of different all-sided perforated bluff bodies.

As the perforated bluff body, ‘Six holes’, has the best galloping suppression performance among the proposed all-sided bluff bodies, further experiments are conducted to investigate the effect of different surface treatments on the overall suppression performance. Figure 6 shows the experimental and numerical results of dynamic responses of the six-hole perforated bluff bodies with different surface treatments, i.e., front perforated, side perforated, back perforated, and all-sided perforated. It can be observed that the all-sided perforated bluff body performs the best in terms of galloping suppression; however, the front- and double-sided perforated bluff bodies can also suppress the vibrations. On the other hand, it is worth noting that the back perforated bluff body even enhances the vibration

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amplitude compared with the original bluff body, which implies that such a design can be inversely applied in flow-induced energy harvesting scenarios. (b)15

8 Original-Exp Original-Num Six-All-Exp Six-All-Num Six-Front-Exp Six-Front-Num Six-Side-Exp Six-Side-Num Six-Back-Exp Six-Back-Num

6 4

RMS Voltage (V)

Beam Tip Deflection (mm)

(a)

2 0

0

1

2

3

Wind Speed (m/s)

4

5

Original-Exp Original-Num Six-All-Exp Six-All-Num Six-Front-Exp Six-Front-Num Six-Side-Exp Six-Side-Num Six-Back-Exp Six-Back-Num

10

5

0

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1

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4

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Fig. 6. Experimental and numerical dynamic responses of six-hole perforated bluff bodies with different surface treatments: (a) Beam tip deflection versus wind speed; (b) RMS voltage versus wind speed.

3.3

Aerodynamic Transverse Force Coefficient (CF y )

In order to figure out the underlying aerodynamic reasons behind the experimental and numerical results, the classic Den Hartog’s criterion [5] and Hu’s findings [7,8] are adopted to analyse the aerodynamic transverse force coefficient for each bluff body. Results of the transverse force coefficients obtained by wind tunnel CFD simulations via ANSYS Workbench 18 are plotted in Fig. 7 and listed in Table 2. According to Den Hartog’s criterion as expressed in Eq. (15), to experience galloping oscillations, the initial slope of CF y versus the angle of attack (α) curve should be positive. Moreover, according to previous research conducted by Hu et al. [7,8], the larger the peak value of CF y and the α corresponding to CF y = 0, the greater the galloping response. Therefore, Fig. 7(a) proves the dynamic results shown in Fig. 4, i.e., increasing the holes of perforated bluff bodies can enhance the performance of galloping oscillations suppression, and the bluff body with vertically parallel six holes performs best in this study. Moreover, according to Eq. (16) which is obtained by assuming a zero equivalent linear damping [27], the cut-in speed (Ucut−in ) is inversely proportional to the linear aerodynamic coefficient (A1 ). Hence, this also validates the observed phenomenon of Fig. 4 that the cut-in wind speed is increased with the increasing number of holes of bluff bodies. Furthermore, Fig. 7(b) also accounts for the dynamic results shown in Fig. 6. From the point of view of cut-in wind speed, it indicates that the bluff body of ‘Six-Back’ has the lowest cut-in speed, followed by the bluff body of ‘Original’, ‘Six-Front’, ‘Six-All’ and ‘Six-Side’. Besides, based on Hu’s findings, the galloping response of ‘Six-Back’ is greater than that of the ‘Original’ bluff body, and the response of ‘Six-Side’ is also greater than that of ‘Six-All’. The above results not just qualitatively validate the experimental and

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numerical results, but also prove the feasibility of the quasi-steady assumption used in the mathematical modelling.  dCF y  = A1 > 0 (15) dα α=0◦ Ucut−in ∝

2c ρa Lbb Dbb A1

0.2

0.2

0

0

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(b) 0.4

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-0.2

(16)

-0.2 CFD

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-0.6 -0.8

0

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Fig. 7. Diagrams of the transverse force coefficient (CF y ) versus the angle of attack (α) for different perforated bluff bodies: (a) original and all-sided perforated bluff bodies; (b) six-hole perforated bluff bodies with four types of surface treatments.

Table 2. The 7th-order polynomial approximation of the transverse force coefficient CF y for different bluff bodies.

4

Surface treatment A1

A3

A5

A7

Original Two holes Four holes Six holes (Six - All) Six - Front Six - Side Six - Back

36.48 34.38 17.2 16.88 39.62 2.54 35.32

161.7 150.1 25.34 40.65 224.8 −82.43 146.7

278.8 252.2 −34.67 19.42 472.8 −279.1 237.6

1.85 1.686 0.9063 0.6053 1.079 0.5813 1.994

Conclusions

In this study, a novel method for suppressing galloping oscillations by using perforated bluff bodies is proposed and validated by experimental and numerical studies. A total of six types of perforated bluff bodies with all-sided or

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non-all-sided surface treatments are proposed in order to compare the galloping mitigation effectiveness. Comprehensive mathematical modelling of the passive vibration suppression system is established based on the extended Hamilton’s principle, Euler-Bernoulli beam theory, and the quasi-steady assumption. Further experimental and numerical results validate the derived modelling and prove that increasing the number of holes on perforated bluff bodies can effectively reduce the vibration amplitude and increase the galloping cut-in wind speed. Moreover, only the all-sided perforated bluff body performs the best in terms of galloping suppression. Therefore, the perforated bluff body with a series of vertically parallel six holes has the best galloping suppression performance among all the proposed bluff bodies. Wind tunnel CFD simulations are also performed to investigate the underlying aerodynamic reasons. Based on the classic Den Hartog’s criterion and Hu’s findings, the results obtained by the wind tunnel experiments and numerical simulations are justified. As the perforated bluff body has a great potential to suppress galloping oscillations, in future, advanced optimization methods [3,21] can be implemented to optimize the structural design of the bluff body, such as the position and number of holes, to better enhance the suppression performance. Acknowledgements. This work was supported by the Hong Kong Innovation and Technology Commission (Project No. MRP/030/21), The Chinese University of Hong Kong (Project ID: 4055178), and the National Natural Science Foundation of China (Grant No. 12202151, No. 12272140).

Appendix A: Mode Shapes and Natural Frequencies The mass-normalized mode shapes and natural frequencies of a cantilever beam with a tip bluff body are respectively [6,15]:    λi λi λi λi φi (x) = Mi cos x − cosh x + σi sin x − sinh x (A.1) Lb Lb Lb Lb  ωi = λ2i

Eb Ib mb L4b

(A.2)

where Mi is the modal amplitude, which can be obtained from orthogonality conditions by the following equation: Lb 0

    Dbb  Dbb  φi (Lb ) Mt φj (Lb ) + φj (Lb ) φi (x)m(x)φi (x)dx + φi (Lb ) + 2 2 (A.3) 



+ φi (Lb )It φj (Lb ) = δij where δij is the Kronecker delta.

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Furthermore, λi is the eigenvalue, which can be obtained from the following characteristic equation:   N1 N3 (1 − cos λi cosh λi ) λi 4 − N1 N2 2 + N3 (sin λi cosh λi + sinh λi cos λi ) λi 3 −2N1 N2 (sin λi sinh λi ) λi 2 + N1 (sinh λi cos λi − cosh λi sin λi ) λi + (1 + cos λi cosh λi ) = 0 (A.4) where N1 = ters.

Mt mb Lb ,

Dbb 2Lb ,

N2 =

and N3 =

It mb L3b

are three dimensionless parame-

Besides, σi is the modal constant of the i-th mode, which can be obtained by: σi =

sin λi − sinh λi + N1 λi [(cos λi − cosh λi ) − N2 λi (sin λi + sinh λi )] (A.5) cos λi + cosh λi − N1 λi [(sin λi − sinh λi ) + N2 λi (cos λi + cosh λi )]

Appendix B: Reduced-order Model Abbreviations The abbreviations in the reduced-order model Eq. (13) are listed as follows: Lb

ωn2

=

Y Iφn  (x)2 dx

(B.1)

0

Lb Anijk =

     φn (x) φi  (x) Y Iφj  (x)φk  (x) dx

(B.2)

0

Lb BnI =

φn (x)θ (x)dx

(B.3)

0

Lb BnII =

φn  (x)θ(x)dx

(B.4)

0

CnijI

1 = 2

CnijII =

Lb

  θ(x)φn (x) φi  (x)φj  (x) dx

(B.5)

0

1 2

Lb

θ(x)φn  (x)φi  (x)φj  (x)dx

(B.6)

0

Dn = φn (Lb ) +

Dbb  φn (Lb ) 2

(B.7)

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References 1. Bibo, A., Abdelkefi, A., Daqaq, M.F.: Modeling and characterization of a piezoelectric energy harvester under combined aerodynamic and base excitations. J. Vib. Acoust. 137(3), 031017 (2015) 2. Blevins, R.D.: Flow-induced Vibration. Van Nostrand Reinhold Company (1977) 3. Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004) 4. Daqaq, M.F., Alhadidi, A.H., Khazaaleh, S.: Suppression of structural galloping by applying a harmonic base excitation at certain frequencies. Nonlinear Dyn. 110(4), 3001–3014 (2022) 5. Den Hartog, J.P.: Mechanical Vibrations. Courier Corporation (1985) 6. Erturk, A., Inman, D.J.: Piezoelectric energy harvesting. John Wiley & Sons (2011) 7. Hu, Gang, Tse, K..T.., Kwok, K..C..S..: Galloping of forward and backward inclined slender square cylinders. J. Wind Eng. Ind. Aerodyn. 142, 232–245 (2015). https:// doi.org/10.1016/j.jweia.2015.04.010 8. Hu, G., Tse, K.T., Wei, M., Naseer, R., Abdelkefi, A., Kwok, K.C.: Experimental investigation on the efficiency of circular cylinder-based wind energy harvester with different rod-shaped attachments. Appl. Energy 226, 682–689 (2018) 9. Hua, X., Wang, C., Li, S., Chen, Z.: Experimental investigation of wind-induced vibrations of main cables for suspension bridges in construction phases. J. Fluids Struct. 93, 102846 (2020) 10. Meesala, V.C.: Modeling and analysis of a cantilever beam tip mass system, Ph. D. thesis, Virginia Tech (2018) 11. Modi, V.J., Seto, M.L.: Passive control of flow-induced oscillations using rectangular nutation dampers. J. Vib. Control 4(4), 381–404 (1998) 12. Mokni, L., Kirrou, I., Belhaq, M.: Galloping of a wind-excited tower under internal parametric damping. J. Vib. Acoust. 136(2), 024503 (2014) 13. Païdoussis, M.P., Price, S.J., De Langre, E.: Fluid-structure Interactions: Crossflow-induced Instabilities. Cambridge University Press (2010) 14. Preumont, A.: Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems. Springer (2006). https://doi.org/10.1007/1-4020-4696-0 15. Rezaei, M., Khadem, S.E., Firoozy, P.: Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects. Int. J. Eng. Sci. 118, 1–15 (2017) 16. Rezaei, M., Talebitooti, R.: Wideband PZT energy harvesting from the wake of a bluff body in varying flow speeds. Int. J. Mech. Sci. 163, 105135 (2019) 17. Rezaei, M., Talebitooti, R.: Effects of higher-order terms in aerodynamic force on the nonlinear response of a galloping PZT energy harvester. J. Theor. Appl. Vibr. Acoust. 6(2), 271–280 (2020) 18. Rezaei, M., Talebitooti, R., Liao, W.H.: Exploiting bi-stable magneto-piezoelastic absorber for simultaneous energy harvesting and vibration mitigation. Int. J. Mech. Sci. 207, 106618 (2021) 19. Rezaei, M., Talebitooti, R., Liao, W.H.: Investigations on magnetic bistable PZTbased absorber for concurrent energy harvesting and vibration mitigation: numerical and analytical approaches. Energy 239, 122376 (2022) 20. Rezaei, M., Talebitooti, R., Liao, W.H., Friswell, M.I.: Integrating PZT layer with tuned mass damper for simultaneous vibration suppression and energy harvesting considering exciter dynamics: an analytical and experimental study. J. Sound Vib. 546, 117413 (2023)

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21. Sigmund, O., Maute, K.: Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013) 22. Simiu, E., Scanlan, R.H.: Wind Effects on Structures: Fundamentals and Applications to Design, vol. 688. Wiley, New York (1996) 23. Teimourian, A., Hacişevki, H., Bahrami, A.: Experimental study on suppression of vortex street behind perforated square cylinder. J. Theor. Appl. Mech. 55(4), 1397–1408 (2017) 24. Teimourian, A., Teimourian, H.: Vortex shedding suppression: a review on modified bluff bodies. Eng. 2(3), 325–339 (2021) 25. Wang, L., Liu, W., Dai, H.: Aeroelastic galloping response of square prisms: the role of time-delayed feedbacks. Int. J. Eng. Sci. 75, 79–84 (2014) 26. Xing, J., Fang, S., Fu, X., Liao, W.H.: A rotational hybrid energy harvester utilizing bistability for low-frequency applications: modelling and experimental validation. Int. J. Mech. Sci. 222, 107235 (2022) 27. Xing, J., Rezaei, M., Dai, H., Liao, W.H.: Investigating the effect of surface protrusions on galloping energy harvesting. Appl. Phys. Lett. 122(15), 153902 (2023) 28. Xing, J., Rezaei, M., Dai, H., Liao, W.H.: Investigating the coupled effect of different aspect ratios and leeward protrusion lengths on vortex-induced vibration (VIV)-galloping energy harvesting: modelling and experimental validation. J. Sound Vib. 568, 118054 (2024)

Motion Planning for Wave-Like-Actuated Manta-Inspired Amphibious Robots Yixuan Wang1,2

, Qingxiang Wu1,2(B)

, Xuebing Wang1,2

, and Ning Sun1,2

1 Institute of Robotics and Automatic Information Systems (IRAIS), College of Artificial

Intelligence, Nankai University, Tianjin 300350, China [email protected] 2 Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China

Abstract. Inspired by manta rays, an amphibious robot is designed by utilizing a wave-like mechanism for propulsion. In terms of robot design, the swimming characteristics of the manta ray are analyzed, and mechanical structures such as flexible biomimetic fins and wave-like propulsion mechanisms are designed, enabling the robot to move both in water and on land. Furthermore, the kinematic model of the wave-like-actuated manta-inspired amphibious robot is established. Then, a path planning method based on the improved Rapidly-exploring Random Tree (RRT) algorithm is proposed, and combined with S-curve acceleration and deceleration planning to achieve velocity planning for robots. Experiments are conducted to validate the robot’s motion performance and the effectiveness of the motion planning algorithm. Keywords: Wave-like propulsion · Amphibious robot · Improved RRT algorithm · S-curve acceleration and deceleration planning

1 Introduction Amphibious robots capable of operating in both terrestrial and underwater environments are essential for various tasks, such as environmental monitoring, resource exploration, topographic mapping, pipeline and cable inspection, and more. However, these tasks often present significant challenges due to the unique characteristics of underwater environments, such as obstructed visibility, wave effects, and signal interference, which can impact the efficiency and accuracy of the robots. To overcome these challenges, research on biomimetic amphibious robots has been conducted worldwide [1], demonstrating significant research and practical applications value [2]. Ma X et al. developed a multimodal amphibious robot with flipper paddles [3], designed as separate modules for water and land operations and ingeniously integrated together. ACM-R5 [4] employs deformable mechanisms and multi-joint mechanical design to achieve adaptability and flexibility in different environments. Biomimetic robots, inspired by natural organisms, exhibit superior adaptability and flexibility. For example, biomimetic robots mimic the hexapod walking of insects [5], the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 529–540, 2024. https://doi.org/10.1007/978-981-97-0554-2_40

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flight of birds [6], and the swimming of fish [7]. In the realm of biomimetic oscillating fins, Katzschmann et al. developed a bio-inspired underwater soft robotic fish called SoFi [8], while Wang Y researched a flexible fin-propelled underwater robot [9]. Motivated by the above discussion, this paper aims to investigate the motion planning problem of wave-like-actuated manta-inspired amphibious robots, as depicted in Fig. 1. First, the swimming posture and movement patterns of manta rays are analyzed and studied, in order to cleverly apply them to the design of robots. Subsequently, by combining the physical model and kinematic equations of the robot, the kinematic model is established. Based on this, the motion planning algorithms for the robot in water and on land are deployed, including aspects such as path planning, velocity planning, etc., followed by the implementation and simulation verification of aforementioned algorithms. Finally, experiments are designed to validate the effectiveness and feasibility of the biomimetic manta ray amphibious robot’s motion planning algorithms, providing new ideas and methods for research and applications in this field.

Fig. 1. Research content flowchart

2 Design of the Manta-Inspired Amphibious Robot 2.1 Analysis of Manta Ray Motion Mode The manta ray is an underwater creature with a unique swimming style, as shown in Fig. 2. Its elongated undulating fins provide excellent maneuverability and high adaptability. It can cruise smoothly in calm waters, swim slowly in complex environments, make turns, and rapidly accelerate from a stationary position. Therefore, biomimetic studies of the manta ray’s swimming style hold significant value for the robot design. In the manta ray’s locomotion, the main propulsive force is generated by its pectoral fins. The pectoral fins extend along a large portion of its body, gradually tapering towards the rear and displaying a substantial number of fin rays. Each fin-like ray oscillates around its equilibrium, and the phase difference between adjacent fin-like rays remains constant during swimming [10]. This continuous undulation represents the overall motion of the fins coordinated by the fish’s nervous system. Unlike other fish species, the manta ray can smoothly transition from forward to backward swimming without turning its head, and vice versa. As a result, one of the unique features of the manta ray’s pectoral fin undulation is its ability to provide bidirectional propulsion simultaneously [11]. Noticing the wave-like characteristics of the elongated fin, it can be effectively parameterized using sinusoidal functions for enabling the design of a wave-like propulsion mechanism. Furthermore, by simply adjusting the parameters, it is easy to achieve bidirectional propulsion and freely switch between forward and backward movement, which

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Fig. 2. The manta ray

has significant implications for designing autonomous underwater vehicles with higher maneuverability and stability. 2.2 Design of Mechanical Structure In the control system architecture of the manta-inspired amphibious robot, the ESP32 acts as the robot’s brain, facilitating communication with the upper computer, driving motors, and capturing motor speed data. The microcontroller receives wireless signals from the upper computer, transmitting speed signals to the drive module for motor rotation control through signal amplification. Encoders capture motor speed information, forming a closed-loop control system that precisely manages motor speed and direction. The motor which driven by the microcontroller rotates the helical rods, enabling wave-like propulsion and flexible biomimetic fin movements for linear advancement, deceleration, acceleration, and turning. The mechanical structure of the wave-like-actuated manta-inspired amphibious robot, as depicted in Fig. 3, consists of multiple components. The entire robot comprises front and rear motor compartments, wave-like propulsion mechanisms, flexible biomimetic fins, a central electronic control compartment located above the middle section, and other fixed parts. Flexible biomimetic fins Electric control cabin

Motor compartment

Joint couplings

Fig. 3. Three-dimensional structure of the manta-inspired amphibious robot

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Motor Compartment and Joint Couplings. The front and rear motor compartments have identical structures, each housing a brushless motor installed on the left and right sides. Each brushless motor’s shaft is connected to a helical rod, where one end attached to the motor shaft serves as the active rod head, and the other end without a motor is connected to the passive rod head, forming a joint coupling. Each helical rod passes through n active linkages connected in series, creating articulated joint couplings that can swing up and down as the helical rods rotate. These joint couplings, as shown in the orange part in Fig. 4, are crucial components for actuating the robot motion. Helical Rods

Joint Couplings

Flexible Biomimetic Fins

Fig. 4. Bottom view of the manta-inspired amphibious robot

Wave-Like Propulsion Mechanism. The middle wave-like propulsion mechanism [12] is also a vital component of the robot. It utilizes two parallel rows of helical rods to create a sinusoidal wave translation of the articulated joint couplings, as illustrated in Fig. 5. This sinusoidal wave enables the robot to move forward and backward on land. The advantages of the wave-like propulsion mechanism include high efficiency, low energy consumption, as well as strong reliability. Compared with traditional wheeled or tracked robots, the manta-inspired amphibious robot exhibits more agile and flexible movement on land, which is beneficial for navigating through complex terrains. Furthermore, through the arrangement of parallel columns and differential drive, the wave-like propulsion mechanism can achieve turning the motion control, providing the robot with all-round mobility.

Fig. 5. Side view of the manta-inspired amphibious robot

Flexible Biomimetic Fins. The design of the flexible biomimetic fins involves simulating and optimizing the fins found in biology, resulting in efficient fins capable of

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propelling the robot efficiently underwater. The use of silicone rubber covering not only protects the supporting bones but also enhances the flexibility and durability of the fins, allowing the robot to move more flexibly and smoothly in water. Besides providing excellent stability and maneuverability, the biomimetic fins achieve efficient propulsion with lower energy consumption, while reducing noise and hydrodynamic resistance in water.

3 Establishment of the Manta-Inspired Amphibious Robot’s System Model The helical rod of the manta-inspired amphibious robot can be represented as a helix with the following expression [12]: x = Db, y = A sin(2π b), z = A cos(2π b),

(1)

where D is the length of the helix pitch, A is the radius of the helix, and b is the independent parameter. The two parallel rows of the wave-like propulsion mechanism convert the helix motion into the sinusoidal wave translation of the articulated joint couplings. The twodimensional projection of the helix on the X-Z plane (y = 0) produces a sinusoidal function curve, as shown in Fig. 6.

Fig. 6. Representation of the helical rod and wave-like propulsion

The two-dimensional projection expression is x = Db,   z = A cos(2π b) = A cos 2πDx .

(2)

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Based on the structural design of the manta-inspired amphibious robot, the relationship for wave speed and frequency, and the relationship between the rotation radius and rotation angle, can be derived as q = [ x y α ]T : ⎡

⎤ ⎡ D cos α D cos α ⎤  vx 120 120 sin α D sin α ⎦ nL q˙ = ⎣ vy ⎦ = ⎣ D120 120 nR D D ω − 120l 120l

(3)

where nL and nR are the respective speeds of the left and right motors, respectively, α represents the pose angle of the robot, and l represents the vertical distance of the robot’s center of mass to the line of wave mechanism velocities on the left and right sides. To describe the robot’s motion patterns more clearly, we set the positive direction of the x-axis as the forward direction of the robot. The motor shaft of the front-end motor rotates clockwise, and the connected helical shaft points to the positive direction of the x-axis, giving the robot a tendency to move towards the positive x-axis. Conversely, the motor shaft of the rear-end motor rotates counterclockwise, and the connected helical shaft also points to the positive direction of the x-axis, similarly giving the robot a tendency to move towards the positive x-axis. When both motors have the same speed, the robot will move in a straight line along the x-axis. However, when the speed of the front-end motor is greater than that of the rear-end motor, the robot will tend to turn towards the negative y-axis. Conversely, when the speed of the rear-end motor is greater than that of the front-end motor, the robot will tend to turn towards positive y-axis. The greater the speed difference between the two motors, the more pronounced the turning tendency of the robot will be.

4 Motion Planning Algorithms, Simulations and Experiments Motion planning is an important research direction in the field of robotics, with the aim of designing algorithms to enable robots to perform specific tasks in complex environments. Over the past few decades, various motion planning algorithms have been proposed [13]. For the motion planning of the bionic aquatic-terrestrial amphibious robot inspired by the manta ray, it is possible to explore a novel motion planning algorithm to enhance the robot’s motion control capability and accuracy for better task completion. Additionally, this research can offer new ideas and innovative points for the development of motion planning in the field of robotics. 4.1 Path Planning Principles of Path Planning Algorithm. The Rapidly-exploring Random Tree (RRT) algorithm [14] is a single-robot path planning algorithm based on a tree-like structure. The basic idea of the RRT algorithm is to randomly sample points in the configuration space and gradually connect these sampled points together through a certain expansion strategy, forming a tree-like structure. Figure 7 illustrates the node expansion process of the Rapidly-exploring Random Tree.

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qrand

qstart

qnear

qnew qgoal

Fig. 7. Illustration of node expansion process in Rapidly-exploring Random Tree

The RRT algorithm is well-suited for solving path planning problems in highdimensional and continuous spaces, and it can perform online path planning during runtime. Nevertheless, it may produce relatively longer paths and cannot guarantee to find the optimal solution. Therefore, in this paper, the RRT algorithm is optimized based on specific conditions. Specific Steps for Implementing Improved RRT Algorithm. The improvement of the RRT path planning algorithm focuses on the termination condition and path optimization. First, the process is initialized by setting the step size and maximum number of iterations, selecting the start and target points, and constructing them in the form of a tree. Random sampling is then performed, introducing a method of random direction. When selecting a parent node, the closest point to the sampled point is chosen as the parent node, avoiding the need for nearest neighbor search during parent node selection, thus saving computational time. Moreover, this method increases the coverage of the search space, enhancing the success rate of path planning. During path planning, collision detection is conducted when extending towards the sampled point. If there is no collision in this distance, and the distance between the new point and all existing points is greater than a certain threshold, the new point is added to the RRT tree. To expedite the search process and conserve resources, two main improvements are made to the basic RRT. Firstly, after each iteration of generating a new point, it checks whether a direct connection can be made between this new point and the target point. If a direct connection is feasible, the algorithm connects this point directly to the target point, resulting in the target path. After reaching the maximum number of iterations, if no solution path is found, the path planning is considered a failure.

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Secondly, considering that the manta-inspired amphibious robot may need to repeatedly go back and forth between the target point and the start point, this paper optimizes the already explored path to obtain a smoother trajectory. The approach is as follows: 1. Start from the initial node as the first node and traverse the entire path step by step. 2. Connect the starting node and the node being traversed (ending node). a. If there is no collision between the starting node and the ending node, directly connect them to eliminate intermediate redundant nodes. b. If a collision occurs, determine the previously traversed start and end nodes (line segment) as part of the new path and update the starting node to be the previous ending node from the previous iteration. 3. Repeat the above iteration until the entire path is traversed. Ultimately, this results in an optimized path consisting of a few line segments. The visualization of path planning is shown in Fig. 8 (where the start and target points are located at [0, 0] and [10] respectively, the green line represents the path obtained from the Rapidly-exploring Random Tree search, the red line represents the optimized path, and the black squares represent the obstacles). 10 9 8 7

y(m)

6 5 4 3 2 1 0 0

2

4

6

8

10

x(m)

Fig. 8. Simulation results of path planning

It can be observed that the original RRT algorithm takes a longer and more winding path, with frequent changes in the robot’s direction, leading to the energy loss. The improved RRT, on the other hand, simplifies the path into a straight line, shortening the distance and allowing the robot to achieve higher speeds on the straight segments, further reducing the overall travel time.

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4.2 Velocity Planning To address the drawbacks of the trapezoidal acceleration/deceleration planning method, the S-curve acceleration/deceleration planning method is chosen. The S-curve acceleration/deceleration planning method is an advanced motion control technique aimed at achieving smoother and more precise motion control by controlling the changes in acceleration and deceleration, resulting in an S-shaped trajectory during the acceleration, constant velocity, and deceleration phases. The advantages of the S-curve acceleration/deceleration planning method are as follows: 1. Smooth motion: The S-curve acceleration/deceleration planning method ensures smooth velocity changes during the acceleration, constant velocity, and deceleration phases, reducing motion impacts and vibrations, thereby achieving smoother motion. 2. High motion efficiency: Compared to the trapezoidal acceleration/deceleration planning method, the S-curve acceleration/deceleration planning method allows for faster velocity changes, leading to shorter motion duration and improved motion efficiency. 3. High control precision: The S-curve acceleration/deceleration planning method enables precise control of velocity and displacement changes, resulting in higher motion control accuracy. In general, the S-curve acceleration/deceleration planning method offers higher motion control precision and efficiency compared to the trapezoidal acceleration/deceleration planning method. However, it requires more advanced algorithms and equipment support, and also places higher demands on the mechanical structure. Fortunately, the mechanical structure and controller of the manta-inspired amphibious robot can meet these requirements effectively. Therefore, the S-curve acceleration/deceleration planning method, which provides better results, is chosen for implementation. According to the method described in reference [15], the displacement profile as a function of time is given by: ⎧ 0 ≤ t < t1 J τ13 /6 ⎪ ⎪ ⎪ ⎪ 2 ⎪ S1 + v1 τ2 + JT1 τ /2 t 1 ≤ t < t2 ⎪ ⎪ ⎪ S + v τ + JT τ22 /2 − J τ 3 /6 t ≤ t < t ⎪ ⎨ 2 2 3 1 3 2 3 3 (4) S(t) = S3 + v3 τ4 t 3 ≤ t < t4 ⎪ ⎪ 3 /6 ⎪ S + v τ − J τ t ≤ t < t 4 4 5 4 5 ⎪ 5 ⎪ ⎪ 2 /2 ⎪ S + v τ − JT τ t ≤ t < t ⎪ 6 1 6 5 5 5 6 ⎪ ⎩ S6 + v6 τ7 − JT1 τ72 /2 + J τ73 /6 t6 ≤ t < t7 After conducting the simulation, the curves for acceleration, velocity, and displacement are obtained, as shown in Fig. 9.

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Fig. 9. Simulation results of velocity planning

4.3 Experiments The experimental prototype is depicted in Fig. 10, and the preliminary experimental results of path planning are shown in Fig. 11.

Actual Path Planned Path

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Fig. 11. Preliminary experimental results of path planning

As seen in the actual path compared to the planned path, the deviation is minimal. This confirms the reasonable design of the robot’s mechanical structure and the effectiveness of the electronic control system, indicating stable motion performance. Furthermore, it demonstrates that the robot can follow the planned trajectory to a satisfactory extent, achieving the preliminary application of path planning and laying a solid foundation for the future application of more advanced algorithms.

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The preliminary experimental results of velocity planning, shown in Fig. 12, demonstrate that the robot can successfully start and stop with smooth acceleration and deceleration. However, during startup, the robot did not fully track the expected S-curve for acceleration due to the maximum static friction being greater than sliding friction. This analysis suggests that improvements can be made in the control algorithm to optimize velocity planning. In this section, through the conducted experiments, the motion performance of the designed bionic amphibious robot and the effectiveness of the motion planning algorithm are verified. In the initial validation experiments without velocity planning, an unexpected situation occurred due to poor motor gear engagement, highlighting the importance of velocity planning. The experimental results show that the designed robot can achieve motion and exhibits good stability and accuracy. The motion planning algorithm enables path and velocity planning, resulting in smoother and more natural robot motion.

5 Conclusion In this paper, a bionic amphibious robot inspired by the manta ray is designed. Overall, the structural design of this manta-inspired amphibious robot is utilized by the principles of wave propulsion and biomimetic fins to achieve amphibious motion. By designing and constructing the robot’s mechanical and circuit structures, successful emulation of the manta ray’s swimming motion is achieved. The model is established, and an improved RRT algorithm is used for path planning. A S-curve acceleration and deceleration planning method is adopted for the speed control, enabling precise robot control. Finally, the experimental results demonstrate that the amphibious biomimetic robot designed in this paper exhibits relatively good motion smoothness and control stability. Acknowledgement. This work was supported in part by the National Natural Science Foundation of China under Grant 52205019 and Grant 62373198, in part by the Guangdong Basic and Applied

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Basic Research Foundation under Grant 2023A1515012669, in part by the China Postdoctoral Science Foundation under Grant 2021M701779, in part the Fundamental Research Funds for the Central Universities under Grant 078-63233098, and in part by the State Key Laboratory of Robotics and Systems (HIT) under Grant SKLRS-2023-KF-13.

References 1. Wang, G., Chen, D., Chen, K., et al.: Current status and development trends of bionic robots. J. Mech. Eng. 51(13), 27–44 (2015) 2. Baines, R., Fish, F., Kramer-Bottiglio, R.: Amphibious robotic propulsive mechanisms: current technologies and open challenges. In: Bioinspired Sensing, Actuation, and Control in Underwater Soft Robotic Systems, pp. 41–69 (2021) 3. Ma, X., Wang, G., Liu, K.: Design and optimization of a multimode amphibious robot with propeller-leg. IEEE Trans. Rob. 38(6), 3807–3820 (2022) 4. Yamada, H.S.: Development of amphibious snake-like robot ACM-R5. In: The 36th International Symposium on Robotics (ISR 2005), Tokyo (2005) 5. Johnson, A.M., Koditschek, D.E.: Toward a vocabulary of legged leaping. In: 2013 IEEE International Conference on Robotics and Automation, pp. 2568–2575. IEEE (2013) 6. Jianlin, X., Bifeng, S., Wenping, S., et al.: Progress of Chinese “dove” and future studies on flight mechanism of birds and application system. Trans. Nanjing Univ. Aeronaut. Astronaut. 37(5) (2020) 7. Berlinger, F., Gauci, M., Nagpal, R.: Implicit coordination for 3D underwater collective behaviors in a fish-inspired robot swarm. Sci. Rob. 6(50), eabd8668 (2021) 8. Katzschmann, R.K., DelPreto, J., MacCurdy, R., et al.: Exploration of underwater life with an acoustically controlled soft robotic fish. Sci. Rob. 3(16), eaar3449 (2018) 9. Wang, Y., Cai, M., Wang, S., et al.: Development and control of an underwater vehicle– manipulator system propelled by flexible flippers for grasping marine organisms. IEEE Trans. Industr. Electron. 69(4), 3898–3908 (2021) 10. Li, F., Hu, T., Wang, G., et al.: Locomotion of Gymnarchus niloticus: experiment and kinematics. J. Bionic Eng. 2, 115–121 (2005) 11. Hu, T., Shen, L., Lin, L., et al.: Biological inspirations, kinematics modeling, mechanism design and experiments on an undulating robotic fin inspired by Gymnarchus niloticus. Mech. Mach. Theory 44(3), 633–645 (2009) 12. Zarrouk, D., Mann, M., Degani, N., et al.: Single actuator wave-like robot (SAW): design, modeling, and experiments. Bioinspir. Biomim. 11(4), 046004 (2016) 13. Liu, H., Yang, J., Lu, J., et al.: A survey of motion planning for mobile robots. China Eng. Sci. 8(1), 85–94 (2006) 14. Song, J., Dai, B., Shan, E., et al.: An improved RRT path planning algorithm. Acta Electron. Sin. 38(2A), 225 (2010) 15. Chen, Y., Wei, H., Wang, Q.: Linear and S-shaped acceleration/deceleration discrete algorithm for numerical control system. China Mech. Eng. 5, 567–570 (2010)

Time-Optimal Anti-swing Trajectory Planning of Double Pendulum Crane Based on Chebyshev Pseudo-spectrum Method Ken Zhong and Yuzhe Qian(B) School of Artificial Intelligence, Hebei University of Technology, Tianjin, China [email protected]

Abstract. As a typical underdrive system, an overhead crane has been widely used in modern industrial production and transportation. However, when the load volume in the crane system is too large or the hook quality is too large, the bridge crane system will show the characteristics of double-pendulum, increasing the difficulty of control. Based on this, this paper proposes a time-optimal trajectory planning method for the double-pendulum bridge crane system, which can be obtained. Specifically, the paper first transforms the system kinematics model; based on this basis Then, considering the various constraints including the twolevel swing angle and the trolley speed and acceleration limit, the optimization problem is transformed into a nonlinear programming problem which is easier to solve, and the trajectory constraints can be considered very conveniently in the conversion process. Solving the nonlinear programming problem yields the timeoptimal trolley trajectory. Finally, the simulation results show that the time-optimal trajectory planning method has satisfactory control performance. Keywords: Double Pendulum Crane · Trajectory Planning · Chebyshev Pseudo-Spectrum Method

1 Introduction In the industrial production process, in order to transport the load to the desired position, various crane systems, including bridge crane, cantilever crane, tower crane, Marine crane and Marine crane, have been widely used. In order to simplify the mechanical structure of the crane system, the load is not often directly controlled, but indirectly drags the load to the target position through the movement of the trolley. As a result of this structure, the control input dimension of the crane system is smaller than the degree of freedom dimension to be controlled. The system with this feature is the so-called underdrive system [1]. Due to the removal of some drivers of the system, increasing the system freedom and improving the system flexibility, the underdrive system is superior to the full drive system in terms of energy saving, price reduction, and enhanced system adaptability. However, external disturbance, crane start and stop, speed change and so on will make the load swing, which will not only reduce the transport efficiency of the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 541–553, 2024. https://doi.org/10.1007/978-981-97-0554-2_41

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crane, but also bring huge safety risks. Therefore, the automatic control method of crane system has practical significance and wide application value, which has received the attention of scholars. In recent years, domestic and foreign scholars have put forward many control methods for the control problem of bridge crane system. In order to simplify the control algorithm design of bridge crane system, a control method based on partial feedback linearization is proposed [2, 3]. Fang et al. proposed a new two-step design strategy, namely motion planning stage and adaptive tracking control stage to control such an underactuated system as an overhead crane [4]. Peng et al. investigated the impact of uncertainty on the crane movement in the stage of trajectory planning, proposed an uncertain method based on interval model [5]. In literature [6], Singhose et al. used the idea of input shaping to control the crane system, which could effectively suppress the load residual swing. In [7], Fang et al. design a series of energy-based controllers to regulate the trolley to a desired position while reducing the pendulation of the payload at the same time. Chen et al. present a time optimal trajectory planning scheme for double pendulum crane systems, which can yield a global time-optimal swing-free trajectory [8]. Sun et al. proposed an optimal trajectory planning method for double pendulum crane based on differential flatness theory, considering a series of constraints such as system swing angle constraints and trolley speed constraints [9]. In [10], Sun et al. proposes a trajectory planning method based on phase plane analysis, which can better suppress load swing and eliminate residual swing. Vaughan et al. proposed a trajectory planning method based on phase plane analysis, which can better suppress load swing and eliminate residual swing [11]. Guo et al. designed an energy-based control method by analyzing the passivity of the crane system [12]. In [13], a linear sliding mode control method was proposed based on the complex dynamic model of double pendulum overhead crane system, which can effectively weaken the system chattering. Zhang et al. proposed a trajectory planning strategy for double pendulum bridge crane based on swing angle constraint [14]. Wang et al. integrated the smooth shaping technology and active disturbance rejection control as an anti-swing control method for doublependulum crane, which can solve the problem of long anti-swing time, low positioning accuracy and poor anti-disturbance ability of double-pendulum overhead crane without payload swing angle sensor [15]. Aiming at the problem that the load swing of double pendulum crane is difficult to be measured directly in the actual production process, Xiao et al. proposed a sliding mode control method of double pendulum crane based on load swing state estimation is proposed [16]. Kang et al. proposed a control strategy combining adaptive neural network and packet fuzzy control based on double pendulum bridge crane system to solve the control problem of underdriven nonlinear system [17]. In [5], Peng et al. proposed an uncertainty research method based on interval model to Reduce the influence of uncertainty on crane movement. Although the above control strategies can realize the control of the double-swing crane system, they are difficult to ensure the maximum operation efficiency of the crane system.

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In order to improve the transport efficiency of the double pendulum crane system, a global time optimal trajectory planning method is proposed, which can realize the control objectives of accurate positioning of the trolley and rapid load elimination at the same time. Different from the existing methods, the proposed method can obtain the global time optimal trolley trajectory. Specifically, the kinematic model of the crane system is transformed first, and an acceleration-driven system model is obtained. Then, by using this model, considering various physical and safety constraints in the crane system, a function to be optimized with transport time as the optimization objective is constructed. In order to facilitate the solution of the optimization problem, Chebyshev pseudospectrum method is used to discretization and approximation the obtained optimization problem and corresponding constraints at Chebyshev-Gauss-Lobatto (CGL) points. Finally, the effectiveness of the proposed method is verified by numerical simulation.

2 Problem Statement The crane model with double pendulum effect is shown in Fig. 1, whose dynamic characteristics is illustrate as follows: (m1 + m2 )l2 cos θ1 x¨ + (m1 + m2 )l12 θ¨1 + m2 l1 l2 cos(θ1 − θ2 )θ¨2 + m2 l1 l2 sin(θ1 − θ2 )θ˙22 + (m1 + m2 )gl1 sin θ1 = 0

(1)

m2 l2 cos θ2 x¨ + m2 l1 l2 cos(θ1 − θ2 )θ¨1 + m2 l22 θ¨2 − m2 l1 l2 θ˙12 sin(θ1 − θ2 )+ m2 gl2 sin θ2 = 0

(2)

where m, m1 , m2 denote the masses of the the trolle, the hook and the load, respectively, l1 denote the length of the rope, l2 represent the equivalent rope length, which denote the distance between the load centroid and the centroid of the hook. x(t) represent the trolley movement, θ1 (t), θ2 (t) describe the angles of the two pendulums, g is the gravity acceleration constant.

Fig. 1. Schematic illustration for the crane model with double-pendulum effects.

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Divide (1) and (2) by (m1 + m2 ) l1 and m2 l2 , respectively, we can obtain: 2 l2 cos θ1 x¨ + l1 θ¨1 + mm1 +m cos(θ1 − θ2 )θ¨2 + 2 m2 l2 2 ˙ m1 +m2 sin(θ1 − θ2 )θ2 + g sin θ1 = 0

(3)

cos θ2 x¨ + l1 cos(θ1 − θ2 )θ¨1 + l1 θ¨2 − l1 θ˙12 sin(θ1 − θ2 ) + g sin θ2 = 0

(4)

Equations (3) and (4) describe the coupling relationship between vehicle position shift x(t) and the two stage swing angles θ1 (t) and θ2 (t) of the system, that is, the influence of trolley motion on load swing. This method is based on the analysis of the coupling relationship and the planning of a trolley trajectory with the ability to reduce the pendulum. Considering the safety, efficiency and physical constraints of the actual crane system, this paper will plan a trolley trajectory with analytical expression for the underactuated crane system with double pendulum effect. The specific control objectives to be achieved are as follows [9]: 1) To reach the target position quickly and accurately, the trolley ought to get the destination xf at time t = T from its initial position x0 at time t = 0, while velocity and acceleration signals ought to be zero. Then we have x(0) = x˙ (0) = x¨ (0)

(5)

x(T ) = xf , x˙ (T ) = x¨ (T ) = 0

(6)

where, T represents the time required for the transportation process and the initial position x0 = 0. 2) In the entire control task, the trolley velocity and acceleration should be kept in suitable ranges, in sense that |˙x(t)| ≤ vmax , |¨x(t)| ≤ amax

(7)

where, vmax , amax represent the permitted trolley velocity and acceleration, respectively. 3) In order to ensure that the load can be processed directly at the end of the task, when the trolley reaches the target position, there should be no residual swing and the angular velocity is zero, that is θ1 (0) = θ˙1 (0) = 0, θ1 (T ) = θ˙1 (T ) = 0

(8)

θ2 (0) = θ˙2 (0) = 0, θ2 (T ) = θ˙2 (T ) = 0

(9)

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4) To avoid the collision caused by the violent swing of the load, the swing angle and angular velocity of the two stage swing should be kept in suitable ranges during the transport process, that is |θ1 (t)| ≤ θ1 max , |θ2 (t)| ≤ θ2 max

(10)

    θ˙1 (t) ≤ ω1 max , θ˙2 (t) ≤ ω2 max

(11)

where, θ1 max , θ2 max , ω1 max , ω2 max represent the permitted payload swing angle and angular velocity amplitudes, respectively. In summary, the following optimization problems can be constructed: min T s.t. x(0) = x˙ (0) = x¨ (0) = 0 x(T ) = xf , x˙ (T ) = x¨ (T ) = 0 |˙x(t)| ≤ νmax , |¨x(t)| ≤ amax θ1 (0) = θ˙1 (0) = 0, θ1 (T ) = θ˙1 (T ) = 0, θ2 (0) = θ˙2 (0) = 0, θ2 (T ) = θ˙2 (T ) = 0, |θ  1 (t)| ≤ θ1 max , |θ2 (t)| ≤ θ2 max θ˙1 (t) ≤ ω1 max , θ˙2 (t) ≤ ω2 max

(12)

Next, we propose a pseudo-spectral method to solve the optimization problem, and a time optimal trajectory is planned for the trolley.

3 Trajectory Planning In this section, a time-optimal trajectory planning strategy based on pseudo-spectrum method is proposed, and the trajectory of the trolley is obtained by solving (12). Specifically, the kinematic model of the crane system is transformed into an acceleration driving model, in which the acceleration of the crane can be regarded as the system input. Then, based on the acceleration driving model, the original optimization problem can be rewritten into a new form. Then the constrained optimization problem is transformed into a series of nonlinear programming problems by using Chebyshev pseudospectrum method. Finally, by solving the nonlinear optimization problem, we can obtain the time-optimal trolley trajectory.

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3.1 Transformation of System Models In order to facilitate subsequent trajectory planning, the double-pendulum crane system model and optimization problem (12) is transformed here. For this purpose, the system full state vector ζ (t) is defined as follows:  T ζ = x x˙ θ1 θ˙1 θ2 θ˙2

(13)

According to the kinematics model (3) and (4) of the system, the acceleration of the vehicle can be taken as the input of the system. At this point, the kinematic model can be transformed into the following form [8]: ζ˙ = f (ζ ) + h(ζ )u

(14)

where, u(t) is the acceleration of the trolley x¨ (t), f (ζ ), h(ζ ) represents the auxiliary function with respect to ζ (t), in the following form: T  f (ζ ) = x˙ 0 θ˙1 A θ˙2 B

(15)

 T h(ζ ) = 0 1 0 C 0 D

(16)

For the convenience of description, auxiliary variables A, B, C, D are defined as follows:  ⎧ m2 C1−2 2 l2 S1−2 C1−2 θ˙22 l1 S1−2 θ˙12 + mm1 +m ⎪ 2 ⎪ A = − l1 (m1 +m2 )−m2 l1 C1−2 2 ⎪  ⎪ ⎪ l2 ⎪ − g(S2 − S1 C1−2 ) − l11 gS1 − l1 (mm12+m S1−2 θ˙22 ⎪ ⎪ 2) ⎪ ⎪ m1 +m2 m2 l2 ⎨ B= l S1−2 × C1−2 θ˙22 + l1 S1−2 θ˙12 2 2 (m1 +m2 )−m2 l2 C1−2 m1 +m2 (17)  ⎪ − g(S2 − S1 C1−2 ) ⎪ ⎪ ⎪ m2 C1−2 1 ⎪ ⎪ C =l ⎪ 2 (C2 − C1 C1−2 ) − l1 C1 ⎪ 1 (m1 +m2 )−m2 l1 C1−2 ⎪ ⎪ m1 +m2 ⎩D = − (C2 − C1 C1−2 ) l (m +m )−m l C 2 2

1

2

2 2 1−2

In formula, the following simplified form is used: S1 = sin θ1 , S2 = sin θ2 , C1 = cos θ1 , C2 = cos θ2 S1−2 = sin(θ1 − θ2 ), C1−2 = cos(θ1 − θ2 ) Using the resulting acceleration driven system model (14), the original optimization problem (12) can be transformed into the following form: min T

(18a)

s.t. ζ˙ = f (ζ ) + h(ζ )u

(18b)

 T ζ (0) = 0 0 0 0 0 0

(18c)

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T  ζ (T ) = xf 0 0 0 0 0

(18d)

|˙x(t)| ≤ vmax , |˙u(t)| ≤ amax

(18e)

|θ1 (t)| ≤ θ1 max , |θ2 (t)| ≤ θ2 max

(18f)

    θ˙1 (t) ≤ ω1 max , θ˙2 (t) ≤ ω2 max

(18g)

By solving this optimization problem, the optimal time T ∗ required to complete the control objective and the corresponding optimal trolley trajectory can be obtained. 3.2 Trajectory Planning Based on Chebyshev Pseudo-spectrum Method In order to obtain the time-optimal vehicle trajectory, the key is how to solve the constrained optimization problem. In this paper, the Chebyshev pseudo-spectrum method is used to deal with the optimization problem, and the time optimal solution and the optimal trajectory can be obtained conveniently. Different from most existing methods, this method can analyze and process the original system more directly, and can obtain the global time optimal solution. In order to adapt to the requirements of the Chebyshev pseudo-spectrum method, it is necessary to use coordinate transformation to transform the time interval corresponding to the trajectory from t ∈ [0, T ] to the interval τ ∈ [−1, 1], that is τ=

2t −1 T

(19)

In Chebyshev pseudo-spectrum method, the selection of interpolation points (i.e. CGL points) is τi = cos((N − k)π/N ), i = 0, · · · , N

(20)

These nodes all fall on the interval [−1, 1] and satisfy τ0 = −1, τN = 1. In particular, they are the extreme value of the N degree Chebyshev polynomial TN (t), and the jth Chebyshev polynomial is 

(21) Tj (t) = cos j cos−1 t , j = 0, · · · , N

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Then, the system state quantity and input quantity to be planned can be discretely expressed in the following form: ζ (τ0 ), ζ (τ1 ), ζ (τ2 ), · · · , ζ (τN ) u(τ0 ), u(τ1 ), u(τ2 ), · · · , u(τN ) Using these nodes, the Lagrange interpolation polynomial is constructed as follows:

(−1)j+1 1 − τ 2 T˙ N (τ )

φj (τ ) = (22) N 2 cj τ − τj where  cj =

2, j = 0, N 1, 1 ≤ j ≤ N − 1

Using Eq. (22) and the values of the system state quantity and input quantity at CGL point, the system state quantity trajectory and input quantity trajectory can be approximated in the following way: ζ (τ ) ≈

N N   ζ τj φj (τ ), u(τ ) ≈ u τj φj (τ ) j=0

(23)

j=0

where ζ τj , u τj represent the system state quantity and input quantity at τ = τj , respectively. Take the derivative of Eq. (23), and use the specific form of the interpolation function in Eq. (22) to calculate and simplify, and the derivative of the trajectory of the state quantity can be obtained as follows: ζ˙ (τi ) =

N N   ζ τj φ˙ j (τi ) = ζ τj Dji (τi ) j=0

(24)

j=0

where, ζ˙ (τi ) represents the derivative of the state locus at τ = τi . Dji (τi ) represents the derivative of φj at τ = τi , in the following form: ⎧ c (−1)i+j i ⎪ ⎪ cj (τi −τj ) ⎪ ⎪ ⎨ − τi 2 1−τi2 Dji = ⎪ 2N 2 +1 ⎪ ⎪ 6 ⎪ ⎩ 2N 2 − 6+1

, i = j ,1 ≤ i = j ≤ N − 1 ,i = j = 0 ,i = j = N

(25)

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By using (24), (25) and the trajectory values at CGL points, the differential equation constraint (18b) in optimization problem (18) can be discretized and approximated. The specific results are as follows: N 

ζ (τi )Dji −

i=0

 T f (ζ (τi )) + h((ζ (τi )))u(τi ) = 0 2

(26)

Next, the boundary condition constraints in the optimization problem also need to be transformed into algebraic constraints, where the formula (18c) can be directly rewritten as follows:  T (27) ζ (τ0 ) = 0 0 0 0 0 0 For the end time, using the Clenshaw-Curtis integral, (18d) can be expressed as: ζ (τN ) = ζ (τ0 ) +

N  T   ωi f (ζ (τi )) + h((ζ (τi )))u(τi ) 2

(28)

i=1

where, ωi is Clenshaw-Curtis weight. When N is even ⎧ 1 ⎪ ⎨ ω0 = ωN = N 2 −1 N /2 4  1 ⎪ cos 2πNks , s = 1, 2, · · · , N2 ⎩ ωs = ωN −s = N 1−4k 2

(29)

k=0

When N is odd ⎧ 1 ⎪ ⎨ ω0 = ωN = N 2 ⎪ ⎩ ωs = ωN −s =

4 N

N −1 2



k=0

1 1−4k 2

cos 2πNks , s = 1, 2, · · · , N 2−1

(30)

To sum up, all constraints in the optimization problem can be expressed in the form of algebraic constraints. Based on this, the original optimization problem can be transformed into a nonlinear programming problem with algebraic constraints, as follows: min T s.t. N    ζ (τi )Dji − T2 f (ζ (τi )) + h((ζ (τi )))u(τi ) = 0 i=0  T ς (τ0 ) = 0 0 0 0 0 0 N    ζ (τ0 ) + T2 ωi f (ζ (τi )) + h((ζ (τi )))u(τi ) = i=1 T  xf 0 0 0 0 0 ζ (τ ) − γ ≤ 0, − ζ (τ ) − γ ≤ 0 u(τ ) − amax ≤ 0, − u(τ ) − amax ≤ 0

(31)

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where, the vector γ has the following form: T  γ = ∞ vmax θ1 max ω1 max θ2 max ω2 max For the above constrained nonlinear programming problem, this paper chooses Sequential quadratic programming (SQP) to solve it, and the following time-optimal state vector sequence can be obtained: ζ (τ0 ), ζ (τ1 ), ζ (τ2 ), · · · , ζ (τN ) The above formula is the optimal sequence of time discrete state vectors. By taking the first two terms of each vector (trolley displacement and trolley velocity) and interpolating them, the corresponding time-optimal trolley displacement and velocity trajectory can be obtained.

4 Simulation Results In this section, to verify the effectiveness of the designed trajectory planning method, we use MATLAB to conduct some simulations. The used parameters are set as M = 6.5 kg, m1 = 2.0 kg, m2 = 0.5 kg g = 9.8 kg, l1 = 0.5 m, l2 = 0.4 m The target position of the trolley is selected as xf = 0.6 m, and the trajectory constraints are selected as follows: θ1 max = θ2 max

= 2 deg, vmax = 0.3

ω1 max = ω2 max = 5 deg,

m/s

amax = 15 m/s2

The simulation results are shown in Fig. 2 and 3. As can be seen from Fig. 2, when the trolley moves according to the planned trajectory, it only takes T* = 5.9895 s to complete the given transport task, and the trolley can accurately reach the target position. In the whole process, the speed of the vehicle does not exceed the given limit vmax = 0.3 m/s. At the same time, it can be seen from Fig. 3 that the maximum swing angle of both the first and the second order swing angles does not exceed the given value of 2°, and there is no residual swing at the end of transportation. This also ensures the safety of the load during transport. Similarly, the angular velocity corresponding to the two stages of oscillation is also kept within the given range. Then we increased the load weight and rope length, and the simulation results are shown in Fig. 4. As can be seen from Fig. 4, when the load weight and rope length increase, the swing angle is still within the restricted range. In summary, the simulation results verify the efficiency and safety of the proposed optimal trajectory planning method.

Time-Optimal Anti-swing Trajectory Planning of Double Pendulum Crane

Fig. 2. Simulation results of trolley position and velocity

Fig. 3. Simulation results of first and second order swing angles and angular velocities

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Fig. 4. Simulation results after increasing load weight and rope length

5 Conclusion In this paper, a time-optimal trajectory planning strategy based on Chebyshev pseudospectrum method is proposed for the overhead crane with double pendulum effect. Specifically, firstly, the kinematic model of the crane system is transformed into an acceleration driving model, and based on this model, a constrained optimization problem is constructed considering various constraints. Then, the optimization problem is processed by Chebyshev pseudo-spectrum method and transformed into nonlinear programming problem which is more convenient to solve. On this basis, the time optimal vehicle trajectory can be obtained. The trajectory planning method proposed in this paper not only considers the object of the pendulum, but also can deal with the actual physical constraints such as the pendulum angle constraint, the angular velocity constraint, the trolley velocity constraint and the acceleration constraint. Different from the existing methods, the proposed method can obtain the global time optimal trolley trajectory. Finally, the effectiveness of the proposed method is verified by numerical simulation. In future work, the angular acceleration constraint will be considered in the trajectory planning process to obtain better control effect.

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References 1. Liu, Y., Yu, N.: A survey of underactuated mechanical systems. IET Control Theory Appl. 7(7), 921−935 (2013) 2. Tuan, A.L., Lee. S., Dang, V.: Partial feedback linearization control of a three-dimensional overhead crane. Int. J. Control Autom. Syst. 11(4), 718−727 (2016) 3. Tuan, A.L.: Partial feedback linearization control of overhead cranes with varying cable lengths. Int. J. Precis. Eng. Manuf. 13(4), 501−507 (2012) 4. Fang, Y., Ma, B., Wang, P.: A motion planning-based adaptive control method for an underactuated crane system. IEEE Trans. Contr. Sys. Techn. 20(1), 241−248 (2012) 5. Peng, J., Shi, Y.: Trajectory planning of double pendulum crane considering interval uncertainty. J. Mech. Eng. 55(02), 204−213 (2019) 6. William, S., Dooroo, K., Michael, K.: Input shaping control of double-pendulum bridge crane oscillations. J. Dyn. Syst. Meas. Control 130(3), 1−7 (2008) 7. Fang, Y.: Nonlinear coupling control laws for an underactuated overhead crane system. IEEE/ASME Trans. Mechatron 8(3), 418−423 (2003) 8. Chen, H.: Pseudospectral method based time optimal anti-swing trajectory planning for double pendulum crane systems. Acta Automatica Sinica 42(01), 153−160 (2016) 9. Sun, N.: Motion planning for cranes with double pendulum effects subject to state constraints. Control Theory Appl. 31(7), 974−980 (2014) 10. Sun, N.: Transportation taskoriented trajectory planning for underactuated overhead cranes using geometric analysis. IET Control Theory Appl. 6(10), 1410−1423 (2012) 11. Vaughan, J.: Control of tower cranes with double-pendulum payload dynamics. IEEE Trans. Control Syst. Technol. 18(6), 1345−1358 (2010) 12. Guo, W., Liu, T.: Double-pendulum-type crane dynamics and passivity based control. J. Syst. Simul. 20(18), 4945−4948 (2008) 13. Wang, J. Qiang, M.: Research on sliding mode control of underactuated double pendulum crane system. J. Ordnance Equip. Eng. 40(12), 193−198 (2019) 14. Zhang, C., Shao, J.: Trajectory planning method based on swing angle constraint for double pendulum crane. Process Autom. Instrum. 40(09), 40−45 (2019) 15. Wang, D., Xiao, G., Li, W.: Anti-swing control of double-pendulum crane based on command smoothing and active disturbance rejection control. J. Railway Sci. Eng. 19(03), 831−840 (2022) 16. Xiao, G., Zhu, Z.: Sliding mode control for double-pendulum overhead cranes with playload swing state observation. J. Central South Univ. (Sci. Technol.) 52(04), 1129−1137 (2021) 17. Liu, K., Sun, W.: Application of grouped adaptive fuzzy neural network on double pendulum crane. Sci. Technol. Eng. Crane 21(15), 6285−6290 (2021)

Nonlinear Inertia and Its Effect Within an X-shaped Mechanism Zhenghan Zhu

and Xingjian Jing(B)

Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China [email protected]

Abstract. This paper presents a new understanding and development related to nonlinear inertia and its effect in coupling with an X-shaped anti-vibration mechanism, which is validated by prototyping and experiments. The new inertial unit integrated in a well-designed X-shaped mechanism allows larger excitation displacements and more adjustable inertia ratios, resulting in significantly lower vibration transmissibility and resonance peak, and produces three different typical nonlinear inertia forms. A key parameter indicator (Ratio of Inertia) is proposed to identify and indicate the different types of nonlinearities, and the resulting beneficial effects are explored. The performance improvement in dynamics with respect to the linear counterpart is evaluated for different nonlinear inertia forms. The results show that: (a) The U-shaped symmetrical nonlinear inertia coupling with X-shaped mechanisms can provide better vibration isolation performance at low frequency; (b) High-frequency transmissibility can be tuned to different level, indicating a tunable band-suppress property, which is a unique property discovered in this study; (c) the nonlinear inertia contributes significantly to tune the interactive force between vibration source and object in the low frequency range (ρi

Then, we utilize an autonomous center selection strategy in [4]. It has three steps to pick out the cluster centers of each class. Firstly, the gamma value of sample i is computed by γi = ρi δi and is sorted in a gamma set γ in decreasing order. Secondly, we start from the first sample in γ to check whether it has both large distance from other centers and high density one by one as shown in (8): δi ≥ 2σ (δi ) ∩ ρi ≥ μ(ρi ).

(8)

Once this condition is no longer met, the check ends immediately. Thirdly, we take all the samples that satisfy the condition as the clustering center. As each centers’ label is available, we can utilize these centers as prototypes of the categories  y to which y  they belong. They are recorded in a set C = C s , · · · , C t , and C y = c1 , · · · , cnc represents nc centers of class y.

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3.3 Memory Network for Replay We assign a self-organizing incremental neural network [23] to each class as memory, which can learn sparse and generalized exemplar Different from tradi y representations. y  tional SOINN, we utilize the centers C y = c1 , · · · , cnc to initialize the network of  y y  class y, where nc nodes are created at first M y = c1 , · · · , cnc . Then, the SOINN learn all samples of class y based on a Hebbian learning rule one by y one. Each sample only appear once in the continual learning way. When a sample xi of class y comes, nearest  two  ythe SOINN  finds the first  nodes b1 and b2 . If the node activation y y conditions xi − wb1  > THb1 and xi − wb2  > THb2 are met simultaneously, xi is a novel sample for current memory, and a new node is created to store this new sample. y Otherwise, we can consider that xi is similar with b1 and b2 is a neighbor of b1 . Thus, we y build a connection between b1 and b2 , then xi is utilized to update the weight of node b1 and its neighbors. The update contributes to obtain generalized exemplar representations. y Besides, we also add a new node for xi when the initial network has less than two nodes. The aim is to deal with the situation where there is only one center point for a class. The online incremental learning ability of the network can help select more representative samples of each class for replay rather than the samples that are only close to the mean and reduced over time.

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When all samples of one class have been learned, nodes that are rarely activated after created in this SOINN will be treated as noise and deleted to simplify the After traversing all new classes, the whole memory set M =   1memory network. M , · · · , M s , M s+1 , · · · , M t is obtained and can be used to replay in the next leaning routine. Algorithm 3 illustrates the details of the memory construct process.

4 Experimental Results We conduct our experiments on the CIFAR-100 dataset [24] to evaluate our method. In the class-incremental scenario, classes come in order and are processes only once. Therefore, we split the dataset in batches of 10, 20, or 50 classes at a time. To verify the effectiveness of two proposed improvements, we compare iCaRL [20], Our method 1 which means iCaRL only improved by density-peaks-based prototype selection strategy and our method 2 which represents the whole algorithm with two improvements. The experiments are executed based on avalanche [25] library in Python. The evaluation criterion is multi-class accuracy on the test set for classification. The results are demonstrated in Fig. 1 and Table 1. We can find that our method 2 achieves higher average multi-class accuracy the iCaRL under three class batch size. Moreover, Fig. 1 demonstrates that this excellent performance is throughout the learning process. Thus, the two improvements can indeed improve the learning effectiveness of iCaRL.

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(a) 10 classes

(b) 20classes

(c) 50classes Fig. 1. Multi-class accuracy on CIFAR-100 with 10, 20 and 50 classes in each learning routine.

Table 1. Average multi-class accuracy on CIFAR-100. Batch size

iCaRL

Our method 1

Our method 2

10 classes

0.611

0.608

0.665

20 classes

0.663

0.652

0.701

50 classes

0.681

0.679

0.704

The accuracy results of our method 1 are only marginally better than less than iCaRL. It seems only using the density-peaks-based prototype selection strategy does not improve the learning effectiveness. However, we can find in Fig. 1 (a) that our method 1 is superior to iCaRL at the beginning of learning and gradually fall behind as classes grow. That indicate that the density-peaks-based prototype selection strategy has a positive effect on learning. It is the fixed exemplar set in iCaRL that limits ResNet to learn comprehensive class datadistribution. Because when learned class increasing, the number of exemplars m = K t for each class decrease. In late learning, only few exemplars are selected for distillation loss to update the parameters of ResNet. Thus, the extracted feature only is associated with a few density peaks, but the final nearest-mean classifier tends to correctly classify samples in the mean value’s spherical neighborhood. When cooperated with our proposed memory based on a modified SOINN, the density-peaks-based prototype selection strategy can make a difference. Because the

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SOINN can learn sparse and generalized representations for each class, which equals to update ResNet with the whole data distribution of each class. At the same time, the density-peaks-based prototype selection strategy can help SOINN to store density-peaks samples, which contributes to learn arbitrary-shape data distributions rather than only spherical distribution of nearest-mean methods.

5 Conclusion In this paper, an improved incremental classifier and representation learning method is proposed for elderly escort robots. This method utilizes a density-peaks-based prototype selection strategy to pick out appropriate cluster centers and prototypes of each class. Moreover, a modified self-organizing incremental neural network is equipped to each class for replay, which is initialized by its prototypes. A series of contrast experiments with iCaRL on CIFAR-100 dataset have demonstrated that the improvements can improve the learning effectiveness. Therefore, our method is an efficient continual learning method. In future work, we will consider applying this method on physical robots such as Nao to test the practical effects of the proposed method. Acknowledgment. This work was supported by the Youth Fund of Natural Science Foundation of Shandong Province under Grant No. ZR2021QF130, the Youth Fund of Shandong Province under Grant No. ZR202102230323, the National Natural Science Foundation of China under Grant No. 62273163, the Outstanding Youth Foundation of Shandong Province Under Grant No. ZR2023YQ056, the Key R&D Project of Shandong Province under Grant No. 2022CXGC010503, and the Innovation and Entrepreneurship Training Program for College Students No. 2023190105.

References 1. Pu, L., Moyle, W., Jones, C., Todorovic, M.: The effectiveness of social robots for older adults: a systematic review and meta-analysis of randomized controlled studies. Gerontologist 59(1), 37–51 (2019) 2. Kim, J., et al.: Companion robots for older adults: Rodgers’ evolutionary concept analysis approach. Intell. Serv. Rob. 14(5), 729–739 (2021) 3. Lesort, T., Lomonaco, V., Stoian, A., Maltoni, D., Filliat, D., Díaz-Rodríguez, N.: Continual learning for robotics: definition, framework, learning strategies, opportunities and challenges. Inf. Fusion 58, 52–68 (2020) 4. Parisi, G.I., Kemker, R., Part, J.L., Kanan, C., Wermter, S.: Continual lifelong learning with neural networks: a review. Neural Netw. 113, 54–71 (2019) 5. De Lange, M., et al.: A continual learning survey: defying forgetting in classification tasks. IEEE Trans. Pattern Anal. Mach. Intell. 44(7), 3366–3385 (2022) 6. Lesort, T., Caselles-Dupré, H., Garcia-Ortiz, M., Stoian, A., Filliat, D.: Generative models from the perspective of continual learning. In: 2019 International Joint Conference on Neural Networks, pp. 1–8. IEEE, Budapest, Hungary (2018) 7. Rolnick, D., Ahuja, A., Schwarz, J., Lillicrap, T.P., Wayne, G.: Experience replay for continual learning. In: 33rd Annual Conference on Neural Information Processing Systems, NeurIPS, Vancouver, Canada (2019)

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8. Wang, Z., Mehta, S.V., Póczos, B., Carbonell, J.: Efficient meta lifelong-learning with limited memory. arXiv preprint arXiv:2010.02500 (2020) 9. Wu, Y., Bighashdel, A., Chen, G., Dubbelman, G., Jancura, P.: Continual pedestrian trajectory learning with social generative replay. IEEE Rob. Autom. Lett. 8(2), 848–855 (2023) 10. Mundt, M., Pliushch, I., Majumder, S., Hong, Y., Ramesh, V.: Unified probabilistic deep continual learning through generative replay and open set recognition. J. Imaging 8(4), 93 (2022) 11. Li, Z., Hoiem, D.: Learning without forgetting. IEEE Trans. Pattern Anal. Mach. Intell. 40(12), 2935–2947 (2018) 12. Kirkpatrick, J., et al.: Overcoming catastrophic forgetting in neural networks. Appl. Math. 114(13), 3521–3526 (2017) 13. Zenke, F., Poole, B., Ganguli, S.: Continual learning through synaptic intelligence. In: 34th International Conference on Machine Learning, pp. 3987–3995. PMLR, Sydney, Australia (2017) 14. Serra, J., Suris, D., Miron, M., Karatzoglou, A.: Overcoming catastrophic forgetting with hard attention to the task. In: 35th International Conference on Machine Learning, pp. 4548–4557. PMLR, Stockholm, Sweden (2018) 15. Mallya, A., Lazebnik, S.: PackNet: adding multiple tasks to a single network by iterative pruning. In: 31st IEEE Conference on Computer Vision and Pattern Recognition, pp. 7765– 7773. IEEE, Salt Lake City, United States (2018) 16. Wiwatcharakoses, C., Berrar, D.: A self-organizing incremental neural network for continual supervised learning. Expert Syst. Appl. 185, 115662 (2021) 17. Parisi, G.I., Tani, J., Weber, C., Wermter, S.: Lifelong learning of spatiotemporal representations with dual-memory recurrent self-organization. Front. Neurorobot. 12, 78 (2018) 18. Sprechmann, P., et al.: Memory-based parameter adaptation. arXiv preprint arXiv:1802.10542 (2018) 19. He, X., Jaeger, H.: Overcoming catastrophic interference using conceptor-aided backpropagation. In: International Conference on Learning Representations (2018) 20. Rebuffi, S.A., Kolesnikov, A., Sperl, G., Lampert, C.H.: iCaRL: incremental classifier and representation learning. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition, pp. 5533–5542. IEEE, Honolulu, United states (2017) 21. He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385 (2015) 22. Rodríguez, A., Laio, A.: Clustering by fast search and find of density peaks. Science 344(6191), 1492–1496 (2014) 23. Nakamura, Y., Hasegawa, O.: Nonparametric density estimation based on self-organizing incremental neural network for large noisy data. IEEE Trans. Neural Netw. Learn. Syst. 28(1), 8–17 (2017) 24. Krizhevsky, A.: Learning multiple layers of features from tiny images. Technical report, University of Toronto (2009) 25. Lomonaco, V., et al.: Avalanche: an end-to-end library for continual learning. In: 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 3595– 3605. IEEE, Virtual, Online, United states (2021)

Exact Dynamic Analysis of Viscoelastic Double-Beam System Using Dynamic Stiffness Method Fei Han1 , Nianfeng Zhong1 , and Tao Yang1,2,3(B) 1 School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical

University, Xi’an 710129, China [email protected] 2 Research and Development Institute of Northwestern Polytechnical University in Shenzhen, 518057 Shenzhen, China 3 Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China

Abstract. Double-beam structure with a viscoelastic core has a wide range of application scenarios in engineering. To investigate the dynamic behavior of this type of structures, the dynamic stiffness method and Wittrick-Williams algorithm are employed to obtain the dynamic characteristics of the double-beam system. Besides, the modal superposition method is utilized to calculate the dynamic response. By comparing with finite element solutions, the accuracy of proposed method is verified. Results show that the damping coefficient of the connection layer and the axial tension of the upper beam have a significant effect on reducing the structural response in a certain range. The methods and conclusions in this paper will be helpful to structural design and vibration suppression of practical structures. Keywords: Double-beam · flexural vibration analysis · dynamic stiffness method · Wittrick–Williams algorithm · dynamic response

1 Introduction A composite beam structure, which is commonly found in the field of engineering, can often be simplified as a double-beam model connected by a distributed spring [1]. Dynamic characteristic analysis of this model is the key to ensuring the quality of the structural design, and is the prerequisite for the analysis of the dynamic stability of such structures [2]. Reference [3] provides a detailed review of the method of dynamic characteristic analysis, and highlights that most of the existing methods are only suitable for the scenarios in which either the two beams are identical or considering the effect of the mass of the connection layer is not required. For more complex cases, such as when the mechanical conditions of the two beams are different or the mass of the connection layer cannot be ignored, it is necessary to apply appropriate measures to analyze the dynamic behavior and characteristics of the composite beam. ICANDVC2023 best presentation paper © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 593–611, 2024. https://doi.org/10.1007/978-981-97-0554-2_45

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The dynamic stiffness method (DSM) is an accurate solution and suitable for any boundary condition and frequency range; therefore, it can be applied in the analysis of the dynamic characteristics of complex structures [4–7]. In this method, the frequency equation of the structure is obtained by establishing the exact DSM of the structure, and then the dynamic characteristics of the structure can be analyzed by solving the frequency equation Therefore, the DSM is employed in this study to analyze the dynamic characteristics of a double-beam structure to obtain accurate results over a wide frequency range [8]. The DSM has a higher accuracy and computational efficiency compared with the finite element method (FEM) [9]; however, its application range is relatively limited owing to the difficulties in solving the frequency equation, which is usually a complex and transcendental equation. With the development of the Wittrick–Williams (W-W) algorithm, the application of the DSM to one-dimensional bars and beam structures has significantly improved [10–19]. This method can ensure the solution accuracy of the frequency equation without losing the roots [20]. However, the application of the W-W algorithm for double-beam structures still requires two solutions, namely, (1) The solution of the vibration mode function. The governing differential equation (GDE) of a double-beam system being a fourth-order differential equation, the determination of its general solution is certainly more difficult than for a general beam structure when considering the effect of the mass of the elastic layer; (2) The solution of the frequency equation. The clamped-clamped frequency equation of a double-beam structure is difficult to obtain; consequently, calculation of clampedclamped frequency count J0 becomes the bottleneck in determining the solution of the frequency equation. Existing studies have analyzed a double-beam system only under specific boundary conditions, and the frequency equation is usually solved by zero root search methods, because of which there is a possibility of missing roots [21, 22]. For example, Li et al. [3] solved the frequency equation of a double-beam system using the Muller root-finding method. This method requires solving each order of the modal frequency stepwise, so that it is difficult to improve the calculation efficiency, and it cannot ensure the absence of missing roots in theory. If the W-W algorithm could be used to solve the frequency equation of a double-beam system, it can be expected to obtain a specified order frequency efficiently, accurately, and without missing roots. In this study, the orthogonality condition of the structure is further derived, then, an analytical solution for the response of the structure under arbitrary loads are given using the modal superposition method.

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Fig.1. Geometrical model of the double-beam elements connected by an elastic layer

2 Motion Equation and General Solution of the Double-Beam System 2.1 Governing Differential Equation of the Double-Beam System The mechanical model of the double-beam system being studied is shown in Fig. 1. The equation of motion for the damping control of forced vibrations of the elastically connected double-beam system can be obtained from Hamilton’s principle.     ∂u2 ∂ 4 u1 ∂ 2 u1 m3 ∂ 2 u1 ∂ 2 u2 ∂u1 + m − + + k(u − u + c E1 I1 4 + ) 1 1 2 ∂x 4 ∂t 2 ∂t 2 ∂t 2 ∂t ∂t +P1 E2 I2

∂ 2 u1 = f1 (x, t) ∂x2

(1)

∂ 2 u1 = f2 (x, t) ∂x2

(2)

    ∂ 4 u2 ∂ 2 u1 m3 ∂ 2 u1 ∂ 2 u2 ∂u1 ∂u2 + m + + − k(u − u − c − ) 2 1 2 ∂x4 4 ∂t 2 ∂t 2 ∂t 2 ∂t ∂t + P2

where k is the stiffness of the distributed spring layer, c is the viscous damping coefficient of the double-beam system, f1 (x, t) and f2 (x, t) represent the external loads per unit length of the upper and lower beams; ui = ui (x, t) is the transverse displacement function; mi is mass per unit length of Beam 1, Beam 2, and spring layer when i = 1, 2, and 3, respectively; x and t are the spatial coordinate and time, respectively; l0 is the length of the beams; subscript i denotes Beam 1 (i = 1), Beam 2 (i = 2), and spring layer (i = 3); Pi is the constant compressive axial force acting through the centroid of the beam crosssection; Ei is the Young’s modulus of elasticity; Ii is the moment of inertia of the beam cross-section; and k is the spring stiffness per unit length. The nodal forces on each discrete component are the dynamic node shear force and dynamic joint moments. The time-varying parts of these quantities are expressed as complex exponents eiωt , where √ i = −1.

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2.2 Solution of the Differential Equations of Vibration Under an axial load, its governing differential equation based on the classical Bernoulli– Euler beam theory can be expressed as [3].   ∂ 4 u1 ∂ 2 u1 ∂ 2 u1 m3 ∂ 2 u1 ∂ 2 u2 + m1 2 + k(u1 − u2 ) + P1 2 = 0 (3) + E1 I1 4 + 2 2 ∂x 4 ∂t ∂t ∂t ∂x   ∂ 4 u2 ∂ 2 u2 ∂ 2 u2 m3 ∂ 2 u1 ∂ 2 u2 + m + − k(u − u = 0 (4) E2 I2 4 + + P ) 2 1 2 2 ∂x 4 ∂t 2 ∂t 2 ∂t 2 ∂x2 By separating the variables, solution um (x, t) of Eq. (3) and (4) can be expressed as um (x, t) = ϕm (x)eiωt , (m = 1, 2)

(5)

where ϕm (x) is the vibration mode function of Beams 1 and 2. By substituting Eq. (5) into Eq. (3) and (4), the undamped modal frequency and mode shape can be solved from following fourth-order differential equations of ϕm (x). m3 2 ω [ϕ1 (x) + ϕ2 (x)] + k[ϕ1 (x) − ϕ2 (x)] + P1 ϕ1 (x) = 0 4 m3 2 E2 I2 ϕ2(4) (x) − m2 ω2 ϕ2 (x) − ω [ϕ1 (x) + ϕ2 (x)] − k[ϕ1 (x) − ϕ2 (x)] + P2 ϕ2 (x) = 0 4 (4)

E1 I1 ϕ1 (x) − m1 ω2 ϕ1 (x) −

(6) (7)

The solutions of Eqs. (6) and (7) can be expressed as ϕ1 (x) = Aeκx

(8)

ϕ2 (x) = Beκx

(9)

By substituting Eqs. (8) and (9) into Eqs. (6) and (7), respectively, the equivalent algebraic eigenvalue equation of the system can be obtained, and the equations have nontrivial solutions when the determinant of the coefficient matrix of A and B vanishes. Setting the determinant as zero yields an eighth-order polynomial characteristic equation in κ. η4 κ 8 + η3 κ 6 + η2 κ 4 + η1 κ 2 + η0 = 0 where η4 = E1 I1 E2 I2 , η3 = E2 I2 P1 + E1 I1 P2 , 1 η2 = (E1 I1 + E2 I2 )k + P1 P2 − (E2 I2 (4m1 + m3 ) + E1 I1 (4m2 + m3 ))ω2 4 1 η1 = k(P1 + P2 ) − (4m2 P1 + 4m1 P2 + m3 (P1 + P2 ))ω2 4 1 η0 = −k(m1 + m2 + m3 )ω2 + (4m1 m2 + (m1 + m2 )m3 )ω4 4

(10)

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First, Eq. (10) can be rewritten as χ 4 + a1 χ 3 + a2 χ 2 + a3 χ + a4 = 0

(11)

where     χ = κ 2 , a1 = η3 η4 , a2 = η2 η4 , a3 = η1 η4 , a4 = η0 η4。 Then Eq. (11) can be decomposed into    κ 2 + p1 κ + q1 κ 2 + p2 κ + q2 = 0

(12)

where  

  

p1 q1 a1 λ1 −2a3 1 1 2

= 2 a1 ± a1 − 4a2 + 4λ1 , = 2 λ1 ± 2 . p2 q2 a1 −4a2 +4λ1 and λ1 is a real root of the following cubic equation:   λ3 − a2 λ2 + (a1 a3 − 4a4 )λ + 4a2 a4 − a32 − a12 a4 = 0

(13)

The four roots of Eq. (13) can be defined as       2 χ1 χ3 p1 p22 p1 p2 − q1 , − q2 =− ± =− ± 2 4 2 4 χ2 χ4

(14)

The general solutions of Eqs. (6) and (7) are expressed as ϕ1 (x)=A1 eκ1 x + A2 e−κ1 x + A3 eκ2 x + A4 e−κ2 x + A5 eκ3 x + A6 e−κ3 x + A7 eκ4 x + A8 e−κ4 x =

4    A2j−1 eκj x + A2j e−κj x

(15)

j=1

ϕ2 (x)=B1 eκ1 x + B2 e−κ1 x + B3 eκ2 x + B4 e−κ2 x + B5 eκ3 x + B6 e−κ3 x + B7 eκ4 x + B8 e−κ4 x =

4  

B2j−1 eκj x + B2j e−κj x



(16)

j=1

where A and B are two sets of independent constants, and their relationship is given by B2j−1 = tj A2j−1

B2j = tj A2j

in which tj =

4k + 4P1 κj2 + 4E1 I1 κj4 − (4m1 + m3 )ω2 4k + m3 ω2

(j = 1 − 4)

Shear forces V1 (x) and V2 (x) bending moments M1 (x) and M2 (x) at different sections of Beams 1 and 2 can be expressed as V1 (x) =

4  j=1

E1 I1 ϕ1 (x) + P1 ϕ1 (x) =

4     E1 I1 κj3 + P1 κj A2j−1 eκj x − A2j e−κj x j=1

(17)

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V2 (x) =

4 

E2 I2 ϕ2 + P2 ϕ2 =

j=1

4     E2 I2 tj κj3 + P2 tj κj A2j−1 eκj x − A2j e−κj x (18) j=1

M1 (x) =

M2 (x) =

4 

E1 I1 ϕ1 (x) =

4 

  −E1 I1 κj2 A2j−1 eκj x +A2j e−κj x

(19)

  −E2 I2 κj2 tj A2j−1 eκj x +A2j e−κj x

(20)

j=1

j=1

4 

4 

j=1

E2 I2 ϕ2 (x) =

j=1

ϕ1 (x) and ϕ2 (x) can be rewritten into the following matrix form: T  (i) (i) (i) (i) (i) (i) (i) ϕ1 (x)=1 (x) · A(i) A A A A A A A 1 3 5 7 2 4 6 8

(21)

T  (i) (i) (i) (i) (i) (i) (i) ϕ2 (x)=1 (x) · B(i) B B B B B B B 1 3 5 7 2 4 6 8

(22)

where   1 (x)= eκ1 x eκ2 x eκ3 x eκ4 x e−κ1 x e−κ2 x e−κ3 x e−κ4 x Let B = LA, where L is a diagonal matrix diag(t1 ,t2 ,t3 , t4 , t1 , t2 , t3 , t4 ) and T  (i) (i) (i) (i) (i) (i) (i) A = A(i) ; 1 A3 A5 A7 A2 A4 A6 A8 T  (i) (i) (i) (i) (i) (i) (i) B= C (i) 1 C3 C5 C7 C2 C4 C6 C8

2.3 Vibration Function Characterized by the Dynamic Boundary Displacements Above, we have obtained vibration mode functions ϕ1 (x) and ϕ2 (x), where coefficient matrices A and B of the double-beam systems can be determined by the dynamic displacement boundary conditions. This model can be used to establish the vibration equations of the systems under different boundary conditions. In Fig. 2, αeiω , θ eiω , Veiω , and Meiω represent the displacement, rotation angle, shear force, and bending moment of the beam segments, respectively. The force and displacement boundary conditions of the double-beam element under an axial force, as shown in Fig. 2, can be expressed as Eqs. (23) and (24). x = 0 : ϕ1 = α1a ϕ1 = θ1a |x = l : ϕ1 = α1b ϕ1 = θ1b x = 0 : ϕ2 = α2a ϕ2 = θ2a |x = l : ϕ2 = α2b ϕ2 = θ2b

(23)

x = 0 : V1 = V1a M1 = M1a |x = l : V1 = −V1b M1 = −M1b x = 0 : V2 = V2a M2 = M2a |x = l : V2 = −V2a M2 = −M2a

(24)

Exact Dynamic Analysis of Viscoelastic Double-Beam System V1ei

V2 ei

M 1ei

V1ei

M 1ei

M 1a ei

M 1bei

P1

P1 i

V2 ei

V1be

M 2 b ei

P2

P2 V2 a e

i

Beam 2

i

l

1a

e

ei

V2be

2b

i

2a

ei

x i

l 1b

1a

M 2 ei

M 2 ei

M 2 a ei

x

Beam 1 V1a e

599

1b

ei

2a

e

ei

i

ei

2b

ei

Fig. 2. Boundary conditions of the double-beam elements

3 Dynamic Stiffness Matrix Considering node displacement vector D(i) = {α1a , α2a , θ1a , θ2a , α1b , α2b , θ1b , θ2b }T as the boundary condition of Eqs. (6) and (7), the matrix expression of undetermined coefficient A can then be obtained. By substituting Eqs. (21) and (22) into Eqs. (23) and (24), node displacement D(i) depicted in Fig. 2 can by expressed in terms of A as D(i) = M(i) · A

(25)

where ⎡

M (i)

1 ⎢ ⎢ t1 ⎢ ⎢ κ1 ⎢ ⎢ t 1 κ1 =⎢ ⎢ e κ1 li ⎢ ⎢ κ1 li ⎢ t1 e ⎢ ⎣ κ1 e κ 1 l i t 1 κ1 e κ 1 l i

1 t2 κ2 t 2 κ2 e κ2 li t 2 e κ2 li κ2 e κ 2 l i t 2 κ2 e κ 2 l i

1 t3 κ3 t 3 κ3 e κ3 li t 3 e κ3 li κ3 e κ 3 l i t 3 κ3 e κ 3 l i

1 t4 κ4 t 4 κ4 e κ4 li t 4 e κ4 li κ4 e κ 4 l i t 4 κ4 e κ 4 l i

1 t1 −κ1 −t1 κ1 e−κ1 li t1 e−κ1 li −κ1 e−κ1 li −t1 κ1 e−κ1 li

1 t2 −κ2 −t2 κ2 e−κ2 li t2 e−κ2 li −κ2 e−κ2 li −t2 κ2 e−κ2 li

1 t3 −κ3 −t3 κ3 e−κ3 li t3 e−κ3 li −κ3 e−κ3 li −t3 κ3 e−κ3 li

⎤ 1 ⎥ t4 ⎥ ⎥ ⎥ −κ4 ⎥ ⎥ −t4 κ4 ⎥ ⎥ e−κ4 li ⎥ ⎥ −κ l 4 i t4 e ⎥ ⎥ −κ l 4 i ⎦ −κ4 e −t4 κ4 e−κ4 li

By substituting Eq. (24) into Eqs. (17), (18), (19) and (20), the node forces shown in Fig. 2 can also be expressed in terms of A as F(i) = R(i) · A

(26)

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where node force vector F(i) ={V1a V2a M1a M2a V1b V2b M1b M2b }T and R(i) is ⎤ ⎡ ˜t1 ˜t2 ˜t3 ˜t4 −˜t1 −˜t2 −˜t3 −˜t4 ⎥ ⎢ ˆt1 ˆt2 ˆt3 ˆt4 −ˆt1 −ˆt2 −ˆt3 − ˆt4 ⎥ ⎢ ⎥ ⎢ ˜ˆ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˆt 2 ˆt 3 ˆt 4 ˆt 1 ˆt 2 ˆt 3 ˆt 4 ⎥ ⎢ t1 ⎥ ⎢ ¯t2 ¯t3 ¯t4 ¯t1 ¯t2 ¯t3 ¯t4 ⎥ ⎢ ¯t1 (i) R =⎢ ⎥ ⎢ −˜t1 eκ1 li −˜t2 eκ2 li −˜t3 eκ3 li −˜t4 eκ4 li ˜t1 e−κ1 li ˜t2 e−κ2 li ˜t3 e−κ3 li ˜t4 e−κ4 li ⎥ ⎥ ⎢ ⎢ −ˆt1 eκ1 li −˜t2 eκ2 li −˜t3 eκ3 li −˜t4 eκ4 li ˜t1 e−κ1 li ˜t2 e−κ2 li ˜t3 e−κ3 li ˜t4 e−κ4 li ⎥ ⎥ ⎢ ⎣ −ˆt1 eκ1 li −˜t2 eκ2 li −˜t3 eκ3 li −˜t4 eκ4 li −˜t1 e−κ1 li −˜t2 e−κ2 li −˜t3 e−κ3 li −˜t4 e−κ4 li ⎦ −ˆt1 eκ1 li −˜t2 eκ2 li −˜t3 eκ3 li −˜t4 eκ4 li −˜t1 e−κ1 li −˜t2 e−κ2 li −˜t3 e−κ3 li −˜t4 e−κ4 li where ˜tj = E1 I1 κj3 + P1 κj ˆtj = E2 I2 κj3 + P2 tj κj ˜ˆt = −E I κ 2 t = −E I t κ 2 (j = 1 − 4) j 1 1 j j 2 2 j j The relationship between the node force and displacement of each element can be obtained by subtracting coefficient matrix A from Eqs. (25) and (26). F(i) = R(i) · M−1 · D(i) =K(i) · D(i)

(27)

where K(i) is the exact dynamic stiffness (DS)matrix of an axially loaded double-beam element. The derivation process of exact DS matrix K(i) does not depend on the boundary conditions. Therefore, its analytical expression is generalized and could be used for any boundary condition. The assembly procedure of the global DS matrix is similar to that of the FEM. First, the dimension of the global DS matrix is determined according to the number of elements. Second, the degree of freedom (DOF) of each element is encoded to formulate the general global DS matrix. Finally, according to the specific boundary condition of the structure, global DS matrix K(0) is obtained by eliminating the corresponding row and column of the restrained DOFs in the general global DS matrix. This assembly procedure is similar to the FEM; therefore, it will not be described here.

4 Analysis of the Free-Vibration Characteristics 4.1 Solution of Modal Frequencies The simplest approach for applying the W-W algorithm to solve the frequency equation is the dichotomy method; for a detailed solution process refer [24]. The dichotomy method only has a first-order accuracy; therefore, Yuan et al. proposed an improved Newton method that has a second-order accuracy. This method first solves a generalized eigenvalue problem: Ka  = μK a , and then extrapolates result ωμ closer to the true value value by obtaining eigenvalue μ each time. If ωa is used to denote the approximate  calculated by the dichotomy method, then we obtain Ka =K(ωa ) and K a =dK(ωa ) dω.

Exact Dynamic Analysis of Viscoelastic Double-Beam System

601

The improved Newton method with a second-order mode accuracy has a major advantage over the previous transcendental eigenvalue solution methods that typically yield modes of much lower accuracy than those of the natural frequencies. Details of the solution process of the Newton method can be found in [25]. 4.2 Solution of the Vibration Modes Because the Newton method is a local convergence method, a reasonable step size can be obtained only when the value of the initial frequency is sufficiently close to the exact solution, ensuring a second-order convergence speed of the method. This significantly limits the application of the Newton method. However, if the W-W algorithm could be fully utilized, then the Newton iteration process could be effectively protected and guided, ensuring the correct direction and proper length of step size μ. Based on this, the Newton method could be eventually transformed into a global convergence method [24]. The complete guidance and protection process of the Newton method is discussed in [24]. Using this elegant and efficient method, both the natural frequencies and vibration modes could be computed accurately and reliably, which has become a standard with the W-W algorithm. In this study, this improved Newton method is employed to calculate the modal frequencies and vibration modes of the double-beam structure.

5 Dynamic Response Analysis 5.1 Solution of the Dynamic Response Once the natural frequencies and modes have been obtained as described above, the problem of forced vibration in general for the elastically connected double-beam system can now be investigated. Based on Eqs. (6) and (7) and the boundary conditions, the following orthogonality conditions can be derived for different mode shapes of the elastically connected doublebeam system, its main derivation process is shown in the Appendix: 

L 0

m1 ϕ1n ϕ1m + m2 ϕ2n ϕ2m +

 m3 (ϕ1n ϕ1m + ϕ2m ϕ2n + ϕ2m ϕ1n + ϕ2n ϕ1m ) dx = mn δmn 4

(28)

where mn is the generalized mass in the n th mode and δmn is the Kronecker-Delta function. Once the orthogonality is derived, the forced vibration responses of the elastically connected double-beam system can be easily obtained using the normal mode method. By superimposing the natural modes, the dynamic responses can be obtained from the following equations: u1 (x, t) =

∞ 

qn (t)ϕ1n (x)

(29)

qn (t)ϕ2n (x)

(30)

n=1

u2 (x, t) =

∞  n=1

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where qn (t) are the generalized time-dependent modal coordinates. Substituting Eqs. (29) and (30) into Eqs. (1) and (2), respectively, and then using Eqs. (6) and (7) yields ⎡ ⎤ m3 ∞ ϕ q ¨ + + ϕ + c(ϕ − ϕ q q m )¨ )˙ (ϕ 1n 2n n 1n 2n n  ⎢ 1 1n n ⎥ 4 (31) ⎣ ⎦ = f1 (x, t) m 3 2 2 ωn (ϕ1n + ϕ2n )qn n=1 +m1 ωn ϕ1n qn + 4 ⎡ ⎤ m3 ∞ m2 ϕ2n q¨ n + (ϕ1n + ϕ2n )¨qn − c(ϕ1n − ϕ2n )˙qn  ⎢ ⎥ 4 (32) ⎣ ⎦ = f2 (x, t) m3 2 2 ωn (ϕ1n + ϕ2n )qn n=1 +m2 ωn ϕ2n qn + 4 Multiplying Eqs. (31) and (32) by ϕ1m and ϕ2m , respectively, and then summing these four equations, integrating from 0 to L and using the orthogonality condition Eq. (28)yields q¨ n (t) + 2ξn ωn q˙ n (t) + ωn2 qn (t) = Fn (t)

(33)

where Fn (t) can be expressed as L  Fn (t) = m1n 0 f1 (x, t)ϕ1n + f2 (x, t)ϕ2n dx, and ξn is a non-dimensional quantity known as the viscous damping factor, which can be expressed as ξn =  Cn =

L

Cn 2mn ωn

c(ϕ1n ϕ1n + ϕ2n ϕ2n − 2ϕ1n ϕ2n )dx

0

The assumptions here has been made to take advantage of the orthogonality condition Eq. (28) in order to avoid having coupling terms q˙ n in Eq. (33). By using Duhamel’s integral, the general solution of Eq. (33) can be obtained as follows: qn (t) = e−ξn ωn t [An cos(ωnd t) + Bn sin(ωnd t)]  t 1 + Fn (τ )e−ξn ωn (t−τ ) × sin[ωnd (t − τ )]d τ ωnd 0

(34)

where ωnd = 1 − ξn2 , An and Bn are coefficients related to the initial conditions. Substitution of Eq. (34) into Eqs. (29) and (30) gives the general solutions for bending deflection u1 (x, t), u2 (x, t) of the upper and lower beams.

6 Numerical Examples The DSM is an exact solution method, and when combined with the W-W algorithm, arbitrary-precision frequencies could be obtained. In order to verify the accuracy of the results, the following two typical boundary conditions are selected as examples:

Exact Dynamic Analysis of Viscoelastic Double-Beam System

603

Case I: upper beam clamped-clamped; lower beam clamped-clamped; Case II: upper beam simply supported-simply supported, lower beam clampedsimply supported. The structural parameters of the double-beam system are as follows: b1 = 0.01m, h1 = 0.005m, b2 = 0.01m, h2 = 0.01m, L=1 m, E1 =2.0 × 1011 N m2 ,     E2 =2.0 × 1011 N m2 , ρ1 =7600kg m3 , ρ2 =7600kg m3 , k = 8.0 × 103 N m2 , c = 100Ns/m, m3 = 0.001kg m. where bi , hi , and ρi are the width, length, and mass density of the ith beam. The mass per unit length, cross-sectional area, and inertial moment of the ith beam are mi = ρi Ai ,  3 Ai = bi hi , and Ii = bi hi 12, respectively. Utilizing the above parameters, the initial six orders of frequencies of the axially loaded double-beam system obtained are listed in Table 1–2. Table 1. Natural frequencies (Hz) of the axially loaded double-beam for Cases I Mode No.

P1 = −700 N & P2 = −1000 N (Tension) P1 = 700 N & P2 = 1000 N (Compression) ANSYS

Case I

W-W

ANSYS

W-W

1

41.34

41.34

23.88

23.88

2

59.47

59.46

51.99

51.99

3

89.07

89.06

60.30

60.31

4

151.42

151.39

127.26

127.28

5

159.44

159.41

140.89

140.92

6

252.90

252.84

218.95

219.00

Table 2. Natural frequencies (Hz) of the axially loaded double-beam for Cases II Mode No.

Case II

P1 = −400 N & P2 = −600 N (Tension)

P1 = 400 N & P2 = 600 N (Compression)

ANSYS

W-W

ANSYS

W-W

1

24.26

24.26

10.81

10.81

2

36.69

36.70

29.97

29.97

3

61.01

61.02

40.26

40.27

4

98.64

98.66

90.28

90.30

5

117.64

117.66

95.43

95.45

6

198.31

198.35

175.83

175.87

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It can be seen from Tables 1–2 that (1) the axial pressure reduces the stiffness of the double-beam structure, thereby decreasing the value of each order of frequency. In contrast, the axial tension increases the stiffness and frequencies of the structures; (2) the effect of the boundary condition on the frequencies of slender beams is extremely significant. Thus, during the dynamic analysis of these engineering structures, a simplification or approximation of the boundary condition will lead to errors that cannot be neglected. After the normalization of the vibration mode vectors of the double-beam system, the initial six orders of the vibration modes corresponding to the Case I are shown in Figs. 3(a) to 3(f). Upper beam

Upper beam

Lower beam

Lower beam

1 0.2

0.8

0

0.6

-0.2

0.4

-0.4 -0.6

0.2 0

-0.8

0

0.2

0.4

0.6

0.8

1

-1

0

0.2

0.4

x/l

(a) 1st order Upper beam

Upper beam

Lower beam

1

0.5

0.5

0

0

-0.5

-0.5

0

0.2

0.4

0.6

0.8

1

-1

0

0.2

0.4

(c) 3rd order

0.5

0.5

0

0

-0.5

-0.5

0.2

0.4

0.6

Upper beam

Lower beam

1

0

Lower beam

0.8

1

(d) 4th order

1

-1

1

x/l

x/l

Upper beam

0.8

(b) 2nd order

1

-1

0.6

x/l

0.6

x/l

(e) 5th order

0.8

1

-1

0

0.2

0.4

Lower beam

0.6

0.8

1

x/l

(f) 6th order

Fig. 3. Initial six normal mode shapes of the double-beam system for Cases I with P1 = −700 N and P2 = −1000 N

Exact Dynamic Analysis of Viscoelastic Double-Beam System

605

The dynamic responses of the elastically connected double-beam systems can be calculated without difficulty based on the natural frequencies and mode shapes. Assuming that the upper beam is subjected to a concentrated simple harmonic force F = 1000 sin ωt in the mid-span. When the load frequency is 50 Hz, the deflections in the mid-span of the upper and lower beams are calculated from 0 to 1 s, and the responses of the beams are calculated as follows: • Case I

(a) Upper beam(c=100)

(c) Upper beam(c=0)

(b) Lower beam(c=100)

(d) Lower beam(c=0)

Fig. 4. The dynamic bending deflections at the mid-span of the double-beam system for Cases I with P1 = −700 N and P2 = −1000 N

As can be seen from Fig. 4, the response results of this paper coincide with the ANSYS results, which proves the accuracy of the calculation method of this paper. In order to investigate the effect of the damping coefficient c on the structural response, the variation of the steady state solution for the mid-span displacements of the upper and lower beams is obtained by varying the value of c in conjunction with the above mentioned example of calculations.

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Fig. 5. The effect of c on the steady state value of mid-span displacement

It can be seen from Fig. 5 that: increasing the parameter c has a significant effect on reducing the displacement of structural vibration when c is within a certain range of values, while beyond this range, the effect is not significant. In order to study the effect of axial tension on the dynamic response of the structure, the change of the steady state solution of the displacement in the span of the upper and lower beams is obtained by changing the values of the axial tension P1 and P2 . The results of the calculations are as follows: Figure 6 shows that (1) the steady state values of the mid-span displacements of the upper and lower beams increase with the increase of the axial tension in the upper beam, reaching a peak value and then showing a decreasing trend.; and (2) The increase in the axial tension of the lower beam has little effect on the steady state value of the mid-span displacements of the upper and lower beams, and the effect of the increase in the axial tension of the lower beam on reducing the steady state value of the mid-span displacements is more pronounced only when the axial tension of the upper beam is in a specific range of values.

Exact Dynamic Analysis of Viscoelastic Double-Beam System

(a) Upper beam

(b) Lower beam

(c) Upper beam Fig. 6. The effect of P1 and P2 on the steady state value of mid-span displacement

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(d) Lower beam Fig. 6. (continued)

7 Conclusion In this study, the dynamic stiffness and frequency equation of a double-beam system are obtained by solving its governing differential equation under different boundary conditions. An improved Wittrick-Williams algorithm is employed to accurately solve the frequency equation of the double-beam system. Then, the modal superposition method is utilized to determine the dynamic response of the double-beam system under simple harmonic loading. The present results are in good agreement with those of the finite element software. In this paper, a parametric analysis of the double-beam system is carried out, and the results show that the structural response can be effectively reduced by increasing the damping coefficient of the connection layer within a certain range; the dynamic response of the structure is not necessarily reduced by increasing the axial tensile force of the beams, and the change in the value of axial tensile force of the upper beams has a more significant effect on the response of the structure. Acknowledgements. This work is supported by the National Nature Science Foundation of China (Grant No. 12002279 and No. 12232015), the Fundamental Research Funds for the Central Universities, NWPU (G2020KY05307), and Guangdong Basic and Applied Basic Research Foundation (Granted No. 2022A1515010967, 2023A1515012821). TY wishes to thank the supports from Hong Kong Scholar. The authors declare that they have no conflict of interest.

Appendix: The orthogonality condition for the double-beam systems is derived as follows: Equations (6) and (7) yield m1 ω2 ϕ1 (x) +

m3 2 (4) ω [ϕ1 (x) + ϕ2 (x)] = E1 I1 ϕ1 (x) + k[ϕ1 (x) − ϕ2 (x)] + P1 ϕ1 (x) 4 (35)

Exact Dynamic Analysis of Viscoelastic Double-Beam System

m2 ω2 ϕ2 (x) +

609

m3 2 (4) ω [ϕ1 (x) + ϕ2 (x)] = E2 I2 ϕ2 (x) − k[ϕ1 (x) − ϕ2 (x)] + P2 ϕ2 (x) 4 (36)

Equations (35) and (36) yield m3 2 (4)  ω [ϕ1n (x) + ϕ2n (x)] = E1 I1 ϕ1n (x) + k[ϕ1n (x) − ϕ2n (x)] + P1 ϕ1n (x) 4 n m 3 2 (4)  m2 ωn2 ϕ2n (x) + ω [ϕ1n (x) + ϕ2n (x)] = E2 I2 ϕ2n (x) − k[ϕ1n (x) − ϕ2n (x)] + P2 ϕ2n (x) 4 n m1 ωn2 ϕ1n (x) +

(37) (38)

Multiplying Eq. (37) and (38) by ϕ1m and ϕ2m , respectively, and adding them together

m3 2 ϕ1n (x)ϕ1m (x) + ϕ2n (x)ϕ1m (x) 2 2 ω m1 ωn ϕ1n (x)ϕ1m (x) + m2 ωn ϕ2n (x)ϕ2m (x) + 4 n +ϕ1n (x)ϕ2m (x) + ϕ2n (x)ϕ2m (x) (4)

(4)

= E1 I1 ϕ1n (x)ϕ1m (x) + E2 I2 ϕ2n (x)ϕ2m (x) + k[ϕ1n (x) − ϕ2n (x)]ϕ1m (x)   −k[ϕ1n (x) − ϕ2n (x)]ϕ2m (x) + P1 ϕ1n (x)ϕ1m (x) + P2 ϕ2n (x)ϕ2m (x)

(39) Similarly, it can be obtained that 2 2 m1 ωm ϕ1m (x)ϕ1n (x) + m2 ωm ϕ2m (x)ϕ2n (x) +



m3 2 ϕ1m (x)ϕ1n (x) + ϕ2m (x)ϕ1n (x) ω 4 m +ϕ1m (x)ϕ2n (x) + ϕ2m (x)ϕ2n (x)

(4) (4) = E1 I1 ϕ1m (x)ϕ1n (x) + E2 I2 ϕ2m (x)ϕ2n (x) + k[ϕ1m (x) − ϕ2m (x)]ϕ1n (x)

(40)

  −k[ϕ1m (x) − ϕ2m (x)]ϕ2n (x) + P1 ϕ1m (x)ϕ1n (x) + P2 ϕ2m (x)ϕ2n (x)

Integrating Eqs. (39) and (40) from 0 to L, respectively, and then subtracting them, using the boundary conditions give 

2 ωn2 − ωm



L

m1 ϕ1n ϕ1m + m2 ϕ2n ϕ2m +

0

 m3 (ϕ1n ϕ1m + ϕ2m ϕ2n + ϕ2m ϕ1n + ϕ2n ϕ1m ) dx = 0 4

(41)

Then, the orthogonality condition can be obtained 

L

m1 ϕ1n ϕ1m + m2 ϕ2n ϕ2m +

0

 m3 (ϕ1n ϕ1m + ϕ2m ϕ2n + ϕ2m ϕ1n + ϕ2n ϕ1m ) dx = mn δmn 4

(42)

where mn is the generalized mass in the n th mode and δmn is the Kronecker-Delta function.

References 1. Kozi´c, P., Pavlovi´c, R., Karliˇci´c, D.: The flexural vibration and buckling of the elastically connected parallel-beams with a Kerr-type layer in between. Mech. Res. Commun. 56(2), 83–89 (2014) 2. Kim, J.H., Choo, Y.S.: Dynamic stability of a free-free Timoshenko beam subjected to a pulsating follower force. J. Sound Vib. 216(4), 623–636 (1998)

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Nonlinear Control Strategy for Tower Cranes with Variable Cable Lengths and Multivariable State Constraints Hui Guo1 , Wei Peng1(B) , Menghua Zhang2(B) , Chengdong Li1 , and Fei Jiao1 1 The School of Information and Electrical Engineering, Shandong Jianzhu University,

Jinan 250101, China [email protected] 2 School of Electrical Engineering, University of Jinan, Jinan 250022, China [email protected]

Abstract. Tower cranes play a crucial role in construction, but their complex dynamics and under-actuation pose significant control challenges. This research proposes a sophisticated multi-variable state-constrained controller for tower cranes with varying cable lengths. By introducing auxiliary terms, the controller effectively constrains the actuated variables, underactuated variables, and specific composite variables, ensuring precise cargo positioning and swing suppression. The control approach for tower cranes in this paper enhances both safety and operational efficiency. Finally, the proposed method’s feasibility and robustness are validated through simulation experiments. Keywords: Unactuated state constraints · Composite variable constraints · Underactuated system · Tower crane · Varying cable length

1 Introduction The tower crane is a typical example of underactuated systems, characterized by having fewer input variables than output variables [1–3]. Despite cost and structural advantages, designing controllers for them is complex due to limited input and high state coupling, posing significant challenges. Current manual operation leads to low efficiency and safety risks. Efficient tower crane controllers are crucial for improved performance and safety [4]. Over the years, tower cranes have seen extensive research in various control methods, including Sliding Mode Control (SMC), adaptive algorithms, and their combinations, aimed at enhancing performance, safety, and efficiency [5–7]. SMC offers robustness, rapid response, and precision, but may generate chattering. To address this, studies combine adaptive methods with SMC for smoother and more adaptable control. Tower crane dynamics are influenced by uncertainties and disturbances, handled by adaptive control algorithms, which adjust parameters in real-time for improved robustness [8, 9]. Additionally, fuzzy control and Observer-Based Nonlinear Control are employed © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 612–623, 2024. https://doi.org/10.1007/978-981-97-0554-2_46

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for systems with imprecise information, estimating states and disturbances for effective monitoring and correction [10–12]. In construction, tower cranes aim for rapid and precise positioning while suppressing cargo swing. Although significant progress has been made in achieving these goals, further improvements are needed. In certain scenarios, operating at different heights leads to variations in cable lengths, inducing swing due to factors like wind and inertia. Dynamic cable length adjustment is essential for stable operations. Strong coupling among state variables requires constraints to ensure precise load control, especially in complex work environments with potential obstacles. Effective constraints prevent collisions and ensure reliable control. Constraints on both actuated and underactuated variables are crucial to prevent excessive swing angles and instability, avoiding accidents. Reasonable constraints on composite load position variables enhance operational efficiency and trajectory tracking. This paper proposes a nonlinear control method for a five-degree-of-freedom tower crane with varying cable lengths, aiming to achieve multi-variable state constraints. The contributions are: 1) Establishing a model for the tower crane with variable cable lengths, enabling precise cargo hoisting, jib rotation, trolley transport, and effective swing suppression. 2) Devising a multi-variable constraint strategy, confining all state variables within well-defined ranges, including actuated, underactuated, and composite variables representing the load position. 3) Applying multi-variable constraints to ensure stable system dynamics, enabling accurate control of the load position and orientation for safe and efficient tower crane operation. The structure of this article is as follows: In Sect. 2, the dynamic model of the system is introduced, and in Sect. 3, the controller design is presented. In Sect. 4, simulation tests using Simulink are performed, comparing the proposed method with the PD (Proportional-Derivative) approach.

2 Problem Formulation 2.1 Tower Crane System Dynamics Based on the Lagrangian modeling technique, we derive the dynamic equations corresponding to the five degrees of freedom (DOF), resulting in five second-order nonlinear differential equations expressed as follows: ˙ q˙ + G(q) = U − Uf M (q)q¨ + C(q, q)

(1)

where q = [φ x l θ1 θ2 ]T denotes the system state vector, l corresponds to the cable length, x and φ represent the displacement of the trolley and the angle of the jib slew, respectively. Additionally, θ 1 and θ 2 refer to the swing angles of the payload, the input vector U=[Fφ Fx Fl 0 0]T consists of three control inputs: the tower rotating torque (Fφ), the trolley driving force (Fx), and the rope tension (Fl). Additionally,

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Uf=[f φ fx fl d 1θ˙ 1 d 2θ˙ 2]T represents friction vector, d 1, d 2 represents the coeffi˙ ∈ R3×3 are inertia and centripetal-Coriolis matrices, cients of friction M (q), C(q, q) respectively, and G(q) = [0 0 mg−mgcosθ1 cosθ2 mglsinθ1 cosθ2 mglcosθ1 sinθ2 ]T is the gravitational force vector.

2

1

Fig. 1. Model of 5-DOF varying-cable-length tower cranes.

The DOF can be categorized into two vectors: actuated states (qa ) and unactuated states (qu ). The actuated states control the jib slew angle, trolley position, and suspension cable length, while the unactuated states correspond to the swing angles of the payload. qa = [φ x l]T , qu = [θ1 θ2 ]T

(2)

Therefore, Eq. (1) is arranged into two equations M11 q¨ a + M12 q¨ u + C11 q˙ a + C12 q˙ u + G1 = u − a q˙ a

(3)

M21 q¨ a + M22 q¨ u + C21 q˙ a + C22 q˙ u + G2 = −u q¨ u

(4)

where M11 , C11 ∈ R3×3 ,M21 , C21 ∈ R2×3 ,M22 , C22 ∈ R2×2 , M12 , C12 ∈ R3×2 , G1 ∈ R3 , G2 ∈ R2 , a ∈ R3×3 ,u ∈ R2×2 ,a q˙ a and u q˙ u represent the part corresponding to the actuated and the underactuated of Uf. The following reformulation of the equations will allow the control inputs to affect the unactuated dynamics. Equations (3) and (4) are rewritten as follows: −1 u q˙ u ) q¨ a = Ma−1 (u − a q˙ a − Ca q˙ a − Cu q˙ u − Ga + M12 M22

(5)

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where Ma , Ca ∈ R3×3 ,Cu ∈ R3×2 ,Ga ∈ R3 ,a ∈ R3×3 , u ∈ R2×2 , denote some auxiliary matrices/vectors defined as follows: −1 −1 Ma = M11 − M12 M22 M21 , Ca = C11 − M12 M22 C21

(6)

−1 −1 Cu = C12 − M12 M22 C22 , Ga = G1 − M12 M22 G2

(7)

2.2 Control Objective For system (1), our objective is to design an appropriate control input to achieve the following goals: Realize the positioning performance of the jib and trolley while eliminating the double-pendulum angles. So that the cantilever and the cart can reach the desired position accurately respectively, the sway angle of the payload can be suppressed to zero, described as lim [φ(t), x(t), l(t), θ1 (t), θ2 (t)]T = [φd , xd , ld 0, 0]T

t→∞

(8)

Throughout the control process, the maximum amplitudes of the actuated and unactuated state variables are limited to a suitable range: qim < qi < qiM , i ∈ {1, 2, ...5}

(9)

where qi (t) represents the ith state variable, while qim and qiM stand for the lower and upper bounds of qi (t),respectively. The payload position must be confined within an appropriate safety range. The expression for the payload position is as follows: φ1 (q) = xp (t) = x + l sin θ1

(10)

φ1m < φ1 (q) < φ1M

(11)

where φ1 represents the state variable, while φ1m and φ1M stand for the lower and upper bounds of φ1 (t), respectively. ˙ (q)/2 − C(q, q) ˙ is skew-symmetric, Property1 [12]:M (q) is positive-definite, and M meaning that ξT [

˙ (q) M ˙ − C(q, q)]ξ = 0, ∀ξ ∈ R5 2

(12)

3 Control System Design In this section, we aim to address the diverse constraints, encompassing both actuated and unactuated state constraints, as well as constraints on specific composite variables. To achieve the objectives specified in Eqs. (8) - (11), we will devise auxiliary terms, incorporating constrained variable signals and actuated velocity signals.

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By examining the comprehensive expression of Ma, it can be demonstrated that Ma is a positive-definite matrix. Moreover, the shaped mechanical energy function of Eq. (1) can be represented as follows: Es (t) =

1 T q˙ M (q)q˙ + mgl(1 − cos θ1 cos θ2 ) 2

(13)

where m denote the mass of trolley. Kinetic energy is a nonnegative scalar quantity and the value of mgl(1 − cos θ1 cos θ2 ) is always nonnegative. It is evident that Es (t) is nonnegative. By calculating the time derivative of Eq. (13) and performing simplifications using Eqs. (1), (12), we can derive the following relationship: 1 ˙ (q)q˙ + mg ˙l(1 − cos θ1 cos θ2 ) E˙ s = q˙ T M (q)q¨ + q˙ T M 2 +mp gl θ˙1 sin θ1 cos θ2 + mp gl θ˙2 cos θ1 sin θ2 = (Fφ − fφ )φ˙ + (Fx − fx )˙x + (Fl − fl + mg)˙l

(14)

The primary objective of the controller is to attain the desired position of the actuated states while eliminating the unactuated states, which increases complexity to the control design. First, we introduce the actuated state error vector as follows: T  ea = qa − qad = φ − φd x − xd l − ld

(15)

 T eu = qu − qud = θ 1 θ 2

(16)

where ea represents the tracking error of the actuated states. The desired positions of the trolley, jib and cable lengths are xd , φd and ld , respectively. On the other hand, the error eu corresponds to the unactuated states, where the desired swing angles are set to zero. Furthermore, the positioning error of qi (t) is defined as ei = qi − qid = μTi ea , i ∈ {1, 2, 3}

(17)

where μi ∈ R3 represents the unit vector as μTi



i−1

  = 0 ··· 0 1 0 ··· 0

(18)

Next, based on Eq. (14), we devise the auxiliary function 1 (t) to address the actuated state constraints. 1 =

3  αi i=1

ei2 2 (qi − qiM )2 (qi − qim )2

(19)

Following that, we define a scalar function as 1 1 V1 = kE Es + kv q˙ aT Ma q˙ a + 1 + eaT κp ea , 2 2

(20)

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where kE and kV ∈ R+ represent positive control gains, while κp ∈ R3×3 signifies a positive definite diagonal control gain matrix. By calculating the time derivative of Eq. (20) and performing certain simplifications, we arrive at the ensuing relationship: 1 ˙ a q˙ a + q˙ aT κp ea +  ˙1 V˙ 1 = −kE q˙ aT a q˙ a − kE q˙ uT u q˙ u + kE q˙ aT u + kv q˙ aT Ma q¨ a + kv q˙ aT M 2 = q˙ aT [(kE + kv )I3 u + B1 ] − (kE + kv )q˙ aT a q˙ a − kE q˙ uT u q˙ u (21) where I3 denotes the 3 × 3 identity matrix. To simplify and facilitate the derivation, an auxiliary term B1 is designed as 1 ˙ −1 ˙ a + kυ M11 M12 B1 = −kv (Ca q˙ a + Cu q˙ u + Ga ) + kv M u q˙ u aq 2 3 (22)  (qi − qiM )(qi − qim ) − ei (2qi − qiM − qim ) T OO + {κp + αi [ ]μ μ } · e O i i a (qi − qiM )3 (qi − qim )3 i=1

By deriving from Eq. (21), we first design a regulation controller: u = [(kE + kv )Im ]−1 · [−B1 − (q˙ uT κd 1 q˙ u + quT κd 2 qu )κd 3 q˙ a ]

(23)

where κd 1 , κd 2 ∈ R2×2 , κd 3 , κp ∈ R3×3 denote positive definite control gain diagonal matrices. The regulation controller is meticulously formulated by iteratively analyzing and combining unactuated variables with actuated variables. During this process, careful consideration is given to the definitions of δi and qi , resulting in an intricate controller design achieved through repetitive trial-and-error analysis [12]. 2 =

5  αi i=4

2

sec[δi (qi − qi )2 ]q˙ aT q˙ a

(24)

which is an elaborate combination of unactuated variables and actuated variables derived by repetitive trial-and-error analysis, where δi and qi are defined as δi =

2π qim + qiM ,q = (qim − qiM )2 i 2

(25)

It is demonstrated that (24) effectively addresses unactuated state constraints in [12]. Based on this, concerning the composite variable φ1 , we design an auxiliary function, denoted as 3 =

β1 2 ln [1 − ε1 (φ1 − φ 1 )2 ]q˙ aT q˙ a 2

(26)

where β1 is a nonnegative control gain, and ε1 , φ 1 are defined as ε1 =

4 φ1m + φ1M ,φ = (φ1m − φ1M )2 1 2

(27)

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Next, a scalar function V3 (t) is constructed based on Eqs. (20), (24), and (26) in the following manner: ˙2+ ˙ 3, V3 = V1 + 2 + 3 ⇒ V˙ 3 = V˙ 1 + 

(28)

˙ 2 and  ˙ 3 are given by where  ˙2 = 

5 

αi q˙ aT {sec[δi (qi − qi )2 ]q¨ a

(29)

i=4

+δi tan[δi (qi − qi )2 ] · sec[δi (qi − qi )2 ](qi − qi )˙qi q˙ a } ˙ 3 = β1 q˙ aT {ln2 [1 − ε1 (φ1 − φ 1 )2 ]q¨ a − ln[1 − ε1 (φ1 − φ 1 )2 ] 

2ε1 (φ1 − φ 1 )φ˙ 1 1 − ε1 (φ1 − φ 1 )2

q˙ a } (30)

Following that, we obtain the following relationship based on Eqs. (21), (22), (28), (29) and (30): V˙ 3 = q˙ aT (Au + B + C + D) − kE q˙ uT u q˙ u

(31)

where A, B, C, D are explicitly given by A = (kE + kv )I3 +{

5 

αi sec[δi (qi − qi )2 ] + β1 · ln2 [1 − ε1 (φ1 − φ1 )2 ]}Ma−1 ,

(32)

i=4

D = Wγ C = −(kv + kE )a q˙ a −{

5 

αi sec[δi (qi − qi )2 ] + β1 ln2 [1 − ε1 (φ1 − φ 1 )2 ]}Ma−1 a q˙ a

(33)

i=4

1 ˙ ˙a B = −kv (Ca q˙ a + Cu q˙ u + Ga ) + kv M aq 2 3  (qi − qiM )(qi − qim ) − ei (2qi − qiM − qim ) +{κp + αi [ ]μi μTi }ea (qi − qiM )3 (qi − qim )3 i=1

−

5 

αi {sec[δi (qi − qi )2 ]Ma−1 (Ca q˙ a + Cu q˙ u + Ga )

i=4

−δi tan[δi (qi − qi )2 ] sec[δi (qi − qi )2 ](qi − qi )˙qi · q˙ a } −β1 {ln2 [1 − ε1 (φ1 − φ 1 )2 ]Ma−1 (Ca q˙ a + Ga + Cu q˙ u ) 2ε1 (φ1 − φ 1 )φ˙ 1 q˙ a } + ln[1 − ε1 (φ1 − φ 1 )2 ] 1 − ε1 (φ1 − φ 1 )2

(34)

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where W ∈ R3×2 , γ ∈ R2 are defined as −1 W = kv M12 M22 +{

5 

αi sec[δi (qi − qi )2 ]

(35)

i=4 −1 +β1 ln2 [1 − ε1 (φ1 − φ 1 )2 ]}Ma−1 M12 M22

γ = [λ4 q˙ 4 λ5 q˙ 5 ]T = Q · θ

(36)

where Q ∈ R2×2 , θ ∈ R2 are given by Q = diag{θ˙1 , θ˙2 }

(37)

θ = [λ4 λ5 ]T

(38)

˙ =W ·Q Y (q, q)

(39)

˙ be defined as Let Y (q, q)

Subsequently, D can be transformed into the following expression: ˙ ·θ D = Y (q, q)

(40)

Utilizing these auxiliary terms, a new controller is formulated as follows:

˙ · θ] u = A−1 [−B − (q˙ uT κd 1 q˙ u + quT κd 2 qu )κd 3 q˙ a − Y (q, q)

(41)

the vector θˆ ∈ R2 represents the approximation for the unknown vector θ, and its update law is as follows: .



˙ T q˙ a θ = −1 θ Y (q, q)

(42)

The matrix θ ∈ R2 represents a positive definite update gain diagonal matrix. The parameters κd1 , κd2 , κd3 , and κp are defined in Eq. (23). Moreover, the parameters αi and β1 are selected as nonnegative values, while kE and kv are chosen from the set of positive real numbers. Due to space constraints, we will not elaborate on the stability proof in this article.

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4 Simulink Results In this section, we conduct simulations using MATLAB/Simulink software to evaluate the performance of the proposed method. Additionally, we select the ProportionalDerivative (PD) method as a comparative approach and present corresponding experimental results for further validation. In this set of experiments, both the proposed controller and a PD controller are employed to accomplish the task of transporting the cargo from its initial position to the desired target location. The simulation platform is set up with the following system parameters: m = 1kg, M = 7kg,J = 6.8kg · m, g = 9.8m/s2 . The initial/final conditions are set as φ(0) = 0deg, x(0) = 0m, l(0) = 1.5m, φd = 30 deg, xd = 0.5 m, and a ld = 0.9 m Figures 2–3 show the simulation results of the jib slew angle φ, trolley translation displacement x, cable length l, payload swing angles θ1 , θ2 and control torque/forces Fφ, Fx, Fl. 4.1 Simulink Group1 In this group, to effectively demonstrate the control performance of the designed control strategy, we include a comparative analysis with the well-established ProportionalDerivative (PD) control method. The PD controller can be written as Fφ = −kpφ eφ − kd φ φ˙ + Ff φ − kφ (θ˙12 + θ˙22 )φ˙

(43)

Fx = −kpx ex − kdx x˙ + Ffx − kx (θ˙12 + θ˙22 )˙x

(44)

Fl = −kpl el − kdl i − mp g + dl ˙l − kl (θ˙12 + θ˙22 )˙l

(45)

In this group, the control gains of the designed control method are tuned as kpφ = 28, kd φ = 0.8, kpx = 24.8, kdx = 8, kpl = 15, kdl = 5, kφ = 200, kx = 200, kl = 200. Figure 2 displays the experimental results of the PD method: at 2 s, the cable length and the angle of the jib slew reach the target positions, and the trolley almost precisely reaches its target position. The swing angle θ1 ∈ (−2deg, 2deg) and stabilizes at 6 s. The swing angle θ2 ∈ (−1.5deg, 1.5deg) and stabilizes at 11 s.

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Fig.2. The results of Simulink group1. (a) Bule line: jib slew angle, trolley translation displacement, cable length, payload swing angles. (b) Red line: Control torque/forces. (Color figure online)

4.2 Simulink Group2 In this group, the multi-variable constraint controller proposed in this paper is used for simulation experiment test. In addition to achieving the target position, this controller also satisfies the variable constraints: φ ∈ (−1deg, 36deg), x ∈ (−0.01m, 0.51m), θ1 ∈ (−4deg, 4deg), θ2 ∈ (−2deg, 2deg), l ∈ (0.8m, 2m). The simulation platform is set up with the following system parameters: kE = 7, kv = 1, kp = diag{120, 150, 120}, kd 1 = diag{3, 7}, kd 2 = diag{7, 3}, kd 3 = diag{103, 94, 85}, α1 = α2 = α3 = α4 = 0.01, β = 0.01, θ = diag{7, 10}. Figure 3 displays the experimental outcomes of the proposed method in this study: at 2 s, the cable length reaches the target length. The trolley and the angle of the jib slew almost satisfy the target requirements. The swing angle θ1 ∈ (−1deg, 1deg) and begins to stabilize at 5 s. The swing angle θ2 ∈ (−0.5deg, 0.5deg) eventually stabilizing at 11 s.

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Fig. 3. The results of Simulink group2. (a) Bule line: jib slew angle, trolley translation displacement, cable length, payload swing angles. (b) Red line: Control torque/forces. (Color figure online)

By comparing Figs. 2 and 3, it is evident that both the jib angle, trolley translation and cable length rapidly and accurately reach their respective desired values. The positioning errors of the jib and trolley are almost negligible. Additionally, the proposed controller effectively suppresses the load swing, resulting in minimal residual oscillation. Both the proposed controller and the PD controller achieve the control objectives effectively. However, the proposed method incorporates constraint variables to prevent the system from exceeding safety limits, which sets it apart from the PD controller. It is also observed that the PD controller achieves faster target positioning, but the proposed method exhibits satisfactory positioning speed as well. Furthermore, the PD method induces larger load swing compared to the proposed method. The swing range in the proposed method is only half that of the PD controller, although it requires slightly more time to reach the balance position. The experimental results demonstrate that the proposed control method excels in precise positioning and effective swing suppression.

5 Conclusion This paper primarily focuses on the constrained actuated variables, underactuated variables, and compound variables in a 5-DOF tower crane. To tackle this issue, we propose a novel controller that incorporates relevant auxiliary terms and constructs a new Lyapunov function to satisfy various variable constraints. The effectiveness of the proposed

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controller is validated through comparative experiments with a traditional PD controller. The experimental results demonstrate the robustness and performance of our method, achieving the goals of precise cargo positioning and effective swing suppression. Moreover, the proposed controller exhibits stable and smooth control outputs for the system states, providing a practical solution for tower cranes with varying cable lengths, and the experimental results further confirm its superiority. Acknowledgements. This work was supported in part by the National Natural Science Foundation of China under Grant No. 62273163, the Outstanding Youth Foundation of Shandong Province Under Grant No. ZR2023YQ056, the Key R&D Project of Shandong Province under Grant No. 2022CXGC010503, and Shandong Provincial Science, Technology SME Innovation Capability Improving Project No. 2022TSGC2157.

References 1. Zhang, M., Zhang, Y., Chen, H., Cheng, X.: Model-independent PD-SMC method with payload swing suppression for 3D overhead crane systems. Mech. Syst. Signal Process. 129, 381–393 (2019) 2. Liu, Q., Zhang, H., Leng, J., Chen, X.: Digital twin-driven rapid individualised designing of automated flow-shop manufacturing system. Int. J. Prod. Res. 57 (12), 3903–3919 (2019) 3. Leng, J., Zhang, H., Yan, D., Liu, Q., Chen, X., Zhang, D.: Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop. J. Amb. Intell. Human. Comput. 10(3), 1155–1166 (2018). https://doi.org/10.1007/s12652-018-0881-5 4. Sun, W., Su, S., Xia, J., Wu, Y.: Adaptive tracking control of wheeled inverted pendulums with periodic disturbances. IEEE Trans. Cybern. 50(5), 1867–1876 (2020) 5. Zhang, M., Jing, X., Zhu, Z.: Disturbance employment-based sliding mode control for 4-DOF tower crane systems. J. Mech. Syst. Signal Process. 161, 107946 (2021) 6. Zhang, M., Zhang, Y., Ji, B., et al.: Adaptive sway reduction for tower crane systems with varying cable lengths. J. Autom. Constr. 119, 103342 (2020) 7. Pham, T.V., Manh, H.C., Quoc, H.D., et al.: Adaptive fractional-order fast terminal sliding mode with fault-tolerant control for underactuated mechanical systems: application to tower cranes. J. Autom. Constr. 123, 105333 (2021) 8. Ouyang, H., Tian, Z., Yu, L., et al.: Adaptive tracking controller design for double-pendulum tower cranes. J. Mech. Mach. Theory. 153, 103980 (2020) 9. Zhang, M., Zhang, Y., Ji, B., Ma, C., Cheng, X.: Modeling and energy-based sway reduction control for tower crane systems with double-pendulum and spherical-pendulum effects. Meas. Control. 53(1–2), 141–150 (2020) 10. Roman, R.C., Precup, R.E., Petriu, E.M.: Hybrid data-driven fuzzy active disturbance rejection control for tower crane systems. Eur. J. Control. 58, 373–387 (2020) 11. Yang, T., Sun, N., Chen, H., Fang, Y.: Observer-based nonlinear control for tower cranes suffering from uncertain friction and actuator constraints with experimental verification. IEEE Trans. Ind. Electron. 68(7), 6192–6204 (2021) 12. He, C., Ning, S.: Nonlinear control of underactuated systems subject to both actuated and unactuated state constraints with experimental verification. IEEE Trans. Ind. Electron. 67, 7702–7714 (2019)

Design and Dynamic Analysis of a Flexible Inertia Device for Vehicle Suspensions Bohuan Tan(B) , Xingui Tan, Jingang Liu, Hai Li, and Yilong Xie School of Mechanical Engineering and Mechanics, Xiangtan University, Xiangtan, China [email protected]

Abstract. The inertia device has gained widespread utilization in vibration isolation systems owing to its capability of effectively changing the inherent frequencies of the system. This paper presents a flexible inertia (FI) device that consists of the transmission device, electromagnetic damping device and flywheel. The inertial characteristics of the FI device can be designed in series with an electromagnetic damper and a flywheel. Through this design, FI device can reduce the rigid impacts during severe vibrations and can set different initial damping to get different equivalent inertance. The dynamic model of the proposed FI device is integrated into the quarter-vehicle suspension. The parametric analysis is carried out to determine the key inertia parameters that influence the vibration isolation performance. Simulation results indicate that the suspension system with the FI device can obtain better vibration isolation performance compared with the passive vibration isolation system. Keywords: Flexible inertia device · Inerter · Vehicle suspension

1 Introduction As people’s demands for driving performance of automobiles continue to rise, the vehicle suspension system has attracted widespread attention from researchers due to its significant potential in improving various driving characteristics of vehicles, such as comfort, smoothness, and stability. In recent years, advanced vehicle suspension systems have become a hot topic of research. Compared to traditional passive suspensions, they can significantly enhance vehicle driving performance. Advanced vehicle suspensions can be divided into semi-active suspensions [1, 2] and active suspensions [3, 4]. The principle of semi-active suspension is to automatically adjust the damping or stiffness according to the road conditions to achieve vibration control of the suspension system. However, the performance of traditional semi-active suspension systems has reached a bottleneck due to the limitations of the inherent structure of springs and dampers. In order to overcome the above shortcomings, many people have carried out research on this. Since Smith [5] proposed the inerter in 2002, it has attracted the attention of many researchers. In a mechanical network, the inerter functions similarly to a capacitor element in an electrical network. It is a dual-ended mechanical element similar to a spring and damper, with the difference that it can generate force proportional to the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 624–637, 2024. https://doi.org/10.1007/978-981-97-0554-2_47

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relative acceleration between its two ends. Another characteristic of the inerter is negative stiffness effect, where under certain conditions, it generates a force that aids its motion rather than just providing resistance [6]. In recent years, the inerter has been applied in vehicle suspensions [7, 8]. Suspension systems equipped with inertia devices generally exhibit better performance than traditional passive suspensions [9–11]. The design of inertia devices typically includes methods such as ball screws [12], gear racks [13]and hydraulics [14], etc. This study proposes a flexible inertia (FI) device with the advantages of reducing the rigid impact and conveniently adjusting equivalent inertance. In addition, the FI device can be easily transformed into a semi-active system. The FI device was integrated into the 1/4 vehicle suspension. Simulation analysis was performed on the FI suspension system, validating the isolation performance of the FI device. The rest of the paper is organized as follows: Sect. 2 show the structure and presents system model of the FI device; In Sect. 3, FI suspension model is established and its parameters are analyzed; The performance of FI suspension is simulated and analyzed in Sect. 4; Finally, the conclusions are drawn in Sect. 5.

2 Inertia Device and System Model 2.1 Configuration and Working Principle Figure 1 shows the structural sketch of the FI device. As shown, The FI device consists of a ball screw mechanism, a motor, and a flywheel. The flywheel is fixed to the motor stator using screws, and it can rotate freely together with the motor stator. The motor rotor is directly connected to the ball screw mechanism through axle connection. The ball screw mechanism consists of a ball screw, ball nut, and bearing balls. It is a linearto-rotary converter that transforms linear motion into the rotational motion of the ball nut and amplifies the movement. In this device, the ball screw mechanism converts the relative linear motion at both ends into the rotation of the ball screw, which drives the rotation of the motor rotor. As a result, there is relative rotation between the motor rotor and the motor stator, leading to electromagnetic effects and generating damping forces. This causes the motor stator and flywheel to start rotating together, producing a significant inertia force to counteract or promote the relative motion at both ends. 2.2 System Model As shown in Fig. 1, θ1 and θ2 represent the rotational angles of the ball screw and the flywheel, respectively, while xd represents the relative displacement at both ends of the new inertia device. Due to the characteristic of the ball screw mechanism to convert linear motion into rotational motion, the rotational angle θ1 of the motor rotor can be expressed as follows: θ1 =

2π xd = rg xd P

(1)

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Fig. 1. Structural diagram of the FI device.

where P and rg are the lead and transmission ratio of the ball screw mechanism, respectively. The schematic diagram of the system circuit for the Fi device is shown in Fig. 2, consisting of a motor and external resistor Ro . i represents the circuit current, and the model of the motor can be represented by an internal inductor Li , an internal resistor Ri , and a voltage source e. This voltage source can generate a voltage proportional to the motor’s rotational speed under the condition of constant motor voltage constant. The principle of generating damping is that the speed difference between the motor rotor and the motor stator generates an electric current, which flows through the circuit resistor and converts into thermal energy, dissipating the vibrational energy in the form of heat. Therefore, the motor is also an electromagnetic damping device. The induced voltage is directly proportional to the relative rotational speed between the motor rotor and the motor stator and can be represented as follows:   (2) e = ke θ˙1 − θ˙2 where ke is the voltage constant. To simplify the system model, we choose to ignore the internal inductance Li , and the circuit current is: e i= (3) Ro + Ri The torque output of the electromagnetic damping device is: Tm = −ki i where ki is the torque constant. And in the motor, ki = ke .

(4)

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Fig. 2. External circuit of electromagnetic damping device.

Therefore, the damping of the electromagnetic damping device is: ki2 Tm = cm = −  Ro − Ri θ˙1 − θ˙2

(5)

The damping of the FI device in the suspension is: c = rg2 cm

(6)

With the damping known, the external impedance of the circuit can be obtained as: Ro =

ki2 rg2 c

− Ri

(7)

When the selected lead of the ball screw mechanism is 0.016 m, the internal resistance is 5, and the voltage constant ke and torque constant ki are both 0.5, the relationship between the external circuit resistance and the damping can be obtained, as shown in Fig. 3. The flywheel is subjected to the torque generated by the electromagnetic damping device, and the dynamic equation of the flywheel can be obtained as:   (8) j1 θ¨2 = cm θ˙1 − θ˙2 where j1 is the moment of inertia of the flywheel.

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Damping (Ns/m)

2500

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0 10

20

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Fig. 3. Damping and external resistance relationship.

3 Suspension with the Inertia Device Vibration Control 3.1 Equivalent Change of Inertia The model of the FI device is shown in Fig. 4, where x1 and x2 are the displacements at both ends of the device, θ1 and θ2 represent the rotational angles of the motor rotor and the flywheel, respectively. ji is the moment of inertia of the flywheel, ci represents the damping of the electromagnetic damping device, and fi represents the force generated by the FI device. Therefore, the dynamic model of the FI device can be written as: θ˙1 = (˙x1 − x˙ 2 )rg

(9)

  fi = rg ci θ˙1 − θ˙2

(10)

  ci θ˙1 − θ˙2 = ji θ¨2

(11)

The transfer function of this system is:   rg2 ci θ˙1 − θ˙2 fi = x˙ 1 − x˙ 2 θ˙1

(12)

In the Laplace domain, it can be represented as: rg2 ci ji s Fi = ji s + ci (X1 − X2 )s

(13)

where Fi (s), X1 (s) and X2 (s) are the Laplace transforms of fi (t), x1 (t) and x2 (t), respectively.

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Fig. 4. FI device model.

The mechanical admittance of the FI device can be obtained as: Y =

b21 ca w2 b21 w2 + ca2

+

b1 ca2 jw b21 w2 + ca2

(14)

where ca = rg2 ci , b1 = rg2 ji . From the mechanical admittance, it can be observed that at a certain angular frequency w, system, and

b1 ca2 b21 w2 +ca2

b21 ca w2 b21 w2 +ca2

represents the equivalent damping in the FI device

represents the equivalent inertance in the FI device system. There-

fore, by changing the damping of the electromagnetic damper, the equivalent inertance of the FI device can be altered. This indicates that by setting different initial damping values, the FI device can have different inertances. The model of the FI device can be further simplified, as shown in Fig. 5, where it can be effectively considered as a series connection of a damper and an inertance. where x1 and x2 represent the displacements at the upper and lower ends of the device, respectively, and x3 represents the displacement at the connecting point. 3.2 1/4 Suspension with the FI Device Model The FI device is placed in a 1/4 suspension system, and the FI suspension model is shown in Fig. 6. In the model, ms represents the sprung mass, mt represents the unsprung mass, ks represents the suspension sprung stiffness, kt represents the equivalent tire stiffness, c1 represents the equivalent damping caused by frictions, ca represents the damping of the FI device, b1 represents the inertia of the flywheel, xs , xt and xz respectively represent the displacements of the vehicle body, the tire and the connecting point. xr is the road excitation.

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Fig. 5. Simplified FI device model.

Therefore, the suspension dynamic model can be written as: ms x¨ s + c1 (˙xs − x˙ t ) + ks (xs − xt ) + ca (˙xs − x˙ z ) = 0

(15)

mt x¨ t + kt (xt − xr ) − c1 (˙xs − x˙ t ) − ks (xs − xt ) − ca (˙xs − x˙ z ) = 0

(16)

ca (˙xs − x˙ z ) = b1 (¨xz − x¨ t )

(17)

In the Laplace domain, the vibration transfer function of the FI suspension model from the road surface to the vehicle body is: Xs kt =     2 Xr B ms s + E mt s2 + kt + B1 − E

(18)

1 1 2 where A = b1 s+c , B = c s+k +Ac 2 , E = c1 s + ks + ca s − Aca s, Xs (s) and Xr (s) are a s a b1 s 1 the Laplace transforms of xs (t) and xr (t), respectively. The vibration transfer function provides an intuitive understanding of the suspension’s isolation performance. In this paper, the analysis of the vibration transfer function can be conducted by varying the initial damping of the electromagnetic damping device.

3.3 Key Inertia Parametric Analysis As mentioned above, the FI device will have different inertances under different initial damping conditions. Taking four different initial damping values as shown in Table 1, the relationship between equivalent inertance and damping can be obtained, as shown in Fig. 7. The other parameters of the suspension ms = 300 kg, mt = 40 kg, ks = 18000 N/s and ks = 200000 N/s. Further analysis of the suspension system. In the frequency domain, the suspension transfer function for the four cases is shown in Fig. 8.

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Fig. 6. Quarter vehicle FI suspension model.

Table 1. Suspension configurations. c1 (Ns/m)

ca (Ns/m)

b1 (Kg)

Case 1

100

500

460

Case 2

100

1000

460

Case 3

100

1500

460

Case 4

100

2000

460

From Fig. 8, it can be observed that the transfer function varies with different frequencies of road excitation. Below a frequency of 2 Hz, due to the coupling effect between damping and inertance, increasing the damping effectively reduces the peak value of the transfer function and shifts the natural frequency of the system to the left. Above a frequency of 2 Hz, increasing the damping leads to a slight increase in the transfer function. Therefore, based on the above analysis, in the FI suspension system, if the road excitation is at low frequencies, it is preferable to select a larger initial damping. If the road excitation is at high frequencies, a smaller initial damping can be chosen. When the road excitation is close to the natural frequency of the system, adjusting the ini-tial damping can avoid resonance frequencies and achieve better damping effects.

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450

Equivalent innertance (kg*m 2 )

400 350 300 250 200 150 100 50 0 10-2

10-1

100

101

Frequency (Hz)

Fig. 7. Equivalent inertance. 5 Case 1 Case 2 Case 3 Case 4

Vibration transmissibility |Xs/Xr|

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8

9

10

Freqency (Hz)

Fig. 8. Vibration transmissibility.

4 Simulation Study In order to verify the vibration isolation performance of the FI device, the FI suspension is compared with the passive suspension. The parameters selected for FI suspension are case 4 above, and sinusoidal road surface and bump road surface are used as inputs. First, the sinusoidal road with amplitude of 0.04 m and frequency of 1.7 Hz is utilized as input. The simulation results are shown in Fig. 9.

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(a)

(b) Fig. 9. Vehicle dynamic responses under sinusoidal road excitation. (a) Vehicle body acceleration. (b) Suspension deflection. (c) Tyre deflection.

Figure 9 shows that, compared to the conventional passive suspension, the FI suspension can significantly reduce vehicle body acceleration, suspension deflection, and tire deflection. This indicates that under this road excitation, the FI suspension can effectively improve the ride comfort and operational stability.

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Fig. 9. (continued)

Under the conditions of the bump road with a bump height of 0.06 m and a bump length of 2 m. The vehicle speed is 20 km/h, and the simulation results are shown in Fig. 10. From Fig. 10, it can be observed that, compared to the traditional passive suspension, the FI suspension shows a slight increase in vehicle body acceleration at the first peak. This is caused by the characteristic of the FI device to reduce rigid impacts. However, after the first peak, there is a significant decrease in vehicle body acceleration. The dynamic deflection of the FI suspension is also slightly reduced, and the condition of tire deflection is similar to the acceleration of the body. Overall, in the above two road conditions, the FI suspension can effectively improve the driving performance of the vehicle. This demonstrates that the proposed FI device has excellent vibration isolation performance.

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(a)

(b) Fig. 10. Vehicle dynamic responses under bump road excitation. (a) Vehicle body acceleration. (b) Suspension deflection. (c) Tyre deflection.

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Fig. 10. (continued)

5 Conclusion This paper has proposed a FI device consisting of a ball screw device, an electromagnetic damping device and a flywheel. The FI device obtains different equivalent inertance by setting the initial damping of the electromagnetic damping device, which has the advantages of easy to change the inertance and reduce the rigid impact. The FI suspension model was established, and its vibration transmissibility was analyzed in frequency domain. The key parameters of the FI device were determined to have excellent vibration isolation performance. Through simulation analysis, compared with the traditional passive suspension, the FI suspension can effectively improve the driving performance of the vehicle. It shows that the FI device has excellent vibration isolation performance.

References 1. Pepe, G., Carcaterra, A.: VFC – variational feedback controller and its application to semiactive suspensions. Mech. Syst. Signal Process. 76–77, 72–92 (2016) 2. Wu, J., Zhou, H., Liu, Z., Gu, M.: A load-dependent PWA-H∞ controller for semi-active suspensions to exploit the performance of MR dampers. Mech. Syst. Signal Process. 127, 441–462 (2019) 3. Shao, X., Naghdy, F., Du, H.: Reliable fuzzy H∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping. Mech. Syst. Signal Process. 87, 365– 383 (2017) 4. FernandoViadero-Monasterio, B.L., Boada, M.J.L., Boada, V.D.: H∞ dynamic output feedback control for a networked control active suspension system under actuator faults. Mech. Syst. Signal Process. 162, 108050 (2022). https://doi.org/10.1016/j.ymssp.2021.108050

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5. Smith, M.C.: Synthesis of mechanical networks: the inerter (in English). IEEE T Autom. Contr. 47(10), 1648–1662 (2002) 6. Lei, L., Duan, Y.-F., Spencer, B.F., Lu, X., Zhou, Y.: Inertial mass damper for mitigating cable vibration. Struct. Control Health Monitor. 24(10), e1986 (2017). https://doi.org/10.1002/stc. 1986 7. Hu, Y.L., Chen, M.Z.Q., Shu, Z.: Passive vehicle suspensions employing inerters with multiple performance requirements (in English). J. Sound Vib. 333(8), 2212–2225 (2014) 8. Xu, T.Y., Liang, M., Li, C., Yang, S.: Design and analysis of a shock absorber with variable moment of inertia for passive vehicle suspensions (in English). J. Sound Vib. 355, 66–85 (2015) 9. Chen, M.Z.Q., Yinlong, H., Chanying, L., Guanrong, C.: Performance benefits of using inerter in semiactive suspensions. IEEE Trans. Control Syst. Technol. 23(4), 1571–1577 (2015) 10. Shen, Y.J., Chen, L., Yang, X.F., Shi, D.H., Yang, J.: Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension (in English). J. Sound Vib. 361, 148–158 (2016) 11. Chen, L., Liu, C., Liu, W., Nie, J., Shen, Y., Chen, G.: Network synthesis and parameter optimization for vehicle suspension with inerter. Adv. Mech. Eng. 9(1), 168781401668470 (2017). https://doi.org/10.1177/1687814016684704 12. Papageorgiou, C., Houghton, N.E., Smith, M.C.: Experimental testing and analysis of inerter devices. J. Dyn. Syst. Measur. Control 131(1), 011001 (2009). https://doi.org/10.1115/1.302 3120 13. Brzeski, P., Lazarek, M., Perlikowski, P.: Experimental study of the novel tuned mass damper with inerter which enables changes of inertance (in English). J. Sound Vib. 404, 47–57 (2017) 14. Wang, F.C., Hong, M.F., Lin, T.C.: Designing and testing a hydraulic inerter (in English). P I Mech. Eng. C-J. Mech. 225(C1), 66–72 (2011)

A Quasi-Zero Stiffness Nonlinear Absorber Based on Centrifugal Force Hulun Guo(B) and Zhiwei Cao Tianjin Key Laboratory of Nonlinear Dynamics and Control, Department of Mechanics, Tianjin University, Tianjin 300072, China [email protected]

Abstract. A design approach for quasi-zero stiffness (QZS) nonlinear absorbers for rotor system vibration suppression is proposed in this paper. An example of the QZS nonlinear absorber is designed for a single-disc rotor system considering gyroscopic. The nonlinear absorber consists of a rigid mass ring and four circumferentially distributed nonlinear springs, which are coaxially mounted and rotated together with the disc. The equations of motion of the rotor system with the QZS nonlinear absorber are derived by the Lagrange’s equation. According to the motion equations, it shows that the rigid mass ring reveals a negative stiffness due to the centrifugal forces generated by its vibration. The quasi-zero stiffness is realized by combining the positive stiffness provided by the nonlinear springs and the negative stiffness generated by the centrifugal forces. The stability of the rotor system is compromised at high rotational speed due to the speed-dependent negative stiffness provided by the centrifugal force. Thus, the influence of the linear and nonlinear stiffness on vibration suppression and stability of the rotor system are discussed. The present results indicate that increasing the linear stiffness appropriately can enhance the decay rate of the peak vibration and improve the stability of the rotor system. Keywords: Rotor system · Quasi-zero stiffness · Vibration suppression

1 Introduction Rotary system has been widely used in various industry machines. Various excitations such as mass eccentricity force, flow induced force, and the harmonic mechanical excitations are applied to rotor system. These require a more delicate design of the components specifically near their dynamic stability margins. Different active and passive methods are available for improving the dynamic behavior of rotor systems. The active approach mostly directly actuates to force the required degrees of freedom. The passive vibration suppression methods mainly include variable stiffness or damping. Generally speaking, passive absorbers with lower prices and less complexity are preferred. Various theoretical and experimental studies evaluated the use of various nonlinear absorbers to reduce system vibrations. The traditional linear dynamic vibration absorbers usually have a relatively narrow vibration suppression frequency range. The nonlinear © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 638–649, 2024. https://doi.org/10.1007/978-981-97-0554-2_48

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absorbers, especial the QZS nonlinear absorbers, are effective in wider frequency range of vibration absorption. The nonlinear energy sinks (NES) is one of nonlinear absorbers which have been studied widely [1]. Many different type of NESs have been designed, such as: track NESs [2, 3], piecewise NESs [4, 5], impact NESs [6–8], rotary impact NESs [9], lever-type NESs [10–12], etc. Several studies evaluated the application of NES for reducing vibration in rotating systems. Guo et al. [13] applied a NES to reduce the shaft whirl motion of a hollow turbine-generator rotor under mass unbalanced force at the critical speeds. They found that the NES is able to efficiently reduce the resonant amplitude of the rotor systems when the stiffness and mass ratios are optimized appropriately. Bab et al. [14] studied the efficiency of a number of smooth NESs on the vibration attenuation of a rotor system under mass eccentricity force. The multiple scales-harmonic balance method was used to examine the effect of the parameters on the NES efficiency and the range of happening of the strongly modulated response. Then, they investigated the efficiency of NESs which were located on the bearing of a rotor system [15]. Furthermore, Bab et al. [16] studied the effects of a number of smooth NESs located on the disk and bearings on the vibration attenuation of a rotor-blisk-journal bearing system under excitation of a mass eccentricity force. A genetic algorithm was used to optimize the parameters of the nonlinear energy sinks and its objective function was considered as minimizing the vibration of the rotating system within an operating speed range. They also presented a study on the effect of a smooth NES on the vibration reduction of a rotor system by considering the rub occurrence in the disk position [17]. Taghipour et al. [18] considered the nonlinear restoring forces, and studied the vibration reduction of a horizontally supported Jeffcott rotor system using linear tuned mass dampers (TMD), NES, and TMD-NES. Yao et al. [19–21] proposed a non-smooth NES with piecewise linear stiffness, and applied it to suppress the vibration of the rotor-blade system under transient and steady state excitations. Tehrani et al. [22, 23] presented a method to optimize the parameters of TMD and NES in order to attenuating the vibrations of a Jeffcott rotor system and a flexible bladed rotor system, and investigated the nonlinear performances of the system. In this paper, a QZS nonlinear absorbers is designed by using a general nonlinear stiffness and the negative stiffness from the centrifugal force. Firstly, the dynamic model of a rotor-QZS nonlinear absorber is obtained by the Lagrange’s equation. Then, the influence of the linear and nonlinear stiffness on vibration suppression and stability of the rotor system is studied in detail.

2 Mathematical Model 2.1 Structural Model of the Rotor System with a QZS Nonlinear Absorber The structure of the rotor system with a quasi-zero stiffness (QZS) nonlinear absorber is shown in Fig. 1(a). The QZS nonlinear absorber is made up of a rigid mass ring co-axial with the disc and four circumferentially distributed springs and dampers that connect the rigid mass ring to the disc. The QZS nonlinear absorber is installed in the grooves on the side of the disc and rotates together with the disc. Figure 1(b) illustrates the equivalent model of the motion of the rotor system. In the same generalized coordinate, the displacements of the rotor system in the x and y

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(a) Three-dimensional model.

(b) Two-dimensional equivalent model

Fig. 1. Schematic diagrams of rotor system with a QZS nonlinear absorber.

directions are x 1 and y1 , and the displacements of the rigid mass ring are x n and yn . The mass of the rotor and ring are denoted as m1 and mn . In addition, cn is the damping coefficient, k c is the linear stiffness of the spring, and k n is the nonlinear stiffness of the nonlinear spring of the absorber. The deformations of the ring are neglected, and the deformations of the springs in the first and fourth quadrants of the coordinate system are defined as δ 1 and δ 2 . According to the equivalent model of the nonlinear absorber, the deformations of the springs can be expressed as δ1 = (xn − x1 ) cos ωt + (yn − y1 ) sin ωt δ2 = (xn − x1 ) sin ωt − (yn − y1 ) cos ωt

(1)

where ωt is the angle between the spring in the first quadrants and the x-axis. Due to the nonlinear absorber rotating with the disc, the angle varies with t. 2.2 Structural Model of the Rotor System with a QZS Nonlinear Absorber In this study, the equations of the motion for the rotor system with a QZS nonlinear absorber are derived using the Lagrange’s equation. And a single-disc rotor system with gyroscopic effects into consideration was employed as an example. For the single-disc rotor system considering gyroscopic, the kinetic energy can be given by T1 =

 1    1  2 m1 x˙ 1 + y˙ 12 + Jd θ˙x2 + θ˙y2 + Jp ω2 − 2Jp ωθ˙y θx 2 2

(2)

in which”.” denotes differentiation with respect to time t, θ x and θ y are the angular displacement of the disc around the x-axis and y-axis, J d is the diametric inertia of the disc, J p is the polar inertia of the disc and ω is the rotational speed of the rotor.

A QZS Nonlinear Absorber Based on Centrifugal Force

The Lagrange’s equation can be written as   ∂T1 d ∂T1 dt ∂ q˙ j − ∂qj = Qj , j = 1, 2, 3, · · · , n

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(3)

The generalized coordinate qj of the rotor system considering gyroscopic is given by q1 = x1 , q2 = y1 , q3 = θx , q4 = θy

(4)

The generalized force Qj can be obtained through the elastic potential energy function of the shafts are Q1 = −krr x1 − krϕ θy , Q2 = −krr y1 + krϕ θx , Q3 = kϕr y1 − kϕϕ θx , Q4 = −kϕr x1 − kϕϕ θy

(5)

By Substituting Eq. (2) and Eq. (5) into Eq. (3), The motion equations of the singledisc rotor system considering gyroscopic can be established as follow ⎧ m1 x¨ 1 + cx x˙ 1 + krr x1 + krϕ θy = m1 eω2 cos ωt ⎪ ⎪ ⎨ m1 y¨ 1 + cy y˙ 1 + krr y1 − krϕ θx = m1 eω2 sin ωt (6) ⎪ Jd θ¨x + ωJp θ˙y − kϕr y1 + kϕϕ θx = 0 ⎪ ⎩ Jd θ¨y − ωJp θ˙x + kϕr x1 + kϕϕ θy = 0 where e is the disc eccentricity. cx and cy represent the damping coefficients of the rotor system in the x and y direction, respectively. The expressions for the bending and torsional stiffness of the rotor k rr , k rϕ , k ϕr and k ϕϕ are as follows krr =

729 EI 8 L3 , krϕ

EI = kϕr = − 81 4 L2 , kϕϕ =

27 EI 2 L

(7)

where L is the span length of the rotor system, E and I are the elastic modulus and the flexural section modulus of the shaft, I = π d 4 /64, d is the diameter of the shaft. For the nonlinear absorber, the kinetic energy T n is written as  1    1  2 2 Tn = mn x˙ n2 + y˙ n2 + Jdn θ˙xn + Jpn ω2 − 2Jpn ωθ˙yn θxn (8) + θ˙yn 2 2 In the present study, the assumption of the diametric inertia and polar inertia of the nonlinear absorber obey J dn = J pn = 0. The potential energy U n and the dissipative energy Rn of the nonlinear absorber are  1   1  Un = kc 2δ12 + 2δ22 + kn 2δ14 + 2δ24 (9) 2 4  1  Rn = cn 2δ˙12 + 2δ˙22 (10) 2 By applying Lagrange’s equation to combine Eqs. (8)–(10), and substituting the resulting expression into the motion equations of the rotor system. Then, the equations of motion of the rotor system with the QZS nonlinear absorber can be obtained as M¨q + C˙q + Kq + f (q) = F

(11)

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The generalized coordinate q = [x 1 , y1 , θ x , θ y , x n , yn ]T , and M, C, K and f (q) are mass matrix, damping matrix, linear stiffness matrix and nonlinear stiffness vectors of the rotor system. F is the non-conservative force acting on the rotor system, which mainly refers to the eccentric forces of the rotor system and the centrifugal force of the rigid mass ring. The detailed expressions of these matrices and vectors are as follows. ⎤ ⎡ m1 0 0 0 0 0 ⎢ 0 m 0 0 0 0⎥ 1 ⎥ ⎢ ⎥ ⎢ ⎢ 0 0 Jd 0 0 0 ⎥ (12) M=⎢ ⎥ ⎢ 0 0 0 Jd 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 mn 0 ⎦ 0 0 0 0 0 mn ⎡ ⎤ cx + 2cn 0 0 0 −2cn 0 ⎢ 0 0 0 −2cn ⎥ 0 cy + 2cn ⎢ ⎥ ⎢ ⎥ 0 0 0 Jp ω 0 0⎥ ⎢ C=⎢ (13) ⎥ ⎢ 0 0⎥ 0 0 −Jp ω 0 ⎢ ⎥ ⎣ −2cn 0 0 0 2cn 0⎦ 0 0 0 2cn 0 −2cn ⎡ ⎤ krr + 2kc 0 0 krϕ −2kc 0 ⎢ 0 −2kc ⎥ 0 krr + 2kc −krϕ 0 ⎢ ⎥ ⎢ ⎥ 0 −kϕr kϕϕ 0 0 0⎥ ⎢ K=⎢ (14) ⎥ ⎢ 0 0 kϕϕ 0 0⎥ kϕr ⎢ ⎥ ⎣ 0 0 0 2kc 0⎦ −2kc 0 0 0 2kc 0 −2kc ⎡ ⎤ −kn δ13 cos ωt − kn δ23 sin ωt ⎢ −k δ 3 sin ωt + k δ 3 cos ωt ⎥ n 2 ⎢ n 1 ⎥ ⎢ ⎥ 0⎥ ⎢ f (q) = ⎢ (15) ⎥ ⎢ 0⎥ ⎢ ⎥ ⎣ kn δ13 cos ωt + kn δ23 sin ωt ⎦ kn δ13 sin ωt − kn δ23 cos ωt ⎡ ⎤ m1 eω2 cos ωt ⎢ m eω2 sin ωt ⎥ ⎢ 1 ⎥ ⎢ ⎥ 0⎥ ⎢ (16) F=⎢ ⎥ ⎢ 0⎥ ⎢ ⎥ ⎣ mn xn ω2 ⎦ mn yn ω2 Based on the motion equations, it can be observed that the centrifugal force of the rigid mass ring is linearly related to its displacement and the ratio of the centrifugal force to the displacement remains constant and negative along the displacement direction. Therefore, the centrifugal force of the rigid mass ring can be regarded as a negative stiffness. In this study, the quasi-zero stiffness is composed of the positive stiffness provided by nonlinear and linear springs, and the negative stiffness induced by centrifugal force.

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3 Vibration Suppression Analysis of the Rotor System Referring to journal articles [20], the parameters of the rotor system are provided as  m1 = 1 kg, Jd = 4 × 10−4 kg · m2 , e = 3 × 10−5 m, cx = cy = 10N · s m, (17) d = 0.01 m E = 210 Gpa, Jp = 8 × 10−4 kg · m2 , L = 0.435 m, It should be noted that the gravity of the rotor and rigid ring is much smaller compared to their centrifugal forces. Consequently, the gravitational forces are neglected in the analysis in this paper. As the system exhibits central symmetry and gravity is not considered, the vibrations of the rotor and rigid ring in the x and y directions are consistent. Therefore, for the numerical discussion, we only focus on the vibrations in the x direction.

Fig. 2. Amplitude frequency response curves of the rotor system with and without the QZS nonlinear absorber.

Comparison of the amplitude frequency response curves between the rotor system without the QZS nonlinear absorber and the rotor system with the QZS nonlinear absorber is shown in Fig. 2. For the rotor system without the absorber, the vibration peaks at a rotational speed of ω1 = 2641.2r/min. The parameters of the QZS nonlinear absorber are given as mn = 0.01 kg cn = 0.8 N · s/m, kn = 5 × 108 N/m, kc = 0

(18)

It can be seen that the incorporation of the nonlinear absorber with linear stiffness k c = 0 resulted in a reduction in the peak value of the rotor system, which decreased from 8.29 × 10−4 m to 5.37 × 10−4 m. The QZS nonlinear absorber can suppress the peak vibration of the rotor system at the peak rotational speed. However, when the rotational speed exceeds 3170.8r/min, the amplitude of the rotor system with the QZS nonlinear absorber is larger than that of the original rotor system, which potentially influences

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the rotor system stability under high speed. In order to analyze this phenomenon, the time-history of the rotor system and the stiffness curve of the QZS nonlinear absorber under different rotational speeds were drawn in Fig. 3 and Fig. 4.

Fig. 3. Time-history response curves of the rotor system with the QZS nonlinear absorber under different rotational speeds. (mn = 0.01kg, cn = 0.8N·s/m, k n = 5 × 108 N/m, k c = 0)

The four corresponding speeds for Fig. 3 and Fig. 4 are as follows: (a) below the peak rotational speed ω1 of the rotor system without the absorber; (b) the peak rotational speed of the rotor system with the absorber; (c) the peak rotational speed ω1 ; (d) the rotational speed at which the absorber’s vibration attenuation effect fails. It can be seen that as the rotational speed increases, the negative stiffness provided by the centrifugal force of the rigid mass ring increases. Consequently, the bistable region in the stiffness curves of the absorber enlarges, which adversely impacts the stability of the rotor system. By increasing the nonlinear stiffness or introducing linear stiffness, the bistable region of the absorber can be reduced. In the following sections, the impact of these two parameters will be discussed. As shown in Fig. 5, when the QZS nonlinear absorber has linear stiffness, a noticeable reduction in amplitude is observed in the rotor system at high rotational speed. Comparing the amplitude frequency response curves of the rotor system without linear stiffness, it is evident that the peak value of the rotor system with a linear stiffness of k c =

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Fig. 4. Stiffness curves of the QZS nonlinear absorber under different rotational speeds. (mn = 0.01kg, cn = 0.8N·s/m, k n = 5 × 108 N/m, k c = 0)

Fig. 5. Comparison of amplitude frequency response curves of the rotor system under different linear stiffness. (mn = 0.01kg, cn = 0.8N·s/m, k n = 5 × 108 N/m).

1/2mn ω1 2 near the peak rotational speed is lower. And when the linear stiffness is excessively large, the amplitude frequency response curves of the rotor system approach that of the original rotor system. The peak vibration suppression rate of the QZS nonlinear absorber can be enhanced by appropriately selecting the linear stiffness. Additionally, with the increase in linear stiffness, the time-history of the rotor system at high rotational speeds tends to be harmonic in Fig. 6. The impact of nonlinear stiffness variations on the vibration suppression performance of the absorber is discussed in Fig. 7. In comparison to linear stiffness, an increase in nonlinear stiffness has a more pronounced effect on the amplitude decay of the rotor

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Fig. 6. Time-history response curves of the rotor system with the QZS nonlinear absorber under different linear stiffness. (mn = 0.01kg, cn = 0.8N·s/m, k n = 5 × 108 N/m)

system at high rotational speeds, while its impact on the peak vibration is relatively minor. In addition, it is worth noting that a greater nonlinear stiffness of the absorber could result in an enlargement of the peak value in the rotor system. Moreover, the impact of nonlinear stiffness on the rotor system’s time-history response curve can be seen in Fig. 8.

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Fig. 7. Comparison of amplitude frequency response curves of the rotor system under different nonlinear stiffness. (mn = 0.01kg, cn = 0.8N·s/m, k c = 0)

Fig. 8. Time-history response curves of the rotor system with the QZS nonlinear absorber under different nonlinear stiffness. (mn = 0.01kg, cn = 0.8N·s/m, k c = 0)

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4 Conclusion A QZS nonlinear absorber is proposed in this paper to suppress the vibration of the rotor system. The absorber is installed on the groove of the disc and rotates together with it. Using the Lagrange’s equation, the motion equations of a single-disc rotor system with the QZS nonlinear absorber considering gyroscopic effects are derived. It shows that the rigid mass ring’s centrifugal force in the absorber exhibits negative stiffness characteristics. Based on this conclusion, the quasi-zero stiffness structure is proposed in this study by combing the positive stiffness from the linear and nonlinear springs and the negative stiffness from the centrifugal force. The following conclusions are obtained by the design and numerical analysis of the QZS nonlinear absorber structure. The QZS nonlinear absorber can effectively suppress the vibration at the peak rotational speed of the rotor system. However, the negative stiffness provided by centrifugal force is associated with the rotor’s rotational speed, which can affect the stability of the rotor system at high rotational speeds. The impacts of both linear stiffness and nonlinear stiffness variations on the decay rate of peak value and the stability of the rotor system were discussed. The peak value of the rotor system is more pronounced with an increase in linear stiffness, while the stability at high rotational speeds is more affected by an increase in nonlinear stiffness. In engineering applications, the stiffness parameters of the QZS nonlinear absorber can be adjusted to meet specific requirements. The findings of this study provide a design approach for realizing quasi-zero stiffness by using centrifugal force to provide negative stiffness. However, the negative stiffness varies with the rotational speed of the rotor system, which causes the vibration suppression of the absorber designed in a limited frequency range. Further research is required to explore how to expand its vibration reduction bandwidth.

References 1. Ding, H., Chen, L.Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100, 3061–3107 (2020) 2. Wang, J.J., Zhang, C., Li, H.B., Liu, Z.B.: Experimental and numerical studies of a novel track bistable nonlinear energy sink with improved energy robustness for structural response mitigation. Eng. Struct. 237, 112184 (2021) 3. Wang, J.J., Wierschem, N., Spencer, B.F., Lu, X.L.: Experimental study of track nonlinear energy sinks for dynamic response reduction. Eng. Struct. 94, 9–15 (2015) 4. Yao, H.L., Cao, Y.B., Wang, Y.W., Wen, B.C.: A tri-stable nonlinear energy sink with piecewise stiffness. J. Sound Vib. 463, 114971 (2019) 5. AL-Shudeifat, M.A.: Nonlinear energy sinks with piecewise-linear nonlinearities. Journal of Computational and Nonlinear Dynamics 14, 124501 (2019) 6. Li, H.Q., Li, A.: Potential of a vibro-impact nonlinear energy sink for energy harvesting. Mech. Syst. Signal Process. 159, 107827 (2021) 7. Al-Shudeifat, M.A., MSaeed, A.S.: Comparison of a modified vibro-impact nonlinear energy sink with other kinds of NESs. Meccanica 56(4), 735–752 (2020). https://doi.org/10.1007/ s11012-020-01193-3 8. Fang, B., Theurich, T., Krack, M., Bergman, L.A., Vakakis, A.F.: Vibration suppression and modal energy transfers in a linear beam with attached vibro-impact nonlinear energy sinks. Commun. Nonlinear Sci. Numer. Simul. 91, 105415 (2020)

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9. Saeed, A.S., AL-Shudeifat, M.A., Vakakis, A.F., Cantwell, W.J.: Rotary-impact nonlinear energy sink for shock mitigation: analytical and numerical investigations. Arch. Appl. Mech. 90, 495–521 (2020) 10. Zang, J., Cao, R.Q., Zhang, Y.W.: Steady-state response of a viscoelastic beam with asymmetric elastic supports coupled to a lever-type nonlinear energy sink. Nonlinear Dyn. 105, 1327–1341 (2021) 11. Zang, J., Zhang, Y.W.: Responses and bifurcations of a structure with a lever-type nonlinear energy sink. Nonlinear Dyn. 98, 889–906 (2019) 12. Zang, J., Yuan, T.C., Lu, Z.Q., Zhang, Y.W., Ding, H., Chen, L.Q.: A lever-type nonlinear energy sink. J. Sound Vib. 437, 119–134 (2018) 13. Guo, C., AL-Shudeifat, M.A., Vakakis, A.F., Bergman, L.A., McFarland, D.M., Yan J.: Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks. Nonlinear Dyn. 79, 527–538 (2015) 14. Bab, S., Khadem, S.E., Shahgholi, M.: Lateral vibration attenuation of a rotor under mass eccentricity force using non-linear energy sink. Int. J. Non-Linear Mech. 67, 251–266 (2014) 15. Bab, S., Khadem, S.E., Shahgholi, M.: Vibration attenuation of a rotor supported by journal brings with nonlinear suspensions under mass eccentricity force using nonlinear energy sink. Meccanica 50, 2441–2460 (2015) 16. Bab, S., Khadem, S.E., Shahgholi, M., Abbasi, A.: Vibration attenuation of a continuous rotor-blisk-journal bearing system employing smooth nonlinear energy sinks. Mech. Syst. Signal Process. 84, 128–157 (2017) 17. Bab, S., Najafi, M., Sola, J.F., Abbasi, A.: Annihilation of non-stationary vibration of a gas turbine rotor system under rub-impact effect using a nonlinear absorber. Mech. Mach. Theory 139, 379–406 (2019) 18. Taghipour, J., Dardel, M., Pashaei, M.H.: Vibration mitigation of a nonlinear rotor system with linear and nonlinear vibration absorbers. Mech. Mach. Theory 128, 586–615 (2018) 19. Yao, H., Wang, Y., Cao, Y., Wen, B.: Multi-stable nonlinear energy sink for rotor system. Int. J. Non-Linear Mech. 118, 103273 (2020) 20. Yao, H., Cao, Y., Ding, Z., Wen, B.: Using grounded nonlinear energy sinks to suppress lateral vibration in rotor systems. Mech. Syst. Signal Process. 124, 237–253 (2019) 21. Cao, Y., Yao, H., Han, J., Li, Z., Wen, B.: Application of non-smooth NES in vibration suppression of rotor-blade systems. Appl. Math. Model. 87, 351–371 (2020) 22. Tehrani, G.G., Dardel, M.: Mitigation of nonlinear oscillations of a jeffcott rotor system with an optimized damper and nonlinear energy sink. Int. J. Non-Linear Mech. 98, 122–136 (2018) 23. Tehrani, G.G., Dardel, M.: Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic absorbers. Commun. Nonlinear Sci. Numer. Simul. 69, 1–30 (2019)

A Cellular Strategy for Eliminating the Failure of Nonlinear Energy Sinks Under Strong Excitation Sun-Biao Li1 and Hu Ding1,2,3(B) 1 School of Mechanics and Engineering Science, Shanghai Key Laboratory of Mechanics in

Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China [email protected] 2 Shaoxing Institute of Technology, Shanghai University, Shaoxing 312074, China 3 Shanghai Institute of Aircraft Mechanics and Control, Zhangwu Road, Shanghai 200092, China

Abstract. Nonlinear energy sinks (NES) have numerous advantages, such as wide vibration bandwidth and excellent vibration reduction performance. However, under high excitation intensity, its high vibration attenuation effect often becomes ineffective. Therefore, exploring methods to address this issue and broaden their application range remains a subject for further research. This paper investigates the dynamic characteristics of systems composed of linear oscillators and multiple NES cells and studies the vibration reduction effect of NES cells using the Complexifiction-Averaging (CxA) method, and the obtained results were numerically verified using the Runge-Kutta (R-K) method. The results show that when NES cells are present in the form of cells, increasing the number of cells can reduce the system’s saddle-node (SN) bifurcation region, especially shrinking the frequency island region produced by the system under strong excitation. When the number of cells reaches a certain value, the frequency island of the system disappears. Additionally, regardless of whether the system generates frequency islands or not, increasing the number of cells generally improves the vibration reduction efficiency of NES cells. Thus, the cellular strategy proposed in this paper effectively addresses the ineffectiveness of traditional NES under strong excitation and expands its application range. Keywords: Nonlinear energy sink · Cells · Complexification-Averaging method · Bifurcation · Strong excitation

1 Introduction Nonlinear energy sinks (NES) are efficient nonlinear passive dampers with several advantages, including wide vibration bandwidth, excellent vibration attenuation, simple structure, and no need for additional energy input [1, 2]. As a result, its structural design, dynamic characteristics, and vibration attenuation properties have been widely and continuously studied [3, 4]. However, under high excitation amplitudes, NES fails to achieve © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 650–660, 2024. https://doi.org/10.1007/978-981-97-0554-2_49

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effective vibration attenuation and may even worsen the vibration of the primary system, hindering its engineering applications [5, 6]. One significant reason for its ineffectiveness is the occurrence of frequency islands in the system composed of NES and the primary oscillator. Therefore, it is crucial to explore ways to address the ineffectiveness of NES under strong excitation. Since Vakakis first proposed the term “nonlinear energy sink” [7], extensive and in-depth research has been conducted on its dynamic and vibration attenuation characteristics. Numerous experimental and theoretical studies have demonstrated that systems composed of NES coupled with the primary oscillator exhibit complex and rich dynamic behaviors [8–12], such as chaos, Hopf bifurcation, Saddle node (SN) bifurcation, as well as various response mechanisms, including strongly modulated responses, weakly modulated responses, and coexistence of multiple responses [13–15]. Furthermore, as a passive vibration attenuator, NES can exhibit highly efficient nonlinear energy pumping phenomena when coupled with the primary system [16–19]. This allows the transfer of energy from the primary system irreversibly to the NES, ultimately dissipated through damping. Notably, compared to linear absorbers, NES can stably absorb energy from the primary system over a wide frequency range [20]. Over the past two decades, various types of NES have been designed, including limited NES [21], tristable NES [22], nonlinear damping NES [23], vibro-impact NES [10], and more. As a nonlinear vibration attenuator, researchers have extensively studied the vibration reduction characteristics of NES and explored various ways to enhance its damping capabilities. Many studies have shown that to achieve higher vibration attenuation efficiency with NES, system parameters should be adjusted to bring the system into the modulated response stage [24], especially the strongly modulated response(SMR) stage [16, 25, 26]. Modifying the damping and stiffness of NES can both improve its vibration reduction efficiency [27–29]. Zhang et al. used multiple parallel NES for vibration control and found that increasing the mass and damping of NES or adjusting its nonlinear stiffness contributes to enhanced damping efficiency [30]. Ding et al. proposed the cellbased distribution strategy for NES in distributed damping and applied it to multi-modal vibration attenuation of elastic beams, theoretically proving its effectiveness [31]. C.N. To et al. pointed out that the appearance of frequency islands in large amplitude vibration within a specific parameter range can be considered a hazardous phenomenon in practical systems [6]. Chen et al. demonstrated that distributed parallel NES on a simply supported beam can suppress higher branch responses (frequency islands) of the system [32]. Song et al. highlighted that parallel NES (PNES) can suppress the occurrence of frequency islands and exhibit superior vibration absorption effects under impact and harmonic excitations [33]. In response to the aforementioned issues, this paper proposes a cellular strategy for NES to suppress the occurrence of frequency islands, further resolving the problem of ineffective vibration reduction of NES under strong excitation. Section 1 will establish the motion differential equations for a system composed of multiple NES cells coupled with linear oscillators and obtain approximate analytical solutions using the Complexification-Averaging method (CxA). In Sect. 2, the inhibitory effect of cells on

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frequency islands and the vibration reduction characteristics of NES cells will be investigated, with validation through the Runge-Kutta method (R-K). Section 3 will provide a comprehensive summary of the entire study.

2 Dynamics Equations and Approximate Analytical Solutions The mechanical model of the system is illustrated in Fig. 1. Linear Oscillator

mee

k1

sin( t) 1 2

M

kN m

... n

c1 x1

NES Cells

cN x2

Fig. 1. System Mechanics Model

The motion differential equations of the system are as follows: ⎧ M x¨ 1 + c1 x˙ 1 + k1 x1 = me eω2 sin(ωt) + c1N (˙x2 − x˙ 1 ) + k1N (x2 − x1 )3 ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ +c2N (˙x3 − x˙ 1 ) + k2N (x3 − x1 ) + ... + cnN (˙xn+1 − x˙ 1 ) + knN (xn+1 − x1 ) , ⎪ ⎪ ⎨ m x¨ + c (˙x − x˙ ) + k (x − x )3 = 0, 1 2 1N 2 1 1N 2 1 ⎪ ⎪ m2 x¨ 3 + c2N (˙x3 − x˙ 1 ) + k2N (x3 − x1 )3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎩ mn x¨ n+1 + cnN (˙xn+1 − x˙ 1 ) + knN (xn+1 − x1 )3 = 0,

(1)

In equation set (1), M, c1 and k 1 represent the mass, stiffness, and damping of the primary oscillator, respectively. n denotes the number of NES cells. mj , cjN and k jN ( j = 1,2,…,n) represent the mass, damping, and nonlinear stiffness of the jth NES cell, respectively. me , e and ω represent the mass, eccentricity, and frequency of the eccentric rotor. x 1 , x j+1 ( j = 1,2,…,n) and t represent the displacements of the primary oscillator, the centroid of the jth NES cell and time, respectively. In this study, the excitation intensity is controlled by varying me. It is assumed in this paper that: ⎧ m1 + m2 + ... + mn = m, ⎪ ⎪ ⎨ m mj = (j = 1, 2, ..., n), (2) ⎪ n ⎪ ⎩ kjN = kpN , cjN = cpN (j, p = 1, 2, ..., n) where m represents the total mass of the NES cell. At this point, under the condition that each NES cell has the same initial conditions, the motion pattern of each cell is entirely

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identical over time. Based on this, the displacement of the cell is uniformly represented by x 2 . For computational convenience, the following variable substitution is introduced: (3) v = Mx1 + mx2 , w = x1 − x2 , ϕ1 eiωt = v˙ + iωv, ϕ2 eiωt = w˙ + iωw, √ where i = −1. In this case, the displacements of the primary oscillator and NES cell can be expressed as: x1 =

|ϕ10 | + m|ϕ20 | v + mw v − Mw (|ϕ10 | − M |ϕ20 |) = cos(ωt), x2 = = cos(ωt), M +m ω(M + m) M +m ω(M + m) (4)

where ϕ 10 and ϕ 20 represent the values of ϕ 1 and ϕ 2 when the system is in steady-state response. By substituting transformation (3) into equation set (1) and multiplying each equation on both sides by eiωt , then averaging the equations over one excitation period, the following slow-variable equation set can be obtained: ϕ˙1 +

ϕ1 + nmϕ2 ϕ1 + nmϕ2 1 iωϕ1 + nc1 − nk1 i = −me eω2 i, 2 2(M + nm) 2ω(M + nm) 2

ϕ1 + mϕ2 ϕ1 + mϕ2 iωϕ2 + mnc1 − mn2 k1 i 2 2(M + m) 2ω(M + m) ϕ2 3(M + m)|ϕ2 |2 ϕ2 i 1 − nkN +(M + m)ncN = −mme eω2 i 2 8ω3 2

(5)

mM ϕ˙2 + mM

(6)

By performing algebraic calculations on the fixed points of the equation set formed by (5) and (6), the following nonlinear equation set (7) satisfied by the steady-state response amplitudes of the system can be obtained. ⎛

⎞ 2 2 2 9c12 n2 kN 9m2 n3 kN 9mMn2 kN ⎛ ⎞ + + ⎜ ⎟ 3c12 mnkN ⎜ 64ω8 ⎟ 64ω6 32ω6 3m2 Mn2 kN 3k1 m2 nkN ⎜ ⎟ − − + ⎜ ⎟ 2 2 2 2 2⎟ ⎜ 9k1 mn2 k 2 ⎟ 8ω4 8ω2 8ω4 N + 9M n kN − 9k1 Mn kN ⎟ + |ϕ20 |4 ⎜ |ϕ20 |6 ⎜ ⎜ ⎟ ⎜− ⎟ ⎝ 3mM 2 nk 3k12 mnkN ⎠ ⎜ 32ω8 64ω6 32ω8 ⎟ 3k1 mMnkN N ⎜ ⎟ − + − ⎝ 9k 2 n2 k 2 ⎠ 8ω2 4ω4 8ω6 + 1 10N 64ω ⎛ 2 2 2 ⎞ k1 mn2 cN k1 Mn2 cN k12 n2 cN − + ⎜− ⎟ 2ω2 2ω2 4ω4 ⎜ ⎟ ⎜ ⎟ 1 1 ⎜ 1 2 2 2 ⎟ ⎜ + m n cN + c1 m2 ncN + c12 m2 ⎟ ⎜ ⎟ m2 m2e e2 ω4 4 2 4 2 ⎟ +|ϕ20 | ⎜ , ⎜ 1 2 n2 c2 ⎟= 4 c ⎜ 2 + 1 M 2 n2 c 2 + 1 N⎟ ⎜ + mMn2 cN ⎟ N ⎜ 2 4 4ω2 ⎟ ⎜ ⎟ 2 m2 ⎝ 1 ⎠ k 1 − k1 m2 M + 1 2 + m2 M 2 ω2 2 4 4ω ⎫ ⎧

2   ⎨ mM ω   3(M + m)|ϕ20 |2 (M + m)ncN 2 ⎬ 2 = mω 2 |ϕ |2 | |ϕ − nkN + 20 10 ⎭ ⎩ 2 2 2 8ω3

(7) Equation set (7) is a set of nonlinear equations that can be solved using the pseudoarclength continuation method. Furthermore, the second equation in equation set (7)

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represents the Slow invariant manifold (SIM) of the system. By substituting the obtained |ϕ 10 | and |ϕ 20 | into Eq. (4), maximum displacement amplitudes x 1max and x 2max of the primary oscillator and NES can be obtained, respectively: x1 max =

|ϕ10 | + m|ϕ20 | |ϕ10 | − M |ϕ20 | , x2 max = ω(M + m) ω(M + m)

(8)

By multiplying the displacement amplitudes by ω2 , the maximum acceleration amplitudes a1max and a2max of the primary oscillator and NES can be obtained, respectively: a1 max =

|ϕ10 | + m|ϕ20 | |ϕ10 | − M |ϕ20 | ω, a2 max = ω M +m M +m

(9)

3 Effects of Cellular Strategy 3.1 Effect of Cellular Strategy on SN Bifurcation The first equation of equation set (7) can be rewritten in the following form: γ3 Z23 + γ2 Z22 +γ1 Z2 =γ4 ,

(10)

where Z2 =|ϕ20 |2 ⎞ ⎛ 2 2 2 k1 Mn2 cN k12 n2 cN k1 mn2 cN 1 2 2 2 1 1 2 2 2 ⎜ − 2ω2 − 2ω2 + 4ω4 + 4 m n cN + 2 c1 m ncN + 4 c1 m ⎟ γ1 = ⎝ ⎠ c2 n2 c2 k 2 m2 1 1 1 1 2 2 + mMn2 cN + M 2 n2 cN + 1 2 N − k1 m2 M + 1 2 + m2 M 2 ω2 2 4 4ω 2 4ω 4 γ2 = −

3c12 mnkN 8ω4

γ3 =

9c12 n2 kN2 64ω8

γ4 =

m2 m2e e2 ω4 , 4

+

−

3m2 Mn2 kN 8ω2

9m2 n3 kN2 64ω6

+

+

3k1 m2 nkN 8ω4

9mMn2 kN2 32ω6

−

−

3mM 2 nkN 8ω2

9k1 mn2 kN2 32ω8

+

+

3k1 mMnkN 4ω4

9M 2 n2 kN2 64ω6

−

−

3k12 mnkN 8ω6

9k1 Mn2 kN2 32ω8

+

9k12 n2 kN2 64ω10

(11) When the system undergoes SN bifurcation, the left-hand side of Eq. (10) and its derivative with respect to Z 2 should both be equal to zero. That is:  γ3 Z23 + γ2 Z22 +γ1 Z2 =γ4 , (12) 3γ3 Z22 + 2γ2 Z2 +γ1 = 0 By eliminating Z 2 from Eq. (12), the system’s SN bifurcation curve can be obtained:  2   γ1 6γ1 γ3 − 2γ22 + 2γ2 (γ1 γ2 + 9γ3 γ4 ) 6γ1 γ3 − 2γ22 +3γ3 (γ1 γ2 + 9γ3 γ4 )2 = 0

(13)

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(a)

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(b)

Fig. 2. Effect of NES cell number on SN bifurcation: (a) me = 0.0181 kg; (b) me = 0.0362 kg.

Based on the SN bifurcation curve, the typical SN bifurcation region can be plotted in the “excitation frequency - number of NES cells” parameter space, as shown in Fig. 2. It can be seen in Fig. 2 that as the excitation intensity increases, the system’s SN bifurcation with respect to the number of NES cells mainly exhibits the following two patterns. (1) The system undergoes SN bifurcation on the right side of the resonance peak, and the bifurcation interval decreases with an increase in the number of NES cells until it disappears. (2) The system exhibits SN bifurcation on both sides of the resonance peak, but the SN bifurcation on the left side of the resonance peak appears in the form of frequency islands. With an increase in the number of NES cells, the SN bifurcation intervals, including the frequency islands, decrease until they disappear. All SN bifurcation intervals, including frequency islands, decrease with an increase in the number of NES cells. 3.2 Effect of Cellular Strategy on Damping Efficiency Figure 3 displays the influence of the number of NES cells on the vibration reduction efficiency, and it corresponds one-to-one with Fig. 2. Simultaneously, Fig. 3 illustrates the specific presence of SN bifurcations shown in the amplitude-frequency curves of Fig. 2. The damping efficiency h is defined as follows: η=

a1 max(withoutcells) − a1 max(withcells) × 100%, a1 max(withoutcells)

(14)

where a1max(withoutcells) and a1max(withcells) represent the amplitude of the acceleration of the primary oscillator under the conditions without NES cells and with NES cells, respectively. From 3(a), it can be observed that as the number of NES cells increases from 1 to 5 and 10, the SN region gradually reduces until it disappears, which aligns well with Fig. 2(a). It is worth noting that increasing the number of NES cells not only reduces the

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SN bifurcation region but also significantly enhances the vibration attenuation efficiency. In Fig. 3(a) and Fig. 3(b), as the number of cells increases to the point where the SN bifurcation region disappears, the vibration attenuation efficiency of NES cells rises from 44.76% and 40.53% to 82.66% and 91.24%, respectively.

(a)

(b)

Fig. 3. Effect of NES Cell Number on Vibration Reduction Efficiency: (a) me = 0.0181 kg; (b) me = 0.0362 kg.

The time-domain responses and phase diagrams of certain specific points on the amplitude-frequency curve shown in Fig. 3 are presented in Fig. 4. The time-domain responses and phase diagrams corresponding to the top of the frequency island in Fig. 3(b) are shown in Fig. 4(a) and Fig. 4(b), respectively. At this point, the system’s response is in a stable state, with the initial conditions set as x 1 = 0.0023, dx 1 /dt = 0, x 2 = 0, dx 2 /dt = 0. When the initial conditions are changed to zero initial conditions, the system’s response returns to the main amplitude-frequency curve, as depicted in Fig. 4(c) and Fig. 4(d). This indicates that, in the SN bifurcation region, different stable responses can be achieved with the same system parameters. The time-domain responses and phase diagrams of the response points corresponding to the frequencies of 7.44 Hz in Fig. 3(b) at n = 1, 4, and 13 are analyzed, as depicted in Fig. 4(e) to Fig. 4(m). It can be observed that at these points, the system and NES are in the stage of strongly modulated response, and the intensity of the modulated response decreases with an increase in the number of NES cells, leading to a more stable system response.

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(a)

(b)

(c)

(d)

(e)

(f)

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Fig. 4. Time course and phase diagrams of primary oscillator: (a) Time course: f = 6.7 Hz, n = 1, Initial conditions: x 1 = 0.0023, dx 1 /dt = 0, x 2 = 0, dx 2 /dt = 0; (b) Phase diagrams: f = 6.7 Hz, n = 1, Initial conditions: x 1 = 0.0023, dx 1 /dt = 0, x 2 = 0, dx 2 /dt = 0; (c) Time course: f = 6.7 Hz, n = 1, Zero initial condition; (d) Phase diagrams: f = 6.7 Hz, n = 1, Zero initial condition; (e) Time course: f = 7.44 Hz, n = 1, Zero initial condition; (f) Phase diagrams: f = 7.44 Hz, n = 1, Zero initial condition; (g) Time course: f = 7.44 Hz, n = 4, Zero initial condition; (h) Phase diagrams: f = 7.44 Hz, n = 4, Zero initial condition; (k) Time course: f = 7.44 Hz, n = 13, Zero initial condition; (m) Phase diagrams: f = 7.44 Hz, n = 13, Zero initial condition.

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(g)

(h)

(k)

(m) Fig. 4. (continued)

4 Conclusion The present study proposes the cellular strategy for NES and conducts an in-depth investigation of the dynamic characteristics of a system comprising multiple cells coupled with linear oscillators using the Complexification-Averaging method and Runge-Kutta method. Based on this, the inhibitory effect of cells on frequency islands generated in the system and the vibration reduction efficiency of NES are discussed. The main conclusions are as follows: (1) The cellular strategy effectively suppresses the generation of frequency islands in the system, and when the number of cells reaches a certain value, the frequency islands disappear. (2) The cellular strategy not only inhibits frequency islands but also enhances vibration attenuation efficiency. The vibration attenuation efficiency of NES increases with an increase in the number of cells. (3) The cellular strategy efficiently addresses the problem of NES failure under strong excitation and broadens the applicability of NES. In conclusion, this study proposes the cellular strategy for NES, overcoming the challenge of failure problem of NES under strong excitation, and laying the theoretical foundation for its broader engineering applications. In the future, the author will explore

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the more in-depth dynamic characteristics of NES with different distribution forms and explore additional methods to overcome its application limitations and enhance its vibration reduction potential. Acknowledgments. The authors gratefully acknowledge the support of the National Science Fund for Distinguished Young Scholars (No. 12025204) and the Program of Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018).

References 1. Ding, H., Chen, L.-Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100(4), 3061–3107 (2020). https://doi.org/10.1007/s11071-020-05724-1 2. Saeed, A.S., Abdul Nasar, R., AL-Shudeifat, M.A.: A review on nonlinear energy sinks: designs, analysis and applications of impact and rotary types. Nonlinear Dyn. 111(1), 1–37 (2023) 3. Wang, J., Zhang, C., Li, H., Liu, Z.: Experimental and numerical studies of a novel track bistable nonlinear energy sink with improved energy robustness for structural response mitigation. Eng. Struct. 237, 112184 (2021) 4. Guo, H.L., Yang, T.Z., Chen, Y.S., Chen, L.Q.: Singularity analysis on vibration reduction of a nonlinear energy sink system. Mech. Syst. Signal Pr. 173, 109074 (2022) 5. Malatkar, P., Nayfeh, A.H.: Authors’ response to the rebuttal by A.F. Vakakis and L.A. Bergman of Steady state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator, vol. 47, 2007, pp. 167–179, Nonlinear Dynam, 53(1–2), 169–171 (2008) 6. To, C.N., Marzbani, H., Qu´ôc, D.V., Simic, M., Fard, M., Jazar, R.N.: Frequency island and nonlinear vibrating systems. In: DePietro, G., Gallo, L., Howlett, R.J., Jain, L.C., Vlacic, L. (eds.) Intelligent Interactive Multimedia Systems and Services. SIST, vol. 98, pp. 140–150. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-92231-7_15 7. Vakakis, A.F.: Inducing passive nonlinear energy sinks in vibrating systems. J. Vib. Acoust. 123(3), 324–332 (2001) 8. Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P., Bergman, L., McFarland, D.M.: Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Phys. D. 204(1–2), 41–69 (2005) 9. Liu, R., Kuske, R., Yurchenko, D.: Maps unlock the full dynamics of targeted energy transfer via a vibro-impact nonlinear energy sink. Mech. Syst. Signal Process. 191, 110158 (2023) 10. Li, S., Wu, H., Chen, J.: Global dynamics and performance of vibration reduction for a new vibro-impact bistable nonlinear energy sink. Int. J. Non-Linear Mech. 139, 103891 (2022) 11. Al-Shudeifat, M.A., Saeed, A.S.: Periodic motion and frequency energy plots of dynamical systems coupled with piecewise nonlinear energy sink. J. Comput. Nonlinear Dyn. 17(4), 041005 (2022) 12. Li, H., Li, A., Kong, X., Xiong, H.: Dynamics of an electromagnetic vibro-impact nonlinear energy sink, applications in energy harvesting and vibration absorption. Nonlinear Dyn. 108(2), 1027–1043 (2022). https://doi.org/10.1007/s11071-022-07253-5 13. Gendelman, O.V., Starosvetsky, Y., Feldman, M.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes. Nonlinear Dyn. 51(1–2), 31–46 (2008) 14. Gendelman, O.V.: Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn. 37(2), 115–128 (2004)

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15. Sui, P., Shen, Y., Wang, X.: Study on response mechanism of nonlinear energy sink with inverter and grounded stiffness. Nonlinear Dyn. 111, 1–23 (2023) 16. Qiu, D., Li, T., Seguy, S., Paredes, M.: Efficient targeted energy transfer of bistable nonlinear energy sink: application to optimal design. Nonlinear Dyn. 92(2), 443–461 (2018). https:// doi.org/10.1007/s11071-018-4067-7 17. Jiang, X., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Steady state passive nonlinear energy pumping in coupled oscillators: Theoretical and experimental results. Nonlinear Dyn. 33(1), 87–102 (2003) 18. Li, T., Seguy, S., Berlioz, A.: On the dynamics around targeted energy transfer for vibroimpact nonlinear energy sink. Nonlinear Dyn. 87(3), 1453–1466 (2016). https://doi.org/10. 1007/s11071-016-3127-0 19. Wang, C., Krings, E.J., Allen, A.T., Markvicka, E.J., Moore, K.J.: Low-to-high frequency targeted energy transfer using a nonlinear energy sink with softening-hardening nonlinearity. Int. J. Nonlinear Mech. 147, 104194 (2022) 20. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., Mojahed, A., Gzal, M.: Nonlinear targeted energy transfer: state of the art and new perspectives. Nonlinear Dyn. 108(2), 711–741 (2022). https://doi.org/10.1007/s11071-022-07216-w 21. Geng, X.-F., Ding, H., Mao, X.-Y., Chen, L.-Q.: A ground-limited nonlinear energy sink. Acta Mech. Sinica. 38(5), 521558 (2022) 22. Zeng, Y.-C., Ding, H.: A tristable nonlinear energy sink. Int. J. Mech. Sci. 238, 107839 (2023) 23. Xu, K.-F., Zhang, Y.-W., Niu, M.-Q., Zang, J., Xue, J., Chen, L.-Q.: An improved nonlinear energy sink with electromagnetic damping and energy harvesting. Int. J. Appl. Mech. 14(06), 2250055 (2022) 24. Amin Moslemi, S.E., Khadem, M.K., Davarpanah, A.: Nonlinear vibration and dynamic stability analysis of an axially moving beam with a nonlinear energy sink. Nonlinear Dyn. 104(3), 1955–1972 (2021). https://doi.org/10.1007/s11071-021-06389-0 25. Li, H., Li, A., Kong, X.: Design criteria of bistable nonlinear energy sink in steady-state dynamics of beams and plates. Nonlinear Dyn. 103(2), 1475–1497 (2021). https://doi.org/10. 1007/s11071-020-06178-1 26. Starosvetsky, Y., Gendelman, O.V.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: optimization of a nonlinear vibration absorber. Nonlinear Dyn. 51(1–2), 47–57 (2008) 27. Yang, K., Zhang, Y.W., Ding, H., Chen, L.Q.: The transmissibility of nonlinear energy sink based on nonlinear output frequency-response functions. Commun. Nonlinear Sci. 44, 184– 192 (2017) 28. Xue, J.R., Zhang, Y.W., Ding, H., Chen, L.Q.: Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation. Appl. Math. Mech. 41(1), 1–14 (2019). https://doi.org/10.1007/s10483-020-2560-6 29. Wang, G.X., Ding, H., Chen, L.Q.: Performance evaluation and design criterion of a nonlinear energy sink. Mech. Syst. Signal Pr. 169, 108770 (2022) 30. Zhang, W.Y., Niu, M.Q., Chen, L.Q.: Vibration reduction of a Timoshenko beam with multiple parallel nonlinear energy sinks. Appl. Sci-Basel. 12(18), 9008 (2022) 31. Ding, H., Shao, Y.: NES cell. Appl. Math. Mech. 43(12), 1793–1804 (2022) 32. Chen, J., He, W., Zhang, W., Yao, M., Liu, J., Sun, M.: Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks. Nonlinear Dyn. 91, 885–904 (2018) 33. Song, W., Liu, Z., Lu, C., Li, B., Fuquan, N.: Analysis of vibration suppression performance of parallel nonlinear energy sink. J. Vibr. Control. 29(11–12), 2442–2453 (2023)

A Stable Adjustable Nonlinear Energy Sink You-cheng Zeng1(B) , Hu Ding1 , and Jinchen Ji2 1 Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied

Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, 200444 Shanghai, China [email protected] 2 School of Mechanical and Mechatronic Engineering, FEIT, University of Technology Sydney, NSW 2007 Ultimo, Australia

Abstract. The nonlinear energy sink (NES) is very sensitive to external excitation intensity, which seriously hinders the application of NES in engineering practice. In this paper, a stable adjustable NES model is proposed, which is consisted of a pair of axially compressed clamped beams, guide rods, mass blocks, and flexible hinges. By adjusting the length of the guide rod, the distance of the clamped beams, and the geometric relationship between the buckling deflection, it is convenient to convert between three different types of NES: monostable, bistable, and tristable. The Lagrange equation is used to derive the dynamic equation of the system, and the approximate analytical solution is obtained by using the harmonic balance method, which is then mutually verified with the numerical solution. Through theoretical calculation and experimental verification, three different types of NES dynamic responses and vibration reduction mechanisms are studied under different excitation intensities. The results show that NES can achieve good vibration reduction effects by strong modulation response (SMR) under appropriate excitation amplitudes; Bistable NES can reduce the energy threshold of NES and effectively suppress small amplitude vibrations by performing chaotic inter well oscillations. The tristable NES can perform chaotic inter well oscillations and eliminate detached resonance curve, which has a good vibration suppression effect on large vibrations. The device proposed in this paper can conveniently adjust the type of NES based on different external excitation intensities, providing a way to address the sensitive issue of excitation intensity in engineering applications of NES. Keywords: Nonlinear energy sink · Bistable · Tristable · Excitation intensity · Vibration control

1 Introduction The nonlinear energy sink (NES) is a passive nonlinear vibration absorber [1, 2]. Compared with linear vibration absorber, the NES does not have linear stiffness and can resonate with the main system over a relatively wide frequency band, possessing the ability to reduce vibration over a wide frequency range [3, 4]. In addition, NES has © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 661–676, 2024. https://doi.org/10.1007/978-981-97-0554-2_50

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inherent nonlinearity and can transfer the energy of the main system unidirectionally to NES, also known as targeted energy transfer (TET) [5, 6]. Therefore, it has received widespread attention due to its excellent vibration reduction performance [7–9]. For the design of NES, achieving nonlinear stiffness is crucial. Based on the types of nonlinearity, various types of NES have been studied, such as vibro-impact NES [10, 11], magnet-based NES [12, 13], rotational NES [14]. In order to meet different engineering practice needs, different types of NES were proposed. Wang et al. [15, 16] studied a track NES for vibration reduction in buildings. Cao et al. [17] proposed a cam-roller NES for vibration suppression of multi-frequency excitation. In order to improve the reliability of the NES, Geng et al. [18] proposed a magnetic-enhanced NES. Some scholars have proposed the concept of series and parallel NES from the perspective of the connection way between NES [19–21]. Boroson et al. [22] found that parallel NES can achieve more robust NES designs. Ding et al.[23] proposed the concept of NES cells, which can improve the universality and adaptability of NES in engineering practice by adjusting the number of cells. Although NES has been studied and applied in various engineering practice fields [24–27], there are still significant limitations. NES has an energy threshold, and when the input energy is below this energy threshold, TET cannot be excited, so the suppression effect of NES on small amplitude vibration is poor [22]. In addition, due to the strong nonlinearity of NES, when the excitation amplitude is too large, the appearance of the detached resonance curve will also lead to a sharp decrease in NES vibration reduction efficiency [28]. Therefore, NES is very sensitive to external excitation amplitudes, and the excitation amplitudes are too small or too large, NES cannot achieve good vibration reduction effect. In order to solve this problem, many scholars have also proposed many solutions. To address the issue of NES failure under low amplitude vibration, Zhang et al. proposed a fractional NES. Yang et al. [29] integrated NES and negative stiffness elements to achieve vibration suppression of fluid conveying pipelines. They found that the proposed NES can achieve low threshold, high energy dissipation efficiency, and higher robustness. Zang et al. [30, 31] found that increasing the NES’s mass can also improve the performance of the NES under large excitation. Chen et al. [21] found that parallel NES can eliminate the high branch response of the system under harmonic excitation. Wang et al. [32] used acoustic black hole effect to enhanced nonlinear performance of NES under large harmonic excitation. Chen et al. [33] proposed a non-smooth NES for eliminating the stable higher response branch and enlarging the effective range of the traditional cubic NES. However, the NES devices proposed in previous work can only individually suppress low amplitude or high amplitude vibrations, so the advantages of NES are still not fully utilized. Inspired by the unique advantages of bistable and tristable characteristics in the field of energy harvesting [34, 35], in this paper, a stable adjustable NES model is proposed, which is consisted of a pair of axially compressed clamped beams, guide rods, mass block, and flexible hinges. By adjusting the geometric relationship between the length of the guide rod, the distance of the clamped beam, and the buckling deflection, it is convenient to convert between three different types of NES: monostable, bistable, and tristable. Based on the characteristics of three different stable NES, used to suppress vibrations of different intensities.

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The remaining part of this paper is organized as follows: the mechanical model is established and the governing equations are obtained in Sect. 2. The equilibrium bifurcation and stability are studied in Sect. 3. The dynamics and vibration reduction analysis are introduced in Sect. 4. The experimental verification is conducted in Sect. 5. The conclusion is given in Sect. 6.

2 Mechanical Model Figure 1 shows a stable adjustable NES model, which consists of a pair of axially compressed clamped beams, guide rods, mass blocks, and flexible hinges. When the axial force P is less than or equal to the critical pressure Pcr of the clamped beam, the clamped beam has only one stable configuration, as shown in Fig. 1(a); When the axial force P is greater than the critical pressure Pcr of the clamped beam, the clamped beam will buckle, resulting in two stable configurations and one unstable configuration, and the midpoint buckling disturbance of the beam is d, as shown in Fig. 1(b). The displacement of the transverse vibration of the beam is represented by w(s,t), and the displacement of the mass m1 along the guide rail is represented by q(t). The length of the guide rod is a, the distance between the clamped beams is 2b, and the length of the clamped beam is L. Then the stable adjustable NES is coupled with a linear oscillator (LO), where m, k, and c are the mass, stiffness, and damping of the LO, respectively. The force excitation is F = F 0 cos(ωt), where F 0 is the excitation force amplitude, and ω is the excitation frequency. According to the generalized Hamilton principle, the governing equation of the coupled system can be written as  t2  t2 Wdt = 0 (1) δ (T − U )dt + δ t1

t1

The kinetic energy of m can be expressed as T1 =

1 m˙x(t)2 2

(2)

The kinetic energy of m1 can be expressed as T2 =

1 m1 (˙q(t) + x˙ (t))2 2

(3)

The total kinetic energy of the coupled system can be expressed as T = T1 + T2

(4)

The potential energy of m can be expressed as U1 =

1 2 kx 2

The bending potential energy of the clamped beam is expressed as  2 2  1 L ∂ w U2 = EI ds 2 0 ∂s2

(5)

(6)

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Fig. 1. Schematic diagram: (a)the stable adjustable NES when P ≤ Pcr ; (b) the stable adjustable NES when P > Pcr ; (c) the stable adjustable NES is coupled with the LO.

where E and I are the modulus of elasticity and the moment of inertia of the clamped beam, respectively. The axial tensile potential energy is denoted as  2  1 L ∂w P ds (7) U3 = − 2 0 ∂s The potential energy generated by the stretching of the neutral surface can be expressed as   2 2 L 1 ∂w EA ds (8) U4 = 8L 0 ∂s where A is the cross-sectional area of the clamped beam. The potential energy of the coupled system U can be expressed as U = U1 + 2(U2 + U3 + U4 )

(9)

For the clamped beams, for the mass m1 acts at the middle of the beam, the first-order mode will be dominant in the deflection of the beam, then according to the Galerkin discretization, the deflection w(s,t) can be represented by w(s, t) = g(t)φ(s)

(10)

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where g(t) and φ(s)=[1 − cos(2π s/L)]/2 are the time-dependent expansion coefficients and the first basic mode shapes. Therefore, g(t) and q(t) satisfy the following relationship:  g(t) = b − a2 − q2 (t) (11) The potential energy of the clamped beams U c are expressed as:  Uc = U1 + U2 + U3 = 2 K1 g(t)4 + K2 g(t)2   4 2     = 2 K1 b − a2 − q2 (t) + K2 b − a2 − q2 (t)

(12)

where EA K1 = 8L 1 K2 = 2



L 0





∂ 2φ EI ∂s2

2

L  ∂φ 2

0

2

ds

∂s 1 ds − 2



L

0

(13) 

∂φ P ∂s

2 ds

(14)

The external work is as follows: W = Fx − cx˙x − c1 q˙q

(15)

Substituting Eqs. (2)–(15) into Eq. (1), Lagrange’s equation for the system is given by

 d ∂(T − U ) ∂(T − U ) ∂W − = dt ∂ x˙ ∂x ∂x

 ∂(T − U ) ∂W d ∂(T − U ) − = dt ∂ q˙ ∂q ∂q

(16)

The differential equation of motion for the coupled system is obtained as follows: m¨x + c˙x + kx − Fm1 (q) − c1 q˙ = F0 cos(ωt), m1 (¨q + x¨ ) + c1 q˙ +Fm1 (q) = 0 The nonlinear restoring force F m1 can be expressed as  3      ∂Uc 2 2 2 2 = 2 4K1 b − a − q (t) + 2K2 b − a − q (t) Fm1 (q) = ∂q

(17)

(18)

The transverse nonlinear restoring force F b of the clamped beam can be expressed as Fb =

 ∂Uc = 2 4K1 g(t)3 + 2K2 g(t) ∂g

(19)

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The critical pressure Pcr of the clamped beam should satisfy that the coefficient of the linear stiffness term g(t) in Eq. (19) is zero, that is, K 2 = 0, thus Pcr can be expressed as  L  2 2  L  2 ∂ φ ∂φ EI ds/ ds (20) Pcr = 2 ∂s ∂s 0 0 When the axial force P is greater than the critical pressure Pcr , the clamped beams buckle and have two stable configurations, where the midpoint buckling deflection d can be calculated: −K2 d= (21) 2K1

3 Static Equilibrium Position Bifurcation The static characteristics of the absorber mainly include the equilibrium bifurcations, the nonlinear restoring force, and potential energy, which can help us to deeply understand the form mechanism of monostable, bistable, and tristable NES. Table 1 is the parameters of the clamped beam, and the distance of the clamped beams b = 120 mm. As shown in Fig. 2, when the axial force P = 0 N, as the length of the guide rod a increases, the NES type changes from monostable to bistable. When the axial force P = 80 N, the clamped beam buckles due to the axial force being greater than the critical pressure of the clamped beam. As the length of the guide rod a increases, the NES type undergoes a transition from monostable, bistable to tristable. Table 1. The parameters of the clamped beams Parameters

Symbol

Value

Unit

Length

L

144

mm

width

bc

20

mm

Thickness

hc

0.2

mm

Young’s modulus

E

203

Gpa

4 Dynamics and Vibration Reduction Performance Analysis Observing Eqs. (17) and (18), to obtain the approximate analytical solutions of the coupled system, the nonlinear restoring force of the NES need be processed by Taylor expansion. Figure 3 shows the Taylor expansion comparison results of TNES nonlinear restoring force and potential energy It can be seen that as the expansion order increases, the results become more accurate. Therefore, in order to balance calculation time and

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(a)

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(b)

Fig. 2. Bifurcation diagrams of the equilibrium positions when (a) P = 0 N, (b) P = 80 N.

20

2

1

TNES b=60 mm d=0.6813 mm a=b+d P=50 N

No expansion 7th-order expansion 9th-order expansion 11th-order expansion

Potential energy (J)

Nonlinear restoring force F m1 (N)

solution accuracy, the 9-order expansion is selected. In order to further verify the accuracy of the 9th order Taylor expansion, the nonlinear recovery forces of NES and NES were verified. Figure 4 shows the comparison results, and it can be seen that the 9th order Taylor expansion fully meets the requirements for solving accuracy.

0

-1

-2 -0.015

10-4

15 10

No expansion 7th-order expansion 9th-order expansion 11th-order expansion

TNES b=60 mm d=0.6813 mm a=b+d P=50 N

5 0

-0.01

-0.005 0 0.005 Displacement q (m)

0.01

0.015

-5 -0.015

-0.01

-0.005 0 0.005 Displacement q (m)

(a)

0.01

0.015

(b)

Fig. 3. Comparison of Taylor expansion orders: (a) Nonlinear restoring force; (b) Potential energy.

Therefore, Eq. 17 can be rewritten as Fm1 (q) = B1 q + B3 q3 + B5 q5 + B7 q7 + B9 q9 The dynamic equation of the coupled system becomes as follows:

 m¨x + c˙x + kx − B1 q + B3 q3 + B5 q5 + B7 q7 + B9 q9 − c1 q˙ = F0 cos(ωt), m1 (¨q + x¨ ) + c1 q˙ +B1 q + B3 q3 + B5 q5 + B7 q7 + B9 q9 = 0

(22)

(23)

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(a)

(b)

Fig. 4. Comparison of nonlinear restoring force when Taylor expansion order is 9: (a) NES; (b) BENS.

4.1 Approximate Analytical Solutions To obtain an analytical solution for the system, the harmonic balance method (HBM) is applied. The responses of the LO and the absorber are assumed to be x(t) = a1,0 + q(t) = a2,0 +

N  n=1 N 

cos(nωt)a1,n + cos(nωt)a2,n +

n=1

N  n=1 N 

sin(nωt)b1,n , (24) sin(nωt)b2,n

n=1

where n is the harmonic order, a1,n and a2,n ,are constant terms, a1,n , b1,n , a2,n , b2,n are the undetermined coefficients for the corresponding harmonic terms. Substituting Eq. (24) into Eq. (23), and using the pseudo arc length method based on Newton’s iterative method, the harmonic term coefficients are extracted. The harmonic order N is set to 3, and Eq. (17) is solved by the Runge–Kutta (RK) method. 4.2 Vibration Reduction Analysis Firstly, the vibration reduction efficiency is defined as follows: ξ=

| max(AT ) − max(AL )| × 100 % , max(AL )

(25)

where AT and AL are the amplitude-frequency responses of the LO coupled with the absorber and the LO uncoupled with the absorber, respectively. Max(AT ) and Max(AL ) are extracted throughout the entire frequency band. The parameters of the LO are m = 2.5613 kg, k = 3066 N/m, c = 0.42 Ns/m. The parameters of the stable adjustable NES are m1 = 2.5613 kg, c1 = 0.12 Ns/m, other parameters are shown in Table 1. As shown in Figs. 5(a)–(c), when the excitation amplitude is very small (F 0 = 0.1 N), SMR cannot be triggered due to the presence

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of the NES energy threshold, resulting in low vibration reduction efficiency. When the excitation amplitude increases to F 0 = 0.2 N, SMR is excited, and the vibration reduction efficiency is improved. Continuing to increase the excitation amplitude to F 0 = 0.25 N, the appearance of detached resonance curve will lead to a decrease in vibration reduction efficiency. Therefore, as shown in Fig. 8(d), as the excitation amplitude increases, the vibration reduction efficiency of NES first increases and then decreases.

3 2

Amplitude of x (m)

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Without NES With NES With NES-RK 0.01

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(a) 0.014

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Without NES With NES-Stable With NES-Unstable With NES-RK

F 0=0.25N

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10-3

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0.02 0.01 0 -0.01 -0.02 80

82 t (s)

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0.004 0.002 0 32

33

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(rad/s) (c)

(d)

Fig. 5. Amplitude frequency response of LO: (a) F 0 = 0.1 N; (b) F 0 = 0.2 N, (c) F 0 = 0.25 N, (d) Vibration reduction efficiency.

When F 0 = 0.1 N, the vibration reduction efficiency of NES is low due to the input energy being lower than the energy threshold of NES. When the geometric distance relationship satisfies a = 60.3 mm, b = 60 mm, and P = 0, it is a BNES. As shown in Fig. 6, when the excitation amplitude F 0 = 0.1 N, the BNES can still perform chaotic inter well oscillation between two stable points, achieving efficient vibration reduction efficiency. When F 0 = 0.25 N, the vibration reduction efficiency of NES is low due to the appearance of the detached resonance curve. When the geometric distance relationship satisfies b = 60 mm, P = 50, d = 0.6813 mm, a = b + d, it is a TNES. As shown in Fig. 7, when the excitation amplitude F 0 = 0.25 N, the BNES can still perform chaotic

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dq/dt (m/s)

0.5

0

-0.5 -0.015

-0.01

-0.005

(a)

0 q (m)

0.005

0.01

0.015

(b)

Fig. 6. The excitation amplitude F 0 = 0.1 N: (a) the amplitude frequency response curve of LO coupled BNES; (b) the phase trajectory of BENS in the resonance region.

inter well oscillation between three stable points, achieving efficient vibration reduction efficiency.

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-1 -0.02

-0.01

0

0.01

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q (m)

(a)

(b)

Fig. 7. The excitation amplitude F 0 = 0.25 N: (a) the amplitude frequency response curve of LO coupled TNES; (b) the phase trajectory of BENS in the resonance region.

5 Experimental Verification 5.1 Experimental Device Figure 8 shows the experimental device of the LO with the stable adjustable NES. The whole experimental device is connected with the base. The stable adjustable NES consists of a pair of axially compressed clamped beams, guide rods, mass, and flexible hinges. The displacement of LO and stable adjustable NES are measured by an acceleration sensor and a laser displacement sensor, respectively. The exciter generates a constant force to push the base 8 to obtain a constant acceleration excitation ax .

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Fig. 8. Experimental device of the LO with the stable adjustable NES

5.2 Experimental Results Firstly, the vibration reduction efficiency and dynamic response of the NES under different excitation acceleration ax are studied experimentally. Figures 9, 10, 11, 12 and 13 study the amplitude-frequency response of the LO and the time-history response of the NES in the resonance region at the excitation accelerations ax = 0.015 g, 0.03 g, and 0.05 g, respectively. Figure 9 shows that when the excitation acceleration ax = 0.015 g, the excitation amplitude is small, the SMR can not be excited because of the energy threshold, and the NES performs a small periodic motion, and vibration reduction efficiency of the NES is very low. As shown in Fig. 10, when the excitation acceleration ax = 0.03 g, the input energy is higher than the energy threshold of NES, and the NES undergoes SMR in the resonance region, thereby improving the vibration reduction efficiency. Figure 11 shows that when the excitation acceleration ax is increased to 0.05 g, the amplitude-frequency response of the LO has a periodic high branching response, and the vibration reduction efficiency of the NES is decreased. However, after adjusting the length of guide rod so that the NES is converted to the BNES, when the excitation acceleration is ax = 0.015 g, compared with Fig. 9(a), it can be seen from Fig. 12(a) that the BNES still have good vibration suppression effect; Fig. 12(b) shows that in the resonance region, the BNES performs chaotic inter-well oscillation between two stable points. By adjusting the axial force to cause the beam to buckle and changing the length of the guide rod, the NES will be converted to the TNES. Compared with Fig. 11(a), Fig. 13(a) shows that the TNES has a good vibration suppression effect when the excitation acceleration is ax = 0.05 g; Fig. 13(b) shows that the TNES performs chaotic inter-well oscillation between three stable points in the resonant region.

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30

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Frequency (Hz)

(a)

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Fig. 9. Excitation acceleration ax = 0.015 g: (a) LO amplitude-frequency response; (b) Timehistory response of NES in resonance region.

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0 5.4

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(a)

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(b)

Fig. 10. Excitation acceleration ax = 0.03 g: (a) LO amplitude-frequency response; (b) Timehistory response of NES in resonance region.

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Fig. 11. Excitation acceleration ax = 0.05 g: (a) LO amplitude-frequency response; (b) Timehistory response of NES in resonance region. 30

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6 Conclusion In this paper, a stable adjustable NES model is proposed. By adjusting geometric relationships, monostable, bistable, and tristable NES are obtained. The stable adjustable NES is analytically, numerically, and experimentally analyzed. The conclusions are as follows. (1) The NES can effectively reduce vibrations by exhibiting strong modulation response when subjected to appropriate excitation amplitudes. However, both excessively low and excessively high excitation amplitudes will diminish the NES’s ability to reduce vibrations efficiently. (2) The bistable NES can perform chaotic inter-well oscillation and reduce the energy threshold of NES, which has a good vibration suppression effect on small vibrations. (3) The tristable NES can perform chaotic inter-well oscillation and eliminate detached resonance curve, which has a good vibration suppression effect on large vibrations. Acknowledgments. The authors gratefully acknowledge the support of the National Science Fund for Distinguished Young Scholars (No. 12025204) and the Program of Shanghai Municipal Education Commission (No. 2019–01-07–00-09-E00018).

References 1. Zhang, Y., Kong, X., Yue, C.: Vibration analysis of a new nonlinear energy sink under impulsive load and harmonic excitation. Commun. Nonlinear Sci. Numerical Simulation 116 (2023) 2. Dekemele, K., Van Torre, P., Loccufier, M.: Design, construction and experimental performance of a nonlinear energy sink in mitigating multi-modal vibrations. J. Sound Vib. 473, 115243 (2020)

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3. Guo, H., Yang, T., Chen, Y., Chen, L.-Q.: Singularity analysis on vibration reduction of a nonlinear energy sink system. Mech. Syst. Signal Process. 173 (2022) 4. Chen, H.-Y., Mao, X.-Y., Ding, H., Chen, L.-Q.: Elimination of multimode resonances of composite plate by inertial nonlinear energy sinks. Mech. Syst. Signal Process. 135, 106383 (2020) 5. Wang, T., Ding, Q.: Targeted energy transfer analysis of a nonlinear oscillator coupled with bistable nonlinear energy sink based on nonlinear normal modes. J. Sound Vibration 556 (2023) 6. McFarland, D.M., Kerschen, G., Kowtko, J.J., Lee, Y.S., Bergman, L.A., Vakakis, A.F.: Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators. J. Acous. Soc. Am. 118, 791–799 (2005) 7. Dekemele, K., Habib, G.: Inverted resonance capture cascade: modal interactions of a nonlinear energy sink with softening stiffness. Nonlinear Dyn. 111, 9839–9861 (2023) 8. Saeed, A.S., Al-Shudeifat, M.A., Cantwell, W.J., Vakakis, A.F.: Two-dimensional nonlinear energy sink for effective passive seismic mitigation. Commun. Nonlinear Sci. Numerical Simul. 99 (2021) 9. Ding, H., Chen, L.-Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100, 3061–3107 (2020) 10. Gendelman, O.V., Alloni, A.: Forced system with vibro-impact energy sink: chaotic strongly modulated responses. Procedia IUTAM 19, 53–64 (2016) 11. Li, W., Wierschem, N.E., Li, X., Yang, T.: On the energy transfer mechanism of the singlesided vibro-impact nonlinear energy sink. J. Sound Vib. 437, 166–179 (2018) 12. Al-Shudeifat, M.A.: Asymmetric magnet-based nonlinear energy sink. J. Comput. Nonlinear Dyn. 10 (2015) 13. Dou, J., Li, Z., Cao, Y., Yao, H., Bai, R.: Magnet based bi-stable nonlinear energy sink for torsional vibration suppression of rotor system. Mech. Syst. Signal Process. 186 (2023) 14. Sigalov, G., Gendelman, O.V., Al-Shudeifat, M.A., Manevitch, L.I., Vakakis, A.F., Bergman, L.A.: Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink. Nonlinear Dyn. 69, 1693–1704 (2012) 15. Wang, J., Wierschem, N., Spencer, B.F., Lu, X.: Experimental study of track nonlinear energy sinks for dynamic response reduction. Eng. Struct. 94, 9–15 (2015) 16. Wang, J., Zheng, Y.: Development and robustness investigation of track-based asymmetric nonlinear energy sink for impulsive response mitigation. Eng. Struct. 286 (2023) 17. Dou, J., Yao, H., Li, H., Cao, Y., Liang, S.: Vibration suppression of multi-frequency excitation using cam-roller nonlinear energy sink. Nonlinear Dyn. (2023) 18. Geng, X., Ding, H., Jing, X., Mao, X., Wei, K., Chen, L.: Dynamic design of a magneticenhanced nonlinear energy sink. Mech. Syst. Signal Process. 185 (2023) 19. Khazaee, M., Khadem, S.E., Moslemi, A., Abdollahi, A.: A comparative study on optimization of multiple essentially nonlinear isolators attached to a pipe conveying fluid. Mech. Syst. Signal Process. 141, 106442 (2020) 20. Wierschem, N.E., et al.: Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment. J. Sound Vib. 331, 5393– 5407 (2012) 21. Chen, J.E., He, W., Zhang, W., Yao, M.H., Liu, J., Sun, M.: Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks. Nonlinear Dyn. 91, 885–904 (2017) 22. Boroson, E., Missoum, S., Mattei, P.-O., Vergez, C.: Optimization under uncertainty of parallel nonlinear energy sinks. J. Sound Vib. 394, 451–464 (2017) 23. Ding, H., Shao, Y.: NES cell. Appl. Math. Mech. (English Edition) 43, 1793–1804 (2022) 24. Dou, J., Yao, H., Cao, Y., Han, S., Bai, R.: Enhancement of bistable nonlinear energy sink based on particle damper. J. Sound Vib. 547, 117547 (2023)

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25. Kremer, D., Liu, K.: A nonlinear energy sink with an energy harvester: transient responses. J. Sound Vib. 333, 4859–4880 (2014) 26. Saeed, A.S., Abdul Nasar, R., Al-Shudeifat, M.A.: A review on nonlinear energy sinks: designs, analysis and applications of impact and rotary types. Nonlinear Dyn. (2022) 27. Lu, Z., Wang, Z., Zhou, Y., Lu, X.: Nonlinear dissipative devices in structural vibration control: a review. J. Sound Vib. 423, 18–49 (2018) 28. Starosvetsky, Y., Gendelman, O.V.: Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system. J. Sound Vib. 312, 234–256 (2008) 29. Yang, T., Liu, T., Tang, Y., Hou, S., Lv, X.: Enhanced targeted energy transfer for adaptive vibration suppression of pipes conveying fluid. Nonlinear Dyn. 97, 1937–1944 (2019) 30. Zang, J., Yuan, T.-C., Lu, Z.-Q., Zhang, Y.-W., Ding, H., Chen, L.-Q.: A lever-type nonlinear energy sink. J. Sound Vib. 437, 119–134 (2018) 31. Zang, J., Cao, R.-Q., Zhang, Y.-W., Fang, B., Chen, L.-Q.: A lever-enhanced nonlinear energy sink absorber harvesting vibratory energy via giant magnetostrictive-piezoelectricity. Commun. Nonlinear Sci. Numer. Simul. 95, 105620 (2021) 32. Wang, T., Tang, Y., Qian, X., Ding, Q., Yang, T.: Enhanced nonlinear performance of nonlinear energy sink under large harmonic excitation using acoustic black hole effect. Nonlinear Dyn. (2023) 33. Chen, J.E., Sun, M., Hu, W.H., Zhang, J.H., Wei, Z.C.: Performance of non-smooth nonlinear energy sink with descending stiffness. Nonlinear Dyn. 100, 255–267 (2020) 34. Rezaei, M., Talebitooti, R.: Investigating the performance of tri-stable magneto-piezoelastic absorber in simultaneous energy harvesting and vibration isolation. Appl. Math. Model. 102, 661–693 (2022) 35. Zhou, S., Cao, J., Inman, D.J., Lin, J., Liu, S., Wang, Z.: Broadband tristable energy harvester: modeling and experiment verification. Appl. Energy 133, 33–39 (2014)

Gait-Planning-Based Path Planning for Crocodile-Inspired Pneumatic Soft Robots Yize Ma1,2 , Qingxiang Wu1,2(B) , Zehao Qiu1,2 , and Ning Sun1,2 1

2

Institute of Robotics and Automatic Information Systems (IRAIS), College of Artificial Intelligence, Nankai University, Tianjin 300350, China Institute of Intelligence Technology and Robotic Systems, Shenzhen Research Institute of Nankai University, Shenzhen 518083, China [email protected]

Abstract. In many cases, soft robots with their inherent flexibility, ease of interaction, and high adaptability in complex environments, receive widespread attention. Among them, gait planning is a key technology to ensure the performance of soft robots. This paper proposes a gaitplanning-based path planning method for crocodile-inspired pneumatic soft robots. Based on the crocodile-inspired pneumatic soft robots, the motion characteristics of the robot with different gaits are analyzed. Then, a path planning method using A* algorithm is proposed according to the environmental characteristics of road width, obstacles, and other road information. The experimental environment is built independently to complete robot path planning and gait selection and verify the effectiveness of the proposed gait-planning-based path planning method. Keywords: Gait planning · Path planning algorithm · Pneumatic robots

1

· Soft robots · A*

Introduction

In recent years, many different types of soft robots have entered people’s vision, and they are widely used in various fields, including production, life, scientific research, and exploration [1,2]. What’s more, soft robots perform excellently in human-computer interaction [3]. Additionally, due to their simple structure, pneumatic soft robots have become increasingly popular [4]. Compared with rigid robots, pneumatic soft robots can avoid irreversible damage when subjected to external impacts. Because soft robots generally have the characteristics of lightweight and softness, they are more suitable for movement in complex environments. In theory, pneumatic soft robots have infinite degrees of freedom [5], making their applications flexible [6]. Flexibility makes it possible for pneumatic soft robots to achieve complex nonlinear movements relying solely on simple inputs. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 677–688, 2024. https://doi.org/10.1007/978-981-97-0554-2_51

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Conventional quadruped robot research has been fully developed [7], while the quadruped soft robots are still worth exploring. Mosadegh et al. have designed pneumatic networks actuating rapidly [8]. Wu et al. designs a quadruped robot inspired by tortoises using pneumatic networks as actuators [9]. The tortoise-inspired robot performs amphibious movements on land and water and is considered capable of many tasks, such as disaster rescue and field exploration, due to its excellent adaptability to the environment. Inspired by these works, this paper proposes a path planning method based on gait analysis for crocodile-inspired pneumatic soft robots, and independently builds a set of experimental platform for verification.

2

Design of Robots and Gaits

2.1

Robots Introduction

A pneumatic soft robot inspired by crocodiles with 8 actuators is established in Fig. 1, which includes four leg actuators, one trunk actuator, and one tail actuator. Due to the interlinked and orthogonal structure, the crocodile’s legs will bend down and back simultaneously if inflated. Alternating inflation and deflation can make the crocodile-inspired pneumatic soft robot move forward. Two traditional bellows pneumatic actuators are placed parallel on both sides of the spine for turning. The tail actuator can swing left and right through the grid chambers arranged on both sides. In addition, tail swinging will affect the center of gravity and stability of crocodiles, which serves as the main organ for swimming in the water.

Fig. 1. Three-dimensional model of the crocodile-inspired pneumatic soft robot.

2.2

Gaits Analysis of Crocodile-Inspired Pneumatic Soft Robots

According to the motion mode of crocodiles, three typical gaits of crocodileinspired pneumatic soft robots are analyzed. (see Fig. 2). In detail, a tort gait is designed for a pneumatic soft crocodile. Due to the low center of gravity of the

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Up Down (a) Crawling

Head

(b) Tort

(c) Galloping

Fig. 2. Three typical gaits of the crocodile-inspired pneumatic soft robot.

crocodile-inspired pneumatic soft robot, it is usually not necessary to consider its stability. Based on the movement of crocodiles in nature state, a crawling gait is designed. When crocodiles leisurely walk, they only move one leg at a time, which consumes less energy for their actions. A galloping gait is usually not considered by rigid robots because it may impact on the driver, while the resilience of soft robots allows them to use this easily controlled gait. Additionally, the crocodileinspired pneumatic soft robots use ubiquitous air to transmit energy without the need to carry additional actuated and control module, making the robot lighter. Specifically, lightweight will help reduce the impact of actuators interacting with the environment during movement. When input different air pressures into the bellows on both sides of the spine, the crocodile-inspired pneumatic soft robot can achieve steering. Several experiments are conducted to measure the characteristics of the crawling gait, the tort gait, and the galloping gait. Table 1 shows that the tort gait runs fastest, galloping gait slower than the tort but rectilinearity is high, and the crawling gait is the slowest. Furthermore, the tort gait with the fastest linear velocity is beneficial for turning. Based on these data, different gaits will be integrated into path planning. 2.3

Gait-Planning-Based Path Planning

A* Algorithm with Orientation Constraint. A* algorithm is mature and widely used [10,11]. In this paper, a constrained A* algorithm is adopted, which considers that crocodile-inspired pneumatic soft robots can only move forward and turn, and introduces a turning radius to determine whether the robot can reach the next point. In other words, the orientation of robots is taken into account. Although the robot position is considered discrete, its orientation is

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Table 1. Characteristics of gaits for the crocodile-inspired pneumatic soft robots Experiment

vCrawling (mm/s) vTort (mm/s) vGalloping (mm/s) Turning radius (mm/s)

1

1.09

4.01

2.15

28.57

2

1.15

4.18

2.29

33.02

3

1.04

4.29

2.01

29.75

4

1.19

4.22

2.39

29.41

5

1.20

4.25

2.33

30.09

Average

1.13

4.19

2.23

30.16

Rectilinearity

High

Low

High

-

continuously changing and constrains the direction of motion. The A * algorithm always chooses the position where the sum of past costs and future costs is the lowest, and this minimum value is usually represented by f (i). In this paper, the direction should be restricted, then f (i) is modified as follows: f (i) = g(i) + h(i) + m · t(i),

(1)

where h(i) is the Euclidean distance meaning the cost estimation between the current cell and the target cell, g(i) is the length of the path from the initial cell to the current cell through the selected path, t(i) is the value brought by turning, and m is the weight. Especially, t(i) is a nonlinear function that depends on the turning radius of the crocodile-inspired pneumatic soft robot. To make t(i) meaningful, cells are given a direction d as the robot passes through it. The turning radius of the crocodile-inspired pneumatic soft robots is 0.3 m (see Table 1), and the side of the cell is determined as 0.1 m. In this paper, t(i) can be expressed as follows:  ek|di+1 −dn | − 1, 0 < |di+1 − dn | < dMax , (2) t(i + 1) = ∞, other. where di+1 is the angle between this cell to the next cell, which is usually π/4, 0, and −π/4, dn is the direction of the robot when it is passing through the cell (see Fig. 3), |di+1 − dn | represents the angle between the upcoming cell and the current direction and k is the weight of |di+1 − dn |, k and m can determine range of t(i), dMax is the maximum turning angle. By increasing f (i) to ∞, the algorithm is restricted from reaching cells the crocodile-inspired pneumatic soft robots cannot reach, thereby making the A* algorithm more in line with the actual situation of crocodile-inspired pneumatic soft robots. In a word, by introducing t(i), constraints on the direction of robot motion can be achieved.

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i+1 0°

Direction of cell 45°

i

di+1

dn Fig. 3. Diagrammatic sketch of di+1 and dn .

At the same time as each iteration, the cell’s directional information of dn is updated. If the cell diagonally ahead is the next to arrive, the crocodile-inspired pneumatic soft robots will make a turn, which action makes the direction change when the crocodile-inspired pneumatic soft robots reach the next cell. The angle increment (Δd) is the distance between two cells divided by the turning radius, which is rewritten as follows:  l(i,i+1) /r, turning right, Δd = (3) −l(i,i+1) /r, turning left, where l(i,i+1) is the distance between this cell and the next cell, and r is the turning radius of the crocodile-inspired pneumatic soft robots. Defining clockwise as the positive direction, then Δd is positive when turning right but is negative when turning left. In addition, expanding obstacles is a common method to avoid the tedious calculations caused by the collision volume of robots. In this paper, two experimental scenes are built: Scene 1: A passage surrounded by four obstacles, with a narrowest point of 0.43 m can be passed through. Scene 2: A passage surrounded by four obstacles, with a narrowest point of 0.30 m can be passed through. Based on the these two scenes, Fig. 4 and Fig. 5 are obtained. Through introducing direction into the A * algorithm and considering the changing angle, when and where the crocodile-inspired pneumatic soft robots should turn can be obtained, which is helpful for gait switching and control. Gaits-Planning-Based Path Planning. In terms of gaits, crocodile-inspired pneumatic soft robots can be considered with a crawling gait for normal cruising

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before receiving a task. After receiving tasks or signals, the tort gait with fast speed is considered. If the path requires the rectilinearity, a galloping gait with better rectilinearity will be adopted. The limbs and tail of the crocodile-inspired pneumatic soft robots are all soft material, so in case the robot collides with an obstacle, as long as the approximate direction is correct, it can return to the predetermined track on its own. Although the crocodile-inspired pneumatic soft

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robots can pass through a restricted narrow passage, when the two sides of the passage are not rigid obstacles but traps, the situation will be different. At this point, it is necessary to consider the rectilinearity of the robot’s actions, such as when crossing a single wooden bridge without guardrails on both sides. In scene 2 (see Fig. 5), the channel through which robots can pass is narrow, and a more linear gait must be adopted not only to ensure that the robot can enter the predetermined channel but also to avoid collision with obstacles. 2.4

Experimental Results

In the experiments, the crocodile-inspired pneumatic soft robots achieve different gait movements by setting with a series of input signals that conform to the predetermined path. The motion capture platform (SLIK PRO 700DX) is applied to obtain the trajectories of the crocodile-inspired pneumatic soft robots. The marked points are located on both sides of the connector which connects the crocodile’s tail and truck (see Fig. 6). In the experiments, an air pump is used to drive the crocodile-inspired pneumatic soft, which is the only energy source of the crocodile-inspired pneumatic soft robots. A computer is used to connect to the MicroLabBox (dSPACE R2021a) for valves control and is also used to operate the motion capture platform. The valves are the components that adjust the air pressure proportionally based on voltage. The MicroLabBox provides a platform for combining software and hardware. The entire system cannot only input sequence signals in advance to operate the crocodile-inspired pneumatic

Fig. 6. The self-fabricated crocodile-inspired pneumatic soft robot.

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soft robots offline, but also operate the crocodile-inspired pneumatic soft robots in real-time through the computer and the MicroLabBox. The experimental environment for crocodile-inspired pneumatic soft robots is shown in Fig. 7.

Fig. 7. Experimental environment for crocodile-inspired pneumatic soft robots.

In scene 1 (Fig. 8, Fig. 9, and Fig. 10), where the road is wide, crawling is chosen to reach the target quickly. According to the path obtained by the A* algorithm, the crocodile-inspired pneumatic soft robot needs to first move forward, turn right, and then go straight. In the long straight motion, the obstacles on both sides are far from the head, tail, and limbs of the crocodile-inspired pneumatic soft robot, so it is possible to directly choose a faster gait. Then the crocodile-inspired pneumatic soft robot needs to turn right to reach the target. Figure 10 shows the trajectory of the crocodile-inspired pneumatic soft robot in a wide channel environment. Based on Fig. 10, it can be acquired that the crocodile-inspired pneumatic soft robot with proposed method can reach the target without collision. In scene 2 (Fig. 11, Fig. 12, and Fig. 13), the road is just enough to accommodate the crocodile-inspired pneumatic soft robots, so we ensure that its route is straight to avoid colliding with obstacles. Galloping gait is the best one for this situation. Then the crocodile robot needs to turn right at an obtuse angle to reach target. Figure 13 shows that the crocodile-inspired pneumatic soft robot with proposed method can maintain straightness in a narrow and long straight channel, enter a predetermined orbit, and reach the target.

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200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

path edge target start obstacle

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

y/cm

Fig. 8. Scene 1: wide road experimental environments.

x/cm

200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

left right target path obstacle target start

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

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Fig. 9. Trajectories acquired A* algorithm in scene 1.

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Fig. 10. Actual trajectories of crocodile-inspired pneumatic soft robot in scene 1.

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0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

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Fig. 11. Scene 2: narrow road experimental environments.

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Fig. 12. Trajectories acquired A* algorithm in scene 2.

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Fig. 13. Actual trajectories of crocodile-inspired pneumatic soft robot in scene 2.

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This paper proposes a gait-plan-based path planning method for a pneumatic crocodile-inspired soft robot. By analyzing three gaits (crawling, tort, and galloping), the characteristics of three gaits are obtained. In detail, the crawling gait consumes less energy, with high rectilinearity, but slow speed, making it suitable for autonomous cruising without tasks. The tort gait is fast and suitable for targeted tasks, but it is limited in narrow environments because of poor rectilinearity. The galloping gait has moderate speed, with high rectilinearity, which can compensate for shortcomings of the tort gait, and also has obstaclecrossing ability. To this end, a path planning algorithm based on A* algorithm is proposed, taking path width and robot mobility into account. Finally, the effectiveness of the proposed method is verified through two scenes of different environmental experiments (the narrow scene and the wide scene). At the same time, the experimental results demonstrate that the proposed method can ensure the pass ability of the crocodile-inspired pneumatic soft robots. Acknowledge. This work was supported in part by the National Natural Science Foundation of China under Grant 52205019 and Grant 62373198, in part by the Guangdong Basic and Applied Basic Research Foundation under Grant 2023A1515012669, in part by the China Postdoctoral Science Foundation under Grant 2021M701779, in part by the Fundamental Research Funds for the Central Universities under Grant 07863233098, and in part by the State Key Laboratory of Robotics and Systems (HIT) under Grant SKLRS-2023-KF-13.

References 1. Armanini, C., Boyer, F., Mathew, A.T., Duriez, C., Renda, F.: Soft robots modeling: a structured overview. IEEE Trans. Rob. 39(3), 1728–1748 (2023) 2. Rus, D., Tolley, M.: Design, fabrication and control of soft robots. Nature 521, 467–475 (2015) 3. Whitesides, G.: Soft robotics. Angew. Chem. 57(16), 4258–4273 (2018) 4. Polygerinos, P., Correll, N., Morin, S., Mosadegh, B., Onal, C., Petersen, K., Cianchetti, M., Tolley, M., Shepherd, R.: Soft robotics: review of fluid-driven intrinsically soft devices; manufacturing, sensing, control, and applications in humanrobot interaction. Adv. Eng. Mater. 19(12), 1700016 (2017) 5. Trivedi, D., Rahn, C., Kier, W., Walker, I.: Soft robotics: biological inspiration, state of the art, and future research. Bionics Biomech. 5(3), 99–117 (2008) 6. James, W., et al.: Soft robotics: a review of recent developments of pneumatic soft actuators. Actuators 9(1), 3 (2020) 7. Li, J., Wang, J., Yang, S.X., Zhou, K., Tang, H.: Gait planning and stability control of a quadruped robot. Comput. Intell. Neurosci. 2016, 9853070 (2016) 8. Mosadegh, B., Polygerinos, P., Keplinger, C., Wennstedt, S., et al.: Pneumatic networks for soft robotics that actuate rapidly. Adv. Func. Mater. 5, 2163–2170 (2014) 9. Wu, M., et al.: A fully 3D-printed tortoise-inspired soft robot with terrainsadaptive and amphibious landing capabilities. Advanced Materials Technologies 7(12), 2200536 (2022)

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10. DuchoE, F., Babinec, A., Kajan, M., BeEo, P., Florek, M., Fico, T., Juriˇsica, L.: Path planning with modified A star algorithm for a mobile robot. Procedia Engineering 96(2014), 59–69 (2014) 11. Abed, M., Lutfy, O., Al-Doori, Q.: A review on path planning algorithms for mobile robots. Eng. Technol. J. 39(5), 804–820 (2021)

Research on Nonlinear Energy Sink Vibration Reduction of Floating Raft System Hong-Li Wang1 and Hu Ding1,2(B) 1 Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Frontier Science

Center of Mechanoinformatics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China [email protected] 2 Shaoxing Institute of Technology, Shanghai University, Shaoxing 312074, China

Abstract. The floating raft vibration isolation system has the problem of multifrequency vibration in the process of vibration. The existence of multi-frequency vibration makes it difficult to apply linear vibration absorption. In this paper, the nonlinear vibration absorption is applied to the floating raft vibration control, and the floating raft vibration control strategy using multiple nonlinear energy sinks (NESs) distributed arrangement is proposed for the first time. Considering that the base of the original floating raft system has elastic support, the natural frequencies and modes of the floating raft system are analyzed, and the dynamic models of the floating raft system with different distribution modes of NES are established. Through dynamic analysis, the influence laws of NES distribution on floating raft vibration control are compared. Based on the vibration reduction of low-order and high-order modes, the NES layout mode is analyzed. The results show that the distributed nonlinear energy sinks arrangement can effectively control the vibration of floating raft, and has good vibration damping effect on all modes of floating raft. The floating raft vibration attenuation for low-order and high-order modes should be arranged in different NES placement modes. In conclusion, the study of this paper provides a new and efficient control strategy for floating raft vibration. Keywords: Vibration · Floating raft system · Nonlinear energy sink

1 Introduction Nowadays, the vibration problem of ships and submarines can not be ignored, and the vibration problem existing in the running process will also affect the comfort and concealment of the ride [1]. In order to control the vibration of ships and submarines, the floating raft system is proposed. The floating raft vibration isolation system has been studied by scholars because of its simple structure, strong practicability and good vibration reduction effect [2, 3]. In the high frequency range, the floating raft vibration isolation system is superior to the traditional first-stage vibration isolation system, which can effectively suppress the vibration and noise transmitted from the main engine and auxiliary engine [4, 5]. Especially in submarines, the further optimization of floating raft system vibration reduction effect has been widely concerned [6]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 689–699, 2024. https://doi.org/10.1007/978-981-97-0554-2_52

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Floating raft vibration isolation system is a special vibration isolation system developed on the basis of traditional two-stage vibration isolation. The floating raft system consists of the upper power equipment, the upper vibration isolator, the middle raft frame, the lower vibration isolator and the lower base [7, 8]. Active vibration isolation and passive vibration isolation are concluded as the two kinds of floating raft vibration isolation system. Most of them are passive vibration isolation. Li et al. [9] proposed a floating raft vibration isolation system constructed by a double-layer quasi-zero-stiffness mechanism and analyzed the mathematical relationship between amplitude and frequency and force stiffness, as well as the influence of mass ratio and damping ratio on the vibration reduction performance. Li et al. [10] used the average method to analyze the dynamic characteristics of the nonlinear vibration isolation floating raft system with feedback control, which provides a certain basis for the design and improvement of the passive control of floating raft system. Song et al. [11] studied the suppression of vibration and noise radiation in raft system by using the theory of periodic structure. Liu et al. [12] studied the influence of floating raft vibration isolation system on the vibration characteristics of Marine pump, providing a basis for the study of floating raft power equipment vibration. Ren et al. [13] discussed the vibration transmission performance of a floating raft system with attached pipes, and concluded that the existence of additional transmission paths caused by pipes would destroy the symmetry of the system. Shi et al. [14] considered the floating raft system under the condition of time-varying liquid mass, and analyzed the influence of mass change on the characteristics of raft displacement, isolator load distribution, mode frequency and so on. On the basis of these studies, the vibration characteristics of the floating raft system have been analyzed in detail. However, the vibration reduction effect of floating raft isolation system still has a large optimization space, and the problem of multi-frequency still exists. Nonlinear energy sink (NES) vibration absorption has been well developed in the last few years [15–17]. NES is a kind of nonlinear vibration absorber, it can achieve a wide band of vibration absorption, and the structure is simple, the additional mass is small. Furthermore, NES has no effect on the resonant frequency of the main system, is now a very popular vibration absorber [18, 19]. Due to the vibration of the NES itself, the vibration energy of the main system can be absorbed, so as to achieve effective vibration reduction. NES vibration damping has been well developed and applied over the past few years [20–23]. In this paper, the application of NES to the vibration reduction of floating raft system is proposed for the first time. NES adopts distributed placement, and the influence of different NES placement on the vibration reduction effect is compared, and the acceleration level and amplitude are used to evaluate the vibration reduction efficiency of the system. Through the mutual verification of the approximate analytical method and the numerical simulation method, the influence of different NES distribution modes on the vibration control effect of the floating raft system is demonstrated. It shows that NES has obvious effect on the further optimization of the vibration damping effect of floating raft system.

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2 Mechanical Model and Modal Analysis of the Floating Raft System In this section, the mechanical model of the floating raft vibration isolation system is introduced, as shown in Fig. 1. Meanwhile, the modal analysis of the floating raft system is carried out.

Fig. 1. Model of the floating raft system

In the floating raft model shown in Fig. 1, the mass and corresponding displacement of the three power units are m3 ,m4 , m5 , x 3 , x 4 and x 5 , respectively. The mass of the base and the raft are m1 , m2 , and the displacement are x 1 , x 2 , respectively. The upper vibration isolator is composed of damping c3 , c4 , c5 and stiffness k 3 , k 4 , k 5 . The lower vibration isolator is composed of damping c2 and stiffness k 2 . The base is regarded as an elastic base with damping c1 and stiffness k 1 below. The external excitation of the system is F 1 = F 2 = F 3 = Aω2 sinωt, in the form of the inertia force generated by the eccentric rotation of the power equipment when working. A is the product of the eccentric mass of the power equipment and the eccentric distance, and ω is the circular frequency of the external excitation. The governing equation of the floating raft system is obtained as m1 x¨ 1 + k1 x1 + c1 x˙ 1 + k2 (x1 − x2 ) + c2 (˙x1 − x˙ 2 ) = 0, m2 x¨ 2 + k2 (x2 − x1 ) + c2 (˙x2 − x˙ 1 ) + k3 (x2 − x3 ) + c3 (˙x2 − x˙ 3 ) + k4 (x2 − x4 ) + c4 (˙x2 − x˙ 4 ) + k5 (x2 − x5 ) + c5 (˙x2 − x˙ 5 ) = 0, m3 x¨ 3 + k3 (x3 − x2 ) + c3 (˙x3 − x˙ 2 ) = F1 (t), m4 x¨ 4 + k4 (x4 − x2 ) + c4 (˙x4 − x˙ 2 ) = F2 (t), m5 x¨ 5 + k5 (x5 − x2 ) + c5 (˙x5 − x˙ 2 ) = F3 (t).

(1)

The selected floating raft system parameters are shown in Table 1, and the product of the eccentric mass of the power equipment and the eccentric distance A = 0.002. The modal analysis of the floating raft system is carried out based on the parameters in Table 1.

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Notation

Value (N/m)

Notation

Value (N·s/m)

Notation

Value (kg)

k1

106

c1

14

m1

30

k2

6.5 × 104

c2

5

m2

7.5

k3

6 × 104

c3

5

m3

7

k4

104

c4

2.5

m4

5

k5

104

c5

2.5

m5

3

Equation (1) is written in matrix form ¨ + CX ˙ + KX = F. MX

(2)

The corresponding differential equation of undamped free vibration motion is written as ¨ + KX = 0, MX where the mass matrix and stiffness matrix are expressed as respectively ⎡ ⎡ ⎤ k1 + k2 m1 0 0 0 0 −k2 0 0 ⎢ −k k + k + k + k −k −k ⎢ 0 m 0 0 0 ⎥ ⎢ ⎢ ⎥ 2 2 2 3 4 3 4 5 ⎢ ⎢ ⎥ M = ⎢ 0 0 m3 0 0 ⎥, K = ⎢ 0 −k3 k3 0 ⎢ ⎢ ⎥ ⎣ 0 ⎣ 0 0 0 m4 0 ⎦ 0 k4 −k4 0 0 0 0 0 0 m5 0 −k5

(3) ⎤ 0 −k5 ⎥ ⎥ ⎥ 0 ⎥, (4) ⎥ 0 ⎦ k5

and the displacement vector is expressed as X = (x1 , x2 , x3 , x4 , x5 )T .

(5)

The solution of simple harmonic vibration is defined as X = X0 sin(ωt),

(6)

where ω is the natural circular frequency of the system, and X0 is the vibration mode vector of each node. Substitute Eq. (6) into (3), we get  K − ω2 M X0 = 0. (7) X0 is not all 0 in free vibration, then there is  det K − ω2 M = 0.

(8)

The natural frequency can be obtained by solving Eq. (8). Then, the vibration mode vector of the system can be solved by substituting ω into Eq. (7). The calculated natural frequency of the floating raft system is shown in Table 2.

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Table 2. The natural frequency of the floating raft system Order

Natural frequency (Hz)

1

6.14

2

8.36

3

10.99

4

24.11

5

30.52

The mode matrix is calculated as ⎡ ⎤ 1.00 1.00 1.00 1.00 1.00 ⎢ 15.70 15.11 14.18 5.79 −0.59 ⎥ ⎢ ⎥ ⎢ ⎥ X0 = ⎢ 18.99 22.28 31.95 −3.45 0.18 ⎥. ⎢ ⎥ ⎣ 61.28 −39.87 −10.26 −0.55 0.03 ⎦ 28.35 87.55 −32.99 −0.98 0.06

(9)

3 NESs in the Floating Raft System Based on the mechanical model of the floating raft system, two kinds of vibration reduction models of floating raft system with NESs are constructed in Fig. 2. In the NES damping model shown in Fig. 2, the mass, damping, cubic nonlinear stiffness and displacement of NES added on the three power units are m6 , m7 , m8 , c6 , c7 , c8 , k 6 , k 7 , k 8 , x 6 , x 7 and x 8 , respectively. The mass, damping, cubic nonlinear stiffness and displacement of NES added on the raft and base are m9 , m10 , c9 , c10 , k 9 , k 10 , x 9 and x 10 , respectively. Take the NES-M-2 as an example, the governing equation is obtained as m10 x¨ 10 + c10 (˙x10 − x˙ 1 ) + k10 (x10 − x1 )3 = 0, m9 x¨ 9 + c9 (˙x9 − x˙ 2 ) + k9 (x9 − x2 )3 = 0, m8 x¨ 8 + c8 (˙x8 − x˙ 5 ) + k8 (x8 − x5 )3 = 0, m7 x¨ 7 + c7 (˙x7 − x˙ 4 ) + k7 (x7 − x4 )3 = 0, m6 x¨ 6 + c6 (˙x6 − x˙ 3 ) + k6 (x6 − x3 )3 = 0, m5 x¨ 5 + c5 (˙x5 − x˙ 2 ) + k5 (x5 − x2 ) + c8 (˙x5 − x˙ 8 ) + k8 (x5 − x8 )3 = F3 (t), m4 x¨ 4 + c4 (˙x4 − x˙ 2 ) + k4 (x4 − x2 ) + c7 (˙x4 − x˙ 7 ) + k7 (x4 − x7 )3 = F2 (t), m3 x¨ 3 + c3 (˙x3 − x˙ 2 ) + k3 (x3 − x2 ) + c6 (˙x3 − x˙ 6 ) + k6 (x3 − x6 )3 = F1 (t), m2 x¨ 2 + c2 (˙x2 − x˙ 1 ) + k2 (x2 − x1 ) + k3 (x2 − x3 ) + c3 (˙x2 − x˙ 3 ) + k4 (x2 − x4 ) + c4 (˙x2 − x˙ 4 ) + k5 (x2 − x5 ) + c5 (˙x2 − x˙ 5 ) + c9 (˙x2 − x˙ 9 ) + k9 (x2 − x9 )3 = 0, m1 x¨ 1 + k1 x1 + c1 x˙ 1 + k2 (x1 − x2 ) + c2 (˙x1 − x˙ 2 ) + c10 (˙x1 − x˙ 10 ) + k10 (x1 − x10 )3 = 0.

(10)

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Fig. 2. (a) NES distribution model 1 (NES-M-1), (b) NES distribution model 2 (NES-M-2)

For strongly nonlinear systems, the harmonic balance method (HBM) can obtain approximate analytical solutions. Odd harmonic terms are selected because the coupled system contains cubic nonlinearity. The order of the harmonic term is set to one and the analytical solution of the displacement response of the NES-M-2 is assumed as xi =

n

j=1

ai,j cos(jωt) +

n

bi,j sin(jωt),

(11)

j=1

where ai,j and bi,j are the harmonic coefficients, and i = 1, 2, …, 10. The harmonic order is expressed as j, and j = 1, 2, …, n. Substitute Eq. (11) into (10). Here, the calculation process of the first-order HBM is taken as an example. Then, the values of the harmonic coefficient are solved by the pseudo-arc-length extension method. The amplitude of the base displacement response (Ax1 ) and acceleration of the base response (ax1 ) are calculated by  2 + b2 , Ax1 = a1,1 1,1 (12) 2 ax1 = −ω a1,1 cos(ωt) − ω2 b1,1 sin(ωt) = D sin(ωt + α),   2 + b2 , α = arctan a1,1 . where D = −ω2 a1,1 1,1 b1,1

Research on Nonlinear Energy Sink Vibration Reduction of Floating Raft System

The acceleration level of the system is defined as

 L = 20 lg max ax1 /a0 = 20 lg|D/a0 |,

695

(13)

where a0 is the reference value, a0 = 10−6 m/s2 . Based on Eq. (10), the Runge-Kutta (RK) numerical simulation technique is used to calculate the response curve of the system numerically. NES parameters added on floating raft system are shown in Table 3, where m6 /m3 = m7 /m4 = m8 /m5 = m9 /m2 = m10 /m1 = 5%. Table 3. NESs parameters Notation

Value (N/m3 )

Notation

Value (N·s/m)

Notation

Value (kg)

k6

6 × 107

c6

12

m6

0.35

k7

8.2 × 106

c7

8

m7

0.25

k8

8.2 × 106

c8

8

m8

0.15

k9

2 × 108

c9

17

m9

0.375

k 10

6 × 108

c10

150

m10

1.5

Based on the parameter values in Table 1 and Table 3, the comparison of amplitudefrequency response curve of the NES distribution model 1 between HBM and RK is shown in Fig. 3, which shows that the analytical method has great accuracy.

Fig. 3. Comparison of amplitude-frequency response curve of the NES distribution model 1 between HBM and RK

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4 Comparison of Two NES Distribution Modes 4.1 (Nes-M-1) The comparison between the response of NES-M-1 and that of the floating raft system is shown in Fig. 4. Only adding NES on the three power units has a good vibration damping effect for the low-order modes, and the amplitude of acceleration level and base decreases to a great extent compared with the model without NES.

Fig. 4. Comparison of the response between NES-M-1 and floating raft system: (a) acceleration level-frequency for low-order modes, (b) acceleration level-frequency for high-order modes, (c) base amplitude-frequency for low-order modes, (d) base amplitude-frequency for high-order modes

4.2 (Nes-M-2) The comparison between the response of NES-M-2 and that of the floating raft system is shown in Fig. 5. It can be seen that NES should be added to the three power units, raft and base as in NES distribution model 2 to achieve effective vibration reduction for all frequency bands. For both low- and high- order modes, NES-M-2 can effectively reduce

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Fig. 5. Comparison of the response between NES-M-2 and floating raft system: (a) acceleration level-frequency for low-order modes, (b) acceleration level-frequency for high-order modes, (c) base amplitude-frequency for low-order modes, (d) base amplitude-frequency for high-order modes

the acceleration level and the amplitude of the base, and the vibration reduction effect is very good. By comparison, it is found that the vibration attenuation of the fourth and fifth order modes in the high frequency band can be realized by NES distribution model 2, and the vibration attenuation effect is obviously better than NES distribution model 1. NES distribution model 1 mainly affects the damping effect of the first three modes of the system, which is consistent with the results of modal analysis.

5 Conclusions In this article, NESs are applied to the floating raft vibration isolation system, and a new floating raft system vibration reduction model is proposed. The external excitation form of the system is in the form of the inertia force generated by the eccentric rotation of the power equipment. Meanwhile, the natural frequency and mode analysis of the floating raft system are carried out. Based on the results of modal analysis, the vibration isolation

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models of three different NES distribution modes are established, and the governing equations are given. The approximate analytical solution of NES damping system is obtained by harmonic balance method and verified by numerical method. Acceleration level and base amplitude are used as the evaluation indexes for vibration reduction of floating raft system. The vibration damping effect of three different NES distribution models on floating raft system is studied and compared. The following main conclusions are drawn. NES can play a great role in damping the vibration of the floating raft system and can be used to reduce the vibration of all the five modes of the system. According to the modal analysis results of the floating raft system, NES should be distributed in order to achieve effective vibration reduction. This means that NES should be placed on the three power units, raft and base. The mass ratio of NES to the main mass is controlled at 5%, which achieves a small additional mass and effectively controls the vibration of the floating raft system. To control the vibration of the floating raft system for the low-order modes (4–13 Hz), NES should be added to the three power devices. On the contrary, to realize the highorder modes (15–40 Hz) vibration reduction effectively, NES can only be added on the raft frame and base. In this paper, NES is first used to optimize the damping effect of floating raft system. In the future work, the use of NES cells to reduce the vibration of floating raft system and optimizing the quality of NES added to different structures are worth further consideration. Acknowledgments. The authors gratefully acknowledge the support of the China National Funds for Distinguished Young Scholars (No. 12025204) and the Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018).

References 1. Fang, Y., Zuo, Y., Xia, Z.: Study on design method and vibration reduction characteristic of floating raft with periodic structure. In: IOP Conference Series: Materials Science and Engineering. 322(4) (2018) 2. Qu, Z.-Q., Chang, W.: Dynamic condensation method for viscously damped vibration systems in engineering. Eng. Struct. 22(11) (2000) 3. Xiong, Y.P., Xing, J.T., Price, W.G.: Power flow analysis of complex coupled systems by progressive approaches. J. Sound Vibration 239(2) (2001) 4. Yang, T., et al.: On active syhchrophasing control of vibration for a floating raft vibration isolation system. In: 22nd International Congress on Sound and Vibration (ICSV), Florence, ITALY (2015) 5. You, W., et al.: Design of a floating raft system by exploiting the nonlinear damping. Advances in Mechanical Engineering 15(5) (2023) 6. Xie, X., Li, M., Wang, J.: Nonlinear vibration behaviors of marine rotor system coupled with floating raft-airbag-displacement restrictor under ship heaving motion. Adv. Mech. Eng. 13(12) (2021) 7. Yang, B., et al.: Improvements of magnetic suspension active vibration isolation for floating raft system. Int. J. Appl. Electromagnetics Mech. 53(2) (2016)

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8. Zhenli, Z., et al.: Research on the effects of raft on vibration isolation. J. Phys. Conf. Ser. 2342(1) (2022) 9. Yingli, L. and X. Daolin.: Force transmissibility of floating raft systems with quasi-zerostiffness isolators. J. Vibration Control 24(16) (2018) 10. Li, Y., et al.: Nonlinear dynamic analysis of 2-DOF nonlinear vibration isolation floating raft systems with feedback control. Chaos Solitons Fractals 45(9–10), 1092–1099 (2012) 11. Song, Y., et al.: Suppression of vibration and noise radiation in a flexible floating raft system using periodic structures. J. Vibration Control 21(2) (2015) 12. Liu, H., et al.: Vibration control of a marine centrifugal pump using floating raft isolation system. J. Low Frequency Noise Vibration Active Control 39(2) (2020) 13. Longlong, R., et al.: Dynamic Modeling and Characteristic Analysis of Floating Raft System with Attached Pipes. Shock and Vibration. 2017 (2017) 14. Liang, S., Guanghui, C., Wenjun, B.: Mechanical characteristics analysis and control algorithm for floating raft system with mass variation. Sci. Rep. 13(1) (2023) 15. Zang, J., et al.: A lever-type nonlinear energy sink. J. Sound Vib.Vib. 437, 119–134 (2018) 16. Ding, H., Chen, L.-Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn.Dyn. 100(4), 3061–3107 (2020) 17. Jiang, X., et al.: Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dyn.Dyn. 33(1), 87–102 (2003) 18. Ji, J.C.: Design of a nonlinear vibration absorber using three-to-one internal resonances. Mech. Syst. Signal Process. 42(1–2) (2014) 19. Pengcheng, Z., et al.: Multi-resonator coupled metamaterials for broadband vibration suppression. Appl. Math. Mech. 42(1) (2020) 20. Ding, H., Chen, L.-Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dynamics. 100(prepublish) (2020) 21. Han, Y., Cao, Q., Ji, J.: Nonlinear dynamics of a smooth and discontinuous oscillator with multiple stability. Int. J. Bifurcation Chaos 25(13) (2015) 22. Lu, Z.-Q., Ding, H., Chen, L.-Q.: Resonance response interaction without internal resonance in vibratory energy harvesting. Mech. Syst. Signal Process. 121, 767–776 (2019) 23. Zhang, Y.-W., et al.: Vibration power flow characteristics of the whole-spacecraft with a nonlinear energy sink. J. Low Frequency Noise Vibration Active Control 38(2), 341–351 (2019)

X-mechanism Guided Elastic QZS Vibration Isolator Design for Beneficial Nonlinear Stiffness Chuanping Liu and Xingjian Jing(B) City University of Hong Kong, Hong Kong, China [email protected]

Abstract. Mechanical metamaterials are emerging vividly in recent decades elevating the limits of mechanical properties. Compared to traditional mechanical materials with classical elastic theory, the state-of-the-art mechanical metamaterials possessed many unconventional mechanical properties, such as negative stiffness, zero stiffness, ultra-toughness, negative Poisson’s ratio, etc. As stiffness being the natural property of all materials, modification on material stiffness to achieve quasi-zero stiffness attracts many research attentions. In this research, an X-mechanism guided design paradigm on elastic isolator with quasi-zero stiffness was explored by integrating both softening and hardening mechanisms from rigid body X-mechanism into soft elastic isolator of small dimension. The synergistic effect of softening and hardening mechanisms was investigated analytically and numerically. While the finite element analysis models illustrated the mechanism on reaching quasi-zero stiffness, experimental test on 3D printed elastic isolator samples demonstrates the promising results of quasi-zero stiffness range and explores favorable engineering features. The printed elastic isolator possessed a size comparable to a coin. Furthermore, the loading capacity of proposed elastic isolator are quantified with both finite element analysis and experimental test. With both finite element analysis and experimental testing, 3D printed elastic isolator samples with promising quasi-zero stiffness behavior illustrates the possibility of performing superior vibration isolation and actuator control within one-piece tiny devices. Keywords: Mechanical Metamaterial · Quasi-zero stiffness · Vibration isolator

1 Introduction Mechanical metamaterials, emerging under the big background of metamaterial, are recognized as a field to transcend the knowledge of classical mechanics of materials by reprogramming the overall mechanical performance through multi-physical engineering process [1]. By harnessing the knowledge of nonlinear sciences, geometry mathematics, biology, optic, electromagnetics, the research community of mechanical metamaterials were focusing on creating unconventional physical properties and increasing the mechanical performance on material-wise. The delicate engineered mechanical metamaterials of the state-of-the-art were investigated to exhibit unconventional physical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 700–710, 2024. https://doi.org/10.1007/978-981-97-0554-2_53

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behaviors of negative stiffness, quasi-zero stiffness (QZS), negative Poisson’s ratio, shape-morphing deformation, ultra-toughness, ultra-softness, etc. Among many interesting potential mechanical properties, QZS property was considered as one unique mechanical property that involves reducing the stiffness of metamaterials approaching to zero at certain conditions. From the practical engineering perspective, vibration transmissivity was approaching to zero and the corresponding systems were idle to vibration of large bandwidth frequency range [2–4]. Meanwhile, periodically working in the QZS zone will automatically dissipate vibrational energy with damping system. The unique nonlinear performance makes QZS property leads to prominent effects in low frequency vibration isolation, vibration protection, and vibration source trapping in engineering applications, such as high precision manufacturing, high precision measurement, MEMS protection, micro-vibration control, energy harvesting, mechanical actuators, etc. [3–12]. The intuitive design of a QZS mechanism was composed by one part of negative stiffness and one counterpart of positive stiffness, where the positive stiffness contributed to support the working payload and the key ingredient of negative stiffness compensated positive stiffness at certain payload generate QZS area with low vibration transmissivity. The classical schema of QZS mechanisms was built on three linear springs with one spring providing positive stiffness in the movement direction and two oblique springs serving for negative stiffness [2, 3]. Following this trend, the state-of-the-art QZS mechanisms included springs-truss structures [13–16], cam-rollers [17–19], buckling beams [20–23], and active magnetic control [24, 25]. Indeed, QZS mechanisms with either passive or active control demonstrated the prominent vibration isolation and reduction performance. However, the structure scale of proposed mechanisms was arranged from dozen centimeters to meters, the mechanism proper weights were considerably large with the rigid structure, and active control system raised difficulties in nonlinear control and auxiliary system. Acquiring QZS property into a small and light weight scale was therefore investigated in the context of mechanical metamaterials. The current solutions to QZS mechanical metamaterials involved using bistable buckling unit stackings [26–30], active magnetic control array [25], origami-based hinge structure [31–33]. These kinds of QZS metamaterial designs were more or less extended from the tradition QZS mechanism, where some engineering deficiencies also remained. In planar configuration, the bow-like buckled beam can be snapped to a symmetric buckling position with a perpendicular disturbance, namely snap-though behavior. By taking the advantage of snap-through behavior with elastic instability, mechanics-wise QZS unit was conceived and a stack of many QZS units performed larger range of QZS property in the bulky level [26, 27]. However, the design with snap-through behavior cannot avoid local vibration when the buckled beam snapped, and simply stacking snap-through units cannot systematically determine which snap-through behavior happened first or together. This issue in snap-through based QZS metamaterial resulted in a zig-zag profile in the stiffness curve, which is not a perfect solution for vibration control [28]. To avoid the sudden stiffness jumps, curved slender beams were considered to replace the pre-buckling structure, where the jumping phenomenon was softened as the structure traveling through the equilibrium point [29]. Furthermore, curved beams with positive stiffness were also introduced to encore reduce

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the jumping in snap-through [30]. Nevertheless, the exact pair of positive-negative stiffness required delicate accommodation on the designing structure. Since the nonlinearity of buckling structure is hard to be predicted and controlled in the exact engineering working scenario, designing such a buckling structure for QZS property relied on trialerrors. Other research tried to avoid this difficulty by adding the active magnetic control device in the lattice structure [25]. The magnetic coil array was mounted in between the lattice structure and controlled to alter the structure stiffness by increasing or releasing the connection in the structure. Although the active magnetic controlled meta-structure increased the possibilities in reshaping the structure stiffness through control system, the weight of coil and peripheric control system limited the flexibility in engineering application. In the nonlinear deformation metamaterials, origami structure opened another thread of exploring QZS meta-structures. Originally, origami was prominent in exhibit high foldability with nonlinear Poisson’s ratio, where this implied nonlinear deformation in the structure. Such a nonlinear folding fashion also influenced the structure’s stiffness response where the joint stiffness took important role [31]. The 3D triclinic origami unit with three stable morphism structures was stacked into a 2D array to perform simultaneously quasi-zero Poisson’s ratio and QZS in stiffness around initial state [32]. Furthermore, a curved origami structure was investigated to consider not only joint stiffness, but also facet bending stiffness, where the most folded joints induced the most deformed facets in the origami structure [33]. This alternative mechanism strengthened the weak stiffness of deformed joints. However, the overall structure stiffness was still limited due to the folding induced fatigue and the selection on materials being soft in folding stiffness, yet strong in bending or torsion stiffness. In this research article, a new design paradigm on elastic meta-structure for QZS performance under loading is proposed to integrate the nonlinear QZS structure from X-mechanism into one elastic device and demonstrate the viable solution for advance vibration isolation in small scale meta-structure. This proposed paradigm, specifically referring to an elastic QZS isolator, inherits the nonlinear QZS performance from the analytical results of X-mechanism. In a straightforward understanding, the proposed structure replaces rigidity of X-mechanism and reduces the structural complexity by harnessing the elastic deformation. Meanwhile, the design principle comes with interpretable designing parameters in the structural design for acquiring different levels of QZS. Therefore, it provides a clear thread of QZS structural design without trial and errors. Also, the proposed meta-structural elastic isolator functions in a passive manner, adapting to a wide scope in engineering applications. The investigation on the proposed elastic QZS isolator is presented in the following content: In Sect. 2, the elastic QZS isolator is presented with the inspiration from X-mechanism and the interpretation on the elastic components. Section 3 contains finite element analysis (FEA) on elastic isolator for exploring designable parameters. Designable QZS property is predicted with the designing parameters. In Sect. 4, the samples of elastic QZS isolators printed with 3D printer are presented in a small size of a coin. Experimental results of static loading test are presented to further illustrate the effectiveness of proposed meta-structure on elastic isolator. Conclusion on the theoretical designing process, FEA simulations, and experimental loading tests of elastic isolator is drawn in the final section. References are provided accordingly by the end.

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2 Design Principle Structural designs of vibration isolation mechanisms with QZS property have been studied as X-mechanisms in previous research [4, 8, 11, 13, 14]. As one of the stateof-the-art QZS mechanisms, X-mechanism involves leveraging nonlinear mechanical structure of beam-joint mechanisms with linear springs, where the springs’ elongation is controlled in a nonlinear fashion. Therefore, X-mechanism can perform a softening stiffness response with QZS property for vibration isolation. The inspiring designs of X-mechanism enlightens the way to design QZS soft isolator possessing high efficiency vibration isolation with one entire design. The discovery procedure of proposed QZS soft isolator structure is shown in following figures:

Fig. 1. Designing process from X-mechanism to soft QZS isolator: (a) Basic X-mechanism, (b) Multi-stage X-mechanism variants, (c) Soft QZS isolator inspired from multi-stage X-mechanism, (d) Structure elements of soft QZS isolator.

In the design of X-mechanisms as shown in Fig. 1(a) and (b), the loading direction is in the vertical direction while springs are connected to the joints moving in the horizontal direction. The X-mechanism does not only change the moving direction of joints, but also, develop a nonlinear relationship between loading displacement and horizontal elongation of springs. This nonlinear relationship is described with trigonometric functions governing the softening process of springs. Stacking X-mechanisms with different spring stiffness ki (i = 1, 2, 3, …, 10) in Fig. 1(b), the mechanism can perform multi-stage QZS in loading process for vibration isolator. In the ultimate fashion as keeping increasing the joint number, two limbs of multi-stage X-mechanism become infinite degree-offreedom mechanical arms without considering the springs. Naturally, the mechanical arm leads to a continuum arc beam like structure, yet placing springs to connect soft limbs is practically impossible. Herein, the design of elastic QZS isolator is proposed

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in Fig. 1(c) to solve this problem by both inheriting a continuous degree-of-freedom of X-mechanism and integrating the spring resilience into one piece. The entire structure of elastic QZS isolator is given in Fig. 1(c) with both outside structure view and section view. In the section view, the lateral shape of soft isolator is conformed from the infinite mechanical beam with itself positioned circularly in 3D space. In Fig. 1(d), the soft isolator is decomposed into elements, with vertical element and horizontal element. The vertical element works as the pillar of continuum structure, and the horizontal element works as the horizontal springs placed in the multi-stage Xmechanisms. Interestingly, the intersection element of vertical and horizontal elements performs as a flexible joint which follows both the bending movement and elongation of vertical and horizontal elements, respectively. The stiffness softening process harnessed by both vertical and horizontal elements through this flexible joint can be simulated with FEA models as follows:

Fig. 2. FEA simulation on illustration of synergistic effect of vertical and horizontal elements: (a) Compression simulation on simple vertical element with stiffness softening and stress concentration, (b) Compression simulation on elastic isolator with predictable QZS and stress curation.

Before getting into the simulation, some designable parameters of elastic QZS isolator are required to be clarified. In Fig. 2(a), four basic parameters of isolator are defined by Rin denoting for the inner radius, d denoting for thickness, l denoting for effective working length, and i denoting for the bottleneck radius. The outer radius Rout is therefore given by (Rin + d) in the basic concept. The horizontal element inner radius Rhin and outer radius Rhout can be obtained as: Rhin = Rin · cos θ,

(1)

Rhout = Rout · cos θ,

(2)

with θ ∈ [arctan(l/2Rin ), − arctan(l/2Rin )]. In the loading simulation performed by Abaqus, the first loading scenario shown in Fig. 2(a) considers a simple arc beam representing vertical element without horizontal element. The simple arc beam is bended as expected as a buckling beam with stiffness

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softening in the vertical direction. However, the bending moment maximizes in the middle of the vertical element, which it is the reason for simple bending structure failure in the general engineering case. In the concept of X-mechanism guided designing paradigm, the flexible joint element located in the middle of curved vertical element gains the maximal rotation, whereas other elements have not yet been effectively touched. Therefore, the synergistic effect of vertical and horizontal elements is introduced to balance this unfavorable effect in the softening process. In Fig. 2(b), the compression result on a semi-structure of elastic QZS isolator is presented. The deformation of horizontal element strengthens the stress concentrated middle segment of a single vertical element, where this resilience force F h can be analytical computed as: h F h = 2EA · (Rh∗ in − Rin ) cos θ

(3)

with E being the Young’s Modulus of elastic material, A being the section area of horizontal element, Rhin and Rh∗ in being inner radius of prior and posterior deformation, respectively. From Eq. (3), taking a moment of θ , F h is performing the same as a spring with stiffness given by (2EA · cos θ ). The result of incorporating F h from horizontal element shown in Fig. 2(b) further strengthens the loading capacity of elastic isolator in early deformation, while the stiffness softening process is not damaged following the X-mechanism like motion. From the stress distribution diagram, horizontal element also contributes to averaging the stress into the entire structure of elastic isolator, avoiding unfavorable stress concentration in the middle segment of vertical direction. However, as approaching to the QZS point of elastic isolator, the equilibrium point of elasticity induces singularity problem for FEM to solve. To qualify the QZS zone for different parameter sets of elastic isolators, compression experiments on 3D printed elastic isolator models will be presented in the following section.

3 Loading Simulations In this section, loading simulations on elastic isolators with different inner radius Rin are firstly presented to guide the 3D fabrication on actual isolator models. Other parameters The designable parameters presented in Sect. 2 including {l, r, d} are currently fixed for exploring the loading performance of elastic isolator relating to Rin . For better guiding the 3D printing model, the considered FEA simulation uses the material property of Elastic 50A resin provided by Formlabs, possessing working Young’s Modulus of 0.94 MPa and Poisson’s Ratio of 0.3. In Fig. 3, the FEA simulation results are calculated on the series of elastic QZS isolators with the variation of inner radius Rin . With other designable parameters defined as l = 20 mm, r = 4 mm, d = 2 mm, the inner radius Rin is chosen in {12 mm, 25 mm, 30 mm, 35 mm, 40 mm} for carefully looking into the influence of inner radius on the loading capacity at QZS performance range. In Fig. 3(a), the reaction force of proposed meta-structure in elastic QZS isolators during loading is given respectively. To standardize the results, the curves are plotted with respect to deformation ratio ε = l l. The simulated loading curves of elastic QZS isolators illustrate the trend to reach QZS zone respectively. Further, from the smallest inner radius to the largest, the QZS levels at maximum loading capacity increase, which ultimately determine the working

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Fig. 3. FEA simulation results on elastic QZS isolators with Rin = 12 mm, 25 mm, 30 mm, 35 mm, 40 mm. Other designable parameters are given as l = 20 mm, r = 4 mm, d = 2 mm: (a) Loading capacity diagram predicted with QZS performance; (b) Stiffness softening curves computed from the loading capacity – deformation ratio results; (c) Maximum loading capacity in relation to inner radius Rin .

performance in supportable dynamic loading in vibration isolation. In Fig. 3(b), the stiffness according to each elastic isolators are computed based on the deformation ratio. The stiffness curves present a clear process of stiffness softening leading to QZS property. The QZS property is reached between deformation ratio from 20% to 25%. For large inner radius samples, the stiffness curve in small deformation ratio maintains almost constant, then drop to quasi-zero stage starting from 10% deformation ratio. The ultimate loading capacity at QZS levels is given in Fig. 3(c), where a larger inner radius led to a large loading capacity. The simulated results are interpolated by a second order trend given by −0.0077R2in + 0.7715Rin − 3.0364. This result will be compared to experimental loading test further.

4 Experimental Loading Performance The proposed design paradigm of elastic isolator has been illustrated both in design principles and numerical simulations with FEA results, the actual performance of such a design in engineering application is crucial to further proof of proposed structure. Here, the test models of elastic isolators are fabricated using Elastic 50A resin in a small dimension with 3D printer. The printed models have inner radius of Rin = 12 mm, 25 mm, 40 mm, respectively. Other designable parameters are controlled as l = 20 mm, r = 4 mm, d = 2 mm. This series of tiny elastic isolators will not only prove the effectiveness of reaching QZS performance under loading, but also demonstrate that the proposed design paradigm allows QZS isolator device to be practically miniaturized into tiny structure and integrated in one piece. The printed elastic isolators are shown in Fig. 4. The printed elastic isolators are tested with the compression mode of the tensile stress testbed provided by INSTRON. The mounted compressor has a maximum loading of 1000 kg, and the load cell sensor has a maximum measurement range of 500N with sensitivity of 0.5N. The actual tensile stress machine and the test tool is shown as (Fig. 5): The compression test was conducted with each model 3 repetitive times to make sure the test results consistently converging and eliminating possible test errors. The

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Fig. 4. 3D printed elastic isolators with Elastic 50A resin, compared to a coin.

Fig. 5. Compression testbed – INSTRON E1000 with compression test tool setup.

compression test results on models of Rin = 12 mm, 20 mm, 40 mm, are given as follows:

Fig. 6. Compression test result on elastic isolator samples with Rin = 12 mm, 25 mm, 40 mm: (a) Diagram of reaction force verse deformation ratio is given, where QZS zone is highlighted in blue, unstable deformation is highlighted in orange, and pure compression stage with proper material is highlighted in yellow. (b) Maximum loading capacity summary for comparison of experimental results to FEA results, with error estimation of 25% at 12 mm, 10% at 25 mm, and 14% at 40 mm.

In the compression test, three models were showing different levels of QZS loading threshold in Fig. 6. In Fig. 6(a), The stiffness softening process before QZS point was conformed with the prediction in FEA analysis. In Fig. 6(b), the loading thresholds were conformed with the FEA numerical simulation results with an averaging error of 16.3%. Among three different levels of QZS loading thresholds, the thresholds of elastic isolators with Rin = 25 mm, 40 mm appeared as predicted in FEA models, yet the threshold of elastic isolator with Rin = 12 mm appeared at deformation ratio around 20%, which is slightly earlier than the numerical result. This difference can happen with the actual nonlinear stiffness of Elastic 50 A resin, where the stiffness at low strain rate was 30%

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less than the stiffness used in the FEA simulations. Furthermore, elastic isolator with Rin = 12 mm shown in Fig. 4 had the geometry with bigger belly, mathematically larger curvature. As the section view from FEA result in Fig. 2(b) showing the compression leading to the inflation in the middle of the elastic isolator, the larger curvature elastic isolator can enter QZS zone much earlier than smaller one, which can be experimentally observed here. As the loading process proceeding further, unstable deformation area appeared in both elastic isolators of Rin = 25 mm, 40 mm. Physically, the loading process contributed to expand the horizontal element of elastic isolators. Therefore, for elastic isolators with larger Rin or smaller inner curvature, the restoring energy in horizontal element of elastic isolator will be released as the deformation process passing through QZS point, resulting in this unstable area. This also experimentally explained the difficulty of convergence in FEA models after QZS point. After this short unstable range, all elastic isolators got compressed further into quasiparallel positive loading curves. In this stage, the structural deformation of elastic isolators reached to the limit state where bending in vertical element and expansion in horizontal element both ended. The pure compression behavior led the loading curve with material’s proper deformation. It is physically reasonable that elastic isolators with smaller Rin or bigger inner curvature reached this stage earlier. The pure compression stage with positive stiffness is essential in QZS device for stopping the backtrack of loading stiffness and assisting in loading heavier loads than expected. In engineering applications, the unstable area and the pure compression stage can create a loading trap that statically supports the loads, meanwhile, it helps to control the dynamical vibration behavior into the QZS zone. This is favorable for engineering applications.

5 Conclusion In this research, an X-mechanism inspired new design paradigm is proposed for elastic QZS device, more specifically for elastic QZS isolator. The design paradigm leads to integrating the beneficial nonlinearity with QZS property with beam-joint centered mechanisms into one piece of tiny dimension. The proposed elastic QZS isolator following this paradigm illustrated structural QZS behavior by leveraging the synergistic mechanism of vertical buckling beam and horizontal ring element. The FEA numerical results on elastic isolators predicted the effectiveness of such a synergistic mechanism. Tailoring the inner radius of elastic isolators resulted in modifying the loading capacity and QZS threshold in the proposed design. To further understand the behavior of elastic QZS isolators during and after QZS zone, experimental studies were conducted on elastic isolators with different inner radius. The elastic QZS isolators were 3D printed into small size comparable to a coin. The expected QZS zone appeared as predicted in FEA simulations. Compression test also revealed realistic response of elastic isolator after QZS zone, discovering stiffness trap developed with unstable deformation and pure compression stage. This can be practically favorable in engineering applications to ensure that the vibration isolation is always around QZS zone. Future exploration of such a design paradigm will be expected in:

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Parameter exploration in proposed design paradigm to enlarge QZS zone. Miniaturization of current structure for repetitive pattern of bulky materials. Exploration on dynamical behaviors of periodic vibration and shocking. Exploration on multi-stage QZS behaviors with proposed design paradigm.

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Design and Vibration Control of Secondary Suspension for Maglev Train Based on Magnetorheological Fluid Damper Yougang Sun1,2(B) , Dandan Zhang1 , Hongyu Ou1 , Guobin Lin2 , and Haiyan Qiang3 1 Institute of Rail Transit, Tongji University, Shanghai 201804, China

[email protected]

2 National Maglev Transportation Engineering R&D Center, Tongji University,

Shanghai 201804, China 3 Shanghai Maritime University, Shanghai 201306, China

Abstract. When the maglev train with a speed of 600 km/h runs at high speeds, the vertical carriage vibrations caused by irregularity of tracks are intensified, affecting passenger comfort and even safety. Therefore, it is necessary to find new damping devices to reduce carriage vibrations. Magnetorheological fluid dampers have advantages such as continuously adjustable damping, low-power consumption, high damping force output, and fast response speed. This paper firstly discusses the feasibility of applying magnetorheological fluid dampers to the secondary suspension system of maglev vehicles. It investigates the working mode, installation position, and stroke of magnetorheological fluid dampers for high-speed maglev trains. Based on a simplified suspension model for maglev trains, a magnetorheological fluid damper control system based on fuzzy logic is designed. The effectiveness of the proposed semi-active secondary suspension system based on magnetorheological fluid dampers is verified through numerical simulations, demonstrating its ability to effectively improve the vibration reduction performance and ride comfort of maglev trains. Keywords: Maglev train · Magnetorheological fluid damper · Secondary suspension · Fuzzy system · Vibration control

1 Introduction Maglev trains are a new type of rail transit that use maglev force to suspend on the track. They offer several advantages over traditional rail systems, including non-contact operation, high speed, low noise, no pollution, and energy saving [1–4]. Currently, the high-speed maglev transportation system is one of the fastest land transportation tools in the world, with a top speed of up to 600 km/h, which can fill the speed gap between highspeed railway and air transportation. However, track irregularities can lead to increased vertical vibration, which can affect the smoothness and ride comfort of maglev trains. To address this issue, a new vibration damping device can be added to the suspension of the secondary suspension of maglev train to restrain train vibration and improve running quality. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 711–724, 2024. https://doi.org/10.1007/978-981-97-0554-2_54

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At present, passive secondary suspension based on air spring is widely used in highspeed maglev trains. However, the stiffness and damping of this type of suspension are not adjustable, which makes it difficult to guarantee optimal vibration reduction effects under different speeds and road conditions [4, 5]. Magnetorheological (MR) dampers are an advanced application of semi-active devices that provide adjustable damping forces using MR fluid [6–8]. One of the main advantages of MR dampers is their controllable damping capacity. By applying a magnetic field to a magnetorheological fluid, the viscosity and stiffness of the fluid can be rapidly changed, allowing for real-time adjustment of the damping force. This enables precise control of the suspension system, improving ride stability and comfort. Another advantage of MR dampers is their ability to provide both passive and semi-active damping characteristics. In passive mode, the damping force is based on the inherent viscosity of the fluid, providing reliable and consistent damping capacity. In semi-active mode, the damping capacity can be actively adjusted to adapt to different road or load conditions, enhancing the dynamics performance of the suspension. Many advancements have been made in the design and optimization of MR fluid dampers for transportation. Lord Company in the United States has been using MR liquid since an early stage and has achieved significant progress in the fields of MR liquid dampers, clutches, and braking. They have developed a series of MR liquid dampers and braking devices that exhibit excellent performance [9, 10]. The University of Nevada developed a semi-active suspension system with MR dampers for the US Army [11]. The automotive semi-active suspension system, Magneride, jointly developed by Delphi and Lord, won the “Global Top 100 Innovations” award. This system can significantly enhance vehicle ride comfort and handling stability [12]. Nguyen and Choi optimized the MR damper design of the vehicle suspension system through finite element analysis [13]. In the design optimization parameters, damping force, dynamic range, and valve size are considered. A linear single-tube MR damper is used as an example of a vehicle suspension system. Mangal and Kumar developed an optimized MR damper model by considering various geometric parameters and utilizing Taguchi coupled finite element method and simulation software ANSYS [14]. Gavin et al. optimized the design of MR Damper based on minimum power consumption and minimum induction time constant, considering multiple parameter conditions such as force capacity, electrical characteristics, and device size [15]. Ferdaus et al. proposed an optimal design for a single-tube linear MR Damper by considering various damper configurations [16]. The semi-active MR suspension system can adjust the damping coefficient based on the track irregularity, allowing for better ride comfort and handling stability. This is particularly important for the secondary suspension of maglev trains to do research in MR damper, which can significantly enhance the running quality of 600 km/h trains. In this paper, we first introduce the secondary suspension structure of the high-speed maglev train, followed by an analysis of its vibration characteristics. We then determine the working mode, installation position and stroke of the dampers. Finally, a control method for MR fluid dampers based on fuzzy logic is proposed, and its effectiveness is verified through numerical simulation.

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2 Structure of the Secondary Suspension of High-Speed Maglev Train The high-speed maglev system with common conductor mainly consists of a levitation frame, secondary suspension, vehicle body, etc. The levitation frame is composed of a frame, primary suspension, and electromagnets. The framework is connected to the levitation electromagnet and guide electromagnet through a primary suspension system. The secondary suspension is composed of air spring, bolster and boom, and each levitation frame is connected to the car body through the air spring. Figure 1 shows the high-speed maglev system with common conductor. AB is the bottom of the car body,

(a) Schematic diagram of the secondary suspension structure

(b) Parameterized model of secondary suspension Fig. 1. Secondary suspension of high-speed maglev train

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BCD is the bottom of the car body connecting piece, and the B end is fixed on the car body. DE is a boom. One end of the boom is connected to the car body and the other end is connected to the bolster. The bolster is connected to the levitation frame via the G spot. Point F is the connection point between the bolster and the levitation frame. FI is an empty spring, and point I is connected to the empty spring support of the levitation frame. If only the vertical vibration of the train is considered, the levitation force is linearly simplified as spring force and damping force, and the suspension is decoupled for analysis. The fluctuation of the levitation gap is simplified as excitation input. The train is composed of the vehicle body and suspension frame, and the simplified model is shown in Fig. 2.

Fig. 2. The diagram of simplified suspension model

The mass m2 of vehicle body is 1875 kg and the mass m1 of suspension frame is 275 kg. There are dampers and springs between the suspension frame and the vehicle body. The spring stiffness k2 is 200000 N/m. The spring stiffness k1 is 2000000 N/m. And the damping coefficient of the first series is 2000 Ns /m. The differential equation of motion is {

m2 x2 = −κ2 (x2 − x1 ) − c2 (x˙2 − x˙1 ) m1 x¨ 1 = k2 (x2 − x1 ) + c(˙x2 − x˙ 1 ) − k1 (x1 − x0 ) − c1 (x˙1 − x˙0 )

(1)

3 Vibration Characteristics Analysis of Secondary Suspension 3.1 Analysis of Vibration Transmissibility Change the secondary damping coefficient and observe the impact on the vibration transmission of the car body and suspension. Change the secondary damping coefficient to 500 N·s/m, 1000 N·s/m, 2000 N·s/m, 5000 N·s/m, 10000N·s/m, and 20000N·s/m and apply a sweeping frequency excitation with a constant amplitude of 0–100 Hz, then the vibration transmission curve was observed from the vehicle body to the foundation as shown in the Figs. 3 and 4.

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Fig. 3. The diagram of vehicle vibration transmission rate

Fig. 4. The diagram of levitation vibration transmission rate

3.2 Selection of Working Mode and Structural Form of MR Damper The vehicle body has two resonance points, one around 1 Hz and the other around 11 Hz. The vehicle body and bogie near 1 Hz in the first resonance zone significantly reduce the vibration transmission rate with increased damping, indicating that increased damping can effectively reduce vibration in the resonance zone. And the resonance point moves to the right continuously with the increase of damping. When the secondary damping coefficient reaches 20000 N s/m, the body is equivalent to being connected with the

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bogie, and the whole system degenerates into single degree of freedom vibration, with only one Formant. In the frequency band near 11 Hz in the second resonance zone, the vibration of the vehicle body and bogie decreases with increasing damping. If only the effect of vibration reduction in the resonance zone is considered, a larger damping should be chosen. The vibration transmission rate in the frequency range of 1–11 Hz between two resonance zones is not significantly affected by changes in damping, but the vibration transmission rate of the suspension frame increases with increasing damping, reducing the smoothness of the train. Excessive damping in the high-frequency range can also increase the vibration of the vehicle body, affecting comfort. In summary, the application of magnetorheological dampers in vibration reduction systems and the use of semi-active algorithms to control vibration can better suppress broadband random vibration of trains while achieving the goal of reducing the transmission rate of vehicle body and the vibration of suspension frame.

4 Bouc-Wen Model of Magnetorheological Damper Bouc-wen model is a widely used magnetorheological damper model, which can accurately reflect the nonlinear characteristics of magnetorheological damper. In this paper, Bouc-Wen model is used to simulate the mechanical behavior of the damper. The model diagram can be descripted in Fig. 5.

Fig. 5. Bouc-Wen model simplified model diagram

The calculation formula is as follows: F = c˙x + k(x − x0 ) + αz z˙ = −γ | x˙ | z(z)n−1 − β x˙ z n + A˙x

(2)

In the formula, F is the damping force, x, x˙ is the displacement and velocity of the piston respectively, c is the viscous damping coefficient, k is the stiffness coefficient, a is the coefficient of the hysteresis operator. A, γ and β are the proportional coefficients of the hysteresis operator, which describes the smoothness and size of the curve, and n

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is generally 2. The three quantities of k, c and a are greatly affected by the current, and the remaining parameters can be regarded as constants. After parameter identification based on the test data of the magnetorheological fluid damper, the Bouc-Wen simulation model is established in Simulink, as shown in the following Fig. 6:

Fig. 6. Bouc-Wen simulation model

When the input frequency is 6 Hz, the amplitude is 10 mm, and the input current is 0A, 1A, 1.5A and 2A, the model simulation curve is shown in Fig. 7:

5 Fuzzy Control Algorithm of Magnetorheological Damper In practical applications, the parameters of the control system of the MR damper are difficult to be accurately measured and identified due to its nonlinear characteristics. However, the traditional model-based control method has great limitations in solving the problems of system nonlinearity and structural time-varying. In view of the above problems, it is necessary to introduce intelligent control algorithm to make the control system more robust. Fuzzy control is a control algorithm that uses fuzzy logic to model uncertain complex systems. Different from the traditional control algorithms that rely on mathematical models, fuzzy control uses language rules and expert knowledge to control the system, which is suitable for the control of magnetorheological dampers. The core of fuzzy control is the design of the fuzzy controller, which follows the following process: (1) Determine the input variables and output variables. Based on the control characteristics and control properties of the maglev train structure, the vehicle acceleration,

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(a) Damping force-displacement relationship diagram

(b)

Damping force-velocity relationship diagram Fig. 7. Bouc-Wen simulation curve

relative velocity between the body and suspension are selected as input variables, and the output variable is the control current of the damper. The fuzzy language sets for the controller’s input and output variables are as follows: x¨ 2 = {NL, NM , NS, ZO, PS, PM , PL}

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x˙ 2 − x˙ 1 = {NL, NM , NS, ZO, PS, PM , PL} I = {ZO, VS, RS, S, M , L, RL, VL} Gaussian type is used for the membership functions of the fuzzy language variables for both input and output. (2) Design fuzzy control rules. The basic objective of rule design is that when the body and relative velocity are greater than 0, the suspension is in the extension stroke, and the damper should output a larger damping force. When the relative velocity is less than 0, the suspension is in the compression stroke, and the damper outputs a smaller damping force. In these two cases, the control current is determined based on the positive or negative value and the magnitude of the acceleration. The control rules table is shown in Table 1, and the surface relationship diagram is shown in Fig. 8. Table 1. Fuzzy Rules Table x˙ 2 − x˙ 1

I

NL x¨ 2

NM

NS

ZO

PS

PM

PL

NL

VS

RS

S

M

ZO

ZO

ZO

NM

ZO

RS

S

M

ZO

ZO

ZO

NS

ZO

VS

RL

ZO

ZO

ZO

ZO

ZO

ZO

ZO

VS

ZO

S

ZO

ZO

PS

ZO

ZO

VS

VS

RL

M

S

PM

ZO

ZO

ZO

S

RL

L

M

PL

ZO

ZO

ZO

RL

VL

RL

L

(3) By using the membership degree of fuzzy sets and the operation of fuzzy logic, the input is fuzzified. Apply fuzzy inference rules to derive the fuzzy output result based on the fuzzy variables. This mapping process can effectively handle incomplete or uncertain information. The fuzzy controller designed in this paper adopts the Mamdani inference method to obtain the output fuzzy set, and uses the centroid method for defuzzification. 5.1 Analysis of Fuzzy Control Algorithm Simulation Results The acceleration and relative speed of the suspension are taken as inputs to the fuzzy controller, and the damper current is output through the fuzzy controller. The damping force is obtained from the MR damper simulation model and input into the suspension model. The fuzzy control algorithm simulation model is shown in Fig. 9:

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Fig. 8. Input-Output Relationship Surface Graph

Fig. 9. Simulation model of fuzzy control algorithm

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When the system inputs a sine signal, the acceleration response of the vehicle body is shown in Fig. 10:

Fig. 10. Sinusoidal excitation vehicle acceleration response

The maximum acceleration of the uncontrolled passive suspension vehicle body is 0.30 m/s2 , and the root mean square value of the acceleration is 0.179 m/s2 . The maximum acceleration of semi Active suspension based on fuzzy control is 0.193 m/s2 , the peak acceleration is reduced by 35.5%, the root mean square value is 1.99 m/s2 , and the root mean square value is reduced by 33.6%. From the simulation results, it can be seen that the designed fuzzy control algorithm can improve the sinusoidal excitation control signal and meet the vibration reduction requirements. When the system inputs a random signal from the track, the acceleration response of the vehicle body is shown in Fig. 11. The peak acceleration of the uncontrolled passive suspension vehicle body is 0.305 m/s2 , and the root mean square value is 0.112 m/s2 . The peak acceleration of the semi Active suspension based on fuzzy control is 0.198 m/s2 , the peak acceleration is reduced by 35.02%, and the root mean square value of the acceleration is 0.078 m/s2 , which is reduced by 30.01%. From the simulation results, it can be seen that the designed fuzzy control algorithm can improve the sinusoidal excitation control signal and meet the vibration reduction requirements. 5.2 Comparison of Simulation Results In the GB/T 5599-2019 national standard, the Sperling stability index method is specified for the quantitative evaluation of the operational quality of railway vehicles. The

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Fig. 11. Random Excitation Vehicle Acceleration Response

calculation formula for the Sperling index vertical stationarity index is:  √ A3 10 Wz = 3.57 F(f ) = 3.57 A3 B3 f Bs = 0.588[

1.911f 2 + (0.25f 2 )

2

(3)

1/2

] (1 − 0.277f 2 )2 + (1.563f − 0.0368f 3 )2

(4)

In the formula, A is the acceleration (m/s2 ); B is represents the sensitivity of humans to vibration, which is a frequency weighting coefficient. Table 2. Comparison of Effects of Different Control Algorithms Maximum accelerationa 2 max (m/s )

Root mean square Vertical Sperling accelerationa rms (m/s2 ) index

Passive suspension

0.30

0.179

2.61

Fuzzy control

0.198

0.078

2.04

Sky-Hook Damper Control

0.229

0.141

2.27

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According to the simulated vehicle acceleration, the Sperling index is calculated. Table 2 lists the peak acceleration, root mean square acceleration, and vertical Sperling index of two different control algorithms. It can be seen that the fuzzy control algorithm has better control effect.

6 Conclusion This paper analyzes the structural characteristics and vibration properties of the secondary suspension system for maglev trains, pointing out the problems that need to be addressed in the vibration reduction design. It demonstrates that magnetorheological dampers are a feasible engineering solution. Based on the structure of the secondary suspension system, suitable operating modes and structures are chosen, and the installation position and stroke determination method for the dampers are specified. The mechanical characteristics of the dampers are simulated by establishing a Bouc-Wen mechanical model based on the parameters of the magnetorheological fluid damper. A fuzzy controller is designed for vibration control simulation, and the results show that magnetorheological fluid dampers can effectively improve the vibration of the maglev train carriage. Our future work will focus on setting up an experimental platform and conducting vibration reduction experiments to validate the proposed solution.

References 1. Li, F., Sun, Y., Xu, J., He, Z., Lin, G.: Control methods for levitation system of EMS-type maglev vehicles: an overview. Energies 16, 2995 (2023) 2. Prasad, N., Jain, S., Gupta, S.: Electrical components of maglev systems: emerging trends. Urban Rail Transit 5, 67–79 (2019) 3. Sun, Y., He, Z., Xu, J., et al.: Dynamic analysis and vibration control for a maglev vehicleguideway coupling system with experimental verification. Mech. Syst. Signal Process. 188, 109954 (2023) 4. Sun, Y., Gao, D., He, Z., Qiang, H.: Influence of electromagnet-rail coupling on vertical dynamics of EMS maglev trains. Mechatronics Intell. Transp. Syst. 1, 2–11 (2022) 5. Yang, P., Sun, Y., Luo, Y., Hongyu, O.: Stability simulation and analysis of maglev vehicle at different speed based on UM. Mechatronics Intell. Transp. Syst. 2, 42–52 (2023) 6. Sun, S., Yang, J., Yildirim, T., et al.: A magnetorheological elastomer rail damper for wideband attenuation of rail noise and vibration. J. Intell. Mater. Syst. Struct.Intell. Mater. Syst. Struct. 31(2), 220–228 (2020) 7. Yang, J., Sun, S., Ezani, S., et al.: New magnetorheological engine mount with controllable stiffness characteristics towards improved driving stability and ride comfort. Smart Mater. Struct. 31(12), 125009 (2022) 8. Chen, Z., Sun, S., Deng, L., et al.: Investigation of a new metamaterial magnetorheological elastomer isolator with tunable vibration bandgaps. Mech. Syst. Signal Process. 170, 108806 (2022) 9. Weiss, K.D., Carlson, J.D., Nixon, D.A.: Viscoelastic properties of magneto- and electrorheological fluids. J. Intell. Mater. Syst. Struct.Intell. Mater. Syst. Struct. 5(11), 772–775 (1994) 10. Carlson, J.D., Catanzarite, D.M., Clair, K.A.: Commercial magnetorheological fluid devices. Int. J. Mod. Phys. B 10, 2857–2865 (1996)

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11. Dogruer, U.: Design and Development of a Magneto-rheological Fluid Damper for a High Mobility Multi-purpose Wheeled Vehicle (HMMWV). University of Nevada. Reno (2003) 12. Lord Corporation Rheologic Linear Damper. Rd-1001/Rd-1004, product. Information Sheet, Lord corp. Pub: (1997) 13. Nguyen, Q.H., Choi, S.B.: Optimal design of a vehicle magnetorheological damper considering the damping force and dynamic range. Smart Mater. Struct. 18(1), 015013 (2008) 14. Mangal, S.K., Kumar, A.: Geometric parameter optimization of magneto-rheological damp-er using design of experiment technique. Int. J. Mech. Mater. Eng. 10, 1–9 (2015) 15. Gavin, H., Hoagg, J., Dobossy, M.: Optimal design of MR dampers. In: Proceedings of the USJapan Workshop on Smart Structures for Improved Seismic Performance in Urban Regions. Seattle, WA, 2001, 14, pp. 225–236 (2001) 16. Ferdaus, M.M., Rashid, M.M., Hasan, M.H., et al.: Optimal design of Magneto-Rheological damper comparing different configurations by finite element analysis. J. Mech. Sci. Technol. 28, 3667–3677 (2014)

Nonlinear Dynamic Analysis of Flywheel Rotor Systems with Multiple Fit Clearances Qinkai Han(B) , Zhaoye Qin, and Fulei Chu Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China [email protected]

Abstract. Dynamic analysis of a flywheel rotor system with multiple bearing clearances is conducted in this study. Firstly, Jones-Harris method is used to deduce the equivalent nonlinear support stiffness of angular contact high-speed ball bearing with preload considered. Ignoring the structural deformation, the contact model of inner and outer rings with clearance is used to describe the clearance fit between the outer ring and the sleeve, and between the sleeve and the bearing pedestal. Based on Hertz contact theory, the discontinuous and nonlinear support stiffness is then obtained. The lumped parameter model of rotor-bearing-pedestal system is established and verified based on dynamic test results. Based on this, the influence of clearance fit parameters and external excitation parameters on the vibration response amplitude of the system is analyzed. When the radial displacement is greater than the fit clearance, the radial support stiffness changes abruptly and increases rapidly. If it increases to a certain value, the increase of radial stiffness would slow down. The vibration response curve appears sub-harmonic resonances and amplitude jump. The increase of the fit clearance aggravates the vibration response level, the resonance speed of the system moves to the low speed, and the jumping phenomenon becomes more prominent, which is not conducive to the stable operation of the rotor system. Keywords: Flywheel rotor system · Dynamic analysis · Fit clearance

1 Introduction In order to facilitate assembly and reduce the adverse effects of working state changes (such as temperature rise) on the operation of the rotating machinery, it is an effective way to design reasonable clearance in the bearing structure [1]. However, the bearing fit clearance brings discontinuous characteristics of bearing stiffness, which might have a negative impact on the dynamic characteristics of high-speed flexible rotor. In order to reasonably design the mating clearance and ensure the stable and reliable operation of the high-speed flexible rotor, it is necessary to study the mechanical behaviors of the fit clearance and its influence on the dynamic characteristics of the flexible rotor. The internal radial clearance of rolling bearings is one of the most concerned fit clearances in the existing literature. Many scholars have carried out nonlinear dynamic analysis [2], and discussed the instability of the rotor system caused by the bearing radial © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 725–732, 2024. https://doi.org/10.1007/978-981-97-0554-2_55

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clearance [3, 4]. It was found that under certain conditions, chaotic bifurcation and other complicated nonlinear phenomena would also occur [5, 6]. In addition, bearing pedestal is generally connected to the ground through bolts. Under severe working conditions, the bolts may be loose, resulting in a clearance between the pedestal and the ground, also known as the pedestal looseness. Many scholars have also conducted rotor dynamic studies considering the pedestal looseness. The coupling effects of pedestal looseness with other faults, such as rotor imbalance and rotor-stator rubbing, on the nonlinear rotor system were also evaluated. There are also multiple fit clearances between the bearing outer ring and the sleeve, and between the sleeve and the pedestal. The existence of these clearances will also have a significant impact on the dynamic characteristics of the rotor system. Therefore, the nonlinear dynamics of rotor-bearing-pedestal systems with multiple fit clearances is studied in this paper. Firstly, using the Jones-Harris model [7], the equivalent nonlinear support stiffness of an angular contact high-speed ball bearings considering the preload is deduced. Neglecting the structural deformation, the contact of inner and outer rings with clearance is adopted to describe the clearance fit between bearing outer rings and sleeves, and between sleeves and pedestals. Then, based on Hertzian contact theory, discontinuous and nonlinear support stiffness considering fit clearance is derived accordingly. Considering various external excitations, including static/dynamic unbalance excitations and bearing waviness excitations, a lumped parameter model of the rotor-bearing-pedestal system is established. Based on a three-axis dynamic fore test platform, dynamic tests are carried out and compared with the simulation results. Based upon this, the effects of various parameters (including clearance length and unbalance mass) on the amplitude of the system vibration response are analyzed. Finally, some conclusions are summarized.

2 Theoretical Modeling and Analysis The schematic diagram of the high-speed rotor-bearing-pedestal system for this study is shown in Fig. 1. A pair of angular contact ball bearings installed face to face are used to support the rotation of the flywheel rotor. Usually, the inner ring of the bearing and the rotor shaft are interference fit, while the bearing outer ring and the cylindrical sleeve are clearance fit. In addition, the outer surface of the sleeve and the pedestal are also clearance fit, which can effectively avoid the bearing jam caused by the thermal expansion of the rotor. In the following research, a nonlinear support stiffness model of angular contact ball bearing under preload, nonlinear stiffness model caused by the clearance fit between the bearing outer ring and sleeve, and between sleeve and pedestal, will be established in turn. 2.1 Nonlinear Support Stiffness of Rolling Bearings Angular contact bearings usually require preloading. Once preloading is applied, the contact angle will change, and with the increase of preloading, the contact angle will continue to increase. Then the load distribution and support stiffness are changed accordingly. The external forces acting on the bearing inner ring can be obtained through the

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Installation clearances Cylindrical sleeve

Pedestal

Cylindrical sleeve

Installation clearances

Pedestal

Right bearing

Left bearing

Flywheel rotor

End cover

End cover

Pedestal

Pedestal

Fig. 1. Schematic diagram of the high-speed rotor-bearing-pedestal system with multiple fit clearances

above analysis when the axial and radial displacements are given. The local linear support stiffness can be obtained by the force-displacement relationship. 2.2 Transversal Stiffness of Clearance Fit As shown in Fig. 1, there are two sources of fit clearance in the rotor system: (1) fit clearance between the bearing outer ring and cylindrical sleeve, and (2) fit clearance between the cylindrical sleeve and pedestal. The two sources of fit clearance could be simplified as the contact problem of inner and outer cylindrical rings with clearances, as shown in Fig. 2. It is assumed that the structure is rigid and only the contact deformation is considered. (a)

(b) Outer ring

Inner ring

do d i r

1 pd 2

max max

Fig. 2. Transversal deformations considering fit clearance: (a) initial state; (b) after deformation

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Under the action of radial load, the contact deformation at any point of the inner ring can be expressed as 1 δψ = δr cosψ − Pd (1) 2 where δr denotes the radial displacement of inner ring, Pd is the diameter clearance between the inner and outer rings, ψ is the azimuth angle at the starting point of the maximum contact deformation, as shown in Fig. 2. Equation (1) could be rewritten as a function of the maximum contact deformation (δmax )   1 (2) δψ = δmax 1 − (1 − cosψ) 2 where  = 21 (1 − distribution as

Pd 2δr ).

From Eq. (1), one can also obtain the angle range of load ψ1 = a cos(

pd ) 2δr

(3)

If the clearance is ignored, then one has ψl = π2 . According to the Hertizan contact theory, the contact force is determined by the contact deformation  10/9 1 Qψ = Qmax 1 − (1 − cosψ) (4) 2 where Qmax = Kl (δmax )10/9 is the maximum contact force, Kl is the contact stiffness of line-contact. For the steel inner and outer rings, one has Kl = 8.06 × 104 l 8/9 , wherein l is the axial length of the rings. Based on the static force balance condition, it is easy to obtain 10/9   +ψ l  1 1 − (1 − cosψ) cosψdψ (5) Fr = Qψ cosψdψ = Qmax 2 −ψl In general, the radial displacement of the inner ring is obtained by combining the two lateral displacements δx , δy as  δr = δx2 + δy2 (6) And the angle θ with respect to the x-axis is ⎧

⎨ acos δx δy ≥ 0 δr

θ= ⎩ 2π − acos δx δy < 0 δr

(7)

Obviously, the radial force components along the x-axis and y-axis are expressed by Fx = Fr cosθ Fy = Fr sinθ

(8)

Therefore, the external force applied on the inner ring can be obtained by the above analysis when the radial displacement is given. The local linear stiffness can be obtained by the force-deformation relationship.

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2.3 Dynamic Model of the Rotor-Bearing-Pedestal System The flywheel rotor is a complex spoke structure. Ignoring the structural vibration of the rotor itself, the rotor, bearing and pedestal are regarded as concentrated masses, and the rigid-flexible coupling dynamic model is established and shown in Fig. 3. Considering five degrees of freedom of the rotor, including translations along three directions u , θ v ), the equations of (ufw , vfw , wfw ) and angular deflections around Xfw , Yfw axes (θfw fw motion for the flywheel rotor can be derived. ul2

ur2

klx2 clx2 1 pl1 2

ml 2

clx1 1 pl 2 2

l

u l1

u r1

klx1 m l1

X k lbx

k lby

cly1

kly1

ml 2 vl2

kly 2

mr 2

m r1

krx1

cly 2

k rbx

wfw v fw

c lby

m l1

v l1

c rbx

crx1

u fw

c lbx

Z c lbz

u fw

1 pl 2 2

Y

krx2 1 pr1 2 1 pr2 2

rbz

k rbz

v fw

1 pl1 2

crx2

1 pr1 2

m r1 kry1

cry1

mr 2

1 pr2 2

kry2

cry2

v r1

vr 2

Fig. 3. Rigid-flexible coupling dynamic model of the rotor-bearing-pedestal system

Through the above analysis, the vibration equations of the subsystems are obtained. The vibration analysis model of the whole rotor-bearing-pedestal system is obtained by assembling the subsystems’ equations M¨q + (C1 + G)˙q + K1 q = Fe + Fg

(9)

T  u , θv , u , v , u , v , u , v , u , v denotes the vecwhere q = ufw , vfw , wfw , θfw r1 r1 r2 r2 l1 l1 l2 l2 fw tor of degrees of freedom, M, K1 , C1 , G are the mass, stiffness, damping and gyroscopic matrices of the system, Fe , Fg denote vectors of the unbalanced excitation and selfweight excitation. Through assembling the coefficient matrices and external excitation vectors of each subsystem, the above matrices and vectors can be obtained, respectively. Obviously, due to the multiple fit clearances, the elements in the stiffness matrix K1 are composed of the nonlinear stiffness of the bearing and the nonlinear stiffness of the supporting system. Since these stiffness elements are all related to the relative motions of the flywheel rotor and the concentrated mass of the support, and the fit clearances between the inner and outer rings of the bearing, sleeve and pedestal structure, K1 is a non-linear matrix related to the degree of freedom vector q of the system. In order to solve the vibration model of the rotor-bearing-pedestal system shown in Eq. (9), numerical iteration integration is needed.

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3 Analysis and Discussions 3.1 Effect of Unbalanced Mass Figure 4 shows the change of RMS value of rotor system vibration response with rotating speed when the unbalanced excitation increases from 0.4 g.cm to 1.2 g.cm. For the xdirection vibration response, the peak values appear at 8000 rpm, 4000 rpm and 2000 rpm. The speed of 8000 rpm is called the primary resonant speed, and the other two are called the sub-harmonic resonant speeds. At these resonance speeds, the phenomenon of amplitude jump is also accompanied, which is mainly caused by both bearing and clearance fit nonlinearities. For the y-direction response, as the gravity force is the dominant force, so the response amplitude changes little at 0–10000 rpm. Except for at the resonant speeds, the amplitude decreases and has a certain jump. With the increase of unbalance mass, the response amplitude changes little in the range of 0–8000 rpm. In the region higher than 8000 rpm, the response amplitude increases (for x-direction vibration) or decreases (for y-direction vibration).

Fig. 4. Change of RMS value of rotor system vibration response with rotating speed: (a)xdirection response and (b) y-direction response

3.2 Effect of Fit Clearances The influence of two fit clearances is considered here. When the fit clearance between the bearing outer ring and the sleeve Pd1 increases from 2e–3 mm to 30e-3 mm, the system responses in both x- and y- directions are shown in Fig. 5. It is shown that with the increase of fit clearance, the response amplitude increases in the whole speed range, especially when the speed is greater than 2000 rpm. This shows that the increase of

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fit clearance aggravates the vibration response level and is not conducive to the stable operation of the rotor system. In addition, with the increase of the fit clearance, the resonance speed of the system moves to the low speed, and the jumping phenomenon becomes more prominent. This is owing to the overall reduction of radial stiffness, and more significant non-linear effect. A similar change of the response amplitude of the system with the increase of fit clearance between the sleeve and pedestal can also be observed. It is noticed that the response amplitude does not increase much when the fit clearance is increased at some speed within 8000 rpm-10000 rpm. For the rotor system

Fig. 5. Change of the response amplitude of the system with the increase of fit clearance between the bearing outer ring and sleeve: (a) x-direction response and (b) y-direction response

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concerned in the study, the working speed is generally within this speed range. The vibration response of the system can be kept at a low level by reasonably designing the fit clearance and working speed.

4 Conclusion In this paper, the dynamics of rotor-bearing-pedestal system with multiple fit clearances is studied. The equivalent nonlinear stiffnesses of both angular contact high-speed ball bearing with preload and the clearance fit are derived respectively. Then, the lumped parameter model of the system is established, and verified by dynamic test. Based on this model, the influence of unbalanced excitation parameters and clearance fit parameters on the vibration response amplitude of the system is analyzed. The conclusions are summarized as follows: (1) When the radial displacement is greater than the fit clearance, the radial support stiffness changes abruptly and increases rapidly. If it increases to a certain value, the increase of radial stiffness would slow down. For the large radial displacement, the change of fit clearance has little effect on the radial stiffness. (2) The vibration response curve appears sub-harmonic resonances and amplitude jump. With the increase of unbalance mass, the response amplitude changes little in the low speed region, and increases (for x-direction) or decreases (for y-direction) in the high speed region. (3) The increase of the fit clearance aggravates the vibration response level, the resonance speed of the system moves to the low speed, and the jumping phenomenon becomes more prominent, which is not conducive to the stable operation of the rotor system.

References 1. Yu, H., Ma, Y., Xiao, S., Hong, J.: Mechanical and dynamic characteristics of bearing with looseness on high-speed flexible rotor. J. Beijing Univ. Aeronautics Astronautics 43(8), 1677– 1683 (2017). (In Chinese) 2. Tiwari, M., Gupta, K.: Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. J. Sound Vib.Vib. 238, 723–756 (2000) 3. Harsha, S.: Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect. Mech. Mach. Theory 41, 688–706 (2006) 4. Li, Y., Luo, Z., Liu, Z., Hou, X.: Nonlinear dynamic behaviors of a bolted joint rotor system supported by ball bearings. Archive of Applied Mechanics, 1–15 (2019) 5. Ganesan, R.: Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances. Mech. Mach. Theory 31, 781–798 (1996) 6. Chavez, J., Hamaneh, V., Wiercigroch, M.: Modelling and experimental verification of an asymmetric Jeffcott rotor with radial clearance. J. Sound Vib.Vib. 334, 86–97 (2015) 7. Harris, T., Kotzalas, M.: Rolling Bearing Analysis. Taylor & Francis, Boca Raton (2007)

Comparison Studies of Dynamic Characteristics for Coupled Bearing-Rotor Systems with Fixed and Pivot-Supported Pads Wennian Yu1,3(B) , Chaodong Zhang1,3 , and Lu Zhang1,2 1 College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044,

China [email protected] 2 CRRC Zhuzhou Electric Locomotive Co., Ltd., Zhuzhou 412000, People’s Republic of China 3 State Key Laboratory of Mechanical Transmission for Advanced Equipments, Chongqing University, Chongqing 400044, China

Abstract. The coupled bearing (including four-pad tilting pad journal bearings and six-pad tilting pad thrust bearings) is mainly used in a nuclear power circulating pump to support the high axial loads and radial loads from a vertical rotary system with minimum power loss, low vibration, and high load capacity. Its lubrication and vibration characteristics have significant effects on the dynamic performance and operation reliability of the nuclear power circulating pump. In this paper, an original mixed-lubrication dynamic model for the coupled bearing-rotor system is proposed to study the dynamic characteristics of the system considering the effects of the pivot clearance, asperity contact, and elastic deformation. The innovation of the proposed model is that it integrates the horizontal-rocking vibrations of the rotor and pivot pads with the mixed lubrication of the coupled bearing. The coupling effects between the dynamic characteristic of the rotor and the lubrication behavior of the coupled bearing are revealed. In addition, the time-dependent dynamic performance of the fixed pad coupled bearing and the tilting pad coupled bearing are compared. Numerical results indicate that the self-adaptive tilt of the pad makes the oil film pressure more evenly distributed on each pad of the journal and thrust bearings. The tilting pad coupled bearing can effectively improve vibration resistance, system stability, and load-sharing capacity compared to the fixed pad coupled bearing. It is expected that the proposed mixed-lubrication dynamic model can provide guidance for improving the lubrication performance of the coupled bearing and the stability of the system. Keywords: Tilting pad coupled bearing · Pivot-supported pads · Fixed pads · Mixed-lubrication dynamic model

1 Introduction Compared to fixed pad sliding bearings, pivot-supported tilting pad coupled bearings are widely used in fluid machinery, such as nuclear power plants, compressors, pumps, and turbines, which can simultaneously bear a large amount of radial and axial load [1, 2]. Tilting journal/thrust pads can generate horizontal-rocking vibration around the pivot. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 733–748, 2024. https://doi.org/10.1007/978-981-97-0554-2_56

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The pivot is a critical structure that realizes the self-adaptive tilt of the pads. However, the unstable working conditions of the rotor and pivot pads can cause the rupture of the hydrodynamic film of the coupled bearing, which may lead to pad surface faults (i.e. surface wear, surface scratch, gluing, etc.). Therefore, it is necessary to investigate the lubrication performance and dynamic characteristics of the tilting pad coupled-bearingrotor system. Due to their own structural limitations, traditional fixed-pad sliding bearings may operate in the mixed lubrication regime when running at heavy load and low speed conditions [3, 4]. In recent years, dynamic characteristics of the tilting pad bearing have become one of the hot research directions. Timothy et al. [5] reviewed the development of dynamic modeling for tilting pad bearings. Benti et al. [6] analyzed the unbalance response of tilting pad journal bearings with four and eight pads through experimental and numerical methods. Haugaard and Santos [7] studied the stability of journal bearings with four elastic tilting pads through the finite element method. Rendl et al. [8] investigated the nonlinear dynamics of tilting pad journal bearings with flexible pivots. They neglect the angular misalignment of the journal and the pads, i.e. the unbalance force and moment of the rotor were not considered in this model. Suh et al. [9] investigated the effects of the pivot design and the angular misalignment on tilting pad journal bearings. The cylindrical pivot design has only a tilting motion and the spherical pivot design has three angular motions (i.e. tilt, pitch, and yaw). Liu et al. [3] analyzed the vibration responses of tilting pad journal bearing-rotor systems with elastic and damped pivots. They found that the bearing significantly improved the system’s stability. The above studies mainly concentrated on the dynamic characteristic of tilting pad journal bearings. Some scholars have also carried out related research work in the dynamic characteristic analysis of tilting pad thrust bearings, and adhered the wedge-shaped rubber to the stainless steel base to improve the lubrication performance and load-sharing capacity of the system [10–14]. Beek et al. [10, 11] discussed the dynamic characteristics of rubber-supported thrust bearings. Sun et al. [12] developed a tribo-dynamic model of the rubber-supported water-lubricated thrust bearing considering the 5-DOF model of the rotor and the 3-DOF model of the thrust bearing. Liang et al. [13] analyzed the lubrication performance of rubber-supported water-lubricated thrust bearings, and the accuracy of the simulation results is verified by the experimental results. Liang et al. [14] conducted a series of experiments considering three types of water-lubricated hydrodynamic thrust bearings (i.e. pivot-supported, rubber-supported, and fixed pad bearings). They found that the pivot-supported thrust bearings have good lubrication performance, load-carrying properties, and wear resistance. The above research on the dynamic and lubrication characteristics of the tilting pad bearing-rotor systems mainly considered the effects of the horizontal motion of the rotor and the circumferential tilt motion of the pad. Due to the rocking motion of the rotor and the axial tilt motion of the pad, the distribution of the pressure and film thickness will be affected. Thus, these factors should be considered in the dynamic modeling of the tilting pad bearing-rotor system, which is seldom studied in the previous work. In the aforementioned works, these studies primarily focused on the lubrication and vibration characteristics of the tilting pad journal or thrust bearings. It should be noted that the bearing applied in a nuclear power circulating pump is a key element

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to support the radial and axial load. However, there are only a few published studies on the dynamic characteristics of coupled bearing-rotor systems (including journal and thrust bearings). Wang et al. [15, 16] developed a mixed lubrication model of coupled journal-thrust-bearing systems considering the effect of mass-conserving cavitation and shaft misalignment to analyze the effects of boundary conditions on the lubrication characteristic of the bearing system. Kim et al. [17] calculated the axial and radial load capacities of the coupled groove bearing. Xiang et al. [18] developed the four-degree of freedom (4-DOF) tribo-dynamic model of coupled journal-thrust water-lubricated bearing systems, which considered the disturbing forces and moments caused by the propeller rotor. They verified the proposed model through the experiment and studied the transient coupling effects between the dynamic feature of the rotor and the mixed lubrication characteristic of the coupled bearing. The disturbing forces and moments are calculated by the quasi-steady method, and the elliptical arc groove is adopted in the coupled bearings. Since the pivot can achieve the self-adaptive tilt of a pad, the tilting action of the pads allows the load component of each pad to pass through the center of the journal. Thus the tilting pad bearing can effectively improve vibration resistance and stability [19, 20]. However, referring to previous research, most researchers mainly focused on the dynamic characteristic analysis of fixed-pad coupled bearing, while the performance analysis of coupled bearings with pivot-supported tilting pad journal and thrust bearings is not considered. Besides, the comparisons of mixed-lubrication dynamic behaviors between the fixed-pad and tilting-pad coupled bearings have not been reported, which constitutes the primary motivation of this study. In this paper, an original mixed-lubrication dynamic model for tilting pad coupedbearing-rotor systems with pivot-supported pads applied in a nuclear power circulating pump is established considering the horizontal-rocking vibration of the rotor and pivot pads as well as the mixed lubrication of the coupled bearing. By the numerical study, the dynamic characteristics of two types of coupled bearing models (i.e. fixed pad coupled bearing and tilting pad coupled bearing) working under various rotational speed conditions are compared and discussed. The coupling effects between the dynamic characteristic of the rotor and the lubrication behavior of the coupled bearing are revealed.

2 Mixed-Lubrication Dynamic Modeling of Coupled Bearing-Rotor Systems The research object of this paper is a tilting pad coupled-bearing-rotor system applied in the nuclear power circulating pump, and its structure diagram is shown in Fig. 1, which consists of a rotor supported by coupled bearings (Fig. 1(a)), four-pad tilting pad journal bearings (Fig. 1(c)), and six-pad tilting pad thrust bearings (Fig. 1(d)). In the tilting pad coupled-bearing-rotor system, there are transient interactions among the horizontalrocking vibration of the rotor and pad, and the lubrication behavior of coupled bearings, as shown in Fig. 1(b).

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Fig. 1. Schematic diagram of tilting pad coupled bearing-rotor systems

2.1 Mixed-Lubrication Model Journal Bearing Based on the average Reynolds equation derived by Patir and Chen [21] and the turbulence model proposed by Ng and Pan [22], the transient hydrodynamic pressure of the m-th pad for the journal bearing can be calculated as:       ∂φs 1 ∂ φθ h3Jm ∂PhJm ∂hJm ∂ φw h3Jm ∂PhJm ω φ + σ (1) + = c R2 ∂θ kθ μ ∂θ ∂w kw μ ∂w 2 ∂θ ∂θ where the subscript J represents the tilting pad journal bearing; the subscript m represents the m-th pad of the journal bearing (m = 1, 2,…, M, the same below); μ is the dynamic viscosity; h is the lubrication gap; Ph is the hydrodynamic pressure; σ is the combined roughness; R is the radius of the journal bearing; φ θ and φ w are the flow factors; φ c is the contact factor; φ s is the shear factor; k θ and k w are the turbulence coefficients; ω is the angular velocity of the rotor (rad/s). In addition to the horizontal motion of the rotor and the circumferential tilt motion of the pad, the rocking motion of the rotor and the axial tilt motion of the pad will also affect the film thickness between the pad and the journal, which is not considered in the traditional model. Therefore, considering the effects of the elastic deformation, geometric gap, and micro-motions of the rotor and the journal pads, the transient lubrication gap of the m-th pad for the journal bearing can be calculated as: hJm =C + ecos(θ − φ) − (ξJm − κC)cos(θ − θm ) − Rp αJm sin(θ − θm )+   B sinβJm cos(θ − θm ) + hJmis + hem w− 2

(2)

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hJmis

   B  −ϕx cos(θ − φ) + ϕy sin(θ − φ) = w− 2

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(3)

where C is the radial clearance; φ is the attitude angle; κ is the preload factor; Rp is the pad radius of the journal bearing, and Rp = R + C; he is the pad elastic deformation; e is the unbalanced eccentricity; B is the axial width of the tilting pad journal bearing; w is the pad polar radius of the journal bearing; w is the pad polar radius of the journal bearing; θ m is the position angle of the m-th pad for the journal bearing; ϕ x and ϕ y are the angular displacements of the rotor, respectively; parameters θ, ξ J , α J , and β J are shown in Fig. 2.

Fig. 2. Coordinate system of the tilting pad journal bearing

Thrust Bearing In the mixed lubrication regime, the hydrodynamic behaviors considering the roughness surface and the turbulence effect are governed by the modified average Reynolds equation, which can be expressed as:       φt h3Tn ∂PhTn ∂hTn ∂hTn ∂ φr rh3Tn ∂PhTn ∂ ωr φc + σ φs (4) + = ∂r kr μ ∂r r∂θ kt μ ∂θ 2 ∂θ ∂t where the subscript T represents the tilting pad thrust bearing; the subscript n represents the n-th pad of the journal bearing (n = 1, 2,…, N, the same below); k r and k t are the turbulence coefficients; φ r and φ t are the flow factors. Similarly, the transient lubrication gap of the n-th pad in the thrust bearing can be calculated as: hTn = h0 + rsin(θn − θ )sinαTn + [Rn − rcos(θn − θ )]sinβTn + hTmis + hen   hTmis = r ϕx sinθ + ϕy cosθ

(5) (6)

where h0 is the geometric gap at the equilibrium position; θ p is the tilting angle of the thrust pad. θ n is the position angle of the n-th pivot pad for the thrust bearing; parameters θ, ξ T , α T , and β T are shown in Fig. 3.

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Fig. 3. Coordinate system of the tilting pad thrust bearing

The hydrodynamic pressure of the tilting pad coupled-bearing is comprised of the film pressure and the asperity contact pressure [23, 24]. The latter can be calculated using the Greenwood and Tripp model [25], which is written as: √    σ 16 2 h (7) Pc = π (Dβσ )2 E f 15 β σ

6.804   h h 4.4056 × 10−5 4 − σh σ ≤4 = f (8) h σ 0 > 4 σ where D and β are the density and radius of asperity, respectively, and E is the composite elastic modulus. 2.2 Dynamic Model The coordinate system of the rotor is shown in Fig. 4, and the coordinate system is established in the center of the fixed shaft. By solving the transient Reynolds equation of coupled bearings, the oil film force and moment between the coupled bearing and the rotor can be obtained, and then the rotor can perform the micro-movements along the horizontal (x), vertical (y), and axial (z) directions and micro-rotation around the x-axis and y-axis within the lubrication gap. Therefore, the dynamic behavior of the rotor can be governed by the following dynamic equations based on a 5-DOF model: ⎧ mr x¨ = FJx (t) − Fub cos(ωt) − Fex ⎪ ⎪ ⎪ ⎪ mr y¨ = FJy (t) − Fub sin(ωt) − Fey ⎪ ⎪ 

⎨  (9) ϕx2 + ϕy2 − Fez − mr g mr z¨ = FTz (t) − Fub sin ⎪ ⎪ ⎪ ⎪ ⎪ J ϕ¨ = MJx (t) + MTx (t) − Mex ⎪ ⎩ rx x Jry ϕ¨y = MJy (t) + MTy (t) − Mey where mr is the mass of the rotor; J rx and J ry represent the rotational inertia of the rotor; g is the gravitational acceleration; F ub is the unbalanced force of the rotor due

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to eccentricity; t is the operating time, F ub = mr e ω2 [26]; F e and M e are the external force and moment, respectively; parameters x, y, z, ϕ x , and ϕ y are shown in Fig. 4. F J , F T , M J , and M T are the transient reaction forces and moments of journal bearings and thrust bearings acting on the rotor.

Fig. 4. Coordinate system and force analysis of the rotor

For a tilting pad journal bearing under a mixed lubrication regime, the transient reaction force and moment acting on the rotor are shown in Fig. 2. The analytical formula can be calculated as:    ¨    B cosθ MJx (t) w− = Rd θ dw (10) (PJh (θ, w, t) + PJc (θ, w, t)) MJy (t) sinθ 2

J     ¨   B cosθ MJx (t) w− = Rd θ dw (11) (PJh (θ, w, t) + PJc (θ, w, t)) MJy (t) sinθ 2

J where Ph and Pc are the transient nodal film pressure and contact pressure, which are obtained by the mixed-lubrication model; Meanwhile, the transient reaction force and moment yielded by the tilting pad thrust bearing can be expressed by: ¨ FTz (t) = (12) (PTh (θ, r, t) + PTc (θ, r, t))rdrd θ 



MTx (t) = MTy (t)

¨

T



 −cosθ rdrd θ (PTh (θ, r, t) + PTc (θ, r, t)) sinθ

T

where r is the pad polar radius of the thrust bearing.

(13)

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Due to the influence of pivot clearance, hydrodynamic pressure, and lubrication clearance, the journal pads allow for radial and swing motions, and the thrust pads allow for axial and swing motions. The motion equations of the m-th pivot pad for the journal bearing can be expressed as:   ⎧ ⎨ mJ ξ¨Jm = FJ ξ m (t) − FJmp + KJm ξJm (14) J α¨ = MJ αm (t) ⎩ J ¨Jm JJ βJm = MJ βm (t) where F p is the pre-tightening force; K is the pivot stiffness; m and J are the pad mass and pad moment of inertia, respectively; the resultant force (F Jξ m ) and moments (M Jαm and M Jβm ) can be obtained by integrating from contact pressure and film pressure loaded in the m-th pad for the journal bearing. Similarly, the motion equations of the m-th pivot pad for the journal bearing ⎧ ⎨ mT ξ¨Tn = FT ξ n − KTn ξTn (15) J α¨ = MT αn (t) ⎩ T ¨Tn JT βTn = MT βn (t) where the resultant force (F Tξ n ) and moments (M Tαn and M Tβn ) can be obtained by integrating from contact pressure and film pressure loaded the n-th pad of the thrust bearing. 2.3 Calculation Procedure For the tilting pad coupled-bearing-rotor system applied in the nuclear power circulating pump, it consists of four pieces of journal pads and six pieces of thrust pads. Thus, a mixed-lubrication dynamic model containing 38 degrees of freedom is established considering the coupling effects of the mixed lubrication behavior of the coupled bearing and the dynamic characteristic of the rotor and the pivot pads. The detailed calculation process is shown in Fig. 5. The initial values of structural parameters for the tilting pad coupled bearing-rotor system are first set. Based on the Reynolds boundary conditions (Eq. (16)), the finite difference method is used to discretize the modified average Reynolds equation (Eqs. (1) and (4)), and the discretized equation is solved by the successive over-relaxation iteration method to obtain the transient lubrication parameters (e.g. film pressure, contact pressure, elastic deformation, etc.) at time t. Then, the transient reaction force and moment of the rotor and pads are calculated as the load excitation of the coupled bearing-rotor system. The integrated governing equations (Eqs. (9), (14) and (15)) of motion of the coupled bearing-rotor system are solved by the Runge Kutta method to obtain the vibration response of the tilting pad coupled-bearing-rotor system, which in turn affects the film force at the next time step (t + t) by updating the lubrication gap. This iterative process continues until the predetermined simulating time t 0 is reached. Therefore, the coupling simulation of lubrication and vibration characteristics of the tilting pad coupled bearing-rotor system is realized. The main parameters of the system are illustrated in Table 1.    p θ, ± B2 = 0, p(θin , w) = 0 (16) p(θon , w) = ∂p/∂θ = 0

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Fig. 5. Calculation flowchart

Table 1. Parameters of the tiling pad coupled bearing-rotor system Parameters

Value

Parameters

Value

Rotational speed (rpm)

500

Number of journal pads

4

Eccentricity (µm)

1.5

Journal bearing radius (mm)

45

Radial clearance (mm)

0.1

Journal bearing width (mm)

44.5

Number of Thrust pads

6

Arc angle of journal pads (°)

70

Thrust pad radius (mm)

45

Pivot radius of thrust pads (mm)

97.5

Pivot offset

0.5

Surface roughness (µm)

0.55

Moment of inertia of the rotor (kg·m2 )

3.1605

Rotor mass (kg)

175.8

Mass of journal pads (kg)

1.55

Moment of inertia of journal pads (kg·m2 )

0.01

Moment of inertia of thrust pads (kg·m2 )

0.061

Mass of thrust pads (kg)

1.145

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3 Results and Discussions Figure 6 compares the vibration responses of the rotor for two types (i.e. fixed-pad and tilting-pad) of coupled bearings when the rotating speed is 400 rpm, 500 rpm, and 600 rpm. It can be seen that as the rotating speed increases, the vibration displacement of the rotor for the fixed-pad and tilting-pad coupled bearings in the x and y directions increases. The amplitudes of vibration displacement of the rotor for the tilting pad coupled bearing are smaller than those of the fixed pad coupled bearing under the same working conditions. It can also be seen from the axis orbit of Fig. 6(c) that the tilting pad coupled-bearing-rotor systems have good stability compared to the fixed-pad coupled bearing.

(a)

(b)

(c)

(d)

Fig. 6. Displacement responses of different coupled bearings working under various speed conditions: (a) x-direction, (b) y-direction, (c) angular displacement around the x-axis, and (d) axis orbit

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Figure 7 compares the time-domain and frequency-domain responses in terms of the y-directional rotor acceleration for the two types of coupled bearing when the rotating speed is 500 rpm. It can be seen that the amplitudes of vibration acceleration of the rotor for the tilting pad coupled bearing are smaller than those of the fixed pad coupled bearing. It can also be seen from Fig. 7(b) that the peak-frequency amplitudes of the rotor acceleration for the fixed pad coupled bearing in the y direction are larger than those of the tilting pad coupled bearing. The peak frequencies for the two cases are mainly located at the rotating frequency f s and its odd harmonics (e.g. 3f s , 5f s , etc.). Therefore, the tilting pad coupled-bearing-rotor system can effectively improve vibration resistance compared to the fixed-pad coupled bearing.

(a)

(b)

Fig. 7. Acceleration responses of the different coupled bearings when the speed is 500 rpm: (a) y-direction, (b) FFT spectra

Figure 8 shows the development of the oil film pressure with time for both the journal and thrust bearings of the fixed-pad and tilting-pad coupled bearings. It can be seen that the maximum oil film pressure increases for the fixed-pad and tilting-pad journal and thrust bearings with the increase of the rotating speed, and the maximum oil film pressure of the fixed-pad coupled bearing (i.e. journal and thrust bearing) is larger than that of the tilting-pad coupled bearing. This means that tilting pad thrust bearings are more stable than fixed pad thrust bearings. Compared to the thrust pad bearing, the rotating speed has a greater influence on the oil film pressure of the journal pad bearing. This phenomenon is caused by the unbalanced force due to the rotor eccentricity. It can be also seen from Fig. 9 that the peak frequencies of transient oil film pressure for the journal bearing are mainly located at the frequency multiplication components 4f s (i.e. M × f s , M is the number of journal pads) and 8f s (i.e. 2M × f s ). The peak frequencies of transient oil film pressure for the thrust bearing are mainly located at rotating frequency f s and its harmonics 6f s (i.e. N × f s , N is the number of thrust pads). Therefore, the peak

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frequency of the oil film pressure for the journal bearing is only related to the frequency caused by the number of its own pads, while the thrust bearing is related to the rotating frequency and the frequency caused by the number of its own pads.

(a)

(b)

Fig. 8. Transient oil film pressure of the fixed-pad and tilting-pad coupled bearing under different speed conditions: (a) journal bearing, and (b) thrust bearing

(a)

(b)

Fig. 9. Comparison of FFT spectra of transient oil film pressure for the fixed-pad and tilting-pad coupled bearings: (a) journal bearing, and (b) thrust bearing

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Figure 10 and Fig. 11 plot the distribution of the oil film pressure at different time instants for both the journal and thrust bearings of the fixed-pad and tilting-pad coupled bearings. It can be seen from Fig. 10 that the pressure peak on each pad of the fixed pad thrust bearing is mainly concentrated on the oil outlet position of the pad. Figure 11 shows that the oil film pressure of the fixed pad journal bearing is mainly concentrated on a single pad due to the misalignment of the rotor. Since the pivot can achieve the self-adaptive tilt of a pad, the tilting action of the pads allows the load component of each pad to pass through the center of the journal. This phenomenon of tilting pad journal and thrust bearings can also be seen in Fig. 10 and Fig. 11. Therefore, the tilting pad coupled bearing can effectively improve the system stability and load-sharing capacity compared to the fixed pad coupled bearing.

(a)

(b) Fig. 10. Distribution of oil film pressure at different time instants: (a) fixed pad thrust bearing, and (b) tilting pad thrust bearing

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(a)

(b) Fig. 11. Distribution of oil film pressure at different time instants: (a) fixed pad journal bearing, and (b) tilting pad journal bearing

4 Conclusions In order to explore the dynamic characteristics of the coupled bearing used in a nuclear power circulating pump, an original mixed-lubrication dynamic model of the tilting pad coupled-bearing-rotor system is proposed considering the effects of dynamic characteristics of the rotor and the pads as well as the mixed lubrication behaviors of the coupled bearing. The time-dependent dynamic behaviors between fixed-pad and tilting-pad coupled bearings working under various rotating speed conditions are compared. The main conclusions can be summarized as: 1. Compared to the fixed pad coupled bearing, the tilting pad coupled bearing can reduce the vibration responses of the rotor and the oil film pressure of the coupled bearing, which effectively improves system stability, vibration resistance, and load-sharing capacity. 2. The self-adaptive tilt of the pad makes the oil film pressure more evenly distributed on each pad of the journal and thrust bearings. 3. The rotating frequency and its odd harmonics exist in the peak frequencies of the rotor acceleration. The peak frequencies of film pressure for the journal bearing are mainly located at M × f s and 2M × f s (M is the number of journal pads), whereas at f s and N × f s (N is the number of thrust pads) for thrust bearing.

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Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant No. 52105086) and the National Key R&D Program of China (Grant No. 2020YFB2010103). We are grateful to the Fundamental Research Funds for the Central Universities (Grant No. 2022CDJGFCG004, 2022CDJKYJH049).

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Nonlinear Dynamic Performance and Analysis Model of Pump Valve System of Diaphragm Pump Hydraulic End Jiameng Zhang1,2 , Wensheng Ma2 , Chunchuan Liu1,2(B) , and Zicheng Zhao2 1 College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001,

China [email protected] 2 Chongqing Machinery Electric Co., Ltd., Chongqing 400055, China

Abstract. The analysis of the dynamic lift performance and valve gap flow characteristics of the diaphragm pump hydraulic end valve has an important impact on the safety and stability of the diaphragm pump valve design. In this paper the dynamic performance of valve and valve gap flow of a new diaphragm pump is calculated and analyzed in Chongqing pump industry. Firstly, the nonlinear motion equation of a new type of cone valve, and the dynamic lift curve of the pump valve is calculated. Calculated valve gap flow of the pump valve and compare it with the actual flow data to verify the correctness of the dynamic analysis model of the pump valve. Based on the dynamic model and the main functions of diaphragm pump valve established in this paper, maximum flow rate, ending speed and hydraulic loss, are calculated and analyzed. Meanwhile, the effects of erosion failure and collision failure on the service life of diaphragm pump cone valve are analyzed. At the same time, the influence of impulse and spring stiffness on the volume efficiency of diaphragm pump is calculated and discussed. Through the above research and analysis, the practical engineering guiding significance is provided for the geometrical optimization and design of the hydraulic end cone valve of the new mine diaphragm pump. Keywords: Diaphragm pump · hydraulic end · cone valve · dynamic lift performance · valve gap flow

1 Introduction Reciprocating diaphragm pump consists of power end [1], hydraulic end, drive device, control system, hydraulic auxiliary system, feed and discharge compensation system, etc. The diaphragm is used to separate the transported pulp from the power end drive system, and the conveying granular pulp medium cannot contact with the reciprocating power end parts, avoiding the serious wear caused by solid particles to the pump [2]. As one of the core mechanical components of reciprocating diaphragm pump, the pump valve is susceptible to the particle size grading of conveying minerals, rheological characteristics and other factors of pulp, and is more prone to failure than other parts of the pump [3, 4]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 749–760, 2024. https://doi.org/10.1007/978-981-97-0554-2_57

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With the increasing application of reciprocating pump in petrochemical industry, the basic theory research of reciprocating diaphragm pump has developed rapidly, but the research of domestic and foreign scholars focuses on the inhalation performance and volume efficiency and other issues, because the pump valve movement involves many factors, and the related basic theory is not perfect [5]. Scholars at home and abroad obtain the lift curve characteristics of the valve disc by establishing the method of the surface pressure difference and flow function of the valve disc, and verify the accuracy of the function [6–11] through experimental methods. For a long time, scholars who study pump valves generally derive according to the classical theory when studying the motion law of pump valves [12–14]. Because the problems involved in the pump operation are very complex, scholars at home and abroad are often limited to study the dynamic characteristics of the hydraulic end pump valve core under certain assumptions. This leads to a serious lag in the study of the valve core of the diaphragm pump. In 1968, German U. Adolph [15] established the second-order nonlinear inhomogeneous motion differential equation of lift with mass valve. The defect of this equation is that although it can be solved by numerical solution method, there is no method to solve the motion singularity. In 1977, The Japanese scholar [16] simulated the results of Adolf’s equation by numerical method. Although this method is relatively accurate, the object of the pump valve studied is relatively single, because the design parameters of the pump valve cannot be involved in the calculation, and must be designed before the calculation. Li Xiangyang [17] using the classical reciprocating pump theory, a new three-port valve plate structure is designed and optimized to adjust the angle of the non-dead point transition zone to obtain the optimal structure. Wang Teng [18] et al. studied the mechanism of the movement of the pump valve plate under the condition of ultra-high pressure, and established a set of special real-time measurement detection system, in order to optimize the life of the pump valve. Jiang’ao Zhao [19] et al. found the contact surface of cylinder block and valve plate as the key design problem to determine the pump performance and service life, conducted an experimental study, and summarized several typical structural optimizations, surface forming and reinforcement methods. About all, the research on the mechanism of the pump valve of the reciprocating diaphragm pump has lasted for quite a long time. The mechanism of the pump valve studied by scholars at home and abroad is basically using different methods to establish equations of motion or using simulation software to try to avoid a series of mathematical problems that cannot be solved. There is no deep pump valve core in the process of movement dynamics and valve gap fluid flow rate problems, resulting in the lack of impact damage and erosion damage foundation. Therefore, the impact dynamic mechanics and valve gap flow rate are necessary in the design and optimization of pump valve structure. According to the basic principles of dynamics and fluid mechanism. The method of numerical calculation is applied, and the dynamic characteristic motion curve of the diaphragm pump valve is obtained. The flow curve flowing through the valve gap and the measured data curve are compared to verify the accuracy of the applied method in this paper. For the practical problems of closer joint engineering, the relationship between the geometrical parameters of pump valve and the characteristic index of diaphragm pump is also calculated and discussed in detail.

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2 Cone Valve Nonlinear Motion Model 2.1 Simplified Model of the Cone Valve The working principle of diaphragm pump is shown in Fig. 1. The motor drives the crankshaft rotation through the reducer, and then transforms the rotating motion into linear motion through the connecting rod and crosshead, which drives the piston to do reciprocating motion. When the piston moves to the left end, the indoor pressure of the hydraulic oil chamber at the front of the piston decreases to/or below the atmospheric pressure. Under the action of atmospheric pressure and the positive pressure of the front pump of the inlet, the diaphragm in the diaphragm chamber moves to the left, and the slurry enters the chamber at the larger end of the diaphragm under the positive pressure until the piston reaches the extreme position on the left side. While in this process the discharge valve closes and the suction valve opens. When the piston moves to the right end, the suction valve is closed, and the piston pushes the hydraulic oil to push the rubber diaphragm in the diaphragm chamber to the right direction. At the same time. The discharge valve is opened with the help of pressure to deliver the slurry to the pipe.

Outlet

Inlet

Fig. 1. Schematic of the working principle of the diaphragm pump

In the schematic, (1) motor (2) crankshaft (3) connecting rod (4) crosshead (5) piston liner (6) piston (7) guide rod (8) probe (9) diaphragm (10) diaphragm chamber (11) discharge valve (12) suction valve. Due to the slurry does not contact the moving parts such as the piston during the work process, the abrasion of these parts is avoided. Therefore, the maintenance times and operation costs are reduced. As the part of the reciprocating diaphragm pump, the pump valve is the important link responsible for the pulp transportation. The design of diaphragm pump valve has an main impact on the flow rate and volume efficiency of pulp conveying. Through the reciprocating movement of diaphragm pump piston, the pump valve realizes periodic closed and opened. So, the diaphragm chamber realizes the

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alternating contact between inhalation, discharge and pipe cycle, as the way to complete the process of pulp inhalation and discharge of diaphragm pump circulation. In Fig. 2, a simplified model diagram of the conical valve structure of a new type of diaphragm pump used in the CQPI. This type of conical valve is also commonly used in the domestic diaphragm pump series products.

Fig. 2. Schematic of the conical valve

Among them, vk -seat hole slurry flow rate, d k -seat hole diameter, h-valve core rise, vx -valve gap slurry flow rate, d f -valve core diameter, θ -valve core cone angle. In the actual operation of the diaphragm pump, the movement law of the pump valve is very complex. Therefore, some small factors in the pump valve work are assumed in the practical engineering. According to the classical pump valve theory, the basic assumptions of diaphragm pump cone valve are as follows: 1) 2) 3) 4) 5)

Liquid is incompressible; Pump body parts are rigid; The valve core quality is negligible; The liquid cylinder block is always fully filled during inhalation; The link is infinitely long.

2.2 Movement Law of the Cone Valve in the Steady State Due to the complex movement law of the diaphragm pump valve, the stable state of the pump valve is generally analyzed and discussed firstly. And the stable state of the pump valve is defined: the speed of the liquid flowing through the valve gap is steady with time. And then, the valve core is in a certain fixed lift h, which is called the stable state. Therefore, in this state, the force analysis of the valve core is shown in Fig. 3:

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G+Fs

P2

P1 Fig. 3. Loading analysis of valve core

   In which: G - The gravity of the cell and spring in the media, G = G0 1 − ρ ρ0 , G0 , ρ 0 and ρ respectively are the weight, density and dielectric density of the core spring in the air; F s -spring force, k corresponds to the spring stiffness coefficient; P1 , P2 -indicates the pressure on the lower and upper surface of the diaphragm pump valve respectively. According to the balance equation of the force on the valve cell of the diaphragm pump, and the area difference between the upper and lower surfaces of the valve core is not enough to affect the movement law of the valve core. Therefore, the balance equation of the valve core movement can be obtained as G + Fs P2 − P1 = ρg ρgAk

(1)

where, Ak -the cross-sectional area of the seat hole. Equation (1) means that the valve cell of the diaphragm pump maintains a stable state through the pressure difference between its upper and lower surfaces. Considering that the upper and lower surfaces of the hydraulic end valve core of the diaphragm pump are two different liquid levels, the Bernoulli equation when the liquid flows through the valve gap can be listed as follows: v2 − v22 P2 − P1 = (z1 − z2 ) + 1 + hL ρg 2g

(2)

where z1 and z2 indicate the liquid height on the upper and lower surfaces of the valve, respectively. hL represents the hydraulic loss. According to the actual working state of the diaphragm pump, compared with the hydraulicloss of the valve, the height difference z1 − z2 and the velocity difference head v12 − v22 2g of the upper and lower surface of the valve core are very small. Therefore, negligible compared with the hydraulic loss of

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the valve, the specific load Hv of the diaphragm pump valve can be obtained Hv =

G + Fs ρgAk

(3)

Based on the assumptions of the classical pump valve theory, the piston discharge flow is equal to the flow through the valve gap or seat. Therefore available based on the continuity conditions between the flows, q = αlf h cos θ vx y(t)

(4)

where, q represents the flow rate through the valve gap. α shows the shrinkage coefficient of the orifice section. From the hole outlet equation, the calculated expression for the valve gap flow velocity vx can be obtained,  1 P2 − P1 (5) 2 vx = √ ρ 1+ξ According to the design of reciprocating diaphragm pump, the theoretical flow calculation formula of valve can be obtained as   λ q = ARω sin φ + sin 2φ (6) 2  where, ω is the crank speed, ϕ is the crank rotation angle, and λ = R L is the link ratio. Replacing Eq. (6) and (5) into Eq. (3) and considering  the link length L is much greater than R, ignoring the influence of the link ratio λ = R L. The calculation expression of the dynamic displacement of the valve core of the diaphragm pump valve in stable state is, y(t) =

ARω sin φ  s μlf cos θ 2g G+F ρgAf

(7)

where, μ - the flow coefficient of the pump valve; The lf - the perimeter of the pump valve cell. Further, we can obtain the lift speed calculation expression of the diaphragm pump valve, y˙ (t) =

ARω2 cos φ  s μlf cos θ 2g G+F ρgAf

(8)

When estimating core displacement, the flow factor μ and specific load H v can be assumed unchanged. The maximum core rise when the crank angle is φ = 90◦ .

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2.3 Movement of Pump Valve in Unstable State As a special case in the steady state is only suitable for flow field calculation of nonautomatic valves. In the flow field calculation of the non-automatic valve, the valve core is added to the flow field as a boundary condition to calculate the flow field characteristics. The diaphragm pump cone valve, as a spring automatic valve, needs to calculate its impact dynamic performance and its flow characteristics through the valve gap. Therefore, it is necessary to calculate the valve core in an unstable state. The movement law of the pump valve in the stable state is not applicable in practical engineering. When the diaphragm pump is working in practice, the valve core rise is changed and is non-stable state. In the process of valve core movement, the liquid involved in the movement will rise less and fall more, which is called Westphal phenomenon. Considered the Westphal phenomenon of the pump valve, the continuous equation of motion of the core will become, q = αlf cos θ vx y ± Af

dy dt

(9)

Considering the characteristics of the Westphal phenomenon and the direction of the core movement speed, the negative sign in the formula can be removed, and the core speed is replaced into the type (7), so the calculation expression of the core lift in the unstable state is obtained as, y(t) =

Af ARω2 cos φ ARω sin φ  −

2  s s μlf cos θ 2g G+F μlf cos θ 2g G+F ρgAf ρgAf

(10)

Because the spring force changes with the expansion of the spring during the movement of the valve stem, it needs to be considered as a variable rather than a constant amount when calculating the motion lift curve. So put the spring force Fs = G + k(y0 + y) into type (10), and arrange the motion control equation of the valve core lift y is, y4 +



2Fsv 3 y + (A cos φ + B)y2 + C cos φ − D sin2 φ + E cos2 φ − F = 0 k

(11)

The coefficients of the terms are as follows: γ A2f ARω3

2 γ A2f ARω2 Fsv Fsv A= , B = 2 2, C = 2 2 2 , μgklf2 cos2 θ k lf μ lf k cos2 θ

D=

γ Af A2 R2 ω2 2μ2 gklf2 cos2 θ

, E=

A2 A2f γ 2 R2 ω4 4g 2 k 2 lf4 μ4 cos4 θ

γ = ρg,

=

, F=

γ Af A2 R2 ω2 Fsv 2μgk 2 lf2 cos2 θ

,

ω . μlf cos θ

Equation (11) is the time domain control equation of the dynamic lift y(t) of a new diaphragm pump cone valve, which is a quartic nonlinear inhomogeneous equation. When the diaphragm pump performance and valve structure parameters are determined A, B, C, D, E, γ, ε and  of Eq. (11) are constant. Since Eq. (11) is a fourth-order

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nonlinear equation, it is difficult to solve by analytical or semi-analytical method. In this paper, we will use the ode45 algorithm in MATLAB to solve Eq. (11), and compare with the existing experimental test results to verify the correctness of the diaphragm pump cone valve motion model established in this paper. On this basis, the dynamic lift performance, valve gap flow rate and its optimization performance will be further discussed.

3 Results and Analysis and Discussion 3.1 Influence of Impact Speed and Spring Stiffness Due to the Westphal phenomenon, the core displacement is not 0 when the crank angle is at 0° or 180°. According to type (13), the lag angle and lag height of the massless valve can be deduced. The angle obtained when h = 180° is the closed lag angle: tan φ0 =

Af ω  s μlf cos θ 2g G+F ρgAf

(12)

At the φ = 180◦ time, the rise value of the valve core is the lag height: Af ARω2 hd =

2  s μlf cos θ 2g G+F ρgAf

(13)

Combined the continuity Eq. (12). When the core is closed, the valve gap flow is 0. So the end speed can obtained: h˙ d =

Af ARω2  s μlf cos θ 2g G+F ρgAf

(14)

Figure 4 shows the valve core lift and Westphal phenomenon flow curves under different strokes. It can be seen from (a) that in the process of impulse change, the difference between the opening lag angle under the three impulse is only 0.33°, while the difference between the closing lag angle is 1.23°. The maximum value can be clearly seen that the higher the impulse, the greater the lift maximum. Since the stroke of the piston is determined by the pump, the time of discharging the same flow rate is changed within a cycle. (b) reflects the flow curve contained in the Westphal phenomenon. The volumetric efficiency of the diaphragm pump is a very important index in the design. The larger the flow rate, the higher the lag height of the valve core. According to the pump valve mechanism, the greater the leakage flow caused by the lag height, the lower the volume efficiency of the diaphragm pump. Figure 5 shows the flow curve of the core rise and Westphal phenomenon with different spring stiffness. When the characteristics of the diaphragm pump have been determined, the greater the spring stiffness, the smaller the lift maximum value. The difference between the core opening lag angle for k = 7000 N/m and k = 5000 N/m

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45rad/min 50rad/min 55rad/min

45rad/min 50rad/min 55rad/min

Westphal flow(m3/s)

Lift(m) Time(s) a) valve core lift curve

Time(s) b) Westphal flow curve

Fig. 4. Two curves under different stroke

5000N/m 6000N/m 7000N/m

Time(s) a) valve core lift curve

Westphal flow (m3/s)

Lift(m) 5000N/m 6000N/m 7000N/m

Time(s) b) Westphal flow curve

Fig. 5. Two kind of curves under different spring stiffness

is 1.53°, but the interpolation between the lag height is only 0.0848 mm. Westphal The difference between flow leakage (i.e., the integral of the curve in (b)) is only 0.0057%. Therefore, it can be considered that the largest factor affecting the volume efficiency of the diaphragm pump is the liquid leakage caused by the opening lag Angle. 3.2 Effect on the Overall Performance of the Hydraulic End of the Diaphragm Pump This subsection mainly uses the calculation method of the massless valve in the unstable state to discuss the relationship between the above parameters for the diaphragm pump volume efficiency, pump valve performance and structure. The pump valve is the only key component in the diaphragm pump except the diaphragm. Its performance such as lift maximum value, lag Angle, lag height, closing speed, closing speed, valve gap flow rate, opening resistance and hydraulic loss when the fluid passes through the valve are important indicators to ensure the normal operation of the diaphragm pump. Some of these performance indicators are related to volumetric efficiency, while others are related

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to the service life of the pump valve itself, such as the erosion damage caused by the flow rate of the valve gap and the collision damage caused by the closing speed. The design of the pump valve and its optimization are discussed in a table below. Table 1. Performance index of and pump valve with different stroke Stroke Maximum Lag (rad/min) of lift (m) height (m)

Lag angle (°)

Opening Volumetric Closing resistance efficiency speed (Pa) (%) (m/s)

Maximum Hydraulic valve gap loss (m) flow (m/s)

45

0.020167

0.002342 6.668316 7729.422

95.869751

0.094393 4.278392

0.232810

50

0.022102

0.002805 7.291771 7729.422

95.472959

0.114789 4.343449

0.287419

55

0.024000

0.003298 7.899422 7729.422

95.011585

0.136920 4.406104

0.347778

As shown in Table 1, the opening pressure is constant, it is not related to the change of impulse. The higher the lag height, the larger the lag angle, the lower the volume efficiency of the diaphragm pump. Volume efficiency includes three factors, namely, seal failure leakage, lag height and lag angle caused by lag leakage and liquid compression. In the case of the determination of the performance of diaphragm pump, the leakage caused by the hysteresis effect becomes the main factor affecting the volume efficiency of diaphragm pump. Closing speed is an important test of the life of the pump valve. The closing speed increases from 45 to 55 times/min by about 0.043 m/s per minute. The greater the closing speed, the stronger the collision with the base when the core is closed, and the easier the sealing performance of the pump valve is to fail. The erosion effect caused by the fluid of the valve gap is the main factor of seat damage. The greater the flow rate, the more likely the seat is to damage. The hydraulic loss also increased by about 0.115m with the increase of the impulse. This increase also brings the suction pressure burden of the pump, which should be mainly considered when designing and optimizing the performance of the diaphragm pump. Table 2. Performance index of pump valve with spring stiffness Stroke Maximum Lag (rad/min) of lift (m) height (m)

Lag angle (°)

Opening Volumetric Closing resistance efficiency speed (Pa) (%) (m/s)

Maximum Hydraulic valve gap loss (m) flow (m/s)

5000

0.024664

0.003480 8.110730 6261.808

94.858869

0.130403 3.977838

0.299031

6000

0.022850

0.002995 7.531906 7450.390

95.293800

0.120985 4.287507

0.299031

7000

0.021411

0.002635 7.069680 8638.973

95.617366

0.113482 4.570986

0.299031

Table 2 clearly shows that the change of hydraulic loss in the spring stiffness is a fixed value, adjusting the spring stiffness does not improve the suction performance of the diaphragm pump. With the increase of spring stiffness, the lag height and lag Angle gradually decrease, the resulting lag effect leakage is reduced, and the volume efficiency of the diaphragm pump is also improved. The closing speed decreases, the

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damage caused by the collision effect decreases, and the service life of the pump valve also increases. The failure caused by the pulp scour effect becomes stronger with the increase of the maximum gap flow rate, and the erosion damage of the valve seat becomes stronger. It can be seen that in the process of the design and optimization of the pump valve, the spring stiffness can improve the volume efficiency of the diaphragm pump, but also consider the damage of the erosion effect.

4 Conclusion This paper uses the classic massless valve method, to establish the nonlinear equation of the movement of the valve core. Using the motion curve, analyze the lag angle, lag height and lift maximum. The theoretical flow curve of 10 cycles is obtained by the peak superposition method, and is compared with the measured data. Through the comparison of two curves, the calculation method is correct. Through the basic theory of reciprocating diaphragm pump, derived the performance of the cone valve, such as valve flow rate maximum, sitting speed and hydraulic loss, and calculate and analyze the relationship between the impulse, spring stiffness and diaphragm pump volume efficiency, further analysis of the following conclusions in this paper, can provide reference value for the design of pump valve in practical engineering. 1. After the cone valve equation of motion derived from the classical plate valve, the theoretical flow curve and the motion curve integral obtained by the numerical analytical method are compared with the measured flow curve, which can be considered to be correct within the range of the error allowed; 2. The higher the opening of the diaphragm pump, the greater the leakage caused by the hysteresis effect, and the lower the volume efficiency of the pump in the process of the design optimization of the diaphragm pump. At the same time, the negative impact of the additional suction pressure of the diaphragm pump should also be considered; 3. The greater the spring stiffness, the smaller the leakage caused by the hysteresis effect, and the higher the volumetric efficiency of the pump. In the working process of the diaphragm pump design and optimization, the spring stiffness can be adjusted to improve the volume efficiency. But at the same time, the maximum flow rate of the valve gap increases, and the erosion and damage effect is also an important indicator.

References 1. Zhou, C.: Failure diagnosis of diaphragm pump check valve in slurry pipeline conveying system. Kunming University of Science and Technology, Yunnan (2020) 2. Lu, X.: A development of diaphragm transfusing pump with high concentration and solidliquid two phases. Chin. Hydraulics Pneumatics (2), 82–85 (2011) 3. Mu, Z., Huang, G., Wu, J., Fan, Y.: Early fault diagnosis of high pressure diaphragm pump check valve based on differential empirical mode decomposition. J. Vib. Meas. Diagn. 5, 758–764 (2018) 4. Chen, Y.: Early fault diagnosis of high-pressure diaphragm pump univalve based on Chaos and stochastic resonance. Kunming University of Science and Technology, Yunnan (2020)

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5. Yan, G., Zhao, J., Dong, Y.: Study on kinematics of reciprocating pimp valve. China Mech. Eng. 15(18), 1617–1619 (2004) 6. Dimitrov, S.: Investigation of static characteristics of pilot operated pressure relief valves. Ann. Faculty Eng. Hunedoara-Int. J. Eng. 11(2), 201–206 (2013) 7. Long, C., Guan, J.: A method for determining valve coefficient and resistance coefficient for predicting gas flowrate. Exp. Thermal Fluid Sci. 35(6), 1162–1168 (2011) 8. McCloy, D., Martin, H.R.: Control of fluid power: analysis and design, 505 p. Chichester Sussex, England, Ellis Horwood, Ltd; New York, Halsted Press (1980) 9. Jeong, H.S., Kim, H.E.: Experimental based analysis of the pressure control characteristics of an oil hydraulic three-way on/off solenoid valve controlled by PWM signal. J. Dyn. Syst. Meas. Control 124(1), 196–205 (2002) 10. Weaver, D.S., Ziada, S.: A theoretical model for self-excited vibrations in hydrauli gates, valves and seals. J. Pressure Vessel Technol. 102(2), 146–151 (1980) 11. Hayashi, S., Hayase, T., Kurahashi, T.: Chaos in a hydraulic control valve. J. Fluids Struct. 11(6), 693–716 (1997) 12. Turkowski, M.: Progress towards the optimisation of a mechanical oscillator flowmeter. Flow Meas. Instrum.Instrum. 14(1), 13–22 (2003) 13. Deane, J.: Flow sensing know -how. Control. Eng. 46(9), 170–172 (1999) 14. Henry, M.P., Clark, C.: Response of a Coriolis mass flow meter to step changes in flow rate. Flow Meas. Instrum.Instrum. 14(3), 109–120 (2003) 15. Adolph, U.: Berechnung des ArbeitsspielsSelbsttaitiger Ventile Sehnellaufender Kolbenpumpen. Maschinebautechnik 17(4), 189–193 (1968) 16. 毛利建太郎, 農業用往復運动ポンプ弁の 動 とそのシミし一シ,《流体工学》 , vol. 13, no. 6 (1977) 17. Li, X., Xi, Y., Xiao, D., Tao, J.: Valve plate structural optimal design and flow field analysis for aviation bidirectional three-port piston pump. J. Energies 14, 3246 (2021) 18. Wang, T., Wang, G., Dai, L., Chen, L., Qiu, S., Li, R.: Motion mechanism study on the valve disc of an ultra-high pressure reciprocating pump. J. Energies 160 (2021) 19. Zhao, J., Fu, Y., Ma, J., Fu, J., et al.: Review of cylinder block/valve plate interface in axial piston pumps: theoretical models, experimental investigations, and optimal design. Chin. J. Aeronaut. 34(01), 111–134 (2021)

Bird-Inspired Nonlinear Oscillator with Triboelectric Nanogenerator for Vibration Control and Energy Harvesting Jiayi Liu1 , Yingxuan Cui1 , Tao Yang1,2,3(B) , and Xingjian Jing2 1

3

Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, China 2 Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China [email protected] Research and Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China

Abstract. A bird-inspired nonlinear oscillator (BINO) is proposed for low-frequency vibration control and energy harvesting of bridges. By integrating the spring into the oscillator, the space utilization rate and reliability are further improved. Theoretical analysis and experimental results show that BINO with triboelectric nanogenerator damper (BINOTENGD) with quasi-zero stiffness (QZS) has excellent low-frequency vibration isolation performance and stable output voltage. In addition, BINO-TENG can produce resonance phenomenon in the frequency range of 3-8 Hz under bistable condition, and obtain high output. Finally, BINO-TENGD under QZS condition is applied to the beam bridge, and the results show that adding BINO-TENGD to the beam bridge is beneficial to suppress vibration. Therefore, BINO has a certain application prospect in the fields of bridge low-frequency vibration control, vibration detection and energy harvesting.

Keywords: Low-frequency vibration isolation Bird-inspired nonlinear oscillator

1

· Energy harvesting ·

Introduction

Bridges, as important transportation infrastructure, play an indispensable role in people’s lives. However, due to long-term use and environmental impact, bridge structures generally suffer from low-frequency vibration problems, which not only affect the safety and reliability of bridges, but also have a negative impact on ride comfort and traffic efficiency [1–3]. Therefore, vibration control technology has become the key to ensuring the safety of bridge operation and improving bridge performance. On the other hand, bridges that meet safety monitoring standards have a lot of sensors, and vibration energy harvesting technology can provide c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024  X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 761–774, 2024. https://doi.org/10.1007/978-981-97-0554-2_58

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power for some low-power sensors [4–9]. Therefore, achieving the integration of low-frequency vibration control and energy harvesting is particularly important [10–13]. Traditional linear isolators are difficult to achieve low-frequency isolation due to their high resonance frequency. In recent years, nonlinear QZS isolation technology has attracted widespread attention as an advanced vibration control method due to its high static and low dynamic stiffness characteristics [14,15]. The core idea is to achieve precise control of vibration by introducing adjustable negative stiffness components. When the amplitude is large, the negative stiffness component will provide a force opposite to the vibration direction, reducing the stiffness of the vibration system and achieving vibration isolation effect. At small amplitudes, the stiffness of negative stiffness components approaches zero, without significant impact on the system, maintaining high stiffness and ensuring structural stability [16]. Xu et al. [17] proposed a nonlinear magnetic low-frequency isolator with QZS characteristics and studied its excellent low-frequency isolation performance. Ding et al. [18] used QZS for vibration isolation of fluid conveying pipelines. Yang et al. [33,34] proposed a high-order QZS oscillator for Super low frequency vibration isolation and vibration energy harvesting. The vibration energy harvesting technology has the potential to achieve self power supply and vibration monitoring of microelectronic devices such as sensors, and has been receiving attention from scholars [19–23]. At present, the main vibration energy harvesting technologies include electromagnetic power generation [24], piezoelectric power generation [25], and frictional power generation [26]. However, electromagnetic and piezoelectric power generation are difficult to effectively collect low-frequency vibration energy. In contrast, triboelectric nanogenerator (TENG) has the advantages of excellent energy conversion efficiency and high output voltage in low-frequency vibration energy harvesting, so it is widely concerned [27]. For TENG, increasing the relative displacement between friction materials can increase the output power, so increasing the amplitude becomes an effective means to improve energy harvesting efficiency. The nonlinear bistable mechanism can significantly improve the response amplitude of the energy harvesting device, thereby improving the low-frequency energy harvesting efficiency [28–30]. Traditional linear vibration energy harvesters often cannot meet the requirements. Huguet et al. [31] studied the characteristics of harmonics and subharmonics and demonstrated that the bistable mechanism can significantly increase amplitude and expand energy acquisition bandwidth. Guan et al. [32] proposed a TENG based on a nonlinear oscillator to improve output power. Biomimetic structures are one of the current research hotspots. Inspired by the airplane mode of birds, Luo et al. [35] proposed a low-frequency broadband TENG. Feng et al. [36] observed the X-shaped support structure of the human leg, designed a vibration isolator, and verified its vibration isolation performance. Yan et al. [37] proposed a biomimetic polygonal skeleton structure inspired by the cat body skeleton. In this study, drawing on the body structure of birds, a bird-inspired nonlinear oscillator (BINO) is proposed to achieve

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low-frequency vibration control and vibration energy harvesting. The effect of TENG damping (TENGD) on vibration isolation performance under quasi zero stiffness conditions was studied, and the effect of rigid link length on energy collection performance under bistable conditions was explored.

2

Static Analysis

Fig. 1. (a) Inspired diagram of bird body structure. (b) Schematic diagram of BINO principle. (c) The resilience curve of BINO. (d) The potential energy diagram of BINO. (e) TENG’s power generation schematic diagram.

The schematic diagram of the principle of BINO device is shown in Fig. 1a. Inspired by the structure of bird bodies, BINO integrates elastic components onto the oscillator. The oscillator simulates the body of a bird, with a horizontal spring in the middle simulating the muscles responsible for flapping the wings on the chest and back of the bird, and the movement of the connecting rods on both sides simulating the flapping motion of the bird. The integrated oscillator avoids rotational displacement caused by the spring, improving space utilization and reliability. BINO does not have load-bearing capacity. When adding linear springs in the vertical direction, BINO can achieve vibration control while also harvesting energy. The simplified mechanical model of BINO parallel vertical linear spring is shown in Fig. 1b. In subsequent theoretical analysis, this model can be reduced to BINO when the stiffness of the vertical is zero. The concentrated mass is M , the stiffness of the vertical spring is K1 , the original length of the horizontal spring is Ls , the stiffness is K2 , and the length of the rigid link is L. The starting position is when the rigid connecting rod is parallel to the x-axis, yΔ

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is the displacement of the concentrated mass relative to the starting position . β is the angle at which the rigid rod rotates from its starting position, and D is the distance between the two ends of the rigid rod along the x-axis direction. y = yΔ L , b = D/L. The equivalent restoring force in the dimensionless form of BINO can be written as [38]   b F (y) = K1 y − 4K2 y 1 −  (1) 1 − y2 The potential energy U (y) and equivalent stiffness K(y) of BINO can be written as  2  K1 y 2 + 2K2 b − 1 − y 2 (2) U (y) = 2   4K 2 by 2 b (3) K(y) = K1 +  3/2 − 4K2 1 −  1 − y2 1 − y2 Make K1 = K2 , K(y = 0) = 0, the quasi zero stiffness condition for BINO can be determined as b=0.75. The potential energies of BINO at different values of b is shown in Fig. 1c. When b = 0.75, BINO has QZS characteristics and the potential energy is monostable. When b < 0.75, the potential energy of BINO is bistable and BINO has negative stiffness characteristics. When b > 0.75, the potential energy of BINO is monostable and its stiffness is nonlinear positive. Figure 1d. shows the QZS curves in the parameter space of (b, K1 ) under different stiffness K2 conditions. Based on different spring stiffness values, corresponding b can be found to meet the QZS condition.

3

Excellent Performance of BINO-TENGD for Vibration Control

TENGD is composed of TENG and a pressure regulating mechanism. The working principle of TENG is shown in Fig. 1f, where rabbit hair and polytetrafluoroethylene (PTFE) are not charged in the initial state. The rabbit hair will be charged after contact with PTFE, as shown in Fig. 1e-I. The Electronegativity of rabbit hair is lower than PTFE, so rabbit hair is positively charged, while PTFE is negatively charged. To maintain electrostatic balance, the copper electrode on the left is negatively charged, while the copper electrode on the right is positively charged. When the rabbit hair slides to the right, as shown in Fig. 1e-II. To balance the potential difference, transfer the positive charge from the right copper electrode to the left copper electrode. As the rabbit hair gradually slides to the far right, the left copper electrode carries a positive charge and the right copper electrode carries a negative charge, as shown in Fig. 1e-III. When the rabbit hair slides from right to left, in order to balance the potential difference between the copper electrodes, positive charges are transferred from the left copper electrode to the right copper electrode, as shown in Fig. 1e-IV. When the left copper electrode only carries negative charges and the right copper electrode only carries

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Fig. 2. (a) The structural schematic diagram of BINO-TENGD (b) The force transfer rate curve of BINO-TENGD under different damping cd. (c) The amplitude response under different friction conditions. (d) Voltage output under different friction conditions.

positive charges, the rabbit hair returns to the left contact state, as shown in Fig. 1e-I, completing a motion cycle. The pressure adjustment mechanism can change the pressure between the two sliders of TENG by adjusting the length of the pressure spring, thereby changing the magnitude of friction force. Therefore, TENGD can be used to adjust damping. According to the principle of equivalent viscous dampers, the dynamic friction force can be expressed as ˙ = y. ˙ f = μc N sgn(y)

(4)

Fourier series expansion of periodic square wave function sgn(x), ˙ and only basic harmonic terms are considered, while considering harmonic excitation, equivalent of damping coefficient of the Coulomb friction model is =

4μc Ks z πωA

(5)

where, μc is the friction coefficient between PTFE and Cu film, and  is the equivalent damping coefficient of the Coulomb friction model. N is the vertical pressure exerted by the pressure spring, which can be expressed as N = Ks z, z is the deformation of the pressure spring.

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The dimensionless equation of BINO-TENGD given by the extended Hamilton principle can be written as y¨ˆ(τ ) + cd yˆ˙ (τ ) − ω02 yˆ(τ ) + Λ1 yˆ3 (τ ) + Λ2 yˆ5 (τ ) + Λ3 yˆ7 (τ ) = f cos(ωτ )

(6)

where τ = ω0 t, cd = Mcω0 , ω02 = (4k2 − k1 − 4bk2 ) , Λ1 = 2bk2 , Λ2 = 32 bk2 , Λ3 = 5 4 bk2 . The excitation is considered as a harmonic vibration signal with f cos(ωτ ), f is the excitation amplitude and ω is the excitation frequency. The multi-scale method can be used to obtain the average amplitude frequency response of BINO-TENG, which can be expressed as 2

(cd ωA) +

 

3Λ1 3 5Λ2 5 35Λ3 7 ω 2 + ω02 A − A − A − A 4 8 64

2 = f2

The force transmitted to the base is written as

 2 3Λ1 3 5Λ2 5 35Λ3 7 2 2 Fr (A) = (cd ωA) + ω0 A − A − A − A 4 8 64

(7)

(8)

Then, the force transfer rate of NO-TENGD with QZS characteristics can be written as

2  2 3 7 (cd ωA) + ω02 A − 3Λ4 1 A3 − 5Λ8 2 A5 − 35Λ 64 A Tf = (9) f2 The effect of damping on the force transfer rate of BINO-TENGD is shown in Fig. 2b. It can be seen in the low-frequency range, damping has little effect on the transmission rate of force. As the damping increases, the peak of the resonance rate can be suppressed, but in the effective isolation zone, the transmission rate will increase. The increase in damping can slightly reduce the initial isolation frequency, which is the frequency at which the force transmittance is less than 1. In other words, the TENGD proposed in this paper can be used to reduce the response formant and the initial vibration isolation frequency during the vibration reduction process, and improve the low-frequency vibration isolation performance of BINO. To further validate the theoretical analysis results, experiments were conducted using the model shown in Fig. 2a as a prototype. In Fig. 2c, z represents the compression length of spring 3 that controls the pressure. z = 0 mm, z = 5 mm, and z = 10mm correspond to the conditions of no friction, small friction, and large friction, respectively. From the experimental results, it can be found that when the excitation frequency is small, the amplitude response of BINO-TENGD is larger under frictionless conditions, and the amplitude response of BINO-TENGD under high friction conditions is smaller than that under small friction conditions. The experimental results indicate that introducing TENGD friction is beneficial for vibration isolation. When the excitation frequency is high, the amplitude response of BINO-TENGD is smaller under

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frictionless conditions, and the amplitude response of BINO-TENGD under high friction conditions is smaller than that under low friction conditions. It can be seen from Fig. 2d, when the frequency is low, the output voltage under high friction conditions is higher than that under low friction conditions. As the frequency increases, the output voltage under high friction conditions gradually decreases compared to that under low friction conditions. Based on the power generation principle of TENGD shown in Fig. 2c and Fig. 1f, it can be concluded that under high friction conditions, the pressure between rabbit hair and PTFE is relatively high. Increasing surface pressure can increase the contact area between rabbit hair and polytetrafluoroethylene material, thereby improving charge transfer efficiency. Therefore, when the frequency is low and the friction force is high, the amplitude of the concentrated mass is smaller and the output voltage is larger. As the vibration frequency increases, the displacement response under small friction conditions becomes more obvious, and the absolute value of the total relative displacement of TENGD gradually increases. Therefore, the output voltage under small friction conditions gradually exceeds that under large friction conditions. BINO-TENGD can achieve both low-frequency vibration control and vibration energy harvesting.

4

BINO-TENG for Low-Frequency Vibration Energy Harvesting Under Bistable Conditions

Figure 3a shows a prototype machine used for the experiment, which explores the effects of connecting link lengths and excitation intensity on energy harvesting performance. The output voltage of BINO-TENG at different frequencies is shown in Fig. 3b. It can be seen the output voltage of BINO-TENG first increases and then decreases with the increase of frequency. When the amplitude is 20 mm, 30 mm, and 40 mm, and the frequency is 6 Hz, the maximum output voltage of BINO-TENG is 759 V, 1086 V, and 1108 V, respectively. When the excitation amplitude is 40 mm, high-energy orbital motion between BINO-TENG large amplitude is achieved at 6 Hz, and low-energy orbital motion in small amplitude wells is achieved at 2 and 9 Hz. This phenomenon also indicates that the proposed BINO-TENG can achieve broadband, low-frequency, and efficient vibration energy harvesting. When the frequency is 6 Hz, the output of BINO-TENG at 30 mm and 40 mm is almost equal to the output at 40 mm, indicating that BINO-TENG can improve the output at a fixed amplitude. The output voltages of BINO-TENG under different connecting rod lengths are shown in Fig. 3c. When the length of the connecting rod is 62.5 mm, the frequency range of output exceeding 600 V is 3.5–4.5 Hz, and the maximum output voltage of 1080V is obtained at 4 Hz. When the link length is 65.0 mm, the frequency range of output exceeding 600 V is 3.0–7.0 Hz, and the maximum output voltage of 1108 V is obtained at 6 Hz. When the link length is 62.5 mm, the frequency range of output exceeding 600 V is 5–8 Hz, and the maximum output voltage of 1112 V is obtained at 6 Hz. It can be seen when the link length is 65.0 mm, BINO-TENG has the widest output frequency and a higher output

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Fig. 3. (a) The structural schematic diagram of BINO-TENG (b) The voltage output of BINO-TENG under different constant amplitude excitations at different frequencies. (c) The variation of BINO-TENG output voltage with frequency under different connecting rod lengths. (d) The short-circuit current of BINO-TENG at different frequencies.

frequency in the frequency range of 3–7 Hz. An increase in the length of the connecting rod will cause the bistable range to shift towards higher frequencies. The short-circuit current and transfer charge of BINO-TENG at different frequencies are shown in Fig. 3d. and Fig. 4a, respectively. The short-circuit current of BINO-TENG ranges from 0.97 µA at 2 Hz to 12.17 µA at 6 Hz. Further increasing the frequency, the current is reduced to 3.14 µA at 9 Hz. The amount of transferred charge is 83.12 nC. The current and output power of BINO-TENG under different loads are shown in Fig. 4b. When the load resistance is 100 MΩ, the maximum output power is 1.25 mW. The charging curves of BINO-TENG capacitors with different capacities are shown in Fig. 4c. BINO-TENG can convert 1.0 µF, 3.3 µF, 4.7 µF, and 10 µF within 60 s, respectively Charge the capacitor to 7.8 V, 3.7 V, 2.5 V, and 1.1 V. As shown in Fig. 4d, during BINO-TENG operation, a total of 50 series connected LED lights can be instantly lit.

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Fig. 4. (a) The transfer charge amount of BINO-TENG at a frequency of 6 Hz. (b) Short circuit current, output voltage, and output power curves of BINO-TENG under different load conditions. (c) BINO-TENG charges capacitors of different capacities. (d) Schematic diagram of 50 LED lights illuminated by BINO-TENG.

5

Application Value in Bridge Vibration Control and Detection as Well as Energy Capture

Taking a multi-span continuous beam bridge as an example, the lateral motion of a uniform, isotropic, and uniform elastic beam bridge is given using the Euler Bernoulli equation, as follow [39] ∂ 4 Y (X, T ) ∂Y (X, T ) ∂ 2 Y (X, T ) + EI + ξ = η(X, T ) d ∂T 2 ∂T ∂X 4

(10)

where Y (X, T ) is the transverse deflection of the beam bridge. ξd is the external viscous damping coefficient of the beam. ρ and E are the density of the beam and Elastic modulus of the beam. S and I are the area and second moment of area of beam’s cross section. η(X, T ) represents all the external loads including the external forces due to the BINO-TENGD under each pier of the beam bridge. The dynamic equation of a simplified mechanical model can be written as 1 ∂4y ∂y ∂2y + + ς + K0 y = p0 cos(ωt) d ∂t2 ∂t π 4 ∂x4

(11)

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Fig. 5. (a) BINO used for bridge vibration control and vibration energy collection model. Figure (b)and (c) are the force transfer rate of bridge vibration with BINOTENGD. (d) The output voltage of BINO-TENGD under constant amplitude excitation of 4mm at different frequencies. 



ξd P0 Ω where y = YL , x = X L , ςd = ρSω0 , p0 = ρSL , ω = ω0 , t = T ω0 In which ω0 =  π2 EI L2 ρS , and K0 is the dimensionless stiffness of the bridge. When BINO-TENGD is added for control, the dynamic equation of the bridge under quasi zero stiffness conditions can be written as

∂2y 1 ∂4y ∂y + + (ς + μ ) + Γ0 y + Γ1 y 3 + Γ2 y 5 + Γ3 y 7 = p0 cos(ωt) d d ∂t2 ∂t π 4 ∂x4 where μd =

cd  , Γ0 ρSω0

=

3ω02 L2  , Γi 2ρSω0

=

3γˆi L2  2ρSω0

(12)

L represents the length of the bridge.

Excitation is a harmonic vibration signal P0 cos(ΩT ), P0 is the excitation amplitude, Ω is the excitation frequency. The boundary conditions under simply supported conditions can be written as  ∂ 2 y(x, t)  y(x, t)|x=0,x=1 = 0 · · · and · · · =0 (13) ∂x2 x=0,x=1 Therefore, the amplitude frequency response and transmission rate can be written as    2 W (Q, ω) = (ς d + μd ωQ) + (Γ

0

2

+ ω1 − ω

2

 Q+

3Γ1 3 5Γ2 5 35Γ3 7 Q + Q + Q 4 8 64

2

2

− p0 = 0

(14)

Tb =

 

((ς + μ ) ωQ)2 + (Γ + ω 2 Q + d  d 0 1 p0 2

3Γ1 3 4 Q

+

5Γ2 5 8 Q

+

35Γ3 7 64 Q

2 (15)

where ω1 is the first-order dimensionless natural frequency of the beam bridge.

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As shown in Fig. 5b, by comparing the force transmissibility of the beam bridge with and without BINO-TENGD, it is found that the frequency of the amplitude resonance peak of the bridge with BINO-TENG moves to low frequency, and the resonance peak decreases. The results indicate that using QZS to install BINO-TENGD can effectively reduce the response amplitude of the bridge. The effects of damping on the force transfer rate of BINO-TENGD is shown in Fig. 5c. As the damping increases, the resonant peak is effectively suppressed. The response amplitude and transmission rate outside the resonant frequency band hardly change with the variation of damping. The results indicate that adding BINO-TENGD to the beam bridge is beneficial for suppressing vibration. The output voltage at different frequencies is shown in Fig. 5d. The output voltage at different frequencies is stable and has significant differences. It can be used for monitoring the vibration frequency of bridges.

6

Conclusion

This article proposes a bird-inspired nonlinear oscillator, and studies its excellent performance in vibration control and vibration energy collection. BINO improves reliability and space utilization by integrating springs onto the oscillator, making it easier to couple with other mechanical structures. Theoretical analysis shows that BINO-TENGD under QZS conditions can achieve low-frequency vibration isolation function. In addition, the damping can be increased by adjusting the pressure of TENGD, which can effectively suppress the resonance peak value of the response amplitude and the force transmissibility, reduce the vibration isolation initial frequency, and improve the vibration isolation performance. The experiment verified the amplitude response and output voltage of BINOTENGD under different excitation frequencies and pressure conditions, verified the theoretical analysis results, and proved that BINO-TENGD has excellent low-frequency vibration isolation performance and stable voltage output, which can be used for vibration detection. Setting the stiffness of the vertical spring in BINO-TENGD to zero can degenerate into BINO-TENG with bistable characteristics for low-frequency vibration energy harvesting. The experimental results indicate that BINO-TENG has a high output power in the wide range of 3–7 Hz. BINO-TENG has the highest output at an excitation frequency of 6 Hz, amplitude of 40 mm, and link length of 65.0 mm. By changing the length of the connecting rod, the frequency range of the bistable range can be adjusted. The short-circuit current of BINO-TENG is 12.17 µA. The output voltage is 1108 V, and the optimal output power is 1.25 mW. BINO-TENG can drive 50 LED lights or charge capacitors. Finally, BINO-TENGD under QZS conditions was installed on the beam bridge, and the response amplitude and force transfer ability were proposed to evaluate its isolation performance, and compared with the isolation performance without isolators. The results indicate that using BINO-TENGD under QZS conditions can achieve low-frequency vibration suppression, and increasing damping can reduce the response amplitude and force transfer rate of the beam bridge

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within the resonant frequency range. Therefore, BINO can be applied to the integration of low-frequency vibration isolation and vibration energy collection for bridges. At the same time, monitoring the relationship between output voltage and frequency to monitor bridge vibration has good engineering application prospects. Acknowledgement. This work was supported by National Natural Science Foundation of China (Granted Nos. 12002272 and 12272293), and Guangdong Basic and Applied Basic Research Foundation (Granted No. 2022A1515010967, 2023A1515012821). TY wishes to thank the supports from Hong Kong Scholar.

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Improvement of Small Target Detection Algorithm Based on YOLOV5 Shoujun Lin, Lixia Deng(B) , Huanyu Chen, Lingyun Bi, and Haiying Liu Qilu University of Technology (Shandong Academy of Sciences), Jinan, China [email protected]

Abstract. Target detection has always been a difficult problem in computer vision. The commonly used target detection algorithms are Two stage and One stage. In order to address small ground targets and reduce the storage of UAV, the paper present an improved method on the basis of YOLOV5. All C3 modules are replaced by in the backbone network. It can improve the speed and accuracy of the network target detection, and improve the efficiency of the model training and inference. A new feature fusion mechanism is proposed, which greatly improves the feature acquisition and fusion capabilities for networks. A 160*160 detection head is added at the end of the network. Experimental Specific improvements resulted in a 3.6% increase in recall, a 3.9% increase in mAP_0.5, and a 2.8% increase in mAP_0.5:0.95. Keywords: target detection · YOLOv5 · UAV

1 Introduction Target detection, as a popular direction of computer vision and image processing, is now widely used in pedestrian detection, face detection, text detection, traffic labeling and traffic light detection, remote sensing target detection and many other aspects [1]. It is of great practical significance to reduce the use of human resources through the application of computer vision. As a fundamental algorithm in the field of identity recognition, target detection plays a crucial role in subsequent tasks such as face recognition, gait recognition, crowd counting and example segmentation. At present, large and medium-sized target detection has achieved remarkable results, but with the rapidly evolving intelligent systems and the wide application of portable photographic devices, many fields need to obtain key information from small targets. This fully demonstrates that small target detection has important research value and application prospects. However, due to the low resolution and limited information of small targets, the key features are easily lost in the down sampling process, which leads to the serious phenomenon of missed detection and false detection. In addition the model performance is also subject to the interference of light intensity changes, image noise, complex background, target occlusion and other factors, which exacerbate the detection difficulty [2]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 775–784, 2024. https://doi.org/10.1007/978-981-97-0554-2_59

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Target detection has two main algorithms in the market, which are one-stage detection algorithm and two-stage detection algorithm. Small target detection is mainly applied to one-stage detection algorithm, one-stage algorithm detection efficiency, real-time is good, but the precisions are not as high as the two-stage algorithm. In 2016, the YOLO algorithm was proposed by Joseph Redmond et al. [3]. The algorithm solves the object detection as a regression problem, the input image after one inference, it can get the position of all the objects in the image and its belonging to the category and the corresponding confidence probability. It is a pioneer in one-stage detection algorithms. After four years of development, the YOLO algorithm has made great progress, and YOLOv4 [4] and YOLOv5 appeared successively in 2020.YOLOv4 replaces the backbone network Darknet53 in YOLOv3 with CSPDarkNet53; the Neck part introduces the SPP (Spatial Pyramid Pooling) [5] The sensory field is expanded, and the PAN (Path Aggregation Network) [7] module is added on top of the FPN (Feature Pyramid Networks) [6] module in YOLOv3 to better fuse the feature map information at different scales.YOLOv4, in addition to maintaining the original detection accuracy, utilizes numerous optimization strategies to improve the detection speed by another big step. After development, the birth of YOLOv5 brings the YOLO series into a new era with the introduction of several YOLOv5l models based on the YOLOv5l model. Compared to YOLOv4, YOLOv5 takes up less memory, is easier to deploy, and is faster to detect. In small target detection, the phenomena of closely arranged objects, different target morphologies, and more noise are important reasons for the difficulty of small target detection. In order to solve these problems, after reviewing the performance of many standard target detection algorithms, we finally chose to use YOLOv5s, a lightweight alternative to YOLOv5s, as the baseline model, and improve the model’s small-target detection capability by optimizing its network structure. The improvement model’s performance was demonstrated on the VisDrone2019 dataset. The main research described is as follows: 1) The modified C4 module is used instead of the C3 module in YOLOV5s backbone network. 2) Improve the network structure of YOLOV5s using deeper feature fusion to enhance the feature extraction and fusion capability of the network. 3) A 160 * 160 detection head is added at the end of the network structure to improve the network detection speed and accuracy.

2 Basic Algorithm Framework and Introduction Taken together, YOLOv5s meets the conditions on computational resources, detection accuracy, deployment difficulty, etc. Therefore, YOLOv5s is chosen as the basic algorithm and improved, and the network architecture of YOLOv5 is illustrated in Fig. 1. The YOLOv5 model consists of four parts, i.e., Input, Backbone (backbone network), Neck network and Output. At the input side YOLOv5 network model is augmented with Mosaic data augmentation with reference to CutMix data augmentation to randomly zoom in, randomly zoom out, randomly crop, and randomly layout the dataset [8]. The backbone network is used to extract image features, including the main structures: the Focus structure, the CSP structure, the C3 module, and the SPPF. Using the

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Fig. 1. YOLOv5 network architecture

Focus slicing operation [9], the planar information of the image is transformed into the channel dimension, and the two-fold down sampling operation can be realized with the guarantee of non-destructive image information, which can help to increase the speed of the network; the CSP is essentially a multiple-use residual network structure. The use of the CSP module allows the learning ability of the network to be improved, making the trained model lightweight, while maintaining high accuracy, and also reducing the difficulty of deployment and budgetary costs [10]; The main module that performs residual learning is C3, which is divided into two main structures. One using multiple bottleneck stacks as described above, and the other going through a base convolution module. And finally the two branches are subjected to stitching operation; the use of C3 module is less computationally intensive and more accurate; SPPF, while retaining the original function of SPP, further fuses feature maps of different sizes and sizes, improves the characterization ability of the feature maps, and the operation speed is further improved. In the feature fusion structure, YOLOv5 uses the structure of FPN spliced PAN as shown in Fig. 2. FPN is a top-down process that combines feature information with the lower level features through the upper level sampling operation in order to compute the predicted feature maps [11]. The YOLOv5 network structure adds a layer of feature pyramid with two bottom-up PAN structures after the FPN level. Predictive feature maps are generated by a sampling process at the lower level, where higher level features are merged with lower level features.

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A top-down FPN structure is used to convey strong semantic features and enhance feature extraction, while a bottom-up feature pyramid is used to convey strong localization features, enhance feature fusion, and perform parameter aggregation for different backbone and detection layers.

Fig. 2. Feature fusion networks

A loss function is set at the output of the YOLOv5 network structure to further optimize the network performance. The loss function of YOLOv5 mainly uses CIOU_Loss [12], which integrates the prediction frame and the target frame to improve the detection speed and accuracy. The calculation is as follows  2 wgt w 4 arctan − arctan π2 hgt h v α= 1 − IoU + v   ρ 2 b, bgt LCIoU = 1 − IoU + + αv c2 v=

(2.1) (2.2) (2.3)

Confidence loss and classification loss use binary cross entropy and sigmoid function. YOLOv5 has a strong feature extraction capability, which is sufficient to deal with various scales of target detection tasks in daily scenes, and it can be said that the algorithm’s detection and recognition performance is already good enough. However, YOLOv5 still has room for improvement when used in embedded devices with insufficient memory and to ensure a certain detection efficiency [13].

3 Improved YOLOv5 3.1 Improved C4 Module In order to better adapt to the detection targets of different scales and sizes, increase the sensory field of the network, and improve the feature extraction ability of the network, this paper proposes a new C4 module on the basis of YOLOV5 C3 instead of the C3 module in the backbone network. The basic idea is to reduce the convolution of two of the first layer to one, and then add a 1 * 1 convolution after the first layer, as shown in Fig. 3.

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Fig. 3. C4 network module

Experiments have proved that the improved C4 module has a great improvement in enhancing the feature extraction ability of the network, and through the superposition of convolutional layers, YOLOV5 can extract efficient and useful target features to accomplish efficient and accurate target detection. 3.2 Feature Fusion Improvement In YOLOv3 and subsequent versions of the YOLO series of algorithms, the features extracted from the backbone network need to be fully fused at the Neck end before they are fed into the detection layer to predict the object category and location information. Therefore, the effect of feature fusion at the Neck side is very important for the detection results. In this paper, the following main points are added to the feature fusion: 1) After the first C4 module of the backbone network, a feature output is added and connected to the Neck end, and the feature fusion is performed at the same scale, which is input to the P4 detection layer for prediction after the C3 module and convolution and other operations. 2) After the second C4 module, a feature output is elicited and it is feature fused to the second C3 module at the Neck end. 3) Add a feature output after the first convolution at the Neck end and fuse it with the sixth convolution for feature fusion. (As indicated by the orange line in Fig. 4) Experiments demonstrate the optimization of the feature fusion structure of YOLOv5s. This fusion method can well fuse deep features with shallow features and enhance the information transfer between feature maps. It reduces the effect of feature information loss due to up sampling and down sampling, and effectively improves the feature fusion capability of the original YOLOv5s.

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3.3 Adding Detection Heads The original YOLOv5 model has 3 output layers, P1, P2 and P3, which are used to detect small, medium and large targets respectively. With the addition of the P4 detection layer, the network is able to do detection on 4 different scales of feature maps, and its network structure is shown in Fig. 4. Since the P4 detection head fuses shallow and large resolution feature maps, it can divert more shallow feature messages to deeper feature maps during the down-sampling procedure. Therefore, the introduction of P4 detection head makes the network extra sensitive to small targets. On the one hand, it increases the depth of the network, so that the network can better learn the multilevel feature information of the target, and enhances the network’s ability to detect multi-scale targets in complex environments. On the other hand, it can pass more shallow features into deep features, so that the network acquires more information about small targets and improves the detection ability of the model for small targets.

Fig. 4. Improved YOLOv5 network structure

4 Experimental Results The dataset used in this paper is VisDrone2019, which is rich in scenarios, including parking lots, highways, residential areas, etc. It simulates as much as possible the conditions that UAVs may encounter in real environments, and it can verify whether the improved algorithm is pervasive and stable. The operating system used for the experiments in this paper is Windows 11, the system running memory is 16 GB, the CPU model is I5–13 generation, the GPU model is NVIDIA-RTX3060, and the video memory is 12G.The algorithm is based on the PyTorch deep learning framework, and uses the Python programming language, and uses CUDA11.1 and the corresponding version of CUDNN for the computing acceleration.

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During the training process of the algorithmic model, the total number of training rounds is set to 200, a stochastic gradient descent strategy is used, the initial learning rate is set to 0.001, momentum and weight decay are set to 0.937 and 0.0005 respectively, and mosaic data augmentation and a priori frame adaptive tuning strategies are used. The metrics of the YOLOV5 and improved YOLOV5 algorithms are shown in Figs. 5 and 6, which show that the training loss has been reduced to a relatively small degree, and the metrics tend to converge after 200 training sessions.

Fig. 5. Results of the classical YOLOv5s algorithm

Fig. 6. Results of improved YOLOv5s algorithm

The performance comparison of the two algorithms on the VisDrone2019 test set is shown in the figure above, from which we can see that the performance of the improved YOLOV5 is improved in all categories, with a 3.6% increase in recall, a 3.9% increase in mAP_0.5, and a 2.8% increase in mAP_0.5:0.95.

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The detection results are shown in Fig. 7 and Fig. 8, which demonstrate the detection results of the traditional YOLOv5s algorithm and the improved algorithm of this paper, respectively. It is not difficult to see that in Fig. 8, not only the missing targets in Fig. 7 are accurately labeled, but also some incorrect labels in the original figure are corrected, and it can be seen that the improved algorithm’s detection accuracy has been greatly improved, and the results of detection are satisfactory.

Fig. 7. Detection results of YOLOv5s

Fig. 8. Detection results of Improve YOLOv5s

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Through the above experimental comparison, we can see that the improved algorithm has made great progress in all indicators compared with YOLOv5s. YOLOv5s algorithm will have a slight fitting phenomenon during training, but the improved algorithm does not have such a phenomenon during training. Obviously, the improved algorithm is more suitable for small target detection and is more in line with real-world applications. Table 1. Comparative results Method

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[email protected]:0.95

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32

17.1

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34.8

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35.9

19.9

In Table 1, It can be seen that the improved algorithm, in terms of all the indicators, has a more obvious improvement compared with YOLOv5s and YOLOv4-tiny.

5 Summarize In this paper, an improved small target detection algorithm based on YOLOV5s is proposed. The main idea is to replace the C3 module in the YOLOV5S backbone network with a C4 module, and then YOLOV5s is used for deeper feature fusion and a P4 detection head is added, which greatly improves the detection performance. The detection effect is good in each performance evaluation index. It can recognize the targets in the image well and has more advantages in small target detection. Obviously, the algorithm is an improved algorithm more suitable for small target detection tasks.

References 1. Liu, L., Ouyang, W., Wang, X., et al.: Deep learning for generic object detection: a survey. Int. J. Comput. Vision 128(2) (2020) 2. Chen, G., et al.: A survey of the four pillars for small object detection: multiscale representation, contextual information, super-resolution, and region proposal. IEEE Trans. Syst. Man Cybern. Syst. 52(2), 936–953 (2022) 3. Redmon, J., Divvala, S., Girshick, R., et al.: You only look once: unified, real-time object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 779–788 (2016) 4. Bochkovskiy, A., Wang, C.Y., Liao, H.Y.M.: Yolov4: optimal speed and accuracy of object detection. arXiv preprint arXiv:2004.10934 (2020) 5. He, K., Zhang, X., Ren, S., et al.: Spatial pyramid pooling in deep convolutional networks for visual recognition. IEEE Trans. Pattern Anal. Mach. Intell. 37(9), 1904–1916 (2015) 6. Lin, T.Y., Dollár, P., Girshick, R., et al.: Feature pyramid networks for object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2117– 2125 (2017)

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7. Liu, S., Qi, L., Qin, H., et al.: Path aggregation network for instance segmentation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 8759–8768 (2018) 8. Lang, X., Ren, Z., Wan, D.: MR-YOLO: an improved YOLOv5 network for detecting magnetic ring surface defects. Sensors 22(24), 9897 (2022) 9. 李鸿. 基于轻量化网络的目标检测算法研究. 中国科学院大学(中国科学院光电技术研 究所) (2022) 10. Spatial pyramid pooling in deep convolutional networks for visual recognition. IEEE Trans. Pattern Anal. Mach. Intell. 37(9), 1904–1916 (2015) 11. Lin, T., Dollár, P., Girshick, B.R., et al.: Feature pyramid networks for object detection. CoRR, abs/1612.03144 (2016) 12. Zheng, Z., Wang, P., Liu, W., et al.: Distance-IoU loss: faster and better learning for bounding box regression. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 07 (2020) 13. Zhang, H., Deng, L., Bi, L., Liu, H.: Small object detection algorithm based on improved YOLOv5. In: 2023 IEEE International Conference on Control, Electronics and Computer Technology (ICCECT), Jilin, China, pp. 280–283 (2023)

Aerodynamic Analysis of a Double Elastic Panel-Cavity with One Side Exposed to Supersonic Flow Hao Liu, Yegao Qu(B) , and Guang Meng School of Mechanical Engineering, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Minhang District, Shanghai 200240, China [email protected]

Abstract. In this paper, the aeroelastic stability and nonlinear response of a structural-acoustic system that consists of two panels with an acoustic cavity between them is investigated. A non-linear higher-order shear deformation zig-zag theory is adopted to model the composite laminated panel, and the linear wave equation is employed for modeling the compressible fluid in the cavity. Based on the Piston aerodynamic theory, a theoretical approach with the finite element method for solving instability modes and the transient response of the panel-cavity aerodynamic system is suggested. The aeroelastic instability boundary under different dynamic pressures is calculated, and the influence of the stiffness ratio between the top and bottom elastic panels is analyzed. Keywords: Fluid-structure-acoustic interaction · Nonlinear supersonic flutter · Panel-cavity system

1 Introduction The aeroelastic problem involving fluid-structure-acoustic interaction is commonly encountered in aerospace and astronautics science and engineering. For example, when one side of a thin-walled composite structure (e.g., aircraft skins and paneling) is exposed to airflow, vibration noise is radiated into the cabin. Simultaneously solving the multi-physics fields involved in aeroelasticity and investigating the complex interactions between the external flow, the structure, and the internal acoustic environment has strong engineering requirements and practical significance. The nonlinear aeroelastic dynamics of isotropic and composite panels in supersonic flow is a focal research topic that has been well understood by experimental, theoretical, and numerical research for decades. The classical panel flutter problem is mainly about determining the critical value of non-dimensional dynamic pressure, beyond which the panel exhibits a limit cycle oscillation due to nonlinear structural restoring forces. The supersonic flow may be modeled based on the piston aerodynamic theory, the linearized potential flow theory, and the quasi-steady aerodynamic theory, which are classified as theoretical modeling. Bolotin [1] was one of the earliest to propose a theoretical model © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 785–796, 2024. https://doi.org/10.1007/978-981-97-0554-2_60

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that considered the effect of different in-plane boundary constraints in the context of nonlinear supersonic fluid-structure stability (flutter and buckling). Surveys and reviews of the early theoretical studies were provided by Dowell [2] and Mei [3]. The majority of theoretical works [4–6] coupled the von Kármán plate equations with a linearized aerodynamic model to analyze the stability of panels in supersonic flow. The nonlinear limit cycle oscillation (LCO) motions of isotropic, functionally graded material [4], and constant or variable stiffness composite laminate (CSCL or VSCL) [5, 6] panels were considered. The aerodynamic solutions can also be achieved by numerical methods based on the Euler equations or the Navier-Stokes equations [7, 8]. Numerical solutions provide more accurate predictions of the flow field but typically require higher computational costs. For an elastic panel backed with an acoustic cavity and subjected to supersonic flow. Ganji and Dowell [9] studied the noise transmission into and radiation from a rectangular cavity through a flexible structure subjected to supersonic flow. Freydin and his co-workers [10, 11] carried out a fully coupled aero-thermal-acoustic-elastic analysis by the theoretical method. The flutter instability region due to the coalescence of structural and cavity modes was observed. The present study intends to extend the conventional flutter analysis of composite panels to fluid-structure-acoustic interactions of a complex double panel-cavity system in supersonic flow. Here, a general higherorder shear deformation zig-zag theory is adopted to model the geometric nonlinearity of the composite laminated panel, and the compressible fluid in the cavity is modeled by an acoustic wave equation. Based on the first-order Piston aerodynamic theory, the aerodynamic, the nonlinear composite panel, and the acoustic cavity are formulated as a strongly coupled aeroelastic system based on the finite element method. The effects of the aerodynamic pressure and stiffness difference between the elastic panels on the flutter dynamics of the panel-cavity aeroelastic system are discussed.

2 Numerical Methodology 2.1 Model Description The physical model of the two-dimensional fluid-structure-acoustic interaction model that consists of two panels with an acoustic cavity between them is illustrated in Fig. 1(a). The upper surface of the top panel is exposed to supersonic airflow with a velocity of U∞ = M∞ c∞ and a compressible stationary fluid is supposed in the cavity. M∞ and c∞ are the Mach number and speed of sound of the freestream. The flow densitycan be calculated with the state equation using the freestream static pressure: ρ∞ = P∞ RT∞ , where R represents the gas constant. The properties of the stationary compressible air in the cavity are set equal to the free stream properties, i.e. the density ρa and the speed of sound ca are set equal to the free stream static conditions. The static pressure differential across the panel is supposed to be equal to zero, and the boundaries of the cavity are treated as rigid wall boundary conditions except for the top and bottom flexible panels. Figure 1(b) depicts the schematic diagram of the laminated panel. In this study, both top and bottom composite laminated panels have a length of L and a thickness of H, with rigidly clamped edges. A local Cartesian coordinate system O-XYZ is defined on the central axis of the laminated panel with Nth layers, where the X-axis is aligned

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along the central line and the Z-axis is in the thickness direction. The isotropic material parameters for the kth layer are: elastic modulus E (k) , shear modulus G (k) , Possion’s ratio v(k) and density (ρs )(k) . The orthotropic material parameters of the kth layer are: (k) (k) (k) (k) (k) (k) elastic moduli E1 and E2 , shear moduli G12 , G13 and G23 , Possion’s ratio v12 and density (ρs )(k) . The thickness of the kth layer of the laminate is Hk = Zk+1 − Zk , where Zk+1 and Zk are the thickness coordinates of the upper surface and lower surface of the layer, respectively. In addition, the ply-angle of the kth layer is defined as θk .

Fig. 1. Schematic diagram of (a) double elastic panel-cavity with top side exposed to supersonic flow, and (b) a laminated panel with several layers.

2.2 Structural Modeling of Composite Laminated Panel The beam assumption is adopted for the panel modeling, and it is assumed that the deformations of the panel are characterized by its center line and only take place in the X-Z plane. The displacement vector [ u˜ s w˜ s ] of an arbitrary point in the composite laminated panel is expressed as:  (X ,t) u˜ s (X , Z, t) = us (X , t) + f (Z) ∂ws∂X + g(Z)ϑs (X , t) + ϕ(Z, k)ηs (X , t) (1) w˜ s (X , Z, t) = ws (X , t) where us and ws represent the displacement components of the panel along the X and Z directions, respectively. ϑs and ηs are the generalized displacement variables of the mid-plane. In this study, Reddy’s theory [12] is used,  therefore, the associated shape functions f (Z) = −4Z 3 3H 2 and g(Z) = Z − 4Z 3 3H 2 . ϕ(Z, k) is a zigzag function, defined as [13]:     2 Zk+1 + Zk 8Z 3 k Z− (2) − ϕ(Z, k) = (−1) Hk 2 3Hk H 2 Following this, the von Kármán strain-displacement assumption is adopted to account for the large deformation of the panel. And for the beam model, neglecting the stresses and strains in the thickness direction (Z-direction) of the beam, the stress-strain relationship yields: ˜ (k) εxz ˜ (k) εxx , σxz = ks Q σxx = Q 11 55

(3)

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 (k) T  (k) (k) −1  (k)  (k) 2  (k) (k) (k) Q12 Q22 Q26 Q12 (k) (k) ˜ ˜ Q11 = Q11 − Q44 (4) (k) (k) (k) (k) , Q55 = Q 55 − Q 55 Q16 Q26 Q66 Q16 (k)

where σxx and σxz are the normal and shear stress components of the kth layer. Qij indicates the material constant with respect to the global coordinate system after the application of the equation of transformation. k s is a shear correction factor, defined as k s = 1 for higher-order shear deformation beam theories. The governing equations of the composite laminated panel are derived based on the virtual work approach. A two-node beam element is developed to discretize the panel here. Linear interpolation functions are used to discretize the generalized displacement variables us , ϑs and ηs , while the Hermite functions are adopted to interpolate the transversal variable ws . The finite element formulation for an undamped composite laminated panel can be written as:

(5) Ms q¨ s + Ksl + Ksnl qs = fa where Ms is the mass matrix of the structure. Ksl and Ksnl are respectively the linear and nonlinear stiffness matrix of the structure. fa represents the internal acoustic force vector applied on the structure nodes. For detailed derivation and explicit forms of the structural matrices, the reader is referred to [14]. 2.3 Acoustic Modeling of the Compressible Fluid in Cavity The propagation of sound in an acoustic domain with density ρa is described by the linear wave equation. Given the speed of sound ca in the acoustic domain, the partial differential equation expressed in acoustic pressure is: ∂ 2 pa − ca2 ∇ 2 pa = 0 (6) ∂t 2 A four-node quadrilateral element is adopted to discretize the acoustic domain, and the bilinear interpolation functions are used to discretize the pressure variables pa . The finite element formulation of Eq. (14) is derived [15]: Ma p¨ a + Ka pa = fs 1 Ma = 2 ca





a

NaT Na dV , Ka

=

(7)

(∇Na ) ∇Na dV , fs = T

a

∂ a

NaT naT ∇pa dS

(8)

where Ma and Ka are the mass matrix and stiffness matrix of the acoustic domain, respectively. fs denotes the boundary force term of the acoustic domain. If part of the boundary is rigid, the boundary condition is imposed by setting ∇pa = 0. If the boundary is interacting with a flexible structure, the following conditions must be fulfilled: ∇pa = −ρa u¨ s , fs = −npa

(9)

where n denotes the surface normal vector ate the structure-acoustic interface pointing into the acoustic domain.

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2.4 Piston Aerodynamic Theory of Supersonic Flow The theoretical aeroelastic analysis of the laminated panel backed with a cavity in a freestream flow with properties ρ∞ , U∞ , and M∞ is considered here. The aerodynamics is approximated by the first-order Piston Theory which dictates the following relation between pressure perturbation and the local motion of the panel:

 ∂ws ∂ws ∂ws ρ∞ U∞ ∂ws pPT = + U∞ = λPT + gPT (10) M∞ ∂t ∂x ∂t ∂x where ws is the transversal deflection of the panel. The resulting aerodynamic damping and stiffness matrix is defined as DPT and KPT , respectively.

3 Theoretical Monolithic Model of the Panel-Cavity Aeroelastic System The monolithic finite element expression for a two-dimensional panel-cavity aeroelastic problem is derived. At the boundary between the structural and acoustic domains, the wall-normal velocity of the beam interface must coincide with the wall-normal component of the acoustic particle velocity, and pressure fluctuations of the acoustic field induce a force on the interface. Using the relation in Eq. (10), the boundary force fs of the acoustic domain can be described in terms of structural acceleration, and the coupling forces fa in Eq. (6) can be written as:    fs = −ρa NaT naT Ns dS u¨ s = Csa Hq¨ s , fa = − 1 ρa HT CTsa pa (11) ∂ sa

Moreover, considering the Piston aerodynamic theory in Eq. (11), the coupled aeroelastic system of equations within the Lagrangian formulation is established as:       q¨ s DPT H 0 q˙ s Ms 0 + Csa H Ma p¨ a 0 0 p˙ a         l nl 0 Ks + Ks + KPT H − 1 ρa HT CTsa qs = + (12) 0 Ka pa 0 where Csa is the structural/acoustic coupling matrix, and H represents the transformation matrix between the transversal displacement and the generalized displacement vector. In this study, a Newmark-β scheme combined with the Newton-Raphson iterative method is used to obtain the nonlinear vibration response of the vibroacoustic system. By choosing the algorithm’s coefficients as β = 1/4 and α = 1/2, this implicit algorithm is unconditionally stable and known as the method of constant acceleration.

4 Verification The modal verification is conducted to validate the proposed monolithic method in the vibroacoustic problem of the panel-cavity system first. The natural frequencies of an isotropic panel-cavity model with fixed supports at both ends are considered. The

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material parameters of the isotropic panel are Young’s modulus E = 1.528 × 1010 Pa, Poisson’s ratio ν = 0.3, and density ρs = 390 kg m−3 . The flexible panel has a length L of 10 m and a height H of 0.1 m, and the cavity has a depth D of 5 m. The remaining walls of the cavity are modeled as rigid. The speed of sound and density of the acoustic fluid are respectively 299.5 m s−1 and 0.39 kg m−3 . The panel is discretized into 50 finite element segments, and the cavity is discretized with four-node quadrilateral elements of size 50 × 10 (50 segments in the span direction and 10 segments in the depth direction). In Table 1, the natural frequencies of the double elastic panel-cavity system are compared with the FEM solutions calculated by the commercial finite element software COMSOL. The present results are in excellent agreement with that of FEM solutions, with a relative error of less than 4% (Fig. 2). Table 1. Natural frequencies of the double elastic panel-cavity system. Mode

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18.262

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18.845

29.891

Fig. 2. The first six modes of the double elastic panel-cavity system.

Then, the aeroelastic behavior of a flexible panel is investigated to validate the proposed method. In this case, the cavity coupling term is neglected thereby not considering the acoustic effect. The simulation condition is set to the same as Gordnier’s work [7].The panel is characterized by a thickness ratio of H/L = 0.002, mass ratio of μ = ρf L ρs H = 0.1, and Poisson’s ratio of 0.3. The supersonic airflow is in the positive x-direction with the Mach numbers M∞ = 1.8. Figure 3 shows the non-dimensional limit cycle oscillation amplitude values ws /H as a function of the non-dimensional dynamic

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 2 L3 D at the measured location x/L = 0.75 in the panel, where pressure λ = ρf U∞  D = EH 3 12(1 − ν 2 ) for isotropic material. Gordnier’s [7], and Herjranfer’s [8] results are compared with the proposed results. The proposed method shows excellent agreement in terms of flutter onset point and amplitude growth.

Fig. 3. Variation of limit cycle oscillation amplitude with dynamic pressure at x/L = 0.75.

5 Results and Discussion 5.1 Effects of Dynamic Pressure on Flutter Dynamics In this section, linear modal analysis and nonlinear transient analyses are performed to investigate the influences of dynamic pressure on the aeroelastic dynamics of the panel-cavity system. The Mach number of the supersonic airflow is 2. Freestream static pressure is equal to 25000 Pa, and static temperature is set as 223 K. The composite panel with clamped ends has a thickness ratio of H/L = 0.002 which is modeled as a four-layered laminated structure with fiber paths [θ / − θ /θ / − θ ], where θ = 20 deg. It is assumed that each layer has the same orthotropic material, and the orthotropic material properties are: E1 , E2 = E1 /10, G23 = G13 = G12 = E1 /20, v12 = 0.3, and mass ratio μ = according to 0.1. In this  study, the values of2 the3 density ρs and modulus3E1 are calculated 2 ) for orthotropic L D1 , where D1 = E1 H 12(1 − ν12 μ = ρf L ρs H and λ = ρf U∞ material. The rectangular cavity between the panels has a length of L and a depth of D = 0.5L. In the numerical analysis, the finite element mesh of the panel-cavity system consists of 50 finite element segments for the panel and 50 × 100 quadratic elements ˆ for the cavity. A non-dimensional time-step size  of t = 0.001 are selected to carry out the following simulations, where ˆt = tU∞ L. The eigenvalue solutions (real and imaginary parts) of the first seven modes with different aerodynamic pressures are shown in Fig. 4. A local mode coalescence phenomenon appears in a small range of aerodynamic pressures, i.e., an acoustic dominant mode coalesces with a structural dominant mode altered by the aerodynamic interaction between λ = 500 and 580. It is noteworthy that a mode veering is also observed in the dynamic pressure range of local mode coalesce, since the mode frequency dominated

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by the bottom panel is hardly unaffected by the dynamic pressure. In addition, Fig. 5 provides the linear mode shapes of the panel-cavity system at aerodynamic pressure λ = 520, 550, and 800. As the dynamic pressure increases further, the first mode frequency increases while the second mode frequency decreases before the onset point of the flutter (λ = 700). When λ is larger than a critical value, the aeroelastic flutter of the panel-cavity system happens due to the coalescence of lower structural dominant modes of the top panel.

Fig. 4. Linear eigenvalue solutions: (a) real part and (b) imaginary part of the flutter mode frequencies as a function of non-dimensional dynamic pressure λ.

Fig. 5. Linear mode solutions with (a) λ = 520, (b) λ = 550, and (c) λ = 800.

Following this, the nonlinear flutter dynamics behaviors of the panel-cavity system are assessed by using the numerical method. Two types of flutter dynamics behaviors under typical aerodynamic pressure conditions (λ = 550, 800) are investigated. The deformation envelope of the panel in an LCO period, the time-domain response, and the Fast Fourier Transform (FFT) curves of the measurement point located at the x/L = 0.75 position of the panel for fifty dimensionless time steps are given in Fig. 6. It can be seen that the deformation envelopes of the panel coincide with the mode solution. In the case

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λ = 550, a primary feature is the occurrence of acoustic resonance in the cavity, which results in the bottom panel showing a large amplitude LCO motion. While in the case λ = 800, the amplitude of the top panel is significantly larger than that of the bottom panel.

Fig. 6. Nonlinear numerical results: the deformation envelope of the panel in an LCO period, the time domain response, and the FFT curve of the transversal displacement at x/L = 0.75 under dynamic pressure (a) λ = 550 and (b) λ = 800.

5.2 Effects of Stiffness Ratio on Flutter Dynamics This section concerns the effects of the stiffness difference between the bottom panels and the top panel on flutter dynamics. The geometrical and material properties of the panel are introduced in the previous section. Several cases are considered with a stiffness ratio E b /E t = 0.4, 4, and 20, where E b and E t denote the elastic modulus of the bottom panel and top panel, respectively. Figure 7, Fig. 8, and Fig. 9 give the eigenvalue solutions (real and imaginary parts) of the first seven modes as a function of aerodynamic pressure with different stiffness ratios. For the case E b /E t = 0.4, the 7th mode frequency which is determined by the bottom panel becomes higher, therefore, the mode veering does not occur in the dynamic pressure range of local mode coalescence. Furthermore, it is interesting to note the local mode coalescence between the acoustic dominant mode and the structural dominant mode does not occur when the stiffness ratio is equal to 4. For the case E b /E t = 20, it is observed that the acoustic resonance has a lower frequency. Based on the linear mode analyses, three typical cases of flutter dynamics behaviors under aerodynamic pressure λ = 550 and different stiffness ratios (E b /E t = 0.4, E b /E t = 1, E b /E t = 20) are investigated. From Fig. 10, as the stiffness ratio increases, the modes of bottom panels the switches to a lower one, and the LCO amplitude of the bottom panel decreases obviously.

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Fig. 7. Linear eigenvalue solutions: (a) real part and (b) imaginary part of the flutter mode frequencies as a function of non-dimensional dynamic pressure λ (E b /E t = 0.4).

Fig. 8. Linear eigenvalue solutions: (a) real part and (b) imaginary part of the flutter mode frequencies as a function of non-dimensional dynamic pressure λ (E b /E t = 4).

Fig. 9. Linear eigenvalue solutions: (a) real part and (b) imaginary part of the flutter mode frequencies as a function of non-dimensional dynamic pressure λ (E b /E t = 20).

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Fig. 10. Nonlinear numerical results: the deformation envelope of the panel in an LCO period, the time domain response, and the FFT curve of the transversal displacement at x/L = 0.75 under dynamic pressure λ = 550 and stiffness ratio (a) E b /E t = 0.4, (b) E b /E t = 1, and (c) E b /E t = 20.

6 Conclusions A monolithic numerical method is developed for predicting the fluid-structure-acoustic coupling behaviors of the supersonic aeroelastic panel-cavity system. The numerical finite element framework couples the Piston aerodynamic theory, the nonlinear structural equations, and wave acoustic equations. The validation of the method is confirmed by comparing the present results with reference solutions in the literature. The aeroelastic stability and transient responses of two panels with an acoustic cavity between them are investigated. The results show that at a small range of aerodynamic pressure, the mode veering phenomenon and local mode coalescence occurs between the acoustic dominant mode and the structural dominant mode of the aeroelastic system. The acoustic resonance flutter significantly decreases the aeroelastic stability bound of the panelcavity system, resulting in a significantly large amplitude LCO motion of the bottom panel. The effect of the stiffness ratio between the top and bottom panels is analyzed. A particular phenomenon is that the acoustic resonance flutter does not occur at a particular stiffness ratio. The acoustic resonance flutter with a lower frequency is observed at a large stiffness ratio. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. U2141244, 11932011, 12121002, 12002203), the Oceanic Interdisciplinary Program of Shanghai Jiao Tong University (Grant No. SL2021ZD104), and the Science and Technology Cooperation project of Shanghai Jiao Tong University & Inner Mongolia Autonomous

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Region-Action Plan of Shanghai Jiao Tong University for “Science and Technology Prosperity” (Grant No.2022XYJG0001-01-08).

References 1. Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability. Macmillan, New York (1963) 2. Dowell, E.H.: Panel flutter - a review of the aeroelastic stability of plates and shells. AIAA J. 8, 385–399 (1970) 3. Mei, C., Abdel-Motagaly, K., Chen, R.: Review of nonlinear panel flutter at supersonic and hypersonic speeds. Appl. Mech. Rev. 52, 321–332 (1999) 4. Navazi, H.M., Haddadpour, H.: Nonlinear aero-thermoelastic analysis of homogeneous and functionally graded plates in supersonic airflow using coupled models. Compos. Struct. 93, 2554–2565 (2011) 5. Stanford, B.K., Jutte, C.V., Wu, K.C.: Aeroelastic benefits of tow steering for composite plates. Compos. Struct. 118, 416–422 (2014) 6. Rahmanian, M., Farsadi, T., Kurtaran, H.: Nonlinear flutter of tapered and skewed cantilevered plates with curvilinear fiber paths. J. Sound Vib. 500, 116021 (2021) 7. Gordnier, R.E., Visbal, M.R.: Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter. J. Fluids Struct. 16, 497–527 (2002) 8. Hejranfar, K., Azampour, M.H.: Simulation of 2D fluid-structure interaction in inviscid compressible flows using a cell-vertex central difference finite volume method. J. Fluids Struct. 67, 190–218 (2016) 9. Ganji, H.F., Dowell, E.H.: Sound transmission and radiation from a plate-cavity system in supersonic flow. J. Aircr. 54, 1877–1900 (2017) 10. Freydin, M., Dowell, E.H.: Fully coupled nonlinear aerothermoelastic computational model of a plate in hypersonic flow. AIAA J. 59, 2725–2736 (2021) 11. Freydin, M., Dowell, E.H., Spottswood, S.M., Perez, R.A.: Nonlinear dynamics and flutter of plate and cavity in response to supersonic wind tunnel start. Nonlinear Dyn. 103, 3019–3036 (2021) 12. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press, Florida (2003) 13. Carrera, E.: On the use of the Murakami’s zig-zag function in the modeling of layered plates and shells. Comput. Struct. 82, 541–554 (2004) 14. Qu, Y., Long, X., Li, H., Meng, G.: A variational formulation for dynamic analysis of composite laminated beams based on a general higher-order shear deformation theory. Compos. Struct. 102, 175–192 (2013) 15. Sandberg, G., Ohayon, R.: Computational aspects of structural acoustics and vibration, Springer, New York, 24-71 (2009). https://doi.org/10.1007/978-3-211-89651-8 16. Dowell, E.H.: Nonlinear oscillations of a fluttering plate. AIAA J. 4, 1267–1275 (1966)

Observer-Based Robust Control for Active Suspension Systems by Employing Beneficial Disturbances and Coupling Effects Menghua Zhang1 , Zengcheng Zhou2 , Qiang Liu3 , and Xingjian Jing4(B) 1 School of Electrical Engineering, University of Jinan, Jinan 250022, China 2 Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong,

China 3 Shaodong Luruan Digital Technology Co., LTD. Smart Energy Branch, Jinan 250000, China 4 Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China

[email protected]

Abstract. In this paper, a novel observer-based robust control method is designed for an active suspension system with inevitable disturbances as well as coupling effects. The difference from the existing control methods is that the designed control method specifically investigates the influences of disturbances as well as couplings. This method effectively mitigates the negative disturbance and coupling effects while retaining the positive effects, leading to enhanced robustness. To achieve this, a nonlinear disturbance observer is employed to accurately estimate both parametric/uncertainties and external disturbances. And on this basis, positive or negative effect indicators that assess the impact of disturbances and couplings on the active suspension system, thereby incorporating them into the controller design. The entire suspension system’s asymptotic stability is ensured by the Lyapunov technique. Experimental results are presented to validate the effectiveness of the proposed control method. Keywords: Active Suspension System · Observer · Robust Control · Coupling

1 Introduction The rapid growth the automotive industry has led to increasingly stringent requirements for ride comfort and driving safety. Suspension systems, as a crucial component of the vehicle chassis, have a significant impact on ride comfort, stability, and road handling, attracting considerable attention [1]. Active suspensions, in particular, offer superior vibration isolation performance when the vehicle encounters road irregularities [2]. Consequently, researchers have been developing various control methods for active suspension systems to enhance their performance, including both linear [3–5] and nonlinear approaches [6–9]. To optimize control performance, it is necessary to thoroughly investigate and consider several practical challenges that arise in active suspension systems. These challenges include unavoidable disturbances (such as parametric uncertainty, unmodeled © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 797–806, 2024. https://doi.org/10.1007/978-981-97-0554-2_61

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uncertainty, and external disturbances), as well as inherent nonlinear coupling effects. Addressing these issues is crucial in designing effective controllers for active suspensions. In the existing literature, several control methods have been proposed to tackle these practical issues. For example, to mitigate the impact of unexpected disturbances, numerous robust control methods [10–12] based on disturbance observers have been designed for active suspension systems. In general, inherent coupling terms in control system is often treated as model uncertainties, resulting in various challenges in controller design. To address these challenges, several robust methods [13, 14], which are insensitive to uncertainties, have been designed. As described in [15, 16], the coupling effects in the system are characterized as either useful or harmful. An alternative strategy in controller design is to enhance control performance by eliminating harmful coupling effects while retaining the useful ones. This approach aims to exploit the inherent coupling effects to improve system performance and resilience to disturbances. As a consequence, these robust control methods and disturbance observer-based control methods provide effective means to attenuate the effects of unexpected disturbances and uncertainties in active suspension systems. They ensure robustness, stability, and improved performance even in the presence of model uncertainties and coupling effects. Until now, the existing literature lacks well-developed approaches to address uncertainties, couplings, and disturbances simultaneously in active suspension systems. None of the existing methods fully consider all these issues to harness the potential beneficial effects. To overcome this gap, an observer-based robust control method is proposed for active suspensions, aiming to exploit the positive effects resulting from couplings, as well as disturbances to achieve superior control performance. The rest of this paper is concluded as follows. In Sect. 2, the mathematical model of active suspension systems is provided. In Sect. 3, the main results including the nonlinear disturbance observer, disturbance and coupling effect indicators, the disturbance observer-based robust control method, as well as the stability analysis, are given. In Sect. 4, several experimental results are illustrated to verify the effectiveness of the designed control method. In Sect. 5, some conclusion remarks are provided.

2 Mathematical Model for Active Suspension Systems The dynamic equations for active suspension systems (shown in Fig. 1) are described as follows: ms z¨s = −Fs − Fd + d1 + u(t)

(1)

mu z¨u = Fs + Fd − Ft − Fb + d2 − u(t)

(2)

with ms as well as mu referring to the sprung and unsprung masses, respectively, zs , zu , and zr standing for the displacements of the sprung mass, unsprung mass, and the road surface, respectively, d1 as well as d2 being the disturbances, u(t) referring to the control input, Fs as well as Fd standing for the spring as well as damper forces, respectively, Ft as well as Fb denoting the tire elasticity as well as damping forces, which are mathematically expressed by: Fs = ks (zs − zu )

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Fd = kd (˙zs − z˙u )

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where ks stands for the stiffness coefficient, kd denotes the damping coefficient, kt and kb refer to the tire stiffness as well as damping coefficients, respectively, expressed by: ks = k s (1 + s )

(7)

kd = k d (1 + d )

(8)

kt = k t (1 + t )

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kb = k b (1 + b )

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within k s , k d , k t , as well as k b standing for corresponding nominal values, s , d , t , as well as b referring to the perturbance ranges of corresponding coefficients. Substituting (7) and (8) into (1), one has ms z¨s = −k s (1 + s )(zs − zu ) − k d (1 + d )(˙zs − z˙u ) + d1 + u(t)

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= −k s zs − k d z˙s − k s zu − k d z˙u − k s s (zs − zu ) − k d d (˙zs − z˙u ) + d1 + u(t) = − k s zs − k d z˙s − k s zu − k d z˙u +  + u(t)

(11)

where  = −k s s (zs − zu ) − k d d (˙zs − z˙u ) + d1 represents the lumped disturbance, −k s zu − k d z˙u stands for the state coupling effect resulting from the unsprung mass acceleration system (2). Assumption 1: Regarding the active suspension system (11), it is known that the lumped disturbance  is bounded, in the sense that || ≤ λ

(12)

where λ refers to the top limitation of .

3 Main Results 3.1 Nonlinear Disturbance Observer Design In order to prompt subsequent disturbance observer design, we define the observation error as follows: ˜ ˆ = −

(13)

˜ stands for the observation error,  ˆ refers to the estimated value of . Based where  on the structure of (11), the following disturbance observer is constructed as follows: ˆ =β 1 + β2

(14)

  β˙1 = − lβ1 − l β2 + u(t) − k s zs − k d z˙s − k s zu − k d z˙u

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β2 = lms z˙s

(16)

with β1 and β2 referring to two auxiliary functions, l standing for positive observer gain. Theorem 1: Given the intentionally designed nonlinear disturbance observer (14)–(16), ˜ will always remain within the predefined it can be concluded that the observation error  permissible range, which can be expressed as follows:   ˜ (17)  ≤ bd ˜ which will be defined later. where bd stands for the top limitation of , Proof: Let’s differentiate (14) and substitute (11), (15) and (16) into the resulting equation, yields ˙ˆ =β˙ + β˙  1 2

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  = − lβ1 − l β2 + u(t) − k s zs − k d z˙s − k s zu − k d z˙u   + l −k s zs − k d z˙s − k s zu − k d z˙u +  + u(t) ˆ + l = − l

(18)

From (18), it can be easily derived that ˙˜ − l  ˜ + ˙ =

(19)

Solving (19), it is derived that −Lt + λ ˜ ≤ (0)e ˜   l    ˜  ˜  λ →   ≤ (0) + l

(20)

≤ bd   ˜  + λ . ˜ ˜ bd = (0) where (0) denotes the initial value of , l 3.2 Coupling and Disturbance Effects Indicators To fully utilize the advantages of beneficial state-coupling and disturbance, two indicators for state-coupling and disturbance effects are proposed and defined as follows. Definition 1: For the active suspension system (11), we define the state coupling effect indicator as follows:    (21) J1 = sgn (˙zs + 2α arctan(zs )) −k s zu − k d z˙u Based on (21), the state-coupling effect on the active suspension system (11) is introduced as ⎧ ⎨ J1 < 0, State - coupling effect is advantageous (22) J > 0, State - coupling effect is disadvantageous ⎩ 1 J1 = 0, State - coupling effect is nil Definition 2: For the active suspension system (11), the disturbance effect indicator is constructed as follows:   ˆ (23) J2 = sgn zs  Based on (23), the disturbance effect on the active suspension system (11) is defined as

⎧ ⎨ J2 < 0, Disturbance effect is beneficial J > 0, Disturbance effect is detrimental ⎩ 2 J2 = 0, Disturbance effect is nil

(24)

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3.3 Observer-Based Robust Control Method Theorem 2: For the active suspension system (11), if the observer-based robust control method is designed as follows: 2kP 2kD u(t) = − arctan(zs ) − arctan(˙zs ) − kS sgn(˙zs + 2α arctan(zs )) π   π ˆ + k s zu +k d z˙u F(J1 ) − F(J 2)

(25)

with kP , kD , kS and α referring to positive control gains, F(J1 ) and F(J2 ) standing for the following functions:

1, J1 ≥ 0 (26) F(J1 ) = 0, J1 < 0

1, J2 ≥ 0 F(J2 ) = (27) 0, J2 < 0 then the sprung mass displacement zs will asymptotically converge to 0, in the sense that lim zs = 0

t→∞

Proof: The Lyapunov function candidate is constructed as follows:   1 2kP 1  2 2 V = ms z˙s + zs arctan(zs ) − ln 1 + zs 2 π 2 + 2αms z˙s arctan(zs )

(28)

(29)

The detailed derivation is omitted due to the page limit.

4 Experimental Results and Analysis In this section, the practical control performance of the designed observer-based control method with respect to sinusoidal road input (shown in Fig. 2) is validated, and the passive suspension system is selected as the comparative method.

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The vehicle body acceleration results of passive suspension system as well as the designed control method concerning sinusoidal road input are shown in Fig. 3. It is shown that, the proposed controller suppresses the vehicle vibration into a smaller range when being compared with the passive suspension system, implying better ride comfort of the designed control method.

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Figures 4, 5 and 6 show the control input, the suspension space, and the dynamic tyre load. It is shown that the suspension space as well as the dynamic tyre load of the active suspension system and the proposed control method remain in the allowed scopes. 5

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5 Conclusions A disturbance observer-based robust control method is designed in this paper to address several practical issues, including robustness, unavoidable disturbance, and inherent state-coupling effect. The aim is to improve vehicle ride comfort by taking these factors into consideration. The key component of this control method is the nonlinear disturbance observer, which is designed to precisely estimate the disturbance in the system. By accurately estimating the disturbance, the control method can effectively compensate for its effects and make appropriate adjustments in real-time. This allows for a more efficient control strategy that maintains good ride comfort. The control method also considers the inherent state-coupling effect of active suspension systems. The coupling effect is often present in real-world systems but are not adequately addressed in existing control methods. By incorporating these effects into the control design, the proposed method can better adapt to system dynamics and improve overall performance. Importantly, this paper claims to differentiate itself from existing literature results by strategically and beneficially addressing estimated disturbance and state-coupling effect. The proposed method aims to provide superior robustness and improved ride comfort, which have been validated through experiments. It is worth noting that this study presents a completely new approach to the design of robust controllers and claims to be the first to employ beneficial state-coupling and disturbance. Acknowledgements. This work was supported in part by the National Natural Science Foundation of China under Grant No. 62273163, the Outstanding Youth Foundation of Shandong Province Under Grant No. ZR2023YQ056, the Startup Fund of City University of Hong Kong under Grant No. 9380140, the Key R&D Project of Shandong Province under Grant No. 2022CXGC010503.

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References 1. Na, J., Huang, Y., Wu, X., Su, S., Li, G.: Adaptive finite-time fuzzy control of nonlinear active suspension systems with input delay. IEEE Trans. Cybern. 50(6), 2639–2650 (2020) 2. Wang, T., Li, Y.: Neural-network adaptive output-feedback saturation control for uncertain active suspension systems. IEEE Trans. Cybern. 52(3), 1881–1890 (2022) 3. Sun, W., Gao, H., Kaynak, O.: Finite frequency H∞ control for vehicle active suspension systems. IEEE Trans. Control Syst. Technol. 19(2), 416–422 (2011) 4. Wang, R., Karimi, H.R., Chen, N.: Robust fault-tolerant H∞ control of active suspension systems with finite-frequency constraint. Mech. Syst. Sig. Process. 62–63, 341–355 (2015) 5. Yin, S., Huang, Z.: Performance monitoring for vehicle suspension system via fuzzy positivistic c-means clustering based on accelerometer measurements. IEEE/ASME Trans. Mechatron. 20(5), 2613–2620 (2015) 6. Zhang, M., Jing, X., Zhang, L., Huang, W., Li, S.: Toward a finite-time energy-saving robust control method for active suspension systems: exploiting beneficial state-coupling, disturbance, and nonlinearities. IEEE Trans. Syst. Man Cybern. Syst. (in press). https://doi.org/10. 1109/TSMC.2023.3277439 7. Zeng, Q., Liu, Y., Liu, L.: Adaptive vehicle stability control of half-car active suspension systems with partial performance constraints. IEEE Trans. Syst. Man Cybern. Syst. 51(3), 1704–1714 (2021) 8. Li, P., Lam, J., Lu, R., Li, H.: Variable-parameter-dependent saturated robust control for vehicle lateral stability. IEEE Trans. Control Syst. Technol. 30(4), 1711–1722 (2022) 9. Liu, Y., Chen, H.: Adaptive sliding mode control for uncertain active suspension systems with prescribed performance. IEEE Trans. Syst. Man Cybern. Syst. 51(10), 6414–6422 (2021) 10. Pan, H., Sun, W., Gao, H., Hayat, T., Alsaadi, F.: Nonlinear tracking control based on extended state observer for vehicle active suspensions with performance constraints. Mechatronics 30, 363–370 (2015) 11. Ginoya, D., Shendge, P.D., Phadke, S.B.: Sliding mode control for mismatched uncertain systems using an extended disturbance observer. IEEE Trans. Industr. Electron. 61(4), 1983– 1992 (2014) 12. Guo, Z., Guo, J., Zhou, J., Chang, J.: Robust tracking for hypersonic reentry vehicles via disturbance estimation-triggered control. IEEE Trans. Aerosp. Electron. Syst. 56(2), 1279– 1289 (2020) 13. Rath, J.J., Defoort, M., Sentouh, C., Karimi, H.R., Veluvolu, K.C.: Output constrained robust sliding mode based nonlinear active suspension control. IEEE Trans. Industr. Electron. 67(12), 10652–10662 (2020) 14. Yan, S., Sun, W., He, F., Yao, J.: Adaptive fault detection and isolation for active suspension systems with model uncertainties. IEEE Trans. Reliab. 68(3), 927–937 (2019) 15. Guo, Z., Zhou, J., Guo, J., Cieslak, J., Chang, J.: Coupling-characterization-based robust attitude control scheme for hypersonic vehicles. IEEE Trans. Industr. Electron. 64(8), 6350– 6361 (2017) 16. Guo, Z., Ma, Q., Guo, J., Zhao, B., Zhou, J.: Performance-involved coupling effect-triggered scheme for robust attitude control of HRV. IEEE/ASME Trans. Mechatron. 25(3), 1288–1298 (2020)

Parametric Study on Performance of Parallel Asymmetric Nonlinear Energy Sinks Huiyang Li1,2 and Jianen Chen1,2(B) 1 Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control,

School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China [email protected] 2 National Demonstration Center for Experimental Mechanical and Electrical Engineering Education, Tianjin University of Technology, Tianjin 300384, China

Abstract. The complexification-averaging method is utilized to study the amplitude-frequency response of a system connected to a parallel asymmetric nonlinear energy sink (NES) under harmonic excitation, and the Runge-Kutta method is used to analyze the change law of the frequency band of the higher branches of response and the strongly modulated responses with the assistance of the complexification-averaging method. Under shock loading, the effect of parallel asymmetric NES parameters on its vibration absorption efficiency is investigated. The results demonstrate that the frequency band of higher branches of response is widened by reducing the stiffness ratio, mass ratio, and damping ratio of the two NES, emphasizing the fact that the frequency band of the strongly modulated response can be widened or narrowed with the reduction of these parameters. Moreover, reducing the mass ratio and damping ratio of the two NESs decreases the vibration absorption efficiency when the system is subjected to a shock load. However, the vibration absorption efficiency increases by reducing the stiffness ratio. Keywords: nonlinear energy sink · asymmetry · higher branches of response · energy dissipation

1 Introduction In order to overcome the shortcomings of traditional linear vibration absorbers, the nonlinear energy sink (NES) with the characteristics of small mass and wide band vibration absorption are proposed, and had become an important research direction for vibration damping and isolation [1, 2]. Scholars connect traditional NES in series or parallel, and optimize its parameters to improve NES’s vibration absorption performance. Fixed the total mass of NES, Li et al. [3] verified that multiple NESs connected in parallel will get better vibration absorption performance when the primary oscillator has a small initial displacement, and Song et al. [4] proposed a parallel NES and proved that its performance is better than that of purely cubic NES. The generation of higher branches of response requires greater excitation amplitude. In addition, the system connected with © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 807–819, 2024. https://doi.org/10.1007/978-981-97-0554-2_62

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parallel NES has shorter attenuation time under shock excitation. Chen et al. [5] connected two parallel NESs through a linear spring, and found that the vibration absorption performance of parallel-coupled NES was higher than that of parallel NES under moderate excitation amplitude, but the performance was slightly worse under small or large excitation amplitude. They also studied the system in which multiple NES are connected in series [6], and the parameters of each NES vary proportionally. It is concluded that each oscillator can produce chaotic responses with different amplitudes at the same excitation frequency. Boroson et al. [7] verified that a higher energy dissipation ratio can be obtained in a wider excitation range with the increase of the number of NES in parallel. In addition, the structural improvement of NES is also a method to enhance its vibration absorption performance and broaden its application range. By introducing electromagnetic system, Xu et al. [8] improved the vibration absorption performance of NES under harmonic excitation and shock excitation, and provided a new strategy for the system with insufficient damping. Bak et al. [9] studied the response of the system connecting multi-degree-of-freedom NES with binary tree structure. Wei et al. [10] combined a cubic NES and vibro-impact NES to form a new NES, which not only retains the performance of the two NESs, but also has better vibration absorption performance than the optimal value of the cubic NES. Wang et al. [11] studied the influence of geometric parameters of piecewise linear NES on multiple solutions, and found that the branching of multiple linear segments is more complicated within the research scope. Zhang et al. [12] described the generating conditions of the system’s strongly modulated response, and verified that NES with the combined nonlinear damping has good vibration absorption performance. Wang et al. [13] used differential evolution algorithm to optimize the double springs NES, and found that the excitation amplitude is easier to change the optimal parameter range of NES. AL-Shudeifat et al. [14] put forward the magnetic NES by using the repulsive force of the permanent magnet, which provided an idea for further studying the vibration reduction performance of the system. Geng et al. [15] put forward an encapsulated NES, which not only has better vibration absorption performance than traditional NES, but also effectively suppresses double peaks. Geng et al. [16] also reinforced the cubic NES by permanent magnets and achieved good vibration absorption effect. In this paper, the higher branches of response and the strongly modulated responses of parallel asymmetric NES, as well as the shock absorption efficiency, are investigated. Using the asymmetry characteristics between the parameters of the two NES, the frequency band of the system generating higher branches of response and strongly modulated responses under harmonic excitation is investigated, and the influence of asymmetry between the parameters under shock load on the energy dissipation ratio is analyzed.

2 Mechanical Model Figure 1 depicts the system consisting of a linear primary oscillator with a single degree of freedom and a parallel asymmetric nonlinear energy sink, and the dynamic equation is derived by Newton’s law: M x¨ 1 + K0 x1 + C0 x˙ 1 + K1 (x1 − x2 )3 + C1 (˙x1 − x˙ 2 ) + K2 (x1 − x3 )3 + C2 (˙x1 − x˙ 3 ) = F cos(t)

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M1 x¨ 2 + K1 (x2 − x1 )3 + C1 (˙x2 − x˙ 1 ) = 0 M2 x¨ 3 + K2 (x3 − x1 )3 + C2 (˙x3 − x˙ 1 ) = 0

(1)

where x1 , x2 and x3 are the displacements of the linear primary oscillator and two purely nonlinear oscillators; K0 and C0 the stiffness coefficient and damping coefficient of the linear primary oscillator; K1 and K2 are the stiffness coefficients of two purely nonlinear oscillators; C1 and C2 are the damping coefficients of two purely nonlinear oscillators; F and  are excitation amplitude and frequency, respectively.

Fig. 1. Mechanical model.

Dimensionless the parameter in Eq. (1): ε1 =

M1 M2 , ε2 = , τ= M M



K0 Ci Ki F , ci = √ t, f = (i = 0, 1, 2), ki = (i = 0, 1, 2) M K0 K0 K0 M

Then, Eq. (1) can be converted to: x¨ 1 + k0 x1 + c0 x˙ 1 + k1 (x1 − x2 )3 + c1 (˙x1 − x˙ 2 ) + k2 (x1 − x3 )3 + c2 (˙x1 − x˙ 3 ) = f cos(τ )

ε1 x¨ 2 + k1 (x2 − x1 )3 + c1 (˙x2 − x˙ 1 ) = 0 ε2 x¨ 3 + k2 (x3 − x1 )3 + c2 (˙x3 − x˙ 1 ) = 0

(2)

Set y1 = x1 − x2 and y2 = x1 − x3 , introduce the following variable substitution: x˙ 1 + ix1 = α1 eit , x˙ 1 − ix1 = α 1 e−it y˙ 1 + iy1 = α2 eit , y˙ 1 − iy1 = α 2 e−it y˙ 2 + iy2 = α3 eit , y˙ 2 − iy2 = α 3 e−it

(3)

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where α1 , α2 , and α3 are functions of t, and α 1 , α 2 , and α 3 represent the conjugation of α1 , α2 , and α3 , respectively, and i is an imaginary unit. From Eq. (3), we obtain: x1 =

α1 eit − α 1 e−it α1 eit + α 1 e−it , x˙ 1 = 2i 2

x¨ 1 =

α˙ 1 eit + iα1 eit + α˙ 1 e−it − iα 1 e−it 2

y1 =

α2 eit − α 2 e−it α2 eit + α 2 e−it , y˙ 1 = 2i 2

y¨ 1 =

α˙ 2 eit + iα2 eit + α˙ 2 e−it − iα 2 e−it 2

y2 =

α3 eit − α 3 e−it α3 eit + α 3 e−it , y˙ 2 = 2i 2

y¨ 2 =

α˙ 3 eit + iα3 eit + α˙ 3 e−it − iα 3 e−it 2

(4)

Substituting Eq. (4) into Eq. (2) and eliminating the fast-variable term, the slowvariable equation is as follows: α˙ 1 + iα1 +

3k2 α22 α 2 3k3 α32 α 3 k1 α1 + c1 α1 + + c α + + c3 α3 = f 2 2 i 4i3 4i3

ε1 (α˙ 1 + iα1 − α˙ 2 − iα2 ) −

3k2 α22 α 2 − c2 α2 = 0 4i3

ε2 (α˙ 1 + iα1 − α˙ 3 − iα3 ) −

3k3 α32 α 3 − c3 α3 = 0 4i3

(5)

Set α1 = a1 + ib1 , α2 = a2 + ib2 , and α3 = a3 + ib3 , where a1 , b1 , a2 , b2 , a3 and b3 are functions of τ , substitute them into Eq. (5), and separate the real and imaginary parts to obtain:     3k2 a22 + b22 b2 3k3 a32 + b23 b3 k1 b1 a˙ 1 = b1 − − c2 a2 − − c3 a3 + f − c1 a1 −  43 43     3k a2 + b22 a2 3k3 a32 + b23 a3 ˙b1 = −a1 + k1 a1 − c1 b1 + 2 2 − c2 b2 + − c3 b3  43 43     3k2 a22 + b22 b2 3k3 a32 + b23 b3 k1 b1 − c1 a1 − a˙ 2 = − − c2 a2 − 3  43  4   2  1 3k2 a2 + b22 b2 −c3 a3 + f + b2 − + c2 a2 ε1 43

Parametric Study on Performance of Parallel Asymmetric

    3k2 a22 + b22 a2 3k3 a32 + b23 a3 a k 1 1 − c1 b1 + − c2 b2 + b˙ 2 =  43 43    2  2 3k2 a2 + b2 b2 1 −c3 b3 − a2 − + c2 a2 − ε1 43     3k2 a22 + b22 b2 3k3 a32 + b23 b3 k1 b1 − c1 a1 − a˙ 3 = − − c2 a2 − 3  43  4   2  2 1 3k3 a3 + b3 b3 −c3 a3 + f + b3 − + c3 a3 ε2 43     3k a2 + b22 a2 3k3 a32 + b23 a3 ˙b3 = k1 a1 − c1 b1 + 2 2 − c2 b2 +  43 43    2  2 3k3 a3 + b3 b3 1 −c3 b3 − a3 − + c3 a3 − ε2 43

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(6)

To obtain the steady-state response solution of the system, set the sum of the a˙ 1 = a˙ 2 = a˙ 3 =  0 and b˙ 1 = b˙ 2 = b˙ 3 = 0, and the amplitudes of the oscillators can be calculated as A =

a12 +a22 . 

3 Characterization Study of the System To ensure the correctness of the calculation process, this paper analyzes the system using the complexification-averaging method and the Runge-Kutta method, and compares the results of the semi-analytical and numerical solutions. The values of the parameters in the system are as follows: ε1 = 0.08, ε2 = 0.02, c0 = 0.01, c1 = c2 = 0.005, k0 = 1, k1 = 0.8, k2 = 0.2. Figure 2 shows the frequency response of the linear primary oscillator at f = 0.003; the dots and circles represent the results of the complexificationaveraging method and the Runge-Kutta method, respectively. The results obtained by the two methods are consistent, which can verify the correctness of the derivation process. In the following study, the relationship between the stiffness, mass, and damping of the two NES is k1 + k2 = 1, m1 + m2 = 0.1, and c1 + c2 = 0.01. 3.1 Band Interval of Higher Branches of Response First, the frequency response of the primary oscillator connected to parallel asymmetric NES is investigated. By varying the stiffness ratio, mass ratio, damping ratio of the two NESs, and excitation amplitude of the system, the variation of the frequency bands that produces higher branches of response is analyzed. When one parameter ratio changes independently, the other two ratios remain constant at 4:1. Figure 3 depicts the effect of various parameters on the frequency bands that produces higher branches of response. As shown by the four subplots in Fig. 3, the frequency bands of the higher branches of response gradually expand as the excitation amplitude increases. As shown in Fig. 3(a), reducing the stiffness ratio, mass ratio, and damping ratio of the two NES

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Fig. 2. Frequency response of the linear primary oscillator at f = 0.003

simultaneously results in a wider frequency band of higher branches of response at the same excitation amplitude, and higher branches of response can be generated at a lower excitation amplitude. Figure 3(b) also reveals that as the stiffness ratio decreases, the higher and lower branch solutions merge at smaller excitation amplitudes, resulting in a significantly reduced vibration absorption performance of the NES and a higher risk for failure. Figures 3(c) and 3(d) depict the aforementioned patterns, which generate higher branches of response in multiple frequency bands simultaneously.   In Fig.  4, the frequency response of the linear primary oscillator at k1 k2 = c1 c2 = 4:1, ε1 ε2 = 1:3 is given, where the black points are the stable solution and the blue points are the unstable solution. In Fig. 4(a), the entire frequency band is a stable solution at f = 0.002. As shown in Fig. 4(b), when f = 0.004, the unstable solutions occur in the band of  = 0.985~1.022, which is generally manifested as a strongly modulated response, and the vibration absorption of NES is better in this band. In Fig. 4(c), when f = 0.006, the amplitude of the primary oscillator increases significantly, and there are two stable solutions and one unstable solution in the frequency band of  = 0.954~0.99. In Fig. 4(d), when the excitation amplitude f = 0.008, a higher branch occurs in the band of  = 0.826~0.897, and the peak value increases significantly, accompanied by a lower branch and an unstable solution. In Fig. 4(e), when the excitation amplitude continues to increase to f = 0.009, another higher branch occurs in the  = 0.734~0.76 band, but the amplitude increases slightly in the entire band. In Fig. 4(f), when f = 0.011, the two higher branch and lower branch tend to merge. The generation of higher branch indicates that the amplitude of the primary oscillator will increase under certain circumstances, jumping from the lower branch to the higher branch can causing the sudden failure of the NES.

f

f

f

813

f

f

f

f

f

Parametric Study on Performance of Parallel Asymmetric

f

f

f

(b) changing the stiffness ratio

f

f

f

f

f

(a) changing the stiffness, mass and damping ratios simultaneously

(c) changing the mass ratio

(d) changing the damping ratios

Fig. 3. Influences of the parameters on the frequency band of higher branches of response

3.2 Frequency Band of Strongly Modulated Response Strongly modulated responses of the system can significantly increase the energy transferred from the primary oscillator to the purely cubic oscillator, thereby enhancing the NES’s vibration absorption performance and decreasing the amplitude of the primary oscillator. The effect of the parameters of parallel asymmetric NES on the frequency band of strongly modulated response is illustrated in Fig. 5. In Fig. 5(a), the stiffness ratio, mass ratio, and damping ratio of the two NES change simultaneously, while the corresponding frequency band of strongly modulated response under the same excitation amplitude remains almost unchanged. In addition, with the increase of the excitation amplitude, the  frequencyof the strongly modulated response gradually widens. In Fig. 5(b), taking k1 k2 = ε1 ε2 = 4:1, as the damping ratio decreases, the frequency

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Fig. 4. Frequency response of the primary oscillator connected to a parallel asymmetric NES.

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f f f f

f

f

f

f

band in which the strongly modulated responses exist is wider, and the excitation amplitude  required  to produce the strongly modulated responses is smaller. In Fig. 5(c), fixed k1 k2 = c1 c2 = 4:1, as the mass ratio decreases, the strongly modulated responses can be produced at a smaller excitation amplitude. When the strongly modulated responses only appeared in one frequency band, the range under the same excitation amplitude becomes wider. When there are two frequency bands, the range under the same excitation amplitude gradually narrows,  and theinterval between the two frequency bands gradually increases. In Fig. 5(d), c1 c2 = ε1 ε2 = 4:1, reducing the stiffness ratio of the two NES under the same excitation amplitude, the frequency band where the strongly modulated responses exists gradually widens when f ≤ 0.003, and the opposite is true when f > 0.003. With the decrease of the stiffness ratio, strongly modulated responses no longer exist at certain excitation amplitudes, and the range of excitation amplitudes with this property increases gradually.

(b) changing the damping ratios

f

f

f

f

f

f

f

f

(a) changing the stiffness, mass and damping ratios simultaneously

(c) changing the mass ratio

(d) changing the stiffness ratio

Fig. 5. The influence of the parameters on the frequency band of the strongly modulated response

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4 Performance under Shock Excitation This section analyzes the effect of the ratios of stiffness, mass, and damping of the two NESs on the energy dissipation ratio under shock load. First, the energy of the linear primary oscillator without NES connection is E1 , and the energy of a linear primary oscillator connected to a parallel asymmetric NES is E2 , as shown in Eq. (7):  1 τ 2 x1 + x˙ 12 dt (7) E1,2 = 2 0 Then, the energy dissipation ratio of the parallel asymmetric NES to the linear primary oscillator is as follows: η=

E1 − E2 E1

(8)

Figure 6 illustrates the change in the energy dissipation ratio of NES with shock excitation amplitude. In Fig. 6(a), the energy dissipation ratio of the NES is almost unchanged when the stiffness, mass, and damping ratios are simultaneously changed. Figure 6(b) demonstrates the change in the energy dissipation ratio of NES with respect to the amplitude of the shock excitation when only the stiffness ratio is changed. Take  ε1 ε2 = c1 c2 = 4:1, although the stiffness ratio is different, the variations of the energy dissipation ratio of NES with the change in the shock amplitude are the same. The energy dissipation ratio of NES increases as the shock amplitude increases, and then it decreases gradually within a certain range of shock amplitude. The excitation amplitude increases further, and the energy dissipation ratio of the NES reaches to a maximum level. Overall, the reduction in the stiffness ratio at both smaller and larger shock amplitudes results in a greater energy dissipation ratio for parallel asymmetric NES. Figures 6(c) and 6(d)  show that the energy dissipation ratio of NES degrade through reducing ε1 ε2 or c1 c2 independently. The system dynamics equation is analyzed by the Runge-Kutta method, and ε1 = 0.08, ε2 = 0.02, c1 = 0.008, c2 = 0.002, and k1 = 0.5, k2 = 0.5, as shown in Fig. 7. The time history of each oscillator under different excitation amplitudes is obtained. In Fig. 7(a), when f = 1, the displacements of both the linear primary oscillator and the two cubic oscillators are small, and the parallel asymmetric NES is not activated. In Fig. 7(b), when f = 3, the heavier cubic oscillator has a small amplitude, and the other cubic oscillator vibrates significantly. Then,  one of the NES is activated, which is represented by the first peak in Fig. 7(b) as k1 k2 = 1:1. In Fig. 7(c), when f = 6, both cubic oscillators vibrate sharply, that is, both NESs are activated, so that the energy dissipation ratio of NES reaches another peak.

Parametric Study on Performance of Parallel Asymmetric

(a) change the stiffness, mass and damping ratios simultaneously

(b) changing the stiffness ratio

(c) changing the mass ratio

(d) changing the damping ratios

Fig. 6. Influence of the parameters on the energy dissipation ratio.

(a)

=1

(b)

=3

(c)

=6

Fig. 7. Time history of each oscillator under different shock excitation amplitudes.

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5 Conclusions In this paper, the frequency response of systems connected to parallel asymmetric NES is studied using a combination of the Runge-Kutta method and the complexificationaveraging method, and the effect of parallel asymmetric NES parameters on its vibration absorption efficiency under shock excitation is investigated. The key findings are as follows: (1) The influence of parameters in parallel asymmetric NES on the higher branches of response when the system is subjected to harmonic excitation is studied. The results demonstrate that by decreasing the stiffness ratio, mass ratio, and damping ratio between the two NES, the frequency band producing higher branches of response becomes wider, and the merging phenomenon of the higher and lower branches under smaller excitation amplitude occurs. Furthermore, when modifying the mass ratio or damping ratio, multiple frequency bands simultaneously produce higher branches of response. (2) The effect of the NES parameters on the strongly modulated response of the system under harmonic excitation is investigated. The results demonstrate that the frequency band of the strongly modulated response remaining almost unchanged when the stiffness ratio, mass ratio, and damping ratio between the two NES are changed simultaneously. When the damping ratio of the two NESs is changed independently, the frequency band of the strongly modulated responses grows. When the stiffness ratio or mass ratio changes alone, the frequency band that produces the strongly modulated responses can be widened or narrowed. (3) The effects of stiffness, mass ratio, and damping ratio of two NESs on the vibration absorption efficiency under shock excitation are investigated. The results indicate that changing the ratios of stiffness, mass, and damping between the two NES simultaneously has no impact on the energy dissipation ratio of the parallel asymmetric NES. When the mass ratio or damping ratio is reduced alone, the energy dissipation ratio of parallel asymmetric NES will be worse; however, when the stiffness ratio is reduced, the energy dissipation ratio of parallel asymmetric NES will be better for both smaller and larger excitation amplitudes.

References 1. Ding, H., Chen, L.Q.: Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100(4), 3061–3107 (2020) 2. Lu, Z., Wang, Z.X., Lü, X.L.: A review on nonlinear energy sink technology. J. Vib. Shock 39(04), 1–16 (2020). (in Chinese) 3. Li, J.W., Ding, W.C., Li, G.F.: Connection and performance of a vibration system with multidegree of freedom nonlinear energy sink. J. Lanzhou Jiaotong Univ. 36(1), 96–101 (2017). (in Chinese) 4. Song, W.Z., Liu, Z., Lu, C., et al.: Analysis of vibration suppression performance of parallel nonlinear energy sink. J. Vib. Control 29(11–12), 2442–2453 (2022) 5. Chen, J.E., Zhang, W., Liu, J., et al.: Vibration absorption of parallel-coupled nonlinear energy sink under shock and harmonic excitations. Appl. Math. Mech. (English Edition) 42(8), 1135–1154 (2021)

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6. Chen, J.E., Sun, M., Zhang, W., et al.: Cross-scale energy transfer of chaotic oscillator chain in stiffness-dominated range. Nonlinear Dyn. 110(3), 2849–2867 (2022) 7. Boroson, E., Missoum, S., Mattei, P., et al.: Optimization under uncertainty of parallel nonlinear energy sinks. J. Sound Vib. 394, 451–464 (2017) 8. Xu, K., Zhang, Y., Niu, M., et al.: An improved nonlinear energy sink with electromagnetic damping and energy harvesting. Int. J. Appl. Mech. 14(06), 2250055 (2022) 9. Bak, B.D., Rochlitz, R., Kalmár-Nagy, T.: Energy transfer mechanisms in binary treestructured oscillator with nonlinear energy sinks. Nonlinear Dyn. 111(11), 9875–9888 (2023) 10. Wei, Y.M., Wei, S., Zhang, Q.L., et al.: Targeted energy transfer of a parallel nonlinear energy sink. Appl. Math. Mech.-English Edn. 40(5), 621–630 (2019) 11. Wang, Q., Liu, H., Liu, Y., et al.: Investigation on multiple periodic solution branches of a system with piecewise linear nonlinear energy sink. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 236(12), 6451–6462 (2022) 12. Zhang, Y.F., Kong, X.R.: Analysis on vibration suppression response of nonlinear energy sink with combined nonlinear damping. Chin. J. Theor. Appl. Mech. 55(04), 972–981 (2023). (in Chinese) 13. Wang, G.X., Ding, H., Chen, L.Q.: Optimization of a nonlinear energy sink with double springs and harmonic excitation. J. Dyn. Control 19(06), 46–51 (2021). (in Chinese) 14. AL-Shudeifat, M.A.: Asymmetric magnet-based nonlinear energy sink. J. Comput. Nonlinear Dyn. 10(1), 014502 2015 15. Geng, X.F., Ding, H.: Two-modal resonance control with an encapsulated nonlinear energy sink. J. Sound Vib. 520, 116667 (2022) 16. Geng, X.F., Ding, H., Jing, X.J., Mao, X.Y., Wei, K.X., Chen, L.Q.: Dynamic design of a magnetic-enhanced nonlinear energy sink. Mech. Syst. Signal Process. 185, 109813 (2022)

Harnessing LSTM for Nonlinear Ship Deck Motion Prediction in UAV Autonomous Landing Amidst High Sea States Feifan Yu1,2,3

, Wenyuan Cong1,2,4 , Xinmin Chen1,2 and Jiqiang Wang1,2(B)

, Yue Lin1,2

,

1 Zhejiang Provincial Engineering Research Centre for Special Aircrafts, Ningbo 315336, China 2 Ningbo Institute of Materials Technology and Engineering, CAS, Ningbo 315201, China

[email protected]

3 University of Chinese Academy of Sciences, Beijing 101408, China 4 Faculty of Electrical Engineering and Computer Science, Ningbo University, Ningbo 315211,

China

Abstract. Autonomous landing of UAVs in high sea states requires the UAV to land exclusively during the ship deck’s “rest period,” coinciding with minimal movement. Given this scenario, determining the ship’s “rest period” based on its movement patterns becomes a fundamental prerequisite for addressing this challenge. This study employs the Long Short-Term Memory (LSTM) neural network to predict the ship’s motion across three dimensions: longitudinal, transverse, and vertical waves. In the absence of actual ship data under high sea states, this paper employs a composite sine wave model to simulate ship deck motion. Through this approach, a highly accurate model is established, exhibiting promising outcomes within various stochastic sine wave combination models. Keywords: Long Short-Term Memory · high sea state · Ship Attitude Composite Prediction

1 Introduction The establishment of a robust maritime nation hinges on a formidable air-sea threedimensional transportation network. Advancing the capabilities of near-sea Unmanned Aerial Vehicles (UAVs) for tasks such as maritime patrol, search and rescue [1], and emergency transport significantly enhances maritime transportation potential. This mode of transport holds a pivotal role in future development strategies. Nonetheless, the landing phase in naval aviation constitutes merely 4% of the entire flight process, yet contributes to 44.4% of all aviation accidents. Remarkably, unmanned aircraft landing incidents contribute up to 80% of these mishaps. As a result, it becomes imperative to pursue autonomous landing navigation control technology [2] to enhance landing success rates. Various techniques exist for drone landings, similar to the deck runway landings of carrier-based aircraft [3]. However, one drawback of this approach is the requirement © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 X. Jing et al. (Eds.): ICANDVC 2023, LNEE 1152, pp. 820–830, 2024. https://doi.org/10.1007/978-981-97-0554-2_63

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for an extended runway, limiting its suitability primarily to large vessels. The crash net landing [4] method is also prevalent, but it escalates the risk of drone damage and leads to considerable maintenance expenses. Furthermore, sky hook [5] and parachute landings [6] are alternative strategies, albeit susceptible to environmental influences and necessitating precise parachute deployment. Given the shortcomings of these landing methods, this study opts for the more advantageous vertical take-off and landing approach of tilt-rotor UAVs [7]. This method not only ensures a smooth, non-detrimental interaction with both UAVs and ships, but also demonstrates distinct merits in facilitating UAV landings amid high sea conditions. Specifically, this advantage is underscored by the ship’s random six-degree-of-freedom movement and the time allocated for the UAV to adjust its landing point during vertical take-off and landing. This adjustment ensures the ship’s tilt remains within an acceptable range upon landing, thereby significantly boosting the success rate of UAV ship landings in challenging sea conditions. Nonetheless, the vertical take-off and landing method is not exempt from certain limitations. Specifically, these limitations manifest during drone landings, where the ship’s inclination due to wave effects must remain within a certain range. Landings are viable only when the ship’s tilt remains within an acceptable angle. The time interval during which the ship’s tilt falls within this acceptable angle is referred to as the "rest period." Safely landing the drone during this rest period is considered secure. As a result, the realm of autonomous UAV landings necessitates an exploration into software-based predictions of a ship’s three-dimensional inclination magnitude induced by wave forces. Building upon the aforementioned context, this paper employs the Long Short-Term Memory (LSTM) [8] network to prognosticate the three-dimensional motion pattern of a vessel subjected to wave influences. Tailoring its focus to the practical application milieu, this study centers on ship motion within sea state 5 conditions. Given the current unavailability of ship motion data for sea state 5, this research utilizes Huang’s [9] sine wave superposition methodology to replicate the deck motion of Knox-class warships, subsequently leveraging this synthesized dataset for model training. Subsequently, a stochastic sine wave combination model is constructed based on sea state data at level 5 to validate the precision of the proposed model presented in this study. The forthcoming thesis can be broadly divided into four parts. Section 2 commences by outlining the dataset construction procedure, followed by an exposition on the significance of each individual indicator. In Sect. 3, the LSTM model is employed. Initially, the model combines indicators across three dimensions and subsequently applies this amalgamation to predict the previously mentioned dataset, ultimately yielding a highprecision predictive model. In Sect. 4, a stochastic sine wave composite model is constructed to assess the trained model’s performance. This evaluation is accomplished by utilizing known sea state parameters corresponding to the 5-level classification. The aim is to affirm the model’s precision and its comparative advantages. Section 5 encapsulates the content discussed thus far, concluding with a summary of the findings and offering insights into prospective avenues for future research.

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2 Data Preparation When a ship traverses the sea under conditions of high sea state, it becomes susceptible to substantial wind and wave influences. Consequently, the ship experiences intricate six-degree-of-freedom movements [10]. Within the context of UAV ship landings, the attainment of relative hovering between the UAV and the ship has been achieved. As a result, it becomes feasible to exclude the longitudinal and transverse lateral displacement components during the deconstruction of this intricate six-degree-of-freedom motion. Consequently, we decompose this intricate six-degree-of-freedom motion into three distinct modes of movement: the ship’s longitudinal, transverse, and vertical oscillations. The longitudinal and transverse oscillations pertain to the ship’s amplitude of movement along its longitudinal or transverse axes due to wind and wave forces. Conversely, vertical oscillations denote alterations in the ship’s vertical displacement as a result of wind and wave influences. This is visually illustrated in the figure presented below (Fig. 1).

Fig. 1. Illustration of the ship rocking model

Given the absence of three-dimensional ship motion data under the impact of waves during high sea states, this study employs the sine wave superposition methodology outlined in Huang’s work to replicate the deck movement of Knox-class warships. The simulated model for combining sine waves is detailed as follows: hs (t) = 0.2172 sin(0.4t) + 0.4714 sin(0.5t) + 0.3592 sin(0.6t) + 0.2227 sin(0.7t) θs (t) = 0.005 sin(0.46t) + 0.00946 sin(0.58t) + 0.00725 sin(0.7t) + 0.00845 sin(0.82t)

ϕs (t) = 0.021 sin(0.46t) + 0.0431 sin(0.54t) + 0.029 sin(0.62t) + 0.022 sin(0.67t)

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In this model, θs (t) represents the longitudinal swing, ϕs (t) signifies the transverse swing, and hs (t) denotes the vertical swing. The measurements for θs (t) and ϕs (t) are expressed in terms of swing inclination units, while hs (t) represents the height of elevation or descent. The visual depiction of this configuration is illustrated below (Fig. 2):

Fig. 2. Sinusoidal combinatorial model

3 Model Training In this section, we will proceed to train the LSTM model with the aim of predicting three key parameters of a ship’s behavior under wave influence: the longitudinal swing, the angle of the transverse swing, and the height of the vertical swing. Initially, we divide each of the sinusoidal wave combination models mentioned above into sets of 2000 data points. These data point sets are subsequently employed as both our training and testing datasets. The initial 70% of data points are designated as the training set, while the remaining 30% are allocated for testing purposes. The crux of our training approach hinges on harnessing the inherent capabilities of LSTM. This involves predicting the magnitude of the forthcoming data point based on the learning of 40 consecutive data points leading up to the predicted data point. This process is reiterated to predict the entire curve depicting the ship’s motion attitude. Furthermore, the model incorporates a composite LSTM neural network prediction architecture that concurrently forecasts three distinct output parameters—namely, the longitudinal, transverse, and vertical sways—as a unified prediction. This feature notably underscores the paper’s strengths.

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For model training, a continuous dataset comprising 2000 data points is employed. Precisely, the initial 70% of the dataset, equating to 1400 points, constitutes the training set used to train our network. Subsequently, the remaining 30% (600 points) is allocated for testing the predictions generated by our network. The outcomes of the training and testing sets are depicted in Figs. 3 and 4.

Fig. 3. Training process training set effect

In Fig. 3, it is evident that the current model exhibits a near-flawless performance in predicting the transverse swing ϕs (t) curve on the training set, demonstrating minimal errors. Although the outcomes for the longitudinal swing θs (t) curve and vertical swing hs (t) curve are marginally less remarkable compared to the ϕs (t) curve, they nevertheless remain highly satisfactory. Figure 4 illustrates that the current model adeptly forecasts all three curves on the test set. However, it’s noteworthy that while the prediction for the hs (t) curve remains highly accurate, there is a relatively minor jitter observed in its performance during the initial 300 data points of the test set. It’s important to highlight that through experimentation, it was observed that increasing the number of hidden neurons in the LSTM correlates with a reduction in the amplitude of this jitter, consequently enhancing the prediction quality.

Harnessing LSTM for Nonlinear Ship Deck Motion Prediction

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Fig.4. Training process test set effect

4 Model Validation The foregoing highlights the exceptional performance of this model across both the training and test sets. The ensuing section proceeds to examine the model’s efficacy in the context of high sea state conditions. Specifically, this study focuses on the validation and generalization capabilities of the model within a class 5 sea state scenario. This particular sea state is characterized by a significant wave height ranging from 2.5 to 4 m and a characteristic period spanning 5.5 to 6.7 Hz. Initial steps involve the random generation of a sinusoidal combination model to emulate the deck movement of a ship within a class 5 sea state. A few of the noteworthy parameters characterizing the class 5 sea state are presented in the subsequent table: Table 1. Selected reference data for sea state 5

hs (t) ϕs (t) θs (t)

amplification

cyclicality

1.0 < Z 1/3 < 1.9   6.3 < ϕ a 1/3 < 12.0

5