Modern Mechanics and Applications: Select Proceedings of ICOMMA 2020 [1 ed.] 9811632383, 9789811632389

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Modern Mechanics and Applications: Select Proceedings of ICOMMA 2020 [1 ed.]
 9811632383, 9789811632389

Table of contents :
Contents
A Dual Approach for Modeling Two- and One-Dimensional Solute Transport
1 Introduction
2 Establishing a 2DV Solute Transport Equation Based on the Dual Approach
3 Comments
4 Results
5 Comments
6 Conclusions
References
Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application
1 Introduction
2 Rayleigh Quotient for Multiple Cracked Bar
3 Application for Calculating Natural Frequencies
4 Numerical Illustration and Discussion
5 Conclusion
References
Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core
1 Introduction
2 Theoretical Study
2.1 Plate Geometry and Assumptions
2.2 Flexural Motion of the Plate
2.3 Boundary Conditions of the Plate-Porous Coupling
2.4 Determination of Sound Transmission Loss
2.5 Modelling the Poroelastic Material
3 Validation
4 Numerical and Experimental Results
4.1 Test Setup
4.2 Sample Description
4.3 Results and Discussion
5 Conclusions
Appendix A. Expressions of the Matrix Equation (58)
References
Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method
1 Introduction
2 Experimental
2.1 Material
2.2 Fabrication Method
2.3 Analysis Methods
3 Result and Discussion
3.1 Dimension Accuracy
3.2 Morphology
3.3 Porosity
3.4 Tensile Test
3.5 Hardness Test
4 Conclusion
References
Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions
1 Introduction
2 Design and Setup the Model of Apparatus
2.1 Design and Setup Experimental Apparatus
2.2 Experimental Operation
3 Results and Discussion
4 Conclusion
References
Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method
1 Introduction
2 Material Properties
3 Formulations
4 Ritz Method
5 Numerical Results and Discussions
6 Conclusions
References
Self-vibrational Analysis of a Tensegrity
1 Introduction
2 Self-vibration of Axially Prestressed Elements
3 Equation of Axial Motion in Vibration
4 Spectral Element Formulation
4.1 Governing Equations in the Frequency Domain
4.2 Spectral Nodal DOFs and Forces
5 Dynamic Shape Function
5.1 Weak Form of Governing Equation
6 Spectral Element Equation
7 Solving the Spectral Element Equation
8 Vibration of Triplex Tensegrity
9 Concluding Remarks
References
Free Vibration of a Bi-directional Imperfect Functionally Graded Sandwich Beams
1 Introduction
2 Problem and Formulation
3 Numerical Results
4 Conclusions
References
Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation
1 Introduction
2 Problem and Formulation
2.1 Grade Functionally Beam
2.2 Government Equations
2.3 Nonlocal Continuum Beam
2.4 Numerical Formulation
3 Numerical Result
4 Conclusions
References
Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism
1 Introduction
2 Configuration of the SMA Damper
3 Modeling of the SMA Damper
3.1 Characterization of the SMA Springs
3.2 Design of the SMA Damper
4 Experimental Test
5 Hysteresis Model of the SMA Damper
6 Conclusions
References
A High-Order Time Finite Element Method Applied to Structural Dynamics Problems
1 Introduction
2 Variational Formulation for Structural Dynamics
3 First-Order Time Finite Element
4 High-Order Time Finite Elements
5 Numerical Example
6 Conclusions
1 The Proof of Equations (24) and (25)
2 Element ``Stiffness'' Matrix of p-TFE for SDOF System
3 MATLAB Code for Obtaining ``Stiffness'' Matrix of bp-TFE for SDOF Systems
References
A Multibody Dynamics Approach to Study an Insect-Wing Structure
1 Introduction
2 Multibody Dynamics Approach
2.1 Hencky-Bar Chain Model
2.2 Equations of Motion
2.3 Validation
3 Beam Model of Hawkmoth Wings
3.1 FEM Model
3.2 Mass and Stiffness Distributions of the Multibody System
4 Results and Discussion
5 Conclusions
References
Natural Ventilation and Cooling of a House with a Solar Chimney Coupled with an Earth – To – Air Heat Exchanger
1 Introduction
2 Problem Formulation and Numerical Method
2.1 The Coupled Solar Chimney and Earth – to – Air Heat Exchanger System
2.2 Numerical Model
3 Results and Discussions
3.1 Changing the Applied Heat Flux in the Solar Chimney
3.2 Changing the Chimney Gap
3.3 Changing the EAHE Pipe Length
3.4 Changing the EAHE Pipe Diameter
4 Conclusions
References
Dynamic Analysis of Curved Beam on Elastic Foundation Subjected to Moving Oscillator Using Finite Element Method
1 Introduction
2 Finite Element Simulation and Dominant Equations
2.1 The Beam Element on Elastic Foundation Subjected to Moving Oscillator
2.2 Governing Differential Equations for Total System
3 Numerical Analysis
3.1 Model Validation
3.2 Calculation Results
4 Conclusions
Appendix
References
Buckling Analysis of FG GPLRC Plate Using a Naturally Stabilized Nodal Integration Meshfree Method
1 Introduction
2 Mathematical Formulations
2.1 Multilayer Functionally Graded Graphene Platelet Reinforced Composites
2.2 Kinematics of FG GPLRC Plate
2.3 Moving Kriging Interpolation Shape Functions
2.4 Nodal Integration Formulations for Plates
2.5 Naturally Stabilized Nodal Integration
3 Numerical Results
3.1 Comparison Results
3.2 Square Plate with a Heart Hole
4 Conclusion
References
Low Temperature Effect on Dynamic Mechanical Behavior of High Damping Rubber Bearings
1 Introduction
2 Experiments
2.1 Specimens and Test Conditions
2.2 Static Equilibrium Hysteresis
2.3 Instantaneous State
2.4 The Temperature Dependence of the Viscosity Property
3 A Calculation to Determin the Viscosity of HDRB from SR Tests
4 Conclusion
References
Effects of Clayey Soil and Hemp Fibers on the Flexural Behavior of Soil Concrete
1 Introduction
2 Materials and Specimens
2.1 Materials
2.2 Specimens
3 Flexural Behavior of Soil Concrete
3.1 Three-Point Bending Test
3.2 Load – Displacement Curve
3.3 Load – Crack Width Curve
4 Conclusions
References
Optimal Design of Functionally Graded Sandwich Porous Beams for Maximum Fundamental Frequency Using Metaheuristics
1 Introduction
2 Frequency Maximization for FG-SWP Beams
2.1 Design Problem
2.2 Analytical Solution for Natural Frequency
3 Implemented Metaheuristics
3.1 Parameter Setting
3.2 Constraint Handling
4 Numerical Results
4.1 Comparison Among Meta-heuristics
4.2 Comparison Among Beam Theories
5 Conclusion
References
Nonlinear Buckling Behavior of FG-CNTRC Toroidal Shell Segments Stiffened by FG-CNTRC Stiffeners Under External Pressure
1 Introduction
2 Fundamental and Established Formulations
3 Validation, Numerical Investigations and Discussions
4 Conclusion
References
Nonlinear Vibration of Shear Deformable FG-CNTRC Plates and Cylindrical Panels Stiffened by FG-CNTRC Stiffeners
1 Introduction
2 Smeared Stiffener Technique and Fundamental Equations
3 Numerical Investigations
4 Conclusion
References
Vibration Characteristics of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Submerged in Fluid Medium
1 Introduction
2 Theoretical Formulation
2.1 Geometry and Material Properties
2.2 Kinematics
2.3 Constitutive Relations
2.4 Formulation of Fluid
2.5 Governing Equations
2.6 Solution Procedure
3 Results and Discussion
3.1 Validation Study
3.2 Parametric Study
4 Conclusions
References
Large Amplitude Free Vibration Analysis of Functionally Graded Sandwich Plates with Porosity
1 Introduction
2 Theoretical Formulation
2.1 FG Sandwich with Porosity Properties and Geometric Configuration
2.2 Displacement Field and Strains
2.3 Constitutive Relations
2.4 Strain Energy of the FG-SPP
2.5 Kinetic Energy
2.6 Finite Element Formulation
3 Results and Discussion
3.1 Convergence and Validation Study
3.2 Parametric Study
4 Conclusions
References
Dynamic Analysis of a Functionally Greded Sandwich Beam Traversed by a Moving Mass Based on a Refined Third-Order Theory
1 Introduction
2 Theoretical Formulations
2.1 The FGSW Beam Model
2.2 Mathematical Model
3 Finite Element for Mulation
4 Numerical Results
4.1 Accuracy and Convergence Studies
4.2 Parametric Study
5 Conclusions
References
Thermal Bending Analysis of FGM Cylindrical Shells Using a Quasi-3D Type Higher-Order Shear Deformation Theory
1 Introduction
2 Theoretical Formulation
3 Solution Procedure
4 Numerical Analysis and Discussion
4.1 Validation
4.2 The Effect of the Boundary Condition
4.3 The Effects of Thickness
5 Conclusions
References
Experimental and Finite Element Analysis of High Strength Steel Fiber Concrete – Timber Composite Beams Subjected to Flexion
1 Introduction
2 Experimental Campaign
2.1 High Strength Steel Fiber Concrete in Compression and Flexion
2.2 Properties of Oak and Beech Timber
3 Constitutive Laws
3.1 Concrete Damaged Plasticity
3.2 Hill Criterion
4 Modelling of Timber-Concrete Composite Beam
4.1 Verification of the Selected Model
4.2 Modelling of the Timber-Concrete Composite Beam
4.3 Parametric Study: Influence of the Steel Connection Diameter on Shear Resistance
5 Conclusions
References
Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions
1 Introduction
2 FGP Cylindrical Shell
3 Theory and Formulation
4 Analytical Solution
5 Numerical Results and Discussions
5.1 Comparison Research
5.2 Numerical Investigations
6 Conclusion
Appendix A
References
Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector
1 Introduction
2 Simulation Modeling
2.1 Mathematical Modeling
2.2 Solenoid Model in Maxwell
2.3 Solenoid Model in Simplorer
3 Simulation Results
3.1 Effects of Plug and Sleeve Shapes on Electromagnetic Force
3.2 Dynamic Response Under the Effects of Plug and Sleeve Shapes
4 Conclusions
References
Practical Method for Tracking Fatigue Damage
1 Introduction
2 Description of the Method
2.1 Variogram
2.2 Smooth Orthogonal Decomposition
3 Numerical Simulation
4 Experimental Validation
5 Discussion
6 Conclusion
References
An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel
1 Introduction
2 Theoretical Basis
2.1 Fatigue
2.2 Theoretical Basis of Heat Treatment
3 Experiment and Results
3.1 Experimental Equipment
3.2 Recommended Sample Parts
3.3 Fabrication of Sample Parts
3.4 Heat Treatment
3.5 Fatigue Test Process
4 Conclusions
References
Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed
1 Introduction
2 Advanced Signal Decomposition Methods
3 Experimental Results
4 Conclusion
References
Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading
1 Introduction
2 Experimental Investigation
2.1 Material
2.2 Test Specimens
2.3 Test Set-Up
3 Compare the Experimental Results with the Results Obtained from the Numerical Model
3.1 3D FEM Element Modeling
3.2 Visco-plastic Regularization on the CDP
3.3 Isotropic Hardening
4 Results and Discussion
4.1 Failure Characteristics
4.2 Measurement Data
5 Conclusions
References
Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact
1 Introduction
2 Fracture Tests
2.1 Dimensions and Material Properties of the Test Models
2.2 Experimental Set-Up
2.3 Test Results
3 Numerical Analysis of Fracture Tests
3.1 Finite Element Modelling
3.2 Material Definition
3.3 Numerical Results
4 Disscussions
4.1 Collision Tests
4.2 Numerical Simulations
5 Conclusions
References
Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion
1 Introduction
2 Methodology to Assess the Strength of Mooring Lines of Floating Offshore Structures
2.1 Line Dynamic Response Analyses
2.2 Safety Factor
2.3 Minimum Breaking Loads of Mooring Lines from Catalogues of Manufacturers [4]
3 Numerical Method for Chain Testing Simulation
3.1 Plastic Models
3.2 Small-Sliding Interaction Between Bodies [9]
3.3 Contact Formulation [9]
3.4 Contact Pressure in Case of Hard Contact [9]
3.5 Stress Analysis by FEM
4 Strength Assessment of Mooring Line System of DH01 FPU
4.1 Input Data
4.2 Chain Testing Simulation
4.3 Numerical Testing Results
4.4 Comparison to Calculation of MBL According to Vicinay [4]
4.5 Strength Checking Results
5 Discussion
References
An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat
1 Introduction
2 Hydration Heat analysis
2.1 Heat Transfer Analysis
2.2 Thermal Stress Analysis
2.3 Boundary Condition of Heat Transfer in Concrete Block
3 Behaviors of Tower Footing of Cua-Dai Extradosed Bridge Due to Hydration Heat
3.1 Modelling of Concrete Block
3.2 Hydration Temperature in Tower-Footing Concrete
3.3 Stress in Concrete Block Due to Heat of Hydration
3.4 Displacement Due to Heat of Hydration
4 Conclusions
References
A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams
1 Introduction
2 Nonlocal Dynamic Stiffness Model of a FGM Timoshenko Nanobeam Element
3 Numerical Results and Discussion
3.1 Comparison of Calculation Results and Published Results
3.2 Effects of Neutral Axis Position and Nonlocal Parameter on the Nondimensional Frequency
3.3 Effect of Boundary Conditions on Nondimensional Frequencies
3.4 Effect of Nanobeam Length on the Nondimensional Frequencies
4 Conclusion
References
Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings
1 Introduction
2 Input Data for the Analysis
2.1 Model of the Bridge
2.2 Model of the HDRB
2.3 Earthquake Ground Motions and Solution Algorithm for Motion Equations
3 Output Data of the Analysis
3.1 Low Temperature Effect
3.2 Modeling Effect
4 Conclusion
References
Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load
1 Introduction
2 Governing Equations
3 Solution Method
4 Numerical Result
4.1 Validation
4.2 Natural Vibration Frequencies
4.3 Nonlinear Response of Variable Thickness FGM Cylindrical Shell
5 Conclusions
References
Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load
1 Introduction
2 Governing Equations
3 Solution Method
4 Results and Discussion
4.1 Validation
4.2 Nonlinear Dynamic Stability of Variable Thickness FGM Shell
5 Conclusions
Appendix 1: Stiffness Coefficients Aij ,Bij ,Dij in Eq. (6, 7)
Appendix 2: The Coefficients Lij (i,j = 1,3 ),Pk (k = 1,5 ) in Eq. (9)
Appendix 3: The Coefficients Iij (i,j = 1,3 ),Rk (k = 1,6 ) in Eq. (11)
References
Vibration and Dynamic Impact Factor Analysis of the Steel Truss Bridges Subjected to Moving Vehicles
1 Introduction
2 Computational Models and Assumptions
2.1 Computational Models and Assumptions
2.2 The Differential Equation of Motion
2.3 Analysis Vibration of the Steel Truss Bridge
3 Conclusion
References
An Experimental Study on the Self-propelled Locomotion System with Anisotropic Friction
1 Introduction
2 Experimental Setup with Varied Anisotropic Friction
2.1 Working Principle of the System
2.2 Experimental Implementation
3 Results and Discussion
4 Conclusion
References
Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation
1 Introduction
2 Governing Equations and Turbulence Modeling
3 Case Study and Coordinate System
4 Virtual Captive Model Test
4.1 Static Drift Test
4.2 Harmonic Test
5 Numerical Modeling
6 Results and Discustions
6.1 Validation
6.2 Hydrodynamic Forces
7 Conclusions
References
Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures
1 Introduction
2 Methods
2.1 Dynamic Magnification Factor for Steady State Acceleration
2.2 Optimization Process for TMD Parameters
3 Results and Discussion
3.1 New Optimum TMD Parameters
3.2 Empirical Expressions for New TMD Parameters
3.3 Evaluating the Efficiency of the Proposed Tuning Solutions
4 Conclusions
References
Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and  Regenerated Power
1 Introduction
2 Mathematical Modelling
3 Optimal Solution of Electrical Damping Ratio Under Harmonic Excitation
4 Numerical Investigation for Harmonic Excitation
5 Optimal Solution of Electrical Damping Ratio Under White Noise Excitation
6 Conclusion
References
Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio
1 Introduction
2 Wind Tunnel Test Investigation
2.1 Experimental Setup
2.2 Wind Attack Angles
2.3 Experimental Model Fabrication
2.4 Experiment Parameters
3 Results and Discussions
3.1 Wind-Induced Vibration of Indented Cable Surface in Low Damping Region
3.2 Wind-Induced Vibration of Axially Protuberated Cable in Low Damping Region
3.3 Axial Flow Near the Wake of Cable
4 Conclusions
References
The Equation Connecting Boyle’s and Bernoulli’s Laws
1 Introduction
2 Connecting Equation
2.1 The Volume Flow Rate of Air Passing Through an Orifice Between Two Compartments
2.2 The Internal Pressure Change of the System Due to Changing External Pressure
2.3 The System with N Compartments
3 Illustrative Example
4 Discussion and Conclusions
References
Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys
1 Introduction
2 Numerical Model
2.1 Assumptions for the Flow and Heat Transfer
2.2 Governing Equations
2.3 Computational Domain and Boundary Conditions
2.4 Computational Mesh and Numerical Setup
2.5 Model Validation
3 Results and Discussions
3.1 Flow and Temperature Fields
3.2 Length of the Reverse Flow
3.3 Induced Flow Rate
3.4 Thermal Efficiency
4 Summaries
References
A Solar Chimney for Natural Ventilation of a Three – Story Building
1 Introduction
2 Problem Formulation and Numerical Method
2.1 The Solar Chimney
2.2 Assumptions and Governing Equations
2.3 Validation of the Numerical Model
3 Results and Discussions
3.1 Base Configuration (Base Case)
3.2 Testing Configurations (Case 2 to 4)
3.3 Selected Configuration (Case 5)
3.4 Changing Inlet Size for Similar Flow Rates at All Inlets (Case 6)
4 Conclusions
References
Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume
1 Introduction
2 Numerical Method
2.1 Studied Solar Chimney
2.2 Numerical Model
2.3 Validation of the Numerical Model
3 Results and Discussions
3.1 Induced Flow Rate Versus Channel Width and Gap
3.2 Induced Flow Rate Versus Channel Volume
3.3 Thermal Efficiency
3.4 Proposed Chimney Configurations for Maximizing Induced Flow Rate
4 Conclusions
References
Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface
1 Introduction
2 Description of the Problem and Numerical Modelling
2.1 Numerical Model
2.2 Model Validation
3 Results and Discussions
3.1 Effects of the Step on the Flow Field and Heat Transfer
3.2 Effects of the Step Length and Height
4 Conclusions
References
The Effect of Ground Surface Geometry on the Wing Lift Coefficient
1 Introduction
2 Panel Method and Calculation Model
2.1 Panel Method
2.2 Model Calculation
2.3 Validations
3 Numerical Results
3.1 The Influence of the Ground Surface Geometry Shape
3.2 The Influence of Wind
4 Conclusions
References
Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach
1 Introduction
2 Wind Tunnel Design
3 Numerical Method
4 Results and Discussion
5 Conclusion
References
A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory
1 Introduction
2 Basic Equations
2.1 Problem Description
2.2 Modified Strain Gradient Elasticity Theory
2.3 Kinematics of FG Microplate
3 FG Microplate Formulation Based on Moving Kriging Interpolation Shape Functions
3.1 Moving Kriging Interpolation Shape Functions
3.2 A MKI-Based Formulation Based on the sFSDT and Modified Strain Gradient Theory
4 Numerical Examples and Discussions
4.1 FG Square Microplate
4.2 FG Circular Microplate
5 Conclusions
References
Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core
1 Introduction
2 Theory and Formulations
2.1 Sandwich Beam and Material Modelling
2.2 Displacement and Strain Fields
2.3 Constitutive Equation
2.4 Energy Expressions
2.5 Hamilton’s Principle and Governing Equations
2.6 Analytical Solutions
3 Results and Discussion
3.1 Validation
3.2 Comprehensive Study
4 Summary and Conclusions
References
Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials
1 Introduction
2 Second Gradient Model Formulation
2.1 Variational Principle
2.2 Equations with Lagrange Multipliers
2.3 Finite Element: First and Second Quadratic Elements
2.4 Constitutive Relations
3 Failure Analysis of Pressurized Hollow Cylinder
3.1 Pressurized Hollow Cylinder Problem
3.2 Failure Analysis
4 Conclusion
References
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
1 Introduction
2 Method
2.1 Literature Review on the Effects of pH on the Soil Strength
2.2 Model and Factors for a Soil-Cement Samples in the Model
2.3 Strategy for Virtual Experiment
3 Results
4 Discussion
5 Conclusion
References
Effects of Test Specimens on the Shear Behavior of Mortar Joints in Hollow Concrete Block Masonry
1 Introduction
2 Experimental Program
2.1 Materials
2.2 Shear Pull-Out Test of Masonry
3 Conclusions and Perspectives
References
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells Under External Pressure
1 Introduction
2 ES-FG Porous Cylindrical Shells
3 Fundamental Equations and Solution of the Problem
4 Numerical Results
4.1 Comparisons
4.2 Effect of Porosity Coefficients e0 and Foundation
4.3 Effect of Geometric Parameters (Effects of hcore/h,R/h, L/R Ratios)
4.4 Effects of Stiffeners and Volume Fraction Index
5 Conclusions
Appendix
References
Bulk Modulus Prediction of Particulate Composite with Spherical Inclusion Surrounded by a Graded Interphase
1 Introduction
2 Model of Multi-coated Sphere Assemblage and Limiting Case
3 Differential Substitution Scheme and the Coating with Radially Variable Moduli
4 Numerical Results
5 Conclusion
References
Post-buckling Response of Functionally Graded Porous Plates Rested on Elastic Substrate via First-Order Shear Deformation Theory
1 Introduction
2 Functionally Graded Porous Plates
3 Theoretical Formulation
4 Buckling and Postbuckling Analysis
5 Numerical Results and Discussion
5.1 Validation Examples
5.2 Parametric Study
6 Conclusions
References
Elastic Buckling Behavior of FG Polymer Composite Plates Reinforced with Graphene Platelets Using the PB2-Ritz Method
1 Introduction
2 Laminated FG-GPLRC Plates
3 Theoretical Formulation
4 Pb2-Ritz Method Based on the FSDT
5 Numerical Results and Discussions
5.1 Convergence Study
5.2 Validation Studies
5.3 Effect of Number of Laminae NL
5.4 Effect of GPL Distribution Type and GPL Weight Fraction
5.5 Effect of Boundary Conditions
5.6 Effect of Plate Length-To-Thickness Ratio
5.7 Effect of Plate Aspect Ratio (b/a)
6 Conclusions
References
Design and Fabrication of Mecanum Wheel for Forklift Vehicle
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusion
References
Muti Object Prediction and Optimization Process Parameters in Cooling Slope Using Taguchi-Grey Relational Analysis
1 Introduction
2 Experimental Procedure
2.1 Materials
2.2 Cooling Slope Casting Process
2.3 Analysis of Samples
2.4 Taguchi’s Experimental Design
3 Grey Relational Analysis
4 Results and Discussion
4.1 Microstructural Investigations
4.2 Analysis of Variance for Grey Relational Grade
4.3 Predicting Optimal Value and Confirmation Experiments
5 Conclusions
References
The Effect of Porosity on the Elastic Modulus and Strength of Pervious Concrete
1 Introduction
2 The Strain and Stress Fields
3 Calculation of the Elastic Modulus of Porous Concrete
4 Calculation of the Compressive Strength of Porous Concrete
5 Conclusions
References
Predicting Capacity of Defected Pipe Under Bending Moment with Data-Driven Model
1 Introduction
2 Finite Element Model
3 Developing a Data-Driven Model with Random Forest Regression
4 FEA Database and the Developed Model
5 Conclusion
References
Image Recognition Using Unsupervised Learning Based Automatic Fuzzy Clustering Algorithm
1 Introduction
2 Some Distance Measurements for Discrete Data
2.1 Measure Distance Between Points
2.2 Clustering Evaluation Criteria
3 The Proposed Technique
3.1 Methods to Extract Image Data
3.2 The Proposed Algorithm
4 Applying in Image Recognition
5 Conclusion
References
The Effect of the Non-uniform Temperature and Length to Height Ratio on Elastic Critical Buckling of Steel H-Beam
1 Introduction
2 Finite Element Model
3 Results
4 Conclusions
References
On the Thin-Walled Theory’s Application to Calculate the Semi-enclosed Core Structure of High-Rise Buildings
1 Introduction
2 Working Characteristics and Deformation of the Semi-enclosed Core Structure Subjected to Torsion
3 Investigation of the Behavior of the Semi-enclosed Core Wall Structure Subjected to Horizontal Load
4 Conclusion
References
Numerical Simulation of Full-Scale Square Concrete Filled Steel Tubular (CFST) Columns Under Seismic Loading
1 Introduction
2 FEA Modeling Program
2.1 Modeling Procedure
2.2 Numerical Validation
2.3 Parametric Studies
3 Numerical Results and Discussions
3.1 Numerical Results
3.2 Effect of Axial Load Level
3.3 Effect of B/t Value
3.4 Effect of Concrete Strength
4 Conclusions
References
Numerical Modeling of Shear Behavior of Reinforced Concrete Beams with Stirrups Corrosion: Finite Element Validation and Parametric Study
1 Introduction
2 Materials Law for Modeling Corroded RC Beams
2.1 Concrete Material Law
2.2 Steel Reinforcement Law
2.3 Constitutive Law of the Steel - Concrete Bond
3 Validation of FE Models for Shear-Critical Corroded RC Beams
3.1 Presentation of the Tested Beam
3.2 Modeling of the RC Corroded Beam with Stirrups Corrosion in Shear Span
3.3 Validation of FE Model
4 Parametric Study
4.1 Effect of Concrete Compressive Strength
4.2 Effect of Stirrup Ratio
4.3 Effect of Bond Strength
5 Conclusions
References
Characterization of Stress Relaxation Behavior of Poscable-86 High-Strength Steel Wire
1 Introduction
2 Methodology
3 Experimental Procedure
4 Results and Discussion
5 Conclusions
References
Seismic Performance of Concrete Filled Steel Tubular (CFST) Columns with Variously Axial Compressive Loads
1 Introduction
2 Experimental Program
2.1 Specimen Design
2.2 Material Properties
2.3 Test Procedure
3 Test Results and Discussions
3.1 Test Results and Observations
3.2 Lateral Load vs. Lateral Displacement Curves
3.3 Lateral Strength and Deformation Capacity
3.4 Lateral Stiffness Degradation
4 Conclusions
References
Damage Simulation Based on the Phase Field Method of Porous Concrete Material at Mesoscale
1 Introduction
2 Finite Element Discretization from the Phase-Field Problem Framework
3 Arrangement of Aggregate Particle Based on Monte-Carlo Simulation
4 Calculation of the Compression Strength of Porous Concrete
5 Conclusions
Referencess
Constitutive Response and Failure Mechanism of Porous Cement-Based Materials Under Triaxial Stress States
1 Introduction
2 Porous Cement-Based Materials Under Triaxial Tests
2.1 DEM Simulation of Triaxial Tests
2.2 The Influence of Confining Stress
2.3 Influence of Intermediate Principal Stress on Octahedral Plane
3 Conclusion
References
Development of Automatic Landing Control Algorithm for Fixed-Wing UAVs in Longitudinal Channel in Windy Conditions
1 Introduction
1.1 Effects of Turbulence on UAV’s Motion in the Longitudinal Channel
2 Development of Automatic Landing Control Algorithm for Fixed Wing UAVs in Longitudinal Channel in Windy Conditions
3 Simulation Results and Conclusion
References
An Assessment of Terrain Quality and Selection Model in Developing Landslide Susceptibility Map – A Case Study in Mountainous Areas of Quang Ngai Province, Vietnam
1 Introduction
2 Methodology
2.1 Study Area
2.2 Data Collection
2.3 DEM Analysis
2.4 Landside Susceptibility Model
2.5 Accuracy Assessment
3 Result and Discussion
3.1 DEM Selection
3.2 Landslide Susceptibility Mapping
3.3 Landslide Susceptibility Assessment
4 Conclusion
References
Analysis of the Impact of Ponds Locations in Flood Cutting from a Developing City in Vietnam
1 Introduction
2 Methodology and Materials
2.1 Study Area
2.2 Scenarios
2.3 Methodology
3 Result and Discussion
4 Conclusion
References
Turning Electronic and Optical Properties of Monolayer Janus Sn-Dichalcogenides By Biaxial Strain
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies
1 Introduction
2 Constitutive Equations of Hyperelastic Material
3 IRBF-Based Meshfree Method for Hyperelasticity
3.1 Integrated Radial Basis Function Approximation
3.2 The Weak Form and Meshfree Method
4 Numerical Examples
4.1 Inhomogeneous Compression Problem
4.2 Curved Beam Problem
5 Conclusion
References
Structural Damage Identification of Plates Using Two-Stage Approach Combining Modal Strain Energy Method and Genetic Algorithm
1 Introduction
2 Modal Strain Energy and Genetic Algorithm-Based Damage Identification Approach
2.1 Finite Element Method for Free Vibration Analysis of Plate Structures
2.2 Modal Strain Energy Method
2.3 Genetic Algorithm
2.4 Flow Chart
3 Numerical Verification
3.1 Properties of Plate
3.2 Modal Analysis
3.3 Identification of the Damage’s Location
3.4 Identification of the Damage’s Extent
4 Conclusions
References
Auxetic Structure Design: A Multi-material Topology Optimization with Energy-Based Homogenization Approach
1 Introduction
2 Homogenization with Periodic Boundary Conditions
3 Alternating Active Phase Algorithm (AAPA) of Topology Optimization
4 Optimization Model
5 Numerical Examples
5.1 Single-material Auxetic Patterns
5.2 Multi-material Auxetic Patterns
5.3 Benchmark Verification
6 Conclusions
References
Fuzzy Structural Identification of Bar-Type Structures Using Differential Evolution
1 Introduction
2 Fuzzy Fe Model Updating
2.1 FE Model Updating Procedure
2.2 Fuzzy Concept of Uncertainty
2.3 Updating Fuzzy Model Parameters
3 Differential Evolution with Nearest Neighbor Comparison
3.1 Basic of Differential Evolution
3.2 Nearest Neighbor Comparison Method
4 Numerical Examples
4.1 2D Truss
4.2 2D Frame
5 Conclusion
References
Application of Artificial Intelligence for Structural Optimization
1 Introduction
2 Problem Description
3 Building the Deep Neural Network Classification Model
4 Structural Optimization Using Differential Evolution
5 Conclusions
References
A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks
1 Introduction
2 The Proposed 3D Haptic Joystick
3 Controller Design for the 3D Haptic Joystick
3.1 Analysis of the 3D Haptic System
3.2 Close Loop Control of the Feedback Force
3.3 Design of PID Controller for the Feedback Force
3.4 Design of SMC Controller for the Feedback Force
4 Results and Discussions
5 Conclusions
References
Experimental and Numerical Investigations on Flexural Behavior of Retrofitted Reinforced Concrete Beams with Geopolymer Concrete Composites
1 Introduction
2 Research Significance
3 Experimental Work
4 Finite Element Modeling
5 Comparison Experimental and Analytical Results
6 Conclusion
References
A Study of Fluid-Structure Interaction of Unsteady Flow in the Blood Vessel Using Finite Element Method
1 Introduction
2 Governing Equations and the Finite Element Method
2.1 Governing Equations
2.2 Finite Element Formulation for FSI Coupling
3 Results and Discussions
3.1 Validation
3.2 Blood Flow in a Carotid Bifurcation
3.3 Blood Flow Through the Aortic Valve
4 Conclusion
References
Stable Working Condition and Critical Driving Voltage of the Electrothermal V-Shaped Actuator
1 Introduction
2 Theoretical Model of EVA
2.1 Configuration of EVA
2.2 Heat Transfer Model
2.3 Thermal Expansion Force
3 Critical Voltages of Mechanical and Thermal Stability
3.1 Critical Voltage for Beam-Buckling Condition
3.2 Critical Voltage for Thermal Stability Condition
4 Safe Working Condition of the V-Shaped Actuator
5 Conclusion
References
Electromechanical Properties of Monolayer Sn-Dichalcogenides
1 Introduction
2 Computational Methods
3 Results and Discussion
4 Conclusions
References
Author Index

Citation preview

Lecture Notes in Mechanical Engineering

Nguyen Tien Khiem Tran Van Lien Nguyen Xuan Hung   Editors

Modern Mechanics and Applications Select Proceedings of ICOMMA 2020

Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini, Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany

Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. Topics in the series include: • • • • • • • • • • • • • • • • •

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More information about this series at http://www.springer.com/series/11236

Nguyen Tien Khiem Tran Van Lien Nguyen Xuan Hung •



Editors

Modern Mechanics and Applications Select Proceedings of ICOMMA 2020

123

Editors Nguyen Tien Khiem Vietnam Academy of Science and Technology Hanoi, Vietnam

Tran Van Lien National University of Civil Engineering Hanoi, Vietnam

Nguyen Xuan Hung Ho Chi Minh City University of Technology (HUTECH) Ho Chi Minh, Vietnam

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-16-3238-9 ISBN 978-981-16-3239-6 (eBook) https://doi.org/10.1007/978-981-16-3239-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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

Contents

A Dual Approach for Modeling Two- and One-Dimensional Solute Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hung Nguyen

1

Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nguyen Tien Khiem, Nguyen Minh Tuan, and Pham Thi Ba Lien

13

Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tran Ich Thinh and Pham Ngoc Thanh

26

Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method . . . . . . . . . . . . . . . . . . . . . . . . Do H. M. Hieu, Do Q. Duyen, Nguyen P. Tai, Nguyen V. Thang, Ngo C. Vinh, and Nguyen Q. Hung Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions . . . . . . Quoc-Huy Ngo, Ky-Thanh Ho, and Khac-Tuan Nguyen Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dang Xuan Hung, Huong Quy Truong, and Tran Minh Tu Self-vibrational Analysis of a Tensegrity . . . . . . . . . . . . . . . . . . . . . . . . Buntara S. Gan

54

63

72 87

Free Vibration of a Bi-directional Imperfect Functionally Graded Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Le Thi Ha

v

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Contents

Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Le Thi Ha Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism . . . . . . . . . . . . . . . . . . 125 Duy Q. Bui, Hung Q. Nguyen, Vuong L. Hoang, and Dai D. Mai A High-Order Time Finite Element Method Applied to Structural Dynamics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Thanh Xuan Nguyen and Long Tuan Tran A Multibody Dynamics Approach to Study an Insect-Wing Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Vu Dan Thanh Le, Anh Tuan Nguyen, Ngoc Thanh Dang, and Binh Van Phung Natural Ventilation and Cooling of a House with a Solar Chimney Coupled with an Earth – To – Air Heat Exchanger . . . . . . . . . . . . . . . . 158 Viet Tuan Nguyen, Y Quoc Nguyen, and Trieu Nhat Huynh Dynamic Analysis of Curved Beam on Elastic Foundation Subjected to Moving Oscillator Using Finite Element Method . . . . . . . . . . . . . . . . 169 Nguyen Thai Chung, Duong Thi Ngoc Thu, and Nguyen Van Dang Buckling Analysis of FG GPLRC Plate Using a Naturally Stabilized Nodal Integration Meshfree Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Chien H. Thai, P. Phung-Van, and H. Nguyen-Xuan Low Temperature Effect on Dynamic Mechanical Behavior of High Damping Rubber Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Nguyen Anh Dung and Le Trung Phong Effects of Clayey Soil and Hemp Fibers on the Flexural Behavior of Soil Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Duc Chinh Ngo and Ngoc Tan Nguyen Optimal Design of Functionally Graded Sandwich Porous Beams for Maximum Fundamental Frequency Using Metaheuristics . . . . . . . . 229 Hoang-Anh Pham, Truong Q. Huong, and Hung X. Dang Nonlinear Buckling Behavior of FG-CNTRC Toroidal Shell Segments Stiffened by FG-CNTRC Stiffeners Under External Pressure . . . . . . . . . 240 Tran Quang Minh, Vu Minh Duc, and Nguyen Thi Phuong Nonlinear Vibration of Shear Deformable FG-CNTRC Plates and Cylindrical Panels Stiffened by FG-CNTRC Stiffeners . . . . . . . . . . 256 Dang Thuy Dong, Tran Quang Minh, and Vu Hoai Nam

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Vibration Characteristics of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Submerged in Fluid Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Duong Thanh Huan, Tran Huu Quoc, Vu Van Tham, and Chu Thanh Binh Large Amplitude Free Vibration Analysis of Functionally Graded Sandwich Plates with Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Tran Huu Quoc, Duong Thanh Huan, and Ho Thi Hien Dynamic Analysis of a Functionally Greded Sandwich Beam Traversed by a Moving Mass Based on a Refined Third-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Le Thi Ngoc Anh, Tran Van Lang, Vu Thi An Ninh, and Nguyen Dinh Kien Thermal Bending Analysis of FGM Cylindrical Shells Using a Quasi-3D Type Higher-Order Shear Deformation Theory . . . . . . . . . 316 Tran Ngoc Doan, Tran Van Hung, and Duong Van Quang Experimental and Finite Element Analysis of High Strength Steel Fiber Concrete – Timber Composite Beams Subjected to Flexion . . . . . 331 Ngoc Tan Nguyen, Van Dang Tran, Viet Duc Nguyen, and Dong Tran Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 347 Tran Huu Quoc, Vu Van Tham, and Tran Minh Tu Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Nguyen Ba Hung and Ocktaeck Lim Practical Method for Tracking Fatigue Damage . . . . . . . . . . . . . . . . . . 373 Nguyen Hai Son An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Tang Ha Minh Quan and Dang Thien Ngon Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed . . . . . . . . . . . . . 393 Nguyen Trong Du and Nguyen Phong Dien Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 N. Van Xuan, N. Canh Thai, N. Ngoc Thang, and T. Van Toan Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact . . . . . . . . . 416 Quang Thang Do, Sang-Rai Cho, and Van Dinh Nguyen

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Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion . . . . . . . 431 Pham Hien Hau and Vu Dan Chinh An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat . . . . . . . . . . . . . . . . . . . . . . 446 Nguyen Van My and Vo Duy Hung A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Tran Van Lien, Ngo Trong Duc, Tran Binh Dinh, and Nguyen Tat Thang Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings . . . . . . . 476 Nguyen Anh Dung and Nguyen Vinh Sang Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load . . . . . . . . . . . . . . . . . . . . . 489 Khuc Van Phu, Dao Huy Bich, and Le Xuan Doan Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load . . . . . . . . . . . . . . . . . . 506 Khuc Van Phu, Dao Huy Bich, and Le Xuan Doan Vibration and Dynamic Impact Factor Analysis of the Steel Truss Bridges Subjected to Moving Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Toan Nguyen-Xuan, Loan Nguyen-Thi-Kim, Thao Nguyen-Duy, and Duc Tran-Van An Experimental Study on the Self-propelled Locomotion System with Anisotropic Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Van-Du Nguyen, Ky-Thanh Ho, Ngoc-Tuan La, Quoc-Huy Ngo, and Khac-Tuan Nguyen Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Thi Loan Mai, Myungjun Jeon, Seung Hyeon Lim, and Hyeon Kyu Yoon Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 Huu-Anh-Tuan Nguyen Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and Regenerated Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 La Duc Viet, Nguyen Cao Thang, Nguyen Ba Nghi, and Le Duy Minh Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio . . . . . . . . . . . . . 584 Hoang Trong Lam and Vo Duy Hung

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The Equation Connecting Boyle’s and Bernoulli’s Laws . . . . . . . . . . . . 596 Nguyen Ngoc Tinh and Nguyen Linh Ngoc Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Trieu Nhat Huynh and Y. Quoc Nguyen A Solar Chimney for Natural Ventilation of a Three – Story Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Tung Van Nguyen, Y Quoc Nguyen, and Trieu Nhat Huynh Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Y Quoc Nguyen and Trieu Nhat Huynh Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Y Quoc Nguyen and Trieu Nhat Huynh The Effect of Ground Surface Geometry on the Wing Lift Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Dinh Khoi Tran and Anh Tuan Nguyen Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Phan Thanh Long A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Lieu B. Nguyen, Chien H. Thai, and H. Nguyen-Xuan Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 Tran Quang Hung, Do Minh Duc, and Tran Minh Tu Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials . . . . . . . . . . . . . . . . . . . . . . . 707 Trung-Kien Nguyen Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Tham Hong Duong, Huan NguyenPhu Vo, and Danh Thanh Tran Effects of Test Specimens on the Shear Behavior of Mortar Joints in Hollow Concrete Block Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 Thi Loan Bui Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells Under External Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Le Kha Hoa, Pham Van Hoan, Bui Thi Thu Hoai, and Do Quang Chan

x

Contents

Bulk Modulus Prediction of Particulate Composite with Spherical Inclusion Surrounded by a Graded Interphase . . . . . . . . . . . . . . . . . . . 755 Nguyen Duy Hung and Nguyen Trung Kien Post-buckling Response of Functionally Graded Porous Plates Rested on Elastic Substrate via First-Order Shear Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Le Thanh Hai, Nguyen Van Long, Tran Minh Tu, and Chu Thanh Binh Elastic Buckling Behavior of FG Polymer Composite Plates Reinforced with Graphene Platelets Using the PB2-Ritz Method . . . . . . 780 Xuan Hung Dang, Dai Hao Tran, Minh Tu Tran, and Thanh Binh Chu Design and Fabrication of Mecanum Wheel for Forklift Vehicle . . . . . . 795 Thanh-Long Le, Dang Van Nghin, and Mach Aly Muti Object Prediction and Optimization Process Parameters in Cooling Slope Using Taguchi-Grey Relational Analysis . . . . . . . . . . . 811 Anh Tuan Nguyen, Dang Giang Lai, and Van Luu Dao The Effect of Porosity on the Elastic Modulus and Strength of Pervious Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Viet-Hung Vu, Bao-Viet Tran, Viet-Hai Hoang, and Thi-Huong-Giang Nguyen Predicting Capacity of Defected Pipe Under Bending Moment with Data-Driven Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 Hieu C. Phan, Nang D. Bui, Tiep D. Pham, and Huan T. Duong Image Recognition Using Unsupervised Learning Based Automatic Fuzzy Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 V. V. Tai and L. T. K. Ngoc The Effect of the Non-uniform Temperature and Length to Height Ratio on Elastic Critical Buckling of Steel H-Beam . . . . . . . . . . . . . . . . 857 Xuan Tung Nguyen, Myung Jin Lee, Thac Quang Nguyen, and Jong Sup Park On the Thin-Walled Theory’s Application to Calculate the Semi-enclosed Core Structure of High-Rise Buildings . . . . . . . . . . . 866 Nguyen Tien Chuong and Doan Xuan Quy Numerical Simulation of Full-Scale Square Concrete Filled Steel Tubular (CFST) Columns Under Seismic Loading . . . . . . . . . . . . . . . . . 875 Hao D. Phan, Ker-Chun Lin, and Hieu T. Phan Numerical Modeling of Shear Behavior of Reinforced Concrete Beams with Stirrups Corrosion: Finite Element Validation and Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 Tan N. Nguyen and Kien T. Nguyen

Contents

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Characterization of Stress Relaxation Behavior of Poscable-86 High-Strength Steel Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 Ngoc-Vinh Nguyen and Viet-Hung Truong Seismic Performance of Concrete Filled Steel Tubular (CFST) Columns with Variously Axial Compressive Loads . . . . . . . . . . . . . . . . 915 Hao D. Phan and Ker-Chun Lin Damage Simulation Based on the Phase Field Method of Porous Concrete Material at Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 Hoang-Quan Nguyen, Ba-Anh Le, and Bao-Viet Tran Constitutive Response and Failure Mechanism of Porous Cement-Based Materials Under Triaxial Stress States . . . . . . . . . . . . . . 935 Thang T. Nguyen, Lien V. Tran, Tung T. Pham, and Long N. Tran Development of Automatic Landing Control Algorithm for FixedWing UAVs in Longitudinal Channel in Windy Conditions . . . . . . . . . . 945 Hong Son Tran, Duc Cuong Nguyen, Thanh Phong Le, and Chung Van Nguyen An Assessment of Terrain Quality and Selection Model in Developing Landslide Susceptibility Map – A Case Study in Mountainous Areas of Quang Ngai Province, Vietnam . . . . . . . . . . . 959 Doan Viet Long, Nguyen Chi Cong, Nguyen Tien Cuong, Nguyen Quang Binh, and Vo Nguyen Duc Phuoc Analysis of the Impact of Ponds Locations in Flood Cutting from a Developing City in Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971 Quang Binh Nguyen, Duc Phuoc Vo, Hoang Long Dang, Hung Thinh Nguyen, and Ngoc Duong Vo Turning Electronic and Optical Properties of Monolayer Janus Sn-Dichalcogenides By Biaxial Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 Vuong Van Thanh, Nguyen Thuy Dung, Le Xuan Bach, Do Van Truong, and Nguyen Tuan Hung A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 Thai Van Vu, Nha Thanh Nguyen, Minh Ngoc Nguyen, Thien Tich Truong, and Tinh Quoc Bui Structural Damage Identification of Plates Using Two-Stage Approach Combining Modal Strain Energy Method and Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Thanh-Cao Le and Duc-Duy Ho Auxetic Structure Design: A Multi-material Topology Optimization with Energy-Based Homogenization Approach . . . . . . . . . . . . . . . . . . . 1018 T. M. Tran, Q. H. Nguyen, T. T. Truong, T. N. Nguyen, and N. M. Nguyen

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Contents

Fuzzy Structural Identification of Bar-Type Structures Using Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 Ba-Duan Nguyen and Hoang-Anh Pham Application of Artificial Intelligence for Structural Optimization . . . . . . 1052 Tran-Hieu Nguyen and Anh-Tuan Vu A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Diep B. Tri, Le D. Hiep, Vu V. Bo, Nguyen T. Nien, Duc -Dai Mai, and Nguyen Q. Hung Experimental and Numerical Investigations on Flexural Behavior of Retrofitted Reinforced Concrete Beams with Geopolymer Concrete Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Do Van Trinh and Khong Trong Toan A Study of Fluid-Structure Interaction of Unsteady Flow in the Blood Vessel Using Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 S. T. Ha, T. D. Nguyen, V. C. Vu, M. H. Nguyen, and M. D. Nguyen Stable Working Condition and Critical Driving Voltage of the Electrothermal V-Shaped Actuator . . . . . . . . . . . . . . . . . . . . . . . 1102 Kien Trung Hoang and Phuc Hong Pham Electromechanical Properties of Monolayer Sn-Dichalcogenides . . . . . . 1113 Le Xuan Bach, Vuong Van Thanh, Hoang Van Bao, Do Van Truong, and Nguyen Tuan Hung Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121

A Dual Approach for Modeling Two- and One-Dimensional Solute Transport The Hung Nguyen(B) Faculty of Water Resources Engineering, University of Science and Technology, The University of Danang, Da Nang, Vietnam [email protected]

Abstract. The solute transport equation of one-dimensional (1D) or twodimensional vertical (2DV) flow is normally constructed by the classic average method. These solute transport equations are integrated from the right to the left river bank; the average values received by this approach therefore do not generalize by means of dual approach. This paper presents the application of a dual approach to establish the 2DV solute transport equation. In particular, the concentration in a 2DV flow is obtained by twice integrals: (i) integration from the right river bank to the intermediate vertical surface layer between the right bank and the left bank, and then (ii) integration from the right bank to the left bank. From the twodimensional horizontal (2DH) [7] and 2DV flow constructed by a dual approach, the researcher receives the 1D flow equation. The average concentration obtained from the dual approach is better than the classical approach, particularly, in the case of mixed solute transport, stratification, and etc. The basic equation obtained is based on the dual approach that describes the solute transport is more accurate than the classical method. In other words, it provides some flexible parameters to adjust based on the field or experimental data. A case study of solute transport (salinity transport) in Huong river system is illustrated. Keywords: Dual approach · Two-dimensional solute transport equation · Two-dimensional vertical average concentration · One-dimensional solute transport equation

1 Introduction The solute transport of the 2DV flow equations is usually obtained by integrating the three-dimensional (3D) flow equation from the right to the left bank of a river cross-session [9]; in the 2DV equations, the average values are known as the global average values (GAV). Based on the document [1], the global-local average values (GLAV) obtained using the dual approach are more general than the GAV. Therefore, in this paper, the 2DV solute transport equation is established based on the dual approach [1, 2], the 3D solute transport equation is integrated twice; the first integration is from the right bank to an intermediate vertical surface layer between the right and the left river

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1–12, 2022. https://doi.org/10.1007/978-981-16-3239-6_1

2

T. H. Nguyen

bank, then the second integration is from the right bank to the left bank. The average concentration value obtained from this 2DV solute transport equation is more accurate than the classical approximation. It can be described many complex physical phenomena of solute transport such as the stratification flow, mixed solute transport, etc. The numerical calculation results compared with the field data of the 1D salinity intrusion problem are also illustrated to further clarify the research problem.

2 Establishing a 2DV Solute Transport Equation Based on the Dual Approach The 3D solute transport equation in the form of conservation reads [9]:       ∂(uc) ∂(vc) ∂(wc) ∂ ∂c ∂ ∂c ∂ ∂c ∂c (D + εx ) (D + εy ) (D + εz ) + + + = + + ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂y ∂z ∂z                    (I )

(J )

(K)

(L)

(M )

(N )

(1)

(P)

Where t is the time; x, y, z are the Cartesian coordinates system; u, v, w are the velocity components in the x, y, and z directions respectively; c is the solute transport conservative scalar variable; D is the molecular diffusion coefficient; εx , εy , εz , are the turbulent diffusion coefficients. Firstly, integrating the term (I ) from the right bank Y1 to the intermediate vertical surface layer Ym located between the right bank Y1 and the left bank Y2 and at a distance b1 from the right bank Y1 (see Fig. 1), we get: Ym TI = Y1

∂ ∂c dy = ∂t ∂t

Ym

Ym cdy −

Y1

c Y1

∂ ¯ ∂(dy) ¯ 1 . ∂b1 = (Cb 1) − C ∂t ∂t ∂t

(2)

In Fig. 1 b is the width of the river section at the free surface; b1 is the width at the free surface from at which the concentration Cm is almost equal to zero; Y1 , Ym and Y2 are the Cartesian coordinates corresponding.

Fig. 1. Sketch of the dual approach of the 2DV solute transport equations

A Dual Approach for Modeling Two- and One-Dimensional Solute

3

Where C¯ and C¯ 1 are the average concentrations at the width b and b1 respectively is given: ⎛ ⎞ Y2 Ym Y2 Ym 1⎜ 1 ⎟ 1 ¯ cdy = ⎝ cdy + cdy ⎠ = cdy (3a) C= b b b Y1

Y1

Ym

Y1

(Because, from Ym to Y2 , C ≈ 0). Posing C1 is average concentration, such that: Ym Y1

∂(dy) = C1 c. ∂t

C1 =

Y1

∂(dy) ∂b1 = C1 ∂t ∂t

Ym

∂b ∂t Y 1

c.

∂(dy) ∂t

(3b)

C¯ 1 = α1 .C¯

(3c)

Y1 1 C¯ 1 ∂(dy) = ∂b c. α1 = ¯ ¯ ∂t C C ∂t

(3d)

Posing:

Where

1

Ym

Y1

Then, the term I in Eq. (1) is integrated the second time from the right bank Y1 to the left bank Y2 , we get: Y2  T 2I = Y1

  Y2  Y2 ∂ ¯ ∂ ¯ ∂b ∂b1 1 ¯ (C.b1 ) − C1 . db1 = (Cb1 ) db1 − C¯ 1 . db1 ∂t ∂t ∂t ∂t Y1 Y1       I1

Y2  T I1 = 2

Y1

I2

 Y2 Y2 ∂ ¯ ∂ ¯ ¯ 1 . ∂ (db1 ) (Cb1 ) db1 = Cb1 .db1 − Cb ∂t ∂t ∂t Y1 Y1       I11

(4a)

(4b)

I12

Where: I11 =

 1 ∂ ¯ 2 . C(Y2 − Y12 ) 2 ∂t

  Y2 ∂ 2 ∂ ∂b1 1 ¯ 2 ∼ ¯ ¯ = Cb1 . (db1 ) = I2 = C1 . db1 = .C12 (Y − Y1 ) ∂t ∂t 2 ∂t 2 Y2 Y1      

(4c)

Y2

I12

I12

I2

(4d)

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T. H. Nguyen

¯ C¯ 12 = α12 .C.

Let:

(4e)

Y2

∂ C¯ 1 . (ydy) ∂t Y1 ⎧ ⎫ ⎪ Y2 ⎪ Y2 ⎨ 1 2 ∂(dy) ⎬ ∂ = ∂ 2 . (ydy) . c. ∂b ⎪ ∂t ⎪ C¯ ∂t (Y2 − Y12 ) ⎩ ∂t ⎭ ∂t

Where: α12 =

2

C¯ ∂t∂ (Y22

− Y12 )

.

Y1

Y1

With: T 2 I = I1 − I2 = (I11 − I12 ) − I2 = I11 − 2I12 T 2I =

(4f)

   ∂ 2 1 ∂ ¯ 2 . C(Y2 − Y12 ) − α12 C¯ (Y2 − Y12 2 ∂t ∂t

(4g) (5a)

In the non-conservative form: 1 1 ∂ ¯ ¯ ∂ (Y22 − Y12 ) − α12 C. T 2I ∼ = (Y22 − Y12 ). (C) 2 ∂t 2 ∂t

(5b)

Similarly, the second term J is integrated the first time from the right bank Y1 to the intermediate vertical surface layers Ym , we get: Ym TJ = Y1

∂ ∂uc .dy = ∂x ∂x

Ym

Ym ucdy −

Y1

uc Y1

∂ ∂ ∂ (dy) = (b1 .UC) − β1 .UC. (b1 ) (6a) ∂x ∂x ∂x

Where: UC =

β1 =

1

1 h1

Ym

Y1 Ym

.

uc

UC ∂b ∂x Y 1

(uc).dz

(6b)

∂ (dy) ∂x

(6c)

The term J of Eq. (1) is integrated the second time from the right bank to the left bank: T 2J =

Y2 Y1

∂ (b .UC)dy − ∂x 1







J1

Where: J1 =

Y2 Y1

∂ ∂x (b1 .UC)dy

β1 .UC.

Y1



∂ (b )dy ∂x 1



(7a)



J2

Y2 Y2 ∂ ∂ = b1 .UCdy − β1 b1 UC (dy) ∂x ∂x Y1 Y1       J11



J1 =

Y2

J12



∂ 1 1 ∂ β2 UC (Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) ∂x 2 2 ∂x       J11

J12

(7b)

A Dual Approach for Modeling Two- and One-Dimensional Solute

5

Noticed that: J2 = J12 .

(7c)

So: T2J = J1 − J1 = (J11 − J12 ) − J2 = J11 − 2J12

(8a)

J1 =

 1 ∂  ∂ β2 UC.(Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) 2 ∂x ∂x Where: β2 =

β3 =

Y2

1

UCb1 dy

2

UC b2

− Y12 )

(8c)

Y1

Y2

1 ∂ UC ∂x (Y22

(8b)

UC Y1

∂ (ydy) ∂x

(8d)

Again, the fourth term L is integrated on the left-hand side of the Eq. (1) the first time from Y1 to Ym , we get: Ym TL = Y1

Ym Ym ∂ ∂ ∂ (wc)dy = (wc)dy − (wc) (dy) ∂z ∂z ∂z Y1 Y1       L1

∂ Where: L1 = ∂z Ym L2 =

(wc) Y1

(9a)

L2

Ym (wc)dy = Y1

∂ (WC.b1 ) ∂z

(9b)

∂ ∂ ∂ (dy) = δ1 .(WC). (Ym − Y1 ) = δ1 .(WC). (b1 ) ∂z ∂z ∂z

(9c)

∂ ∂ (WC.b1 ) + δ1 .(WC). (b1 ) ∂z ∂z

(9d)

Thus: TL =

1 Where: WC = b1

δ1 =

1

Ym (wc)dy Y1

Ym

∂ (WC). ∂z (b1 ) Y

(WC) 1

(10a)

∂ (dy) ∂z

(10b)

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T. H. Nguyen

Similarly, the term L of the Eq. (1) is integrated the second time, we obtain: Y2

Y2 ∂ ∂ T L= (WC.b1 )dy − δ1 .(WC). (b1 )dy ∂z ∂z Y1 Y1       2

L11

Y2 L11 = Y1

L22

Y2 Y2 ∂ ∂ ∂ (WC.b1 )dy = WC.b1 .dy − (WC).b1 (dy) ∂z ∂z ∂z Y1 Y1       L111

L11

(11a)

L112

  ∂ 1 2 1 ∂ 2 δ2 .WC. (Y2 − Y1 ) − δ3 .WC. . (Y22 − Y12 ) = ∂z 2 2 ∂z      

(11b)

L112

L111

We have: L22 ≈ L112 . Therefore: T2L = L11 − L22 = L111 − L112 − L22 = L111 − 2L112

(11c)

The correction coefficients δ2 , δ3 are calculated as follows: 2 δ2 = 2 (Y2 − Y12 ) δ3 =

2 ∂ 2 ∂z (Y2

− Y12 )

Y2 bdy

(11d)

Y1

Y2 Y1

∂ (bdy) ∂z

(11e)

The third term K is integrated on the left-hand side of Eq. (1), the first time, we obtain: Ym TK = Y1

∂(vc) dy = 0 (noted that v ≈ 0) ∂y

(12)

The first term M is integrated on the right-hand side of Eq. (1), the first time, we get:      Ym ∂ ∂ ∂c ∂ C¯ (D + εx ). dy = γ1 TM = (13a) (D + εx ). .(Ym − Y1 ) ∂x ∂x ∂x Y1 ∂x

A Dual Approach for Modeling Two- and One-Dimensional Solute

7

Where: γ1 =

∂ ∂x



  ∂ ∂c  (D + ε dy . ). x ¯ ∂x (D + εx ). ∂∂xC .(Ym − Y1 ) Y ∂x Ym

1

(13b)

1

The term M of Eq. (1) is integrated the second time, we receive: Y2 T 2M = Y1

    ∂ ∂ ∂ C¯ 1 ∂ C¯ γ1 (D + εx ). .b1 dy = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) ∂x ∂x 2 ∂x ∂x (14a)

The coefficient γ2 is calculated by: 2 γ2 = 2 . (Y2 − Y12 ) Thus:

Y2 b.dy

(14b)

Y1

  ¯ ∂ 1 ∂ C T 2 M = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x

(14c)

Similarly, the third term P on the right-hand side of Eq. (1) is integrated, the first time, we get: Ym TP = Y1

    ∂ ∂ ∂c ∂ C¯ (D + εz ). dy = θ1 (D + εz ). .(Ym − Y1 ) ∂z ∂z ∂z ∂z

(15a)

Where: θ1 =

∂ ∂z



Ym

1 ¯



(D + εz ). ∂∂zC .(Ym − Y1 )

. Y1

  ∂ ∂c (D + εz ). dy ∂z ∂z

(15b)

The term P of Eq. (1) is integrated, the second time, we obtain: Y2 T P= 2

Y1

    ∂ ∂ ∂ C¯ 1 ∂ C¯ θ1 (D + εz ). .b1 dy = .θ1 .θ2 (D + εz ). .(Y22 − Y12 ) ∂z ∂z 2 ∂z ∂z (15c)

The coefficient θ2 is calculated as: 2 θ2 = 2 . (Y2 − Y12 )

Y2 bdy Y1

(15d)

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T. H. Nguyen

Therefore:

  ∂ 1 ∂ C¯ T P = .θ1 .θ2 (D + εz ). .(Y22 − Y12 ) 2 ∂z ∂z 2

(15e)

The second term N on the right-hand side of Eq. (1) is integrated, the first time, we get: Ym TN = Y1

    ∂ ∂c ∂c  ∂c  (D + εy ). dy = (D + εy ). y − (D + εy ). y ∂y ∂y ∂y m ∂y 1

(16a)

The term N is integrated the second time, we obtain: Y2 T N= 2

Y1



In case:

∂c  ∂y y Z

= m

  Y2 ∂c  ∂c  (D + εy ). y dy − (D + εy ). y dy ∂y m ∂y 1

(16b)

Y1



∂c  ∂y y Z

, then the term: T 2 N = 0 1

Therefore, the 2DV solute transport equation based on the dual approach can be expressed as follows: T 2I + T 2J + T 2K + T 2L = T 2M + T 2N + T 2P     1 ∂ ¯ 2 ∂ 2 1 ∂  2 2 ¯ . C(Y2 − Y1 ) − α12 C (Y2 − Y1 ) + β2 UC.(Y22 − Y12 ) 2 ∂t ∂t 2 ∂x   ∂ 1 ∂ ∂ WC.(Y22 − Y12 ) − δ1 δ3 .WC (Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) + δ2 . ∂x 2 ∂z ∂z 1 ∂ ∂ C¯ = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x   1 ∂ ∂ C¯ + .θ1 .θ2 (17) (D + εz ). .(Y22 − Y12 ) 2 ∂z ∂z From Eq. (17), we received the 1D solute transport equation based on the dual approach as follows:    ∂ 2 1 ∂ ¯ 2 2 2 ¯ . C(Y2 − Y1 ) − α12 C (Y − Y1 ) 2 ∂t ∂t 2  ∂ 1 ∂  UC.(Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) + β2 (18) 2 ∂x ∂x   1 ∂ ∂ C¯ = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x

A Dual Approach for Modeling Two- and One-Dimensional Solute

9

3 Comments It is easy to see that the 2DV solute transport equation averaged by the classic method is a special case of the general form Eq. (17). In fact, if Y2 − Y1 = b ∼ = const, so: ∂ 2 − Y 2 ) ∼ 0, ∂ (Y 2 − Y 2 ) ∼ 0, ∂ (Y 2 − Y 2 ) ∼ 0 (the bank width changes (Y = = = 1 1 1 ∂t 2 ∂x 2 ∂z 2 ∼ insignificantly  over the time and space) and the coefficients (αi , βi , γi , δi , θi ) = 1, ∂c  ∼ ∂c  ∂y y m = ∂y y 1 then Eq. (17) becomes the 2DV solute transport equation averaged by the classic method Eq. (19):    ∂ ¯ 2 ∂  ∂  C(Y2 − Y12 ) + UC.(Y22 − Y12 ) + WC.(Y22 − Y12 ) ∂t ∂x   ∂z  ¯ ∂ ∂C ∂ ∂ C¯ 2 2 = (D + εx ). .(Y2 − Y1 ) + (D + εz ). .(Y22 − Y12 ) ∂x ∂x ∂z ∂z From Eq. (19), we receive the 1D Eq. (20) as follows:      ¯ ∂ ¯ 2 ∂ ∂ ∂ C C(Y2 − Y12 ) + UC.(Y22 − Y12 ) = (D + εx ). .(Y22 − Y12 ) ∂t ∂x ∂x ∂x

(19)

(20)

If this equation is combined with Eq. (20) in [6], we have the 1D solute transport equation based on the dual approach more generally as follows:     ∂ ¯ 2 ∂ ¯ 2 ∂  ∂  C(Zs − Zb2 ) + C(Y2 − Y12 ) + UC.(Zs2 − Zb2 ) + UC.(Y22 − Y12 ) ∂t ∂t ∂x ∂x    ¯ ∂ ∂ C¯ ∂ ∂ C = (D + εx ). .(Zs2 − Zb2 ) + (D + εx ). .(Y22 − Y12 ) ∂x ∂x ∂x ∂x (21) In the case Zs − Zb = h ∼ = const and Y2 − Y1 = b ∼ = const, Eq. (21) is simplified to:   ∂ ¯ ∂   ∂ ∂ C¯ C + UC = (D + εx ). (22) ∂t ∂x ∂x ∂x Equation (22) is the familiar classical 1D solute transport equation [4, 9].

4 Results To illustrate the generality of the solute transport Eqs. (17) or (21) established by the dual approach in this paper, a case study of the salinity intrusion of Huong river system in the dry season is presented [5]. There are two branches: Ta-Trach and Huu-Trach confluencing at Nga-Ba-Tuan, upstream of the Huong river; the water flows along the mainstream of Huong river to the Thao-Long barrage at downstream [5], see Fig. 2.

10

T. H. Nguyen

Fig. 2. A simplified sketch of the Huong river system

Boundary conditions: (i) Upstream boundary conditions: QBinhÐieu + QDu,o,ngHoa + QKL = (0.98 + QKL ) m3 /s; (ii) Downstream boundary condition: tidal water level (frequency P = 25%) at ThaoLong barrage, see Fig. 3.

Fig. 3. Water level boundary condition at downstream

To determine the reasonable size of Khe-Lu reservoir used for salinity repelling on the Hu,o,ng river system, all step discharge from Khe-Lu reservoir are assumed in the document [5].

A Dual Approach for Modeling Two- and One-Dimensional Solute

11

Fig. 4. Salinity at La-Y in a tidal period corresponding with the Khe-Lu discharge QKL = 10.00 m3 /s

Algorithm and Program: We apply the finite difference method using the four-point Preissmann scheme, the weight θ = 0.66, to solve the 1D Saint -Venant equations system [3–6, 8, 9], then solve the solute transport Eq. (21) and (22) using fractional steps: transport equation and diffusion equation. The characteristic method was applied to solve the transport equation and the weighted finite difference scheme for the diffusion equation [5]. The calculated results of the salinity at La-Y corresponding with the Khe-Lu discharge QKL = 10.00 m3 /s, are plotted in Fig. 4.

5 Comments Figure 4 shows that the numerical solution results based on the dual approach Eq. (21) are more accurate than the ones of the classical method Eq. (22). In addition, in the general Eq. (18) based on the dual approach, the calculation results can be better by changing the coefficients (αi , βi , γi , δi , θi ). Due to the complicated changing of the river-bed topography and the surface roughness of Huong river-bed, so the simulation results based on the dual approach still cannot capture all the observed data.

6 Conclusions The solute transport Eqs. 2DV (17) and 1D (21) derived from the dual approach are more general than those received from the classic approach with the appearance of new terms in the Eqs. (17) and (21); they could be used to describe the complex physic phenomena of solute transport better than the equations established by classical method. On the other hand, the calculated results can be better obtained by changing corrected coefficients (αi , βi , γi , δi , θi ).

12

T. H. Nguyen

References 1. Anh, N.D.: Dual approach to averaged values of functions. Vietnam J. Mech. 34(3), 211–214 (2012) 2. Anh, N.D.: Dual approach to averaged values of functions: advanced formulas. Vietnam J. Mech. 34(4), 321–325 (2012) 3. Cunge, J.A., et al.: Practical Aspects of Computational River Hydraulics. Pitman Publishing INC (1980) 4. Nguyen, T.D.: One-dimensional mathematical model of salinity intrusion in rivers network, Physical Mathematical Dissertation, Hanoi (1987) 5. Nguyen, T.H.: Salinity intrusion in Huong river network and the measure of hydraulic construction, J. Sci. Technol. (Five University of Technology), (2), 17–21 (1992) 6. Nguyen, T.H.: A dual approach to modeling solute transport. In: The International Conference on Advances in Computational Mechanics, ACOME 2017 August 02–04, Phu quoc, Vietnam, Lecture Notes in Mechanical Engineering, pp. 821–834 (2017) 7. That, T.T., Nguyen, T.H., Nguyen, D.A.: A dual approach for model construction of twodimensional horizontal flow. In: Viet, N.T., Xiping, D., Tung, T.T. (eds.) APAC 2019: Proceedings of the 10th International Conference on Asian and Pacific Coasts, 2019, Hanoi, Vietnam, pp. 115–119. Springer Singapore, Singapore (2020). https://doi.org/10.1007/978-98115-0291-0_17 8. Preissmann, A.: Propagation des intumescences dans les canaux et Les Rivieres. Congres de l’Association Francaise de Calcule, Grenoble, France (1961) 9. Wu, W.: Computational River Dynamics. Taylor & Francis Group, London, UK (2008)

Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application Nguyen Tien Khiem1,2(B) , Nguyen Minh Tuan1 , and Pham Thi Ba Lien3 1 Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

{ntkhiem,nmtuan}@imech.vast.vn

2 CIRTECH Institute, HUTECH, Ho Chi Minh City, Vietnam

[email protected]

3 University of Transport and Communications, Hanoi, Vietnam

[email protected]

Abstract. The Rayleigh quotient is an attractive relationship between natural frequency and mode shape of structural vibration. However, it would be useful only for approximately calculating natural frequencies of a structure by using the properly chosen trial functions of the mode shapes in case when both the modal parameters are unknown. This is completely appropriate for the case of cracked structures when an explicit expression of natural frequencies is needed for some purpose such as structural health monitoring. The present report is devoted first to establish Rayleigh quotient for longitudinal vibration in multiple cracked bars and then to involve it for calculating natural frequencies of the cracked structure by using mode shapes of uncracked one. Numerical examples are accomplished for accessing usability of various approximate expressions of the obtained Rayleigh quotient. Keywords: Rayleigh quotient · Natural frequencies · Multiple cracked bar · Structural health monitoring

1 Introduction The frequency-based method has proved to be an efficient technique for solving the structural damage detection problem. This is because natural frequencies are the most typical characteristic of a structure that can be easily and accurately measured in the practice. One of the drawbacks of the method is that the characteristic equations established for determining natural frequencies of damaged structures are usually in implicit form. The lack of explicit expression of natural frequencies through damage parameters obstructs solving the inverse problem of damage detection. Thus, seeking explicit expression of the characteristic equation or the natural frequencies themself with purpose to apply for solving the damage detection problem is vitally needed. A lot of studies were devoted to establish explicit expression of characteristic equation of cracked structures, but there are a few publications concerned with finding explicit expression for natural frequencies of the structures. The first formula for calculating fundamental frequency of a mechanical system was proposed by Rayleigh and it has been named Rayleigh quotient. The © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 13–25, 2022. https://doi.org/10.1007/978-981-16-3239-6_2

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Rayleigh quotient was first applied for calculating fundamental frequency of cracked beam [1] and then it has been developed for multiple cracked beam [2, 3]. The present study is devoted to establish Rayleigh quotient for multiple cracked bars and apply it for calculating natural frequencies of longitudinal vibration in the structures. Longitudinal vibration of bar with single crack was first studied in [4–8] and then it has been developed for multiple cracked bars in [9–12]. General procedure for crack detection in vibrating bars was theoretically proposed by Morassi and his coworkers [7, 8, 13], who revealed that using only natural frequencies cannot uniquely resolve the problem of crack detection even in case of single crack. The authors proposed to use additionally anti-resonant frequencies for obtaining unique solution of the crack detection problem [14–16]. General theory and application of anti-resonant frequencies for multi-crack detection in bars were proposed and discussed recently in [17] by the authors of this study. Nevertheless, as shown below, the Rayleigh quotient established in this work would be helpful to solve the non-uniqueness problem of crack detection in bars without involving the anti-resonant frequencies but using only natural frequencies. The structure of this report is arranged as follows: First, general Rayleigh quotient of multiple cracked bars is established (Subsect. 2). Then, the established formula is applied for calculating natural frequencies of multiple cracked bar (Subsect. 3) using mode shapes of uncracked one. Finally, numerical illustration and discussion are provided in Subsect. 4 for singly cracked bars with fixed-free and free-free ends.

2 Rayleigh Quotient for Multiple Cracked Bar Let’s consider an uniform bar with material and geometrical parameter E, ρ, A, L that contains n cracks at positions e1 , ..., en modeled by translational spring of stiffness K j , j = 1, ..., n, calculated from the crack depth a1 , ..., an by using the formulas [18] (Fig. 1). γ = EA/KL = 2π(1 − ν 2 )(h/L)f (z), z = a/h, f (z) = z 2 (0.6272 − 0.17248z + 5.92134z 2 − 10.7054z 3 + 31.5685z 4 − 67.47z 5 + 139.123z 6 − 146.682z 7 + 92.3552z 8 ),

Fig. 1. Model of a multiple-cracked bar

Rayleigh Quotient for Longitudinal Vibration

For the bar, natural mode of vibration is governed by the equation   (x) + λ2 (x) = 0, x ∈ (0, 1), λ = ωL ρ/E,

15

(1)

that is solved together with boundary conditions at the bar ends x = 0, x = 1 and compatibility conditions at the cracked sections 





Kj [(ej + 0) − (ej − 0)] = EA (ej ) = EA (ej − 0) = EA (ej + 0), j = 1, ..., n or (ej + 0) = (ej − 0) + γj  (ej ), γj = EA/LKj , j = 1, ..., n.

(2)

Let solution of Eq. (1) in the segment (ej−1 , ej ), j = 1, ..., n + 1, e0 = 0, en+1 = 1 be denoted by j (x), j = 1, ..., n + 1, that must satisfy conditions (2) rewritten as. j+1 (ej + 0) = j (ej − 0) = j (ej ), j+1 (ej + 0) = j (ej − 0) + γj j (ej ), j = 1, ..., n.

(3)

Since every function j (x) satisfies Eq. (1) in the segment (ej−1 , ej ) one has got j (x) + λ2 j (x) = 0, x ∈ (ej−1 , ej ).

(4)

Multiplying both sides of Eq. (1) by j (x)and taking integration along the segment (ej−1 , ej ) lead to ej

ej

ej



j (x)j (x)dx + λ2 ej−i

ej

   2j (x)dx = B ej−1 , ej − j2 (x)dx + λ2 2j (x)dx = 0 ej−i ej−i ej−i

(5)

or ej λ

ej 2j (x)dx

2



=

ej−1

j2 (x)dx − B(ej−1 , ej ),

(6)

ej−1

  where B(ej−1 , ej ) = j (ej − 0)j (ej − 0) − j (ej−1 + 0)j (ej−1 + 0) . Now, summing both sides of Eq. (3) yields 1 λ

1  (x)dx =

2 0



 2 (x)dx −

2

0

n+1  j=1

B(ej−1 , ej ).

(7)

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N. T. Khiem et al.

Taking account of conditions (3) the last sum in Eq. (4) can be calculated as. n+1 







B(ej−1 , ej ) = 1 (e1 − 0)1 (e1 − 0) − 1 (0)1 (0) + 2 (e2 − 0)2 (e2 − 0)

j=1 





− 2 (e1 + 0)2 (e1 + 0) + 3 (e3 − 0)3 (e3 − 0) − 3 (e2 + 0)3 (e2 + 0) 

..



+ ... + n (en − 0)n (en − 0) − n (en−1 + 0)n (en−1 + 0) 







+ n+1 (1)n+1 (1) − n+1 (en + 0)n+1 (en + 0) = n+1 (1)n+1 (1) − 1 (0)1 (0) −

n 



γj  2 (ej ).

j=1

So that we obtain 1 1 n      2 2  (x)dx =  2 (x)dx + γj  2 (ej ) −  (1)(1) +  (0)(0) λ 0

or



λ =⎣

1

2

 (x)dx +

2

0

j=1

0

n 

⎤ 2





γj  (ej ) +  (0)(0) −  (1)(1)⎦/

j=1

1 2 (x)dx, (8) 0

that is so-called Rayleigh quotient for multiple cracked bars. Obviously, the obtained quotient is general for arbitrary boundary conditions that is represented by the term     B(0, 1) =  (0)(0) −  (1)(1) It’ easily to verify that the latter term would be disappeared for the conventional boundary conditions (free-free ends; fixed-free ends or fixed-fixed ends), so that the Rayleigh quotient is reduced to ⎡ 1 ⎤ 1   n    γj  2 (ej )⎦/ 2 (x)dx. (9) λ2 = ⎣  2 (x)dx + j=1

0

0

In case of uncracked bar, the latter quotient becomes 1 λ20

=

2

1

0 (x)dx/ 0

20 (x)dx,

(10)

0

where ω0 , 0 (x) are natural frequency and mode shape of the uncracked bar that are provided below (Table 1) for further application. It can be verified that the quotient (10) is consistently fulfilled with the exact natural frequencies and mode shapes given in Table 1. Therefore, the Rayleigh quotient would be meaningless if both the exact frequency and mode shape are available. But it will be greatly useful for calculating natural frequencies by using trial functions of unknown mode shapes. Obviously, the natural frequencies are more accurately calculated as the trial functions are chosen more closed to the exact mode shapes. This technique is applied below for calculating natural frequencies of cracked bars.

Rayleigh Quotient for Longitudinal Vibration

17

Table 1. Natural frequencies and mode shapes of uncracked bar with conventional boundary conditions Boundary conditions

Natural frequencies

Natural mode shapes

Free-Free Ends

λ0k = kπ, k = 1, 2, 3, ....

0k (x) = Ck cos kπ x

Fixed-Free Ends

λ0k = (2k + 1)π/2, k = 0, 1, 2, ....

0k (x) = Ck sin(k + 1/2)π x

Fixed-Fixed Ends

λ0k = kπ, k = 1, 2, 3, ....

0k (x) = Ck sinkπ x

3 Application for Calculating Natural Frequencies In this section, natural frequencies of multiple cracked bars are calculated by properly choosing trial mode shapes. Namely, the mode shapes are chosen in the form j (x) = 0 (x) + Aj x + Bj , x ∈ (ej−1 , ej ), j = 1, ..., n + 1

(11)

with 0 (x) being mode shape of uncracked bar satisfying given boundary conditions and constants Aj , Bj . Substituting expression (8) into conditions (3) leads to 

Aj+1 = Aj ; Bj+1 = Bj + γj 0 (ej ), j = 1, ..., n The latter relationships allow one to have got Aj = A1 ; Bj = B1 +

j−1 



γk 0 (ek ), j = 1, ..., n

(12)

k=1

and, as a particularity, An+1 = A1 ; Bn+1 = B1 +

n 



γk 0 (ek ), j = 1, ..., n.

(13)

γk 0 (ek ), x ∈ (ej−1 , ej ), j = 1, ..., n + 1

(14)

k=1

Consequently, j (x) = 0 (x) + A1 x + B1 +

j−1 



k=1

and 1 (x) = 0 (x) + A1 x + B1 ; n+1 (x) = 0 (x) + A1 x + B1 +

n 



γk 0 (ek ), x ∈ (ej−1 , ej ).

(15)

k=1

Substituting expressions (12) into boundary conditions allow the constants A1 and B1 to be determined, for example, for free-free (a), fixed-free (b) and fixed-fixed (c) end bar one gets respectively (a) A1 = 0, B1 = 0; (b)A1 = B1 = 0; (c) A1 = −

n  k=1



γk 0 (ek ), B1 = 0 .

(16)

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Now, using expressions (11) we can calculate the following integrals 1

2

 (x)dx =

e n+1  j 



j2 (x)dx

j=1 ej−1

0

=

e n+1  j 

2

1



[0 (x) + 2A1 0 (x) + A21 ]dx

=

j=1 ej−1



02 (x)dx + 2A1 [0 (1) − 0 (0)] + A21 ; 0

(17) 1  (x)dx = 2

e n+1  j 

2j (x)dx

j=1 ej−1

0

=

e n+1  j 

[20 (x) + A21 x2 + Bj2 + 2A1 x0 (x) + 2Bj 0 (x) + 2A1 Bj x]dx

j=1 ej−1

1 =

+

n+1 

    20 (x)dx − 2 A1 /λ20 ej 0 (ej ) − 0 (ej ) − ej−1 0 (ej−1 ) + 0 (ej−1 )

0 n+1 

j=1 n+1        Bj2 ej − ej−1 − 2/λ20 Bj 0 (ej ) − 0 (ej−1 )

j=1

j=1

n+1 n+1     3 2 ej3 − ej−1 + A1 Bj ej2 − ej−1 + A21 /3

j=1

1 = 0

j=1

20 (x)dx + 2λ−2 0

n 



−2 γj 02 (ej ) + λ−2 0 0 + λ0 1 + 2 ,

(18)

j=1

where          0 = A21 /3 + (A1 + B1 )B1 λ20 − 2A1 0 (1) − 0 (1) + 0 (0) − 2B1 0 (1) − 0 (0) ; 1 =

n 

     bj γj 0 (ej ); bj = 2 λ20 2B1 + A1 − 2B1 ej − A1 ej2 − 0 (1) , j = 1, ..., n;

j=1

2 =

n  j,k=1

    ajk γj γk 0 (ej )0 (ek ); aij = 1 − ej , ajk = 1 for j  = k

(19)

Rayleigh Quotient for Longitudinal Vibration

19

So, we obtain  λ

2

/λ20

1 0

=

n 

2

0 (x)dx + 1

λ20

0

 γj 

2

j=1

20 (x)dx

+2

(ej ) + 2A1 [0 (1) − 0 (0)] + A21

n  j=1

. 2

γj 0 (ej ) + 0 + 1 + λ20 2 

Recalling Eq. (13) it can be assumed that γj 2 (ej ) ≈ γj 02 (ej ), the latter equation can be rewritten as   n 1  2  2 2 0 (x)dx + γj 0 (ej ) + 2A1 [0 (1) − 0 (0)] + A1 λ2 /λ20 =

j=1

0

λ20

1

20 (x)dx

0

+2

n  j=1

.

(20)

2

γj 0 (ej ) + 0 + 1 + λ20 2

On the other hand, from Eq. (7) one gets

1 0



02 (x)dx = λ20

intact mode shape 0 (x) normalized by condition λ20

1 0

1 0

20 (x)dx and in case of

20 (x)dx = 1, the quotient (20)

would be simplified to 1+ λ2 /λ20 =

n  j=1



γj 02 (ej ) + 2A1 [0 (1) − 0 (0)] + A21

1+2

n  j=1

.



(21)

γj 02 (ej ) + 0 + 1 + λ20 2

Particularly, the obtained Rayleigh’s quotient for conventional boundary conditions: fixed-free and free-free end bars gets the form 1+ λ2 /λ20 =

n  j=1

1+2

n  j=1



γj 02 (ej ) n 

2

γj 0 (ej ) + λ20

j,k=1



(22)



ajk γj γk 0 (ej )0 (ek )

with natural frequencies and mode shapes of uncracked bars given in Table 1. This formula is similar to that obtained by Khiem et al. [2, 3] for multiple cracked beams and provides an efficient tool for modal analysis and identification of cracked bars. Also, expanding the function in the right-hand side of Eq. (19) in Taylor’s series of crack magnitude vector γ = (γ1 , ..., γn ) and keeping first three order terms allow one to obtain λ2 /λ20 = 1 −

n  j=1



γj 02 (ej ) − (λ20 /2)

n  j,k=1





ajk γj γk 0 (ej )0 (ek ),

(23)

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N. T. Khiem et al.

Fig. 2. Three lowest frequency ratios (cracked/intact) of fixed-free end bar calculated by Rayleigh quotient and first approximation compared to the ones obtained from exact characteristic equation (solid lines – exact solution of frequency equation; dash line - first approximation; dash-dot lines – second approximation; dot lines – Rayleigh quotient)

Rayleigh Quotient for Longitudinal Vibration

21

Fig. 2. (continued)

Moreover, the first order asymptotic approximation of the formula (19) λ2 /λ20 = 1 −

n 



γj 02 (ej )

(24)

j=1

was obtained earlier by Morassi in [7] and Shifrin in [11, 12].

4 Numerical Illustration and Discussion For illustration of the proposed above theory, natural frequencies in longitudinal vibration of singly cracked bar with fixed-free and free-free ends are computed by using the conducted Rayleigh quotient mutually with its first and second approximations. The approximately computed frequencies are compared with those obtained by solving the exact frequency equations that allows evaluating the usefulness of the quotient for cracked bar. Results shown in Figs. 2 and 3 are ratio of natural frequencies of cracked bar to those of intact (uncracked) one computed as function of crack position along the bar length in various crack depth. For a given crack depth, there are depicted in the Figures four curves that represent the frequency ratio computed from exact frequency equation [17] and Eqs. (19)–(21) respectively. Figure 2 and Fig. 3 show the first three dimensionless frequencies of fixed-free and free-free end bar respectively. The graphs given in the Figures demonstrate clearly that discrepancy between natural frequencies computed from the Rayleigh quotient and the exact ones increases with

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N. T. Khiem et al.

Fig. 3. Three lowest frequency ratios (cracked/intact) of free-free end bar calculated by Rayleigh quotient and first approximation compared to the ones obtained from exact characteristic equation (solid lines – exact solution of frequency equation; dash line - first approximation; dash-dot lines – second approximation; dot lines –Rayleigh)

Rayleigh Quotient for Longitudinal Vibration

23

Fig. 3. (continued)

crack depth and mode number. Namely, for crack depth less than 20% bar thickness fundamental natural frequency computed from either Rayleigh quotient or exact frequency equation are almost identical and all the frequencies computed from the first approximation (Eq. 21) of Rayleigh quotient are most closed to the exact ones in comparison with those computed from the second approximation (Eq. (20)) and even with the original Rayleigh quotient (Eq. (19)). However, the frequencies computed from Eq. (19) and (20) abolish the symmetrical effect of the cracks located symmetrically about the middle of bar with symmetric boundary conditions (Fig. 3). This is helpful for solving the non-uniqueness problem in identification of crack in bar with symmetric boundary conditions [7]. All the discussions allow one to make a concluding remark as follows: The first approximation of Rayleigh quotient is best employed for calculating natural frequencies of cracked bar, while the second approximation is more useful for crack detection in bar from natural frequencies.

5 Conclusion In the present report, so-called Rayleigh quotient is established for longitudinal vibration in multiple cracked bar that provides a simple tool for engineers to calculate natural frequencies of the damaged structures. This is then used for obtaining an explicit expression of the natural frequencies in terms of crack parameters such as position and depth of the cracks. The obtained expression not only simplifies calculating natural frequencies of cracked bars but also offers a useful tool for crack detection in bar by using natural

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frequencies. First and second asymptotic approximations of the Rayleigh formula with respect to crack depth have been derived for more effective use in both analysis and identification of the cracked structures. Comparison of natural frequencies computed by using different approximations with those computed from the well-known exact frequency equation shows that first approximation of Rayleigh formula is well appropriate for calculating natural frequencies of cracked bar while the second one would be more useful for crack localization from measured natural frequencies. The use of the established Rayleigh quotient for the crack detection problem has not been investigated in this work, but it will be subject of further study of the authors. Acknowledgement. This work was completed with support from VAST under Grant of number NCVCC03.04/20–20.

References 1. Fernandez-Saez, J., Rubio, L., Navarro, C.: Approximate calculation of the fundamental frequency for bending vibration of cracked beams. J. Sound Vib. 225(2), 345–352 (1999) 2. Khiem, N.T., Toan, L.K.: A novel method for crack detection in beam-like structures by measurements of natural frequencies. J. Sound Vib. 333, 4084–4103 (2014) 3. Khiem, N.T., Tran, H.T., Ninh, V.T.A.: A closed-form solution to the problem of crack identification for a multistep beam based on Rayleigh quotient. Int. J. Solids Struct. 150, 154–165 (2018) 4. Adams, R.D., Cawley, P., Pye, C.J., Stone, B.J.: A vibration technique for non-destructively assessing the integrity of structures. J. Mech. Eng. Sci. 20, 93–100 (1978) 5. Narkis, Y.: Identification of crack location in vibrating simply supported beams. J. Sound Vib. 172, 549–558 (1994) 6. Gladwell, G.M.L., Morassi, A.: Estimating damage in a rod from changes in node positions. Inverse Probl. Eng. 7, 215–233 (1999) 7. Morassi, A.: Identification of a crack in a rod based on changes in a pair of natural frequencies. J. Sound Vib. 242(4), 577–596 (2001) 8. Rubio, L., Fernandez-Saez, J., Morassi, A.: Crack identification in non-uniform rods by two frequency data. Int. J. Solids Struct. 75–76, 61–80 (2015) 9. Ruotolo, R., Surace, C.: Natural frequencies of a rod with multiple cracks. J. Sound Vib. 272, 301–316 (2004) 10. Khiem, N.T., Toan, L.K., Khue, N.T.L.: Change in mode shape nodes of multiple cracked rod. Vietnam J. Mech. 35(3), 175–188 (2013) 11. Shifrin, E.J.: Inverse spectral problem for a rod with multiple cracks. Mechanical Systems and Signal Processing 56–57, 181–196 (2015) 12. Shifrin, E.I.: Identification of a finite number of small cracks in a rod using natural frequencies. Mech. Syst. Signal Process. 70–71, 613–624 (2016) 13. Rubio, L., Fernandez-Saez, J., Morassi, A.: Identification of two cracks in rod. J. Sound Vib. 339, 99–111 (2015) 14. Rubio, L., Fernandez-Saez, J., Morassi, A.: Identification of two cracks in a rod by minimal resonant and antiresonant frequency data. Mech. Syst. Sig. Process. 60–61, 1–13 (2015) 15. Dilena, M., Morassi, A.: Structural health monitoring of rods based on natural frequency and antiresonant frequency. Struct. Health Monit. 8(2), 149–173 (2009)

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16. Dilena, M., Morassi, A.: Reconstruction method for damage detection in beam based on natural frequency and anti-resonant frequency measurements. J. Eng. Mech. 136(3), 329–344 (2010) 17. Lien, P.T.B., Khiem, N.T.: Resonant and anti-resonant frequencies of multiple cracked bar. Vietnam J. Mech. VAST 41(2), 157–170 (2019) 18. Chondros, T.G., Dimarogonas, A.D., Yao, J.: Longitudinal vibration of a continuous cracked rod. Eng. Fract. Mech. 61, 593–606 (1998)

Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core Tran Ich Thinh1(B) and Pham Ngoc Thanh2 1 Hanoi University of Science and Technology, 1 Ðai Co Viet, Ha Noi, Viet Nam

[email protected] 2 Viettri University of Industry, Viet Tri, Phu Tho, Viet Nam

Abstract. In this study, based on a modal superposition method and Biot’s theory, an analytical model on vibroacoustic behavior of clamped and simply supported orthotropic rectangular composite sandwich plates with foam core has been derived. Theoretical predictions of sound transmission loss (STL) across finite composite sandwich plates with poroelastic material agree well with the experimental results in most frequency ranges of interest. Basing on the numerical results obtained, the influence of different parameters of the two thin laminated composite sheets and polyurethane foam core layer on STL of a sandwich plate is quantitatively evaluated and discussed in detail. Keywords: Composite sandwich plate · Poroelastic material · Sound transmission loss · Sound insulation

1 Introduction The composite sandwich plate with a foam core is widely used in many industries, including aircraft, building, motor vehicle and ships because of the advantages like high strength, light weight, low cost and good sound insulation. The sound transmission loss (STL) of sandwich structures has been the subject of many studies. As far as sound insulation improvement is concerned, poroelastic materials among various sandwich cores have been widely applied in double-wall structures due to their excellent sound absorption capability. By employing Biot’s theory [1] on wave propagation in a poroelastic medium, Bolton et al. [2, 3] modelled analytically the problem of sound transmission through infinite double-wall sandwich panels lined with poroelastic materials and validated the theoretical model against their experimental results. Panneton and Atalla [4] studied the sound transmission loss through multilayer structures made from a combination of elastic, air, and poroelastic materials. The presented approach is based on a three-dimensional finite element model. It uses classical elastic and fluid elements to model the elastic and fluid media and uses a two-field displacement formulation derived from the Biot theory [1]. Kropp and Rebillard [5] investigated the possibilities of the sound insulation of double-wall constructions at low frequencies. The expression ‘low frequencies’ implies a frequency range where the fundamental resonance frequencies of double wall constructions or the critical frequency of stiff single leaves (e.g. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 26–53, 2022. https://doi.org/10.1007/978-981-16-3239-6_3

Vibroacoutic Behavior of Finite Composite Sandwich Plates

27

concrete walls) are normally situated. Both effects lead to a reduced sound insulation which can cause problems to fulfill requirements. Sgard et al. [6] researched the lowfrequency diffuse field transmission loss through double-wall sound barriers with elastic porous linings. The studied sound barriers are made up from a porous-elastic decoupling material sandwiched between an elastic skin and a septum. The prediction approach is based on a finite element model for the different layers of the sound barrier coupled to a variation boundary element method to account for fluid loading and calculated efficiently using a Gauss integration scheme. Lee and Kondo [7] investigated the analytical and experimental studies of noise transmission loss of a three-layered viscoelastic plate. An improved equation of motion for sandwich plates with simply supported boundary condition is devisedand implemented to the finite plate noise transmission problem by using Rayleigh integral. Antonio et al. [8] studied the acoustic insulation provided by infinite double panel walls, when subjected to spatially sinusoidal line pressure loads, is computed analytically. The model calculation does not involve limiting the thickness of any layer, as the Kirchhoff or Mindlin theories require. Carneal √ and Fuller [9] proposed to increase of the panel stiffness by an empirical factor of 2 for the analytical solution of the simply supported boundary condition in order to approximate the experimental results of the clamped boundary condition. Wang et al. [10] presented theoretical modeling of the sound transmission loss through double-leaf lightweight partitions stiffened with periodically placed studs by assuming that the effect of the studs can be replaced with elastic springs uniformly distributed between the sheathing panels, a simple smeared model is established. Tanneau et al. [11] proposed a method for optimizing acoustical linings is described and applied to multilayered panels including solid, fluid, and porous components. This optimization is based on an analytical simulation of the insulation properties and a genetic algorithm. Lee et al. [12] simplified Bolton’s formulation by retaining only the energetically strongest wave among those propagating in the poroelastic material because the contribution of the shear wave to the transmission loss was found to be always negligible. Chazot and Guyader [13] studied sound transmission loss through double panels used a patch-mobility approach. The model is excitation by blocked patch pressures that take into account room geometry and source location. In recent years, D’Alessandro et al. [16] reviewed the most significant works in the literature about the acoustic response of sandwich panels. The focus is on presenting an exhaustive list of dedicated and validated models, which are able to predict the sound transmission through sandwich panels according to their specific configuration. Xin and Lu [17] adopted the equivalent fluid model to study sound transmission through an infinite double-wall panel with and without rib-stiffened core, respectively, filled with porous sound absorptive materials. Enhanced sound insulation properties of the double-wall sandwich panels have been observed in all these previous works. Shojaeifard et al. [18] studied power transmission through infinite double-walled laminated composite panels using the classical laminated plate theory and Biot’s theory [1] for wave propagation in the porous sandwich layer. Petrone et al. [19] investigated numerical and experimental on the acoustic power, radiated by Aluminium Foam Sandwich panels used a Finite Element model and formulation of the Rayleigh integral. Liu [21] extended the theoretical model of Bolton et al. [3] to address sound transmission across

28

T. I. Thinh and P. N. Thanh

triple-wall panels with poroelastic linings. Zhou et al. [14, 15] investigated the effects of an external mean ow and extended the vibroacoustic problem to a three-dimensional one, and Liu and Sebastian and Liu [22] further considered the effects of an internal gap mean flow. Hassan et al. [20] studied the acoustic behavior of double-walled panels, with sandwiched layer of porous materials is presented within Classical laminated plate theory for laminated composite panels. Equations of wave propagation are firstly extracted based on Biot’s theory for porous materials, then the transmission loss of the structure is estimated in a broadband frequency. Liu and Daulin [23, 24] studied the vibroacoustic problem of sound transmission across a rectangular double-wall sandwich panel clamp mounted on an infinite acoustic rigid baffle and lined with poroelastic material based on Biot’s theory [1] and the coupling methods between the poroelastic core and the panel determine the various configurations and associated boundary conditions. In this study, an analytical model on vibroacoustic behavior of clamped and simply supported orthotropic rectangular composite sandwich plates with foam core has been derived. Biot’s theory is employed to describe wave propagation in elastic porous media. The two face composite plates are modeled as classical thin plates. By using the modal superposition theory, a double series solution for the sound transmission loss of the structure is obtained with the help of the Galerkin method. Theoretical predictions of sound transmission loss (STL) across finite composite sandwich plates with poroelastic material agree well with the experimental results in most frequency ranges of interest. Basing on the numerical results obtained, the influence of different parameters of the two thin laminated composite sheets and polyurethane foam core layer on STL of sandwich plate is quantitatively evaluated and discussed in detail. Specifically, we have compared sound transmission losses between theoretical and experimental by influencing factors such as: the core density, the surface plates thickness and the core thickness for the composite sandwich plate with clamped boundary condition and consider the influence of the porous layer thickness, the faceplate thickness and the incident angles of the composite sandwich plate with simply supported boundary condition. Thereby, we consider the effect of two clamped and simply supported conditions on the sound transmission loss.

2 Theoretical Study 2.1 Plate Geometry and Assumptions A double-laminated composite plate lined with poroelastic materials consists of two rectangular, orthotropic, homogeneous and sufficiently thin plates commonly made of laminated composite, as illustrated in Fig. 1. The two elastic plates are fully clamped along their edges to an infinite rigid acoustic baffle. This double-composite plate is filled with porous materials. The bottom plate is excited by √a plane harmonic sound wave with ω being the angular frequency and the symbol j = −1. Without loss of generality, an incident sound wave of unit amplitude is assumed, and its incident direction is determined by the elevation angle ϕ and the azimuth angle θ (see Fig. 1). The vibrations of the upper plate are transmitted through the structure via the poroelastic medium to the bottom plate which induces pressure variations and thus a transmitted sound wave. The incidence field and transmission field separated by the infinite

Vibroacoutic Behavior of Finite Composite Sandwich Plates

29

Fig. 1. Schematic of a clamped double- laminated composite plate lined with poroelastic materials: (a) global view, (b) side view.

rigid baffle are assumed to be semi-infinite with identical air properties, i.e. air density ρ 0 and speed of sound in ambient air c0 . The panel dimensions are chosen as follows: the length and width of the plate are a, b; the bottom and upper plates thickness are h1 , h2 , and H is the core thickness of the composite sandwich (porous materials) in between the two plates. 2.2 Flexural Motion of the Plate The vibro-acoustic response of an orthotropic symmetric double- composite plate with poroelastic materials (Fig. 1) induced by sound excitation for the bottom and upper plates, respectively: D11 ∂

+ 2(D12 + 2D66 ) ∂

4 w (x,y;t) 1 ∂x4 4 ∂ w1 (x,y;t) 22 ∂y4

+D

∗ ∂ D11

+ m∗1 ∂∂tw21 = jωρ0 1 + σzs + σz  4  ∗ ∗ ∂ w2 (x,y;t) + 2 D12 + 2D66 ∂x2 ∂y2

4 w (x,y;t) 2 ∂x4 4 ∗ ∂ w2 (x,y;t) 22 ∂y4

+D

4 w (x,y;t) 1 ∂x2 ∂y2

2

(1)

f

+ m∗2 ∂∂tw22 = −jωρ0 3 − σzs − σz 2

f

(2)

where: w1 , w2 are the transverse displacements of the upper and bottom plates, respectively. m∗i = ρp hi (i = 1, 2) is the mass per unit area of the bottom and upper plates. ρ p is the density of the upper or bottom plates; ρ 0 is the density of air. F1 and F3 are the f acoustic velocity potential in the incident sound field and the transmission field. σzs ; σz are the normal stresses in the z-direction in the exterior fluid field and solid field. Dij and Dij∗ are the flexural rigidities (see any textbook of Mechanics of composite materials):  1  k 3 Qij zk+1 − zk3 For the bottom plate : Dij = 3 n

k=1

(3)

30

T. I. Thinh and P. N. Thanh

For the upper plate : Dij∗ =

 1  ∗k  ∗3 Qij zk+1 − zk∗3 3 n

(4)

k=1

2.3 Boundary Conditions of the Plate-Porous Coupling Boundary conditions representing the continuity of the normal velocity and displacement are applied at the surfaces of the porous layer and the facing plate, dependent on the coupling method. Since the plates are assumed to be homogeneous, orthotropic and sufficiently thin compared with the lateral dimensions, the in-plane deformations (and shear stresses) are commonly neglected according to the classical Kirchhoff-Love plate theory. For the coupling method as shown in Fig. 2, three boundary conditions must be satisfied at the bonded interface of the porous layer and facing plate, i.e., one normal velocity condition and two normal displacement conditions: v*z = jωw, usz = w, ufz = w

(5)

where w is the transverse (normal) displacement of the elastic plate that contains the convention eiωt , usz and ufz are the solid and fluid displacement vectors and v*z is the normal acoustic particle velocity in the exterior fluid field defined in Eq. (16).

Fig. 2. Different configurations of the double-laminate composite plate

Moreover, the boundary conditions of the bonded coupling in the Eq. (5) must be applied at the interface of the plate and the porous lining: at z = h1 −

∂1 = jωw1 , usz = w1 , ufz = w1 ∂z

atz = H + h1 −

∂3 = jωw2 , usz = w2 , ufz = w2 ∂z

(6) (7)

Under the excitation of harmonic sound waves, the transverse deflection of the upper and bottom plates can be expressed in a form of modal decomposition: w1 (x, y, t) =

∞  m,n=1

φmn (x, y)α1,mn ejωt ; w2 (x, y, t) =

∞  m,n=1

φmn (x, y)α2,mn ejωt

(8)

Vibroacoutic Behavior of Finite Composite Sandwich Plates

with φmn is the modal function for the clamped boundary can be written as:    2nπ y 2mπ x 1 − cos φmn (x, y) = 1 − cos a b and the simply supported boundary takes the form:  mπ x   nπ y  ϕmn = sin sin a b

31

(9)

(10)

and α 1,mn ; α 2,mn are the modal coefficients of the plate displacement and will be determined by applying the clamped and simply supported boundary conditions. The fully clamped boundary of the bottom and upper plates, the transverse displacement and the rotation moment must be equal to zero along the plate edges, i.e. the following clamped boundary conditions should be satisfied: x = 0, a, ∀ 0 < y < b, w1 = w2 = 0,

∂w1 ∂w2 = =0 ∂x ∂x

(11)

y = 0, b, ∀ 0 < x < a, w1 = w2 = 0,

∂w2 ∂w1 = =0 ∂y ∂y

(12)

and the simply supported boundary conditions should be satisfied: x = 0, a, ∀ 0 < y < b, w1 = w2 = 0,

∂ 2 w1 ∂ 2 w2 = =0 ∂x2 ∂x2

(13)

y = 0, b, ∀ 0 < x < a, w1 = w2 = 0,

∂ 2 w1 ∂ 2 w2 = =0 2 ∂y ∂y2

(14)

2.4 Determination of Sound Transmission Loss The acoustic velocity potential in the incident sound field (field 1 in Fig. 1) consists of an incident wave and a reflective wave with the amplitudes I and β, respectively, and it can be expressed in a harmonic form as: 1 = Ie−j(kx x+ky y+kz z−ωt ) + βe−j(kx x+ky y−kz z−ωt )

(15)

The transmission field (field 3 in Fig. 1) with the amplitude γ in this field and its velocity potential can be written as: 3 = γ e−j(kx x+ky y+kz z−ωt )

(16)

The poroelastic medium is excited by an incident plane harmonic sound wave. For a given incidence direction (ϕ, θ ), the components of the acoustic wavenumber are defined as: kx = k0 sin φ cos θ ; ky = k0 sin φ cos θ ; kz = k0 cos φ

(17)

32

T. I. Thinh and P. N. Thanh

where k x , k y , k z are wave number in the x, y, and z directions and k0 = ω/c0 is the acoustic wave number in air and c0 is the acoustic speed in the air. The amplitudes of the reflected and transmitted waves, β(x,y) and γ (x,y) are dependent on the local positions on the upper and bottom plates, respectively. These acoustic velocity potentials are related to the acoustic pressure p and the normal acoustic particle velocity v*z in the fluid field and solid field: p = ρ0

∂ ∂ = jωρ0 ; v*z = − ∂t ∂t

(18)

The sound power of the incident or transmitted wave per unit area (i.e. acoustic intensity) is defined as p = Re(pv ∗ ) 2, and the acoustic particle velocity is related to the sound pressure through p = p (ρ0 c0 ) for harmonic waves, ρ 0 is the density air and c0 is the acoustic speed in the air. The sound power can be expressed as [26] 1  = Re 2

b a 0

pvz∗ dA

0

1 = 2ρ0 c0

b a p2 dA 0

(19)

0

Considering the relation (18) and the velocity potential definitions (15) and (16), the sound power of incident and transmitted are determined by. 1 =

ω2 ρ0 2c0

b a 21 dxdy; 3 = 0

0

ω2 ρ0 2c0

b a 23 dxdy 0

(20)

0

The power transmission coefficient for a single incident wave with fixed direction angles is defined as τ (φ, θ ) =

3 1

(21)

For diffuse incident sound, the averaged transmission coefficient can be derived by integration as [3, 5, 9, 17]: 2π φ lim

τdiff =

0

τ (φ, θ ) sin φ cos θ d φd θ

0 2π φ lim 0

(22) sin φ cos θ d φd θ

0

where ϕlim is the limiting angle defining the diffuseness of the incident field. The sound transmission loss is defined as: STL(φ, θ ) = 10 log[1/τ (φ, θ )]

(23)

Vibroacoutic Behavior of Finite Composite Sandwich Plates

33

2.5 Modelling the Poroelastic Material According to Biot’s theory [1], an elastic porous material is assumed statistically isotropic and has the solid phase (elastic frame) as well as the fluid phase (air contained in pores). The model of Bolton [3] allows three waves in the porous material, two longitudinal waves, and one shear wave. Meanwhile, Lee et al. [7] shown that the shear wave is always negligible compared to the longitudinal waves. Hence, there are four wave components within the porous material, i.e. positive and negative propagating components of the two most energetically waves along the z-axis, which are characterised by the complex amplitudes D1 -D4 . Thus, the displacements of the solid and fluid phase are obtained as:

k1z −jk1z z k1z jk1z z k2z −jk2z z k2z jk2z z s −j(kx x+ky y−ωt ) D1 2 e − D2 2 e + D3 2 e − D4 2 e uz = je k1 k1 k2 k2 (24) ⎛ ⎞ k1z −jk1z z k1z − c1 D2 2 ejk1z z ⎟ ⎜ c1 D1 k 2 e k1 ⎜ ⎟ f 1 uz = je−j(kx x+ky y−ωt ) ⎜ (25) ⎟ k2z −jk2z z k2z jk2z z ⎠ ⎝ +c2 D3 2 e − c2 D4 2 e k2 k2 The expressions of the normal stresses in the z-direction are derived as      σzs = e−j(kx x+ky y−ωt ) ε1s D1 e−jk1z z + D2 ejk1z z + ε2s D3 e−jk2z z + D4 ejk2z z

(26)

     f f f σz = e−j(kx x+ky y−ωt ) ε1 D1 e−jk1z z + D2 ejk1z z + ε2 D3 e−jk2z z + D4 ejk2z z

(27)

f

f

for the solid and fluid phases, respectively, where the parameters ε1s , ε2s , ε1 , ε2 are in the form: ε1s = 2N f ε1

2 k1z

k12

+ A + c1 Q, ε2s = 2N

= Q + c1 S,

f ε2

2 k2z

k22

+ A + c2 Q

(28)

= Q + c2 S

where, the first Lame coefficient A and the elastic shear modulus N are defined as A=

νs Es Es ; N= 2(1 + νs ) (1 + νs )(1 − 2νs )

(29)

here, E s is the real static Young’s modulus of solid phase, ν s is the Poision’s ratio. The coefficients Q and S represent the coupling between the volume changes of the solid and fluid phases and are defined as: Q = (1 − β)Ef and S = βEf

(30)

where β is the porosity of the porous material and E f denotes the bulk modulus of the  −1 elasticity of the fluid in the pores (assumed to be air), Ef = E0 1 + 2(χ − 1)Tc (λ) λ

34

T. I. Thinh and P. N. Thanh

where E0 = ρ0 c02 is the bulk modulus of air, and χ is the ratio of specific heats; √ 1 2 2 / λ = P r λc −j, with Pr the Prandtl number, λc = 8ωρ 0 ϑ βψ, ϑ the geometrical structure factor and ψ the flow, resistivity. Tc (λ) = J1 (λ) J0 (λ), with J0 and J1 the zero- and first-order Bessel functions of the first kind.   2 2 − k 2 + k 2 , and the = k1,2 The wavenumber component in the z-direction, k1,2z x y coefficients are c1 = b1 − b2 k12 , c2 = b1 − b2 k22  ∗   ∗  ∗ ∗ b1 = ρ11 S − ρ12 Q ρ22 Q − ρ12 S     ∗ ∗ b2 = PS − Q2 ω2 ρ22 Q − ρ12 S

(31)

Under the assumption of plane harmonic waves, their wavenumber are expressed as  A1 ± A21 − 4A2 2 (32) k1,2 = 2 where, the parameters are defined as      ∗  ∗ ρ∗ − ρ∗ 2 ∗ Q + ρ∗ P ω4 ρ11 S − 2ρ12 ω2 ρ11 22 12 22 A1 = , A2 = 2 2 PS − Q PS − Q with P = A + 2N, where the complex equivalent densities are ∗ ρ11 = ρ11 + a jω, ρ11 = ρs + ρa ∗ ρ12 = ρ12 − a jω, ρ12 = −ρa ∗ ρ22 = ρ22 + a jω, ρ22 = ρ2 + ρa

(33)

(34)

With ρ s and ρ 2 = β.ρ 0 being the bulk densities of the solid and fluid phases, respectively; ρa = ρ2 (ψ − 1) is an inertial coupling coefficient between the fluid and solid, and a is viscous coupling factor:       −1   a = iωψρ2 ρc∗ ρ0 − 1 with ρc* = ρ0 1 − 2 λc −j Tc λc −j

(35)

Besides, two auxiliary parameters can be defined as f

f

1 = ε1s + ε2s and 2 = ε1 + ε2

(36)

which will be used in the subsequent derivations. f Substituting the expressions of F1 , F3 in Eqs. (15) and (16), uzs , uz in Eqs. (24) and (25) and w1 , w2 in Eq. (8) into the boundary condititions (9) and (10), one obtains: atz = h1 e−j(kx x+ky y) kz (I − β) = ω1

k1z k1z k2z k2z je−j(kx x+ky y) D1 2 − D2 2 + D3 2 − D4 2 k1 k1 k2 k2

(37a)

= 1

(37b)

Vibroacoutic Behavior of Finite Composite Sandwich Plates

je

−j(kx x+ky y)

k1z k1z k2z k2z c1 D1 2 − c1 D2 2 + c2 D3 2 − c2 D4 2 k1 k1 k2 k2

35

= 1

atz = H + h1 e−j(kx x+ky y+kz (H +h1 )) kz γ = ω2

(37c) (38a)



⎞ k1z −jk1z (H +h1 ) k1z −jk1z (H +h1 ) D e − D e + 2 2 ⎜ 1 k2 ⎟ k1 ⎜ ⎟ 1 je−j(kx x+ky y) ⎜ ⎟ = 2 k2z −jk2z (H +h1 ) ⎠ ⎝ k2z −jk2z (H +h1 ) D3 2 e − D4 2 e k2 k2 ⎛ ⎞ k1z −jk1z (H +h1 ) k1z −jk1z (H +h1 ) − c1 D2 2 e +⎟ ⎜ c1 D1 k 2 e k1 ⎜ ⎟ 1 je−j(kx x+ky y) ⎜ ⎟ = 2 k2z −jk2z (H +h1 ) k2z −jk2z (H +h1 ) ⎠ ⎝ c2 D3 2 e − c2 D4 2 e k2 k2 where the parameters G1 , G2 are in the form:   1 = φmn (x, y)α1,mn and 2 = φmn (x, y)α2,mn m,n

(38b)

(38c)

(39)

m,n

Combine Eqs. (37a) and (38a), it is straightforward to solve the amplitudes of the reflected and transmitted wave as. ω ω β(x, y) = I − 1 ej(kx x+ky y) ; γ (x, y) = 2 ej(kx x+ky y+kz (H +h1 )) (40) kz kz Substituting the expressions of β and γ back into Eqs. (19) and (20), the acoustic velocity potentials F1 and F3 on the surface of the upper and bottom plates can be rewritten as. 1 (x, y, 0, t) = 2Ie−j(kx x+ky y−ωt ) −

ω ω 1 ejωt ; 3 (x, y, H , t) = 2 ejωt kz kz

(41)

Combining Eqs. (37b), (37c), (38b) and (38c) yields a set of four algebraic equations for four unknown (1.e. D1 – D4 ) which can be rearranged into a matrix equation: MD = R

(42)

where M is a 4 x 4 transfer matrix: ⎤ ⎡ −kI kII −kII kI ⎥ ⎢ c1 kI −c1 kI c2 kII −c2 kII ⎥ M=⎢ ⎣ e−jk1z (H +h1 ) kI −ejk1z (H +h1 ) kI e−jk2z (H +h1 ) kII −ejk2z (H +h1 ) kII ⎦ (43) c1 e−jk1z (H +h1 ) kI −c1 ejk1z (H +h1 ) kI c2 e−jk2z (H +h1 ) kII −c2 ejk2z (H +h1 ) kII with the auxiliary coefficients   kI = jk1z k12 and kII = jk2z k22

(44)

36

T. I. Thinh and P. N. Thanh

T  the 4 x 1 unknown vector D = D1 D2 D3 D4 , and the forcing vector  T R = ej(kx x+ky y) 1 1 2 2

(45)

The unknown vector D can be soved simultaneously from the matrix Eq. (42) as D = V.R, where V = M −1 is the inverse of the transfer matrix M. Substituting the solutions of D1 – D4 into Eqs. (26) and (27), the expressions of the normal stresses in the solid f and fluid phases, σzs , σz can be obtained as ! −jk1z z " e [(V11 + V12 )1 + (V13 + V14 )2 ] s,f = ε1 ejωt +ejk1z z [(V21 + V22 )1 + (V23 + V24 )2 ] ! −jk2z z " [(V31 + V32 )1 + (V33 + V34 )2 ] s,f jωt e + ε2 e +ejk2z z [(V41 + V42 )1 + (V43 + V44 )2 ] s,f

σz

(46)

where V ij (i, j = 1, 2, 3, 4) are the elements of the inverse matrix V. In order to determine the unknown α 1,mn and α 2,mn , applying the Galerkin method to the plate motion Eqs. (1) and (2) leads to: ⎛ ⎞ ∂ 4 w1 (x, y; t) ∂ 4 w1 (x, y; t) + 2(D12 + 2D66 )

b a ⎜ D11 ⎟ ∂x4 ∂x2 ∂y2 ⎜ ⎟ ⎜ ⎟.φmn (x, y)dxdy = 0 2w ⎝ ∂ 4 w1 (x, y; t) ∂ 1 f⎠ ∗ s +D22 + m1 2 − jωρ0 1 − σz − σz 0 0 ∂y4 ∂t (47) ⎛ ⎞ 4  4  ∗ ∗ ∂ w2 (x, y; t) ∗ ∂ w2 (x, y; t) + 2 D + 2D

b a ⎜ D11 ⎟ 12 66 ∂x4 ∂x2 ∂y2 ⎜ ⎟ ⎜ ⎟.φmn (x, y)dxdy = 0 4 w (x, y; t) 2w ⎝ ⎠ ∂ ∂ 2 2 f ∗ ∗ s +D22 + m2 2 + jωρ0 3 + σz + σz 0 0 ∂y4 ∂t (48) f

After substituting the expressions of w1 , w2 , F1 , F3 and σzs , σz , i.e. Equations (10), (40) and (46) into Eqs. (47) and (48) and integrating over the entire plate surface with laborious algebraic manipulations, one obtains two sets of infinite simultaneous equations: at z = 0, !#  m 4  n 4  m 2  n 2 $ α1,mn + 3D22 + 4(D12 + 2D66 ) 4π 4 ab 3D11 a b a b %  n 4  m 4   + 2D22 α1,kn + 2D11 α1,ml b a k l (49) 9ab  3ab  + R1,mn + R1,kn 4 m,n 2 k    3ab  + R1,ml + ab R1,kl = 2jωρ0 Ifmn kx , ky , (k = m, l = n) 2 l

k,l

Vibroacoutic Behavior of Finite Composite Sandwich Plates

37

at z = H + h1 , $ !#  4  4  m 2  n 2  ∗ ∗ m ∗ n ∗ α2,mn + 3D22 + 4 D12 + 2D66 4π 4 ab 3D11 a b a b %  4  4   ∗ n ∗ m + 2D22 α2,kn + 2D11 α2,ml b a k

l

 3ab  3ab  9ab  R2,mn + R2,kn + R2,ml + ab R2,kl = 0, (k = m, l = n) + 4 m,n 2 2 k

l

k,l

(50) where the coefficients Qi,mn (i = 1, 2) are in the form: R1,mn = −α1,mn R11 − α2,mn R12 , R2,mn = α1,mn R21 + α2,mn R22

(51)

  ρ0 ∗ R11 = 1 (V11 + V12 + V21 + V22 ) + 2 (V31 + V32 + V41 + V42 ) + m1 − j ω2 , kz R12 = 1 (V13 + V14 + V23 + V24 ) + 2 (V33 + V34 + V43 + V44 ),   R21 = 1 (V11 + V12 )e−jk1z (H +h1 ) + (V21 + V22 )ejk1z (H +h1 ) +   + 2 (V31 + V32 )e−jk2z (H +h1 ) + (V41 + V42 )ejk2z (H +h1 ) ,   R22 = 1 (V13 + V14 )e−jk1z (H +h1 ) + (V23 + V24 )ejk1z (H +h1 ) +   + 2 (V33 + V34 )e−jk2z (H +h1 ) + (V43 + V44 )ejk2z (H +h1 ) , (52) The constants f mn are produced through the integration process combined with clamped boundary condition   fmn kx , ky =

 

b a  2mπ x 2nπ y −j(kx x+ky y) 1 − cos 1 − cos e dxdy a b 0

(53)

0

with simply supported boundary condition   fmn kx , ky =

b a sin 0

 mπ y  a

sin

 nπ y  e−j(kx x+ky y) dxdy b

(54)

0

The infinite algebraic equation system specified in Eqs. (47) and (48) has the modal coefficients α 1,mn and α 2,mn as the unknowns. This set of infinite equations can be sovle numerically by taking an appropriate truncation: 1 ≤ m, n ≤ M, where the maximum

38

T. I. Thinh and P. N. Thanh

mode number M is a compromise between accuracy and computational cost. Therefore, the simultaneous equation system can be rearranged as a 2M 2 × 2M 2 matrix equation: $ ! " # " ! α1,mn T11,mn T12,mn Fmn = (55) T21,mn T22,mn 2M 2 x2M 2 α2,mn 2M 2 x1 0 2M 2 x1 where the elements of the displacement coefficient vectors α i,mn (i = 1, 2) and the generalized force vector Fmn take the form: & '  αi,mn M 2 x1 = αi,11 αi,21 ... αi,M 1 (56) T α1,12 α1,22 ... αi,M2 ... αi,MM (i = 1, 2)  T {Fmn }M 2 x1 = 2iωρ0 f11 f21 ... fM 1 f12 f22 ... fM 2 ... fMM (57) and the matrix elements are expressed as 4 4 3        ∗i T11,mn M 2 xM 2 = 4π 4 ab ∗i − R  ; T = −R ∗i 11 12,mn M 2 xM 2 12 1 2 2; i=1

i=1

i=1

4 4 3        4 ∗∗i T21,mn M 2 xM 2 = R21 ∗i ; T = 4π ab  − R ∗i 22,mn M 2 xM 2 11 2 1 2; i=1

i=1

i=1

(58) ∗i where the expressions of the M 2 x M 2 submatrices ∗i 1 (i = 1, 2, 3), 2 (i = 1, 2, 3, 4) ∗∗i and 1 (i = 1, 2, 3) are given in Appendix A. After sovling this matrix Eq. (55), the unknown modal coefficients α i,mn (i = 1, 2) of the plate displacements will be determined and therefore all the dependent variables such as w1 , w2 , β, γ will be known.

3 Validation For the first validation study, the sound transmission loss (STL) across a finite clamped double-aluminium plate with poroelastic material is calculated and compared with result of Bolton et al. [3]. The properties of aluminium plates and poroelastic material as follows: For aluminium plate: length × width of plate, a × b = 1.2 m × 1.2 m; upper plate thickness, h1 = 1.27 mm; bottom plate thickness, h2 = 0.762 mm; porous layer thikness, H = 27 mm; Young’s moduluas, E p = 70 GPa; Bulk density, ρ p = 2700 kg/m3 ; Poisson’s ratio, ν p = 0.33. For poroelastic material: Bulk density of the solid phase, ρ s = 30 kg/m3 ; In vacuo bulk Young’s modulus, E s = 8.105 Pa; Bulk Poisson’s ratio, ν s = 0.40; flow resistivity, ψ = 5.103 MKS Rayls m−1 ; Porosity, β = 0.9; geometrical structure factor, ϑ = 0.78; density, ρ 0 = 1.21 kgm−3 , speed of sound, c0 = 343 ms−1 , ratio of specific heats, χ = 1.4; Prandtl number, Pr = 0.71. It is obvious from Fig. 3 that there is an acceptable agreement between present analytical and theoretical results shown in [3].

Vibroacoutic Behavior of Finite Composite Sandwich Plates

39

Fig. 3. Comparison between the predicted STL curve and result of Bolton et al. [3].

For the second validation study, the STL across a finite simply supported doublecomposite plate with poroelastic material is calculated and compared with result of Lee et al. [7]. The properties of aluminium plates and poroelastic material as follows: For aluminium plate: Length × width of plate, a × b = 0.303 m × 0.203 m; upper plate thickness, h1 = 0.5 mm; bottom plate thickness, h2 = 0.5 mm; porous layer thikness, H = 1 mm; Young’s moduluas, E p = 73.2 GPa; Bulk density, ρ p = 2720 kg/m3 ; Poisson’s ratio, ν p = 0.33. For poroelastic material are: Bulk density of the solid phase, ρ s = 1600 kg/m3 ; In vacuo bulk Young’s modulus, E s = 4.12. GPa; Poisson’s ratio, ν s = 0.40; flow resistivity, ψ = 25.103 MKS Rayls m−1 ; Porosity, β = 0.9; geometrical structure factor, ϑ = 0.78; density, ρ 0 = 1.21 kgm−3 , speed of sound, c0 = 343 ms−1 , ratio of specific heats, χ = 1.4; Prandtl number, Pr = 0.71.

Fig. 4. Comparison between the predicted STL curve and results of Lee and Kondo [7].

Figure 4 shows a comparison between the predicted STL curve and the experimental curve of Lee and Kondo in [7]. The curves are presented in third-octave bands in the frequency range of 100–10000 Hz. The agreement between the present analytical method and experimental curves is good. The results illustrated the capability of the proposed method to predict STL curves for sandwich plates. It is seen that the minimal STL is

40

T. I. Thinh and P. N. Thanh

obtained in the low frequency range where the modal behavior of the plate is important. The first mode of the plate produces an important resonant dip in the STL.

4 Numerical and Experimental Results 4.1 Test Setup In this section, to measure the sound transmission loss through the sandwich structure, we proceed through a system of two rooms (Source room and Receiver room) meeting ISO 3741–1988 [24] as shown in Fig. 5. Measurement experiments were conducted at the Experimental Testing Center – Nha Trang University in cooperation with Vietnam Metrology Institute. Sound transmission loss is determined by the sound pressure level measurement method, in accordance with ASTM E2249–02 (E) [25, 27].

Fig. 5. The transmission suite

The test samples are clamped between two reverberation rooms: the source room is (2,06 m × 3,7 m × 2,1 m) and the receiving room is (1,54 m × 3,7 m × 2,1 m) [26, 27], as shown in Fig. 6. The sound transmission loss is given by [26]: STL = L1 − L2 + 10 log(

AP T ) 0.161 V2

(59)

where L 1 and L 2 are the average sound pressure levels in the source and receiving room, respectively. Ap is the test-specimen area; V 2 is the volume of the receiving room; T is the reverberation time of the receiving room when the sandwich plate is clamped between the two rooms. To conduct the experiment, first of all, it is necessary to determine the position of the speaker (sound source) as described in Fig. 6. Omnidirectional speaker 4292L B&K with a large capacity will create a strong sound source. Microphone 4189 (3) is placed 0.2 m from the face of the sample board to measure the sound pressure level in the source room. Microphone 4189 (4) is placed at a distance of 0.2 m from the test surface attached to the partition wall to measure the sound pressure level in the receiving room. The reverberation time (T60) is calculated by the average decay data from 16 points equidistant from the sample.

Vibroacoutic Behavior of Finite Composite Sandwich Plates

41

Fig. 6. Test arrangement for samples.

The composite sandwich plates have an area (1.2 m × 1.2 m) are divided into 16 squares (0.3 m × 0.3 m), corresponding to 16 discrete measurement points. According to experimental, the distance between the measuring head and the center of the sound source must be 2 to 3 times the distance between the two microphones to ensure the measurement error is less than 1 dB. After a number of tests, the distance of the microphone (3) was selected to be 0.2 m and the probe (4) in the studio was located 0.17 m from the sample surface. Sound transmission loss (STL) through composite sandwich plates has been calculated and measured according to ASTM E 1289–91 (E) [25, 27]. Sound leaks on the side of the samples were detected at some high frequencies and treated by sealing with a sealant. Test equipment has been calibrated and reinstalled for subsequent measurements. Then, to assess the role of component materials for sound transmission loss (STL), we compare STL values across three structural groups. 4.2 Sample Description According to the standards of Vietnam Register: QCVN 56–2013/BGTVT. The upper and lower skin layers of the sandwich composite panel are made of fiberglass (WR800)/polyester with a symmetrical configuration in the 0°/90° direction; core is polyurethane foam - PU.

42

T. I. Thinh and P. N. Thanh

Fig. 7. Fabrication of sandwich composite materials.

Configuration and geometry parameters of composite sandwich plates are given in Table 1. Table 1. Configuration and geometry parameters of composite sandwich plates Plate

Configuration

a

b

tf

tc

tf

h

ρs

Surface mass

m

m

m (×10–3 )

m (×10–3)

m (×10–3 )

m

kg/m3

kg/m2

A

3WR800 + PU1 + 3WR800

1,2

1,2

2,53

30

2,53

0,0351

46,88

9,502

B

3WR800 + PU2 + 3WR800

1,2

1,2

2,53

30

2,53

0,0467

57,87

9,832

C

3WR800 + PU3 + 3WR800

1,2

1,2

2,53

40

2,53

0,0451

79,86

11,290

D

4WR800 + PU3 + 4WR800

1,2

1,2

3,37

40

3,37

0,0467

79,86

13,978

E

4WR800 + PU3 + 4WR800

1,2

1,2

3,37

30

3,37

0,0359

79,86

13,173

4.3 Results and Discussion Firstly, the effects of core density, skin thickness and core thickness on STL of a clamped composite sandwich plates are investigated theoretically and experimentally. The geometry and material properties of plates are given in Table 2.

Vibroacoutic Behavior of Finite Composite Sandwich Plates

43

Table 2. Geometric parameters and mechanical properties of the face plate and core materials Plate

Dimension (m)

Thickness t f /t c /t f (x103 m)

Density of skin ρ f (kg/m3 )

Core density ρ s (kg/m3 )

E1 (GPa)

E2 (GPa)

G12 (GPa)

v12

Ec (GPa)

vc

Surface mass kg/m2

A

1.2 x 1.2

2.53/30/2.53

1600

46.88

10.58

2.64

1.02

0.17

0.0570

0.25

B

1.2 x 1.2

2.53/30/2.53

1600

57.87

10.58

2.64

1.02

0.17

0.0580

0.25

9.832

C

1.2 x 1.2

2.53/40/2.53

1600

79.86

10.58

2.64

1.02

0.17

0.0585

0.25

11.290

D

1.2 x 1.2

3.37/40/3.37

1600

79.86

10.58

2.64

1.02

0.17

0.0585

0.25

13.978

E

1.2 × 1.2

3.37/30/3.37

1600

79.86

10.58

2.64

1.02

0.17

0.0585

0.25

13.173

9.502

a. Effect of the Core Density In this section, we consider the effect of core density to STL when they have the same skin layer thickness and core thickness for material pair (A – B). Pair (A - B) changes the density of core layer ρ s = 46.88 kg/m3 and ρ s = 57.87 kg/m3 , the remaining parameters are the same in Table 2. In Fig. 8, the STL curves across clamped plate A and D are plotted. In the lowfrequency region (f < 125 Hz), at 100 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 0.35 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 0.13 dB. In the medium-frequency region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 0.34 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 0.84 dB. In the high-frequency region (f > 2000 Hz), at 10000 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 2.07 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 2.58 dB.

Fig. 8. Comparision of sound transmission loss through clamped plates A and B.

The density of the core layer has the greatest effect on the sound transmission loss, as the density of the core layer increases, the theoretical and experimental STL values increase. The theoretical average increase is 2.124 dB, which is empirically 1,988 dB for most of the frequency band in 1/3 of the octave band.

44

T. I. Thinh and P. N. Thanh

b. Effect of the Surface Plates Thickness In this section, we consider the effect of skin thickness to STL when they have the same skin layer thickness and core thickness for material pair (C – D). Pair (C – D) increased skin thickness from t f = 2.53.10−3 m to t f = 3.37.10−3 m and core thickness t c = 40.10−3 m, the remaining parameters are the same in Table 2.

Fig. 9. Comparision of sound transmission loss through clamped plates C and D.

Figure 9 show, the STL curves across clamped plate C and D are plotted. In the low-frequency region (f < 125 Hz), at 80 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 0.09 dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 0.83 dB. In the medium-frequency region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 1.60dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 2.19 dB. In the high-frequency region (f > 2000 Hz), at 5000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 5.26 dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 6.37 dB. The thickness of the skin layer has a significant effect on the sound transmission loss through the composite sandwich panels. For example, when t f increases from t f = 2.53.10−3 m to t f = 3.37.10−3 m, the other parameters do not change, the STL theoretically increases by 1.476 dB, empirically the STL increases 1.774 dB corresponding to the 1/3 Octave frequency range. c. Effect of the Core Thickness In this section, we consider the effect of core thickness to STL when they have the same skin layer thickness and the density of the core for material pair (D – E). Group (D – E) increased core thickness from t c = 30.10−3 m to t c = 40.10−3 m and skin thickness t f = 3.37. 10−3 m, the remaining parameters are the same in Table 2. In Fig. 10, the STL curves across clamped plate D and E are plotted. In the lowfrequency region (f < 125 Hz), at 80 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 0.32 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 0.13 dB. In the medium-frequency

Vibroacoutic Behavior of Finite Composite Sandwich Plates

45

Fig. 10. Comparision of sound transmission loss through clamped plates D and E.

region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 1.64 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 2.19 dB. In the high-frequency region (f > 2000 Hz), at 3125 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 3.14 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 4.13 dB. The core layer thickness does not significantly affect sound transmission loss through composite sandwich plates. Specifically, when t c increases from 30.10−3 m to t c = 40.10−3 m, other parameters do not change, STL theoretically only increases 0.560 dB, experimental STL curve increases 0.567 dB corresponding to 1/3 Octave frequency range. In this next subsection, the effects of the porous layer thickness, the faceplate thickness and the incident angle on STL of a simply supported composite sandwich plates are investigated. The bottom and upper Glass/Epoxy orthotropic laminated composite plates configuration is [90/0/0/90]s . The geometry and material properties of plates are given in Table 3. Table 3. Composite materials properties and the geometrical dimensions Composite

E1 (GPa) E2 (GPa) G12 (GPa) ν 12

Glass/Epoxy 40.851

10.097

3.788

ρ a x b h1 (m) (Kg/m3 ) (m2 )

0.27 1946

h2 (m)

H (m)

1.2 x 1.27.10–3 0.762.10–3 27 1.2

d. Effect of the Porous Layer Thickness In order to explore the effects of the core cavity thickness on sound transmission loss through a finite simply supported double-composite plate calculated with changed values of the porous layer thickness: H = 17, 27, 37 and 47.10−3 m, as shown in Fig. 11.

46

T. I. Thinh and P. N. Thanh

From Fig. 11, increasing the thickness of the core layer leads to a reduction the coincidence frequency because the porous layer plays a major role in reducing the noise of the sound. On the other hand, we also see that the sound transmission loss increases as the core porous layer thickness increases and also shows the soundproofing capacity of the double-plates filled poroelastic material.

Fig. 11. Effect of the porous layer thickness on STL of a simply supported composite sandwich.

e. Effect of the Faceplate Thickness To quantify the effects of the faceplate thickness, the STL versus frequency curve is presented in Fig. 12 for a finite system. Four pairs values of the faceplate thickness are selected: (h1 , h2 ) = (1.0,1.0), (1.0, 2.0), (2.0, 1.0) and (2.0, 2.0).10−3 m. According to Fig. 12, the effect of the plate thickness parameter on the sound transmission loss is considered. It is observed that increasing the thickness of the plate will increase the sound transmission loss, which means that sound waves transmitted through thicker sheets will be less spreading and more reflective, and will also lead to reduced frequency coincidence.

Fig. 12. Effect of the skin thickness on STL a simply supported composite sandwich.

Vibroacoutic Behavior of Finite Composite Sandwich Plates

47

f. Effects of the Incident Angles The influence of sound incident angles (elevation angle and azimuth angle) on the sound transmission loss of a finite simply supported orthotropic laminated double-composite plate is considered. Shown in Fig. 13 for the double-plate excited by sound waves incident at a series of incident angles ϕ = 0°; 30°; 45° and 60° and a fixed azimuth angle θ = 45°. Generally, the STL spectra of the oblique incidence cases drop considerably as the elevation angle increases and this STL reduction grows at higher frequencies. In other words, more oblique incident waves are easier to transmit through the double-wall sandwich panel especially in the high-frequency range. This is because constructive interference between the incident sound waves and structural bending waves may occur at oblique incidence.

Fig. 13. Effect of the incident angle on STL of a simply supported composite sandwich.

Similarly, Fig. 14 shows the effect of the azimuth angle on the STL spectra for the double-plate by sound waves incident at a series of azimuth angles θ = 15°; 30°; 45° and 60° and a fixed incident angle ϕ = 45°. As shown in Fig. 14, the azimuth angle has a limited influence on STL behaviors. In fact, the STL spectra of all azimuth angle (θ) values coincide with one another at frequencies below 1000 Hz, i.e. a negligible effect of the azimuth angle. The effect of the azimuth angle (θ) on STL, however, become more obvious and complicated above this frequency. The overall trend of the STL levels increases with larger azimuth angles despite the complex modal behaviors at high frequencies. The limited effect of the azimuth angle on the sound transmission characteristics of the finite clamped plate may be attributed to the fairly symmetrical system of the square panel, which is in accordance with the observation in previous work. In the final study, we investigated the effect of boundary conditions on STL of the composite sandwich plates. g. Effects of Boundary Conditions In this section, we evaluate the influence of boundary conditions (clamped and supported) on sound transmission loss.

48

T. I. Thinh and P. N. Thanh

Fig. 14. Effect of the azimuth angle on STL of a simply supported composite sandwich.

Fig. 15. Effect of the clamped and simply supported boundary conditions on STL of composite sandwich plate

From Fig. 15, other dips determined mainly by the natural vibration of the radiating panel are significantly shifted when the boundary conditions are changed. Additionally, it can be seen that the STL values of the clamped system are distinctly higher than those of the simply supported system in the lower frequency range. This is attributed to the more rigorous constraint provided by the clamped condition than that by the simply supported condition, which is equivalent to increasing the plate stiffness.

5 Conclusions An analytical model on sound transmission through a finite composite sandwich plates with foam core has been derived on the basis of previous theories [3, 24]. The vibroacoustic problem is formulated and solved by employing a modal decomposition to account for the clamped and simply supported boundary condition, and the Galerkin method for the plate motion equations. We draw some of the following conclusions: • Confirm the correctness, reliability and excellent agreement between current research and theories [3, 7].

Vibroacoutic Behavior of Finite Composite Sandwich Plates

49

• The sound transmission loss through a clamped composite sandwich is highly dependent on the density of the core layer, or in other words, the core density is the strongest factor affecting the sound insulation of the composite sandwich structure. The experimental data above is a useful scientific basis for simulation studies, numerical calculations of sound transmission loss through the composite sandwich, and an important contribution in the calculation, design, and fabrication of the composite sandwich structures has the best sound insulation. • The sound transmission loss across a simply supported composite sandwich increases as the core porous layer thickness increases. This is one of the parameters that greatly affect the sound insulation of double-plates. • The sound transmission loss across a simply supported composite sandwich increase as the thickness of the faceplate increase while greatly affecting the resonant frequency. • When the elevation angle of the incident sound waves increases, the sound transmission loss decreases. On the other hand, the incident azimuth angle has negligible influence on the STL of a finite simply supported composite sandwich plate. • Obtained results at oblique sound incidence for the two different boundary condition cases suggest that, the sound insulation of the double composite-plate with a clamped boundary condition is better than that of the double-plate with a simply supported boundary condition.

Appendix A. Expressions of the Matrix Equation (58) ∗i The expressions of the M 2 x M 2 submatrices ∗i 1 (i = 1, 2, 3), 2 (i = 1, 2, 3, 4) and ∗∗i 1 (i = 1, 2, 3), as appearing in the matrix elements T ij,mn (i, j = 1, 2; m, n = 1, 2, …, M), are as flows: ⎡ ⎤ ⎡ ⎤ λ∗1 C1 1,1i ⎢ ⎥ ⎢ ⎥ λ∗1 C2 ⎢ ⎥ 1,2i ⎢ ⎥ ∗1 ⎢ ⎥ 1 = ⎢ ; C = (i = 1,2,...,M); ⎥ i . . ⎢ ⎥ .. .. ⎣ ⎦ ⎣ ⎦

λ∗1 1,mn = 3D11

 m 4

CM

a

M 2 xM 2

+ 3D22

 n 4 b

+ 4(D12 + 2D66 )

λ∗1 1,Mi MxM  m 2  n 2 a

b (A.1)

⎡ ⎢ ⎢ ⎢ = ∗2 1 ⎢ ⎣

λ∗2 1,1



⎤ λ∗2 1,2

..

.

λ∗2 1,M

⎥ ⎥ ⎥ ⎥ ⎦

; λ∗2 1,n M 2 xM 2

0 ⎢ ⎢  n 4 ⎢ 1 ⎢ .. = 2D22 ⎢ b ⎢. ⎢. ⎣ ..



1 ··· ··· 1 .. ⎥ . 0 .. .⎥ ⎥ . . . . . . .. ⎥ . . . .⎥ ⎥ ⎥ .. . 0 1⎦

1 ··· ··· 1 0

MxM

(A.2)

50

T. I. Thinh and P. N. Thanh



∗3 1

⎤ λ∗3 1 .. ⎥ . ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ ∗3 λ ⎦

0 λ∗3 1 ··· ··· ⎢ ∗3 .. ⎢λ . ⎢ 1 0 ⎢ .. . . . . . . =⎢ . . . . ⎢ ⎢ . .. ⎣ .. . 0 ∗3 λ1 · · · · · · λ∗3 1

⎡ ; λ∗3 1 = 2D11

 m 4 ⎢ ⎢ ⎢ a ⎣



14 24

..

. M4

1

0

⎥ ⎥ ⎥ ⎦ MxM

M 2 xM 2

(A.3) ⎤



∗1 2

1 ⎢ 9ab ⎢ 1 = ⎢ .. 4 ⎣ .

⎥ ⎥ ⎥ ⎦ 1

⎡ ∗2 2 =

9ab ⎢ ⎢ ⎢ 4 ⎣



λ∗2 2

..

.

⎥ ⎥ ⎥ ⎦ λ∗2 2

, λ∗2 2 M 2 xM 2

∗3 ⎤

λ∗3 2

(A.4)

M 2 xM 2



⎤ λ∗2 2

;

⎤ 0 1 ··· ··· 1 ⎢ .. ⎥ ⎢1 0 ... .⎥ ⎢ 3ab ⎢ . . . . . ⎥ ⎥ = ⎢ .. . . . . . . .. ⎥ ⎢ ⎥ 2 ⎢. ⎥ . . . ⎣. . 0 1⎦ 1 ··· ··· 1 0

(A.5)

MxM

0 · · · · · · λ2 ⎤ ⎡ ⎢ ∗3 .. ⎥ .. 1 ⎢λ ⎥ . 0 . ⎥ ⎢ 2 ⎥ 3ab ⎢ ⎥ ⎢ .. . . . . . . .. ⎥ ⎢ 1 ∗3 ∗3 = ; λ = ⎢ ⎥ ⎢ . . . . ⎥ .. ⎥ 2 2 ⎢ . ⎦ 2 ⎣ . ⎢ . ⎥ .. ⎣ .. ⎦ . 0 λ∗3 1 MxM 2 ∗3 0 λ∗3 · · · · · · λ 2 2 2 2 M xM ⎡ ⎡ ⎤ ⎤ ∗4 ∗4 0 λ2 · · · · · · λ 2 0 1 ··· ··· 1 ⎢ ∗4 ⎢ .. ⎥ .. ⎥ .. ⎢λ ⎢1 0 ... . .⎥ . ⎥ ⎢ 2 0 ⎢ ⎥ ⎥ ⎢ .. . . . . . . .. ⎥ ⎢ .. . . . . . . .. ⎥ ∗4 ∗4 2 = ⎢ . ; λ = ab ⎢ ⎥ ⎥ . . . . ⎥ 2 ⎢ ⎢. . . . .⎥ ⎢ . ⎢ ⎥ ⎥ . .. .. ⎣ .. ⎣ .. . 0 1⎦ . 0 λ∗4 ⎦ 2

(A.6)

(A.7)

∗4 0 1 · · · · · · 1 0 MxM λ∗4 2 · · · · · · λ2 M 2 xM 2 ⎡ ⎤ ⎤ ⎡ ∗ λ∗∗1 C1 1,1i ⎢ ⎥ ⎥ ⎢ λ∗∗1 C2∗ ⎢ ⎥ 1,2i ⎥ ⎢ ∗∗1 ∗ ⎢ ⎥ 1 = ⎢ ; Ci = ⎢ (i = 1,2,...,M); ⎥ . . ⎥ . . ⎦ ⎣ . . ⎣ ⎦ ∗ CM λ∗∗1 1,Mi MxM M 2 xM 2   m 4  n 4   2 n 2   m ∗ ∗ ∗ ∗ λ∗∗1 + 3D22 + 4 D12 + 2D66 1,mn = 3D11 a b a b (A.8)

Vibroacoutic Behavior of Finite Composite Sandwich Plates

⎡ ∗∗2 1

⎢ ⎢ =⎢ ⎢ ⎣

λ∗∗2 1,1

..

.

λ∗∗2 1,M

⎤ 0 1 ··· ··· 1 ⎢ .. ⎥ .. ⎢ . .⎥ ⎥  n 4 ⎢ 1 0 ⎢ .. . . . . . . .. ⎥ ∗ = 2D22 ⎢. . . . .⎥ ⎥ b ⎢ ⎥ ⎢. .. ⎣ .. . 0 1⎦ ⎡

⎤ λ∗∗2 1,2

51

⎥ ⎥ ⎥ ⎥ ⎦

; λ∗∗2 1,n M 2 xM 2

1 ··· ··· 1 0

MxM

(A.9) ⎡

∗∗3 1

∗3 ⎤

0 λ∗3 1 ··· ··· ⎢ ∗3 .. ⎢λ . ⎢ 1 0 ⎢ .. . . . . . . =⎢ . . . . ⎢ ⎢ . .. ⎣ .. . 0 λ∗3 · · · · · · λ∗3 1 1

λ1 .. . .. .



⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗3 λ ⎦

; λ∗∗3 1 = 2D11

 m 4 ⎢ ⎢ ⎢ a ⎣



14 24

..

. M4

1

0

⎥ ⎥ ⎥ ⎦ MxM

M 2 xM 2

(A.10) The expression of the constants f mn (k x , k y ) is in the form:   fmn kx , ky =

b a 0

ϕmn (x, y)e−j(kx x+ky y) dxdy

0

 

b a  2mπ x 2nπ y −j(kx x+ky y) 1 − cos 1 − cos e = dxdy a b 0 0 ⎧ ⎪ ab for kx = 0, ky = 0 ⎪  ⎪ ⎪ 2 π 2 a 1−e−jbky ⎪ 4in ⎪ ⎪   for kx = 0, ky  = 0 ⎪ ⎪ ⎨ ky ky2 b2 −4n2 π 2   = 4im2 π 2 b 1−e−jakx ⎪ for kx = 0, ky = 0 ⎪ kx (kx2 a2 −4m2 π2 ) ⎪  ⎪   ⎪ −jbk ⎪ ⎪ 16m2 n2 π 4 1−e−jakx 1−e y  ⎪ ⎪ ⎩ − k k (k 2 a2 −4m2 π 2 ) k 2 b2 −4n2 π 2 for kx = 0, ky = 0 x y x y   fmn kx , ky =

b a 0

0

b

a

=

ϕmn (x, y)e−j(kx x+ky y)

sin 0

0

 mπ x  a

sin

 nπ y  e−j(kx x+ky y) dxdy b

52

T. I. Thinh and P. N. Thanh

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

ab { 1−(−1)m −(−1)n +(−1)m+n } if kx mnπ 2 ' & mab 1−(−1)m e−jakx −(−1)n +(−1)m+n e−jakx 2 mnπ , nab 1−(−1)m −(−1)n e−jbky +(−1)m+n e−jbky

= 0 and ky = 0 if kx = 0 and ky = 0

if kx = 0 and ky = 0 ⎪ ⎪ mnπ 2 , ⎪ ⎪ 2 1−(−1)m e−jakx −(−1)n e−jbky +(−1)m+n e−j (akx +bky ) ⎪ mnabπ ⎪ ⎪   ⎪ if kx = 0 and ky = 0 ⎩ (k 2 a2 −m2 π 2 ) k 2 b2 −n2 π 2 x

y

References 1. Biot, M.A.: Theory of propagation of elastic waves in a uid-saturated porous solid I. lowfrequency range. II. higher frequency range. J. Acoust. Soc. Am. 28(2), 168–191 (1956) 2. Bolton, J.S., Green, E.R.: Normal incidence sound transmission through double-panel systems lined with relatively stiff, partially reticulated polyurethane foam. Appl. Acoust. 39, 23–51 (1993) 3. Bolton, J.S., Shiau, N.M., Kang, Y.J.: Sound transmission through multi-panel structures lined with elastic porous materials. J. Sound Vib. 191(3), 317–347 (1996) 4. Panneton, R., Atalla, N.: Numerical prediction of sound transmission through finite multilayer systems with poroelastic materials. J. Acoust. Soc. Am. 100(1), 346–354 (1996) 5. Kropp, W., Rebillard, E.: On the air-borne sound insulation of double wall constructions. Acta Acust. 85(5), 707–720 (1999) 6. Sgard, F.C., Atalla, N., Nicolas, J.: A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings. J. Acoust. Soc. Am. 108(6), 2865–2872 (2000) 7. Lee, C., Kondo, K.: Noise transmission loss of sandwich plates with viscoelastic core. American Institute of Aeronautics & Asreonautics AIAA-99–1458, pp. 2137–2147 (1999) 8. Antonio, J., Tadeu, A., Godinho, L.: Analytical evaluation of the acoustic insulation provided by double infinite walls. J. Sound Vib.. 263(1), 113–129 (2003) 9. Carneal, J.P., Fuller, C.R.: An analytical and experimental investigation of active structural acoustic control of noise transmission through double panel systems. J. Sound Vib. 272(3), 749–771 (2004) 10. Wang, J., Lu, T.J., Woodhouse, J., Langley, R.S., Evans, J.: Sound transmission through lightweight double-leaf partitions: theoretical modelling. J. Sound Vib. 286, 817–847 (2005) 11. Tanneau, O., Casimir, J.B., Lamary, P.: Optimization of multilayered panels with poroelastic components for an acoustical transmission objective. J. Acoust. Soc. Am. 120(3), 1227–1238 (2006) 12. Lee, J.S., Kim, E.I., Kim, Y.Y., Kim, J.S., Kang, Y.J.: Optimal poroelastic layer sequencing for sound transmission loss maximization by topology optimization method. J. Acoust. Soc. Am. 122(4), 2097–2106 (2007) 13. Chazot, J.D., Guyader, J.L.: Prediction of transmission loss of double panels with a patchmobility method. J. Acoust. Soc. Am. 121(1), 267–278 (2007) 14. Zhou, J., Bhaskar, A., Zhang, X.: Sound transmission through a double-panel construction lined with poroelastic material in the presence of mean ow. J. Sound. Vib. 332(16), 3724–3734 (2013) 15. Zhou, J., Bhaskar, A., Zhang, X.: Optimization for sound transmission through a double-wall panel. Appl. Acoust. 74(12), 1422–1428 (2013) 16. D’Alessandro, V., Petrone, G., Franco, F., De Rosa, S.: A review of the vibroacoustics of sandwich panels: Models and experiments. J. Sandw. Struct. Mater. 15(5), 541–582 (2013)

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17. Lu, T.J., Xin, F.X.: Vibro-acoustics of Lightweight Sandwich, Science Press Beijing and Springer-Verlag Berlin Heidenberg (2014) 18. Shojaeifard, M.H., Talebitooti, R., Ranjbar, B., Ahmadi, R.: Power transmission through double-walled laminated composite panels considering porous layer-air gap insulation. Appl. Math. Mech. 35(11), 1447–1466 (2014). https://doi.org/10.1007/s10483-014-1877-7 19. Petrone, G., DAlessandro, V., Franco, F., De Rosa, S.: Numerical and experimental investigations on the acoustic power radiated by aluminium foam sandwich panels. Compos. Struct.118, 170–177 (2014) 20. Hassan, S., Roohollad, T., Reza, A., Behzad, B.: A study on acoustic behavior of poroelastic media bonded between laminated composite panels. Latin Am. J. Solid Struct. 11, 2379–2407 (2014) 21. Liu, Y.: Sound transmission through triple-panel structures lined with poroelastic materials. J. Sound Vib. 339, 376–395 (2015) 22. Liu, Y., Sebastian, A.: Effects of external and gap mean flows on sound transmission through a double-wall sandwich panel. J. Sound Vib. 344, 399–415 (2015) 23. Daudin, C., Liu, Y.: Vibroacoustic behaviour of clamped double-wall panels lined with poroelastic materials. In: Proceedings of the 23rd International Congress on Sound and Vibration, vol. 341. Athens, Greece (2016) 24. Liu, Y., Daudin, C.: Analytical modeling of sound transmission through finite clamped doublewall sandwich panels lined with poroelastic materials. Compos. Struct. 172, 359–373 (2017) 25. Acoustics-Determination of sound power levels of noise sources—Precision methods for broad-band sources in reverberation rooms, International Standard ISO 3741–88(E), International Organization of Standardization, Geneva, Switzerland (1998) 26. ASTM standard method for laboratory measurement of airborne sound transmission loss of building partitions and elements using sound intensity, American Standard ASTM 2249– 02(E) (2002) ´ truyên ` âm qua tâm ´ composite sandwich và u´,ng du.ng vào 27. Tien. D.D.: Nghiên cu´,u tôn thât ` tàu thuy. Ða.i ho.c Nha Trang (2019) giam ôn ij

ij

ij

Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method Do H. M. Hieu, Do Q. Duyen, Nguyen P. Tai, Nguyen V. Thang, Ngo C. Vinh(B) , and Nguyen Q. Hung(B) ` Mô.t, Binh Duong, Vietnam Faculty of Engineering, Vietnamese-German University, Thu Dâu {hieu.dhm,duyen.dq,tai.np,thang.nv,vinh.nc,hung.nq}@vgu.edu.vn ij

Abstract. This research focuses on introduction and evaluation of machine parts manufactured by bound powder deposition method, a new approach in metal 3D printing. After a review of metal 3D printing technology, a new approach for metal printing, called bound powder deposition, is introduced. Several specimens manufactured by the bound powder deposition are then conducted for experimental works. In the experimental works, the crystal structure of the printed part is investigated. In addition, typical mechanical proprieties of the printed part produced by the bound powder deposition are tested and compared with other those produced by other 3D printing and metallurgy methods. Keywords: Metal 3D printing · Additive manufacturing · Bound powder deposition · Bound powder extrusion · Binder jetting

1 Introduction Additive manufacturing (AM), commonly called as 3D printing, has become increasingly popular in various industrial sectors such as aerospace, medical applications, transportation, consumer products. This AM has also been considered as a significant factor in development of the fourth industrial revolution. The AM processes are classified into seven groups, which is defined by the standard terminology of ISO/ASTM 52900 [1], a cooperation between ISO/TC 261 and ASTM Committee F42. The groups include directed energy deposition (DED), powder bed fusion (PBF), material extrusion, material jetting, binder jetting, sheet lamination, and VAT photopolymerization process. These processes can produce complex shapes, high-strength lightweight designs, scalable rapid prototypes on various types of materials such as metals, ceramics, composites, polymers [2]. Among these processes, PBF processes such as electron beam melting (EBM) or selective laser melting (SLM), and DED processes like laser engineering net shape (LENS) are the most common approaches in industry to create 3D printed metallic parts [3, 4]. Such processes employ electron beams or high laser energy to fuse metallic powders or wires resulting in layer by layer of 3D objects. Although the electron beams or high-power lasers can provide parts with high density (99%), which results in superior © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 54–62, 2022. https://doi.org/10.1007/978-981-16-3239-6_4

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properties to castings; the two common approaches of metallic additive manufacturing have low production volumes with slow processing speed, high material cost, and high machining cost [4]. Moreover, outer shapes, uniform sizes, and chemical compositions of metal powders must be controlled well, which makes them become costly. Therefore, the systems using PBF or DED approach are commonly costly, approximately in a region of $250,000 to more than $1.5 million [5]. Additionally, PBF processes exist some limitations in design related to enclosed cavities. After 3D printing, unprocessed powders or trapped powders, which are remained inside the printed component cavities, needs to be removed by additionally mechanical operations [6]; as a result, PBF processes do not work effectively with the enclosed cavities. Moreover, metallic powders at micro sizes are hazardous to human health, and need to be manipulated correctly with proper safety protections [7]. Recently, a new metal 3D printing system based on material extrusion approach, called bound powder deposition (BPD) or bound powder extrusion (BPE), has been developed with a base cost of $99,500 [8]. A filament which is made of a combination between metallic powders and polymer is thermally extruded through a nozzle to deposit layer by layer of desired 3D objects. Therefore, the process does not lose any metal powders. Additionally, filament costs are more economic than metal powder costs [9]. This 3D printing system is an advanced fused filament fabrication (FFF) 3D printer, which is ranked in number one of additive manufacturing techniques according to their utility function values [10]. The presented system can reduce capital costs, compared to the ones of SLS and EBM. Moreover, during BPE procedure, metal powders kept by polymer are not easy to be diffused into the ambient air. Therefore, toxicity and flammability risks can be minimally decreased. In our experiment, several 17-4 PH stainless steel specimens were first fabricated by the BPE technique. Several typical mechanical proprieties of the 3D printed stainless steel (17-4 PH) parts are then investigated and are compared to the properties of 17-4 PH stainless steel parts, which are produced by conventional manufacturing techniques. Such examination approaches like dimension accuracy, morphology, surface roughness, porosity test, tensile test and hardness test were carried out. The obtained results can open feasibility for practical applications of 17-4 PH stainless steel metal, which is produced by the new, economic, and friendly 3D metal printing system.

2 Experimental 2.1 Material A filament comprised of metal powder and plastic binder (17-4 PH stainless steel, Markforged, USA) was used in this research. The composition of 17-4 PH stainless steel includes chromium (15–17.5%), nikel (3–5%), copper (3–5%), silicon (1% max), manganese (1% max), niobium (0.15–0.45%), carbon (0.07% max), phosphorus (0.04%), sulphur (0.03% max), and iron (the rest).

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2.2 Fabrication Method Bound powder extrusion includes several steps: 3D model design, slicing the model into digital layers using a conversion to STL file, set-up of printing parameters, layer-bylayer printing, debinding, sintering, and post processing. Most of the steps are similar to other additive manufacturing processes, except the layer-by-layer printing, debinding and sintering, which are unique characteristics of BDE. A Metal X system (Markforged, USA) used in this research consists of three machines, which can represent for the three main unique steps of BPE technique, as shown in Fig. 1. The printer, with a maximum printing volume of 300 × 220 × 180 mm3 , has a heated print chamber, a heated bed with vacuum-sealed print sheet and two extruders, one for supplying metal powders mixed with wax polymer and one for supplying ceramic as release material. The washing machine with a washing volume of 18.356 cm3 is employed to dissolve the binding material using Opteon SF79. The sinter machine is a high-performing furnace with a sintering workload of 3.020 cm3 to transform lightly bound-metal-powder part to a full

Fig. 1. A bound powder extrusion system.

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metal part. An argon gas is inserted into the chamber of the sinter machine during the sintering process. In the beginning of printing steps, a filament consisting metal powders enclosed in a thermoplastic polymer is heated and softened to create layers of 3D object by the printer. Depending on design of the printed part, supports can be printed to prevent any deformation or any damages of the main printed parts. A ceramic layer is deposited between the supports and the printed parts during the construction at 1:10 ratio to metal spools. This layer supports detaching and removing the support structures easily. After that, a debinding process is carried out by using the washing machine, following by using a sintering process to create the final 3D parts. The printed specimens are exanimated to evaluate typical mechanical properties. The fabrication parameters (recommended by the manufacturer) are shown in Table 1. Table 1. Fabrication parameters (recommended by the manufacturer) Mass of printed part

58.99 g

Post-sintered layer height 0.125 mm Printing time

4 h 16 m

Debinding time

12 h

Sintering time

27 h

Material cost

7.62 USD

2.3 Analysis Methods An optical microscope (BX53M, Olympus, Japan) was used to observe morphology of the printed part. To measure surface roughness, a portable surface roughness tester (SJ210, Mitutoyo, Japan) was used. Porosity of the printed part was tested by observation of the printed part’s cross-section surface. The cross section was prepared by the following steps: the printed part was cut into a small pieces by an abrasive cutter (Metacut 251, Metkon, Turkey); an automatic mounting press (Ecopress 100, Metkon, Turkey) was then used to hold the small piece for the next polishing process, which employed a grinding–polishing machine (Forcipol 1V, Metkon, Turkey). Moreover, tensile test was carried out on printed specimens, which had been produced with ASTM E8 Standard [11], by a tensile tester (AGX-Plus 50 kN, Shimadzu, Japan). Additionally, hardness test was performed on printed samples with ASTM E18 standard [12], by a Rockwell hardness testing machine (250 MRS, Affri, USA).

3 Result and Discussion 3.1 Dimension Accuracy The measured dimensions (length × width × thickness) of the 3D printed part are 151.32 mm × 20.13 mm × 3.04 mm while the nominal ones are 150 mm × 20 mm ×

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3 mm. The actual dimensions are greater than the design’s ones, with errors less than 1–2%. This can prove for the high accuracy of the BPD system. Additionally, the printed part with these dimensions only costs approximately 7.62 USD. This is cost-effective compared to other 3D metal printing approaches like PBF or DED, which is in good agreement with the IDTechEx report [9]. 3.2 Morphology Figure 2 illustrates morphology of the printed specimen, which includes not only the appearance but also the surface morphology. The design model was shown in Fig. 2a with a taken photo of the specimen during the printing process attached in the top right corner. After printing, debinding, and then sintering, top view and front view of the final specimen were shown in Fig. 2b and 2c. In the top view, the metallic filament and its printing direction could be observed by naked eyes. The boundaries around the inside metallic filaments, which created the shape of an object, were also seen. The inside metallic filament played an important role in the printed object’s mechanical properties. In this research, the printing direction had a 45°-tilted angle (red arrow in Fig. 2b), which could strongly enhance for the strength of the specimen. Each filament after heating and depositing on the support has a width of approximately 500 µm, as shown in Fig. 2d. Line by line was deposited on each layer of the printed specimen resulting in surface roughness, which is approximately 2.68 µm. In the front view, layer-by-layer structure of the printed specimen can be clearly observed. The thickness of each layer was approximately 125 µm. The connections between layers were sufficiently good. Additionally, bending issue almost did not appear on the printed specimen.

Fig. 2. Morphology of the printed specimen.

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3.3 Porosity Open porosity was not observed on the outer surface of the 3D printed specimen. Therefore, appearance of the 3D printed product is acceptable compared to the one made by conventional techniques. However, enclosed porosity can be observed on the cross section, as shown in Fig. 3. The porosity of the printed one, roughly 0.50–5 µm in diameter, was quite large compared to that of conventionally fabricated 17-4 PH stainless steel, which is approximately 0.45–1.80 µm in diameter. The largest porosity, which appeared in BPD technique, was roughly 2.8 times in diameter to the one appeared in the traditional technique. The conventionally fabricated 17-4 PH stainless steel showed small sizes of porosities because it was produced by top-down approach, where the initial material was a bulk metal with optimized fabrication parameters like heat treatment time. Conversely, the 3D printed specimen was fabricated by bottom-up approach, where the starting material was a metal filament. The connection between filaments could cause large gaps, which would become porosities after sintering. To reduce the quantities and the sizes of porosities, several fabrication parameters like sintering times, which have not been optimized yet in this research, should be investigated. Although the obtained results showed large sizes of porosities, BPD technique is still advantageous compared to other 3D printing techniques using fused filament fabrication approaches. For instance, in Thompson’s research [13], the sizes of porosities were up to 10 µm in diameter or even greater than 30 µm in diameter. The samples also existed more porosities than the ones using this BPD technique.

Fig. 3. Porosity of: (a) conventionally fabricated 17-4 PH stainless steel, and (b) 3d printed specimen.

3.4 Tensile Test A comparison in ultimate tensile strength (UTS) between 3D printed specimen and conventionally fabricated specimen is presented in Fig. 4. The UTSs are 985 MPa in the 3D printed specimen and 1163 MPa in the conventionally fabricated specimen. The strength of the printed one was reduced roughly 15%. Sintering time and porosity

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may cause difference between the two specimens. The insufficient sintering time or the large sizes of porosities made the connection between filaments be not highly strong. Therefore, during tensile test, the samples could be broken at a lower value of UTS, compared to those using conventional techniques. Interestingly, the 3D printed specimen had no necking, a mode of tensile deformation. After its ultimate tensile strength, the 3d printed specimen was immediately broken, which might represent for a brittle property. On the other hand, the conventionally fabricated specimen still existed its typical necking after applying the ultimate tensile strength. The specimen clearly showed a deformation in the neck position, which can present for a ductile property. The disappearance of necking in the 3D printed specimen made its strains at the applied ultimate tensile strength moment and at the fracture moment same; whereas, the strain at the fracture point was greater than that at the applied ultimate tensile strength point in the conventionally fabricated specimen. Although the fabrication parameters have not been optimized in this research, the obtained UTSs are sufficiently high, similar as 17-4 PH stainless steel H1150 [14]. This makes the printed specimens satisfy for applications required good strengths such as end-of-arm tools, lightweight brackets, and pump shafts. The BPD technique demonstrates not only for its advantages like other 3D printing techniques, such as complexity, simple operation steps; but also for its ability to produce similar mechanical properties as the ones made by conventional techniques.

Fig. 4. Tensile test of the conventionally fabricated specimen and of the 3D printed specimen.

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3.5 Hardness Test The 3D printed specimen showed a hardness value of 29.5 HRC, as shown in Fig. 5. Although this value is not remarkable compared to the value of conventionally fabricated 17-4 PH stainless steel (37.1 HRC), the hardness of the printed specimen using BPE approach is sufficiently strong. The measured results of the 3D printed specimen were in good agreement with the printed specimen’s porosity. When the porosity of the printed specimen increases, its hardness can be decreased. Additionally, in this primarily research, the printed specimen was just as sintered, and the sintering time has not been optimized yet.

Fig. 5. Hardness test of the 3D printed part: (a) before the test, and (b) after the test.

4 Conclusion The new and low-cost approach in metal 3D printing, bound powder deposition, is introduced. Several 17-4 PH stainless steel specimens were prepared to investigate typical mechnical properties. The printed dimensions have small errors compared to desired ones. The morphology was also obsersed. The porosity of 3D printed specimens has its diameters larger than the one of conventionally fabricated specimens. The ultimate tensile strengths between the printed one and the conventional one were compared. The 3D printed specimen showed a low value of the strength with the interesting disappearance of the neck, a typical phenomenon of metal material during tensile test. Additionally, the hardness number of the 3D printed specimen was smaller than that of the traditional specimen. The studies in this research will open a new way for applications of 17-4 PH stainlesss steel with a low 3D printing fabrication cost and a friendly approach. Acknowlegement. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.01–2018.335.

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References 1. A. Standard. ISO/ASTM 52900: 2015 Additive manufacturing-General principlesterminology. ASTM F2792-10e1 (2012) 2. Gibson, I., Rosen, D.W., Stucker, B.: Additive Manufacturing Technologies, vol. 17. Springer, Boston, MA (2014) 3. Gu, D.D., Meiners, W., Wissenbach, K., Poprawe, R.: Laser additive manufacturing of metallic components: materials, processes and mechanisms. Int. Mater. Rev. 57(3), 133–164 (2012) 4. Frazier, W.E.: Metal additive manufacturing: a review. J. Mater. Eng. Perform. 23(6), 1917– 1928 (2014) 5. Garcia-Colomo, A., Wood, D., Martina, F., Williams, S.W.: A comparison framework to support the selection of the best additive manufacturing process for specific aerospace applications. Int. J. Rapid Manuf. 9(2–3), 194–211 (2020) 6. Hunter, L.W., Brackett, D., Brierley, N., Yang, J., Attallah, M.M.: Assessment of trapped powder removal and inspection strategies for powder bed fusion techniques. Int. J. Adv. Manuf. Technol. 106(9–10), 4521–4532 (2020). https://doi.org/10.1007/s00170-020-0430-w 7. Bours, J., Adzima, B., Gladwin, S., Cabral, J., Mau, S.: Addressing hazardous implications of additive manufacturing: complementing life cycle assessment with a framework for evaluating direct human health and environmental impacts. J. Ind. Ecol. 21(S1), S25–S36 (2017) 8. Campbell, R.I., Wohlers, T.: Markforged: Taking a different approach to metal additive manufacturing. Metal AM 3(2), 113–115 (2017) 9. Richard, C., Jonathan, H., Ivan De, B.: Metal Additive Manufacturing 2020–2030. IDTechEx (2020) 10. Sgambatia, A., et al.: URBAN: conceiving a lunar base using 3D printing technologies. In: Proceedings of the 69th International Astronautical Congress, pp. 1–5. Bremen, Germany (2018) 11. Astm, E.: Standard test methods for tension testing of metallic materials. Annual Book of ASTM Standards, ASTM (2001) 12. Astm, A.: ASTM E18-03: Standard test methods for rockwell hardness and rockwell superficial hardness of metallic materials. Annual Book of ASTM Standards (2003) 13. Thompson, Y., Gonzalez-Gutierrez, J., Kukla, C., Felfer, P.: Fused filament fabrication, debinding and sintering as a low cost additive manufacturing method of 316L stainless steel. Additive Manuf. 30, 100861 (2019) 14. Steel, A.K.: 17-4 PH Stainless Steel: Product Data Bulletin. AK Steel Corporation, West Chester, OH, USA (2015)

Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions Quoc-Huy Ngo1 , Ky-Thanh Ho1 , and Khac-Tuan Nguyen2(B) 1 Faculty of Mechanical Engineering,

Thai Nguyen University of Technology, Thai Nguyen, Vietnam {ngoquochuy24,hkythanh}@tnut.edu.vn 2 Faculty of Automotive Engineering, Thai Nguyen University of Technology, Thai Nguyen, Vietnam [email protected]

Abstract. This report presents some experimental results of the effect of the Coulomb friction on the resonant frequency of the two-mass system connected by a nonlinear leaf spring. The experimental apparatus designed and built has an ability to vary the excitation frequency, the excitation force and friction force. Experimental data show that the resonance frequency of the system tends to decrease when increasing the friction force. The resonant frequency of the system can be expressed as the functions depending both on the amplitude of excitation force and on the Coulomb friction force. The experimental results serve as the basic premise for the development of studies applied in self-movement structure operating under different resistant environments. Keywords: Resonance frequency · 2DOF · Vibration-driven locomotion · Friction force · Excitation force

1 Introduction The nonlinear oscillator consists of two masses connected by one spring is the basic model used to theoretically study and applied to a variety of research fields, such as vibrations, multi-body systems, structural dynamics and transportation… especially in the operability of a self-moving mechanism (new locomotion system) [1–6]. Generally, the mathematical model of the system contains two ordinary conjugate differential equations with cubic non-linearity. Some researchers have presented several techniques for solving analytically a second order differential equation with various strong non-linear characteristic. S.K. Lai and C.W. Lim [5] used an analytical approach developed for non-linear free vibration of a conservative, two degrees of freedom mass-spring system having linear and non-linear stiffness (with model in Fig. 1). Max–Min Approach (MMA) is applied to obtain an approximate solution of three practical cases in terms of a non-linear oscillation system [3]. Recently, many researchers based on two-mass one-spring model developed the mobile devices employing vibration for motion, also © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 63–71, 2022. https://doi.org/10.1007/978-981-16-3239-6_5

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called vibration-driven locomotion systems; such as designing, modeling and experimental validation; dynamical analysis; optimal progression and motion control [7–13]. Figure 2 is the physical model of the system employing vibration for motion. However, most of these studies have not yet evaluated the effect of friction on the mechanical movement. In previously theoretical studies, the friction usually is not fully considered. In fact, the self-moving models, such as biomedical applications of capsule and rehabilitation robots in medical, pipe capsule robots… have to work in different environments, where the environmental resistance is different. Therefore, the study of self-movement under different resistance conditions is very important. Recently, Christoph Kossack et al. [14] reported frequency response function (FRF) determined by either experiment or simulation for a dynamic oscillator with a sliding friction contact (Coulomb friction). The results of this study have been just used for single degree of freedom (SDOF) system, thus haven’t applied to capsule robots.

Fig. 1. Basic system of the two masses connected by linear or non-linear spring [2, 3, 5]

Fig. 2. Physical model of the system

With vibration-driven locomotion systems based on two masses one spring, phenomenon resonance plays an important role. When an oscillating force is applied at a resonant frequency of a dynamical system, the system will oscillate at a higher amplitude than when the same force is applied at the others, i.e. non-resonant frequencies. Mathematically, when a system operates at resonant frequency, there is a 90° phase between the exciting force and the system’s response. Therefore, it’s possible to say that the force is in phase with the speed response of the system. It means that the sense of the force is the same as the sense of the motion [15, 16]. Consequently, the exciting force applies permanently in the direction and does positive work-done on the system body. This can enhance the effectively changing energy of the system. This study uses the model of two masses connected by a nonlinear spring to evaluate oscillation of mechanical system, specifically in terms of vibration resonance. The research results are very significant in assessing the ability of vibration-driven system under the different resistance conditions.

2 Design and Setup the Model of Apparatus 2.1 Design and Setup Experimental Apparatus Design and prototype are two cornerstone aspects in studying vibration-driven locomotion systems. Generally, the design of vibration-driven locomotion systems is based

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on the mechanics of the interaction between the inertial mass and the system body. In such system, the inertial mass is excited and controlled to periodically move inside the system body, creating the inertial force when changing either its direction of motion, or acceleration, or velocity. The locomotion thus can be generated with the presence of inertial force inside and resistant force from surrounding environment.

Fig. 3. (a) The experimential diagram and (b) experimential apparatus

This experimental study uses an apparatus based on the model of two masses and a non-linear leaf spring system, as shown in Fig. 2 and Fig. 3. A mini electro-dynamical shaker (1) is placed on a slider of a commercial linear bearing guide (3), providing a tiny friction force. An additional mass (2) was clamped on the shaker shaft. Generally, a sinusoidal current applying to the shaker leads to relative linear oscillation of the shaker shaft with the inertial mass added on, creates the excitation force F e . This force depends on the current supplied and can be adjusted the sinusoidal voltage supplying to the amplifier. The moveable mass, combined by the addition mass and the shaker shaft, is assigned as inertial mass m1 , playing the role of the internal mass of the capsule robot. A non-contact position sensor (6), model Kaman KD-2306, was used to measure the relative motion of the inertial mass and the shaker body, i.e. measuring X 1 –X 2 . The movement of the shaker body was recognized by a linear variable displacement transformer (LVDT), i.e. absolute distance X 2 . A carbon tube (4) is connected with the shaker body by means of a flexible joint, avoiding any misalignment when moving. As shown on Fig. 4(a), the carbon tube is able to slide between two aluminum pieces in the form of a V-block (5). The two V-blocks are fixed on two electromagnets (7). The body shaker, including the sensors LVDT and the carbon tube, was referred as the mass m2 of the mass-spring model. The total weight of additional mass and the shaker shaft was considered as the inertial mass m1 . A set of tests was carried out to determine the stiffness of leaf spring connecting the movable shaft with the shaker body. It is noted that the shaker body was fixed on the slider during these tests. A string was attached to the shaker shaft and rode over a pulley while a series of certain masses were hung on the other end of the string. The gravitational force of the masses pulled the shaft moving forward. The displacement of the shaker shaft, corresponding to each level of masses, was measured by using the non-contact position sensor. A nonlinear curve fitted with the cubic function was then applied to determine the relationship between the gravitational force and the movement. The experiment tests revealed that the spring is a nonlinear spring with hard characteristics (the cubic

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term is positive). In addition, the damping coefficient, c was determined by logarithmic decrement method. 2.2 Experimental Operation Supplying a certain value of electrical current to the electromagnets provides a desired clamping force on the tube and thus receiving a corresponding value of sliding friction. The friction force was measured by pushing or pulling the body to move at a steady speed V s . Experimental data revealed that the friction force F f depends on the applied voltage V by fitted curve in the following relationship: Ff = 1.52889 + 0.13591 ∗ e1.25642∗V

(1)

This is configuration allows varying the friction force without changing the body mass and thus can experimentally check with the locomotion capacity in different friction levels. The signals from the sensors were captured by a data acquisition system (DAQ) and then stored and analyzed.

Fig. 4. Varying the friction force: (a) apparatus structure and (b) the dependency of friction force on supplied voltage

A supplementary experiment was implemented to determine the relationship of the magnetic force and the supplied current. A load-cell was used as an obstacle resisting the shaker movement and thus to measure the magnetic force induced. A DC voltage was supplied to the shaker to generate the magnetic force. Varying the voltage, several pairs of the current passing the shaker and the force were collected. Experimental data revealed that the force is proportional to the current supplied to the shaker (see Nguyen et al. 2017 [7] for detailed information of how to determine this relationship). In this study, the amplitude of supplied current was setup with three values, as 0.5 A, 0.75 A and 1.0 A, providing the magnitude of the exciting force, as 5.4 N, 8.1 N and 10.8 N, respectively. During each experiment, these values were kept without changing. All of the experimental parameters are summarized on Table 1.

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Table 1. Parameters of experiments. Parameter

Notation

Value

Unit

Internal mass

m1

0.518

Kg

Body mass

m2

1.818

Kg

Linear stiffness

k1

2988.388

N/m

Cubic stiffness

k2

52158700

N/m3

Damping

c

8.893542

Ns/m

Friction force

Ff

0; 3.56; 8.31; 16.63; and 28.5

N

Excitation force

Fe

5.4; 8.1; and 10.8

N

Excitation frequency

f exc

[2–30]; swept by 1 Hz step in 2 s

Hz

3 Results and Discussion Theoretically, resonance occurs when the phase difference between the excitation force F e and the X 1 –X 2 shift is 90 degrees. At that time, the excitation current is minimum and the swept amplitude of oscillation X 1 –X 2 is maximum. Figure 5 shows an experimental data received by sweeping from 2 Hz to 30 Hz with 3.56 N of friction force and 5.4 N of excitation force. The parameters included sweeping step as 1 Hz and sweeping time as 3 s each step. The experimental data was firstly conducted in time regime (right figure) then changed into frequency regime (right figure). The maximum amplitude of oscillation X 1 –X 2 was revealed at 14 Hz. At this frequency, the excitation force is the smallest. Others experimental data were carried out similarly by changing the excitation forces and friction forces.

Fig. 5. Determining the experimental frequency resonance

Figure 6 is the experimental data showing the relationship of resonance frequency and friction force at three levels of excitation force 5.4 N, 8.1 N and 10.8 N, respectively. Firstly, as can be seen in all sub-plots, the amplitude relative displacement of the internal mass and the shaker body increases when raising the excitation frequency to maximum,

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then decreases if continuously raising. The excitation frequency, where the relative displacement reaches maximum value, is the resonant frequency. Secondly, at each level of excitation force, when increasing the friction force, the resonance frequency tends to decrease. Thirdly, without changing the level of resistant force, the resonance frequency increases as the amplitude of excitation force F e increases from 5.4 N to 10.8 N. Theoretical studies of the dynamics of the mechanical system with linear springs show that the natural frequency f n of this system can be determined by following:   1 k1 ∗ (m1 + m2 ) 1 2988.388 ∗ (0.518 + 1.818) = 13.703Hz = fn = 2π m1 ∗ m2 2π 0.518 ∗ 1.818 The results showed that the resonance frequency of the system was different from the natural frequency f n . The causes of this difference may be from the nonlinearity of the system, the friction force and damping applied to the system.

Fig. 6. The variation of resonance frequency under different friction force without changing the exciting force, respectively: (a) F e = 5.4 N; (b) F e = 8.1 N; and (c) F e = 10.8 N

Another view of relationship of resonance frequency and friction force is shown on Fig. 7 to support the above mentioned ideas. The amplitude of oscillation of the mechanical system at resonance increases with increasing the amplitude of excitation force at all investigated resistance levels F f . Besides, with the same value of the excitation force, the magnitude of oscillation (i.e. amplitude of X 1 –X 2 ) at resonance decreases when increasing the resistant force F f . As shown on Fig. 8(a), the resonant frequency f res of the system at each amplitude of excitation force can be represented by a quadratic polynomial function depending on the Coulomb friction force as following: fres = A + B ∗ Ff + C ∗ Ff2

(2)

By fitting the plot data (Fig. 8(a)), the detailed expressions describe the relationship between resonant frequency and friction force as following: fres1 = 12.98197 − 0.38692 ∗ Ff + 0.01123 ∗ Ff2 fres2 = 13.98197 − 0.38692 ∗ Ff + 0.01123 ∗ Ff2

(3)

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Fig. 7. The variation of resonant frequency under the same Coulomb friction when varying the excitation force, respectively: (a) F f = 3.56 N; (b) F f = 16.63 N; and (c) F f = 28.5 N

fres3 = 15.03625 − 0.32791 ∗ Ff + 0.00898 ∗ Ff2 Where f res1 , f res2 , f res3 were the resonant frequency of the system depending on friction force at exctitation force as 5.4 N, 8.1 N and 10.8 N, respectively. The experimental data also show that the coefficients of Eq. (2) depend on excitation force, as shown on the Fig. 8(b). This relation can be expressed as following equations: y = a + b ∗ Fe A = 10.91864 + 0.38042 ∗ Fe B = −0.45577 + 0.01093 ∗ Fe

(4)

C = 0.01386 − 4.167 ∗ 10−4 ∗ Fe

Fig. 8. Fitted curves of the Frequency Response Function with the Coulomb friction force and excitation force.

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4 Conclusion The effects of the Coulomb friction on the resonant frequency of the two-mass system connected by a nonlinear leaf spring were experimentally investigated by sweeping in the range from 2 Hz to 30 Hz. Experimental data show that the resonant frequency of the system seemed to decrease when increasing the resistant force. Under the different friction, the resonant frequencies have been changed quite in comparing with the natural frequency of the system. The resonant frequency of the system can be expressed as the functions depending both on the amplitude of excitation force and on the Coulomb friction force. The experimental results serve as the basic premise for the development of studies applied in self-movement structure operating under different resistant environments. Acknowledgement. This research was funded by Vietnam Ministry of Education and Training, under the grant number B2019-TNA-04. The authors would like to express their thank to Thai Nguyen University of Technology, Thai Nguyen University for their supports. Technical guidances and supports from Dr. Van-Du Nguyen from Thai Nguyen University of Technology are very much appreciated.

References 1. Cveticanin, L.: Vibrations of a coupled two-degree-of-freedom system. J. Sound Vib. 247(2), 279–292 (2001) 2. Cveticanin, L.: The motion of a two-mass system with non-linear connection. J. Sound Vib. 252(2), 361–369 (2002) 3. Ganji, S.S., Barari, A., Ganji, D.D.: Approximate analysis of two-mass–spring systems and buckling of a column. Comput. Math. Appl. 61(4), 1088–1095 (2011) 4. Guo, S., et al.: Development of Multiple Capsule Robots in Pipe. Micromachines (Basel) 9(6) (2018) 5. Lai, S.K., Lim, C.W.: Nonlinear vibration of a two-mass system with nonlinear stiffnesses. Nonlinear Dyn. 49(1–2), 233–249 (2006) 6. Xu, J., Fang, H.: Improving performance: recent progress on vibration-driven locomotion systems. Nonlinear Dyn. 98(4), 2651–2669 (2019). https://doi.org/10.1007/s11071-019-049 82-y 7. Nguyen, V.-D., et al.: The effect of inertial mass and excitation frequency on a Duffing vibro-impact drifting system. Int. J. Mech. Sci. 124–125, 9–21 (2017) 8. Liu, L., et al.: Reliability of elastic impact system with Coulomb friction excited by Gaussian white noise. Chaos Solitons Fractals 131, 109513 (2020) 9. Liu, Y., et al.: Bifurcation analysis of a vibro-impact experimental rig with two-sided constraint. Meccanica 55(12), 2505–2521 (2020). https://doi.org/10.1007/s11012-020-011 68-4 10. Nguyen, V.-D., et al., A new design of horizontal electro-vibro-impact devices. J. Comput. Nonlin. Dyn. 12(6), 061002 (2017) 11. Safaeifar, H., Farshidianfar, A.: A new model of the contact force for the collision between two solid bodies. Multibody Syst. Dyn. 50(3), 233–257 (2020). https://doi.org/10.1007/s11 044-020-09732-2

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12. Yan, Y., Liu, Y., Manfredi, L., Prasad, S.: Modelling of a vibro-impact self-propelled capsule in the small intestine. Nonlinear Dyn. 96(1), 123–144 (2019). https://doi.org/10.1007/s11 071-019-04779-z 13. Duong, T.-H., et al.: A new design for bidirectional autogenous mobile systems with two-side drifting impact oscillator. Int. J. Mech. Sci. 140, 325–338 (2018) 14. Christoph Kossack, J.Z., Schmitz, T.L.: The sliding friction contact frequency response function. Procedia Manufacturing 34, 73–82 (2019) 15. Polesel-Maris, J.R.M., et al.: Experimental investigation of resonance curves in dynamic force microscopy. Nanotechnology 14(9), 1036–1042 (2003) 16. Bleck-Neuhaus, J.: Mechanical resonance 300 years from discovery to the full understanding of its importance (2018)

Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method Dang Xuan Hung, Huong Quy Truong(B) , and Tran Minh Tu National University of Civil Engineering, 55 Giai Phong Road, Hai Ba Trung District, Hanoi, Vietnam {hungdx,truonghq,tutm}@nuce.edu.vn

Abstract. This paper deals with the nonlinear bending response of functionally graded porous beams reinforced by graphene platelets (GPLs) with various boundary conditions using the Ritz method. Based on the trigonometric shear deformation beam theory and the von Kárman type of geometrical nonlinearity strains, the system of nonlinear governing equations is derived using the minimum total potential energy principle. This system of nonlinear equations is then solved by the Newton–Raphson method. The comparison with the available published results validates the obtained results. The effects of the porosity distribution patterns, the porosity coefficient, the GPL reinforcements, the slenderness ratios and the boundary conditions on the nonlinear deflection of the FGP porous beam are also investigated. Keywords: Porous beam · Nonlinear bending · Ritz method · Trigonometric shear deformation beam theory · Graphene platelet reinforcement

1 Introduction Functionally graded materials (FGMs) are a class of composites that are firstly invented by Japanese scientists in 1984. FGMs are characterized by a continuous variation in both composition and material properties in one or more directions, thus eliminating interface problems and diminishing the stress concentration that normally exists in conventional laminated composites [1, 2]. Porous materials (metal foams) are a new class of FGMs characterized by low density, lightweight, good stiffness and excellent energy absorption, Porosities inside metal foams can be distributed in different manners. Their mechanical properties are significantly influenced by the amount of porosities and their pattern of distribution. To archive desire material properties, graded non-uniform porosities are introduced to produce FG metal foams which have been proved to have better structural response than the normal uniform foams and are investigated by the increasing number of researchers [3–10]. Introducing nanofillers into porous materials is a practical way to strengthen their mechanical properties and maintain their potential for lightweight structures simultaneously. The properties of such materials can be significantly improved with the addition of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 72–86, 2022. https://doi.org/10.1007/978-981-16-3239-6_6

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nanofillers in the form of carbon nanotubes (CNTs) and graphene platelets (GPLs). Thus, carbon nanotube reinforced beams and plates have been extensively studied to examine their static and dynamic response [11–15]. Until now, static and dynamic analysis of GPL reinforced structures is still at the beginning stage [16–20]. A nonlinear static analysis is required for any static application in which the stiffness of the entire structure changes during the loading scenario. Many works focused on nonlinear behaviours of beams and plates are reported in the existing literature. Feng et al. [21] studies the nonlinear bending behavior of Timoshenko multi-layer polymer nanocomposite beams reinforced with GPLs using Ritz method. Shen et al. [22] studied the large amplitude vibration of functionally graded graphene-reinforced composite laminated plates resting on an elastic foundation and in thermal environments based on a higher-order shear deformation plate theory and perturbation technique. Wu et al. [12] analyzed nonlinear vibration of imperfect FG-CNTRC beams is based on the firstorder shear deformation beam theory and von Kármán geometric nonlinearity. Ke et al. [15] investigated the nonlinear free vibration of FG nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory. Barati et al. [20] used a refined beam model to investigate the post-buckling behavior of geometrically imperfect porous beams reinforced with graphene platelets. Yas and Rahimi [23] presented thermal buckling of FG porous nanocomposite beams subjected to a thermal gradient using the generalized differential quadrature method. This paper deals with the geometrically nonlinear bending analysis of functionally graded porous beams reinforced by graphene platelets with various boundary conditions using the Ritz method. GPLs are distributed in the thickness direction with uniform and nonuniform patterns. Uniform, symmetric, and asymmetric distributions of porosity have been considered. After conducting the validate example, the effects of the porosity distribution patterns, the porosity coefficient, the GPL reinforcements, the slenderness ratios and the boundary conditions on the nonlinear deflection of the FGP porous beam are investigated in detail.

2 Material Properties Consider a beam with thickness h, width b, and length L defined in the Cartesian coordinate system (x, z) as shown in Fig. 1. The porosity distributes within the thickness according to the symmetric, asymmetric and uniform laws below [24]. ⎧ ⎨ ψ(z) = cos(π  z/h)  (1) ψ(z) = cos π z/2h + π4 ⎩ ψ(z) = ψ0 This porosity distribution leads to a variety of material properties as follows. ⎧ ⎪ ⎨ Ez = Ec [1 − e0 ψ(z)] Gz = Ez /[2(1 + νz )] ⎪ ⎩ ρz = ρc [1 − em ψ(z)]

(2)

Where Ec and ρc denote the Young modulus and the mass density of the material without porosity, respectively; E1 , E2 (ρ1 , ρ2 ) denote the maximum, minimum Young’s

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Porosity distribution 1

Porosity distribution 2

Uniform distribution

Fig. 1. Porosity distribution: symmetric, asymmetric and uniform laws

modulus (mass density) without GPLs of the non-uniform porosity distribution and that  of the uniform law is denoted by E ; The porosity coefficient e0 and the mass density coefficient em are defined by e0 = 1 − E2 /E1 and em = 1 − ρ2 /ρ1 . Based on the relationship between the mechanical properties of closed-cell cellular solids (3) that is fitted from statistical Gaussian Random Fields model [25], Ez = Ec



2.3 ρz + 0.121 /1.121 ρc

ρz 1. There is an intersection between the frequencies of different porosity volume fractions. Comparison between Fig. 4 (nx = 0.5) and (nx = 2), shows that when nx is low, the first frequency parameter of Fig. 4 (nx = 0.5) is higher than Fig. 4 (nx = 2). The effect of the layer thickness ratio on the frequency parameter can also be seen from Fig. 5, it shows the first frequency parameter of imperfect beam for different values of the layer thickness ratio versus the FG (nz) power indexes. It is observed that increasing nz decrease the frequency parameter of the sandwich beam. It is seen that the first frequency parameter of imperfect FG beam, increases with the increment of the layer thickness ratio value. But the effect of the layer thickness ratio on the first frequency parameter of the imperfet FG beam is complicated. When nz is low, increasing the layer thickness ratio decreases the frequency parameter of beam, while when nz is more, the frequency parameter increases with the increment of the layer thickness ratio.

4 Conclusions The vibration of bi-dimensional imperfect FG sandwich beam has been investigated in the present work. The beam consists of three layers, the upper face of the sandwich beam is made of porous 2D-FG, the lower face is made of ceramic and the core is made of one-dimensional functionally graded (1D-FG) porous. The material properties

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are assumed to vary in both the thickness and longitudinal directions by the power-law functions. Based on a third-order shear deformation theory, a beam element was formulated and employed in computing the vibration characteristics. The obtained numerial result reveals that the variation of the material properties plays an important role in the frequencies. A parametric study has been carried out to highlight the effects of the material distribution and the layer thickness ratio on the frequencies of the sandwich beam. Moreover, the effect of porosity volume fraction vp is studied on the vibrational behavior of the sandwich beam. The influence of the aspect ratio on the vibration behaviour of the beam has also been examined and discussed.

References 1. Zenkour, A.M., Allam, M.N.M., Sobhy, M.: Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations. Acta Mech. 212(3–4), 233–252 (2010) 2. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Maheri, A., Lee, J.: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng. Struct. 64, 12–22 (2014) 3. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Inam, F., Lee, J.: A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos. Struct. 119, 1–12 (2015) 4. Zhu, S., Jin, G., Wang, Y., Ye, X.: A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations. Acta Mech. 227(5), 1493–1514 (2016) 5. Simsek, M., Al-Shujairi, M.: Static. Free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads. Compos. B: Eng. 108(3), 18–34 (2017) 6. Vo-Duy, T., Ho-Huu, V., Nguyen-Thoi, T.: Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method. Frontiers Struct. Civil Eng. 13(2), 324–336 (2018) 7. Nemat-Alla, M., Noda, N.: Edge crack problem in a semi-infinite FGM plate with a bidirectional coefficient of thermal expansion under two-dimensional thermal loading. Acta Mech. 144(3–4), 211–229 (2000) 8. Lu, C.F., Chen, W.Q., Xu, R.Q., Lim, C.W.: Semi-analytical elasticity solutions for bi-directional functionally graded beams. Int. J. Solids Struct. 45(1), 258–275 (2008) 9. Simsek, M.: Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Compos. Struct. 133, 968–978 (2015) 10. Wang, Z.-H., Wang, X.-H., Xu, G.-D., Cheng, S., Zeng, T.: Free vibration of twodirectional functionally graded beams. Compos. Struct. 135, 191–198 (2016) 11. Karamanlı, A.: Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3D shear deformation theory. Compos. Struct. 174, 70–86 (2017) 12. Nguyen, D., Nguyen, Q., Tran, T., Bui, V.: Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Acta Mech. 228(1), 141–155 (2016) 13. Tran, T.T., Nguyen, D.K.: Free vibration analysis of 2D-FGM beams in thermal environment based on a new third-order shear deformation theory. Viet. J. Mech. 40(2), 121–140 (2018) 14. Ha, L.T.: Free vibration of sandwich beams with bi-directional functionally graded core. Transport Commun. Sci. J. 68, 9–16 (2019)

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15. Ha, L.T., Tram, T.T.: Vibration analysis of sandwich beam with functionally graded (FG) skins subject to a moving load in high temperature environment. Transport Commun. Sci. J. 58, 34–40 (2017) 16. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media (1987) 17. Sallica-Leva, E., Jardini, A., Fogagnolo, J.: Microstructure and mechanical behavior of porous Ti–6Al–4V parts obtained by selective laser melting. J. Mech. Behav. Biomed. Mater 26, 98–108 (2013) 18. Ebrahimi, F., Ghasemi, F., Salari, E.: Investigating thermal effects on vibration behavior of temperature-dependent compositionally raded Euler beams with porosities. Meccanica. https://doi.org/10.1007/s11012-0150208-y 19. Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331– 1342 (2015) 20. Ha, L.T., Khue, N.T.K.: Free vibration of functionally graded (FG) porous nano beams. Transport Commun. Sci. J. 70(2), 98–103 (2019). https://doi.org/10.25073/tcsj.70.2.32 21. Shafei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M.: Vibration of twodimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Method Appl. Mech. Eng. 322, 615–632 (2017) 22. Simsek, M.: Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions. Compos. Struct. 149, 304–314 (2016) 23. Shi, G., Lam, K.Y.: Finite element formulation vibration analysis of composite beams based on higher-order beam theory. J. Sound Vib. 219, 707–721 (1999) 24. Nemat-Alla, M.: Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int. J. Solids Struct. 40, 7339–7356 (2003) 25. Reddy, J.N.: A simple higher-order theory for laminated composite plates. ASME-J. Appl. Mech. 51(4), 745–752 (1984) 26. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29(8), 901–916 (1991) 27. Soldatos, K.P.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 94(3–4), 195–220 (1992) 28. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40(6), 1525–1546 (2003) 29. Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct. 89, 94–101 (2009)

Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation Le Thi Ha(B) University of Transport and Communications, No.3 Cau Giay Street, Lang Thuong Ward, Dong Da District, Hanoi, Vietnam [email protected]

Abstract. In this paper, the free vibration of prestress two-dimensional imperfect functionally graded (2D-FG) nano beam partially resting on a Winkler foundation is investigated by finite element method. The material properties of 2D-FG nano beam are assumed vary in both axial and thickness directions according to a power law. Based on Eringen nonlocal elasticity theory, the governing equations of motion are derived. A parametric study in carry out to show the effect of material distribution, nonlocal effect, prestress and elastic support on the natural frequencies of the beams. The finite element method is employed to establish the equations and compute the vibration characteristics of the beam. Keywords: Two-dimensional imperfect functionally graded · Nanobeams · Nonlocal model · Elastic foundation · Free vibration · Finite element method

1 Introduction Functionally graded materials (FGMs) have wide applications in modern industries including aerospace, mechanical, electronics, optics, chemical, biomedical, nuclear and civil engineering. Besides, nanotechnology is primarily concerned with fabrication of FGMs and engineering structures at a nanoscale. The structures such as nanoplate and nanobeam used in systems and devices such as nanowires, nano-probes, atomic force microscope (AFM), nanoactuators and nanosensors. Investigations on mechanical behavior of nanostructures in general and nanobeams in particular have received much attention from researchers recently. The nonlocal field theory, one of the talented continuum models in nanomechanics considering size-dependent effects, was first developed by Eringen [1–3]. Most classical continuum theories are based on hyperelastic constitutive relations which assume that the stress at a point is functions of strain at the point. On the other hand, the nonlocal continuum mechanics assumes that the stress at a point is a function of strains at all points in the continuum. By using this theory, the equilibrium differential and motion equations for nanostructures can be derived. Several investigations on vibration of nanobeams have been reported in the literature. Reddy [4] studied the bending, bending, buckling and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 112–124, 2022. https://doi.org/10.1007/978-981-16-3239-6_9

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vibration of homogenous nanobeams. A compact generalized beam theory is used by Aydogdu [5] to analyze the bending, buckling and vibration of nanobeams. Simsek and Yurtcu [6] derived analytical solutions for bending and buckling of FG nanobeams. Based on the finite element method, Eltaher et al. [7, 8] studied free vibration of FG sizedependent nanobeams. On the basis of the nonlocal differential constitutive relations of Eringen, Zemri, Houari, Bousahla and Tounsi [9] proposed a nonlocal shear deformation beam theory to study the bending, buckling, and vibration of FG nanobeams. In addition to the functionally graded materials, porous materials are also a new type of material with emerging applications. Thus, these kinds of materials have been the focus of many mechanical studies [10, 11]. Besides, similar to the functionally graded materials, the dynamic and vibration of the structures that are made of porous materials are interesting for scientists. Ebrahimi et al. [12] studied thermo-mechanical vibration of functionally graded (FG) beams made of porous material subjected various thermal loadings are carried out by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Wattanasakulpong [13] studied flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory by using the Chebyshev collocation method. Based on Bernoulli beam theory, the free vibration of functionally graded (FG) porous nano beams is studied by Ha and Khue [14]. The finite element method is used to discretize the model and to compute the vibration characteristics of the beams. A parametric study in carry out to show the effects of the nonlocal parameter and porous parameter, material distribution on the natural frequencies of the beams are examined and discussed. Besides, similar to the functionally graded materials, the dynamic and vibration of the structures that are made of porous materials are interesting for scientists. Guastavino and Goransson [15] studied the vibration of anisotropic porous foam materials. Takahashi and Tanaka [16] investigated on a method of theoretical approach of acoustic coupling due to flexural vibration of porous plates made of elastic materials. In addition, two-dimensional functionally graded (2D-FG) porous nanobeam was studied by Shafiei [17]. Based on Timoshenko beam theory, Shafiei studied the vibration behavior of the two-dimensional functionally graded (2D-FG) nano and microbeams which are made of two kinds of porous materials for the first time. In this paper, the free vibration of prestress two-dimensional imperfect functionally graded (2D-FG) nanobeam partially resting on a Winkler is studied. Euler–Bernoulli beam theory incorporated with nonlocal differential equation of Eringen is used to derive the nonlocal differential equations of motion and the finite element method is employed to compute the frequencies of the beam. The effect of nonlocal parameter, a porosity volume fraction, FG power indexes and the foundation support on the vibration characteristics is examined and discussed.

2 Problem and Formulation 2.1 Grade Functionally Beam Figure 1 shows a simply supported beam with length L subjected to axial compression force Pe , width b and height h, resting on a Winkler elastic foundation with ratio stiffness k W and length L e . The 2D-FG nanobeams is composed of metal and ceramic with varying

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z Em, Ec, ρm , ρc, L, h Pe

porous 2D- FG

Pe

h x

kw, Le Fig. 1. 2D-FG nanobeam partially resting on a Winkler elastic foundation.

material volume fraction along x and z directions. So the mechanical properties of the nanobeams such as Young’s modulus ‘E’, mass density ‘ρ’ vary along axial (x-axis) and the thickness directions (z-axis). The porous 2D-FG nanobeam is considered to have even porosity distributions across the beam thickness. Consider the imperfect 2D-FG beam with a porosity volume fraction, α (α  1 ), distributed equally in the ceramic and metal phases, the modified rule of mixture is proposed as [17]:   α α + Pc Vc − (1) P(x, z) = Pm Vm − 2 2 the ()m and ()c subscripts respectively denote the ceramic and metal, now, the total volume fraction of the metal and ceramic is 1 and the power law of volume fraction of the ceramic is given as [17] Vm + Vc = 1  nz  x nx Vc (x, z) = 21 + hz L

(2)

where z is measured from the mid-plane; Vc and Vm are the volume fractions of ceramic and metal, respectively; ‘nx’ and ‘nz’ are the FG power indexes (along the length and thickness), respectively, are ascribed to the volume fraction change of the material composition. The material properties can be evaluated by a simple rule [17]   z nz  x nx α 1 + P(x, z) = Pm + (Pc − Pm ) − (Pc + Pm ) (3) 2 h L 2 When the beam is perfect (α = 0), the material of the beam is 100% ceramic when nx and nz are set to be zero. Thus, Young’s modulus ‘E’, mass density ‘ρ’ equations of the imperfect nanobeam can be expressed as: nz  x nx α  − (E + Em ) E(x, z) = Em + (Ec − Em ) 21 + hz  1 z nz  xLnx α2 c (4) ρ(x, z) = ρm + (ρc − ρm ) 2 + h − L 2 (ρc + ρm ) 2.2 Government Equations The axial displacement u and transverse displacement w at any point based on Euller Bernoulli beam theory are given by 0 u(x, z, t) = u0 (x, t) − z ∂w ∂x , w(x, z, t) = w0 (x, t)

(5)

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where u0 and w0 are the axial and transverse displacements at the mid plane, and t is the time. By using the small deformation, the axial strain reads εxx =

∂u0 ∂ 2 w0 ∂u 0 = − z 2 = εxx − zκ 0 ∂x ∂x ∂x

(6)

0 = ∂uo , κ 0 = ∂ w0 are the extensional strain and the bending strain, where εxx ∂x ∂x2 respectively. Hamilton’s principle has the form 2

t2 (δU + δV − δT )dt = 0

(7)

t1

where δU , δT , δV are the virtual strain and kinetic energies, and virtual potential. L   0 N δεxx δU = − M δκ 0 dx,

(8)

0

in which, N b

h/2 −h/2

=

b

h/2 −h/2

σxx (x, z) dz is the axial normal force and M

=

zσxx (x, z) dz is the bending moment.  L  ∂w0 ∂δw0 δV = −b Pe + kW w0 δw0 dx ∂x ∂x L

δT =

 I11

0

0

∂u0 ∂δu0 ∂w0 ∂δw0 + ∂t ∂t ∂t ∂t



 − I12

∂δu0 ∂ 2 w0 ∂u0 ∂ 2 δw0 + ∂t ∂x∂t ∂t ∂x∂t

(9)

+ I22

∂ 2 w0 ∂ 2 δw0 dx ∂x∂t ∂x∂t

(10) where I11 , I12 , I22 are the mass moments, defined as h/2 (I11 , I12 , I22 ) = b

  ρ(x, z) 1, z, z 2 dz

(11)

−h/2

Substituting Eqs. (8), (9) and (10) into Eq. (7), we obtained the following equations of motion ∂N ∂ 2 u0 ∂ 3 w0 = I11 2 − I12 ∂x ∂t ∂x∂t 2

(12)

∂ 2 w0 ∂ 3 u0 ∂ 4 w0 ∂ 2 w0 ∂ 2M = I + I − I + k w − P 11 12 22 W 0 e ∂x2 ∂t 2 ∂x∂t 2 ∂x2 ∂t 2 ∂x2

(13)

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2.3 Nonlocal Continuum Beam According to Eringen [1, 2], the nonlocal stress tensor components σij at point x are expressed as          (14) σ = K x − x, τ t(x ) d x V

where t(x) are the components x  at point  ofthe classical macroscopic stress tensor         and the kernel function K x − x, τ represents the nonlocal modulus, x − x is the distance (in Euclidean norm) and τ is a material constant that depends on internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). The macroscopic stress t at a point x in the Hookean solid is related to the strain at the point by the generalized Hooke’s law t(x) = C(x) : ε(x) tij = Cijmn εmn

(15)

where C is the fourth-order elasticity. The integral constitutive relation in Eq. (14), however makes the elasticity problems difficult to solve, in addition to possible lack of determinism. It is possible [2] to represent the integral constitutive relations in an equivalent differential form as   1 − τ 2 l 2 ∇ 2 σij = Cijmn εmn (16) τ = e0l a where e0 is a material constant, a, l are the internal and external characteristic lengths, respectively. Equation (16) can be expressed as L(σij ) = Cijmn εmn L = 1 − μ∇ 2 μ = e02 a2

(17)

Nonlocal stress resultants for beams is determined by using (17). The nonlocal theory results in differential relations involving the stress resultants and the strains, and for 2D-FG beam, stress resultants can be written by. L(σxx ) = E(x, z)εxx , where L = 1 − μ

∂2 ∂x2

(18)

and E(x, z) is elastic modulus. If μ = 0, the relations of the local theories are resumed. For Euler - Bernoulli theory beam, Eq. (18) can be written as σxx − μ

∂ 2 σxx = E(x, z)εxx ∂x2

(19)

From Eq. (19), we have the constitutive relations N −μ

∂ 2N 0 = A11 εxx − A12 κ 0 ∂x2

(20)

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M −μ

∂ 2M 0 = A12 εxx − A22 κ 0 ∂x2

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(21)

where A11 , A12 and A22 are respectively the axial, axial-bending coupling and bending rigidities, and they are defined as    (22) (A11 , A12 , A22 ) = E(x, z) 1, z, z 2 dA A

Substituting Eq. (12) into Eq. (20) and Eq. (13) into Eq. (21) we obtain the expressions for axial force and moment of the beam   ∂u0 ∂ 2 w0 ∂ 3 u0 ∂ 4 w0 (23) − A12 2 + μ I11 − I12 2 2 , N = A11 ∂x ∂x ∂x∂t 2 ∂x ∂t ∂u0 ∂ 2 w0 ∂ 2 w0 ∂ 3 u0 ∂ 4 w0 ∂ 2 w0 M = A12 + μ I11 2 + I12 −I22 2 2 + kW w0 − Pe . − A22 ∂x ∂x2 ∂t ∂x∂t 2 ∂x ∂t ∂x2

(24) Finally, we can obtain the differential equations of motion for the beam in the forms ∂ 2 u0 ∂ 3 w0 ∂ 4 u0 ∂ 5 w0 ∂ 2 u0 ∂ 3 w0 A11 2 − A12 3 + μ I11 2 2 − I12 3 2 = I11 2 − I12 (25) ∂x ∂x ∂x ∂t ∂x ∂t ∂t ∂x∂t 2   3 4 4 5 6 4 2 A12 ∂∂xu30 − A22 ∂∂xw40 + μ I11 ∂x∂ 2w∂t02 + I12 ∂x∂ 3u∂t02 − I22 ∂x∂ 4w∂t02 −Pe ∂∂xw40 + kW ∂∂xw20 ∂ u0 ∂ w0 ∂ w0 = I11 ∂∂tw20 + I12 ∂x∂t 2 − I22 ∂x 2 ∂t 2 + kW w0 − Pe ∂x 2 2

3

4

2

(26)

2.4 Numerical Formulation This section is devoted to drive the finite element model for nonlocal Euler–Bernoulli beam. Based on the Hamilton principle, by substituting Eq. (23, 24) into Eq. (8) and the results produced is substituted into Eq. (7), the following variational statement of nonlocal Euler–Bernoulli beam can be deduced: t2 L  ∂ 2 w0 ∂ 2 δw0 ∂ 2 w0 ∂δu0 ∂u0 ∂ 2 δw0 0 ∂δu0 A11 ∂u ∂x ∂x + A22 ∂x2 ∂x2 − A12 ∂x2 ∂x − A12 ∂x ∂x2 t1 0  2 2 ∂ 2 δw0 ∂ 2 w0 ∂ 2 δw0 0 ∂δw0 − I11 μ ∂∂tw20 ∂ ∂xδw2 0 + k1 w0 δw0 + Pe ∂w − μk w + μP w 0 e 2 2 2 ∂x ∂x ∂x ∂x ∂x   ∂ 3 u0 ∂δu0 ∂ 4 w0 ∂δu0 ∂ 3 u0 ∂ 2 δw0 ∂ 4 w0 ∂ 2 δw0 − μI12 ∂x∂t + μ I11 ∂x∂t 2 ∂x − I12 ∂x 2 ∂t 2 ∂x 2 ∂x 2 + μI22 ∂x 2 ∂t 2 ∂x 2    2 ∂w0 ∂δw0 ∂δu0 ∂ 2 w0 ∂ 2 w0 ∂ 2 δw0 0 ∂δu0 0 ∂ δw0 + I12 ∂u − I11 ∂u ∂t ∂t + ∂t ∂t ∂t ∂x∂t + I12 ∂t ∂x∂t −I22 ∂x∂t ∂x∂t dxdt = 0 (27) The finite element method is employed herein to solve the equations of motion. To this end, the beam is assumed being divided into a number of two-node beam elements

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with length of l. The vector of nodal displacements (d) for the element considering the transverse shear rotation γ0 as an independent variable contains six components as d = {u1 , w1 , θ1 , u2 , w2 , θ2 }

(28)

where u1 , w1 , θ1 , u2 , w2 , θ2 are the values of u0 , w0 and θ0 at the node 1 and at the node 2. In Eq. (28) and hereafter, a superscript ‘T ’ is used to denote the transpose of a vector or a matrix. u0 = NuT d, w0 = NwT d.

(29)

where Nu, Nw denote the matrices of shape functions for u0 , w0, respectively. In the present work, using linear and cubic Hermite polynomials to interpolate the axial and transverse displacements   Nu = Nu1 0 0 Nu2 0 0   (30) Nw = 0 Nw1 Nw2 0 Nw3 Nw4 with Nu1 = l−x l , x Nu2 = l , 2 Nw1 = 1 − 3xl 2 +

2x2 l + 3 3x2 − 2xl 3 , l2 −x2 x3 l + l2 .

Nw2 = x −

Nw3 = Nw4 =

2x3 , l3 x3 , l2

(31)

Putting Eq. (29) into the variational statement form Eq. (27), performing integration, the following element equation is obtained: ¨ + KD = 0 (M + Mnl )D

(32)

where D is the total vector displacement, M and K are the structural mass and stiffness matrices assembled from the element mass and stiffness matrices over the total elements, respectively; and Mnl is the nolocal mass over the total element. When the nonlocal parameter μ = 0, Eq. (32) returns to the free vibration equation for the conventional beams. In addition, Eq. (32) leads to an eigenvalue problem for determining the frequency ω as   ¯ =0 K − ω2 (M + Mnl ) D (33) ¯ is the vibration amplitude. Equation (33) leads with ω is the circular frequency and D to an eigenvalue problem, and its solution can be obtained.

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3 Numerical Result The 2D-FG nanobeams is composed of steel (SUS304) and alumina (Al2 O3 ) where its properties vary along axial (x-axis) and the thickness directions (z-axis) with (SUS304, Em = 210 Gpa, ρm = 7800 kg/m3 , alumina (Al2 O3 ), Ec = 390 Gpa, ρc = 3960 kg/m3 ). The bottom surface of the beam is pure steel, whereas the top surface of the beam is pure alumina. The beam geometry has the following dimensions: L (length) = 10, b (width) = 1 and h (thickness). The frequency is normalized (λi ) as [7]  2 ρc A (34) λi = ωi L Ec I where ωi is the i th natural frequency of the beam and A = b.h, 3 I = bh 12 .

(35)

In (35), I is the moment of inertia of the cross section of the beam, A is the cross section area of the beam. The foundation support parameter Le (0 ≤ Le ≤ 1) is defined as a ratio of the supported length to the total beam length, and the foundation stiffness parameter k 1 is introduced as. k1 = kW

L4 Em I

(36)

and axial compression force Pe [18] Pe =

−π 2 Em I L2

(37)

Table 1. Comparison of fundamental frequency paraneter for simply supported 1D-FG nanobeams (nx = 0, k 1 = 0, Pe = 0, α = 0). μ n=0 Present [7]

n=1

n=5

Present [7]

Present [7]

1 9.4062 9.4238 6.6669 6.7631 5.6639 5.7256 2 9.0102 9.0257 6.3863 6.4774 5.4255 5.4837 3 8.6604 8.6741 6.1384 6.2251 5.2148 5.2702 4 8.3483 8.3607 5.9172 6.0001 5.027

5.0797

5 8.0678 8.0789 5.7184 5.7979 4.858

4.9086

The validation of the derived formulation is necessary to confirm before computing the vibration characteristics of the beam. In Table 1, the fundamental frequencies

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Table 2. Comparison of fundamental frequency parameter for porous 2D-FG nanobeams [17] (α = 0.1; μ = 0.15, k 1 = 0, Pe = 0). nz

Source

L/h = 12 nx = 0

nx = 1

nx = 5

nx = 0

nx = 1

nx = 5

1

Present

13.5034

11.5727

10.3789

13.7486

11.6084

10.4099

[17]

13.5487

11.5981

10.4459

13.8728

11.8787

10.6968

Present

11.4050

10.6151

10.0909

11.4385

10.6460

10.1205

[17]

11.3922

10.7048

10.2581

11.6787

10.9691

10.5079

Present

10.8555

10.3432

10.0001

10.8865

10.3731

10.0294

[17]

10.8931

10.4660

10.1917

11.1672

10.7241

10.4399

5 10

L/h = 18

parameter of a simply supported nanobeam with an aspect ratio L/h = 20 and nx = 0, k 1 = 0, Pe = 0, α = 0 obtained in the present paper are compared with the results by Eltaher et al. [7]. A good agreement can be noted from Table 1, irrespective of the nolocal parameter. The frequency factor in the Table 2 is obtained by setting k1 and Pe to zero, α = 0.1, μ = 0.15 and using the geometric and material data of the corresponding references [17]. A very good agreement between the results of the present work with those of the references [17] is noted from Table 2.

Fig. 2. Non-dimensional natural frequency parameter of imperfect 2D-FGM nanobeams when L/h = 20, α = 0.3, k1 = 50, Le = L/3.

Figure 2 shows the non-dimensional frequency of simply supported FG, porous nanobeam for different values of nonlocal parameter (μ) versus the FG power indexes when L/h = 20, α = 0.3, k 1 = 50, L e = L/3, respectively. It is observed that increasing nx

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and nz decrease the frequency parameter of the nanobeam which is because increasing the nz and nx power indexes decrease the stiffness of the nano beam. It is seen that the non-dimensional frequency of FG nanobeam, decreases with the increment of the nonlocal value. In addition, the decrease of the fundamental frequency parameter is more significant for nx < 1 and nz < 1. The influence of the foundation stiffness on the fundamental frequency parameter of the nanobeam can be seen from Fig. 3, where the grading index nx and nz versus the fundamental frequencies is depicted for various values of the foundation stiffness parameter k 1 when L/h = 20, α = 0.3, μ = 0.3, L e = L/2. Regardless of the index nx and nz, the fundamental frequency parameter is increased with the increase of the foundation stiffness k 1 which is because increasing the foundation stiffness k1 increase the stiffness of the nanobeam.

Fig. 3. Grading index nx and nz versus on fundamental frequency parameter with different foundation stiffness parameter k1 for imperfect 2D-FG nano beam when L/h = 20, α = 0.3, μ = 0.3, Le = L/2.

Figure 4 shows the non-dimensional frequency of simply supported nano beam made of porous materials versus the nz and nx power indexes. Figure 4 shows that increasing the nz power indexes decrease the frequency which is because increasing the nz decrease the stiffness of the nano beam. In addition, it is seen in Fig. 4 that the dependency of the frequency on the nz power indexes increase with the porosity volume fraction (α), and this is more intense for porous, as it is seen that the decrement of the frequency due to the FG power indexes is so high that when nz  1, there is an intersection between the frequencies of different porosity volume fractions. In fact, the effect of the porosity volume fraction on the frequency of the porous beams highly depends on nz as in higher values of nz, the frequency decreases with the increment of α but when nz are low, the increment of α increases the frequency.

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Fig. 4. Non-dimensional natural frequency of imperfect 2D-FGM nanobeams when L/h = 20, μ = 0.5, Pe = 0, Le = L/2, nx = 0.5.

Fig. 5. Effect of foundation parameter Le on fundamental frequency parameter of imperfect 2DFGM nano beam when L/h = 20, μ = 0.5, α = 0.3, k1 = 50, nx = 0.5

The effect of the foundation support on the fundamental frequency parameter can be seen from Fig. 5, where the frequencies of the nano beam are given for various values of the foundation parameter Le . The frequencies are clearly seen to be increased by raising the foundation parameter Le , regardless of the axial compression forces (Pe = 2 0 or Pe = −πL2Em I ) which is because increasing the foundation parameter Le increase the stiffness of the nano beam. Besides, the frequency parameter is increased with the increase of the axial compression force Pe . Similar to Fig. 3, it is seen that increasing nz decreases the non-dimensional frequency of the nano beam.

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The grading index nz versus the fundamental frequency parameters of the nanobeams is illustrated in Fig. 6 for L/h = 20, α = 0.3, L e = L/2, nx = 0.5, respectively. As seen from the figures, for a give value of the nonlocal parameter, the fundamental frequencies decreases when the nonlocal parameter μ increase, regardless of the axial compression force and the foundation stiffness parameter. It is observed that as the value of Pe increases and k1 decreases, the fundamental frequencies increase, regardless of the nonlocal parameter.

Fig. 6. Grading index nz versus fundamental frequency parameter with different parameter μ of porous 2D-FG nano beam when L/h = 20, α = 0.3, Le = L/2, nx = 0.5.

4 Conclusions The vibration characteristics of porous 2D-FG Euler-Bernoulli nanobeam partially resting on a Winkler elastic foundation are evaluated. The nonlocal Eringen model considering the scale effect is taken into account in the derivation of the equations of motion. The finite element method is employed to discretize the model and to compute the fundamental frequency parameters. A parametric study has been carried out to highlight the effect of nonlocal parameter, foundational support, axial compression force and the foundation stiffness on fundamental frequency parameters of the nano beam. The obtained numerical results show that, the nonlocal parameter plays an important role in the frequencies of nanobeam, and the frequencies increase with increasing the nonlocal parameter. Besides, the fundamental frequency parameters increase with increasing the axial compression force and the foundation stiffness.

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References 1. Eringen, A.C., Edelen, D.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972) 2. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer-Verlag, New York (2002) 3. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983) 4. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007) 5. Aydogdu, M.: A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Phys. E 41, 1651–1655 (2009) 6. Simsek, M., Yurtcu, H.H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013) 7. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012) 8. Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F.: Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37(7), 4787–4797 (2013) 9. Zemri, A., Houari, M.S.A., Bousahla, A.A., Tounsi, A.: A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct. Eng. Mech. 54, 693–710 (2015) 10. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media (1987) 11. Sallica-Leva, E., Jardini, A., Fogagnolo, J.: Microstructure and mechanical behavior of porous Ti–6Al–4V parts obtained by selective laser melting. J. Mech. Behav. Biomed. Mater 26, 98–108 (2013) 12. Ebrahimi, F., Ghasemi, F. Salari, E.: Investigating thermal effects on vibration behavior of temperature-dependent compositionally raded Euler beams with porosities. Meccanica., https://doi.org/10.1007/s11012-015-0208-y 13. Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331– 1342 (2015) 14. Ha, L.T., Khue, N.T.K.: Free vibration of functionally graded (FG) porous nano beams. Transport and Communications Science Journal 70(2), 98–103 (2019). https://doi.org/10. 25073/tcsj.70.2.32 15. Guastavino, R., Göransson, P.: Vibration dynamics modeling of anisotropic porous foam materials. In: Proceedings of Forum Acusticum, Budapest Hungary, pp. 123–128 (2005) 16. Takahashi, D., Tanaka, M.: Flexural vibration of perforated plates and porous elastic materials under acoustic loading. J. Acoust. Soc. Am. 112, 1456–1464 (2002) 17. Shafei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M.: Vibration of twodimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Method Appl. Mech. Eng. 322, 615–632 (2017) 18. Gere, J.M., Timoshenko, S.P.: Mechanics of materials. Chapman & Hall, Third SI Edition (1989)

Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism Duy Q. Bui1,2 , Hung Q. Nguyen3(B) , Vuong L. Hoang2 , and Dai D. Mai4 1 Faculty of Civil Engineering, HCMC University of Technology and Education,

Ho Chi Minh City, Vietnam [email protected] 2 Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam {buiquocduy,hoanglongvuong}@iuh.edu.vn 3 Faculty of Engineering, Vietnamese-German University, Thu Dâu ` Mô.t, Binh Duong, Vietnam [email protected] 4 Faculty of Mechanical Engineering, HCMC University of Technology and Education, Ho Chi Minh City, Vietnam [email protected] ij

Abstract. This study aims at design and hysteresis modeling of a novel damper featuring shape memory alloy (SMA) to mitigate structural vibrations. The damper consists of SMA springs for actuation and a wedge mechanism to amplify and convert the actuating force into friction against the inner cylindrical face of the damper housing. From the friction between the wedges and housing, the damping force is archived. Following an introduction of SMA spring actuators and SMA dampers, the proposed SMA damper is configured. Experiments are then conducted on SMA springs to obtain their performance characteristics such as transformation temperature, heating time and actuating force. Based on the experimental data, design of the proposed SMA damper is performed and a damper prototype is fabricated. Experimental tests are then conducted to evaluate the damper performance. From the experimental results, hysteresis phenomenon of the SMA damper is presented and investigated. To predict the damper behavior, several hysteresis models are adopted and validated with comparisons and discussions. Keywords: SMA actuator · SMA spring · SMA damper · Hysteresis model · Structural vibrations

1 Introduction Material science has attained considerable improvements over the past decades. From the demand for light weight, powerfulness and controllability, smart materials have entered society. Smart materials additionally supply functions such as actuating and sensing to address strict engineering problems by convert reciprocally between mechanical (e.g., © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 125–136, 2022. https://doi.org/10.1007/978-981-16-3239-6_10

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force, displacement) and non-mechanical (e.g., voltage, temperature) responses [1]. Due to these unique characteristics, smart materials have received numerous interest in the field of structural vibration control. It is noticeable that conventional suspension systems can help mitigate most engineering oscillations at low resonant frequency. However, they possess unchangeable damping level causing the exciting force to be transmitted greater to contiguous parts and floor at high frequencies, which may lead to noises, uncomfortableness and step–by– step failure of the devices using them. Therefore, semi–active suspension systems that can reasonably adjust damping coefficient corresponding to frequency excitations are desired. A type of smart material that has been widely explored in semi-active suspension systems is magneto-rheological fluid (MRF). MRF contains microscopic magnetic particles suspended within carrier oil. A change in the magnetic field applied to the MRF associates with a change in the fluid viscosity, which leads to damping ability. There are many study works of MRF dampers to solve engineering vibration problems [2–6]. As compared with conventional passive dampers, MRF dampers have remarkable advantages of high damping force, easy control and high reliability. However, the off–state force of the dampers is to some extent high resulting in unwanted formidable oscillations at high frequency excitations. In addition, sealing up the MRF during operating process is a challenging issue as it gives rise to structural complexity of the dampers. High cost of MRF is also a major impediment for the broad application of such dampers in practice. Another smart material that has been recently investigated to develop damper is shape memory alloy (SMA). SMA has two phases possessing different crystal structures and properties called martensite M (lower temperature) and austenite A (higher temperature). When subjected to particular excitations, SMA experiences a reversible transformation between the two phases to dissipate or absorb energy, which respectively provides applicability for actuating or sensing. In actuating function, SMA can return to its memorized original shape as heated over Af point (austenite finish temperature), generating a useful force. Inspired by this unique characteristic, several scholars have designed SMA dampers for seismic isolation of engineering structures such as Graesser and Cozzareli [7], Clark et al. [8], Wilde et al. [9], Han et al. [10] and Zuo et al. [11]. The satisfactory research results have shown the promise and potential of such SMA damper kind. Aiming at simplicity and compactness for installability in small spaces (e.g., suspension systems of washing machine, vehicle), this paper deals with a new damper featuring SMA spring actuator that can produce a high damping force to suppress structural oscillations at low resonant frequency while keeping off-state force small to lessen the force transmission at high frequencies. The characteristics of the SMA springs are first determined through experimental works to serve as an input for calculation, then a damper prototype is designed, fabricated and evaluated. Hysteresis behavior of the proposed damper is analyzed and simulated by some idealized models with comparisons and discussions.

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2 Configuration of the SMA Damper Figure 1 shows configuration of the proposed SMA damper. As shown in the figure, a sleeve shaft slides along a cylindrical housing. Inside the sleeve shaft, a wedge mechanism including an actuating head and four wedges is used to generate friction force against the housing.

Fig. 1. Schematic configuration of the SMA damper.

When the SMA springs are heated, they extend and push the actuating head to the right. This makes the four wedges move outward in contact with the inner face of the housing. A damping force is produced from the friction between the housing and wedges. Noteworthily, the SMA spring actuator has an inherent disadvantage of low actuating force. In this study, by using a wedge mechanism actuated by an appropriate number of SMA springs, the actuating force from the wedges to the inner face of the housing can be magnified to yield a required damping force. When the SMA springs are cool down, they are contracted to their original shapes and a return spring is employed to push the actuating head back to initial position. The four wedges then move inward and the friction between the wedges and the cylindrical housing is significantly reduced. The damping force and off-state force (the force when the SMA is at ambient temperature of 25 °C) of the SMA damper can be adjusted by using the adjustor.

3 Modeling of the SMA Damper 3.1 Characterization of the SMA Springs Figure 2 shows the experimental setup to specify characteristics of SMA springs. Two heat insulating pads are used to prevent direct contact between the tested SMA spring and adjacent parts. A DC power supply is employed to provide the spring with a high enough current so that the austenite phase transformation occurs to the finish. Data of force and temperature are transfered from a force sensor and a thermo sensor to a computer through an A/D converter.

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Fig. 2. Experimental setup to specify characteristics of SMA springs.

In this work, three specimens of the SMA springs made by SAES® Getters Group (SmartFlex ® SMA spring) are considered for testing. The geometric dimensions of the springs are given as follows: – Spring 1: mean diameter 9 mm, wire diameter 0.8 mm, length 15 mm – Spring 2: mean diameter 6 mm, wire diameter 1.2 mm, length 20 mm – Spring 3: mean diameter 10 mm, wire diameter 2 mm, length 25 mm

(a) spring 1

(b) spring 2

(c) spring 3

Fig. 3. Experimental responses of the three SMA spring specimens.

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The experimental force–temperature–time responses of the three SMA springs are presented in Figs. 3(a–c). As shown in the figures, the actuating force increases with the induced temperature, which is time transient response. At steady state, the maximum actuating forces of the springs 1–3 are respectively about 8.3, 13.8 and 28.1 N. The force saturation of the three springs are almost reached at 20, 26 and 53 s (response time), indicating that the Af points are approximately 50, 60 and 80 °C, respectively. It is noted that the low response of the SMA spring actuators come from both the time for phase transformation of the SMA material and the time transient response of the excitation temperature. From the results, it can be realized that the spring 1 exhibits the fastest response ability but lowest actuating force while the spring 3 is contrary. In regard to damper size, the spring 2 is the best choice. Considering these data, the design process of the SMA damper is next implemented. 3.2 Design of the SMA Damper In this section, the SMA damper is designed based on the equilibrium of forces acting on the damper components when the SMA springs are active, as shown in Fig. 4.

Fig. 4. Equilibrium of forces acting on the actuating head (left) and wedges (right).

As the SMA springs are heated, the actuating force F SMA is generated making the atuating head slide to the right. The sleeve shaft is lubricated on the inner cylindrical face to minimize the friction against the actuating head. By neglecting this friction force, the axial equilibrium equation of the actuating head is given as following FSMA − Fsp − FW sin α = 0

(1)

where F W is the total force caused by the four wedges and α is the cone angle of each wedge. The restoring force of the return spring F sp is obtained by Fsp = ksp = k

w tgα

(2)

where k is the return spring stiffness, Δsp and Δw are respectively the displacements of the return spring and wedges. As the four wedges are pushed outward, the reaction force F N at the contact between the wedges and housing is generated. By neglecting the very small friction between the four wedges and sleeve shaft, the radial equilibrium equation of the four wedges is then obtained by FN − FW cos α = 0

(3)

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The friction force between the four wedges and housing is determined by Ff = μFN

(4)

in which μ is the friction coefficient. This friction force is also the on-state damping force F d of the damper which can be derived by combining the Eqs. (1–3) as follows   w Fd = μ FSMA − k ctgα (5) tgα In this study, the material of the housing and four wedges is commercial C45 steel, the cone angle of each wedge α is 10°, the return spring stiffness k is 5 N/mm and the initial thickness of the gap between the wedges and housing is 0.2 mm. With respect to devices owning small assembly spaces, the desired damping force is about 80 to 100 N. From the view of balance between the actuating force, state changing time and damper size, the SMA spring 2 is chosen for our design. To achieve the desired damping force, two SMA springs should be used, which is able to produce a damping force up to 80.8 N.

4 Experimental Test To verify the proposed SMA damper, a damper prototype is fabricated and its performance is experimentally evaluated. The SMA damper prototype and its components are shown in Fig. 5.

Fig. 5. SMA damper prototype and its decomposed components.

Figure 6 shows the test rig to assess the SMA damper performance. In the system, the linear motion of the shaft is created from the turning motion of a motor through a crank–slider mechanism. The servo motor is controlled by a computer and provides the crank shaft with different constant angular velocities. Once the experiment process is stated, a current of 4 A is applied to the SMA springs. A linear variable differential transformer (LVDT) is employed to record the displacement and a force sensor is used to measure the damping force. From the sensors, the output signals are then sent to the computer via a data acquisition (DAQ).

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Fig. 6. Test rig to assess the SMA damper performance.

Figure 7(a) represents the responses of the SMA damper prototype in damping force-time relation when the motor rotates constantly at an angular velocity of 4π rad/s (2 Hz). This is also the frequency that the resonance usually occurs. From the figure, it is observed that when the SMA damper is not heated (no current is applied to the SMA springs), the off-state force is about 8 N. This is mainly the friction force between the housing and the shoulders of the sleeve shaft. When the SMA springs are electrically powered, the average damping force is 76.5 N at the steady state, which is about 95% of the theoretical one (80.8 N). Thus, a good correlation is achieved. Moreover, the figure indicates that the rising time to change from the off-state force to the steady value of the damping force is around 25 s, which is consistent to the experimentally measured data of the spring 2 (26 s).

(a) force – time

(b) force – displacement

Fig. 7. Experimental step responses of the SMA damper under 2 Hz frequency.

To clearly describe the damping characteristics of the SMA damper at steady state (at which the temperature is around 50 °C), the relation between the damping force and the displacement of the damper shaft over two cycles under the same frequency excitation is obtained, as presented in Fig. 7(b). The hysteresis of the SMA damper, particularly at

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the traveling stroke ends, is clearly shown in the figure. It is noted that there is a sudden change of damping force when the shaft motion changes its direction. This is obvious because the damping force of the proposed SMA damper is generated by the friction between the cylindrical housing and the wedges which is almost Coulomb friction.

(a) 3 Hz

(b) 5 Hz

Fig. 8. Experimental step responses of the SMA damper under higher frequencies.

The same remarks can be derived from Figs. 8(a–b) as testing the damper under higher frequency excitations of 3 and 5 Hz. Minor differences in the off–state force and maximum damping force are admitted as they tend to raise a little with the motor spindle speed, which can be basically explained by the inertial effect of the damper shaft.

5 Hysteresis Model of the SMA Damper In this research, three hysteresis models including Bingham model [12], Bouc–Wen model [13, 14] and Bui’s model [15] are adopted to predict the hysteresis phenomenon of the SMA damper. The three models are in turn mathematically expressed as follows. Bingham: Fd = c˙x + fc sgn(˙x) + f0

(6)

Fd = c˙x + kx + αz

(7a)

z˙ = −γ |˙x|z|z|(n−1) − β x˙ |z|n + A˙x

(7b)

Bouc–Wen:

Bui’s model:

   Fd = c˙x + kx + f0 + D sin C arctan B(1 − E)z + E arctan(Bz) + H arctan7 (Bz) (8a)

Design and Hysteresis Modeling

 z=

x˙ + Sa x for x¨ ≥ 0 x˙ + Sb x for x¨ < 0

133

(8b)

where x is the displacement, x˙ is the velocity, c and k are respectively the damping and stiffness coefficients, f 0 is the constant bias force, z is an independent variable, and f c , α, β, γ , n, A, S a , S b , B, C, D, E, H are the factors characterizing the hysteresis curve shape. Based on the experimental data, the model parameters are estimated using the curve fitting combined with least square method. The objective function OBJ is to minimize the sum of squared errors between the experimental and predicted damping forces, which is represented by OBJ = min

p

Fm.i − Fexp.i

2

(9)

i=1

where F m.i , F exp.i are respectively the ith predicted and experimental forces, and p denotes the number of calculation points. The estimated parameters of the three models for the 2 Hz frequency excitation are listed in Table 1. Table 1. Estimated parameters of the three models for the 2 Hz frequency excitation. Model

Parameter

Bingham model

c = 142.1 N.s/m, f c = 41.5 N, f 0 = 0 N

Bouc–Wen model

c = 67.6 N.s/m, k = 277.6 N/m, α = 1041.5 N/m, β = 0 m–2 , γ = 278.7 m–2 , n = 2, A = 0.88

Bui’s model

c = 74.7 N.s/m, k = 308.2 N/m, S a = 4.63 s−1 , S b = 6.64 s−1 , B = 13.7 s/m, C = 0.93, D = 57.1 N, E = –0.6, H = 2.11

Fig. 9. Comparisons between the predicted and experimentally measured responses under a frequency of 2 Hz.

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Using the above optimized parameters, the damper responses predicted by the three models are obtained and compared with the experimental data. Figure 9 show the comparisons between the experimental and predicted damping forces in relation to displacement under 2 Hz frequency. From the figure, it is observed that the three models well adapt the experimentally measured data. Against the others, the Bingham model cannot thoroughly depict the hysteresis behavior of the SMA damper at the two stroke ends; however, it possesses structural simplicity which is beneficial in cases of requiring quick modeling with relative precision such as design for an expected damping force or initial estimation of damper characteristics. On the contrary, the Bouc–Wen model and Bui’s model can track the damping force better but are more complicated at the same time; they are thus appropriate to strict cases such as feedback or control designs. More evidences can be found in Figs. 10(a–b) for the higher frequency excitations.

(a) 3 Hz

(b) 5 Hz

Fig. 10. Comparisons between the predicted and experimentally measured responses under higher frequencies.

To further assess performance of the three models, errors between the experimentally measured data and each model are quantitatively analyzed. The normalized error of damping force in displacement domain E x is given by

T

2  

Fexp − Fm  dx dt dt

0

(10) Ex = 2    T

Fexp − μexp  dx dt dt 0

where μexp are the average experimental forces over the cycle T. The results under different frequency excitations are shown in Table 2. It can be realized that the Bouc–Wen model and Bui’s model have higher accuracy in capturing the hysteresis phenomenon of the SMA damper, which mainly comes from the effective and close control of the model physical parameters.

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Table 2. Normalized errors between the experimental responses and each model. Model

2 Hz

3 Hz

5 Hz

Bingham model

0.262 0.268 0.261

Bouc–Wen model 0.074 0.086 0.06 Bui’s model

0.03

0.029 0.034

6 Conclusions This research work contributed to a novel SMA damper for structural vibration control featuring SMA spring actuator and wedge mechanism. After a description of the damper configuration, three SMA springs were tested to specify their characteristics. Modeling of the proposed damper was then conducted based on the experimental data and a damper prototype was designed, manufactured and validated. The results showed that the damper behaves almost like a Coulomb friction damper. The average damping force could reach up to 95% of the calculated value while the off–state force is only about 8 N, which is small enough to prevent most force transmissibility at high frequencies. However, with the state changing time of about 25 s, the damper should be further investigated for effective applicability in closed-loop control systems. The proposed SMA damper also exhibited the hysteresis behavior in damping force– displacement relation, especially at the stroke ends. To capture this phenomenon, three idealized models consisting of Bingham model, Bouc–Wen model and Bui’s model were adopted. Although the Bingham model has lowest accuracy, its simple structure facilitates the modeling which is advantageous to design or premilinary characterization. The Bouc–Wen model and Bui’s model control the hysteresis curve more effectively relying on more physical parameters and hence usually prove their abilities in feedback or control designs. In the next stage of this research, many types of SMA springs and various heating methods will be studied to reduce the actuating time of the damper. In addition, the hysteresis with temperature variable will be investigated for dynamic modeling of the SMA damper. Acknowledgement. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.01–2018.335.

References 1. Lagoudas, D.C. (ed.): Shape Memory Alloys, Modeling and Engineering Applications. Springer, New York (2008). https://doi.org/10.1007/978-0-387-47685-8 2. Carlson, J.D.: Low–cost MR fluid sponge devices. J. Intell. Mater. Syst. Struct. 10, 589–594 (1999) 3. Nguyen, Q.H., Nguyen, N.D., Choi, S.B.: Optimal design and performance evaluation of a flow–mode MR damper for front–loaded washing machines. Asia Pac. J. Comput. Eng. 1, 3 (2014)

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4. Nguyen, Q.H., Choi, S.B., Woo, J.K.: Optimal design of magnetorheological fluid–based dampers for front–loaded washing machines. Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci. 228, 294–306 (2014) 5. Bui, D.Q., Hoang, V.L., Le, H.D., et al.: Design and evaluation of a shear–mode MR damper for suspension system of front–loading washing machines. In: Proceedings of the International Conference on Advances in Computational Mechanics, Phu Quoc Island, Vietnam, pp. 1061– 1072 (2017) 6. Bui, Q.D., Nguyen, Q.H., Nguyen, T.T., et al.: Development of a magnetorheological damper with self–powered ability for washing machines. Appl. Sci. 10, 4099 (2020) 7. Graesser, E.J., Cozzarelli, F.A.: Shape memory alloys as new materials for seismic isolation. J. Eng. Mech. 117, 2590–2608 (1991) 8. Clark, P.W., Aiken, I.D., Kelly, J.M., et al.: Experimental and analytical studies of shapememory alloy dampers for structural control. In: Proceedings of SPIE 2445, San Diego, CA, USA, pp. 241–251 (1995) 9. Wilde, K., Gardoni, P., Fujino, Y.: Base isolation system with shape memory alloy device for elevated highway bridges. Eng. Struct. 22, 222–229 (2000) 10. Han, Y.L., Li, Q.S., Li, A.Q., et al.: Structural vibration control by shape memory alloy damper. Earthq. Eng. Struct. Dyn. 32, 483–494 (2003) 11. Zuo, X.B., Li, A.Q., Chen, Q.F.: Design and analysis of a superelastic SMA damper. J. Intell. Mater. Syst. Struct. 19, 631–639 (2007) 12. Stanway, R., Sproston, J.L., Stivens, N.G.: Non-linear modeling of an electrorheological vibration damper. J. Electrost. 20, 167–184 (1987) 13. Bouc, R.: Modele mathematique d’hysteresis. Acustica 24, 16–25 (1971) 14. Wen, Y.K.: Method of random vibration of hysteretic systems. J. Eng. Mech. 102, 249–263 (1976) 15. Bui, Q.D., Nguyen, Q.H., Bai, X.X., et al.: A new hysteresis model for magneto–rheological dampers based on Magic Formula. Proc. Inst. Mech. Eng. C–J. Mech. Eng. Sci. (2020)

A High-Order Time Finite Element Method Applied to Structural Dynamics Problems Thanh Xuan Nguyen1(B) and Long Tuan Tran2 1

National University of Civil Engineering, Hanoi, Vietnam College of Urban Works Construction, Hanoi, Vietnam

2

Abstract. The article proposes a high-order time finite element method based on the well-posed variation formulation that is equivalent to the conventional strong form of governing equations in structural dynamics. Three cases related to the term “high-order” include: the time finite element that is analogous to the spatial second-order beam element; the p-power of the time-to-go (T − t) in the formulation of “stiffness” matrix and “nodal force” vector; and the combination of both of them. In each case, the element “stiffness” matrix and “nodal force” vector are established and shown in details with notes on practical implementations.

Keywords: High-order time finite element Variation formulation

1

· Structural dynamics ·

Introduction

Dynamic response of a structure to an external excitation is of great concern in practical analysis and design. When dealing with a dynamic problem, usually the finite element method or a modal superposition approach is used to spatially discretize the structure, hence to reduce the problem to a set of ordinary differential equations in time that can be solved with one of many time stepping approaches [1,2]. This kind of procedure is widely used in practice and fairly well understood. Generally, for solving a set of ordinary differential equations in time, there are mainly two classes of direct time integration methods: explicit and implicit. Implicit methods (such as ones in β-Newmark family, Houbolt’s method, and Runge-Kutta method) possess unconditional stability, but may require much more computations than that needed for an explicit method. On the other hand, explicit methods are conditionally stable. When coupled with the conventional finite element (FE) computation, the step size in any explicit method depends on the FE spatial mesh size and thus requires more computational effort. For both implicit and explicit methods, a priori error analysis is often not easily available, since they are all derived in the spirit of finite difference. A review of implicit and explicit ones can be found in [3–5], just to name a few. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022  N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 137–148, 2022. https://doi.org/10.1007/978-981-16-3239-6_11

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A different approach for dealing with dynamic problem, being less popular than the methods mentioned above, is to use of the time (or temporal) finite element method (TFEM). The time finite element (TFE) formulation has several potential advantages as it can be applicable to both energy equation, and directly to the equations of motion. It is the straightforward derivation of higher order approximations in time. A key advantage of time finite element method, and the one often overlooked in its past applications, is the ease in which the sensitivity of the transient response with respect to various design parameters can be obtained. Usually, the TFEM approximation yields an accuracy superior to that of more conventional time stepping schemes at same computational cost [6]. Furthermore, the formulation is easy and convenient for computer implementation. The pioneer approaches, based on Hamilton’s Law of varying action, can date back to the research of Argyris and Scharpf, who employed Hermite cubic interpolation polynomials (akin to the beam finite element) to express the response over each time finite element [7]. The method, based on the Hamilton’s principle, was applied to a single-degree-of-freedom system but no numerical examples were considered. Fried [8] applied this approach to study the transient response of a damped system and transient heat conduction in a slab. Fried used a step by step approach to avoid storing and working with large matrices. Zienkiewicz and Parekh [9] used a time finite element approach to solve heat conduction problems. The formulation was based on Galerkin procedure over a time interval. In [10], Hulbert also employed the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. A general convergence theorem can be proved in a norm stronger than the energy norm. French and Peterson [11] proposed a time-continuous finite element method by transforming the second-order differential equations into first-order ones. Some other researchers have presented the variational formulation by allowing the TFE solution to be discontinuous at the end of each time element interval. Tang and Sun [12] introduced a unified TFE framework for the numerical discretization of ordinary differential equations based on TFE methods. In [6], Park used a bi-linear formulation for developing the time finite element method to obtain transient responses of both linear, nonlinear, damped and undamped systems. The sensitivity of the response with respect to various design parameters was also established. Results for both the transient response and its sensitivity to system parameters, when compared to a previously available approach that employs a multi-step method, are excellent. Recently, Wang and Zhong [13] proposed a time continuous Galerkin finite element method for structural dynamics. Its convergence property was proved through an a priori error analysis. To the best of our knowledge, the time finite elements already available until now are just of the first-order kind under the view of time discretization. Thus the authors will propose in this article some developments of high-order time finite elements based on the well-posed variational formulation. Three cases related to the term “high-order” include: the time finite element that is analogous to the spatial second-order beam element; the p-power of the time-to-go (T − t) in the formulation of “stiffness” matrix and “nodal force” vector; and the combination of both of them. The rest of the article is as follows. In Sect. 2, variational

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formulation for structural dynamics is presented, followed by the first-order time finite element in Sect. 3. In Sect. 4, we propose high-order time finite elements in the above mentioned three cases. A numerical example is shown in Sect. 5, to illustrate the use of proposed high-order time finite elements in solving an SDOF dynamic problem. The article finalize with conclusions in Sect. 6.

2

Variational Formulation for Structural Dynamics

Consider a structure having n degrees of freedom in the time interval LT = ]0, T [. The mass matrix M ∈ Rn×n , and stiffness matrix K ∈ Rn×n are, generally, symmetric and positive definite. The damping matrix C ∈ Rn×n is, generally, symmetric non-negative definite. The structure is subjected to initial conditions u0 and u˙ 0 , in addition to an external load F ∈ L2 (LT ), where L2 (LT ) is denoted for the Hilbert space on the time interval LT . The governing equations with the unknown u is u (0) = u0 , u˙ (0) = u˙ 0

L u ≡ M¨ u + Cu˙ + Ku = F,

(1)

Denote H (LT ) as the Sobolev space of order two. We introduce the two spaces as follows   2 (2) (LT ) = u ∈ H 2 (LT ) : u (0) = u0 , u˙ (0) = u˙ 0 H0p   2 2 H00 (LT ) = u ∈ H (LT ) : u (0) = 0, u˙ (0) = 0 (3) 2

Also, an energy norm vE : H 2 (LT ) → R of a vector v is defined as   T 1 T 1 T 2 vE = v˙ Mv˙ + v Kv dt 2 2 0

(4)

The following two theorems were proved in [13]. Theorem 1 (Variational formulation for structural dynamics). The strong form of structural dynamics in the above equation is equivalent to the following for2 (LT ) such that mulation: find u ∈ H0p B (u, v) =  (v) ,

2 ∀v ∈ H00 (LT )

(5)

where  B (u, v) =

T

0

  (v) =

0

T

 

s

v˙ T (M¨ u + Cu˙ + Ku) dt ds =

0 s 0

v˙ T F dt ds =



T

(T − t) v˙ T (M¨ u + Cu˙ + Ku) dt

(6)

0



T

(T − t) v˙ T F dt

(7)

0

Theorem 2 (Solution estimation). The following estimate for solution from the variational formulation uE ≤ C (F0 + |u0 | + |u˙ 0 |)

(8)

holds, where C denotes a positive constant (independent of mesh size or time step size), ·0 is the usual norm of L2 (LT ) and |·| is defined as the usual length of a vector.

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First-Order Time Finite Element

Let the interval [0, T ] be divided into a finite number N of non-overlap sub2 2 intervals each of which has the length of Ti . Denote H0p,τ (LT ) and H00,τ (LT ) as 2 2 the finite dimensional sub-spaces of H0p (LT ) and H00 (LT ), respectively. Usually, 2 2 (LT ) and H00,τ (LT ) are assumed to be of polynomial forms in each eleH0p,τ ment with degree p ≥ 2. Then, the time (Galerkin) finite element formulation is 2 (LT ) established by referring to the variational formulation (5): find uτ ∈ H0p,τ such that 2 ∀vτ ∈ H00,τ (LT ) (9) B (uτ , vτ ) =  (vτ ) , Well-posedness of the time FEM (9) is directly implicated in Theorem 1 and 2. Generally, in structural dynamics, we consider both displacement and velocity responses, and moreover, uτ ∈ H 2 (LT ). Thus, the Hermitian interpolation functions are suitable for establishing the time finite element. In [13], the firstorder time finite element was obtained using Hermitian interpolation of degree p = 3, that is, for the ith element uτ = H1 ub + H2

Ti Ti u˙ b + H3 ue + H4 u˙ e = Hq, 2 2

τ ∈ [0, Ti ]

(10)

where ub , u˙ b and ue , u˙ e are, respectively, nodal (displacement and velocity) responses at the beginning and at the end of the time interval [0, Ti ] of that element. Vector q collects all the nodal responses in element i. The Hermitian interpolation functions - the shape functions - are as usual given as follows 1 (1 − ξ)2 (2 + ξ) 4 1 H3 (ξ) = (1 + ξ)2 (2 − ξ) 4 H1 (ξ) =

1 (1 + ξ)(1 − ξ)2 4 1 H4 (ξ) = − (1 + ξ)2 (1 − ξ) 4 H2 (ξ) =

(11) (12)

where ξ = (−1 + 2τ /Ti ) ∈ [−1, 1]. Then, element stiffness matrix K and nodal element load vector f are obtained as follows  Ti   ˙ T MH ¨ + CH ˙ + KH dt (13) K= (T − t) H 0

 f=

Ti

˙ T F dt (T − t) H

(14)

0

Now, by usual assembly process as seen in conventional finite element method, a set of algebraic equations is established. Also, the initial displacement and velocity are taken into account and entered to the formulation as the boundary conditions of the problem. After these steps, the nodal displacement and velocity at each instant of time can be obtained by solving Ksys qsys = fsys

(15)

It is noteworthy that the global stiffness matrix Ksys is a block diagonal matrix and therefore, Eq. (15) can be efficiently solved by several usual algorithms, including the parallel method [13].

High-Order Time Finite Element Method

4

141

High-Order Time Finite Elements

In this section, we propose three high-order time finite elements. The first one, denoted as 2-order b-TFE, is a direct extension of the first-order time finite element above, and adapted analogously to the second-order beam element with three equidistant nodes (see Fig. 1). Normally, the finite elements of order larger than two are not commonly used in practice since they might destroy the simple philosophy of the finite element method. In stimulating the concept of secondorder beam element in conventional method, the following Hermitian interpolation functions are chosen as   H2 (ξ) = ξ − 6ξ 2 + 13ξ 3 − 12ξ 4 + 4ξ 5 Ti   H4 (ξ) = −8ξ 2 + 32ξ 3 − 40ξ 4 + 16ξ 5 Ti   H6 (ξ) = −ξ 2 + 5ξ 3 − 8ξ 4 + 4ξ 5 Ti

H1 (ξ) = 1 − 23ξ 2 + 66ξ 3 − 68ξ 4 + 24ξ 5 H3 (ξ) = 16ξ − 32ξ + 16ξ 2

3

4

H5 (ξ) = 7ξ 2 − 34ξ 3 + 52ξ 4 − 24ξ 5

(16) (17) (18)

where ξ = τ /Ti ∈ [0, 1], and Ti is the “length” of the ith element. The three nodes in this element might not be equidistantly located. However, under practical view, the use of elements having non-equidistantly located nodes are not common. Also, we could otherwise use hierarchical functions as interpolation ones. However, if the isoparametric element is intended, then the ξ-t relationship is complex enough so that the advantage of using hierarchical polynomials in saving computational efforts might be lost.

With the element nodal responses qT = uTb u˙ Tb uTm u˙ Tm uTe u˙ Te , the response uτ for the ith element can be interpolated from the above Hermitian functions. The formula for the matrix H in Eq. (10) varies case by case, since it depends on the number of degrees of freedom. For example, if the system is single-degree-of-freedom, then matrix H in Eq. (10) is defined as

(19) H = H 1 H2 H3 H4 H5 H6 meanwhile if the system has n degree-of-freedom, then ⎡ ⎢ ⎢ ⎢ H=⎢ ⎢ ⎣

H1

H2

H1 ..





H6

H2 ..

. H1 





···

. H2 

diag. matrix of size n diag. matrix of size n

⎤ H6 ..





. H6 

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(20)

diag. matrix of size n

Now, we follow Eqs. (13) and (14) for all time elements in the temporal mesh. In these equations, the time derivatives of H are given as ˙ = 1 dH , H Ti dξ

2 ¨ = 1 d H H Ti2 dξ 2

(21)

The explicit formulae for the resulting element “stiffness” matrix and element “equivalent force” vector also vary case by case, since they are functions of the

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Fig. 1. (a) Conventional (spatial) three-node beam finite element; and (b) Corresponding three-node time finite element

M, C, and K matrices obtained from spatial discretization as well. We observe here that a large intermediate manipulations of matrices for the element matrices (stiffness matrix and equivalent force vector) should be conducted. However, in practical implementation, we can take the advantage of sparsity in matrix H in these computations. In addition, these computations are prepared just once. For an n-DOF system, the resulting element time “stiffness” matrix is 6n × 6n, and that of “equivalent force” vector is 6n × 1. When all element matrices are already determined, the global stiffness matrix and global equivalent force vector can be found by usual assembling procedure as in conventional finite element method. The second time finite element, denoted as p-TFE, is based on the following observations. As we can see from Theorem 1, Eq. (5) is simply posed to be the ˙ which is a residual L u − F weighted by a time dependent function (T − t) v, ˙ It raises to product of the time-to-go (T − t) and the test on-going velocity v. the idea that another time dependent function can be used as the weight. This idea is also supported by the fact (see proof at the Appendix)  0

T

 

s1

0

 ··· 

sp

0

p folds



 f dt dsp . . . ds1 =

T

p

(T − t) f (t) dt

(22)

0

Thus, Eq. (5) in Theorem 1 holds with the bi-linear form B : H 2 (LT ) × H 2 (LT ) → R and linear functional  : H 2 (LT ) → R expressed as below  B (u, v) =

T

0

0

 = 0



T

s1

 ··· 

p folds

0

sp



v˙ T (M¨ u + Cu˙ + Ku) dt dsp . . . ds1

p (T − t) v˙ T (M¨ u + Cu˙ + Ku) dt

(23)

(24)

High-Order Time Finite Element Method





T

 (v) =



0

0

s1

 ··· 

p folds

0

sp

T



v˙ f dt ds =



T

p (T − t) v˙ T f dt

143

(25)

0

Supported by this, as before, we have the element “stiffness” matrix and the element “equivalent force” vector where the time-to-go (T − t) is raised to the power of p as follows  K=

Ti

  p ˙ T ¨ + CH ˙ + KH dt MH (T − t) H

(26)

p ˙ T (T − t) H F dt

(27)

0

 f=

Ti

0

We see that, in this formulation, the specific types of “shape” functions will give different variants of the element matrices. For the second type of time finite element proposed in this article, the common Hermitian interpolation polynomials of degree 3 are used. For SDOF system n = 1, the element “stiffness” matrix is shown in the Appendices section. When the six “shape” functions in Eqs. (16)–(18) are used, then we have the third type of time finite element, which is a combination of the second-order beam analogous element and the “p-power of the time-to-go” element. This type of element is denoted as bp-TFE for later reference. It is too lengthy to show the formulas of element “stiffness” matrix and “equivalent force” vector here. Instead, MATLAB code to obtain these quantities are given in the Appendices section.

5

Numerical Example

In this section, we will illustrate the use of proposed high-order TFEs in solving an undamped single-degree-of-freedom (SDOF) dynamic problem. Out of the accuracy aspect, we will not explore all other aspects of numerical performance of the proposed method through this very example. For comparison, the problem is also solved by Newmark-β method - the most popular numerical integration method for transient structural dynamics - with three different sets of parameters (γ, β), namely (1/2, 0), (1/2, 1/6), and (1/2, 1/4). The most accurate results obtained by using Newmark-β methods with these sets of parameters are shown in Table 1 and Table 2. The results offered by [13] are also taken into the comparison of accuracy. The statement of the problem is as follows. Given a SDOF system having the mass of m = 1, the stiffness of k = π 2 /4, and there is no damping c = 0. The system is at rest when it is suddenly enforced by a pulse force f (t) given below  1, 0 < t < 1 f (t) = 0, t ≥ 1

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The exact displacement and velocity responses of the system are found to be  4 (1 − cos πt/2) /π 2 , 0 σˆ¯ 3 , ˆ ˆ ˆ of where  the effective stress tensor. It can be easily shown   σ¯ 1 , σ¯ 2 and σ¯ 3 are eigenvalues 2 = 3 q¯ − p¯ and σˆ¯ max = 13 q¯ − p¯ , along the tensile and compressive that σˆ¯ max TM CM meridians, respectively.

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If σˆ¯ max < 0, the corresponding yield conditions are:

Let Kc = q¯ (TM ) /¯q(CM ) for any given value of the hydrostatic pressure p¯ with σˆ¯ max γ +3 < 0, then Kc = 2γ +3 . The fact that K c is constant does not seem to be contradicted by experimental c) evidence [13]. The coefficient γ is, therefore, evaluated as γ = 3(1−K 2Kc −1 . A value Kc = 2/3, which is typical for concrete, gives γ = 3. If σˆ¯ max > 0, the yield conditions along the tensile and compressive meridians reduce to:

Let Kt = q¯ (TM ) /¯q(CM ) for any given value of the hydrostatic pressure p¯ with σˆ¯ max β+3 > 0, then Kt = 2β+3 . σ¯ ) The plastic-damage model assumes non-associated potential flow, ε˙ pl = λ˙ ∂G( ∂ σ¯ . The flow potential G chosen for this model is the Drucker-Prager hyperbolic function as Eq. (9) where ψ is the dilation angle measured in the p-q plane at high confining pressure, σ t0 is the uniaxial tensile stress at failure, and ∈ is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero).  (9) G = (∈ σt0 tan ψ)2 + q¯ 2 − p¯ tan ψ

3.2 Hill Criterion It can be assumed that the anisotropic plasticity of timber progresses according to the Hill yield criterion (Hill R. 1948) [15]. The Hill’s potential function can be expressed in terms of rectangular Cartesian stress components as: f (σ ) =



2 + 2M σ 2 + 2N σ 2 F(σ22 − σ33 )2 + G(σ33 − σ11 )2 + H (σ11 − σ22 )2 + 2Lσ23 31 12

(10)

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Where: F, G, H, L, M and N are constants obtained by tests of the material in different orientations. They are defined as:



 0 2 σ 1 1 1 1 1 1 1 F= + + + 2 − 2 = (11) 2 2 2 2 2 R222 R33 R11 σ22 σ33 σ11



 0 2 σ 1 1 1 1 1 1 1 + + + 2 − 2 G= = (12) 2 2 2 2 2 R233 R11 R22 σ33 σ11 σ22



 0 2 σ 1 1 1 1 1 1 1 + + + 2 − 2 H= = (13) 2 2 R211 R22 R33 σ2 σ2 σ2 11

22



33

τ0

2

3 = σ23 2R223 2 3 τ0 3 M = = 2 σ13 2R213 2 3 τ0 3 N= = 2 σ12 2R212 3 L= 2

(14)

(15)

(16)

Where, each σ¯ ij is the measured yield stress value when σij∞ is applied as the only √ nonzero stress component; σ 0 is the user-defined reference yield stress and τ 0 = σ 0 / 3. The six yield stress ratios such as R11 , R22 , R33 , R12 , R13 and R23 corresponding to σσ110 , σ22 σ33 σ12 σ13 , , , and στ230 , respectively, which have to be defined in the simulation. σ0 σ0 τ0 τ0 The flow rule is described in Eq. (17) as follows, with b from the definition of f above. ⎡ ⎤ −G(σ33 − σ11 ) + H (σ11 − σ22 ) ⎢ F(σ − σ ) − H (σ − σ ) ⎥ 22 33 11 22 ⎥ ⎢ ⎢ ⎥ d λ ∂f −F(σ − σ − σ + G(σ ) ⎢ 22 33 33 11 ) ⎥ (17) = b, with b = ⎢ d εpl = d λ ⎥ ⎢ ⎥ 2N σ12 ∂σ f ⎢ ⎥ ⎣ ⎦ 2M σ31 2Lσ23

4 Modelling of Timber-Concrete Composite Beam 4.1 Verification of the Selected Model The models have been constructed based on the experimental tests, 3-point bending test for the concrete and 4-point bending test for the timber. The Concrete Damaged Plasticity model and the Hill criterion have been considered for the concrete and the timber behavior, respectively. The selected models were previously used by many authors in the literature. The parameters of the selected models are adjusted by many calculations

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Dilation angle ψ (degree)

Eccentricity ∈ (non-unit)

σb0 /σc0 (non-unit)

K (non-unit)

Viscosity parameter (second)

35

0.1

1.16

0.667

0.007985

and assessing the difference between the experiment and the model, namely comparing the Force/Displacement curve from the model with experiment. The parameters of the Concrete Damaged Plasticity model are summarized in Table 4 as below. The evolution of the damage in tension and compression is shown in Fig. 6. The concrete grade in compression and tension is 45 MPa and 5.5 MPa, respectively.

Compression

Tension

Fig. 6. Damaged evolution of the concrete in compression and tension

For the timber, the elasto-plastic anisotropic model with Hill criterion has been applied. The elastic and plastic parameters of timber have been presented in Table 5 as below. The comparison between the experiment and the numerical model has been demonstrated in Fig. 7. It is observed that the numerical curve can meet effectively the experimental curves. A high precision in the elastic term and also in the damaged moment can be achieved. 4.2 Modelling of the Timber-Concrete Composite Beam In the context of timber-concrete composition, a timber-concrete composite T-beam has been considered under finite element analysis. The selected models have been integrated in the global model. The concrete flange and the timber web are connected by steel dowels. The steel dowels are considered to be 2D beam elements. The steel dowel gets the yield strength of 450 MPa at 0.2% of strain. The elastic modulus of the steel dowel is equal to 210 GPa.

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Table 5. Properties of timber model Timber elastic properties

Hill parameters of timber

Longitudinal Young’s Modulus (YM)

E L (MPa)

14788

σ c,90 (MPa)

7

Radial YM

E R (MPa)

1848

F

0.99

Tangent YM

E T (MPa)

1087

G

0.09

Shear Modulus (SM) in LR plane

GLR (MPa)

1260

H

0.01

SM in TL plane

GTL (MPa)

971

N

1.7

SM in RT plane

GRT (MPa)

366

L=M

1.5

RT plane Poisson’s Ratio

ν RT

0.67

LT plane Poisson’s Ratio

ν LT

0.46

LR plane Poisson’s Ratio

ν LR

0.39

Fig. 7. Comparison of the experimental curves with FE model: (a) Concrete; (b) Timber

These beam elements are placed within the 3D elements of the concrete and timber environment. The constraint called “embedded region” available in ABAQUS software has been used in this case of connection. That means the steel dowel is embedded in the concrete and the timber. The contact between the timber and the concrete is “hard” contact that does not allow each other to penetrate. The friction coefficient of the contact is 0.2. On haft of the composite beam is presented in Fig. 8.

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Fig. 8. 3D FE model of one half of the timber-concrete composite beam

Figure 9 presents the numerical results of the model such as the Mises Stress, the damaged status of concrete in tension, the Mises stress of steel dowels. This model may predict effectively the behavior of the composite beam. By the Mises stress distribution, the concrete and the timber are significantly affected in the middle area. But the visual deformation of steel dowels shows the important shear stress in the extreme zone of beam. That can be explained by the important shear force distributed around the supported zones. It is noted that the damage of concrete in tension is observed when the connection is deformed and the concrete is loaded in tension.

Damage of concrete in tension

Deformation of steel dowels

Fig. 9. Results of the numerical composite beam model

In this study, the different contacts between the timber – the steel dowels and the concrete – the steel dowels haven’t been distinguished because of a more complex calculation. Normally, the contact with timber is more flexible than with concrete. Consequently, the concrete is theoretically more loaded in tension. It is recommended to reinforce the steel dowels. It can be concluded that the role of steel dowels is important. In order to optimize the dimension of the steel dowel, a parametric study presented in the next section has been mentioned.

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4.3 Parametric Study: Influence of the Steel Connection Diameter on Shear Resistance In this extended section, the influence of the steel dowel diameter on shear resistance in the context of the composition is focused. Three different diameters of dowel such as 5 mm, 10 mm and 20 mm have been studied. Figure 10 presents the relative sliding between the timber and the concrete in the function of the different diameters of steel dowel. It is noted that the diameter of 5 mm is weak for maintaining the composition. Nevertheless, the diameter of Fig. 10. Relative sliding between timber and concrete in 20 mm presents a high shear the function of dowel diameter resistance. It can be observed that the dowel with a diameter larger than 20 mm may not necessary

5 Conclusions This research presented the experimental results with regard to the mechanical properties of the high strength steel fiber concrete and beech and oak timber subjected to flexion. The experimental results concerned the bending resistance and the elastic modulus. It can be concluded that the component of steel fiber can openly affect the bending resistance of the global concrete beam. With reference to the timber, the beech takes up a high bending resistance that can achieve around 71 MPa. In the context of concrete-timber composite beam, a finite element model has been constructed that allows predicting the behavior of the concrete-timber composite beam. The selected models have been verified by experiment for all timber and concrete. The results presented a high precision of the numerical model. An extended study has been carried out in order to study the influence of the dowel diameter on the shear resistance of the composite beam. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01–2019.11.

References 1. Holschemacher, K., Klotz, S., Weisse, D.: Application of steel fibre reinforced concrete for timber-concrete composite constructions. Leipz. Annu. Civil Eng. Rep. 7, 161–170 (2002)

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2. Holschemacher, K., Klotz, S., Klug, Y., Weisse, D.: Application of steel fibre reinforced concrete for the revaluation of timber floors. In: Proceedings of the Second International Structural Engineering and Construction Conference, Rome, Italy, pp. 1393–1398 (2003) 3. Heiduschke, A., Kasal, B.: Composite cross sections with high performance fibre reinforced concrete and timber. For. Prod. J. 53(10), 74–78 (2003) 4. Kieslich, H., Holschemacher, K.: Composite constructions of timber and high-performance concrete. Adv. Mater. Res. 133–134, 1171–1176 (2010) 5. Le Roy, R., Pham, H.S., Foret, G.: New wood composite bridges. Eur. J. Environ. Civil Eng. 13(9), 1125–1139 (2009) 6. Pham, H.S.: Optimization and fatigue behavior of wood-UHPC connection for new composite bridges. PhD thesis, Ecole Nationale des Ponts et Chaussées (2007). (in French) 7. Schafers, M., Seim, W.: Investigation on bonding between timber and ultra-high performance concrete (UHPC). Constr. Build. Mater. 25, 3078–3088 (2011) 8. Caldova, E., Blesak, L., Wald, F., Kloiber, M., Urushadze, S., Vymlatil, P.: Behaviour of timber and steel fibre reinforced concrete composite construction with screwed connections. Wood Res. 59(4), 639–660 (2014) 9. Auclair, S.C., Sorelli, L., Salenikovich, A.: A new composite connector for timber-concrete composite structures. Constr. Build. Mater. 112, 84–92 (2016) 10. Schanack, F., Ramos, Ó.R., Reyes, J.P., Low, A.A.: Experimental study on the influence of concrete cracking on timber concrete composite beams. Eng. Struct. 84, 362–367 (2015) 11. ISO 13061–1: 2014 Physical and mechanical properties of wood – test method for small clear wood specimens – Part 1: Determination of moisture content for physical and mechanical tests (2014) 12. European standard EN 408: Timber structures. Structural timber and glued laminated timber. Determination of some physical and mechanical properties. European Committee for Standardization (2010) 13. Lubliner, J., Oliver, J., Oller, S., Oñate, E.: A plastic-damage model for concrete. Int. J. Solids Struct. 25 (3): 299–326 (1989) 14. Lee, J., Fenves, G.L.: Plasticdamage model for cyclic loading of concrete structures. J. Eng. Mech. 124(8), 892–900 (1998) 15. Hill, R.: A theory of Yielding and Plastic Flow of Anisotropic Metals, p. 281. Royal Society Publishing, London (1948)

Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions Tran Huu Quoc, Vu Van Tham(B) , and Tran Minh Tu National University of Civil Engineering, Hanoi, Vietnam {quocth,thamvv,tutm}@nuce.edu.vn

Abstract. This paper presents an analytical approach to analyze the free vibration of stiffened cylindrical shells made of porous functionally graded (FGP) materials. The governing equations are derived by using Hamilton’s principle based on the first-order shear deformation theory (FSDT) in conjunction with Lekhnitsky smeared technique. Two types of porosity distributions including symmetric and non-symmetric are considered. Applying the Galerkin method and the axial displacement field functions, the natural frequencies of the cylindrical shell under different boundary conditions are determined. The accuracy of the present formulation is made by comparing the obtained results with those available in published reports. The influences of material parameters such as porosity coefficient, porosity distribution types, and shell geometrical parameters on the natural frequencies are investigated and discussed. Keywords: Cylindrical shell · Free vibration · Functionally graded porous material · FSDT

1 Introduction Functional graded porous materials (FGPMs) are new material, which is invented by the combination of the concept of FGMs and porous materials. In FGPMs, microstructure and porosities are graded along a specific direction according to a defined rule. Due to their excellent properties such as lightweight, heat-resistant property, crackresistant wealth, and energy-absorbing capability, FGPMs have been widely used in many engineering fields. There are numerous studies on the vibrational characteristics of FGP shells [1–10]. Among those, the free vibration response of functionally graded porous cylindrical shells (FGP-CS) is investigated by Wang and Wu [5] using sinusoidal shear deformation theory and the Rayleigh-Ritz method. Survey results indicate that the porosity distribution types, porosity coefficient considerably affect the natural frequency of the FGP-CS. Using the FSDT and Rayleigh-Ritz method, Li et al. [6, 7] predicted natural frequencies of FGP spherical and cylindrical shells subjected to elastic restraints. The free vibration of FGP doubly-curved shallow shell has been examined by Zhao et al. [8, 9] in the framework of FSDT. Li et al. [10] analyzed the vibration of FGP stepped CS in the thermal environment. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 347–361, 2022. https://doi.org/10.1007/978-981-16-3239-6_26

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One of the most effective solutions for increasing stiffness of the structures is to stiffen them with stiffeners. The stiffness of the structure is increased significantly, while the structural weight increases slightly when the stiffeners are appropriately strengthened. For this reason, many studies have been made on the dynamic behaviour of shells stiffened by various materials, e.g. [11–14]. Mustafa and Ali [15] determined natural frequencies of isotropic cylindrical shells stiffened by stringers and rings using the energy method. Lee and Kim [16] presented the optimization of the orthogonally stiffened cylindrical shell under axial compression according to the minimum weight criterion. At a later stage, when the FGMs are applied, free vibration characteristics of FG cylindrical shell continue to gain significant attention of researchers. Tu et al. [17, 18] examined free vibration of the rotating FG-CS in a thermal environment based on classical shell theory and FSDT. Through reviewing the literature and to the best of the authors’ knowledge, there are no studies on the free vibration of stiffened FGP-CS subjected to different boundary conditions. Thus, based on the FSDT and smeared stiffener technique, the free vibration of FGP-CS reinforced by both stringers and rings is studied in this paper. Using the Galerkin method with the axial displacement field of beam functions, the free vibration responses of the stiffened FGP-CS under different boundary conditions are investigated.

2 FGP Cylindrical Shell Considered a circular cylindrical shell (CS) with the reference cylindrical coordinate system (x, θ, z) is placed on the shell’s middle surface, as shown in Fig. 1. The cylindrical shell has length L, uniform thickness h, radius R, and made of FGPMs. The FGP-CS is stiffened by isotropic rings and stringers. The symbols (hr , br ) and (hs , bs ) denote the rings and stringers’ height and width, while ss and sr are the distance between two adjacent rings and stringers, respectively. Two porosity distribution patterns named symmetric (Fig. 1b) and non-symmetric (Fig. 1c) through the thickness direction are considered in this study. The effective material properties of the FGP-CS are defined by [5]: Type I (symmetric distribution): E(z) = E1 [1 − e0 cos(π z/h)], ρ(z) = ρ1 [1 − em cos(π z/h)] Type II (non-symmetric distribution):       1 1 1 1 z z E(z) = E1 1 − e0 cos π + , ρ(z) = ρ1 1 − em cos π + 2 h 2 2 h 2

(1)

(2)

in which, e0 is the porosity coefficient and em is the porosity coefficient for mass density: e0 = 1 −

E0 G0 =1− E1 G1

(0 < e0 < 1), em = 1 −

 ρ0 = 1 − 1 − e0 ρ1

(0 < em < 1)

(3) where (E0 , ρ0 ) and (E1 , ρ1 ) are the minimum and maximum values of effective parameters that occur at the bottom and the top surfaces, respectively.

Free Vibration of Stiffened FGP Cylindrical Shell

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Fig. 1. Configuration of the stiffened FGP-CS: (a) Stiffened cylindrical shell; (b) symmetric porosity distributions; (c) non-symmetric porosity distributions.

3 Theory and Formulation According to the FSDT, the displacements of a point in the FGP-CS are assumed as [19]: u(x, θ, z, t) = u0 (x, θ, t) + zϕx (x, θ, t), θ (x, θ, z, t) = v0 (x, θ, t) + zϕθ (x, θ, t), w(x, θ, z, t) = w0 (x, θ, t)

(4)

where u0 , v0 and w0 are the displacements of a certain point located on the midsurface of the shell and ϕx (x, θ, t), ϕθ (x, θ, t) are the rotations of the normal to the mid-surface of the shell about the θ -and x-axis, respectively. The strain-displacement relation of the shell can be expressed as: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ 0 ⎪ ⎪ ⎪ εxx ⎪ ⎪ κxx ⎪ ⎪ ⎪ ⎪ ⎪ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ εθθ ⎨ κθθ ⎪ ⎬ ⎪ ⎬ ⎨ εθθ ⎪ ⎬ 0 + z κxθ (5) γxθ = γxθ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γθz ⎪ ⎪ γθz ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩γ0 ⎪ ⎩ 0 ⎪ ⎭ ⎪ ⎭ ⎩γ ⎪ ⎭ xz xz where w0 ∂u0 v0 ∂u0 ∂v0 ∂v0 ∂w0 0 0 0 ; εθθ + ; γxθ + ; γθz − ; = = = ϕθ + ∂x R∂θ R ∂x R∂θ R∂θ R ∂ϕ ∂w ∂ϕ ∂ϕ ∂ϕ 0 x θ x θ γxz0 = ϕx + ; κxx = ; κθθ = ; κxθ = + (6) ∂x ∂x R∂θ R∂θ ∂x

0 εxx =

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The constitutive equations of FGP-CS are obtained by using Hooke’s law as follows: ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ εxx ⎪ Q11 Q12 0 0 0 ⎪ σxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε ⎪ ⎪ ⎢Q Q ⎪ ⎪ ⎥⎪ ⎪ ⎨ θθ ⎪ ⎬ ⎢ 12 11 0 0 0 ⎥⎪ ⎬ ⎨ σθθ ⎪ ⎢ ⎥ (7) σxθ = ⎢ 0 0 Q66 0 0 ⎥ γxθ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪σ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 Q44 0 ⎦⎪ γθz ⎪ θz ⎪ ⎪ ⎪ ⎪ ⎩γ ⎪ ⎭ ⎭ ⎩σ ⎪ 0 0 0 0 Q xz

xz

55

where Q11 = Q22 =

E(z) νE(z) E(z) ; Q12 = ; Q44 = Q55 = Q66 = 2 2 1−ν 1−ν 2(1 + ν)

(8)

The stresses resultants of the stiffened FGP-CS are obtained by using the smeared stiffener technique: ⎤⎧ ⎧ ⎫ ⎡¯ ¯ 0 ⎫ A11 A12 0 B¯ 11 B¯ 12 0 ⎪ εxx Nxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ A¯ A¯ ⎪ ⎥⎪ 0 ⎪ ⎪ Nθθ ⎪ εθθ ⎪ ⎪ ⎢ 12 22 0 B¯ 12 B¯ 22 0 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎬ ⎥ 0 ¯ ¯ 0 0 B 0 0 A ⎢ ⎥ Nxθ γxθ 66 66 =⎢¯ ¯ (9) ⎥ ¯ 11 D ¯ 12 0 ⎥⎪ ⎢ B11 B12 0 D ⎪ Mxx ⎪ κxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪κ ⎪ ⎪M ⎪ ⎪ ⎣ B¯ B¯ ⎪ ¯ 12 D ¯ 22 0 ⎦⎪ ⎪ 12 22 0 D ⎪ ⎪ ⎩ θθ ⎪ ⎩ θθ ⎪ ⎭ ⎭ ¯ 66 Mxθ κxθ 0 0 B¯ 66 0 0 D      Qθθ 0 A¯ γθz = ks 44 ¯ (10) Qxx γxz 0 A55 where A¯ 11 = A11 + Ess As s ; A¯ 12 = A12 ; B¯ 11 = B11 + Es Asss zs ; B¯ 12 = B12 ; ¯ 11 = D11 + Es As Is ; D ¯ 12 = D12 ; D ss h/2  A¯ 44 = ks Q44 dz + Gr Ar /sr ; Gr = Gs =

E0 1−2ν

−h/2

A¯ 22 = A22 + ErsrAr ; A¯ 66 = A66 ; B¯ 22 = B22 + Er Asrr zr ; B¯ 66 = B66 ; ¯ 22 = D22 + Er Ar Ir ; D ¯ 66 = D66 ; D A¯ 55 = ks

h/2  −h/2

for the external stiffeners and Gr = Gs =

sr

Q55 dz + Gs As /ss ; E1 1−2ν

for the internal stiffeners

(11) with A11 = A22 = A12 = A66 =

 h/2 −h/2

 h/2

−h/2

 h/2

 h/2  h/2 E(z) E(z) E(z) 2 dz; B = B = zdz; D = D = z dz 11 22 11 22 2 2 2 1 − ν 1 − ν 1 −h/2 −h/2 −h/2 − ν  h/2  h/2 νE(z) νE(z) νE(z) 2 dz; B = zdz; D = z dz; 12 12 2 2 1 − ν2 1 − ν 1 −h/2 −h/2 − ν  h/2  h/2 E(z) E(z) E(z) 2 dz; B66 = zdz; D66 = z dz; 2(1 + ν) 2(1 + ν) 2(1 + ν) −h/2 −h/2

Free Vibration of Stiffened FGP Cylindrical Shell Is =

bs h3s br h3r hs + h hr + h + As zs2 ; Ir = + Ar zr2 ; zs = ± ; zr = ± 12 12 2 2

351

(12)

in which As , Ar and zr ,zs are the cross-sectional areas and distances from the midsurface of the shell to the centroid of each ring and stringer, respectively. k s = 5/6 is the shear correction factor. Using Hamilton’s principle, the equations of motion of the stiffened FGP-CS are obtained as: ∂Nxθ ∂Nxx + = J0 u¨ 0 + J1 ϕ¨x ∂x R∂θ ∂Nθθ Qθθ ∂Nxθ + + = J0 v¨ 0 + J1 ϕ¨θ ∂x R∂θ R ∂Qθθ Nθθ ∂Qxx + − = J0 w ¨0 ∂x R∂θ R ∂Mxθ ∂Mxx + − Qxx = J1 u¨ 0 + J2 ϕ¨x ∂x R∂θ ∂Mθθ ∂Mxθ + − Qθθ = J1 v¨ 0 + J2 ϕ¨θ R∂θ ∂x

(13)

in which, Ji are the moments of inertia:  h/2 ρr Ar ρs As i + Jj = ρ(z)z ¯ dz (i = 0, 1, 2); ρ(z) ¯ = ρ(z) + ss h sr h −h/2

(14)

where ρs = ρr = ρ1 for the outer stiffeners and ρs = ρr = ρ0 for the inner stiffeners. Using Eqs. (5)–(7), and (9)–(10), the equations of motion (13) can be rewritten as χ11 u0 + χ12 v0 + χ13 w0 + χ14 ϕx + χ15 ϕθ χ21 u0 + χ22 v0 + χ23 w0 + χ24 ϕx + χ25 ϕθ χ31 u0 + χ32 v0 + χ33 w0 + χ34 ϕx + χ35 ϕθ χ41 u0 + χ42 v0 + χ43 w0 + χ44 ϕx + χ45 ϕθ χ51 u0 + χ52 v0 + χ53 w0 + χ54 ϕx + χ55 ϕθ

=0 =0 =0 =0 =0

(15)

where differential operators χij (i, j = 1, 2,…,5) are defined in Appendix A.

4 Analytical Solution The Galerkin technique is implemented to solve the system of differential equations to determine the natural frequencies of stiffened FGP-CS under various boundary conditions (BC). The displacement components are chosen as [5, 20]: u0 (x, θ, t) = w0 (x, θ, t) = ϕθ (x, θ, t) =

∞ ∞   m=1 n=1 ∞ ∞  

Umn φ1m (x)φ1n (θ) cos(ωt), v0 (x, θ, t) =

m=1 n=1 ∞ ∞   m=1 n=1

∞ ∞  

Vmn φ2m (x)φ2n (θ) cos(ωt),

m=1 n=1 ∞ ∞  

Wmn φ3m (x)φ3n (θ) cos(ωt), ϕx (x, θ, t) =

m=1 n=1

m (x)φ n (θ) cos(ωt), mn φφx φx

m (x)φ n (θ) cos(ωt). mn φφθ φθ

(16)

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with n n φ1n (θ ) = φ3n (θ ) = φφx (θ ) = cos(nθ ), φ2n (θ ) = φφθ (θ ) = sin(nθ ).

(17)

in which: Umn , Vmn , Wmn , mn , mn are unknowns, m and n are the numbers of half-wave in the longitudinal and circumferential directions, respectively. It is noted that φkm (x) (k = 2, 3) is the function which satisfied boundary conditions:  γ x γ x γ x  γ x  m m m m + H2 cos − μm H3 sinh + H4 sin φkm (x) = H1 cosh L L L L (18) ∂φ m (x)

k The remaining functions φim (x) (i = 1, φx) can be chosen such that φim (x) = ∂x . Coefficients H1 , H2 , H3 , H4 , m and µm in Eq. (16) for three types of boundary condition are listed in Table 1 [5]:

Table 1. Values of H i , γm and μm for several boundary conditions Hi (i = 1, 2, 3, 4)

γm

μm

Clamped–clamped (C–C)

H 1 = 1, H 2 = −1, H 3 = 1, H 4 = −1

(2m+1)π 2

coshγm −cosγm sinhγm −sinγm

Clamped–simply supported (C–S)

H 1 = 1, H 2 = −1, H 3 = 1, H 4 = −1

(4m+1)π 4

coshγm −cosγm sinhγm −sinγm

Simply supported (S–S)

H 1 = 0, H 2 = 0, H 3 = 0, H 4 = −1 mπ

BC

1

Next, substituting Eq. (16) into Eq. (15), and then applying the Galerkin method, one obtained the following equations [20]:  (χ11 u0 + χ12 v0 + χ13 w0 + χ14 ϕx + χ15 ϕθ )u0 dxd θ dt = 0 t θ x (χ21 u0 + χ22 v0 + χ23 w0 + χ24 ϕx + χ25 ϕθ )v0 dxd θ dt = 0 t θ x (χ31 u0 + χ32 v0 + χ33 w0 + χ34 ϕx + χ35 ϕθ )w0 dxd θ dt = 0 (19) t θ x (χ41 u0 + χ42 v0 + χ43 w0 + χ44 ϕx + χ45 ϕθ )ϕx dxd θ dt = 0 t θ x (χ51 u0 + χ52 v0 + χ53 w0 + χ54 ϕx + χ55 ϕθ )ϕθ dxd θ dt = 0 t θ x

Integrating and rearranging terms in Eq. (19), the obtained equations can be expressed in the form:   (20) [K] − ω2 [M ] {q} = {0} in which {q} = {Umn , Vmn , Wmn , mn , mn }T . The natural frequencies of the shell can be obtained by solving the eigenvalue problem of Eq. (20).

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5 Numerical Results and Discussions In the following examples, the geometric parameters of shell and stiffeners and material properties are listed in Table 2. Table 2. Geometrical and structural properties of stiffened shells Model

R1 R2 (stringers) R3 (rings) (unstiffened)

Number of stiffeners

R4 (stringers) R5 (orthogonal rings/stringers)

60

19

04

13/20

Radius (cm)

100

24.2

49.759

19.45

20.3

Thickness (cm)

5

6E-2

0.165

4.64E-2

0.204

Length (cm)

20E3

60.96

39.45

98.68

81.3

Height of stiffeners (cm)

0.702

0.5334

01.01

0.6/0.6

Width of stiffeners (cm)

0.2554

0.3175

0.104

0.4/0.8

E 0 (N/m2 )

6.9 × 1010

6.9E × 1010 2.00 × 1010

2.07 × 1011

v0

0.3

0.3

0.3

0.3

ρ o (kg/m3 )

2714

2762

7770

7430

Stiffeners type

External

External

Internal

Internal

5.1 Comparison Research The accuracy of the proposed model is validated through numerical examples for stiffened isotropic cylindrical shells, FGP cylindrical shells under some different boundary conditions. The present results are compared with the available data in the literature. 5.1.1 Example 1 Firstly, a simply-supported isotropic CS reinforced by various stiffeners (models R2, R3, R4, and R5) is considered. The comparison presented in Table 3 shows a good agreement between the present results and those of Mustafa and Ali [15]. The results also reveal that the cylindrical shell frequencies with stringers decrease. In contrast, those of the cylindrical shell with rings increase as the circumferential wave number increases, and the frequencies of the cylindrical shell with both stringers and rings first decrease and then increase.

354

T. H. Quoc et al. Table 3. The comparison of the natural frequency f (Hz) of the stiffened ICS

Mode number m

f (Hz) n R2 (Stringers) Ref. [15]

1

1 1141

R3

R4

R5

(Rings)

(Stringers)

(Orth. stiffened)

Present Ref. [15]

Present Ref. [15]

Present Ref. [15] Present

1144

1204

1226

778

778

942

936

2

674

673

1587

1585

317

317

439

437

3

427

425

4462

4396

159

159

337

331

4

296

295

8559

8328

100

102

482

477

5

225

223

13780

13227

92

91

740

738

5.1.2 Example 2 The second example is conducted for the unstiffened ICS under different boundary conditions. The shell parameters are set equal to h/R = 1/100  and L/R = 20. Five first non-dimensional circumferential frequencies ω¯ = ωR 1 − ν 2 ρ/E(m = 1) are calculated and compared with those in Ref. [5]. It can be seen that the results of this study match well with those obtained by Wang and Wu based on the sinusoidal shear deformation theory and the Rayleigh-Ritz method (Table 4). Table 4. Comparison of non-dimensional natural frequency for the unstiffened ICS (m = 1, R/h = 100, L/R = 20) n BC S-S [5]

C-C Present [5]

C-S Present [5]

Present

1 0.0161 0.0160 0.0340 0.0340 0.024830 0.024721 2 0.0094 0.0094 0.0119 0.0119 0.008411 0.008379 3 0.0221 0.0221 0.0072 0.0072 0.005897 0.005884 4 0.0421 0.0420 0.0090 0.0090 0.008717 0.008704 5 0.0680 0.0679 0.0137 0.0138 0.013682 0.013663

5.1.3 Example 3 The next validation is made for an un-stiffened cylindrical shell with √ different porosity coefficients. The dimensionless frequency parameter ωˆ = ωR ρ1 /E1 of simplysupported (S-S) FGP-CS with m = 1 and various n are listed in Table 5. The excellent

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agreement between the obtained results by the present approach and Wang and Wu [5] model confirms the accuracy of the current solution. Table 5. Comparison of natural frequencies for S-S unstiffened CS with different PC (m = l, R/h = 500, L/R = 20) e0

Model

n 1

2

3

4

5

6

7

8

0

Wang [5]

1.2429

1.2387

1.2325

1.2256

1.2195

1.2159

1.2165

1.2228

Present

1.2449

1.2407

1.2346

1.2277

1.2217

1.2183

1.2190

1.2256

0.2

Wang [5]

1.2155

1.2118

1.2064

1.2006

1.1958

1.1937

1.1959

1.2038

Present

1.2176

1.2139

1.2086

1.2028

1.1982

1.1962

1.1985

1.2068

0.4

Wang [5]

1.1893

1.1862

1.1818

1.1772

1.1740

1.1736

1.1777

1.1876

Present

1.1914

1.1883

1.1840

1.1796

1.1765

1.1763

1.1806

1.1909

0.6

Wang [5]

1.1677

1.1653

1.1620

1.1590

1.1577

1.1595

1.1660

1.1785

Present

1.1701

1.1677

1.1645

1.1617

1.1605

1.1626

1.1694

1.1823

0.8

Wang [5]

1.1633

1.1617

1.1599

1.1591

1.1601

1.1648

1.1745

1.1906

Present

1.1661

1.1647

1.1630

1.1623

1.1637

1.1689

1.1791

1.1958

5.2 Numerical Investigations In this section, some parametric investigations are conducted to explore the effects of boundary conditions, porosity coefficients, the number of stiffeners, length-to-radius ratio, and thickness-to-radius ratio on the free vibration response of stiffened FGP-CS (R5). 5.2.1 Natural Frequencies of FGP-CS with Several Boundary Conditions (BCs) The effect of BCs on the vibration characteristics of the FGP-CS under three types of BCs is studied here. The boundary conditions including simply supported at both ends (S-S), clamped at both ends (C-C), and the clamped-simply supported at two ends (C-S). Both symmetric and non-symmetric porosity distributions are investigated. In Table 6, the first eight natural frequencies of the orthogonal stiffened FGP-CS under three boundary conditions are calculated and presented. These frequencies are also shown graphically in Fig. 2. The results show that at the low value of n, the influence of boundary conditions is significant. The C-C FGPCS has the highest frequencies, followed by the corresponding C-S and S-S ones. The frequencies decrease when n increases from 0 to 3, and frequencies converge when n > 4. The convergence of the frequencies indicates that the effect of BCs is insignificant for the high circumferential modes of vibration of cylindrical shells.

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Table 6. Natural frequencies f (Hz) of the orthogonal stiffened (external) FGP-CS with for several boundary conditions (e0 = 0.4, m = l, L/R = 0.2, R/h = 100, nr = ns = 10) BCs

n 1

2

3

4

5

6

7

8

Type I S-S

823.399 345.535 268.500 418.564 663.924 971.432 1335.698 1755.440

C-C

1146.592 602.457 400.950 461.161 677.682 976.795 1338.228 1756.872

C-S

1032.318 487.669 331.101 435.650 669.027 973.353 1336.613 1755.990

Type II 1146.560 601.958 397.268 449.946 657.965 947.576 1298.072 1704.232

C-S

1032.302 487.176 326.805 423.873 649.115 944.075 1296.444 1703.356

f (Hz)

823.396 344.981 263.331 406.380 643.917 942.149 1295.551 1702.839

C-C

f (Hz)

S-S

Fig. 2. Natural frequencies of stiffened FGP-CS with S-S, C-C, and C-S boundary conditions

5.2.2 Effect of the Porosity Distribution and Porosity Coefficient The effects of the porosity coefficient e0 and two types of porosity distribution on the natural frequencies of the C-C orthogonal stiffened FGP-CS with different PC (m = l, L/R = 5, h/R = 0.01) are shown in Table 7 and Fig. 3. It is observed that the frequencies of the FPG-CS with symmetric porosity distribution are higher than those of the shell with non-symmetric porosity distribution. The results also indicate that the frequencies of FGP-CS with symmetric porosity distribution decrease at a lower rate than those of FGP-CS with non-symmetric porosity distribution. 5.2.3 Effect of the Stiffener An FGP-CS with various stiffeners is considered in this example. Material properties are set as in the model R5 (m = l, L/R = 5, R/h = 100). Natural frequencies of the unstiffened cylindrical shell and the CS reinforced by ten rings and ten stringers are shown in Fig. 4a for the symmetrical porosity distribution and Fig. 4b for non-symmetrical

Free Vibration of Stiffened FGP Cylindrical Shell

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Table 7. Natural frequencies of the C-C orthogonal stiffened FGP-CS with different PC (m = l, L/R = 5, R/h = 100) e0

n 1

2

3

4

5

6

7

8

Type I 0

1237.125 646.994 417.680 453.253 652.461

936.461 1281.801 1682.552

0.2

1193.241 625.301 409.126 456.098 663.138

953.773 1306.095 1714.584

0.4

1146.592 602.457 400.950 461.161 677.682

976.795 1338.228 1756.872

0.6

1098.186 579.154 394.155 470.284 699.004 1009.763 1383.998 1817.003

0.8

1054.077 558.925 392.245 488.954 735.655 1065.101 1460.416 1917.268

Type II 1237.125 646.994 417.680 453.253 652.461

936.461 1281.801 1682.552

0.2

1193.229 625.101 407.651 451.485 654.953

941.619 1289.381 1692.665

0.4

1146.560 601.958 397.268 449.946 657.965

947.576 1298.072 1704.232

0.6

1098.117 578.175 386.945 448.956 661.878

954.868 1308.613 1718.234

0.8

1053.936 557.084 378.745 450.349 669.150

967.004 1325.827 1741.037

f (Hz)

f (Hz)

0

a)

b)

Fig. 3. Variation of fundamental frequencies versus porosity coefficients

distribution. These figures indicate that the frequencies of the shell increase as the number of stiffener increase. However, the increment is not significant when n < 3. It is getting more significant when n > 4. The effect of the width-to-thickness ratio of stiffeners on the stiffened FGP-CS is calculated and presented in Fig. 5. As expected, the natural frequencies of the FGP-CS increase as the width-to-thickness ratio increase.

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Fig. 4. Effect of the number of stiffener on the natural frequencies of the FGP-CS

Fig. 5. Effect of the geometrical parameter of stiffener on the natural frequencies of the FGP-CS

Fig. 6. Effect of the L/R ratio on the natural frequencies of the FGP-CS

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5.2.4 Effect of L/R Ratio The variation of fundamental frequencies versus length-to-radius ratio L/R is given graphically in Fig. 6a and b for type I and type II of the FGP-CS, respectively. The shell properties are set equal to R = 1m, h/R = 0.02. It is observed that the frequencies of the CS decrease as the L/R ratio increases and strongly decrease for L/R ≤ 5. Furthermore, the figures also show that the frequencies of the un-stiffened type I porous cylindrical shell and the stiffened type II porous cylindrical shell converge when L/R > 5 for different porosity coefficients.

6 Conclusion The present study developed an analytical solution for the free vibration analysis of the stiffened FGP-CS. The governing equation is derived by using the FSDT and smeared stiffener technique. The free vibration responses of the stiffened FGP-CS under different boundary conditions are determined by using the Galerkin method. The comparison results validate the accuracy of the present model. The numerical investigation shows that the porosity configuration, porosity coefficients, the number of the stiffener, the width-to-thickness of stiffener significantly affect the vibration behaviour of the FGP-CS. Furthermore, a few conclusion can be withdrawn as: (1) The boundary condition has a remarkable influence on the vibration response of FGP-CS at a low circumferential number. For all types of boundary conditions, the frequency first decreases and then increases as the circumferential number n increases. (2) The stiffness of the FGP-CS decreases as the L/R ratio increases. As shorter cylindrical shell as the fundamental frequency more sensitive to the L/R ratio. (3) The non-symmetric porosity distribution has a more considerable effect on the free vibration behaviour of the FGP-CS than the symmetric porosity distribution. (4) The stiffened FGP cylindrical shell always vibrate with higher frequency than the un-stiffened FGP cylindrical shell. Thus, by proper stiffener arrangement, the stiffness of the cylindrical shell increases significantly while the weight gain is insignificant.

Acknowledgements. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant No. 107.02-2018.322). The financial support is gratefully acknowledged.

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Appendix A 2 2 A¯ +A¯ ∂2 ; χ = A ¯ 12 ∂ ; χ14 = B¯ 11 ∂ 2 + B¯ 66 ∂ 2 ; χ11 = A¯ 11 ∂ 2 + A¯ 66 2∂ 2 ; χ12 = 12 R 66 ∂x∂θ 13 R∂x ∂x R ∂θ ∂x2 R2 ∂θ 2 ¯B12 +B¯ 66 ∂ 2 χ15 = R ∂x∂θ ;   2 2 A¯ χ21 = χ12 ; χ22 = A¯ 66 ∂ 2 + A¯ 22 2∂ 2 − ks 44 ; χ23 = A¯ 22 + ks A¯ 44 2∂ ; ∂x R ∂θ R2 R ∂θ   ∂2 2 2 A¯ ; χ25 = B¯ 66 ∂ 2 + B¯ 22 2∂ 2 + ks R44 ; χ24 = B¯ 12 + B¯ 66 R∂x∂θ

∂x

R ∂θ

χ31 = −χ13 ; χ32 = −χ23 ;     2 2 k A¯ B¯ 12 ∂ B¯ 22 A¯ ∂ ¯ ¯ χ33 = ks A¯ 55 ∂ 2 + s 244 ∂ 2 − 22 2 ; χ34 = ks A55 − R ∂x ; χ35 = ks A44 − R R∂θ ; ∂x

R

∂θ

R

2 2 2 ¯ 11 ∂ − ks A¯ 55 + D ¯ 66 ∂ ; χ45 = D¯ 12 +D¯ 66 ∂ ; χ41 = χ14 ; χ42 = χ24 ; χ43 = −χ34 ; χ44 = D 2 2 2 R ∂x∂θ

∂x

2

R ∂θ

2

¯ 66 ∂ − ks A¯ 44 + D ¯ 22 ∂ χ51 = χ15 ; χ52 = χ25 ; χ53 = −χ35 ; χ54 = −χ45 ; χ55 = D ∂x2 R2 ∂θ 2

References 1. Tran, T.T., Pham, Q.-H., Nguyen-Thoi, T.: Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method. Defence Technol. (2020) 2. Salehipour, H., Shahsavar, A., Civalek, O.: Free vibration and static deflection analysis of functionally graded and porous micro/nanoshells with clamped and simply supported edges. Compos. Struct. 221, 110842 (2019) 3. Salehipour, H., et al.: Static deflection and free vibration analysis of functionally graded and porous cylindrical micro/nano shells based on the three-dimensional elasticity and modified couple stress theories. Mech. Based Des. Struct. Mach. 2020, 1–22 (2020) 4. Ghadiri, M., SafarPour, H.: Free vibration analysis of size-dependent functionally graded porous cylindrical microshells in thermal environment. J. Therm. Stresses 40(1), 55–71 (2017) 5. Wang, Y., Wu, D.: Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory. Aerosp. Sci. Technol. 66, 83–91 (2017) 6. Li, H., et al.: Free vibration characteristics of functionally graded porous spherical shell with general boundary conditions by using first-order shear deformation theory. Thin-Walled Struct. 144, 106331 (2019) 7. Li, H., et al.: Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos. B Eng. 164, 249–264 (2019) 8. Zhao, J., et al.: A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions. Compos. B Eng. 156, 406–424 (2019) 9. Zhao, J., et al.: Vibration behavior of the functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method. Compos. B Eng. 157, 219–238 (2019) 10. Li, Z., et al.: The thermal vibration characteristics of the functionally graded porous stepped cylindrical shell by using characteristic orthogonal polynomials. Int. J. Mech. Sci. 182, 105779 (2020) 11. Bahmyari, E., et al.: Vibration analysis of thin plates resting on Pasternak foundations by element free Galerkin method. Shock. Vib. 20(2), 309–326 (2013) 12. Damnjanovi´c, E., Marjanovi´c, M., Nefovska-Danilovi´c, M.: Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements. Compos. Struct. 180, 723–740 (2017) 13. Duc, N.D.: Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory. Eur. J. Mech. A/Solids 58, 10–30 (2016)

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14. Qin, X., et al.: Static and dynamic analyses of isogeometric curvilinearly stiffened plates. Appl. Math. Model. 45, 336–364 (2017) 15. Mustafa, B., Ali, R.: An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput. Struct. 32(2), 355–363 (1989) 16. Lee, Y., Kim, D.: A study on the optimization of the stiffened cylindrical shell. Trans. KSME 13(2), 205–212 (1989) 17. Tu, T.M., Loi, N.V.: Vibration analysis of rotating functionally graded cylindrical shells with orthogonal stiffeners. Latin Am. J. Solids Struct. 13(15), 2952–2969 (2016) 18. Tran, M.-T., et al.: Free vibration of stiffened functionally graded circular cylindrical shell resting on Winkler–Pasternak foundation with different boundary conditions under thermal environment. Acta Mechanica (2020) 19. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton (2004) 20. Lam, K., Loy, C.: Influence of boundary conditions for a thin laminated rotating cylindrical shell. Compos. Struct. 41(3–4), 215–228 (1998)

Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector Nguyen Ba Hung1(B) and Ocktaeck Lim2 1 School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi,

Vietnam [email protected] 2 School of Mechanical Engineering, University of Ulsan, Ulsan, Republic of Korea [email protected]

Abstract. The development of fast response injectors is one of the potential research trends, which plays an important role to improving the performance and reducing exhaust emission of the internal combustion engine. A model-based study is conducted to examine the effects of design parameters on electromagnetic force as well as dynamic performance of a solenoid applied for gaseous fuel injectors. Operation of the solenoid is depicted by mathematical models including a mechanical model and an electrical model. A 2D model of the solenoid is created in Maxwell software to calculate electromagnetic force. This 2D model is created in a symmetric shape to reduce the computational cost. Afterwards, the 2D model of solenoid with the electromagnetic force calculated is imported into a Simplorer software to simulate the dynamic performance. The establishment of the solenoid model in Simplorer is based on the mathematical models presented previously. The shapes of plug and sleeve in the solenoid are changed as input variables to examine their effect on the electromagnetic force and dynamic response of the solenoid. The simulation results show that the change of plug and sleeve shapes can improve the electromagnetic force and minimize dynamic response of the solenoid, which can be a useful contribution for designing a high performance solenoid gas injector. Keywords: Solenoid · Gas injector · Electromagnetic force

1 Introduction Solenoid is an electromechanical device, which consists of an electromagnetic subsystem (coil, magnetic parts) and a mechanical subsystem (spring, plunger). When the coil of the solenoid is supplied an input voltage, an electromagnetic force is generated to attract the plunger. Then the plunger will move to the initial position by an elastic force of the spring as the supply of the input voltage is stopped. One of the applications of solenoid is fuel injector [1, 2]. For fuel injection systems, the solenoid plays an important role in controlling the valve opening time of an injector, as well as deciding the engine © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 362–372, 2022. https://doi.org/10.1007/978-981-16-3239-6_27

Effects of Design Parameters on Dynamic Performance

363

combustion performance [3]. The valve opening time of a solenoid injector is decided by the electromagnetic force, which can be improved by changing designs parameters. Yujeong et al. [3] used Ansys Maxwell software combined with a commercial optimization software PIAnO to simulate electromagnetic dynamics and minimize the response time under the effects of dimension parameters of a direct-acting solenoid valve applied for a fuel pump. A study of a high-speed solenoid valve for a common rail injector [4] showed that the electromagnetic force was affected by drive current and the air gap between the armature and iron core. They showed that, the reduced air gap resulted in increasing the electromagnetic force. Besides, they showed that the increment of the electromagnetic force first increased and then decreased as the drive current increased. Hwang et al. [5] examined the influence of input voltage, winding numbers and wire diameters on the response time of a solenoid applied for diesel injector. Their results showed that the winding number of the wire had significant effect on the response time of the solenoid. Yin and Wu [6] established mathematical models to simulate operating characteristics of a solenoid valve used for the fuel injection system of a gas engine. They investigated the influences of voltage, air gap and number of the coil turns on the open and close response time of the solenoid valve. Their simulation results showed that when the number of coil turns increased, the electromagnetic force increased; however, they also indicated that an increase in coil turns number resulted in an increase in the close response time of the solenoid valve which were similar to simulation results shown in [1, 7]. Liu et al. (2014) analyzed the effects of key factors on the electromagnetic force of a high-speed solenoid valve. Cvetkovic and his research group [8] used a modelling approach to investigate the influence of various plunger pole shapes on the operating performance of a solenoid applied for a fuel injector. Their results showed that the initial size of the fuel injector could be reduced by 35%, the attraction force increased by 26% and the response time reduced by 76% by using the developed approach. Besides the studies presented above, some other studies also concentrated on improving the electromagnetic force as well as dynamic response of the solenoid based on the influence of design parameters such as number of coil turns, pole radius, plunger shape, armature radius [9–13]. Among design parameters, dimensions of plug and sleeve in a solenoid are considered key parameters affecting the electromagnetic force as well as dynamic response of the solenoid, which were rarely mentioned in the previous studies. This paper presents a model-based study to improve the electromagnetic force and dynamic response of a solenoid applied for a gaseous fuel injector. The solenoid consists of main components such as a plunger, a plug, a spring, a coil, a bobbin, a casing and a sleeve inserted between the plunger and the coil. Mathematical models are established to describe operation of the solenoid. A two-dimension (2D) symmetric model of the solenoid is built in Maxwell software to simulate the electromagnetic force under the effects of air gap between the plunger and plug. Then, the 2D model of the solenoid is imported into a Simplorer software to simulate dynamic response. The establishment of the solenoid model in Simplorer is based on the mathematical models presented previously. The electromagnetic force and dynamic response of the solenoid are investigated under the effect of changing the dimension parameters of the plug and sleeve.

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2 Simulation Modeling 2.1 Mathematical Modeling A free body diagram of the solenoid is depicted in Fig. 1. When the coil is provided an input voltage, an electromagnetic force is generated to attract the plunger. The plunger attracted by the electromagnetic force moves up and compress the spring. Then, the plunger will be returned to the initial position by the spring elastic force as the supply of input voltage is stopped. Based on the operating principle of solenoid, mathematical models are established to describe the operation of the solenoid, including mechanical and electrical models. In the mechanical model, the motion of plunger is obeyed the Newton’s second law: 

F =m

d 2x dt 2

Fe − Fs − Fd − Fg = m

(1) d 2x dt 2

(2)

where, m is the mass of the plunger, x is the displacement of the plunger, t is the time, F e is the electromagnetic force, F s is the spring force, F d is the damping force and F g is the gravitational force. For the electrical model, the input voltage providing to the coil is defined by: v0 = Ri +

dλ dt

(3)

where vo is the input voltage, R is the resistance in the coil, i is the current, and λ is the total flux linkage of the winding. The λ is also defined by: λ = Nϕ

(4)

where N is the number of coil turns and Φ is the total magnetic flux. By combining Eqs. (3) and (4), the input voltage is rewritten as follow: v0 = Ri + N

dϕ dt

(5)

The λ is also defined as a function of the inductance and the current: λ = L(x)i

(6)

where L(x) is the inductance of the system. Based on Eqs. (3) and (6), the input voltage and current providing to the coil can be derived: v0 = Ri + L(x)

dL(x) dx di +i dt dx dt

(7)

Effects of Design Parameters on Dynamic Performance

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Fig. 1. Free body diagram of the solenoid

  1 di dL(x) dx = v0 − Ri − i dt L(x) dx dt

(8)

The electromagnetic force in Eq. (2) is a variable connecting the mechanical model and the electrical model, which is derived based on the following equations [14]: 

i

Wm (i, x) =

i λ(i, x)di =

0

L(x).idi =

1 2 i .L(x) 2

(9)

0 

Fe =

1 dL(x) ∂Wm (i, x) = i2 . ∂x 2 dx

(10)



where Wm (i, x) is called the co-energy [14], which is defined as a function of the current in the coil and inductance. The inductance of the system is calculated by the following relationship [15]: L(x) =

N2 n  i

(11)

1

where

n 

i is the total reluctance of the system.

1

2.2 Solenoid Model in Maxwell In order to reduce the computational cost, a 2D symmetric model of the solenoid is created in Maxwell software shown in Fig. 2a to simulate the electromagnetic force, which

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uses the initial conditions shown in Table 1. The material properties of the components in the solenoid are described in Table 2, which are also utilized as input parameters for the simulation. The initial conditions and material properties are selected based on specifications of a real solenoid gas injector. A boundary condition is applied on the plunger, assigning it as a moving object by effect of the electromagnetic force. The coil is assigned as an excited element when provided by current to generate the electromagnetic force acting on the plunger. In this paper, the effects of plug and sleeve shapes on the electromagnetic force are investigated, which are considered through the variation of their dimension parameters, as shown in Fig. 2b.

Fig. 2. (a) Model of the solenoid injector in Maxwell and (b) design parameters of plug and sleeve

Table 1. Initial conditions of the solenoids Input voltage, V

12

Resistance of the coil, 

8.5

Coil turns

412

Max. stroke of the plunger, mm 0.2

Effects of Design Parameters on Dynamic Performance

367

Table 2. Material properties of the solenoid Components Material type Coil

Copper

Bobbin

Polyamide 66

Casing

Stainless steel S430

Sleeve

Stainless steel S410

Plug

Stainless steel S416

Plunger

Stainless steel S416

2.3 Solenoid Model in Simplorer The solenoid operates based on mechanical and electrical models as described in Fig. 1, in which the mechanical model depicts the motion of the plunger under the effects of forces applied on it, while the electrical model describes the input voltage and current providing to the coil to create the electromagnetic force acting on the plunger. Based on the mechanical and electrical models, a model of the solenoid is created in Simplorer, in which this model uses initial conditions shown in Table 1 and the simulated 2D model of the solenoid exported from Maxwell software as input parameters to simulate the dynamic response of the solenoid under the effects of design parameters as mentioned in Fig. 2b. Model of the solenoid in Simplorer is shown in Fig. 3

Fig. 3. Model of the solenoid in Simplorer

3 Simulation Results 3.1 Effects of Plug and Sleeve Shapes on Electromagnetic Force The shape of plug is varied by changing the dimension L1 , as illustrated in Fig. 2b. This implies that the shape of sleeve is also changed according to L1 . Besides, the shape of sleeve is also changed in dimension L2 and L3 , as described in Fig. 2b. Effects of these dimension parameters on the electromagnetic force are shown in Fig. 4. The simulation results show that the electromagnetic force increases when the air gap between the

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plunger and plug is reduced. The reduced air gap leads to increasing the magnetic flux density in the plunger, thus it causes the electromagnetic force to increase. When L1 is increased from 0 mm (original version) to 0.25 mm, the electromagnetic force is increased accordingly, as observed in Fig. 4a. This can be explained by the increased magnetic flux density in the plug as the dimension L1 is increased. The increased speed of electromagnetic force is clearly observed when the air gap is reduced from 0.2 mm to 0.02 mm, as shown in Fig. 4a. When L1 is increased to 0.25 mm, the electromagnetic force obtains the highest value, thus it is selected as an input parameter for studying the effects of L2 on the electromagnetic force. Figure 4b depicts the effects of L2 on the electromagnetic force, in which L2 is varied at 0 mm, 0.1 mm and −0.1 mm. It can be seen that the increase of L2 in the positive direction (L2 = 0.1 mm) results in decreasing the electromagnetic force. Conversely, when L2 is reduced in the negative direction (L2 = −0.1 mm), the electromagnetic force is increased due to the increased magnetic field strength in the air gap. Figure 4c shows the effects of L3 on the electromagnetic force, in which L1 = 0.25 mm and L2 = −0.1 mm are selected as input parameters which resulted in the highest electromagnetic force. It can be found that the increase of L3 in the positive direction (L3 = 2 mm) does not significantly affect the electromagnetic force. On the contrary, the electromagnetic force is significantly increased when L3 is reduced to − 3 mm in the negative direction, as observed in Fig. 4c. The magnetic field strength in the air gap increases when L1 , L2 and L3 are selected at 0.25 mm, −0.1 mm and −3 mm, respectively. This can be seen through a comparison of magnetic field strength between the original and developed versions, as shown in Fig. 5.

Fig. 4. Electromagnetic force under the effects of (a) L1 , (b) L2 and (c) L3

3.2 Dynamic Response Under the Effects of Plug and Sleeve Shapes Effects of dimension parameters L1 , L2 and L3 on the electromagnetic force of the solenoid in transient mode are described in Fig. 6. The simulation results show that the increased speed of the electromagnetic force is faster during 2 ms when L1 is increased from 0 mm to 0.25 mm, as shown in Fig. 6a. This is due to the increased electromagnetic force as presented in Fig. 4a. Similarly, the increase of electromagnetic force by increasing L2 described in Figs. 4b causes the electromagnetic force to increase faster

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Fig. 5. Magnetic field strength of (a) original version and (b) developed version of the solenoid

in the transient mode, as observed in Fig. 6b. For the case of changing L3, the variation of electromagnetic force in the transient mode is not significant as L3 is increased from 0 mm to 2 mm in the positive direction. Conversely, the electromagnetic force starts to show an increased trend when L3 is reduced from 0 mm to −2 mm in the negative direction, which is shown by blue dash line in Fig. 6c. Furthermore, by reducing L3 to −3 mm, the increased speed of the electromagnetic force is significantly improved, as shown by green dash line. This is explained by the increased electromagnetic force in Fig. 4c. As the results of increasing the electromagnetic force, the plunger displacement is increased faster in the transient mode, as shown in Fig. 7. The electromagnetic force increases faster within 2 ms, which causes the plunger to quickly moves to the maximum stroke.

Fig. 6. Effects of (a) L1 , (b) L2 and (c) L3 on electromagnetic force in transient mode

Based on the simulation results obtained in Fig. 7c, the response time of the plunger is derived under the effects of L3 . It can be found that the lowest response time (1.15 ms) is obtained at L3 = −3 mm, as can be seen in Fig. 8. By combining with the dimension parameters optimized previously, a developed version of the solenoid is created to compared with the original version, as shown in Fig. 9.

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Fig. 7. Effects of (a) L1 , (b) L2 and (c) L3 on plunger displacement in transient mode

Fig. 8. Effects of L3 on response time of the plunger

Figure 9 shows comparison results in electromagnetic force and plunger displacement of the original and developed versions of the solenoid, in which the initial conditions, such as resistance of the coil, input voltage and number of the coil, are retained at 8.5 , 12 V and 412, respectively. The simulation results show that the electromagnetic force of the developed version is much faster increased than that of the original version. As the result, the plunger displacement in the developed version is increased faster, which implies that the response time of the developed version is significantly improved when compared with the original version.

Effects of Design Parameters on Dynamic Performance

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Fig. 9. Comparison results between original version and developed version of the solenoid

4 Conclusions Operation of a solenoid applied for gas injector was described through the mathematical models. The solenoid was modelled and simulated in Maxwell and Simplorer to calculate the electromagnetic force and dynamic response. Effects of design parameters of plug and sleeve, including dimensions L1 , L2 and L3 , on the electromagnetic force and dynamic response of the solenoid were investigated. The simulation results indicated that the electromagnetic force and response time of the solenoid were strongly affected by L1 and L3 , while the effect of L2 was not significant. The dynamic performance of the developed solenoid was significantly improved when compared with the original solenoid, which was obtained at L1 = 0.25 mm, L2 = −0.1 mm and L3 = −3 mm. This study could be a useful reference for designing high-performance solenoid injectors and other applications related to the solenoid. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2020.17.

References 1. Hung, N.B., Lim, O.T.: A simulation and experimental study on the operating characteristics of a solenoid gas injector. Adv. Mech. Eng. 11(1), 1–14 (2018)

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2. Hung, N.B., Lim, O.T., Yoon, S.: Effects of structural parameters on operating characteristics of a solenoid injector. Energy Procedia 105, 1771–1775 (2017) 3. Yujeong, S., Seunghwan, L., Changhwan, C., Jinho, K.: Shape optimization to minimize the response time of direct-acting solenoid valve. J. Magn. 20(2), 193–200 (2015) 4. Zhao, J., Fan, L., Liu, P., Grekhov, L., Ma, X., Song, E.: Investigation on electromagnetic models of high-speed solenoid valve for common rail injector. Math. Probl. Eng. 2017, 1–10 (2017) 5. Hwang, J.W., Kal, H.J., Park, J.K., Martychenko, A.A., Chae, J.O.: A study on the design and application of optimized solenoid for diesel unit injector. KSME Int. J. 13, 414–420 (1999) 6. Yin, L., Wu, C.: The characteristic analysis of the electromagnetic valve in opening and closing process for the gas injection system. J. Electromagn. Anal. Appl. 8, 152–159 (2016) 7. De, X., Hong, F., Peng, L., Wei, Z., Yun, F.L.: Electromagnetic force on high-speed solenoid valve based on correlation analysis. Int. J. Smart Sens. Intell. Syst. 8, 2267–2285 (2015) 8. Cvetkovic, D., Cosic, I., Subic, A.: Improved performance of the electromagnetic fuel injector solenoid actuator using a modeling approach. Int. J. Appl. Electromagn. Mech. 27, 251–273 (2008) 9. Liu, P., Fan, L., Hayat, Q., Xu, D., Ma, X., Song, E.: Research on key factors and their interaction effects of electromagnetic force of high-speed solenoid valve. Sci. World J. 2014, 1–13 (2014) 10. Yoon, S., Hur, J., Chun, Y., Hyun, D.: Shape optimization of solenoid actuator using finite element method and numerical optimization technique. IEEE Trans. Magn. 33, 4140–4142 (1997) 11. Subic, A., Cvetkovic, D.: Virtual design and development of compact fast-acting fuel injector solenoid actuator. Int. J. Veh. Des. 46, 309–327 (2008) 12. Grekhov, L., Zhao, J., Ma, X.: Fast-response solenoid actuator computational dimulation for engine fuel systems. In: International Conference on Industrial Engineering, Applications and Manufacturing, Russia (2017) 13. Liu, H., Zhao, S., Wang, J.: Research on the influencing factors of dynamic characteristics of electronic injector. Appl. Mech. Mater. 44–47, 260–264 (2011) 14. Woodson, H.H., Melcher, J.R.: Electromechanical Dynamics. Wiley, New York (1968) 15. Giurgiutiu, V., Lysheyski, S.E.: Micromechatronics: Modeling, Analysis, and Design with MATLAB, 2nd edn. CRC Press, Boca Raton (2009)

Practical Method for Tracking Fatigue Damage Nguyen Hai Son(B) School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam [email protected]

Abstract. Estimating and tracking dynamics of fatigue damage is essential for fatigue failure prediction. Many recent developments in the area of damage diagnosis are mainly aimed at detecting damage in variable environmental or operating conditions. These methods do not provide a comprehensive framework that can be used to determine the time evolution in damage. In this paper, a reliable and practical methodology for tracking the evolution of the fatigue damage from the measurement of structural vibration is presented. The proposed approach is independent of any particular model of the system and is only based on the variogram of the measured data in conjunction with the smooth orthogonal decomposition. Validation of the method is demonstrated for both synthetic and experimental data. Keywords: Fatigue dynamics · Variogram · Smooth orthogonal decomposition

1 Introduction Fatigue is an omnipresent problem in all oscillating mechanical and structural systems that can lead to catastrophic injuries or loss of life. Fatigue process is driven by the structural dynamics and environmental factors. However, fatigue itself also affects these dynamics by altering structural parameters. The fundamental problem in developing effective damage predictive model is that it is very hard to measure the needed damage variable due to its hidden nature. Therefore, tracking fatigue damage is one of the critical tasks in the field of structural health monitoring (SHM). In general, SHM techniques can be classified into two classes: model-based [1] and data-based approaches [2]. The model-based methods require a priori knowledge of the system’s properties [3]. On the other hand, data-based (or model-free) approaches can be applied to the measured responses of the system without any prior information [4]. Most of the approaches are based on features that are sensitive enough to the damage, for example, the statistics in time or frequency domain [5–7]. Nonlinear time series methods tend to characterize dynamical systems through long-time invariant quantities such as Lyapunov exponents or fractal dimensions. The application of these metrics in SHM is presented in [8, 9]. Other interesting features that describe a change in the geometric properties of an attractor has been employed to identify damage [10, 11]. Although these quantities are able to detect sudden changes in a system caused by damage, they are ill-suited for continuous damage tracking. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 373–381, 2022. https://doi.org/10.1007/978-981-16-3239-6_28

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In [12], a simple nonlinear feature called characteristic distance (CD) was introduced to characterize changes in slow-time variability in the fast-time flow. A one-to-one relationship between CD-based tracking vectors and actual damage states causing these changes has been demonstrated [12]. However, this procedure requires the reconstruction of the phase space [13], which cannot process in real-time and is sensitive to noise. To address this problem, the metric that is fast to calculate and robust to noise, namely the variogram [14] of the structural responses is introduced as the feature vector. This metric is applicable in situations where fast online conputation is needed. In the next section, the variogram of a signal is introduced. The independence of the evolution of the variogram from a Gaussian white noise is demonstrated. Then the variogram coupled with the smooth orthogonal decomposition are used to track fatigue damage variable in both simulations and an experimental system.

2 Description of the Method In material, a fatigue crack usually grows over thousands of load cycles. Hence, the fatigue damage is viewed as a slow-time state variable evolving in a hierarchical dynamical system [17]: x˙ = f(x, t; φ),

φ˙ = g(φ, x),

and

y = h(x),

(1)

where x ∈ Rn is a dynamical state variable, f : Rn × R → Rn is a flow, over-dot indicates differentiation with respect to time t, and φ ∈ R is a hidden slow-time damage variable, which alters the system’s dynamics. 0 <   1 is a small rate constant defining the time scale separation. A time series y is a measurement of the fast time variable. In practical situations, the functions f and g are not provided, we only have access to the scalar time series of measurements {yi }N i=1 . In the following, the variograms of the observed fast time variable y are utilized to characterize the variations of the slow damage variable φ. 2.1 Variogram A variogram is a function describing the spatial continuity of the data. The variogram of a scalar time series {yi }N i=1 is defined [14] as γ (h) =

2 1  E y(i + h) − y(i) , 2

(2)

where E is the expected value function. By the Birkhoff ergodic theorem [15], the variogram of the scalar time series is an invariant measure of the system described by N Eq. (1). In practice, the measurements {yi }N i=1 is corrupted by noise {μi }i=1 . Assume N {μi }i=1 is a Gaussian white noise. Hence, the variogram becomes 2 1  E (y(i + h) + μ(i + h)) − (y(i) − μ(i)) 2  2 1  1  γ (h) = E y(i + h) − y(i) + E μ(i + h)2 + μ(i)2 − 2μ(i + h)μ(i) 2 2

γ (h) =

Practical Method for Tracking Fatigue Damage

  + E (y(i + h) − y(i))(μ(i + h) − μ(i))

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(3)

Since {μi }N i=1 is a Gaussian white noise, E[μ(i)] = 0, therefore γ (h) =

2 1  E y(i + h) − y(i) + E[μ(i)]2 . 2

(4)

Equation (4) shows that all variograms are shifted by the power of the noise. This component can be eliminated by subtracting the variogram with the mean of all variograms. It is assumed that the state variable φ changes slowly. Thus, φ is considered as approximately constant for a fast-time data set collected over an intermediate time interval. The variogram γ (h) is a function of φ. For tracking purposes, the variograms are calculated for each data recorded over intermediate time interval k and are assembled into a feature vector as: vk = [γ (1), γ (2), . . . γ (m)]. Then the feature vectors are constructed into a tracking matrix Y = [y1 ; y2 ; . . . ; yN ]. This matrix Y embeds the changes in the parameters of the fast-time subsystem and can be projected onto actual damage state. In particular, there is a projection q that yields a damage tracking coordinate ϕ = Yq. It is hypothesized that ϕ is a linear combination of components of φ. In practical situations, q and ϕ are not known. However, they can be estimated from Y by the smooth orthogonal decomposition (SOD) [16, 17] technique. 2.2 Smooth Orthogonal Decomposition SOD can be viewed as a generalization of the proper orthogonal decomposition (POD) [18, 19]. In a time-series reduction context, the problem is stated as decomposition of a multivariate time series arranged into a matrix Y ∈ Rm×n - where n states are measured/recorded at m time instances (m > n) into coordinates and modes: Y = QXT =

n 

qi xiT ,

(5)

i=1

where the modal coordinates {qi }ni=1 ∈ Rm are the columns of Q ∈ Rm×n , and the corresponding modes {xi }ni=1 ∈ Rn are in X ∈ Rn×n . Before the decomposition, the mean is removed from each column of Y to make results unbiased. In POD, the modal matrix is orthogonal and the coordinates are capturing maximal (in the least squares k  qi xiT . It is sense) signal energy in each successive k-dimensional approximation Y ≈ i=1

usually presented as an eigenvalue problem for an auto-covariance matrix Y = 1n YT Y: Y xi = λi xi .

(6)

Since the auto-covariance matrix is symmetric, the resulting eigenvectors are orthonormal. However, if this matrix is ill-conditioned than it is better to solve it by a singular value decomposition: Y = UXT ,

(7)

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where U is an orthogonal matrix and has the same size as Y,  ∈ Rn×n is a diagonal matrix with nonnegative diagonal elements in decreasing order called singular values (square roots of eigenvalues), and Q = U. The square of a singular value is called a proper orthogonal value and corresponds to the variance or energy in the corresponding coordinate. Given the fact that proper orthogonal modes are orthonormal, the model reduction does not require pseudo-inverse of the modal subspace P = [x1 , . . . , xk ]: q˙ = PT f(Pq, t; φ).

(8)

The SOD generalizes the POD by adding a constraint on the variances in the velocity ˙ = DY ∈ Rp×n , where D ∈ Rp×m is a temporal discrete field of the original matrix: Y differential operator. This has an effect of imposing smoothness on the time coordinates q, and the corresponding modes are obtained from the generalized eigenvalue problem ˙ T Y: ˙ of the auto-covariance matrices Y and  ˙ = 1 Y Y

n

Y zi = λi Y˙ zi ,

(9)

where z are smooth projective modes (SPMs), and the corresponding modal matrix Z = [z1 , z2 , . . . , zn ] is no longer orthogonal. The columns of Q = YZ now contain smooth orthogonal coordinates (SOCs), and smooth orthogonal modes (SOMs) are given by columns of Z−T . Alternatively, a generalized singular value decomposition can be used to obtain SOD: Y = UXT , Y˙ = VXT ,

(10)

where U ∈ Rm×n and V ∈ Rp×n are orthogonal,  ∈ Rn×n and  ∈ Rn×n are diagonal matrices, and the generalized singular values diag()/diag() are square root of the smooth orthogonal values (SOVs) λi from Eq. (9). The corresponding SOCs are given by Q = U, and the smooth orthogonal modes are X = Z−T . Larger SOVs correspond to smoother SOCs. Since the slow-time state variable is a product of a smooth deterministic process, these variables are expected to be embedded in the smoothest SOCs.

3 Numerical Simulation In the following, we will present an example with the time series derived from a Rössler equation to validate the proposed damage identification algorithm. In the simulations, a damage process is introduced by slowly changing a parameter in the equations x˙ = −y − z y˙ = x + ay z˙ = b + z(x − c),

(11)

where b is fixed at 0,6; c is fixed at 6,0 and a is varied sinusoidally from 0,1 to 0,4. Here, a is considered a damage variable. For each value of a, 100 000 steady-state is generated with time step 0,06. The bifurcation diagram is shown in Fig. 1. Then 300 variograms

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Fig. 1. Bifurcation diagram for Rössler equation

Fig. 2. (a) Parameter a versus time; (b) plot of the dominant SOC1 corresponding to the largest SOV; (c) plot of SOC1 versus parameter a (blue dots) with least square linear fit (red line)

of time series y are estimated. Finally, the 500 × 300 tracking matrix Y is assembled. The tracking result after applying SOD to this matrix is shown in Fig. 2. In Fig. 2c, the blue dots are almost on the red line which is the least square linear fit of the blue dots. This indicates that there is a strong linear relationship between the smoothest SOC and the parameter a. Now, we consider a two-well Duffing equation x¨ + γ x˙ + αx + βx3 = f cos t,

(12)

where the parameters and forcing frequency are fixed to γ = 0, 25; α = −1; β = 1; = 1. f is used as a slow damage variable, is varied sinusoidally from 0,373 to 0,4 (Fig. 3.a). In this range of f , the response of the system is in the chaotic region.

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In the calculations, the first 50 cycles of data are dropped, and 100,000 steady state points are recorded for each forcing amplitude using a sampling time t = π/100. The tracking result for 700 variograms of time series x is shown in Fig. 3 b and c. Again, we obtain a linear relationship between the smoothest SOC1 and the slowl variable f .

Fig. 3. (a) Parameter f versus time; (b) plot of the dominant SOC1 corresponding to the largest SOV; (c) plot of SOC1 versus parameter f (blue dots) with least square linear fit (red line)

4 Experimental Validation

Fig. 4. (a) Photograph of the system; (b) schematic of the apparatus. 1. Shaker; 2. Granite base; 3. Slip table; 4. Back mass; 5. Specimen supports; 6. Pneumatic cylinder supports; 7. Rails; 8. Front cylinder; 9. Front mass; 10. The specimen; 11; Linear bearings for the masses; 12. Central rail for the masses; 13. Linear bearings for the slip table; 14. Back cylinder; 15. Flexible axial coupling.

The experimental system was first introduced in [20]. A schematic of the test rig is shown in Fig. 4. The specimen is a single-edge notched beam, which is simply supported by pins on each end. The masses are kept in contact with the specimen by using two pneumatic cylinders. The main operating principle of the rig is that inertial forces of masses generated by the electromagnetic shaker provide dynamic loads to the specimen. In this experiment, the shaker is driven by random signals.

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Fig. 5. Tracking results for 3 tests (left); The corresponding fatigue damages (right)

The crack growth is monitored by an Alternating Current Potential Difference (ACPD) crack growth monitor. ACPD is a nondestructive testing technique which utilize the skin effect of current flow on the surface of metallic objects. First, a high-frequency AC voltage (20 kHz) is injected into the beam. Then a voltage (potential difference) across the notch is measured. If the crack grows, the current flowing along the beam’s surface will travel further and the potential difference will be greater. The structural response is measured using a single-axis accelerometer mounted to the front mass. All data from the sensors are low-pass filtered with 400 Hz cut off frequency and are recorded at 1 kHz sampling rate. In what follows, the variogram-based damage

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identification method is applied to the time series of the acceleration of the front mass to identify the hidden damage and to track its evolution. The acceleration time series is split into small records. Each record contains 20 000 points (20 s). 160 variograms are calculated for each record. Then the tracking matrix Y is assembled. The tracking results are shown in Fig. 5 for 3 different tests, where the right figures show the ACPD signals and the left figures show the corresponding tracking results. There is a strong linear relationship between the real fatigue crack and the smoothest SOC as seen in Fig. 6 where all the blue dots are close to the red lines.

Fig. 6. Plots of the dominant SOC1 versus actual fatigue damages (blue dots) with least square linear fit (red line)

5 Discussion Rössler oscillator was used to demonstrate how the variograms can identify and track linearly varying parameters that put the oscillator through complicated bifurcations. In Duffing simulation, the tracking result is not as clean as of Rössler simulation. However, we still have the linear relationship between the smoothest SOC and the actual damage variable. Experimental validation of the proposed method showed that the variograms can be used to track and identify slow damage changes in realistic scenarios (Fig. 5). While other metrics (i.e., characteristic distances [12]) can also track these parameter changes, the new metrics required several orders of magnitude fewer calculation time, making them suitable for online applications.

6 Conclusion The variogram originated from linear geostatistics is used for tracking fatigue damage variable in conjunction with the smooth orthogonal decomposition. We have shown that this feature is an invariant measure of a dynamical system. The main advantage of the variograms was that they were fast to calculate and did not require large data or computational resources, making them suitable for online, real-time applications. Numerical simulations and experiments were used to show how the slow fatigue damage variable can be continuously tracked and identified using the proposed metrics. Acknowledgment. This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-215.

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References 1. Park, K., Reich, G.W.: Model-based health monitoring of structural systems: progress, potential and challenges. In: Proceedings of the 2nd International Workshop on Structural Health Monitoring, pp. 82–95 (1999) 2. Farrar, C.R., Worden, K.: Structural Health Monitoring: A Machine Learning Perspective. Wiley, Hoboken (2012) 3. Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W.: Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Technical report, Los Alamos National Lab., NM, United States (1996) 4. Doebling, S.W., Farrar, C.R., Prime, M.B., et al.: A summary review of vibration-based damage identification methods. Shock Vibr. Dig. 30(2), 91–105 (1998) 5. Luo, G., Osypiw, D., Irle, M.: Real-time condition monitoring by significant and natural frequencies analysis of vibration signal with wavelet filter and autocorrelation enhancement. J. Sound Vib. 236(3), 413–430 (2000) 6. Sampaio, R., Maia, N., Silva, J.: Damage detection using the frequency-response-function curvature method. J. Sound Vib. 226(5), 1029–1042 (1999) 7. Tian, J., Li, Z., Su, X.: Crack detection in beams by wavelet analysis of transient flexural waves. J. Sound Vib. 261(4), 715–727 (2003) 8. Ghafari, S., Golnaraghi, F., Ismail, F.: Effect of localized faults on chaotic vibration of rolling element bearings. Nonlinear Dyn. 53(4), 287–301 (2008) 9. Wang, W., Chen, J., Wu, X., Wu, Z.: The application of some non-linear methods in rotating machinery fault diagnosis. Mech. Syst. Sig. Process. 15(4), 697–705 (2001) 10. Todd, M., Nichols, J., Pecora, L., Virgin, L.: Vibration-based damage assessment utilizing state space geometry changes: local attractor variance ratio. Smart Mater. Struct. 10(5), 1000 (2001) 11. Hively, L.M., Protopopescu, V.A.: Channel-consistent forewarning of epileptic events from scalpeeg. IEEE Trans. Biomed. Eng. 50(5), 584–593 (2003) 12. Nguyen, S.H., Chelidze, D.: New invariant measures to track slow parameter drifts in fast dynamical systems. Nonlinear Dyn. 79(2), 1207–1216 (2014). https://doi.org/10.1007/s11 071-014-1737-y 13. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, vol. 7. Cambridge University Press, Cambridge (2004) 14. Armstrong, M.: Basic linear geostatistics. Springer, Heidelberg (1998). https://doi.org/10. 1007/978-3-642-58727-6 15. Pollicott, M., Yuri, M.: Dynamical Systems and Ergodic Theory, vol. 40. Cambridge University Press, Cambridge (1998) 16. Chelidze, D., Zhou, W.: Smooth orthogonal decomposition-based vibration mode identification. J. Sound Vib. 292(3), 461–473 (2006) 17. Chelidze, D.: Identifying multidimensional damage in a hierarchical dynamical system. Nonlinear Dyn. 37(4), 307–322 (2004) 18. Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000) 19. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(1), 539–575 (1993) 20. Falco, M., Liu, M., Chelidze, D.: A new fatigue testing apparatus model and parameter identification. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 44137, pp. 1007–1012 (2010)

An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel Tang Ha Minh Quan1(B) and Dang Thien Ngon2 1 Faculty of Engineering, Van Lang University, Ho Chi Minh, Vietnam

[email protected]

2 Faculty of Mechanical Engineering, University of Technology and Education, Ho Chi Minh,

Vietnam [email protected]

Abstract. Duplex stainless steel (Duplex) is a new material that has superior properties compared to other stainless steels in terms of avoiding corrosion stress and extremely good yield strength, high flow resistance and cheaper than austenite stainless steel series. However, fatigue strength that always leads to destruction occurs during the working process of axle-shaped machine parts under phase changes. In this paper, the evaluation of fatigue strength of Duplex steel to predict the longevity, maintenance plan for axial-shaped machine parts as well as the advanced mechanical heat treatment mode for Duplex steel are investigated. Samples are made from Duplex SAF 2205 (ISO 1143: 2009) to run fatigue tests by “Staircase” method to determine fatigue strength for Duplex 2205. In addition, this study also provides heat treatment mode for Duplex 2205: heating temperature 950 °C, heat retention time of 15 min and cooling down in the same furnace. The results indicate that supplied-stage Duplex 2205 steel will not be destroyed by fatigue under load under phase changes with 360 MPa stress, which is about 50% higher than AISI 304 (240 MPa) stainless steel. Duplex 2205 steel after heat treatment is enhanced mechanical properties and better durability than Duplex 2205 steel in the supply state. Keywords: Fatigue strength · Duplex stainless steel · DSS 2205 · Duplex heat treatment

1 Introduction Duplex stainless steel is also known as two-phase steel because it has a structure consisting of ferrite phases and austenite phases with approximately equal proportions and intermingling. With such a two-phase construction they will have the best performance combination of the two phases. In the chemical composition of steel contains a high content of chromium (21–23%), it has twice the strength of austenite stainless steel and has a significantly better ductility than ferrite stainless steel. This steel can be mentioned as LDX 2101, SAF 2304, 2205, 253MA [1].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 382–392, 2022. https://doi.org/10.1007/978-981-16-3239-6_29

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The axle-shaped machine parts are commonly used in the mechanical industry, it is often subjected to corrosion due to friction and stressed under load. During the working process, there is little downtime so machine parts may be damaged, especially shaft parts…. causing the situation of stopping production to carry out maintenance, repair, and replacement of machine parts. The replacement of shaft parts is inevitable, the shaft replacement process can be time consuming, costly, and reduces the machine’s performance resulting in low productivity. Therefore, we need to research more new materials such as two-phase steel (Duplex), with higher durability and longevity to put into the manufacture of machine parts, to meet the requirements of high-strength operation of machines, increasing stability and longevity of machines are essential in the development of science and technology. Duplex stainless steel solves the above problems, it has good corrosion resistance, high durability, and cheaper price than other common stainless steels. Stainless steel manufacturing technology often involves forming methods such as rolling, forging, casting, and machining. Materials such as Duplex stainless steel will have a different internal microstructure after forming processes due to the different properties of the two phases. Duplex steel is used in the paper industry, automobile, aviation, and ship industries… [2] to improve the durability of shaft-shaped machine parts that operate at high frequency and intensity, easily damaged and destroy. Cast Duplex Steel was first produced in Finland in 1930 and patented in France in 1936 [3]. Duplex stainless steel is gradually being used in Europe to replace traditional steels according to Japanese JIS standards such as: SUS 201, SUS 304, SUS 316… European countries are always at the forefront of research. and development of this steel. There have been studies on fatigue strength when changing the ferrite phase ratio of Duplex steel [4], the results showed that when changing the composition of the ferrite phase, the fatigue strength of steel is affected. To aid in predicting the fatigue life of a component, fatigue tests are carried out using coupons to measure the rate of crack growth by applying constant amplitude cyclic loading and averaging the measured growth of a crack over thousands of cycles. However, there are also a number of special cases that need to be considered where the rate of crack growth obtained from these tests needs adjustment. Such as: the reduced rate of growth that occurs for small loads near the threshold or after the application of an overload; and the increased rate of crack growth associated with short cracks or after the application of an underload [5]. This paper focuses on the fatigue strength of Duplex 2205 steel. In addition, the heat treatment mode to improve the mechanical properties for this steel is also studied.

2 Theoretical Basis 2.1 Fatigue Fatigue is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. Once a fatigue crack has initiated, it will grow a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack

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exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure. The fatigue limit of a material (Sr ) under a given condition is the maximum value of the time-varying stress corresponding to a number of fundamental stress cycles without the material being destroyed. Each material has its own number of fundamental stress cycles (N0 ) [6]. The fatigue curve is based on the results of fatigue tests, the fatigue curve is designed to show the relationship between the varying stresses (σ) with the corresponding number of stress cycles (N) (Fig. 1). If the stress is reduced to a certain limit σr for some materials, the life of N can be greatly increased without the part of the machine being destroyed. The value σr is called the fatigue strength (long term) limit of the material.

Fig. 1. Wöhler fatigue curve

2.2 Theoretical Basis of Heat Treatment In theory, the metal is heated at a specified temperature, then the temperature is maintained for a certain period. After that, the metal is cooled down at a predetermined speed to achieve the required texture and properties of the metallic material. (Fig. 2) [7].

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Fig. 2. Characteristic parameters of the heat treatment process

3 Experiment and Results 3.1 Experimental Equipment Fatigue strength tests were performed on the machine of the research group of mechanical engineering and environment (REME Lab) of the Ho Chi Minh City University of Technology Education. Fatigue testing machine as shown in Fig. 3 (Table 1).

Fig. 3. Fatigue testing machine

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Value

Engine power (kW)

2

Spindle speed (rpm)

500–10000

Force exerted through the loadcell (N) 20–2000 Sample size (mm)

12–15

3.2 Recommended Sample Parts The standard specifies the part of a rotating bend fatigue test sample of nominal diameter 5–12,5 (mm) and specifies that the machining process does not residual stress concentration on the test sample. The purpose of fabricating a rotating bend fatigue test sample is to determine the fatigue properties of the material represented on an S-N (stress-cycle) curve. Based on ISO 1143: 1975 and actual fatigue testing machine conditions, sample parts of fatigue strength test proposed as shown in Fig. 5 (Fig. 4).

Fig. 4. Rotating bend fatigue test sample

3.3 Fabrication of Sample Parts The test sample is machined from two-phase stainless steel (Duplex), grade 2205. Chemical composition after being analyzed by Spectro Max spectrometer (Germany) at Nam Long Construction Engineering Co., Ltd is shown in Table 2. The parts are machined simultaneously on the Miyano LK-01 CNC lathe (Fig. 6) with the following cutting modes [7]: – Cutting depth: t = 0,5 mm – Feed rate: s = 0,2 mm/r – Cutting speed: Roughing turning v = 1200 rpm, Finishing turning v = 2000 rpm

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Fig. 5. (a) Machined parts on CNC lathes; (b) Check by clock comparison; (c) Check by panme Table 2. Specifications of fatigue resistance testing machine Steel grade Chemical composition (% mass) 2205

Cr

Ni

Mo

Mn

Si

P

S

C

Fe

21.48

5.54 3.57 1.63 0.529 0.0021 0.0053 0.045 66.5

Fig. 6. Test sample in supply status of steel

3.4 Heat Treatment – Metallographic specimen preparation The parts used for metallographic specimen is a cylinder with the size 12 × 20 (mm), this size is suitable for the actual furnace conditions for easy implementation during the inspection. The microscopic organization test sample in the supply state is checked at two location: RD (Rolling Direction) and TD (Transverse Direction) positions. The purpose of this double-sided inspection is to know what the grain structure of the steel in the supply state looks like, thereby determining the appropriate heat treatment regime to uniformize the grain structure to serve the process of running the steel fatigue test in the heat treatment state (Fig. 7).

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Fig. 7. (a) 3D sample design; (b) Metallographic specimen sample

– Etchant for Duplex steel 2205 Duplex 2205 steel has a high corrosion resistance so the use of etchanting solutions must be appropriate. The composition of the etchanting solution used includes distilled water 100 ml, ethanol 100 ml, HCl 100 ml, CuCl2 5 g, the etchanting time is 5 min [8]. – Metallographic specimen result of Duplex steel 2205 in supply state Duplex 2205 steel in the supply state after etchanting (surface corrosion) was examined for microscopic organization on an optical microscope IMS-300. The results were observed in two cross sections (Transverse Direction - TD) and vertical section (Rolling Direction - RD) as shown in Fig. 8.

Fig. 8. (a) Transverse Direction – TD; (b) Rolling Direction - RD

In the supply state in the cross-section, the particles are round and evenly distributed, in the vertical section, the grain structure is elongated, this affects the machining process as well as the steel’s cyclic load capacity. Therefore, processing the stretched grain structure by heat treatment, making the steel grain becomes more spherical so that the steel can be easily machined, improving the mechanical properties and durability of steel. – Metallographic specimen result of Duplex steel 2205 in heat treatment state

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The sample is subjected to heat treatment in many states, but in this study, the author proposes 3 states of heat treatment as follows: State 1: The sample is heated at 500 °C for 6 h, cooled with air and the microscopic structure of Duplex 2205 steel after heat treatment has the shape and structure as shown in Fig. 9.

Fig. 9. (a) Heat treatment diagram; (b) Microstructure in state 1

In state 1, there was a transformation of the steel grain to the shape, but some particles still did not achieve the desired shape. State 2: the sample is heated at 500 °C for 6 h, cooled with the furnace, the microstructure of Duplex steel 2205 after heat treatment in state 2 has the shape and structure as shown in Fig. 10.

Fig. 10. (a) Heat treatment diagram; (b) Microstructure in state 2

In state 2, the majority number of steel grains had a pronounced change in shape and were more evenly arranged than in state 1. But in this state, the degree of steel recrystallization has not been completely. some large seeds. State 3: the sample is heated at 950 °C for 15 min, cooled with the furnace, the microstructure of Duplex steel 2205 after heat treatment in state 3 has the shape and structure as shown in Fig. 11.

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Fig. 11. (a) Heat treatment diagram; (b) Microstructure in state 3

In state 3, all steel grains have been small, with nearly the same shape and size, evenly distributed in the organization of steel. After performing heat treatment in many states, we see that in state 3, the steel will achieve a stable state after restructuring the grain, small steel grains, more evenly distributed to improve the mechanical properties and durable for steel. From that result, we choose heat treatment mode in state 3 to heat treatment for the parts that needs to fatigue test. 3.5 Fatigue Test Process A machine, which is called the four-point bending fatigue testing machine, was designed to examine the fatigue strength of the specimens (Fig. 3). There are four clamps in the machine to keep rotating bending specimen fixed. While the two external clamps force the specimen’s movement vertically, the expected specimen’s bending was created by the vertical sinusoidal of the two inside clamps. Fatigue was tested by a measured force mode, which was calculated by a load cell. The number of cycles to failure was recorded by an encoder near the motor spindle. Fatigue tests were carried out under rotating bending conditions with a stress ratio R of −1 and a loading frequency of 50 Hz. The fatigue strength is determined when specimen fracture occured or the number of cycles reaches 107 . Fracture of specimens is defined as the increase in vibration amplitude above a preset level. Fatigue test process of Duplex steel 2205 are carried out in two states: supply state and heat treatment state. The parts are destroyed (Fig. 12), this is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. From the above results, we plot the fatigue curve according to the collected data. The fatigue curve is a representation of the relationship between stress and cycle shown in Fig. 13. The fatigue curve of the sample (Fig. 13) shows the cyclic load capacity of Duplex 2205 steel in the supply state and heat treatment state. In the supply state, when at the stress of 360 MPa the parts is not destroyed, reaching the value 107 . Thereby, we find that the Duplex 2205 steel in the supply state has about 50% higher strength than the other common stainless steels (AISI steels 107 at a stress level of 240 MPa). In the state

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Fig. 12. Parts are destroyed

Fig. 13. Wöhler fatigue curve

of heat treatment, the parts reach 107 with a stress level of 400 Mpa, so it has a higher strength than Duplex 2205 steel in the supply state about 11%.

4 Conclusions The results achieved in this study: Proposing sample parts for fatigue testing in accordance with ISO 1143: 1975 and proposed parts of tensile test specimens made in accordance with ISO 6892-1: 2009. The study has performed a mechanical review of Duplex steel 2205 in the heat treatment mode as follows: temperature 950 °C, time 15 min and cooling with the furnace.

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Determining the fatigue strength of shaft parts made from Duplex steel 2205 in the supply state with a value of 360 MPa, the parts reach 107 cycles, about 50% higher than the common stainless steels. Determining the fatigue strength of the shaft parts made from Duplex steel 2205 in the state of heat treatment, with a value of 400 MPa, the parts reach the cycle of 107 . The results show that, through heat treatment will make increased mechanical strength and fatigue strength by about 11% for Duplex 2205 steel, which is also worth considering in the part design process. Acknowledgements. This study is supported by the Research Group of Mechanical and Environmental Engineering (REME Lab) of Ho Chi Minh City University of Technology and Education in Viet Nam.

References 1. Ta´nski, T., Brytan, Z., Labisz, K.: Fatigue behaviour of sintered duplex stainless steel. Procedia Eng. 74, 421–428 (2014). https://doi.org/10.1016/j.proeng.2014.06.293 2. Propelling the boating world. International Molybdenum Association, The voice of the molybdenum industry 3. Dr. Sunil, D.K.: Duplex stainless steels-an overview. Int. J. Eng. Res. Appl. 07(04), 27–36 (2017). https://doi.org/10.9790/9622-0704042736 4. Johansson, J.: Residual Stresses and Fatigue in a Duplex Stainless Steel. Linkoping Studies in Science and Technology, Sweden (1999) 5. Suresh, S.: Fatigue of Materials. Cambridge University Press. ISBN 978–0–521–57046–6 6. Van Quyet, N.: Fatigue Theory Basis. Vietnam Education Publising House Limited Company, Vietnam (2000) 7. Dang, V.N.: Fatigue Theory Basis. Vietnam National University Press, Ho Chi Minh City 8. Co-ordinating working group “Classification Societies – Diesel”. Guidance for evaluation of Fatigue Tests. Conseil International Machines a Combustion (2009)

Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed Nguyen Trong Du(B) and Nguyen Phong Dien School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam [email protected]

Abstract. Effective signal processing methods are essential for machinery fault diagnosis. Most conventional signal processing methods lack adaptability, thus being unable to extract meaningful diagnostic information. Signal decomposition methods have excellent adaptability and high flexibility in describing arbitrary complicated signals. They can extract rich characteristic details and revealing the underlying physical nature. In recent years, several signal decomposition methods are applied to rotating machines operating steady rotating speed. However, it is challenging to use these methods in the case of variable rotation speed. This study proposes a novel signal analysis procedure that combines several advanced signal decomposition methods such as order tracking, spectrum analysis, and Hilbert transform to eliminate the influence of speed changing in vibration signals. It is convenient to create different feature vectors from the measured vibration signal. A classification algorithm is finally applied to detect gear faults automatically. The experimental test rig in the case of rotating machinery having cracked gear demonstrates the usefulness of the proposed signal processing procedure. Keywords: Signal decomposition · Fault diagnosis · EMD · Wavelet packet · Neural networks

1 Introduction The fault diagnosis of rotating machines has attracted increasing interest in the last decades due to their ubiquity and importance in the industry. Recent research trends focus on developing a diagnostic system that works automatically with the application of artificial intelligence. According to this, the diagnostic procedure usually involves the following steps: – Diagnostic data collection, – Diagnostic feature selection to determine condition indicators, – Evaluating the instant values of these indicators from measured diagnostic signals, corresponding to different technical conditions of the diagnostic object © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 393–400, 2022. https://doi.org/10.1007/978-981-16-3239-6_30

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– Implementation of an algorithm for pattern recognition and classification, or a classifier with supervised learning such as artificial neural networks (ANN) [1] and support vector machine (SVM) [2]. Reliable diagnostic systems often require a large enough set of diagnostic data to ensure accurate results. However, a low-cost and applicable diagnostic system should require only a limited amount of diagnostic data while ensuring acceptable diagnostic accuracy. The problem here is to find solutions that reduce the amount of diagnostic data needed for the system. It is believed that the diagnostic feature selection using signal processing methods is very promising to solve the problem. There are many methods for selecting a diagnostic feature, and the signal decomposition method is one of the most appreciated ways. However, most of these methods only give diagnostic features in steady operation condition of the rotating machine. This study evaluates existing signal decomposition methods and proposes a novel method for selecting diagnostic features that can apply for variable rotating speed. This novel method is called advanced signal decomposition. These selected parameters constitute a set of input data for training and implementing a classification algorithm to classify fault in the rotating machines in non-stationary conditions.

2 Advanced Signal Decomposition Methods The raw vibration signals are usually decomposed into different parts using various decomposition methods to extract the diagnostic features. The results of these methods allow a more in-depth insight into the signal components. Thus, it quickly realizes diagnostic symptoms, enhances damage detection, and selects features closely related to different types of damages when failures occur in the considered machine. Signal decomposition methods proposed for fault diagnosis can be divided into three groups: The conventional technique from the first group have been well verified and already in industrial applications such as spectral analysis, time-synchronous averaging, cepstrum analysis, etc. Other methods based on time-frequency transforms are proposed, published, and used for individual tasks, but not in common use, such as wavelet analysis. The last group includes approaches and methods that are relatively new, complicated, and not tested in large scale projects. Besides, machines often operate under fluctuating load conditions and varying rotational speeds (non-stationary conditions). The conventional analysis methods suitable for analyzing stationary vibrations cannot be well applied in such situations. A simple but effective decomposition method is the time-synchronous averaging (TSA) [3–5]. TSA has become a standard technique to detect gear faults in the time domain for many years. Using TSA, a vibration signal measured at a rotating machine can be divided into a periodic part and a non-periodic part, including the random noise components. However, TSA is just appropriate to rotational varying slowly, and it just seems a preprocessing step to reduce measured noise [6]. Wavelet packet decomposition (WPD) is a generalization of the wavelet transform that is firstly introduced by Coifman [7, 8]. In this method, the original signal is decomposed into several signals containing one approximation and some details. It divides the

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time-frequency space into various parts and allows better time-frequency localization of signals versus the wavelet transform. The generalized synchrosqueezing transform (GST) [9, 10] has been developed to overcome the limitation of WPD in analyzing signals containing close frequencies. However, a sophisticated signal processing tool such as GST requires a complex numeric algorithm with a higher computation cost. Cepstrum transform [11] is the inverse Fourier transform of the logarithm of the Fourier transform of the signal. It is readily decomposed into a primary source and noise source. The disadvantages of cepstrum analysis are several Fast Fourier transforms (FFTs) and Inverse Fast Fourier transform (iFFTs are used on each window, which can be computationally expensive [12]. Also, cepstrum analysis essentially low-pass filters the spectrum to get the spectral envelope, which potentially averages some of the spectrum’s spectral peaks. For fault diagnostics, this can be undesirable [13]. Empirical mode decomposition (EMD) is also called Huang trans-form, which can effectively separate frequency components of the signal in the form of intrinsic mode function (IMF) [2, 14] from high frequency to low frequency. And the original signal can be reconstructed by decomposed components. This method, which has been researched widely, is very suitable for nonlinear and non-stationary signal processing. The above methods are all effective with steady rotating speeds. However, there are still many limitations in the case of the variable rotation speed. Therefore, the novel analysis procedure is proposed as follows: Step 1. Taking computed order tracking of the raw vibration signal using a reference signal and a standard interpolation algorithm. After this step signal is called a resampled signal in domain angle to eliminate the effect of various rotating speed. Step 2. Calculation of IMFs of the resampled signal based on the EMD method. Step 3. Using the Hilbert spectrum to look for gear fault symptoms. If there are any symptoms, go to the next step. If not, go back to the previous step. Step 4. Calculation standard deviation of IMFs to generate feature vectors as inputs for neural networks to classify gear fault.

3 Experimental Results This example demonstrates the effects of the proposed method in a gearbox operating at variable rotating speeds. Before selecting diagnostic features, the vibration signal is needed preprocessing by many different techniques. Firstly, resampling the original signal with equal angle increment to eliminate the affection of variable rotating speed. Secondly, the resampled signal is analyzed by the EMD method to generate IMFs mode with the different frequency ranges. Finally, each IMFs signal’s standard deviation is chosen to create a feature vector that is an input parameter for an Artificial neural network. The experimental setup consists of a single-stage gearbox presented in the article [13]; as shown in Fig. 1, this gearbox has a gear ratio of 14/40. The input shaft is driven by a motor controlled by an AC inverter to change the rotating speed from 500 RPM to 1000 RPM. There is a tooth crack on the pinion of the gearbox. The vibration signal is measured by an accelerometer positioned at the closest position to the test gear. An

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Crack Tooth

Fig. 1. Gearbox test rig and pinion with a crack tooth

optical tachometer mounted in proximity to the input shaft is used to generate a onepulse-per-revolution signal called a tacho signal. The vibration signals were sampled at 10 kHz. The following equation determines the meshing frequency: GMF = 14 × f shaft (GMF is gear meshing frequency, and F shaft is input shaft frequency) [15]. Applying conventional methods such as spectral analysis (FFT) to measured vibration signal from this gearbox is given in Fig. 2a. It’s seen that the frequency spectrum is significantly obscured by spectral smearing. Thus, this is a significant problem with conventional analysis methods, and it has less effective with the vibration signal is sampled in the time domain under variable rotating speed conditions.

Fig. 2. The spectrum of the original signal (a) and order spectrum of the resampled signal in the angle domain (b)

The angular resampling technique is applied to the original vibration signal, then FFT analysis for the resampled signal to achieve the order spectrum as Fig. 2b. The order

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spectrum is dominated by the gear meshing frequency and its harmonics (shown 14, 28, 42 order). In the second step, the EMD algorithm is applied to decompose vibration signals into modes. Figure 3 displays the empirical mode decomposition in nine IMFs of the angle domain resampled signal from IMF1 to IMF9 represents the frequency components excited by the gear crack fault, IMF10 is the residual, respectively. Mode IMF1 contains the highest signal frequencies, mode IMF2 the next higher frequency band, and so on.

Fig. 3. IMFs of the resample signal with gear crack fault

The Hilbert-Huang transform is then applied to each of the above IMFs. The Hilbert spectrum is shown in Fig. 4. Many transient vibrations are repetitive with a 2π rad interval, corresponding signal vibration of the cracked gear. This symptom only appears when the signal is resampled in the angle domain. After finding symptoms of cracked gear as mentioned above, the last step of the proposed method, taking the standard deviation of each IMF mode, constitutes feature vectors. Those features are the input data sets for the classification of faults based on the neural network. Classification results between healthy gear and cracked tooth gear using selected features based on the proposed method, as in Fig. 5b. In which the correct classification and testing results are the same 100%. This result is a better classification of fault without it, as in Fig. 5a only with 83% correct classification and 92.3% correct testing. This good result has eliminated the effect of variable rotating speed by order tracking and using EMD to decompose resampled signal to achieve diagnostic features that sensitively responded with gear crack fault.

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2π (1 revolution)

Total revolutions Fig. 4. HHT spectrum of the resampled signal with gear crack fault

Fig. 5. Result of classification based on selected feature vectors by proposed method (b) and without the proposed method (a)

4 Conclusion The application of advanced and sophisticated signal decomposition tools such as the time-frequency analysis using GST or EMD allows us to get more information about the gear fault condition from vibration signals. Also, these decomposition methods

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are applicable for fault detection of machines operating under non-stationary conditions. However, these methods require a more complex numeric algorithm with a higher computation cost. This study proposes a novel signal analysis procedure that combines many signal decomposition methods such as order tracking, spectrum analysis, and Hilbert transform to eliminate the influence of speed changing in vibration signals. It is convenient to create different feature vectors from the measured vibration signal. A classification algorithm is finally applied to detect gear faults automatically. The gear fault classification results in a gearbox operating at a variable rotating speed are also presented using a gearbox test rig. The continued investigation on the application of these decomposition methods is therefore highly recommended. Acknowledgment. This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-214.

References 1. Rafiee, J., Arvani, F., Harifi, A., Sadeghi, M.H.: Intelligent condition monitoring of a gearbox using artificial neural network. Mech. Syst. Signal Process. 21, 1746–1754 (2007) 2. Dien, N.P., Du, N.T.: On a diagnostic procedure to automatically classify gear faults using the vibration signal decomposition and support vector machine. In: Fujita, H., Nguyen, D.C., Vu, N.P., Banh, T.L., Puta, H.H. (eds.) ICERA 2018. LNNS, vol. 63, pp. 425–432. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04792-4_55 3. Bechhoefer, E., Kingsley, M.A.: A review of time synchronous average algorithms. Paper presented at the Annual Conference of the Prognostics and Health Management Society, San Diego, CA (2009) 4. Combet, F., Gelman, L.: An automated methodology for performing time synchronous averaging of a gearbox signal without speed sensor. Mech. Syst. Signal Process. 21, 2590–2606 (2007) 5. Braun, S.: The synchronous (time domain) average revisited. Mech. Syst. Signal Process. 25(4), 1087–1102 (2011) 6. Zhang, G., Isom, J.: Gearbox vibration source separation by integration of time synchronous averaged signals. Paper presented at the Proceedings of the annual conference of the prognostics and health management society, Montreal, QC, Canada 25–29 Sept 2011 7. Coifman, R.R.: Signal Processing and Compression with Wavelet Packets (1990) 8. Feng, Y., Schlindwein, F.S.: Normalized wavelet packets quantifiers for condition monitoring. Mech. Syst. Signal Process. 23, 712–723 (2009) 9. Li, C., Liang, M.: A generalized synchrosqueezing transform for enhancing signal time– frequency representation. Mech. Syst. Signal Process. 92, 2264–2274 (2012) 10. Oberlin, T., Meignen, S., Perrier, V.: The fourier-based synchrosqueezing transform. Paper presented at the 2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP), France (2014) 11. Randall, R.B.: A history of cepstrum analysis and its application to mechanical problems. Mech. Syst. Signal Process. 97, 3–19 (2016) 12. Hao, T., Xiao-yong, K., Yong-jian, L., Jun-nuo, Z.: Fault diagnosis of gear wearing based on order cepstrum analysis. Appl. Mech. Mater. 543–547, 922–925 (2014)

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13. Dien, N.P., Du, N.T.: Fault detection for rotating machines in non-stationary operations using order tracking and cepstrum. In: Sattler, K.-U., Nguyen, D.C., Vu, N.P., Tien Long, B., Puta, H. (eds.) ICERA 2019. LNNS, vol. 104, pp. 349–356. Springer, Cham (2020). https://doi. org/10.1007/978-3-030-37497-6_41 14. Cheng, J., Yu, D., Tang, J., Yang, Y.: Application of frequency family separation method based upon EMD and local Hilbert energy spectrum method to gear fault diagnosis. Paper presented at the Mechanism and Machine Theory vol. 43 (2007) 15. Meltzer, G., Dien, N.P.: Fault diagnosis in gears operating under non-stationary rotational speed using polar wavelet amplitude maps. Mech. Syst. Signal Process. 18(5), 985–992 (2004)

Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading N. Van Xuan1(B) , N. Canh Thai2 , N. Ngoc Thang2 , and T. Van Toan2 1 Directorate of Water Resources, 2 Ngoc Ha, Ba Dinh, Hanoi, Vietnam 2 Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam

{ncanhthai,nnthang}@tlu.edu.vn, [email protected]

Abstract. In recent years, the pillar dam is a built efficient water regulation facility in Vietnam. For the pillar dam using the radial gate, all the load and the impact from the radial gate to the pier trunnion are concentrated in the area of the mandrel wheel. During the operation and operation, the pier trunnion area has many potential dangers if not calculated properly. For the types of specialized structures, as the structure just stated, simulation work is most tested and is the basis for the numerical model [1–3]. In experiments, depending on each stage of loading, stress-strain values have differences compared with theoretical calculations [4–6]. In the scope of the paper, the authors will analyze the effect of the choice of concrete grades on the material nonlinearity in this area, the effect of the dynamic load with the fluctuation amplitude of the wave cycle, thereby determining the changes in stress versus strain allowed of the material on the basic cross-section. Keywords: Monotonic loading · Cyclic loading · Nonlinear material · Dynamic · Pillar dam

1 Introduction The impact of the circulating load has a great influence on the waterworks in general and the Pillar dam in particular. Due to the influence of wave vibrations, the pressure creates stress-strain relationships that change at the position of the load. With a Pillar dam using the Radial gate, all wave pressure is applied to the valve and transmitted to the pier trunnion. The change in the amplitude of the wave oscillation, characterized by the Jonswap wave spectrum, produces a cyclic load, affecting the strength at this site. With the cylindrical structure of the pier trunnion of the radial gate, the design and operation process has many potential risks if not paid adequate attention. The 1986 Dau Tieng hydroelectric pier trunnion accident has left many harmful consequences as proof [7]. Designing the structure of the reinforced concrete cellar pillar structure following TCVN 10400:2015, TCVN 8218:2009 standards often use the material to work on the linear elastic phase, not considering the nonlinear stress-strain relationship the actual material [8, 9].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 401–415, 2022. https://doi.org/10.1007/978-981-16-3239-6_31

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Some constructions in the 50s of the 20th century, such as the Folsom dam, USA, had a breakdown in 1995, due to the error of not considering the effect of friction on the stress distribution in the ear region, pin valve, which has a nonlinear effect on geometry and material, leads to poor design stress distribution, bending stresses in the compression member area of the valve gate system [10]. In the world, the problem of structural analysis in the nonlinear behavior has been studied and included in the standards in the form of additional formulas for determining internal force, such as EN 1992 1-1 of the European Union, JSCE No15 of Japan, ACI 318-95 of the United States,… but just stop at some basic forms [4, 11, 12, 13, 14]. Therefore, empirical research is very necessary. The effect of the cyclic oscillation of the wave creates a varying pressure on the radial gate. Depending on the geographical region, the Jonswap wave spectrum has different values, with different cycles and amplitudes. In Vietnam, wave fluctuations are usually measured by either the pressure method or the resistance method, see Fig. 1 and 2. The wave model in the study applies to the pillar dam. Pillar dams are built a lot in the estuary area, in addition to keeping fresh water, it also has a very important task to limit storm surge, rising water,… Therefore, the influence of waves, especially waves in the bays and estuaries, is very important.

Fig. 1. Wave spectrum at KimSon estuary, Ninh Binh, Vietnam

Fig. 2. Wave spectrum at Danang bay, Vietnam

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There are two solutions to design the trunnion area of the Pillar dam using the radial gate. The first design option is to select the reinforcement of the circular arch, the trunnion girder is located deep in the supporting pillar body. This is a commonly used solution in Vietnam today, as the newly built Van Phong dam also uses this solution (Fig. 3). The second design option is to select the reinforcement placed parallel, the trunnion girder is usually located on the outer edge of the concrete pillar body (Fig. 4).

r ei n f o r c emen t st eel

t r u n n io n g i r d er

pu t r ei n f o r c ed st eel i n c i r c u l a r sh a pe

a) Common steel laying

b) VanPhong Dam, BinhDinh, Vietnam

Fig. 3. Typical design in Vietnam

t r u n n i o n g i r d er

Ten d o n Gr o u ps

pu t r ei n f o r c i n g st eel i n pa r a l l el

a) Common steel laying

b) John Redmond Dam, Texas, US

Fig. 4. The trunnion is at the boundary of pier (EM 1110-2-2702, 2000)

In this paper, the effects of the trunnion girder design according to the first option are studied and discussed. The compressive stress of material is considered to be the main property suitable for concrete materials. Conducting studies of reinforced concrete structures under different load conditions and repeated cycles is often done by experimental methods. Feedback results are analyzed, compared to draw problems to overcome. At the same time, the results are also used as a basis for the correction of numerical models. For each type of specialized structure, a lot of research has been done, Mirmiran and Shahawy studied the behavior of specimens subjected to the static effects of compressive loads, and in terms of the load loop cycle – unloading [15]. Chen et al., studied the behavior of the reinforced concrete structure’s different joints [16]. Experimental results pointed out that the connective joints have the ability to efficiently dissipate energy. Tong et al., have studied the behavior of H-shaped reinforced concrete steel beams [17]. Chen et al., have studied with many samples to give different solutions for the combination of steel and concrete.

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Recently, Lam Thanh Quang Khai in his thesis, also analyzed the steel-fiber reinforced concrete dome structure [18]. Experimental research and analysis on numerical model linear and nonlinear values in static load. In this study, the experiment was performed on a cylindrical-dam body model using a bow valve. The results are evaluated based on analyzing the relationship of linear and nonlinear load-displacement, stress-strain. The test also considers that the difference in the load cycle affects the stress-strain distribution.

2 Experimental Investigation 2.1 Material Materials used are concrete M200, M300, M400 designed according to the natural conditions in Vietnam. Material quality meets TCVN 8219: 2009, ASTM C39. Concrete using PC40 cement. The composition of materials for 1 m3 of concrete is shown in Table 1. A comparison between grades of concrete according to Vietnamese and EU standards is shown in Table 2. Table 1. Material distribution norms for 1 m3 of concrete No.

Material

Unit

Concrete grade 200

300

400

1

Cement PC 40

kg

296

394

470

2

Yellow sand

m3

0.489

0.447

0.427

3

Crushed stone (0.5 × 1)cm

m3

0.888

0.87

0.86

4

Water

litre

195

195

186

5

Chemical additives

Chemical

Table 2. Converted concrete grade according to EU standards Durability level EC-2

Compressive strength level

Concrete grade

B15

M200

90

B20

M250

110

B22.5

M300

130

C20/25

B25

M350

155

150

C25/30

B30

M400

170

180

C16/20

Computational strength

Computational strength EC-2 120

Concrete was cast with 6 samples and maintained at its design strength after 28 days of age. The stress-deformation relationship of concrete is shown in Fig. 5. The physical and mechanical properties of the tested concrete sample are given in Table 3. Experimental cube samples and destructive material samples are illustrated in Fig. 6.

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Fig. 5. Stress - strain curve of the concrete sample to be tested

Fig. 6. Damage model of the experimental concrete

Table 3. Properties of the materials used in the specimens Concrete grade

Rn kN/m2

Rk kN/m2

E kN/m2

1c

1cu

f cm,tt kN/m2

f cm, LT kN/m2

200

9120

740

2.41E + 07

0.0018

0.0035

17000

20000

300

13680

1030

2.91E + 08

0.0019

0.0035

21500

30000

400

17630

1210

3.28E + 07

0.0022

0.0035

25500

40000

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2.2 Test Specimens The experimental Specimens are designed in a steel frame system, performed at the Structural mechanics and strength of materials Laboratory in Thuyloi University, Vietnam. In collaboration with The National Key Laboratory of River and Coastal Engineering (KLORCE), the address in the University area. Test and measurement equipment: using strain gauge, conductor, bonding glue, electronic displacement LVDT, load cell, data logger, of TML (Tokyo Measuring Instruments Laboratory Co., Ltd), Japan. Experimental models are performed according to the instructions of national standards of Vietnam [19–21]. The frame system used this test is designed with full capacity of up to 400 kN by the hydraulic jack. Loads can be applied in terms of force/time or displacement/time. Figure 7 shows the steel frame and test specimen in the laboratory. The design of the position of the steel frame and the load cell, hydraulic jack is shown in Fig. 8.

Fig. 7. Framework of the tests run in the laboratory

The research model is a hydraulic construction model built on the basis of the crosssection of the pier dam using the arch gate. The purpose of the study is load interaction and the impact on the structure. The wave load is converted to the basic form for analysis as the cyclic loading acts directly on the mandrel wheel. Designing cross-section model with scale of 1:35, similar to main cylindrical model of Van Phong dam, Phuoc Hoa dam built. The selection of the test Pillar dam size is based on a representative basic cross-section dimension of a Pillar dam using a radial gate. The goal is to analyze the difference in stress values when calculated analytically, linearly, and the actual results are nonlinear, experimentally. The load exerted at the valve ear position, simulating the working of the gate valve to beat the pillar. The designs comply with TCVN 10400: 2015, EM 1110-2-2702. The steel frame is designed to load the specimen in the direction of the effect of the valve gate pressure on the ear of the battery cylinder. The loading is performed in increments of increments and is carried out in different repetition cycles. Figure 10 shows the attachment of the sensors to the surface of a strain gauge test piece under load.

Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading

Fig. 8. Loading framework

407

Fig. 9. Loading - unloading step

Fig. 10. Paste the Strain Gauge sensor

Measure displacement by electronic device CDP-25 (accuracy: 0.002 × 10–6 mm), mechanical displacement meter (Indicator, accuracy: 0.001 mm); Deformation measurement with resistance sensor (Strain Gauge, Rosette paste); The devices are connected to the data logger and the computer records the load parameters in each step. 2.3 Test Set-Up The experimental model is built based on a sample segment of a miniature pillar dam with the size 800 × 600 × 80 mm, linking the hard clamp to the ground, and bearing concentrated load in the direction of transmission from the Radial arch valve. In the test of the equal load on both sides of the valve, the stages are performed in sequence as follows, see Fig. 9: A. The load increases gradually from 0–5 kN. Check the operation of load systems, sensors, gauges. Eliminate peripheral effects such as tilting,… B. Unloading gradually decreases to 0. Return to the initial state. C. The load increases gradually from 0–8 kN. Get experimental data, check calculation results. D. Unloading gradually decreases to 0. Return to the initial state. E. The load increases gradually to 10 kN, the load increases gradually until the sample fails. Monitor residual deformation, check destructive phenomena

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In the 1st load test procedure, the cyclic loading is simple. Exact determine the value of stress and strain corresponding to each load level, from there are experimental data compared with the linear calculation results. Specimens were tested under different kinds of cyclic loading to investigate the hysteretic behavior and constitutive characteristics. The cyclic load forms are simulating some basic wave phases, usually occurring in the waters of Vietnam. In Fig. 11, the load period is to simulate the wave action acting on the radial gate of the Pillar dam.

a ) load cycle type 1

b ) load cycle type 2

Fig. 11. Cyclic load patterns

3 Compare the Experimental Results with the Results Obtained from the Numerical Model 3.1 3D FEM Element Modeling The purpose of this paper is actual experimental results, numerical results are for reference only. Linear values can be calculated using algebraic formulas, by mechanics of continuous medium [22]. So these details are a brief introduction. The paper is done based on analysis by the finite element method (FEM), with Abaqus software. The model uses Concrete Damaged Plasticity (CDP). Destruction mechanism of the model that are based on tensile cracking and compression crushing. In the CDP model, the damage law is based on the formula of Lubliner and Lee/Fenves, which is the basis for the Abaqus plastic destruction model for concrete [23, 24]. The mentioned yield surface revised by Lee and Fenves (1998) to take into account the varying tensile and compressive strength of concrete. The present yield surface is determined with equation:         ∼pl ∼pl  ∼pl 1 (1) q − 3αp + β ε σmax − γ −σmax − σc εc F σ, ε = 1−α 



Where p is the effective hydrostatic pressure stress and q is the Mises equivalent effective stress. These two parameters are stress invariants of the effective stress tensor which are used by the yield surface and plastic flow potential function [23], σmax is the algebraically 

∼pl

maximum eigenvalue of the deviatory part of effective stress tensor (σ ), ε is equivalent plastic strains. The parameters α, β and γ are dimensionless material constants.

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The model is made with dimensions similar to the actual dam design. Concrete uses C3D8R element, which is a 3-dimensional, 8-node solid element. Steel uses T3D2 element, embedded in concrete with the assumption that adhesion between reinforcement and concrete is absolute. The displacement of the x, y, z directions of the bottom is zero. The input parameters and the stress measurement point are performed similarly to the experimental model. The element size is 4 mm at the main Pillar dam area and 1 mm around the pier trunnion area. 3.2 Visco-plastic Regularization on the CDP The process of adjusting viscosity-plasticity to the CDP model parameters is done through the continuous correction of finite elements that are closely observed in reality. The viscosity parameter was adjusted according to the behavior of the concrete material, and refer to Lee [25]. Viscosity parameters are also adjusted for non-linearity for each grade of concrete, see Wosatko et al., Weiren et al., [26, 27]. 3.3 Isotropic Hardening Isotropic hardening describes the variation of the size of yield surface σ0 , which is a function of equivalent plastic strain εp , and is shown in the following equation: σ0 = σ|0 + Q∝ (1 − e−biso ε ) p

(2)

where σ |0 is the yield stress at zero plastic strain and Q∞ and b are material parameters. Q∞ is the maximum change in the size of the yield surface, and b defines the rate at which the size of the yield surface changes as plastic straining develops. When the equivalent stress defining the size of the yield surface remains constant (σ 0 = σ |0 ), the model reduces to a nonlinear kinematic hardening model.

4 Results and Discussion 4.1 Failure Characteristics In the test on both sides of the valve, the phases are performed sequentially according to the load levels, gradually increasing until the sample is damaged. In all cases, cracks appear oblique as shown in Fig. 12. Alternating are small cracks, which begin to open when the load exceeds 6 kN. This phenomenon is relatively consistent with the process of pulling and breaking of the material. The area around the valve ear separates the tension and the compression zone, the process of deformation increases gradually, appears cracks. The forcedisplacement relationship in the load test 1 at procedure is illustrated in Fig. 13 and 14. In terms of linear behavior analysis, solving the problems can be done by analytic formulas in material strength, and continuous environmental mechanics. However, in practice, the linear relationship between stress and strain accounts for only a fraction of the entire working time of the material (linear elastic phase). In the nonlinear analysis

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Fig. 12. Failure patterns of the specimens and FEM result

problem, the dangerous stress value is different from the linear analysis problem, and this value reflects more accurately the actual work of the project. The force-displacement relationship under cyclic loads is depicted in Fig. 15. The peak point of the graph in the helices is similar to that of the graph under normal loads. With the wave spectrum tending to grow, the graph is broken into many short, continuous foot beats. With an equal wave spectrum, simulating transverse waves, the plot is broken into multiple long legs and tends to be relatively more evenly interrupte.

Fig. 13. Force-displacement relation at point I

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Fig. 14. Force-displacement relation at point II

Fig. 15. Compare the relationship between cyclic load types

4.2 Measurement Data Table 4, 5 and 6 are the results of experimental measurements, with comparison of calculated results from numerical models, where: – Phase 1: Elastic deformation phase – Phase 2: Stress starts to cause cracking – TT: linear result; Pt: numerical result, nonlinearity; TN: experimental results

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Phase 1 - σ Principal σ TT

σ Pt

σ TN

kN

kN/m2

kN/m2

kN/m2

%

I

1.602

3.881

4.051

4.08

4.88

II

2.628

70.508

67.493

66.8

−5.55

III

9.007

365.408

351.336

348.6

−4.82

IV

9.984

263.286

258.75

258.12

−2

Measuring point

Load

Phase 2 - σ Principal σ TT

σ Pt

σ TN

kN

kN/m2

kN/m2

kN/m2

%

I

2.045

4.955

6.294

6.2

20.08

II

2.935

78.745

65.913

64.7

−21.71

Measuring point

III

Does not form phase 2

IV

Does not form phase 2

Different rates

Different rates

Table 5. Measurement results ε, U Measuring point

Load

Phase 1 - ε εTT

Different rates εPt

εTN

kN

%

I

1.6

4.3762E-06

4.374E-06

0.00000459

4.66

II

2.63

5.0118E-06

4.64E-06

0.00000452

−10.88

Measuring point

Load

Phase 2 - ε εTT

Different rates εPt

εTN

kN I

2.05

% 5.607E-06

5.595E-06

0.00000588

4.64

4.53E-06

0.00000442

−26.75

II

2.94

5.6025E-06

Measuring point

Load

Phase 1 - U U TT

Different rates U Pt

U TN

kN

mm

mm

mm

%

I

1.6

0.00205763

0.002058

0.00216

4.74

II

2.63

0.00267112

0.0026545

0.00252

−5.997

Measuring point

Load

Phase 2 - U U Pt

U TN

U TT

Different rates

kN

mm

mm

mm

%

I

2.05

0.002636

0.0026401

0.00277

4.83

II

2.94

0.00298597

0.0029298

0.00273

−9.38

Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading

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Table 6. Measurement results at point I with grades of concrete Load

Phase 1 - σ Principal σ TT

σ Pt

σ TN

kN

kN/m2

kN/m2

kN/m2

%

M200

2.5

6.057

7.52

7.896

23.29

M300

2.5

6.057

7.106

7.461

18.82

M400

2.5

6.057

6.793

7.133

15.08

Concrete grade

Different rates

5 Conclusions When the concrete works in the elastic stage, the difference between the linearly calculated value and the experiment is not large, ≤6%, depending on the position near or far from the applied point. When the concrete works in the next stage, and when cracking begins to appear, there is a big change in the geometrical size of the cylinder, the difference between linear and experimental values is ~22%. This result is similar to the study of L.T.Q.Khai when studying the roof of concrete cover [21, 28–31]. The higher the grade of concrete, the lower the difference between the linear and experimental values. The cracks begin to crack at the end of the transition, and gradually expand to complete failure as the load continues. This stage of work while maintaining this partial crack is relatively common in old buildings that need upgrading today. The article has pointed out the need for experimentation, noting nonlinear analysis for new structural design, and upgrading and repairing old works effectively.

References 1. Xuan, N.V., Thai, N.C., Thang, N.N.: Anh huong cua mac be tong den tinh phi tuyen vat - tru do. - Hoi nghi Khoa hoc Thuy loi toan quoc; Ha noi, pp. 16–18 (2017). lieu tai tai van dap . ISBN 978-604-82-2273-4. (N.V. Xuan, N.C. Thai, N.N. Thang. Influence of concrete grade on material nonlinearity at the pier trunnion of Pillar dam. National irrigation science conference; Ha noi, pp. 16–18 (2017). ISBN 978-604-82-2273-4) 2. Sakai, J., Kawashima, K.: Unloading and reloading stress–strain model for confined concrete. J. Struct. Eng. 132(1), 112–122 (2006) 3. MacGregor, J.G., Wight, J.K.: Reinforced Concrete: Mechanics. Prentice-Hall, Hoboken (2005) 4. Ferguson, H.A., Blokland, P., et al.: The Haringvliet sluices. The Hague: RIJKSWATERSTAAT Communications (1970) 5. US Army Corps of Engineers. EM 1110-2-1604 Hydraulic Design of Navigation Locks. Washington, DC (2006) 6. The U.S Army Corps of Engineers. Ohio River & Tributaries Navigation System Five Year Development Plan. FY11-FY15 (2010)

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7. Phan, S.K.: Su co mot so cong trinh thuy loi tai Viet nam va cac bien phap phong tranh. Ha noi: Nha xuat ban nong nghiep; 2000 Phan Sy Ky. Breakdown of some water works in Vietnam and preventive measures. Agricultural Publishing Company, Hanoi (2000) 8. TCVN 10400:2015 Cong trinh thuy lo.,i-Dap tru do-Yeu cau ve thiet ke. Ha Noi: Bo Khoa ho.c va Cong nghe; 2015 (TCVN 10400:2015 Hydraulic Structures–Pillar Dam–Technical requirement for Design. Hanoi: Ministry of science and technology; 2015) 9. TCVN8218:2009. Tieu chuan quoc gia. Be tong thuy cong. Yeu cau ky thuat. Ha Noi: Bo Khoa ho.c va Cong nghe; 2009 (TCVN8218:2009. Hydraulic concrete–Technical requirements. Hanoi: Ministry of science and technology; 2009) 10. Ishii, N., Anami, K., Knisely, C.W.: Retrospective consideration of a plausible vibration mechanism for the failure of the folsom dam tainter gate. Int. J. Mech. Eng. Robot. Res. 3(4), 314–345 (2014) 11. EM 1110-2-2702. Engineering and Design: DESIGN OF SPILLWAY TAINTER GATES. U.S. Army Corps of Engineers, Washington, DC (2000) 12. EN 1992 1-1. Design of concrete structures. European Standard (2004) 13. Japan Society of Civil Engineers. JSCE No.15 Standard specifications for concrete structures. Tokyo (2007) 14. American Concrete Institute. ACI 318-95 Building Code Requirements for Structural Concrete (1995) 15. Mirmiran, A., Shahawy, M.: Behavior of concrete columns confined by fiber composites. J. Struct. Eng. 123, 583–590 (1997) 16. Chen, C.C., Lin, K.T.: Behavior and strength of steel reinforced concrete beam–column joints with two-side force inputs. J. Constr. Steel Res. 65, 641–649 (2009) 17. Tong, L., Liu, B., Xian, Q., Zhao, X.-L.: Experimental study on fatigue behavior of Steel Reinforced Concrete (SRC) beams. Eng. Struct. 123, 247–262 (2016) 18. Lam, T.Q.K.: Nghien cuu trang thai ung suat bien dang cua mai vo thoai be tong cot thep cong hai chieu duong nhieu lop. Chuyen nganh: Ky thuat xay dung cong trinh dan dung va cong nghiep. DH Kien truc Ha Noi ed.: Luan an Tien si ky thuat. 2019, p. 137–138. (L.T.Q.Khai. Stress strain study of multilayer positive two-dimensional curved reinforced concrete shell roof. Specialized: Civil construction and industry. Hanoi University of Architecture, pp. 137– 138. Ph.D. thesis (2019) 19. TCVN 8214:2009. Hydraulics physical model test of water headworks. Ministry of Science and Technology, Ha Noi (2009) 20. TCVN 12196:2018. Hydraulics structures-Physical model test of rivers. Ministry of Science and Technology, Ha Noi (2018) 21. Thang, N.N., Van Thuan, B., Khai, L.Q., Chuyen, N.V.: Lectures subjects experimental works. Thuyloi University, Ha Noi (2012) 22. Thu, D.V., Oanh, N.N.: Co hoc moi truong lien tuc. Hà nô.i: Nha xuat ban tu dien bach khoa. D. V. Thu, N. N. Oanh. Mechanics of continuous medium. The Publisher of the Encyclopedia, Hanoi (2007) 23. Lubliner, J., Oliver, J., Oller, S., Oñate, E.: A plastic-damage model for concrete. Solids Struct. 25(3), 299–326 (1989) 24. Lee, J., Fenves, G.L.: Plastic-damage model for cyclic loading of concrete structures. Eng. Mech. 124(8), 892–900 (1998) 25. Lee, J.: Theory and implementation of plastic-damage model for concrete structures under Cyclic and Dynamic Loading. Ph.D. thesis (in English). University of California, Berkeley, USA (1996) 26. Wosatko, A., Pamin, J., Polak, M.A.: Application of damage-plasticity models in finite element analysis of punching shear. Comput. Struct. 151, 73–85 (2015)

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27. Ren, W., Sneed, L.H., Yang, Y., He, R.: Numerical simulation of prestressed precast concrete bridge deck panels using damage plasticity model. Int. J. Concr. Struct. Mater. 9(1), 45–54 (2015) 28. Sarti, F., et al.: Development and testing of an alternative dissipative posttensioned rocking timber wall with boundary columns. J. Struct. Eng. 142(4), E4015011 (2016) 29. Kankam, C.K., Meisuh, B.K., Sossou, G., Buabin, T.K.: Stress-strain characteristics of concrete containing quarry rock dust as partial replacement of sand. Case Stud. Constr. Mater. 7, 66–72 (2017). https://doi.org/10.1016/j.cscm.2017.06.004 30. Abdulrezzak, B.A.K.I.S, ¸ Mesut, Ö.Z.D.E.M.˙IR., Ercan, I.SI.K., ¸ EL Alev, A.K.I.L.L.I.: The impact of concrete strength on the structure performance under repeated loads. Bitlis Eren Univ. J. Sci. Technol. 6(2), 87–87 (2016) 31. Dere, Y., Koroglu, M.A.: Nonlinear FE modeling of reinforced concrete. Int. J. Struct. Civil Eng. Res. 6(1), 71–74 (2017)

Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact Quang Thang Do1(B) , Sang-Rai Cho2 , and Van Dinh Nguyen3 1 Department of Naval Architecture, Nha Trang University, Nha Trang, Vietnam

[email protected]

2 School of Naval Architecture and Ocean Engineering, University of Ulsan, and UlsanLab Inc.,

Ulsan 44610, Korea [email protected] 3 Department of Mechatronics, Nha Trang University, Nha Trang, Vietnam [email protected]

Abstract. Tubular member structures are the major component structures of jacket of fixed floating offshore platforms, tension leg platforms (TLPs), or drilling jack-up rigs. During its operations, these structures are constantly serviced by support vessels. Collisions between them are unavoidable. One key considers during the design and safety of these structures is to make sure that they have good safety in the scenario of ship collisions. This research presents a series of fracture tests and numerical study results of T-joints tubular structures subjected to dynamic mass impact. The collision scenarios considered in this study were the collisions between T-joints tubular members and support vessels or floating objects. Eight T-joints of H-shaped tubular members were fabricated with different dimensions and tested under dynamic mass impact by applying the drop test machine. The detailed explanations of test setups and test results are reported. Finite element analyses (FEA) of the collision behaviors of the experimental models were performed using the ABAQUS software. For describing the fracture of T-joint tubular members, the Hosford-Coulomb fracture model was applied. A good agreement between the test results and numerical results was obtained. Furthermore, two ductile fracture modes of the T-joints tubular structures are also discussed. Keywords: Tubular T-joint · Impact test · Numerical simulation · Fracture response

1 Introduction Tubular member structures are widely used in FPSOs (Floating Production Storage and Offloading), jackets/legs of fixed offshore platforms and bracings of floating offshore installations. The main benefits of tubular structures are good resistance for axial compression/tension loadings and small drag force of passing fluid. Moreover, tubular member structures are also beneficial for fabrication and construction. During its operations, these structures are constantly serviced by support vessels. Collisions between © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 416–430, 2022. https://doi.org/10.1007/978-981-16-3239-6_32

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them are unavoidable. The collision between tubular members and support vessels is the most important reason for damaged offshore structures. It can lead to serious consequences for economic losses and even human life or loss of structural strength [1]. Thus, the reparations of these structures are very difficult and time-consuming [2]. Nowadays, collision accidents between ship and offshore tubular members are the key consideration for designing tubular structural performance and safety. Therefore, assessing the effect of collision damages on these structure’s strength is important for designing procedures. Several researchers have studied experiments on the collision of tubular structures. Furnes and Amdahl [3], Søreide et al. [4], Ellinas and Walker [5] reported the experimental investigations on quasi-static impact responses of tubular. Cho [2] presented the dynamic impact tests on small-scale single tubes. Jones et al. [6] performed quasi-static and dynamic impact experiments on fully fixed boundary conditions for mild steel pipes. Pendersen et al. [7] performed on the estimation of collision force between merchant vessels’ bow and gravity-supported offshore installations. Amdahl and Johansen [8] provided the collision of the ship’s bow and rigid-jacket tubular, while Amdahl and Eberg [9] studied a rigid ship’s bow collision with deformable tubular structures. Recently, Cerik et al. [10] and Cho et al. [11] presented extensive test data of the dynamic impact tests on H-shape tubular members for damage assessment at T-joint of bracing and chord. The impact locations were performed at mid-span and 200 mm away from mid-span. More recently, Do et al. [12–19] reported the test and numerical results on ring- and stringer-stiffened cylinder subjected to dynamic mass impact. Then, these models were tested under hydrostatic pressure for assessing their residual strength. It is noted that most of tests and numerical results have been reported in the literature, which was only considered the plastic deformation domain. In an actual case, with the large impact velocities, the fracture will occur. However, until now, there is a lack of test results for the fracture tubular members subjected to dynamic mass impact. Thus, it is necessary to provide several test results of this loading. This paper may useful for researchers to understand deeply the fracture behavior of T-joins tubular members under the dynamic mass impact. Nowadays, Recently, NFEM (Nonlinear Finite Element Methods) is a powerful tool to forecast the residual strength of offshore tubular structures during ship collisions. Because NFEM is convenient to use and the cheapest way to perform the full dimensions of actual ocean engineering structures [19, 21]. Thus, NFEM is the best-chosen way for assessing the collision responses of tubular structures during ship collision accidents. However, one of the main challenges when modeling the collisions between tubular structures and supporting vessels is the ductile fracture criterion to forecast the fracture propagation in the ductile material. Until recently, these problems are not yet answered accuracy and it is still an intensive research topic. When performing NFEM of collision, it is necessary to input a suitable fracture criterion because it is strongly affected by the membrane stretching, crushing, and tearing of shells [20]. These progressive failures would be influenced by global structural collapse modes [21]. In this paper, the HosfordCoulomb ductile fracture material models provided by Cerik et al. [22–25] were applied. The Hosford-Coulomb ductile fracture model has recently become quite popular because it was the ease of calibration, simplicity, flexibility, and accuracy.

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On this background, the objective of this study is to provide the test results of fracture T-joins tubular members under dynamic mass impact. Then, the numerical simulation was performed by using a Hosford-Coulomb ductile fracture model. Finally, the deformed shape and failure modes, and stress states obtained for various models were also discussed.

2 Fracture Tests 2.1 Dimensions and Material Properties of the Test Models The models are H-shape tubulars which consisted one brace and two chords joined to the brace ends by welding. In actual offshore tubular structures, the ranges of D/t ratio are generally used from 20 to 60. In this study, the dimensions of test models are also chosen based on this range of D/t ratio. There were eight H-shape tubular models which were fabricated. The dimensions of chords were the same for all models but different dimensions for braces. The experimental models were classified as E, G and H series following their bracing dimensions. The detail dimensions of each model are indicated in Table 1. It is noted that the design dimensions and fabrication procedures of test models were following the design code of API [26] and ABS [27]. In this table, L, D and t are the total length, outside diameter and thickness of tubular, respectively. It is noted that the thicknesses of chord and brace were carefully measured at several points on each chord and brace using ultrasonic measuring instruments and determined to be an average value, which is commonly less than nominally value. The material of test models was fabricated from the general-purpose structural steel. The material properties were achieved by carry out the quasi-static tensile tests. The average value of quasi-static tensile test results is summarized in Table 2.

Table 1. Measured scantlings of experimental models (unit: mm) Model Dimension of chord

Dimension of bracing tc

Lb

Db

L c /Dc Dc /t c L b /Db Db /t b

Lc

Dc

tb

E3

1300

114 4.05 886

89.1 2.10 11.40

28.15

9.94

42.43

G1

1300

114 6.02 866

76.0 1.79 11.40

18.94 11.39

42.46

G2

1300

114 6.02 866

76.0 1.80 11.40

18.94 11.39

42.22

G3

1300

114 6.00 866

76.0 1.80 11.40

19.00 11.39

42.22

G4

1300

114 6.05 866

76.0 1.80 11.40

18.84 11.39

42.22

G6

1300

114 6.05 866

76.0 1.79 11.40

18.84 11.39

42.46

H3

1300

114 6.04 886

90.0 2.08 11.40

18.87

9.84

43.27

H4

1300

114 6.07 886

90.0 2.08 11.40

18.78

9.84

43.27

Experimental and Numerical Investigations on the Fracture Response

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Table 2. Material properties for each series of experimental models. Model series

E-series G-series H-series

Yield strength, σY (MPa)

360.3

319.7

317.3

Ultimate tensile strength, σT (MPa) 413.5

418.7

391.1

Young’s modulus, E (GPa)

207

206

206

Hardening start strain, εHS

0.0279

0.0186

0.0245

Ultimate tensile strain, εT

0.1447

0.1723

0.1743

2.2 Experimental Set-Up The experiments were performed using a drop test machine which was successful in earlier dynamic mass impact tests [10–19]. The details of the experimental arrangements are indicated in Fig. 1. On the top of the drop test machine, there is a pulley which guides and keeps the striking mass by an electromagnet. The striking mass was attached to a rigid knife-edge striker. During the drop test, drop height can be controlled for obtaining the expected collision energy level. When electromagnet is disconnected, the striking mass falls freely due to its gravity. In this study, the different impact energy levels and striking mass types were applied for each model. The shapes and mass arrangement of strikers were designed to minimize the imbalances in free-fall process. Furthermore, to investigate the fracture response at T-joint of tubular, the impact locations were selected around 200 mm aside from the mid-length of bracing. Therefore, the shape of the striker was also required that it cannot touch the chord of tubular during the test. The radius value of knife edge header is 10 mm for the striker type I and II. The dimensions of striker and its indenter header shape are depicted in Fig. 2. The details of collision test conditions are shown in Table 3. For the experimental boundary conditions, the model was firmly clamped and inserted rubber pads grip at both ends of the chord using the support structures, as shown in Fig. 1. The support structures of clamps were bolted to a rigid foundation of the loading frames. It is guaranteed that the chords are fully fixed and not allowed any translations and rotations of both ends of chord. In addition, 2D strain gauge was attached in eight positions at both T-joint regions and impact areas. Among them, four strain gauges were attached in outer of the shell at the collision area, and the next four strain gauges were placed at both T-joint of the tubular model. The impact location was moved with various distance values away from the midlength of bracing members to investigate the behavior of T-joint. During the impact test, when the striking mass hit the brace of model, the rebounded phenomenon was commonly occurred. Therefore, for protecting the model from unexpected deformation after the striking mass rebounded, the rubber pads were used for covering all surfaces of model except impact region as indicated in Fig. 1.

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2.3 Test Results In this section, the results of the series dynamic mass impact tests were presented. The aim of these series test models was investigated the damages of H-shape tubular member under dynamic mass impact from initial fracture to complete fracture at T-joint. It means that the impact energies were increased gradually until complete fracture occur by changing the striking masses or drop highs. Therefore, the deformations of the tested models were classified into two groups: partial fracture and complete fracture damages.

Fig. 1. Dynamic mass impact test setup

Fig. 2. Dimensions of striker mass: (a) Type I; (b) Type II

When stiffness of bracing member is quite smaller compared to that of the chord members, the plastic deformation occurs only on bracing. In these cases, if the collision energy level was increased, the fracture will take account on bracing at T-joint location which near the impact line. Furthermore, if the length of bracing is short, the deformation angle α as in Fig. 3 is increased. It means that the overall bending damage on bracing causes to the increase of the rotational deformation angle on bracing member. It is a significant reason leading to the value of shear strain to be increased. When this shear value reached to the shear failure criterion of material properties, fracture damage was taken a place at T-joint location, as shown in Fig. 4. When the brace ruptures at the T-joint, it will not absorb the collision energy anymore. Owing to overall bending, high tension forces at the welded joints can reason to fracture on the bracing. It is noted that the large plastic deformation of bracings which were absorbed the impact energy, leading to decrease the damages on chord. It is improved the safety of entire structures. The partial fracture damages at T-joint locations are depicted in Figs. 5 and 6 for models E3, G6, H3 and H4, respectively. For all models, the fracture was occurred near the welded joint of bracing and chord. After obtained the partial fracture damages, the collision energies were increased to achieve the complete fracture. Figure 7 is illustrated

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Table 3. Impact test conditions for tubular members Model

Velocity (m/s)

Striker mass (Kg)

Kinetic energy (J)

Striker type

Impact location

E3

5.25

460

6339

I

200 mm aside from mid-length

G1

5.13

673

8856

II

200 mm aside from mid-length

G2

4.93

673

8179

II

200 mm aside from mid-length

G3

4.52

673

6875

II

200 mm aside from mid-length

G4

3.81

673

4885

II

200 mm aside from mid-length

G6

2.58

673

2240

II

200 mm aside from mid-length

H3

2.94

673

2909

II

200 mm aside from mid-length

H4

2.39

673

1922

II

200 mm aside from mid-length

the complete fracture damage of model G4 before releasing boundary condition. The complete fracture damages and each cross-section of models G1 and G2 are described in Fig. 8, respectively. It is noted that cross-section at fracture lines were not smooth and it has some tearing parts. The rotation of striker when it was released from the electromagnet, was one of the most important reason for tearing parts of the cross-section. The summary of experimental results is shown in Table 4. In this table, the measured permanent extents of damages including the dent depth (d d ) and overall damages (d 0 ) were presented. The failure mode of each model is also provided in this table.

Fig. 3. Idealised profiles of a damage on both of bracing and chord tubular

d0 = sin(α + β)

  a2 + b2

tan α =

a b

(1) (2)

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Q. T. Do et al.

tan β =

c d

(3)

Fig. 4. Idealised stress distribution in T-joint of tubular member after denting test

Fig. 5. Partial fracture damage of models: (a) E3; (b) G6

3 Numerical Analysis of Fracture Tests 3.1 Finite Element Modelling It is confirmed that the proposed numerical simulation method has been verified and assessed the accuracy by comparing it with available test data, which could be found in references [12–18]. Thus, in the current paper has only presented the summary of NFEA (Nonlinear Finite Element Analyses) for fracture tubular members. NFEA was carried out by applying the ABAQUS explicit method. The dynamic/explicit solution was used for collision analysis. The shell element S4R was used for all structural modeling. For the striker, R3D4 element type with the rigid body was applied. The general contact with penalty method was defined for the contact between striker mass outside surface and bracing of tubular. The friction coefficient at contact region was assumed with 0.3. Before performing the numerical simulations on test model, the convergence tests were checked for selecting the optimum mesh size. The mesh size of the contact area was 5 × 5 mm, while that for the outside of the collision region was 10 × 10 mm. The selected mesh size is suitable to capture the deformed shape and fracture behavior accurately. For the boundary conditions, the ends of two chords of the tubular member

Experimental and Numerical Investigations on the Fracture Response

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were fixed with all degrees by using the high tensile steel bolt as the test setups. The boundary conditions and finite element modelling were described in Fig. 9. Furthermore, the model was generally vibrated during the collision process. Thus, the Rayleigh damping was applied for reducing these vibrations and quickly obtain a static equilibrium state. Rayleigh damping equation was presented as Eq. (4): C = αM + βK

(4)

where M and K are the mass matrix and stiffness matrix, respectively. α is a coefficient, which can be calculated as the lowest natural frequency of model. The lowest natural frequency can be obtained by the first mode of a Subspace Eigen solver in ABAQUS. β is a coefficient of the stiffness proportional damping factor.

Fig. 6. Partial fracture damage of models: (a) H3; (b) H4

Fig. 7. Complete fracture damage of model G4 before releasing boundary condition

3.2 Material Definition For impact simulation, the equations suggested by Do et al. [12] were applied for determining the material definition. These equations were developed based on a large database of dynamic tensile test results of various marine steels. The true stresses and strains were determined as Eqs. (5) and (6) using the static tensile test results in Table 2. When considering the effects of the strain hardening and yield plateau, the equations from (7) to

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(11) were used. when considering the strain-rate hardening effects, the revised equations reported by Do et al. [12] were used. The dynamic yield strength (σ YD ), dynamic ultimate tensile strength (σ TD ), dynamic hardening start strain (εHSD ) and dynamic ultimate tensile strain (εTD ), which are calculated relation to the strain rate ε˙ as shown in Eqs. (12)–(15). In this paper, the strain rate was created with different values (10 s−1 , 20 s−1 , 50 s−1 , 70 s−1 , 100 s−1 and 150 s−1 ).

Fig. 8. Complete fracture damage of models: (a) G1; (b) G2

Table 4. Results of measured damage for each test models Model

Bracing

Damage mode

dd

do

E3

59.90

64.50

Partial fracture

G1, G2, G3, G4





Complete fracture

G6

50.0

45.90

Partial fracture

H3

56.0

33.20

Partial fracture

H4

49.50

23.20

Partial fracture

Fig. 9. Finite element model for collision analysis

Experimental and Numerical Investigations on the Fracture Response

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σtr = σ (ε + 1)

(5)

εtr = ln(ε + 1)

(6)

σtr = Eεtr when 0 < εtr ≤ εY ,tr

(7)

 εtr − εY ,tr  σtr = σY ,tr + σHS,tr − σY ,tr when εY ,tr < εtr ≤ εHS,tr εHS,tr − εY ,tr

(8)

σtr = σHS,tr + K(εtr − εHS,tr )n when εHS,tr < εtr

(9)

where n=

  σT ,tr εT ,tr − εHS,tr σT ,tr − σHS,tr σT ,tr − σHS,tr K= n εT ,tr − εHS,tr

εTD εT

σYD = 1 + 0.3(E/1000σY )0.5 (˙ε )0.25 σY 0.35  σTD = 1 + 0.16(σT /σYD )3.325 (˙ε )1/15 σYD εHSD = 1 + 0.1(E/1000σY )1.73 (˙ε )0.33 εHSS   = 1 − 0.117 (E/1000σT )2.352 (σT /σY )0.588 (˙ε )0.2

(10) (11) (12) (13) (14) (15)

It is noted that there is fracture occurred in the impact test. Therefore, when generating the true stress-plastic strain curves which were extended after the initiation of necking. For describing the fracture on T-joint tubular members, the Hosford-Coulomb fracture model was applied. The Hosford-Coulomb fracture model defines the ductility limit of structures with different stress states. In particular, this model can be understood as a mesh-independent. In this model, the initial crack was occurred by the onset of shear localization. The fracture stress criterion can be described as the Hosford equivalent stress σ¯ HF , and the normal stress reaches the maximum shear stress at critical value b. The detail of Hosford-Coulomb ductile fracture model formulations was presented as Eqs. (16) to (23). σ¯ HF + c(σ1 + σ3 ) = b  σ¯ HF =

  (1/a) 1 a a a (σ1 − σ2 ) + (σ1 − σ3 ) + (σ2 − σ3 ) 2

(16) (17)

Equation (17) can be rewritten into cylindrical coordinates as below:

b σ¯ f η, θ¯ =   1 a a a (1/a) + c(2η + f1 + f3 ) 2 (f1 − f2 ) + (f1 − f3 ) + (f2 − f3 )

(18)

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where  2 π  1 − θ¯ f1 θ¯ = cos 3 6  2 π  f2 θ¯ = cos 3 + θ¯ 3 6 

2 π  f3 θ¯ = − cos 1 + θ¯ 3 6

(19) (20) (21)

The mixed stress-strain can be expressed by applying the inverse of isotropic powerlaw hardening as follows: ε¯ f

1



η, θ¯ = b(1 + c) nf g η, θ¯

pr

(22)

where

g η, θ¯ =



1/nf 

(1/a) 1 a a a + c(2η + f1 + f3 (f1 − f2 ) + (f1 − f3 ) + (f2 − f3 ) 2 (23)

3.3 Numerical Results Figures 10 and 11 show the comparison between numerical deformed shape of models E3 and G4 with experimental results. The damage on model E3 is a partial fracture while that of model G4 is a complete fracture, respectively. The deformed shapes observed in tests are captured well by numerical analysis. As mention in the previous section, when the collision energy level was significantly increased, the fracture will take account on bracing at the T-joint location which near the impact line. The energy dissipation of axially restrained tubular is limited by fracture at joints due to excessive membrane straining. It is a typical example of tensile fracture at joint. In the numerical model, the shear fracture criteria were used as fracture criteria. When this shear value reached to the shear failure criterion of material properties, fracture damage occurred at T-joint location. The comparisons of test results and numerical results are summarized in Table 5. The mean of the uncertainty modelling factor X m is 1.027 for local shell denting and 1.059 for overall bending damage, respectively. The predicted local shell denting and overall bending damage are commonly larger than those of test results. It should be concluded that the numerical results are quite accuracy and reliable when compared to test results. Furthermore, it is noted that the complete fracture result is not included in this table because when complete a fracture occurs the bracing tube is released and it is not contributed to the whole structure’s strength.

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Fig. 10. Comparison between numerical result and experimental result for model E3

Fig. 11. Comparison between numerical result and experimental for model G4

4 Disscussions 4.1 Collision Tests A series of dynamic impact tests included four partial fracture models and four complete fracture models, were successfully provided in this research. The detailed explanations for test setup, test procedure and test results were also presented in this study. The main goal of these tests was to investigate the damage responses at T-joint tubular members. In general, the behavior of tubular members under dynamic mass impact is very complicated because the combination between local shell denting and the overall bending damage. Thus, it is difficult to propose to classify all deformation modes of these structures. The failure modes of experimental results are depicted in Fig. 12. Where the energy parameter λE is defined as the ratio of kinetic energy of striker (E k ) to energy absorption capacity of struck structure (E a ), as defined in Eq. (24). The kinetic energy of the striking mass and the static energy absorption capacity is described in Eq. (25) and (26), respectively. Where σ Y and σ T are yield strength and ultimate tensile strength,

428

Q. T. Do et al. Table 5. Comparison of test results and numerical results

Model

Db (mm)

Experiment results

Numerical results

Bias (FEA/Test), X m

dd/Db (1)

d0/Db (2)

dd/Db (3)

d0/Db (4)

(3)/(1)

(4)/(2)

E3

89

0.67

0.72

0.71

0.70

1.060

0.972

G6

76

0.66

0.60

0.71

0.64

1.076

1.067

H3

90

0.62

0.37

0.57

0.40

0.919

1.081

H4

90

0.55

0.26

0.58

0.29

1.055

1.115

1.027

1.059

Mean

respectively. εT and V str are the ultimate tensile strain and structure volume, respectively. In this figure, the failure mode of tubular member was divided into two groups: Mode I-Partial fracture at T-joint and Mode II- Complete fracture at T-joint. When the ratio of energy (λE ) was small, the partial fracture was occurred. And when the ratio of energy (λE ) was then increased further, the complete fracture at T-joint was occurred. It means that the bracing tube is released and it is not contributed to the whole structure’s strength. Ek Ea

(24)

1 2 mv ; Kinetic energy 2

(25)

σY + σT εT Vstr ; Strain energy absorption capacity 2

(26)

λE = Ek = Ea =

Fig. 12. Plotting of failure mode of test models against basic parameters Ek /Ea and Lb /Db ,and Db /tb

Experimental and Numerical Investigations on the Fracture Response

429

4.2 Numerical Simulations Based on the numerical results, it is concluded that a quite good accuracy was observed between the numerical results and test results. The mean of X m (the uncertainty modelling factor) is 1.027 and 1.059 for local shell denting and overall bending damage, respectively. The predicted local shell denting and overall bending damage are commonly larger than those of test results. The reason can be related to the sensitive of local damage due to the rotation angle and indenter direction during the collision tests. Another reason can be explained by the uncertainty of the dynamic material modelling, where the strain rate effects have the important role. Additionally, the Hosford-Coulomb ductile fracture model which applied in numerical analysis are reasonable and accuracy for predicting the fracture responses of tubular T-joint members under dynamic mass impact. Therefore, the suggested NFEM can be applied to perform further case studies for developing the design equations.

5 Conclusions The main goal of the present study was to provide the series of experimental and numerical results of H-shape tubular member under dynamic mass impact. These results were shown insights understanding of the fracture behavior at T-joint of tubular member during a ship collision. The numerical simulations have a good agreement with the experimental results. The predicted local shell denting and overall bending damage are commonly larger than those of test results. The rotation angle during the free fall of striking mass and the uncertainty of the dynamic material modelling were the main reason. The failure modes of tubular member under dynamic mass impact were very complex owning to the combination of local shell denting and the overall bending damage. It is not straightforward to suggest the simple parameter for classifying all deformation modes and responses of tubular members. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2019.333.

References 1. DNV. Accident statistics for fixed offshore units on the UK Continental Shelf 1980–2005: Research Report RR566. HSE Books (2007) 2. Cho, S.R.: Design approximations for offshore tubulars against collisions. Ph.D. thesis, Faculty of Engineering. University of Glasgow, Glasgow, UK (1987) 3. Furnes, O., Amdahl, J.: Ship collisions with offshore platforms, pp. 310–318. Intermaritec, Hamburg, Germany (1980) 4. Søreide, T.H., Moan, T., Amdahl, J., Taby, J.: Analysis of ship/platform impacts. In: 3rd International Conference on Behaviour of Offshore Structures, Boston, USA, pp. 257–278 (1982) 5. Ellinas, C.P., Walker, A.C.: Damage on tubular bracing members. In: Proceedings of IABSE Colloquium on Ship Collision with Bridges and Offshore Structures, Copenhagen, Denmark, pp. 253–261 (1983)

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6. Jones, N., Birch, S.E., Birch, R.S., Zhu, L., Brown, M.: An experimental study on the lateral impact of fully clamped mild steel pipes. Proc. Int. Mech. Eng. Part E J. Process. Mech. Eng. 206, 111–127 (1992) 7. Pedersen, P.T., Valsgard, S., Olsen, D.: Ship impact: bow collisions. Int. J. Impact Eng. 13, 163–187 (1993) 8. Amdahl, J., Johansen, A.: High-energy ship collision with jacket legs. In: Proceedings of the International Offshore and Polar Engineering Conference (2001) 9. Amdahl, J., Eberg, E.: Ship collision with offshore structure. In: Moan (ed.) Structural Dynamics – EURODYN 1993, Balkema, Rotterdam (1993) 10. Cerik, B.C., Shin, H.K., Cho, S.R.: A comparative study on damage assessment of tubular members subjected to mass impact. Mar. Struct. 46, 1–29 (2016) 11. Cho, S.R., Le, D.N.C., Jeong, J.H., Frieze, P.A., Shin, H.K.: Development of simple designoriented procedure for predicting the collision damage of FPSO caisson protection structures. Ocean Eng. 142, 458–469 (2017) 12. Do, Q.T., Muttaqie, T., Shin, H.K., Cho, S.R.: Dynamic lateral mass impact on steel stringerstiffened cylinders. Int. J. Impact Eng. 116, 105–126 (2018) 13. Cho, S.R., Do, Q.T., Shin, H.K.: Residual strength of damaged ring-stiffened cylinders subjected to external hydrostatic pressure. Mar. Struct. 56, 186–205 (2017) 14. Do, Q.T., Muttaqie, T., Park, S.H., Shin, H.K., Cho, S.R.: Predicting the collision damage of steel ring-stiffened cylinders and their residual strength under hydrostatic pressure. Ocean Eng. 169, 326–343 (2018) 15. Do, Q.T., Muttaqie, T., Park, S.H., Shin, H.K., Cho, S.R.: Ultimate strength of intact and dented steel stringer-stiffened cylinders under hydrostatic pressure. Thin-Walled Struct. 132, 442–460 (2018) 16. Do, Q.T., Le, D.L.C., Seo, B.S., Shin, H.K., Cho, S.R.: Fracture response of tubular T-joints under dynamic mass impact. In: ICCGS-2019, Lisbon, Portugal, pp. 75–84 (2019) 17. Do, Q.T., Huynh, V.V., Vu, M.T., Tuyen, V.V., Pham, T.N.: A new formulation for predicting the collision damage of steel stiffened cylinders subjected to dynamic lateral mass impact. Appl. Sci. 10, 3856 (2020) 18. Do, Q.T., Huynh, V.N., Tran, D.T.: Numerical studies on residual strength of dented tension leg platforms under compressive load. J. Sci. Technol. Civil Eng. 14, 96–109 (2020) 19. Do, Q.T.: Deriving formulations for forecasting the ultimate strength of locally dented ringstiffened cylinders under combined axial. Sci. Technol. Dev. J. 23, 640–654 (2020) 20. Moan, T.: Development of accidental collapse limit state criteria for offshore structures. Struct. Saf. 31, 124–135 (2009) 21. Calle, M.A.G., Alves, M.: A review-analysis on material failure modeling in ship collision. Ocean Eng. 106, 20–38 (2015) 22. Cerik, C.B., Lee, K., Park, S.J., Choung, J.: Simulation of ship collision and grounding damage using Hosford-Coulomb fracture model for shell elements. Ocean Eng. 173, 415–432 (2019) 23. Park, S.-J., Lee, K., Cerik, B.C., Choung, J.: Ductile fracture prediction of EH36 grade steel based on Hosford–Coulomb model. Ships Offshore Struct. 14(sup1), 219–230 (2019) 24. Cerik, B.C., Park, B., Park, S.J., Choung, J.: Modeling, testing and calibration of ductile crack formation in grade DH36 ship plates. Marine Struct. 66, 27–43 (2019) 25. Cerik, B.C., Choung, J.: Rate-dependent combined necking and fracture model for predicting ductile fracture with shell elements at high strain rates. Int. J. Impact Eng. 146, 103697 (2020) 26. API. Bulletin on Stability Design of Cylindrical Shells, API Bulletin 2U, 3rd edn, Washington (2004) 27. ABS, Guide for Buckling and Ultimate Strength Assessment for Offshore Structures (2014)

Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion Pham Hien Hau(B) and Vu Dan Chinh National University of Civil Engineering, Hanoi, Vietnam {hauph,chinhvd}@nuce.edu.vn

Abstract. Strength of mooring lines of floating offshore platforms are specified by Minimum Breaking Loads (MBL). The MBL are determined by experimental formula depended on diameters of mooring lines in catalogues of manufacturer. For existing floating offshore platforms, after an operating duration the diameters are reduced due to corrosion. It means that strength of the mooring lines are reduced, too. There are some studies about the problem. In our previous researches, the theoretical basis and fundamental steps for evaluating and re-evaluating the durability of mooring lines of floating offshore structures were studied. However, the corrosion is not uniform for different locations of the lines, so it is not easy to estimate accurately the reduced strength. In this paper, we propose a method to assess reduced strength of the existing mooring chains based on numerical models. It will be applied to evaluate strength of mooring line system of DH-01 FPU Platform which is operating in Dai Hung field in Viet Nam sea. The research results can be used as a reference for similar projects in Vietnamese conditions. Keywords: Floating offshore platforms · Existing mooring lines · Reduced strength · Corrosion · Numerical models

1 Introduction During the exploitation process, the mooring system of floating offshore platforms have been degraded due to environmental factors (corrosion, marine growth…). When the platforms are changed exploitation position or extended life time, it is necessary to re-evaluate the strength of existing mooring lines, taking into account the influence of the environmental factors. One of the most important factors is corrosion. It causes the reduction of diameter, weight and strength of mooring chains. Strength condition of mooring lines are depended on the maximum tension force and the Minimum Breaking Loads (MBL) of the lines [1, 2]. The Minimum Breaking Loads can be simply determined through the relationship with sections of the segments from catalogues of manufacturers by experimental formula [4]. However, in case of lack of survey data, the breaking loads can be predicted by numerical testing simulations of mooring lines. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 431–445, 2022. https://doi.org/10.1007/978-981-16-3239-6_33

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There are some studies in the world about the problem, such as [5, 6]. In our previous researches [7, 8], the theoretical basis and fundamental steps for evaluating and reevaluating the durability of mooring lines of floating offshore structures were studied. In this paper, we establish different models using ABAQUS program to determine MBL of corroded chain links based on survey data. The results will be compared with MBL in Vicinay’s Catalogue to verify the accuracy of the models. Finally, we evaluate strength conditions of a mooring line system through the MBL results and tension forces in these lines due to external loading by ARIANE program.

2 Methodology to Assess the Strength of Mooring Lines of Floating Offshore Structures 2.1 Line Dynamic Response Analyses A methodology based on time domain simulation has been developed and proved by Bureau Veritas [3] for mooring lines design and assessment using ARIANE software. In our research, we have to perform the analyses in quasi-dynamic of the mooring line. All of the response simulations in the mooring line systems are executed in time domain. The calculation procedure was proposed in H.H Pham (2015) [7].

Fig. 1. Time domain simulation of tension force of mooring lines using ARIANE software

For any considered sea-state, n simulations in 3h (a short-term sea-state duration) should be performed (Fig. 1). The response signals are built up depended on time steps. For each mooring line, the maximum tension at fairlead (connection point between mooring line and floating structure) during each simulation is obtained by using different “seeds”, i.e. different sets of elementary waves, wind components. Means and the standard deviations of the maximum values in n simulation are obtained. A design line tension corresponding to n simulations is defined by Bureau Veritas (2004) [2]:   n n  1  1  Tk ; TS =  (1) TD = TM + aTS ; TM = (Tk − TM )2 n n−1 k=1

where : n is the number of simulation;

k=1

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433

Tk is the maximum tension at fairlead obtained during the k-th simulation; TM is the mean of Tk obtained by the n simulations; TS is the (n-1) standard deviation; TD is the design tension of the line. a is a factor, given in [2] as a function of n et the type of analysis. With quasi-dynamic method, n =5 => a=1.8; n=20 => a=0.5; n≥30 => a=0.4. 2.2 Safety Factor To assess strength condition of mooring lines of floating offshore structures, we have to evaluate the Safety Factor (SF) for line components. According to Bureau Veritas (2004) [2], SF for mooring lines is expressed by the formula as below:  (2) SF = TBr TD ≥ [SF] where: TBr is the Minimum Breaking Load (MBL) of the mooring line; TD is the design tension of the line given by Eq. (1); [SF] is the minimum Safety Factor, depend on analysis conditions and analysis method. For quasi-dynamic method in the intact condition [SF] is 1.75 and in the survival condition (one mooring line is broken) [SF] is 1.25 [2]. 2.3 Minimum Breaking Loads of Mooring Lines from Catalogues of Manufacturers [4] According to Vicinay Mooring Lines Catalogue [4], Minimum Breaking Loads of mooring chain are expressed by formulas given in Tables of Mechanical properties, depends on the grade of chain (Fig. 2). For example, the MBL of chain links Grade R4S is: TBr = 0.0304 Z (kN) with Z = d2 (44 − 0.08d) where d is diameter of the chain (mm).

Fig. 2. Mechanical properties of chain from catalogues of manufacturers [4]

(3)

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The formula (3) can be used to determine MBL of uniform corroded chains with arbitrary diameters. However, in the next sections, authors will present a method to evaluate the MBL by numerical models, using Abaqus software, in case of non-uniform corroded chains for different locations of the lines.

3 Numerical Method for Chain Testing Simulation 3.1 Plastic Models

Fig. 3. Illustration of stress – strain curve of a metal sample belonging to over-stress power law [10]

Fig. 4. Slave nodes interacting with a two-dimensional master surface [9]

To determine breaking loads of chain mooring line, it is necessary to analyse chain links in plastic stage. The plastic models are usually used is over-stress power law with stress - strain relationship as below (Fig. 3) [9, 10].  n σ (4) ε˙ pl = D σyoy − 1 for σy ≥ σyo Where εpl is the equivalent plastic strain rate; σy is the yield stress at nonzero plastic strain rate; σyo is the static yield stress which may depend on the plastic strain εpl via isotropic hardening and other parameters; D and n are material parameters that can be functions of temperature and, possibly, of other predefined state variables. 3.2 Small-Sliding Interaction Between Bodies [9] An “anchor” point on the master surface, Xo, is computed for each slave node so that the vector formed by the slave node and Xo coincides with the normal vector N(Xo) (Fig. 4). Suppose that a search for the anchor point, Xo, of slave node 103 reveals that Xo is on segment 1–2. Then, we find that: Xo = X (uo ) = (1 − uo )X1 + uo X2

(5)

Where X1 and X2 are the coordinates of nodes 1 and 2, respectively, and uo is calculated so that Xo – X103 coincides with N(Xo). Moreover, the contact plane tangent direction, vo , at Xo is chosen so that it is perpendicular to N(Xo); i.e., ∂(Xo ) = T (X2 − X1 ) ∂u where T is a (constant) rotation matrix. vo = T

(6)

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435

3.3 Contact Formulation [9] At each slave node that can come into contact with a master surface we construct a measure of over closure h (penetration of the node into the master surface) and measures of relative slip. These kinematic measures are then used, together with appropriate Lagrange multiplier techniques, to introduce surface interaction theories for contact and friction. In three dimensions the over closure h along the unit contact normal n between a slave point XN+1 and a master plane M(ξ1 , ξ2 ), where ξi parametrize the plane, is determined by finding the vector (M− XN+1 ) from the slave node to the plane that is perpendicular to the tangent vectors v1 and v2 at M. Mathematically, we express the required condition as: h.n = M (ξ1 , ξ2 ) − XN +1

(7)

v1 .(M (ξ1 , ξ2 ) − XN +1 ) = 0 v2 .(M (ξ1 , ξ2 ) − XN +1 ) = 0

(8)

When

If at a given slave node h < 0, there is no contact between the surfaces at that node, and no further surface interaction calculations are needed. If h ≥ 0, the surfaces are in contact. 3.4 Contact Pressure in Case of Hard Contact [9] In this case. p = 0 for h < 0 (open), and h = 0 for p > 0 (close) The contact constraint is enforced with a Lagrange multiplier representing the contact pressure in a mixed formulation. The virtual work contribution is: δ = δp.h + p.δh

(9)

and the linearized form of the contribution is: d δ = δp.dh + dp.δh

(10)

3.5 Stress Analysis by FEM In case of two bodies contact together and have pressure interaction, stress will be analysed by FEM as similar as a rigid body when suffer loads [11].

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4 Strength Assessment of Mooring Line System of DH01 FPU The section presents a strength assessment of mooring lines of DH01 (Fig. 5) based on the numerical method and Vicinay’s formula as mentioned above. These results will be compared to clarify applicability and suitability of numerical models for the problem. The contents are detailed in items as below. 4.1 Input Data 4.1.1 DH01 FPU Data [12]

Fig. 5. Position and general layout of Dai Hung Field, Vietnam

DH-01 FPU Structural data are assumed in the Table 1: Table 1. Dai Hung FPU (DH-01) structural data Items

Unit Value

Main particulars pontoon length m

108.2

Moulded Breadth

m

67.36

Main deck length

m

68.6

Main deck elevation

m

36.6

Main deck breadth

m

60.92

Maximum draught

m

21.3

Lightship weight

T

10715

Displacement

T

20543

Draft - Operational

m

21.34

Draft – Survival

m

19.51

Draft – Transit

m

6.7

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4.1.2 Mooring Input Data Mooring lines configuration and arrangement are presented in Fig. 6.

Fig. 6. Dai Hung FPU mooring lines profile and arrangement

As original design, one mooring line includes two segments. Static segments are laid out on seabed with the length of 685m, Dynamic segments with the length of 465m. The mooring systems of DH-01 is modeling in ARIANE program (Fig. 7).

Fig. 7. Modeling of DH-01’s mooring systems in ARIANE software

The original and corroded configurations of chain links are presented in Fig. 8. Whereas, based on site survey report in 2015 [13], corroded diameters, Dc, include values of 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm, respectively. Chain link material has Grade R4S, with mechanical properties are listed in the table as below.

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a) Original Chain Link

b) Corroded Chain Links

Fig. 8. Configuration of chain links Table 2. Mechanical properties of Chains Yield strength (MPa)

Ultimate strength (MPa)

Elongation (%)

MBL (kN)

700

960

12%

10600

4.1.3 Environmental Data [14] Environmental data are assumed in the Table 3 as below: Table 3. Summaries of environmental data Items

Unit Value

Highest astronomical tide HAT

m

10 min. mean speed (1 year period)

m/s

111.5 23.4

10 min. mean speed (100 year period)

m/s

34.6

Significant wave height (1 year period)

m

4.8

Peak period (1 year period)

s

10.7

Significant wave height (100 year period)

m

10.0

Peak period (100 year period)

s

16.0

Surface total steady current (1 year period)

m/s

1.53

Surface total steady current (100 year period) m/s

1.97

4.2 Chain Testing Simulation The number of chain links should be odd to satisfy the symmetric. Then the models have more than 3 links to change corroded chain links at different locations. So in this paper we choose testing model of 5 links joined together using Abaqus as the Fig. 9 as below. Herein, two outermost chains are connected with reference points which are positions of tension forces.

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Fig. 9. Testing model in Abaqus

439

Fig. 10. Meshing of chain testing models

The first model uses original dimension for all chain links. In the next models, for one case of corrosion in relation with Dc = 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm, the corroded chain link is located at location numbers of 2, 3, respectively. Meshing models are shown in Fig. 10. Contact model between two chain links are hard contact with small sliding interaction (Fig. 11). Displacements are assigned at Reference Points RP1 and RP2 to make tension forces. The displacements are increased step by step. At the time the first chain link meets fully plastic condition, the related tension force is Minimum Breaking Load (MBL).

Fig. 11. Contact interaction model between chain links

4.3 Numerical Testing Results 4.3.1 Minimum Breaking Load of Original Model of Chain As the analysis result of numerical model using Abaqus, the Minimum Breaking Load of original model of chain is 10689 kN. The deviation from the designed MBL in Table 2 is 89 kN (0.84%). The stresses distribution result and tension force chart are expressed in the Figs. 12 and 13.

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Fig. 12. Stress distribution on original model of chain

Fig. 13. Tension force chart of original model

4.3.2 Minimum Breaking Loads of Corroded Chain Models As the input data, corroded diameters Dc = 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm are used to make numerical testing models of chain. For a case of Dc, there are 2 models in relation with the corroded chain link at location numbers of 2 and 3. So, total of testing models are 6x2 = 12. The results of MBL are summarized in the Table 4. Table 4. Minimum Breaking Load (MBL) analysis results Dc (mm)

MBL (kN) Corroded chain link at location No 2

Corroded chain link at location No 3

84

7995

8043

86

8352

8303

87

8572

8553

88

8674

8667

90

8948

8966

91

9201

9203

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The stresses distribution results, tension force charts and deformations of chain testing models in relation with Dc = 91 mm and 84 mm are expressed in the Figs. 14, 15, 16 and 17.

Fig. 14. Stress and deformation distribution on corroded chain model, Dc = 91 mm at location No. 2

Fig. 15. Tension force chart of corroded chain model, Dc = 91 mm at location No. 2

Fig. 16. Stress and deformation distribution on corroded chain model, Dc = 84 mm at location No.3

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Fig. 17. Tension force chart of corroded chain model, Dc = 84 mm at location No.3

4.4 Comparison to Calculation of MBL According to Vicinay [4] As the formula (3), according to Vicinay the Minimum Breaking Loads of corroded chain models are expressed in the table below. Table 5. Minimum Breaking Load (MBL) results according to Vicinay Dc (mm)

MBL (kN)

Deviation from the numerical model results (%)

84

7997

0.03 – 0.57

86

8346

0.07 – 0.51

87

8553

0 – 0.22

88

8667

0 – 0.08

90

8966

0 – 0.2

91

9203

0 – 0.02

The comparison results can be expressed in the chart as Fig. 18.

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Fig. 18. Chart of MBL results comparison

4.5 Strength Checking Results According to [8], the maximum tension force at Fairlead location of mooring lines of DH01 FPU is 5422kN for extreme condition and 6826 kN for survival condition. The strength checking results, follow the expression (2), are expressed in Tables 6, 7 as below. Table 6. Strength checking of corroded chain – Extreme Condition Dc (mm)

MBL (kN)

Tension force (kN)

SF

[SF]

Conclusion

84

7997

5422

1.47

1.75

Failed

86

8346

5422

1.54

1.75

Failed

87

8553

5422

1.58

1.75

Failed

88

8667

5422

1.60

1.75

Failed

90

8966

5422

1.65

1.75

Failed

91

9203

5422

1.70

1.75

Failed

As the results in Tables 6 and 7, according to Bureau Veritas (2004) [2], formula (2), safety conditions are not satisfied for almost case of corroded chains. So it is necessary to replace the mooring line by the new one. In case of re-using the mooring line, the owner have to increase platform draft or limit the allowable environmental parameters for operating.

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MBL (kN)

Tension force (kN)

SF

[SF]

Conclusion

84

7997

6826

1.17

1.25

Failed

86

8346

6826

1.22

1.25

Failed

87

8553

6826

1.25

1.25

Failed

88

8667

6826

1.27

1.25

Pass

90

8966

6826

1.31

1.25

Pass

91

9203

6826

1.35

1.25

Pass

5 Discussion Assessment of reduced strength of the mooring lines taking into account corrosion is an important recommend for owners to decide solutions to extend life or to re-use. In the recent, the experimental formulas of Vicinay are usually used to estimate strength of corroded mooring lines in approximately with assumption that corrosion is uniform. However, the corrosion in fact is non-uniform. The paper suggests a method to estimate strength of non-uniform corroded chain links by numerical testing simulations. Based on survey data of mooring lines of DH01 FPU, there are 12 numerical testing models are performed in relation with six different types of corroded chain links. The results of tension forces at the time of failure of chain links are compared together and compared with MBL according to Vicinay. So, it can be seen that: - MBL of corroded mooring chains of numerical models and Vicinay’s formulas are the same. The maximum deviation is only 0.57%. - In testing models, we can notice that there is a very small change of MBL when corroded chain links lay at different locations. However, the models only consider to 5 chain links. The effects of number of chain links to actual strength of mooring lines may be considered in the next studies. - The results show the suitability of numerical models for determination of reduced strength of mooring chains. So, in general the numerical models can be used instead of experimental testing models with high cost. Then, the method can be applied for actual projects in Vietnamese conditions. However, due to survey data only provide corrosion diameters at intersection location of two adjacent chain links, so the testing models are only simple. Moreover, in case of assessment of chain links with local corrosions, the numerical method can be used although Vicinay formulas were not available. Authors will publish studied results of the problem in the next papers. Acknowledgements. This research is funded by National University of Civil Engineering (NUCE) under grant number 35–2019/KHXD-TÐ. We would like to thank NUCE for this funding.

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References 1. American Petroleum Institute: Recommended practice for design and analysis of stationkeeping systems for floating structures. API RP 2SK 3rd edition (2005) 2. Bureau Veritas: Classification of mooring systems for permanent offshore units, Guidance Note NI 493 DTM R00 E, Paris (2004) 3. Bureau Veritas: Quasi-dynamic analysis of mooring systems using ARIANE software, Guidance Note NI 461 DTO R00 E, Paris (1998) 4. Vicinay: Mooring lines catalogue. http://www.vicinaycadenas.net/brochure/#/30/zoomed 5. Crapps, J., et al.: Strength assessment of degraded mooring chains. Offshore Technology Conference, USA (2017) 6. Gordon, R.: Considerations for Mooring Life Extension. Society of Naval Architects and Marine Engineers, USA (2015) 7. Pham, H.H.: FPSO - Fiabilité des lignes d’ancrage avec prise en compte de fatigue, ISBN: 978–3–8381–7928–5, p. 336. Presses Académiques Francophones (2015) 8. Pham, H.H., Vu, D.C.: Re-evaluation of mooring system durability for renewal and reuse of floating offshore structures. In: Proceeding of APAC 2019 - International Conference on Asian and Pacific Coasts, pp. 1115–1122 (2019) 9. Dassault Systèmes Simulia Corp. Abaqus Theory Manual (2011) 10. Keller, C., Herbrich, U.: Plastic instability of rate dependent materials a theoretical-approach in comparison to FE analyses. In: 11th Europan LS-Dyna Conference 2017, Austria (2017) 11. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, Seventh Edition. Butterworth-Heinemann (2013) 12. PVEP. DH 01-FPU Structural and Mooring Systems Drawings (2006) 13. PVEP. DH 01-FPU Mooring Chain Survey Reports (2015) 14. PVEP: Final meteocean and environmental design criteria for Dai Hung fields, VSP.VN.2006, p. 157 (2006)

An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat Nguyen Van My and Vo Duy Hung(B) University of Science and Technology, The University of DaNang, 54 Nguyen Luong Bang Street, Danang City, Vietnam [email protected]

Abstract. The effects of hydration heat can cause the potential of cracks in towerfooting. This paper presents a case study in which the construction of mass concrete bridge foundation of Cua-Dai Extradosed Bridge in Quang Ngai, Vietnam was investigated through FEM software. The 3D-simulation will be conducted to predict the thermal performance. The temperature development profile, temperature difference, tensile stress and displacement were predicted in detail. Results showed that the heat of hydration in Cua-Dai Bridge was very high, which can cause early cracks in concrete structure. The investigation also provided clear insight into the temperature development of concrete block with complicated compositions and ambient conditions. In addition, hydration heat induced tensile stress and displacement also investigated thoroughly. Finally, the critical comments will be given. Keywords: Mass concrete · Tower-footing · Cua-Dai Bridge · Temperature · Tensile stress · Displacement

1 Introduction The early-age temperature development in mass concrete structures has a significant impact on their durability and quality. A high temperature differential can result in large temperature-induced tensile stresses that can cause early cracking. [1–4] The large thermal differential is primarily caused by a large amount of heat generated due to hydration of cement in the core of the structure that is dissipated at a very slow rate. [5–8] The proper design and construction of mass concrete can help prevent disasters such as crack or over displacement. In Vietnam, many concrete dam and long span bridge projects are actively planned or in progress. Due to effect of hydration heat, mass concrete construction and maintenances have great concern from designers and operators. The main difference between mass concrete and other types is its heat of hydration behavior. Depending on thickness of concrete block, the temperature generates from the hydration of cement can reach a true adiabatic condition in the interior. At a high temperature, the internal concrete tends to expand, while the external concrete tends to shrink and resist the expansion of internal concrete. The temperature difference between the center and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 446–457, 2022. https://doi.org/10.1007/978-981-16-3239-6_34

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the surface can cause thermal cracks in concrete block. Therefore, one of the challenges in mass concrete design and construction is analysis and control early crack due to hydration heat. Many thermal stress analyses in mass concrete have been performed by the finite element method (FEM). Midas Civil software is a multipurpose finite element analysis package that can perform 3-D analysis [9]. These components can accurately simulate the behavior of different kind of structures, from small structures to large and complex models. [10, 11] In this paper, concrete behavior due to hydration heat of Cua-Dai extradosed bridge will be examined. Firstly, 3D simulation of tower footing will be established by solid element with similar dimensions of Cua-Dai tower footing. The boundary condition of footing will be considered carefully at this stage. Then, heat convection and conduction will be simulated under considering the characteristic of concrete and ambient condition. Finally, behaviors of Cua-Dai tower footing due to hydration heat will be investigated thoroughly. The characteristic of temperature, stress, displacement due to hydration heat will be elucidated in detail.

2 Hydration Heat analysis Hydration heat analysis includes heat transfer analysis and thermal stress analysis. Heat transfer analysis is the process of calculating temperature changes over time related to heat generation, convection, thermal conductivity that occur during cement hydration. Thermal stress analysis provides stress calculations for mass concrete according to each construction phase based on changes in temperature distribution over time obtained from heat transfer analysis. Thermal stress analysis also considers changes in material properties as well as shrinkage and creep. Heat transfer analysis consists of two main parts: thermal conductivity analysis and thermal convection analysis [12]. 2.1 Heat Transfer Analysis Heat Transfer Analysis Relate to Below Definition

a) Heat Conduction Heat conduction is a form of heat transfer related to the exchange of energy from a high temperature area to a low temperature area. The heat conduction rate is proportional to the area perpendicular to the direction of conduction and the thermal gradient. According to Fourier’s law [13, 14]: Qx = −kA

∂T ∂x

Where: Qx : The amount of heat transferred, the unit is kcal / h.m.° C. k: The thermal conductivity coefficient, for saturated concrete, k = 1.21–3,11. A: The heat conductive area, perpendicular to the thermal conductivity direction. ∂T ∂x : Thermal gradient.

(1)

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b) Heat Convection The amount of heat transmitted by convection on a solid block area unit is calculated as follows [13, 14]: q = hc (T − T∞ )

(2)

Where. hc : Heat transfer coefficient between solids and liquids or air. hc depends on many factors such as the geometry of the surface, the physical properties of the liquid / air, the average temperature of the liquid surface, when calculating the concrete blocks in the air, hc can be calculated as follows: hc = 5.2 + 3.5v (v is wind speed; m/s) [13]. T: The solids surface temperature. T∞: The liquid / air ambient temperature. The amount of heat inside the concrete arises due to hydration in a unit of time and volume is [13, 14]: g=

1 rcKae−at/24 24

(3)

Where. K is the maximum temperature rise (° C). a is the reaction rate coefficient. r is the volume of concrete (kg / m3 ). t is time (day). 2.2 Thermal Stress Analysis Stress in mass concrete at each construction stage is calculated using the results of heat transfer analysis, node temperature distribution, as well as consideration of changes in material properties by time. These calculations relate to some concepts such as the equivalent age of concrete by temperature and time and cumulative temperature. The compressive strength of concrete is calculated according to the equivalent age and cumulative temperature, calculated by the following formula [14, 15]: fc (t) =

t f  (28) a + bteq c

Where. fc (t) is the concrete strength at the time of calculation. fc (28) is concrete strength at age 28 days. a, b are coefficients depending on the type of cement. t is the calculation time (day). t eq is the equivalent age of concrete.

(4)

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2.3 Boundary Condition of Heat Transfer in Concrete Block The detail boundary condition of heat transfer in tower footing of Cua-Dai Bridge shown as Fig. 1. The construction of tower footing divide into two main stages. During the construction, steel formwork is used, and the concrete block put on ground with specific heat around 25°. In first stage of construction, heat transfer through steel formwork in horizontal direction. The heat convection between concrete block and ambient air through the top surface. For second construction stage, boundary condition is quite similar to the first construction stage. The only difference is that the bottom of second concrete block will transfer heat through first block in boundary of concrete layers. Ambient Steel formwork Ambient Ambient

Steel formwork

Boundary of concrete layers

Ambient Steel formwork

Ambient

Soil ground with specific heat

Fig. 1. Boundary condition of heat transfer in bridge tower footing

3 Behaviors of Tower Footing of Cua-Dai Extradosed Bridge Due to Hydration Heat 3.1 Modelling of Concrete Block Cua-Dai extradosed brige across the Tra-Khuc River in Quang Ngai Province. It is the biggest extradosed bridge ever built in Vietnam. The largest tower-footing has dimmension as Fig. 2 and Fig. 3. Therefore, the concrete for its tower footing is very huge, more than 3000 m3 . To figure out the characteristic of hydration heat in tower-footing of Cua-Dai bridge, the tower-footing with dimension 25 m × 25 m × 5 m will be simulated by Midas Civil. The heat hydration analysis of Midas Civil already proved its reliability. The temperature by field measurement and FEM analysis are very close in term of value and temperature development [16, 17]. Figure 2, Fig. 3 and Fig. 4 illustrate the tower footing simulation by Midas Civil software. In heat hydration analysis, concrete block is simulated by solid element of 0.5 × 0.5 × 0.5 m. Two stages of construction can be seen as Fig. 2 and Fig. 3. The first construction stage relate to tower footing while the second construction stage is for tower shaft. The detail of physical characteristic of concrete and ambient environment can be seen Table 1. [14, 18]. The cement contents is 463 kg/m3 and the strength of concrete of tower footing is 35 MPa [18]. The heat convection coefficient of steel formwork and air are 12 kcal/m2 °C and 15 kcal/m2 °C respectively [14]. The detail of other paramenters can be seen in Table 1.

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Fig. 2. First construction stage of tower footing

Fig. 3. Second stage of foundation blocks

Fig. 4. Location of typical nodes

3.2 Hydration Temperature in Tower-Footing Concrete According to the analysis results, the temperature in concrete blocks is a characteristic result of the hydrothermal analysis. Due to the hydration process, the hydration temperature in the first construction stage increases sharply and then decreases with time as shown in Fig. 5. The highest temperature in first construction stage is 72.7 °C at 72 h after pouring in the center of footing-block. The high temperature zone is concentrated in the center and gradually decreases around the concrete block.

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Table 1. Characteristic of concrete [14, 15, 18] Parameters

Value

Specific heat capacity (kcal.g/kg °C)

0,25

Specific weight kg/m3

2400

Heat convection coefficient of steel formwork (kcal/m2 °C) 12 Heat convection coefficient of air (kcal/m2 °C)

15

Ambient temperature (°C)

25

Temperature at concrete pouring (°C)

20

Concrete strength 28 days (kgf/m2 )

3.5×106

Elastic modulus of concrete (kgf/m2 )

2,8.109

Coefficient of thermal expansion

1,0.10–5

Poisson coefficient

0,18

Cement content (kg/m3 )

463

From the analysis results at the temperature of 5 nodes in Fig. 5, the highest temperature is concentrated in the core of concrete block (Node N2727 and N2745). Node N4759 and node N4772 exhibit similar heat distribution after pouring 130 h, then the temperature at the node N4772 started to increase when pouring concrete of tower. Node N4759 and N6801 are located on the open surface, therefore the temperature increases rapidly after 20 h pouring concrete and gradually decreases to the ambient temperature. On the block surfaces, the temperature increases a little, then quickly decreases to the ambient. The temperature difference between surface and the core of mass is up to 40 °C at 60 h after pouring. The finding absolutely agreed with field measurement in [17, 18]. Figure 6, Fig. 7, and Fig. 8 show the hydration temperature change during time in tower footing of Cua-Dai Bridge. At first, temperature was evenly distributed in the concrete block. Later, it increases strongly in the core of block. This could be because that when concrete pours, the concrete mortar is still in liquid form, the convection and heat transfer ability is good, so the temperature is evenly distributed. Later, the heat emits from the cement hydration reaction will accumulate in the block when the concrete begins to harden. Consequently, temperature increase highly around the core on mass concrete. 3.3 Stress in Concrete Block Due to Heat of Hydration Figure 9 shows that the stress in the concrete block. At pouring time, stress is mainly the compressive stress, while the tensile stress only appears mainly in the surface and around the edges of the formwork. In this period, temperature inside of concrete block increases, then the inner concrete tends to heat up, so the stress generated mainly compression. Whereas temperature at the block-surface was decreased rapidly. The reduction of temperature makes concrete tend to shrink, but this shrink will be kept by higher temperatures of the inner layers. This is the main cause of the tensile stress appearance.

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Fig. 5. Temperature distribution of five nodes

Fig. 6. Temperature at 10 h in first construction stage

Fig. 7. Temperature at 72 h in first construction stage

When the tensile stress exceeds the allowable value, early cracks will be occurred. After 80 h, the tensile stress appears nearly all the block surface (Fig. 10). The development of the tensile strength in center of block can be seen in Fig. 11, in which the green line is allowable stress and the red line is hydration heat stress. It was found that in the temperature increase phase (60–80 h after pouring) the compressive

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Fig. 8. Temperature after 140 h in second construction stage

stress at the center increased gradually. However, during the temperature decrease phase (after 90–100 h) the compressive stress reduces due to the balance with the tensile stress in the center of block. The value of tensile stress increases gradually depending on the difference of temperature between the center and the surface of the concrete block. In parallel, the stress distribution in the concrete surface is shown as Fig. 12, in which the green and yellow are allowable stress. It is realized that at 10 h, the stress at the surface starts exceeding the allowable stress, leading to cracks occurrence. This period lasts until about 820 h. The highest stress of 4.3 MPa coincides with the time when the temperature in the concrete block reaches the highest temperature (80 h). The maximum tensile stress occurs at the concrete surface because the temperature difference in this position is the highest. Nevertheless, it should pay attention that the reinforcement did not be considered in current model. Placement of reinforcement in mass concrete can reduce the stress on surface. Hence, the tensile stress in current model might be higher when compared to reality. 3.4 Displacement Due to Heat of Hydration The displacement due to heat of hydration shows in Fig. 13. Fig. 14. Fig. 15. Fig. 16 The largest displacement in the block occurs when the temperature in the concrete block is at the strongest heat development stage, then it will reduce by time. In detail, the largest value occurred at 50 h with 5.4 mm of displacement. Then, the displacement decreases with decreasing temperature in the concrete block. This proves that the temperature change in the concrete block greatly affects the displacement of the concrete block.

Fig. 9. Tensile stress after 10 h

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Fig. 10. Tensile stress after 80 h

Fig. 11. Stress in the center of block

Fig. 12. Stress in block surface

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Fig. 13. Displacement field at 10 h

Fig. 14. Displacement field at 50 h

Fig. 15. Displacement field at 140 h

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Fig. 16. Displacement field at 1030 h

4 Conclusions The impact of heat hydration on the mass concrete of Cua-Dai Bridge’s tower-footing has been discussed in detail. The hydration temperature can cause early crack on concrete surface, so it should be carefully considered during construction not only this bridge but also the similar structure. Current study allows the following conclusions: • The maximum temperature is caused by hydration heat of cement at the center of the block is over 70°C at around 80h after pouring concrete. The temperature difference between the core and the surface of block is more than 40°C. • The largest tensile stress appearing during pouring occurred at the tower-footing surface. This tensile stress exceeded the allowable stress in early age concrete. In addition, heat hydration also created quite large displacement in concrete block of Cua-Dai Bridge tower footing, especially in the conner of concrete block. • The heat of hydration in tower-footing of Cua-Dai Bridge can cause early cracks and displacement in concrete structure. Hence, it should pay attention during construction of mass concrete of tower. The mitigation countermeasures should be consider applying to reduce the effect of hydration heat.

References 1. Ballim, Y.: A numerical model and associated calorimeter for predicting temperature profiles in mass concrete. Cem. Concr. Compos. 26, 695–703 (2004). https://doi.org/10.1016/S09589465(03)00093-3 2. Nili, M., Salehi, A.M.: Assessing the effectiveness of pozzolans in massive high-strength concrete. Constr. Build. Mater. 24, 2108–2116 (2010). https://doi.org/10.1016/j.conbuildmat. 2010.04.049 3. Choktaweekarn, P., Tangtermsirikul, S.: Effect of aggregate type, casting, thickness and curing condition on restrained strain of mass concrete. Songklanakarin J. Sci. Technol. 32, 391–402 (2010) 4. Kolani, B., et al.: Hydration of slag-blended cements. Cem. Concr. Compos. 34, 1009–1018 (2012). https://doi.org/10.1016/j.cemconcomp.2012.05.007

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5. ACI (American Concrete Institute) Guide to Mass Concrete, ACI 207–06 (2006) 6. Riding, K.A., Poole, J.L., Schindler, A.K., Juenger, M.C.G., Folliard, K.J.: Statistical determination of cracking probability for mass concrete. J. Mater. Civ. Eng. 26, 04014058 (2013). https://doi.org/10.1061/(ASCE)MT.1943-5533.0000947 7. Folliard, K.J., et al.: MeadowsPrediction model for concrete behavior: final report, Texas Dep. Transp. (2002) 8. Riding, K.A., Poole, J.L., Schindler, A.K., Juenger, M.C.G., Folliard, K.J.: Evaluation of temperature prediction methods for mass concrete members. ACI Mater. J. 103, 357–365 (2006) 9. MIDAS Information Technology (2004), Heat of Hydration- Analysis Manual (2004) 10. Kim, J.Y.: Heat Transfer Analysis 4th (Ed.) Tae Sung Software Engineering. INC (2005) 11. Akin, J.E.: Finite Element for Analysis and Design, Academic Press (1994) 12. Gebhart, B.: Heat Condition and Mass Diffusion, McGraw-Hill (1993) 13. Midas IT. Analysis Reference- Midas Civil. (2005–2006) 14. Ngo, D.Q., et al.: Modeling and analysis of bridge structures with MIDAS Civil. Construction Publishing House, 2 (2016) 15. ACI (American Concrete Institute) Cement and Concrete Terminology, ACI 116R (2000) 16. Ho, N.K, Vu, C.C.: The analysis of temperature fields and thermal stresses in mass concrete structures, J. Constr. Sci. Technol. (2002) 17. Sargam, Y., et al.: Predicting thermal performance of a mass concrete foundation- A field monitoring case study. Case Stud.Constr. Mater. 11, e00289 (2019) 18. The Design of Cua-Dai Extradosed Bridge, Project management board of transportation construction investment, Quang Ngai province (2017)

A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams Tran Van Lien1(B) , Ngo Trong Duc2 , Tran Binh Dinh1 , and Nguyen Tat Thang1 1 National University of Civil Engineering, Hanoi, Vietnam

{LienTV,dinhtb,thangnt}@nuce.edu.vn 2 Fujita Corporation Vietnam, Ho Chi Minh City, Vietnam

Abstract. This paper presents a nonlocal dynamic stiffness model (DSM) for free vibration analysis of Functionally Graded (FG) nanobeams. The nanobeam is investigated on the basis of the Nonlocal Elastic Theory (NET). The NET nanobeam model considers the length scale parameter, which can capture the small scale effect of nano structures considering the interactions of non-adjacent atoms and molecules. Material characteristics of FG nanobeams are considered nonlinearly varying throughout the thickness of the beam. The nanobeam is modelled according to the Timoshenko beam theory and its equations of motion are derived using Hamilton’s principle. The DSM is used to obtain an exact solution of the equation of motion taking into account the neutral axis position with different boundary conditions. The DSM is validated by comparing the obtained results with published results. Numerical results are presented to show the significance of the material distribution profile, nonlocal effect, and boundary conditions on the free vibration of nanobeams. It is shown that the study can be applied to other FG nanobeams as well as more complex of framed nanostructures. Keywords: DSM · Nonlocal · FG · Nanobeam · Nondimensional frequency

1 Introduction Functionally Graded Material (FGM) [1, 2] is a new generation composite material that is made up of two or more component materials with a continuous variation in the ratio of components in one or more directions. FGMs are employed in micro/nano electro-mechanical systems (MEMS/ NEMS) to archive high sensitivity and desired performance. Nano-sized structures as plates, sheets and beams are widely used in NEMS devices. Nanobeams are specially attracting more and more attention due to their various potential applications. Because of the size effect, classical elasticity theory cannot fully and accurately investigate the mechanical behaviours of nanostructures. Therefore, Nonlocal Elasticity Theory (NET) was first proposed by Eringen [3] assuming that the stress tensor at one point is not only a function of deformation but also includes all surrounding ones. Currently, NET is widely used to formulate differential equations of motion of nanostructures using homogeneous materials [4–6] and FGM materials [7]. Reddy [8] established © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 458–475, 2022. https://doi.org/10.1007/978-981-16-3239-6_35

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the equations of vibration and stability of homogeneous nanobeams according to the NET for Euler – Bernoulli, Timoshenko, Reddy and Levinson beam theories. Many other authors have developed analytical methods [9–12], Rayleight – Ritz method [13], Finite Element Method (FEM) [14–19], differential transform method [20], differential quadrature method [21],.. to consider the bending, stability and free vibration problems of nanobeams from homogeneous materials. Simsek and Yurtcu [22], Rahmani and Pedram [23] simultaneously studied bending and buckling of Timoshenko FGM beams using analytical methods. In addition, Mechab at el [24] studied on free vibration, while Uymaz [25] researched on forced vibration of nanobeams, both using the higher-order shear deformation theory. Ebrahimi and Salari [26] exploited a semi-analytical method to study the vibrational and buckling analysis of Euler – Bernoulli FGM nanobeams considering the physical neutral axis position. A spectral finite element formulation was indicated by Narendar and Gopalakrishnan [27] to investigate the vibration of nonlocal continuum beams. The analytical solutions found above are all in Navier’s series, therefore, they are limited for simply supported beams. For other boundary conditions, the authors applied FEM to analyze free vibration and buckling of FGM nanobeams according to Euler - Bernoulli beam theory [28, 29], and Timoshenko theory [30–32]. Recently, the authors of [33] solved for natural frequencies and mode shapes of microbeams under various boundary conditions using the state-space concepts. As the FEM is formulated on the base of frequency independent polynomial shape function, it could not be used to capture all necessary high frequencies and mode shapes of interest. An alternative approach called dynamic stiffness method fulfilled the gap of FEM by using frequency-dependent shape functions that are found as exact solution of vibration problem in the frequency domain [34, 35]. Although exact solutions of the vibration problem are not easily constructed for complete structures, but they, if were available, enable to study exact response of the beam in arbitrary frequency range. Adhikari et al. [4, 36] obtained the dynamic stiffness matrix of a nonlocal rod in closed form. The frequency response function obtained using the proposed DSM shows an extremely high modal density near the maximum frequency. To the best of the authors’ knowledge, the DSM based approach to the nonlocal nanobeams is a gap that has to be fulfilled. In the present work a nonlocal dynamic stiffness model is developed to investigate the free vibration characteristics of FGM nanobeams on the basis of NET and Timoshenko beam theory. Comparison between obtained results and published results shows the reliability of the method. The effect of actual neutral axis position, nonlocal size effects, material distribution profile and geometry parameters on the vibration frequency of nanobeams with different boundary conditions has been studied in detail.

2 Nonlocal Dynamic Stiffness Model of a FGM Timoshenko Nanobeam Element For a FGM nanobeam (Fig. 1), the material properties vary along the z thickness direction, and are assumed to take the form [1]:

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Fig. 1. A FGM beam

 P(z) = Pb + (Pt − Pb )

z 1 + h 2

n

h h − ≤z≤ 2 2

(1)

where P stands for elasticity modulus E, shear modulus G and mass density ρ, respectively; the subscripts t and b refer to the corresponding values of top and bottom layer materials; n is volume fraction index, and z is the coordinates from the mid-plane of the beam. The displacement field of the Timoshenko beam is given by: u(x, z, t) = u0 (x, t) − (z − h0 )θ (x, t) ; w(x, z, t) = w0 (x, t)

(2)

where u0 (x,t), w0 (x,t) are axial displacement, deflection of a point on neutral axis, respectively; h0 is the distance from neutral axis to x axis; θ is the angle of rotation of the cross-section around the y axis. From there, we get the strain components: εxx = ∂u0 /∂x − (z − h0 )∂θ/∂x ; γxz = ∂w0 /∂x − θ = −ϕ

(3)

Using notations:

A33

   (A11 , A12 , A22 ) = E(z) 1, z − h0 , (z − h0 )2 dA; A     = η G(z)dA; (I11 , I12 , I22 ) = ρ(z) 1, z − h0 , (z − h0 )2 dA A

(4)

A

where η is the shear correction factor, η = 5/6 for a rectangular cross-section. Because the modulus of elasticity of FGM material changes with height, the neutral plane of the FGM beam may not coincident with its geometrical mid-plane. Neglecting the influence of axial displacement and the nonlocality effect, while using the relations σxx = Eεxx ; σxz = Gγxz [30], neutral axis position can be determined when the axial force at cross-section vanishes: ⎡ ⎤   ∂θ ∂θ = −A12 =0 σx dA = −⎣ E(z)(z − h0 )dA⎦ ∂x ∂x A

A

This leads to A12 = 0. The position of the neutral axis h0 and I 12 can be written by: h0 =

n(RE − 1)h nbh2 ρb Rρ − RE ρt Et ; I12 = . ; Rρ = ; RE = 2(n + 2)(n + RE ) 2(2 + n) n + RE ρb Eb

(5)

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Based on Hamilton’s principle: T (δT − δU )dt = 0

(6)

0

where T and U are kinetic energy and strain energy, respectively. They can be described as:   

L  ∂u0 ∂δu0 ∂u0 ∂δθ ∂w0 ∂δw0 ∂θ ∂δu0 ∂θ ∂δθ I11 + − I12 + + I22 dx δT = ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t 0

(7) L δU =

N 0

∂δu0 ∂δθ ∂δw0 −M +Q − Qδθ dx ∂x ∂x ∂x

(8)

where N, M, Q are the axial normal force, the bending moment and the shear force, respectively:    N = σxx dA ; M = (z − h0 )σxx dA ; Q = σxz dA (9) A

A

A

Substituting Eqs. (7)–(9) into Eq. (6), we obtained: ∂ 2 u0 ∂ 2 θ ∂Q ∂ 2 w0 ∂M ∂ 2 u0 ∂ 2θ ∂N = I11 2 − I12 2 ; = I11 2 ; − Q = I12 2 − I22 2 ∂x ∂t ∂t ∂x ∂t ∂x ∂t ∂t

(10)

and the boundary conditions are: u0 = 0 or N = 0; w0 = 0 or Q = 0; θ = 0 or M = 0

(11)

The nonlocal constitute equations for nanobeams can be written in form [3]: σxx − μ

∂ 2 σxx ∂ 2 σxz = Eε ; σ − μ = Gγxz xx xz ∂x2 ∂x2

(12)

where μ = e02 a2 is a nonlocal parameter; e0 is a constant associated with each material; a is an internal characteristic length. Taking the derivative Eq. (10) and using Eqs. (3),(12), we obtain:     ∂w0 ∂u0 ∂θ ∂ 3 u0 ∂ 3θ ∂ 3 w0 − A12 + μ I11 − θ + μI11 N = A11 − I ; Q = A 12 33 2 2 ∂x ∂x ∂x ∂x∂t ∂x∂t ∂x∂t 2   ∂u0 ∂θ ∂ 2 w0 ∂ 3 u0 ∂ 3θ M = A12 − I22 − A22 + μ I11 2 + I12 2 ∂x ∂x ∂t ∂x∂t ∂x∂t 2

(13)

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Substituting Eq. (13) into Eq. (10) leads to the equations of free vibration of the FGM nanobeam:

A22

∂2θ ∂x2

  ∂ 2 u0 ∂2θ ∂ 2 u0 ∂2θ ∂ 4 u0 ∂4θ A11 2 − A12 2 − I11 2 + I12 2 + μ I11 2 2 − I12 2 2 = 0 ∂x ∂x ∂t ∂t ∂x ∂t ∂x ∂t   2 2 2 4 ∂w0 ∂ u0 ∂ θ ∂ u0 ∂ u0 ∂4θ − θ − I22 2 + I12 2 − μI12 2 2 + μI22 2 2 = 0 − A12 2 + A33 ∂x ∂x ∂t ∂t ∂x ∂t ∂x ∂t   ∂ 2 w0 ∂ 2 w0 ∂ 4 w0 ∂θ − I11 2 + μI11 2 2 = 0 A33 − ∂x ∂x2 ∂t ∂x ∂t

and the corresponding boundary conditions (11) can be written in form:   ∂u0 ∂θ ∂ 3 u0 ∂ 3θ =0 − A12 + μ I11 − I12 u0 = 0 or A11 ∂x ∂x ∂x∂t 2 ∂x∂t 2   ∂w0 ∂ 3 w0 w0 = 0 or A33 − θ + μI11 =0 ∂x ∂x∂t 2   ∂u0 ∂θ ∂ 2 w0 ∂ 3 u0 ∂ 3θ θ0 = 0 or A12 − I =0 − A22 + μ I11 2 + I12 22 ∂x ∂x ∂t ∂x∂t 2 ∂x∂t 2

(14)

(15)

When A12 = 0 and I 12 = 0, which means n = 0 or RE = Rρ , the first equation of (14) is independent to the others two equations. Meanwhile, the axial vibration of the FGM beam is uncoupled with the bending vibration similar to that of a homogeneous beam. The following two equations are similar in form to the bending vibration equation of the Timoshenko beam, but the coefficients depend on the parameters of FGMs. Setting: ∞ {U , , W } =

{u0 (x, t), θ (x, t), w0 (x, t)}e−iωt dt

(16)

−∞

Equation (14) in the frequency domain can be obtained as:   d 2U   d 2 − A12 − μI12 ω2 + I11 ω2 U − I12 ω2 = 0 A11 − μI11 ω2 dx2 dx2   d 2U   2   2 d + A dW − I ω2 U + I ω2 − A + A − μI ω =0 − A12 − μI12 ω2 22 22 33 12 22 33 dx dx2 dx2  d 2W  d A33 − μI11 ω2 + I11 ω2 W = 0 − A33 dx dx2

(17)

Putting into the matrices and vectors are defined as following:   ⎛ ⎞ 2 2 0   A11 − μI11 ω 2  − A12 − μI12 ω2  ⎠ A = ⎝ − A12 − μI12 ω A22 − μI22 ω 0 0 0 A33 − μI11 ω2 ⎧ ⎫ ⎛ ⎞ ⎞ ⎛ 0 0 0 −I12 ω2 0 I11 ω2 ⎨U ⎬      B = ⎝ 0 0 A33 ⎠ ;  C = ⎝ −I12 ω2 I22 ω2 − A33 0 ⎠ ; {z} = ⎩ ⎭ 0 −A33 0 0 0 I11 ω2 W (18)

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Equation (17) now can be described in form of:          A z +  B z +  C {z} = {0}

(19)

Choosing the solutions of Eq. (19) in the form of {z0 } = {d}eλx leads to        λ2  A +λ  B +  C {d} = {0}

(20)

Equation (20) have non-trivial solutions when  ! " ! " ! " ˜ =0 det λ2 A˜ + λ B˜ + D

(21)

We receive cubic equations of η = λ2 : η3 +aη2 +bη +c = 0. Using notations η1 , η2 , η3 are solutions of the cubic equations and λ1,4 = ±k1 ; λ2,5 = ±k2 ; λ3,6 = ±k3 ; kj =



ηj ; j = 1, 2, 3

Then the general solutions of Eq. (19) are in form as {z0 (x, ω)} =

(22)

6  # dj eλj x . From j=1

the first and the third equations of (20), we yield ⎡

α1 C 1 ⎢ {z0 (x, ω)} = ⎣ C1 β1 C 1

α2 C 2 C2 β2 C 2

α3 C 3 C3 β3 C 3

α4 C 4 C4 β4 C 4

α5 C 5 C5 β5 C 5

⎤ α6 C 6 'T & ⎥ C6 ⎦ · ek1 x ek2 x ek3 x e−k1 x e−k2 x e−k3 x β6 C 6

(23) where {C} = (C1 , ..., C 6 )T are independent constants and   A12 − μI12 ω2 k12 + I12 ω2 A33 k1   α1 =  = α4 ; β1 =  = −β4 A11 − μI11 ω2 k12 + I11 ω2 A33 − μI11 ω2 k12 + I11 ω2 (24) Similarly, α2 = α5 ; β2 = −β5 ; α3 = α6 ; β3 = −β6 , Eq. (23) can be rewritten in the form: {z0 (x, ω)} = [G(x, ω) ]{C}

(25)

⎤ α1 ek1 x α2 ek2 x α3 ek3 x α1 e−k1 x α2 e−k2 x α3 e−k3 x [G(x,ω)] = ⎣ ek1 x ek2 x ek3 x e−k1 x e−k2 x e−k3 x ⎦ k x k x k x −k x −k x 1 2 3 1 2 β1 e β2 e β3 e −β1 e −β2 e −β3 e−k3 x

(26)

where ⎡

Let’s consider an FGM nanobeam element as shown in Fig. 2. Nodal displacements and forces of the element are introduced as & ' ˆ e = {U1 , 1 , W1 , U2 , 2 , W2 }T ; {P e } = {N1 , M1 , Q1 , N2 , M2 , Q2 }T U (27)

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where U1 = z1 (0, ω) ; 1 = z2 (0, ω) ; W1 = z3 (0, ω); U2 = z1 (L, ω) ; 2 = z2 (L, ω) ; W2 = z3 (L, ω)   "  " !  ! ; Q1 = − A33 − μI11 ω2 ∂x z3 − A33 z2 N1 = − A11 − μI11 ω2 ∂x z 1 − A12 − μI12 ω2 ∂x z2 x=0 x=0 !    " M1 = − A12 − μI12 ω2 ∂x z1 − A22 − μI22 ω2 ∂x z2 − μI11 ω2 z3 x=0   "  " !  ! N2 = A11 − μI11 ω2 ∂x z 1 − A12 − μI12 ω2 ∂x z2 ; Q2 = A33 − μI11 ω2 ∂x z3 − A33 z2 x=L x=L !    " 2 2 2 M2 = A12 − μI12 ω ∂x z1 − A22 − μI22 ω ∂x z2 − μI11 ω z3 x=L

(28)

y Q2

Q1 L

N1

W1

W2 U1

Θ1

N2

j

i M1

x M2

U2 Θ2

Fig. 2. Node co-ordinates, nodal loads of a nanobeam element

Substituting Eq. (25) into Eq. (28), we get   & ' [G(0, ω)] −B (G)x=0 · {C} Ue = .{C} ; {Pe } =  F BF (G)x=L [G(L, ω)] (

with [BF ] is the matrix operator    ⎤ ⎡ 2 ∂ − A − μI ω2 ∂ − μI ω 0 A 11 11 x 12 12 x     2 ⎦ [BF ] = ⎣ A12 − μI12 ω2 ∂x − A22 − μI22 ω2 ∂x  −μI11 ω 2  A33 − μI11 ω ∂x 0 −A33

(29)

(30)

Eliminating constant vector {C} in Eq. (29) leads to {Pe (ω)} =

 

−1 & ' ! "& ' −BF (G)x=0 · [G(0, ω)] . Ue = Ke (ω) . Ue BF (G)x=L [G(L, ω)] (

(

(

(31)

! " where Kˆ e is dynamic stiffness matrix of the FGM nanobeam element. For a given structure that consists of a number of FGM nanobeam elements like above, by means of balancing all the internal forces at every nodes of the structure, there

A Nonlocal Dynamic Stiffness Model for Free Vibration (

465

(

will be obtained total dynamic stiffness matrix K(ω) and nodal load vector P. Letting U be the total DOF vector of the structure, equation of motion of which conducted by the dynamic stiffness method is ! "& ' & ' ˆ = Pˆ K (ω) . U (32) (

(

Therefore, natural frequencies {ω} = {ω1 ω2 ... ωn } can be found from of the equation ˆ det[K(ω)] =0

(33)

and mode shape related to the natural frequency ωj is

  & '  0  ϕj (x) = Cj G x, ωj Uj (

(34)

where ˆ [G(x, ω)] = [G(x, ω)] ·



[G(0, ω)] [G(L, ω)]

−1 (35)

& ' ˆ j is the normalized solution corresponding to ωj . Cj0 is an arbitrary constant and U

3 Numerical Results and Discussion 3.1 Comparison of Calculation Results and Published Results In this subsection, the numerical results are compared with the published results to validate the present study. Firstly, the first five natural frequencies of a simply supported FGM beam are compared with results of Simsek [37] for the following geometric and material properties: L = 0.5 m, h = 0.125 m, E t = 70 Gpa, ρ t = 2707 kg/m3 , E b = 210 Gpa, ρ b = 7850 kg/m3 ,vt = vb = 0.3 and volume fraction index n = 1. As seen from Table 1, the present frequencies are in good agreement with the results of Simsek and the difference between the frequencies of the two studies is very small. Table 1. Comparison of the first five natural frequencies (rad/s) of a simply supported FGM beam 1st freq 2nd freq

3rd freq

4th freq

5th freq

Simsek [37] 6443.08 21470.95 39775.55 59092.37 78638.36 Present

λ01

6443.09 21470.95 39775.55 59092.37 78637.78

The second * comparison is made for the nondimensional fundamental frequencies ) ) 2 = ω1 .L h. ρt Et with various boundary conditions for the following geomet-

ric and material properties: L/h = 5, E t = 70 GPa, ρ t = 2702 kg/m3 , E b = 280 GPa,

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ρ b = 3960 kg/m3 , vt = vb = 0.3. The results of the proposed model are compared to the results for five different higher order beam theories: parabolic shear deformation theory (PSDBT) [37], higher order shear deformation (HSDBT) [38], Reddy-Bickford beam theory (RBT) [39], a quasi-3D theory [40] and first order shear deformation theory (FSDBT) [32]. As seen from Table 2, the present dynamic stiffness model is in a good agreement with the two analytical [37, 38] and three finite element [32, 39, 40] models. The*third comparison is made for the nondimensional fundamental frequencies λ1 = ) ω1 .L2 . ρt A Et I of homegenous nonlocal beams with four nonlocal beam theories: the Euler-Bernoulli (EBT), the Timoshenko beam theory (TBT), the Reddy beam theory (RBT) and Levinson beam theory (LBT). The results from present nonlocal dynamic stiffness model are compared to analytical solutions of Reddy [8] in Table 3 with different nonlocal parameters. As seen from Table 3, the proposed model is very accurate estimates for the nondimensional fundamental frequencies of homegenous nonlocal nanobeams. Finally, consider a simply supported FGM nanobeam with geometric and material parameters as follows: L = 10 nm, b = h = L/10 = 1 nm; E t = 70 Gpa, Gt = 26 Gpa, 3 The calculated ρ t = 2700 kg/m3 ; E b = 393 Gpa, Gb = 157 Gpa, ρ b = 3960 kg/m * [30]. ) 2 results for three first nondimensional frequencies λi = ωi .L . ρt A Et I ; i = 1, 2, 3 with regard to the neutral axis (NA) and mid-plane axis (MA) are compared to the results published by Eltaher et al. [30] which used FEM with 100 elements shown in Table 4. It can be seen that the results are quite consistent, especially when taking into account the actual position of the neutral axis, the error is less than 0.5%. Comparisons of calculation results and published results show the reliability of the proposed nonlocal dynamic stiffness model. 3.2 Effects of Neutral Axis Position and Nonlocal Parameter on the Nondimensional Frequency Figure 3 illustrates the variation of nondimensional frequency of the simply supported beam considering the neutral axis and the mid-plane axis regarding different nonlocal coefficient μ. The first nondimensional frequency is greatly influenced by the volume fraction index n and the neutral axis position. Simutaneostly, the first nondimensional frequency calculated according to the actual neutral axis position is always lower than this in the mid-plane axis case. Besides, the relation (5) shows that the neutral axis position is independent from the nolocal coefficient μ. In later calculation, we only consider the case that the neutral axis is corresponding to the actual position. Figure 4 shows the variation of 15 first nondimensional frequencies of a simply supported FGM Timoshenko beam with different nonlocal coefficient μ with volume fraction index n = 5. It is obvious that the nondimensional frequencies decrease significantly as the non-local coefficient μ increases, especially for high nondimensional frequencies, and thus the small scale effect cannot be neglect. The same effect is showed for any volume fraction index. In sum, the nonlocal beam theory should be used if one needs accurate predictions of high frequencies of nanobeams.

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+ * ) Table 2. Comparison of the nondimensional fundamental frequencies λ01 = ω1 .L2 h. ρt Et of FGM beams with various boundary conditions and volume fraction index n Pinned – Pinned 1/n

PSDBT [37]

HSDBT [38]

RBT [39]

Quasi-3D [40]

FSDBT [32]

Present

0

5.15274

5.1528

5.1528

5.1618

5.1525

5.1524

0.5

4.41108

4.4102

4.4019

4.4240

4.2312

4.2449

1.0

3.99042

3.9904

3.9716

4.0079

3.9708

3.9903

2.0

3.62643

3.6264

3.5979

3.6442

3.7051

3.7248

5.0

3.40120

3.4009

3.3743

3.4133

3.3605

3.3722

10

3.28160

3.2815

3.2653

3.2903

3.1307

3.1362



2.67732

-

-

-

2.6771

2.6772

Fixed- Fixed 1/n

PSDBT [37]

HSDBT [38]

RBT [39]

Quasi-3D [40]

FSDBT [32]

Present

0

10.07050

10.0726

10.0678

10.1851

9.9975

9.9974

0.5

8.74670

8.7463

8.7457

8.8641

8.4251

8.4251

1.0

7.95030

7.9518

7.9522

8.0770

7.8998

7.8998

2.0

7.17670

7.1776

7.1801

7.3039

7.3228

7.3228

5.0

6.49349

6.4929

6.4961

6.5960

6.5579

6.5579

10

6.16515

6.1658

6.1662

6.2475

6.0695

6.0695



5.23254

-

-

-

5.1946

5.1947

Fixed- Free 1/n

PSDBT [37]

HSDBT [38]

RBT [39]

Quasi-3D [40]

FSDBT [32]

Present

0

1.89523

1.8957

1.8952

1.9055

1.8944

1.8942

0.5

1.16817

1.1612

1.6180

1.6313

1.5547

1.5548

1.0

1.46328

1.4636

1.4633

1.4804

1.4627

1.4628

2.0

1.33254

1.3328

1.3326

1.3524

1.3674

1.3674

5.0

1.25916

1.2594

1.2592

1.2763

1.2401

1.2402

10

1.21834

1.2187

1.2184

1.2308

1.1504

1.1540



0.98474

-

-

-

0.9843

0.9843

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* ) Table 3. Comparison of the nondimensional fundamental frequencies λ1 = ω1 .L2 . ρt A Et I of a simply supported beam with various nolocal parameter (L/h = 10) μ (10−18 ) EBT [8] TBT [8] RBT [8] LBT [8] Present 0

9.8696

9.7454

9.7454

9.7657 9.7451

0.5

9.6347

9.5135

9.5135

9.5333 9.5132

1.0

9.4159

9.2973

9.2974

9.3168 9.2971

1.5

9.2113

9.0953

9.0954

9.1144 9.0951

2.0

9.0195

8.9059

8.9060

8.9246 8.9057

2.5

8.8392

8.7279

8.7279

8.7462 8.7277

3.0

8.6693

8.5601

8.5602

8.5780 8.5599

3.5

8.5088

8.4017

8.4017

8.4193 8.4015

4.0

8.3569

8.2517

8.2517

8.2690 8.2515

4.5

8.2129

8.1095

8.1095

8.1265 8.1093

5.0

8.0761

7.9744

7.9744

7.9911 7.9742

Table 4. Comparison of nondimensional frequencies of the beam with results of Eltaher et al. Frqs μ (10-18) ( )

0.0

2.0

4.0

MA (Present)

1 2 3

9.7075 37.0962 78.1547

1 2 3

17.0278 64.1874 136.4469

1 2 3

8.8713 27.7303 46.9034

1 2 3

15.5611 48.0080 81.9216

1 2 3

8.2196 23.0989 36.6272

1 2 3

14.4179 39.9980 63.9788

NA MA NA MA NA MA (Present) ([30]) ([30]) (Present) (Present) ([30]) n=0.0 n=0.5 9.7075 9.7032 9.7032 14.5998 13.6072 14.8669 37.0962 37.0382 37.0382 51.8360 52.1179 52.7163 78.1547 77.9135 77.9135 110.6544 110.1423 119.2343 n=5 n=10 16.7305 17.1057 16.7366 17.6526 17.5472 17.6845 64.2634 65.6963 64.3491 67.2760 67.3039 67.8750 136.2956 139.3094 136.6522 142.5265 142.4755 143.8072 n=0.0 n=0.5 8.8713 8.8674 8.8674 13.3416 12.4351 13.5863 27.7303 27.6870 27.6870 38.8463 38.9595 42.3969 46.9034 46.7622 46.7622 66.5345 66.1005 71.5567 n=5 n=10 15.2894 15.6323 15.2950 16.1321 16.0358 16.1613 48.0385 49.1097 48.1026 50.3002 50.3114 50.7383 81.7958 83.6044 82.0098 85.5480 85.5045 86.3037 n=0.0 n=0.5 8.2196 8.2160 8.2160 12.3611 11.5216 12.5883 23.0989 23.0628 23.0628 32.3883 32.4527 35.3159 36.6272 36.5169 36.5169 51.9778 51.6184 55.8791 n=5 n=10 14.1662 14.4840 14.1714 14.9470 14.8578 14.9741 40.0154 40.9076 40.0687 41.9022 41.9086 42.2642 63.8749 65.2873 64.0420 66.8071 66.7711 67.3952

NA ([30]) 13.6100 52.1595 110.3186 17.5542 67.4009 142.8763 12.4377 38.9905 66.2061 16.0421 50.3839 85.7451 11.5241 32.4785 51.7008 14.8637 41.9690 66.6590

A Nonlocal Dynamic Stiffness Model for Free Vibration

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Fig. 3. Variation of first nondimensional frequency of the simply supported FGM Timoshenko beam with different volume fraction index n and nonlocal parameter μ

Fig. 4. Variation of nondimensional frequencies of the simply supported FGM Timoshenko beam with different nonlocal parameter μ in case volume fraction index n = 5

3.3 Effect of Boundary Conditions on Nondimensional Frequencies Tables 5–7 show the results of three first frequencies considering with the neutral axis for a beam with simply support - clamped (S-C) boundary condition (Table 5), clamped ends (C-C) boundary conditions (Table 6) and cantilever (C-F) boundary conditions (Table 7), respectively. It can be seen that cantilever nanobeam has smaller nondimensional frequencies than the simply supported nanobeam with the same geometric and material characteristics. Simultaneously, nondimensional frequencies of simply supported nanobeam are smaller than the nondimensional frequencies of clamped ends beam as well as normal macro-beams. Especially, the frequencies of FGM Timoshenko

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nanobeam decrease gradually when the nonlocal coefficient increases, except the case of the fundamental frequency of cantilever beam, which gradually increases. This finding was also mentioned in [14, 41] when considering the vibration of Euler-Bernoulli and Timoshenko homogenenous nanobeam, respectively. Table 5. Nondimensional frequencies of FGM Timoshenko beam with S-C boundary conditions μ Frqs n (10−19 ) (λi ) n = 0.0 0.0

1.0

2.0

3.0

4.0

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

n = 10

1

14.8361

18.8361

20.8406

22.3482

23.7961

25.6493

26.8765

2

45.3006

57.5081

63.8553

68.6619

73.2147

78.8466

82.4600

3

87.7203 111.3415 124.1009 133.8410 142.9432 153.7930 160.5041

1

14.0555

17.8442

19.7415

21.1688

22.5404

24.2969

25.4604

2

37.9963

48.2260

53.5290

57.5488

61.3649

66.0974

69.1389

3

63.3329

80.3602

89.5092

96.5040 103.0665 110.9269 115.8056

1

13.3841

16.9911

18.7965

20.1549

21.4609

23.1341

24.2427

2

33.3665

42.3453

46.9929

50.5178

53.8679

58.0278

60.7035

3

52.0901

66.0861

73.5934

79.3369

84.7328

91.2057

95.2275

1

12.7989

16.2477

17.9731

19.2716

20.5204

22.1209

23.1815

2

30.1021

38.2000

42.3879

45.5654

48.5873

52.3424

54.7589

3

45.2911

57.4563

63.9763

68.9663

73.6574

79.2890

82.7897

1

12.2831

15.5925

17.2476

18.4934

19.6918

21.2281

22.2464

2

27.6430

35.0777

38.9206

41.8369

44.6118

48.0615

50.2821

3

40.6156

51.5227

57.3657

61.8386

66.0455

71.0976

74.2389

3.4 Effect of Nanobeam Length on the Nondimensional Frequencies Figures 5a–c illustrate the variation of three first natural frequencies of the simply supported FGM Timoshenko beam with different nonlocal coefficient μ in case of volume fraction index n = 5 and with the ratios: L/h = 10, 20, 30, 40, 50, respectively. As the nonlocal coefficient μ increases, there is a significant decrease in the natural frequencies, especially when the ratio L/h in a range of 10–30. It is obvious that non-local coefficient shows to be more influenced on the high frequencies compared to the lower ones.

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Table 6. Nondimensional frequencies of Timoshenko FGM beam with C-C boundary conditions μ Frqs n (10−19 ) (λi ) n = 0.0 0.0

1.0

2.0

3.0

4.0

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

n = 10

1

20.9722

26.6388

29.5363

31.7150

33.7859

36.3901

38.0894

2

53.7468

68.2727

76.0277

81.9004

87.3889

94.0134

98.1736

3

97.1430 123.3937 137.9543 149.0651 159.3058 171.1992 178.3833

1

19.8080

25.1564

27.8870

29.9418

31.8977

34.3602

35.9682

2

44.6699

56.7189

63.1132

67.9657

72.5217

78.0500

81.5340

3

69.6280

88.3834

98.6814 106.5632 113.8842 122.4700 127.6931

1

18.8121

23.8890

26.4778

28.4272

30.2848

32.6257

34.1551

2

38.9900

49.4959

55.0549

59.2783

63.2530

68.0883

71.1408

3

57.1223

72.4901

80.9002

87.3453

93.3479 100.4092 104.7140

1

17.9486

22.7904

25.2570

27.1153

28.8876

31.1228

32.5836

2

35.0220

44.4527

49.4343

53.2218

56.7911

61.1397

63.8874

3

49.6190

62.9594

70.2486

75.8388

81.0521

87.1936

90.9413

1

17.1910

21.8268

24.1867

25.9655

27.6630

29.8052

31.2056

2

32.0537

40.6814

45.2339

48.6968

51.9633

55.9464

58.4647

3

44.4842

56.4391

62.9654

67.9730

72.6468

78.1571

81.5212

Table 7. Nondimensional frequencies of cantilever (C-F) Timoshenko FGM beam μ (10−19 ) Frqs (λi ) n 0.0

1.0

2.0

3.0

4.0

1

n = 0.0

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

3.4893

4.4277

4.8875

5.2327

5.5679

n = 10

6.0059

6.3008

2

20.9127 26.5248 29.3562 31.4989 33.5595 36.1810

37.9040

3

54.4253 69.1555 77.2754 83.2275 88.8537 95.7577 100.1455

1

6.0307

6.3269

2

19.6114 24.8746 27.5311 29.5415 31.4744 33.9325

35.5475

3

45.4655 57.6334 64.0013 68.8667 73.4910 79.1785

82.7965

1

3.5037

18.4917 23.4547 25.9605 27.8567 29.6796 31.9971

33.5194

3

39.6347 50.2509 55.8058 60.0472 64.0769 69.0319

72.1833 6.3808

2

17.5142 22.2150 24.5888 26.3852 28.1118 30.3067

31.7483

3

35.6308 45.1771 50.1736 53.9879 57.6109 62.0641

64.8954

3.5489

4.5032

4.9711

5.2992

5.6146

6.0823

1

4.9494

5.2765

5.5910

2

4.4836

4.9282

5.2544

6.3535

3.5334

4.4645

4.9076

6.0562

1

3.5184

4.4459

5.3225

5.6388

5.6636

6.1090

6.4089

2

16.6503 21.1194 23.3763 25.0842 26.7258 28.8123

30.1827

3

32.6801 41.4369 46.0221 49.5222 52.8461 56.9298

59.5252

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Fig. 5. Variation of first three nondimensional frequencies of the simply supported FGM Timoshenko beam with different nonlocal parameter μ and ratios L/h in case of n = 5.

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4 Conclusion In this work, free vibration of FGM Timoshenko nanobeams is investigated. Based on Haminton’s principle and Nonlocal Elasticity Theory, the differential equations of vibration of a FGM nanobeam element is derived. A nonlocal dynamic stiffness model is developed to obtain the exact solution of these equations to taking into account the actual neutral axis. Comparison with published results of other authors shows the reliability of the proposed model. On that basis, the influence of nonlocal parameters, materials, geometric parameters and boundary conditions on the vibration frequencies of nanobeams is investigated. The analysis results show that the first frequency counting on the actual neutral axis position is always lower than the one counting on the mid-plane axis. The frequencies of Timoshenko FGM nanobeams decrease gradually as the nonlocal coefficient increases in all cases of boundary conditions except the first frequency of cantilever beams. Simutaneously, the effect of nonlocal coefficients on high frequencies is much greater than lower ones. In this regards, the studies can be extended to other types of FGM materials as well as more complex types of beam structures. Acknowledgement. This research is funded by National University of Civil Engineering under grant number 35–2020/KHXD-TÐ.

References 1. Shen, H.-S.: Functionally graded materials: nonlinear analysis of plates and shells. CRC press (2016) 2. Mahamood, R.M., Akinlabi, E.T.: Functionally graded materials. Springer (2017) 3. Eringen, A.C.: Nonlocal continuum field theories. Springer Science & Business Media (2002) 4. Karlicic, D., et al.: Non-local structural mechanics. John Wiley & Sons (2015) 5. Polizzotto, C.: Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38(42–43), 7359–7380 (2001) 6. Eltaher, M., Khater, M., Emam, S.A.: A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40(5–6), 4109–4128 (2016) 7. Salehipour, H., Shahidi, A., Nahvi, H.: Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015) 8. Reddy, J.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007) 9. Wang, C., Zhang, Y., He, X.: Vibration of nonlocal Timoshenko beams. Nanotechnol. 18(10), 105401 (2007) 10. Li, C., et al.: Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11(02), 257–271 (2011) 11. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E. 41(9), 1651–1655 (2009) 12. Wang, C., et al.: Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech. 134(6), 475–481 (2008) 13. Chakraverty, S., Behera, L.: Free vibration of non-uniform nanobeams using Rayleigh-Ritz method. Phys. E. 67, 38–46 (2015)

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14. Eltaher, M., Alshorbagy, A.E., Mahmoud, F.: Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37(7), 4787–4797 (2013) 15. Eltaher, M., et al.: Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl. Math. Comput. 224, 760–774 (2013) 16. Pradhan, S.: Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finite Elem. Anal. Des. 50, 8–20 (2012) 17. de Sciarra, F.M.: Finite element modelling of nonlocal beams. Phys. E. 59, 144–149 (2014) 18. Tuna, M., Kirca, M.: Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Compos. Struct. 179, 269–284 (2017) 19. Alotta, G., Failla, G., Zingales, M.: Finite element method for a nonlocal Timoshenko beam model. Finite Elem. Anal. Des. 89, 77–92 (2014) 20. Ebrahimi, F., Nasirzadeh, P.: A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method. J. Theor. Appl. Mech. 53(4), 041–1052 (2015) 21. Jena, S.K., Chakraverty, S.: Free vibration analysis of variable cross-section single layered graphene nano-ribbons (SLGNRs) using differential quadrature method. Frontiers Built Environ. 4, 63 (2018) 22. Sim¸ ¸ sek, M., Yurtcu, H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013) 23. Rahmani, O., Pedram, O.: Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci. 77, 55–70 (2014) 24. Mechab, I., El Meiche, N., Bernard, F.: Free vibration analysis of higher-order shear elasticity nanocomposite beams with consideration of nonlocal elasticity and poisson effect. J. Nanomech. Micromech. 6(3), 04016006 (2016) 25. Uymaz, B.: Forced vibration analysis of functionally graded beams using nonlocal elasticity. Compos. Struct. 105, 227–239 (2013) 26. Ebrahimi, F., Salari, E.: A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. CMES: Comput. Model. Eng. Sci. 105(2), 151–181 (2015) 27. Narendar, S., Gopalakrishnan, S.: Spectral finite element formulation for nanorods via nonlocal continuum mechanics. J. Appl. Mech. 78(6), 061018 (2011) 28. Eltaher, M., Alshorbagy, A., Mahmoud, F.: Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos. Struct. 99, 193–201 (2013) 29. Eltaher, M., Emam, S.A., Mahmoud, F.: Free vibration analysis of functionally graded sizedependent nanobeams. Appl. Math. Comput. 218(14), 7406–7420 (2012) 30. Eltaher, M., et al.: Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl. Math. Comput. 235, 512–529 (2014) 31. Eltaher, M., et al.: Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014) 32. Aria, A., Friswell, M.: A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos. B Eng. 166, 233–246 (2019) 33. Trinh, L.C., et al.: Size-dependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions. Compos. B Eng. 134, 225–245 (2018) 34. Su, H., Banerjee, J.: Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput. Struct. 147, 107–116 (2015) 35. Van tran, L., Ngo, D.T., Nguyen, K.T.: Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method. (2018)

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Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings Nguyen Anh Dung(B) and Nguyen Vinh Sang 175 Tay Son, Dong Da, Hanoi, Vietnam

Abstract. Use of high damping rubber bearings (HDRBs) in seismic isolation is a viable practice to protect bridges from earthquakes. The behavior of HDRB under cyclic load is dominated by nonlinear rate-dependent elasto-plastic responses and low temperatures. Rate-dependency effect due to viscosity depends strongly on the current strain but the effect is vividly different in loading/unloading phases. A rheology model, proposed recently for HDRB, can reproduce the behaviors of HDRB under cyclic loads. In this paper, the effects of incorporating the rheology model in simulating the seismic responses of a bridge superstructure-pier-foundation (S-PF) system are investigated. In addition, the change of the model parameters at low temepratures affacting to the dynamic responses of the bridge is also investigated in this study. To this end, nonlinear dynamic analysis of a six span continuous seismically isolated bridge is conducted for two different strong earthquake ground motions. The nonlinear hysteretic behaviors of bridge piers are considered using Takeda tri-linear model. The nonlinearity is restricted to be lumped in plastic hinges located at the bottom of piers. Three temperature conditions are considered in the analysis: the room temperature (+23 °C) and the two low temperatures (−10 °C and −30 °C) for checking the low temperature effect. Two models for the isolation bearings are considered for comparison: the conventional bilinear model [1, 2] used in design practice and the ratedependent rheology model [3]. To evaluate the time-history response of the bridge system, a solution algorithm has been developed to solve the equations of motion of the system and the first order differential equation governing the nonlinear rate-dependent behavior of HDRB. The solution algorithm is successfully implemented in general purpose finite element code. The implication of using the rheology model in response prediction of the S-P-F system is studied by comparing the rotation of pier and shear strain of the bearing obtained using the bilinear model and the rheology model. The comparison suggests that the modeling of HDRB considering the nonlinearities due to elasticity and viscosity effects is vital for rational prediction of the seismic response of highway bridges. Keywords: Dynamic analysis · Low temperature effect · Modeling effect · High damping rubber bearings

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 476–488, 2022. https://doi.org/10.1007/978-981-16-3239-6_36

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1 Introduction Bridges are a crucial part of the overall transportation system as they play very important roles in evacuation and emergency routes for rescues, first-aid, firefighting, medical services and transporting disaster commodities to expatriates. In this regard, bridges serve as a transportation lifeline of modern society. In view of the importance of the bridge structure, it is a contemporary key issue to minimize as much as possible the loss of the bridge functions against earthquakes to enhance continued functioning of the community life. However, bridges may pose a serious threat to public safety during extreme events like earthquakes. In the recent major earthquakes such as the 1994 Northridge earthquake, the 1995 Kobe earthquake, the 1999 Chi-Chi earthquake, the 2008 Wenchuan earthquake, and the 2010 Chile earthquake, highway bridges suffered from a variety of structural damages [4–7]. To reduce the damage concentration at the piers of an ordinary bridge, an effective way is to adopt an isolated bridge, which utilizes a seismic dissipation device that limits forces transferred from the superstructure to the substructure while accommodating the concomitant displacement [8]. Rubber bearings is the one of most effective isolators, economical laminated-rubber bearings featuring alternate layers of rubber and steel sheets have been extensively adopted in highway bridges. The bearings possess the capacities of sustaining large vertical loads and accommodating thermal movements of the superstructure with low maintenance [9]. There are three types of laminated rubber bearings: natural rubber bearings, lead rubber bearings, and high damping rubber bearings (HDRB). More recently, high damping rubber bearings have been widely used in Japan due to their high flexibility and high damping capability. In the manufacture process of high damping rubber material, a large amount of fillers (about 30%) including carbon black, silica, oils and some other particles is added during the vulcanization process [10, 11] in order to improve the desirable material properties, such as the strength and damping capabilities. HDRB consist of alternative layers of high damping rubber and steel plates bonded by vulcanization. They have some similar behavior to other elastomeric bearings, such as being able to support vertical loads with limited or negligible deflection and horizontal loads with large deflections. However, HDRB are characterized by some special properties that are very different from standard elastomeric bearings. They present high damping capability, and bearings defined as HDRB should provide an equivalent viscous damping of at least 10% [12]. HDRB also present a very useful property for the application in the base isolation of structures. They are very stiff for small deformation and soft for large deformation. This property allows the structure to respond rigidly to small excitations like braking forces and provides high flexibility for large excitations like earthquakes. Moreover, high damping rubber material is very durable. Gu and Itoh (2006) [13] estimated that the change of the equivalent horizontal stiffness of HDRB is about 10–25% after 100 years. Therefore, there is little or no maintenance requirement for HDRB. However, even after years of practice, the mechanical behaviors of HDRB, such as the rate dependence and the low temperature dependence behaviors, are still difficult and unsolved issues in engineering practice.

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Experimental observations [14, 15] showed that mechanical properties of HDRB are dominated by the nonlinear rate dependence. Furthermore, experimental results also indicated that HDRB possess strong nonlinear strain hardening at high strain levels. These aspects are more obvious at low temperatures. The experimental investigations of Lion (1997) [16] indicated that the rate sensitivity of the reinforced rubber strongly depends on temperature, especially at low temperatures. In addition, the experimental observations in [17] showed that the low temperature dependence on the shear stress-strain responses of HDRB is more significant than that of natural rubber bearings and lead rubber bearings. However, the bilinear design model in some guide specifications [1, 2] omit the rate dependence behavior and the strain hardening behavior of HDRB, and this omission may cause major errors in predicting the seismic responses of the structures isolated by HDRB. Therefore, it is a great concern for engineers designing isolation structures in the areas where are not only very cold but also seismically active such as Hokkaido of Japan (1968 Tokachi Earthquake), Alaska of US (1964 Alaska Earthquake). In order to overcome the limitations of the current design model, Bhuiyan et al. (2009) [3] have developed a rate-dependent rheology model. The underlying key approach in this work is an additive decomposition of the total stress response into two contributions including rate-independent equilibrium stress and rate-dependent overstress. In this approach, the rate-independent behavior relating to the equilibrium response and the rate-dependent behavior relating to the overstress response can be obtained separately from experimental data then the equilibrium parameters and overstress parameters are determined directly from experiments. In this paper, a nonlinear dynamic analysis of a highway bridge isolated with HDRB is conducted to investigate the effect of the HDRB’s mechanical behavior on the dynamic performance of the bridge. In order to study about the temperature effect of HDRB, there are three temperature conditions considered in the analysis: the room temperature (+23 °C); and the two low temperatures (−10 °C and −30 °C). In order to investigate the modeling effect of HDRB, there are two analytical models of HDRB considered for comparison: the bilinear design model in [1, 2] and the rate-dependent rheology model in [3].

2 Input Data for the Analysis 2.1 Model of the Bridge The highway bridge used in this analysis is similar to the bridge in [18], the differece between them only is the kind of rubber bearings. Figure 1(a) shows the physical model of a five-span continuous reinforced concrete (RC) deck-steel girder bridge isolated by laminated rubber bearings provided at top of the RC piers. The laminated rubber bearings comprised is high damping rubber bearings. The isolation bearings are positioned at between the steel girders and top of the piers. The superstructure consists of continuous reinforced concrete slab, covered by 8 cm of asphalt with 0.26 × 18 m section, supported on two continuous steel girders. The substructures consist of RC piers and footings supported on pile foundations. The geometric and material properties of the bridge deck, piers with footings are given in Table 1.

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(a)

(b)

Fig. 1. Description of the target bridge (a) longitudinal view of a multi-span continuous highway bridge, (b) analytical model of the bridge; all dimensions are in mm

Table 1. Geometric and material properties of the piers Properties Cross-section of the pier cap (mm2 ) B1 × W1 Cross-section of the pier body (mm2 ) B2 × W2 Cross-section of the footing (mm2 ) B3 × W3 Number of piles/pier

Specifications Pier S1 & S2

Pier P1 & P4

3300 × 9600

2000 × 9600

3300 × 6000

2000 × 6000

5000 × 8000

5000 × 8000 4

Young’s modulus of concrete (MPa)

25000

Young’s modulus of steel (MPa)

200000

The analytical model of the bridge system is shown Fig. 1(b). The entire structural system is approximated as a continuous 2-D frame. A mathematical model of discrete form is used to approximate the continuous system. This form of modeling leads to a system with a finite number of degrees of freedom. Following the standard finite element procedure, these degrees of freedom are chosen as the nodal displacements of the discrete finite element model. The superstructure and substructure of the bridge are

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modeled as a lumped mass system divided into a number of small discrete segments. Each adjacent segment is connected by a node and at each node two degrees of freedom are considered: horizontal translation and rotation. The masses of each segment are assumed to be distributed between the two adjacent nodes in the form of point masses. The dynamic analysis has been successfully implemented in commercially available finite element software [19] in order to compute the seismic responses of bridges. 2.2 Model of the HDRB In order to characterize the mechanical behavior of isolation bearing two types of analytical models of the bearings are used in the study: the rate-dependent rheology model as developed by [3] and the bilinear design models as specified in [1, 2]. These two models are briefly discussed in the following sub-sections. 2.2.1 Rate-dependent Rheology Model The structure of the rate-dependent rheology model in [3] is shown in Fig. 2. The mathematical description of the model is briefly stated in Eqs. (1a)–(1e).

Fig. 2. The rate-dependent rheology model in [3] for HDRB in the analysis.

τ = τep (γa ) + τee (γ ) + τoe (γc )   γ˙s = 0 for τep  = τcr = C1 γa with γ˙s = 0 for τep  < τcr   τee = C2 γ + C3 γ m  sgn(γ )

(1a)



τep

 n  γ˙d  τoe = C4 γc with τoe = A  sgn(γ˙d ) γ˙o

(1b) (1c) (1d)

Dynamic Performance of Highway Bridges

and A =

1 1 (A1 exp(q|γ |) + Au ) + (A1 exp(q|γ |) − Au ) tanh(ξ τoe γd ) 2 2

481

(1e)

where C i (i = 1 to 4), τ cr , m, Al , Au , q, n, and ξ are parameters of the model to be determined from experimental data. Table 2. The rate-dependent rheology model parameters are determined from experimental data

In order to investigate the low temperature effect, the test data were carried out at three tempertures: 23 °C; −10 °C; −30 °C. The experimetal results were presented in [20]. These data are used to determine the parametarers in this analysis. The identification parameters of the rheology model are presented in Table 2. 2.2.2 Bilinear Design Model The bilinear design model of the bearings can be represented using a simplified structure of the rheology model. The simplified structure of the model is formed by using a linear elastic response (τee ) and an elasto-plastic response (τep ) only and hence the third branch (τoe ) of the model is discarded. In this case three parameters are required to represent the bilinear relationships of stress-strain responses of the bearings: C 1 parameter corresponds to the initial shear modulus, C 2 , the post yield shear modulus and τ cr , the characteristic yield strength of the bearings. The structure of this model is presented in Fig. 3.

Fig. 3. The bilinear design model in [1, 2] for HDRB in the analysis.

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The bilinear parameters are determined from sinusoidal loading tests with frequency of 0.5 Hz. The determination of these parameters at different temperatures are presented in Table 3. Table 3. The bilinear design model parameters are determined from experimental data

2.3 Earthquake Ground Motions and Solution Algorithm for Motion Equations 2.3.1 Earthquake Ground Motions The uncertainty characteristics of the earthquake ground motions regarding ground type, intensity and frequency contents have a great effect on nonlinear time history responses of members. In designing bridges [2], two levels of design ground motions should be considered, i.e. ground motion that is likely to occur during the service period of the bridge and destructive ground motion that is less likely to occur during the service period. These design ground motions are termed as Level 1 ground motion, and Level 2 ground motion, respectively. Level 2 ground motion contains the ground motion resulting from a large plate-boundary earthquake (type-I) and that from an inland earthquake (type-II). Type-I earthquake is characterized by a ground motion with large amplitude and long duration such as the Kanto earthquakes (Tokyo, 1923) and the type-II earthquake is characterized with low probability of occurrence, strong acceleration and short duration such as the Kobe earthquake (Kobe, 1995). Figure 4(a) and (b) show the time histories of type-I and type-II earthquake ground motions, respectively, used in the analysis. 2.3.2 Motion Equations and Solution Algorithm Equations that govern the dynamic response of the bridge can be derived by considering the equilibrium of all forces acting on it using the d’Alermbert’s principle. In this case, the internal forces are the inertia forces, the damping forces, and the restoring forces, while the external forces are the earthquake induced forces. To allow incorporation of nonlinear hysteretic models of the isolation bearings and the piers, the equations of motion of the the superstructure-pier-foundation (S-P-F) system can be written as ¨ − Ct U ˙ − t+t Fb − t Fs ¨ + CU ˙ + t KU = t+t P − Mt U MU

(2)

where M and C are the mass and damping matrices, respectively; t K is the tangent stiffness matrix at time t; U = t+t U − t U stands for the increment of displacement vector; t+t Fb the internal force vector derived from the isolation bearings at time t; t Fs ,

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(a)

Fig. 4. The ground acceleration histories used in the seismic response analysis bridge pier system (a) type-I, and (b) type-II earthquake ground motions; the acceleration histories correspond to the standard acceleration histories designed for level-2 earthquakes [2].

the restoring force of the bridge excluding isolation bearing at time t; and t+t P, the external force vector at time t + t. A single-step solution algorithm is developed to implement the rheology models in the nonlinear dynamic analysis for computing the responses of the bridge system. The solution algorithm includes the solution of equations of motion using the unconditionally stable Newmark’s constant-average-acceleration method and the solution of the differential equation governing the behavior of isolation bearings. Furthermore, Newton-Raphson iteration procedure consisting of corrective unbalanced forces is employed within each time step until equation condition is achieved. All calculations are implemented in a commercially available finite element software (Resp-T software) [19].

3 Output Data of the Analysis The dynamic analysis is implemented in commercially available Resp-T software [17]. The two designed earthquake waves: type-I and type-II specified by JRA (2004) are applied to the model bridge in the longitudinal direction. Due to symmetry of the bridge structure shown in Fig. 1(a) and due to space limitation, only one pier’s results P1 (=P4) using HDRB at three different temperature conditions are graphically presented and discussed herein. Seismic responses of the bridge system are presented in Figs. 5, 6, 7, and 8. 3.1 Low Temperature Effect Figure 5 shows the moment-rotation relations of the plastic hinges of the pier and Fig. 6 represents the shear stress-strain responses of the isolation bearings at room temperature (23 °C) and low temperatures (−10 °C, −30 °C). The low temperature effect is very clear in Fig. 6, the difference between the room temperature responses and the low temperature responses in Fig. 5 is quite large.

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Fig. 5. Moment-rotation responses of plastic hinge of the pier P1 (=P4) obtained using HDRB for Level-2 earthquake ground motions, (a) type-I earthquake and (b) type-II earthquake. Moment ratio (M/My) is the bending moment (M) at the level of plastic hinge divided by the yield moment (My) and rotation ductility is the rotation occurred at the plastic hinge level divided by the yield rotation

Fig. 6. Shear stress-strain responses of the HDRB at the top of the P1 (=P4) piers for Level-2 earthquake ground motions, (a) type-I earthquake and (b) type-II earthquake.

3.2 Modeling Effect In Figs. 7 and 8, the dynamic responses of the bridge are shown for a typical earthquake ground motion (level 2 type II earthquake ground motion). Figures 7 and 8 present the moment-rotation hysteresis at the plastic hinge level and the shear stressstrain responses of the HDR at three temperatures, respectively. Each figure presents the dynamic responses of the bridge obtained for the bilinear design model and the ratedependent rheology model of the isolation bearings. In general, the effect of modeling of isolation bearing on the seismic responses is clearly appeared in all figures. However, at low temperature (−10 °C, −30 °C), this effect is somewhat more significant than at the room temperature (+23 °C). The similar observations are also obtained for the type I earthquake ground motion but with different magnitudes.

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Fig. 7. Moment-rotation responses of plastic hinge of the pier P1 (=P4) obtained using HDRB for Type II earthquake ground modtion at (a) 23 °C (b) −10 °C (c) −30 °C; moment ratio (M/My ) is the bending moment (M) at the level of plastic hinge divided by the yield moment (My ) and rotation ductility is the rotation occurred at the plastic hinge level divided by the yield rotation

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Fig. 8. Shear stress-strain responses of the HDRB at the top of the P1 (=P4) piers for Type II earthquake ground modtion at (a) 23 °C (b) −10 °C (c) −30 °C.

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4 Conclusion A nonlinear dynamic analysis of a seismically isolated five-span continuous highway bridge was carried out. Commercially available software was used to implement the algorithm and to compute the seismic responses of the bridge. The seismic responses of the bridge were computed using the two analytical models of the isolation bearings at three different temperatures (at 23 °C, −10 °C, and −30 °C). The dynamics responses of the highway bridge, moment-rotation relationships of plastic hinges of the piers and shear stress-strain responses of the bearings, are compared at different temperatures. The effect of low temperatures on these responses is very clear. Moreover, the comparison of the dynamic responses of the bridge with different models for bearings is also conducted. In general, the effect of modeling of HDRB is clearly appeared in all earthquake ground motions for the isolation bearings in three temperatures. However, at low temperature, this effect is more significantly appeared in the responses than that at room temperature. This may be attributed to an overestimation of the second shear modulus of HRDB using the bilinear model in compared with the rheology mode in simulating the experimental data. Acknowledgements. The experimental works were conducted by utilizing the laboratory facilities and bearings-specimens provided by Japan Rubber Bearing Association. Author indeed gratefully acknowledge the kind cooperation extended by them in performing the tests. This work was supported by JSPS KAKENHI (Grants-in-Aid for Scientific Research) Grant Number 26289140 and Prof. Okui in Saitama University, Japan.

References 1. American Association of State Highways and Transportation Officials (AASHTO): Guide Specification for Seismic Isolation Design, 2nd edn. Washington D.C., USA (2010) 2. Japan Road Association: Specifications for Highway Bridges, Part V: Seismic Design, Tokyo, Japan (2004) 3. Bhuiyan, A.R., Okui, Y., Mitamura, H., Imai, T.: A rheology model of high damping rubber bearings for seismic analysis: identification of nonlinear viscosity. Int. J. Solids Struct. 46, 1778–1792 (2009) 4. Xiang, N., Goto, Y., Makoto Obata, M., Alam, S.: Passive seismic unseating prevention strategies implemented in highway bridges: a state-of-the-art review. Eng. Struct. 194, 77–93 (2019) 5. Di Sarno, L., da Porto, F., Guerrini, G., Calvi, O.M., Camata, G., Prota, A.: Seismic performance of bridges during the 2016 Central Italy earthquakes. Bull. Earthq. Eng. 17, 5729–5761 (2019) 6. Kilanitis, I., Sextos, A.: Impact of earthquake-induced bridge damage and time evolving traffic demand on the road network resilience. J. Traffic Transp. Eng. 6, 35–48 (2019) 7. Zhang, Y., Shi, Y., Liu, D.: Seismic effectiveness of multiple seismic measures on a continuous girder bridge. Appl. Sci. 10, 624–648 (2020) 8. Yi, J., Yang, H., Li, J.: Experimental and numerical study on isolated simply-supported bridges subjected to a fault rupture. Soil Dyn. Earthq. Eng. 127, 01–11 (2019) 9. Xiang, N., Shahria Alam, M., Li, J.: Shake table studies of a highway bridge model by allowing the sliding of laminated-rubber bearings with and without restraining devices. Eng. Struct. 171, 583–601 (2018)

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10. Kelly, J.M.: Earthquake Resistant Design with Rubber, 2nd edn. Springer, London (1997). https://doi.org/10.1007/978-1-4471-0971-6 11. Yoshida, J., Abe, M., Fujino, Y.: Constitutive model of high-damping rubber materials. J. Eng. Mech. 130(2), 129–141 (2004) 12. Marioni, A.: The use of high damping rubber bearings for the protection of the structures from the seismic risk. In: Jornadas Portugesas de Engenharia de Estruturas, Lisbon, Portugal (1998) 13. Gu, H., Itoh, Y.: Ageing effects on high damping bridge rubber bearing. In: Proceeding of the 6th Asia-Pacific Structural Engineering and Construction Conference, Kuala Lumpur, Malaysia (2006) 14. Dall’Asta, A., Ragni, L.: Experimental tests and analytical model of high damping rubber dissipating devices. Eng. Struct. 28, 1974–1984 (2006) 15. Hwang, J.S.: Evaluation of equivalent linear analysis methods of bridge isolation. J. Struct. Eng. 122, 972–976 (1996) 16. Lion, A.: On the large deformation behavior of reinforced rubber at different temperatures. J. Mech. Phys. Solids 45, 1805–1834 (1997) 17. Imai, T., Bhuiyan, A.R., Razzaq, M.K., Okui, Y., Mitamura, H.: Experimental studies of rate-dependent mechanical behavior of laminated rubber bearings. In: Joint Conference Proceedings 7CUEE & 5ICEE, 3–5 March 2010, Tokyo Institute of Technology, Tokyo, Japan, pp. 1921–1928 (2010) 18. Razzaq, M.K., Okui, Y., Bhuiyan, A.R., Amin, A.F.M.S., Mitamura, H., Imai, T.: Application of rheology modeling to natural rubber and lead rubber bearings: a simplified model and low temperature behavior. J. Jpn. Soc. Civ. Eng. 3(68), 526–541 (2012) 19. Resp-T: User’s Manual for Windows, Version 5 (2006) 20. Nguyen, D.A., Dang, J., Okui, Y., Amin, A.F.M.S., Okada, S., Imai, T.: An improved rheology model for the description of the rate-dependent cyclic behavior of high dampung rubber bearings. Soil Dyn. Earthq. Eng. 77, 416–431 (2015)

Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load Khuc Van Phu1 , Dao Huy Bich1 , and Le Xuan Doan2(B) 1 Vietnam National University, Hanoi, Vietnam 2 Tran Dai Nghia University, Ho Chi Minh City, Vietnam

Abstract. The paper investigates nonlinear vibration of FGM cylindrical shells with varying thickness subjected to mechanical load. The novelty of the paper is to establish motional equations of the structure. Since then, the natural frequency of vibration and the dynamic responses of the structure have been determined. Effect of material (coefficient k) and geometric parameters on natural frequency and nonlinear dynamic responses of the shell are also examined. Keywords: Nonlinear vibration · Variable thickness · FGM cylindrical shell · Dynamic responses

1 Introduction Variable thickness structures help to reduce the weight of produces and save materials without losing the bearing capacity, there for, this is the most favorite structure of designers. Variable thickness FGM shells are special structures and frequently used in modern industries such as: aerospace, aircraft, space vehicles and in other important engineering structures. Recently, nonlinear vibration problems of FGM shell in general and in particular variable-thickness FGM shells are attracting attention of many scientists. On nonlinear vibration of constant thickness FGM shell, Loy et al. [1, 2] based on Love’s shell theory to study vibration of FGM cylindrical shells subjected to mechanical load in thermal environment. The governing equations were derived by using Rayleigh– Ritz method. Haddadpour et al. [3] cared about natural frequencies and vibration of FGM cylindrical shells subjected to mechanical load including thermal effects with four sets of in-plane boundary conditions. Equilibrium equations were derived based on Love’s shell theory and solved by using Galerkin method. Matsunaga [4] presented an analysis for nonlinear vibration of FGM circular cylindrical shells based on 2D higherorder deformation theory. The power series expansion method and Hamilton’s principle were employed to solve the problem. Sofiyev [5] found out fundamental frequencies of clamped FGM conical shells subjected to uniform pressures. The governing equations were established based on Donnell’s theory and solved by Galerkin’s method. Mollarazi [6] investigated free vibration of cylindrical shell made of FGM by using mesh less method. Weak form of equilibrium equation and squares approximation were © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 489–505, 2022. https://doi.org/10.1007/978-981-16-3239-6_37

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used to establish motion equations of structure. By using analytical method Bich and Nguyen [7] examined effects of axial and transverse loads on nonlinear vibration of FGM cylindrical shell. The governing equations are derived based on improved Donnell shell theory. Nonlinear dynamic responses of structure are obtained by using Galerkin method, the Volmir’s assumption and fourth-order Runge–Kutta method. Also based on Donnell shell theory, Avramov [8] analyzed nonlinear vibrations of FGM cylindrical shell. The nonlinear vibration modes were determined by using the harmonic balance method and Galerkin method. Malekzadeh et al. [9] investigated free vibration of rotating FGM truncated conical shells subjected to mechanical load with various boundary conditions. Dynamic equilibrium equations and motion equations of the shell were derived according to the first-order shear deformation theory (FSDT). Take into account effect of Coriolis forces and centrifugal forces. By the same way, effect of internal pressure on natural frequencies of structures was examined by these authors [10]. Kim [11] analyzed free vibration and natural frequencies of FGM shells partially resting on elastic foundations with an inclined edge by using analytical method. Equilibrium equations of structure were established according to the FSDT, variational approach and Rayleigh–Ritz method. Quan et al. [12] solved nonlinear vibration problems of imperfect FGM shallow shells surround by elastic foundations subjected to mechanical, thermal and damping loads. The Reddy’s third-order shear deformation theory was employed to establish governing equations. Natural frequencies and dynamic responses of structure were obtained by using Galerkin method and fourth-order Runge–Kutta method. Also using analytical method, Phu et al. [13, 14] studied on nonlinear vibration of sandwich-FGM and stiffened sandwich FGM cylindrical shell filled with fluid subjected to mechanical loads in thermal environment. The classical shell theory with geometrical nonlinearity in von Karman-Donnell sense and smeared stiffener technique were used to define motion equations of structure. Natural frequencies and dynamic responses of the shell were determined by using Galerkin’s method and Runge–Kutta method. Nonlinear vibration problems of FGM elliptical cylindrical shells reinforced by carbon nanotubes embedded in elastic mediums in thermal environments were dealt with by Dat et al. [15]. Research is carried out based on classical shell theory, nonlinear dynamic responses of structure were found out by using the Galerkin method, RungeKutta method and Airy stress function. Han et al. [16] predicted free vibration of FGM thin cylinder shells filled inside with pressurized fluid based on Flügge shell theory. In governing equations, internal static pressure is regarded as the pre-stress term. On vibration of variable thickness shell structures, there are some typical research as: Ganesan et al. [17] employed semi-analytical finite element to analyze vibration characteristic of variable thickness circular cylindrical shells made of steel based on Love’s first approximation shell theory. By the same way, natural frequencies and vibration of laminated conical shells and variable-thickness cylindrical shells made of composite were also considered by these authors [18, 19]. Abbas et al. [20] predicted fundamental frequencies and vibration mode of open circular cylindrical shells with the thickness varies in axial direction in high temperature environment. Transfer matrix method and the fourth-order Runge–Kutta method were used to solve problem. Kang et al. [21, 22] employed 3-D method and Ritz method to examine fundamental frequencies and vibration mode of spherical shell segments and variable thickness conical-cylindrical

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shells. Also study on variable thickness spherical shells, Efraim et al. [23] based on two-dimensional theory and dynamic stiffness method to determine natural frequencies and vibration mode shapes of structure with different boundary conditions. Ataabadi et al. [24] presented an analysis on vibration of variable thickness shells in parabolic or circular profile. The linear shell theory, energy method and spline functions were used to survey. Nonlinear vibrations investigation of complex-shape shallow shells with variable thickness under various boundary conditions has been proposed by Awrejcewicz et al. [25]. The classical theory and first-order shear deformation theory are used to establish governing equations. Natural frequencies of structure were obtained by using R-functions theory, Ritz’s method, Galerkin method and the fourth-order Runge–Kutta method. About variable thickness FGM structure, Ghannad et al. [26] based on the FSDT and the perturbation theory to analyze static problems of variable thickness FGM cylindrical shells subjected to internal pressure. The distribution of displacement and stress in axial and radial direction were determined by using inverse matrix method. Free vibration problems of variable thickness sandwich-FGM shells were solved by Tornabene et al. [27]. Higher-order shear deformation theory (HSDT) and local generalized differential quadrature method were used to determine natural frequencies of structure. By using finite element method, Golpayegani et al. [28] analyzed free vibrations of FGM cylindrical shells under different boundary conditions. The thickness of structure varies linearly along the axial direction. Effects of varying thickness, the length and radius of structure on frequencies were also considered. From above review, According to authors’ knowledge, there is no investigation on nonlinear vibration of FGM cylindrical shell with variable thickness subjected to axial compression load and external pressure published yet. In present paper, the governing equations of variable thickness structure are established based on classical shell theory, take into account the geometrical nonlinearity in von Karman-Donnell sense. Natural frequencies and nonlinear dynamic responses of the shell are obtained by using RungeKutta method and Galerkin method. Effect of material and geometric dimension on natural frequencies and nonlinear dynamic responses of the shell are also considered.

2 Governing Equations Consider the variable-thickness FGM cylindrical shell with the length L and the radius R subjected to uniform pressure q(t) and axial compression load p(t) as shown in Fig. 1. Assume that radius is much larger than the thickness of the shell (R >> h), Thickness 0 of shell h can be determined as: h(x) = ax + b in which: a = h1 −h L ; b = h0 . The effective properties of material can be expressed as follow: P(z) = Pm · Vm (z) + Pc · Vc (z) = Pm + (Pc − Pm )Vc (z)

(1)

In which Pm and Pc are material and ceramic properties, V c and V m are volume fractions of ceramic and metal, respectively, and are related by V c + V m = 1. Ceramic volume fractions in the structure are distributed as following law:   z k 1 + (2) Vc (z) = 2 h(x)

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Fig. 1. Variable-thickness FGM cylindrical shell

Poisson’s ratio is assumed to be constant (ν = constant). According to classical shell theory [29], the strain-displacement relationship of the shell: εij = εij0 + zkij with (i, j = x, y) in which 0 = εxx

kxx =

  ∂u0 ∂v0 1 ∂w 2 0 w ∂x + 2 ∂x ; εyy = ∂y − R 2 2 − ∂∂xw2 ; kyy = − ∂∂yw2 ;

+

1 2



∂w ∂y

2

(3)

0 = ; γxy

∂u0 ∂y

+

∂ w kxy = −2 ∂x∂y

∂v0 ∂x

2

Applying Hooke’s law for FGM shell subjected to mechanical load: ⎫ ⎫ ⎧ ⎡ ⎤ ⎧ 1ν 0 ⎨ σxx ⎬ ⎨ εxx ⎬ E(z) ⎣ ⎦ · εyy = σ ν1 0 2 ⎭ ⎩ yy ⎭ 1 − υ(z) ⎩ τxy γxy 0 0 (1 − ν)/2

+

∂w ∂w ∂x ∂y

(4)

(5)

Integrating Stress-Strain relationship through the thickness of the shell, we obtain the governing equations of the variable thickness FG cylindrical shell:       0  ε N [A] [B]  ij  = (6) .  ij  with (i, j) = (xx, yy, xy) Mij [B] [D] kij Equation (6) can be rewritten as following: ⎫ ⎡ ⎧ N ⎪ ⎪ ⎪ xx ⎪ A11 ⎪ ⎪ ⎪ ⎢A ⎪ ⎪ Nyy ⎪ ⎪ ⎪ 21 ⎢ ⎪ ⎪ ⎪ ⎬ ⎢ ⎨N ⎪ xy ⎢ 0 =⎢ ⎢ B11 ⎪ Mxx ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ B21 ⎪ ⎪ ⎪ Myy ⎪ ⎪ ⎪ ⎭ ⎩ 0 M xy

A12 A22 0 B12 B22 0

0 0 A66 0 0 B66

B11 B12 0 B21 B22 0 0 0 B66 D11 D12 0 D21 D22 0 0 0 D66

⎧ 0⎫ ⎤ ⎪ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ εyy ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎥ ⎪ ⎨ 0⎪ ⎬ ⎥ ⎪ γxy ⎥ ⎥· ⎪ ⎥ ⎪ ⎪ kxx ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ k yy ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ kxy

in which: A11 =A22 =

E1 · h(x) ν · E1 · h(x) ; A12 = A21 = ; 2 1−ν 1 − ν2

(7)

Nonlinear Vibration Analysis for FGM Cylindrical Shells

B11 =B22 = D11 = D22 = A66 =

E2 · h2(x) 1 − ν2

; B12 = B21 =

E3 · h3(x) 1 − ν2

493

ν · E2 · h2(x)

; D12 = D21 =

E2 · h2(x) E1 · h(x) ; B66 = ; D66 2(1 + ν) 2(1 + ν)

1 − ν2 ν · E3 · h3(x)

; 1 − ν2 E2 · h2(x) = 2(1 + ν)

Ec − Em (Ec − Em )k ; E2 = ; (k + 1) 2(k + 1)(k + 2)   1 1 1 Em + (Ec − Em ) − + E3 = 12 k + 3 k + 2 4(k + 1)

E1 =Em +

Internal forces and moment resultants can be inferred from Eq. (7) as follows:   ⎧   0 + υ · ε0 + B ⎪ Nxx = A11 εxx 11 kxx + ν · kyy ⎪ yy ⎪   ⎪   ⎪ ⎪ 0 + υ · ε0 + B ⎪ Nyy = A11 εyy 11 kyy + ν · kxx ⎪ xx ⎪ ⎪ ⎨N = A · γ0 + B · k xy 66  xy 66 xy   0 0 +D ⎪ Mxx = B11 εxx + ν · εyy ⎪ 11 kxx + ν · kyy ⎪ ⎪   ⎪   ⎪ 0 + ν · ε0 + D ⎪ Myy = B11 εyy ⎪ 11 kyy + ν · kxx xx ⎪ ⎪ ⎩ 0 +D ·k Mxy = B66 · γxy 66 xy

(8)

Nonlinear motion equations of FGM cylindrical shell with variable thickness based on classical shell theory are [29]: ⎧ ∂Nxy ∂Nxx ∂2u ⎪ ⎪ ∂x + ∂y = ρ1 ∂t 2 ⎪ ⎪ ∂Nxy ∂Nyy ⎪ ∂2v ⎪ ⎪ ⎨ ∂x + ∂y = ρ1 ∂t 2 ∂ 2 Myy ∂ 2 Mxy ∂ 2 Mxx ∂ 2w ∂ 2 w Ny ∂ 2w (9) + + Nyy 2 + +2 + Nxx · 2 + 2Nxy ⎪ ⎪ 2 2 ∂x ∂x∂y ∂y ∂x ∂x∂y ∂y R ⎪ ⎪ ⎪ ⎪ ∂ 2w ∂w ∂ 2w ⎪ ⎩ −p · h(x) 2 + q = ρ1 2 + 2ερ1 ∂x ∂t ∂t   −ρm in which: ρ1 = ρm + ρck+1 h(x) = ρ1∗ h(x) . Substituting Eq. (4) and Eq. (8) into Eq. (9) we obtain: ⎧ ∂2u ⎪ ⎪ ⎪ L11 (u) + L12 (v) + L13 (w) + P1 (w) = ρ1 ∂t22 ⎪ ⎪ ⎨ L21 (u) + L22 (v) + L23 (w) + P2 (w) = ρ1 ∂ 2v ∂t (10) L31 (u) + L32 (v) + L33 (w) + P3 (w) + P4 (u, w) + P5 (v, w) ⎪ ⎪ 2 2 ⎪ ⎪ ∂ w ∂w ∂ w ⎪ ⎩ +q + p · h(x, y) 2 = ρ1 2 + 2ερ1 ∂x ∂t ∂t in which: L11 (u) =A11

∂ 2 u ∂A11 ∂u ∂ 2u + A + . ; 66 ∂x2 ∂x ∂x ∂y2

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L12 (v) =A66 ·

∂A11 ∂v ∂ 2v ∂ 2v +ν· + ν · A11 ; ∂x∂y ∂x ∂y ∂x∂y

∂ 3w ∂ 3w ∂ 3w ∂B11 ∂ 2 w − ν · B11 − 2B66 · − 3 2 2 ∂x ∂x∂y ∂x∂y ∂x ∂x2 1 ∂w ∂B11 ∂ 2 w ∂A11 w − ν · A11 ; −ν · −ν· ∂x ∂y2 ∂x R R ∂x    2   ∂w ∂w ∂ 2 w ∂w 2 1 ∂A11 ∂w ∂ 2 w P1 (w) = +ν· + ν · + A11 2 ∂x ∂x ∂y ∂x ∂x2 ∂y ∂x∂y   2 ∂ w ∂w ∂w ∂ 2 w ; +A66 · + ∂x∂y ∂y ∂x ∂y2 L13 (w) = −B11

∂A66 ∂u ∂ 2u + ; ∂x∂y ∂x ∂y ∂ 2v ∂ 2 v ∂A66 ∂v L22 (v) =A66 · 2 + + A11 2 ; ∂x ∂x ∂x ∂y

L21 (u) =(A66 + A11 ν) ·

∂B66 ∂ 2 w 1 ∂ 3w ∂w ∂ 3w · − A11 ; − (νB11 + 2B66 ) · 2 − 2 3 ∂y ∂x ∂y ∂x ∂x∂y R ∂y   ∂w ∂ 2 w ∂A66 ∂w ∂w ∂w ∂ 2 w ∂ 2 w ∂w ∂w ∂ 2 w + A + A + ; P2 (w) = A11 + ν · · 66 66 ∂y ∂y2 ∂x ∂x∂y ∂x2 ∂y ∂x ∂x∂y ∂x ∂x ∂y L23 (w) = −B11

I31 (u) = B11

∂ 3u ∂B11 ∂ 2 u ∂ 2 B11 ∂u ∂ 3u + + ν · B +2 + (2B ) 66 11 ∂x3 ∂x∂y2 ∂x ∂x2 ∂x2 ∂x ∂u 1 ∂B66 ∂ 2 u + ν · A11 ; +2 2 ∂x ∂y R ∂x

∂B11 ∂ 2 v ∂B66 ∂ 2 v ∂ 3v ∂ 3v + 2ν · + 2 + + ν · B (2B ) 66 11 ∂y3 ∂x2 ∂y ∂x ∂x∂y ∂x ∂x∂y 2 1 ∂v ∂ B11 ∂v + A11 ; +ν 2 ∂x ∂y R ∂y  4  ∂ w ∂ 4w ∂ 4w L33 (w) = −D11 + 4 − 2(D11 ν + 2D66 ·) 2 2 4 ∂x ∂y ∂x ∂y  3   2  3 2 ∂ D11 ∂ w ∂D11 ∂ w ∂ w ∂ 2w − −2 + ν · − ν · ∂x ∂x3 ∂x2 ∂y ∂x2 ∂x2 ∂y2

L32 (v) = B11

∂ 2 B11 w 1 2 w − A11 − ν · 2 ∂x R R R R ∂ 2w ∂ 2w 2 ∂D66 2 − ν · B11 2 − B11 2 − 4 R ∂x R ∂y ∂x −ν ·

∂B11 ∂w ∂x ∂x ∂ 3w · ; ∂x∂y2

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     2 ∂w 2 w ∂ 2w 1 ∂w ∂ 2w A11 P3 (w) = +ν· −ν· 2 − A11 2R ∂y ∂x R ∂y2 ∂x       2 2 ∂w ∂w 1 ∂ 2 B11 ∂w ∂ 2 w ∂B11 ∂w ∂ 2 w + + ν · + ν · + 2 2 ∂x2 ∂x ∂y ∂x ∂x ∂x2 ∂y ∂x∂y   2 2 ∂w ∂ 3 w ∂w ∂ 3 w ∂w ∂ 3 w ∂w ∂ 3 w ∂ w +B11 2ν · + ν + + + ν · ∂x∂y ∂y ∂y3 ∂x ∂x3 ∂y ∂x2 ∂y ∂x ∂x∂y2   ∂ 2w ∂ 2w ∂B66 ∂ 2 w ∂w ∂w ∂ 2 w + −2B11 ν 2 +2 ∂y ∂x2 ∂x ∂x∂y ∂y ∂x ∂y2    2 ∂ 3 w ∂w ∂ 2 w ∂ 2 w ∂ 2w ∂w ∂ 3 w +2B66 · − + ; + ∂x2 ∂y ∂y ∂x2 ∂y2 ∂x∂y ∂x ∂x∂y2      2  2 ∂w 2 ∂w 2 ∂ 2w ∂ 2w 1 ∂w ∂w 1 P4 (w) = A11 2 +ν· +ν· + A11 2 2 ∂x ∂x ∂y 2 ∂y ∂y ∂x +2A66 ·

∂w ∂w ∂ 2 w ; ∂x ∂y ∂x∂y

∂u ∂ 2 w ∂u ∂ 2 w ∂u ∂ 2 w ; + ν · A11 + 2A66 · 2 2 ∂x ∂x ∂x ∂y ∂y ∂x∂y ∂v ∂ 2 w ∂v ∂ 2 w ∂v ∂ 2 w P6 (v, w) =A11 + ν · A11 + 2A66 · . 2 2 ∂y ∂y ∂y ∂x ∂x ∂x∂y

P5 (u, w) =A11

Equation (10) can be used to investigate nonlinear dynamic vibration of simply supported circular cylinder shell with variable thickness under mechanical load.

3 Solution Method This paper considers the variable thickness FGM cylindrical shell with simply supported at both ends, subjected to external uniformly distributed pressure q(t) and axial compression force in term of time p(t). The boundary conditions are: w = 0, Nxy = 0, Mxx = 0, Nxx = −p · h at x = 0 and x = L Displacement components of the cylindrical shell can be expanded as: u = Umn (t) cos αx sin βy; v = Vmn (t) sin αx cos βy; w = Wmn (t) sin αx sin βy (11) in which: α = respectively).

mπ L ;

β =

n R

(m, n - the half-waves number in x and y direction,

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Substituting Eq. (11) into Eq. (10) then applying Galerkin procedure yields ⎧ 2 ⎪ I11 U + I12 V + I13 W + R1 W 2 = ρ1∗∗ ddtU2 ⎪ ⎪ ⎪ ⎪ ∗∗ d 2 V 2 ⎪ ⎪ ⎨ I21 U + I22 V + I23 W + R2 W = ρ1 dt 2

  pm2 π 3 (La + 2b)R 2 3 ·W I U + I V + I W + R W + R W + R U + R V + 32 33 3 4 6 5 ⎪ 31 ⎪ 8L ⎪ ⎪ ⎪ ⎪ 4δn δm RLq d 2W dW ⎪ ⎩ + = ρ1∗∗ 2 + 2ερ1∗∗ mnπ dt dt (12) in which: ρ1∗ L(La + 2b)π R (−1)n − 1 (−1)m − 1 ; δn = ; δm = ; 2 2  8 E1 π (ν − 1)n2 L2 − 2π 2 R2 m2 (La + 2b) ; = 16RL(1 − ν 2 )

ρ1∗∗ = I11

E1 mπ 2 n(La + 2b) ; 16(ν − 1)   E1 νmπ 2 (La + 2b) E2  π 2 R2 m2 + L2 n2 ν E2 n2      ; = − + 24(1 + ν)mR 8 ν2 − 1 24L2 Rm ν 2 − 1   2E1 2π 2 R2 m2 − L2 n2 ν (La + 2b) E1 n(La + 2b)   R1 = ; − 9(1 + ν)R 9L2 Rn ν 2 − 1   E1 π (La + 2b) 2L2 n2 − R2 m2 (ν − 1)π 2 I22 = ; 16(ν 2 − 1)LR

I12 =I21 = I13

E1 π Ln(La + 2b)   8R 1 − ν 2      E2 n 3L4 a2 n2 − n2 R2 m2 π 2 + L2  − 3L2 a2 + 3(2ν − 1)m2 π 2 L2 R2 a2 ; + 24(ν 2 − 1)R2 Lm2 π   2E1 2L2 n2 − π 2 R2 m2 ν (La + 2b) E1 mπ(La + 2b) R2 = − ; 9R2 Lmπ(ν 2 − 1) 9(1 + ν))L    νE1 mπ 2 (La + 2b) E2 m2 π 2 R2 + L2 n2 ν  − 3L2 a2 − 3L2 a2 E2 n2      ; =− − + 24m(1 + ν)R 8 1 − ν2 24L2 Rm ν 2 − 1   E2 nπ E1 Lnπ(La + 2b) E2 L  − 6L2 a2 n3   − I32 = + ; 24m2 π R2 (ν 2 − 1) 24(1 + ν)L 8R ν 2 − 1 I23 = −

I31

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E1 L(La + 2b)π   8R ν 2 − 1  !   " E2 3L4 a2 n2 − νR2 m2 π 2 + 2n2 L2  − 3L2 a2 − 6m2 π 2 (2R − 3)νR2 L2 a2 I33 =

+

+ 12R2 Lπ m2 (ν 2 − 1)   ! " !   " 4 4 2 2 2 2 2 2 ∗ 2 E3 (La + 2b) m π R R m π + 2n L ν  − 3L R2 a2 + L4 n4 m2 π 2 ∗ + 6νR2 a2 − 3L2 a2 16π m2 (ν 2 − 1)R3 L3 −

E3 π(m2 ∗ π 2 + 3L2 a2 )(La + 2b) ; 8(1 + ν)LR

    2E1 5L2 n2 − 3π 2 R2 m2 ν (La + 2b) 8a2 E2 L2 n2 ν − 11R2 m2 π 2 + R3 = − 9LR2 mnπ(ν 2 − 1) 27LRmnπ(ν 2 − 1) #  4 4 4   $ 4 4 2 2 ∗ 2 2 4E2 9 R m π + L n m π  − 4L a 7R4 m4 π 4 − 10L4 n4 + 81L3 R3 m3 π 3 n(ν 2 − 1)    2 2 ∗  8E2 nν 9m2 ∗ π 2 − 34L2 a2 4E2 nν( 9m π  − 44L2 a2 + + 27LRmπ(ν 2 − 1) 81LRmπ(ν 2 − 1)   2E2 n( 4L2 a2 − 9m2 π 2 ∗ 16E2 a2 Ln + + 27Rmπ(1 + ν) 81LRmπ(1 + ν)   4 4 4 3E1 π π R m + 2L2 π 2 R2 m2 n2 ν + L4 n4 (La + 2b) E1 m2 π 3 n2 (La + 2b) + R4 = 256R3 L3 (ν 2 − 1) 128L(1 + ν)R  2 2 2  8E1 π R m + L2 n2 ν (La + 2b) 2E1 n(La + 2b) + ; R5 = − 9L2 Rn(ν 2 − 1) 9(1 + ν)R  2 2 2  8E1 νR m π + L2 n2 (La + 2b) 2E1 mπ (La + 2b)   R6 = ; + 9L(1 + ν) 9R2 Lmπ 1 − ν 2      = 2π 2 m2 L2 a2 + 3Lab + 3b2 + 3L2 a2 ; ∗ = L2 a2 + 2Lab + 2b2 ; According to Volmir’s assume [30] by ignoring the inertial components along x and y axes (u ω0 ), the relationship between velocity and deflection of variable thickness FGM shell is complex curves. When increase the amplitude of excitation force to great value, the velocity-deflection relationship become disturbed curves (Fig. 10).

Fig. 10. The dw/dt-w relationship of the shell

5 Conclusions In present article, motion equations of FGM cylindrical shells with variable thickness is presented based on the classical shell theory, taking into account the nonlinear geometry of von Karman-Donnell. Nonlinear differential equation of the shell is obtained by

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using Galerkin method. Nonlinear dynamic responses of the shell are determined by the fourth-order Runge-Kutta method and Matlab software. Some conclusions can be inferred from results of present analysis: • Fundamental frequencies of variable thickness FGM shell depend on volume fraction index (k) and the vibration mode (m, n). The lowest natural frequencies of the shell corresponding to mode shape (m, n) = (1, 7). • Geometric parameters of the shell (L, R, h1 , h0 ) are factors effect on vibration amplitude of the structure. When geometric dimensions of the shell increase (L, R), vibration amplitude of the shell will increases, it means, the stiffness of structure decreases. • When the frequencies of external force are near to the natural frequencies of the shell, the beating phenomenon is observed, the deflection-velocity relationship is closed curve. • When the frequency and amplitude of excitation force are far from the natural frequency of the shell, the velocity-deflection relationship becomes disturbed curves.

Acknowledgments. This research is funded by National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant number 107.02-2018.324.

References 1. Loy, C.T., Lam, K.Y., Reddy, J.N.: Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324 (1999) 2. Pradhan, S.C., Loy, C.T., Lam, K.Y., Reddy, J.N.: Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl. Acoust. 61, 111–129 (2000) 3. Haddadpour, H., Mahmoudkhani, S., Navazi, H.M.: Free vibration analysis of functionally graded cylindrical shells including thermal effects. Thin Walled Struct. 45, 591–599 (2007) 4. Matsunaga, H.: Free vibration and stability of functionally graded circular cylindrical shells according to 2D higher-order deformation theory. Compos. Struct. 88, 519–531 (2009) 5. Sofiyev, A.H.: On the vibration and stability of clamped FGM conical shells under external loads. J. Compos. Mater. 45, 771–788 (2011) 6. Mollarazi, H.R., Foroutan, M., Dastjerdi, R.M.: Analysis of free vibration of functionally graded material (FGM) cylinders by a meshless method. J. Compos. Mater. 46, 507–515 (2011) 7. Bich, D.H., Nguyen, N.X.: Nonlinear vibration of functionally grade circular cylindrical shells based on improved Donnell equations. J. Sound Vib. 331(25), 5488–5501 (2012) 8. Avramov, K.V.: Nonlinear modes of vibrations for simply supported cylindrical shell with geometrical nonlinearity. Acta Mech. 223, 279–292 (2012) 9. Malekzadeh, P., Heydarpour, Y.: Free vibration analysis of rotating functionally graded truncated conical shells. Compos. Struct. 97, 176–188 (2013) 10. Heydarpour, Y., Malekzadeh, P., Aghdam, M.M.: Free vibration of functionally graded truncated conical shells under internal pressure. Meccanica 49(2), 267–282 (2013) 11. Young-Wann, K.: Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge. Compos. Part B Eng. 70, 263–276 (2015)

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Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load Khuc Van Phu1(B) , Dao Huy Bich1 , and Le Xuan Doan2 1 Vietnam National University, Hanoi, Vietnam 2 Tran Dai Nghia University, Ho Chi Minh city, Vietnam

Abstract. The main purpose of this paper is to study on nonlinear dynamic stability of FGM cylindrical shell with variable thickness under mechanical load. The governing equations of this structure are established based on the classical shell theory and take into account the geometrical nonlinearity in von KarmanDonnell sense. Nonlinear dynamic responses of the shell are obtained by using the fourth-order Runge-Kutta method and Galerkin method. Dynamic critical loads are determined according to Budiansky-Roth criterion. Effect of material and geometric dimension on dynamic critical load and nonlinear dynamic responses of the shell are also considered. Keywords: Dynamic stability · Variable thickness · FGM cylindrical shell · Dynamics responses

1 Introduction FGM shells are special structures and frequently used in modern industries such as: aerospace, aircraft, space vehicles and in other important engineering structures. Recently, Variable thickness structure becomes more and more popular in important industries due to its preeminent properties. It helps to reduce the weight of structure and saves materials while ensuring the bearing capacity. Study on the dynamic stability of FGM cylindrical shell is necessary to ensure the structure works efficiently and reliably. This topic has attracted the interest of many scientists in research and discussion. The nonlinear static and dynamic stability of FGM shell structure has been analyzed by several scientists such as Shen [1] studied the post-buckling of finite-length FGM cylindrical thin shell subjected to axial compression loads in thermal environments. Research was carried out based on the classical shell theory. Buckling critical loads were determined by using singular perturbation technique. Sofiyev et al. [2] solved dynamic stability problems of FGM cylindrical thin shells subjected to linear torsional load. Galerkin’s method and Lagrange–Hamilton principle were employed to determine dynamic critical load. Also by using Galerkin method, Sofiyev [3–5] investigated dynamic buckling of FGM truncated conical and cylindrical shells resting on Pasternak foundations under axial compressive load, axial tension and hydrostatic pressure. Dynamic critical loads were obtained by using Galerkin method and superposition © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 506–521, 2022. https://doi.org/10.1007/978-981-16-3239-6_38

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method. Golchi et al. [6] based on the first-order shear deformation theory (FSDT) and Hamilton’s principle to analyze thermal buckling of stiffened FGM truncated conical shells. The obtained governing equations were solved by using differential quadrature method (DQM). Dynamic buckling problem of suddenly-heated FGM cylindrical shell subjected to mechanically load was studied by Shariyat. [7]. Dynamic buckling load was detected according to modified Budiansky criterion. In this work, the effect of initial geometrical imperfection was taken into account. Darabi et al. [8] based on large deflection theory to investigate dynamic stability of FGM shells under periodic axial load using Galerkin method and Bolotin’s method. Sheng et al. [9] presented an analysis of dynamic stability of FGM cylindrical shells embedded in elastic media based on FSDT. Effect of transverse shear strains and rotary inertia were taken into account. Bagherizadeh et al. [10] studied the mechanical buckling of simply-supported FG cylindrical shell surrounded by elastic media and subjected to mechanical loads. The governing equations are established based on a higher-order shear deformation theory (HSDT). Nonlinear static and dynamic buckling problems of FGM cylindrical shells subjected to mechanical load were solved by Huang et al. [11–13]. The Donnell shell theory, large deflection theory, energy method and the four-order Runge–Kutta method are employed to investigate nonlinear dynamic responses of structures, the critical loads were determined according to Budiansky-Roth criterion. Najafizadeh et al. [14] presented an analysis which is concerned with elastic stability of stiffened FGM circular cylinder shells subjected to axial compression loading using the Sander’s assumption. Dung et al. [15–19] focused on using semi-analytical approach to investigate nonlinear dynamic buckling of stiffened FGM thin cylindrical shells surrounded by elastic foundations and subjected to mechanical load in thermal environments. Equilibrium equations were derived based on the classical shell theory, Donnell’s shell theory and smeared stiffeners technique. Nonlinear dynamic responses of structures were obtained by using Galerkin method and fourth order Runge–Kutta method. Nonlinear dynamic critical loads were determined according to Budiansky–Roth criterion. By similar approach, Nam et al. [20, 21] studied the nonlinear buckling of sandwich-FGM cylindrical shells reinforced by spiral stiffeners under mechanical-thermal load. Khoa et al. [22] studied the nonlinear dynamic buckling of imperfect piezoelectric S-FGM cylindrical shells subjected to electrical and mechanical loads, resting on elastic medium in thermal environment. The Reddy’s third-order shear deformation theory and Galerkin method were employed to solve problems. Heydarpour et al. [23] proposed an analysis into dynamic stability of rotating FGM shells reinforced by carbon nanotube under mechanical load based on FSDT and DQM. Phu et al. [24, 25] investigated dynamic buckling of sandwich-FGM shells and stiffened sandwich-FGM cylindrical shells filled with fluid, surrounded by elastic foundations in thermal environment based on the classical shell theory and the smeared stiffener technique. Dynamic responses of structures were obtained by using Galerkin method and Runge–Kutta method. The dynamic critical load was determined according to Budiansky-Roth criterion. Recently, Zhang et al. [26] analyzed the dynamic buckling of FGM cylindrical shells under thermal shock based on the Hamiltonian principle. Thermal buckling problem of FGM cylindrical shells with properties of material depending on temperature was solved by Huang et al. [27].

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Until this time, there are very little publications investigate on dynamic stability of variable thickness structures. Some typical authors such as Luong et al. [28, 29] based on the small deflection theory and the thin shell theory to investigate buckling of simply supported cylindrical panel with small changes in the shell thickness subjected to mechanical load. On dynamic stability of variable thickness shell made from FGM, Shariyat et al. [30] proposed an analysis of nonlinear thermal buckling of imperfect, variable thickness cylindrical shells made of bidirectional FGM subjected to thermal load. Governing equations were established by using the third-order shear-deformation theory and nonlinear finite element method. Buckling critical load was determined according to modified Budiansky’s criterion. Next, Thang et al. [31] presented an investigation on effects of variation of the thickness on buckling of imperfect S-FGM cylindrical panels under axial compressive load and external distributed pressure. The governing equations were derived by using classical shell theory and Galerkin procedure. From above reviews, according to authors’ knowledge, analysis on nonlinear dynamic stability of variable thickness FGM cylindrical shell subjected to mechanical load has no investigation published yet. In present article, the governing equations of this structure are established based on the classical shell theory and take into account the geometrical nonlinearity in von Karman-Donnell sense. Nonlinear dynamic responses of the shell are obtained by using Runge-Kutta method and Galerkin method. Dynamic critical loads are determined according to Budiansky-Roth. Effect of material, geometric dimension and loading speed on dynamic critical loads and nonlinear dynamic responses of the shell are also considered in detail.

2 Governing Equations Consider a variable thickness FGM cylindrical shell with geometric dimensions and bearing as shown in Fig. 1. Assume that the radius of shell is much larger than the thickness (R >> h), Thickness of the shell h can be determined as: h(x) = a.x + b. in which: a = (h1 −h0 )/L; b = h0 . Assumes that V c and V m are volume fractions of ceramic and metal, respectively, are related by V c + V m = 1. In which, the volume fraction of ceramic is expressed as:   z k 1 + (1) Vc (z) = 2 h(x) Effective properties of materials (E, ρ, ν…etc.) can be expressed as: P(z) = Pm .Vm (z) + Pc .Vc (z) = Pm + (Pc − Pm )Vc (z);

(2)

According to classical shell theory [28], the strain-displacement relationship of the shell in form as: 0 + 2z.kxy εx = εx0 + z.kx ; εy = εy0 + z.ky ; γxy = γxy

in which 0 = εxx

kxx =

  ∂u0 1 ∂w 2 0 ∂x + 2 ∂x ; εyy 2 − ∂∂xw2 ; kyy

= =

∂v0 w ∂y − R 2 − ∂∂yw2 ;

+

1 2



∂w ∂y

2

0 = ; γxy

∂u0 ∂y

+

∂v0 ∂x

∂2w

kxy = −2 ∂x∂y

(3)

+

∂w ∂w ∂x ∂y

(4)

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Fig. 1. Variable thickness FGM cylindrical shell

Hooke’s law for FGM shell subjected to mechanical load is defined as: ⎫ ⎫ ⎧ ⎧ εx + νεy ⎬ ⎨ σxx ⎬ E(z) ⎨ = σ νεx + εy 2 ⎩ ⎭ ⎩ yy ⎭ 1 − υ(z) τxy (1 − ν).γxy /2

(5)

Integrating Stress-Strain relationship through the thickness of the shell, internal force and moment resultants of variable FGM shell can be obtained as follows:   ⎧   0 0 ⎪ ⎪ ⎨ Nxx = A11 εxx + υ.εyy  + B11 kxx + ν.kyy   0 + υ.ε 0 + B (6) Nyy = A11 εyy 11 kyy + ν.kxx xx ⎪ ⎪ ⎩ 0 + B .k Nxy = A66 .γxy 66 xy   ⎧   0 0 +D ⎪ ⎪ Mxx = B11 εxx + ν.εyy 11 kxx + ν.kyy ⎨     0 + ν.ε 0 + D (7) M ε = B yy 11 11 kyy + ν.kxx yy xx ⎪ ⎪ ⎩ 0 + D .k Mxy = B66 .γxy 66 xy Stiffness coefficients Aij , Bij , Dij in Eq. (6, 7) see Appendix 1. Nonlinear motion equations of FGM cylindrical shell with variable thickness subjected to external pressure q(t) and an axial compression p(t) based on the classical shell theory [33] are expressed as: ⎧ ∂Nxy ∂Nxx ∂2u ⎪ ⎪ ∂x + ∂y = ρ1 ∂t 2 ⎪ ⎪ ⎪ ⎨ ∂Nxy + ∂Nyy = ρ1 ∂ 22v ∂x ∂y ∂t (8) ∂ 2 Mxy ∂ 2 Myy ∂ 2 Mxx ∂2w ∂2w ⎪ + 2 + ⎪ 2 2 + Nxx . ∂x 2 + 2Nxy ∂x∂y ∂x∂y ⎪ ∂x ∂y ⎪ ⎪ ⎩ + N ∂ 2 w + Ny − p.h(x) ∂ 2 w + q = ρ ∂ 2 w + 2ερ ∂w yy ∂y2 1 ∂t 2 1 ∂t R ∂x2   −ρm h(x) = ρ1∗ h(x) . in which: ρ1 = ρm + ρck+1 Substituting Eq. (4), Eq. (6) and Eq. (7) into Eq. (8) we obtain: ⎧ 2 L11 (u) + L12 (v) + L13 (w) + P1 (w) = ρ1 ∂∂t 2u ⎪ ⎪ ⎪ 2 ⎨ L21 (u) + L22 (v) + L23 (w) + P2 (w) = ρ1 ∂∂t 2v (9) ⎪ L31 (u) + L32 (v) + L33 (w) + P3 (w) + P4 (u, w) ⎪ ⎪ ⎩ 2 2 ∂w + P5 (v, w) + q + p.h(x, y) ∂∂xw2 = ρ1 ∂∂tw 2 + 2ερ1 ∂t

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The coefficients Lij (i, j = 1, 3), Pk (k = 1, 5) in Eq. (9) see Appendix 2. Equation (9) are governing equations used to investigate nonlinear dynamic stability of variable thickness FGM circular cylinder shell subjected to mechanical load.

3 Solution Method The present article considers a variable thickness FGM cylindrical shell with simply supported at both ends, subjected to uniformly pressure q(t) and axial compression force p(t). The boundary conditions are: w = 0, Nxy = 0, Mxx = 0, Nxx = −p.h at x = 0 and x = L. Displacement components of the cylindrical shell can be expanded as follows: u = Umn (t)cosαx sin βy; v = Vmn (t) sin αxcosβy; w = Wmn (t) sin αx sin βy (10) n in which: α = mπ L ; β = R (m, n - the half-waves number in x and y direction, respectively). Substituting Eq. (10) into Eq. (9) then applying Galerkin procedure yields: ⎧ 2 ⎪ I11 U + I12 V + I13 W + R1 W 2 = ρ1∗∗ ddtU2 ⎪ ⎪ ⎪ ⎨ I21 U + I22 V + I23 W + R2 W 2 = ρ ∗∗ d 2 V 1 dt 2   2 3 2 ⎪ I31 U + I32 V + I33 W + R3 W + R4 W 3 + R5 U + R6 V + pm π (La+2b)R .W ⎪ 8L ⎪ ⎪ ⎩ 4δn δm RLq d 2W dW ∗∗ ∗∗ + mnπ = ρ1 dt 2 + 2ερ1 dt (11)

The coefficients Iij (i, j = 1, 3), Rk (k = 1, 6) in Eq. (11) see Appendix 3. According to Volmir’s assumption [34], by ignoring inertial components along x and y axes (u F f− ). It is contrasted to the experimental study in [10, 11], where F f+ < F f− . Keeping the preset friction force, F f0 unchanged, the anisotropic friction condition will increase when raising the value of inclined angle. In this study, θ was considered as an adjustable parameter. 2.2 Experimental Implementation The above model (Fig. 2(b)) was realized as shown in Fig. 3, as developing from the apparatus built at TNUT’s laboratory by Nguyen et al. [3, 8, 12].

Fig. 3. (a) Experimental diagram and (b) a photograph of the experimental apparatus

A mini electro-dynamical shaker (1) was placed on a slider of a commercial linear bearing guide (4), providing a tiny rolling friction force. Since the friction force inside the slider is much smaller than the preset friction, the effect of slider type on the experimental results is also small and can be neglected. The slider support can be adjusted to provide a certain inclined angle θ. An additional mass (2) was clamped on the shaker shaft with the support of sheet springs. Generally, applying a sinusoidal current with frequency f exc to the shaker leads to relative linear oscillation of the shaker shaft with the mass added on. Hereafter, the moveable mass, combined by the addition mass and the shaker shaft, is assigned as inertial mass, m1 , playing the role of the internal mass of the system. A noncontact position sensor (7) was used to measure the relative motion of the inertial mass m1 . The motion of the shaker body was recognized by a linear variable displacement transformer. The body shaker, including the sensors and the carbon tube, is referred as the mass m2 . A force sensor (3) was used as the obstacle block to measure the impact force. In order to preset the friction force when remaining the body mass m2 , a carbon tube (5) is connected with the shaker body by means of a flexible joint, avoiding any misalignment when moving. The detailed mechanism of providing preset friction is depicted in Fig. 4. As depicted in detail on Fig. 4a, the carbon tube is able to slide between two aluminium pieces in the form of a V-block (6). The two V-blocks are fixed on two electromagnets (7). Supplying a certain value of electrical current to the coupled electromagnets provides a desired clamping force on the tube and thus a corresponding value of sliding

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Fig. 4. Varying the friction force: (a) apparatus structure and (b) the dependency of friction force on the supplied voltage.

friction (Fig. 4b). The preset friction force was measured by pulling the body to move at a steady speed. By adjusting the voltage supplied to the coupled electromagnets, the corresponding friction force can be obtained (See [3] for more detailed). The shaker is powered by a sinusoidal current generated by a laboratory function generator and then amplified by a commercial amplifier. The application of a sinusoidal current to the shaker leads to oscillation of the shaker shaft and the mass attached on it (i.e. inertial mass m1 ). As provided by the shaker supplier, the magnetic force, F m is solely depended on the current supplied. Adjusting the sinusoidal supplying to the amplifier can provide a desired excitation force. A supplementary experiment was implemented to verify the relation of the magnetic force and the current supplied. A load cell was used as an obstacle resisting the shaker movement and thus to measure the magnetic force induced. A DC voltage was supplied to the shaker to generate the magnetic force. Varying the voltage, several pairs of the current passing the shaker and the force were collected. Experimental data confirmed that the excitation force is proportional to the current supplied to the shaker (See [8] for detailed information of how to determine this relation). For operational parameters, two levels of the preset friction force, F f0 were set as 6.8 N and 13.6 N. With respect to the total weight of the two masses as 2.336 kg, such levels of friction force correspond to two levels of friction coefficient as approximately be 0.3 and 0.6. In addition, two levels of the inclined angle θ were implemented at 2.5° Table 1. Parameters of the apparatus. Parameter

Notation

Value

Unit

Inertial mass

m1

0.518

kg

Body mass

m2

1.818

kg

Preset resistant force Ff0

6.8 and 13.6 N

Force ratio

α = Fm/Ff0 0.59

Inclined angle

θ

2.5 and 5.0

°

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and 5.0°. The force ratio, α is the ratio between excitation force amplitude F m and the preset resistant force F f0 . In this study, the force ratio was chosen, as 0.59, and unchanged during all experiments. This value was practically found to effectively drive the system moving. The overall experimental parameters are given in Table 1.

3 Results and Discussion Firstly, the progression velocities of the whole system are considered. Figure 5 presents the average velocities of the system body when varying the preset friction force F f0 , corresponding to two levels of the inclined angle θ.

Fig. 5. The average velocities of the locomotion system for: (a) θ = 2.5°; (b) θ = 5.0°.

As can be seen, when the excitation frequency f exc is less than 10 Hz, at each value of the inclined angle θ, i.e. each friction force ratio, the body mass m2 almost moves forward slowly. Also, larger preset friction force resulted in faster moving forward of the system. When increasing the excitation frequency larger than 10 Hz, the average velocity of the system significantly changed. The following interesting issues can be remarked: – In the case of applying the smaller friction ratio, corresponding to the inclined angle θ = 2.5° (Fig. 5a), with larger preset friction force (F f0 = 13.6 N), increasing excitation frequency in the range f exc ∈ [10, 17] Hz resulted in faster forward motion of the system. However, for smaller preset friction (F f0 = 6.8 N), this relationship was only true for the range of f exc ∈ [10, 13] Hz. Operation with larger excitation frequency reduced the moving forward velocity of the system. With the excitation larger than 19 Hz for large preset friction force, and 16 Hz for small preset friction force, the system started moving backward. – A similar phenomenon can be observed with higher inclined angle, θ = 5.0° (Fig. 5b). However, the range of the excitation frequency where the forward velocity increased with higher frequency become narrower compared to that in previous case. The highest forward velocity appeared at f exc = 15 Hz with high preset friction force (F f0 = 13.6 N). For smaller preset friction force (F f0 = 6.8 N), the system started moving backward if the excitation frequency is equal to or larger than 11 Hz.

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In order to give deeper insight into the system behavior, Fig. 6 illustrates the time history of the two masses’ motion for two cases: one for forward movement and another for backward movement. The two sub-plots have the same inclined angle (θ = 5.0°) and other parameters, except the preset friction force and the excitation frequency. In each sub-plots, the oscillation period of both masses m1 and m2 on each sub-plot were limitted by two impact instants. The oscillation of the mass m2 between two impact instants was divided into four areas: area (I) where the body mass m2 moved forward just after impact; area (II) where the body mass m2 moved backward; area (III) where body mass m2 may move forward or backward direction; and area (IV) where the body mass m2 moved backward. In Fig. 5, two impact instants are denoted by two red-dashed vertical lines; the areas between impact instants are separated by red-dot vertical lines. (a) Ff0 = 13.6 N; fexc = 15 Hz

(b) Ff0 = 6.8 N; fexc = 19 Hz

Fig. 6. Time history of motions of the internal mass X1 (black short dashed curve) and of the body X2 (blue solid curve) with θ = 5.0°: (a) Ff0 = 13.6 N; fexc = 15 Hz; and (b) Ff0 = 6.8 N; fexc = 19 Hz.

As can be seen on the Fig. 6, the impacts occurred at the instant of the peaks of the relative motion between the inertial mass m1 and the system body, m2 . The body mass m2 moved forward just after obtaining impact force from the inertial mass m1 . The both two masses started moving forward together (area I). Under the action of the preset friction force, the body mass m2 was quickly stopped. As a result of the gravity force and the inertial force of mass m1 , the system then started moving backward significantly in dynamic friction condition (area II). In the area III on the Fig. 5(b), the backward moving trend of the body mass applied by the smaller preset friction force seemed continuing. This could be suggested that at this instant, the inertial mass m1 would reached its limitted position of it amplitude, hence its velocity was slowdown, the inertial force of mass m1 acting on the mass m2 increased significantly. However, in the case of larger preset friction force, the velocity in backward direction of the inertial mass m1 increased, thus the inertial force acting body mass m2 decreased. This inertial force may be the reason make the body mass m2 moving forward in area III (Fig. 5(a)). In the first stage of area IV, both the gravity force and inertial force of mass m1 had the trend to pull the body mass m2 moving backward, so m2 moved very fast in backward direction. The width of the first stage of area IV is not the same, it is smaller on the Fig. 5(a) than that on the Fig. 5(b), maybe depending on the levels of preset friction force. In the second

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stage of area IV on the Fig. 5(a), the resultant of inertial and gravity force may be equal to the larger preset friction force. Thus it is seemed that the body mass m2 did not move. Even the body mass m2 tended to move forward because of the forward direction of the mass m1 . Decreasing the preset friction force, in the second stage of area IV, the mass m2 continued moving backward until the mass m1 moved forward (Fig. 6(b)). From these practical results, it could be concluded that the forward motion of the locomotion system in the case of anisotropic friction may strongly depend on the level of the preset friction force. The larger preset friction force is, the easier for the system to move forward. It is worth noting that, the mechanism behind of moving forward motion of the whole system in anisotropic frictional condition is still an open dynamic problem. The data obtained would be a good resource for further researches to explain why the system moves forward or backward direction in anisotropic frictional condition.

4 Conclusion This paper presented experimental results and several initial analyses on a vibrationdriven locomotion apparatus in anisotropic frictional condition. The experimental apparatus provided an ability of presetting the friction force and adjusting the difference between backward friction and forward friction by mean of an inclined angle of the sliding guide, while maintaining the weight of the whole system unchanged. A series of experimental tests were implemented, providing a deep insight of the locomotion system responses, both in progression velocity of the system and in the relative motions of the masses. The directional motion of the system in anisotropic friction condition was experimentally found to strongly depend on the level of the preset friction force. The larger initial friction force is, the easier and faster in forward direction of the locomotion system moves, even with high anisotropic friction condition, i.e. with larger inclined angle. The experimental results would be useful for further researches on the design and operation of vibration-driven locomotion systems. Acknowledgement. This research was funded by Vietnam Ministry of Education and Training, under the grant number B2019-TNA-04. The authors would like to express their thank to Thai Nguyen University of Technology, Thai Nguyen University for their supports.

References 1. Liu, P., Yu, H., Cang, S.: Modelling and analysis of dynamic frictional interactions of vibrodriven capsule systems with viscoelastic property. Eur. J. Mech. A. Solids 74, 16–25 (2019) 2. Yan, Y., et al.: Proof-of-concept prototype development of the self-propelled capsule system for pipeline inspection. Meccanica 53(8), 1997–2012 (2017). https://doi.org/10.1007/s11012017-0801-3 3. Nguyen, V.-D., La, N.-T.: An improvement of vibration-driven locomotion module for capsule robots. Mech. Based Des. Struct. Mach. 1–15 (2020) 4. Chernous’ko, F.L.: Analysis and optimization of the motion of a body controlled by means of a movable internal mass. J. Appl. Math. Mech. 70(6), 819–842 (2006)

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5. Pavlovskaia, E., et al.: Modelling of ground moling dynamics by an impact oscillator with a frictional slider. Meccanica 38(1), 85–97 (2003) 6. Nguyen, V.-D., Woo, K.-C., Pavlovskaia, E.: Experimental study and mathematical modelling of a new of vibro-impact moling device. Int. J. Non-Linear Mech. 43(6), 542–550 (2008) 7. Liu, Y., et al.: Modelling of a vibro-impact capsule system. Int. J. Mech. Sci. 66, 2–11 (2013) 8. Nguyen, V.-D., et al.: The effect of inertial mass and excitation frequency on a Duffing vibro-impact drifting system. Int. J. Mech. Sci. 124–125, 9–21 (2017) 9. Xu, J., Fang, H.: Stick-slip effect in a vibration-driven system with dry friction: sliding bifurcations and optimization. J. Appl. Mech. 81, 061001 (2014) 10. Xu, J., Fang, H.: Improving performance: recent progress on vibration-driven locomotion systems. Nonlinear Dyn. 98(4), 2651–2669 (2019). https://doi.org/10.1007/s11071-019-049 82-y 11. Du, Z., et al.: Experiments on vibration-driven stick-slip locomotion: a sliding bifurcation perspective. Mech. Syst. Sig. Process. 105, 261–275 (2018) 12. Nguyen, V.-D., et al.: Identification of the effective control parameter to enhance the progression rate of vibro-impact devices with drift. J. Vibr. Acoust. 140(1), 011001 (2017)

Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation Thi Loan Mai, Myungjun Jeon, Seung Hyeon Lim, and Hyeon Kyu Yoon(B) Department of Naval Architecture and Marine Engineering, Changwon National University, Gyeongsangnam-do, Republic of Korea [email protected]

Abstract. Many studies have been done to estimate the maneuvering characteristics of commercial ships. Studies on a high-speed catamaran, there was a great number of theoretical, numerical studies, and experimental investigation, most of them have focused on the resistance performance. In addition, a CFD (Computational Fluid Dynamics) method has become a possible tool to predict hydrodynamics. This study focuses on predicting hydrodynamic maneuvering characteristics of high-speed catamaran by utilizing RANS (Reynold-Averaged Navier-Stokes) solver. The Delft 372 catamaran model was selected as the target hull to analyze hydrodynamic characteristics. Due to the high-speed condition and changeable attitude, the motion of the catamaran was complex. The comparisons of the obtained CFD results including the free surface effects in resistance performance with experimental data were shown a relatively good agreement and it could demonstrate that the presented method could be used for predicting hydrodynamic coefficients of high-speed catamaran. Then virtual captive model tests were performed to obtain hydrodynamic coefficients. The lower-order Fourier coefficients were applied to get the hydrodynamic coefficients in dynamic motion. The linear fitting to Fourier coefficients was observed very well using least square method in the harmonic motion. Keywords: Delft 372 catamaran · RANS solver · High-speed · Virtual captive model test · Hydrodynamic characteristics · Maneuvering simulation

1 Introduction The demand for high-speed catamaran has been increasing significantly in recent years for variety of purposes such as passenger, military, and commercial applications. Studies on high-speed catamaran, there was a great number of theoretical, numerical studies, and experimental investigation, however, most of them have focused on the resistance performance and seakeeping performance. Broglia et al. (2011) simulated to analyze the interference phenomena between monohull with focus on its dependence on the Reynold number using CFD numerical method. The flow around a high-speed vessel in both catamaran and monohull configuration was observed, in addition, the wave © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 546–559, 2022. https://doi.org/10.1007/978-981-16-3239-6_41

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patters, wave profiles, streamlines, surface pressure, and velocity fields were analyzed. In addition, they performed an experimental investigation of hull interference effects on the total resistance by changing the distance between monohull of a catamaran in calm water in 2014. On the other hand, the seakeeping investigation for high-speed catamaran has been done by this author in order to emphasize the influence of Froude on the maximum response of the vertical ship motions and the role of the nonlinear effects both on the ship motions and on the added resistance. Wave resistance for high-speed catamaran was investigated by Moraes et al. (2004), they applied two methods were the slender-body theory and the 3D panel method using Shipflow software to estimate the effects of catamaran hull spacing included the effects of shallow water on the wave resistance component. Regarding CFD simulation for estimating hydrodynamic characteristics, Su et al. (2012) utilized RANS numerical method to predict the motion and analyze the hydrodynamic performance of planning vessels at high speed. Liu et al. (2018) simulated a virtual captive model tests for KCS (KRISO Container Ship) using CFD to estimate linear and nonlinear hydrodynamic coefficients in the 3rd-order Abkowiz model. After that maneuvering simulation was predicted. Applying CFD in marine vehicles using STARCCM+ (Hajivand and Mousavizadegan 2015), OpenFOAM (Islam and Soares 2018), and Ansys FLUENT (Nguyen et al. 2018) have been done for predicting maneuvering characteristics. This paper focuses on estimating the maneuvering characteristics of Delft 372 catamaran at high-speed. Ansys FLUENT 20.1 is used for solving RANS equation. Resistance performance is conducted in order to validate the numerical method. The comparisons of obtained results in resistance performance with experimental data were demonstrated that the presented numerical method was appropriate. Therefore, virtual captive model tests as static drift, pure sway, pure yaw, and combined pure yaw with drift test is conducted in order to obtain hydrodynamic coefficients. The Fourier series is applied to analyze hydrodynamic coefficients in harmonic motion. Especially, the coupling coefficients are found at combined pure yaw with drift angle.

2 Governing Equations and Turbulence Modeling The homogenous multiphase Eulerian fluid approach is adopted in this study to describe the interface between the water and air, mathematically. Assumption, the flow around the ship is incompressible. The governing equations that need to be solved are the mass continuity equation and momentum equations, which are given in Eqs. (1) and (2) respectively. ∂ui =0 ∂xi

(1)

∂τij ∂ui ∂ui 1 ∂p ∂ 2 ui + uj =− +ν − + fi ∂t ∂xj ρ ∂xi ∂xi ∂xj ∂xj

(2)

where ui and uj are the average velocity components; p is the average pressure; ν is the kinematic viscosity; xi and xj are the ith and j th coordinates in the fluid domain

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respectively; and ρ is the water density; τij = ui uj is so-called the Reynolds stress tensor; ui and uj are the fluctuating components; fi is the external forces. In order to capture the wave pattern of the free surface, the volume of fluid (VOF) method is implemented. A transport equation in Eq. (3) is then solved for the advection of this scalar quantity, using the velocity files obtained from the solution of the NavierStokes equations at the last time step. ∂q  u) = 0 + ∇(q ∂t 2 

(3)

Equation (3) gives the volume fraction q for each phase in all computation cell where qk = 1; u is the velocity; ∇ is the gradient.

k=1

Furthermore, a k −ω SST (Shear Stress Transport) model turbulence model is applied to consider the viscous effects. In this turbulence model, k is the turbulence kinetic energy and ω is the dissipation rate of the turbulent energy. The two equation model of a k − ω SST is given by the following:   ∂k ∂ ∂k ∂k ∗ (4) = Pk − β ωk + + ui (ν + σk νt ) ∂t ∂xi ∂xi ∂xi   ∂ω ∂ω γ ∂ ∂ω σω2 ∂k ∂ω ∗ 2 = Pk − β ω + (5) + 2(1 − F1 ) + ui (ν + σω νt ) ∂t ∂xj μt ∂xi ∂xi ω ∂xi ∂xi

3 Case Study and Coordinate System The candidate ship presented in this study is Delft 372 catamaran model with symmetrical demi-hulls shape that originally used in TU-Delft by Van’t Veer (1988). The model hull of Delft 372 catamaran is depicted in Fig. 1 and the main particulars are given in Table 1.

Fig. 1. Delft 372 catamaran geometry

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Table 1. Main particular of Delft 372 catamaran Main particular

Symbol

Scale

Unit

Full scale

Model scale



1.00

33.33

Length perpendiculars

Lpp

m

100.00

3.00

Beam overall

BOA

m

31.33

0.94

Beam demi-hull

b

m

8.00

0.24

Distance between center of the demi-hull

H

m

23.33

0.70

Separation distance

s

m

15.33

0.46

Draft

T

m

5.00

0.15

Vertical center of gravity

KG

m

11.33

0.34

Longitudinal center of gravity

LCG

m

47.00

1.41

Wetted surface area

S

m2



1.945

Two right-hand coordinate system are adopted to define the kinematic and hydrodynamic forces acting on the Delft 372 catamaran as shown in Fig. 2. The earth-fixed coordinate system o0 –x 0 y0 is set up to define the ship motion. The body-fixed coordinate system o–xy is used for computing the hydrodynamic forces. The origin is located at the intersection of the waterline plane and the center-line plane at the mid-ship section. The equation of motion for maneuvering ship in 3DOF become: m(˙u − vr − xG r 2 ) = X m(˙v + ur + xG r˙ ) = Y I˙z r˙ − mxG (˙v + ur) = N

(6)

where m is ship mass; u, v, and r are the surge velocity, sway velocity, and yaw rate, respectively; u˙ , v˙ , and r˙ are the corresponding surge acceleration, sway acceleration, and angular acceleration; Iz is the moment of inertia about the z-axis; X , Y and N are the resultant forces acting on ship in surge force, sway force, and yaw moment, √ respectively; U is the ship speed defined as U = u2 + v2 ; β is the drift angle defined by β = tan−1 (−v / u). In addition, the hydrodynamic forces acting on the ship hull are expressed as the below equation: XH = X0 + Xu u + Xuu u2 + Xuuu u3 + Xvv v2 + Xrr r 2 + Xvr vr YH = Yv˙ v˙ + Yr˙ r˙ + Yv v + Yvvv v3 + Yr r + Yrrr r 3 + Yvvr v2 r + Yvrr vr 2 NH = Nv˙ v˙ + Nr˙ r˙ + Nv v + Nvvv v3 + Nr r + Nrrr r 3 + Nvvr v2 r + Nvrr vr 2

(7)

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Fig. 2. Coordinate system

4 Virtual Captive Model Test The maneuvering characteristics of the high-speed catamaran are estimated by performing a virtual captive model test in order to get the hydrodynamic force. The computational conditions are depicted in Table 2. Table 2. The computational conditions. Case

Froude number (-)

β(deg.)

vmax (m/s)

rmax (rad/s)

Resistance

0.1–0.8 *interval 0.05







Static Drift

0.45

−10°–10o *interval 2o





Pure Sway

0.45



0.179, 0.269, 0.358, 0.448



Pure Yaw

0.45





0.366, 0.448, 0.529, 0.610

Combined Pure Yaw-Drift

0.45

2°, 4°, 6°, 8o



0.366, 0.448, 0.529, 0.610

4.1 Static Drift Test The ship travels through the tank in oblique flow due to a given drift angle β in static drift test. The configuration in Fig. 3 and mathematical model Eq. (8) described static drift test: X = X0 + Xvv v2 Y = Y0 + Yv v + Yvvv v3 N = N0 + Nv v + Nvvv v3

(8)

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Fig. 3. Configuration of static drift test

4.2 Harmonic Test The harmonic test includes pure sway, pure yaw, and combined yaw with drift test. Pure sway test, the ship moves through the tank on straight-ahead course while it is oscillated from side to side. The pure sway motion expresses in terms of the velocities u = constant while r = 0 and v oscillates harmonically. The linear acceleration coefficients such as Yv˙ and Nv˙ will be estimated in this test. The configuration in Fig. 4 and hydrodynamic forces in Eq. (9) determined for pure sway motion: X = X0 + Xvv v2 Y = Y0 + Yv˙ v˙ + Yv v + Yvvv v3 N = N0 + Nv˙ v˙ + Nv v + Nvvv v3

(9)

Fig. 4. Configuration of pure sway test

In harmonic motion, the Frourier series method (Sakamoto et al. 2012) is ultilized to simplify the mathematical models Eq. (9). Since hamonic motion are prescribed by sine and cosine function, hence hydrodynamic forces can be rewritten as Fourier series with angular frequency ω: f = f0 +

3 

fcn cos(nωt) +

n=1

3 

fsn sin(nωt)

(10)

n=1

On the other hand, the harmonic forms are determined by replacing the motion equation (v = −vmax cos ωt, v˙ = v˙ max sin ωt) into Eq. (9). X = X0 + Xc2 cos 2ωt Y = Yc1 cos ωt + Ys1 sin ωt + Yc3 cos 3ω N = Nc1 cos ωt + Ns1 sin ωt + Nc3 cos 3ω

(11)

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Pure yaw test, the ship moves through the tank while it conducts a pure yaw motion, where it is forced to follow the tangent of the oscillating path. The term of velocities is expressed that v = 0, while r and u oscillate harmonically. The hydrodynamic coefficients as Xrr , Yr˙ , Yr , Yrrr , Nr˙ , Nr , and Nrrr are estimated in the test. The configuration Fig. 5 and hydrodynamic forces in Eq. (12) defined for pure yaw motion: X = X0 + Xrr r 2 Y = Y0 + Yr˙ r˙ + Yr r + Yrrr r 3 N = N0 + Nr˙ r˙ + Nr r + Nrrr r 3

(12)

Fig. 5. Configuration of pure yaw test

Combined yaw and drift is described the same as with pure yaw test, however, a drift angle is set on the motion in order to obtain a drift angle relative to the tangent of the oscillating path. The terms of velocity represents v = 0 and constant, while r and u oscillate harmonically. The coupling hydrodynamic coefficients as Xvr , Yvvr , Yvrr , Nvvr , and Nvrr are estimated in this test. The combined pure yaw with drift test is described in Fig. 6 and hydrodynamic forces are shown in the below equation: X = X0 + Xvv v2 + Xrr r 2 + Xvr vr Y = Y0 + Yv v + Yvvv v3 + Yr r + Yrrr r 3 + Yvvr v2 r + Yvrr vr 2 N = N0 + Nv v + Nvvv v3 + Nr r + Nrrr r 3 + Nvvr v2 r + Nvrr vr 2

(13)

Fig. 6. Configuration of combined yaw and drift

Corresponding to pure sway test, the harmonic forms of pure yaw and combined yaw and drift are obtained by substituting motion equation (r = rmax sin ωt, r˙ = r˙max cos ωt) into Eqs. (12) and (13). Finally, coefficient of hamonic test can obtain from Fourier seriers as shown in Table 3.

3 Ys1 = Yr r + Yrrr r 3 4 Yc1 = Yr˙ r˙ 1 Ys3 = − Yrrr r 3 4 1 Y0 = Y∗ + Yv v + Yvvv v3 + Yvrr vr 2 2 3 1 Ys1 = Yr r + Yrrr r 3 + Yvvr v2 r 4 2 Yc1 = Yr˙ r˙ 1 Yc2 = − Yvrr vr 2 2 1 Ys3 = − Yrrr r 3 4

1 X0 = X∗ + Xrr r 2 2 1 Xc2 = − Xrr r 2 2

1 X0 = X∗ + Xvv v2 + Xrr r 2 2 Xs1 = Xvr vr 1 Xc2 = − Xrr r 2 2

Pure yaw

Combined pure yaw with drift

1 N0 = N∗ + Nv v + Nvvv v3 + Nvrr vr 2 2 3 1 Ns1 = Nr r + Nrrr r 3 + Nvvr v2 r 4 2 Nc1 = Nr˙ r˙ 1 Nc2 = − Nvrr vr 2 2 1 Ns3 = − Nrrr r 3 4

3 Ns1 = Nr r + Nrrr r 3 4 Nc1 = Nr˙ r˙ 1 Ns3 = − Nrrr r 3 4

Ns1 = Nv˙ v˙ 1 Nc3 = − Nvvv v3 4

Ys1 = Yv˙ v˙ 1 Yc3 = − Yvvv v3 4

1 X0 = X∗ + Xvv v2 2 1 Xc2 = Xvv v2 2

Pure sway

N component   3 Nc1 = − Nv v + Nvvv v3 4

Y component   3 Yc1 = − Yv v + Yvvv v3 4

X component

Harmonic test

Table 3. Fourier coefficients for harmonic test Investigation of Maneuvering Characteristics of High-Speed Catamaran 553

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The obtained hydrodynamic forces are non-dimensionalized by ship speed, ship length and water density as follows, u  v rL ; v = ; r = U U U X Y N   X = ;Y = ; N = 2 2 2 2 0.5ρU L 0.5ρU L 0.5ρU 2 L3

u =

(14)

5 Numerical Modeling In order to simulate the virtual captive model tests, the CFD software ANSYS Fluent 2020R1 is applied. Incompressible unsteady RANS with k − ω SST turbulence model is used for simulating two-phase volume of the fluid technique. The volume of fluid (VOF) method is applied to capture the position of the free surface. In addition, a SIMPLE algorithm solves for pressure-velocity coupling. The y + value of 30 is estimated for Reynolds number of 7.3E+6. The catamaran is covered by a rectangular domain to simulate the captive model test. According to Practical Guidelines for ship CFD Application (ITTC 2011) the dimensions of the domain are chosen to be able to avoid backflow and side flow. Additionally, boundary conditions are needed to assign for ensuring the physical characteristics of a fluid problem. The wall boundary condition represents the object’s surface, it is so-called no-slip wall. Pressure-inlet with open channel flow is required to start the calculation. Pressure-outlet with open channel flow is usually treated as a far-field condition, where flow properties are almost unchanged and hydrostatic pressure is specified. The symmetry condition indicates that the normal velocity and gradients of all variables at the symmetry plane is zero. The mesh generation for calculation is shown in Fig. 7.

Fig. 7. Mesh generation

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6 Results and Discustions 6.1 Validation First at all, in order to estimate well the calculation of catamaran, a validation of the numerical method is conducted by comparing the results of CFD simulation with experimental results performed by Broglia et al. (2014). Resistance performance is simulated with a range of Froude numbers from 0.1–0.8. The comparison of the results in Figs. 8, 9, 10 and 11 shown the consistency between CFD and the experiment. Consequently, the CFD numerical method is appropriate to simulate the captive model tests.

Fig. 8. The total resistance

Fig. 9. The resistance coefficient

Fig. 10. The sinkage

Fig. 11. The trim

6.2 Hydrodynamic Forces After validation the virtual captive model test is performed. The Froude number of 0.45 is selected for simulating high-speed catamaran. The reason for this selection due to fast increasing resistance and trim angle at this point as illustrated in Figs. 8, 9, 10 and 11. Moreover, sinkage is largest at Froude number 0.45 in Fig. 10. The below figures show the results of hydrodynamic forces acting on catamaran for all the computed cases, which cover static drift, pure sway, pure yaw, and combined pure yaw with drift tests. The catamaran is symmetrical about the vertical center plane, the hydrodynamic forces acting on the hull in case of static drift are symmetrical in both positive and negative drift angles as shown in Fig. 12. The hydrodynamic forces in the case of harmonic motion are analyzed using the Fourier series. The added mass coefficients are obtained from

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Fig. 12. Hydrodynamic forces of static drift test

Fig. 13. Fitting in-phase values of hydrodynamic forces versus lateral acceleration in pure sway test

Fig. 14. Fitting in-phase values of hydrodynamic forces versus yaw rate in pure yaw test

in-phase values and damping coefficients are obtained from outphase values. On the other hand, the coupling coefficients are determined from the combined yaw with drift by subtracting forces acting on the hull in the single motions from the force acting on the hull in the combined motions. The obtained hydrodynamic forces are approximated using least square regression for each mathematical model in order to get hydrodynamic coefficients. Figures 13, 14, 15, 16 describe the hydrodynamic forces acting on the catamaran and fitting curves in the harmonic motion. It can be seen that the linear

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Fig. 15. Fitting out-phase values of hydrodynamic forces versus yaw rate in pure yaw test -2.5E-3

-5.0E-4

-3.0E-4 -2.0E-4 -1.0E-4 -0.0E+0

-2.0E-3 β= β= β= β=

-2.0E-3

Y'out-phase (-)

X'out (-)

-4.0E-4

2 deg. 4 deg. 6 deg. 8 deg.

2 deg. 4 deg. 6 deg. 8 deg.

-1.5E-3 -1.0E-3 -5.0E-4

0

0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)

0.0E+0

β= β= β= β=

-1.6E-3

N'out-phase (-)

β= β= β= β=

2 deg. 4 deg. 6 deg. 8 deg.

-1.2E-3 -8.0E-4 -4.0E-4

0

0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)

0.0E+0

0

0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)

Fig. 16. Fitting out-phase values of hydrodynamic forces versus yaw rate in combined yaw with drift test

Fig. 17. Wave pattern at various drift angle

reactions are fitting very well to the hydrodynamic forces. The wave pattern of static drift test in Fig. 17 illustrates cause asymmetry wave pattern around the ship which leads to sway force and yaw moment and as drift angle increases the asymmetry of the wave pattern is clearly observed. Finally, the hydrodynamic coefficients obtained from CFD simulation in this study are listed in Table 4.

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Coefficient X0 Xu  Xuu  Xuuu  Xvv Xrr  Xvr

Values

Coefficients

−2.41E−3

Yv˙ Yr˙ Yv  Yvvv Yr  Yrrr  Yvvr  Yvrr

6.60E−3 −5.42E−3 6.98E−4 −6.05E−2 3.07E−3 −9.15E−6

Values

Coefficients

Values

−4.92E−4

Nv˙ Nr˙ Nv  Nvvv Nr  Nrrr  Nvvr  Nvrr

2.71E−4

−2.45E−4 −1.36E−2 −4.18E−1 −3.01E−2 3.50E−2 5.72E−4 −7.64E−2

−8.33E−5 −5.84E−3 −6.82E−2 −1.34E−2 1.48E−2 5.81E−4 −1.04E−1

7 Conclusions In this paper, the CFD numerical method has been utilized for simulating the virtual captive model test of Delft 372 catamaran at the high-speed conditions. The straight and oblique motion was simulated in the stationary reference frame. The harmonic motions were implemented in the dynamic mesh motion using a user-defined function written by c programming. The resistance performance was conducted to validate the CFD numerical method, an agreement was observed between the CFD method and experiment results. Additionally, the fitting resistance result using the least square method was found in first-order, secondorder, and third-order terms. Harmonic motions were performed for obtaining the added mass coefficient in pure sway and pure yaw test. Especially, the coupling coefficients were estimated in combined yaw with drift test. The Fourier series was applied for predicting the hydrodynamic coefficients in the harmonic test. The linear fitting using the least square method was observed very well to the hydrodynamic forces in the harmonic test. The estimated coefficients will be used for predicting maneuvering characteristics in further study. Acknowledgement. Following are results of a study on the “Leaders in INdustry-university Cooperation + Project, supported by the Ministry of Education and National Research Foundation of Korea.

References Broglia, R., Bouscasse, B., Jacob, B., Olivieri, A., Zaghi, S., Stern, F.: Calm water and seakeeping investigation for fast catamaran. In: 11th International Conference on Fast Sea Transportation 2011, pp. 336–344 (2011) Broglia, R., Jacob, B., Zaghi, S., Stern, F., Olivieri, A.: Experimental investigation of interference effects for high-speed catamarans. Ocean Eng. 76, 75–85 (2014)

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Castiglione, T., Stern, F., Bova, S., Kandasamy, M.: Numerical investigation of the seakeeping behavior of a catamaran advancing in regular head waves. Ocean Eng. 38, 1806–1822 (2011) Ghadimi, P., Mirhosseini, S.H., Dashtimanesh, A.: RANS simulation of dynamic trim and sinkage of a planning hull. Appl. Math. Phys. 1, 6 (2013) Hajivand, A., Mousavizadegan, S.: Virtual simulation of maneuvering captive tests for a surface vessel. Int. J. Naval Archit. Ocean Eng. 7, 848–872 (2015) Islam, H., Soares, C.: Estimation of hydrodynamic derivatives of a container ship using PMM simulation in OpenFOAM. Ocean Eng. 164, 414–425 (2018) International Towing Tank Conference (ITTC): ITTC-Recommended Procedures and Guidelines Practical Guidelines for Ship CFD Application (2011) Jeon, M.J., Nguyen, T.T., Yoon, H.K.: A study on verification of the dynamic modeling for a submerged body based on numerical simulation. Int. J. Eng. Technol. Innov. 10, 107–120 (2020) Liu, Y., Zou, Z.J., Guo, H.P.: Predictions of ship maneuverability based on virtual captive model tests. Eng. Appl. Comput. Fluid Mech. 12, 334–353 (2018) Mai, T.L., Nguyen, T.T., Jeon, M.J., Yoon, H.K.: Analysis on hydrodynamic force acting on a catamaran at low speed using RANS numerical method. J. Navigat. Port Res. 2, 53–64 (2019) Milanov, E., Chotukova, V., Stern, F.: System based simulation of Delft372 catamaran maneuvering characteristics as function of water depth and approach speed. In: 29th Symposium on Naval Hydrodynamics (2012) Moraes, H.B., Vasconcellos, J.M., Latorre, R.G.: Wave resistance for high-speed catamarans. Ocean Eng. 31, 2253–2282 (2004) Nguyen, T.T., Yoon, H.K., Park, Y.B., Park, C.J.: Estimation of hydrodynamic derivatives of full-scale submarine using RANS solver. J. Ocean Eng. Technol. 32, 386–392 (2018) Su, Y.M., Chen, Q.T., Shen, H.L., Lu, W.: Numerical simulation of a planning vessel at high speed. J. Mar. Sci. Appl. 11, 178–183 (2012) Sakamoto, N.M., Carriace, P.M., Stern, F.: URANS simulations of statics and dynamic maneuvering for surface combatant: part 1. Verification and validation for forces, moment, and hydrodynamic derivatives. J. Mar. Sci. Technol. 17, 422–445 (2012) Van’t Veer, R.: Experimental results of motions, hydrodynamic coefficients and wave loads on the 372 catamaran model. Delft University Report 1129 (1998)

Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures Huu-Anh-Tuan Nguyen(B) Faculty of Civil Engineering, University of Architecture Ho Chi Minh City, Ho Chi Minh City, Vietnam [email protected]

Abstract. A tuned mass damper (TMD) is a type of auxiliary energy dissipation device mounted in a structure to mitigate the vibrations of the structure. Optimum TMD’s parameters including the natural frequency and damping ratio have been suggested in the literature, which normally focus on attenuating the displacement response of the primary structure. While excessive displacements can affect the safety and integrity of a structure, limiting acceleration response levels are more important at serviceability limit state as these relate to the functionality of nonstructural components and occupier comfort. The current paper aims at developing optimum TMD parameters for minimizing the steady state acceleration of structures such as building floors subjected to harmonic forces, especially considering the inherent damping present in the main structure. For each mass ratio and structure damping value considered, numerical simulations were performed to search for a combination of TMD tuning frequency and damping that minimize the peaks observed on the response spectra of the combined structure-TMD system. The new optimum TMD parameters were found for a range of mass ratios and primary structural damping ratios commonly used in practice. Empirical expressions for the tuning parameters were also derived by means of curve fitting to facilitate design work. The reliability of the fitting models was satisfactory with the calculated 95-th percentile error being as low as 0.2% and the correlation coefficient being as high as 0.999. Compared with various tuning formulas in relevant literature, the new proposal for TMD parameters was found to provide higher reduction in the structural acceleration response, hence more effective in terms of acceleration control. Keywords: Tuned mass damper · Optimum parameters · Harmonic excitation · Acceleration · Curve fitting · Empirical formula

1 Introduction A tuned mass damper (TMD) which conventionally consists of a mass, a spring, and a dashpot is a type of auxiliary energy dissipation device attached to a structure to attenuate the vibrations of the structure. The TMD’s frequency is tuned to a particular natural frequency of the primary structure. When that frequency is excited, the TMD will resonate out of phase with the structural motion and dissipate energy via its inertia force © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 560–572, 2022. https://doi.org/10.1007/978-981-16-3239-6_42

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acting on the structure. In the past few decades, TMDs and their variations have been widely applied in buildings, chimneys, bridges and other industrial facilities. Passive and active TMDs have been proved to be very effective in controlling horizontal motion of tall buildings under wind gust loadings and earthquake excitations [1–3]. Different types of TMDs ranging from conventional spring-mass-dashpot systems to pendulum TMD have also been explored for human-induced floor vibration applications with some degree of success [4, 5]. Regarding office floors whose occupants would even be sensitive to low level of vibrations, an innovative configuration of viscoelastic TMD in the form of grouped sandwich beams has recently been developed to reduce the floor response to human walking [6, 7]. A number of solutions for optimum TMD dynamic parameters have been suggested in the literature, which usually focus on attenuating the displacement response of the primary structure. The loading scenarios considered commonly include harmonic excitation or white noise random excitation applied as either input force or base acceleration on the primary structure [8–14]. This paper limits the investigation to structures subjected to harmonic forces such as building floors excited by unbalanced rotating machines or walking activities [15, 16]. Figure 1 shows a two degree of freedom (2DOF) system which is a simplified model of a primary structure attached to a TMD. The symbols m, k, c, x represent the mass, stiffness, damping coefficient and displacement with subscripts s and d referring to the primary structure and the TMD, respectively. Let f be the ratio of the TMD’s natural frequency to that of the main system and ζ d be the TMD’s damping ratio. The TMD design is essentially the process of specifying optimum values for f and ζ d to effectively reduce the motion of the primary structure.

Fig. 1. Simplified model of a structure with a TMD

Let m be the ratio of the TMD mass and the main mass. When the primary structure is undamped, the TMD parameters that minimize the peak displacement of the main mass can be determined using the classical expressions (1) and (2) developed by Den Hartog, which are functions of only the mass ratio [8]. f =

1 1+m

(1)

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 ζd =

3m 8(1 + m)

(2)

In order to gain better damping for the primary structure and limit the relative motion of the damper mass, Krenk [9] suggested replacing the factor 3/8 in (2) by the factor 1/2 whilst retaining (1) for the tuning frequency. Their formula for optimum TMD damping ratio therefore becomes:  m (3) ζd = 2(1 + m) When the primary structure has a certain amount of damping, a closed form solution for TMD parameters cannot be obtained analytically. Tsai and Lin [11] performed numerical simulations to find the optimum values of tuning frequency ratio and TMD damping ratio that minimize the structure’s displacement response. They then derived the fitted regression formulas for f and ζ d as given by Eqs. (4) and (5) where ζ s is the damping ratio of the main structure.      √ √ 1 + 1 − 2ζs2 − 1 − 1.398 + 0.126 m − 2.004m ζs m f = 1+m  √  √ (4) − 0.362 − 5.897 m + 8.533m ζs2 m  ξd =

3m + 0.157ζs − 0.321ζs2 + 0.195ζs m 8(1 + m)

(5)

Another extension of the Den Hartog solution to minimize the maximum displacement of the damped main mass was introduced by Abubakar and Farid [12] with the fitted Eqs. (6) and (7) being functions of both m and ζ s . More recently, Salvi and Rizii [13] produced Eqs. (8) and (9) for best tuning of TMD with regard to displacement control as well.     m 1 1 − 1.5906ζs (6) f = 1+m 1+m  0.1616ζs 3m ζd = + (7) 8(1 + m) (1 + m)   √ 1√ f = 1 − 3m (8) m + ζs 2 ζd =

1 3√ m + ζs 5 6

(9)

While excessive displacements can affect the safety and integrity of a structure (so these need be controlled to acceptable limits in design), limiting acceleration response levels are more important at serviceability limit state as these relate to the functionality of non-structural components and occupier comfort. Current guidelines on floor vibrations

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normally specify acceptable vibration levels for human comfort, relating to various human activities in different environments, in terms of acceleration [16, 17]. A solution for the TMD parameters that could minimize the structure’s acceleration response was recommended by Warburton [14] with expressions (10) and (11) for undamped structure, i.e. ζ s = 0. 1 f =√ 1+m  3m ζd = 8(1 + m/2)

(10)

(11)

The present paper aims at developing optimum TMD parameters for minimizing the steady state acceleration response of damped structures subjected to harmonic forces. The numerical search for tuning parameters is followed by the derivation of empirical expressions that well fit the numerical results hence useful for practical design. Comparisons with various tuning formulas in the literature are also made to verify the efficiency of the newly proposed solution.

2 Methods 2.1 Dynamic Magnification Factor for Steady State Acceleration The governing equation of motion of the 2DOF system of Fig. 1 when the main mass is subjected to an external harmonic force F(t) can be written as: M¨x + C˙x + Kx = F

(12)

in which the mass matrix M, stiffness matrix K, damping matrix C, displacement vector x and forcing vector F are expressed as follows: M=

ms 0 0 md

K=

ks + kd −kd −kd kd



C=

cs + cd −cd −cd cd

x = {xs xd }T F = {F 0}T

(13)

The steady-state solution for Eq. (12) can be found by using exponential representation. The harmonic√force is expressed as F = F 0 ejωt in which ω is the forcing circular frequency and j = −1. Let the steady-state solution of x be of the form: x = Xejωt x˙ = jωXejωt x¨ = −ω2 Xejωt

(14)

where X = {X s X d }T is the vector of displacement amplitude for the combined 2DOF system. Moreover, the following conventional terms are used:   ks kd cs cd ωd = ζs = ζd = ωs = ms md 2ωs ms 2ωd md md ωd ω m= f = r= (15) ms ωs ωs

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The solution of (12) can then be obtained in the form of (16) with associated coefficients defined by (17). Xs −(a1 + jb1 ) = F0 /ks a2 + jb2

(16)

a1 = f 2 − r 2 b1 = 2ζd rf a2 = r 2 − r 4 + f 2 r 2 (1 + m) − 1 + 4r 2 f ζs ζd 

 b2 = 2r ζs r 2 − f 2 + f ζd r 2 + mr 2 − 1

(17)

The dynamic magnification factor for steady state acceleration response, D, of the primary structure in the combined system is now defined as:    2 2 X¨ s  2 a1 + b1 D= =r (18) F0 /ms a22 + b22   in which X¨s  is the magnitude (modulus) of the complex acceleration amplitude X¨s . 2.2 Optimization Process for TMD Parameters A Matlab code [18] was developed to perform numerical simulations that searched for optimal values of f and ζ d with regard to minimizing the maximum value of the steady state acceleration magnification factor D of a damped primary structure. Table 1 shows the investigated range for various parameters where each parameter would vary from its lower to upper values in the numerical search. Table 1. Investigated range of parameters Parameter

Lower Upper Increment

Mass ratio, m

0.0025 0.1

0.0025

Main structure damping ratio, ζ s 0

0.1

0.005

Tuning frequency ratio, f

1.2

0.00025

0.8

TMD damping ratio, ζ d

0

0.3

0.0005

Forcing frequency ratio, r

0.8

1.2

0.0005

For a given mass ratio m and primary structural damping ratio ζ s , the search procedure was as follows:

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(a) For a tuning frequency ratio value, f , generate the acceleration response spectra corresponding to different values of TMD damping ratios ζ d in the investigated range. Each response spectrum was calculated using Eq. (18), showing the relationship between the dynamic magnification and the forcing frequency ratio. An example of the generated response spectra is presented in Fig. 2 where various response spectra with different ζ d were plotted for the same m = 0.02, ζ s = 0.05 and f = 1. Two peaks can normally be observed in each response spectrum. Identify the ζ d value that minimized the peak response at the tuning frequency ratio being considered, f . (b) Repeat step (a) for the range of tuning frequency ratio specified in Table 1. (c) Select the combination of f and ζ d that minimized the peak acceleration response. This combination defined the optimum TMD parameters for the mass ratio and primary structural damping ratio under consideration. The search procedure was then repeated for another combination of m and ζ s from which the corresponding optimum combination of f and ζ d was found. The investigation covered a range of m and ζ s normally used in civil engineering structures.

Fig. 2. Different response spectra with varying ζ d for the same m = 0.02, ζ s = 0.05, and f = 1

3 Results and Discussion 3.1 New Optimum TMD Parameters The numerical search resulted in 840 combinations of optimum tuning frequency f and optimum TMD damping ratio ζ d for the range of mass ratio m and structure damping ζ s investigated. Examples of the newly proposed optimum TMD parameters attuned to minimum structure acceleration are presented in Tables 2 and 3. The lower the inherent damping present in the primary structure, the lower the required TMD’s damping value

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and the lower the tuning frequency ratio would be. Moreover, for the same value of the main structure damping, the tuning frequency ratio decreases and the TMD’s damping increases when the mass ratio increases. Table 2. Optimum tuning frequency ratio m

f for ζ s = 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.01 0.9958

0.9968 0.9978 0.9993 1.0010 1.0028 1.0048 1.0070 1.0098 1.0125

0.02 0.9910

0.9923 0.9935 0.9953 0.9970 0.9990 1.0013 1.0040 1.0068 1.0100

0.03 0.9863

0.9875 0.9895 0.9913 0.9930 0.9953 0.9978 1.0008 1.0038 1.0065

0.04 0.9818

0.9830 0.9850 0.9870 0.9890 0.9913 0.9938 0.9968 0.9998 1.0028

Table 3. Optimum TMD damping ratio m

ζ d for ζ s = 0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.01 0.0630

0.0640 0.0670 0.0675 0.0680 0.0700 0.0720 0.0740 0.0740 0.0755

0.02 0.0885

0.0900 0.0925 0.0935 0.0955 0.0975 0.0995 0.1000 0.1020 0.1025

0.03 0.1080

0.1105 0.1100 0.1120 0.1150 0.1170 0.1185 0.1190 0.1205 0.1245

0.04 0.1235

0.1265 0.1270 0.1285 0.1310 0.1335 0.1360 0.1370 0.1395 0.1425

Figures 3 and 4 present a comparison of various solutions for optimum f and ζ d where the mass ratio was fixed at 2% and the main structure damping varied from 0 to 10%. Compared with the solutions for displacement optimization (Den Hartog, Krenk, Tsai & Lin, Abubakar & Farid, Salvi & Rizii), the ones developed for acceleration control (Warburton, this paper) suggested a higher value for the tuning frequency ratio. In response to an increase in the main structure damping, the tuning frequency ratio decreases for displacement control but increases for acceleration control. The new tuning frequency ratio can even be greater than 1.0 for a certain level of the main mass damping. Regarding the TMD damping, whilst the ζ d value proposed by this paper is bit higher than that suggested by Den Hartog and Tsai & Lin, the highest required value for ζ d comes from the solution of Krenk for ζ s less than about 7.5% as observed in Fig. 4.

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Fig. 3. Various proposals for tuning frequency ratio (m = 0.02)

Fig. 4. Various proposals for TMD damping ratio (m = 0.02)

3.2 Empirical Expressions for New TMD Parameters The relationship between the new optimum TMD parameters (f, ζ d ) and the input parameters (m, ζ s ), as found from the numerical search, is graphically illustrated in Fig. 5. Attempts were made to fit empirical models to the numerical data to facilitate the TMD

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design process. It was proposed that the fitting model consisted of three terms with the first one being a function of the mass ratio, m, as in the case of Warburton’s suggestion for undamped structures. The second and third terms would represent the effect of the damping of the main structure, ζ s , and the interaction between m and ζ s respectively. The empirical expressions for optimum tuning frequency and optimum TMD damping ratio were obtained by employing data fitting with Matlab, as given by Eqs. (19) and (20). f =√  ξd =

1 1+m

√ + 0.8860ζs1.8 + 0.3916ζs m

√ 3m + 0.0362ζs0.7 + 0.6493ζs m 8(1 + m/2)

(19)

(20)

Fig. 5. Optimum TMD parameters

To examine the quality of the proposed curve fitting functions, the TMD parameters acquired from the numerical search were plotted against those calculated using Eqs. (19) and (20), as shown in Fig. 6. The fitting model demonstrated very good accuracy with the scattered plot being well fitted on a straight line. The R-squared correlation coefficient was found to be as high as 0.999 for both of the proposed empirical formulas. Furthermore, Fig. 7 shows the cumulative distribution of error of the fitting models. The error defined here was the ratio of the absolute difference between the numerical search result and the fitted value to the former. The average error was only 0.03% for the fitted tuning frequency ratio and 0.60% for the fitted absorber damping ratio. The 95-th percentile error was also found to be very minimal at just 0.08% and 1.54% for the approximation of f and ζ d respectively, which indicated a high degree of accuracy of the fitting models.

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Fig. 6. Fitting model vs Numerical simulation

Fig. 7. Cumulative distribution of error of fitting models

3.3 Evaluating the Efficiency of the Proposed Tuning Solutions The dynamic magnification factor for the steady state acceleration response of the original structure (without TMD) can be calculated using the following equation [19]: r2 D0 =  2 1 − r 2 + (2ζS r)2

(21)

As an example of the optimized TMD’s efficiency, Fig. 8 presents a comparison of the acceleration response of the original structure and the structure with TMD for m = 2% and ζ s = 5% where the TMD parameters were selected using various formulas. It can be seen that, among the optimum tuning solutions examined, only the TMD formula developed by this paper resulted in the same response levels at the two peaks observed on the associated response spectrum. The spectra in accordance with the other absorber solutions all had maximum accelerations exceeding the response level resulted from the new tuning parameters. The percentage of reduction in acceleration response, Ra , due to the addition of TMD to the primary mass, can now be defined as Ra = (D0.max − Dmax )/D0.max where D0.max and Dmax are the maximum dynamic magnification factors of the original structure

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H.-A.-T. Nguyen

Fig. 8. Acceleration response spectra (m = 0.02; ζ s = 0.05)

and the structure with TMD respectively. Table 4 presents the Ra values derived from different tuning solutions when the mass ratio was taken as 2%. Compared with the popular Den Hartog’s method, the solutions offered by Tsai & Lin and Abubakar & Farid provided a little lower reduction whilst those suggested by Krenk and Warburton resulted in a higher reduction in the acceleration response. On the other hand, the new TMD parameters developed in this paper were proved to be most effective, resulting in the greatest reduction in acceleration. For instance, the maximum steady state response of a structure with a damping ratio of 5% can be alleviated by 47.4% by an optimally tuned TMD that has a mass ratio of 2%. Another observation was that the Ra value was higher in structures with lower inherent damping. Table 4. Acceleration response reduction (m = 2%) ζs

Percentage of reduction, Ra , in accordance with TMD solutions Den Hartog Krenk

Tsai & Lin

Abubakar & Farid

Salvi & Rizii

Warburton

This paper

0.01

81.6%

81.7%

81.3%

81.3%

81.6%

82.9%

83.1%

0.02

68.1%

68.5%

67.4%

67.4%

67.7%

70.0%

70.5%

0.03

57.8%

58.4%

56.7%

56.7%

57.0%

60.0%

60.9%

0.04

49.8%

50.5%

48.2%

48.3%

48.6%

52.0%

53.4%

0.05

43.3%

44.2%

41.3%

41.6%

41.8%

45.5%

47.4%

0.06

38.0%

38.9%

35.6%

36.1%

36.2%

40.0%

42.4%

0.07

33.5%

34.5%

30.9%

31.6%

31.6%

35.5%

38.3%

0.08

29.8%

30.8%

26.9%

27.8%

27.8%

31.6%

34.9%

0.09

26.5%

27.6%

23.5%

24.6%

24.5%

28.2%

31.9%

0.10

23.7%

24.8%

20.6%

21.8%

21.8%

25.3%

29.3%

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4 Conclusions The present paper discussed a numerical simulation procedure to search for optimum TMD parameters that can minimize the acceleration response of a damped structure subjected to harmonic excitations. New optimum values for the tuning natural frequency ratio and the absorber’s damping ratio were proposed for the mass ratio and the structure’s damping ratio of up to 10%. Empirical formulas for the optimum absorber parameters were developed and the accuracy of the regression equations was successfully evaluated. For a given mass ratio and primary structural damping ratio, the tuning frequency ratio for acceleration optimization was found to be higher than that for displacement control, i.e. the TMD’s natural frequency should be closer to the primary structure’s natural frequency. When the damping inherent in the main structure increases, the optimum tuning frequency ratio for acceleration control shows an increasing trend whereas that for displacement control shows a decreasing trend. The absorber’s damping, on the other hand, should be increased with the increase of the structure’s damping for either displacement or acceleration optimization. The new solution for optimum TMD parameters was benchmarked against the classical Den Hartog method and a number of its modifications. In the event that acceleration response is of concern, a TMD with parameters tuned to the newly proposed values can flatten the transfer function of the damped structure in the vicinity of resonance. Compared with various tuning formulas in relevant literature, the proposed solution was found to provide higher reduction in the structural acceleration response, thus more effective in terms of minimizing acceleration due to harmonic excitation.

References 1. Tuan, A.Y., Shang, G.Q.: Vibration control in a 101-storey building using a tuned mass damper. J. Appl. Sci. Eng. 17(2), 141–156 (2014) 2. Elias, S., Matsagar, V.: Research developments in vibration control of structures using passive tuned mass dampers. Ann. Rev. Control 44, 129–156 (2017) 3. Ikeda, Y., Yamamoto, M., Furuhashi, T., Kurino, H.: Recent research and development of structural control in Japan. Jpn. Archit. Rev. 2(3), 219–225 (2019) 4. Setareh, M., Ritchey, J., Baxter, A., Murray, T.: Pendulum tuned mass dampers for floor vibration control. J. Perform. Const. Facil. 20(1), 64–73 (2006) 5. Nyawako, D., Reynolds, P.: Technologies for mitigation of human-induced vibrations in civil engineering structures. Shock Vibr. Digest 39(6) (2007) 6. Nguyen, T.H., Saidi, I., Gad, E.F., Wilson, J.L., Haritos, N.: Performance of distributed multiple viscoelastic tuned mass dampers for floor vibration applications. Adv. Struct. Eng. 15(3), 547–562 (2012) 7. Nguyen, T., Gad, E., Wilson, J., Haritos, N.: Mitigating footfall-induced vibration in long-span floors. Aust. J. Struct. Eng. 15(1), 97–109 (2014) 8. Den Hartog, J.P.: Mechanical Vibrations, 4th edn. McGraw-Hill (1956) 9. Krenk, S.: Frequency analysis of the tuned mass damper. J. Appl. Mech. 72(6), 936–942 (2005) 10. Krenk, S., Høgsberg, J.: Tuned mass absorbers on damped structures under random load. Probab. Eng. Mech. 23(4), 408–415 (2008)

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11. Tsai, H.-C., Lin, G.-C.: Explicit formulae for optimum absorber parameters for force-excited and viscously damped systems. J. Sound Vib. 176(5), 585–596 (1994) 12. Abubakar, I., Farid, B.: Generalized Den Hartog tuned mass damper system for control of vibrations in structures. Earthq. Resistant Eng. Struct. 104, 185–193 (2009) 13. Salvi, J., Rizzi, E.: A numerical approach towards best tuning of Tuned Mass Dampers. In: Proceedings of the International Conference on Noise and Vibration Engineering ISMA2012, pp. 2419–2433 (2012) 14. Warburton, G.B.: Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Eng. Struct. Dynam. 10(3), 381–401 (1982) 15. Bachmann, H., Ammann, W.: Vibrations in Structures Induced by Man and Machines. IABSE, Zürich, Switzerland (1987) 16. Smith, A., Hicks, S., Devine, P.: SCI P354 - Design of floors for vibration: a new approach. The Steel Construction Institute, Ascot (2009) 17. ISO. ISO 10137:2007: Bases for Design of Structures - Serviceability of Buildings and Walkways Against Vibrations, 2nd edn. International Organization for Standardization, Geneva (2012) 18. The MathWork Inc.: Matlab Data Analysis. The MathWork Inc., Massachusetts, USA (2014) 19. Chopra, A.K.: Dynamics of Structures - Theory and applications to Earthquake Engineering, 4th edn. Prentice Hall (2012)

Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and Regenerated Power La Duc Viet(B) , Nguyen Cao Thang, Nguyen Ba Nghi, and Le Duy Minh Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam [email protected]

Abstract. The automotive industry has started to show interest in harvesting energy from the motion of the suspension. Unlike conventional shock absorber which dissipates energy by passing a fluid, generally oil, through small orifices, thus generating heat, the regenerative shock absorber transforms the linear motion of the suspension into a rotation of an electric generator. The electrical load of the generator should be optimized because the too large value suppresses the useful electric current while the too small value gains too little energy. This paper presents the analytical solutions of the optimal electrical load of the generator for two objectives: ride comfort and regenerated power, under two cases of excitation: harmonic and random. Keywords: Vibration energy harvesting · Analytical optimization · Regenerative shock absorber · Regenerative damper

1 Introduction The traditional shock absorber of vehicle dissipates vibration energy as heat. This traditional design has the advantage of simplicity but the disadvantages are the overheat problem and the wasted energy. The regenerative absorber converts the wasted energy to the useful form. As stated in [1], there is only about 10–20% of the fuel energy used to ride the vehicle, while a significant amount of energy is wasted in the shock absorber, about 10% (Fig. 1). Base on the operation principle, the regenerative shock absorber can be divided into two groups: mechanical and electromagnetic. The mechanical regenerative shock absorber is constructed from the normal hydraulic or air absorber. However, the kinetic energy transferring to the absorber is converted to the potential energy of the fluid or air in the chamber. These types of absorber have many disadvantages including the complex pipe system, the additional weight and the remarkable installation space. The leak problem can affect to the system. The operation frequency range of the hydraulic system is narrow. Moreover, the usable ability of the potential energy of the fluid or air is limited because the modern automotive industry aims to the electric vehicles. Therefore, the studies on the regenerative hydraulic or air absorber are quite limited. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 573–583, 2022. https://doi.org/10.1007/978-981-16-3239-6_43

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Conversely, the electromagnetic regenerative absorber converts the impact energies to the electrical energy, which is more convenient in storing and reuse, has high efficiency, requires a small installation space and many more advantages [2]. In the recent years, electromagnetic regenerative absorber has attracted many interests. The motor in the electromagnetic absorber system can be used to provide the control force in the control task and concurrently provide the absorb force in the regenerative task.

Fig. 1. Energy distribution provided by the car engine

The regenerative absorber can use the ball screw to convert the translation to the rotation motion. The reference [3] proposed the regenerative absorber (Fig. 2) and analyzed its dynamic and efficiency in regenerating energy.

Fig. 2. Device regenerating energy by ball screw system

Another mechanism can be used is the rack and pinion mechanism to convert the translation to the rotation (Fig. 3) [4].

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Fig. 3. Electromagnetic regenerative absorber using rack and pinion

The simplest model for the regenerative absorber is the single-degree-of-freedom (SDOF) mass-spring-damper system integrated with the regenerative energy circuit through the electromagnetic transductions [5]. Because the mechanical and electrical systems are connected through the transduction, the mechanical and electrical parameters influence each other in the energy extraction. To optimize the extracted energy, the parameters of two systems should be considered simultaneously. The electrical load of the motor is a key parameter needs to be optimized. It affects both the absorptive and regenerative efficiencies. The too large value of the electrical load can suppress the useful electric current and also reduce the absorptive efficiency. The too small value of electrical load, however, extracts too small energy and results in the overdamped situation, which also reduce the absorptive efficiency. This paper presents the analytical optimal solution for the electrical load of the regenerative absorber. The optimal solution guarantees both the absorptive and regenerative efficiencies.

2 Mathematical Modelling We consider a simplest model of the regenerative absorber. Figure 4 shows a SDOF oscillator with a electromagnetic transduction. The system contains a mass-spring-damper system which is base excited by the ambient vibration.

Fig. 4. SDOF regenerative energy system under base excitation, electromagnetic transduction.

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The motion equation of this model can be easily obtained by the second Newton’s law as [6] ⎧ ⎨ m¨x + cm (˙x − y˙ ) + k(x − y) + fe = 0 (1) k k (˙x − y˙ ) ⎩ fe = t e RH In the Eq. (1), f e is the force provided by the electromagnetic transduction (electromotive force), RH is the resistance of the electric load, k t and k e respectively are the motor constant and the electromotive force coefficient. Moreover, cm is the inherent mechanical damping coefficient of the system, k is the stiffness of the leaf spring and m is the vehicle’s mass. At last, x and y respectively are the displacements of the mass and the base, respectively. Base on the equation, the electromotive force can be seen as a viscous damping force. Therefore, the equivalent electrical damping of the regenerative circuit can be defined as: ce =

kt ke RH

(2)

The acceleration of m represents the absorptive efficiency while the extracted energy of the electromagnetic transduction expresses the regenerative efficiency. Use the following   normalized variables: natural frequency ωn = k m, mechanical damping ratio ζm =   cm (2mωn ), electrical damping ratio ζe = kt ke (2mωn RH ), we change the Eq. (1) to: x¨ + 2(ζm + ζe )ωn (˙x − y˙ ) + ωn2 (x − y) = 0

(3)

The regenerative energy is the energy extracted through electrical load. The average energy can be written in the form: 1 P= T

T

2mωn fe (˙x − y˙ )dt = T

0

T ζe (˙x − y˙ )2 dt

(4)

0

in which T is a certain time.

3 Optimal Solution of Electrical Damping Ratio Under Harmonic Excitation Let us consider the rough road described as harmonic form and is represented in the complex form: y=

  iωt  −iωt e cos ωt = + e ω2 2ω2

(5)

in which  is an amplitude independent of the excitation frequency ω. The amplitude of the rough (/ω2 ) is inversely proportional to the square of frequency. It implies the

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constant acceleration spectral, which is commonly available in daily lives. The stationary response has form: 1  iωt xp e + xp e−iωt (6) x= 2 where x p is the complex amplitude of the vehicle’s displacement. Substitute (5) and (6) into Eq. (3) yields:

 i  iω −ω2  iωt xp e + xp e−iωt + 2(ζm + ζe )ωn xp eiωt − xp e−iωt − eiωt − e−iωt 2 2 2ω

  1  xp eiωt + xp e−iωt − eiωt + e−iωt =0 +ωn2 2 2ω2

(7) Let the term relating to eiωt be equal to zero and solve the resulting equation we obtain the complex amplitude of the vehicle’s displacement as: xp =

ω2 + i2(ζm + ζe )ωn ω n

 ω2 ωn2 − ω2 + i2(ζm + ζe )ωn ω

(8)

Substitute the formula (8) into (6), then into the formula of average power (4), in which the time of the integration is one period, we have: mωωn P= π

2π/ω 

ζe



i  2 iω  iωt xp e − xp e−iωt − eiωt − e−iωt dt 2 2ω

0

Substitute the formula (8) into (9) and integrate, note that 2π/ω 

(9)

2π/ω 

eiωt dt

=

0

e−iωt dt = 0, after some manipulations, we have:

0

2 mωn ω2 ζe P=

2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2

(10)

Moreover, the ride comfort is characterized as the acceleration transmissibility ratio as:

     ω2 xp  ωn2 + 4(ζm + ζe )2 ω2 = ωn 

2  ω2 − ω2 + 4(ζm + ζe )2 ω2 ω2 n

(11)

n

We have some remarks: – When ζe is very large, the regenerative power in (10) tends to 0 and the transmissibility ratio in (11) tends to 1. That implies there is no regeneration of energy and there is also no absorptive efficiency (base acceleration is transmitted totally to the mass).

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– When ζe is very small, the regenerative power in (10) also tends to 0, which implies no regenerative efficiency. ωn . Espe– When ζe and ζm is very small, the transmissibility ratio in (11) tends to |ω−ω n| cially, when the resonance occurs (ω = ωn ), the transmissibility ratio tends to infinity and the absorber is totally broken down. The above remarks show that the electrical damping ζe (depending on the electrical load RH ) needs to be optimal calculated. The optimal problem at this time is multiobjective: maximize P in (10) and minimize transmissibility ratio in (11). To combine the objectives, we consider the following objective function:    2 ω2 xp  Pωn J =γ − − γ (1 ) m2 

ωn2 ωn2 + 4(ζm + ζe )2 ω2 γ ωn2 ω2 ζe = − (1 − γ ) (12)

2

2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 in which the first term is the normalized power while the second term is the square of transmissibility ratio. The weighting γ (0 < γ < 1) defines the importance between two objectives: regenerate energy and shock absorption. When γ = 0 or 1, the objective is totally shock absorption or regenerating energy, respectively. The problem is to maximize J with respect to ζ e . Simplify J, we have:

γ ω2 ζe + (γ − 1) ωn2 + 4(ζm + ζe )2 ω2 J = ωn2 (13)

2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 Letting the derivative of J with respect to ζ e be equal to 0 results in the equation to calculate the optimal damping ratio ζ e :

γ ω2 ζe + (γ − 1) ωn2 + 4(ζm + ζe )2 ω2 γ + 8(ζm + ζe )(γ − 1) = (14)

2 2 2 2 2 2 8(ζm + ζe )ωn2 + 4(ζm + ζe ) ω ωn ωn − ω Simplifying (14) leads to a quadratic equation of ζe :  8α(ζm + ζe )(γ ζe + (γ − 1)(2 − α)) = (1 − α)2 + 4(ζm + ζe )2 α γ

(15)

where we define α as the square of frequency ratio: α=

ω2 ωn2

(16)

The optimal solution of ζ e depends on the ratio α and weighting value γ . Especially, when γ = 1 and at the resonance (α = 1), solving (15) gives: ζe = ζm

(17)

Therefore, if the objective is totally regenerate energy then the electrical damping ratio should be chosen as the mechanical damping ratio. This result was stated in [6, 7].

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4 Numerical Investigation for Harmonic Excitation We study the change of optimal electrical damping ratio versus the changes of α and γ. The plot of ζe versus α is show in Figs. 5, 6 with varying values of γ and ζm .

Fig. 5. Change of optimal value of ζe versus α, for ζm = 10%

The results lead to some following remarks: – When γ is near 1 implying the objectives is more important in regenerating energy, the optimal damping ratio changes not much. At the resonance, the optimal damping is equal to ζm as discussed above, Outside the resonance point (the frequency ration is smaller or larger than 1), the optimal damping ratio should be increased. – When γ decreases implying the objective is more important in shock absorption, the clear trend is that: the damping ratio needs to be large at low frequency and vice verse. That means the damping should be very small to isolate the vehicle at the high frequency. – The results have no remarkable change when the mechanical damping ratio is changed. – At small α, i.e. stiff dampers, the optimal ζe is very sensitive to γ.

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Fig. 6. Change of optimal value of ζe versus α, for ζm = 1%

It is important to note that there is a special point in Fig. 5 or 6, which is fixed when γ varies. This point can be found by substituting the extreme cases of γ = 0 and γ = 1 into (15). This yields:  8α(ζm + ζe )(α − 2) = 0 (18) 8α(ζm + ζe )ζe = (1 − α)2 + 4(ζm + ζe )2 α Solving (18) gives quadratic equation: 16(ζm + ζe )ζe = 1 + 8(ζm + ζe )2 which results in the value of ζe independent of γ:  2 + 16ζm2 ζe = 4

(19)

(20)

The value (20) is a good trade-off electrical damping regardless of the value of weighting γ. To demonstrate it, with ζm = 0.1, we plot the objective function (13) with two values of γ and with the trade-off damping (20) (=0.3674), low damping ζe = 0.1 and high damping ζe = 1. The resulting curves are shown in Fig. 7. It is seen that the trade-off value (20) has the good effective in raising the performance curves, regardless of the value γ as expected.

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Fig. 7. Objective function J versus α, for ζm = 1% and γ = 0.6 (left), γ = 0.9 (right)

5 Optimal Solution of Electrical Damping Ratio Under White Noise Excitation Let the acceleration y¨ is modeled as a white noise with intensities of σ. The stationary response of the system (3) has following form [8]: 

(x − y)2



=

σ2 ; 4(ζm + ζe )ωn2



(˙x − y˙ )2



=

σ2 4(ζm + ζe )

(21)

where • denotes the stationary stochastic averaging operator. As mentioned above, the absolute acceleration of the vehicle represents the absorption efficiency. From (21) we calculate the average of square of acceleration as:    2  2 2 x¨ = ωn (x − y) + 2(ζm + ζe )ωn (˙x − y˙ )     = ωn4 (x − y)2 + 2(ζm + ζe )ωn3 (x − y)(˙x − y˙ ) + 4(ζm + ζe )2 ωn2 (˙x − y˙ )2 (22) Because the displacement and velocity are two independent stochastic processes, we have:

Then

(x − y)(˙x − y˙ ) = 0

(23)

      x¨ 2 = ωn4 (x − y)2 + 4(ζm + ζe )2 ωn2 (˙x − y˙ )2

(24)

Using (21) in (24) gives:

  1 ω2 σ 2 x¨ 2 = n + 4(ζm + ζe ) 4 ζm + ζe

(25)

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The acceleration transmissibility is:  2

x¨ 1 ωn2 = + 4(ζm + ζe ) σ2 4 ζm + ζe Use (21), the average power has form:   mωn σ 2 ζe P = 2mωn ζe (˙x − y˙ )2 = 2(ζm + ζe )

(26)

(27)

The same combined objective function as (12) is considered here:  2

x¨ Pωn 1 ωn2 ωn2 ζe J =γ − (1 − γ ) − (1 − γ ) 2 = γ + 4(ζm + ζe ) mσ 2 σ 2(ζm + ζe ) 4 ζm + ζe

2γ ζe + (γ − 1) 1 + 4(ζm + ζe )2 (28) = ωn2 4(ζm + ζe ) The problem is to maximize J with respect to ζ e . Letting the derivative of J with respect to ζ e be equal to 0 results in the equation to calculate the optimal damping ratio ζ e:

2γ ζe + (γ − 1) 1 + 4(ζm + ζe )2 2γ + (γ − 1)8(ζm + ζe ) (29) = 4(ζm + ζe ) 4 Solving (29) leads to the simple solution of ζ e :  1 2γ ζm 1+ ζe = 2 1−γ

(30)

It is seen from (30) that the optimal electrical damping ratio increases with the weighting value γ . That means if the importance of the regenerated power increases, the electrical damping ratio should be large. In the limit case, when ζm = 0, the optimal solution is ζe√= 1/2. Return to (20), in the harmonic excitation case, the optimal solution is ζe = 1/ 8, which is a bit smaller.

6 Conclusion This paper investigates the optimal electrical damping ratio for a regenerative shock absorber. The studied model is a single-degree-of freedom vehicle model subjected to road excitation. The paper studies the cases of harmonic and random excitation and considers an index combining the regerative energy and the acceleration transmissbility ratio. The results show that the electrical damping, which is inversely proportional to the electrical load of the motor, should be optimized to achieve both two objectives. The optimal solution is calculated from a quadratic equation in the case of harmonic excitation or from a simple explicit formula in the case of random excitation. In the limit case, √ where the mechanical damping is zero, the optimal electrical damping should be 1/ 8 and 1/2 for the harmonic and random excitation, respectively. To verify the solution, in some specific cases, the optimal value returns to the known solutions in literature. The numerical investigation is also carried out to see the effects of the weighting value.

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Acknowledgement. This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “08/2018/TN".

References 1. Pei, S.Z.: Design of Electromagnetic Shock Absorbers for Energy Harvesting from Vehicle Suspensions. Master Degree thesis, Stony Brook University (2010) 2. Gysen, B.L.J., Tom, P.J., Paulides, J.J.H., et al.: Efficiency of a regenerative direct-drive electromagnetic active suspension. IEEE Trans. on Vehicular Technology 60(4) (2011) 3. Zuo, L., Scully, S., et al.: Design and characterization of an electromagnetic energy harvester for vehicle suspensions. Smart Material and Structure 19 (2010) 4. Graves et al.: Experimental verification of energy-regenerative feasibility for an automotive electrical suspension system. In: ICVES IEEE International Conference on Vehicular Electronics and Safety. IEEE, pp. 1–5 (2007) 5. Stephen, N.: On energy harvesting from ambient vibration. J. Sound Vib. 293(1), 409–425 (2006) 6. Tai, W.-C., Zuo, L.: On optimization of energy harvesting from base-excited vibration. J. Sound Vib. 411, 47–59 (2017) 7. Tang, X., Zuo, L.: Enhanced vibration energy harvesting using dual-mass systems. J. Sound Vib. 330(21), 5199–5209 (2011) 8. Lutes, L.D., Sarkani, S.: Random Vibration: Analysis of Structural and Mechanical Systems. Elsevier, Amsterdam (2004)

Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio Hoang Trong Lam and Vo Duy Hung(B) The University of DaNang – University of Science and Technology, 54 Nguyen Luong Bang Street, Danang city, Vietnam [email protected]

Abstract. Cable vibration due to wind is one of key issue in design of stayed-cable bridge. To control the rain wind induced vibration, indented surface and axially protuberated cables have been applied in Japan for many years. Nevertheless, it is also figured that those methods are still defective in mitigating cable dry galloping. Moreover, vibration characteristics of indented surface and axially protuberated cables with low damping ratio have not fully investigated yet. The aim of this study is to investigate the aerodynamic responses of low damping indented surface and axial protuberated cable. Firstly, the WTT will examine cables vibration in dry and rain condition for indented cables. Then, aerodynamic responses of axial protuberated cables will be elucidated fully. Finally, the axial flow in the wake of cables will be measured and discussed in detail. Keywords: Aerodynamic responses · Indented surface · Axially protuberated cables · Low damping ratio · Axial flow

1 Introduction Since stay cables can be vulnerable due to wind or rain-wind excitation [1–5], many control methods have been developed to mitigate cable vibration so far. Those methods can simply be categorized in two groups. The first is aerodynamic control methods, which related to modification the aerodynamic characteristic of stay cable. The other group is mechanical control methods, which typically install the external dampers or use crosstie system. Consequently, the damping of cable will be enhanced, or natural frequency be decreased and thus mitigate the vibration of cable. Aerodynamic control is a passive control method that modifies aerodynamic feature of the cable cross-section. It is most effective in controlling rain-wind-induced cable vibration by preventing accumulation of rain to form water stream on the surface of cable. In Japan, indented cable surface and axial protuberated cable surface were applied in some bridges. The use of axially protuberated cable was developed for Higashi-Kobe cable-stayed bridge. The idea was to have these deep axial protuberated cables along the cable, which would drive the water down without allowing any transverse movements [6]. In a preliminary investigation, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 584–595, 2022. https://doi.org/10.1007/978-981-16-3239-6_44

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cable with parallel, spiral, wave, and other forms were examined. Consequently, axial protuberated cable were found as the most effective solution in restraining rain-wind induced vibration [7]. However, it was reported on site that this kind of cable still exhibited large vibration to some extent [8]. In parallel, for controlling rain-wind induced vibration, indented surface was firstly developed by Miyata et al. in 1994 [9]. This type of cable was applied firstly in Tatara Stayed Cable Bridge in Japan, which is the longest stay cable at completion time. This control method was effective in suppressing rain wind with low drag force coefficient as smooth surface. Its effectiveness in real scale confirmed as no rain wind induced vibration occurred after opening about one year [10]. However, this mitigation method still exhibited the large amplitude vibration in the original stage. Indented cable with Scruton number around 21, started vibrating with large amplitude from 10 m/s of wind speed and became divergent around 30 m/s of wind speed [11]. In addition, it was reported by Katsuchi and Yamada recently that dry galloping could occur for indented cable in low damping ratio [10]. Furthermore, the health monitoring of Tatara cablestayed bridge recently revealed that indented surface cable exhibited limited vibration [12]. In parallel, the detail of vibration of these type of cables in low damping region have not investigated fully yet. Above discussion have motivated this study, in which vibrations of two cable surface modification in low damping region were investigated by wind tunnel tests. One of cable model is a circular cylinder with indented surface. Another is axially protuberated surface. Rainfall was simulated by rain simulator system in wind tunnel. Finally, mechanism of axial flow in span-wise and stream-wise direction will be compared between smooth surface, indented surface, and axially protuberated surface. The paper is structured into three main sections. Section 2 shows the detail of wind tunnel test setup and aerodynamic responses of low damping indented-cable and axially protuberated-cable in both rain and dry condition. In addition, the discussion about axial flow mechanism near the wake of cable also elucidate clearly in this section. Finally, critical conclusions are provided in Sect. 3.

2 Wind Tunnel Test Investigation 2.1 Experimental Setup Tests were conducted in the open-circuit wind tunnel of the Yokohama National University. A cable model was supported by a single degree of freedom spring system in vertical plane and small wire system was used in horizontal plane to keep cable model unmoved laterally. Cable’s position can be changed by flow angle and inclined angle as shown in Fig. 1. A uniform flow was used with turbulence intensity of about 0.6%. In this experiment, rain can be simulated by an overhead nozzle system. Rain volume at the cable model position was 40–50 mm/h in the wind-speed range (around 8–15 m/s), which was adjusted to create the critical rain-wind induced vibration amplitude. Because of the limitation of wind tunnel capacity, maximum wind speed is up to 20 m/s. Cable diameters 158 mm with damping ratio ranges from 0.08 to 0.25%, which is very low damping region of stay cables compare to the on-site cables damping. The natural frequency is around 0.8–1 Hz.

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2.2 Wind Attack Angles Wind attack angles are determined through α, β, and β* defined as in Fig. 2. In this study, the inclined angle was fixed at 9° 25° and 40° in combination with the flow angle of 0°, 15°, 30°, 45° and 60°. These inclinations of 25° and 40°were examined to consider the longest and medium stay cables angles, while the inclined angle of 9° to consider the stay cables located in aviation restriction area. 2.3 Experimental Model Fabrication Two kind of models were examined, included a cable with axially protuberated surface and a cable with indented surface. The model samples were fabricated with same scale to real bridge cables. Cable diameters are 158 mm with an effective model length of 1.5 m and aspect ratio is 9.5. The indented model was fabricated same pattern of stayed cables of Tatara stayed cable Bridge as Fig. 3, while axially protuberated model was fabricated similarly to Higashi Kobe Bridge cables by adding the twelve rubber fillets axially. The detail of axially protuberated surface modification shown as Fig. 4.

β∗

β

α

Wind

Fig. 1. Cable model orientation in wind tunnel

Fig. 2. Wind attack angles

Fig. 3. Indented surface cable

2.4 Experiment Parameters Table 1 shows the detail of experiment parameters in which cable diameters are 110 mm and 158 mm. Damping ratio ranges from approximately 0.08% to 0.25% and natural

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υ

Fig. 4. Axially protuberated cables cable

frequency is around 0.77–1.02 Hz in considering typical stay cables values. Due to the limitation of wind tunnel capacity, maximum wind speed is up to 20 m/s equivalent to Reynolds number around 2.1 × 105 . Nevertheless, according to previous studies cable galloping of inclined cable could be observed in the subcritical Reynolds number regime as well as in the transition and critical Reynolds number regime. Therefore, above range of Reynolds number enable to reproduce the dry galloping. Table 1. Conditions of experiments Diameter: D (mm)

158

Effective length (mm)

1,500

m (kg/m)

14.55–16.13

Natural frequency (Hz)

0.82–1.02

Damping ratio

0.08%–0.25%

Scruton number (2mδ/ρD2 ) 5.1–15.6 Reynolds number

0–2.1 × 105

3 Results and Discussions 3.1 Wind-Induced Vibration of Indented Cable Surface in Low Damping Region Figure 5, Fig. 6 and Fig. 7 show the galloping amplitude versus wind speed in dry and rain condition of indented cable surface. In case of inclination angle at 40°, most of vibration amplitude is small except cases of 40°–45° and 40°–60° in rain condition. Large amplitude start occurring at wind speed around 15 m/s. In dry condition, large vibration only occur in case of 40°–60°, the highest amplitude occurred started at wind speed 12 m/s, the remain cases exhibits small amplitude ( P2t and ‘−’ when P1t < P2t . 2.2 The Internal Pressure Change of the System Due to Changing External Pressure Consider the system (consisting of only one compartment) with n orifices ventilating out, Ai is the area of each orifice ventilating out, i = 1..n, see Fig. 3.

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Fig. 3. The system has one compartment

During airflow, the atmospheric pressure change ventilates the air in and out of the system. Pressure Pt and volume V t are related by Eq. (1): Pt Vt = P0 V0

(8)

in which: P0 is the internal atmospheric pressure of the compartment at the beginning t = 0; V 0 = V is the volume of the system; Pt is the internal atmospheric pressure of the system at time t; V t is the atmospheric volume corresponding to Pt : Vt = V + Vt

(9)

where ΔV t is the volume of air circulating (into, out of) through the system per interval of time t. Thus: Vt =

n 

Vit

(10)

i=1

in which: ΔV it is the volume of air circulating through orifice i. According to Eq. (7), ΔV it can be expressed as 

t Vit = ±

Ai 0

2 | Pτ − piτ | d τ ρ

(11)

in which: piτ or pit is the external pressure at orifice Ai . From Eqs. (9), (10), (11), V t becomes: t  n  2 | Pτ − piτ | d τ Vt = V + (12) ± Ai ρ i=1

0

Replace V t from Eq. (12) in Eq. (8):  n  P0 V = V + ± Ai Pt i=1

0

t

 2 | Pτ − piτ | d τ ρ

(13)

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When external pressure changes the same (pit = pt for all orifices), Eq. (13) is represented as: t  2 P0 A | Pτ − pτ | d τ (14) = 1 ± Pt V ρ 0

in which A is the total area of all the orifices of the system: A = Replace pt from Eq. (2) in Eq. (14) P0 A = 1 ± Pt V

n  i=1

Ai .

 t    2  1 2  dτ P ρu + − C τ t   ρ 2

(15)

0

Get the derivative of Eq. (15)



    dPt 1 2 A Pt2 2  Pt + ρut − C  − =±  dt V P0 ρ 2

2

2 4 dPt 2 A Pt 1 2 P ρu = + − C t dt ρ V 2 P02 2 t

(16) (17)

in which C is the pressure of the field when the flow velocity is zero (ut = 0). 2.3 The System with N Compartments An isothermal system consists of N compartments and contains orifices, see Fig. 4, during airflow, the change in atmospheric pressure causes air to go in and out of the system and also circulate between compartments. Consider compartment e and name parameters of this compartment as follows: V e is the volume of compartment e, e = 1..N; Pet is internal atmospheric pressure of compartment e; ne is the number of orifices ventilating out; Aei is the area of orifice ventilating out, i = 1..ne ; Aej is the area of the orifice which ventilates with other compartments, j = 1..N, j = e; case of compartments e and j do not open to each other, then Aej = 0. Pressure P and volume V are related by Boyle-Mariotte: Pet Vet = P0 Ve

(18)

in which: P0 is the internal atmospheric pressure of compartment e at the beginning t = 0; Pet is the internal atmospheric pressure of compartment e at time t; V et is the atmospheric volume corresponding to Pet : Vet = Ve + Vet

(19)

where ΔV et is the volume of air circulating (into, out of) through compartment e per interval of time t. Vet = Vit + Vjt

(20)

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Fig. 4. Connection diagram of the N compartments which the system consists of

where: ΔVit is the volume of air circulating through orifice ventilating out Aei ; ΔVjt is the volume of air circulating through orifices which ventilate with other compartments Aej . Thus: Vit =

ne 

Veit

(21)

Vejt

(22)

i=1

Vjt =

N  j=1

in which: ΔV eit is the volume of air circulating through orifice i; ΔV ejt is the volume of air circulating within compartment j. According to Eq. (7), ΔV eit and ΔV ejt can be expressed as  

t Veit =

Aei 0

Aej 0

(23)

 

t Vejt =

 2 |Peτ − peiτ | d τ ± ρ   2  Peτ − Pjτ  d τ ± ρ

(24)

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in which: Pjτ or Pjt is the internal pressure of compartment j; peiτ or peit is the external pressure at orifice Aei . From Eqs. (19), (20), (21), (22), (23), (24), Eq. (19) becomes: t  t  ne N    2  2 | Peτ − peiτ | d τ Vet = Ve + Peτ − Pjτ  d τ + ±Aej ±Aei ρ ρ j=1

i=1

0

0

(25) Replace V et from Eq. (25) in Eq. (18):  N  P0 Ve = Ve + ±Aej Pet j=1

t



 ne   2  ±Aei Peτ − Pjτ  d τ + ρ i=1

0

t

 2 | Peτ − peiτ | d τ ρ

0

(26)

When external pressure changes the same, Eq. (26) is represented as: t  t  N   P0 Ve 2  2 | Peτ − pτ | d τ Peτ − Pjτ  d τ ± Ae = Ve + ±Aej Pet ρ ρ j=1

0

0

(27) in which Ae is the total area of the orifices of the compartment e: Ae = Replace pt from Eq. (2) in Eq. (27):  N  P0 Ve = Ve + ±Aej Pet j=1

t

0



 2  Peτ − Pjτ  d τ ± Ae ρ

ne  i=1

 t    1 2 2   dτ + − C ρu P eτ  ρ 2 t

Aei .

(28)

0

Get the derivative of Eq. (28) ⎞ ⎛     N 2      dPet Pet 2 2 1  Pet + ρu2 − C  ⎠  Pet − Pjt  ± Ae ⎝ = − ±Aej  dt Ve P0 ρ ρ 2 t j=1

(29) Thus, the internal pressure of the system included N compartments is represented as a system of first-order differential equations with N unknowns Pet . In the case of the closed system included N + 1 compartments in which the pressure of the compartment N + 1 takes to change initiative (the gas flow field is the same as the compartment N + 1), the Eq. (27) is rewritten as below after getting the derivative: ⎞ ⎛   N 2      P 2 2 dPet = − et ⎝ Pet − Pjt  ± Ae(N +1) Pet − P(N +1)t  ⎠ ±Aej dt Ve P0 ρ ρ j=1

(30) or 2 P0 P˙ et = −Pet

N +1  j=1

 ±αej

 2  Pet − Pjt  ρ

(31)

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A

where αej = Veje ; e = 1..N. Likewise, in the case of the closed system included N + M compartments in which the pressure of M compartments take to change initiative, the internal pressure of the system is expressed: ⎛ ⎞   N N +M 2      dPet Pet 2 2  Pet − Pjt  +  Pet − Pjt  ⎠ ⎝ = − ±Aej ±Aej dt Ve P0 ρ ρ j=1

j=N +1

(32) or 2 P0 P˙ et = −Pet

N +M j=1

 ±αej

 2  Pet − Pjt  ρ

(33)

The Eq. (33) is a system of first-order differential equations that describes the internal pressure of compartment e (e = 1..N) of the isothermal system included N + M compartments in which the pressure of M compartments take to change initiative (known parameter).

3 Illustrative Example An isothermal system (with orifices ventilating out) that is immersed in the velocity of the airflow field (abide by Bernoulli’ law). The airflow longitudinal velocity fluctuates between 10 m/s and 42 m/s (corresponding with umin and umax ) with a frequency of 1 Hz (T = 1), see Fig. 5. V is the volume of the system, A is the total area of the orifices of the system. Study differential pressure between inside and outside of the system, i.e. ΔPt .

Fig. 5. Graph of the airflow longitudinal velocity

The differential pressure, W = ΔPt is expressed. 1 Pt = Pt − pt or Pt = Pt + ρut2 − pA 2

(34)

here, pA is the pressure of the field when the airflow velocity equaled to zero, pA = 1013 hPa. Figure 6 shows the result for solving Eqs. (16) and (34) with different values of the A/V ratio, in which the differential pressure values also fluctuate with the

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corresponding frequency of 1 Hz. In the case of A/V = 0 the peak value of differential pressure in the T period, W p reaches the maximum W 0 = 350 N/m2 and does not change direction (W ≥ 0). When A/V = 0 the value of W p decreases rapidly as the A/V ratio increases, the value of W p is only 16.3%W 0 when the A/V = 8.10–4 that is still very small.

Fig. 6. Graph of the differential pressure between inside and outside of the system, Pt

4 Discussion and Conclusions In addition to the isothermal conditions, to apply the Boyle formula, the condition for the static state of air inside the system must be satisfied. In order to satisfy this, the total area of the orifices must be very small compared to the volume of the whole system (very small S/V ratio) so that a significant dynamic state of the air takes place only at the orifices. The inside volume of the system is considered large, resulting in a large circulation area, which greatly reduces the air movement within the system. Hence, the air velocity in it can be considered negligible. This is equivalent to the air pressure in the system (having one compartment) being the same at all points. In summary, the internal pressure of the system with N compartments is represented as a system of first-order differential equations with N unknowns. In practice, there are many processes very close to the isothermal process that needs to be controlled which this equation can solve. For example, controlling the pressure difference between the inside and outside of a building (loading) during a tornado by installing ventilation openings of sufficient area and located at appropriate locations [8], etc. The connecting equation has not been able to fully describe the physical phenomena occurring in nature, for specific situations it needs to be studied alongside other impact conditions. The authors hope that the equation connecting Boyle’s and Bernoulli’s laws will pave the way for more complex analyses in relevant fields.

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References 1. https://en.wikipedia.org/wiki/Boyle’slaw 2. https://en.wikipedia.org/wiki/Bernoulli’sprinciple 3. Betz, A.: Introduction to the Theory of Flow Machines, 1st edn. Elsevier Ltd., Rio de Janeiro (1966) 4. Loitsyanskii, L.G.: Mechanics of Liquids and Gases, 2nd edn. Elsevier Ltd., Rio de Janeiro (1966) 5. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Elsevier Ltd., Rio de Janeiro (1987) 6. https://en.wikipedia.org/wiki/Newton’slawsofmotion 7. Nguyen, N.T.: Force on building due to changing atmospheric pressure during tornadoes. J. Sci. Technol. Vietnam 44(5), 91–100 (2006) 8. Nguyen, N.T., Nguyen, L.N.: Tornado-induced loads on a low-rise building and the simulation of them. In: Huang, Y.-P., Wang, W.-J., Quoc, H.A., Giang, L.H., Hung, N.-L. (eds.) GTSD 2020. AISC, vol. 1284, pp. 543–555. Springer, Cham (2021). https://doi.org/10.1007/978-3030-62324-1_46

Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys Trieu Nhat Huynh(B) and Y. Quoc Nguyen Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {trieu.hn,y.nq}@vlu.edu.vn

Abstract. Solar chimneys absorb solar energy to create stack effects which induce air flow for natural ventilation of buildings. Numerical models based on Computational Fluid Dynamics (CFD) have been increasingly utilized to simulate air flow and heat transfer in solar chimneys. One of the factors influencing the accuracy of the CFD models for solar chimneys is the size of the computational domain. In this study, effects of the sizes of the computational domain for a vertical solar chimney were investigated. Two sizes of the domain were tested, as suggested in the literature: A small domain which has the same physical size as a cavity inside the solar chimney (Domain S) and an extended one that covers both the cavity and the ambient air (Domain L). The flow structure, induced air flow rate, and thermal efficiency of the chimney were predicted and compared between the two domains. It is seen that Domain S offered identical results to those of Domain L at low gap – to – height ratios. At higher gap – to – height ratios, Domain S over – predicted the reverse flow region, and under – predicted the induced flow rate and the thermal efficiency. The critical gap – to – height ratios increased with the chimney height. The results can be used to determine whether Domain S, which requires less computational cost than Domain L, can be used to predict performance of a solar chimney with acceptable accuracy. Keywords: Natural ventilation · Solar chimney · Thermal effect · Computational domain · CFD

1 Introduction Solar heat gain on buildings can cause two opposite effects. For normal buildings, solar heat gain increases cooling load; hence the energy consumptions. For buildings with sustainable design, solar heat gain can be a benefit for natural heating or cooling of the building and help to reduce the energy usage. Solar chimney is one of the most common methods based on solar radiation for ventilating the building naturally. In solar chimneys, solar energy is converted into the flow energy. A typical solar chimney has an open cavity enclosed by surfaces which can absorb solar radiation. The absorbed heat warms the air in the cavity and induces the thermal, or stack effect which draws the air through the openings of the cavity. With suitable designs, the air flow can help to circulate air through the building for ventilation or cooling [1]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 606–616, 2022. https://doi.org/10.1007/978-981-16-3239-6_46

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Studies of solar chimney have been conducted with experiments [2, 3], mathematical analysis [4] or numerical simulations [5–8]. Numerical models based on Computational Fluid Dynamics (CFD) have been increasingly utilized to simulate air flow and heat transfer in solar chimneys [5–8]. One of the factors influencing the accuracy of the CFD models for solar chimneys is the size of the computational domain. Employing smaller domains reduces the computational grid elements; hence lessening the computational cost. As the simulated flow variables are determined from the conditions applied at the boundaries of the domains, different selections of the computational domain can change the results significantly [5, 9, 10]. Zhang et al. [9] and Liu et al. [10] tried to reduce the computational domains for natural convection flow in a cavity [9] or for an external flow with special treatments of the flow variables at the boundaries. However, they did not report the benefits of using small domains over the additional costs due to the additional treatment at the boundaries. For simulating solar chimneys, Gan [5] tested two kinds of the computational domain. The smaller domain covered only the air cavity inside the chimney while the larger one included both the air cavity and surrounding air space. He showed that the air flow rate through the chimney predicted with the extended domain was closer to the experimental data than that obtained with the small domain. From the results, he suggested different minimum extensions of the domain from the chimney cavity. Gan [5] also claimed that for chimneys with low ratios of the cavity width (channel gap) and height, the small domain can offer acceptable results. However, he did not report how the computed flow variables changed with the size of the domain. In this study, we investigated effects of the size of the computational domain on the predicted air flow through a vertical solar chimney. The examined factor included the induced air flow rate and thermal efficiency through the chimney, flow structure, temperature fields, and particularly the reverse flow at the outlet of the chimney as the size of the domain changed.

2 Numerical Model 2.1 Assumptions for the Flow and Heat Transfer The following assumptions were assumed for the air flow and heat transfer in solar chimneys, as suggested in the literature [5–8]: • • • •

The air flow can be described in two dimensions in a vertical plane. The air flow and heat transfer are statistically steady. The air flow is incompressible. Convective heat transfer dominates in the air channel of the chimney. Conduction heat loss through the walls of the chimney and radiative heat transfer can be ignored. • Air properties follow the Boussinesq approximation where only the air density varies with the air temperature according to Eq. (1) [11]. ρ = ρ0 (1 − β(1 − T0 ))

(1)

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where ρ and ρ0 are the air densities at the flow temperature T and the reference temperature T0 , respectively; β is the air thermal expansion coefficient, β = 1/T . T0 is the ambient temperature, which is 293 K. 2.2 Governing Equations The equations governing the air flow and heat transfer in solar chimneys describe the conservations of mass, momentum, and energy. To obtained time – averaged quantities of the flow variables, the Reynolds – Averaged Navier – Stokes equations can be applied. Their two – dimensional forms for steady turbulent flows using vector notation are as follows. ∂uj =0 ∂xj

    ∂ ui uj ∂ui 1 ∂p ∂   ν =− − gi β(T − T0 ) + − ui uj ∂xj ρ ∂xi ∂xj ∂xj     ∂ Tuj ν ∂T ∂  = − T  uj ∂xj ∂xj Pr ∂xj

(2) (3) (4)

where u and u , and T and T  indicate the time – averaged and fluctuating velocity, and temperature, respectively; p is the pressure; ρ, ν, and Pr stand for the air density, kinematic viscosity, and the Prandtl number, respectively; g is the gravitational acceleration; presents a time – averaged quantity. In the Eqs. (3) and (4), the turbulence model RNG k− ∈ was employed for solving    the turbulence stress ui uj and the turbulent heat flux T  uj , as suggested by Gan [5]. 2.3 Computational Domain and Boundary Conditions The considered solar chimney associated with two types of the computational domain is displayed in Fig. 1. The chimney was a typical one with a straight vertical air channel. Absorbed solar radiation was stored and transferred to the air in the channel on one side. The length of the channel was H while its gap was G. For the small domain (Domain S, Fig. 1a), the sizes of the domain coincided with that of the air channel. The large domain (Domain L, Fig. 1b) was extended from Domain S. We followed the findings from Gan [5] for the minimum requirement of the extension, as shown in Fig. 1b. He suggested that the domain should cover a space of 5.0 × G to the sides and the bottom, and 10.0 × G above the outlet of the air channel. On the domain S, the walls of the air channel were applied with no – slip conditions. At the inlet and outlet, atmospheric pressure conditions were employed. For thermal boundary conditions, the air entered the domain at the inlet at the ambient temperature which was assumed at 293 K. A uniform heat flux of 600 W/m2 was applied on the heated wall. The other wall of the air channel was assumed adiabatic. On the domain L, the whole boundary of the domain was exposed to the ambient air; hence applied with atmospheric pressure. Air entering the domain from the ambient was

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also at 293 K. The heated wall inside the air channel was also assigned with a uniform heat flux of 600 W/m2 . All other walls were adiabatic. With Domain S, the flow pressure, direction, and gradient were constrained right at the inlet and the outlet of the air channel. For example, the velocity vector must be normal to the inlet and outlet line in Fig. 1a. With Domain L, the flow was allowed to adapt freely to the flow dynamics at the inlet and outlet of the air channel. Consequently, the flow pressure, direction, and gradient may deviate from those applied on the Domain S.

Fig. 1. Computational domains and the according mesh: Domain S (a) and Domain L (b).

2.4 Computational Mesh and Numerical Setup The structures of the computational mesh for both domains are also presented in Fig. 1. Structured rectangular mesh cells were utilized for the whole domain. The mesh density was finer near the solid surface and at the inlet and outlet of the air channel, as the gradients of the flow quantities are strong at those locations. To check effects of the mesh resolution, we used the non – dimensional distance y+ of the first grid cell near the solid surfaces in the air channel. It was found that the induced flow rate and the temperature rise of the air through the chimney changed within 1.0% as the maximal y+ was less than 1.0. Details of the mesh – independent tests can be seen in [7, 8]. Finite Volume Method was utilized to discretize the governing equations, Eq. (2)–(4). The commercial CFD software ANSYS Fluent (Academic Version 2019R3) was used for

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the numerical setup. For the coupling between the continuity and momentum equations, the SIMPLEC scheme was selected. The PRESTO! method was used for interpolation of the pressure from the cell center to the cell faces. Other settings included second order upwind scheme for the momentum and energy equations and first order upwind scheme for k and ∈ equations. For the convergence criteria, the scaled residuals of all variables (velocities, pressure, temperature, turbulent kinetic energy, and turbulent dissipation rate) were set to 10−6 and achieved in all cases. 2.5 Model Validation The CFD model was validated against the CFD results by Gan [5] and the experimental data by Chen et al. [2]. The tested solar chimney was a vertical one whose height and width were 1.5 m and 0.62 m, respectively. The air gap changed from 0.1 m to 0.6 m. A uniform heat flux of 400 W/m2 was applied on one side of the air channel.

Fig. 2. Comparison of the air flow rate obtained with our CFD model, the CFD results by Gan [5], and the experimental data by Chen et al. [2].

Figure 2 compares the induced flowrate among the three methods. The CFD results by Gan [5] were obtained for both domains. Compared to the CFD results by Gan [5], our CFD results agreed well for the Domain S. Good agreement was also seen for the Domain L at G = 0.4 m and 0.6 m. At smaller gaps with the Domain L, our predicted data were lower than those by Gan [5]; however, they matched better with the measured data at G = 0.1 m. The difference between the two CFD results might be due to different boundary conditions of the turbulence which were not reported by Gan [5].

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Compared to the measured data, both CFD results agreed only at G = 0.1 m. As the gap increased, the CFD data with the Domain S decreased while both the measured data and the CFD results with the Domain L increased. The difference between the measured flow rate and the predicted flow rate with the Domain L was within 20.0% which should be within the measurement uncertainties [2, 5].

3 Results and Discussions It is observed in Fig. 2 that the flow rate predicted with the Domain S matched with that with the Domain L only at G = 0.1 m. As the gap increased, the CFD model with the Domain S underpredicted the induced flow rate. Therefore, it is deduced that both domains may offer similar results at small air channel gap, or in general at low gap – to – height ratio G/H. By changing that ratio for a given solar chimney, one can find the range of G/H where the Domain S can work acceptably. In this section, solar chimneys with different height, ranging from 0.5 m to 2.0 m, and gaps, ranging from G/H = 0.01 to 0.2 were examined with both domains. The performance in terms of the flow and temperature fields, induced flow rate, and thermal efficiency was then analyzed. In all cases, the heat flux was kept to 600 W/m2 . 3.1 Flow and Temperature Fields Examples of the predicted flow and temperature fields at the outlet of a solar chimney of H = 0.5m but with two different G/H ratios of 0.05 and 0.1 for both types of the

Fig. 3. Velocity and temperature fields at the outlet of a solar chimney with H = 0.5 m and two values of G/H predicted with both Domain S and Domain L. Lines inside the air channel indicate the regions of the reverse flow.

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computational domain. At G/H = 0.05 (Fig. 3a), Domain S predicted a reverse flow at the outlet. The reverse flow with Domain S brought air at the ambient temperature of 293 K into the air channel from the outlet and yielded a lower temperature area near the outlet. In contrast, a reverse flow was not observed with the domain L. The velocity and thermal boundary layers developed from the heated surface and occupied the whole channel gap at the outlet. In Fig. 3b, both domains predicted a reverse flow at the outlet of the chimney but with different sizes. The flow reversal region obtained with Domain S was about three times larger than that with Domain L. Though with a reverse flow, the thermal boundary layer with Domain L extended to the whole air gap at the outlet of the chimney. With Domain S, the velocity and thermal boundary layers for the case of G/H = 0.1 were even thinner than those for the case of G/H = 0.05. 3.2 Length of the Reverse Flow Figure 4 shows the length l where the reverse flow penetrated from the outlet of the chimney predicted with both domains. As sketched in Fig. 2a, the penetration area is the region with negative y – velocity component, as suggested by Khanal and Lei [12]. For comparison, the penetration length l was normalized by the channel height H.

Fig. 4. Length of the reverse flow normalized by the channel height predicted by Domain L and Domain S.

With the Domain S, a reverse flow appeared at G/H = 0.03 and H = 0.5 m. There was no reverse flow for other heights at G/H = 0.03. With the Domain L, flow reversals started at G/H ≈ [0.05 – 0.1]. At a given G/H ratio, the penetration length predicted with the Domain S was always higher than that with the Domain L.

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With the Domain L, the normalized penetration length l/H was the same for different heights at G/H = 0.1. At G/H = 0.15, l/H is seen to decrease with the height. In contrast, for the Domain S, a clear trend of l/H versus H is not observed. Khanal and Lei [12] used domain size similar to the Domain S in this study and reported the penetration length for a chimney with G/H = 0.137 and at different Rayleigh number, Ra, which is defined by Eq. (5). Ra =

gβqi H 4 vαk

(5)

where qi is the heat flux applied on the absorber surface; v, α, and k are the kinematic viscosity, thermal diffusivity, and thermal conductivity of air, respectively. They (Khanal and Lei 2012) [12] showed that l/H increased with Ra. In contrast, our results, for example at G/H = 0.05 with Domain S, showed that l/H decreased with H; hence decreased with the Rayleigh number according to Eq. (5). This difference may be due to the fact that in Khanal and Lei [12] simulations, the chimney geometries were fixed, and the heat flux changed, in contrast to our settings. 3.3 Induced Flow Rate The induced mass air flow rate through the tested solar chimneys with both domains is presented in Fig. 5. Figure 5a shows the variation of the mass flow rate as the chimney height and gap changed with both domains. For a given H, the flow rates obtained with both domains were identical when G/H was below a critical value. When G/H increased beyond that value, the flow rate predicted by the Domain S was significantly lower than that of the Domain L. Furthermore, the flow rate with the Domain S decreased while that of the Domain L increased as G/H increased.

Fig. 5. Induced air flow rate obtained with both domains for different chimney heights and gaps.

The flow rate obtained with the Domain L was non-dimensionalized by that with the Domain S and plotted in Fig. 5b. The critical values of G/H below which the difference between the two flow rates was within ±10.0% were 0.03, 0.05, 0.15, and 0.2 for H = 0.5 m, 1.0 m, 1.5 m, and 2.0 m, respectively. Therefore, the critical value increased with the chimney length, and according to Eq. (5), with the Rayleigh number.

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3.4 Thermal Efficiency Thermal efficiency of a solar chimney is defined by Eq. (6) which is the ratio of the heat absorbed by the air flow and the heat supplied to the air channel.

Fig. 6. Thermal efficiency obtained with both domains for different chimney heights and gaps.

η=

Qm cp (To − Ti ) Qi

(6)

where Qm is the mass flow rate; cp is the air specific heat capacity; Ti and To are the air temperature at the inlet and outlet of the air channel, respectively; Qi is the heat supplied to the air channel. Thermal efficiencies of the tested chimneys are plotted in Fig. 6. In Fig. 6a, it is seen that both domains predicted that the thermal efficiency decreased as the gap increased. At the lowest G/H ratio, 0.01, both domains offered identical values of η which was close to 1.0. This can be explained from the data in Fig. 4 and Fig. 5. Firstly, from Fig. 4, there was no flow reversal at G/H = 0.01 with both domains. At this very low G/H ratio, velocity and thermal boundary layer are expected to fully occupy the channel gap at the outlet [2, 5, 7, 8]. Therefore, both domains could result in identical temperature rises, T = To − Ti . Secondly, the data in Fig. 5 show similar air flow rate at G/H = 0.01. Consequently, Eq. (6) yielded identical thermal efficiencies at this gap – to – height ratio. As the gap increased further, the Domain S always offered lower η than the Domain L did. This fact again can be seen from the data in Figs. 4 and 5. The Domain S resulted in larger reverse flow regions at the outlet of the air channel (Fig. 4). As shown in Fig. 3, a larger reverse flow region led to lower average air temperature at the outlet; hence lower T. It also resulted in lower air flow rate, as seen in Fig. 5. Therefore, lower thermal efficiency obtained with the Domain S as the air gap increased can be expected. Figure 6b shows the ratio between the thermal efficiency predicted with Domain S and Domain L, ηL /ηS . All data of ηL /ηS for H = 1.5 m and 2.0 m were within [0.8–1.2], i.e. the difference of ηL and ηS was always within ±20.0%. For H = 0.5 m and 1.0 m, the ratio ηL /ηS within that range was when G/H < 0.03 and 0.05, respectively. Therefore, the critical G/H ratio at which the difference in the predicted thermal efficiency was within an acceptable range also increased with the chimney height, similar to the trend of the induced flow rate.

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4 Summaries Domain S and Domain L were utilized to compute the air flow and heat transfer in solar chimneys at different heights and gaps. It is seen that both domains offered similar results in terms of the flow structure, induced air flow rate, and thermal efficiency only at low gap – to – height ratios, which was below 0.03. As the gap increased, Domain S over – predicted the reverse flow at the outlet of the air channel. For a given chimney height, Domain S resulted in initiation of flow reversal at lower gap – to – height ratio. Once a flow reversal was observed with both domains, Domain S offered larger flow reversal regions. Both domains resulted in induced air flow rate which differed from each other below 10.0% when the gap – to – height ratio was below a critical value for a given height. The critical gap – to – height was 0.03, 0.05, 0.15, and 0.2 for H = 0.5 m, 1.0 m, 1.5 m, and 2.0 m, respectively; hence increased with the chimney height. Similarly, both domains also offered thermal efficiencies with difference below 20.0% when the gap – to – height ratio was below 0.03 and 0.05 for H = 0.5 m and 1.0 m, respectively. For H = 1.5 m and 2.0 m, the difference was always below 20.0%. In summary, the results from this study show that the small domain, which saves computational cost, can predict performance of solar chimneys at acceptable accuracy for low gap – to – height ratios. The critical gap – to – height ratios increased with the chimney height. A simple rule is that once there is no reverse flow predicted with the small domain, the results obtained with the small domain are identical to those with the large one.

References 1. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 2. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 3. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe Walls. Energy Build. 39(2), 128–135 (2007) 4. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 5. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 6. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 7. Nguyen, Y.Q., Wells, J.C.: Effects of wall proximity on the airflow in a vertical solar chimney for natural ventilation of dwellings. J. Build. Phys. 44(3), 225–250 (2020) 8. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020) 9. Zhang, T., Yang, H.: Flow and heat transfer characteristics of natural convection in vertical air channels of double-skin solar façades. Appl. Energy 242, 107–120 (2019)

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10. Liu, X., Xie, Z., Dong, S.: On a simple and effective thermal open boundary condition for convective heat transfer problems. Int. J. Heat Mass Transf. 151, 119355 (2020) 11. Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (1988) 12. Khanal, R., Lei, C.: Flow reversal effects on buoyancy induced air flow in a solar chimney. Sol. Energy 86(9), 2783–2794 (2012)

A Solar Chimney for Natural Ventilation of a Three – Story Building Tung Van Nguyen(B) , Y Quoc Nguyen, and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {tung.nv,y.nq,trieu.hn}@vlu.edu.vn

Abstract. Energy consumption of buildings can be significantly reduced with appropriate design for natural ventilation, particularly based on solar energy. Solar heat gained on the building envelope can be converted into thermal effects to induce air flow for natural ventilation or cooling of the building interior and the building envelope. For the ventilation application, the solar chimney should provide sufficient air flow rate. In this study, we investigated a solar chimney for ventilation of a three – story building. The chimney was assumed to be integrated into the building envelope and connected to all three stories of the building. To predict the air flow rate induced by the chimney through each story, a computational model was built based on the Computational Fluid Dynamics (CFD) method. By changing the design parameters (locations and dimensions of the air inlets) of the chimney, the ventilation rate through each story was computed. Temperature and velocity distributions in the chimney were also obtained for different design scenarios. From the results, the optimal design of the chimney which can provide nearly equal ventilation rate for all three stories was found. This study can serve as a demonstration for applications of the CFD technique in design of sustainable buildings. Keywords: Solar chimney · Flow rate · Thermal efficiency · Natural ventilation · CFD

1 Introduction With appropriate designs for natural ventilation, energy consumption of a building can be significantly reduced. Methods for natural ventilation can be based on effects induced by external wind flow or heat [1]. The methods based on heat usually employ structures absorbing solar radiation which can be converted into thermal effects to induce air flow for natural ventilation or cooling of the building interior and the building envelope. The structures creating such effects are classified as Solar Chimneys [2]. Many studies have reported effectiveness of solar chimneys in reducing energy consumption on buildings. In Japan, Miyazaki et al. [3] showed that a solar chimney integrated into an office building in Tokyo could save up to 50% of annual energy consumption. That number for a solar chimney attached to a test room in Qatar was 8.8% [4]. In China, Hong et al. [5] reported that a solar chimney for a two–story building saved up to 77.8% of energy for ventilation. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 617–630, 2022. https://doi.org/10.1007/978-981-16-3239-6_47

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When integrated into multi–story buildings, different configurations of solar chimneys have been examined [6–8]. Punyasompun et al. [6] compared two types of solar chimney for a three – story building: separate solar chimneys for each floor and a combined one connected to all floors. They reported that the combined solar chimney outperformed the separate ones with lower air temperature in each room and higher induced air flow rate. Asadi et al. [7] compared the induced air flow rate obtained with seven different configurations of solar chimneys for a seven–story building. Their results showed that the solar chimney on the southeast corner of the building offered the highest 24 h ventilation rate while the one in the middle of the south wall offered the highest flow rate in day time. Similar conclusions were also reported by Mohammed et al. [8] for a two – story building. One of the factors influencing most the performance of solar chimneys is the location of the air inlet [9, 10]. Effects of the inlet location of the air inlet of the solar chimney for a single room were reported by Shi et al. [10]. Those for multi – story buildings

Fig. 1. Skechmatic of a solar chimney for ventilation of a three – story building (H = 9.4 m, G = 0.2 m) with different configurations together with an example of the computational domain and mesh of Case 2.

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have not been reported thoroughly in the literature. Although Mohammed et al. [8] also considered different locations of the solar chimney inlet in their simulation, their model is for a small–scale building with dimensions of 1 m × 1 m × 1 m of each story. They did not report the scaling of the results for real size buildings. In this study, design of a solar chimney for a three – story building is examined. By changing the location, the number of inlets of each floor, and the size of inlet, it is aimed to find the best configurations of the solar chimney to achieve nearly equal ventilation rate for all floors. In the following sections, the problem is described together with the numerical method. Next, effects of the above factors are presented and discussed. Finally, main findings are summarized.

2 Problem Formulation and Numerical Method 2.1 The Solar Chimney A solar chimney is assumed to be integrated into a three – story building, as sketched in Fig. 1. The solar chimney consists of an air channel enclosed by an outer wall and a wall of the building. For typical designs of solar chimneys, the outer wall is usually a glass plate which receives solar radiation directly. Solar radiation is then transmitted through the outer wall and absorbed by the building wall. With proper insulation of the wall inside the building, the absorbed heat is not transferred into the building but supplied to the air inside the air channel of the solar chimney. Once heated, the air rises due to the stack effects and tends to induce an air flow through the chimney. If the solar chimney is connected to the rooms of the building and its upper end is open to the atmosphere, the induced air flow can help to ventilate the building naturally. In this study, the solar chimney was examined numerically under four configurations, as shown in Fig. 1. In Fig. 1a, the chimney has two openings at the lower and upper ends. An induced air flow enters and exits the air channel at the lower and upper openings, respectively. The solar chimney was then connected to the rooms by the inlets near the floor (lower inlet) in Fig. 1b while in Fig. 1c, the inlet of each floor was near the ceiling (upper inlet). In Fig. 1d, there were two inlets of each floor, both the lower inlet and the upper inlet. In Figs. 1b–d, two cases of the lower end of the solar chimney were considered: being open or closed. In all cases, the upper end of the solar chimney was open. The lower end of the air channel, in general, can be open to the atmosphere. Therefore, when it is open, ambient air can enter the air channel. It is assumed that the total height of each floor is 3 m. The total height H of the solar chimney is 9.4 m including the heights of three floors and the top ceiling. The width G of the air channel was fixed to 0.2 m. The inlets in Fig. 1b–d have the same height of 0.2 m while its length L is 0.5 m. 2.2 Assumptions and Governing Equations Performance of the solar chimney was assessed in term of the induced air flow rate induced through each inlet and the total flow rate at the outlet of the chimney. In practice, solar radiation approaching the chimney changes during daytime. However, for simplicity, a quasi–steady two–dimensional problem was considered in this study. Therefore, the

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air flow and heat transfer in the chimney were assumed to be steady and two – dimensional. In addition, as the air speed in typical solar chimneys have an order of about 1.0 m/s, the air flow was expected to be incompressible. For heat transfer, as the most important process is from the absorbed surface, which is the building wall in Fig. 1, to the air in the channel and the opposing glass plate, both convective and radiative heat transfer were considered. The convection and radiation are from the absorber surface to the air in the channel and the outer wall, respectively. Conduction loss through the building wall and the glass plate was not modeled for simplicity. For obtaining the induced air flow rate through the chimney together with the assumption of steady flow, time – averaged flow variables can be described with the Reynolds – Averaged Navier Stokes equations [9, 11]. The equations represent the conservation principles of the mass, momentum, and energy of the flow. Using the Einstein summation notation, the equations are as follows. ∂uj =0 ∂xj

    ∂ ui uj ∂ui 1 ∂p ∂   ν =− − gi β(T − T0 ) + − ui uj ∂xj ρ ∂xi ∂xj ∂xj     ∂ Tuj ν ∂T ∂  = − T  uj ∂xj ∂xj Pr ∂xj

(1) (2) (3)

In Eq. (1) to (3), the velocity u and temperature T are time – averaged while u and indicate their fluctuating components, respectively. p, ν, ρ, and Pr are the pressure, air kinematic viscosity, density, and the Prandtl number, respectively. β = 1/T0 and is the air thermal expansion coefficient. g is the gravitational acceleration; is the time – averaged operator. Air properties are taken at T0 . In Eq. (2), the Boussinesq approximation is used for variation of the air density with temperature. As the temperature rise of the air flow through typical solar chimneys is reported to be not high, this approximation has been widely used for simulations of air flows in solar chimneys in the literature [9, 11–13].    The turbulence tress ui uj in Eq. (2) and the turbulent heat flux T  uj in Eq. (3) were solved with the standard k–ω model with modification for low – Reynolds number effects by a damping coefficient for the eddy viscosity. This turbulent model was also employed in previous studies of solar chimneys [9, 13]. For the solar chimney in Fig. 1, the gap – to – height ratio, G/H, is 4.25%. Gan [14] showed that for solar chimneys with very low G/H, a computational domain fitted with the air channel was sufficient. Accordingly, in this study, the computational domain covered only the air channel for Fig. 1a, and the air channel and the inlet sections for other cases. An example of the computational domain is presented in Fig. 1b. For discretizing the governing equations, Eqs. (1), (2) and (3), Finite Volume Method was used on a structured rectangular mesh with the ANSYS Fluent CFD Code (Academic version 2019R3). In the numerical setup, the SIMPLE method was employed for the coupling between the continuity and momentum equations; the PRESTO! Method was applied for interpolation of the pressure value from the cell center to the cell faces on T

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the staggered mesh; the second order upwind scheme was employed for the momentum and energy equations while first order upwind discretization was used for the k and ω equations. In all cases, the convergence criterion was set for the scaled residual to be lower than 10−6 . It was achieved in al tests. The mesh density was higher near the walls of the chimney where the gradients of the flow variables are strong. An example of the computational mesh is also displayed in Fig. 1b. To check effects of the mesh resolution, we ran the preliminary numerical model for the solar chimney in the experiment by Yilmaz and Fraser [15]. The number of elements in each direction was increased gradually and the according induced flow rate was plotted in Fig. 2. It is seen that a mesh with 20,000 elements is sufficient for this test. Denser meshes resulted in a change of the flow rate below 1.0%. The equivalent required non-dimensional distance of the first grid point from the solid surfaces, or y+ was below 1.0, which agrees with the findings by Zamora and Kaiser [9]. For the boundary conditions, atmospheric pressure was applied at the inlets of the rooms and at the outlet of the solar chimney. At the lower end of the chimney, section a–b in Fig. 1, when it was closed, a no–slip wall condition was applied. When it was open, the atmospheric pressured was assumed. Other walls were treated with no–slip conditions. The air temperature was set to 293 K at the inlets. A uniform heat flux of 600 W/m2 was assigned on the left wall (the absorber surface) in Fig. 1. The right wall was allowed to receive radiative heat transfer from the heated surface which was described by the S2S model in Fluent. When the lower end of the air channel was closed, it was treated as an adiabatic wall. At all inlets, turbulent kinetic energy and a length scale were used. As the turbulence should be low [9], it was assumed a turbulent intensity of 2.0% and a length scale of half of the inlet height, i.e. 0.1 m.

Fig. 2. Variation of the flow rate versus the number of elements for simulation of the solar chimney in the experiment by Yilmaz and Fraser [15] (Case of 100 °C). The measured flow rate by Yilmaz and Fraser [15] was also plotted. The selected mesh is indicated with the open marker.

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2.3 Validation of the Numerical Model The CFD model was applied to predict the induce flow rate through the solar chimney in the experiment by Yilmaz and Fraser [15]. The solar chimney is similar to the base case (Fig. 2a). The height and the gap of the chimney were 3.0 m and 0.1 m, respectively. In the experiment, uniform heat fluxes were applied on a surface of the air channel and the corresponding wall temperatures were in the range of 60–130 °C. Uniform wall tempratures were also applied in the CFD model. The predicted and measured flow rates are presented in Fig. 3 and Table 1. It is seen that the model slightly over–predicted the flow rate. However, the maximum discrepancy is less than 10.0% and should be within measurement uncertainty. Therefore, the model is acceptably accurate for predicting the induced flow rate.

Fig. 3. Comparison of the computed flowrate versus the measured data for the experiments by Yilmaz and Fraser [15]. Table 1. Comparison of the computed flowrate versus the measured data for the experiments by Yilmaz and Fraser [15] Wall temperature (°C)

Measurement (kg/s)

CFD (kg/s)

Difference (%)

60

0.0599

0.0603

0.79

75

0.0674

0.0736

9.16

90

0.0792

0.0846

6.76

100

0.0854

0.0912

6.76

130

0.1062

0.1091

2.71

3 Results and Discussions In this section, the induced flow rate through different cases of the solar chimney in Figs. 1a to 1d is presented. The heat flux applied on the heated wall was from 200 W/m2 to 800 W/m2 . From the results, the configuration which offers the highest flow rate was proposed and further examined.

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3.1 Base Configuration (Base Case) In the base case shown in Fig. 1a, the chimney had one inlet at the lower end of the air channel. This configuration has been studied widely in the literature [15–18]. The flow and velocity fields are plotted in Fig. 4. In Fig. 4a, the thermal boundary layers are seen to develop from both walls of the air channel. While the heat source was on the left wall, the right wall was heated due to radiative heat transfer from the left one. At the outlet, both the thermal and velocity boundary layers have occupied the whole channel gap. No flow reversal appears at the outlet. Gan [14] argued that with a reverse flow at the outlet, an extension of the computational domain outside the air channel is required to achieve reliable results. Furthermore, without a reverse flow, using a domain covering only the air channel can offer results as good as those of an extended one. As there is not a flow reversal at the outlet seen in Fig. 4a, using the computational domain consisting only the air channel in this study should be sufficient. The flow rate was also obtained with different heat fluxes on the heated wall and plotted in Fig. 4c. The flow rate increased with the applied heat flux. This trend can be physically expected, as the thermal effect; hence induced flow rate, is boosted with more heat input in the air channel.

Fig. 4. Temperature and flow fields, and the induced flow rate of the base case.

3.2 Testing Configurations (Case 2 to 4) The induced flow rate through the configurations 2 to 4 was predicted and presented in Fig. 5. For comparison, the flow rate in each case was normalized by that of the base

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case, Qtotal /Qbase . Though the data seem to increase with the heat flux, the variations are within only about 2.0% for each case. Except for the Case 3 (closed), all other cases have Qtotal /Qbase ≈ 1.0. The flow rate of Case 3 (closed) was about 94.0%–95.0% of the flow rate of the base case. Lower flow rate in Case 3 may be due to shorter elevation difference between the inlet of the first floor and the outlet of the air channel. Therefore, the configurations of Case 2 to Case 4 offered similar flow rate to that of the base case. Effects of the location and configurations of the inlets on the total flow rate through the chimney were seen negligible. Figures 6, 7 and 8 show the temperature and velocity fields in the air channel and the induced flow rate when the lower end was closed and open. The flow rate through each inlet is normalized by the total flow rate at the outlet of the air channel, Q/Qtotal , in each case. The temperature fields in Figs. 6a, 7a, and 8a show that when the lower end of the air channel was closed, heat was trapped below the inlet of the first floor. The portion of the air channel attached to the first floor had higher temperature. As heat transport took place only at the upper part of this portion, and moreover, the air flow only circulated inside that space, the trapped heat may result in such high temperatures. However, on the upper part of the channel, both cases of closed and open lower ends had very similar temperature distributions. Figure 6b, 7b, and 8b show difference only at the first floor inlet of the closed lower end cases where the velocity was higher than that of the open lower end cases. The flow fields on the upper part of the chimney in both cases of the lower end were identical.

Fig. 5. Comparison of the total flow rate of Case 2 to 4 to that of the base case, when the lower end of the channel was closed (closed) or open (open).

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Fig. 6. Temperature and flow fields, and the induced flow rate of Case 2.

Figures 6c, 7c, and 8c show that the Q/Qtotal of an inlet almost does not change with the heat flux. For both cases of the lower end, the induced flow rate at each inlet decreased from the first to the third floor. This point can be expected, as the higher floor has lower elevation difference between the inlet and the outlet of the air channel; hence weaker thermal effects and accordingly lower induced flow rate. For all configurations, at each inlet, Q/Qtotal of the closed lower end cases was always higher than that of the open lower end cases. For the open lower end cases, flow rates at all three inlets were below 30.0% of the total flow rate. About from 30% to 40.0% of the total flow rate was from the lower inlet of the air channel. In contrast, when the lower end was closed, the first floor inlet had the highest flow rate with more than 50% of the total flow rate. The other two inlets had flow rates similar to those of the open lower end cases. Particularly, in Case 3 with the closed lower end, the Q/Qtotal at the first floor inlet was up to about 70% (Fig. 7c). In contrast, those ratios of the third floor were only about 5.0%. This low flow rate is possibly because of short distance between the third floor inlet and the outlet of the air channel. 3.3 Selected Configuration (Case 5) From the induced flow rate obtained in Cases 2 to 4 in Fig. 6c, 7c, and 8c, it is seen that higher flow rates through the inlets in the floors are obtained when the lower end of the chimney was closed. In addition, the following configurations of the inlets offered the highest flow rate: • For the first floor: The inlet of Case 3 which yielded Q/Qtotal ≈ 70.0%.

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Fig. 7. Temperature and flow fields, and the induced flow rate of Case 3.

• For the second and third floor: The inlets of Cases 2 and 4 offered similar ratios of Q/Qtotal which are higher than those in Case 3. However, the ratios of Q/Qtotal of Case 4 are slightly higher than those of Case 2. Aimed at achieving both high and uniform flow rates at the inlets of all rooms, the selected configuration (Case 5) of the chimney has inlets similar to that of Case 3 for the first floor and those of Case 4 for the second and the third floor. Characteristics of the selected configuration are presented in Fig. 9. The total flow rate of Case 5 was similar to those of Cases 2 to 4 which was about 98.0% of that of the base case. The temperature and velocity fields in Figs. 9a and 9b show that on the second and third floors, among two inlets, more air was induced into the lower one. The flow rate through the upper inlet was about 20.0% of that of the lower one. The total flow rate through each room is plotted in Fig. 9c. The ratio of Q/Qtotal also changed negligibly with the heat flux. The highest flow rate is seen at the first floor with Q/Qtotal ≈ 41.0% and the lowest flow rate was at the third floor with Q/Qtotal ≈ 24.0%. Therefore, the flow rates among the floors were less different from each other, compared to those of Cases 2 to 4. As seen in Fig. 9, the flow rate through the first floor is about twice that through the third floor. In order to reduce the difference in the flow rates among the floors, different inlet sizes were tested (Case 6). Totally, there were five combinations of the inlet sizes which are denoted in Fig. 10a. Table 2 shows the total flow rates as the inlet sizes changed. The heat flux was kept to 600 W/m2 . In Table 2, except for the Case 6b, all other cases offered the same flow

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Fig. 8. Temperature and flow fields, and the induced flow rate of Case 4.

rate of 0.35 kg/s and about 6.0% higher than that of the base case. However, Case 6b resulted in 34.0% less flow rate compared to that of the base case. It is seen in Fig. 10b that the flow rates of Cases 6a and 6c–6d are similar to each other. Similar flow rates can be achieved for the first and the second floors. The flow rates through the third floor were always lower than those through the other floors. The flow rates of Case 6b differed from those of the others at the first and the third floors. It is seen that the main difference of Case 6b to the other cases is the inlet size i4. Between the inlet i4 and i5, the flow rate through i5 was minor, as seen from the flow fields in Figs. 7c, 8c, and 9c. Therefore, changing the inlet size i5 might not change the flow rate through the third floor much. However, reducing the inlet size i4 resulted in lower flow rate, as expected. From the pressure distributions of Cases 6a and 6b, it is seen that lower flow rate at the inlet i4 in Case 6b yielded higher pressure in the air channel around that inlet; hence lower pressure differences between that area and the inlets i1–i3, and consequently lower flow rate through those inlets in Case 6b. Therefore, the size of the lower inlet of the third floor is critical as it affects not only the total flow rate (Table 2) but also the flow rates through the floors 1 and 3. To check the ventilation performance of the proposed solar chimney, it was assumed that the chimney in Case 6 had a width of 1.0 m. The ACH (Air Changes per Hour) of that chimney for the building in Fig. 1 with a floor area of 8.0 m × 4.0 m is also plotted in Fig. 10b. ACH is defined as the number of times in one hour the air in a room is replaced, as seen in Eq. (4). ACH =

Q(kg/s) × 3600 ρ Vol.

where Vol. (m3 ) is the volume of the room.

(4)

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From Fig. 10b, except for Case 6b, all other cases offered ACH values greater than 3.0, which is the minimum requirement for ventilation of residential buildings [ASHREA 62.2].

Fig. 9. Temperature and flow fields, and the induced flow rate of Case 5.

3.4 Changing Inlet Size for Similar Flow Rates at All Inlets (Case 6) Table 2. Flow rates as the inlet sizes changes. Case

Qtotal (kg/s)

Case 6a (i1 = i2 = i3 = 0.2 m, i4 = 0.3, i5 = 0.1 m)

0.35

Case 6b (i1 = i2 = i3 = 0.2 m, i4 = 0.25, i5 = 0.15 m)

0.27

Case 6c (i1 = i2 = i3 = 0.2 m, i4 = 0.35, i5 = 0.05 m)

0.35

Case 6d (i1 = 0.2, i2 = i3 = 0.15, i4 = 0.35, i5 = 0.05 m)

0.35

Case 6e (i1 = 0.15, i2 = i3 = 0.1, i4 = 0.35, i5 = 0.05 m)

0.35

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Fig. 10. The induced flow rate at different inlet sizes.

4 Conclusions Different configurations of a solar chimney for natural ventilation of a three–story building have been examined. The results show that effects of the location and configurations of the inlets on the total flow rate through the chimney were seen negligible. While in all tested configurations, the induced flow rates varied within 6.0% around the value obtained with the base case, higher flow rates through the rooms were obtained when the lower end of the air channel was closed. The maximum difference was seen at the first floor where the flow rate with the closed lower end was up to about three times of that with the open lower end (Case 3). The best configuration for achieving the highest flow rates and the most similar flow rates through all three floors was with one inlet in the first floor and two inlets in the second and the third floors. By changing the inlet sizes, it was possible to achieve similar flow rates through the first and the second floors but the flow rate through the third floor was always less. The size of the lower inlet of the third floor is critical to both the total flow rate and the flow rates through the other floors. This study demonstrates possible applications of the CFD technique in design of sustainable buildings. In future works, optimization techniques may be applied to further optimize the inlet sizes to achieve more similar flow rates among the floors.

References 1. Awbi, H.: Ventilation of Buildings, 2nd edn. Spon Press, London (2003) 2. Bansal, N.K., Mathur, R., Bhandari, M.S.: Solar chimney for enhanced stack ventilation. Build. Environ. 28(3), 373–377 (1993)

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3. Miyazaki, T., Akisawa, A., Kashiwagi, T.: The effects of solar chimneys on thermal load mitigation of office buildings under the Japanese climate. Renew. Energy 31(7), 987–1010 (2006) 4. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 5. Hong, S., He, G., Ge, W., Wu, Q., Lv, D., Li, Z.: Annual energy performance simulation of solar chimney in a cold winter and hot summer climate. Build. Simul. 12(5), 847–856 (2019) 6. Punyasompun, S., Hirunlabh, J., Khedari, J., Zeghmati, B.: Investigation on the application of solar chimney for multi-storey buildings. Renew. Energy 34(12), 2545–2561 (2009) 7. Asadi, S., Fakhari, M., Fayaz, R., Mahdaviparsa, A.: The effect of solar chimney layout on ventilation rate in buildings. Energy Build. 123, 71–78 (2016) 8. Mohammed, H.J., Jubear, A.J., Obaid, H.: Natural ventilation in passive system of vertical two-stores solar chimney. J. Adv. Res. Fluid Mech. Therm. Sci. 69(2), 130–146 (2020) 9. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 10. Shi, L., et al.: Interaction effect of room opening and air inlet on solar chimney performance. Appl. Therm. Eng. 159, 113877 (2019) 11. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 12. Khanal, R., Lei, C.: An experimental investigation of an inclined passive wall solar chimney for natural ventilation. Sol. Energy 107, 461–474 (2014) 13. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018) 14. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 15. Yilmaz, T., Fraser, S.M.: Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating. Int. J. Heat Mass Transf. 50(13–14), 2612–2623 (2007) 16. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 17. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 18. Khanal, R., Lei, C.: Flow reversal effects on buoyancy induced air flow in a solar chimney. Sol. Energy 86(9), 2783–2794 (2012)

Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume Y Quoc Nguyen(B) and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {y.nq,trieu.hn}@vlu.edu.vn

Abstract. Ventilation of buildings can be based on mechanical systems, such as fans, or natural driving forces, such as wind or heat. Of common natural ventilation methods, solar chimneys convert solar heat gained on the envelope of buildings into flow energy to ventilate or to cool the buildings. As solar chimneys are typically integrated into the building envelope, i.e. walls or roofs, architects determine shapes and sizes of a chimney mainly based on available space on the envelope. This raises a need for maximizing the ventilation performance of a solar chimney for a given space on a building envelope. In this study, ventilation performance of a typical vertical solar chimney was assessed in term of the induced flow rate that it can provide per volume. A three-dimensional numerical model based on the Computational Fluid Dynamics method was built to predict the induced air flow rate through the chimney as its dimensions changed. The tested dimensions included the height, the width, and the gap of the chimney. The induced air flow rate was obtained with different volumes of the chimney. The results show that the induced air flow rate nominated by the chimney volume changed with all three dimensions. Higher flow rate per volume was achieved with the chimneys with shorter heights and lower gap – to – height ratios. Therefore, it is suggested that to maximize the air flow rate per chimney volume, smaller chimneys are preferred. These findings agree with the results in the literature. Keywords: Natural ventilation · Solar chimney · Thermal effect · CFD · Flow rate per volume

1 Introduction Ventilation of buildings is to supply fresh air into the buildings to achieve desired levels of indoor air quality and thermal comfort. For driving the air flow, mechanical systems, such as fans, or natural driving forces, such as wind or heat can be used [1]. The methods based on thermal effects create air flows through a confined space with two conditions: temperature difference between the air inside the space and elevation difference between the inlet and the outlet of the space.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 631–640, 2022. https://doi.org/10.1007/978-981-16-3239-6_48

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Solar chimneys [2] are of the common systems for natural ventilation of building based on thermal effects. The main component of a solar chimney is the air channel which had open inlet and outlet at the bottom and the top, respectively. Solar radiation is absorbed on either the front or the rear wall of the channel. When the front wall is a glazing (glass) plate, solar radiation is transmitted through the front wall and absorbed on the rear wall of the channel. In contrast, when the front wall is opaque, it absorbs solar radiation directly. The absorbed heat is then transferred to the air in the channel. As the air temperature rises, the pressure inside the air channel deviates from that of the ambient and induces an air flow through the channel. When the inlet of the channel is connected to a room, stalled air inside the room is withdrawn. Fresh air is then supplied to the room through suitable arrangement of windows or doors. A typical solar chimney is illustrated in Fig. 1 where it is integrated into the front wall of the building.

Fig. 1. Illustration of a solar chimney for ventilation of a building.

As the thermal effects depend on the temperature rise in the air channel and the elevation difference between the inlet and the outlet, the absorbed heat and the height of the channel have been shown to influence most the performance of a solar chimney [3–9]. Those studies reported that the induced air flow rate increases with both the heat flux on the surface of the channel and the height of the channel. In addition, the flow rate also increases with the cross – sectional area of the channel which is the product of the channel gap and width (Fig. 2) [3, 4]. Therefore, increasing either the gap or the width results in higher flow rate. As solar chimneys are typically integrated into the building envelope, i.e. walls or roofs, architects determine shapes and sizes of a chimney based on not only the required induced flow rate sufficient for ventilation of the building but also the available space on the envelope. This raises the need for maximizing the ventilation performance of a solar chimney for a given space on a building envelope. For maximizing induced flow rate for a roof solar chimney, Hirunlabh et al. [10] compared different configurations with different lengths. They found that increasing the length resulted in higher induced air flow rate but decreased the ratio of the induced flow rate and the area of the absorber surface. Therefore, they suggested, for the same available length, to use multiple short chimneys instead of a long one.

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The research by Hirunlabh et al. [10] showed effects of the length of the chimney on the induced flow rate per surface area. However, as the chimney gap and width were fixed in their experiment, effects of the gap and the width have not been reported. In this study, we studied effects of all three main dimensions of the chimney, i.e. height, gap, and width, on the ratio of the induced flow rate and the volume of the chimney numerically. The configurations of the examined solar chimney together with the numerical model are described in section “Numerical Method”. Effects of the dimensions of the chimney on the induced flow rate are presented in section “Results and Discussions”. Finally, section “Conclusions” summarizes the main findings of this study.

2 Numerical Method 2.1 Studied Solar Chimney To focus on effects of the dimensions of the chimney, the examined chimney was a simple vertical one as depicted in Fig. 2. The air inlet and outlet are at the bottom and the top of the air channel, respectively. With this configuration, influences of other configurational factors can be eliminated. For example, the solar chimney illustrated in Fig. 1 has horizontal inlet and outlet. Zamora and Kaiser [11] and Shi et al. [12] showed that the size and the location of the inlet strongly affect the induced flow rate. By using straight vertical inlet and outlet, such undesired influences can be removed. As denoted in Fig. 2, three main dimensions of the chimney are the height H, the gap of the air channel G, and the width W. Besides the inlet and the outlet of the air channel, which are openings, other sides of the channel are rigid walls.

Fig. 2. The studied problem: a) Modeled solar chimney and b) Computational mesh.

The air flow and heat transfer were modeled inside the air channel. Consequently, the computational domain had the same size as the air channel. Absorbed solar radiation was transferred to the air in the channel from one wall. Accordingly, a heat source with uniform distribution was assumed on the left wall in Fig. 2a.

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2.2 Numerical Model The air flow and heat transfer in the air channel are described by conservation equations for the mass, momentum, and energy. As the air speed and the temperature rise through the air channel were not high [3], it was assumed that the air flow in the channel is incompressible; the air flow and heat transfer are statistically steady; and the Boussinesq approximation can be used for the change of air properties with temperature. Besides, convection heat transfer, which was reported to dominate in solar chimneys [3, 5], together with radiative heat transfer were considered. Conduction heat transfer and heat loss through the wall of the channel to ambient environment, was not modeled. As the flow in solar chimneys with practical sizes are in early turbulent regime, a suitable turbulence model was also required. The Reynolds – Averaged Navier – Stokes equations were employed in this study. Their forms for steady turbulent flows are as follows. ∂uj =0 ∂xj

    ∂ ui uj 1 ∂p ∂ ∂ui   =− − gi β(T − T0 ) + − ui uj v ∂xj ρ ∂xi ∂xj ∂xj     ∂ Tuj v ∂T ∂  = − T  uj ∂xj ∂xj Pr ∂xj

(1) (2) (3)

In Eq. (1) to (3), time – averaged velocity and temperature are denoted as u and T. u and T  indicates the fluctuating component of the velocity and temperature, respectively. p is the pressure. ν, ρ, and Pr are the air kinematic viscosity, density, and the Prandtl number, respectively. g stands for the gravitational acceleration; indicates a time – averaged quantity.    For the turbulence tress ui uj and the turbulent heat flux T  uj , the standard k–ω model with modifications for effects of low – Reynolds number was utilized. This turbulent model was also employed in previous studies of solar chimneys [11, 13]. Finite Volume Method was used to discretize the governing equations, Eq. (1) to (3) on the structured staggered mesh presented in Fig. 2b. The mesh was distributed more near the walls of the chimney where there were strong gradients of the flow variables. To find the most suitable mesh which offered mesh – independent solutions, the number of grid cells was gradually increased for different test cases. Plots of the induced flow rate versus the non-dimensional distance of the first grid point from the solid surfaces, or y+ , showed the range of y+ where the change of the induced flow rate was below 1.0%. It was found that the required range of y+ was below 1.0. Details can be seen in [8, 9]. For the boundary conditions, it was assumed that the flow enters and leaves the domain at the atmospheric pressure. Therefore, zero – gage pressure was applied at the inlet and the outlet of the domain. The walls were treated as no – slip and adiabatic ones except for the heated wall where a uniform heat flux of 600 W/m2 was assigned. For the coupling between the continuity and momentum equations, Eq. (1) and (2), the SIMPLE method was employed. In addition, to interpolate the pressure value from

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the cell center to the cell faces, the PRESTO! Method was applied. The above numerical schemes were conducted with the ANSYS Fluent CFD code (Academic version). Radiative heat transfer among the surfaces of the channel was modeled with the S2S model in ANSYS Fluent. 2.3 Validation of the Numerical Model For validation, the CFD model was applied to simulate the solar chimney in the experiment by Burek and Habeb [4]. The chimney was 1.0 m high and 0.92 m wide. The gap of the air channel changed in the range of 20 mm–110 mm. A uniform heat flux was applied on one wall of the channel whose dimensions were 1.0 m × 0.92 m. The measured induced flow rate through the chimney in the experiment and the one obtained with our CFD model at the total heat input of 600 W are plotted together in Fig. 3. It is seen that the numerical results are close to the measured ones. The CFD model slightly over – predicted the induced flow rate at the gaps of 100 mm and 110 mm. However, the maximum difference was only 6.5% which should be within the measurement errors.

Fig. 3. Comparison of the computed flow rate versus the measured data for the experiments by Burek and Habeb [4].

3 Results and Discussions In this section, the induced flow rate through solar chimneys with different heights, widths, and gaps obtained with the CFD model are presented. The chimney heights were 0.5 m, 1.0 m, and 1.5 m, which is the typical range in practical applications of solar chimney for natural ventilation of buildings [2]. The width and gap at each H changed and resulted in the gap – to – height ratio of 0.05, 0.1, and 0.15. Similarly, the width – to – gap ratio at each height was from 1.0 to 15.0. The heat flux was fixed at 600 W/m2 in all cases.

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3.1 Induced Flow Rate Versus Channel Width and Gap In Fig. 4, the induced flow rate is plotted as functions of the width and the gap of the examined chimneys. It is seen in Fig. 4a that the induced flow rate increased linearly with the width at any gap. In general, at a given width, the flow rate increased with the height of the chimney. However, at a given height, the trend of the flow rate with the gap was not consistent. At H = 1.5 m, the flow rate was enhanced as G/H increased from 0.05 to 0.1 and 0.15. At the other heights (H = 1.0 m and 0.5 m), the flow rate did not always increase with G/H.

Fig. 4. Induced flow rate as functions of the channel width and gap: (a) Flow rate vs. channel width, (b) Flow rate per volume vs. width – to – gap ratio.

To evaluate the effectiveness of increasing the width and the gap, the flow rate was normalized by the volume, as seen in Fig. 4b. The width was also non – dimensionalized by the gap. As the width – to –gap ratio, W/G, increased beyond a certain value, the flow rate per volume approached to a constant. The critical W/G was in the range of 5 to 7 for all tested cases. The flow rate per volume is plotted versus G/H for W/G = 15 in Fig. 5. For a given height, the flow rate per volume decreased with the gap – to – height ratio. As G/H increased from 0.05 to 0.15 for H = 0.5 m with W/G = 15, the flow rate per volume was reduced from 0.91 to 0.36. This trend is consistent with the experimental results by Chen et al. [3] and Burek and Habeb [4] which are also presented in Fig. 5. In Fig. 5, at G/H = 0.05 and 0.1, the flow rate per volume always decreased as the height increased from 0.5 m to 1.5 m. Hirunlabh et al. [10] reported that the flow rate per area of the absorber surface always decreased in their chimneys with the height up to 3.0 m and the gap of 0.14 m; hence G/H = 0.047. Therefore, the trend in Fig. 5 for G/H = 0.05 is consistent with that by Hirunlabh et al. [10] for G/H = 0.047. It is noticed in Fig. 5 that the trend at G/H = 0.15 is different from that of the other gap – to – height ratios. The flow rate per volume of H = 1.0 m dropped below that of H = 1.5 m. Examining the flow velocity at the outlet of this case reveals that the boundary layers of the H = 1.0 m case did not develop to the channel center and yielded an area with very low velocity. In contrast, in the cases of H = 0.5 m and 1.5 m, the boundary layers occupied the whole channel gap, as seen in Fig. 6. For the cases with G/H = 0.05

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Fig. 5. Induced flow rate per volume as functions of the channel gap for W/G = 15 together with the experimental results by Chen et al. [3] and Burek and Habeb [4].

and 0.1, the boundary layers always occupied the whole channel gaps at all heights. Accordingly, the trend of the flow rate per volume at G/H = 0.15 was different from the other cases.

Fig. 6. Velocity field at the outlet of the chimney with G/H = 0.15 and W/G = 15: a) H = 0.5 m, b) H = 1.0 m, c) H = 1.5 m.

3.2 Induced Flow Rate Versus Channel Volume The induced flow rate is plotted versus the volume of the chimney in Fig. 7. Figure 7a shows that the flow rate increased linearly with the volume with similar slopes for all chimneys. The experimental data by Chen et al. [3] and Burek and Habeb [4] also have similar trend. While the slope of the data by Burek and Habeb [4] is similar to that of our data, the slope of the data by Chen et al. [3] is much lower. This may be because of the same applied heat flux in our simulations and the experiments by Burek and Habeb [4], which was 600 W/m2 , and different heat flux in Chen et al. [3], which was 400 W/m2 . The trends of the data in Fig. 7a shows that at a given volume, the flow rate decreased as both the gap – to – height ratio and the chimney height increased. This observation

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Fig. 7. Induced flow rate as functions of the volume: a) Flow rate versus volume and b) Flow rate per volume versus volume.

applies to all cases, except for the case of H = 1.0 m and G/H = 0.15. The reason for this exception may be the same as that discussed in Sect. 3.1. To examine the effectiveness of increasing the chimney volume, the flow rate is normalized by the volume and plotted in Fig. 7b. Among three groups of G/H, for G/H = 0.05, the flow rate per volume increased with the volume. In contrast, for G/H = 0.1 and 0.15, the flow rate per volume decreased with the volume. In all cases, the flow rate per volume approached to constants as the volume increased. The flow rate per volume also decreased with the gap – to – height ratio and the chimney height. These trends are similar to those in Fig. 4b and 5. The experimental data by Chen et al. [3] and Burek and Habeb [4] are also plotted in Fig. 7b. In these experiments, the height and width of the chimneys were fixed while the gap changed. Therefore, in Fig. 7b, the increase of the volume of their chimneys is merely due to the increase of the air gap and associated with the decrease of the G/H ratio. The trends of variation of our results are seen to be similar to their data, as the flow rate per volume decreases with the gap – to – height ratio. 3.3 Thermal Efficiency Thermal efficiency of solar chimneys is the ratio between the heat absorbed by the air flow and the heat supplied to the flow from the wall of the channel [4, 8, 9], as defined by Eq. (4). η=

Qcp (To − Ti ) . Qi

(4)

where Q is the mass flow rate; cp is the air specific heat capacity; Ti and To are the air temperature at the inlet and outlet of the air channel, respectively; Qi is the heat supplied to the air channel. Thermal efficiencies of the examined chimneys decreased and approached constants as the volume increased, as seen in Fig. 8. Higher efficiencies were achieved with lower gap – to – high ratios for a given chimney height. The highest efficiency was obtained

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Fig. 8. Thermal efficiency.

with H = 0.5 m and G/H = 0.05. For this chimney height with W/G = 15, increasing G/H from 0.05 to 0.1 and 0.15 resulted in the reductions of the thermal efficiency from 0.96 to 0.56 and 0.34, respectively. However, a consistent trend of variation of the thermal efficiency with the chimney height was not seen in Fig. 8. 3.4 Proposed Chimney Configurations for Maximizing Induced Flow Rate From the data in Fig. 4, 5, 7, and 8, it is concluded that to achieve higher flow rate per volume, using shorter chimneys with lower gap – to – height ratios is preferred for an available volume. For example, for H = 0.5 m and W = 0.375 m, increasing the gap from 0.025 m (G/H = 0.05) to 0.05 m (G/H = 0.1) resulted in an increase of the flow rate from 4.3 g/s to 5.3 g/s. Therefore, the volume increases 200.0% but the increase of the flow rate is only 23.3%. For this case, if the architectural space is available, instead of utilizing single chimney with G = 0.05 m, one can design two chimneys with G = 0.025 m and get a total flow rate of 2 × 4.3 g/s = 8.6 g/s, which is 62.3% higher than that of the single chimney. Another example is considered for an available space where one can design a chimney with H = 1.5 m, W = 0.375 m, and G = 0.075 m and achieve a flow rate of 18.4 g/s. Instead, with the same volume, nine chimneys with the same W but H = 0.5 m and G = 0.025 m can be arranged. The total flow rate of nine smaller chimneys is 9 × 4.3 g/s = 38.7 g/s, which is 210% higher than that of the single chimney. Similar configurations for roof solar chimney was also proposed by Hirunlabh et al. [10]. They compared flow rate through chimneys for a given length of 3.5 m: a single one and three chimneys whose lengths were 1.0 m, 1.0 m, and 1.5 m. The total induced flow rate by three chimneys was up to 176% higher than that by the single one.

4 Conclusions The induced air flow rate through the chimneys with different heights, widths, and gaps have been predicted numerically. Performance effectiveness of the chimneys was

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assessed by the flow rate per volume. The results show that the flow rate per volume changes with the chimney width and approaches constant as the width – to – gap ratio is above about from 3 to 5. The flow rate per volume decreases as the gap – to – height ratio increases. When plotted as the function of the chimney volume, the flow rate per volume is higher for lower gap – to – height ratio of a given chimney height, and lower height for a given gap – to – height ratio. The thermal efficiency is also higher for lower gap – to – height ratio. From the results, it is proposed that to maximize the induced flow rate, with an available architectural space of a building, arrangement of many small chimneys with lower height and gap – to – height ratio is preferred over a big chimney which occupies the whole space. Acknowledgement. We appreciate Mr. N. V. Tung of CARE Research Group at Van Lang University for preparing the sketch in Figure 1.

References 1. Awbi, H.: Ventilation of Buildings, 2nd edn. Spon Press, London (2003) 2. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 3. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 4. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 5. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 6. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 7. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 8. Nguyen, Y.Q., Wells, J.C.: Effects of wall proximity on the airflow in a vertical solar chimney for natural ventilation of dwellings. J. Build. Phys. 44(3), 225–250 (2020) 9. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020) 10. Hirunlabh, J., Wachirapuwadon, S., Pratinthong, N., Khedari, J.: New configurations of a roof solar collector maximizing natural ventilation. Build. Environ. 36(3), 383–391 (2001) 11. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 12. Shi, L., Cheng, X., Zhang, L., Li, Z., Zhang, G., Huang, D., Tu, J.: Interaction effect of room opening and air inlet on solar chimney performance. Appl. Therm. Eng. 159, 113877 (2019) 13. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018)

Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface Y Quoc Nguyen(B) and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {y.nq,trieu.hn}@vlu.edu.vn

Abstract. Maximizing the utilization of renewable energy is one of the important points for designing sustainable buildings. Among the natural energy resources applied in buildings, solar radiation can be harnessed with solar chimneys. These devices absorb solar radiation for heating air in an enclosed channel. The thermal effects associated with the heated air can induce an air flow which can be used for ventilation, heating, or cooling of the connected buildings. This method can help to reduce the energy consumption of a building significantly. As solar chimneys have been attracting a number of studies in the literature, research interests in this topic have been focusing on enhancing the ventilation performance of typical solar chimneys by testing different shapes of the absorber surface. In this study, a novel type of a vertical solar chimney with a stepped absorber surface, unlike a straight one in typical chimneys, was studied with a numerical model. The air flow and heat transfer inside the air channel were computed with a CFD (Computational Fluid Dynamics) model. Performance of the chimney in terms of the induced air flow rate and thermal efficiency through the chimney, and the Nusselt number inside the air channel was investigated under different dimensions of the step and at different heat fluxes. The results show that the step strongly disturbed the distribution of the Nusselt number on the absorber surface and enhanced the induced air flow rate up to 11%, the air temperature rise through the chimney, and the thermal efficiency of the air flow up to 225% compared to those of a typical solar chimney. Therefore, the effectiveness of the proposed stepped absorber surface has been seen. Keywords: Computational fluid dynamics · Solar chimney · Natural ventilation · Stepped surface

1 Introduction According to the World Green Building Council [1], one of the main features of a green building is to maximize the usage of renewable energy resources to reduce negative impacts on the natural environment. Among the renewable energy resources available for buildings, solar energy has many applications, such as heating water, lighting, and particularly for ventilation with solar chimneys. Solar chimneys are referred to separate systems or structures of a building which can develop air flows based on solar radiation. A solar chimney consists of a confined channel © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 641–652, 2022. https://doi.org/10.1007/978-981-16-3239-6_49

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whose surfaces absorb solar radiation to heat the air in the channel. As the air is heated, thermal effects cause the internal pressure gradient to deviate from the ambient one which induces an air flow through the openings of the channel. With suitable arrangement of the openings of the air channel on the buildings, the induced air flow can be used for different purposes, such as heating the building, cooling the building facades, or ventilating the building naturally [2, 3]. For ventilation purpose, performance of a solar chimney is evaluated mainly by the speed of the air flow through the channel or the induced air flow rate [3]. As the main driving force of the air flow is buoyancy, the induced air speed is proportional to the heat supplied to the air channel, or the solar radiation flux that the air channel absorbs [3]. The induced air flow rate is reported to increase with the chimney height and the gap of the air channel [4, 5]. In addition, nearby walls also strongly influence performance of solar chimneys [6]. Several attempts have been made in the literature to improve ventilation performance of solar chimneys. Hirunlabh et al. [7] examined several configurations of a solar chimney with the same height to maximize the induced flow rate. They concluded that using several isolated chimneys offers more flow rate than a single one did. The flow rate is enhanced by special designs of the inlet of the air channel [8] or the absorber surface [9, 10]. Al-Kayiem et al. [8] reported that for a roof – top solar chimney, a vertical inlet of the air channel offers more flow rate up to 84% compared to that of a horizontal one. Higher flow rates are also achieved with a perforated absorber surface [9] compared to those of a smooth surface. Chorin et al. [10] examined flow field and heat transfer in a closed cavity with an obstacle on the heated surface. They showed that the heat transfer rate is strongly enhanced around the obstacle. However, as the cavity in their study was closed, no induced flow rate was reported. As an attempt to increase the heat transfer rate on the absorber surface, hence the induced flow rate through the channel, we examined a solar chimney with a stepped absorber surface in this study by numerical simulations. The problem is described in Sect. 2 together with the numerical model. Section 3 presents the results and discussions. Finally, the main findings are summarized in the section Conclusions.

2 Description of the Problem and Numerical Modelling A typical solar chimney for ventilation of a building is sketched in Fig. 1a. The chimney is integrated into a building wall. The air channel is confined between the building wall and a cover plate which absorbs solar radiation. With proper thermal insulation, the building wall can be adiabatic. The absorbed heat is transferred into the air in the channel. Heated air rises in the channel and escapes through the top of the channel. Air is sucked into the channel from the room through the opening at the bottom of the channel. Accordingly, the room is then ventilated. To study air flow and heat transfer, a two – dimensional model of the typical solar chimney is employed, as also presented in Fig. 1a. The modeled chimney is also confined by the cover and the building walls. The opening on the building wall is represented by a horizontal inlet. The absorbed heat is modeled as a heat source on the absorber wall. This type of model is similar to those in previous studies [11–13].

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The solar chimney in Fig. 1a has a smooth absorber surface while in Fig. 1b, the absorber surface has a step on the upper part. The main objectives of the step are to enhance heat transfer on the absorber surface and to suppress flow reversal at the chimney outlet. Khanal and Lei [14] reported that as the gap – to – height ratio of a chimney exceeds a certain value, a flow reversal exists at the chimney outlet and obstructs the increasing rate of the flow rate. They proposed an inclined wall to eliminate the flow reversal. Similar effects can also be expected with the step shown in Fig. 1b. 2.1 Numerical Model To assess ventilation performance of the solar chimney in Fig. 1, Computational Fluid Dynamics (CFD) technique was employed. It was assumed that the air flow and heat transfer inside the chimney are steady. Accordingly, time – averaged quantities of the flow field can be obtained with the Reynolds Averaged Navier – Stokes equations. Other assumptions of the governing equations are as follows. • The air flow is incompressible, as the average air speed in the air channel is typically quite low, which was about 1.0 m/s [4, 5, 17]. • The fluid properties follow the Boussinesq approximation [11, 12]. • Convective heat transfer dominates in the air channel. Radiative heat transfer was not modeled [12]. • Heat loss to the ambient through conduction of the channel walls was also not modeled. Besides, Chen et al. [4] reported that the air flow in typical solar chimneys with the heights of about from 1.0 m to 3.0 m was turbulent. Therefore, a turbulence model is also required. The governing equations for conservations of mass, momentum, and energy are described by Eq. (1) to (3), using the Einstein notation. ∂uj =0 ∂x

(1)

Fig. 1. Schematics of a typical solar chimney with a plane wall absorbing solar radiation (a) and the proposed chimney with a stepped absorber surface (b).

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ρuj

  ∂ui ∂ui ∂p ∂   μ =− − ρfi β(T − T0 ) + − ρui uj ∂xj ∂xi ∂xj ∂xj   μ ∂T ∂T ∂  ρuj = − ρT  uj ∂xj ∂xj Pr ∂xj

(2) (3)

In the above equations, u, p, and T indicate velocity vector, pressure, and temperature, respectively.  and − stand for the fluctuating and time – averaged components, respectively. ρ and μ are respectively the fluid density and dynamic viscosity. f is the gravitational acceleration. Pr is the Prandtl number.   In Eqs. (2) and (3), a turbulence model is required for the Reynolds tress − ρui uj 

and the turbulent heat flux − ρT  uj . Previous studies of solar chimneys have employed different turbulence models, such as the standard k – ε model [2, 15], standard k – ω model [16], and low Reynolds number k–ω model [11]. In this study, we tested the CFD model for the solar chimney in the experiment by Yilmaz and Fraser [17]. It is found that the low Reynolds number k–ω model offered the best agreement between the predicted and measured temperature distribution across the chimney gap, as seen in Fig. 3a. Therefore, it is adopted in this study. The computational domain is displayed in Fig. 2a which covers both the solar chimney and the ambient air. As suggested by Gan [12], the domain was extended to include the space surrounding the inlet and outlet of the chimney to allow the air flow adapt to local flow dynamics at those locations. For this chimney, we adopted the minimum extensions of ten times and five times of the air gap above and to the sides of the chimney, respectively. It was also found that including an extension at the inlet had negligible effects on the simulation results. Therefore, for saving computational cost, the domain was not extended at the inlet. The computational mesh is presented in Fig. 2b. Non-uniform rectangular mesh was used. The mesh density was higher near the solid surfaces where there were high gradients of the flow variables. To find the most suitable mesh density and pattern, the number of grid cells and different grow rates of the mesh size from the solid surface were tested for the solar chimney in the experiment by Yilmaz and Fraiser [17]. The induced flow rate and temperature rise through the chimney was plotted against the nondimensional distance of the first grid point from the solid surfaces, or y+ . The results show that for y+ ≤ 1.0, and the according mesh of more than 80 × 200 elements in the air channel, the results were mesh – independent. This agrees with the findings by Zamora and Kaiser [11]. For the boundary conditions, the atmospheric pressure was assigned at the open boundaries of the domain, as indicated in Fig. 2a. On the walls, no – slip conditions were applied for the velocity and adiabatic conditions for the heat transfer, except for the heated wall where a uniform heat flux was distributed. Finite Volume Method was used for discretizing the governing equations. For the coupling between the continuity and momentum equations, Eq. (1) and (2), the SIMPLE method was employed. In addition, to interpolate the pressure value from the cell center to the cell faces on staggered mesh, the PRESTO! Method was applied. The above numerical schemes were conducted with the ANSYS Fluent CFD code (Academic version 2019R3).

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Fig. 2. Schematics of the computational domain (a) and mesh (b). H and G are the height and the gap of the air channel, respectively. L and g are respectively the length from the outlet and the height of the step.

2.2 Model Validation The CFD model was validated against the experiment by Yilmaz and Fraser [17] and the CFD model by Zhang and Yang [15]. The experiment was conducted on a solar chimney whose height, gap, and width were 3.0 m, 0.1m, and 1.0 m, respectively. A uniform temperature of 100 °C was applied on one side of the air channel. Distributions of the flow velocity and temperature were measured across the channel gap at half the channel height and at the outlet, respectively. The CFD simulation by Zhang and Yang [15] was also conducted with a two – dimensional RANS model and the standard k – ε model.

(a) Temperature profile.

(b) Velocity profile.

Fig. 3. Comparison of the distributions of temperature and velocity profiles across the channel gap at H/2 among our CFD results, measured data by by Yilmaz and Fraser [17], and the CFD results by Zhang and Yang [15].

Comparisons of the temperature and velocity profiles are presented in Fig. 3a and 3b, respectively. Good agreement between our results and those by Zhang and Yang

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[15] for both profiles can be seen. Compared to the measured data, our results are closer than those by Zhang and Yang [15] for the temperature distribution. For the velocity distribution, both of the CFD results slightly underestimated the velocity near the heated surface and overestimated near the opposite wall. However, the difference of the induced flow rate between our data and the measured one was only less than 6.7%. Therefore, our CFD model is acceptably accurate.

3 Results and Discussions The validated CFD model was used to simulate the flow field and heat transfer inside the solar chimney sketched in Fig. 1 with and without the step on the absorber surface. In all cases, the chimney height H and the channel gap G were kept to 2.0 m and 0.2 m, respectively. The inlet height was also kept to 2G, as shown in Fig. 2a. Bassiouny and Koura [18] reported that effects of the inlet height on performance of a solar chimney were negligible when it was equal to or more than twice the channel gap. On the heated wall of the air channel, a heat flux of 600 W/m2 was assigned. This value is around the average solar radiation used in previous studies [4, 12]. The Rayleigh number based on the channel height and the heat flux is 4.1013 . Therefore, the flow in the air channel is expected to be turbulent [4]. For the step, the length L and the height g changed to yield the ratios of g/G and L/H up to 0.5. Effects of the step on the velocity and temperature fields in the air channel, the distributions and the averaged value of the Nusselt number on the heated wall, the induced flow rate, and the thermal efficiency were obtained and discussed in this section. 3.1 Effects of the Step on the Flow Field and Heat Transfer Figure 4 shows examples of the velocity and flow fields near the outlet of the solar chimney without a step on the heated surface (base case) and with the step with two different step heights. Without a step, the velocity and thermal boundary layers developed along the heated surface. At the outlet, both layers expanded to only about half of the channel gap. There existed a reverse flow at the outlet of the channel; hence the flow area was significantly reduced. The reverse flow also brought air from the ambient at lower temperature into the air channel and resulted in lower temperature distribution on the region of the reverse flow. With the step, in both cases of the step length, the reverse flow was successfully suppressed. The flow was guided toward the unheated wall by the step. Accordingly, the thermal layer almost occupied the whole channel gap. However, with the shorter step length (L/H = 0.025), a new reverse flow appeared near the outlet on the heated side. The flow area was reduced even more than that of the base case. With the longer step length (L/H = 0.1), the flow occupied the whole channel gap at the outlet. Distributions of the Nusselt number on the heated surface of the cases in Fig. 4 are plotted in Fig. 5. The local Nusselt number is defined based on the channel gap as in Eq. (4). Nu =

qw G (Tw − Ta )λ

(4)

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(a) Flow field.

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(b) Temperature distribution.

Fig. 4. Velocity and temperature distributions near the outlet of the air channel with the heated plane sureface (a), the stepped surface with L = 0.1 m (L/H = 0.025) (b) and L = 0.2 m (L/H = 0.1) (c), with g = 0.05 m (g/G = 0.25)

Fig. 5. Distributions of the Nusselt number on the heated surface without and with the step with L = 0.1 m and L = 0.2 m (g = 0.05 m).

Where qw = 600 W/m2 and Tw were the heat flux and temperature of the wall, respectively; Ta is the air temperature at the inlet of the air channel (293 K); and λ is the air thermal conductivity. It is seen in Fig. 5 that, compared to the base case, the presence of the step caused a significant drop of the local Nusselt number in front of the step. This may be because of a small flow separation zone at that location, which is not visible in Fig. 5a due to the scale of the plot. Higher wall temperature in that separation zone, as seen in Fig. 4b, resulted in lower Nusselt number (Eq. (4)). At further downstream of the step, the local Nusselt number gradually increased and achieved a peak in both cases of the step length. Chorin et al. [10] also reported an increase of the Nusselt number downstream of the circular cylinder in their experiment. However, a trend of variation similar to those

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with the step in Fig. 5 was not reported. It may be because the number of measurement points along the heated surface close to the obstacle in their experiment was not high enough to capture such a trend. 3.2 Effects of the Step Length and Height From the local Nusselt number distributions, such as those in Fig. 5, the averaged Nusselt number is obtained by Eq. (5).

Fig. 6. Normalized averaged Nusselt number as functions of the step length and height.

H Nuav =

(Nu dl)/H

(5)

0

In Fig. 6, the averaged Nusselt numbers for three cases of the step height g are normalized by the averaged Nusselt number of the base case and plotted as functions of the step length. It is seen that the ratio of Nuave /Nubase was less than 1.0 and varied slightly about 92.0% for all three cases of g and at any value of L/H. Therefore, with a step, the average Nusselt number on the heated surface decreased compared to that of the base case. Effects of the step height and length on the Nusselt number was also negligible. The induced flow rate was also normalized by that of the base case and plotted in Fig. 7. The flow rate changed with both the step height and length. At L/H < 0.3, compared to the flow rate of the base case, the case with g = 0.1 m always offered lower flow rates while the case with g = 0.025 m resulted in higher values. The maximum reduction of the flow rate for g = 0.1 m was up to 25.0% while the maximum increase for g = 0.025 m was about 11.0%. For the case with the intermediate step height, g = 0.05 m, the flow rate first decreased and then increased to a peak of about 9.0% higher than that of the base case.

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Fig. 7. Normalized flow rate as functions of the step length and height.

As L/H exceeded about 0.3, the induced flow rate was almost constant with the step length. The flow rate was about 5.0% higher than that of the base case only for g = 0.05 m. For two other step heights, the flow rates were similar to that of the base case. As observed in Fig. 6, the Nusselt number always slightly decreased with the presence of the step compared to that of the base case. However, the induced flow rate was enhanced in some cases. Therefore, the increase of the flow rate was not due to an increase of the heat transfer on the heated surface, which was the case reported by Lei et al. [9]. They showed that the increase of the flow rate was mainly due to better heat transfer with a perforated absorber surface. In this study, the increase of the flow rate was possibly due to the suppression of the reverse flow at the outlet. This point can be seen in Fig. 4a and 4b for g = 0.05 m and L/H = 0.1 and in Fig. 7 where with that step height and length, the ratio of Q/Qbase is equal to 1.09. This effect was similar to that reported by Khanal and Lei [13]. They suppressed the reversed flow at the outlet of their chimney by using a wall inclined toward the outlet of the air channel and obtained an enhancement of the induced flow rate. The reduction of the flow rate for g = 0.05 m and L/H < 0.1, and g = 0.1 m and L/H < 0.3 is expected due to severe contraction of the flow area. The steps in such cases served as obstacles to the flow; hence increased the flow resistance. Accordingly, the flow rate was less compared to that of the base case. The temperature rise (Eq. (6)) and the thermal efficiency (Eq. (7)) were compared to those of the base case and plotted in Fig. 8. T = To − Ti

(6)

In Eq. (6), To and Ti are the temperatures at the outlet and the inlet of the air channel, respectively.   (7) η = Qcp T /Ii In Eq. (7), Q is the induced flow rate; cp is the specific thermal capacity of air; and I i is the total heat delivered from the heated surface.

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(a) Temperature rise.

(b) Thermal efficiency.

Fig. 8. Temperature rise of the air flow through the chimney and thermal efficiency as functions of the step length and height.

It is seen in Fig. 8 that the temperature rise and the thermal efficiency were always higher than those of the base case. T and η are seen to increase with the step height. The increase of the temperature rise was up to 2.4 times of that of the base case while the relative increase of the thermal efficiency was up to 2.25. As the L/H ratio increased, the ratios of T/Tbase and η/ηbase approached constants, which were always above 1.0. As seen in Fig. 4b, the enhancement of the temperature rise was due to the elimination of the reverse flow at the outlet of the channel with the step. As the flow rate did not always increase or increased only up to 11.0% with the presence of the step, the highly enhanced thermal efficiency in Fig. 8 is expected mainly due to the increase of the temperature rise, according to Eq. (7). This conclusion is further confirmed as the trends of variations of the data in Fig. 8a and 8b are very similar to each other.

(a) Induced flow rate.

(b) Temperature rise and thermal efficiency.

Fig. 9. Effects of the step height on the flow rate, temperature rise, and thermal efficiency for two different step lengths: L = 0.05 m (L/H = 0.025) and L = 0.2 m (L/H = 0.1).

To further examine effects of the step height, two cases of L/H with the peak flow rates in Fig. 7, i.e. L/H = 0.025 and 0.1, were selected to test with different step heights. The results are presented in Fig. 9 where the ratio of g/G changed from 0.05 to 0.5 (g

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= 0.01 m to 0.1 m). The Nuave /Nubase ratio slightly varied from about 0.93 to 0.89 while the ratio of Q/Qbase changed in the ranges of 0.95–1.08 and 0.75–1.1 for L = 0.2 m and 0.05 m, respectively. The peak flow rates were similar to those in Fig. 7. The ratio of g/G corresponding to the peak flow rates were 0.125 and 0.5 for L = 0.05 m and 0.2 m, respectively, as seen in Fig. 9a. In Fig. 9b, both the temperature rise and the thermal efficiency increased with the step height. Therefore, the trends of T /T base and η/ηbase are consistent with those in Fig. 8. For L = 0.05 m, the peak η/ηbase of 1.85 is seen at g/G = 0.3125. For L = 0.2 m, the η/ηbase seems to grow steadily. Similar points are also observed for the T/Tbase ratio in Fig. 9b.

4 Conclusions Effects of a step on the absorber surface of a solar chimney were investigated numerically. While the chimney height, air gap, and the heat flux were fixed, the step height g and length L changed. Two main effects are observed from the results. Firstly, the step guided the induced air flow toward the unheated wall; hence suppressed the reverse flow at the outlet. This effect is seen with low step height. Secondly, the higher step height also blocked the air flow and increased the flow resistance. While the presence of the step slightly reduced the averaged Nusselt number about 8.0% in average, the induced flow rate, and particularly the temperature rise and the thermal efficiency were strongly influenced. Compared to the base case, higher flow rate was achieved with g = 0.05 m and up to 11.0%, but lower flow rate is seen for g = 0.1 m and up to 25.0%. The highest Q/Qbase of about 1.1 was achieved with g/G = 0.0125 and L = 0.05 m. The temperature rise and thermal efficiency were always higher than those of the base case up to 2.4 and 2.25 times, respectively. For possible extensions of this study, three – dimensional steps or other types of the obstacles may be examined with three – dimensional CFD models. For verifications of the simulations, experiments can be conducted.

References 1. World Green Council: What is green building? (2020). https://www.worldgbc.org/whatgreen-building. Accessed 21 Sep 2020 2. Gan, G.: Simulation of buoyancy-induced flow in open cavities for natural ventilation. Energy Build. 38(5), 410–420 (2006) 3. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 4. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 5. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 6. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020)

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7. Hirunlabh, J., Wachirapuwadon, S., Pratinthong, N., Khedari, J.: New configurations of a roof solar collector maximizing natural ventilation. Build. Environ. 36(3), 383–391 (2001) 8. Al-Kayiem, H.H., Sreejaya, K.V., Chikere, A.O.: Experimental and numerical analysis of the influence of inlet configuration on the performance of a roof top solar chimney. Energy Build. 159, 89–98 (2018) 9. Lei, Y., Zhang, Y., Wang, F., Wang, X.: Enhancement of natural ventilation of a novel roof solar chimney with perforated absorber plate for building energy conservation. Appl. Therm. Eng. 107, 653–661 (2016) 10. Chorin, P., Moreau, F., Saury, D.: Heat transfer modification of a natural convection flow in a differentially heated cavity by means of a localized obstacle. Int. J. Therm. Sci. 151, 106279 (2020) 11. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 12. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 13. Khanal, R., Lei, C.: A numerical investigation of buoyancy induced turbulent air flow in an inclined passive wall solar chimney for natural ventilation. Energy Build. 93, 217–226 (2015) 14. Khanal, R., Lei, C.: An experimental investigation of an inclined passive wall solar chimney for natural ventilation. Sol. Energy 107, 461–474 (2014) 15. Zhang, T., Yang, H.: Flow and heat transfer characteristics of natural convection in vertical air channels of double-skin solar façades. Appl. Energy 242, 107–120 (2019) 16. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018) 17. Yilmaz, T., Fraser, S.M.: Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating. Int. J. Heat Mass Transf. 50(13–14), 2612–2623 (2007) 18. Bassiouny, R., Koura, N.S.A.: An analytical and numerical study of solar chimney use for room natural ventilation. Energy Build. 40(5), 865–873 (2008)

The Effect of Ground Surface Geometry on the Wing Lift Coefficient Dinh Khoi Tran(B) and Anh Tuan Nguyen Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Vietnam [email protected]

Abstract. The studies of wing aerodynamics in ground effect generally consider the ground surface to be either absolutely flat or wavy in a form of a sinusoidal function. However, in practice, ground surfaces may have complex geometry, which affects the aerodynamic characteristics of wings flying over them. The aim of the paper is to analyze the influence of some basic ground geometry parameters on the lift force coefficient of a wing model. The mirror-image method is integrated into the unsteady panel code to solve the ground effect problem with the complex geometry. The program of the panel method, which was developed based on the potential-flow theory, is written in FORTRAN with the use of parallel computation techniques to reduce the run time. The calculation results show that the variation of the ground surface geometry may have profound effects on the unsteady lift coefficient properties of the wing. The phase, the shape and the amplitude of the lift coefficient variation may be altered significantly when we change the geometry of the ground surface. Moreover, the effect of the ground geometry in the presence of horizontal wind is also analyzed. This paper provides close insight into these phenomena with physical explanations, which have not been addressed by other researchers. Keywords: Ground effect · Unsteady panel method · Unsteady aerodynamics

1 Introduction Ground effect is of great importance for vehicles flying at low altitude [1]. This effect needs considering while studying wing-in-ground (WIG) craft or airplanes during takeoff and landing. In many previous studies of ground effect, the surface is normally considered to be flat [2]. At the same time, many researchers have investigated the ground effects of sinusoidal surfaces [3, 4]. In reality, ground surfaces can be much more complex; and therefore, their effect may change significantly as a function of the geometry. Tang and Wang [5] have shown that a wind field can be strongly affected by complex terrain. However, these researchers did not indicate to what degree the flight characteristics of an airplane may be altered while flying over terrain. There is the fact that the impact of hilly ground on the aerodynamics of an airplane may be insignificant. Nevertheless, this phenomenon could be magnified due to the presence of horizontal wind. Liu et al. [6] have obtained the flow around a 3D hill through large eddy simulations in the case © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 653–662, 2022. https://doi.org/10.1007/978-981-16-3239-6_50

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of horizontal wind. It should be noted that, this kind of Computational Fluid Dynamics (CFD)-based method requires large computer resources. Hence, the simulation for the combined model of a hilly surface and a flying vehicle is challenging. In our previous paper [7], we have introduced a novel model based on the unsteady panel method and the mirror-image technique for the problem of wing and curved ground surface interaction. The validity of the model was analyzed and confirmed for various cases. In here, the model is applied to answer the question to what extend the ground surface geometry influence the wing lift coefficient, which is regarded as an important indication of wing aerodynamic characteristics.

2 Panel Method and Calculation Model 2.1 Panel Method According to [8], the following assumptions are made during the development of the unsteady panel method: (1) the flow is attached on the surfaces of solid bodies; (2) viscous effects are negligible; (3) the flow is irrotational, which means there is no vorticity everywhere in the computational domain except for the surfaces of solid bodies and wake sheets. The panel method used in this paper is similar to that in [9] with the distribution of sources and doublets. More details of the present panel method can be found in [7]. 2.2 Model Calculation To study the effects of the geometry of the ground surface on the wing lift coefficient in the ground effect, the authors assume that the wing flies over an infinite flat surface, on which there is a curved surface located within ABCD. Then we can model the wing and curved surface located within ABCD using sources and doublets. The whole model will be mirrored symmetrically over the flat surface EFGH. The mathematical basis of this model is depicted in Fig. 1. The sources and doublets of the wings are taken in mirror symmetry in the usual way. The mirrored values of the sources and the doublets on the curved surface are σmirror = σ

Fig. 1. Modeling of curved limit surface

(1)

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μmirror = −μ

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(2)

As explained in [7], the sources on the flat plane ABCD have the zero strength according to the Neumann boundary condition. At the same time, the doublets and those on the mirrored surface A’B’C’D’ have the same strength but in the opposite direction. Therefore, when the surface ABCD is in proximity to its mirrored counterpart, singularities on these two surfaces will cancel out each other. This approach can eliminate computational errors due to the finite size of the ground surface, which happens in [10]. Figure 2 shows the wing model flying at a speed V over a stationary curved surface without wind.

Fig. 2. Computational model without wind

In the presence of horizontal wind, the calculation model will include the wing flying over a stationary curved surface and the incoming flow at a speed Vwind . In this case, the curved geometry of the ground surface will alter the velocity direction of the fluid around it; therefore affect the aerodynamic properties of the wing. 2.3 Validations To verify, the authors build a model with sinusoidal limit surface corresponding to the conditions as [3, 4]. The results of calculation and comparison are presented in detail in [7]. They have confirmed the validity of the model and computational program.

3 Numerical Results 3.1 The Influence of the Ground Surface Geometry Shape 3.1.1 Effects of Ground Curved in the Flight Direction Here, we study a rectangular wing with an aspect ratio AR of 4, the length of chord c = 0,1(m), the flight velocity V = 20(m/s), the curve amplitude a = 0.5c, the wavelength λ = 5c. The elevation of the ground surface is defined as for various shapes [11]   x  a tanh C sin 2π (3) z(x) = tanh C λ x is the longitudinal coordinate. As C approaches 0, function z(x) becomes sinusoidal; and as C increases, the function approaches the step form.

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Fig. 3. Dimension curved ground

Fig. 4. The wing model and its wake patterns over several types of ground surface geometry (a); and lift coefficient corresponding to various ground surface geometry parameters (b).

The ground surface has dimensions of 5c × 6c (Fig. 3). Results obtained with different C values are shown in Fig. 4. The calculation results show that the variation of the ground surface geometry may have profound effects on the unsteady lift coefficient properties of the wing. The phase, the shape and the amplitude of the lift coefficient variation may be altered significantly when we change the geometry of the ground surface. In order to obtain the details of these

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effects, the result is presented in the domain of the nondimensional spatial frequency F through the Fourier transform. As shown in Fig. 4, the variation of the lift coefficient can be given as a function of the nondimensional travelled distance, which is obtained through a division by the chord length. Therefore, the spatial frequency is normalized by 1/c. Figure 5 exhibits the amplitude spectrum of the variation shown in Fig. 4 for various values of C. It is seen that, all of the spectrums have a peak at a frequency of 0.2 that is considered to be a fundamental frequency. The magnitude of this peak increases with C. Similarly, the second peak is observed at a frequency of 0.6. However, the trend at this peak is not as apparent as the first one. When we compare the amplitude spectrum of the lift coefficient that that of the ground elevation (Fig. 5), we can find consistency at the first peak (the major one); the consistency at the second peak is not so clear. However, it is possible to state that the lift coefficient follows the elevation of the ground in terms of amplitude (Fig. 6). A similar analysis is conducted for the phase of the variation. While Fig. 7 shows the phase differences between cases at a frequency of 0.2, those are not observed for the ground elevation (Fig. 8). Hence, these phase differences cannot come from the ground elevation, rather they are due to the unsteadiness of the flow. When C increases from 0 to 3, the lift coefficient variation is delayed by 5° in phase. At a large C, the slope of the ground is more significant, and so is the unsteadiness. According to Wrights and Cooper [12], the unsteadiness causes a delay in the lift force development, which is seen in Fig. 7.

Fig. 5. Amplitude spectrum of the lift coefficient in the frequency domain

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Fig. 6. Amplitude spectrum of the ground elevation in the frequency domain

Fig. 7. Phase of the lift coefficient in the frequency domain

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Fig. 8. Phase of the ground elevation in the frequency domain

3.1.2 Effects of Laterally Curved Ground Figure 9 shows the definitions of longitudinally and laterally curved ground problems. In this subsection, we study sinusoidal curved surfaces with the number of waves N varying from 1 to 5 (Fig. 10).

Fig. 9. Longitudinally (a) and laterally (b) curved ground

Fig. 10. The wake patterns with different numbers of waves.

The wing and surface geometry parameters are similar to those above. Wing velocity of 20 m/s. The calculation results are obtained and shown in Fig. 11.

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Fig. 11. Change of wing lift coefficient when the ground surface has different horizontal number of waves

Fig. 12. Distribution of the lift coefficients along wing span with different values of N

There are significant differences between the cases in Fig. 11. It is found that in general, the lift coefficient increases with the number of waves. This trend can be attributed to the air trapped between waves, which plays a roll of cushions to augment the lift force of the wing. Figure 12 provides clear evident of this trend. 3.2 The Influence of Wind To study the effect of wind on the wing lift coefficient in the ground effect, we use the above wing model flying over a longitudinal curved surface. Several wind speed values are included in the computation and the results are presented in Fig. 13. The calculation results show that wind has a significant influence on the wing lift coefficient when it is combined with curved ground. For a positive wind value, a drop

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Fig. 13. The variation of the lift coefficient on the wings when flying over a limited surface with different wind speeds

in the lift coefficient when the wing above the leeward region is followed by an increase trend above the windward side of the wave. To provide more details of this phenomenon, the distributions of the pressure coefficient on the wing at the entrance (x = 0) and at the exist (x = 5c) of the wave are plotted and compared (Figs. 14 and 15).

Fig. 14. Pressure distribution on the wing when wind speed = −2 m/s

Fig. 15. Pressure distribution on the wing when wind speed = 2 m/s

4 Conclusions In this paper, the mirror image method is combined with the unsteady panel method to efficiently simulate the aerodynamic interaction between the wing and the curved ground surface. Furthermore, this method allows to study the effect of wind on the wing lift coefficient when it flies over a curved ground surface. The calculation results show the influence of the ground surface geometry on the change of the lift force coefficient. When analyzing the result in the frequency domain, it is observed that the amplitude spectrum of the lift coefficient follows that of the ground elevation. However, due to the unsteadiness of the flow, there is phase differences between cases. In general, the unsteadiness causes a delay in the phase of the lift force

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development. The effect of laterally curved ground is also studied in this paper. The results show that the lift coefficient tends to increase with the number of lateral waves. When there is a combination of curved ground and wind effects, the change in the lift coefficient of the wing becomes considerable. Further research is needed for this problem.

References 1. Federal Aviation Administration: Airplane Flying Handbook, p. 348. U.S. Dep. Transp. (2016) 2. Qu, Q., Jia, X., Wang, W., Liu, P., Agarwal, R.K.: Numerical simulation of the flowfield of an airfoil in dynamic ground effect. J. Aircr. 51(5), 1659–1662 (2014) 3. Im, Y.H., Chang, K.S.: Unsteady aerodynamics of a WIG airfoil flying over a wavy wall. In: 38th Aerospace Sciences Meeting and Exhibit, vol. 37, no. 4, pp. 690–696 (2000) 4. Zhi, H., Xiao, T., Chen, J., Wu, B., Tong, M., Zhu, Z.: Numerical analysis of aerodynamics of a NACA4412 airfoil above wavy water surface, no. June, pp. 1–29 (2019) 5. Tang, C., Wang, L.: Numerical simulation of wind field over complex terrain and its application in Jiuzhaihuanglong airport. Procedia Eng. 99, 891–897 (2015) 6. Liu, Z., Hu, Y., Fan, Y., Wang, W., Zhou, Q.: Turbulent flow fields over a 3D hill covered by vegetation canopy through large eddy simulations. Energies 12(19), 1–18 (2019) 7. Tran, D.K., Nguyen, A.T.: The application of the panel method to predict the unsteady aerodynamic coefficients of a wing model flying over bumpy ground. In: ICOMMA (2020) 8. Katz, J., Plotkin, A.: Low-Speed Aerodynamics From Wing Theory to Panel Methods, 2nd edn. Cambridge University Press, New York (2001) 9. Nguyen, A.T., Han, J.S., Han, J.H.: Effect of body aerodynamics on the dynamic flight stability of the hawkmoth Manduca sexta. Bioinspiration Biomimetics 12(1), 016007 (2017) - nh d˘ -ac tru,ng khí dô - ng lu,c cua cánh quay tru,c th˘ang xét d-ên ´ su., 10. Pham, T.D.: Nghiên cu´,u xác di . . . . . tu,o,ng tác vo´,i thân và m˘a.t gio´,i ha.n. Le Quy Don Technical University (2020) 11. Berman, G.J., Wang, Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168 (2007) 12. Wright, J.R., Cooper, J.E.: Introduction to Aircraft Aeroelasticity and Loads, 2nd edn. Wiley, Hoboken (2015) ij

Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach Phan Thanh Long(B) Faculty of Transportation Mechanical Engineering, The University of Danang – University of Science and Technology, Danang, Vietnam [email protected]

Abstract. Closed-loop small wind tunnels, with many advantages, play an essential role in experimental research on aerodynamics in many fields. In this study, to design a small, low-speed, closed-loop wind tunnel for researching and teaching, the author carried out a systematic investigation of flow in the wind tunnel using Computational Fluid Dynamics (CFD). The required objective is a uniform flow in the test section of the wind tunnel and a low-pressure loss. The effect of guide vane configurations in the big corners on the flow quality in the test section was evaluated. This analysis showed the flow in the test section was more affected by the presence of upstream guide vanes. Finally, the simulation results demonstrated that uniformity of flow in test section is well obtained in the case of full configuration of wind tunnel. Keywords: Small wind tunnel · Aerodynamics · CFD

1 Introduction Experimental research on low-speed aerodynamics plays an important role in many fields such as studying the basic problems of fluid mechanics, automotive design, wind turbines, and building design. This research is usually carried out in a small wind tunnel, combined with measuring equipment to obtain data on drag force, lift force, and moment of the airflow acting on an object. In the case of the use of complex measuring equipment, these experiments can also reveal the velocity field and structure of flow in detail [1]. Although with the development of numerical simulation methods, especially the CFD (Computational Fluid Dynamics) method and the computational capacity of the computer, several studies involving low-speed aerodynamics can carry out using CFD simulation to achieve the required parameters [2]. However, the experimental studies in the wind tunnel still play an important role, helping to obtain detailed results and make final decisions in the design process [1]. The design of a wind tunnel is relatively complicated, while the use of CFD to support wind tunnel design is still limited. Several studies on evaluating the wind tunnel system using numerical methods have been performed. Moonen et al. [3] established a methodology of modeling the airflow in a closed-loop wind tunnel, using the CFD method and a standard k-ε turbulence model to determine pressure loss and flow rate in © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 663–672, 2022. https://doi.org/10.1007/978-981-16-3239-6_51

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the test section. Gordon and Imbali [4] used CFD to evaluate the flow conditions within the test section of a closed-loop wind tunnel at the University of Aberdeen. Besides, Calautit et al. [5] also performed a numerical study to investigate the effect of the guide vanes on flow quality in a small closed-loop wind tunnel at the University of Leeds. In this study, to build a small wind tunnel for research and teaching in the technical university of Vietnam, a design of a small closed-loop wind tunnel was introduced. Total pressure loss in the wind tunnel was determined from the analytical model, then used directly as a boundary condition in a numerical simulation. The quality of the flow in the wind tunnel, especially in the test section, was investigated, from which the performance of the designed wind tunnel was evaluated.

2 Wind Tunnel Design Small low-speed wind tunnels are generally divided into two types: open-loop and closed-loop. The advantages of open-loop wind tunnel are the low construction cost and the use of smoke for visualization of the flow. The disadvantages of the open-loop wind tunnel are that the flow quality, especially in the test section is not high, the required power of the fan is higher and therefore it tends to be noisy. On the other hand, the closedloop wind tunnel has the advantage of being able to create a uniform flow in the test section, the required power of the fan is lower, so less noise when operating. However, the fabrication cost of the closed-loop wind tunnel is higher than that of the open-loop wind tunnel [6]. A schematic view of the closed-loop wind tunnel is shown in Fig. 1.

Fig. 1. CAD model of a closed-loop wind tunnel 1 – Test section; 2 – Small diffuser; 3, 4 – Small corners; 5 – Fan; 6 – Big diffuser; 7, 8 – Big corners; 9 – Expansion; 10 – Setting chamber; 11 – Contraction;

The test section is the place where test models and the measuring equipment are installed. The size of the test section as well as the flow velocity determines the size of the other components and the overall dimensions of the wind tunnel. In this study, the test section has a rectangular shape, with a height/ width ratio of about 4/3 [1]. Figure 2 illustrates the geometric sizing of the test section.

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Fig. 2. Geometric sizing of the test section: (a) side view; (b) front view

In the closed-loop wind tunnel, the diffusers, with an increasing cross-section area along its length, are used to recover the static head of the flow by decreasing the flow velocity, therefore the energy loss of system reduces. The small diffuser is placed at the downstream end of test section and the big diffuser is at outlet of the fan. The small diffuser was designed with an inlet area of 0.6 × 0.32 m2 and an outlet area of 0.85 × 0.6 m2 . The length of small diffuser is 1.94 m. For the big diffuser, the sizes are 0.85 × 0.6 m2 ; 1.2 × 0.85 m2 and 2.76 m, respectively. The geometric sizing of small diffuser is shown in Fig. 3.

Fig. 3. Geometric sizing of small diffuser: (a) side view; (b) front view

The corners in closed-loop wind tunnel have an important task to turn the flow by 90° without separation and recirculation. Behind the small diffuser, the small corners have dimensions of 0.85 × 0.6 m2 in the inlet and outlet areas. This corner is integrated with 7 guide vanes to reduce the flow separation occurring it the turn (Fig. 4a). The big corner, with 8 guide vanes and 1.2 × 0.85 m2 cross-section area, is located after the second diffuser (Fig. 4b).

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Fig. 4. Geometric sizing of (a) small corner and (b) big corner

3 Numerical Method To evaluate the performance of the designed wind tunnel, the flow characteristics in the wind tunnel are considered and predicted by CFD numerical simulation method, using commercial software ANSYS Fluent [7]. This is a software used to solve fluid dynamics problems through the finite volume method. To simplify the calculation, the flow through the fan is ignored. Instead, the pressure loss in the wind tunnel is identified, and then is assigned to the intake fan boundary condition. The fan power required to maintain a steady flow in the wind tunnel is equal to the total pressure loss of flow while moving through the system. The pressure loss coefficients obtained at different components of wind tunnel are summarized in Table 1. The pressure loss of this wind tunnel was calculated at 105.22 Pa. Figure 5 shows the 3-D model of wind tunnel, in which an intake fan condition was applied to inlet surface and a zero gauge pressure was set for outlet surface. Table 1. Loss coefficients and pressure loss of wind tunnel components No.

Component

Loss coefficient k [Ref.]

Pressure loss p = 1 2 2 kρv [Pa]

1

Test section

0.023 [8]

12.25

2

Small diffuser

0.079 [8]

23.8

3

Big diffuser

0.079 [8]

4

Small corner

0.152 [8]

40.5

5

Big corner

0.158 [8]

10.52

6

Setting chamber

0.22 [1]

10.65

7

Contraction

0.008 [8]

1.54

5.96

The mesh used in this simulation is an unstructured type, in which a high mesh resolution was applied at guide vanes and setting chamber to capture exactly the flow passing

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Fig. 5. 3-D CAD model of wind tunnel: (1) Intake flow; (2) Pressure outlet

through. The number of elements was varied from 2,596,431 (coarse mesh), 4,950,075 (medium mesh) and 6,267,618 (fine mesh) respectively for grid-independence evaluation. Variation of the velocity of airflow along the centerline of the test section is observed and shown in Fig. 6. This figure shows that the difference in velocity is independent of grid elements beyond 4,950,075 elements. Medium mesh was then adopted for further study. For this mesh configuration, the average orthogonal quality and skewness were 0.717 and 0.28, respectively (Fig. 7). A approriate average value of 54.3 for y+ of wall bounded cells has been determined.

Fig. 6. Grid independence test

In the CFD method, the three-dimensional flow analysis is solved by approximating the governing equations of the flow. These equations represent the conservation of mass and conservation of momentum. For the steady, incompressible flow, the conservation of mass in a control volume is expressed in terms of the continuity equation as follows [9]:   ∂u ∂v ∂w + + =0 (1) ∂x ∂y ∂z

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Fig. 7. Unstructured mesh of 3-D wind tunnel model

where ρ is the density of air, u, v, and w are the velocity components in the x, y, and z directions, respectively. The conservation of momentum, commonly known as the Navier-Stokes equation, in a three-dimensional, steady, incompressible flow, has the following form [9]:    2  ∂u ∂u ∂p ∂ u ∂ 2u ∂ 2u ∂u + ρfx +v +w =− +μ + + (2) ρ u ∂x ∂y ∂z ∂x ∂x2 ∂y2 ∂z2     2 ∂p ∂v ∂v ∂v ∂ v ∂ 2v ∂ 2v + ρfy =− ρ u + + (3) +v +w +μ ∂x ∂y ∂z ∂y ∂x2 ∂y2 ∂z2    2  ∂w ∂w ∂p ∂ w ∂ 2w ∂ 2w ∂w + ρfz +v +w =− +μ + + (4) ρ u ∂x ∂y ∂z ∂z ∂x2 ∂y2 ∂z2 Where p is the pressure, fx , fy , and fz are the body force per unit mass in the x, y, and z directions, respectively. To solve the governing equations applied to turbulent flow, CFD method is often used in combination with the turbulence models to close the Navier-Stokes equations. In this study, the two-equation model k-ω SST is used because it has the ability to model both near-surface and far-field flow [10]. This turbulence model utilizes the transport equations based on turbulent kinetic energy (k) and specific dissipation rate (ω), similar to the standard k-ω model. In addition, in this model, a blending function is also introduced to activate the turbulence model according to the proximity to the surface. This makes the k-ω SST turbulence model suitable and highly reliable for flows with adverse pressure gradients and separation. The transport equations for turbulence model are given by:   ∂ ∂k ∂ ∂ k + Gk − Yk + Sk (ρk) + (5) (ρkui ) = ∂t ∂xi ∂xj ∂xj   ∂ ∂ ∂ω ∂ ω + Gω − Yω + Sω + Dω (ρω) + (6) (ρωui ) = ∂t ∂xi ∂xj ∂xj where  k and  ω represent the diffusivity of turbulent kinetic energy and ω, Gk and Gω symbolize the generation rate of k and ω, Yk and Yω are the dissipation rates of k and ω, respectively; Sk and Sω are user-defined source terms, respectively.

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4 Results and Discussion This study investigated two different configurations of the wind tunnel. The first configuration is a wind tunnel without guide vanes in two big corners, and the other is a full configuration of the wind tunnel. The numerical results evaluated the influence of guide vanes in the big corners on the test section flow quality. The result in Fig. 8 shows that in the absence of the guide vanes, the flow separation zones appear in two big conners and therefore, the flow in test section is unsteady. The maximum velocity that can be reached in this section is about 22 m/s.

Fig. 8. Contours of velocity magnitude for wind tunnel without guide vanes in big corners

For a more detailed investigation of the flow inside the wind tunnel in this case, the streamline of flow throughout wind tunnel is investigated as presented in Fig. 9. The results show more clearly the vortices formed in the upper big corner. Meanwhile, in the small corner at the downstream of flow, because of the presence of guide vane, the flow through this section is very uniform.

Fig. 9. Streamlines of circulating flow in the wind tunnel without guide vanes in big corners

For the second configuration, in which the guide vanes are integrated for both small and big corners, the numerical results are shown in Fig. 10 and Fig. 11. Figure 10 presents the contours of velocity in the wind tunnel, and a uniform flow is obtained in the test

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section. However, the maximum flow velocity is lower (−21 m/s) as compared to the first configuration without guide vanes in the big corners. The improvement in the test section flow quality demonstrated the influence of the guide vanes on wind tunnel design. The disadvantage of the presence of the guide vanes is an increase in energy loss, therefore the maximum flow velocity slightly decreases.

Fig. 10. Contours of velocity magnitude for wind tunnel with guide vanes in corners

The streamlines of flow in wind tunnel are presented in Fig. 11, in which the vortices only appear at a few points in big corners, and the flow is steady throughout the wind tunnel.

Fig. 11. Streamlines of circulating flow in the wind tunnel with guide vanes in corners

In addition, the velocity flow distributions at inlet, middle, and outlet areas in the test section are considered as shown in Fig. 12. The results show that the velocity distribution at the center of the test section is very uniform.

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Fig. 12. Velocity flow distribution in test section of wind tunnel (a) inlet; (b) middle; (c) outlet

The velocity profiles of airflow at middle of test section of the wind tunnel with and without guide vanes are presented in Fig. 13. It is possible to observe that the importance of guide vanes in the corner to obtain a uniformity of airflow velocity in the test section of wind tunnel. This result also demonstrated the obtained quality of flow in the test section.

Fig. 13. Velocity profile of the flow at middle of the test section (a) width; (b) height

5 Conclusion A small, low-speed closed-loop wind tunnel was designed. The airflow characteristics were then evaluated by the numerical CFD simulation method using ANSYS Fluent software. The results demonstrated the importance of the guide vanes in the corners of wind tunnel. In the case of presence of the guide vanes, a good overall flow distribution

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was observed in the wind tunnel system, especially in the test section. These results showed that the design of the wind tunnel is suitable and may be further studied for perfection. Acknowledgment. This work was supported by The University of Danang, University of Science and Technology, code number of Project: T2019-02-08.

References 1. Barlow, J.B., Rae Jr., W.H., Pope, A.: Low-Speed Wind Tunnel Design. Wiley (1999) 2. Turkel, E.: Preconditioning techniques in computational fluid dynamic. Annu. Rev. Fluid Mech. 31(1), 385–416 (1999) 3. Moonen, P., Blocken, B., Roels, S., Carmeliat, J.: Numerical modelling of the flow conditions in a closed circuit low-speed wind tunnel. J. Wind Eng. Ind. Aerodyn. 94, 699–723 (2006) 4. Gordon, R., Imbabi, M.S.: CFD simulation and experimental validation of a new closed circuit wind/water tunnel design. J. Fluids Eng. 120(2), 311–318 (1998) 5. Calautit, J.K., Chaudhry, H.N., Hughes, B.R., Sim, L.F.: A validated design methodology for a closed-loop subsonic wind tunnel. J. Wind Eng. Ind. Aerodyn. 125, 180–194 (2014) 6. Mehta, R.D., Bradshaw, P.: Design rules for small low speed wind tunnels. Aeronaut. J. 83, 443–453 (1979) 7. https://www.ansys.com/. Accessed 10 Oct 2020 8. González, M.H.: Design Methodology for a Quick and Low-Cost Wind Tunnel. Polytechnic University of Madrid, Spain (2013) 9. Annderson, J.D.: Computational Fluid Dynamics: The Basics with Application. McGraw-Hill (1995) 10. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA 32, 1598–1605 (1994)

A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory Lieu B. Nguyen1(B) , Chien H. Thai2,3 , and H. Nguyen-Xuan4 1 Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education,

Ho Chi Minh City, Vietnam [email protected] 2 Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam [email protected] 3 Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam 4 CIRTech Institute, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam [email protected]

Abstract. In this work, we present a size-dependent numerical approach for free vibration analysis of functionally graded (FG) microplates based on the modified strain gradient theory (MSGT), simple first-order shear deformation theory (sFSDT) and moving Kriging meshfree method. The present approach decreases one variable compared with the original first-order shear deformation theory (FSDT). Moreover, it only uses three material length scale parameters to capture the size effects. The effective material properties as Young’s modulus, Poisson’s ratio and density mass are homogenized by a rule of mixture. Thanks to the principle of virtual work, the discrete system equations solved by the moving Kriging meshfree method, are derived. In addition, due to satisfying a Kronecker delta function property of the moving Kriging integration shape function, the essential boundary conditions are easily enforced similar to the standard finite element method. Rectangular and circular FG microplates with different boundary conditions, material length scale parameter and volume fraction are exampled to evaluate natural frequencies. Keywords: Moving Kriging meshfree method · Functionally graded microplate · Modified strain gradient theory · Simple first-order shear deformation theory

1 Introduction Experimental studies have recently been indicated that size effects are obviously occurred in micro/nano-structures [1], in which the effect of the strain-gradient tensor is significant compared with the strain tensor. To solve micro/nanostructures, there are several modelling approaches consisting of atomistic, hybrid atomistic continuum mechanics and continuum mechanics. In practical problems, the continuum mechanic model © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 673–690, 2022. https://doi.org/10.1007/978-981-16-3239-6_52

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is simply and easily implemented compared to those models. However, it cannot be explained mechanical behavior of micro/nanostructures due to ignoring material length scale parameters (MLSPs). For this reason, many various size dependent continuum theory models including the couple stress [2], modified couple stress [3], nonlocal elasticity [4], modified nonlocal elasticity [5], surface elasticity [6], strain gradient [7] and general strain gradient [8] have been proposed to investigate size effects of micro/nanostructures. By using the symmetric condition of the couple-stress tensor for equilibrium equation of couple moments, Lam et al. [1] proposed the modified strain gradient theory (MSGT) with three MLSPs instead of five as in [8]. Thus, it is suitable to use in practical computations. Moreover, the size-dependent model based on MSGT has been applied to analyze the behaviors of micro-beam, micro-plate and micro-shell structures. A micro-scale Kirchhoff model was firstly presented by Movassagh and Mahmoodi [9] for bending analysis of isotropic plate. This model was developed for analyzing of FG microplate [10] and multi-layer [11]. In addition, a size dependent model based on FSDT was presented by Ansari et al. [12, 13] to study bending, buckling and free vibration behaviors of FG microplates. Moreover, a size dependent higher-order shear deformation theory model for analyzing of FG microplates was reported in Refs. [14, 15]. Using this model, Thai et al. [16] studied for bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite microplates. Zhang et al. [17, 18] presented a size dependent refined model by using the MSGT and a refined higher-order shear deformation theory (RPT). That model combining with isogeometric analysis (IGA) for the static bending, free vibration and buckling analyses of FG microplates was also developed by Thai et al. [19, 20]. After that, that model was applied for the thermal analysis of FG microplates by Farzam and Hassani [21]. In the other hand, Salehipour and Shahsavar [22] developed a size dependent three dimensional elasticity theory model for free vibration analysis of FG microplates. Moreover, a size dependent cylindrical thin shell model was presented by Zeighampour and Tadi [23]. As concluded in a review paper [24], it can be seen that the size-dependent models have been developed in the last five years, in which the couple stress and nonlocal elasticity theory models are larger than strain gradient theory models. In addition, a number of papers published based on the analytical solutions are very large compared to the numerical solutions. For reasons, the development of new size-dependent numerical models as finite element method, isogeometric analysis, meshfree method, etc. is really necessary. This paper concentrates on a moving Kriging interpolation (MKI) meshfree method. It was firstly applied for mechanical problems by Gu [25]. An advantage of the present meshfree method is satisfy the kronecker delta property of the shape functions. Therefore, the essential boundary condition is imposed similar to the finite element method. The meshfree method combined with modified couple stress theory was presented in Refs. [26, 27]. In this study, a new size-dependent meshless computational model based on the MSGT and sFSDT for analyzing of FG microplates is presented. The displacement field of the proposed model decreases one variable compared to FSDT. Moreover, it is simple and free of shear locking. Several examples are given in order to demonstrate the efficiency of proposed method.

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2 Basic Equations 2.1 Problem Description The FG plate is made of a mixture of ceramic and metal, in which ceramic-rich and metalrich surfaces are distributed at the top (z = h/2) and bottom (z = −h/2), respectively. The effective material properties are computed by the rule of mixture as follow     z n 1 h h + ; Vm = 1 − Vc Vc (z) = ,z ∈ − , (1) 2 h 2 2 where the symbols c, m andn are the ceramic, metal and power index, respectively. The effective material properties based on the rule of mixture are defined by: Ee = Ec Vc (z) + Em Vm (z); νe = νc Vc (z) + νm Vm (z); ρe = ρc Vc (z) + ρm Vm (z) (2)

2.2 Modified Strain Gradient Elasticity Theory The MSGT which only contains three material length scale parameters, is proposed by Lam et al. [1]. The virtual strain energy U in an isotropic linearly elastic material according to the MSGT can be expressed by:  1 U = (3) (σ : ε + m : χ + p : ζ + τ : η)dV 2 V where ε, χ, ζ and η are the strain tensor, the symmetric rotation gradient tensor, the dilatation gradient tensor and the deviatoric stretch gradient tensor, respectively; σ is the Cauchy stress tensor; m, p and τ are high-order stress tensors corresponding with strain gradient tensors χ, ζ and η, respectively. The components of the strain, symmetric rotation gradient and dilatation gradient tensors are written as:  1 ¯ T (4) ε = ∇ u¯ + (∇ u) 2  1 χ = ∇θ + (∇θ)T (5) 2 ⎧ ⎫ ⎧ ⎫ ⎨ ζx ⎬ ⎨ εxx,x + εyy,x + εzz,x ⎬ ζ = ζy = εxx,y + εyy,y + εzz,y (6) ⎩ ⎭ ⎩ ⎭ ζz εxx,z + εyy,z + εzz,z where

∇=



∂ ∂ ∂x ∂y

⎫ ⎧  ⎧ ⎫ ⎧ ⎫ ⎪ 1 ∂ w¯ − ∂ v¯ ⎪ ⎪ ∂z ⎪ ⎨ u¯ ⎬ ⎨ θx ⎬ ⎨ 2  ∂y  ⎬ ∂ 1 ∂ u ¯ ∂ w¯ ¯ ; u = = ; θ = − θ v ¯ y ∂z 2  ∂z ∂x  ⎪ ⎩ ⎭ ⎩ ⎭ ⎪ ⎪ ⎭ ⎩ 1 ∂ v¯ − ∂ u¯ ⎪ θz w¯ 2

∂x

∂y

(7)

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The deviatoric stretch gradient components are expressed by     ηxxx = εxx,x − 15 ζx − 25 εxx,x + εxy,y + εxz,z ; ηyyy = εyy,y − 15 ζy − 25 εxy,x + εyy,y + εyz,z ;     ηzzz = εzz,z − 15 ζz − 25 εxz,x + εyz,y + εzz,z ; ηyyx = ηyxy = ηxyy = 13 ηxxx + 2εxy,y + εyy,x − εxx,x     1 1 ηzzx = ηzxz = ηxzz = 3 ηxxx + 2εxz,z + εzz,x − εxx,x ; ηxxy = ηxyx = ηyxx = 3 ηyyy + 2εxy,x + εxx,y − εyy,y     ηzzy = ηzyz = ηyzz = 13 ηyyy + 2εyz,z + εzz,y − εyy,y ; ηxxz = ηxzx = ηzxx = 13 ηzzz + 2εxz,x + εxx,z − εzz,z   ηyyz = ηyzy = ηzyy = 13 ηzzz + 2εyz,y + εyy,z − εzz,z ;   ηxyz = ηyzx = ηzxy = ηxzy = ηzyx = ηyxz = 13 εyz,x + εxz,y + εxy,z

(8)

where symbols ‘,x’, ‘,y’ and ‘,z’ define the derivative of any function following x, y and z directions, respectively. The constitutive relations are given by σ = Cε; m = 2G 21 Iχ; p = 2G 22 Iζ and τ = 2G 23 Iη

(9)

where C, G, 1 , 2 , 3 , I are the stiffness tensor or elasticity tensor, shear modulus, three material length scale parameters and corresponded identity matrices, respectively. 2.3 Kinematics of FG Microplate Let us consider a plate of total thickness h. The mid-plane surface denoted by is a function of x and y coordinates and, the z-axis is taken normal to the plate. The displacement field of any points in the plate according to the first-order shear deformation theory is described by: u¯ (x, y, z) = u(x, y) + zβx (x, y) v¯ (x, y, z) = v(x, y) + zβy (x, y) w(x, ¯ y, z) = w(x, y)

(10)

where u, v, w, βx and βy are in-plane, transverse displacements and two rotation components in the y–z, x–z planes, respectively. To eliminate the shear locking phenomenon in FSDT, a hypothesis is introduced as follows b b ; βy = −w,y w = wb + ws ; βx = −w,x

(11)

Inserting Eq. (11) into Eq. (10), the displacement fields of the FSDT which is called the simple first-order shear deformation theory (sFSDT) are described by: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ b b u¯ = u − zw,x u ⎨ u¯ ⎬ ⎨ ⎬ ⎨ −w,x ⎬ b or u b ¯ = v¯ = + z −w,y = u1 + zu2 v¯ = v − zw,y v ⎩ ⎭ ⎩ b ⎭ ⎩ ⎭ s b s w¯ w +w w¯ = w + w 0

(12)

Inserting Eq. (12) into Eq. (4), strain components according to sFSDT are presented by   b ; ε = v − zw b ; ε = 1 γ = 1 u + v b εxx = u,x − zw,xx yy ,y xy ,x − zw,xy ; ,yy 2 xy 2 ,y s ; ε = 1 γ = 1 ws εxz = 21 γxz = 21 w,x yz 2 yz 2 ,y

(13)

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Bending and shear strains can be described as: T T   = ε1 + zε2 and γ = γxz γyz = εs ε = εxx εyy γxy

(14)

where T T   T  s b w b 2w b s ws ε1 = u,x v,y u,y + v,x ; ε2 = − w,xx ; ε = w ,yy ,xy ,x ,y

(15)

Similarly, the rotation vector is rewritten by inserting Eq. (12) into Eq. (7) by      ∂ u¯ 1 1 b 1 1 ∂ v¯ s b s θx = 2w,y + w,y ; θy = −2w,x − w,x ; θz = − = v,x − u,y 2 2 2 ∂x ∂y 2 (16) Substituting Eq. (16) into Eq. (5), the rotation gradient components are described as     1 b + ws b − ws ; χ −2w χxx = 21 2w,xy = ,xy ,xy ,xy ; 2  yy  ; b − wb + 1 ws − ws (17) χxy = 21 w,yy ,xx ,yy ,xx 2     χxz = 41 v,xx − u,xy ; χyz = 41 v,xy − u,yy ; χzz = 0 The rotation gradient components can be rewritten under a compact form by ⎧ ⎫ ⎫ ⎧ b + 1 ws   w,xy ⎪ ⎪ 2 ,xy ⎨ ⎨ χxx ⎪ ⎬ ⎬ ⎪ χ 1 b s xz s −w,xy χb = χyy =   − 2 w,xy  ;χ = χ ⎪ yz ⎩ 1 wb − wb + 1 ws − ws ⎪ ⎩χ ⎪ ⎭ ⎭ ⎪ ,xx ,yy ,xx 4  xy  2 ,yy v − u,xy = 14 ,xx v,xy − u,yy

(18)

Substituting the strains in Eq. (13) into Eq. (6), the dilatation tensor is expressed by:     b b b b ; ζy = v,yy + u,xy − z w,yyy ; ζx = u,xx + v,xy − z w,xxx + w,xyy + w,xxy   (19) b + wb ζz = − w,xx ,yy The dilatation tensor can be rewritten under a compact form by: T  = ζ1 + zζ2 ζ = ζ x ζy ζz

(20)

where   T T b − wb b + wb b b ζ1 = u,xx + v,xy v,yy + u,xy −w,xx ; ζ2 = − w,xxx ,yy ,xyy w,yyy + w,xxy 0

(21)

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L. B. Nguyen et al.

Similarly, substituting the strains in Eq. (13) into Eq. (8), the deviatoric stretch gradient components are defined as   b b ; + 35 w,xyy ηxxx = 25 u,xx − 15 u,yy − 25 v,xy + z − 25 w,xxx   b b ηyyy = 25 v,yy − 15 v,xx − 25 u,xy + z − 25 w,yyy ; + 35 w,xxy   3 4 8 3 b b ηyyx = ηyxy = ηxyy = − 15 u,xx + 15 u,yy + 15 v,xy + z 15 w,xxx − 12 w,xyy 15   3 4 8 3 b b v,yy + 15 v,xx + 15 u,xy + z 15 w,yyy − 12 w ηxxy = ηxyx = ηyxx = − 15 15 ,xxy ;  3 1 2 3 b 3 b ηzzx = ηzxz = ηxzz = − 15 u,xx − 15 u,yy − 15 v,xy + z 15 w,xxx + 15 w,xyy ; (22)   3 1 2 3 b 3 b ηzzy = ηzyz = ηyzz = − 15 v,yy − 15 v,xx − 15 u,xy + z 15 w,yyy + 15 w,xxy ;     1 4 b b + wb s s ηzzz = 15 w,xx ,yy − 5 w,xx + w,yy ; ηxxz = ηxzx = ηzxx = − 15 w,xx 1 b 4 s 1 s + 15 w,yy + 15 w,xx − 15 w,yy ; 4 b 1 b 4 s 1 s ηyyz = ηyzy = ηzyy = − 15 w,yy + 15 w,xx + 15 w,yy − 15 w,xx ; 1 b s ; ηxyz = ηyzx = ηzxy = ηxzy = ηzyx = ηyxz = − 3 w,xy + 13 w,xy

These components are also rewritten under compact forms by:  T T  η¯ = ηxxx ηyyy ηyyx ηxxy ηzzx ηzzy = η¯ 1 + z η¯ 2 ; η˜ = ηzzz ηxxz ηyyz ηxyz

(23)

where ⎫ ⎧ 2 b b ⎫ 2 1 2 − 5 w,xxx + 35 w,xyy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5 u,xx − 5 u,yy − 5 v,xy ⎪ ⎪ 2 b 3 b ⎪ ⎪ ⎪ ⎪ 2 1 2 − w + w ⎪ ⎪ ⎪ v − v − u ,yyy ,xxy ⎪ ⎪ ⎪ 5 5 5 ,yy 5 ,xx 5 ,xy ⎪ ⎪ ⎪ ⎬ ⎨ 12 3 b b ⎬ 3 4 8 w − w − 15 u,xx + 15 u,yy + 15 v,xy ,xxx ,xyy 15 15 ¯ ; η η¯ 1 = = 2 3 4 8 12 b ⎪; 3 b ⎪ − 15 ⎪ v,yy + 15 v,xx + 15 u,xy ⎪ ⎪ ⎪ ⎪ 15 w,yyy − 15 w,xxy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 1 2 3 b ⎪ 3 b ⎪ ⎪ ⎪ − 15 u,xx − 15 u,yy − 15 v,xy ⎪ w,xxx + 15 w,xyy ⎪ ⎪ ⎪ ⎪ ⎪ 15 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 3 ⎩ 3 b 1 2 3 b ⎭ − 15 v,yy − 15 v,xx − 15 u,xy w + w ,yyy ,xxy 15 ⎧ 1 ⎫ 15 b + 1 wb − 1 ws − 1 ws w,xx ⎪ ⎪ ,yy ,xx ,yy ⎪ ⎪ 5 5 5 5 ⎪ 4 b ⎨ 1 b 4 s 1 s ⎪ − 15 w,xx + 15 w,yy + 15 w,xx − 15 w,yy ⎬ η˜ = 4 b 1 b 4 s 1 s ⎪ ⎪ − 15 w,yy + 15 w,xx + 15 w,yy − 15 w,xx ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1 b 1 s − 3 w,xy + 3 w,xy ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(24)

The classical and higher-order stress elastic constitutive relations are expressed as: ⎧ ⎫ ⎡ ⎫ ⎤⎧ ⎪ σxx ⎪ εxx ⎪ Q11 Q12 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎨ σyy ⎪ ⎨ εyy ⎪ ⎬ ⎢ Q21 Q22 0 0 0 ⎥⎪ ⎬ ⎢ ⎥ (25) ςxy = ⎢ 0 0 Q66 0 0 ⎥ γxy ⎢ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ς 0 γ 0 0 0 Q xz xz 55 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ς ⎪ ⎭ 0 0 0 0 Q44 ⎩ γyz ⎭ yz ⎧ ⎫ ⎫ ⎡ ⎤⎧ ⎪ mxx ⎪ χxx ⎪ 10000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 0 1 0 0 0 ⎥⎪ ⎪ ⎨ myy ⎪ ⎨ χyy ⎪ ⎬ ⎬ ⎢ ⎥⎪ ⎢ ⎥ (26) mxy = 2Gl12 ⎢ 0 0 1 0 0 ⎥ χxy ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ 0 0 0 1 0 m χ xz xz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩m ⎪ ⎭ 0 0 0 0 1 ⎩ χyz ⎭ yz

A Size-Dependent Meshfree Approach for Free Vibration Analysis

679

⎧ ⎫ ⎤⎧ ⎫ ⎡ 1 0 0 ⎨ ζx ⎬ ⎨ px ⎬ = 2Gl22 ⎣ 0 1 0 ⎦ ζy (27) p ⎩ ⎭ ⎩ y⎭ 001 pz ζz ⎫ ⎫ ⎧ ⎡ ⎤⎧ ⎪ ηxxx ⎪ τxxx ⎪ 100000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎫ ⎪ ⎡ ⎤⎧ ⎪ ⎪ ⎪ ⎢ 0 1 0 0 0 0 ⎥⎪ ⎪ ⎪ ⎪ ⎪ τ η 1000 ⎪ ηzzz ⎪ yyy yyy ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ τzzz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎬ ⎨ ⎢ ⎥⎨ ⎬ ⎨ ⎢ 0 1 0 0 ⎥ ηxxz ⎬ τyyx τxxz ⎢ 0 0 1 0 0 0 ⎥ ηyyx ⎥ = 2Gl32 ⎢ ; = 2Gl32 ⎢ ⎥ ⎣ 0 0 1 0 ⎦⎪ ηyyz ⎪ (28) ⎪ ⎢ 0 0 0 1 0 0 ⎥⎪ ⎪ τxxy ⎪ ηxxy ⎪ τyyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎢ ⎥ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 0 1 0 ⎦⎪ τzzx ⎪ ηzzx ⎪ τxyz ηxyz 0001 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ 000001 τzzy ηzzy where Ee , 1−νe2

Q11 = Q22 =

Q12 = Q21 = G=

νe Ee , Q66 1−νe2 Ee s 2(1+νe ) ; k

= =

Ee 2(1+νe ) ; 5 6

Q55 = Q44 =

k s Ee 2(1+νe ) ;

(29) in which Ee and νe are the effective Young module and Poisson’s ratio, respectively. The discrete Galerkin weak form for the free vibration analysis of the FG microplate are described by: $ h/2 $   σxx δεxx + σyy δεyy + ςxy δγxy + ςxz δγxz + ςyz δγyz d dz + . . .

−h/2

$ h/2 $   mxx δχxx + myy δχyy + 2mxy δχxy + 2mxz δχxz + 2myz δχyz d dz + . . .

−h/2

$ h/2 $   τxxx δηxxx + τyyy δηyyy + 3τyyx δηyyx + 3τxxy δηxxy + 3τzzx δηzzx + 3τzzy δηzzy d dz + . . .

−h/2

$ h/2 $ 

−h/2

+

 τzzz δηzzz + 3τxxz δηxxz + 3τyyz δηyyz + 6τxyz δηxyz d dz + . . .

$ h/2 $  $ $ h/2  ¯ e u¨¯ d dz = 0 δ uρ px δζx + py δζy + pz δζz d dz +

−h/2

−h/2

(30) The Eq. (30) can split into two independent integrals following to middle surface and z-axis direction. Substituting Eqs. (25)–(28) into Eq. (30), the discrete Galerkin weak form can be rewritten as follows:   T  T δ χb Dbc cb χb d + δ χs Dsc cs χs d + ...



 



 T de de T di T dev dev ¨ˆ ˆ ˆ ˆ ˆ ˆ ˜ ˆ ud

¯ + + δ ζˆ D ζˆ d + δ η¯ D  ηd

δ η˜ D  ηd

δ uˆ T m =0



ˆ εd + δ εˆ T Dˆ





 T δ εs Ds εs d +







(31)



where T T T    εˆ = ε1 ε2 ; ζˆ = ζ1 ζ2 ; ηˆ¯ = η¯ 1 η¯ 2 ;        de de 0 Adi Bdi Ade Bde  Ab Bb di de ˆ ˆ ˆ ˆ ;D = ;D = ; = D= Bb Db Bdi Ddi Bde Dde 0  de 

(32)

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L. B. Nguyen et al.

in which 

⎡ ⎤   Q11 Q12 0 $ $ h/2 Q44 0    h/2 b b b 2 s ⎣ ⎦ dz; A , B , D = −h/2 1, z, z Q21 Q22 0 dz; D = −h/2 0 Q55 0 0 Q66 $ h/2 $ h/2   Dbc = −h/2 2Gl12 I3×3 dz; Dsc = −h/2 2Gl12 I2×2 dz; Adi , Bdi , Ddi $ h/2   ; = −h/2 2Gl22 1, z, z 2 I3×3 dz $ h/2  $ h/2    Ddev = −h/2 2Gl32 I4×4 dz; Ade , Bde , Dde = −h/2 2Gl32 1, z, z 2 I6×6 dz; cb = diag(1, 1, 2); cs = diag(2, 2);  dev = diag(1, 3, 3, 6);  de = diag(1, 1, 3, 3, 3, 3)    1 h/2 $   I 1 I2 u ˆ ; m = ρe 1, z, z 2 I3×3 dz; uˆ = (I1 , I2 , I3 ) = u2 I2 I3 −h/2 (33)

in which I2×2 , I3×3 , I4×4 , I6×6 are the identity matrices of size 2 × 2, 3 × 3, 4 × 4 and 6 × 6, respectively.

3 FG Microplate Formulation Based on Moving Kriging Interpolation Shape Functions 3.1 Moving Kriging Interpolation Shape Functions To construct the moving Kriging shape functions for interpolation kinematic variables, the domain of problem is discretized into a set of nodes xI (I = 1,…, NP), in which NP denotes the number of nodes in the problem domain. The nodal displacement vector is defined as uh (x) =

NP %

NI (x)uI

(34)

I =1

where uI and φI are respectively the unknown coefficient associated with node I and the moving Kriging shape function which is described as follows NI (x) =

m % j=1

pj (x)AjI +

n % k=1

rk (x)BkI or NI (x) = pT (x)A + rT (x)B

(35)

A Size-Dependent Meshfree Approach for Free Vibration Analysis

681

in which n is a number of nodes in a support domain x ∈ and m is the dimension of a polynomial basis function space. In addition, A, B, p(x) and r(x) are expressed as: A = (PT R−1 P)−1 PT R−1 , B = R−1 (I − PA) T  p(x)= 1 x y x2 xy y2 ... and r(x) = [ R(x1 , x) R(x2 , x) · · · R(xn , x) ]T

(36)

where I is the identity matrix of size n × n. The cubic polynomial functions are chosen for the present work by:  T p(x)= 1 x y x2 xy y2 x3 x2 y xy2 y3 (m = 10) Besides, P and R are written as follows ⎡ ⎤ ⎡ ⎤ p1 (x1 ) ... pm (x1 ) R(x1 , x1 ) ... R(x1 , xn ) ⎦ P =⎣ ... ... ... ⎦ and R =⎣ ... ... ... p1 (xn ) ... pm (xn ) R(xn , x1 ) ... R(xn , xn )

(37)

(38)

 2  is a correlation function, in which E denotes where R(xI , xJ )= 21 E uh (xI ) − uh (xJ ) an expected value of a random function. In this study, the Gaussian function is used and it is defined as R(xI , xJ ) = e

 2 βr − a IJ 0

(39)

where rIJ = xI −xJ , the correlation parameter β is related to the variance σ 2 of the normal distribution function by β 2 = 1/2σ 2 . The scale factor a0 is used to normalize the distance. Regularly distributed nodes, a0 is taken as the length of two adjacent nodes, and it is chosen to be the maximum distance between a pair of nodes in the support domain for the irregularly distributed nodes. In addition, the correlation parameter β = 1 is chosen for the present work. The size of the support domain to build the shape functions can be defined as dm = αdc

(40)

where dc and α are an average distance between nodes and a scale factor, respectively. In this study, the scale factor is taken equal to 3.0 (α = 3.0). 3.2 A MKI-Based Formulation Based on the sFSDT and Modified Strain Gradient Theory According to the MK interpolation, the displacement field can be approximated as u (x, y) = h

n %

NI (x, y)qI

I =1

where qI =



uI vI wIb wIs

T

are degrees of freedom of node I.

(41)

682

L. B. Nguyen et al.

Substituting Eq. (41) into Eq. (15), the strain components can be rewritten as n n n % %  T %  1 2 T εˆ = ε1 ε2 = BsI qI Bˆ I qI and εs = BI BI qI = I =1

I =1

in which, ⎡

⎡ ⎤ NI ,x 0 0 0 0 0 NI ,xx B1I = ⎣ 0 NI ,y 0 0 ⎦; B2I = −⎣ 0 0 NI ,yy NI ,y NI ,x 0 0 0 0 2NI ,xy

(42)

I =1

⎤   0 0 0 0 NI ,x s ⎦ 0 ; BI = 0 0 0 NI ,y 0

(43)

Similarly, the curvatures are obtained by substituting Eq. (41) into Eq. (18) as follows χb =

n %

BbcI qI ; χs =

I =1

n %

BscI qI

(44)

I =1

in which, ⎡ ⎤ 1   00 NI ,xy 2 NI ,xy 1 −NI ,xy NI ,xx 0 0 b s 1 ⎣ ⎦ (45) ; BcI = BcI = 0 0  −NI ,xy   − 2 NI ,xy  4 −NI ,yy NI ,xy 0 0 0 0 21 NI ,yy − NI ,xx 41 NI ,yy − NI ,xx Substituting Eq. (41) into Eq. (20), the dilatation components can be rewritten as n  n T %  T % ζ ζ1 ζ2 ζˆ = ζ1 ζ2 Bˆ I qI = qI = BI BI I =1

(46)

I =1

where ⎡

ζ1

BI

0 NI ,xx NI ,xy = ⎣ NI ,xy NI ,yy 0 0 0 −NI ,xx − NI ,yy

⎡ ⎤ 0 0 0 −NI ,xxx − NI ,xyy 2 ζ ⎣ ⎦ = ; B 0 0 0 −NI ,yyy − NI ,xxy I 0 00 0

⎤ 0 0 ⎦; 0

(47)

Substituting Eq. (41) into Eq. (23), the dilatation stretch gradient components can be rewritten as n  n n T % %  T % η¯ η˜ η¯ η¯ ηˆ¯ = η¯ 1 η¯ 2 = BI qI Bˆ I qI ; η˜ == BI 1 BI 2 qI = I =1

I =1

(48)

I =1

where ⎡

⎡ ⎤ 2N 1 0 0 − 25 NI ,xxx + 35 NI ,xyy − 25 NI ,xy 00 5 I ,xx − 5 NI ,yy 2 2 1 2 3 ⎢ ⎢ ⎥ − 5 NI ,xy ⎢ ⎢ 0 0 − 5 NI ,yyy + 5 NI ,xxy 5 NI ,yy − 5 NI ,xx 0 0 ⎥ ⎢ 3 ⎢ ⎥ 4 3 12 8 + 15 NI ,yy 0 0 ⎥ η¯ 2 ⎢ 0 0 15 NI ,xxx − 15 NI ,xyy η¯ ⎢− N 15 NI ,xy BI 1 = ⎢ 15 I ,xx ⎥ ; BI = ⎢ 8 N 3 N 4 ⎢ ⎢ 0 0 3 NI ,yyy − 12 NI ,xxy − 15 I ,yy + 15 NI ,xx 0 0 ⎥ 15 15 ⎢ 3 15 I ,xy1 ⎢ ⎥ 2 ⎣ − NI ,xx − NI ,yy ⎣ 0 0 3 NI ,xxx + 3 NI ,xyy ⎦ − N 0 0 15 15 15 I ,xy 15 15 2 N 3 N 1 3 N 3 − 15 0 0 15 − 15 I ,xy I ,yy − 15 NI ,xx 0⎤0 I ,yyy + 15 NI ,xxy ⎡ 1 1 1 1 0 0 5 NI ,xx + 5 NI ,yy − 5 NI ,xx − 5 NI ,yy ⎢ ⎥ 4 N 1 1 4 ⎢ 0 0 − 15 η˜ I ,xx + 15 NI ,yy 15 NI ,xx − 15 NI ,yy ⎥ BI = ⎢ ⎥ 4 N 4 N 1 N 1 N ⎣ 0 0 − 15 ⎦ + − I ,yy 15 I ,xx 15 I ,yy 15 I ,xx 1 1 00 − 3 NI ,xy N I ,xy 3

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥; 0⎥ ⎥ 0⎦ 0

A Size-Dependent Meshfree Approach for Free Vibration Analysis

683

Substituting Eq. (41) into Eq. (12), u1 and u2 can be described as follows n n % T %  1 2 T  ˆ I Nq = uˆ = u1 u2 NI NI qI = I =1

(49)

I =1

Substituting Eqs. (42), (44), (46) and (48) into Eq. (31), the weak form of the free vibration analysis of the FG microplate is rewritten as   K − ω2 M q¯ = 0 (50) where K (K = Kε + Kχ + Kζ + Kη ) and F are the global stiffness matrix and force vector, respectively, in which $ $  T $  T ˆTD ˆ Bd

ˆ + (Bs )T Ds Bs d ; Kχ = Bbc Dbc  bc Bbc d + Bsc Dsc  sc Bsc d ;       T de de η¯ T di ζ T $ $ $ η˜ ˆ B ˆ d ; Kη = ˆ ˆ Bˆ d + ˆζ D ˆ η¯ D Kζ = B Ddev  dev Bη˜ d

B

B $ T iωt ˆ m ˆ ˆ Nd ; ¯ q = qe M= N

Kε =

$

B

(51)

where ω and q¯ in Eq. (51) are the natural frequency and corresponded mode shapes.

4 Numerical Examples and Discussions In this study, three length scale parameters which are assumed to be equal through the thickness of plate l1 = l2 = l3 = l = 15 × 10−6 m, are taken the same as in [17]. The FG microplate is made from the bottom metal surface (Aluminum-Al) to the top ceramic surface (alumina-Al2 O3 ). The material properties for Al are Em = 70 GPa, vm = 0.3, 3 ρm = 2702 kg/m3 and for Al2 O3 Ec = 380 GPa, vc = 0.3, & . The & ρc = 3800 kg/m non-dimensional natural frequency is normalized by ω¯ = ωh

ρc Ec

or ω¯ = ωR2

hρc Dc ,

in

Ec h3 ; a and R are the length of square plate or the radius of circular plate, 12(1−vc2 )

which Dc = respectively. The square and circular microplates are modeled as in Fig. 1, respectively.

a) Simply supported boundary condition (BC)

b) Roller/simply supported BC

c) Fully clamped BC.

Fig. 1. Distribution nodes of square and circular microplates.

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4.1 FG Square Microplate Let us consider a simply supported FG square microplate with a distribution of node as shown in Fig. 1. Different length-to-thickness ratios, power index and material length scale-to-thickness ratio are studied. Table 1 tabulates the first two non-dimensional natural frequencies of the simply supported isotropic FG square microplate. Obtained results are compared to those proposed by Zhang et al. [17] using the analytical solution based on the RPT (4 DOFs) and MSGT, Thai et al. [14] using the IGA based on the TSDT (5 DOFs) combined to MSGT and Thai et al. [19] using the IGA based on the RPT (4 DOFs) and MSGT. As in Table 1, it is observed that the non-dimensional natural frequencies predicted by the classical plate model (l/h = 0) are always smaller than those of the size dependent microplate model (0 < l/h ≤ 1). The non-dimensional natural frequencies decrease when increasing the length-to-thickness ratio and power index. In addition, for the case of l/h = 0, it can be seen that an excellent agreement is achieved when comparing with those results. In the case of l/h = 0, obtained results are smaller than those compared ones. It can be explained that the present model only considers FSDT instead of HSDT as those referenced solutions. Besides, an increase of the material length scale ratio l/h leads to a rise of the non-dimensional natural frequencies. For this reason, it can be concluded that the stiffness of FG microplate increases when considering the size-dependent effect. Table 1. Comparison of non-dimensional natural frequencies ω¯ of simply supported isotropic FG square microplate. a/h n

References

l/h 0.0

5

0

0.1

0.2

0.5

1.0

Exact-RPT [17] 0.2113 0.2273 0.2698 0.4688 0.7011 IGA-TSDT [14] 0.2113 0.2264 0.2666 0.4566 0.8344 IGA-RPT [19]

0.2113 0.2271 0.2687 0.4630 0.8477

Present

0.2112 0.2243 0.2566 0.3756 0.5433

0.5 Exact-RPT [17] 0.1804 0.1953 0.2344 0.4131 0.6290 IGA-TSDT [14] 0.1807 0.1948 0.2318 0.4032 0.7406

1

IGA-RPT [19]

0.1807 0.1955 0.2338 0.4085 0.7491

Present

0.1804 0.1931 0.2240 0.3346 0.4866

Exact-RPT [17] 0.1631 0.1774 0.2144 0.3823 0.5832 IGA-TSDT [14] 0.1631 0.1766 0.2118 0.3729 0.6867

2

IGA-RPT [19]

0.1631 0.1772 0.2136 0.3778 0.6948

Present

0.1630 0.1751 0.2046 0.3088 0.4504

Exact-RPT [17] 0.1472 0.1603 0.1944 0.3490 0.5270 IGA-TSDT [14] 0.1472 0.1596 0.1920 0.3402 0.6266 IGA-RPT [19]

0.1471 0.1601 0.1935 0.3449 0.6360

Present

0.1477 0.1586 0.1850 0.2789 0.4068 (continued)

A Size-Dependent Meshfree Approach for Free Vibration Analysis Table 1. (continued) a/h n

References

l/h 0.0

5

0.1

0.2

0.5

1.0

Exact-RPT [17] 0.1378 0.1470 0.1764 0.3117 0.4553 IGA-TSDT [14] 0.1358 0.1463 0.1740 0.3029 0.5540 IGA-RPT [19]

0.1357 0.1466 0.1753 0.3081 0.5656

Present

0.1379 0.1462 0.1669 0.2436 0.3524

10 Exact-RPT [17] 0.1358 0.1402 0.1666 0.2896 0.4150 IGA-TSDT [14] 0.1301 0.1394 0.1642 0.2807 0.5111

10

0

IGA-RPT [19]

0.1299 0.1397 0.1655 0.2861 0.5230

Present

0.1323 0.1393 0.1569 0.2240 0.3220

Exact-RPT [17] 0.0577 0.0619 0.0730 0.1258 0.2309 IGA-TSDT [14] 0.0577 0.0617 0.0725 0.1240 0.2268 IGA-RPT [19]

0.0577 0.0619 0.0729 0.1254 0.2297

Present

0.0577 0.0616 0.0719 0.1155 0.1790

0.5 Exact-RPT [17] 0.0489 0.0529 0.0632 0.1113 0.2057 IGA-TSDT [14] 0.0490 0.0528 0.0629 0.1098 0.2023

1

IGA-RPT [19]

0.0490 0.0529 0.0633 0.1110 0.2047

Present

0.0490 0.0527 0.0624 0.1026 0.1602

Exact-RPT [17] 0.0442 0.0480 0.0578 0.1028 0.1907 IGA-TSDT [14] 0.0442 0.0478 0.0573 0.1013 0.1873

2

IGA-RPT [19]

0.0442 0.0479 0.0577 0.1024 0.1896

Present

0.0442 0.0478 0.0569 0.0946 0.1483

Exact-RPT [17] 0.0401 0.0435 0.0524 0.0933 0.1731 IGA-TSDT [14] 0.0401 0.0434 0.0520 0.0918 0.1698

5

IGA-RPT [19]

0.0401 0.0435 0.0523 0.0930 0.1722

Present

0.0401 0.0433 0.0516 0.0855 0.1340

Exact-RPT [17] 0.0377 0.0405 0.0478 0.0825 0.1514 IGA-TSDT [14] 0.0377 0.0403 0.0474 0.0810 0.1482 IGA-RPT [19]

0.0377 0.0404 0.0477 0.0822 0.1508

Present

0.0379 0.0404 0.0469 0.0750 0.1161

10 Exact-RPT [17] 0.0364 0.0388 0.0453 0.0764 0.1390 IGA-TSDT [14] 0.0364 0.0387 0.0449 0.0750 0.1359 IGA-RPT [19]

0.0363 0.0387 0.0451 0.0761 0.1384

Present

0.0366 0.0387 0.0444 0.0692 0.1062

685

686

L. B. Nguyen et al.

4.2 FG Circular Microplate Let us consider a FG circular microplate of the radius R subjected to roller, simply supported and fully clamped BCs. Similarly, various values of power index and material length scale-to-thickness ratio l/h are also studied. Table 2 gives the first non-dimensional natural frequency of the FG circular microplate. Again, results are compared to those of given by Zhang et al. [18] based on the exact-RPT solution using MSGT, Thai et al. [14] based on the IGA-TSDT solution using MSGT and Thai et al. [19] based on the IGA-RPT solution using MSGT. As observed in Table 2, it can be seen that the non-dimensional natural frequency increases when changing boundary conditions from roller to fully clamped. Besides, the non-dimensional natural frequencies obtained from the present solution in the case of l/h = 0 are always smaller than those referenced ones because of using FSDT. Moreover, the difference of non-dimensional natural frequency between MSGT and classical theory is much significant as l/h = 1, however, this difference decreases when the thickness of the microplate becomes larger. The first six mode shapes of fully clamped FG circular microplate are plotted in Fig. 2. Table 2. Comparison of normalized frequency ω¯ of simply supported Al/Al2 O3 circular microplate with R/h = 5. n

Reference

l/h 0.0

0.1

0.2

0.5

1.0

Roller 0

0.5

2

Exact-RPT [18]

4.7369

5.1587

6.0483

10.3040

18.8251

IGA-TSDT [14]

4.7787

5.1089

5.9896

10.2008

18.6325

IGA-RPT [19]

4.7787

5.1112

5.9961

10.2134

18.6459

Present

4.7787

5.0945

5.9188

9.4435

14.4214

Exact-RPT [18]

3.9449

4.3098

5.1568

9.0789

16.7701

IGA-TSDT [14]

4.0578

4.3697

5.1915

9.0335

16.6167

IGA-RPT [19]

4.0569

4.3709

5.1964

9.0408

16.6140

Present

4.0557

4.3569

5.1341

8.3864

12.9085

Exact-RPT [18]

3.0603

3.3293

4.0844

7.5050

14.0837

IGA-TSDT [14]

3.3185

3.5868

4.2909

7.5573

13.9654

IGA-RPT [19]

3.3167

3.5864

4.2932

7.5649

13.9725

Present

3.3204

3.5780

4.2396

6.9873

10.7901 (continued)

A Size-Dependent Meshfree Approach for Free Vibration Analysis

687

Table 2. (continued) n

Reference

l/h 0.0

5

10

0.1

0.2

0.5

1.0

Exact-RPT [18]

2.9573

3.1333

3.7489

6.6380

12.3157

IGA-TSDT [14]

3.1228

3.3396

3.9174

6.6810

12.2095

IGA-RPT [19]

3.1206

3.3383

3.9185

6.6910

12.2299

Present

3.1346

3.3363

3.8641

6.1337

9.3556

Exact-RPT [18]

2.8568

3.1021

3.6326

6.1304

11.3096

IGA-TSDT [14]

3.0159

3.2046

3.7118

6.1833

11.1987

IGA-RPT [19]

3.0135

3.2030

3.7125

6.1940

11.2236

Present

3.0285

3.2008

3.6567

5.6608

8.5548

Exact-RPT [18]

4.7369

5.1587

6.0483

10.3041

18.8253

IGA-TSDT [14]

4.7787

5.1090

5.9898

10.2016

18.6339

Simply supported 0

0.5

2

5

10

IGA-RPT [19]

4.7787

5.1112

5.9961

10.2134

18.6459

Present

4.7787

5.0945

5.9188

9.4435

14.4214

Exact-RPT [18]

4.1906

4.5444

5.3535

9.1899

16.8296

IGA-TSDT [14]

4.1695

4.4740

5.2801

9.0861

16.6500

IGA-RPT [19]

4.1686

4.4753

5.2853

9.0936

16.6484

Present

4.1671

4.4604

5.2204

8.4299

12.9248

Exact-RPT [18]

3.8259

4.0600

4.7081

7.8705

14.2831

IGA-TSDT [14]

3.6642

3.9131

4.5742

7.7347

14.0839

IGA-RPT [19]

3.6622

3.9129

4.5770

7.7419

14.0932

Present

3.6677

3.9036

4.5160

7.1320

10.8457

Exact-RPT [18]

3.5844

3.7315

4.2734

6.9620

12.4965

IGA-TSDT [14]

3.4042

3.6086

4.1573

6.8400

12.3195

IGA-RPT [19]

3.4018

3.6073

4.1586

6.8501

12.3428

Present

3.4200

3.6061

4.0972

6.2606

9.4057

Exact-RPT [18]

3.1929

3.4342

3.9258

6.3122

11.4127

IGA-TSDT [14]

3.1729

3.3551

3.8468

6.2738

11.2620

IGA-RPT [19]

3.1704

3.3535

3.8476

6.2849

11.2893

Present

3.1872

3.3510

3.7868

5.7324

8.5833

Exact-RPT [18]

9.4248

10.2145

12.0955

20.9426

38.8453

IGA-TSDT [14]

9.2672

9.9484

11.7378

20.1819

36.9736

IGA-RPT [19]

9.2705

9.9680

11.7958

20.3440

37.2740

Present

9.2643

9.8476

11.3035

16.7967

24.4090

Fully clamped 0

(continued)

688

L. B. Nguyen et al. Table 2. (continued)

n

Reference

l/h 0.0

0.5

2

5

10

0.1

0.2

0.5

1.0

Exact-RPT [18]

8.3094

9.0188

10.7161

18.6309

33.9181

IGA-TSDT [14]

7.9253

8.5595

10.2094

17.8361

32.8279

IGA-RPT [19]

7.9261

8.5766

10.2619

17.9658

33.0175

Present

7.9131

8.4752

9.8615

14.9591

21.8619

Exact-RPT [18]

7.4473

7.9817

9.3874

16.0942

29.4843

IGA-TSDT [14]

6.4726

7.0300

8.4722

15.0740

27.9396

IGA-RPT [19]

6.4711

7.0398

8.5083

15.1851

28.1224

Present

6.4934

6.9760

8.1615

12.4815

18.2847

Exact-RPT [18]

6.6452

7.2463

8.4811

14.3555

25.9336

IGA-TSDT [14]

5.9806

6.4559

7.6895

13.4247

24.7246

IGA-RPT [19]

5.9784

6.4620

7.7195

13.5515

24.9871

Present

6.0685

6.4405

7.3719

10.9099

15.8414

Exact-RPT [18]

6.2216

6.6676

7.7952

13.1556

23.9134

IGA-TSDT [14]

5.7278

6.1522

7.2532

12.4215

22.7102

IGA-RPT [19]

5.7254

6.1570

7.2824

12.5619

23.0236

Present

5.8192

6.1335

6.9290

10.0331

14.4730

a) Mode 1

b) Mode 2

d) Mode 4

e) Mode 5

c) Mode 3

f) Mode 6

Fig. 2. The first six mode shapes of fully clamped FG circular microplate with n = 10, l/h = 1.

A Size-Dependent Meshfree Approach for Free Vibration Analysis

689

5 Conclusions A size-dependent numerical approach based on MSGT, sFSDT and moving Kriging meshfree method was presented for analyzing free vibration of FG microplates. The advantages of the present approach are only having four unknowns and three material length scale parameters. Thus, it is suitable for computation of size-dependent practical problems. The rule of mixed is used to homogenize material properties as Young’s modulus, Poison’s ratio and density mass. In addition, when taking zero of all material length scale parameters, the classical sFSDT model was retrieved from the present model. Numerical results pointed out that the non-dimensional natural frequencies obtained from the present solutions are always smaller than those referenced ones in the case of l/h = 0. In addition, the stiffness of microplate can be increased as considering small scale effects leading to a rise of the natural frequencies. Acknowledgment. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2019.35.

References 1. Lam, D., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003) 2. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968) 3. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002) 4. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972) 5. Salehipour, H., Shahidi, A.R., Nahvi, H.: Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015) 6. Gurtin, M.E., Weissmuller, J., Larche, F.: The general theory of curved deformable interfaces in solids at equilibrium. Philis. Mag. A 78, 1093–1109 (1998) 7. Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 1–4 (1999) 8. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964) 9. Movassagh, A.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech.-A/Solids 40, 50–59 (2013) 10. Mirsalehi, M., Azhari, M., Amoushahi, H.: Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method. Eur. J. Mech.-A/Solids 61, 1–13 (2017) 11. Li, A., Zhou, S., Wang, B.: A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Compos. Struct. 113, 272–280 (2014) 12. Ansari, R., Gholami, R., Shojaei, M., Mohammadi, V., Sahmani, S.: Bending buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur. J. Mech.-A/Solids 49, 251–267 (2015) 13. Ansari, R., Hasrati, E., Shojaei, M., Gholami, R., Mohammadi, V., Shahabodini, A.: Sizedependent bending, buckling and free vibration analyses of microscale functionally graded mindlin plates based on the strain gradient elasticity theory. Latin Am. J. Solids Struct. 13, 632–664 (2016)

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14. Thai, S., Thai, H.T., Vo, P.T., Patel, V.I.: Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput. Struct. 190, 219–241 (2017) 15. Sahmani, S., Ansari, R.: On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct. 95, 430–442 (2013) 16. Thai, H.C., Ferreira, A.J.M., Rabczuk, T., Nguyen-Xuan, H.: Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory. Eur. J. Mech.A/Solids 72, 521–538 (2018) 17. Zhang, B., He, Y., Liu, D., Shen, L., Lei, J.: An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl. Math. Model. 39, 3814–3845 (2015) 18. Zhang, B., He, Y., Liu, D., Lei, J., Shen, L., Wang, L.: A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates. Compos. B Eng. 79, 553–580 (2015) 19. Thai, C.H., Ferreira, A.J.M., Nguyen-Xuan, H.: Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory. Compos. Struct. 192, 274–288 (2018) 20. Thai, C.H., Ferreira, A.J.M., Phung-Van, P.: Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory. Eng. Anal. Bound. Elem. 117, 284–298 (2020) 21. Farzam, A., Hassani, B.: Size-dependent analysis of FG microplates with temperaturedependent material properties using modified strain gradient theory and isogeometric approach. Compos. B Eng. 161, 150–168 (2018) 22. Salehipour, H., Shahsavar, A.: A three dimensional elasticity model for free vibration analysis of functionally graded micro/nano plates: modified strain gradient theory. Compos. Struct. 206, 415–424 (2018) 23. Zeighampour, H., Beni, Y.T.: Cylindrical thin-shell model based on modified strain gradient theory. Int. J. Eng. Sci. 78, 27–47 (2014) 24. Thai, H.T., Vo, P.T., Nguyen, T.K., Kim, S.E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017) 25. Gu, L.: Moving Kriging interpolation and element-free Galerkin method. Int. J. Numer. Meth. Eng. 56, 1–11 (2003) 26. Thai, C.H., Ferreira, A.J.M., Lee, J., Nguyen-Xuan, H.: An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on the modified couple stress theory and moving Kriging-based meshfree method. Int. J. Mech. Sci. 142, 322–338 (2018) 27. Tran, D.T., Thai, C.H., Nguyen-Xuan, H.: A size-dependent functionally graded higher order plate analysis based on modified couple stress theory and moving Kriging meshfree method. Comput. Mater. Continua 57, 447–483 (2018)

Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core Tran Quang Hung1 , Do Minh Duc1(B) , and Tran Minh Tu2 1 The University of Da Nang - University of Science and Technology, Da Nang, Vietnam

[email protected]

2 National University of Civil Engineering, Hanoi, Vietnam

[email protected]

Abstract. In this report, the static response of sandwich beam with fluidinfiltrated porous core and two face sheets made of functionally graded materials is investigated. Variation in mechanical properties of the sandwich beam is assumed to be continuous along the thickness direction. Various beam theories for one-dimensional modelling of the beam are considered. The relationship between stress and strain obeys Biot’s theory of linear poroelasticity. The governing equations of the beam are derived by applying Hamilton’s principle and solved analytically by Navier’s solution. Comparative and comprehensive studies are conducted to examine both the accuracy and the effects of various parameters, such as power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam. Keywords: Functionally graded material · Fluid-infiltrated porous material · Porous sandwich beam · Navier’s solution · Beam theories

1 Introduction Due to the distribution of pores in their body, porous materials (i.e., metal foams, ceramic foams and polymer foams) own a series of excellent properties, such as low density, the efficient capacity of energy dissipation, low thermal conductivity. These advanced materials, therefore, is a reasonable selection for engineering applications of aerospace, automotive industry, civil construction, ship building, etc. [1–3]. In porous materials, the pores with different shapes and sizes are non-/uniformly distributed and can be either infiltrated with liquid or empty. Porous materials with liquid phase in their pores are classified as fluid-infiltrated/saturated porous materials. Due to the pores together with the liquid phase inside, mechanical behavior of structures made of porous materials may be different from that of conventional structures made of monolithic materials and needs much more research efforts to reveal. The stress-strain relation in porous materials is first developed by Biot [4]. This relation is an extension of Hooke’s law and called the linear theory of poroelasticity. This theory is commonly used to simulate the physical phenomena of porous media. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 691–706, 2022. https://doi.org/10.1007/978-981-16-3239-6_53

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Up to date, there have been many theoretical and experimental studies investigating the mechanical responses of porous structures. Some authors have focused on the effects of porosities without the liquid phase in the pores on static, vibration and buckling response of structures, such as beams [5, 6], plates [7, 8] and shells [9–11]. If the pores are infiltrated with liquid, the fluid pressure could appear in the pores. Many studies have considered this additional pressure to provide a more realistic prediction of structural behavior. For example, Galeban et al. [12] used Euler-Bernoulli assumptions to study free vibration of functionally graded (FG) thin beams made of saturated porous materials. Rad et al. [13] employed both first-order and third-order shear deformation theories to study elastic buckling of fluid-infiltrated porous plates. Ebrahimi and Habibi [14] predicted the deflection and vibration characteristics of rectangular plates made of saturated porous FG materials with the framework of third-order shear deformation plate theory. Based on 3D elasticity theory, Babaei et al. [15] investigated natural frequencies and dynamic response of thick annular sector plates and cylindrical panels made of saturated porous materials. Stress analysis of FG saturated porous rotating truncated cone by finite element modeling and 2-D axisymmetric elasticity theory was presented by Babaei and Asemi [16]. Rezaei and Saidi [17] attempted to study free vibration response of fluid-saturated porous annular sector plates based on Mindlin plate theory. Heshmati et al. [18] studied the vibration behavior of poroelastic thick curved panels with graded open-cell and saturated closed-cell porosities. Using classical plate theory, Panah et al. [19] examined pore fluid pressure and porosity effects on the bending and thermal postbuckling behavior of FG saturated porous circular plates. Sandwich structures with porous core, also called porous sandwich (PS) structures, is an ideal solution for lightweight structures while maintaining structural performance. The two face sheet layers are thin, but strong and stiff to withstand the bending and compression/tension in the plane. Meanwhile, the thick core mainly sustains the transverse shear and connects the face sheet layers. Moreover, the thick inner core with porosity reduces self-weight and plays roles as a heat diffuser, energy absorber, sound insulation sheet or vibration damper. With all the mentioned advantages, PS structures have become more and more popular in structural design. Mechanical behavior of PS structures has received lots of attention from researchers. Chen et al. [20] investigated nonlinear free vibration of shear deformable sandwich beams with an FG porous core based on the context of Timoshenko beam theory. Bamdad et al. [21] investigated free vibration and buckling of sandwich Timoshenko beam with porous core and composite face sheets subjected to electric and magnetic fields and resting on elastic foundation. Mu and Zhao [22] analyzed the natural vibration of composite sandwich beams with FG face sheets and a metal foam core. Jasion et al. [23] studied the global and local buckling-wrinkling of the face sheets of sandwich beams and sandwich circular plates with metal foam core. Using three modified Timoshenko hypotheses, Magnucka-Blandzi [24] presented the dynamic stability and static stress state of a simply supported sandwich beam with a metal foam core. Qin and Wang [25] studies the large deflections of fully clamped and simply supported slender sandwich beams with a metallic foam core subjected to transverse loading by a flat punch. Zhang et al. [26] investigated the large deflection response and the optimal design of slender multilayer metal sandwich beams with metal foam core subjected to low-velocity

Static Behavior of Functionally Graded Sandwich Beam

693

impact. Wang et al. [27] proposed a high-order shear deformation theory to investigate the transient response of a sandwich beam with FG face sheets and FG porous core under moving mass. Chinh et al. [28] studied the static flexural of sandwich beam with FG face sheets and FG porous core by third-order shear deformation beam theory. Based on the first-order shear deformation plate theory, Chen et al. [29] examined the buckling and bending behavior of sandwich plates with uniform and FG porous cores. Akbari et al. [30] employed differential quadrature method and third-order shear deformation theory to study the free vibration of sandwich cylindrical panels made of an FG saturated porous core and two similar homogenous face sheets. The literature reviews show that: (i) researches on PS structures taking pore pressure of fluid phase into account are extremely rare when compared with single-layer porous structure. There has been one work by Akbari et al. [30] for sandwich cylindrical panels available in the open literature; (ii) there has been a few reports, i.e., [20, 27–29], regarding the structural layers so that ensuring mechanical continuity at the interfaces, which is a requirement for designing sandwich structures in order to eliminate the stress concentration phenomenon at the contact between layers; (iii) the beam theories are employed almost individually in each of the studies. There has not been a comparison of their effects on the results of simulating the behavior of the sandwich structures with porous core yet. This issue is worth examining, particularly for shear deformation beam theories because shear stress distribution in the porous core may be different from that in monolithic one. This study attempts to analyze the static behavior of sandwich beams with FG face sheets and an FG fluid-infiltrated porous core. The material of the layers is tailored so that its mechanical properties vary continuously along the beam depth in order to avoid the stress concentration at the interfaces between layers. Different beam theories are considered for 1-D beam modeling and their effects on the analysis results are examined. The constitutive equation is based on Biot’s theory of linear poroelasticity. The governing equations of the beams are derived from Hamilton’s principle and solved analytically by Navier’s solution. Comparative studies are carried out to validate the analysis. Effects of power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam are investigated in detail.

2 Theory and Formulations 2.1 Sandwich Beam and Material Modelling A PS beam of dimensions L × b × h as shown in Fig. 1 is considered in this report. The z-axis is taken along the height (h) of the beam, and the x-axis coincides with the mid-plane. The two similar face sheet layers are made of an FG material, whereas a fluid-infiltrated FG porous material is adopted to construct the core. For convenience, the thickness ratio among the face sheet-core-face sheet is used to name the sandwich beam. For instance, 1-8-1 beam denotes the sandwich beam whose core thickness is eight times the face thickness (hc /hf = 8). Young’s modulus of two FG face sheets is assumed to alter according to the power-law form, whereas that of FG porous core varies according to a cosine rule as follows:

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z

FG material

q(x)

hf hc hf

x

o FG porous material

FG material

h4 = h/2 h3 h2

h

h1=-h/2

b

L Fig. 1. Geometric parameters and coordinate system of PS beams

⎧   h −z k ⎪ + Ec , z ∈ [h3 , h4 ] (top face sheet layer) E (3) (z) = (Em − Ec ) h 4−h ⎪ ⎪ 4 3 ⎨   z E (2) (z) = Em 1 − eo cos h π−h , z ∈ [h2 , h3 ] and h2 = −h3 (porous core layer) , 3 2 ⎪ ⎪   ⎪ (1) ⎩ z−h1 k E (z) = (Em − Ec ) h −h + Ec , z ∈ [h1 , h2 ] (bottom face sheet layer) 2

(1)

1

where E c and E m are, respectively, Young’s moduli of the constituents; k is the powerlaw index, k ≥ 0; e0 is the porosity coefficient (0 ≤ eo < 1). Equation (1) shows that Young’s modulus E varies smoothly along the thickness and is symmetric with respect to the mid-plane (z = 0); the super script (i) denotes the i-th layer. It should be noted that Poisson’s ratio v is assumed to be constant for each layer and shear modulus G(z) = E(z)/(2 + 2ν). 2.2 Displacement and Strain Fields Based on quasi-3D beam theory, the displacement components u(x, z), w(x, z), which are, respectively, along the x- and z-directions, can be expressed by the following equations [31]:

o (x) + Φ(z)ϕos (x) u(x, z) = uo (x) − z ∂w∂x (2) w w(x, z) = wo (x) + ∂Φ(z) oz (x) ∂z In Eq. (2), uo , wo are the axial and transverse displacements on the mid-plane (z = 0), respectively; ϕos is the transverse shear strain of any point on the x-axis; Φ is the shape function which characterizes the distribution of the shear strain through the beam depth, Φ = z − 4z3 / 3h2 is selected for this study. The strain-displacement relations can be given by ⎧ ∂ϕos ∂uo ∂ 2 wo ∂u ⎪ ⎪ ⎨ εx = ∂x = ∂x2 − z ∂x2 + Φ ∂x ∂ Φ εz = ∂w (3) ∂z = ∂z 2 woz   ⎪ ⎪ ∂w ∂u ∂w ∂Φ oz ⎩ γxz = ϕos + + = ∂z

∂x

∂z

∂x

It is observed that Eqs. (2) and (3) are expressed for quasi-3D; they can, however, be simplified to other beam theories by adjusting functions Φ and woz . For example, Φ

Static Behavior of Functionally Graded Sandwich Beam

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= 0 for classical beam theory (CBT); Φ = z and woz = 0 for first-order beam theory (FBT); Φ = z − 4z3 / 3h2 and woz = 0 for third-order beam theory (TBT). In other words, quasi-3D theory could be considered as a general form of the four typical beam theories mentioned. 2.3 Constitutive Equation For fluid-filled porous materials, the stress-strain relations obey the linear poroelasticity theory of Biot, which can be express as [4] σij = 2Gεij + λθ δij − pαδij

(4)

in which σij and εij are the stress and strain components, respectively; δ ij is Kronecker delta; θ is the volumetric strain; α is the Biot coefficient of effective stress; p is the pore fluid pressure; λ is Lame parameter. The quantities p and α are defined as p = M (ξ − αθ ); α = 1 − GG0 , 2G(νn −ν) Eo ; M = α 2 (1−2ν ; νn = ν+αB(1−2ν)/3 Go = 2(1+ν) 1−αB(1−2ν)/3 )(1−2ν)

(5)

n

where ξ is the variation of fluid volume content; M is Biot’s modulus; ν n is undrained Poisson’s ratio; B is the Skempton coefficient; E o is the elastic modulus of the material with no porosity (eo = 0). For the undrained condition (ξ = 0) and quasi-3D theory with plane stress condition (σ y = 0), Eq. (4) can be written in the following matrix form: ⎧ ⎫ ⎡ ⎤⎧ ⎫ Q11 Q12 0 ⎨ εx ⎬ ⎨ σx ⎬ {σ } = σz = ⎣ Q21 Q22 0 ⎦ εz , (6) ⎩ ⎭ ⎩ ⎭ τxz γxz 0 0 Q33 where elastic coefficients are given as Q11 = Q22 =

2G(z) ; 1 − νn

Q12 = Q21 =

2νn G(z) = Q11 νn ; Q33 = G(z) 1 − νn

(7)

The other beams theories (CBT, FBT and TBT) can also use Eq. (6) by setting the coefficients as Q11 = 2(1 + νn )G(z);

Q12 = Q21 = 0;

Q33 = G(z)

(8)

If pore fluid pressure is neglected (p = 0), Eq. (4) reduces to conventional Hook’s law. Thus, Eqs. (4–8) can be used for the monolithic material of the face sheets by setting B = 0 or ν n = ν. 2.4 Energy Expressions The variation of the strain energy can be stated as L  (σx δεx + σz δεz + τxz δγxz )dAdx

δU = 0 A

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  L  ∂δuo ∂ 2 δwo ∂δϕos ∂δwoz N − Mb + Rs δwoz + Q δϕos + dx, δU = + Ms ∂x ∂x2 ∂x ∂x 0

(9) where the stress resultants are defined as N = Q=

 A A

σx dA;

Mb =

τxz ∂Φ ∂z dA



zσx dA; Ms =

A



Φσx dA; Rs =

A

 A

∂2Φ σ dA; ∂z 2 z

(10)

In Eqs. (9) and (10), A is the area of the beam cross-section, and L is the length of the beam. By substituting Eqs. (3) and (6) into Eq. (10), the stress resultants can be expressed by matrix form ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ ⎪ ∂uo /∂x A1 B1 Bs C1 0 ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎥ 2 2 ⎪ ⎪ ⎪ ⎪ D Ds C2 0 ⎥⎨ −∂ wo /∂x ⎬ ⎨ Mb ⎬ ⎢ ⎢ ⎥ , (11) Ms = ⎢ ∂ϕos /∂x Hs C3 0 ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ R 0 C w s ⎪ 4 oz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩Q ⎪ sym. As ⎩ (ϕos + ∂woz /∂x) ⎭ where ⎧     2 A1 = Q11 dA; B1 = zQ11 dA; Bs = ΦQ11 dA; C1 = Q12 ∂∂zΦ2 dA ; ⎪ ⎪ ⎪ ⎪ A A A A  2  ⎪ ⎪ ⎪ ⎪ ⎨ D = z Q11 dA; Ds = zΦQ11 dA; A A    2 2 C2 = Q12 z ∂∂zΦ2 dA ; Hs = Φ 2 Q11 dA; C3 = Q12 Φ ∂∂zΦ2 dA; ⎪ ⎪ ⎪ ⎪ A A A ⎪  2 2 ⎪    2 ⎪ ⎪ ⎩ C4 = Q22 ∂∂zΦ2 dA; As = G ∂Φ dA ∂z A

(12)

A

The variation of the potential energy under a transverse load q acting on the top surface of the beam can be stated as L δV = −

qδwo dx

(13)

0

2.5 Hamilton’s Principle and Governing Equations Hamilton’s principle is used to derive the governing equations of the beam. This principle can be expressed for the case of static equilibrium as t2 (δU + δV )dt = 0 t1

(14)

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Substituting the expressions δU and δV from Eqs. (9) and (13) into Eq. (14) and then integrating by parts, and collecting the coefficients of δuo , δwo , δϕ os and δwoz , the governing equations of the beam are obtained:  δuo :

∂N ∂ 2 Mb ∂Ms ∂Q = 0; δwo : + Q = 0; δwoz : Rz − =0 + q = 0; δϕos : − ∂x ∂x2 ∂x ∂x

(15)

Next, substituting Eq. (11) into Eq. (15), the governing equations can be expressed as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

oz A1 ∂∂xu2o − B1 ∂∂xw3o + Bs ∂∂xϕ2os + C1 ∂w ∂x = 0 2

B1 ∂∂xu3o − D ∂∂xw4o + Ds ∂∂xϕ3os +C2 ∂ ∂xw2oz +  q=0 ∂ 2 ϕos ∂ 3 wo ∂woz ∂woz − Ds ∂x3 + Hs ∂x2 + C3 ∂x − As ϕos + ∂x = 0   2 ∂ϕos ∂ 2 woz os =0 + C2 ∂∂xw2o − C3 ∂ϕ − C w + A + 4 oz s ∂x ∂x ∂x2 3

⎪ Bs ∂∂xu2o ⎪ ⎪ ⎪ ⎪ ⎩ −C1 ∂uo ∂x 2

2

3

3

4

2

(16)

2.6 Analytical Solutions Navier’s technique based on Fourier series is used to determine the analytical solution for a simply supported beam. Assuming each of the displacement components uo , wo , ϕ os and woz is trigonometric series, which satisfies the boundary conditions, as follows ⎧ ∞ ∞   ⎪ ⎪ ⎪ u U cos(λx); w Wn sin(λx) = = (x) (x) ⎪ n o ⎪ ⎨ o n=1 n=1 (17) ∞ ∞ ⎪   ⎪ ⎪ ⎪ ϕos (x) = Φn cos(λx); woz (x) = Wzn sin(λx) ⎪ ⎩ n=1

n=1

where U n , W n , Φ n and W zn are the unknown coefficients which need to be determined; (λ = nπ/L). The uniformly distributed load qo is also expanded in Fourier series as q(x) =

∞  n=1

Qn sin(λx); Qn =

4qo nπ

(n = 1, 3, 5, ...),

(18)

Substituting Eqs. (17) and (18) into Eq. (16) yields the following analytical solution: ⎤⎛ ⎞ ⎛ ⎞ ⎡ Un 0 s11 s12 s13 s14 ⎟ ⎜ ⎜ ⎟ ⎢ s22 s23 s24 ⎥ ⎥⎜ Wn ⎟ = ⎜ −Qn ⎟, ⎢ (19) ⎦ ⎠ ⎝ ⎝ ⎣ s33 s34 Φn 0 ⎠ sym. s44 Wzn 0 where ⎧ s11 ⎪ ⎪ ⎨ s22 ⎪ s ⎪ ⎩ 33 s44

= −λ 2 A1 ; s12 = λ 3 B1 ; s13 = −λ 2 Bs ; s14 = λC1 ; = −λ 4 D; s23 = λ 3 Ds ; s24 = −λ 2 C2 ; = −λ 2 Hs − As ; s34 = λC3 − λAs ; = −C4 − λ 2 As

(20)

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The unknown series coefficients are obtained by solving Eq. 19. Having known the coefficients, the displacements and stresses can be straightforwardly calculated by using Eqs. 6 and 17.

3 Results and Discussion Based on mathematical modelling, numerical examples for the simply supported beam subjected to uniform load qo are conducted to validate and comprehend the static responses. Four different beam theories (CBT, FBT, TBT and quasi-3D) are considered in the analysis. In the case of FBT, shear correction factor is taken to be 5/6 for the beam of rectangular cross-section. In this study, assuming that the FG face sheets are composed of ceramic (E c = 380 MPa, ν c = 0.3) and metal (E m = 70 MPa, ν m = 0.3) [32], whereas the porous soft-core are made of a metal foam (E m = 70 MPa, ν m = 0.3) - referred from [16]. Following non-dimensional parameters are used to present all numerical results: ⎧       ⎨ wˆ = 100Em4bh3 w L , 0 ; wˆ oz = 100Em4bh3 woz L ; uˆ = 100Em4bh3 u 0, − h ; 2 2 2 qo L  qo L qo L (21)   ⎩ σˆ x = bh σx L , h ; σˆ z = bh σz L , z ; τˆxz = bh τxz (0, 0) qo L 2 2 qo L 2 qo L 3.1 Validation Firstly, the validation is done for the specific case of the investigated beam having a hardcore without porosities by setting the porosity coefficient eo to zero and interchanging the roles of E m and E c in Eq. (1). This problem was studied by Vo and his co-workers [32] using the analytical solution and the finite element method, and by Sayyad et al. [33] using Navier’s solution. It is seen that the obtained results given in Table 1 are almost identical to those of Vo et al. [32] and agree very well with Sayyad et al. [33]. For further validation, another example is carried out for PS beam without pore fluid pressure by setting the Skempton coefficient B = 0 or ν n = ν for the porous core. The present results are compared with those of Chinh et al. [28] which are based on TBT and mesh-free method. The comparison of results in Table 2 shows an excellent agreement between them. Table 1. Comparison of the displacement, axial stress and shear stress for 1-2-1 hardcore beam with different beam theories (L/h = 5, eo = 0) Theory CBT FBT

Source

k=1

k=5



σˆ x

τˆxz



σˆ x

τˆxz

Present

5.0798

1.2192



8.1409

1.9538



Vo et al. [32]

5.0798





8.1409





Present

5.4408

1.2192

0.6268

8.5762

1.9538

0.7559

Sayyad et al. [33]

5.3807

1.2192

0.6183

8.5036

1.9539

0.7457

Vo et al. [32]

5.4408

1.2192

0.7507

8.5762

1.9538

0.9053 (continued)

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Table 1. (continued) Theory TBT Quasi-3D

Source

k=1

k=5



σˆ x

τˆxz



σˆ x

τˆxz

Present

5.4122

1.2329

0.8124

8.5136

1.9705

0.8926

Vo et al. [32]

5.4122

1.2329

0.8123

8.5137

1.9705

0.8925

Present

5.3611

1.2315

0.7993

8.4276

1.9672

0.8763

Vo et al. [32]

5.3612

1.2315

0.7993

8.4276

1.9672

0.8763

Table 2. Comparison of the displacement wˆ for 1-1-1 PS beam (k = 0.5, TBT, B = 0) Source

L/h eo 0

Present

Chinh et al. [28]

0.4

0.6

0.8

5

6.3807 6.4530 6.5342 6.6263 6.7316

20

5.3496 5.3586 5.3682 5.3785 5.3896

Chinh et al. [28] Present

0.2

6.3807 6.4530 6.5342 6.6263 6.7316 5.3496 5.3586 5.3682 5.3785 5.3896

3.2 Comprehensive Study Table 3 is used to present the displacement, axial stress and shear stress values of the 1-81 beam with fluid-infiltrated FG porous core. Two values of span-to-height ratio (L/h = 5, which represents for thick beams, and L/h = 20, which represents for thin ones), different values of porosity coefficient eo and different beam theories are considered. As expected, CBT, without shear deformation effect, gives the smallest displacements wˆ of all beam theories; the displacements (ˆu, w) ˆ calculated by quasi-3D are slightly smaller than those calculated by TBT. Those two effects are more pronounced for the thick beam (L/h = 5) than for the thin one (L/h = 20). Interestingly, CBT and FBT give the same values of uˆ as well as of σˆ x . In addition, due to the different shape functions Φ, the values of shear stress obtained by FBT are distinctly different from those obtained by TBT/quasi-3D. Moreover, as increasing the value of the porosity coefficient eo , both the displacements uˆ and wˆ increase. This is because it reduces the stiffness of the beam. Besides, the deviation of the calculated results among the beam theories increases with the increase in eo . For example, for eo = 0 and L/h = 5, the relative difference of wˆ between FBT and TBT is (wˆ TBT − wˆ FBT )/wˆ TBT × 100% = (10.1349 − 9.9604)/9.9604 × 100% = 1.75%, but for the same situation with eo = 0.9 that is (13.6431 − 11.4876)/11.4876 × 100% = 18.76% (the data are taken from Table 3). Unlike CBT, FBT and TBT, the displacement wˆ oz exists for quasi-3D theory, which including thickness stretching effect. This displacement versus L/h ratio for 1-1-1 and 1-8-1 schemes with different values of eo is illustrated in Fig. 2. It is seen that eo has little effect, whereas L/h and sandwich schemes have a strong effect on wˆ oz . The displacement

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Table 3. Values of displacements, axial stress and shear stress of 1-8-1 PS beam (B = 0.5, k = 0.5) L/h = 5

L/h = 20

eo

Theory

0

CBT

8.7560 2.8019 11.4078 –

FBT

9.9604 2.8019 11.4078 0.3852 8.8313

TBT

10.1349 2.8659 11.5866 0.6819 8.8423

0.7015 45.6760 0.6967

Quasi-3D 10.0671 2.8276 11.5907 0.6745 8.8383

0.7009 45.6770 0.6949



0.3 CBT



σˆ x

τˆxz

9.0211 2.8867 11.7532 –





8.7560

0.7005 45.6313 –

9.0211

σˆ x

τˆxz

0.7005 45.6313 0.3852

0.7217 47.0127 –

FBT

10.3866 2.8867 11.7532 0.3057 9.1064

0.7217 47.0127 0.3057

TBT

10.7751 2.9643 11.9702 0.6099 9.1309

0.7229 47.0669 0.6241

Quasi-3D 10.7062 2.9248 11.9685 0.6041 9.1265

0.7222 47.0303 0.6228

0.6 CBT

9.3092 2.9789 12.1285 –

9.3092

0.7447 48.5141 –

FBT

10.8855 2.9789 12.1285 0.2017 9.4077

0.7447 48.5141 0.2017

TBT

11.7392 3.0804 12.4134 0.4848 9.4614

0.7464 48.5853 0.4973

Quasi-3D 11.6691 3.0396 12.4110 0.4808 9.4563

0.7456 48.5187 0.4964

0.9 CBT

9.6235 3.0795 12.5380 –

9.6235

0.7699 50.1522 –

FBT

11.4876 3.0795 12.5380 0.0596 9.7400

0.7699 50.1522 0.0596

TBT

13.6431 3.2361 12.9802 0.2010 9.8754

0.7724 50.2627 0.2071

Quasi-3D 13.5721 3.1941 12.9903 0.1996 9.8697

0.7716 50.1808 0.2068

wˆ oz decreases and approaches zero when increasing L/h. Furthermore, 1-1-1 beam with higher stiffness has smaller wˆ oz than that of 1-8-1 one. 1-8-1 beam 1-8-1 beam 1-1-1 beam

1-1-1 beam

L/h

L/h

Fig. 2. The displacement wˆ oz of 1-1-1 and Fig. 3. The displacement wˆ of 1-1-1 and 1-8-1 1-8-1 beams with respect to L/h (B = 0.5, k = beams with respect to L/h (B = 0.5, k = 1) 1)

Variation of the non-dimensional displacement wˆ at the center of the beam with respect to L/h ratio is depicted in Fig. 3. Two schemes of the sandwich beam (1-1-1 and 1-8-1 beams), together with four different beam theories, are considered. As can be seen,

Static Behavior of Functionally Graded Sandwich Beam

701

1-8-1 beam

1-1-1 beam

eo

Fig. 4. The displacement wˆ of 1-1-1 and 1-8-1 beams with respect to eo (B = 0.5, k = 0.5, L/h = 5)

k

Fig. 5. The displacement wˆ of 1-8-1 beam with respect to k (quasi-3D, B = 0.5, L/h = 5)

B Fig. 6. The displacement wˆ of 1-8-1 beam with respect to B (quasi-3D, k = 0.5, L/h = 5)

the effect of the different beam theories on the obtained results is important, especially for the thick beams (small L/h). This effect, however, can be neglected for the thin beams. Figure 3 also shows that the results calculated using TBT are not significantly discrepant with those using quasi-3D, but clearly different from those using FBT. In other words, there is a significant difference in the outcome between FBT and TBT/quasi-3D. In Fig. 4, the displacement wˆ versus the porosity coefficient eo for 1-1-1 and 1-8-1 beams is displayed. The displacement increases with increasing eo . This trend is more pronounced in thick core beam (1-8-1 beam) than in thin core one (1-1-1 beam). It is also unveiled that for thick core beam, the computed results employing TBT/quasi-3D increases more rapidly with respect to eo than employing FBT/CBT. Figure 5 portrays the transverse displacement wˆ versus power-law index k with different values of eo . Only quasi-3D theory and 1-8-1 beam with L/h = 5 are performed for this investigation. It can be seen that for each of eo , the greatest displacement is obtained when k = 0. This is because the beam is full of metal and has the smallest stiffness. As k increases, the volume fraction of the ceramic phase constituting the face sheets, which has much higher Young’s modulus, increases. Thus, it reduces quickly the displacement.

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hc/hf

hc/hf b) L/h = 20

a) L/h = 5

Fig. 7. The displacement wˆ with respect to hc /hf (quasi-3D, k = 0.5, B = 0.5)

Effect of pore fluid pressure through the Skempton coefficient B on the transverse displacement wˆ is depicted in Fig. 6. It is revealed that increasing B results in a slight reduction of w, ˆ except for eo = 0. However, this effect is not significant. Effect of the layer thickness ratio (hc /hf ) on the transverse displacement wˆ of PS beams with different values of eo is displayed in Fig. 7. It is pointed out that the displacement increases with increasing hc /hf . The reason is that the beam becomes softer as the porous core is thicker. Besides, the gap between the two adjacent curves increases with the increase in L/h. It proves that the effect of eo on the response of the beam is more sensitive for the thick porous core beams (high hc /hf ) than for the thin core ones. It is also observed that this gap is greater for the thick beam (Fig. 7a) than for the thin one (Fig. 7b).

k

L/h

Fig. 8. Effect ofk and L/h on transverse displacement wˆ of 1-1-1 and 1-8-1 beams (quasi-3D, eo = 0.6, B = 0.5)

Effect of the interaction between k and L/h ratio on the displacement wˆ is plotted in 3 dimensions in Fig. 8. Again, the strong effect belongs to the region having small values of k or L/h.

Static Behavior of Functionally Graded Sandwich Beam

z/h

703

z/h

a) Axial stress

b) Shear stress

Fig. 9. Variation of stresses σˆ x , τˆxz over the beam depth (1-8-1 beam, eo = 0.6, B = 0.5, k = 2, L/h = 5)

Axial and shear stresses through the beam depth for different beam theories are illustrated in Fig. 9. As expected, the stresses are distributed continuously throughout the beam depth. It is clear that in Fig. 9, the influence of the beam theories considered on the axial stress is negligible. However, due to the difference of the shape functions Φ, the shear stress obtained by FBT is entirely different to that obtained by TBT/quasi-3D. It is interesting to note that the maximum shear stress belongs to the face sheets instead of the core.

z/h

Fig. 10. Variation of normal stress σˆ z over the beam depth (1-8-1 beam, B = 0.5, k = 2, L/h = 5)

For quasi-3D theory, the variation of the normal stress caused by the thickness stretching effect is shown in Fig. 10. As expected, this stress exists but is small when compared with the axial stress in Fig. 9a. Besides, this stress focuses on the face sheets and is anti-symmetrically distributed with respect to the mid-plane (z = 0) for the studied beam.

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4 Summary and Conclusions Analytical solution together with four different beam theories (i.e., classical, first-order, third-order and quasi-3D theories), which are unified in quasi-3D theory, is presented for static behavior analysis of porous sandwich beam. The beam is constructed of two FG face sheets and a fluid-infiltrated FG porous core so that mechanical properties vary smoothly in the thickness direction. Stress-strain relations obeys Biot’s theory of linear poroelasticity. Hamilton’s principle is used to derive the governing equations of the beam. The accuracy of the study is validated. Comprehensive studies are conducted to investigate the effects of parameters, including power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam. Some major conclusions can be reached from the study: (1) Sandwich beam with face sheets made of FG materials shows the effectiveness in both increasing the stiffness and satisfying the continuous condition of mechanical properties at the interface of layers; (2) Pore fluid pressure has no significant effect on the static response of the sandwich beam with fluid-infiltrated FG porous core; (3) For thick beams, the effects of different beam theories on the obtained results of the analysis of displacements and shear stress are significant. However, their effects on axial stress are negligible; (4) Static response predicted by third-order shear deformation beam theory and that by quasi-3D shear deformation beam theory is slightly different; (5) There is a considerable difference in the calculated outcomes between first-order and third-order/quasi-3D beam theories, especially for the thick beam with thick core and high porosity.

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Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials Trung-Kien Nguyen(B) Faculty of Civil Engineering and Industrial Construction, National University of Civil Engineering, 55 Giai Phong Road, Hanoi, Vietnam [email protected]

Abstract. An innovative approach that combines the Finite Element Method (FEM) and Discrete Element Method (DEM) has been applied for investigating the failure of a pressurized hollow cylinder made of cohesive-frictional granular materials. At the microscopic level, a Volume Element (VE) by granular assembly is used to describe the discrete nature of granular media. DEM-based model is defined through this VE. At the macroscopic level, we use FEM to simulate the boundary value problem. Two levels are then bridged through numerical homogenization. In this way, the mechanical problem could be studied at both macro-scale (finite element mesh) and micro-scale (grain interaction). It is well-known in continuum modeling by FEM that when bifurcation occurs, the numerical solutions lose their uniqueness if only first gradient model is used. To preserve the objectivity of the solution, a local second gradient is employed and shown to efficiently regularize the problem. The numerical results by the combined FEM×DEM modeling indicate that when the failure occurs, the shear band is reproduced numerically in the case of pressurized hollow cylinder. Moreover, the second gradient parameter gives an implicit internal length which is directly related to the width of the shear band. Keywords: FEM · DEM · Second gradient model · Pressurized hollow cylinder · Granular materials

1 Introduction An innovative approach that combines the Finite Element Method (FEM) and the Discrete Element Method (DEM) (FEM×DEM approach) has been developed since the last decade [1–4]. The main interest of this method lies in a numerical homogenization process that bridges the gap between macro- and micro-scales. At the microscopic level, DEM-based model is used to describe the constitutive relation of the material while at the macroscopic level, FEM model is used. By the way, continuum modeling can take into account the discrete nature of the materials. This method has been successfully applied in modeling the behavior of granular materials and real-scale boundary value problem made of granular materials. However, most of them used first gradient model at the macroscopic level, which shows the non-objectivity of the solution. Especially when strain localization is observed, the result shows a strong mesh dependency. This is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 707–715, 2022. https://doi.org/10.1007/978-981-16-3239-6_54

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because the first gradient model does not consider the microstructure of the material. In this case, an enriched model such as the second gradient model has to be considered to overcome this limitation. Several second gradient approaches have been developed in the literature [5–8]. They are mainly classified into non-local and local families. Regarding the local approach, Chambon and co-workers [9, 10] developed a local second gradient model and applied to 2D finite element modeling. This local approach is believed to be coupled with various first gradient constitutive models. Pressurized hollow cylinder is a well-known problem that its basic solution could be found in every textbook. However, the numerical solution that accounts for the complex behavior of a hollow cylinder made of geomaterials remains an interesting question, which was not fully answered in the literature. The problem of a hollow cylinder can be related to borehole stability or gallery behavior in the field of geotechnical engineering. To demonstrate the capabilities of the FEMxDEM approach to reproduce the complexities of geomaterials in a real scale problem and the advantages of local second gradient in preserving the objectivity of the solution, this paper aims at simulating and analyzing such phenomenon through the bias of failure of a pressurized hollow cylinder made of cohesive-frictional granular materials. Moreover, as the strain localization is believed to suffers in both first and second gradient finite element models, the paper aim also to demonstrate the effect of the second gradient model in regularizing mesh dependency problems. The rest of the paper is structured as follows: After the introduction, Sect. 2 briefly introduces local second gradient model formulation, coupled with the FEM×DEM approach. Section 3 analyzes the failure of the hollow cylinder made of cohesive-frictional granular materials with an emphasis on the influence of the material-made density and second gradient parameters. Conclusion and remarks are given in the last section.

2 Second Gradient Model Formulation Before entering the text of second gradient model formulation, we briefly summarize the principle of FEM×DEM approach. At the microscopic level, a Volume Element (VE) by the granular assembly is used to describe granular materials and served as DEMbased constitutive law. At the macroscopic level, we use FEM to resolve the boundary value problem. Unlike conventional FEM, numerical resolution by FEMxDEM approach required two levels of numerical convergence. One is numerical convergence at microscale, i.e. convergence of DEM calculation. This DEM convergence criterion requires that the granular assembly is stable enough after being submitted to deformation applied. It is controlled through the total kinetic of the system. When the scale is bridged from the micro-scale (DEM) to macro-scale (FEM) through the numerical homogenization process, traditional convergence on residual force and displacement are required at the macro-scale (FEM). These two levels of convergence ensure that the whole problem (both macro- and micro-scales) reaches an equilibrium state at the end of each loading step. Local second gradient has been developed in laboratory 3SR, Grenoble over two decades. For more details about the formulation and model, the readers can refer to [9, 10]. We recall hereafter the principal elements with emphasis on the first gradient

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constitutive relation based on DEM. It can be noted that the current DEM model is fully developed for 3D cohesive-frictional granular materials and it can be used either in 3D FEM modeling or 2D FEM plane stress/strain problems. We limit this paper to the 2D problem at the macroscopic level. 2.1 Variational Principle We present the weak form of the balance equations for a given body  as follows [10].    ∗ 2 ∗ t ∂ui t ∂ ui (1) σij t + ijk t t d t − Pe∗ = 0 ∂xj ∂xj ∂xk t     ∂u∗ ρ t fit ui∗ d t + (2) Pe∗ = pit ui∗ + Pit i nk d  t ∂xk t ∂t t are Cauchy and double stress tensors, respectively; P ∗ is the external Where σijt , ijk e ∗ virtual work. ui is a virtual displacement field; ρ t , fit are mass density and body force; pit , Pit are classical and double external forces; ∂ is the boundary of .

2.2 Equations with Lagrange Multipliers To avoid the difficulties while solving the boundary value problems by finite element with second gradient models, it commonly weakens the constraint by introducing Lagrange multipliers, noted λtij . Equation (1) is therefore re-written:       ∗ ∗ ∗ 2 u∗ ∂u ∂u ∂ ∂u t i i λtij − it d t − Pe∗ = 0 (3) σijt it + ijk d t − ∂xj ∂xjt ∂xkt ∂xjt ∂xj t t  Moreover, the fields ui and νij = ∂ui ∂xj have to meet the following weak form constraint:    t ∂uit ∗ ∂ui λij − t d t = 0 (4) ∂xjt ∂xj t The external virtual power could be expressed as:    t ∗ pi ui + Pit νik∗ ntk ds ρ t fit ui∗ d t + Pe∗ = t

∂t

(5)

This result leads to assuming that the constitutive relation of the first and second t function of ν . gradient is as: σijt function of ui and ijk ij 2.3 Finite Element: First and Second Quadratic Elements Finite elements used in this study are depicted in Fig. 1, including the first gradient (FG) and the second gradient (SG) quadrilateral elements. The FG element has eight nodes for ui while SG element has eight nodes for ui , four nodes for νij , and one node for λij . Both types of elements are used in the numerical simulations in the following section.

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Fig. 1. First and Second gradient quadrilateral elements (left) and transformed elements (right)

2.4 Constitutive Relations a. First gradient part Regarding the first gradient part, we used Numerical Homogenized Law based on Discrete Element Modeling (DEM-based model). Spherical particles are used as described in Fig. 2. The normal and tangential contact forces are noted fn and ft , respectively. They are calculated as follows: fn = −kn · δ + fc

(6)

ft = −kt · ut

(7)

Where δ is the overlap between two contacted particles; ut is the relative tangential displacement. To reproduce cohesive-frictional granular materials, a local cohesion fc is introduced. Tangential contact force is limited by Coulomb threshold as: ft ≤ μ · |fn | = tan ϕ · |fn | where μ is the intergranular coefficient of friction and ϕ is the friction angle. The stress tensor σ is determined by the homogenized formula: 1   σ = f ⊗l V

(8)

(9)

C

where f is contact force and l is the branch vector joining the centers of two particles in contact; V is the volume of granular assembly and C is the contact list; f ⊗ l denotes the tensor product of two vectors f and l.

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Fig. 2. Contact model

b. Second gradient part For the second gradient part, we use the particular case of isotropic linear relation obtained by [9, 11]. This latter involving only one parameter noted D. This parameter is linked implicitly to the internal length of the model [10]. ∇ ∂ v˙ ij (10)  = [D] ∂xk ijk ∇

where  is Jaumann derivative of double stress. ijk

3 Failure Analysis of Pressurized Hollow Cylinder 3.1 Pressurized Hollow Cylinder Problem The cross-section behavior of the pressurized hollow cylinder problem is simulated in this section. Thanks to the axisymmetric conditions, only one-fourth of the crosssection is considered. The problem is 2D plane strain, stress-controlled by external (pe ) and internal (pi ) pressures. In order to investigate the behavior of the hollow cylinder when increasing the inner pressure, in these simulations, the external pressure is kept constant and equal to initial isotropic stress pe = σ0 = const. The inner pressure is increasing from initial pressure (σ0 ) up to the failure of the specimen (Fig. 3). The inner and outer radius of the hollow cylinder are ri = 10 mm and re = 40 mm. Regarding finite element mesh, first gradient and second gradient quadratic elements are used. The domain is discretized in 870 elements (FG or SG element). At the microscopic level, a granular assembly of 1000 spherical grains is used to build the first gradient constitutive relation. These grains interact via contact model as described in  Sect. 2.  Microscopic parameters are chosen based on dimensional analysis: kn kt = 1; κ = kn σ0 = 500 and μ = 0.5. The influence of the internal length of the second gradient model is investigated thanks to the second gradient parameter D.

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Fig. 3. Pressurized hollow cylinder problem

3.2 Failure Analysis a. Influence of material-made density: Dense (DS) and loose samples (LS) In this section, first gradient finite element model is used to highlight the influence of material-made density on the macroscopic behavior of the hollow cylinder. The density of the material is characterized by the density of granular assembly (i.e. Volume Element at Gauss point). It is classified into dense and loose samples. Both samples have the same grain size distribution but different solid fractions. The dense sample has a solid fraction of 0.641 while the loose sample has a solid fraction of 0.560. Figure 4 illustrates the equivalent strain in DS and LS cases atfailure. The failure occurred earlier in the case of loose sample. It can be seen that at pi pe = 3.0, the shear band was formed in LS case. In DS case, as the internal pressure continued to increase,  the failure developed and strain localization was also observed at pi pe = 4.0. In both cases, the shear bands were initiated and propagated from the inner surface. It can be noted that in first gradient modeling, as failure develops if strain localization occurs,

Fig. 4. Influence of density: dense and loose samples

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the width shear band shrinks to mesh size element. The shear band in spiral form is in agreement with numerical/experimental works performed by [12–16]. b. Influence of internal length Parametric studies have been done by varying the second gradient parameter D. This latter requires that D give an (implicit) internal length greater than the internal length of finite element mesh. On the contrary, the regularization will not take effect because the width band reduces to the mesh size. To highlight the effect of second gradient parameters, dense sample cases were chosen for performing the FEMxDEM calculations. Numerical parameters were similar for all simulations except the second gradient parameter that takes the value of D = 0.01 and D = 0.001. Shear band distributions of simulated cases are displayed in Fig. 5. The result shows that internal length correlates with the width of the shear band. By decreasing D, the width of the shear band decreases as well.

Fig. 5. Influence of internal length parameters on the width of shear band.

The result in Fig. 6 shows that the mesh dependency problem suffered in the first gradient model has been regularized by the second gradient model. Two finite element meshes of 780 SG and 1280 SG elements were used. Shear bands with similar width, orientation, and distribution were observed in both cases. This result demonstrates that the second gradient model has been preserved the objectivity of the problem. It means that the behavior of the problem is independent of finite element mesh (geometry characteristic) but representing the physics internal length of material-made (second gradient parameter). Careful calibration of internal length by the second gradient parameter is mandatory.

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Fig. 6. Second gradient model regularizes mesh dependency

4 Conclusion Failure analysis of pressurized hollow cylinder problems under 2D plane strain condition was presented in this paper. In order to represent the cohesive-frictional granular materials, a DEM-based model was used to describe the first gradient constitutive relation. To preserve the objectivity of the problem, a local second gradient was employed within the framework of FEM×DEM multiscale modeling. In all cases, failure of the pressurized hollow cylinder was presented in form of shear band distribution. The results of the numerical simulations showed a great influence of loose/dense sample and second gradient parameter on strain localization which occurred at the inner boundary of the hollow cylinder. The failure occurred earlier in the loose case than in the dense case. Regarding the effect of the second gradient model, the second parameter control implicitly the internal length of the problem. This latter is correlated to the width of the shear band. The result also demonstrated that the second gradient model preserves the objectivity of the solution when bifurcation occurs. Thus selecting an appropriate second parameter, which has a strong influence on the failure phenomenon of the problem, for such analysis should be carefully considered. Finally, to highlight the advantages of multi-scale continuum-discrete approach, future works should be placed in a detailed comparison of current pressurized hollow cylinder solutions by FEM×DEM computations with the use of other classics/advanced constitutive models for geomaterials or with other concurrences multi-scale modeling such as FEM2 .

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References 1. Nguyen, T.K., Combe, G., Caillerie, D., Desrues, J.: FEM×DEM modelling of cohesive granular materials: numerical homogenisation and multi-scale simulations. Acta Geophys. 62(5), 1109–1126 (2014) 2. Nguyen, T.K., et al.: FEM× DEM: a new efficient multi-scale approach for geotechnical problems with strain localization. In: EPJ Web of Conferences, vol. 140, p. 11007. EDP Sciences (2017) 3. Desrues, J., Argilaga, A., Dal Pont, S., Combe, G., Caillerie, D., Kein Nguyen, T.: Restoring mesh independency in FEM-DEM multi-scale modelling of strain localization using second gradient regularization. In: International Workshop on Bifurcation and Degradation in Geomaterials, pp. 453–457. Spring