Vibration Engineering for a Sustainable Future Active and Passive Noise and Vibration Control, Vol. 1 9783030476175, 9783030476182

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Vibration Engineering for a Sustainable Future Active and Passive Noise and Vibration Control, Vol. 1
 9783030476175, 9783030476182

Table of contents :
Preface
Acknowledgement
Organization
Local Organizing Committee
Steering Committee
Scientific Committee
UTS Support Staff and Volunteers
Reviewers
Organiser APVC 2019
Sponsors and Exhibitors, APVC 2019
Accelerate Discovery With the NI Platform
Special Sessions
Topic: Recent Advances on Vibration Control of Engineering Structures
Topic: Active Noise Control for a Quieter Future
Topic:12pt Noise, Vibration and their Applications in Electricity Power Systems
Topic: Applications and Advances in Laser Doppler Vibrometry
Contents
Part I Noise and Vibration Control
Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate
1 Introduction
2 Deriving Equations of Motion
3 Eliminating Singularity in the Electrostatic Force
4 Transient Behavior
5 Concluding Remarks
References
Elimination of High-Frequency Whistle Noise in a Residential Ducted Air-Conditioning System Using a Dedicated Pipe Muffler Design
1 Introduction
2 Whistle Noise Measurement and Analysis
3 Muffler Design and Its Noise Attenuation Performance Study
4 Conclusion
References
A Novel and Intelligent Multi-Mode Switching Control Strategy for Active Suspension Systems with Energy Regeneration
1 Introduction
2 The Full-Vehicle Dynamic Model
3 The Design of the Optimal Controller in Active Suspension
4 A Novel Multi-Mode Switch Control Strategy
5 Conclusion
References
Design of a Circular-Type Pod Silencer with Annular Two-Layered Air Passages for a High-Pressure Axial Flow Fan
1 Introduction
2 Analysis
2.1 Noise Characteristics
2.2 Finite Element Analysis
3 Design
3.1 Design Variables
3.2 Optimization
4 Result
5 Conclusions
References
Optimal Control of Acoustic Radiation Power for a Triple-Walled Structure
1 Introduction
2 Theory
3 Numerical Analysis
4 Conclusion
References
Part II Vehicle System Dynamics and Control
Towards Overcoming the Challenges of the Prediction of Brake Squeal Propensity
1 Introduction
2 Examples of CEA Success and Limitations
2.1 CEA Success
2.2 Limitations of CEA Due to Nonlinearity
3 Stochastic Approach Incorporating Uncertainty Analysis
4 Conclusions
References
Handling Dynamics of an Ultra-Lightweight Vehicle During Load Variation
1 Introduction
2 Problem Formulation
2.1 Vehicle Specification and Multi-Body Simulation Model
2.2 Consequence of Load Variation
3 Control Implementation
4 Simulation and Results
5 Conclusion
References
A Hybrid Electromechanical Engine Mount Design
1 Introduction
2 New Engine Mount Design Concept
3 Numerical Simulations
4 Simulation Results/Conclusions
References
A Hybrid Modeling Approach to Accurately Predict Vehicle Occupant Vibration Discomfort
1 Introduction
2 Method
2.1 Prediction of Vehicle Seat Frame Vibration in Occupied Condition
2.2 Prediction of Vibration at Occupant-Seat Contact Points
3 Results and Discussion
4 Conclusion
References
Pass-By Noise Synthesis from Transfer Path Analysis Using IIR Filters
1 Introduction
2 IIR Filter Design
2.1 Transfer Path Analysis (TPA) and Airborne Source Quantification (ASQ)
2.2 Preprocessing the Measured Noise Transfer Functions (NTF)
2.3 Implementation of Time-Varying IIR Filter Coefficients
3 Automotive Example
4 Conclusions
References
Motion and Vibration Control of Automotive Drivetrain with Control Cycle Limitation
1 Introduction
2 Controlled Object
2.1 Plant Model of a Simplified Drivetrain
2.2 Mathematical Model and Nonlinear Parameters
3 Control System Design
3.1 Sampled-Data H∞ Control
3.2 Weighting Function to Realize Servo System
4 Compensation for Backlash
5 Control Simulations
6 Conclusion
References
Measuring Road Conditions with an IMU and GPS Monitoring System
1 Introduction
2 Methodology
2.1 Measuring Road Conditions
2.2 Emissions Measurement
2.3 Study Routes
3 Results and Discussion
4 Conclusion
References
A Novel Controllable Electromagnetic Variable Inertance Device for Vehicle Vibration Reduction
1 Introduction
2 EMVI Device Model
3 EMVI Device Characteristics Analysis
4 EMVI Suspension Performance
4.1 Sinusoidal Excitation to EMVI Suspension
4.2 Random Excitation to EMVI Suspension
5 Conclusion
References
Conceptual Design Model of Road Noise on Automotive Bodies in White Based on Energy Propagation
1 Introduction
2 Statistical Energy Analysis
2.1 SEA Equations
2.2 Analytical SEA and Optimization
2.3 Representation of Automotive BIW in the ASEA Model
3 ASEA Prediction
3.1 Calculating Frequency Response from the ASEA Model
3.2 Calculating Frequency Response from FEM
3.3 Comparison of Frequency Responses by ASEA and FEM
4 Optimization with ASEA
4.1 BIW Optimization with the SEA Model
4.2 Verification with FEM Redesigned Models
5 Conclusions
References
Modeling and Measuring of Generated Axial Force for Automotive Drive Shaft Systems
1 Introduction
2 Measuring of GAF of a Drive Shaft System
3 Modeling of a Drive Shaft System
3.1 Modeling Friction and Contact Force Between the Roller and the Track
3.2 Calculation of the Contact Stiffness and the Nominal Force Exponent
3.3 Identification of Contact and Friction Parameters
4 Results
5 Conclusions
References
Optimization on Energy Management Strategy with Vibration Control for Hybrid Vehicles
1 Introduction
2 Dynamic Modelling
2.1 Mechanical Model of a Drive Line Torsional Vibration
2.2 Control Model of the Power Devices
2.3 Energy Management
3 Controller Design
4 Numerical Simulations and Analysis
5 Conclusions
References
Modelling and Vibration Characteristics Analysis of a Parallel Hydraulic Hybrid Vehicle
1 Introduction
2 Dynamic Modelling of the PHHV
3 Results and Analysis
4 Conclusion
References
Part III Vibration and Control of Beams, Plates and Shells
Model Validation of a Vehicle Fuel Tank for Modal Analysis
1 Introduction
2 Fuel Tank and Finite Element Model
3 Response Surface Model
4 Model Validation
4.1 Parameter and Its Distribution
4.2 Model Validation in Terms of Bayesian Factor
5 Summary
References
Development of a Suspension Seat Using a Magneto-Spring and Free Play Damper
1 Introduction
2 Experimental Method
3 Experimental Results
4 Discussion
5 Conclusions
References
Virtual Sensing Application Cases Exploiting Various Degrees of Model Complexity
1 Introduction
2 Case 1: Load Torque Estimation on an Electromechanical Powertrain Using a Multi-physical 1D Model
3 Case 2: Load and Full-Field Strain Estimation on a Twistbeam Rear Suspension Using a Linear FE Model
4 Case 3: Wheel Center Load Estimation on a MacPherson Suspension Using a Non-linear, Flexible Multibody Model
5 Conclusions
References
Efficiency Analysis of a Dual-Motor Electric Vehicle Powertrain
1 Introduction
2 Layout and Work Mode
2.1 Layout of Powertrains
2.2 Work Mode Analysis
2.3 Powertrain Parameter Matching
3 Work Mode Optimization
4 Simulation Results
5 Conclusion
References
Part IV Active and Passive Vibration Control
Numerical Analysis of Dynamic Hysteresis in Tape Springs for Space Applications
1 Introduction
2 Model Setup
3 Model Validation
4 Amplitude Sweeps
5 Conclusion
References
Adaptive Control of a String-Plate Coupled System
1 Introduction
2 Ideal Displacement Control System
2.1 Derivation of Control Law
2.2 The Frequency Characteristics of the Coupled System
2.3 Acoustic Radiation Power of the Coupled System
3 Adaptive-Filter-Based Approach of the Displacement Control System
4 Summary
References
Time-Delay-Based Direct Wave Control of the Phononic Beam
1 Introduction
2 Model of the Time-Delay-Controlled Phononic Beam
3 Time-Delay-Induced Nonreciprocal Wave Propagation
4 Conclusions
References
Free Vibration Analysis of Multilayer Skew Sandwich Spherical Shell Panels with Viscoelastic Material Cores and Isotropic Constraining Layers
1 Introduction
2 System Configuration and Mathematical Formulation
3 Finite Element Formulation
4 Results and Discussions
5 Conclusions
References
A Size-Dependent Variable-Kinematic Beam Model for Vibration Analysis of Functionally Graded Micro-beams
1 Introduction
2 Theoretical Formulation
3 Numerical Examples
3.1 Convergence Study
3.2 Validation Study
3.3 Parameter Study
4 Conclusions
References
Vibration Analysis of a Viscoelastic Beam Equipped with a Resilient Impact Damper
1 Introduction
2 Dynamic Modeling
3 Numerical Simulations and Analysis
4 Conclusions
References
Vibration Analysis of a Beam with Both Ends Fixed Using Molecular Dynamics Method
1 Introduction
2 Analytical Model of a Beam with Both Ends Fixed
2.1 Molecular Dynamics Method Using MEAM Potential
2.2 Construction of MD Model of a Beam with Both Ends Fixed
3 Result of Contact Simulation
4 Conclusion
References
Numerical Investigation of Vibration Characteristics and Damping Properties of CNT-Based Viscoelastic Spherical Shell Structure
1 Introduction
2 Mathematical Modeling
2.1 Shell Formulation [6]
2.1.1 Strain Displacement Relations
2.1.2 Determination of Equation of Motion and Finite Element Modeling
3 Material Fabrication and DMA-8000 Test
4 Results and Discussion
4.1 Analysis of MWCNT-Based Nanocomposite Sample in Scanning Electron Microscopy (SEM)
4.2 Numerical Results Based on First Five Natural Frequencies
4.3 Numerical Results Based on Frequency Response and Transient Response
5 Conclusion
References
Part V Recent Advances on Vibration Control of Engineering Structures
Semi-active Vibration Suppression of a Structure by a Shear-Type Damper Using Magnetorheological Grease
1 Introduction
2 Methods
2.1 Shear-Type MR Grease Damper
2.2 Two-Degree-of-Freedom Vibration System
2.3 Vibration Suppression Test
3 Results and Discussion
3.1 Transmissibility of Displacement
3.2 Discussion on the Control Law
4 Conclusion
References
A Comparison of Vibration Control Performance for the Electromagnetic Damper with Various Control Strategies
1 Introduction
2 Theoretical Background
2.1 Numerical Model for Stay Cable
2.2 Dynamic Model for the EM Damper
3 Numerical Simulations and Results
4 Conclusion
References
Experiment and Numerical Investigations on a Vertical Isolation System with Quasi-Zero Stiffness Property
1 Introduction
2 Static Force-Displacement Relation
2.1 The Construction of Vertical QZS Isolation System
2.2 Static Test Verification
3 Shake Table Test and Numerical Simulation
3.1 Comparison Between Simulink and OpenSees Results
3.2 Comparison Between OpenSees and Shake Table Test Results
4 Conclusions
References
Robust Vibration Control of an Overhead Crane by Elimination of the Natural Frequency Component
1 Introduction
2 Overhead Crane Model
3 Control Strategy
3.1 Elimination of Natural Frequency Component
3.2 Construction of Target Trajectory
3.3 Robustness Improvement Conditions
4 Results and Discussion
5 Conclusion
References
Development of an Active Mass Damper Driven by an Amplitude-Modulated Signal
1 Introduction
2 Theoretical Description of the Damper Model
2.1 Amplitude Modulation of a Signal and Its Demodulated Components
2.2 Equation of Motion for a Vibration System with an AMD Driven by the AM Signal
3 Dynamic Characteristics Test for the Active Mass Damper
4 Conclusions
References
Part VI Active Noise Control for a Quieter Future
Building Vibration Suppression Through a Magnetorheological Variable Resonance Pendulum Tuned Mass Damper
1 Introduction
2 Device Design and Modeling
2.1 Device Structure
2.2 Dynamic Modeling and Validation
3 Scale-Building Experiments
3.1 Experimental Setup
3.2 Results
4 Conclusion
References
Dynamic Property Optimization of a Vibration Isolator with Quasi-Zero Stiffness
1 Introduction
2 Dynamic Properties Investigation
3 Dynamic Property Optimization
4 Conclusion
References
Study on the Influence of Structural Nonlinearity on the Performance of Multiunit Impact Damper
1 Introduction
2 Simulation Method
3 Performance Study
4 Conclusion
References
Design of a Quasi-Zero Stiffness System Based on Electromagnetic Vibration Isolation
1 Introduction
2 Design of the Isolator
3 Dynamic Analysis of the Isolator
4 Conclusions
References
Affine Combination of the Filtered-x LMS/F Algorithms for Active Control
1 Introduction
1.1 The Proposed Affine Combination Algorithm
2 Simulations
2.1 Case I
2.2 Case II
2.3 Case III
3 Conclusion
References
An Experimental Study on Virtual Sound Barrier Performance in Workplaces
1 Introduction
2 Experiments and Discussions
3 Conclusions
References
Active Control of Sound Transmission Through an Aperture in a Thin Wall
1 Introduction
2 Theory
3 Numerical Simulations
3.1 Verification of the Theoretical Model
3.2 Incident Directions of the Primary Sound
3.3 Configurations of Secondary Sources
3.4 Configurations of Error Microphones
4 Conclusions
References
Real-Time Active Noise Control of Multi-tonal Noise Based on Multiply Connected Single Adaptive Notch Filters
1 Introduction
2 Methods
2.1 A Basic Algorithm
2.2 The Proposed Tracking-Type FX SAN ANC Algorithm
2.3 Numerical Simulation
2.4 The Experimental Setup
3 Results
4 Conclusions
References
A New Frequency Domain Adaptive Filter Coefficients Updating Method and Its Steady-State Performance in Frequency and Time Domain
1 Introduction
2 Related Works
3 Constrained NFBLMS and Its Steady-State Behavior
3.1 Frequency Domain
3.2 Time Domain
4 The Framework of CFBLMS
5 Numerical Simulation and Analysis
6 Conclusion
References
Effects of Reverberation on Active Noise Control Headrest Performance
1 Introduction
2 Evaluation System
2.1 Experimental Setup
2.2 Feedforward and Feedback Structures in the ANC Headrest
3 Results and Discussions
4 Conclusions
References
Zero Control Power Phenomena in the Minimization of Sound Power Using Multiple Control Sources
1 Introduction
2 Zero Control Power Phenomenon Using One CSS
3 Zero Control Power Phenomenon Using Multiple CSSs
3.1 Minimization of Total Sound Power in the Case of Two CSSs
3.2 Zero Power Control Law in the Case of Multiple CSSs and Its Control Effect
4 Conclusions
References
Author Index
Subject Index

Citation preview

Sebastian Oberst Benjamin Halkon Jinchen Ji Terry Brown  Editors

Vibration Engineering for a Sustainable Future Active and Passive Noise and Vibration Control, Vol. 1

Vibration Engineering for a Sustainable Future

Sebastian Oberst • Benjamin Halkon • Jinchen Ji Terry Brown Editors

Vibration Engineering for a Sustainable Future Active and Passive Noise and Vibration Control, Vol. 1

Editors Sebastian Oberst Centre for Audio, Acoustics and Vibration Faculty of Engineering and IT University of Technology Sydney Sydney, NSW, Australia

Benjamin Halkon Centre for Audio, Acoustics and Vibration Faculty of Engineering and IT University of Technology Sydney Sydney, NSW, Australia

Jinchen Ji School of Mechatronic and Mechanical Engineering, Faculty of Engineering and IT University of Technology Sydney Sydney, NSW, Australia

Terry Brown School of Mechatronic and Mechanical Engineering, Faculty of Engineering and IT University of Technology Sydney Sydney, NSW, Australia

ISBN 978-3-030-47617-5 ISBN 978-3-030-47618-2 (eBook) https://doi.org/10.1007/978-3-030-47618-2 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved 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 Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

These proceedings, presented in three volumes, contain a selection of papers presented at the 18th Asia-Pacific Vibration Conference (APVC 2019) held at the University of Technology Sydney (UTS) in Sydney, Australia, from 18 to 20 November, 2019. Vibration and associated phenomena are all around us every day but are often overlooked and/or not fully understood. However, it is of fundamental importance to the engineering of the systems we continually interact with in our daily lives. This conference enabled experienced vibration engineering researchers and practitioners, along with the experts of the future, to come together to present and discuss their latest interests and activities in the domain. Additionally, five international leaders in the field from across the region and beyond presented keynote plenary sessions. The APVC is a long-standing technical conference with a proud history. It was first held in Japan in 1985 and since then every two years in several different countries including Korea, China, Australia, Malaysia, Singapore, New Zealand, Hong Kong and Vietnam. At APVC 2019, we had 219 delegates from 15 different countries including some from outside the region (Germany, Great Britain, France, Czech Republic, Brazil and United Arab Emirates). Thank you to all the APVC 2019 sponsors whose financial support and presence in the exhibition area helped us deliver a vibrant and successful event. We especially acknowledge our Platinum and Gold sponsors: Polytec GmbH, Warsash Scientific Pty Ltd, Bestech Australia Pty Ltd and Siemens Digital Industries Software. We also thank UTS for hosting the conference and UTS Tech Lab for its generous support. The papers presented in these proceedings encompass fundamental and applied research, theoretical approaches, computational methods and simulation, and experimentation in vibration engineering. The authors, from 18 different countries, are researchers and practitioners, including professors, students, engineers and scientists from academia and industry. The three volumes, each with papers organized into parts aligned with the conference oral presentation technical sessions, are:

v

vi

Preface

Vol. 1 – Active and Passive Noise and Vibration Control Vol. 2 – Experiments, Materials and Signal Processing Vol. 3 – Numerical and Analytical Methods to Study Dynamical Systems Contributions were invited from across the region and beyond with a total of 245 extended abstracts submitted with the local organizing committee, which accepted 183 after review, representing a rejection rate of 25%. Authors of accepted extended abstracts were then invited to submit full papers with a maximum length of six pages. A total of 183 full papers were submitted and 145 were selected (21% rejection rate) for the proceedings following a rigorous review process involving world-leading experts in their fields as external reviewers. At least two reviewers considered each paper. Selected reviewers were active researchers in the relevant fields and we sincerely thank them for providing their expert opinion, valuable time and effort. The Local Organizing Committee compiled the reviews and sent them to authors to assist them with refining and improving their papers before final submission and editorial approval. We would also like to thank all authors for their excellent work and significant contribution. Finally, we would like to thank Springer for their support in producing and publishing these proceedings. Sydney, NSW, Australia

Sebastian Oberst Benjamin Halkon Jinchen Ji Terry Brown

Acknowledgement

On behalf of the Scientific Steering Committee, I would like to express my sincere gratitude to the Local Organizing Committee for their greatest contribution to the APVC2019. The LOC has been organized by Professor Jinchen (JC) Ji and Professor Benjamin (Ben) Halkon in UTS. I also would like to sincerely thank UTS students and staff, session chairs and external sponsors and exhibitors, who worked extremely hard to deliver an excellent event. I also thank the Steering Committee for its contribution. In delivering APVC 2019, many firsts were achieved, including: invitations and participants specifically extended to beyond the AsiaPacific region; engineers from industry actively engaged with the conference; the APVC conference financially supported by a government department (the New South Wales Government through the Office of the Chief Scientist and Engineer); best student paper awards rigorously panel reviewed; and peer-reviewed selected papers officially published in three volumes by international publisher Springer. Tokyo Metropolitan University, Japan

Takuya Yoshimura

vii

Organization

Local Organizing Committee Jinchen (JC) Ji Benjamin (Ben) Halkon Sebastian Oberst Terry Brown Liya Zhao Philippe Blanloeuil Paul Walker Yancheng Li Hamed Kalhori

University of Technology Sydney, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia University of New South Wales, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia

Chair Chair Technical Chair Program Chair Ordinary Member Ordinary Member Ordinary Member Ordinary Member Ordinary Member

Steering Committee Takuya Yoshimura Shigehiko Kaneko Toshihiko Komatsuzaki

Tokyo Metropolitan University, Japan (Chairman) University of Tokyo, Japan Kanazawa University, Japan (continued) ix

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Organization

Youngjin Park

Korea Advanced Institute of Science and Technology, Korea Yonsei University, Korea Beijing Institute of Technology, China Hong Kong Polytechnic University, Hong Kong Nanjing University of Aeronautics and Astronautics, China Tsinghua University, China Qingdao University of Technology, China University of Technology, Malaysia University of Science, Malaysia Hanoi University of Science and Technology, Vietnam University of Canterbury, New Zealand Curtin University, Australia University of Technology Sydney, Australia University of Technology Sydney, Australia Hokkaido University, Japan Hanyang University, Korea University of Canterbury, New Zealand Northeastern University, China

No-Cheol Park Haiyan Hu Li Cheng Jinhao Qiu Zhichao Hou Tianran (Terry) Lin M. Salman Leong Zaidi Mohd, Ripin Nguyen Van Khang Stefanie Gutschmidt Ian Howard Jinchen Ji Benjamin Halkon Oshihiro Narita (Honourable) Hong Hee Yoo (Honourable) Athol J. Carr (Honourable) Ban Chun Wen (Honourable)

Scientific Committee

Richard Markert Xiaojun Qiu Rodney Entwistle Yong-Hwa Park Con Doolan Benjamin Cazzolato Jaspreet Singh Pietro Borghesani Tamas Kalmar-Nagy Mohammad Fard Guilin Wen Jin Zhou

Darmstadt University of Technology, Germany University of Technology Sydney, Australia Curtin University, Australia Korea Advanced Institute of Science and Technology, Korea University of New South Wales, Australia University of Adelaide, Australia The University of Auckland, New Zealand University of New South Wales, Australia Budapest University of Technology and Economics, Hungary Royal Melbourne Institute of Technology, Australia Hunan University, China Shanghai University, China (continued)

Organization CW Lim Dongping Jin Xinwen Wang Jian Xu Shaopu Yang Hu Ding Qinsheng Bi Hailin Wang Qingyun Wang Jie Huang Jie Yang Haiping Du Linke Zhang Sam Han

xi City University of Hong Kong, Hong Kong Nanjing University of Aeronautics and Astronautics, China China University of Mining & Technology, Beijing Tongji University, China Shijiazhuang Tiedao University, China Shanghai University, China Jiangsu University, China South China Agricultural University, China Beihang University, China Beijing Institute of Technology, China Royal Melbourne Institute of Technology, Australia University of Wollongong, Australia Wuhan University of Technology, China ActronAir, Australia

UTS Support Staff and Volunteers

Abbasnejad, Behrokh Boeni, Alison Chong, Hong Kit Darwish, Abdel Hayati, Hasti

Ho, Ngoc Thao Han (Sophie) Huynh, Timothy Li, Wenjie Lu, Shuixiu Lym, Martin

Ni, Qing Pereira, Kyle Sansom, Travers Xiao, Tong Ye, Kan

Zhang, Ying Zheng, Jingyang

Reviewers

Abdrrahim, Houmat Abu Bakar, Abdul Rahim Adams, Christian Adams, George G. Agoston, Katalin

Huang, Dongmei Hui, Kar Hoou Huston, Dryver Inoue, Tsuyoshi Irvine, Tom

Matsuzaki, Kenichiro Matthews, David Melnikov, Anton Min, Cheonhong Mitchell, Sean

Walker, Paul Wang, Shuping Wang, Yuxing Wang, Xu Wang, Feng (continued)

xii Aihara, Tatsuhito Alkmim, Mansour Bai, Shipeng Baydoun, Suhaib Koji Bi, Kaiming Bianciardi, Fabio Biswal, Deepak Kumar Blanloeuil, Phillipe Bonneau, Oliver Borghesani, Pietro Brown, Terry Buchwald, Patrick Butlin, Tore Carpenter, Harry Cazzolato, Benjamin Cheer, Jordon Chen, Tong Chiang, Yan Kei Choudhury, Madhurjya Dev Christie, Matthew Dai, Wei Daniel, Christian Darwish, Abdel Davy, John De Ryck, Laurent Denimal, Enora Dhupia, Jaspreet Singh Ding, Qian Ding, Hu Dong, Xufeng Dubbini, Janet L Dubey, Manish Kumar Fard, Mohammad Fisher, Joeffrey Forrier, Bart Fowler, Deborah Fujita, Satoshi Furuya, Kohei

Organization Ishikawa, Satoshi Ito, Atsuhiro Iwamoto, Hiroyuki Ji, Jinchen

Moreau, Danielle Joy Morishita, Shin Nakano, Yutaka Nakashima, Itsuki

Wang, Qiang Wang, Yuning Wang, Kuoting Wang, Shiliang

Jin, Dongping Jung, Hyung-Jo Kalhori, Hamed

Nerse, Can Ng, Alex Ning, Donghong

Wantanabe, Seiji Watterson, Peter Wen, Hao

Kang, Hooi Siang

Nitzschke, Steffen

Wen, Guilin

Karimi, Hamid Reza Karimi, Mahmoud Kawamura, Shozo Kessissoglou, Nicole Shen, Jianwei Kawamura, Shozo Kessissoglou, Nicole

Oberholster, Abrie Oberst, Sebastian Ota, Shinichiro Papangelo, Antonio Patnaik, S Srikant Pradhan, Somanath Prasad, Marehalli

Woschke, Elmar Wu, Lifu Wu, Helen Xiao, Tong Xiao, Tong Xu, Jian Xu, Daolin

Kil, Hyun Gwon Kim, Dong Joon Kim, Dong Hyeon Kingan, Micheal Joseph

Qiu, Xiaojun Qu, Jiao Rahnejat, Homer Rose, Francis

Yabui, Shota Yamada Keisuke Yamamoto, Hiroshi Yamazaki, Toru

Komatsu, Tadashi Komatsuzaki, Toshihiko Kondo, Eiji Kondou, Takahiro Koo, Bonsoo Koutsovasillis, Panagiotis Krueger, Timm Kundu, Pradeep

Rudorf, Martin Ruppert, Michael S, Bala Murugan Saeed, Omear Saito, Takashi Sanliturk, Kenan Y. Sasaki, Takumi Seering, Warren

Yamazumi, Mitsuhiro Yan, Han Yang, Jian Yang, Guidong Yao, JianChun Ye, Kan Yonezawa, Heisei Yoo, Hong Hee

Kurihara, Kai Kuroda, Katsuhiko Lai, Joseph Lee, Doo Ho Lee, Chan

Shah, JayKumar Shangguan, Wenbin Shiiba, Taichi Shin, Eung-Soo Smith, Wade

Yoshida, Tatsuya Yue, Xiaole Zhang, Hua Zhang, Xiaoxu Zhang, Xiaozhu

Lei, Jiazhen Lei, Gang Li, Weihua Li, Huan Li, Wei Lidfors Lindqvist, Anna

Sowa, Nobuyuki Spannan, Lars Stender, Merten Stone, Brian Su, Zhu Sueda, Miwa

Zhang, Guoqiang Zhang, Linke Zhang, Kai Zhao, Sipei Zhao, Feng Zhao, Liya (continued)

Organization Geng, Xiaofeng Gong, Sanpeng Hahn, Eric Han, Tian Halkon, Benjamin Hansen, Kristy Hauret, Patrice Hayati, Hasti Hirano, Yakashi Hisano, Shotaro Hosoya, Naoki Hossain, Mahbub Hou, Zhichao

xiii Lin, Susanna Liu, Pengfei Logan, Patrick Lu, Shuixiu Lu, Yun Luo, Liang Luo, Quantian Luo, Lin Lv, Ling Maheo, Lauret Makki Alamdari, Mehrisadat Mao, Xin Matsuyama, Marin

Sun, Xiuting Taji, Shoichi Tao, Jiancheng Terashima, Osamu Terumichi, Yoshiaki Tian, Fangbao Tomoda, Akinori Tsuchida, Takahiro Tsujiuchi, Nobutaka Ura, Kentaro Vahidi, Ardalan Veidt, Martin Vio, Gareth

Zheng, Minyi Zhong, Jiaxin Zhou, Jiaxi Zhou, Yulong Zhou, Shilei Zhou, Liangqiang Zhu, Qiaoxi Zhu, Chendi Zou, Kun Zou, Hai-Shan

Organiser APVC 2019

Centre for Audio, Acoustics and Vibration

xv

Sponsors and Exhibitors, APVC 2019

Polytec has been bringing light into the darkness for 50 years. With more than 400 employees worldwide, we develop, produce and distribute optical measurement technology solutions for research and industry. Our quality innovative products have an excellent reputation internationally among the expert community. We find solutions tailored to our customers’ requirements. The development and production of innovative measurement systems, especially for our core technology Laser Doppler Vibrometry (LDV), has kept our customers and us at the forefront of dynamic characterization. The implementation of LDV extends from basic vibration measurement tasks to advanced modal analysis / FE correlation. Ultimately, our solutions are meant to help companies to assert and build upon their technological leadership. The effective use of the laser technology allows a non-contact, non-invasive test method for vibration, which is widely appreciated across industries such as Automotive, Aerospace, Semiconductor, Consumer Electronics, etc. The decades of experience have allowed Polytec to expand the technology and our product line-up, which could be generally summarized as follows: Micro to macro sample dimensions Single point or full-field scanning 1D or 3D axis data FRF, transient, mode and operational deflection shape analysis capable xvii

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Sponsors and Exhibitors, APVC 2019

Max frequency BW up to 2.5 GHz Max distance up to 300 m Sub-pm resolution Together with Warsash Scientific, our long-standing partner in the ANZ region for over 40 years, Polytec would welcome visitors to inspect the latest Polytec VibroFlex series range of research grade single point laser Doppler vibrometers at the APVC 2019 exhibition. The modular concept of VibroFlex combines the versatility of a universal front-end with a selection of special sensor heads, tailored to the needs of your measuring task, including the latest Xtra IR sensor head option. For more information, please visit www.polytec.com

Bestech Australia, an ISO9001 certified company, supplies state-of-the-art test and measurement sensors for measurement of physical parameters, data acquisition systems as well as technical teaching equipment from world leading manufacturers. Our constantly expanding product portfolio is suitable for university research, teaching and R&D laboratories as well as for demanding high precision measuring applications in industrial environments. We are proud to complement this with our own manufacture. We offer full local technical support throughout the entire product lifecycle including product specification, commissioning, training and repair. This is delivered by our team of factory trained application engineers and product specialists. We pride in delivering excellent service for ultimate customer satisfaction.

Digitization is rapidly gaining ground. Today’s manufacturers develop new product architectures and material types, offer consumers customization options and massively introduce smart functionalities. These innovations are enabled by capabilities such as mechatronics, additive manufacturing, and concepts like cloud or the internet of things. Engineers need to master this additional complexity, which is often related to an ever-increasing demand for energy efficiency, while still dealing with classic performance requirements, such as noise, vibrations and durability. This evolution urges companies to dramatically transform their classical verification-centric development processes. Instead, the Digital Twin paradigm is on a rise. In this new approach, manufacturers associate every individual product to a set of ultra-realistic, multi-physics models and data, which stay in-sync, and can predict its real behaviour throughout the lifecycle, starting from the very early stages.

Sponsors and Exhibitors, APVC 2019

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To achieve this, simulation needs to gain realism to become capable of taking up a predictive role, while the combination of increased validation workload and the exploration of uncharted design territories requires more productive and innovative testing methodologies. On top of that, manufacturers will need to deploy an infrastructure that helps them remove the traditional barriers between departments, even letting product development continue after delivery. That is exactly the core business of Siemens Digital Industries Software. Siemens Digital Industries Software offers manufacturers across the various industries a comprehensive environment that helps their engineering departments create and maintain a Digital Twin. Within this offering, the Simcenter™ solutions portfolio focuses on performance engineering. Simcenter uniquely integrates physical testing with system simulation, 3D CAE and CFD, and combines this with design exploration and data analytics. Simcenter helps engineers accurately predict vehicle performance, optimize designs and deliver innovations faster and with greater confidence.

At Brüel & Kjær, we help our customers solve sound and vibration challenges and develop advanced technology for measuring and managing sound and vibration. We ensure component quality, optimize product performance and improve the environment. Founded in 1942, Brüel & Kjær Sound & Vibration Measurement A/S has grown to become the world’s leading supplier of advanced technology for measuring and managing the quality of sound and vibration. The sound and vibration challenges facing our customers are diverse, including vibration in car engines; evaluation of building acoustics; mobile telephone sound quality; cabin comfort in passenger airplanes; production quality control; wind turbine noise; and much more. Our innovative and highly practical solutions have made us the first choice of engineers and designers from around the world. Many of our researchers and developers are recognized as world experts, who aid the scientific community and teach at renowned centres. By applying their thorough knowledge and experience, we can help you at every stage of your product’s life cycle: ensuring quality from design to manufacture, and efficiency throughout deployment and operations. Brüel & Kjær maintains a network of sales offices and representatives in 55 countries. That means local-language help is always at hand during office hours. A global group of engineering specialists supports our local teams. They can advise on and help solve all manner of sound and vibration measurement, analysis and management problems. To further support our customers worldwide, we regularly hold courses and road-shows, and participate in sound and vibration focused trade shows and conferences worldwide.

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Sponsors and Exhibitors, APVC 2019

MathWorks is the leading developer of mathematical computing software. MATLAB, the language of engineers and scientists, is a programming environment for algorithm development, data analysis, visualization, and numeric computation. Simulink is a block diagram environment for simulation and Model-Based Design for multidomain dynamic and embedded engineering systems. Engineers and scientists worldwide rely on these product families to accelerate the pace of discovery, innovation, and development in automotive, aerospace, electronics, financial services, biotech-pharmaceutical, and other industries. MATLAB and Simulink are also fundamental teaching and research tools in the world’s universities and learning institutions.

Accelerate Discovery With the NI Platform Researchers are driving time-critical, ambitious innovation while addressing grand engineering challenges in the broad areas of transportation, wireless communications, medicine, energy, and climate change. Across each of these application areas, researchers need to easily acquire measurements, scale to complex multidisciplinary systems, and rapidly prototype a scalable solution. For more than 40 years, NI is central to accelerating researcher innovation by providing the technology and support to prototype systems, publish findings and secure funding from 5G Wireless and Communications all the way to Autonomous and Electrical Vehicles.

John Morris Scientific was founded In April 1956 to service consumables and instrumentation throughout the South Pacific Science industry. Today we employ over 85 talented sales and service professionals across 12 key locations – to ensure you receive unrivalled customer service in your location. John Morris Group offers technical/application advice to Laboratory, R&D, Nano-Fabrication, High Energy Physics, Synchrotron, Industrial, Process and Food/Packaging users. Although we have the largest range, we believe ‘It’s not about the box’ and we look forward to delivering your team with solutions that add value to your process. As you face increasing pressures (budget, time and results), the success of our offering

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is based on more than just supplying you with the ‘box’ or its price. At John Morris Group, we focus on satisfying your end-to-end needs . . . today and over the longer term.

Special Sessions

Topic: Recent Advances on Vibration Control of Engineering Structures Organizers: Dr Yancheng Li, University of Technology Sydney, Australia, email: [email protected] Dr Kaiming Bi, Curtin University, Australia, email: [email protected] A/Prof Xufeng Dong, Dalian University of Technology, China, email: [email protected]

Topic: Active Noise Control for a Quieter Future Organizers: Dr Sipei Zhao, University of Technology Sydney, Australia, email: [email protected] Dr Shuping Wang, University of Technology Sydney, Australia, email: [email protected] Associate Professor Lifu Wu, Nanjing University of Information Science & Technology, email: [email protected]

Topic: Noise, Vibration and their Applications in Electricity Power Systems Organizers: A/Professor Linke Zhang, Wuhan University of Technology, China, email: [email protected] xxiii

xxiv

Special Sessions

A/Professor Yuxing Wang, Zhejiang University, China, email: [email protected] Professor Tianran Lin, Qingdao University of Technology, China, email: [email protected]

Topic: Applications and Advances in Laser Doppler Vibrometry Organizers: Dr Ben Halkon, University of Technology Sydney, Australia, email: [email protected] Dr Philippe Blanloeuil, University of New South Wales, Australia, email: [email protected] Professor Enrico Primo Tomasini, Universita Polytecnica delle Marche, Italia, email: [email protected]

Contents

Part I Noise and Vibration Control Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hamed Kalhori, Ben Halkon, Behrokh Abbasnejad, Bing Li, and Alireza Shooshtari

3

Elimination of High-Frequency Whistle Noise in a Residential Ducted Air-Conditioning System Using a Dedicated Pipe Muffler Design Sam Han, J. C. Ji, and Kan Ye

11

A Novel and Intelligent Multi-Mode Switching Control Strategy for Active Suspension Systems with Energy Regeneration . . . . . . . . . . . . . . . . . . . . . . Hang Wu and Ling Zheng

21

Design of a Circular-Type Pod Silencer with Annular Two-Layered Air Passages for a High-Pressure Axial Flow Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chan Lee, Hyun Gwon Kil, Jong Jin Park, Dong Hyun Kim, and Sang Ho Yang Optimal Control of Acoustic Radiation Power for a Triple-Walled Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hiroki Tanaka, Hiroyuki Iwamoto, and Shotaro Hisano

29

37

Part II Vehicle System Dynamics and Control Towards Overcoming the Challenges of the Prediction of Brake Squeal Propensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhi Zhang, Sebastian Oberst, and Joseph C. S. Lai

47

Handling Dynamics of an Ultra-Lightweight Vehicle During Load Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Lidfors Lindqvist and Paul D. Walker

55

xxv

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Contents

A Hybrid Electromechanical Engine Mount Design . . . . . . . . . . . . . . . . . . . . . . . . . Nader Vahdati, Esmaail Farah, and Oleg Shiryayev A Hybrid Modeling Approach to Accurately Predict Vehicle Occupant Vibration Discomfort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianchun Yao, Mohammad Fard, and Kazuhito Kato Pass-By Noise Synthesis from Transfer Path Analysis Using IIR Filters . . Mansour Alkmim, Fabio Bianciardi, Guillaume Vandernoot, Laurent De Ryck, Jacques Cuenca, and Karl Janssens Motion and Vibration Control of Automotive Drivetrain with Control Cycle Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Yonezawa, I. Kajiwara, C. Nishidome, T. Hatano, M. Sakata, and S. Hiramatsu Measuring Road Conditions with an IMU and GPS Monitoring System. . Enoch Zhao, Paul D. Walker, Albert Ong, and Fatma Al-Widyan

63

71 79

87

95

A Novel Controllable Electromagnetic Variable Inertance Device for Vehicle Vibration Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Pengfei Liu, Minyi Zheng, Donghong Ning, Liang Luo, and Nong Zhang Conceptual Design Model of Road Noise on Automotive Bodies in White Based on Energy Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Toru Yamazaki, Keita Suwabe, Kousuke Nakanishi, Hirotaka Shiozaki, and Junichi Yanase Modeling and Measuring of Generated Axial Force for Automotive Drive Shaft Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Huayuan Feng, Wen-Bin Shangguan, and Rakheja Subhash Optimization on Energy Management Strategy with Vibration Control for Hybrid Vehicles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Yifan Wei, Yuning Wang, and Zhichao Hou Modelling and Vibration Characteristics Analysis of a Parallel Hydraulic Hybrid Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Shilei Zhou, Paul Walker, and Nong Zhang Part III Vibration and Control of Beams, Plates and Shells Model Validation of a Vehicle Fuel Tank for Modal Analysis . . . . . . . . . . . . . . 145 Shuyu Wang, Peibao Wu, Zhichao Hou, Xuehong Chen, and Shuai Wang Development of a Suspension Seat Using a Magneto-Spring and Free Play Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Masahiro Mashino, Etsunori Fujita, Shigeyuki Kojima, Yumi Ogura, and Shigehiko Kaneko

Contents

xxvii

Virtual Sensing Application Cases Exploiting Various Degrees of Model Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Karl Janssens, Bart Forrier, Roberta Cumbo, Enrico Risaliti, Bram Cornelis, Tommaso Tamarozzi, and Wim Desmet Efficiency Analysis of a Dual-Motor Electric Vehicle Powertrain . . . . . . . . . . 169 Bing Wang, Jinglai Wu, Xianqian Hong, Nong Zhang, and Daisheng Zhang Part IV Active and Passive Vibration Control Numerical Analysis of Dynamic Hysteresis in Tape Springs for Space Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Richard Martin, Merten Stender, and Sebastian Oberst Adaptive Control of a String-Plate Coupled System . . . . . . . . . . . . . . . . . . . . . . . . . 185 Marin Matsuyama, Hiroyuki Iwamoto, Shotaro Hisano, and Nobuo Tanaka Time-Delay-Based Direct Wave Control of the Phononic Beam . . . . . . . . . . . . 193 Xiaoxu Zhang, Jian Xu, and Hongbin Fang Free Vibration Analysis of Multilayer Skew Sandwich Spherical Shell Panels with Viscoelastic Material Cores and Isotropic Constraining Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Deepak Kumar Biswal and Sukesh Chandra Mohanty A Size-Dependent Variable-Kinematic Beam Model for Vibration Analysis of Functionally Graded Micro-beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Zhu Su, Kaipeng Sun, and Jie Sun Vibration Analysis of a Viscoelastic Beam Equipped with a Resilient Impact Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Xiao-Feng Geng, Hu Ding, and Li-Qun Chen Vibration Analysis of a Beam with Both Ends Fixed Using Molecular Dynamics Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Akinori Tomoda, Masahiro Yamanaka, and Taiyo Takashima Numerical Investigation of Vibration Characteristics and Damping Properties of CNT-Based Viscoelastic Spherical Shell Structure . . . . . . . . . . . 231 S. Srikant Patnaik, Tarapada Roy, and D. Koteswar Rao Part V Recent Advances on Vibration Control of Engineering Structures Semi-active Vibration Suppression of a Structure by a Shear-Type Damper Using Magnetorheological Grease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Shuto Nagamatsu and Toshihiko Shiraishi

xxviii

Contents

A Comparison of Vibration Control Performance for the Electromagnetic Damper with Various Control Strategies . . . . . . . . . . . . . 249 Hyung-Soo Kim, Seungkyung Kye, and Hyung-Jo Jung Experiment and Numerical Investigations on a Vertical Isolation System with Quasi-Zero Stiffness Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Peng Chen and Ying Zhou Robust Vibration Control of an Overhead Crane by Elimination of the Natural Frequency Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Kai Kurihara, Takahiro Kondou, Hiroki Mori, Kenichiro Matsuzaki, and Nobuyuki Sowa Development of an Active Mass Damper Driven by an Amplitude-Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Toshihiko Komatsuzaki, Tetsuma Sadaoka, and Haruhiko Asanuma Part VI

Active Noise Control for a Quieter Future

Building Vibration Suppression Through a Magnetorheological Variable Resonance Pendulum Tuned Mass Damper . . . . . . . . . . . . . . . . . . . . . . . . 281 Matthew Daniel Christie, Shuaishuai Sun, and Weihua Li Dynamic Property Optimization of a Vibration Isolator with Quasi-Zero Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Huan Li, Jianchun Li, Yancheng Li, and Yang Yu Study on the Influence of Structural Nonlinearity on the Performance of Multiunit Impact Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Zheng Lu, Naiyin Ma, and Hengrui Zhang Design of a Quasi-Zero Stiffness System Based on Electromagnetic Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Yu Chen, Hao Wen, and Dongping Jin Affine Combination of the Filtered-x LMS/F Algorithms for Active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Somanath Pradhan, Xiaojun Qiu, and Jinchen Ji An Experimental Study on Virtual Sound Barrier Performance in Workplaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Sipei Zhao and Xiaojun Qiu Active Control of Sound Transmission Through an Aperture in a Thin Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Shuping Wang and Xiaojun Qiu

Contents

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Real-Time Active Noise Control of Multi-tonal Noise Based on Multiply Connected Single Adaptive Notch Filters . . . . . . . . . . . . . . . . . . . . . . . 335 Shun Hirose, Toshihiko Komatsuzaki, Naoki Kimura, Keita Tanaka, and Taisei Yamaguchi A New Frequency Domain Adaptive Filter Coefficients Updating Method and Its Steady-State Performance in Frequency and Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Xin Mao, Yang Xiang, Si Qin, and Yangxing Liu Effects of Reverberation on Active Noise Control Headrest Performance Lifu Wu, Chiming Fang, and Zhuang Cheng

351

Zero Control Power Phenomena in the Minimization of Sound Power Using Multiple Control Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Yuta Ogasawara, Hiroyuki Iwamoto, and Shotaro Hisano Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Part I

Noise and Vibration Control

Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate Hamed Kalhori, Ben Halkon, Behrokh Abbasnejad, Bing Li, and Alireza Shooshtari

1 Introduction An electrically actuated microplate forms one side of the variable capacity airgap capacitor and the other side is a stationary electrode connected to the output circuit, as shown in Fig. 1. The capacitive microelectromechanical systems (MEMS) microplates are the actuation components in many micropumps, micromirrors, microphones, microsensors, etc., and are used to open or close an electric circuit by an electrostatic force [1]. The actuation is achieved by applying a DC polarization voltage to the microplate to deflect it to contact the output circuit, at which the switch is in the on-state. When removing the DC voltage, the microplate is released to its original position, at which the switch is in the off-state. Applying a DC voltage to the structure creates an impulsive force, resulting in transient vibration of the system. As the electrostatic force deforms the microplate, the force changes with

H. Kalhori () School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran e-mail: [email protected] B. Halkon · B. Abbasnejad School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected]; [email protected] B. Li School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi, China A. Shooshtari Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_1

3

4

H. Kalhori et al.

b a

x y

Microplate

z

V h

Electrode

d

Fig. 1 A schematic of a rectangular air gap capacitive microplate with its geometry

the plate deflection, resulting in the coupling of the electrical and mechanical forces. This creates a complex equation of motion. In capacitive microplates, the electrostatic load has an upper limit beyond which the mechanical restoring force of the plate can no longer resist its opposing force, thereby leading to a continuous increase in the microstructure deflection and, accordingly, an increase in the electric forces in a positive feedback loop. This behavior continues until a physical contact is made with the stationary electrode. This structural instability phenomenon is known as ‘pull-in’. Typically, in capacitive microplates, the microplate is actuated by DC voltages higher than the pull-in voltage in order to snap it in a fast impact. To the best of the author’s knowledge, none of the papers available in the literature dealt analytically with static and transient deflection of a capacitive rectangular microplate, neither did they look at deriving a parametric relationship among design parameters. So there is a need to find a closed form answer for transverse displacement in order to determine the effects of various characteristic parameters on transverse displacement. In this chapter, the nonlinear equations of motion for a rectangular capacitive microplate based on a combination of the classical plate theory and the von Kármántype nonlinearity were derived. The nonlinear terms were due to the electric force nonlinearity and the mid-plane stretching of the plate. A significant geometric nonlinearity appeared when the plate deformed approximately one-half of its thickness, and so mid-plane stretching was of great importance. By introducing a force function, these equations were reduced to a set of coupled nonlinear partial differential equations (PDEs) and a compatibility equation. Due to the singularity in

Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate

5

the electrostatic force, applying the Galerkin procedure to the PDEs was impossible. To overcome this, two methods to treat the singularity were proposed. By using the Galerkin method, a nonlinear ordinary differential equation (ODE) including nonlinear inertia and stiffness terms was obtained. The multiple time scales method was implemented to offer an analytical expression for the transient vibration of the microplate. This expression can create an effective design tool useful in design optimization, simulation, and prediction of the mechanical behavior of MEMS devices.

2 Deriving Equations of Motion It is assumed that the span-to-thickness ratio is greater than 20 and so the transverse shear deformation, the rotary inertia, and the transverse normal stress are neglected. The microplate physical dimensions and material properties are listed in Table 1. The linear classical plate theory is based on the Kirchhoff hypothesis. That is, tractions on surfaces parallel to the reference plane are negligibly small compared to the in-plane stresses, and the in-plane displacements are linear functions of z. Following on some mathematical procedures, the partial differential equation of motion of the plate is given by D∇ 4 w + ρhw¨ = w,xx ψ,yy − 2w,xy ψ,xx + w,yy ψ,xx + I2 ∇ 2 w¨ + P ,

(1)

where dot denotes derivation with respect to time, ∇ 4 is the differential operator, w is deflection of the plate, and D is the flexural rigidity defined as D=

Eh3 ,  12 1 − ν 2

(2)

where, E is elastic modulus, h is thickness, and ν is Poisson’s ratio of the plate. The compatibility equation in terms of stress functions is defined as   2 − wxx wyy . ∇ 4 ψ = Eh wxy

(3)

Equations (1) and (3) form the equations of motion. The electric pressure P is given as

Table 1 Physical dimensions and material properties of the microplate a 230 μm

b 230 μm

h 2 μm

d 4 μm

υ 0.3

E 160 Gpa

ρ 2230 kg/m3

6

H. Kalhori et al.

p=

V (t)2 1 ε0 , 2 (d − w)2

(4)

where ε0 is the dielectric constant, and V(t) is the applied voltage. It should be noted that due to the limitation of the theoretical model, the effect of the fringe electrical field near the plate edges on the applied force is neglected. Introducing Eq. (4) into Eq. (1) gives the equation of motion as ∇ 4w =

 ρh 1 V (t)2 h  w,xx F,yy − 2w,xy F,xx + w,yy F,xx − w¨ + ε0 , D D 2 D(d − w)2 (5)

where ψ = hF. Here, V(t) = Vp and the electrostatically excited microplate is considered with the geometry specified in Fig. 1.

3 Eliminating Singularity in the Electrostatic Force According to Eq. (5), there is a singularity in the electrostatic force at w = d. Therefore, the Galerkin procedure cannot be applied due to the impossibility of integrating the PDE over the domain. Two methods are suggested to treat the term 1/(d−w)2 in the electrostatic force in the discretization procedure. Beginning with the first method, this term is expanded in a Taylor series up to fifth order. In the second method of treating the electrostatic force, both sides of Eq. (5) are multiplied by (d−w)2 so that the electrostatic force is presented precisely. The drawback of this method is that it is more complicated to numerically integrate the equation compared to the method of Taylor series. The Galerkin procedure is applied by means of the following equation using Maple   Lw dxdy,

(6)

A

where A is the area of the rectangular plate and L is Eq. (5). The following equations are obtained for both methods of Taylor series and Multiplication, respectively. ¨ + B1 Z(t) + B2 Z(t)2 + B3 Z(t)3 + B4 Z(t)4 + B5 Z(t)5 + B6 = 0, Z(t)

(7a)

¨ + A2 Z(t)2 Z(t) ¨ + A1 Z(t)Z(t) ¨ + A3 Z(t) + A4 Z(t)2 + A5 Z(t)3 + Z(t) 4 5 2 A6 Z(t) + A7 Z(t) + A8 V (t) = 0,

(7b)

wmax

Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

7

Using Eq. (7b)

Using Eq. (7a)

0

50

100

150

200

Non-dimensional voltage Fig. 2 Variation of the maximum nondimensional deflection wmax with respect to the nondimensional electrostatic load

where Ai (i = 1...8) and Bi (i = 1...6) are constant coefficients and Z(t) is a variable of time. Fig. 2 shows the maximum plate deflection wmax = w(0.5, 0.5) with the electrostatic load using Eq. (7a) and (7b) setting the time derivatives equal to zero. As seen, for nondimensional voltages lower than 75 both curves are in good agreement.

4 Transient Behavior Since there is no closed form solution for the problem, a Multiple Time Scales approach [2–5] is used to obtain the approximate solution. The approximate solution of Eq. (7b) is assumed as a second-order expansion in terms of a small positive parameter ε, which is a measure of the amplitude of the motion. Z (τ0 , τ1 , τ2 , ε) = Z0 (τ0 , τ1 , τ2 ) + εZ1 (τ0 , τ1 , τ2 ) + ε2 Z2 (τ0 , τ1 , τ2 )

(8)

where Z0 , Z1 , and Z2 are three unknown functions, and the multiple independent time scales are defined as τ0 = t, τ1 = εt, and τ2 = ε2 t The coefficients in Eq. (7b) are ordered to show up O (ε2 ) as follows

(9)

H. Kalhori et al.

Deflection

8

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10 15 Nondimensional time

20

Fig. 3 Transient response of the system

 ¨ + A2 Z(t)2 Z(t) ¨ + ω2 Z(t) + ε2 A1 Z(t)Z(t) ¨ + A4 Z(t)2 + A5 Z(t)3 + Z(t)  ˙ A6 Z(t)4 + A7 Z(t)5 + μZ(t) + A8 Vp 2 = 0. (10) The one-term approximate solution of Eq. (10) is Z=−

A8 Vp2 A3

+ a0 e−1/2μτ 2 cos

+

A2 A28 Vp2 a 2A3

+ 38 a 3 A2 A3

+

A4 A8 Vp2 a A3



12A6 A8 Vp2 a 3 8A3



3A5 A28 Vp4 a 2A23 10 5 32 A7 a



√ A3 τ0 +

5A7 A48 Vp8 a 2A43



√1 − A3 a



30A7 A28 Vp4 a 3 8A23

  τ2 + β0

− 38 A5 a 3 − 12 aA1 A8 Vp2

+

2A6 A38 Vp6 a A33

+

(11) The transient dynamic behavior of the system is presented in Fig. 3. Now, as time goes by, the time-dependent part of the approximate solution goes to zero. The remaining equation is independent of time and represents the maximum static deflection of the microplate. So, the relation for maximum static deflection is Ws = −

A8 Vp2 A3

,

(12)

Nonlinear Vibration of an Electrostatically Excited Capacitive Microplate

9

5 Concluding Remarks Two methods for treating the singularity in the electrostatic voltage were compared. It was shown that for small magnitudes of voltage both methods led to the same level of deflection, though the accuracy of the Taylor method depends on the order of expansion. Moreover, an analytical expression for transient vibration of the microplate was derived using the method of multiple time scales. The static deflection was obtained by removing the time-dependent parts of the transient response of the system providing an effective design tool for predicting the mechanical behavior of MEMS devices for small applied voltages and up to pull-in phenomenon.

References 1. Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12(5), 672–680 (2003) 2. Shooshtari, A., Hoseini, S.M., Mahmoodi, S.N., Kalhori, H.: Analytical solution for nonlinear free vibrations of viscoelastic microcantilevers covered with a piezoelectric layer. Smart Mater. Struct. 21(7), 075015 (2012) 3. Hosseini, S.M., Kalhori, H., Shooshtari, A., Mahmoodi, S.N.: Analytical solution for nonlinear forced response of a viscoelastic piezoelectric cantilever beam resting on a nonlinear elastic foundation to an external harmonic excitation. Compos. Part B. 67, 464–471 (2014) 4. Hosseini, S.M., Shooshtari, A., Kalhori, H., Mahmoodi, S.N.: Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers. Nonlinear Dyn. 78(1), 571–583 (2014) 5. Mareishi, S., Kalhori, H., Rafiee, M., Hosseini, S.M.: Nonlinear forced vibration response of smart two-phase nano-composite beams to external harmonic excitations. Curved Layer. Struct. 2(1), (2015)

Elimination of High-Frequency Whistle Noise in a Residential Ducted Air-Conditioning System Using a Dedicated Pipe Muffler Design Sam Han, J. C. Ji, and Kan Ye

1 Introduction Besides providing high energy efficiency at comfortable temperature and humidity levels, providing quiet working and living conditions has become one of the key factors for air conditioner manufacturers to compete in an increasingly competitive market. Yet, with increased living standards, customers’ expectations on the noise level of any appliances, including air-conditioning units, have also steadily increased. Low noise level with good sound quality is a must-have attribute for the development of a new product. In general, the noise of a residential ducted air-conditioning unit mainly comes from the compressor and the cooling fan. Compressor noise can be categorized into three groups, namely, mechanical noise, electric-magnetic noise, and fluid flow-induced noise. Flow-induced noise stems from the pressure fluctuation of the refrigerant (fluid-state or gas-state) while circulating in system. Scroll compressors are often used, which have acoustic cavities within the compressor to reduce discharge noise propagating to the airconditioning pipe system [1, 2]. However, for some cavity designs, the high pressure and the high velocity discharge refrigerant becomes the cause for unpleasant highfrequency flow-induced noise as an additional noise source when passing through the cavity. Noise issues from an abnormal compressor were studied by Zhang et al. [3]. The noise can be propagating and amplified to some extent inside the pipe. One part of the noise is radiated to the outside directly through the pipe loops. The other

S. Han () Actron Engineering Pty Ltd, Marsden Park, NSW, Australia e-mail: [email protected] J. C. Ji · K. Ye School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_2

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part of the noise causes the pipe to vibrate. The pipe vibration can be transferred to the supports, which, subsequently, at strong enough vibrations, also start radiating noise. A controlling method using a damper was discussed by Wu and Wei [4]. Recently a high-frequency whistle noise issue of the ducted air-conditioning systems has been reported. In the present study, such a noise issue is identified through onsite noise tests firstly in Sect. 2. Although muffling devices can be used to reduce noise, commonly used mufflers for vehicle engines and industrial ducts are found not suitable for air conditioning pipe system due to the high fluid velocity that goes through. Due to the high fluid velocity within its pipe system and the restriction of space and cost, a dedicated pipe muffler design is proposed in Sect. 3. Both theoretical calculations and onsite tests of the newly designed muffler show that it is superior to the old design. Lastly, Sect. 4 summarizes the presented work.

2 Whistle Noise Measurement and Analysis Initial investigation is conducted onsite to localize the noise source. During the measurement, it is found that the whistle noise occurred under relatively higher discharge pressure condition. The reason could be that the velocity of the refrigerant inside the discharge pipe is different under different values of discharge pressure. The flow noise level inside the pipe depends on the velocity of the refrigerant. Figure 1 shows the indoor noise spectrum under 3000 kPa operating condition and the sound quality metric total-tone-to-noise ratio (TNR) analysis. From Fig. 1, it can be seen that the frequency content of the whistle noise is over 2000 Hz, mainly in the range of 2200 Hz~4000 Hz. The TNR analysis also shows that the noise contains several tonal components between 2000 Hz and 4000 Hz. The maximum TNR is over 20 dB, which means the tone component is very prominent and dominant. By comparison of removing the noise isolation jacket at the compressor and switching the working conditions, the whistle noise source is

Fig. 1 Problem indoor noise spectrum and its sound quality TNR for onsite measurement

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localized at the pipe section from the compressor discharge port to the reversal valve. The noise generated inside this pipe section propagates in line wave to other pipes and induces the vibration of pipes and eventually radiates audible noise inside the building. In order to develop solutions to address the whistle noise issue, the noise was duplicated on another unit in an anechoic room for further study.

3 Muffler Design and Its Noise Attenuation Performance Study From both the initial onsite investigations and the measurement in the anechoic room, the pipe section from the compressor discharge port to the reversal valve is identified as the noise source. Muffling devices are commonly used to reduce noise associated with high-pressure gas. A muffling device allows the passage of fluid while at the same time restricts the free passage of sound. A muffler is able to suppress the generation of noise and also attenuate noise already produced. To reduce the flow noise inside the pipe, a small expansion type muffler is first chosen for the test. The muffler is inserted in the pipe section just after the compressor discharge port, as shown in Fig. 2. The simple expansion-cylinder-type muffler has dimensions of 110 mm length and 54 mm diameter. The refrigerant inlet and outlet diameter of the muffler is 10 mm. According to theoretical calculation [3–6], when R410A as sound

Fig. 2 Muffler inserted in the pipe system

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Fig. 3 Noise spectrum comparison with and without adding muffler in the anechoic room

propagation medium at zero mean flow, the muffler up-limit cutoff frequency (first radial mode) is 4067 Hz. Below this frequency, the muffler should provide certain noise reduction except for pass-through frequencies. However, the test results, as shown in Fig. 3, reveal that adding the simple expansion-type muffler does not reduce the outdoor unit noise in the concerned frequency range even though there is slight amplitude decrease at a few peak frequencies. The reason the muffler does not demonstrate any reduction of the high-frequency noise could be that the flow velocity inside the muffler is high. The Mach Number of the mean flow of the muffler is not zero. A higher Mach Number generally lowers the cutoff frequency slightly [4]. At the same time, a higher velocity flow can generate turbulence inside the muffler and self-excited noise at the outlet of muffler. In order to further confirm that the simple expansion muffler does not have the capability to reduce the high-frequency noise in the pipe system with highvelocity mean flow, the muffler noise insertion loss performance is examined in a reverberant room using high-pressure air as medium to replace the refrigerant inside the air conditioner. The theoretical transmission loss calculation by air is also carried out. Figure 4 shows the tested insertion loss and the calculated transmission loss. It can be easily noticed that the tested insertion loss has decreased dramatically after the third wavenumber (over 4000 Hz). The calculated cutoff frequency is 3687 Hz, which is close the actual testing results. Based on the test results with high-pressure air, the muffler loses its noise attenuation ability after the third wavenumber. Based on the refrigerant R410A fluid property [5], the sound speed of the refrigerant R410A is about 0.53 times of the

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Fig. 4 Noise attenuation performance of a simple muffler

sound speed of air. The actual cutoff frequency of the muffler with high-pressure refrigerant R410A is around 1954 Hz. This means that the muffler will not reduce the noise level over 2000 Hz when the simple expansion-type muffler is installed in the air-conditioning discharge pipe system, which is evidenced from the actual testing results shown in Fig. 3. As such, mufflers used in air-conditioning pipe systems cannot be similar to those commonly used for vehicle engines and industrial ducts. Those expansion mufflers are filled with acoustic absorption material to help improve high-frequency noise reduction performance. Due to the restriction of space and cost, it is not feasible to design multichamber mufflers for air-conditioning applications. In terms of the above limitations, the key to solve the high-frequency whistle noise of airconditioning systems is to design one muffler with low cost and easy manufacturing. A large number of experiments are carried out to examine several design configurations, including muffler size; penetration pipe length; the size, location, and quantity of the holes; and the combination of two different sizes. It is found that the diameter and the quantity of the holes had a significant effect on the muffler noise reduction performance for the concerned frequency. A shorter muffler did not have the higher cutoff frequency. An increase in the diameter of muffler decreases the cutoff frequency. The penetration pipe at inlet improves the high-frequency noise reduction performance but degrades the low-frequency noise attenuation performance. Under a certain perforation ratio, the perforated holes have little effect on noise reduction. Holes located close to muffler inlet end give better noise

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Fig. 5 Prototype of the new muffler and penetration pipe

Fig. 6 Insertion loss comparison for mufflers

attenuation performance compared to holes located at the tip of the penetration pipe. Figure 5 shows the muffler and its insert perforated pipe. Figure 6 shows the insertion loss of the designed muffler. Compared to the muffler without the insert pipe, the insertion loss performance under low wavenumbers has a slight decrease; however, the muffler with the perforated insert pipe exhibits 10~15 dB higher insertion loss over 4200 Hz. This shows that the designed muffler has better noise attenuation performance at a high frequency range than the simple muffler and good noise reduction performance at all frequency ranges. The designed muffler is fitted into the air-conditioning pipe. In an anechoic room, the noise of the air-conditioning outdoor unit is measured again under the same running conditions. Figure 7 shows that the noise level above 2000 Hz is reduced. There is no prominent tonal component in the noise spectrum. At the same time, the

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Fig. 7 Noise and vibration comparison in an anechoic room using the new muffler

Fig. 8 Onsite noise comparison with new muffler

acceleration level on the gas pipe which connects the outdoor unit and the indoor unit is decreased by 70% with the designed muffler installed. The anechoic room testing results proved that the designed muffler can reduce the high-frequency whistle noise of the air-conditioning unit. The muffler is then installed in the onsite unit which was tested in the initial investigations. Under the same operating discharge pressure condition, the noise inside the property is measured again. The noise differences before and after the muffler was installed are shown in Fig. 8. With the designed muffler fitted in, the high-frequency whistle noise disappears, except the peaks at 3350 Hz and 4030 Hz. Over 4500 Hz, the noise spectrum is very smooth and all peaks are eliminated.

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Fig. 9 Onsite noise spectrum with new muffler and pipe reinforced

It should be noted that the higher noise level peaks at 3350 Hz and 4030 Hz are not observed in the anechoic room testing. Further inspection inside the building ceiling confirms that the pipe is in contact with the steel frame rigidly at one location. The steel frame resonates and radiates noise at its resonant frequencies under the pipe vibration excitation. Even though the pipe is wrapped using thermal insulation foam, the foam has lost its vibration isolation performance since the compression is very high under the bending force (moments). Then the pipe is properly bent to remove the bending force (moment) load and a new piece of foam is placed between the pipe and the steel frame. Noise measurement is taken again under the same operating condition. Figure 9 illustrates the final noise spectrum compared with the original complained noise spectrum. It can be clearly seen that the high-frequency whistle noise is eliminated completely from the noise spectrum. The air-conditioning noise is almost inaudible inside the property.

4 Conclusion In this study, a whistle noise issue caused by ducted air-conditioning units in high discharge pressure working condition is investigated. Both noise spectrum and TNR analysis for onsite test and in anechoic room state that such noise contains several tonal components at high frequency between 2000 Hz and 4000 Hz. However, the commonly used expansion type muffler is tested and found not suitable for air-

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conditioning due to its high fluid velocity. Theoretical calculations and experiment results pointed that a simple muffler would not reduce the noise level over 2100 Hz considering a ducted air-conditioning pipe system. To overcome the noise issue, a novel pipe muffler is designed and tested for its noise reduction performance. The muffler with the perforated inlet penetration pipe is found to have the best noise attenuation performance at high frequencies without sacrificing its noise reduction performance at the range of low frequencies. Both anechoic room test results and onsite test results state that the high-frequency whistle noise was eventually eliminated. The overall noise level was greatly reduced and the sound quality was greatly improved as well.

References 1. Hagiwara, S.: Development of scroll compressor of improvement high-pressure-housing. International Compressor Engineering Conference 1998, Purdue University, West Lafayette, Indiana, USA (1998) 2. Lee J.K., Lee S.J., Lee D.S., Lee B.C., Lee U.S.: Identification and reduction of noise in a scroll compressor. International Compressor Engineering Conference. Purdue University, West Lafayette, Indiana, USA (2000) 3. Jinquan, Z., Rongting, Z., Huanhuan, G., Zhiming, W., Jia, X.: Abnormal compressor noise diagnosis using sound quality evaluation and acoustic array method. International Compressor Engineering Conference,. International Compressor Engineering Conference. Purdue University, West Lafayette, Indiana, USA (2012) 4. Wu, C., Lei, X.: Noise control and sound quality evaluation of outdoor unit of split airconditioner. J. Meas. Eng. 4(2), 58–69 (2016) 5. Ramanathan, S.K.: Linear acoustic modelling and testing of exhaust mufflers. Master of science thesis. KTH, Stockholm (2007) 6. Munjal, M.L.: Acoustics of Ducts and Mufflers, 2nd edn. Wiley, New York (2014)

A Novel and Intelligent Multi-Mode Switching Control Strategy for Active Suspension Systems with Energy Regeneration Hang Wu and Ling Zheng

1 Introduction In recent years, the energy harvesting potential based active suspension system in vehicles has been investigated extensively. Zuo et al. [1] demonstrated that the potential energy of a typical passenger car is between 100 and 400 W considering that the car is traveling on good and average roads at a speed of 97 km/h approximately. The potential energy of the harvestable power is summarized in a very broad range from 46 to 7500 W for different vehicle categories and operating conditions [2]. In [3], an energy regenerative seat suspension with a variable external resistance is proposed and the vibration energy is directly harvested from the rotary movement of the suspension’s scissors structure. Zhang et al. [4] designed active and energy-regenerative controllers for the suspension system with DC motor which operate in two modes: one is active control to enhance ride comfort, the other is energy-regenerative control for energy harvesting. Huang et al. [5] developed a controller based on model predictive control in which two control modes, such as the active control mode for full energy consumption and the semi-active mode for energy regeneration, were designed in order to reduce energy consumption. The active control mode aims at improving ride comfort, while the semi-active mode enhances energy regeneration by virtue of vibration energy caused by the uneven road. However, the potential regenerative energy and the dynamic performance vary greatly and depend on driving conditions involving vehicle speed, road surface, and traffic condition. Unfortunately, the influence of these driving conditions is seldom considered comprehensively. In this chapter a comfortable factor is introduced and a novel and intelligent multi-mode switch control strategy is developed in order

H. Wu · L. Zheng () College of Automotive Engineering, Chongqing University, Chongqing, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_3

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to achieve an overall improvement of the dynamic performance and the energy regenerative capability in vehicles according to the preference of the driver and the driving conditions.

2 The Full-Vehicle Dynamic Model A full vehicle dynamic model with 7 DOF is used to develop this new multi-mode switch control strategy for active suspension systems with permanent magnet (PM) linear motors as active actuators. It is shown in Fig. 1. According to Fig. 1, the dynamic equations are generated as follows [5]: ⎧ mb z¨b = c1 (˙z1 − z˙ 5 ) + k1 (z1 − z5 ) + u1 + c2 (˙z2 − z˙ 6 ) + k2 (z2 − z6 ) + u2 + ⎪ ⎪ ⎪ ⎪ c3 (˙z3 − z˙ 7 ) + k3 (z3 − z7 ) + u3 + c4 (˙z4 − z˙ 8 ) + k4 (z4 − z8 ) + u4 ⎪ ⎪ ⎪ ⎪ IP θ¨ = [c3 (˙z3 − z˙ 7 ) + k3 (z3 − z7 ) + u3 + c4 (˙z4 − z˙ 8 ) + k4 (z4 − z8 ) + u4 ] b ⎪ ⎪ ⎪ ⎪ ⎪ − [c1 (˙z1 − z˙ 5 ) + k1 (z1 − z5 ) + u1 + c2 (˙z2 − z˙ 6 ) + k2 (z2 − z6 ) + u2 ] a ⎪ ⎪ ⎨ ¨ Ir φ = [c2 (˙z2 − z˙ 6 ) + k2 (z2 − z6 ) + u2 + c4 (˙z4 − z˙ 8 ) + k4 (z4 − z8 ) + u4 ] lr ⎪ − [c1 (˙z1 − z˙ 5 ) + k1 (z1 − z5 ) + u1 + c3 (˙z3 − z˙ 7 ) + k3 (z3 − z7 ) + u3 ] ll ⎪ ⎪ ⎪ ⎪ m z ¨ = k5 (z01 − z1 ) + k1 (z5 − z1 ) + c1 (˙z5 − z˙ 1 ) − u1 1 1 ⎪ ⎪ ⎪ ⎪ m2 z¨2 = k6 (z02 − z2 ) + k2 (z6 − z2 ) + c2 (˙z6 − z˙ 2 ) − u2 ⎪ ⎪ ⎪ ⎪ ⎪ m3 z¨3 = k7 (z03 − z3 ) + k3 (z7 − z3 ) + c3 (˙z7 − z˙ 3 ) − u3 ⎪ ⎩ m4 z¨4 = k8 (z04 − z4 ) + k4 (z8 − z4 ) + c4 (˙z8 − z˙ 4 ) − u4 (1)

z5

b

a

c1

k1

z8 Sprung muss zb

lr

c4

ll

ay

z1

k2

(wheel 4) Pitch Axis

k5 c2

k6

(wheel 2)

k8 z04

unsprung muss

z2

z x

z7

z4

θ

z6

ll

φ z01

lr k4

ax

Roll Axis

y

z02 road

Fig. 1 7-DOF full vehicle model with active suspension system

c3

k3

z3 (wheel 3) k7

z03

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The displacements of four end points with sprung mass can be written when the small pitching angle θ and rolling angle ∅ are assumed: z5 = zb − ll φ − aθ z6 = zb + ll φ − aθ z7 = zb + ll φ + bθ z8 = zb − ll φ + bθ

(2)

3 The Design of the Optimal Controller in Active Suspension For the full vehicle model, the body acceleration, the pitch acceleration, the roll acceleration, the suspension, and tire deflections are very important characteristics and they should be chosen as the output variables of active suspension systems. So the state variables X and the output variables Y can be defined as: 

˙ θ˙ , z1 , z2 , z3 , z4 , z˙ 1 , z˙ 2 , z˙ 3 , z˙ 4 , z01 , z02 , z03 , z04 XT = z, φ, θ, z˙ , φ, 

¨ θ¨, z5 −z1 , z6 −z2 , z7 −z3 , z8 −z4 , z1 −z01 , z2 −z02 , z3 −z03 , z4 −z04 YT = z¨ , φ, (3) Therefore, the state space equations are generated as ˙ = AX + BU + GW X Y = CX + DU

(4)

where, W is the input matrix of road disturbances with white Gaussian noise, such W = [z01 z02 z03 z04 ]T . U is the input matrix of the active control force in suspension systems, which is expressed as U = [u1 u2 u3 u4 ]T . The linear quadratic regulator (LQR) controller is designed to improve ride comfort and road holding as well as maintain suspension working space within a reasonable range. The objective function is defined as follows:  ∞ J = 12 0 q1 z¨b2 + q2 φ¨ 2 + q3 θ¨2 + q4 (z5 − z1 )2 + q4 (z6 − z2 )2 + q4 (z7 − z3 )2 + q4 (z8 − z4 )2 + q5 (z1 − z01 )2 +

 q5 (z2 − z02 )2 + q5 (z3 − z03 )2 + q5 (z4 −z04 )2 +q6 u21 +q6 u22 +q6 u23 +q6 u24 dt   = 12 YT QY + uT Ru dt (5)

where, qi (i = 1–6) is the weighted coefficient of the body vertical acceleration, the roll angular acceleration, the pitch angular acceleration, the suspension deflection, the tire deflection, and the control force. Furthermore, Eq. (5) can be deduced as 1 J = 2



∞

 ˆ + 2XT Nu + uT Ru ˆ dt XT QX

0

(6)

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ˆ = CT QC, N = CT QD, and R ˆ = R + DT QD. Theoretically, the In Eq. (5), Q solution to the optimal control problem is a state feedback law U = −KX where the feedback gain K is determined by solving the Riccati equation. In order to obtain the optimal weight coefficients of each evaluation indicator for the LQR controller, the corresponding weight coefficients are optimized by means of the particle swarm optimization (PSO) method. A set of the optimal weight coefficients such as q1 = 9.8815, q2 = 9.885, q3 = 9.8986, q4 = 9.8825, q5 = 3.26 × 105 , and q6 = 2 × 10−5 are thus obtained. Finally, the optimal feedback gain matrix K is obtained. The full-vehicle model parameters are listed in Table 1.

4 A Novel Multi-Mode Switch Control Strategy In this chapter, permanent magnet (PM) linear motors are used as actuators in the proposed active suspension systems. The working states of the PM linear motor can be described in Figs. 2 and 3. Figure 2 shows that in electromotor state, the PM linear motor converts the electrical energy into the mechanical energy and consume energy from the power source. Figure 3 shows that the PM motor operates in electromagnetic generator state in which the motor converts mechanical energy into electrical energy to charge the power source. In general, if the vehicle is in low energy state, it is possible to sacrifice the comfort in vehicle to regenerate energy by virtue of vibration energy caused by

Table 1 Parameters of full vehicle dynamic model Parameter C1 , C2 , C3 C4 /N∗ s/m M1 , M2 , M3 , M4 / Kg K1 , K2 , K3 , K4 /N/m K5 , K6 , K7 , K8 / N/m

Value 1100 40 19600 200000

Parameter Ms / Kg Ix / Kg∗ M2 Iy / Kg∗ M2 a/m

Value 1245 480 1356 1.08

Parameter b/m lr/m ll/m

Fig. 2 The electromotor state

i

Value 1.62 0.775 0.775

R

v F + -

Fig. 3 The generator state

U

i

E

+ -

Re v F

+ -

U

E

+ -

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the uneven road. However, when the vehicle is full of energy and ride comfort is not good enough for passengers, it is required to consume a part of the energy and improve the comfort. In order to achieve this goal, a comfortable factor is considered and a novel and intelligent multi-mode switching control strategy is proposed and developed. The control block diagram is shown in Fig. 4. Figure 4 shows the multi-mode switch control logic in active suspension systems. fdamp denotes the damping force; fa is the active control force expected; comfort is the comfortable factor to be induced; gapmax is the limited switch value. It is seen that when fa and fdamp are in opposite directions, active mode is triggered, and a large amount of electric energy from power sources is consumed to improve the ride comfort in the vehicle. However, when fdamp and fa are in the same direction, three situations can be considered. If the absolute value of fdamp is greater than the absolute value of fa , it means the extra damping force over the active control force is obtained, in this situation, the suspension can operate in semi-active mode to harvest vibration energy and ensure a good vibration isolation. If the absolute difference between fa and fdamp is less than Gap, it implies that the active control force fa is not greater than the damping force by much. A small sacrifice of ride comfort is possible to regenerate vibration energy. On the other hand, if the absolute difference between fa and fdamp is greater than Gap, it implies that the active control force fa is much greater than the damping force, it is impossible to sacrifice ride comfort anymore, and an active mode has to be triggered again to maintain minimum requirement of ride comfort in the vehicle. In Fig. 4, the switch value gapmax needs to be determined through experiments on the road for different vehicle categories. The proposed strategy is tested on the road surface of B type and the passenger car drives at a speed of 20 m/s. The simulation results are shown in Figs. 5 and 6. Energy regeneration Ere and energy consumption Eco can be expressed as follows:

fdamp, fa

Fdamp, fa same direction?

No fa

Active mode

fdamp

Semi-active mode

Yes 0≤ – comfort ≤ –1

|fa|Gap? No

Fig. 4 Intelligent control strategy

fout

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300 Ere Eco Ere Eco Eco

Energy(J)

250 200

comfort=0 comfort=0 comfort=1 comfort=1 whole active

150 100 50 0 0

1

2

3

4

5

Time (s) Fig. 5 Energy regeneration and energy consumption

Ratio of energy regeneration (%)

80 Comfort=0 Comfort=1 60

40

20

0 0

1

2

3 Time (s)

Fig. 6 Ratio of energy regeneration

4

5

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Table 2 RMS comparisons of suspension performance in vehicles Dynamics performances z¨b /(m·s2 ) 2) ¨ ∅/(rad/s 2 ¨ θ/(rad/s ) (z1 − z01 )/(mm) (z2 − z02 )/(mm) (z3 − z03 )/(mm) (z4 − z04 )/(mm)

Ere =

4  

Comfort0 0.4037 0.6342 0.4878 0.0028 0.0027 0.0027 0.0026

Comfort1 0.3952 0.5737 0.4758 0.0028 0.0027 0.0027 0.0026

fi damp · (˙z4+i − z˙ i ) dt

i=1

Passive suspension 0.5066 0.8548 0.5410 0.0028 0.0028 0.0027 0.0027

Eco =

4  

Effects (%) Comfort 0 25.495 34.7912 10.8878 2.4462 3.5111 2.2440 2.9070

Effects (%) Comfort 1 28.1857 49.0121 13.6975 3.0012 3.6484 2.4083 3.5287

fi a · (˙z4+i − z˙ i ) dt

(7)

i=1

It is seen in Fig. 5 that when the comfortable factor is equal to 0 (maximum energy regeneration state), almost all recovery energy is consumed to ensure minimum ride comfort and handling performance in the vehicle. In contrast, when the comfortable factor is equal to 1 (maximum comfort state), a larger amount of energy than the recovery energy alone is consumed to improve the ride comfort in vehicle. The ratio of recovery energy in semi-active mode to active mode is shown in Fig. 6. Similar results as in Fig. 5 are demonstrated. This implies that nearly 50% of energy consumption in active mode is regenerated in maximum energy regeneration state. In maximum comfort state, only 38% of energy consumption in active mode is regenerated due to the requirement of the ride comfort improvement related to the preference of the driver. Table 2 demonstrates the influence of semi-active regeneration mode on the performance of the suspension system in vehicles. It is shown in Table 2 that no matter which type of strategy is adopted in the semi-active mode, the overall properties of the suspension system can clearly be improved, although the recovery energy is reduced in the maximum comfort state.

5 Conclusion In this chapter, a LQR controller of active suspension system with PM linear motors is designed and a set of the optimal weight coefficients is obtained by means of the PSO method. A novel and intelligent multi-mode switch control strategy is proposed to regenerate vibration energy and maintain an excellent dynamic performance. The simulation results show that the proposed multi-mode switch control strategy can improve ride comfort significantly and achieve considerable energy regeneration.

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References 1. Zuo, L., Zhang, P.-S.: Energy harvesting, ride comfort, and road handling of regenerative vehicle suspensions. J. Vib. Acoust. 135(011002), (2013) 2. Abdelkareem, M.A.A., et al.: Vibration energy harvesting in automotive suspension system: a detailed review. Appl. Energy. 229, 672–699 (2018) 3. Ning, D., Sun, S., Du, H., et al.: Vibration control of an energy regenerative seat suspension with variable external resistance. Mech. Syst. Signal Process. 106, 94–113 (2018) 4. Zhang, G., Cao, J., Yu, F.: Design of active and energy-regenerative controllers for DC-motorbased suspension. Mechatronics. 22(8), 1124–1134 (2012) 5. Huang, K., Yu, F., Zhang, Y.: Active controller design for an electromagnetic energyregenerative suspension. Int. J. Automot. Technol. 12(6), 877–885 (2011)

Design of a Circular-Type Pod Silencer with Annular Two-Layered Air Passages for a High-Pressure Axial Flow Fan Chan Lee, Hyun Gwon Kil, Jong Jin Park, Dong Hyun Kim, and Sang Ho Yang

1 Introduction Axial flow fans are widely used in low pressure air handling systems such as cooling, air-conditioning, or ventilating equipment. But subway ventilation systems require axial flow fans with relatively high pressure at high flow capacity. These generate a high noise level. The noise consists of two components: discrete frequency noise component at blade passing frequency (BPF) due to rotating impellers and broadband noise component due to turbulence in inflow and exhaust jet mixing [1]. The main contribution to the high noise level is from the discrete frequency noise component. Silencers are needed to reduce the high noise level. Rectangular silencers in subway ventilation systems have been widely used. But those silencers generate relatively high pressure loss. Therefore, there have been industrial needs for reducing high noise level with circular-type pod silencers effective for axial flow fan performance with lower pressure loss. The circular-type pod silencer was analyzed by using a transfer matrix method with plane wave approximation [2]. The design curves for performance evaluation of passive pod silencers were provided by simulating the acoustic performance of the silencers with commercial FEA software [3]. In this chapter, a circular-type pod silencer with annular two-layered air passages has been designed to reduce

C. Lee () · H. G. Kil · D. H. Kim University of Suwon, Hwaseong-si, Gyeonggi-do, Republic of Korea e-mail: [email protected]; [email protected]; [email protected] J. J. Park DAS, Research Center, Gyeonggi-do, Republic of Korea S. H. Yang Samwon E&B, Gyeonggi-do, Republic of Korea e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_4

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high noise level generated by a high-pressure axial flow fan. In the design process, design parametric study and design optimization have been performed with ANSYS acoustic code [4] and optimization code PIAnO [5].

2 Analysis 2.1 Noise Characteristics The noise source considered in this research is an axial flow fan with high pressure (780 Pa) and blade out diameter 1.778 m in Fig. 1a. It generates a high noise level. The noise consists two kinds of noise components: discrete frequency noise at blade passing frequency (BPF) and the broadband noise distributed over wide frequency range. BPF noise is produced mainly due to rotating steady fan blade thrust and blade interaction. Broadband noise is produced over the entire frequency range due to the turbulent boundary layer on blade surface, inflow turbulence, and blade wake. Figure 1b shows the typical pattern of noise spectrum measured from the regenerative blower. Here BPF corresponds to 198 Hz.

2.2 Finite Element Analysis The model is a circular-type pod silencer with annular two-layered air passages in Fig. 2a. Pod and annular tubes consist of perforated tubes except on the outside surface. The tubes are filled with sound-absorbing materials. The acoustic domain of the silencer is divided into elements, as shown in Fig. 2b, c. FEA is applied to solve the wave equation and to predict the transmission loss (TL), which corresponds to the logarithmic ratio between the incident sound power at the inlet of the silencer and the transmitted power at the outlet of the silencer.

Fig. 1 (a) Axial flow fan and (b) measured noise spectrum of the axial flow fan

Design of a Circular-Type Pod Silencer with Annular Two-Layered Air. . .

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Fig. 2 (a) Circular-type pod silencer model and (b) finite element model of a circular-type pod silencer with inside modeling as (c)

Fig. 3 (a) Silencer model and (b) comparison of the predicted the TL of silencer models (without a pod [model A], with a rigid pod [model B], and with an sound-absorbing pod [model C], respectively) with the results of reference [8] for analysis verification

For a more efficient way to model perforation of the silencer, meshes on the perforated tube are replaced by the two inner and outer concentric surfaces with acoustic transfer admittance. For the acoustic transfer admittance, the transfer admittance of the perforated plate [6] with the same perforation pattern of the perforation tube is used. In order to model the sound-absorbing material, the Miki equivalent fluid model [7] with complex characteristic impedance has been implemented. The simulation has been verified by comparing the simulated TL of the circular-type pod silencer with annular one-layered air passages with the result of the reference [8]. The silencer models are circular-type silencer without a pod, circular-type silencer with a rigid pod, and circular-type silencer with soundabsorbing pod (flow resistance R = 16,000 Ns/m4 ) in Fig. 3a. These have the dimensions h = 0.15m, Dp = 0.3m, d = 0.2m, and L = 1.2m. The predicted numerical results for the TL have been compared with the corresponding results in [8] in good agreement, as shown in Fig. 3b. They showed that the FEA approach can be used to evaluate the TL of the circular-type pod silencer with sound-absorbing layers and a pod.

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3 Design 3.1 Design Variables The design model is the circular-type pod silencer with annular two-layered air passages as shown in Fig. 2a. Design variables are pod diameter Dp , two air passage gaps h1 and h2 , thickness of the sound-absorbing material d, silencer length L, type and density ρ of the sound-absorbing material, and porosity of the perforated tubes σ . Considering a flame-retardant material, glass wool (GW) is selected as the sound-absorbing material. The design parametric study was performed by evaluating the TL. Figures 4, 5, and 6 show the influence of changing corresponding design variable on the TL of the silencer. Figure 4a shows little influence of changing the thickness of the absorbing outer layer on the TL if it is more than about 0.20 m. Figure 4b shows that as the air flow gap decreases, the TL increases in all frequency regions and the frequency at which the maximum TL value occurs also increases. Figure 5a shows that as the length of the silencer increases, the TL value increases in the main noise reduction frequency band. Figure 5b shows that changing the pod diameter has little influence on the TL. Figure 6a shows that as the density of the sound-absorbing material increases, the main noise reduction frequency band increases but the maximum TL value decreases. Figure 6b shows little influence of changing the porosity of the perforated tube on the TL, if it is more than about 46.2%. The parametric study results show that the most sensitive design parameter corresponds to the air passage gap.

3.2 Optimization The optimization process has been performed with three free design continuous variables such as two air passage gaps h1 and h2 and pod diameter Dp with

Fig. 4 Influence of corresponding design variable change on the TL of the silencer as (a) soundabsorbing layer thickness d and (b) inner air passage gap h1

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Fig. 5 Influence of corresponding design variable change on the TL of the silencer as (a) silencer length L and (b) sound-absorbing pod diameter Dp

Fig. 6 Influence of corresponding design variable change on the TL of the silencer as (a) soundabsorbing material density ρ in unit of kg/m3 and (b) porosity of the perforated tubes σ

consideration of sensitivity and coupled effects of the design variables. It has been also performed for each of three discrete variables of density of the sound-absorbing material that correspond to the commercially available 32 kg/m3 , 48 kg/m3 , and 64 kg/m3 . The other less-sensitive variables are regarded to be constant. The silencer installation space condition in subways provides the constraints for the outer diameter D0 and length L as D0 ≤ 3.3 m and L ≤ 4.3 m. Considering the fan casing inner diameter 1.8 m of the axial flow fan, constraint as inner diameter Di of the silencer, Di ≥1.8 m, is also given. The objective function of the present design optimization problem is defined as the overall pressure level at the outlet of the silencer, which is determined by calculating the TL of the silencer and finally subtracting the TL from the measured fan noise spectrum. As an optimization algorithm, the progressive quadratic response surface method (PQRSM) [5] has been used. This method uses a meta-model generated based on the design of experiment (DOE) and provides more reliable optimum solutions than conventional gradient-based optimization algorithms [5]. The design constraints for feasible ranges of the three free design variables – 0.6 m ≤ Dp ≤ 1.2 m, 0.2 m ≤ h1 ≤ 0.5m, and 0.2 m ≤ h2 ≤ 0.5 m – are considered.

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Fig. 7 (a) Sensitivity analysis results of optimal variables and (b) reduced noise spectrum of the axial flow fan with the silencer

4 Result Sensitivity of the free design optimal variables is shown in Fig. 7a. In Fig. 7a, Y/Yref and X/Xref express the change in the overall SPL relative to the reference value of the overall SPL and the change in the design variable relative to the reference value of the corresponding design variable, respectively. It shows that the most sensitive design parameter corresponds to the outer air passage gap h2 . Through the present design optimization process, optimal design variables are: each air passage gap 0.2 m, the sound-absorbing material thickness 0.2 m, and the pod diameter of 0.8 m, that lead to the outside diameter 2.4 m. Noise attenuation by the optimal silencer with sound-absorbing material density 32 kg/m3 , 48 kg/m3 , and 64 kg/m3 is 22 dB(A), 23 dB(A) and 24 dB(A), respectively. Sound-absorbing material GW 48 is selected considering material cost. Noise attenuation by the final optimum silencer is shown to be the overall pressure level reduction of 23 dB(A) as shown in Fig. 7b.

5 Conclusions A circular-type pod silencer with annular two-layered air passages has been designed to reduce the high noise level generated from an axial flow fan in subways. In order to effectively reduce the noise, design parametric study and design optimization have been performed. The overall SPL of 106 dB(A) has been expected to be reduced to 83 dB(A). Noise attenuation with the optimal silencer is shown to be an overall pressure level reduction of 23 dB(A).

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Acknowledgment This work was supported by the Energy Technology Development Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry and Energy, Republic of Korea (No.20172010106010).

References 1. Dixon, S.L.: Fluid Mechanics and Thermodynamics of Turbomachinery, 7th ed. Butterworth & Heinemann (2014) 2. Munjal, M.L.: Analysis and design of pod silencers. J. Sound Vib. 262(3) (2003) 3. Ramarkrishnan, R.: Performance analysis of annular psssive silencers. J Can. Acoust. Assoc. 43(3) (2016) 4. ANSYS: ANSYS user manual, ver. 18.0, ANSYS Inc. (2018) 5. PIAnO (Process, Integration, Automation and Optimization) PIAnO user manual, Ver. 2019 (2019) 6. Mechel, F.P.: Formulas of Acoustics, 2nd ed. Springer (2008) 7. Miki, Y.: Acoustical properties of porous materials-modifications of Delany-Bazley models. J. Acoust. Soc. Japan,11(1) (1990) 8. Ver, I.L., Beranek, L.L.: Noise and Vibration Control Engineering-Principles and Applications, 2nd edn. Wiley, Hoboken (2006)

Optimal Control of Acoustic Radiation Power for a Triple-Walled Structure Hiroki Tanaka, Hiroyuki Iwamoto, and Shotaro Hisano

1 Introduction In recent machine designs such as automobiles, aircraft, and so on, the material of the structure tends to be thin and have high flexibility. A thin-walled, highly flexible structure tends to form a structural acoustic coupling field in which the closed space sound field inside the cavity and the structural vibration of the outer wall are coupled, and the structural acoustic coupling field changes to a complex characteristic different from a single structure or a rigid-walled cavity. Although a multiple-walled structure has excellent sound insulation, there are some frequencies whose performance deteriorates due to resonance. In order to solve this problem, active control can be introduced [1, 2]. In the previous research [2], only the case where the control force was installed in the middle plate of the triple-walled structure was examined, and the other cases were not examined. Therefore, in this research, we focus on the other types of control for the triple-walled structure and compare their control effects. Furthermore, based on the numerical results, the suppression mechanism of acoustic radiation power is discussed. First, the target structure is analytically modeled using the modal coupling method that utilizes modal amplitudes of the uncoupled subsystems [3]. Next, the optimal control law for minimizing the acoustic radiation power of the triple-walled structure is derived by using the elementary radiator method, which calculates the acoustic radiation power based on the minute and divided elements of the panel. Finally, numerical simulations of the control system are conducted, confirming the effectiveness and suppression mechanism of the proposed method.

H. Tanaka · H. Iwamoto () · S. Hisano Seikei University, Tokyo, Japan e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_5

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2 Theory First, we derive the theoretical solution for modeling a triple-walled structure depicted in Fig. 1 using the modal coupling method. The modal amplitude vectors for plates 1, 2, and 3 and spaces 1 and 2 can be obtained as follows: 

a1 = Za1 (q1 + C1 b1 − C2 b2 ) , a2 = Za2 (q2 + C3 b2 − C4 b3 )       , b1 = Ys1 g1 − CT1 a1 , b2 = Ys2 g2 + CT2 a1 − CT3 a2 , b3 = Ys3 g3 + CT4 a2 (1)

where a are acoustic mode amplitude vectors, b are vibration mode amplitude vectors, C is coupling coefficients matrix, q is volume velocity vector, g is an external force vector, Za is the matrix depending on spatial information, and Ys is the matrix depending on plate information. The calculation process of the above vectors is omitted because of space limitation. The sound pressure and structural vibration velocity can be found by Eq. (1). Second, we derive the optimal control input that minimizes the acoustic radiation power of the triple-walled structure based on the elementary radiator method [4]. The external force is assumed to act on one point of plate 1, and the acoustic radiation power of plate 3 is minimized by control input acting at each subsystem except for plate 1. The acoustic radiation power of plate 3 can be expressed as Pw3 = VH 3 RV3 ,

(2)

where V3 is the vibration velocity vector of plate 3 described by V3 =

M3 

b3 3 = T3 b3 .

m3 =1

Fig. 1 Model with external force and control source added to the triple-walled structure

(3)

Optimal Control of Acoustic Radiation Power for a Triple-Walled Structure

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Eq. (1) is simultaneously calculated first, and by solving simultaneously Eq. (1) the acoustic mode amplitude a2 of space 2 is derived as     a2 = Q2 q2 + C3 Ys2 g2 + C3 Ys2 CT2 X1 q1 + C1 Ys1 g1 − C2 Ys2 g2 − C4 Ys3 g3 , (4) where X1 , X2 , and Q2 are written as 

 −1  Xj,k,l,m = I+Zaj,k Cj,l Ysj,k CTj,l +Ck,n Ysk,l CTk,n Zaj,k (j = 1, k = 2, l = 3, n = 4)  −1 T T Q2 = I−X2 C3 Ys2 C2 X1 C2 Ys2 C3 X2

(5) Then, when Eq. (4) is substituted into Eq. (1), the vibration mode velocity amplitude b3 of plate 3 is derived as      b3 = Ys3 CT4 Q2 C3 Ys2 CT2 X1 C1 Ys1 g1 + Ys3 CT4 Q2 C3 Ys2 − C3 Ys2 CT2 X1 C2 Ys2 g2        + Ys3 I − CT4 Q2 C4 Ys3 g3 + Ys3 CT4 Q2 C3 Ys2 CT2 X1 q1 + Ys3 CT4 Q2 q2

(6) Substituting the above equation into Eq. (3) gives the following equation:      V3 = T3 Ys3 CT4 Q2 C3 Ys2 CT2 X1 C1 Ys1 g1 + T3 Ys3 CT4 Q2 C3 Ys2 − C3 Ys2 CT2 X1 C2 Ys2 g2        + T3 Ys3 I − CT4 Q2 C4 Ys3 g3 + T3 Ys3 CT4 Q2 C3 Ys2 CT2 X1 q1 + T3 Ys3 CT4 Q2 q2 ≡

3  i=1

Gi gi +

2  j =1

Gj +3 qj ≡

3  i=1

2  Gi ’i fi + Gj +3 ψj’ qj j =1

(7) ’ is an acoustic mode function vector of spaces 1 and 2 and ’ is a where ψ1,2 2,3 vibration mode function vector of plates 2 and 3. where f1 is an external force acting on plate 1, f2 and f3 are control forces given to plates 2 and 3, and q1 and q2 are control sources of the acoustic space. Furthermore, by substituting Eq. (5) into Eq. (2), the acoustic radiation power of plate 3 is represented as a quadratic function of each control force. Therefore, the control force that satisfies the following equation is optimal for the suppression of the acoustic radiation power of plate 3: ∂Pw3 ∂Pw3 ∂Pw3 ∂Pw3 = 0, = 0, = 0, =0 ∂f2 ∂f3 ∂q1 ∂q2

(8)

From the above equations, the optimal feedforward control input is derived as ⎧ ⎨

−1    H GH RG ’ ’ H gH fj,k = − ’j,k (i = 1, j = 2, k = 3) j.k j,j j,k i,i Gi,i RGj,k j,k −1    ’ H ⎩ ql,m = − ψ’H GHn,o RGn,o ψ’ gH (l = 1, m = 2, n = 4, o = 5) l,m l,m l,l Gl,l RGn,o ψl,m

(9)

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Based on Eq. (9), numerical analysis on acoustic radiation power optimum control is performed.

3 Numerical Analysis In this section, numerical analysis on active minimization of acoustic radiation power of the triple-walled cavity is conducted to investigate the relation between the control effect and the type of control method. The software used for the calculation is MATLAB. The physical parameters of cavities and plates are all assumed to be equal as shown in Table 1. Control effects are compared with respect to the type of control input. Acoustic modes and structural modes are calculated up to the (3, 3, 3) mode and the (3, 3) mode, respectively. Also, the target frequency band is from 0 [Hz] to 200 [Hz], and the resonant frequencies of spaces 1 and 2 and plates 1, 2, and 3 are listed in Tables 2 and 3. In addition, the control force was installed at a position that is not on the (1,2) mode node. Figure 2a, b show the acoustic radiation power when the control sound source is placed in space 1 or space 2. Figure 2c, d show the acoustic radiation power when control force is placed on plate 2 or plate 3. In all figures, both uncontrolled and controlled cases are plotted. In the case of noncontrol, although three peaks were confirmed in the vicinity of 80 [Hz], the number of peaks is suppressed to two when the control sound source is placed in space 2 and when the control force is applied to plate 3. Moreover, when the control sound source was placed in space 1 and when the control force is applied to plate 2, the number of peaks was suppressed to one. Furthermore, when comparing the baseline of frequency characteristics, it was also confirmed that the control effect is higher as the control input is closer to the disturbance point regardless of the control force and the control sound source. Therefore, the finding that the active noise control is more effective in suppressing the acoustic radiation power than the active vibration control shown in the previous research on the double wall structure is not a general result Table 1 Cavity and flat plate specifications

Material Thickness of plate Density of plate Poisson’s ratio   Lx , Ly , Lz1,2 Air density Velocity of sound Young’s modulus Disturbance point Control force point Control source point

SUS304 0.8 [mm] 7900 [kg/m3 ] 0.29 (0.18, 0.38, 0.43) 1.21 [kg/m3 ] 340 [m/s] 200 × 109 [Pa] (0.01, 0.01, 0) (0.18, 0.38) (0.18, 0.38, 0.215)

Optimal Control of Acoustic Radiation Power for a Triple-Walled Structure Table 2 Resonant frequency for each mode in plates 1 to 3

x axis direction 1 2 1 2 3

y axis direction 1 1 2 2 1

41 Frequency [Hz] 72.07 111.70 177.73 248.66 288.29

Table 3 Resonant frequency for each mode in spaces 1 and 2 x axis direction 0 0

y axis direction 0 0

z axis direction 0 1

Frequency [Hz] 0 392.61

(a) One sound source (space 1)

(b) One control force (plate 2)

(c) One sound source (space 2)

(d) One control force (plate 3)

Fig. 2 Frequency response of the acoustic radiation power of the triple-walled structure with and without control

in the multiple-walled structure. Next, we will consider the suppression mechanism of the acoustic radiation power when the control sound source is placed in space 1, which has the highest control effect with respect to the baseline. Figures 3 and 4 show the sound pressure distribution and the vibration velocity distribution with and without control at 70.80 [Hz] and 77.08 [Hz]. Focusing on the vibration velocity distribution and sound pressure distribution with control at all frequencies, the vibration shape of the plate adjacent to space 1 in which the control sound source is installed is in the form of the (1, 2) mode in which energy is difficult to transmit. Therefore, the influence of the vibration of plate 2 on space 2 is very small compared to that before control. As a result, it is considered that the input from space 2 to plate 3 also becomes small, and the

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(a) Non-control

(b) Control

Fig. 3 Sound pressure and vibration velocity distribution at 70.80 [Hz]

(a) Non-control

(b) Control

Fig. 4 Sound pressure and vibration velocity distribution at 77.08 [Hz]

acoustic radiation power is suppressed. Next, we consider the difference in control effect at each frequency. As apparent from Fig. 4a, a resonant peak appears only at 70.80 [Hz] after control, while three peaks around 80 [Hz] existed before control. This can be explained by the type of contribution to each mode in space 1, that is, at 77.08 [Hz], spaces 1 and 2 behave like air springs when not controlled. On the other hand, at 70.80 [Hz], the air space mainly gives the mass effect to the coupled mode because plates 1 and 2 adjacent to space 1 have the same phase and no large difference in vibration amplitude. Therefore, even if the sound pressure in space 1 is suppressed by active noise control, it only suppresses the slight spring effect that space 1 has, and as a result, the control effect is inferior compared to the case of other frequencies.

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4 Conclusion This chapter compared various acoustic radiation power control in a triple-walled structure and clarified its suppression mechanism. First, modeling of the triplewalled structure was performed using the modal coupling method. Next, the optimal control law of the acoustic radiation power was derived using the elementary radiator method. Furthermore, the numerical analysis on various acoustic radiation power optimal control methods was conducted, and their control effects were compared. As a result, in all control methods, it was found that the acoustic radiation power is suppressed, and the control effect is larger when the control input is closer to the disturbance. It was also confirmed that when one control source is placed in space 1, the best control effect is obtained, and the number of the peaks around 80 [Hz] are reduced from three to one after control. At the peak frequency after control, the air in space 1 mainly gives the mass effect to the coupled mode because plates 1 and 2 adjacent to space 1 have the same phase and no large difference in vibration amplitude. Therefore, even if the sound pressure in space 1 is suppressed, the control effect is small, and peaks appear after control.

References 1. Okada K., Iwamoto H., Tanaka N.: Active noise control and active vibration control to suppress double-walled radiation (in Japanese). Proceedings of Design and Dynamics Conference 2016 (2016) 2. T. Mizobuchi, Active control of sound radiated from a multiple-walled structure (in Japanese), Seikei University Bachelor thesis, (2013) 3. Elliott, S.J., Johnson, M.E.: Radiation modes and the active control of sound power. J. Acoust. Soc. Am. 94(4), 2194–2204 (1993) 4. Kim, S.M., Brennan, M.J.: A compact matrix formulation using the impedance and mobility approach for the analysis of structural-acoustic systems. J. Sound Vib. 223(1), 97–113 (1999)

Part II

Vehicle System Dynamics and Control

Towards Overcoming the Challenges of the Prediction of Brake Squeal Propensity Zhi Zhang, Sebastian Oberst, and Joseph C. S. Lai

1 Introduction Prediction and analysis of brake squeal has been a concern to the automotive industry since the early 1900s [1, 2]. With the advent of computer technology, numerical analysis and experimental methods, research into brake squeal has exploded in the last two decades (Fig. 1) and is not showing any signs of slowing down. Despite the progress being made in better describing brake squeal as a phenomenon, reliable numerical prediction of brake squeal propensity is as difficult as ever because there are many challenges, which include nonlinearity, uncertain material properties, contact and boundary conditions and accurate friction models. In this chapter, we will use some of our works [3–5] to illustrate the success and limitations of the traditional complex eigenvalue analysis (CEA) when used with noise dynamometer tests and propose a stochastic approach incorporating uncertainty analysis with CEA to overcome some of these challenges [6–8].

Z. Zhang · J. C. S. Lai School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT, Australia S. Oberst () Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney, Sydney, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_6

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Fig. 1 Number of papers published in English language journals and proceedings with keywords “brake squeal”, according to ISI Web of Science (accessed on 20 July 2019)

2 Examples of CEA Success and Limitations 2.1 CEA Success Traditional linear CEA combined with noise dynamometer tests has seen some success [1, 2]. In this example [3], a brake system was found by noise dynamometer tests to squeal at 9.4 kHz, corresponding to the second tangential in-plane rotor mode. A finite element (FE) model of the brake rotor and pads is shown in Fig. 2a together with the modifications to the hat section in Fig. 2b, c respectively with the addition of stiffeners and conical sections to eliminate the squeal. CEA applied to the FE model using Abaqus v6.4 shows in Fig. 2d that as the friction coefficient μ increases from 0, the doublet modes near 6.4 kHz merge at the critical friction coefficient of 0.2 where it becomes unstable and the negative damping is about 0.15% (Fig. 2e). Subsequent CEA applied to the modified rotors in Fig. 2b, c indicates that as μ increases, the doublet modes do not merge and the damping is non-negative. Noise dynamometer tests reveal that neither modified rotors squeal at 9.4 kHz, hence demonstrating the success of using CEA to first identify unstable vibration modes and then to design countermeasures.

2.2 Limitations of CEA Due to Nonlinearity It has long been suspected that the transient and fugitive character of brake squeal is caused by nonlinearity in the brake system, which could have its origin in boundary and contact conditions or material properties [1]. By applying nonlinear dynamics analysis techniques to sound pressure time series data recorded in a brake noise dynamometer, Oberst & Lai [4] were the first to identify the transition of the dynamics of a full brake system from a limit cycle to an unstable torus attractor.

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Baseline 9520 9510

Frequency (Hz)

9500 9490 9480 9470 9460 9450 9440 9430 9420 0

0.1

0.2

0.3

0.4

0.5

0.6

Friction coefficient

(a) FE model of rotor and pads

(d) Modal frequencies

(b) Stiffeners added to swan neck

Negative damping

0.20%

0.15%

0.10%

0.05%

0.00% 0

0.1

0.2

0.3

0.4

0.5

0.6

Friction coefficient

(c) Conical section of hat

(e) Negative damping level

Fig. 2 Model of brake rotor and pads and CEA near the second tangential in-plane mode [3]

The role of nonlinearity in brake squeal was examined by Oberst and Lai [5] for a pad-on-disc finite element model of a simplified brake system. While inplane pad-mode instabilities are not detected by the CEA, the dissipated energy obtained by harmonic pressure excitation is negative at frequencies of these pad modes, indicating potential instabilities. Transient nonlinear time domain analysis of the pad and disc dynamics reveals that in-plane pad vibrations excite a dominant out-of-pane disc mode. As shown in Fig. 3 for the out-of-plane disc displacement with μ = 0.6, shortly after contact is made between the disc and pad, a limit cycle is formed (Fig. 3a). A vortex regime due to a secondary Hopf-bifurcation follows (Fig. 3b), leading to the formation of a quasi-periodic 2-torus (Fig. 3c) and evolving into a toroidal attractor (Fig. 3d). Detailed nonlinear dynamic analysis confirms that the evolved torus is unstable and will lead to squeal. Furthermore, it has been shown that irregularity in the friction data and the information contained in measured data can be used to predict squeal from the non-squealing part of a time series [9, 10]. However, the numerical prediction requires a full brake system to be simulated using time domain methods, which is currently not a viable technical solution.

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Uz 1 (t + 2τ) × 10−7 m

50

Dp ≈ 2.57 × 10−7 m

3

0

-3 -3

Uz 1(

3

0

Uz 1 (t) × 10−7 m

0 t) ×

5 10 − 5 m

-2 0 2 Uz 1 (t + τ) × 10−5 m

2 0 Dp ≈ 3.77 × 10−5 m

-2

Uz 1(t +

2 0 -2 τ ) × 10 − 5 m

-2

0 1 t) Uz (

Uz 1 (t + 2τ) × 10−5 m

Uz 1 (t + 2τ) × 10−5 m

Dp ≈ 5.8×5 10−5 m

2 0 -2

2 −5

× 10

m

-2

0

2

Uz 1 (t) × 10−5 m

Fig. 3 Evolution of the dynamics of the disc with time for μ = 0.6 soon after contact is made between disc and pad: (a) limit cycle; (b) vortex; (c) periodic 2-torus; (d) after initial transients [5]

3 Stochastic Approach Incorporating Uncertainty Analysis While nonlinear time domain analysis of the dynamics involved in pad-disc interactions can overcome the deficiency of the linear CEA in predicting brake squeal propensity, such an analysis [5, 11] is computationally very expensive compared to the current industry practice of using CEA combined with noise dynamometer/road tests. In addition, the operating conditions (such as pressure and temperature), contact conditions between disc and pad, surface roughness, boundary conditions and even material properties are not known with sufficient details and accuracies. Accurate modelling of the friction at the interface between contact surfaces is another challenge. Consequently, there have been a number of studies of brake squeal using uncertainty analysis [6–8, 12–15] to account for these uncertainties. In order to address the challenges posed by nonlinearities, we have shown that incorporating the uncertainty analysis with the efficient but linear CEA can improve the reliability of prediction of unstable vibration modes due to nonlinearities that are not too severe [8].

Towards Overcoming the Challenges of the Prediction of Brake Squeal Propensity

c1x

c1z

51

k1a

z

m1

k1x

k 1c

Zoom in

c px P1

k py

c py

k pz

c zp

k px

y

x

v

Fig. 4 3 × 3 spring-mass-damper oscillators coupled through friction to a rigid sliding plate [9] 1 0.95

Coulomb CEA Coulomb friction work Velocity dependent CEA Velocity dependent friction work LuGre CEA LuGre friction work

0.85 0.75

Probability

0.65 0.55 0.45 0.35 0.25 0.15 0.05 -0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mode No. Fig. 5 Probability of a positive real part (CEA) or positive friction work [6]

We have used an analytical model of 3 × 3 spring-mass-damper oscillators coupled to a rigid sliding plate through friction (Fig. 4) to study explicitly the influence of contact condition (by disconnecting some connectors), sliding speed, friction models and a statistical distribution of friction coefficients on the prediction of unstable vibration modes. As an illustration, we used three different friction models (Coulomb-Amonton, Velocity dependent and LuGre), with randomly generated model parameter values (106 for each friction model, obtained from the statistical distributions determined from experimental brake tests) and conducted Monte Carlo simulations using CEA to study the effect of the uncertainty in friction modelling on instability prediction [6]. The probability of the real part of a complex eigenvalue being positive and that of positive linearized friction work is calculated. As shown in Fig. 5, although there are a number of unstable vibration modes (most prominently 9, 13, 18 and 22 with probability greater than 50%), mode 9 is the only unstable mode that has a probability greater than 50% for positive real part of the complex

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eigenvalue and positive friction work and which is independent of the friction model used. We consider an identified unstable mode with probability greater than 50% and independent of the friction model used being “robustly” unstable and likely to initiate squeal.

4 Conclusions In this chapter, we illustrated the initial success of the linear CEA when combined with noise dynamometer/road tests. We outlined the challenges faced by CEA for the reliable prediction of brake squeal: uncertain operating and contact conditions; material properties and friction models. We report that the use of experimental information enables prediction of instabilities in the pre-squealing regime but that there is yet a viable numerical prediction technique to be developed. Finally, we presented a stochastic approach by incorporating uncertainty analysis (to account for uncertainties present in a brake system) and CEA to identify “robustly” unstable vibration modes that are likely to squeal independent of the uncertainties. This approach shows great potential to be applied to a full brake system.

References 1. Kinkaid, N.M., O’Reilly, O.M., Papadopoulos, P.: Automotive disc brake squeal. J. Sound Vib. 267, 105–166 (2003) 2. AbuBakar, A.R., Ouyang, H.J.: A prediction methodology of disk brake squeal using complex eigenvalue analysis. Int. J. Veh. Des. 46, 416–435 (2008) 3. Papinniemi, A., Lai, J.C.S.: A study on in-plane vibration modes in disc brake squeal noise. Proceedings of Inter-Noise 2005, Rio de Janeiro, 10 pp (2005) 4. Oberst, S., Lai, J.C.S.: Chaos in brake squeal noise. J. Sound Vib. 330, 955–975 (2011) 5. Oberst, S., Lai, J.C.S.: Nonlinear transient and chaotic interactions in disc brake squeal. J. Sound Vib. 2015, 272–289 (2015) 6. Zhang, Z., Oberst, S., Lai, J.C.S.: Instability analysis of friction oscillators with uncertainty in the friction law distribution. IMechE J. Mech. Eng. Sci. 230, 948–958 (2016) 7. Oberst, S., Lai, J.C.S.: Uncertainty modelling for detecting friction-induced pad-mode instabilities in disc brake squeal. Proceedings of ICA 2010, Sydney, 12 pp (2010) 8. Zhang, Z., Oberst, S., Lai, J.C.S.: On the potential of uncertainty analysis for prediction of brake squeal propensity. J. Sound Vib. 377, 123–132 (2016) 9. Stender, M., Tiedemann, M., Hoffmann, N., Oberst, S.: Impact of an irregular friction formulation on dynamics of a minimal model for brake squeal. Mech. Syst. Signal Process. 107, 439–451 (2018) 10. Stender, M., Oberst, S., Tiedemann, M., Hoffmann, N.: Complex machine dynamics: systematical recurrence analysis of disk brake vibration data. Nonlinear Dyn. (2019). https://doi.org/ 10.1007/s11071-019-05143-x 11. Sinou, J.J.: Transient non-linear dynamic analysis of automotive disc brake squeal - on the need to consider both stability and non-linear analysis. Mech. Res. Commun. 37, 96–105 (2010) 12. Butlin, T., Woodhouse, J.: Friction-induced vibration: quantifying sensitivity and uncertainty. J. Sound Vib. 329, 509–526 (2010)

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13. Culla, A., Massi, F.: Uncertainty model for contact instability prediction. J. Acoust. Soc. Am. 126, 1111–1119 (2009) 14. Renault, A., Massa, F., Lallemand, B., Tison, T.: Experimental investigations for uncertainty quantification in brake squeal analysis. J. Sound Vib. 367, 37–55 (2016) 15. Tison, T., Heusaff, A., Massa, F., Tupin, I., Nunes, R.: Improvement in the predictivity of squeal simulations: uncertainty and robustness. J. Sound Vib. 333, 3394–3412 (2014)

Handling Dynamics of an Ultra-Lightweight Vehicle During Load Variation Anna Lidfors Lindqvist

and Paul D. Walker

1 Introduction Lightweight vehicles are popular due to their high efficiency and low energy consumption [1, 2]. Even though these are major benefits, there are safety concerns as load variation can significantly change the handling dynamics [2]. The weight of a driver, passenger/s and/or any additional luggage naturally increases the load-to-curb weight ratio. This ratio is the weight of the loaded vehicle compared to an empty vehicle. Thus, a lighter vehicle will have a greater load-to-curb weight ratio than a standard vehicle with the same additional weight. Common commercial passenger vehicles with an empty weight of 800–2000 kg usually show no more than 20% increase in weight even with two passengers. Whilst for lightweight vehicles, like smaller urban vehicles of 450–550 kg, increase by at least 30% and with additional luggage included, this can reach 40% [1]. However, for ultra-lightweight vehicles, with a curb weight of 200–300 kg, the load-to-curb weight ratio with two passengers can be as high as 80%, if an 80 kg per person is considered. If an additional 40 kg luggage is included, the ratio increases to 100%. The dynamic handling problem occurs due to the shift of the center of gravity, inertia and the direct influence on the tire properties [2–5]. Drivers can perceive the variation in handling dynamics as unsafe and uncomfortable [2]. In most studies, vehicle parameters are considered as constants, but in reality, all vehicles have a slight change in parameters each time they are driven [4]. The variation in parameters is highly dependent on the load scenario, especially in lighter vehicles as the load-to-curb weight ratio is greater. In general, vehicle parameters can differ by as much as 30% [1]. However, ultra-lightweight, four-wheeled vehicles were not considered. In [2], direct yaw control with the objective of compensating for A. Lidfors Lindqvist () · P. D. Walker University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_7

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the influence of luggage load was presented. Stability was improved, however, the load-to-curb weight ratio was below 30%. The purpose of this study is to investigate how the load-to-curb weight ratio affects vehicle handling and the direct yaw moment control implemented on an ultra-lightweight vehicle without updating the vehicle parameters. The problem formulation, including vehicle specification and modelling, and the consequence of load variation is presented in Sect. 2, then control implementation in Sect. 3, followed by simulation and results in Sect. 4, and lastly conclusion in Sect. 5.

2 Problem Formulation 2.1 Vehicle Specification and Multi-Body Simulation Model The research in this chapter is based on the ATN solar car. This is a two-seated solarelectric vehicle, designed to compete in the Bridgestone World Solar Challenge 2019. The vehicle is rear wheel driven with in-wheel electric motors and can reach a top speed of 130 km/h. In-wheel motors allow for yaw control directly via control of the wheel torque. Information in regards to the vehicles parameters are presented in Tables 1 and 2. Simcenter Amesim™ was used to realize the computational multi-body vehicle model. The simulation model takes the expected values of the parameters into

Table 1 Description of driving scenario and its influence on the load-to-curb weight ratio

Description Empty Driver Driver and passenger Driver, passenger and luggage

Scenario S1 S2 S3

Additional weight (kg) 0 80 160

Total weight (kg) 282 362 442

Load-to-curb weight ratio (%) 0 28 57

S4

200

482

71

Table 2 Change in parameters due to load case scenario Scenario S1 S2 S3 S4

Lf (m) 1.516 1.394 1.316 1.331

Lr (m) 1.484 1.606 1.684 1.669

Ixx (kg.m2 ) 141.557 302.144 426.211 433.878

Iyy (kg.m2 ) 263.584 275.623 282.286 285.650

Izz (kg.m2 ) 566.580 771.05 903.61 903.990

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account; in the future they may vary from the actual vehicle since final aspects of the designs are yet to partake.

2.2 Consequence of Load Variation The curb weight of the vehicle is compared to the additional load. In [3], 600 load scenarios, with a variation of passenger and additional weights, was calculated. Due to the limited space and 80 kg driver/passenger weight restriction of the ATN solar car, only limited scenarios are considered in this study. However, in practicality it can never be fully determined since each time the vehicle is loaded, the scenario is slightly different, hence real-life scenarios surpass those in this study. The load reference naming of the scenarios and the effect on the load-to-curb weight are displayed in Table 2. In comparison to the lightweight vehicle in [2], the vehicle in this study already experiences a higher curb-to-load weight ratio at S2, more than their heaviest considered scenario. The ultra-lightweight vehicle presents a more extreme case, as the load-to-curb ratio rapidly increases at low curb weights (Fig. 1). As a result of the additional weight, the vehicle’s internal parameters, inertia and center of gravity change (Table 2). Lf and Lr are the distance from front and rear wheel to centre of gravity. Ixx, Iyy and Izz being the moment of inertia of respective axis.

CURB-TO-LOAD RATIO

120%

S2

100%

S3

80%

S4

60% 40% 20% 0% 200

400

600

800 1000 1200 1400 1600 1800 2000 CURB WEIGHT (KG)

Fig. 1 Load-to-curb weight ratio, during S2, S3 and S4 in respect to vehicle curb weight

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Fig. 2 Flowchart of control strategy used

3 Control Implementation Direct yaw moment control was implemented via the rear wheels to improve yaw and sideslip. The logics of the controller is displayed in the flowchart (Fig. 2). In [6, 7] dynamic curvature (k) was introduced as a control variable. A similar strategy has been adopted in this chapter. The reference model has been derived from the steady state of the bicycle model and is used to realize the desired handling of the vehicle. A PI control is implemented to realize the additive torque. The torque is then distributed between the left and right rear wheel.

4 Simulation and Results In this chapter, a co-simulation of Simcenter Amesim™ and MATLAB Simulink was used. The desired values for the controller have been calculated in relation to an empty vehicle (S1). This is done so that the effectiveness of the control can be studied when the internal parameters of the vehicle change. Due to lack of tire data, the tire properties have been approximated and updated values for each scenario were not considered. An open-loop driver configuration was employed. Double lane change procedure was performed as per ISO 3888-1:1999(E). Naturally the yaw rate decreases as the weight of the vehicle increases, which is displayed for the uncontrolled vehicle. The controller can successfully reduce the yaw rate for all cases; however, it does not decrease as efficiently as weight is increased (Fig. 3). Similarly, sideslip decreases as the vehicle becomes heavier. A slight overshoot in controlled sideslip is displayed in scenario 3, which may be due to the use of dynamic curvature as a control variable (Fig. 4). By overlaying the controlled scenarios for sideslip, unstable behavior can be displayed. The increase in loaded weight prolongs the controller to settle. However, it also displays mostly smoother transitioning and at times lower sideslip for S4

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Fig. 3 Yaw rate for scenarios 1–4, controlled and uncontrolled

Fig. 4 Sideslip for scenarios 1–4, controlled and uncontrolled

compared to S1–S3 (Fig.5). This may be explained by (a) S4 already displayed lower sideslip (b) better weight distribution balance, due to loading at rear. The results suggest that load variations introduce uncertainty in control, causing errors. Since the desired variables consider the parameters of an empty vehicle, the error is not completely unfounded. However, it implies that change in the internal parameters of a vehicle needs to be accounted for to minimize error.

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Fig. 5 Controlled sideslip for scenarios 1–4 overlaid at time frame 7–10 s

5 Conclusion In this chapter, the change in handling dynamics of an ultra-lightweight, the ATN solar car, during four load scenarios was studied. Although ignoring the load variation in the control strategy, the controller can, in fact, reduce both yaw and slip. However, the results suggest that the controller becomes less efficient when the loadto-curb weight ratio increases and uncertainty is introduced as a result of ignoring the change in parameters. To minimize possible control errors, the change in internal vehicle parameters needs to be considered. This can be a challenge as in reality the load scenario can vary and each ride can be different. The more the vehicle parameters deviate from base vehicle, the uncertainty of the control increases if further load is introduced; this could potentially cause even more disruption. Future research includes robust control strategy development and physical testing. Acknowledgments This work was supported by the Australian Technology Network solar car project; the ATN solar car team. It was also supported by the Australian Research Council Discovery Early Career Research Award (DE170100134).

References 1. Kohlhuber, F., Lienkamp, M.: Load Problem of Lightweight Electric Vehicles and Solution by Online Model Adaptation. Springer Fachmedien Wiesbaden, Wiesbaden (2014) 2. Raksincharoensak, P., et al.: Robust vehicle handling dynamics of light-weight vehicles against variation in loading conditions. In 2017 IEEE International Conference on Vehicular Electronics and Safety (ICVES) 2017 3. Kohlhuber, F., Lienkamp, M.: Online estimation of physical vehicle parameters with ESC sensors for adaptive vehicle dynamics controllers. In Internationales Stuttgarter Symposium Automobil-und Motorentechnik 2013

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4. Wei, Z., Xu, J., Halim, D.: Braking force control strategy for electric vehicles with load variation and wheel slip considerations. IET Electr. Syst. Transp. 7(1), 41–47 (2017) 5. Kim, H., et al.: Robust design optimisation of adaptive cruise controller considering uncertainties of vehicle parameters and occupants. Veh. Syst. Dyn. 1–19 (2019) 6. Okajima, H., et al.: Direct yaw-moment control method for electric vehicles to follow the desired path by driver. In Proceedings of SICE Annual Conference 2010 2010 7. Jang, Y., et al.: Lateral handling improvement with dynamic curvature control for an independent rear wheel drive EV. Int. J. Automot. Technol. 18(3), 505–510 (2017)

A Hybrid Electromechanical Engine Mount Design Nader Vahdati

, Esmaail Farah, and Oleg Shiryayev

1 Introduction Engine mounts are essential components in the automotive and aerospace industries for control of engine-transmitted noise and vibration. Engine mounts or motor mounts have three main functions: (1) support engine weight, (2) reduce the transmitted engine disturbance forces to the vehicle structure, and (3) limit engine motion caused by shock excitations. Not all commercially available engine mounts can perform all three tasks but an ideal engine mounting system should have these functions. By delivering the three above-mentioned benefits or functions, the engine mounting system not only provides passenger comfort and reduced cabin noise and vibration, but also helps to increase the life of the engine and the structure, by reducing engine-transmitted fatigue loads. Fluid or hydraulic engine mounts, due to their superior performance over elastomeric mounts, are widely used in aerospace and helicopter applications due to the engine operating at constant RPM for major part of the flight. Fluid or hydraulic engine mounts [1, 2] consist of two fluid chambers (upper and lower chambers) connected to one another through a hose or a pipe called an inertia track, see Fig. 1. At the upper chamber, there is a stiff rubber component that is capable of supporting the engine weight, acting like a piston pumping fluid between the two chambers, and containing the top chamber’s fluid. At the lower chamber, there is a soft rubber diaphragm containing the bottom chamber’s fluid. The engine N. Vahdati () · E. Farah Khalifa University, Abu Dhabi, UAE e-mail: [email protected]; [email protected] O. Shiryayev University of Alaska Anchorage, Anchorage, AK, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_8

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Vin

Fluid

6

2

Rubber Upper Chamber

Sf : Vin

Lower Chamber

1

R8: R0

C : Ct

4

1

Apt .. TF

5

8 0

7

1

3

10

I10 : If

9

Rubber Diaphragm

Vo

R4 : Br

C9 : Cb

Fig. 1 Single pumper hydraulic engine mount design and its bond graph [2, 4]

1.2K Peak Frequency Notch Frequency

K*

Mag

N/mm

Notch Depth

0

30 HZ

60

Fig. 2 Typical dynamic stiffness of a hydraulic engine mount versus frequency [2]

is connected to the upper stiff rubber component and as the engine vibrates, the engine induces a displacement or velocity, Vin , as shown in Fig. 1. The bottom of the hydraulic engine mount is connected to the chassis and if the chassis moves, its velocity is represented by a parameter, Vo , as shown in Fig. 1. The movement of the fluid between the two chambers is due to the relative motion between the engine and the chassis. As the frequency of the engine motion approaches what is called the notch frequency, the maximum reduction in the transmitted force or dynamic stiffness occurs. Figure 2 shows a typical dynamic stiffness of a typical hydraulic engine mount. The notch and peak frequencies occur due to the fluid, having mass, acting like a tuned vibration absorber, bouncing between the upper rubber and lower rubber diaphragm. It can be noted from Fig. 2 that the dynamic stiffness reaches its minimum at the notch frequency. It is at this frequency that the greatest cabin vibration and noise

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Electronics

Vin 1

Mechanical Transformer 2

Pivoted

2

Rubber Stiffness

1 Rubber Damping

Br

Kr Gy Coil

N Permanent Magnet

S

Fig. 3 Fluid-less engine mounting system [3]

reduction is achieved. But designing the hydraulic engine mount notch frequency to be at a desired location is not easy. Due to manufacturing process variations, the notch frequency will rarely coincide with the desired frequency at the first manufacturing pass. Resolving this problem involves redesigning the hydraulic engine mount subcomponents in an iterative process, until the notch frequency is at the desired location, which is very costly. The aim of this research study is and has been to design a new engine mounting system, a fluid-less electronic engine mounting system, in a way that its notch frequency can be easily tuned, using electronics, during the manufacturing process or in the field. A fluid-less electronic engine mounting system was studied and proposed by [3], and this work is the only fluid-less engine mount design reported in the literature. In a typical hydraulic engine mounting system, the engine input motion (mechanical power) is transformed to hydraulic power and vice versa, and in the language of bond graph [4], this transformation is modeled by a transformer element, TF (see Fig. 1). In [3], the authors used bond graph modeling technique to create a bond graph model of a typical hydraulic engine mounting system and then tried to use that bond graph model to find an equivalent electro-mechanical engine mount design. In reference [3], fluid-less engine mount design, the mechanical energy was converted to the electrical domain and vice versa using a voice coil and two lever arms, as shown in Fig. 3. The authors could not find an exact equivalent electro-mechanical model of a hydraulic engine mount. In the bond graph modeling technique, the voice coil does not act as a transformer but as a gyrator; so, an exact analogy could not be found. But in the mount design of [3], energy transformation from mechanical to electrical domain successfully took place. But the friction between the coil and the magnet,

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metal springs and their guides, and friction at the pin joints resulted in reducing the notch depth dramatically. The other problem with the design of reference [3] was weight. The design was too heavy and large in size; thus, not practical for real applications. So here in this chapter, a new fluid-less engine mount design concept is introduced. Using the bond graph modeling technique, the mathematical model and state space equations of this new design are developed. MATLAB is used to simulate the system. The simulation results indicate that this new design is feasible and a prototype should be designed and tested to verify simulation results.

2 New Engine Mount Design Concept Many different conceptual designs for fluid-less electronic engine mount were considered and the authors believe the rotary design shown in (Fig. 4) has the best potential. In this design, a rack and pinion mechanism is introduced to account for the mechanical advantage instead of levers that were used in the design of Fig. 3. The implementation of this mechanism results in a more compact design. It also reduces the number of moving parts in the system, thereby reducing the damping in the system. As the top rubber component moves due to engine motion, the rack moves vertically, thus creating sinusoidal rotational motion to the pinion. A rotary voice coil or a DC motor is used to convert mechanical rotational power to electrical power. In between the pinion and the rotary voice coil, there is a rubber disk. This rubber disk was added to make the bond graph model in Fig. 5 to be similar to the bond graph model in Fig. 1. The voice coil is connected to electronics and these electronics are to be used to create an electrical resonance (electrical tuned vibration absorber effect) and to tune the engine mount’s notch frequency. The bond graph parameters are described below: Vin

Input velocity, m/s

C 2 , R4

rp

Radius of pinion, m

I6 , R9

C10 , R13

Rubber disk compliance & damping Coil inductance, resistance

I14 , R16

α, I18 , R20

C21, I23, R25 & C24

Rubber compliance & damping Pinion MOI & damping Voice coil MOI and damping Electronics added to the coil

The below state space equations were used in MATLAB to plot the engine mount dynamic stiffness. In addition to dynamic stiffness, the current in the coil and the angular displacement of the rotary voice coil were also included in the output equation since the voice coil has a limited stroke and the coil has a maximum

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Top Metal Plate

Pinion Top Rubber

Rotary Voice Coil or a motor I1

Rack Rubber Disk

If

Inductor Inductor Resistor

Capacitor

Pinion

Cb

Capacitor

R1

Ct Resistor Rf

Fig. 4 Schematic of the rotary electromagnetic mount design

Fig. 5 Bond graph model in Fig. 4 [4]

current- carrying capacity at resonance, and the induced current in the coil cannot surpass the maximum limit. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

q˙2 q˙10 P˙14 P˙18 q˙21 P˙23 q˙24





0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ = ⎢0 ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢ ⎦ ⎣0 0

0 0 1 C10

0 0 0 0

0

0 0

0 0 0

−1 I14 −R13 −R14 −α I14 I18 −R20 −1 α I14 I18 C21 1 0 0 −1 I18 I23 0 0 C121 RI2325 0 0 0 I123

Fin =

1 1 R13 − 0 0 c2 rp c10 rp I14

⎤ ⎡ 1 q2 ⎢ 1 ⎥⎢ ⎥ ⎥ ⎢ q10 ⎥ ⎢ rp ⎢ R13 ⎥⎢ ⎥ ⎢ P14 ⎥ ⎥ ⎢ ⎢ rp ⎥⎢ ⎥ ⎥ ⎢ P18 ⎥ + ⎢ ⎢ 0 ⎥⎢ ⎢ ⎥⎢q ⎥ ⎥ 21 ⎥⎢ 0 ⎥ ⎢ ⎥ −1 ⎣ P23 ⎦ ⎢ ⎣ ⎦ 0 C24 q24 0 0 0 0 0 0 0

⎤⎡

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ vin ⎥ ⎥ ⎥ ⎥ ⎦

(1)

⎤ q2 ⎢q ⎥ ⎢ 10 ⎥ ⎥ ! ! ⎢ ⎢ P14 ⎥ R13 +Rq I6 ⎥ ⎢ 0 0 ⎢ P18 ⎥ + +R4 vin + 2 v˙in ⎥ ⎢ rp2 rp ⎢ q21 ⎥ ⎥ ⎢ ⎣ P23 ⎦ q24 (2) ⎡



0 0 0 0

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3 Numerical Simulations The simulation started with the selection of the best voice coil as it is the main component used to produce the notch. More than 40 different commercial voice coils were considered in our MATLAB simulations. This was followed by a parametric study to identify the main voice coils parameters that affect the notch depth, width, and location. The same process was used to identify which electronics had the most effect on the notch. Finally, the rest of the components needed to build the prototype were selected according to availability and compatibility. For our work, the rotary voice coil of Table 1 was selected, due to its high torque sensitivity (α), which enabled it to outperform other voice coils in terms of better transferring the energy from mechanical to electrical and vice versa.

4 Simulation Results/Conclusions The MATLAB simulation results show that our proposed design is feasible. The maximum coil current and rotary voice coil angular displacement were checked and found to be within the operating range of the voice coil, if input displacement is equal or less than ±1.112 mm. A good notch (i.e., depth and width) was generated at 44.5 Hz. Simulation results indicate that this engine mount design is more suitable for fixed wing applications since engine input motions are small. Figure 6 shows that the notch can be varied using a variable capacitance, and thus can be tuned in the field. To allow engine mount lateral motion, the rack can be pin-joined to the top metal. It is important to mention that when large lateral motions are present in the engine mount, there is a possibility of disengagement of the rack from the pinion.

Baseline values 4.89e-3 kg.m2

1.15 N-m/amp

3.7 mH

3

Parameters J of the motor (I14 )

Motor torque constant α

Inductance of the motor

R of the motor

% Change in parameters +20% −20% +20% −20% +20% −20% +20% −20%

Notch depth (×106 ) 0.586 (−7.88%) 0.8405 (10.9%) 0.8085 (8.55%) 0.5607 (−9.7%) 0.7177 (1.86%) 0.6584 (−2.51%) 0.7104 (1.32%) 0.6444 (−3.54)

Table 1 Sample of the parametric sensitivity analysis for the voice coil parameters Notch location (Hz) 49.36 (7.3%) 58.06 (−9%) 51.21 (−3.8%) 54.88 (3.1%) 53.02 (0.43%) 53.48 (−0.43%) 53.59 (−0.64%) 52.9 (0.66%)

Width (±10% of notch depth) 46.8–51.2 (16.4%) 54.3–60.8 (−23.6%) 48.5–53.3 (7.6%) 52.1–56.9 (8.76%) 49.9–55.3 (−1.9%) 50.5–55.6 (2.86%) 50.5–55.9 (−2.86%) 50.0–54.9 (6.29%)

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4.5

Dynamic Stifness vs. Frequency

´106

C=300?F C=1000?F

4

C=2000?F C=5000?F C=10000?F

Dynamic Stifness (N/m)

3.5 3

C=20000?F

2.5 2 1.5 1 0.5 0

0

20

40

60

80

100 120 Frequency (Hz)

140

160

180

200

Fig. 6 Dynamic stiffness versus capacitance (300 μF to 20 K μF)

References 1. Vahdati, N.: A detailed mechanical model of a double pumper fluid mount. ASME J Vib. Acoust. 120, 361–370 (1998) 2. Vahdati, N., Ahmadian, M.: Variable volumetric stiffness fluid mount design. Shock. Vib. 11, 21–32 (2004) 3. Vahdati, N., Mashayekhi, M.: Fluidless electronic engine mount. In: Crocker, M., Pawelczyk, M. (eds.) International Congress on Sound and Vibration (ICSV22), Florence Italy, vol. 1, pp. 735–742. The International Institute of Acoustics and Vibration (IIAV), Auburn (2015) 4. Rosenberg, R.C., Karnopp, D.C.: Introduction to Physical System Dynamics. McGraw Hills, New York (1983)

A Hybrid Modeling Approach to Accurately Predict Vehicle Occupant Vibration Discomfort Jianchun Yao, Mohammad Fard, and Kazuhito Kato

1 Introduction The vibration discomfort of the seated occupant can be evaluated using the accelerations measured at the contact between the seated occupant body and seat surface. The dynamics response of the seat and the vibrational discomfort of the vehicle occupant are influenced by the non-linear dynamics of the human body [1] and the interactions between the seat and the vehicle floor [2]. Several Finite Element (FE) models of coupled occupant body-seat [3] have been developed to predict vibration transmission of the automotive seat when coupled to the occupant body. However, the FE model of the occupant body is challenging to develop, optimize, and use in predicting the seating comfort of the occupant [4]. The vibration discomfort of the occupant is usually evaluated without consideration of the structural dynamics of the vehicle body. When the resonant frequencies of the seat are close to that of the vehicle body, it will provide a poor ride comfort. However, there are a few studies about the combined vehicle-seat-occupant model in prediction of the vibration transmission to the vehicle occupants [5]. In studying the vibration transmission between the vehicle body and the seat, the seat base (vehicle floor panel around the S/MTGs) is often assumed to be a rigid body [2]. However, Qiu et al. [2] found that the seat base does not always perform as a rigid plate, especially at a high frequency. This chapter aims to develop a hybrid modelling approach to accurately predict the vibration discomfort of the vehicle occupant by combining the vibration transfer J. Yao () · M. Fard School of Engineering, RMIT University, Bundoora, VIC, Australia e-mail: [email protected]; [email protected] K. Kato NHK Spring Co., Ltd., Yokohama, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_9

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matrices of sub-systems in the frequency domain. The sub-systems can be sources from physical tests or FE models.

2 Method The vehicle body vibration caused by road inputs transmits to the seat via seat mounting points, which then transfer to the occupant body through the seat foam. The dynamics of vehicle body-seat-occupant body coupled system are obtained by combining transfer matrices of sub-systems in the frequency domain. The structural dynamics of the vehicle body is represented by a transfer matrix from the force input at the shock absorber (S/ABS) mounts to the response at the seat mounting (S/MTG) points. The structural dynamics of the seat frame are modelled as an acceleration transmissibility matrix from the multi-axial accelerations at the S/MTG points to the multi-axial accelerations at different points of the seat frame. The dynamics of the occupant body plus the seat foam are derived from a physical test as an acceleration transmissibility matrix from the points of the seat frame to the occupant-to-foam contact points (See Fig. 1). The interactions between the sub-systems (the vehicle body, the seat frame, and the occupant body-seat foam coupled system) are shown in Fig. 2, where FS/ABS is the vertical force input applied at the S/ABS mounting points, and aS/Frame is the acceleration obtained at a point of the seat frame.

Fig. 1 Schematic illustration of the hybrid model developed to represent the vibration transmission from the vertical load input at the shock absorber mounts to the fore-and-aft acceleration at the occupant-seat contact point, where n is the number of S/MTG points. Identical models are developed to predict the vibration transfer functions from the vertical load input at S/ABS mounts to the lateral acceleration and the vertical acceleration at the occupant-seat contact point

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Fig. 2 Schematic illustration of sub-structure synthesis between vehicle body, automotive seat frame, and the seated occupant body at the contact points: (a) the vehicle body-seat-occupant body coupled system, (b) the vehicle body, (c) the automotive seat, and (d) the occupant body-seat foam coupled system

2.1 Prediction of Vehicle Seat Frame Vibration in Occupied Condition The structural dynamics of the vehicle body is defined as a transfer function HVehicle , a complex ratio between the acceleration at the S/MTG points on the vehicle floor and the vertical force at the S/ABS mounts in the frequency domain. The dynamics of the automotive seat is expressed as acceleration transmissibility, TSeat , which is from the S/MTG point to a point on the seat frame. HVehicle =

a(1) S/MTG FS/ABS

, TSeat =

aS/Frame (2) FS/MTG



(2) FS/MTG (2) aS/MTG

=

aS/Frame

(1)

(2)

aS/MTG

Considering the force equilibrium at the connecting points between two sub(1) (2) systems, the S/MTG points force FS/MTG and the reaction force FS/MTG are equal (1)

(2)

in magnitude but opposite in direction, while the accelerations aS/MTG and aS/MTG at the seat mounting points are same. Therefore, the transfer matrix of the vehicle body-seat frame coupled system from the S/ABS mounts to the point of the seat frame are defined in Eq. (2). [HVehicle−Seat ] = [HVehicle ] × [TSeat ]

(2)

2.2 Prediction of Vibration at Occupant-Seat Contact Points The vehicle floor vibration transmitted to the occupant body through the seat cushion, the seatback, and the occupant’s feet. The transmissibility of the seat frame

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Fig. 3 Measurement locations on the seat frame (P1) and the occupant-to-foam contact points on the seat surface (Point I). Tri-axial accelerometers were attached to the seat foam surface following Ittianuwat’s [6] configuration of the seat surface measurement location

TSeat from the S/MTG point to the points of the seat frame and the transmissibility of the occupant-foam system TOccupant from the point of the seat frame to the occupantto-foam contact point are defined in Eq. (3) and Eq. (4). TSeat =

TOccupant =

aOccupant (2) FS/Frame

a(1) S/Frame

(3)

aS/MTG

(2)



FS/Frame (2) aS/Frame

=

aOccupant (2)

,

(4)

aS/Frame

(1)

where aS/MTG , aS/Frame , and aOccupant are the accelerations at the front left seat mounting point, the seat foam-frame connecting points of the seat frame, and the occupant-to-foam contact point, respectively. Due to a force equilibrium, the action (1) (2) force FS/Frame and the reaction force FS/Frame at the foam-frame connection points are equal in magnitude but opposite in direction. The accelerations at the foam(1) (2) frame connecting point aS/Frame and aS/Frame are same. Finally, the transfer matrix of the seat-occupant coupled system is computed by multiplying the transmissibility matrices of two sub-systems, defined in Eq. (5).  

Tseat−Occupant = [TSeat ] • TOccupant

(5)

The transmissibility matrix of occupant-seat foam system between the inputs at a nearby seat-foam connection point and the responses at an occupant-to-foam contact point shown in Fig. 3 are derived using the cross-spectral density function method.

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Fig. 4 Frequency Response Functions in (a) fore-and-aft cross-axis FRF from S/ABS mounts to the point P1 of seatback frame; (b) lateral cross-axis FRF from S/ABS mounts to the point P1 of seatback frame; (c) vertical in-line FRF from S/ABS mounts to the point P1 of seatback frame (—, values from vehicle body FE model; ·····, predicted value)

3 Results and Discussion Firstly, the predicted Frequency Response Functions (FRFs) between the shock absorber mounts and the points of the seat frame obtained from the developed hybrid model are compared with the values obtained from the Finite Element model of vehicle body plus seat system. The magnitude and the phase of the predicted FRFs of the vehicle seat obtained from the developed hybrid model are compared with the values obtained from the vehicle body FE model, shown in Fig. 4. The predicted results show a close agreement with the values obtained from the vehicle body FE model below 100 Hz. The results show that the developed modelling method provided a reasonable estimation of the seat vibration when coupled with the vehicle body. Secondly, the predicted acceleration transmissibilities from the front left S/MTG point to the occupant-to-foam contact point using the developed method are compared with the values obtained from the seat dynamics test. The magnitude and the phase of the predicted accelerometer transmissibilities, the occupant-to-foam contact points, and the front left S/MTG point are compared with the measured value obtained from the seat dynamics test shown in Fig. 5. The predicted transmissibilities from the S/MTG point to the occupant-to-foam contact points on seatback surface provided reasonable fits to the measured values, although some differences can be seen in values along the fore-and-aft and lateral direction in a frequency range of 50–100 Hz. The results show that the developed modelling method provided a reasonable estimation of the vibration discomfort of the occupant.

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Fig. 5 Accelerometer transmissibilities in (a) fore-and-aft in-line transmissibilities from the S/MTG point to the occupant-to-seat foam contact point I; (b) lateral in-line transmissibilities from the S/MTG point to the occupant-to-seat foam contact point I; (c) vertical in-line transmissibilities from the S/MTG point to the occupant-to-seat foam contacts point I (—, values from physical test; ·····, predicted values)

4 Conclusion A hybrid model is developed to predict the vibration discomfort of the vehicle occupant by combining the transfer matrices of sub-systems. The structural dynamics of the vehicle body and the seat frame are obtained from FE models and parametrically represented by a transfer matrix and a transmissibility matrix, respectively. The dynamics of the seated occupant body plus the seat foam are derived from the physical test as a transmissibility matrix. The results show that the developed model provided a reasonable estimation of the vibration transmission from the vehicle body to the seated occupant. This chapter presented ongoing research in developing a hybrid modelling approach to predict vibration transmission to a vehicle occupant. A more detailed prediction method will be published in future.

References 1. Lo, L., Fard, M., Subic, A., Jazar, R.: Structural dynamic characterization of a vehicle seat coupled with human occupant. J. Sound Vib. 332, 1141–1152 (2013) 2. Qiu, Y., Griffin, M.J.: Transmission of roll, pitch and yaw vibration to the backrest of a seat supported on a non-rigid car floor. J. Sound Vib. 288, 1197–1222 (2005)

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3. Grujicic, M., Pandurangan, B., Arakere, G., Bell, W.C., et al.: Seat-cushion and soft-tissue material modeling and a finite element investigation of the seating comfort for passenger-vehicle occupants. Mater. Des. 30, 4273–4285 (2009) 4. Zhang, X., Qiu, Y., Griffin, M.J.: Developing a simplified finite element model of a car seat with occupant for predicting vibration transmissibility in the vertical direction. Ergonomics. 58, 1220–1231 (2015) 5. Zhou H, Qiu Y (2015) A simple mathematical model of a vehicle with seat and occupant for studying the effect of vehicle dynamic parameters on ride comfort 6. Ittianuwat, R., Fard, M., Kato, K.: Evaluation of seatback vibration based on ISO 2631-1 (1997) standard method: the influence of vehicle seat structural resonance. Ergonomics. 60, 82–92 (2017)

Pass-By Noise Synthesis from Transfer Path Analysis Using IIR Filters Mansour Alkmim, Fabio Bianciardi, Guillaume Vandernoot, Laurent De Ryck, Jacques Cuenca, and Karl Janssens

1 Introduction Pass-by noise (PBN) engineering is a well-defined procedure in the development process of vehicles. Automotive original equipment manufacturers (OEMs) and component suppliers are looking into techniques that allow setting realistic noise and vibration harshness (NVH) targets for the contributing noise sources and, by doing so, frontloading the pass-by noise performance into the design process. More specifically, the indoor PBN test is becoming an attractive alternative to the exterior PBN due to the easy reproducibility and significantly reduced constraints due to changing environmental conditions. Although the PBN sound levels can be obtained by energetic approach, the current solutions rely on mixing and interpolation techniques [1] which does not provide satisfactory virtual PBN audio signal. Audio synthesis for sound quality assessment has increased in importance due to the electrification of vehicles and for sound quality assessment using psychoacoustic metrics. The objective of this paper is to develop and implement a time-domain TPA synthesis technique. The sound synthesis of a pass-by run with inclusion of the Doppler effect and accurate sound pressure levels is conducted by representing the measured transfer functions as time-varying IIR filters that are updated at

M. Alkmim () Siemens Digital Industries Software, Leuven, Belgium Department of Mechanical Engineering, KU Leuven, Heverlee, Belgium e-mail: [email protected] F. Bianciardi · L. De Ryck · J. Cuenca · K. Janssens Siemens Digital Industries Software, Leuven, Belgium G. Vandernoot Siemens Digital Industries Software, Chatillon, France © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_10

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Fig. 1 Pass-by noise synthesis diagram

sampling rate speed. The methodology is based on the TPA-ASQ (airborne source quantification) method, which allows to separate the airborne noise sources of a vehicle and evaluate their individual contribution to the overall pass-by noise level [2–4]. The advantages of implementing an IIR filter design are the provided flexibility, the efficiency in implementation, and the possibility of real-time audio applications. The main underlying challenge of the sound synthesis is to obtain a computationally efficient procedure and click-free (no artifacts) solution.

2 IIR Filter Design The IIR filter design procedure is summarized in Fig. 1. The first step consists in obtaining the noise transfer functions (NTF) at discretized target microphone positions (refer to Fig. 2b). The second step consists of the decomposition of the NTFs into a minimum phase and all-pass filter [5]. The minimum phase magnitude is employed in the IIR filter design, while the all-pass phase provides the time delay information which can be used to implement the Doppler effect.

2.1 Transfer Path Analysis (TPA) and Airborne Source Quantification (ASQ) TPA is a source-transfer path technique to assess the contributions of multiple sources in a system toward specific targets, for instance, the driver’s ears in a car. The process exploits the availability of transfer functions between source points and measurement (indicator) points. In the pass-by noise context, targets appear as a

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linear microphone array in the far field; and sources are airborne, radiating outward from, e.g., the powertrain, the tailpipe, and the front and rear tires. The set of NTFs between each source and target points represents changes in propagation distance, directivity, and angle of incidence. The ASQ method aims at identifying and separating the airborne noise sources from each component. By assuming that the noise-producing components can be represented by a set of monopole sources, the operational acoustic loads can be identified from independent component measurements by means of inverse techniques. Two types of ASQ are commonly employed, linear phase-based pressure inversion method and power-based energetic approach [6]. While the linear approach accounts for the source phase information, the energetic approach assumes that all sources are uncorrelated.

2.2 Preprocessing the Measured Noise Transfer Functions (NTF) Two properties are exploited to reduce the computational cost of the sound synthesis while maintaining the main audio cues [5]. The first property is the pole-zero filters, which states that every stable filter can be decomposed into minimum phase and all-pass filters associated in series. The second property is that all-pass filters can be approximated by pure delays. This property is used to estimate the time delay from the sources to the target microphones. The decomposition of the NTF is given by [5] H (j ω) = |Hmin (j ω)Hap (j ω)|ej φmin (j ω) ej φap (j ω) ,

(1)

where φmin (j ω) denotes the minimum phase, φap (j ω) denotes the excess phase, |Hmin (j ω)| is the minimum phase magnitude, and the |Hap (j ω)| = 1 is the all-pass magnitude. The phase and magnitude of a minimum phase system are uniquely related by the inverse Hilbert transform φmin (j ω) = H −1 {ln(−|Hmin (j ω)|)}. Two additional operations, the smoothing and warping, are performed to improve the filter fitting at low frequencies. The smoothing consists of a convolution of the NTF with a frequency-dependent Hann window. The warping has the effect of resampling the NTF by replacing in the Z-domain the unit delay z−1 by an all-pass filter. Then, the IIR filter is least square fitted on the warped NTF magnitude employing the YuleWalker algorithm [5]. Lastly, the resultant filter is de-warped in frequency domain.

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2.3 Implementation of Time-Varying IIR Filter Coefficients The IIR filter with time-varying coefficients is implemented by solving the difference equation described in Eq. (2), which is also known as direct-form I [7]. For a general, causal, and linear time invariant (LTI) difference equation, the output y(n) (pass-by response) at discrete time n is computed from the present input x(n) (sources) and the past output samples as y(n) =

M 

bi (n)x(n − i) −

i=0

N 

aj (n)y(n − j ),

(2)

j =1

where M is the feedforward filter order and N is the feedback filter order. Note that the IIR coefficients are changing at the source audio sample rate. The final step in the synthesis is the inclusion of the Doppler effect as a propagation time delay, which can be inferred from the slope of the all-pass excess phase (refer to Fig. 1). Each source-microphone path is characterized by a propagation time delay. A single time delay curve is obtained for the vehicle by averaging over all its source propagation time delay curves. The averaged time delay curve is linearly interpolated at the audio sample rate level before being applied to the time-varying implementation output.

3 Automotive Example The PBN synthesis is applied to an automotive indoor PBN measurement example. An energetic ASQ approach was used to identify and separate the source contributions from each component. In this example, six main components were identified, and each one was characterized as consisting of a given number of sources. A linear array of microphones 7.5 m away from the nominal vehicle center line is employed to record the target responses and the propagated NTFs. The target microphones and near-field indicators near components were measured synchronously at the audio sampling frequency during an operational test on the chassis dyno. A comparison of the NTF against the IIR filters for a single source and two target microphone positions is considered. This comparison is done to evaluate the ability of the IIR filter to reproduce the sound pressure levels (SPL) of the NTF in a static scenario. Figure 2 shows the frequency response function of the measured NTF and the estimated 16th order IIR filter for the two selected positions. Figure 2 shows that the IIR filters capture the main trends of the NTF. Improvements in the fit can be achieved by increasing the filter order at the expense of instabilities and increase in computational time. Figure 3 shows the SPL calculated reciprocally in the center of the array from a moving source by applying a time-

|H(w)| [dB]

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60 40

|H(w)| [dB]

20 60 40 20 3

10

10

4

Frequency [Hz]

(b)

(a)

Fig. 2 (a) Frequency response function of the measured NTF ( ) and the estimated 16th order IIR filter at two target microphone positions: position A ( ) and position B ( )

90

SPL [dB(A)]

80 B 70 A 60 IIR filter A IIR filter B time-varying IIR NTF (FIR)

50 40 0

0.2

0.4

0.6

0.8

1

1.2

Time [s] Fig. 3 (Color online) Equivalent SPL estimated at the center of the array from a moving source, by applying a time-varying IIR compared to the SPL from IIR and NTF (FIR) filters at two target microphone positions

varying IIR filter. The SPL calculated at two target microphone positions with both the NTF (FIR) and the IIR filters are shown for reference. It can be noticed that the SPL from the IIR filter at both fixed positions are in agreement with the SPL from the NTFs. Moreover, the time-varying SPL value matches the respective SPL of the target microphones at the instant where the virtually moving source crosses the target microphone position.

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SPL [dB(A)]

OAL measured TPA total Component 1 Component 2 Component 3 Component 4 Component 5 Component 6 Energetic TPA

10 dB(A)

position [m] Fig. 4 (Color online) SPL of time-varying IIR filter for each component contribution, the total contribution, the measured overall level (OAL), and the energetic TPA from Simcenter Testlab ( ) at their respective component color scheme

From an indoor PBN test, Fig. 4 shows the SPL employing the time-varying IIR filter of each component contribution, as well as the summed (total) contribution, the measured overall level, and the energetic TPA results from Simcenter Testlab as references. The measured overall level (OAL) provides a reference value for the time-varying filter, and it is obtained by energetic superposition of the target microphone signals. Figure 4 demonstrates that the total TPA contribution derived from the IIR implementation, the measured OAL, and the energetic TPA result are in agreement with each other. At the component level, the time-varying IIR matches with the energetic TPA, except for the central position of components 4 and 5. The proper match of SPL demonstrates the feasibility of using IIR filters for PBN sound synthesis. Additionally, the time-varying IIR filter implementation yields, in a computationally fast manner, the PBN time signals which allow for complementary listening tests and sound quality assessment.

4 Conclusions In this paper we presented a formulation for the sound synthesis of pass-by noise from a TPA approach. We validated the methodology in an automotive example, first at a single source and two target microphones and later in time-varying scenarios. The new approach was shown to match the SPL from Simcenter Testlab TPA while additionally providing the synthesized pass-by noise audio. Next step consists of validation of the synthesized audio against an exterior PBN test.

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Acknowledgments We gratefully acknowledge the European Commission for its support of the Marie Sklodowska Curie program through the H2020 ETN PBNv2 project (GA 721615).

References 1. Matsumoto, M., Tohyama, M., Yanagawa, H.: Acoust. Sci. Technol. 24(5), 284 (2003). https:// doi.org/10.1250/ast.24.284 2. Janssens, K., Aarnoutse, P., Gajdatsy, P., Britte, L., Deblauwe, F., Van der Auweraer, H.: In SAE 2011 Noise and Vibration Conference and Exhibition (2011). https://doi.org/10.4271/2011-011609 3. Verrecas, B., de Ponseele, P.V., Britte, L., Roudey, M., Converset, P., Fréjacques, M.: In: Automotive Acoustics Conference 2nd International. ATZ-Conference, p. 19 (2013) 4. Corbeels, P.: In: Aachen Acoustics Colloquium – AAC (2017) 5. Jot, J.M., Larcher, V., Warusfel, O.: In: Audio Engineering Society Convention 98 Audio Engineering Society (1995) 6. Janssens, K., Bianciardi, F., Britte, L., de Ponseele, P.V.: In: ISMA, p. 18 (2014) 7. Smith, J.O.: Introduction to Digital Filters: With Audio Applications. Center for Computer Research in Music and Acoustics, Stanford (2007)

Motion and Vibration Control of Automotive Drivetrain with Control Cycle Limitation H. Yonezawa, I. Kajiwara, C. Nishidome, T. Hatano, M. Sakata, and S. Hiramatsu

1 Introduction To improve the comfort of automobiles, vibrations of the automotive drivetrain must be reduced. Additionally, the automotive drivetrain includes backlash as a structural nonlinearity in differential gears, which degrades the vibration control characteristics. Under the driving condition of tip-in and tip-out, in which the engine torque becomes a step input, a dead-zone characteristic of backlash causes an overshoot in the body vibration. Previous studies have reported the control of automobile drivetrains with backlash. In particular, many outcomes are related to model predictive control (MPC) [1, 2]. These are broadly considered effective techniques. Furthermore, tuning the design parameter of a control system was studied [3]. Another problem with the vibration control is that an engine used as an actuator has a limitation on the control period. That is, a torque in the engine can only be updated at the moment when an explosion occurs in the combustion chamber and the crankshaft rotates by a predetermined angle. The essential problem of this limitation is that stability and vibration suppression performance are lost upon extending

H. Yonezawa () · I. Kajiwara Division of Human Mechanical Systems and Design, Hokkaido University, Sapporo, Japan e-mail: [email protected]; [email protected] C. Nishidome CATEC Inc., Tokyo, Japan e-mail: [email protected] T. Hatano · M. Sakata · S. Hiramatsu Integrated Control System Development Division, Mazda Motor Corporation, Hiroshima, Japan e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_11

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(longer) control cycles due to the engine mechanism. However, vibration control of vehicle powertrains considering the control cycle limitation has yet to be studied. The discrete approximation errors of traditional controllers expanded by the control cycle limitation should be compensated. This research studies the vibration control of an automotive drivetrain with backlash considering the control cycle limitation. First, a plant model is presented which abstracts a real vehicle structure to focus on the effect due to backlash while reflecting only the basic components of a drivetrain. Next, we propose an approach to apply the sampled-data controller to an actuator with longer control cycles like the engine [4]. The sampled-data controller can explicitly optimize responses between sampling points, allowing stability and control performance to be ensured with long control cycles. The sampled-data H∞ controller is designed as a servo system by using frequency shaping. In addition, a simple and practical control mode switching technique is constructed to compensate for backlash. Finally, the effectiveness of the proposed method is validated by control simulations.

2 Controlled Object 2.1 Plant Model of a Simplified Drivetrain Figure 1a shows an original automotive drivetrain. This is a rotary drive system, which transmits the torque from the engine to the tires. In this study, a simplified model, which abstracts an actual vehicle structure to focus on backlash, is considered as the controlled object (Fig. 1b). Although the original system is a rotary model, this research employs a translational model transmitting force to easily develop an experimental equipment. In Fig. 1b, backlash

Fig. 1 Controlled object model simplifying a real drivetrain structure. (a) an original drivetrain model and (b) the simplified model to focus on the influence due to backlash

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Table 1 Parameters of the plant model Symbol M3 m2 M1 K3

Value 0.232 kg 0.039 kg 1.04 kg 660.0 N/m

Symbol K1 C3 C2 C1

Value 2.7 × 104 N/m 0.31 Ns/m 5.55 Ns/m 20.12 Ns/m

Symbol K2 C11

Value 1.5 × 104 N/m 3.58 Ns/m

is described as the dead zone between m2 and M1 . In this chapter, positioning control is validated for the vehicle body displacement. Table 1 shows the plant parameters.

2.2 Mathematical Model and Nonlinear Parameters The plant shown in Fig. 1(b) is modeled for the control system design and simulations. The state equation and the output equation of the plant are written as x˙ = Ax + B 1 w + B 2 u y = Cx + D1 w + D2 u ⎡

0 0 0

⎢ ⎢ ⎢ ⎢ A=⎢ − ⎢ ⎢ ⎣

(K2 +K3 ) M3 K2 m2

0 ⎡ ⎢ ⎢ ⎢ ⎢ B 1= ⎢ ⎢ ⎢ ⎣

0 0 0 0 1 m2



1 M1



0 0 0 K2 M3 − (SwKm12+K2 ) SwK1 M1



⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , B 2= ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

0 0 0 0 0

0 0 0 0 SwK1 m2 1 − SwK M1



(1)

1 0 0

0 1 0

C2 m2

C2 M3 − (SwCm12+C2 ) SwC1 M1

3) − (C2M+C 3

0

0 0 1 0 SwC1 m2 (SwC 1 +C11 ) − M1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎥ ⎥ , C= [ 1 0 0 0 0 0 ] , D1 =0, D2 =0, w=OKG ⎥ ⎥ ⎦

1 M1

(2) The nonlinear characteristics of backlash are represented by the switching nonlinear parameters included in the state equation. The switching parameters Sw in A- matrix and OKG of the disturbance provide a dead-zone effect with the model. The relative displacement of M1 and m2 is X = X1 − x2 , and the relationship between the nonlinear parameters and the transferred force F is expressed as follows. OKG is the constant.

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F = Sw · K1 · ΔX + OKG = Sw · K1 · (X1 − x2 ) + OKG ⎧ ⎧ ⎨ − |K1 × |B|| , X1 − x2 > |B| ⎨ 1, X1 − x2 > |B| Sw = |K1 × |B|| , X1 − x2 < − |B| 1, X1 − x2 < − |B| , OKG = ⎩ ⎩ |X1 − x2 | ≤ |B| 0, |X1 − x2 | ≤ |B| 0, (3)

3 Control System Design An engine has variable intervals to update the control input depending on the speed. As this is a basic study for the problem, using a model with fixed intervals to update the input, we aim to construct a control system that can suppress performance deterioration even when the control cycle is made longer. An approach with the sampled-data control is effective to solve this problem.

3.1 Sampled-Data H∞ Control The sampled-data control system is a digital control system to control a continuoustime plant G(s) using a discrete-time controller K[z], interposing the ideal sampler Sh and the zeroth-order holder H(θ ) = I [5]. Figure 2 shows the generalized plant used to design the sampled-data H∞ controller. Specifically, it is a controller design problem using the performance index based on the continuous-time input/output signals, as shown in Fig. 2. Therefore, a controller which explicitly optimizes responses between sampling points can be obtained without discrete approximations. This feature suggests that the sampled-

Fig. 2 Block diagram of the control system

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data controller can maintain the control performance even when it is used with longer control cycles.

3.2 Weighting Function to Realize Servo System The controlled output must follow a target signal without offsets while suppressing vibrations in the control frequency range. Under the condition where the control cycles are extended, traditional servo systems with an approximate integrator cannot achieve the performances of both vibration suppression and tracking since the chances of updating the control input decrease. Hence, a frequency weighting function W1 described as follows is used to design a servo controller (Fig. 2) [4]. M(s) =

s+(2π ×ε2 ) , 1.0

W˜ 1 (s) =

2.481×105 s 3 +125.7s 2 +7896s+2.482×105

×

1.0 s+(2π ×ε1 )

W1 (s) = W˜ 1 (s) · M(s) (ε1 ε2 < controlled frequency : 4 H z)

(4)

W1 (s) realizes an integral characteristic around 0 Hz, while a flat passband of a low-pass filter is maintained in the vibration suppression band (1 Hz – 10 Hz).

4 Compensation for Backlash In real vehicles, there is a dead time before driver’s operations are reflected on the vehicle body behavior. In this study, vibrations due to backlash are improved by processing to reduce backlash in advance during this dead time [5]. A control mode switching of the sampled-data H∞ controller (Asd , Bsd , Csd , Dsd ) compensates for backlash. During the dead time, the state-space representation of the controller is described as epre (k) = rpre (k) − X3 (k) = rpre − X3 (k) x AW (k + 1) = Asd x sd (k) + B sd epre (k) x AW (k) = z−1 [x AW (k + 1)] x sd (k + 1) = x AW (k) u(k) = C sd xsd (k) + Dsd epre (k)

(5)

This mode achieves gradual coupling upstream and downstream of the system by reducing backlash during the dead time. Specifically, the control inputs calculated by switching the target signal to a small positive value rpre reduce backlash. In addition, anti-windup, which temporarily stops updating the state variable xsd (k) in the controller, is also applied to reset an accumulation of errors due to an uncontrolled time zone.

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5 Control Simulations We consider the essential condition where the input update intervals are extended by forcibly holding the control input value calculated from the digital controller with a zeroth-order hold for a long constant period. Considering the actual vehicle property, the control period is determined to be the reciprocal of a frequency, which is 5 times the natural frequency (4 Hz). This period is also used for implementation of the controller. Additionally, a traditional controller obtained by discretizing a continuous-time H∞ controller with the control period is compared with the sampled-data controller. Figure 3 denotes the vehicle body vibrations obtained by the simulations. See the legend in Fig. 3 for the result indicated by each color. The blue line shows the result obtained by the sampled-data controller without the backlash compensation. All the control systems greatly reduce the vibration compared to that without control (red line). Especially, the proposed technique shown in the magenta line provides the highest control performance. The residual oscillation remains in the green line because of the control cycle limitation. The traditional digital controller with a discrete approximation does not take into account the responses between the timings updating the control input. Therefore, the influence of the discrete approximation error increases as the control periods are extended, resulting in a deteriorated vibration suppression performance. The improvement, thanks to the backlash compensation, is observed from the comparison of the blue and magenta lines. Without the control mode switching (blue line), an overshoot occurs after the target signal rises due to the adverse effect of backlash. The dead-zone effect of backlash produces an uncontrolled time zone, causing an accumulation of control errors. Thus, an unnecessary large control input is calculated, resulting in the large overshoot.

Fig. 3 Control simulation results

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6 Conclusion This study proposed the method to suppress the adverse effect caused by backlash under the control cycle limitation for motion and vibration control of an automobile drivetrain. As a future work, we will consider the time-varying control period. The future work of this study is to verify the effectiveness of the proposed control system by applying it to real vehicles.

References 1. Baumann, J., Torkzadeh, D.D., Ramstein, A., Kiencke, U., Schlegl, T.: Model-based predictive anti-jerk control. Control. Eng. Pract. 14, 259–266 (2006) 2. Rostalski, P., Besselmann, T., Bari´c, M., Van Belzen, F., Morari, M.: A hybrid approach to modeling, control and state estimation of mechanical systems with backlash. Int. J. Control. 80(11), 1729–1740 (2007) 3. Berriri, M., Chevrel, P., Lefebvre, D.: Active damping of automotive powertrain oscillations by a partial torque compensator. Control. Eng. Pract. 16, 874–883 (2008) 4. Yonezawa, H., Kajiwara, I., Nishidome, C., Hiramatsu, S., Sakata, M., Hatano, T.: Vibration control of automotive drive system with backlash considering control period constraint. J Adv. Mech. Des. Syst. Manufact. 13(1), 1–15., Paper No. 18-00430 (2019) 5. Chen, T.C., Francis, B.A.: H2 -optimal sampled-data control. IEEE Trans. Autom. Control. 36(4), 387–397 (1991)

Measuring Road Conditions with an IMU and GPS Monitoring System Enoch Zhao

, Paul D. Walker

, Albert Ong

, and Fatma Al-Widyan

1 Introduction The objective of real-time road condition measurement is to estimate factors in traffic conditions based on measurement data obtained. Most road condition sensors available measures road surface conditions and are typically fixed in place (embedded pavement sensors or mounted on the road side). However, such sensors are expensive to implement and maintain [7]. Mobile sensors generally require Global Positioning System (GPS) positioning and must be in the range of satellites. Although these sensors are cheaper than fixed sensors, accuracy is compromised in urban areas. The ISO 8608:2016 [6] standard specifies a uniform method of reporting measured vertical profile data taken on roads; however, measurement methods and processing equipment were not included. There is a need to observe and measure road conditions in a cost-effective and accurate manner. This can be achieved by utilizing an IMU: MPU-6050 MicroElectromechanical System (MEMS) module, 3-axis gyroscope, 3-axis accelerometer, and 400 kHz I2 C communication. An accurate IMU can precisely determine and track a vehicle’s dynamics on road surfaces. This is a continuous measurement approach with the data continuously collected to the test vehicle itself. Validation of the vehicle’s exact location is required to verify the accuracy of the results obtained by the IMU. A GPS unit with a Global Navigation Satellite System (GNSS) receiver

E. Zhao () · P. D. Walker · F. Al-Widyan School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected]; [email protected]; [email protected] A. Ong Faculty of Transdisciplinary Innovation, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_12

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(Broadcom BCM47752) was therefore selected to quantify the vehicle’s precise longitudinal and lateral locations. Its specific features meets the measurement needs: low procurement and operating costs, low energy consumption, compact in size, simplicity in installation and deployment, and sufficient computation capabilities [7]. Real-world driving vehicles’ ride quality, fuel consumption, and GHG emissions are sensitive to road conditions and road grade [3, 4]. For instance, a vehicle travelling on hilly roads consumes more fuel and produces more emissions than when travelling on straight, levelled roads [5, 10]. A vehicle’s Real World Driving Emission (RDE) is measured with a PEMS unit. The data collection process involves temporarily attaching a PEMS unit to the vehicle and collecting, analyzing, and recording the vehicle’s emissions and fuel consumption. The raw data collected is a direct representative of the real-world driving carried out under everyday driving conditions [2]. In this aspect, however, the results represent the driving conditions of the individual test and differs from laboratory test results [12]. Nevertheless, it is due to the direct influence of real-world conditions such as weather, traffic, and road conditions that on-road emissions testing provides valuable tools for linking emission rates to specific driving conditions, and hence the deficiency in the control of GHG and pollutants can be identified [8, 9]. Vehicles travelling on poor road conditions will experience cyclic loads [1]. Proper road condition measurement will assist with the classification of roads with similar vehicle excitation or damages into the respective categories. Furthermore, road condition monitoring defines and quantifies the severity of road usage at any given road section. This contributes to predicting road degradation that may prevent road vehicles from potential damages [11]. Therefore, the purpose of this study is as follows: 1. To measure road surface conditions with an externally installed IMU and quantifying the vehicle’s precise longitudinal and lateral locations with GPS 2. To measure and compare the emissions produced by a vehicle travelling in various road conditions The observations, results, and discussion of this project is presented and then concludes this paper.

2 Methodology The following methods are for measuring road surface conditions with the on-board GPS, an externally installed IMU, and a PEMS unit.

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2.1 Measuring Road Conditions The vehicle’s positioning was determined by using an externally installed GPS unit and IMU. An additional Garmin GPS sensor connected to the PEMS unit was also included for data comparison (model 16x™, positioning accuracy to within 3 m without using an external Differential GPS (DGPS) beacon receiver). The IMU was firmly fixed mid-way along the wheelbase of the vehicle, as close as possible to the vehicle’s center of gravity and in the direction of travel. The expected test outcomes were the collection of data of vehicle linear acceleration and rate of change: x-axis (front of vehicle), y-axis (side of vehicle), and z-axis (top of vehicle).

2.2 Emissions Measurement The data collection process involves temporarily attaching a PEMS unit to the vehicle and collecting, analyzing, and recording the vehicle’s emissions (see Fig. 1). The raw data collected is a direct representative of the real-world driving carried out under everyday driving conditions. The vehicle was installed with the following PEMS components used to simultaneously record emissions data: gas analyzer, particle analyzer, exhaust filter, Exhaust Flow Meter (EFM), PEMS controller, battery pack, charge master, GPS, and sensors (ambient temperature, humidity, exhaust gas temperature, and ambient pressure). The emissions measured complied with the EC 2016/427 requirements. These emissions include Carbon Monoxide (CO), Carbon Dioxide (CO2 ), Nitrogen Oxides (NOx ), Total Hydrocarbons (HC), and Particulate Matter (PM).

Fig. 1 PEMS Unit and Study Vehicle

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Fig. 2 GPS-tracked selected study route

2.3 Study Routes A local series of routes were planned in the Sydney, New South Wales, Australia. The boundaries of the test include time duration (10~15 min), stop duration (5~10 min), trip distance (≥ 5 km each for urban and suburban), speed range (25~50 km/h urban and 60~80 km/h suburban), altitude (0~700 m), and altitude difference (≤100 m). The selected route was a 2 km long road over an 80 m rise with an average road grade of 3–4% and data collection was repeated over 3 runs on the selected route. The selected route is shown in Fig. 2. The selected vehicle was a MY05 Toyota Prius with a mileage of 131,500 km. The vehicle was readily available and the PEMS unit was already installed and calibrated from previous projects.

3 Results and Discussion The vehicle altitude and road dynamics (acceleration – particularly vertical acceleration and rate of change) data was plotted with respect to time (see Fig. 3). Because the Broadcom BCM47752 GNSS receiver recorded data at 1 Hz (one pulse per second) and actual driving route was heavily influenced by road conditions, the measured route was not expected to agree with the estimated route generated by digital maps. However, the GPS tracked route (see Fig. 2) were precise enough to be nearly identical to the digitally estimated route. The altitude data obtained by the GPS receiver was consistent with the road gradient mentioned; however, minimal elevation change was not detected. The response of the vehicle’s suspension to the imperfections of the road surface was measured by the IMU’s accelerometer and gyroscope. It was observed that the lower vertical acceleration rate and gyroscope rate of change, the smoother the road surface (better road surface conditions).

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Fig. 3 Vehicle dynamics graphical data

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It was possible to mathematically integrate the inertial sensor acceleration measurements (dead-reckoning) twice to obtain positioning and orientation data. However, doing so has introduced accumulated measurement errors and drifts, and gravity must be removed from the acceleration. The problem could be rectified with algorithmic smoothing and filtering, such as utilizing a Kalman filter, which will be planned for future works. By comparing the IMU and emissions data, it was observed that the vehicle produced more GHG emissions travelling on rougher roads than on smooth surfaces. Additionally, the vehicle produced more emissions on traveling hilly and steeply inclined roads than on straight, levelled roads.

4 Conclusion The estimation of road grade based on a stand-alone GPS receiver was observed to be accurate, but was unable to measure minimal elevation change. The IMU was able to detect minute changes in road surface conditions. Thus, the IMU’s sensitivity and low-cost deployment allows more practical utilization for monitoring road surface conditions. It was observed that the IMU presented the potential to measure road conditions, which will assist with the classification of roads with similar vehicle excitation or damages into the respective categories [6]. Further monitoring and analysis can define and quantify the severity of road usage at any given road section. As vehicles travelling on poor road conditions will experience cyclic loads, this IMU and GPS road condition monitoring system may contribute to predicting road degradation to prevent road vehicles from potential damages. The vehicle’s exact longitudinal and lateral coordinates were validated with a GPS receiver. Consequently, it can be concluded that this project provided evidence that demonstrated the sensitivity of real-world driving vehicles’ ride quality, fuel consumption, and GHG emissions to road conditions and road grade. Furthermore, utilizing a 6 DOF IMU can fulfil the objective of real-time road condition measurement to estimate factors in traffic conditions.

References 1. Adlinge, S.S., Gupta, A.K.: Pavement deterioration and its causes. IOSR J Mech Civil Eng (IOSR-JMCE). 9–15 (2004) 2. Australian Automobile Association Real-World Driving Emissions Testing in Australia, Australia. (2017) 3. Boriboonsomsin, K., Barth, M.: Impacts of road grade on fuel consumption and carbon dioxide emissions evidenced by use of advanced navigation systems. Transp. Res. Rec.: J. Transp. Res. Board. 2139(1), 21–30 (2009) 4. Boroujeni, B.Y., Frey, H.C., Sandhu, G.S.: Road grade measurement using in-vehicle, standalone GPS with barometric altimeter. J. Transp. Eng. 136(6), 605–611 (2013)

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5. Cicero-Fernândez, P., Long, J.R., Winer, A.M.: Effects of grades and other loads on on-road emissions of hydrocarbons and carbon monoxide. J. Air Waste Manage. Assoc. 47(8), 898–904 (1997) 6. International Standard 2016. Mechanical Vibration – Road Surface Profiles – Reporting of Measured Data, ISO 8608:2016, Switzerland 7. Lei, T., Mohamed, A.A., Claudel, C.: An IMU-based traffic and road condition monitoring system. HardwareX. 4, 1–12 (2018) 8. Ligterink, N.: Real-world vehicle emissions. Int Transport Forum. 1–16 (2017) 9. Mock, P., German, J.: The future of vehicle emissions testing and compliance. Int Counc Clean Transport. 11 (2015) 10. Pastorello, C., Mellios, G.: Explaining Road Transport Emissions: A Non-technical Guide, p. 15, Copenhagen (2016) 11. Pawar, P.R., Mathew, A.T. & Saraf, M.R.: IRI (International Roughness Index): an indicator of vehicle response. International Conference on Materials Manufacturing and Modelling, pp. 11738–11750. Elsevier Ltd., Vellore (2018) 12. Williams, M., Minjares, R.: A technical summary of Euro 6/VI vehicle emission standards. Int Counc Clean Transport. 1–12 (2016)

A Novel Controllable Electromagnetic Variable Inertance Device for Vehicle Vibration Reduction Pengfei Liu, Minyi Zheng, Donghong Ning, Liang Luo, and Nong Zhang

1 Introduction An inerter is a two-terminal mechanical device with the same unit as the mass element, which has attracted attention since its first introduction. The force applied to an inerter is proportional to the relative acceleration of its two terminals [1], and some research has proved that the mechanical network of a vehicle suspension with constant inertance can improve the ride comfort of a vehicle [2, 3]. The inerter consists of a ball screw and a flywheel [4], which when added to the vehicle suspension structure can improve its damping performance, especially at the natural frequency of a vehicle body. Recently, research on variable stiffness and variable damping suspension for vibration control is becoming popular [5]. A damping-controllable seat suspension with variable external resistance [6] and a mechanical system that can control the equivalent stiffness is formed by connecting the variable damping device in series with a spring [7]. Also, many investigations are ongoing on the semi-active system with an inerter for to improve vehicle suspension performance [8, 9]. The

P. Liu · M. Zheng () · L. Luo Hefei University of Technology, Hefei, China e-mail: [email protected] D. Ning () Hefei University of Technology, Hefei, China University of Wollongong, Wollongong, NSW, Australia e-mail: [email protected] N. Zhang Hefei University of Technology, Hefei, China University of Technology Sydney, Sydney, NSW, Australia © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_13

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aforementioned researches are all about the study of vehicle performance with constant inertance integrated into passive and semi-active suspensions. In this chapter, we propose an electromagnetic variable inertance (EMVI) device. The basic characteristics of an EMVI device and the vibration reduction performance of EMVI suspension are studied by numerical simulation. The remainder of this chapter is organized as follows: Section 2 introduces the composition and model of EMVI; the EMVI device dynamic characteristics are researched in Sect. 3; Section 4 studies the vehicle EMVI suspension performance based on simulation; Finally, Section 5 presents the conclusions of this study.

2 EMVI Device Model The structure of the EMVI is shown in Fig. 1. It includes a ball screw, a DC motor, and a resistance controllable unit (RCU). The motor is an energy exchange medium, its rotor is connected with a screw, and its stator has a large inertance that can rotate freely like a flywheel. The ball screw converts the axial motion at both ends of the EMVI into a rotary motion. The electromagnetic torque of the DC motor is controlled by the RCU, and the stator of the motor is rotated by the electromagnetic torque. The schematic circuit diagram of the direct current (DC) motor and RCU is shown in Fig. 2a where the DC motor is equivalent to a voltage source e, a resistor ri , and an inductor Li . The Rm is resistance within the Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) switch, when the MOSFET switch is turned on. When the rotor and stator of the DC motor rotate relative to each other, the current ip in the coil will generate an electromagnetic torque Te which drives the stator to rotate. The induced voltage e of the DC motor is proportional to the rotational speed ω, and according to Kirchhoff’s voltage law (KVL), we can conclude that ⎧ Rm Re ⎪ ⎨ RE = D Rm +Re + (1 − D) Re di e = ke ω = (ri + RE ) ip + Li dtp ⎪ ⎩ T e = k i ip

(1)

where D is the PWM duty cycle, Re is the external resistor, ip is the current in the circuit, and ke is the voltage constant of the DC motor. Nut

Ball screw

DC motor

RCU PWM

Fig. 1 Structure of the controllable EMVI device

A Novel Controllable Electromagnetic Variable Inertance Device for Vehicle. . .

(a)

(b)

DC motor

ri li

RCU

ip

Fd

PWM

Te+Tmf ωd

Jb , l ω z

Tsf

Jm

Re

up

External MOSFET resistor

e

105

Jw

Screw Nut

z

Rotator Stator

Fig. 2 Simplified model diagram of the EMVI device

The simplified mechanical model of the EMVI device is shown in Fig. 2b. Assume the friction torque between the nut and screw is Tsf = 0.03 ∗ sgn (ωz ), and the air gap resistance torque between the rotor and stator is Tmf = 0.01 ∗ sgn (ωz − ωd ). Also, consider the moment of inertia of the screw and the motor rotor are Jb and Jw , respectively. The motion equation of the motor stator and the axial force Fd can be written as    2π T Fd = 2π + T + T + + J z ¨ (J ) e sf mf m b l l (2) Jw ω˙ d = Te + Tmf where z is the axial motion displacement between the nut and the screw, l is the pitch of the ball screw, ωz is the angular velocity of the screw, Jw is the calculated moment of inertia of the motor stator, ωd is the angular velocity of the motor stator.

3 EMVI Device Characteristics Analysis A time and frequency domain simulation is carried out to study the variable inertance characteristic of the EMVI device. In the time-domain simulation, it is assumed that an external force acts on the EMVI device for a sinusoidal motion with a frequency of 2 Hz and an amplitude of 0.03 m. Also, it ignores the friction of the ball screw and the friction inside the motor; the Laplace transform is performed on Eqs. (1) and (2), and we get Fd (s) = z(s)s 2

"

2π l

#2 $

Jw ki2 ki2 + Jw (RE + ri + Li s) s

% + (Jm + Jb )

(3)

where Fd (s) and z(s) are the Laplace transforms of Fd (t) and z(t), respectively. Fd (s) As the inertance force is proportional to the relative acceleration, z(s)s 2 represents

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Table 1 Basic parameters of the EMVI device Parameter Torque constant ki Internal resistance ri Internal inductance Li External resistance Re

Value 0.245 Vs/rad 1.41 Ohm 0.644 mH 50 Ohm

Parameter Pitch of the screw L Inertia of the screw Jb Inertia of the rotor Jm Inertia of the stator Jw

Value 0.020 m 400 gcm2 1340 gcm2 15000 gcm2

Fig. 3 The time and frequency domain simulation of the EMVI device

the equivalent inertance of the EMVI device. The selected DC motor is MAXON 353301, and the ball screw is TBI SFH1620. Their parameters are listed in Table 1. As shown in Fig. 3a, as the duty cycle D of the PWM wave increases, the kinetic energy stored in the EMVI increases, which means that the magnitude of the inertance of the EMVI device can be controlled by changing D. In Fig. 3b, the maximum value of the equivalent inertance can reach 160 kg, and the equivalent inertance value decreases as the excitation frequency increases.

4 EMVI Suspension Performance The EMVI suspension is used to connect the EMVI device in parallel with the spring and damper of a conventional vehicle suspension. To verify the EMVI suspension performance, the vehicle’s vertical acceleration, suspension deflection, and tire dynamic load are used as control targets to design a quadratic optimal controller for EMVI suspension. The energy of the EMVI device is derived from the vibration of the suspension and there is no external energy supply to EMVI suspension, so the value of D should be judged by the electromagnetic torque and equivalent inertance. Table 2 lists the basic parameters of EMVI suspension.

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Table 2 The basic parameters of EMVI suspension Value 320 kg 45 kg 22 kN/m

Traditional suspension EMVI suspension

6 4 2 0 -2 -4

Parameter Suspension damping cs Tire stiffness kt Tire damping ct (b)

10

Traditional suspension EMVI suspension

5 0 -5

Value 700 Ns/m 190 kN/m 100 Ns/m (c)

3 Tire dynamic load (kN)

(a)

8

Suspension travel (cm)

Traditional suspension EMVI suspension

2 1 0 -1 -2

2 Time (s)

4

0

2 Time (s)

4

0

2 Time (s)

4

Duty cycle (%)

0

Force (N)

Sprung mass acceleration (m/s 2)

Parameter Sprung mass ms Unsprung mass mt Spring stiffness ks

Fig. 4 Suspension performance comparison curve under sinusoidal excitation

4.1 Sinusoidal Excitation to EMVI Suspension A sinusoidal excitation with an amplitude of 0.03 m and a frequency of 1.5 Hz is applied to the tire to simulate and analyze the EMVI suspension performance. The simulation results are shown in Fig. 4a, b, and c; the performance of the EMVI suspension is significantly better than the conventional suspension at 1.5 Hz. As Fig. 4d, e show, under sinusoidal excitation, the maximum value of the motor stator speed tends to a stable value, and the output force generated by the EMVI device can account for 30% of the ideal active power.

4.2 Random Excitation to EMVI Suspension Random road profile can be implemented in terms of road roughness grade classified by ISO 8608. Shown in Fig. 5a, b and c is the performance comparison curve of the suspension, when the road surface roughness level is C and the speed is 15 m/s.

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4 2 0 -2 -4

(b)

6 Suspension travel (cm)

Traditional suspension EMVI suspension

Traditional suspension EMVI suspension

4 2 0 -2 -4

0

10 20 Time (s)

30

(c)

3 Tire dynamic load (kN)

(a)

6

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2 1 0 -1 -2

0

10 20 Time (s)

30

0

10 20 Time (s)

30

Value

Sprung mass acceleration (m/s2)

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Fig. 5 Suspension performance comparison curve under random excitation

As shown in Fig. 5d, compared to traditional suspension, the parameters’ values of EMVI suspension are decreased by 14.6%, 22.6%, and 14.9% regarding Root Mean Square (RMS), frequency-weighted RMS (FW-RMS), and vibration dose value (VDV), respectively. To clearly express the distribution effect of EMVI suspension on the vibration reduction of the vehicle, the comparison of the power spectral density (PSD) of the acceleration is shown in Fig. 5e. It can be seen that the designed EMVI suspension can reduce the vibration acceleration of the sprung mass at low frequencies.

5 Conclusion This chapter presents an EMVI device for vehicle suspension vibration reduction. Time and frequency domain simulations show that the inertance of the EMVI device could be controlled by the PWM duty cycle and has better variable inertance characteristics at low frequencies. In addition, compared to conventional suspensions, EMVI suspension vehicles have improved performance in sprung acceleration, suspension travel, and tire dynamics at low-frequency excitation. From comparison of the PSD of the sprung acceleration, it is clear that EMVI suspension can significantly reduce the vibration of the vehicle body in the frequency range of 1–3 Hz.

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Acknowledgments The research is supported by the National Natural Science Foundation of China (51675152), the National Key Research and Development projects of China (2018YFB0105505-03), and the Anhui New Energy Automobile and Intelligent Networking Automotive Industry Technology Innovation Project (IMIZX2018001).

References 1. Smith, M.C., Wang, F.C.: Performance benefits in passive vehicle suspensions employing inerters. Veh. Syst. Dyn. 42(4), 235–257 (2004) 2. Hu, Y., Chen, M.Z.Q.: Passive vehicle suspensions employing inerters with multiple performance requirements. J. Sound Vib. 333(8), 2212–2225 (2014) 3. Wang, R., et al.: Design and test of vehicle suspension system with inerters. Proc. Inst. Mech. Eng. 228(15), 2684–2689 (2014) 4. Shen, Y.J., et al.: Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension. J. Sound Vib. 361, 148–158 (2016) 5. Sun, S.S., et al.: A compact variable stiffness and damping shock absorber for vehicle suspension. IEEE/ASME Trans. Mechatronics. 20(5), 2621–2629 (2015) 6. Ning, D.H., et al.: Vibration control of an energy regenerative seat suspension with variable external resistance. Mech. Syst. Signal Process. 106, 94–113 (2018) 7. Sun, S.S., et al.: A new generation of magnetorheological vehicle suspension system with tunable stiffness and damping characteristics. IEEE Trans. Ind. Inf. (2019) 8. Hu, Y., et al.: Inerter-based semi-active suspensions with low-order mechanical admittance via network synthesis. Trans. Inst. Meas. Control. 40(15), 4233–4245 (2018) 9. Chen, M.Z.Q., et al.: Performance benefits of using inerter in Semiactive suspensions. IEEE Trans. Control Syst. Technol. 23(4), 1571–1577 (2015)

Conceptual Design Model of Road Noise on Automotive Bodies in White Based on Energy Propagation Toru Yamazaki, Keita Suwabe, Kousuke Nakanishi, Hirotaka Shiozaki, and Junichi Yanase

1 Introduction In the development of machine products, it is important to specify the required performance for high-speed development with low cost at early design stages [1]. As for automobiles, noise and vibration performance is generally not considered in early design stages. There is thus a need to develop a formulated functional model for road-noise problems in early design stages. For road-noise problems, the vibration energy propagation characteristic is important. So, such a functional model can be constructed based on statistical energy analysis (SEA), especially on analytical SEA (ASEA) with analytical formulated parameters. SEA is a method for predicting mid- and high-frequency noise and vibration transmission through large, complex systems by considering vibration energy propagation through the system, which is viewed as an aggregate of simple parts. Vibrational energy is easily calculated in ASEA by simple mathematical equations formulated from only the basic information of the subsystems, such as plate thicknesses and material properties. ASEA can thus be used in the early stages of automobile design, and vibrational behaviors can be approximately captured. Concepts for lower noise and vibration can be obtained through optimization of the model.

T. Yamazaki () Kanagawa University, Yokohama, Japan e-mail: [email protected] K. Suwabe · K. Nakanishi Undergraduate School of Kanagawa University, Yokohama, Japan H. Shiozaki · J. Yanase Mitsubishi Motor Corporation, Okazaki, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_14

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There has been some research about applying ASEA to early design [2] [3]. In this study we consider an automotive body in white (BIW) with doors and windows represented by SEA plates, and with ASEA applied to determine how target parameters should be set to reduce body vibration. ASEA thus provides design concepts for lowering noise and vibration.

2 Statistical Energy Analysis 2.1 SEA Equations SEA is an analysis method based on balancing vibration energies between a few or many subsystems. In the SEA model, the system is regarded as an assembly of subsystems. It is assumed that energy dissipation in subsystems is proportional to the vibration energy thereof. Transfer energy between subsystems is also assumed to be proportional to the vibration energy between two subsystems. Considering the power balance leads to a set of equations. The basic SEA equation is P = ωLE

(1)

where P is the external power input vector, E is the subsystem vibration energy vector, and ω is the band center angular frequency. Matrix L is the matrix of loss factors, written as ⎡

−η21 η1 + η12 + η13 + · · · ⎢ η + η − η 12 2 21 + η23 + · · · L=⎣ .. .. . .

⎤ ··· ···⎥ ⎦ .. .

(2)

where ηi is the internal loss factor (ILF) of subsystem i and ηij is the coupling loss factor (CLF) from subsystem i to subsystem j. Loss factors are dependent on the frequency.

2.2 Analytical SEA and Optimization In analytical SEA, the CLF is evaluated using Eq. (3) if the subsystem is like a thin plate. 2Lij τij ηij = π Si

&

hi ω

' 4

Ei   12ρi 1 − vi 2

(3)

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Here, hi, Si, Vi, Ei, ρi, and νi are, respectively, the plate thickness, surface area, volume, Young’s modulus, mass density, and Poisson’s ratio of subsystem i, and Lij and τ ij are, respectively, the coupling length and energy transmission efficiency between subsystems i and j. If two plate subsystems are orthogonally coupled, the energy transmission efficiency for a bending wave is given by  τij =

−5/4 + −3/4 + 3/4 + 5/4 −2 /2 + −1/2 + 1 + 1/2 + 2 /2

2 (4)

where  = hj/hi is the plate thickness ratio. From the above, the CLF is evaluated by poor subsystem information. Optimization for minimizing a target subsystem’s vibrational energy can thus be carried out with an SEA model regarding the subsystem’s fundamental information as design variables.

2.3 Representation of Automotive BIW in the ASEA Model Test Object The target object is the automotive BIW with doors and windows shown in Fig. 1 (body of an Outlander, Mitsubishi Motor Corporation). All parts are made of steel except subsystem 21 (windshield), which is made of glass. We constructed an SEA model for this structure. SEA Modeling In SEA modeling, this structure was approximately subdivided into the 32 SEA subsystems shown in Figs. 1 and 2. Note that structural complexity (e.g., component shapes and dispersion of the component thicknesses) cannot be considered in SEA modeling.

3 ASEA Prediction 3.1 Calculating Frequency Response from the ASEA Model To calculate the vibration energy of road noise, we considered power input of 1 W into subsystem 26 (the front suspension cross member), which seems to receive an exciting force from the road through the tires. Regarding subsystems as plates, CLF was calculated by Eq. (3), and the energy transmission efficiency was calculated by Eq. (4). All subsystems except the windshield are made of the same material, so their ILF value was assumed to be the same (0.001 or 0.01). Vibration energy of each subsystem was calculated according to Eq. (1).

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Fig. 1 Target object, an automotive BIW with doors and windows

Fig. 2 Connections between subsystems

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Fig. 3 FEM model of the automotive BIW

Fig. 4 Excited structures in FEM analysis (front floor structures are omitted)

3.2 Calculating Frequency Response from FEM Figure 3 shows the FEM model for the target automotive BIW. We conducted frequency response analysis using this model to calculate kinetic energy, which was regarded as vibration energy. A sinusoidal excitation force of 1 N was applied to subsystem 26 (the front suspension cross member; see Fig. 4), and kinetic energy was calculated at multiple points in each subsystem. The structural damping was set as 0.1.

3.3 Comparison of Frequency Responses by ASEA and FEM Figure 5 shows the results of the above-described calculations of vibration energy for some subsystems. In each figure, continuous lines indicate kinetic energy normalized by the power input at some points on the subsystem evaluated by FEM, and dotted lines indicate vibration energy calculated from ASEA analysis. Vibration energy as calculated by the FEM model largely varies through all frequencies and among selected points, but counterpart calculations from ASEA

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Fig. 5 Comparisons of subsystem kinetic energy by ASEA and FEM

are stable across all frequencies. This demonstrates that the ASEA method can obtain baseline response for vibrational energy and can capture approximate mean vibrational behavior in the subsystem.

4 Optimization with ASEA 4.1 BIW Optimization with the SEA Model Target Subsystem We used this ASEA model for optimization to derive concepts for low noise and vibration. The input subsystem is subsystem 26, and the objective function is to minimize vibration energy in subsystem 22, which seemed to contribute to road noise in the front compartment. The design variables are thickness

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Table 1 Optimization results Subsystem No. 11 12 13 14 15 16 17 18 19 20

Thickness change rate 1.02 1.20 2.00 1.53 1.23 2.00 0.61 0.62 1.32 2.00

Subsystem No. 21 22 23 24 25 27 28 29 30 32

Thickness change rate 1.00 2.00 0.50 0.50 0.50 2.00 1.72 2.00 1.14 0.55

Fig. 6 Optimization result. Color strength indicates degrees of thickness change

change rates (hi-changed/hi-original) in subsystems 11–25, 27–30, and 32. Each thickness change rate is constrained to between 0.5 and 2.0. Results and Interpretation Table 1 and Fig. 6 show the results of thickness change optimization. A concept for lowering vibration energy in subsystem 22 was obtained from these results. From Eq. (4), ηij depends on hi, so the transfer energy from subsystem i to j seems to be larger if hi is bigger than hj. In the results, thickness change rates in front-structure subsystems were close to 0.5, which means, to pass the vibration energy in subsystem 26, the subsystem excited to other front-structure subsystems, reducing the energy transferred to subsystem 22.

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4.2 Verification with FEM Redesigned Models Vibration energies in subsystem 22 were calculated by two FEM models, an original and a redesigned model, whose thicknesses are changed as above. As Fig. 7 shows, a sinusoidal excitation force of 1 N was separately applied at four points in subsystem 26, either vertical or horizontal to the front suspension. Vibration energy was calculated as the average of the eight cases. Figure 8 shows the results of calculating vibration energy for subsystem 22 using FEM and ASEA. ASEA calculations were conducted as in sect. 3.1 except for the ILF value, considering differences in the calculation method. Figure 8 shows that dashboard vibration was reduced across nearly the full frequency range, except at a few frequencies associated with local natural modes. We also calculated the vibration energy ratio. Figure 9 shows vibration energy ratio of the two FEM models. The energy ratio in each subsystem was calculated as the subsystem energy divided by the total of all subsystem energies. Ratios for front structure subsystems in the redesigned model are larger than the original model across nearly the entire frequency range. In contrast, the ratios for subsystem 22, the target subsystem, and subsystem 26 in the redesigned model show decreased subsystem excitation.

Fig. 7 Excited points in the front suspension and directions of exciting forces

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Fig. 8 Comparison of vibration energies in subsystem 22

Fig. 9 Comparisons of vibration energy contributions between the original and the redesigned models

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These suggest that vibration in subsystem 22 was reduced by passing the energy in subsystem 26 to other front-structure subsystems in accordance with the concept obtained from the ASEA optimization.

5 Conclusions We applied ASEA modeling to predict vibration energy behavior in an automotive BIW and performed optimization to lower subsystem energy. Through this procedure, a concept for fulfilling the required performance can be obtained at early design stages. The following summarizes this research: 1. Vibration behavior in an automotive BIW could be broadly grasped using an ASEA plates model. 2. A concept for lowering vibration could be obtained through optimization with an ASEA model, using the dashboard in an automotive BIW as the target subsystem. The concept for lowering vibration energy is as follows: input energy is passed from the suspension to front subsystems to the greatest extent possible. 3. The concept obtained from optimization was confirmed through FEM verification. Vibration energy in the redesigned model was smaller than in the original one.

References 1. Malen, D.E.: Fundamentals of Automobile Body Structure Design. SAE International (2011) 2. Ikeda, A.T.K., Nakamura, H., Yamazaki, T.: Subsystems’ Layout Change Method based on Analytical SEA for Vibration Reduction; Utilization for an Injection Pump of an Engine, Proceedings of inter-noise 2019, 1613.pdf , 2019–6, Madrid, Spain (2019) 3. Kataoka, D, Nakamura, H, Yamazaki, T.: Vibration reduction with additional subsystems as absorber or bridge by using analytical SEA. Proceedings of inter-noise 2019, 1553.pdf , 2019– 6, Madrid, Spain (2019)

Modeling and Measuring of Generated Axial Force for Automotive Drive Shaft Systems Huayuan Feng, Wen-Bin Shangguan, and Rakheja Subhash

1 Introduction The drive shaft system is an important subsystem of the automotive transmission system used to transfer the input torque from the transmission to the wheels, and it can ensure the same rotational speed of the output end as that of the input end within certain articulation angles. A tripod joint is a plunging-type joint that is widely used in the drive shaft system. Due to the friction and kinematic characteristics inside the tripod joint, the generated axial force (GAF) will be generated, and this axial force has third-order characteristics of shaft rotational speed [1–5]. When a car moves abruptly, the GAF may cause NVH (Nosie, Vibration, and Harshness) problems in a car, namely, “take off shudder.” Since the GAF of a tripod joint is generated due to internal friction of the tripod joint, Lee CH et al. [1] developed a test bench for measuring friction characteristics under various running conditions, and carried out modeling and analysis of the friction characteristics. Lee KH et al. [2] established a non-rotating-type GAF test bench to measure the GAF of a drive shaft system under different running condition. However, the drive shaft system was not run in dynamic rotating condition and it was different from the actual motion. To investigate the idle vibration problem caused by a drive shaft system with a tripod joint, Sa JS et al. [3] developed a test bench to measure the plunging force and GAF of the drive shaft system. Serveto S et al. [4, 5] established multi-body dynamic models integrating Coulomb friction and an impact function-based contact force model for estimating GAF of a drive shaft system. The contact force model is defined as the relation between normal contact force, contact stiffness, force exponent, and penetration

H. Feng · W.-B. Shangguan () · R. Subhash South China University of Technology, Guangzhou, Guangdong Province, China e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_15

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displacement, etc. The key issuers for estimating contact force are to determine contact stiffness and force exponents, but the method was not given in the paper. Peak deviations of up to 50% were observed between the estimated and the measured GAF [5]. Jo GH et al. [6] presented an analytical model for calculating GAF using pure sliding and rolling-sliding friction models. Different from the works in Refs. [4, 5], the friction model proposed by Refs. [1] was used to calculate the friction force in the paper [7], and the contact stiffness, penetration displacement, and damping ratio of the contact pair were used to characterize the normal contact force. However, there are no experiments in this study to validate the models. In summary, the validity of some aforementioned models has not been established or the models revealed relatively large prediction errors. Effective prediction of models of a drive shaft system with a tripod joint necessitates accurate estimations of the friction and the contact pairs model parameters, such as the friction coefficient, the contact stiffness, and the force exponent. These parameters are considered constants and evaluated assuming circular contact geometry in the aforementioned studies, irrespective of the operating conditions. The contributions of the study are summarized below. A GAF test bench for measuring GAF of a drive shaft system under different running conditions is presented. Based on the finite element analysis, a method for calculating the contact stiffness and the initial force exponent between rollers and tracks is presented. A methodology is proposed for identifying the friction coefficients, and the force exponent for the roller-track contact pairs under different operating conditions.

2 Measuring of GAF of a Drive Shaft System The schematic and test bench for measuring GAF of a drive shaft system are shown in Fig. 1. During the measurement, the shaft rotational speed is controlled by the electric motor and the loaded end connected tripod joint is connected to a hydraulic actuator. Four force sensors are installed at the loaded end to measure GAF. The GAF is measured and analyzed under different friction coefficients, articulation

Fig. 1 (a) Schematic diagram and (b) pictorial view of the drive shaft system test bench

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Table 1 Test matrix Series A1 A2 A3

Shaft rotational speed N (rpm) 200 200 100–1000

Input torque T0 (Nm) 300 100 to 700 600

Lubricant None, G1, G2 G2 G2

Articulation angle δ (degrees) 0 to 16 4, 8, 12, 16 4, 8, 12, 16

Fig. 2 Effects of operating factors on the measured GAF

angles δ, input torque T0 , and shaft rotational speed N on GAF, as summarized in Table 1. The measured GAF versus the articulation angles under different friction coefficients (Lubricants) are shown in Fig. 1a. It is seen that GAF increases with an increase in the articulation angle δ. The magnitude of GAF decreases with the use of both types of lubricant in the tripod joint. Figure 2b, c illustrate variations in the measured GAF with variations in the shaft rotational speed and the input torque, respectively. The results suggest that the magnitude of GAF increases nearly linearly with increase in the input torque. The magnitude of GAF, however, decreases with increase in the shaft rotational speed.

3 Modeling of a Drive Shaft System A multibody dynamic model for estimating GAF of a drive shaft system with a tripod joint and a ball joint is shown in Fig. 3. It is seen that the model consists of five rigid body parts: roller, needle, tripod, housing, and drive shaft. In the model, needles are simplified into cylinders and the ball joint is modeled by a ball pair. Constraints between the rigid parts are as follows: the housing is connected to the ground by a revolute pair; one end of the drive shaft is connected to the ground by a ball pair and the other end is rigidly connected to the tripod; rollers and needles, and needles and the tripod are both connected by frictionless cylindrical joints; the motion between rollers and tracks is restrained by contact pairs with friction. In

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Fig. 3 (a) Multibody dynamic model of the drive shaft system and (b) constraints and inputs

calculation, the rotation speed is applied at the drive shaft, and the input torque is applied to the tripod joint end.

3.1 Modeling Friction and Contact Force Between the Roller and the Track A velocity-based friction model [8] is used to characterize the friction coefficient μ between the roller and the track, and is expressed as:     μ = μs sin Ctan−1 Bυ − E Bυ − tan−1 (Bυ)

(1)

where μs is the static friction coefficient, υ is relative velocity between the roller and the track, and C, B, and E are constants to be determined from the measured data. The relation of the normal contact force Fn and penetration displacement σ between the roller and the track is characterized by the impact function [5], given by: Fn = kσn + cσ˙

(2)

where k is the contact stiffness, n is the force exponent, c is the damping coefficient of the contact pair, and σ˙ is the time derivative of the penetration displacement.

3.2 Calculation of the Contact Stiffness and the Nominal Force Exponent Owing to the structural symmetry of the tripod joint, a quasi-static finite element (FE) model is only formulated to derive the force-deflection relation for a single

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Fig. 4 (a) Loading of the roller; (b) 6-degrees-of-freedom constraint applied to housing; and (c) variations in the identified contact stiffness and nominal force exponent with input torques

roller-track contact pair together with the housing, as shown in Fig. 4a, b. The simulation relation is used to identify the contact stiffness and the nominal force exponent by fitting the relation in Eq. (2) using the least squares error minimization method, while neglecting the damping term (equivalent to the Hertz contact model [8]). Figure 4c illustrates variations in k and n with the input torque in the 200– 600 Nm range. It is seen that both the contact stiffness and the nominal force exponent decrease with increase in the input torque. This can be attributed to variations in the contact geometry and penetration with the applied torque.

3.3 Identification of Contact and Friction Parameters According to Hertz contact theory [5, 8], it is assumed that the contact stiffness is only related to the input torque (Fig. 4c). However, the changes in the articulation angle and the shaft rotational speed can lead to variations in the contact geometry and thus the force exponent. Furthermore, variations in the friction coefficient caused by changes in the internal temperature and state of the lubricant can alter the contact geometry and thus the force exponent. An identification methodology is proposed for identifying the friction coefficients and force exponent for the tripod joint model under different shaft rotational speeds, articulation angles, and input torques: Min f (q, xi ) = Σi | (FE (q) –FM (xi )) /FM (xi ) |

(3)

Subject to : μs –μs l ≥ 0, μs u –μs ≥ 0, μd –μd l ≥ 0, μd u –μd ≥ 0, n–nl ≥ 0 and nu –n ≥ 0. where q = (μs , μd , n) is the parameter vector to be identified; xi defines the operating factors; FM (xi ) is the measured GAF under the xi operating condition; FE (q) is the computed GAF corresponding to the same conditions; μs is the static friction

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Fig. 5 Comparisons of model-predicted and measured GAF magnitudes of an alternate drive-shaft system as functions of the articulation angle (N = 200 rpm)

coefficient; μd is defined as the dynamic friction coefficients from Eq. (1). The friction coefficients are constrained to vary within the defined lower (μs l , μd l ) and upper (μs u , μd u ) bounds. nl and nu define the lower and upper bounds, respectively, of the force exponent n. The nominal force exponent values identified as a function of the input torque (Fig. 4(c)) served as the starting values.

4 Results Based on the identified contact and friction parameters, the GAF of other drive shaft systems with different tripod joints can be calculated by the multibody dynamic model. The model validity was examined for the drive shaft system employing an alternate design of a tripod joint. The drive shaft system employed an identical ball joint, while the size of the tripod joint was smaller than that used for the model identification and it consisted of elliptical rollers. Figure 5 compares the GAF response of the identified model with the measured data over the entire range of the articulation angle. The peak relative error between the model results and the measured data is below 15%, indicating a high calculation accuracy of the GAF calculation model.

5 Conclusions The GAF of the tripod joint in the drive shaft system increased nearly linearly with increase in the articulation angle, while the rate of increase was strongly affected by

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the type of lubricant and thus the friction. The GAF also increased nearly linearly with the input torque but decreased with increase in the shaft speed. The force-deflection characteristics of the roller-track contact pair, obtained from the finite element model, revealed good correlation with the Hertz contact model with the contact stiffness and the force exponent decreasing with the input torque. Based on the identified friction coefficient and the contact parameters, the GAF predicted from the model showed very good agreements with the measured data.

References 1. Lee, C.H., Polycarpou, A.A.: A phenomenological friction model of tripod constant velocity (CV) joints. Tribol. Int. 43(4), 844–858 (2010) 2. Lee, K.H., Lee, D.W., Chung, J.H., et al.: Design of generated axial force measurement tester for tripod constant velocity joints under shudder condition. J. Mech. Sci. Technol. 28(10), 4005– 4010 (2014) 3. Sa, J.S., Kang, T.W., Kim, C.M.: Experimental study of the characteristics of idle vibrations that result from axial forces and the spider positions of constant velocity joints. Int. J. Automot. Technol. 11(3), 355–361 (2010) 4. Serveto, S., Mariot, J.P., Diaby, M.: Modelling and measuring the axial force generated by tripod joint of automotive drive-shaft. Multibody Sys. Dyn. 19(3), 209–226 (2008) 5. Lim, Y.H., Song, M.E., Lee, W.H., et al.: Multibody dynamics analysis of the drive-shaft coupling of the ball and tripod types of constant velocity joints. Multibody Sys. Dyn. 22(2), 145–162 (2009) 6. Jo, G.H., Kim, S.H., Kim, D.W., et al.: Estimation of generated axial force considering rollingsliding friction in tripod type constant velocity joint. Tribol. Trans. 61(5), 889–900 (2018) 7. Cai, Q.C., Lee, K.H., Song, W.L., et al.: Simplified dynamics model for axial force in tripod constant velocity joint. Int. J. Automot. Technol. 13(5), 751–757 (2012) 8. Johnson, K.L.: Contact Mechanics. University of Cambridge (1985)

Optimization on Energy Management Strategy with Vibration Control for Hybrid Vehicles Yifan Wei

, Yuning Wang

, and Zhichao Hou

1 Introduction Hybrid electric vehicles (HEVs) are considered to be a future vehicle trend because of their potential to save energy and reduce emissions. For parallel hybrid vehicles, Kim designed a fuzzy control strategy to improve engine efficiency and reduce emissions [1]. Chen proposed an online adaptive control strategy, obtaining the most energy-saving solution through a dynamic programming algorithm [2]. Panday summarized mainstream energy management strategies [3]. It can be noted that none of the previous strategies take vibration control into consideration. In the field of vehicles’ power system vibration analysis, a lot of researches have been conducted. Doughty used the extended transfer matrix to analyze the vibration of a damped crankshaft and found a solution with an iterative method [4]. Wakabayashi achieved analysis on the torsional vibration, the axial vibration, and lateral vibration of an engine crankshaft using a three-dimensional model [5]. Motivated by such an observation, this study introduces a novel energy management method which takes vibration into consideration as well as energy consumption. A multi-objective optimal problem was defined and solved based on Toyota Prius 4. By incorporating vibration control into energy economy optimization, the vehicle driving management system is improved.

Y. Wei · Y. Wang · Z. Hou () School of Vehicle and Mobility, Tsinghua University, Beijing, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_16

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2 Dynamic Modelling 2.1 Mechanical Model of a Drive Line Torsional Vibration Figure 1 shows a schematic diagram of the transmission system of Toyota Prius 4, while the overall vibration model of the vehicle is demonstrated as Fig. 2. The overall system consists of an engine, a planetary gear train, a final drive, 2 motors, and a drive shaft. The engine is directly connected to the planet carrier. One motor is attached to the sun gear, and another one to the ring gear. The final drive and rim are also linked to the ring gear. Power is transited to the wheels through the differential from the final drive. The final drive and drive shaft can be simplified to spring damping structures. The pivotal problem of vibration analysis should be focused on the planetary gear system. In this study gear meshing is modelled by the international stiffness simplification formula as kr = k (0.75εα + 0.25)

(1)

where k is the stiffness of single gears, and εα is gear face coincidence. Section status and parameters are shown in Table 1 [6].

Fig. 1 Power system of Toyota Prius 4

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Fig. 2 Vibration model of the transmission system Table 1 Status and parameters of the vibration model Variable I1 Jc I3 Jp I4 I5 I6 I7

Section Crankshaft Gear carrier Sun gear Plane gear Ring gear Final drive Wheels Car body

Value 0.28 kg · m2 0.06 kg · m2 0.03 kg · m2 0.01 kg · m2 0.1 kg · m2 0.1 kg · m2 0.5 kg · m2 120 kg · m2

Variable k1 k2 k3 k4 c1 c5 c6

Section Shock absorber Gear stiffness Gear stiffness Suspension Shock absorber Final drive Wheels

Value 620 Nm/rad 401 kNm/rad 5830 Nm/rad 8600 Nm/rad 4.77 Nm · s/rad 47.8 Nm · s/rad 286.5 Nm · s/rad

2.2 Control Model of the Power Devices The model is based on the energy management strategy (EMS) [7]. In normal working speed range, the temperature, open circuit voltage, and charge impedance of the battery can be regarded basically constant. As a result, the dynamic characteristics of the battery SOC could be expressed as in paper [8]. Study has indicated that the engine steady-state response is completely sufficient for estimation of the fuel consumption [9]. The dynamics of motors are not a critical concern in this paper, so it is simplified as a first-order process. Generally, the longitudinal dynamic equation and resistance can be described as in paper [7]. For the selected Toyota Hybrid System, the relationship between the planetary gear components can be expressed

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as in the paper [10]. The dynamic model of the whole vehicle can thus be described as ⎧   ω ωEM1 ωEM2 ⎪ + Tbrake Tveh = ηT ωEng T + T + T ⎨ Eng EM1 EM2 ωr ωr r   2 2 2    ωEng ωEM1 ωEM2 2 +I + ⎪ I + I + I ⎩ Iveh = mveh rW 6 Eng EM1 EM2 ωr ωr ωr (2) where Tr is the output torque on the ring gear; ωr is the ring gear speed; ωEng , ωEM1 , ωEM2 are output speeds of the engine, motor 1, and motor 2; and TEng , TEM1 , and TEM2 are output torques of the three devices, respectively. Other parameters used for analysis include tire radius rW = 0.343 m, vehicle mass mveh = 1800 kg, and transmission efficiency ηT = 0.8.

2.3 Energy Management For energy management, a cost function is defined with three items, namely, the fuel consumption rate, equivalent motor power, and loss of battery power during discharging and charging, as shown below ˙ Jf uel = Peq (t) + λPbat (t) = Ql m˙f (t) + PMG (t) + λ(t)SOC(t)V OC Q

(3)

where Peq (t) is the equivalent power of engine and motor, Pbat (t) is the power of the battery, and λ= λ(t) is the coefficient related to the state of battery. Ql is the low calorific value of fuel, m ˙ f is the rate of fuel consumption, PMG (t) is the power of the two motors. Apart from the torque and speed range restricted by MAP figures of the k max engine and motor, we introduce other limitations, namely, TMG ·ωMG ·ηMG ≤ Pbat and 20 % ≤ SOC ≤ 80%, to ensure the battery is working in the linear interval [11].

3 Controller Design Ride comfort experienced by passengers in a car is mainly related to the vibration acceleration transferred from the car body to the human body. Therefore, in the intended energy management strategy, the optimization target should include the following vibration control index besides total energy consumption JN V H =

 tf  2 t0 a1 ϕ¨ dt

(4)

where a1 is a weighting constant, while tf and t0 represent the beginning and ending time of test.

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Fig. 3 New energy management strategy algorithm

A practical and feasible method to combine energy and vibration targets is local optimal control. The main idea is as follows. Derive a solution according to energy, then relax the solution at the expense of energy economy, so as to find the condition for the least JNVH . Here, the priority of fuel consumption is higher than the priority of vibration control. The new energy management strategy can be described as shown in Fig. 3. As shown in the figure, the new power management system is achieved using several steps. In the first step, external conditions are given by driving demand and road conditions, including the demand torque Tr and speed ωr . An original switch torque Tsw0 is then derived according to a traditional energy management strategy. At the third step, loosen the requirement on energy consumption, set the largest energy sacrifice that could be accepted, and figure out the available switch range [Tsw0 + ΔT, Tsw0 − ΔT]. Finally, try every switch torque strategy by traverse algorithm, and find the solution which has the lowest JNVH .

4 Numerical Simulations and Analysis In order to find the mode-switch boundary for global driving, numerical simulations were made using Simulink and Carsim. With the built model of Prius’s planetary train and engine, the new energy management system described in Sect. 3 was applied to optimize the vibration of the vehicle. A set of switch points are recalculated according to the JNVH result, as demonstrated in Tables 2 and 3 and Fig. 4. In traditional energy management where fuel consumption is the only target, the switch torque is 59 Nm. When adding the influence of vibration to the strategy, the

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Table 2 Vehicle vibration under different switch torque (ωr = 50 rad/s) Tsw (Nm) JNVH

58 3.1

59 2.8

60 2.6

61 2.6

62 2.7

63 2.7

64 2.7

65 2.9

66 3.1

67 3.4

58 3.5

59 3.4

Table 3 Vehicle vibration under different switch torque (ωr = 150 rad/s) Tsw (Nm) JNVH

55 5.0

56 4.9

57 4.9

58 4.6

59 4.1

55 3.8

56 3.8

57 3.6

Fig. 4 (a, b) Results of JNVH under ωr = 50rad/s when Tsw = 58 and 60Nm

Fig. 5 (a, b) Carsim-Simulink results

switch torque is changed to 62 Nm. It is worth noting that at Tsw = 57 Nm, it is likely to attribute a possible system resonance. Generally speaking, the switch time is slightly delayed. With the sacrifice of energy consumption, the total vibration level (shown by JNVH ) can be reduced to 83.6% and 95% when required speed is 150 rad/s and 50 rad/s. Vehicle vibration and engine torque after simulation are presented in Fig. 5, which clearly demonstrate the benefit of the strategy with vibration control. From Fig. 5a, one can note that the vehicle without vibration control is superior in fuel economy and dynamic response. This is because the output torque is relatively higher, and the target speed can be reached faster. Fig. 5b on the other hand shows that no vibration control in energy management results in much bigger fluctuation with vehicle longitudinal acceleration. Combining both figures, one can attribute the

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effect of vibration control to the delaying of torque switch point. The observation is consistent with the results shown in Tables 2, 3, and Fig. 4.

5 Conclusions In this chapter, a novel energy management strategy is introduced, combining vibration control and energy consumption reduction. The main idea is a trade-off between engine energy management and vibration control. This multi-objective optimal problem was exemplified with a model of Toyota Prius 4. Simulation results reveal that the new strategy makes the torque switch occur later than the one without vibration control, and can clearly reduce vehicle vibration with sacrificing fuel economy to certain level.

References 1. Kim, J., Kang, J., Choi, W., Park, J., Byun, S., ..., Kim, H.: Control algorithm for a power split type hybrid electric vehicle. In SPEEDAM 2010, pp. 1575–1580. IEEE (2006) 2. Chen, Z., Mi, C.C.: An adaptive online energy management controller for power-split HEV based on dynamic programming and fuzzy logic. In 2009 IEEE Vehicle Power and Propulsion Conference, pp. 335–339. IEEE (2009) 3. Panday, A., Bansal, H.O.: A review of optimal energy management strategies for hybrid electric vehicle. Int. J. Vehic. Technol. 2014, (2014) 4. Doughty, S.: Transfer matrix eigensolutions for damped torsional systems. J. Vib. Acoust. Stress. Reliab. Des. 107(1), 128–132 (1985) 5. Wakabayashi, K., Honda, Y., Kodama, T., Shimoyamada, K., Iwamoto, S.: The effect of typical torsional viscous-friction damper on the reduction of vibrations in the three dimensional space of diesel engine shaftings. SAE Trans. 1852–1872 (1993) 6. Prokhorov, D.V.: Toyota Prius HEV neurocontrol and diagnostics. Neural Netw. 21(2–3), 458– 465 (2008) 7. Li, L., Wang, X.: Fuel consumption optimization for smart hybrid electric vehicle during a car-following process. Mech. Syst. Signal Process. 87, 17–29 (2017) 8. Hu, X., Murgovski, N., Johannesson, L., Egardt, B.: Energy efficiency analysis of a series plugin hybrid electric bus with different energy management strategies and battery sizes. Appl. Energy. 111, 1001–1009 (2013) 9. Sciarretta, A., Serrao, L., Dewangan, P. C., Tona, P., Bergshoeff, E. N. D., Bordons, C., . & Hubacher, M. A control benchmark on the energy management of a plug-in hybrid electric vehicle. Control. Eng. Pract., 29, 287–298 (2014) 10. Liu, J., Peng, H., Filipi, Z.: Modeling and analysis of the Toyota hybrid system. J. Perform. Constr. Facil. 23(6), 399–405 (2005) 11. Kim, N., Cha, S.: Optimal control of hybrid electric vehicles based on Pontryagin’s minimum principal. IEEE Trans. Veh. Technol. 19(5), 1279–1287 (2011)

Modelling and Vibration Characteristics Analysis of a Parallel Hydraulic Hybrid Vehicle Shilei Zhou, Paul Walker, and Nong Zhang

1 Introduction Hydraulic hybridization provides a promising solution to reduce the vehicles’ fuel consumption and emission in addition to the electrification and electric hybridization [1–3]. For these city-used medium- and heavy-duty vehicles such as the delivery vehicles and the refuse collection vehicles, the hydraulic driving system has more advantages due to its high power density and low cost. Parallel hydraulic hybrid vehicles (PHHV) could be refitted from the conventional rear driving vehicles by adding a hydraulic driving system on the driveshaft so that the design cost and system complexity are both reduced. In the PHHV, a dual functional swashplate hydraulic pump/motor (HPM) is added into the vehicle driveline via the vehicle driving shaft [4]. The HPM is used to recover the braking energy and reuse it for vehicle launching and driving, by which the vehicle fuel efficiency is improved [5]. Conventional vehicle powertrains are designed to improve noise, vibration and harshness (NVH) performance, especially under the engine excitation. With the addition of hydraulic driving system to the driveline, the vibration characteristics of the driveline change. Furthermore, the HPM also introduces excitations caused by fluid fluctuation [6]. The vibration characteristics of the PHHV powertrain should be studied to avoid powertrain damage and improve the vehicles’ driving comfortability [7]. In this chapter, a multibody dynamic model is built by using the lumped mass method. The natural frequencies and mode shapes of the PHHV are attained from the dynamic model and compared with a conventional vehicle.

S. Zhou () · P. Walker · N. Zhang University of Technology Sydney, Sydney, NSW, Australia e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_17

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2 Dynamic Modelling of the PHHV PHHV structure is shown in Fig. 1 [8]. Based on a conventional rear-wheel drive vehicle, the hydraulic driving system is installed on the driveshaft. The refit doesn’t change the original engine driveline. The engine power is transmitted to the wheels via the engine clutch, the automated manual transmission (AMT), the driveshaft and the halfshafts. In the hydraulic driving system, the accumulator is used to store high-pressure oil, and the reservoir is used to store the low-pressure oil. During vehicle driving, the HPM works as a motor, and the oil flows from the accumulator to the reservoir. The oil is pumped from the reservoir to the accumulator during regenerative braking. To protect the HPM from overspeeding operations, a HPM clutch is adopted to disengage the HPM from the driveline when HPM speed reaches its maximum operating speed. A multibody dynamic model when the engine clutch and the HPM clutch are engaged is established to study the vibration characteristics of the PHHV, as shown in Fig. 2. PHHV dynamic model is described by differential equations according to Newton’s second law. The differential equations are written in matrix form Jθ¨ + Kθ = 0

Valve

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J15

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K1 −K1 ⎢ − K1 K1 + K2 −K2 ⎢ ⎢ ⎢ ...... ⎢ ⎢ −K6 K6 + K7 −K7 ⎢ ⎢ ⎢ −K7 K7 + K8 −K8 ∗ i1 ⎢ ⎢ −K8 ∗ i1 ⎢ K=⎢ ⎢ −K9 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤

−K9 −K10 −K11/ i3

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K9 + K10 −K10 K10 + K11/ i3ˆ2 + K15 ∗ i2ˆ2 −K11/ i3 K11 + K12 −K12

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In the inertial matrix J, J1 toJ15 represent the inertias of the engine accessory, piston 1, piston 2, piston 3, piston 4, flywheel, clutch, transmission input shaft, transmission output shaft, driveshaft part 1, driveshaft part 2, differential driving gear, differential driven gear, halfshaft and HPM, respectively. In the stiffness matrix K, K1 to K15 represent the stiffnesses of the shaft of engine accessory, shaft between piston 1 and piston 2, shaft between piston 2 and piston 3, shaft between piston 3

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and piston 4, shaft between piston 4 and flywheel, clutch spring, transmission input shaft, transmission output shaft, driveshaft 1, driveshaft 2, shaft between driveshaft and reducer, shaft of differential gears, halfshaft, tyre and HPM clutch. i1 , i2 andi3 are the gear ratio of the AMT, HPM gear and the final gear.

3 Results and Analysis Through the eigenvalue method [9], the frequency and the mode shapes can be attained. The system matrix A is built by A = J−1 K

(5)

PHHV frequencies are listed in Table 1. Table 1 also gives the natural frequencies of the conventional vehicle which does not include the HPM. The natural frequencies and the mode shapes of the two vehicles are attained by the same eigenvalue method. From the results, the HPM brings a new natural frequency of 21.27 Hz to the powertrain and moves the natural frequencies of 30.84 Hz and 54.02 Hz to 33.97 Hz and 73.44 Hz. The higher-order natural frequencies are not changed. Mode shapes of the PHHV powertrain are compared with the mode shapes of the conventional vehicle powertrain, as shown in Fig. 3. The results show that with the same frequencies, the mode shapes are also similar except for there is one more degree of freedom which indicates the HPM vibration. The other powertrain components keep the same vibration mode as in the conventional vehicle. It’s worthy to notice that under the additional natural frequency of 21.27 Hz, the HPM has the maximum vibration among the powertrain. This natural frequency and its corresponding mode shape should be carefully considered to avoid excessive vibration of the HPM. Table 1 Natural frequencies of PHHV and conventional vehicle PHHV powertrain Order Frequency 1 0.41 2 2.08 3 21.27 4 33.97 5 73.44 6 106.19 7 141.91 8 254.08

Order 9 10 11 12 13 14 15

Frequency 268.94 290.9 760.43 1259.5 1409.6 1753 2094.6

Conventional vehicle powertrain Order Frequency Order 1 0.41 8 2 2.08 9 3 30.84 10 4 54.02 11 5 106.18 12 6 138.96 13 7 253.96 14

Frequency 268.94 290.9 760.43 1259.5 1409.6 1753 2094.6

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4 Conclusion In the PHHV, the additional HPM mainly changes the lower-order natural frequencies. The mode shapes of these natural frequencies are also slightly changed. The higher-order natural frequencies and mode shapes are unaffected. The HPM also introduces a new natural frequency and mode shape. The HPM has the biggest vibration under this additional natural frequency. Efforts should be made to design vibration reduction systems for the PHHV powertrain considering the new natural frequency and mode shape.

References 1. Zhang, Z., Chen, J., Wu, B.: The control strategy of optimal brake energy recovery for a parallel hydraulic hybrid vehicle. Proc. Inst. Mech. Eng. D J. Automob. Eng. 226(11), 1445–1453 (2012) 2. Vu, T.-V., Chen, C.-K., Hung, C.-W.: A model predictive control approach for fuel economy improvement of a series hydraulic hybrid vehicle. Energies. 7(11), 7017–7040 (2014) 3. Bender, F.A., Bosse, T., Sawodny, O.: An investigation on the fuel savings potential of hybrid hydraulic refuse collection vehicles. Waste Manag. 34(9), 1577–1583 (2014) 4. Ning, X., et al.: Optimization of energy recovery efficiency for parallel hydraulic hybrid power systems based on dynamic programming. Math. Probl. Eng. 2019, 1–11 (2019) 5. Wu, B., Lin, C.-C., Filipi, Z., Peng, H., Assanis, D.: Optimal power management for a hydraulic hybrid delivery truck. Veh. Syst. Dyn. 42(1–2), 23–40 (2004) 6. Huang, J., Yan, Z., Quan, L., Lan, Y., Gao, Y.: Very important characteristics of delivery pressure in the axial piston pump with combination of variable displacement and variable speed. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 229(7), 599–613 (2015) 7. Nguyen, T., Elahinia, M., Wang, S.: Hydraulic hybrid vehicle vibration isolation control with magnetorheological fluid mounts. Int. J. Veh. Des. 63(2–3), 199–222 (2013) 8. Zhou, S., Walker, P., Tian, Y., Zhang, N.: Parameter design of a parallel hydraulic hybrid vehicle driving system based on regenerative braking control strategy. In: SAE World Congress Experience, Detroit, MI, United States, 19–21 April 2019, no. 2019-01-0368 9. Tang, X., Jin, Y., Zhang, J., Zou, L., Yu, H.: Torsional vibration and acoustic noise analysis of a compound planetary power-split hybrid electric vehicle. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 228(1), 94–103 (2013)

Part III

Vibration and Control of Beams, Plates and Shells

Model Validation of a Vehicle Fuel Tank for Modal Analysis Shuyu Wang, Peibao Wu, Zhichao Hou, Xuehong Chen, and Shuai Wang

1 Introduction With increase in global vehicle production and use of lightweight materials, plastic fuel tanks gain momentum in the passenger vehicle market. The finite element method has been widely used to simulate plastic fuel tank performance. The vibration characteristics of the plastic fuel tank in vehicle are affected by many factors, such as the material properties, complex geometry or boundary conditions, fluid-structure interaction, and so on. Incorrectness and uncertainty with these factors will render discrepancy between simulation and test results. It is thus of great value to perform study to improve the model accuracy and to understand the impact of parameter uncertainty. Experimental data have been widely used to validate or correct simulation model for engineering structures [1]. As structure complexity increases, uncertainties in the structural system must be addressed. The concept of model verification and validation (V&V) was put forward in the 1950s. After several decades, the concept has gradually developed into an attractive approach for model checking and updating with a systematic consideration about possible uncertainty [2–4]. In an actual engineering structure, both aleatory and epistemic uncertainties can exist with key parameters. To incorporate and analyze the impact of parameter uncertainty, a finite element model can be used with the Monte Carlo method [5]. The combined approach is attractive due to its high accuracy, but its efficiency is

S. Wang · P. Wu · Z. Hou () School of Vehicle and Mobility, Tsinghua University, Beijing, China e-mail: [email protected] X. Chen · S. Wang YAPP Automotive Systems Co., Ltd, Yangzhou, China e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_18

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a big problem for a complex structure with huge degrees of freedom due to many times running of the finite element model. The hypothesis test and Bayesian factor are two commonly used methods for model validation. By assuming that the experimental results and model predictions have the same distribution related to parameter uncertainty, the hypothesis test mainly judges whether the test data rejects a simulation model. The Bayesian factor method, on the other hand, is usually used to judge whether the test data supports the simulation model [6]. Taking modal analysis on a plastic fuel tank as an example, a method for model validation is presented in this paper. A finite element model of the fuel tank was established first, where fluid-structure interaction is incorporated. Response surface models were then constructed in terms of main design parameters, by means of orthogonal experimental design and support vector machine. The Lilliefors test and a double-layer sampling were combined to figure out the distribution of the parameters with uncertainty. Instead of the finite element model, the constructed response surface models were used to achieve analysis on uncertainty transfer. The Bayesian factors were finally calculated and used for model validation.

2 Fuel Tank and Finite Element Model The plastic fuel tank under investigation is presented in Fig. 1. The fuel tank contains the tank body and some accessories. The thin-walled body of the tank is made of multiple layers of a high-density polyethylene (HDPE) material, and the main accessories are a fuel pump and two steel strips. The steel strips connect the lower part of the fuel tank to the vehicle chassis, and the tank is connected to the vehicle

Fig. 1 A plastic fuel tank

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Fig. 2 The fuel tank test

body at upper surface through two rubber cushions. Figure 2 shows the tank under modal test. After making necessary simplification with geometry and attachments, a finite element model was established for the tank. During the modeling, an equivalent wall thickness is used according to constant tank mass, and oil pump is described as an equivalent elastic shell. The fuel tank has a nominal volume of 50 L, while the maximum volume is 63 L. In order to understand the fluid-structure interaction of the fuel tank, various amounts of water will be filled into the tank. Using water instead of oil is to match physical tests according to related standards. The fluid domain is divided by tetrahedral elements. There exist three types of fluid surfaces, i.e., the surface between water and the tank wall, the surface between water and the pump wall, and the free surface. For the first two surfaces, the TIE constraint was defined to capture the relation between fluid nodes and structure elements on the walls. The free surface is described by setting zero sound pressure on the surface. Computational modal analyses were separately conducted on the fuel tank with various water fillings and under the installation status. Experimental modal analyses were performed correspondingly by means of hammer impact, as shown in Fig. 2. The experimental results are used to validate the finite element models once necessary.

3 Response Surface Model In order to achieve model validation with a balance between accuracy and efficiency, we chose to construct a response surface model for the eigenvalues of the fuel tank based on Latin square sampling.

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Table 1 Factors and levels of Latin square sampling Factors Young’s modulus (MPa) Tank thickness (mm) Density (g/cm3 ) Strip thickness (mm)

Level 1 800 5 0.8 1

Level 2 866.7 5.667 0.9 1.333

Level 3 933.3 6.333 1.0 1.667

Level 4 1000 7 1.1 2

Considering the material and manufacturing technology of a plastic fuel tank, we chose four design parameters as the factors, each with four levels, as listed in Table 1. The sample numbers in the training set and in the test set were chosen as 16 and 9, respectively. With the parameter values defined by each sampling, corresponding modal properties of the fuel tank were calculated using the finite element model. These calculated modal properties are then used to construct the response surface model. In view of small samples available and complexity of the target problem, the support vector machine approach [7] was adopted to construct the response surface model of the modal property of the fuel tank with respect to the four chosen design parameters. The natural frequencies predicted by the response surface model are in good agreement with those calculated based on the finite element model. The mean square error (MSE) and the R2 coefficients between the surrogate model and the original finite element model, of the first six modes related to both the training and test sets, clearly demonstrate the accuracy and generalization ability of the response surface model.

4 Model Validation In this section, we conduct model validation by checking uncertainty transfer from parameter to natural frequency. As a demonstration, only one design parameter, the Young’s modulus of the fuel tank material, is assumed to be of uncertainty. During the study, the response surface model is used instead of the finite element model to ensure a higher efficiency. The Bayesian factor is adopted as the uncertainty metric and used to judge the suitability of the finite element model for modal analysis.

4.1 Parameter and Its Distribution Without loss of generality, the Young’s modulus of the fuel tank material, HDPE, is assumed to be of uncertainty. Mechanical tests were conducted on the HDPE specimens for the Young’s modulus, and the results are given in Table 2. As the

Model Validation of a Vehicle Fuel Tank for Modal Analysis Table 2 Young’s modulus from experiment and the Lilliefors test

Specimen E (MPa)

1 889

149 2 747

3 863

4 770

5 849

Average 823

Standard deviation, 55.16; Lilliefors metric, 0.2598; metric threshold, 0.3431

specimen number is quite small, we performed the Lilliefors test [8] to check whether the Young’s modulus is a normal distribution. With a confidence level of 95%, the statistics about the test are also listed in Table 2. It is clear that the Lilliefors metric is smaller than the threshold. This indicates that the test data does not refuse a normal distribution. As the exact distribution parameters can’t be figured out from just five measured data, we settled to accept the Young’s modulus has a normal distribution but regard the mean value and the standard deviation vary within certain intervals. In the following subsection, the mean and the standard deviation are set by experience within [800, 846] MPa and [40, 60], respectively.

4.2 Model Validation in Terms of Bayesian Factor For model validation, we introduce two hypotheses: Hypothesis 1, notated as M1 : the established finite element model is a right model for modal analysis on the fuel tank. Hypothesis 2, notated as M2 : the established finite element model is not suitable for modal analysis on the fuel tank. For a concerned parameter with uncertainty y with a prior probability density function f (y), the Bayesian factor can be finally deduced as   f T |y 0 P (T |M1 : y = y0 ) B (y0 ) = = P (T |M2 : y = y0 ) m(T )

(1)

In the equation, P(T|M1 : y = y0 ) and P(T|M2 : y = y0 ) are, respectively, the posterior probability of assumptions M1 and M2 related to event T. y0 is a predefined  data of y, f (T|y0 ) is the posterior probability density of y at y0 , and m(T) = f (T|y)f (y)dy. With a physical variable that may have mixed uncertainty, a double-layer sampling technique is adopted to quantify model uncertainty. For a concerned input parameter which satisfies a normal distribution, the double-layer sampling is achieved by two layer circulations, to get exact samples and to analyze uncertainty transfer. In this study, corresponding to the Young’s modulus with uncertainty, the loop numbers of the outer and inner circulations were set by trial and errors to 10 and 50, respectively. As a result, a total of 500 Monte Carlo sampling were performed

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Fig. 3 Prior and posterior probability density of frequency Table 3 The Bayesian factors of the first six natural frequencies Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Average

0L 1.269 1.334 1.082 1.368 0.569 1.367 1.165

21 L 1.274 1.405 0.697 1.294 1.366 1.221 1.212

25 L 0.736 1.412 0.752 1.414 1.410 1.050 1.129

42 L 0.037 1.201 0.752 1.207 1.342 1.349 0.981

50 L 0.026 1.342 1.032 1.159 1.412 1.376 1.058

63 L 1.398 1.317 1.161 1.293 1.374 0.347 1.148

Average 0.792 1.335 0.913 1.289 1.245 1.118 1.116

using the constructed response surface model. For the first mode of the fuel tank with 21L water inside, the prior and posterior probability density of the frequency is presented in Fig. 3. For all the first six modes of the fuel tank with various water fillings, the obtained Bayesian factors are listed in Table 3. From Fig. 3 and Table 3, it can be noted that at model confirmation point of 25.15 Hz, the prior probability of the model is 0.3697 and the posterior probability of the model is 0.4711, and the Bayesian factor B = 1.274. This indicates that the model is suitable for modal analysis on the fuel tank. In other words, Hypothesis 1 holds. According to Table 3, one can find that among a total of 36 Bayesian factors, 28 were greater than 1.0, and the overall average of the Bayesian factor is 1.116. As a result, the established finite element model for the fuel tank can be roughly regarded as suitable for modal analysis on the tank. However, with mode 1, mode 3, and all modes with a water filling of 42 L, the average Bayesian factors are smaller than 1. Furthermore, with mode 1, the Bayesian factor is quite small with water fillings of 42 L or 50 L. It is thus necessity to improve the finite element model under these conditions.

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5 Summary A procedure is presented in this paper to validate finite element models of a plastic fuel tank with various water fillings for modal analysis. Fluid-structure interaction and actual installation status were incorporated in the finite element model. According to orthogonal experimental design and simulation results from the finite element models, response surface models were constructed by means of supporting vector machine. Assuming the Young’s modulus of the tank material is of uncertainty, a double-layer sampling and the Monte Carlo method were combined to calculate the probability density functions of the first six natural frequencies. The Bayesian factors were computed using the prior and posterior probability functions, so as to validate the concerned finite element models. It is observed that the established finite element model of the fuel tank is suitable for modal analysis.

References 1. Carri, A.D., Weekes, B., Maio, D.D., et al.: Extending modal testing technology for model validation of engineering structures with sparse nonlinearities: a first case study. Mech. Syst. Signal Proc. 84, 97–115 (2017) 2. Roy, C.J., Oberkampf, W.L.: A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput. Methods Appl. Mech. Eng. 200(25– 28), 2131–2144 (2011) 3. Sargent, G.R.: Verification and validation of simulation models. J. Simul. 7(1), 12–24 (2013) 4. Durst, P.J., Anderson, D.T., Bethel, C.L.: A historical review of the development of verification and validation theories for simulation models. Int. J. Model. Simul. Sci. Comput. 08(02), 191– 209 (2017) 5. Deng, J., Li, X., Gu, D.: Reliability analysis on pillar structure using a new Monte-Carlo finite element method. Chin. J. Rock Mech. Eng. 21(4), 459–465 (2002) 6. Tallman, A.E., Blumer, J.D., Wang, Y., et al.: Multiscale model validation based on generalized interval Bayes’ rule and its application in molecular dynamics simulation. In: ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014, V01AT02A042 (2014) 7. Hearst, M.A., Dumais, S.T., Osman, E., et al.: Support vector machines. IEEE Intell. Syst. 13(4), 18–28 (1998) 8. Lilliefors, H.W.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc. 62(318), 399–402 (1967)

Development of a Suspension Seat Using a Magneto-Spring and Free Play Damper Masahiro Mashino, Etsunori Fujita, Shigeyuki Kojima, Yumi Ogura, and Shigehiko Kaneko

1 Introduction Some passenger seats on trucks and earth-moving machines have a suspension mechanism. There are basically two types of the earth-moving machines, a crawler version or one equipped with tires, each having the required performance to suit their application. In general, the seat suspension unit for an earth-moving machine has a stroke of around 160 mm, and the damping force of the oil damper is set low to absorb whole-body vibration and any impact forces. At this time, the damping ratio is set at around 0.35. However, it is difficult to pass ISO 7096:2000 codes with this design specification [1]. Therefore, the damper characteristics and the spring characteristics are adjusted depending on each set of test conditions. From previous research, we developed spring characteristics and damping characteristics incorporating dead zones [2, 3]. In this report, the speed of bottoming and topping was decreased by a quadric crank chain mechanism. The optimal formula that makes use of the two dead zones was examined using the general solution of the previous research [4].

M. Mashino () · E. Fujita · S. Kojima · Y. Ogura Delta Kogyo Co., Ltd, Hiroshima, Japan e-mail: [email protected]; [email protected]; [email protected]; [email protected] S. Kaneko Major in Mechanical Engineering, Waseda University, Tokyo, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_19

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2 Experimental Method Figure 1 shows the suspension seat, with the seat and magneto-suspension having a quadric crank chain mechanism placed in series. The second illustration shows the 40 mm stroke on the seat and the magneto-suspension unit’s movement behavior. The seat has a rotational hinge mechanism, with a 40 mm stroke in the vertical plane and an 11 mm stroke front-to-rear. The magneto-suspension unit also has a rotational hinge mechanism and the same 40 mm vertical stroke, but the maximum front-rear movement is set at 8 mm. The parameters of the seat suspension and magneto-unit related to vibration characteristics are set so that these suspensions move in opposite directions. The target value of frictional force is set at 80 N from our preceding study. This target value is cancelled by frictional force caused by input waves. The subject is 171 cm tall and weighs 65 kg. The excitation wave was EM7, which meets the criteria laid out in ISO 7096:2000. This experiment was conducted after approval of the Research Ethics Review Board in Delta Kogyo Co., Ltd., and with informed consent of the subjects in compliance with the Declaration of Helsinki.

Fig. 1 Suspension seat

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3 Experimental Results Figure 2 shows a diagram of the vibration transmissibility of EM7 when the 65 kg subject is sitting on the seat. We calculated the seat effective amplitude transmissibility value (hereinafter referred to as SEAT value) according to ISO 7096:2000. The SEAT value was 0.57 – lower than the 0.6 reference value outlined in ISO 7096:2000. The resonance frequency was 0.8 Hz; the gain of the resonance peak was 1.09. The vibration transmissibility of the damping characteristics is also shown in Fig. 2. The gain of the resonance peak was 1.09, while the resonance frequency was 1.2 Hz. The gain in the resonance peak was lower than the reference value of 1.5 stated in ISO 7096. A 75 kg rubber weight was used as a loading mass for the damping test. The excitation wave of the damping test had a displacement amplitude of 16 mm. From these experiments, the vibration transfer characteristics to pass ISO 7096:2000 are becoming clear. In addition, the vibration transfer characteristics obtained by analysis are also described. Figure 3 shows the load-displacement characteristics of the magneto-suspension used for the experiment, while Fig. 4 shows a Lissajous diagram of the free play damper’s load-displacement.

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4 Discussion The analysis approach was based on previous research [2]. The analysis was carried out with the spring constant of the suspension unit set at 4500 N/m, the spring constant of the seat suspension at 3000 N/m, and the spring constant of the seat at 25,000 N/m, as shown in Fig. 1. We assumed and calculated that the seat and the suspension unit are arranged as series systems and that the seat suspension and the suspension unit are in parallel. In addition, the dead zone of the damper was used to provide an approximate anti-phase for the seat suspension and suspension unit on the resonance point. As a result, the analytical value and the experimental value have almost the same transfer function. Then, the spring constant of the vibration system was 800 N/m. The reason for the difference in transfer function is that the structural damping force exceeds the upward excitation force, and it functions as a rigid body until 0.6 Hz. We designed two types of structure by using the knowledge garnered from these experiments: One type is a structure in which a damper acts on the torsion bar installed in the upper frame body (Type A), while the other has a damper acting on the spring characteristics, featuring a dead zone, and installed in the lower frame body (Type B). In addition, we calculated the analytical values of the Type A structure, in which a damper acts on the torsion bar installed in the upper frame body. Figure 5a and b shows suspension units incorporating the two structures. Figure 6 shows the vibration transmissibility of EM7 on the two types of suspension unit and the analytical values of the Type A structure. This experiment was performed using the subject mentioned beforehand. In the Type A model, the damper acts on the spring structure in the upper frame body (with a dynamic spring constant of about 16,000 N/m), while the damper acts a small amount on the spring structure of lower frame body, having a dead zone. Therefore, the impact force of topping and bottoming is reduced by the damping ratio, which is created by the springing characteristics of the torsion bar and the damping characteristics of the damper. In the Type B model, vibration is damped in a low-frequency band by a damper that has a dead zone and spring characteristics. In the case of topping and bottoming, the damper reduces the velocity before reaching the stroke end. In addition, as the input increases, the stroke amount also increases. The dead zone is 15 mm. When there is an input under the spring, low-speed movement occurs in the dead zone. The spring starts swinging when the vibration energy input from under the spring overcomes the frictional force. The dynamic spring constant of the dead zone was about 4500 N/m when the amplitude was ±7.5 mm, and the friction force was 76–114 N. The swinging caused by the dynamic spring constant of 4500 N/m reduced the structural damping, which enabled phase control from the low-frequency band.

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Free play damper

Torsion bar

Torsion bar (a) Type A

Torsion bar Free play damper

Torsion bar (b) Type B Fig. 5 Newly designed suspension unit using these research results

5 Conclusions The vibration transfer characteristics in vertical vibration, which satisfy ISO 7096:2000 guidelines, were clarified experimentally, and the needed parameters became clear from this analysis.

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Fig. 6 Comparison of vibration transmissibility of two types of structure and analytical result: vertical

1. When two types of suspension mechanism were coupled, the spring constant of the suspension seat was reduced by phase control, and the resonance frequency could be reduced. 2. The spring constant calculated by analysis was 800 N/m, with a structural damping force of 76–114 N starting to swing from 0.6 Hz. In addition, apparent structural damping was approximately 0 N, and a non-linear suspension structure, in which the analytical values and the experimental values are almost the same, was achieved.

References 1. ISO7096: Earth-Moving Machinery—Laboratory Evaluation of Operator seat Vibration. International Organization for Standardization, Geneva (2000) 2. Fujita, E., Nakagawa, N., Ogura, Y., Kojima, S.: An experimental study for a nonlinear combination spring using a magnet-spring. Trans. Jpn. Soc. Mech. Eng. Ser. C. 66(645), 1445– 1452 (2000). (in Japanese) 3. Fujita, E., Nakagawa, N., Ogura, Y., Ohshimo, H., Sugimoto, E., Kojima, S.: Experimental research on a vibration isolator device utilizing a magnet-spring. J. Des. Eng. 36(2), 126–135 (2001). (in Japanese) 4. Mashino, M., Makita, S., Kuwano, Sugimoto, E., Ogura, Y., Fujita, E., Kaneko, S.: Development of Seat Suspension Mechanism USING Magneto-Spring and Free Play Damper, Japan Society for Design Engineering the Chugoku Branch, vol. 35, pp. 61–66 (2019). (in Japanese)

Virtual Sensing Application Cases Exploiting Various Degrees of Model Complexity Karl Janssens, Bart Forrier, Roberta Cumbo, Enrico Risaliti, Bram Cornelis, Tommaso Tamarozzi, and Wim Desmet

1 Introduction Several approaches exist to obtain insight in the performance of a product. A first approach is the analysis of measurement data gathered in a test campaign. The major advantage of this approach is that measurements reflect the performance of the real system in operating conditions. However, setting up a full measurement campaign can be costly and time-consuming, and not all locations of interest are always accessible. Alternatively, numerical simulations can be used to analyze the system. While simulations provide clear insight in the underlying physical processes and generate full-field information, they often only approximately describe the real behavior of the system. Virtual sensing is a new emerging approach which attempts to combine the best of both worlds by coupling numerical simulation models with easily attainable measurement data. This enables the model-based estimation or virtual measurement of physical quantities at any desired location and blends the real-world accuracy of measurements with the insights and flexibility of models. The fusion of measurement data and simulation models in an estimator framework creates tremendous application potential for the industry, both in the product design cycle and product usage phase.

K. Janssens () · R. Cumbo · B. Cornelis Siemens Industry Software NV, Leuven, Belgium e-mail: [email protected] B. Forrier · E. Risaliti · T. Tamarozzi Siemens Industry Software NV, Leuven, Belgium Katholieke Universiteit Leuven, Heverlee, Belgium W. Desmet Katholieke Universiteit Leuven, Heverlee, Belgium © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_20

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In this work, the wide scope and capabilities of virtual sensing are illustrated on three industrial setups targeting different applications and system (model) complexities: (1) An electromechanical powertrain: usage of a lumped parameter (1D) model (2) A twistbeam rear suspension: usage of a linear finite element (FE) model (3) A MacPherson suspension: usage of a non-linear, flexible multibody model

2 Case 1: Load Torque Estimation on an Electromechanical Powertrain Using a Multi-physical 1D Model In the first application example, a model-based estimation framework was developed for indirect load torque measurements on an electromechanical powertrain. Dynamic load torque characterization is instrumental for efficiency, noise and vibration, and durability analysis in a broad range of industrial applications. Figure 1 (top) shows the electromechanical powertrain test rig, consisting of two nominally identical 5.5 kW induction machines (IM) in a back-to-back mounting. The IM are connected mechanically via a double-cardan transmission and a torque sensor. Both IM are inverter-fed with a common DC bus, allowing four-quadrant operation. The modeled IM is speed-controlled and operates in motor

Fig. 1 (top) Electromechanical powertrain test rig, with indication of measured inputs and outputs. The unknown load torque TL is measured for validation. (bottom) Validation of the virtual load torque sensor (red, estimated; blue, measured) during a speed transient (4 s time segment, left plot) and in constant speed regime (0.5 s time segment, right plot)

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mode; the opposing IM is torque-controlled in generator mode. The double-cardan transmission is Z-configured, by offsetting the modeled IM laterally. This introduces non-linear torsional dynamics. To obtain an accurate virtual sensor for the load torque (TL ) during transient and dynamic operation, this non-linearity must be taken into account. The model-based estimator is an extended Kalman filter (EKF) in which a non-linear lumped-parameter (1D) electromechanical powertrain model yields the predictions of model states and sensor outputs. The correction step is based on the comparison of the actual output measurements with their predicted counterparts. The modeled powertrain includes a transient electrical IM model, coupled to a torsional dynamic mechanical model of the cardan transmission. The variables at the coupling interface are the rotor angle (θ M ) and speed (ωM ) and the electro-magnetic (EM) torque acting on the rotor of the modeled IM. The inputs of the electrical model are the (measured) stator phase voltages (vabc ) and the rotor angle (θ M ) and speed (ωM ) at the coupling interface. The outputs of the electrical model are the stator line currents (iabc ) and the EM torque. The intermediate shaft of the cardan transmission is modeled as a linear (torsional) spring-damper. A lumped inertia is placed on either end of the intermediate shaft, with an angle-dependent transmission ratio in between, corresponding to the nonlinear cardan-joint kinematics. This combination introduces not just a torsional resonance but also non-linear torsional dynamics at the second shaft order of rotation. The load torque TL is an unknown input to the electromechanical model. Its a priori prediction in the EKF is that of a zero-order hold (ZOH) behavior (TL remains constant). The uncertainty attributed to this input model should allow an accurate estimate after the EKF’s measurement-based correction. The selection of sensor data to be used in the EKF’s correction step was optimized based on an analysis of the linearized closed-loop estimator behavior. This bandwidth analysis considers the uncertainty values (covariance matrices) and the model of the EKF for several combinations of sensor types and locations. It yields the steady-state frequency response function (FRF) between the actual load torque TL and the virtual sensor output, i.e., the EKF estimate of TL , for each set of sensors considered. This FRF would ideally be the unity value over the whole frequency bandwidth; however in practice a low-pass effect is observed. Thus, different sensor sets can be evaluated and compared easily based on their estimator bandwidths. The optimized instrumentation set of Fig. 1 (top), with two acceleration measurements (α M and α L ), leads to the highest bandwidth of the virtual torque sensor. It is also most robust against variations in uncertainty attributed to the ZOH input model. The resulting virtual sensor for load torque was validated against the torque signal obtained with an accurate torque flange in transient and steady-state conditions. Figure 1 (bottom) shows a validation result for transient operation and constant speed regime and confirms the accuracy of the virtual torque sensor. Multi-sine

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excitation in steady-state conditions has shown that the virtual sensor has a useful bandwidth of nearly 200 Hz, i.e., far beyond the torsional resonance frequency of 78 Hz and well above the frequency range of the second shaft order. More details are presented in [1].

3 Case 2: Load and Full-Field Strain Estimation on a Twistbeam Rear Suspension Using a Linear FE Model In the second application example, the virtual sensing approach was extended to flexible systems. A twistbeam rear suspension demonstrator was developed, aiming the model-based estimation of the load inputs and the resulting strain field of the structure from a reduced set of sensors combined with a linear FE model. The physical setup is shown in Fig. 2 (top left). The twistbeam is mounted on a rigid frame by clamped bushings. The right tire is in free conditions and excited by a 6 DoF hydraulic shaker which allows a full control on the desired excitation spectrum up to 200 Hz in six directions independently. The left tire is clamped. Thirty-eight strain gauges are installed over the structure, some of which are used as

Fig. 2 (top left) Twistbeam rear suspension test rig. (bottom) 6 DoF load input estimations and (top right) augmented strain field estimations from a reduced set of strain sensors (random vs. optimal configuration after OSP) in a test with 1 Hz sine excitation

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input for the estimations, while others are used for validation. A load cell is installed between the right tire and twistbeam to validate the model-based estimations of the input forces and moments. A twistbeam FE model was developed and correlated against measurements in free-free conditions. The clamped bushings were characterized from operational measurements under dynamic excitation. Optimal matching between strain sensors (location and orientation) and FE model (nodes) was obtained from static tests under pre-load. Operational tests were carried out under sine and pseudo-random excitations, covering a frequency range up to 50 Hz. A coupled state-input estimation was performed to estimate the 6 DoF load inputs and resulting strain field of the twistbeam. The estimations were done from a reduced set of 12 strain sensors. Two sets of strain sensors were evaluated: (i) a random set and (ii) an optimal set selected by an OSP (optimal sensor placement) strategy ensuring a maximum observability of the loads. Figure 2 (bottom) shows the 6 DoF load input estimations for a 1 Hz sine excitation compared to measured data for estimation cases with random and optimal set of strain sensors. Figure 2 (top right) validates the full-field strain estimations at the 38 instrumented strain sensors against measurements in the test with 1 Hz sine excitation. The OSP strategy results in more accurate estimations of loads and responses. More details are presented in [2].

4 Case 3: Wheel Center Load Estimation on a MacPherson Suspension Using a Non-linear, Flexible Multibody Model The dynamic behavior of vehicle suspension systems becomes even more complex in durability applications when the suspension shows a non-linear response to large road input displacements. This high level of complexity was tackled in a third application example, targeting a model-based estimation of wheel center loads (WCLs) in durability tests. The measurement of WCLs is the key to assess durability performance. In current practice, OEMs perform direct measurements of the WCLs during so-called Road Load Data Acquisition (RLDA) testing campaigns in which the prototype vehicle is heavily instrumented and then driven on proving grounds or public roads. Besides strain gauges, string pots, and accelerometers, the instrumentation includes wheel force transducers (WFTs) which enable an accurate direct measurement of WCLs. However, due to customization in combination with reduced time to market, an increasing number of vehicles must be tested at the same time, while the test equipment available is typically limited, especially for what concerns the WFTs. Hence, an approach which allows to indirectly measure the WCLs in situations where WFTs are not available is of great practical value. A state estimation framework was developed to perform the information fusion between measurements coming from a reduced instrumentation and a non-linear,

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Fig. 3 (top left) MacPherson suspension test rig. (top right) Flexible multibody model of the suspension. (bottom) Estimated vs measured WCLs under road excitation

flexible multibody model of the vehicle suspension system. Such type of system simulation model is needed to accurately reconstruct the time domain signal for all six WCLs. The reduced instrumentation set includes strain gauges on the suspension components. The framework allows to generate virtual sensors for WCLs and for the full strain field on the suspension components. The virtual sensors developed are validated on a MacPherson suspension test rig. Figure 3 (top left) shows the setup. The MacPherson suspension, which is composed of a control arm, a tie rod, a lower strut, and an upper strut all connected to a steering knuckle, is attached to a fixed frame on the vehicle body side and loaded by a hydraulic shaker capable of reproducing operational road loads at the tire patch. Figure 3 (top right) shows instead the flexible multibody model of the suspension where the wheel center (WC) is the input location where the WCLs are applied. Over 30 strain gauges are placed on the control arm and on the knuckle, some of which are used as input to the estimation procedure, while the rest are used for validation. A load cell is installed between the knuckle and the tire rim to measure the WCLs for validation.

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The estimation scheme developed is based on an extended Kalman filter (EKF), which is able to cope with the non-linear model employed. Thanks to a OSP strategy, six optimal sensors were chosen among the ones installed on the structure. The EKFbased procedure combines the flexible multibody model with the six strain gauges data to generate virtual WCLs and full strain field measurements. Figure 3 (bottom) shows the 6 DoF WCL estimation results for a road reproduction case. The comparison between the estimated signals and the ones measured with the load cell shows the high degree of matching achieved. More details are presented in [3].

5 Conclusions The capabilities of virtual sensing were successfully demonstrated on industrial setups. Three demonstrators were presented targeting different applications and model complexities, ranging from 1D models to linear FE and complex, flexible multibody models. Good estimation results were obtained in the three use cases. As a general conclusion, one can state that virtual sensing requires a fairly accurate model which properly describes the dynamics of the system. The accuracy requirements are high when inputs have to be estimated, whereas the model accuracy can be relaxed if responses are to be estimated. Optimal sensor placement significantly improves the accuracy of the results.

References 1. Forrier, B., Naets, F., Desmet, W.: Broadband load torque estimation in mechatronic powertrains using nonlinear kalman filtering. IEEE Trans. Ind. Electron. 65(3), 2378–2387 (2018) 2. Cumbo, R., Tamarozzi, T., Janssens, K., Desmet, W.: Kalman-based load identification and full-field estimation analysis on industrial test case. Mech. Syst. Signal Process. 117, 771–785 (2019) 3. Risaliti, E., Tamarozzi, T., Vermaut, M., Cornelis, B., Desmet, W.: Multibody model based estimation of multiple loads and strain field on a vehicle suspension system. Mech. Syst. Signal Process. 123, 1–25 (2019)

Efficiency Analysis of a Dual-Motor Electric Vehicle Powertrain Bing Wang, Jinglai Wu, Xianqian Hong, Nong Zhang, and Daisheng Zhang

1 Introduction Electric vehicle (EV) powertrain design and optimization have played an important role in improving the overall efficiency. The single-gear reducer occupies a major market share due to its simple structure and low cost. In order to improve the efficiency of an electric driving system, a single motor with two-speed transmission system has been proposed, and two-speed dual-clutch transmission (DCT)- and continuously variable transmission (CVT)-based EVs were compared in reference [1]. As known, traditional automated manual transmission (AMT) gains more popularity because of its simple structure and high efficiency. But it always has the problem of torque interruption during gear shifting process because of the disengagement of synchronizer. A two-speed AMT was presented in reference [2] which was called as inverse automated manual transmission (I-AMT), since the clutch was located at the rear of the transmission so that the traction interruption could be avoided. Compared with the single-speed transmission, the two-speed IAMT shows greater performance in terms of power and economy [3].

B. Wang · J. Wu () · X. Hong · D. Zhang School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, Anhui, China e-mail: [email protected]; [email protected]; [email protected]; [email protected] N. Zhang School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, Anhui, China School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_21

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Though single-motor input powertrains have been used and investigated widely, dual-motor input powertrains were implemented to eliminate the power interruption during gear shifting. When two downsized motors are applied as a replacement for the original large power single motor, the torque utilization factor of the motor driving can be increased, and potentially the operating efficiency can be improved [4]. To improve the energy efficiency and dynamic performance of EVs, a novel dualmotor input with three-speed powertrain is proposed. Corresponding gear ratios are designed to meet the dynamic performance requirement, and the most efficient driving mode will be selected by the energy management strategy to reduce the energy loss. Three speeds and five driving modes can be realized by integrated control of the two motors and the synchronizer position, and the energy efficiency of EVs can be improved by optimizing the speed ratio and mode.

2 Layout and Work Mode 2.1 Layout of Powertrains This part provides two layouts of EV powertrains, including the widely used and studied single motor with two-speed transmission (SM2ST) which has two work modes and a novel dual motor with three-speed transmission (DM3ST) seen in Figs. 1 and 2. DM3ST has the similar structure with SM2ST except for its two power inputs. In order to better evaluate the economy performance of the two transmission systems, the summed power of EM1 and EM2 in DM3ST is set to be the same as EM0 in SM2ST, and the power distribution ratio between the two motors is noted by Fig. 1 Layout of powertrain with SM2ST

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Fig. 2 Layout of powertrain with DM3ST

Fig. 3 Optimal work mode of DM3ST

Table 1 Parameters of vehicle and powertrain

mv 1379 kg

A 2.2m2

R 0.3 m

CD 0.25

f 0.011

the symbol β, Pm1 = βP0, Pm2 = (1 − β)P0 ; we set the power distribution ratio as β = 0.6 with maximized overall efficiency among the three values:β = 0.4,0.6,0.8. The efficiency maps of EM1 and EM2 have the same shapes of EM0 seen in Fig. 3.

2.2 Work Mode Analysis The two powertrains share the same vehicle parameters in Table 1; the load of cruising vehicle is expressed by the following equation:

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Table 2 DM3ST work modes



Work mode Mode1 Sub-mode1_1 Mode2 Sub-mode1_2 – Mode3 Sub-mode3_1 Sub-mode3_2

SY sleeve Left Left Neutral Right Right

EM1 On On Off On On

Freq = mv gfcosα + mv gsinα + CD Av 2 /21.15 Treq (α, v) = Freq R

EM2 Off On On Off On

(1)

where the Treq (a, v) is the vehicle load with variates of incline angle of the road α and vehicle speed v, Freq is the resistance, and the values of these parameters are shown in Table 1. There are two work modes of SM2ST: Mode 1 has a larger gear ratio of i1 i0 , and Mode 2 works in high-speed driving phase with a smaller ratio of i2 i0 . The two work modes are analyzed in the following equations: 

Ir ω˙ + Im0 ωm0 ˙ ik if = Tm0 ik i0 − ωm0 = ik i0 ωr



Treq

, k = 1, 2

(2)

 where Ir is the total wheel moment of inertia; Im0 is the EM0 rational inertia; ωr , ωm0 denote the speed of wheel and EM0, respectively; and Tm0 is EM0 torque. Regarding DM3ST, we define the three sets of gear ratios as ia , ib , if . DM3ST can realize three gear ratios according to change the positions of synchronizer sleeve, i.e., iEM1 _ 1 = ia ib if , iEM1 _ 2 = ib if , iEM1 _ 3 = if ; three-mode ratio is the output ratio of EM1. The output ratio of EM2 is fixed iEM2 = ib if ; the combination of different synchronizer positions and motor states can achieve five modes shown in Table 2. When EM1 is off, Tm1 = 0; when EM2 is off, Tm2 = 0; when EM1 and EM2 are both on, the total output torque is equal to the torque coupling of Tm1 iEM _ x and Tm2 iEM2 , x = 1, 2 or 3; the dynamic equation is expressed as follows: 

Ir ω˙ r + Im1 ω˙ m1 iEM _x + Im2 ω˙ m2 iEM2 = Tm1 iEM _x + Tm2 iEM2 − ωm1 = iEM1_x ωr , ωm2 = iEM2 ωr



Treq (3)

2.3 Powertrain Parameter Matching The gear ratios should be constrained to meet three dynamic requirements which are the maximum grade ability of 40% at 10 km/h, vehicle top speed of 150 km/h, and vehicle speed achieving 100 km/h after 10 s acceleration. The basic principle of the

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constraints is the maximum output torque must be higher than the required torque; the constraints of maximum gradeability and maximum speed should be satisfied as follows: iEM1_1 Tm1_max (ωm1 ) + iEM2 Tm2_max (ωm2 ) ≥ Treq (arctan(0.4), v1 )

(4)

iEM1_3 Tm1_max (ωm1 ) + iEM2 Tm2_max (ωm2 ) ≥ Treq (arctan(0), v2 )

(5)

Solving Eqs. (4) and (5), the gear ratios of SM2ST can be selected as i1 , i2 , if as 2.67, 1.33, and 3 which are the same parameters as reference [4] with good performance. In terms of DM3ST, we choose the gear ratio ia, ib , and i0 as 1.7, 1.8, and 3.5. Speed at 10 s is 116 km/h and 118 km/h, respectively, which is over 100 km/h, so the acceleration time is satisfied:  v10 =

10 

 iEM1_x Tm1_max + iEM2 Tm2_max − Treq /mv R )) dt, v0 = 0

(6)

0

3 Work Mode Optimization The overall efficiency is optimized which serves as the basis for an energy management strategy (EMS). The efficiency optimization target function is expressed as max ηsys , where ηsys = Pout /Pin . Both powertrains share the same output power expressed as Pout = Freq v/1000. The input power is as follows: Pin_SM2ST = ωmo Tmo + Pm0_loss

(7)

Pin_DM3ST = ωm1 Tm1 + Pm1_loss + ωm2 Tm2 + Pm1_loss

(8)

where Pm1 _ loss and Pm2 _ loss are the power loss of EM1 and EM2. Motor speed ω and torque T are valid within the maximum. The overall efficiency is mainly affected by vehicle speed and acceleration, while the vehicle dimensions are constrained. The optimized work modes of DM3ST and SM2ST are shown in Figs. 3 and 4 with different color faces. In the single-motor driving modes, the overall efficiency is fixed, and work modes are fully determined by the target speed and target acceleration. However, the speed of the two motors are proportional, and torques can be combined discretionarily within the constraints in the torque coupling modes; the essence of efficiency optimization is to find the optimal EM1 torque and EM2 torque to enhance the overall efficiency ηsys .

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Fig. 4 Optimal work mode of SM2ST

Fig. 5 EM0, EM1, and EM2 torque in NEDC

4 Simulation Results This section compares the efficiency of the two powertrains under two different driving cycles, namely, NEDC and HWFET. A facing-backward transmission simulation model was developed by Simulink. Figure 5 shows the speed and torque of motors during NEDC, SM2ST works in the Mode1 when the vehicle accelerates and decelerates, while it works in Mode2 when the torque demand is low. DM3ST mostly works in single-motor driving mode; when the demand power is high, DM3ST mostly works in dual-motor driving mode. Figure 6 shows the speed and torque of motors during HWFET.

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Fig. 6 EM0, EM1, and EM2 torque in HWFET

Fig. 7 Overall efficiency in NEDC

Figures 7 and 8 show the efficiency of the two powertrains with the time duration points of the NEDC and HWFET driving cycles. It proved that the overall efficiency of DM3ST is increased by 12.7% over SM2ST under NEDC and increased by 7.6% under HWFET. We can see that DM3ST enables better efficiency than SM2ST and its benefit is summarized in Table 3.

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Fig. 8 Overall efficiency in HWFET Table 3 Efficiency comparison in different drive cycles

ηSM2ST ηDM3ST Gain(%)

NEDC 75.51 85.13 ↑12.7

HWFET 84.31 90.79 ↑7.6

5 Conclusion This paper proposes a novel powertrain DM3ST and a traditional one SM2ST for comparison. Motors and gear ratio are selected by dynamic performance constraints. The energy management strategies (EMS) for gear and mode selecting are also investigated which are based on the optimization model by maximizing the overall efficiency, the economy performance is analyzed through the simulation of two driving cycles, and the results indicated that the DM3ST gains higher efficiency than SM2ST. No power interruption shift strategy will be continued in subsequent studies. Acknowledgments This research is supported by the National Key Research and Development Program of China (No. 2018YFB0105505-03).

References 1. Ruan, J., Walker, P.D., Watterson, P.A., Zhang, N.: The dynamic performance and economic benefit of a blended braking system in a multi-speed battery electric vehicle. Appl. Energy. 183, 1240–1258 (2016) 2. Gao, B., et al.: Gear ratio optimization and shift control of 2-speed I-AMT in electric vehicle. Mech. Syst. Signal Process. 50–51, 615–631 (2015) 3. Hu, M., Zeng, J., Xu, S., et al.: Efficiency study of a dual-motor coupling EV powertrain. IEEE Trans. Veh. Technol. 64(6), 2252–2260 (2015) 4. Jinglai, W., Jiejunyi, L., Jiageng, R., et al.: Efficiency comparison of electric vehicles powertrains with dual motor and single motor input. Mech. Mach. Theory. 128, 569–585 (2018)

Part IV

Active and Passive Vibration Control

Numerical Analysis of Dynamic Hysteresis in Tape Springs for Space Applications Richard Martin, Merten Stender, and Sebastian Oberst

1 Introduction Small satellites like CubeSats have become increasingly popular since the turn of the millennium [1]. They are often equipped with self-deployable appendages. For selfdeploying antennas, metal tape springs – the same as carpenter tape measures – are a very convenient design choice. They are lightweight, can fit in very confined spaces, and unfold easily by using stored strain energy. Their disadvantage, however, is that with increasing length external influences, like residual drag from the atmosphere, solar radiation or deliberate changes of the attitude of the satellite easily excite vibrations [2]. Despite the immanent issues of unwanted vibrations, not much attention has been paid to their principal understanding. Thin-walled structures like those used for space applications often show complex dynamics and require more sophisticated approaches for investigation. Recent studies by Oberst et al. [3] have investigated the complex dynamics of tape spring antennas. Static bending tests and dynamic experiments have been conducted, and model updating techniques were used to

R. Martin () Centre for Audio, Acoustics and Vibration, University of Technology Sydney, Ultimo, NSW, Australia Dynamics Group, Hamburg University of Technology, Hamburg, Germany e-mail: [email protected]; [email protected] M. Stender Dynamics Group, Hamburg University of Technology, Hamburg, Germany e-mail: [email protected] S. Oberst Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney, Sydney, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_22

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obtain an appropriate finite element model to be studied as dynamical system. In the experiments a hysteretic behavior of the system response for an amplitude sweep of the excitation was observed [2–4]. In the following, this hysteretic behavior is recreated using an equivalent finite element model of a metal tape spring which is excited near the clamping point by a stepped sweep of a harmonic concentrated force [3].

2 Model Setup Following the experiments by Oberst and Tuttle, a vertically hung up tape spring of length L = 500 mm is considered [3, 4]. The tape spring is modeled by extruding a section of a circle, neglecting the secondary curvature of a real-world tape spring [4] (Fig. 1). A variation in thickness was not considered (even though the effect on mode shape fluctuations and instability predictions has shown to be large [3, 4]). The upper end is clamped and the point of excitation remains consistently at LF = 10 mm. However, to excite torsional modes as they were observed by Oberst and Tuttle, the concentrated load acts eccentric at one quarter of the arc length of the cross section (Fx ) instead of at the center (Fo ). For real-world tape springs of a certain length, torsional modes will always occur due to the low torsional stiffness and the sensitivity to small perturbations of the system. The torsional modes are emphasized by introducing a twist of 1° across the lengthwise direction (z) of the tape. The model is pretensioned in z-direction by a gravitational load of 1 g. The material model is linear elastic, using Rayleigh damping with experimentally determined parameters [4]. Displacements u2 , velocities v2 , and accelerations a2 in y-direction are measured at the centerline at a point Lm = 100 mm from the clamped end (see also Table 1 and Fig. 1). To model the tape spring antenna, the FEM software Abaqus was used to find an appropriate element size and type, a model for a static bending of a tape spring, similar to Seffen and Pellegrino [5], was set up. The bending moment was set close to the buckling point of opposite-sense bending. The results for the resulting displacement are depicted for different element types and numbers of elements in Fig. 2. Because of its relatively quick convergence and acceptable computation time,

Table 1 Material and geometry parameters, following Oberst et al. [3, 4] Material property Density Young’s modulus Poisson’s ratio Damping parameters

Value  = 7815.42 mkg3 E = 202.34 GPa ν = 0.254 αR = 0.01473 s−1 βR = 2.12 · 10−5 s

Geometry property Width Height Thickness Overall length Point of measurement Point of excitation

Value w = 17.5 mm h = 2.75 mm t = 0.11 mm L = 500 mm Lm = 100 mm LF = 10 mm

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Fig. 1 (a) Cross section of the tape spring, (b) geometry and boundary conditions, following Oberst et al. [3, 4] (for experimental setup cf. Fig. 5 in [3])

Fig. 2 Mesh convergence for a static bending test, following Seffen and Pellegrino [5], with the resulting rotational displacements θmax of the ends for different numbers of elements in lengthwise direction nL and in crosswise direction nR

the linear S4 element was chosen for the dynamical model. Here, the mesh of the dynamical model consists of 2,250 linear general shell elements, resulting in an approximate element length of 2 mm. For the integration in thickness direction, the Simpsons integration rule with 15 points is used, cf. [4] which used reduced quadratic eight-node shell elements (S8R) instead. After the static pretensioning step, an implicit time integration step is executed, using the Hilber-Hughes-Taylor implementation of Abaqus. The sampling rate of the output is set to fs = 5 kHz for the step response (Fig. 3) and fs = 1 kHz for the investigation of the hysteresis for amplitude sweeps (Fig. 4).

3 Model Validation The model is compared to experiments conducted by Oberst and Tuttle [3] in the linear regime for small excitations. The power spectral density estimates of the forced response are obtained by simulating the response to a step of 1 N in opposite-

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Fig. 3 Step responses for different configurations of the model, compared to the experimental data (shifted by −40 dB) by Oberst (Fig. 6 in [3]). The modes 1, 2, 3, 4, 7, and 9, which are determined using the Lanczos implementation of Abaqus, are marked with arrows

Fig. 4 Amplitude sweep of the exciting force F and system response, measured by the velocities v at Lm = 100 mm. The examples A, B, C, and D are marked with gray bars

sense bending direction in time domain for T = 5 s and applying a FFT to the obtained time series v2 (t). The results are depicted in Fig. 3. With regard to the spectrum, mode shapes, and frequencies, we found a good qualitative agreement of the experimental data and the simulations. The step response of the FE model is slightly stretched in frequency direction. One reason could be that for the experiments, a tape spring coated with paint was used which mass loaded the structure [4]. Astonishingly, we encountered large difference of about 40 dB. These differences are likely attributed to a scaling/amplification factor being wrongly chosen in experiments or finite elements. Also other error sources might have contributed such as numerical and modelling issues including the different boundary conditions (clamping point, connection to exciter) and different settings in the PSD especially using Welch spectrum in experiment just over the whole time series length and windowed PSD calculations, disregarding nonuniform thickness and the use of

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a step input instead of a random burst. Further, experimental uncertainties and deviations in scaling factors of sensors or amplifiers cannot be ruled out. To fully validate the results presented here, it is required to repeat some of the experiments conducted in [3] which is planned for the near future. Nevertheless, qualitative agreement could be achieved which is a promising result of simulating the tape spring using finite elements.

4 Amplitude Sweeps Oberst and Tuttle observed a hysteresis in the velocity response for amplitude sweeps of a harmonic exciting force of f = 15.5 Hz for a L = 500 mm long tape spring antenna [3]. This phenomenon is reproduced in this study using the FE model. The excitation frequency here is f = 70 Hz; the amplitude is swept from A = 0.5 N to A = 4 N in steps of ΔA = 0.5 N and back to A = 0.5 N with the same step size. Each step is integrated for T = 10 s because it is not known a priori how long the system would take to reach a potential steady state. The last step is T = 15 s long to give the system more time to decay, though the system is too weakly damped to settle in a steady state even for this prolonged integration time. For small increasing (↑) excitation amplitudes A < 3 ↑ N, the system settles in a steady state. For A = 3 ↑ N there is a very slow growth visible in u and v. This growth is likely caused by a Hopf bifurcation in the system, i.e., a stable fixed point became unstable. This increase accelerates with A = 3.5 ↑ N, and the system jumps at t ≈ 68 s to a chaotic orbit. With the decrease of the amplitude, the system undergoes several bifurcations. Regular vibrations can be observed for decreasing (↓) excitation amplitudes A = 3.5 ↓ N. The system becomes more irregular again for A = 3 ↓ N. For A = 1 . . . 0.5 ↓ N the system settles in a harmonic vibration with a main frequency of f ≈ 9.5 Hz, which corresponds to the first bending mode.

Fig. 5 Root-mean-square values of u and v during the amplitude sweep. The examples A and C are marked at the respective points. B and D correspond to transient states and are therefore marked between the points. The (u, v) phase spaces are displayed corresponding to A, B, C, and D

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The hysteresis can be visualized more clearly by calculating the root-meansquare (rms) values of the displacements u and velocities v for the last two seconds of each step. The results are displayed in Fig. 5 for every step A of the amplitude sweep. A linear relation between the excitation amplitude and the systems response is visible for 0.5 ↑ N ≤ A ≤ 2.5 ↑ N. The length of the time integration is however too short for the system to reach a steady state again at A = 0.5 ↓ N. Therefore, it can be observed that the hysteresis does not form a closed loop.

5 Conclusion The finite element model is suitable to model the dynamical system of a thinwalled tape spring antenna. A hysteresis in the system response for an amplitude sweep, as it was experimentally observed by Oberst and Tuttle [3], could be replicated. However, whether the same qualitative behavior in equal- and oppositesense bending direction as observed in the experiments can also be replicated using the finite element model still needs to be validated. The large difference in the power spectral density estimate of the system could be addressed by conducting a further model refinement and by running validating experiments. The model can be used to conduct further studies, e.g., to investigate the influence of geometry and load parameters and to detect bifurcations in the system. By applying model order reduction and system identification techniques on time series generated by the model, a numerically efficient reduced model could be derived, which would expedite further studies. Here it is important to generate the same attractor, i.e., the same invariant set, so that the model’s dynamics are preserved relative to the experimental dynamics. Robust invariant measures such as experimentally validated attractor templates could provide an unambiguous quality criterion [6].

References 1. Swartwout, M.: The first one hundred cubesats: a statistical look. J. Small Satell. 2(2), 213–233 (2013) 2. Oberst, S., Griffin, D., Tuttle, S., Lambert, A., Boyce, R.R.: Analysis of thin curved flexible structures for space applications. In: Proceedings of Acoustics 15–18 Nov 2015, Hunter Valley (2015) 3. Oberst, S., Tuttle, S.: Nonlinear dynamics of thin-walled elastic structures for applications in space. Mech. Syst. Signal Process. 110, 469–484 (2018) 4. Oberst, S., Tuttle, S.L., Griffin, D., Lambert, A., Boyce, R.R.: Experimental validation of tape springs to be used as thin-walled space structures. J. Sound Vib. 419, 558–570 (2018) 5. Seffen, K.A., Pellegrino, S.: Deployment dynamics of tape springs. Proc. R. Soc. A Math. Phys. Eng. Sci. 455(1983), 1003–1048 (1999) 6. Gilmore, R., Lefranc, M.: Topology Analysis of Chaos. Wiley VCH Verlagsges, Weinheim, Germany (2002). https://doi.org/10.1002/9783527617319

Adaptive Control of a String-Plate Coupled System Marin Matsuyama, Hiroyuki Iwamoto, Shotaro Hisano, and Nobuo Tanaka

1 Introduction There are various systems in which different structures are combined. One of them is a coupled system of strings and plates. A typical example is a stringed instrument. Acoustic stringed instruments are difficult for the player to suppress the volume. Hence, the sound from the instrument may be a noise to nearby residents in a housing complex. In order to overcome this problem, some approaches are attempted. For example, Benacchio et al. [1] proposed to alter the sound radiated from the guitar by actively changing the modal parameter of the structure. Miyake [2] presented formulation of the string-plate coupled system that mimics the stringed instruments and investigated the effects of the direct velocity feedback (DVFB) applied to the plate structure. As a result, it was clarified that the control effect of the DVFB is relatively low for the string-dominated vibration mode. Therefore, in this study, the purpose is to suppress the sound radiation of the system such as a stringed instrument. For simplicity, that kind of a system is regarded as a coupled system of the string and plate with a spring connection. The string is considered as a simply supported beam receiving the axial force, and the plate is assumed to be simply supported. Therefore, the target system is linear. First, the modal equation of the coupled system is derived based on the equations of motion of each subsystem. Secondly, a feedforward control law for suppressing the displacement at arbitrary position of the plate structure is derived. This is followed by the numerical analysis of the displacement control system. The control effects are evaluated by displacement distributions and acoustic radiation power. It is clarified that if control actuators are placed around the spring connection, sufficient control effect is

M. Matsuyama () · H. Iwamoto · S. Hisano · N. Tanaka Department of System Design, Seikei University, Tokyo, Japan e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_23

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obtained. Thirdly, in order to simulate more practical case, usage of an adaptive filter is discussed. The numerical analysis on adaptive-filter-based approach is carried out in the time domain under the condition where the disturbance is multitone, clarifying that the proposed method reduces the plate-dominated mode as well as the stringdominated mode.

2 Ideal Displacement Control System 2.1 Derivation of Control Law First, in order to derive a control law, a mode equation of a coupled system of a string and plate is determined. From the equations of motion of the string and plate, the equations of motion of the system in which the string and plate are coupled are obtained. When W1 is a displacement of a string and W2 is that of a plate, the equations of motion when a string and a plate are coupled are expressed by the following equations: EI

∂4 ∂2 ∂2 W W W1 (x, t) t) − ρA t) + T (x, (x, 1 1 ∂x 4 ∂t 2 ∂x 2

(1)

= −k {W1 (xa , t) − W2 (xs , ys , t)} δ (x − xs ) + fd δ (x − xd ) , D∇ 4 W2 (x, y, t) + ρh

∂2 W2 (x, y, t) ∂t 2

= −k {W2 (xs , ys , t) − W1 (xa , t)} δ (x − xs ) δ (y − ys ) + fc δ (x − xc ) (y − yc ) , (2) where E is Young’s modulus, I is the second moment of area, ρis the density, A is the cross-sectional area of the string, T is the tension of the string,k is the spring constant, D is the bending stiffness, h is the thickness of the plate, fd is the disturbance force, and fc is the control force. The excitation point is the point xd on the string, xa is the connection point of the string, (xs , ys ) is the connection point of the plate, and (xc , yc ) is the control point of the plate. The general solution is expressed as follows: W1 =

M 

ϕi1 (x)η1,i1 (t),

(3)

i1 =1

W2 =

M2 M1   i2 =1 i3 =1

ϕi2 i3 (x, y) η2,i2 i3 (t),

(4)

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where ϕi1 (x) and η1,i1 (t) are coupled eigenfunction and coupled modal coordinate of the string and ϕi2 i3 (x, y) and η2,i2 i3 (t) are those of the plate. Substituting Eqs. (3) and (4) into Eqs. (1) and (2) and using the orthogonality of the modes, the following equation is obtained: "

η¨ 1 (t) η¨ 2 (t)

#

" +

−kϕ1 ϕT 1 + kϕ1 ϕT 1 2 T − kϕ2 ϕ1 2 + kϕ2 ϕT 2

#"

η1 (t) η2 (t)

#

" =

fd ϕ1d fc ϕ2c

# .

(5)

Furthermore, Eq. (5) is arranged and expressed as ¨ + Dη(t) = q(t). η(t)

(6)

When harmonic vibration is assumed as η(t) = ηejωt , the eigenvalue problem is formulated as   (7) D − ω2 I η = 0, where I is the identity matrix. Letting V be the mode matrix consisting of normalized eigenvectors derived from Eq. (7), the coupled modal coordinate vector is expressed as η(t) = Vξ (t) .

(8)

Since D is a symmetric matrix, it can be diagonalized by the orthogonal matrix V. Therefore, it is expressed as follows: ξ¨ (t) + VT DVξ (t) = VT q.

(9)

Next, from the equation of motion of the coupled system, the control law for nullifying the displacement at any point is derived. If VT in Eq. (9) is taken as VT = (H1 H2 ), the right side becomes VT q = fd H1 ϕ1 (xd ) + fc H2 ϕ2 (xc , yc ) .

(10)

When assuming harmonic vibration as ξ (t) = ξ ejωt , Eq. (9) is expressed as follows:  −1 ξ = −ω2 I + VT DV VT q.

(11)

Substituting Eq. (10) into Eq. (11) and considering that VT DV is a diagonal matrix with square of the natural angular frequencies of the coupled system ωi2 , the above equation is rewritten as follows:

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⎛ ⎜ ξ=⎜ ⎝

1 ω12 −ω2

0

0

0

1 ω22 −ω2

0

0



  ⎟ 0 ⎟ fd H1 ϕ1 (xd ) + fc H2 ϕ2 xc , yc . ⎠ .. .

(12)

Assuming Eq. (12) as ξ = afd + bfc and substituting this into Eq. (8), it is expressed as ˜ c. η = Vξ = V (afd + bfc ) = a˜ fd + bf

(13)

Equation (13) further expands to "

η1 η2

#

" =

a˜ 1 a˜ 2

#

" fd +

b˜ 1 b˜ 2

# fc .

(14)

The displacement is set to zero, and substituting η2 into Eq. (4), the control force expression for zero displacement is derived as follows: −1    ϕT2 (xs , ys ) a˜ 2 fd . fc = − ϕT2 (xs , ys ) b˜ 2

(15)

2.2 The Frequency Characteristics of the Coupled System In this subsection, the frequency characteristics of the coupled system with and without control are analyzed. In the simulation, the material of the string and plate is assumed to be a steel, the dimension of the plate is 0.52 [m] × 0.31 [m] × 0.001 [m], the dimension of the string is 0.001 [m] × 0.001 [m] × 0.9 [m], the position of the connecting spring is set at x = 0.1 of the string and (x, y) = (0.1, 0.1) of the plate, and its spring constant is 10 [N/m]. The disturbance is placed at x = 0.9 on the string, and the control force and the displacement control point are placed to (x, y) = (0.1, 0.1) on the plate. The results of the analysis are shown in Fig. 1. The upper graph shows the frequency response of the string and the lower graph shows that of the plate. As shown in the figure, the frequency characteristic of the string is hardly changed after control; however the response of the plate decreases at all frequencies. Therefore, the plate response is reduced even at the peak frequencies derived from the uncoupled string.

2.3 Acoustic Radiation Power of the Coupled System The acoustic radiation power of the plate of the coupled system is described. Acoustic radiation power is the energy that the sound source radiates per unit time.

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Without control With control

Gain[dB]

0 -20 -40 -60 0

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Ferequency[Hz]

Gain[dB]

0 -100 -200 -300 -400

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Frequency[Hz] Fig. 1 Frequency characteristics of the displacement of the coupled system with and without control Fig. 2 Acoustic radiation power of the plate with and without control

0

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10

Without control With control

-40

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Ferequency[Hz]

Figure 2 shows the result of acoustic radiation power of the coupled system with and without control. Control point is placed at the spring connection point as with the previous subsection. In this case, the most significant control effect is obtained. In the target system, the spring connection point is the disturbance point for the plate. Therefore, the condition considered here indicates the coincidence of the disturbance and control points, resulting in the significant reduction of the acoustic radiation power.

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3 Adaptive-Filter-Based Approach of the Displacement Control System Adaptive filter based on the LMS algorithm [3] was used to make more practical consideration. The five tones signal is given as a disturbance, and adaptive feedback control is used to suppress plate vibration. The block diagram of the adaptive control performed is shown in Fig. 3, in which the primary path indicates the transfer function from the disturbance to the error sensor and secondary path indicates the transfer function from the control input to the error sensor. Figure 4 shows the frequency responses of the primary path and secondary path. Considering Figs. 3 and 4, the peaks that exist only in the primary path coincide with the peak of the string, and the peaks of the secondary path coincide with the peak existing only in the plate. Therefore, the periodic sound of the peak of the string and the periodic sound of the plate were given as the disturbances. Figure 5 shows the frequency response of the error signal of the adaptive control system. The second and fourth peaks are due to the plate resonances, and the first, third, and fifth peaks are due to the string resonances. From this result, it was found that although the vibration derived from the string is not much suppressed, the vibration derived from the plate can be suppressed. In the time response, the amplitude of the error signal with control was reduced to about 25% compared to the case without control. In this case, the control input is updated by using the information about the secondary path characteristics. However, the effect of the strings does not appear in the secondary path characteristics. Therefore, it is not possible to sufficiently control the peak of vibration due to the string.

Fig. 3 Adaptive feedback control system

Adaptive Control of a String-Plate Coupled System Fig. 4 Frequency characteristics of the primary path and secondary path

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0 Primary path Secondary path

-20 -40

Gain[dB]

-60 -80 -100 -120 -140 -160 -180 -200

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Frequency[Hz] Fig. 5 Frequency response of an error signal of the adaptive control system

-110

Without control With control

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Amplitude [/]

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0

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Frequency [Hz]

4 Summary This paper has presented displacement control of the spring-plate coupled system using the adaptive feedback control. The main results are summarized as follows: 1. In the ideal displacement control system, significant control effect is obtained when the control point is set at the connection point of the spring. This is because this controlled state indicates the coincidence of the disturbance and control forces for the plate.

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2. In the adaptive-filter-based approach of the displacement control system, it was found that although the vibration derived by the string is not much suppressed, the vibration derived by the plate can be suppressed.

References 1. Benacchio, S., Mamou-Mani, A., Chomette, B., Finel, V.: Active control and sound synthesistwo different ways to investigate the influence of the modal parameters of a guitar on its sound. J. Acoust. Soc. Am. 139, 1411 (2016) 2. Miyake, N.: Vibration isolation of a string-panel coupling system, Bachelor’s thesis, Seikei University (2015) 3. Farhang-Boroujeny, B.: Adaptive Filters Theory and Applications 2nd. Wiley, USA (2013)

Time-Delay-Based Direct Wave Control of the Phononic Beam Xiaoxu Zhang

, Jian Xu

, and Hongbin Fang

1 Introduction The phononic crystals (PCs) are synthetic materials or structural systems that exhibit a certain form of spatial periodicity. Extensive studies [1, 2] found that the elastic/acoustic wave propagation at certain frequency bands could be damped when the PCs show negative material properties, e.g., negative mass density, negative elastic modulus, or both. To achieve the negative properties, local resonators in either form of mechanical structures [3] or R-L coupled shunting circuits [4], active tuners such as NC coupled shunting circuits [5], and hybrid architectures [6] have been proposed. Beyond the researches on bandgap tuning and vibration damping, a cutting-edge study is designing new phononic structures to achieve the property of nonreciprocal wave propagation [7, 8]. These phononic structures mainly utilize the nonlinearity of electric or magnetic cells to break the symmetry of the system’s transmissibility. However, shortcomings like high complexity and weak programmability restrict further applications of these structures. Recently, a new technique, called the digital synthetic impedance (DSI), arises great concern from researches on the advanced control of vibration damping [9, 10]. Basically, the DSI consists of a programmable digital signal processor (DSP) and a pair of sensor and actuator. The embedded DSP obtains the reference signal from the sensor and then generates a control signal to drive the actuator. Because of the powerful programmability of the DSP, the DSI can easily achieve complicate

X. Zhang () · H. Fang Institute of AI and Robotics, Fudan University, Shanghai, China e-mail: [email protected]; [email protected] J. Xu School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_24

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control goals. Therefore, it is promising to be widely used in the active control of the phononic structures. In addition, time delay always exists in the digital circuits. The studies presented in [11, 12] have shown that the time delay can be used as a parameter to tune the system’s frequency response. Motivated by this property, the delay may also be used as a parameter in the DSI to tune the phononic structure’s bandgap. Furthermore, if the delayed control is unidirectional, the wave propagation in the phononic structure will probably be nonreciprocal. To figure out whether the time delay can change the property of wave propagation, this paper presents some preliminary analysis. The rest of this note is organized as follows. In Sect. 2, the method of lattice approximation is proposed to model the time-delay-controlled beam. In Sect. 3, the time-delay-induced nonreciprocal wave propagation is demonstrated by using the derived model. In the last section, some conclusions are given.

2 Model of the Time-Delay-Controlled Phononic Beam The time-delay-controlled phononic beam is a new architecture for wave control. By assuming that the wavelength is much larger than the dimension of the phononic cell, the lattice approximation method is proposed to model this system. Figure 1 gives the configuration of the time-delay-controlled phononic beam, where the sensing and actuation piezoelectric patches are collocated distributed on the elastic beam, the circuit of the OPA and the capacitance acts as the charge amplifier, the microcontroller converts the signal between analog and digital formats, and the time delay is programmed in the built-in digital signal processor. Technical details of the time-delay programming can be found in [11, 12]. The digital signal processor buffers the signal from the left sensing patch for a short while and then transmits it to the right actuation patch to generate an interference wave. As can be expected, the incident wave will be suppressed when the interference wave has a phase shift of 180 degrees. On the other hand, nonreciprocal wave propagation will occur because the delayed control is unidirectional.

y

Sensing Patch

Lp

Lb

Hp

B

Hb

x C

C

z Actuation Patch

OPA

MC

OPA

Fig. 1 Configuration of the time-delay-controlled phononic beam

MC

OPA

Time-Delay-Based Direct Wave Control of the Phononic Beam Fig. 2 Lattice approximation of the phononic beam

195

wi–1,r , µi–1,r

wi ,l ,µi ,l

wi ,r , µi ,r

wi+1,l , µi+1,l

i –1, r

i, l

i, r

i + 1, l

As shown in Fig. 2, there are four nodes that geometrically relate to the piezoelectric patches of the ith beam cell. Supposing that the nodes at the left and right ends of the piezoelectric patches can be denoted as (i, l) and (i, r), then the other two nodes can be represented as (i − 1, r) and (i + 1, l) by considering the continuity of the cell. Furthermore, supposing that the deflections and rotations at nodes (i, l) and (i, r) are wi, l , wi, r , θ i, l , and θ i, r , then the deflections and rotations at the other two nodes can be similarly represented as wi − 1, r , wi + 1, l , θ i − 1, r , and θ i + 1, l , respectively. Since the wavelength is much larger than the dimension of the phononic cell, the deformation between the nodes can be approximately represented by polynomial interpolation of the deformations of the nodes, namely, ψi (x) = (N1 (ξ ) N2 (ξ ) N3 (ξ ) N4 (ξ )) ri ,

(1)

where N1 (ξ ) = 1 − 3ξ 2 + 2ξ 3 , N2 (ξ ) = (xi, r − xi, l )(ξ − 2ξ 2 + ξ 3 ), T  N3 (ξ ) = 3ξ 2 − 2ξ 3 , N4 (ξ ) = (xi, r − xi, l )(−ξ 2 + ξ 3 ), ri = wi,l θi,l wi,r θi,r , ξ = (x − xi, l )/(xi, r − xi, l ), and x ∈ [xi, l , xi, r ]. xi, l and xi, r are the horizontal coordinates of nodes (i, l) and (i, r). The driven signal on the ith phononic cell is measured from the (i − 1)th phononic cell. The total electrical charge on the (i − 1)th sensing patch can be calculated by Q=−

  e31 B Hb + Hp 2

xi−1,r xi−1,l

ψ

(x)dx =

  e31 B Hb + Hp T F ri−1 , 2

(2)

where e31 is the piezoelectric constant; B, Hb , and Lb denote the width, thickness, and length of the beam; Hp and Lp represent the thickness and length of the piezoT   T electric patches; F = 0 1 0 −1 ; and ri−1 = wi−1,l θi−1,l wi−1,r θi−1,r . According to Kirchhoff’s current law, the electrical charge on the amplifier’s capacitor, with a capacitance of C, should be −Q. It further implies that the voltage output of the charge amplifier is −Q/C. Supposing that the microcontroller amplifies the voltage signal with a gain of g and then delays the output with a time delay of τ , then the voltage applied on the actuation patch of the ith phononic cell should be   ge31 B Hb + Hp T Q (t − τ ) =− F ri−1 (t − τ ) . V3 = −g C 2C

(3)

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Supposing that the mass densities of the beam and the piezoelectric patches are ρ b and ρ p , then the kinetic energy of the system can be written as 

B T = 2

xi+1,l

ρb Hb



˙2

ψ (x)dx + 2ρp Hp

xi−1,r

xi,r

! ˙2

ψ (x)dx .

(4)

xi,l

Supposing that the elastic modulus of the beam is χ b , then the co-energy [13] of the system can be represented as W∗

=

B 2

"  −

 − Hb + H2b −

2

Hb 2 H − 2b

−Hp

 xi+1,l

2 xi−1,r χb S1 dxdy





Hb 2 Hb 2

+Hp  xi,r 2 xi,l χ11 S1 dxdy

#  xi,r   2 2 xi,l ε33 E3 − 2e31 E3 S1 − χ11 S1 dxdy ,

(5)

where E3 = V3 /Hp is the electrical field applied on the actuation patch. Substituting Eqs. (3), (4), and (5) into Lagrange’s equation yields     Mα + Mp r¨ i + Kα + Kp ri + Mβ r¨ i± + Kβ ri± + gKθ ri−1 (t − τ ) = 0, (6) T  where ri± = wi+1,l θi+1,l wi−1,r θi−1,r , ⎞ 0 156 −22Lb 0 2 BLb ρb Hb ⎜ 0 0 ⎟ ⎟, ⎜ − 22Lb 4Lb Mα = ⎝ 0 0 156 22Lb ⎠ 420 0 0 22Lb 4L2b ⎛

⎛ Kα =

BHb3 χb 6L3b

6 −3Lb 0 ⎜ − 3Lb 2L2 0 b ⎜ ⎝ 0 0 6 0 0 3Lb

⎞ 0 0 ⎟ ⎟, 3Lb ⎠ 2L2b



⎞ 0 0 54 13Lb BLb ρb Hb ⎜ 0 −13Lb −3L2b ⎟ ⎜ 0 ⎟, Mβ = ⎝ 54 −13Lb 0 0 ⎠ 420 0 0 13Lb −3L2b ⎛ Kβ =

BHb3 χb 6L3b

0 ⎜ 0 ⎜ ⎝ −6 − 3Lb

0 0 3Lb L2b

⎞ −6 −3Lb 3Lb L2b ⎟ ⎟, 0 0 ⎠ 0 0

Time-Delay-Based Direct Wave Control of the Phononic Beam

⎛ Mp =

Kp =

B 6L3p



  BLp ρb Hb + 2ρp Hp ⎜ ⎜ ⎜ ⎝ 420

156 22Lp 54 − 13Lp

197

⎞ 22Lp 54 −13Lp 4L2p 13Lp −3L2p ⎟ ⎟ ⎟, 13Lp 156 −22Lp ⎠ −3L2p −22Lp 4L2p

⎛ 6   ⎜ 3Lp 3 2 2 Hb χb + 2Hp χ11 3Hb + 6Hb Hp + 4Hp ⎝

and Kθ = −



3Lp −6 3Lp 2L2p −3Lp L2p ⎟ ⎠, − 6 −3Lp 6 −3Lp 2 2 3Lp Lp −3Lp 2Lp

2 H +H B 2 e31 ( b p) 4C

2

FFT .

3 Time-Delay-Induced Nonreciprocal Wave Propagation Equation (6) gives the equation of motion of the time-delay-controlled beam. In this section, we will use it to study the property of wave propagation via a special case. The dimensions and properties of the phononic beam are given in Table 1. As shown in Fig. 3a, the number of periods of the beam is set as 35, and the property of wave propagation is analyzed from two directions. Firstly, the multiperiod beam is cantilevered at the left end with a displacement excitation, and the response is measured at the right end. The transmissibility is calculated to evaluate the wave propagation from left to right. Then, the multi-period beam is cantilevered at the right end with a displacement excitation, and the response is measured at the left end. The transmissibility is calculated to evaluate the wave propagation from right to left. As can be seen in Fig. 3b, within the range of 8000 Hz to 10,000 Hz, the wave can freely pass the beam from the left to the right, but is blocked in the opposite direction. This property indicates that the time-delay strategy can be used as a new technique other than the nonlinear acoustic diode to acquire broadband nonreciprocal propagation. We should mention that the nonreciprocal wave propagation always exists, at least in a special frequency range, when the timedelayed control is activated. The propagation is usually amplified from one end but is inhibited from the other end. The reason we choose τ = 72 μs for demonstration is that it is a critical condition where the wave passes uniformly from the left but is inhibited from the right. Table 1 The dimensions and properties of the phononic beam

Parameter B Lp Hp Lb Hb C

Value 10 mm 20 mm 0.8 mm 20 mm 1 mm 10 nF

Parameter ρp ρb χ 11 χb e31 ε33

Value 7600 kg/m3 8900 kg/m3 50 GPa 100 GPa −7.5 C/m2 1.593∗ 10−8 F/m

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Left source 1

2

34

35

delay

delay

Right source 1

2

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35 delay

delay (a)

(b)

Fig. 3 Property of wave propagation of the time-delay-controlled phononic beam: (a) scenario of the numerical experiment; (b) wave transmissibility corresponding to sources at different ends (g = − 10, τ = 72 μs)

4 Conclusions This paper reported a new technique for wave control of the one-dimensional phononic beam. Because the time-delayed control between the phononic cells is recursive, the method of lattice approximation was proposed to model the system. It can be concluded from the wave propagation analysis that the new technique can acquire broadband nonreciprocal wave propagation easily. Noting that this paper only presented a preliminary study of the new technique, further studies will be made to get a deeper insight into its characteristics and possible applications. For example, the problem of stability caused by the time delay will be considered, and experimental demonstrations will be carried out. Acknowledgments This research is supported by the National Natural Science Foundation of China (No. 11902077 and No. 11772229) and the Shanghai Sailing Program (No. 19YF1403000).

References 1. Lakes, R.S., Lee, T., Bersie, A., et al.: Extreme damping in composite materials with negativestiffness inclusions. Nature. 410(6828), 565–567 (2001) 2. Mei, J., Ma, G., Yang, M., et al.: Dark acoustic metamaterials as super absorbers for lowfrequency sound. Nat. Commun. 3, 756 (2012) 3. Chen, Y.Y., Barnhart, M.V., Chen, J.K., et al.: Dissipative elastic metamaterials for broadband wave mitigation at subwavelength scale. Compos. Struct. 136, 358–371 (2016) 4. Deü, J.F., Larbi, W., Ohayon, R., et al.: Piezoelectric shunt vibration damping of structuralacoustic systems: finite element formulation and reduced-order model. J. Vib. Acoust. 136(3), 031007 (2014) 5. Tateo, F., Collet, M., Ouisse, M., et al.: Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control. J. Intell. Mater. Syst. Struct. 26(8), 952–964 (2015)

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6. Chen, Y.Y., Hu, G.K., Huang, G.L.: An adaptive metamaterial beam with hybrid shunting circuits for extremely broadband control of flexural waves. Smart Mater. Struct. 25(10), 105036 (2016) 7. Wang, Y., Yousefzadeh, B., Chen, H., et al.: Observation of nonreciprocal wave propagation in a dynamic phononic lattice. Phys. Rev. Lett. 121(19), 194301 (2018) 8. Gliozzi, A.S., Miniaci, M., Krushynska, A.O., et al.: Proof of concept of a frequency-preserving and time-invariant metamaterial-based nonlinear acoustic diode. Sci. Rep. 9(1), 9560 (2019) 9. Neˇcásek, J., Václavík, J., Marton, P.: Digital synthetic impedance for application in vibration damping. Rev. Sci. Instrum. 87(2), 024704 (2016) 10. Yan, B., Wang, K., Hu, Z., et al.: Shunt damping vibration control technology: a review. Appl. Sci. 7(5), 494 (2017) 11. Zhang, X.X., Xu, J., Huang, Y.: Experiment on parameter identification of a time delayed vibration absorber. IFAC-PapersOnLine. 48(12), 57–62 (2015) 12. Zhang, X.X., Xu, J., Ji, J.C.: Modelling and tuning for a time-delayed vibration absorber with friction. J. Sound Vib. 424, 137–157 (2018) 13. Preumont, A.: Vibration Control of Active Structures: an Introduction, 3rd edn. Springer, Heidelberg (2011)

Free Vibration Analysis of Multilayer Skew Sandwich Spherical Shell Panels with Viscoelastic Material Cores and Isotropic Constraining Layers Deepak Kumar Biswal

and Sukesh Chandra Mohanty

1 Introduction Vibration suppression of shell-like structures using VEM as core layer and constrained by elastic stiffer layer is one of many control techniques widely adopted by vibration engineers. The efficiency of FSDT-based B-spline Raleigh-Ritz method (RRM) over analytical RRM for any type of material anisotropy was established by Wang [1] for the vibration study of a laminated fiber-reinforced composite skew plate. Woo et al. [2] investigated the free vibration characteristics of skew isotropic Mindlin plates with or without cutouts using FEM based on Legendre polynomial and referred to the method as p-FEM. Garg et al. [3] investigated the free vibration behavior of skew isotropic, FRC plates and sandwich laminates with orthotropic core based on HSDT using a C0 isoparametric finite element. A FEMbased approach was followed by Reddy and Palaninathan [4] with the use of a high-precision triangular element for the free vibration study of laminated skew plates and effectively avoided any matrix inversion or numerical integration for the consistent mass matrix. The static stability and free vibration analyses of a skew viscoelastic sandwich plate with a FGM constraining layer were carried out by Joseph and Mohanty [5] using FEM based on FSDT. Though few works on the skew sandwich shell with orthotropic material core and un-skewed multilayered sandwich plates had been carried out in the past, the

D. K. Biswal () Department of Design and Automation, School of Mechanical Engineering, Vellore Institute of Technology, Vellore, Tamilnadu, India e-mail: [email protected] S. C. Mohanty Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela, Odisha, India e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_25

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study on natural vibration multilayered viscoelastic sandwich skew shell panels has never been attempted. The novelty of the present research is the consideration of independent transverse displacements of the elastic layers and the inclusion of energy dissipation due to strain deformations in the longitudinal and transverse normal directions of the VEM cores along with that due to the transverse shear deformation. An in-house finite element-based code is developed in the MATLAB platform to carry out the numerical analysis which also takes care of the skew coordinates.

2 System Configuration and Mathematical Formulation A multilayer doubly curved un-skewed sandwich panel schematic is illustrated in Fig. 1a. The bottom (1), central (3), and top (5) layers are isotropic material and the core layers 2 and 4 are VEM. The motions of the VEM layers are constrained by elastic layers. The constraining layers assume FSDT and the core layers follow the assumptions given by Biswal and Mohanty [6]. The sandwich panel mid-surface with sectional planes is shown Fig. 1b. The skew shell panel is obtained by cutting the un-skewed sandwich shell with parallel sectional planes which are at an angle of α with the y-axis. The schematic diagram of the plan of the skew sandwich panel is illustrated in Fig. 1c with sides denoted as a and b and the skew angle as α. To implement the boundary restraints on the skew sandwich shell panel, it is pertinent to transform the Cartesian coordinate system (x, y) into oblique coordinate system(r, s). The transformation of the coordinates is given by Eq. (1), and the rotations components in the oblique coordinates are governed by Eq. (2). The relation between the partial derivatives of the above two coordinate systems is given in Eq. (3): y = s cos α x = r + s sin α

a z

a

x

y

Sandwich panel mid-surface

(1)

y

s α

α

L ayer 5 ( T CL ) L ayer 4 ( A V L ) L ayer 3 ( ICL ) L ayer 2 ( PV L ) L ayer 1 ( B L )

b

b

b

a

a

x,r a

(a)

(b)

(c)

Fig. 1 (a) Sandwich panel schematic; (b) sandwich panel mid-surface with sectional planes; (c) plan of the mid-surface of skew sandwich panel

Free Vibration Analysis of Multilayer Skew Sandwich Spherical Shell Panels. . . i i θx0 (x, y, t) = θr0 (r, s, t) cos α i i i θy0 (x, y, t) = −θr0 (r, s, t) sin α + θs0 (r, s, t)

∂(•) ∂x ∂(•) ∂y

= ∂(•) ∂r = − tan α ∂(•) ∂r +

1 ∂(•) cos α ∂s

203

(2)

(3)

3 Finite Element Formulation A sandwich shell finite element [6] with eight nodes on the element boundary is used for the skew multilayer sandwich panel analysis. The finite element equations of motion of free vibration of multilayer skew sandwich panel after globalization can be written as

rs  K −  Mrs urs = 0

(4)

where Mrs and Krs are the global inertia and stiffness matrices with appropriate transformation from skew coordinates. urs is the generalized displacement vector in the global coordinates.  is the vector of complex natural√frequencies given by  = ω2 (1 + jηs ). ω is the modal frequency defined as  = Re (), and ηs is the associated structural loss factor given by ηs = Im ()/ Re (). The structural loss factor indicates the amount of damping, i.e., the energy dissipated to that of induced vibration.

4 Results and Discussions Since there is no concurrent literature pertaining to the dynamic analysis of skew sandwich shell panels with VEM core, the finite element code for the vibration of skew laminated plate is validated first, and then it is further developed for the vibration analyses of viscoelastic sandwich panel. The nondimensional modal frequencies of antisymmetric cross-ply, 0◦ /90◦ /0◦ /90◦ laminated composite skew plates with simply supported and clamped restraints on all sides are calculated using the present FEM code and compared with those carried out by Wang [1] and Garg et al. [3] as shown in Table 1. It is found that the results obtained using the present FEM formulation are very close with the established results. The sandwich panel material properties are taken from Biswal and Mohanty [6]. Figure 2a, b presents the effect of boundary conditions on the first two modal frequencies and associated structural loss factors, respectively, with the variation of core layer thickness of multilayer skew spherical sandwich panels. The frequencies are observed to be decreasing with the increase in core thickness with all the end conditions as the addition of VEM layer increases the flexibility of the skew

CCCC

Boundary condition SSSS

Mode no. 1 2 3 1 2 3

Skew angle 0◦ Present Ref. [1] 1.4938 1.5119 2.4657 2.4656 2.4657 2.4656 2.3246 2.3947 3.8195 3.9532 3.8195 3.9532 Ref. [3] 1.5119 2.4656 2.4656 2.3947 3.9533 3.9533

30◦ Present 2.1847 2.8354 3.5039 2.6854 3.8758 4.8125 Ref. [1] 1.941 2.9063 3.6124 2.7796 4.1564 4.9237

Ref. [3] 1.9439 2.9389 3.6131 2.7798 4.1566 4.924

45◦ Present 2.9519 3.4469 3.9681 3.2227 4.3612 5.602

Ref. [1] 2.6652 3.7126 4.2757 3.443 4.8219 6.085

Ref. [3] 2.6752 3.3131 4.2772 3.4434 4.8823 6.0858

Table 1 Nondimensional modal frequencies of antisymmetric cross-ply laminated composite skew plates, 0◦ /90◦ /0◦ /90◦

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Free Vibration Analysis of Multilayer Skew Sandwich Spherical Shell Panels. . . CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

200 160 120 80 40

0.18

Modal System loss factor

Modal Natural frequency

240

205 CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02

0 0.0

0.4

0.8

1.2

1.6

2.0

0.0

2.4

Core layer thickness ratio (t2/t1)

0.4

0.8

1.2

1.6

2.0

2.4

Core layer thickness ratio (t2/t1)

(a)

(b)

Fig. 2 Variation of first two (a) modal frequencies, (b) structural loss factors versus core layer thickness with various end restraints α = 30o CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

240 200 160 120 80 40 0 0.5

1.0

1.5

2.0

2.5

Aspect ratio (a/b)

(a)

3.0

3.5

0.24

Modal System loss factor

Modal Natural frequency

280

CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

0.20 0.16 0.12 0.08 0.04 0.00 0.5

1.0

1.5

2.0

2.5

3.0

3.5

Aspect ratio (a/b)

(b)

Fig. 3 Variation of first two (a) modal frequencies and (b) associated structural loss factors versus aspect ratio with various end restraints. α = 30o

sandwich structure. Also, the increase in boundary constraints increases the natural frequencies of the system. The structural loss factors are observed to be initially decreasing for a very thin thickness of the core layer and increase afterward as the latter increases for all boundary conditions except CFFF, where it only increases with the increase of the latter. The boundary conditions follow the sequence of CCCC, SSSS, SFSF, CFCF, and CFFF in the increasing order of first-mode structural loss factor for all thicknesses of the core layer. For the second mode, in the increasing order of loss factors, the end conditions have the same sequence as that for the first mode up to t2 /t1 = 0.8, and with core layer thickness more than this value, the sequence follows the order CCCC, SSSS, CFCF, SFSF, and CFFF. The influence of end restraints on the modal frequencies and structural loss factors with the variation of thickness of the constraining layers of multilayer skew spherical sandwich panels are depicted, respectively, in Fig. 3a, b. With an increase in constraining layer thickness up to t5 /t1 ≤ 0.4, the frequencies are observed to decrease, and it increases with the further increase of thickness. The reason being,

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Modal Natural frequency

320 280 240 200 160 120 80

0.20

Modal System loss factor

CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

360

CCCC 1 CFCF 1 CFFF 1 SSSS 1 SFSF 1 CCCC 2 CFCF 2 CFFF 2 SSSS 2 SFSF 2

0.16 0.12 0.08 0.04

40 0

0.00 0.0

0.4

0.8

1.2

1.6

2.0

Constraining layer thickness ratio (t5/t1)

(a)

2.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

Constraining layer thickness ratio (t5/t1)

(b)

Fig. 4 Variation of first two (a) modal frequencies, (b) associated structural loss factors versus constraining layer thickness with various end restraints. α = 30o

when the constraining layer thickness is very small, it adds up to the flexibility of the multilayer skew sandwich panel, and as the thickness increases, the rigidity overtakes its flexibility. The modal system loss factors are seen to first increase for a very thin layer of constraining layer thickness and then decrease as the constraining layer thickness increases. This happens due to the same reason cited above. In the decreasing order of first-mode system loss factor, the boundary conditions follow the sequence of CFFF, CFCF, SFSF, SSSS, and CCCC for constraining layer thicknesses of t5 /t1 > 0.6, and at thicknesses of t5 /t1 ≤ 0.6, the sequence follows CFFF, CFCF, SFSF, CCCC, and SSSS. The structural loss factor corresponding to second mode in the decreasing order follows the sequence of end conditions CFFF, SFSF, CFCF, SSSS, and CCCC for all thicknesses of the constraining layer. Figure 4a, b depicts the effect of boundary conditions on the first two modal natural frequencies and associated structural loss factors, respectively, for different aspect ratio of multilayer skew spherical sandwich panels. It is observed that the first mode frequency of the system under CCCC and SSSS decreases as the aspect ratio increases, whereas that for CFCF, SFSF, and CFFF, it increases with an increase in aspect ratio. The frequencies corresponding to the second mode under CCCC, SSSS, and CFFF are seen to be decreasing with the increase of aspect ratio. The second mode frequency under SFSF is observed to be increasing as the aspect ratio increases. Under CFCF, it initially decreases with the increase in aspect ratio and increases afterward. For all the end conditions, the first-mode structural loss factors are seen to increase with the increase in aspect ratio. The same is true for secondmode structural loss factors under CFFF, SSSS, and CCCC, whereas the secondmode structural loss factor under CFCF and SFSF decreases with the increase in aspect ratio.

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5 Conclusions The frequencies of multilayer skew sandwich panels, respectively, decrease and increase as the core and constraining layer thicknesses increase. The structural loss factors of multilayer skew sandwich structures initially decrease for thinner core layers and increase with a further increase of VEM core thickness. And they first increase and then decrease as the thickness of the constraining layer increases. Increase in boundary constraints increases the frequencies and decreases the structural loss factors. The frequencies with all boundaries fixed and simple support decrease as the aspect ratio increases, and under other boundary conditions, they increase with the increment of the latter.

References 1. Wang, S.: Free vibration analysis of skew fibre-reinforced composite laminates based on firstorder shear deformation plate theory. Comput. Struct. 63(3), 525–538 (1997) 2. Woo, K.S., Hong, C.H., Basu, P.K., Seo, C.G.: Free vibration of skew Mindlin plates by pversion of F.E.M. J. Sound Vib. 268(4), 637–656 (2003) 3. Garg, A.K., Khare, R.K., Kant, T.: Free vibration of skew fiber-reinforced composite and sandwich laminates using a shear deformable finite element model. J. Sandw. Struct. Mater. 8(1), 33–53 (2006) 4. Krishna Reddy, A.R., Palaninathan, R.: Free vibration of skew laminates. Comput. Struct. 70(4), 415–423 (1999) 5. Joseph, S.V., Mohanty, S.C.: Buckling and free vibration analysis of skew sandwich plates with viscoelastic core and functionally graded material constraining layer. Proc. Inst. Mech. Eng. Part G J Aerosp. Eng. 0(0), 1–13 (2017) 6. Biswal, D.K., Mohanty, S.C.: Free vibration study of multilayer sandwich spherical shell panels with viscoelastic core and isotropic/laminated face layers. Compos. Part B Eng. 159(September 2018), 72–85 (2019)

A Size-Dependent Variable-Kinematic Beam Model for Vibration Analysis of Functionally Graded Micro-beams Zhu Su, Kaipeng Sun, and Jie Sun

1 Introduction Functionally graded materials (FGMs) have wide applications in many engineering fields due to unique material properties. Recently, FGMs have been utilized to build microstructures. Since beams are one of fundamental structures, many efforts have been focused on dynamic characteristics of FGM micro-beams. Sim¸ ¸ sek and Reddy [1] studied bending and free vibration of FGM micro-beams where the modified couple stress theory (MCST) in conjunction with different shear deformation beam theories was adopted. An analytical solution for static and vibration problems of FGM micro-beams was presented by Salamat-talab et al. [2] using MCST with third-order shear deformation beam theory. Vibration analysis of FGM curved micro-beams was carried out by Zhang et al. [3] using strain gradient theory (SGT) and nth-order shear deformation theory. It is noted that in the classical or shear deformation beam theory, the transverse displacement is assumed to be constant in the thickness direction, which may be inappropriate due to a strong variation of material properties along thickness for FGM structure [4]. Then, in order to provide more accurate prediction for FGM micro-beams, the thickness stretching effect has been considered by some investigators [5, 6]. The purpose of our work is to develop a size-dependent variable-kinematic beam model. Based on this model, the influence of transverse shear and normal deformations on vibration behavior is investigated.

Z. Su () College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China e-mail: [email protected] K. Sun · J. Sun Shanghai Institute of Satellite Engineering, Shanghai, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_26

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2 Theoretical Formulation Figure 1 shows a typical two-phase FGM micro-beam with coordinate system and dimensions. Using Taylor expansion, a variable displacement field is defined as [7] ui (x, z, t) =

Ni n=0

zn uin (z, t) , i = 1, 3

(1)

where u1 and u3 are the axial and transverse displacement components, respectively. Ni denotes the sum of Taylor expansion terms which control the accuracy of the method. According to the MCST, the components of strain, rotation, and curvature can be defined as     εij = ui,j + uj,i /2, θi = (curl(u))i /2, χij = θi,j + θj,i /2

(2)

Then, the Cauchy stress and couple stress are defined as σij = λδ ij εmm + 2με ij , mij = 2μl 2 χij

(3)

where l is introduced to take into account size effect. Based on Mori-Tanaka estimate, the locally effective bulk and shear moduli K and μ are (K − K2 ) / (K1 − K2 ) = V1 / {1 + V1 (K1 − K2 ) / (K2 + 4μ2 /3)}

(4)

(μ−μ2 ) / (μ1 −μ2 ) =V1 / (1+V1 (μ1 −μ2 ) / {μ2 +μ2 (9K2 +8μ2 ) / [6 (K2 +2μ2 )]}) (5) where the subscripts 1 and 2 represent the different constituents in a typical twophase FGMs. V1 is the fraction volume and defined as V1 = (1/2 + z/ h)p

b

a x, u1

Fig. 1 Schematic of a FGM micro-beam

h

z, u3

(6)

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where p denotes gradient index. Then, the Lamé constants λ can be obtained by λ = K − 2μ/ 3. The energy functional is related to strain energy (U) and kinetic energy (T), given as L=U −T

(7)

where 1 U= 2

 V

  1 σij εij + mij χij dV , T = 2

$"

 ρ V

∂u1 ∂t

#2

" +

∂u3 ∂t

#2 % dV

In order to solve the problem, with the assumption of a harmonic motion, the displacement variables are uniformly represented by series of Chebyshev orthogonal polynomials using a linear coordinate transformation with ξ = 2x/a − 1 .  M uin (ξ, t) = F (ξ ) Am Tm (ξ ) ej ωt m=0

(8)

where ω is the angular frequency and M is the truncated number. Tm (ξ ) denotes the m-order Chebyshev orthogonal polynomial of the first kind, defined as T0 (ξ ) = 1, T1 (ξ ) = ξ, Tm (ξ ) = 2ξ Tm−1 (ξ ) − Tm−2 (ξ ) m ≥ 2

(9)

F(ξ ) is the boundary characteristic function which is defined as F (ξ ) = (1 − ξ )g (1 + ξ )q

(10)

It is noted that g and q can be set to 0, 1, and 2 for different boundary conditions [8]. Then, minimizing the energy functional (L) with respect to unknown coefficients, one can obtain the eigen equation as   [K] − ω2 [M] {A} = {0}

(11)

3 Numerical Examples Unless otherwise stated, the FGMs are assumed to be fabricated from Al2 O3 and Al with material properties given as E1 = 380 GPa, μ1 = 0.3, ρ 1 = 3960 kg/m3 for Al2 O3 , and E2 = 70 GPa, μ2 = 0.3, ρ 2 = 2702 kg/m3 for Al. The material length scale parameter is defined as l = 15 μm. The nondimensional frequency parameter is defined as /  = ωa 2 / h ρ2 /E2

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Table 1 Convergence of nondimensional frequency parameters of simply supported FGM micro-beam with different Ni and M N1 = 1, N3 = 0

N1 = 3, N3 = 0

N1 = 3, N3 = 3

N1 = 5, N3 = 5

M 8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14

1 9.355 9.355 9.355 9.355 9.355 9.355 9.355 9.355 9.205 9.205 9.205 9.205 9.205 9.205 9.205 9.205

2 35.032 35.032 35.032 35.032 35.028 35.028 35.028 35.028 34.622 34.622 34.622 34.622 34.621 34.621 34.621 34.621

3 50.138 50.138 50.138 50.138 50.136 50.136 50.136 50.136 45.695 45.695 45.695 45.695 45.694 45.694 45.694 45.694

4 72.422 72.418 72.418 72.418 72.414 72.411 72.411 72.411 71.886 71.882 71.882 71.882 71.882 71.878 71.878 71.878

5 99.908 99.908 99.908 99.908 99.888 99.888 99.888 99.888 90.954 90.954 90.954 90.954 90.949 90.949 90.949 90.949

6 117.942 117.831 117.829 117.829 117.937 117.825 117.824 117.824 117.337 117.225 117.224 117.224 117.330 117.218 117.217 117.217

3.1 Convergence Study Table 1 presents the first six nondimensional frequency parameters of simply supported FGM micro-beams with different numbers of Taylor expansion Ni and truncated number M. The used parameters are a/h = 10, h/l = 1, and p = 1. The results show that the present formulation has an excellent convergence characteristic.

3.2 Validation Study Table 2 presents fundamental frequency parameters of FGM micro-beams. The obtained solution is in good agreement with available solution based on FBT [9] and TBT [10]. Table 3 presents nondimensional fundamental frequency parameters of FGM beam. A quasi-3D solution [11] based on finite element method also listed here as a reference. A good agreement can be found from comparison.

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Table 2 Fundamental frequency parameters of FGM micro-beams (p = 2, a/h = 12) BC C-C

S-S

FBT [9] N1 = 1, N3 TBT [10] N1 = 3, N3 FBT [9] N1 = 1, N3 TBT [10] N1 = 3, N3

=0 =0 =0 =0

h/l = 1 1.6022 1.6569 1.6246 1.6761 0.7625 0.7768 0.7854 0.7790

h/l = 1.5 1.2194 1.2611 1.2291 1.2712 0.5784 0.5821 0.5903 0.5832

h/l = 2 1.0416 1.0772 1.0642 1.0838 0.4968 0.4949 0.5042 0.4956

h/l = 3 0.8885 0.9188 0.8904 0.9218 0.4285 0.4212 0.4304 0.4215

h/l = 6 0.7807 0.8067 0.7769 0.8047 0.3812 0.3697 0.3787 0.3695

h/l = 10 0.7548 0.7803 0.7454 0.7761 0.3662 0.3577 0.3701 0.3573

Table 3 Quasi-3D nondimensional fundamental frequency parameters of FGM beams (a/h = 5) BC C-C S-S C-F

Ref. [11] p = 0.5 8.8641 4.4240 1.6313

p=1 8.0770 4.0079 1.4804

p=2 7.3039 3.6442 1.3524

p=5 6.5960 3.4133 1.2763

Present p = 0.5 8.8922 4.4188 1.6389

p=1 8.0821 3.9981 1.4821

p=2 7.2787 3.6313 1.3492

p=5 6.5749 3.4044 1.2747

3.3 Parameter Study Figure 2 depicts variation of fundamental frequency parameters of FGM microbeams against different h/l. For simply supported boundary conditions, it is observed that increase of h/l leads to the decrease of frequency parameters. It is also seen that the results from the case of N1 = 1 and N3 = 0 are in good agreement with results from N1 = 3 and N3 = 0, which validate that the FBT can provide accepted solutions for moderately thick micro-beam. In addition, the transverse normal deformation effect (N3 = 0) plays a significant role. By relative error analysis, it can be found that the transverse normal deformation effect becomes strong as the h/l increases. The similar behavior can be found for the case of clamped boundary conditions.

4 Conclusions A size-dependent variable-kinematic beam model for free vibration of FGM micro-beams is presented. In the formulation, the variable-kinematic description is obtained by expanding the displacement fields in terms of thickness coordinate by any desired order Taylor expansion. The modified couple stress theory is adopted to take size effect into account. The solution is obtained by using ChebyshevRitz method where boundary function is introduced to deal with various boundary conditions. The numerical examples demonstrate the present formulation has good accuracy and stability. The size effect (h/l) and stretching effect (N3 = 0) are significant for vibration behaviors of FGM micro-beam.

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10

22 N =1, N =0

S-S

9

1

N =3, N =3

18

N =5, N =5

16

1

3

7 6

N =1, N =0 1

3

N1=3, N3=0 N =3, N =3 3

N =5, N =5 1

3

14 12

5 4

C-C

1

3

1

1

W

1

8 W

20

3

N1=3, N3=0

10 2

4

6 h/l

8

10

8

1 2 3 4 5 6 7 8 9 10 h/l

Fig. 2 Variation of fundamental frequency parameters of FGM micro-beams against different h/l (p = 1)

Acknowledgment This work was supported by National Natural Science Foundation of China (Grant No. 51805250, 11602145), Natural Science Foundation of Jiangsu Province of China (Grant Nos. BK20180429), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References 1. Sim¸ ¸ sek, M., Reddy, J.N.: Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int. J. Eng. Sci. 64, 37–53 (2013) 2. Salamat-Talab, M., Nateghi, A., Torabi, J.: Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int. J. Mech. Sci. 57(1), 63–73 (2012) 3. Zhang, B., He, Y., Liu, D., et al.: A novel size-dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos. Struct. 106, 374– 392 (2013) 4. Carrera, E., Brischetto, S., Cinefra, M., et al.: Effects of thickness stretching in functionally graded plates and shells. Compos. Part B. 42(2), 123–133 (2011) 5. Trinh, L.C., Nguyen, H.X., Vo, T.P., et al.: Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory. Compos. Struct. 154, 556–572 (2016) 6. Yu, T., Hu, H., Zhang, J., et al.: Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory. Thin-Walled Struct. 138, 1–14 (2019) 7. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, New York (2003) 8. Wang, Y.Q., Teng, M.W.: Vibration analysis of circular and annular plates made of 3D graphene foams via Chebyshev-Ritz method. Aerosp. Sci. Technol. 95, 105440 (2019) 9. Ke, L.L., Wang, Y.S., Yang, J., et al.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50, 256–267 (2012)

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10. Ansari, R., Shojaei, M.F., Gholami, R.: Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method. Compos. Struct. 136, 669–683 (2016) 11. Vo, T.P., Thai, H.T., Nguyen, T.K., et al.: A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos. Struct. 119, 1–12 (2015)

Vibration Analysis of a Viscoelastic Beam Equipped with a Resilient Impact Damper Xiao-Feng Geng, Hu Ding, and Li-Qun Chen

1 Introduction The flexible beam structure has many practical applications in various fields. The flexible structure has multiple resonant frequencies, which seriously affects the stability of flexible structures. Many passive control techniques have been proposed to suppress the vibration of beam structures, such as tuned mass damper [1], nonlinear energy sink [2], pounding tuned mass damper [3], pendulum mass damper [4], and so on. Although there are many passive vibration reduction technologies, the demand for new technology is endless. The impact damper, one of the passive technologies, is also widely used to suppress vibration of the primary structures due to its simple structure, low cost, and the ability to work in harsh environments [5]. Because the collision is discontinuous, the main damping mechanism of the impact damper is the momentum exchange between the collision bodies [6], and the collusion gap and mass ratio of the impact damper are discussed [7]. The bifurcation phenomenon of the impact damper also exists due to high nonlinearity during collision [8]. The damping device consisting of multiple impact dampers is also used to reduce the vibration response of main system [9]. In order to improve the damping effect, the improvement of impact damping has been continuously carried out. The particle impact damper is proven to have excellent damping effect [5]. The combination of nonlinear energy sink and impact damper has also been studied to improve vibration damping capacity [10]. There also have been many studies on impact damping for viscoelastic beams. The nonlinear behavior of the cantilever beam is studied under impact damper [11]. The vibration of the rotating

X.-F. Geng () · H. Ding · L.-Q. Chen Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, China e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_27

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beam is suppressed by the impact damper [12]. The acoustic characteristics of the impact damper are considered [13]. Usually,the differential equation of motion of discontinuous system is calculated by the Newmark-beta or the Runge-Kutta method [14]. The particle impact dampers are also investigated to reduce the response of the beam [15]. What is more, the impact damper has been widely used in robots [16], buildings [17], and so on. Although some researchers paid more attention to apply impact damper to suppress vibration of the continuum system, most the works equate the continuum system to a single degree of freedom system. However, the vibration suppression of continuums for multiple-order modes is less investigated, especially the high-order mode of the cantilever beam. In order to reduce the vibration amplitude of the cantilever beam in multiple modes, the cantilever beam attached the impact damper is investigated. The discontinuous differential equations are given to analyze the effect of the weight of the vibrator and the collision gap on vibration suppression of the cantilever beam.

2 Dynamic Modeling Figure 1 shows that the cantilever beam attached an impact damper at the free end. The ball can move freely. The collision process between the ball and the cantilever beam is simulated a spring and a damping. d is the collision gap. Friction is equivalent to viscous damping with the coefficient of cμ . According to Hertz contact theory, the contact force between the ball and the cantilever beam can be described by the following: Fn =

4 / 3 Q l 3π

where l is the equivalent contact radius;  is the extrusion displacement; Q is the material parameters, which can be described as 1/Q = (1 − μ1 2 )/π E1 + (1 − μ2 2 )/ π E2 ; and μ1 , μ1 and E1 , E1 are the Poisson’s ratio and modulus of elasticity of the ball and the cantilever beam, respectively. In order to simplify the calculation, the linear model is selected [18]:

y d k c

d



F

k c

Fig. 1 The diagram of the cantilever beam with impact damping

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Fn =

219

4 √ Q lΔ · Δ = KΔ 3π

where K is the equivalent contact stiffness. According to energy consumption, damping can be equivalent to the following [19]: ' 2 ln e

"

c = − 0  K π 2 + (ln e)2

m1 m2 m1 + m2

#

where e is the recovery coefficient and m1 and m2 are collision bodies, respectively. Therefore, the collision process can be described as Pn = Fn + cΔ˙ = KΔ + cΔ˙

(1)

Here, the governing equation of the cantilever beam attached the impact damper or not can be obtained as follows: EI uxxxx (x, t) + αEI uxxxx,t (x, t) + ρAutt (x, t) = F (t)δ (x − x0 ) + H {K [y(t) − u (L, t)] + c [y(t) ˙ − u˙ (L, t)]} + cμ [y(t) ˙ − u˙ (L, t)] my(t) ¨ + H {K [y(t) − u (L, t)] + c [y(t) ˙ − u˙ (L, t)]} + cμ [y(t) ˙ − u˙ (L, t)] = 0 (2) where u(x, t) and y(t) are displacement of the cantilever beam and the ball, respectively. ρA is the unit mass of the cantilever beam, L is the length, EI is the bending stiffness of the cantilever beam, F(t) is the external force, δ is the Dirac function, and cμ is the viscous damping coefficient. H is the Heaviside function, which can be expressed as  H =

1 0

f or |w(t) − u (x, t)| ≥ d f or |w(t) − u (x, t)| ≤ d

(3)

where d is the collision gap. The result is calculated by using the Runge-Kutta algorithm. The parameters of the cantilever beam and pounding stiffness are given, shown as in Table 1. Table 1 The parameters of cantilever beam and impact damping

Notation E ρ L A

Value 70GPa 2800 kg/m3 1m 10−4 m2

Notation I α cμ e

Value 3.125 × 10−10 0.0002 0.08 0.6

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3 Numerical Simulations and Analysis The vibration response of the cantilever beam with impact damper or not is plotted in Fig. 2. The mass of the free vibrator m = 0.02 kg; the collision gap d = 0.003 m, 0.005, and 0.015 m; the pounding stiffness k = 4 × 106 N/m. The first- and secondorder mode vibration amplitude of the cantilever beam is also reduced. As can be seen from Fig. 2, the effect of vibration suppression is different due to the change of gap. The damping effect is different at the same gap for the first and second mode. For each mode, there is an optimal gap value to achieve the best damping effect, and too small or larger gap weakens the suppression of vibration. The velocity response of the cantilever beam and ball is investigated to illustrate the difference in vibration reduction, as shown in Figs. 3 and 4. As depicted in Figs. 3 and 4, when the gap is small, the velocity curves of the ball and the beam are similar, with the same velocity direction and small momentum exchange, as shown in Fig. 3a, b. The excellent damping effect occurs when the ball contacts the cantilever beam at the opposite velocity, which can maximize momentum exchange, as described in Figs. 3c and 4a and b. However, the extra larger gap prevents the ball from contacting the cantilever beam, and the ball only works through damping to consume the vibration energy of the cantilever beam, as plotted in Fig. 4c. The effect of different ball mass on vibration suppression is also analyzed, as described in Fig. 5. The large mass is beneficial to the exchange of momentum in contact, so the amplitude of the cantilever beam decreases with the increase of the weight of the ball. Due to the introduction of additional mass, the resonant frequency of the beam is reduced. The resonant peak value of the cantilever beam is recorded, and the threedimensional diagram is given under the action of different mass and gap. It can be seen from Fig. 6 that smaller gap is not conducive to vibration suppression. In a large gap, the reduction of vibration amplitude is due to the

(a)

(b)

Fig. 2 The amplitude-frequency response curves for m = 0.02 kg, d = 0.03 m, 0.005 m, 0.015 m, k = 4 × 106 N/m: (a) the first-order mode, (b) the second-order mode

Vibration Analysis of a Viscoelastic Beam Equipped with a Resilient Impact Damper

(a)

221

(b)

(c) Fig. 3 The first-mode velocity response of the cantilever beam and ball: (a) d = 0.003 m, (b) d = 0.005 m, (c) d = 0.015 m

damping consumption of vibration energy, and there is no exchange of momentum. For the second mode, the weight of the ball is too large to have self-excitation, which makes the vibration amplitude of the cantilever beam suddenly increase.

4 Conclusions The main purpose is that the multiple modal vibration of cantilever beam can be suppressed by the impact damper. The conclusions can be summarized as follows: 1. Multi-order modal vibration of the cantilever beam can be suppressed by the impact damper, and the large vibrator weight improves the suppression of multiorder modal vibration of the cantilever beam. 2. For each mode, there is an optimal gap to maximize the damping effect. However, for different modes, the optimal gap is different.

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(a)

(b)

(c) Fig. 4 The second-mode velocity response of the cantilever beam and ball: (a) d = 0.003 m, (b) d = 0.005 m, (c) d = 0.015 m

Fig. 5 The amplitude-frequency response curves with different m, k = 4 × 106 N/m: (a) the firstorder mode, (b) the second-order mode

Vibration Analysis of a Viscoelastic Beam Equipped with a Resilient Impact Damper

(a)

223

(b)

Fig. 6 The resonant peak value of the cantilever beam with different mass and gap: (a) the first mode, (b) the second mode

References 1. Li, C., Zhuang, T., Zhou, S.T., et al.: Passive vibration control of a semi-submersible floating offshore wind turbine. Appl. Sci-Basel. 7, 509 (2017) 2. Ahmadabadi, Z.N., Khade, S.E.: Nonlinear vibration control of a cantilever beam by a nonlinear energy sink. Mech. Mach. Theory. 50, 134–149 (2014) 3. Tian, L., Rong, K.J., Zhang, P., Liu, Y.P.: Vibration control of a power transmission tower with pounding tuned mass damper under multi-component seismic excitations. Appl. Sci-Basel. 7, 447 (2017) 4. Roffel, A.J., Narasimhan, S., Asce, M., Haskett, T.: Performance of pendulum tuned mass dampers in reducing the responses of flexible structures. J. Struct. Eng. 139, 04013019 (2013) 5. Yao, B., Chen, Q., Xiang, H.Y., Gao, X.: Experimental and theoretical investigation on dynamic properties of tuned particle damper. Int. J. Mech. Sci. 80, 122–130 (2014) 6. Cheng, C.C., Wang, J.Y.: Free vibration analysis of a resilient impact damper. Int. J. Mech. Sci. 45(4), 589–604 (2003) 7. Park, J., Wang, S., Crocker, M.: Mass loaded resonance of a single unit impact damper caused by impacts and the resulting kinetic energy influx. J. Sound Vib. 323, 877–895 (2009) 8. Peterka, F.: Bifurcations and transition phenomena in an impact oscillator. Chaos, Solitons Fractals. 7, 1635–1647 (1996) 9. Gharib, M., ASCE, A.M., Karkoub, M.: Experimental investigation of linear particle chain impact dampers in free-vibration suppression. J. Struct. Eng. 143(2), 04016160 (2016) 10. Li, T., Qiu, D., Seguy, S., Berlioz, A.: Activation characteristic of a vibro-impact energy sink and its application to chatter control in turning. J. Sound Vib. 405, 1–18 (2017) 11. Wouw, V.D., Bosch, V.D., Kraker, D.A., Campen, D.H.: Experimental and numerical analysis of nonlinear phenomena in a stochastically excited beam system with impact. Chaos, Solitons Fractals. 9, 1409–1428 (1998) 12. Cheng, J.L., Xu, H.: Inner mass impact damper for attenuating structure vibration. Int. J. Solids Struct. 43, 5355–5369 (2000) 13. Afsharfard, A., Farshidianfar, A.: Design of nonlinear impact dampers based on acoustic and damping behavior. Int. J. Mech. Sci. 65, 125–133 (2012) 14. Afsharfard, A., Farshidianfar, A.: An efficient method to solve the strongly coupled nonlinear differential equations of impact dampers. Arch. Appl. Mech. 82, 977–984 (2012) 15. Du, Y.C., Wang, S.L.: Modeling the fine particle impact damper. Int. J. Mech. Sci. 52, 1015–1022 (2010)

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16. Jam, J.E., Fard, A.A.: Application of single unit impact dampers to reduce undesired vibration of the 3R robot arms. Int. J. Aerosp. Sci. 2(2), 49–54 (2013) 17. Lu, Z., Lu, X.L., Lu, W.S., Masri, S.F.: Experimental studies of the effects of buffered particle dampers attached to a multi-degree-of-freedom system under dynamic loads. J. Sound Vib. 331(9), 2007–2022 (2012) 18. Misra, A., Cheung, J.: Particle motion and energy distribution in tumbling ball mills. Powder Technol. 105, 222–227 (1999) 19. Jankowski, R.: Analytical expression between the impact damping ratio and the coefficient of restitution in the non-linear viscoelastic model of structural pounding. Earthq. Eng. Struct. Dyn. 35(4), 517–524 (2016)

Vibration Analysis of a Beam with Both Ends Fixed Using Molecular Dynamics Method Akinori Tomoda, Masahiro Yamanaka, and Taiyo Takashima

1 Introduction Contact portions and gaps in mechanical systems such as a magnetic head and platter in a hard disk drive, a cantilever beam in an atomic force microscope, and a contact beam in a magnetic relay generate noise and damage of important parts under vibration input. Impact and friction phenomenon probably cause an amplitude increase of the system. Therefore, the estimation of the vibration characteristics of the system is important to prevent severe damage and noises due to an amplitude increase. The vibration characteristics of the system with complex shape can be calculated using FEM (finite element method). In the previous study, the authors proposed the FE model of the thin plate structure with fixed end and a contact portion [1]. The FE model is described as a two-dimensional beam element for the plate and a nonlinear spring element based on Hertzian contact stress for a contact portion. However, the vibration response of the structure depends not only on a contact stress but also on friction and impact phenomenon in a contact portion and gap. Accurately estimation of the vibration characteristics of the system with contact portion using FEM is difficult because of parameter variations in contact surface under vibration input. In this study, the application of the MD for modeling of contact portion in the thin plate structure is investigated. MD is able to calculate positions and momentums of each atom using Newton’s second law of motion and interatomic potential [2].

A. Tomoda () Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, Fukuoka, Japan e-mail: [email protected] M. Yamanaka · T. Takashima Graduate School of Engineering, Fukuoka Institute of Technology, Fukuoka, Japan e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_28

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MEAM (modified embedding atom method) potential is adopted for this numerical analysis for describing metallic bonds of atoms [3]. MEAM potential can treat a crystal structure with fractures, surfaces, impurities, and alloying addition. In this paper, MD model of a beam made of single-crystal Cu (copper) with both ends fixed is proposed, and the vibration response of the beam after contacting the small block of single-crystal Cu is estimated. The vibration analysis method using MD can obtain the vibration response of many mechanical systems with nonlinear elements without vibration test.

2 Analytical Model of a Beam with Both Ends Fixed 2.1 Molecular Dynamics Method Using MEAM Potential MD is the method of tracking atomic behavior by applying Newton’s second law of motion based on atomic force on each atom [2, 4]. The equation of motion for each atom in the system is mi

d 2 Ri = Fi dt 2

(1)

where mi is the mass and Ri is the position vector of atom i. Fi is the force acting atom i and can be expressed as the following equation using the potential energy Etot of the system: Fi = −∇i Etot = −

∂Etot ∂Ri

(2)

where ∇ i is the differential operator with respect to the atomic position Ri (xi , yi , zi ). ∇ i is expressed as follows: ∇i = ex

∂ ∂ ∂ ∂ + ey + ez = ∂xi ∂yi ∂zi ∂Ri

(3)

where ex , ey , ez are unit vectors of positive direction of the x, y, z axes in Cartesian coordinate system. Many interatomic potentials for determining the potential energy of a system which correspond to the types of atoms and bonds to be analyzed were developed. In this paper, the analysis target is a beam with both ends fixed consisting of copper atoms, and the analysis model is developed using MEAM potential. The total energy of a system of atoms is expressed as the sum of many body terms of energy due to electrons in the vicinity of an atom and the terms of the two-body interaction [3]. The potential energy Etot is given by

Vibration Analysis of a Beam with Both Ends Fixed Using Molecular Dynamics. . .

Etot =

⎫   1   ⎬ φij Rij F ρ + ⎭ ⎩ i i 2

⎧ ⎨ i

227

(4)

j ( =i)

where ρ i is the electron density at atom i, Fi is the embedding energy to embed an atom i into the background electron density ρ i , and φ ij is a pair interaction between atoms i and j whose distance is given by Rij . We can track behavior of atoms by calculating the force acting on each atom using Eqs. (1), (2), (3), and (4).

2.2 Construction of MD Model of a Beam with Both Ends Fixed The MD model for calculating vibration response of a beam with both ends fixed is shown in Fig. 1. The model consists of the beam and the block for impactor made of the FCC (face-centered cubic) Cu lattice using MEAM potential. We determined the initial position of 4451 atoms by taking into consideration the equilibrium interatomic distance 2.56 Å and the lattice constant 3.62 Å in Cu atoms. The temperature in the system is calculated as follows:  mi vi 2 3 Nk B T = 2 2 N

(5)

i

where N is the number of atoms existing in the system, kB is the Boltzmann constant (1.38 × 10−23 J/K), T is the temperature in the system, and vi is the velocity of atom i. After setting initial conditions, the force acting on each atom is calculated by Eqs.

Block (Cu) Fixed end

Fixed end Contact portion

Beam (Cu)

22.8 nm

45.6 nm

Fig. 1 The MD model of a beam and a block for calculating vibration response

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(2) and (4). The present study adopts the Velocity Verlet algorithm for numerical integration [5]. Considering the vibration period of Cu atom, we use an integration time step of 1 fs. Total steps of numerical integration is 100,000 (100 ps). We assume that the number of atoms N, the volume of the system V, and total energy in the system E are constant during MD calculation. We use the software called LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) developed by Sandia National Laboratories for MD calculation [6].

3 Result of Contact Simulation The result of MD calculation is shown in Fig. 2a, b, c, and d. Initial velocity of atoms is 1000 m/s and initial temperature of the system is 0.01 K. Figure 2a shows the initial state of the beam and the block. After contacting the block and the beam, vibration of the first mode of the beam with both ends fixed occurred (Fig. 2b–d). The block was deformed after contacting faces between the lower of the block and the upper of the beam. Natural period of the first mode of the beam T1 can be calculated by the time difference between Fig. 2c and d. T1 and natural frequency of the first mode f1 is calculated as follows:   T1 = 2 × 62.5 × 10−12 − 16.0 × 10−12 = 93.0 × 10−12 [s] (6) f1 =

1 ≈ 10.7 [GHz] T1

(7)

The MD method can calculate contact phenomenon without coefficient of restitution. However, the accuracy of the result of MD calculation depends on the initial condition of atom such as momentum, controlling for the temperature of the fixed ends, and the number of atoms in the system. Especially, the surface area of the beam affects the equilibrium position of each atom and accuracy of the construction of the surface of the beam. Results of the large-scale simulation will be needed for the verification of MD calculation for vibration analysis.

4 Conclusion The application of the MD for modeling of contact portion in the thin plate structure is investigated. Analytical model of a beam with both ends fixed and a block for impactor using MD method is proposed. The first mode of the beam is observed in the MD calculation. The natural frequency of the first mode f1 is 10.7 GHz when initial velocity of atoms is 1000 m/s and initial temperature of the system is 0.01 K. Results of the large-scale simulation will be needed for the verification of MD calculation for vibration analysis.

Vibration Analysis of a Beam with Both Ends Fixed Using Molecular Dynamics. . .

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Block (Cu) Beam (Cu)

(a)

(b)

(c)

(d) Fig. 2 Vibration response of the first mode of the beam. (a) t = 0 [ps], (b) t = 3 [ps], (c) t = 16 [ps], (d) t = 62.5 [ps]

Acknowledgments This work was supported by JSPS KAKENHI Grant Number JP19K04284. The computation was carried out using the computer resource offered under the category of General Projects by Research Institute for Information Technology, Kyushu University.

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References 1. Kikuchi, K., Tomoda, A., Watanabe, T.: Study on the Vibration Characteristics of a Thin Plate Structure with Contact Area. Mechanical Engineering Congress 2015, Sapporo (2015) 2. The Japan Society of Mechanical Engineers ed.: Foundation of Atomic/Molecular Simulation of Solids and Liquids (Series of Computer analysis on dynamics). 2nd edn. CORONA Publishing Co., Tokyo (2001) 3. Baskes, M.I.: Modified embedded-atom potentials for cubic materials and impurities. Phys. Rev. B. 46(5), 2727–2742 (1992) 4. Ueda, A.: Molecular Simulation - from Classical to Quantum Methods. SHOKABO Co., Tokyo (2003) 5. Swope, W.C., Andersen, H.C., Berens, P.H., Wilson, K.R.: A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: application to small water clusters. J. Chem. Phys. 76(1), 637–649 (1982) 6. LAMMPS Molecular Dynamics Simulator Homepage, https://lammps.sandia.gov/. Last accessed 25 July 2019

Numerical Investigation of Vibration Characteristics and Damping Properties of CNT-Based Viscoelastic Spherical Shell Structure S. Srikant Patnaik

, Tarapada Roy

, and D. Koteswar Rao

1 Introduction The carbon nanotubes have the exceptional mechanical and thermal properties. Lin et al. [1] depicted the energy dissipation between the polymer and carbon nanotubes by mathematical formulations. The hybrid composite made by nanocomposite is studied by Khan et al. [2], and they depicted improved damping characteristics for the hybrid composites. Latibari et al. [3] investigated the interfacial interaction (stick-slip mechanism) between the CNTs and polymer resin and concluded that by increasing volume fractions of CNT, the natural frequency decreases significantly because of damping effect. Ajayan et al. [4] described a technique for the alignment of carbon nanotubes for the nanocomposite. Thess et al. [5] investigated singlewalled nanotubes at a high temperature of 1200◦ C by the method of X-ray diffraction and electron microscopy. The author also depicted the effect of different agglomeration parameters on damping of the shell structure. In the present article, 5% and 10% volume fractions of the CNTs are reinforced in the polymer and obtained the viscoelastic properties of such nanocomposites with the help of DMA8000. Further, based on such viscoelastic material, vibration analysis of a doubly curved shell structure is carried out and determined the resonant frequency.

S. S. Patnaik · T. Roy () · D. Koteswar Rao National Institute of Technology Rourkela, Rourkela, Odisha, India e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_29

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2 Mathematical Modeling 2.1 Shell Formulation [6] As shown in Fig. 1, the midsurface of the shell structure in the Cartesian coordinate system is mapped through the suitable exact parametrization. Two independent coordinates in the parametric space have been considered as the midsurface curvilinear coordinates of the shell. The normal direction coordinate to the middle surface of the shell has been represented by z. The shell midsurface in the global Cartesian coordinates in terms of the position vector is r (α1 , α2 ) = X (α1 , α2 ) i + i + (α1 , α2 ) j + Z (α1 , α2 ) k ∧∧

(1)



where i ,j , and k are unit vectors along the X, Y, and Z axes, respectively. The tangent to the isoparametric curves sα1 and sα2 , respectively, are ∂r ∂r and, r,2 = ∂α1 ∂α2

(2)

√ √ r,1 .r,1 and A2 = r,2 .r,2

(3)

r,1 =

A1 =

Defining above A1 and A2 as Lame’s parameters or measure numbers that are fundamentals for the understanding of curvilinear coordinates, ds 2 = A1 2 dα1 2 + A2 2 dα2 2

Fig. 1 Geometry of layered composite shell in Cartesian coordinate system

(4)

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The unit tangent vectors to the isoparametric curve sα1 and sα2 are ∧

e1 = ∧

e2 =

r,1 A1 r,2 A2

4 (5)

The subscripts 1 and 2 in r,1 andr,2 indicate the derivative with respect to α 1 and α 2 , respectively. The unit normal vector to the tangent plane of any point on the reference surface can be expressed as r,1 × r,2 5 eˆn = 5 5r,1 × r,2 5

(6)

The normal curvatures of the shell midsurface can be expressed as 1 eˆn .r,11 1 eˆn .r,22 =− and =− 2 R1 R2 A1 A2 2

(7)

and the twist curvatures of the shell midsurface can be expressed as 1 eˆn .r,12 =− R12 A1 A2 2.1.1

(8)

Strain Displacement Relations

Neglecting normal strain component in the thickness direction, the five strain components of a doubly curved shell may be expressed as T 

T 0 ε0 γ 0 γ 0 γ 0 εxx εyy γxy γyz γxz = εxx yy xy yz xz (9) T  + z kxx kyy kxy 0 0 0 where ε0 xx , ε0 yy and γ 0 xy are the in-plane strains of the midsurface in the Cartesian coordinate system and kxx , kyy and kxy are the bending strains (curvatures) of the midsurface in the Cartesian coordinate system.

2.1.2

Determination of Equation of Motion and Finite Element Modeling

After energy calculations, equation of motion is obtained from Hamilton’s principle: t2 [δ (T − U ) + δW ] dt = 0; t1

(10)

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t1 and t2 are the time intervals and equation of motion can be written as  

e  

e  e   e  e  + Cu d + Ku {d} = f e ; Mu d

(11)

The mass matrix is expressed as

e Mu =

⎧ h ⎪  1 ⎪ ⎨ 2 −1

⎪ ⎪ ⎩− h 2

⎫ ⎪ ⎪ ⎬

ρ[N]T [N] dz |J | dξ dη ; ⎪ ⎪ ⎭

(12)

The elemental structural stiffness matrix is expressed as

e Ku =



V

e T

 Bu [C] Bue dV ,

(13)

where

   Kue = Kbe + Kse ;

(14)

  e Bb [0]

e ; [0] Bs

(15)

e Bu =

 HC  Db [0] 

; [C] = [0] DsH C

(16)

Subscripts “e” and “u” are elemental and structural notations, respectively. Bue is the strain displacement matrix and D is the rigidity matrix.

3 Material Fabrication and DMA-8000 Test As shown in Fig. 2, the fabrication of nanocomposite is carried out by reinforcing the CNTs into the polymer with the help of ultrasonic probe sonicator as per ASTMD4065. The nanocomposite samples are tested in DMA-8000 with frequency scan mode under single cantilever support to obtain viscoelastic properties (storage modulus and loss factor) of nanocomposite (NC) material.

Numerical Investigation of Vibration Characteristics and Damping Properties. . .

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Fig. 2 (a) Flow diagram of constituents of NCs, (b) DMA-8000

Fig. 3 (a) Image of 5% vol. CNT. (b) Image of 10% vol. CNT

4 Results and Discussion 4.1 Analysis of MWCNT-Based Nanocomposite Sample in Scanning Electron Microscopy (SEM) Few samples fabricated with 5% and 10% by volume CNTs inclusion in the polymer. With the help of SEM at scaling 10 μm, SEM images are captured which show the surface texture after dispersion of CNTs in the epoxy for NC samples. From images the dispersion of the samples has been observed as shown in Fig. 3a, b. The higher CNTs are agglomerated and producing weak interphase; hence up to 10% of CNTs volume fractions are considered for further DMA tests and obtained storage modulus and loss factor.

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Fig. 4 Natural frequencies, (a) R/a = 10 and (b) R/a = 100, of the NCs spherical shell structure

4.2 Numerical Results Based on First Five Natural Frequencies The viscoelastic properties obtained experimentally from DMA-8000 are used in numerical mathematical model in MATLAB environment. As shown in Fig. 4a, b, it is clear from the results that the higher CNTs volume fraction reinforcement for such structures leads to higher natural frequencies.

4.3 Numerical Results Based on Frequency Response and Transient Response Frequency responses are depicted in Fig. 5a, b for the 5% and 10% CNTs volume fraction-reinforced nanocomposite-based viscoelastic doubly curved shell structures. As shown in Fig. 6a, b, the transient responses are depicting the lower settling time with the higher volume fraction of CNTs for such structures.

5 Conclusion Microscopic analysis is done using SEM and observed higher agglomerations at higher CNTs volume inclusion. The material property of such NCs material system is obtained with the help of DMA-8000. First-order shear deformation theory is used for shell structure where SERENDEPITY element is considered with five degrees of freedom per node. A MATLAB code is generated for this formulation by incorporating Rayleigh damping model to depict transient response of the structure.

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Fig. 5 Frequency responses, (a) R/a = 10 and (b) R/a = 100, of the NCs spherical shell structure

Fig. 6 Impulse responses, (a) R/a = 10 and (b) R/a = 100, of NCs spherical shell structure

Transient response shows that the significant decrease in the settling time as the CNTs volume fraction increases and significant increase in the resonant frequency are also observed. Higher agglomeration is observed with increase in volume fraction of the CNTs, hence proposing the 10% CNTs volume fraction-reinforced nanocomposite-based doubly curved shell structure for the real-life applications because of its better damping properties. Acknowledgments The authors kindly acknowledged the IMPRINT cell of the Ministry of Human Resource Development (MHRD) and Department of Science and Technology (SERBDST), Government of India, for a project grant (F. No. IMPRINT-6292) under which the research work was carried out.

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References 1. Lin, R.M., Lu, C.: Modeling of interfacial friction damping of carbon nanotube-based nanocomposites. Mech. Syst. Signal Process. 24(8), 2996–3012 (2010) 2. Khan, S.U., Li, C.Y., Siddiqui, N.A., Kim, J.-K.: Vibration damping characteristics of carbon fiber-reinforced composites containing multi-walled carbon nanotubes. Compos. Sci. Technol. 71(12), 19 (2011) 3. Latibari, S.T., Mehrali, M., Mottahedin, L., Fereidoon, A., Metselaar, H.S.C.: Investigation of interfacial damping nanotube-based composite. Compos. Part B. 50(2013), 354–361 (2013) 4. Ajayan, P.M., Stephan, O., Colliex, C., Trauth, D.: Aligned carbon nanotube arrays formed by cutting a polymer resin–nanotube composite. Science. 265, 1212–1214 (1994) 5. Thess, A., Lee, R., Nikolaev, P., Dai, H., Petit, P., Robert, J., Xu, C., Lee, Y.H., Kim, S.G., Rinzler, A.G., Colbert, D.T., Scuseria, G.E., Tomanek, D., Fischer, J.E., Smalley, R.E.: Crystalline ropes of metallic carbon nanotubes. Science. 273, 483–487 (1996) 6. Thomas, B., Roy, T.: Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech. 227(2), 581–599

Part V

Recent Advances on Vibration Control of Engineering Structures

Semi-active Vibration Suppression of a Structure by a Shear-Type Damper Using Magnetorheological Grease Shuto Nagamatsu and Toshihiko Shiraishi

1 Introduction A magnetorheological (MR) fluid is a suspension of micron-sized magnetic particles in a carrier fluid which was first reported in 1948 [1]. When you apply a magnetic field, it changes its rheological properties within few milliseconds. As the state changes from liquid to semisolid, it generates a large yield shear stress up to 100 kPa. This change is caused by MR effect, which is a phenomenon that magnetically polarized particles form chain-like structures along the field. Due to the quick and drastic change of properties, MR fluids are used in dampers for semiactive vibration suppressions [2, 3]. However, conventional dampers using MR fluids have two problems. The sedimentation and the sliding friction degrade their damping performance. A sedimentation is a settlement of particles arising from the density mismatch between particles and oils. It affects the initial damping performance after a long-term standing. The sliding friction between a cylinder and sealing elements also affects the damping performance. It increases the minimum damping force to narrow the dynamic range of the damper. In order to solve these problems, a shear-type damper using MR grease was developed [4]. A magnetorheological (MR) grease is a suspension of magnetic particles in a grease which is a lubricant composed of base oils, thickeners, and additives [5]. The thickener has a three-dimensional network structure. We hope that the particles in MR greases are held by the structure. The shear-type MR grease damper has a wide dynamic range because it doesn’t require sealing elements to confine fluids. You can use this type of damper since the semisolid grease is

S. Nagamatsu () · T. Shiraishi Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_30

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less likely to leak out during operation. With such a mechanism, the shear-type MR grease damper solves the problems. Additionally, a past study showed that a semi-active method using the damper can suppress vibrations of a single-degree-offreedom structure in all the frequency range [6].

2 Methods 2.1 Shear-Type MR Grease Damper The proposed shear-type MR grease damper generates a current-controllable damping force. The brass plate moves in the brass box filled with the MR grease. In order to apply a magnetic field perpendicular to the vibration direction, the iron core and the electromagnetic coil are placed as shown in Fig. 1a and form a closed magnetic circuit. When you apply the current to the coil, a magnetic field is generated inside the coil. Then, in the gap between the box and the plate, MR effect arises by the formed magnetic circuit through the iron core. Finally, with the movement of the plate, the MR grease generates a force to resist the motion of the plate. This force includes not only the viscous damping force but also the frictional damping force by MR effect. The latter is the product of the yield shear stress of the MR grease and the area of the plate which contacts with the MR grease. In past studies, it was confirmed that the yield shear stress of MR fluids and MR greases increases in proportion to the square of the magnetic flux density [5, 7]. Hence, this damper generates the damping force which increases in proportion to the square of the external current.

Fig. 1 Schematic diagrams of the experimental devices

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2.2 Two-Degree-of-Freedom Vibration System The vibration system is composed of a structure model and the shear-type MR grease damper. To investigate the performance of the damper in a two-degreeof-freedom structure, a new structure model was constructed. It consisted of the base floor, the middle floor, and the target floor as shown in Fig. 1b. The first and second natural frequencies were 3.0 Hz and 7.9 Hz, respectively. These values were calculated using the specifications of the weight, 1.5 kg (each floor), and the stiffness: 1400 N/m (total of leaf springs). The feature of this system is the positional relationship between the damper and the target floor. The damper is located between the middle and base floor; thus it cannot exert forces to the target directly. In the vibration suppression test which is described below, the damping performance was verified by evaluating the reduction of the response of the top floor. Therefore, the novelty of this study is the vibration suppression of the target floor by a damper placed apart from the target.

2.3 Vibration Suppression Test In the vibration suppression test, we evaluated the damping performance of the damper under forced displacement excitations to the base. The experimental setup was arranged as shown in Fig. 2. A forced displacement was applied to the base floor by the electromagnetic actuator. The input waveform was produced by the function generator. Then, the displacements of three floors were measured by the laser displacement sensors. The experiment was conducted with the condition shown in Table 1. The frequency of input sinusoidal wave was in the range from 2.0 Hz to 12.0 Hz which includes the two natural frequencies. In the semi-active method, the current was varied in the range from 0 A to 4 A in response to the vibration of the structure. It was calculated in the controller based on the velocity-proportional control proposed in a previous study [6]. In this control law based on skyhook damper scheme, the output current Ic [A] according to the input relative velocity between the target and base floor x˙r [m/s] was calculated by the following equations: 5 5 / 5 −1.91x˙ − 0.15+ (1.91x˙ −0.11)2 −4 (0.56x˙ +0.15) (−0.09−36.59) x˙ 5 r r r r5 5 Ic = 5 5 5 5 2 (0.56x˙r + 0.15) (1)  I=

Ic 0

if x˙t x˙r > 0 if x˙t x˙r ≤ 0

"

x˙t : Absolute velocity of the target floor x˙r = x˙t − x˙b : Relative velocity

# (2)

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Fig. 2 An overview of the experimental setup Table 1 The experimental condition for the vibration suppression test Input wave Displacement amplitude of input Frequency of input Current Current (control) Control law Coil turns Temperature

Single sinusoidal wave 0.5 mm 2.0 Hz ~ 12.0 Hz 0 A (off-state), 4 A (on-state) Variable in the range from 0 A to 4 A Velocity-proportional control 125 40 ◦ C

In skyhook damper scheme, damping forces proportional to the absolute velocity of the target floor can reduce responses in all the frequency range. On the other hand, the actual forces are depending on the current and the relative velocity between the middle and base floor. In Eq. (1), forces proportional to the relative velocity x˙r were generated by changing the current according to x˙r . By using it in conjunction with Eq. (2), the current-dependent forces were exerted only when they can reduce the response.

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3 Results and Discussion 3.1 Transmissibility of Displacement The passive methods: off-state and on-state showed the different damping performance as shown in Fig. 3. Focusing on off-state, its transmissibility increased rapidly at two frequencies. It suggested that 3.4 Hz and 9.6 Hz were the experimental natural frequencies. Both of them were higher than the calculated values due to the effect of the viscosity of the MR grease and the actual stiffness which was different from expected one. On the other hand, on-state reduced vibrations at the natural frequencies. The reduction rates at 3.4 Hz and 9.6 Hz were 90% and 86%, respectively. It was due to so large damping force by the constant current 4 A that fixed between two layers of the base and middle floor. However, the result showed a demerit of on-state. In the middle frequency range, the transmissibility of on-state was larger than that of off-state. In that range, the small damping ratio is better than the large one to reduce the response. The limitation of passive methods was shown in this experiment. In contrast to these results, the semi-active method using the velocityproportional control obtained better damping performance in the frequency range from 5.0 Hz to 8.0 Hz and in the high frequency range such as 11.0 Hz and 12.0 Hz compared to on-state. However, at the natural frequencies, there was a problem that the vibration responses were larger than that of on-state. The reduction rates for off-state at 3.4 Hz and 9.6 Hz were 76% and 83%, respectively. As a result of the trying to suppress vibrations in the middle or higher frequency range, the damping performance at the natural frequencies got worse. To suppress vibrations in all

Fig. 3 The resonance curve under the single sinusoidal excitation

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the frequency range, the damping performance of the semi-active method around 3.4 Hz must be improved.

3.2 Discussion on the Control Law As shown in Fig. 3, the vibration response at the first natural frequency was not reduced sufficiently by the proposed semi-active method. In order to suppress vibrations around 3.4 Hz, a larger damping ratio is required. Then, we propose the following equation: / (3) Ic = α |x˙t | By taking the square root of x˙t , we hope that the generated force will be approximately proportional to x˙t since the force is proportional to the square of the current. The value of the coefficient α was 0.5 A/(mm/s)1/2 which was experimentally determined as the maximum current at 9.6 Hz was 4.0 A. Equation (3) gives you a maximum control current which is larger than the present value from Eq. (1): 3.0 A as shown in Fig. 4. Around the first natural frequency, x˙t is larger than x˙r since there is little phase difference. So, Eq. (3) realizes a larger damping ratio at 3.4 Hz. In order to see the effect of this equation in other frequencies, we did the same calculation. At the middle frequency, 6.0 Hz, the maximum value of the current was smaller than the value from (1). It means the reduction of the damping ratio, or the improvement of the damping performance. In this frequency range, x˙t is smaller than x˙r since the vibrations of the base and target floors are in antiphase. So, this calculation result makes sense. At the second natural frequency, 9.6 Hz, the maximum current was almost the same as the value from Eq. (1) since the calculated control currents are asymptotically approaching 4 A independent of the phase differences. From this result, Eq. (3) will reduce the vibration response by the maximum damping ratio. It is true that the proposed equation does not exert forces proportional to the absolute velocity, but it can generate currents that can solve the problem which we have now.

Fig. 4 The time series of the calculated current by Eqs. (1) and (3)

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4 Conclusion In order to verify the damping performance of the proposed shear-type MR grease damper, the vibration suppression test was conducted. The suppression target was a two-degree-of-freedom structure. Compared to the case without control, the velocity-proportional control using the damper reduced the vibration response by 76% and 83% at the first and second natural frequency, respectively. By changing the control scheme, the damping performance would be improved in all the frequency range.

References 1. Rabinow, J.: The magnetic fluid clutch. Trans AIEE. 67, 1308–1315 (1948) 2. Dyke, S.J., Spencer, B.F., Sain, M.K., Carlson, J.D.: An experimental study of MR dampers for seismic protection. Smart Mater. Struct. 7(5), 693–703 (1998) 3. Yao, G.Z., Yap, F.F., Chen, G., Li, W.H., Yeo, S.H.: MR damper and its application for semiactive control of vehicle suspension system. Mechatronics. 12(7), 963–973 (2002) 4. Shiraishi, T., Sugiyama, T.: Development of a shear type controllable damper with magnetorheological grease. Mech. Eng. J. 2(3), 14–00551 (2015) 5. Shiraishi, T., Miida, Y., Sugiyama, S., Morishita, S.: Typical characteristics of magnetorheological grease and its application to a controllable damper. Trans. Jpn. Soc. Mech. Eng. C. 77(778), 2193–2200 (2011). (in Japanese) 6. Shiraishi, T., Misaki, H.: Vibration control by a shear type semi-active damper using magnetorheological grease. J. Phys. Conf. Ser. 744, 012012 (2016) 7. Shiraishi, T., Morishita, S.: Measurements of typical characteristics of MR fluids and their application to design of MR devices considering working modes. Trans. Jpn. Soc. Mech. Eng. C. 70(696), 128–134 (2004). (in Japanese)

A Comparison of Vibration Control Performance for the Electromagnetic Damper with Various Control Strategies Hyung-Soo Kim, Seungkyung Kye

, and Hyung-Jo Jung

1 Introduction Stay cables are susceptible to vibrations because of their relatively small damping ratios and flexibility. Due to this reason, many studies have been conducted to increase the damping performance of cables for preventing unexpected disasters such as installing dampers or adjusting cable surface treatment. However, the existing methods using a cable damper for vibration reduction or an accelerometer for tension estimation have some limitations in that they operate separately and require an additional power source. To solve these drawbacks, a new cable vibration control system, which is capable of performing three functions simultaneously, vibration control, energy harvesting, and tension estimation, was developed by using an electromagnetic (EM) damper. An EM damper has been studied in many areas for its ability to harvest energy simultaneously with vibration control. Palomera-Arias (2005) made an attempt to apply the EM damper to civil engineering structures [1], and Shen and Zhu (2015) applied it to the bridge cable for harvesting energy from cable vibration [2]. In this study, the vibration control performance of an EM damper which is applied in various control strategies is evaluated. The EM damper can be used as an adaptable control device due to its characteristic that the damping force varies depending on the resistance of the connected external circuit. The effectiveness of semi-active control, passive control, and two-mode (with energy harvesting (EH) circuit) control strategies for the EM damper was numerically investigated.

H.-S. Kim · S. Kye · H.-J. Jung () Department of Civil and Environmental Engineering, KAIST, Daejeon, South Korea e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_31

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2 Theoretical Background 2.1 Numerical Model for Stay Cable For the numerical analysis, numerical modeling of the target cable and the EM damper was performed. Figure 1 and Eq. (1) show numerical model for stay cable with damper. The terms m, g, l, T0 , and λ denote mass per unit length, acceleration of gravity, cable length, tension force acting on the cable, and sag parameter, respectively. In this model, deflection of cable due to the sag was considered, and shape functions based on the deflection due to a static force at the damper were used. d2 λ2 m · v¨t (x, t) + c · v˙t (x, t) − T0 2 v (x, t) + 3 T0 dx l



l

v (x, t) dt = fw (x, t)

0

(1) Target cable model was made based on the Seohae Bridge cable resource. Figure 2 shows cable properties and simulation setup. The cable damper is installed at 3% of cable length, and response of cable is measured at 1/2 L and 1/4 L point.

2.2 Dynamic Model for the EM Damper The dynamic model is based on the Bouc-Wen model, which describes the behavior of the nonlinear hysteresis curve, and the model is shown in Fig. 3. This model was designed to simulate the behavior of the magnetorheological (MR) damper [3]. Equation (2) represents the damping force which is generated by damper. In this session, it was demonstrated that this dynamic model also effectively simulates the behavior of an EM damper. The firefly algorithm was applied as an optimization method to specify the parameters [4]. The fitted parameter values of EM damper connected to 0 Ohm are shown in Table 1.

Fig. 1 Numerical model for stay cable

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Fig. 2 Properties of target cable and simulation setup

Fig. 3 Phenomenological model for MR damper [3] Table 1 Parameter of dynamic model for the EM damper connected to 0 Ohm Parameter α n γ

Value 450.06 90.12 3404.3

Parameter B A c0

Value 9495.6 123.94 163.33

Parameter c1 k0 k1

Value 1964.2 3563.1 39.18

F = αz + c0 (x˙ − y) ˙ + k0 (x − y) + k1 (x − x0 ) = c1 y˙ + k1 (x − x0 ) ˙ |z|n + A (x˙ − y) ˙ z˙ = −β |x˙ − y| ˙ z|z|n−1 − γ (x˙ − y)

(2)

(3)

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Experimental results Simulation results

Damping force (N)

200 100 0 -100 -200 -300 -400 -500 -0.015

-0.01

-0.005

0

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0.01

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Fig. 4 Comparison of simulation and experimental results of an EM damper

y˙ =

1 (αz + c0 x˙ + k0 (x − y)) c0 + c1

(4)

The dynamic model of the damper was fitted to the external resistance of 0 Ohm, 27 Ohm, and EH circuit, respectively, and its behavior was sufficiently simulated with an error of about 10%. Figure 4 shows comparison of simulation and experimental behavior by a dynamic model and an actual damper.

3 Numerical Simulations and Results A numerical simulation based on the designed damper model was conducted to compare vibration attenuation performance under actual wind load conditions. A proper wind load model is needed to evaluate the vibration control performance in the wind load acting on the actual cable. The Kaimal spectrum which is a representative statistical turbulence model was used as input wind loads. The Kaimal wind spectrum is given by Eq. (5) where f is the frequency, h is the height of the wind hub, vw is the mean wind speed, z0 is the roughness length, and l is the turbulence length scale.    −1 2 ln zh0 · l · vw

s (f) =   1 + 1.5 f ·

l vw

 5 3

(5)

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50 40

Uncontrolled Passive control Two-mode control

Acceleration (m/s2)

30 20 10 0 -10 -20 -30 -40 -50 0

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Fig. 5 Acceleration response at midpoint Table 2 Maximum and RMS response at midpoint

Case Uncontrolled Passive control Two-mode control

Acceleration (m/s2 ) Maximum 42.0165 (100%) 31.5967 (75.20%) 27.0891 (64.47%)

RMS 5.5093 (100%) 3.9037 (70.86%) 3.5388 (64.23%)

According to the Korea Hydrographic and Oceanographic Agency (KHOA), the mean wind speed in 2017 was 3.73 m/s at the Seohae Bridge. The input wind load scenario was made based on the mean wind speed data at the Seohae Bridge investigated by KHOA [5]. The response of the cable with various control strategies was obtained through the numerical simulation. Figure 5 shows the acceleration response of the midpoint of the cable in time history. As shown in Table 2, the responses of passive control and two-mode control were reduced by 25% and 36%, respectively, compared with the uncontrolled case.

4 Conclusion In this paper, numerical model of cable with the EM damper is proposed, and its suitability is verified. In addition, numerical simulations were performed to compare the vibration control performance when various vibration control strategies were applied. This result shows the feasibility of the EM damper for cable structure, and the EM damper can be used as semi-active controller. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2019R1A2C2007835).

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References 1. Palomera-Arias, R.: Passive Electromagnetic Damping Device for Motion Control of Building Structures. Doctoral dissertation, Massachusetts Institute of Technology (2005) 2. Shen, W., Zhu, S.: Harvesting energy via electromagnetic damper: application to bridge stay cables. J. Intell. Mater. Syst. Struct. 26(1), 3–19 (2015) 3. Spencer Jr., B.F., Dyke, S.J., Sain, M.K., Carlson, J.: Phenomenological model for magnetorheological dampers. J. Eng. Mech. 123(3), 230–238 (1997) 4. Yang, X.S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press, Frome (2010) 5. COPYRIGHT© 2013 ~ 2016 KOREA HYDROGRAPHIC and OCEANOGRAPHIC AGENCY

Experiment and Numerical Investigations on a Vertical Isolation System with Quasi-Zero Stiffness Property Peng Chen

and Ying Zhou

1 Introduction √ The linear vibration isolators are effective when ωex /ωn > 2 (ωex is the frequency of the external input and ωn is the frequency of the isolated system). However, for excitations such as environment vibration, shock, earthquake, etc., low-frequency components exist that are unfavorable to the isolation system. Excessive isolation deformation and even amplified acceleration response are prone to occur under these types of excitations. To improve the performance of isolation under low-frequency vibrations, engineers and scientists have been investigating the nonlinear isolation devices. For the nonlinear stiffness isolation devices, they can be divided into two categories. The first type is based on geometric nonlinearity which means to realize the stiffness nonlinearity through the geometric orientation arrangement of a single element and its geometric deformation. Thus, the natural frequency of the isolator can be controlled at a small level. This type of devices was firstly proposed to reduce the ground micro-vibration for the use of gravitational wave detection, which includes torsional bar spring [1–3], X-shape pendulum [4], conical pendulum [5], Euler spring isolator [6], etc. The other type of stiffness nonlinear devices is obtained based on the combination of negative stiffness components and positive stiffness component. Most of these types of devices are referred to as quasi-zero stiffness (QZS) system. Alabuzhev firstly defined the QZS concepts and documented several constructions of the negative and positive stiffness element combination [7]. The QZS systems are characterized by “high static and low dynamic” stiffness, and they are supposed to obtain low natural frequency while restraining excessive isolation deformation. A P. Chen () · Y. Zhou () Tongji University, Shanghai, China e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_32

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variety of QZS devices have been proposed and investigated including pre-loaded linkage and linear spring [8], inclined spring and linear spring [9], folded pendulum [10], disc spring and linear spring [11], etc. Building on many previous works, for the QZS system composed of disc spring and linear helical spring, the nonlinear static and dynamic behaviors have been studied theoretically [12]. This paper mainly designed static and shake table tests to verify the corresponding theories and to evaluate the isolation performance of this type of vertical QZS isolator. Two different simulation approaches are proposed and verified by comparing the results of the two methods and the experimental results. The comparison indicates that both the two simulation approaches, the Simulink package and the open-source platform – OpenSees – can accurately calculate the dynamic response of the system.

2 Static Force-Displacement Relation 2.1 The Construction of Vertical QZS Isolation System The QZS isolation system is obtained by connecting the disc spring and linear helical spring as shown in Fig. 1. The two springs provide the restoring force in parallel, and the total restoring force FQZS is given as FQZS = Fd + Fl = Fd + kx

(1)

where Fl is the linear spring force, k is the spring stiffness, and Fd is the disc spring force, and it is strongly nonlinear which can generate negative tangent stiffness when

Fig. 1 Cross-sectional schematic diagram of the QZS system

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deformed at its flat position. The force-displacement relation of disc spring has been given by previous study [11].

2.2 Static Test Verification The static tests were performed using the MTS 809 loading system in Tongji University. The design parameters of the three different tested QZS systems are listed in Table 1. The experimental and test comparison results including the forcedisplacement and tangent stiffness-displacement curves are shown in Fig. 2. It is proved that the test results correlate well with the theoretical equations.

Table 1 Design parameters of the tested QZS system Parameters r1 r2 t h k

Value 50 mm 26 mm 2.2 mm; 1.8 mm; 1.5 mm 5.3 mm; 5.0 mm; 6.13 kN/mm

Parameters d1 d2 C = d1 /d2 r = h/t

Value 47 mm 29 mm 1.50 2.41; 2.78; 3.00

Fig. 2 Static experimental and theoretical comparison of the (a–c) force-displacement and (d–f) tangent stiffness-displacement

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3 Shake Table Test and Numerical Simulation 3.1 Comparison Between Simulink and OpenSees Results To accurately calculate the dynamic response of the vertical QZS isolation system, two numerical approaches are proposed. The equation of motion of the system under ground motions can be expressed as   u¨ˆ + 2ωn ξ u˙ˆ + 2pλ − r 2 λ + 1 ωn2 uˆ + λr 2 ωn2 uˆ 3 = −x¨g (t)/h

(2)

where uˆ is the non-dimensional displacement and x¨g (t) is the ground acceleration; other definitions can be found in [12]. The first method is to solve the above equation using Simulink package, and the second is to create a new QZS element in the open-source calculation platform – OpenSees – which can depict the force-displacement relation shown in Eq. (1) and calculate the dynamic response under ground motions. To numerically compare the two methods, an artificial earthquake record with a peak acceleration of 70 gal is selected which is shown in Fig. 3a. The equivalent viscous damping ratio is chosen as 10%. The acceleration and displacement time history response are compared in Fig. 3b and c. It is shown that the two methods lead to almost the same results. For the Simulink method, it can only solve the system that is exactly at the minimum stiffness condition (strict QZS system) due to the coordinate transform. However, the OpenSees method can calculate the response of the system at any condition, and the gravity applied is considered separately.

3.2 Comparison Between OpenSees and Shake Table Test Results Dynamic tests for the system with C = 1.5 and r = 2.41 listed in Table 1 are conducted to verify the accuracy of the numerical simulations. The electromagnetic table is used to perform harmonic shaking as shown in Fig. 4. To obtain the transmissibility of the system, the shake table performs harmonic vibrations with different frequency for each case in vertical direction. The system response is recorded by accelerometers, and the transmissibility is calculated accordingly for each input case. The transmissibility under different input frequency is presented in Fig. 5, and the corresponding OpenSees numerical simulation results are also plotted. The maximum displacement amplitude is controlled for the harmonic shaking, with 0.25 mm for DP1–12 and 0.5 mm for DP13–23. SDP1–12 and SDP13–23 are the corresponding simulation results. For predicting the transfer law of the system, the numerical method is more accurate in the resonant and high-frequency range, while there is a relatively large

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Fig. 3 Time history comparison of the two calculation method, (a) input wave, (b) acceleration response, and (c) displacement response

error in low-frequency range. The reason for the large relative error in low-frequency band is that the experiment and the simulation are performed using displacement control approach, which means the absolute acceleration value is small when the input frequency is small. Even though the relative error in low-frequency band is large, the overall absolute error is small. The comparison between numerical and simulation results verifies the accuracy of the OpenSees simulation method.

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Fig. 4 Electromagnetic shake table tests for QZS isolation system

Fig. 5 Transmissibility comparison between shake table (DP1–23) and numerical results (SDP1– 23)

4 Conclusions This paper introduces a novel QZS vertical isolation system using disc spring and helical spring connected in parallel. Static tests are performed to verify the mechanic equations of the system. Two numerical simulation approaches for dynamic time history calculations are proposed and compared and verified by shake table tests. Results prove that the static theoretical equations correlate well with

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the experiment and the proposed simulation method can accurately calculate the dynamic transmissibility of the QZS system. Acknowledgments The authors acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51878502) and Shanghai Science and Technology Commission Project (19XD1423900).

References 1. Winterflood, J., Blair, D.G.: A long-period vertical vibration isolator for gravitational wave detection. Phys. Lett. A. 243(1–2), 1–6 (1998) 2. Blair, D.G., Ju, L., Peng, H.: Vibration isolation for gravitational wave detection. Classical Quantum Gravity. 10(11), 2407 (1993) 3. De Salvo, R., Gaddi, A., Gennaro, G., et al.: A proposal for a pre-isolator stage for the Virgo super attenuator. VIRGO Note NTS. 96, 034 (1996) 4. Demetriades, G.F., Constantinou, M.C., Reinhorn, A.M.: Study of wire rope systems for seismic protection of equipment in buildings. Eng. Struct. 15(5), 321–334 (1993) 5. Winterflood, J., Barber, T.A., Blair, D.G.: A long-period conical pendulum for vibration isolation. Phys. Lett. A. 222(3), 141–147 (1996) 6. Winterflood, J., Barber, T.A., Blair, D.G.: Mathematical analysis of an Euler spring vibration isolator. Phys. Lett. A. 300(2–3), 131–139 (2002) 7. Alabuzhev, P.M.: Vibration Protection and Measuring Systems with Quasi-Zero Stiffness. CRC Press, Boca Raton (1989) 8. Platus, D.L.: Negative-stiffness-mechanism vibration isolation system. Vibration control in microelectronics, optics, and metrology. In: Gordon, C.G. (ed.) International Society for Optics and Photonics, vol. 1619, pp. 44–45. SPIE, Washington, DC (1992) 9. Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristics. J. Sound Vib. 301(3–5), 678–689 (2009) 10. Blair, D.G., Liu, J., Moghaddam, E.F., et al.: Performance of an ultra-frequency folded pendulum. Phys. Lett. A. 228(4–5), 243–249 (1994) 11. Meng, L., Sun, J., Wu, W.: Theoretical design and characteristics analysis of a quasi-zero stiffness isolator using a disk spring as negative stiffness element. Shock Vib. 2015, 1109–1137 (2015) 12. Zhou, Y., Chen, P., Mosqueda, G.: Analytical and numerical investigation of a quasi-zero stiffness vertical isolation system. J. Eng. Mech. 145(6), 04019035 (2019)

Robust Vibration Control of an Overhead Crane by Elimination of the Natural Frequency Component Kai Kurihara, Takahiro Kondou, Hiroki Mori, Kenichiro Matsuzaki, and Nobuyuki Sowa

1 Introduction When an overhead crane is used to move cargo, residual vibration often occurs at the end of the load transit, which significantly lowers efficiency. Many methods have been proposed to control transport with an overhead crane in a way that does not induce residual vibration [1]. These methods can be divided broadly into two categories: closed-loop and open-loop control. A previous paper [2] proposed a new type of open-loop control for damped nonlinear systems that uses the relation between residual vibration and the natural frequency component [3, 4]. In this study, to improve the robustness of the estimation error for the natural frequency, new conditions are assigned to design the trolley trajectories. Two approaches are proposed, where one is to increase the number of the eliminated frequency components and the other is to make the frequency derivatives of the frequency component zero at the estimated natural frequency. The effectiveness of these approaches for robustness was verified by numerical simulations, and the two methods were compared.

K. Kurihara () · T. Kondou · H. Mori · N. Sowa Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan e-mail: [email protected] K. Matsuzaki Department of Mechanical Engineering, Kagoshima University, Kagoshima, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_33

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2 Overhead Crane Model Figure 1 shows the analytical model for an overhead traveling crane. The model consists of a trolley of mass M, an inelastic massless rope of constant length l, and a cargo of point mass m. The trolley is driven by an input force fx in the x direction along a straight rail, and the cargo and the rope are assumed to move along a vertical plane containing the rail under the gravitational acceleration g. The damping coefficients for the damping forces acting on the trolley and the cargo are, respectively, denoted by C and c, and the translational displacement of the trolley and the angular displacement of the cargo (cargo sway) are, respectively, denoted by x and θ . The aim of the control is to use an input force fx to move the trolley from x = 0 to x = d during a transit time Twhile suppressing residual vibration. The overhead crane is in static equilibrium at τ = 0, where τ is defined as dimensionless time τ = t/T. The equations of motion for x and θ are derived as follows:   2 (1+μ) ξ

+μρ cos θ ·θ

+2μ ζξ +ζθ ωa ξ +2μρζ θ ωa cos θ · θ −μρ sin θ · θ =σξ , (1) ρ 2 θ

+ρ cos θ · ξ

+2ρ 2 ζθ ωa θ +2ρζ θ ωa cos θ · ξ +ρ 2 ωa2 sin θ =0,

(2)

where “ ’ ” = d/dτ . The other variables and parameters are as follows: x ξ= , L

m μ= , M

ωa =T ω˜ a ,

& g , ω˜ a = l

ζξ =

C , 2mω˜ a

ζθ =

c l , ρ= , 2mω˜ a L (3)

where L is the characteristic length and ωa is the actual value of the dimensionless natural angular frequency of the cargo sway. To ensure positioning accuracy of the trolley, the dimensionless control input σ ξ is set as follows: σξ = G (ξt − ξ ) ,

(4)

where ξ t is the target trolley trajectory to be determined and G is the feedback gain. Fig. 1 Analytical model of overhead crane

x

fx

M, C

Rail

O

Trolley q

g

l m, c

Cargo

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3 Control Strategy 3.1 Elimination of Natural Frequency Component To determine the target trajectory ξ t in Eq. (4) for the crane model, which has nonlinear and damping terms, we rearrange Eq. (2) as follows: θ

+ωe2 θ =σθe ,

σθe =ωe2 (θ − sin θ ) −

cos θ · ξ

+ 2ρζ θ ωe θ + 2ζθ ωe cos θ · ξ

, ρ (5)

where ωe is the estimated value of the dimensionless natural angular frequency that includes the estimation error and σ θe is the apparent external force that accounts for the influences of nonlinearity and damping. To verify the influence of the estimation error, the target trajectory is determined as ξ te (including the displacement error) from Eq. (1) using the estimated frequency ωe instead of the actual frequency ωa and Eq. (5). Numerical simulations were carried out using Eqs. (1), (2), and (4) and an assigned ξ te with the actual value ωa . Here, the relationship between the actual value ωa and the estimated value ωe is defined as follows: ωa = (1 + ε) ωe ,

(6)

where ε is a dimensionless parameter that represents the magnitude of the estimation error. σ θe is regarded as an external force acting on an undamped linear system having dimensionless frequency (and error) ωe . To suppress the residual vibration, σ θe must satisfy the following conditions [2]: 

1

 σθe cos ωe τ dτ = 0

0



1

σθe sin ωe τ dτ = 0.

(7)

0

3.2 Construction of Target Trajectory To move the trolley to the target position δ = d/L at τ = 1, it starts from a standstill, accelerates and decelerates smoothly, and comes to a standstill at the end of the control process. To accomplish this, we introduce the following six conditions for ξ : ξte (0) = 0 ,

ξ(1) = δ ,

ξte (0) = ξ (1) = 0 ,

ξte

(0) = ξ

(1) = 0.

(8)

Because the control input σ ξ is given by Eq. (4), the trolley motion does not perfectly track the target trajectory. Therefore, the conditions at τ = 1 in Eq. (8) are imposed on ξ .

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To determine the target trajectory, we represent it as follows [2]: ξte (τ ) =

N 

an Pn (τ ) ,

Pn (τ ) =

n=0

n  1 1 d n  6 2 τ τ −τ . n 3 n! τ dτ

(9)

This formula is composed of the basis functions Pn (τ ) (n = 0, 1, · · · , N), which are determined by modifying Legendre polynomials to satisfy Pn (0) = 0, Pn (0) = 0, and Pn

(0) = 0. Because ξ te defined in Eq. (9) preliminarily satisfies the conditions at τ = 0 in Eq. (8), the number of coefficients N + 1 is determined to be 5, resulting in N = 4. Newton’s method is employed to obtain the values of an that satisfy these five conditions.

3.3 Robustness Improvement Conditions To improve robustness of the estimation error for the natural frequency, the frequency component of σ θe around ωe needs to decrease. Here, we propose two methods for improving robustness. The first approach is to eliminate two frequency components near ωe : 

1

  σθe cos 1 ± εp ωe τ dτ = 0

 ∧

0

1

  σθe sin 1 ± εp ωe τ dτ = 0.

0

(10) The second approach is to make the frequency derivatives of the frequency component contained in σ θe zero at ωe : d dε



1 0

5 5 σθe cos (1+ε) ωe τ dτ 55

=0 ∧ ε=0

d dε



1 0

5 5 σθe sin (1+ε) ωe τ dτ 55

=0. ε=0

(11) For convenience, the control strategy using only Eq. (7) is referred to as ZF1 , single-zero-frequency component; that using only Eq. (10) as ZF2 , double-zerofrequency component; and that using both Eqs. (7) and (11) as D1 ZF1 , first-order differential and single-zero-frequency component. The number of conditions of ZF2 and D1 ZF1 increases to seven from the five conditions of ZF1 . Therefore, N in Eq. (9) is set at 6 when using ZF2 and D1 ZF1 strategies.

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4 Results and Discussion The simulations were numerically integrated using the Runge-Kutta-Gill method, and the interval 0 ≤ τ ≤ 1 was divided into 1024 intervals. The parameters used for the simulations are shown in Table 1. ζ ξ is determined based on the catalog value of a control equipment that is used in our experiments. The transit speed was described by the dimensionless parameter ν: ν = ωe /2π,

(12)

where ν is the ratio of the control time period τ = 1 to the dimensionless estimated natural period 2π /ωe . The numerical simulations for ν = 1.2 are shown in Fig. 2, where the trolley position ξ and sway angle θ of the cargo are plotted with respect to time. The black, blue, and red curves, respectively, represent the results for ZF1 , ZF2 (εp = 0.1), and D1 ZF1 , and the results without and with estimation error (ε = 0, 0.15) are, respectively, shown in panels (a) and (b). As shown in Fig. 2a, when ε is zero, the ZF1 and D1 ZF1 strategies completely suppress the residual vibration, but that remains in the case of ZF2 . In contrast, if ε has a nonzero value, the residual vibration amplitudes for ZF2 and D1 ZF1 are considerably smaller than that for ZF1 , as shown in Fig. 2b. Table 1 Dimensionless simulation parameters

μ 1.0

ζξ 5.00

ζθ 0.05

G 1.0 × 105

δ 1.0

ρ 1.0

Fig. 2 Simulations using ZF2 and D1 ZF1 control strategies for ν = 1.2. Trolley travel distance is shown on the left, and cargo sway is shown on the right

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Fig. 3 Effect of εon residual energy Eres for ν = 1.2

Furthermore, Fig. 3 shows the effect of ε on residual vibration for ν = 1.2. To evaluate the magnitude of the residual vibration, the residual vibration energy Eres is defined as follows:  5 5 2 (13) Eres = θ /2ωa2 + (1 − cos θ ) 5 . τ =1

The dashed blue curve represents the result for ZF2 (εp = 0.05), and the other curves represent the same cases as in Fig. 2. In Fig. 3, D1 ZF1 shows consistently smaller Eres than ZF1 , and both ZF2 cases reduce Eres when |ε| is large. Therefore, the conditions in Eqs. (10) and (11) are shown to be effective for improving estimation robustness.

5 Conclusion In addition to preventing residual vibration, the overhead crane control strategies developed in this study improved the robustness of the estimation error for the natural frequency. To improve the robustness, two approaches are proposed. One is to increase the number of eliminated frequency components, and the other is to make the frequency derivatives of the frequency component zero at the estimated natural frequency. The effectiveness of the methods was examined by numerical simulations, which showed that these approaches can improve the robustness to similar degrees. However, the latter approach is easier to use because it does not need the additional parameter εp to be determined.

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References 1. Abdel-Rahman, E.M., Nayfeh, A.H., Masoud, Z.N.: Dynamics and control of cranes: a review. J. Vib. Control. 9(7), 863–908 (2003) 2. Kurihara, K., Furuse, D., Kondou, T., Mori, H., Matsuzaki, K., Sowa, N.: Vibration control of overhead traveling crane by elimination of the natural frequency component (Optimization of trolley trajectory), The 17th Asia Pacific Vibration Conference, 2017, Nanjing, China, 13–15 November 3. Bhat, S.P., Miu, D.K.: Precise point-to-point positioning control of flexible structures. J. Dyn. Syst. Meas. Control. 112(4), 667–674 (1990) 4. Yamaura, H., Ono, K., Nagase, T.: Robust access control for a positioning mechanism with mechanical flexibility. Trans. Jpn. Soc. Mech. Eng..C. 58(549), 1399–1405 (1992). (in Japanese)

Development of an Active Mass Damper Driven by an Amplitude-Modulated Signal Toshihiko Komatsuzaki, Tetsuma Sadaoka, and Haruhiko Asanuma

1 Introduction Vibrations should be reduced in order to avoid structural failure and to provide safety and comfort to humans [1]. In recent mechanical systems, the problems have become obvious as the size and working speed of the equipment become smaller and faster. Much attention has been paid to develop vibration control devices, e.g., the vibration isolators and absorbers [2]. The use of the dynamic vibration absorbers (DVAs) offers a relatively simple and low-cost solution to mitigate vibrations in structures. They are classified into three types depending on their working principle: passive, active, and semi-active absorbers. The size of the absorbers is expected to be smaller in view of space saving and weight reduction. However, constitutive physical characteristics limit realization of the compact design without sacrificing the required damping performance. For example, in the case of the active devices, small-sized actuators may not be suitable to directly generate an inertial force at low frequency. Therefore, it is usually difficult to attenuate lowfrequency vibration by a miniaturized vibration control device. In acoustics, a sound production theory known as “parametric array” has brought the development of the high-directional loudspeaker. The parametric array utilizes the high directive characteristics of ultrasonic sound, where the embedded difference component of two ultrasonic carrier waves is expanded by the nonlinear acoustic effect as audible sound [3]. A numerical model of the parametric array was further expanded to

T. Komatsuzaki () · H. Asanuma Institute of Science and Engineering, Kanazawa University, Kanazawa, Ishikawa, Japan e-mail: [email protected]; [email protected] T. Sadaoka Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_34

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the amplitude-modulated primary carrier source expression, where an equivalent difference component is demodulated as an audible sound [4]. In the present study, being inspired by the acoustical research, we propose a novel active-type mass damper for vibration control of structures. By using a relatively small, highfrequency-driven actuator to drive the mass with an amplitude-modulated (AM) control signal, the low-frequency damping force can be indirectly demodulated. Mathematical formulation of the driving principle, both numerical and experimental verification of the demodulated low-frequency acceleration component to be used for the inertial control force of an active mass damper (AMD), is shown.

2 Theoretical Description of the Damper Model In this section, a mathematical formulation of the theory to drive the damper mass by the low-frequency component demodulated from the amplitude-modulated highfrequency carrier signal is shown.

2.1 Amplitude Modulation of a Signal and Its Demodulated Components Amplitude modulation (AM) is a method used to transmit information in the form of radio, acoustic, and light waves. The AM technique is commonly used in electro-communication, specifically in radio broadcasting. In AM, the strength of the message signal being transmitted is transformed into the magnitude of the monotonic carrier wave amplitude, whose frequency is usually much higher than frequencies contained in the message. For the purpose of illustration, we define the message signal with a single frequency ωs and an amplitude As , as fs = As sinωs t. The carrier wave with an amplitude Ac and frequency ωc is also given as fc = Ac sinωc t. By using these notations and by superimposing the former signal on the amplitude of the latter signal, the amplitude-modulated carrier signal, fAM , is expressed by the following equation: fAM = (Ac + As sin ωs t) sin ωc t

(1)

If we define the modulation index, α = As /Ac , Eq. (1) can be rewritten as follows: fAM = Ac (1 + α sin ωs t) sin ωc t c = Ac sin ωc t − αA 2 {cos (ωc + ωs ) t − cos (ωc − ωs ) t}

(2)

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Fig. 1 The amplitude-modulated signal used to drive an active mass damper: (a) an example of the waveform and (b) its frequency spectrum Fig. 2 A mathematical model of an active mass damper

Equation (2) signifies that two harmonic components other than the carrier frequency component ωc appear in the AM signal, namely, the sum and difference components, ωc + ωs and ωc − ωs . These are referred to as the sidebands. An example of the AM waveform and its frequency spectrum is schematically shown in Fig. 1.

2.2 Equation of Motion for a Vibration System with an AMD Driven by the AM Signal We consider a force-excited one-degree-of-freedom vibration system with an AMD attached to the primary system, as shown in Fig. 2. The AMD is expected to generate the control force in order to counteract the external force and to reduce vibration. The AMD driving signal, u(t), is assumed to be supplied in the form of the displacement command. The equation of motion for the system in Fig. 2 is expressed as follows: mx¨ + cx˙ + kx = −mu u(t) ¨ + F sin ωt

(3)

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Fig. 3 Frequency responses of a vibration system constituting AMD and the amplitude-modulated signal driving an actuator

In the above equation, m and mu represent the primary mass and the damper mass, c the damping coefficient, and k the spring constant. The amplitude-modulated driving signal to be supplied to the actuator is given by the following equation: u(t) = Ac {1 + α sin (ωc + ωs ) t} sin ωc t

(4)

The definitions of parameters in Eq. (4) are the same as in the previous subsection. Note here that demodulated component with a frequency ωs is of particular interest to be used as the control force. By the sum identity, Eq. (4) can be rewritten in the following form: u(t) = Ac sin ωc t −

αAc {cos (2ωc + ωs ) t − cos ωs t} 2

(5)

The third term in Eq. (5) signifies that the harmonic frequency component with the lowest frequency value is demodulated from the AM driving signal. By differentiating the equation twice with respect to time, the acceleration which is accordingly used to generate the inertial control force is obtained as u(t) ¨ = −Ac ωc2 sin ωc t +

αAc αAc 2 ω cos ωs t. (2ωc + ωs )2 cos (2ωc + ωs ) t − 2 2 s (6)

Although not shown in Fig. 2, the generated inertial force is assumed to be transmitted to the primary mass through a spring and a dashpot inserted in parallel between an actuator and the mass. If the carrier frequency ωc is large enough in comparison to the target frequency ωs and also if the spring and dashpot are designed so that the resonance of the adjacent system is located near ωc , these auxiliary elements are expected to work as a mechanical low-pass filter. Therefore, the first and the second terms on right-hand side of Eq. (6) have little influence on the response of the target vibration system having low-frequency resonance (ωc and 2ωc + ωs in Fig. 3). By neglecting these terms, the equation of motion can conclusively be obtained as follows:

Development of an Active Mass Damper Driven by an Amplitude-Modulated Signal

mx¨ + cx˙ + kx = mu

αAc 2 ω cos (ωs t + φ) + F sin ωt 2 s

275

(7)

3 Dynamic Characteristics Test for the Active Mass Damper Based on the theoretical and numerical verification, we developed an amplitudemodulated active mass damper (AM-AMD) whose size and weight are approximately 50 mm and 50 g. Detail of the design is summarized in Table 1. A voice coil motor is used to drive the damper mass. As shown in Fig. 4a, the mass is softly supported by the four leaf springs made of phosphor bronze so that the equilibrium position is ensured. Driving the actuator by the carrier signal, the fundamental test was conducted to demodulate the target frequency component. The test setup is schematically shown in Fig. 4b. An amplitude-modulated driving signal was edited by PC and supplied to the voice coil motor via a waveform generator and a servo driver. The acceleration of the mass was measured by an accelerometer attached to the mass. The waveform and its frequency spectrum were observed by a fast Fourier transform (FFT) analyzer. In the test, we investigated the frequency value and the magnitude of the demodulated components by changing combination of the target and carrier frequencies. Among the cases tested, a few response examples are shown in Figs. 5 and 6. For responses shown in Fig. 5, the carrier frequency was fixed at 1 kHz, while the target demodulated frequencies were set to 50 Hz and 200 Hz, respectively. In contrast, in Fig. 6, the target frequency was fixed at 50 Hz, and the carrier frequencies were set to

Table 1 Dimensions of the active mass damper components Vibrating part Plate spring (phosphor bronze)

Dimension (length, width, height) Mass Dimension (length, width, thickness)

(20, 20, 25) mm 45.0 g (23, 3, 0.2) mm

Fig. 4 An amplitude-modulated AMD device: (a) a photo of the fabricated device and (b) the fundamental test setup to drive and evaluate the AMD response

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Fig. 5 With the fixed carrier frequency at 1 kHz, frequency responses of AMD are compared for changing the target frequency fs : in (a), fs was set to 50 Hz, and in (b), fs was 200 Hz

Fig. 6 With the fixed target frequency at 50 Hz, frequency responses of AMD are compared for changing the carrier frequency fc : in (a), fc was set to 250 Hz, and in (b), fc was 1 kHz

250 Hz and 1 kHz. For all combinations of the carrier and modulation frequencies, the demodulated difference and sum acceleration components were observed whose frequency values were perfectly identical to the numerically predicted results. However, the amplitudes of the demodulated components did not correspond well with predictions. The modeling accuracy of the overall mechanical system can be improved, specifically by considering the response characteristic of the voice coil motor.

4 Conclusions In the present research, we developed a novel active-type mass damper for structural vibration control whose mass is actively driven by an amplitude-modulated signal. Fundamental test results have shown that the target difference frequency acceleration component for any combination of the carrier and modulation frequencies was successfully reproduced, which can accordingly be used to generate the inertial

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damping force of the active mass damper. We believe that the theory proposed herein contributes to downsize the existing mass dampers. Acknowledgments This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI, grant no. 18 K04014. The support is gratefully acknowledged.

References 1. Mohan, D.R.: Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J. Sound Vib. 262, 457–474 (2003) 2. Palm, W.J.: Mechanical Vibration. Wiley, Hoboken (2007) 3. Westervelt, P.J.: Parametric acoustic Array. J. Acoust. Soc. Am. 35(4), 535–537 (1962) 4. Yoneyama, M., Fujimoto, J.: The audio spotlight: an application of nonlinear interaction of sound waves to a new type of loudspeaker design. J. Acoust. Soc. Am. 73(5), 1532–1536 (1983)

Part VI

Active Noise Control for a Quieter Future

Building Vibration Suppression Through a Magnetorheological Variable Resonance Pendulum Tuned Mass Damper Matthew Daniel Christie, Shuaishuai Sun, and Weihua Li

1 Introduction Given the eternal prevalence of natural disasters, particularly those induced by seismic activity, there is a need for continued innovation and development in the field of seismic vibration control. With magnetorheological elastomer (MRE) tuned mass dampers [1, 2] and base isolators [3] showing much promise in protection of buildings and other civil structures through variable resonance, the stroke limitation of these devices is their primary weakness [2]. More-conventional magnetorheological fluid (MRF) dampers have been shown to attenuate vibration in buildings through damping control [4], however lacking the resonance shifting property which can further improve performance, particularly in tuned-mass-damper resonance seeking. Within the scope of magnetorheological-based devices, for a variety of applications such as vehicle shock absorbers [5] and legged robots [6], MRF has been repurposed from variable damping to variable stiffness through mechanical design. This has served these applications well in solving both the issues of the limited stroke achieved in MRE devices and the lack of stiffness or resonance control with MRF devices. Furthermore, for the field of seismic vibration control, MR dampers have a vaster research history and have already seen widespread application. As such, adoption of technology based on MRF should come with fewer logistical challenges in implementation than alternative MRE devices.

M. D. Christie () · W. Li School of Mechanical, Materials, Mechatronic, and Biomedical Engineering, University of Wollongong, Wollongong, NSW, Australia e-mail: [email protected] S. Sun New Industry Creation Hatchery Center, Tohoku University, Sendai, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_35

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Inspired by the exciting possibilities for MRF-based variable resonance in seismic vibration control, this work details the experimental studies and model validation of a variable resonance magnetorheological-fluid-based pendulum tuned mass damper (MR-PTMD). Through a differential mechanism, a rotary magnetorheological damper governs the flow of power between the pendulum mass and a torsional spring. Governing of this behavior produces a controllable shift in TMD resonance of up to 104% from the off-state value through control of damper torque. This is achieved through control of input current to the internal electromagnetic coils of the damper over a 0 A to 3 A range. Under the benchmark seismic records of the 1994 Northridge, 1940 El Centro, and 1885 Mexico City earthquakes, improved vibration suppression is demonstrated via semi-active control of the device in fivestory scale-building vibration experiments.

2 Device Design and Modeling 2.1 Device Structure The MR-PTMD (see Fig. 1) is based on a conventional PTMD, with a swinging mass of 1.75 kg on a moment arm lp of 70 mm, however connected to a planetary gearbox which differentially couples it to a rotary MR damper and torsional spring kt of 1.67 N·m/rad. This coupling mechanism allows the controlled addition of spring stiffness through the control of damper torque. This damper torque is varied through the supply of electric current of up to 3 A to the internal coils of the device, providing a magnetic field which increases the MR fluid viscosity from its off-state viscosity at 0 A.

Fig. 1 MR-PTMD CAD illustration (dimensions in mm)

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2.2 Dynamic Modeling and Validation Following from the simulation data presented in [7] and the rheological properties provided by the manufacturer of the MRF-140CG used (LORD Co.), the yield torque produced by the rotary MR damper TMR can be established. Key parameters for the MR-PTMD are illustrated in Fig. 2. In particular, the pendulum displacement θ p , the absorber displacement xa , and the gear teeth numbers of the ring gear NR = 41 and the sun gear NS = 9 are used to establish the model. Through the planetary gearbox, the torque as seen by the pendulum mass TGB is then defined as " TGB = kt

NR NS

#2

  θp + θp,yield − θp,max

(1)

when the damper is in the pre-yield state, i.e., TMR  ≤ TMR, yield , and   TGB = TMR,yield sgn θ˙p

"

NR NR + NS

# (2)

once the damper yields, i.e., TMR  > TMR, yield . From the masses of the pendulum mp and absorber ma , the pendulum length lp , fixture stiffness ka and damping ca , and internal gearbox losses cGB , the model is derived through a Euler-Lagrange approach, resulting in the two following equations:     ma + mp x¨a + mp lp θ¨p + ca x˙a + ka xa = −x¨g ma + mp ,

Fig. 2 MR-PTMD dynamic model: (a) front view and (b) side view

(3)

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Fig. 3 Resonant frequency comparison of experimental and simulated results

and mp lp x¨a + mp lp2 θ¨p + cGB θ˙p + TGB + mp glp θp = −x¨g mp lp .

(4)

From the experimental transmissibility characterization presented in [7] and simulations conducted in MATLAB Simulink, the MR-PTMD resonant frequency mapping to damper input current is shown in Fig. 3. Evident through similar trends in saturation and identical inflexion, the model fits the experimental result quite well.

3 Scale-Building Experiments 3.1 Experimental Setup To evaluate the performance of the prototype in a lab setting, the conventional approach of exciting a scale building was followed. In this case, a five-story building was constructed (see Fig. 4), following a 1:20 length scale factor, based on a 20 m tall building. Consequently, a 1:203 mass scale led to a floor mass of 5.0 kg. This building was excited with a linear-actuator-based shaker platform,

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excited along a single horizontal axis. Across multiple tests, accelerometers (ADXL 203 EB) measured acceleration of all floors, while displacement relative to ground was measured with a laser displacement sensor (Micro-Epsilon ILD1302-100). All information was recorded with a DAQ (NI cDAQ-9174/ NI9201), with the MRPTMD being controlled through a short-time Fourier transform algorithm running on an NI myRIO-1900. This controller adjusted the control current supplied through an amplifier to the damper of the MR-PTMD, such that the resonant frequency of the device matched the excitation frequency in real time, purposed to achieve optimal vibration attenuation.

3.2 Results To investigate the vibration suppression capability of the semi-active controlled MRPTMD, it was compared against two passive cases: passive-off, with a 0 A current supplied to the device, and passive-on, with a 1.8 A (optimal) current supplied. Evident in the multi-floor data of the building in Fig. 5, the semi-active case tends to best reduce vibration. In Fig. 5a, b, over the relatively wideband frequency range of the 1994 Northridge earthquake (Mw = 6.7), the lower- and higher-stiffness passive tunings more often result in exaggerated vibration. Meanwhile, semi-active control of the MR-PTMD results in overall optimal performance, particularly in achieving the lowest RMS displacement, at best reducing the fifth-floor RMS displacement by 15.2%. In Fig. 5c, d the relatively high dominant frequency of the 1940 El Centro record (Mw = 6.9) becomes evident, with the semi-active control leading to an identical result to the higher-stiffness passive-on tuning, in this case with the

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semi-active control resulting in a 23.6% displacement reduction. However, due to the lower dominant frequency present in the 1985 Mexico City case (Mw = 8.0) of Fig. 5e, f, the stiffer passive-on tuning is outperformed by the passive-off case, mirrored by the semi-active control, which adapts to these frequency shifts, albeit with a smaller 4.3% reduction here. Identical performance can be observed in the RMS acceleration.

4 Conclusion Demonstrated through scale-building seismic experiments, the designed and modelvalidated magnetorheological-fluid-based pendulum tuned mass damper is capable of achieving variable resonance and outperforming passive control modes. Through semi-active control, the device showed up to a 15.2% reduction in fifth-floor RMS displacement in the 1994 Northridge tests, 23.6% in the 1940 El Centro tests, and a 4.3% reduction in the 1985 Mexico City tests. With similar performance in terms

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of RMS acceleration, the MR-PTMD has proved to offer a useful approach to semiactive resonance control for seismic-protection applications.

References 1. Sun, S., et al.: Development of magnetorheological elastomers–based tuned mass damper for building protection from seismic events. J. Intell. Mater. Syst. Struct. (2018). https://doi.org/ 10.1177/1045389X17754265 2. Sun, S., et al.: An adaptive tuned vibration absorber based on multilayered MR elastomers. Smart Mater. Struct. 24(4), (2015) 3. Gu, X., et al.: Semi-active control of magnetorheological elastomer base isolation system utilising learning-based inverse model. J. Sound Vib. 406, 346–362 (2017) 4. Spencer Jr., B.F., et al.: Phenomenological model for magnetorheological dampers. J. Eng. Mech. 123(3), 230–238 (1997) 5. Sun, S.: Development of a novel variable stiffness and damping magnetorheological fluid damper. Smart Mater. Struct. 24(8), 085021 (2015) 6. Christie, M.D., et al.: A highly stiffness-adjustable robot leg for enhancing locomotive performance. Mech. Syst. Signal Process. 126, 458–468 (2019) 7. Christie, M.D., et al.: A variable resonance magnetorheological-fluid-based pendulum tuned mass damper for seismic vibration suppression. Mech. Syst. Signal Process. 116, 530–544 (2019)

Dynamic Property Optimization of a Vibration Isolator with Quasi-Zero Stiffness Huan Li, Jianchun Li, Yancheng Li, and Yang Yu

1 Introduction Vibration isolation, as an effective solution of structural vibration protection, has been widely investigated. The √ vibration isolation effect occurs only when its frequency ratio is larger than 2, which requires the system with low stiffness to realize low natural frequency. However, it is accompanied by sacrificing the loading support capacity of the system with large deformation. This problem can be avoided by utilizing QZS vibration isolator (QZSVI) since its nonlinear force-displacement characteristic endows it with high-static and low-dynamic stiffness that makes it realize small transmissibility and in the meantime provide sufficient loading support to the isolated structure. One of the typical QZSVIs, a three-spring mechanism designed by Molyneaux [1], is shown in Fig. 1. The two oblique springs contribute to negative stiffness, and the vertical spring is used to keep the system stable with positive stiffness. Niu et al. [2] designed a novel QZSVI by utilizing one disk spring instead of coil springs to provide negative stiffness. Compared with Molyneaux’s device, this device can support more load with small displacement at equilibrium position. In addition to the spring, pre-buckled beams also work as effective energy storage elements to generate negative stiffness. By comparing the properties of QZSVIs

H. Li · J. Li () · Y. Yu School of Civil and Environmental Engineering, Faculty of Engineering and Information Technology, University of Technology Sydney, Ultimo, Australia e-mail: [email protected] Y. Li School of Civil and Environmental Engineering, Faculty of Engineering and Information Technology, University of Technology Sydney, Ultimo, Australia College of Civil Engineering, Nanjing Tech University, Nanjing, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_36

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using mechanical springs and buckled beams, Mori et al. [3] demonstrated that utilizing buckled beams could reduce the ratio of local stiffness and static stiffness. Robertson et al. [4] developed magnetic QZSVI that used the magnetic levitation force to support load and generate negative stiffness, which could reduce the inherent frequency of system without scarifying the load-carrying capacity. Zhou et al. [5] adopted rollers, cams and springs, etc. to design and fabricate a novel QZSVI with two working states, which endows it the capacity to work under both smaller and larger amplitude excitation. Besides, a compact and easily mountable honeycombs QZSVI was fabricated in nylon 11. It can generate negative stiffness when external load is added and recover to its original shape after removing the external load. Virk et al. [6] and Izard et al. [7] proposed similar recoverable QZSVIs called cellular structures using metamaterials which can realize extremely high Young’s modulus and damping simultaneously. In this paper, the dynamic properties of Molyneaux’s QZSVI are investigated in depth. During this process, three optimization requirements are proposed, and then a comprehensive objective function is proposed to optimize the stiffness ratio. The optimization results are obtained by ensuring minimization of the objective function with genetic algorithm.

2 Dynamic Properties Investigation In Fig. 1, the three-spring model is adopted as the analytical model in this paper since it is the simplest prototype of a series of QZSVIs. By adjusting the values of the stiffness of springs and the structural parameters, the QZSVI can realize zero stiffness at the equilibrium position. Its static property analysis and optimization have been finished by Carrella [8], who proposed that the desirable initial angle

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θ should be 37◦ ~ 66◦ and the stiffness ratio α = kkhv is from 0.35 to 2.00. The dynamic properties of this QZSVI include resonance frequency, amplitudefrequency response (AFR), and force transmissibility. Its equation of motion under sinusoidal force F is expressed by Eq. (1). It can be further simplified as Eq. (2): mx¨ + cx˙ + F (x) − mg = F = f cos (ωe t)

(1)

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0 c , xˆ = xh , uˆ = hu , fˆ = kfv h . Note that whereτ = ωn t, γ = ωωne , ωn = kmv , ξ = 2mω n ωn is the natural frequency of the QZSVI around equilibrium position. The nonlinear coefficient μ = (2α+1)(4α+1) (4.16 ~ 1.41) is defined to evaluate the nonlinearity 8α 2 of the QZSVI. This equation is the Duffing equation of the nonlinear system under harmonic force excitation, the approximate analytic solution of which can be obtained by harmonic balance method [9]. The AFR and force transmissibility of the QZSVI are given by Eq. (3) and Eq. (7), respectively: "

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each γ and a peak value arising when γ equals to 1. Compared with that, the resonance frequency of the QZSVI is changeable, and there are several steady-state response solutions between its jump-down frequency γ res and jump-up frequency γ up [9], which means it is unstable within this region. The jump phenomenon can be avoided when damping ratio is larger than the threshold damping ratio in Eq. (6). A maximum AFR and a peak force transmissibility arise when the QZSVI is under resonance frequency, which are described by Eqs. (4), (8), and (5), respectively.

3 Dynamic Property Optimization A desirable vibration isolator should be designed with a low resonance frequency to expand its working frequency region and be able to isolate the transmission of external excitation without sacrificing the stability of the whole system, i.e., avoiding excessive deformation and deformation fluctuation. As the major contributor of a vibration isolation system, the performance of the QZSVI needs to be carefully tailored and optimized. Hence, in this section, a comprehensive optimization for QZSVI is conducted by considering all these requirements. The optimization parameters are the stiffness ratio α (or nonlinear coefficient μ), the damping ratio ξ , and the amplitude of excitation f. The relationships between them and property parameters are listed below (Table 1). The amplitude of force f is an external parameter, hence not regarded as an optimization parameter, which is set to 0.01. The situations with ξ = 0.01, 0.05, 0.10, and threshold damping ratio are investigated. Three weight coefficients are defined to present the importance of each optimization requirement. Besides, as Eq. (9) describes, the dimensionless method is adopted to make the three values in the same order of magnitude from 0 to 1. Therefore, the optimization aim is to gain the best nonlinear coefficient or stiffness ratio to make the objective function (9) to be minimization under different damping ratios: Oi = c1

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(9)

where c1 , c2 , and c3 are the weight coefficients. c1 + c2 + c3 =1, 0 ≤ c1 , c2 , c3 ≤ 1. Table 1 Relationships between optimization parameters and optimization requirements f Optimization requirements Small √ (resonance frequency)min √ (maximum AFR)min √ (peak force transmissibility)min √ Note: - no effect, good effect, × bad effect No. a b c

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In this paper, the genetic algorithm is used to solve the optimization problem by ensuring the minimum values of the objective function [10]. The optimization results for different cases are shown in Fig. 4. Figure 4a indicates that when c2 is close to 0, the optimal stiffness ratio is equal to 2, while it equals to 0.35 when c2 approaches to 1. According to line 1, the optimal stiffness ratio decreases with c1 increasing. In Fig. 4b, d, a larger stiffness ratio is profitable for obtaining better optimization result. However, when damping ratio is set to the threshold damping ratio, the optimal stiffness ratio is about 0.8 under case 1 and 0.35 under case 3. For case 2, when damping ratio increases to the threshold damping ratio, the objective function reaches to a minimum value at α = 0.35. The optimal stiffness ratios are 0.68, 0.53, and 0.40, respectively, when the values of damping ratio are 0.01, 0.05, and 0.10. That is because a small maximum AFR is the most important constraint condition and the effect of stiffness ratio on the maximum AFR, resonance frequency, as well as peak force transmissibility are contrary.

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4 Conclusion This paper conducts a comprehensive optimization of the dynamic properties of a typical QZSVI. An objective function is proposed by considering multiparameters, including the resonance frequency, maximum AFR, and peak force transmissibility, the importance of which is decided by three weight coefficients. The increasing of the weight coefficients c1 and c2 results in the decreasing of optimal stiffness ratio. The values of the objective function decrease with the increase of damping ratio. The optimal stiffness ratios of the cases with constant damping ratio and the threshold damping ratio are significantly different. According to the working condition, reasonable weight coefficients and damping ratio are necessary for designing a desirable QZSVI.

References 1. Molyneaux, W.: Supports for Vibration Isolation, ARC/CP-322 (1957) 2. Niu, F., Meng, L., Wu, W., et al.: Design and analysis of a quasi-zero stiffness isolator using a slotted conical disk spring as negative stiffness structure. J. Vibroeng.16(4), (2014) 3. Mori, H., Waters, T., Saotome, N., et al.: The effect of beam inclination on the performance of a passive vibration isolator using buckled beams. J. Phys. Conf. Ser. 744,. IOP Publishing, 012229 (2016) 4. Robertson, W.S., Kidner, M.R.F., Cazzolato, B.S., et al.: Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. J. Sound Vib. 326(1–2), 88–103 (2009) 5. Zhou, J., Wang, X., Xu, D., et al.: Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam–roller–spring mechanisms. J. Sound Vib. 346, 53–69 (2015) 6. Virk, K., Monti, A., Trehard, T., et al.: SILICOMB PEEK Kirigami cellular structures: mechanical response and energy dissipation through zero and negative stiffness. Smart Mater. Struct. 22(8), (2013) 7. Izard, A.G., Alfonso, R.F., McKnight, G., et al.: Optimal design of a cellular material encompassing negative stiffness elements for unique combinations of stiffness and elastic hysteresis. Mater. Des. 135, 37–50 (2017) 8. Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 301(3–5), 678–689 (2007) 9. Brennan, M., Kovacic, I., Carrella, A., et al.: On the jump-up and jump-down frequencies of the Duffing oscillator. J. Sound Vib. 318(4–5), 1250–1261 (2008) 10. Yu, Y., Li, J., Li, Y., et al.: Comparative investigation of phenomenological modeling for hysteresis responses of magnetorheological elastomer devices. Int. J. Mol. Sci. 20(13), 3216 (2019)

Study on the Influence of Structural Nonlinearity on the Performance of Multiunit Impact Damper Zheng Lu, Naiyin Ma, and Hengrui Zhang

1 Introduction In recent years, particle impact damper, with the advantages of robustness, has attracted much more attention in the field of vibration control in civil engineering [1]. The impact damper can effectively reduce the dynamic response of the primary structure by exchanging momentum and dissipating energy through the collision between solid particle and the cavity [2]. However, the impulsive force in the collision is great, with the attendant noise and potential local deformations. Therefore, some researchers have proposed multiunit impact damper (MUID) to replace impact damper. Some theoretical, numerical, and experimental studies have been carried out on the characteristics of the MUID attached to a single-degree-of-freedom linear structure [3–5]. However, the structure will inevitably enter the nonlinear state in extreme cases such as major earthquakes. Hence, there is an urgent need to study the influence of structural nonlinearity on the performance of the MUID. The contents of this paper are arranged as follows: Sect. 2 presents the model of a MUID and the governing equation of a primary structure with the MUID under random excitation. In Sect. 3, the performance of MUID under stationary random excitation is studied, based on both elastic and nonlinear benchmark structures. The influence of structural nonlinearity on the performance of MUID is also analyzed.

Z. Lu () Department of Disaster Mitigation for Structures, Tongji University, Shanghai, China State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China e-mail: [email protected] N. Ma · H. Zhang Department of Disaster Mitigation for Structures, Tongji University, Shanghai, China © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_37

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2 Simulation Method The MUID contains n units, and each unit consists of a mass mpi that is coupled to the primary structure by piecewise dashpot cpi and spring kpi , as shown in Fig. 1. The motion governing equation of the primary structure controlled by MUID under random excitation is as follows:     [M] U¨ + [C] U˙ + [K] {U } = {FI } + {FP }

(1)

where [M], [K], and [C] represent the mass, stiffness, and damping matrices of the primary structure, respectively. The structural responses, including displacement, velocity, and acceleration matrices, are shown in Eq. (1) as {U}, {U˙ }, and {U¨ }. The inertial force {FI } generated by the random excitation and nonlinear control force {FP } generated by the MUID are expressed as follows: {FI } = −u¨ g [M] {1I } {FP } = Φ

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where {1I }(N × 1) = [1 1 · · · 1]T and the u¨ g represents the ground acceleration. Φ denotes the location  vector of the MUID with n units. For the ith unit, the nonlinear control force {FP i during the collision consists of the nonlinear stiffness force and nonlinear damping force, which can be emulated by nonlinear functions, referred to as G(yi ) and H (yi , y˙i ) shown in Fig. 2. Therefore, the nonlinear control force can be calculated by cpi and kpi , which can be expressed as follows: kpi = mpi ωp2 = mpi (λω1 )2 = λ2 mpi ω12

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where ω1 and ωp represent the first-order frequency of the primary structure and the particle, respectively. λ is the coefficient of the rigid collision between the particle and the cavity, which equals to 20 based on previous studies [6]. ξ represents the damping ratio, which can be used to simulate inelastic impacts. In fact, the damping ratio is related to the well-known coefficient of restitution, as shown in Fig. 2c.

3 Performance Study In this section, the vibration control effect of MUID under stationary random excitation is analyzed, based on both elastic and nonlinear benchmark structures. The benchmark structure is a steel frame, which can calculate the nonlinear response by considering the material nonlinearity at the ends of the beam and column [7]. Furthermore, the stationary random excitation is obtained by filtering the normal (Gaussian) distribution by 0 to 5 Hz. To reach a stationary response, the simulations are done for at 4000 s, as shown in Fig. 3, in which μ, ξ , N, and d/σ x0 represent mass ratio, damping ratio, number of units, and gap clearance ratio, respectively. The root-mean-square (RMS) displacement σ x of the top layer is selected to quantitatively analyze the control performance of MUID, which is normalized by dividing by uncontrolled condition σ x0 . The effects of different system parameters on the RMS response level of the benchmark structure are plotted in Figs. 4, 5, and 6. It should be noted that these points on the figures are the results of numerical simulation and these lines are the results of the polynomial fitting of these points. Figure 4 shows the RMS response levels of the benchmark structure controlled by the MUID with different units. One can note that the control effect of the MUID on RMS displacement is significantly larger than the single-unit impact damper with the identical μ and ξ . However, when the other parameters are identical, the response of the benchmark structure controlled by the 10-unit impact damper and the 50-unit impact damper is very close. Hence, increasing the number of units after

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exceeding an appropriate value will not lead to higher control performance. Another important conclusion worth mentioning is that the robustness of the MUID is better than the single-unit impact damper with more stable control effect in a wider gap clearance. Figure 5 plots the RMS response levels of the benchmark structure controlled by the MUID with different mass ratios. It is observed from the diagram that the higher mass ratio of particles can reduce more RMS response of the controlled benchmark structure, but the attenuation of the RMS response is not directly proportional to the increase in mass ratio μ. Furthermore, with the improvement of the mass ratio, the vibration control effect of per unit mass ratio will be reduced in a nonlinear way. Figure 6 plots the RMS response levels of the structure controlled by MUID with different damping ratios. One can note that the lower values of damping ratio ξ lead

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0.80

0.80

0.75

0.75 2

4

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8 d/ σx0

(a)

10

12

14

2

4

6

8 d/ σx0

10

12

14

(b)

Fig. 6 RMS response levels for the benchmark structure (effect of damping ratio ξ , μ = 0.02, N = 10): (a) elastic structure; (b) nonlinear structure

to a more vibration reduction of the structure controlled by the MUID. The reason is that a lower ξ is corresponded to a higher coefficient of restitution e, and a higher e can lead to a higher relative velocity following the impact, which results in more collisions. The comparison between Figs. 4a, 6a and 4b, 6b shows that the RMS response level of the nonlinear structure is higher than the elastic structure. In other words, the structural nonlinearity will lead to a decrease in the effectiveness in the MUID to control the benchmark structure. In order to explore the reasons for the performance degradation of MUID, the effective momentum exchange [8] and energy dissipation during the operation of the MUID are calculated, as shown in Fig. 7. As can be seen, for the nonlinear structure, the effective momentum exchange and energy dissipation of the MUID are significantly lower than those of elastic

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8

10

x 10

9 8

2

Energy dissipation (J)

Effective momentum exchange (kg·m/s)

x 10 2.5

1.5 1

6 5 4 3 2

0.5 Elastic structure Nonlinear structure

0

7

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4

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(a)

10

12

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Elastic structure Nonlinear structure

1 0

2

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(b)

Fig. 7 (a) Effective momentum exchange; (b) energy dissipation of a MUID with 10 units, ξ = 0.2, and μ = 0.02

structures. Hence, the MUID has a better control performance on the vibration of the elastic structure, while the control performance on the nonlinear structure will be reduced.

4 Conclusion The MUID is generated by distributing the particle mass in the impact damper. In this paper, the vibration control effect of MUID under stationary random excitation is studied, and the influence of structural nonlinearity on the performance of the MUID is analyzed. The following conclusions can be drawn: 1. The control effect and robustness of the MUID is better than the single-unit impact damper, but increasing the number of units after exceeding an appropriate value will not lead to higher control performance. 2. The dynamic response of the benchmark structure can be significantly reduced by attaching a lightweight MUID with reasonable parameters. 3. The structural nonlinearity will lead to a decrease of the vibration control performance of the MUID. The reasons for this phenomenon are that the effective momentum exchange and energy dissipation of the MUID will decrease when the benchmark structure responds in a nonlinear state. Acknowledgments Financial supports from National Key Research and Development Program of China (2018YFC0705602, 2017YFC1500701) are highly appreciated. Financial support from the National Natural Science Foundation of China (51922080) is also highly appreciated. This work is also supported by Program of Shanghai Academic Research Leader (18XD1403900) and the Fundamental Research Funds for the Central Government Supported Universities (11080).

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References 1. Lu, Z., Wang, Z.X., Masri, S.F., Lu, X.L.: Particle impact dampers: past, present, and future. Struct. Control. Health. Monit. 25, 25 (2018) 2. Masri, S.F., Caughey, T.K.: On the stability of the impact damper. J. Appl. Mech. 33, 586–592 (1966) 3. Cempel, C.: The multi-unit impact damper: equivalent continuous force approach. J. Sound Vib. 34, 199–IN191 (1974) 4. Bapat, C.N., Sankar, S.: Multiunit impact damper—re-examined. J. Sound Vib. 103, 457–469 (1985) 5. Nayeri, R.D., Masri, S.F., Caffrey, J.P.: Studies of the performance of multi-unit impact dampers under stochastic excitation. J. Vib. Acoust.-Trans. Asme. 129, 239–251 (2007) 6. Masri, S.F., Ibrahim, A.M.: Response of the impact damper to stationary random excitation. J. Acoust. Soc. Am. 53, 200 (1973) 7. Ohtori, Y., Christenson, R.E., Spencer, B.F., Dyke, S.J.: Benchmark control problems for seismically excited nonlinear buildings. J. Eng. Mech. 130, 366–385 (2004) 8. Lu, Z., Masri, S.F., Lu, X.L.: Parametric studies of the performance of particle dampers under harmonic excitation. Struct. Control. Health. Monit. 18, 79–98 (2011)

Design of a Quasi-Zero Stiffness System Based on Electromagnetic Vibration Isolation Yu Chen, Hao Wen, and Dongping Jin

1 Introduction Nowadays, the requirements for stability of the future space-based missions are getting higher and higher. The satellite itself contains various types of sources of vibration, which will seriously affect the imaging quality of the optical payload [1]. In order to solve this problem, many researchers have put their efforts into the design of an effective vibration isolator. Some researchers focused on how to reduce the stiffness of the isolator so that it can isolate the vibration with low frequency. For example, Kamesh designed a low-frequency flexible space platform [2]. Notably, the quasi-zero stiffness (QZS) system is also a research topic of great concern for many other researchers. Carrella designed a simple QZS stiffness system that consists of one vertical linear spring and two inclined linear springs [3]. Some researchers introduced nonlinear springs to the system to improve the performance of vibration isolation [4]. Some other researchers analyzed a QZS system that is slightly different from the previous system three-spring QZS system in that the system is composed of one vertical spring and two horizontal springs [5]. QZS characteristics can also be provided by many other mechanisms. Some researchers combined springs and magnets to form a QZS system [6]. A cam-roller-spring mechanism was introduced to form a QZS system [7]. There are also some researchers also focusing on the torsion QZS system [8]. Usually, the QZS system is hard to design; however, in the space environment, the satellites are subjected to microgravity, which is beneficial to design a simpler but more effective system [9]. There are also some researchers putting their efforts into the design and control of an electromagnetic actuator for vibration isolation [10, 11]. But different from these existing electromagnetic

Y. Chen · H. Wen () · D. Jin State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_38

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isolators or actuators [10, 11], the isolator proposed in this paper has an improved structure which endows it with a natural equilibrium point, where the output force is zero without any active control. This paper aims at designing a QZS isolator based on electromagnetic vibration isolation. To this end, a nonlinear electromagnetic vibration isolator which has a softening stiffness is designed first, and then it is combined with a linear stiffness component to form a QZS isolator. The remainder of the paper is organized as follows: Section 2 introduces the design process of the QZS isolator and the effects of parameters are discussed. Section 3 builds up the dynamic model of the system and demonstrated its performance. And in Sect. 4, some conclusions on the performance of the isolator are given.

2 Design of the Isolator This section is aimed at designing an isolator that has remarkable softening stiffness. The structure of the proposed isolator is shown in Fig. 1. The isolator is combined with two coaxial cylindrical magnets and rectangle coils. When the coil is within the magnetic field of the magnet and electrified, Ampere’s force will generate on the coil. Stiffness can be calculated by differentiating Ampere’s force with respect to the displacement, which isk = dF/dx, where k denotes the stiffness, F Ampere’s force, and x the relative displacement between the two parts of the isolator. When x = 0 mm, forces applied to four sides of the coil will be equal and opposite, which endows the isolator a natural equilibrium point. The fictitious charge distribution of a cylindrical permanent magnet is the same as that of a thin coil [12], so the magnetic field of a cylindrical permanent magnet can

Fig. 1 Side view (left) and top view (right) of the isolator

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be calculated via Biot-Savart Law [13]. Providing that the equivalent plane current of the magnet is J, the height of the magnet is h, and the radius of the magnet is r0 , magnetic flux density at point p(x, z) can be written as μ0 J Bz = − 4π

 h 0



$

0

r0 (x − r0 cos θ ) cos θ − (r0 sin θ )2

%

3/2 dθ dz0 (x − r0 cos θ )2 + (r0 sin θ )2 + (z − z0 )2 (1)

where x = 0 represents the magnet’s centerline and z = 0 represents the magnet’s bottom. When relative displacement between coils and magnets appears, Ampere’s force applied to the coil is F = Jc

 i

a/2 −a/2



(b+d)/2 "

Bz

"0 # (x + q)2 + p2 , zi

(b−d)/2

"0 ## dqdp −Bz (x − q)2 + p2 , zi

(2)

where a, b, and d represent the length, width, and line width of the coil, p and q are the variables of integration, zi represents the height of the i-th layer, and Jc represents the current density of the coil. Jc is calculated by dividing total current I with respect to coil width d, which is Jc = I/d, and in the following discussion, Jc is assumed to be 1. The parameter Δk(N/m), which equals to k(x = 0 mm)-k(x = 8 mm), is chosen as the evaluation criteria to measure the nonlinearity of the isolator. Varying length a from 30 to 60 mm and b from 30 to 50 mm, the change of Δk is shown in Fig. 2. One can see in Fig. 2 that the size of Δk increases with a increasing. But with b increasing, the size of Δk increases first and then falls down. It is notable that the lowest point always appears around b = 25 mm. Considering that the real size of the isolator is limited, so the parameters are eventually selected as a = 60 mm and b = 25 mm. The fitting result of the stiffness with respect to the aforementioned parameters shows that it is a quadratic nonlinear system, which is kI = −kc + kx 2 = −450.9 + 3.621x 2

(3)

Combining it with an electromagnetic positive linear stiffness of suitable size kQZS = kc , then a QZS system will yield.

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Fig. 2 Stiffness when d = 4 mm, 0 ≤ a ≤ 60 mm, and 0 ≤ b ≤ 50 mm

3 Dynamic Analysis of the Isolator If the positive stiffness component is also a noncontact type, the entire isolator will become a noncontact isolator, and a system similar to the Disturbance-Free-Payload (DFP) system [14] will yield. The satellite is separated into a support module (SM) and payload module (PM) by the proposed isolator. The dynamic model of the system is mx¨ + cx˙ + kx 3 = −mx¨2

(4)

where x = x1 -x2 represents the relative displacement between PM and SM, x1 and c and m represent the displacement, damping, and mass of PM, and x2 represents the displacement of SM. Assuming that the base undergoes the excitationx2 = A2 cos (t), the dimensionless dynamic equation can be expressed as x¨ˆ + 2ξ x˙ˆ + a xˆ 3 = 2 cos (τ )

(5)

where 2n = xˆ =

kQZS m ;ξ

x A2 ; a

=

c x¨ x˙ ¨ ˙ 2mn ; xˆ = A2 2n ; xˆ = A2 n k/m k A 2 = kQZS A2 2 ;  = n ; τ 2n 2

=

= n t

(6)

Applying harmonic balance method and assuming thatxˆ = A1 cos (τ + ϕ), the motion absolute transmissibility will yield that

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Fig. 3 Transmissibility when ζ = 0.06 and A2 = 1, 1.4, 1.5, 2.5 mm and the corresponding linear system k = kQZS

' T =

1 + A1 2 +

2A1 2 2

"

3 aA1 2 − 2 4

# (7)

With Eq. (7), transmissibility with respect to  yields as shown in Fig. 3. Figure 3 shows the cases where ζ = 0.06 and the amplitude of base excitation varies from 1 mm to 2.5 mm. It is obvious that with excitation amplitude decreasing, the jump-down frequency and the transmissibility both decrease. Compared with the linear system with only the linear stiffness kQZS , the isolation capability of the case A2 = 2.5 mm is worse due to the large jump-down frequency. But it will be improved when excitation amplitude gets small, for example, smaller than 1.5 mm. The conclusion can be drawn that if the proposed system is to provide effective isolation, the excitation amplitude should be small enough, or in other words, it can effectively cope with micro-vibration. So far, the influences of input current I have not been discussed. As has been claimed in Sect. 2, kI is in proportion to I and so are k and kQZS , which means that the change of input current will not affect parameter a.√The only parameter related to input√current is = ω/ωn , and ωn is proportional to k, so  is inversely proportional to I . Assuming that when I=Ic , ωn is equal to 1 rad/s. The curve with respect to ω is shown in Fig. 4. The jump-down frequencies of the three cases are 0.082 rad/s, 0.26 rad/s, and 0.82 rad/s, respectively, which is obviously inversely proportional to the input current. So the characteristic of the proposed isolator is adjustable with input current

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Fig. 4 Transmissibility when A2 = 1.5 mm, ζ = 0.06, and I=Ic /10, Ic , Ic ∗ 10

even after the structure has been fixed, which outperforms some traditional QZS systems.

4 Conclusions This paper proposes a QZS system based on electromagnetic vibration isolation. First, this paper introduces the designing process of the negative stiffness component of the system. Second, the dynamic behavior of the proposed system is analyzed. The dynamic model is build up, and the motion absolute transmissibility is obtained by applying harmonic balance method. The influences of base excitation amplitude and input current are discussed in detail. Simulation results show that the proposed QZS system is significantly superior to traditional linear systems when coping with micro-vibration. And most importantly, it is easily adjustable through adjustment of the constant supply of coil current even after the structure has been fixed, which makes it surpass some QZS systems using traditional mechanical structures. Acknowledgments This is a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. This work was supported by the National Natural Science Foundation of China under Grants 11732006 and 11702146 and the Equipment Pre-research Foundation under Grant 6140210010202 and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures under Grant NUAA MCMS-0118G01.

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References 1. Li, M., Zhang, Y., Wang, Y.: The pointing and vibration isolation integrated control method for optical payload. J. Sound Vib. 438, 441–456 (2019) 2. Kamesh, D., Pandiyan, R., Ghosal, A.: Modeling, design and analysis of low frequency platform for attenuating micro-vibration in spacecraft. J. Sound Vib. 17(329), 3431–3450 (2010) 3. Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 3–5(301), 678–689 (2007) 4. Kovacic, I., Brennan, M.J., Waters, T.P.: A study of a nonlinear vibration isolator with a quasizero stiffness characteristic. J. Sound Vib. 3(315), 700–711 (2008) 5. Wang, X., Liu, H., Chen, Y.: Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis. Int. J. Mech. Sci. 142–143, 235–244 (2018) 6. Xu, D., Yu, Q., Zhou, J.: Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 14(332), 3377–3389 (2013) 7. Zhou, J., Wang, X., Xu, D.: Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam–roller–spring mechanisms. J. Sound Vib. 346, 53–69 (2015) 8. Zheng, Y., Zhang, X., Luo, Y.: Analytical study of a quasi-zero stiffness coupling using a torsion magnetic spring with negative stiffness. Mech. Syst. Signal Process. 100, 135–151 (2018) 9. Liu, C., Jing, X., Daley, S.: Recent advances in micro-vibration isolation. Mech Syst Signal Process. 56–57, 55–80 (2015) 10. Wu, Q., Yue, H., Liu, R.: Parametric design and multiobjective optimization of maglev actuators for active vibration isolation system. Adv. Mech. Eng. 6, 215358 (2015) 11. Gong, Z., Ding, L., Yue, H.: System integration and control design of a maglev platform for space vibration isolation. J. Vib. Control. 11(25), 1720–1736 (2019) 12. Ravaud, R., Lemarquand, G., Babic, S.: Cylindrical magnets and coils: fields, forces, and inductances. IEEE T Magn. 9(46), 3585–3590 (2010) 13. Agashe, J.S., Arnold, D.P.: A study of scaling and geometry effects on the forces between cuboidal and cylindrical magnets using analytical force solutions. J. Phys. D. Appl. Phys. 10(41), 105001 (2008) 14. Pedreiro, N.: Spacecraft architecture for disturbance-free payload. J. Guid. Control. Dyn. 5(26), 794–804 (2003)

Affine Combination of the Filtered-x LMS/F Algorithms for Active Control Somanath Pradhan, Xiaojun Qiu, and Jinchen Ji

1 Introduction The filtered-x least mean square (FxLMS) algorithm commonly employed in active noise control (ANC) systems uses an adaptive filter for generating the control signal, and the performance of the systems is highly reliant on the transient- and steadystate behavior of the control filter [1]. Despite having a simple structure, the FxLMS algorithm with a fixed step size maintains a balance between convergence speed (CS) and noise reduction (NR). Furthermore, the imperfect secondary path model and long-tap control filter affect the convergence. In order to circumvent the above issue, variable step size technique has been incorporated with FxLMS algorithm with moderately increased computational load [2]. Convex combination of adaptive filters has been proposed to achieve improved performance in transient and steady state, in which the overall adaptive filter exploits the abilities of the component filters [3]. Several researchers have integrated the convex combination into the ANC systems, in which multiple control signals are combined to produce the necessary cancelling signal. A convex combination of the FxLMS algorithms with different step sizes has been proposed for singlechannel and multiple-channel feedforward ANC systems, which is shown to outperform the VSS-FxLMS algorithm [4]. The convex combination strategy of the filtered-x least mean fourth (FxLMF) algorithms with different step sizes has been introduced in [5]. A filtered-x generalized mixed norm (FxGMN) algorithm and its convex combination have been reported in [6]. Recently, a FxLMS/F algorithm is introduced for ANC applications [7]; with an aim to boost the performance of the

S. Pradhan () · X. Qiu · J. Ji School of Mechatronic and Mechanical Engineering, Faculty of Engineering and IT, University of Technology Sydney, Sydney, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_39

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FxLMS/F, a convex combination strategy is integrated. The combining parameter in the convex combination strategy is restricted to lie in the range (0,1). There has been little research integrating the affine combination of adaptive filters [8] into the ANC applications. This paper proposes the affine combination of the FxLMS/F algorithms with different step sizes to achieve faster CS in transient state and lower residual error in steady state. A gradient-based update rule is developed for the combining parameter, which is not confined to lie in any interval. Simulations have been carried out to test the efficacy of the affine combination for white noise and multitone noise. Furthermore, the affine combination is investigated for a case of imperfect secondary path model.

1.1 The Proposed Affine Combination Algorithm To achieve improved control performance for an ANC system, the FxLMS/F algorithm has been proposed in [7], and the cost function of which is written as ζ(n) =

  1 2 1 e (n) − γ ln e2 (n) + γ , 2 2

(1)

where e(n) is the instantaneous residual error signal with n denoting the discrete time index and γ is a threshold to control the adaptation speed and NR performance. The controller weight update of the FxLMS/F algorithm is given by w (n + 1) = w(n) − μ





e3 (n) e2 (n) + γ

x (n) = w(n) −

μ e(n)x (n) 1 + γ /e2 (n)

(2)

where x (n) = [x (n), x (n − 1), . . . , x (n − L + 1)]T , obtained by filtering the reference signal x(n) through the model of the secondary path, and μ is the step size. It can be noticed that if γ  e2 (n), the algorithm in Eq. (2) acts as the FxLMF algorithm and if e2 (n)  γ , it acts as the FxLMS algorithm. Therefore, Eq. (2) has advantages of both the algorithms. The parameters μ and γ play a vital role in deciding the CS and NR. With an objective to achieve the conflicting requirement of faster convergence and lower residual noise, an affine combination of the FxLMS/F algorithms for a feedforward ANC system is proposed. The schematic block diagram of the affine combination algorithm for a singleˆ channel ANC system is depicted in Fig. 1. P(z), S(z), S(z) denote the primary path, secondary path, and estimated secondary path, respectively, and M is the length of secondary path and its estimate. L1 and L2 are the lengths of control filters w1 (n) and w2 (n). The combined control filter output can be written as y(n) = λ(n)y1 (n) + [1 − λ(n)]y2 (n), where λ(n) denotes the combining parameter for the affine combination, y1 (n) = w T1 (n)x 1 (n) and y2 (n) = w T2 (n)x 2 (n) with xi (n) = [xi (n), xi (n − 1), . . . , xi (n − Li + 1) ]T , i = 1, 2. The signal y(n) propagates

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Fig. 1 Schematic block diagram of the proposed affine combination algorithm

through the secondary path and acoustically combined with p(n), the unwanted primary disturbance. The residual error signal captured by the error microphone is e(n) = p(n) + s(n), where s(n) is the cancelling signal. Individual error signals e1 (n) and e2 (n) are required for updating the control filters w1 (n) and w2 (n), respectively, which in turn requires the availability of primary disturbance. However, the primary disturbance signal is not available independently in practical ANC systems. On ˆ the assumption that the secondary path model S(z) is obtained offline, p(n) can be estimated as p(n) ˆ = e(n) − sˆ ∗ {λ(n)y1 (n) + [1 − λ(n)] y2 (n)} , where sˆ denotes the impulse response vector of secondary path model and ∗ is the convolution operation. The individual error signals are obtained as e1 (n) = p(n) ˆ + sˆ ∗ y1 (n) = e(n) + [1 − λ(n)] [s1 (n) − s2 (n)] ande2 (n) = p(n) ˆ + sˆ ∗ y2 (n) = e(n) + λ(n) [s2 (n) − s1 (n)]. The individual control filters are updated using Eq. (2) as wi (n + 1) = wi (n) − μi

ei3 (n) ei2 (n) + γ

x i (n), i = 1, 2.

(3)

The combining parameter is updated by minimizing the cost function in Eq. (1) as λ (n + 1) = λ(n) − μa

  e3 (n) ∂ζ (n) = λ(n) + μa 2 (s2 (n) − s1 (n)) , ∂λ(n) e (n) + γ

(4)

where μa is the step size for updating λ(n). It has been reported that the affine combination can provide improved performance if the power-normalized least mean

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square (PN-LMS) algorithm is employed to update the combining parameter [9]. Hence, Eq. (4) is modified as 

μa λ (n + 1) = λ(n) + ps (n) + ε



 e3 (n) (s2 (n) − s1 (n)) , e2 (n) + γ

(5)

where ps (n) is the power of the signal s2 (n) − s1 (n), which can be estimated recursively as ps (n) = ρps (n − 1) + (1 − ρ)[s2 (n) − s1 (n)]2 , ε is a small positive constant to evade large step size if ps (n) becomes very small, and ρ is the forgetting factor (0.9 < ρ < 1). The overall control filter of the affine combination algorithm can be given by weq (n) = λ(n)w1 (n) + [1 − λ(n)]w2 (n). The above described affine combination of the FxLMS/F algorithms is hereafter termed as the A-FxLMS/F. It is to be noted that the affine combination strategy can be applied to any component algorithm with complementary performance, e.g., one algorithm with different step sizes, different algorithms, and algorithm with different filter lengths.

2 Simulations The efficacy of the proposed algorithm is evaluated through simulations in MATLAB environment. The primary path and secondary path considered here are FIR filters of length 128 each, which are obtained from the data in [10]. The estimated secondary path is of length M = 128. The two component filters are of length L1 = L2 = 128. The averaged noise reduction, ANR(n) = 20 log10 [Ae (n)/Ap (n)], is used as the metric for comparison, where Ae (n) = ρAe (n − 1) + (1 − ρ)|e(n)| and Ap (n) = ρAp (n − 1) + (1 − ρ)|p(n)| with Ae (0) = 0 and Ap (0) = 0. All the results obtained are averaged over 50 independent trials.

2.1 Case I The reference signal considered in this case is a white noise with zero mean and unit variance. The ANR curves are depicted in Fig. 2a. One can notice that the FxLMS/F algorithm with a large step size μ1 exhibits faster CS with higher steadystate residual noise, whereas the algorithm with a small step size μ2 exhibits slower convergence with lower steady-state residual noise. The A-FxLMS/F algorithm exploits the complementary ability of both the component algorithms, and therefore it achieves faster convergence in the transient state and lower residual error in the steady state. Figure 2b shows the variation of the combining parameter for the affine combination, which decides the amount of contribution of the component filters towards the overall performance. It can be seen that the affine combination follows the FxLMS/F with μ1 when the combining parameter is far from zero, and it follows the FxLMS/F

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Fig. 2 Case I: (a) ANR for white noise, (b) variation of combining parameter

with μ2 when λ(n) is close to zero. Unlike the convex combination, the combining parameter in affine combination does not have to lie in any restricted range, i.e., it can be more than one or negative value. The parameters used in the simulations for this case are μ1 = 0.0008, μ2 = 0.00002, μa = 0.005, γ = 0.00001, ε = 0.00001, and ρ = 0.999. The parameters are chosen by trial and error to maintain system stability and yet provide the best performance.

2.2 Case II The reference signal considered in this case is a multitone signal with variance 2, comprising frequencies of 150, 300, and 450 Hz. A white Gaussian additive noise with variance 0.001 is mixed with it. The learning curve for ANR is depicted in Fig. 3. One can observe that the proposed algorithm outperforms its components (the FxLMS/F algorithm with μ1 and μ2 ) in terms of faster convergence and lower residual noise. The reasons are the same as Case I. The parameters used in the simulations for this case are μ1 = 0.0002, μ2 = 0.000004, μa = 0.02, γ = 0.00001, ε = 0.00001, and ρ = 0.999.

2.3 Case III In this case, the integration of the affine combination strategy for the case of an imperfect secondary path model is investigated. The input signal is the same as Case I. The secondary path is changed at the middle of simulation, whereas the estimated secondary path is maintained invariant throughout the simulation. The ANR curve is depicted in Fig. 4. One can observe from the ANR curve that after secondary path change, the A-FxLMS/F algorithm behaves in a similar way to the one with

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Fig. 3 Case II: ANR for multitone noise

Fig. 4 Case III: ANR for white noise with an imperfect secondary path model

the perfect secondary path model (before path change), which agrees well with the results reported in [4]. It is worth mentioning that the phase deviation between S(z) ◦ ˆ and S(z) after secondary path change is less that ±90 . The parameters used in simulation for this case are μ1 = 0.0005, μ2 = 0.00005, μa = 0.005, γ = 0.01, ε = 0.00001, and ρ = 0.999. The proposed A-FxLMS/F algorithm outperforms its component FxLMS/F algorithms for various noise scenarios by taking the advantages of both the complementary algorithms. It is also effective for an imperfect secondary path model ◦ with a phase deviation less than ±90 ; however, the computational burden of the A-FxLMS/F algorithm is greater than twice that of the FxLMS/F algorithm. Since the component control filters are independent of each other, the inherent advantages of parallelism can be used on multiple processor core to reduce computational load per microprocessor core.

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3 Conclusion An affine combination of the FxLMS/F algorithms is proposed to achieve faster convergence in the transient state and lower residual error in the steady state. The combining parameter is updated using a gradient-based update rule. The simulation results illustrate that the A-FxLMS/F algorithm is effective for controlling white noise and multitones, and it is also effective when a perfect model of the secondary path is not available. Future work includes applying the affine combination strategy to frequency domain adaptive filters for active control operation.

References 1. Ardekani, I.T., Abdulla, W.H.: Theoretical convergence analysis of FxLMS algorithm. Signal Process. 90(12), 3046–3055 (2010) 2. Chang, D.C., Chu, F.T.: A new variable tap-length and step-size FxLMS algorithm. IEEE Signal Process. Lett. 20(11), 1122–1125 (2013) 3. Arenas-Garcia, J., Azpicueta-Ruiz, L.A., Silva, M.T., Nascimento, V.H., Sayed, A.H.: Combinations of adaptive filters: performance and convergence properties. IEEE Signal Process. Mag. 33(1), 120–140 (2015) 4. Ferrer, M., Gonzalez, A., de Diego, M., Pinero, G.: Convex combination filtered-x algorithms for active noise control systems. IEEE Trans. Audio Speech Lang. Process. 21(1), 156–167 (2012) 5. Al Omour, A.M., Zidouri, A., Iqbal, N., Zerguine, A.: Filtered-x least mean fourth (FXLMF) and leaky FXLMF adaptive algorithms. EURASIP J. Adv. Signal Process. 2016(1), 1–20 (2016) 6. Song, P., Zhao, H.: Filtered-x generalized mixed norm (FXGMN) algorithm for active noise control. Mech. Syst. Signal Process. 107, 93–104 (2018) 7. Song, P., Zhao, H.: Filtered-x least mean square/fourth (FXLMS/F) algorithm for active noise control. Mech. Syst. Signal Process. 120, 69–82 (2019) 8. Bershad, N.J., Bermudez, J.C.M., Tourneret, J.Y.: An affine combination of two LMS adaptive filters−transient mean-square analysis. IEEE Trans. Signal Process. 56(5), 1853–1864 (2008) 9. Candido, R., Silva, M.T., Nascimento, V.H.: Transient and steady-state analysis of the affine combination of two adaptive filters. IEEE Trans. Signal Process. 58(8), 4064–4078 (2010) 10. Sen, K.M., Morgan, D.R.: Active Noise Control Systems: Algorithms and DSP Implementations. Wiley, New York (1996)

An Experimental Study on Virtual Sound Barrier Performance in Workplaces Sipei Zhao

and Xiaojun Qiu

1 Introduction A virtual sound barrier (VSB) is an active noise control system that uses an array of loudspeakers and microphones to create a useful size of quiet zone. It can be used to reduce sound propagation, radiation, or transmission from noise sources or to reduce noise level around people in a noisy environment [1]. Various types of VSB systems have been investigated for different applications in the past decades. Zou et al. performed theoretical and experimental studies on a 16-channel cylindrical VSB system, where an average of more than 10 dB noise reduction up to 550 Hz was observed inside a cylindrical region with 0.2 m height and 0.2 m radius [2]. In a further study, Zou and Qiu investigated the effect of the human’s head on the performance of the cylindrical VSB system by assuming the human’s head to be a rigid sphere [3]. Numerical simulations and experimental results showed the tendency of the control performance with respect to the configuration with a scattering sphere to be similar to that without the sphere [3]. Similarly, Epain and Friot utilized 30 loudspeakers and microphones to create a quiet zone inside a sphere with a radius of 0.3 m, and their results showed that broadband noise can be cancelled in a frequency range up to 500 Hz [4]. In addition to the abovementioned three-dimensional VSB systems, planar VSB systems have been studied recently. Wang et al. developed a planar VSB system to reduce the acoustic radiation of a sound source inside a cavity through the baffled opening [5]. Experimental results demonstrated the planar VSB to be effective in reducing sound transmission through the opening below 500 Hz, except at some special frequencies that depend on the position of the secondary source

S. Zhao () · X. Qiu Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_40

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plane [5]. Tao et al. studied an independent system consisting of multiple singlechannel modules to reduce the complexity of the above planar VSB system, which achieves similar performance to the coupled system when the independent system is inherently stable [6]. To further increase the accessibility and light and air circulation through the opening, Wang et al. also explored the planar VSB systems with secondary sources and error microphones on the edges of the opening, where it was found that there exists an upper-limit frequency for the systems that depends on the size of the opening [7, 8]. Based on the abovementioned theoretical and experimental research in laboratories, the VSB systems have been widely applied to reduce noise in different scenarios [9]. Zou et al. [10] explored the VSB systems on the power transformer noise reduction. Zhao et al. employed a VSB system to attenuate traffic noise and create a small quiet zone at a parkland near a highway [11, 12]. This chapter investigates the performance of VSB systems for indoor workplace noise reduction. An eight-channel circular VSB system is built inside a small room to create a small quiet zone, and an eight-channel planar VSB system is mounted to the door of the small room to attenuate noise propagation from outside to inside. The experimental setup and measurement results are presented and discussed.

2 Experiments and Discussions In a VSB system, multiple secondary sources are used to produce control sound that cancels the primary noise to create a quiet zone. Therefore, the total sound pressure at the quiet zone is the superposition of the primary noise and the control sound, i.e., pt (ω) = pp (ω) + Z (ω) x (ω) q (ω)

(1)

where pp (ω) = [pp,1 (ω), pp,2 (ω), . . . , pp,N (ω)]T and pt (ω) = [pt,1 (ω), pt,2 (ω), . . . , pt,N (ω)]T denote the primary noise and the total sound pressure at angular frequency ω, respectively, and N is the number of the evaluation points in the quiet zone, x(ω) is the reference signal, Z(ω) is the transfer function matrix from the secondary sources to the error microphone, and q(ω) denotes the control coefficients [13]. The performance of the VSB systems is evaluated by the noise reduction (NR), which is defined as the ratio of the sum of the squared sound pressure inside the evaluation area without and with active control, i.e., $N % N 5 52  2 5pp (ri )5 / |pt (ri )| NR = 10log10 i=1

(2)

i=1

where N is the number of evaluation points, ri denotes the coordinate of the ith evaluation points, and pp and pt are the primary and total sound pressure, respectively [2].

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To investigate the noise attenuation performance of the VSB system in workplaces, experiments were conducted in a small room with sound-absorbing material finish on the floor and the walls, where a circular VSB and a planar VSB were installed. In the experiments, the eight-channel circular VSB system places eight loudspeakers uniformly along a circle with a diameter of 1.4 m to create a quiet zone with a diameter of 0.8 m inside the small room, as shown in Fig. 1. The error microphones were placed 0.3 m in front of each corresponding loudspeaker. Both the loudspeakers and the error microphones were 1.5 m above the ground. The primary noise was played back by two loudspeakers, which are 2.2 m from the VSB system. The reference microphone is between the VSB system and the noise sources, about 0.2 m in front of the noise sources. The TigerANC-II controller from Antysound was used to acquire the reference and error signals and update the control sound signals that are reproduced through the loudspeakers. The sound pressure level (SPL) at ten points inside the error microphone array was measured in 1/3 octave bands with and without active noise control (ANC), respectively, and the noise reduction is calculated according to Eq. (2). The measured SPL and NR are presented in Fig. 2a,b, respectively, where the error bar indicates the standard deviation of the SPL at ten measurement points. It demonstrates that the eight-channel circular VSB system reduces the low-frequency noise between 80 Hz and 250 Hz effectively, with a maximum noise reduction of 8 dB at 100 Hz. The eight-channel planar VSB system mounts eight loudspeakers to the door of the small room to prevent noise from propagating inside from the outside, as illustrated in Fig. 3. The interval distances between the loudspeakers were 0.4 m. The error microphones were 0.3 m in front of each corresponding loudspeaker. The primary noise was played back by a loudspeaker that was 2.2 m from the door outside the room, and the reference microphone was 0.2 m in front of the noise

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Fig. 3 (a) Diagram and (b) photo of the experimental setup for the planar VSB

source. The same ANC controller and experimental procedure as the circular VSB were used for the planar VSB. The SPL at ten locations inside the room were measured with and without ANC, respectively, and the averaged SPL are compared in Fig. 4a. The NR is calculated according to Eq. (2) and is shown in Fig. 4b. The experimental results demonstrate that the eight-channel planar VSB system is effective in reducing the noise below 250 Hz, and the noise reduction can reach about 6 dB at 80 Hz.

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Unfortunately, Figs. 2 and 4 show that both the circular and planar VSB systems cannot reduce the higher-frequency noise above 400 Hz due to the small number of secondary sources. The effective frequency range is determined by the distance between the secondary sources. When the distance between the secondary sources is larger than one half of the noise wavelength, the system is ineffective. For the circular VSB system, the distance is about 0.55 m, which corresponds to an effective frequency range below 309 Hz. For the planar VSB system, the distance is 0.4 m, corresponding to an effective frequency range of 425 Hz. To reduce noises above 400 Hz, a larger number of secondary sources are needed to achieve a better performance. This will be investigated in the future work.

3 Conclusions This chapter investigates the performance of the VSB systems for noise attenuation in workplaces by presenting the measurement results of an eight-channel circular VSB inside a small room and an eight-channel planar VSB system mounted to the door of the room. Firstly, eight loudspeakers are uniformly placed along a circle with a diameter of 1.4 m to form a circular VSB to create a quiet zone with a diameter of 0.8 m. The experimental results show that the circular VSB system is effective in reducing the low-frequency noise from 100 Hz to 250 Hz, and the noise reduction can reach 8 dB at 100 Hz. Then, the loudspeakers are mounted at the door of the room to form a planar VSB system to reduce the noise propagating from outside to inside. The planar VSB system is found to be effective in reducing the low-frequency noise below 250 Hz, and the noise reduction in the room can reach about 6 dB at 80 Hz. However, neither the circular nor the planar VSB system was able to attenuate the higher-frequency noise above 400 Hz, which will be further studied with a larger number of secondary sources in the future.

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Acknowledgments Dr. Jordan Lacey and Mr. Simon Maisch of RMIT University are sincerely appreciated for their help in the experiments.

References 1. Qiu, X.: An Introduction to Virtual Sound Barrier. CRC Press, New York (2019) 2. Zou, H., Qiu, X., Lu, J., Niu, F.: A preliminary experimental study on virtual sound barrier system. J. Sound Vib. 307, 379–385 (2007). https://doi.org/10.1016/j.jsv.2007.06.042 3. Zou, H., Qiu, X.: Performance analysis of the virtual sound barrier system with a diffracting sphere. Appl. Acoust. 69, 875–883 (2008). https://doi.org/10.1016/j.apacoust.2007.06.002 4. Epain, N., Friot, E.: Active control of sound inside a sphere via control of the acoustic pressure at the boundary surface. J. Sound Vib. 299, 587–604 (2007). https://doi.org/10.1016/ j.jsv.2006.06.066 5. Wang, S., Tao, J., Qiu, X.: Performance of a planar virtual sound barrier at the baffled opening of a rectangular enclosure. J. Acoust. Soc. Am. 138, 2836–2847 (2015). https:// doi.org/10.1016/j.apacoust.2015.12.019 6. Tao, J., Wang, S., Qiu, X., Pan, J.: Performance of an independent planar virtual sound barrier at the opening of a rectangular enclosure. Appl. Acoust. 105, 215–223 (2016). https://doi.org/ 10.1016/j.apacoust.2015.12.019 7. Wang, S., Tao, J., Qiu, X., Pan, J.: Mechanisms of active control of sound radiation from an opening with boundary installed secondary sources. Cit. J. Acoust. Soc. Am. 143, 3345 (2018). https://doi.org/10.1121/1.5040139 8. Wang, S., Tao, J., Qiu, X., Pan, J.: A boundary error sensing arrangement for virtual sound barriers to reduce noise radiation through openings. Cit. J. Acoust. Soc. Am. 145, 3695 (2019). https://doi.org/10.1121/1.5112502 9. Qiu, X., Zou, H.: Recent progress in research on virtual sound barriers. In: Proceedings of ACOUSTICS 2016. pp. 1–10. Brisbane, Australia (2016) 10. Zou, H., Huang, X., Hu, S., Qiu, X.: Applying an active noise barrier on a 110 KV power transformer in Hunan. In: Inter-Noise 2014. pp. 5561–5567 (2014) 11. Zhao, S., Cheng, E., Qiu, X., Lacey, J., Maisch, S.: A method of configuring fixed coefficient active noise controllers for traffic noise reduction. In: Proceedings of INTER-NOISE 2017. p. 8 (2017) 12. Zhao, S., Qiu, X., Lacey, J., Maisch, S.: Configuring fixed-coefficient active control systems for traffic noise reduction. Build. Environ. 149, 415–427 (2019). https://doi.org/10.1016/ j.buildenv.2018.12.037 13. Nelson, P.A., Elliott, S.J.: Active Control of Sound. Academic, London (1992)

Active Control of Sound Transmission Through an Aperture in a Thin Wall Shuping Wang and Xiaojun Qiu

1 Introduction Much research has been carried out on the active control of sound radiation through openings due to its better performance than traditional passive noise control methods at low frequencies [1]. Elliott et al. investigated the active control of incident sound through apertures and found that only a few secondary sources are needed if the aperture size is compatible with the acoustic wavelength, while more secondary sources are required for larger openings [2]. To reduce the effect of the active noise control (ANC) system on the lighting and natural ventilation of the opening, a double-layer secondary source arrangement at the boundary of the opening was proposed, which works effectively up to an upper-limit frequency determined by the opening size [3]. Active control of sound transmission through small openings has also been investigated in previous work. Qiu et al. proposed to install an ANC system inside a hole of 6 cm × 6 cm × 30 cm and found that a system with a single-channel ANC can achieve 20 dB noise reduction at frequencies below 2700 Hz, while a system with four secondary sources can increase the upper-limit frequency to 3900 Hz [4]. ANC on a shorter aperture is investigated in this chapter. There are many potential applications of this research, including designing of noise control enclosures and improving the transmission loss of thin structures where holes are required for ventilation, cabling, and/or access.

S. Wang () · X. Qiu University of Technology Sydney, Sydney, NSW, Australia e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_41

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2 Theory A schematic diagram of the model of an aperture on an infinitely large rigid wall is shown in Fig. 1. The size of the aperture is lx × ly × lz . The origin of the coordinate is located at one of the corner of the aperture. The primary source (the red star in Fig. 1) is on one side of the aperture, and the objective is to reduce sound transmission through the aperture to the receiver side. The theoretical sound field for such a model is investigated with the modal superposition method. The sound pressure at r on the source side can be expressed as [5] 5 5 5 5 −jk 55r−r’s 55

p1 (r) =

jρ0 ωq 0 e jρ0 ωq 0 e−jk|r−rs | 5 5 + 5 5 4π |r − rs | 4π 5r − r’s 5

 +

G1 (r, r0 )

∂p1 (r0 ) dS0 , ∂z

S1

(1) where ρ 0 is the air density, ω is the angular frequency, k is the wavenumber, q0 is the strength of the sound source located at rs , and rs ’ is the location of the image source regarding z = 0 plane, and G1 (r, r0 ) =

e−jk|r−r0 | . 2π |r − r0 |

(2)

The sound pressure inside the aperture is p2 (r) =

  PNi e−jkNz z + PNr ejkNz z ϕn (x, y) ,

(3)

n

where " ϕn (r) = cos

# " # ny π nx π x cos y , lx ly

(4)

and the sound pressure on the receiver side is p3 (r) = −

1 2π



∂p3 (r0 ) ∂z

"

# exp (−jk · r) dS0 . r

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S2

By combing the boundary conditions at the two ends of the aperture, which are the continuity of the sound pressure and particle velocity, the coefficients PNi and PNr can be obtained, which are used to calculate the sound pressure in each part of the system.

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The sum of the squared sound pressures at error points (where the error microphones are located at) plus the weighted control source power is defined as the cost function [6] J = |p|2 + βqH s qs

(6)

where p is the vector consisting of the sound pressures at error points, β is a positive real number to constrain the control effort, and qs is the vector consisting of the strengths of the secondary sources [7]. By minimizing Eq. (6), the optimal strengths of secondary sources can be obtained, which is  −1 q s = − ZH Z + βI ZH se se se Ze qp ,

(7)

where Zse is the matrix of acoustic transfer functions from the secondary sources to the error microphones, Ze is the vector of the acoustic transfer function from the primary source to the error microphones, and qp is the strength of the primary source. The noise reduction performance of the active noise control system is evaluated with the sound power level calculated with the sound pressures at far-field evaluation points on a semi-sphere at the center of the end of the aperture, which are shown as “x” in Fig. 1.

Fig. 1 The model of an aperture on an infinitely large thin wall

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3 Numerical Simulations 3.1 Verification of the Theoretical Model In the numerical simulations, the size of the aperture is 0.06 m × 0.06 m × 0.08 m. The sound pressure levels at (0.03, 0.03, 0.05) m, which is inside the aperture, and (0.04, 0.05, 0.12) m, which is on the receiver side when the sound source is located at two different positions, are shown in Fig. 2. It is clear that they agree well with the results obtained with the finite element method (FEM) software Virtual.Lab at every frequency below 4000 Hz, and it demonstrates the validity of the theoretical method. Due to the less computation time of the theoretical method, the acoustic transfer functions are calculated with the theoretical method in this chapter.

3.2 Incident Directions of the Primary Sound The primary source is assumed to be a monopole point source with the strength of 10−4 m3 /s, and the distance between the primary source and the center of the aperture is 0.1 m. The secondary source is at (0.03, 0.03, 0.06) m inside the aperture. The sound power levels with and without ANC when the incidence direction (θ in Fig. 1) is 0◦ , 30◦ , and 60 ◦ are shown in Fig. 3. The sum of the squared sound pressures at the evaluation points is applied as the cost function to optimize qs as it provides the upper-limit performance for the specified configuration of secondary sources, and β is set as 0.1 in the simulations. It can be observed from Fig. 3 that when the primary source is located at the direction of 0◦ , such a system with one secondary source at the center of the aperture 90

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Fig. 3 The sound power levels of the system with and without ANC with the primary sound incident from different directions

can achieve more than 20 dB noise reduction at all the frequencies below 4000 Hz, while the upper-limit frequency of effective control is about 2900 Hz for 30◦ and 2700 Hz for 60◦ .

3.3 Configurations of Secondary Sources Different configurations of secondary sources are investigated when the primary source is fixed at (0.03, 0.03, −0.1) m. The sound power levels with and without ANC are shown in Fig. 4. All the secondary sources are located at z = 0.06 m plane inside the aperture. It can be seen from Fig. 4 that the system with a secondary source at the center of the aperture performs significantly better than that with one at the corner of the aperture in this case. Adding another secondary source at the corner of the aperture improves the performance a little, and when there are four secondary sources at the corners, with one at each corner, the noise reductions are significantly increased.

3.4 Configurations of Error Microphones The effect of error microphones on the noise reduction performance of the ANC system is also investigated. In this case, the strengths of secondary sources are

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optimized by minimizing the sum of the squared sound pressures at the error points. The primary source is located at (0.03, 0.03, −0.1) m, and the secondary source is at the center of the aperture, which is at (0.03, 0.03, 0.06) m. Two configurations of error microphones are compared: one error microphone 0.01 m in front of the secondary source and four error microphones at the four corners of the aperture, which are also in the plane 0.01 m in front of the secondary source. The results are shown in Fig. 5.

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It can be seen that a single error microphone in front of the secondary source at the center of the aperture can only provide effective noise reduction below 1500 Hz, while the system with four error microphones at the four corners of the aperture reduces the sound power level by more than 10 dB at all the frequencies below 4000 Hz.

4 Conclusions The performance of an ANC system applied at an aperture in a thin wall to reduce sound transmission through the aperture is investigated in this chapter. It is found that an ANC system with a single secondary source at the center of the 0.06 m × 0.06 m aperture with a thickness of 0.08 m can provide more than 20 dB noise reduction below 4000 Hz when the incident sound is from the direction of 0◦ . Secondary sources at the corner of the aperture do not perform as well as that at the center, but the performance can be improved by increasing the number of them. It is also found that implementing a single error microphone in front of the secondary source at the center can only provide effective noise reduction below 1500 Hz, while introducing four error microphones at the four corners of the aperture can increase the frequency of effective control to 4000 Hz.

References 1. De Salis, M., Oldham, D., Sharples, S.: Noise control strategies for naturally ventilated buildings. Build. Environ. 37(5), 471–484 (2002) 2. Elliott, S., Cheer, J., Bhan, L., Shi, C., Gan, W.: A wavenumber approach to analysing the active control of plane waves with arrays of secondary sources. J. Sound Vib. 419, 405–419 (2018) 3. Wang, S., Tao, J., Qiu, X., Pan, J.: Mechanisms of active control of sound radiation from an opening with boundary installed secondary sources. J. Acoust. Soc. Am. 143(6), 3345–3351 (2018) 4. Qiu X., Qin M., Zou H.: Active control of sound transmission through a hole in a large thick wall. In: Proceedings of Acoustics 2017, Perth, Australia (2017) 5. Sgard, F., Nelisse, H., Atalla, N.: On the modeling of the diffuse field sound transmission loss of finite thickness apertures. J. Acoust. Soc. Am. 122(1), 302–313 (2007) 6. Nelson, P., Elliott, S.: Active Control of Sound. Academic, London (1992) 7. Kirkeby, O., Nelson, P., Hamada, H.: Local sound field reproduction using digital signal processing. J. Acoust. Soc. Am. 100(3), 1584–1593 (1996)

Real-Time Active Noise Control of Multi-tonal Noise Based on Multiply Connected Single Adaptive Notch Filters Shun Hirose, Toshihiko Komatsuzaki, Naoki Kimura, Keita Tanaka, and Taisei Yamaguchi

1 Introduction Noise problems have become obvious in various environments due to the massive development of industrial machineries. In general, the soundproof materials, soundabsorbing materials, and damping materials are used to alleviate sound emission to environments. However, these materials are usually not effective for low-frequency sounds. Despite the difficulties in solving low-frequency noise problems, they have drawn a particular interest in mechanical engineering field, because they cause an adverse effect on a human body. In the meantime, a lot of attention have been paid to the active noise control (ANC). ANC is a technology for reducing noise according to the interference with the sound output from the control source, which is thought to be effective especially for low-frequency noise. The ANC technique was first proposed in 1936 [1], and many studies regarding ANC have been made thus far along with the development of the digital signal processors. The objective of the present study is to reduce low-frequency muffled noises together with multi-tonal noises inside a cabin of a construction machine, which changes its frequency with the engine speed. We propose an ANC system which can estimate changes of the target noise according to the detected engine speed and generate internal reference

S. Hirose Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa, Japan T. Komatsuzaki () Institute of Science and Engineering, Kanazawa University, Kanazawa, Ishikawa, Japan e-mail: [email protected] N. Kimura · K. Tanaka · T. Yamaguchi KOMATSU Ltd., Komatsu, Ishikawa, Japan e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_42

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signals that makes system stable. In accordance with the frequency tracking, the secondary path filter coefficients are adaptively switched and interpolated based on the predefined table.

2 Methods 2.1 A Basic Algorithm In the present study, a simple but adaptive active noise controller, namely, a singlefrequency adaptive notch filter ANC (SAN ANC) [2], was introduced. Additionally, the filtered-X least mean square (FxLMS) algorithm [3] is adopted to compensate for the signal delay in the secondary path [4]. The FxLMS algorithm was combined with the SAN ANC to conform the FX SAN ANC algorithm. The block diagram of the FX SAN ANC is shown in Fig. 1. The control signal is expressed as y(n) = w0 cos (ω0 n) + w1 sin (ω0 n) ,

(1)

where w0 and w1 are the adaptive filter coefficients and n represents the sampling number. The reference signal frequency, ω0 , is expressed with respect to the sampling frequency Fs and noise frequency f0 as ω0 =

2πf0 . Fs

(2)

A residual error e(n) is defined as the sum of the noise d(n) and control output y(n) as

Fig. 1 The block diagram of the FX SAN ANC

Real-Time Active Noise Control of Multi-tonal Noise Based on Multiply. . .

e(n) = d(n) + y(n).

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(3)

The coefficients, w0 and w1 , are updated according to the FxLMS algorithm [3] as ⎧   ⎨ w0 (n + 1) = w0 (n) − μe(n) SˆR (ω0 ) cos (ω0 n) + SˆI (ω0 ) sin (ω0 n)   . (4) ⎩ w1 (n + 1) = w1 (n) − μe(n) SˆR (ω0 ) sin (ω0 n) − SˆI (ω0 ) cos (ω0 n) In Eq. (4), μ represents the step-size parameter. The real part of the secondary path function S(z) is expressed as SˆR (ω0 ), and the imaginary part is expressed as SˆI (ω0 ). SˆR (ω0 ) and SˆI (ω0 ) are adopted as secondary path coefficients.

2.2 The Proposed Tracking-Type FX SAN ANC Algorithm The proposed tracking-type FX SAN ANC algorithm is based on the FX SAN algorithm. The multiply connected FX SAN processors work in parallel in order to track a multi-tonal noise. A simulated engine noise d(n) whose fundamental frequency changed gradually with time was generated from a sound source. d(n) is expressed as d(n) =

N  i=1

# i ∗ 2πf0 n , sin Fs "

(5)

where f0 represents simulated engine speed (in Hz). The control signal, y(t), in the block diagram of Fig. 2 is written as y(n) =

N −1 

  wi,0 cos (ωi n) + wi,1 sin (ωi n) ,

(6)

i=0

where ωi is determined according to the following formula: ωi =

2πf0 i . Fs

(7)

In addition, the secondary path coefficients are adaptively switched and interpolated according to the predefined table.

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Fig. 2 The block diagram of the tracking-type FX-SAN ANC Table 1 Parameters in numerical simulation

Noise signal level SˆR (case of 70 Hz, distance 200 mm) SˆI (case of 70 Hz, distance 200 mm) Fs M f0

1V 0.3495 −0.1198 10 kHz 0.01 100 Hz

2.3 Numerical Simulation Numerical simulations are performed prior to experiments in order to check operation of the FX SAN ANC algorithm. Table 1 shows parameters used in simulations. Single-frequency noise is adopted as a target sound to be attenuated.

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Fig. 3 Experimental setup: (a) illustration of the setup and (b) placement of the loudspeakers and error microphone inside an anechoic room

2.4 The Experimental Setup The setup shown in Fig. 3a is composed of a primary sound source producing the noise, a secondary source generating the controlling sound, and a microphone detecting interfered sound. Experiments were performed in an anechoic room (Fig. 3b). A simple harmonic noise was chosen for checking performance of the FX SAN ANC. The frequency was varied from 60 Hz to 400 Hz. The distance between the secondary source and the error microphone was chosen among 200 mm, 300 mm, and 400 mm, whereas the distance between the primary source and the microphone was fixed at 500 mm. The experiments were performed for the cases with and without filtering the reference signal with the secondary path filter coefficients. A simulated engine noise was generated based on Eq. (5) inside the digital signal processor (DSP) and output from the primary source. The fundamental speed, f0 , was increased from 16.7 Hz (1000 rpm) to 33.3 Hz (2000 rpm) at a rate of 0.83 Hz/s (50 rpm/s). In this experiment, the 3rd, 5th, 6th, 9th, and 12th harmonic frequency components of the engine rotation (lie approximately between 51 Hz and 100 Hz) are targets to be attenuated. The simulated engine sound is expressed as follows: d(n) =

5  i=1

" sin

i ∗ 2πf0 n Fs

# (8)

3 Results Figure 4 shows the time history of the error signal obtained in numerical simulation. The signal converged to almost zero. The numerical simulation result shows that the FX SAN ANC algorithm worked successfully on reducing noise without problems. Figure 5 shows the time histories of the error signal for the case when the distance between the secondary source and the error microphone was set at 200 mm and the noise frequency was 70 Hz. In this experiment, the influence of the reference signal

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Fig. 4 Time history of the error signal obtained in numerical simulation

Fig. 5 Time history of error signal for the case 200 mm, 70 Hz: (a) without secondary path filtering and (b) with secondary path filtering

filtering with the secondary path coefficients to the noise reduction was investigated. The result shows that the filtering had a significant effect on the control performance. Figure 6 shows the time-frequency analysis diagram obtained in experiment using the simulated engine noise. In Fig. 6b, the diagram is shown for the case when the ANC was activated. The tracking-type FX SAN ANC is found to be effective for reducing the engine noise having multi-tonal component. Due to the internally generated and mutually independent tonal reference signals along with the adaptive switching of the secondary path filter, the noise could be cancelled with an excellent stability. However, the harmonic distortion of the primary wave in the control sound deteriorated the performance, especially at the higher-order components. The problem of the harmonic distortion at low-frequency component is primarily attributed to the sound reproduction performance of the loudspeaker. The ability to produce low-frequency sound is essentially determined by the size of the diaphragm and the enclosure design. The size might be limited by the site of application, which would force us to compromise on designing the ANC system.

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Fig. 6 Time-frequency analysis diagram obtained for simulated engine sound: (a) without ANC and (b) with ANC

4 Conclusions In the present study, the tracking-type FX SAN ANC is proposed as a method for mitigating an engine sound incorporating multi-tonal component. Both numerical simulation and experimental results showed that the single-frequency and multitonal noise can be reduced effectively by using the FX SAN ANC algorithm. By tracking the frequency change of the target multi-tonal noise incorporating five frequency components and by automatically switching and interpolating the secondary path filters according to the predefined table, the proposed ANC system can effectively reduce noise with the great stability and performance.

References 1. Lueg, P.: Process of silencing sound oscillations. U.S. Patent, 2043416 (1936) 2. Widrow, B., et al.: Adaptive noise canceling: principles and applications. Proc. IEEE. 63(12), 1692–1716 (1975) 3. Morgan, D.R.: An analysis of multiple correlation cancellation loops with a filter in the auxiliary path. IEEE Trans. Acoust. Speech Signal Process. 28(4), 454–467 (1980) 4. Burgess, J.C.: Active adaptive sound control in a duct: a computer simulation. J. Acoust. Soc. Am. 70, 715–726 (1981)

A New Frequency Domain Adaptive Filter Coefficients Updating Method and Its Steady-State Performance in Frequency and Time Domain Xin Mao, Yang Xiang, Si Qin, and Yangxing Liu

1 Introduction Adaptive filters are used in a wide variety of applications [1, 2]. The most common algorithm is the least mean square (LMS) algorithm. However, when the input signal to the filter is highly correlated, the algorithm become slow down. In addition, with a high-order filter, the computational burden is too heavy in many applications [3, 4]. Frequency domain adaptive filter is used as an substitution, attempting to reach the optimal solution faster and in more efficient. However, the commonly used bin-normalized frequency domain block LMS (NFBLMS) algorithm suffers from deterioration of least mean square especially in non-causal conditions [3–5]. To alleviate the limitations of the NFBLMS, we propose a new frequency domain filter coefficients updating method, which add an extra constraint on the bin-normalized frequency domain algorithm. It is theoretically proven that the new algorithm can converge to the optimal solution in any conditions.

X. Mao · Y. Xiang () School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, China e-mail: [email protected]; [email protected] S. Qin School of Foreign Languages, Central China Normal University, Wuhan, China Y. Liu Wuhan TCL Research Co., Ltd, Wuhan, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_43

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2 Related Works The well-known LMS algorithm has the form w (n + 1) = w(n) + 2μe(n)x(n)

(1)

where x(n) is reference signal, typically consists of x(n) = [x(n), x (n − 1) , · · · , x (n − N + 1)]T

(2)

μ is the time domain step size. N is the filter size. The error at sample n is given by e(n) = d(n) − x(n)T w(n)

(3)

where d(n) is desired signal. A block implementation of LMS (BLMS) algorithm was proposed to gain computational efficiency [4, 6, 7]. The idea is to accumulate a block of data and update the time domain vector w once every block, namely, at the k th block w (k + 1) = w(k) +

2μ X(k)T e(k) L

(4)

where L represents the block length. X(k) = [x ((k − 1) L + 1) , x ((k − 1) L + 2) , · · · , x(kL)]T

(5)

e(k) = [e ((k − 1) L + 1) , e ((k − 1) L + 2) , · · · , e(kL)]T

(6)

e ((k − 1) L + i) = d ((k − 1) L + i) − x((k − 1) L + i)T w(k)

(7)

μ is the step size [6] shows that for stationary signals, the mean square error (MSE) of BLMS is equal to optimal solution. NFBLMS is an efficient way to implement BLMS in frequency domain by using the overlap-save method [7]. The NFBLMS algorithm can be written as wf (k + 1) = wf (k) + 2GμMf XH f (k)Vef (k)

(8)

where

T wf (k) = wf,1 (k), · · · , wf,2N (k) = F In the equation above, we denote by G and V



w(k) 0N ×1

 (9)

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V = FG0,N F−1

(10)

where #.  " 2π p, q = 1, 2, . . . , 2N F = F exp −j (p − 1) (q − 1) 2N

(11)

is the 2 N-point FFT matrix; GN, 0 , G0, N are diagonal matrices representing weight vector constraint and error vector constraint, respectively  GN,0 =

IN ×N 0N ×N 0N ×N 0N ×N



 G0,N =

0N ×N 0N ×N 0N ×N IN ×N

 (12)

Xf (k) is the input signal matrix in the frequency domain Xf (k) = FX(k)F−1  X(k) =

X1 X2 X2 X1

(13)

 =



x(kN) x ((k − 1) N ) x ((k − 1) N + 1) · · · ⎢ x ((k − 1) N − 1) x ((k − 1) N) x − 1) · · · (kN ⎢ ⎢ . .. .. . .. .. ⎣ . . x ((k + 1) N + 1) x ((k − 1) N + 2) · · · x (kN + 1)

⎤ · · · x ((k + 1) N − 1) · · · x ((k + 1) N − 2) ⎥ ⎥ ⎥ .. .. ⎦ . . ···

x ((k − 1) N ) (14)

is a circulant matrix whose first column is the reference signal vector, ef (k) = df (k) − Xf (k)wf (k)

(15)

is the error vector in the frequency domain, df (k) is the 2 N-point FFT of the desired signal and Mf is a nonsingular matrix controlling the convergence rate,  1 1 1 = FMF−1 Mf = diag , ,..., p0 p1 p2N −1 

(16)

where pi represents the power of the i th frequency bin. M = F−1 Mf F =



M 1 M2 M2 M1

 (17)

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is a circulant matrix whose first column is the inverse Fourier transform of the normalizing vector.

3 Constrained NFBLMS and Its Steady-State Behavior Add an extra constraint on the NFBLMS, (8) becomes wf (k + 1) = wf (k) + 2GμMf GXH f (k)Vef (k)

(18)

(18) is the Constrained NFBLMS algorithm (CNFBLMS). In order to understand the characteristics of the algorithm in more detail, two ways both in frequency and in time domain are given to prove whether the steady-state behavior of CNFBLMS can convergence to the optimal option in any condition or not.

3.1 Frequency Domain Find wf ∈ C2N , it can be known from (18) wf ∈ range [G]

(19)

range[G] is the subspace spanned by the columns of G. The rank of G is N, since ∼

G = GH , there exists a full column rank matrix G ∈ C 2N ∗N ∼ ∼H

G = GG

(20) ∼

Then for every wf ∈ range[G], there exists a unique wf ∈ C N , such that ∼ ∼

wf = Gwf

(21) ∼

The optimization problem can be recast in terms of wf [5] deduced that a unique optimal solution exists, which is given by  H   ∼ −1 ∼ H   ∼ ∼ wf o = G E XH (k)VX (k) G G E XH f f f (k)Vdf (k)

(22)

For CNFBLMS, substituting (20) and (21) into (18) ∼



∼H

∼ ∼H

wf (k + 1) = wf (k) + 2G μMf GG XH f (k)Vef (k)

(23)

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where ∼ ∼

ef (k) = df (k) − Xf (k)Gwf (k)

(24) ∼

Taking the expected value of both sides of (23), assuming that Xf (k) and wf (k) are independent, it can be shown that if the algorithm converges then, satisfy

∼ wf ∞ (k)

will

 H      ∼ −1  ∼ H ∼ ∼ −1 ∼ H ∼ ∼H ∼ wf ∞ (k) = G E XH (k)VX (k) G G M G G M G G f f f f   E XH (k)Vd (k) f f (25) (25) can be expressed as  H   ∼ −1 ∼ H   ∼ ∼ H wf ∞ (k) = G E XH (k)VX (k) G G E X (k)Vd (k) f f f f

(26)

Comparing (22) and (26), it can be found that the frequency steady-state solution of CNFBLMS is equal to optimal solution which means CNFBLMS can convergence to optimal solution unconditionally.

3.2 Time Domain Substituting (10) into (18), CNFBLMS becomes −1 wf (k + 1) = wf (k) + 2μFGN ,0 F−1 Mf FGN ,0 F−1 XH f (k)FG0,N F ef (k) (27)

Substituting (14) and (17) into (27) and applying inverse Fourier transformation on both sides of (27) leads to 

        w (k + 1) w(k) M1 M2 X1 X2 e(k) = + 2μGN,0 GN,0 0N ×1 M2 M1 X2 X1 0N ×1 0N ×1

(28)

e(k) is given in (6) in which L is changed by N. e(k) = [e ((k − 1) N + 1) , e ((k − 1) N + 2) , . . . , e(kN)]T = d(k) − XT2 w(k) (29)

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Dealing with (28) by some simple matrix operations yield 

     w (k + 1) w(k) M1 X2 e(k) = + 2μ 0N ×1 0N ×1 0N ×1

(30)

Taking the expected value of (30), if algorithm converges, w∞ (k) will satisfy   w∞ (k) = E −1 X2 XT2 E {X2 d(k)}

(31)

From (31) the time domain steady-state of the CFBLMS algorithm is the same with the one of the BLMS algorithm which can convergence to the optimal solution [6].

4 The Framework of CFBLMS Block diagram of the CFBLMS is given in Fig. 1. For conventional CFBLMS there are 7 times Fourier transformation and 3 times complex multiplication, the total

xf(k) Reference signal

IFFT

-

Save last block

x(t)

Desired signal

wf(k) w(t) Concatenate two blocks

Delay Delay

wf(k+1) w(t+1)

FFT FFT Save first block

Make second block zeros

Insert zero block

Conjugate FFT IFFT

2μMf FFT

Make second block zeros

IFFT

Fig. 1 Block diagram of CNFBLMS, dotted and dashed line indicate the signal flow of conventionl CNFBLMS and its delayless version, solid line indicates both algorithm

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number of real number multiplication times is 14Nlog2N + 38 N, for the same analysis, the number of NFBLMS and LMS is 10Nlog2N + 34 N, 2N2 . When N = 512, the real number multiplication times is 82968, 63488, 524288. CFBLMS costs relative less computation burden. Delayless version makes filtering operation per sample saving waiting time, it can be used in a compact active noise control application.

5 Numerical Simulation and Analysis

MSE(dB)

To demonstrate the effectiveness of the proposed CNFBLMS, a simulation is given to compare the MSE with NFBLMS in a prediction problem. The reference signal is generated by passing Gaussian white noise with unit variance through a lowpass filter with transfer function H(z) = [(1 − 0.5z−1 )/(1 − 0.6z−1 )]16 . The desired signal is one sample ahead of the reference signal. Figure 2 show the convergence of the MSE of two algorithms of different filter size. When N is 16, 32, 64, For NFBLMS the attenuation of MSE is 15.82 dB, 13.18 dB, 10.02 dB. For CNFBLMS the value is 18.82 dB, 18.65 dB, 18.83 dB. The simulation demonstrates the superior steadystate behavior of the proposed CNFBLMS algorithm in solving prediction problem.

(a) Filter size N is 16

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Fig. 2 Convergence of MSE in non-causal condition

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6 Conclusion This paper proposes CNFBLMS algorithm and proves that it can converge to the optimal solution in any case both in frequency and time domain. This algorithm has wide potential application in compact active noise control systems.

References 1. Bai, M.R., Pan, W., Chen, H.: Active feedforward noise control and signal tracking of headsets: electroacoustic analysis and system implementation. J. Acoust. Soc. Am. 143(3), 1613–1622 (2018) 2. Chien, Y., Li-You, J.: Convex combined adaptive filtering algorithm for acoustic echo cancellation in hostile environments. IEEE Access. 6, 16138–16148 (2018) 3. Haykin, S.: Adaptive Filter Theory, 4th edn. Prentice Hall, Upper Saddle River (2001) 4. Feure, A., Rafaely, B.: On the steady state performance of frequency domain algorithms. IEEE Trans. Signal Process. 41, 419–423 (1993) 5. Elliott, S.J., Rafaely, B.: Frequency domain adaptation of causal digital filters. IEEE Trans. Signal Process. 48(5), 1354–1364 (2000) 6. Clark, G.A., Mitra, S.K., Parker, S.R.: Block implementation of adaptive digital filters. IEEE Trans Circuits Syst. CAS-28, 584–592 (1981) 7. Shynk, J.J.: Frequency-domain and multirate adaptive filtering. IEEE Signal Process. Mag. 9(1), 14–37 (1992)

Effects of Reverberation on Active Noise Control Headrest Performance Lifu Wu, Chiming Fang, and Zhuang Cheng

1 Introduction Active noise control has been present in research, development, and practical applications since its first patent is proposed in the 1930s [1, 2]. The ANC headphones have already become commercial products due to the small enclosure where noise reduction is achieved [3]. But wearing headphone sometimes brings discomfort or pressure to the human ear; hence it is necessary to explore the technology to generate quiet zone near the human ear, and one possible solution is the active headrest which places the secondary source and sensors around a headrest to control the disturbance at the human ears. The concept of active headrest was proposed in the 1950s by Olson and May [4], but due to the limitation of electronic technology at that time, it was difficult to apply to practice. Until the 1990s, Rafaely et al. built an active headrest prototype based on the feedback ANC system and analyzed its performance [5]. Since then, there have been much work related to active headrest. For instance, the direct installation of error sensors in the desired positions is inconvenient in the practical application, then the remote microphone techniques [6] and virtual microphone arrangements [7] were proposed to control the noise at a remote location. To accommodate the noise control performance degradation due to the head movement, the location information of human head is required, which can be realized by three-dimensional head tracking system; the sound field transfer function corresponding to different head positions is measured and stored in advance;then the moving position of human

L. Wu () · C. Fang · Z. Cheng School of Electronic & Information Engineering, CICAEET, Nanjing University of Information Science and Technology, Nanjing, China e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_44

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head is tracked in real time by the head tracking system; and the corresponding transfer function is called to calculate and control the noise [8, 9]. Active headrest has a wide range of application scenarios, especially in complex noise environments with high noise level and numerous noise sources, for example, in vehicle cabins [6], high-speed train [9], and open office environment [10]. However, so far active headrest is not as popular as the ANC headphone; one reason is that the performance of active headrest is not so satisfactory when it is used in real complex noise environments rather than the laboratory environments. In this chapter, the influence of reverberation on active headrest performance is investigated by real-time ANC experiments in the anechoic chamber and the reverberant chamber, respectively. The motivation is that active headrest is a local control system, and its target application scenarios such as ship cabins and workshops are usually with obvious reverberation.

2 Evaluation System 2.1 Experimental Setup An active headrest is built with two channels (left and right channels) as shown in Fig. 1. The reference microphones and the secondary sources are installed on a “U” structure, and their locations are fixed in the experiment. The error microphones are installed on the tripods, and their locations are changed in the experiment. In order to satisfy the causality of the ANC system, the primary source is placed in front of the reference microphones with 1.2 m distance, and thus the primary noise arrives at the reference microphones earlier than the error microphones.

Fig. 1 The experiment settings, (a) the devices’ connection in the experiments, (b) the situation in the anechoic chamber

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The wires of the reference microphones and error microphones in the headrest are passed through the microphone preamplifier and then connected to the ANC controller. The secondary sources (loudspeakers) in the headrest are connected to the ANC controller to generate the secondary signals. B&K Pulse is the central device which provides the noise signal to the primary source (loudspeaker) and also captures the signals from the reference microphones, error microphones, and secondary sources. The laptop is used to monitor the working states of the devices and save the data. The ANC controller is implemented on Xilinx Zynq-7000 fieldprogrammable gate array, and the sampling frequency is 16,000 Hz.

2.2 Feedforward and Feedback Structures in the ANC Headrest The feedforward structure is composed of one reference microphone to pick up the reference noise x(k), one error microphone to measure the residual noise e(k), and one secondary source to generate the cancelling signal yf (k) for attenuation of the primary noise d(k). Here the reference signal x(k) is ˆ filtered through S(z), the so-called estimation of the secondary path S(z), and the control filter Wf (z) is represented as a tap-weight vector of length L, i.e.,  T wf (k) = wf0 (k), wf1 (k), . . . . . . , wfL−1 (k) . By using the Z transforms of the ˆ signals and assuming S(z) =S(z)and D(z) = X(z)P(z) in Fig. 2a, if only the feedforward structure works, the residual noise is E(z) = [P (z) + S(z)Wf (z)] X(z)

(1)

In order to make residual noise E(z) in Eq. (1) tend to be zero, the optimal feedforward control filter is Wf (z) = − P(z)/S(z). The feedback structure used in this chapter is based on the internal model control (IMC) introduced in [9] and demonstrated in Fig. 2b. The feedback structure consists of the control filter Ws (z), and the reference signal xs (k) is synthesized as

Fig. 2 Block diagram of (a) the feedforward structure and (b) the feedback structure

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ˆ Xs (z) = E(z) − Ys (z)S(z)

(2)

By using the Z transforms of the signals in Fig. 2b, if only the feedback structure works, the residual noise is E(z) =

ˆ 1 + S(z)W s (z)   D(z) ˆ 1 − S(z) − S(z) Ws (z)

(3)

ˆ Under ideal condition where S(z)= S(z), the feedback structure controller aims to make E(z) tend to be zero and Ws (z) = − 1/S(z).

3 Results and Discussions In the experiments, the band-limited white noise with 400 Hz–800 Hz is played as the primary noise, and the power spectral density (psd) is used to evaluate the noise reduction performance related to different frequency. In addition, the psd is calculated using the signal captured by the error microphone. It is also found that the performance of the left channel is similar to the performance of the right channel for the active headrest, so in the rest, only the results of the left channel are given due to the limited space. The reverberant chamber is of 7.8 m length, 5.5 m width, and 4.3 m height, and the frequency-dependent reverberation time is listed in Table 1. The critical distance in room acoustics is the distance at which the sound pressure level of the direct sound and the reverberant sound is equal; a reverberant room generates a short critical distance, and an anechoic chamber generates a longer critical distance. According to the equation in [11], the critical distance is about 0.3 m in the reverberant chamber, so in the experiment, the distance between the error microphone and the secondary source is set as 0.10 m (case 1) and 0.4 m (case 2), respectively. The noise reduction performance of the two cases is shown in Fig. 3, where “A-FF” and “A-FB” denote the curves of feedforward and feedback ANC in the anechoic chamber, respectively, and “R-FF” and “R-FB” denote the curves of feedforward and feedback ANC in the reverberant chamber, respectively. It is found in Fig. 3 that (1) the noise reduction of feedforward structure is more than 25 dB for the two cases in the anechoic chamber, which not only confirms that the real-time ANC system works well but also indicates that the

Table 1 The frequency-dependent reverberation time in the reverberant chamber Frequency (Hz) T60 (s)

125 10.0

250 9.1

500 8.9

1000 6.6

2000 4.8

4000 3.4

8000 1.9

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Fig. 3 The noise reduction performance of the two cases, (a) case 1 and (b) case 2

“A-FF” performance degradation is not so severe when the distance between the error microphone and the secondary source changes from 0.1 m to 0.4 m and (2) the “A-FB” noise reduction decreases as the distance between the error microphone and the secondary source increases from 0.1 m to 0.4 m, and the best noise reduction of “A-FB” is about 10 dB in Fig. 3a, while that is almost 0 dB in Fig. 3b, and the main reason is that the 0.4 m distance is too far, i.e., the delay of the secondary path is too long to achieve certain noise reduction for the feedback structure. When comparing the noise reduction performance between the anechoic chamber and the reverberant chamber, it is noticed from Fig. 3 that (1) the noise reduction performance of the feedforward structure in reverberant chamber is much worse than that in anechoic chamber for the two cases; (2) the performance difference of the feedback structure between the reverberant chamber and the anechoic chamber is not so big for the two cases; and (3) the curves of “A-FF” and “A-FB” are more smooth than that of “R-FF” and “R-FB”, i.e., the noise reduction fluctuates with the frequency more severely in the reverberant chamber. The effects of the reverberation on the ANC performance can be initially explained by comparing the primary paths and the secondary paths shown in Fig. 4. It is clear that there are much more echoes in the late part of the primary path in Fig. 4b than that in Fig. 4a, and these echoes are caused by the strong reflections in the reverberant chamber, which leads to the worse noise reduction of feedforward structure in Fig. 3. The secondary paths in Fig. 4a and b are similar, although there are also many echoes in Fig. 4b, and their amplitude is relatively small, and this may be the reason for the similar performance of the feedback structure in Fig. 3.

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Fig. 4 The primary paths and secondary paths of the case 2, (a) anechoic chamber, (b) reverberant chamber

4 Conclusions The real-time ANC headrest system is implemented, and experiments are performed in the anechoic chamber and the reverberant chamber, respectively. The results show that the performance of the feedforward structure in reverberant chamber is much worse than that in anechoic chamber, while the performance difference of the feedback structure between them is not so big; in addition, the noise reduction fluctuates with the frequency more severely in the reverberant chamber than that in the anechoic chamber. Initial analysis demonstrates that the echoes caused by the strong reflections in the reverberant chamber may lead to the worse noise reduction of feedforward structure, and it is necessary to do more detailed experiments and analysis about the effects of the reverberation on the active headrest performance in the future.

References 1. Kuo, S.M., Morgan, D.R.: Active Noise Control Systems: Algorithms and DSP Implementations. Wiley, New York (1996) 2. Elliott, S.J.: Signal Processing for Active Control. Academic Press, London (2001) 3. Chang, C.Y., Li, S.T.: Active noise control in headsets by using a low-cost microcontroller. IEEE Trans. Ind. Electron. 58, 1936–1942 (2011) 4. Olson, H.F., May, E.G.: Electronic sound absorber. J. Acoust. Soc. Am. 25(6), 1130–1136 (1953) 5. Rafaely, B., Elliott, S.J.: H2 /H∞ active control of sound in a headrest: design and implementation. IEEE Transactions on Control System Technology. 7, 79–84 (1999) 6. Jung, W., Elliott, S.J., Cheer, J.: Local active control of road noise inside a vehicle. Mech. Syst. Signal Process. 121, 144–157 (2019) 7. Moreau, D., Cazzolato, B., Zander, A., Petersen, C.: A review of virtual sensing algorithms for active noise control. Algorithms. 1(2), 69–99 (2008)

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8. Jung, W., Elliott, S.J., Cheer, J.: Combining the remote microphone technique with headtracking for local active sound control. J. Acoust. Soc. Am. 142(1), 298–307 (2017) 9. Han, R., et al.: Combination of robust algorithm and head-tracking for a feedforward active headrest. Appl. Sci. 9, 1760–1770 (2019) 10. Sujbert, L., Szarvas, A.: Noise-canceling office chair with multiple reference microphones. Appl. Sci. 8, 1702–1720 (2018) 11. Kuttruff, H.: Room Acoustics. Taylor & Francis, New York (2009)

Zero Control Power Phenomena in the Minimization of Sound Power Using Multiple Control Sources Yuta Ogasawara, Hiroyuki Iwamoto, and Shotaro Hisano

1 Introduction The active noise control that introduces secondary sound sources may achieve sufficient control effect for a low-frequency noise problem. In this method, there is a restriction that the control range is limited to the narrow region around the sensor point since the conventional active noise control aims to nullify the sound pressure measured by microphones. In order to obtain a sufficient control effect over a wide region, it is necessary to install a large number of sensors and actuators in the entire control region. In order to overcome the problem described above, the minimization of the total sound power [1, 2] that is the summation of sound power of each point source is focused in this study. The sound power minimization technique requires only the information of each sound sources unlike the sound pressure reduction described above. Although discussions on active control of radiated power from multiple primary sound sources using multiple CSSs have been made in the previous studies [3], the objective of this chapter is to derive analytical solution of the zero control power law in the case of multiple CSSs under the condition that the CSSs are arranged at equal intervals on the circumference. It was shown in Ref. [4, 5] that such arrangement results in the best control performance. First, the zero control power phenomenon in the case of one CSS is discussed. Next, the zero control power phenomenon in the case of multiple CSSs is discussed. Furthermore, the analytical solution of the zero control power law in the case of multiple CSSs is derived. Finally, it is numerically shown that as the number of CSSs increases, the control effect of the total acoustic power is improved.

Y. Ogasawara () · H. Iwamoto · S. Hisano Seikei University, Musashino, Tokyo, Japan e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-47618-2_45

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2 Zero Control Power Phenomenon Using One CSS It is assumed that one primary sound source and one CSS in a certain plane, and the distance between them is defined as d. The volume velocity of primary sound source and CSS are defined as qp and qs , respectively. Then the total sound power can be expressed as the following equation [1, 2]: ωρ 0 k pw = 8π

" ## " 5 52   5qp 5 + |qs |2 + qs qp ∗ + qp qs ∗ sin kd , kd

(1)

where ω is the angular frequency, ρ o is the air density, and k is the wave number. Equation (1) is a quadratic form for qs , and therefore the control law for minimizing the total acoustic power is derived as qs = −qp sinc kd.

(2)

The total sound power is minimized, when the value of the sound power emitted from the CSS becomes zero, which is referred to as a zero control power phenomenon. This can be proved by substituting Eq. (2) into Eq. (1). Acoustic intensity is a vector quantity indicating the magnitude of sound energy considering the direction of the sound. Therefore, it can be used for sign of acoustic power of sound sources. Figure 1 shows the distribution of acoustic intensity when total acoustic power is minimized by one CSS. As shown in the figure, in the vicinity of the primary sound source, arrows representing acoustic intensity are blown out radially. On the other hand, the arrows in the vicinity of the CSS cannot be seen

Fig. 1 Sound intensity map when total acoustic power is minimized by one control source

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as blowout or absorption and can be observed as if the arrows are passing. From this characteristic, it can be concluded that the control power is neither positive nor negative, and hence the zero control power phenomenon occurs.

3 Zero Control Power Phenomenon Using Multiple CSSs 3.1 Minimization of Total Sound Power in the Case of Two CSSs The zero power phenomenon when two CSSs are used is considered based on the idea of Sect. 2. As shown in Fig. 2, a primary sound source p and two CSSs, sa and sb , exist in a certain plane. It is hypothesized that the distances between these respective sound sources are d1 , d2 , and d3 . In order to minimize the total sound power under the conditions shown in Fig. 2, the equation of the total sound power is determined as in the case of one CSS. Therefore, the volume velocity of each CSS that realizes minimizing the total acoustic power is derived as 

qsa qsb



  $ −qp (A−BC) % −1  1 C qp A 1−C 2 =− = −qp (−AC+B) , qp B C 1 2

(3)

1−C

where A = sinc kd 1 , B = sinc kd 2 , C = sinc kd 3 .

(4)

Figure 3 shows the distribution of acoustic intensity when total acoustic power is minimized by two CSSs described in Eq. (3). As shown in the figure, it was found that a zero control power phenomenon occurs even if multiple control sources are

Fig. 2 Arrangement of point sound source at two control points

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Fig. 3 Intensity level map when the total acoustic power is minimized using two CSSs

used. In this case, compared to the case of one control source, the reduction of total acoustic power is improved by 13.6 dB.

3.2 Zero Power Control Law in the Case of Multiple CSSs and Its Control Effect It was clarified that a larger suppression effect could be obtained by increasing the number of CSSs. As a result, it can be considered that the suppression effect becomes higher as the number of CSSs increases. The condition is that the CSSs are equally spaced on the same circumference whose center is the primary sound source position. When three CSSs are arranged at equal intervals around the primary sound source, the optimal volume velocity for minimizing the total sound power is derived as ⎡ ⎤ ⎡ ⎤−1 ⎡ ⎤⎫ qsa 1 AA qp B ⎪ ⎬ ⎣ qsb ⎦ = −⎣ A 1 A ⎦ ⎣ qp B ⎦ , (5) ⎪ ⎭ AA 1 qsc qp B where A = sinc kd, B = sinc

√ 3kd,

(6)

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where d is the radius of the circumference. The following two things can be held to the characteristics of the above equation. First, all control laws at each control point are equal. Second, one contains an n-by-n square matrix, in which n is the number of CSSs. In the square matrix, the arguments of the sinc function include the product of the wave number in the sound field and the distance between the primary sound source and each CSS. This is completely apparent in the arrangement of three control points where the distances from the primary sound source to the CSSs are equal as shown in Eq. (5). Based on these results, the optimal control law in the case where n CSSs are equally spaced around the radius of the primary sound source is derived as q = −C−1 D ⎫ ⎡ ⎤ ⎤ ⎡ ⎪ qp sinc kd ⎪ ⎪ qs1 ⎪ ⎢ ⎥ ⎪ .. ⎪ ⎢ ⎥ ⎥ ⎢ q ⎪ ⎪ . ⎢ ⎥ ⎢ s2 ⎥ ⎪ ⎪ ⎢ ⎥ ⎢ . ⎥ ⎪ . ⎪ ⎥ ⎥, D = ⎢ ⎪ . . q=⎢ ⎪ ⎢ ⎥ ⎢ . ⎥ . ⎪ ⎪ ⎢ ⎥ ⎥ ⎢ ⎪ ⎪ .. ⎢ ⎥ ⎣ qs(n−1) ⎦ ⎪ ⎪ ⎣ ⎦ ⎪ . ⎪ ⎬ qsn qp sinc kd ⎡ ⎤⎪. 1 sinc kd m(1) ··· sinc kd m(n−2) sinc kd m(n−1) ⎪ ⎪ ⎪ ⎪ ⎢ sinc kd 1 sinc kd ··· sinc kd m(n−2) ⎥ ⎪ m(n−1) m(1) ⎢ ⎥⎪ ⎪ ⎪ ⎢ ⎥ ⎪ .. .. .. ⎪ ⎢ ⎥ ⎪ . C=⎢ ⎥⎪ 1 . . sinc kd m(n−1) ⎪ ⎢ ⎥⎪ ⎪ .. ⎢ ⎥⎪ .. .. ⎪ ⎣ sinc kd m(n−2) ⎦ . . ⎪ sinc kd m(1) ⎪ . ⎪ ⎪ ⎭ sinc kd m(n−1) sinc kd m(n−2) ··· sinc kd m(1) 1 (7)

The distance d, which is an argument of the sinc function in Eq. (8), is described as " dm(n) = 2d cos

# π (n − 2m) , 2n

(8)

where m represents the identity number of the CSS counted from the certain CSS as the reference point. As shown in Eq. (7), the matrix C is a circulant matrix or Toeplitz matrix [4, 5]. The eigenvectors of the circulant matrix are equal to the column vectors of the discrete Fourier transform matrix (DFT matrix), and therefore it is possible to diagonalize the circulant matrix with the DFT matrix. As a result, Eq. (7) can be simplified as q = −F−1 L−1 FD,

(9)

where F is the DFT matrix and L is the diagonal matrix whose elements are eigenvalue of the matrix C. Furthermore, F is an orthogonal matrix, so that the inverse matrix is equivalent to its transposed matrix. In addition, the inverse of L is easily calculated since it is diagonal. Therefore, defining the eigenvectors of the

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Fig. 4 Intensity level map when the total acoustic power is minimized using three CSSs

matrix C as ai, the optimal volume velocity vector is rewritten as ⎡

⎤ ⎛ a1 T −a1 T L−1 a1 ⎢ a2 T ⎥ ⎢ ⎥ ⎜ .. q = − ⎢ . ⎥ L−1 [a1 a2 · · · an ] D = ⎝ . ⎣ .. ⎦ T L−1 a − a n 1 an T

⎞ . . . −a1 T L−1 an ⎟ .. .. ⎠ D, . . T −1 · · · −an L an (10)

Next, control effects and zero control power phenomenon are confirmed by numerical simulations. Using the optimal control laws shown in Eqs. (7) and (8), the distribution of acoustic intensity in the sound field is calculated as shown in Fig. 4. The arrows in the vicinity of three CSSs cannot be seen as blowout or absorption and can be observed as if the arrows are passing. Thus, the zero control power phenomenon occurs. Furthermore, total acoustic power with and without control and the corresponding control effect are calculated as shown in Fig. 5. Although the numerical results are limited to the condition that the CSSs are installed at a constant interval of 0.4 m from the primary source and the frequency is fixed at 100 Hz, higher suppression effect is obtained when the number of the CSSs increases.

4 Conclusions In this study, based on the concept of using multiple control sound sources (CSSs), the zero power phenomenon and the minimization of the total sound power were

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Fig. 5 Control effect and sound pressure level in the generation of zero power phenomenon at 1–3 control source and uncontrolled

discussed. Under the condition that the CSSs are arranged at equal intervals on the circumference, it was found that the formula of the optimal control law in the case of multiple CSSs is analytically derived based on a property of a circulant matrix.

References 1. Tanaka, N., Snyder, S., et al.: Acoustic power minimization by active noise and vibration control (On zero control power output) (in Japanese). Trans. JSME Ser. C. 59(569), 86–93 (1994) 2. Kobayashi, K., Tanaka, N.: Minimization of acoustic potential energy in enclosure using both active noise control and active vibration control. JSME Int. J. 47-4, 1133–1139 (2004) 3. Colin, H., Snyder, S., et al.: Active Control of Noise and Vibration, vol. II. CRC Press Taylor and Francis Group, Boca Raton (2012) 4. Ogasawara, Y., Iwamoto, H., Hisano, S.: Generalization of the zero power phenomenon in multiple sound sources (Minimize sound power and condition of occurrence of zero power phenomenon) (in Japanese). In: 2019th Dynamics and Design Conference, No301. JSME, Fukuoka, Japan, 27–30 August (2019) 5. Gray, R.M.: Toeplitz and Circulant matrices: a review. Found. Trends Commun. Inf. Theory. 2(3), 155–239 (2006)

Author Index

A Abbasnejad, Behrokh, 3 Alkmim, Mansour, 79 Al-Widyan, Fatma, 95 Asanuma, Haruhiko, 271

B Bianciardi, Fabio, 79 Bing, Wang, 169 Biswal, Deepak Kumar, 201

C Chen, Li-Qun, 217 Chen, Peng, 255 Chen, Xuehong, 145 Chen, Yu, 305 Cheng, Fang, 351 Christie, Matthew, 281 Cornelis, Bram, 161 Cuenca, Jacques, 79 Cumbo, Roberta, 161

D Daisheng, Zhang, 169 Desmet, Wim, 161 Ding, Hu, 217

F Fang, Hongbin, 193 Farah, Esmaail, 63 Fard, Mohammad, 71

Feng, Huayuan, 121 Forrier, Bart, 161 Fujita, Etsunori, 153 G Geng, Xiao-Feng, 217 H Halkon, Benjamin, 3 Han, Sam, 11 Hatano, Takashi, 87 Hiramatsu, Shigeki, 87 Hirose, Shun, 335 Hisano, Shotaro, 37, 185, 359 Hou, Zhichao, 129, 145 I Iwamoto, Hiroyuki, 37, 185, 359 J Janssens, Karl, 79, 161 Ji, J.C., 11, 313 Jin, Dongping, 305 Jinglai, Wu, 169 Jung, Hyung-Jo, 249 K Kajiwara, Itsuro, 87 Kalhori, Hamed, 3 Kaneko, Shigehiko, 153 Kato, Kazuhito, 71 Kil, Hyun Gwon, 29

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368 Kim, Hyung-Soo, 249 Kimura, Naoki, 335 Komatsuzaki, Toshihiko, 271, 335 Kondou, Takahiro, 263 Kurihara, Kai, 263 Kuwano, Ryuji, 153 Kye, Seung-Kyung, 249

L Lai, Joseph C.S., 47 Lee, Chan, 29 Li, Bing, 3 Li, Huan, 289 Li, Jianchun, 289 Li, Weihua, 281 Li, Yancheng, 289 Lindqvist, Anna Lidfors, 55 Liu, Pengfei, 103 Liu, Yangxing, 343 Lu, Zheng, 297 Luo, Liang, 103

M Ma, Naiyin, 297 Makita, Soichi, 153 Mao, Xin, 343 Martin, Richard, 179 Mashino, Masahiro, 153 Matsuyama, Marin, 185 Matsuzaki, Kenichiro, 263 Mohanty, Sukesh Chandra, 201 Mori, Hiroki, 263

N Nagamatsu, Shuto, 241 Nakanishi, Kousuke, 111 Ning, Donghong, 103 Nishidome, Chiaki, 87 Nong, Zhang, 169

O Oberst, Sebastian, 47, 179 Ogasawara, Yuta, 359 Ogura, Yumi, 153 Ong, Albert, 95

P Patnaik, S Srikant, 231 Pradhan, Somanath, 313

Author Index Q Qin, Si, 343 Qiu, Xiaojun, 313, 321, 327

R Rao D, Koteswar, 231 Risaliti, Enrico, 161 Roy, Tarapada, 231 Ryck, Laurent De, 79

S Sadaoka, Tetsuma, 271 Sakata, Masato, 87 Shangguan, Wen-Bin, 121 Shiozaki, Hirotaka, 111 Shiraishi, Toshihiko, 241 Shiryayev, Oleg, 63 Shooshtari, Alireza, 3 Sowa, Nobuyuki, 263 Stender, Merten, 179 Su, Zhu, 209 Subhash, Rakheja, 121 Sugimoto, Eiji, 153 Sun, Jie, 209 Sun, Kaipeng, 209 Sun, Shuaishuai, 281 Suwabe, Keita, 111

T Takashima, Taiyo, 225 Tamarozzi, Tommaso, 161 Tanaka, Hiroki, 37 Tanaka, Keita, 335 Tanaka, Nobuo, 185 Tomoda, Akinori, 225

V Vahdati, Nader, 45 Vandernoot, Guillaume, 79

W Walker, Paul, 55, 95, 137 Wang, Shuai, 145 Wang, Shuping, 327 Wang, Shuyu, 145 Wang, Yuning, 129 Wei, Yifan, 129 Wen, Hao, 305 Wu, Hang, 21

Author Index Wu, Lifu, 351 Wu, Peibao, 145 X Xiang, Yang, 343 Xianqian, Hong, 169 Xu, Jian, 193 Y Yamaguchi, Taisei, 335 Yamanaka, Masahiro, 225 Yamazaki, Toru, 111 Yanase, Junichi, 111 Yao, Jianchun, 71

369 Ye, Kan, 11 Yonezawa, Heisei, 87

Z Zhang, Hengrui, 297 Zhang, Nong, 103, 137 Zhang, Xiaoxu, 193 Zhang, Zhi, 47 Zhao, Enoch, 95 Zhao, Sipei, 321 Zheng, Ling, 21 Zheng, Minyi, 103 Zhou, Shilei, 137 Zhou, Ying, 255

Subject Index

A Acoustic radiation power coupled system, 188–189 displacement distributions, 185 machine designs, 37 numerical analysis, 40–42 sound source, 321 theory, 38–40 triple-walled structure, 37 Acoustic guitar, 185 Active control adaptive, 185–191 energy consumption, 21 phononic structures, 194 in suspension systems, 23 Active mass damper (AMD) DVAs, 271 dynamic characteristics, 275–276 frequency-driven actuator, 272 parametric array, 271 Active noise control (ANC) adaptive notch filters, 335–341 block diagram, 336 FxLMS/F, 314 FX SAN, 337–338 headphone, 352 multi-tonal noise, 335–341 noise reduction, 323, 329 reduction performance, 331 reverberation, 351–356 sound power levels, 330, 332 source, 40 transmission, 327–333 SPL, 324

time-frequency analysis, 341 VSB, 321 Active suspension, 21–25, 27 Adaptive control acoustic stringed instruments, 185 displacement control system, 190, 191 effects, 185–186 frequency response, 191 ideal displacement system acoustic radiation power, 188–189 derivation, 186–188 frequency characteristics, 188 strategy, 129 Adaptive filters active control operation, 319 affine combination, 314 ANC, 313 convex combination, 313 displacement control system, 190 frequency domain, 343 LMS algorithm, 343 plate-dominated mode, 186 time domain, 186 Adaptive notch filter ANC, 335–336 basic algorithm, 336–337 experimental setup, 339–341 FX SAN ANC algorithm, 337, 338 numerical simulation, 338 Affine combination ANC, 313, 314 FxLMS, 313 proposed algorithm, 314–316 simulations, 316–318 Airborne source quantification (ASQ), 80–82

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372 Amplitude modulation (AM) actuator, 274 AMD device (see Active mass damper (AMD)) demodulated components, 272–273 source expression, 272 Analytical statistical energy analysis (ASEA) BIW automotive, 113 optimization, 116–117 comparison of frequency responses, 115–116 FEM, 115 frequency response, 113 redesigned models, 118–120 target parameters, 112 Aperture active control, 327 ANC system, 327 numerical simulations error microphones, 331–333 incident directions, 330–331 secondary sources, 331 theoretical model, 330 theory, 327–329 Automated manual transmission (AMT), 138, 140, 169 Automobile drivetrain compensation for backlash, 91 controlled object mathematical model, 89–90 nonlinear parameters, 89–90 plant model, 88–89 control simulations, 92 vibration control, 87 Automotive body in white (BIW), 112–117, 120

B Backlash compensation, 91 control cycle limitation, 88 nonlinear characteristics, 89 structural nonlinearity, 87 vehicle structure, 88 Bin-normalized frequency domain block LMS (NFBLMS) algorithm, 344–345 frequency domain, 346–347 least mean square, 343 prediction problem, 349 time domain, 347–348

Subject Index Brake squeal CEA (see Complex eigenvalue analysis (CEA)) ISI Web of Science, 47, 48 stochastic approach, 50–52 uncertainty analysis, 50–52 Building protection, 281–287

C Cable vibration control system, 249 Capacitive microplate DC polarization, 3 equations of motion, 5–6 PDEs, 4–5 rectangular air gap, 3, 4 singularity elimination, 6–7 static and transient deflection, 4 transient behavior, 7–8 Carbon nanotubes (CNTs) hybrid composite, 231 mathematical modeling, 232–234 MWCNT-based nanocomposite sample, 235, 236 numerical results, 236 viscoelastic material, 231 CFBLMS algorithm, 348–349 Circular-type Pod silencer axial flow fans, 29 design process, 30 FE analysis, 30–31 noise characteristics, 30 transfer matrix method, 29 Circulant matrix, 345, 346, 363, 365 Collision gap, 218–220 Complex eigenvalue analysis (CEA) nonlinearity, 48–50 success, 48 Conceptual design model ASEA, 111 machine products, 111 SEA, 112–114 Control cycle limitation, 88, 92, 93 Control system design sampled-data, 90–91 weighting function, 91 Convex combination, 313, 314, 317 Coupled system, 72–74, 185–191, 322 Coupling coefficients, 38 controlled addition, 282 electrical and mechanical forces, 4 energy transmission efficiency, 113 external force and control source, 38

Subject Index rotor angle, 163 structural acoustic, 37

D Damper, 153–159 AMD (see Active mass damper (AMD)) conventional vehicle suspension, 106 MUID (see Multi-unit impact damper (MUID)) theoretical description AM, 272–273 equation of motion, 273–275 viscoelastic beam, 217–223 Damping effect active control force, 25 cantilever beam, 218 characteristics, 153 CNT-based, 231–237 contact pair, 122 non-negative, 48 parameters, 180 ratio, 246 spring structures, 130 suspension structure, 103 Design optimization objective function, 33–34 sensitivity analysis, 34 variables, 32, 33 Digital signal processor (DSP), 193, 339 Digital synthetic impedance (DSI), 193, 194 Direct yaw moment control, 56, 58 Displacement control system acoustic radiation power, 188–189 derivation, 186–188 frequency characteristics, 188 Drive shaft systems automotive transmission system, 121 contact, 125–126 force and friction, 124 stiffness, 124–125 friction parameters, 125–126 GAFs measurement, 121–123 nominal force exponent, 124–125 tripod joint, 121 Dual motor powertrain AMT, 169 energy efficiency, 170 EV, 169 layout of, 170, 171 parameter matching, 172–173 simulation results, 174–176 work mode optimization, 173–174

373 Ducted air-conditioning unit, 12, 18, 19 Dynamic characteristics AMD, 275–276 battery SOC, 131 EMVI device, 104 FGM micro-beams, 209 Dynamic hysteresis amplitude sweeps, 183 CubeSats, 179 model setup, 180–181 validation, 181–183 root-mean-square values, 183 thin-walled structures, 179 Dynamic properties investigation, 290–293 optimization, 293–294 QZSVI, 289–290 Dynamic vibration absorbers (DVAs), 271

E Earth-moving machine, 153 Efficiency optimization, 173 Electric vehicle, 56, 129, 169–176 Electromagnetics (EM) actuator, 243, 305 adaptable control device, 249 dynamic model, 250–252 isolator design, 306–308 dynamic analysis, 308–310 QZS stiffness system, 305 numerical model for stay cable, 250 simulations, 252–253 results, 252–253 suspension random excitation, 107–108 sinusoidal excitation, 107 vibration reduction, 249 Electromagnetic variable inertance (EMVI) damping-controllable seat suspension, 103 device characteristics analysis, 105–106 model, 104–105 suspension performance, 106–108 two-terminal mechanical device, 103 Electromechanical hybrid (see Hybrid) MEMS, 3 powertrain, 162–164 torque estimation, 162–164

374 Energy management, 132 algorithm, 133 EMS, 129, 131 fuel consumption, 133 gear and mode selecting, 176 Energy propagation, 111 Energy regeneration and consumption, 25, 26 harvesting, 21 ratio, 25, 26 seat suspension, 21 Engine mounts components, 63 dynamic stiffness, 64 fluid/hydraulic, 63 fluid-less, 65, 66 mounting system, 63 new design concept, 66–67 single pumper hydraulic, 64

F Filtered-x least mean square (FxLMS) algorithm, 313, 317, 319 combination, 316 step sizes, 313, 314 Finite element (FE), 48, 71, 75, 76, 124, 164–165, 182, 225 Finite element method (FEM) and ASEA, 115–116 automotive BIW, 115 excited structures, 115 frequency response, 115 FSDT, 201 parameter variations, 225 verification, 118–120 Friction coefficient, 48 and contact force, 124 contact surfaces, 50 cylinder and sealing elements, 241 function-based contact force model, 121 Laplace transform, 105 models, 47 parameters, 125–126 pin joints, 66 probability, 51 sliding plate, 51 viscous damping, 218 Flexible multibody model, 162, 165–167 Free play damper, 156 Frequency domain analysis, 349 CFBLMS, 348–349

Subject Index LMS, 343 numerical simulation, 349 related works, 344–346 Fuel tank experimental data, 145 FEM, 146–147 global vehicle production, 145 modal analysis, 146 parameter, 148–149 response surface model, 147–148 Functionally graded materials (FGMs), 201, 209–214 FX SAN ANC algorithm, 336–341 G Generated axial force (GAF) contact force, 122 drive shaft system (see Drive shaft systems) magnitudes, 126 measuring, 122–123 multi-body dynamic models, 121 non-rotating-type, 121 NVH, 121 rolling-sliding friction models, 122 tripod joint, 121 Global positioning system (GPS) lateral locations, 96 range of satellites, 95 receiver, 100 selected study route, 98 sensor, 97 H Headrest, 351–356 High-noise level, 29, 30, 34, 352 Human body vibration, 71, 132, 335 Hybrid electromechanical engine bond graph modeling technique, 66 dynamic stiffness vs. capacitance, 68, 70 engine mount design concept, 66–67 engine-transmitted noise, 63 fluid/hydraulic engine mounts, 63 fluid-less engine mounting system, 65 numerical simulations, 68 parametric sensitivity analysis, 68, 69 simulation results, 68 single pumper, 63, 64 vehicles (see Hybrid vehicles) vibration discomfort discussion, 75–76 FE models, 71, 72

Subject Index occupant-seat contact points, 73–75 results, 75–76 structural dynamics, 71 sub-structure synthesis, 73 transmission, 72 vehicle seat frame vibration, 73 Hybrid vehicles analysis, 133–135 controller design, 132–133 dynamic modelling control model, 131–132 energy management, 132 line torsional vibration, 130–131 mechanical model, 130–131 power devices, 131–132 HEVs, 129 numerical simulations, 133–135 PHHV (see Parallel hydraulic hybrid vehicle (PHHV)) power system vibration analysis, 129 Hydraulic pump/motor (HPM) halfshaft, 140 motor, 138 overspeeding operations, 138 vehicle driveline, 137 vibration, 142 I IIR filter design ASQ, 80–81 NTF, 81 time-varying, 82 TPA, 80–81 Impact damper acoustic characteristics, 218 cantilever beam, 219 primary structures, 217 vibration response, 217 Industrial applications, 162 Inertia measuring unit (IMU) accelerometer, 98 gyroscope, 98 road surface conditions, 96 sensitivity, 100 vehicle’s dynamics, 95 ISO 7096:2000, 153–155 L Lightweight vehicles benefits, 55 control implementation, 58 dynamic handling problem, 55–56 load-to-curb weight ratio, 55

375 load variation, 57 multi-body simulation model, 56–57 results, 58–60 simulation, 58–60 vehicle specification, 56–57 Load variation consequence, 57 uncertainty in control, 59 Low-frequency, 15, 108, 157, 255, 272, 305, 335, 340, 359

M Magnetorheological fluid (MRE) dampers, 281 device structure, 282 dynamic modeling, 283–284 MRF-based variable resonance, 282 scale-building experiments experimental setup, 284–285 results, 285–286 validation, 283–284 Magnetorheological (MR) grease, 241–247, 250, 251, 281 Magneto-spring, 153–159 Material fabrication, 234, 235 Microelectromechanical systems (MEMS), 3, 5, 9, 95 Microplate air-gap capacitor, 3 DC voltage, 3 material properties, 5 physical dimensions, 5 rectangular air gap capacitive, 4 static deflection, 8 transient vibration, 5 Modal coupling, 37, 38, 43 Model based testing, 161–165, 209 Model validation and verification Bayesian factor, 149–150 experimental data, 145 FEM, 146–147 fuel tank, 146–147 global vehicle production, 145 modal analysis, 146 parameter, 148–149 response surface model, 147–148 Modified couple stress theory, 213 Modified embedding atom method (MEAM), 226–227 Molecular dynamics method construction, 227–228 contact simulation, 228, 229 MEAM potential, 226–227

376 Muffler design air-conditioning pipe systems, 15 expansion-cylinder-type, 13 insertion loss comparison, 16 Mach number, 14 noise attenuation performance, 14, 15 spectrum comparison, 14 and vibration comparison, 17 onsite noise comparison, 17–18 pipe system, 13 prototype, 16 Multi-mode switching strategy active suspension system, 21, 23–25 control modes, 21–22 electromotor state, 24 energy regeneration, 25, 26 full-vehicle dynamic model, 22–23 generator state, 24 intelligent control strategy, 25 optimal controller, 23–24 RMS comparisons, 27 Multiple-walled structure, 37, 41 Multi-unit impact damper (MUID) performance study, 299–302 simulation method, 298–299 single-degree-of-freedom linear structure, 297

N Nanocomposites, 231, 234–237 Natural frequency component, 263–268 Natural vibration, 202 Noise transfer functions (NTF), 80–83 Nonlinear vibration DC polarization, 3 equations of motion, 5–6 PDEs, 4–5 rectangular air gap, 3, 4 singularity elimination, 6–7 static and transient deflection, 4 transient behavior, 7–8 Nonreciprocal wave propagation, 193, 194, 197–198 Numerical simulation, 8, 104, 133–135, 220–221, 252–253, 258, 267, 321, 349 configurations, 331–333 incident directions, 330–331 theoretical model, 330

Subject Index O Optimal control acoustic radiation, 37–43 in active suspension, 23–24 CSSs, 363 numerical simulations, 364 Optimal solution, 343, 344, 346–348, 350 Overhead crane, 263–268

P Parallel hydraulic hybrid vehicle (PHHV) analysis, 140 dynamic modelling, 138–140 hybridization, 137 natural frequencies, 140 powertrains, 137, 140 results, 140 Partial differential equations (PDEs), 4–6 Pass-by noise (PBN) automotive example, 82–84 IIR filter design, 80–82 reproducibility, 79 synthesis diagram, 80 time-domain TPA synthesis, 79 TPA-ASQ, 80 Passive control, 217, 249, 253, 286 Phononic beam DSI, 193 nonreciprocal wave propagation, 197–198 synthetic materials, 193 time-delay-controlled, 194–197 Phononic crystals (PCs), 193, 275 Portable emissions measurement system (PEMS), 96–98 Powertrain modelling, 163

Q Quasi-zero stiffness (QZS) dynamic property optimization, 289–295 electromagnetic vibration isolation, 305–310 vertical isolation system, 255–261 QZS vibration isolator (QZSVI) cellular structures, 290 Duffing equation, 291 magnetic, 290 nonlinear force-displacement, 289 peak force transmissibility, 293 resonance frequency, 291

Subject Index R Random excitation EMVI suspension, 107–108 motion governing equation, 298 MUID, 297 operational tests, 165 Real-world driving emissions, 96, 97, 100 Residual vibration, 263–265, 268 Reverberation active headrest, 351, 352 discussions, 354–356 experimental setup, 352–353 feedback structures, 353–354 feedforward, 353–354 results, 354–356 Road condition measurement discussion, 98–100 GPS, 95 methodology emissions, 97 study routes, 98 vehicle’s positioning, 97 purpose, 96 real-world driving vehicles, 96 results, 98–100 Robust control analytical model, 264 discussion, 267–268 improvement conditions, 266 natural frequency component elimination, 265 open-loop control, 263 overhead crane, 263, 264 results, 267–268 target trajectory, 265–266 S Sampled-data control, 88, 90–92 Sandwich shell, 201–203 Seismic-control, 281, 282, 286, 287 Semi-active vibration suppression control law, 246 MR, 241 shear-type MR grease damper, 242 test, 243–244 transmissibility of displacement, 245–246 two-degree-of-freedom system, 243 Servo system, 88, 91 Shake table test and OpenSees, 258–259 Simulink and OpenSees Results, 258, 259

377 Shear-type damper, 241–243, 247 Shell formulation determination of equation, 233–234 strain displacement relations, 233 Simulation case I, 316–317 case II, 317, 318 case III, 317–318 circular-type pod silencer, 31 control, 92 EMVI device, 108 energy regeneration, 25, 26 MATLAB Simulink, 284 method, 298–299 numerical, 68, 133–135, 220–221, 252–253, 330–333, 338, 349 physical processes, 161 and results, 58–60, 174–176 time-domain, 105 vehicle specification, 56–57 Sound synthesis, 79–81, 84 Spherical shell structure, 201–207, 236, 237 Static force-displacement relation QZS, 256–257 test verification, 257 Statistical energy analysis (SEA) automotive BIW, 113, 114 BIW optimization, 116–117 equations, 112 optimization, 112–113 Strain gradient theory (SGT), 209 Structural dynamics, 71–73, 76 Structural loss factor, 203, 205–207 Structural nonlinearity, 87 performance study, 299–302 simulation method, 298–299 single-degree-of-freedom linear structure, 297 Suspension seat discussion, 157–159 earth-moving machines, 153 experimental method, 154, 155 results, 155, 156 ISO 7096:2000, 153

T Tape springs, 179–183 Target trajectory, 265–266

378 Time-delay nonreciprocal wave propagation, 194, 197–198 phononic beam, 194–197 Torque electromagnetic, 104, 106 engine, 87 estimation, 162–164 in NEDC, 174 planetary gearbox, 283 ring gear, 132 sensitivity, 68 vehicle vibration, 134 wheel, 56 Transfer matrix, 29, 72–74, 76, 129 Transfer path analysis (TPA), 80–81, 84 Tripod joints, 121–124, 126 Twistbeam rear suspension, 162, 164–165

U Uncertainty analysis, 47, 50–52 Upper-limit frequency, 322, 327, 331

V Variable-kinematic beam model FGMs, 209 numerical examples convergence, 212 parameter, 213, 214 validation, 212, 213 SGT, 209 theoretical formulation, 210–211 Vehicle dynamics, 22–24, 99 Vertical isolation system linear vibration isolators, 255 nonlinear stiffness, 255 numerical simulation, 258–260 QZS, 255–256 shake table test, 258–260 static force-displacement relation, 256–257 Vibration automotive drivetrain, 87–93 control (see Vibration control) engine-transmitted noise, 63 hybrid modeling approach (see Hybrid) nonlinear (see Nonlinear vibration) reduction (see Vibration reduction) variable-kinematic beam (see Variablekinematic beam model) velocity, 38, 41 Vibration analysis analytical model, 226–228

Subject Index contact simulation, 228, 229 FEM, 225 MD, 225–226 pivotal problem, 130 planetary gear system, 130 Vibration control automotive drivetrain (see Automotive drivetrain) devices, 271 double wall structure, 40 electromagnetic damper, 249–253 energy management, 129–135 hybrid vehicles (see Hybrid vehicles) MUID, 299 overhead crane, 263–268 robust, 263–268 variable damping suspension, 103 variable stiffness, 103 Vibration isolation, 18, 25, 289, 293, 305–310 Vibration reduction cable damper, 249 EMVI device, 104 MUID, 301 PHHV powertrain, 142 vehicle, 108 velocity response, 220 Vibration suppression deteriorated, 92 MRE, 281–286 multiple-order modes, 218 servo controller, 91 shell-like structures, 201 stability, 87 Virtual sensing applications and system, 162 coupling numerical simulation, 161 electromechanical powertrain, 162–164 flexible multibody model, 165–167 full-field strain estimation, 164–165 linear FE model, 164–165 McPherson suspension, 165–167 multi-physical 1D model, 162–164 torque estimation, 162–164 twistbeam rear suspension, 164–165 Virtual sound barrier (VSB) coupled system, 322 discussions, 322–325 experiments, 322–325 numerical simulations, 321 Viscoelastic beam analysis, 220–223 damping mechanism, 217 dynamic modeling, 218–219

Subject Index numerical simulations, 220–223 practical applications, 217 Viscoelastic material (VEM) discussions, 203–206 elastic stiffer layer, 201 formulation FE, 203 mathematical, 202–203 orthotropic material, 201 results, 203–206 system configuration, 202–203 W Wave control, 194, 198 Weighting function, 91

379 Whistle noise air conditioner manufacturers, 11 analysis, 12–13 cavity designs, 11 high-frequency, 12 measurement, 12–13 Muffler design, 13–18 noise attenuation performance, 13–18 pipe vibration, 12 Z Zero control power phenomena CSSs, 359–361 multiple CSSs, 362–365 sound minimization, 359, 361–362