Vibration engineering for a sustainable future : experiments, materials and signal processing. Vol. 2 9783030481537, 3030481530

Citation preview

Sebastian Oberst Benjamin Halkon Jinchen Ji Terry Brown  Editors

Vibration Engineering for a Sustainable Future Experiments, Materials and Signal Processing, Vol. 2

Vibration Engineering for a Sustainable Future

Sebastian Oberst • Benjamin Halkon Jinchen Ji • Terry Brown Editors

Vibration Engineering for a Sustainable Future Experiments, Materials and Signal Processing, Vol. 2

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

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

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

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

ISBN 978-3-030-48152-0 ISBN 978-3-030-48153-7 (eBook) https://doi.org/10.1007/978-3-030-48153-7 © 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 future experts, 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). We thank all the APVC 2019 sponsors whose financial support and presence in the exhibition area helped us to 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 chapters organized into parts aligned with the conference’s 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; a total of 245 extended abstracts were submitted to the local organizing committee, which accepted 183 after review, representing a rejection rate of 25%. Authors of the 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 leading experts 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. University of Technology Sydney, Australia

Sebastian Oberst

University of Technology Sydney, Australia

Benjamin Halkon

University of Technology Sydney, Australia

Jinchen Ji

University of Technology Sydney, Australia

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 was organized by Prof. Jinchen (JC) Ji and Prof. Benjamin (Ben) Halkon at the UTS. I would also like to sincerely thank the 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 Asia-Pacific region; engineers from the industry actively engaged with the conference; the APVC conference was financially supported by a government department (the New South Wales Government through the Office of the Chief Scientist and Engineer); best student paper awards were rigorously panel reviewed; and peer-reviewed selected papers were officially published in three volumes by international publisher Springer. Tokyo Metropolitan University, Hachioji, Japan

Takuya YOSHIMURA

vii

Organization

Local Organizing Committee Jinchen (JC) Ji University of Technology Sydney, Australia (Chair) Benjamin (Ben) Halkon University of Technology Sydney, Australia (Chair) Sebastian Oberst University of Technology Sydney, Australia (Technical Chair) Terry Brown University of Technology Sydney, Australia (Program Chair) Liya Zhao University of Technology Sydney, Australia (Ordinary Member) Philippe Blanloeuil University of New South Wales, Australia (Ordinary Member) Paul Walker University of Technology Sydney, Australia (Ordinary Member) Yancheng Li University of Technology Sydney, Australia (Ordinary Member) Hamed Kalhori University of Technology Sydney, Australia (Ordinary Member)

Steering Committee Takuya Yoshimura Shigehiko Kaneko Toshihiko Komatsuzaki Youngjin Park No-Cheol Park Haiyan Hu Li Cheng

Tokyo Metropolitan University, Japan (Chairman) University of Tokyo, Japan Kanazawa University, Japan Korea Advanced Institute of Science and Technology, Korea Yonsei University, Korea Beijing Institute of Technology, China Hong Kong Polytechnic University, Hong Kong ix

x

Organization

Jinhao Qiu

Nanjing University of Aeronautics and Astronautics, China Zhichao Hou Tsinghua University, China Tianran (Terry) Lin Qingdao University of Technology, China M. Salman Leong University of Technology, Malaysia Zaidi Mohd Ripin University of Science, Malaysia Nguyen Van Khang Hanoi University of Science and Technology, Vietnam Stefanie Gutschmidt University of Canterbury, New Zealand Ian Howard Curtin University, Australia Jinchen Ji University of Technology Sydney, Australia Benjamin Halkon University of Technology Sydney, Australia Oshihiro Narita (Honourable) Hokkaido University, Japan Hong Hee Yoo (Honourable) Hanyang University, Korea Athol J. Carr (Honourable) University of Canterbury, New Zealand Ban Chun Wen (Honourable) Northeastern University, China

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 CW Lim Dongping Jin Xinwen Wang Jian Xu Shaopu Yang Hu Ding Qinsheng Bi

Technical University Darmstadt, 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 City University of Hong Kong, Hong Kong Nanjing University of Aeronautics and Astronautics, China China University of Mining and Technology, Beijing Tongji University, China Shijiazhuang Tiedao University, China Shanghai University, China Jiangsu University, China

Organization

Hailin Wang Qingyun Wang Jie Huang Jie Yang Haiping Du Linke Zhang Sam Han

xi

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 Böni, 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

Matsuzaki, Kenichiro Matthews, David Melnikov, Anton Min, Cheonhong Mitchell, Sean Moreau, Danielle Joy Morishita, Shin Nakano, Yutaka Nakashima, Itsuki Nerse, Can Ng, Alex Ning, Donghong Nitzschke, Steffen Oberholster, Abrie Oberst, Sebastian Ota, Shinichiro Papangelo, Antonio Patnaik, S Srikant Pradhan, Somanath Prasad, Marehalli

Walker, Paul Wang, Shuping Wang, Yuxing Wang, Xu Wang, Feng Wang, Qiang Wang, Yuning Wang, Kuoting Wang, Shiliang Wantanabe, Seiji Watterson, Peter Wen, Hao Wen, Guilin Woschke, Elmar Wu, Lifu Wu, Helen Xiao, Tong Xiao, Tong Xu, Jian Xu, Daolin

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

Huang, Dongmei Hui, Kar Hoou Huston, Dryver Inoue, Tsuyoshi Irvine, Tom Ishikawa, Satoshi Ito, Atsuhiro Iwamoto, Hiroyuki Ji, Jinchen Jin, Dongping Jung, Hyung-Jo Kalhori, Hamed Kang, Hooi Siang Karimi, Hamid Reza Karimi, Mahmoud Kawamura, Shozo Kessissoglou, Nicole Shen, Jianwei Kawamura, Shozo Kessissoglou, Nicole

xii 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 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

Organization Kil, Hyun Gwon Kim, Dong Joon Kim, Dong Hyeon Kingan, Micheal Joseph Komatsu, Tadashi Komatsuzaki, Toshihiko Kondo, Eiji Kondou, Takahiro Koo, Bonsoo Koutsovasillis, Panagiotis Krueger, Timm Kundu, Pradeep

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

Yabui, Shota Yamada Keisuke Yamamoto, Hiroshi Yamazaki, Toru

Rudorf, Martin Ruppert, Michael

Yamazumi, Mitsuhiro Yan, Han

S, Bala Murugan Saeed, Omear Saito, Takashi Sanliturk, Kenan Y.

Yang, Jian Yang, Guidong Yao, JianChun Ye, Kan

Sasaki, Takumi Seering, Warren

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 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

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

Zhang, Guoqiang Zhang, Linke Zhang, Kai Zhao, Sipei Zhao, Feng Zhao, Liya

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

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

Veidt, Martin Vio, Gareth

Organiser APVC 2019

xiii

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 & Operational Deflection Shape analysis capable Max frequency BW up to 2.5GHz Max distance up to 300m Sub-pm resolution.

xv

xvi

Sponsors and Exhibitors, APVC 2019

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, that stay in-sync, and can predict its real behavior throughout the lifecycle, starting from the very early stages. 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,

Sponsors and Exhibitors, APVC 2019

xvii

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.

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,

xviii

Sponsors and Exhibitors, APVC 2019

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 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] xix

xx

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 Measurement Techniques and Sensors 1

2

3

4

5

Simulation and Measurement of an Electric Driven Turbocharger Test Rig with Full Floating Ring Bearing . . . . . . . . . . . . . . . Christian Daniel, Elmar Woschke, and Steffen Nitzschke

3

Visualization of Strain Distribution in Tire Tread Block Using Intermittent Digital Camera System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Matsubars, I. Kohei, S. Kawamura, and F. Tomonari

11

Field Measurements of the Attenuation of Vibration Between an Underground Tunnel and Ground Surface Through Sydney Sandstone and Shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Weber, H. Puckeridge, and P. Karantonis

21

Experimental Study on Rail Corrugation Development with 1/10 Scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masayoshi Okita and Yoshiaki Terumichi

29

Measurement and Dynamic Mode Analysis of Flow-Induced Noise with Combined Proper Orthogonal Decomposition . . . . . . . . . . . . . Osamu Terashima, Ayumu Inasawa, and Reon Nishikawa

35

Part II Experimental Modal Testing and Analysis 6

Accuracy Improvement to the Identified Modal Parameters of Systems with General Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ningsheng Feng, Eric Hahn, and Minli Yu

7

Classification of Characteristic Modes for Vibration Reduction . . . . . . Itsuki Nakashima, Takumi Inoue, and Ren Kadowaki

8

Vision-Based Modal Testing of Hyper-Nyquist Frequency Range Using Time-Phase Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Donghyun Kim and Youngjin Park

45 51

59

xxi

xxii

9

10

Contents

Experimental Investigation on the Effect of Tuned Mass Damper on Mode Coupling Chatter in Turning Process of Thin-Walled Cylindrical Workpiece. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Nakano, T. Kishi, H. Takahara, L. Croppi, and A. Scippa Eliminating the Influence of Additional Sensor Mass on Structural Natural Frequency in the Modal Experiment . . . . . . . . . . Feng Zhao, Wenliao Du, and Hongwei Li

65

73

Part III Dynamic Behaviour of Materials Including Nano-composite Structures and Vibration Absorbing Materials 11

12

13

A Study of the Vibration Reduction Effect of Sound Absorbing Material Within Acoustic Box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiajun Hong, Takuya Yoshimura, and Makoto Takeshita Experimental Study on the Effects of Pickguard Material on the Sound Quality of Electric Guitars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Osamu Terashima, Taisei Ito, Hiroyuki Yamada, Shota Mizukami, Ryoma Morisaki, and Toshiro Miyajima Influence of Pulverized Material on Vibration and Sound Characteristics of an Operating Ball Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Takuya Ito, Tatsuya Yoshida, and Fumiyasu Kuratani

83

91

99

14

Three-Dimensional Strain Calculation of Rubber Composite with Fiber-Shaped Particles by Feature Point Tracking Using X-Ray Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 M. Matsubars, T. Shinnosuke, N. Asahiro, S. Kawamura, T. Ise, T. Nobutaka, I. Akihito, K. Masakazu, and F. Shogo

15

Experimental Study on a Passive Vibration Isolator Utilizing Dynamic Characteristics of a Post-Buckled Shape Memory Alloy . . . 115 Takumi Sasaki and Yuta Kimura

16

Vibration Reduction of a Composite Plate with Inertial Nonlinear Energy Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Hong-Yan Chen, Hu Ding, and Li-Qun Chen

17

Vibration Analysis of Harmonically Excited Antisymmetric Cross-Ply and Angle-Ply Laminated Composite Plates . . . . . . . . . . . . . . . . 129 Chendi Zhu and Jian Yang

18

Analysis of Influence of Multilayer Ceramic Capacitor Mounting Method on Circuit Board Vibration . . . . . . . . . . . . . . . . . . . . . . . . . 137 Dongjoon Kim, Wheejae Kim, Joo Young Yoon, Eunho Lee, and No-Cheol Park

Contents

xxiii

Part IV Dynamic Model Updating and System Identification 19

Comparison of the Input Identification Methods for the Rigid Structure Mounted on the Elastic Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Shigeru Matsumoto and Takuya Yoshimura

20

Iterative Learning Control for Vision-Based Robotic Grasping . . . . . . 153 Yun Lu, Zheng Huang, Hao Wen, and Dongping Jin

21

Floor Response Spectrum of Nuclear Power Plant Structure Considering Soil-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Yuree Choi, Heekun Ju, and Hyung-Jo Jung

22

Real-Time Identification of Vehicle Motion-Modes . . . . . . . . . . . . . . . . . . . . . 167 Tong Chen, Minyi Zheng, Nong Zhang, Liang Luo, and Yishan Pan

23

Estimation of Normalized Eigenmodes and Natural Frequencies by Using the Effect of Accelerometers Mass . . . . . . . . . . . . . . 175 Junichi Hino, Satoshi Ooya, and Yuka Shigenai

Part V Structural Health and Machine Condition Monitoring 24

Pitting Fault Severity Diagnosis of Spur Gears Using Vibration and Acoustic Emission Sensor Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Pradeep Kundu, Ashish K. Darpe, and Makarand S. Kulkarni

25

Loosening Detection of a Bolted Joint Based on Monitoring Dynamic Characteristics in the Ultrasonic Frequency Region . . . . . . . . 191 Takanori Niikura, Naoki Hosoya, Shinji Hashimura, Itsuro Kajiwara, and Francesco Giorgio-Serchi

26

Gas Turbine Fault Detection Using a Self-Organising Map . . . . . . . . . . . 197 Kar Hoou Hui, Ching Sheng Ooi, Meng Hee Lim, Mohd Dasuki Yusoff, and Mohd Salman Leong

27

A Comparative Analysis Between EMD- and VMD-Based Tacho-Less Order Tracking Techniques for Fault Detection in Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Madhurjya Dev Choudhury, Liu Hong, and Jaspreet Singh Dhupia

28

An Effective Indicator for Defect Detection in Concrete Structures by Rotary Hammering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Y. Hasebe, F. Kuratani, T. Yoshida, and T. Morikawa

29

A Vibration-Based Strategy for Structural Health Monitoring with Cosine Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 C. H. Min, S. G. Cho, J. W. Oh, H. W. Kim, and B. M. Kim

xxiv

30

Contents

A New Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Ren Kadowaki, Takumi Inoue, Kentaro Kameda, and Kazuhisa Omura

Part VI

Vibration Isolation and Reduction

31

Vibration Isolation Performance of an LQR-Stabilised Planar Quasi-zero Stiffness Magnetic Levitation System . . . . . . . . . . . . . . . . . . . . . . . 237 Nur Afifah Kamaruzaman, William S. P. Robertson, Mergen H. Ghayesh, Benjamin S. Cazzolato, and Anthony C. Zander

32

Proposition of Isolation Table Considering the Long-Period Earthquake Ground Motion (Method of Changing Natural Frequency of Isolation System with Additional Spring) . . . . . . . . . . . . . . . 245 Shozo Kawamura, Tetsuhiko Owa, Tomohiko Ise, and Masami Matsubara

33

Experimental Vibration Analysis of Seismic Isolation System Using Inertial Mass Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Ryo Masano, Nanako Miura, and Akira Sone

34

Development of Sliding-Type Semi-active Dynamic Vibration Absorber Using Active Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Eiji Kondo, Tsubasa Matsuzaki, Noriyoshi Kumazawa, Mitsuhiro Oda, and Kazumasa Kono

35

Development of a Vibration Isolator Using Air Suspensions with Slit Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Hiroshi Yamamoto, Haruki Nakanozo, and Terumasa Narukawa

36

Development of a Tuning Algorithm for a DynamicVibration Absorber with a Variable-Stiffness Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Tappei Kawai, Toshihiko Komatsuzaki, and Haruhiko Asanuma

37

Design Approach of Laminated Rubber Bearings for Seismic Isolation of Plant Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 J. H. Lee, T. M. Shin, and G. H. Koo

Part VII Noise, Vibration and Their Applications in Electricity Power Systems 38

Sound Transmission of Beam-Stiffened Thick Plates. . . . . . . . . . . . . . . . . . . 297 K. Zhang and T. R. Lin

39

On the Feasibility of Transformer Insulation Aging Detection with Vibration Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 J. H. Yang, X. Cai, H. H. Jin, J. L. Hou, and Y. X. Wang

Contents

xxv

40

Study on the Effectiveness of Transformer Equivalent to Point Source in Substation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Ling Lv, Linke Zhang, and Li Wang

41

Transformer Acoustic Equivalent Model in Engineering Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Dakun Li, Wei Li, Li Wang, Linke Zhang, and Zhixing Li

42

Simulation Study on Noise Reduction Effect of Substation Noise Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Xuan Cai, Xuelei Zhan, Yong Cai, and Li Wang

43

Fault Recognition of Induction Motor Based on Convolutional Neural Network Using Stator Current Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Tian Han, Ze Wang, Zhongjun Yin, and Andy C. C. Tan

44

A Numerical Study on Active Noise Radiation Control Systems Between Two Parallel Reflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Jiaxin Zhong, Jiancheng Tao, and Xiaojun Qiu

Part VIII Applications and Advances in Laser Doppler Vibrometry 45

Characterization of Active Microcantilevers Using Laser Doppler Vibrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Michael G. Ruppert, Natã F. S. De Bem, Andrew J. Fleming, and Yuen K. Yong

46

Experimental Investigation on Generation Mechanism of Friction Vibration in Toner Fixing Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Y. Nakano, Y. Matsumura, T. Hase, and H. Takahara

47

Using a Laser Doppler Vibrometer to Estimate Sound Pressure in Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Tong Xiao, Xiaojun Qiu, and Benjamin Halkon

48

Experimental and Numerical Modal Analysis of an Axial Compressor Blisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Patrick Buchwald, Christian U. Waldherr, Jochen Schell, Heinrich Steger, and Damian M. Vogt

49

Effectiveness of Using Damping as a Parameter to Detect Impact Damages in GFRP Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Shaheen M. P., Klaudiusz Holeczek, Ashish K. Darpe, and S. P. Singh

50

Debonding Growth Monitoring Through Ultrasonic Guided Waves Field Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 P. Blanloeuil, L. R. F. Rose, M. Veidt, and C. H. Wang

xxvi

51

Contents

Nonlinear Ultrasonic Guided Waves for Damage Detection . . . . . . . . . . . 405 C. T. Ng and A. Kotousov

Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Part I

Measurement Techniques and Sensors

Chapter 1

Simulation and Measurement of an Electric Driven Turbocharger Test Rig with Full Floating Ring Bearing Christian Daniel, Elmar Woschke, and Steffen Nitzschke

1.1 Introduction The support system of high speed rotating systems is often based on fluid bearings. Due to their simple and cheap design as well as a good damping characteristics, they are usually used in automotive turbochargers (TC). To enlarge the damping of the fluid bearings, a floating ring is introduced to realise an additional outer fluid bearing. This design has some advantages like higher damping but also some disadvantages like nonlinear behaviour caused by the two fluid films which can both induce oil-whirl excitations. The authors presented simulations of different bearing designs for an automotive turbocharger [2]. Nevertheless, the full-floating ring design is widely used in automotive turbochargers, since the production costs are low in case of pure cylindrical bearings. Besides the mentioned advantages, this design leads to a highly non-linear behaviour. An integral state variable of these bearings is the ring speed. It depends on different bearing parameters like clearance, bearing width and of course the resulting pressure distribution in the fluid films. For comparison of simulation and measurement, the ring speed is essential. Other researcher groups have also performed measurements of the floating ring speed. One of the first [6] was published in 1954 and contains stroboscopical determination of the ring speed on a relative large test rig for a single bearing. Later, [4] used a simple rotor, which was smaller but still larger than an automotive turbocharger. Köhl et al. [7] built a transparent printed housing to get the possibility to arrange a high speed camera on the area of the ring’s oil feed holes. By image processing the passing oil feed holes are counted and so the ring speed could be determined.

C. Daniel () · E. Woschke · S. Nitzschke Institute of Mechanics, Otto von Guericke University Magdeburg, Magdeburg, Germany 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/10.1007/978-3-030-48153-7_1

3

4

C. Daniel et al.

1.2 Simulation of Run-Up The equation of motion of the rotor dynamic system includes the rigid and elastic bodies. The shaft is considered as finite Timoshenko beam elements and the housing as general elastic body reduced with 50 eigenvectors. This forms a system of second order equations, which is solved numerically in time domain using appropriate semi-implicit time integration solvers like [8]. This is done in our own multibody simulation software EMD, which was developed for simulation of non-linear dynamic systems [1] including the bearing interaction. The fluid bearings are considered by Reynolds equation, which describes the pressure distribution in hydrodynamic bearings ∂ ∂x



h3 ∂p η ∂x



∂ + ∂y



h3 ∂p η ∂y

 = 6(U1 + U2 )

∂h ∂h + 12 . ∂x ∂t

(1.1)

Equation (1.1) is solved numerically in every time step for the input quantities h and ∂h/∂t for the inner and outer oil film. For this sake, the finite volume method is used to build a system of equations. The numerical solution in every time step is very time consuming, but necessary to cover all non-linear effects of the bearing system. This part is implemented as force routines in EMD. The friction torque consists of two parts, the shear stress due to the velocity gradient and the shear stress due to the pressure gradient. Felscher [5] showed that the gradient becomes important for high eccentricities and should not be neglected. The friction torque at the cylindrical surface area is given by  Mf r =

τ (x, y) · rs dA .

(1.2)

The friction torque Mf r is acting on the inner side of the floating ring, which accelerates the ring; the torque at the outer surface decelerates the ring. In addition to the friction torque at the radial surfaces, the friction torque at the axial side of the floating ring is present. Domes has mentioned this equation in his thesis [4] and has used it for comparison with measurements at a sample rotor. Here the sum of both acting friction torques on the axial sides of the floating ring are formulated as function of the axial gaps s1 and s2 .

Mf a =

   1 π  4 1 · Ωring . · ra − ri 4 · η · + 2 s1 s2

(1.3)

The resulting ring speed Ωring depends on all acting torques and inertias, which are a result of the numerical time integration process. The ring speed could hardly be estimated a priori because it’s dependency on the pressure distribution and vice versa.

1 Simulation and Measurement of an Electric Driven Turbocharger Test Rig. . .

5

1.3 Electric Driven Test Rig for TC The test rig used in the present study consists of a simplified turbocharger, where the turbine and compressor wheel are substituted by equivalent cylindrical masses to reduce the flow loss during run-up by neglecting aerodynamic effects acting on the blades (Fig. 1.1). The parameter of the shaft are mentioned in [3]. The run-up is realised by an electric synchronous motor with approx. 2000W and an idle speed of 50,000 rpm. This does not cover the whole speed range of this turbocharger, but it is enough to show the relevant effects like oil-whirling motions of the shaft. The shaft displacement is measured at the compressor side by using a laser triangulation sensor. The shaft speed is measured by an optical speed sensor at the turbine side. The speed of one floating ring is measured by an eddy current sensor placed at the compressor side bearing – Fig. 1.2. The turbine side bearing was not measured because the housing is unfavourable for direct access with a tapped hole. Figure 1.2 shows the installation of the eddy current sensor EU05 from micro-epsilon in the sickle groove of the oil inlet. This ensures that the tapped hole for the sensor has no hydrodynamical influence on the system.

1.4 Measurement of Floating Ring Speed The speed of the ring is measured by counting the oil feed holes in the floating ring. This is realised by measuring the position of the floating ring in the midplane using an eddy current sensor. The signal is disturbed by the oil feed holes in the

Fig. 1.1 Electric driven test rig for TC with high speed coupling

6

C. Daniel et al.

Fig. 1.2 Ring speed measurement at bearing on the compressor side in the sickle groove

floating ring. Figure 1.3 illustrates the signal of the eddy current sensor. The signal can be easily converted into a rectangular signal in order to count the edges for speed measurement.

1 Simulation and Measurement of an Electric Driven Turbocharger Test Rig. . .

7

Fig. 1.3 Raw signal of the eddy current sensor and modification for edge counting

1.5 Comparison of Simulation and Measurement The boundary conditions of the system are well known. For the simulation, the temperature and the pressure of the oil are set to the values, which are pretended at the test rig. The oil (FUCHS 600639273 10W-40) was fed with 3 bar and 60◦ C. The change of clearance due to the different bearing materials is regarded in the simulation. The temperatures are calculated with an analytical heat transfer model during the run-up, which is necessary to meet the correct sub2 entry frequency. Figure 1.4 shows the results of measurement of shaft and ring speed. The compassion of the shaft displacement at the compressor side using a spectrogram to analyse the frequency of the vibration content during run-up is shown in Fig. 1.5. In the Measurement the sub 1 starts at fD = 350 Hz an vibrates with fsub1 = 200 Hz and sub 2 fD = 820 Hz an vibrates with fsub2 = 414 Hz. The simulation fD = 344 Hz an vibrates with fsub1 = 200 Hz this is close to the measurement. The sub 2 entry speed in the simulation is fD = 800 Hz an approx. 20 Hz to low but the predicted frequency of the sub 2 fsub2 = 420 Hz meets the measurement quiet very good.

8

C. Daniel et al.

Fig. 1.4 Comparison between measured and simulated floating ring speed

1.6 Conclusion The paper showed a method for ring speed measurement of automotive turbochargers by using a small sized eddy current sensor. The oil feed holes in the floating ring influence the signal of displacement measurement by passing the sensor. The advantage of using an eddy current sensor compared to other speed sensor concepts is the duality of measurement results and the possibility to use it on small bearings without disturbing the bearings behaviour. The comparison to the nonlinear simulation with EMD shows a good prediction of the ring speed and the frequency of subharmonic vibration. The start frequency of subharmonic vibration can be predicted very good for sub 1 and sub 2.

1 Simulation and Measurement of an Electric Driven Turbocharger Test Rig. . .

9

Fig. 1.5 Spectrogram of shaft displacement at compressor (measurement/simulation)

References 1. Daniel, C.: Simulation von gleit- und wälzgelagerten Systemen auf Basis eines Mehrkörpersystems für rotordynamische Anwendungen. Ph.D. thesis, Otto-von-Guericke Universität Magdeburg (2013) 2. Daniel, C., Göbel, S., Nitzschke, S., Woschke, E., Strackeljan, J.: Numerical simulation of the dynamic behaviour of turbochargers under consideration of full-floating-ring bearings and ball bearings. In: ICOVP 2013 – 11th International Conference on Vibration Problems (2013)

10

C. Daniel et al.

3. Daniel, C., Woschke, E., Nitzschke, S.: Simulation and measurement of ring speed of full floating ring bearing in an automotive turbocharger. In: Santos, I. (ed) Proceedings of the 13th International Conference Dynamics of Rotating Machinery – SIRM, Copenhagen (2019) 4. Domes, B.: Amplituden der unwucht- und selbsterregten Schwingungen hochtouriger Rotoren mit rotierenden und nichtrotierenden schwimmenden Büchsen. Ph.D. thesis, Universität Karlsruhe (1980) 5. Felscher, P.: Rückwirkung des Gleitlagermoments auf die Drehbewegung des Rotors. Ph.D. thesis, Technische Universität, Darmstadt (2016). http://tuprints.ulb.tu-darmstadt.de/5545/ 6. Kettleborough, C.: Frictional experiments on lightly-loaded fully floating journal bearings. Aust. J. Appl. Sci. 5, 211–220 (1954) 7. Köhl, W., Kreschel, M., Filsinger, D.: Modellabgleich eines Turboladerrotors in Schwimmbuchsenlagerung anhand gemessener Schwimmbuchsendrehzahlen, in Proceedings SIRM2015 11. Internationale Tagung Schwingungen in Rotierenden Maschinen, Magdeburg, pp. 1–10 (2015). http://tubiblio.ulb.tu-darmstadt.de/72538/ 8. Shampine, L.F., Reichelt, M.W., Kierzenka, J.A.: Solving index-1 daes in matlab and simulink. SIAM J. Sci. Comput. 41(3), 538–552 (1999)

Chapter 2

Visualization of Strain Distribution in Tire Tread Block Using Intermittent Digital Camera System M. Matsubars, I. Kohei, S. Kawamura, and F. Tomonari

2.1 Introduction Automobile tires are one of the important components of automobiles. In addition to supporting the weight of the vehicle, they transmit the driving and braking force to the road surface through their contact with it. Furthermore, tires play a vital role in steering the vehicle. The part of the tire surface that is in direct contact with the road surface is called the tread. The tread has engraved patterns consisting of grooves, blocks, and sipes (very fine grooves). In evaluating the tire performance, the design of the tread is of particular interest. The EU’s tire noise regulation ECE-R117 has led to increased demands for quiet tires; thus, they have been the subject of extensive research [1–5]. Against this backdrop, it is important to grasp the deformation state of the tread block that that makes repeated intermittent contact with the road surface. Several methods that do not require contact have been proposed to measure tire deformation and strain. One of such methods utilizes images captured using charge-coupled device (CCD) or high-speed camera based on the digital image correlation or sampling moire method [2, 6]. In addition, the use of a Doppler vibrometer has been proposed [7]. However, the deformation of the block occurs during repeated penetration into the ground plane and detachment from it on high speed. Therefore, it is difficult to capture the deformation state of the block using conventional methods due to the relationship between the sampling frequency and resolution. In high-speed photography, there is a reciprocal relationship between

M. Matsubars · I. Kohei () · S. Kawamura Toyohashi University of Technology, Toyohashi-Shi, Aichi, Japan e-mail: [email protected]; [email protected]; [email protected] F. Tomonari Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_2

11

12

M. Matsubars et al.

the frame rate and the pixel number. Therefore, there is no commercially available device for capturing the deformation of the tread block, especially pertaining to minute deformation. Thus, we propose a measurement system with high sampling frequency and high pixels that can obtain images in a pseudo manner. The system was achieved by manipulating a strobe and a high-pixel commercial digital single-lens camera at high speed. Based on the signal from the external trigger device attached to the wheel, we synchronized the shutter operation of the camera and the flash duration of the strobe. By performing exposure based on the flash duration of the strobe, the system achieves less image blurring. This paper presents the outline of the system and the measurement results.

2.2 Measurement System The proposed system consists of a digital single-lens camera, a strobe, and an external trigger device, which are controlled by a field-programmable gate array (FPGA). Figure 2.1 shows the outline of the system, and Fig. 2.2 shows the operation image of each apparatus. The FPGA receives the signal from the external trigger and operates the shutter of the digital single-lens camera. The shutter speed is set to fast, and the room is underexposed. By firing the flash precisely when the shutter is fully open, it is possible to capture an instantaneous image based on the flash duration, thereby improving the signal-to-noise (S/N) ratio. Although the deformation of the tread block is an unpredictable phenomenon, it is presumed to be a periodic phenomenon that corresponds to the frequency of tire rotation. In this study, pseudo-continuously captured images were obtained by varying the

Fig. 2.1 System overview

2 Visualization of Strain Distribution in Tire Tread Block Using Intermittent. . .

13

Fig. 2.2 Operation image of each apparatus

shooting timing each rotation cycle. The advantage of this method is that it can be effected using a digital single-lens camera; furthermore, for subsequent shooting, it does not require resetting. A digital single-lens camera with a high pixel count was selected, because it functions independently of the transfer time of the image data to the external storage device. Furthermore, the imaging timing can be adjusted with the time resolution based on the performance of the FPGA, and a high sampling frequency can be realized. The proposed system can provide measurements with high spatial resolution and time resolution for periodic phenomena. Thus, it is suitable for capturing the minute deformation of the tread block of a tire rotating at high speed.

2.3 Measurement Experiment 2.3.1 Experiment Outline We obtained shots of the rotating tire using the proposed measurement system. Figure 2.3 shows the placement of each apparatus. The sample tires are commercially available, and their size was 195/80R15. The characteristics of the tires are as follows: internal pressure, 230 kPa; rotational speed, 50 km/h; and axial load, 4 kN. Furthermore, as shown in Fig. 2.4, a marker point was created on the tire block using a laser processing machine and putty. Then, we prepared two patterns with a view to performing comparison based on block size. Patterns with large combinations of adjacent blocks are called pattern A, and patterns with small combinations are called pattern B.

14

M. Matsubars et al.

Fig. 2.3 Placement of each apparatus Fig. 2.4 Marker point

A photoelectric sensor was attached to the frame, to be used as the external trigger, and a reflector was attached to the wheel. Consequently, one pulse wave, the trigger signal, was output per rotation. The pulse wave was received by the FPGA, and a signal was sent to prompt the camera and strobe to shoot. Then, the camera takes 100 shots at the same position. The next position was captured by delaying the signal output from the FPGA by 0.12 msec. The camera was programmed to stop shooting when a total of 3500 shots had been taken. And the strobe flash time is 1.62 μsec. In addition, idle operation was performed for 10 min after the commencement of the test to stabilize the testing machine and warm the tires. The camera used was a commercially available Nikon D5500. The shooting mode was manual, shutter speed was 1/200, pixel count was 6000 × 4000 pixels, ISO sensitivity was 100, and f-number was 5.6.

2 Visualization of Strain Distribution in Tire Tread Block Using Intermittent. . .

15

Fig. 2.5 Recognized marker points by Hough transform

Fig. 2.6 Delaunay triangle

2.3.2 Image Processing The photographed marker points, as shown in Fig. 2.5, were round. Therefore, round-shape fitting was performed using the Hough transform, and the center of the circle was designated the coordinate point of the marker point. It was necessary to eliminate unnecessarily recognized points. Next, the average value at all the positions of the point on the root side of the block where deformation was expected to be minimal was calculated. A photo was captured near the average at each position and used as a representative photo at that position. Next, the method of calculate strain is described. Delaunay triangle is illustrated as shown in Fig. 2.6 based on the point group data of the representative photograph obtained by the above method. At that time, we use only point data that recognized in common through each position. Furthermore, among the illustrated triangles, those with a large aspect ratio are removed. Next, the strain is calculated based on the theory of the finite element method of linear triangular elements. In addition, when evaluating with strain energy, the modulus of longitudinal elasticity E of the tread block and the Poisson’s ratio v are necessary. This time, the calculation was performed with E = 1.5 × 106 Pa and ν = 0.46, which are reference values for general rubber material.

16

M. Matsubars et al.

Fig. 2.7 Standard deviation at each position (blue line shows x direction and red line shows y direction)

2.3.3 Measurement Result Figure 2.7 plots the standard deviation of the shooting positions of 100 images at each position of each block pattern. The standard deviation of pattern A was 21.029 pixels in the x direction and 5.042 pixels in the y direction. For pattern B, the standard deviation was 12.371 pixels in the x direction and 7.766 pixels in the y direction. In this study, the reference was 0.015 mm/pixel. From Fig. 2.7, it can be observed that there was a point when the movement of the standard deviation in the y direction exceeded the one in the x direction. It was confirmed from the image that contact began from this position. Both patterns exhibited large standard deviations

2 Visualization of Strain Distribution in Tire Tread Block Using Intermittent. . .

17

Fig. 2.8 Strain in each direction and strain energy (pattern A)

in the x direction, because the tire rotates in the circumferential direction whose variation exceeds that in the radial direction. Furthermore, the standard deviation in the x direction tended to be larger for pattern A and smaller for pattern B. This was attributed to the largeness of both the block and the deformation range. Figure 2.8 shows the strain distributions in the x and y directions and the strain energy distribution of pattern A. From the top to bottom, the following conditions are represented: before ground, forward ground, rear ground, and in the ground plane. Figure 2.9 shows the same calculation for pattern B. By continuously varying the imaging timing this way, it was possible to obtain pseudo-continuous images, that is, images with higher pixel count and field of view, compared to a high-speed camera. The tire used in this experiment had a step, as shown in Fig. 2.4. Thus, when the image of the step area was captured with a camera, the angle will vary. Consequently, an abnormal value was displayed for the strain around the step area. Therefore, this part was excluded. It can be confirmed that the strain distribution in the x direction slightly increased as the patterns A and B penetrated the ground plane. Thus, it was assumed that the block expanded in the lateral direction. Furthermore, the distribution was sparse overall. This was attributed not to the varied strain but to the large blur in the x direction.

18

M. Matsubars et al.

Fig. 2.9 Strain in each direction and strain energy (pattern B)

Then, we observed the strain distribution in the y direction. In this case as well, as patterns A and B both entered into the ground plane, the strain value increased in the negative direction from the point of contact. In particular, it can be confirmed that the distortion on the outer peripheral side of the block increased when the patterns penetrated the ground plane. This was due to the fact that the outer peripheral side of the block had a notch, as can be seen from Fig. 2.4. Furthermore, the tire underwent a compressive strain before penetrating the ground plane. This was attributed to the deformation of the tire belt due to rotation. In addition, similarly to the x direction, the distribution in the y direction was mottled, as a result of the image blurring. It was confirmed from observing the strain energy distribution that the strain energy was generated from the point where contact began. Furthermore, it was also verified that large strain energy was generated on the outer peripheral side of the block in the ground plane. This was similar to the manifestation of distortion in the y direction. Thus, it was concluded that the strain generated during the rotation of the tire tread indicated that the strain in the y direction was dominant.

2 Visualization of Strain Distribution in Tire Tread Block Using Intermittent. . .

19

2.4 Conclusion In this study, we tried to visualize the deformation of tread block by using a system that captures pseudo-continuous images. For that purpose, the strain distribution was calculated and the validity of the operation of the proposed system could be shown. And the visualization of deformation could be reached. In the future, it is necessary to improve the system and reduce blurring and to use a twin-lens camera so that the tread part with steps can be photographed.

References 1. Kindt, P., Bianciardi, F.: Tire/road noise - characterization and potential further reductions of road traffic noise. Proc. Inter. Noise. 16, 2254–2264 (2016) 2. Tsujiuchi, N., Matsuda, A., Seki, H., Takahashi, H.: Developing evaluation model of Tire pattern impact noise. Proc. Inter. Noise. 16, 2969–2979 (2016) 3. Garcia, D.C., Sanchez, E.V., Davo, N.C., Vicente, H.C., Lozano, M.S.: A new methodology to assess sound power level of Tyre/road noise under laboratory controlled conditions in drum test facilities. Appl. Acoust. 110, 23–32 (2016) 4. Chiu, J.T., Tu, F.Y.: Application of a pattern recognition technique to the prediction of tire noise. Appl. Acoust. 110, 23–32 (2016) 5. Matsubara, M., Tajiri, D., Ise, T., Kawamura, S.: Vibrational response analysis of tires using a three-dimensional flexible ring-based model. J. Sound Vib. 408, 368–382 (2017) 6. Hanada R., Miyazawa M.: Development of tire tread block displacement measurement method with non-contact shape measurement method (First Report:Verification of Measurement Possibility), Proceedings of Transactions of the Society of Automotive Engineers of Japan, Inc. No.143-17, 878–882 (2017) 7. Diaz, C.G., Kindt, P., Middelberg, J., Vercammen, S., Thiry, C., Close, R., Leyssens, J.: Dynamic behavior of a rolling tyre:Experimental and numerical analyses. J. Sound Vibration. 364, 147– 164 (2016)

Chapter 3

Field Measurements of the Attenuation of Vibration Between an Underground Tunnel and Ground Surface Through Sydney Sandstone and Shale C. Weber, H. Puckeridge, and P. Karantonis

3.1 Introduction Due to the many challenges associated with constructing aboveground road and rail infrastructure in major cities, many new projects are constructed underground. One advantage of tunnels is a reduction in environmental (airborne) noise. However, during construction, vibration from tunnel boring machines, roadheaders, rock hammers, drilling machines, and other sources generate vibration which can be transmitted through the ground and into nearby buildings. During operations, vibration is also produced by trains and to a lesser extent by road traffic. The vibration from underground construction and operations has the potential to disturb building occupants in the form of tactile vibration or ground-borne noise. The satisfactory operation of vibration-sensitive equipment may also be impaired. To predict vibration impacts at sensitive receivers and manage potential impacts, it is necessary to understand the source, propagation, and receiver system [1]. This includes: the amplitude and frequency content of the source vibration levels, the vibration attenuation between the tunnel and the ground surface, and how vibration propagates into and throughout receiver buildings. On a recent project in Sydney, field measurements were undertaken at seven sites to investigate how vibration attenuates between the tunnel and ground surface. This study presents the results of the ground attenuation measurements where the lithology between the tunnel and ground surface comprised Hawkesbury Sandstone and/or Ashfield Shale. The results are intended to be used in empirical models to improve ground-borne noise and vibration predictions on underground projects with similar ground conditions.

C. Weber () · H. Puckeridge · P. Karantonis Renzo Tonin & Associates, Strawberry Hills, 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/10.1007/978-3-030-48153-7_3

21

22

C. Weber et al.

An examination of how surface vibration levels propagate into and within buildings (building coupling loss and amplification) is presented in [2].

3.2 Ground Attenuation Measurements 3.2.1 Measurement Sites Vibration measurements were undertaken at seven sites near newly constructed twin-bored tunnels in Sydney. Both tunnels are segmentally lined with an internal diameter of approximately 6 m. The purpose of the measurements was to quantify the vibration attenuation between the tunnel and ground surface. A summary of the measurement distances and ground conditions at two representative sites is shown in Table 3.1. Measurement distances represent the radial (slant) distance between the tunnel and surface locations. At site 5, the tunnel was founded in Hawkesbury Sandstone. At site 6, the tunnel was founded in Hawkesbury Sandstone with Ashfield Shale above.

3.2.2 Vibration Source in Tunnel Source vibration levels were generated in one of the rail tunnels using an excavator with a hydraulic rock hammer attachment. This source generated strong and steady vibration levels which were measurable in the opposite tunnel and on the ground surface, providing a strong signal-to-noise ratio in the 1/3 octave frequency range between 31.5 Hz and 315 Hz. This range corresponds with the predominant frequencies associated with construction and train vibration sources, tactile vibration, and audible ground-borne noise. Additional details are provided in [2]. Measurements were performed at up to five distances at each site (see Table 3.1). This was achieved by varying the source position in the tunnel, with a fixed measurement position on the surface. The near distances at each site ranged from 30 m to 42 m and the maximum distances at each site ranged from 59 m to 66 m. Simultaneous vibration measurements were performed in the adjacent tunnel (near-field, 9 m away, radial direction) and on the ground surface near sensitive buildings (far-field, vertical direction).

Table 3.1 Measurement distances and ground types at each site Site 5 6

Measurement distances (m) 66, 47, 31, −, 41 50, 37, 34, 43, 59

Ground types Hawkesbury Sandstone Hawkesbury Sandstone/Ashfield Shale

3 Field Measurements of the Attenuation of Vibration Between. . .

23

3.2.3 Measurements on Surface Above the tunnels, measurements were performed on the ground surface approximately 5 m from buildings. Test locations were selected in areas with low background vibration levels (away from major roads). The vibration measurement time histories were reviewed to assist in identifying periods where vibration levels were steady and not significantly influenced by extraneous events. Excluded results are marked as “– ” in Table 3.1. 1/3 octave band vibration levels were excluded if the signal-to-noise ratio was not greater than 10 dB.

3.3 Results 3.3.1 Comparison of Near-Field and Far-Field Vibration Levels Sample measurement results from one measurement position at site 5 (31 m radial distance) are provided in Fig. 3.1. The results show that the signal-to-noise ratio is greater than 10 dB at 1/3 octave frequencies of 25 Hz and above. At 40 Hz and below, vibration levels on the wall of the adjacent tunnel (nearfield) are similar to vibration levels on the ground surface (far-field), indicating negligible attenuation. At 125 Hz and above, vibration levels in the near-field are much higher than vibration levels in the far-field, indicating significant attenuation (around 30 dB).

Surface, Z

Tunnel, R

Surface, Background

Tunnel, Background

Vibration Levels - dB re 1nm/s

100 90 80 70 60 50 40 30 20 20 25 31.5 40 50 63 80 100 125 160 200 250 315

1/3 Octave Centre Frequency - Hz Fig. 3.1 Sample near-field and far-field vibration results at one measurement position at site 5 (31 m radial distance) and ambient background vibration levels

24

C. Weber et al.

Fig. 3.2 Vibration attenuation results versus radial distance at all measurement sites. Results are shown for 1/3 octave frequency bands 40 Hz, 80 Hz, 160 Hz, and 315 Hz. Vibration attenuation is expressed as far-field vibration levels minus near-field vibration levels

3.3.2 Vibration Reduction Between Tunnel and Ground Surface The results in Fig. 3.1 can be expressed in terms of the vibration loss between the near-field (opposite tunnel) and far-field (ground surface) measurement locations. Sample measurement results in the 40 Hz, 80 Hz, 160 Hz, and 315 Hz 1/3 octave frequency bands are shown in Fig. 3.2. The results in the remaining 1/3 octave frequency bands (not shown) provide similar trends. The attenuation results from all sites are included in Fig. 3.2. The plots show the measured attenuation at radial distances from the vibration source in the tunnel. Consistent with the results in § 3.1, little or no attenuation occurs at low frequencies (40 Hz), and some measurement positions indicate higher vibration levels measured on the ground surface. Significant attenuation occurs at higher frequencies. It is not well understood why amplification is present at some low frequencies data points, but similar results have been observed on other projects [3–4]. Meng et al. [5] note that amplification can occur on the ground surface away from tunnels at low frequencies and could be due to multi-reflections of shear waves in soft soils near the surface and because of the underground source. The amplification effect is greatest when the horizontal distance is similar to the tunnel depth. ISO 1483732 [6] notes that the distance attenuation is monotonic in the near-field (at low frequencies).

3 Field Measurements of the Attenuation of Vibration Between. . .

25

3.4 Empirical Ground Attenuation Models Numerous models have been proposed to predict the vibration attenuation between tunnels and receivers on the ground surface. Common prediction models are discussed in [6–9] and typically account for geometric losses, material losses, or both. The following formula has been used to describe the attenuation losses: V (r) = V (ro )

 r γ o

r

e−

πf η c .(r−ro )

(3.1)

Where: V(r) is the vibration amplitude at the ground surface. V(ro ) is the vibration amplitude at the reference position (opposite tunnel). r and ro represent the far-field and near-field distances. γ is the geometric attenuation loss term (0 = no loss, 1 = underground point source). f is frequency in Hz. η is the soil loss factor. c is the wave speed. The results in Fig. 3.2 show three curves which represent the predicted geometric damping loss, material damping loss, and both (reference distance of 9 m). At low frequencies, the geometric attenuation term (frequency invariant) predicts higher losses than the measurement results. At higher frequencies, the geometric attenuation predicts lower losses than the measurement results. The measurement data shows that the attenuation loss in Sydney Sandstone and Shale is frequency dependent, indicating that prediction models which consider only the source-receiver distance would be unsuitable for accurate predictions. The material damping curve in Fig. 3.2 is based on an average shear wave velocity (c = 600 m/s) and soil loss factor (η = 0.06). Some engineering judgment is required to ensure that the selected parameters remain conservative, but do not result in significant overdesign. The use of automated curve fitting algorithms is not recommended. In practice, the measurement results at frequencies greater than 80 Hz are likely to include a combination of geometric and material losses and both terms are normally considered. However, in the current study, material damping losses alone were found to provide the best correlation with the measurement results for radial distances of 30 m to 66 m. Figure 3.3 provides a summary of the measurement results at site 6, together with the material damping loss curves for the near and far locations.

26

C. Weber et al.

50 m

37 m

34 m

59 m

A

B

43 m

Vibration Reduction - dB

10 0 -10 -20 -30 -40 -50 31.5 40

50

63

80 100 125 160 200 250 315

1/3 Octave Centre Frequency - Hz Fig. 3.3 Sample vibration attenuation results from site 6. Curve A shows predicted ground attenuation at 59 m distance for an average shear wave velocity (c = 600 m/s) and soil loss factor (η = 0.06). Curve B shows predicted ground attenuation at 37 m

3.5 Conclusion Vibration measurements were undertaken to investigate how vibration attenuates between tunnels and the ground surface in lithology comprising Hawkesbury Sandstone and/or Ashfield Shale. The results indicated little or no vibration attenuation at low frequencies, but significant attenuation at higher frequencies. Common prediction models typically account for geometric losses, material losses, or both. While the measurement results in this study (at frequencies greater than 80 Hz) are likely to include a combination of geometric and material losses, an empirical model which considers material damping losses alone was found to provide the best correlation with the measurement results (radial distances of 30 m to 66 m). An acceptable correlation was obtained at most test sites with an average shear wave velocity (c = 600 m/s) and soil loss factor (η = 0.06). Because the vibration attenuation is frequency dependent, prediction models which consider only the source-receiver distance are unsuitable for accurate predictions. The results indicate that geometric attenuation should not be used at low frequencies, as this may result in predicted levels lower than the measurement results.

3 Field Measurements of the Attenuation of Vibration Between. . .

27

References 1. International Organization for Standardization (ISO): ISO 14837-1:2005 - Mechanical vibration – Ground-borne noise and vibration arising from rail systems (2005) 2. Karantonis, P., Weber, C. and Puckeridge, H.: Ground-borne noise & vibration propagation measurements and prediction validations from an Australian Railway Tunnel Project. In: Proceedings of International Workshop on Rail Noise 13, Ghent, Belgium, (2019) 3. Bullen, R.: Propagation of vibration from rail tunnels: comparison results for two ground types, In: Proceedings of Acoustics 2004, Gold Coast, Australia (2004) 4. Kiwamu, T., Furuta, M.: Attenuation properties of ground vibration propagated from subway tunnels in soft ground. In: Proceedings of International Workshop on Rail Noise 12 (IWRN12), Terrigal (2016) 5. Meng, M. et al.: Reasons and laws of ground vibration amplification induced by vertical dynamic load, J. Central South Univ. Technol. (2014) 6. International Organization for Standardization (ISO): ISO 14837-32:2015 - Mechanical vibration – Ground-borne noise and vibration arising from rail systems - Measurement of dynamic properties of the ground (2015) 7. Davis, D.: A review of prediction methods for ground-borne noise due to construction activities. In: Proceedings of 20th International Congress on Acoustics, ICA (2010) 8. California Department of Transportation Division of Environmental Analysis: Transportation and Construction Vibration Guidance Manual (2013) 9. Burgemeister, K., Fisher, K., Franklin, K.: Measurement and prediction of construction vibration affecting sensitive laboratories. In: Proceedings of Acoustics 2011, Gold Coast, Australia (2011)

Chapter 4

Experimental Study on Rail Corrugation Development with 1/10 Scale Model Masayoshi Okita and Yoshiaki Terumichi

4.1 Introduction Rail corrugation is a pattern of periodic irregularities that forms continuously on the top of a rail as railways run over it repeatedly. Corrugation causes noise and vibration and then decreases ride comfort, traveling safety, and environmental friendliness [1]. Application of a friction modifier and rail correction are taken as measures against corrugation development, but these require a great deal of cost and labor [2]. Therefore, elucidation of the rail corrugation mechanism is required. Field surveys, numerical simulations, and model experiments are approaches for clarifying the wear phenomena. The problem with field surveys is that management of the measurement environment is difficult. Numerical simulation is less expensive compared to field surveys and experiments, but the wear development model used for analysis and the contact model between wheels and rails are still evolving, and model validation is still insufficient. Although scale model experiments can be expensive, they have the advantage of enabling management of the measurement conditions. Also, they can simulate the contact condition between wheels and rails in an actual railway and they easily reproduce actual phenomena. In this study, a scale model experiment was conducted to clarify the mechanism of corrugation development.

M. Okita () · Y. Terumichi Sophia University, Tokyo, 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/10.1007/978-3-030-48153-7_4

29

30

M. Okita and Y. Terumichi

4.2 Experimental Setup and Measurement 4.2.1 Bogie and Track This experiment was conducted using a 1/10 scale model of an actual railway. Figure 4.1 shows the bogie and track model. The bogie was a 1/10 scale model of a general subway car bogie in terms of gauge, wheel diameter, and wheel tread shape. The wheelsets and bogie frame were connected by links and springs, and the tread shape was an arc. The track consisted of straight and curved tracks and two turn tracks. The curvature radius of the curved track could be changed from the equivalent of R100 m to 300 m in the actual system. The sleepers were arranged at regular intervals, and rubber pads were interposed between the rail and the sleepers and between the sleepers and the track; therefore, the track structure simulated a solid-bed track with resilient sleepers. The rail was a 1/10 scale model of a 50-kg N rail.

4.2.2 Physical Parameters The experiment was conducted with R10 m, the steepest curve that can be simulated by this model. The physical parameters are shown in Table 4.1.

Fig. 4.1 Bogie and track 1/10 scale model

Table 4.1 Physical parameters for model

Parameter Radius of curvature Bogie mass Wheel base Velocity of bogie Total laps Gauge Sleeper spacing Cant

Units m kg mm km/h mm mm mm

Value 10 26.25 190 4.11 15,000 143.5 77 2.1

4 Experimental Study on Rail Corrugation Development with 1/10 Scale Model

31

4.2.3 Measurement Method In this experiment, the rail surface condition, rail surface shape, contact force, rail acceleration, wheel/rail shape, and contact position were measured. The rail surface condition was photographed using a digital camera every 250 laps. The rail surface shape was measured using a portable roughness tester. The rail acceleration was measured by attaching a three-axis accelerometer to the back of the rail. The acceleration data were subjected to fast Fourier transform (FFT) analysis to calculate the frequency distribution. The contact force was observed at a fixed point using a single-axis strain gauge. The wheel/rail shape and contact position were measured using a laser displacement sensor.

4.3 Results and Discussion In general, because contact between metals results in adhesion wear, Archard’s wear equation [3] was used to model wear in this experiment: W =

kσ δ H

(4.1)

where W is the wear amount in cubic meters, k is the wear coefficient, σ is the contact force [N], δ is the slip distance [m], and H is the Vickers hardness [N/m2 ]. We focused on the contact force and slip as physical factors that affect wear and promote corrugation development.

4.3.1 Rail Surface Condition and Shape The rail surface condition was checked visually and photographed every 250 laps. Corrugation was visually recognized at 6250 laps, but it was not possible to evaluate it photographically because it was so small that it could be seen only by adjusting the amount of light hitting the rail and the observation angle. As the bogie continued to run, the surface change became a little more visible, but this could not be seen in the pictures. The rail surface shape was measured every 1000 laps and 250 laps as needed. Figure 4.2 shows the initial shape of the rail and the shape at 6250 laps. The horizontal axis in the figure is the rail position and the vertical axis is relative depth where the initial position of the stylus for measuring surface roughness is 0. As shown in Fig. 4.2, the rail surface shape before bogie travel does not show periodic irregularities, but the initial roughness of a machine surface is remarkable. This roughness could also be confirmed at any position on the rail. It was confirmed that the roughness gradually decreased with repeated bogie travel. The change in

M. Okita and Y. Terumichi 4

4

3

3

2

2

Depth [m]

Depth [m]

32

1 0 -1

1 0 -1

-2

-2

-3

-3

-4

0

5

10

15

Position [mm]

20

-4

25

0

5

(a)

10

15

Position [mm]

20

25

(b)

Fig. 4.2 Rail surface shape after (a) 0 laps and (b) 6250 laps

the rail surface condition was not significant, and it was possible to observe wear to such an extent that the passage marks of the wheel were slightly visible after repeated runs. The measurement results at 6250 laps show that the rail surface contains periodic irregularities, and similar irregularities were observed to persist after 6250 laps. These periodic irregularities were not seen in the initial shape. The corrugation wavelength is shown by the vertical red dotted lines; this wavelength was almost constant at an average value of about 3.43 mm.

4.3.2 Rail Acceleration and Contact Force The rail acceleration was measured every 250 laps. The FFT analysis results for the vertical acceleration of the inner rail are shown in Fig. 4.3. It can be seen that various frequency bands from 0 to 1000 Hz exist in the range of 0.5 m/s2 at 0 laps. A band around 280 Hz is confirmed temporarily at 6000 laps and stably after 9500 laps. The frequency was calculated to be 332 Hz from the wavelength determined by the surface roughness and the bogie speed of 4.11 km/h. This is very close to 280 Hz and thus in good agreement with the accelerometer results. Also, when an eigenvalue analysis was performed in consideration of the contact rigidity between the wheel and rail [4], it was found that the frequency of the mode of the vertical motion of the wheelset corresponded closely to the experimental results. Therefore, the elastic contact between the wheel and rail was found to have a great influence on the formation of rail corrugation. In the measurement of the contact force, the amount of strain in the wheel load direction and lateral force direction for both the inner and outer rails were measured every 250 laps. The amount of strain in the wheel load and lateral force directions did not change significantly after repeated passes. Also, because the amount of strain corresponds to the amount of change caused by the wheel load and lateral force, the

1.5

1.0

0.5

0.0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Amplitude spectrum [m/s2]

Amplitude spectrum [m/s2]

4 Experimental Study on Rail Corrugation Development with 1/10 Scale Model

33

1.5

1.0

0.5

0.0 0

200

400

600

Frequency [Hz]

(a)

800

1000

1200

1400

1600

1800

2000

Frequency [Hz]

(b)

Fig. 4.3 Rail acceleration after (a) 0 laps and (b) 6000 laps

amount of strain and the magnitude of contact force are proportionate. Therefore, the contact force did not change, even after repeated passes, and the development of wear is not considered to affect the contact force.

4.3.3 Wheel/Rail Shape and Contact Position We designed this experiment to focus on the wear of the rails rather than the wheels. Therefore, the wheels were made of wear-resistant material. When the initial shape and shape after 15,000 laps by the wheels were measured, it was found that the shapes matched. In the rail shape measurement, the range considered to change with traveling was measured and the amount of wear was calculated from the change in the shape at that point. Figure 4.4 shows the amount of outer rail worn per lap. It can be seen that the wear increases until 6000 laps and remains constant from 6000 laps. It was also found that inner rail wear was very small compared to outer rail wear. The wheel contact position tended to converge to a constant point after 6000 laps. The change in the contact position for the front axle was within 100 μm on the inner and outer tracks, based on measurements performed each laps. The change in the contact position for the rear axle was within 300 μm, and the contact position did not converge to a constant point, compared with the front axle. The rail with corrugation was found to be the inner track. Therefore, to consider the relationship with corrugation, the rail contact position was measured only on the inner rail. The front axle was stable within about 100 μm after 6000 laps. Conversely, compared with the front axle, the rear axle had a large fluctuation in the contact position and contact at a specific point cannot be considered. Therefore, the amount of change in the contact position is smaller for the front axle than the rear axle, and contact at a specific point at the front axle can be considered.

34

M. Okita and Y. Terumichi

Fig. 4.4 Amount of wear to outer rail

4.4 Conclusion From the experimental results for rail acceleration, it was found that a specific frequency of the wavy surface of the rail is prominent in the vertical direction when corrugation occurs. According to an eigenvalue analysis in consideration of the contact rigidity between the wheel and the rail, it is considered that the frequency is mainly caused by the vertical vibration of the wheelset due to contact rigidity. Therefore, it can be said that the elastic contact between the wheel and the rail greatly affects corrugation development. Also, because the number of laps at which corrugation occurs is close to the number of laps at which the total wear amount and the contact position between the wheel and the rail have converged, the contact state becomes stable and the action of the contact force and the occurrence of slip in the same area are considered to contribute to rail corrugation development.

References 1. Grassie, S.L., Kalousek, J.: Rail corrugation: Characteristics, causes and treatments. Proc. Instit. Mech. Eng. Part F. 207, 57–68 (1993) 2. Lewis, R., Dwyer-Joyce, R.S., Lewis, S.R., Hardwick, C., Gallardo-Hernandez, E.A.: Tribology of the wheel-rail contact: The effect of third body materials. Int. J. Railway Technol. 1(1), 167 (2012) 3. Kato, K., Adachi, K.: Wear mechanisms, modern tribology handbook, Volume One, Chapter 7. CRC Press (2001) 4. Shaba, A.A., Zaazaa, K.K., Sugiyama, H.: Railroad vehicle dynamics: A computational approach, pp. 127–137. CRC Press, Boca Raton (2008)

Chapter 5

Measurement and Dynamic Mode Analysis of Flow-Induced Noise with Combined Proper Orthogonal Decomposition Osamu Terashima, Ayumu Inasawa, and Reon Nishikawa

5.1 Introduction Even though the measurement of wind noise in low-noise wind tunnels and its numerical predictions are extensively and commonly performed nowadays, identification of the noise source and understanding the mechanism of its generation is potentially time-consuming because it is difficult to correlate the velocity field and far-field acoustic sound information precisely. Therefore, simple techniques have been developed to correlate these variables easily based on proper orthogonal decomposition (POD) [1, 2] and linear stochastic estimation (LSE) [3]. In this study, modal analyses were performed first using POD to extract the information of the flow field. Subsequently, LSE or extended POD was performed on the obtained data, and the dominant mode of the far-field sound pressure was determined. Using these methods, the speed, efficiency of identification, and the understanding of the generation mechanism are expected to improve considerably. This is the first attempt to apply these flow-field analysis methods to analyze flow-induced sound based on each mode of the velocity field. Correspondingly, the results of the modal analyses elicited using the present method for the flow-induced sound emitted from a circular cylinder are discussed.

O. Terashima () · R. Nishikawa Toyama Prefectural University, Toyama, Japan e-mail: [email protected]; [email protected] A. Inasawa Tokyo Metropolitan 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/10.1007/978-3-030-48153-7_5

35

36

O. Terashima et al.

5.2 Experimental Method 5.2.1 Experimental Setup The experiments were performed in a circuit-type, low-noise wind tunnel. The turbulent intensity and the uniformity of the streamwise velocity were determined at the nozzle exit which had an area of 500 × 500 mm2 . The test section was in an anechoic chamber whose length, width, and height were 7.0 m, 3.9 m, and 3.0 m, respectively. The background noise level in the test section was less than 25 dB [A], where [A] represents the A-weighted sound pressure level. Figure 5.1 shows a perspective view of the test section and coordinate system. The coordinate system was defined as follows: the axial (streamwise) coordinate was x, the vertical (cross-streamwise) coordinate was y, the spanwise coordinate was z, and the origin o was set at the center of the nozzle exit, as shown in Fig. 5.1. A circular cylinder whose diameter d and length l were 10 and 500 mm, respectively, was installed so that its center was at the position of x = 100 mm. An 11-point simultaneous measurement of the streamwise velocity behind the cylinder was performed with a hot-wire probe that had 11 hot wires, as shown in Fig. 5.2.

y x

25

(b) z

Strut

Flow

2

U∞

1600

27

experiment, respectively. (a) (a)

Single hot-wire Plastic thin plate

1220

Fig. 5.1 (a) Schematic view of the test section and the coordinate system and (b) a hot-wire probe for the 11-point simultaneous measurement of the streamwise velocity Fig. 5.2 Profile of eigenvalue λ(n) at x/d = 14

5 Measurement and Dynamic Mode Analysis of Flow-Induced Noise. . .

37

The diameter of the hot wire was 5 μm and the distance between the hot wires was 2 mm. Hot wires were connected to the constant temperature circuit [4]. A microphone (Brüel & Kjær, Inc., Model 4138) connected to the signal conditioner (Brüel & Kjær, Inc., Model 2690) was set at x = 100, y = 0, and z = 1000 mm and was used to measure far-field flow-induced sound emitted from the cylinder simultaneously. Signals of the hot wire and microphone were saved on a PC using an analog-digital converter. The sampling number and frequency of the hot wires and microphone in this experiment were 1,048,576 and 20 kHz, respectively. The mean streamwise velocity at the nozzle exit U∞ and Reynolds number based on the diameter of the cylinder were 22 m/s and 14,200 in this experiment, respectively.

5.2.2 Combined Proper Orthogonal Decomposition In the present method, modal analysis of the streamwise velocity fluctuations (i.e., the information of the flow field) was performed first. Then, LSE or extended POD was performed using the POD results to determine the dominant mode of the farfield sound pressure that was strongly related to that of the streamwise velocity fluctuation. The procedure used to determine the dominant mode of the streamwise velocity fluctuation with LSE [5] is outlined as follows. First, the eigenfunction of the streamwise velocity fluctuation u for the cross-streamwise direction φ (y) was calculated, as shown in Eq. (5.1). The symbol λ is the eigenvalue, and S is the domain to be calculated, and y’ is the other position for obtaining the two-point correlation. 

    Ruu y, y  φ y  dy  = λφ(y)

(5.1)

S

Herein, Ruu (y, y’) is the two-point correlation tensor defined in Eq. (5.2).    

Ruu y, y  = u(y)u y 

(5.2)

Using the eigenfunctions obtained thus far, each mode can be expressed using Eq. (5.3). u(n) (y, t) = a (n) (t)φ(n) (y, t)

(5.3)

In Eq. (5.3), n is mode number and a (n) (t) is the modal coefficient expressed in accordance to Eq. (5.4).  a (n) (t) =

u (y, t) φ(n) (y, t) dy S

(5.4)

38

O. Terashima et al.

Subsequently, the far-field (domain ) sound pressure P (Y) correlated to each mode of p with LSE (Y ∈ ) was predicted. The far-field sound pressure fluctuation for each mode of u, P (n) (Y, t), is expressed by Eq. (5.5). P (n) (Y, t) = A (Y, y1 ) u(n) (y1 , t) +A (Y, y2 ) u(n) (y2 , t) + . . . +A (Y, yN ) u(n) (yN , t)

(5.5)

where, y1 , y2 , . . . , yN are the measurement position of the streamwise velocity fluctuation and N is the number of measurements. Each modal coefficient A (Y, yi ) is calculated from Eq. (5.6). 

   u(n) (yi , t) u(n) yj , t A (Y, yi ) = u(n) (yi , t) P (Y, t)

(5.6)

Hence, modal coefficients are obtained by solving N-simultaneous equations. Similarly, predict P (Y) with extended POD from following equations [6, 7]. 

a (n) (t) • a (n) (t) = λ(n)

φ

(n)

(n) a (t) • u(y) (y) = λ(n)

ψu

(5.7)

(n)

a (n) (t) • P (Y ) (Y ) = λ(n)

(5.8)



P (n) (Y, t) = a (n) (t)ψu (n) (Y )

(5.9)

(5.10)

5.3 Results and Discussions Figure 5.2 shows the energy contribution ratio for each fluctuating mode of the streamwise velocity at x/d = 14 obtained by POD. The abscissa and ordinate represent the mode number (1–11) and energy contribution ratio, respectively. Figure 5.2 shows that the 1st mode had nearly half fluctuating energy (48%) and the 2nd mode was half that of the 1st mode (24%). Figure 5.3 shows the contour map of the reconstructed instantaneous streamwise velocity fluctuation u/U∞ measured at x/d = 14. Figure 5.3a and b show the 1st and 2nd modes, respectively. The abscissa and ordinate represent the time t and

5 Measurement and Dynamic Mode Analysis of Flow-Induced Noise. . .

39

Fig. 5.3 Contour map of the reconstructed instantaneous streamwise velocity fluctuation u/U∞ measured at x/d = 14. (a) and (b) show 1st and 2nd mode, respectively

Fig. 5.4 Power spectra of the measured and estimated far-field sound pressure fluctuation at x/d = 14. (a) and (b) show the comparison of the 1st and 2nd mode by LSE and that of each method (LSE and e-POD), respectively

cross-streamwise position y normalized by d, respectively. Figure 5.3a reveals that positive and negative velocity fluctuations existed at symmetrical positions with respect to the cylinder center. Figure 5.3b shows that strong velocity fluctuations existed downstream of the cylinder center. These coherent velocity fluctuations were caused by the Kármán vortex street. Figure 5.4 shows the power spectra of the measured and estimated far-field sound pressure fluctuation at x/d = 14. Figure 5.4a and b shows the comparison of the 1st and 2nd mode by LSE and that of each method (LSE and e-POD), respectively. The red and black lines show the power spectra of the far-field sound pressure directly obtained from the measured data with and without the hot-wire probe, respectively. The effect of installing the probe in the wake of the circular cylinder on the far-field sound pressure was observed but this effect occurred in higher-frequency regions. Figure 5.4a reveals that the estimated far-field sound pressure of the 1st and 2nd modes had a peak at 460 Hz. This peak was also confirmed in the power spectra of the far-field sound pressure directly obtained from the measured data shown as a red

40

O. Terashima et al.

line. Based on the general Strouhal number of the circular cylinder (approximately 0.20), this peak was caused by the Kármán vortex street. In addition, the peak value of the 1st mode at 460 Hz was nearly the same value as that directly obtained from the measured data. Therefore, a dominant sound pressure fluctuation of 460 Hz was nearly the result of the 1st mode streamwise velocity fluctuation. The 2nd mode had a peak at 460 Hz, but its contribution to the sound generation was small. Similarly, the 2nd mode also had a peak at 920 Hz, which was due to the probe installation, but its contribution was also small. Therefore, the 2nd mode of the streamwise velocity fluctuation had 24% of the fluctuating energy, but it did not strongly contribute to the sound generation. Figure 5.4b reveals that some differences exist between estimated results by the LSE and extended POD methods. Each method could estimate the far-field sound pressure derived from the 1st and 2nd mode streamwise velocity fluctuation. However, since e-POD is simpler than LSE as shown in Sect. 5.2, authors recommend to use e-POD.

5.4 Summary POD analysis was examined with LSE and e-POD analysis to study a combined analysis method for comprehensively evaluating fluid information and generated far-field sound information. Experimental results showed that there were minor differences between the two analyzed sets of results obtained by the two techniques. Further, the results showed that the radiated peak sound pressure was strongly related to the first mode of the velocity field in the wake near the cylinder with a pair of fluid lumps having positive and negative streamwise velocity fluctuations on opposite sides of the centerline of the wake. In a future study, this method will be applied to various flow fields in which flow-induced sound is an issue, and the relationship between each mode present in the flow field and the generated sound pressure fluctuation will be investigated. These analysis methods for the flow field have validated its usefulness in our previous research [7] on the turbulent jet whose flow field has irregularity. Therefore, it can be expected to analyze the brad-banded flow-induced sound. Acknowledgment This work was partially supported by JSPS Kakenhi Grant Number 18 K13691.

References 1. Lumley, J.: The structure of inhomogeneous turbulent flows, atmospheric turbulence and radio wave propulsion, pp. 168–178. Nauka, Moscow (1967) 2. Moin, P., Moser, R.D.: Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509 (1989) 3. Adrian, R. J.: On the role of conditional averages in turbulence theory, turbulence in liquids. In: Zakin, J. L. and Patterson, G. K Ed. Proceeding of the fourth Biennial Symposium on Turbulence in Liquids September 1975, pp. 323–332 (1977)

5 Measurement and Dynamic Mode Analysis of Flow-Induced Noise. . .

41

4. Terashima, O., Sakai, Y., Onishi, K., Nagata, K., Ito, Y., Miura, K.: Improvement of the constant temperature anemometer and measurement of energy spectra in a turbulent jet. Flow Meas. Instrum. 35, 92–98 (2014) 5. Bonnet, J.P., Cole, D.R., Delville, J., Glauser, M.N., Ukeiley, L.S.: Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structure. Exp. Fluids. 17, 307–314 (1994) 6. Maurel, S., Borée, D.J., Lumley, J.L.: Extended proper orthogonal decomposition: Application to jet/vortex interaction. Flow Turbulence Combustion. 67, 125–136 (2001) 7. Terashima, O., Sakai, Y., Goto, Y., Onishi, K., Nagata, K., Ito, Y.: On the turbulent energy transport related to the coherent structures in a planar jet. Exp. Thermal Fluid Sci. 68, 697–710 (2015)

Part II

Experimental Modal Testing and Analysis

Chapter 6

Accuracy Improvement to the Identified Modal Parameters of Systems with General Viscous Damping Ningsheng Feng, Eric Hahn, and Minli Yu

6.1 Introduction Correct modelling of the foundation of a rotor bearing foundation system (RBFS) is still problematic [1]. One approach for such modelling uses motion measurements of the rotor and foundation at selected monitoring points to identify relevant modal parameters for an equivalent foundation (one which when substituted for the actual foundation reproduces the vibration behaviour of the RBFS over the operating speed range of interest). If successful, such an identification technique would be applicable to existing turbomachinery. In earlier work, an approach was developed which identified the relevant modal parameters for an equivalent foundation of a simple RBFS [2]. However, that work assumed a diagonalisable foundation damping matrix. Here it is assumed that the damping matrix is symmetric; a generalisation that greatly increases the difficulty of identifying the relevant modal parameters. Thus, attention has been directed to identifying a simple two degrees of freedom (DOF) lumped parameter system investigated in refs [3, 4]. Even for such a simple system, fully satisfactory identification could not to be achieved when measurement data was only accurate to two significant digits. Whereas theory requires that the system eigenvalues and eigenvectors be complex conjugates, this did not turn out to

N. Feng () Shandong University, School of Mechanical Engineering, Ji’nan, Shandong, China E. Hahn The University of New South Wales, School of Mech. and Man. Eng., Sydney, Australia e-mail: [email protected] M. Yu Department of Applied Mechanics and Engineering., Guangzhou, Sun Yat-sen University, Guangdong, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_6

45

46

N. Feng et al.

be the case [4]. This paper investigates the influence of measurement data accuracy on this anomaly using two basically different identification approaches. The relevant theory is summarized.

6.2 Notation A 2n × 2n modified system modal matrix with elements ajk ; AT = −1 . C,K,M n × n symmetric system damping, stiffness and mass matrices. f, F 2n × 1 vector of excitation forces, 2n × 1 vector of their amplitudes. g n × 1 vector of excitation forces. K*,M* 2n × 2n matrices defined by eqns (6.4) and (6.3). k 2n × 2n diagonal matrix defined by eqn (6.8). k mode number; k = 1, 2, ...2n. m 2n × 2n diagonal matrix defined by eqn (6.7) with diagonal elements mk . n number of system degrees of freedom. q,Q n × 1 vector of displacements, n × 1 vector of their amplitudes. x, X 2n × 1 vector of velocities and displacements as defined by eqn (6.5), 2n × 1 vector of their amplitudes. λ 2n × 2n diagonal matrix of system eigenvalues with elements λk . Φ, 2n × 1 eigenvector, 2n × 2n system modal matrix with elements Φ jk . Ω excitation frequency, rad/s.

6.3 Theory For a general n DOF lumped parameter system subjected to periodic force excitation with frequency Ω, the equations of motion may be written as: Mq¨ + Cq˙ + Kq = g

(6.1)

Eqn (6.1) may be written in state space form as: M∗ x˙ + K∗ x = f

(6.2)

where:

M 0 M = 0 −K ∗

CK K = K 0 ∗

 (6.3)

 (6.4)

6 Accuracy Improvement to the Identified Modal Parameters of Systems. . .

47

x = [q˙ q]T

(6.5)

 T f= g 0

(6.6)

Since M* and K* are symmetric, the equation set obtained when f = 0 constitutes an eigenvalue problem [3]. Assuming distinct eigenvalues , in the absence of rigid body modes, the eigenvalues and corresponding eigenvectors will occur as complex conjugate pairs so that λ2i−1 = λ2i , i = 1 . . . n and 2i−1 = 2i , i = 1 . . . n. Also, the eigenvectors are orthogonal with respect to the M* and K* matrices, so that [3]: Φ T M∗ Φ = m

(6.7)

Φ T K∗ Φ = k

(6.8)

m−1 k = −λ

(6.9)

Assuming periodic response with fundamental frequency Ω, one can write eqn (6.2) as: i M∗ X + K∗ X = F

(6.10)

Premultiplication of eqn (6.10) by T and setting AT = −1 gives: [i I − λ]AT X − m−1 Φ T F = 0

(6.11)

Eqn (6.11) yields 2n identification equations (k = 1 . . . 2n): (i − λk )

2n  j =1

aj k Xj −

2n 

Φj k Fj /mk = 0

(6.12)

j =1

For any k, one can write eqn (6.12) using the measurements of Xj and Fj corresponding to a particular excitation frequency Ω. Provided measurements are available for at least as many frequencies as unknowns in eqn (6.12), one can solve for the unknowns λk , ajk and Φ jk /mk . The procedure adopted is to first solve for λk ; then substitute this eigenvalue into eqn (6.12) and solve for the remaining unknowns. On repeating this for the other k’s, one can obtain the matrices λ, A,  and m. These parameters suffice for an equivalent foundation, enabling one to compute the original mass, damping and stiffness matrices. Thus, accurate solutions

48

N. Feng et al.

for the eigenvalues predicates the accuracy of the solutions for all the other modal parameters. Two approaches, termed BISECTION-MIN and BISECTION-SI, have been developed in house for solving for the eigenvalues. Both approaches used bisection to find the zeros of a function of a complex variable; the former uses a minimisation approach which requires determinant evaluations and differentiations, while the latter uses a simple iteration formulation which requires the solution of linear simultaneous equations [4].

6.4 Numerical Experiments and Discussion All numerical experiments were for the simple 2 DOF system already considered in ref. [4]. The schematic of the system is shown in Fig. 6.1. Using Laplace transforms, for the given excitation force, the system response amplitudes for the displacements q1 and q2 were calculated over the range of frequencies from 0.05 to 3.5 rad/s in steps of 0.15 rad/s, giving 24 data sets in total. These computed amplitudes constituted the ‘measurements’. These measurements were available to 15-digit accuracy. As such, they were used to validate the softwares. Identification approach evaluations used ‘measurements’ rounded off to two-digit accuracy. For the system in Fig. 6.1, measurements for at least five frequencies (i.e. five data sets) are required to solve for the unknowns in eqn (6.12). Also, with five data sets one can differentiate analytically [4] and one can avoid the need for linear regression, hence eliminating computation errors from these sources. The data sets selected were those corresponding to frequencies of 0.50 rad/s, 0.65 rad/s, 1.55 rad/s, 2.45 rad/s and 2.75 rad/s. For two-digit input data accuracy, using these five data sets gave better identification than using all 24 data sets [4]. Eigenvalue identifications were then carried out for 15-, 5-, 4-, 3- and 2digit input data accuracies using the minimization approach with both analytical and numerical differentiations (BISECTION-MIN ANAL and BISECTION-MIN

f1

f2

q1

q2 c

c

m

k

2m

4k

Fig. 6.1 2 DOF system [3]. (m=1 kg, c=0.2 N.s/m, k =1N/m, f1 = sin Ωt N, f2 =0)

6 Accuracy Improvement to the Identified Modal Parameters of Systems. . .

49

Table 6.1 Effect of input data accuracy on eigenvalues λ1 and λ2 (in rad/s). Complex conjugate eigenvalues are italicised for easier interpretation. Actual values are -.222596 ± 2.57826i rad/s. DATA ACCURACY λ1 (15 digit) λ2 (15 digit) λ1 (5 digit) λ2 (5 digit) λ1 (4 digit) λ2 (4 digit) λ1 (3 digit) λ2 (3 digit) λ1 (2 digit) λ2 (2 digit)

BISECTION-MIN ANAL -.222596+2.57826i -.222596-2.57826i -.222596+2.57826i -.221159-2.57885i -.222585+2.57821i -.241489-2.61216i -.222663+2.57851i Cannot find root -.224822+2.57995i Cannot find root

BISECTION-MIN NUML -.222596+2.57826i -.222500-2.57813i -.222596+2.57826i -.221094-2.57875i -.222585+2.57821i -.241422-2.61211i -.222663+2.57851i Cannot find root -.224822+2.57995i Cannot find root

BISECTION-SI -.222596+2.57826i -.222596-2.57826i -.222596+2.57826i -.221159-2.57885i -.222585+2.57821i -.241490-2.61216i -.222663+2.57851i Cannot find root -.224822+2.57995i Cannot find root

NUML) and the simple iteration approach (BISECTION-SI). The actual eigenvalues were obtained using the eigenvalue solver in Matlab [5]. The results are given to six-digit accuracy in Table 6.1 for the first eigenvalue pair (λ1, λ2 ) and in Table 6.2 for the second eigenvalue pair (λ3, λ4 ). The eigenvalues in the lower half plane are italicised for easier interpretation. For this particular system, regardless of input data accuracy or identification approach, the odd numbered eigenvalues are identified more accurately (often significantly so) than their complex conjugates; and identification accuracy decreases, as expected, with decreasing input data accuracy. The errors in the real parts of the odd-numbered eigenvalues with two-digit data accuracy are around 1.5%. Though small, they lead to errors at resonances in subsequent frequency response predictions [4]. Further improvement in accuracy can probably only be achieved by using more data sets, though less accuracy was achieved when using all 24 data sets [4]. The identifications obtained with BISECTION-MIN ANAL and BISECTION-SI are identical to at least six significant digits regardless of input data accuracy; those obtained with BISECTION-MIN NUM are occasionally not quite as accurate as those obtained with the other two approaches for the even-numbered eigenvalues. The BISECTION-SI approach involves solution of simultaneous equations. The condition numbers associated with these equations are always significantly larger (usually by an order of magnitude or more) for the even-numbered eigenvalue solutions, suggesting that for these solutions, the equations are more ill conditioned. Hence it appears reasonable to set the even-numbered eigenvalues as being simply the complex conjugates of the identified odd-numbered ones.

6.5 Conclusions 1. For this system, eigenvalues in the upper half plane can be identified more accurately than their complex conjugates.

50

N. Feng et al.

Table 6.2 Effect of input data accuracy on eigenvalues λ1 and λ2 (in rad/s). Complex conjugate eigenvalues are italicised for easier interpretation. Actual values are -.0274040 ± .545795i rad/s. DATA ACCURACY λ3 (15 digit) λ4 (15 digit) λ3 (5 digit) λ4 (5 digit) λ3 (4 digit) λ4 (4 digit) λ3 (3 digit) λ4 (3 digit) λ3 (2 digit) λ4 (2 digit)

BISECTION-MIN ANAL -.0274040+.545795i -.0274040-.545795i -.0274037+.545793i -.0273577-.546701i -.0273974+.545803i -.0251692-.544651i -.0275424+.545913i Cannot find root -.0270275+.546698i Cannot find root

BISECTION-MIN NUML -.0274040+.545795i -.0274032-.545796i -.0274037+.545793i -.0273511-.546702i -.0273974+.545803i -.0251666-.544651i -.0275424+.545913i Cannot find root -.0270275+.546698i Cannot find root

BISECTION-SI -.0274040+.545795i -.0274040-.545795i -.0274037+.545793i -.0273577-.546701i -.0273974+.545803i -.0251691-.544650i -.0275424+.545913i Cannot find root -.0270275+.546698i Cannot find root

2. Identifications obtained with BISECTION-MIN ANAL and BISECTION-SI are identical to at least six significant digits regardless of input data accuracy. 3. BISECTION-SI and BISECTION-MIN NUML approaches gave equally accurate identifications for upper half plane eigenvalues (i.e. the odd-numbered eigenvalues). 4. It appears reasonable to set the even-numbered eigenvalues as being simply the complex conjugates of the identified odd-numbered ones.

References 1. Lees, A., Sinha, J., Friswell, M.: Model-based identification of rotating machines. Mech. Syst. Signal Process. 23(6), 1884–1893 (2009) 2. Yu, M., Feng, N. and Hahn, E.: On the identification of damped foundations in rotating machinery using modal parameters, ISMA2016 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 19-21 September, (2016). 3. Meirovitch, L.: Computational methods in structural dynamics. Sijthoff & Noordhoff, Rockville (1980) 4. Feng, N., Hahn, E., Yu, M.: Identification of the modal parameters of systems with general damping. In: ISMA2018 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 17–19 September, (2018) 5. MATLAB, Version 5.1,The Maths Works Inc. , Natick, (1997).

Chapter 7

Classification of Characteristic Modes for Vibration Reduction Itsuki Nakashima

, Takumi Inoue, and Ren Kadowaki

7.1 Introduction Detailed FEM models contribute to accurate vibration predictions and provide large numbers of vibration modes from vibration analysis. Consequently, when it comes to large-scale models such as automobile bodies, it is time consuming to evaluate and reduce each vibration. In order to improve the efficiency of vibration analysis, there are several methods proposed to reduce calculation time and process of vibration analysis [1–3]. However, these methods require advanced experience and complicated calculations. Moreover, they do not present an effective approach to reduce vibrations of coupled mode which consists of multiple types of deformations. The purpose of this study is to develop a mode classification method for efficient vibration reduction and obtain rational strategy for reducing vibrations of coupled modes. Macroscopically, characteristic modes of automobile bodies, which consist of frame structure and attached panels, are regarded as combinations of global and local deformations. Depending on these deformation scales, each characteristic mode is classified. Furthermore, based on the classification, we propose a rational approach for vibration reduction of each mode group.

I. Nakashima () Graduate School of Integrated Frontier Sciences, Kyushu University, Fukuoka, Japan e-mail: [email protected] T. Inoue · R. Kadowaki Kyushu University, Fukuoka, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_7

51

52

I. Nakashima et al.

7.2 Mode Classification Method 7.2.1 Simplified FEM Model for Demonstration Figure 7.1 shows the demonstrative FEM model of automotive underbody. This model consists of 3095 nodes with frame and attached panels. In addition, this model has 18 eigenvalues in 0 ~ 200 Hz frequency range.

7.2.2 Approach for Mode Classification Several characteristic modes of the underbody FEM model are shown in Fig. 7.2. Seen in Fig. 7.2, mode 1 has only large deformation of the whole structure. We define this structural deformation as global deformation. Mode 2 and 3 also have this global deformation, with combination of deformations of the panels. This partial deformation is defined as local deformation. Mode 4 and 5 have only local deformations of the panels whereas the whole structures do not deform largely. In this way, characteristic modes can be classified by focusing on its global and local deformations. In order to classify each characteristic mode according to these deformation scales, we extracted global and local deformations from each original mode by means of our proposed calculation method [4] based on the idea of Guyan’s static reduction. The concept of the extraction is shown in Fig. 7.3. The red dashed line indicates the boundary area between the frame and panels. First, the original displacement of the boundary area is extracted. Then we approximated the static displacements of other components caused by the displacement of the boundary area. This is how the global deformation is extracted from original displacement. The local deformation is the remained partial displacement of components. The specific calculation process is explained as follows. Equation (7.1) shows the equation of motion of the model divided in boundary area and other components.

Fig. 7.1 A simplified FEM model of automobile underbody and its components

Fig. 7.2 Characteristic modes of underbody FEM model

7 Classification of Characteristic Modes for Vibration Reduction

53

Fig. 7.3 Concept of extracting global and local deformation from original characteristic mode

In the equation, M indicates mass matrix, C for damping matrix, K for stiffness matrix, F for force vector and x for displacement vector. The subscript b indicates the boundary area, and c for other components.

M b M bc M cb M c



      x¨ b C b C bc x˙ b K b K bc xb Fb + + = x¨ c x˙ c C cb C c K cb K c xc Fc (7.1)

We approximate the static displacement of components which follows the displacement of the boundary area by ignoring the inertial force, damping force and external force in the equation. Equation (7.2) shows the relation between the original displacement of boundary area xb and static displacement of other components x’c .

K b K bc K cb K c



xb x c

 =

 0 0

(7.2)

From Equation (7.2), the static displacement of the components caused by the original displacement of the boundary area is derived as shown in Equation (7.3). x ’ c = −K c −1 K cb x b = T xb

(7.3)

The T matrix indicates the transformation matrix. By applying this equation to rth modal displacement Xr , we approximate the global deformation of each mode Xr g as shown in Equation (7.4). The global deformation consists of original displacement of boundary area and static displacement of other components derived by the transformation matrix.

Xbr Xr = Xcr



,

g Xr

X br = X cr



I = T

 Xbr

(7.4)

On the other hand, the local deformation is derived by subtracting the global deformation Xr g from the original mode displacement Xr .

54

I. Nakashima et al.

Table 7.1 Result of mode classification Mode groups Characteristic mode

Global modes 1

Local modes 4 ~ 6, 8, 9, 11 ~ 18

Coupled modes 2, 3, 7, 10

7.2.3 Mode Classification By focusing on the extracted global and local deformations, we define three types of mode groups. One is the global modes, such as mode 1 which mainly consists of large global deformation without local deformations of components. Another is the local modes which mainly consists of local deformation with very little global deformation such as mode 4 and 5. The other is coupled modes which consists of both apparent global and local deformations such as mode 2 and 3. This point of view enables to classify each mode efficiently. In this way, we classified 18 characteristic modes up to 200 Hz into global modes, local modes and coupled modes. The result of classification is tabulated in Table 7.1.

7.3 Vibration Reduction Method 7.3.1 Approaches for Vibration Reduction In order to reduce vibration of each characteristic mode, it is effective to reinforce each deforming component. The vibrations of global modes are reduced by reinforcing the deforming frame structure. Likewise, the vibrations of local modes are reduced by reinforcing the vibrating panel. The vibrations of coupled modes are reduced by reinforcing both deforming frame and vibrating panel as well; however, the balance between the two reinforcement is essential for vibration reduction. This is because incorrect reinforcement contributes to vibration increase. We demonstrate this problem with characteristic mode 2 and 3 as examples of coupled modes. As seen in Fig. 7.2, these two modes have similar global and local deformations; thus it is expected that the same reinforce process is effective. In order to reduce the vibrations of these modes, the coloured elements indicated in Fig. 7.4 are selected as reinforce targets. The red-coloured elements in Fig. 7.4a is where the stress is concentrated on the frame. These elements are the reinforce target for global deformation. On the other hand, the black-coloured elements on the panel in Fig. 7.4b are the reinforce targets for local deformation. This layout of reinforce target elements simulates an addition of ribs on the vibrating panel. Moreover, the excitation point on the frame edge is also indicated in Fig. 7.4a. The reinforcement is conducted by increasing the stiffness of the target elements. Specifically, the Young’s modulus is increased first and then the shear modulus is also increased to keep the original Poisson ratio. For examples, two reinforcement conditions are conducted as shown in Table 7.2. The changes of response power

7 Classification of Characteristic Modes for Vibration Reduction

55

Fig. 7.4 Reinforce target elements for coupled modes 2 and 3 (a) Reinforce target elements for global deformation (b) Reinforce target elements for local deformation Table 7.2 Reinforcement condition Reinforcement condition A B

Multiplying factor of Young’s modulus of frame 1.8 1.4

Multiplying factor of Young’s modulus of panel 1.4 1.8

Fig. 7.5 Frequency response changes in each reinforcement settings

spectra caused by each reinforcement condition are shown in Fig. 7.5. Harmonic oscillations and structural damping with same values are applied for each response calculation. The response power spectra are calculated as square sum of translational velocity of all nodes. As shown in Fig. 7.5, the reinforcement condition A decreases both peak value of mode 2 and 3. On the other hand, reinforcement condition B increases the peak value of mode 2. Therefore, there exist an incorrect reinforcement that contribute to vibration increase.

56

I. Nakashima et al.

Fig. 7.6 Peak value changes of mode 2 and 3 (a) Peak value changes of mode 2 (b) Peak value changes of mode 3 (c) Product of peak value changes of mode 2 and 3

7.3.2 Rational Strategy for Vibration Reduction of Coupled Modes It is essential for coupled modes to reinforce correctly for vibration reduction. In order to give correct reinforcement conditions, we have traced the change of the peak value in each multiplying factor of frame and panel reinforcement. The peak value in each multiplying factor of reinforcement is calculated, and the change of the peak values before and after the reinforcement is plotted as contour figure in Fig. 7.6. The grey-coloured area in the figure indicates that the vibration peak is decreased by the reinforcement whereas uncoloured white area indicates the vibration has increased. Figure 7.6a shows the peak value change in mode 2 and (b) shows the change in mode 3. The reinforcement conditions A and B which are mentioned in Table 7.2 are also plotted as a dot and a triangle in each figure. As it is shown in Fig. 7.5, the reinforcement condition B reduced the vibration of mode 3 but increased the vibration of mode 2. In order to derive correct reinforcement conditions that reduces both coupled modes 2 and 3 at a glance, the product of values in Fig. 7.6a and b are plotted in Fig 7.6c. The coloured area in Fig. 7.6c indicates that the vibrations of both mode 2 and 3 are reduced. This shows that the vibrations of coupled modes are reduced only when they are reinforced in limited correct conditions. As a result, the strategy for reducing vibrations of coupled mode is indicated as coloured area in Fig. 7.6c.

7.4 Conclusion In this paper, we have proposed a mode classification method based on the different types of deformations. By means of this method, characteristic modes of automobile body model are classified into three groups. We have also provided effective approach for vibration reduction of each mode group. Moreover, we have

7 Classification of Characteristic Modes for Vibration Reduction

57

demonstrated that the coupled modes require careful reinforcement for vibration reduction, because effective reinforce condition is limited and incorrect reinforcements contribute to vibration increase. Therefore, a reasonable strategy for vibration reduction of coupled modes is suggested.

References 1. Koizumi, T., Tsujiuchi, N., Nakahara, S., Nakamura, Y., Oshima, H.: The proposal of mode classification method for efficient vibration analysis. Trans. JSME (in Japanese). 75(754), 1543– 1549 (2009) 2. Mochizuki, T., Hagiwara, I.: A comparison between modal differential sub-structure method and conventional component modal synthesis methods. J. Syst. Des. Dyn. 5(2), 320–331 (2011) 3. Nagamatsu, A., Okuma, M.: Component mode synthesis (in Japanese), Baifukan, Tokyo (1991) 4. Tanaka, S., Kawano, T., Inoue, T., Kadowaki, R.: Selection method of representative modes to improve vibration analysis efficiency of automotive body by removing local vibration (in Japanese). In: Dynamics & design conference 2016, No.16–15, p. 363. JSME, Tokyo (2016)

Chapter 8

Vision-Based Modal Testing of Hyper-Nyquist Frequency Range Using Time-Phase Transformation Donghyun Kim and Youngjin Park

8.1 Introduction Mechanical structures such as automobiles, building structures, ships, etc. are exposed to vibration under operating conditions. Especially under resonance, small excitations cause large deformation of structures, which is directly related to human safety. Therefore, determining the dynamic characteristics and carrying out a safety diagnosis of a structure are essential and can be accomplished with a modal analysis. The modal testing system can be utilized in a variety of applications with diverse methods depending on the sensor that measures vibration. Recently, there have been many studies in the field of noncontact measurement techniques. Among the proposed techniques, vibration measurement using a camera has a significant advantage that the displacement response of the entire surface can be measured at one time, in contrast with the contact-type sensor. However, to perform a practical modal testing using the vision information of a camera, it is necessary to extend the measurable frequency band, which is limited to the existing low-frequency band. One key problem is that the band of the measurable vibration frequency is limited to low frequency in the modal testing using the image information of the camera. Low-cost cameras have a limited frame rate of about 100 to 150 Hz. According to Nyquist theory, high-frequency vibration above 50–75 Hz causes data loss due to aliasing. A mode with a high natural frequency, such as a car bonnet, has limitations that cannot be measured. Therefore, it is necessary to study methods of measuring the high-frequency vibration shape beyond the Nyquist frequency under a limited

D. Kim () · Y. Park Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea 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/10.1007/978-3-030-48153-7_8

59

60

D. Kim and Y. Park

frame rate. To overcome the aliasing problem caused by the limited frame rate, we propose a time-phase transformation method based on the phase of the sine excitation signal with a single frequency.

8.2 Theory of Hyper-Nyquist Frequency Range Measurement by Dynamic Photogrammetry Using Time to Phase Transformation 8.2.1 Excitation Signal for Modal Testing The importance of proper excitation method selection in experimental modal testing has already been discussed in several studies [1, 2]. In earlier works, only the modal testing using a signal with a single harmonic excitation (pure harmonic excitation) was possible. In recent years, the most common method to scan the entire frequency range is to use continuous sweeping of the frequency. If the input frequency changes slowly enough, the quasi-steady state condition can be met. There are various excitation methods such as random excitation using a white noise signal, transient excitation using an impact hammer, and periodic chirp using a fast sweep sine signal. In this study, we used a method of changing the excitation frequency stepwise among various excitation methods. For single frequency excitation signals, the steady-state response of a linear system is characterized by the same single frequency component characteristics. Using this feature, we propose a method of measuring vibration in the hyper-Nyquist frequency range.

8.2.2 Time to Phase Transformation Harmonic inputs have the advantage of being able to meet steady-state conditions and measuring the response of the system at a single frequency of interest. That is, the response of a linear system can be measured according to a specific input frequency that is known precisely. When the number of sampled data for one period of the input signal is N, the N points sampled data of the system response signals contain information of one cycle of the response signal. The Hilbert transform is used for the time-phase transformation, and the transformation process can be easily understood from Fig. 8.1. Graph (a) shows the excitation signal (x(t)) with exactly known frequency. Graph (b) shows that the analytic signal (xA (t)) was applied to the Hilbert transform. Graph (c) shows the phase signal (ψ(t)) of the original excitation signal. The phase signal of the excitation signal from 0 to 2 pi can be easily obtained by applying the Hilbert transform which converts the excitation signal of a specific frequency into a signal having a real part and an imaginary part. Since the

8 Vision-Based Modal Testing of Hyper-Nyquist Frequency Range Using. . .

61

Fig. 8.1 The time to phase transformation process of the excitation signal of a specific frequency: (a) Excitation signal with exactly known frequency (x(t)) (b) Analytic signal (xA (t) ≡ x(t) + j x(t) ˆ → Aej ψ(t) ) (c) Phase signal (ψ(t))

Fig. 8.2 Comparing the time-phase transformation-based measurement to the conventional measurement: (a) Original signal and time-based capture (general method with 8 frame rates) (b) Phase-based capture (suggested method with 8 frame rates)

steady-state response of the system changes the phase with the same period as the excitation signal, we can use this phase signal to prevent the aliasing phenomenon by creating a trigger signal. The set of images collected in synchronization with the generated trigger signal is processed according to the phases of known frequencies. Figure 8.2 presents an example of measuring the response of a system caused by a 10 Hz excitation signal using only 8 frame rates (sampling rate). Graph (a) shows the measurement results on the general time axis, and the aliasing phenomenon is conceived. Graph (b) shows how to measure and trigger on the proposed phase axis. In this proposed method, the one-period response can be accurately measured with the desired phase resolution without aliasing.

62

D. Kim and Y. Park

8.3 Experiment An experiment was conducted to confirm the feasibility of vibration measurement using the proposed time-phase transformation method.

8.3.1 Setup The measurement object is a cantilever aluminum beam having a length of 500 mm, a height of 40 mm, a thickness of 1 mm, E = 70Gpa, and density of 2.7 g/cm3 . Camera settings were 60 μm/pixel, 75 frames per rate, and shutter speed of 0.01 ms. A vibration is realized with a single excitation point at 30 mm from the fixed end. The state of vibration is 80μm and 80 Hz, which is higher than the Nyquist frequency. The camera is an inexpensive daily use camera, a Flea® 3 FL3-U313Y3M-C, which can take images with a maximum 100fps. The resolution was 1280 pixels × 1024 pixels. In the present work, a FPGA development board (Altera DE2–115) was used for the proposed image acquisition on the phase axis. We implemented internal software that generates a PWM signal to create a triggering signal for image acquisition. The PWM signal was generated by synchronizing the divided excitation signal to the pre-calculated phase clock using the base clock of the board. The displacement of vibration was extracted from the photographed images using various methods [3, 4]. The purpose of this study is to confirm the possibility of measuring vibration in the hyper-Nyquist frequency range using a limited frame rate. A description of the image post-processing for extracting the displacement information from the image is thus omitted. In this study, we use the dynamic moiré method. A detailed description of the process and the theory can be found in [4].

8.3.2 Results Experimental verification of the ability to measure vibration above the Nyquist frequency using a limited frame rate was performed before extracting modal parameters from the measured displacement vibration data. Since the frame rate used in the experiments is fixed at 75 fps, it is theoretically impossible to measure vibrations above 37.5 Hz, which is the Nyquist frequency. In this experiment, we compared two cases of measurement results. The first case is the conventional time-based measurement using the fixed 75fps. The other case is the suggested method using the time-phase transformation, which is a phase-based measurement using the same fixed frame rate. The following two graphs show the results for an arbitrary point of the surfaces of the measured cantilever beam. Figure 8.3a shows the time-displacement response data obtained using the conventional approach with

8 Vision-Based Modal Testing of Hyper-Nyquist Frequency Range Using. . .

63

Fig. 8.3 Experimental results for an arbitrary specific point. (a) The conventional time-based measurement. (b) The suggested phase-based measurement

a fixed sampling rate. We extracted the time-displacement response from 150 images measured at 80 Hz vibration for 2 seconds. Since the frame rate was insufficient, an aliasing phenomenon occurred and the vibration appeared as 5 Hz. This is a false measurement. Figure 8.3b shows the phase-displacement response data obtained by phase-triggered measurement applying the proposed method in this paper. The same number of images (150 frames) were acquired and used. The number of phase points triggered in this measurement was set to 15. Thus, in the graph, the green dots represent all raw data measured ten times for the same phase moment. This is considered a measurement error due to the influence of the internal jitter present in the camera sensor. However, the error could be reduced through the averaging effect. The red dots represent the average value of the results measured ten times. From the measured displacement value, it can be confirmed that the displacement information of one period of 0 to 2pi can be accurately measured. It shows a sinusoidal shape because the steady-state response of the forced excitation with a single frequency has a sinusoidal waveform in one-period response.

8.4 Conclusion Vibration measurement for modal testing with a camera causes aliasing and insufficient resolution problems. For insufficient resolution problems, various postprocessing processes have already been studied. In this study, we extract the displacement from the image using the dynamic moiré method. However, there are still limitations in applying this method to various industrial environments. To solve the aliasing problem due to the limited frame rate, we proposed a triggering method based on the phase of the input signal. For modal testing, various signals can be used in the excitation. The proposed triggering method is based on the feature that the steady-state response of a linear system has the same single frequency component

64

D. Kim and Y. Park

when using a single frequency excitation. Therefore, it can be applied only when the excitation signal has an exactly known single frequency. Therefore, the excitation signal for the modal parameter extraction uses a sine stepped method. After reaching a steady state for a particular single frequency excitation, the camera is triggered to take an image at a specific phase interval of the excitation signal rather than at a constant time interval. By using the proposed method, the displacement response of high mode can be obtained even at a limited frame rate. Finally, the feasibility of this approach was demonstrated through experimental vibration measurement of a cantilever beam. The modal testing method proposed in this research provides the same frequency resolution over all measurement frequency ranges, and no-leakage measurements can be performed over the entire frequency range. Acknowledgments This work was supported by the Brain Korea 21 Plus Project.

References 1. Ewins, D.J.: Modal testing: Theory and practice. Research Studies Press, Letchworth (1984) 2. Gloth, G., Sinapius, M.: Analysis of swept-sine runs during modal identification. Mech. Syst. Signal Process. 54(4), 1421–1441 (2004) 3. Choi J.: A study on visual modal analysis using dynamic moire method. KAIST Master thesis (2016) 4. Li, W., Su, X., Liu, Z.: Large-scale three-dimensional object measurement: A practical coordinate mapping and image data-patching method. Appl. Opt. 40, 3326–3333 (2001)

Chapter 9

Experimental Investigation on the Effect of Tuned Mass Damper on Mode Coupling Chatter in Turning Process of Thin-Walled Cylindrical Workpiece Y. Nakano, T. Kishi, H. Takahara, L. Croppi, and A. Scippa

9.1 Introduction In recent years, the rigidity of a thin-walled cylindrical workpiece such as a jet engine turbine case decreases to reduce the weight of an aircraft. As a result, chatter vibration is more likely to occur during the turning process of the workpiece. The chatter vibration causes the reduction of machining accuracy, wear of tools, noise, and so on. Chatter vibration is classified into forced chatter vibration and self-excited chatter vibration. Self-excited chatter occurs by regenerative effect and mode coupling [1–3]. While the various approaches have been proposed to suppress regenerative chatter [4], there is little research on the countermeasure against mode coupling chatter. One of the effective countermeasures against regenerative chatter is the application of the tuned mass damper (hereafter referred to as TMD) [5, 6]. Using the TMD can increase the critical width of cut at the onset of regenerative chatter. However, there are few studies which investigate the design method of TMDs to suppress mode coupling chatter in a turning process of the thin-walled cylindrical workpiece effectively. In the present work, the effect of the TMDs attached to the rotating thin-walled cylindrical workpiece on mode coupling chatter generated in a turning process is experimentally investigated. The natural frequency and the mounting position of TMDs are determined according to the natural frequency of the workpiece and the workpiece vibration mode during mode coupling chatter generation. The present study shows three 1.5% mass ratio TMDs can totally control mode coupling chatter.

Y. Nakano () · T. Kishi · H. Takahara School of Engineering, Tokyo Institute of Technology, Tokyo, Japan e-mail: [email protected] L. Croppi · A. Scippa Department of Industrial Engineering, University of Firenze, Firenze, Italy © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_9

65

66

Y. Nakano et al.

9.2 Experimental Setup and Vibration Characteristic of Workpiece Figure 9.1a shows the dimension of the cylindrical workpiece made of SUS304. The dimensional ratio of the outer diameter to the axial length ratio of the workpiece is approximately equal to that of a jet engine turbine case. The workpiece is fixed to a lathe using a ring jig as shown in Fig. 9.1b and 9.1c. An impact test of the workpiece is conducted in order to obtain the natural frequencies and the natural modes using an accelerometer (Ono sokki, NP-2106) and an impact hammer (PCB, 086C03). The accelerometer is attached to the tip of the workpiece in the radial direction. The frequency response function (hereafter referred to as FRF) of the workpiece is measured by hammering the tip of the workpiece in the radial direction. Figure 9.2 shows the FRF of the workpiece and the natural mode of the workpiece. The second mode, the third mode, and the fourth mode have respectively two, three, and four antinodes in the circumferential direction. The natural frequencies of the second mode, the third mode, and the fourth mode are respectively 2090 Hz, 2240 Hz, and 3520 Hz. Since the first bending mode of the workpiece is 4340 Hz, the bending mode is much higher than the circumferential modes. Therefore, only the circumferential modes are focused in this study.

Fig. 9.1 Workpiece dimension and fixing method for workpiece to lathe. (a) Workpiece dimension. (b) Fixing method for workpiece. (c) Workpiece attached to lathe

Fig. 9.2 FRF and vibration mode of workpiece

9 Experimental Investigation on the Effect of Tuned Mass Damper on Mode. . .

67

Next, an impact test is conducted to investigate the FRF of the tool. The overhang length of the tool from the tool stand to the tip of the tool is 25 mm. As a result of the tool impact test, it was confirmed that the lowest natural frequency of the tool was about 5500 Hz which was much higher than that of the workpiece.

9.3 Vibration Mode of Workpiece During Mode Coupling Chatter The present study focuses on mode coupling chatter generated in the finishing process of the longitudinal turning of the thin-walled cylindrical workpiece. The conditions where mode coupling chatter occurred were determined to refer to the conditions indicated in the work by Yamamoto et al. [3]. The width of cut is 0.025 mm, the cutting speed is 150 m/min, and the feed rate is 0.05 mm/rev. As a result of conducting the stability analysis in advance using the zero order analysis [7], it has been confirmed that the selected width of cut is sufficiently smaller than the critical width of cut at the onset of regenerative chatter and that it is a condition where regenerative chatter does not occur. As a result of the longitudinal turning test, the chatter frequency was 2230 Hz. This frequency was close to the natural frequency in the third mode of the workpiece as shown in Fig. 9.2. In order to measure the workpiece vibration mode in the radial direction during the chatter generation, six eddy current sensors (PU-05, Applied Electronics Corporation) were arranged at a regular pitch of 20 degrees in the radial direction of the workpiece as shown in Fig. 9.3a. The sensors were positioned 30 mm in the axial direction from the tool. The vibration displacements of measuring points from #13 to #18 in Fig. 9.3b were measured by six eddy current sensors. The vibration displacements in the points from #1 to #12 in Fig. 9.3b were predicted by interpolating considering the symmetry of the workpiece vibration

Fig. 9.3 Eddy current sensor arrangement and chatter vibration mode of workpiece. (a) Measuring points of workpiece. (b) Chatter vibration mode of workpiece

68

Y. Nakano et al.

Fig. 9.4 Vibration displacement of workpiece from #13 to #18 during chatter generation

Fig. 9.5 TMD and how to attach to workpiece (a) TMD dimension. (b) Workpiece with TMDs

mode. Figure 9.3b shows the predicted vibration mode of the workpiece during the chatter generation. The chatter vibration mode shown in Fig. 9.3b is in good agreement with the third mode shown in Fig. 9.2. Figure 9.4 shows the vibration displacements at the measuring points from #13 to #18. From Fig. 9.4, the traveling wave is observed. The workpiece vibration mode during the chatter generation is not fixed in space.

9.4 Effect of TMD on Mode Coupling Chatter Since the workpiece vibration mode during mode coupling chatter generation was dominant, we decided to attach TMDs to the rotating workpiece directly. Figure 9.5a shows the dimension of the TMD. The TMD was designed to minimize the mass of the TMD in order to minimize the deformation of the workpiece due to the rotation. The mass of the TMD was 1.5% of the mass of the workpiece. The length of the cantilever beam of the TMD was determined so that the natural frequency of the TMD was almost equal to the chatter frequency. In the present study, three TMDs were attached to the inner surface of the workpiece using a double sided tape at a regular interval of 120 degrees which coincides with the interval of antinode of the third vibration mode of the workpiece as shown in Fig. 9.5b. The three TMDs were manufactured to have equal mass and natural frequency. As a result of the impact test of the TMD, the natural frequency of the TMD was 2040 Hz and the damping

9 Experimental Investigation on the Effect of Tuned Mass Damper on Mode. . .

69

ratio of the TMD was 2.0%. In addition, instead of the TMDs, three masses with the same mass as the TMD were attached to the workpiece to compare the vibration characteristics of the workpiece and the chatter suppression effect. Figure 9.6 shows the original workpiece FRF, the workpiece FRF with three TMDs, and the workpiece FRF with three masses. From Fig. 9.6, the peak amplitude related to the third mode of the workpiece greatly decreased by the application of the TMDs. When the masses were attached, the peak amplitude also decreased. But the reduction rate was less than the case of the application of the TMDs. Each natural frequency and damping ratio of the workpiece vibration mode was shown in Table 9.1. From Table 9.1, the damping ratio of the third mode of the workpiece was largest when the TMDs were attached to the workpiece. Next, the longitudinal turning tests were conducted when the TMDs and the masses were attached to the workpiece. Figure 9.7 shows the vibration displacements of the workpiece without the TMDs or the masses, with the TMDs and with the masses. From Fig. 9.7, the vibration displacement was greatly decreased when the TMDs were attached to the workpiece. In addition, there was no chatter mark as shown in Fig. 9.8. Therefore, the mode coupling chatter was suppressed when three TMDs were attached to the cylindrical workpiece. From Fig. 9.7, the vibration displacement was decreased when the three masses were attached to the workpiece. However, the chatter occurred yet. Compared with the TMDs, the suppression effect

Accelerance m/s2/N

10000 original workpiece workpiece with TMDs workpiece with masses

1000

100

10

1 0

500

1000

1500 2000 2500 Frequency Hz

3000

3500

4000

Fig. 9.6 FRF of original workpiece, workpiece with TMDs, and workpiece with masses Table 9.1 Natural frequency and damping ratio of workpiece Original workpiece Workpiece with masses Workpiece with TMDs

Second mode 2090 Hz 0.14% 1980 Hz 0.36% 1960 Hz 0.66%

Third mode 2240 Hz 0.052% 2050 Hz 0.35% 2260 Hz 1.1%

Fourth mode 3520 Hz 0.039% 4220 Hz 1.1% 3360 Hz 0.86%

Displacement μm

70

Y. Nakano et al. 20

original workpiece workpiece with TMDs workpiece with masses

10 0 -10 -20 0

1

2

3

4

5

Time ms Fig. 9.7 Vibration displacement of original workpiece, workpiece with TMDs, and workpiece with masses

Fig. 9.8 Machining surface of workpiece with chatter and without chatter (a) Original workpiece. (b) Workpiece with TMDs

on the chatter was low. It was made clear that the TMDs could control the mode coupling chatter even if the masses and the damping of the TMDs were small.

9.5 Conclusion In the present work, the effect of the TMDs on mode coupling chatter in the turning process of a thin-walled cylindrical workpiece was experimentally investigated. It was confirmed that the chatter vibration mode of the workpiece was the circumferential vibration mode with three nodal diameters by measuring the vibration of the workpiece during the chatter generation using six eddy current sensors. According to the measured chatter vibration mode of the workpiece, three TMDs were attached to three antinode positions of the third mode of the workpiece. A turning process test verified experimentally that mode coupling chatter was completely suppressed by the proposed TMDs. Mode coupling chatter could be suppressed even if the damping of the TMD was small and the total mass of three TMDs was less than 4.5% of the mass of the cylindrical workpiece.

9 Experimental Investigation on the Effect of Tuned Mass Damper on Mode. . .

71

References 1. Marian, W., Erhan, B.: Sources of nonlinearities, chatter generation and suppression in metal cutting. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 359(1781), 663–693 (2001) 2. Ehsan, J., Mohammad, R.M.: Numerical simulation of interaction of mode-coupling and regenerative chatter in machining. J. Manuf. Process. 27, 252–260 (2017) 3. Yamamoto S., Kurita Y., Oura Y., Tomida K., Kawata M., Matsumoto T.: Chatter vibration of workpiece deformation in cutting thin cylindrical workpiece(Effects of workpiece size on occurrence of chatter vibration), Dynamics and Design Conference 2015, Hirosaki, Japan, 25– 28 Aug 2015 (2015) 4. Munoa, J., Beudaert, X., Dombovari, Z., Altintas, Y., Budak, E., Brecher, C., Stepan, G.: Chatter suppression techniques in metal cutting. CIRP Ann. Manuf. Technol. 65, 785–808 (2016) 5. Wang, M., Zan, T., Yang, Y., Fei, R.: Design and implementation of nonlinear TMD for chatter suppression: An application in turning process. Int. J. Mach. Tools Manuf. 50(5), 474–479 (2010) 6. Nakano Y, Takahara H, Kobayashi A: Design of tuned mass damper for forced chatter and regenerative chatter in end milling process of low-regidity workpiece, 25th International Congress on Sound and Vibration, Hiroshima, Japan, 8–12 July 2018 (2018) 7. Altintas, Y.: Manufacturing automation: Metal cutting mechanics, machine tool vibration, and CNC design 2nd edition. Cambridge University Press, Cambridge (2012)

Chapter 10

Eliminating the Influence of Additional Sensor Mass on Structural Natural Frequency in the Modal Experiment Feng Zhao, Wenliao Du, and Hongwei Li

10.1 Introduction Contact sensors that are additional mass of original structures are usually used in modal experiments. When a contacting sensor is located at a certain position, its effect on the natural frequencies will be different for individual modes since the effective mass of the sensor for each mode will be different [1–4]. It is a well-known fact that the natural frequency of a mode is least affected by the mass loading effect of the transducer if the transducer is installed near a nodal line (point) of a mode. This influence feature can be used to modify structure modes [5, 6]. Wasik et al. [7] shifted the mode frequency successfully by using additional mass according to the requirement. However, it was adverse for the results of modal experiment tests. Some measures, such as contact sensors with extremely light mass and laser vibrometers without contact [8] or microphone to measure the sound, are used in tests to eliminate the influence of additional mass on tested results. Some theories [9–11] have also been developed to eliminate the influence of additional masses. In this paper, a simple and effective experiment method of eliminating the influence of additional mass on modal test will be developed by using the mass influence feature of modes.

F. Zhao · W. Du · H. Li Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou University of Light Industry, Zhengzhou, China F. Zhao () 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/10.1007/978-3-030-48153-7_10

73

74

F. Zhao et al.

10.2 A Cantilever Beam and Its Accurate Mode Frequencies A cantilever beam model and its first three order mode shapes and LMS testing system are shown in Fig. 10.1. The physical parameters of this cantilever beam are 0.659 m of length, 0.03 m of width, 0.006 m of height, and 7850 kg/m3 of density. The first three order mode frequencies calculated by theory are 11.47 Hz, 71.88 Hz, and 201.26 Hz. Like as conventional modal experiments, the method of moving force hammer is conducted on this model attached the contact sensor with 61 g mass located at the measuring point, as shown in Fig. 10.1. The influences of additional mass on the first three order mode frequencies are plotted in Fig. 10.2 to show location relations between the measuring points and mode nodes. The relation of influence of the additional mass location on mode nodes is used herein to identify the accurate modal parameters.

Fig. 10.1 LMS testing system, the cantilever beam, and its first three order frequencies

Fig. 10.2 Influence of additional mass location on the first three order frequencies under 61 g additional mass

10 Eliminating the Influence of Additional Sensor Mass on Structural Natural. . . 





 J1

J1 $PSOLWXGH  

V VY V V V V V V V V V V V V V V VY V VY Y YY YY YV YV YV YV V VY YY Y Y R



$PSOLWXGH

Y Y V Y Y Y Y Y R



75

 

9  84

/LQHDU

H 



+]

/LQHDU 11.1



+]

(b)

(a)

Fig. 10.3 Stabilization plot of the 1st order mode for the additional mass located at: (a) the 10th measuring point away from mode nodes, (b) the 1st measuring point near the 1st mode node



62.91

Y R Y Y Y Y Y Y Y R R R R

V Y V Y Y Y Y Y Y Y Y R

V Y Y Y V Y Y Y Y V Y Y Y R

V Y V Y Y Y Y R



Y YV Y Y



 

R

Y Y Y R

J1

J1

$PSOLWXGH



V V V V V V V V V V V V V V V V V V V V Y Y V V

$PSOLWXGH

Y Y V V VVVV Y Y V V VVYY Y Y Y Y VVYV Y Y Y VYYY V V VYY Y Y Y VYY Y V V VYV Y Y Y VVY Y Y Y VYY Y Y VYV Y VY Y VV Y VV V VV Y VV Y VY Y Y VV Y VY Y VY Y VV Y YY YY YVV YYY YY VV VV YY YY YY Y R



Y V Y Y Y Y Y Y R

Y R

R Y Y R V R R Y Y Y R

68  82

R



/LQHDU

+]





(a)

/LQHDU

+]

VVVV VVVV VVVY VVVY VVVV VVYY VYYY VYYY VYY VYY VYY VYV VYV VVV VVV VYV YYY VYY VV YY YY VY VY VY YY YY V Y Y V Y Y R

  



(b)

Fig. 10.4 Stabilization plot of the 2nd-order mode for the additional mass located at: (a) the 10th measuring point away from mode nodes, (b) the 8th measuring point close to the 2nd mode node

10.2.1 Accurate First-Order Frequency Moving the contact sensor to the 1st and 10th measuring points, respectively, the identified mode frequencies are shown in Fig. 10.3. It is evident that the result of the sensor located at the 1st measuring point is accurate. The stabilization plots present following figures are calculated from the FRFs by LMS software.

10.2.2 Accurate 2nd-Order Frequency Moving the contact sensor to the 10th and 8th measuring points, respectively, the identified mode frequencies are shown in Fig. 10.4. It is evident that the result of the sensor located at 8th measuring point is accurate.

76

F. Zhao et al.

10.2.3 Accurate 3rd-Order Frequency Moving the contact sensor to the 10th and 9th measuring points, respectively, the identified mode frequencies are shown in Fig. 10.5. It is evident that the result of the sensor located at the 9th measuring point is accurate.

10.3 A Suspended Beam and Its Accurate Mode Frequencies A suspended beam model is shown in Fig. 10.6, which is also used to verify the method. The physical parameters of the suspended beam are 0.7 m of length, 0.03 m of width, 0.006 m of height, and 7850 kg/m3 of density. The first three-order mode frequencies of this model calculated by theory are 64.68 Hz, 178.3 Hz, and 349.58 Hz. The location relations between the measuring points and mode nodes are shown in Fig. 10.6.



VY V VV V VV V VV V VV V VY V VY V VY V VY V VY V VV V VV V VY V VV V VY V V YV Y YV V V V V V V V V V V VV VY Y Y YV YY YV V Y Y R

V V Y V Y V Y V Y V V V V V Y V Y V V Y V Y Y Y V Y Y V V V Y Y Y Y Y

181.87 

V Y V V Y V V Y V V Y V V Y Y Y Y Y V Y Y V Y Y V Y Y Y Y Y V V V Y Y Y Y Y V Y Y V Y Y Y Y Y V Y Y Y Y Y Y Y Y Y V Y R

V V V V V V V V V V V V V V V V Y Y Y Y Y Y

R R

V V V V V V V V V V V V V V V V V V V Y V Y Y Y Y Y Y Y

Y Y V Y V Y V Y V Y V Y V Y V Y Y Y Y Y Y Y V Y Y Y Y Y Y Y Y Y Y Y Y R

Y Y Y Y Y Y Y Y Y R R



 

R



R

J1 $PSOLWXGH

J1

$PSOLWXGH



 /LQHDU +]

(a)



Y Y Y Y Y Y V Y Y Y Y Y Y Y Y R

V V Y V Y V Y V V V V Y Y V Y Y Y Y Y Y



VY Y VY Y V V V Y Y Y V Y Y Y V V Y Y Y Y Y

V V VY V V V YV YV V V V V V V V V V V V V Y Y V Y Y V Y Y R

V V V V V V V V Y Y Y Y Y Y Y Y Y Y Y

V V V V Y Y V Y Y

Y Y Y Y V Y V Y Y Y Y Y Y Y Y

V V V V V V V Y Y Y Y Y Y Y Y

R

VY Y Y Y Y Y Y Y V V Y Y V V Y Y V Y Y Y Y R R

V Y V Y V Y V Y V V V V V V V V V V V V V Y V Y Y Y Y Y Y Y Y

YV Y YV V YVY Y YY Y V Y VY V V V Y VY VY Y VY VYY V Y VY Y VY VY Y VY Y VY Y YY Y VV VV VY YY YY Y Y Y V Y Y V V R

/LQHDU +]

Y V Y Y Y V Y Y Y

V V V V Y Y V Y V Y V V Y Y Y Y Y Y Y Y Y Y Y

Y Y Y Y Y

  

194  88



(b)

Fig. 10.5 Stabilization plot of the 3rd-order mode for the additional mass located at: (a) the 10th measuring point away from mode nodes, (b) the 9th measuring point near the 3rd mode node

Fig. 10.6 LMS testing system, suspended beam, and its first three-order frequencies

10 Eliminating the Influence of Additional Sensor Mass on Structural Natural. . .

J1 $PSOLWXGH





60.1

VY Y VY Y VY Y VY Y Y V Y V V V V V Y V Y V V Y Y V V Y V Y V Y Y R

Y Y Y Y Y

V V V Y Y Y Y

Y V V Y Y V Y Y Y R Y Y Y Y Y Y Y Y R

V V V V Y V Y Y Y Y Y Y R

 

Y Y Y R

/LQHDU +]





Y V Y V Y Y



 J1 $PSOLWXGH

Y Y Y Y Y Y Y Y Y Y Y Y Y Y R

77



65.64



(a)

/LQHDU +]

V V V V V V Y V V V V V V V V V V V V V V V V V Y V Y R



 



(b)

Fig. 10.7 Stabilization plots of the 1st-order mode for the additional mass located at: (a) the 1st measuring point away from mode nodes, (b) the 3rd measuring point near the 1st mode node



YVYY YVYY VYY VYY YYY VY YY VV YY VY YY YY VY VY YY YY YY VY VV VV VY VV YV YV YV YY Y V V V Y R

Y Y Y Y Y

168  01



Y Y V V Y Y Y Y Y V Y Y Y Y Y R R



Y Y R

  

J1 $PSOLWXGH

J1 $PSOLWXGH



 /LQHDU +]





(a)

180.61

/LQHDU +]

V V V V V V V V V V V V V V V V V V V V V V V V V V V V R



 



(b)

Fig. 10.8 Stabilization plots of the 2nd-order mode for the additional mass located at: (a) 1st measuring point away from mode nodes, (c) 6th measuring point near the 2nd mode node

10.3.1 Accurate 1st-Order Frequency Moving the contact sensor to the 1st and 3rd measuring points, respectively, the identified mode frequencies are shown in Fig. 10.7. It is evident that the result of the sensor located at 3rd measuring point is accurate.

10.3.2 Accurate 2nd-Order Frequency Moving the contact sensor to the 1st and 6th measuring points, respectively, the identified mode frequencies are shown in Fig. 10.8. It is evident that the result of the sensor located at 6th measuring point is accurate.

78

F. Zhao et al.



V Y V Y Y V Y Y Y Y Y

V V V V V V V V V V V V Y V V Y Y V V Y R

Y Y Y Y Y Y Y Y Y Y Y Y Y Y

Y Y

Y Y Y Y Y Y Y Y Y Y Y Y Y R R R

Y Y V R

Y Y Y Y V V Y Y Y Y Y Y Y V Y Y

332  47



/LQHDU +]

YY Y Y V V V Y Y Y Y Y Y Y Y Y Y Y Y V Y Y Y Y Y R

Y Y

Y Y Y V Y V Y Y Y V Y Y Y Y Y Y Y

Y V V V V V V V V V V Y V V V V Y Y Y Y Y Y Y Y Y Y R





 

 J1 $PSOLWXGH

J1 $PSOLWXGH



YYY VVY YVY YVY YV YV YY YY Y Y V YY YV YY YY YY V V V Y V V Y Y V V V V V V Y R



353.55



(a)

/LQHDU +]

V V V V V V V V V V V V V V V V V V V V V V V V Y V Y V Y R

  



(b)

Fig. 10.9 Stabilization plots of the 3rd-order mode for the additional mass located at: (a) the 1st measuring point away from mode nodes, (d) the 10th measuring point near the 3rd mode node

10.3.3 Accurate 3rd-Order Frequency Moving the contact sensor to the 1st and 10th measuring points, respectively, the identified mode frequencies are shown in Fig. 10.9. It is evident that the result of the sensor located at 10th measuring point is accurate.

10.4 Conclusions The contact sensor affects the mode frequencies of lightweight and low stiffness structures significantly. An experiment method by using the contact sensors with big mass is applied to identify the accurate mode parameters. The key point of this method is the relation of influence of additional mass on modes. The tested results for the cantilever beam and the suspended beam show that the identified mode frequencies by placing the contact sensor near the modal node are accurate as compared to the theoretical results and the tested results by placing the contact sensor away from the node have errors more than 10%. Acknowledgments This research was funded by the National Nature Science Foundation of China (U1804141), the Program for Science/Technology Innovation Talents in Universities of Henan Province (17HASTIT028), and the Key Research Projects of Henan Higher Education Institutions (19A130003).

References 1. Huseyinoglu, M., Cakar, O.: Determination of stiffness modifications to keep certain natural frequencies of a system unchanged after mass modifications. Arch. Appl. Mech. 87, 1629–1640 (2017) 2. Sinha, J.K., Singh, S., Rao, A.R.: Added mass and damping of submerged perforated plates. J. Sound Vib. 260, 549–564 (2003)

10 Eliminating the Influence of Additional Sensor Mass on Structural Natural. . .

79

3. Valentin, D., Presas, A., Egusquiza, E., Valero, C.: Experimental study on the added mass and damping of a disk submerged in a partially fluid-filled tank with small radial confinement. J. Fluids Struct. 50, 1–17 (2014) 4. Wang, L., Li, Y.Q., Shen, Z.Y.: Experimental investigation on the added mass of membranes vibrating in air. J. Vibr. Eng. 24(2), 125–132 (2011) 5. Cakar, O.: Mass and stiffness modifications without changing any specified natural frequency of a structure. J. Vib. Control. 17, 769–776 (2011) 6. Ouyang, H.J., Zhang, J.F.: Passive modifications for partial assignment of natural frequencies of mass–spring systems. Mech. Syst. Signal Process. 50-51, 214–226 (2015) 7. Wasik, M., Lis, K., Lehrich, K., Mucha, L.: Model-based dynamic structural modification of machine tools. Shock and Vibration, Article ID 3469171, 9 pages (2018) 8. Warren, C., Niezrecki, C., Avitablile, P., Pingle, P.: Comparison of FRF measurements and mode shapes determined using optically image based, laser, and accelerometer measurements. Mech. Syst. Signal Process. 25, 2191–2202 (2011) 9. Cakar, O., Sanliturk, K.Y.: Elimination of transducer mass loading effects from frequency response functions. Mech. Syst. Signal Process. 19(1), 87–104 (2005) 10. Bi, S.S., Ren, J., Wang, W., Zong, G.H.: Elimination of transducer mass loading effects in shaker modal testing. Mech. Syst. Signal Process. 38, 265–275 (2013) 11. Zamani, P., Anbouhi, A.T., Ashory, M.R., Khatibi, M.M., Nejad, R.M.: Cancelation of transducer effects from frequency response functions: Experimental case study on the steel plate. Adv. Mech. Eng. 8(4), 1–12 (2016)

Part III

Dynamic Behaviour of Materials Including Nano-composite Structures and Vibration Absorbing Materials

Chapter 11

A Study of the Vibration Reduction Effect of Sound Absorbing Material Within Acoustic Box Jiajun Hong, Takuya Yoshimura, and Makoto Takeshita

11.1 Introduction Sound absorbing materials have played a key role as a passive solution for reduction of vibration and noise in automobile design. In order to improve the vehicle ride comfort, a large variety of poroelastic materials is used as the sound absorbing materials in floor coverings, rear shelves, and luggage areas to reduce the vibration and noise effectively in the car [1]. The noise inside vehicle cabin can be divided into airborne noise (propagated through the air) and structure-borne noise (propagated through the structure) [2]. Both airborne and structure-borne noise should be controlled for a higher ride comfort in the passenger compartment. In the science of acoustical control, two classes of acoustical treatments are used to cope with airborne noise, namely, barriers and absorbers. Similarly, two classes of acoustical treatments are used to cope with structure-borne noise, namely, isolators and dampers [3]. Poroelastic materials play the role as absorbers in acoustical treatments; acoustic waves propagating through cellular structures are dissipated by energy loss caused by heat exchange [4, 5]. And it also acts as a damper for the structure to reduce the structure-borne noise. The noise inside the cabin is known to have a relatively high contribution from the vibration of floor parts. This paper experimentally investigates the vibration reduction effect of the sound absorbing materials within an acoustic box. The shift and amplitude attenuation at certain natural frequency was discussed. A simplified numerical model with coupled J. Hong () · T. Yoshimura Tokyo Metropolitan University, Tokyo, Japan e-mail: [email protected]; [email protected] M. Takeshita Toyota Boshoku Corporation, Aichi, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_11

83

84

J. Hong et al.

acoustic-structural characteristics was constructed for predicting the vibration reduction effect caused by sound absorbing materials.

11.2 Experiment 11.2.1 Experimental Model and Sound Absorbing Material The experimental model of this research is shown in Fig.11.1a. The acoustic box is a simplified cabin-like structure and consists of seven aluminum panels (bottom, rear, top, two front, and two side panels). The dimensions are 810 mm in length × 450 mm in width × 400 mm in height. The thickness is 3 mm at the bottom, 10 mm at the rear and the top, 20 mm at the front, and 25 mm at both sides. The panels at the bottom, the rear, and the top are fixed by bolts, and they are replaceable. The acoustic box has a coupled acoustic-structural characteristic [6] where the aluminum plates are structural systems and the air inside the cabin is an acoustic system. In this study, the floor panel is more flexible than other panels. Therefore, it has a relatively high contribution to the structure-borne noise inside the cabin. The felt-type poroelastic material is shown in Fig.11.1b. The dimension of the felt layer is 350 mm × 350 mm × 5 mm. Single layer of felt weighs 65 g. Single or multiple layer(s) of sound absorbing materials were attached to the bottom plate of an acoustic box to reduce the vibration of floor panel and noise in the acoustic box.

11.2.2 Vibration Experiment Impact hammering test was conducted to measure the frequency response functions (FRFs) of the acoustic box. The frequency range is 0 to 500 Hz with the resolution

Fig. 11.1 Experiment model: (a) acoustic box, (b) sound absorbing material

11 A Study of the Vibration Reduction Effect of Sound Absorbing Material. . .

85

Fig. 11.2 Excitation point and response point

Fig. 11.3 Acoustic box in three attachment cases

of 0.5 Hz and five averages applied. The excitation location and the response measurement point are shown in Fig. 11.2. The excitation point is set on the bottom panel and is excited by an impact hammer (PCB 086B03). Vibration is measured by using an accelerometer (PCB 352C66) set on the bottom panel. Noise is measured by using a microphone (PCB 378B02) set in the internal space of the acoustic box. Since the bottom panel of the acoustic box is more flexible than the others, many bottom-dominant modes are identified. The acoustic box in three attachment cases is shown in Fig. 11.3. Case 1 is used to observe the whole noise and vibration reduction effect caused by felt layer(s) attached to the inside of bottom panel. While case 2 is used to observe the vibration reduction effect, as the felt layer(s) just acts as a damper but no absorber when it is attached to the outside of bottom panel. And case 3 is used to observe the additional mass effect by attaching clay blocks with the same mass as felt layer(s).

11.2.3 Experimental Results The sound pressure FRFs of the acoustic box without attachment and with 1, 2, and 3 felt layer(s) attachment (inside) are shown in Fig. 11.4. Modal analysis results are also shown in the figure. As can be seen, with the increase of sound absorbing materials, all the peaks continue to attenuate and move toward low frequencies.

86

J. Hong et al.

Fig. 11.4 FRFs of the acoustic box with several felt layers and mode analysis results

Fig. 11.5 The (a) frequency shift and (b) amplitude attenuation of the first peak in three attachment cases

Among seven natural modes, only the fourth mode is an acoustic resonant mode which indicates the generation of high levels of airborne noise. The amplitude attenuation of the fourth peak is caused by the noise attenuation and the shift toward low frequencies is caused by the change of flow velocity and volume in the acoustic box. All other modes are vibration modes which indicates the generation of high levels of structure-borne noise. These peaks are attenuated by damping effect and the frequencies are shifted by additional mass effect. In this work, we just consider the vibration reduction effect caused by sound absorbing materials. The amplitude attenuation and frequency shift of the first peak (around 112.4 Hz) in three attachment cases are shown in Fig. 11.5. The figure shows that additional mass effect plays a major role in the frequency domain, but it has no effect on reducing vibration (clay blocks vs. felt layer inside). And when the damping effect is added in case 2, the degree of reduction is basically the same as the felt layer(s) attached to the inside of bottom panel (felt layer outside vs. felt layer inside).

11 A Study of the Vibration Reduction Effect of Sound Absorbing Material. . .

87

11.3 Numerical Simulations In order to predict the vibration reduction effect caused by sound absorbing materials, numerical analysis is carried out by commercial FE software MSC Patran/Nastran. A simplified numerical model was constructed as shown in Fig. 6(a) for predicting the frequency shift and amplitude attenuation of the first peak. It models the acoustic space and has the coupled acoustic-structural characteristics. For a well definition of assembly conditions, 48 bolts were defined as shown in purple. The acoustic box consists of seven aluminum panels. Aluminum material parameters were assumed as follows: Young’s modules of 69 Gpa, Poisson’ ratio of 0.33, and mass density of 2700 kg/m3 . Air material parameters were assumed as follows: mass density of 1.205 kg/m3 and speed of sound of 343 m/s. The FEM model for this analysis consists of 173,396 eight-noded hexahedral elements (HEX8) with element length 10 mm. Frequency response analysis was performed to predict the vibration reduction effect of sound absorbing material within acoustic box. Figure 11.6b shows comparison of acceleration FRFs between FE modelling results and experimental results. It shows the simulation results below 250 Hz are in excellent agreement with the experimental results. It also exists some small differences at natural frequencies above 250 Hz, which caused by the complex assembly conditions between the panels of acoustic box, but in reasonable range. Correctness of the simulation model has been confirmed. Figure 7a shows the simplified numerical model for simulating the equivalent mass and additional damping effects caused by 1, 2, and 3 felt layer(s)  attached to the inside of the bottom panel; 13 concentrated masses (as shown in mark) with 5 g, 10 g, and 15 g were defined on the nodes at the same location of clay blocks; and 40 dampers (as shown in yellow line) with scalar damping coefficient 0.2%, 0.4%, and 0.6% were defined between these nodes. Figure 11.7b and c shows the simulation results with concentrated masses, scalar dampers, and both. The simulation results are consistent with the experimental results for the additional mass effects on the shift of natural frequency and for the additional damping effects

Fig. 11.6 (a) numerical model of the acoustic box and (b) comparison of frequency response

88

J. Hong et al.

Fig. 11.7 (a) model with concentrated masses and scalar dampers, (b) frequency shift, and (c) amplitude attenuation of the first peak

on the amplitude attenuation of the peak. The difference between simulation and experimental results within an acceptable region (maximum difference of 0.7 Hz and 2.8 dB).

11.4 Conclusions The main objective of this research was to investigate the effects of sound absorbing materials on the vibration reduction. Impact hammering test was conducted to measure the FRFs of the acoustic box in several attachment cases. With the introduction of sound absorbing materials, the structure-borne peaks are attenuated and shifted toward low frequencies. For vibration modes with the contribution of the bottom panel, results show the additional mass effects on the shift of natural frequency and additional damping effects on the amplitude attenuation of the peak. A simplified numerical model with coupled acoustic-structural characteristics was constructed for predicting the frequency shift and amplitude attenuation. The correctness of the simulation model was confirmed. The simulation results were in good agreement with the experimental results. The constructed simplified model can be used to predict the vibration reduction effect caused by sound absorbing materials. To investigate the noise reduction at acoustic resonant frequency in detail is a part of future work.

References 1. Saha P, Chahine J.: Sound package materials in automobiles. ASME noise control and acoustic division, NCA 22(Book No. G01027), ASME International (1996) 2. Gardonio, P.: Review of active techniques for aerospace vibro-acoustic control. J. Aircr. 39(2), 206–214 (2002)

11 A Study of the Vibration Reduction Effect of Sound Absorbing Material. . .

89

3. Parikh, D.V., Chen, Y., Sun, L.: Reducing automotive interior noise with natural fiber nonwoven floor covering systems. Text. Res. J. 76(11), 813–820 (2006) 4. Sagartzazu, X., Hervella-Nieto, L., Pagalday, J.: Review in sound absorbing materials. Arch. Comput. Methods Eng. 15, 311–342 (2008) 5. Hoang, M.T., Bonnet, G., Luu, H.T., Perrot, C.: Linear elastic properties derivation from microstructures representative of transport parameters. J. Acoust. Soc. Am. 135, 3172–3185 (2014) 6. Yoshiro I.: Study of identification of coupled mode properties of coupled acoustic-structural system. Graduation thesis of Tokyo Metropolitan University (2014)

Chapter 12

Experimental Study on the Effects of Pickguard Material on the Sound Quality of Electric Guitars Osamu Terashima, Taisei Ito, Hiroyuki Yamada, Shota Mizukami, Ryoma Morisaki, and Toshiro Miyajima

12.1 Introduction An electric guitar is composed of many elements, such as pegs for winding strings, bridges for supporting strings, a neck, a body, fingerboards, and so on [1]. A pickguard (PG), which is used for the prevention of the guitar body from being damaged by the pick, is additionally a typical element of this instrument. In the past, the PG has frequently been made of plastic; however, in recent years we have attempted to use metal to differentiate our designs from existing products. However, in our preliminary experiments, metals altered the original functional sound performance of the guitar. In addition, when a copper PG was installed, the glossiness, chromaticity, and texture of the metal bestowed positive changes upon the guitar. While a PG made of metal achieves the above benefits, the cause for the change in the generated sound is still not clear from an engineering standpoint. Recent studies of guitars have involved the pickup for converting string vibrations into voltage signals [2] or guitar timbre [3]; however, there are no examples of studies on PGs. Furthermore, there are no existing design and production guidelines for manufacturing PGs. Consequently, whenever the PG material is changed, it frequently involves significant trial and error for an artisan to perfect the design and production process. In this study, we conducted experiments to analyze how different materials used in the PG affect the sound quality of electric guitars with the goal of establishing

O. Terashima () · T. Ito · R. Morisaki · T. Miyajima Toyama Prefectural University, Toyama, Japan e-mail: [email protected] H. Yamada · S. Mizukami Kaishindo-music, Toyama, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_12

91

92

O. Terashima et al.

the design and production guidelines for manufacturing PGs. First, we tested PGs produced from different materials; the difference in sound quality of each attached PG was compared with the conventional plastic pickguard. The material of the PG we selected for further testing was copper with a thickness of 1.0 mm. Next, the vibrational characteristics of the PG and its effect on the sound quality was investigated. Finally, we examined the vibrational characteristics of the PG when each guitar string was played. We experimentally examined the relationship of the vibrational characteristics (vibrational modes) of the PGs and the sound they generated, to identify the cause for the changes in sound performance produced by different materials used for the producing of PG.

12.2 Experimental 12.2.1 Measurement of Performance Sounds In this study, we used the regular double-cutaway guitar shown in Fig. 12.1(a). We examined the differences in sound performance when the PG for this guitar [white area in Fig. 12.1(a)] was plastic (thickness of 2.5 mm and mass of 0.11 kg) and copper (thickness of 1.0 mm and mass of 0.27 kg). Factors other than the PG, such as the performer, were always identical. We acquired guitar output voltage signals from the jack shown by the red circle in Fig. 12.1(a). In addition, we placed a microphone at the ears of the performer to acquire sound pressure signals to validate the correlation of the output voltage signals and the sound pressure. The microphone placement position was 1.0 m in the horizontal direction from the front of an amplifier, at a height of 1.1 m from the ground. For the microphone,

Fig. 12.1 (a) A photograph of an ordinary double-cutaway electric guitar with a plastic pickguard and (b) measurement points of the vibrational acceleration

12 Experimental Study on the Effects of Pickguard Material on the Sound. . .

93

we used a PCB 130F20 and used an OROS FFT Analyzer OR34 to acquire the above output voltage signals and sound pressure signals. The sampling frequency was 51.2 kHz and the sampling time was 20 s. Sound performance was based upon open strings and single tones. We compared the sounds when using a plastic PG and copper PG with the 3rd string (196 Hz natural frequency).

12.2.2 Measurement of Vibrational Modes For the experiment, we used the same double-cutaway guitar shown in Fig. 12.1(a). We used a guitar PG made of plastic (thickness of 2.5 mm and mass of 0.11 kg) and one made of copper (thickness of 1.0 mm and mass of 0.27 kg); we measured the output voltage of the guitar when plucking each string, which is the source signal of the sound generated by a guitar. When this was completed, we measured the vibrational acceleration perpendicular to the surface of the PG in nine points on the PG surface shown in Fig. 12.1(b). If we consider measurement efficiency, simultaneously measuring the vibrational acceleration in the nine points would be desirable; however, instead we installed multiple measurement sensors and collected measurements at each point considering changes in vibrational characteristics. In addition, we measured the vibrational acceleration perpendicular to the surface of the PG at the pegs of plucked strings simultaneously with the surface vibrational acceleration. For measurements, we used a data logger (Keyence NR500/NR-CA04/NR-HA08); we simultaneously measured the signals of vibrational acceleration sensors (PCB 352C22, mass 0.5 g) positioned at each measurement point and guitar output voltage. During measurements, the sampling frequency was 100 kHz and sampling time was 40 s. In this study, to ensure that the force used to pluck the strings was constant consistently, we positioned a weight on one end of a thread, as shown in Fig. 12.2; after fastening the other end to a string, we cut the middle of the thread and freely vibrated the string. This was done to facilitate vibrations being generated in identical order as a close facsimile to the vibrations generated when the strings were plucked by the thumb of an adult male; the mass of the weight was 80 g.

Strings

Thread Guitar

Jack Working stand Fig. 12.2 Experimental setup to pluck a string

Cutting position Mass

94

O. Terashima et al.

12.3 Results and Discussions 12.3.1 Measurement of Performance Sounds Figure 12.3 shows the time series variation of output voltage signals when playing the 3rd string as an open string. Figure 12.3(a) shows the results 2 s after plucking strings; Fig. 12.3(b) shows the results at 0.25 s. The vertical axis in each figure displays the values of output voltage signals normalized by their largest value; the horizontal axis shows the time. In addition, the red exhibits the results when using a plastic PG; and the green displays the results when using a copper PG. According to the figures, when playing these strings in an open state, the rate of voltage damping at the beginning of performance, that is, after the initialization of voltage output (time change in voltage amplitude), differs; further, damping occurs rapidly when using a copper PG. Hence, sound damping occurs more rapidly than when using a plastic PG; moreover, a sharp and clear sound is produced. Figure 12.4(a) and (b) each show the results of time-frequency analysis of the output voltage when plucking three strings after installing a copper-constructed PG and a plastic-constructed one. The horizontal axis represents the time; the vertical axis represents the frequency. In addition, shading in the figure represents the effective value, Vrms , of voltage. The window type is Hanning, with a window size of 32,768 points (approximately 0.32 s). According to Fig. 12.4(a), we determined that when a copper PG was installed, there was a strong output voltage of 196 Hz, which is the natural frequency of the 3rd string, as well as 392 Hz, which is two times that. Conversely, according to Fig. 12.4(b), when a plastic PG was installed, we determined a voltage output of 196 Hz, as well as 98 Hz and 147 Hz, which are 0.5 and 0.75 times that voltage, respectively; rarely was the strong 392 Hz voltage output observed with copper, detected with the plastic PG. From the above, it is clear that the sound generated changes depending upon differences in the material used in the construction of the PG. Furthermore, although we have only included the results for 3rd string in this paper out of space considerations, similar trends were seen for other string was played (2nd string and 4th string).

12.3.2 Measurement of Vibrational Modes The cause of changes in output voltage due to differences in PG material indicated in Sect. 12.3.1 were discussed from the standpoint of PG vibrational mode. We determined the vibrational acceleration at each measurement point on the PG when plucking the 3rd string, the frequency response function at each measurement point from the simultaneous measurement results for vibrational acceleration generated at the pegs, and examined the vibrational mode of the PG at 196 Hz based upon amplitude and phase information. For analysis and visualization of the vibrational mode, we used ME’scopeVES operational modal analysis software. We used 32,768

Normalized output voltage Normalized output voltage

12 Experimental Study on the Effects of Pickguard Material on the Sound. . .

95

2.0 (a) 1.0 0.0 –1.0

Plastic

–2.0 2.0 1.0 0.0 –1.0

Cu

–2.0

1

0

2

Normalized output voltage Normalized output voltage

Time [s] 2.0 1.0

Plastic

(b)

0.0 –1.0 –2.0 2.0 Cu 1.0 0.0 –1.0 –2.0

0.00

0.05

0.10

0.15

0.20

0.25

Time [s]

Fig. 12.3 Time series of the output voltage when the 3rd string is played

points (approximately 0.32 s) of measurement data, similar to the output voltage analysis, to compute the response function. Figures 12.5 and 12.6 show the results of visualizing the vibrational mode when installing a copper PG and plastic PG, respectively. The green areas in the figures represent the PG. In addition, (a) and (b) in both figures are the results of a phase

Frequency [Hz]

O. Terashima et al.

400

Frequency [Hz]

96

400

2.0 㽢 10-4

2.0 㽢 10-2

(a)

200 0 0.0

0.5

Time [s]

1.0

1.5

1.0

1.5

(b)

200

0 0.0

0.5

Time [s]

Fig. 12.4 Results of time-frequency analysis of the output voltage with (a) copper-constructed and (b) plastic-constructed one

Fig. 12.5 Results of the visualization of the 196 Hz vibrational acceleration with cooperconstructed PG. The phases of (a) and (b) are 0 and π, respectively

Fig. 12.6 Results of the visualization of the 196 Hz vibrational acceleration with plasticconstructed PG. The phases of (a) and (b) are 0 and π, respectively

of 0 (2π) and π, respectively. From the figures, when a copper-constructed PG was installed, the vibrations close to measurement points, 8 and 9 in Fig. 12.1(b), were strong, with vibration in a direction opposite to that of other points. In

12 Experimental Study on the Effects of Pickguard Material on the Sound. . .

97

addition, the direction of the vibrations at measurement points other than 8 and 9 primarily corresponded with each other. Conversely, when a plastic-constructed PG was installed, vibrations near measurement points, 4 and 9 in Fig. 12.1(b), were noticeable; however, these were weaker in intensity than when a copper-constructed PG was used and differed in vibrational direction depending upon the measurement point. From the above, it can be concluded that when a copper-constructed PG was installed, the high hardness of copper results in the PG and the body of the guitar itself vibrating strongly at the excitation frequency of the string. This results in stronger excitation frequency output voltage (sound), as shown in Fig. 12.4.

12.4 Summary Based on the results of examining the effects of differences in PG material upon sound performance using a regular double-cutaway guitar, we determined that using a metal PG provided a sharp, clear sound with rapid damping. In addition, a copperconstructed PG produced a sound with an emphasis on its natural frequency. The cause of this change was produced by the high hardness of copper which results in the PG and the body of the guitar itself vibrating strongly at the excitation frequency of the string. As an additional benefit, the above reveals the possibility that using a metal PG could render it possible to provide a new timber previously not observed with electric guitars. Acknowledgment This work was partially supported by the research fund of Toyama Prefectural University. The authors express their gratitude to it. The authors also express their gratitude to Mr. Hajime Hosoi and Mr. Tatsuya Yamaguchi for performing the experiment and discussing the measurement results with the authors.

References 1. Fletcher, N.H., Thomas, D.R.: The physics of musical instruments, pp. 207–232. Springer Science & Business Media (2012) 2. Mohamad, Z., Dixon, S., Harte, C.: Pickup position and plucking point estimation on an electric guitar via autocorrelation. J. Acoust. Soc. Am. 142(6), 3530–3540 (2017) 3. Paté, A., Le Carrou, J.L., Fabre, B.: Predicting the decay time of solid body electric guitar tones. J. Acoust. Soc. Am. 135(5), 3045–3055 (2014)

Chapter 13

Influence of Pulverized Material on Vibration and Sound Characteristics of an Operating Ball Mill Takuya Ito, Tatsuya Yoshida, and Fumiyasu Kuratani

13.1 Introduction A ball mill, which is a type of grinding equipment, is used widely in the mineral and chemical processing industry because it has a simple structure and the ability to grind material finer than other devices. Hard balls such as steel, which are called media, are placed into a mill cylinder with material to be ground, and the mill is rotated. The material is ground by collision force between the material and the balls or the mill wall. Operators usually cannot monitor the conditions of the material during mill operations. Moreover, the time of a grinding operation depends on desired pulverized size and material. Vibration and radiated sound are generated from the mill wall during operations. The vibration and radiated sound of the operating mill provide a significant amount of information about the operating conditions and the material conditions in the mill. The understanding of the vibration and radiated sound characteristics of operating mills is needed to estimate the pulverized material size from the measurement data of the vibration and radiated sound. Therefore, the vibration and radiated sound during mill operation have been analyzed in many studies [1, 2]. Moreover, an automation of ball mills based on measured vibrations has been studied experimentally [3]. In recent years, many studies using the distinct element method (DEM) have been conducted to understand internal conditions of mills [4, 5]. In this study, we simulated mill operations to understand the influence of the operation conditions on the vibration and radiated sound. From the simulation results, we examine the influence of the difference in operating conditions (size

T. Ito · T. Yoshida () · F. Kuratani Department of Mechanical Engineering, University of Fukui, Fukui, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_13

99

100

T. Ito et al.

and properties of the material particle) on the motion inside the mill, the vibration, and radiated sound of the mill wall.

13.2 Analysis Method [6] 13.2.1 Distinct Element Method (DEM) In this study, we analyzed the motion of the balls and material using the distinct element method (DEM). When the sphere element i contacts the sphere element j, the normal force Fnj and the shear force Fsj between the elements are calculated based on the Voigt model shown in Fig. 13.1. In the normal direction, the spring constant kn is given by the Hertz’s contact theory and the elastic modulus, and the viscous damping coefficient ηn is given by the coefficient of restitution of the spheres [7]. In the tangential direction, the spring constant ks , the damping coefficient ηs , and the sliding friction coefficient μ are given. The motion equations about the translational and rotational direction for the sphere i are expressed by the Eqs. (13.1) and (2), and the motion of elements is calculated by numerical integration. mi

d 2 xi = mi g + ΣFnj + ΣFsj dt 2

(13.1)

d 2 θi = ΣTi dt 2

(13.2)

Ii

In the above equations mi is the mass of the sphere i, xi is the position vector of the sphere i, g is the gravity vector, Ii is the moment of inertia of the sphere i, θi is the rotation angle vector of the sphere i, and Tj is the vector of the moment by the shear force.

Fig. 13.1 Modeling the contact between spheres in the DEM

13 Influence of Pulverized Material on Vibration and Sound Characteristics. . .

101

13.2.2 Vibration and Radiated Sound Analysis In the vibration analysis, the mill wall was modeled using shell elements based on the finite element method. The side of the mill cylinder is divided into 100 elements in the circumferential direction and 25 elements along the rotational axis direction of the mill. The nodes on the edges of the cylinder were given fixed boundary conditions because the actual cylinder edges are welded to flanges and bolted. The contact forces obtained by DEM analysis are given as the input force to the mill wall. In this time, the transient analysis was performed by numerical integration with the Newmark-β method. The mode superposition method was used, and vibration modes of 20 kHz or less were adopted. The modal damping ratio was set to 0.1% for all modes. These values were set based on the measuring results for the mill which authors measured in privies study [6]. In the radiated sound analysis, the sound pressure at an observation point was calculated by regarding an element of the finite element as a point source. When an element represented by the position vector ri from the element vibrates at acceleration x¨ (r, t) at time t, the sound pressure P(t) at the observation point is expressed by Eq.(13.3). P (t) =

n  ρ x¨ (ri , t − di /c) A 2π di

(13.3)

i=1

In the equation, n, A, ρ, c, and di, respectively, indicate the number of elements, the area of the elements, the density of air, the speed of sound, and the distance from an element to the observation point. The sound pressure at the observation point is predicted by the summation the sound pressure from each element.

13.3 Analysis Conditions As a fundamental principle of this analysis model, the material particles are not pulverized by the mill operation, and pseudo grinding operations are simulated by changing the material particle diameters. At this time, only one particle diameter of material particle is selected. Table 13.1 shows the ball diameter Db , material particle diameter Dm , and the number of the balls and material particles. The volume ratio of the balls and materials was set to 7:3 and maintain this ratio in constant under the all simulation conditions. The DEM simulations were conducted with the parameters shown in Table 13.2. The measuring point of the vibration displacement was placed at the position the mill wall at 45◦ from the horizontal surface as shown in Fig. 13.2. Moreover, the measuring point of the sound pressure was set on the point which is the extension connecting the mill center and the measuring point of the vibration. Furthermore, the rotational speed is set to 92.3 rpm because previous author’s study [6] confirmed that the vibration response increases as the rotational

102

T. Ito et al.

Table 13.1 Diameters and number of balls and material particles Media ball (steel) Material particle (steel) Material particle (ceramic) Material particle (polyethylene)

Particle diameter Db [mm] Number of particles Particle diameter Dm [mm] Number of particles Particle diameter Dm [mm] Number of particles Particle diameter Dm [mm] Number of particles

19.8 147 19.8 63 19.8 63 19.8 63

10.3 144 10.3 144 10.3 144

4.18 7662 4.18 7662 4.18 7662

Table 13.2 DEM parameters Steel (mill wall, media ball, and material particle) Ceramic (material particle) Polyethylene (material particle) Coefficient of friction μ

Density [kg/m3 ] Young’s modulus E [GPa] Poisson’s ratio ν Density [kg/m3 ] Young’s modulus E [GPa] Poisson’s ratio ν Density [kg/m3 ] Young’s modulus E [GPa] Poisson’s ratio ν Steel – steel Steel – ceramic Ceramic – ceramic Steel – polyethylene Polyethylene – polyethylene

7870 205 0.3 3600 330 0.23 940 0.45 0.38 0.4 0.4 0.35 0.2 0.3

speed increases. The experiment, which cannot be descripted due to the limitation of space, was conducted with ceramic particles, and the validation of the simulation was confirmed.

13.4 Analysis Results Figures 13.3, 13.4, and 13.5 show frequency responses of the vibration and radiated sound with steel, ceramic, and polyethylene as material particles. Here, the frequency range is determined 1 to 6 kHz. The peak frequency response of the radiated sound caused by collision between the particles is not included in this range. The decrease in the particle sizes of the material particle reduces the peak amplitude value. This is considered that the smaller material particle decreases the collision force with the wall. However, in the cases of the polyethylene, there is no clear difference of the peaks between 19.8 mm and 10.3 mm. Table 13.3 shows overall values of the frequency spectra on each of the material particles and decreasing rates of overall values, which are based on 19.8 mm on each material.

13 Influence of Pulverized Material on Vibration and Sound Characteristics. . .

103

Fig. 13.2 Measuring points of vibration and radiated sound

10

10

4

19.8 mm

10.3 mm

Sound pressure [Pa]

Velocity [m/s]

10

4.18 mm

2

0

1

2

3

4

5

10 10 10 10

1

19.8 mm

10.3 mm

4.18 mm

-1

-3

-5

6

1

2

3

4

5

Frequency [kHz]

Frequency [kHz]

(a) Velocity

(b) Sound pressure

6

Fig. 13.3 Frequency responses with steel particles

10

10

4

19.8 mm

10.3 mm

Sound pressure [Pa]

Velocity [m/s]

10

4.18 mm

2

0

1

2

3

4

5

6

10 10 10 10

1

19.8 mm

10.3 mm

4.18 mm

-1

-3

-5

1

2

3

4

5

Frequency [kHz]

Frequency [kHz]

(a) Velocity

(b) Sound pressure

6

Fig. 13.4 Frequency responses with ceramic particles

The decreasing rates of the steel and ceramic show similar tendency. However, there is clear difference in the decreasing rates of the ceramic and the polyethylene. Here, focusing on the mechanical property values of the material particles shown in Table

104

T. Ito et al.

10

10

4

19.8 mm

10.3 mm

Sound pressure [Pa]

Velocity [m/s]

10

4.18 mm

2

0

1

2

3

4

5

10 10 10 10

1

19.8 mm

10.3 mm

4.18 mm

-1

-3

-5

6

1

2

3

4

5

Frequency [kHz]

Frequency [kHz]

(a) Velocity

(b) Sound pressure

6

Fig. 13.5 Frequency responses with polyethylene particles Table 13.3 Overall values of velocity and sound pressure Material Steel

Ceramic

Polyethylene

Velocity [μm/s] (decreasing rate) Sound pressure [Pa] (decreasing rate) Velocity [μm/s] (decreasing rate) Sound pressure [Pa] (decreasing rate) Velocity [μm/s] (decreasing rate) Sound pressure [Pa] (decreasing rate)

Material diameter Dm [mm] 19.8 10.3 4.18 2.07 1.34 0.68 (−) (−35%) (−67%) 1.73 1.11 0.52 (−) (−36%) (−70%) 2.10 1.43 0.79 (−) (−32%) (−62%) 1.81 1.20 0.59 (−) (−34%) (−67%) 0.85 0.77 0.13 (−) (−9%) (−85%) 1.17 1.25 0.11 (−) (+7%) (−91%)

13.2, Young’s modulus of the polyethylene is much smaller than the ones of the steel and ceramic. Therefore, the decreasing rate would depend on the Young’s modulus of the material particle. From the above, the vibration and radiated sound are influenced by changing in the size and the property of the material particles.

13.5 Conclusions We examined the influence of the size and properties of the material on vibration and radiated sound characteristics of a ball mill with a simulation model. From the results, the decrease in the material particle size reduces the levels of the vibration and radiated sound response of the mill wall due to the decrease in the collision force between the material and the wall. Moreover, in terms of the material property, the decreasing rate of the overall value would depend on the Young’s modulus of the material particles.

13 Influence of Pulverized Material on Vibration and Sound Characteristics. . .

105

References 1. Martins, S., Li, W., Radziszewski, P., Caron, S., Aguanno, M., Bakhos, M., Petch, E.L.: Validating the instrumented ball outputs with simple trajectories. Miner. Eng. 21(11), 782–788 (2008) 2. Zeng, Y., Forssberg, E.: Monitoring grinding parameters by signal measurements for an industrial ball mill. Int. J. Miner. Process. 40(1–2), 1–16 (1993) 3. Tang, J., Zhao, L., Zhou, J., Yue, H., Chai, T.: Experimental analysis of wet mill load based on vibration signals of laboratory-scale ball mill shell. Miner. Eng. 23(9), 720–730 (2010) 4. Mishra, B.K., Rajamani, R.K.: The discrete element method for the simulation of ball mills. Appl. Math. Model. 16(11), 598–604 (1992) 5. Hosseini, P., Martins, S., Martin, T., Radziszewski, P., Boyer, F.R.: Acoustic emissions simulation of tumbling mills using charge dynamics. Miner. Eng. 24(13), 1440–1447 (2011) 6. Yoshida, T., Kuratani, F., Ito, T., Taniguchi, K.: Vibration characteristics of an operating ball mill. Proceedings of resent advances in structural dynamics. J. Phys. 1264 (2019) 7. Tsuji, Y., Tanaka, T., Ishida, T.: Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 71(3), 239–250 (1992)

Chapter 14

Three-Dimensional Strain Calculation of Rubber Composite with Fiber-Shaped Particles by Feature Point Tracking Using X-Ray Computed Tomography M. Matsubars, T. Shinnosuke, N. Asahiro, S. Kawamura, T. Ise, T. Nobutaka, I. Akihito, K. Masakazu, and F. Shogo

14.1 Introduction In recent years, lightweight and miniaturized mechanical structures have been promoted owing to their functional capabilities and other benefits. Consequently, flexibility must be ensured in such mechanical structures. Increase in the vibration sensitivity and the reduction in space owing to the arrangement of a vibration damping material leads to the excitation of vibration that is sufficient to hinder the original mechanical function. Therefore, it would be advantageous to develop a vibration damping material exhibiting high vibration damping properties when used a small amount vibration damping material. Complexing of fine particles with viscoelastic material for improving damping properties has been previously investigated [1–4]. However, the underlying mechanism for such an improvement is not yet known. To apply the phenomenon of improving damping characteristics to damping material production, we first attempt to elucidate the underlying mechanism.

M. Matsubars · T. Shinnosuke () · S. Kawamura · K. Masakazu · F. Shogo Toyohashi University of Technology, Toyohashi-Shi, Aichi, Japan e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] N. Asahiro Hyogo Prefectural Institute of Technology, Hyogo, Japan e-mail: [email protected] T. Ise Kindai University, Osaka, Japan e-mail: [email protected] T. Nobutaka · I. Akihito Doshisha University, Kyoto, 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/10.1007/978-3-030-48153-7_14

107

108 Table 14.1 Compounding ratio of composite rubber

M. Matsubars et al. NR Filler Zinc oxide Stearic acid Sulfur Vulcanization accelerator

[phr] [phr] [phr] [phr] [phr] [phr]

100 5 6 0.5 3.5 0.7

14.2 Composite Material Fabrication A damping material was prepared, in which fibrous polyethylene terephthalate (PET) fine particles were compounded using natural rubber (NR). The NR used was Ribbed Smoked Sheet Grade 1. As a compounding agent for rubber, stearic acid, zinc oxide, sulfur, and a vulcanization accelerator (sulfenamide-type accelerator BBS) were used. The compounding ratio was based on the standard formulation (pure rubber compound 2) described in JIS K 6352, as listed in Table 14.1. We prepared the PET fine particles with a fiber length of 3 mm. For the composite of PET fine particles and NR, a closed type mixer (Laboplast Mill 10C 100 type, manufactured by Toyo Seiki Seisakusho Co., Ltd.) was used. Using an open roll (manufactured by Nippon Roll Co., Ltd.), a rubber sheet was produced from the obtained composite rubber. Consequently, compression and shear flows were applied to the sheet, such that the fine particles are oriented in the sheetdischarging orientation (referred to as 0◦ orientation). This sheet was vulcanized at 150 ◦ C for about 10 minutes using a compression molding machine to prepare vibration damping material sheet having thickness 2 mm. A test piece of width 5 mm was carved from this sheet. Three types of test pieces were fabricated: NR only and composites with 0◦ - and 90◦ -oriented fibrous particles.

14.3 Dynamic Viscoelasticity Test 14.3.1 Measurement Method The storage elastic modulus E , loss elastic modulus E, and loss coefficient η were evaluated using a dynamic viscoelasticity measuring device (Rheogel-E 4000 HP, manufactured by UBM). The test piece had a distance between the clamps of 20 mm. The excitation frequency was 10 Hz, the room temperature was in the range 24–26 ◦ C, and the initial strain was 10% of the distance between the chucks. E , E, and η were measured when the strain amplitude was varied under the aforementioned conditions.

14 Three-Dimensional Strain Calculation of Rubber Composite with Fiber. . . NR + Fibrous PET (0˚)

NR + Fibrous PET (90˚)

109

NR only

Loss factor[-]

0.035

0.025

0.015 0

0.5

1

1.5 2 Strain [%]

2.5

3

3.5

Fig. 14.1 Dynamic viscoelasticity test results comparing the loss factor and showing damping characteristics with varying strain amplitude

14.3.2 Measurement Result Figure 14.1 shows the comparison of the loss factors under different compounding conditions. It can be seen that the loss factor is high when the particles are compounded. In the NR and the test pieces having 90◦ -oriented fibrous particles, the loss factor remains constant and uninfluenced by the strain amplitude. Additionally, the test piece having 0◦ -oriented fibrous particles exhibits a greater loss factor than that with 90◦ -oriented fibrous particles, and the former’s strain amplitude dependency is remarkable. Therefore, it can be said that when fibrous particles are compounded, the strain amplitude dependency becomes stronger as the strain amplitude direction coincides with the longitudinal direction of the fiber.

14.4 Calculation of 3D Strain Distribution We have hypothesized that the increase of strain locally promotes more energy dissipation, thereby serving as a factor for the improvement of the damping characteristics. Energy dissipation is due to the strain energy which is obtained from the square of the strain. If there is a region that is locally distorted than if the entire material is distorted uniformly, the resulting strain energy is higher. Therefore, a large local strain is considered to be a factor that improves the attenuation characteristics. Verification was attempted by calculating the 3D strain distribution in the test piece using X-ray computed tomography (CT) images.

14.4.1 X-Ray CT Method CT achieves nondestructive and 3D measurement and visualization of the interior of a test piece by utilizing the variation in the X-ray absorption coefficient of a substance. The images were captured at the large synchrotron radiation facility

110

M. Matsubars et al.

Fig. 14.2 CT setup in SPring-8 Fig. 14.3 Illustration of the reconstruction procedure of a CT image

Tensile load Z

Test piece

Y X

x-y section image 0.5 m

2048 layers

x-z section image Fixed end

SPring-8 in Japan. The photographs were obtained before and after the application of a tensile load. Figure 14.2 shows a schematic diagram when the photographs were being captured. These photographs were obtained while rotating the test piece on the rotating stage, and each image was acquired by reconstructing the acquired photographic data. Figure 14.3 depicts an outline of the 3D CT image. The data acquired were 0.5 [μm] per 1 [pixel], the tensile direction was along the z-axis, and the plane perpendicular to the z-axis was the xy-plane. By performing image reconstruction, it was possible to generate 2048 layers of xy-plane images having 2048 × 2048 pixels. The center of the xy-plane image was carved with 1000 × 1000 pixels and images of the xz-plane and yz-plane of 1000 × 2048 pixels were produced with 1000 layers. Filtered convolution back projection [5] was used for image reconstruction.

14.4.2 Strain Distribution Calculation Method Using 3D data obtained by X-ray CT before and after the application of the tensile load, we attempted to calculate the 3D strain distribution by tracking feature points

14 Three-Dimensional Strain Calculation of Rubber Composite with Fiber. . .

111

in the test piece. The strain distribution was calculated for composites with 0◦ - and 90◦ -oriented fibrous particles. First, the tracking range of feature points is arbitrarily specified. A cross-sectional image at the top and bottom of the range visually searched such that the same cross section is obtained before and after tensile load application. The feature point is a high-brightness point in the image, and it arbitrarily determines the range of values determined as high brightness. Subsequently, binarization processing is performed using the determined threshold of brightness to extract feature points and thereafter, quantitative analysis is performed in the specified tracking range. Next, the same feature points before and after deformation are matched, and a match list of feature points is created to perform the tracking. Using this match list and feature point data quantitatively analyzed, it is possible to calculate the amount by which a feature point moved owing to the application of a tensile load, thereby enabling strain calculation. Strain can be calculated strain in each axial direction, equivalent strain, and main strain 1, 2, and 3. The calculation results can be plotted on a 3D coordinates to create a 3D strain distribution.

14.4.3 Strain Distribution Calculation Result Table 14.2 presents the calculation results of equivalent strain using the feature point tracking method described in the previous section for both 0 ◦ and 90 ◦ orientations. Moreover, the whole of strain value was divided by the average value of strain, followed by normalization and comparison. The strain plot of each orientation is shown in Fig. 14.4. The histogram of strain frequency is shown in Fig. 14.5. Figure 14.4 shows that there is both a small and large strain in the 0 ◦ orientation and a large strain region locally. On the other hand, the 90 ◦ orientation shows almost uniform distribution. Furthermore, from the normalized histogram of equivalent strain in Fig. 14.5, the 0 ◦ orientation has a wide and nonuniform distribution. It can be seen that the 90 ◦ orientation has a narrow and uniform distribution. From both figures, it was confirmed that large distortion occurred locally in the 0 ◦ orientation. Table 14.2 Calculation result of equivalent strain

Fiber orientation Minimum value Maximum value Average value

0◦ 0.0202 1.3240 0.1055

90◦ 0.0174 0.7605 0.0360

112

M. Matsubars et al. 0

2

z

z

x

x

y

y

(a) 0° orientation

(b) 90° orientation

Fig. 14.4 Three-dimensional plot of normalized equivalent strain distribution

Frequency

3000

2000 0 deg orientation

1000

0

90 deg orientation

0

1

2

3

Normalized equivalent strain [-]

Fig. 14.5 Histogram of normalized equivalent strain

14.5 Conclusion In this study, the interior of the composite material was studied using X-ray CT. Moreover, an attempt was made to verify that more energy is dissipated by increasing the strain locally by compounding the filler. 3D strain was calculated by tracking feature points inside the test piece from X-ray CT data. By creating a 3D plot image of strain distribution, it was confirmed that there are sparsely located regions where the strain is extremely large and small inside the material in the test piece having 0◦ -oriented fibrous particles. In addition, it was found that the distribution of the strain value was wide and sparsely even when compared with the one with 90◦ -oriented fibrous particles. As the test piece with 0◦ -oriented fibrous particles with sparse distribution of strain values showed high damping

14 Three-Dimensional Strain Calculation of Rubber Composite with Fiber. . .

113

characteristics during the dynamic viscoelasticity test, the hypothesis that local strain generation promotes more energy dissipation is therefore correct. Acknowledgments This work was supported by JSPS KAKENHI (grant numbers 16 K 18041 and 18 K 13715). This work was also financially supported by an advanced technological research project conducted by the Research and Development Center for Advanced Composite Materials of Doshisha University and a MEXT (the Ministry of Education, Culture, Sports, Science and Technology, Japan)-supported Program for the Strategic Research Foundation at Private Universities (2013-2017, the project S1311036).

References 1. Muraoka, K., Ohta, T., Ohkubo, H., Yagi, N., Masuda, T.: Effects of deformation history on viscoelasticity of filled rubber compounds. Soc. Rubbers Sci. Technol. Japan. 76(11), 405–409 (2003) 2. Shohara, K., Uotani, K., Yamane, H.: Influence of surface treatment and crosslinking density on the rheological and mechanical properties of particle filled elastomers. Soc. Rheol. Japan. 32(2), 79–84 (2004) 3. Ranganathan, R., Ozisik, R., Keblinski, P.: Viscoelastic damping in crystalline composites: A molecular dynamics study. Compos. Part B. 93, 273–279 (2016) 4. Kulhavy, P., Petru, M., Syrovatkova, M.: Possibilities of the additional damping of unidirectional fiber composites by implementation of viscoelastic neoprene and rubber layers. Shock. Vib. 2017, 1–15 (2017) 5. Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. Society of Industrial and Applied Mathematics, Philadelphia (2001)

Chapter 15

Experimental Study on a Passive Vibration Isolator Utilizing Dynamic Characteristics of a Post-Buckled Shape Memory Alloy Takumi Sasaki and Yuta Kimura

15.1 Introduction Passive isolators remain the most commonly adopted form of isolator due to their simplicity, stability, and low cost. A key trade-off in the selection of most passive mounts (isolators for vertical direction) is their static load bearing capacity, requiring high static stiffness, versus their isolation bandwidth that requires dynamically low stiffness. These conflicting requirements can be reconciled by designing a softening nonlinearity into the force-deflection curve. In previous studies, in order to realize such nonlinearity, many systems have been proposed [1–3]. One commonly used technique makes use of a post-buckled beam. In this study, we proposed a passive vibration isolator for vertical direction using a post-buckled SMA beam and experimentally investigated the isolation performance of the proposed system. By utilizing the negative stiffness characteristics of a post-buckled SMA that had been newly detected by the authors, the fundamental natural frequency could be considerably lowered and isolation performance improved while keeping the structure of the isolator simple and small. The details of the isolator and its measured motion transmissibility are presented.

T. Sasaki () · Y. Kimura University of Kitakyushu, Kitakyushu, 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/10.1007/978-3-030-48153-7_15

115

116

T. Sasaki and Y. Kimura

15.2 Conceptual Design of the Isolator and Experimental Realization The passive vibration isolator investigated in this study incorporated a post-buckled shape memory alloy (SMA) beam, a coil spring and constraint elements (linear guides) that constrained the isolated object to vertical motion. The conceptual schematic of the isolator is shown in Fig.15.1. The SMA beam was oriented vertically with one end fixed to a base and the weight of the isolated object acting at the other end. If the weight of the isolated object is larger than the buckling load of the SMA beam, the SMA beam will buckle. In the post-buckled state, it will exhibit negative tangent stiffness (the gradient of the force-displacement curve becomes negative), a phenomenon that was previously observed by the authors [4]. By installing in parallel to the post-buckled beam a coil spring whose spring constant just cancels the negative tangent stiffness of the post-buckled SMA beam, in principle, quasi-zero-stiffness (QZS) will be achieved. Figure 15.2 shows a photograph of the experimental setup. Due to consideration of rig fabrication, the experimental test rig was slightly different from the conceptual model shown in Fig. 15.1. Figure 15.2 shows that two SMA beams were used and the coil spring acted as a tension spring for supporting the isolated object. However, the dynamic effects of these elements were the same as those in Fig. 15.1. The “dummy” isolated object was mounted on two post-buckled SMA beams and the coil spring was installed between the isolated object and the upper common base frame. A linear guide was used as a constraint element. The base frame was mounted rigidly to an electrodynamic shaker that applied vertical harmonic motion to the isolator. The SMA beams could be heated by Joule heat that was generated by applying electric current directly to the SMA beam. The temperature of the SMA beams was monitored by a thermocouple set at the middle of the SMA beam and kept

Constraint element

Isolated object

Coil spring

Post-buckled SMA beam

Base Fig. 15.1 The conceptual schematic of the isolator using a post-buckled SMA beam

15 Experimental Study on a Passive Vibration Isolator Utilizing Dynamic. . .

117

Coil spring Dummy of an isolated object

Linear guide

Post-buckled SMA beam Common base frame Electrodynamic shaker Fig. 15.2 Photograph of an experimental setup showing the dummy of an isolated object is mounted on the post-buckled SMA beam and the coil spring, which is installed between the isolated object and the upper common base frame Table 15.1 Specification of SMA used in the experiment

Material composition Heat treatment condition

Dimension

Ti-50.2at%Ni Temperature Time Cooling condition Width Thickness Length

480 [◦ C] 40 [min] Water cooing 5.5 [mm] 0.5 [mm] 110 [mm]

to a set temperature by controlling the electric current with a PID control system (in Fig.15.2, the thermocouple is detached). A titanium-nickel-based SMA, with specifications summarized in Table 15.1, was used for the post-buckled beams in the test rig.

15.3 Static Restoring Force of Post-Buckled SMA Firstly, the static force-displacement curve of post-buckled SMA beam was measured. In this experiment, the isolated object and the coil spring were not installed in the rig. A load cell was set on a servo linear slider and a quasi-static longitudinal compressive displacement was imposed at one end of the SMA beam. The displacement was measured using a laser displacement meter and the force was measured using the load cell. The temperature of the SMA was monitored by a thermocouple set at the middle of the SMA beam. Measured force-displacement curves of the post-buckled SMA at 30 ◦ C, 40 ◦ C, 50 ◦ C, 60 ◦ C, and 70 ◦ C are shown in Fig. 15.3.

T. Sasaki and Y. Kimura

Static restoring force [N]

118

20

Compression process 15

Shape recovery process : 70 : 60 : 50

10 0

: 40 : 30

5 10 Longitudinal displacement ratio x / L [ 㸣]

Fig. 15.3 Force-displacement curve of post-buckled SMA

The ordinate and the abscissa show the static restoring force and the normalized longitudinal displacement x/L at the end of the beam (where L is the beam length). From this figure, the following characteristics were confirmed: (i) The static restoring force of the post-buckled SMA varied depending on the temperature of the SMA beam. (ii) The hysteretic characteristic of the static restoring force, that is, the static restoring force in the compression process and the shape recovery process were different. (iii) In some post-bucked states, the gradient of the force-displacement curve of SMA beam became negative; that is, the post-buckled SMA beam showed negative tangent stiffness. Therefore, by utilizing the negative tangent stiffness, a passive vibration isolator with quasi-zero-stiffness (QZS) will be achieved.

15.4 Transmissibility of the Isolator The isolator with an isolated object mass of 2.494 kg and a coil spring with stiffness 0.083 N/mm was mounted on an electrodynamic shaker and subjected to small amplitude harmonic base motion in order to measure the motion transmissibility

Transmissibility [dB]

15 Experimental Study on a Passive Vibration Isolator Utilizing Dynamic. . .

0

119

Applied harmonic excitation ; 2 5m/s (Acceleration amplitude)

–20 Applied harmonic excitation ; 0.5mm (Displacement amplitude)

–40 10

0

1

10 Frequency [Hz]

10

2

Fig. 15.4 Motion transmissibility of the isolator using a post-buckled SMA beam

of the system. Displacement amplitude of 0.5 mm was applied to the base frame in the frequency range from 2 Hz to 16 Hz and acceleration amplitude of 5.0 m/s2 was applied in the frequency range from 17 Hz to 100 Hz. The motions of the isolated object and base were measured using accelerometers. The measured acceleration signals were converted into the frequency domain with FFT and the frequency component of base excitation frequency was used for calculating the transmissibility. The temperature of the SMA beams was controlled to be 70 ◦ C and the system was stabilized at a static equilibrium position of x = 11 mm. (These conditions produce SMA negative tangent stiffness as shown in Fig. 15.3.) Measured transmissibility of the isolator indicates that the first resonance of the system occurs at about 3 Hz (Fig. 15.4.). This corresponds to the first natural frequency of the system and it is supposed that QZS was not realized in this experiment owing to miss tuning of the spring constant or error in temperature control of the SMA. The benefit of the negative stiffness was confirmed by comparing to the natural frequency of the linear system, with 11 mm static displacement due to gravity, which is 4.75 Hz according to Eq. (15.1). 1 fn = 2π



g xst

(15.1)

Beyond 3 Hz, no other resonance appeared in the frequency range less than 100 Hz. However, beyond 20 Hz the transmissibility was not lowered and stayed at about −30 dB. Investigation to lower the first natural frequency and transmissibility beyond 20 Hz should be conducted for future optimization of the isolator.

120

T. Sasaki and Y. Kimura

15.5 Conclusion and Future Work This paper investigated a passive vibration isolator using a post-buckled SMA beam. The static restoring force of the post-buckled SMA beam was measured and special characteristics for post-buckled SMA were confirmed. By utilizing these characteristics, we designed and made an experimental test rig of the proposed isolator and measured the motion transmissibility experimentally. The potential benefit of the negative stiffness of a post-buckled SMA beam was confirmed. However, it was compromised by miss tuning of QZS, resulting in the positive stiffness of the coil spring not being sufficiently canceled out by the negative stiffness of the SMA beam. In addition, the transmissibility in the frequency range beyond 20 Hz was not lowered as much as expected. Improving these phenomena needs to be addressed in future work. Acknowledgment This work was supported by JSPS Grant-in-Aid for Scientific Research (C) number 18 K04025.

References 1. Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314, 371–452 (2008) 2. Sasaki, T., Waters, T.P.: Analysis of a vibration isolation table comprising post-buckled shaped beam isolators. J. Phys. Conf. Ser. 744(1), 1–8 (2016) 3. Chiyojima, T., Sasaki, T.: Development of a passive vibration isolator using a post-buckled -shaped beam (Improvement of the isolation performance by considering mode shapes of a -shaped beam). APVC2017 conference Nanjing, China, 11–16 Nov 2017 (2017) 4. Akamatsu, T., Sasaki, T., et al.: Study on a passive vibration isolator for vertical direction using post-buckled shape memory alloy beam, MoViC2018 conference Daejeon, Korea, 6–8 Aug 2018 (2018)

Chapter 16

Vibration Reduction of a Composite Plate with Inertial Nonlinear Energy Sink Hong-Yan Chen, Hu Ding, and Li-Qun Chen

16.1 Introduction Composite plate structures are very common in the life and the engineering, which will inevitably produce excessive resonance leading to the structural damage. Considering the vibration control of composite plate structures is a hot issue in the engineering field. Vakakis [1] proposed a nonlinear energy sink (NES), which transfers energy from the primary structure to the NES [2]. The nonlinear vibration absorber can reduce the vibration in a wide frequency band [3]. At present, NES has become an important research content in the civil engineering [4], aerospace [5], machinery [6], and vibration energy acquisition [7]. These studies show that the nonlinear energy sink has become a promising vibration absorber. Generally, the mass of the NES provides the inertance of the absorber, but the composite structure has a strict requirement for lightweight; otherwise it will affect the safety of the system. Smith [8] presented an inertial element named inerter. It can provide inertia many times greater than its mass [9]. Zhang [10] put forward an inertial NES. In this chapter, the combination of the NES and the inertial NES has played a very good role in vibration reduction. However, this chapter only deals with a discrete system. The concept of inertial NES can also be extended to continuous systems. In this chapter, the inertial NES is connected to the composite plate. The vibration response of the inertial NES system is compared with that of the system without the

H.-Y. Chen · 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] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_16

121

122

H.-Y. Chen et al.

z o

x F

y NES

k

c

inerter

Fig. 16.1 Composite plate with an inertial NES

NES. The results show that the inertial NES can eliminate the vibration of all modes more effectively with a lighter added mass.

16.2 Dynamic Model The simply supported composite plate considering a NES is depicted in Fig. 16.1 with the length a, the width b, and the thickness h. Displacement of an arbitrary point is derived by using the higher-order shear deformation theory [11] as: u (x, y, z, t) = u0 (x, y, t) + zθα (x, y, t) + z2 u∗0 (x, y, t) + z3 θα∗ (x, y, t) , v (x, y, z, t) = v0 (x, y, t) + zθβ (x, y, t) + z2 v0∗ (x, y, t) + z3 θβ∗ (x, y, t) , w (x, y, z, t) = w0 (x, y, t) , (16.1) where u, v, and w represent displacements in x, y, and z directions, respectively. u0 , v0 , and w0 are displacements of the midplane. The normal and shear strains and normal and shear stresses of the composite plate could be referred to Ref. [12]. The governing equation of the plate can be obtained by using the Hamilton’s principle: 

t2

(δU + δW − δK) dt = 0,

(16.2)

t1

Then the motion equations of the composite plate with NES can be arranged:

16 Vibration Reduction of a Composite Plate with Inertial Nonlinear Energy Sink ∂Nxy ∂Nxx ¨ 0 + I1 θ¨α + I2 u¨ ∗0 + I3 θ¨α∗ , ∂y + ∂x = I0 u ∂Nxy ∂Nyy ∗ ¨ ¨∗ ∂x + ∂y = I0 v¨0 + I1 θβ + I2 v¨0 + I3 θβ , 2 2 ∂Nyz 2Nxy ∂ w0 N ∂ w0 ∂Nxz ∂w0 ∂Nxy ∂w0 ∂Nyy ∂w0 ∂Nxy + yy ∂y + ∂x∂y ∂y∂y + ∂x ∂y + ∂y ∂y + ∂y ∂x + ∂x 2 ∂ w0 ∂w0 ∂Nxx + Nxx ˙ 0 − p (w0 , wN ) + F cos ( t) = I0 w¨ 0 , ∂x∂x + ∂x ∂x + μw ∂Mxy ∂Mxx ¨ 0 + I2 θ¨α + I3 u¨ ∗0 + I4 θ¨α∗ , ∂y − Nxz + ∂x = I1 u ∂Mxy ∂Myy ∗ ¨ ¨∗ ∂x − Nyz + ∂y = I1 v¨0 + I2 θβ + I3 v¨0 + I4 θβ , ∂Oxy ∂Oxx ¨ 0 + I3 θ¨α + I4 u¨ ∗0 + I5 θ¨α∗ , ∂y − 2Mxz + ∂x = I2 u ∂Oxy ∂Oyy ∗ ¨ ¨∗ ∂x − 2Myz + ∂y = I2 v¨0 + I3 θβ + I4 v¨0 + I5 θβ , ∂Pxy ∂Pxx ¨ 0 + I4 θ¨α + I5 u¨ ∗0 + I6 θ¨α∗ , ∂y − 3Oxz + ∂x = I3 u ∂Pxy ∂P + ∂yyy = I3 v¨0 + I4 θ¨β + I5 v¨0∗ + I6 θ¨β∗ , ∂x − 3O  yz  M w¨ N + k(wN − w0 )3 + c (w˙ N − w˙ 0 ) = 0,

123

(16.3)

The force resultant and the moment of inertia of the plate can be written as [11]. Based on the dual-trigonometric series, displacements can be assumed as follows: u0 = w0 = θβ = v0∗

=

θβ∗ =

m ˜  n˜  m=1 n=1 m ˜  n˜ 

u0i (t) cos mπa x sin nπy b , v0 =

m=1 n=1 m ˜  n˜ 

w0i (t) sin mπa x sin nπy b , θα =

φ0i (t) sin

m=1 n=1 m ˜  n˜  m=1 n=1 m ˜  n˜  m=1 n=1

mπ x a

∗ (t) sin mπ x v0i a

cos

nπy b ,

u∗0

=

cos

nπy b ,

θα∗

=

m ˜  n˜  m=1 n=1 m ˜  n˜ 

v0i (t) sin mπa x cos nπy b ,

m=1 n=1 m ˜  n˜  m=1 n=1 m ˜  n˜  m=1 n=1

θ0i (t) cos mπa x sin nπy b , u∗0i (t) cos mπa x sin nπy b ,

∗ (t) cos mπ x sin nπy , θ0i a b

∗ (t) sin mπ x cos nπy , φ0i a b

(16.4)

The nonlinear equation of the system is as follows: η11 w¨ 0i + η12 w0i − η13 f (t) + η14 μw˙ 0i  3 m ˜  n˜  +k w0i (t)Φ (m, n) · δ (x0 , y0 ) − wN · Φi (m, n) δ (x0 , y0 ) m=1 n=1   m ˜  n˜  w˙ 0i (t)Φ (m, n) · δ (x0 , y0 ) − w˙ N · Φi (m, n) δ (x0 , y0 ) = 0, +c m=1 n=1



m ˜  n˜ 

3

M w¨ N + k wN − w0i (t)Φ (m, n) · δ (x0 , y0 ) m=1 n=1   m ˜  n˜  +c w˙ N − w˙ 0i (t)Φ (m, n) · δ (x0 , y0 ) = 0. m=1 n=1

(16.5)

124

H.-Y. Chen et al.

Fig. 16.2 Comparison with two methods

where, k, c, and M denote the stiffness, damping, and inertance of the NES and wij and wN denote the amplitude of the plate and the NES, f (t) and μ denote the excitation and damping of the plate, ηij are the coefficient terms, and Φ(m, n) is the mode shape function. In order to verify the result by the RK method [13], the HBM method [14, 15] is employed in Fig. 16.2. It can be observed that analytical solutions have a good agreement with the simulation.

16.3 Numerical Simulations and Analysis The material properties and geometry parameters are introduced in Table 16.1 and Table 16.2. Epoxy and graphene nanoplatelets (GPLs) are chosen as the polymer matrix and the reinforcements [16].

16.3.1 The Mode of the Plate The model shapes of the plate are shown as in Fig.16.3. The coordinates of black solid points represent the locations of the large deformation of each mode.

16 Vibration Reduction of a Composite Plate with Inertial Nonlinear Energy Sink

125

Table 16.1 Geometry parameters of the plate and the NES Plate Item Length Width Height Damping

Notation a b h μ

Value 1m 1m 0.01 m 15 N s/m

NES Item Excitation Damping Stiffness Inertance

Notation F c k M

Value 100 N 50 N s/m 1•107 N/m3 1.2 kg

Table 16.2 Material parameters of the plate Matrix Item Young’s modulus Density Poisson ratio

Notation EM ρM νM

(a)

(c)

Value 3.0 GPa 1.2 g/cm3 0.34

Reinforcement Item Young’s modulus Density Poisson ratio

Notation EG ρG νG

Value 1.01 TPa 1.06 g/cm3 0.186

(b)

(d)

Fig. 16.3 Modal shapes of the plate: (a) the 3-D surf plot at ω1 = 110.56 rad/s, (b) the 2-D contour plot at ω1 = 110.56 rad/s, (c) the 3-D surf plot at ω2 = 551.96 rad/s, (d) the 2-D contour plot at ω2 = 551.96 rad/s

126

H.-Y. Chen et al.

16.3.2 The Location and Number of the NES Figure 16.4 and 16.5 show the vibration reduction effect of NES with different positions and different numbers. Connect the NES at a location where large deformations occur in each mode. The relationship between the location of the NES and the modal shapes of the plate can be seen. The NES can effectively absorb various modes and achieve good vibration reduction effect. The effect of different locations is depicted in Fig. 16.4. It is observed that the efficiency becomes better when the NES is closing to the middle point. Besides, the influence of the NES on the vibration of the plate is almost identical at the symmetrical position of the midpoint. Different numbers of the NES are discussed based on certain locations in Fig. 16.5. 1 NES, 2 NESs, and 3 NESs mean fixing NESs at the point y0 = 0.5 m,

(a)

(b)

Fig. 16.4 Vibration reduction under different locations of the NES: (a) ω1 = 110.56 rad/s, (b) ω2 = 551.96 rad/s

(a)

(b)

Fig. 16.5 Vibration reduction under different numbers of NES: (a) ω1 = 110.56 rad/s, (b) ω2 = 551.96 rad/s

16 Vibration Reduction of a Composite Plate with Inertial Nonlinear Energy Sink

127

the points y0 = 0.17 m and y0 = 0.5 m, and the points y0 = 0.17 m, y0 = 0.5 m, and y0 = 0.83 m, respectively. It is quite natural that the efficiency increases when the number of the NES increases. As a result, the appropriate number of the NES should be adopted according to the need for the engineering design.

16.4 Conclusion For composite plate, there are many concentrated modes. In this chapter, the inertial NESs are suspended at the large deformation of the composite plate structure, which can effectively reduce the vibration of the structure. It is found that when the NES hanging at the midpoint, the performance is the best. The more NES, the better vibration control effect. However, it also increases the mass of the system. Therefore, the appropriate number of the NES should be set according to the engineering requirements.

References 1. Vakakis, A.F.: Inducing passive nonlinear energy sinks in vibrating systems. J. Vib. Acoust. 123, 324–332 (2001) 2. Gourc, E., Seguy, S., Michon, G., Berlioz, A., Mann, B.P.: Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink. J. Sound Vib. 355, 392–406 (2015) 3. Georgiades, F., Vakakis, A.F.: Dynamics of a linear beam with an attached local nonlinear energy sink. Commun. Nonlinear Sci. 12, 643–651 (2007) 4. Wierschem, N.E., Hubbard, S.A., Luo, J., Fahnestock, L.A., Spencer, B.F., McFarland, D.M., Quinn, D.D., Vakakis, A.F., Bergman, L.A.: Response attenuation in a large-scale structure subjected to blast excitation utilizing a system of essentially nonlinear vibration absorbers. J. Sound Vib. 389, 52–72 (2017) 5. Zhang, W.F., Liu, Y., Cao, S.L., Chen, J.H., Zhang, Z.X., Zhang, J.Z.: Targeted energy transfer between 2-D wing and nonlinear energy sinks and their dynamic behaviors. Nonlinear Dyn. 90, 1841–1850 (2017) 6. Bab, S., Khadem, S.E., Shahgholi, M., Abbasi, A.: Vibration attenuation of a continuous rotorblisk-journal bearing system employing smooth nonlinear energy sinks. Mech. Syst. Signal Process. 84, 128–157 (2017) 7. Pennisi, G., Mann, B.P., Naclerio, N., Stephan, C., Michon, G.: Design and experimental study of a nonlinear energy sink coupled to an electromagnetic energy harvester. J. Sound Vib. 437, 340–357 (2018) 8. Smith, M.C.: Synthesis of mechanical networks: The inerter. IEEE Trans. Autom. Control. 47, 1648–1662 (2002) 9. Chen, M.Z.Q., Papageorgiou, C., Scheibe, F., Wang, F.C., Smith, M.C.: The missing mechanical circuit element. IEEE Circuits Syst. Mag. 9, 10–26 (2009) 10. Zhang, Z., Lu, Z.Q., Ding, H., Chen, L.Q.: An inertial nonlinear energy sink. J. Sound Vib. 450, 199–213 (2019) 11. Wang, A.W., Chen, H.Y., Hao, Y.X., Zhang, W.: Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets. Results Phys. 9, 550–559 (2018)

128

H.-Y. Chen et al.

12. Song, M.T., Yang, J., Kitipornchai, S.: Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos. Part B-Eng. 134, 106–113 (2018) 13. Zhang, Y.W., Zhang, Z., Chen, L.Q., Yang, T.Z., Fang, B., Zang, J.: Impulse-induced vibration suppression of an axially moving beam with parallel nonlinear energy sinks. Nonlinear Dyn. 82, 61–71 (2015) 14. Luo, A.C.J., Huang, J.Z.: Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. J. Vib. Control. 18, 1661–1674 (2012) 15. Dai, H.H., Jing, X.J., Wang, Y., Yue, X.K., Yuan, J.P.: Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mech. Syst. Signal Pr. 105, 214–240 (2018) 16. Li, X.Q., Song, M.T., Yang, J., Kitipornchai, S.: Primary and secondary resonances of functionally graded graphene platelet-reinforced nanocomposite beams. Nonlinear Dyn. (2019)

Chapter 17

Vibration Analysis of Harmonically Excited Antisymmetric Cross-Ply and Angle-Ply Laminated Composite Plates Chendi Zhu and Jian Yang

17.1 Introduction Laminated composite plates have been widely used in engineering structures such as aircraft wings and wind turbines because of the lightweight, high strength-to-weight ratio and high stiffness-to-weight ratio. To exploit the potential benefits of such structures, it is necessary to have a deep understanding of the vibration behaviour. Much work has been done on the free vibration behaviour of conventional composite plates with specific configurations and boundary conditions [1, 2]. Moreover, the free vibration analysis of composite plate with curvilinear fibre laminates has been investigated [3]. Theoretical solutions may be available for the free vibration of symmetric laminated composite plate composited of general orthotropic laminates and that in specific configurations such as antisymmetric cross-ply and angle-ply in certain types of boundary conditions [4]. The angle-ply laminated composite plates provide a very flexible design of the structure by changing the lamination angle. For thin laminated plates, the classical plate theory (CPT) can be employed, while for moderately thick or thick laminates, the first-order shear deformation theories may be used to obtain more accurate results [5, 6]. In engineering applications, composite structures are usually subjected to external loads. When the excitation frequencies are close to the structural resonances, there may be significant increase in the vibration level. This may cause structural failure, destruction of the equipment or very high noise level. The vibration characteristics of composite plates under harmonic loading have not been fully revealed. Moreover, the effects of structure parameters such as fibre orientations and stacking sequence on the dynamic behaviour still need further exploration. In view of this, this chapter investigates the vibration

C. Zhu · J. Yang () Department of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham Ningbo China, Ningbo, People’s Republic of 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/10.1007/978-3-030-48153-7_17

129

130

C. Zhu and J. Yang

behaviour of antisymmetric cross-ply and angle-ply composite plates subjected to a harmonic excitation. The Navier’s solution technique will be employed to obtain dynamic response. Compared with finite element method, the theoretical approach can be used for vibration analysis of laminated composite plates with various fibre orientations at much lower computational cost. Results are presented and discussed for the demonstration of the analysis approach. The influence of important structural parameters on the vibration behaviour will be discussed.

17.2 Mathematical Modelling Figure 17.1 shows a rectangular plate of side lengths of a and b and total thickness h. The plate composes N orthotropic layers perfectly bonded together. Each layer of the plate is made up of linearly elastic orthotropic materials. A harmonic force with amplitude of F and frequency ω is applied to the plate at geometric center of the plate. The fibre orientation θ is defined as the angle measured from the positive direction of OX to the material principle axis of laminates in the k-th layer. According to the FSDT, the displacements and rotations are defined by the following equations as: u (x, y, z, t) = u0 (x, y, t) + zφx (x, y, t) , v (x, y, z, t) = v 0 (x, y, t) + zφy (x, y, t) ,

(17.1)

w (x, y, z, t) = w 0 (x, y, t) , where u, v, and w are the displacements in the OX, OY and OZ directions, respectively, and φ x and φ y are the shear rotations on the OX and OY directions, respectively. The relationship between internals forces or moments and laminate stiffness and the displacements has been previously defined in ref. [1] and used in this chapter. The equations of motion are derived from the Hamilton’s principle. Substituting the corresponding internal forces and moments into the equations of motion, the solutions of the natural frequency and displacements are obtained based on the Navier’s approach.

Fig. 17.1 Plate geometry and layer numbering laminates

17 Vibration Analysis of Harmonically Excited Antisymmetric Cross-Ply. . .

131

For an orthotropic laminate with a single layer or multiple layers symmetrically arranged about the laminate middle surface, they solely have laminate stiffness of A11 , A12 , A22 , A66 , D11 , D12 , D22 , D66 . The procedure of vibration analysis based on the Navier’s approach has been demonstrated in ref. [6]. For antisymmetric cross-ply laminates, they have extensional stiffness A11 , A12 , A22 = A11 ,and A66 ; bending-extension coupling stiffness B11 and B22 = − B11 ; and bending stiffness D11 , D12 , D22 = D11 . Because of existence of the B11 and B22 , the equilibrium differential equations are coupled. In the simply supported edge boundary condition (SS-1) in Nx = v = w = Mx = 0, at x = 0, x = a and Ny = u = w = My = 0, at y = 0, y = b, they can be solved. For antisymmetric angleply laminates, they have A11 , A12 , A22 and A66 , bending-extension coupling stiffness B16 and B26 ; and bending stiffnesses D11 , D12 , D22 = D11 . The dynamic governing equations can be solved for the simply supported edge boundary condition (SS-2) in Nxy = u = w = Mx = 0, at x = 0, x = a and Nxy = v = w = My = 0, at y = 0, y = b [4]. Based on Navier’s approach, the solutions are assumed as: w (x, y, t) =



Wmn (t) sin αx sin βy, m  n ⎫  Umn (t) cos αx sin βy ⎬ u (x, y, t) = m  n  (antisymmetric cross − ply) , Vmn (t) sin αx cos βy ⎭ v (x, y, t) = m  n ⎫  Umn (t) sin αx cos βy ⎬ u (x, y, t) = m  n  (antisymmetric angle − ply) . Vmn (t) cos αx sin βy ⎭ v (x, y, t) = m n

(17.2) ∼

The transverse load q is also expanded in the double-Fourier sine series as: q˜ (x, y, t) =

∞ ∞  

˜ mn (t) sin αx sin βy. Q

(17.3)

m=1 n=1

For a concentrated transverse load located at the point(xe , ye ), the coefficients Qmn are as follows: ∼

∼

4F (t) sin αxe sin βye , Qmn (t) = ab ∼

(17.4)

∼

where F (t) = F 0 eiωt is the harmonic external excitation force with frequency ω ∼

and complex amplitude F 0 . The analytical solution of the natural frequencies can also be obtained from Eq. 17.5:

132

⎛⎡ ⎜⎢ ⎜⎢ ⎜⎢ ⎝⎣

C. Zhu and J. Yang k11 k12 0 k14 k15

k12 k22 0 k24 k25

0 k14 k15 0 k24 k25 k33 k34 k35 k34 k44 k45 k35 k45 k55

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ − ω2 ⎢ ⎦ ⎣

I0 0 0 I0 00 00 00

000 000 I0 0 0 0 I2 0 0 0 I2

⎤⎞ ⎧ ⎪ ⎪ ⎨ ⎥⎟ ⎪ ⎥⎟ ⎥⎟ ⎦⎠ ⎪ ⎪ ⎪ ⎩

Umn Vmn Wmn Amn Bmn

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

=

⎧ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 ⎪ ⎬ Qmn

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ⎪ ⎭

,

0

(17.5) where: k11 =A11 α 2 +A66 β 2 , k12 = (A12 +A66 ) αβ, k22 =A66 α 2 +A22 β 2 , k24 =k15 , ) k14 = B11 α 2 + B66 β 2 , k15 = (B12 + B66 ) αβ for antisymmetric cross-ply, k25 = B66 α 2 + B22 β 2 ) k14 = 2B16 αβ, k15 = B16 α 2 + B26 β 2 for antisymmetric angle-ply, k25 = 2B26 αβ k33 = κA55 (α)2 + κA44 (β)2 , k34 = κA55 α, k35 = κA44 β, k44 = D11 (α)2 + D66 (β)2 + κA55 , k45 = (D12 + D66 ) αβ, k55 = D66 (α)2 + D22 (β)2 + κA44 . (17.6)

17.3 Results and Discussion Here a four-layered laminated composite square plate is considered with dimensions of 0.2m by 0.2m with total thickness of 0.0008 m. The material used is T300/934 CFRP and the material properties are set as E11 = 120Gpa, E22 = E33 = 7.9Gpa, G12 = G13 = 5.5GPa, G23 = 1.58GPa, ν 12 = ν 13 = 0.33, ν 23 = 0.022, and ρ = 1580kg/m3 . The structural damping coefficient is 0.01. The force with the amplitude of 1 N was applied at the centre point (xe = 0.1m, ye = 0.1m). For orthotropic composite laminated plates and antisymmetric cross-ply composite plates, the boundary condition is all edges of simply supported (SS-1). For antisymmetric angle-ply composite plates, the boundary condition is SS-2 as defined in the previous section. Figure 17.2 shows the vibration response of the centre point of the composite plate with all edge of the simply supported (SS-1) in frequency range from 50 Hz to 400 Hz. The solid and dash lines represent the theoretical results of the orthotropic composite plate in [0]4 and antisymmetric cross-ply plate in [0/90]2 , respectively, for comparison. The symbols ‘◦’ and ‘’ represent FEA results from ANSYS simulations. The figure shows that the theoretical results are consistent with numerical ANSYS results. Thus the theoretical approach based on FSDT is verified. For the orthotropic laminated plate, the displacement amplitude reaches maxima at two resonant frequencies corresponding to first and third structural modes, respectively. When the lamination scheme is changed to cross-ply configuration, the number of the resonant peak is reduced to one and the resonant frequency reduces

17 Vibration Analysis of Harmonically Excited Antisymmetric Cross-Ply. . .

133

Fig. 17.2 Displacement amplitude of the central point of the composite laminated plates. —, ◦ ◦ ◦ orthotropic plate in [0 ]4 ; ——-, antisymmetric cross-ply plate in [−0 /90 ]2 ; ‘◦’ and ‘’ represent FEA results

Fig. 17.3 Displacement amplitude of the central point of the antisymmetric angle-ply laminated ◦ ◦ ◦ ◦ ◦ ◦ ◦ composite plates. —, [0 ]4 ; ——-, [−15 /15 ]2 ; -·-·- [−30 /30 ]2 ; - - - - - [−45 /45 ]2

to a lower value. The vibration response at the original second resonant frequencies at 230 Hz of the cross-ply composite plate is much lower than that for the plate ◦ with [0 ]4 case. The figure reveals that the analytical approach can be employed to investigate the vibration response of the plate with different lamination schemes under various loading conditions according to practical requirements. Figure 17.3 presents the vibration response of the antisymmetric angle-ply lami◦ ◦ ◦ ◦ ◦ nated composite plates with lamination scheme of [0 ]4 , [−15 /15 ]2 , [−30 /30 ]2 , ◦ ◦ and [−45 /45 ]2 . The results are obtained from the theoretical method based on FSDT. The boundary conditions have been assumed as SS-2. The figure shows that the first resonant frequencies increase with the fibre orientation angle θ in the form ◦ ◦ of [−θ /θ ]2 . For the plate orientated in [−45 /45 ]2 , the peak value of the vibration response is lower than others. The figure shows that the resonant frequencies are significantly changed by changing the fibre orientation. This is due to the variations of stiffness of the composite plate caused by the change of fibre orientation. When the excitation frequency is at 90 Hz, the response value changes with the fibre angle. When the plate is excited at the second resonant frequencies at 230 Hz,

134

C. Zhu and J. Yang

Fig. 17.4 Effects of fibre orientation of the antisymmetric angle-ply laminated composite plates. (a) First resonant frequency and (b) displacement amplitude of the central point excited at 90Hz. ◦

the vibration response of the plate with [0 ]4 is much larger than that for the plate ◦ ◦ with [−45 /45 ]2 case. Figure 17.4(a) and (b) presents the effects of fibre orientation θ of the antisymmetric angle-ply plate [−θ /θ ]2 on the first resonant frequency ω1 and the vibration response at an excitation frequency of 90 Hz. In Fig. 17.4(a), it is observed that with the increase of the fibre orientation θ , the first resonant frequency ω1 in the frequency range also increases. The resonant frequency increases substantially ◦ ◦ between θ = 10 and θ = 35 , but there is smaller increment beyond that range. The figure demonstrates that at prescribed excitation frequency, the dynamic response amplitude could be reduced by changing the fibre orientation. In Fig. 17.4(b), it is found that the dynamic response amplitude at 90 Hz reduces with the increase ◦ of fibre angle. The displacement response amplitude for plate with θ > 30 is much lower than that for θ = 0 case. This characteristic again shows that the fibre orientation has a large effect on the forced vibration response and thus should be carefully designed in practical applications for vibration suppression.

17.4 Conclusion This chapter investigated the vibration behaviour of harmonically excited laminated composite plates with different lamination schemes including general orthotropic laminated composite plates and antisymmetric cross-ply and angle-ply composite plates. The equations of motion based on FSDT were solved theoretically to obtain the dynamic response of any points on composite plates. The results demonstrated that the resonant peaks of dynamic response can be changed substantially by changing the fibre orientation. Under certain conditions, implementation of the antisymmetric cross-ply configuration could reduce the number of peaks in a prescribed frequency range and shift the resonant frequencies. Furthermore, the effects of fibre orientation of a harmonically forced antisymmetric angle-ply plate on

17 Vibration Analysis of Harmonically Excited Antisymmetric Cross-Ply. . .

135

the resonance frequencies and the vibration response were investigated. The results demonstrated that vibration response of the composite plate could be suppressed by tailoring the fibre orientation at prescribed loading conditions.

References 1. Reddy, J.N.: Mechanics of Laminated Composite Plates: Theory and Analysis. CRC Press, Boca Raton (1997) 2. Khandan, R., Noroozi, S., Sewell, P., Vinney, J.: The development of laminated composite plate theories: A review. J. Mater. Sci. 47, 5901–5910 (2012) 3. Honda, S., Narita, Y.: Natural frequencies and vibration modes of laminated composite plates reinforced with arbitrary curvilinear fiber shape paths. J. Sound Vib. 331, 180–191 (2012) 4. Thai, H.-T., Choi, D.-H.: A simple first order shear deformation theory for laminated composite plates. Compos. Struct. 106, 754–763 (2013) 5. Sayyad, A.S., Ghugal, Y.M.: On the free vibration of angle-ply laminated composite and soft core sandwich plates. J. Sandw. Struct. Mater. 0(00), 1–33 (2016) 6. Zhu, C.D., Yang, J.: Free and forced vibration analysis of composite laminated plates. In: The 26th International Congress on Sound and Vibration 2019 (ICSV26). IIAV, Montreal, Canada (2019)

Chapter 18

Analysis of Influence of Multilayer Ceramic Capacitor Mounting Method on Circuit Board Vibration Dongjoon Kim , Wheejae Kim, Joo Young Yoon, Eunho Lee, and No-Cheol Park

18.1 Introduction Many previous studies have reported that high-pitched noise generated from printed circuit boards (PCBs) such as circuit boards of solid-state drives (SSDs) and smartphones are mainly caused by the vibration of multilayer ceramic capacitors (MLCCs). MLCC consists of ferroelectric material such as barium titanate (BT; BaTiO3 ) and therefore, it has piezoelectric characteristics due to the perovskite structure of BT. So MLCC changes its volume following applied electric field which is generated inside the inner electrodes. When a user applies an AC electric field or a pulse signal, MLCC starts to oscillate and its vibrations are transmitted to the PCB through the solder. When the frequency of the applied electric field coincides with the PCB’s resonance frequency, the PCB starts to vibrate with large amplitudes, which may lead to high-pitched acoustic noise. As MLCC acts like an actuator, it is important to reveal the vibration transfer characteristics from MLCC to PCB. MLCC is a kind of surface-mounting device (SMD), so the solder mount is the main vibration transfer path. Therefore, to minimize the vibration transmission, multiple factors such as mounting direction, angle, and shape of solder should be controlled. In this paper, various mounting conditions of MLCC are studied using the finite element (FE) analysis. Commercial X5R ceramic capacitors were used to build a reliable FE model. Then, influential parameters were selected, and their effects were analyzed by conducting parametric studies.

D. Kim · W. Kim · J. Y. Yoon · E. Lee · N.-C. Park () Yonsei University, Seoul, 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/10.1007/978-3-030-48153-7_18

137

138

D. Kim et al.

18.2 Finite Element Model Construction In previous studies [1, 2], a detailed FE model including the internal and external electrodes was constructed. However, since the test MLCC specimen has a total of 720 layers of internal electrodes and dielectric layers, a detailed model needs many elements and it brings a lot of execution time or CPU time. To overcome these problems, Kim et al. [3] reduced the number of FE model elements by simplifying the inner structures to one solid body. They assumed that MLCCs are almost isotropic in the transverse direction as dielectric layers are polarized by a z-direction. Based on the above assumptions, they applied the equivalent transverse isotropic properties to the inner structure elements. In this study, the same modeling method and material properties were used to construct the bulk MLCC model. The PCB of M.2 SSD was chosen as test specimen for sketching FE model and modal testing. As PCB is also a kind of complex substrate, modal testing was performed to calculate equivalent properties of copper and epoxy resin layers. Also, MPC contact was used to describe the bolt mounting and the connection between connector on PCB and M.2 socket. Below figure represents the constructed MLCC and PCB FE model (Fig. 18.1).

18.3 Numerical Simulations and Analysis In this study, height of the solder, mounting direction, and location of MLCC were chosen as variables for parametric studies. Each variable was set to the design points with a certain range, and the output variable was set as the averaged vibration amplitude at the center point region of the PCB where the largest deformation occurs. First, the effect of height of the solder was studied. The range was set to 80 ~ 120% of reference solder height (0.6 mm) and the results are shown in Fig. 18.2. As explained in [4], the solder close to the head side of MLCC is the main vibration transmission path. It can be explained that the smaller solder height results

Fig. 18.1 Finite element model of (a) test specimen PCB and (b) bulk MLCC

18 Analysis of Influence of Multilayer Ceramic Capacitor Mounting Method. . .

139

Fig. 18.2 Variation of vibration amplitude according to the change of solder height 1

3 9

×10–7 1

10

0.5 0 –10

×10–8 1

10

0.5 5 –8

–6

–4 X

–2

0 0

Mode shape @ 2.3 kHz L mode

Y

0 –10

×10–8 1

10

0.5 5 –8

–6

–4 X

–2

0 0

Mode shape @ 8.4 kHz W mode

Y

0 –10

5 –8

–6

–4 X

–2

0 0

Y

Mode shape @ 17.3 kHz LW mode

Fig. 18.3 Parametric study results for three modes when MLCCs are mounted in horizontal direction

in reduced vibration transmission path and decreased vibration amplitude. In other words, lower solder height has an advantage in acoustic noise reduction. Next, the effects of mounting position and direction were analyzed. As shown in Fig. 18.2, large vibration peaks occur at the resonance frequencies and each peak level differs significantly. It means that the mounting effect should be analyzed considering target peak’s mode shape. For this reason, in this study, three mode shapes were chosen as reference mode shapes considering wave propagation direction: L mode (2.3 kHz), W mode (8.4 kHz), and LW mode (17.3 kHz). The location was set for 9 points and angle was set as 0 or 90 degree. The mode shapes of PCB and results of parametric study are shown in Figs. 18.3 and 18.4. Figure 18.3 represents the results when MLCCs are mounted in horizontal direction (L direction). As L mode’s wave propagation direction coincides with MLCCs’ mounting direction, large vibration amplitude was observed compared to

140

D. Kim et al. 1

3 9

10–8 1

10

10–8 5

10

0.5 0 –10

10–8 1

10

0.5 5 –8

–6

–4 X

–2

0 0

Y

Mode shape @ 2.3 kHz L mode

0 –10

5 –8

–6

–4 X

–2

0 0

Mode shape @ 8.4 kHz W mode

Y

0 –10

5 –8

–6

–4 X

–2

0 0

Y

Mode shape @ 17.3 kHz LW mode

Fig. 18.4 Parametric study results for three modes when MLCCs are mounted in vertical direction

other modes. The largest vibration occurred when MLCCs are mounted at point 7 where the largest deformation occurs in L mode shape. The smallest vibration occurred when MLCCs are mounted at point 6 where the smallest deformation occurs in L mode shape. It means that it is easy to suppress vibration when mounting MLCCs around the nodal points of target mode. These features can be observed in other modes as well, and these characteristics can be summarized as follows. First, mounting MLCC at the nodal point is effective for vibration reduction. However, in this case, the cross correlation between nodal points of several modes should be preanalyzed. Second, mounting MLCC in vertical direction to the wave propagation direction leads to less vibration of PCB. In this case, the wave propagation direction of each mode should be analyzed prior to MLCC mounting.

References 1. Ko, B.H., Jeong, S.G., Ahn, Y.G., Park, K.S., Park, N.C., Park, Y.P.: Analysis of the correlation between acoustic noise and vibration generated by a multi-layer ceramic capacitor. Microsyst. Technol. 20(8–9), 1671–1677 (2014) 2. Wang, Y.Q., Ko, B.H., Jeong, S.G., Park, K.S., Park, N.C., Park, Y.P.: Analysis of the influence of soldering parameters on multi-layer ceramic capacitor vibration. Microsyst. Technol. 21(12), 2565–2571 (2015) 3. Kim, H., Kim, D., Park, N.C., Park, Y.P.: Acoustic noise and vibration analysis of solid state drive induced by multi-layer ceramic capacitors. Microelectron. Reliab. 83, 136–145 (2018) 4. Kim, D., Kim, W., Kim, W.C.: Dynamic analysis of multilayer ceramic capacitor for vibration reduction of printed circuit board. J. Mech. Sci. Technol. 33(4), 1595–1601 (2019)

Part IV

Dynamic Model Updating and System Identification

Chapter 19

Comparison of the Input Identification Methods for the Rigid Structure Mounted on the Elastic Support Shigeru Matsumoto and Takuya Yoshimura

19.1 Introduction In order to reduce vibration, transfer path analysis is commonly used, which evaluates the vibration at the evaluation point as the product of the frequency response function and the excitation force. When transfer path analysis (TPA) is used, structural change guidelines for reducing vibration noise can be obtained by identifying the excitation force and transfer path with large contribution. Although a load cell is used to measure the excitation force, when the internal structure is complicated, it is often difficult to measure directly because the installation space cannot be secured. Therefore, it is necessary to identify the excitation force indirectly. Since the accuracy of TPA depends on the accuracy of input identification, a more accurate method is required. Also, using window function to reduce leakage error may affect input identification. As an input identification method, matrix inversion method is commonly used. In this method, since the propagation of the measurement error by calculation of the inverse matrix cannot be avoided, the accuracy may be insufficient. On the other hand, apparent mass method [1, 2] which has been presented by one of the authors for improving the identification accuracy can avoid that calculation by directly estimating the inverse matrix of the accelerance matrix. In this chapter, two methods of matrix inversion method and apparent mass method are used for input identification method. In the compressor used in this research, the excitation force is generated by rotation of the spiral part (scroll) during operation, but this part is difficult to excite because it exists inside. Therefore, assuming that the compressor is a rigid

S. Matsumoto () · T. Yoshimura Tokyo Metropolitan University, Tokyo, 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/10.1007/978-3-030-48153-7_19

143

144

S. Matsumoto and T. Yoshimura

body, the excitation force on the surface is contracted to the center of gravity, and the excitation force acting on the center of gravity is identified by the frequency response function (FRF) between the center of gravity and the response measurement point. In this chapter, input identification methods for the rigid body mounted on the elastic supported are proposed and verified not only in the frequency domain but also in the time domain. In addition, the effect of the window function on input identification is also verified.

19.2 Theory 19.2.1 Input Identification Method In order to identify the excitation force, the FRF matrix is estimated by the vibration experiment as the first step. Next, the acceleration response during operation is measured, and the excitation force is identified by multiplying it with the FRF matrix estimated by experiment. This section describes two methods, matrix inversion method and apparent mass method.

19.2.1.1

Matrix Inversion Method

In this method, the accelerance (acceleration/force) is used as the FRF. The frequency is ω, the number of excitation points is m, and the number of acceleration measurement points is n. The excitation force is given by Eq. (19.1) and pseudoinverse matrix is given by Eq. (19.2). Where F is the identified excitation force matrix (m x m), Y is the acceleration matrix (n x m), and H is accelerance matrix (n x m). F and Y are Fourier spectrum. From Eq. (19.1), the excitation force is identified by product of pseudo-inverse of accelerance matrix and the error contained in the measurement data may be propagated by this calculation: F (ω) = H (ω)+ Y (ω)  −1 H + = H HH HH

19.2.1.2

(n > m)

(19.1) (19.2)

Apparent Mass Method

Apparent mass method estimates apparent mass (force/acceleration) directly that is equivalent to the inverse of accelerance matrix. By this method, the calculation of the inverse of accelerance matrix is avoided. Therefore, apparent mass method can be expected to improve the accuracy comparing with the inverse method. The excitation force is given by Eq. (19.3), where G is the apparent mass matrix (m x n): F (ω) = G (ω) Y (ω)

(19.3)

19 Comparison of the Input Identification Methods for the Rigid Structure. . .

145

19.2.2 Window Function Since the FFT assumes that the signal is periodic, the signal that contains discontinuity tends to have analysis error. To suppress the discontinuity between the end and the beginning of the time data, the window function is applied before the FFT is conducted. In this chapter, the force window is used for the force signal and the exponential window is used for the acceleration response in the impact testing.

19.2.3 Contraction of the Excitation Force to the Center of Gravity The excitation force is transformed from the excitation point to the center of gravity according to Eq. (19.4) using a transformation matrix. The external force due to hammering is only the translational component. Fx , Fy , Fz , Mx , My , Mz are the components of the excitation force acting on the center of gravity, xi , yi , zi are the relative displacement from the center of gravity to the point i, and Fxi , Fyi , Fzi represent the components of the excitation force at point i. By contracting to the center of gravity, the excitation force has a total of 6 degrees of freedom of force and moment components: ⎧ ⎫ ⎡ ⎪ 1 Fx ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 0 ⎪ ⎪ ⎪ ⎪ F y ⎢ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ Fz ⎢ 0 =⎢ ⎢ 0 ⎪ ⎪ M x⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ zi My ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Mz − yi

0 1 0 −zi 0 xi

0 0 1 yi − xi 0

⎤ ⎥⎧ ⎥ F ⎫ ⎥ ⎨ xi ⎬ ⎥ ⎥ Fyi ⎥⎩ ⎭ ⎥ Fzi ⎦

(19.4)

19.2.4 Frequency Averaged Error The frequency averaged error [3] is an error index obtained from the average of identification errors at frequencies in a specific section and is used to evaluate the identification accuracy. Designating the relative error of the excitation force as εave when the data measured in the excitation experiment is taken as the true value, the frequency averaged error is expressed by Eq. (19.5) where M is number of frequency lines: εave

M 1  = εrel (ω)2 M l=1

(19.5)

146

S. Matsumoto and T. Yoshimura

19.3 Excitation Experiment As shown in Fig.19.1, the compressor is fixed on a steel plate with the rubber inserted in between, and the hammering test was conducted. A total of nine triaxial accelerometers (PCB, 356A32) were installed at points 1 to 9, points 1 to 13 were excited using an impact hammer (PCB, 086B03), and the excitation force and acceleration were measured using the FFT spectrum analyzer (NI PXI-1042). For the points 1 to 9, the vicinity of the sensor was excited in the normal direction, for the points 10 to 12 in the vertical direction and for the point 13 in the circumferential direction of the cylindrical body. Table 19.1 summarizes the experimental conditions. The same excitation experiment was conducted twice to obtain data for estimating FRF and data for input identification. In addition, since the coherence is deteriorated below 5 [Hz], 5–500 [Hz] was targeted for analysis.

19.4 Input Identification Result Input identification was performed in both the frequency domain and the time domain, and in each case the errors with and without the window function were

Fig. 19.1 Compressor and measurement model. (a) Top of Compressor (b) Compressor (c) Compressor model Table 19.1 Table captions should be placed above the tables

Number of samples Measurement time [s] Frequency resolution [Hz] Maximum frequency [Hz] Number of averages

4096 3.2 0.3125 500 5

19 Comparison of the Input Identification Methods for the Rigid Structure. . .

147

Fig. 19.2 Identified force Fx with window (P7)

evaluated and compared. Window function is used in FRF measurement and operation measurement. In this chapter, we show the translational component (Fx ) in the x direction when the point is excited as the identification result.

19.4.1 Input Identification Result in the Frequency Domain The error in the frequency domain is evaluated by averaging over frequency and then averaging over all excitation points. The evaluation is divided into whole range and low and high frequency range. First, Fig.19.2 shows the identification result when the window function is not used. Both methods can be identified with good accuracy, but at 20 [Hz] or less, the estimates are larger than the true value. Apparent mass method tends to estimate smaller than the matrix inversion method because it minimizes the error included in the excitation force. Therefore, in this case, it is considered that they are close to true value. Figure 19.3 shows the error averaged over all excitation points. Figure 19.3 (a) is the translation, (b) is the rotation component, and W, L, and H are whole, low, and high frequency, respectively, range. From (a), although the accuracy of the matrix inversion method is high regardless of the frequency in the translation, that of apparent mass method is high in the rotation. Figure 19.4 shows the identification result when using the window function. Compared with Fig. 19.2, there is no significant change in the apparent mass method, but it was greatly deteriorated in the matrix inversion method. Figure 19.5 shows the comparison of the accuracy with and without the window function. In matrix inversion method, the accuracy is degraded in all cases, but in apparent mass method, that is slightly improved.

148

S. Matsumoto and T. Yoshimura

Error of rotation[-]

Error of translation [-]

0.04 0.03 0.02 0.01 0

Matrix Inversion 0.8 Apparent mass 0.6 0.4 0.2 0

0 W1 L2 H3 4

0 W 1 2L H 3 4

Fig. 19.3 Comparison of error of each method: (a) translation, (b) rotation

Fig. 19.4 Identified force Fx without window (P7)

19.4.2 Input Identification Result in the Time Domain The excitation force in the time domain is identified by inverse Fourier transform of the excitation force identified in the frequency domain. In the time domain, the errors are averaged over time and evaluated as in the frequency domain. Also, it is divided into the whole time and the impact duration (0–0.03 [s]) at which the excitation force is applied. Figure 19.6 shows the identification result of impact duration when the window function is not used. It can be identified with good accuracy, but the error of matrix inversion method is slightly larger. Figure 19.7

19 Comparison of the Input Identification Methods for the Rigid Structure. . .

149

Fig. 19.5 Comparison of error with and without window: (a) matrix inversion, (b) apparent mass

Fig. 19.6 Identified force Fx with window (P7)

shows the error in the time domain. M and A on the horizontal axis are matrix inversion method and apparent mass method, respectively. In the time domain, the accuracy of apparent mass method is high in all cases, and the difference in impact duration is large. Figure 19.8 shows the identification result when the window function is used. The accuracy of the matrix inversion method was deteriorated. Figure 19.9 shows the comparison of the accuracy with and without the window function. Although there is no significant change in the total time, it is greatly deteriorated at the impact duration of the matrix inversion method.

150

S. Matsumoto and T. Yoshimura

0.001 Error of moment[N]

Error of force [N]

0.012 0.008 0.004

0.0008 0.0006 0.0004 0.0002

0

0

M 1 A 2

total me impact duraon

0

3

0

M 1

A 2

3

Fig. 19.7 Comparison of error of each method: (a) translation, (b) rotation

Fig. 19.8 Identified force Fx without window (P7)

0.05

Error of force [N]

Error of moment[N]

0.05 0.04 0.03 0.02 0.01 0

no window window

0.04 0.03 0.02 0.01 0

0

M 1

A 2

3

0

M 1

A 2

3

Fig. 19.9 Comparison of error with and without window: (a) impact duration, (b) total time

19 Comparison of the Input Identification Methods for the Rigid Structure. . .

151

19.4.3 Discussion It is known that Tikhonov’s regularization is a technique for reducing errors in the calculation of pseudo-inverses, but it is only effective when the accelerance matrix is ill-conditioned. In contrast, apparent mass method is superior in that it can improve the accuracy regardless of the accelerance matrix condition because the inverse of accelerance matrix is directly obtained by minimizing the error included in the force.

19.5 Conclusion In the translation of the frequency domain, the input identification estimate by matrix inversion method is superior to the estimate by apparent mass method, but in all other cases, apparent mass is superior to matrix inversion. When the window function was used, the change is slight in the apparent mass method, and it was deteriorated in the matrix inversion method. And in the time domain with window, the change in the total time is small and the accuracy was deteriorated in the impact duration.

References 1. Kobayashi, S., Yoshimura, T.: Identification of excitation force spectra using apparent-mass matrix. The Japan Society of Mechanical Engineers, Series C. 75(753), 1500–1507 (2009) 2. Kubota, H., Yoshimura, T., Fukui, K.: Identification of internal force of a compressor generated in operating. Dynamics & Design Conference. 16-15, 350 (2016) 3. Itofuji, T., Yoshimura, T.: Accuracy improvement of vibrational input identification using dynamic strain measurement. The Japan Society of Mechanical Engineers. 81(829), 14–00698 (2015)

Chapter 20

Iterative Learning Control for Vision-Based Robotic Grasping Yun Lu, Zheng Huang, Hao Wen, and Dongping Jin

20.1 Introduction Repetitive tasks are common in industrial production, such as assembly, grasping, carrying, welding, and so on. Among these tasks, grasping is the most common one and is the first step of many other tasks. Hence, great efforts have been made to develop control schemes of robots to accomplish grasping tasks, such as deep Q-learning [1], convolutional neural network (CNN) [2], and so on. In this paper, the iterative learning control (ILC) method, which is one of the most effective methodologies in solving repeated tracking control problems [3], is proposed to realize robotic grasp. The ILC method, which was proposed and researched for the first time in 1984 [4], makes use of a sequence of trial iterations to generate the control signals in tracking a desired trajectory. The superiority of ILC in trajectory tracking enables it to be applied in many fields. In [5], precise tracking of desired reference trajectories in gasoline engines was realized by multivariable model-based normoptimal parallel ILC strategies. For a manipulator system with uncertain parameters and external disturbances, Liu et al. developed an adaptive ILC approach based on disturbance estimation, which provides rapid convergence of trajectory tracking errors [6]. Not only for industry robots but also mobile robots, ILC was found to be effective in improving the tracking trajectory performance [7, 8]. In general, ILC has a wide range of applications in different domains and is an essential control method in trajectory tracking.

Y. Lu · Z. Huang · 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]; [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_20

153

154

Y. Lu et al.

In this paper, we present a PD closed-loop ILC method which is able to solve the problem of robot grasping effectively. We use the UR5 as the robotic manipulator system to accomplish an easy grasping task. Numerical simulations and hardware experiments are conducted to certify that the proposed ILC method is valid in tracking a given trajectory to the target grasping position. The paper is organized as follows. Section 20.2 presents the kinematic models of UR5. In Sect. 20.3 the PD closed-loop ILC is formulated. The results of numerical simulations and hardware experiments are shown in Sect. 20.4 and Sect. 20.5, respectively. The conclusion is presented in Sect. 20.6, as well as future work.

20.2 Kinematics Models In this work, a UR5 robotic arm from Universal Robots is employed for the grasping task. UR5 arm is a well-known 6-degree-of-freedom robotic manipulator [9], which is famous for its lightweight, speed, easy to program, flexibility, and safety. With the Debavit-Hartenberg parameters obtained from [10], both forward and inverse kinematic models of UR5 can be established. For the forward kinematics, the six joint angles can be used to get the transformation matrix from the base coordinate system to the end-effector coordinate system. The transformation matrix from the joint i-1 to i is given in the following form: ⎤ cos θi − sin θi cos αi sin θi sin αi ai cos θi ⎢ sin θi cos θi cos αi − cos θi sin αi ai sin θi ⎥ ⎥ =⎢ ⎣ 0 sin αi cos αi di ⎦ 0 0 0 1 ⎡

Tii−1

(20.1)

The transformation matrix from the base to the end-effector is obtained in the form of: T = T10 T21 T32 T43 T54 T65

(20.2)

For the inverse kinematics, the pose of the end-effector is known and then a set of joint configurations q = [θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 ] needs to be found. Actually, UR5 has six degrees of freedom, which guarantees that the solutions of the inverse kinematics exist in its workspace. Derivation of the inverse kinematics in this section is adopted from [9]. Generally, the sets of joint configurations obtained from inverse kinematics are not unique, so which one of them should be chosen? To solve the problem, the work follows the design philosophy of limiting the spatial configuration of the manipulator and considering the minimum travel from the initial preset pose to the target pose.

20 Iterative Learning Control for Vision-Based Robotic Grasping

155

20.3 Iterative Learning Control In general, the ILC scheme is effective when a system satisfies: the controlled system is constant, the tracking task is repetitive, and the initial state is invariable. The manipulator system in this paper is considered to satisfy the above three conditions, so designing an ILC law is a nice choice for us to solve the problem of robot grasp. The joint angles of the manipulator here can be regarded as a discrete system, which can be described as: x (k + 1) = x(k) + un (k)Δk

(20.3)

where x ∈ R6 × 1 is the state variable and represents the six joint angles of the arm physically, un is a 6 × 1 vector of controller input and physically denotes the six joint angular velocities, and k is a constant length of the discrete time step. Considering the working environment of the manipulator is complex and full of various uncertain disturbances, a PD closed-loop ILC is selected in this article: * + un+1 (k) = un (k) + K e QP en+1 (k) + [en+1 (k) − en+1 (k − 1)]

(20.4)

where k is the k-th discrete-time node, un (k) is the controller output of the n-th iteration and u0 (k) is set to 0 all the time, Ke is the differential gain, and Ke × QP is the proportional gain of the control law; the error en + 1 (k) of the (n + 1)-th iteration is defined as en + 1 (k) = xd (k) − xn + 1 (k), where xd (k) is the desired joint angles of the robotic arm and xn + 1 (k) is the current joint angles in the (n + 1)-th iteration.

20.4 Simulation In this section, a simulation case is implemented to verify that the UR5 can successfully grasp the target along a given trajectory under the proposed PD closedloop ILC law. We denote two poses of the end-effector of UR5, pose A and pose B:     A = 0.5 0.45 0.35 0.1 −1 −0.2 B = 0.25 0.25 0.05 0 −π/2 0 as the poses of the initial and target points, respectively. In order to obtain the joint angles by using inverse kinematics, we describe the poses with SO(4) transformation matrices by employing the method mentioned in [11]. According to the design principle mentioned in Sect. 20.2, we obtain two sets of joint configurations qA and qB . Thereafter the trajectory of each joint is constructed via Hermite interpolation. The whole grasping process takes 10 s, which is discretized into 101 nodes. By a trial-and-error strategy employed for parameter selection with the aim to improve

156

Y. Lu et al.

Fig. 20.1 The iterative process of the robot system under the PD closed-loop ILC. (a) The first three joints (b) The last three joints

Fig. 20.2 The maximum absolute error of each joint versus iterations

the robustness and convergence rate of the scheme, we get Ke = − 4 × I6 × 6 and QP = 0.2 × I6 × 6 . The iteration is set to stop after the eighth iteration. Figure 20.1 shows the results of the simulation, where the red curve represents the expected trajectories and the blue curve represents the actual trajectories in the first, third, and eighth iteration. The maximum absolute error enmax = max (en (k)) of each joint versus iterations is revealed in Fig. 20.2. It is easy to find that the error of each joint gradually decreases and almost equals to zero after the sixth iteration, which illustrates that the robot has a good performance in tracking the desired trajectories and can successfully reach to the target grasp position.

20 Iterative Learning Control for Vision-Based Robotic Grasping

157

20.5 Experiment Hardware experiments (Fig. 20.3) are also conducted to verify the proposed scheme for tracking a given trajectory to a target position and grasping an object successfully by the PD closed-loop ILC. Prior to the experiment, camera calibration is performed to obtain the transformation matrix between the camera frame and the robot base framecam base T . During the experiment, the pose of the target in the camera frame target cam P is obtained with the camera and then is sent to the computer. On the computer, the pose of the target in the camera frame is transformed to the robot base frame target base P and a set of joint configurations qt corresponding to the target pose is obtained by the same method mentioned in the simulation section. After that, the desired trajectories of each joint are planned from the initial configuration to the target configuration and are tracked by the PD ILC controller. Finally, the control output is transmitted to the robot control box through the workstation to manipulate the robot to the target position. In the experiment, we just keep and utilize the data of the last iteration of the PD ILC controller, as shown in Fig. 20.4, where Fig. 20.4(a) gives the desired trajectories (red line) and actual trajectories (blue dotted line) of the first three joints while Fig. 20.4(b) gives these of the last three joints. It is easy to find that the robot can well track the desired trajectories and the errors are small enough to be ignored, which means that the end-effector of the robot arm can reach the target grasp point successfully.

Fig. 20.3 The hardware experiment platform

158

Y. Lu et al.

Fig. 20.4 Desired trajectories and actual trajectories of six joints in the last iteration of the experiment. (a) the first three joints (b) the last three joints

20.6 Conclusions In this paper, we have presented a PD closed-loop ILC to realize the robot grasping task. We established the forward and inverse kinematics models of a UR5 manipulator to plan the desired trajectory of each joint. Numerical simulations and hardware experiments are performed to demonstrate the outstanding performance of ILC in tracking given trajectories and for grasping tasks, which facilitates the development of the automated industrial system. There are some limitations to our experiment platform that can form directions for future work. Actually, the visual system brings some errors, resulting in a distance of about 1–2 cm between the end-effector and the grasp point on the target. In the future, some methods will be explored to solve this problem, such as online visual feedback. Besides, some other methods may be employed to compare with the ILC method in terms of convergence. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11832005).

References 1. Chen, R., Dai, X.Y.: Robotic grasp control policy with target pre-detection based on deep Q-learning. In: 3rd International Conference on Robotics and Automation Engineering, pp. 29–33. IEEE, Guangzhou (2018) 2. Zhang, H., Lan, X., Zhou, X., Wang, J., Zheng, N.: Vision-based robotic grasp success determination with convolutional neural network. In: 7th Annual International Conference on CYBER Technology in Automation, Control, and Intelligent Systems, pp. 31–36. IEEE, Hawaii (2018) 3. Xu, J.X., Bien, Z.Z.: Iterative Learning Control, 1st edn. Springer, Boston (1998)

20 Iterative Learning Control for Vision-Based Robotic Grasping

159

4. Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of dynamic systems by learning: A new control theory for servomechanism of mechatronics systems. In: 23rd IEEE Conference on Decision and Control, pp. 1064–1068. IEEE, Las Vegas (1984) 5. Heinzinger, D., Fenwick, B., Paden, B., Miyazaki, F.: Stability of learning control with disturbances and uncertain initial conditions. IEEE Trans. Autom. Control. 37(1), 110–114 (1992) 6. Liu, K., Chai, Y., Sun, Z., Li, Y.: An adaptive iterative learning control approach based on disturbance estimation for manipulator system. Int. J. Adv. Robot. Syst. 16(3), 1–13 (2019) 7. Maniarski, R., Paszke, W., Patan, M.: Application of iterative learning methods to control of a LEGO wheeled mobile robot. In: 23rd International Conference on Methods and Models in Automation and Robotics, pp. 126–131. IEEE, Miedzyzdroje (2018) 8. Xing, X., Liu, J.: Modeling and robust adaptive iterative learning control of a vehicle-based flexible manipulator with uncertainties. International Journal of Robust and Nonlinear Control. 29(8), 2385–2405 (2019) 9. Kebria, P.M., et al.: Kinematic and dynamic modelling of UR5 manipulator. In: 2016 IEEE International Conference on Systems, Man, and Cybernetics, pp. 4229–4234. IEEE, Budapest (2016) 10. Spong, M.W., Vidyasagar, M.: Robot Dynamics and Control, 1st edn. Wiley, New York (2008) 11. John, J.: Craig: Introduction to Robotics - Mechanics and Control, 3rd edn. Springer, Heidelberg (2005)

Chapter 21

Floor Response Spectrum of Nuclear Power Plant Structure Considering Soil-Structure Interaction Yuree Choi, Heekun Ju, and Hyung-Jo Jung

21.1 Introduction Considering soil-structure interaction (SSI) effect on nuclear power plant (NPP) structure is becoming more important nowadays. For the structures constructed on the rock, there is no rocking motion arisen at the base and the seismic response of the structure only depends on the properties of the structure. However, in the case of structures on the soft soil site, the SSI induces a rocking motion and greatly affects the seismic response of the structure. It can cause frequency shifts and amplitude changes of a floor response spectrum (FRS) of the structure [1]. Therefore, SSI effect should be considered in constructing the structural model for the realistic simulation of the seismic response. ACS SASSI is adopted for SSI analysis, which conducts SSI analysis in frequency domain. SSI analysis can be performed in two methods, direct method and substructure method. In direct method, SSI analysis is computed in a single step. In substructure method, on the other hand, SSI analysis is conducted step by step. Free-field analysis and interaction analysis are done separately, so it is easy to manage each system. In this paper, the seismic responses of the NPP structure with and without SSI effects are compared to investigate SSI effect. A simplified beam-stick model with a rigid base mat is adopted for reducing computational costs.

Y. Choi · H. Ju · H.-J. Jung () Department of Civil and Environmental Engineering, KAIST, Daejeon, South Korea 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/10.1007/978-3-030-48153-7_21

161

162

Y. Choi et al.

21.2 Numerical Modelling 21.2.1 Structural Model A simplified beam-stick model with a base mat is adopted as a numerical model, in order to reduce computing time (see Fig. 21.1). This model is based on an in-service reactor containment building structure in Korea. The reactor containment building is composed of the containment shell and the internal structure. In this paper, only the containment shell is used for numerical analysis. It is designated as a 66-m-tall beam-stick model with 14 lumped masses. Four different heights of the structure are selected for the comparison, bottom and top of the structure and two points between them.

21.2.2 Input Motion As an input ground motion, the seismic design response spectrum of the US NRC R.G. 1.60 is applied. Time history acceleration generated by using ACS SASSI module EQUAKE is applied as an input motion in x direction. Their peak ground accelerations (PGA) are anchored to 0.1 g (see Fig. 21.2).

Fig. 21.1 Simplified beam-stick model

21 Floor Response Spectrum of Nuclear Power Plant Structure Considering. . .

163

Fig. 21.2 Time histories of input motion (x direction) [2]

Table 21.1 Soil properties of soft soil and rock site [2] P-wave velocity (m/s) S-wave velocity (m/s) Unit weight (kN/m2 ) Damping ratio

Soft soil site (w/ SSI) 1200 600 6.22 0.02

Rock site (w/o SSI) 60,960 30,480 7.18 0.01

21.2.3 Soil Profile For the comparison between fixed model and SSI model, two different soil profiles are adopted. Soil profiles from ACS SASSI example are used. For analysis without SSI, rock site with shear wave velocity of 2400 m/s or greater can be assumed to be fixed base [3]. The following Table 21.1 gives soil properties of soft soil site and rock site which used for numerical analysis.

21.3 Soil-Structure Interaction Methodology SSI analysis can be performed both in the frequency and time domain. The timedomain SSI analysis method would require huge computational costs, whereas the frequency-domain SSI analysis is less computationally expensive. In frequencydomain analysis, it is possible to consider frequency-dependent properties of the soil. Also, SSI analysis can be conducted in two methods, direct and substructure method. In the direct method, the whole soil-structure system is analyzed in a single step. On the other hand, the substructure method, SSI analysis, is separately done through free-field analysis and structural analysis. In this paper, SSI analysis is conducted in frequency domain by ACS SASSI, which is based on the substructure method. This method can be divided into four different approaches, rigid boundary, flexible boundary, flexible volume, and subtraction. Flexible volume (FV) substructure method, which consists of the free-field system, the structural system, and the excavated soil system, is adopted.

164

Y. Choi et al.

21.4 Result Comparison between the input response spectrum and response spectrum of the bottom of the structure is shown in Fig. 21.3. For the fixed model, two response spectra are similar because soil does not have an influence on the structure. While the SSI model shows different trend. At certain frequency range, the acceleration of the SSI model significantly amplified due to SSI effects. In order to investigate the SSI effects on the structure, floor response spectra of the structure are also compared at four different heights, including the bottom of the structure (point 1), top of the structure (point 4), and two points between them (point 3 and 4). Acceleration time histories at point 2 and point 4 are shown in Fig. 21.4. Both results show that fixed model has a larger acceleration than a model considering SSI effect, especially for the higher level of the structure. Acceleration time histories show that fixed model has a bigger acceleration than SSI model. However, as shown in the Fig. 21.5 and Fig. 21.6, floor response spectrum of SSI model has much bigger amplitude than a fixed model at 2.477 Hz.

Fig. 21.3 Response spectrum (x dir.) at point 1 of fixed base (left) and soft soil site (right)

Fig. 21.4 Acceleration time history at point 2 (left) and point 4 (right)

21 Floor Response Spectrum of Nuclear Power Plant Structure Considering. . .

165

Fig. 21.5 FRS (x dir.) at point 1 (left) and point 2 (right)

Fig. 21.6 FRS (x dir.) at point 3 (left) and point 4 (right) Table 21.2 Resonances of the structure

First peak Second peak

SSI model 2.477 Hz 12.328 Hz

Fixed model 4.642 Hz 13.219 Hz

In this reason, FRS of the structure should be investigated for the precise analysis (Table 21.2). As shown in Figs. 21.5 and 21.6, the peak acceleration frequency of the SSI model becomes smaller than fixed model. It changed from 4.642 Hz to 2.477 Hz. At the higher level of the structure, the peak spectral acceleration of the SSI model is bigger than that of the fixed model. For this reason, considering SSI effect is important for the structures constructed on the soft soil site due to frequency shifts and amplitude changes.

166

Y. Choi et al.

21.5 Conclusion In this paper, SSI effect is numerically investigated by comparing seismic response of two models with two different soil profiles. Acceleration time history shows that the acceleration of the structure becomes smaller when SSI effect is considered. However, the peak frequency of the FRS is reduced from 4.642 Hz to 2.477 Hz due to soil flexibility. At this frequency, spectral acceleration of SSI model is much bigger than that of the fixed model. The spectral acceleration of SSI model is about 6 times bigger than fixed model at the top of the structure. It is due to the SSI effect which causes a rocking motion and it has a significant effect on the seismic response of the structures. Hence, SSI effects need to be considered when a structure is constructed on the soft soil site, especially for an important structure like NPP. Acknowledgment This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry and Energy (MOTIE) of the Republic of Korea (No. 201810102410).

References 1. Wang, X., et al.: Analysis of Seismic Soil-Structure Interaction for a Nuclear Power Plant (HTR10). Sci. Technol. Nuclear Installat. (2017) 2. Ghiocel, D. M.: ACS SASSI NQA Version 3.0.0., User Manuals (2018) 3. Kennedy, R.: Seismic analysis of safety related nuclear structures. ASCE Standard. 4–98

Chapter 22

Real-Time Identification of Vehicle Motion-Modes Tong Chen, Minyi Zheng, Nong Zhang, Liang Luo, and Yishan Pan

22.1 Introduction Vehicle suspension system plays a vital role in vehicle’s ride comfort and safety and has been deeply researched for years [1, 2]. Active suspension systems that commonly use four independent actuators to generate control force are able to provide better ride comfort than passive suspensions. Accurate control on active suspensions and improvement of vehicle dynamic performance are strongly based on vehicle motion-mode identification. The motion-mode energy method (MEM) has been proposed to identify the dominant vehicle motion-mode at each time instant [3]. All vehicular states and road inputs are required for the traditional MEM (based on the 7-DOF vehicle model), while the road inputs and unsprung mass motions are

T. Chen School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, China Automotive Research Institute, Hefei University of Technology, Hefei, China M. Zheng () School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, China e-mail: [email protected] N. Zhang School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei, China School of Mechatronics and Intelligent Systems, Faculty of Engineering, University of Technology Sydney, Sydney, NSW, Australia L. Luo · Y. Pan Automotive Research Institute, Hefei University of Technology, Hefei, China © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_22

167

168

T. Chen et al.

inaccessible in real time. A real-time motion-mode identification method based on feedforward neural networks was proposed recently [4], which only uses suspension deflections to recognize the vehicle body mode energy ratio. However, the networks have trouble in wheel dominant motion-mode identification. The sprung mass acceleration signal at four suspension connected points and four unsprung mass acceleration signals contain individual motion information of sprung mass and unsprung mass, which are alternative input signals. However, direct dependencies between accelerations and energy at instant time are not significant. Thus, a long short-term memory (LSTM) is used to learn long-term dependencies between accelerations and motion-mode energy. LSTM, a kind of recurrent neural networks (RNNs) with LSTM units, has been widely used in solving time sequence-related problems [5, 6]. Acceleration and suspension deflection signals are typical time sequence data. Thus LSTM is more suitable than feedforward neural networks for motion-mode classification. In this study, LSTM classification networks are proposed to identify the dominant motionmode in real time.

22.2 Vehicle Modelling To generate training data for vehicle motion-mode classification and to analyse the motion-modes, a linear 7-DOF vehicle body model is established. The DOFs of vehicle model include sprung mass vertical motions, roll and pitch rotations and four vertical motions of the unsprung masses. Anti-roll bars are not used in this study. Nomenclature zb θ,ϕ ms fl,fr,rl,rr Iθ , Iϕ lf , lr tf , tr zbi zti ksi csi kt mu

Vertical displacement of sprung mass Pitch angle and roll angle of the sprung mass Sprung mass Front left, front right, rear left, rear right Sprung mass pitch and roll moment of inertia Distance from sprung mass CG to the front axis and rear axis Half width of the front axis and rear axis Sprung mass displacements at four suspension contact points (i = fl, fr, rl, rr) Unsprung mass displacements (i = fl, fr, rl, rr) Spring stiffness for each suspension (i = fl, fr, rl, rr) Damping coefficient for each suspension (i = fl, fr, rl, rr) Vertical tire stiffness 1/4 unsprung mass

Assuming that the pitch angle θ and the roll angle ϕ are small enough, then the motions of equations of the vehicle body model can be formed as:

22 Real-Time Identification of Vehicle Motion-Modes

MS O O MU



169

   Z¨ M LCS LT −LCS Z˙ M + − CS L T C S Z¨ U Z˙ U  

ZM LKS LT −LKS =F + − KS LT KS + KT ZU

(22.1)

where: T  ZM = [zb θ ϕ]T , ZU = ztfl ztfr ztrl ztrr ,     MS = diag ms Iθ Iϕ , MU = diag mu mu mu mu ,     KS = diag ksfl ksfr ksrl ksrr , KT = diag kt kt kt kt , ⎡ ⎤ 1 1 1 1   CS = diag csfl csfr csrl csrr , L = ⎣ lf lf −lr −lr ⎦ . tf −tf tr −tr and F represents the sprung mass and unsprung mass external force.

22.3 Motion-Mode Energy Method The target data sets are generated from motion-mode energy method [3]. According to the modal analysis theory, 7-DOF vehicle model has seven motion-modes. Kinetic energy eki and potential energy epi in each mode can be obtained in Eq. (22.2) and (22.3). The motion-mode energy ei is defined as sum of these energies. The motion between each component vector Xi is uncoupled; therefore ei in each motion-mode are also isolated:  

 MS O 1 Z˙ Mi (22.2) eki = Z˙ Mi Z˙ Ui Z˙ Ui O MU 2 epi

T  

1 ZSi − ZUi LKS LT −LKS ZSi − ZUi = − KS LT KS + KT ZUi − ZGi 2 ZUi − ZGi ei = eki + epi

(22.3) (22.4)

T  where ZG = zgfl zgfr zgrl zgrr is the ground input vector. Finally, the total energy of all modes and the mode energy ratio are written as: E=

7  i=1

ei ,

ηi =

ei E

(22.5)

170

T. Chen et al.

22.4 Motion-Mode Classification by LSTM Alternative signals such as accelerations and suspension deflections are accessible easily in real time. However, suspension deflections are coupled with sprung mass motions and unsprung mass motions, and acceleration signals need long-term sequence inputs. Thus, a LSTM layer is used to learn the long-term dependencies. If the total energy is lower than a threshold, the motion-mode is classified into ‘0’. Otherwise, the classification is following the mode which owns the highest energy ratio. The class number and its corresponding motion-mode are listed in Table 22.1. In LSTM unit, input gate, output gate and forget gate are used to address longterm dependencies. LSTM overcomes the gradient vanish problem that occur in RNN training [7]. Thus, LSTM classification networks are proposed to classify vehicle dominant motion-mode in real time. The structure of the LSTM classification networks is described as follows. The input layer has 12 nodes. The second layer is LSTM layer with 30 nodes. Layer 3 is a fully connected layer with eight nodes, and layer 4 is a soft max layer. These two layers transfer the LSTM outputs into probabilities that the inputs belong to each target class. Finally, the classification layer gives the class of dominant motionmode according to soft max function output probabilities. Before the neural network training, the training data sets are generated by the vehicle model simulation. A flow chart of generation of the training data sets is shown in Fig. 22.1.

Table 22.1 Motion-mode classification ClaClass number 0 1 2 3

Excitation

Dominant motion-mode Static Body bounce motion-mode Body roll motion-mode Body pitch motion-mode

Vehicle model

Class number 4 5 6 7

Suspension deflection acceleration

Dominant motion-mode Wheel bounce motion-mode Wheel roll motion-mode Wheel pitch motion-mode Wheel articulation motion-mode

Input Training data set

target Vehicle state vector

MEM method

Fig. 22.1 Flow chart of generation of the training data sets

Energy in each mode

NNs training

22 Real-Time Identification of Vehicle Motion-Modes

171

22.5 Numerical Simulations In this section, the proposed LSTM classification networks method is compared with traditional 7-DOF vehicle MEM in simulation. The classification accuracy of the developed method is validated by comparison of the mode energy. In the speed bump test, a vehicle drives over a speed bump at a speed of 20 km/h. Figure 22.2 shows the simulation results. When the front wheels hit the bump at 0.1 s, the mode 4 becomes the dominant mode. With the wheel motion transferring to vehicle through suspension, mode 1 becomes the dominant motionmode at 0.26 s. When the rear wheels hit the bump at 0.6 s, the dominant mode switches into mode 6 rapidly. Finally, at 0.75 s, mode 3 becomes the dominant motion-mode. For there is no roll or articulation motion, mode 2, mode 5 and mode 7 are not presented in this figure. Figure 22.2(f) shows the LSTM classification results compared with MEM method. LSTM networks are able to classify different dominant motion-modes with acceptable sensitivity and accuracy. The classification accuracy (proportion of right classification time to total simulation time) arrives 94.8%. s

Fig. 22.2 Simulation results. (a) Sprung mass accelerations, (b) unsprung mass accelerations, (c) suspension deflections, (d) total energy, (e) mode energy ratio, (f) class

172

T. Chen et al.

Fig. 22.3 Simulation results. (a) Sprung mass accelerations, (b) unsprung mass accelerations, (c) suspension deflections, (d) total energy, (e) mode energy ratio, (f) class

The steering and braking test studies the combined operation of the steering and braking of a vehicle when avoiding crash. Figure 22.3 shows the simulation results. The vehicle drove straight line at initial speed of 40 km/h, then applying a step brake force (corresponding to a longitudinal deceleration of 0.6 g) from 0.4 s to 0.8 s ◦ and applying a 90 step steering wheel input from 0.6 s to 0.8 s. Thus, during the period of 0.6–0.8 s, vehicle drove under a combined braking and steering operation. The LSTM networks classify a dominant body pitch motion-mode during the pure braking operation (from 0.4 s to 0.6 s). After that, during the combined operation (from 0.6 s to 0.8 s), the mode energy ratio of body pitch motion is overtaken by body roll motion. The proposed LSTM networks capture the sequence of the change of these dominant motion-modes, and the classification results well match the MEM’s. The classification accuracy is 98.7%.

22 Real-Time Identification of Vehicle Motion-Modes

173

22.6 Conclusion This paper proposes a real-time vehicle dominant motion-mode identification method based on the LSTM classification networks and MEM. This method only uses suspension deflection and vertical acceleration as inputs. The dominant motionmode can be obtained and defined as eight classes. Thus, compared with traditional method that needs difficultly approachable full vehicle state and road input signals, the developed method requires less inputs and is simply applicable to vehicles realtimely. A 7-DOF vehicle model with body roll and pitch moment excitation and road input is used to generate network training data sets. In the different driving scenarios, accuracy of the LSTM networks classification is over 90% compared with traditional MEM in simulation. This method benefits design of vehicle dynamic control strategy and the active suspension control performance. Acknowledgment This research is supported by the National Natural Science Foundation of China (51675152), the National Key Research and Development projects of China (2018YFB0105505-03), and Anhui New Energy Automobile and Intelligent Networking Automotive Industry Technology Innovation Project (IMIZX2018001).

References 1. Cao, D.: Editors’ perspectives: Road vehicle suspension design, dynamics, and control. Veh. Syst. Dyn. 49(1–2), 3–28 (2011) 2. Hu, Y.: Comfort-oriented vehicle suspension design with skyhook inerter configuration. J. Sound Vib. 405, 34–47 (2017) 3. Zhang, N.: Motion-mode energy method for vehicle dynamics analysis and control. Veh. Syst. Dyn. 52(1), 1–25 (2013) 4. Wang, L.: Real-time identification of vehicle motion-modes using neural networks. Mech. Syst. Signal Process. 50-51, 632–645 (2015) 5. Liu, J.: VibroNet: Recurrent neural networks with multi-target learning for image-based vibration frequency measurement. J. Sound Vib. 457, 51–66 (2019) 6. Luo, H.: Integration of multi-Gaussian fitting and LSTM neural networks for health monitoring of an automotive suspension component. J. Sound Vib. 428, 87–103 (2018) 7. Wen, L.: Optimal load dispatch of community microgrid with deep learning based solar power and load forecasting. Energy. 171, 1053–1065 (2019)

Chapter 23

Estimation of Normalized Eigenmodes and Natural Frequencies by Using the Effect of Accelerometers Mass Junichi Hino, Satoshi Ooya, and Yuka Shigenai

23.1 Introduction Operational modal analysis is employed to extract experimental modal characteristics of a structure. Then the excitation force is unknown; normalized eigenmodes cannot be obtained. Derivation of the normalized eigenmodes is one of the important issues in operational modal analysis [1]. Therefore, several procedures have been proposed for determining the normalized eigenmodes to overcome the problem [2– 5]. The procedures give known changes to the structure and detect fluctuation of dynamic characteristics. Generally, it is easier to change the dynamic characteristics due to additional masses than the change of the rigidity. In practice, changes in dynamic characteristics due to additional masses are mainly used. Therefore, the method with additional masses is called the mass change method. The mass change method specifies scaling factors to derive normalized eigenmodes. The mass change method has been widely applied to civil engineering structures [6]. Contrary, there are not many applications to machine structures. In mechanical structures, the influence of the added mass on the dynamic characteristics is relatively large. Accelerometer mass can also be considered to have an effect on dynamics. Then, a procedure of using an accelerometer as an additional mass is proposed. On the

J. Hino () Tokushima University, Tokushima, JAPAN e-mail: [email protected] S. Ooya Graduate School of Engineering, Tokushima University, Tokushima, JAPAN e-mail: [email protected] Y. Shigenai Department of the Mechanical Engineering, Faculty of Engineering, Tokushima University, Tokushima, JAPAN e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_23

175

176

J. Hino et al.

other hand, piezoelectric accelerometers are usually used to measure the vibration of structures. The conventional mass change method changes the natural frequency by using an additional mass in addition to the accelerometer. However, the masses of the accelerometers can be regarded as the additional masses. The number of accelerometers used in measurement is less than the measurement points. The accelerometers are moved over all measurement points, which detects changes in the natural frequency at each location of the accelerometers. Additionally, it is possible to derive correct natural frequencies excluding the influence of the accelerometers masses. Numerical calculations and experiments are carried out on an aluminum plates. The measurements are carried out by using three accelerometers, and each accelerometer has a mass of 2 g. In the experiment, the mass of accelerometer may be considered to have little influence on the dynamic characteristics of the plate. Thus, three kinds of additional mass of 3 g, 8 g, and 13 g are prepared to investigate the effect of additional mass.

23.2 Mass Change Method The original eigenvalue problem without damping can be written as: Mϕω2 = Kϕ

(23.1)

where M and K are the mass and the stiffness matrix, respectively. ϕ is a eigenmode vector, ω denotes the eigenvalue. The conventional mass change method derives the normalized eigenmodes by adding the known masses to the original system. However, in this study, since it is assumed that the accelerometers already work as additional masses, the eigenvalue problem in the first location of accelerometers is described as: (M + ΔM 1 ) ϕ1 ω12 = Kϕ1

(23.2)

where ΔM1 is the mass matrix for the first accelerometer location. Subscript 1 indicates the first accelerometer location. Then, the eigenvalue problem in the second location of accelerometers can be rewritten as follows: (M + ΔM 2 ) ϕ2 ω22 = Kϕ2 .

(23.3)

where ΔM2 is the mass matrix for the second accelerometer location. Similarly, the accelerometers are moved overall measurement points. In each measurement, we provide a reference point in which the accelerometer is placed on a fixed location. Thus, the eigenmodes of the whole structure can be estimated. We assume the each eigenmode does not change before and after adding masses [2, 3]. The following is written with subscripts 1 and 2 for simplicity: ϕ1 ∼ = ϕ2 = ϕ

(23.4)

23 Estimation of Normalized Eigenmodes and Natural Frequencies by Using. . .

177

Equation (23.3) is subtracted from eq. (23.2) using the assumption in eq. (23.4) in which the eigenmode is invariant:     Mϕ ω12 − ω22 + ΔM 1 ω12 − ΔM 2 ω22 ϕ = 0

(23.5)

Equation (23.5) is premultiplied by ϕT :     ϕ T Mϕ ω12 − ω22 + ϕ T ΔM 1 ω12 − ΔM 2 ω22 ϕ = 0

(23.6)

Here, the relationship between unnormalized eigenmodes and the normalized ones can be written using scaling factor α as follows: ψ = αϕ

(23.7)

Then M-orthogonality as a constraint is applied: ψT Mψ = 1

(23.8)

Equation (23.6) is rewritten as:     ω12 − ω22 + α 2 ϕ T ΔM 1 ω12 − ΔM 2 ω22 ϕ = 0

(23.9)

The scaling factor is estimated by measurement results from the first and the second accelerometer locations: ,/ / -/ / 2 − ω2 -/ ω / 1 2 (23.10) α = ./ 2 / / ω2 ϕ T ΔM 2 ϕ − ω12 ϕ T ΔM 1 ϕ / Here, the absolute value is taken in the square root so that the scaling factor does not become an imaginary number. The scaling factors are calculated for all combinations of accelerometer locations and the mean value is derived. Next, the natural frequency is estimated without the influence of the accelerometer mass. Eigenvalue problems without additional masses for eq. (23.1) are rewritten as: Mϕ0 ω02 = Kϕ0

(23.11)

where the subscript 0 denotes no additional mass. From eq. (23.11) and eq. (23.2), the natural frequency excluding the influence of the additional mass can be estimated as follows: ω20 = ω21 + α2 ϕT ΔM 1 ϕω21

(23.12)

178

J. Hino et al.

23.3 Numerical Example In order to confirm the validity of the proposed procedure, it applied to a finite element model of an aluminum plate (300 mm square, 8 mm thickness). The plate shown in Fig. 23.1 is modeled by triangular plate elements. Then the number of elements is 450; the boundary conditions of four sides are free. The measurements are carried out by using three accelerometers, and each accelerometer has a mass of 2 g. Three accelerometers are moved over 12 measurement points. Point 1 as a reference point is selected. The eigenmodes can be measured with fewer accelerometers than the number of measurement points. It is not necessary to mount accelerometers at all measurement points and acquire data simultaneously. We acquire modal characteristics at 11 measurement points except the reference point by using two accelerometers in this procedure. The combination of accelerometer location shown in Table 23.1 is employed in this example. Then six sets of natural frequencies and eigenmodes are obtained and the 15 (= 6 C2 ) scaling factors are calculated. Table 23.2 shows the mean value of scaling factors in each mode order. The natural frequencies and eigenmodes before and after mass addition were obtained by solving the eigenvalue problem. The M-orthogonality of scaled eigenmodes is shown in Table 23.3. Table 23.4 shows the natural frequencies excluding the influence of the additional masses. The estimated natural frequencies coincide the theoretical ones. Fig. 23.1 An aluminum square plate model

23 Estimation of Normalized Eigenmodes and Natural Frequencies by Using. . . Table 23.1 Combination of measurement position

Position of accelerometers 1

2

4

1

3

7

1

6

8

1

5

10

1

9

11

1

9

12

1 2 3 4 5 6

Table 23.2 Average scaling factors

179

Mode 1st 2nd 3rd 4th 5th

Scaling factor 15.112 18.227 16.414 29.865 29.256

Table 23.3 M-orthogonality of scaled eigenmodes Mode 

1st 2nd 3rd 4th 5th

 1st 1.0078 4.47e–14 8.48e–14 –1.39e–15 3.98e–15

Table 23.4 Natural frequencies of mass change method (MCM) and FEM

2nd 4.47e–14 1.0006 8.68e–16 –4.34e–16 –2.35e–15

3rd 8.48e–15 8.68e–16 1.0179 –1.85e–14 9.35e–15

4th –1.39e–15 –4.34e–16 –1.85e–14 0.98551 2.10e–13

Mode 1st 2nd 3rd 4th 5th

5th 3.98e–15 –2.35e–15 9.35e–15 2.10e–13 0.93719

Natural frequency (Hz) MCM FEM 287.94 287.94 416.76 416.76 534.33 534.31 738.33 738.33 748.41 748.42

23.4 Experiment In experiment, the estimation of scaling factors is carried out for an aluminum square plate which has same dimension of the simulation model. Four corners of the plate are supported by rubber sponge to approximate boundary conditions similar to numerical calculations. Three accelerometers (PCB, 352C65) are used here. The accelerometer mass is 2 g and the measurement procedure is the same as in the simulation. The combinations of measurement locations also follow Table 23.1. However, the mass of accelerometer may be considered to have little influence on the dynamic characteristics of the plate. Thus, three kinds of additional mass of 3 g, 8 g, and 13 g were prepared to investigate the effect of additional mass. The mass

180

J. Hino et al.

is used attached to the accelerometer. After all, the effects for estimation results are considered in four types additional mass of 2 g (accelerometer mass only), 5 g (=2 g + 3 g), 10 g (2 g + 8 g), and 15 g (2 g + 13 g). Since it is necessary to give an excitation force in the experiment, an excitation force is given by an impulse hammer. In this experiment, we excited the plate by an impulse hammer. The excitation force signal cannot be used because we assume the application in the operational state. Since an impulse force includes a broadband frequency component, the vibration responses sufficiently describe the dynamic characteristics. In this research, we solve the realization problem by the subspace identification method using only the output responses and extract the natural frequencies and eigenmodes. In order to confirm the orthogonality of the normalized modes, we used the norm of orthogonality of normalized eigenmodes proposed by Matsumura et al. [7]. Table 23.5 shows the investigation results of the orthogonality of eigenmodes. The additional mass should be at least 5 g for this specimen. It is difficult to estimate accurate scaling factors in experiments due to the influence of measurement noise and so on. Table 23.6 shows the average scaling factors for each additional mass. The appropriate results for low-order modes were obtained, while inadequate results were derived for higher-order modes. When comparing the mode shapes Table 23.5 M-orthogonality in XORviaGDOP

Table 23.6 Average of scaling factors

MCM Mode 1st 2nd (a) Additional mass of 2 g FEM 1st 1.00572 –0.02422 2nd 0.00162 0.49249 3rd 0.00252 0.67195 (b) Additional mass of 5 g FEM 1st 1.01712 0.28221 2nd 0.00134 0.83571 3rd 0.00486 –0.25498 (c) Additional mass of 10 g FEM 1st 0.98780 0.01153 2nd 0.00214 1.13641 3rd 0.00224 –0.0146 (d) Additional mass of 15 g FEM 1st 1.00132 –0.02044 2nd 0.00226 1.04112 3rd 0.00710 –0.05566

Mode 1st 2nd 3rd

3rd –0.08750 –0.01168 1.06036 0.095377 –0.07886 0.884989 –0.00253 0.00190 0.94560 0.00800 0.00432 0.91573

Scaling factors for additional mass 2g 5g 10 g 15 g 3.678 1.114 1.875 1.890 0.748 0.240 0.302 0.372 0.879 0.455 0.327 0.758

23 Estimation of Normalized Eigenmodes and Natural Frequencies by Using. . . Table 23.7 Natural frequencies without additional masses

Mode 1st 2nd 3rd

181

Natural frequencies for additional mass [Hz] 2g 5g 10 g 15 g 296.8 289.5 297.4 294.8 431.6 430.3 427.8 428.6 544.2 543.3 541.9 544.0

obtained by experiments with those of the finite element method, the shapes are similar in low-order mode, but the difference is distinct in higher-order modes. Higher-order eigenmodes will be affected by the additional mass more than loworder eigenmodes. Finally, the natural frequency without the influence of the added mass is shown in Table 23.7. It is difficult to compare the natural frequency results excluding the influence of the accelerometer mass accurately. In the future, it is desirable to measure the natural frequencies by using a noncontact transducer such as a laser Doppler vibrometer.

23.5 Conclusion We proposed a mass change method that uses accelerometers as the additional mass to obtain the scaling factor which is an important parameter estimating normalized eigenmode. Numerical calculations and experiments were carried out on an aluminum plates. The adequate estimation results were derived by the simulation. The experiment shows that the normalized eigenmodes strongly depends on the accuracy of the measured mode shapes. Since the assumption that the shape of eigenmode does not change before and after mass addition in not adequate, the change of eigenmode after addition of mass will be considered.

References 1. Bernal, D., Gunes, B.: “Damage localization in output-only systems” A flexibility based approach. In: Proceedings of the International Modal Analysis Conference (IMAX) XX, Los Angeles, California, pp. 1185–1191, February, (2002) 2. Parlooo, E., Verboven, P., Guillaume, P., Van Overmeire, M.: Sentivitiy-based operational mode shape normalisation. Mech. Syst. Signal Process. 16(5), 757–767 (2002) 3. Brincker, R., Andersen, P.: A way of getting scaled mode shapes in output only modal analysis. In: Proceedings of the International Modal Analysis Conference (IMAC) XX, paper 141, February, (2003) 4. Khatibi, M. M., Ashory, M. R., Albooyeh, A. R.: Determination of scaling mode shapes in response only modal analysis. In: Proceedings of the IMAC-XXVII, February, (2009) 5. Bernal, D.: A receptance based formulation for modal scaling using mass perturbations. Mech. Syst. Signal Process. 25, 621–629 (2011)

182

J. Hino et al.

6. Khatibi, M.M., Ashory, M.R., Malekjafarian, A., Brincker, R.: Mass-stiffness change method for scaling of operational mode shapes. Mech. Syst. Signal Process. 26, 34–59 (2012) 7. Matsumura, Y., Tsujiuchi, N., Koizumi, T.: Cross orthogonality checks of modal vectors(1st report, coordinate cross Orthogonality checks without a mass matrix). Trans. JSME Ser. C. 65(634), 2271–2278 (1999) (in Japanese)

Part V

Structural Health and Machine Condition Monitoring

Chapter 24

Pitting Fault Severity Diagnosis of Spur Gears Using Vibration and Acoustic Emission Sensor Measurements Pradeep Kundu, Ashish K. Darpe, and Makarand S. Kulkarni

24.1 Introduction Gear health can be monitored using multiple sensors such as vibration, acoustic emission (AE), etc. Different types of sensors have different sensitivity toward fault progression and it is important to investigate their comparative performance for fault diagnosis. The vibration signal contains rich information related to the gear dynamics and the changes in the dynamic response due to gear tooth degradation. However, the measurements from the vibration sensor are affected by the structural resonance and mechanical background noise (under 20 kHz). Suitable signal processing methods are needed to extract the damage-related information using the vibration signal. Comparatively, the frequency content of the AE-based sensor is between 20 kHz and 1 MHz and hence is not significantly affected by the structural resonance and other kinds of noises (under 20 kHz). The AE-generated stress waves are generally at the microscopic level and they may identify the defect at an early stage [1]. For gear pitting fault diagnosis, many HIs are extracted using raw vibration and AE signal in the literature. For example, the HI NA4 is specially designed for monitoring the progressive and severe pitting damage on multiple numbers of gear teeth [2, 3]. Recently, the HI CCR is proposed for monitoring the natural pitting progression on the gear tooth surface [4]. The indicators based on RMS [5], ensemble empirical mode decomposition [6], wavelet [7], etc. are developed using the AE signal. The present work compares the performance of some of the time domain-based HIs extracted from AE and vibration sensor signal. The time P. Kundu () · A. K. Darpe Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, India e-mail: [email protected]; [email protected] M. S. Kulkarni Indian Institute of Technology Bombay, Powai, India e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_24

185

186

P. Kundu et al.

domain-based HIs such as NA4 and CCR are extracted using the raw vibration signal whereas the HI such as absolute energy is extracted from the raw AE signal. Most of the studies in the literature presented HIs based on vibration and AE signal using the seeded gear fault data set. However, this work presents the HI performance using run to failure natural pitting progression experimental data set. This detailed comparison between the vibration and AE sensor-based indicators for observing the natural pitting progression can form an important input for the health monitoring strategy of industrial gearboxes.

24.2 Experimental Setup Figure 24.1(a) shows the developed experimental setup used to perform run to failure pitting experiments. Single-stage spur gearbox with a step-down gear ratio of 1.96 (27 and 53 teeth) is used in the experiments. The gears were subjected to 35 Nm torque and rotating at 2400 rpm during experimentation. The vibration and AE signal was collected at a sampling rate of 20 kHz and 100 kHz, respectively. The vibration signal was acquired for 10 seconds and AE signal was acquired for 1 second after each six-minute interval. The pitting progression on the gear tooth

Fig. 24.1 (a) Developed experimental setup, (b) pitting progression images for a pinion tooth during experiment 2

24 Pitting Fault Severity Diagnosis of Spur Gears Using Vibration. . .

187

surface was measured using a visual inspection camera as shown in Fig. 24.1(b). The oil of low dynamic viscosity and mild steel gears without any surface treatment were used to fail the gear because of pitting failure mode in a shorter span. More details related to experimental setup and pitting stages identification can be seen in [4].

24.3 Fault-Sensitive Health Indicators Many HIs can be extracted from both vibration and AE sensor signals using time domain (TD), frequency domain (FD), and time-frequency domain (TFD) processing. However, the present work is restricted to the extraction of only the time domain-based HIs. With the increase in pitting fault severity, the sideband families or amplitudes increase in the vibration signal due to amplitude or frequency modulation of the gear carrier signal. HIs extracted using residual vibration signal that only contains the sidebands are helpful for monitoring the pitting fault. In this study, NA4 and CCR HI are extracted using residual vibration signal. The HI NA4 is calculated by the ratio of the fourth moment of residual vibration signal to the square of the average variance of the residual vibration signal, whereas HI CCR represents correlation coefficient of the residual vibration signal in faulty stage with the residual vibration signal obtained in healthy stage. The mathematical expression for these indicators is available in [4, 8]. AE is the structure-borne propagating elastic stress waves, which are generated in the gearbox due to the rapid release of energy during the interaction of a pair of gears [9, 10]. The AE signal energy changes with an increase in defect size and hence HI such as absolute energy is extracted from the raw AE signal. The HI absolute energy represents the energy amount released by the material or a structure. This HI is calculated by taking sum of square of amplitude of all sample points in a raw vibration signal.

24.4 Results and Discussion The HIs obtained using raw vibration and AE signal are shown in Figs. 24.2 and 24.3, respectively. Based on the visual inspection images, three stages of damage are considered: initial medium and severe pitting. From Fig. 24.2, the HI CCR extracted using the vibration signal has better trending compared to the NA4 indicator. From Fig. 24.3, HI extracted using AE signal shows a consistent monotonic increasing trending. In Fig. 24.3, for experiment 2, the initial 400-minute data was not found useful for processing mainly due to issues related to sensor mounting. Hence, for experiment 2, the value of the HI observed at the 400th minute is considered to be the starting point and all the previous values are treated at this level. In Figs. 24.2

188

P. Kundu et al.

Fig. 24.2 HIs extracted from raw vibration signal

Fig. 24.3 HI extracted from raw AE signal

and 24.3, “I”, “M,” and “S,” respectively, stand for the initial, medium, and severe pitting observed on the gear tooth surface based on visual inspection images. The visual trending of HIs as observed from Figs. 24.2 and 24.3 needs to be quantified based on some metrics. The metrics such as trendability and interstage sensitivity (ISS) can be used for HI performance comparison. Trendability checks

24 Pitting Fault Severity Diagnosis of Spur Gears Using Vibration. . . Table 24.1 Metric trendability value

Feature name NA4 CCR Feature name Absolute energy

189

Experiment 1 0.33 0.95 Experiment 1 0.92

Experiment 2 0.49 0.93 Experiment 2 0.93

Table 24.2 Metric interstage sensitivity value HI Name Vibration based NA4 CCR

Exp 1 Exp 2 Exp 1 Exp 2

Acoustic emission based Absolute energy Exp 1 Exp 2

H to I zone

I to M zone

M to S zone

−24.31 20.15 −51.09 −46.84

4.53 4.44 −52.41 −49.18

−13.23 −13.20 −108.42 −238.10

73.96 36.49

118.36 188.78

99.74 28.08

the HI trending/correlation with the time progression or fault progression in a complete life cycle. The metric ISS measures/quantifies the HI variation in different health stages of the gear. This metric calculates the % change in the HI value between consecutive health stages of the gear. The indicator for fault prognosis application should have a high value of trendability whereas for fault diagnosis application the HI with a high value of the ISS is desirable. The trendability value calculated for HIs extracted from the raw vibration and AE signal is shown in Table 24.1. Among the indicators extracted using vibration signal, the indicator CCR has almost similar trendability compared to the absolute energy indicator extracted using AE signal. Table 24.2 shows the ISS metric value obtained for HIs extracted from both vibration and AE sensor-based measurement. In Table 24.2, negative sensitivity value indicates that indicator value decreases in the next gear health stage compared to the current health stage of the gear and vice versa for a positive value. For both the experiments, the HI extracted from the AE sensor data shows significant increase among different health stages of the gear. Based on the ISS metric value, the NA4 HI extracted using vibration signal shows the conflicting trend for different experiments in consecutive pitting severity stages. For example, the NA4 HI value is found to have increased by 20.15% for healthy (H) to I zone and increased by 4.44% for the I to M zone, whereas it shows a decreasing trend from the M to S zone (−13.20%) for experiment 2. Similarly, for experiment 1, this HI value decreased from H to I zone (−24.31%), increased from I to M zone (4.53%), and then decreased from M to S zone (−13.23%). However, the CCR HI value is consistently decreasing among different health zones of the gear. In summary, CCR HI extracted using the raw vibration signal and absolute energy HI extracted using AE signal shows a consistent change and monotonic trend with an increase in pitting severity level.

190

P. Kundu et al.

24.5 Conclusion This study compared the performance of the time-domain data-based HIs extracted from the measured vibration and AE sensor data for pitting fault severity diagnosis. In previous work of authors, it was observed that based on the vibration signal, CCR is the best indicator to monitor pitting fault progression compared to other vibration sensor-based indicators. The CCR HI extracted from the vibration signal and absolute energy HI extracted from AE signal has comparable sensitivity and trendability to monitor the pitting fault progression. However, for extracting the CCR indicator using raw vibration signal, few steps such as performing time synchronous averaging (TSA) and then filtering the sidebands from the TSA vibration signal are used. This process is time consuming and requires signal processing background knowledge. In addition, for performing the TSA on raw vibration signal, the reference speed signal is required from additional sensors such as tachometer or proximity probe which increases the cost of performing the gear condition monitoring. Comparatively, the absolute energy HI extracted from the raw AE signal is very straightforward to extract with no special signal processing background knowledge or additional sensor is necessary. For pitting fault diagnosis, these preliminary studies point to a better promise with the AE sensorbased measurement data in terms of background knowledge required and cost of performing condition monitoring. However further work and more measurements are necessary to establish the consistency of the results which is underway.

References 1. Loutas, T.H., Sotiriades, G., Kalaitzoglou, I., Kostopoulos, V.: Condition monitoring of a single-stage gearbox with artificially induced gear cracks utilizing on-line vibration and acoustic emission measurements. Appl. Acoust. 70(9), 1148–1159 (2009) 2. Zakrajsek, J.J., Townsend, D.P., Decker, H.J.: An analysis of gear fault detection methods as applied to pitting fatigue failure data. NASA Tech. Memo. 105950, 1–10 (1993) 3. Dempsey, P. J.: Integrating oil debris and vibration measurements for intelligent machine health monitoring. University of Toledo (2003) 4. Kundu, P., Darpe, A.K., Kulkarni, M.S.: A correlation coefficient based vibration indicator for detecting natural pitting progression in spur gears. Mech. Syst. Signal Process. 129, 741–763 (2019) 5. Sharma, R.B., Parey, A.: Condition monitoring of gearbox using experimental investigation of acoustic emission technique. Proced. Eng. 173, 1575–1579 (2017) 6. Liang, J., Zhong, J.H., Yang, Z.X.: Correlated EEMD and effective feature extraction for both periodic and irregular faults diagnosis in rotating machinery. Energies. 10(10), 1652 (2017) 7. Scheer, C., Reimche, W., Bach, F.: Early fault detection at gear units by acoustic emission and wavelet analysis. J. Acoustic Emission. 25, 331–340 (2007) 8. Sharma, V., Parey, A.: A review of gear fault diagnosis using various condition indicators. Proced. Eng. 144, 253–263 (2016) 9. Loutas, T.H., Roulias, D., Pauly, E., Kostopoulos, V.: The combined use of vibration, acoustic emission and oil debris on-line monitoring towards a more effective condition monitoring of rotating machinery. Mech. Syst. Signal Process. 25(4), 1339–1352 (2011) 10. Hamzah, R.R., Mba, D.: The influence of operating condition on acoustic emission (AE) generation during meshing of helical and spur gear. Tribol. Int. 42(1), 3–14 (2009)

Chapter 25

Loosening Detection of a Bolted Joint Based on Monitoring Dynamic Characteristics in the Ultrasonic Frequency Region Takanori Niikura, Naoki Hosoya and Francesco Giorgio-Serchi

, Shinji Hashimura, Itsuro Kajiwara

,

25.1 Introduction In typical structures and machines [1], bolts and nuts are used for easy assembly, disassembly, and maintenance. However, axial force control is critical because a decrease in the axial force of bolted joints results in fatigue fractures due to the applied vibrations on the bolted joints and chronic degradation. To assess these issues, (a) hammering tests and (b) torque tests are currently used. Hammering tests can be used for bolts with almost no applied axial force, but they cannot assess the level of existing axial force. In addition, the test results depend on the tester’s technical maturity. Torque tests, which control the axis force using a calibrated wrench, are widely employed. This method can be easily conducted, but precisely evaluating the axial force is difficult because the frictional property of the screw seating surface varies according to every degree of fastening [2–4]. These methods have some limitations because the accuracy of the axial force assessment depends on multiple factors, including the degree of flatness of the bolt’s end face. Some approaches have been developed to assess the axial force on bolted joints. Axial force assessment methods that can directly assess the force by incorporating a sensor into bolts include strain gauges and PZTs [5, 6]. Additionally, a method that incorporates a thin plate sensor into bolted joints has been considered [7, 8]. These methods require additional processing when a great number of bolts are used in

T. Niikura · N. Hosoya () · S. Hashimura Shibaura Institute of Technology, Tokyo, Japan e-mail: [email protected] I. Kajiwara Hokkaido University, Tokyo, Japan F. Giorgio-Serchi University of Edinburgh, Scottish Microelectronics Centre, Edinburgh, UK © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_25

191

192

T. Niikura et al.

structures. Furthermore, bolt status observations using this method are impractical because it also requires countless electrical cables. Our study assesses the axial force on bolted joints by investigating the changes in the natural frequency of a protruding thread part to axial force. The rigidity of the jointed area at the fixed end of the protruding thread part decreases in accordance with the decrease in the axial force. To investigate the usability of our method on strength classification of bolts, we assessed the loosening detection performance in three strength classifications (10.9, 8.8, and 4.8).

25.2 Bolted Joint Figure 25.1 shows the bolted joint. The bolt size and the nominal length are M10 and 50 mm, respectively. We used commercially available hexagon head bolts with three strength classifications: 10.9, 8.8, and 4.8. As shown in Fig. 25.1(b), the axial force was measured by four strain gauges that were attached to points just behind the bolt head in the front and back of the bolt. For each point, two gauges were attached to measure the axial force along and across the axis. The appropriate axial force was defined to be 60% of the 0.2% proof stress of each M10 bolt: 31 kN for strength classification 10.9, 22 kN for strength classification 8.8, and 11 kN for strength classification 4.8. We consider these values to represent 100% of the axial force.

Bolted joint 3

2 6

5

1 2–1.5

Impulse hammer

11.6

(a)

(b)

Measurement direction M10

8.4

4

15 20 50

5

Fig. 25.1 Experimental setup of the bolted joint testbed: (a) an image of the bolted joint and the location (referenced with number from 1 to 6) of the impact points of the hammer testing; (b) schematic location and orientation of the strain gauges within the bolted joint

25 Loosening Detection of a Bolted Joint Based on Monitoring Dynamic. . .

193

25.3 Experimental Modal Testing in the Ultrasonic Frequency Region 25.3.1 Measurement of the Frequency Response Functions The natural frequencies and the vibration mode shapes of its protruding thread part were investigated by conducting an impact test using an impulse hammer (see Fig. 25.1(a)). Point 1 was on the protruding thread part and points 2–6 were on the cube surface. Each point was excited in sequence using an impulse hammer and the responses were measured using an accelerometer attached directly behind point 6 and recorded by a spectrum analyzer. From the relationship between these inputs and output, six frequency response functions (FRFs) were obtained. The average count (the number of hits by the impulse hammer) for one identical fastened state was also ten. The target frequency region was set to 40 kHz. The bolt (strength classification, 10.9) was attached to the bolted joint at an appropriate axial force of 31 kN (100%). Based on the measured six FRFs, the modal parameters were identified.

25.3.2 Loosening Detection of a Bolted Joint We investigated whether the natural frequency of the protruding thread part changed in response to a change in the axial force of the bolted joint. Figure 25.2(a) shows the cross-FRFs at points 6–1 for the bolt (strength classification, 10.9). The FRFs with axial forces of 31 kN (100%), 22 kN (70%), and 12 kN (30%) are overlaid for comparison. The coherence function, the phase characteristics, and the amplitude of the cross-FRF are also shown. The most relevant outcome of this test consists in the shift in the natural frequency around 35 kHz in response to the decrease in axial force from 70% to 30% of the appropriate axial force. This result agrees with the reports from previous discussions [9]. These results confirm that the effect of loosening of a bolted joint becomes apparent at higher frequency range such as the ultrasonic frequency region [9]. This result brings evidence that the observation of the relationship between the decreased natural frequency of the protruding thread part and the decreased axial force in the ultrasonic frequency region provides the basis to easily estimate the degree of a loosening of a bolted joint. The axial force does not influence the natural frequency near 20 kHz (see Fig. 25.2(a)). Based on these findings, we aim to devise a methodology to monitor the natural frequency of the protruding thread part in the high-frequency range to accurately detect the loosening of a bolted joint. Figure 25.2(b) and 25.2(c) shows the crossFRFs at points 6–1 for the bolt (strength classification, 8.8) and for the bolt (strength classification, 4.8). The FRFs (Fig. 25.2(b)) for the bolt (strength classification, 8.8) with axial forces of 22 kN (100%), 11 kN (50%), and 6.7 kN (30%) and the FRFs (Fig. 25.2(c)) for the bolt (strength classification, 4.8) with axial forces

(a)

1 0.5 0 180 90 0 -90 -180 4 10 10 10

Phase Amplitude [m/Ns2] [deg.] Coherence

T. Niikura et al.

Phase Amplitude [m/Ns2] [deg.] Coherence

194 (b)

Strength classification: 10.9

2

0

0

5

15 20 25 30 35 40 Frequency [kHz] 12 kN (30%) 22 kN (70%) Phase Amplitude [m/Ns2] [deg.] Coherence

31 kN (100%)

10

(c)

1 0.5 0 180 90 0 -90 -180 4 10 10 10

1 0.5 0 180 90 0 -90 -180 4 10 10 10

Strength classification: 8.8

2

0

0

5

22 kN (100%)

10

15 20 25 30 35 40 Frequency [kHz] 6.7 kN (30%) 11 kN (50%)

Strength classification: 4.8

2

0

0

5

11 kN (100%)

10

15 20 25 30 35 40 Frequency [kHz] 3.4 kN (30%) 6.7 kN (60%)

Fig. 25.2 Cross-FRFs of the bolted joint. The shift of the resonant peak in the 27 to 35 kHz highlights the relationship between the decreased axial force and the decreased natural frequency of the protruding thread part (strength classification: (a) 10.9, (b) 8.8, and (c) 4.8)

of 11 kN (100%), 6.7 kN (60%), and 3.4 kN (30%) are overlaid for comparison, respectively. If structures or machines have different strength classification bolts, we can find the loosening of a bolted joint using the relationship between the decreased natural frequency of the protruding thread part and the decreased axial force in the ultrasonic frequency region; however we cannot compare the different strength classification bolts such as the strength classification 10.9 and 8.8. The strength classification of bolts yields a different appropriate axial force, resulting in a different natural frequency of a protruding thread part. In our method, we used an impact hammer to vibrate the bolted joint with M10. If a natural frequency of a protruding thread part has over 40 kHz, our method might not detect a loosening of the bolted joint. One solution may be to employ laser excitation technology such as vibration tests based on laser ablation [10–16], laserinduced plasma shock wave [17–21], or spherical projectile impact system [22] but this needs to be further examined.

25 Loosening Detection of a Bolted Joint Based on Monitoring Dynamic. . .

195

25.4 Conclusions We demonstrated that loosening detection of bolted joints can be performed by monitoring the relationship between the bending mode natural frequency of the protruding thread part and the axial force of a bolt. This experiment employed the M10 hexagon head bolts whose strength classifications are 10.9, 8.8, and 4.8 and accounts for random fluctuations by performing averaging over repeated trials of the same test. The bending mode and the natural frequency of the protruding thread part were investigated via an impact test using an impulse hammer and FEA. The results confirm the existence of a clear relationship between the bending mode frequency and the axial force, showing that the frequency decreases in accordance with the decrease in the axial force. While this work succeeds at highlighting the correlation between axial force and natural frequency of the protruding thread part, it also highlights the need to define a criteria of classification to determine the threshold at which bolt loosening occurs. This future work, together with the results presented here, will pave the way to the employment of this technique for contactless bolt axial force measurement in real-world applications. Combining our method, laser excitation technique, spherical projectile impact system, and laser Doppler vibrometer to measure responses, we might realize a remote assessment system. Acknowledgments This work was partly supported by the Precise Measurement Technology Promotion Foundation (PMTP-F). We thank the Japan Society for the Promotion of Science for support under Grants-in-Aid for Scientific Research programs (Grant-in-Aid for Challenging Exploratory Research, Project No. JP17K18858, and Grant-in-Aid for Scientific Research (B), Project No. JP19H02088).

References 1. Giorgio-Serchi, F., Weymouth, G.D.: Drag cancellation by added-mass pumping. J. Fluid Mech. 798, R3 (2016) 2. Zhang, X., Wang, X., Luo, Y.: An improved torque method for preload control in precision assembly of miniature bolt. J. Mech. Eng. 58, 578–586 (2012) 3. Hosoya, N., Hosokawa, T., Kajiwara, I., Hashimura, S., Huda, F.: Evaluation of the clamping force of bolted joints using local mode characteristics of a bolt head. J. Nondestruct. Eval. 37(4), 75 (2018) 4. Hashimura, S., Miki, T., Otsu, T., Komatsu, K., Inoue, S., Yamashita, S., Omiya, Y.: Robustness of polyisobutylene for friction coefficients between bearing surfaces of bolted joints. J. Eng. Tribol., in press. (2019). https://doi.org/10.1177/1350650119854245 5. Ritdumrongkul, S., Fujino, Y.: Identification of the location and level of damage in multiplebolted-joint structures by PZT actuator-sensors. J. Struct. Eng. 132, 304–311 (2006) 6. Shao, J., Wang, T., Yin, H., Yang, D., Li, Y.: Bolt looseness detection based on piezoelectric impedance frequency shift. Appl. Sci. 6, 298 (2016) 7. Okugawa, M., Egawa, K.: Study on smart washer using piezoelectric material for bolt loosening detection. J Jpn Soc Non-Destr Inspection. 52, 511–516 (2003) 8. Wang, B., Huo, L., Chen, D., Li, W., Song, G.: Impedance-based pre-stress monitoring of rock bolts using a piezoceramic-based smart washer – A feasibility study. Sensors. 17, 250 (2017)

196

T. Niikura et al.

9. Huda, F., Kajiwara, I., Hosoya, N., Kawamura, S.: Bolt loosening analysis and diagnosis by non-contact laser excitation vibration tests. Mech. Syst. Signal Process. 40, 589–604 (2013) 10. Hosoya, N., Kajiwara, I., Inoue, T., Umenai, K.: Non-contact acoustic tests based on nanosecond laser ablation: Generation of a pulse sound, source with a small amplitude. J. Sound Vib. 333, 4254–4264 (2014) 11. Hosoya, N., Kajiwara, I., Umenai, K.: Dynamic characterizations of underwater structures using non-contact vibration test based on nanosecond laser ablation in water: Investigation of cavitation bubbles by visualizing shockwaves using the Schlieren method. J. Vib. Control. 22, 3649–3658 (2016) 12. Hosoya, N., Umino, R., Kajiwara, I., Maeda, S., Onuma, T., Mihara, A.: Damage detection in transparent materials using non-contact laser excitation by nano-second laser ablation and high-speed polarization-imaging camera. Exp. Mech. 56, 339–343 (2016) 13. Hosoya, N., Terashima, Y., Umenai, K., Maeda, S.: High spatial and temporal resolution measurement of mechanical properties in hydrogels by non-contact laser excitation. AIP Adv. 6, 095223 (2016) 14. Hosoya, N., Umino, R., Kanda, A., Kajiwara, I., Yoshinaga, A.: Lamb wave generation using nanosecond laser ablation to detect damage. J. Vib. Control. 24, 5842–5853 (2018) 15. Hosoya, N., Kajiwara, I., Umenai, K., Maeda, S.: Dynamic characterizations of underwater structures using noncontact vibration tests based on nanosecond laser ablation in water: Evaluation of passive vibration suppression with damping materials. J. Vib. Control. 24, 3714– 3725 (2018) 16. Hosoya, N., Ozawa, S., Kajiwara, I.: Frequency response function measurements of rotational degrees of freedom using a non-contact moment excitation based on nanosecond laser ablation. J. Sound Vib. 456, 239–253 (2019) 17. Hosoya, N., Nagata, M., Kajiwara, I., Umino, R.: Nano-second laser-induced plasma shock wave in air for non-contact vibration tests. Exp. Mech. 56, 1305–1311 (2016) 18. Hosoya, N., Mishima, M., Kajiwara, I., Maeda, S.: Non-destructive firmness assessment of apples using a non-contact laser excitation system based on a laser-induced plasma shock wave. Postharvest Biol. Technol. 128, 11–17 (2017) 19. Zhang, Y., Hiruta, T., Kajiwara, I., Hosoya, N.: Active vibration suppression of membrane structures and evaluation with a non-contact laser excitation vibration test. J. Vib. Control. 23, 1681–1692 (2017) 20. Hosoya, N., Yoshinaga, A., Kanda, A., Kajiwara, I.: Non-contact and non-destructive lamb wave generation using laser-induced plasma shock wave. Int. J. Mech. Sci. 140, 486–492 (2018) 21. Kajiwara, I., Akita, R., Hosoya, N.: Damage detection in pipes based on acoustic excitations using laser-induced plasma. Mech. Syst. Signal Process. 111, 570–579 (2018) 22. Hosoya, N., Kato, J., Kajiwara, I.: Spherical projectile impact using compressed air for frequency response function measurements in vibration tests. Mech. Syst. Signal Process. 134, 106295 (2019)

Chapter 26

Gas Turbine Fault Detection Using a Self-Organising Map Kar Hoou Hui , Ching Sheng Ooi, Meng Hee Lim, Mohd Dasuki Yusoff, and Mohd Salman Leong

26.1 Introduction The unscheduled downtime of turbomachinery in critical industries, such as oil and gas, power generation, petrochemical, and aviation, results in large financial losses to the industries. In addition, component failure, such as blade, bearing, shaft, and gear failure, may lead to a catastrophic failure that threatens human life. Oil and gas plants in Malaysia are implementing machinery condition monitoring and fault detection by observing the equipment parameters, such as temperature, pressure, vibration levels, and operating speeds, at various machine locations. This chapter presents a case study of a gas turbine that was operating within its original equipment manufacturer (OEM) limits but was found to have obvious damage on multiple blades during its periodic and borescope inspection, as depicted in Fig. 26.1. Gas turbine condition monitoring and fault detection solely based on the OEM limits are thus deemed to be insufficient. The hypothesis of this study is that the faulty machine parameters are deemed to be outliers in a self-organising map (SOM) when the SOM is generated by using all historical machine parameters. This chapter explores the feasibility of SOM to provide insight into gas turbine health conditions. The SOM concept will be introduced in the following section.

K. H. Hui () · C. S. Ooi · M. H. Lim · M. S. Leong Institute of Noise & Vibration, Universiti Teknologi Malaysia, Kuala Lumpur, Malaysia M. D. Yusoff PETRONAS Gas Berhad, Kuala Terengganu, Terengganu, Malaysia © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_26

197

198

K.H Hui et al.

Fig. 26.1 Multiple instances of blade damage were found during the periodic and borescope inspection of the gas turbine

26.2 Self-Organising Map A self-organising map (SOM) is an unsupervised machine learning technique inspired by an artificial neural network (ANN), which mimics the human brain in processing signals. This map was developed for clustering applications by grouping data based on their similarity. As the name implies, an SOM, represented in a two-dimensional graphical layout, consists of neurons arranged in contact with one another. It varies from a typical ANN-correlated input-output; using hidden layers via error feedback minimisation, an SOM applies competitive learning with the objective of visualising a high-dimension complex space in a straightforward manner. An SOM consists of a competitive layer, which can classify a dataset with any number of features into as many classes as the layer has neurons. An SOM recognises input data with similar characteristics or patterns and assembles identified groups as neighbourhood neurons. The associated topology-preserving maps are established by assigning a unique weighting factor to each neuron corresponding to the similarities. For every iterative training cycle, an SOM refines the input neuron locations, weighing factor, and weighing equations in expectation of a mature form of cluster mapping. The overview of the SOM training process is displayed in Fig. 26.2. Self-organising maps have been implemented in various applications such as machinery health assessments [1–5], as well as in the fields of medicine [6–8], finance [9], and energy [10]. On the one hand, the adaptive nature of an SOM has recently been recognised in a range of applications, particularly in non-linear structure mapping, dimensional reduction, and clustering illustrations. On the other hand, an unsupervised SOM could not only potentially reduce premature output class labelling but also be capable of generating feature maps for visual aid purposes. For instance, an SOM has been found to be notably effective in candidate evaluation for genetics pool, topological collaborative clustering, and harmony memory embedded multilayer deep learning [11–13]; cost-effective data collection tracking [14]; intrusion dynamic systems accommodating dynamic vehicle ad hoc network model [15]; potential disaster risk assessment [16]; and contaminated water image processing corresponding to Caenorhabditis elegans activity [17].

26 Gas Turbine Fault Detection Using a Self-Organising Map

199

Fig. 26.2 Overview of SOM training cycle

26.3 Data Collection and Machine Trending This section provides a brief summary of the data collection methodology relating to the data used in this research. The machine parameters used in this study are gas generator rotor speed (RPM), power turbine rotor speed (RPM), compressor vibration displacement (micron), compressor vibration velocity (mm/s), compressor inlet temperature (◦ C), compressor discharge temperature (◦ C), compressor discharge pressure (bar), power turbine inlet temperature (◦ C), and exhaust temperature (◦ C). The parameters were logged half-hourly into the turbomachinery condition monitoring system from 1 October 2016 to 31 October 2017. The SOM used in this study consists of 10 neurons in each dimension. The twodimensional map is capable of classifying the weekly machine parameters collected on a half-hourly basis into 100 categories based on their similarity of characteristics. The machine parameters during machine outage were first omitted. Given that the machine parameters during machine operation fluctuate and the fluctuation range between machine parameters is rather inconsistent, the machine parameters were then normalised by scaling between 0 and 1 before they were input into the SOM algorithm. The SOM sample hits obtained from the SOM algorithm are described in the next section.

200

K.H Hui et al.

26.4 Results and Discussion The purpose of this study was to evaluate the collected machine parameters on a weekly basis to identify the change in the data characteristics, thereby allowing for early machinery faults to be determined. The hypothesis of this study is that the faulty machine parameters are deemed to be outliers in an SOM when the SOM is generated by using all historical machine parameters. Figure 26.3 (figure on the left) presents a typical SOM sample hit; the number in each hexagon represents the number of machine parameters that were classified into the neuron. The machine parameters that were classified into a neuron or its neighbouring neurons have higher similarity compared to machine parameters that were classified into neurons located further away. Therefore, a dataset with similar characteristics will be distributed evenly across all neurons, as illustrated in Fig. 26.3 (figure on the left). When a dataset consists of outliers, then an SOM shall present a map with the majority of data distributed across most of the neurons. However, some data were concentrated in a smaller number of neurons, as depicted in Fig. 26.3 (figure on the right). The initial objective of this study was to identify the abnormality of a gas turbine from its trending machine parameters. In the SOM of the first week’s dataset shown in Fig. 26.3 (figure on the left), the dataset was found to be distributed across all neurons. This indicated that all data in the first week have similar characteristics, and this was assumed to be the baseline of the gas turbine. On the other hand, a significant change in the SOM is observed in Fig. 26.3 (figure on the right). The SOM was constructed based on data collected from 1 October 2016 to 23 January 2017. Figure 26.3 (figure on the right) demonstrates that a small cluster was formed in the SOM, and this indicates that outliers existed in the dataset. It can thus be suggested that the gas turbine was operating in a condition that was different from the initial operating condition in October 2016.

Fig. 26.3 SOM based on data collected from 1 to 7 October 2016 (figure on the left) and from 1 October 2016 to 23 January 2017 (figure on the right)

26 Gas Turbine Fault Detection Using a Self-Organising Map

201

Fig. 26.4 SOM based on data collected from 1 October 2016 to 29 June 2017 (figure on the left) and from 1 October 2016 to 26 October 2017 (figure on the right)

Another major change of the SOM was found in the SOM constructed with data collected up to 29 June 2017, as shown in Fig. 26.4 (figure on the left). The collected data were clearly split into two classes. A possible explanation for this might be that the gas turbine was operating in a condition that was different from its baseline and that this continued for a certain period of time. Thus, the number of outliers in Fig. 26.3 (figure on the right) was increased and grew into a larger group, as illustrated in Fig. 26.4 (figure on the left). Figure 26.4 (figure on the right) presents an SOM constructed with the whole year’s dataset; the data were clustered into two groups. These findings suggest that the gas turbine’s operating condition was changed as early as 23 January 2017, and it may indicate that machine faults were developed at this stage. If the operator was triggered by this change, then the operator may take a step forwards to thoroughly analyse the data collected to identify the machine faults, rather than assuming that the machine is running at a normal condition as all collected data were within the OEM limits. This study has been verified by the periodic and borescope inspection that was conducted on 15 November 2017, when blade damage was found on multiple blades.

26.5 Conclusion This study set out to identify significant changes in the data collected over a year, even though all collected data were within the OEM limits. This study identified that an initial change in the characteristics of the collected data took place on 23 January 2017. The results of this study indicate that an SOM is capable of machine health monitoring, which would allow an operator to take early action.

202

K.H Hui et al.

Acknowledgments The authors would like to extend their deepest gratitude to the Institute of Noise and Vibration UTM for funding the study under the Higher Institution Centre of Excellence (HICoE) Grant Scheme (R.K130000.7809.4 J226, R.K130000.7843.4 J227, and R.K130000.7843.4 J228).

References 1. Li, Z., Fang, H., Huang, M., et al.: Data-driven bearing fault identification using improved hidden Markov model and self-organizing map. Comput. Ind. Eng. 116, 37–46 (2018) 2. Germen, E., Ba¸saran, M., Fidan, M.: Sound based induction motor fault diagnosis using Kohonen self-organizing map. Mech. Syst. Signal Process. 46, 45–58 (2014) 3. Yu, J.: A hybrid feature selection scheme and self-organizing map model for machine health assessment. Appl. Softw. Comput. J. 11, 4041–4054 (2011) 4. Adiletta, G.: Self-organizing maps for surveying lubrication within squeeze film dampers. Tribol. Int. 41, 1255–1266 (2008) 5. Huang, R., Xi, L., Li, X., et al.: Residual life predictions for ball bearings based on selforganizing map and back propagation neural network methods. Mech. Syst. Signal Process. 21, 193–207 (2007) 6. Wu, H., Kato, T., Yamada, T., et al.: Personal sleep pattern visualization using sequence-based kernel self-organizing map on sound data. Artif. Intell. Med. 80, 1–10 (2017) 7. Moshou, D., Hostens, I., Papaioannou, G., Ramon, H.: Dynamic muscle fatigue detection using self-organizing maps. Appl. Softw. Comput. J. 5, 391–398 (2005) 8. Wong, G.: Visualization of gene expression data using SOM.pdf. 15, 953–966 (2002) 9. Chen, M.Y.: Visualization and dynamic evaluation model of corporate financial structure with self-organizing map and support vector regression. Appl. Softw. Comput. J. 12, 2274–2288 (2012) 10. Räsänen, T., Ruuskanen, J., Kolehmainen, M.: Reducing energy consumption by using selforganizing maps to create more personalized electricity use information. Appl. Energy. 85, 830–840 (2008) 11. Santos, I.G., Carneiro, V.Q., Silva Junior, A.C., et al.: Self-organizing maps in the study of genetic diversity among irrigated rice genotypes. Acta Sci. Agron. 41, 1–9 (2019) 12. Rastin, P., Matei, B., Cabanes, G., et al.: Impact of learners’ quality and diversity in collaborative clustering. J. Artif. Intell. Softw. Comput. Res. 9, 149–165 (2019) 13. Hasan, S., Shamsuddin, S.M.: Multi-strategy learning and deep harmony memory improvisation for self-organizing neurons. Soft. Comput. 23, 285–303 (2019) 14. Faigl, J.: Data collection path planning with spatially correlated measurements using growing self-organizing array. Appl. Softw. Comput. J. 75, 130–147 (2019) 15. Liang, J., Chen, J., Zhu, Y., Yu, R.: A novel intrusion detection system for vehicular ad hoc networks (VANETs) based on differences of traffic flow and position. Appl. Softw. Comput. J. 75, 712–727 (2019) 16. Chen, N., Chen, L., Ma, Y., Chen, A.: Regional disaster risk assessment of China based on self-organizing map: Clustering, visualization and ranking. Int. J. Disaster Risk Reduct. 33, 196–206 (2019) 17. Jeong, I.S., Lee, S.R., Song, I., Kang, S.H.: A biological monitoring method based on the response behavior of caenorhabditis elegans to chemicals in water. J. Environ. Inf. 33, 47–55 (2017)

Chapter 27

A Comparative Analysis Between EMDand VMD-Based Tacho-Less Order Tracking Techniques for Fault Detection in Gears Madhurjya Dev Choudhury

, Liu Hong

, and Jaspreet Singh Dhupia

27.1 Introduction Fault detection in gears using spectral analysis of measured vibration signal is based on detecting equally spaced sidebands around the gear mesh frequency (GMF) and its harmonics. These sidebands, which occur at the fault characteristic frequencies (FCFs), appear due to the presence of gear tooth defect-induced amplitude and phase modulations of the gear mesh vibration signal. However, extracting fault information from the measured vibration signal is challenging as the FCFs and GMF are related to the shaft rotating frequency, which varies during operation [1]. This operating speed fluctuation smears the resultant vibration spectrum. In order to achieve a better detection result under speed variation, order tracking (OT) techniques are adopted, which are based on the means of extracting instantaneous shaft speed information with the help of supplementary equipment, like a tachometer [1]. However, due to non-availability of such auxiliary sensors in most industrial settings, tacho-less OT is gaining popularity. The existing tacho-less OT methods use traditional bandpass filtering to extract the signal around the shaft speed harmonics from a complex multicomponent gearbox signal for estimating the instantaneous speed [1]. However, selecting accurate filter parameters, like the center frequency and the bandwidth, especially under considerable speed variation is difficult, which may lead to an inaccurate estimation of the shaft speed harmonics. In this study, a more adaptive approach is considered to handle such complex gearbox signals using adaptive decomposition methods [2]. EMD is one such AMD that M. D. Choudhury · J. S. Dhupia () University of Auckland, Auckland, New Zealand e-mail: [email protected]; [email protected] L. Hong Wuhan University of Technology, 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/10.1007/978-3-030-48153-7_27

203

204

M. D. Choudhury et al.

can adaptively decompose any complicated multicomponent signal into a sum of intrinsic mode functions (IMFs). However, as EMD is based on recursive decomposition, there still exists many deficiencies such as mode mixing [2]. In order to avoid such problems and obtain better decomposition performance, a non-recursive adaptive decomposition method known as variational mode decomposition (VMD) [2] is considered in recent times. VMD adopts a joint-optimization scheme to extract the signal modes concurrently by solving a constrained variational problem that can adaptively select the filter center frequency and update the corresponding bandwidth until the appropriate mode is identified [2]. In this study, EMD/VMD is used to decompose the gearbox vibration signal into different IMFs in order to extract a component corresponding to the gear mesh vibration signal. Thereafter, the selected IMF is demodulated using the Hilbert transform (HT) to extract its instantaneous phase from where the shaft speed profile is estimated. The estimated speed profile is then used to resample to the original gearbox vibration signal in order to construct the order spectrum to detect the presence of any localized gear teeth fault. The capability of the proposed tacho-less OT method is demonstrated using a synthetic gearbox signal. The rest of the paper is organized as follows. Section 27.2 gives a brief background of the theories used in this study followed by the introduction of the proposed tacho-less OT method. Section 27.3 investigates the effectiveness of the proposed detection method on a synthetic gearbox signal. Finally, the paper is concluded in Sect. 27.4.

27.2 Theoretical Background This section briefly discusses the gear mesh vibration model and the underlying theory of VMD for the ease of understanding of the readers.

27.2.1 Gear Mesh Vibration Signal The vibration signal of a gear having Ng teeth and connected to a shaft rotating with fs (t), in the presence of a localized defect, can be modelled as [1] xg (t) =

M 

  Am (t) (1 + am (t)) cos 2π mf m (t)t + bm (t) + ϕi ,

(27.1)

m=1

where fm (t) is the GMF, M is the highest harmonic order of the GMF, and am (t) and bm (t) are the amplitude and phase modulation functions introduced due to the presence of gear fault, respectively. These modulation functions can be expressed as

27 A Comparative Analysis Between EMD- and VMD-Based Tacho-Less. . .

am (t) =

bm (t) =

N n=1

N n=1

205

  amn cos 2π nf s (t)t + δmn ,

(27.2)

  bmn cos 2π nf s (t)t + βmn .

(27.3)

Now in order to extract the gear shaft speed fs (t), fm (t) must be estimated from where the instantaneous shaft speed profile can be calculate, as in the case of gear, fm (t) = Ng fs (t). The instantaneous phase of Eq. (27.1) can be extracted by the Hilbert transform (HT) as ψi = 2πfm (t)t + b(t) + ϕ.

(27.4)

Now, dividing ψ i by 2π t (consider b(t)/2π t and ϕ/2π t to be negligible for an increase in t), the GMF can be approximated, which can then be utilized to estimate the shaft speed. However, HT-based demodulation gives efficient result in the case of mono-component signals only, and as typical gearbox vibration signal is multicomponent in nature, therefore, to extract a mono-component corresponding to the fundamental GMF, adaptive mode decomposition methods like VMD and EMD can be used. The next section briefly describes the fundamentals of the VMD algorithm only, in context to the present study and skips a discussion on EMD as it has already been extensively discussed in literature [2].

27.2.2 VMD Basics VMD decomposes a complex multicomponent signal x(t) into a discrete number of narrow-band IMFs, mk (t), and centered around a frequency ωk . For each of the obtained IMFs, VMD constructs an analytic signal by means of HT and calculates its unilateral frequency spectrum. Then the IMF’s spectrum is shifted to baseband. After that, the bandwidth is estimated through L2 -norm of the gradient of the shifted signal. The goal of optimization is to minimize the sum of the spectral widths of all the IMFs as follows [2], 0 min

{mk },{ωk }

12 2   K 1  K   1 1 j −j ω t k 1 1∂t δ(t) + ∗ m (t) e mk = x(t), s.t. k 1 1 πt 2 k=1

k=1

(27.5) where {mk } = {m1 , . . . ,mK } and {ωk } = {ω1 , . . . ,ωK } are, respectively, the IMF vector and their corresponding center frequency vector; δ(.) is the Dirac delta function; * denotes the convolution function; K is the number of IMFs to be extracted; and the constraint indicates that the sum of the obtained IMFs is equal

206

M. D. Choudhury et al.

to the original signal x(t). Readers may refer to [2] for a detailed understanding. In the context of the present work, VMD can be used to decompose the gearbox vibration signal into its constituent IMFs, which will help in isolating the required gear mesh vibration signal.

27.2.3 Tacho-Less Fault Detection Algorithm The steps involved in the proposed tacho-less gear fault detection algorithm are as follows: Step 1: Decompose the gearbox vibration signal into its constituent IMFs using adaptive mode decomposition methods like EMD and VMD. Step 2: Select the IMF corresponding to the fundamental GMF. Step 3: Demodulate the selected IMF using HT to calculate the instantaneous shaft speed profile as explained in Sect. 27.2.1. Step 4: Resample the original vibration signal to generate the order spectrum for fault detection.

27.3 Simulation Analysis This section discusses a MATLAB-based simulation model of a faulty gearbox signal in order to assess the effectiveness of the proposed tacho-less OT method. It also provides a comparison of the signal decomposition achieved using both VMD and EMD. For this analysis, a gearbox vibration signal is considered as [1] x(t) = xg (t) + n(t),

(27.6)

where xg (t) is the gear mesh vibration signal described in Eq. (27.1) and n(t) is a Gaussian white noise of SNR = 5 dB, which is added to the vibration signal x(t) to account for the challenges encountered in practical measurements. The virtual gear is considered to have Ng = 20, and a speed variation function, fs (t) = fs + 1.5sin(2π t/16), is used to simulate a speed fluctuation around the nominal gear shaft rotational frequency fs = 20 Hz. Figure 27.1a plots the speed profile fs (t) used for this simulation. Without the loss of generality, the first gear mesh harmonic (m = 1) is considered, and its corresponding amplitude and initial phase are taken as A1 = 5 and ϕ1 = π/4. The harmonic order (n) of the modulation functions am (t) and bm (t) is assumed to be 3, with their corresponding amplitudes set as a11 = 0.25, a12 = 0.30, a13 = 0.12, b11 = 0.18, b12 = 0.10, and b13 = 0.15. This study is done considering a sampling frequency of 4096 Hz. The spectrum of the simulated signal is shown in Fig. 27.1b. However, the sidebands around the gear mesh harmonics due to the presence of fault

27 A Comparative Analysis Between EMD- and VMD-Based Tacho-Less. . .

207

Fig. 27.1 (a) Speed fluctuation profile used in the simulation and (b) spectrum of the gearbox vibration signal x(t) in fault condition in the presence of speed fluctuation and noise

Fig. 27.2 (a) IMFs generated using VMD and (b) comparison between the true and the VMD estimated gear shaft speed

cannot be distinguished from the spectrum because of the smearing phenomenon caused by shaft speed fluctuation. Therefore, in order to alleviate smearing and improve the spectrum, the steps of the proposed algorithm are applied. Now, according to the steps of the proposed algorithm discussed in Sect. 27.2.3, the gearbox signal is decomposed using both VMD and EMD. Figure 27.2a shows the IMFs generated using VMD for K = 5 (estimated according to the criteria provided in [3]). Here, IMF 2 is found to have an IF centering the GMF; therefore, it is selected to estimate the shaft speed profile. Figure 27.2b shows the shaft speed profile estimated using IMF 2 and is found to have a close match with the true speed profile used in this simulation. However, it is observed that for the gearbox signal decomposed using EMD, the obtained shaft speed profile has a mismatch as shown in Fig. 27.3b. This may be attributed to the fact that the EMD-generated IMFs, shown in Fig. 27.3a, suffer from mode mixing, which results in the generation of IMFs that are not strictly mono-component in nature, and hence the obtained instantaneous phase of the selected IMF using HT doesn’t correspond to the exact phase of the gear mesh vibration signal. Now, the estimated shaft speed profiles are used to generate the order spectrum of the original gearbox vibration signal x(t). Figure 27.4a shows that the spectrum obtained using VMD has improved as compared to Fig. 27.1b, and the fault-induced

208

M. D. Choudhury et al.

Fig. 27.3 (a) First five IMFs generated using EMD and (b) comparison between the true and the EMD estimated gear shaft speed

Fig. 27.4 Order spectrum of original signal using (a) VMD and (b) EMD

sidebands can be clearly identified. As expected, the order spectrum of Fig. 27.4b was obtained using EMD, even though it has lesser smearing than Fig. 27.1b, but it cannot clearly highlight the sidebands and the smearing problem persists. Thus, the VMD-based tacho-less OT method helps in alleviating the smearing problem in order to generate a clearer spectrum, which aids in reliable fault detection in the case of gears. However, one important observation is that the position of the obtained peaks of the FCFs and the GMF in the order spectrum (Fig. 27.4a) deviate from their nominal values (the GMF peak should have been at the 20th order). This suggests that the order spectrum still has unwanted effects due to the speed fluctuation. It can be further improved by using the fast dynamic time warping (FDTW) [1], which can help in squeezing the time-dependent frequencies, further towards the correct constant rotational speed orders.

27.4 Conclusion and Future Work In this paper, a tacho-less OT method for gear fault detection using adaptive mode decomposition is proposed. Two decomposition methods, VMD and EMD, are analyzed in this study to understand their signal decomposition performances such

27 A Comparative Analysis Between EMD- and VMD-Based Tacho-Less. . .

209

that the generated modes can be utilized to accurately estimate the instantaneous shaft speed information in the absence of supplementary sensors like a tachometer. An analytical simulation model of a fixed-axis gearbox is used to investigate the performance of the proposed detection algorithm. The simulation results indicate that VMD has better capabilities than EMD in adaptively extracting a sensitive mode from a noisy gearbox vibration signal, which can then be exploited for order spectrum analysis in order to eliminate the smearing effect caused by speed fluctuation. Future work of this study includes improving the VMD-based tacho-less OT using FDTW and extending it to detect faults in practical gearboxes.

References 1. Hong, L., Qu, Y., Dhupia, J.S., Sheng, S., Tan, Y., Zhou, Z.: A novel vibration-based fault diagnostic algorithm for gearboxes under speed fluctuations without rotational speed measurement. Mech. Syst. Signal Process. 94(2017), 14–32 (2017) 2. Feng, Z., Zhang, D., Zuo, M.J.: Adaptive mode decomposition methods and their applications in signal analysis for machinery fault diagnosis: A review with examples. IEEE Access. 5, 24301–24331 (2017) 3. Feng, Z., Zhang, D., Zuo, M.J.: Planetary gearbox fault diagnosis via joint amplitude and frequency demodulation analysis based on variational mode decomposition. Appl. Sci. 7(8), 1–19 (2017)

Chapter 28

An Effective Indicator for Defect Detection in Concrete Structures by Rotary Hammering Y. Hasebe, F. Kuratani, T. Yoshida, and T. Morikawa

28.1 Introduction The hammering test is one of the most common methods for inspection of internal defects in concrete structures such as voids and delaminations. In this method, an inspector strikes the concrete surface by an inspection hammer and distinguishes the defective from the healthy parts based on the hammering sounds. Although this method is easy to implement, it takes a long time for large structures, and the hammering force variation is large. Furthermore, the inspection results depend on the inspector’s experience. To improve the inspection efficiency, a rotary hammering test was investigated [1]. To replace the human inspection with an automated inspection, quantitative evaluation indexes of hammering test have been proposed [2], and recently, methods based on machine learning have been studied [3]. In this chapter, we use a rotary hammer instead of a conventional inspection hammer to improve the inspection efficiency and the hammering force variation. Then, we present an indicator for effectively identifying the defects. Furthermore, we propose a method based on the k-means clustering for determining a threshold to distinguish the defective from the healthy parts. We conduct the hammering test experiments of concrete specimens with artificial defects. The hammering sounds are measured with a microphone moving with the rotary hammering device. Three indicators are compared, and the method proposed for determining the threshold is validated.

Y. Hasebe · F. Kuratani () · T. Yoshida · T. Morikawa Department of Mechanical Engineering, University of Fukui, Fukui, 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/10.1007/978-3-030-48153-7_28

211

212

Y. Hasebe et al.

28.2 Experiment 28.2.1 Experimental Method Figure 28.1 shows the experimental setup used in this study. A rotary hammering device is attached to the aluminum frame via sliders. It moves in the longitudinal direction and strikes the concrete surface. A microphone (B&K Type 4958) is mounted on the hammering device to measure the hammering sound near an impact location. The measurements are made using a spectrum analyzer (A&D AD-3651, 24-bit A/D converter) with a sampling frequency of 12.8 kHz and 65,536 data points. In this study, a rectangular reinforced concrete block (1150 mm × 200 mm × 150 mm) with internal artificial defects was used as a specimen. Figure 28.2 shows a cross section of the concrete specimen. To simulate voids, three Styrofoam boards (150 mm × 90 mm × 30 mm) were buried, where the defect depth (a distance between the upper surfaces of the concrete block and the artificial defect) was varied at 15 mm, 20 mm, and 30 mm. As a fundamental study to develop a method based on the k-means clustering for defect detection, we used the voids larger than actual size, producing the difference between the sounds at the defective and the healthy parts.

Fig. 28.1 Experimental setup

A

B I

II

Void Fig. 28.2 Cross section of the concrete specimen

III

Reinforcing steel

28 An Effective Indicator for Defect Detection in Concrete Structures. . .

213

The rotary hammering device was moved from point A to B (a distance of 950 mm) shown in Fig. 28.2 at a speed of 200 mm/s. Since the rotary part at the tip of the device has a hexagonal cross section, it strikes the surface six times per rotation. As a result, the sounds were measured at 55 different impact locations. The rotary part at the tip of the device is pressed by a spring, and a pressing force is adjustable. We examined the effect of pressing force on the sound pressure by varying pressing force at 10 N, 27 N, and 40 N. The pressing force 27 N increased the sound pressures at the defective parts, and consequently, the difference between the sounds at the defective and the healthy parts was clearly distinguished. This implies that the pressing force is important in using the rotary hammering and needs to be adjusted appropriately.

28.2.2 Effect of Defect Depth on Time Waveform We examine the effect of defect depth on the time waveform of hammering sound. Figure 28.3 shows the time waveforms of sound pressure measured from the three defect depth specimens. (a), (b), and (c) are the results for 15 mm, 20 mm, and 30 mm defect depth specimen. The sound pressures measured at the three defective parts, which are indicated between the two blue dashed lines, are obviously higher than those at the healthy parts in all the defect depths. In addition, as the defect depth increases, the sound pressures at the defective parts decrease. This is because the thickness between the upper surfaces of the concrete block and the artificial defect increases with the defect depth, and consequently, the vibration of the concrete surface decreases.

28.2.3 Effect of Defect Depth on Frequency Spectrum

Sound pressure [Pa]

We examine the effect of defect depth on the frequency spectrum of hammering sound. Typical time waveforms of the sound pressure at the healthy and the

100

I

III

II

I

100

50

III

II

100

0

0

0

-50

-50

1

2

3

4

Time [s]

(a) 15 mm defect depth

5

III

50

50

-50 -100 0

II

I

-100 0

1

2

3

4

5

-100 0

(b) 20 mm defect depth

1

2

3

4

Time [s]

Time [s]

(c) 30 mm defect depth

Fig. 28.3 Time waveforms of sound pressure for different defect depths

5

214

Y. Hasebe et al.

detective parts were selected, and their frequency spectra were calculated by fast Fourier transform. Figure 28.4 shows the frequency spectra of sound pressure at the defective part for the three defect depths. (a), (b), and (c) are for 15 mm, 20 mm, and 30 mm defect depth. A dominant peak frequency varies depending on the defect depth. Its frequency shifts to a higher frequency, and its amplitude decreases with the defect depth. This is because the thickness of concrete over the defect increases with the defect depth. Consequently, the bending stiffness of the concrete section increases, and the resulting natural frequency increases. For the healthy parts, a dominant peak appears near 2 kHz in all the defect depths though the figures are not shown here.

28.3 Effective Indicator

Sound pressure [Pa]

We present an indicator for effectively identifying the defects by distinguishing the difference between the sounds at the defective and the healthy parts. The three indicators shown in Fig. 28.5 are compared: (a) the maximum amplitude value of the time waveform of sound pressure, (b) the maximum peak value in the frequency spectrum of sound pressure, and (c) the overall value (the total power of the sound over a specified frequency range). In the overall, X[k]is the discrete Fourier transform of discrete time signals, and N is the number of data. The overall value is capable of measuring the differences between the amplitudes and the damping properties of the time waveforms by one indicator.

3

3

3

2

2

2

1

1

1

0

2

4

6

0

2

(a) 15 mm defect depth

4

0

6

2

4

Frequency [kHz]

Frequency [kHz]

(b) 20 mm defect depth

(c) 30 mm defect depth

Frequency [kHz]

6

75 50 25 0 -25 -50 -75 0

Sound pressure [Pa]

Sound pressure [Pa]

Fig. 28.4 Frequency spectra of sound pressure at the defective part for different defect depths

0.02

0.04

Time [s]

(a) Maximum amplitude Fig. 28.5 Three indicators

2.5 2 1.5

OA 

1

1

k =1

0.5 0

N

2

4

6

Frequency [kHz]

(b) Maximum peak

2

  N X [k ]

(c) Overall

I

III

II

75 50 25 0 0

1.5

10

20

30

40

50

Impact number

(a) Maximum amplitude

I

III

II

1 0.5 0 0

40

Overall [Pa2]

100

Max. peak [Pa]

Max. amplitude [Pa]

28 An Effective Indicator for Defect Detection in Concrete Structures. . .

10

20

30

40

50

Impact number

(b) Maximum peak

215 I

III

II

30 20 10 0 0

10

20

30

40

50

Impact number

(c) Overall

Fig. 28.6 Comparison of the indicators (15 mm defect depth)

We calculated the three indicators from the sound pressure measured at each impact location. Figure 28.6 shows the comparison of the three indicators. The horizontal axis indicates the impact number (impact location). All indicators show high values at the defective parts. For the healthy parts, the maximum amplitude values and the maximum peak values vary widely with impact location, whereas the overall values have a small variation. Its small variation at the healthy parts enables us to clearly distinguish the difference between the sounds at the defective and the healthy parts. This indicates that the overall value is the most effective indicator.

28.4 The k-Means Clustering-Based Defect Detection We propose a method based on the k-means clustering for determining a threshold to distinguish the defective from the healthy parts. This clustering is an unsupervised learning method and partitions the dataset into some clusters. We use the overall values as a dataset, where each overall value corresponds to each impact location. First, the two-dimensional dataset (overall value and impact location) is reduced to a one-dimensional dataset by extracting the overall values. Next, the one-dimensional dataset is partitioned into the clusters, and their centers are obtained by the k-means clustering. Since the smaller overall values are expected to correspond to the healthy parts, we find the smallest cluster center. Finally, the data points belonging to the cluster with the smallest center are removed, and the rest of the data points are reconstructed to a two-dimensional dataset. In the k-means clustering, the number of clusters needs to be determined. We perform the k-means clustering by varying its number from 2 to 10 clusters and determine the number. Furthermore, in basic k-means clustering, initial cluster centers are selected on a random basis, and however, we use the maximum and the minimum values of the dataset as the largest and the smallest centers of the initial cluster centers. Figure 28.7 shows the smallest cluster center according to the number of clusters. The smallest center for each defect depth remains stable when the number of clusters exceeds 7.

216

Y. Hasebe et al.

Fig. 28.7 Smallest cluster center according to the number of clusters I

II

III

(a) 15 mm defect depth

I

II

III

(b) 20 mm defect depth

I

II

III

(c) 30 mm defect depth

Fig. 28.8 Results of defect detection

Figure 28.8 shows the data points after removing the data points belonging to the cluster with the smallest center. The data points at the defective parts are clearly extracted, and the locations of the defective parts are identified for 15 mm, 20 mm, and 30 mm defect depth. In addition, the longitudinal size of the defective parts is estimated for 15 mm defect depth. This enables effective and accurate defect detection in concrete structures.

28.5 Conclusions In this chapter, we presented the indicator for effectively identifying the defective parts in using the rotary hammering. Furthermore, we proposed the method based on the k-means clustering for determining the threshold to accurately distinguish the defective and the healthy parts. We conducted the hammering test experiments of the concrete block specimens with the artificial defects, and the three indicators were compared. The results show that the overall value of the sound is the most effective indicator because it clearly indicates the difference between the sounds at

28 An Effective Indicator for Defect Detection in Concrete Structures. . .

217

the defective and the healthy parts. The threshold for the overall value is properly determined by finding the smallest cluster center as the number of clusters increases. This enables effective and accurate defect detection in concrete structures. Acknowledgement This work was supported by JSPS KAKENHI Grant Number JP18K04015.

References 1. Sonoda, Y., Nakayama, A., Miyoshi, A.: A fundamental study on diagnostic mechanism of the rotary hammering test by acoustic analysis. J. Struct. Eng. 54A, 599–606 (2008) (in Japanese) 2. Sonoda, Y., Okamura, M., Tamai, H.: A Fundamental Study on Hammering Sound Test of Deteriorated Concrete Structures, Proceedings of the Annual International Conference on Architecture and Civil Engineering, (216379) (2018). https://doi.org/10.5176/2301-394X_ACE18.118 3. Kasahara, L., Fujii, H., Yamashita, A., Asama, H.: Unsupervised learning approach to automation of hammering test using topological information. ROBOMECH J. 4(13), 1–10 (2017). https://doi.org/10.1186/s40648-017-0081-7

Chapter 29

A Vibration-Based Strategy for Structural Health Monitoring with Cosine Similarity C. H. Min, S. G. Cho, J. W. Oh, H. W. Kim, and B. M. Kim

29.1 Introduction Offshore structures are likely to be damaged due to long-term exposure to extreme marine environments (waves, current, sea wind, water pressure, vortex vibration, salinity, etc.). Especially after the Macondo accident in the Gulf of Mexico in April 2010, interest in the structural safety of offshore structures has been increasing. In the case of offshore industry, almost every offshore structure is customized for their mission and environmental condition in the design and manufacturing process, which in turn makes structural characteristics of the individuals very distinctive. This makes accumulated data from other past cases less applicable. Moreover, most offshore fields are far from onshore network. Therefore, an advantage of online network system would not be applied to manage massive amounts of data. Structural health monitoring (SHM) has developed with a wide range of applications to industrial fields, e.g., buildings, bridges, wind turbines, and aircraft. One of the very popular approaches of the traditional SHM is vibration-based damage detection using modal parameters such as natural frequencies and mode shapes [1–3]. In recent years, the classical technique goes on to be faced with the beginning of the big data which is still quite challenging [4] but may be more applicable to the real-time SHM during operation. For the big data-relevant SHM, several pieces of research have been lately performed. Above all, concept and

C. H. Min () · S. G. Cho · J. W. Oh · H. W. Kim Offshore Industries R&BD Center, Korea Research Institute of Ships & Ocean Engineering, Geoje, Republic of Korea e-mail: [email protected]; [email protected]; [email protected] B. M. Kim Department of Convergence Study on the Ocean Science and Technology, Ocean Science and Technology School of Korea Maritime and Ocean University, Busan, 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/10.1007/978-3-030-48153-7_29

219

220

C. H. Min et al.

procedure of the SHM were comparatively analyzed in aspects of the big data, and the process of the big data-relevant SHM was summarized: data cleansing, dimensionality reduction, data fusion and feature extraction, feature classification, outlier detections, and prediction [5]. Conversely speaking, however, it could be unfavorable for complicated structures in real. In detail, the sizes of the big data for the SHM can be exorbitantly large depending on the number of members of the structure due to the fact that output-only system identification (SI), a.k.a. operational modal analysis (OMA), and algorithms generally require computational costs increasing cubically and linearly with sensor channels and samples, respectively. Moreover, most of the offshore structures consist of lots of members, which means that the number of required sensors would be a lot, as well. Consequently, it causes a huge computing cost to increase proportionally with the structural complexity. In order to solve these challenges, Min et al. proposed a cosine similaritybased single damage detection method [6]. The method estimates cosine similarity compared to the database consisting of sensitivities of rates of change of natural frequencies corresponding to decreases of elastic modulus of each element. However, the ability to figure out multi-damages is required for damage detection with a view to application for real structures. Therefore, Kim et al. proposed the improved method capable of monitoring multiple damages simultaneously [7]. This paper based on Kim et al. [7] presents an effective vibration-based structural health monitoring using cosine similarity and vibration data from sensors. This vibration-based strategy uses the rate of changes of natural frequencies before and after the occurrence of various damage scenarios including not only single but also multiple damages. The proposed method alerts the occurrence of damage with normalized warning index which facilitates workers to manage risks of damage. A numerical example is considered to verify the proposed method.

29.2 Methodology There are various types of damages ranging from corrosion and degradation to crack, and all the types of the damages cause loss of mass, change of material properties, and/or some kinds of irreversible deformations. Note that only changes in stiffness parameters are considered since the change in mass is negligible in common structural damage (e.g., cracks, time-dependent degradation in concrete structure, and loosen connections in steel structure), and the effect of change in damping parameters on change in spectral information is negligibly small.

29.2.1 Damage Estimation Vector and Matrix In this paper, natural frequencies are used to evaluate damage. Particularly, the rate of change of a natural frequency before and after damage of a pre-updated FE model

29 A Vibration-Based Strategy for Structural Health Monitoring with Cosine. . .

221

is considered as Eq. (29.1), and a vector composed of them from first to nth mode, which is named damage estimation vector (DEV) in this study, is utilized as a similarity comparison target, as Eq. (29.2). DEVs are calculated from the natural frequencies and constructed as a matrix named a damage estimation matrix (DEM) as Eq. (29.3). In other words, the DEM is a matrix made up of DEVs and plays a role in a sort of damage indices: zi,j =

∗ −f fi,j j

fj

(29.1)

,

/ /T s i = / zi,1 zi,2 · · · zi,j · · · zi,n / , ⎡ / /T ⎢ / / ⎢ S = / s1 s2 · · · sm / = ⎢ ⎣

z1,1 z1,2 z2,1 z2,2 .. .. . . zm,1 zm,2

· · · z1,n · · · z2,n .. z . i,j

(29.2) ⎤ ⎥ ⎥ ⎥, ⎦

(29.3)

· · · zm,n

Here, i and m are an arbitrary index number and the total number of the designed damage conditions, respectively; j and n are an arbitrary and the maximum available mode number, respectively; fj is the jth natural frequency of the undamaged FE model; fi,j * is the jth natural frequency of the FE model in the ith damage case; zi,j is the rate of change of jth natural frequency under the ith damage condition; si is the DEV for the ith damage case; and S is the DEM for all the designed damage conditions.

29.2.2 Normalized Warning Index and Damage Reflection Vector During operation, a maintenance system of an offshore structure would be monitoring any kind of structural responses, such as strain or acceleration. The sets of natural frequencies obtained from sensors by inspection times are expressed by Eq. (29.4). Then, the rate of change of natural frequencies between two sequential inspection cycles can be calculated as Eq. (29.5), which is used as warning index (WI) in this study. The normalized warning index (NWI) can be expressed by Eq. (29.6). Damages would be perceived if the NWI is larger than the threshold value at dth inspection, and in turn, the rates of changes of natural frequencies between d-1th and dth inspections are vectorized, named damage reflection vector (DRV) in this study, as Eq. (29.7):

222

C. H. Min et al.

⎡

f11 /T / ⎢ f2 / / ⎢ 1 D = / d1 d2 · · · dk · · · dp / = ⎢ . ⎣ .. p f1 p

p WIj

=

/ / /T / / / d d d h = / W I 1 W I 2 · · · W I n / = //

=

j

(29.4)

p

· · · fn

p−1

fj − fj

(29.5)

,

p−1

fj p

p NW I j

⎤ · · · fn1 · · · fn2 ⎥ ⎥ . ⎥, f k .. ⎦

f21 f22 .. . p f2

p−1

W Ij − μj p−1

(29.6)

,

σj

f1d −f1d−1 f2d −f2d−1 f1d−1 f2d−1

···

fnd −fnd−1 fnd−1

/T / / , /

(29.7)

Here, k and p are an arbitrary cycle number and the most recent cycle number of the inspections, respectively; fj k is the jth natural frequency at the kth inspection cycle; dk is the vector of natural frequencies at the kth inspection cycle; D is a matrix consisting of all the natural frequencies by the most recent inspection, WIj p is the warning index of the jth natural frequency at the pth inspection cycle; d is the inspection cycle number when a damage is recognized; and h is the DRV at the time.

29.2.3 Damage Identification with Cosine Similarity Once the occurrence of damage is recognized, its locations and severities are identified as soon and accurate as possible. Basically, if the damage is exactly same as one of the damage scenarios, the DEV and DRV will be identical as well. That is, the similarity between the predefined DEM and DRV can be used as a key parameter for damage identification. It is employed in this study for similarity assessment of the DEM and DRV as Eq. (29.8). If fortunately, the cosine similarity value is exactly 1, the angle between two vectors is 0, meaning that they have the same direction. Therefore, accurate damage locations and severities might be directly identified: si · h CS i = =3 s i  h n

n

d j =1 zi,j W I j

 2 ,  2 n  d j =1 zi,j j =1 W Ij

(29.8)

Here, CSi is the cosine similarity value of the ith DEV for the damage. The entire procedure can be summarized below:

29 A Vibration-Based Strategy for Structural Health Monitoring with Cosine. . .

1. 2. 3. 4. 5. 6. 7. 8.

223

Select a real structure. Make an FE model corresponding real structure. Perform eigenvalue analysis to obtain the natural frequencies of the FE model. Compute the natural frequencies change ratios matrix. Measure the acceleration data of the structure under operational condition. Estimate natural frequencies of the structure from monitoring data. Assess the damage through the warning index. Finally, detect location and severity of damages in the structure using cosine similarity.

29.3 Verification and Conclusions This study validates the proposed method for a 2D cantilever beam in Fig. 29.1. The material properties information are listed in Table 29.1. For examining the accuracy and efficiency of the proposed damage detection method, two scenarios listed in Table 29.2 are investigated. In both scenarios, damage is inflicted in the structure by reducing the Young modulus of the appropriate elements. In this numerical study, eigenvalue analysis replaces system identification to obtain the natural frequencies. It is assumed that the natural frequencies computed by eigenvalue analysis are estimated from the obtained data. The proposed method

Fig. 29.1 2D cantilever beam Table 29.1 Property information of the cantilever beam

Table 29.2 Damage scenarios

Property Modulus of elasticity Mass density Diameter Length Damage scenario Case 1 Case 2

Damage location E2 E2, E7

Value 2×1011 N/m2 7850 kg/m3 0.024 m 1m

Damage severity (%) 20 20, 40

224

C. H. Min et al.

Fig. 29.2 Result of damage detection of Case 1 (left) and Case 2 (right)

correctly detects damaged elements of both cases and estimates the damage severity. The results of the damage detection are presented in Fig. 29.2. This research proposes a multiple damage detection method using cosine similarity. The proposed method uses natural frequency change ratios between nondamaged structures and damaged structures. The accuracy of the proposed method was numerically verified by the damage scenario. The best benefit of this method is that any extra task and professional knowledge is not required at all for workers in real offshore field. Such a simple and intuitive system to treat damages would absolutely improve the applicability and usability of SHM in offshore operation. A more extensive study on monitoring systems, including usable sensors in offshore, is needed to apply the proposed method to structures in a real situation. Acknowledgments The authors are grateful for the full support shown for this research work. This research was supported by a grant from Endowment Project of “Technology development of digital twin in oscillating water column type for smart operation and maintenance service” funded by Korea Research Institute of Ships and Ocean engineering (PES3090).

References 1. Lee, K., Jeong, S., Sim, S.H., Shin, D.H.A.: Novelty detection approach for tendons of Prestressed concrete bridges based on a convolutional autoencoder and acceleration data. Sensors. 19, 1663 (2019) 2. Fritzen, C.P., Klinkov, M., Kraemerbbs, P., Koranyi, B.: Vibration-based damage diagnosis and monitoring of external loads. In: Ostachowicz, W., Güemes, J.A. (eds.) New Trends in Structural Health Monitoring, pp. 149–208. Springers, Vienna (2013) 3. Fassois, S.D., Kopsaftopoulos, F.P.: Statistical time series methods for vibration based structural health monitoring. In: Ostachowicz, W., Güemes, J.A. (eds.) New Trends in Structural Health Monitoring, pp. 209–264. Springers, Vienna (2013)

29 A Vibration-Based Strategy for Structural Health Monitoring with Cosine. . .

225

4. Gulge, N.S., Shahidi, G.S., Matarazzo, T.J., Pakzad, S.N.: Current challenges with BIGDATA analytics in structural health monitoring. In: Niezrecki, C. (ed.) Structural Health Monitoring & Damage Detection, Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics 2017, pp. 49–54. Springer Nature, Cham, Switzerland (2017) 5. Cremona, C., Santos, J.: Structural health monitoring as a big-data problem. Struct. Eng. Int. 28(3), 243–254 (2018) 6. Min, C.H., Kim, H.W., Oh, J.W., Cho, S.G., Sung, K.Y., Yeu, T.K., Hong, S., Yoon, S.M., Kim, J.H.: Similarity-Based Damage Detection Method: Numerical Study. The 3rd International Conference on Advanced Engineering-Theory and Applications 2016, vol. 415, pp. 428–435. Springer, Busan (2016) 7. Kim, B.M., Min, C.H., Kim, H.W., Cho, S.G., Oh, J.W., Ha, S.H., Yi, J.H.: Structural health monitoring with sensor data and cosine similarity for multi-damages. Sensors. 19(14), 3047 (2019)

Chapter 30

A New Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse Ren Kadowaki

, Takumi Inoue, Kentaro Kameda, and Kazuhisa Omura

30.1 Introduction Bolt loosening may lead to severe accidents in many fields, such as in machines, architectures, and medical implants. It is required to find the early loosening. There are some testing methods for the bolt loosening with ultrasound [1–3], electrical impedance [4], thermal impedance [5], and dynamic response to laser excitation [6]. However, the previous methods have some restrictions on bolt types or bolt materials. The purpose of this study is to develop a new nondestructive testing method. This study uses an ultrasonic pulse and projects it to a target bolted joint from the side of the bolt; for example, this method is applicable for a bolt and nut assembly. We observe variations of transmission pulse received at the opposite side of the bolted joint. It is expected that the pulse is affected by the bolt loosening while the pulse travels in the bolted joint. In this paper, as the first step, we demonstrate the proposed method using a simple metal block and a bolt and discuss its mechanism through an identification of the transmission paths of the pulses.

R. Kadowaki () · T. Inoue · K. Omura Kyushu University, Fukuoka, Japan e-mail: [email protected]; [email protected] K. Kameda Graduate School of Engineering, Kyushu University, Fukuoka, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_30

227

228

R. Kadowaki et al.

30.2 Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse This study proposes a nondestructive testing method for bolt loosening with the transmitted ultrasonic pulse. A schematic drawing of the proposed method is shown in Fig. 30.1. This method requires two ultrasonic transducers as a transmitter and a receiver, a pulser/receiver, and a data logger. The two transducers are set on the side of a bolted joint. An ultrasonic pulse is sent from the transmitter. It travels across flanks of the bolt. The flanks are contact interfaces between the bolt and the fastened part. Reflection and refraction of the ultrasonic pulse occur on the flanks. Furthermore, reflectance and transmittance are affected by contact pressure on the flanks. Because the contact pressure changes depending on bolt loosening, we can detect the bolt loosening through observation of the transmitted ultrasonic pulse.

30.3 Experiment 30.3.1 Experimental Setup A test piece consists of a stainless steel bolt and an aluminum block with 80 mm square and 20 mm height as shown in Fig. 30.2. Two transducers (A544S-SM, OLYMPUS) whose diameter is 6 mm are attached to the block and are connected to a pulser/receiver (5073PR, OLYMPUS). The nominal frequency of the transducers is 10 MHz. The bolt which is longer than the height of the block is fastened to the center of the block. The nominal size of the bolt is JIS-M16 with a 16 mm diameter and a 2 mm pitch. Before the bolt is fastened, the bolt surface is coated with bolt tension stabilization (Fcon, Tohnichi). Figure 30.3 represents a waveform of a sent pulse from the transducer. This pulse is observed through an 80 mm-thickness aluminum block without a hole. The sampling frequency is 2.5 GHz. The number of data points is 8192. In this experiment, we observe various transmitted pulses when the bolt is tightened with four kinds of torque, which are 40, 60, 80, and 100 Nm.

Data logger

Pulser/Receiver

Transducer (Transmitter)

Bolted joint

Ultrasonic pulse

Transducer (Receiver)

Fig. 30.1 A schematic drawing of our proposed testing method. Two transducers face each other across the bolt. An ultrasonic pulse is sent and received with the transducers

30 A New Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse

08

Transducer (Receiver)

A

A’

Cross section A-A’ Base block Bolt (M16)

20 6

Top view Transducer (Transmitter)

229

80

Fig. 30.2 Test piece for the experimental study. It consists of a rectangular base block made with aluminum alloy and a bolt (JIS, M16) made with stainless steel. A couple of transducers are set on both sides of the base block

Fig. 30.3 Ultrasonic pulse sent to the block

30.3.2 Result Figure 30.4 represents transmitted pulses when the bolt is tightened with abovementioned torques. All curves consist of two pulses although we sent a single pulse shown in Fig. 30.3. The amplitude of the first pulse in 100 Nm torque condition is about 8% larger than 40 Nm torque condition. About the second pulse, the difference of amplitudes is 65%. This result indicates that the tightening torque can be evaluated by observing the transmitted ultrasonic pulse.

30.4 Discussion 30.4.1 Mode Conversion on a Flank Only the ultrasonic pulses which travel across the bolt are received in this experiment because the bolt diameter is larger than the receiver diameter. The ultrasonic pulses are reflected or refracted on the flanks as shown in Fig. 30.5. In this figure, longitudinal wave and transverse wave are distinguished by subscripts L or T. Superscripts A and B denote the mediums. A longitudinal wave may be converted to a transverse wave. It is called mode conversion. Equation (30.1) represents relationships between the incident angle θ 0 , the reflection angle of the longitudinal wave θ 1 , the reflection angle of the transverse wave θ 2 , the refraction

230

R. Kadowaki et al.

B

Waveform (V)

A

Tightening torque 40 Nm 60 Nm 80 Nm 100Nm

A

2

B

0 -2 12.8

13.2 13.6 Time (s)

14.0

Fig. 30.4 Received pulses in four kinds of tightening torques which are 40 Nm (solid line), 60 Nm (dashed line), 80 Nm (dotted line), and 100 Nm (dotted-dashed line). Two pulses appear although one pulse is sent. Their amplitudes increase as the tightening torque increases

Incident L or T-wave

Reflected T-wave Reflected L-wave

0

Material A Sound velocity: Material B Sound velocity:

c LA ,

1 2

c TA

c LB , c TB

Boundary surface 4

3

Refracted L-wave

Refracted T-wave Fig. 30.5 Reflection and refraction of ultrasound on a boundary interface. One incident pulse splits into a maximum of four pulses

angle of the longitudinal wave θ 3 , and the refraction angle of the transverse wave θ 4. Material properties of the bolt and the block are shown in Table 30.1: sin θ0 sin θ1 sin θ2 sin θ3 sin θ4 = A = A = B = B . A cL or T cL cT cL cT

(30.1)

30 A New Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse

231

Table 30.1 Material property of each component Component Base block (aluminum alloy) Bolt (stainless steel)

Density (kg/m3 ) 2690 7910

Sound velocity (m/s) Longitudinal wave Transversal wave 6420 3040 5790 3100

Block

Bolt Path (c)

Block 55.4 deg

47.9 deg Block

Block 60.0 deg

Bolt

51.3 deg

47.9 deg

Path (a) 51.3 deg

51.3 deg

Path (d) 51.3 deg

Path (b) 60.0 deg

Load flank

55.4 deg

51.3 deg Path (e) Path (f) 51.3 deg

54.7 deg 54.7 deg

Clearance flank

Fig. 30.6 Transmission paths which the received pulses travel along. The pulses through paths (a) and (b) remain longitudinal pulse. Paths (c)–(f) include two mode conversions

30.4.2 Identification of the Transmission Paths A transmission path of the ultrasonic pulse splits into a maximum of four paths by the reflection and the refraction on a flank. However, only paths (a)–(f) shown in Fig. 30.6 reach the receiver because the diameter of the receiver is only 6 mm. Figure 30.6 illustrates the transmission paths in the center section of the bolt with a schematic drawing. In this figure, the flanks are classified into the load flanks, which are the upper side of the screw thread of the bolt and the clearance flank. Paths (a) and (b) are the transmission paths of the refracted longitudinal pulses. A difference between the two paths is the order of the flanks which the pulse travels across. Paths (c) and (d) are the transmission paths with mode conversion at the left thread. The sent longitudinal pulse is reflected and converted to a transverse pulse. This transverse pulse is converted to a longitudinal pulse again at the next thread. Differences between the two paths are the order of the flanks, which the pulse travels across, and the number of the crossing of the load flanks. Paths (e) and (f) are the other transmission paths with mode conversion. There are two mode conversions in these paths. Again, differences between the two paths are the order of the flanks and the number of the crossing of the load flanks. The time to travel each transmission path is calculated from the path length and the sound velocity written in Table 30.1. It takes 12.7 μs to travel along paths (a) and (b). On the other hand, it takes 13.4 μs to travel along paths (c)–(f). It means

232

R. Kadowaki et al.

that the pulses which travel along paths (a) and (b) and the pulses which travel along paths (c)–(f) are received separately.

30.4.3 Relationships Between the Tightening Torque and the Transmitted Pulse As the tightening torque increases, the contact pressure on the load flank also increases. It causes increase of the transmittance on the load flank [7]. On the other hand, the transmittance on the clearance flank is smaller than the transmittance of the load flank and independent of the tightening torque because the bolt and the block do not make contact on the clearance flank. In the transmission paths mentioned above, the ultrasonic pulses through paths (c) and (e) travel across the load flank twice. Therefore, it is expected that these pulses are most sensitive to the tightening torque, and their amplitude increases as the tightening torque increases. It corresponds to the feature of the second pulse in Fig. 30.4. The increase of the amplitude of the second pulse means the increase of the tightening torque of the bolt. Thus, although this experimental setup is now almost ideal, it is concluded that the change of the tightening torque of the bolt, which corresponds to the bolt loosening, is evaluated using the proposed method. This relationship is now qualitative. We attempt to elucidate the quantitative relationship between the tightening torque and the transmitted pulses in the proposed method and to focus on the reflected pulse to extend its application.

30.5 Conclusion This study proposed a nondestructive testing method for the bolt loosening with the transmitted ultrasonic pulse. To demonstrate the proposed method, we sent an ultrasonic pulse to an aluminum block in which a stainless bolt was tightened with various torque levels. The received ultrasonic wave consisted of two pulses although one pulse was sent. And the amplitude of the second pulse increased with increase of the tightening torque. The behavior of the transmitted wave was discussed based on the identification of the transmission path in the block. Six transmission paths of the received pulses were identified through the calculation of reflection angle and refraction angle on the flank. Two paths of them traveled across load flanks twice and were affected by the tightening torque strongly. Expected features of the pulses through these transmission paths corresponded with the features of the second pulse obtained in the experiment. This result suggested that the proposed method was qualitatively demonstrated with the experimental approach.

30 A New Testing Method for Bolt Loosening with Transmitted Ultrasonic Pulse

233

References 1. Yamamoto, E., Motegi, R.: Measurement of bolt tension by ultrasonic technique. J. Mar. Eng. Soc. Jpn. 13(10), 781–785 (1978) (in Japanese) 2. Jhang, K.Y., Quan, H.H., Ha, J., Kim, N.Y.: Estimation of clamping force in high-tension bolts through ultrasonic velocity measurement. Ultrasonics. 44, e1339–e1342 (2006) 3. Kim, N., Hong, M.: Measurement of axial stress using mode-converted ultrasound. NDT & E Int. 42(3), 164–169 (2009) 4. Argatov, I., Sevostianov, I.: Health monitoring of bolted joints via electrical conductivity measurements. Int. J. Eng. Sci. 48(10), 874–887 (2010) 5. Jalalpour, M., Kim, J.J., Reda Taha, M.M.: Monitoring of L-shape bolted joint tightness using thermal contact resistance. Exp. Mech. 53(9), 1531–1543 (2013) 6. Huda, F., Kajiwara, I., Hosoya, N., Kawamura, S.: Bolt loosening analysis and diagnosis by non-contact laser excitation vibration tests. Mech. Syst. Signal Process. 40(2), 589–604 (2013) 7. Krolikowski, J., Szczepek, J.: Prediction of contact parameters using ultrasonic method. Wear. 148, 181–195 (1991)

Part VI

Vibration Isolation and Reduction

Chapter 31

Vibration Isolation Performance of an LQR-Stabilised Planar Quasi-zero Stiffness Magnetic Levitation System Nur Afifah Kamaruzaman, William S. P. Robertson, Mergen H. Ghayesh, Benjamin S. Cazzolato, and Anthony C. Zander

31.1 Introduction A maglev system studied in this paper uses a combination of fixed permanent magnets exerting upward repulsing (positive stiffness) and attracting (negative stiffness) forces on a set of magnets levitating between them. Such arrangement of positive and negative stiffness mechanisms in parallel creates an isolator with quasizero stiffness (QZS) characteristics, enabling low-frequency vibration isolation and large static load support [1]. One of the most important aspects of the maglev system is the design of an active stability control as the QZS position is only marginally stable; the added control stiffness must be sufficiently large to stabilise the system but not so large that the vibration isolation performance suffers. Various control strategies have been presented to stably levitate maglev systems in general, but only few consider vibration isolation performance [2]. In structural control, linear quadratic regulator (LQR) is regarded as the most effective, yet simple in comparison to other common algorithms [3]. Currently lacking in the literature is the investigation of an LQR design for maglev vibration isolation systems, as most studies targeted on trajectory tracking performance [4]. The LQR aims to minimise the cost function  J =

∞4

5 x(t)T Qx(t) + u(t)T Ru(t) dt

(31.1)

0

N. A. Kamaruzaman () · W. S. P. Robertson · M. H. Ghayesh · B. S. Cazzolato · A. C. Zander School of Mechanical Engineering, The University of Adelaide, Adelaide, SA, Australia e-mail: [email protected]; [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/10.1007/978-3-030-48153-7_31

237

238

N. A. Kamaruzaman et al.

by finding a reasonable balance between reducing the deviations of the state variable x from their equilibrium and the control action u through the weighting matrices Q and R. A linear state-feedback control law u(t) = −Kx(t) provides the solution, where K = −R−1 BT P is the feedback gain matrix obtained by solving the algebraic Riccati equation AT P + PA − PBR−1 BT P + Q = 0. The selection of the LQR weighting matrices remains a challenge to-date; it is likely that different requirements will produce varying solutions. Metaheuristic algorithms such as genetic algorithm (GA) have been used extensively to establish optimal LQR weighting matrices in various applications [5]. Though not as popular as GA, pattern search is more efficient for a small search area as it uses comparatively fewer steps to converge to a global solution. In this paper, pattern search is adopted to find the LQR weighting matrices that will lead to an optimal isolation performance and control cost. The design parameters of the pattern search algorithm such as its objective and constraints are outlined. The interplay between the LQR weighting matrices and the vibration isolation performance is examined, which helps to assess the suitability of the controller design approach.

31.2 Maglev System Design and Modelling The design and modelling of the QZS maglev system constrained in the xz-plane are outlined here to enable LQR design with pattern search algorithm in Sect. 31.3. The permanent magnet spring design shown in Fig. 31.1a follows that of our previous studies [6]. A particular emphasis was placed on the normalised horizontal gap p

Fig. 31.1 The designed maglev system. The QZS operating position remains at z = 0 for all normalised horizontal gap p with appropriate position offset ht and hb . (a) Schematic of the planar maglev system with six cuboid permanent magnets. Large arrows show the magnetisation direction. (b) 6-DOF actuation with four vertical V and horizontal H voice coils. Note only 3-DOF actuation is required for the planar system

31 Vibration Isolation Performance of an LQR-Stabilised Planar Quasi-zero. . .

239

Fig. 31.2 Rot. natural frequency f at z = 0 for varying gap p. Low-frequency isolator performance improves with decreasing f

(= cp/c); it was shown that the rotational stability of the system improves with increasing gap p, that is, the rotational 6 stiffness kββ increases, thus reducing control cost. Yet, the natural frequency f (= kββ /Iyy /2π ) decreases favourably due to the (relatively) large influence of the lever arm cp as shown in Fig. 31.2. The moment of inertia Iyy is approximated as M/3 · (cp/2)2 . The equations of motion of the system (without external excitations) are given by 7 M x¨ = Fx (x, z, β) − cx x˙ + Fsx ) M z¨ = Fz (x, z, β) − cz z˙ + Fsz − Mg , Iyy β¨ = Ty (x, z, β) − cβ β˙ + Tsy

(31.2)

where the nonlinear magnetic forces Fx , Fz , and torque Ty are functions of the displacement of the levitating mass M in the horizontal x, vertical z, and rotational β DOFs. The (passive) magnetic force, torque, and stiffness expressions have been outlined in [7]. Considering eddy current and other energy losses, a damping ratio of 20% is used to calculate the viscous damping ci for all axes. The stability control forces Fsx , Fsz , and torque Tsy are expressed as 7 Fsx = FH1 + FH3 ) Fsz = FV1 + FV2 + FV3 + FV4 , Tsy = L(FV2 − FV4 )

(31.3)

where FHj and FVj for j ∈ (1, 2, 3, 4) are the electromagnetic forces of the horizontal Hj and vertical Vj voice coils, and L is half the distance between the voice coils V2 and V4 (Fig. 31.1b). The force constant of each horizontal and vertical

240

N. A. Kamaruzaman et al.

Fig. 31.3 3-D point cloud indicates 125 positions of linearisation; ‘×’ marks the centred, QZS position

1

¯ (°)

0.5 0

–0.5

5 0

–1 –5

0 x (mm)

5

–5 z (mm)

voice coil is 20 N/A and 10 N/A, respectively, with a maximum current supply of 2A. The design of the voice coil imposes displacement/rotational limits li to the mass, that is, ±4.5 mm, ±6.0 mm, and ±1.0° from the centred position in the horizontal x, vertical z, and rotational β directions, respectively. The nonlinear equations given in Eq. (31.2) were linearised at 125 operating positions as illustrated in Fig. 31.3. Using a Taylor series expansion, the linearised model is obtained as ⎡ ⎡ ⎤ 0 x˙ ⎢ (j ) ⎢ x¨ ⎥ ⎢− kMxx ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎢ z˙ ⎥ ⎢ ⎢ ⎥ = ⎢ kzx (j ) ⎢ z¨ ⎥ ⎢ − M ⎢ ⎥ ⎢ ⎣β˙ ⎦ ⎢ 0 ⎣ k (j ) β¨ − βx Iyy

1

0

0

(j ) cx −M − kMxz

0

0

0 − kMzz 0 0 k

βz 0 − Iyy

0 1

0

0

k (j ) − Mxβ

0 k

(j )

k

(j )

(j )

cz −M − Mzβ 0 0

(j )

0 − Iββ yy

0 0 0 1 c

− Iyyβ

⎤

⎡ ⎤ ⎡ 0 ⎥ x ⎥ ⎢ x˙ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢M ⎥⎢ ⎥ ⎢ 0 ⎥ ⎢z⎥ ⎢ ⎥⎢ ⎥ + ⎢ ⎥ ⎢ z˙ ⎥ ⎢ 0 ⎥⎢ ⎥ ⎢ ⎥ ⎣β ⎦ ⎣ 0 ⎦ 0 β˙

0 0 1 M 0 0 0 1 0 M 0 0 0 0

⎤⎡

⎤ F H1 ⎥ ⎢F ⎥ ⎥ ⎢ H3 ⎥ ⎥⎢ ⎥ ⎥ ⎢ FV1 ⎥ ⎥, 1 1 1 ⎥⎢ ⎢ ⎥ F M M M ⎥ ⎢ V2 ⎥ ⎥ 0 0 0 ⎦ ⎣ FV3 ⎦ L L FV4 Iyy 0 − Iyy 0 0 0 0 0 0

0 0 0

(31.4) where is the spring stiffness in υ direction with ψ displacement/rotation at the position j . Gain scheduling is adopted to account for the dynamic changes. (j ) kυψ

31.3 LQR with Pattern Search Algorithm The selection of the LQR weighting matrices begins with Bryson’s rule, that is, the elements of the diagonal matrices are defined as the inverse of the square of the maximum value of the corresponding state x and input u. For clarity, the maximum displacement/rotation of the mass and control force in each DOF are as discussed in Sect. 31.2, while the maximum velocity are defined as the fraction of the corresponding peak-to-peak displacement/rotation limit li over the simulation sampling time of ts = 0.025 s. Two scaling penalties q and r are introduced as the design parameters for the weighting matrices, thus

31 Vibration Isolation Performance of an LQR-Stabilised Planar Quasi-zero. . .



Q = q diag 8 R=r diag



1

1

1

1

1

241



1

, , , , , (xmax )2 (x˙max )2 (zmax )2 (˙zmax )2 (βmax )2 (β˙max )2 1

1

1

1

1

1

, , , , , (FH1max )2 (FH2max )2 (FV1max )2 (FV2max )2 (FV3max )2 (FV4max )2

, (31.5) 9 .

(31.6) With the above definition of LQR weighting matrices, the two scaling penalties are dependent and linked by a weighting ratio q/r. The pattern search algorithm is applied to find the minimum weighting ratio q/r (accuracy of 2 decimal places) that would give a stable solution. This objective is decided based on the assumption that a (relatively) small controller gain K (∝ q/r) is preferred for low-frequency vibration isolation. An online search method is proposed, where the weighting ratio q/r is updated until it converges such that the Gaussian random base excited system is stable (nonlinear constraint) with a small, but sufficiently large control effort. The excited system is considered to be stable if the mass levitates within the displacement/rotational limits li . Also, the electromagnetic (control) forces are constrained during the MATLAB/Simulink simulation. For each iteration of the pattern search, the weighting ratio q/r remains constant across all 125 operating positions.

31.4 Numerical Results and Discussion Unless otherwise mentioned, the analyses presented here is based on the maglev system with the normalised horizontal gap of p = 2. Figure 31.4 shows the resultant weighting ratio q/r for varying excitation amplitude factor Λ, where the standard deviation of the Gaussian random base excitation for each DOF is defined as σi = Λ3 | li |. As the excitation amplitude increases, a larger control signal u is required to stabilise the system, corresponding to a higher weighting ratio q/r. The vibration isolation performance of the system with the resultant minimum weighting ratio of q/r = 1.44 (for Λ = 0.3) are compared against that of 5 and 10 times said q/r to assess the effectiveness of the designed objective of the pattern search. An increasing ratio q/r implies the mass deviates less from its equilibrium displacement and velocity, which translates as having smaller resonance peaks (Fig. 31.5a). The frequency at which vibration isolation begins however shifts towards the higher frequency region. This reflects the importance of minimising the ratio q/r, especially for low-frequency vibration isolation. The peak resonance can further be suppressed by manipulating the velocity terms of the LQR weighting matrices, or additional absolute damping control.

242

N. A. Kamaruzaman et al.

Fig. 31.4 Resultant weighting ratio q/r for varying base excitation amplitude obtained from the pattern search with accuracy of 2 decimal places

(a)

(b)

Fig. 31.5 Vibration transmissibility with random base excitations (Λ = 0.3). (FFT properties: 210 NFFT, Hanning window, 50% overlap, 10 avg., 40 Hz sampling frequency). dB referenced against mm or degree. (a) Effects of increasing weighting ratio q/r; excitation amplitude factor is fixed to Λ = 0.3. (b) LQR robustness to changes in rot. stability; q/r = 1.44 for all gap p, Λ = 0.3

The robustness of the LQR controller to changes in the rotational characteristics of the maglev system has been demonstrated. Figure 31.5b shows the isolation performance of the base excited (Λ = 0.3) system for the normalised horizontal gap of p = [2, 5, 8], using the weighting ratio q/r solved for p = 2. The designed LQR controller not only stabilises the system but effectively applies less control effort and suppresses the resonance peaks as the rotational stability of the system improves with the normalised horizontal gap p.

31 Vibration Isolation Performance of an LQR-Stabilised Planar Quasi-zero. . .

243

31.5 Conclusion This paper has proposed a novel LQR design for a quasi-zero stiffness maglev system using pattern search algorithm. The main advantage of the proposed online pattern search method is that the nonlinear system is guaranteed to be stable when subjected to base excitations since external excitations are included in the simulation. The robustness of the designed LQR controller with gain scheduling has been demonstrated for the nonlinear, position-varying QZS maglev vibration isolation system at 125 points across the operating envelope and multiple sets of normalised horizontal gap p.

References 1. Liu, Y., Xu, L., Song, C., Gu, H., Ji, W.: Dynamic characteristics of a quasi-zero stiffness vibration isolator with nonlinear stiffness and damping. Arch. Appl. Mech. 89(9), 1743–1759 (2019) 2. Yalçin, B., Sever, M., Erkan, K.: Observer-based H2 controller design for a vibration isolation stage having hybrid electromagnets. J. Low Freq. Noise Vib. Active Control 37(4), 1134–1150 (2018) 3. Saaed, T., Nikolakopoulos, G., Jonasson, J.-E., Hedlund, H.: A state-of-the-art review of structural control systems. J. Vib. Control 21(5), 919–937 (2015) 4. Maji, D., Biswas, M., Bhattacharya, A., Sarkar, G., Mondal, T., Dey, I., MAGLEV system modeling and LQR controller design in real time simulation. In: 2016 International Conference on WiSPNET, pp. 1562–1567 (2016) 5. Wongsathan, C., Sirima, C.: Application of GA to design LQR controller for an inverted pendulum system. In: 2008 IEEE International Conference on Robotics and Biomimetics (2009) 6. Kamaruzaman, N.A., Robertson, W.S.P., Ghayesh, M.H., Cazzolato, B.S., Zander, A.C.: Improving passive stability of a planar quasi-zero stiffness magnetic levitation system via lever arm. In: 2018 IEEE INTERMAG, pp. 1–5 (2018) 7. Kamaruzaman, N.A., Robertson, W.S.P., Ghayesh, M.H., Cazzolato, B.S., Zander, A.C.: Quasizero stiffness magnetic levitation vibration isolation system with improved passive stability: a theoretical analysis. In: Acoustics (2018)

Chapter 32

Proposition of Isolation Table Considering the Long-Period Earthquake Ground Motion (Method of Changing Natural Frequency of Isolation System with Additional Spring) Shozo Kawamura, Tetsuhiko Owa, Tomohiko Ise, and Masami Matsubara

32.1 Introduction Currently, various types of isolation systems are being developed, such as the horizontal isolation system [1], three-dimensional isolation system [2], and small isolation system especially for artwork [3]. And many of them are designed for the short-period ground motion. However, in recent years, it is known that the short-period ground motion appears first, and then, the long-period ground motion appears, depending on the characteristics of the ground. Long-period ground motion contains periodic components close to the natural frequency of a typical isolation system. Therefore, there is a problem that the response of the isolation system increases significantly because of resonance when the long-period ground motion acts onto the isolation system [4]. For the above reasons, some isolation systems have been proposed for long-period ground motions. For example, when the isolation system approaches resonance, the restoring force or damping force is temporarily applied in some way. In addition, there are other methods that add the restoring force and damping force by active control [5]. In this study, a new type passive isolation system is proposed for not only the short-period ground motion but also the long-period one. The basic concept is that when the ground motion shifts from a short-period one to a long-period one and the amplitude of the isolation table exceeds the threshold, the natural frequency changes continuously with additional spring. In the previous studies, the stiffness or friction may be added only during the large amplitude of the response, while S. Kawamura · T. Owa () · M. Matsubara Toyohashi University of Technology, Toyohashi, Aichi, Japan e-mail: [email protected]; [email protected]; [email protected] T. Ise Kindai University, Osaka, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_32

245

246

S. Kawamura et al.

this study is characterized by adding stiffness continuously. Firstly, the proposed isolation system is modeled as a one-degree-of-freedom system, and the validity of the proposed method is checked by numerical simulation with sine sweep. The system parameters are determined by the experimental equipment, and one of the earthquakes is obtained from an actual earthquake in Japan. Secondly, based on the simulation results, the isolation system is constructed. Finally, the vibration suppression effect is confirmed by the experiment when the sine sweep wave and seismic wave act on the system. In addition, since ground motion does not shift from a long-period to a short-period, the isolation system does not resonate after the additional spring is connected.

32.2 Outline of Proposed Isolation System In the isolation system proposed in this research, an additional spring is attached to the isolation table when the vibration amplitude of the isolation table exceeds the threshold. Figure 32.1 shows a single-degree-of-freedom model of the isolation system. Here, the isolation system is composed of a consistent mass m, the initial spring with spring constant k1 , and the damper with damping coefficient c. The spring constant of the additional spring is k2 . Also, the distance between the isolation table and the additional spring is set as the clearance Xa . When the relative displacement z = x − y exceeds the clearance Xa , an additional spring is connected to the isolation table. Figure 32.2 shows the concept of resonance suppression when the earthquake motion shifts from a short-period to a long-period in the singledegree-of-freedom model. When the vibration of the isolation table becomes large, because the dominant frequency of the ground motion approaches to the natural frequency fa , the natural frequency of the isolation system moves from fa to fb at the intersection of the resonance curves. Therefore, it is possible to suppress the resonance of the isolation table caused by long-period ground motion. Fig. 32.1 A single-degree-of-freedom system model of isolation system

32 Proposition of Isolation Table Considering the Long-Period Earthquake. . .

247

Fig. 32.2 Suppression mechanism of resonance

32.3 Validity Check by Numerical Simulation 32.3.1 Parameter Setting In order to carry out the simulation, the parameters of the model are determined according to the actual ground motion. Firstly, the natural frequency of the initial state that suppresses short-period ground motions is fa =1.0 (Hz), and the natural frequency of the additional spring state that suppresses long-period ground motions is fb =5.0 (Hz). The damping ratio is ζ a = ζ b =0.05 (−) in both states. Then, the intersection point of the resonance curve is determined; the frequency is 1.39 (Hz), and the magnification factor of amplitude is 1.08 (−). The clearance Xa is determined so that the additional spring is connected at this frequency. Secondly, the amplitude of the input acceleration of the base excitation is determined using actual ground motion. The root-mean-square value of the 2007 Niigataken Chuetsuoki earthquake (Minami Uonuma Rokkamachi NS direction from Meteorological Agency HP in 2018), which is a long-period ground motion, is 0.31 (m/s2 ). Therefore, the amplitude of the acceleration of the base excitation a is 0.3 (m/s2 ). Lastly, the amplitude of displacement of the base excitation Y and the one of displacement of the table X at the switching frequency are determined, respectively, as follows: Y =

a (2πf )2

= 3.93 × 10−3 [m]

X = 1.08 × Y = 4.25 × 10−3 [m] .

(32.1)

(32.2)

When the frequency is f =1.39[Hz], the phase in the initial state is ϕ = −163.61[◦ ]. Therefore, the maximum value of relative displacement (x − y) in steady-state vibration is as follows:

248

S. Kawamura et al.

|x − y|max = |Xsin (ωt) − Y sin (ωt + ϕ)|max = 8.10 × 10−3 [m] .

(32.3)

Therefore, the clearance Xa is determined to be 8.0 (mm).

32.3.2 Simulation Conditions The base excitation is simulated for the model in Fig. 32.1. Specifically, the base is oscillated with a logarithmic sine sweep signal of acceleration amplitude A=0.3 (m/s2 ), maximum frequency fmax =5.0 (Hz), minimum frequency fmin =0.5 (Hz), sweep time T=60 (s), and sweep speed b = −0.075 (Hz/s). The simulation is carried out on the following two isolation systems: (A) Isolation system A (corresponding to short-period ground motion). The equation of motion is Eq. (32.4) for all simulation times. Also, the relative displacement z = x − y is introduced, and Eq. (32.5) is obtained: mx¨ + c (x˙ − y) ˙ + k1 (x − y) = 0

(32.4)

z¨ + 2ζa ωa z˙ + ωa 2 = −y¨

(32.5)

(B) Isolation system B (corresponding to both short-period and long-period ground motion). The equation of motion is the same as Eq. (32.5) as long as the magnitude of the relative displacement z = x − y does not exceed Xa . When the relative displacement z = x − y exceeds Xa and the additional spring is connected, Eqs. (32.6) and (32.7) are obtained: mx¨ + c (x˙ − y) ˙ + k1 (x − y) + k2 (x − y − Xa ) = 0   ¨ z¨ + 2ζb ωb z˙ + ωb 2 Z − ωb 2 − ωa 2 Xa = −y.

(32.6) (32.7)

32.3.3 Simulation Results Figures 32.3 and 32.4 show the simulation results of isolation systems A and B, respectively. In both cases, the response decreases around 0–30 s, corresponding to short-period ground motion. On the other hand, after 35 s, the results are different. As shown in Fig. 32.3 of isolation system A, the response after 30 s greatly increases

32 Proposition of Isolation Table Considering the Long-Period Earthquake. . .

249

Fig. 32.3 Results of the simulation isolation system A

Fig. 32.4 Results of the simulation isolation system B

due to resonance, and the response becomes maximum around 46 s. However, Fig. 32.4 shows that the response of isolation system B reduces after 35 s. This is because the additional spring is connected around 35 s. However, at the time of additional spring connecting, the acceleration response is temporarily increased to about 1 m/s2 . The reason is considered that a large restoring force acts on a small displacement after connecting. Furthermore, since the additional spring has a natural length, the vibration center moves around 8 mm after connecting, which is a little smaller than Xa . From the above results, it is confirmed that isolation system B has the effect of suppressing the resonance when the base excitation shifts from a shortperiod to a long-period.

250

S. Kawamura et al.

Fig. 32.5 Overall view of the isolation system

32.4 Outline of Experimental Equipment of Isolation Table Figure 32.5 shows a photograph of the constructed isolation system. The upper plate corresponds to the isolation table, and the lower plate corresponds to the base surface. A weight is mounted on the upper plate, and an initial spring is attached to the plate. Then, an additional spring connecting mechanism using a neodymium magnet and silicone rubber sponge is installed on one side of the upper plate. This system is mounted on a vibrating table and vibrated horizontally to evaluate the vibration reduction effect.

32.5 Results of Experiment 32.5.1 Sine Sweep Excitation Experiment The system is excited by the sine sweep signal. Figures 32.6 and 32.7 show the experimental results of isolation systems A and B, respectively. Both results show the same tendency as the simulation results. Therefore, it can be seen that the isolation table produced exhibits an effective response reduction effect.

32.5.2 Actual Ground Motion Excitation Experiment The system is excited by the actual earthquake ground motion, which is the 2007 Niigata Prefecture Chuetsu-oki earthquake (Minami Uonuma Rokkamachi NS direction from Meteorological Agency HP in 2018). The dominant period of the short-period excitation is about 0.4 s from 13 to 20 s, while the one of the long-period excitation is about 1.0 s after 20 s. Figures 32.8 and 32.9 show

32 Proposition of Isolation Table Considering the Long-Period Earthquake. . .

251

Fig. 32.6 Results of the experiment isolation system A (sine sweep wave)

Fig. 32.7 Results of the simulation isolation system B (sine sweep wave)

the experimental results of isolation systems A and B, respectively. As shown in Fig. 32.8 of isolation system A, the response increases from 25 to 60 s due to resonance. On the other hand, Fig. 32.9 shows that the response is reduced after 25 s. This is because the additional spring is connected around 25 s. From the above results, it can be seen that the isolation system proposed in this study is effective for actual ground motion.

252

S. Kawamura et al.

Fig. 32.8 Results of the experiment isolation system A (actual earthquake motion)

Fig. 32.9 Results of the experiment isolation system B (actual earthquake motion)

32.6 Conclusions In this study, we proposed a new type passive isolation system considering longperiod ground motion. First, the proposed isolation system is modeled with a single-degree-of-freedom model, and simulation is performed using sine sweep waves. As a result, it is shown that the vibration can be reduced by the proposed isolation system. In addition, an isolation system with an additional spring connecting mechanism by a neodymium magnet and a sponge is created, and an excitation experiment is performed on that system. As a result, it is shown that the vibration of the isolation table can be reduced even when the sine sweep waveform and the

32 Proposition of Isolation Table Considering the Long-Period Earthquake. . .

253

actual earthquake ground motion act on the system. Therefore, it is found that the isolation system proposed in this study can suppress the vibration of the isolation table even if the ground motion shifts from a short-period to a long-period. Acknowledgments This work was supported by JSPS KAKENHI Grant Number JP18K04019.

References 1. Matsuoka, T., Takayanagi, T., Niwa, T., Ohmata, K.: Study on small seismic isolation table using coil springs. Trans. JSME (in Japanese). 81, 825 (2015). https://doi.org/10.1299/transjsme.1400607 2. Fujita, T., Kuramoto, S., Omi, T.: Three-dimensional earthquake isolation system. Trans. Jpn. Soc. Mech. Eng. Ser. C. 51(471), 2768–2778 (1985) (in Japanese) 3. Ueda, M., Ohmata, K., Yamagishi, R., Yokoo, J.: Vertical and three-dimensional seismic isolation tables with bilinear spring force characteristics (For which a A-shaped link mechanism is used). Trans. Jpn. Soc. Mech. Eng. Ser. C. 73(735), 2932–2939 (2007) (in Japanese) 4. M.R. Shekari, N. Khaji, M.T. Ahmadi, On the seismic behavior of cylindrical base-isolated liquid storage tanks excited by long-period ground motions. Soil Dyn. Earthq. Eng., 30, 968– 980 (2010) 5. Lua, L.-Y., Linb, G.-L., Kuoa, T.-C.: Stiffness controllable isolation system for near-fault seismic isolation. Eng. Struct. 30, 747–765 (2008)

Chapter 33

Experimental Vibration Analysis of Seismic Isolation System Using Inertial Mass Damper Ryo Masano, Nanako Miura, and Akira Sone

33.1 Introduction Nowadays, there are a lot of damages in structures (bridge, building, road, etc.) which are caused by earthquakes in Japan. In addition, the damage of computers leads to losing important information. There is a seismic isolation system installed, but it is reported that computers fall over. Because of some reasons including the seismic isolation system resonates and the displacement response increases, the range of motion is exceeded. Also, in the case of active seismic isolation, there is a problem that the active control system does not work when the power supply is stopped by the earthquake. In fact, in the Tohoku earthquake in 2011, it was reported that power was cut off due to a large-scale blackout, and active seismic isolation did not work [1]. Recently, the dependence of information systems has been rapidly expanding in all fields. Thus, the earthquake countermeasures for computers should be improved. Therefore, we improve a new design of a seismic isolation system using power generation function installed under a large computer as earthquake countermeasure. In this research, an inertial mass damper (inerter) is applied for this seismic isolation system. An inerter gives a reaction force proportional to relative acceleration, and it is one of the damping devices effective for reducing the response. Until now, there are some researches about using inertial mass damper to control the vibration. Isoda et al. [2–4] have investigated the response characteristics of the response reduction mechanism using an analysis model. Sugimoto et al. [5] have introduced an inerter into a full-scale building and studied it by a full-scale

R. Masano () · N. Miura · A. Sone Kyoto Institute of Technology, Kyoto, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_33

255

256

R. Masano et al.

excitation experiment. Regarding power generation function, Miura et al. [6] studied two-layer seismic isolation: one is a layer to reduce vibration (upper layer), and the other is a layer to regenerate energy (lower layer). In this research, we design a seismic isolation system with response reduction effect and power generation performance by incorporating the power generation mechanism into the inerter and conduct the vibration experiment. In addition, the effects of parameters are evaluated by simulation.

33.2 Model The analysis model is shown in Fig. 33.1, and the equation of motion is shown in Eq. (33.1):       M x¨ + C x˙ − x˙f + K x − xf + ψ x¨ − x¨f = 0

(33.1)

where M represents the mass of the computer, C represents the damping, K represents the stiffness, ψ represents the inertial mass, x represents the absolute displacement of the isolation device, and xf represents the absolute displacement of the ground. We incorporate inertia mass ψ in parallel into a single mass system which consists of M, C, and K. Next, we describe the features of an inerter which we actually designed. The outline diagram of an inerter is shown in Fig. 33.2. This inerter consists of the ball screw, the ball nut, the gears, and the bearing. It can convert the displacement due to vibration into the rotation angle of gear1 where x represents the axial displacement of the ball screw. When the flywheel with rotational inertia moment Iθ rotates θ , the obtained torque T is defined as Eq. (33.2): T = Iθ θ¨1 =

   2 D1 2π 1 π m D1 2 m x¨ = x¨ 2 2 Ld Ld 2

M

x xf Fig. 33.1 Analysis model

K

C

ψ

(33.2)

33 Experimental Vibration Analysis of Seismic Isolation System Using. . .

Transducer

θ2

257

Gear

Bearing Ball nut

Gear

(Flywheel)

Ball screw x Ld θ1 Fig. 33.2 Inerter

where Ld represents the ball screw lead, D1 represents the flywheel diameter, and m represents the flywheel mass. Also, the reaction force F can be expressed by Eq. (33.3): F =

T Iθ θ¨1 π 2 D1 2 mx¨ = ψ x¨ = = Ld /2π Ld /2π 2Ld 2

(33.3)

From Eq. (33.3), it is generally considered that the inertial mass ψ is about 100–1000 as heavy as the flywheel mass m and a large inertial mass effect can be obtained. Also, we incorporate a generation mechanism in the inerter. The flywheel rotates at high speed when the inerter generates an axial reaction force. We propose a mechanism to charge with transducer using the rotation. θ˙1 and θ˙2 represent the angular velocity of gear1 and gear2, respectively. Gear1 is attached directly to the ball screw; gear2 is attached to the transducer. Z1 represents the number of gear1 teeth; Z2 represents the number of gear2 teeth. θ˙2 is shown in Eq. (33.4): θ˙2 =

Z1 2π Z1 x. ˙ θ˙1 = Z2 Ld Z2

(33.4)

In this research, we incorporate the inerter using power generation function in the seismic isolation system. The outline diagram of the seismic isolation system is shown in Fig. 33.3. A mechanism which can convert the displacement due to vibration into the rotation angle of the gear and charge the transducer is designed. Tension springs, k = 40 (N/m), were added from both ends. The seismic isolation layer can be displaced by 45 mm from the neutral plane and is stopped by the stopper above that. In addition, a weight was attached to the nut of the ball screw. By adjusting the weight M and changing inertia mass ψ, we tried to design to be able to conduct an experiment by adjusting various parameters. Table 33.1 shows the values of seismic isolation system parameters.

258

R. Masano et al.

Ball nut

Bearing Tension spring

Gear

Ball screw

Liner guide

Aluminum frame 476 [mm]

Fig. 33.3 Model of a seismic isolation system using an inertial mass damper Table 33.1 Seismic isolation system parameters

Parameter Mass of the structure Mass of the rotor Spring constant Diameter of gear1 Diameter of gear2 Lead Number of gear1 teeth Number of gear2 teeth Inertial mass Inertial mass ratio

Symbol M m K G1 G2 Ld Z1 Z2 ψ γ

Value 1.99 24.7 × 10−3 80 50 × 10−3 25 × 10−3 10 × 10−3 50 25 3.05 1.53

Unit Kg Kg N/m m m m

Kg

Using Eq. (33.3), the inertial mass ψ is obtained as follows: ψ=

π 2 D1 2 2Ld 2

m.

(33.5)

Also, the inertial mass ratio γ can be expressed as the following equation using the mass of the structure M and the inertial mass ψ: γ =

ψ M

(33.6)

33.3 Experiment Method We actually made the seismic isolation system shown in Fig. 33.4 and verified it using sine wave. Figure 33.4 shows the situation of the experiment. We measured the input wave from the vibration table and the acceleration in the seismic isolation layer and the voltage obtained by the motor at that time.

33 Experimental Vibration Analysis of Seismic Isolation System Using. . .

259

Fig. 33.4 The situation of the experiment

Fig. 33.5 Acceleration response magnifications

33.4 Result 33.4.1 Analysis Result In Fig. 33.1, the Fourier transforms of x and xf is defined as X and XF , respectively. The absolute acceleration response magnification can be expressed as Eq. (33.7): X −ω2 ψ + iωc + k . = 2 XF −ω (M + ψ) + icω + k

(33.7)

From Eq. (33.7), Fig. 33.5 shows acceleration response magnification at γ = 0.5, 1.0, 1.5, and 3.0, respectively. The analysis results in Fig. 33.5 show that the inerter works in a longer period. In addition, it was found that the resonance region became longer in the period as the mass ratio γ became larger, while the response reduction region became larger as the mass ratio γ became smaller.

260

R. Masano et al.

33.4.2 Experimental Result Next, we will describe the results of vibration experiments. Input wave used the sine wave of frequency 3.0 Hz. The time history response of the input wave and the acceleration in the seismic isolation layer at that time are shown in Fig. 33.6. Also, the time history response of the voltage obtained from the motor at that time is shown in Fig. 33.7. Figure 33.6 shows that the seismic isolation layer hardly moves to the sine wave of frequency 3.0 (Hz), but it numerically decreases by about 6.2% on average. Figure 33.7 shows that the relative acceleration causes the motor to rotate, and the maximum voltage of approximately 1.4 V is obtained. In this experiment, it was considered that the inerter hardly worked because the input wave used the sine wave of frequency 3.0 Hz. However, according to Fig. 33.5, we think that it is possible to

Fig. 33.6 Absolute acceleration response

Fig. 33.7 The time history response of the voltage

33 Experimental Vibration Analysis of Seismic Isolation System Using. . .

261

reduce the response by adjusting the parameters, finding them that the inerter works more effectively, and applying them to the experiments.

33.5 Conclusion In this research, we designed the isolation system with power generation performance using inertial mass and conducted vibration experiments. At this stage, the inerter was not functioning well and could hardly obtain the response reduction effect, but from the examination of the analysis model, it was found that response reduction and power generation performance could be expected by adjusting the parameters. Future works are to clarify the effect of parameter adjustment and response reduction effect when the long-period wave is used as input wave by the experiment. Also, it is necessary to conduct verification using seismic waves.

References 1. Nakajima, K., Giron, N., Kohiyama, M., Watanabe, K., Yoshida, M., Yamanaka, M., Inaba, S., Yoshida, O.: Seismic Response Analysis of a Semi-Active-Controlled Base-Isolated Building During the 2011 Great East japan earthquake, Proceedings of the International Symposium on Engineering Lessons Learned from the 2011 Great East Japan Earthquake, Tokyo, Japan (2012) 2. Isoda, K., Hanzawa, T., Tamura, K.: A study on response characteristics of a sdof model with rotating inertia mass dampers. J. Struct. Constr. Eng. AIJ. 74(642), 1469–1476 (2009) (in Japanese) 3. Isoda, K., Hanzawa, T., Tamura, K.: Basic study on vibration control system by rotating inertial mass dampers concentrated in the lower stories. J. Struct. Constr. Eng. AIJ. 78(686), 713–722 (2013) (in Japanese) 4. Nakamura, Y., Hanzawa, T., Isoda, K.: Performance-Based Placement Design of Tuned Inertial Mass Damper, 13th World Conference on Seismic Isolation, Energy Dissipation and Active Vibration Control of Structures, Sendai, Japan (2013) 5. Sugimoto, K., Fukukita, A., Hanzawa, T., Okamoto, M.: Full-scale dynamic vibration test and dynamic model of inertial mass damper. J. Struct. Constr. Eng. AIJ. 83(749), 997–1007 (2018) (in Japanese) 6. Miura, N., Sone, A.: Fundamental study on vibration mode-based design of two-layer seismic isolation for energy harvesting and vibration suppression. J. Struct. Constr. Eng. AIJ. 64B, 173–178 (2018) (in Japanese)

Chapter 34

Development of Sliding-Type Semi-active Dynamic Vibration Absorber Using Active Electromagnetic Force Eiji Kondo, Tsubasa Matsuzaki, Noriyoshi Kumazawa, Mitsuhiro Oda, and Kazumasa Kono

34.1 Introduction Semi-active dynamic vibration absorbers (DVAs) can effectively suppress the vibrations of a controlled object subjected to external harmonic forces at various frequencies and do not require an advanced control system like active DVAs do. The natural frequency and damping coefficients of semi-active DVAs can be adjusted to suppress the forced vibrations of the controlled object. The spring constant of a semi-active DVA is usually tuned by a mechanical device with a complex structure [1–3], and the damping coefficient is tuned mainly by adjustable oil dampers and partly by magnetic dampers [4]. However, the semi-active DVAs developed thus far are large because mechanical devices are used to adjust the spring constants and damping coefficients in these devices. In contrast, the motion of the mass of an active DVA is commonly controlled by an active electromagnetic force according to signals proportional to the displacement and velocity of the mass measured by sensors on the DVA and the controlled object [5]. Thus, the purpose of this study was to develop a compact and easily tunable sliding-type semi-active DVA controlled by an active electromagnetic force.

E. Kondo () · N. Kumazawa Kagoshima University, Kagoshima-shi, Japan e-mail: [email protected] T. Matsuzaki · M. Oda · K. Kono Makino Milling Machine Co., Ltd., Aikawa-machi, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_34

263

264

E. Kondo et al.

34.2 Configuration of the Dynamic Vibration Absorber System Figure 34.1 shows the proposed DVA system, which is composed of a DVA with a voice coil motor (VCM), a controller for tuning proportional and derivative gains, a strain amplifier, and a current amplifier. The relative displacement between the mass of the DVA and the frame of the VCM in Fig. 34.2b is measured by the strain amplifier as the strain of a parallel flat spring deformed by the displacement of the mass.

34.2.1 Structure of the Dynamic Vibration Absorber with a Voice Coil Motor Figure 34.2 shows a photograph and schematic of a DVA with a commercially available VCM. The mass of the DVA is supported by a pair of parallel flat springs, which determines the minimum natural frequency and keeps the mass in the neutral position without an active electromagnetic force. The mass is also fixed to the moving coil.

Strain amp Control signal Vp Vd

Controller

ebi eai

kp

Dynamic vibration absorber (DVA)

kd ka

ebo

Current amp

e Voice coil motor (VCM)

Fig. 34.1 Schematic of sliding-type semi-active DVA system Vibration direction

Mass Parallel flat springs Strain gauge Support Moving coil Frame Permanent magnet

(a) Side view Fig. 34.2 Structural drawing of a DVA with a VCM

Vibration direction

(b) Over view

34 Development of Sliding-Type Semi-active Dynamic Vibration Absorber. . .

265

Fig. 34.3 Model of DVA/VCM

Table 34.1 Free vibration parameters of the DVA with a VCM Spring constant k 3.1 kN/m

Mass m 0.31 kg

Damping coefficient c 7.92 N·s/m

Counter electromotive damping coefficient cm 3.05 N·s/m

34.2.2 Calculation Model of Dynamic Vibration Absorber with Voice Coil Motor Figure 34.3 shows the calculation model of the DVA with a VCM. The motion of the mass of the DVA is described by mx¨ + cx˙ + kx = −fc ,

fc = Kc i,

(34.1)

where x is the displacement of the mass, m is the mass, c is the damping coefficient, k is the spring constant, fc is the magnetic force, i is the current, Ke is the counter electromotive force coefficient, R is the resistance of the coil, and Kc is the force coefficient. The input voltage ebo and gain Ki of the current amplifier are related as e = Ki ebo . From this, the following relationship can be derived from Eq. (34.1): mx¨ + (c + cm ) x˙ + kx + Kf ebo = 0, cm =

Kc Ke Kc Ki , Kf = . R R

(34.2)

The modal parameters of the DVA without an active electromagnetic force were measured through impact testing. The spring constant was obtained from the displacement of the mass and the force applied to the mass. The estimated free vibration parameters of the DVA are given in Table 34.1.

34.3 Controller for Tuning Proportional and Derivative Gains 34.3.1 Controller Overview The input voltage of the controller is proportional to the displacement of the mass of the DVA, measured by the strain amplifier as the strain of the parallel flat spring. The output voltage is a combination of the input voltage multiplied by a proportional

E. Kondo et al. 0.010

12 10 8 6 4

Vd [V] 0.8 1.7 2.5 3.3 4.2 5.0

1.88Vp + 0.103

2 0 0

1

Estimated derivative gain kd

Estimated proportional gain kp

266

2

3

4

5

6

0.006 0.004

0.00151Vd + kd0

0.002 0.000 0

Proportional gain control signal Vp [V]

(a) Estimated proportional gain

Vp [V] 1.0 2.0 3.0 4.0 5.0

0.008

1 2 3 4 5 Derivative gain control signal Vd [V]

(b)

Estimated derivative gain

Fig. 34.4 Estimated gains plotted against the control signal

gain and the rate of change of the input voltage multiplied by a derivative gain. The spring constant and damping ratio of the proposed semi-active DVA can be tuned by adjusting the proportional and derivative gains, respectively, provided by analog electric circuits; these adjustments are determined by the control signals through the external terminals.

34.3.2 Proportional and Derivative Gains The equation for the electric controller circuits with time delay L is expressed as ebo = kp ebi (t − L) + kd e˙bi (t − L) ,

(34.3)

where ebi is the input voltage, ebo is the output voltage, kp is the proportional gain, and kd is the derivative gain. Figure 34.4a shows the estimated proportional gain kp , which increased linearly as the proportional gain control signal Vp was increased, regardless of the derivative gain control signal Vd , as described by kp = 1.88Vp + 0.103.

(34.4)

Figure 34.4b shows the estimated derivative gain kd , which also increased linearly as its corresponding control signal Vd was increased. However, as shown in Fig. 34.4b, there was also some dependence on the proportional gain control signal Vp . This behavior can be described by the following formula: kd = 0.00151Vd + kd0 , kd0 = 1.52 × 10−4 Vp + 3.28 × 10−4 ,

(34.5)

where kd0 is the derivative gain when Vd is equal to 0 V and is a function of Vp .

34 Development of Sliding-Type Semi-active Dynamic Vibration Absorber. . .

267

34.4 Natural Frequency and Damping Ratio 34.4.1 Equation of Motion of Mass with Active Electromagnetic Force The displacement x can be converted to the output voltage of the strain amplifier, which is the input voltage ebi of the control circuit, using a linear conversion coefficient Ks . This substitution into Eq. (34.3) yields the output voltage as a function of x. If the delay time L is sufficiently short relative to the period of vibration, the following approximation can be derived from Eq. (34.3):   ebo = Ks kp x (t) + Ks kd − Ks kp L x(t) ˙ − Ks kd Lx(t). ¨

(34.6)

Consequently, substituting Eq. (34.6) into Eq. (34.2) yields the equation of motion of the mass subjected to an active electromagnetic force fc as     (m − Kd L) x¨ + c + cm + Kd − Kp L x˙ + k + Kp x = 0,

(34.7)

where Kp = Kf Ks kp and Kd = Kf Ks kd . The coefficient Kp is called the electromagnetic spring constant, and the quantity given by Kd – Kp L is called the electromagnetic damping coefficient in this study. Equations for the natural frequency fn and damping ratio ζ could then be derived from Eq. (34.7) as : 1 fn = 2π

c + cm + Kd − Kp L k + Kp ,ζ = . m − Kd L 4π (m − Kd L) fn

(34.8)

34.4.2 Measurement of Natural Frequency and Damping Ratio The natural frequency and damping ratio of the DVA were estimated from the frequency response function in the case where a separate input voltage eai of the controller, which describes the external force to the mass, is considered as the input and the output voltage of the strain amp, which describes the displacement of the mass shown in Fig. 34.1, is considered as the output. Figure 34.5a shows the predicted and estimated values of the electromagnetic spring constant Kp (referred to as the EM spring constant in figures hereafter) plotted against the control signals. The predicted value of Kp was calculated from the following equation: Kp = Kf Ks kp = 3675Vp + 201.4,

(34.9)

which is derived from Eq. (34.4) with Kf = 6.3 and Ks = 310, and the resulting curve is plotted as a solid line in Fig. 34.5a. The estimated values of Kp were calculated as

E. Kondo et al. 50

20000 15000

Kp = 4 2mfn2 k Predicted eq.

10000 5000

Modification

Vd [ V ] 0.2 1.0 1.8 2.6 3.4 4.2 5.0

0 0 2 4 6 8 Proportional gain control signal Vp [V]

Natsural frequency [Hz]

EM spring constant Kp [N/m]

268

40

Predicted eq.

30 20 Modification 10

Vd [ V ] 0.2 1.0 1.8 2.6 3.4 4.2 5.0

0 0 2 4 6 8 Proportional gain control signal Vp [V]

(a) Electromagnetic spring constant

(b) Natural frequency

Fig. 34.5 Electromagnetic spring constant and natural frequency plotted against the proportional gain control signal

Kp = 4π mfn2 − k,

(34.10)

where the term Kd L in Eq. (34.8) was neglected because the estimated values of Kp were not affected by Vd . The estimated Kp values are plotted as symbols, and they increase with increasing Vp at the same rate as the predicted Kp values. However, Eq. (34.9) requires modification as Kp∗ = 3675Vp − 1561 because the predicted value of Kp0 at Vp = 0 V was different from the value of Kp0 indicated by the dashed line extrapolated from the estimated values. Figure 34.5b shows the natural frequency fn plotted against Vp . The measured natural frequency, plotted as symbols, nearly coincided with the natural frequency calculated from the modified electromagnetic spring constant Kp * , plotted as a dashed curve. The predicted electromagnetic damping coefficient Kd – Kp L was calculated as   Kd − LKp =Kf Ks kd −Lkp =2.95Vd + (0.29 − 3671L) Vp + (0.64 − 201L) , (34.11) which was derived from Eqs. (34.4) and (34.5) with Kf = 6.3 and Ks = 310. Figure 34.6a shows the estimated values of Kd – LKp , calculated as Kd − LKp = 4π 2 mfn ζ − c − cm ,

(34.12)

which was derived from the second expression in Eq. (34.8) with the term Kd L neglected. The estimated values of Kd – LKp , plotted as symbols in Fig. 34.6a, increased with increasing Vd at the same rate as predicted by the coefficient of 2.95 in Eq. (34.11), as shown by the solid lines in the figure. Consequently, Eq. (34.11) must be modified as ∗  Kd − LKp = 2.95Vd − 2.07 − 1.43Vp .

(34.13)

20

c cm Vp [V] 0.0 1.0 2.0 3.0 5.0

10

0

-10 0

269

40

Kd LKp = 4 mfn

Daping ratio [%]

EM damping coefficient Kd Kp L [Ns/m]

34 Development of Sliding-Type Semi-active Dynamic Vibration Absorber. . .

30 20

Vp [V] 1.0 2.0 3.0 4.0 5.0

10

2.95Vd + Kd0 1 2 3 4 5 6 7 Derivative gain control signal Vd [V]

(a) Electromagnetic damping coefficient

0

0

1

2

3

4

5

Derivative gain control signal Vd [V]

(b) Damping ratio

Fig. 34.6 Electromagnetic damping coefficient and damping ratio plotted against the derivative gain control signal

Figure 34.6b shows the damping ratio plotted against the derivative gain control signal Vd at different values of the proportional gain control signal Vp . The measured damping ratio, plotted as symbols, almost coincided with the calculated damping ratio, plotted as solid lines, when the proportional gain control signal Vp was greater than 3.0 V but diverged from the calculated damping ratio with increasing Vd when the proportional gain control signal Vp was less than 2.0 V.

34.5 Conclusions The electromagnetic spring constant was found to linearly increase with the control signal for the proportional gain of the controller, and it reached approximately 17 kN/m. Consequently, the natural frequency can be approximately tuned within the range of 14–40 Hz. The electromagnetic damping coefficient also linearly increased with the control signal for the derivative gain in the range of 3–35 Ns/m; however, the electromagnetic damping coefficient when the control signal of the derivative gain was 0 V varied with the control signal for the proportional gain.

References 1. Nemoto, K., et al.: Characteristics of adjustable dynamic vibration absorber. Trans. JSME. 84(862), 18-00062 (2018). https://doi.org/10.1299/transjsme.18-00062. (in Japanese) 2. Nekomoto, Y., Fujita, K.: Development of adaptive tuned mass damper ATMD. Trans. JSME Ser. C. 71(705), 1463–1470 (2005) (in Japanese)

270

E. Kondo et al.

3. Tanaka, N., Kikushima, Y.: A study of semi-active damper (Verification of the adaptive controllability). Trans. JSME Ser. C. 54(497), 160–165 (1988) (in Japanese) 4. Sunakoda, K., et al.: Study on hybrid magnetic damper with effect of magnetization and demagnetization. Trans. JSME Ser. C. 73(736), 3179–3184 (2007) (in Japanese) 5. Seto, K., Sawatari, K., Fujita, K., Furuishi, Y.: A study on active dynamic absorbers. Trans. JSME Ser. C. 57(538), 1834–11841 (1991) (in Japanese)

Chapter 35

Development of a Vibration Isolator Using Air Suspensions with Slit Restrictions Hiroshi Yamamoto, Haruki Nakanozo, and Terumasa Narukawa

35.1 Introduction It is important for a vibration isolator to expand a frequency band in which its vibration transmissibility is reduced less than one. For this purpose, many kinds of suspension which have low stiffness are used to lower a resonance frequency of the vibration isolator. Since an air suspension can realize low stiffness and can adjust its equilibrium position in the vertical direction independently of the mass of a supported object, it is used in the vibration isolator for many situations [1]. In order to lower its stiffness, its volume is enlarged using a reservoir tank connected to a main cylinder with an airflow channel. In this case, the damping effect of the air suspension is due to fluid resistance through the channel. Formerly, its resistance is given by a small hole called an orifice restriction in the channel. However, the damping coefficient of the air suspension with the orifice restriction decreases as vibration amplitude decreases [3, 4, 6]. For this reason, the air suspension with the orifice restriction cannot reduce the resonance amplitude if its oscillating amplitude becomes small. In order to improve its damping characteristics, one of the authors proposes an air suspension with a slit restriction. The slit restriction is fabricated as a narrow airflow channel connected between the cylinder and the reservoir tank. Since a clearance of the slit is very small compared to its length in the flow direction, the airflow in the slit can be considered as a viscous flow. We have fabricated the air suspension with the slit restriction and have made clear that its damping coefficient does not depend on its vibration amplitude [4].

H. Yamamoto () · T. Narukawa Saitama University, Saitama, Japan e-mail: [email protected]; [email protected] H. Nakanozo IMJ Corporation, Tokyo, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_35

271

272

H. Yamamoto et al.

When the vibration isolator uses air suspensions, vibration modes of the supported object are a superposition of a translational mode along the vertical axis and a rotational mode around the horizontal axis. An excited vibration mode mainly depends on the oscillating mode of a foundation. However, vibration characteristics of the multi-degrees-of-freedom vibration isolator supported with air suspensions are still not made clear enough. The purpose of this study is to develop a multi-degrees-of-freedom vibration isolator using four air suspensions with slit restrictions. For this purpose, we apply an optimum design procedure to minimize the resonance amplitude for the fabricating vibration isolator and measure its characteristics in order to validate the advantages of the proposed vibration isolator.

35.2 Configuration of the Proposed Vibration Isolator Figure 35.1a shows detail of the air suspension with slit restriction. This air suspension consists of the cylinder, the piston which is connected to the supported object, the slit restriction fabricated as the narrow airflow channel, and the air reservoir tank. In order to avoid air leak between the cylinder and the piston, a rubber rolling seal is installed between them. Its stiffness and damping in the moving direction of the piston are negligible, and a variation of the volume of the cylinder due to the out-of-plane deformation of the seal is also negligible. The narrow airflow channel corresponding to the slit restriction is composed of a shallow and relatively long groove fabricated on a flat surface of a plate and another flat plate faced on the grooved side of the plate. To form the narrow airflow channel with two plates makes the slit restriction to be realized easily. The damping coefficient corresponding to the equivalent damping ratio can be tuned by tuning the depth of groove of the plate, which corresponds to the clearance of the slit restriction. Figure 35.1b shows the proposed vibration isolator which supports the object by four air suspensions whose characteristics are all the same. This vibration isolator is symmetric about two orthogonal axes through its center of gravity.

Piston Rubber rolling seal

Supported object

Cylinder Slit restriction

Foundation

Reservoir tank

Air suspension (a)Air suspension

(b)Vibration isolator

Fig. 35.1 Configuration of the air suspension with slit restriction and the vibration isolator

35 Development of a Vibration Isolator Using Air Suspensions with Slit Restrictions

273

35.3 Numerical Analysis If the damping force is proportional to the velocity and the damping coefficient is constant, the air suspension can be modeled as a three-element system [2, 4] which consists of two springs and a damper as shown in Fig. 35.2a. In this dynamical model, k(=κA2 p0 /V) is equivalent stiffness, c(=A2 D ) is equivalent damping coefficient, and η(=Vr /V) is volume ratio. The cross section of the piston is A, the whole volume of the air suspension is V(=Vc + Vr ), Vc is volume of the cylinder, Vr is volume of the reservoir tank, p0 is initial pressure, and κ is specific heat. The restriction factor of the slit restriction is D(=12μl/bh3 ), where μ is viscosity, l is slit length, b is slit width, and h is slit clearance. Since the proposed isolator is symmetry as shown in Fig. 35.1b, we obtain a simplified symmetric dynamical model of the 2-DOF system [5] as shown in Fig. 35.2b, which consists of translational and rotational mode. From these models, the equations of motion for translational and rotational mode can be obtained as follows: m¨z +

I θ¨ +

k k k ˙ + (z − w) = 0, c (w˙ − u) (w − z) + (w − u) = 0 1−η 1−η η (35.1)

  k k k lt 2 (θ − ψ) = 0, c ψ˙ − φ˙ + (ψ − θ ) + (ψ − φ) = 0 1−η 1−η η (35.2)

Substitutingz = Zejωt , w = Wejωt , u = Uejωt , θ = Θejωt , ψ = Ψ ejωt , and φ = Φejωt into Eqs. (35.1) and (35.2), transfer functions which describe the relation between the amplitude of the supported object and that of the foundation of the vibration isolator can be obtained as follows: Htra =

2j ηζp + 1 Z = , U −2j η (1 − η) ζp3 − p2 + 2j ηζp + 1

Fig. 35.2 Equivalent dynamical model of the air suspension and the vibration isolator

(35.3)

274

H. Yamamoto et al.

Hrot =

2j ηζp + 1 " = . # −2j α 2 η (1 − η) ζp3 − α 2 p2 + 2j ηζp + 1

(35.4)

In these equations, p(=ω/ωn ) is a nondimensional oscillating angular frequency,  √  √  ωn k/m is natural frequency of the translational mode, ζ = c/2 mk is √ equivalent modal damping ratio of the translational mode, and α = I /m/ lt is a nondimensional radius of gyration. Defining the oscillating center of foundation as a point O as shown in Fig. 35.2b, oscillating mode of the foundation can be described by oscillating mode factor β(=lO /lt ). Only translational mode is oscillated when β = ∞, only rotational mode is oscillated when β = 0, and both modes are oscillated in another case. In order to evaluate characteristics of the vibration isolator for arbitrary oscillating mode as a same manner, using zA = ZA ejωt and uA = UA ejωt corresponding supporting point A, the transfer function can be obtained as follows: H =

ZA Hrot + βHtra . = UA 1+β

(35.5)

Using Eq. (35.5), the optimum equivalent damping ratio which gives the minimum resonance amplitude in arbitrary oscillating mode of the foundation can be obtained. The optimum equivalent damping ratio and corresponding dimensions of the slit restriction in order to fabricate the vibration isolator can be determined. The optimum equivalent damping ratios corresponding to minimize the resonance amplitude in two particular cases can be obtained as follows: ζtra

√ : 2 2−η , ζrot = αζtra . = 4η 1 − η

(35.6)

Equivalent damping ratio ζ tra gives minimum resonance amplitude when translational mode is only oscillated (β = ∞), and ζ rot gives it when rotational mode is only oscillated (β = 0).

35.4 Experimental Apparatus and Procedure Figure 35.3 shows an experimental apparatus for measuring transfer function of the proposed vibration isolator. Its amplitudes of the supported object and the  foundation corresponding to supporting points A and A shown in Fig. 35.2b are measured. The distance lO can be varied to set β( = lO /lt ). The isolator is installed on an electromagnetic oscillator directory when translational mode (β = ∞) is only oscillated.

35 Development of a Vibration Isolator Using Air Suspensions with Slit Restrictions

275

Laser displacement sensor Supported object

FFT analyzer

Rotating foundation

Electromagnetic oscillator

Vibration isolator

Fig. 35.3 Experimental apparatus 3.5

3.5 Exp.(h = 0.10mm) Exp.(h = 0.15mm) Exp.(h = 0.20mm) Theo.(h = 0.10mm) Theo.(h = 0.15mm) Theo.(h = 0.20mm)

Transmissiblity

2.5

2

Exp.(h = 0.15mm) Theo.(h = 0.10mm) Theo.(h = 0.15mm) Theo.(h = 0.20mm)

3 2.5

Transmissibility

3

1.5 1

2 1.5

1

0.5

0.5 (a) β = ∞

0

0

5

10 15 20 Oscillating frequency[Hz]

(b) β = 0

0 25

0

5

10 15 20 Oscillating frequency[Hz]

25

Fig. 35.4 Transfer function of translational mode (β = ∞) and rotational mode (β = 0)

35.5 Experimental and Numerical Results Figure 35.4 shows experimental and numerical results of transfer function where the translational mode (β = ∞) or the rotational mode (β =0) √ is only oscillated,  respectively. Since the nondimensional radius of gyration α = I /m/ lt of this apparatus is 0.87, the natural frequency of rotational mode is a little larger than that of translational mode. As shown in Fig. 35.4a, the numerical results almost coincide with the experimental results. When the slit clearance h is 0.15 mm, the maximum transmissibility of translational mode becomes the smallest as shown in Fig. 35.4a. From Eq. (35.6) and related relationships, the optimum slit clearance for rotational mode corresponding to Fig. 35.4b is about 0.157 mm, which is a little larger than the actual value, h = 0.15 mm, in Fig. 35.4b. Even if it is not the optimum value for rotational mode, the maximum transmissibility becomes small. Because the nondimensional radius of gyration α is 0.87 and close to one.

276

H. Yamamoto et al. 2

3

Transmissibility

2.5

Exp.(β = 0) Exp.(β = 1.7) Exp.(β = 3.4) Exp.(β = ∞)

Maximum Transmissibility

Theo.(β = 0) Theo.(β = 1.7) Theo.(β = 3.4) Theo.(β = ∞)

2

1.5 1

1.5

β=∞ β=0

1

0.5

0.5

0

0

0

5

10 15 20 Oscillating frequency[Hz]

25

0

0.01

0.02 0.03 0.04 Oscillating amplitude[mm]

0.05

Fig. 35.5 Effect of oscillating mode and oscillating amplitude on transmissibility

Left-hand side of Fig. 35.5 shows the effects of the oscillating mode on the transmissibility where the slit clearance h is 0.15 mm. In all cases, the maximum transmissibility is about 1.6 and relatively small. Right-hand side of Fig. 35.5 shows the effects of the oscillating amplitude on the transmissibility the slit clearance h is 0.15 mm. The maximum transmissibility of both the translational mode and the rotational mode is almost independent of the oscillating amplitude of the foundation.

35.6 Conclusions From our obtained results, a procedure in order to reduce the maximum transmissibility of the vibration isolator is as follows: At first, its natural frequency of the rotational vibration mode should be brought close to that of the translational vibration mode. For this purpose, a distance between the connecting position of the supported object to the air suspension and its center of mass should be brought close to a radius of gyration of the supported object. Under the above-described condition, the equivalent damping ratio corresponding to the translational vibration mode should be tuned to be obtained the minimum resonance amplitude in the translational mode. We can tune the equivalent damping ratio to the optimum value by tuning the clearance of the slit restriction. The maximum transmissibility corresponding to the resonance amplitude of the fabricated vibration isolator can be kept about 1.6 in case if not only the translational vibration mode or the rotational vibration mode is oscillated but also both modes are oscillated at the same time. The maximum transmissibility is not varied even if the oscillating amplitude is varied because of the linearity of the damping characteristics of the applied slit restriction. Acknowledgments This work was supported by JSPS KAKENHI Grant Number JP16K06148.

35 Development of a Vibration Isolator Using Air Suspensions with Slit Restrictions

277

References 1. Matsudaira, T.: Air spring for vehicles. J. Soc. Mech. Eng. 60(464), 908–915 (1957) 2. Oda, N., Nishimura, S.: Vibration of air suspension vehicles and their design. Trans. Jpn. Soc. Mech. Eng. 35(273), 996–1002 (1969) 3. Koyanagi, S.: Optimum design methods of air spring suspension systems. Trans. Jpn. Soc. Mech. Eng. Ser. C. 49(439), 410–415 (1983) 4. Yamamoto, H., et al.: Effects of restriction characteristics on characteristics of air suspension. Trans. Jpn. Soc. Mech. Eng. Ser. C. 74(739), 521–528 (2008) 5. Yamamoto, H., et al.: Optimum design method for a two degree of freedom system supported with air suspensions. Trans. Jpn. Soc. Mech. Eng. Ser. C. 74(741), 1129–1136 (2008) 6. Asami, T., et al.: Theoretical and experimental analysis of the non-linear characteristics of an air spring with an orifice. Trans. Jpn. Soc. Mech. Eng. Ser. C. 77(777), 1674–1685 (2011)

Chapter 36

Development of a Tuning Algorithm for a DynamicVibration Absorber with a Variable-Stiffness Property Tappei Kawai, Toshihiko Komatsuzaki, and Haruhiko Asanuma

36.1 Introduction Dynamic vibration absorbers (DVAs), known as typical vibration control devices, are highly effective for harmonic vibration but not for unsteady disturbances. Therefore, semi-active DVAs have been proposed whose natural frequency is tuned to follow the change of disturbance frequency by adjusting stiffness. Many pieces of research have been conducted on DVAs whose natural frequency can be adjusted according to the external magnetic field using an elastomer composite with a controllable stiffness: a magnetorheological elastomer (MRE) [1, 2]. In the studies, the zero crossing method was used to estimate the frequency from the waveform of a reference signal. However, as the frequency became high, inaccurate estimation of the disturbance frequency significantly degraded the damping performance. The estimation accuracy and speed for real-time tracking are issues to be improved in order to maximize the performance of the semi-active DVA. In this study, we propose a frequency estimation method based on an adaptive line enhancer algorithm (ALE) [3] to be used for changing the properties of a frequency-tuneable DVA.

T. Kawai Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa, Japan e-mail: [email protected] T. Komatsuzaki () · H. Asanuma Institute of Science and Engineering, Kanazawa University, Kanazawa, Ishikawa, 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/10.1007/978-3-030-48153-7_36

279

280

T. Kawai et al.

36.2 MRE-Based Variable-Stiffness DVA 36.2.1 Summary of an MRE An MRE, known as a functional material, is an elastomeric composite incorporating ferrous particles. Under the presence of a magnetic field, magnetically polarized particles form chains with interparticle attractive forces causing changes in the apparent viscoelasticity of the composite. Among the attempts that have been made to enhance the changes in their viscoelasticity, the modulus could be changed by several tens to several hundred times by a relatively weak field of about 300 mT [3]. By applying MREs to DVAs, the natural frequency of the DVAs can be adjusted by the external command signal. Accordingly, adaptive-tuned DVAs can be developed that work in a wide frequency range. In this study, we used the heat-curable silicone elastomer and the permalloy particles to form an MRE. The particles with an approximate diameter of 10 μm are randomly dispersed within the host elastomer by 80 wt%. The natural frequency of the DVA developed in this study changes up to 300% to the nominal state without the magnetic field in the case of the smaller excitation level below 0.2 G.

36.2.2 The MRE-Based Variable-Stiffness DVA Figure 36.1 shows the photo and the schematic of the variable-stiffness DVA developed in this study. The DVA consists of duralumin, steel, brass, coil, and MREs. Two ring-shaped MREs are located between two concentric steel rings at upper and lower parts of the device to form a closed magnetic circuit. The MRE stiffness changes according to the magnetic field strength. Brass is attached to compensate for the moving mass.

Fig. 36.1 Appearance and internal structure of variable-stiffness dynamic absorber. (a) A photo of the absorber and (b) schematic of internal structure. When an electric current is applied, a magnetically closed circuit is formed internally by the action of the coil

36 Development of a Tuning Algorithm for a DynamicVibration Absorber. . .

281

36.3 Frequency Estimation Method for Tuning 36.3.1 Zero Crossing Method In past studies, the zero crossing method was conventionally used to estimate the frequency from the waveform of a reference signal [1, 2]. The instantaneous frequency is calculated based on the period between zero crossover points in the time history. However, as the frequency becomes high, inaccurate frequency estimation degrades the damping performance. The covered frequency range is also dependent on the system’s sampling frequency. Moreover, poor noise tolerance may cause false detection of the dominant frequency. A disturbance frequency estimation method with high accuracy and stability is necessary for signals containing noise.

36.3.2 Frequency Estimation Method Based on an ALE In this study, an ALE algorithm [4] is used to estimate the frequency from the waveform of a reference signal. In the proposed tuning algorithm using ALE, the frequency is estimated from the time series of the delayed input signal, based on the fact that the autocorrelation function of noise converges rapidly to zero when the phase difference for several sampling periods is set. As shown in Fig. 36.2, the control system with ALE consists of delay elements and linear predictors. For the estimation, we use acceleration data xk measured in real time from the target vibration system and delayed past acceleration data xk-Δ . The error function εk is defined as the difference between the output of the predictor xk ’ and the measured signal xk . The variable filter coefficient wk is corrected according to the optimization algorithm so that the error is minimized. The disturbance frequency fk is successively estimated from the modified predictor output at every sampling period by the following Eq. (36.1):

Fig. 36.2 Block diagram of ALE for frequency estimation. Filter coefficients that are used to estimate signal frequency are optimized according to an adaptive algorithm so as to minimize the error signal

282

T. Kawai et al.

fk

1 cos−1 = 2π



wk 1 + r2

 Fs .

(36.1)

In the above equation, r denotes the pole radius that determines the bandwidth of the filter. In our case, r = 0.95. Fs represents the sampling frequency. The estimated value fk ’ is then used as an input for the controller to change stiffness of the DVA.

36.4 Fundamental Test of the Variable-Stiffness DVA 36.4.1 Natural Frequency and Damping Ratio Changeability The basic characteristics were evaluated by measuring the variable range of the natural frequency and damping ratio of the variable-stiffness DVA. The DVA is excited by a shaker while changing the excitation level and frequency. Two accelerometers were attached to a vibrator head, and the DVA movable mass and their signals were fed into a fast Fourier transform (FFT) analyzer. The frequency response function was calculated, and the natural frequency was read from the curve. The damping ratio was obtained from the function using the half-power bandwidth method. The measurement results are shown in Fig. 36.3. The natural frequency of the DVA was found to elevate with increasing electric current. The bandwidth is larger when the acceleration level is smaller, and the natural frequency increased up to 240 Hz (150 Hz bandwidth) at an excitation level of 0.2 G. The damping ratio showed little tendency to the applied current irrespective of the excitation level, because a reliable damping ratio value is assumed to be less than 0.1 in the halfpower bandwidth method.

Fig. 36.3 Fundamental properties of the variable-stiffness dynamic absorber: (a) natural frequency variation and (b) damping ratio

36 Development of a Tuning Algorithm for a DynamicVibration Absorber. . .

283

36.4.2 Damping Performance Evaluation as a Passive Damper By applying a constant electric current and fixing the elastic modulus of MRE, the damping performance of the DVA working as a passive device was evaluated. The target vibration system composed of an acrylic plate having dimensions of 390 mm × 490 mm × 8 mm was clamped at the top of a rigid box. The box was made of duralumin plates with 20 mm thickness and inner dimensions of 350 mm × 450 mm × 500 mm. The plate was excited acoustically by a loudspeaker placed inside the box, and a swept-sine excitation signal was generated by a function generator. The system responses of the swept-sine excitation of frequency varying from 40 to 250 Hz at a speed of 32 Hz/s were obtained. The sampling frequency of the signal processor was set at 640 Hz, and the measurement time was set to 6.4 s. Figure 36.4 shows the acceleration response of the plate. When the current was low, the response was reduced in the low frequency range. The reduction range shifted to the higher frequency as the applied current increased. These results imply that the damping performance can be dramatically improved by tuning the absorber’s natural frequency and tracing the minimum response at each applied current.

Fig. 36.4 Frequency response of the vibration system obtained in the experiment. The response is reduced by the action of the dynamic absorber with respective electric current values applied

284

T. Kawai et al.

36.5 Real-Time Control Using the Proposed Tuning Algorithm The frequency is detected in real time from the reference signal using two types of frequency estimation methods described above, and the current value is automatically controlled to tune the natural frequency of the DVA. As shown in Fig. 36.5, the input signal is subjected to the frequency estimation by the DSP controller, and the corresponding appropriate current value is calculated and supplied by the DC power supply. The applied current value is determined with reference to the previously defined correspondence table between the natural frequency of the DVA and the output voltage. Two reference signals were adopted and compared: (i) the sweep signal generated by the signal generator and (ii) the acceleration signal of the DVA. Obviously, the use of the latter signal is preferred for practical applications. The experimental conditions are the same as in Sect. 36.4.2, but the measurement time was extended to 25.6 s. Figures 36.6 and 36.7 show the estimated frequency histories and the vibration system responses when respective reference signals were used. The responses show

Fig. 36.5 Experimental setup for the real-time control: (a) input signal is obtained from the function generator, and (b) input signal is obtained from the acceleration signal of the absorber

Fig. 36.6 The real-time control results when the disturbance source signal was used for frequency estimation: (a) the estimated frequency and (b) frequency responses

36 Development of a Tuning Algorithm for a DynamicVibration Absorber. . .

285

Fig. 36.7 The real-time control results when the DVA acceleration signal was used for frequency estimation: (a) the estimated frequency and (b) frequency responses

that the proposed method has an excellent performance compared to the zero crossing method. In the zero crossing method, the response is disturbed due to the influence of the noise in the frequency estimation process. On the other hand, the proposed method estimated almost accurately the change of the frequency. The system response was found to reduce and follow the minimum value by the real-time frequency tracking control. The reduction rate is approximately 1/5 of the response without DVA. However, the estimation is corrupted in the low frequency band; in the proposed method, the filter coefficients are updated using the past data in order to suppress noise. Therefore, the sudden change of the disturbance frequency is not tractable by the proposed method. Fortunately, such an abrupt change seems to be rare in practical applications.

36.6 Conclusion The frequency estimation method using ALE has better accuracy than the zero crossing method, and the damping performance of the tuneable DVA is greatly improved in a wide frequency range. Further enhancement could be expected by improving the efficiency of the magnetic circuit of the DVA and stiffness changeability of MRE.

References 1. Sun, S., Deng, H., Yang, L., Li, W., Du, H., Alici, G., Nakano, M.: An adaptive tuned vibration absorber based on multilayered MR elastomers. Smart Mater. Struct. 24(4), 045045 (2015) 2. Komatsuzaki, T., Inoue, T., Terashima, O.: Broadband vibration control of a structure by using a magnetorheological elastomer-based tuned dynamic absorber. Mechatronics. 40, 28–136 (2016) 3. Chertovich, A.V., Stepanov, G.V., Kramarenko, E.Y., Khokhlov, A.R.: New composite elastomers with giant magnetic response. Macromol. Mater. Eng. 294(5), 336–341 (2010) 4. Hush, D., Ahmed, R., David, R., Stearns, S.: An adaptive IIR structure for sinusoidal enhancement, frequency estimation, and detection. IEEE Trans. Acoust. Speech Signal Process. 34(6), 1380–1390 (1986)

Chapter 37

Design Approach of Laminated Rubber Bearings for Seismic Isolation of Plant Equipment J. H. Lee, T. M. Shin, and G. H. Koo

37.1 Introduction Seismic isolation is one of the widely used technologies. And it has already been applied to the NPP. Recently, there has been a considerable earthquake in South Korea. Then, concerns about the seismic performance of NPP increased during operation. The Korean-type NPP represented as APR1400 is designed to withstand 0.3 g of safety shutdown earthquake (SSE) and is known to have a performance of 0.5 g. APR1400 contains a variety of structures and equipment, among which some may be vulnerable to earthquakes. Upgrading the target seismic performance of APR1400 to more than 0.6 g may increase the range of vulnerable equipment. So, this research has been launched to improve the seismic performance of NPP for beyond-design-basis earthquakes. The standards and requirements related to seismic isolation is mainly concerned with large-scale seismic isolators for buildings or civil structures so far. It is confirmed that the relevant requirements for small-sized isolators for equipment subject to this study have not yet been established. For this reason, the design, fabrication, and test of small-sized isolators for equipment are carried out in reference to the existing standards and studies for the large-scale isolator design and applications. Besides seismic enhancement of equipment in beyond-designbasis earthquakes, the objective of the study is to develop adequate requirements for the design, fabrication, and testing of small-sized isolators with 10 tons or less as unit bearing weight. It is initiated by applying the existing code requirements J. H. Lee · T. M. Shin () Korea National University of Transportation, Chungju, South Korea e-mail: [email protected] G. H. Koo Korea Atomic Energy Research Institute, 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/10.1007/978-3-030-48153-7_37

287

288

J. H. Lee et al.

for large-scale isolators directly to the design of small-sized ones and carried on gradually by revising the design parameters through seismic analysis and discussion with fabricators.

37.2 Characteristics of Equipment Isolation The seismic technology using large LRBs for the structures, such as buildings or bridges, is already well known and increasingly used worldwide so far. Unlike large-scale structural seismic isolators, there are not many studies for the application or test of small LRBs for equipment, especially for the NPP equipment subject to this study. In addition, the relevant technical code and standards have not yet been established. Design parameters of the small LRBs affecting the equipment seismic performance, for example, are weight and stiffness of the target equipment, horizontal and vertical stiffness, the shear strain, natural frequencies, damping, shape factor of the isolator, and the input earthquake. In the study, to efficiently review and analyze the applicability of the seismic isolator to various types of equipment not yet selected, plant equipment is classified into some weight groups and assumed to be rigid. And most equipment is assumed to have their weights in the range of about 4 ~ 100 tons and supported by approximately 4 ~ 10 isolators. Therefore, the design equipment weight per unit seismic isolator is assumed to have a representative value of four different kinds (1, 2, 5, 10 tons). In this paper, only the contents of the 1 ton test specimen are covered.

37.3 Design of Small LRBs for NPP Equipment A design strategy is tried so that the maximum shear strain and displacement may be initially picked up as the preferred design variables for the small LRB for equipment. For the design purpose, the target maximum shear strain for the designbasis earthquake like safe shutdown earthquake (SSE) is set to be 100% of the pure rubber portion in the isolator bearing, and 200% of shear strain is set to be a maximum to withstand seismic loading beyond-design-basis earthquake (0.6 g). In large LRB design in general, as the long-term compressive deformation of rubber layer, thickness has been mainly taken into account for high loading. And the requirements for shape factors, which assure the stability of isolators and structures, have not been sensitive. In general, the primary shape factor for the large-scale isolators is designed to be at least 25 and the secondary shape factor at least 5, which meet the code requirements with no big difficulties. However, this is different from small LRBs for equipment. In the case of small LRB design for equipment isolation, the shape factor of the bearing is very limited by the fabrication constraints of a thin laminated rubber sheet. Difficulties in

37 Design Approach of Laminated Rubber Bearings for Seismic Isolation. . .

289

making relatively thin rubber layers satisfying the target performance of horizontal flexibility and vertical stiffness deteriorate the shape factors of the small LRBs for equipment, though they mitigate the problem of long-term compressive deformation. This shows a similar trend when designing small LRBs for NPP equipment. For the reason, in this study, the objective primary shape factor is designed to be 6 or higher and the secondary shape factor to be 2.5 or higher. And the tentative prototype design is reviewed by static performance test. The vertical deformation of the rubber layer for the design pressure is turned out to be very small as expected (Poisson ratio = 0.5: volumetric immovable characteristic).

37.3.1 Design of Horizontal Stiffness The design horizontal stiffness KH of the LRB is determined as follows by considering the shear and P-delta effects on the design vertical surface pressure: 1 KH = K1 +K 2 ·K3 . K1 = HB /Ss , K2 = H3B /12 · Sb , K3 = (1 + P/Ss )2

(37.1)

HB is a total height of LRB, P is a design vertical load, and it is determined in the following equation: Sb = Eb I [ntR + (n − 1) ts ] /ntR .

(37.2)

Ss = GAs [ntR + (n − 1) ts ] /ntR .

(37.3)

n, tR , and ts represent the number of rubber layers, the thickness of the rubber layer, and the thickness of the steel layer, respectively. And Eb is the apparent compressive modulus of rubber considering volumetric compression of the bending of the laminated rubber. This is determined by the following equation: Eb = Eb E∞ / (Eb + E∞ ) .

(37.4)

Eb is an apparent compression coefficient without considering the compressive properties of rubber and is determined as follows:     2 2 Eb = Eo 1 + kS12 ≈ 3G 1 + kS12 . 3 3

(37.5)

K and Eo represent the hardness modification factor and the elastic modulus of rubber, respectively. In addition, S1 is the primary shape factor, expressed as follows in the case of a circular seismic isolator [1, 2]:

290

J. H. Lee et al.

S1 = (Do − Di ) /4tR .

(37.6)

37.3.2 Design of Vertical Stiffness For design vertical loads of 1ton, the design vertical stiffness is determined in the following equation:   KV = As • Ec / (n • tR ) .

(37.7)

Ec is apparent Young’s modulus corrected, if necessary, by allowing for compressibility. And it is determined in the following equation [1, 2]. Ec is apparent Young’s modulus corrected for bulk compressibility depending on S1 : Ec = (E∞ • Ec ) / (E∞ + Ec ) .

(37.8)

37.3.3 Specification of the Prototype Design The design specifications of the prototype small LRB, determined using the aforementioned design conditions and formulas, are shown in Table 37.1. And Fig. 37.1 shows the shape of the LRB for upper mass 1 ton. Table 37.1 Specification of LRBs

Sort Outer diameter (mm) Total rubber height (mm) Horizontal stiffness (kN/m) Shape factor S1 Shape factor S2

Fig. 37.1 Shape of LRB (design vertical load 1ton)

Upper mass 1 ton 65 24 31.4 6.7 2.7

37 Design Approach of Laminated Rubber Bearings for Seismic Isolation. . .

291

According to the design, the maximum horizontal displacement of LRB should be within 24 mmv100%) fordesign-basis earthquake (0.3 g) and within 48 mm (200%) for beyond-design-basis earthquake (0.6 g).

37.4 Static Performance Test and Analysis In this chapter, it is reviewed for the static performance tests of small LRBs. Tests are carried out to check the performance of the LRBs designed and fabricated through the above process. The test method and sequence are based on the large LRBs according to the relevant criteria [3]. The static performance test was carried out with 50%, 100%, and 200% of the design load in vertical load and 50%, 100%, 150%, 200%, and 300% of the total rubber height in horizontal displacement. The static test was carried out for 11 cases. Test cases 1–4 and 5–8 were the same test cases and repeated for verification of the repeatability of the design values. The vertical load of test cases 1–4 (5–8) is the same as the design weight of 1 ton, and the horizontal displacement is changed to 50%, 100%, 150%, and 200% of the total rubber height. Test cases 9–11 are subjected to extreme conditions, cases 9 and 10 are 1 ton vertical load, and horizontal displacement is 250% and 300% of the total rubber height, respectively. And case 11 has a vertical load of 2 ton and a horizontal displacement of 300% of the total rubber height. As a result, hysteresis curves and effective horizontal stiffness are obtained. From Fig. 37.2, we can see a slight decrease in effective horizontal stiffness, but this is estimated to be due to the effects of test case 4, and the repeatability problems for the design values are not significant. It should also be noted that test case 4 may have caused local damage inside the LRB. The effective horizontal stiffness at 100% horizontal displacement and 100% vertical load is 40 kN/m and at 200% horizontal displacement and 100% vertical load is 29 kN/m. This has an error of about 27% and 8%, respectively, compared to the design horizontal stiffness. In general, considering the 100% horizontal displacement and 100% vertical load conditions for the design values, it can be found that the conditions are beyond 20% of the tolerances of the design values specified in ASCE 4–16 [4]. However, it can be seen that the performance of the beyond-design-basis earthquake (200% H-Disp.) is within the tolerances of the design value, and it is necessary to improve the design values later. Figure 37.3 shows the shape of the hysteresis curve of 1 ton specimen at 100% vertical load by varying horizontal displacement. The graph shows that the slope of the hysteresis curve gradually decreases and then reverses. This phenomenon is assumed to be due to poor shape factors. This can cause excessive bending motion on the edges of the LRB. The hysteresis curves for more than 200% horizontal displacement reveal suspicion of local buckling, which can be observed by some deformation of steel plates after the test in Fig. 37.4. It is also possible to verify that the horizontal alignment of the laminated steel plate is not perfect and that there is a slight deviation in the thickness of the laminated rubber.

292

J. H. Lee et al. Case1 VS. Case5

Case2 VS. Case6

6.00E-01

1.00E+00 5.00E-01

2.00E-01

Lateral Force (kN)

Lateral Force (kN)

4.00E-01

0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01

0.00E+00 -5.00E-01 -1.00E+00

-8.00E-01 -1.00E+00 -2.00E+01

-1.00E+01 0.00E+00 1.00E+01 Lateral Displacement (mm)

-1.50E+00 -3.00E+01

2.00E+01

3.00E+01

Case4 VS. Case8

1.50E+00

2.00E+00

1.00E+00

1.50E+00 Lateral Force (kN)

Lateral Force (kN)

Case3 VS. Case7

-1.50E+01 0.00E+00 1.50E+01 Lateral Displacement (mm)

5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00

1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00

-2.00E+00 -4.00E+01

-2.00E+01 0.00E+00 2.00E+01 Lateral Displacement (mm)

4.00E+01

-2.00E+00 -6.00E+01

-3.00E+01 0.00E+00 3.00E+01 Lateral Displacement (mm)

6.00E+01

Fig. 37.2 Comparison of hysteresis curves for the same tests

Hysteresis Curves

1.50E+00

Case10 Case9

-5.00E-01

Case8 Lateral Force (kN)

5.00E-01

Case7 Case6 Case5 Case4 Case3

-1.50E+00

Case2 Case1 -2.50E+00 -8.00E+01

Lateral Displacement (mm) -4.00E+01 0.00E+00 4.00E+01

8.00E+01

Fig. 37.3 Hysteresis curve with 100% vertical load cases

37.5 Conclusion and Future Research In conclusion, the effective horizontal stiffness obtained from the hysteresis curve is quite close to the design value (allowable range ± 20%). And the reversal in the

37 Design Approach of Laminated Rubber Bearings for Seismic Isolation. . .

293

Fig. 37.4 Cross-sectional shape before (left) and after (right) the test (1ton)

slope of the hysteresis curve obtained from the test is judged to have been caused by local buckling due to poor shape factor, which was caused by the limitation of thin rubber layer fabrication. To improve the shape factor, a new test specimen is being fabricated by changing the design by raising the target horizontal frequency. And static performance test and shaking table test will be carried out soon using a new test specimen. Acknowledgement The research was supported by the Ministry of Trade Industry and Energy through KETEP (No. 201800030003) and by a grant from the Academic Research Program of Korea National University of Transportation (KNUT) in 2019.

References 1. Koo, G.H., Lee, J.H., Yoo, B.: Mechanical characteristics of laminated rubber bearings for seismic isolation. J. EESK. 1(2), 79–89 (1997) 2. Lee, J.H., Shin, T.M., Koo, G.H.: Feasibility study for seismic performance enhancement of NPP based on equipment base isolation. Trans. KPVP. 14(02), 88–95 (2018) 3. ISO 22762-1: Elastomeric Seismic Protection Isolators Part1 Test Methods (2010) 4. ASCE 4: Seismic Analysis of Safety-Related Nuclear Sturctures, ASCE Standard (2016)

Part VII

Noise, Vibration and Their Applications in Electricity Power Systems

Chapter 38

Sound Transmission of Beam-Stiffened Thick Plates K. Zhang and T. R. Lin

38.1 Introduction Stiffened plate structures are one of the most important classes of engineering structures. This type of structures is often found in engineering structures such as ship hulls, aircraft fuselages, or power transformers. Several works [1–7] have been devoted to understanding the sound transmission property of beam-stiffened plates in the last few decades. However, the analytical approaches developed this far were mostly based on Kirchhoff thin plate and Euler-Bernoulli thin beam theories. It is well known that thin plate/thin beam models can only be used for noise and vibration analysis in low frequencies, whereas a large prediction discrepancy can be induced at high frequencies due to the neglecting of transverse deformation and rotatory inertia in the plate and beams. Recently, Zhang and Lin [8] presented an analytical solution on the vibration of a beam-stiffened Mindlin plate in which the condition of using a thin plate/thin beam model is clearly defined. It was found that using a thin plate/thin beam model will yield a less than 5% discrepancy when the plate bending wavelength is 12 times greater than the plate thickness. Sound radiation or transmission of beam-stiffened plates was not considered in their study. This work extends the analytical study of the previous work [8] to investigate sound radiation of a beam-stiffened Mindlin plate under mechanical and sound excitations. The rest of the paper is arranged as follows: in Sect. 38.2, an analytical solution on the vibration response and sound transmission of a beam-stiffened thick plate enclosed by an infinite rigid baffle under an external point force, and a plane sound wave excitation is presented. The effectiveness of the model in the

K. Zhang · T. R. Lin () Center for Structural Acoustics and Machine Fault Diagnosis, Qingdao University of Technology, Qingdao, Shandong, China e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_38

297

298

K. Zhang and T. R. Lin

vibration response and sound transmission prediction of the beam-stiffened plate is evaluated in Sect. 38.3 in a comparison study. The main findings from this study are summarized in Sect. 38.4.

38.2 The Mathematical Model Figure 38.1 shows the beam-stiffened Mindlin plate baffled in an infinite plane and the associated coordinate system considered in this study. For simplicity, all edges of the beam-stiffened plate are assumed to be simply supported. A Timoshenko beam model is used to describe the flexural vibration of the beam. It is also assumed that the beam is symmetrical with respect to the midplane of the plate so that the flexural vibration and torsional vibration of the beam are uncoupled. Assuming that the beam-stiffened plate is in steady-state harmonic motion, the time-dependent term ejωt can then be suppressed [9]. Using a Mindlin plate model, the governing equations of the bending (W) and rotational displacements (ϕx , ϕy ) of the plate due to a point force and external sound pressure excitation can be written as follows [8]: ∂ 2W ∂x 2

+

∂ 2W ∂y 2

∂ 2 ϕx ∂x 2

+

−

∂ϕ x ∂ϕ y ρp hω2 W 2Pi − 2Pr Q M − + = − δ (x − xb ) − δ  (x−xb ) , ∂x ∂y C C C C (38.1)

C 1 − ν ∂ 2 ϕx 1 + ν ∂ 2 ϕy + + 2 2 ∂y 2 ∂x∂y D



 ρp Ip ω2 ϕx ∂W − ϕx + = 0, ∂x D (38.2)

Fig. 38.1 Model description of the beam-stiffened Mindlin plate in an infinite baffle

38 Sound Transmission of Beam-Stiffened Thick Plates

∂ 2 ϕy ∂y 2

+

1 − ν ∂ 2 ϕy 1 + ν ∂ 2 ϕx C + + 2 2 ∂x 2 ∂x∂y D



299

 ρp Ip ω2 ϕy ∂W − ϕy + = 0, ∂y D (38.3)

where D and C are the bending and shearing rigidity, h is the thickness, ν is the Poisson’s ratio, Ip is the area moment of inertia, and ρ p is the mass density of the plate. ω is the angular frequency, Q and M are the coupling force and moment per unit length at the beam/plate interface, and Pi andPr are the sound pressures of the incident and radiated sound fields, respectively. δ represents Dirac delta function,  and δ represents the first-order derivative of the delta function with respect to the x coordinate. Using a Timoshenko beam model, the coupled differential equations for the transverse (U) and the rotation displacement (ψ) of the beam are given as follows [8]: 

∂ 2ψ ∂y 2

∂ψ ∂ 2U − ∂y ∂y 2

+

κGA Eb Ib



 +

Q ρb ω2 U= , κG κGA

 ∂U ρb ω2 −ψ + ψ = 0, ∂y Eb

(38.4)

(38.5)

where ρ b is the mass density, κ is the shearing coefficient, Eb and G are the Young’s and shear modulus, and A and Ib are the cross-sectional area and the moment inertia of the beam. The governing equation for the torsional displacement (θ ) of the beam is given by [8] ∂ 2θ ∂y 2

− kt2 θ =

M , T

(38.6)

where kt is the torsional wave number and T is the torsional stiffness of the beam. The sound pressure is governed by the classical homogeneous Helmholtz equation: ∇ 2 p (x, y, z) + k 2 p (x, y, z) = 0.

(38.7)

The continuity between the vibration displacement normal to the plate and the fluid particle is given by Euler’s equation: / ∂Pr (x, y, z) // = ω2 ρ0 W (x, y) . / ∂z z=0

(38.8)

300

K. Zhang and T. R. Lin

The unknown modal coupling forces (Qn ) and moments (Mn ) at the beam/plate interface can be determined using the following continuity and compatibility conditions: U = W |x=xb , θ = ϕx |x=xb ,

(38.9)

From the above equations, the plate modal vibration velocity can be determined using a modal expansion solution approach as   i − $m sin (km xb ) Qn − $m km cos (km xb ) Mn , Vmn = Ymn 2Pmn

(38.10)

where Ymn is the modal admittance of the Mindlin plate and m and n are the modal indices corresponding to the two plate edge directions. The sound transmission coefficient can now be calculated using τ (α, β) =

∞ ∞  / /  (ρ0 c0 )2 / i /2 Zmn |Ymn |2 /Pmn / , Lx Ly cos (α)

(38.11)

m=1 n=1

where Zmn is the modal radiation impedance.

38.3 A Numerical Analysis It is assumed that both the plate and the beam are made of steel with the following material properties: Young’s modulus E = 1.95e11 N/m2 , density ρ = 7700kg/m3 , and Poisson’s ratio ν = 0.3. The plate has a surface area of 2 × 1 m2 and is 5 mm thick. The beam is assumed to have a rectangular cross-sectional area of A = 0.005 (width) × 0.05(height) m2 . A uniform internal loss factor η = 0.01 is used for both the plate and the beam which is incorporated into the complex Young’s modulus E(1 + jη) where j is the imaginary number. The density and sound speed of the air are ρ 0 = 1.225kg/m3 and c0 = 343m/s. The beam is located at xb = 0.5 m on the plate, and the point force is applied at (x0 , y0 ) = (0.2 m, 0.3 m) on the plate in the simulation. The frequency increment used in the simulation is 1 Hz.

38.3.1 Plate Vibration Response (In Vacuo) Figure 38.2a compares the calculated input mobilities of the based plate using Mindlin thick plate and Kirchhoff thin-plate models (1 Hz resolution). The difference between the results of the two models is mainly in the high-frequency region. The difference increases as the frequency increases. It is shown that results calculated using thin-plate model is only good for low-frequency vibration

38 Sound Transmission of Beam-Stiffened Thick Plates

301

Fig. 38.2 Input mobilities of unribbed base plates/ribbed plates due to a point force excitation

prediction [8]. A similar result is also found for the calculated input mobilities of the beam-stiffened plate using either a thick plate/thick beam or thin plate/thin beam model as shown in Fig. 38.2b.

38.3.2 Sound Transmission Figure 38.3 compares the TLs of the infinite baffled plate due to a normal incident plane sound wave excitation applied on one side of the plate calculated using the following four plate models, i.e., thin base plate, thick base plate, (thin) beamstiffened thin plate, and (thick) beam-stiffened thick plate for a better understanding of the effect of the stiffener on the sound transmission loss (TL) of the plate. Similar to the vibration response shown in Fig. 38.2, the deviation of the TLs between the thick plate and the thin-plate models mainly reflects in the frequency deviation in the high frequencies but not on the transmission loss of the plate. Figure 38.3c shows a general decreasing of the TL at higher frequencies after the beam insertion, which is believed to be due to the increased structural impedance. Further investigation will be conducted in the following study.

38.3.3 Sound Radiation Vibration and sound transmission of engineering structures such as transformers are either excited by external sound source or mechanical source. Figure 38.4 compares the sound radiation of the beam-stiffened thick plate due to a normal incident plane sound wave and a point force excitation. It is shown that the radiation sound power under the point force excitation is low at low frequencies and then increases as the frequency increases. It is vice versa for the sound excitation case. The averaged radiation efficiency of the beam-stiffened plate due to the sound and the point force excitations is shown in Fig. 38.4c. It is found that the plate radiation efficiency due to the sound excitation is higher than that due to the point force excitation prior to the critical frequency.

302

K. Zhang and T. R. Lin

Fig. 38.3 A comparison of the sound transmission loss

Fig. 38.4 A comparison of the sound radiation of ribbed thick plate due to the excitation of the external sound source and mechanical force source

38.4 Conclusions An analytical model was presented in this paper for the prediction of vibration and sound transmission of a beam-stiffened Mindlin plate under either a plane sound wave or a point force excitation. The vibration response, sound transmission, and sound radiation characteristics of the stiffened thick plates under the point force and

38 Sound Transmission of Beam-Stiffened Thick Plates

303

the sound source excitation are compared in the study. It is found that the vibration velocity and radiated sound power of the beam-stiffening plate due to the point force excitation increases in general as the frequency increases, which is vice versa for the sound excitation case. It is shown that the averaged sound radiation efficiency of the plate due to the sound wave excitation is higher than that due to the point force excitation prior to the critical frequency. The analytical tool presented in this study can guide the design of noise and vibration control of relevant engineering structures such as power transformer tanks or machine covers. Acknowledgements The financial support from Shandong Provincial Government of the People’s Republic of China through the “Taishan Scholar” program for this work is gratefully acknowledged.

References 1. Maidanik, G.: Vibrational and radiative classification of modes of a baffled finite panel. J. Sound Vib. 34, 447–455 (1974) 2. Wallace, C.E.: Radiation resistance of a rectangular panel. J. Acoust. Soc. Am. 51(3), 946–952 (1972) 3. Heckl, M.: Radiation from plane sound sources. Acust. 37, 155–166 (1977) 4. Leppington, F.G., Broadbent, E.G., Heron, K.H.: The acoustic radiation efficiency of rectangular panels. Proc. R. Soc. Lond. A Math. Phys. Sci. 382(1783), 245–271 (1982) 5. Williams, E.G.: A series expansion of the acoustic power radiated from planar sources. J. Acoust. Soc. Am. 73(5), 1520–1524 (1983) 6. Maxit, L.: Wavenumber space and physical space responses of a periodically ribbed plate to a point drive: A discrete approach. Appl. Acoust. 70(4), 563–578 (2009) 7. Shen, C., Xin, F.X., Cheng, L., Lu, T.J.: Sound radiation of orthogonally stiffened laminated composite plates under airborne and structure borne excitations. Compos. Sci. Technol. 84, 51–57 (2013) 8. Zhang, K., Lin, T.R.: An analytical study of vibration response of a beam stiffened Mindlin plate. Appl. Acoust. 155, 32–43 (2019) 9. Lin, T.R.: An analytical and experimental study of the vibration response of a clamped ribbed plate. J. Sound Vib. 331, 902–913 (2012)

Chapter 39

On the Feasibility of Transformer Insulation Aging Detection with Vibration Measurements J. H. Yang, X. Cai, H. H. Jin, J. L. Hou, and Y. X. Wang

39.1 Introduction Solid insulation of a power transformer is of vital importance for its normal operation. Since the degradation of insulation paper is irreversible, it has been regarded as a key factor to evaluate the remaining life of a transformer. As the main composition of transformer insulation system, fiber-based insulation papers were massively utilized to construct its longitudinal insulation of winding assemblies. Winding faults due to insulation breakdowns account for approximately 15–20% of the total number of transformer faults [1]. Thermal aging, which is regarded as the main form of insulation aging, has been thoroughly discussed under different temperature, moisture conditions [2]. Recently, mechanical vibration as a fatigue load has been taken into consideration during the aging process [3]. Indeed, insulation paper is simultaneously subjected to thermal load, electrical load, and mechanical load during the operation of the transformer. Previous studies have shown that all these loads contribute to the degradation of the insulation paper, which leads to discharges, short-circuits, and eventually breakdown of the whole unit [4]. However, there is still a lack in the study on the combined effect of aging under thermal, electrical, and mechanical loads. In this paper, we designed a series of material aging tests, in which the thermal, mechanical, and electrical aging factors were considered for the 0.8 mm insulation paper specimens. Consequently, the authors proposed a dynamic model that could

J. H. Yang · H. H. Jin · J. L. Hou · Y. X. Wang () Key Laboratory for Biomedical Engineering of Ministry of Education, Zhejiang University, Hangzhou, Zhejiang, China e-mail: [email protected] X. Cai State Grid Hubei Electric Power Research Institute, Wuhan, Hubei, China © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_39

305

306

J. H. Yang et al.

quantitatively link these influence factors with insulation paper’s mechanical properties. The feasibility analysis of detecting these aging events was realized with a coupled beam model.

39.2 Multi-Physical Aging of Insulation Papers 39.2.1 Experimental Setup In order to simulate the degradation process of the insulation paper under thermal load, electrical load, and mechanical load, a multi-physical aging test rig was designed as shown in Fig. 39.1. The whole experimental setup contains 3 subsystems: (1) Temperature control system; (2) Mechanical aging system; and (3) Partial discharge system. By keeping temperature unchanged, or stopping the shaker, this test rig could realize different aging conditions. Temperature control system is composed of an electrical incubator, which could provide high temperature environment, i.e., 130 ◦ C. A mini shaker and its driving devices are employed to vibrate the clamped insulation specimens. Rigidly connected with the shaker is an impedance head, which could record the dynamic load of each test. For the electrical aging, the test specimen can be fitted in an Electro-Static Discharge (ESD) simulator, which could generate 5000 Hz electric pulse with adjustable voltage outputs.

Fig. 39.1 Experiment setup of the multi-physical aging test

39 On the Feasibility of Transformer Insulation Aging Detection. . .

307

Fig. 39.2 Tensile curve of the insulation paper under different aging time

39.2.2 Thermal Aging of Insulation Papers In this experiment, insulation specimens (10 × 70 × 0.08 mm) were aged 7 days, 14 days, and 21 days under the thermal load, which is defined as 130 ◦ C constant temperature. The specimens were then dried and sent to measure the tensile strength and stiffness by a universal material testing machine (Zwick BZ 2.5/TNIS, range 2.5kN). The tensile curves of the insulation papers are shown in Fig. 39.2, where the tensile strength was found decreasing with the aging procedure, while the stiffness of the insulation paper was found increasing after thermal aging, which had been reported by Jin et al. [5]. Indeed, the cellulose chains break when the insulation paper ages in high temperature, resulting in a decrease in the strength of the insulation paper [6]. Similar to the description of cellulose hydrolysis process [7, 8], we deduce equations as follows: 4  5 T S = T S 0 1 − εT∗ S 1 − e−kT S t (39.1) 4  5 K = K0 1 ± εk∗ 1 − e−kK t

(39.2)

where TS, Kare the tensile strength and stiffness of the sample after aging, TS0 , K0 is the initial tensile strength and stiffness; εT∗ S is the coefficient of insulation loss rate; εk∗ is the maximum value of the stiffness change ratio; kTS is the falling rate constant of TS and is the maximum value of TS loss rate; kk is the rate constant of stiffness

308

J. H. Yang et al.

Fig. 39.3 Mechanical properties of insulation paper in heat aging experiment

change; tis aging time. In Eq. (39.2), “+” means the stiffness increases after aging, and “-” means the stiffness decreases after aging. Based on the stiffness and tensile strength data collected from the thermal aging experiment, coefficients of Eqs. (39.1) and (39.2) are fitted as presented in Fig. 39.3. It is clearly shown that the proposed models could well reflect the degradation rules, which illustrates the rationality of our modeling assumptions.

39.2.3 Multi-physical Aging of Insulation Papers Multi-physical aging is achieved by applying thermal load, electrical load, and mechanical load to the same insulation specimens. For the electrical aging, the discharge impulse (50,000 times) was applied to a 30 × 10 mm area in the middle of each specimen. For the mechanical aging, the dynamic load was 2.07 N with 0.5 kg static load at 100 Hz. For the thermal aging, all specimens were aged under the same thermal load as described in 2.2. After the aforementioned multi-physical aging, the degradation trends of tensile strength and stiffness of the insulation papers are presented in Fig. 39.4 based on Eqs. (39.1) and (39.2). It is found that the ultimate tensile force of the insulation paper is further reduced compared with thermal aging due to mechanical vibration and partial discharge. This trend is completely different compared with thermal aging. Figure 39.4b further confirms this phenomenon, where the stiffness of insulation paper increased after multi-physical aging, although the increasing magnitude is significantly reduced.

39 On the Feasibility of Transformer Insulation Aging Detection. . .

309

Fig. 39.4 The (a) tensile force and (b) stiffness variation of insulation paper after mechanicalelectrical-thermal aging

Fig. 39.5 (a) Experimental setup of the coupled beam model and (b) its validation FRFs

39.3 Vibration Variation Due to Insulation Aging To verify the feasibility of whether transformer insulation aging can be detected with vibration measurement, we designed a coupled beam model, which contains two parallel beams connected by 10 uniformly distributed springs. These springs can be replaced according to the stiffness measured from the aged insulation papers. With regard to the simulation procedure, we firstly validate the finite element code with a coupled beam model, where two beams were welded at the middle and beam ends. Each beam is 0.9 × 0.01 × 0.03 m, and is made of steel. Figure 39.5a shows the experimental rigs to measure the vibration frequency response functions (FRFs). By keeping the same boundary conditions, the vibration response of this beam was calculated with the FE codes. The good agreement between predicted and measured FRFs is shown in Fig. 39.5b, which provides direct validation for the FE model. Based on the validated simulation procedure, the FRFs of coupled beams with 4 different stiffness values are calculated as presented in Fig. 39.6. It is shown that the

310

J. H. Yang et al.

Fig. 39.6 Vibration FRFs of beams coupled with aged insulation papers

bending modes at 313 Hz gradually moved to the higher frequency after thermal aging. In other words, the insulation degradation can be reflected by vibration features and could probably be detected by vibration measurement, which has been scheduled in the future work. This work is supported by the National Natural Science Foundation of China Grant No. 61803332 and 11,574,269, the Natural Science Foundation of Zhejiang Provence Grant No. LQ18E050001, the Fundamental Research Funds for the Central Universities (2019FZJD005), and the State Grid research project (5200201922098A-0-0-00).

References 1. Wang, M.: Review of condition assessment of power transformers in service. IEEE Electr. Insul. Mag. 18(6), 12–25 (2002) 2. Tao, C.: Study on reducing acid value of vegetable oil for transformer insulation. Insul. Mater. 5, 15–18 (2005) 3. Li, C.: Experiments research on the deterioration of inverter transformer insulation paper mechanical properties with mechanical-thermal synergy. Proc. CSEE. 39(2), 612–620 (2019) 4. Emsley, A.: Review of chemical indicators of degradation of cellulosic electrical paper insulation in oil-filled transformers. IEEE Proc. Sci. Meas. Technol. 141(5), 324–334 (1994) 5. Jin, M.: Effects of insulation paper ageing on the vibration characteristics of a transformer winding disk. IEEE Trans. Dielectr. Electr. Insul. 22(6), 3560–3566 (2015) 6. Hill, D.J.T.: A study of degradation of cellulosic insulation materials in a power transformer, part 1: Molecular weight study of cellulose insulation paper. Polym. Degrad. Stab. 48, 79–87 (1995) 7. Hill, D.J.T.: A study of degradation of cellulosic insulation materials in a power transformer, part 2: tensile strength of cellulose insulation paper. Polym. Degrad. Stab. 49, 429–435 (1995) 8. Ding, H.: On the degradation evolution equations of cellulose. Cellulose. 15(2), 205–224 (2008)

Chapter 40

Study on the Effectiveness of Transformer Equivalent to Point Source in Substation Ling Lv, Linke Zhang, and Li Wang

40.1 Introduction The acoustic simulation of substation equipment radiated noise has become an important auxiliary means to analyze the mechanism of substation equipment. In the prediction of noise radiation in substation, it is very important to establish the sound source model of substation. According to the actual structure and noise mechanism of the substation, it is a technical difficulty to simulate the radiated noise of the substation. At home and abroad, some researches on substation acoustic models have been carried out. The first computational model was proposed by Gordon C.G. in 1979[1]. Gordon C.G. puts forward that the transformer is equivalent to an ideal point sound source, which will make the calculation more simple, but the practical application is limited. Tian [2] et al. proposed to simplify the noise equipment such as transformer into point source and area source, respectively, so as to calculate the distribution characteristics of the sound source. Ramsis [3] proposed three different noise simulation software to predict and compare the radiated noise. Mo j [4] uses three prediction software to establish the noise model of substation equipment and makes the acoustic prediction of the radiated sound field of transformer. R.S. Ming [5] proposed to predict the noise radiation direction of transformer based on the measured acceleration model of the surface of transformer shell. Hu [6] and others

L. Lv School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, People’s Republic of China L. Zhang · L. Wang () School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, People’s Republic of China Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan, People’s Republic of China © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_40

311

312

L. Lv et al.

further proposed a new equivalent acoustic model of transformer. The transformer surface is considered to be a mixture of many plane sound sources. Kozupa [7] proposed a discrete method to calculate the acoustic radiation of transformer. The idea of this algorithm is to divide the surface of transformer shell into small rectangular elements. Each element assumes that the rigid baffle piston radiates noise outward. Based on the above analysis, the idea of transformer equivalent sound source model is adopted in this paper. In the study of acoustic simulation of substation sound field, the transformer is simulated by a cube surrounded by six planes, ignoring the influence of the bottom, and the other five planes are equivalent to plane sound source. The radiated sound field of transformer plate is simulated and compared with that of point source. The effectiveness of point source is determined.

40.2 Modeling and Simulation 40.2.1 Transformer Equivalent Model Taking the transformer as an example, it is simplified into a hexahedron model. Neglecting the influence of the bottom to the ground, the vibration of the transformer is equivalent to five surface sound sources. In order to simulate the sound field of substation, a flat plate is taken as the research object. One plane of the transformer is equivalent to a flat plate, and the acoustic simulation of the flat plate is carried out. The specific figure is shown in Fig. 40.1. The actual noise field of transformer is simulated by applying certain excitation force and boundary conditions to the plate, and the actual sound field of point source is simulated.

Fig. 40.1 Transformer plate model

40 Study on the Effectiveness of Transformer Equivalent to Point Source in Substation

313

40.2.2 Modal Analysis The modal response analysis of the plate model of the transformer is carried out to analyze the dynamic response of the plate structure under the simple harmonic load at different frequencies. The plate is set to a square plate. The length of the plate is 1 m, the width is 1 m, and the thickness of the plate is set to 0.01 m. The material parameter properties of the plate are set as follows: Young’s modulus of the plate is 7e10, Poisson’s ratio is 0.25, and the density is 2400 kg / m3. The damping of the plate is set to 1%. The displacement boundary condition is applied to the edge of the plate. The mesh size of the plate model is 0.05 m. The finite element model is shell element. The modal frequency of the plate is calculated and compared with the theoretical value. The comparison value of modal frequency calculation of flat plate is shown in Table 40.1. According to the formula, D is the bending stiffness of the plate, ρ is the material density, e is the young’s modulus, μ is the Poisson’s ratio, and H is the thickness of the plate.  mπ 2

 nπ 2

a

b

wmn =

+ 3

ρh D

D=

Eh3   12 1 − μ2

(40.1)

The six natural modes of plate obtained from the modal analysis are shown in Figs. 40.2, 40.3 and 40.4. Table 40.1 Comparison value of Natural frequencies of plate (unit:Hz)

Fig. 40.2 1st and 2nd mode shapes

Modal order 1 2 3 4 5 6

Simulation 50.5437 126.752 202.583 255.050 330.377 437.287

Theoretical 50.584 126.461 202.338 252.922 328.799 431.142

Error 0.0403 0.2910 0.2450 2.1280 1.5780 6.1450

314

L. Lv et al.

Fig. 40.3 3rd and 4th mode shapes

Fig. 40.4 5th and 6th mode shapes Fig. 40.5 Point load

40.2.3 Harmonic Response Analysis In order to get the sound radiation of the plate, the harmonic response analysis of the plate is done. First, a load of the excitation force is 10 N. The excitation frequency is 0–500hz, and the set step is 1 Hz. The location of the excitation force (−0.25, 0, −0.25) is shown in Fig. 40.5.

40 Study on the Effectiveness of Transformer Equivalent to Point Source in Substation

315

Fig. 40.6 Displacement frequency response function

Fig. 40.7 Different radius measurement points

The displacement frequency response function of a node on the plate is shown in Fig. 40.6. It can be seen from the figure that the peak point of the frequency response function corresponds to the seventh-order natural frequency of the plate.

40.2.4 Acoustic Directivity Analysis Take the geometric center point O of the plate as the origin, and set the acoustic directivity measurement points with different radii, as shown in Fig. 40.7. In total, four acoustic directivity measuring points with different radius range are set, with radii of 5 m, 10 m, 30 m, and 50 m, respectively. In the process of acoustic simulation, we take the mode corresponding to the peak frequency of frequency response function as the vibration form of the plate to calculate the acoustic directivity in this case.

316

L. Lv et al.

40.3 Results and Analysis Figures 40.8, 40.9 and 40.10 show the acoustic directivity figure when the vibration frequency of the seventh-order natural mode is taken as the vibration form in the harmonic response analysis of the plate.

Fig. 40.8 Acoustic directivity at 50 Hz and 126 Hz

Fig. 40.9 Acoustic directivity at 202 Hz and 253 Hz

Fig. 40.10 Acoustic directivity at 328 Hz and 431 Hz

40 Study on the Effectiveness of Transformer Equivalent to Point Source in Substation

317

It can be seen from Fig. 40.8 that when the vibration mode is the first natural mode and the vibration frequency value is 50 Hz, the flat plate can be regarded as a point sound source in any radius range; when the vibration mode is the second natural mode and the vibration frequency value is 126 Hz, the acoustic directivity figure obtained is close to the acoustic directivity figure of the point sound source. When the value of the vibration frequency increases gradually, the acoustic directivity diagram changes significantly from the vibration frequency of 202 Hz to 253 Hz. When the vibration frequency is 202 Hz, the flat plate can be equivalent to a point source; when the vibration frequency is 253 Hz, the flat plate can be equivalent to a point source only in the 45 ◦ – 135 ◦ range of the flat plate.

40.4 Conclusion Through the above analysis, the following conclusions can be drawn: 1. When the plate is in low-order vibration mode, the plate can be equivalent to a point source in all space fields. In this case, the vibration of the plate can be regarded as a point sound source. 2. When the plate is in the state of medium-frequency vibration, the plate cannot be regarded as a point source in the whole space-time. 3. In the medium-frequency vibration mode, if only the sound radiation in some directions is considered, the high-frequency vibration plate can be regarded as the point sound source. 4. Whether the vibration body is regarded as a point sound source, it is not measured according to the distance between the measuring point and the vibration body, but according to whether the vibration body itself is in high-order or low-order vibration mode. Acknowledgments This work was supported by the State Grid Corporation headquarters technology project under Grant Nos. 5200-201922098A-0-0-00.

References 1. Xiao’an, G., Rongying, S., Jitai, X.: Trends in noise research of transformers abroad [J]. Transformers. 39(6), 33–38 (2002) 2. Haoyang, T., Min, H., Wei, P. et al.: Research on noise evaluation of substation station boundary based on acoustic field simulation Technology. Comput. Simul. (1), (2017) 3. Kai, S.: Research on noise characteristics, prediction and regulation technology of UHV substation [D]. (2016) 4. Mo, J., Su, J., Liu, J., et al.: Radiated noise prediction for large power transformer tank[J]. Gaoya Dianqi/High Voltage Apparatus. 50(6), 32–37+44 (2014) 5. Ming, R.S., Pan, J., Norton, M.P., et al.: The sound-field characterisation of a power transformer[J]. Appl. Acoust. 56(4), 257–272 (1999)

318

L. Lv et al.

6. Hu, J., Liao, Q., Yan, Y., et al.: Study on calculation method of transformer multipoint source model [J]. J. Comput. Inf. Syst. 11(8), 2885–2893 (2015) 7. Kozpa, M, Kolber, K.: Radiation efficiency as a crucial parameter in transformer noise evaluation [C]// Forum Acusticum. (2014)

Chapter 41

Transformer Acoustic Equivalent Model in Engineering Application Dakun Li, Wei Li, Li Wang, Linke Zhang, and Zhixing Li

41.1 Introduction Due to the acceleration of urbanization process, in spite of urban substations or suburban substations, the surrounding land resources are becoming increasingly tense, and the building and population density around substations are gradually increasing, which makes the problem of transformer noise more and more serious [1]. How to reduce the noise of substation [2] has become a critical problem for the government. The prediction of noise radiation from transformers is necessary for substation noise control engineering. Finite element or boundary element methods are traditional methods for noise analysis [3]. However, because of the complex internal structure of the transformer, it is difficult to build an accurate acoustic model using these methods [4], and the computation load required for these two methods is relatively large. Because of these two reasons, the application of finite element and boundary element methods is limited in engineering. If the transformer acoustic model is replaced by a single point sound source [5], the error of calculation results will be too large to meet the engineering requirements. How to build a simplified transformer acoustic model suited for engineering applications has always been a problem.

D. Li () Beijing Xiaomi Mobile Software Co., Ltd, Beijing, China W. Li Wuhan Radioactive and Dangerous Solid Waste Pollution Management Center, Wuhan, China L. Wang · L. Zhang · Z. Li Wuhan University of Technology, 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/10.1007/978-3-030-48153-7_41

319

320

D. Li et al.

Fig. 41.1 Principle of the equivalent source method

In this paper, the beamforming and equivalent source method are applied to the transformer equivalent acoustic model’s building, and the model is verified on a specific substation.

41.2 Equivalent Source Method The equivalent source method is an acoustic modeling method that has emerged in recent years [6]. The main idea is to use a group of point sound sources to simulate an object with complex internal structures. The principle of the equivalent source method is shown in the following figure (Fig. 41.1). S is the surface of the vibrating body, S is the virtual boundary, D is the inside of the vibrating body, E is the outside of the vibrating body, and a set of virtual sources are set in D. If the equivalent source is a monopole source with no directivity and is in a free field, the group of equivalent sources’ integral formula can be discretized to the equation P = H × σ . ⎛

H11 H21 H31 ⎜ H12 H22 H32 · · · ⎜ ⎜ H13 H23 H33 ⎜ ⎜ .. .. ⎝ . . H1n H2n H3n · · ·

Hm1 Hm2 Hm3 .. . Hmn

⎞⎛

σ1 ⎟ ⎜ σ2 ⎟⎜ ⎟ ⎜ σ3 ⎟⎜ ⎟⎜ . ⎠ ⎝ .. σn

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎠ ⎝

p1 p1 p1 .. . p1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(41.1)

41 Transformer Acoustic Equivalent Model in Engineering Application

321

In Eq. 41.1, m is the number of receiving points, n is the number of virtual sources, and m is greater than n. H is the relative position of the receiving point and virtual source, σ is the test value of the receiving point, P is the equivalent source value. If each H and σ are determined, the equivalent source values can be obtained. At present, the equivalent source method is still in the theoretical research stage [7]. The main problem is how to choose the number and locations of equivalent sources, which will have a great influence on the calculation results. Mature guidelines about this problem have not been formed yet, which limit its applications in engineering.

41.3 Beamforming Analysis of Transformer Noise Beamforming technology is a method, which can obtain highly directional beam by a microphone array [8]. A self-developed sound array is used to test a 220KV outdoor transformer, and the calculated sound pressure map of the transformer is shown below. In Fig. 41.2, it can be seen that the sound pressure is the highest at the central part of the transformer and it gradually decreases outward. We can see that the sound pressure contour is similar to the distribution of the vibration characteristics on the surfaces of the transformer.

Fig. 41.2 Transformer sound pressure map

322

D. Li et al.

Fig. 41.3 Transformer equivalent acoustic model

41.4 Transformer Equivalent Acoustic Model According to the beamforming analysis of transformer noise and combined with the equivalent source method, the acoustic simplified model of the transformer is equivalent to the following form. As shown in Fig. 41.3, each surface of the transformer is equivalent to 9 point sound sources. The virtual source surfaces have similar shape to the corresponding transformer surface, and the virtual source surface is in the transformer, parallel to the transformer plane and is away from 1 m from the transformer plane. If the equivalent source is a monopole source without directivity, the transmission equation of sound energy can be expressed as the following form: 

− → − → I • d S = I • 4π r 2

W =

(41.2)

S

W is the equivalent source sound power, r is the transmission distance, I is the sound intensity at the receiving point. Combined with Eq. 41.1, if the number of test points on each surface is greater than 9, each strength of the equivalent sources can be calculated. The test points are 1 m away from the transformer surface.

41.5 Equivalent Model Verification A 220 KV substation is selected as the verification object. The substation is shown in the following figure. As shown in Fig. 41.4, the substation consists of two main transformers, a control room, and a sound barrier. Points 1 and 2 are selected behind the noise barrier and near the substation boundary. If the difference between the test and simulation values at these two points is small, the validity of the equivalent model can be proved.

41 Transformer Acoustic Equivalent Model in Engineering Application

323

Fig. 41.4 Substation’s buildings and electrical equipment’s layout

Fig. 41.5 Main transformer1 noise spectrum

The noise spectrum at a point 1 m away from the transformer is shown below (Fig. 41.5). In the spectrum, the sound pressure level is relatively high at 300 Hz and 600 Hz. Therefore, the two frequency bands are used as research bands. The sound ray method is used for simulation [9]. The 300 Hz and 600 Hz simulation results are shown below (Fig. 41.6). The values of points 1 and 2 which are behind the noise barrier and near the substation boundary can be obtained. The simulation values and test values at 2 points are shown in Table 41.1. As shown in Table 41.1, the difference between the simulated values and the test values are within 1.5db, which indicates that the transformer equivalent model is suitable for the sound field prediction in substation noise control engineering.

41.6 Conclusion In this paper, the beamforming method is used to solve the problems when the equivalent source method is applied to the transformer. What’s more, the transformer equivalent model suitable for engineering applications is established

324

D. Li et al.

Fig. 41.6 300 Hz and 600 Hz simulation results of the substation Table 41.1 Simulation values and test values Frequency Point 1 2

300 Hz 600 Hz Test value Simulation value Difference Test value Simulation value Difference 36.8 dB 36.5 dB 0.3 dB 38.3 dB 38.4 dB 0.1 dB 41.5 dB 40.1 dB 1.4 dB 40.4 dB 41.8 dB 1.4 dB

and verified by simulation and test in a specific substation. By using 9 equivalent sources for each of the transformer surfaces, we can get similar sound pressure levels to the experimental results at the test points, but, whether this model is suitable for indoor substation noise prediction remains to be investigated in the future.

41 Transformer Acoustic Equivalent Model in Engineering Application

325

References 1. Gange, M.: Low-frequency and tonal characteristics of Transformer noise. Proceedings of Acoustics 2011, Gold Coast, Australia, 2–4 November 2011 2. Bishop, E.A.: Mitigation of substation noise. Electr. Eng. 50(5), 351–352 (2013) 3. Gladwell, G.M.L.: A variational formulation of damped acousto structural vibration problems. J. Sound Vib. 4(2), 172–186 (1966) 4. Zhu, L., Yang, Q., Yan, R., et al.: Numerical computation for a new way to reduce vibration and noise due to magnetostriction and magnetic forces of transformer cores. J. Appl. Phys. 113(17), 17A333 (2013) 5. Gordon, C.G.: A method for predicting the audible noise emissions from large outdoors power transformers. IEEE Trans. Power Syst. PAS-98(3), 1109–1112 (1979) 6. Talotte, C., Gautier, P.-E., Thompson, D.J., et al.: Identification, modelling and reduction potential of railway noise sources: A critical survey. J. Sound Vib. 267(3), 447–468 (2003) 7. Zawieska, M.W.: Power transformer as a source of noise. Int. J. Occup. Saf. Ergon. 13(4), 381– 389 (2007) 8. Hald, J., Christensen, J.J.: A novel beamformer array design for noise source location from intermediate measurement distances. J. Acoust. Soc. Am. 112(5), 2448–2448 (2002) 9. Xuan, C., Li, W., Sheng, W., Na, Y., Li, D.: Radiation noise prediction of outdoor substation based on ray tracing method. The 46th international congress and exposition on noise control engineering, Hong Kong, China, 27–30 Aug 2017

Chapter 42

Simulation Study on Noise Reduction Effect of Substation Noise Barrier Xuan Cai, Xuelei Zhan, Yong Cai, and Li Wang

42.1 Introduction Noise barrier is widely used in noise control of substation as an important means to prevent noise propagation. Nowadays, many of the design of noise barrier in substation are based on past experience and lack of scientific guidance, which cannot maximize the benefits of noise barrier in noise control engineering. Noise barrier has been widely used as the main control method. Watts [1] et al. found that adding any simple structure to the top of the noise barrier could increase the noise reduction of the noise barrier. Jin [2] et al. studied the inclined noise barrier to improve its insertion loss. It was found that the insertion loss of the inclined noise barrier was 1–4 dB higher than that of the normal vertical noise barrier. Osman [3] et al. first proposed that the noise barrier with T-shaped top structure has more obvious noise reduction effect than that with vertical structure. Hothersall [4] et al. studied the insertion loss of the T-shaped top barrier by numerical calculation and scale model experiments. The results show that the insertion loss of the T-shaped top barrier increases by 1–3 dB compared with the traditional vertical barrier with the same height. Okubo [5, 6] et al. studied the “soft surface” of a waterwheel-shaped cylinder on the top of the sound barrier. Two-dimensional boundary element calculations and proportional model tests show that the noise reduction can be improved at high frequencies, but the noise reduction effect at low frequencies is decreasing.

X. Cai State Grid Hubei Electric Power Research Institute, Wuhan, Hubei, China X. Zhan · Y. Cai State Grid Hubei Electric Power CO., LTD, Wuhan, Hubei, China L. Wang () School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, Hubei, China © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_42

327

328

X. Cai et al.

The current research of noise barrier mainly focuses on road noise reduction. Mun [7] et al. present a novel method to design a desired noise barrier using the global optimization of a simulated annealing algorithm. This method focuses on minimizing the barrier dimensions, which are related to material and construction costs, as well as satisfying the target sound pressure levels at receiver points. Kumar [8] et al. indicate that artificial neural network can be useful to determine the height of noise barrier accurately, which can effectively achieve the desired noise level reduction. This paper takes four-type typical outdoor substation noise barriers as an example; the noise reduction effect of sound barrier is studied when the distance between sound source and sound barrier changes. The analysis process and conclusions can provide reference for the optimal design of the noise barrier.

42.2 Modeling In this paper, four typical types of sound barrier structures are analyzed and compared. They are I type, T type, 7 type, and Y type. The cross-sectional profile of the sound barrier and the geometric dimensions of the profile are shown in Fig. 42.1. The total height of the four types of sound barriers is 5 m. Figure 42.1 also illustrates the location of the sound source. The sound source is a point source, 2 m from the ground and L from the sound barrier. The sound insulation effects of the sound barrier with L = 1 m, L = 3 m, and L = 5 m are studied, respectively. According to the structure of the noise barrier, the acoustic analysis model is established by using finite element acoustic method. The acoustic fluid is air. The air sound velocity is 343 m/s and density is 1.225 kg/m3 . The acoustic mesh

Fig. 42.1 Noise barrier structure type and noise source position

42 Simulation Study on Noise Reduction Effect of Substation Noise Barrier

329

Fig. 42.2 Acoustic model and field point model

model of noise barrier is shown in Fig. 42.2. The typical operating frequencies of transformers in substations were 100 Hz, 200 Hz, and 300 Hz. The mesh size is 0.1 m, for the maximum frequency of 300 Hz; the acoustic wavelength is 1.14 m, which is more than 10 times the mesh size of the acoustic finite element model. The finite element mesh meets the acoustic simulation requirements. The interface between acoustic fluid and structure (ground and sound barrier) is acoustic hard boundary. The internal finite element mesh is acoustic finite elements, while the twodimensional meshes on the surface of the acoustic fluid domain are acoustic infinite elements, which is used to truncate the infinite air space and act as a nonreflective acoustic interface. In Fig. 42.2, rectangular acoustic field points are set up in XY plane to visualize the spatial distribution of sound pressure. Meanwhile, 180 virtual microphone acoustic field points with radius of 7 m distributed in the XY plane are used to represent the acoustic directivity for sound barrier sound insulation performance. In the simulation, the sound source is a point source whose source intensity is 1Pa.

42.3 Result and Analysis 42.3.1 Sound Pressure Map Figures 42.3, 42.4 and 42.5 shows the sound pressure map of the noise barrier at different frequencies for L = 1 m, L = 3 m, and L = 5 m, respectively. The range of the sound pressure is from 0 dB to 120 dB. The dark color represents the area with higher sound pressure, and the light color represents the area with lower sound pressure. Sound pressure map can be divided into three parts with sound barrier as the interface: sound source area on the left, noise reduction area on the right, and sound diffraction area on the top of the barrier. As can be seen from the Figs. 42.3, 42.4 and 42.5, no matter what type of sound barrier is used and at what frequency, the sound pressure level of the sound source area is similar. In the area of sound diffraction area

330

X. Cai et al.

Fig. 42.3 Sound pressure map (L = 1 m)

Fig. 42.4 Sound pressure map (L = 3 m)

Fig. 42.5 Sound pressure map (L = 5 m)

at the top of the barrier, the noise distribution is different because of the different top structure. Compared with the I-type barrier, the three barriers, T type, 7 type, and Y type, can significantly concentrate the higher sound pressure region on the top and deflect toward the sound source region, thereby reducing the noise in the diffraction

42 Simulation Study on Noise Reduction Effect of Substation Noise Barrier

331

region behind the barrier. The Y-type noise barrier has a larger noise reduction area than the type I noise barrier. It can also be seen that when the distance between the sound source and the noise barrier is constant for the same noise barrier, the higher frequency noise can get a larger noise reduction area than the lower frequency behind the noise barrier. With the distance increase between the sound source and the noise barrier, the noise reduction for different noise barrier tends to be the same distribution.

42.4 Sound Pressure Directivity Figure 42.6 showed the sound pressure directivity of the sound barrier for L = 1 m, L = 3 m, and L = 5 m, respectively. It can be seen from Fig. 42.6 that the directivity curve can also be divided into three parts. The noise reduction area is between 0 and 90 degree, the sound source area is between 90 and 180 degree, and the top area of the sound barrier is near 90 degree. It can be seen from Fig. 42.6 that the acoustic directivity for different structural-type noise barriers was the same in the sound source area. In the top area of the noise barrier, the Y-type noise barrier has obvious lower sound pressure value. In the noise reduction area, the directional distribution for different top structural noise barriers was similar, and the Y-type noise barrier has lower sound pressure level. With the distance increase between the sound source and the noise barrier, the directional distribution of the sound barrier with different top structures tends to be the same

42.5 Conclusion Through the analysis of the research results, the following conclusions can be obtained: 1. Noise barriers with different top structures have different sound insulation performances and Y-type sound insulation performance is the best. 2. Noise barrier can get better noise reduction effect for high-frequency noise than for low-frequency noise. 3. When the distance between sound source and noise barrier is closer, the more noise reduction can be obtained, and the noise reduction effect of Y-type noise barrier is more obvious than that of I-type noise barrier.

332

X. Cai et al.

Fig. 42.6 Sound pressure directivity

Acknowledgments This work was supported by the State Grid Corporation headquarter technology project under Grant Nos. 5200-201922098A-0-0-00.

42 Simulation Study on Noise Reduction Effect of Substation Noise Barrier

333

References 1. Watts, G.R., Morgan, P.A.: Acoustic performance of an interference-type noise-barrier profile. Appl. Acoust. 49(1), 1–16 (1996) 2. Jin, B.J., Kim, H.S., Kang, H.J., et al.: Sound diffraction by a partially inclined noise barrier. Appl. Acoust. 62(9), 1107–1121 (2001) 3. May, D.N., Osman, M.M.: The performance of sound absorptive, reflective, and T-profile noise barriers in Toronto. J. Sound Vib. 71(1), 65–71 (1980) 4. Hothersall, D.C., Crombie, D.H., Chandler-Wilde, S.N.: The performance of t-profile and associated noise barriers. Appl. Acoust. 32(4), 269–287 (1991) 5. Okubo, T., Fujiwara, K.: Efficiency of a noise barrier on the ground with an acoustically soft cylindrical edge. J. Sound Vib. 216(5), 771–790 (1998) 6. Okubo, T., Fujiwara, K.: Efficiency of a noise barrier with an acoustically soft cylindrical edge for practical use. J. Acoust. Soc. Am. 105(6), 3326–3335 (1999) 7. Mun, S., Cho, Y.H.: Noise barrier optimization using a simulated annealing algorithm[J]. Appl. Acoust. 70(8), 1094–1098 (2009) 8. Kumar, K., Parida, M., Katiyar, V.K.: Optimized height of noise barrier for non-urban highway using artificial neural network. Int. J. Environ. Sci. Technol. 11(3), 719–730 (2014)

Chapter 43

Fault Recognition of Induction Motor Based on Convolutional Neural Network Using Stator Current Signal Tian Han, Ze Wang, Zhongjun Yin, and Andy C. C. Tan

43.1 Introduction As an indispensable power device in industry, induction motor failure will not only affect the overall production efficiency of the system and economic losses but may also result in human casualty in severe catastrophic accidents. In order to ensure safe operation of the production process, finding and detecting incipient problems is crucial to ensure timely repair of the defective component and avoid unexpected failure with consequential loss of production capacity. Although the replacement cost of induction motor in the production system is small, its failure could lead to major shutdown of the whole production system and result in significant downtime. It is therefore essential to continuously monitoring the operating condition of the motor to detect symptom of failure in advance with efficient fault diagnosis system. The basic components of an induction motor are rotor, stator, a set bearings, fan, and housing. It is an integrated electrical device which exhibits a range of failure modes associated to the basic components. According to the latest data on inductor motor faults from the Institute of Electrical and Electronics Engineers (IEEE), bearing faults account for 44%, stator faults account for 26%, rotor faults account for 8%, and other faults are 22% [1]. Current induction motor fault monitoring and diagnostic technology can be divided into two groups, namely, model based and signal analysis [2]. Empirical mode decomposition (EMD) to decompose and filter the motor bearing vibration signal to improve the bispectrum changes and extract fault frequency is reported in

T. Han () · Z. Wang · Z. Yin School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, China e-mail: [email protected] A. C. C. Tan LKC Faculty of Engineering, University Tunku Abdul Rahman, Kajang, Selangor, Malaysia © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_43

335

336

T. Han et al.

[3]. Talhaoui et al. [4] used a combination of short-time Fourier transform (STFT) and discrete wavelet transform (DWT) techniques to diagnose broken-bar faults of rotor in induction motors. Fang et al. [5] proposed a rotor fault diagnosis method for asynchronous motor based on motor current signal spectrum analysis and support vector machine. This method makes use of support vector machine to correlate the characteristic information and fault mode of motor current spectrum signal. All the above methods are based on the signal processing and adopt a method of manually extracting features. Therefore, abundant prior knowledge is required to apply appropriate signal processing strategy and extensive practical experience is necessary to analyze a large amount of data and is very time consuming. In recent years, deep learning has become a research hotspot, which was first proposed by Hinton et al. [6] in 2006. It conducts feature learning through layers from bottom to top and, finally, obtains nonlinear expression of features. It has been widely used in many fields, such as handwriting zip code digital recognition system with convolutional neural network (CNN) [7], the speech features of Parkinson’s patients using. Therefore, its potential application in the field of fault diagnosis is widely valued and led to many scholars exploring the technique. Gan et al. [8] used wavelet packet node energy as an input data of deep belief network (DBN) to identify the pattern of rolling bearing faults. In literature [9], the wavelet packet energy diagram was used as input data of deep CNN to perform fault diagnosis of rotating bearing. Chen et al. [10] reconstructed the two-dimensional input image of vibration signal of gearbox as the input of CNN by extracting statistical features in the time and frequency domains for gearbox fault diagnosis. In this paper, the convolutional neural network (CNN) is applied for fault recognition of the induction motor using current signals. The current signal of different condition of motors is measured and converted into images as CNN’s inputs. In order to achieve high accuracy for fault diagnosis, the model coefficients of CNN are discussed in detail. The results illustrate the feasibility and effectiveness of CNN in motor fault recognition.

43.2 Induction Motor Fault Recognition System In this paper, single-phase stator current signal of induction motor is acquired. The time domain image obtained through preprocessing is directly used as the input to the convolutional neural network. The model structure of the convolutional network is established including an input layer, four hidden layers (convolutional layer + pooling layer), two fully connected layers, and a SoftMax output layer. The induction motor fault recognition system is illustrated in Fig. 43.1. In this paper, the induction motor fault recognition process is divided into the following steps: 1. The collected current signal of motors is converted into time domain image and creates data sets.

43 Fault Recognition of Induction Motor Based on Convolutional Neural. . .

337

Get the signal of motors Data preprocessing Converted into time domain image

CNN model (empirical method sets each parameter)

Model training and testing Model training

Whether the model meets requirements?

Modify the model and parameters N

Y Save the model for induction motor fault diagnosis

Fig. 43.1 Induction motor fault recognition system

2. Construct a convolutional neural network model, initialize parameters, and determine hyper-parameters. 3. The created training data set is used as the model input and the output is obtained through forward propagation. The model is trained by the optimization method. Through numerous iterations, the ideal training output is obtained. 4. Verify that the network model meets the actual requirements through the test data set. If it does not meet step (3), modify the model and parameters and continue the training and verification process. 5. Save the model for induction motor fault diagnosis.

43.3 Experiments and Analysis 43.3.1 Calculation Environment TensorFlow 1.6.0 and python 3.6.4 are used as a deep learning framework under Windows 10. The training process is carried out by hardware environment with Intel Core i7 7700, 16G DDR4, and GPU (NVIDIA GTX 1060 6G).

338

T. Han et al.

Fig. 43.2 Time waveform of faulty motors

43.3.2 Experimental Test and Results The data used in this chapter are acquired from test rig including an induction motor driving an open fan through pulleys and belt system. The fan with adjustable blade inclination is connected via a shaft to the pulley. The induction motor has four poles operating at 60 HZ with an output of 0.5 kW. Six induction motors (one normal and six faults) were used in the experiment to generate the required data set under full-load conditions including unbalance rotor (MUN), broken rotor bar (BRB), faulty bearing (FBO), angular eccentricity (AMIS), parallel eccentricity (PMIS), bowed rotor (BR), and phase unbalance (PUN). The normal motor was used as a benchmark for comparison with experimental data of other faulty motors. Three AC current probes were used to measure the motor current and each fault was collected under the same working conditions. The sampling frequency was 5KHz, and 2540 sets of time domain signal data were collected for each type of fault. Each set of data has 256 points. The collected data was converted into an image sample, 2048 samples in each condition were randomly selected and the corresponding tag was used as a training set, and the rest of data set was test sets. Figure 43.2 shows the time domain waveforms of the stator current signal under different fault conditions, which show no significant different and is impossible to differentiate the type of faults.

43 Fault Recognition of Induction Motor Based on Convolutional Neural. . . Table 43.1 Accuracy of different network structure

Kernels Layers 2 3 4 5

339

3×3

5×5

7×7

9×9

99.57% 99.82% 99.88% 99.84%

99.78% 99.90% 99.98% 99.93%

99.76% 99.90% 99.94% 99.87%

99.13% 99.73% 99.70% 99.38%

43.3.3 The Influence of Structure and Hyper-parameter on CNN 1. The Influence of Structure on CNN In a CNN, both the number of convolutional layers and size of convolution kernel have certain influence on the training and final fault recognition accuracy. In order to determine the network structure with the highest performance, the number of network convolution layers is set to 2, 3, 4, and 5 layers, respectively. The numbers of convolution kernels are set to (3 × 3), (5 × 5), (7 × 7), and (9 × 9) for each convolutional layer, and 16 kinds of model structures were obtained. Each structure has three training and test data with the same learning rate (0.0005), batch size (128), and iterations (100). Finally, the test results are averaged and the results are shown in Table 43.1. From Table 43.1, we can see that the highest accuracy is 99.98% with four convolution layers and (5 × 5) convolution kernel size. 2. The Effect of Learning Rate on CNN The optimization process of the neural network directly determines the quality of the model and is a very important step. The gradient descent method is applied to optimize the parameters of the neural network according to the defined loss function. Therefore, the loss function of the neural network model on the training set is as small as possible. The learning rate controls the update speed of the parameters and determines the magnitude of each update of the parameters. Therefore, in order to improve the efficiency of network training, different learning rates are studied. The results are shown in Fig. 43.3. As can be seen from Fig. 43.3, the model can converge by increasing the number of iterations, but at the same time the training time will also increase. Generally, the model cannot converge by increasing the number of iterations in this case. Therefore, in order to speed up the convergence of CNN networks and improve the accuracy of the network, the learning rate selected in this chapter is 0.0005. 3. The Effect of Batch Size on CNN The batch size is one of the important hyper-parameters of the neural network. Therefore, choosing a reasonable batch size will also have an effect on improving the performance of the neural network. In this section, different batch sizes (the other parameters are kept the same) are explored to determine the influence on the

340

T. Han et al.

Fig. 43.3 The effect of different learning rate on CNN training Table 43.2 Effect of different batch sizes on CNN training Batch size 1 2 4 Iterations 100 Training time/min Non convergence Training loss Training accuracy (%) Test loss Test accuracy (%)

8

16

32

64

128

256

512

40.00 0.067 98.02 0.15 96.75

32.71 0.025 99.37 0.04 98.63

29.75 0 100 0 99.97

27.56 0 100 0 99.97

26.66 0 100 0 100

26.25 0 100 0.01 99.87

26.32 0 100 0.01 99.78

performance of the neural network. In addition, taking into account the settings of the computer hardware and the way of using, the batch size is set to a power of 2; the CPU/GPU can operate with a better performance. Furthermore, due to the hardware limitations, the maximum batch size is set to 512. The results are shown in Table 43.2. It can be seen from Table 43.2 that when the batch size is too small, the model does not converge within 100 iterations. With the increase of batch size, the training time decreases, at the same time the test accuracy rate increases for the same amount of data. However, when the size of the batch size exceeds 128, the training time does not decrease significantly while the test accuracy rate decreases. This is because when the batch size increases, the number of iterations for full data set decreases,

43 Fault Recognition of Induction Motor Based on Convolutional Neural. . .

341

Fig. 43.4 Training results of five experiments Table 43.3 Results of five experiments Number of experiments Training accuracy/% Test accuracy/%

1 100.00 99.95

2 100.00 99.95

3 100.00 99.97

4 100.00 100.00

5 100.00 100.00

and the total number of iterations of the model decreases. Thus, the accuracy will decrease with the same number of iterations. According to above analysis, the batch size is set to 128. 4. Experimental Results and Analysis Based on the above analysis, the CNN parameters are determined: with 0.0005 learning rate, 128 batch size and number of iterations of 100. After preprocessing (remove titles, axes, blanks, etc.), the stator current signal is converted into image as the model input. Each experiment was repeated five times to obtain consistency. The curve of loss and accuracy during the training process is shown in Fig. 43.4, and the test results are presented in Table 43.3. It can be seen from Fig. 43.4 that both the loss value and the accuracy rate of the five experiments converged after 100 iterations. And the recognition accuracy can reach up to 100%. Based on the above results, it indicates that the stator current signal in time domain image can be used as the input to the convolutional neural network to effectively identify the conditions of induction motor. 5. Comparison of Experimental Results In order to fully verify the validity of the proposed method, the data set used in this chapter is compared with the published motor fault diagnosis methods [11, 12]. The results are shown in Table 43.4. The method used in this chapter has accurately diagnosed motor faults with the mapping of multilayer networks and supervised parameter adjustment. The accuracy of the proposed method is 100% and the complexity of feature processing is avoided. Based on the hardware and software environments, the average training time for training is 16,384 samples which is 26.66 min, and the average testing time for

342

T. Han et al.

Table 43.4 Motor diagnostic results of different methods Methods MI + SVM [11] GA + ANN [12] This chapter CNN

Input vector Multi-scale entropy of vibration signal Current signal features Current signal time domain waveform

Accuracy 96.25% 98.89% 100%

testing is 3936 samples which is 3.68 s. Hence, it is evidence that the proposed method is high efficiency for induction motor fault diagnosis.

43.4 Conclusion This chapter concludes that CNN-based fault diagnosis for induction motor fault can be implemented using stator current signal. The advantage of the proposed CNN-based method outperforms the traditional “feature+classifier” with “imageto-results” method. The complexity of feature processing is avoided. This chapter also provides a detail analysis to improve the accuracy, the influence of neural network structure, and hyper-parameter on the performance of CNN. Comparing with the traditional method, the current signal time domain image is used as the CNN input, which reduces the need for prior knowledge of the fault signal, signal processing technology, and theoretical and practical experience. At the same time, the diagnosis calculation cost is lowered with high recognition accuracy. The validation results show that the proposed method has a recognition accuracy of 100% for the eight operating states of the motor including normal, unbalance rotor, broken rotor bar, faulty bearing, angular eccentricity, parallel eccentricity, bowed rotor, and phase unbalance. Therefore, the CNN-based method is feasible and effective for motor fault condition recognition. However, it should be stressed that the data set used in this chapter is steady current signal collected under laboratory-specific experimental conditions. In real practical situations industrial motors mostly operate under nonstationary and complex conditions. Therefore, the performance of the model used in this chapter has yet to be verified with actual industrial operation data. Unfortunately, most companies are reluctant to share their real-life data, and this future work will be persistently pursued. Acknowledgment The authors gratefully acknowledge the financial support by the National Key Research and Development Program of China (Grant No. 2018YFC0810500).

43 Fault Recognition of Induction Motor Based on Convolutional Neural. . .

343

References 1. Vitor, A L O., Scalassara, P. R., Endo, W., et al.: Induction motor fault diagnosis using wavelets and coordinate transformations[C]// IEEE international conference on industry applications, IEEE, 1–8 (2017) 2. Blodt, M., Granjon,P., Raison, B., et al.: Models for bearing damage detection in induction motors using stator current monitoring[C]// IEEE international symposium on industrial electronics, IEEE, 1813–1822 (2006) 3. Yang, J., Zhao, M.: Fault diagnosis of traction motor bearings using modified bispectrum and empirical mode decomposition. Proc. Csee. 32(18), 116–122 (2012) 4. Talhaoui, H., Menacer, A., Kessal, A., et al.: Fast Fourier and discrete wavelet transforms applied to sensorless vector control induction motor for rotor bar faults diagnosis. ISA Trans. 53(5), 1639–1649 (2014) 5. Fang, R., Zheng, L., Ma, H., et al.: Fault diagnosis for rotor of induction machine based on MCSA and SVM. Chin. J. Sci. Instrum. 02, 252–257 (2007) 6. Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science. 313(5786), 504–507 (2006) 7. Lecun, Y., Boser, B., Denker, J.S., et al.: Backpropagation applied to handwritten zip code recognition. Neural Comput. 1(4), 541–551 (2014) 8. Gan, M., Wang, C., Zhu, C.: Construction of hierarchical diagnosis network based on deep learning and its application in the fault pattern recognition of rolling element bearings. Mech. Syst. Signal Process. 72–73(2), 92–104 (2016) 9. Ding, X., He, Q.: Energy-fluctuated multiscale feature learning with deep ConvNet for intelligent spindle bearing fault diagnosis. IEEE Trans. Instrum. Meas. 66(8), 1926–1935 (2017) 10. Chen, Z.Q., Li, C., Sanchez, R.V.: Gearbox fault identification and classification with convolutional neural networks. Shock. Vib. 2015(2), 1–10 (2015) 11. Pan, S., Han, T., Tan, A.C.C., et al.: Fault diagnosis system of induction motors based on multiscale entropy and support vector machine with mutual information algorithm. Shock. Vib. 2016(7), 1–12 (2016) 12. Han, T., Yang, B.S., Choi, W.H., et al.: Fault diagnosis system of induction motors based on neural network and genetic algorithm using stator current signals. Int. J. Rotating Mach. 2006(3), 644–657 (2006)

Chapter 44

A Numerical Study on Active Noise Radiation Control Systems Between Two Parallel Reflecting Surfaces Jiaxin Zhong

, Jiancheng Tao, and Xiaojun Qiu

44.1 Introduction In some noise control cases, there are reflecting surfaces around the noise sources such as sound barriers or walls around transformers [1]. For active noise control (ANC) systems, the noise reduction performance is affected by nearby reflecting surfaces due to the source radiation resistance change caused by the surfaces [2, 3]. The effects of an infinitely large reflecting surface [4], a finite size reflecting surface [5], and two vertically placed reflecting surfaces perpendicular to each other [6] on ANC systems have been investigated. However, the performance of an ANC system with two vertically reflecting surfaces that are parallel to each other is still not known. For single channel ANC systems, it is found that the noise reduction increases significantly when the primary and secondary sources are placed along a line perpendicular to a reflecting surface [4]. For multichannel ANC systems with the primary source on the reflecting surface, the noise reduction can be maximally increased if the secondary sources are placed as far apart as possible to each other and the ground [3]. The mechanism is that the additional reflecting surface

J. Zhong () · X. Qiu School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Ultimo, NSW, Australia e-mail: [email protected]; [email protected] J. Tao Key Laboratory of Modern Acoustics and Institute of Acoustics, 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/10.1007/978-3-030-48153-7_44

345

346

J. Zhong et al.

produces more image secondary sources which can enhance the performance of ANC systems. For the case of ANC systems near two reflecting surfaces with the right included angle (90◦ ), the noise reduction of the system has been investigated only with simulations [6]. Numerical results show that higher noise reduction can be achieved for the system compared with that case with only one reflecting surface after optimizing the locations of the sources and surfaces. However, there is no report on the effects of using two reflecting surfaces parallel to each other around ANC systems. This chapter investigates the feasibility of increasing the noise reduction performance of ANC systems on ground by introducing two vertically placed reflecting surfaces which are parallel to each other. The noise reduction of the ANC system inside two infinitely large reflecting surfaces is computed analytically first according to the normal mode theory inside a flat space; then the effects of the included angle between the source line and the normal line of the surfaces as well as the surface interval are investigated. The numerical results are presented and the mechanisms for the performance improvement are discussed.

44.2 Theory Figure 44.1 shows a single channel ANC system on ground, where the distance between the primary source and the secondary source is d and the separation distance between two vertically placed and parallel reflecting surfaces is D. The size of the reflecting surfaces is assumed to be sufficiently large compared with the wavelength of sound waves for simplicity. The included angle between the source line and the normal line of the surfaces, i.e., the line perpendicular to the parallel surfaces, is denoted by θ . The ground plane and the additional reflecting surfaces are assumed to be perfectly reflective throughout the chapter. A cylindrical coordinate system (ρ, ϕ, z) is established with the origin, O, being located at the projection of the primary source on the nearest reflecting surface shown in Fig. 44.1. The z axis direction is perpendicular to the surfaces and pointing to the other reflecting surface. When the source is on the ground plane (y = 0), the point monopole and its image coincide. In this chapter, all of the sources are assumed to be on ground to focus on the effects of the two parallel reflecting surfaces on ANC systems. The locations of the primary and the secondary sources are then rp = (ρ p , 0, zp ) and rs = (ρ s , 0, zs ), respectively. The sound pressure at the point r = (ρ, ϕ, z) generated by a point monopole at r0 = (ρ 0 , 0, z0 ) between two infinitely large and parallel reflecting surfaces can be obtained by [7]

44 A Numerical Study on Active Noise Radiation Control Systems Between. . .

347

Fig. 44.1 An ANC system on ground with two vertically placed and parallel reflecting surfaces

p (r; r0 ) =

ρair ωkq 0 2π

 nπ  jmϕ , D z0 e m=0 n=0 3  3  (1) × Jm k 2 − (nπ/D)2 ρ< Hm k 2 − (nπ/D)2 ρ> ∞  ∞ 



nπ π kD εn cos D z



cos

(44.1) where the effects of the ground have been taken into account, j is the imaginary unit; ρ air is the air density; k is the wavenumber; ω is the angular frequency; q0 is the source strength; ρ < = min (ρ, ρ 0 ); ρ > = max (ρ, ρ 0 ); εn is the Neumann factor, i.e., εn = 1 (n = 0) and εn = 2 (n = 1, 2, 3, . . . ); Jm (·) is the Bessel function of the (1) first kind of order m; and Hm (·) is the Hankel function of the first kind of order m. The sound radiation power of a single ANC channel system consisting of one primary source and one secondary source can be formulated as [8] W = A|qs |2 + qs∗ b + b∗ qs + c,

(44.2)

where qs is the complex source strength of the secondary source, * denotes complex conjugation, A = Rs /2, Rs is the self-radiation resistance of the secondary source, b = qp Rps /2, Rps is the mutual radiation resistance between the primary source and the secondary source, c = |qp |2 Rp /2, and qp and Rp are the complex source strength and the self-radiation resistance of the primary source, respectively. The resistance can be obtained by using Eq. (44.1) as Ri = Re [p (ri ; ri ) /qi ] , i = p, s,

(44.3)

348

J. Zhong et al.

    Rps = Re p rs ; rp /qp ,

(44.4)

where Re[·] denotes the real part of the quantity inside the square brackets. The optimal secondary source strength and the sound radiation power under optimal control are [8] qs,opt

  2 Rps Rps 1 // //2 =− qp , Wopt = qp Rp − , Rs 2 Rs

(44.5)

The noise reduction is defined as  NR ≡ −10 lg

 W . W0

(44.6)

where lg means the common logarithm with the base 10 and the sound radiation power of the primary source on ground W0 = (ρ air ωk|qp |2 )/(4π) is used as the reference. This defined noise reduction is 0 dB without active noise control if there are no additional reflecting surfaces around the system. For a constant volume primary source on ground, its sound radiation power (without ANC) varies after introducing reflecting surfaces near it. For example, the sound radiation power is increased by 3 dB when an infinitely large reflecting surface is introduced near the primary source at low frequencies [9]. Therefore, the noise reduction defined by Eq. (44.6) can then be nonzero (or even negative) without ANC when there are additional reflecting surfaces.

44.3 Numerical Results In this section, the source interval, d, is set to 0.1 m throughout the simulations and the frequency of interest ranges from 315 Hz to 5 kHz. There are many geometric configurations of the sources and surfaces. For simplicity, only one case where the midpoint of the source line is on the middle line of the surfaces is considered. Then, the effects of the included angle between the source line and the normal line of the surfaces as well as the surface intervals are simulated and analyzed. The noise reduction with different included angles when the surface interval D is set to 0.1 m is shown in Fig. 44.2a. It can be observed that the noise reduction is significantly affected by the included angle. At the frequencies below 1715 Hz, which is the corresponding frequency when the surface interval equals to the half wavelength, the system with a smaller included angle has higher noise reduction. For example, the noise reduction at 500 Hz increases from 7.8 dB to 19.6 dB as the included angle decreases from 20◦ to 5◦ . The noise reduction is sensitive to the included angle. For example, the noise reduction is infinitely large when θ = 0◦ at

44 A Numerical Study on Active Noise Radiation Control Systems Between. . .

349

Fig. 44.2 Noise reduction of the ANC system on ground with two parallel reflecting surfaces: (a) with the fixed surface interval D = 0.1 m at different included angles θ, and (b) at the fixed included angle θ = 0◦ with different surface intervals D

500 Hz; then it decreases to 19.6 dB when θ increases to 5◦ . It is noted that19.6 dB is larger (more than 13.6 dB) than that without any reflecting surfaces (only ground). It should be noted that the noise reduction performance with the parallel reflecting surfaces can be worse than that without the surfaces (only ground) if the angle is large at low frequencies. For example, the noise reduction without the surface is 6.0 dB at 500 Hz, but it decreases rapidly to 1.8 dB if the included angle is 45◦ . Figure 44.2b shows the effect of the surface interval D on the noise reduction of the ANC system when the included angle θ = 0◦ . The noise reduction varies significantly from infinity to a finite value at 1715 Hz, 858 Hz, and 429 Hz for the source interval D being 0.1 m, 0.2 m, and 0.4 m, respectively. The turning frequency is defined as the frequency where the noise reduction varies from infinity to a finite value. The turning frequency is equal to c0 /(2D) where c0 is the sound speed in the air and it decreases as the surface interval increases. For the surface interval D = 2 m which is larger than the wavelength at 315 Hz (1.1 m) to 5 kHz (0.07 m), the noise reduction curve in Fig. 44.2b fluctuates around the one without any surfaces (only ground) because the surfaces are far enough away from the ANC systems. To have effects at high frequencies, the two surfaces need to be installed closely to each other. The secondary source strength qs,opt and the self-radiation resistances of two sources and their mutual-radiation resistance are shown in Fig. 44.3 for the configuration with D = 0.1 m at θ = 0◦ . When the frequency is less than c0 /(2D), which is 1715 Hz in Fig. 44.3, the secondary source strength is exactly opposite to that of the primary source and the mutual radiation resistance between sources Rps and the self-radiation resistance of the secondary source Rs are exactly the same. Here only the zeroth normal mode exists inside the two parallel rigid boundaries. The radiation caused by the resistances at the primary (or secondary) source is exactly zero which means the overall sound power of the total system is zero. However, in real applications, the size of the reflecting surfaces cannot be infinitely

350

J. Zhong et al.

Fig. 44.3 For D = 0.1 m, (a) noise reduction (left y-axis) and the ratio of the source strengths qs,opt /qp (right y-axis) where qs,opt and qp are the source strength of the secondary and the primary source, respectively; and (b) the self-radiation resistances of the primary and secondary sources, Rp and Rs , and the mutual radiation resistance between the two sources Rps , where R0 = (ρ air ωk)/(2π ) is the self-radiation resistance of the primary source on ground without any surfaces

large which can deteriorate the noise reduction performance to a limited noise reduction.

44.4 Conclusions This chapter demonstrates that the noise reduction performance of a single channel active noise control system on ground can be significantly improved by introducing two vertically placed and parallel reflecting surfaces. If the reflecting surfaces are infinitely large, the sound radiation of the primary source can be completely suppressed in principle by a secondary source provided that the surface distance is less than the half wavelength. This is because only the zeroth normal mode exists inside the two parallel rigid boundaries, which can be completed cancelled if another source with 180 degrees phase is introduced. Future research includes exploring the noise reduction performance with finite size reflecting surfaces, the optimal geometrical shape, and the optimal configurations of the error sensors and secondary sources for multiple channel ANC systems.

References 1. Qiu, X., Lu, J., Pan, J.: A new era for applications of active noise control. INTER-NOISE and NOISE-CON congress and conference proceedings (2014). 2. Boodoo, S., Paurobally, R., Bissessur, Y.: A review of the effect of reflective surfaces on power output of sound sources and on actively created quiet zones. Acta Acust United Acust. 101(5), 877–891 (2015)

44 A Numerical Study on Active Noise Radiation Control Systems Between. . .

351

3. Tao, J., Wang, S., Qiu, X., Pan, J.: Performance of a multichannel active sound radiation control system near a reflecting surface. Appl. Acoust. 123, 1–8 (2017) 4. Pan, J., Qiu, X., Paurobally, R.: Effect of reflecting surfaces on the performance of active noise control. Proc. ACOUSTICS. (2006) 5. Zhong, J., Tao, J., Qiu, X.: Increasing the performance of active noise control systems on ground with a finite size vertical reflecting surface. Appl. Acoust. 154, 193–200 (2019) 6. Xue, J., Tao, J., Qiu, X.: Performance of an active control system near two reflecting surfaces. Proceedings of 20th international congress on sound and vibration. (2013). 7. Brekhovskikh, L.M.: Waves in Layered Media, 2nd edn. Academic, New York (1980) 8. Nelson, P.A., Elliott, S.J.: Active Control of Sound. Academic, San Diego (1992) 9. Ingard, U., Lamb Jr., G.L.: Effect of a reflecting plane on the power output of sound sources. J. Acoust. Soc. Am. 29(6), 743–744 (1957)

Part VIII

Applications and Advances in Laser Doppler Vibrometry

Chapter 45

Characterization of Active Microcantilevers Using Laser Doppler Vibrometry Michael G. Ruppert, Natã F. S. De Bem, Andrew J. Fleming, and Yuen K. Yong

45.1 Introduction The microcantilever is a microelectromechanical systems (MEMS) device and is a key enabling technology for scanning probe microscopes, scanning probe lithography systems, and probe-based data storage systems [1]. One of the most celebrated uses of the microcantilever is for atomic force microscopy (AFM) [2]. Due to its unprecedented resolution, AFM has driven major breakthroughs in material science, surface science, and bio-nanotechnology. In dynamic AFM, the cantilever is actively driven at one of its resonance frequencies, while a sample is scanned underneath a sharp tip. By controlling changes in the oscillation amplitude of the cantilever using a feedback control loop, a 3D image of the surface as well as of nanomechanical sample properties can be obtained [1]. Most conventional AFM systems employ piezoacoustic base-excitation and an optical beam deflection sensor to induce and measure the corresponding vibrations. However, this excitation scheme yields distorted frequency responses (compare Fig. 45.3d) and the optical measurement can require tedious laser alignment and relies on a sufficiently large cantilever with a reflective surface for optimal transduction [3]. As a result, substantial research efforts have gone into active cantilevers with integrating actuation and sensing on the cantilever chip level [4–6]. In this work, we investigate three different cantilever geometries with integrated piezoelectric actuation and sensing which are shown in Fig. 45.1. Each cantilever consists of a device layer of single-crystal silicon with a thickness of 10 μm, one or two 0.5 μm thick regions of piezoelectric aluminum-nitrate (AlN) for actuation and M. G. Ruppert () · N. F. S. De Bem · A. J. Fleming · Y. K. Yong The University of Newcastle, Callaghan, NSW, Australia 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/10.1007/978-3-030-48153-7_45

355

356

M. G. Ruppert et al.

Silicon Oxide Piezo Metal

50μm

100μm

(a)

(b)

100μm

(c)

Fig. 45.1 Design layout of the three cantilever designs investigated in this work with (a) rectangular, (b) triangular, and (c) stepped rectangular geometry

sensing and a 1μm layer of aluminum for electrical connections. The top surface of the device layer is n-doped and serves as the ground return for the piezoelectric transducers. With this layout, the devices can be fabricated with the rapid MEMS prototyping process PiezoMUMPS® (MEMSCAP Inc.).

45.2 Dynamic Analyses Using Laser Doppler Vibrometry In this section, the analysis and characterization of the deflection mode-shapes, system identification including actuator and sensor calibration, and the calibration of the dynamic parameters of the cantilever eigenmodes is performed with laser Doppler vibrometry (LDV) using a Polytec MSA-100-3D. LDV has demonstrated significant influence in the field of experimental structural analysis of MEMS [7].

45.2.1 Modal Analysis In order to experimentally obtain the mode shapes using LDV, the piezoelectric layers are excited with a low-voltage periodic chirp signal, while the cantilevers are scanned underneath the laser. Depending on the geometry of the cantilever, an appropriate scan point layout is chosen with 150–300 scan points as shown in Fig. 45.2. For each scan point, the frequency response from actuation voltage to deflection is obtained by taking the FFT of the velocity and reference signal with 25 complex averages and performing the integration in the frequency domain. When the magnitude spectrum is plotted against the scan point location, the deflection mode shapes as shown in Fig. 45.2 are obtained. It can be noticed that while the first three flexural modes of the rectangular cantilever follow the theoretical Euler Bernoulli mode shapes, the third modes of the triangular and stepped rectangular show deviations. Due to the large base of the stepped rectangular cantilever, it’s third mode follows the mode shape predicted by Mindlin plate theory [8].

45 Characterization of Active Microcantilevers Using Laser Doppler Vibrometry

357

Fig. 45.2 Modal analysis of the first three flexural eigenmodes of three different cantilever geometries using a laser Doppler vibrometer

Phase [deg]

Mag. [dB(m/V)]

-100

-150

-200 180

0

0

-2k

-180

-4k FRF Model

-360 10

100

500

Frequency [kHz] (a)

10

100

1000

10

100

Frequency [kHz] (b)

Frequency [kHz] (c)

-6k 500 100

1000

Frequency [kHz] (d)

Fig. 45.3 Frequency response function (FRF) and identified model of the cantilevers. (a) Rectangular. (b) Triangular. (c) Stepped. (d) Stack-actuated

45.2.2 System Identification The frequency response from piezoelectric actuation voltage to the tip deflection is obtained by considering only one scan point near the tip. The measured responses up to the third flexural mode as well as the identified models are shown in Fig. 45.3a–c. The identified models are of the form [9] Gvib (s) =

3  i=1

αi ωi2 s2 +

ωi Qi s

+ ωi2

+ D,

(45.1)

where αi , ωi , Qi are the actuation gain, natural resonance frequency, and quality factor of the i-th mode. If two consecutive αi are of opposing sign, a non-collocated system is obtained (Fig. 45.3a and b); otherwise the system is said to be collocated (Fig. 45.3c) [10]. As the magnitude responses are in units of dB(m/V), the actuation gains at each resonance can be extracted directly (compare Table 45.1). Note that the response shown in Fig. 45.3d obtained from a conventional piezoacoustic base-

1 2 3 (b) FEA

Mode

1 2 3 (a) Experiment

Mode

363 614 400

35.5 209.8 572

[kHz]

− − −

Rectangular fi Qi

41.9 235 645

[kHz]

Rectangular fi Qi

42.8 1.02 k 12.1 k

N m

k4i 5

42.8 1.28 k 10.1 k

N m

k4i 5

9.30 1.86 −0.56

mV

αi  nm 

8.00 0.66 −0.30

mV

αi  nm  556 850 904

91.6 558 1434

− − −

[kHz]

Triangular fi Qi

101 609 1548

[kHz]

Triangular fi Qi

Table 45.1 Cantilever parameters obtained from (a) experiment and (b) FEA

132 3.30 k 48.5 k

N m

k4i 5

117 4.4 k −

N m

k4i 5

4.9 1.26 −0.02

mV

αi  nm 

3.30 0.75 −0.01

mV

αi  nm  376 364 187

37.8 116 354

[kHz] − − −

Stepped fi Qi

36.4 112 345

[kHz]

Stepped fi Qi

32.0 217 2.35 k

N m

k4i 5

30.4 315 3.57 k

N m

k4i 5

7.48 3.60 0.32

mV

αi  nm 

4.95 2.82 0.19

mV

αi  nm 

358 M. G. Ruppert et al.

45 Characterization of Active Microcantilevers Using Laser Doppler Vibrometry

359

excited cantilever is unsuitable for system identification due to the numerous additional dynamics unrelated to the cantilever.

45.2.3 Thermal Stiffness Calibration The thermal noise method is particularly useful for the calibration of the dynamic spring constants of multiple eigenmodes even for arbitrary geometries [11]. It is based on measuring the cantilever vibrations due to Brownian motion caused by thermal noise. In this case, the potential energy of the harmonic oscillator is equal to the thermal energy 12 kB T = 12 ki x¯i2 where kB denotes the Boltzmann constant, T the equilibrium temperature, ki the spring constant, and x¯i2 the mean squared displacement of the i-th mode. Experimentally, this is performed by measuring the velocity power spectrum at the end of the cantilever using LDV with the piezoelectric layers grounded and integrating the area underneath the resonance peak. This can be done by performing a Lorentzian function fit of the form for each mode [11] Si (f ) =

Ai fi4 Q2i (f 2 − fi2 )2 + f 2 fi2

+ A0 ,

(45.2)

where Ai is a fitting parameter and A0 is the white background noise. From the fit, the mean squared velocity can be extracted as v¯i 2 =

πfi Ai 2Qi

(45.3)

which allows to calculate the spring constant as ki = (2πfi )2

kB T . v¯i2

(45.4)

The results are shown in Fig. 45.4. The spectra are obtained by recording the thermal vibrations in the time domain and obtaining the power spectrum density estimates using Welch’s method with 64 averages, no overlap, and using the Hanning window. It can be seen how the maximum power in each resonance mode decreases with increasing mode number, leading to increasing stiffnesses in the higher eigenmodes.

360

M. G. Ruppert et al. 25

100 800

80

600

20

60

400

40

200

15

20

0

10 41

41.5

42

42.5

233 40

234

235

236

237

640

645

650

3.5

400

200

30

3

20

2.5

10

0 98

100

102

104

1000

600 200

605

610

40

150 100

500

2 1545 1546 1547 1548 1549 1550

615

30

50 20

0 35.5

36

36.5

Frequency [kHz]

37

111

112

113

Frequency [kHz]

114

342

344

346

348

Frequency [kHz]

Fig. 45.4 Thermal noise spectra of the first three eigenmodes of the cantilevers

45.3 Conclusion The results of the experimental characterization of all three cantilevers are stated in Table 45.1 together with the values obtained from a finite element analysis (FEA) using ANSYS Workbench 19.2 and CoventorWare 10.3. For details of the finite element models of piezoelectric cantilevers, the reader is referred to [12]. While the simulation tools allow for the direct extraction of the resonance frequencies and sensitivities, the dynamic stiffnesses are calculated using customized code by equating potential and kinetic energy. It can be noted that the triangular cantilever achieves the highest Q factors across all higher eigenmodes as well as the farthest spacing between resonance frequencies. The stepped rectangular cantilever has the closest spaced higher eigenmodes with lower Q factors, an enhanced second eigenmode sensitivity [6] as well as significantly reduced higher eigenmode stiffnesses. In general, the experimental values agree with the ones obtained from FEA under acceptable tolerances. Note that a full three-dimensional numerical analysis of air damping, as the biggest contributor to the Q factor, is computationally very expensive and time-consuming [13]. Therefore, it is common practice to estimate the Q factors from experimental data.

45 Characterization of Active Microcantilevers Using Laser Doppler Vibrometry

361

References 1. Bhushan, B.: Scanning Probe Microscopy in Nanoscience and Nanotechnology. Springer, Berlin/Heidelberg (2010) 2. Binnig, G., Quate, C.F., Gerber, C.: Atomic force microscope. Phys. Rev. Lett. 56, 930–933 (1986) 3. Ruppert, M.G., Moheimani, S.O.R.: High-bandwidth multimode self-sensing in bimodal atomic force microscopy. Beilstein J. Nanotechnol. 7, 284–295 (2016) 4. Rangelow, I.W., et al.: Review article: active scanning probes: a versatile toolkit for fast imaging and emerging nanofabrication. J. Vac. Sci. Technol. B 35(6), 06G101 (2017) 5. Ruppert, M.G., Fowler, A.G., Maroufi, M., Moheimani, S.O.R.: On-chip dynamic mode atomic force microscopy: a silicon-on-insulator MEMS approach. IEEE J. Microelectromech. Syst. 26(1), 215–225 (2017) 6. Ruppert, M.G., et al.: Multimodal atomic force microscopy with optimized higher eigenmode sensitivity using on-chip piezoelectric actuation and sensing. Nanotechnology 30(8), 085503 (2019) 7. Rothberg, S., et al.: An international review of laser doppler vibrometry: making light work of vibration measurement. Opt. Lasers Eng. 99, 11–22 (2017) 8. Moore, S.I., Ruppert, M.G., Yong, Y.K.: Multimodal cantilevers with novel piezoelectric layer topology for sensitivity enhancement. Beilstein J. Nanotechnol. 8, 358–371 (2017) 9. Ruppert, M.G., Moheimani, S.O.R.: Multimode Q control in tapping-mode AFM: enabling imaging on higher flexural eigenmodes. IEEE Trans. Control Syst. Technol. 24(4), 1149–1159 (2016) 10. Ruppert, M.G., Yong, Y.K.: Note: guaranteed collocated multimode control of an atomic force microscope cantilever using on-chip piezoelectric actuation and sensing. Rev. Sci. Instrum. 88(8), 086109 (2017) 11. Sader, J.E., et al.: Spring constant calibration of atomic force microscope cantilevers of arbitrary shape. Rev. Sci. Instrum. 83(10), 103705 (2012) 12. Moore, S.I., Ruppert, M.G., Yong, Y.K.: An optimization framework for the design of piezoelectric AFM cantilevers. Precis. Eng. 60, 130–142 (2019) 13. Kaajakari, V.: Practical MEMS. Small Gear Publishing, Las Vegas (2009)

Chapter 46

Experimental Investigation on Generation Mechanism of Friction Vibration in Toner Fixing Device Y. Nakano, Y. Matsumura, T. Hase, and H. Takahara

46.1 Introduction Friction force can cause friction-induced vibration, which is a self-excited vibration. Regarding an initial stage of the generation of self-excited vibration as the divergent vibration of the linearized system and considering the stability analysis of the linearized system, we can understand in a mathematical perspective that frictioninduced instability is caused by a negative damping and the asymmetry of the stiffness matrix due to the friction. The example of the self-excited vibration caused by the asymmetry of the stiffness matrix due to the friction is the selfexcited vibration caused by Coulomb’s friction with a constant friction coefficient independently of a sliding velocity such as brake squeal [1, 2] and friction-induced vibration in a paper calender [3]. The example of the self-excited vibration caused by negative damping is friction-induced instability caused by a friction coefficient which has a negative slope with respect to a relative velocity such as friction-induced vibration in windscreen wiper contacts [4] and in a marine water lubricated bearing [5]. Recently, the problem of noise occurred in a toner fixing device of a printer has been reported. In fixing process of an image forming apparatus such as a copier and a printer, the toner is fixed to a paper by heat and pressure. To improve energy efficiency of the printing system, the toner fixing device with low heat capacity has been widely developed. Figure 46.1. shows the configuration of the toner fixing device. Figure 46.1a, b shows the sectional and the detail view of the actual toner

Y. Nakano () · Y. Matsumura · H. Takahara School of Engineering, Tokyo Institute of Technology, Tokyo, Japan e-mail: [email protected] T. Hase Office Printing Business Group, Ricoh Company Ltd., Ebina, Kanagawa, Japan © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_46

363

364

Y. Nakano et al.

Fixing sleeve t=0.2mm

Sliding sheet Pad Nip area

Fixing sleeve Pressure roller

Pad Sliding sheet Heat source

Toner

Motor

Pad holder

Pad holder

Paper

(a)

(b)

Fig. 46.1 Configuration of toner fixing device. (a) Sectional view of fixing device. (b) Detailed view of the actual toner fixing device

fixing device, respectively. As shown in Fig. 46.1, the toner fixing device comprises a fixing sleeve, a pressure roller with an elastic layer covering a core metal, a pad, a pad holder, and a heat source. When a paper with toner passes through between the sleeve and the pressure roller, the toners are fused and fixed to the paper. The sleeve rotates following the pressure roller driven by a motor and is pressed against the roller through the pad in order to form a nip area which means a distributed contact area between the sleeve and the roller indicated by a broken line in Fig. 46.1a. The pad and the inner surface of the sleeve are in sliding contact via a sliding sheet impregnated with a lubricant. In some operational conditions, uncomfortable noise occurs by friction between the sleeve and the sliding sheet attached to the pad. The previous studies have also reported noise problems generated in a toner fixing device [6, 7]. The patent presented by Miyahara [6] stated that stick-slip phenomenon, which is a kind of friction-induced vibration, could be suppressed by increasing the loss elastic modulus of a sliding layer. In addition, the patent presented by Omori [7] proposed the drive torque, which leads to noise, could be reduced by changing the property of a sliding layer and a lubricant. However, the generation mechanism of friction-induced vibration in a toner fixing device has not been clear yet. Countermeasures against friction-induced vibration depend on the generation mechanism. It is important to first identify the generation mechanism. The purpose of the present study is to clarify the generation mechanism of friction-induced vibration in a toner fixing device. The present study investigates two matters experimentally. The one is to clarify the relationship between the vibration occurrence conditions and the friction torque between the sleeve and the sliding sheet attached to the pad. The other is to identify the main vibration sources.

46 Experimental Investigation on Generation Mechanism of Friction. . .

365

46.2 Relationship Between Friction Torque and Noise Occurrence We investigated the friction torque of the sliding part between the sleeve and the sliding sheet in order to clarify the generation mechanism of friction-induced vibration. By measuring the relationship between the driving torque of the pressure roller and the circumferential velocity of the pressure roller, we obtained the relationship between the friction torque and the sliding velocity. Figure 46.2 shows the friction torque with respect to the sliding velocity at the different numbers of printing sheets. Generally, the life span is 100,000 to 400,000 sheets. From Fig. 46.2, it is observed that there is the area where the slope of the curve is negative when the total number of printing sheets is small. On the other hand, the curve is almost a positive slope in 100,000 sheets. We think the increase in the number of printing sheets caused the wear of the sliding sheet and the friction characteristic of the sliding sheet was changed. Figure 46.3 shows the relationship between the total number of printing sheets and the number of noise occurrence incidents. From Fig. 46.3, noise is likely to occur more frequently when the number of printing sheets is not many. From the dependency of the friction torque on velocity and the noise generation tendency with respect to the number of printing sheets, the noise is likely to occur in a region where the torque has a negative slope with respect to the sliding velocity. However, when the torque was measured at the less number of printing sheets than 20,000, the friction vibration occurred. Previous study [8] reported that friction-induced instability with a constant friction coefficient could cause a decrease of macroscopic friction coefficient with velocity due to vibrations. We will investigate the relation between the torque and the friction coefficient by measuring the torque without a noise generation.

Torque [Nm]

0.40 Several hundreds sheets 20,000 sheets 40,000 sheets 100,000 sheets

0.35 0.30 0.25 0.20

0

50

100

150

200

250

Circumferential velocity of pressure roller [mm/s] Fig. 46.2 Relation between friction torque and velocity at different printing sheets

Y. Nakano et al.

Noise occurence incidents

366

25 20 15 10 5 0

~10,000 ~20,000 ~30,000 ~40,000 40,000~

Total number of the printing sheets Fig. 46.3 Relation between total number of printing sheets and number of noise occurrence incident Precision sound level meter Measurement points of Eddy current sensors 160mm B A Fixing sleeve Pressure roller C

Pressure roller Motor

a

b

40°

c

D

30mm

Fixing sleeve

E F

(a)

G

(b)

Fig. 46.4 Measurement method of sound pressure, sleeve vibration, and pressure roller vibration. (a) Measurement of sound level and displacement. (b) Measurement of vibration velocity

46.3 Identification of Vibration Mode Causing Noise 46.3.1 Vibration Mode of the Sleeve and the Pressure Roller in Actual Operation We measured the sound pressure of noise using a precision sound level meter (Ono Sokki Co., Ltd., MI-1233) during the noise generation. Figure 46.4a shows the experimental setup for the measurement point of the sound level meter. As a result of the frequency analysis of the sound pressure, the noise frequency was about 200 Hz. Next, we measured the vibration of the sleeve and the pressure roller in the radial direction near the end in the longitudinal direction during the noise generation using a laser Doppler vibrometer (Ono Sokki Co., Ltd., LV-1800) respectively. In addition,

0.060

0.060

0.030

0.030

Amplitude [m/s]

Amplitude [m/s]

46 Experimental Investigation on Generation Mechanism of Friction. . .

0

-0.030

(a)

0.005

0.010

Time [s]

0.015

0

-0.030 Radial vibration Circumferential vibration

Radial vibration Circumferential vibration

-0.060 0.000

367

0.020

-0.060 0.000 (b)

0.005

0.010

0.015

0.020

Time [s]

Fig. 46.5 Vibration velocity of sleeve and pressure roller. (a) Sleeve. (b) Pressure roller

the circumferential vibration of the sleeve was measured using a laser Doppler surface velocity meter (Ono Sokki Co., Ltd., LV-7000) simultaneously with the radial vibration of the sleeve or the pressure roller. Figure 46.4b shows measuring points of the laser Doppler vibrometer and the laser Doppler surface velocity meter. In the point (a) and the point (b), the vibration velocity of the sleeve or the roller in the radial direction was measured by the laser Doppler vibrometer. In the point (c), the vibration velocity of the sleeve in the circumferential direction was measured by the laser Doppler surface velocity meter. Figure 46.5a, b shows the vibration velocity of the sleeve and the pressure roller in the radial direction with solid lines and the vibration velocity of the sleeve in the circumferential direction with dotted lines. Although Fig. 46.5a, b has different measurement timings, in comparison with the circumferential vibration of the sleeve as a reference, the vibration amplitude of the sleeve in the radial direction is much larger than that of the pressure roller. Therefore, it can be seen that the sleeve was the main vibration source. We measured the radial vibration displacement of the sleeve using seven eddy current sensors (AEC, PU-05) arranged in the circumferential direction of the sleeve as shown in Fig. 46.4a. The measurement points were arranged in the point A to the point G at a regular pitch of 40 degrees shown in the Fig. 46.4a near the center of the sleeve in the longitudinal direction. Figure 46.6b shows the actual operating vibration mode of the sleeve at 200 Hz derived from the amplitude ratio and the phase difference of the frame vibration and the sleeve vibration. Since the entire toner fixing device vibrated during the noise generation, the vibration of the frame of the toner fixing device measured by an accelerometer (Ono Sokki Co., Ltd., NP2106) was set to a reference point shown in Fig. 46.6a. In Fig. 46.6b, the circle symbols represent the vibration displacement plotted at intervals of 1/4 cycle at each eddy current sensor position. From Fig.46.6, it can be seen that the sleeve vibration mode during the noise generation has three nodes in circumferential direction of the sleeve.

368

Y. Nakano et al.

40°(B)

Eddy current sensor × 7

1/4T 2T/4 3T/4 4T/4

0°(A)

80°(C)

320° Nip area

120°(D)

Accelerometer

Fixing sleeve

(a)

280° 160°(E)

(b)

200°(F)

240°(G)

Fig. 46.6 Measurement of actual operating vibration mode of sleeve. (a) Measurement point of accelerometer. (b) Radial vibration mode of sleeve

46.3.2 Effect of Damper Applied to Sleeve on Noise We tried to reduce the noise concerned with friction-induced vibration of the sleeve. We measured the radial vibration displacement of the sleeve when a cylindrical ethylene propylene rubber is brought into frictional contact with the sleeve in order to reduce the sleeve vibration. Figure 46.7a shows the measurement point of an eddy current sensor and the attachment point of the rubber. The sleeve displacement was measured using the eddy current sensor at the point (C) in Fig. 46.7a near the center of the sleeve in the longitudinal direction. Figure 46.7b shows the short-time Fourier transform results of the sleeve vibration when the rubber was pressed against the sleeve near the point (B) shown in Fig. 46.7a. In Fig. 46.7b, the rubber is brought into frictional contact with the sleeve from around 1.7 s during noise generation. From Fig. Fig. 46.7b, it can be seen that the noise of 200 Hz disappears while pressing the rubber against the sleeve. From some additional experimental results of the application of a damper to the sleeve, it was confirmed that the noise was suppressed when the rubber was pressed against the sleeve at the antinode position of the actual operating vibration mode of the sleeve.

46.4 Conclusion The present study experimentally investigated the generation mechanism of noise occurred in the toner fixing device. The conclusions are as follows: 1. We investigated the dependency of the friction torque on velocity and the noise generation tendency with respect to the number of printing sheets and confirmed that the noise is likely to occur in a region where the torque has a negative slope with respect to velocity.

46 Experimental Investigation on Generation Mechanism of Friction. . .

Direction of rotation

80°(C) 320°

Measurement point

300

-40

0°(A)

Nip area

120°(D) 280°

Frequency [Hz]

40°(B)

-60

250

-80

200

-100 -120

150

Duration of damping

160°(E) 200°(F)

240°(G)

Sectional view of fixing device

100

0

1

2

-140

Power/Frequency [dB/Hz]

Rubber Damping point

369

3

Time [s]

Fig. 46.7 Effect of damper on noise. (a) Measuring and damping point. (b) STFT results of sleeve with and without rubber

2. It was clarified that the vibration amplitude of the sleeve is much larger than that of the pressure roller as a result of vibration measurement in actual operation. 3. The actual operating vibration mode of the sleeve was measured, and it was identified that the vibration mode of the sleeve during the noise generation was the radial vibration mode with three nodes. 4. It was confirmed that the noise of 200 Hz generated in the toner fixing device disappears by adding a damper to the sleeve.

References 1. Kinkaid, N.M., O’Rreilly, O.M., Papadopoulos, P.: Automotive disc brake squeal. J. Sound Vib. 267(1), 105–166 (2003) 2. Brunetti, J., Massi, F., D’Ambrogio, W., Berthier, Y.: A new instability index for unstable mode selection in squeal prediction by complex eigenvalue analysis. J. Sound Vib. 377, 106–122 (2016) 3. Spelsberg-Korspeter, G., Hochlenert, D., Hagedorn, P.: Self-excitation mechanism in paper Calenders formulated as a stability problem. Tech. Mech. 31(1), 15–24 (2011) 4. Reddyhoff, T., Dobre, O., Rouzic, J.L., Gotzen, N.A., Parton, H., Doni, D.: Friction induced vibration in windscreen wiper contacts. J. Vib. Acoust. 137(4), 041009–0411-7 (2015) 5. Huanjie, W., Zhenglin, L., Li, Z., Jun, Y.: Influence of both friction and wear on the vibration of marine water lubricated rubber bearing. Wear. 376-377, 920–930 (2017) 6. Miyahara, Y.: United States patent application publication. US 2014/0105659A1 (2014) 7. Omori, K., Okabayashi, Y., Mizoguchi, A.: United States patent application publication. US 2014/0341621 (2014) 8. Laurent, B., d’Errico, S., Yves, B.: Influence of sliding contact dynamics on macroscopic friction coefficient variation. Revue Européenne des Eléments Finis. 305–321 (2005)

Chapter 47

Using a Laser Doppler Vibrometer to Estimate Sound Pressure in Air Tong Xiao

, Xiaojun Qiu

, and Benjamin Halkon

47.1 Introduction Microphones have been used to measure sound pressure extensively for over a century [1]. Although the development of microelectromechanical systems (MEMS) enables microphones to be highly miniaturised, there are still demands from practice for smaller microphones or non-contact sound pressure measurements, such as measuring sound pressure at a person’s ears or measuring sound pressure in a remote location. Laser Doppler vibrometers (LDVs) make use of interferometer optical arrangements to detect the Doppler frequency shift of a vibrating body, resulting in surface √ velocity measurements [2]. The typical resolution can be down to 10 nm/s/ Hz. Due to its accuracy and the non-intrusiveness, the technique and the device are used in an increasingly wide range of industrial and research application domains [2, 3]. In particular, they are used to measure the resonance frequency and loss factor of a microphone diaphragm [4] and its sensitivity [5]. In this chapter, the two fundamental aspects of LDVs (high sensitivity and noninvasiveness) are combined to form an “optical microphone” solution. The probe laser beam from the LDV can be directed to any visible location where a small, vibrating membrane is placed to estimate the sound pressure. Firstly, an LDV is used to measure the vibration of a microphone diaphragm caused by a loudspeaker, where the sound pressure from the microphone provides a reference signal for verifying the idea by comparing it with the velocity measurement from the LDV. The coherence between the loudspeaker signal and the two signals is made respectively to examine the feasibility of using the velocity from the LDV to recover the sound

T. Xiao () · X. Qiu · B. Halkon Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT University of Technology Sydney, Sydney, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2021 S. Oberst et al. (eds.), Vibration Engineering for a Sustainable Future, https://doi.org/10.1007/978-3-030-48153-7_47

371

372

T. Xiao et al.

pressure. Later, a customised membrane is constructed to replace the microphone for practical use, and the coherence between the loudspeaker signal and the velocity measurement from the LDV is assessed again. This coherence decides the quality of using the LDV and the customised membrane to estimate the sound pressure in air.

47.2 Theory The membrane vibration equation in the cylindrical coordinates can be written as [6, 7]: ∇ 2 η (r, θ ) + K 2 η (r, θ ) = −

p (r, θ, 0) pi + , 0