Dynamic Modeling and Active Vibration Control of Structures 9402421181, 9789402421187

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Moon Kyu Kwak

Dynamic Modeling and Active Vibration Control of Structures

Dynamic Modeling and Active Vibration Control of Structures

Moon Kyu Kwak

Dynamic Modeling and Active Vibration Control of Structures

Moon Kyu Kwak Department of Mechanical, Robotics and Energy Engineering Dongguk University Seoul, Korea (Republic of)

ISBN 978-94-024-2118-7 ISBN 978-94-024-2120-0 (eBook) https://doi.org/10.1007/978-94-024-2120-0 © Springer Nature B.V. 2022 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 B.V. The registered company address is: Van Godewijckstraat 30, 3311 GX Dordrecht, The Netherlands

Preface

In 1985, I went to Virginia Tech in the USA to study vibration and met Prof. Meirovitch as an academic advisor. At that time, Prof. Meirovitch, an eminent scholar of vibration theory, shocked me when he told me that he was no longer studying vibration and was studying dynamics and control. I knew dynamics to some extent, but the field of control was a field that was unfamiliar to me. So, the first meeting with “control” began. What I learned while studying feedback control was that control is a way to make a system return to a specific state in the end, and that analyzing differential equations and stability using a mathematical tool is the main task of control theory. In this respect, it is not very different from the vibration analysis process. However, since control design using Laplace transformation, linear algebra using matrix, and optimal control theory were combined, combining control theory with dynamics was recognized as a new challenge for those who study dynamics in 1980s. Because the most of control theories was not developed for the motion control of the structure, it was not easy to apply even a well-established control theory to the motion control of the structure. Therefore, various control theories have been newly developed and applied for maneuvering and vibration suppression of structures. The opportunity to apply the control theory of structures, which was only approached as a theory, to actual structures began with the study of smart structures at the US Airforce Research Lab. It was a novel experience to actually control the vibration of a simple beam structure using piezoceramic sensors and actuators. In addition, I studied the characteristics of smart materials such as shape memory alloy, magneto-restrictive material, and magneto-rheological fluid, and while doing active vibration control experiments using these smart materials, I realized that it was not easy to apply a simple control algorithm in practice. Since then, my research activities have focused on control theory that can be applied in practice. Vibration is a relatively simple field of dynamics problems. The vibration of a simple single-degree-of-freedom vibration system composed of spring-mass-damper is expressed as a second-order ordinary differential equation, and the vibration of a complex multi-degree-of-freedom vibration system is a second-order matrix ordinary differential equation. The vibration of the continuous system is expressed as a partial differential equation, and an approximate multi-DOF system is derived and v

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expressed through discretization methods such as Rayleigh-Ritz method, assumedmode method, and the finite element method to enable numerical analysis. When a dynamic model is derived, qualitative analysis is performed using classical tools such as mathematics or quantitative analysis is performed using numerical analysis methods. It is natural for vibration engineers to derive a dynamic model and perform analysis to theoretically analyze the vibration generated in the structure. Most of the vibration analysis is to calculate the response when there is an external disturbance. As this problem includes control power, the field of active vibration control has been created. The level of vibration predicted using the dynamic model of the vibration system is used as a criterion for judging the safety of the structure and the installed equipment and the comfort of humans living in the structure. However, in recent years, industrial engineers or people in the general society do not request vibration analysis results or vibration measurement results, but demand that the vibration be lowered to a certain level or less. And since these vibration levels are difficult to reach with methods such as resonance avoidance or passive dampers, active methods of using sensors and actuators have been sought. That is, a method of suppressing vibration using a control device has been developed. The core of active vibration control can be said to be the development of a control force calculation algorithm that makes the response suppressed and does not make the system unstable for the dynamic model given by the second-order differential equation. Therefore, it does not end with the derivation of a dynamic model for the vibration system and performing vibration analysis, but it is necessary to design a controller that suppresses vibration by including the control force. There are not many books explaining the dynamic modeling of a vibration system and the process of designing a controller suitable for this model. This book was written to meet this need. The structure of this book is as follows. Chapter 1 contains a literature survey on active vibration control. In particular, research results on smart structures are introduced. And, the results of active vibration control research applied to actual structures such as automobiles, railway vehicles, elevators, building structures, aircrafts, space structures, and robots are introduced. Among the active control theories, practically applicable theories were collected and introduced separately. Of course, it is expected that this book will not cover all the research results on active vibration control. I ask for your understanding. Chapter 2 briefly deals with the analysis of the single-degree-of-freedom vibration system, which is the basis of vibration analysis. A mathematical method for free and forced vibration analysis of a linear induction vibration system composed of springmass-damper, and a numerical analysis method using MATLAB/Simulink™ are introduced. Chapter 3 expands the theory of the single-degree-of-freedom vibration system in Chap. 2 and deals with the free and forced vibration analysis of the multi-degree-offreedom vibration system. The motion of a multi-degree-of-freedom vibration system is expressed as a matrix differential equation. As a method of solving the secondorder matrix ordinary differential equation, the transformation into the state-space equation, which is a first-order matrix ordinary differential equation, is explained.

Preface

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This conversion process is indispensable in order to calculate the response using MATLAB/Simulink. Since the controller design is also based on the state-space equation, this transformation process is described in detail. Chapter 4 deals with vibration analysis of continuous system. Continuous systems such as beam, plate, and shell are expressed by a partial differential equation, and it is extremely difficult to obtain an exact solution except for the uniform case. So, it uses discretization methods such as Rayleigh-Ritz method, assumed-mode method, and finite element method to make an approximate multi-degree-of-freedom vibration system and perform analysis. Chapter 4 explains this process in detail. Chapter 5 introduces the active vibration control algorithm. Although various control theories have been developed, the characteristics of sensors and actuators must also be considered in order to be applied to actual structures. Chapter 5 discusses the limitations and application problems of the control theories developed so far and introduces representative control theories that have been successfully applied to actual structures. Chapter 6 introduces sensors, actuators, and controllers used for active vibration control. The development of the control algorithm must take into account the characteristics of the available sensors and the operating principle of the actuator. If these points are not taken into account, it may be possible with numerical calculations, but practical application may not be possible. In order to verify whether the developed control algorithm can be applied to an actual structure, a controller must be made. To this end, basic knowledge of electronic circuits, micro-controllers, and DSP programming is required. Recently, the control algorithm is generally programmed using a digital controller. To program the control algorithm, the control algorithm expressed by the continuous time equation must be converted into a discrete-time equation. Chapter 6 introduces this process and controller types and usage. Chapter 7 introduces examples that have proven validity by applying them to actual structures. Active vibration control of structures such as beams, plates, and shells with bonded piezoceramic sensors and actuators, active vibration control of building structures using an active mass damper, and active vibration control of automobiles, railway vehicles, and elevators are introduced. Seoul, Korea (Republic of)

Moon Kyu Kwak

Acknowledgements

Writing a book is a lengthy and difficult task. I would like to acknowledge the support of many people. First, I am indebted to all the professors I have had in my life from Dr. K. C. Kim of Seoul National University of Korea to my advisor in graduate school, Dr. Leonard Meirovitch, who had provided me motivation for research in vibration control. I am very grateful to my employer, Dongguk University-Seoul. Many graduate students helped correct errors and improve the text. I thank my beloved wife Soon Hee Lee for all her support and encouragement. I am thankful to my dear parents, Byung-Ki Kwak, Joo-Soon Kang, and parents-inlaw Duk-Soo Lee, and Hyun-Sook Kim, who worked hard so their children could get a good education. I am also thankful to my sons, Albert and Simon, and daughterin-law, Stephanie, for their support to my scholastic activity. While both the author and the publisher have made every effort to produce an error-free book, there may nevertheless be errors. I would be most appreciative if a reader who spots an error or has a comment about the book would bring it to my attention. My current e-mail address is [email protected].

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Contents

1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Smart Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Grid Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Positive Position Feedback (PPF) Control . . . . . . . . . . . . . . 1.2.6 Shunt Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Dynamic Modeling of Piezoelectric Sensors and Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Smart Structure Applications . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Automobile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Railway Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Elevator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Building Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Aircrafts and Space Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Control Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Feedback Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Virtual Tuned Mass Damper (VTMD) Control . . . . . . . . . . 1.9.3 Semi-active Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.5 Active Mass Damper (AMD) and Negative Acceleration Feedback (NAF) Control . . . . . . . . . . . . . . . . . 1.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Vibration Analysis of Single-Degree-of-Freedom System . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Newton’s Second Law and Equation of Motion . . . . . . . . . . . . . . . . 2.3 Undamped Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 5 7 8 9 10 11 11 13 14 15 17 18 18 19 20 20 21 21 24 24 39 39 39 40 xi

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2.4 Damped Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Transfer Function and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Response Calculation by Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Base Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 46 48 54 56 59 65 66

3 Vibration of Multi-degree-of-freedom System . . . . . . . . . . . . . . . . . . . . . 67 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Newton’s Second Law and Equations of Motion . . . . . . . . . . . . . . . 68 3.3 Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Free Vibration of Undamped MDOF System . . . . . . . . . . . . . . . . . . 70 3.5 Free Vibration of Damped MDOF System . . . . . . . . . . . . . . . . . . . . 76 3.6 Reduced-Order Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.7 State-Space Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.9 Arbitrary Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 Vibration Analysis of Continuous System . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Beam Bending Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Vibration of Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Vibration of Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Calculation of Vibration Response of Continuous System . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 107 114 125 138 144 144

5 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Vibration Control of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Direct Velocity Feedback Control for SDOF System . . . . . . . . . . . . 5.4 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Positive Position Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Higher Harmonic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Virtual Tuned Mass Damper Control . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Active Mass Damper and Negative Acceleration Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Filtered-X LMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 150 156 157 158 167 169 175 180 183 189 191

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6 Sensors, Actuators, and Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Piezoelectric Sensor and Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Laser Displacement Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Voice-Coil Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Pneumatic Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Linear Actuator and AMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Vibration Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Magnetorheological Fluid Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Analog Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 dSPACE Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 DSP and Digital Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 195 200 203 203 205 206 208 209 210 211 214 215 216

7 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Beam Equipped with Piezoceramic Sensors and Actuators . . . . . . . 7.3 Grid Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Plate Equipped with Piezoelectric Sensors and Actuators . . . . . . . . 7.5 Plate with AMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Shell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Elevator Vibration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Automobile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Railway Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Building Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 217 224 234 248 255 279 303 310 325 365 365

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Chapter 1

Literature Survey

1.1 Introduction We study vibrations to prevent human and structure from excessive vibrations which may cause failure and to provide comfortable environment to workers, automobiles, and home. Therefore, a theoretical model for predicting vibration has been derived and the work of analyzing the characteristics of vibration has been performed using this. However, as well as predicting vibration, the demand for suppressing vibration that causes discomfort to structures or humans has also increased, and thus, a method to effectively control vibration has also been sought. In order to theoretically analyze the vibration problem of a structure, it is essential to deduce the equation of motion. Therefore, many studies on the vibration problem are related to the derivation of the motion equation and the natural vibration analysis using it. In particular, since the reason why the amplitude of the vibration increases is related to the resonance, it is very important to predict the natural frequency of the structure. When designing a structure, it is a traditional way to cope with vibration to understand the vibration force applied to the structure and make sure that the frequency of the vibration force does not overlap the natural frequency of the structure. However, as more quietness of the structure was required, the vibration treatment method of this degree was not sufficient. The control of mechanical and structural vibration has significant applications in manufacturing, infrastructure engineering, and consumer products. In the machine tool industry, mechanical vibration degrades both the fabrication rate and quality of end products. In civil engineering constructs, structural vibration degrades human comfort. In automotive and aerospace fields, vibration reduces component life, and the associated acoustics noise annoys passengers. There are two methods to suppress the vibration of the structure: passive method and active method. As a passive method, there is a method to avoid resonance by adjusting mass and stiffness. In addition, traditionally, passive isolators and dampers are used to attenuate mechanical vibrations. For example, installing rubber mounting

© Springer Nature B.V. 2022 M. K. Kwak, Dynamic Modeling and Active Vibration Control of Structures, https://doi.org/10.1007/978-94-024-2120-0_1

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1 Literature Survey

between the machines and foundations or adding passive dampers to structures are common practices in vibration isolation and attenuation. One of the well-known passive methods is dynamic absorber (DA) or Tuned Mass Damper (TMD), which absorbs the vibration of the main structure using additional spring-mass-damper. However, when the passive method does not satisfy the vibration tolerance, the active method should be used. This book mainly explains the active method. When using an active method, a model that can predict vibration, a sensor model, an actuator model, and an appropriate vibration control algorithm are required. So, not only vibration theory, but also control theory and electronic circuits must be studied together. It is difficult to acquire additional knowledge about electronic circuits and controls from mechanical or structural engineers studying vibration theory. In particular, control theory or electronic circuits were not developed with vibration control in mind, so it is recognized as a difficult part to study for those who want to actively control vibration. The process of designing an active control system for the attenuation of vibration in machines and structures generally involves many steps. A typical scenario is as follows. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Analyze the vibratory system, e.g. machine or structure, to be controlled. Derive a dynamic model of the system and sensor equation using analytical or numerical methods. Identify natural vibration characteristics if some natural modes are to be controlled. Derive a reduced-order model to simplify the control design. Analyze the type of external disturbance and determine the control strategy. Design control algorithm that meets the specifications. Verify the control algorithm using the simple model. Choose hardware and software and integrate the components on a pilot plant. Implement controller and carry out system test to evaluate the overall performance of the system. Repeat some or all of the above steps if necessary.

Among these steps, it can be said that the design of the control algorithm is the most important part of the design of the active vibration controller. So, most of the studies related to active vibration control are about the development of a stable control algorithm. Since sensors and actuators are used for active vibration control, a sensor equation and actuator dynamic model are required. Many studies are conducted without understanding the sensors and actuators that are actually used for vibration measurement and control, and are often not applicable to actual structures. In the following, the main research results of active vibration control until recently are reviewed. Rao and Sunar (1994) discussed piezoelectricity in the context of distributed sensing and control of flexible structures. Kwak (1995, 2001) reviewed the piezoceramic applications to active vibration control of smart structures. Lee and Han (1997) reviewed research trend in vibration control of smart structures using piezoelectric materials. Goodall (1997) reviewed the active railway suspensions.

1.1 Introduction

3

Ahmadian and Deguilio (2001) studied piezoceramic (PZT) systems for controlling structural vibration. Chopra (2002) and Frecker (2003) reviewed smart structure technology. Tanifuji et al. (2002) discussed mechatronics technology involved in Japanese railway vehicles. Alkhatib and Golnaraghi (2003) reviewed essential aspects involved in the design of an active vibration control system. Hurlebaus and Gaul (2006) gave an overview of research in the area of smart structure dynamics. Song et al. (2006) reviewed vibration control of civil structures. Sinan (2011) reviewed the active structural control. Hudson and Reynolds (2012) provided a comprehensive state-of-the-art review of active vibration control for human-induced vibrations in floor structures. Aridogan and Basdogan (2015) reviewed current state-of-the-art of active vibration and noise suppression systems for plate and plate-like structures with various kinds of boundary conditions. Kolekar (2019) reviewed vibration control of structure with ER and MR fluids, and MR elastomer.

1.2 Smart Structure There are structures in which sensors and actuators are attached to the surface of the structure or inserted as part of the structure and a control system is attached. Such structures are called Adaptive Structures, Sensory Structures, Controlled Structures, Active Structures, Intelligent Structures, Smart Structures, etc. Wada et al. (1989) summarized these terms and defined them as follows. Adaptive Structures are structures that have only an actuator and can change their shape as desired by the user. Sensory Structures are structures that can listen to structural defects or damages in the structure through a detector. The combination of these two types is called Controlled Structures. The control structure is a structure that has both a sensor and an actuator, and it can sense the state or displacement of the system and simultaneously perform control through the feedback control circuit. Control structures are generally terms used when structures, sensors, and actuators are clearly distinguished from structures, whereas when sensors and actuators are included as part of a structure, they are referred to as active structures. Intelligent Structures (Smart Structures) refer to more intelligent structures that can actively cope with changes in situations because they have advanced controllers such as Neural Network and Fuzzy Logic. In recent years, the term smart structure is commonly used. A smart structure is defined as one that contains distributed sensors, actuators and a control scheme to achieve vibration suppression in close cooperation. Conceptually, the smart structure should be able to cope with external disturbances and internal changes. There are many materials that have been tested for actuators and sensors. They are piezoelectric materials, shape memory alloys, electrostrictive materials, magnetostrictive materials, electro-theological fluids, and fiber optics. These materials can be inserted into or bonded with structures thus acting as sensor or actuator. Among them, piezoelectric materials have become popular because of high strength, temperature insensitivity, and ease of implementation. If a voltage is applied to the

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piezoelectric material, then it undergoes deformations. On the contrary, a charge is produced when deformation occurs. Hence, it can be used both sensor and actuator. The most popular piezoelectric material is the piezoceramic material consisting of lead zirconate titanite. In particular, the piezoceramic plate can be easily glued to the surface of structures. The initial studies on modelling and control for smart structures were confined to a simple beam with piezoelectric sensors and actuators. Subsequent studies have broadened their scope to include a variety of structures, such as plates and shells. Piezoelectric property has been well explained in many books (Nisse 1967; Jaffe et al. 1971; Mason 1981; Preumont 2011; Yang 2018). Rao and Sunar (1994) summarized employment of piezoelectric materials as sensors and actuators for control of flexible structures up to that date. They presented a general framework for structural control and discussed applications of such systems in various engineering fields. One of the most common application areas was stated as the aerospace industry, since aerospace structures are flexible and lightweight, and aerospace components are exposed to severe vibrations during their operation. Later, Loewy (1997) reviewed the key applications of smart structures in aeronautical applications with potential uses. Having categorized the smart materials in terms of their energy-exchange capabilities, Loewy (1997) also stated the benefits of using smart structures in aeronautical applications. Brennan et al. (1999) investigated the actuation performance of piezoelectric actuator, magnetostrictive actuator, and electrodynamic actuator for active vibration control.

1.2.1 Beam Bailey and Hubbard (1985) used distributed piezoelectric polymer for the active vibration control of a cantilever beam. The paper by Crawley and de Luis (1987) was one of the earliest papers on piezoelectric actuators, in which they presented two one-dimensional static models describing the ductility between piezoelectric materials and elastic structures. The first model assumes that the piezoelectric material is perfectly bonded to the structure, and the second model assumes that there is an adhesive surface having a certain thickness between the piezoelectric material and the structure. Its effect was investigated and theoretical dynamic models and experimental results were presented. Burke and Hubbard (1988) applied a piezoelectric film actuator the active vibration control of beams and used Lyapunov’s direct method to derive the control laws. Crawley and Anderson (1990) discussed a modeling technique for the case where a one-dimensional piezoelectric material and a beam structure were combined, and compared the theoretical values through experiments. Hagwood et al. (1990) presented a more generalized approach to derive the equation of motion for an electro-mechanical system, a system in which structures, piezoelectric materials, and electronics are combined. Fanson and Caughy (1990) applied the Positive Position Feedback (PPF) controller to a cantilever beam with piezoceramic sensor and actuator. Lee et al. (1991) considered distributed piezofilm sensors and actuators capable of sensing and controlling the modal displacement

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of beam. Hanagud et al. (1992) used the finite element method to model the smart structure and suppressed the vibration with rate-modal feedback and optimal control. Kim (1992) carried active vibration control of beam using piezoelectric actuators. Kim (1993) carried out active vibration control of beam using piezoelectric films. Hwang and Park (1993a) carried out research on the active vibration control of beam with integrated piezoelectric sensors and actuators. Park et al. (1993) investigated the effect of piezoelectric actuators when the bending and torsion of a beam are ductile. Kim et al. (1994) conducted active vibration control of a cantilever beam using a piezoelectric film using Bang-Bang control methods. Yoon et al. (1994) carried out experiment on composite cantilever beam using piezoceramic actuator and polymer sensor. Denoyer and Kwak (1996) dealt with in detail the dynamic modeling of cantilever beams with piezoceramic detectors and actuators bonded, showing that the theoretical frequency response function agrees well with the frequency response obtained from experiments. Lee et al. (1996) experimentally investigated active vibration control of a composite beam with piezoelectric sensors and actuators. Barboni et al. (2000) built special dynamic influence functions for the calculation of optimal position of PZT actuators bonded to beam. Bruant et al. (2001) proposed the method to find the optimal actuator location of beam structure by minimizing the mechanical energy integral. Trindade et al. (2001) presented the design and analysis of the piezoelectric active control of damped sandwich beams using LQR/LQG control. Choi et al. (1995) applied the velocity feedback control to the vibration suppression of a cantilever beam using distributed piezofilm sensors and actuators. Vasques and Rodrigues (2006) compared classical control strategies, constant gain and amplitude velocity feedback, and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller for the active vibration control of smart piezoelectric beams.

1.2.2 Grid Structure Smart grid structure hasn’t been of interest to researchers. Kwak and Heo (2007) used the multi-input multi-output (MIMO) PPF controller and the modified Linear Quadratic Regulator (LQG) controller in order to suppress vibrations of grid structure equipped with piezoceramic sensors and actuators.

1.2.3 Plate Tzou and Tseng (1990), Tzou et al. (1990) developed a finite element method that considers the distribution of piezoelectric materials and investigated the dynamic characteristics of a plate combined with a piezoelectric sensor and actuator. Burke and Hubbard (1991) investigated the use of distributed sensor and actuator for the active vibration control of thin plates. Crawley and Lazarus (1991) investigated the dynamic

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modeling of plate equipped with strain actuator such as piezoelectric actuator. Paige et al. (1993) introduced a method of controlling flutter by attaching piezoelectric materials to a composite plate, and Linear Quadratic Regulator (LQR) was used as a control technique. Nam et al. (1993) deals with the application of piezoelectric materials to the aircraft wing considering the case of attaching the piezoelectric material to the inner layer of the composite plate, and uses the Rayleigh-Ritz method to derive the equation of motion. D’Crus (1993) actively control panel vibrations through the action of piezoceramic actuators bonded to the surface of the panel. Falangas (1994) designed H∞ controller for active vibration control of plate with PZT actuators. Agrawal and Tong (1994) derived the dynamic model of plate with embedded piezoelectric actuator. Lazarus et al. (1996) carried out theoretical modelling and multiinput multi-output (MIMO) LQG control for a rectangular plate by using an assumed modes method. Li et al. (2003a, b) have investigated the use of a piezoelectric wafer attached to a rectangular plate using µ control theory. Song (1996) carried out active vibration control of a thin plate in vehicles. Hwang et al. (1997a) applied LQG control to a plate with piezoelectric sensors and actuators. Rew et al. (1997) carried out experiments on active vibration control of composite plate using piezoelectric sensors and actuators. Yang and Huang (1997) derived piezoelectric constitutive equations for a plate shape sensor and actuator. Valoor et al. (2000) investigated the active vibration control of smart composite plates using self-adaptive neuro-controller. Shen and Homaifar (2001) used rate-feedback control, hybrid fuzzy- proportional-integralderivative (PID) control, genetic algorithms-designed PID control, and LQG/loop transfer recovery control methods for the active vibration control of plate bonded with piezoelectric sensors and actuators. Lee et al. (2002) carried out numerical simulation on active vibration control of rectangular plate embedded with piezoelectric actuators. Kwak et al. (2003) derived the dynamic model of a rectangular plate equipped with piezoceramic sensors and actuators and used the MIMO PPF control. Liu et al. (2004) developed robust controller for the active vibration control of plate by using constrained layer damping. Zhang et al. (2004) designed a µ-synthesis controller to suppress multi-mode vibrations of the clamped plate with self-sensing piezoelectric actuators. Fein et al. (2005) considered the active vibration control of a fluid-loaded plate. Hu and Ng (2005) derived the dynamic model of a circular plate with integrated piezoelectric sensors and actuators and designed a robust controller for the vibration suppression. Kumar (2007) considered the performance of LQR controller as objective function and found optimal location of sensor-actuator pairs on a plate. Sharma et al. (2007) developed fuzzy logic based independent modal space control (IMSC) and fuzzy logic based modified IMSC for the active vibration control of plate. Qiu et al. (2007) proposed control method by combining positive position feedback and proportional-derivative control for the active vibration control of cantilever plate. Chellabi et al. (2009) designed direct optimal control for a plate equipped with piezoelectric actuators. Kwak and Yang (2013a, b) have developed a dynamic model for a submerged rectangular plate equipped with piezoelectric sensors and actuators and designed the active vibration control. Kwak and Yang (2015) derived the dynamic model of a submerged rectangular plate equipped with piezoelectric sensors and actuators and applied MIMO PPF controller. Most of

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these studies were focused on vibration control by means of piezoelectric wafers. Zippo et al. (2015), Ferrari and Amabili (2015) carried out experiment on the active vibration control of a free-edge rectangular sandwich plate equipped with Macro Fibre Compsite (MFC) actuators and PZT sensors. Shin et al. (2020) considered the Active Mass Damper (AMD) as an actuator and proposed MIMO modal-space negative acceleration feedback (NAF) control and decentralized MIMO NAF control for the vibration suppression of a rectangular plate.

1.2.4 Shell Clark and Fuller (1991) demonstrated that the sound radiated from a long, thin aluminum cylinder can be attenuated by using PZT(Lead Zirconate Titanate) actuators and PVDF(Polyvinylidene Fluoride) sensors along with the filtered-X LMS(Least Mean Square) algorithm. Lester and Lefebvre (1991) discussed the bending and in-plane piezoelectric actuation models for a thin cylindrical shell. Sonti and Jones (1991) presented a simple analytical model for a piezoelectric actuator surface-bonded to a cylindrical shell. The active vibration control of a cylindrical shell using piezoelectric sensors and actuators was investigated by Lester and Lefebvre (1991), Zhou et al. (1994), Chaudhry et al. (1995), Sohn et al. (2006), and Kwak et al. (2009). Zhou et al. (1994) also developed a dynamic model of a distributed PZT actuator-driven thin cylindrical shell. Tzou et al. (1994) proposed distributed orthogonal convolving modal actuators designed for a circular ring shell. Chaudhry et al. (1995) derived a dynamic model for a cylindrical shell with surface-bonded piezoelectric actuators. Laplante et al. (2002) studied the active control of vibration and noise radiation from fluid-loaded cylinder using active constrained layer damping. Oh et al. (2002) investigated passive control of the vibration and sound radiation from submerged shell. Sohn et al. (2006) analyzed the natural vibration characteristics of a cylindrical shell equipped with an MFC (Macro Fiber Composite) actuator (http:// www.smart-materials.com/) by using the finite element code and proved theoretically that the LQG (Linear Quadratic Gaussian) controller can be used as an active vibration controller. Pan et al. (2008) studied the active control of radiated pressure of a submarine hull. Kwak et al. (2009) designed the MIMO PPF controller for the theoretical model obtained by the Rayleigh-Ritz method and proved that the vibrations of an aluminum cylindrical shell can be suppressed by MFC actuators. Kwak et al. (2009) employed the result of Kwak and Heo (2007) for the expression of the actuating force by the circumferentially arranged MFC actuator. Caresta (2011) investigated the use of inertial actuators to suppress the sound radiated by a submarine hull, which is modeled as a fluid loaded cylindrical shell with ring stiffeners. Cao et al. (2012) studied active control of low-frequency sound radiation by cylindrical shell with piezoelectric stack force actuators in axial modes. Loghmani et al. (2016a, b) carried out research on active control of radiated sound of a cylindrical shell using piezoelectric sensors and actuators. Kim et al. (2017) applied the NAF control to the vibration of rigid cylindrical shell subjected to internal disturbance.

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Various control algorithms have been developed for structures equipped with piezoelectric sensors and actuators. Most of these control algorithms such as PPF and LQG controls developed for the active vibration control of a structure are concerned with active damping, which implies the suppression of resonant amplitudes. However, these algorithms are no longer effective when dealing with vibrations caused by an external harmonic disturbance with non-resonant excitation frequency. This paper is concerned with the suppression of vibrations caused by a harmonic disturbance with non-resonant frequency, where active damping control loses its advantage. To this end, higher harmonic control (HHC) (Shaw and Albion 1981; Wood et al. 1985), which was developed for reducing helicopter vibration, is considered. Sievers and von Flotow (1992), and Lovera et al. (2003) found that the HHC is in fact a band rejection filter. Yang et al. (2010) investigated the application of the HHC to the active vibration control of a cantilever subjected to a harmonic disturbance. They found that beat phenomena may occur when the filter frequency is slightly different from the excitation frequency. Hence, they proposed a modified HHC (MHHC), which includes the filter damping at the excitation frequency, to expand the control bandwidth. However, they did not provide any theoretical proof for the MHHC. The efficacy of the MHHC for the vibration and radiated sound of the submerged shell was investigated theoretically and it was applied to the control of a fully-submerged cylindrical shell by Kwak and Yang (2013a, b).

1.2.5 Positive Position Feedback (PPF) Control Fanson and Caughy (1990) proposed the use of the Positive Position Feedback (PPF) controller based on the modal displacement signal. The PPF control has been very effective when the position sensor and the force actuator are used to suppress vibrations, which is suitable to piezoelectric sensor and actuator. Poh and Baz (1990) improved the PPF controller for the multi-degree-of-freedom system using the independent modal space control concept. Baz et al. (1992) studied the application of PPF control to equations of motion in modal space based on Independent Modal Space Control (IMSC). Griffin and Denoyer (1992) presented a method of miniaturizing the PPF control circuit, which has been successfully used in piezoceramic actuators. Leo and Inman (1994) used the PPF control to suppress vibrations of slewing structure such as solar arrays and space trusses. Baz and Hong (1997) developed an adaptive modal positive position feedback (AMPPF) method to make the controller follow the performance of an optimal reference model. Han et al. (1997) developed multi-input multi-output PPF controller for smart structure subjected to harmonic disturbances. Kwak et al. (1997) proposed the combination of PPF and SRF controls to suppress vibrations of structures more effectively using piezoceramic sensors and actuators. Kwak (1998) compared MIMO PPF controller with modified LQG controller for active vibration control of smart structure using piezoelectric sensors and actuators. The relationship between the PPF controller and the output feedback controller was investigated by Friswell and Inman (1999). Shin and Kwak (2000) proposed the

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real-time tuning method for the PPF controller using genetic algorithm. Heo and Kwak (2001) applied the genetic algorithm for the multiple parameter tuning of PPF controller. Rew (2002) developed multi-modal vibration control using adaptive positive position feedback control algorithm. Heo et al. (2004a, b) investigated the use of analog circuit and microcontroller for the implementation of PPF controller. Kwak et al. (2004) studied the stability condition, the performance, and the design methodology of the PPF control. Heo et al. (2004a, b) applied the adaptive PPF controller in real time to smart structure. Kwak (2005) proposed the use of block-inverse technique for MIMO PPF controller to suppress more natural modes than the number of sensoractuator pairs available. Kwak and Heo (2007) used the MIMO PPF controller in order to suppress vibrations of grid structure equipped with piezoceramic sensors and actuators. Han (2018) developed adaptive PPF controller using frequency tracking algorithm. In addition, there are many research results that applied PPF controller to vibration control of smart structure. Zippo et al. (2015), Ferrari and Amabili (2015) developed MIMO PPF control methodology for the vibration suppression of a free-edge sandwich plate by using MFC actuators and PZT sensors.

1.2.6 Shunt Method When a piezoelectric material is attached to a structure, the force generated by vibration deforms the piezoelectric ceramic, resulting in a voltage difference. This voltage, or electrical energy, can be dissipated using a shunt circuit. The advantage of the shunt circuit is that it does not require a separate power supply. The shunt circuit may generally be composed of a resistor, an inductive element, a capacitive element, and a switching rectifier. In the shunt circuit, the resistance dissipates energy by heat to give a damping effect, the induction coil has the same effect as a mechanical vibration absorbing device, and the capacitor gives the effect of changing the rigidity of the piezoelectric element. Hagwood and von Flotow (1991) performed the first theoretical analysis of piezoelectric shunt circuits, which were resistance circuits and resistance-induced coil circuits. They showed that vibration can be suppressed by generating electrical resonance similar to the vibration absorbing device when the piezoelectric ceramic is connected to the resistance and the frequency characteristic that appears when the piezoelectric ceramic is connected to the resistance is similar to that of the viscoelastic material. The implementation of attenuation through such a simple shunt circuit continued after that, followed by Aldrich et al. (1993), Edberg et al. (1991) showed how to implement attenuation. Davis and Lesieutre (1995) proposed a method for predicting modal attenuation when shunted by resistance. Hollkamp (1994), Hollkamp and Gordon (1996) studied the implementation of a shunt circuit capable of controlling several modes. Wu (1996, 1997) proposed piezoelectric shunt with parallel R-L circuit. Lesieutre (1998) studied vibration damping and control using shunted piezoelectric materials. In addition, there are many research results on attenuation using a shunt circuit.

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1.2.7 Dynamic Modeling of Piezoelectric Sensors and Actuators Modeling and performance prediction of piezoelectric sensors and actuators as elements of smart structures have been a research interest for building feasible and reliable vibration control systems over the last three decades. Allik and Hughes (1970) applied the finite element method to piezoelectric vibration. McDearmon (1984) discussed the addition of piezoelectric properties to structural finite element program. Hagwood et al. (1990) discussed the modelling of piezoelectric actuator for smart structure. Hwang and Park (1993b) showed the finite element formulation on piezoelectric sensors and actuators. Akella et al. (1994) derived the equations of motion of smart structures bonded with piezoelectric sensors and actuators by applying Hamilton’s principle. Chee et al. (1998) reviewed modeling approaches for piezoelectric materials. As discussed in detail in their review article, analytical and finite element models of piezoelectric materials have been mostly built using linear constitutive piezoelectric equations. They stated that the linear modeling approach is only reliable at low applied electric fields (i.e. low actuation voltages); in the case of large electric fields (i.e. high actuation voltages), nonlinear behavior of piezoelectric materials must be taken into account in analytical and numerical models for more accurate predictions (Chee et al. 1998). Banks and Smith (1993) dealt with a modeling technique when piezoceramic pieces are bonded to shell, plate, beam, etc. Shah et al. (1993), Mitchell and Reddy (1993), Chandrashekhara and Agarwal (1993), Birman (1993), Singh and Vizzini (1993), Robbins and Reddy (1993), Sun (1993), Hwang et al. (1993c) published papers on the modeling of piezoceramic plates inserted in some layers of composite plates using the finite element method. Lin and Abatan (1994) discussed how to use finite element programs currently on the market when piezoelectric materials are used as detectors or actuators. Chen et al. (1997) formulated a finite element by using a new piezoelectric plate element. Benjeddou (2000) summarized and published research trends on finite element modeling of structures including piezoelectric materials. Narayanan and Balamurugan (2003) formulated the finite element model for laminated structures with distributed piezoelectric sensor actuator layers and control electronics. Aoki et al. (2008) derived the dynamic model of a piezoceramic patch actuator. Jamalabadi and Kwak (2019) derived the dynamic model for a galloping structure equipped with piezoelectric wafers. Optimal positioning of piezoelectric sensors and actuators was reviewed and presented in a tabular form for beam and plate structures by Gupta et al. (2011). Singhal et al. (2017) investigated the optimal placement of piezoelectric patches.

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1.2.8 Smart Structure Applications Crawley and de Luis (1987) and de Luis and Crawley (1990) found that in order to realize the vibration control of space structures, the piezoelectric actuators were used for the composite materials with piezoelectric materials attached to the surface, and piezoelectric ceramics. The effect on the structure was examined experimentally. Poh and Baz (1990) suppressed the vibration of the truss structure by using a piezoelectric ceramic stacked actuator. Edberg et al. (1991) produced an Active-Damping Strut using piezoelectric materials. Fanson et al. (1991) experimented with vibration control by inserting actuator members made of piezoelectric materials into the truss structure of JPL. Shibuta et al. (1991), Fujita et al. (1991) introduced a method of making truss members into active actuators using piezoelectric materials. Bronowicki et al. (1993a, b) fabricated a structure in which piezoelectric ceramics were inserted into a composite material structure to be used as a mast of a satellite, and a vibration control experiment was performed by applying an active control technique. Dosch et al. (1992) first introduced a self-sensing piezoelectric actuator. A piezoelectric wafer is normally used as either sensor or actuator, but with the aid of electronic circuitry it can be used as both sensor and actuator at the same time. Ambur and Rinderknecht (2016) mounted a self-sensing piezoelectric actuator at the bearings of rotor to reduce vibrations. He et al. (2018) applied a piezoelectric self-sensing actuator (SSA) to the motorized spindle of CNC machine tools to its vibrations. Neural networks and fuzzy logic have been considered as a control algorithm for smart structures. Kwak and Sciulli (1996) proposed the fuzzy-logic controller for smart structures using piezoceramic sensors and actuators. Jha and Rower (2002), Jha and He (2003, 2004) used a neural network based adaptive controller for the active vibration control of structure with piezoelectric actuators. Abdeljaber et al. (2016) utilized a neurocontroller along with a Kalman Filter to compute the appropriate actuator command. The neurocontroller was trained based on an algorithm that incorporates a set of emulator neural networks which were also trained to predict the future response of the cantilever plate. In applying the piezoelectric wafers to vibration suppression, the high-voltage amplifier is necessary to drive piezoelectric actuator. Kwak et al. (2005) considered Pulse Width Modulation (PWM) to drive the piezoceramic actuator instead of high-voltage amplifier. Lee et al. (2012b) developed a low-cost high-voltage power amplifier for piezoelectric actuation.

1.3 Automobile The existing engine mount design method is designed to isolate the vibration of the engine from the automobile main frame, and for this purpose, a rubber or hydraulic mount is used. The engine mount using rubber is effective in the high frequency band, but there is a limit to suppressing the vibration in the low frequency band generated

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by the engine. Therefore, since it has become difficult to expect any more effects with the existing passive engine mount, a new concept of active engine mount has been introduced to improve the comfort of passengers on board a vehicle. Kowalczyk et al. (2006) carried out active vibration control by attaching a voice-coil type actuator to an automobile frame and applying control force to the 180° phase of the signal transmitted from the engine using an adaptive controller. Leo and Inman (1999) dealt with the problem of optimal control for passive-active isolated vibration systems. Ishihama et al. (1992, 1994, 1995) performed vibration control when the vehicle was idle by using the phase delay control of the hydraulic engine mount, Kadomuka et al. (1995) conducted a study to suppress the vibration of the crankshaft, engine and vehicle by making the fluctuations of the DC motor torque and the engine torque cancel each other in order to reduce the vibration caused by the engine torque fluctuation of the idling vehicle. Hwang et al. (1997b) developed a control logic for the semi-active suspension system, and used a quarter-car model and HILS. Kim et al. (2003) proposed a hybrid mount composed of a piezoelectric actuator laminated with an elastic rubber material and a piezoelectric material, and conducted a study to evaluate the vibration control performance and force transmission rate of the beam structure through experimental implementation. Batterbee and Sims (2007) employed the HILS method and a quarter car model to investigate the effectiveness of the MR damper. Choi et al. (2008) proposed an active hybrid mount using MR fluid and a piezoelectric actuator for effective vibration isolation of the active mount, and a system through simulation using a controller designed for a robust sliding mode control that takes into account the uncertainty of the system. A study was conducted to evaluate the acceleration and transmission power. Oh et al. (2010) measured the vibration generated by the unmanned aerial vehicle engine on the ground for an accurate vibration insulation test, selected a piezoelectric actuator. Quoc et al. (2010) evaluated the performance of an active hybrid mount system consisting of piezostack actuator. The method of implementing the active mount introduced above includes the method of inserting an actuator into the mount itself and adding an active dynamic reducer separately while leaving the passive mount as it is. In the latter case, there is an advantage that the active mounting effect can be implemented by simply attaching a dynamic absorber to the subframe without the need to replace the passive mount. However, since the device is added to the subframe compared to the active mount in which the actuator is inserted, the structural change of the subframe is required. Lee et al. (2009) derived a theoretical model for the dynamic characteristics of Active Linear Actuator (ALA) developed for active vibration control of automobiles and investigated the operating principle.

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1.4 Railway Vehicles Railway technology tries to meet customers’ consistent requirements of riding comfort, safety and low track maintenance cost for fast trains. However, the conventional suspension system can no longer provide adequate performance for high-speed trains. Hence, the concept of an active engine mount or an active suspension system introduced to automobiles has been applied to railway vehicles to reduce excessive vibrations. Various kinds of suspension systems have been designed and applied to railway vehicles. In general, the active system replaces the secondary suspension between the carbody and the bogie, so that it can provide actuating force to the carbody and the bogie based on sensor measurements. The active system was first applied to the lateral suspension system to create the active lateral suspension (ALS). The ALS has been studied analytically and tested experimentally since 1970. The mechatronics system for the ALS consists of sensors, actuators and a controller. The mechatronics systems have already proven performance in tilting trains. The sensors used for the ALS are in general low-frequency accelerometers. The type of actuator used determines the type of system, either a semi-active or an active lateral suspension. Among the various actuators that have been used for trains, the solenoidtype actuator, which is in fact an electro-magnetic actuator, has been popularly used because of its faster response time and wider frequency bandwidth compared to those of hydraulic and pneumatic actuators. The semi-active suspension system using a hydraulic-solenoid valve was adopted for the Shinkansen of Japan in 1995 and the active system was partly introduced in 2002. However, problems of a tank addition and oil leakage arose in application of the hydraulic or pneumatic system. Hence, the ball-screw type and linear magnetic type actuators have been developed recently. Peiffer et al. (2005) reviewed active vibration control of high-speed train. Hirata et al. (1995) applied the H∞ control theory to the active suspension to reduce yaw, lateral and roll motions and showed that these motions were reduced significantly both in simulations and experiments. Roth and Lizell (1996) presented a semi-active damping system to reduce the lateral motion of a railway vehicle and showed that it improved the comfort considerably. The actuator consists of the hydraulic rotary valve (screw pump) and the electro-magnetic type miniature brake. Liu- Sasaki (2000) and Sasaki and Nagai (2003) employed a semi-active suspension system for a tilting train by using a variable damper and a sky-hook control algorithm and showed that the system reduced lateral vibrations by 20–30%. Liu-Henke et al. (2002) showed that the active suspension in combination with a tilt device using hydraulic system provided good ride comfort and cornering safety. The active suspension was achieved by adjusting the spring and damping forces. The initial values of spring constant and damping value were generated by means of the poleplacement and optimized by means of vectorial criteria optimization. Umehara et al. (2007) proposed bilinear robust control for vertical vibration in railway vehicle with semi-active suspensions. Ha et al. (2008) applied the semi-active control based on the magneto-rheological (MR) fluid to a railway vehicle and showed its effectiveness. Simulation and experimental results showed that the ALS system can offer ride

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comfort, safe running behavior and curving ability, all at the same time. Wang and Liao (2009a, b) investigated the semi-active suspension system for railway vehicles using MR dampers and showed numerically that the MR damper can be effectively used to suppress lateral, yaw, and roll accelerations of the carbody. Watanabe et al. (2008) developed oil damper test equipment capable of simulating the actual conditions of railway vehicles. Maki et al. (2009) developed a HILS system for vibration control of a railway vehicle. Conde Mellado et al. (2009) proposed a lateral active suspension operated by pneumatic actuators placed between the bogie and the carbody with the centering strategy. Zhou et al. (2011) investigated the application of an active suspension for tilting enhancement and showed numerically that it provided improved riding quality as well as effective tilt control. Orvnäs et al. (2011) investigated the influence of the H∞ and sky-hook damping control algorithm for lateral ride comfort and showed that both control algorithms are effective. Hudha et al. (2011) investigated the performance of semi-active control of a lateral suspension system attenuating the effect of track irregularities. They used the Bounc-Wen model for the MR damper and tested the sky-hook control algorithm. Ebrahimi et al. (2011) investigated the electromagnetic dampers used in vehicle suspension systems. They proposed novel hybrid electromagnetic dampers. Lee et al. (2012a) developed a HILS system for the testing of active vibration control of lateral vibration of railway vehicle. Kwak et al. (2014) carried out experiment using HILS system for semi-active vibration control of lateral vibrations of railway vehicle by MR fluid damper.

1.5 Elevator With the recent increase in skyscrapers worldwide, the demand for a high-speed elevator system that can provide rapid movement within these buildings is constantly increasing. The vibration of elevators mainly occurs during driving, and this has emerged as a major problem as the elevator speed increases. In particular, vibration generated during driving not only provide unpleasant feelings and uneasy ride comfort to passengers, but can also develop into a big problem when transmitted to residences. Since noise and vibration of elevators are not only important factors that determine ride comfort, but can also be treated as a measure of quality, many elevator manufacturers are developing various anti-vibration measures and technologies to improve them. Vibrations that impede the ride comfort of an elevator are generally classified into lateral, longitudinal, and vertical vibrations. Passengers board the cage, and the cage is designed in a form in which the frame and the anti-vibration rubber are connected to reduce vibration transmitted from the inside or outside. The roller guide is in contact with the guide rail and acts like a wheel. In addition, a damping device is attached to absorb the vibration transmitted from the guide rail. Rail steps and wind pressure are examples of external factors that cause lateral vibration of elevators running at high speed. Since the elevator is a system that runs vertically along the guide rail installed in the hoistway, lateral vibration may occur

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in the elevator when driving on the connected part between the guide rail and the rail or on a curved rail. And when two elevators cross each other, lateral vibration may occur due to the influence of wind pressure. Therefore, in order to minimize these problems, the accurate installation of the guide rail by the skilled person or by attaching a damping device to the guide roller has been tried to suppress the lateral vibration of the elevator due to the step of the guide rail, and the effect of wind pressure is minimized by designing a streamlined cage. However, there are limiting factors in achieving a high-speed elevator and a comfortable ride through this passive method. Therefore, there is a need for a vibration control device that actively responds to disturbances such as wind pressure and rail levels. Many elevator companies are developing their own lateral vibration control technology by applying various actuators (AC Servo Motor, Electromagnet, Voice Coil Actuator, etc.) to elevators. Inaba et al. (1994) studied a device that controls the vibration of the frame by attaching an electromagnet to an elevator frame close to the guide rail and measuring the distance between the guide rail and the frame. Yamazaki et al. (1994) developed a new vibration control mechanism and control method for suppressing horizontal vibration in super-high-speed elevator. The actuator consists of an AC servomotor and a ball screw, and the vibration of the cabin, detected by an acceleration sensor. Mutoh et al. (1999) discussed the lateral vibration modeling method of high-speed elevators and presented the results of numerical analysis for the case where the rail has a sinusoidal curve. In addition, the effectiveness of active vibration control was demonstrated by presenting the design of the active vibration controller and actual experimental results. Teshima et al. (1999, 2000) studied a vibration control device that actively operates against disturbances by implementing an AMD (Active Mass Damper) system using an AC servo motor and a ball screw on the floor of an elevator frame. Funai et al. (2001) proposed a vibration reduction device using an electromagnet instead of an AC servo motor that is complex in hardware. Utsunomiya et al. (2005) studied a device that effectively suppresses vibration transmitted from the guide rail by attaching a self-developed VCA (Voice Coil Actuator) to a roller guide. Noguchi et al. (2006) proposed an active vibration control device that controls transverse motion by attaching an AC thermomotor and a ball screw to a roller guide instead of a VCA. Kwak et al. (2011) derived dynamic modeling for controlling the lateral vibration of an elevator using the energy method, and analyzed the vibration characteristics by calculating the natural frequency and natural mode of the elevator through numerical simulation. Based on the dynamic model, an LQR (Linear Quadratic Regulator) controller was designed and used as an active vibration controller. Baek et al. (2011) conducted a study on the back and forth vibration.

1.6 Building Structure Structural building control and protection of human occupants from natural hazard to scale down earthquake or wind damages have been investigated by many researchers

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and various strategies have been proposed. However, the development of new vibration control technique is still an on-going process to provide more economical, simple and reliable solution to vibration control. Tremendous efforts have been expended in work related to Tuned Mass Damper (TMD), Semi-active mass damper (SMD) and Active Mass Damper (AMD). McNamara (1977) developed a tuned mass damper for buildings. Miller et al. (1988) surveyed control methods for large civil structures. Chung et al. (1989) developed active tendon system driven by hydraulic servo system with optimal control law. Nishimura et al. (1992) developed active tuned mass damper for vibration suppression of building structure. Watanabe et al. (1994) proposed LQR for active vibration control of a building by using reduced-order model in combination with a low-pass filter for spillover problem. However, it required complicated computation. Ankireddi and Yang (1996) investigated the design of ATMD for vibration control of building subjected to wind load using full-state feedback control. Kamada et al. (1997) carried out active vibration control of a four-story building model using thirty-two piezoelectric actuators and examined model-matching method and the H∞ control theory. Cao et al. (1998) studied and designed AMD for a tall TV tower using linear quadratic regulator (LQR) and a nonlinear feedback control algorithm. Baoya and Chunxiang (2000) applied H ∞ control to AMD and reported that vibrations could be suppressed and stability could be improved at the same time. Watakabe et al. (2001) proposed a hybrid mass damper applied to a tall building. Pinkaew and Fujino (2001) developed an optimal control law for semi-active tuned mass damper. Li and Liu (2002) and Li et al. (2003a, b) also used multiple ATMD to attenuate vibrations of structures due to the ground excitation. They demonstrated that multiple AMD could significantly reduce vibration of structures. Ricciardelli et al. (2003) investigated the vibration suppression performance of a TMD that consisted of a spring and a damper and an AMD with an actuator that can reduce the buffeting response of a tall building. Samali et al. (2003) investigated the performance of an AMD with a fuzzy logic controller in a five-storey building undergoing earthquake excitations. Varadarajan and Nagarajaiah (2004) proposed variable stiffness tuned mass damper for building structure. Lin and Chung (2005) studied semi-active control of building structures with semiactive tuned mass damper. Guclu (2006) investigated the sliding mode and PID control used for the active vibration control of building-like structure. Nagarajaiah and Sonmez (2007) proposed a semi-active variable stiffness tuned mass damper for structural vibration control. Pourzeynali et al. (2007) combined genetic algorithms and fuzzy logic to optimize the parameters of the AMD for a building under earthquake excitations. Wang and Lin (2007) developed variable structure control (VSC) and fuzzy sliding mode control (FSMC) for building with AMD. They proved that both methods could control vibrations of the building structure successfully and verified that FSMC was more economical and practical than VSC. Ikeda (2009) introduced practical applications of active and semi-active control of buildings in Japan and reports control verification methods through observation data recorded in controlled buildings. Li et al. (2010) developed an optimum design methodology of active tuned mass damper for asymmetric structures. Mitchell et al. (2012) proposed a wavelet-based fuzzy neuro control algorithm for the hazard mitigation of seismically

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excited buildings equipped with a hybrid control system. Kwak et al. (2015) proposed a semi-active dynamic absorber using electromagnet to change the spring constant. Lin et al. (2015) developed a tuned mass damper with resettable variable stiffness. Yang et al. (2017) proposed a simple control algorithm called negative acceleration feedback (NAF) controller utilizing the tracking ability of an AC servo motor and accelerometer signal directly. They applied the NAF controller to building-like structure and proved that NAF control could be effectively used for AMD operated by the AC servo motor. Talib et al. (2019) proposed the use of multiple AMDs driven by linear motors in order to suppress multiple natural modes effectively.

1.7 Aircrafts and Space Structures One of the most common application areas was stated as the aerospace industry, since aerospace structures are flexible and lightweight, and aerospace components are exposed to severe vibrations during their operation. Forward (1979) carried out experiments on active vibration control of space membrane mirror with piezoelectric actuators and showed that vibrations of the membrane were substantially reduced. Leo and Inman (1994) used the PPF control to suppress vibrations of slewing structure such as solar arrays and space trusses. Agrawal and Bang (1993) studied the active vibration control of flexible space structures by using piezoelectric sensors and actuators. Pearson et al. (1994) described how active control techniques can be applied to vibration reduction of helicopter. Dongi et al. (1996) investigated the panel flutter suppression using self-sensing piezoactuators. Loewy (1997) reviewed the key applications of smart structures in aeronautical applications with potential uses. Having categorized the smart materials in terms of their energy-exchange capabilities, Loewy (1997) also stated the benefits of using smart structures in aeronautical applications. Grewal et al. (2000) investigated the use of multiple piezoelectric actuators for the active control of cabin noise and vibration of aircraft. Patt et al. (2005) studied Higher-Harmonic Control (HHC) algorithm for helicopter vibration reduction. Han et al. (2006) carried out active flutter suppression using piezoelectric actuation. Mahmoodi and Ahmadian (2010) proposed modified acceleration feedback controller for the vibration suppression of aerospace structures. Prakash et al. (2016), implemented an AVC algorithm based on fxLMS on an FPGA board to control vibration of full-scale wing of an all composite two seater transport aircraft. They tested several closed loop configurations like single channel and multi-channel control. Prakash et al. (2016) investigated the use of adaptive controller for active vibration control of aircraft wing using MFC actuator and PZT sensor. Xu et al. (2018) presents a novel control strategy for the tip position and vibration control of a class of space flexible structures.

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1.8 Robots Recently, flexible parallel robots have been intensively studied because of their excellent performance. However, the elastic vibration problem of such robots is serious due to their inertial and driving forces. As a result, suppressing the unwanted elastic vibration is currently a very significant and challenging problem. Collins et al. (1992) described the development and manufacture of distributed piezoelectric film strain sensors, that are spatially shaped for space robots. Kwak et al. (1995) investigated active vibration control of slewing structure equipped with piezoceramic sensors and actuators. Zhang et al. (2001) have performed an experimental study of active vibration control of 3-PRR flexible parallel robots, for which the elastic vibration of flexible links during motion is suppressed by an SRF controller utilizing the KED assumption. Zhang et al. (2010) address the dynamic modeling and efficient modal control of a planar parallel manipulator (PPM) with three flexible linkages actuated by linear ultrasonic motors (LUSM). Qiu et al. (2019) constructed a two-link flexible manipulator and used fuzzy neural network controller for vibration suppression.

1.9 Control Techniques The PPF control has been successfully applied to smart structure equipped with piezoelectric sensors and actuators. The well-known LQG/LQR control has been also applied to active vibration control. LQR/LQG control is a controller based on optimal control algorithm and Kalman filtering. It is a well-established control algorithm developed for the control of MIMO systems. The LQR/LQG control algorithm was derived based on the state equation, which is a first-order differential equation. Therefore, in order to apply the LQR/LQG controller to the active vibration control of a structure, the equation of motion expressed by the second-order system of differential equations must be converted into a first-order state equation. In general, the equation expressing the vibration of a structure is a multi-degree of freedom equation, and when it is converted into a state equation, the state consists of displacement and velocity. LQR is full state feedback control. Therefore, the state must be measurable. In the case of active vibration control, it is impossible to measure all displacements and velocities of a structure using a limited sensor. Therefore, observer is needed, and the algorithm developed for this is Kalman filter. The Kalman filter allows state reconstruction with limited sensor measurement signals. It even allows configuring displacement and velocity from acceleration signals. Although it has these advantages, the following problems arise when applied to real problems. Above all, an accurate system model is required. In other words, accurate vibration system mass matrix, damping matrix, stiffness matrix, and force participation matrix are required for derivation of the optimal gain matrix and Kalman filter design. And because of the complex control algorithm, it

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takes a lot of computation time compared to other control algorithms. When implementing and using an active vibration control algorithm in a digital controller, the shorter the sampling time is, the better. In this respect, LQR/LQG is disadvantageous. Numerous control algorithms have been proposed for the vibration suppression of structures.

1.9.1 Feedback Controls Balas (1979) introduced the concept of direct velocity control. Meirovitch and Baruh (1982) proposed the Independent Modal Space Control (IMSC) for the vibration suppression of continuous system. Bailey and Hubbard (1985) considered Lyapunov control and velocity feedback control. Goh and Caughey (1985) investigated the stability problem caused by finite number of actuators used for large space structures. Burke and Hubbard (1987) derived control methods for beams with various types of boundary conditions and presented design guidelines using Lyapunov’s direct method. Babu and Hanagud (1991) designed a robust controller using µ Synthesis technique for vibration control of structures including piezoelectric sensors/actuators. Tzou and Gadre (1988) used a control technique to change the phase angle and gain in vibration suppression using piezoelectric polymer. Babu and Hanagud (1991) dealt with the problem of robustness of controllers using piezoelectric sensors and actuators. Fanson et al. (1991) discussed the Impedance Matching technique. Hanagud et al. (1992) used Rate Feedback, Modal Feedback, and Optimal Output Feedback control techniques as active control techniques. Pota et al. (1993) investigated the use of H∞ control technique. Betros et al. (1993) showed that the Impedance Matching control technique can be used as a vibration control technique using piezoelectric materials. Lazarus and Crawley (1996) present experimental results using Linear Quadratic Gaussian (LQG) control and Optimal Projection Compensator. Chen et al. (2003) designed active vibration controller using Linear Quadratic Gaussian (LQG) control for the vibration suppression of circular saws. Olgac et al. (1997) proposed the delayed resonator for active vibration control. Dong et al. (2006) investigated the efficiency of a system identification technique known as observer/Kalman filter identification (OKID) technique in the numerical simulation and experimental study of active vibration control of piezoelectric smart structures. Aoki et al. (2008, 2010) used velocity feedback control using piezoceramic actuators. Mahmoodi and Ahmadian (2009) proposed the modified positive position feedback controller based on a first- and second-order compensator. Xu et al. (2018) proposed a new finite frequency H∞ controller working in the inner feedback loop to suppress vibration modes and external disturbances, and developed a new fractional-order PD controller in the outer feedback loop to guarantee the desired position tracking performance.

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1.9.2 Virtual Tuned Mass Damper (VTMD) Control This section introduces the Virtual Tuned Mass Damper (VTMD) control algorithm, which is a control algorithm that directly uses acceleration signals. VTMD control can produce a force equivalent to the inertia force of the TMD. The VTMD was first proposed by Wu (2002), Wu and Shao (2007), and Wu et al. (2007) for a flexible structure subject to harmonic disturbances of uncertain frequency. The VTMD algorithm was called a virtual vibration absorber, and it is mathematically equivalent to a passive TMD. Wu (2002) showed that this virtual absorber not only rejects a given harmonic disturbance but also provides some active damping. The active vibration control that emulates the tuned mass damper algorithmically is inherently stable, because the actual structure equipped with the tuned mass damper is stable and can absorb the effect of disturbance just as the actual tuned mass damper does. Hence, researchers have proposed the direct use of the acceleration signal to produce a proper control action. Sim and Lee (1992) proposed the acceleration feedback control and proved its stability. Nishimura et al. (1994) developed a feedback control algorithm using acceleration. Dyke et al. (1996a) showed that H2/LQG frequency domain control methods employing acceleration feedback can be effectively used for the vibration suppression of seismic structures. Dyke et al. (1996b) developed a state feedback control algorithm using acceleration as the sensor output. Christenson et al. (2003) proposed the active coupled building control using acceleration feedback and used the H2/LQG approach to obtain the control algorithm. Mahmoodi et al. (2011) proposed the modified acceleration feedback control for collocated piezoelectric actuators and accelerometer. Enriquez-Zarate et al. (2016) proposed the positive acceleration feedback control when one beam-column of a building-like structure is coupled with a PZT stack actuator. These controllers are more realistic than the other controllers mentioned earlier because they use acceleration to generate the control signal for piezoelectric actuators. However, the above studies did not deal with the AMD actuated by an AC servo motor that can provide accurate position tracking rather than control force. Shin et al. (2019) developed a virtual tuned mass damper control algorithm that uses acceleration signal directly to generate force.

1.9.3 Semi-active Control Algorithms Semi-active vibration control implies that the damping property of the suspension can be controlled. Therefore, it is not a direct force applied to the structure, but a method of measuring the vibration of the structure and applying an appropriate damping value. Orifice control of automobile shock absorber, Electrorhelogical (ER) fluid and Magnetorheological (MR) fluid are used to electrically or magnetically control the damping value of the fluid inside the shock absorber. Because the semi-active method is not a method of applying a direct force, the control performance is inferior to that

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of the full active method. However, since the control system can be implemented relatively easily, it is more applied to actual structures than the active method. The most popular semi-active control algorithm is the sky-hook control algorithm that was originally developed by Karnopp (1995) for the semi-active control of automobile suspension. Dyke and Spencer (1997) evaluated a number of recently proposed semi-active control algorithms for use with the magnetorheological (MR) damper. Kolekar (2017) applied the MR fluid to the semi-active vibration control of sandwich beam.

1.9.4 Feedforward Control Filtered-x Least Mean Square (fxLMS) algorithm is widely used in the field of active noise cancellation (ANC). Clark and Fuller (1991) carried out acoustic control experiments using a piezoceramic actuator, microphone, PVDF sensor mounted on an aluminum shell and the Filtered-x LMS control technique. Sievers and Flotow (1992) concluded that the fxLMS algorithm is useful as disturbance rejection method, but can be analyzed from a classical linear time invariant feedback perspective. Gupta et al. (2006) considered the fxLMS algorithm for the active vibration control. Pu et al. (2014, 2019) proposed a new variable step size fxLMS algorithm with an auxiliary noise power scheduling strategy for online secondary path modeling, which has a low computational complexity. Shin et al. (2018) designed MIMO fxLMS algorithm for active vibration control of structures subjected to non-resonant excitation.

1.9.5 Active Mass Damper (AMD) and Negative Acceleration Feedback (NAF) Control The control algorithms introduced above are algorithms that receive displacement, velocity or acceleration as inputs and calculate appropriate force as outputs. In practice, there are not many actuators that can directly generate the force required for vibration control. However, among the actuators in use, there is an inertia type actuator. Such an actuator does not directly apply a force to the structure, but is an actuator that moves its own mass to generate an inertial force. In the case of actively controlling the vibration of a large structure such as a building structure, the capacity of the actuator becomes even more problematic. So, instead of using direct force, a method of using inertia force was proposed. The Tuned Mass Damp (TMD) is a popular vibration control method for large structure such as building (McNamara 1977). The TMD that can be easily attached to the structure of interest was developed to suppress the vibration of the primary structure. The TMD is a passive system that suppresses the vibration of the primary structure by tuning its natural frequency to the excitation frequency (Den Hartog

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1984; Inman 2017). However, the TMD has limited performance due to fixed damper parameters, a narrow suppression frequency range, ineffective reduction of nonstationary vibration, and a sensitivity problem because of detuning. In order to overcome the limitations of the traditional TMD, the Semi-active Tuned Mass Damper (STMD) and the Active Tuned Mass Damper (ATMD) or the Active Mass Damper (AMD) were proposed. Both technologies utilize the electronic circuit and control board to sense the motion of structure and to activate the control force. However, the STMD does not produce a direct force but rather changes its natural frequency by changing either stiffness or damping properties to suppress vibrations. This is advantageous compared to the AMD when the electrical power is not available since it can still provide existing damping or stiffness to structure. The AMD applies a control force computed by using a sensor signal and control algorithm, thus resulting in active vibration control. The AMD is capable of suppressing vibrations due to the frequently varying external environment since it uses actuators, sensors, and a feedback control algorithm. However, it may destabilize a main structure if structural parameters change. Hence, the reliability of the control system should be guaranteed before implementation. The AMD is generally actuated by the AC servo motor and a ball-screw mechanism, which appears to be the most feasible mechanism for the real application of AMD when the target natural frequency is low. Also, the proposed AMD system doesn’t require high-voltage power source to activate large control force. Control algorithm is very important when applying AMD technology to real structures because instability may occur. Many control algorithms have been proposed and validated both theoretically and experimentally. Kobori et al. (1991a, b) proposed the design method of the active mass driver system and simplified the control algorithm obtained by applying the optimal control theory. Nishimura et al. (1994a, b) developed a feedback control algorithm by using acceleration feedback to improve the performance of AMD. Watanabe et al. (1994) proposed LQR for active vibration control of a building by using reduced-order model in combination with a low-pass filter for spillover problem. However, it required complicated computation. Chang and Yang (1995) examined single-degreeof-freedom (SDOF) system using AMD with velocity feedback and a full-state feedback control that utilizes displacement, velocity and acceleration measurements. Their results showed that vibrations could be reduced by using both feedback controls. Ankireddi and Yang (1996) investigated the design of AMD for vibration control of building subjected to wind load using full-state feedback control. Adhikari and Yamaguchi (1997) developed filtering compensation in sliding-mode control (SMC) algorithm to eliminate undesired excitation due to AMD-borne vibration. Yang et al. (1997) proposed continuous SMC with a compensator. Cao et al. (1998) applied the AMD to a tall TV tower using the LQR and a nonlinear feedback control algorithm. Baoya and Chunxiang (2000) applied H ∞ control to AMD and reported that vibrations could be suppressed and stability could be improved at the same time. Cao and Lin (2004) developed a robust control algorithm simpler than the LQR for AMD. Samali and Al-Dawod (2003) investigated the use of fuzzy logic controller (FLC) and LQR for ATMD and proved that FLC has potential as

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control methodology for ATMD. Wang and Lin (2007) developed variable structure control (VSC) and fuzzy sliding mode control (FSMC) for building with ATMD. They proved that both methods could control vibrations of the building structure successfully and verified that FSMC was more economical and practical than VSC. Research on control algorithms for the AMD mentioned above assumed that the displacement, velocity and acceleration are measurable. However, it is difficult to directly measure the displacement and velocity of a real vibrating structure since the most popular sensor for vibration measurement is an accelerometer. The displacement and velocity can be thought to be obtainable by integrating the acceleration signal. However, bias and drift involved in the acceleration signal may cause problems in the integration process. Hence, researchers have proposed the direct use of the acceleration signal to produce a proper control action. Yang et al. (2017) proposed Negative Acceleration Feedback (NAF) control to control moving mass based on servo mechanism. The advantage of a servo control system is that movement of the mass can be accurately controlled by the AC servo motor. Yang et al. (2017) have proposed an NAF control that uses an accelerometer signal and produces desired displacement of the moving mass. The proposed NAF control is different from other control algorithms that produce control force. The NAF control theory that directly uses an accelerometer signal as a sensor input is different from other control theories that require displacement or velocity. Yang et al. (2017) have demonstrated the effectiveness of NAF control experimentally by using a building-like structure. Talib et al. (2019) have designed multi-input multioutput modal-space negative acceleration feedback control and verified the control law experimentally by using a building-like structure. They used linear servo motors to move masses. Efforts have been made to combine the passive TMD and AMD to maximize the advantage of both methods. Nishimura et al. (1994a, b) proposed the active–passive composite tuned mass damper, in which the AMD was mounted on the TMD. The two resonant peaks that resulted from the application of the TMD were suppressed by the AMD and feedback control algorithm. Wang et al. (1999) studied the optimization of the composite TMD-AMD proposed by Nishimura et al. (1998) and determined the optimal feedback gains and damper parameters. Balendra et al. (2001) proposed an active tuned liquid column damper system (ATLCD), in which the tuned liquid column damper is fixed on a movable platform, whose movement is controlled by a spring, a dashpot, and a servo actuator. So, in this case, the TMD is mounted on the AMD. Ricciardelli et al. (2003) designed the linear quadratic regulator for the composite TMD-AMD proposed by Nishimura et al. (1998). Shin et al. (2020) proposed TLD-AMD method to suppress building vibrations.

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1.10 Summary Although roughly, the research trends and research results on active vibration control were examined. Active vibration control is an interdisciplinary study that combines vibration engineering, control engineering, and electronic engineering. Therefore, it is impossible to control the vibration of the actual structure without understanding each field. It is necessary to understand the vibration characteristics of the structure to be controlled to determine which vibration to control, and to develop an appropriate control algorithm only when there is an accurate understanding of the sensor and actuator. The emergence of sensors and actuators using smart materials has developed a new academic field called smart structure, and the results have been applied to other fields to enable active vibration control for various structures. Among the sensors/actuators used for active vibration control, piezoelectric ceramics have several advantages, so research on their application is actively progressing, and application to actual structures is being made. Vibration control using piezoelectric ceramics is suitable for vibration control of local structures such as thin plates, but its use is limited because it is still expensive. Vibration control has been mainly studied on structures having simple geometric shapes such as cantilever, flat plate, and cylindrical shell. In addition, the control algorithms developed in the electric control field were applied to the vibration control of the structure. Considering that the degree of freedom of the structure is greater than that of the system handled in control engineering, the development of a control technique suitable for active vibration control of the structure is still required. This book introduces a control algorithm that has been successfully used for active vibration of a structure, and it is a field that requires much research in the future. In performing active vibration control, an actuator capable of producing an appropriate force is required. However, there are not many actuators that can be used in practice. There are also not many sensors that can measure the displacement, velocity, and acceleration of structures. The vibration of the structure inevitably includes a large number of natural vibration modes, but there are actually not many actuators and sensors that can be used to control vibrations effectively. Therefore, a control algorithm that can effectively suppress the vibration with a small number of actuators and sensors still needs to be developed.

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Chapter 2

Vibration Analysis of Single-Degree-of-Freedom System

2.1 Introduction The vibration analysis begins with a simple spring-mass-damper system. This chapter introduces the role of the mass, spring, and damper constituting the vibration system, and the process of deriving the equation of motion. In particular, the process of drawing a free body diagram and deriving the equation of motion using Newton’s second law is explained in detail. For more information, please refer to the textbook on vibration (Meirovitch 1967, 1975, 2010; Inman 2017).

2.2 Newton’s Second Law and Equation of Motion Let us consider a one degree of freedom vibration system consisting of spring, mass, and damper as shown in Fig. 2.1. where m, c, k are the mass, damping coefficient, and spring constant. x is the displacement, d is the disturbance. If we draw a free-body diagram (FBD), it will look like the following. Applying the Newton’s second law of motion to Fig. 2.2, we can obtain + − →



Fx = −kx − c x˙ + d = m x¨

(2.2.1)

where ·= d/dt is the time-derivative, ··= d2 /dt 2 is the second-order time-derivative. Rearranging Eq. (2.2.1) leads to the following equation of motion. Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/978-94-024-2120-0_2) contains supplementary material, which is available to authorized users.

© Springer Nature B.V. 2022 M. K. Kwak, Dynamic Modeling and Active Vibration Control of Structures, https://doi.org/10.1007/978-94-024-2120-0_2

39

40

2 Vibration Analysis of Single-Degree-of-Freedom System

Fig. 2.1 Single-degree-of-freedom vibration system

Fig. 2.2 Free body diagram

m x¨ + c x˙ + kx = d

(2.2.2)

This equation of motion depicts the vibration of a single-DOF system consisting of spring, mass, and damper, but it will be shown later that the vibration of a multiDOF system has a similar equation of motion in the form of a matrix equation. Mathematically, Eq. (2.2.2) is a non-homogeneous linear ordinary differential equation (ODE), and the solution can be obtained in the same way as to solve the ODEs in mathematics textbooks.

2.3 Undamped Free Vibration Let us consider the free vibration problem first, that means that in the equation of motion, Eq. (2.2.2), has no external disturbance and no damping. In this case, the equation of motion is reduced as follows. m x¨ + kx = 0

(2.3.1)

Equation (2.3.1) is a linear homogeneous second-order ODE. Dividing Eq. (2.3.1) by m and introducing ωn2 = k/m, the following equation can be derived. x¨ + ωn2 x = 0

(2.3.2)

√ In vibration theory, ωn = k/m is called the natural frequency because the solution of Eq. (2.3.2) consists of harmonic functions of time, and the natural frequency is the most important parameter in vibration analysis. According to theory of ODE, the solution of Eq. (2.3.2) has the general solution that takes the following form. x(t) = Aest

(2.3.3)

2.3 Undamped Free Vibration

41

where A is the constant determined by initial conditions, s is the parameter related to time. Inserting Eqs. (2.3.3) into (2.3.2) results in the following equation. As 2 est + ωn2 Aest = (s 2 + ωn2 )Aest = 0

(2.3.4)

In Eq. (2.3.4), est cannot be zero. If A becomes 0, then x = 0, which means a trivial solution. The trivial solution implies that the mass doesn’t move, which is not our interest in vibration analysis. Therefore, A should not be zero. The only algebraic equation that satisfies Eq. (2.3.4) is the following equation. s 2 + ωn2 = 0

(2.3.5)

The algebraic equation, Eq. (2.3.5), is called the characteristic equation. The solution of Eq. (2.3.5) determines the characteristics of the differential equation. In the case of free vibration, the solution of the characteristic equation is then written as follows. s = ±iωn

(2.3.6)

√ where i = −1. Inserting Eqs. (2.3.6) into (2.3.3), the general solution of Eq. (2.3.2) is expressed as x = A1 eiωn t + A2 e−iωn t

(2.3.7)

˙ Unknowns A1 and A2 are determined by initial conditions, that are x(0) and x(0) because Eq. (2.3.2) is the second-order ordinary differential equation. The solution can be easily solved by applying the initial conditions to Eq. (2.3.7), but it is easier to find the solution using Euler’s formula for complex numbers. The Euler formula is as follows. eiωn t = cos ωn t + i sin ωn t

(2.3.8)

Using Eq. (2.3.8), the solution, Eq. (2.3.7) can be expressed in terms of harmonic functions. x = B1 cos ωn t + B2 sin ωn t

(2.3.9)

where B1 = A1 + A2 , B2 = i(A1 − A2 ). Differentiating Eq. (2.3.9) with respect to time, the following equation is obtained. x˙ = −B1 ωn sin ωn t + B2 ωn cos ωn t

(2.3.10)

42

2 Vibration Analysis of Single-Degree-of-Freedom System

Applying initial conditions, x(0) = x0 , x(0) ˙ = v0 into Eqs. (2.3.9) and (2.3.10), the following solution can be obtained. x(t) = x0 cos ωn t +

v0 sin ωn t = A cos(ωn t − φ) ωn

(2.3.11)

where  A=

 x02

+

v0 ωn

2 , φ = tan

−1



v0 ωn x0

 (2.3.12a,b)

Equation (2.3.11) shows that it will continue to vibrate in sine or cosine forever if there is an initial condition in the case of free vibration of undamped system. The natural frequency, ωn has a unit of rad/s, and in the actual engineering field, the unit of Hz is used, which is a unit of cycle/s. When the natural frequency is very low, the unit of time, period in this case, is used instead. The relation between units for natural frequency is as follows. ωn (Hz) 2π

(2.3.13a)

1 2π = (s) fn ωn

(2.3.13b)

fn = Tn =

2.4 Damped Free Vibration In a real vibration system, if there is an initial displacement or initial velocity, the free vibration does not vibrate forever, unlike the Eq. (2.3.11). The reason is that there is damping. Now let’s solve the free vibration problem with damping. The equation of motion for the damped free vibration can be written as m x¨ + c x˙ + kx = 0

(2.4.1)

Dividing Eq. (2.4.1) by m and introducing c/m = 2ζ ωn , then Eq. (2.4.1) can be rewritten as x¨ + 2ζ ωn x˙ + ωn2 = 0

(2.4.2)

Comparing Eqs. (2.4.1) with (2.3.2), it can be readily seen that the damping term 2ζ ωn x˙ is added. ζ is called the damping factor. Equation (2.4.2) is also a homogeneous linear ODE, so that the general solution takes the same form of Eq. (2.3.3). Inserting Eqs. (2.3.3) into (2.4.2) results in the following equation.

2.4 Damped Free Vibration

43

(s 2 + 2ζ ωn s + ωn2 )Aest = 0

(2.4.3)

As with the reason mentioned earlier, Aest = 0, so that the following algebraic equation must be satisfied. s 2 + 2ζ ωn s + ωn2 = 0

(2.4.4)

In the case of the undamped free vibration, it can be seen that the root of the characteristic equation is made up of pure imaginary numbers, and the solution of the characteristic Eq. (2.4.4) varies depending on the value of ζ , which is the damping factor and positive. However, the types of roots vary from the threshold of ζ = 1. This book is written for active vibration control, so it is intended for cases where the damping of the structure itself is small. In other words, we will only deal with the case of system with small damping, i.e., underdamped case. In this case, the roots of Eq. (2.4.4) are obtained as follows.    s1,2 = −ζ ± i 1 − ζ 2 ωn

(2.4.5)

As Eq. (2.4.5) shows, the root is composed of negative real and imaginary parts. Therefore, the general solution of Eq. (2.4.2) is expressed as x = e−ζ ωn t (A1 cos ωd t + A2 sin ωd t)

(2.4.6)

 where ωd = 1 − ζ 2 ωn , which is called the damped natural frequency. A1 and A2 are determined by initial conditions as in the case of undamped free vibration. Let us ˙ = v0 . Differentiating obtain the solution when initial conditions are x(0) = x0 , x(0) Eq. (2.4.6) with respect to time results in the following equation. x˙ = −ζ ωn e−ζ ωn t (A1 cos ωd t + A2 sin ωd t) + ωd e−ζ ωn t (−A1 sinωd t + A2 cosωd t) (2.4.7) Applying initial conditions leads to the following simultaneous equation. A1 = x0 , −ζ ωn A1 + ωd A2 = v0

(2.4.8a,b)

We can obtain the following solutions for unknowns by solving the simultaneous equation, Eq. (2.4.8a, b). A1 = x 0 , A2 =

1 (v0 + ζ ωn x0 ) ωd

(2.4.9a,b)

Inserting Eq. (2.4.9a, b) into Eq. (2.4.6), the solution of Eq. (2.4.2) with initial conditions can be written as

44

2 Vibration Analysis of Single-Degree-of-Freedom System



1 x = e−ζ ωn t x0 cos ωd t + (v0 + ζ ωn x0 ) sin ωd t ωd

(2.4.10)

Equation (2.4.10) can be rewritten by using phase as x = Ae−ζ ωn t cos(ωd t − φ)

(2.4.11)

where A is the amplitude and φ is the phase, which are expressed as 

(v0 + ζ ωn x0 )2 ωd2   −1 v0 + ζ ωn x 0 φ = tan x0 ωd

A=

x02 +

(2.4.12a) (2.4.12b)

Comparing the solution of the damped free vibration, Eq. (2.4.11) with the solution of the undamped free vibration, Eq. (2.3.11), we can see that e−ζ ωn t is added. Of course, the amplitude and phase change under the influence of damping. However, it is this exponential function, e−ζ ωn t , that reflects the effect of damping the most. Because ζ , ωn are positive numbers, this exponential function converges to zero as time goes by. The vibration of the actual structure also stops over time due to the influence of damping and ζ is a parameter that reflects this phenomenon. Example 2.1 Let’s calculate the damped free vibration response for the case of x0 = 1m, v0 = 1m/s, ωn = 10rad/s, ζ = 0.05. Matlab program for Eqs. (2.4.11) and (2.4.12a, b) is as follows: clear close all clc zt = 0.05; wn = 10; x0 = 1; v0 = 1; wd = sqrt(1-ztˆ2)*wn; A = sqrt(x0ˆ2+(v0+zt*wn*x0)ˆ2/(wdˆ2)) ph = atan2(v0+zt*wn*x0,x0*wd) t= 0:0.01:5; x = A*exp(-zt*wn*t).*cos(wd*t-ph); Xp0 = A*cos(-ph); Xm0 = -A*cos(-ph); Xpt = Xp0*exp(-zt*wn*t); Xmt = Xm0*exp(-zt*wn*t); figure plot(t,x,t,Xpt,’--k’,t,Xmt,’--k’,’LineWidth’,2.0)

2.4 Damped Free Vibration

45

Fig. 2.3 Time response of damped free vibration

set(gca,’LineWidth’,2.0) axis([0 5 -1.2 1.2]) xlabel(’t(s)’) ylabel(’x(m)’) text(2.5,0.5,’X_0 eˆ{-\zeta\omega_n t}’) text(2.5,-0.5,’-X_0 eˆ{-\zeta\omega_n t}’)

It turns out that A = 1.0112, φ = 0.1491 rad and the response is as shown in Fig. 2.3. The exact solution of the ODE given by Eq. (2.4.2) can be also directly obtained by using Matlab symbolic math tool. clear close all clc syms x(t) zt = 0.05; wn = 10; x0 = 1; v0 = 1; eqn = diff(x,t,2) == -2*zt*wn*diff(x,t) - wnˆ2*x; Dx = diff(x,t); cond = [ x(0)==x0, Dx(0)==v0 ]; xSol(t)=dsolve(eqn,cond) t=linspace(0,5,1000); xSol = eval(vectorize(xSol)); plot(t,xSol)

The following solution is obtained.

46

2 Vibration Analysis of Single-Degree-of-Freedom System xSol(t) = (exp(-t/2)*(133*cos((399ˆ(1/2)*t)/2) + 399ˆ(1/2)*sin((399ˆ(1/2)*t)/2)))/133

For vibration engineers, damping is a very good thing. This is because if there is vibration, it is an element that extinguishes the vibration. Let’s look again at the characteristic equation of the system with damping, Eq. (2.4.4). The roots of this equation appear as in Eq. (2.4.5), and it was confirmed that the roots are composed of real and imaginary parts. Based on Eq. (2.4.5) and Fig. 2.3, we can find out the most important fact in controlling vibrations. The imaginary part eventually appears as a sine or cosine term to represent the vibration, and the real part indicates the decay rate of the vibration. However, the vibration disappears only when the real part becomes a negative value. What if the real part is positive? For example, assume that is negative. Of course, there is no passive damper with negative values in the real world. If ζ is negative, then it can be seen that the vibration does not decrease exponentially, but the amplitude increases over time. This phenomenon is referred to as system destabilization in control theory. In active vibration control, if the controller is incorrectly designed, the system is destabilized and the amplitude is not reduced, but rather the amplitude increases. So, when the root of the characteristic equation is obtained including the controller, if the root has a positive real part, it means that the system is unstable. The case of 0 < ζ < 1 is called the underdamped case, the case of ζ = 1 is called the critically damped case, and the case of ζ > 1 is called the overdamped case. See other vibration textbooks (Meirovitch 1975) for solutions to critically damped and overdamped cases.

2.5 Harmonic Excitation Now let’s analyze the forced vibration problem. In vibration theory, forced vibration analysis means analyzing vibration when there is external disturbance. In general, the initial conditions are ignored when performing forced vibration analysis. In case of external disturbance, the equation of motion appears in the form of a nonhomogeneous ODE. In general, the solution of the non-homogeneous ODE consists of a homogeneous solution and a particular solution. However, only a particular solution is dealt with when a forced vibration analysis is performed. Let us consider a damped system subjected to external disturbance, which is expressed as the following equation of motion. m x¨ + c x˙ + kx = d

(2.5.1)

Introducing d = k d¯ and dividing Eq. (2.5.1) by m, the following ODE is derived. x¨ + 2ζ ωn x˙ + ωn2 x = ωn2 d¯

(2.5.2)

2.5 Harmonic Excitation

47

If the external disturbance has an arbitrary shape in time, it is very difficult to find the solution mathematically. It is also difficult to calculate the transient response in closed form for non-periodic external disturbances such as shock. In the case of forced vibrations, in general, attention is paid to the steady-state response. After a very long time, the initial transient response disappears and only the response to external disturbance remains. Therefore, obtaining a steady-state response for periodic external disturbances accounts for most of the forced vibration analysis. Among the periodic external disturbances, the case of harmonic excitation is the most easily solved mathematically. Harmonic excitation means that external disturbances are given as periodic functions in the form of sine or cosine. It is convenient to use complex numbers because we can solve sine excitation or cosine excitation problems at once. Let’s express the displacement and external disturbance displacement as follows. ¯ iωt x = X eiωt , d¯ = De

(2.5.3a,b)

where ω is the excitation frequency. Inserting Eqs. (2.5.3a, b) into (2.5.2), the following equation is derived. ¯ iωt (−ω2 + 2iζ ωn ω + ωn2 )X eiωt = ωn2 De

(2.5.4)

Dividing Eq. (2.5.4) by ωn2 eiωt and introducing a non-dimensional frequency ratio r = ω/ωn , the following transfer function is obtained. X 1 = 2 ) + i(2ζ r ) ¯ (1 − r D

(2.5.5)

where r represents the ratio of the excitation frequency over the natural frequency. Using complex number theory, Eq. (2.5.5) can be rewritten as follows. X = Ge−iφ D¯

(2.5.6)

where G=

1 (1 − r 2 )2 + (2ζ r )2

, φ = tan

−1



2ζ r 1 − r2

 (2.5.7a,b)

G is the magnitude and φ is the phase. Using Eq. (2.5.6), the solution of Eq. (2.5.2) in the case of harmonic excitation can be written as ¯ i(ωt−φ) x = DGe

(2.5.8)

48

2 Vibration Analysis of Single-Degree-of-Freedom System

Therefore, the response of the system subjected to the excitation force of d = ¯ i(ωt−φ) . Using the property of complex numbers, if ¯ iωt turns out to be x = DGe k De ¯ cos(ωt − φ), and if d = k D¯ sin ωt, then d = k D¯ cos ωt, then the response, x = DG ¯ sin(ωt − φ). The magnitude and phase plots are as shown in Fig. 2.4. x = DG The magnitude in Fig. 2.4 is expressed in dB, which is defined by the following equation. X d B = 20 log10 (X )

(2.5.9)

As can be seen from Fig. 2.4, when ζ is small, the resonance amplitude when r = 1 increases. Therefore, it is advantageous to make the damping larger if the excitation frequency is close to the natural frequency. Figure 2.4 is called the frequency response curve because Fig. 2.4 shows how the system response changes as the excitation frequency of the external harmonic excitation force changes.

2.6 Transfer Function and Matlab In the case of harmonic excitation, it was shown that the analytical solution, which is a steady-state solution, can be obtained using complex numbers. The advantage of the analytical method is that it enables qualitative analysis above all. However, the use of digital computers and the advent of software such as Matlab made it easy to solve vibration problems through numerical calculations. In the previous section, it was confirmed that when the damping is small and resonance occurs, vibration of a large amplitude can occur even with a small excitation. In this section, we want to show that such a phenomenon can be analyzed using Matlab. Let’s use Eq. (2.5.2) again. This linear ODE is an equation that predicts the response of the vibration system displacement when an external disturbance is applied. If you use the numerical analysis method, you can calculate the response by time-integrating for a given time interval. If you use Matlab, you can easily calculate the response without writing a numerical analysis program yourself, and test the active vibration controller using a dedicated function for control. If Eq. (2.5.2) is analyzed by applying linear system theory, it can be diagrammed as the following figure. Figure 2.5 means that d¯ is an input to a system which results an output of x. There are two ways to define system in Matlab. One is to use the transfer function and the other is to use the state-space form. First, let’s look at how to use the transfer function. Let’s apply Laplace Transform with all initial conditions set to 0 in Eq. (2.5.2). 

 ¯ L x¨ + 2ζ ωn x˙ + ωn2 x = ωn2 d¯ ⇒ s 2 + 2ζ ωn s + ωn2 X¯ (s) = ωn2 D(s)

(2.6.1)

Therefore, the transfer function, which shows the relationship between input and output, is expressed as follows.

2.6 Transfer Function and Matlab

49

(a) Magnitude

(b) Phase Fig. 2.4 Frequency response curve of SDOF system

50

2 Vibration Analysis of Single-Degree-of-Freedom System

Fig. 2.5 Linear system with input and output

T (s) =

X¯ (s) ωn2 = 2 ¯ s + 2ζ ωn s + ωn2 D(s)

(2.6.2)

In order to define the transfer function in Matlab, you have to assign numerical values to ωn and ζ . As shown in the following Matlab script, if you type them in the Matlab command window, you will see that the transfer function is defined. When the transfer function is defined in Matlab, the frequency response curve obtained earlier can be easily obtained as shown in Fig. 2.6. clear; close all wn = 1; zt = 0.05; Tf=tf([wn^2],[1 2*zt*wn wn*wn]); [mag,phase,wout]=bode(Tf); Mag = 20*log10(mag(:)); Phase = phase(:); figure(1) semilogx(wout,Mag,'LineWidth',2.0) set(gca,'LineWidth',2.0) xlabel('Frequency (rad)') ylabel('Magnitude (dB)') print -dpsc2 bode_mag figure(2) semilogx(wout,Phase,'LineWidth',2.0) set(gca,'LineWidth',2.0) xlabel('Frequency (rad)') ylabel('Phase (deg.)')

Comparing the bode plot of Fig. 2.6 with the frequency response curve of Fig. 2.4 reveals similarity and difference. The phase plot, Fig. 2.4b has been made on the assumption that there is a positive delay in time as shown by Eq. (2.5.8), but the Matlab bode plot is made by treating it as negative, so we need to be careful when

2.6 Transfer Function and Matlab

51

(a) Magnitude

(b) Phase Fig. 2.6 Magnitude and phase curves obtained by Matlab

52

2 Vibration Analysis of Single-Degree-of-Freedom System

interpreting Matlab results. It can be said that the Matlab approach is more suitable for control system analysis than vibration analysis because phase lead may occur when designing the controller. In order to represent the system in state-space form, auxiliary variables are introduced as follows. x1 = x x2 = x˙ = x˙1

(2.6.3a,b)

Using Eq. (2.6.3a, b), (2.5.2) can be rewritten as follows. x˙2 + 2ζ ωn x2 + ωn2 x1 = ωn2 d¯

(2.6.4)

By rearranging Eqs. (2.6.3b) and (2.6.4), the following system of simultaneous ODE can be obtained.  x˙1 = x2 (2.6.5) x˙2 = −ωn2 x1 − 2ζ ωn x2 + ωn2 d¯ Equation (2.6.5) can be expressed in matrix form as follows. 

x˙1 x˙2



=

0 1 −ωn2 −ζ ωn



x1 x2



+

0 ¯ d ωn2

(2.6.6)

In the state-space equation, an additional required equation is the sensor equation. Suppose that we can measure displacement, i.e. x, and that the external force does not affect the sensor. In this case, it means that x1 of the state vector can be measured, the sensor equation can be written as follows.    x1 + [0]d¯ x = x1 = 1 0 x2

(2.6.7)

Equations (2.6.6) and (2.6.7) can be expressed as follows by introducing a matrix and a vector. x˙ = As x + Bs d¯

(2.6.8a)

y = Cs x + Ds d¯

(2.6.8b)

where x = [x1 x2 ]T and y = x = x1 , and As =



0 1 0 , B = s −ωn2 −2ζ ωn ωn2

(2.6.9a,b)

2.6 Transfer Function and Matlab

53

 Cs = 1 0 , Ds = [0]

(2.6.9c,d)

The Matlab program that defines the system of state-space form and obtains the bode plot is as follows. As Bs Cs Ds

= = = =

[ [ [ [

0 1; -wnˆ2 -2*zt*wn ]; 0; wnˆ2 ]; 1 0 ]; 0 ];

Gs = ss(As,Bs,Cs,Ds); [mag,phase,wout]=bode(Gs);

The system Gs thus defined is the same as the system Tf obtained using the tf function. If there are multiple states that can be measured, or if the sensor value is to be configured as a state vector, then y, Cs , Ds are changed. For example, suppose that the displacement cannot be measured, but the velocity can be measured. Then, we have  y = x˙ = x2 , Cs = 0 1 , Ds = [0]

(2.6.10a–c)

If we can measure both displacement and velocity, then y, Cs , Ds will be changed to.  

x 10 0 y= = x, Cs = , Ds = x˙ 01 0

(2.6.11a–c)

The most popular sensor for vibration measurement is an accelerometer. In this case, the sensor equation cannot be constructed directly, and the sensor equation must be reconstructed instead. Using Eq. (2.5.2), the equation for acceleration can be written as x¨ = −ωn2 x − 2ζ ωn x˙ + ωn2 d¯

(2.6.12)

Therefore, y, Cs , Ds are rewritten as follows.   y = x, ¨ Cs = −ωn2 −2ζ ωn , Ds = ωn2

(2.6.13a–c)

As can be seen from Eqs. (2.6.7), (2.6.10a–c), (2.6.11a–c), and (2.6.13a–c), the sensor equation is composed variously according to the type and number of sensors. Since the considered motion equation does not include the control force, when the control is included, the state equation becomes different. It will be covered again when studying control design.

54

2 Vibration Analysis of Single-Degree-of-Freedom System

2.7 Response Calculation by Simulink Various mathematical tools are used to analyze the forced vibration of the vibration system. As described above, we studied that in the case of harmonic excitation, the response can be derived in a closed-form form. This response represents the forced vibration response called the steady-state solution after a considerable period of time. If the external disturbance is not a harmonic function but a periodic function, the external disturbance can be expressed as the sum of the harmonic functions using the Fourier series. Since the superposition principle can be used if the system that we are dealing with is a linear system, the response can be expressed as the sum of the responses for each harmonic excitation. If the external disturbance is a non-periodic function, the convolution integral can be used, but the case where the response can be obtained mathematically is limited. This is well explained in the classic vibration textbooks. Due to the development of computer software, it is more common to use engineering analysis software such as Matlab and Simulink rather than using math tools. By using Matlab/Simulink, you can be free from the limitations of the mathematical method mentioned above. However, the analytical method still has the advantage that qualitative analysis is possible because it can provide a closed-form solution or an exact solution. And even if software develops, since algorithm is eventually developed based on mathematical logic, it can be said that the importance of mathematics cannot be overemphasized. Simulink has made it easier for researchers and engineers to program for numerical calculations. In particular, it made it possible to see the result immediately without going through a complicated mathematical derivation process in calculating the vibration response. With Simulink, the response to any type of external force can be obtained quickly. For example, let us consider a case in which harmonic excitation is applied to a SDOF vibration system. It is assumed that all initial conditions are 0. The Simulink program is shown in Fig. 2.7 for the case of ωn = 10 rad/s, ζ = 0.05, and d¯ = sin 10t. If you run the Simulink program of 2.7.1, the oscilloscope shows the display as shown in Fig. 2.8. The analytical solution to this problem is obtained using Eqs. (2.5.7a, b) and (2.5.8) as follows. x = 10 sin(10t − π /2)

Fig. 2.7 Simulink program for harmonic excitation

(2.7.1)

2.7 Response Calculation by Simulink

55

Fig. 2.8 Results by Simulink program

Comparing the solution given by Eq. (2.7.1) and the result of Simulink program, Fig. 2.8, it can be seen that the initial response at which the vibrating force begins to be applied is different. Note that analytical solution Eq. (2.7.1) is a steady-state solution, and the response obtained by Simulink is the response from when the vibratory force is applied in the stationary state, so this difference comes out because a transient response is included. However, after a long period of time, we can see that the two responses agree. The Simulink program of Fig. 2.7 uses a transfer function. If we use the state-space equation, we need to use Eqs. (2.6.7) and (2.6.8) and construct As , Bs , Cs , Ds as follows.



 0 1 0 As = (2.7.2a–d) , Bs = , Cs = 1 0 , Ds = [0] −100 −1 100 The Simulink program using this value is shown in Fig. 2.9.

Fig. 2.9 Simulink program for state-space equation

56

2 Vibration Analysis of Single-Degree-of-Freedom System

If you double-click on the State-Space block and enter the values for As , Bs , Cs , Ds obtained earlier, the simulation program is completed. It can be seen that the result obtained by executing the Simulink program of Fig. 2.9 is the same as the result obtained by executing the Simulink program of Fig. 2.8.

2.8 Base Excitation In the previous section, the response to external disturbance was investigated. If there is no external disturbance, but what will the vibration response be when the base moves? This vibration problem is related to vibration of semiconductor equipment for floor vibration, vibration of automobiles against irregularities on the road surface, vibration of railway vehicles against irregularities of rails, and the like. Let us consider the case of the spring-mass-damper system moving along the uneven road surface as shown in Fig. 2.10. Assuming that the gravity is ignored and x > y, the free body diagram is shown in Fig. 2.11. Applying Newton’s second law, we can write +↑



Fx = - k(x − y) − c(x˙ − y˙ ) = m x¨

(2.8.1)

Then the equation of motion can be derived as follows. m x¨ + c x˙ + kx = c y˙ + ky Fig. 2.10 Base excitation

Fig. 2.11 Free body diagram for base excitation problem

(2.8.2)

2.8 Base Excitation

57

Dividing Eq. (2.8.2) by m and introducing parameters used earlier, the following equation can be written. x¨ + 2ζ ωn x˙ + ωn2 x = 2ζ ωn y˙ + ωn2 y

(2.8.3)

If the displacement of the base is given in the form of a harmonic function, we can find the steady-state response. Let’s use complex numbers for this task. Substituting y = Y eiωt , x = X eiωt into Eq. (2.8.3), we can write



 - ω2 + 2iζ ωn ω + ωn2 X = 2iζ ωn ω + ωn2 Y

(2.8.4)

Therefore, the relation between the input displacement and output displacement can be written as 1 + i(2ζ r ) X = = Ge−iφ Y 1 − r 2 + i(2ζ r )

(2.8.5)

where  G=

  2ζ r 3 1 + (2ζ r )2 -1 , φ = tan (1 − r 2 )2 + (2ζ r )2 1 − r 2 + (2ζ r )2

(2.8.6a,b)

in which G is the magnification factor and φ is the phase. Then, the response can be expressed as follows. x = Y Gei(ωt−φ)

(2.8.7)

So, if G is greater than 1, it means that the amplitude is greater than the displacement of the ground, and if G is less than 1, it means that the amplitude is smaller than that of the ground. φ denotes the delay with the movement of the ground, that is, the phase difference. Figure 2.12 shows the magnitude and phase using Eq. (2.8.6). The process of deriving the exact solution Eq. (2.8.6) is a bit complicated. This process can be omitted by using Eq. (2.8.3) to find the transfer function and using Matlab. When Laplace transform is applied to Eq. (2.8.3), the transfer function is simply as follows. 2ζ ωn s + ωn2 X = 2 Y s + 2ζ ωn s + ωn2

(2.8.8)

Using the Matlab functions tf() and bode(), Fig. 2.12 can be easily obtained.

58

2 Vibration Analysis of Single-Degree-of-Freedom System

(a) Magnitude

(b) Phase Fig. 2.12 Magnitude and phase plots for base excitation

2.9 Impulse Response

59

2.9 Impulse Response If the external disturbance is a harmonic function, the steady-state response can be solved mathematically. However, in the case of non-periodic excitation, it is not easy to solve mathematically. The term non-periodic excitation means that excitation no longer repeats the same pattern. As a method of obtaining the response to nonperiodic excitation, a method using the response to unit impulse, that is, unit impulse response, has been proposed. The unit impulse response can be used to obtain a response to arbitrary excitation. The unit impulse is defined as follows. δ(t − a) = 0 for t = a 

∞

−∞

(2.9.1)

δ(t − a)dt = 1

(2.9.2)

where δ(t − a) is the Dirac delta function. If the response when unit impulse is called h(t), the differential equation for impulse response can be written as ¨ + ch(t) ˙ + kh(t) = δ(t) m h(t)

(2.9.3)

˙ initial conditions are h(0) = h(0) = 0. Integrating Eq. (2.9.3) over time period [0, ε] results in the following. ε lim

ε→0+

 ¨ + ch(t) ˙ + kh(t) dt = lim m h(t) +

ε δ(t)dt = 1

ε→0

0

(2.9.4)

0

And each integration leads to ε

 ˙ +) ¨ ˙ ε = h(0 h(t)dt = lim+ h(t) 0

(2.9.5)

˙ h(t)dt = lim+ h(t)|ε0 = lim+ [h(ε) − h(0)] = 0

(2.9.6)

lim+

ε→0

ε→0

0

ε lim

ε→0+

ε→0

ε→0

0

ε h(t)dt = lim+ ε[h(ε) + h(0)] = 0

lim

ε→0+

ε→0

(2.9.7)

0

Taken together, the impulse response can be replaced by a free vibration problem with the following initial conditions.

60

2 Vibration Analysis of Single-Degree-of-Freedom System

Fig. 2.13 Arbitrary excitation and impulse

1 ¨ + ch(t) ˙ + kh(t) = 0, h(0) ˙ m h(t) = , h(0) = 0 m

(2.9.8)

Using the free vibration response formula of the underdamped system obtained earlier, the impulse response is expressed as follows.  h(t) =

1 e−ζ ωn t mωd

0

sin ωd t, t ≥ 0 t 2

(2.9.14)

The disturbance plot is shown in Fig. 2.14 and the impulse response using Eq. (2.9.12) is shown in Fig. 2.15. The Simulink program shown in Fig. 2.16 can be easily constructed. The simulation result is stored in sim_result.mat file to be compared to other results.

Fig. 2.14 Disturbance

62

2 Vibration Analysis of Single-Degree-of-Freedom System

Fig. 2.15 Response

Fig. 2.16 Simulink program for non-periodic excitation

Matlab program using the convolution operation for this example are as follows.

2.9 Impulse Response

clear close all % load simulink_result load sim_result t_s = sim_res(1,:); x_s = sim_res(2,:); figure plot(t_s,x_s,'k-','LineWidth',2.0) xlabel('t(s)','LineWidth',2.0) set(gca,'LineWidth',2.0) xlabel('t(s)'); ylabel('x_d (m)') wn = 10; zt = 1/20; wd = sqrt(1-zt^2)*wn; sys = tf([wn^2],[1 2*zt*wn wn^2]); dt = 0.01; Tf = 10; Tif = 10; t=0:dt:Tf; Nt = length(t); % disturbance info d=(t xi (i = 1, 2, . . . , n − 1). In this case, all springs except the (n + 1)th damper and spring are stretched. Applying Newton’s Law for each mass, the following equations are derived. + − →



+ − →

+ − →

Fx1 = −k1 x1 + k2 (x2 − x1 ) − c1 x˙1 + c2 (x˙2 − x˙1 ) + d1 = m 1 x¨1 (3.2.1a)





Fx2 = −k2 (x2 − x1 ) + k3 (x3 − x2 ) − c2 (x˙2 − x˙1 ) + c3 (x˙3 − x˙2 ) + d2 = m 2 x¨2

(3.2.1b)

Fxn = −kn (xn − xn−1 ) − kn+1 xn + cn (x˙n − x˙n−1 ) − cn+1 x˙n + dn = m n x¨n (3.2.1c)

By rearranging Eq. (3.2.1a, b, c), the following equation of motion (EOM) in matrix form can be obtained.

Fig. 3.1 Multi-degree-of-freedom system

Fig. 3.2 Free body diagram for multi-degree-of-freedom system

3.2 Newton’s Second Law and Equations of Motion

69

¨ + Cx ˙ + Kx = Bd d Mx

(3.2.2)

T T   where x = x1 x2 . . . xn , and d = d1 d2 . . . dn ⎡

0 ··· m2 · · · .. . . . . 0 0 ···

m1 ⎢ 0 ⎢ M=⎢ . ⎣ ..

0 0 .. .

⎤ ⎥ ⎥ ⎥, ⎦

(3.2.3a)

mn

⎡

⎤ c1 + c2 −c2 · · · 0 ⎢ −c2 c2 + c3 · · · ⎥ 0 ⎢ ⎥ C=⎢ . ⎥, . . . .. .. .. ⎣ .. ⎦ 0 0 · · · cn + cn+1 ⎡ ⎡ ⎤ k1 + k2 −k2 · · · 1 0 ··· 0 ⎢ −k2 k2 + k3 · · · ⎢ ⎥ 0 ⎢ ⎢0 1 ··· ⎥ K=⎢ . ⎥, Bd = ⎢ . . . .. .. .. ⎣ .. ⎣ .. .. . . ⎦ . . . 0

0

· · · kn + kn+1

(3.2.3b) ⎤ 0 0⎥ ⎥ .. ⎥ .⎦

(3.2.3c,d)

0 0 ··· 1

in which, the Bd matrix is a matrix that reflects the participation of external disturbances, and this matrix is introduced because external disturbances may not act on all masses. Although Eq. (3.2.2) is a matrix ODE, it can be seen that the form of the equation is similar to the EOM derived for the SDOF vibration system.

3.3 Lagrange Equation As shown in the previous section, the FBD and Newton’s second law can be used to derive an EOM such as Eq. (3.2.2). The EOM can also be derived using the energy method. Let’s look at the use of the Lagrange equation. The Lagrange equation is great help in deriving the motion equation of a continuous system later and the use of approximate methods such as the assumed modes method. Let’s suppose that the general coordinate system representing the motion of a dynamic system with n degrees of freedom consists of q1 , q2 , . . . , qn and kinetic energy and potential energy are expressed by the following equation. T = T (q˙1 , q˙2 , . . . , q˙n ), · · · V = V (q1 , q2 , . . . , qn )

(3.3.1a,b)

where T is the kinetic energy and V is the potential energy. In addition, let us suppose that virtual work is expressed as δ W¯ = Q 1 δq1 + Q 2 δq2 + · · · + Q n δqn

(3.3.2)

where Q i (i = 1, 2, . . . , n) is the generalized force in each coordinate. Then, the Lagrange equation is expressed as

70

3 Vibration of Multi-degree-of-freedom System

d dt



∂L ∂ q˙i

−

∂L ∂D + = Q i , i = 1, 2, . . . , n ∂qi ∂ q˙i

(3.3.3)

where L = T −V is and D is Rayleigh’s dissipation function. The Lagrange equation for each generalized coordinate can also be expressed as the following vector–matrix equation d dt



∂L ∂ q˙

−

∂L ∂D + =Q ∂q ∂ q˙

(3.3.4)

 T T  where q = q1 q2 . . . qn , is Q = Q 1 Q 2 . . . Q n . With this in mind, let’s express the kinetic, potential, and dissipation energies for the above MDOF vibration system as follows. T =

1 1 1 T ˙ ˙ x˙ Mx, V = xT Kx, D = x˙ T Cx 2 2 2

(3.3.5a-c)

 T   ¯ = dT δx, and d = d1 d2 . . . dn T . where, x = x1 x2 . . . xn . Virtual work is δ W In order to apply the Lagrange equation given by Eq. (3.3.4), we use q = T T   x1 x2 . . . xn = x, and Q = d1 d2 . . . dn = Bd d. And the following matrix operation is needed.    ∂ 1 T ∂ 1 T ˙ ∂ 1 T ˙ ˙ ˙ x˙ Mx = Mx, x Kx = Kx, x˙ Cx = Cx ∂ x˙ 2 ∂x 2 ∂ x˙ 2

(3.3.6a-c)

Using Eqs. (3.3.4) and (3.3.6a–c), the same EOM as Eq. (3.2.2) is derived.

3.4 Free Vibration of Undamped MDOF System The EOM for the free vibration of an undamped MDOF system with n masses is obtained from Eq. (3.2.2) as follows: ¨ + Kx = 0 Mx

(3.4.1)

Unlike the SDOF vibration system, since Eq. (3.4.1) is a matrix equation, the natural frequency is not calculated immediately. However, the general solution to Eq. (3.4.1) can be written as follows, similar to the case of a SDOF vibration system x = Aeiωt

(3.4.2)

T  where A = A1 A2 . . . An is a constant vector determined by the initial conditions. We used est for the SDOF vibration system, but used eiωt in Eq. (3.4.2). The

3.4 Free Vibration of Undamped MDOF System

71

reason is that we already know that the problem of undamped free vibration given by Eq. (3.4.1) is that the solution vibrates in sine or cosine over time. Substituting Eq. (3.4.2) into Eq. (3.4.1) yields the following equation. 

2 −ω M + K Aeiωt = 0

(3.4.3)

Since it is eiωt = 0, the equation that satisfies the above equation is as follows. 

2 −ω M + K A = 0

(3.4.4)

Putting λ = ω2 in Eq. (3.4.4) and rearranging it, we can rewrite as follows. (K − λM)A = 0

(3.4.5)

Equation (3.4.5) is a homogeneous simultaneous equation with all zeros on the right. One of the solutions that satisfies this equation is the case of A = 0, which means x = 0. Since static equilibrium without motion is a trivial solution, this solution is meaningless. So, we have to think about the case where Eq. (3.4.5) is satisfied while being A = 0. Using linear algebra, it can be seen that in order for Eq. (3.4.5) to have a meaningful solution, only the following condition, in which the determinant is 0, must be satisfied. |K − λM| = 0

(3.4.6)

In this case, the inverse matrix of matrix (K − λM) cannot exist. If this condition is satisfied, a unique solution of A cannot be obtained. Rather, the number of A that satisfies the Eq. (3.4.5) increases to infinite number. However, for each element in A, Ai are related to each other. Solving Eq. (3.4.6) yields the characteristic equation for λ. And we obtain the solution by solving this characteristic equation. In the case of a MDOF system, of course, the same number of degrees as the order is obtained. The lowest natural frequency is called the fundamental frequency, and the relationship between each element of A constitutes the natural vibration mode. In general, one element is set to 1 and the other elements are calculated to describe the natural vibration mode. For example, in the case of a two-DOF system, solving Eq. (3.4.6) yields a quadratic algebraic equation for λ. And if we use the root formula, we can find the root by hand. However, in the case of a 3-DOF system, a cubic algebraic equation is derived from Eq. (3.4.6), and it is not easy to find the roots of this algebraic equation by hand. So, we resort to the numerical analysis method. If the degree of freedom is larger, it is not easy to obtain the characteristic equation from Eq. (3.4.6). Let’s reorganize Eq. (3.4.5) into the following equation. KA = λMA

(3.4.7)

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3 Vibration of Multi-degree-of-freedom System

This equation is called an eigenvalue problem in mathematics. If we use Matlab, we can find the natural frequencies and the natural modes very easily without the process of solving the determinant to find the characteristic equation. If the mass matrix and stiffness matrices are defined, we can use the following Matlab command to calculate eigenvalues and eigenvectors. [U,Lambda]=eig(K,M)

The output of the Matlab function eig() is the eigenvector matrix, U and the eigenvalue matrix, . The diagonal of  is composed of ω12 , ω22 , · · · , ωn2 . ⎡

ω12 ⎢ 0 ⎢  = 2n = ⎢ . ⎣ ..

0 ω22 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(3.4.8)

0 0 · · · ωn2 Therefore, the first column of U is the eigenvector for ω1 , the second column is the eigenvector for ω2 , and the n th column is the eigenvector for ωn . Eigenvector matrix has a very important property called orthonormality condition given by the following equation. UT MU = I, UT KU = 

(3.4.9a,b)

where I is the identity matrix. Fortunately, this orthonormality condition is automatically satisfied when the eigenvector matrix is obtained using the Matlab eig function. This condition is very important in the analysis of a MDOF system, and is also very important when designing active vibration control for the MDOF system. Example 3.1 Let us consider a three-degree-of-freedom system shown below. The properties of the 3-DOF system shown in Fig. 3.3 are m 1 = m 2 = m 3 = 10 kg and k1 = k2 = k3 = k4 = 1000 N/m. Let’s find the natural frequencies and the natural modes of the system. Matlab program can be written as follows.

Fig. 3.3 Free body diagram for three-degree-of-freedom system

3.4 Free Vibration of Undamped MDOF System

73

Clear m1 = 10; m2 = 10; m3 = 10; k1 = 1000; k2 = 1000; k3 = 1000; k4 = 1000; M = diag([ m1 m2 m3 ]); K = [ k1+k2 -k2 0 ; -k2 k2+k3 -k3 ; 0 -k3 k3+k4 ]; [U, Lambda] = eig(K,M); Om = diag(sqrt(Lambda)); disp(Om) disp(U) disp(U’*M*U)

If we run the above program, we can get the following results. 7.6537 14.1421 18.4776 0.1581 -0.2236 -0.1581 0.2236 -0.0000 0.2236 0.1581 0.2236 -0.1581 1.0000 0.0000 0.0000 0.0000 1.0000 -0.0000 0.0000 -0.0000 1.0000

With the help of Matlab, we have seen that the eigenvalue problem can be solved very easily and the eigenvector calculated by Matlab satisfies the orthonormality condition. Let’s solve the free vibration problem of Undamped MDOF vibration system. Suppose that the initial conditions of Eq. (3.4.1) are given as x(0) = x0 , x˙ (0) = v0

(3.4.10a,b)

It is almost impossible to solve the system of differential Eqs. (3.4.1) with these initial conditions. The proposed method for solving a system of differential equations uses modal transformation. Modal transformation is given by the following equation. x = Uq

(3.4.11)

where q is called the modal coordinate. U is the eigenvector matrix obtained by solving the eigenvalue problem previously, and it is assumed that U satisfies the orthonormality condition, Eq. (3.4.9a,b). Equation (3.4.11) means that the conversion between the actual coordinates x and the modal coordinates q is possible. Substituting Eq. (3.4.11) into Eq. (3.4.1) yields the following equation. MUR q + KUq = 0

(3.4.12)

Multiplying UT in front of Eq. (3.4.12) gives: ¨ + UT KUq = 0 UT MUq

(3.4.13)

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3 Vibration of Multi-degree-of-freedom System

Applying the orthonormality condition, Eq. (3.4.9a,b), Eq. (3.4.13) results in the following equation. q¨ + q = 0

(3.4.14)

If we look closely at Eq. (3.4.14), we can see that the equations of motion for each modal coordinate are independent of each other. Hence, it can be seen that Eq. (3.4.14) can be rewritten as mutually independent modal equation of motion. q¨1 + ω12 q1 = 0 q¨2 + ω22 q2 = 0 .. . q¨n + ωn2 qn = 0

(3.4.15)

It can be seen that the equations of motion for each modal coordinate in Eq. (3.4.15) are independent from each other, and each equation of motion has the same form as the equation of motion for a SDOF system. Therefore, the solution obtained in Chap. 2 can be used immediately. The solutions to Eq. (3.4.15) are as follows. q˙1 (0) sin ω1 t ω1 q˙1 (0) q2 = q2 (0) cos ω2 t + sin ω2 t ω2 .. . q˙1 (0) qn = qn (0) cos ωn t + sin ωn t ωn q1 = q1 (0) cos ω1 t +

(3.4.16)

Equation (3.4.16) can be expressed as a matrix equation as follows. ˙ q = Ct q(0) + St −1 q(0)

(3.4.17)

where ⎡

0 cos ω1 t ⎢ 0 cos ω2 t ⎢ Ct = ⎢ . .. . ⎣ . . 0 0

··· ··· .. .

0 0 .. .

· · · cos ωn t

⎤ ⎥ ⎥ ⎥ ⎦

(3.4.18a)

3.4 Free Vibration of Undamped MDOF System

75

⎡

⎤ sin ω1 t 0 ··· 0 ⎢ 0 sin ω2 t · · · 0 ⎥ ⎢ ⎥ St = ⎢ . .. .. ⎥ .. ⎣ .. . . . ⎦ 0 0 · · · sin ωn t ⎡ ⎤ ω1 0 · · · 0 ⎢ 0 ω2 · · · 0 ⎥ ⎢ ⎥ =⎢ . . . .. ⎥ . . . ⎣ . . . . ⎦

(3.4.18b)

(3.4.18c)

0 0 · · · ωn In order to complete the solution of Eq. (3.4.17), the initial conditions of each modal coordinate system need to be computed. Applying the modal transformation to the initial conditions of the original coordinate system, we can obtain the following relations: ˙ x(0) = Uq(0), v(0) = Uq(0)

(3.4.19a,b)

Multiplying Eq. (3.4.19a,b) by UT M and applying the orthonormality condition leads to the following equation. ˙ = UT Mv(0) q(0) = UT Mx(0), q(0)

(3.4.20a,b)

Substituting these initial conditions into Eq. (3.4.17) completes the solution in the modal coordinates. q = Ct UT Mx(0) + St  - 1 UT Mv(0)

(3.4.21)

Again, using modal transformation to obtain the solution in the original coordinate system, we can obtain the following free vibration response finally. x = UCt UT Mx(0) + USt −1 UT Mv(0)

(3.4.22)

Example 3.2 Let us consider a three-degree-of-freedom system of Example 3.1.  T  T Initial conditions are as follows: x(0) = 1 0 0 ,v(0) = 0 0 0 .

76

3 Vibration of Multi-degree-of-freedom System

The Matlab program to obtain the free vibration response is as follows. clear m1 = 10; m2 = 10; m3 = 10; k1 = 1000; k2 = 1000; k3 = 1000; k4 = 1000; M = diag([ m1 m2 m3 ]); K = [ k1+k2 -k2 0 ; -k2 k2+k3 -k3 ; 0 -k3 k3+k4 ]; [U, Lambda] = eig(K,M); Om = sqrt(Lambda); om = diag(Om); x0 = [ 1 0 0 ]'; v0 = [ 0 0 0 ]'; tdata = 0:0.01:10; nt = length(tdata); for i = 1:nt t = tdata(i); Ct = diag([ cos(om(1)*t) cos(om(2)*t) cos(om(3)*t) ]); St = diag([ sin(om(1)*t) sin(om(2)*t) sin(om(3)*t) ]); xdata(:,i) = U*Ct*U'*M*x0 + U*St*inv(Om)*U'*M*v0; end plot(tdata,xdata,'linewidth',2.0) set(gca,'linewidth',2.0) xlabel('$t\rm{(s)}$','interpreter','latex') ylabel('$Disp.\rm{(m)}$','interpreter','latex') legend('\itx_1','\itx_2','\itx_3')

Matlab’s figure output is as follows (Fig. 3.4).

3.5 Free Vibration of Damped MDOF System In the free vibration analysis of the damped MDOF system, the difficulty of analysis is determined by the nature of damping. The equation for the free vibration of the damped MDOF system can be expressed as follows from Eq. (3.2.2). ¨ + Cx ˙ + Kx = 0 Mx

(3.5.1)

Unlike the undamped case, the eigenvalue problem of damped free vibration given by Eq. (3.3.2) is not directly constructed. To make the analysis easier, let’s introduce the following assumptions.

3.5 Free Vibration of Damped MDOF System

77

Fig. 3.4 Free vibration response of three-degree-of-freedom system

UT CU ∼ = 2Zn

(3.5.2)

where U is the eigenvector matrix of the undamped system, and n is the natural frequency matrix of the undamped system and ⎡

ζ1 ⎢0 ⎢ Z=⎢ . ⎣ ..

0 ζ2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(3.5.3)

0 0 · · · ζn is the damping factor matrix. Although all damping matrices do not allow the same conditions as in Eq. (3.5.2), this assumption assumes that the diagonal term of the UT CU matrix is dominant compared to the off-diagonal term and thus the off-diagonal term can be ignored. In fact, if these assumptions are not made, the effort to decouple the equation of motion using modal transformation is meaningless. Another assumption is that in the dynamic analysis of a real structure, damping itself is almost impossible to approach theoretically. Therefore, it is more advantageous to assume and analyze the damping factor than to construct the damping matrix. We start with the same assumptions in experimental modal analysis. Introducing the modal transformation x = Uq into Eq. (3.5.1), multiplying the equation by UT , and using the orthonormality condition, Eq. (3.4.9a,b), damping matrix assumption, Eq. (3.5.2), we can derive the following modal equation of motion.

78

3 Vibration of Multi-degree-of-freedom System

q¨ + 2Zn q˙ + q = 0

(3.5.4)

Assuming that Eq. (3.5.2) is satisfied, we can see that the original equation of motion given by Eq. (3.5.1) is transformed into the decoupled modal equations of motion given by Eq. (3.5.4). And it can be seen that each modal equation of motion is a SDOF vibration system given by the following equation. q¨i + 2ζi ωi q˙i + ωi2 qi = 0, i = 1, 2, . . . , n

(3.5.5)

 where ωdi = 1 − ζi2 ωi (i = 1, 2, . . . , n) are the damped natural frequencies. Of course, this expression includes the assumption of an underdamped system. The damped natural frequency matrix can be written as ⎡

ωd1 ⎢ 0 ⎢ d = ⎢ . ⎣ .. 0

0 ωd2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(3.5.6)

0 · · · ωdn

If the assumption of Eq. (3.5.2) is established, the eigenvector matrix is the same as the eigenvector matrix of the undamped MDOF system. In addition, the damped natural frequencies of the damped MDOF system are the same as Eq. (3.5.6). The damping factor for each natural mode is given by Eq. (3.5.3). An approximation like this can be used when the damping is very small. If the damping factors are obtained by either free-decay experiment or frequency response curve, then the damping matrix can be constructed by using the orthonormality condition to Eq. (3.5.2). C = 2MUZn UT M

(3.5.7)

Example 3.3 Let us consider a damped three-degree-of-freedom system shown below. The damping coefficients are as follows. The property values of the 3-DOF system with spring-mass-damper shown in Fig. 3.5 are m 1 = m 2 = m 3 = 10 kg, c1 = c2 = c3 = c4 = 10 Ns/m and

Fig. 3.5 Free body diagram for spring-mass-damper system

3.5 Free Vibration of Damped MDOF System

79

k1 = k2 = k3 = k4 = 1000 N/m. Let’s find the damped natural frequency of the system. Matlab program can be written as follows. clear

m1 = 10; m2 = 10; m3 = 10; k1 = 1000; k2 = 1000; k3 = 1000; k4 = 1000; c1 = 10; c2 = 10; c3 = 10; c4 = 10; M = diag([ m1 m2 m3 ]); K = [ k1+k2 -k2 0 ; -k2 k2+k3 -k3 ; 0 -k3 k3+k4 ]; C = [ c1+c2 -c2 0 ; -c2 c2+c3 -c3 ; 0 -c3 c3+c4 ]; [U, Lambda] = eig(K,M); Om = sqrt(Lambda); om = diag(Om); Z = 0.5*U'*C*U*inv(Om); for i = 1:3 omd(i) = sqrt(1-Z(i,i)^2)*om(i); end disp(Z) disp(omd)

Running this program produces the following results for the damping factor matrix: 0.0383 0.0000 0.0000

0.0000 0.0000 0.0707 -0.0000 -0.0000 0.0924

Fortunately, we can see that the damping factor matrix appears as a diagonal matrix. The reason is that the damping coefficients have the same change as the spring constants. This damping is often called proportional damping. The eigenvector is in fact the eigenvector of the undamped case. Let’s solve the initial value problem. The initial conditions are x(0) = x0 , x˙ (0) = v0 . The solution to Eq. (3.5.4) for these initial conditions can be expressed as follows using the solution for the SDOF system derived in Chap. 2. 

 ˙ q = Et Cdt q0 + −1 d Sdt q0 + Zn q0 where

(3.5.8)

80

3 Vibration of Multi-degree-of-freedom System

⎡

⎤ e−ζ1 ω1 t 0 ··· 0 ⎢ 0 e−ζ2 ω2 t · · · 0 ⎥ ⎢ ⎥ Et = ⎢ . .. .. ⎥ .. ⎣ .. . . . ⎦ −ζn ωn t 0 0 ··· e ⎡ cos ωd1 t 0 ··· 0 ⎢ 0 0 cos ωd2 t · · · ⎢ Cdt = ⎢ .. .. .. . . ⎣ . . . .

(3.5.9a) ⎤ ⎥ ⎥ ⎥ ⎦

· · · cos ωdn t ⎤ ⎡ 0 ··· 0 sin ωd1 t ⎥ ⎢ 0 0 sin ωd2 t · · · ⎥ ⎢ Sdt = ⎢ ⎥ .. .. .. .. ⎦ ⎣ . . . . 0 0 · · · sin ωdn t 0

(3.5.9b)

0

(3.5.9c)

If we transform the solution in the modal coordinate, Eq. (3.5.8) into the solution in the original coordinate system using modal transformation, then we obtain 

T  T x = UEt Cdt UT Mx0 + −1 d Sdt U Mv0 + Zn U Mx0

(3.5.10)

As for the damping matrix that satisfies Eq. (3.5.2), it can be seen that the diagonal matrix is derived in the same way as in the orthonormal condition and Eq. (3.4.9a,b), and free vibration analysis can be easily performed. As mentioned earlier, it is not easy to find a damping matrix in a mechanical system. It is not easy to obtain damping by experiment, so several methods have been proposed and used. The simplest method can be said to be a method of estimating the damping factor by obtaining the logarithmic decrement from the free decay experiment. In this case, there is a disadvantage that only the first vibration mode is obtained because it appears dominant. By curve fitting the frequency response curve, the damping factor for each natural mode can be obtained. If you have active vibration control in mind, a system with small damping will be the target. Assuming that the damping is small in the design of the controller is not a big problem. Example 3.4 Let us consider a damped three-degree-of-freedom system of Example  T  T 3.3. Initial conditions are as follows: x(0) = 1 0 0 ,v(0) = 0 0 0 . Matlab program that calculates free vibration response is as follows and resulting plot is given in Fig. 3.6.

3.5 Free Vibration of Damped MDOF System

Fig. 3.6 Free vibration response of damped case clear; close all; clc; m1 = 10; m2 = 10; m3 = 10; k1 = 1000; k2 = 1000; k3 = 1000; k4 = 1000; c1 = 10; c2 = 10; c3 = 10; c4 = 10; M = diag([ m1 m2 m3 ]); K = [ k1+k2 -k2 0 ; -k2 k2+k3 -k3 ; 0 -k3 k3+k4 ]; C = [ c1+c2 -c2 0 ; -c2 c2+c3 -c3 ; 0 -c3 c3+c4 ]; [U, Lambda] = eig(K,M); Om = sqrt(Lambda); om = diag(Om); Z = 0.5*U'*C*U*inv(Om); z = diag(Z); for i = 1:3 omd(i) = sqrt(1-Z(i,i)^2)*om(i); end

81

82

3 Vibration of Multi-degree-of-freedom System

Omd = diag(omd); x0 = [ 1 0 0 ]'; v0 = [ 0 0 0 ]'; tdata = 0:0.01:10; nt = length(tdata); for i = 1:nt t = tdata(i); Et = diag([ exp(-z(1)*om(1)*t) exp(-z(2)*om(2)*t) z(3)*om(3)*t) ]); Cdt = diag([ cos(om(1)*t) cos(om(2)*t) cos(om(3)*t) ]); Sdt = diag([ sin(om(1)*t) sin(om(2)*t) sin(om(3)*t) ]); xdata(:,i) U*Et*(Cdt*U'*M*x0+inv(Omd)*Sdt*(U'*M*v0+Z*Om*U'*M*x0)); end

exp(-

=

plot(tdata,xdata,'linewidth',2.0) set(gca,'linewidth',2.0) xlabel('$t\rm{(s)}$','interpreter','latex') ylabel('$Disp.\rm{(m)}$','interpreter','latex') legend('\itx_1','\itx_2','\itx_3')

3.6 Reduced-Order Analysis The main reason for using modal transformation is that MDOF systems with many degree of freedom can be analyzed by reducing them to independent modal equations of motion. In the vibration analysis, the influence of the natural vibration mode decreases with order, so that including all higher-order modes are not preferred in the analysis because there is no significant difference in the accuracy of the analysis. In the past, when the computer’s memory and CPU speed were limited, it was not possible to analyze an MDOF system with a many degree of freedom, so it was practical to perform the analysis with a lower degree of freedom. It is also effective to use a reduced-order model in active vibration control because it is not easy to control higher natural modes in the real applications. From the full-order model, let us consider only lowest r natural modes, where r < n. In this case, the modal transformation equation given by (3.4.11) is also reduced as follows and expressed using r modal coordinate systems. x = Ur qr

(3.6.1)

T  where Ur is matrix n × r and is a subset of matrix U and qr = q1 q2 · · · qr . This means that the natural vibration modes from the first order to the r th order are

3.6 Reduced-Order Analysis

83

considered. The orthonormality conditions are maintained as follows. UrT MUr = Ir , UrT KUr = r

(3.6.2a,b)

where Ir is the r × r identity matrix and r is the eigenvalue matrix of r × r . Equation (3.5.2) can be written as follows. UrT CUr = 2Zr r

(3.6.3)

where ⎡

ζ1 ⎢0 ⎢ Zr = ⎢ . ⎣ ..

0 ζ2 .. .

··· ··· .. .

0 0 .. .

⎤

⎡

ω1 ⎥ ⎢ 0 ⎥ ⎢ ⎥, r = ⎢ . ⎦ ⎣ ..

0 0 · · · ζr

0 ω2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(3.6.4a,b)

0 0 · · · ωr

Let’s perform the free vibration analysis of the Damped MDOF system using a reduced-order model. Substituting Eq. (3.6.1) into Eq. (3.5.1), multiplying by UrT and using Eq. (3.6.2) to (3.6.3), the following equation is derived. q¨ r + 2Zr r q˙ r + r qr = 0

(3.6.5)

Applying the reduced modal transformation equation to the equation for the initial conditions are as follows. x(0) = Ur qr (0), x˙ (0) = Ur q˙ r (0)

(3.6.6a,b)

Multiplying Eq. (3.6.6a,b) by UrT M leads to the initial conditions for the modal coordinates that are expressed as follows. qr (0) = UrT Mx0 , q˙ r (0) = UrT Mv0

(3.6.7a,b)

Therefore, the response in the original coordinates is given by the following equation. 

T  T x = Ur Er t Cdr t UrT Mx0 + −1 dr Sdr t Ur Mv0 + Zr r Ur Mx0

(3.6.8)

where ⎡

e−ζ1 ω1 t 0 ⎢ 0 e−ζ2 ω2 t ⎢ Er t = ⎢ . .. ⎣ .. . 0 0

··· ··· .. .

0 0 .. .

· · · e−ζr ωr t

⎤ ⎥ ⎥ ⎥, dr ⎦

⎡

ωd1 ⎢ 0 ⎢ =⎢ . ⎣ .. 0

0 ωd2 .. .

··· ··· .. .

0 0 .. .

0 · · · ωdr

⎤ ⎥ ⎥ ⎥ ⎦

(3.6.9a,b)

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3 Vibration of Multi-degree-of-freedom System

⎡

⎤ cos ωd1 t 0 ··· 0 ⎢ ⎥ 0 0 cos ωd2 t · · · ⎢ ⎥ Cdr t = ⎢ ⎥ .. .. . . . . ⎣ ⎦ . . . . 0 0 · · · cos ωdr t ⎡ ⎤ sin ωd1 t 0 ··· 0 ⎢ 0 0 ⎥ sin ωd2 t · · · ⎢ ⎥ Sdr t = ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . . 0 0 · · · sin ωdr t

(3.6.9c)

(3.6.9d)

Example 3.5 Let us consider a damped 3-DOF vibration system of Example 3.3. We are going to consider only two natural modes in the free vibration analysis. The Matlab program for reduced-order analysis is as follows and the response is shown in Fig. 3.7. Figure 3.7. shows that there is no significant difference compared to Fig. 3.6.

Fig. 3.7 Free vibration response by reduced-order model

3.6 Reduced-Order Analysis clear close all clc

m1 = 10; m2 = 10; m3 = 10; k1 = 1000; k2 = 1000; k3 = 1000; k4 = 1000; c1 = 10; c2 = 10; c3 = 10; c4 = 10;

M = diag([ m1 m2 m3 ]); K = [ k1+k2 -k2 0 ; -k2 k2+k3 -k3 ; 0 -k3 k3+k4 ]; C = [ c1+c2 -c2 0 ; -c2 c2+c3 -c3 ; 0 -c3 c3+c4 ];

[U, Lambda] = eig(K,M); Om = sqrt(Lambda); om = diag(Om);

Z = 0.5*U'*C*U*inv(Om); z = diag(Z);

for i = 1:3 omd(i) = sqrt(1-Z(i,i)^2)*om(i); end Omd = diag(omd);

r = 2; Ur = U(:,1:r); Omr = Om(1:r,1:r);

85

86

3 Vibration of Multi-degree-of-freedom System Omdr = Omd(1:r,1:r); Zr = Z(1:r,1:r);

x0 = [ 1 0 0 ]'; v0 = [ 0 0 0 ]';

tdata = 0:0.01:10; nt = length(tdata);

for i = 1:nt t = tdata(i); Ert = diag([ exp(-z(1)*om(1)*t) exp(-z(2)*om(2)*t) ]); Cdrt = diag([ cos(om(1)*t) cos(om(2)*t) ]); Sdrt = diag([ sin(om(1)*t) sin(om(2)*t) ]); xdata(:,i) = Ur*Ert*(Cdrt*Ur'*M*x0 + ... inv(Omdr)*Sdrt*(Ur'*M*v0+Zr*Omr*Ur'*M*x0)); end plot(tdata,xdata,'linewidth',2.0) set(gca,'linewidth',2.0) xlabel('$t\rm{(s)}$','interpreter','latex') ylabel('$Disp.\rm{(m)}$','interpreter','latex') legend('\itx_1','\itx_2','\itx_3')

3.7 State-Space Equation Assuming that the orthogonal property holds for the damping matrix, that is UT CU  2Zn , the equations of motion are transformed into decoupled modal equations of motion, and the response of the MDOF system is calculated using the solution of the SDOF vibration system. This method can be said to be the only calculation method in the past, when the MDOF problem could not be solved immediately. However, in recent years, when computer performance is advanced, it is possible to directly solve the damped MDOF problem by converting the equation of motion, Eq. (3.5.1), to a first-order differential equation so called a state-space equation. Modern control theories are based on the state-space equation for a MDOF control problem and Matlab was developed originally as a software to analyze control systems so that library was designed largely based on the state-space equation. Let’s introduce the following new coordinates for the displacement and velocity in Eq. (3.5.1).

3.7 State-Space Equation

87

z1 = x

(3.7.1)

z2 = x˙ = z˙ 1 (3.7.2) Equations (3.7.1) and (3.7.2) mean that displacement and velocity are expressed as z1 and z2 . Substituting Eqs. (3.7.1) and (3.7.2) into Eq. (3.5.1), we can rewrite the equation as follows. M˙z2 + C2 z2 + Kz1 = 0

(3.7.3)

Equation (3.7.3) can again be rewritten as M˙z2 = −M−1 Kz1 − M−1 Cz2

(3.7.4)

Combining Eqs. (3.7.2) and (3.7.4), we can write the following matrix equation. 

M˙z1 M˙z2





0 I = −M−1 K −M−1 C



z1



z2

(3.7.5)

Equation (3.7.5) can be simply expressed as z˙ = As z

(3.7.6)

T  where z = z1 z2 and 

0 I As = −M−1 K −M−1 C

 (3.7.7)

The solution of Eq. (3.7.6) is simply expressed as follows. z = eAs t z(0)

(3.7.8)

where eAs t = I + As t +

A2s t 2 + ··· 2!

(3.7.9)

Matlab provides this calculation as a function expm(), which simplifies programming. Also, by executing eig(As ), the damped natural frequency can be calculated. The imaginary part is the natural frequency and the real part represents the exponential decay part that is very important to stability analysis. Example 3.6 Let us consider a 10-degree-of-freedom vibration system as an example. This system is analyzed to verify the validity of the modal damping matrix assumption and the validity of the response obtained through modal transform. Each mass, spring constant, and damping coefficient are as follows.

88

3 Vibration of Multi-degree-of-freedom System

m i = 2 kg (i = 1, 2, · · · , 10)   ki = 4 4 4 4 2 2 2 2 3 3 × 100 N/m   ci = 1 1 1 2 2 2 2 1 1 1 Ns/m The initial conditions are  T x0 = 1.0 0.5 2.0 −1.0 −1.0 1.0 0.0 1.0 2.0 0.0  T v0 = 0.1 −0.1 −0.1 0.2 −0.3 −0.1 0.0 0.01 −0.2 0.2 For the free vibration problem of the 10 DOF vibration system, modal analysis ignoring the off-diagonal term of the modal damping matrix, reduced-order modal analysis considering only five natural vibration modes, and full-order analysis using state-space equations are performed. The following matlab program contains all of these calculations. clear close all clc n m k c

= = = =

10; % DOF [ 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 ]; [ 400 400 400 400 200 200 200 200 300 300 300 ]; [ 1 1 1 2 2 2 2 1 1 1 1 ];

% Initial condition x0 = [ 1.0 0.5 2.0 -1.0 -1.0 1.0 0 1.0 2.0 0]'; v0 = [ 0.1 -0.1 -0.1 0.2 -0.3 -0.1 0.0 0.01 -0.2 0.2 ]'; M = diag(m);

% Mass matrix

C = diag([ c(1:n) + c(2:n+1)]); % Damping matrix for i=1:9 C(i,i+1) = -c(i+1); C(i+1,i) = -c(i+1); end K = diag([ k(1:n) + k(2:n+1)]); % Stiffness matrix for i=1:9 K(i,i+1) = -k(i+1); K(i+1,i) = -k(i+1); end

3.7 State-Space Equation [U,Lambda]=eig(K,M); % Solve Eigenvalue problem om = sqrt(diag(Lambda))'; Om = diag(om); C_modal = U'*C*U % neglect off-diagonal terms for i = 1:n zt(i) = 0.5*C_modal(i,i)/om(i); % damping factor end Z = diag(zt); omd = zeros(n,1); for i=1:n omd(i) = sqrt(1-zt(i)^2)*om(i); end Omd = diag(omd); tspan=0:0.01:10; % time nt = length(tspan); x = zeros(n,nt); x(:,1) = x0; for it = 2:nt t = tspan(it); Expm = diag(exp(-zt.*om*t)); Cosm = diag(cos(omd*t)); Sinm = diag(sin(omd*t)); x(:,it) = U*Expm*( Cosm*U'*M*x0 ... + inv(Omd)*Sinm*(U'*M*v0+Z*Om*U'*M*x0)); end % reduce-order analysis nr = 5; Ur = U(:,1:nr); ztr = zt(1:nr); Zr = Z(1:nr,1:nr); omr = om(1:nr); Omr = Om(1:nr,1:nr); omdr = omd(1:nr); Omdr = Omd(1:nr,1:nr); xr = zeros(n,nt);

89

90

3 Vibration of Multi-degree-of-freedom System xr(:,1) = x0; for it = 2:nt t = tspan(it); Expmr = diag(exp(-ztr.*omr*t)); Cosmr = diag(cos(omdr*t)); Sinmr = diag(sin(omdr*t)); xr(:,it) = Ur*Expmr*( Cosmr*Ur'*M*x0 ... + inv(Omdr)*Sinmr*(Ur'*M*v0+Zr*Omr*Ur'*M*x0)); end % State Eq. As = [ zeros(n) eye(n); -inv(M)*K -inv(M)*C ]; z0 = [ x0; v0 ]; z = zeros(2*n,nt); z(:,1) = z0; for it=2:nt t = tspan(it); z(:,it) = expm(As*t)*z0; end xs = z(1:n,:);

np = 301; plot(tspan(1:np),x(6,1:np),'k--', ... tspan(1:np),xr(6,1:np),'k:', ... tspan(1:np),xs(6,1:np),'k-','LineWidth',2.0) set(gca,'LineWidth',2.0) legend('Full-order','Reduced-order','State Eq.') xlabel('$t\rm{(s)}$','interpreter','latex') ylabel('$x_6\rm{(m)}$','interpreter','latex')

If we calculate UT CU, we will get the following result. 0.0488 −0.0143 −0.0511 −0.0189 0.0457 0.0206 −0.0001 0.0059 0.0053 −0.0082

−0.0143 0.2515 0.0213 −0.1428 −0.0259 −0.1150 −0.0445 −0.0575 −0.0246 −0.0342

−0.0511 0.0213 0.5586 0.0832 −0.3280 0.0253 0.0952 −0.1182 −0.0234 −0.0361

−0.0189 −0.1428 0.0832 1.0006 −0.1469 −0.4768 −0.0935 −0.0833 0.0668 0.0474

0.0457 −0.0259 −0.3280 −0.1469 1.3191 0.0371 −0.5792 0.3137 −0.0556 0.2383

0.0206 0.0253 −0.1150 −0.4768 0.0371 1.7118 0.4982 0.5525 0.0591 0.2473

−0.0001 −0.0445 0.0952 −0.0935 −0.5792 0.4982 2.7965 0.4240 0.3896 0.1396

0.0059 −0.0575 −0.1182 −0.0833 0.3137 0.5525 0.4240 2.1931 0.1143 0.6438

0.0053 −0.0246 −0.0234 0.0668 −0.0556 0.0591 0.3896 0.1143 1.8355 0.1239

−0.0082 −0.0342 −0.0361 0.0474 0.2383 0.2473 0.1396 0.6438 0.1239 2.2846

Fortunately, from the above results, it can be seen that the diagonal terms have larger values than the off-diagonal terms. Figure 3.8 shows the responses calculated by modal analysis ignoring off-diagonal terms, Eq. (3.5.9a, b, c), reduced-order modal analysis considering five natural vibration modes, Eq. (3.6.8), state equation,

3.7 State-Space Equation

91

Fig. 3.8 Free vibration response (full, reduced, state)

Eq. (3.7.6). Figure 3.8 also shows that the responses of the initial time period are slightly different but it becomes almost the same as time goes by. This is because the influence of the higher-order modes disappears over time due to the damping effect, and only the influence of the lower-order vibration mode remains. Table 3.1 shows the comparison of the eigenvalues obtained by ignoring the off-diagonal term and the eigenvalue values of the state equation. It can be seen from Table 3.1 that all the eigenvalues obtained by ignoring off-diagonal are close to the exact eigenvalues. Table 3.1 Eigenvalues of approximate and original models Order

Modal damping assumption

State equation

Real

Imaginary

Real

Imaginary

1

−0.0244

3.6610

-0.0244

3.6611

2

−0.1257

6.5347

-0.1257

6.5353

3

−0.2793

9.5871

-0.2792

9.5914

4

−0.5003

13.0003

-0.5004

13.0123

5

−0.6596

15.5654

-0.6583

15.5832

92

3 Vibration of Multi-degree-of-freedom System

3.8 Harmonic Excitation We studied that if harmonic excitation acts on a SDOF system, a steady-state solution can be obtained in an exact form. In the case of MDOF, it is not easy to obtain an analytical solution, but using modal transformation we can obtain a solution close to the exact solution. Let’s reintroduce modal transformation, Eq. (3.4.11) to Eq. (3.2.2). MUq¨ + CUq¨ + KUq = Bd d

(3.8.1)

Multiplying Eq. (3.8.1) by UT leads to the following equation. UT MUq¨ + UT CUq¨ + UT KUq = UT Bd d

(3.8.2)

Using the orthonormality condition, Eq. (3.4.9a,b) and modal damping assumption, Eq. (3.5.2), the following equation is obtained. q¨ + 2Zn q˙ + q = UT Bd d

(3.8.3)

Let us define the right side of Eq. (3.8.3) as UT Bd d = d∗

(3.8.4)

Then, using Eq. (3.8.4), Eq. (3.8.3) is expressed as follows. q¨ + 2Zq˙ + q = d∗

(3.8.5)

Equation (3.8.5) becomes a decoupled modal equation of motion, unlike the fully coupled equation of motion given by Eq. (3.2.2). So, Eq. (3.8.5) can be written separately as follows. q¨i + 2ζi ωi q˙i + ωi2 qi = ωi2 di∗ , i = 1, 2, · · · , n

(3.8.6)

From Eq. (3.8.6), the transfer function for each modal coordinate is expressed as follows. q¯i (s) =

ωi2 d¯ ∗ (s) = G i (s)d¯i∗ (s) s 2 + 2ζi ωi s + ωi2 i

(3.8.7)

where the initial conditions are assumed to be 0. We can derive the following equation from Eq. (3.8.4) d∗ = −1 UT Bd d

(3.8.8)

Therefore, the response by harmonic excitation can be expressed as follows.

3.8 Harmonic Excitation

93

¯ ¯ q(s) = G(s)−1 UT Bd d(s)

(3.8.9)

where  G(s) = diag

ωi2 2 s + 2ζi ωi s + ωi2

(3.8.10)

Modal transformation is again used to find the original displacement and the transfer function. ¯ ¯ x¯ (s) = UG(s)−1 UT Bd d(s) = T(s)d(s)

(3.8.11)

where the transfer function is expressed as follows. T(s) = UG(s)−1 UT Bd

(3.8.12)

In the case of harmonic excitation, the modal coordinates near the excitation frequency are most affected. Therefore, it can be said that calculation of all coordinates is not necessary. If the degree of freedom is low, the transfer function can be obtained using Eq. (3.8.12), but as the degree of freedom increases, it becomes difficult to obtain the transfer function.

3.9 Arbitrary Excitation The state-space equation makes it easy to define transfer functions, making it suitable for Matlab and Simulink programming. Equation (3.2.2) can be converted into the following state-space equation. z˙ = As z + Bs d y = Cs z + Ds d

(3.9.1)

 T where, z = xT x˙T ,    0 0 I As = , Bs = −M−1 K −M−1 C M−1 Bd 

(3.9.2a,b)

If we measure all the displacements, then y = x, then we may obtain   Cs = I 0 , Ds = 0 If we measure all the velocities, then y = x˙ , then we may obtain

(3.9.3a,b)

94

3 Vibration of Multi-degree-of-freedom System

  Cs = 0 I , Ds = 0

(3.9.4a,b)

If we measure all the accelerations, then y = x¨ , then we may obtain   Cs = −M−1 K −M−1 C , Ds = M−1 Bd

(3.9.5a,b)

The structure of the Cs and Ds matrices vary depending on the number of available sensors and the states to be measured as mentioned in Chap. 2. The following Matlab subprogram allows us to transform the second-order matrix ordinary differential equation into the first-order state-space equation. function [As, Bs, Cs, Ds ] = MK_to_SS(M,C,K,nF,nx,nxd,nxdd)

[n,~]=size(M); n2 = 2*n;

nf = 1; BF = eye(n2,1); if nF ~= 0 nf = length(nF); BF = zeros(n,nf); for i=1:nf BF(nF(i),i)=1; end end

As=[zeros(n) eye(n); -inv(M)*K -inv(M)*C]; Bs=[zeros(n,nf); inv(M)*BF];

Cs = []; Ds = []; Csx = []; if nx ~= 0

3.9 Arbitrary Excitation

95

ns = length(nx); Csx = zeros(ns,n); for i=1:ns Csx(i,nx(i))=1; end Cs = [ Csx zeros(ns,n) ]; Ds = zeros(ns,nf); end

Csxd = []; if nxd ~= 0 ns = length(nxd); Csxd = zeros(ns,n); for i=1:ns Csxd(i,nxd(i))=1; end Cs = [ Cs; zeros(ns,n) Csxd ]; Ds = [ Ds; zeros(ns,nf) ]; end

Csxdd = []; if nxdd ~= 0 ns = length(nxdd); Csxdd = zeros(ns,n); for i=1:ns Csxdd(i,nxdd(i))=1; end Cs = [ Cs; -Csxdd*inv(M)*K -Csxdd*inv(M)*C ]; Ds = [ Ds; Csxdd*inv(M)*BF ]; end

end

where nF represents the index matrix that are subjected to excitation force, and nx, nxd, and nxdd represent the index matrix containing the sensor locations for displacement, velocity, and acceleration. Example 3.7 For the MDOF system, suppose that pulse acts on the 3rd mass for 2 s and the accelerometers are attached to the first and fourth masses. The Matlab program to calculate responses is as follows (Fig. 3.9).

96

3 Vibration of Multi-degree-of-freedom System

Fig. 3.9 Responses calculated using state-space equation clear close all clc n = 5; M=eye(n); C = 3*eye(n); K=200*eye(n); for i=1:n-1 C(i,i+1) = -1.5; C(i+1,i) = -1.5; K(i,i+1) = -100; K(i+1,i) = -100; end

3.9 Arbitrary Excitation

97

ind_F = [ 3 ]; ind_sensor_x = 0; ind_sensor_xd = 0; ind_sensor_xdd = [ 1 4 ]; [As, Bs, Cs, Ds ] = MK_to_SS(M,C,K,ind_F,ind_sensor_x, ind_sensor_xd, ind_sensor_xdd) G=ss(As,Bs,Cs,Ds); % construct system transfer function

z0 = zeros(2*n,1);

% zero initial condition

dt = 0.01; tf = 10; t=0:dt:tf; % define time span F = (tn a , a pseudo-inverse can be used to calculate the control force. In this case, the required control force may not be accurately calculated. When designing a controller based on Eq. (5.2.10) and deriving a control algorithm, the input of the controller is modal displacement q. Therefore, an algorithm that extracts modal displacement from the sensor value is also required. The following equation is obtained from Eq. (5.2.5). qr = UrT Mx

(5.2.12)

From Eq. (5.2.12), it can be seen that in order to extract the modal displacement qr , all x must be measured first. Since it is impossible to measure all x, we need to reconstruct x using a small number of sensor signals. This is also not an easy task. Once the system model has been derived, a controller that can suppress vibration needs to be designed. The feedback control loop for vibration control can be simply constructed as shown in Fig. 5.4. The feedback controller algorithms used in active vibration suppression control can be thought of active damping control, which suppresses vibration around resonant frequencies (natural modes). In Fig. 5.4, D(s) is the disturbance, F(s) is the control force, G(s) is the transfer function of structure, H(s) is the compensator that implements the control algorithm, and X(s) is the displacement. If structure is modeled, then the transfer function, G(s), is determined, which can be later verified by experimental result. The disturbance, D(s), may or may not be known prior to control design. The system output, X(s), is assumed to be displacement but it can be velocity or acceleration depending on sensor. Our objective is to suppress vibration so that X(s) is to be suppressed. We will

154

5 Control Design

Fig. 5.4 Feedback loop for active vibration control

seek the appropriate compensator H(s) that can make X(s) to be as small as possible without causing the closed-loop system instability. Let us consider a SISO system first. For the open-loop system that implies the structure without control force, the transfer function can be written as X (s) = G(s) D(s)

(5.2.13)

In the case of the equation of motion given by Eq. (5.2.2), the open-loop transfer function for the SDOF system is as follows. G(s) =

ωn2 X¯ (s) = 2 ¯ s + 2ζ ωn s + ωn2 D(s)

(5.2.14)

As can be seen from Eq. (5.2.2), the SDOF system is a SISO system. Classical control method provides a way to design a feedback controller for this system transfer function. If we include feedback control, then the closed-loop transfer function becomes G(s) X (s) = D(s) 1 + G(s)H (s)

(5.2.15)

Large values of G(s)H(s) are required to reduce the effect of external disturbances in the frequency range where the disturbance has considerable effect. From Eq. (5.2.15), G(s)H(s)  1 entails that X(s) approaches zero. In general, a more intricate approach involving a mathematical model of the system is needed to achieve such an objective. At best, a mathematical model can only be a reduced order approximation of the actual system G(s). Therefore, the control bandwidth, ωc , and effectiveness of the control are restricted by the accuracy of the model. Unmodeled structural dynamics (residual modes) outside ωc may destabilize the system. Previously, we have seen that independent modal equations of motion can be obtained through several assumptions lacking in reality for the MIMO system. Since

5.2 Vibration Control of Structures

155

each of the modal equations of motion is a SISO system, one can apply the vibration control theory of the SDOF system. However, for practical application, it is desirable to start the controller design with the reduced-order modal equations of motion given by Eq. (5.2.6). For Eq. (5.2.6), it is not easy to find the open-loop transfer function using Laplace transformation. So, rather than using Eq. (5.2.6) directly, it is necessary to change it to the state-space equation which is mainly used in the latest control methodology. Previously, the state-space equation was introduced for the response analysis of a MDOF vibration system using Matlab, but in control theory, it was introduced for the control of a MDOF system. Converting Eq. (5.2.6) to state-space equation is as follows. p˙ r = Asr pr + Bsr f + dr

(5.2.16a)

y = Csr pr + Dsr f

(5.2.16b)

T  where pr = qrT q˙ rT , x is assumed to be measurable, and    0 0 0 I , Bsr = Asr = , dr = −r −2Zr r B¯ a r d¯ 

  Csr = Ur 0 , Dsr = [0]

(5.2.17a,b,c) (5.2.17d,e)

where 0 is the zero matrix of n r × n r and I is the identity matrix of n r × n r . Modern control theory was developed based on the state-space equation (5.2.16a, b). For a Linear Quadratic Regulator (LQR), the control force f is given as f = −Gr pr

(5.2.18)

in which Gr is the control gain matrix. Substituting Eq. (5.2.18) into Eq. (5.2.16a), the closed-loop system is expressed as p˙ r = (Asr − Bsr Gr )pr + dr

(5.2.19)

By examining the eigenvalues of the closed-loop system, the stability and the damping factor of each mode can be determined. The controller design process using the state-space equation may seem simple, but it is not.

156

5 Control Design

5.3 Direct Velocity Feedback Control for SDOF System Considering a SDOF vibration system can be thought a case of controlling only one natural vibration mode in a MDOF vibration system. The easiest control algorithm to increase the damping of a SDOF vibration system given by Eq. (5.2.2) is probably the following equation. 2ζa x˙ f¯ = − ωn

(5.3.1)

where ζa is the damping factor to be added to the structure. Substituting Eq. (5.3.1) into Eq. (5.2.2) gives the following equation. x¨ + 2(ζ + ζa )ωn x˙ + ωn2 x = ωn2 d¯

(5.3.2)

It can be seen from Eq. (5.3.2) that the damping value of the structure is instantly increased by introducing the controller given by Eq. (5.3.1), thus naturally reducing the resonant amplitude. The type of controller given by Eq. (5.3.1) is called Direct Velocity Feedback Control (DVFC) [Balas (1979)]. Let’s design the DVFC for the MIMO system. First, let’s assume that the number of actuators and the number of control target natural vibration modes are the same, that is, nr = n a and B¯ a is invertible. Then, for the decoupled modal equations of motion given by Eq. (5.2.11), MIMO DVFC can be expressed similarly to Eq. (5.3.1) as follows. Q = −2G−1 Za r−1 q˙

(5.3.3)

Inserting Eq. (5.3.3) into Eq. (5.2.6) results in the following equation. q¨ + 2(Zr + Za )r q˙ + r q = r d¯

(5.3.4)

As can be seen from Eq, (5.3.4), the damping factor of each mode can be increased as much as desired. Let’s check the stability for MIMO DVFC using the Lyapunov direct method. As the Lyapunov function for Eq. (5.3.4), the energy equation can be constructed as follows. E=

1 1 T q˙ q˙ + qT r q 2 2

(5.3.5)

The time derivative of E without disturbance is derived as follows.   E˙ = q˙ T (q¨ + r q) = q˙ T −2(Zr + Za )r q˙ = −2q˙ T (Zr + Za )r q˙ < 0 (5.3.6)

5.3 Direct Velocity Feedback Control for SDOF System

157

˙ So, it Since (Zr + Za )r is a positive definite matrix, Eq. (5.3.6) holds for all q. can be said that the closed-loop MIMO system by MIMO DVFC given by Eq. (5.3.3) is asymptotically stable. Theoretically, the MIMO DVFC can be as simple as SISO DFVC and can certainly increase damping, but it is not easy to apply in practice. Because we have to measure the velocity. In general, an accelerometer or displacement sensor is used for vibration measurement. When an accelerometer is used, the absolute acceleration of the part where the accelerometer is attached is measured. In theory, we can compute the velocity by integrating this acceleration signal. However, it is almost impossible to compute the velocity through integration of the actual acceleration signal. This is because the measured accelerometer signal inevitably contains bias and drift. In particular, when the signal containing the bias is integrated, the integration result is infinitely increasing positively or negatively over time. To remove this bias, a highpass filter (HPF) can be used. The use of HPF causes distortion of the low-frequency signal. Since the natural frequency of the structure is lower than that of the electric system, it is not desirable to use an HPF for vibration control. If the displacement of the structure is to be measured, then, theoretically, the displacement signal can be differentiated with respect to time to obtain the velocity. In the case of digital control, it can be obtained using the difference equation [x(t + t)−x(t)]/t, but the noise included in the displacement signal causes amplification of the high frequency band component. That is, when the signal containing noise is differentiated, the high-frequency signal becomes larger than the main signal, making it difficult to extract only the velocity signal component of the main signal. So, this simple form of derivative calculation is susceptible to noise. A low-pass filter (LPF) can be used to remove the noise component in the high-frequency band, but it is difficult to obtain an accurate velocity signal because the signal becomes distorted and phase delay may occur. Due to these problems, the DVFC is the easiest control algorithm that can be considered in the vibration control of structures, but it is not easy to apply in reality.

5.4 PID Control The proportional-integral-derivative (PID) control is the most popular controller used in industrial control systems. The PID control consists of a control force proportional to the error with a desired value, a control force proportional to the time change rate of the error, and a control force proportional to the time integration of the error. The PID control used in the industry is generally SISO control and has been used to control relatively simple systems. Since PID control is covered in almost all textbooks on control theory, this is not going to be covered in this book. However, there have been attempts to apply PID control to the active vibration control of structures, so the limitations of the PID control will be explained. When the PID control is used as an active vibration controller, the desired value is zero because a state without vibration is desired. So, if the displacement and velocity

158

5 Control Design

of SISO vibration system given by Eq. (5.2.2) are to be used for the active vibration control, then the PID controller can be expressed as follows. f¯ = −k p x − ki

xdτ − kd x˙

(5.4.1)

where k p , ki , kd are the gains of proportional, integral, and derivative controls. From Eq. (5.4.1), if k p = ki = 0, then the controller becomes D control. And it can be regarded as the DVFC introduced earlier. Let’s consider the case of ki = 0, that is, PD control. In this case, if Eq. (5.4.1) is inserted into Eq. (5.2.2), the equation of motion of the closed-loop system becomes as follows. x¨ + (2ζ ωn + kd ωn2 )x˙ + (ωn2 + k p ωn2 )x = ωn2 d¯

(5.4.2)

It can be seen from Eq. (5.4.2) that the PD control increases the damping of the structure like the DVFC and also increases the natural frequency of the closedloop system. Integral action is a term introduced in consideration of the case where it cannot converge to 0, but it can be said that it is not an action that has a great influence on active vibration control but tries to maintain the static equilibrium. When the PID controller given by Eq. (5.4.1) is applied to the active vibration control of a structure, it has the same problem as in the case of the DVFC. That is, in the case of measuring and using a displacement signal, the velocity term must be obtained through time-derivative and thus a high frequency band signal amplification problem due to noise occurs. And displacement integration causes a signal divergence problem due to bias included in the displacement signal. The problem becomes more serious when using acceleration signals. This is because problems that occur while integrating the acceleration signal are accumulated. It seems that the PID control design for the MIMO system is not achievable when considering application problems. In conclusion, the displacement measurement-based PID controller is not suitable for active vibration control of actual structures.

5.5 Positive Position Feedback Control The Positive Position Feedback (PPF) control was initially proposed by Fanson and Caughey (1990) and it has been used successfully for the active vibration of smart structure equipped with piezoelectric sensor and actuator. The PPF controller is an algorithm that calculates the control force required for vibration suppression when displacement or a signal proportional to the displacement can be measured. If a piezoelectric sensor is bonded to the surface of structure, then a strain signal proportional to the displacement can be measured. If a piezoelectric actuator is also bonded to the surface of structure, then the control force can be applied to structure. Unlike the PID controller, it does not require velocity, so there is no need for a differentiation process. The PPF controller is found to be very effective in suppressing

5.5 Positive Position Feedback Control

159

a specific vibration mode, thus maximizing the damping in a targeted frequency bandwidth without destabilizing other modes. The transfer function of the SISO PPF controller is equal to the one of a simple low-pass filter circuit, allowing it to be easily realized by either an operational amplifier or a microprocessor. For this reason, the PPF controller has been widely used as the active vibration controller for smart structures using piezoelectric sensors and actuators. However, the natural vibration characteristics should be known a priori either theoretically or experimentally in order to successfully apply the PPF controller. Let us study the SISO PPF controller. The SISO PPF controller was originally developed to cope with the SDOF vibration system. Introducing f¯ = g Q into Eq. (5.2.2) results in x¨ + 2ζ ωn x˙ + ωn2 x = gωn2 Q + ωn2 d¯

(5.5.1)

The PPF compensator equation for Eq. (5.5.1) is given by [Fanson and Caughey (1990)] Q¨ + 2ζ f ω f Q˙ + ω2f Q = gω2f x

(5.5.2)

where g is the positive scalar gain, Q is the control command, and ω f and ς f are the filter frequency and the damping factor of the PPF controller. The PPF compensator equation, Eq. (5.5.2) looks almost the same as the structural equation given by Eq. (5.5.1). The PPF terminology stems from the fact that the control command is calculated by using the displacement and positively fed back into the Eq. (5.5.1). The transfer function of the SISO PPF controller given by Eq. (5.5.2) can be expressed in terms of Laplace variable: gω2f Q(s) = 2 = H (s) X (s) s + 2ζ f ω f s + ω2f

(5.5.3)

which is in fact a second-order low pass filter, so that we can utilize an equivalent analog circuit easily (Kwak et al. 2004). Figure 5.5 shows the frequency response plot of the PPF compensator when ω f = 1 rad/s, ζ f = 0.1, g = 1. It can be readily understood from Fig. 5.5 that the cut-off frequency of the PPF controller has 90degree phase shift with slightly higher magnitude. This is why the PPF control can act as an active damping in the vicinity of the filter frequency thus needs fine-tuning. The magnitude of the frequency response plot shown in Fig. 5.5 indicates that the SISO PPF controller has less spillover to higher frequency modes as the magnitude rolls off above the cut-off frequency. Combining Eqs. (5.5.1) and (5.5.2) leads to matrix equation. 

x¨ Q¨





2ζ ωn 0 + 0 2ζ f ω f



x˙ Q˙





ωn2 −gωn2 + −gω2f ω2f



x Q



 =

ωn2 d¯ 0

(5.5.4)

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5 Control Design

Fig. 5.5 Bode plot of PPF compensator

Laplace transform of this matrix equation gives 

s 2 + 2ζ ωn s + ωn2 −gωn2 2 2 −gω f s + 2ζ f ω f s + ω2f



X¯ (s) ¯ Q(s)



 =

¯ ωn2 D(s) 0

(5.5.5)

Hence, the closed-loop transfer function between the disturbance and response can be derived as

 ωn2 s 2 + 2ζ f ω f s + ω2f X¯ (s) (5.5.6) = G c (s) = ¯ (s) D(s) where (s) = s 4 + 2(ζ ωn + ζ f ω f )s 3 + (ωn2 + ω2f + 4ζ ζ f ωn ω f )s 2 +2ωn ω f (ζ ω f + ζ f ωn )s + ωn2 ω2f (1 − g 2 )

(5.5.7)

It can be found that the absolute stability is guaranteed for 0 < g < 1 [Fanson and Caughey (1990)] by applying the Routh-Hurwitz criteria. Therefore, if the gain is less than 1, the closed-loop system is unconditionally stable. And its stability is static stability. That is, the stability depends only on the gain regardless of the frequency. Considering the structure of ζ = 0.01 and ωn = 1 rad/s and setting it to ω f = 1 rad/s, the PPF filter frequency is tuned to the natural frequency. The damping factor of the PPF controller is set as ζ f = 0.3. This damping factor is known to provides a relatively wide frequency band that can be controlled. Figure 5.6 shows

5.5 Positive Position Feedback Control

161

Fig. 5.6 Frequency Response Plots for Closed-Loop System with SISO PPF Controller

the results of comparing open-loop and closed-loop frequency response plots for various gain values. It can be seen from Fig. 5.6 that if the filter frequency, ω f of the SISO PPF controller, is tuned to ωn , then resonance amplitude decreases as the control gain increases but the response outside the natural frequency increases. This occurs because the stiffness of the PPF controller decreases due to the positive position feedback in the region lower than the filter frequency. In other words, if the gain is too large, static instability may occur. Let’s simplify the transfer function obtained earlier by introducing a dimensionless variable. For this, if s = iω, r = ω/ωn , p = ω f /ωn are introduced, the transfer function is then expressed as follows. G(ω) =

p 2 − r 2 + i(2ζ f pr ) W

(5.5.8)

where   W = r 4 − (1 + p 2 + 4ζ ζ f p)r 2 + p 2 (1 − g 2 )   + 2i −(ζ + ζ f p)r 2 + (ζ p 2 + ζ f p)r

(5.5.9)

If the PPF controller is tuned to the natural frequency, it amounts to p = 1, and if the resonance amplitude is of interest, it becomes r = 1. Therefore, the resonance amplitude can be derived as follows.

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5 Control Design

1 |G| = 2 ζ+

g2 4ζ f



(5.5.10)

Considering the relationship between the amplitude at resonance and the damping factor, the damping factor of the closed loop system by the PPF controller is expressed as follows. ζc = ζ +

g2 4ζ f

(5.5.11)

where ςc is the resultant damping ratio by the SISO PPF controller. If g 2 = 0.1, ς f = 0.3, the active damping factor of 8.3% can be added to the target natural frequency. This represents a reduction of about 20 dB on the board diagram. As mentioned earlier, the reason why ζ f = 0.3 is commonly used in PPF control is to expand the frequency band that can be controlled as much as possible because it is difficult to accurately tune the PPF filter frequency to the target natural frequency where the phase difference of the PPF controller is to be 90 degrees. The gain can be adjusted while checking the performance of the closed loop system, but if it is excessive, the control performance in the low frequency band will deteriorate. If the filter frequency of the PPF controller in Eq. (5.5.2) is less than the natural frequency, that is, ω f < ωn , the signal with a frequency higher than that of the PPF frequency is filtered out. If so, the calculated PPF control force becomes very small and does not affect the vibration characteristics of the SDOF system. If the filter frequency of the PPF controller is higher than the natural frequency, that is, ω f > ωn , the control force can be approximated from Eq. (5.5.2) as follows. Q  gx

(5.5.12)

Inserting Eq. (5.5.12) into Eq. (5.5.1) yields the following equation.   x¨ + 2ζ ωn x˙ + 1 − g 2 ωn2 x = ωn2 d¯

(5.5.13)

Therefore, it can be seen that the natural frequency of the natural vibration mode in the frequency band lower than the PPF filter frequency decreases as the gain of the PPF controller increases. Specifically, The SISO PPF controller may lead to spillover problem to lower frequency modes when the feedback gain is close to 1. As seen before, when the SISO PPF controller is applied to the SDOF system, the active damping effect for a specific frequency can be obtained like DVFC. Therefore, if the PPF controller is applied to a structure equipped with a piezoelectric sensor and a piezoelectric actuator, vibration can be suppressed by increasing the damping of the target natural frequency. Suppose that the MIMO system with multiple natural frequencies is equipped with a piezoelectric sensor and one piezoelectric actuator pair. The damping can be increased for only one natural vibration mode by applying the SISO PPF controller. In the case of structures, since the first natural vibration

5.5 Positive Position Feedback Control

163

mode has the greatest effect, it is reasonable to target the first natural frequency when using one pair of piezoelectric sensors and piezoelectric actuators. If there are multiple actuator and sensor pairs, multiple natural vibration modes can be controlled. The easiest way to think about it is to have each piezoelectric sensor and piezoelectric actuator pair control the assigned natural vibration mode. However, in this case, the control force is distributed and the control effect may be reduced. In control theory, such control is called decentralized control. Assuming that n r piezoelectric sensors and piezoelectric actuators are mounted and n r natural vibration modes are to be controlled and other natural vibration modes are to be ignored, the decoupled modal motion equation, Eq. (5.2.11) can be used. And the modal MIMO PPF control algorithm for Eq. (5.2.11) is expressed by the following equation. ˙ r + r Qr = Gr qr ¨ r + 2Z f r Q Q

(5.5.14)

As shown in Eq. (5.5.14), the modal MIMO PPF controller is tuned to the natural frequency of each modal coordinate system. In fact, modal equations of motion given by Eq. (5.2.11) are decoupled so that each modal equation can be treated as a SDOF system. The modal PPF controllers given by Eq. (5.5.14) are also decoupled so that they can be separately calculated. Equations (5.2.11) and (5.5.14) are the same form as SISO PPF control for SDOF system externally for each modal coordinate, but internally, sensor signal and actuator command are interconnected. After calculating the modal coordinate value using Eq. (5.2.12) with the piezoelectric sensor value for the physical coordinate and calculating the modal control force for each modal coordinate system using the PPF control algorithm, the modal control force conversion equation, Eq. (5.2.10) The control force is calculated and provided to the piezoelectric actuator. Since there is a coordinate conversion process, several piezoelectric sensors and piezoelectric actuators cooperate with each other, and through this process, control efficiency is improved. This control method is called centralized control. Let us discuss the design method and stability of the MIMO PPF controller. First, consider n r natural vibration modes, and consider the case of controlling using n r actuators and n r sensors. For now, let us ignore the higher-order modes and assume that each PPF controller is tuned to each natural mode. We can write the coupled structure-compensator equation in matrix form by combining Eqs. (5.2.11) and (5.5.14). 

q¨ r ¨r Q



 +

0 2Zr r 0 2Z f r



q˙ r ˙r Q



 +

r −Gr −Gr r



qr Qr



 =

r d¯ 0 (5.5.15)

The stability of such a system becomes stable if the next matrix is positive definite. In other words,

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5 Control Design

 stable if

r −Gr −Gr r

 >0

(5.5.16)

Therefore, the stability condition is derived as   I − G2 r2 > 0

(5.5.17)

From Eq. (5.5.17), the closed-loop system is stable when G2 < I. If this condition is satisfied, the system is unconditionally stable. However, this condition only applies when the number of natural modes to be controlled is the same as the number of actuators and sensors. For this control technique to be successful, it must be possible to calculate the modal coordinate system to be controlled by Eq. (5.2.10), and B¯ a must be invertible. Since the number of natural modes of a structure is infinite, even if a reduced-order model (ROM) is made, all the natural modes cannot be controlled by the finite number of sensor and actuator. Let us consider the case where the order of ROM is greater than the number of actuators. That is, n r > n a , where n a is the number of actuators. And suppose you control the same number of low-order natural vibration modes as the actuator. This means that low-order natural vibration modes are controlled but high-order natural vibration modes are left uncontrolled. If the vibration mode is divided into modes to be controlled and modes to be uncontrolled, qr can be divided as follows.   qc qr = (5.5.18) qu where qc is the control target modal coordinate of n c × 1 vector and qu is the uncontrolled modal coordinate of (n r − n c ) × 1 vector. And the subscripts c and u are introduced to distinguish the modes to be controlled and the modes to be uncontrolled, respectively. Using Eq. (5.5.18), the modal equation of motion, Eq. (5.2.6) can be divided as follows. q¨ c + 2Zc c q˙ c + c qc = B¯ ac f + c d¯

(5.5.19a)

q¨ u + 2Zu u q˙ u + u qu = B¯ au f + u d¯

(5.5.19b)

¯ ac is an n c × n a matrix. The modal PPF controller for Eq. (5.5.19a) can be where B written as ˙ c + c Qc = Gc qc ¨ c + 2Z f c Q Q

(5.5.20)

In order to decouple the equation of motion, Eq. (5.5.19a), the modal control force must be as follows.

5.5 Positive Position Feedback Control

165

¯ ac f = Gc Qc B

(5.5.21)

Fortunately, since the number of actuators and the number of modal control forces are assumed to be the same, the following relationship holds. −1 Gc Qc f = B¯ ac

(5.5.22)

Inserting Eq. (5.5.22) into Eq. (5.5.19a, b) yields the following equation. q¨ c + 2Zc c q˙ c + c qc = Gc Qc + c d¯

(5.5.23a)

−1 q¨ u + 2Zu u q˙ u + u qu = B¯ au B¯ ac Gc Qc + u d¯

(5.5.23b)

As shown in Eq. (5.5.23a, b), if a PPF controller is designed with the same number of actuators as the number of natural vibration modes to be controlled, control spillover occurs in higher-order modes to be uncontrolled. However, because the PPF controller itself is a low-pass filter, the PPF controller tuned to the low-order natural vibration mode does not have a frequency component that makes the high-order natural vibration mode unstable, so it can be regarded as Qc ≈ 0 in Eq. (5.5.23b). Therefore, the non-control target mode qu does not become unstable by the modal control force Qc . This is the advantage of the MIMO PPF controller using the modal coordinate system. Equations (5.5.20) and (5.5.23a, b) can be combined in matrix form as follows. ⎧ ⎫ ⎡ ⎤⎧ ⎫ 0 0 2Zc c ⎨ q˙ c ⎬ ⎨ q¨ c ⎬ ¨ c + ⎣ 0 2Z f c ˙ Q 0 ⎦ Q ⎩ c⎭ ⎩ ⎭ q¨ u q˙ u 0 0 2Zu u ⎫ ⎡ ⎤⎧ ⎫ ⎧ c −Gc 0 ⎨ qc ⎬ ⎨ c d¯ ⎬ + ⎣ −Gc (5.5.24) c 0 ⎦ Qc = 0 ⎩ ⎭ ⎩ ¯⎭ −1 qu 0 −B¯ au B¯ ac Gc u u d When using a piezoelectric sensor, the measured signal is proportional to q. If the number of piezoelectric sensors is the same as the number of piezoelectric actuators, the following equation can be derived. ¯ sd q = C ¯ sdc qc + C ¯ sdu qu vs = C

(5.5.25)

where vs is the voltage signal of the piezoelectric sensor. qc is required to calculate the MIMO PPF control force given by Eq. (5.5.20), and the following approximation can be used. ¯ −1 vs qˆ c = C sdc

(5.5.26)

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5 Control Design

where qˆ c is the approximate control target modal displacement calculated from the piezoelectric sensor signal. Substituting Eq. (5.5.25) into Eq. (5.5.26) yields the following equation. ¯ ¯ −1 C qˆ c = qc + C sdc sdu qu

(5.5.27)

And the modal PPF controller is modified as follows. ˙ c + c Qc = Gc qˆ c ¨ c + 2Z f c Q Q

(5.5.28)

Inserting Eq. (5.5.27) into Eq. (5.5.28) yields the following equation. ˙ c + c Qc = Gc qc + Gc C ¯ −1 C ¯ ¨ c + 2Z f c Q Q sdc sdu qu

(5.5.29)

Therefore, observer spillover problem occurs. Fortunately, since the PPF control algorithm is the kind of LPF, it is not significantly affected by natural modes other than the tuned natural mode. The influence of the observer spillover problem by the higher-order mode is not large. Assuming that the natural vibration mode is controlled by the number of piezoelectric sensor and piezoelectric actuator pairs, the modal MIMO PPF control calculation algorithm can be rewritten as follows by combining Eqs. (5.5.20), (5.5.22), and (5.5.26). −1 ¯ −1 V(s) H pp f (s) C F(s) = B¯ ac sdc

(5.5.30)

where  H pp f (s) = diag

gi2 ω4f i s 2 + 2ζ f i ω f i s + ω2f i

(5.5.31)

However, if we try to control more natural vibration modes than the number of actuators, a problem arises. This problem amounts to the case of controlling a large number of natural vibration modes with a small number of control force. If the number of natural vibration modes to be controlled is n c , it is n c > n a . In this case, qc in Eq. (5.5.19a) becomes an n c × 1 vector and qu in Eq. (5.5.19b) becomes an (n r − n c ) × 1 vector. When this happens, B¯ ac in Eq. (5.5.19a) becomes an n c × n a matrix and is no longer a square matrix. Therefore, it becomes impossible to derive Eq. (5.5.22). The only way is to use a pseudo-inverse [Kwak et al. (2004)]. In this case, Eq. (5.5.21) is expressed as † Gc Qc f = B¯ ac

(5.5.32)

  T † T ¯ ac −1 B¯ ac where B¯ ac B = B¯ ac . Inserting Eq. (5.5.32) into Eq. (5.5.23a, b) yields the following equation.

5.5 Positive Position Feedback Control

167

† q¨ c + 2Zc c q˙ c + c qc = B¯ ac B¯ ac Gc Qc + c d¯

(5.5.33a)

† q¨ u + 2Zu u q˙ u + u qu = B¯ au B¯ ac Gc Qc + u d¯

(5.5.33b)

First, it can be seen from Eq. (5.5.33b) that the control spill-over problem to the higher-order mode to be uncontrolled is not our concern because the MIMO PPF controllers given by Eq. (5.5.32) operate independently of each other, so interference † does not become an identity matrix in the in other modes is small. Rather, since B¯ a B¯ ac motion equations for the controlled modes, a problem arises that the modal control force is spilled over to another controlled modal displacement. Therefore, stability is not completely guaranteed.

5.6 Higher Harmonic Control Most of control techniques for vibration suppression are interested in increasing the damping for the natural mode to be controlled, so they are sometimes called an active damping. However, the active damping lost its efficacy in coping with a persistent harmonic disturbance having a non-resonant excitation frequency. Hence, the suppression of the response due to a persistent disturbance has been of consistent interest. Let us introduce the following closed-loop transfer function again. G X = D 1 + GH

(5.6.1)

It was explained earlier that If G H is large, the effect of disturbance on the sensor output can be minimized. As a control algorithm capable of responding to harmonic disturbances, there is a disturbance accommodating control called Higher Harmonic Control (HHC). HHC was originally developed for vibration control of helicopter rotor blades (Shaw and Albion 1981). In order to apply the HHC, it is necessary to go through a little complicated calculation process, such as performing time integration with the sine function and cosine function. However, Lisa and Andreas (1992) proved that HHC can be expressed as a transfer function in the form of a band-rejection filter. Therefore, it is possible to apply the HHC algorithm more easily. The HHC can be represented by the following transfer function. H(s) = −

2 as + bω T s 2 + ω2

(5.6.2)

where a and b are the real and imaginary parts of the transfer function, i.e., a = Re[G(iω)], b = Im[G(iω)] and T is the period of the external harmonic disturbance. The HHC is in fact the band rejection filter (Lisa and Andreas 1992) so the effect of the harmonic disturbance on the structure can be cancelled by the HHC in theory. For

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5 Control Design

the control law represented by Eq. (5.6.2) to be more effective, the transfer function between the sensor and actuator should be accurately known either theoretically or experimentally prior to the application of the HHC, i.e. a and b need to be accurately determined. Yang et al. (2010) found that the beat phenomenon may appear instead of vibration suppression if the excitation frequency is slightly different from the HHC filter frequency. To overcome the beat problem, Yang et al. (2010) proposed a Modified Higher Harmonic Control (MHHC) heuristically by incorporating damping factor into the existing HHC to cope with harmonic disturbances in a wider frequency band. The transfer function of the MHHC is as follows H(s) = −

gω (as + bω) + 2ζ f ωs + ω2

s2

(5.6.3)

where ζ f is introduced to widen the control bandwidth and g is the control gain. The introduction of the filter damping ratio ζ f may deteriorate the control performance but it can increase the robustness. This implies that the control algorithm given by Eq. (5.6.3) is in fact a band-rejection filter with a certain amount of robustness around the filter frequency. Yang et al. (2010) applied the MHHC to active vibration control of a cantilever beam and showed that MHHC is effective for both resonance and non-resonant excitation. Figure 5.7 shows the feedback control loop for MHHC. At the excitation frequency, we have G(iω) = a + ib, H(iω) =

g(a − ib) 2ζ f

(5.6.4a,b)

Hence, the closed-loop transfer function at the excitation frequency becomes Gc (iω) =

Fig. 5.7 Feedback control loop for MHHC

1 1+

g (a 2 +b2 ) 2ζ f

(5.6.5)

5.6 Higher Harmonic Control

169

It can be readily seen from Eq. (5.6.5) that the magnitude of the closed-loop transfer function becomes zero if the filter damping is zero, which implies exact tuning. The control performance at the excitation frequency also depends on the gain, which may affect the stability. Let us investigate the stability and performance of the MHHC by considering a SDOF vibration system described by the following equation. G(s) =

ωn2 s 2 + 2ζ ωn s + ωn2

(5.6.6)

We can obtain the closed-loop transfer function by inserting Eqs. (5.6.3) and (5.6.6) into Eq. (5.6.1). Then, the stability of the closed-loop transfer function is determined by the following denominator:    den(s) = s 2 + 2ζ f ωs + ω2 s 2 + 2ζ ωn s + ωn2 − gωωn2 (as + bω)

(5.6.7)

The gain margin can be obtained by analyzing Eq. (5.6.7). The real and imaginary parts of the transfer function are obtained by inserting s = iω.  1 − r2 −2ζ r  , b =   a =  2 2 2 1 − r 2 + (2ζ r ) 1 − r 2 + (2ζ r )2 

(5.6.8a,b)

The non-dimensionalized frequency variable, r = ω/ωn , is introduced. Then, the closed-loop transfer function is expressed as Gc (r ) =

1 1+

 2ζ f

g

(1−r 2 ) +(2ζ r )2 2



(5.6.9)

Figure 5.8 shows the gain margin when ζ = 0.01 and ζ f = 0.005, 0.05. As shown in Fig. 5.8, the gain margin for the lightly damped system is not affected by ζ f except in the resonance region. However, the closed-loop gain is greatly affected by ζ f as shown in Fig. 5.9. As shown in Fig. 5.9, the performance of the HHC is degraded as the filter damping increases.

5.7 Virtual Tuned Mass Damper Control In the previous section, it was explained that the SISO PPF controller is an algorithm that calculates the control force required for vibration suppression when the measurement of displacement or a signal proportional to the displacement is possible. Piezoelectric sensors that can be attached to the surface of a structure generate a signal proportional to the strain when used with a charge amplifier. Since the displacement

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5 Control Design

Fig. 5.8 Maximum gain for MHHC

Fig. 5.9 Closed-loop gain for one DOF model by MHHC

is calculated from this signal, the PPF control algorithm can be applied. However, the sensor mainly used in vibration measurement and control is an accelerometer. This sensor measures the absolute acceleration of the attached position. In this case, the

5.7 Virtual Tuned Mass Damper Control

171

Fig. 5.10 Two DOF system

controller is to calculate the necessary control force based on the measured acceleration. As seen earlier, most control algorithms are composed of absolute displacement and absolute velocity. We normally use the Kalman filtering to estimate absolute displacement and absolute velocity using the measured acceleration because they cannot be measured directly. Hence, a control algorithm using the acceleration signal directly is preferred in practical applications. Let us consider an auxiliary spring-mass-damper system attached to a primary spring-mass-damper system as shown in Fig. 5.10, where m, c, k are the mass, damping, and spring constant of the primary system; m a , ca , ka are the mass, damping, and spring constant of the auxiliary system; and x, xa are the displacements of the primary system and the auxiliary system, respectively. d is the external disturbance that is acting on the primary system. The equations of motion can be expressed as m x¨ + (c + ca )x˙ − ca x˙a + (k + ka )x − ka xa = d

(5.7.1a)

m a x¨a − ca x˙ + ca x˙a − ka x + ka xa = 0

(5.7.1b)

It is well known that vibration of the primary system can be absorbed by the auxiliary system if the natural frequency of the auxiliary system is tuned to the excitation frequency of the external harmonic disturbance (Inman 2017). The auxiliary system made for this objective is called a dynamic absorber (DA) or a Tuned Mass Damper (TMD). The traditional passive TMD has limited performance due to fixed damper parameters, a narrow suppression frequency range, ineffective reduction of non-stationary vibration, and a sensitivity problem because of detuning. Let us consider again a SDOF system with an actuator as shown in Fig. 5.11, where f is the control force. The equation of motion is simply expressed as m x¨ + c x˙ + kx = f + d

(5.7.2)

Comparing Eq. (5.7.2) with Eq. (5.7.1a), the control force equivalent to the force generated by the TMD can be derived as

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5 Control Design

Fig. 5.11 A spring-mass-damper system with an actuator

f = −ca x˙ + ca x˙a − ka x + ka xa

(5.7.3)

Using Eq. (5.7.1b), Eq. (5.7.3) can be re-written as f = −m a x¨a

(5.7.4)

Equation (5.7.4) implies that the force made by the TMD is a counter-acting inertial force. If we apply the control force given by either Eq. (5.7.3) or Eq. (5.7.4), then the actuating force can have the same effect on the system as the mechanical TMD does. Such a force can be thought of as the force generated by a virtual tuned mass damper (VTMD) for the system of Fig. 5.11. Both Eqs. (5.7.3) and (5.7.4) are simple control algorithms, but they still require the displacement, velocity, and acceleration of the VTMD, which cannot be measured in the real world, because the VTMD exists only in a virtual world. However, it can be observed in Eq. (5.7.1b) that the displacement of the VTMD is related to the displacement of the primary system. Applying the Laplace transform to Eq. (5.7.1b), the transfer function between the displacement of the primary system and the displacement of the VTMD can be obtained as ca s + ka X a (s) = X (s) m a s 2 + ca s + ka

(5.7.5)

Using Eqs. (5.7.4) and (5.7.5), the control force amount to the inertial force caused by the VTMD can be rewritten as F(s) m a (2ζa ωa s + ωa2 ) =− 2 s + 2ζa ωa s + ωa2 X¨ (s)

(5.7.6)

where ζa , ωa are the damping factor and the natural frequency of the VTMD. Equation (5.7.6) represents a new control algorithm emulating the VTMD [Shin et al. (2019)]. It should be noted that this new control algorithm given by Eq. (5.7.6) uses only the acceleration of the primary system, and only three parameters need to be tuned. Hence, this new VTMD control algorithm is practical and simple, because it uses only the acceleration of the primary system. The m a in Eq. (5.7.6) can be regarded

5.7 Virtual Tuned Mass Damper Control

173

as a gain of the VTMD controller, ζa determines the control frequency bandwidth, and ωa is the filter frequency of the VTMD. ωa needs to be tuned to the excitation frequency of the external harmonic disturbance or the main frequency component of the disturbance. It can be readily expected that the effect of the harmonic disturbance on the motion of the primary system will be mitigated if the control force calculated by the VTMD control algorithm is applied to the primary structure, just as with the passive TMD. Figure 5.12 shows the transfer functions of the VTMD control algorithm given by Eq. (5.7.6) versus the damping factor when m a = 1kg, ωa = 1rad/s. If the damping factor is small, then the control effectiveness becomes larger, as with the passive TMD. However, the precise tuning is necessary because the control bandwidth becomes small. Of course, large damping deteriorates the control effectiveness. Equation (5.7.6) may also be considered as a notch filter just like MHHC. The VTMD control given by Eq. (5.7.6) is similar to the MHHC control given by Eq. (5.6.3) but basically different because the VTMD control uses acceleration signal instead of displacement and it does not need the transfer function of the main structure. The closed-loop system using the VTMD controller is inherently stable because the original system with the TMD given by Eq. (5.7.1a, b) is mechanically stable. In summary, the VTMD control algorithm given by Eq. (5.7.6) has the following advantages: • It is absolutely stable, because a structure with a TMD is mechanically stable. • It can accommodate resonant and non-resonant harmonic disturbances. The only necessary information is the excitation frequency. • It can provide active damping if it is tuned to the natural frequency. Resonance amplitude can be greatly suppressed. • Control parameters do not depend on the dynamic model of the target structure. • It does not need sensors other than an accelerometer. It also does not need additional computation for the estimation of the velocity and displacement, because it uses only the accelerometer signal. • Its gain is equivalent to the mass of the VTMD so that the gain can be easily calculated. • It can be easily implemented by either a digital controller or an analog circuit. • It is suitable for a vibration isolation system. Let us design a MIMO VTMD controller for a MIMO system. Using the modal equations of motion and the design process of the MIMO PPF control, the MIMO VTMD control can be expressed as the following equation. −1 ¯ −1 ¨ (s) F(s) = B¯ ac Hvtmd (s) C sac x

where

(5.7.7)

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5 Control Design

(a) Magnitude

(b) Phase Fig. 5.12 Transfer function of the VTMD controller

5.7 Virtual Tuned Mass Damper Control

175

⎤ gi 2ζ f i ω f i s + ω2f i ⎦ Hvtmd (s) = diag⎣ 2 s + 2ζ f i ω f i s + ω2f i ⎡

(5.7.8)

The MIMO VTMD control can be applied to structure as the MIMO PPF control can.

5.8 Active Mass Damper and Negative Acceleration Feedback Control Active Mas Damper (AMD) is a device that generates inertia force by moving a heavy mass, but it is not an actuator that can generate direct force. To move the actual mass, an AC servo motor combined with a ball-screw mechanism or a linear servo motor can be used. Many researchers have conducted research and proposed new control algorithms without understanding such motors and mechanisms. As a result, it was difficult to apply such a control algorithm to an actual AMD system. Let us study how to control and drive the servo motor first. The basic operation of a servo motor is to accurately follow a given displacement or speed or torque, and this tracking control is possible because the desired PID control algorithm is already installed inside the servo motor driver. The commands that can be input to most servo motor drivers are the desired position, speed, and torque values. When the desired position is used as an input, the servomotor moves to that position accurately. The speed mode makes the motor rotate at the desired speed. The torque mode likewise controls the servomotor to maintain a given torque. Among these, displacement mode and torque mode are controlled regardless of time. For this reason, it can be said that it is not suitable for active vibration control that must be performed in real time. Yang et al. (2017) proposed the real-time position tracking control of the moving mass using the speed control mode of AC servomotor driver and additional feedback control. Yang et al. (2017) also proposed an acceleration-based control algorithm for AMD control, which was called Negative Acceleration Feedback (NAF) control. To illustrate the AMD and NAF control algorithms, let’s use a SDOF model equipped with the AMD as shown in Fig. 5.13. As explained earlier, it is assumed that the displacement of the actuator such as a ball-screw mechanism or linear motor Fig. 5.13 Schematic diagram for single-degree-of-freedom

176

5 Control Design

is used to accurately control the position of the movable mass, which means the displacement of the moving needs to be the control output. In Fig. 5.13, m, c, k are the mass, damping, and spring constant of the primary system and m a is the mass of the movable mass. x is the displacement of the primary mass and u a is the relative displacement of the additional mass with respect to the primary mass. It should be noted that the relative displacement of the moving mass, u a is going to be controlled precisely by the servo motor. d¯ is the external disturbance acting on the primary mass. The equation of motion for this system can be written as follows: (m + m a )x¨ + c x˙ + kx = −m a u¨ a + d¯

(5.8.1)

Dividing the equation of motion by m + m a and introducing non-dimensional constants and variables, Eq. (5.8.1) can be rewritten as x¨ + 2ζ ωn x˙ + ωn2 x = −g u¨ + ωn2 d

(5.8.2)

where g is the gain for the control.  ωn =

k c ma d¯ ua , d = (5.8.3a–d) , ζ = , u= m + ma 2(m + m a )ωn g(m + m a ) k

The Negative Acceleration Feedback (NAF) controller for Eq. (5.8.2) is expressed by the following equation like the PPF control. u¨ + 2ζc ωn u˙ + ωn2 u = −g x¨

(5.8.4)

where ζc is the damping factor for the NAF controller. As mentioned above, an accelerometer is a popular sensor for vibration measurement in practice. Hence, the implementation of the NAF control algorithm given by Eq. (5.8.4) can be realized by the following simple transfer function if the accelerometer is used as a sensor. U (s) g =− 2 s + 2ζc ωn s + ωn2 X¨ (s)

(5.8.5)

where U (s) and X¨ (s) are the Laplace transforms of u(t) and x(t). ¨ Equation (5.8.5) implies that the desired position for the movable mass can be determined by the acceleration of the primary mass. The transfer function of the control law is in fact a low-pass filter as in the case of the PPF control that has been used in the vibration suppression of structures using piezoelectric sensors and actuators. The PPF control implies that it utilizes the position sensor and positive feedback loop, and its output is the desired force. However, the NAF controller proposed by Yang et al. (2017) negatively feedback the desired displacement of the active mass as its output using acceleration as input. The bode plot for the transfer function given by

5.8 Active Mass Damper and Negative Acceleration Feedback Control

177

Eq. (5.8.5) is shown in Fig. 5.14. g = 0.3, ωn = 1 rad/s are used. It can be seen from Fig. 5.14 that the NAF control has peak magnitude at the filter frequency and 90-degree phase shift similar to the PPF control. Let us investigate the stability of the NAF proposed in this study. By applying the Laplace transform to Eqs. (5.8.2) and (5.8.4), we can obtain (s 2 + 2ζ ωn s + ωn2 )X + gs 2 U = D

(5.8.6a, b)

gs 2 X + (s 2 + 2ζc ωn s + ωn2 )U = 0 In matrix form, we can obtain 

gs 2 s 2 + 2ζ ωn s + ωn2 2 2 gs s + 2ζc ωn s + ωn2

(a) Magnitude

(b) Phase Fig. 5.14 Bode plot for NAF control



X U



 =

D 0

(5.8.7)

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5 Control Design

Fig. 5.15 Frequency response plot for the SDOF system

Therefore, the transfer function between the disturbance and the displacement of the primary mass can be derived as (s 2 + 2ζc ωn s + ωn2 )ωn2 X = D (1 − g 2 )s 4 + 2(ζ + ζc )ωn s 3 + 2(1 + 2ζ ζc )ωn2 s 2 + 2(ζ + ζc )ωn3 s + ωn4 (5.8.8) By applying the Routh-Hurwitz criteria to the denominator, the following stability condition is obtained. stable if 0 < g < 1

(5.8.9)

It can be readily seen that the above stability condition is static, which implies that the stability of the closed-loop system does not depend on the frequency as in the case of the PPF control. Let us investigate the effect of the NAF control on the resonant amplitude. By applying s = iωn to Eq. (5.8.8) and deriving the magnitude, we can obtain   X 1    = (5.8.10) D g2 s=iωn 2 ζ + 4ζ c

5.8 Active Mass Damper and Negative Acceleration Feedback Control

179

It can be readily seen from Eq. (5.8.10) that the active damping can be achieved by choosing a proper gain and damping factor for the controller. Like the PPF control, it seems practical to use 0.3 for ζc . If we use 0.3 for ζc and g 2 = 0.1, then the active damping increases by 0.083. Figure 8.15 shows the frequency response plot for the SDOF system. ωn = 1 rad/s, ζ = 0.01, ζc = 0.3 are used for this figure. Unlike the PPF control, static instability does not appear as the gain increases. Let us consider the case that many AMDs are installed on the MIMO vibration system, then equations of motion for this system including disturbance can be written as M x¨ + CP x + Kx = −Ba u¨ a + Bd d

(5.8.11)

If we apply the modal transformation, x = Ur qr , into Eq. (5.8.11) and use the orthonormality condition along with the modal damping assumption, then we can obtain the reduced-order modal equations of motion. q¨ r + 2Zr r q˙ r + r qr = −B¯ a u¨ a + r d¯

(5.8.12)

There are always more natural modes than available sensors as in the case of previous PPF and VTMD controls even though Eq. (5.8.12) is a reduced-order model. It is also not desirable to control all modes with only a few AMDs. If we divide the generalized displacement vector into generalized displacements to be controlled by the AMD and the remaining displacements, such that qr = [qcT quT ]T , then equations of motion for the modes to be controlled can be rewritten as q¨ c + 2Zc c q˙ c + c qc = −B¯ ac u¨ a + c d¯

(5.8.13)

The size of the modes to be controlled is the same as the number of AMDs ¯ ac becomes a square matrix. It is also assumed that B¯ ac is invertavailable, so that B ible. Then, the multi-input multi-output (MIMO) modal-space negative acceleration feedback (NAF) control can be obtained by the following relation B¯ ac ua = GQ

(5.8.14)

where Q is an n a × 1 modal control force vector, and G is an n a × n a diagonal matrix whose diagonal is the gain for each mode. Hence, the desired displacement of AMD can be obtained by the following: −1 GQ ua = B¯ ac

(5.8.15)

As in the case of SISO NAF controller, we are going to use accelerometer signal. Hence, we have sensor equation as follows. ¯ −1 q¨ c = C sac vs

(5.8.16)

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5 Control Design

¯ sac should be invertible, meaning that sensors should not be placed Of course, C at the node of natural modes of interest. Inserting Eq. (5.8.14) into Eq. (5.8.12), we can obtain the following: ¨ + c d¯ q¨ c + 2Zc c q˙ c + c qc = −GQ

(5.8.17)

The MIMO modal-space NAF control for Eq. (5.8.17) can then be designed as ˙ + c Q = −Gq¨ c ¨ + 2Z f c Q Q

(5.8.18)

where Z f is the damping matrix for NAF control. Combining Eqs. (5.8.17) and (5.8.18), we can obtain the following: 

I G G I



q¨ c ¨ Q





0 2Zc c + 0 2Z f c



q˙ c ˙ Q





c 0 + 0 c



qc Q



 =

c d¯ c 0 (5.8.19)

Hence, the stability condition can be obtained as stable if G2 < I

(5.8.20)

The stability condition, Eq. (5.8.20), is static, meaning that the stability does not depend on frequency. Equation (5.8.20) also implies that the stability is guaranteed if the gain matrix is small enough. By using Eqs. (5.8.14), (5.8.16), and (5.8.18), the MIMO modal-space NAF controller using the acceleration measurement can be written as  −1 −1 −1 2 2 ¯ sac Va (s) G s I + 2sZ f c + c C Ua (s) = −B¯ ac

(5.8.21)

However, control and observer spill-over problems can occur because we take only parts of B¯ a . Yang et al. (2010) showed that uncontrolled modes remain stable if the gain is kept small enough.

5.9 Optimal Control Modern control techniques are based on first-order matrix differential equations called state-space equation. Previously, we studied a method of reducing the equations of motion having n degrees of freedom into a reduced modal equation of motion of n r degrees of freedom using modal transformation. And we also studied a method of converting the second-order matrix ordinary differential equation having n r degrees of freedom into a linear differential equation of 2n r degrees of freedom. The state-space equation derived in this way is as follows.

5.9 Optimal Control

181

p˙ r = Asr pr + Bsr f + dr

(5.9.1a)

y = Csr pr + Dsr f

(5.9.1b)

T  where pr = qrT q˙ rT    0 0 0 I , Bsr = Asr = , dr = −r −2Zr r B¯ a r d¯ 

  Csr = Ur 0 , Dsr = [0]

(5.9.2a,b,c) (5.9.2d,e)

Csr was constituted assuming that qr is measurable, b,ut,it may,change depending on the type of sensor and the installation location. Let us consider the following performance index for the state-space equation given by Eq. (5.9.1a, b). J=

1 2

0

∞

 T  pr Qs pr + f T Rs f dt

(5.9.3)

where Qs , Rs are the weighting matrices. The optimal controller called the Linear Quadratic Regulator (LQR) that is designed based on the performance index given by Eq. (5.9.3) is expressed as follows: f = −Rs−1 BTs Ps pr = −Gs pr

(5.9.4)

where Ps is the solution of the steady-state matrix algebraic Riccati equation. ATs Ps + Ps As − Ps Bs Rs−1 BTs Ps + Qs = 0

(5.9.5)

Inserting Eq. (5.9.4) into Eq. (5.9.1a), the closed-loop system is expressed as follows. p˙ r = (As − Bs Gs )pr + dr

(5.9.6)

Equation (5.9.4) implies that the control force is not calculated by multiplying the measured value y by the gain matrix, but by multiplying all states, pr , by the gain matrix. It can be seen from Eq. (5.9.4) that this is possible only when all modal displacements and modal velocities of the vibrating structure are measurable. Therefore, it is impossible to apply the optimum controller directly to the actual problem. The most commonly used method to solve this problem is the Linear Quadratic Gaussian (LQG) control method. In the LQG control, Eq. (5.9.1) are transformed as follows by adding noise.

182

5 Control Design

p˙ r = Asr pr + Bsr f + dr + vr

(5.9.7a)

y = Csr pr + Dsr f + wr

(5.9.7b)

where vr is white Gaussian system noise and wr is white Gaussian measurement noise. In the LQG controller, the control force is expressed as follows. f = −Gs pˆ r

(5.9.8)

where Gs is the gain matrix obtained by solving the matrix Riccati equation and Eq. (5.9.5) in LQR, and pˆ r is the estimated state vector, not the actual state vector. Estimated state vector is obtained through the following equation.   p˙ˆ r = As pˆ r + Bs f + Ls y − Cs zˆ

(5.9.9)

Ls = Ss CTs Wr−1

(5.9.10)

where

And Ss is obtained from the following matrix Riccati equation. ATs Ss + Ss As − Ss CTs Wr−1 Cs Ss + Vr = 0

(5.9.11)

where two intensity matrices, Vr , Wr is a matrix related to white Gaussian noises, vr , wr . Therefore, in order to apply the LQG controller, Qs , Rs and Vr , Wr must be determined in addition to the system matrix, As , Bs , Cs . In the vibration control of the structure, the system matrix, As , Bs , is determined from dynamic modeling, and Cs is determined by the sensor type and attachment location. Remember that the Qs , Rs and Vr , Wr matrices are the matrices that the controller designer will determine arbitrarily. LQG/LQR controller calculates the estimated state vector using a sensor and multiplies this value by the gain matrix and then obtain the required control force. However, compared to the previous PPF, VTMD, NAF controllers, the calculation algorithm is complex. Note that the most essential information for designing the PPF controller is the natural frequency of the natural mode to be controlled. On the other hand, in order to design the LQG/LQR controller, we need a relatively accurate dynamic model including the control force and also need weighting matrices and noise matrices. In addition, the amount of calculation is much larger in calculating the control force compared to the control algorithms such as PPF control and VTMD control, so it is a burden to program a digital control. If the number of states is small, it is alright, but if the number of states increases, the sampling rate may need to be lowered due to the amount of calculation, which deteriorates the control performance.

5.10 Filtered-X LMS

183

5.10 Filtered-X LMS Feedforward control methods can significantly improve performance over simple feedback control whenever control designer has direct access to information about the disturbance causing vibrations. For example, in rotating machinery, vibration is typically associated with the angular position, velocity, and acceleration. A tachometer or encoder signal can be used in implementing feedforward adaptive filtering for disturbance rejection. The original development of the technique was intended to cancel noise actively. But this technique can be also used to cope with vibrations caused by external disturbance. The scheme of feedforward control is shown in Fig. 5.2. In the most ideal situation, feedforward control can completely eliminate the effect of the measured disturbance on the system. An adaptive filter manipulates the signal that is correlated to the primary disturbance and the output is applied to the system by the actuator. The filter coefficients are adapted in such a way that an error signal at one or several critical points is minimized. The idea is to generate a secondary disturbance, which destructively interferes with the effect of the primary disturbance at the location of the error sensor. Unfortunately, there is no guarantee that the global response is also reduced at other locations, which represents a drawback to the method. If the structural response is not dominated by a single mode, the actuator may reduce the vibration at the point of error measurements but introduce higher levels of vibration in other parts of the structure. Thus, the method is considered to be a local technique, in contrast to feedback which is global. The classical adaptive active control model is a feed-forward control model based on the filtered-X least mean square (FXLMS) algorithm. The FXLMS algorithm requires a reference signal filtered in the secondary path as the feedforward input to the controller. The secondary path is the vibration transfer path extending from the controller output to the error sensor. The flow chart when applying the adaptive feedforward control algorithm to a structure is drawn in more detail as shown in Fig. 5.16. Fig. In 5.16, it is assumed that disturbance and control force were acting on the same structure, and each response is generated and combined through different paths. In the feedforward control algorithm, the control force is expressed as follows using disturbance information. f k = wkT dk = dkT wk

(5.10.1)

where f k is the control force, wk ≡ [wk1 wk2 . . . wkm ] is a filter composed of m coefT  ficients, and dk = dk dk−1 · · · dk−m+1 is a vector composed of m disturbance data in discrete time. Responses due to disturbance and control force are expressed using impulse responses as follows. xdk = hdT dk , x f k = hTf fk

(5.10.2a,b)

184

5 Control Design

Fig. 5.16 Adaptive feedforward control for vibration suppression

Fig. 5.17 Finite impulse response

T  where fk = f k f k−1 · · · f k−m+1 and hd , h f are the transfer function of the structure to disturbance and control force given by the finite impulse response (FIR) of G d (z), G f (z). The control force vector, fk , is then expressed as following equation.

5.10 Filtered-X LMS

fk =

185

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎫ ⎪ ⎪ ⎪ ⎬

wkT dk T wk−1 dk−1 .. . T dk−m+1 wk−m+1

⎪ ⎪ ⎪ ⎭

=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

dkT wk T dk−1 wk−1 .. . T wk−m+1 dk−m+1

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(5.10.3)

The final response is the sum of two responses, which is expressed as follows. xk = xdk + x f k

(5.10.4)

The main purpose of feedforward vibration control is to minimize this final response. Therefore, the index we will minimize is as follows. Jk = xk2

(5.10.5)

The FXLMS algorithm updates the filter coefficient of the control force equation using the gradient decent. wk+1 = wk − μ

∂ xk ∂ Jk = wk − 2μxk ∂wk ∂wk

(5.10.6)

where μ is the constant that determines the update rate. The partial derivate can be derive as follows. 

  T ∂ h f f k ∂ xdk + x f k ∂x f k ∂ xk = = = (5.10.7) ∂wk ∂wk ∂wk ∂wk The assumption used here is that the response by the disturbance is not changed by the control force. As can be seen from the Eq. (5.10.3), the control force vector contains the filter weight of the past. If we assume that the weight changes gradually during the considered time frame, we can write wk = wk−1 = · · · = wk−m+1 Then the control force vector can be rewritten as ⎫ ⎡ ⎧ ⎤ dkT dkT wk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dT wk ⎬ ⎢ dT ⎥ k−1 ⎢ k−1 ⎥ = ⎢ . ⎥wk = DTk wk fk = . .. ⎪ ⎪ ⎣ .. ⎦ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ T T dk−m+1 wk dk−m+1 Therefore,

(5.10.8)

(5.10.9)

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5 Control Design

 ∂ hTf DTk wk ∂ xk = = Dk h f ∂wk ∂wk

(5.10.10)

Inserting Eq. (5.10.10) into Eq. (5.10.6), the weight update algorithm can be derived as follows. wk+1 = wk − 2μxk Dk h f

(5.10.11)

As Eqs. (5.10.1) and (5.10.2) show, in order to use the weight update algorithm, information on the transfer function between the control force and the sensor measurement, and the disturbance is necessary. The transfer function of the structure for control can be obtained through sweeping test or random input test, but information on disturbance itself cannot be not easily obtained through vibration measurement. In real applications, the FXLMS control uses filtered disturbance vector dˆ k instead of original disturbance vector dk , which will not be covered in this book. Unlike Active Noise Cancellation (ANC), feedforward active vibration control is not easy to use in active vibration control of actual structures because it is difficult to obtain information about external disturbance, that may be distorted by the applied control force. Let us take an example of feedforward active vibration control of the SDOF vibration system. Since both the disturbance and the control force act on the same mass, there is no difference in FIR according to path. So, it is hd = h f in this example. Let us consider the case of natural frequency, ωn = 2π rad/s, damping factor, ζ = 0.1. And suppose that disturbance of d¯ = sin(2π t) is acting on this system. The transfer function of this SDOF is simply expressed as T (s) =

ωn2 s 2 + 2ζ ωn s + ωn2

(5.10.12)

The FIR can be calculated using this transfer function as shown in the following figure. The following Matlab program contains information on the operation of Feedforward control for the addressed problem.

5.10 Filtered-X LMS

187

.

188

5 Control Design

5.10 Filtered-X LMS

189

Figure 5.18 shows uncontrolled and controlled responses. Figure 5.19 shows disturbance and control force. As can be seen from 5.19, the feedforward control force acts out-of-phase to external disturbance as time goes by.

5.11 Summary This chapter introduced some of the various control algorithms used for active vibration control. These control algorithms are algorithms whose validity have been proven not only by numerical calculation but also by experiments on real structures. Of course, there are numerous other vibration control algorithms. However, the vibration control algorithm must meet the prerequisite that it should not make the system unstable and can be implemented in real world. Therefore, the reliability must be thoroughly verified before applying it to the actual structure and the applicability of the control algorithm need to be checked. In these days, the control algorithm is generally implemented using a digital controller except for some very simple control algorithms. Therefore, if the control algorithm is too complex, the sampling rate may be slowed due to the amount of calculation, which may cause a delay problem. Therefore, the control algorithm needs to be as simple as possible.

190

Fig. 5.18 Uncontrolled and controlled responses

Fig. 5.19 Disturbance and control force

5 Control Design

References

191

References Balas, MJ (1979) Direct velocity feedback control of large space structures. Journal of Guidance and Control 2(3) 252–253. Brogan, W (1991) Modern Control Theory, 3rd Ed., Prentice-Hall, Inc. Inman, DJ (2017) Vibration with Control 2nd Ed, Wiley. Fanson, JL, Caughey, TK (1990) Positive Position Feedback Control for Large Space Structures. AIAA Journal 28 717–724. Golnaraghi, F, Kuo, BC (2017) Automatic Control Systems, 10th Ed., McGraw-Hill Education. Kwak, MK, Han, SB, Heo, S (2004) The Stability Conditions, Performance and Design Methodology for the Positive Position Feedback Controller. Transactions of the Korean Society of Noise and Vibration Engineers 14(3) 208–213. Lisa, AA, Andreas, HF (1992) Comparison and Extensions of Control Methods for Narrow-Band Disturbance Rejection. IEEE Trans. on Signal Processing 40(10) 2377–2391. Meirovitch, L (1989) Dynamics and Control of Structures. John Wiley & Sons. Palazzolo, AB (2016) Vibration Theory and Applications with Finite elements and Active Vibration Control, Wiley. Shaw, J, Albion, N (1981) Active control of the helicopter rotor for vibration reduction. J. Amer. Helicopter Soc. 26(3) 32–39. Shin, JH, Lee, JH, You, WH, Kwak, MK (2019) Vibration suppression of railway vehicles using a magneto-rheological fluid damper and semi-active virtual tuned mass damper control. Noise Control Engineering Journal 67(6) 493–507. Yang, DH, Kwak, MK, Kim, JH, Park, WH, Sim, HS (2010) Active vibration control experiment of cantilever using active linear actuator for active engine mount. Transactions of the Korean Society for Noise and Vibration Engineering 20(12) 1176–1182. Yang, DH, Shin, JH, Lee, HW, Kim, SK, Kwak, MK (2017) Active vibration control of structure by Active Mass Damper and Multi-Modal Negative Acceleration Feedback control algorithm. Journal of Sound and Vibration 392 18–30.

Chapter 6

Sensors, Actuators, and Controllers

6.1 Introduction In active vibration control, a control algorithm that can suppress vibration while not destabilizing the target structure is important. We should have knowledge of the hardware we are going to implement. Many active vibration control theories are designed without taking into account their practical application, so it is hard to implement them in real system. The first step in implementing active vibration control is, above all, the right sensor selection. Most active vibration control theories are based on the assumption that displacement and velocity can be measured. However, in practice, it is often impossible to measure absolute displacement or absolute velocity. It is possible to measure displacement with a laser displacement sensor, but it should be noted that the displacement relative to the point where the laser sensor is fixed is measured. In the case of a train, for example, if a laser sensor is attached to the bogie and the displacement of the carbody is measured, the signal of the laser displacement sensor is proportional to the relative displacement between the bogie and the carbody. This must be taken into consideration when using non-contact or contact-type displacement sensors. Velocity measurement is even more difficult. It can be thought that the velocity can be obtained by measuring the displacement and applying a differential algorithm to this signal, but it is not possible due to the electrical noise contained in the actual sensor signal. It can be thought that the acceleration signal can be measured and the velocity signal can be obtained through integration, but this is also not easy if the bias or drift included in the actual sensor signal is present as explained earlier. Typical velocity sensors measure acceleration and use a high-pass filter to remove the DC component to obtain a velocity signal. However, using a filter makes it difficult to use the filtered signal for control because a phase difference occurs. The most commonly used sensor for vibration measurement is an accelerometer. However, since many control theories are not based on acceleration signals, problems arise in

© Springer Nature B.V. 2022 M. K. Kwak, Dynamic Modeling and Active Vibration Control of Structures, https://doi.org/10.1007/978-94-024-2120-0_6

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practical application. In addition, in the case of a MDOF system, the control algorithm is often based on all states. However, since so many sensors are not available, it is virtually impossible to measure all necessary displacements and velocities. Sensors that can be used for vibration control can be largely divided into displacement sensors, velocity sensors, and acceleration sensors. Each sensor has a measurable frequency range, sensitivity, and accuracy depending on the application. Therefore, after grasping the characteristics of the vibration to be controlled, a suitable sensor must be selected. In particular, when selecting a sensor, it is necessary to examine the sensitivity and measurement range, frequency characteristics, and price. After selecting the sensor, it is necessary to attach it to the structure and check that the vibration signal to be controlled is well measured. This is because the correct input signal should be provided to the vibration control algorithm. Otherwise, the vibration cannot be controlled with poorly measured sensor signal. In general, noise is the biggest problem in sensor signals. In particular, noise from AC power can be the biggest problem, but electrical noise is always a problem in electrical signal and control. The so-called signal to noise ratio (SNR) is an important factor in vibration control. A low-pass filter (LPF) is often used to remove high frequency noise and a high-pass filter (HPF) is often used to remove bias components. A band-rejection filter can be used to reject AC noise of 60 Hz component, or a band-pass filter (BPF) can be used to capture only specific frequency components. The output of the control algorithm is a value calculated using a sensor signal, in some cases it can be a signal proportional to the force or torque, or it can be the required displacement. There are not many actuators available for active vibration control. Actuators can be divided into actuators that can generate direct force, inertial actuators that can generate indirect forces, and semi-active actuators that can change the damping coefficient. There are actuators that can generate direct force such as Voice-Coil Actuator (VCA), pneumatic actuator, and hydraulic actuator. There are actuators that can create indirect forces such as inertia force such as Active Mass Damper (AMD). In operating the AMD, displacement needs to be controlled rather than force. Actuators such as ball-screw type combined with a possible motor or linear motor are also used for active vibration control. When a displacement actuator employing a ball-screw mechanism is applied to vibration control, it cannot be said that it is suitable as a vibration control actuator because there is a high risk of damage due to excessive reaction force applied to the actuator. In addition, there is a semi-active damper that can control the damping value, such as a damper capable of orifice control or a magneto-rheological fluid damper. In order to apply such an actuator that generates force or generates displacement to the active vibration control of an actual structure, the actuator driving method and problems associated with the implementation must be considered in advance when designing the active vibration control. Once the sensors, actuators and control algorithms are decided, there must be a controller that can implement the control algorithm. The controller receives the voltage signal from the sensor and needs to create a command voltage based on the control algorithm to generate the control force required for the actuator. If the control algorithm is simple, it can be implemented with an analog circuit using an OP amp.

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So, to implement an analog controller, knowledge of electronic circuits including op amps is required. In recent years, the use of digital controllers has become common, and control algorithms can be mounted on the controller through programming. The digital controller calculates the control command based on the sensor signal during a certain sampling period. Therefore, the control algorithm expressed in continuous time must be converted to a discrete time format. When implementing a digital controller for a designed control algorithm, basic knowledge about the method of implementing the control algorithm in discrete-time format, programming method, and electronic circuit is required. This chapter introduces the types and implementation methods of sensors, actuators, and controllers that can be used for active vibration control.

6.2 Accelerometer The accelerometer is the most commonly used sensor for vibration measurement. So, it is good if the active vibration control algorithm is calculated based on the acceleration signal. Accelerometers currently in use can be largely divided into piezoelectric type and MEMS accelerometers. Also, it can be divided into low-frequency and highfrequency type based on the measurable frequency band. In addition, it can be divided into a high-sensitivity accelerometer that measures fine motion and a low-sensitivity accelerometer that measures large acceleration. So, when using an accelerometer for active vibration control, we should look at the two most important specifications. One is sensitivity and the other is frequency bandwidth. The accelerometer’s sensitivity is usually expressed as mV/g and can sense small acceleration as this value increases. Of course, the price also goes up with sensitivity value. Frequency bandwidth refers to the frequency range that the accelerometer can measure. Since not all frequency components can be measured, accelerometers with various frequency bands exist. Active vibration control mainly targets the natural mode in the low frequency range, so a low frequency accelerometer is normally used. Let us take a look at the piezo-type accelerometer. Piezo-type accelerometers are in the form of attaching mass on a piezoelectric material as shown in Fig. 6.1. When vibration occurs, an inertia force is generated by the attached mass, resulting in tensile and compressive deformation of the piezo, and thus charge is generated. This charge is converted into an electric signal and the acceleration is measured. A popularly-used piezo-type accelerometer shown in Fig. 6.2 has a built-in charge amplifier that converts these charges into voltages. This type of accelerometer is called an ICP® 1 (Internal Charge Preamplifier) accelerometer. Recently, MEMS-type accelerometers are also widely used. Because MEMS is small, it can also be installed in cell phones. It has the advantage of being a little cheaper than piezo-type accelerometers. However, since vibration control is mainly performed for vibrations in a low frequency range, an accelerometer of a relatively large size is used. Below are some of the accelerometers that have been mainly used for vibration control of structures. 1

ICP® is a registered trademart of PCB Piezotronics, Inc.

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Fig. 6.1 Piezoelectric accelerometer

Fig. 6.2 ICP accelerometer (courtesy of PCB)

The frequency bandwidth of the MEMS accelerometer shown in Fig. 6.3 is 0.5– 1000 Hz and the sensitivity is 1000 mV/g, and the frequency bandwidth of the servo accelerometer in Fig. 6.3 is 0–100 Hz and the sensitivity is 2500 mV/g. It can be Fig. 6.3 Low-cost accelerometer (courtesy of logtech)

6.2 Accelerometer

197

seen that the servo accelerometer is for low-frequency vibration measurement, thus the sensitivity is also high. The accelerometers from Figs. 6.4, 6.5, and 6.6 are low-frequency accelerometers. These accelerometers have a frequency bandwidth of approximately 0.1–200 Hz and a sensitivity of 10 V/g. So, if we use these accelerometers, we can measure very lowfrequency acceleration with high sensitivity, so that we can measure even very small movement. ICP accelerometers are generally simpler in wiring than MEMS or Servo accelerometers. Connection is generally made using a BNC cable, but a signal conditioner dedicated to an accelerometer as shown in Fig. 6.7 is used together. In this case, acceleration is converted to a bipolar signal (±10V). However, when using a MEMS accelerometer or Servo accelerometer, the connection with other equipment is a little complicated because external power must be supplied. These accelerometers often have more than 4 connection terminals. Electronics majors are familiar with this connection, but for mechanical engineers Fig. 6.4 Servo accelerometer (courtesy of logtech)

Fig. 6.5 Ultra low-frequency accelerometer (PCB 393B31) (courtesy of PCB)

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Fig. 6.6 Low-frequency accelerometer (courtesy of Jewell Instruments)

Fig. 6.7 Signal conditioner for accelerometer (courtesy of PCB)

majoring in vibration, the connection of electronic equipment is a little unfamiliar. For example, the wiring of the MEMS accelerometer in Fig. 6.3 is shown in Fig. 6.8. The external power supply is 9 to 24VDC and the vibration output signal is a biased signal of 2500 mV, which means that 2500 mV is zero acceleration. So, in general, a signal conditioner like Fig. 6.9 is used together. Taking Jewell Instruments’ LCF 200 accelerometer shown in Fig. 6.6 as an example, there are 4 pins. A terminal is + 12 to + 18 V, B terminal is − 12 to − 18 V, and C terminal is a common ground. This means that bipolar power supply is needed. The common ground shares the ground of the power and the ground of the signal. The most common error for mechanical engineers in wiring connection is the sharing of the electrical ground. Therefore, for the accelerometers in Figs. 6.4 and 6.6, the acceleration signal can be obtained only when external power is connected. Only some of the accelerometers on the market are introduced, but the principle of operation is the same. If you want to detect and control vibration, you must first

6.2 Accelerometer

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Fig. 6.8 Wiring diagram for MEMS accelerometer (courtesy of logtech)

Fig. 6.9 MEMS accelerometer and signal conditioner (courtesy of Control Factory)

grasp the vibration characteristics of the target structure and select an accelerometer that can measure the vibration of the target structure effectively.

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6.3 Piezoelectric Sensor and Actuator Piezoelectricity, the basic property of piezoelectric materials, was discovered over 100 years ago by Jacques Curie and Pierre. By using piezoelectric properties, mechanical energy can be converted into electrical energy or, conversely, electrical energy can be converted into mechanical energy. So piezoelectric materials can be used both as sensors and actuators. Piezoelectricity occurs in natural crystals, but also occurs in materials such as piezoceramic (represented simply as PZT) made of materials such as lead, zirconium, and titanium. In order to generate a piezoelectric phenomenon in a mixed metal such as a piezoelectric ceramic, the material is first heated to a Curie temperature, and then a sufficient voltage field is applied in a desired pole direction so that the ions are aligned in the polarization axis direction. After this process, after cooling, the ions of the piezoelectric ceramic remember the polarity and act accordingly. Readers are referred to Yang’s book (2018) and ANSI/IEEE Std 176-1988 for the fundamental theory on piezoelectricity and electromechanical transformation. A piezoelectric material is an electro-mechanical material, and it is a material that generates electricity when a force is applied to cause deformation. Due to the property of generating electricity when deformation occurs, it has been used in various sensors for a long time. The disadvantage of piezoelectric sensors is that static deformation cannot be measured, because charges generated by deformation are released over time. Since vibration is a dynamic deformation, it can be measured using a charge amplifier. The accelerometer introduced above is a vibration measurement sensor made using the properties of such a piezoelectric material. However, by attaching a piezoelectric material to the surface of the structure, the signal generated by the dynamic strain of the structure can be measured. In this case, a charge amplifier must also be used. As a piezoelectric material that can be attached to the surface, a PZT wafer or a PVDF film can be used in smart structure applications. Figure 6.10 shows a sensor in which a circular PZT wafer is attached to a copper plate, which is mainly used as a humidifying device of a humidifier. piezoceramics have the advantage that they can generate a large force for the same applied voltage and are less sensitive to temperature than piezoelectric polymers. In addition, their high strain sensitivity, relatively low noise, and temperature insensitivity make them Fig. 6.10 PZT sensor

6.3 Piezoelectric Sensor and Actuator

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Fig. 6.11 Beam with bonded piezoceramic sensor and actuator

an acceptable choice for sensing strain as well. When using a piezoelectric ceramic wafer as a sensor by attaching it to the surface of a structure as shown in Fig. 6.11, care should be taken because excessive strain can destroy the piezoelectric ceramic wafer. Recently, as part of an effort to maximize the piezoelectric effect, Macro Fiber Composite (MFC) actuators have been developed and used. Figure 6.12 shows the Polymeric Piezoelectric Polyvinylidence Fluoride (PVDF) sensor, which is one of the piezoelectric materials, is limited in use due to the disadvantages of having a lower force generating power as an actuator and being sensitive to temperature compared to piezoelectric ceramics. However, it has the advantage that it can be easily manufactured and used as a distribution detector. Using the distribution sensor, each vibration mode of the structure can be selected and detected. The piezoelectric sensor introduced above is a transducer that converts mechanical strain into an electrical signal. On the contrary, it can be used as an actuator by using a piezoelectric effect that causes deformation when voltage is applied to a piezoelectric material. However, The PVDF actuator can’t produce enough force to suppress vibrations. Instead, PZT actuators have been found to work very well as distributed strain actuators for the vibration control applications because of their high stiffness, Fig. 6.12 PVDF sensor

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Fig. 6.13 Amplified piezo actuator (courtesy of Cedrat Technologies)

good linearity, relative temperature insensitivity, and ease of implementation. Thus, piezoelectric actuators are actuators that reverse the electro-mechanical properties used in piezoelectric sensors. Piezoelectric actuators are actuators that use the property of being deformed according to electric signals, and as shown in Fig. 6.11, they can be attached to the surface of the flat structure to produce bending moment to the structure. The amplified PZT actuator is also sold in the form as shown in Fig. 6.13. In order to drive the piezoelectric actuator, the voltage from the controller must be amplified to a high voltage. Hence, a high-voltage power amplifier (HVA) as shown in Fig. 6.14 is required. The reason why piezoelectric actuators haven’t been put into practical use is because of poor control force, expensive PZT wafer, and high-voltage amplifiers. Refer to Kwak (2001), Lee and Han (1997), Ahmadian and DeGuilio (2001) for more details on piezoelectric applications. They summarized and presented the research results on active vibration control using piezoelectric ceramics.

Fig. 6.14 High-voltage amplifier (courtesy of Eliezer)

6.4 Laser Displacement Sensor

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Fig. 6.15 Triangulation principle of laser displacement sensor

6.4 Laser Displacement Sensor Laser displacement sensor is a non-contact sensor and is an expensive sensor that can measure movement without damaging the target structure. In addition, movement can be measured very precisely and high-speed measurement is possible. The laser sensor consists of an emitting element and a receiving element as shown in Fig. 6.15. When the emitted beam is detected by the receiving element, the motion of the surface hit by the beam is calculated using triangulation. Using the laser displacement sensor such as optoNCDT 2300-20LL, vibration can be measured with a resolution of 0.3 µm within the measuring range of 20 mm. When using such a displacement sensor, it is necessary to recognize that the relative displacement between the location where the displacement sensor is attached and the target location is measured. It can be effectively used for vibration control of vibration isolation table. It is expensive as it has high precision.

6.5 Voice-Coil Actuator The voice coil actuator (VCA) has the same structure as the vibration shaker in Fig. 6.16 used in the vibration test. It is an actuator that uses the force generated by the current flowing through the coil and the magnetic field of the permanent magnet. It is suitable for vibration control because it can generate a relatively large force. However, it is expensive and requires a current amplifier.

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Fig. 6.16 Shaker (courtesy of logtech)

The inside of the voice-coil actuator (VCA) is composed of copper wire and permanent magnet as shown in Fig. 6.17, and the operating principle is the same as that of a general speaker. As shown in the picture, the magnet becomes a stator and the wired part acts as a rotor, and the force generated by Faraday’s law moves the rotor. The strength of the current flowing through the coil appears as a force, which works like an audio speaker. To drive the VCA, you need a high-current amplifier (HCA). HCA is a device that converts voltage into current, and it sends a current proportional to the command voltage to VCA. Therefore, when implementing control power using VCA, a relational expression between the control power generated by VCA versus the input voltage of HCA is required. When HCA is used, a large current flow, so heat is naturally generated in VCA. Hence, the cooling problem needs to be considered at the same time. If the VCA cannot be used by simply attaching it to the structure, a stator and rotor need to be installed on a linear guide to make it move in one direction. Figure 6.18 shows a mini shaker with a built-in amplifier, so additional amplifier is not necessary to drive the shaker. Fig. 6.17 Voice coil actuator

6.6 Pneumatic Actuator

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Fig. 6.18 Mini shaker with an amplifier

6.6 Pneumatic Actuator There are not many examples of using pneumatic or hydraulic actuators for active vibration control. The reason is, first of all, that these actuators have a low frequency band. That means, it cannot be used for vibration control that requires fast movement. The bellow-type pneumatic actuator as shown in Fig. 6.19 can be used for vibration control in the low frequency band. Recently, a servo valve capable of up to several hundred Hz (Fig. 6.20) has been on the market as a commercial product, so active vibration control of less than 10 Hz has become possible. However, it has not been universalized yet. Pneumatic pressure is especially used for air spring of the vibrationisolation table. Because the natural frequency of the vibration-isolation table is low, active vibration control using pneumatic pressure is possible. Fig. 6.19 Bellow-type pneumatic actuator (courtesy of Festo)

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Fig. 6.20 Proportional servo valve (courtesy of Festo)

6.7 Linear Actuator and AMD Motors are used in may mechanical structures such as toys, machine tools, automobiles and aircraft. However, since it is a rotating machine due to its structural characteristics, it is necessary to convert the rotational motion into a linear motion in order to control the vibrations occurring in linear direction. A common way to convert rotational motion into linear motion is the ball-screw mechanism as shown in Fig. 6.21. However, it is difficult to control vibration by connecting this mechanism between two structures because it may be locked when current does not flow. A typical linear actuator that utilizes a servo motor as shown in Fig. 6.21 is an active mass damper, a system that generates inertial force by moving mass. Actually, AC servomotor is used to drive AMD, which is used to control the vibration of the building. AC servomotor is an actuator specialized in precise position control for factory automation. Unlike simple DC motors that run when current is applied, driving AC servo motors requires specialized knowledge.

Fig. 6.21 AC servomotor and ball-screw mechanism

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In recent years, linear motors using Halbach arrays have appeared, making axial actuation possible. However, it is limited to use for vibration control because it is more expensive than a motor that performs rotational motion. A linear motor shown in Fig. 6.22 is less noisy and can produce more stroke compared to the ball-screw mechanism during linear motion. Looking at the research content of an active mass damper, it is sometimes assumed that a force can be applied to a moving mass. However, VCA is the only actuator that can apply a force equal to the force calculated by the control algorithm to the moving mass. Since VCA cannot have a large stroke, its control force is limited. Therefore, the actuator that can drive the moving mass with a large stroke can be said to be an AC Servomotor. However, since AC servomotor is not a device made for vibration control, we have to think about the control logic differently. With this in mind, Yang et al. (2017) developed a vibration control algorithm that calculates the desired moving mass position and made it track the position using an AC servomotor. If the vibration control algorithm is implemented using a digital controller, the position required to perform the calculation is going to be control command in every sampling time. However, the position control mode of the AC servomotor driver is designed not to be driven in every sampling time. Therefore, Yang et al. (2017) proposed a control algorithm that tracks the position by adjusting the speed after setting the AC servomotor driver as the speed mode. The position tracking control was designed using PID control based on encoder output as shown in Fig. 6.23. The AMD turns out be very useful in active vibration control of structures because we don’t have to modify the target structure in order to mount actuators as in the case of VCA and Motors. It can be simply attached to the structure at the position in which vibrations need to be suppressed. However, there are not many commercially available AMD that can be readily applied to structure. Recently, a small AMD consisting of VCA, displacement sensor, and built-in controller is developed as shown in Fig. 6.24.

Fig. 6.22 Linear motor

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Fig. 6.23 Tracking control algorithm for AMD actuated by servomotor

Fig. 6.24 Active linear actuator (courtesy of Control Factory)

6.8 Vibration Compensator If the structure is excited by a single harmonic disturbance, then the counteracting force may be generated by using unbalanced mass. Figure 6.25 shows the vibration compensator used in ships to suppress vibrations of deck house. There are four unbalanced masses whose shapes are semi-circle and rotating in different directions. Each unbalanced mass is rotated by separate AC servo motor and phases are controlled by software. In this way, the magnitude and phase of the counteracting force can be controlled. The working principle of the vibration compensator is simple, but

6.8 Vibration Compensator

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Fig. 6.25 Vibration compensator (courtesy of Control Factory)

the vibration compensator requires accurate measurement of the angular position and velocity of the engine shaft. If that information is available, then the vibration compensator is very useful because we can generate high force with relatively simple and small device.

6.9 Magnetorheological Fluid Damper The Magnetorheological Fluid (MRF) damper is not an actuator that generates force, but a semi-actuator that adjusts the force by changing the damping factor. Instead of a mechanical damper that controls the orifice to adjust the damping coefficient, the MRF damper uses properties that change by the magnetism of the MR fluid. Unlike passive dampers, a magnetic field is formed when electrical current flows. The MRF damper is a typical semi-active damper. Control using this kind of semi-active damper is called semi-active vibration control because we don’t apply direct control force to structure but only control the damping coefficient according to the situation. Compared to the actuator that generates direct force, the control performance is slightly inferior, but it is highly compatible because it can be controlled with a simple electronic control device. Hence, the MRF damper is generally used in connection with a current amp, and the driver’s price is relatively inexpensive because it does not require a high current amplifier that drives the VCA. Figure 6.26 shows a typical MRF damper.

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Fig. 6.26 MRF damper (courtesy of RMS Technology Co., Ltd.)

Fig. 6.27 Schematics of MR damper

The structure of the MRF damper is simple as shown in Fig. 6.27. The MR fluid flowing through the gap is controlled by an electromagnet, which is controlled by applied current. Hence, the applied current determines the damping factor of the MR damper. The sky-hook control algorithm is a well-known control algorithm used for semi-active vibration control.

6.10 Analog Controller If the control algorithm is implemented as an analog circuit, there is an advantage that there is no noise problem due to quantization error or time-delay problem due to digital control. However, once completed, it is not easy to modify because controller was implemented using an operational (OP) amplifier and fixed electric components. This field is especially unfamiliar to mechanical engineers. In Chap. 5, we introduced the PPF controller used in an active vibration control of a smart structure using piezoceramic sensors and actuators. The PPF controller is given by the following equation.

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Fig. 6.28 Analog PPF control circuit

H (s) =

ωc2 s 2 + 2ζc ωc s + ωc2

(6.10.1)

This equation has the same equation as the low-pass filter in electronic circuits. Referring to the reference Heo et al. (2004a; b) that implements LPF using an op amp, Eq. (6.10.1) can be implemented with the following circuit. Applying the electronic circuit theory to the circuit of Fig. 6.28, the following equation is derived between the input voltage and the output voltage. Vo =

1 × C 1 R1 C 2 R2 s2 +

1 C2 (R1 +R2 ) s C 1 R1 C 2 R2

+

1 C 1 R1 C 2 R2

Vi

(6.10.2)

Comparing Eqs. (6.10.1) and (6.10.2), it can be seen that the filter frequency and damping coefficient of the electronic circuit are derived as follows.  ωc =

1 C2 (R1 + R2 ) , 2ζc ωc = C 1 R1 C 2 R2 C 1 R1 C 2 R2

(6.10.3a,b)

Therefore, we can make the PPF transfer function by adjusting the resistance and capacitor values. The advantage of the analog circuit is that there is no need to worry about the sampling rate that inevitably comes in when implementing a digital controller. Therefore, there is no need to worry about instability caused by time delay. Figure 6.29 shows the analog PPF controller implemented on a breadboard.

6.11 dSPACE Controller Even if the sensor, actuator, and vibration control algorithm are ready, we need a controller that can calculate the appropriate control force with the given control

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Fig. 6.29 Analog PPF controller

algorithm using sensor signals and deliver it to the actuator. The previous section introduces how to implement the control algorithm using the analog circuit, but when the algorithm becomes complex, it is not easy to implement using analog circuit. It is also not easy to program control algorithms using generic digital controller such as DSP (Digital Signal Processor) or microcontroller or PLC (Programmable Logic Controller). There are systems that allow people who are not good at many analog circuits or digital programming to easily implement control algorithms, such as DS1104 as shown in Fig. 6.30 or MicroLabBox as shown in Fig. 6.31 of dSPACE

Fig. 6.30 DS1104 R&D controller board (courtesy of dSPACE, Inc.)

6.11 dSPACE Controller

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Fig. 6.31 MicroLabBox (courtesy of dSPACE, Inc.)

Inc. With these products, we can implement control algorithm using Simulink, download and instantly activate the control system. With this rapid prototyping system, researcher who develops control algorithm can easily test his control algorithm using ControlDesk software as shown in Fig. 6.32. Using this software, we can monitor the variables of the control algorithm in real time. Let us consider again the PPF controller implemented by using an analog circuit in the previous section. The Simulink version will look like the block diagram shown

Fig. 6.32 ControlDesk (courtesy of dSPACE, Inc.)

Fig. 6.33 SPACE and Simulink

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Fig. 6.34 C program for DSP board

in Fig. 6.33. It can be readily seen by comparing Fig. 6.33 with Fig. 6.29 how fast control algorithm can be implemented using dSPACE rapid prototyping system. The dSPACE system is a controller that can easily verify the control algorithm in development stage. Especially, it is suitable to researchers who are not familiar with electronics and programming because Simulink blocks can be used without the need for complex programming. However, the control algorithm needs to be executed with a precise sampling time. Hence, when implementing control algorithm by using a dSPACE system with Simulink, control program needs to be set in discretetime mode. We really don’t have to know the conversion of the control algorithm constructed by Simulink blocks to C programs. Once the C program is constructed, the dSPACE system compiles it and produces an executable file, that is downloaded into DSP chip. Since the price of the dSPACE system is high and it is dedicated to prototyping of the control algorithm in early stage, the control board introduced in the next section would be considered for practical applications (Fig. 6.34).

6.12 DSP and Digital Controller After verifying the control algorithm using the dSPACE system, it is desirable to use a low-cost DSP board or a micro-controller for practical use or commercial applications. The DSP board is capable of performing calculation every sampling period at a high-speed sampling rate. Recently, high-performance microcontrollers such as Arduino and mbed are being used in robot control and mechatronic products. If the controller is a SISO control and its target is low frequency vibration (less than 10 Hz), then these low-cost microcontrollers may be used. Since timer interrupt is also built-in, it is possible to implement a control algorithm that must accurately maintain the sampling period. When using a digital controller, the control algorithm expressed in continuous form must be converted to a discrete time base. If the control algorithm is given in the form of Laplace transformation, it can be transformed into an algorithm in the form of discrete time using z-transform. For example, the following bilinear transform can be considered for that purpose. s=

z−1 2 × Ts z+1

(6.12.1)

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Applying Eq. (6.12.1) to the PPF control algorithm given by Eq. (6.10.1), the following discrete-time PPF controller is obtained. H (z) =

b2 z 2 + b1 z + b0 z 2 + a1 z + a0

(6.12.2)

where b2 = ωc2 /, b1 = 2ωc2 /, b0 = b2 a1 = (2ωc2 − 8/Ts2 )/, a0 = (4/Ts2 − 4ζc ωc /Ts + ωc2 )/  = 4/Ts2 + 4ζc ωc /Ts + ωc2

(6.12.3a–c) (6.12.3d,e) (6.12.3f)

Equation (6.12.2) can be expressed as the finite-time difference equation u i = −a1 u i−1 − a0 u i−2 + b2 (xi + 2xi−1 + xi−2 )

(6.12.4)

When using a digital controller, a part of C program for Eq. (6.12.4) is as follows. Unlike the dSPACE system, the DSP board itself does not have an I/O interface suitable for vibration control. Therefore, additional electronic circuits must be added. For example, the analog input of DSP2812 board is 0–3.3 V. In order to accept a bipolar signal such as − 10 to + 10 V signal, we need an additional analog circuit that converts this signal to 0–3.3 V signal. The analog output of the cheap microcontroller has the form of PWM output. The PWM signal is an effective method for speed control of a motor, but analog signal is preferred for active vibration control. In this case, we might need Digital to Analog Converter (DAC), that requires additional circuit. Therefore, it is not desirable to practice control algorithm with the DSP board or micro-controllers in the development stage.

6.13 Summary In this chapter, sensors, actuators, and controllers that have been used in active vibration control are introduced. Electronics in the field of control is rapidly growing to meet the need in industrial revolution. Although mechanical engineers are not accustomed to electric circuits and wiring, active vibration control requires such knowledge in addition to computer programming. Fortunately, there are useful devices that vibration engineers can handle without much effort. Sooner or later, there will be new hardware and software that can be used for the active vibration control. But the knowledge on the basic working principles of sensors, actuators, and controllers will still be required.

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References Ahmadian, M, DeGuilio, AP (2001) Recent Advances in the Use of Piezoceramics for Vibration Suppression. The Shock and Vibration Digest 33(1) 15-22. Heo, S, Kim, KY, Kwak, MK (2004) Implementation of PPF Controller Using Analog Circuit and Microprocessor. Transactions of the Korean Society of Noise and Vibration Engineers 14(6) 455-462 Heo, S, Lee, SB, Kwak, MK, Baek, KH (2004) Real-Time Active Vibration Control of Smart Structure Using Adaptive PPF Controller. Transactions of the Korean Society for Noise and Vibration Engineering 14(4) 267-275. Kwak, MK (2001) Active vibration control of smart structure using piezoceramics. Transactions of the Korean Society of Precision Engineering 18(12) 30-46. Lee, I, Han, JH (1997) Research Trend in Vibration Control of Smart Structures Using Piezoelectric Materials. Journal of the Korean Society for Aeronautical & Space Sciences 25(3) 168-176. Yang, DH, Shin, JH, Lee, HW, Kim, SK, Kwak, MK (2017) Active vibration control of structure by Active Mass Damper and Multi-Modal Negative Acceleration Feedback control algorithm. Journal of Sound and Vibration 392 18-30. Yang, J (2018) An Introduction to the Theory of Piezoelectricity. Advances in Mechanics and Mathematics Book 9 2nd Edition, Springer.

Web sites http://encarta.msn.com/ http://trekinc.com/ http://www.agilent.com/ https://www.cedrat-technologies.com/en/products/product/APA120S.html http://www.controlfactory.co.kr/ http://www.couriertronics.com/docs/MEL/DB-M3-E.pdf http://www.dSPACEinc.com/ http://www.festo.com/ http://www.jewellinstruments.com/linear-angular-accelerometers http://www.lmsintl.com/testlab http://www.logtech.co.kr/ http://www.mathworks.com http://www.mdstec.com https://www.micro-epsilon.com/ http://www.mitsubishielectric.com/fa/ http://www.rmstech.co.kr/ http://www.sensortech.ca/ http://www.smart-material.com/MFC-product-main.html http://www.te.com/

Chapter 7

Application Examples

7.1 Introduction Although active vibration control is theoretically possible, it is necessary to investigate whether it is possible in actual structures. Usually, researchers who study dynamic modeling try to derive as accurate models as possible. In contrast, researchers who design controllers try to develop a control algorithm that works without an accurate model, that is, a robust controller. In particular, they try to design a controller that doesn’t fall into instability in any case. Since active vibration control of a structure targets a large DOF system that is not covered in general control theory, a commonly known control algorithm cannot be applied directly to an actual structure. In addition, the equation of the system is different depending on the type of sensor that measures vibration or actuator that transmits force, so a new control algorithm suitable for this case needs to be developed. Active vibration control has been applied to various structures and proven its effectiveness as introduced in Chap. 1. In this chapter, some of works on the active vibration control of structures will be introduced.

7.2 Beam Equipped with Piezoceramic Sensors and Actuators The work of Denoyer and Kwak (1996) concerning the active vibration control of the beam equipped with piezoceramic sensors and actuators will be briefly summarized. Vibration control of a beam by bonded piezoelectric sensors and actuators is a good example of active vibration control. In the reference (Denoyer and Kwak 1996), a beam with surface bonded piezoelectric sensors and actuators shown in Fig. 7.1 was considered and the equations of motion were derived using the energy approach. The cross-sectional dimensions of the beam are assumed to be small compared to © Springer Nature B.V. 2022 M. K. Kwak, Dynamic Modeling and Active Vibration Control of Structures, https://doi.org/10.1007/978-94-024-2120-0_7

217

218

7 Application Examples

Fig. 7.1 Beam with surface mounted piezoelectric sensors and actuators

the length, and elastic deformation is primarily caused by bending, we can regard the beam structure as an Euler–Bernoulli beam with surface bonded piezoelectric wafers as shown in Fig. 7.1. The axis 3 is defined as the poling direction. As introduced in the dynamic modeling, the elastic displacement, v(x, t), can be discretized using the assumed modes method such that v = v(x, t) =

n 

φ j (x)q j (t) = q

(7.2.1)

j=1

where  = [φ1 φ2 · · · φn ] , q = [q1 q2 · · · qn ]T and φ j (x) ( j = 1, 2, . . . , n) are the assumed mode shapes or admissible functions which must satisfy the geometric boundary conditions of the problem and be differentiable half as many times as the system order and q j (t) ( j = 1, 2, . . . , n) are a set of generalized coordinates. The total kinetic energy of the active structure can then be expressed as T = Tb +

np 

T pi

(7.2.2)

i=1

where Tb represents the kinetic energy of the substructure, T pi represents the kinetic energy of the i th piezoelectric plate, and n p is the number of piezoelectric plates. The kinetic energies are as follows: 1 Tb = 2

L

1 m¯ b (x) v˙ dx, T pi = 2

xi +h i

m¯ pi (x) v˙ 2 dx

2

(7.2.3a, b)

xi

0

where L is the length of the substructure, m¯ b is the mass per unit length of the substructure, m¯ pi is the mass per unit length of the i th piezoelectric plate, xi is the starting x-coordinate of the piezoelectric plate, and h i is the length of the piezoelectric plate. Using Eqs. (7.2.1) and (7.2.2), the total kinetic energy can be written as: T =

1 T q˙ Mq˙ 2

(7.2.4)

7.2 Beam Equipped with Piezoceramic Sensors and Actuators

219

where L m¯ b   dx +

M=

T

n p xi +h i 

m¯ pi T  dx

(7.2.5)

i=1 x i

0

We need to combine the stress–strain relationship from Euler–Bernoulli beam theory with the piezoelectric constitutive equation. The mechanical–electrical equation for the piezoelectric wafer bonded to the beam surface can be written as    T   2 d31 E p D3 ε3 − d31 E3 = (7.2.6) T1 −d31 E p E p S1 where D3 represents the electric displacement along the 3-axis, E 3 represents the applied electrical field density in the 3-axis, S1 represents strain in the 1-axis, T1 represents stress in the 1-axis, ε3T is the permittivity of the piezoelectric material, d31 is piezoelectric charge constant, and E p is defined as the short circuited modulus of elasticity in the direction of the poling axis. The total work for the smart structure is given by W = Wb +

np 

W pi

(7.2.7)

i=1

where Wb is the work done by the beam substructure, and W pi is the work done by the ith piezoelectric plate. The work done by the beam can be expressed as 1 Wb = −Vb = − 2



L E b Ib 0

∂ 2v ∂x2

2

1 dx = − qT Kb q 2

(7.2.8)

where Vb is potential energy of the beam, E b is modulus of elasticity for the substructure, Ib is the moment of inertia for the substructure, and Kb is the beam stiffness matrix given by L Kb =

E b Ib   dx T

(7.2.9)

0

The work done by the i th piezoelectric plate is expressed as an integral over the volume of the piezoelectric such that W pi =

1 2

 (−T1i S1i + D3i E 3i )dVi Vi

220

7 Application Examples

1 = w pi 2

xi +h i y i +t pi 

xi

yi

D3i T1i

T 

1 0 0 −1



 E 3i dydx S1i

(7.2.10)

where yi is the starting point of the piezoelectric as measured from the neutral axis of the beam. Using Eq. (7.2.6) and making the notational change εxi = S1i , Eq. (7.2.10) can be expressed as

W pi

1 = w pi 2

xi +h i y i +t pi

xi

 T

2 2 E p E 3i2 + 2d31 E p E 3i εxi − E p εxi ε3 − d31 dydx

yi

(7.2.11) Assuming that the piezoelectric plates are perfectly bonded to the beam, we can obtain W pi =

1 2 1 γi v − qT bi vi − qT K pi q 2 i 2

(7.2.12)

where vi is the electrode voltage and w pi h i  T 2 ε3 − d31 E p , vi = t pi E 3i , t pi  xi +h i t pi T bi = d31 E p w pi yi +  dx, 2

γi =

xi

 K p = w pi t pi E p

yi2

+ yi t pi +

2 t pi

 xi +h i

3

  dx T

(7.2.13a–d)

xi

Substituting Eqs. (7.2.8) and (7.2.12) into Eq. (7.2.7), the total work can now be expressed as W =

1 1 T v Cv − qT Bv − qT Kq 2 2

C = diag[γi ], K = Kb +

np 

K pi ,

(7.2.14)

(7.2.15a,b)

i=1

B = [b1 b2 · · · bn p ], v = [v1 v2 · · · vn p ]T

(7.2.15c,d)

Using the assumed-mode method, the equations of motion for a structure with surface bonded piezoelectric sensors and actuators can be obtained by using Eqs. (7.2.4) and (7.2.14). Since some of the piezoelectric plates will be used as sensors

7.2 Beam Equipped with Piezoceramic Sensors and Actuators

221

and some as actuators, some will not have actuator voltage inputs and some will not have sensor voltage outputs. Therefore, the B and C matrices can be broken down in sensor and actuator parts corresponding to the sensor and actuator voltages, vs and va . Then, equations of motion and sensor equation can be written as Mq¨ + Kq = −Ba Ga va

(7.2.16)

T vs = Gs C−1 s Bs q

(7.2.17)

where Ga and Gs have been added to represent sensor and actuator amplifier gains. Any dynamics associated with these amplifiers are neglected in this model. The free vibration problem can be solved for the eigenvalues and eigenvectors which satisfy the orthonormality conditions, UT MU = I and UT KU = , where U is a matrix with columns consisting of orthonormal eigenvectors and  = diag[ωi2 ] is a matrix with the eigenvalues along the diagonal. Using the modal transformation q = Uη and the orthonormality relations, Eqs. (7.2.16) and (7.2.17) can be rewritten in terms of modal coordinates as η¨ + 2Zn η˙ + η = −UT Ba Ga va

(7.2.18a)

T vs = Gs C−1 s Bs Uη

(7.2.18b)

The modal damping was added in Eq. (7.2.18a), where Z = diag[ζi ]

(7.2.19)

in which ζi is the modal damping factor. we can also construct state equation using Eq. (7.2.18a, b). x˙ = As x + Bs va

(7.2.20a)

vs = Cs x + Ds va

(7.2.20b)

where the state vector x = [η η˙ ]T , and    0 I 0 , Bs = As = − −2Zn −UT Ba Ga

(7.2.21a, b)



T Cs = Gs C−1 s Bs U 0 , Ds = 0

(7.2.21c, d)



The beam was made from aluminum and PZT-5A piezoelectric ceramic plates were bonded to the surface of the beam. The piezoelectric plates were wired in a configuration that used four plates as a single actuator with one smaller plate

222

7 Application Examples

in between functioning as a sensor. This configuration was chosen so that the sensor/actuator pairs would have co-located characteristics for the modes of interest. The geometry of the structure is illustrated in Fig. 7.2 while the material properties and dimensions are as follows: L = 23.25 in, wb = 2.5 in, tb = 0.0625 in, m¯ b = 0.27318 kg/m E p = 61 GPa, d31 = −171 × 10−12 m/V, t pi = 10−3 in h i = 1.5 in, w pi = 2.5 in, m¯ pi = 0.12419 kg/m h i = 0.5 in, w pi = 1.5 in, m¯ pi = 0.07452 kg/m The actuator wafers were oriented such that an applied voltage would cause the plates on one side of the beam to contract while those on the other side expand, producing bending in the structure. Piezoelectric materials produce a charge on their electrodes when they are deformed. However, to get the desired sensitivity, frequency response, and minimization of loading effects, an interface circuit is required. One such circuit is a charge amplifier. Digital control and data acquisition were performed using an AC-100 digital control computer from Integrated Systems, Inc. Controllers were implemented using MatrixX/SystemBuild software from Integrated Systems running on a VAX 3100 workstation. Frequency response data were collected using an HP 35665A dynamic signal analyzer in the swept-sine mode. The theoretical and experimental frequency responses are shown in Fig. 7.3. This figure shows that the theoretical frequency response agree well with the experimental frequency response over the frequency range of interest, meaning that the theoretical model is good enough for control design. The frequency responses between the other sensor and actuator pairs showed similar correlation between model and experiment. PPF controllers were designed using an 8-mode model of the active structure. The first eight mode shapes of a cantilever beam with the same properties as the substructure of the smart structure were used as the assumed modes or admissible functions. Three independent PPF controls the first three modes of vibration. Each PPF controller uses a separate sensor/actuator pair as shown below. The transfer functions for the three PPF controllers are

Fig. 7.2 Beam equipped with piezoceramic sensors and actuators

7.2 Beam Equipped with Piezoceramic Sensors and Actuators

223

Fig. 7.3 Theoretical and experimental open-loop frequency responses

947.88 s 2 + 18.47s + 947.88

(7.2.22a)

177219.0 s 2 + 63.146s + 177219.0

(7.2.22b)

H1 (s) = H2 (s) =

224

7 Application Examples

Fig. 7.4 Experimental open-loop frequency response and closed-loop frequency response by three mode PPF control

H3 (s) =

24674.0 s 2 + 78.54s + 24674.0

(7.2.22c)

For digital implementation, these transfer functions were again emulated using the bilinear transformation with pre-warping and a sampling rate of 1 kHz. Figure 7.4 shows the uncontrolled and controlled frequency response curves experimentally obtained by applying the PPF controllers given by Eq. (7.2.22a–c).

7.3 Grid Structure The work of Kwak and Heo (2007) concerning the active vibration control of the grid structure will be briefly introduced. The grid structure equipped with piezoceramic sensors and actuators shown in Fig. 7.5 is similar to a solar panel mounted on satellites. The vibration of the solar panel must be controlled to maintain the attitude and pointing accuracy as well as to protect electronic devices sensitive to vibrations. The grid structure built for the demonstration of the active vibration control technique consists of two piezoceramic sensors and two pairs of piezoceramic actuators, which are glued to the surface of the composite grid structure. The composite grid structure was designed and manufactured using Carbon/Epoxy fabric prepreg (Hankook Carbon Co. CF-3327) and Glass/Epoxy fabric prepreg (Hankook Fiber Co. G635). During the lay-up procedure, 4 plies of Carbon/Epoxy were stacked and insulated by a ply of Glass/Epoxy layer on top and bottom surfaces. Copper foil wires were placed on both surfaces for electrodes. Piezoceramic wafers were then bonded on the surface of Glass/Epoxy outer layer using 90-min cure epoxy. Silver epoxy (Daejoo Precision Co. DS-7276-A) was then used as bond between

7.3 Grid Structure

225

Fig. 7.5 Smart grid structure

electrode of ceramic and copper foil wire. Large piezoceramic wafers (63.5 mm L × 38.1 mm W) were used as actuators and small piezoceramic wafers (25.4 mm L × 38.1 mm W) were used as sensors. In order to induce a larger moment to suppress the vibration of main structure, each piezoceramic actuator is bonded on both sides of the structure which acts adversely thus doubling the actuator force. The resulting grid structure is the system equipped with two sensors and two actuators as shown in Fig. 7.5, in which actuators are bonded in the root of the grid structure and small sensors are bonded close to the actuators to make a nearly collocated system. The grid structure, which is in fact a continuous system, has an infinite number of natural frequencies and modes. As can be seen from Fig. 7.5, the structure is actually a three dimensional structure. However, it can be modeled by the grid finite element with two angular displacements and a transverse displacement if lower modes are of prime interest for control. To this end, the equivalent bending stiffness was estimated based on the composite beam theory. The followings are element mass and stiffness matrices used in the finite element formulation. ⎤ ⎡ 2b 0 0 b 0 0 ⎢ 0 156g 22I g 0 54g −13I g ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0 22I g 4I 2 g 0 13I g −3I 2 g ⎥ (7.3.1a) me = ⎢ ⎥ ⎢ b 0 0 2b 0 0 ⎥ ⎥ ⎢ ⎣ 0 54g 13I g 0 156g −22I g ⎦ 0 −13I g −3I 2 g 0 −22I g 4I 2 g

226

7 Application Examples

⎡

d ⎢ 0 ⎢ ⎢ ⎢ 0 ke = ⎢ ⎢ −d ⎢ ⎣ 0 0

0 12a 6La 0 −12a 6La

0 6La 4L 2 a 0 −6La 2L 2 a

−d 0 0 d 0 0

0 −12a −6La 0 12a −6La

⎤ 0 6La ⎥ ⎥ ⎥ 2L 2 a ⎥ ⎥ 0 ⎥ ⎥ −6La ⎦ 4L 2 a

(7.3.1b)

where b = J¯ L/6 and g = m¯ I /420, a = E I /L 3 , d = G J/L in which J¯ is the polar mass moment of inertia per unit length, L is the element length, m¯ is the mass per unit length, I is the sectional area moment of inertia, E I is the bending stiffness, and G J is the torsional stiffness, respectively. The actuation moment resulting from two piezoceramic wafers bonded to the both sides of the grid structure can be approximated by the couple acting on its ends. Based on this assumption, each moment can be written as M p = E p w p tb d31 va

(7.3.2)

where E p is Young’s modulus of the piezoceramic wafer, w p is the width of the piezoceramic actuator, tb is the thickness of the grid structure, d31 is the piezoelectric constant, and va is the actuator voltage, respectively. The piezoceramic wafer can be also used as a sensor. In this case, the charge induced by the deformation turns into the sensor voltage after the signal is processed by a charge amplifier, which can be approximately expressed as follows: vs = −

E p w p tb d31 (α2 − α1 ) 2Cc

(7.3.3)

where Cc is the capacitance of the piezoceramic sensor and α1 , α2 are the slope of the grid at both ends of the piezoceramic sensor, respectively. The resulting equations of motion and sensor equation for the smart grid structure can be written as M¨x + C¨x + Kx = Ba va

(7.3.4a)

vs = Cs x

(7.3.4b)

where x is an m × 1 nodal displacement vector, M, C, K are m × m mass, damping, and stiffness matrices, m is the total degrees of freedom for the finite element model, Ba is an m × actuator participation matrix, Cs is an × m sensor participation matrix, is the number of actuators and sensors, respectively. In our case, is equal to 2. Equation (7.3.4a) represents the structure equation associated with the actuator forces and Eq. (7.3.4b) represents the sensor equation, respectively. Natural frequencies and natural mode shapes can then be obtained by solving the eigenvalue problem. First four mode shapes are plotted in Fig. 7.6. As can be seen

7.3 Grid Structure

227

Fig. 7.6 Natural frequencies and modes of the grid structure

from Fig. 7.6, the grid structure contains bending modes which are symmetric about the middle line and torsional modes which are axisymmetric about the middle line. In fact, the first and the third natural modes are the bending modes similar to the bending modes of the beam and the second and the fourth natural modes are the torsional modes. The experimental natural frequencies were obtained by FFT signal analyzer. Figure 7.7 shows experimental setup to obtain the transfer function between the actuator and sensor. As shown in the figure, the random noise generated by the dynamic signal analyzer is added to the control voltage by the adder circuit consisting of an operational amplifier. The sensor signal was then fed to the dynamic signal analyzer and the A/D port of the controller board. In this way, we can measure the frequency response function and control the system simultaneously. Using modal transform, x = Uq, orthonormality condition, and modal damping assumption, we can transform Eq. (7.3.4a, b) into the reduced-order modal equations as in the case of smart beam: q¨ + 2Z q˙ + q = B¯ a va

(7.3.5a)

¯ sq vs = C

(7.3.5b)

where I is an n × n identity matrix,  is an n × n eigenvalue matrix, and Z is an ¯ s = Cs U, respectively, in which B¯ a n × n damping factor matrix, B¯ a = UT Ba , C ¯ s is an × n sensor participation is an n × actuator participation matrix and C matrix based on the modal coordinates. The input–output relation of Eq. (7.3.5a, b)

228

7 Application Examples

Fig. 7.7 Experimental setup

is in fact the transfer function between the sensor and actuator and can be derived by using Laplace transform. The theoretical and experimental frequency response plots of the transfer function are shown in Fig. 7.8. As shown in Fig. 7.8, there exists a good agreement between the theoretical predictions and experimental results up to the sixth mode. The damping factor of 0.005 is assumed for the all modes in the theoretical model. The MIMO PPF controller was designed to tackle the four natural modes of interest. The numerical data for the matrices shown in Eq. (7.3.5a, b) are as follows:  = diag(3637.1, 18625.8, 95662.8, 219085.6) 2Z = diag(0.603, 1.365, 3.903, 4.681) ⎡ ⎤ −0.0033 −0.0033 ⎢ 0.0063 −0.0063 ⎥ ⎥ B¯ a = ⎢ ⎣ −0.0175 −0.0175 ⎦ −0.0250 0.0250   −10693 15733 −20881 −18038 ¯s = C −10693 −15733 −20881 18038

7.3 Grid Structure

229

Fig. 7.8 Theoretical and experimental frequency response plots

As implied by Eq. (7.3.5a), the control force is to be calculated based on the modal space. In this case, the MIMO modal PPF controller can be written as 1

˙ +  f Q = G 2  f qˆ ¨ + 2Z f  f Q Q

(7.3.6)

where Q is an n × 1 coordinate vector of the modal PPF controller, Z f is an n × n damping factor matrix of the MIMO PPF controller,  f is an n × n compensator frequency matrix,  f = 2f , and G is an n×n gain matrix, respectively. Q represents the modal control force vector and each modal control force is to be computed by the SISO PPF compensator based on the corresponding modal displacement. Equation (7.3.6) is decoupled but the actual control force, va is to be computed from modal control forces. qˆ is the estimation of modal displacement and introduced because they are to be computed based on the sensor measurement. Hence, the control system is internally decoupled but externally coupled. Even though the structural equations of motion given by Eq. (7.3.5a) is a reduced-order system, there are still two problems involved in applying the MIMO PPF controller to the real system. The first problem is the computation of the actuation voltage using the modal control forces. The second problem is the estimation of the modal displacement based on the sensor measurements. Let us discuss the first problem. In order for structural equation of motion, Eq. (7.3.5a) to be compatible with the MIMO PPF controller, Eq. (7.3.6), the following relation should hold.

230

7 Application Examples 1

B¯ a va = G 2  Q

(7.3.7)

If n = , i.e., the number of actuators is equal to the number of modes to be controlled, then B¯ a becomes a square matrix. If B¯ a is non-singular, then the actuator voltage, va can be obtained from the modal control force vector, Q by inverting B¯ a . However, if n > , i.e., there are more modes to be controlled than the number of available actuators, we come across the over-determined case. This amounts to the case that we try to control many modes with a limited number of actuators. We can think of the pseudo-inverse technique to estimate the actuation voltages from the modal PPF control forces as follows:  † 1 va = B¯ a G 2 Q

(7.3.8)

 †  −1 where B¯ a = B¯ aT B¯ a B¯ aT represents the pseudo inverse. Even though the control force estimated by the above formula can successfully suppress vibrations of four natural modes with two actuators and two sensors, it has been observed that the pseudo-inverse technique results in uneven distribution of control authority on each mode. As a result, the control performance cannot be predicted accurately. Let us discuss this problem further by inserting Eq. (7.3.8) into Eq. (7.3.5a).  † 1 q¨ + 2Zq˙ + q = B¯ a B¯ a G 2 Q

(7.3.9)

In order for Eq. (7.3.9) to be a pair with the MIMO PPF controller given by  † Eq. (7.3.6), the resultant of matrix multiplication B¯ a B¯ a should be at least close to identity matrix. However, for the numerical data obtained for the smart grid structure, we obtained. ⎡

⎤ 0.035 0 0.183 0  † ⎢ 0 0.059 0 −0.236 ⎥ ⎥ B¯ a B¯ a = ⎢ ⎣ 0.183 0 0.905 0 ⎦ 0 −0.236 0 0.941

(7.3.10)

It becomes evident from Eq. (7.3.10) that the diagonals of the matrix are not unity. Hence, the pseudo- inverse technique used for the estimation of actuation voltage does not result in the expected control authority as the SISO PPF controller does. To avoid the control authority problem, a new approach based on a block-inverse technique was proposed by Kwak and Heo (2007), which is applicable to the case of n = 2 , n = 4 , . . ., i.e., the number of controlled modes is even number times the number of available actuators and each block matrix is non-singular. We resort our case to n = 2 in order to simplify the implementation. To this end, let us decompose the actuator participation matrix into two sub matrices as follows:

7.3 Grid Structure

231

  B¯ a1 ¯ Ba = ¯ Ba2

(7.3.11)

Each sub matrix is an × square matrix and assumed to be non-singular. Let us then express the actuator voltage as follows 1

va = Ba∗ G 2 Q

(7.3.12)

where a new matrix consists of inverse of each sub matrix    −1 −1 Ba∗ = B¯ a1 B¯ a2

(7.3.13)

Hence, the equations of motion for this case can be rewritten as 1

ˆ 2 Q q¨ + 2Zq˙ + q = BG

(7.3.14)

where  Bˆ =

B¯ a Ba∗

=

  −1    ¯ a2 I B12 I B¯ a1 B =   −1 B¯ a2 B¯ a1 I B21 I

(7.3.15)

For the numerical data of the smart grid structure, we obtained. ⎡

⎤ 1 0 0.190 0 ⎢ 0 1 0 −0.250 ⎥ ⎥ Bˆ = ⎢ ⎣ 5.277 0 1 0 ⎦ 0 −3.995 0 1

(7.3.16)

Let us compare Eq. (7.3.16) with Eq. (7.3.10). It can be readily seen that the matrix of Eq. (7.3.16) holds the unity diagonals. This is a very desirable property from the control point of view. The off-diagonal elements of the matrices shown in Eqs. (7.3.10) and (7.3.16) represent the control spillover effects. Remembering the fact that the PPF controller has control spillover to lower modes, the upper off-diagonal elements should be smaller than the diagonal elements. The lower offdiagonal elements do not play a major role in the MIMO PPF control because the PPF controller has no spillover problem to higher modes as explained earlier. On the contrary to the computation of actuator voltage, the upper off-diagonal elements of the matrix do not degrade the performance of the PPF controller since the PPF controller is in fact a low-pass filter. However, the lower off-diagonal elements should be smaller than the diagonal elements because the lower mode components of the measurement signal can be fed into the PPF controller designed for the higher mode. In view of this fact, it can be readily understood that the new methodology of

232

7 Application Examples

implementing the MIMO PPF controller given by Eq. (7.3.12) is more effective than the pseudo-inverse technique. The second problem is the observer spillover problem as stated earlier. The number of sensors is always not enough to measure all the modal coordinates. The MIMO PPF controller equation given by Eq. (7.3.6) is valid only if true modal displacements are either obtainable or measurable. In reality, we can only estimate the modal displacements based on the sensor measurements. Our problem amounts to the case of estimating more modal displacements with less sensor measurements, which can be said mathematically the under-determined case. The pseudo-inverse technique can also be applied to estimate the modal displacements based on Eq. (7.3.5b) as follows.  † ¯ s vs qˆ = C

(7.3.17)

 †  ¯s = C ¯ sC ¯ Ts −1 represents the pseudo-inverse. Inserting Eq. (7.3.5b) ¯ Ts C where C into (7.3.17), we obtain  † ¯s C ¯ sq qˆ = C

(7.3.18)

 † ¯s C ¯s It can be easily understood that the product of matrix multiplication C should be a unity matrix to make a one-to-one matching. However, the pseudoinversion formula causes the same unbalanced matrix problem as in the estimation of actuator voltage from the modal PPF control forces. The block-inverse technique can be also used for the estimation of modal displacements from the sensor measurements. We used the same assumption that the number of modes to be controlled is twice the number of sensors. Let us first decompose the sensor participation matrix into two sub-matrices.

¯s = C ¯ s1 C ¯ s2 C

(7.3.19)

Then, the estimate of modal coordinates can be written as qˆ = C∗s vs

(7.3.20)

where C∗s

  −1  ¯ C =  s1 −1 ¯ s2 C

(7.3.21)

Inserting Eq. (7.3.20) into Eq. (7.3.5b), we can obtain ˆ qˆ = Cq where

(7.3.22)

7.3 Grid Structure

233

 ˆ = C

¯s C∗s C

=

 ¯ s2 C

I −1

¯ s1 C

  −1 ¯ s1 C ¯ s2 C I

(7.3.23)

As can be seen from Eq. (7.3.23), the matrix multiplication results in the unity diagonals on the contrary to the pseudo-inverse formula, which is a desirable property from the measurement point of view. If we use the numerical data, we obtain ⎡

⎤ 0.208 0 0.406 0 ⎢  † 0.432 0 −0.495 ⎥ ⎥ ¯s C ¯s = ⎢ 0 C ⎣ 0.406 0 0.792 0 ⎦ 0 −0.495 0 0.568 ⎡ ⎤ 1 0 1.953 0 ⎢ 1 0 −1.146 ⎥ ⎥ ˆ =⎢ 0 C ⎣ 0.512 0 1 0 ⎦ 0 −0.872 0 1

(7.3.24a)

(7.3.24b)

On the contrary to the computation of actuator voltage, the upper off-diagonal elements of the matrix do not degrade the performance of the PPF controller since the PPF controller is in fact a low-pass filter. However, the lower off-diagonal elements should be smaller than the diagonal elements because the lower mode components of the measurement signal can be fed into the PPF controller designed for the higher mode. In view of this fact, it can be readily understood that the blockinverse methodology of implementing the MIMO PPF controller is more effective than the pseudo-inverse technique. The inputs to the MIMO PPF controller are the piezoelectric sensor measurement voltages and the outputs of the MIMO PPF controller are the actuation voltages to the piezoceramic actuators. Hence, we need the input-output transfer function of the MIMO PPF controller to be practically implemented. Using Eqs. (7.3.6), (7.3.12) and (7.3.20), the following equation for the MIMO PPF controller can be obtained in the form of transfer function. 1

1

Va (s) = Ba∗ G 2  H pp f (s) G 2 C∗s Vs (s)

(7.3.25)

⎤ 0 0 H1 (s) 0 ⎢ 0 H2 (s) 0 0 ⎥ ⎥ H pp f (s) = ⎢ ⎣ 0 0 H3 (s) 0 ⎦ 0 0 0 H4 (s)

(7.3.26)

where ⎡

in which

234

7 Application Examples

Fig. 7.9 SIMULINK Block Diagram for the MIMO PPF controller

Hi (s) =

ω2f i s 2 + 2ς f ω f i s + ω2f i

, i = 1, 2, 3, 4

(7.3.27)

We chose the damping factor, ς f = 0.3 and the gain matrix, G = diag(0.1, 0.045, 0.018, 0.012). The filter frequency of each PPF controller is tuned to the natural frequency of target mode. The frequency shift caused by the gain turned out to be small, so that it was neglected in the implementation. The MIMO PPF controller given by Eq. (7.3.25) was realized by the dSPACE DSP board, DS1102 along with Simulink. The Simulink block diagram that amounts to Eq. (7.3.25) is shown in Fig. 7.9, which was downloaded to the DSP board. The sampling rate for the A/D and D/A was set to 50 kHz, which is fast enough to control four modes of interest. Figure 7.10 shows the uncontrolled and controlled frequency response plots obtained by experiments. It can be confirmed by experiment that the control performance is close to the one expected theoretically, which validates the proposed control methodology. Figure 7.11 shows the uncontrolled and controlled time responses of the sensor output. After the grid structure was excited, the MIMO PPF controller was activated at t  1 s. It can be concluded from the experiment that the proposed MIMO PPF controller is capable of suppressing vibrations of the grid structure successfully.

7.4 Plate Equipped with Piezoelectric Sensors and Actuators The research work done by Kwak et al. (2015) for the submerged rectangular plate equipped with piezoelectric sensors and actuators will be briefly introduced. Let us consider a rectangular plate with side lengths a in the x direction and b in the z direction, as shown in Fig. 7.12 and assume that the PZT wafers are glued to the plate as shown in Fig. 7.13. The kinetic and potential energies belonging to the k-th PZT wafer are then expressed as

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

Fig. 7.10 Uncontrolled and controlled frequency response plots from experiments

Fig. 7.11 Uncontrolled and controlled time responses of sensor output

235

236 Fig. 7.12 Partially submerged cantilever plate

Fig. 7.13 PZT wafer glued on the plate

7 Application Examples

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

1 Dz b T ¯ ¯ zk q, ˙ VZ k = ρz h z ab q˙ T M q Kzk q 2 2a 3

TZ k =

237

(7.4.1a, b)

where ρz , h z are the mass density and the thickness of the PZT wafer and ¯ zk = Xzk ⊗ Zzk M

(7.4.2a)

¯ zk =X ˆ zk ⊗ Zzk + α 4 Xzk ⊗ Zˆ zk K   ˜ Tzk ⊗ Z˜ zk + X ˜ zk ⊗ Z˜ Tzk + α 2 νz X ¯ zk ⊗ Z¯ zk + α 2 (1 − νz )X

(7.4.2b)

 1 2 Ez 1 2 3 h hh Dz =  h + + h z z 2 z 1 − νz2 4

(7.4.2c)

in which E z is the Young’s modulus and x¯ zk +a¯ zk

Xzk =

¯ zk =   dξ, X

x¯ zk +a¯ zk

T

x¯ zk

ˆ zk = X

x¯ zk +a¯ zk

˜ zk =   dξ, X  T



x¯ zk

(7.4.3a, b)

T  dξ,

(7.4.3c, d)

    dζ ,

(7.4.3e, f)

x¯ zk

z¯ zk+b¯ zk

  dζ , Z¯ zk =

z¯ zk+b¯ zk

T

z¯ zk

Zˆ zk =

T

x¯ zk

x¯ zk +a¯ zk

Zzk =

  dξ,

T

z¯ zk

z¯ zk+b¯ zk



 T

 dζ, Z˜ zk = 

z¯ zk

z¯ zk+b¯ zk

 T   dζ,

(7.4.3g, h)

z¯ zk

where x¯ zk = x zk /a, a¯ zk = azk /a, z¯ zk = z zk /b, b¯ zk = bzk /b, the subscript z represents the PZT wafer, and k represents the k-th PZT wafer. Refer to Chap. 4 for detailed expressions of the non-dimensionalized mass and stiffness matrices. The virtual work performed by the piezoelectric actuators can be written as: δW = δqT B pa v pa

(7.4.4)

where v pa is the actuator voltage vector and

T



B pa = BTpa1 · · · BTpar v pa = v pa1 · · · v par

(7.4.5a)

238

7 Application Examples

B pk



= Bz  (x¯ zk + a¯ zk ) −  (x¯ zk ) ⊗

z¯ zk+b¯ zk

 dζ z¯ zk

x¯ zk +a¯ zk

+

(7.4.5b)



 dξ ⊗   (¯z zk + b¯ zk ) −   (¯z zk )

x¯ zk

Bz =

d31 E z (h + h z ) 2(1 − νz )α

(7.4.5c)

in which d31 is the piezoelectric constant and pai , (i = 1, 2, . . . , r ) is the index of the actuator among the PZT wafers. The sensor equation can be written as: v ps =

1 B ps q = C ps q cch

(7.4.6)

where v ps is the sensor voltage vector, cch is the capacitance value of the charge amplifier and ⎤ ⎤ ⎡ v ps1 B ps1 ⎥ ⎥ ⎢ ⎢ = ⎣ ... ⎦, B ps = ⎣ ... ⎦ ⎡

v ps

v ps p

(7.4.7a, b)

B ps p

in which psi , (i = 1, 2, . . . , p) is the index of the sensor among the PZT wafers. B psi has the same form as B pai . The total kinetic and potential energy for the partially submerged rectangular plate equipped with piezoelectric wafers can be expressed as T =

1 1 T ˙ V = qT Kt q q˙ Mt q, 2 2

(7.4.8a, b)

where ¯ f + ρz h z ab ¯ p + ρ f ab2 M Mt = ρ p hab M



¯ zk M

(7.4.9a)

k

Kt =

Db ¯ Dz b  ¯ Kp + 3 Kzk 3 a a k

(7.4.9b)

¯ f in which Mt , Kt represent the total mass and stiffness matrices, respectively. M is the fluid added mass matrix. Refer to Kwak and Yang (2013a) for the detailed derivation of the fluid added matrix. Using Lagrange’s Equation and introducing the

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

239

damping matrix, equations of motion and sensor equation can be expressed as Mt q¨ + Ct q˙ + Kt q = B pa v pa

(7.4.10a)

v ps = C ps q

(7.4.10b)

where Ct is the damping matrix. Equation (7.4.10a, b) form the actuator and sensor equations. Using the reduced-order modal transformation, q = Up, and the orthonormality condition, to Eq. (7.4.9a, b), then the modal equations of motion and the corresponding sensor equation can be written as ¯ pa v pa p¨ + 2Z  p˙ +  p = B

(7.4.11a)

v ps = C ps p

(7.4.11b)

where Z is the damping factor matrix whose diagonals are the damping factors for each mode and ¯ ps = C ps U  = diag(ω), B¯ pa = UT B pa , C

(7.4.12a–c)

The modal equations of motion given by Eq. (7.4.11a) are desirable from the point of view of control design since we can reduce the order of the equations. We can rewrite Eq. (7.4.11a) by replacing the right-hand side as follows 1

p¨ + 2Z  p˙ +  p == G 2  P

(7.4.13)

Then, the corresponding MIMO PPF controller in modal coordinates can be written as: 1 P¨ + 2Z f  p˙ +  P = G 2  p

(7.4.14)

where G is a gain matrix. If the modal control force is computed using Eq. (7.4.14), then the real control voltage, v pa , can be computed by using the following equation. 1 B¯ pa v pa = G 2  P

(7.4.15)

However, a problem arises if the number of modes that are to be controlled is ¯ ps are not square matrices as in larger than the number of actuators since B¯ pa , C the case of the grid structure presented in the previous section. Using the method of block-inverse technique (Kwak and Heo 2007), the actuator conversion equation and sensor matrix can be written as

240

7 Application Examples

v pa = B∗pa G 2 r Pr , p = C∗ps v ps 1

(7.4.16a, b)

where B∗pa

 −1  −1  ∗ , C ps = = B¯ pa1 B¯ pa2

  ¯ ps1 −1 C  ¯ ps2 −1 C

(7.4.17a, b)

Then, the resulting MIMO PPF controller can be expressed in the form of transfer function as follows: V pa (s) = B∗pa G r H(s) C∗ps V ps (s)

(7.4.18)

where G is the control gain matrix and H(s) is the PPF control matrix, for which the diagonal element represents the transfer function of each PPF control. Hence, the control strategy is almost the same as the one used in the active vibration control of grid structure. In order to verify the theoretical propositions, an experimental test bed was built as shown in Fig. 7.14. The water tank was made of acrylic plate having 10 mm thickness. The size of the water tank is 450 × 450 × 300 mm. Two pairs of piezoceramic sensors and actuators were attached to the rectangular plate, as shown in Fig. 7.15. As a result, four PZT wafers were glued on the aluminum plate as shown in Fig. 7.15. The PZT wafers are placed on where the maximum strain occurs as in the case of cantilever beam. This leads to the position close to the clamped end. In this way, symmetric and axi-symmetric natural modes can be controlled. However, the location of the PZT wafers should be carefully chosen when we deal with the plate having either free or simply-supported boundary conditions. Silicon was placed on Fig. 7.14 Experimental setup on hanged cantilever plate with PZTs

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

241

Fig. 7.15 Dimensions of the aluminum plate and PZTs

the piezoceramic wafers to provide waterproofing. An aluminum plate has the dimensions as shown in Fig. 7.15. The thickness of the aluminum plate is h = 2 mm. The material constants for the aluminum plate are ν = 0.33, ρ p = 2700 kg/m3 , E p = 68.9 GPa. A water density of ρ f = 1000 kg/m3 is used for the submerged case. The thickness of the PZT wafer is h z = 0.5 mm. The material constants for the PZT wafer are νz = 0.31, ρz = 5440 kg/m3 , E p = 30.336 GPa, d31 = 270 × 10−12 N/V. Figure 7.16 shows the schematic diagram of the connections made in this experiment for concurrent measurement and control. An LMS measurement device was used to measure the frequency response between the PZT sensors and actuators. The dSPACE DS 1104 system was used to implement the MIMO PPF control algorithm. The piezoceramic sensors were connected to the A/D ports of the dSPACE DS 1104

242

7 Application Examples

Fig. 7.16 Schematic block diagram for measurement and control

board through the use of charge amplifiers. A 47nF capacitor was used in the charge amplifier to control the amplification factor of the charge amplifier. The sensor signal was also connected to the input port of the LMS measurement device. The output of the LMS measurement device, which contains random signals, was also connected to one of the A/D ports of the dSPACE device. This signal was digitally added to the control signal, as shown in Fig. 7.16. The D/A ports of the DS 1104 board that consists of the controls and the random signal were connected to the piezoceramic actuators through high-voltage power amplifiers. The LMS testlab software was used to generate the frequency response curves during the control experiment. In this way, we can compute the uncontrolled and controlled frequency response curves and the time responses. The MIMO PPF controller given by Eq. (7.4.18) was implemented by using the Simulink block diagram, as shown in Fig. 7.17 and was downloaded to the dSPACE DS 1104 system. There are four PPF controllers in the Simulink block diagram of Fig. 7.17. Two sensor signals were distributed to four PPF controllers and the outputs of the four PPF controllers were combined to two control outputs after multiplying gains. Two control signals were multiplexed to one vector signal by using the mux block of Simulink. Control gain and On/Off blocks were then applied to the mux signal. In this way, we can control both signals at the same time. ControlDesk software was used for this mission. The demux block of Simulink was then used to distribute the mux signal into two separate signals. In Fig. 7.17, ADC block(DS1104ADC_C8) was connected to the LMS signal, DAC blocks(DS1104DAC_C5 and DS1104DAC_C7) indicate the control outputs for the

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

243

Fig. 7.17 Simulink block diagram for control algorithm in air

PZT actuators. As shown in Fig. 7.17, ADC_C8 signal is added to the control signal and outputted to DAC_C5. The sampling rate of the digital PPF controller was of 20 kHz which adequate for the given problem. The transfer function measured between the actuator #1 and the sensor #1 was compared to the theoretical transfer function when the plate was in air and submerged into water, as shown in Figs. 7.18 and 7.19, which confirms the equation of motion given by Eq. (7.4.11a, b). Damping factors, ζ1 = 0.02, ζ2 = ζ3 = 0.005, ζ4 ∼ ζ10 = 0.004 are used for the theoretical model. However, the transfer function at a higher frequency range has a different magnitude, which implies that the modelling accuracy deteriorates for higher modes or the experimental results are not good for higher modes. Figure 7.19 clearly indicates that the theoretical predictions are consistent with the results of the experiment, which validates the adequacy of the added virtual mass matrix proposed by Kwak and Yang (2013a). Even though Kwak and Yang (2013a) formulated the fluid–structure interaction problem for the cantilever plate submerged into the unbounded fluid, their result is valid for the cantilever plate submerged into the bounded fluid considered in the experiment. Figure 7.20 shows the natural mode shapes of the plate in the water that were theoretically obtained. Figure 7.21 shows that the 1st, 3rd and 5th modes are symmetric about the z axis. On the other hand, the 2nd, 4th and 6th modes are axi-symmetric about the z axis. It was found that the natural mode shapes in air are almost the same as the ones in water shown in Fig. 7.20, which implies that the natural mode shapes do not change discernibly as a result of the presence of water. This information on the natural mode shapes was used to design the MIMO PPF controller. The MIMO PPF controller that is given by Eq. (7.4.18) was designed by considering the four natural modes using the block-inverse technique. Since the dynamic characteristics change by the presence of the water, different gain matrices and filter frequencies were used for the MIMO PPF controller given by Eq. (7.4.18).

244

7 Application Examples

Fig. 7.18 Transfer functions of the plate in air

Fig. 7.19 Transfer functions of the partially submerged plate (H = 300 mm)

7.4 Plate Equipped with Piezoelectric Sensors and Actuators Fig. 7.20 Natural mode shapes in water (H = 200 mm)

(a) 1st natural mode

(b) 2nd natural mode

(c) 3rd natural mode

(d) 4th natural mode

245

246

Fig. 7.21 Transfer functions of the plate in air

7 Application Examples

7.4 Plate Equipped with Piezoelectric Sensors and Actuators

Fig. 7.22 Transfer functions of the partially submerged plate (H = 300 mm)

247

248

7 Application Examples

Fig. 7.23 Time-responses of the plate in air subjected to impact

Figures 7.21 and 7.22 show the theoretical and measured transfer functions of the plate in air and submerged in water. As shown in Figs. 7.21 and 7.22, the theoretical predictions for the control performance are in good agreement with the measured results, which again validates the dynamic model and the MIMO PPF controller. Vibration reduction for the first natural modes of each case is about 10 dB. Figures 7.23 and 7.24 show the uncontrolled and controlled time responses of the sensor that were subjected to impact. Figures 7.23 and 7.24 clearly indicate that the vibrations of the plate in air and of that submerged in water were successfully suppressed by the MIMO PPF controller.

7.5 Plate with AMDs The work of Shin et al. (2019a) on active vibration control of a rectangular plate by using AMDs will be briefly summarized. Even though control techniques developed in the work of Shin et al. (2019a) has not been tested experimentally, the use of AMD to control vibrations of continuous structure such as plate needs to be mentioned because the AMD is a good candidate as an actuator. Let us consider a rectangular plate with side lengths a in the x direction and b in the y direction and that it is equipped with AMDs as shown in Fig. 7.25. The

7.5 Plate with AMDs

249

Fig. 7.24 Time-responses of the piezoelectric sensor of the submerged plate subjected to impact (H = 300 mm)

Fig. 7.25 Rectangular plate with AMD

250

7 Application Examples

assumed modes method is used to express the deflection of the plate as w(x, y, t) = W(x, y) q(t)

(7.5.1)

where W(x, y) = [W1 (x, y) W2 (x, y) . . . Wm (x, y)] is a 1 × m matrix consisting of admissible functions, and q(t) = [q1 q2 . . . qm ]T is an m × 1 vector consisting of generalized coordinates, for which m is the number of admissible functions used to approximate the deflection. Kinetic and potential energies of the rectangular plate can then be expressed as follows: TP =

ρ p hab T ¯ Db ¯ ˙ V P = 3 qT K q˙ M p q, pq 2 2a

(7.5.2a, b)

where ρ p is mass density, h is thickness, D = Eh 3 /12(1 − v 2 ), E is the Young’s ¯ p are non-dimensionalized mass and ¯ p and K modulus, and ν is Poisson’s ratio. M stiffness matrices, respectively. Subscript p represents the plate. Refer to Chap. 4 for detailed expressions of the non-dimensionalized mass and stiffness matrices. Kinetic energy of moving and supporting masses of the active mass damper located at (xak , yak ) can be expressed as follows: Tak =

1

m s w˙ 2 (xak , yak ) + m a (w(x ˙ ak , yak ) + u˙ ak )2 , k = 1, 2, . . . , n a (7.5.3) 2

where m s and m a are supporting and moving masses of the AMD, u ak is the relative displacement of the kth AMD, and n a is the number of AMDs. It is assumed that the same AMDs are used for control. Using Eq. (7.5.1), Eq. (7.5.3) can be expressed as follows: 1 T q˙ (m s + m a )WT (xak , yak )W(xak , yak )q˙ + q˙ T m a WT (xak , yak )u˙ ak 2 1 2 + m a u˙ ak , k = 1, 2, . . . , n a 2 (7.5.4)

Tak =

Hence, the kinetic energy of all AMDs can be expressed as Ta =

1 ¯ a q˙ + m a q˙ T BaT u˙ + 1 m a u˙ T u˙ (m s + m a )q˙ T M 2 2

(7.5.5)

where u = [u a1 u a2 · · · u ana ]T and ¯a= M

na  k=1

WT (xak , yak )W(xak , yak ),

(7.5.6a)

7.5 Plate with AMDs

251

⎡ ⎢ ⎢ Ba = ⎢ ⎣

W(xa1 , ya1 ) W(xa2 , ya2 ) .. .

⎤ ⎥ ⎥ ⎥ ⎦

(7.5.6b)

W(xana , yana ) Using kinetic and potential energies given by Eqs. (7.5.2) and (7.5.5), equations of motion for this system including disturbance can be written as Mq¨ + Kq = −m a BaT u¨ a + d

(7.5.7)

where ¯p ¯ p + (m s + m a )M ¯ a , K = Db K M = ρ p hab M a3

(7.5.8a, b)

are mass and stiffness matrices, and d is the disturbance vector. Applying the modal transformation, q = Ur, into Eq. (7.5.7), and using the orthonormality condition and the modal damping, then we can obtain modal equations of motion. r¨ + 2Zq˙ + r = −m a B¯ aT u¨ a + d¯

(7.5.9)

where B¯ a = Ba U, d¯ = UT d and ⎡

ω1 ⎢ 0 ⎢ =⎢ ⎢ ... ⎣

0 ··· ω2 · · · .. . . . . .. 0 0 .

0 0 .. . ωn

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎡

⎤ 0 ··· 0 ζ2 · · · 0 ⎥ ⎥ .. . . .. ⎥ . . . ⎥ ⎦ .. 0 0 . ζn

ζ1 ⎢0 ⎢ Z=⎢ ⎢ ... ⎣

(7.5.10a, b)

Displacement at the mounting point can be written as wak = w(xak , yak , t) = W(xak , yal ) q(t), k = 1, 2, . . . , n a

(7.5.11)

Hence, vector equation for displacement measurement can be expressed as wa = Ba q = Ba Ur = B¯ a r

(7.5.12)

where wa = [wa1 wa2 · · · wana ]T . In reality, displacements cannot be measured, although accelerations can be measured by accelerometers. The AMD proposed Yang et al. (2017) uses acceleration at the mounting point. Thus, measurement equation in the real world can be written as ¨ a = B¯ a r¨ w

(7.5.13)

252

7 Application Examples

There are always more natural modes than available sensors. It is not desirable to control all modes with only a few AMDs. Let us consider a case in which the same number of AMDs as the number of natural modes to be controlled is used. Hence, we ¯ au ]. This results in a reduced-order system for control design. may write B¯ a = [B¯ ac B If we divide the generalized displacement vector into generalized displacements to be controlled by the AMD and the remaining displacements, such that r = [rcT ruT ]T , then controlled and uncontrolled equations of motion and the sensor equation can be written as T u¨ a + d¯ c r¨ c + 2Zc c r˙ c + c rc = −m a B¯ ac

(7.5.14)

T r¨ u + 2Zu u r˙ u + u ru = −m a B¯ au u¨ a + d¯ u

(7.5.15)

¨ a = B¯ ac r¨ c + B¯ au r¨ u w

(7.5.16)

The MIMO modal-space NAF control for Eq. (7.5.14) can then be designed as ¨ + 2Z f c q˙ + c Q = −Gm r¨ c Q

(7.5.17)

where Z f is the damping matrix for NAF control. The MIMO modal-space NAF control can be obtained by the following relation T ua = Gm Q m a B¯ ac

(7.5.18)

where Q is an n a × 1 modal control force vector, and Gm is an n a × n a diagonal matrix whose diagonal is the gain for each mode. Hence, the control displacement of AMD can be obtained by the following: ua =

1 ¯ −T B Gm Q m a ac

(7.5.19)

Let us neglect residual modes for the control design, then Eq. (7.5.20) can be derived from Eq. (7.5.13). −1 ¨a w r¨ c = B¯ ac

(7.5.20)

Of course, B¯ ac should be invertible, meaning that AMDs should not be placed at the node of natural modes of interest. Inserting Eq. (7.5.19) into Eq. (7.5.14), we can obtain the following: ¨ + d¯ c r¨ c + 2Zc c r˙ c + c rc = −Gm Q Combining Eqs. (7.5.17) and (7.5.21), we can obtain the following:

(7.5.21)

7.5 Plate with AMDs



253

     Ina ×na Gm r¨ c r˙ c 2Zc c 0na ×na ¨ + 0na ×na 2Z f c Q q˙ Gm Ina ×na      rc d¯ c c 0na ×na = + 0na ×na c Q 0na ×1

(7.5.22)

Hence, the stability condition can be obtained as stable if Gm2 < I

(7.5.23)

The stability condition, Eq. (7.5.23), is static, meaning that the stability does not depend on frequency. Equation (7.5.23) also implies that the stability is guaranteed if the gain matrix is small enough. By using Eqs. (7.5.17), (7.5.19), and (7.5.20), the MIMO modal-space NAF controller using the acceleration measurement can be written as Ua (s) = −

−1 −1 1 ¯ −T 2 2 ¨ a (s) B G s I + 2sZ f c + c B¯ ac W m a ac m

(7.5.24)

However, control and observer spill over problems can occur because we take only parts of B¯ a . Let us take a look at uncontrolled modes. The control force is spilled into the remaining modes and the resulting equation can be written as T ¯ −T Bac Gm Q + d¯ u r¨ u + 2Zu u r˙ u + u ru = −B¯ au

(7.5.25)

Hence, higher uncontrolled modes are disturbed because of spill over problems. It should be noted here that PPF control does not cause this kind of spill over problem. In addition, modal displacements to be controlled can be calculated based on measured relative displacements of the plate at positions of AMDs. Hence, we may write −1 wa rˆ c = B¯ ac

(7.5.26)

where rˆ c represent the measured modal coordinate. Inserting Eq. (7.5.12) into Eq. (7.5.26), it can be seen that computed modal displacements of natural modes to be controlled consist of actual modal displacements to be controlled and uncontrolled actual modal displacements. −1 Bau ru = Rr rˆ c = rc + B¯ ac

(7.5.27)

−1 ¯ where R = [ I B¯ ac Bau ]. It can be readily seen from Eq. (7.5.27) that uncontrolled modal displacements spill into controlled modal displacements to be used for MIMO modal-space NAF control. −T ¯ aT B¯ ac Gm Q + d¯ r¨ + 2Zq˙ + r = −m a2 B

(7.5.28)

254

7 Application Examples

¨ + 2Z f c q˙ + c Q = −Gm r¨ˆ c = −Gm R¨r Q

(7.5.29)

Equations (7.5.28) and (7.5.29) represent fully coupled equations of motion with MIMO modal-space NAF control that include both controlled and the uncontrolled natural modes. Combined equations can be written in matrix form as 

     −T Gm r¨ 2Z 0n×na r˙ In×n m a2 B¯ aT B¯ ac + ¨ Q Gm R Ina ×na 0na ×n 2Z f c q˙      r d¯  0n×na = + 0na ×n c Q 0na ×1

(7.5.30)

Hence, the stability condition becomes −T 2 ¯ aT B¯ ac Gm R < I stable if m a2 B

(7.5.31)

The stability condition given by Eq. (7.5.31) when considering control and observer spill overs are still static. However, it tells us that the gain should be small enough to satisfy this condition. The control law given by Eq. (7.5.24) represents a fully coupled MIMO modalspace NAF controller. This implies that modal accelerations are computed from accelerations at each point and the modal control is then converted into the real command using matrix computations. For simplicity, one can design each AMD to cope with each natural mode in a decentralized way. Hence, we may write ¨ a = −Gs B¯ a r¨ u¨ a + 2Z f c u˙ a + c ua = −Gs w

(7.5.32)

Equation (7.5.32) implies that the desired displacement of the AMD is calculated by the acceleration measured at that point, and each AMD is assigned to tackle a certain mode. Combining Eqs. (7.5.9) and (7.5.32), we can obtain 

     r¨ 2Z 0n×na r˙ In×n m a B¯ aT + Gs B¯ a Ina ×na 0na ×n 2Z f c u¨ a u˙ a      r d¯  0n×na = + 0na ×n c ua 0na ×1

(7.5.33)

Hence, the stability condition can be obtained as stable if m a B¯ aT Gs B¯ a < I

(7.5.34)

The stability condition, Eq. (7.5.34), is also static. Equation (7.5.34) implies that the stability is guaranteed if the gain matrix is small enough. In addition, control or observer spill over problems would not occur in this case.

7.5 Plate with AMDs

255

Using Eq. (7.5.32), the decentralized MIMO NAF controller using the acceleration measurement can be written as

−1 ¨ a (s) Ua (s) = −Gs s 2 I + 2sZ f c + c W

(7.5.35)

Unlike the fully coupled MIMO modal-space NAF controller given by Eq. (7.5.24), the decentralized MIMO NAF controller given by Eq. (7.5.35) does not require matrix computation. Thus, it is simple to use. As a numerical example, a rectangular aluminum plate having a = 4 m, b = 3 m, h = 3 mm, ρ = 2770 kg/m3 , and E = 70 GPa was considered. Total mass of the plate was 99.72 kg. Three AMDs having support mass of 0.5 kg and moving mass of 1 kg were mounted at (2m, 1.5m), (1m, 1.5m), (2m, 0.75m), respectively. These positions were selected because they are the anti-nodal points of three natural modes, resulting in maximum efficiency for each mode. Six admissible functions were considered in each direction. Thus, the total degree-of-freedom was 36, which was high enough for computing natural frequencies. Natural frequencies of the plate without AMDs were 1.2446, 2.5887, 3.6341, 4.8289, 4.9782, 7.2185, 7.6167, 7.9652, 8.9608, and 10.3548 Hz. Natural frequencies of the plate with AMDs were 1.1752, 2.5145, 3.5245, 4.5713, 4.9782, 6.9414, 7.2806, 7.9652, 8.7104, and 10.3548 Hz, respectively. There were slight decreases in natural frequencies because of added masses of AMDs. The three lowest natural frequencies were used as filter frequencies of NAF controllers. Figures 7.26 and 7.27 show natural mode shapes of the plate with and without AMDs. As shown in Figs. 7.26 and 7.27, AMDs affected only higher modes, particularly the 6th and 7th modes. The modal damping factor was assumed to be 0.6% and filter damping factor of 0.3 was chosen. The MIMO modal-space NAF controller was designed with Gm = 0.2 I3×3 and the decentralized MIMO NAF controller was designed with Gs = 1.5 I3×3 to suppress the three lowest natural modes. In order to obtain transfer function, the excitation point, (2.4242, 0.9091) m, was chosen. Figure 7.28 shows frequency response function at the excitation point when the MIMO modal-space NAF controller is used. As shown in Fig. 7.28, the introduction of AMD increased the damping of the three lowest modes. Figure 7.29 shows impulse response at the excitation point. As can be seen from Fig. 7.29, vibrations were quickly suppressed by the fully coupled MIMO modal-space NAF controller. Figure 7.30 shows frequency response function at the excitation point when the decentralized MIMO NAF controller is used. Figure 7.31 shows impulse response. It can be said that the decentralized MIMO NAF controller is equally effective in suppressing vibrations.

7.6 Shell Structure The work of Kwak and Yang (2013b) on active vibration control of a cylindrical shell with ring stiffeners equipped with piezoelectric sensors and actuators, and submerged into water will be briefly summarized.

256

7 Application Examples

(a) 1st mode

(b) 2nd mode

(c) 3rd mode

(d) 4th mode

(e) 5th mode

(f) 6th mode

(g) 7th mode

(h) 8th mode

Fig. 7.26 Natural modes of a plate without AMDs

There have been many researches on the vibration of cylinderial shell with stiffeners. Galletly (1955) derived equations of motion for a simply-supported ringstiffened cylindrical shells. Al-Najafi and Warburton (1970) investigated the free vibration of ring-stiffened cylindrical shell. Laulagnet and Guyader (1989) studied

7.6 Shell Structure

257

(a) 1st mode

(b) 2nd mode

(c) 3rd mode

(d) 4th mode

(e) 5th mode

(f) 6th mode

(g) 7th mode

(h) 8th mode

Fig. 7.27 Natural modes of a plate with AMDs

shell’s acoustic radiation in light and heavy fluids. Harari and Sandman (1990) investigated radiation and vibrational properties of submerged stiffened cylindrical shells. Yuan and Dickinson (1994) investigated the free vibration of circular cylindrical shell and plate systems. Mattei (1995) derived sound radiation model by using a baffled shell. Kim et al. (2004) investigated the vibration of partially fluid-filled cylindrical

258

7 Application Examples

Fig. 7.28 Frequency response function by MIMO modal-space NAF controller

7.6 Shell Structure

259

Fig. 7.29 Impulse response by MIMO modal-space NAF controller

shell with ring stiffeners. Jafari and Bagheri (2006) investigated free vibration of ring-stiffened cylindrical shell. Iakovlev (2008) investigated the interaction between a submerged cylindrical shell and a shock wave. Pan et al. (2008) studied the free vibration of a ring-stiffened thin circular cylindrical shell with arbitrary boundary conditions. Leblond et al. (2009) derived a dynamic model of a submerged circular cylindrical shell subjected to a weak shock wave. However, the active vibration of submerged cylindrical shell with ring stiffeners has not been studied until the work of Kwak and Yang (2013b). Let us consider a ring-stiffened circular cylindrical shell shown in Fig. 7.32, where R is the radius of the cylindrical shell, h is the thickness, L is the length, θ is the angle with respect to the vertical axis, x is the axis along its length, u, v, w are the displacements in the x, θ and z, directions, and L k , dk and bk are the location, thickness and the width of the kth ring-stiffener, respectively. We utilized the result obtained by Kwak et al. (2010, 2011, 2009) for a cylindrical shell without ring stiffeners, where the displacement in each direction is expressed as a series of functions with n circumferential nodes: u(x, θ, t) = w(x, θ, t) =

∞  n=0 ∞ 

u n (x, θ, t), v(x, θ, t) =

∞ 

vn (x, θ, t),

n=0

wn (x, θ, t).

(7.6.1a–c)

n=0

As in Kwak et al. (2009, 2010), we neglected the case of n = 0 since the natural frequencies of the natural modes without circumferential nodal points are higher than those of the natural mode with circumferential nodal points. To simplify the analysis, each circumferential mode was expressed by admissible functions as follows (Kwak et al. 2009):

260

7 Application Examples

Fig. 7.30 Frequency response function by decentralized MIMO NAF controller

7.6 Shell Structure

261

Fig. 7.31 Impulse response by decentralized MIMO NAF controller

Fig. 7.32 Coordinates and dimensions of ring-stiffened circular cylindrical shell



u n (x, θ, t) = u (x) cos nθ qnuc (t) + sin nθ qnus (t) ,

(7.6.2a)



vn (x, θ, t) = v (x) sin nθ qnvs (t) − cos nθ qnvc (t) ,

(7.6.2b)



wn (x, θ, t) = w (x) cos nθ qnwc (t) + sin nθ qnws (t) ,

(7.6.2c)

262

7 Application Examples

where u (x), v (x), w (x) represent 1 × m matrices consisting of admissible functions, qnuc (t), qnus (t), qnvs (t), qnvc (t), qnwc (t), qnws (t) are m ×1 generalized coordinate vectors corresponding to the cosine and sine modes, and m is the number of admissible functions used for the longitudinal expansion, respectively. As pointed out by Kwak et al. (2009), sets of cosine and sine modes should be considered for response calculations and control designs. The total kinetic and potential energies of the cylindrical shell can be also represented as the sum of the kinetic and potential energies of the nth circumferential mode, as is done for the displacements (Kwak et al. 2009, 2010, 2011): Ts =

∞  n=1

Tsn , Vs =

∞ 

Vsn .

(7.6.3a, b)

n=1

Using the expressions, Eq. (7.6.1) combined with Eq. (7.6.2a–c), and using the Sanders shell theory (Amabili 1997), the following kinetic and potential energies corresponding to the n(≥ 1)th circumferential mode can be obtained in vector–matrix form: 1 T q˙ Muu q˙ nuc + q˙ Tnvs Mvv q˙ nvs + q˙ Tnwc Mww q˙ nwc 2 nuc T + q˙ nus Muu q˙ nus + q˙ Tnvc Mvv q˙ nvc + q˙ Tnws Mww q˙ nws ,

Tsn =

1 T 1 T 1 T qnuc Knuu qnuc + qnvs Knvv qnvs + qnwc Knww qnwc 2 2 2 T T T + qnuc Knuv qnvs + qnvs Knvw qnwc + qnuc Knuw qnwc 1 T 1 T 1 T + qnus Knuu qnus + qnvc Knvv qnvc + qnws Knww qnws 2 2 2 T T T + qnus Knuv qnvc + qnvc Knvw qnws + qnus Knuw qnws

(7.6.4a)

Vsn =

(7.6.4b)

where the mass and stiffness matrices are expressed as Muu = uu , Mvv = vv , Mww = ww ,

Knww

(7.6.5a–c)

 β2 (1 − ν)α 2 n 2 ¯ 1+ uu , Knuu = uu + (7.6.5d) 2 48   β2 ¯ β2 (1 − ν) vv + 1+ Knvv = α 2 n 2 1 + (7.6.5e) vv , 12 2 16   ˆ ww β2  2 2 4 2 ˜ 2 ¯ = α ww + + α n ww − 2νn ww +2(1 − ν)n ww , 12 α 2 (7.6.5f)

7.6 Shell Structure

263

Knuv

 β2 ˆ (1 − ν)αn ˜ 1− = νnα uv − uv , 2 16

˜ uw + Knuw = να 

(1 − ν)n 2 αβ 2 ˆ uw , 24

(7.6.5h)

 n2β 2 nνβ 2 ˜ (1 − ν)nβ 2 ¯ vw − =α n 1+ vw + vw , 12 12 8 2

Knvw

(7.6.5g)

(7.6.5i)

in which ν is the Poisson’s ratio, and α = L/R, β = h/R and 1 uu =

1 Tu u dξ, vv

=

0

˜ uw = 

1

=

0

1

1

1 0

¯ vv =  u dξ,  

T ˜ uv  v  w dξ, 

=

0 T ˜ ww  u w dξ, 

1

= 1

=

 v  v dξ,

(7.6.6d–f)

T

1

 u v dξ,

(7.6.6g–i)

v  w dξ,

(7.6.6g–i)

 w  w dξ.

(7.6.6j–l)

=

T

0 T ˜ vw  w w dξ, 

1 =

0

ˆ uw Tu  v dξ, 

(7.6.6a–c)

0

1

0

ˆ uv = 

T  u

0 T ¯ vw  w  w dξ, 

Tw w dξ, 0

1

¯ uu Tv w dξ, 

0

¯ ww = 

=

0

1 vw =

1 Tv v dξ, ww

0

ˆ ww Tu  w dξ, 

1 =

0

T

0

in which ξ = x/L, the prime and double prime represent the first and the second derivative with respect to ξ , respectively. Refer to Kwak et al. (2011) for the detailed derivations of the above mass and stiffness matrices. Let us derive the kinetic and potential energies of a ring stiffener in matrix form. The total kinetic and potential energies of ring stiffeners are the sum of the kinetic and potential energies belonging to each stiffener as in Eq. (7.6.3). Hence, we can write Tr =

nr  k=1

Tr k , Vr =

nr 

Vr k

(7.6.7a, b)

k=1

where n r is the number of ring stiffeners. We used the expression for the kinetic and elastic potential energies for the kth ring stiffener derived by Wang et al. (1997)

264

7 Application Examples

    2π  ∂u r k 2 ∂vr k 2 ∂wr k 2 + + bk dk ∂t ∂t ∂t 0   2 2 ∂ wr k + (Ixk + Izk ) (R + ek )dθ ∂t∂ x

1 Tr k = ρk 2

2π 

 Vr k = 0

(7.6.8a)

2 ∂wr k 1 ∂ 2 ur k − ∂x R + ek ∂θ 2  2 E k Ixk ∂ 2 wr k w + + rk ∂θ 2 2(R + ek )3 2  ∂vr k E k Ak + wr k + 2(R + ek ) ∂θ  2 2  ∂ wr k 1 ∂u r k G k Jk + + dθ 2(R + ek ) ∂ x∂θ R + ek ∂θ E k Izk 2(R + ek )



(7.6.8b)

where the effect of rotary inertia is neglected, ρk , E k , G k represent the mass density, Young’s modulus and shear modulus of the ring stiffener, bk , dk are the width and height of the ring stiffener as shown in Fig. 7.32, Ak = dk bk is the sectional area of the ring stiffener, Izk , Ixk , Jk are the area moments of inertia in each direction, and h + dk for the internally eccentric stiffener 2  ∂wk ek  ek ∂wk , vr k = vk 1 + − , wr k = wk = u k − ek ∂x R R ∂θ ek = −

ur k

(7.6.9a) (7.6.9b–d)

in which u k = u(x, t)|x=L k , vk = v(x, t)|x=L k , vk = v(x, t)|x=L k

(7.6.10a–c)

where L k is the x position of the ring stiffener as shown in Fig. 7.32. As in the case of displacements, we have Tr k =

∞  n=1

Tr kn , Vr k =

∞ 

Vr kn

(7.6.11a, b)

n=1

Following the same approach introduced earlier, we can obtain the kinetic and potential energies of the kth ring stiffener as follows: 1 T q˙ Mknuu q˙ nuc + q˙ Tnvs Mknvv q˙ nvs + q˙ Tnwc Mknww q˙ nwc 2 nuc T T + 2q˙ nuc Mknuw q˙ nwc +2 q˙ Tnvs Mknvw q˙ nwc +q˙ nus Mknuu q˙ nus

Tr kn =

7.6 Shell Structure

265

+ q˙ Tnvc Mknvv q˙ nvc + q˙ Tnws Mknww q˙ nws

T + 2q˙ nus Mknuw q˙ nws +2 q˙ Tnvc Mknvw q˙ nws ,

(7.6.12a)

1 T 1 T 1 T q Kknuu qnuc + qnvs Kknvv qnvs + qnwc Kknww qnwc 2 nuc 2 2 1 T 1 T T T + qnvs Kknvw qnwc + qnuc Kknuw qnwc + qnus Kknuu qnus + qnvc Kknvv qnvc 2 2 1 T T T + qnws Kknww qnws + qnvc Kknvw qnws + qnus Kknuw qnws (7.6.12b) 2

Vr kn =

where Mknuu = ρk Ak π(R + ek )Tuk uk ,  ek 2 T Mknvv = ρk Ak π(R + ek ) 1 + vk vk , R    ek 2 (Izk + Ixk ) Mknww = ρk Ak π(R + ek ) + L Ak L 2     ne 2 k T  T wk wk ,  wk  wk + 1 + R Mknuw = −

ρk Ak π(R + ek )ek T  uk  wk , L

ρk Ak π(R + ek )2 nek T vk wk , R2  π E k Izk n 4 + G k Jk n 2 Tuk uk , = 3 (R + ek ) Mknvw =

Kknuu

Kknvv =

π E k Ak n 2 (R + ek ) T vk vk , R2

(7.6.13a) (7.6.13b)

(7.6.13c) (7.6.13d)

(7.6.13e) (7.6.13f)

(7.6.13g)

  π E k Ak (R + n 2 ek )2 (R + ek )2 2 2 E Twk wk I (1 − n ) + k xk R2 (R + ek )3 

T 1

+ 2 E k Izk (R + (1 − n 2 )ek )2 + G k Jk n 2 R 2  wk  wk , L (7.6.13h)

π = E k Izk n 2 (R + (1 − n 2 )ek ) + G k Jk n 2 R Tuk  wk , (7.6.13i) 3 (R + ek ) L

Kknww =

Kknuw

Kknvw =

π E k Ak n(R + n 2 ek ) T vk wk R2

(7.6.13j)

266

7 Application Examples

Fig. 7.33 Baffled circular cylindrical shell

where uk = u (L k ), vk = v (L k ), wk = w (L k ),  wk =  w (L k ) (7.6.14a, b, c) For the added-mass computation of a fluid, we utilized the result of Kwak (2010), which is based on the baffled shell assumption, as shown in Fig. 7.33. In solving the fluid–structure interaction problem, the fluid is assumed to be inviscid, incompressible and irrotational. The governing equation for the fluid in contact with the shell is the Laplace equation which is given by ∇2φ = 0

(7.6.15)

The fluid is assumed to be inviscid, incompressible and irrotational. In the case of the baffled shell, the boundary condition at the fluid–structure interface is as follows: ∂φ = ∂r



−w(x, ˙ θ, t) 0

at r = R, 0 ≤ x ≤ L , at r = R, otherwise.

(7.6.16)

Equation (7.6.16) shows why infinitely long rigid cylinders are connected to both ends of the cylindrical shell. Otherwise, the addressed problem cannot be solved by the analytical approach. The velocity potential must also satisfy the radiation conditions which imply that the velocity potential should converge to zero: φ,

∂φ ∂φ , → 0 as x, r → ∞ ∂r ∂ x

(7.6.17)

7.6 Shell Structure

267

If we express the velocity potential in terms of the series expansion of the velocity potential for each circumferential mode as we do for the shell deflection in Eq. (7.6.1), we can write φ(r, x, θ, t) =

∞ 

φn (r, x, θ, t)

(7.6.18)

n=1

where

φn (r, x, θ, t) =  n (r, x) cos nθ q˙ nwc (t) + sin nθ q˙ nws (t)

(7.6.19)



in which  n (r, x) = n1 n2 · · · nm represents the vector of the admissible potential function corresponding to each generalized coordinate. Each velocity potential should also satisfy the Laplace equation so that we can obtain the following equation by inserting Eq. (7.6.19) into Eq. (7.6.15): 1 ∂ni ∂ 2 ni ∂ 2 ni n2 +  + = 0, − ni ∂r 2 r ∂r r2 ∂x2

i = 1, 2, . . . , m

(7.6.20)

Applying the Fourier transform into Eq. (7.6.20) and boundary conditions, Eq. (7.6.16), the general solution of Eq. (7.6.20) is found to be (Kwak 2010) −1 ni (r, x) = 2π

∞ −∞

L

Kn (|ξ |r ) dKn (|ξ |r )/dr |r =R

wi (x)e ¯ −iξ x¯ dx¯ eiξ x dξ

(7.6.21)

0

√ where i = −1. Using the symmetric condition, we can express the above equation as follows: −1 ni (r, x) = π

∞ 0

Kn (ξr ) dKn (ξr)/dr |r =R

L wi (x) ¯ cos ξ(x − x)d ¯ xdξ ¯

(7.6.22)

0

Equations (7.6.19) and (7.6.22) can be used to compute the velocity potential at an arbitrary location in three-dimensional fluid space. Let us derive the kinetic energy of the fluid due to the vibration of the shell. The kinetic energy of a fluid is expressed as follows: 1 Tf = − ρ f 2

2π L  ∂φ φ ∂r 0

=

1 ρf 2

0

2π L

dx Rdθ

r =R

φ(R, x, θ, t) w(x, ˙ θ, t)dx Rdθ 0

0

(7.6.23)

268

7 Application Examples

where ρ f is the mass density of the fluid. Using Eqs. (7.6.19), (7.6.1) and (7.6.22), Eq. (7.6.23) turns out to be Tf =

∞ 

T f n,

(7.6.24)

n=1

where Tfn =

1 T 1 q˙ nwc M f nww q˙ nwc + q˙ Tnws M f nww q˙ nws , 2 2

(7.6.25)

in which ¯ n f ww M f nww = ρ f R L 2 π M

(7.6.26)

represents the added virtual mass matrix due to the fluid. The element of the nondimensionalized added virtual mass matrix is expressed as (Kwak 2010)  ¯ f nww = 1 M ij π 1 1 0

∞ 0

 −Kn ξ¯    −ξ¯ Kn+1 ξ¯ + nKn ξ¯

wi (η)wj (η) ¯ cos ξ¯ (x − η)d ¯ ηdηd ¯ ξ¯ .

(7.6.27)

0

where Kn is the Bessel function of the second kind of order n. We only consider the case in which a piezoelectric actuator and a sensor are mounted in the circumferential direction, as shown in Fig. 7.34. Fig. 7.34 Piezoelectric wafer attached to the cylindrical shell

7.6 Shell Structure

269

The MFC actuator used in the experiments can be assumed as a unidirectional actuator acting in the circumferential direction as shown in Fig. 7.34. Hence, the virtual work done by the actuator for the nth (n ≥ 1) circumferential mode can then be expressed as (Kwak et al. 2009)  i T i  i T i i T T bCanvs V pa bCanwc V pa = δqnvs + δqnwc δWCn  i T i  i T i T T + δqnvc + δqnws bCanvc V pa bCanws V pa

(7.6.28)

i where V pa is the actuating voltage supplied to the ith piezoelectric actuator and i

x pae i i biCanvs = E p d33 (sin nθ pae − sin nθ pas )

v dx

(7.6.29a)

x ipas i i i i cos nθ pae − θ pas cos nθ pas ) biCanwc = E p d33 [(θ pae i

x pae

nh i i + (sin nθ pae − sin nθ pas )] 2R

w dx

(7.6.29b)

x ipas i

x pae i i − cos nθ pas ) biCanvc = −E p d33 (cos nθ pae

v dx

(7.6.29c)

x ipas i i i i biCanws = E p d33 [(θ pae sin nθ pae − θ pas sin nθ pas ) i

nh i i − (cos nθ pae − cos nθ pas )] 2R

x pae w dx

(7.6.29d)

x ipas

in which E p is the Young’s modulus of the piezoelectric material, and d 33 is the piezoelectric constant. The piezoelectric sensor equation can be expressed as (Kwak et al. 2009) i =− V ps

∞ 1  i i i i b (7.6.30) q + b q + b q + b q nvs nwc nvc nws Csnvs Csnwc Csnvc Csnws Cci n=1

i where V ps is the measured voltage of the ith piezoelectric sensor, CCi is the capacitance of a charge amplifier connected to the ith piezoelectric sensor, and

270

7 Application Examples i

x pse i i biCsnvs = E p d33 (sin nθ pse − sin nθ pss )

v dx

(7.6.31a)

x ipss i i i i biCsnwc = E p d33 [(θ pse cos nθ pse − θ pss cos nθ pss ) i

nh i i + (sin nθ pse − sin nθ pss )] 2R

x pse w dx

(7.6.31b)

x ipss i

x pse i i biCsnvc = −E p d33 (cos nθ pse − cos nθ pss )

v dx

(7.6.31c)

x ipss i i i i sin nθ pse − θ pss sin nθ pss ) biCsnws = E p d33 [(θ pse i

nh i i − (cos nθ pse − cos nθ pss )] 2R

x pse w dx

(7.6.31d)

x ipss

Using Eqs. (7.6.4a, 7.6.4a), (7.6.5a–i), (7.6.14), (7.6.15), (7.6.25), (7.6.28) and (7.6.30), we can obtain the equations of motion of the cylindrical shell for the nth circumferential mode and the corresponding sensor equation. M∗ q¨ n + Kn∗ qn = B∗n V pa , n = 1, 2, . . . V ps =

∞ 

C∗n qn

(7.6.32a)

(7.6.32b)

n=1 T T T qn2 ] is the generalized coordinate vector consisting of qn1 (t) = where qn (t) = [qn1 T T T T T T T 1 2 na T qnvc qnws ]T , Vipa = [ V pa V pa . . . V pa ] , [qnuc qnvs qnwc ] and qn2 (t) = [qnus i 1 2 ns T V ps = [ V ps V ps . . . V ps ] , n a and n s are the numbers of actuators and sensors, and     Mn 0 Kn 0 , Kn∗ = (7.6.33a, b) Mn∗ = 0 Mn 0 Kn

7.6 Shell Structure

271

⎡

0

T ⎢ b1 ⎢  Canvs T ⎢ 1 ⎢ b B∗n = ⎢ Canwc ⎢ 0 ⎢ 1 ⎣ bCanvc T  1 T bCanws

⎡

1 1 /Cc1 bCsnwc /Cc1 0 bCsnvs 2 ⎢ 0 b2 ∗ Cn = −⎣ Csnvs /C c2 bCsnwc /C c2 .. .. .. . . .

⎤ ... ...⎥ ⎥ ⎥ ...⎥ ⎥ ...⎥ ⎥ ...⎦ ...

(7.6.33c)

⎤ 1 1 /Cc1 bCsnws /Cc1 0 bCsnvc 2 2 0 bCsnvc /Cc2 bCsnws /Cc2 ⎥ ⎦ .. .. .. . . .

(7.6.33d)

0

 2 T b  2Canvs T bCanwc 0  2 T b  2Canvc T bCanws

B∗n is the force participation matrix, which reflects the effect of the applied voltage on each assumed mode, and C∗n is the mode influence matrix, which reflects the effect of each mode on the sensor voltage. In addition, ⎤ nr nr   M + M 0 M knuw ⎥ ⎢ uu k=1 knuu k=1 ⎥ ⎢ n r ⎥ ⎢  ⎥, ⎢ 0 M + M M Mn = ⎢ vv knvv nvw ⎥ k=1 ⎥ ⎢ nr nr nr ⎦ ⎣    T T Mknuw Mknvw Mww + M f nww + Mknww ⎡

k=1

k=1

k=1

(7.6.34a) ⎤

⎡

nr nr nr    K + K K + K K + Kknuw ⎥ nuu knuu nuv knuv nuw ⎢ k=1 k=1 k=1 ⎥ ⎢ n n n ⎥ ⎢ T r T r r ⎢ Knuv Knvv + Kknvv Knvw + Kknvw ⎥ Kn = ⎢ Knuv + ⎥ k=1 k=1 k=1 ⎥ ⎢ nr nr nr ⎦ ⎣ T    T T T Knuw + Knuw Knvw + Knvw Knww + Kknww k=1

k=1

(7.6.34b)

k=1

It can be readily seen from Eq. (7.6.34a, b) that the mass and stiffness matrices reflect the effects of the ring stiffener and the external fluid. Identical matrices appear in the mass and stiffness matrices of Eq. (7.6.33a–d) because they belong to the sine and cosine modes, respectively. Figure 7.35 shows the ring-stiffened cylindrical shell without water-proof end caps. Welding was not considered to guarantee the accordance with the theoretical model. The ring-stiffened cylindrical shell was manufactured by a lathe from a single aluminum cylindrical block. The cylindrical shell has the following material properties: L = 0.7 m, h = 0.002 m, R = 0.15 m, E = 200 GPa, v = 0.3, ρ = 7850 kg/m3 , ρ f = 1000 kg/m3 . The ring stiffener has the width: 5 mm, height: 5 mm. Figure 7.36 shows the dimension of the ring-stiffened cylindrical shell. Five MFC actuators are glued to the inner surface of the cylindrical shell as shown in Fig. 7.37. The MFC actuator used in this study is the M8557-P1 actuator of Smart Materials Inc. whose size is 103 mm × 64 mm. The MFC sensor used in this study is

272

7 Application Examples

Fig. 7.35 Ring-stiffened aluminum cylindrical shell

Fig. 7.36 Dimensions of ring-stiffened aluminum cylindrical shell

the M4010-P1 whose size is 54 mm × 22 mm. The material properties of the MFC actuator are E 1 = 30.336 GPa, E 2 = 15.857 GPa(Electrode direction), v12 = 0.31, v21 = 0.16, G 12 = 5.515 GPa. The piezoelectric constant of the MFC actuator found in the specification is d 33 = 4.6 × 102 . In the control experiment, the MFC actuators 2 and 3 were connected in parallel and used as one actuator. Since the MFC sensor is located between the MFC actuators 2 and 3, the configuration may be considered as that of collocated control. The MFC actuator 1 was used as the disturbance source. The MFC actuator 4 and 5 were not used for the experiment. The boundary condition for the stiffened shell was considered to be the shear diaphragm condition based on the result of Kwak et al. (2009). Hence, the admissible functions that satisfy the boundary conditions was expressed as

7.6 Shell Structure

273

Fig. 7.37 Macro-composite actuators attached to the cylindrical shell

ui (x) =

√

2 cos

√ iπ x iπ x , vi (x) = wi (x) = 2 sin , i = 1, 2, . . . , m. L L (7.6.35)

Inserting Eq. (7.6.35) into Eqs. (7.6.5a–i) and (7.6.13a–j), we can calculate the mass and stiffness matrices for each nth circumferential mode. Furthermore, inserting Eq. (7.6.35) into Eq. (7.6.27), we can calculate the added virtual mass matrix due to the fluid effect. The infinite integral limit of Eq. (7.6.27) was replaced by 10 as recommended by Kwak (2010). We can then compute the natural frequencies and natural modes of the ring-stiffened cylindrical shell in vacuo or in contact with unbounded external fluid by solving the eigenvalue problem consisting of the mass and stiffness matrices given by Eq. (7.6.35). Figure 7.38 show the theoretical natural frequencies and mode shapes in air. The natural mode shapes in water are almost the same as the one in air but the natural frequencies in water are about one-third of the natural frequencies in air. The external fluid has a huge effect on the natural frequencies but little effect on the mode shapes for the fully-submerged shell. In addition, the first and second natural frequencies in water are not well separated so the control technique should be able to cope with this case. The dynamic signal analyzer, HP35670A, was used to measure the transfer function between the MFC actuator and the MFC sensor, which is shown in Fig. 7.39. Also, the modal test was carried out using the impact hammer, PCB 086C03, the accelerometer, PCB 353B15, the LMS Scadas Mobile, and Test Lab. The natural frequencies and mode shapes in air identified by the modal software are shown in Fig. 7.40. However, the modal testing method could not be used when the ringstiffened shell was immersed into water as shown in Fig. 7.41. Hence, the dynamic signal analyzer was again used to measure the natural frequencies in water, which were identified from the transfer function shown in Fig. 7.42. Table 7.1 summa-

274

7 Application Examples

(a) 458 Hz (3,1)

(c) 725 Hz (4,1)

(b) 523 Hz (2,1)

(d) 916 Hz (4,2)

Fig. 7.38 Theoretical natural frequencies and mode shapes in vacuo

rizes the theoretical and experimental natural frequencies. Table 7.1 shows that the theoretical natural frequencies in vacuo are in good agreement with ones in air and the theoretical natural frequencies in water are slightly lower than the experimental natural frequencies in water except for the (5,1) mode. It is due to the fact that the baffled shell theory for the fluid–structure interaction constrains the fluid motion because of artificial rigid cylinders attached to both ends of the cylindrical shell, thus increases the kinetic energy of the fluid, and in turn increases the virtual fluid mass. Equation (7.6.32a, b) consists of the equation of motion for each circumferential mode. Using the orthonormality condition and the modal transformation, qn = Un pn , Eq. (7.6.32a, b) can be converted to the modal equations of motion and the modal sensor equation p¨ n + n pn = B¯ ∗n V pa , n = 1, 2, . . . , V ps =

∞ 

¯ ∗n pn C

(7.6.36a, b)

n=1

¯ ∗n = C∗n Un . After rearranging Eq. (7.6.36) in the ascending where B¯ ∗n = UnT B∗n and C order of natural frequency and considering a small number of natural modes to be controlled, we can form the reduced-order modal equations of motion and the modal sensor equation. ξ¨ + 2Zξ˙ + ξ = Ba V pa , V ps = Cs ξ

(7.6.37)

7.6 Shell Structure

Fig. 7.39 Transfer function measurement of the shell in air

275

276

7 Application Examples

(a) 482 Hz (3,1)

(c) 715 Hz (4,1)

(b) 520 Hz (2,1)

(d) 936 Hz (4,2)

Fig. 7.40 Experimental natural mode shapes in air

Fig. 7.41 Experimental setup for the stiffened cylindrical shell submerged in water

7.6 Shell Structure

Fig. 7.42 Transfer function between the piezoelectric sensor and actuator in water

277

278

7 Application Examples

Table 7.1 Comparison of natural frequencies in air and water Mode

Theoretical natural frequencies in vacuo (Hz)

Experimental natural frequencies in air (Hz)

Theoretical natural frequencies in water (Hz)

Experimental natural frequencies in water (Hz)

(3,1)

458

482

162

173

(2,1)

523

520

167

178

(4,1)

725

715

283

291

(4,2)

916

936

366

388

(3,2)

958

948

353

365

(5,1)

1141

1084

483

477

where ξ represents the vector consisting of a generalized coordinate to be controlled, Z is the matrix consisting of modal damping factors,  is the diagonal matrix consisting of natural frequencies,  is the diagonal matrix consisting of the squares of the natural frequencies, Ba and Cs are the matrices of force participation and mode influence, respectively. It should be noted here that the damping term was added in Eq. (7.6.37) on purpose. Based on the result for the one-degree-of-freedom system, the filter damping ratio, ζ f = 0.01, for the modified HHC was considered for the active vibration control of the submerged shell subjected to harmonic disturbance. The single-input single-output modified HHC was then applied to the dynamic model derived in the previous chapter. The parameters a and b of the modified HHC were extracted from the transfer function shown in Fig. 7.43. The active vibration controller was implemented digitally by using the DS1104 of dSPACE Inc with 20 kHz sampling rate. Figure 7.44 shows the experimental setup for the active vibration control. The performance of the active vibration control was also measured by the hydrophone. Figure 7.45 shows the hydrophone positions. The sound levels were measured by the hydrophone arrayed around 1-m and 2-m radii circles. Each position was marked by h1 through h8(1-m radius) and H1 through H8(2-m radius). Three excitation frequencies were considered. The first excitation frequency is equal to the first natural frequency and the third excitation frequency is equal to the second natural frequency. The second excitation frequency is the nonresonant frequency between the first and second natural frequencies. Figures 7.46 shows the uncontrolled and controlled time responses of the piezoelectric sensor and hydrophone output. The vibration responses were suppressed rapidly after the control was applied. Figures 7.47 shows the uncontrolled and controlled peak amplitudes of the sound levels measured around the 2-m radius circle. It shows that about 10-dB reductions on average are possible for resonant and non-resonant modes. This demonstrates that the modified HHC can be effectively used for the vibration suppression as well as sound level reduction of the fully submerged ring-stiffened cylindrical shell equipped with piezoelectric sensors and actuators.

7.7 Elevator Vibration Control

279

Fig. 7.43 Closed-loop gain for the shell by MHHC

Fig. 7.44 Measurement and control flow diagram for active vibration control

7.7 Elevator Vibration Control Vibration that hinders the ride comfort of an elevator is generally divided into lateral vibration and longitudinal vibration, and the main structural component of an elevator consists of a cage, a frame, and a roller guide as shown in Fig. 7.48. Anti-vibration

280

7 Application Examples

Fig. 7.45 Hydrophone positions for sound level measurement

rubber is inserted between the frame and the cage to reduce the vibration generated inside the elevator and the vibration transmitted from the outside. The anti-vibration rubber attached to the lower part of the cage plays a role in reducing the vertical and lateral vibration of the cage, and the back and forth vibration, and the anti-vibration rubber attached to the upper part of the cage mainly serves to prevent the cage from shaking. The roller guide contacts the guide rail and plays the same role as the wheel of a general vehicle. Each roller guide is composed of three wheels as shown in Fig. 7.49 and is operated in close contact with the T-shaped guide rail. Each wheel is supported by a spring. In general, a damping device is attached to the roller guide to absorb vibration transmitted from the guide rail. As an external factor that causes lateral vibration and back and forth vibration of an elevator running at high speed, as shown in Fig. 7.48, it is possible to think of the rail irregularity and the wind pressure generated when the elevator crosses. Since the elevator is a system that runs vertically along the guide rails installed on the hoist way, vibrations inevitably occur in the elevator when driving on a connection part between the guide rails or a curved guide rail. In order to minimize such vibration, the irregularity of the guide rail must be minimized above all, which is a very difficult problem in reality. In general, a damping device is attached to the roller guide to

7.7 Elevator Vibration Control

281

Fig. 7.46 Time responses of piezoelectric sensor and hydrophone output by 178 Hz excitation

282

Fig. 7.47 Hydrophone(H) sound level by 178 Hz excitation Fig. 7.48 Structure of elevator

7 Application Examples

7.7 Elevator Vibration Control

283

Fig. 7.49 Roller guide system (RGS) of elevator (courtesy of Hyundai Elevator Co.)

reduce vibration transmitted to the cage. Recently, high-speed elevators are being introduced for skyscrapers. High-speed elevators used in skyscrapers aim to operate at a speed of 1080 m or more per minute. When operating at such a speed, the vibration of the elevator naturally becomes large, and excessive vibration causes anxiety to passengers. Elevators move along rails like trains, and rubber wheels and springs are used for roller guides corresponding to train wheels in order to secure ride comfort against rail bending or other disturbances. There is inevitably a limit to the vibration suppression of high-speed elevators as a passive vibration suppression method using anti-vibration rubber and dampers. Hence, the active roller guide has been developed by many elevator companies. Elevator companies have tried various actuators (AC Servo Motor, Electromagnet, Voice Coil Actuator, etc.) and controlled the vibration of the elevator successfully to reduce the vibration. This section introduces the research results of Kwak et al. (2011), Baek et al. (2011), and Lee et al. (2012) on the active vibration control of elevator. The dynamic model for controlling the lateral vibration of the elevator is shown in Fig. 7.50. Since the outer ring of the guide roller is covered with rubber, it was assumed to be a spring-damper along with the anti-vibration rubber that supports the cage. The equation of motion for the model in Fig. 7.50 is derived as follows (Kwak et al. 2011) M¨x + C¨x + Kx = Bc fc + fr + Bw fw

(7.7.1)

284

7 Application Examples

Fig. 7.50 Mathematical model for elevator vibration

where, x = [xc θc xb θb θ1 θ2 θ3 θ4 ]T represents a vector consisting of the frame, the lateral displacement and rotation angle of the cage, and the rotation angle of each roller guide. fc = [M1 M2 M3 M4 ]T is the control force vector composed of the control moment acting on each roller guide, fr is the external force due to deformation of the guide rail, and fw is the disturbance due to air pressure fluctuations. And each matrix and vector are as follows. ⎡ ⎤ M11 M12 0 0 −M15 M15 M15 −M15 ⎢ M12 Jct 0 0 Jgt −Jgt Jgs −Jgs ⎥ ⎢ ⎥ ⎢ 0 0 mb 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎥ 0 0 Jb 0 ⎢ 0 (7.7.2) M=⎢ ⎥ ⎢ −M15 Jgt 0 0 Jgg 0 0 0 ⎥ ⎢ ⎥ ⎢ M15 −Jgt 0 0 0 Jgg 0 0 ⎥ ⎢ ⎥ ⎣ M15 Jgs 0 0 0 0 Jgg 0 ⎦ 0 0 Jgg −M15 −Jgs 0 0 0

7.7 Elevator Vibration Control

285

⎤ C11 C12 C13 C14 C15 C16 C17 C18 ⎢ C12 C22 C23 C24 C25 C26 C27 C28 ⎥ ⎥ ⎢ ⎥ ⎢C C C C ⎢ 13 23 33 34 0 0 0 0 ⎥ ⎥ ⎢ 0 0 0 0 ⎥ ⎢C C C C C = ⎢ 14 24 34 44 ⎥ ⎢ C15 C25 0 0 C55 0 0 0 ⎥ ⎥ ⎢ ⎢ C16 C26 0 0 0 C66 0 0 ⎥ ⎥ ⎢ ⎣ C17 C27 0 0 0 0 C77 0 ⎦ C18 C28 0 0 0 0 0 C88 ⎡ ⎤ K 11 K 12 K 13 K 14 K 15 K 16 K 17 K 18 ⎢ K 12 K 22 K 23 K 24 K 25 K 26 K 27 K 28 ⎥ ⎢ ⎥ ⎢K K K K ⎥ ⎢ 13 23 33 34 0 0 0 0 ⎥ ⎢ ⎥ 0 0 0 0 ⎥ ⎢K K K K K = ⎢ 14 24 34 44 ⎥ ⎢ K 15 K 25 0 0 K 55 0 0 0 ⎥ ⎢ ⎥ ⎢ K 16 K 26 0 0 0 K 66 0 0 ⎥ ⎢ ⎥ ⎣ K 17 K 27 0 0 0 0 K 77 0 ⎦ K 18 K 28 0 0 0 0 0 K 88   0 Bc = 4 , Bw = [ 0 0 0 0 1 − h w 0 0 ]T I4 ⎫ ⎧ 4 ⎪  ⎪ ˙ ⎪ (cr + c f ) Ri + (kr + k f )Ri ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ fr θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎬ ⎨ fr = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ fr 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ ⎪ r 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ ⎪ r 3 ⎪ ⎪ ⎭ ⎩ fr 4 ⎡

(7.7.3)

(7.7.4)

(7.7.5)

(7.7.6)

and fr θ = − cr (h p + l g )( R˙ 1 + R˙ 2 ) + kr (h p + l g )(R1 + R2 ) + cr (h q + l g )( R˙ 3 + R˙ 4 ) + kr (h q + l g )(R3 + R4 ) − c f (h p ( R˙ 1 + R˙ 2 ) − h q ( R˙ 3 + R˙ 4 )) − k f (h p (R1 + R2 ) − h q (R3 + R4 ))

,

(7.7.7)

where m c , m b , m g represent the mass of the frame, cage, and roller guide, and Jc , Jb , Jg represent the mass moment of inertia about the center of mass of the frame, cage, and guide roller. ks , kr , k f ,kb1 , kb2 , kb3 are shown in Fig. 7.49 shows the stiffness of the spring, and cs , cr , c f ,cb1 , cb2 , cb3 represent the viscos damping coefficient.

286

7 Application Examples

As in the case of lateral vibration, a damping device is attached to the guide roller to reduce the back and forth vibration. The guide roller is installed twice as much as the horizontal direction in the transverse direction, so the level of the transverse vibration is observed to be relatively low compared to the lateral vibration. However, when the elevator is operated at high speed, since the small bend of the guide rail has a great influence on the overall vibration, it is necessary to try to minimize the transverse vibration. Refer to the reference Kwak et al. (2011), and Baek et al. (2011) for the dynamic modeling of elevator transverse vibration. For numerical simulation, the following parameter values were extracted based on the specifications of Tower Unit 5 installed in Hyundai Elevator Co., Ltd. The damping factor was assumed to be proportional damping as follows. m c = 2268 kg, m b = 1858kg, m g = 5 kg, Jc = 10829 kg m2 , Jb = 3922 kg m2 , Jg = 0.2 kg m2 , Jcx = 9627 kg m2 , Jbx = 3350 kg m2 , r g = 0.14 m, rb = 1.1 m, rs = 1.2 m, l g = 0.153 m, ls = 0.223 m, h p = 3.309 m h q = 2.961 m, h b = 1.622 m, h c = 1.405 m, lb = 1.074 m, lc = 1.291 m ks = 12.46 × 103 N/m, kr = 65 × 104 N/m, k f = 25 × 104 N/m, kb1 = 12 × 105 N/m, kb2 = 20 × 105 N/m, kb3 = 65 × 104 N/m, kb4 = 65 × 104 N/m, cs = 0.02ks , cr = 0.001kr , c f = 0.02k f , cb4 = 0.02kb4 , cb1 = 0.02kb1 , cb2 = 0.02kb2 , cb3 = 0.02kb3 . As a result of performing free vibration analysis using this numerical value, the first four natural frequencies were calculated as 2.5, 4.3, 9.1 and 12.7 Hz, respectively. Figure 7.51 shows the natural vibration mode for each natural frequency. Looking at Fig. 7.51a, which is the first natural mode, it can be seen that the primary mode of the elevator is the lateral movement, and the frame and the cage move in the same direction. Figure 7.51b, showing the second natural mode, shows the in-phase rotational vibration in which the cage and the frame rotate in the same direction. Figure 7.51c, which shows the third natural mode, shows out-of-phase rotational vibration. Figure 7.51d also shows the rotational mode. It was observed that these natural frequencies almost coincide with the actual elevator’s natural frequencies. Equation (7.7.1) is not suitable for numerical calculation or controller design using matlab. Converting to an equation of state for numerical calculation, it can be written as follows. ¯ + B¯ f c + d z˙ = Az

(7.7.8)

where ¯ = A



     0 0 I 0 ¯ = , d = , B M−1 (fr + Bw fw ) −M−1 K −M−1 C M−1 Bc (7.7.9a, b)

7.7 Elevator Vibration Control

(a) 1st mode : 2.5Hz

(c) 3rd mode : 9.1Hz

287

(b) 2nd mode : 4.3Hz

(d)4th mode: 12.7Hz

Fig. 7.51 Natural modes and frequencies of elevator lateral motion

Figure 7.52 shows the rail deformation data measured at the elevator tower site. The flexural displacement over the total rail length of 180 m was measured and it shows a deviation of about 1 mm. The active vibration controller considered in this study was a single input/output PID controller. First, it was assumed that an accelerometer that can measure the acceleration of the frame is attached 1.5 m below the center of mass of the elevator frame. And it was assumed that the control forces are two actuators mounted on the bottom RGSs that can be simultaneously driven. In addition, the driving condition of the elevator was assumed to accelerate to reach a speed of 1080 m/min in 9 s, maintain this speed for 1 s, and then decelerate. With the rail deformation data given by Fig. 7.52 and the driving condition, the disturbance data was generated. Figure 7.53 shows the Simulink model without control and with PID control algorithm. Figure 7.54 shows the comparison between the uncontrolled and controlled responses. From Fig. 7.54, it can be seen that the maximum amplitude occurs in the section where the operating speed of the elevator is maximized. In

288

Fig. 7.52 Rail deformation data

7 Application Examples

7.7 Elevator Vibration Control

289

Fig. 7.53 Simulink block diagram for elevator vibration simulation

addition, it can be seen that the maximum amplitude is reduced by about 50% when the controller is driven. Based on the numerical studies Kwak et al. (2011), and Baek et al. (2011) on the lateral vibration and back and forth vibration control of elevators, the validity of the active vibration control was verified through an active vibration control experiment for an actual elevator. The vibration of the elevator to be controlled in this study is a vibration having a frequency band of 3–10 Hz with an acceleration size of 10 mg or less. In particular, the research goal was set to suppress the 3–4 Hz component that most affects the ride comfort. First, in order to control the lateral vibration of the elevator, an active roller prototype system as shown in Fig. 7.55 was designed and manufactured. The active

290

7 Application Examples

Fig. 7.54 Uncontrolled and controlled responses of elevator by rail irregularity

Fig. 7.55 1-axis active roller guide (courtesy of Hyundai Elevator Co.)

roller system in Fig. 7.55 consists of a VCA, roller, lever and support spring supported by a linear guide. The external magnet cylinder and the internal coil cylinder of the linear actuator VCA are non-contact, so a linear guide as shown in Fig. 7.56 is essential. The actuation mechanism shown in Fig. 7.55 is in a form in which moment is transmitted to the cage through a link and hinge structure. The actuation mechanism fits into the existing RGS without changing its configuration. In order to check the performance of the active roller system as shown in Fig. 7.55, a feedback control system as shown in Fig. 7.57 was constructed. The system shown

7.7 Elevator Vibration Control

291

Fig. 7.56 Voice coil actuator mounted on linear guide (courtesy of Hyundai Elevator Co.)

Fig. 7.57 Controller and power amplifier system for active roller guide (courtesy of Hyundai Elevator Co.)

in Fig. 7.57 consists of an accelerometer, a digital signal processing (DSP) board, and a VCA driving amplifier. Endveco’s accelerometer with a sensitivity of 500 mV/g was used as a sensor, and TMS320 F2812 was used as a controller to implement the control algorithm. After designing and testing the active roller system shown in Fig. 7.55, the passive roller guides installed at the bottom of the elevator were replaced with new twoaxis RGSs to control the lateral and transverse vibrations as shown in Fig. 7.58. A total of 6 VCAs were used, 2 VCAs for lateral vibration control, and 4 VCAs for transverse vibration control. In addition, two accelerometers were used, respectively,

292

7 Application Examples

Fig. 7.58 2-axis active roller guide system (courtesy of Hyundai Elevator Co.)

for the lateral and transverse directions. The lateral and transverse active RGSs were operated independently. In other words, SISO control system is implemented for each direction, so that two VCAs are driven in one direction at the same time based on the accelerometer signal in the lateral direction, and four VCAs are programmed to drive in the transverse direction at the same time based on the accelerometer signal in the transverse direction. The active roller system is almost located at the center of gravity of the elevator and is designed to control the pure translational mode. The controller and power amplifier circuit in Fig. 7.57, which was temporarily manufactured and used to drive the 1-axis controller, was manufactured as a PCB as shown in Fig. 7.59 and mounted in an actual elevator. The control system shown in Fig. 7.59 consists of a control unit, a drive unit, and a power supply unit, and the input

7.7 Elevator Vibration Control

293

Fig. 7.59 Controller board (courtesy of Hyundai Elevator Co.)

unit of the control unit is designed to receive accelerometer signals. To measure lowfrequency vibrations of less than 10 mg, a high-sensitivity low-frequency accelerometer is preferred. Hence, Jewell’s servo accelerometer with a sensitivity of 10 V/g was used. The output terminal of the driving part is connected to the VCA. TMS 320F2812 DSP was used as a controller to implement the control algorithm. The ADC inside this DSP has 12-bit resolution, and the controller can execute the control algorithm at a 1 kHz sampling rate using an internal timer. As mentioned above, since it tries to control the vibration component of 10 Hz or less, it was determined that the digital control algorithm could be sufficiently implemented with a sampling rate of 1 kHz. The biggest problem encountered when attempting actual control is the VCA reaction speed, and through the experiment, it has been confirmed that it can be driven up to 15 Hz without phase delay. The active roller guide system is equipped with 6 VCAs, so a number of PWM signals are required to control them independently. The DSP board considered as a controller has enough number of 12 bit PWM (Pulse Width Modulation) ports. Due to the driving characteristics of PWM, noise due to pulses may occur. The DSP chip considered could be programmed to generate a 20 kHz square wave, so there is no noise when driving the VCA. PWM signal is passed to VCA through current amplifier. The control system periodically receives the elevator’s up-down operation signal through CAN (Controller Area Network) communication. This signal is used to operate the active roller system to minimize power consumption. In addition, the control system is equipped with various monitoring functions to protect passengers and the system. The DSP applies an operation signal to the Watchdog System periodically (10 Hz). If this signal is blocked by an error in the DSP, the Watchdog System sends a reset signal to the controller to protect the controller and at the same time cuts off the power supplied to the actuator. The controller also monitors the current applied to the actuator and executes a protection program to prevent damage to the actuator that may occur if overcurrent flows. In addition, the program was designed to stop the operation when excessive vibration occurs for a certain period of time.

294

7 Application Examples

A simple proportional control was considered. In other words, it means that each active roller guide feeds back a value proportional to the acceleration in each direction. Since the measured acceleration signal in actual vibration control contains noise, a narrow band pass filter (BPF) was considered in this study. For this, HPF (HighPass Filter) and LPF (Low-Pass Filter) are combined, and the transfer function is as follows. ω12 ω22 s 2 + c2 s 2 + c3 s + c4

(7.7.10)

c1 = 2ζ1 ω1 + 2ζ2 ω2 , c2 = 4ζ1 ζ2 ω1 ω2 + ω12 + ω22

(7.7.11a, b)

c3 = 2ζ1 ω12 ω2 + 2ζ2 ω1 ω22 , c4 = ω12 ω22 ,

(7.7.11c, d)

H (s) =

s4

+ c1

s3

where

in which ω1 and ω2 are the filter frequencies of HPF and LPF, and ζ1 and ζ2 are the damping factors of HPF and LPF. Figure 7.60 shows the frequency response curve of the BPF given by Eq. (7.7.10). Where ζ1 and ζ2 are 0.6 and the filter frequencies are ω1 = 2π and ω2 = 50π . As can be seen from Fig. 7.60, it can be seen that the signal between ω1 and ω2 passes. The transfer function given by Eq. (7.7.10) cannot be directly transferred to the DSP, which is a digital controller. So, the discretized control equation was derived using z-transform. The execution file of the control program written in C language was downloaded and used by DSP controller. In the

Fig. 7.60 Bode diagram for BPF filter

7.7 Elevator Vibration Control

295

experiments, the vibration control performance was monitored while increasing the gain value of the proportional controller. Through this, the optimum value, before the controller becomes unstable, was selected as the proportional control gain. The active vibration control experiment using active RGSs was conducted at Asan Tower Units built in Hyundai Elevator Co., Ltd. Each tower has an elevator operating speed of 600 m/min. Figures 7.61 shows the experimental results of the acceleration in X (transverse) direction and Y (lateral) direction when the cage moved in the upward or downward direction. As can be seen from Fig. 7.61, even without active vibration control, the acceleration of the elevator does not exceed ± 4 mg in both the transverse and lateral directions. However, it shows that even such small vibrations can be suppressed when the active vibration control system is activated. Table 7.2 shows the peak-topeak values and 95% values, and from this table it can be seen that vibration level was reduced by about 30% (Fig. 7.62). Experiments using elevator towers after mounting the active RGS on the frame required huge men power, cost, and time. Hence, the HILS system was considered to test the control algorithm before installing the active RGS on a real elevator. The HILS system shown in Fig. 7.63 was built for this purpose. The HILS system shown in Fig. 7.63 consists of a real RGS and the disturbance realization system driven by the AC servo motor. Table 7.3 shows the list of equipment used to compose the above equipment. The laser sensor was used to accurately measure the displacement of the AC servo motor. The AC servo motor connected to a ball-screw mechanism produced linear motion. The AC servo motor was driven using speed mode to trace the given position. MicroLabBox of dSPACE Inc. was used for the operation of the HILS system as well as the vibration control algorithm. The force transducer installed on the RGS side is to measure the reaction force generated by the RGS and the rail displacement device. The measured force was fed into the numerical model to calculate the motion of the elevator. Figure 7.63 shows the overall structure of the HILS system constructed. In an actual elevator, a total of 4 RGS units are installed on both the upper and lower sides. However, one real RGS was actually used, and a dynamic model was used for the rest as shown in Fig. 7.64. In the case of excitation with the HILS device, the reaction force caused by the actual RGS is processed with the theoretical model, and the response of the system generated at this time can be calculated. Figure 7.65 shows the schematic diagram for the constructed HILS system. The AC servo motor is driven to excite the RGS, and the AC servo motor is controlled using the signal received from the laser displacement sensor. The reaction force generated by RGS is measured by a force transducer, and this signal is input to the MicroLabBox. Disturbances and reaction forces are fed into the elevator theoretical model, and finally the response of the elevator cage, which is assumed to be a two degree of freedom system, is calculated. For the experiment using the 2 DOF HILS device, a dynamic model was constructed as shown in Fig. 7.66. The reaction force of the actual RGS was used as F in the numerical model. The state-space equation for this model can be written as

296

7 Application Examples

Fig. 7.61 Elevator vibration responses of No. 4 elevator of Hyundai Elevator Co.

7.7 Elevator Vibration Control

297

Table 7.2 Comparison of accelerations of No. 4 elevator in X and Y directions Direction

X

Acc (mg)

Max Pk/Pk

A95

Max Pk/Pk

Y A95

Uncontrolled

5.7

4.5

6.9

5.3

Controlled

4.5

3.7

3.7

3.3

Fig. 7.62 Frequency responses of No. 4 elevator with active RGS of Hyundai Elevator Co.

z˙ = As z + Bs f y = Cs z + Ds f

(7.7.16)

where z = [x θ x˙ θ˙ ]T and   

I2×2 02×2 02×2 As = , Bs = −1 −1 −1 , Cs = I2×2 02×2 , Ds = 02×2 −M K −M C M (7.7.17a–d) 

in which

298

7 Application Examples

Fig. 7.63 Overall of the HILS system with RGS (courtesy of Hyundai Elevator Co.) Table 7.3 List of equipment used in the HILS system

Fig. 7.64 HILS system with single RGS and mathematical model

Equipment

Company

Model No.

Laser sensor

WenglorMEL

DB-M3-E

AC servo motor

Mitsubishi

HF-MP73

Servo driver

Mitsubishi

MR-J3-70A

Force transducer

Curiosity Tech

CMNT-50L

Control board

dSPACE

MicroLabBox

7.7 Elevator Vibration Control

299

Fig. 7.65 Schematic diagram for the HILS system with RGS

Fig. 7.66 Two-degree-of-freedom model for the HILS system

   3c c(L m 0  2b − 2L 2t ) ,C = c(L b − 2L t ) c L b + 2L t 0 IG   − 2L 3k k(L ) b t , f = C y˙ y˙ + K y y + B f F  K= k(L b − 2L t ) k L 2b + 2L 2t 

M=

(7.7.18a, b)

(7.7.18c, d)

where y = [yb yt ]T      1 2 1 2 1 , Ky = k , Bf = Cy = c L b −2L t L b −2L t Lb 

(7.7.19a–c)

300

7 Application Examples

The schematic block diagram of the 2-DOF HILS system is shown in Fig. 7.67 and its Simulink version is shown in Fig. 7.68. The displacements and velocities of the rail at the top and bottom are calculated in advance and stored in workspace of Matlab. Reaction force of the RGS was measured by the force transducer and fed into the A/D channel of MicroLabBox. The translational and angular displacements were calculated using the numerical model given by Eq. (7.7.16) and then the relative displacement of the bottom was calculated and fed into the position tracking algorithm of the AC servo motor. Figures 7.69 shows the translational displacement of the cage. The simulation result is based on the numerical model, in which reaction force is replaced by a theoretical spring-damper model. It can be seen from Fig. 7.69 that HILS experimental result are similar to the numerical simulation result. Semi-full scale HILS system was constructed as shown in Fig. 7.70 to test the performance of the elevator lateral vibration controller. The elevator simulator is suspended from the ceiling and floated from the ground. The total weight of the

Fig. 7.67 Block diagram for the HILS system based on two-degree-of-freedom model

Fig. 7.68 Simulink block diagram for the HILS system based on two-degree-of-freedom model

7.7 Elevator Vibration Control

301

Fig. 7.69 Absolute displacement calculated from two-degree-of-freedom system (simulation vs. HILS system)

Fig. 7.70 HILS experimental setup (courtesy of Hyundai Elevator Co.)

simulator is about 500 kg, and the size is 1.2 m in width and 1 m in length. And the length of the rope used to hang the simulator is 1.2 m. In order to operate the simulator, four active RGS driven and the linear displacement-tracking system driven by the AC servo motor were installed at each corner as shown in Fig. 7.70. This excitation device is the same as the HILS device mentioned above. Accelerometers were installed in the upper and lower centers to measure the vibration of the frame.

302

7 Application Examples

In order to investigate the dynamic characteristics of the simulator, vibration experiments were performed by exciting the bottom linear motor as shown in Fig. 7.71. LMS SCADAS MOBILE equipment was used to measure the Frequency Response Function (FRF). After excitation of the simulator with periodic chirp of 0 ~ 16 Hz, FRF was measured by simultaneously measuring the acceleration installed in the upper and lower parts. Figure 7.72 shows the FRFs. Through FRF measurement, the natural frequencies of the simulator were found to be 5.1 and 6 Hz. In addition, it was observed that the natural mode corresponding to 5.1 Hz is the rotation mode, and

Fig. 7.71 Schematic diagram for measuring the frequency response function

Fig. 7.72 FRF of simulator between third exciter and the bottom and top accelerometer

7.7 Elevator Vibration Control

303

Fig. 7.73 Time history of the acceleration at the bottom with or without VTMD control

the natural mode corresponding to 6 Hz is the translation mode. When the elevator is placed in hoist way, the rotation mode is expected to appear after the translation mode, but different result was observed. This seems to be due to the simulator hung on a cable. The control algorithm used for this experiment was the VTMD control. Rail displacement data were used for numerical calculation, and the result of the lower acceleration responses is shown in Fig. 7.73. It can be seen from Fig. 7.73 that vibrations were successfully suppressed by the VTMD control algorithm. It is expected that the VTMD controller can also be effectively applied to the real elevator vibration control.

7.8 Automobile This section introduces the study on the active vibration control of automobiles done by Lee et al. (2009a, b). The objective of the active vibration control was to suppress the vibration generated by the engine of the vehicle. The excitation force generated by the engine is close to a harmonic disturbance. In particular, it can be observed that several harmonic disturbances correlated with the engine revolutions per minute (RPM) overlap. Therefore, the active vibration controller which can handle the harmonic disturbance of a specific excitation frequency related to the engine RPM is preferred.

304

7 Application Examples

As a control algorithm capable of responding to harmonic disturbances, the HHC was introduced in Chap. 5. HHC was originally developed for vibration control of helicopter rotor blades. Yang et al. (2010) went one step further and developed a modified HHC (MHHC) incorporating damping factor into the existing HHC to cope with harmonic disturbances in a wider frequency band. Experiments on active vibration control were carried out by applying MHHC, which has proven its performance for a simple cantilevered structure, to an actual vehicle in idling state. Experimental results show that MHHC can be effectively used for real vehicles. The Active Linear Actuator (ALA), that is in fact an inertialtype actuator (Lee et al. 2009a, b), and accelerometers are attached to the vehicle’s subframe as shown in Fig. 7.74. The ALA was mounted on the left and right of the engine subframe while using the existing passive mount as it is. Transfer Pass Analysis (TPA) was performed to determine the ALA attachment location. As a result, it was found that the position where the engine’s contribution to the excitation force was greatest was under the left and right passive mounts. That is where the ALAs were attached. Adding more ALA increases the vibration suppression effect, but it was decided to attach two ALAs due to the spatial constraints of the actual vehicle. For active vibration control, the wiring diagram of the hardware for active vibration control of the vehicle is shown in Fig. 7.75. To implement the control algorithm, an Auto Box of dSPACE was used. The algorithm was implemented using Simulink and downloaded to Autobox. An accelerometer signal was connected to the A/D input terminal of the Autobox, and a tachometer pulse signal was connected to the digital input terminal. The engine RPM was calculated using the tachometer pulse signal, and the main frequency of the engine disturbance related thereto was calculated. Using the main excitation frequency information on the engine disturbance, the MHHC algorithm is designed to respond to the changing frequency of harmonic

Fig. 7.74 Active Linear Actuator mounted on the subframe

7.8 Automobile

305

Fig. 7.75 Control signal connection

disturbance. The control signal calculated by the MHHC algorithm was delivered to the ALA through the power driver. Through initial experiments, a value of 0.09 for the damping factor of the MHHC was found to be the most effective. It should be mentioned again that the MHHC algorithm requires not only information about the frequency to be controlled, but also the real and imaginary values of the transfer function between the actuator and the sensor at this frequency So, HP Dynamic Signal Analyzer 35670A was used to measure the transfer function between the sensor and the actuator. Figure 7.76 shows the measured results for the transfer function between the right and left actuators and accelerometers. The left side shows the driver’s seat position, and the right side shows the passenger’s seat position. It can be seen from Fig. 7.76 that the transfer function between the left ALA and the left accelerometer and the transfer function between the right ALA and the right accelerometer are similar. In addition, it can be seen that the size of the transfer function increases at a high frequency, because the local natural frequency of the subframe is about 300 Hz. First, active vibration control was attempted to the idle state of the vehicle. As a result of analyzing the acceleration signal, it was confirmed that the single frequency component (third order component) was the largest. Based on this, MHHC, which controls a single frequency, was applied to the left and right subframe controllers, respectively. The single frequency MHHC program using the Simulink block was implemented as shown in Fig. 7.77. Figures 7.78 and 7.79 show the time histories of accelerometer signals and frequency response curves of them. When the vehicle is running, unlike the idle state, several frequency components appear. Therefore, unlike the case of idle, a control algorithm capable of controlling multiple frequencies simultaneously is required. The block diagram as shown in Fig. 7.80 was constructed to control multiple frequencies by connecting multiple MHHCs in parallel. From the accelerometer signal measurement result, it can be seen that the vibration of the vehicle during driving has the frequency components

306

Fig. 7.76 Transfer function between ALA and accelerometer

7 Application Examples

7.8 Automobile

307

Fig. 7.77 Simulink block diagram for HHC subsystem

in the order of 1.5th, 3rd, 4.5th, and 6th order of engine RPM. Therefore, MHHC that can correspond to each frequency component from the measured RPM was implemented as shown in Fig. 7.80. Figure 7.81 shows the accelerometer signal when the controller is activated after about 5 s when the vehicle is running. From Fig. 7.81, it can be seen that the acceleration of the subframe and seat rail is reduced. Figure 7.82 shows the result of spectrum analysis for each signal. From Fig. 7.82, it can be seen that in the case of subframes, the 1.5th order component decreases the most, followed by the 3rd, 4.5th, and 6th order components. In the case of seat rails, the influence of higher order components is insignificant. The 1.5th order component is the largest, and it can be seen that the 1.5th order component of seat rail vibration is reduced by about 50% through the control of the subframe. When the vehicle’s RPM increases, the vibration frequency also increases, but the vibration level decreases. Therefore, it becomes difficult to confirm the vibration suppression effect. A case of 1200 RPM where the impact during driving is relatively large is presented. The ALA actuators were applied to an actual vehicle and an experiment was performed. The excitation generated by the engine can be expressed in the sum of harmonic disturbances related to the engine RPM. As the engine RPM changes, the frequency component of the harmonic disturbances also changes. The MHHC that can cope with harmonic disturbances was considered, and the control algorithm was modified so that it could be linked with engine RPM. The MHHC algorithm is implemented using Simulink block diagram. In order to suppress multiple frequency components related to engine RPM at the same time, MHHCs corresponding to each frequency component were connected in parallel and applied to the vehicle. As a result of the actual vehicle experiment, it was confirmed that the acceleration level of the sub-frame was reduced by about 1/3 in the case of the idle state, and that the acceleration level of the sub-frame was decreased by about 1/2 when the vehicle was driving.

308 Fig. 7.78 Time histories of acceleration at idle condition

7 Application Examples

7.8 Automobile Fig. 7.79 Spectral analysis of time responses at idle condition

309

310

7 Application Examples

Fig. 7.80 Simulink block diagram for MHHC

7.9 Railway Vehicle This section introduces briefly the research work on the active vibration control of tube train. The dynamic model of other railway vehicles can be found in references Garg and Dukkipati (1984), Lee et al. (2012) and Kwak et al. (2014). The following dynamic model for the capsule train shown in Fig. 7.83 was considered as the numerical simulation and HILS (Shin et al. 2019b). By using the HILS system, it is possible to reduce the error that occurs when only the theoretical model is used, and to solve the cost and safety problems incurred when performing an experiment using a real vehicle. Various types of dampers or actuators can be easily tested prior to experiments on real vehicle by the HILS system. It is also possible to verify control algorithms using real actuator, thereby increasing research efficiency. In Fig. 7.83, the vehicle body of the tube train is supported by springs and dampers mounted on the bogie, which is floated by magnetic forces that are modeled by springs and dampers. Also, vertical and lateral actuators are included to suppress vibrations. In the HILS experiment, the lateral actuator will be replaced by real actuator and its reaction force is fed into the mathematical model. In Fig. 7.83, xc , yc , and θc represent the lateral, vertical, and roll motions of the carbody, and xb , yb , and θb represent the lateral, vertical, and roll motions of the bogie, respectively. In addition, m c , m b are the masses of the body and bogie, Icr , Ibr are the mass moments of inertia about roll motions. kal , kav are spring constants of lateral and vertical springs, k g is the spring constant of the lateral spring located between the bogie and the wall, klv is the spring constant of the vertical spring located between the bogie and the bottom, cv , cal , cav , cg , and clv are the damping coefficients

7.9 Railway Vehicle Fig. 7.81 Time histories of acceleration at driving condition (1200 RPM)

311

312 Fig. 7.82 Spectral analysis for time responses at driving condition (1200 RPM)

7 Application Examples

7.9 Railway Vehicle

313

Fig. 7.83 Train lateral-vibration model

of dampers corresponding to each motion. fl is the lateral actuation force, that can be replaced by the real force measured by load cell in the HILS experiment, f vl , f vr are left and right vertical actuation forces. The carbody and bogie have three-degree-of freedoms, respectively, so that the dynamic model consists of six-degree-of-freedom model. The equations of motion for the capsule train shown in Fig. 7.83 are as follows: Ml x¨ i + Cl x¨ i + Kl xl = Bal Ul + Dal dl

(7.9.1)

where xl ,Ul , dl are displacement vector, control force vector, and disturbance vector, respectively, in which xl = [xc yc θc xb yb θb ]T

(7.9.2a)

Ul = [ fl f vl f vr f ml f mr ]T

(7.9.2b)

dl = [x˙wl x˙wr y˙w xwl xwr yw ]T

(7.9.2c)

314

7 Application Examples

Ml = diag([m c m c Icr m b m b Ibr ])    Cl1 Cl2 Kl1 Kl2 , Kl = Cl2 Cl3 Kl2 Kl3 ⎤ ⎡ −1 0 0 0 0 ⎡ ⎢ 0 −1 −1 0 0 ⎥ 03×6 ⎥ ⎢ ⎥ ⎢ ⎢ cg cg 0 k g k g 0 ⎢ −h lc wv −wv 0 0 ⎥ Cl = ⎢ ⎥, Dal = ⎢ ⎣ 0 0 2clv 0 0 2klv ⎢ 1 0 0 −1 1 ⎥ ⎥ ⎢ ⎣ 0 01×6 1 1 0 0 ⎦ h lb −wv wv −h m h m

(7.9.2d)



Cl =

(7.9.2e) ⎤ ⎥ ⎥ ⎦ (7.9.2f,g)

where ⎡

⎤ 0 2cal h aclc 2cal ⎦ Cl1 = ⎣ 0 2(cav + cv ) 0 2 0 2(cal h ac + cav wa2 + cv wv2 ) 2cal h ac ⎡ ⎤ 0 2cal h ab −2cal ⎦ Cl2 = ⎣ 0 −2(cav − cv ) 0 2 2 0 2(cal h ab h ac − cav wa + cv wv ) −2cal h ac ⎡ ⎤ cal + cg 0 −cal h ab + cg h g ⎢ ⎥ 0 cav + clv + cv  0 ⎥ Cl3 = 2⎢ 2 2 2 ⎦ ⎣ cal h ab + cg h g + cav wa 0 −cav h ab + cg h g +clv wl2 + cv wv2 ⎡ ⎤ 0 2kal h ac 2kal ⎦ Kl1 = ⎣ 0 2kav 0 2kal h ac 0 ⎡

2 2(kal h ac

+

(7.9.3a)

(7.9.3b)

(7.9.3c)

(7.9.3d)

kav wa2 )

⎤ 0 2kal h ab −2kal ⎦ Kl2 = ⎣ 0 −2kav 0 2 −2kal h ac 0 2(kal h ab h ac − kav wa ) ⎡ ⎤ kal + k g 0 −kal h ab + k g h g ⎢ ⎥ 0 kav + klv  0 ⎥ Kl3 = 2⎢ 2 2 ⎣ ⎦ kal h ab + k g h g 0 −kav h ab + k g h g 2 2 +kav wa + klv wl

(7.9.3e)

(7.9.3f)

The second-order matrix differential equation is transformed into the state-space equation to calculate responses and to perform HILS.

7.9 Railway Vehicle Table 7.4 Parameters for the train model

315 Parameters

Value

mc

16361 kg

I cr

16700 kg m2

mb

20934 kg

I br

10,000 kg m2 s

hac

−0.102 m

wa

0.4 m

hlc

−0.05 m

wv

0.2 m

vab

0.497 m

hlb

0.24 m

wl

0.63 m

hg

0.07 m

hm

0.035 m

k av

745.2 kN/m × 2

cav

7.45 kNs/m × 2

k al

333.4 kN/m × 2

cal

3.33 kNs/m × 2

k lv

4.02 MN/m × 2

clv

0 Ns/m

kg

2.34 MN/m × 2

cg

0 Ns/m

cv

6 kNs/m

csky

200 kNs/m

z˙ sl = Asl zsl + Bsl usl .

(7.9.4a)

ysl = Csl zsl + Dsl usl .

(7.9.4b)

 

T where zsl = xlT x l , usl = UlT dlT ,    06×6 06×6 06×6 05×6 , B = sl −Ml−1 Kl −Ml−1 Cl −Ml−1 Bal −Ml−1 Dal     I12×12 012×6 012×5 , Dsl = Csl = −Ml−1 Bal Ml−1 Dal −Ml−1 Kl −Ml−1 Cl 

Asl =

(7.9.5a, b)

(7.9.5c, d)

For the numerical simulation and HILS, parameters shown in Table 7.4 and disturbances shown Figs. 7.84 and 7.85 were used. Natural frequencies and mode shapes

316

Fig. 7.84 Disturbance in lateral direction

Fig. 7.85 Disturbance in vertical direction

7 Application Examples

7.9 Railway Vehicle

317

Table 7.5 Natural frequencies and mode shapes Mode # Freq. (Hz)

1

2 0.82

3 1.29

4

5

1.94

3.64

6 4.30

4.89

xc

−0.100

1.000

0.000

−0.179

0.066

0.000

yc

−0.000

−0.000

1.000

0.000

−0.000

−0.239

θc

1.000

−0.094

0.000

0.015

−0.048

−0.000

xb

−0.000

0.145

−0.000

1.000

−0.022

0.000

yc

−0.000

−0.000

0.187

−0.000

0.000

1.000

θb

0.067

−0.108

−0.000

0.067

1.000

−0.000

are tabulated in Table 7.5. The HILS system configured for experiment in this study is shown in Fig. 7.86. As shown in Fig. 7.86, AC servo motor (Mitsubishi HG-KR73), motor driver (Mitsubishi MR-J4-70A), load cell (Curiotec CLS-500L/500kgf), and MR damper (RMS Technology) or Linear Synchronous Motor (LSM) (AI Korea) was used to construct a HILS experimental setup. The basic configuration is the same as the one used for the elevator HILS. The LSM used in this study receives ± 10 V input and produces a maximum force of 3kN. However, as a result of theoretical calculation, it was confirmed that the force of 3 kN was not necessary, so it was set to produce a force of 2kN according to the maximum 10 V input. When using the HILS system, a feedback loop system must be constructed in which the reaction force by the actual damper or actuator is measured and the vibration response is calculated by inputting it into the theoretical model. Accordingly, the lateral relative displacement between the vehicle body and the bogie calculated in the

Fig. 7.86 Experimental setup of the HILS system with LSM

318

7 Application Examples

Fig. 7.87 Simulink program for the HILS experiment with VTMD control

theoretical model was implemented using an AC servo motor and a ball screw mechanism. When the relative displacement is implemented, the damper or the actuator is simultaneously operated, and the reaction force generated by the damper or actuator is measured by the load cell and input to the theoretical model. Hence, controller that is capable of real-time processing of a series of tasks such as vibration response calculation using theoretical models, input of actual load cell data and output of control signals is essential in implementing the HILS. To this end, MicroLabBox of dSPACE, Inc. was used. The Simulink program used for the HILS system is as shown in Fig. 7.87. The following VTMD control algorithm was considered as the control algorithm for railway vehicle. Fvtmd (s) = −

m a (2ζa ωa s + ωa2 ) ¨ X c (s) (s 2 + 2ζa ωa s + ωa2 )

(7.9.6)

where Fvtmd (s) is the Laplace transform of f vtmd (t), and m a is the mass of the virtual tuned mass damper, which can be considered as a gain of the controller. In order to transform the control force calculated by the VTMD control algorithm into the command voltage that is to be applied to current amplifier, the following equation was used. Vc = f vtmd × 0.005

(7.9.7)

The high-current amplifier as shown in Fig. 7.88 is used to drive the LSM.

7.9 Railway Vehicle

319

Fig. 7.88 Current driver for LSM

For active vibration control using the LSM, three VTMD controllers were combined in parallel, where the target frequencies were set to 0.8, 1.5, and 3.8 Hz, and the damping factor was all set equal to 0.3. The gains of the controllers were set to 150, 600, 600 according to each frequency. Figures 7.89 and 7.90 show the MR damper and the current amplifier, respectively. Because we can control only the damping force of the MR damper, the following semi-active virtual tuned mass damper (SAVTMD) control algorithm was developed by Shin et al. (2019b) and used. ⎧ ⎨

f vtmd (Vcl − Vbl ) ≥ 0, | f vtmd | < f max f vtmd f = f max sgn( f vtmd ) f vtmd (Vcl − Vbl ) ≥ 0, | f vtmd | ≥ f max ⎩ 0 otherwise

Fig. 7.89 MR damper (courtesy of RMS technology)

(7.9.8)

320

7 Application Examples

Fig. 7.90 Current amplifier for the MR damper (courtesy of RMS technology)

f max was set to 5 kN in the HILS experiment. Equation (7.9.8) implies that the control force is not fully active, since the controller for the MR damper can only change the damping coefficient of the damper. Therefore, in theory, it can be said that semi-active control is less effective than active control. In applying the SAVTMD control, the following algebraic equation between the applied current and the damping force was derived by curve fitting and used (Shin et al. 2019b). −857.14I 2 + 3914.29I + 791.43 = | f |

(7.9.9)

Hence, if the control force is determined by Eq. (7.9.8), then the required current is calculated by Eq. (7.9.9). Because the current amplifier shown in Fig. 7.90 is normally used to activate the MR damper, we need the command voltage. The relation between the command voltage and produced current of the current amplifier shown in Fig. 7.90 is simply given as Vc = 5 × I

(7.9.10)

Using this configuration, The MR damper can adjust the damping force according to the command voltage input to the current amplifier. Figures 7.91 through 7.92 show the time histories of lateral and roll accelerations of the body and the frequency response curves. Overall, it can be seen that the active vibration control method is the most effective to suppress the vibration of the vehicle body. Regardless of the control method, the control effect is insignificant for frequency components around 4 Hz. The cause of this phenomenon can be inferred through the eigenvector information of Table 7.5. In the case of the natural mode around 4 Hz, the lateral motion of the bogie is larger than the lateral motion of the carbody, and the direction of the lateral motion of the carbody is opposite to the direction of the lateral motion of the bogie. Therefore, when this natural mode of the vehicle body is to be controlled by high control gain, the motion of the carbody may become unstable by control spillover because the controller tries to suppress the lateral and roll motions of the bogie instead of the lateral and roll motions of the carbody. Figures 7.93 and 7.94 show time-histories of lateral and roll motions

7.9 Railway Vehicle

321

Fig. 7.91 Time history of the lateral acceleration of the carbody

Fig. 7.92 Amplitude spectrum of the lateral acceleration of the carbody

of the bogie corresponding to the lateral and roll motions of the carbody shown in Figs. 7.91 and 7.93. It can be seen from Figs. 7.95 and 7.96 that the motions of the bogie are not greatly affected by the applied control force.

322

7 Application Examples

Fig. 7.93 Time history of the roll acceleration of the carbody

Fig. 7.94 Amplitude spectrum of the roll acceleration of the carbody

Figure 7.97 represents the control forces generated by the active and semi-active control methods. It can be seen that the control force calculated by the active control method is smaller than the control force calculated by the semi-active control method. As a result of the experiment, it was confirmed that approximately 10 kN of force is sufficient. Figure 7.98 shows the command voltage to control the MR damper

7.9 Railway Vehicle

Fig. 7.95 Time history of the lateral acceleration of the bogie

Fig. 7.96 Time history of the roll acceleration of the bogie

323

324

Fig. 7.97 Time history of the control forces

Fig. 7.98 Time history of the command voltages

7 Application Examples

7.9 Railway Vehicle

325

and the LSM. The command voltage for the LSM appears to be large because the two devices have different capacities and the command voltage for the MR damper appears to be always positive as expected.

7.10 Building Structure The works of Yang et al. (2017), Talib et al. (2019), and Shin et al. (2020) will be briefly summarized. The building-like structure can be considered as a cantilever structurewise. However, it is not easy to suppress vibrations of building structure because it is a huge structure and lacks adequate actuator. The AMD has been considered as an actuator for building-like structure in the works of Yang et al. (2017), Talib et al. (2019), and Shin et al. (2020). Let us consider a MDOF system that is an n-story building with an AMD at the top as shown in Fig. 7.99. The equations of motion for this system can be written as M¨x + C¨x + Kx = −Ba u¨ a + d Fig. 7.99 Schematic diagram for multi-degree-of-freedom with AMD

(7.10.1)

326

7 Application Examples

where x = [ x1 x2 . . . xn ]T and ⎡

⎡

⎤ c1 + c2 −c2 · · · 0 ⎢ −c2 c2 + c3 · · · 0 ⎥ ⎥ ⎢ ⎥ ⎥ , ⎥, C = ⎢ . .. . . .. ⎥ ⎣ .. ⎦ . . ⎦ . 0 0 · · · mn 0 0 · · · cn ⎡ ⎤ ⎤ ⎡ k1 + k2 −k2 · · · 0 0 ⎢ −k2 k2 + k3 · · · 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ ⎢ , B K=⎢ . = ⎥ ⎢ . ⎥, a . . .. . ⎦ .. ⎣ .. ⎣ .. ⎦ . .

m1 ⎢ 0 ⎢ M=⎢ . ⎣ ..

0 m2 .. .

··· ··· .. .

0 0 .. .

⎤

· · · kn ⎫ ¨ d1 − m 1 dg ⎪ ⎪ ⎪ ⎬ d2 − m 2 d¨g d= .. ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ ⎭ ¨ dn − (m n + m a )dg 0

⎧ ⎪ ⎪ ⎪ ⎨

(7.10.2a, b)

ma

0

(7.10.2c–e)

in which m i , ci , ki (i = 1, 2, . . . , n) are the mass, damping, and spring constant of ith floor. m a is the active mass, di (i = 1, 2, . . . , n) is the disturbance acting on the ith floor and dg represents the ground motion. Solving the free vibration problem of Eq. (7.10.1) results in the eigenvalue matrix, , and the eigenvector matrix, U, that satisfies the orthonormality condition. If we apply the modal transformation, x = Uq, into Eq. (7.10.1) and use the orthonormality condition and modal damping assumption, then we can obtain the modal equations of motion. q¨ + 2Zq˙ + q = −B¯ a u¨ a + d¯

(7.10.3)

where B¯ a = UT Ba , d¯ = UT d and ⎡

ω1 ⎢ 0 ⎢ =⎢ ⎢ ... ⎣

··· ··· .. . .. 0 0 . 0 ω2 .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎡

ζ1 ⎢0 ⎢ Z=⎢ ⎢ ... ⎣

ωn

0 ⎡

⎢ ⎢ =⎢ ⎢ ⎣

ω12 0 .. .

⎤ ··· 0 ··· 0 ⎥ ⎥ . . .. ⎥ . . ⎥ ⎦ .. 0 . ζn ⎤ 0 0 ⎥ ⎥ .. ⎥ . ⎥ ⎦

0 ζ2 .. .

··· ··· .. . .. 2 0 0 . ωn 0 ω22 .. .

(7.10.4a, b)

(7.10.4c)

In reality, there are more natural modes than available sensors and it is not desirable to control all modes with a single AMD. Of course, we can install more AMDs to

7.10 Building Structure

327

control many modes but this is not economical. Let us consider a case of controlling some modes with a single AMD. In this case, it is assumed that the control force can be represented by GQ = B¯ ac u a

(7.10.5)

where Q is an n c × 1 modal control force vector, G is an n c × n c diagonal matrix whose diagonals are the gain for each mode, and n c is the number of natural modes to be controlled. B¯ ac is the submatrix of B¯ a which is obtained by considering the natural T ¯T T Bau ] . The pseudo-inverse modes to be controlled. Hence, we may write B¯ a = [B¯ ac technique is used for the solution of Eq. (7.10.5). Then, we may write † GQ u a = B¯ ac

(7.10.6)

† where B¯ ac is the pseudo-inverse of B¯ ac . Inserting Eq. (7.10.6) into Eq. (7.10.5), we can obtain † ¯ ac ¨ + d¯ GQ q¨ + 2Zq˙ + q = −B¯ a B

(7.10.7)

The multi-modal NAF control for Eq. (7.10.7) can then be designed as  † T ¨ + 2Zc c q˙ + c Q = G B¯ a B¯ ac q¨ Q

(7.10.8)

where Zc is the damping matrix for the NAF control. Combining Eqs. (7.10.7) and (7.10.8), we can obtain 

     ¯ ¯† q¨ 2Z 0n×n c q˙ In×n † Ba Bac G + ¨ Q 0n c ×n 2Zc c G B¯ a B¯ ac In c ×n c q˙      q d¯  0n×n c = + 0n c ×n c Q 0n c ×1

(7.10.9)

Hence, the stability condition can be obtained as  † † T ¯ a B¯ ac stable if I − B¯ a B¯ ac G2 B >0

(7.10.10)

If the gain matrix is sufficiently small, then the closed-loop system is stable. Numerical simulation was carried out to investigate the performance of the MMNAF control proposed in this study. The building model employed for the simulation has forty floors with 40 m width and 200 m height. Parameters for the building model are listed in Table 7.6. It was assumed that the base was excited by an earthquake and the El Centro earthquake data shown in Fig. 5 was used. The natural frequencies for the building were found to be 0.25, 0.76, 1.27, 1.77, and 2.27 Hz. The MMNAF controller which can tackle the first and second natural modes was designed. The following gain matrix was used

328 Table 7.6 Parameters for numerical simulation

7 Application Examples Parameters

Numerical values

Number of stories

40

Height of building

200 m

Width of building

40 m

Mass of each floor Stiffness of each floor

8 × 105 kg ' 1.35 GN m

Damping coefficient

ς = 0.01

Mass of movable mass

105 kg



0.4 0 G= 0 0.15



The displacements at the fortieth floor with and without MMNAF control are shown in Fig. 7.101. Even though the peak amplitude caused by the earthquake cannot be reduced by the MMNAF control, the response decreases rapidly by the introduction of the MMNAF control. Figure 7.100 confirms that the building vibrations can be controlled by the proposed MMNAF control. Figure 7.101 shows the Frequency Response Function (FRF) for the top displacement of the building with and without control. It can be seen from Fig. 7.101 that the resonant amplitudes of the first and second natural modes are reduced by the MMNAF control. It can be concluded from the numerical simulation that the MMNAF control can suppress the vibrations of the MDOF system.

Fig. 7.100 Time history of top displacement for the building with and without control

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329

Fig. 7.101 FRF of top displacement for the building with and without control

To validate the MMNAF control proposed in this study, the test bed was built as shown in Fig. 7.102. The test bed consists of two floors and the active mass is connected to the AC servo motor by the ball-screw mechanism. The parameters of the test bed are summarized in Table 7.7. The first and second natural frequencies for the test bed were found to be 0.95 and 2.70 Hz, respectively. They were measured experimentally. The AC servo motor, Mitsubishi HC-KFS13 and a servomotor driver, Mitsubishi MR-J2S-10A were used. The active mass was mounted on the linear motion guide. The ball-screw had a pitch of 5 mm. The vibration at the top was measured by the accelerometer, LCF-200-0.5 g from Jewell Instruments. The sensitivity of this sensor was 10 V/g. Figure 7.103 shows the connection diagram for the test bed. The accelerometer signal was fed into the ADC terminals of the DS 1104 controller from dSPACE Inc. Although the MMNAF controller given by Eqs. (7.10.6) and (7.10.8) compute the desired displacement of the movable mass, it was found that the real-time tracking control of the desired position of the movable mass by the AC servomotor was not possible using the position command mode of the AC servomotor deriver. Hence, the velocity command mode that accepts analog signal was used instead, so that the additional PID controller was introduced as shown in Fig. 7.104a. The PID controller was designed to provide proper velocity command to reduce the error between the desired position and the measured position. Also, the internal PID controller of the AC servomotor driver was utilized to follow the desired velocity as shown in Fig. 7.104b. The MMNAF control algorithm and the AC servomotor control algorithm were built using the Simulink software shown in Fig. 7.104. The MMNAF controller consists of two NAF controllers. Each NAF controller was tuned

330

7 Application Examples

Fig. 7.102 Test bed for MMNAF control experiment

Table 7.7 Parameters of test bed

Parameters

Numerical values

Number of stories

2

Height of test bed

1.15 m

Width of test bed

0.4 m

Mass of top floor

6.9 kg

Mass of middle floor

20 kg

Damping coefficient of test bed

ς = 0.0063

to the first and second natural frequencies. The gains for each NAF controller were 1 and 0.1, which were determined experimentally. Figure 7.105 shows the time history of acceleration when the two natural modes of the structure were excited. It can be seen from Fig. 7.105b that the MMNAF control is very effective in suppressing multi-modal vibrations. It was proven experimentally that the proposed MMNAF control can successfully suppress multi natural modes of structures.

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331

Fig. 7.103 Schematic diagram for experimental set up

In the work by Yang et al. (2017), an n-story building with an AMD at the top was considered and multi-modal NAF control was proposed and tested. This implies that a single AMD is used to suppress many natural modes. However, vibration suppression of many natural modes by a single AMD is not an easy task. Talib et al. (2019) tried to suppress two natural modes by two AMDs and used new actuators that were linear servo motors. A structure shown in Fig. 7.106 consists of two floors with vertical side beams and results in a two-degree-of-freedom model. Two linear motors are mounted on each floor. The linear motor consists of the coil part (Mitsubishi LM-H3P2A-07P-BSSO) and the stator (LM-H3S20-288-BSSO). The linear motor driver (Mitsubishi MR-J440A-RJ) was used to drive the linear servo motor. Incremental-type encoder was used to measure the position of the moving part. The moving part of the linear motor was mounted on linear guide. The moving part was heavy enough to be used as AMD for the structure. As shown in Fig. 7.107, an AC servo motor (Mitsubishi HC-KFS13) is connected to the base of the structure through a ball-screw mechanism. The base is also mounted on linear motion guide. The AC servo motor was driven by the servomotor driver (Mitsubishi MR-J2S-10A). Vibration at each floor was measured by servo accelerometer (LCF-200-0.5 g from Jewell Instruments). The input range and sensitivity of this accelerometer are ±0.5g and 10 V/g, respectively. This servo accelerometer is designed to measure ultra-low frequency vibrations of geophysical system, railway vehicle, ocean buoy, aircraft, and elevator. A laser sensor (M3L/20 of MEL Mikroelektronik, GmbH) was used to measure displacement of the base. The resolution of the laser sensor was 0.02 mm. A signal analyzer (Siemens LMS SCADAS Mobile) was then used to measure the transfer function between the base displacement and the acceleration of each floor. DS1104 controller of dSPACE, Inc.

332

7 Application Examples

Fig. 7.104 Control scheme of MMNAF control

was used to implement control algorithms using Simulink blocks. The accelerometer signals were fed into analog-to-digital terminals of the DS 1104 controller. Output voltage signals of digital-to-analog terminals are velocity commands of AMDs calculated by the MIMO modal-space NAF control algorithm, which were delivered to command ports of the linear motor driver. Figure 7.108 shows a simplified building model that consists of two floors equipped with AMD on each floor, vertical beams and torsional springs for the test structure shown in Fig. 7.106. A simple two-degree-of-freedom model consisting of spring-mass-damper was derived first. However, theoretical natural frequencies were not in good agreement with experimental natural frequencies. Hence, we resorted to the beam modeling technique. Based on Fig. 7.108, total kinetic energy is expressed as 1 T = 2

L 0

 2  2

2  1 L 1

,t m¯ b x˙ g + u(t) ˙ dx + m 1 x˙ g + u˙ + m 2 x˙ g + u(L ˙ , t) 2 2 2

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333

Fig. 7.105 Time history of acceleration for the two modes excitation

 2 

2 1 L 1

, t + u˙ a1 + m a x˙ g + u(L + m a x˙ g + u˙ ˙ , t) + u˙ a2 2 2 2

(7.10.11)

where L is the total height, m¯ b is the mass per unit length of the vertical beam, m1 = m2 = mi is the mass of each floor, ma is the mass of the movable mass, x i is the displacement of the floor, x g is the base motion, and uai (i = 1, 2) is the relative displacement of movable mass. The displacement of the vertical beam can be discretized by introducing admissible functions and generalized displacements. u(x, t) = (ξ )q(t)

(7.10.12)

where (ξ ) = [1 (ξ ) 2 (ξ ) · · · n (ξ )] is the matrix consisting of admissible functions, q(t) = [q1 (t) q2 (t) · · · qn (t)]T is the vector consisting of generalized coordinates, ξ = x/L and n is the number of admissible functions. Inserting Eq. (7.10.15) into Eq. (7.10.14a–f) results in T =

1 1 2 2 ¯ t q˙ + 1 q˙ T Mt q˙ + 1 m a u˙ a1 m t x˙ g2 + x˙ g  + m a u˙ a2 + m a x˙ g (u˙ a1 + u˙ a2 ) 2 2 2 2

334

7 Application Examples

Fig. 7.106 Test structure with active mass dampers

+ m a q˙ T T

 1 u˙ a1 + m a q˙ T T (1)u˙ a2 2

(7.10.13)

where ¯ = m t = m¯ b L + 2m i + 2m a , m b = m¯ b L , 

1

1  dξ , Mb =

0

T  dξ 0

(7.10.14a–d)    1 ¯ ¯ + (1) t = m b  + (m i + m a )  2     1 1 Mt = m b Mb + (m i + m a ) T  + T (1)(1) 2 2 The potential energy of the system is expressed as

(7.10.14e) (7.10.14f)

7.10 Building Structure

Fig. 7.107 Schematic diagram for the experimental setup Fig. 7.108 Theoretical model of the test structure

335

336

7 Application Examples

1 V = EI 2

L  o

∂ 2u ∂x2

2

 2  2 ∂u ∂u 1 1 dx + kt + kt 2 ∂ x x=L/2 2 ∂ x x=L

(7.10.15)

where EI is the bending rigidity of the vertical beam and k t is the torsional spring constant. Inserting Eq. (7.10.12) into Eq. (7.10.15) results in V =

1 T q Kt q 2

(7.10.16)

where Kt = Kb + K1 + K2

(7.10.17)

in which EI Kb = 3 L

1

T  dξ , K1 = kt T

  1 1  , K2 = kt T (1) (1) 2 2

0

(7.10.18a–c) The equation of motion can be derived by using the Lagrangian equation d dt



∂L ∂ q˙

−

∂L =0 ∂q

(7.10.19)

where Lagrangian L = T − V . Inserting Eqs. (7.10.16) and (7.10.19) into Eq. (7.10.22) results in Mt q¨ + Kt q = −Ba u¨ a + Bd x¨ g

(7.10.20)

where ua = [u a1 u a2 ]T and    ¯ Tt , Ba = m a T 1 T (1) Bd = − 2

(7.10.21a, b)

By solving the free vibration problem of Eq. (7.10.20), natural frequencies and mode shapes can be obtained. Assuming that the eigenvector of matrix U satisfies the orthonormality condition and using modal transformation and modal damping assumption, q = Ur, Eq. (7.10.20) is transformed into modal equation of motion. ¯ d x¨ g r¨ + 2Z˙r + r = −B¯ a u¨ a + B ¯ t , B¯ a = UT Ba and where B¯ d = −UT  T

(7.10.22)

7.10 Building Structure

337

    2  ω1 0 ω1 0 ζ1 0 , = , = Z= 0 ζ2 0 ω2 0 ω22 

(7.10.23a, b)

In reality, the relative displacement of the floor cannot be measured but estimated by the difference between the measured accelerations of the adjacent floors. Because only accelerations of floors can be measured, The absolute displacement of the floor can be expressed as xa = Ig x g + a q = Ig x g + a Ur

(7.10.24)

where xa = [xa1 xa2 ]T is the absolute displacement vector of floors and

T

 Ig = [1 1]T , a =  21 (1)

(7.10.25a, b)

From Eq. (7.10.27), the following relation holds  r = Na xa − Ig x g

(7.10.26)

 −1 Na = UT aT a U (a U)T

(7.10.27)

where

Inserting Eq. (7.10.26) into Eq. (7.10.22) results in ¯ a u¨ a + da Ma x¨ a + Ca x˙ a + Ka xa = −NaT B

(7.10.28)

where Ma = NaT Na , Ca = 2NaT ZNa , Ka = NaT Na  da = NaT B¯ d + Na x¨ g + 2NaT ZNa x˙ g + NaT Na x g

(7.10.29a–c) (7.10.29d)

Equation (7.10.28) is expressed in terms of absolute displacements of floors. By solving the free vibration problem, we can obtain the same eigenvalue matrix but different eigenvector matrix, which satisfy the orthonormality condition. UaT Ma Ua = I, UaT Ka Ua = 

(7.10.30a, b)

Using the modal transformation, xa = Ua qa , Eq. (7.10.28) is transformed into the modal equations of motion q¨ a + 2Zq˙ a + qa = −UaT NaT B¯ a u¨ a + d¯ a

(7.10.31)

338

7 Application Examples

where d¯ a = UaT da . Equation (7.10.31) is more suitable than Eq. (7.10.22) because absolute displacements of floors can be suppressed if we suppress qa . Let us introduce modal control vector defined as GQ = UaT NaT B¯ a ua

(7.10.32)

where Q is the modal control force vector and G is the gain matrix 

g 0 G= 1 0 g2

 (7.10.33)

The displacement of AMD is related to the modal control vector such that −1  ua = UaT NaT B¯ a GQ

(7.10.34)

Using Eq. (7.10.32), Eq. (7.10.31) can be re-written as ¨ + d¯ a q¨ a + 2Zq˙ a + qa = −GQ

(7.10.35)

The MIMO modal-space NAF controller can be designed as ˙ + Q = −Gq¨ a ¨ + 2Zc Q Q

(7.10.36)

where, Zc is the damping matrix for MMNAF control algorithm that is expressed as  Zc =

ζc 0 0 ζc

 (7.10.37)

By combining Eqs. (7.10.31) and (7.10.36), the coupled equation can be written as 

I G G I



q¨ a ¨ Q





2Z 0 = 0 2Zc 



q˙ a q˙





 0 + 0 



qa Q



 =

d¯ a 0

 (7.10.38)

Hence, the closed-loop system is stable if and only if I−G2 > 0. This implies that the closed-loop system is stable if the gain is sufficiently small. The stability condition is static because it doesn’t depend on frequency. However, the uncontrolled modes are neglected in deriving the stability condition. Hence, the control and observer spillover problem may occur when applying the control given by Eq. (7.10.34) to real structure. The stability condition for the fully coupled system needs further investigation. To implement the MIMO Modal-Space NAF control, desired displacements of the AMDs need to be expressed in terms of accelerations. Using Eqs. (7.10.34), (7.10.36) and modal transformation, the following control law can be derived in the

7.10 Building Structure

339

form of Laplace transform. −1  ¨ a (s) Ua (s) = − UaT NaT B¯ a H(s)UaT Ma X

(7.10.39)

where H(s) is the transfer function matrix of MMNAF control ⎡ H(s) = ⎣

g12 s 2 +2ζc ω1 s+ω12

0

⎤ 0 g22 s 2 +2ζc ω2 s+ω22

⎦

(7.10.40)

Model parameters for the building structure are listed in Table 7.8. The torsional spring constants of Table 7.8 are obtained manually by comparing the theoretical frequency response curves with the experimental frequency response curves. Frequency response curves shown in Figs. 7.109 and 7.110 are theoretical and experimental frequency response curves between base displacement and acceleration of each floor. As shown in Figs. 7.109 and 7.110, theoretical results are in good agreement with experimental results, thus validating the dynamic model derived in this study. It was found that the first natural frequency of the test structure was 2.18 Hz while the second natural frequency of the structure was 6.13 Hz. The following MIMO modal-space NAF control was obtained numerically.  H(s) =

0.316 s 2 +8.176s+185.7

0

0



0.2 s 2 +23.4s+1520

    −0.673 −1.059 −2.219 −3.467 ¨ Ua (s) = − H(s) Xa (s) −1.051 0.652 −3.495 2.151 Application of the above MIMO modal-space NAF control to the theoretical model resulted in theoretical frequency response curves as shown in Figs. 7.111 and 7.112. It can be seen that uncontrolled peak-amplitudes could be reduced significantly when the system was controlled by the MIMO modal-space NAF control. Table 7.8 Parameters of test building structure

Parameters

Values

Height

0.86 m

Thickness of side beam

3 mm

Width of each side beam

100 mm

Young’s Modulus

69 GPa

Mass density of side beam

2720 kg/m3

Mass of floor

20 kg

Movable mass

3.3 kg

Damping coefficient (ζ1 , ζ2 )

0.004

Torsional spring constant

4000 Nm/rad

340

7 Application Examples

Fig. 7.109 Theoretical and experimental frequency response curves for the first floor

7.10 Building Structure

Fig. 7.110 Theoretical and experimental frequency response curves for the second floor

341

342

Fig. 7.111 Theoretical frequency response curve for the first floor

7 Application Examples

7.10 Building Structure

Fig. 7.112 Theoretical frequency response curve for the second floor

343

344

7 Application Examples

To successfully obtain predicted closed-loop frequency response curves, positions of real AMDs should follow the desired position calculated by the above MIMO modal-space NAF control. In our previous study (Yang et al. 2017), PID control was used to make the AC servo motor follow the desired position obtained by the NAF control and the servo motor driver was operated in velocity mode. However, it was found experimentally that the same PID control was inadequate for linear servo motor control because tuning gains of the PID control was found to be an uneasy task. In addition, the linear servo motor ran into instability sometimes. Hence, the sliding-mode control (SMC) algorithm was considered in this study to make the linear motor follow desired position accurately. The basic control law of SMC was used in the experiment. u i = α sat(si ), i = 1, 2

(7.10.41)

where α is a positive constant parameter and sat is a saturation function defined as: ⎧ 1, si > σ ; ⎨ ' sat(si ) = si σ, −σ ≤ si ≤ σ ; ⎩ −1, si < −σ ;

(7.10.42)

in which σ is the thickness of boundary layer and si is a switching variable determined by control designer. The sliding surface was defined as si = ei + K e˙i , i = 1, 2

(7.10.43)

Parameter ei (i = 1, 2) is the difference between actual and desired movements of AMDs and e˙i (i = 1, 2) is the time-derivative of error. In this experiment, α = 0.1, σ = 5, and K = 0.015 were used. It was proven that it effectively controlled the tracking movement of the moving mass of the AMD. Simulink block diagram for the SMC controller is shown in Fig. 7.113. Position-tracking control for the linear servo motor is shown in Fig. 7.114. The AC servo motor used to shake the base was operated by the PID control algorithm. As mentioned earlier, the measurement of frequency response curve between the base displacement and the acceleration was carried out by the frequency analyzer. The output of the frequency analyzer is the base displacement but this cannot be

Fig. 7.113 Simulink block diagram for SMC controller

7.10 Building Structure

345

Fig. 7.114 Position tracking control for the linear servo motor

realized by directly connecting the output of the frequency analyzer to the AC servo motor driver. Hence, the output of the frequency analyzer was fed into the DS1104 controller and the additional position tracking algorithm was used to shake the base according to the output of the frequency analyzer. To obtain closed-loop response curve, MIMO modal-space NAF controls along with the base-tracking control were combined into one program as shown in Fig. 7.115. Figures 7.116 and 7.117 show theoretical and experimental frequency response curves when MIMO modal-space NAF control was applied. Experimental results are in good agreement with predicted results as shown in Figs. 7.111 and 7.112. This demonstrates that vibration of the building structure can be effectively controlled by using the proposed MIMO modal-space NAF control algorithm both theoretically and experimentally. Figures 7.118 and 7.119 show experimental results of uncontrolled and controlled

Fig. 7.115 Simulink block diagram for the MMNAF control and the base-tracking control

346

7 Application Examples

Fig. 7.116 Theoretical and measured frequency response curves for the first floor by the MIMO Modal-Space NAF control

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347

Fig. 7.117 Theoretical and measured frequency response curves for the second floor by the MIMO Modal-Space NAF control

348

7 Application Examples

Fig. 7.118 Time history of acceleration measured at the first floor subject to the arbitrary excitation

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349

Fig. 7.119 Schematic diagram for multi-degree-of-freedom with the TLD and AMD

time-responses of accelerations of each floor when the first natural mode is excited. Figures 7.120 and 7.121 show experimental results of uncontrolled and controlled time-responses of accelerations of each floor when the second natural mode is excited. Figures 7.122 and 7.123 show experimental results of uncontrolled and controlled time-responses of accelerations of each floor when both the first and the second natural modes are excited at the same time. As can be seen in these figures, accelerations of the building structure are reduced significantly by the proposed MIMO modal-space NAF control. Hence, MIMO modal-space NAF control combined with SMC is suitable to AMD using the linear motor. This is clearly proven experimentally. The prerequisite to the NAF control is that the position of the moving mass should be precisely controlled. It was found experimentally that the PID control which was effective in controlling the AC servo motor was no longer effective in the case of a linear servo motor. Hence, the SMC technique was considered to trace desired position in this study. Experimental results showed that the SMC was effective in tracking control of the moving mass of the AMD. The performance of the active control system proposed in this study was validated both theoretically and numerically. Experimental results showed that with moderated

350

Fig. 7.120 Transfer functions

Fig. 7.121 El-Centro earthquake displacement data

7 Application Examples

7.10 Building Structure

351

Fig. 7.122 Time histories of top-floor displacement

control gain, vibrations of the test structure could be effectively suppressed by the proposed MIMO modal-space NAF controller and equipped linear servo motors. The Tuned Liquid Damper (TLD) is a fluid form of the TMD. The passive TLD can be combined with the AMD to enhance the vibration suppression capability of the TLD. Shin et al. (2020) proposed the use of TLD-AMD to exploit the advantages of both systems at the same time. Figure 7.119 shows a multi-degree-of-freedom model that is an n-story building equipped with the TLD and AMD, where m i , ci , ki (i = 1, 2, . . . , n) are the mass, damping coefficient, and spring constant of ith floor; xi (i = 1, 2, . . . , n) is the displacement of ith floor relative to the inertial axis; m a is the mass of the movable mass; di (i = 1, 2, . . . , n) is the disturbance acting on the ith floor; u a is the relative displacement of the additional mass with respect to the displacement of the nth floor; x g is the displacement of the ground; x f and y f are the horizontal and vertical axes moving with the nth floor. It is assumed that the position of the movable mass is accurately controlled by a ball-screw mechanism or linear motor. The kinetic and potential energies, and the energy dissipation function of the MDOF system excluding the TLD can be written as 1 1 1 1 m i x˙i2 + m a (x˙n + u˙ a )2 = x˙ T M˙x + x˙ T Ba u˙ a + m a u˙ a2 Ts = 2 i=1 2 2 2 n

(7.10.44)

352

7 Application Examples

Fig. 7.123 Experimental setup

2 1  1  1 1 k1 x 1 − x g + ki (xi − xi−1 )2 = xT Kx − xT Kg x g + k1 x g2 2 2 i=2 2 2 n

Vs =

(7.10.45) 2 1  1  1 1 c1 x˙1 − x˙ g + ci (x˙i − x˙i−1 )2 = x˙ T C¨x − x˙ T Cg x˙ g + c1 x˙ g2 2 2 i=2 2 2 n

Ds =

(7.10.46) where x = [ x1 x2 . . . xn ]T and

7.10 Building Structure

⎡

m1 ⎢ 0 ⎢ M=⎢ . ⎣ ..

0 m2 .. .

353

··· ··· .. .

⎡

⎤

0 0 .. .

⎥ ⎥ ⎥, ⎦

0 0 · · · mn + ma ⎡ k1 + k2 −k2 ⎢ −k2 k2 + k3 ⎢ K=⎢ . .. ⎣ .. . 0

c1 + c2 −c2 · · · ⎢ −c2 c2 + c3 · · · ⎢ C=⎢ . .. .. ⎣ .. . . 0 0 ··· ⎡ ⎤ ⎤ 0 ··· 0 ⎢ 0 ⎥ ··· 0 ⎥ ⎢ ⎥ ⎥ , B = ⎢ . ⎥, ⎥ a . .. . ⎦ . ⎣ . ⎦ . .

0 0 .. .

⎤ ⎥ ⎥ ⎥, ⎦

(7.10.47a, b)

cn

ma · · · kn ⎡ ⎤ ⎡ ⎤ k1 c1 ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ Kg = ⎢ . ⎥, Cg = ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ 0 0

(7.10.47c, d)

0

(7.10.47e, f)

It is assumed that the TLD is a rectangular box whose dimensions are L × h × b, where L, h, and b are the length, height, and width of the box. It is also assumed that the fluid is irrotational and incompressible, so that the fluid motion can be expressed in terms of velocity potential. Then, the kinetic and potential energies of the fluid can be written as 1 Tf = ρ f b 2

0 L 

−h 0

∂φ x˙n + ∂x f

2

1 V f = ρ f gb 2

 +

∂φ ∂y f

2  dx f dy f

(7.10.48)

L η2 dx f

(7.10.49)

0

where ρ f is the fluid density, φ is the velocity potential, g is the gravitational constant, and η is the vertical deviation from the mean free surface. The linearized kinematic boundary condition on the free surface can be written as ( ∂φ (( ∂η = ∂t ∂ y f ( y f =0

(7.10.50)

The assumed-mode method is employed in this study. Hence, the velocity potential and the vertical deviation can be written as φ(x f , y f , t) = (x f , y f )r(t), η(x f , t) = (x f )p(t)

(7.10.51a, b)

wherer(t) = [r1 (t) r2 (t) · · · rm (t)]T , p(t) = [ p1 (t) p2 (t) · · · pm (t)]T are generalized coordinate vectors, m is the number of admissible functions, and

354

7 Application Examples

(x f , y f ) = [1 (x f , y f ) 2 (x f , y f ) · · · m (x f , y f )] (x f ) = [1 (x f ) 2 (x f ) · · · m (x f )] are matrices of admissible functions, in which  √ iπ y f + h iπ x f cosh , i = 1, 2, . . . , m i (x f , y f ) = 2 cos L L i (x f ) =

√ iπ x f , i = 1, 2, . . . , m 2 cos L

(7.10.52) (7.10.53)

(7.10.54) (7.10.55)

These admissible functions are in fact the eigenvectors of the sloshing motion. The displacement of nth floor can be written as xn = E x

(7.10.56)

where E = [ 0 0 · · · 1 ]. Inserting Eq. (7.10.51) into Eq. (7.10.48) and (7.10.49) and using Eq. (7.10.56), the kinetic and potential energies of the fluid can be rewritten as 1 Tf = ρ f b 2  +rT

0 L  x˙ T ET E x˙ + 2˙xT ET

−h 0

∂ r ∂x f

 ∂T ∂ ∂T ∂ r dx f dy f , + ∂x f ∂x f ∂y f ∂y f

1 V f = ρ f gb pT 2

(7.10.57)

L  T dx f p

(7.10.58)

0

Using Eqs. (7.10.54) and (7.10.55) and carrying out integrations, the kinetic and potential energies of the fluid can be written as   √ 1 2 2L T T T T −1 T T f = ρ f bL h x˙ E E˙x + x˙ E Em N Cs Sh r + π r NCh Sh r (7.10.59) 2 π Vf = where

1 T p Kpp 2

(7.10.60)

7.10 Building Structure

⎡

355

⎡ ⎤ 0 −2 0 · · · ⎢ 0 0 ··· 0⎥ ⎢ ⎥ .. ⎥, Cs = ⎢ .. .. . . ⎣ . . . ⎦ . 0 0 ··· m 0 0 ··· ⎡ sinh π h¯ 0 ··· ⎢ 0 ¯ sinh 2π h · · · ⎢ Sh = ⎢ .. .. .. ⎣ . . .

1 ⎢0 ⎢ N =⎢. ⎣ ..

0 2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

cos mπ − 1 ⎤ 0 ⎥ 0 ⎥ ⎥ .. ⎦ . 0 0 · · · sinh mπ h¯ ⎡ ⎤ cosh π h¯ 0 ··· 0 ⎢ ⎥ 0 cosh 2π h¯ · · · 0 ⎢ ⎥ Ch = ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . . ¯ 0 0 · · · cosh mπ h Em = [ 1 1 · · · 1 ], K p = ρ f gbL Imm

(7.10.61a)

(7.10.61b)

(7.10.61c)

(7.10.61d, e)

in which, h¯ = h/L is the fluid height to length ratio and Imm is an m × m identity matrix. Because Eqs. (7.10.59) and (7.10.60) are expressed in terms of r and p, it is necessary to express equations in terms of one set of generalized coordinates. To this end, we use the linearized free-surface condition given by Eq. (7.10.50). Inserting Eq. (7.10.51) into Eq. (7.10.50), the following equation holds: (x f )p˙ =

π (x f ) N Sh r L

(7.10.62)

Pre-multiplying Eq. (7.10.62) by  T (x f ) and integrating 0 to L, results in: r=

L −1 −1 N Sh p˙ π

(7.10.63)

Hence, we can express the kinetic energy of the fluid in terms of p by inserting Eq. (7.10.63) into Eq. (7.10.59): Tf =

1 1 m f x˙ T ET E˙x + x˙ T ET B p p˙ + p˙ T M p p˙ 2 2

(7.10.64)

where, mf = bhL, and: Bp = where:

√ 2 1 p f bL 2 Em N−2 Cs , M p = p f bL 2 N−1 T−1 h π2 π

(7.10.65a, b)

356

7 Application Examples

⎡

tanh π h¯ 0 ⎢ 0 tanh 2π h¯ ⎢ Th = ⎢ .. .. ⎣ . . 0 0

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(7.10.66)

· · · tanh mπ h¯

The total kinetic and potential energies can be obtained by using Eqs. (7.10.44), (7.10.45), (7.10.60), and (7.10.64): Ttotal = Ts + T f 1 1 1 1 ˙ p p˙ = x˙ T M˙x + x˙ T Ba u˙ a + m a u˙ a2 + m f x˙ T ET E˙x + x˙ T ET B p p˙ + pM 2 2 2 2 (7.10.67) Vtotal = Vs + V f 1 1 1 = xT Kx − xT Kg x g + k1 x g2 + pT K p p 2 2 2

(7.10.68)

The total Rayleigh’s dissipation function is assumed to be Dtotal =

1 T 1 1 x˙ C¨x − x˙ T Cg x˙ g + c1 x˙ g2 + p˙ T C p p˙ 2 2 2

(7.10.69)

where Cp is the fluid damping matrix. Lagrange’s equations for the addressed problem can be written as d dt d dt

 

∂L ∂ x˙ ∂L ∂ p˙

−

∂ Dtotal ∂L + =0 ∂x ∂ x˙

(7.10.70a)

−

∂L ∂ Dtotal + =0 ∂p ∂ p˙

(7.10.70b)



where L = T total − V total . Inserting Eqs. (7.10.67) and (7.10.68) into Eq. (7.10.70a, b), the coupled equations of motion are obtained for the structure and the TLD when subjected to the ground motion and the AMD action.

where

Mt x¨ + ET B p p¨ + C¨x + Kx = Bd dg − Ba u¨ a

(7.10.71)

BTp E¨x + M p p¨ + C p p˙ + K p p = 0

(7.10.72)

7.10 Building Structure

357

⎡

m1 ⎢ 0 ⎢ Mt = M + m f E T E = ⎢ . ⎣ ..

0 m2 .. .

··· ··· .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(7.10.73a)

0 0 · · · mn + ma + m f

Bd = [ 1 0 · · · 0 ]T , dg = c1 x˙ g + k1 x g

(7.10.73b, c)

Equations (7.10.71) and (7.10.72) can be combined into the following matrix equation of motion. Mz z¨ + Cz z˙ + Kz z = Bdz dg − Baz u¨ a

(7.10.74)

where z = [ xT pT ]T and      Mt ET B p C 0nm K 0nm , C , K , = = z z BTp E M p 0mn C p 0mn K p         Cg Kg Bd Ba , Kgz = , Bdz = , Baz = = 0m1 0m1 0m1 0m1 

Mz = Cgz

(7.10.75a–c)

(7.10.75d–g)

in which 0nm , 0mn , 0m1 are n × m, m × n, m × 1 zero matrices. As can be seen from Eq. (7.10.75a), the TLD is inertially coupled with the structure and the effect of the TLD is determined by ET B p . It can be also found from Eqs. (7.10.61e), (7.10.65b), and (7.10.72) that the sloshing frequency matrix can be easily obtained as  p = 2p =

g −1 πg Mp = NTh L L

(7.10.76)

This result is exactly the same as the analytical result of sloshing analysis. In addition, it is assumed that the fluid damping matrix satisfies the following relation: C p = 2M p Z p  p

(7.10.77)

where, Z p is the fluid damping factor matrix. ⎡

ζf1 ⎢ 0 ⎢ Zp = ⎢ ⎢ ... ⎣ 0

⎤ ··· 0 ··· 0 ⎥ ⎥ . . .. ⎥ . . ⎥ ⎦ .. 0 . ζfm

0 ζf2 .. .

(7.10.78)

in which ζ f i (i = 1, 2, . . . , m) is the damping factor of ith sloshing motion.

358

7 Application Examples

If the TLD is lightly damped and tuned to one of the natural frequencies of the target structure, the target resonant peak may clearly decrease, but the typical saddleback phenomenon appears, which means one peak is split into two small peaks. So, the resonant peak may be suppressed but two small peaks appear in the vicinity of the original resonant peak. This is good when the disturbance is harmonic or has dominant excitation frequency. But the use of TLD loses its advantage when subjected to random disturbance or initial condition. In addition, higher modes of the building structure still need to be suppressed but the effect of the TLD tuned to the fundamental frequency of the structure on the higher natural modes of the structure has not been investigated. In order to improve the performance of the TLD by decreasing two peaks, the fluid with high damping coefficient may be used. But it may not be desirable from the practical point of view, because a fluid other than water may not be used. The addition of one AMD that can tackle two small peaks resulted from the TLD as well as higher modes by using independently designed NAF controllers was proposed by Shin et al. (2020). In this way, we can use the advantages of both TLD and AMD. Based on the new concept of using additional AMD, the final displacement of the AMD is to be the sum of the desired displacements of NAF controllers that can tackle the resonant peaks including two peaks resulting from the TLD and higher modes. Hence, we may write u a = Guc

(7.10.79)

where G = [gc1 gc2 · · · gc ], uc = [u c1 u c2 · · · u c ]T , in which gci , u ci (i = 1, 2, . . . , ) are the gains of each controller and commands calculated by the NAF controllers. is the number of NAF controllers considered. The NAF controller is then expressed as: u¨ c + 2Zc c u˙ c + c uc = −Ec x¨

(7.10.80)

where ⎡

⎡ ⎤ ⎤ ωc1 0 · · · 0 0 ··· 0 ⎢ 0 ωc2 · · · 0 ⎥ ζc2 · · · 0 ⎥ ⎢ ⎥ ⎥ .. . . .. ⎥ .. .. . . .. ⎥ , c = ⎢ ⎢ ⎥ . . ⎦ . . ⎥ . ⎣ . . ⎦ .. .. 0 0 . ζc 0 0 . ωc ⎡ 2 ⎤ ωc1 0 · · · 0 ⎢ 0 ω2 · · · 0 ⎥ c2 ⎢ ⎥ ⎥ c = 2c = ⎢ ⎢ ... ... . . . ... ⎥, Ec = [ 0 (n−1) I 1 ] ⎣ ⎦ .. 2 0 0 . ωc ζc1 ⎢ 0 ⎢ Zc = ⎢ ⎢ ... ⎣

(7.10.81a, b)

(7.10.81c, d)

7.10 Building Structure

359

in which ζci , ωci (i = 1, 2, . . . , ) are the damping factors and the target frequencies for the controllers, 0 (n−1) is an × (n − 1) zero matrix, and I 1 = [1 1 · · · 1 ]T . It is assumed that the accelerometer is mounted on the top floor. As Yang et al. (2017) points out, the input to the NAF controller is an acceleration. Considering the fact that the accelerometer is the most popular sensor for vibration measurement, the implementation of the NAF control algorithms given by Eq. (7.10.80) can be easily realized. Combining Eqs. (7.10.71), (7.10.72), (7.10.79) and (7.10.80), the closed-loop matrix equations of motion can be written as: Mc z¨ c + Cc z˙ c + Kc zc = Bdc dg

(7.10.82)

where zc = [ xT pT ucT ]T and ⎡

⎡ ⎤ ⎤ Mt ET B p Ba G C 0nm 0n Mc = ⎣ BTp E M p 0m ⎦, Cc = ⎣ 0mn C p 0m ⎦, 0 n 0 m 2Zc c E 0 m I ⎡ ⎡ ⎤ ⎤ K 0nm 0n Bd Kc = ⎣ 0mn K p 0m ⎦, Bdc = ⎣ 0m1 ⎦ 0 n 0m c 0 1

(7.10.83a, b)

(7.10.83c, d)

in which 0m , 0 m , 0n , 0 n , 0 1 are m × , × m, n × , × n, × 1 zero matrices, and I is an × identity matrix. As can be seen from Eq. (7.10.83a), the AMD is also inertially coupled with the structure. We can use Eq. (7.10.82) to compute the closed-loop transfer function between the ground motion and the structural vibration when the TLD and AMD are both active. In order to investigate the performance of the hybrid control methodology proposed in this study, numerical simulation was carried out using the dynamic model. The numerical building model consists forty floors with 40 m width and 200 m height. Parameters for the building model are listed in Table 7.9. The natural frequencies for the building were found to be 0.25, 0.76, 1.26, 1.76, and 2.26 Hz. The fundamental frequency of the TLD is tuned to the first natural frequency of the building. The length, width, and height of the TLD are 8, 2, and 5 m, respectively. The water density is 1000 kg/m3 . 10 sloshing modes were considered and the sloshing frequencies of the TLD were found to be 0.25, 0.42, 0.54, 0.62, 0.70 Hz, and so on. It can be seen that the natural frequencies of the TLD are more densely populated than those of the building. The first sloshing frequency is tuned to the first natural frequency of the building. As a result, the first natural frequency is split into two natural frequencies, 0.246 and 0.260 Hz. It is interesting to see the effect of fluid damping factor on the transfer function between the ground motion and the displacement of the top floor. As can be seen from Eq. (7.10.75a), the building and the TLD are inertially coupled and the coupling effect can be predicted by B p . In our numerical example,

360 Table 7.9 Parameters for numerical simulation

7 Application Examples Parameters

Numerical values

Number of stories

40

Height of building

200 m

Width of building

40 m

Mass of each floor

8 × 105 kg

Stiffness of each floor

1.35 GN/m

Damping coefficient of each floor

105 Ns/m

Fluid damping factor

0.0035

Length of TLD

8m

Height of TLD

2

Width of TLD

5

Mass of movable mass

105 kg

B p = −104 [9.171 0.000 1.019 0.000 0.367 ] . It can be readily understood that the effect of higher sloshing modes decreases rapidly, which means that a single-degree-of-freedom model for the TLD can be sufficient for control design. The fluid damping factor is actually low, so that 0.35% is considered in the remaining numerical study. The NAF control was designed to suppress three peaks of the transfer function with the TLD, which are 0.246, 0.260, and 0.758 Hz. ζc1 = ζc2 = ζc3 = 0.3, Gc = [3 3 3] are chosen for control. Figure 7.120 shows the transfer function without the TLD and the unpowered AMD, the transfer function with the TLD only, and the transfer function with the TLD and the powered AMD. As can be seen from Fig. 7.120, two peaks resulting from the application of the TLD and the second natural mode of the building are successfully suppressed. The El Centro earthquake data shown in Fig. 7.121 was used in numerical simulation. The displacements at the fortieth floor for each case are shown in Fig. 7.122. Overall amplitude decreases during earthquake and the response after earthquake decreases rapidly by the introduction of the NAF control. Figure 10.122 shows that the proposed hybrid control method can be effectively used to suppress the building vibrations. Figure 7.123 shows the experimental apparatus that was made to validate the TLDAMD combined control method proposed in this study. The experimental apparatus consisted of one floor, and the TLD and AMD were mounted on the same floor. The active mass of the AMD and the based floor were motioned by AC servo motors. Ball-screw mechanism and linear guide were used to accurately control the position of the active mass and the floor. The acceleration of the top floor was measured by the accelerometer, LCF-200-0.5 g from Jewell Instruments [39], and the absolute displacement of the top floor was measured by the laser Sensor M3L/20 from MEL. The sensitivity of the accelerometer was 10 V/g. MicroLabBox of dSPACE Inc. was used to implement the control algorithms. Table 7.10 summarizes the parameters of the test bed. The fundamental frequency of the test bed without water was found to be 1.03 Hz. The damping factor of the

7.10 Building Structure Table 7.10 Parameters of test bed

361 Parameter

Value

ms

18.8 kg

ks

779.77 N/m

ζs

0.0025

ωn

6.44 rad/s

mf

2.62 kg

ζf

0.0314

h

0.131 m

L

0.5 m

b

0.04 m

ρf

1000 kg/m3

ma

1.2 kg

structure was calculated by using the logarithmic decrement. The height of the TLD was determined based on the sloshing frequency that was made equal to the fundamental frequency. The damping factor of the water was also calculated by using the logarithmic decrement. To this end, image processing meth-od was employed. Blue dye was put into water and white back screen was located behind the TLD. Digital Camera (Sony alpha 5100) was used to record the sloshing motion, and the video file was transferred to PC. The Region of Interest (ROI) was set up as shown in Fig. 7.124. The ROI image is (101 581) pixels, and the pixel resolution is 0.25 mm/pixel. Table 7.10 shows that the damping factor of the TLD is small. Hence, it can be expected that two peaks will appear by the TLD action. The accelerometer was connected to the ADC terminals of the MicroLabBox from dSPACE Inc.. The proposed NAF controller given by Eqs. (7.10.79) and (7.10.80)

Fig. 7.124 Image processing

362

7 Application Examples

computes the desired displacement of the movable mass based on the measured acceleration. In order to make the movable mass trace the desired position, the tracking control algorithm used in Yang et al. (2017) was again employed. Refer to the work of Yang et al. (2017) for detailed description of using the AC servo motor, motor driver, and tracking control. The gain and damping factor for the NAF controllers were 0.2526 and 0.3, respectively. Numerical simulation and experiment were carried out to investigate the validity of the dynamic model and the performance of the combined control methodology proposed in this study. Figure 7.125 shows the free vibration response of the system with the TLD. It can be seen from Fig. 7.125 that the simulation result is in good agreement with the measured result. It is evident that the TLD alone cannot suppress vibrations effectively. Figure 7.126 shows the free vibration of the system with the TLD and the AMD. It can be readily seen from Fig. 7.126 that the simulation result is in good agreement with the measured result and the combined TLD-AMD control can suppress vibrations more effectively than the TLD alone does. Figures 7.127 through 7.129 show the theoretical and experimental transfer functions between the ground motion and the acceleration of the top floor. Figures 7.127, 7.128 and 7.129 show that the dynamic model derived in this study can accurately predict the behavior of the system, and the two small peaks that resulted from the application of the TLD can be successfully suppressed by the additional AMD. The hybrid methodology that combines TLD-AMD control was proposed to effectively suppress structural vibrations of the MDOF system. The advantage of the combined TLD-AMD control is that it can utilize the advantages of both the TLD and AMD. The TLD is in charge of suppressing the main target mode, while the

Fig. 7.125 The free vibration response of the system with the TLD

7.10 Building Structure

Fig. 7.126 Free vibration response of the system with the TLD and AMD

Fig. 7.127 Transfer function without TLD and AMD

363

364

Fig. 7.128 Transfer function with TLD only

Fig. 7.129 Transfer function with TLD and AMD

7 Application Examples

7.10 Building Structure

365

AMD is in charge of suppressing the two peaks that resulted from the application of the TLD as well as some other modes of interest. Hence, the proposed hybrid control increases the frequency bandwidth and the performance of the vibration control. NAF controllers that use the acceleration signal were adopted for the AMD control. Numerical results show that higher sloshing modes can be neglected when designing the TLD. It was found in numerical studies that the addition of the AMD can effectively increase the control performance without introduction of a heavy mass. It was proven both theoretically and experimentally that the proposed TLD-AMD control methodology can effectively suppress the vibrations of structures.

7.11 Summary In this chapter, some of examples concerning active vibration control are presented. As can be seen from numerical and experimental results, even the simplest control algorithm is hard to apply in practice. The process of applying the active vibration requires knowledges on sensor, actuator, control algorithm, and implementation method including signal conditioning and amplification of command voltage. The dynamic model is essential to numerically estimate the performance of the designed controller. However, in designing the active vibration controller, practical limitations should be considered a priori, often ignored by researchers.

References Al-Najafi, AMJ, Warburton, GB (1970) Free Vibration of Ring-Stiffened Cylindrical Shells. Journal of Sound and Vibration 13 9–25. Amabili, M (1997) Flexural vibration of cylindrical shells partially coupled with external and internal fluids. ASME Journal of Vibration and Acoustics 119 476–484. Baek, KH, Kim, KY, Kwak, MK (2011) Dynamic Modeling and Controller Design for Active Control of High-speed Elevator Front-Back Vibrations. Transactions of the Korean Society for Noise and Vibration Engineering 21(1) 74–80. Denoyer, KK, Kwak, MK (1996) Dynamic Modelling and Vibration Suppression of a Slewing Active Structure Utilizing Piezoelectric Sensors and Actuators. Journal of Sound and Vibration 189(1) 13–31. Galletly, GD (1955) On the in-vacuo vibrations of simply supported, ring-stiffened cylindrical shells. Proc. 2nd U.S. Nat. Congr. Appl. Mech., Am. Soc. Mech. Engrs., New York, N.Y. 225–231. Garg, VK, Dukkipati, RV (1984) Dynamics of Railway Vehicle Systems. Academic Press. Harari, A, Sandman, BE (1990) Radiation and vibrational properties of submerged stiffened cylindrical shells. Journal of Acoustical Society of America 88(4) 1817–1830. Iakovlev, S (2008) Interaction between a submerged evacuated cylindrical shell and a shock wave - Part 1, Diffraction-radiation analysis. Journal of Fluids and Structures 24 1077–1097. Jafari, AA, Bagheri, M (2006) Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods. Thin-Walled Structures 44 82–90. Kim, YW, Lee, YS, Ko, SH (2004) Coupled vibration of partially fluid-filled cylindrical shells with ring stiffeners. Journal of Sound and Vibration 276 869–897.

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7 Application Examples

Kwak, MK (2010) Free vibration analysis of a finite cylindrical shell in contact with unbounded external fluid. Journal of Fluids and Structures 26 377–392. Kwak, MK, Heo, S (2007) Active vibration control of smart grid structure by multiinput and multioutput positive position feedback controller. Journal of Sound and Vibration 304 230–245. Kwak, MK, Heo, S, Jeong, M (2009) Dynamic modelling and active vibration controller design for a cylindrical shell equipped with piezoelectric sensors and actuators. Journal of Sound and Vibration 321 510–524. Kwak, MK, Heo, S, Lee, MI (2003) Dynamic modelling and control of rectangular plate with piezoceramic sensors and actuators. SPIE’s 10th Annual International Symposium on Smart Structures and Materials, San Diego, California USA 5049 361–370. Kwak, MK, Kim, KY, Baek, KH (2011) Dynamic Modeling and Active Controller Design for Elevator Lateral Vibrations. Transactions of the Korean Society for Noise and Vibration Engineering 21(2) 154–161. Kwak, MK, Lee, JH, Yang, DH, You, WH (2014) Hardware-in-the-loop simulation experiment for semi-active vibration control of lateral vibrations of railway vehicle by magneto-rheological fluid damper. Vehicle System Dynamics 52 891–908. Kwak, MK, Yang, DH (2013a) Free vibration analysis of cantilever plate partially submerged into a fluid. Journal of Fluids and Structures 40 25–41. Kwak, MK, Yang, DH (2013b) Active vibration control of a ring-stiffened cylindrical shell in contact with unbounded external fluid and subjected to harmonic disturbance by piezoelectric sensor and actuator. Journal of Sound and Vibration 332 4775–4797. Kwak, MK, Yang, DH (2015) Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators. Journal of Fluids and Structures 54 848–867. Laulagnet, B, Guyader, JL (1989) Modal analysis of a shell’s acoustic radiation in light and heavy fluids. Journal of Sound and Vibration 131(3) 397–415. Leblond, C, Iakovlev, S, Sigrist, JF (2009) A Fully Elastic Model for Studying Submerged Circular Cylindrical Shells Subjected to a Weak Shock Wave. Mécanique & Industries 10 275–284. Lee, HD, Kwak, MK, Kim, JH, Song, YC, Shim, JH (2009) Dynamic Characteristics of ALA and Active Vibration Control Experiment. Transaction of the Korean Society for Noise and Vibration Engineering 19(8) 781–787. Lee, HW, Kwak, MK, Kim, KY, Lee, HD (2009) Development of Linear Magnetic Actuator for Active Vibration Control. Transactions of the Korean Society for Noise and Vibration Engineering 19(7) 667–672. Lee, JH, Kwak, MK, Yang, DH, Yoo, WH (2012) Development of Dynamic Modeling and Control Algorithm for Lateral Vibration HILS of Railway Vehicle. Transactions of the Korean Society for Noise and Vibration Engineering 22(7) 634–641. Mattei, PO (1995) Sound radiation by a baffled shell: comparison of the exact and an approximate solution. Journal of Sound and Vibration 188(1) 111–130. Pan, Z, Li, X, Ma, J (2008b) A study on free vibration of a ring-stiffened thin circular cylindrical shell with arbitrary boundary conditions. Journal of Sound and Vibration 314 330–342. Preumont, A (2011) Vibration Control of Active Structures 3rd edition. Berlin, Springer. Shin, JH, Han, SB, Kwak, MK (2019a) Active vibration control of plate by active mass damper and negative acceleration feedback control algorithms. Journal of Low Frequency Noise, Vibration and Active Control 40 413–426. Shin, JH, Lee, JH, You, WH, Kwak, MK (2019b) Vibration suppression of railway vehicles using a magneto-rheological fluid damper and semi-active virtual tuned mass damper control. Noise Control Engineering Journal 67(6) 493–507. Shin, JH, Kwak, MK, Kim, SM, Baek, KH (2020) Vibration control of multi-story building structure by hybrid control using tuned liquid damper and active mass damper. Journal of Mechanical Science and Technology 34(12) 5005–5015. https://doi.org/10.1007/s12206-020-1105-4.

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Index

A Accelerometer ICP, 195–197 low-frequency, 13, 197, 198, 293 MEMS, 195–199 Active mass damper (AMD), 7, 15–17, 20– 23, 175, 179, 194, 206–208, 248– 257, 325, 326, 331, 332, 334, 338, 344, 349, 351, 356, 358–360, 362– 365 Active vibration control aircraft, 17 automobile, 12, 303 building structure, vi, vii elevator, 283 railway vehicle, 14, 310 robots, vi space structures, 17 Actuator, 2–24, 143, 147, 150–153, 156, 158, 159, 162–166, 168, 171, 172, 175, 176, 183, 194, 195, 200–212, 215, 217, 218, 220–222, 224–228, 230–234, 237–243, 248, 255, 268– 273, 277, 278, 283, 287, 290, 291, 293, 304, 305, 307, 310, 317, 318, 325, 331, 365 Analog circuit, 9, 159, 173, 194, 210–213, 215 Assumed mode method, 6, 69, 106, 109, 112, 114, 138, 144, 218, 250

B Block diagram, 213, 234, 242, 243, 289, 300, 305, 307, 310, 344, 345 Bode magnitude plot, 48 Bode phase plot, 48, 50, 58

Bode plot, 50, 53, 101, 160, 176, 177

C Characteristic equation, 41, 43, 46, 71, 72 Closed-loop control, 154

D Damping factor, 42, 43, 68, 77–80, 107, 138, 139, 144, 152, 155, 156, 159, 160, 162, 168, 172, 173, 176, 179, 186, 209, 210, 221, 227–229, 234, 239, 243, 255, 278, 286, 294, 304, 305, 319, 357, 359–362 proportional, 79, 107, 138, 139, 286 Damping factor matrix, 77, 79, 152, 227, 229, 239, 357 Direct velocity feedback (DVF) control, 156 DSP, 212, 214, 215, 234, 293, 294 DSPACE controller, 211

E Eigenvalues of a matrix, 72, 83, 138, 227, 326, 337 Eigenvectors of a matrix, 72, 73, 77, 78, 98, 138, 152, 326, 337 Electro-magnetic actuator, 13 Euler’s formula, 41

F Feedback control, 3, 5, 6, 9, 16, 18–20, 22, 23, 147–149, 153, 154, 156, 158, 168, 175, 183, 290

© Springer Nature B.V. 2022 M. K. Kwak, Dynamic Modeling and Active Vibration Control of Structures, https://doi.org/10.1007/978-94-024-2120-0

369

370 Feedforward control, 21, 147, 148, 183, 184, 186, 189 Filtered-x LMS, 7, 21, 183 Finite element method (FEM), 5, 10, 106, 109, 138 Forced response arbitrary excitation, 59–61 base excitation, 56 harmonic excitation, 46, 48, 54, 92 Free vibration damped, 42, 44, 45, 76 undamped, 40, 43, 44, 71 Frequency response, 5, 48–50, 78, 80, 98, 101, 159, 161, 178, 179, 222–224, 227–229, 234, 235, 241, 242, 255, 258, 260, 294, 302, 305, 320, 328, 339–347 Frequency response function (FRF), 5, 227, 255, 258, 260, 302, 328, 329

H Hardware-in-the-loop-simulation (HILS), 12, 14, 295, 298–301, 310, 313–315, 317, 318, 320 Higher harmonic control (HHC), 8, 17, 167– 169, 278, 304, 307 High-voltage amplifier, 11, 202

I Impulse response, 59–61, 140, 183, 184, 255, 259, 261

K Kalman filter, 11, 18, 19

L Lagrange’s equations, 106, 107, 125, 238, 356 Laser displacement sensor, 193, 203, 295 Linear actuator, 12, 206, 208, 290, 304

M Magneto-Rheological (MR) damper, 12, 14, 194, 210, 317, 319, 320, 322, 325 Matlab, 44, 45, 48, 50–54, 57, 62, 64, 65, 72, 73, 76, 79, 80, 84, 86–88, 93–95, 98, 101, 102, 110, 119, 135, 140, 149, 155, 186, 286, 300 Microcontroller, 9, 212, 214

Index Modal analysis, 67, 77, 88, 90, 138 Modal coordinates, 73, 75, 83, 93, 98, 138, 151, 152, 221, 227, 232, 239 Modal damping matrix, 67, 87, 88, 107, 138 Modal transformation, 73, 75, 77, 80, 82, 83, 92, 93, 138, 139, 151, 152, 179, 180, 221, 239, 251, 274, 326, 336–338 Multi-degree-of-freedom (MDOF) system, 8, 65–68, 70–72, 76, 78, 82, 83, 86, 95, 98, 107, 138, 152, 155, 194, 325, 328, 351, 362 N Negative acceleration feedback (NAF) control, 7, 17, 21, 23, 175–180, 252, 254, 327, 328, 331, 332, 339, 344, 345, 349, 359, 360 Newton’s second law of motion, 39 O Optimal control, 5, 6, 12, 16, 18, 22, 180 Orthonormality condition, 72–75, 77, 78, 83, 92, 138, 139, 152, 179, 221, 227, 239, 251, 274, 326, 336, 337 P Piezoelectric actuator, 4–7, 10–12, 16, 17, 20, 158, 162, 163, 165, 166, 202, 237, 268, 269 Piezoelectric sensor, 4–8, 10, 17–19, 158, 159, 162, 163, 165, 166, 169, 176, 200–202, 217, 218, 220, 233, 234, 249, 255, 269, 277, 278, 281 Plate vibrations, 114 Pneumatic actuator, 13, 14, 194, 205 Positive position feedback (PPF) control, 4, 8, 9, 17–19, 158, 159, 162, 163, 165, 166, 170, 175, 176, 182, 211, 215, 230–232, 240, 241, 253 Proportional-integral-derivative (PID) control, 6, 16, 157, 158, 175, 207, 287, 344, 349 Pseudoinverse, 153, 166, 230, 232, 233, 327 R Rayleigh-Ritz method (RRM), 6, 7, 106, 109, 126, 133 Reduced-order analysis, 82, 84, 98 Reduced-order model, 2, 16, 22, 82–84, 101, 102, 151, 164, 179 Riccati equation, 181, 182

Index S Semi-active vibration control, 14, 20, 21, 209, 210 Shunt method, 9 Simulink, 54–56, 61, 62, 64, 65, 93, 101, 102, 213, 214, 234, 242, 243, 287, 289, 300, 304, 305, 307, 310, 318, 329, 332, 344, 345 Single-degree-of-freedom (SDOF) system, 22, 49, 67, 74, 79, 92, 149, 152–156, 162, 163, 171, 178, 179 Single-input, single-output (SISO) system, 154, 155 Smart structure beam, 4 grid structure, 5 plate, 5 shell, 7 Spillover control, 151, 165, 231, 320 observer, 151, 166, 232, 338 Spring-mass-damper system, 39, 56, 78, 171, 172 State space equation, 52, 55, 67, 86, 88, 93, 94, 96, 97, 139, 149, 155, 180, 181, 295, 314

371 Steady-state response, 47, 57, 59

T Transfer function, 47, 48, 50, 55, 57, 92, 93, 98, 149, 153–155, 159–161, 167– 169, 172–174, 176, 178, 184, 186, 211, 222, 224, 227, 228, 233, 240, 243, 244, 246–248, 255, 273, 275, 277, 278, 294, 305, 306, 331, 339, 350, 359, 360, 362–364

V Vibration beam, 140 cylindrical shell, 7, 125, 135, 255, 259 rectangular plate, 6, 114, 248 Vibration compensator, 208, 209 Virtual tuned mass damper (VTMD) control, 20, 172, 173, 175, 179, 182, 303, 318 Voice-coil actuator (VCA), 15, 194, 203, 204, 207, 209, 290–293