Active Control of Structures

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Printer Name: Yet to Come

Active Control of Structures

Active Control of Structures A. Preumont and K. Seto © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-03393-7

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Active Control of Structures ´ Preumont Andre Université Libre de Bruxelles, Brussels

Kazuto Seto Nihon University, Japan

A John Wiley & Sons, Ltd, Publication

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This edition first published 2008 C 2008 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Preumont, Andre. Active control of structures / Andre Preumont, Kazuto Seto. p. cm. Includes bibliographical references and index. ISBN 978-0-470-03393-7 (cloth) 1. Structural control (Engineering) I. Seto, Kazuto. II. Title. TA654.P745 2008 624.1 71–dc22 2008039309

A catalogue record for this book is available from the British Library. ISBN: 978-0-470-03393-7 Set in 10.5/12.5pt Palatino by Aptara Inc, New Delhi, India Printed in UK by TJ International Ltd, Padstow, Cornwall.

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Contents About the Authors

xi

Preface

xiii

Acknowledgements

xv

1 Active Damping 1.1 Introduction 1.1.1 Why Suppress Vibrations? 1.1.2 How can Vibrations be Reduced? 1.2 Structural Control 1.3 Plant Description 1.3.1 Error Budget 1.4 Equations of Structural Dynamics 1.4.1 Equation of Motion Including Seismic Excitation 1.4.2 Modal Coordinates 1.4.3 Support Reaction, Dynamic Mass 1.4.4 Dynamic Flexibility Matrix 1.5 Collocated Control System 1.5.1 Transmission Zeros and Constrained System 1.5.2 Nearly Collocated Control System 1.5.3 Non-Collocated Control Systems 1.6 Active Damping with Collocated System 1.6.1 Lead Control 1.6.2 Direct Velocity Feedback 1.6.3 Positive Position Feedback 1.6.4 Integral Force Feedback 1.6.5 Duality between The Lead and IFF Controllers 1.7 Decentralized Control with Collocated Pairs 1.7.1 Cross-Talk 1.7.2 Transmission Zeros (Case 1) 1.7.3 Transmission Zeros (Case 2) References

1 1 1 2 2 3 4 6 6 8 10 12 15 18 20 21 24 25 29 31 35 44 46 46 47 52 55

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Contents

2 Active Isolation 2.1 Introduction 2.2 Relaxation Isolator 2.2.1 Electromagnetic Realization 2.3 Sky-hook Damper 2.4 Force Feedback 2.5 Six-Axis Isolator 2.5.1 Decentralized Control 2.5.2 Leg Design 2.5.3 Model of the Isolator 2.5.4 Six-Axis Transmissibility 2.6 Vehicle Active Suspension 2.6.1 Quarter-Car Model 2.7 Semi-Active Suspension 2.7.1 Semi-Active Devices 2.7.2 Narrow-Band Disturbance 2.7.3 Quarter-Car Semi-Active Suspension References

57 57 60 62 64 66 69 73 76 80 82 89 91 106 106 107 108 113

3 A Comparison of Passive, Active and Hybrid Control 3.1 Introduction 3.2 System Description 3.3 The Dynamic Vibration Absorber 3.3.1 Single-d.o.f. Oscillator 3.3.2 Multiple-d.o.f. System 3.3.3 Shear Frame Example 3.4 Active Mass Damper 3.5 Hybrid Control 3.6 Shear Control 3.7 Force Actuator, Displacement Sensor 3.7.1 Direct Velocity Feedback 3.7.2 First-Order Positive Position Feedback 3.7.3 Comparison of the DVF and the PPF 3.8 Displacement Actuator, Force Sensor 3.8.1 Comparison of the IFF and the DVF References

117 117 119 120 120 123 124 126 131 133 135 136 137 138 140 142 144

4 Vibration Control Methods and Devices 4.1 Introduction 4.2 Classification of Vibration Control Methods

147 147 148

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4.3 4.4 4.5 4.6

4.7

Construction of Active Dynamic Absorber Control Devices for Wind Excitation Control in Civil Structures Real Towers Using the Connected Control Method Application of Active Dynamic Absorber for Controlling Vibration of Single-d.o.f. Systems 4.6.1 Equations of Motion and State Equation 4.6.2 Representation of a Non-Dimensional State Equation 4.6.3 Control System Design 4.6.4 Similarity Law between Dimensional and Non-dimensional System 4.6.5 Analysis of Vibration Control Effect 4.6.6 Experiment Remarks References

5 Reduced-Order Model for Structural Control 5.1 Introduction 5.2 Modeling of Distributed Structures 5.2.1 Equation of Motion for Distributed Structures 5.2.2 Conventional Modeling of Structures 5.3 Spillover 5.4 The Lumped Modeling Method 5.4.1 A Key Idea for Deriving a Reduced-Order Model 5.4.2 Relationship Between Physical and Modal Coordinate Systems 5.4.3 Modification of Normalized Modal Matrix 5.5 Method of Equivalent Mass Estimation 5.5.1 Meaning of Equivalent Mass 5.5.2 Eigenvector Method 5.5.3 Mass Response Method 5.6 Modeling of Tower-like Structure 5.6.1 Two-d.o.f. Model 5.6.2 Dimension and Dynamic Characteristics of the Tower-Like Structure 5.6.3 Calculation of Parameters of Two-d.o.f. Model 5.6.4 Comparison between the Distributed Parameter and Two-d.o.f. Systems

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151 154 156 158 159 160 162 163 165 173 175 176 179 179 180 180 181 183 185 185 187 188 190 190 191 193 197 197 198 201 203

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5.7

Modeling of Plate Structures 5.7.1 Dimensions of a Plate Structure 5.7.2 Three-d.o.f. Model 5.7.3 Calculation of Parameters of the Three-d.o.f. Model 5.7.4 Comparison between Real System and Three-d.o.f. Systems 5.8 Modeling of a Bridge Tower 5.8.1 Dimensions of a Scaled Bridge Tower 5.8.2 Construction of a Four-d.o.f. Model 5.8.3 Calculation of Parameters of the Four-d.o.f. Model 5.8.4 Comparison between Real System and Four-d.o.f. Systems 5.9 Robust Vibration Control for Neglected Higher Modes 5.10 Conclusions References 6 Active Control of Civil Structures 6.1 Introduction 6.2 Classification of Structural Control for Buildings 6.3 Modeling and Vibration Control for Tower Structures 6.3.1 One-d.o.f. Model 6.3.2 Two-d.o.f. Model for Tower-Like Structures and Its LQ Control 6.3.3 Three-d.o.f. Model for Broad Structures and Its H∞ Robust Control 6.3.4 Four-d.o.f. Model for Bridge Tower and Spillover Suppression Using Filtered LQ Control 6.4 Active Vibration Control of Multiple Buildings Connected with Active Control Bridges in Response to Large Earthquakes 6.4.1 Construction of Four Model Buildings 6.4.2 Characteristics of the Tower Structures 6.4.3 Reduced-order Model of the Four Tower Structures Connected by Four Actuators 6.4.4 Control System Design 6.4.5 Simulated Results of Seismic Response Control 6.4.6 Experiment 6.5 Vibration Control for Real Triple Towers Using CCM 6.5.1 Outline of the Triple Towers 6.5.2 Modeling of Towers

203 203 206 207 208 209 209 210 212 213 217 217 219 221 221 222 222 222 225 228 239 249 249 251 252 254 257 259 264 264 265

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6.5.3 Control System Design 6.5.4 Simulation of the Triple Towers Using CCM 6.5.5 Realization of the CCM 6.6 Vibration Control of Bridge Towers Using a Lumped Modeling Approach 6.6.1 Vibration Problem of Bridge Towers Under Construction 6.6.2 Controlled Object and Its Dynamic Characteristics 6.6.3 Five-d.o.f. Modeling of a Scaled Bridge Tower Structure with a Crane Tower 6.6.4 LQ Control System Design 6.6.5 Simulations 6.6.6 Experiments 6.6.7 H∞ Robust Control 6.7 Conclusion References Index

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266 269 270 274 274 277 278 278 283 283 286 290 291 293

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About the Authors André Preumont André Preumont was born in 1951. He obtained his PhD from the University of Liège (Belgium) in 1981 while he was working in earthquake engineering at Belgonucléaire. In 1985–6, he was visiting professor in the Aerospace and Ocean Engineering Department at Virginia Tech (USA). He was appointed to the chair of Mechanical Engineering and Robotics of ULB (Université Libre de Bruxelles) in 1987. He created the Active Structures Laboratory in 1995. He is also part-time professor at the University of Liège. He is the author of several books on active vibration control and random vibration. His current research interests are in mechatronics, active vibration control, adaptive optics and precision engineering. He has had visiting positions at UTC, Compiègne and INSA-Lyon (France). Kazuto Seto Kazuto Seto was born in 1938. He graduated from a doctoral course in Engineering at Tokyo Metropolitan University in 1971 and received a Dr. Eng. degree from Tokyo Metropolitan University in the same year. He worked as an associate professor and professor in the Department of Mechanical Engineering, National Defense Academy from 1973 until 1993. From 1993, he worked as a professor in the Department of Mechanical Engineering, Nihon University. He retired in 2007 and is now president of Seto-Vibration Control Laboratory. He is an honorary member of Japan Society of Mechanical Engineers (JSME) and a fellow of IE Australia. He was given JSME Awards for his research in 1984 and 1989, and the Dynamics, Measurement and Control Award from JSME in 1994 and 1996. His research interests are in the areas of structural vibration control, system modeling and identification, motion and vibration control for multistructural systems with multi-controlled modes.

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Preface This book is the result of 20 years of friendship which started in Tsuchiura, Japan, in 1989. The two authors come from different backgrounds and continents: André Preumont comes from Europe and does most of his research in space structures and precision engineering, while Kazuto Seto comes from Japan and is a specialist in the control of civil engineering structures. They share the mechatronics (or system) approach, and the desire to make control systems work. This book consists of two parts; Chapters 1–3 written by André Preumont and Chapters 4–6 written by Kazuto Seto. They were written independently, with the authors’ own sensitivity and personal experience, but we believe that they are complementary and they will help young scientists who enter the field as well as more experienced people, at a different level of reading. This introductory text tries to avoid complicated concepts; the only prerequisite for reading this book is some background in automatic control and linear system theory, and in structural dynamics.

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Acknowledgements The authors wish to express their appreciation to their students and collaborators who, over the past 20 years, have spent so much of their time trying to cross the bridge between theory and practice, and to make systems work in the laboratory. And, last but not least, they wish to express their deep gratitude to their wives, Yvette and Mitsu, for their love and patience.

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1 Active Damping 1.1 Introduction 1.1.1 Why Suppress Vibrations? Mechanical vibrations span amplitudes from meters (civil engineering) to nanometers (precision engineering). Their detrimental effects on systems may be of various kinds: Failure. Vibration-induced structural failure may occur as a result of excessive strain during transient events (e.g. building response to earthquake), instability due to particular operating conditions (flutter of bridges under wind excitation), or simply fatigue (mechanical parts in machines). Comfort. Examples where vibrations are detrimental to comfort are numerous: noise in helicopters, car suspensions, wind-induced sway of buildings. Operation of precision devices. Numerous systems in precision engineering, especially optical systems, put severe restrictions on mechanical vibrations. Precision machine tools, DVD readers, and telescopes are typical examples. Moore’s law on the number of transistors on an integrated circuit could not hold without a constant improvement of the wafer stepper accuracy. The performance of large interferometers such as the VLTI is limited by microvibrations affecting the various parts of the optical path. Active Control of Structures A. Preumont and K. Seto © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-03393-7

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Active Damping

Lightweight segmented telescopes (in space as well as earth-based) will be impossible to build in their final shape with an accuracy of a fraction of a wavelength, because of the various disturbance sources such as thermal gradients (which dominate the space environment). Such systems will not exist without the capability to actively control the reflector shape. 1.1.2 How can Vibrations be Reduced? Vibration reduction can be achieved in many different ways, depending on the problem; the most common are stiffening, damping and isolation. Stiffening consists of shifting the resonance frequency of the structure beyond the frequency band of excitation. Damping consists of reducing the resonance peaks by dissipating the vibration energy. Isolation consists of preventing the propagation of disturbances to sensitive parts of the systems. Damping may be achieved passively, with fluid dampers, eddy currents, elastomers or hysteretic elements, or by transferring kinetic energy to dynamic vibration absorbers. One can also use transducers as energy converters, to transform vibration energy into electrical energy that is dissipated in electrical networks, or stored (energy harvesting). Recently, semi-active devices (also called semi-passive) have become available; they consist of passive devices with controllable properties. The magneto-rheological fluid damper is a famous example; piezoelectric transducers with switched electrical networks are another. When high performance is needed, active control can be used; this involves a set of sensors (strain, acceleration, velocity, force, . . .), a set of actuators (force, inertial, strain, . . .) and a control algorithm (feedback or feedforward). Active damping is the main focus of this chapter. The design of an active control system involves many issues such as how to configurate the sensors and actuators (map of strain energy or kinetic energy) and how to secure stability and robustness (collocated actuator/sensor pairs); the power requirement will often determine the size of the actuators and the cost. An alternative which will be discussed later in this text is the so-called hybrid control; this combines active and passive features to achieve performance at reduced cost.

1.2 Structural Control As compared to other control problems, structural control has a number of specific features. Firstly, the systems involved generally have a large number of degrees of freedom (d.o.f.) and a large number of modes (in fact infinitely many, but in most cases it will be more than enough to consider

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Plant Description

3

a discrete approximation of the distributed system). In most cases, finite elements will tend to produce models with too many d.o.f. This problem is usually solved by working in modal coordinates and truncating the model beyond the frequency band of interest. Attention must be paid, however, to the fact that the high-frequency modes outside the frequency band of interest influence the position of the open-loop zeros of the system and call for a quasi-static correction. Secondly, many structures involved in structural control are lightly damped (ξ ∼ 0.001 to 0.05). This means that the stability margin of the uncontrolled modes is small, sometimes very small, and that they are subject to spillover, which means that the control system always tends to destabilize the flexible modes just outside the control bandwidth and this has to be handled adequately; the only margin against spillover instability is provided by damping of the residual modes. The combination of a large number of modes with a small stability margin calls for specific control strategies emphasizing robustness, with respect to the residual dynamics (high-frequency modes) and also with respect to the changes in the system parameters. Control systems with collocated (dual) actuator/sensor pairs exhibit special properties which are especially attractive in this respect. They are analyzed in detail here. In this chapter, we try to relate familiar concepts of structural dynamics and classical concepts of linear systems and control theory; an elementary knowledge of both fields is assumed.

1.3 Plant Description Consider the block diagram of Figure 1.1, in which the plant consists of the structure and its actuator and sensor. w is the disturbance applied to the structure, z is the controlled variable or performance metric (which it Disturbance

w

Control input

u

Plant

z

Performance metric

y

Output measurement

H(s) Figure 1.1 Block diagram of the control system

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Active Damping

is desirable to keep as close as possible to 0), u is the control input and y is the sensor output (they are all assumed scalar for simplicity). H(s) is the control feedback law, expressed in the Laplace domain (s is the Laplace variable). We define the open-loop transfer functions G zw (s) between w and z, G zu (s) between u and z, G yw (s) between w and y, and G yu (s) between u and y. From the definition of the open-loop transfer functions, y = G yw w + G yu Hy

(1.1)

y = (I − G yu H)−1 G yw w.

(1.2)

u = Hy = H(I − G yu H)−1 G yw w = Tuw w.

(1.3)

or

It follows that

On the other hand, z = G zw w + G zu u.

(1.4)

Combining the two foregoing equations, one finds the closed-loop transmissibility between the disturbance w and the control metric z: z = Tzw w = G zw + G zu H(I − G yu H)−1 G yw w. (1.5)

1.3.1 Error Budget The frequency content of the disturbance w is usually described by its power spectral density (PSD), w (ω), which describes the frequency distribution of the mean-square (MS) value σw2 =

0

∞

w (ω)dω

(1.6)

(the units of w is readily obtained from this equation – it is expressed in units of w squared per (rad/s)). From (1.5), the PSD of the performance metric z is given by z (ω) = |Tzw |2 w (ω).

(1.7)

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Figure 1.2 Error budget distribution for various control configurations

z (ω) gives the frequency distribution of the MS value of the performance metric. Even more interesting for design is the cumulative PSD, defined by the integral of the PSD in the frequency range [0, ω]: σz2 (ω) =

ω

0

z (ν)dν =

ω

0

|Tzw |2 w (ν)dν.

(1.8)

It is a monotonically increasing function of frequency and describes the contribution of all the frequencies below ω to the MS value of the error budget. σz (ω) is expressed in the same units as the performance metric z; a typical plot is shown in Figure 1.2 for a hypothetical system with four modes. For lightly damped structures, the diagram exhibits steps at the natural frequencies of the modes and the magnitude of the steps gives the contribution of each mode to the error budget, in the same units as the performance metric. It is very helpful to identify the critical modes in a design, at which the effort should be targeted. This diagram can be used to assess the control laws as well as actuator and sensor configurations. In a similar way, the control budget can be assessed from σu2 (ω)

= 0

ω

u (ν)dν =

0

ω

|Tuw |2 w (ν)dν.

(1.9)

σu (ω) describes how the RMS control input is distributed over the various modes of the structure and plays a critical role in the actuator design.

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As one can see, the frequency content of the disturbance w, described by w (ω), is essential in the evaluation of the error and control budgets and it is very difficult, even risky, to attempt to design a controller without prior information on the disturbance.

1.4 Equations of Structural Dynamics 1.4.1 Equation of Motion Including Seismic Excitation In this section, we recall the equations governing the dynamics of a linear structure subjected to a point force f and a single-axis seismic excitation of acceleration x¨0 (Figure 1.3(a)). The seismic response has some special features which need to be treated with care. We start from Hamilton’s principle, which states that the variational indicator t2 t2 (δL + δWnc )dt = [δ(T ∗ − V) + δWnc ]dt = 0 (1.10) t1

t1

x

1

y

Figure 1.3 (a) Structure subjected to a single-axis support excitation. (b) Fictitious shaking table representation highlighting the reaction force f0 . (c) Partition of the global displacement into rigid body motion and flexible motion relative to the base

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Equations of Structural Dynamics

7

vanishes for arbitrary variations of the path between two instants t1 and t2 compatible with the kinematics, and such that the configuration is fixed at t1 and t2 . In this equation, T ∗ is the kinetic coenergy (often simply called kinetic energy), V the strain energy, and δWnc is the virtual work associated with the non-conservative forces (the external force f and the support reaction f 0 in this case). One may decompose the global displacements x (generalized displacements including translational as well as rotational d.o.f.) into the rigid body mode and the flexible motion relative to the base (Figure 1.3(c)), x = 1x0 + y,

(1.11)

where x0 is the support motion, 1 is the unit rigid body mode (all translational d.o.f. along the axis of excitation are equal to 1 and the rotational d.o.f. equal to 0). The virtual displacements satisfy δx = 1δx0 + δy,

(1.12)

where δx0 is arbitrary and δy satisfies the clamped boundary conditions at the base. Let b be the influence vector of the external loading f (for a point force, b contains all 0 except 1 at the d.o.f. where the load is applied), so that bT x is the generalized displacement of the d.o.f. where f is applied. The energy terms involved in Hamilton’s principle are 1 1 T ˙ x˙ Mx, V = yT K y, 2 2 T δWnc = f 0 δx0 + f b δx = f 0 + f bT 1 δx0 + f bT δy. T∗ =

(1.13) (1.14)

One notices that the kinetic coenergy depends on the absolute velocity, while the strain energy depends on the flexible motion alone, that is, the motion relative to the base. For a point force, bT 1 = 1. Substituting into the variational indicator, one finds

t2 t1

x˙ T Mδ x˙ − yT K δy + ( f 0 + f )δx0 + f bT δy dt = 0

(1.15)

and, upon integrating the first term by parts, taking into account that δx(t1 ) = δx(t2 ) = 0, t1

t2

(−¨xT M1 + f 0 + f )δx0 − (¨xT M + yT K − f bT )δy dt = 0

(1.16)

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for arbitrary δx0 and δy. It follows that M¨x + K y = b f

(1.17)

f 0 = 1T M¨x − f

(1.18)

and

The presence of f on the right-hand side of this equation is necessary to achieve static equilibrium; it is often ignored in the formulation. Combining (1.11) with (1.17) gives the classical equation expressed in relative coordinates, Mÿ + K y = −M1x¨0 + b f.

(1.19)

Also combining with (1.17), (1.18) can alternatively be written f 0 = −1T K y.

(1.20)

The first term on the right-hand side of (1.19) is simply the inertia forces associated with a rigid body acceleration x¨0 of the support. The foregoing analysis can be generalized to a single support multi-axis excitation rather easily (x0 becomes a vector in this case); the extension to several supports with differential motion is more difficult and beyond the scope of this text (e.g. Clough and Penzien, 1975; Preumont, 1994). The structural damping has been ignored for simplicity; if a viscous damping is assumed on the relative coordinates y, a contribution C y˙ is added to the left-hand side of (1.19), and (1.20) must be changed accordingly. 1.4.2 Modal Coordinates Consider the homogenous equation governing the free response (no base motion x0 or external force f ) of the conservative (undamped) system: Mÿ + K y = 0.

(1.21)

If one tries a solution y = φi e jωi t , the mode shape φi and the natural frequency ωi must satisfy the eigenvalue problem (K − ωi2 M)φi = 0.

(1.22)

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9

Because M and K are non-negative definite, the mode shapes are real and the eigenvalues ωi2 ≥ 0. The number of modes is equal to the number of d.o.f. of the system, but the structural response is often dominated by the first ones. We will address the modal truncation in detail a little later. Equation (1.22) being homogeneous, the amplitude of the mode shape φi can be scaled arbitrarily; the φi satisfy the orthogonality conditions φiT Mφ j = μi δi j , φiT K φ j = μi ωi2 δi j ,

(1.23) (1.24)

where δi j is the Kronecker delta index (δi j = 1 if i = j, δi j = 0 if i = j). μi is the modal mass of mode i; it may be chosen arbitrarily, to normalize the mode shapes, μi = 1 is often used for simplicity. The matrix of mode shapes is defined as = (φ1 , φ2 , . . . , φn ); it satisfies the orthogonality conditions T M = diag(μi ), T K = diag(μi ωi2 ).

(1.25) (1.26)

In what follows, we will assume normal (or modal) damping; this means that the matrix T C is diagonal. By analogy with the single-d.o.f. oscillator, we define the modal damping ratio ξi by T C = diag(2ξi μi ωi )

(1.27)

Now, let us consider equation (1.19) again (with damping), Mÿ + C y˙ + K y = −M1x¨0 + b f,

(1.28)

and let us perform a change of variables from physical coordinates y (motion relative to the base) to modal coordinates z according to y = z;

(1.29)

z is the vector of modal amplitudes. Substituting into the foregoing equation, multiplying on the left by T and using the orthogonality relationships, one gets a set of decoupled equations μi zï + 2ξi μi ωi z˙i + μi ωi2 zi = −φiT M1x¨0 + φiT b f.

(1.30)

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In this equation, Γi = −φiT M1

(1.31)

is known as the modal participation factor of mode i; it is simply the work on mode i of the inertia forces associated with a unit acceleration of the support. φiT b is the modal displacement of the point where the force is applied. We define the vector of modal participation factors Γ = (1 , 2 , . . . , i , . . .)T as Γ = −T M1.

(1.32)

1.4.3 Support Reaction, Dynamic Mass Now let us consider the reaction force f 0 due to the seismic motion x¨0 (we assume f = 0 in what follows). From (1.18) and (1.11), ¨ f 0 = 1T M¨x = 1T M(1x¨0 + y) T T = 1 M1x¨0 + 1 M¨z f 0 = mT x¨0 − ΓT z¨ ,

(1.33)

where (1.32) has been used and mT = 1T M1 is the total mass of the system. The fact that mT is the total mass of the system can be seen from the expression for the kinetic energy (1.13): if a rigid body velocity x˙ = 1x˙ 0 is applied, the total kinetic energy is 1 1 1 T x˙ Mx˙ = x˙ 0 1T M1x˙ 0 = mT x˙ 02 . 2 2 2

(1.34)

The dynamic mass of the system is defined as the ratio between the complex amplitude of the harmonic force applied to the shaker, F0 , and the ¨ 0 , for every excitation amplitude of the acceleration of the shaking table, X frequency: ¨ 0 e jωt , x¨0 = X

f 0 = F0 e jωt ,

z = Ze jωt .

(1.35)

Assuming no damping for simplicity, the relationship between the ampli¨ 0 and Z follows from (1.30): tude X Zi =

i ¨ 0. X − ω2 )

μi (ωi2

(1.36)

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Combining with (1.33), the dynamic mass is given by n i2 F0 = mT + ¨0 μi X

ω2 2 ωi − ω2

i=1

.

(1.37)

Before discussing this result, let us consider the alternative formulation based on (1.20); using (1.22) gives f 0 = −1T K z = −1T M diag(ωi2 ) z = ΓT diag(ωi2 ) z

(1.38)

and, combining with (1.36), one finds that n i2 F0 = ¨0 μi X

i=1

ωi2 ωi2 − ω2

.

(1.39)

Comparing (1.37) and (1.39) at ω = 0, one finds that mT =

n 2 i

i=1

μi

(1.40)

,

where the sum extends to all the modes. i2 /μi is called the effective modal mass of mode i; it represents the part of the total mass which is associated with mode i for this particular type of excitation (defined by the vector 1). Equations (1.37) and (1.39) are equivalent if all the modes are included in the modal expansion. However, if (1.39) is truncated after m < n modes, the modal mass of the high-frequency modes is simply ignored, making this result statically incorrect. A quasi-static correction can be applied, by assuming that the high-frequency modes respond in a quasi-static manner (i.e. as for ω = 0); this leads to m i2 F0 = ¨0 μi X i=1

ωi2 ωi2 − ω2

+

n i2 μ i=m+1 i

(1.41)

and, upon using (1.40), m i2 F0 = ¨0 μi X i=1

ωi2 ωi2 − ω2

+ mT −

m 2 i

i=1

μi

= mT +

m 2 i

i=1

μi

ω2 ωi2 − ω2

. (1.42)

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Thus, one recovers the truncated form of (1.37), which is statically correct. If the damping in included, equation (1.37) becomes m i2 F = mT + ¨0 μi X i=1

ω2 ωi2 + 2 jξi ωi ω − ω2

.

(1.43)

It can be truncated after the m modes which belong to the bandwidth of the excitation without any error on the static mass. 1.4.4 Dynamic Flexibility Matrix Having considered the seismic excitation, we now turn to the steadystate harmonic response to a vector excitation f = Fe jωt . The governing equation is M¨x + C x˙ + K x = f,

(1.44)

the steady-state response is also harmonic, x = Xe jωt , and the amplitude of F and X are related by X = [−ω2 M + jωC + K ]−1 F = G(ω)F,

(1.45)

where the matrix G(ω) is called the dynamic flexibility matrix; it is a dynamic generalization of the static flexibility matrix, G(0) = K −1 . The modal expansion of G(ω) can be obtained by transforming (1.44) into modal coordinates x = z as we did earlier. The modal response is also harmonic, z = Ze jωt , and it is easy to see that

1 Z = diag 2 μi (ωi + 2 jξi ωi ω − ω2 )

T F,

(1.46)

T F.

(1.47)

leading to

X = Z = diag

1 2 μi (ωi + 2 jξi ωi ω − ω2 )

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Comparing with (1.45), one finds the modal expansion of the dynamic flexibility matrix, G(ω) = [−ω2 M + jωC + K ]−1 =

n i=1

φi φiT , μi (ωi2 + 2 jξi ωi ω − ω2 )

(1.48)

where the sum extends to all the modes. G lk (ω) expresses the complex amplitude of the structural response of d.o.f. l when a unit harmonic force e jωt is applied at d.o.f. k. G(ω) can be rewritten G(ω) =

n φi φT i

i=1

Di (ω),

(1.49)

1 + 2 jξi ω/ωi

(1.50)

μi ωi2

where Di (ω) =

1−

ω2 /ωi2

is the dynamic amplification factor of mode i. Di (ω) is equal to 1 at ω = 0, it exhibits large values in the vicinity of ωi , |Di (ωi )| = (2ξi )−1 , and then decreases beyond ωi (Figure 1.4). According to the definition of G(ω), the Fourier transform of the response X(ω) is related to the Fourier transform of the excitation F(ω) by X(ω) = G(ω)F(ω).

(1.51)

This equation means that all the frequency components work independently, and if the excitation has no energy at one frequency, there is no energy in the response at that frequency. From Figure 1.4, one can see that when the excitation has a limited bandwidth, ω < ωb , the contribution of all the high-frequency modes (i.e. such that ωk ωb ) to G(ω) can be evaluated by assuming Dk (ω) 1. As a result, G(ω) =

m φi φT i 2 μ ω i i i=1

Di (ω) +

n φi φiT . 2 i=m+1 μi ωi

(1.52)

The first term on the right-hand side is the contribution of all the modes which respond dynamically, and the second term is a quasi-static

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F

Excitation bandwidth

(a)

ωb

ω

Di

1 2ξi

Mode outside the bandwidth

1

(b) 0

ωi

ωb

ω

ωk

Figure 1.4 (a) Fourier spectrum of the excitation F with a limited frequency content ω < ωb. (b) Dynamic amplification of mode i such that ωi < ωb and ωk ωb

correction for the high-frequency modes. Taking into account that G(0) = K −1 =

n φi φT i

i=1

μi ωi2

(1.53)

,

Equation (1.52) can be rewritten in terms of the low-frequency modes only: G(ω) =

m φi φT i 2 μ ω i i i=1

Di (ω) + K −1 −

m φi φT i

i=1

μi ωi2

.

(1.54)

The quasi-static correction of the high-frequency modes is often called the residual mode, denoted by R. Unlike all the terms involving Di (ω) which reduce to 0 as ω → ∞, R is independent of the frequency and introduces a feedthrough (constant) component in the transfer matrix. We will shortly see that R has a strong influence on the location of the transmission zeros and that neglecting it may lead to substantial errors in the prediction of the performance of the control system.

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1.5 Collocated Control System A collocated control system is a control system where the actuator and the sensor are attached to the same d.o.f. It is not sufficient to be attached to the same location; they must also be dual, that is, a force actuator must be associated with a displacement (or velocity or acceleration) sensor, and a torque actuator with an angular (or angular velocity) sensor, in such a way that the product of the actuator signal and the sensor signal represents the energy (power) exchange between the structure and the control system. Such systems have very interesting properties. The open-loop frequency response function (FRF) of a collocated control system corresponds to a diagonal component of the dynamic flexibility matrix. If the actuator and sensor are attached to d.o.f. k, the open-loop FRF is given by G kk (ω) =

m φ2 (k) i

i=1

μi ωi2

Di (ω) + Rkk .

(1.55)

If one assumes that the system is undamped, the FRF is purely real: G kk (ω) =

m i=1

φi2 (k) + Rkk . μi (ωi2 − ω2 )

(1.56)

All the residues are positive (square of the modal amplitude) and, as a result, G kk (ω) is a monotonically increasing function of ω, which behaves as illustrated in Figure 1.5. The amplitude of the FRF goes from −∞ at the Gkk(ω)

resonance

Gkk(0) = K−kk1 static response

ωi

zi

ω i+1

Rkk residual

ω mode

antiresonance

Figure 1.5 Open-loop FRF of an undamped structure with a collocated actuator/sensor pair (no rigid body modes)

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Active Damping

Im(s)

(a) jz i jω i

Im(s)

x

x

x

x

x

Re(s)

x

(b)

Re(s)

Figure 1.6 Pole/zero pattern of a structure with collocated (dual) actuator and sensor: (a) undamped; (b) lightly damped. (Only the upper half of the complex plane is shown – the diagram is symmetrical with respect to the real axis)

resonance frequencies ωi (corresponding to a pair of imaginary poles at s = ± jωi in the open-loop transfer function) to +∞ at the next resonance frequency ωi+1 . Since the function is continuous, in every interval, there is a frequency zi such that ωi < zi < ωi+1 where the amplitude of the FRF vanishes. In structural dynamics, such frequencies are called anti-resonances; they correspond to purely imaginary zeros at ± j zi in the open-loop transfer function. Thus, undamped collocated control systems have alternating poles and zeros on the imaginary axis (Martin, 1978). The pole/zero pattern is that of Figure 1.6(a). For a lightly damped structure, the poles and zeros are just moved a little in the left half-plane, but they are still interlacing (Figure 1.6(b)). If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero, zi , the amplitude of the response of the collocated sensor vanishes. This means that the structure oscillates at zi according to the shape shown by the dotted line on Figure 1.7(b). We will establish in the next section that this shape and the frequency zi are actually a mode shape and a natural frequency of the system obtained by constraining the d.o.f. on which the control system acts. We know from control theory that the open-loop zeros are asymptotic values of the closed-loop poles, when the feedback gain goes to infinity (Franklin et al., 1986; Kailath, 1980). The natural frequencies of the constrained system depend on the d.o.f. where the constraint has been added (it is indeed well known in control theory that the open-loop poles are independent of the actuator and sensor configuration, while the open-loop zeros do depend on it). However, from

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u (a) y

(b)

(c) g

Figure 1.7 (a) Structure with collocated actuator and sensor. (b) Structure with additional constraint. (c) Structure with additional stiffness along the controlled d.o.f

the foregoing discussion, for every actuator/sensor configuration, there will be one and only one zero between two consecutive poles, and the interlacing property applies for any location of the collocated pair. Referring once again to Figure 1.5, one can easily see that neglecting the residual mode in the modeling amounts to translating the FRF diagram vertically in such a way that its high-frequency asymptote becomes tangent to the frequency axis. This produces a shift in the location of the transmission zeros to the right, and the last one even moves to infinity as the feedthrough component disappears from the FRF. Thus, neglecting the residual modes tends to overestimate the frequency of the transmission zeros. As we shall see shortly, the closed-loop poles which remain at finite distance move on loops joining the open-loop poles to the openloop zeros; therefore, altering the open-loop pole/zero pattern has a direct impact on the closed-loop poles. The open-loop transfer function of an undamped structure with a collocated actuator/sensor pair (Figure 1.6(a)) can be written 2 2 (s /z + 1) G(s) = G 0 i 2 i2 j (s /ω j + 1)

(ωi < zi < ωi+1 ).

(1.57)

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Active Damping

Im(G) ω=0

Re(G)

G dB

ω = zi

ω

ωi zi

φ 0° −90°

ω = ωi

ω

−180°

Figure 1.8 Nyquist diagram and Bode plots of a lightly damped structure with collocated actuator and sensor

For a lightly damped structure (Figure 1.6(b)), it reads 2 2 (s /z + 2ξi s/zi + 1) G(s) = G 0 i 2 2i . j (s /ω j + 2ξ j s/ω j + 1)

(1.58)

The corresponding Bode and Nyquist plots are represented in Figure 1.8. Every imaginary pole at ± jωi introduces a 180◦ phase lag and every imaginary zero at ± j zi a 180◦ phase lead. In this way, the phase diagram is always contained between 0 and −180◦ , as a consequence of the interlacing property. For the same reason, the Nyquist diagram consists of a set of near-circles (one per mode), all contained in the third and fourth quadrants. Thus, the entire curve G(ω) is below the real axis (the diameter of every circle is proportional to ξi−1 ). 1.5.1 Transmission Zeros and Constrained System We now establish that the transmission zeros of the undamped system are the poles (natural frequencies) of the constrained system. Consider the undamped structure of Figure 1.7(a) (a displacement sensor is assumed for simplicity). The governing equations are M¨x + K x = b u

(1.59)

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for the structure and y = bT x.

(1.60)

for the output sensor. Here u is the actuator input (scalar) and y is the sensor output (also scalar). The fact that the same vector b appears in the two equations is due to collocation. For a stationary harmonic input at the actuator, u = u0 e jω0 t ; the response is harmonic, x = x0 e jω0 t , and the amplitude vector x0 is the solution of (K − ω02 M)x0 = b u0 .

(1.61)

The sensor output is also harmonic, y = y0 e jω0 t , and the output amplitude is given by y0 = bT x0 = bT (K − ω02 M)−1 b u0 .

(1.62)

Thus, the transmission zeros (anti-resonance frequencies) ω0 are solutions of bT (K − ω02 M)−1 b = 0.

(1.63)

Now, consider the system with the additional stiffness g along the same d.o.f. as the actuator/sensor (Figure 1.7(c)). The stiffness matrix of the modified system is K + gbbT . The natural frequencies of the modified system are solutions of the eigenvalue problem [K + gbbT − ω2 M]φ = 0.

(1.64)

For all g the solution (ω, φ) of the eigenvalue problem is such that (K − ω2 M)φ + gbbT φ = 0

(1.65)

bT φ = −bT (K − ω2 M)−1 gbbT φ.

(1.66)

or

Since bT φ is a scalar, this implies that 1 bT (K − ω2 M)−1 b = − . g

(1.67)

Taking the limit as g → ∞, one can see that the eigenvalues ω satisfy bT (K − ω2 M)−1 b = 0,

(1.68)

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Active Damping

u s a

y

Figure 1.9 Structure with nearly collocated actuator/sensor pair

which is identical to (1.63). Thus, we have established that ω = ω0 ; the imaginary zeros of the undamped collocated system, solutions of (1.63), are the poles of the constrained system (1.64) at the limit, when the stiffness g added along the actuation d.o.f. increases to ∞: lim (K + gbbT ) − ω02 M x0 = 0.

g→∞

(1.69)

1.5.2 Nearly Collocated Control System In many cases, the actuator and sensor pair are close to each other without being strictly collocated. This situation is examined here. Consider the undamped system of Figure 1.9 where the actuator input u is applied at a and the sensor y is located at s. Returning to the modal expansion of the dynamic flexibility matrix (1.48), the open-loop FRF of the system is no longer given by a diagonal term, but rather by G(ω) =

n φi (a )φi (s) y = , 2 2 u i=1 μi (ωi − ω )

(1.70)

where φi (a ) and φi (s) are the modal amplitudes at the actuator and the sensor locations, respectively (the sum includes all the normal modes in this case). Comparing with (1.56), the residues of (1.70) are no longer guaranteed to be positive; however, if the actuator location a is close to the sensor location s, the modal amplitudes φi (a ) and φi (s) will be close to each other, at least for the low-frequency modes, and the corresponding residues will again be positive. The following result can be established in this case: If two neighboring modes are such that their residues φi (a )φi (s) and φi+1 (a )φi+1 (s) have the same sign, there is always an imaginary zero between the two poles (Martin, 1978). Since G(ω) is continuous between ωi and ωi+1 , this result will be established if one proves that the sign of G(ω) near ωi is opposite to that near

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ωi+1 . At ω = ωi + δω, G(ω) is dominated by the contribution of mode i and its sign is φi (a )φi (s) = −sign[φi (a )φi (s)]. sign ωi2 − ω2

(1.71)

At ω = ωi+1 − δω, G(ω) is dominated by the contribution of mode i + 1 and its sign is

φi+1 (a )φi+1 (s) sign 2 ωi+1 − ω2

= sign[φi+1 (a )φi+1 (s)].

(1.72)

− is Thus, if the two residues have the same sign, the sign of G(ω) near ωi+1 + opposite to that near ωi . By continuity, G(ω) must vanish somewhere − . Note, however, that when in between, at zi such that ωi+ < zi < ωi+1 the residues of the expansion (1.70) are not all positive, there is no guarantee that G(ω) is an increasing function of ω, and one can find situations where there is more than one zero between two neighboring poles.

1.5.3 Non-Collocated Control Systems Since the low-frequency modes vary slowly in space, the sign of φi (a )φi (s) tends to be positive for low-frequency modes when the actuator and sensor are close to each other, and the interlacing of the poles and zeros is maintained at low frequency. This is illustrated in the following example: Consider a simply supported uniform beam of mass per unit length m and bending stiffness E I . The natural frequencies and mode shapes are respectively ωi2 = (iπ)4

EI ml 4

(1.73)

φi (x) = sin

iπ x l

(1.74)

and

(the generalized mass is μi = ml/2). Note that the natural frequency increases as the square of the mode order. We assume that a force actuator

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Active Damping

Sensor motion u

s>a y

a = l/10

Mode 1 sin(πx/l)

Mode 2 sin(2πx/l)

l/2

2l/3

l/3

l/4

l/2

3l/4

Mode 3 sin(3πx/l)

Mode 4 sin(4πx/l)

Figure 1.10 Uniform beam with non-collocated actuator/sensor pair. Mode shapes 1 to 4

is placed at a = 0.1 l and we examine the evolution of the open-loop zeros as a displacement sensor is moved to the right from s = a (collocated), towards the end of the beam (Figure 1.10). The evolution of the open-loop zeros with the sensor location along the beam is shown in Figure 1.11; the plot shows the ratio zi /ω1 , so that the open-loop poles (independent of the actuator/sensor configuration) are at 1, 4, 9, 25, . . . . For s = a = 0.1 l, the open-loop zeros are represented by ◦; they alternate with the poles. Another position of the actuator/sensor pair along the beam would lead to a different position of the zeros, but always alternating with the poles. As the sensor is displaced from the actuator, s > a , the zeros tend to increase in magnitude as shown in Figure 1.11, but the low-frequency ones still alternate. When s = 0.2 l, z4 becomes equal to ω5 and there is no zero anymore between ω4 and ω5 when s exceeds 0.2 l. Such a situation, where an imaginary zero comes from below an imaginary pole to a larger

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23

Actuator

z4

25

Mode 5

s 0.2= l

z3 Mode 4

16

z2

0.25

z1 Mode 3

9

0.33 Mode 2

4

0.5

Mode 1

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

s l

Figure 1.11 Evolution of the imaginary zeros when the sensor moves away from the actuator along a simply supported beam (the actuator is at 0.1 l ). The abscissa is the sensor location, and the ordinate is the frequency of the transmission zero

value, is called a pole/zero flipping. Similarly, z3 flips with ω4 for s = l/4, z2 flips with ω3 for s = l/3, and z1 flips with ω2 for s = l/2. Examining the mode shapes, one notices that the pole/zero flipping always occurs at a node of the mode shapes, and this corresponds to a change of sign in φi (a )φi (s), as discussed above. This simple example illustrates the behavior of the pole/zero pattern for nearly collocated control systems: the poles and zeros are still interlacing at low frequency, but not at higher frequency, and the frequency where the interlacing stops decreases as the distance between the actuator and sensor increases. A more accurate analysis (Spector and Flashner, 1989; Miu, 1993) shows that for structures such as bars in extension, shafts in torsion or simply connected spring-mass systems (non-dispersive), when the sensor is displaced from the actuator, the zeros migrate along the imaginary axis towards infinity. The imaginary zeros are the resonance frequencies of the two substructures formed by constraining the structure at the actuator and sensor (this generalizes the result of Section 1.5.1).

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Active Damping

Imaginary zeros migrate towards± j∞

Real zeros come from ± j∞

Figure 1.12 Evolution of the zeros of a beam when the sensor moves away from the actuator. Every pair of imaginary zeros which disappears at infinity reappears on the real axis

For beams with specific boundary conditions, the imaginary zeros still migrate along the imaginary axis, but every pair of zeros that disappears at infinity reappears symmetrically at infinity on the real axis and moves towards the origin (Figure 1.12). Systems with right half-plane zeros are called non-minimum phase. Thus, non-collocated control systems are always non-minimum phase, but this does not cause difficulties if the right half-plane zeros lie well outside the desired bandwidth of the closed-loop system. When they interfere with the bandwidth, they put severe restrictions on the control system, by significantly reducing the phase margin.

1.6 Active Damping with Collocated System In this section, we will use the interlacing properties of collocated (dual) systems to develop single-input, single-output (SISO) active damping schemes with guaranteed stability. By active damping, we mean that the primary objective of the controller is simply to increase the negative real part of the system poles, while maintaining the natural frequencies essentially unchanged. This will simply attenuate the resonance peak in the dynamic amplification (Figure 1.13). Recall that the relationship

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Active Damping with Collocated System Im(s)

(a)

25 Dynamic Amplification (dB)

(b)

Open-loop

φ

Damping

sin φ = ξ

0

ω

Re(s)

Figure 1.13 Role of damping: (a) system poles; (b) dynamic amplification

between the damping ratio ξ and the angle φ with respect to the imaginary axis (Figure 1.13(a)) is sin φ = ξ , and that the dynamic amplification at resonance (Figure 1.13(b)) is 1/2ξ . By analogy with a swing which can be driven fairly easily at its natural frequency, this strategy will often require relatively little control effort. This is why it is also called low authority control (LAC), by contrast with other control strategies which fully relocate the closed-loop poles (natural frequency and damping) and are called high authority control (HAC). A remarkable feature of the LAC controllers discussed here is that the control law requires very little knowledge of the system (at most the knowledge of the natural frequencies). However, guaranteed stability does not mean guaranteed performance; good performance does require information on the system as well as on the disturbance applied to it, for appropriate actuator/sensor placement, actuator sizing, sensor selection and controller tuning. Actuator placement means good controllability of the dominant modes; this will be reflected by well-separated poles and zeros, leading to wide loops in the root-locus plots. In order to keep the formal complexity to a minimum, we will assume no structural damping and perfect actuator and sensor dynamics throughout most of this section. The impact of the actuator and sensor dynamics on stability, and the beneficial effect of passive damping is discussed in (Preumont, 2002). 1.6.1 Lead Control Consider an undamped structure with a collocated, dual actuator/sensor pair. We assume that the open-loop FRF G(ω) has no feedthrough

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Active Damping

(constant) component, so that G(ω) decays at high frequency as s −2 (the high-frequency decay is called roll-off ; the roll-off rate is −40 dB/decade in this case). The open-loop transfer function of such a system is given by G(s) =

n i=1

T 2 b φi , μi s 2 + ωi2

(1.75)

where bT φi is the modal amplitude at the actuator/sensor location. This corresponds, typically, to a point force actuator collocated with a displacement sensor, or a torque actuator collocated with an angular sensor. The pole/zero pattern is that of Figure 1.14 (where three modes have been assumed); there are two more structural poles than zeros, to provide a Im(s)

jz i jωi

–p

Lead

–z

Structure

Re(s)

Figure 1.14 Open-loop pole/zero pattern and root locus of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function with two more poles than zeros). Different scales are used on the real and imaginary axes

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+ –

g s+ z s+ p

u

G(s) =

27

∑ i

2 (b Tφ i)

y

2 μi ( s 2 + ω )

i

Figure 1.15 Block diagram of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function G(s) with two more poles than zeros)

roll-off rate s −2 (a feedthrough component would introduce an additional pair of zeros). This system can be damped with a lead compensator: H(s) = g

s+z s+p

( p z).

(1.76)

The block diagram of the control system is shown in Figure 1.15. This controller takes its name from the fact that it produces a phase lead in the frequency band between z and p, bringing active damping to all the modes belonging to z < ωi < p. Figure 1.14 also shows the root locus of the closed-loop poles when the gain g is varied from 0 to ∞. The closedloop poles which remain at finite distance start at the open-loop poles for g = 0 and eventually go to the open-loop zeros for g → ∞. Since there are two poles more than zeros, two branches go to infinity. The controller does not have any roll-off, but the roll-off of the structure is enough to guarantee gain stability at high frequency.1 Note that since the asymptotic values of the closed-loop poles for large gains are the open-loop zeros zi , which are the natural frequencies of the constrained system, they are therefore independent of the lead controller parameters z and p. For a structure with well-separated modes, the individual loops in the root locus (Figure 1.14) are to a large extent independent of each other, and the root locus for a single mode can be drawn from the lead controller and the asymptotic values ωi and zi of that mode only (Figure 1.16). The characteristic equation for this simplified system

1

For the sake of clarity, throughout this chapter, the root-locus plots often use different scales on the real and imaginary axes, so that the angles are not relevant.

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Active Damping

Im(s) jzi ξ max

jωi

Re(s) –p

–z

Figure 1.16 Structure with well-separated modes and lead compensator; root locus of a single mode

can be written from the pole/zero pattern: 1+α

(s 2 + zi2 )(s + z) = 0, (s 2 + ωi2 )(s + p)

(1.77)

where α is the variable parameter going from α = 0 (open-loop) to infinity. This can be written alternatively as 1+

1 (s 2 + ωi2 )(s + p) = 0. α (s 2 + zi2 )(s + z)

If z and p have been chosen in such a way that z ωi < zi p, this can be approximated in the vicinity of jωi by 1+

p (s 2 + ωi2 ) = 0. α s(s 2 + zi2 )

(1.78)

This characteristic equation turns out to be the same as that of the integral force feedback (IFF) controller discussed in Section 1.6.4. It can be shown

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that the maximum achievable modal damping is given by ξmax =

zi − ωi 2ωi

(ωi > zi /3);

(1.79)

see Preumont (2002).2 1.6.2 Direct Velocity Feedback The direct velocity feedback (DVF) is the particular case of the lead controller as z → 0 and p → ∞. Returning to the the basic equations – M¨x + K x = bu

(1.80)

y = bT x˙

(1.81)

for the structure,

for the output (velocity sensor), and u = −gy

(1.82)

for the control – one easily obtains the closed-loop equation M¨x + gbbT x˙ + K x = 0.

(1.83)

Upon transforming to modal coordinates x = z and taking into account the orthogonality condition (1.25) and (1.26), one gets diag(μi )¨z + gT bbT z˙ + diag(μi ωi2 )z = 0,

(1.84)

where z is the vector of modal amplitudes. The matrix T bbT is in general fully populated. For small gains, one may assume that it is diagonally dominant, diag{(bT φi )2 }. This assumption leads to a set of decoupled equations. Mode i is governed by μi zï + g(bT φi )2 z˙i + μi ωi2 zi = 0. 2

(1.85)

This results from (1.126), the pole in (1.78) playing the same role as the zero in (1.125), and vice versa.

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By analogy with a single-d.o.f. oscillator, one finds that the active modal damping ξi is given by 2ξi μi ωi = g(bT φi )2

(1.86)

or ξi =

g(bT φi )2 . 2μi ωi

(1.87)

Thus, for small gains, the sensitivity of the closed-loop poles to the gain (i.e. the departure rate from the open-loop poles) is controlled by (bT φi )2 , the square of the modal amplitude at the actuator/sensor location. Now let us examine the asymptotic behavior for large gains. For all g, the closed-loop eigenvalue problem (1.83) is (Ms 2 + gbbT s + K )x = 0.

(1.88)

It follows that x = −(K + Ms 2 )−1 gsbbT x or bT x = −gsbT (K + Ms 2 )−1 bbT x.

(1.89)

Since bT x is a scalar, one must have 1 g

(1.90)

sbT (K + Ms 2 )−1 b = 0.

(1.91)

sbT (K + Ms 2 )−1 b = − and, taking the limit as g → ∞,

The solutions of this equation are s = 0 and the solutions of (1.63), that is, the eigenvalues of the constrained system. The fact that the eigenvalues are purely imaginary, s = ± jω0 , stems from the fact that K and M are symmetric and positive semi-definite. Typical root-locus plots for a lead controller and a DVF controller are compared in Figure 1.17. As for the lead controller, for well-separated modes, those which are far enough from the origin can be analyzed independently of each other.

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Im(s)

(a)

31

Im(s)

(b)

Structure

Structure }

}

Re(s)

Re(s) Lead

DVF

Figure 1.17 Collocated control system. (a) Root locus for a lead controller. (b) DVF controller

In this way, the characteristic equation for mode i is approximated by 1+g

s(s 2 + zi2 ) =0 (s 2 + ω12 )(s 2 + ωi2 )

(besides the poles at ± jωi and the zeros at ± j zi , we include the zero at s = 0 and the poles at ± jω1 ) which in turn, if ωi > zi ω1 , may be approximated by 1+g

s 2 + zi2 =0 s(s 2 + ωi2 )

(1.92)

in the vicinity of mode i. This root locus is again identical to that of the IFF, (1.125), and the formula for the maximum modal damping (1.126) applies: ξmax =

ωi − zi 2zi

(1.93)

1.6.3 Positive Position Feedback There are frequent situations where the open-loop FRF does not exhibit a roll-off of −40 dB/decade as above. In fact, a feedthrough component may arise from the truncation of the high-frequency dynamics, as in (1.54), or because of the physical nature of the system (e.g. beams or plates covered with collocated piezoelectric patches, piezoelectric truss).

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+ –

−g 2 s +2ξf ωf s+ω2f

u

G(s)=

T

2

φi) ∑ μ (s(b2+ω 2) i

i

y

i

Figure 1.18 Block diagram of the second-order PPF controller applied to a structure with collocated actuator and sensor (the open-loop transfer function has the same number of poles and zeros)

In these situations, the degree of the numerator of G(s) is the same as that of the denominator and the open-loop pole/zero pattern has an additional pair of zeros at high frequency. Since the overall degree of the denominator of H(s)G(s) must exceed the degree of the numerator, the controller H(s) must have more poles than zeros. Positive position feedback (PPF) was proposed to solve this problem (Goh and Caughey, 1985; Fanson and Caughey, 1990). The second-order PPF controller consists of a second-order filter H(s) =

−g s 2 + 2ξ f ω f s + ω2f

(1.94)

where the damping ξ f is usually rather high (0.5–0.7), and the filter frequency ω f is adapted to target a specific mode. The block diagram of the control system is shown in Figure 1.18; the negative sign in H(s), which produces a positive feedback, is the origin of the name of this controller. Figure 1.19 shows typical root loci when the PPF poles are targeted to mode 1 and mode 2, respectively (i.e. ω f close to ω1 or ω2 , respectively). One can see that the whole locus is contained in the left half-plane, except for one branch on the positive real axis, but this part of the locus is reached only for large values of g, which are not used in practice. The stability condition can be established as follows: the characteristic equation of the closed-loop system is n bT φi φiT b g =0 ψ(s) = 1 + g H(s)G(s) = 1 − 2 s + 2ξ f ω f s + ω2f i=1 μi (s 2 + ωi 2 ) or

n T T b φ φ b i i = 0. ψ(s) = s 2 + 2ξ f ω f s + ω2f − g 2 + ω 2) μ (s i i i=1

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33 Im(s)

Im(s)

jzi

Structure

jωi PPF (mode2)

PPF (mode 1)

Re(s) Stability limit

Figure 1.19 Root locus of the PPF controller applied to a structure with collocated actuator and sensor (the open-loop transfer function has the same number of poles and zeros). (a) Targeted at mode 1. (b) Targeted at mode 2. (For the sake of clarity, different scales are used for the real and the imaginary axes)

According to the Routh–Hurwitz criterion for stability, if one of the coefficients of the power expansion of the characteristic equation becomes negative, the system is unstable. It is not possible to write the power expansion ψ(s) explicitly for an arbitrary value of n; however, one can see easily that the constant term (in s 0 ) is a n = ψ(0) = ω2f − g

n bT φi φT b i

i=1

μi ωi 2

.

In this case, a n becomes negative when the static loop gain becomes larger than 1. The stability condition is therefore n bT φi φiT b g gG(0)H(0) = 2 < 1. (1.95) ω f i=1 μi ωi 2 Note that it is independent of the structural damping in the system. Since the instability occurs for large gains which are not used in practice, the PPF can be regarded as unconditionally stable. Unlike the lead controller of the previous section which controls all the modes which belong to z < ωi < p and even beyond, the PPF filter must be tuned on the targeted mode (it is therefore essential to know the natural frequency accurately), and the

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authority on the modes with very different frequencies is substantially reduced. Several PPF filters can be used in parallel, to target several modes simultaneously, but they must be tuned with care, because of the cross coupling between the various loops. A sensitivity analysis3 shows that, for small gains, the closed-loop poles are such that (φT b)2 ∂si 1 = −j i . (1.96) 2 2 ∂g g=0 2μi ωi ω f − ωi + 2 jξ f ω f ωi The departure rate from the open-loop poles is, as for the DVF, proportional to the square of the modal amplitude at the actuator/sensor location, and also to the transfer function of the compensator evaluated at s = jωi . This contribution is maximized if the controller is tuned on the targeted mode, ω f ωi . In this case,

∂si ∂g

= −j g=0

(φiT b)2 1 2μi ωi 2 jξ f ωi2

(1.97)

which is purely real, meaning that the root locus leaves the open-loop pole orthogonally to the imaginary axis in this case. The corresponding modal damping is given by ξi = g

(φiT b)2 . 4μi ωi4 ξ f

(1.98)

Although this result indicates that the departure rate increases for small values of the filter damping, one must keep in mind that the tuning will be more difficult to achieve for small ξ f . A value of 0.5 ≤ ξ ≤ 0.7 is generally recommended for practical applications. The following first-order PPF controller is an alternative to the secondorder controller (1.94) (Baz et al., 1992; Høgsberg and Krenk, 2006): H(s) =

−g . 1 + τs

(1.99)

A typical root locus is shown in Figure 1.20. As compared to the secondorder controller, this one does not have to be tuned on targeted modes; the roll-off is reduced to −20 dB/decade instead of −40 dB/decade with the second-order controller. The part of the locus on the real axis also becomes 3

See Preumont (2002, p. 234), taking into account that it is a positive feedback in this case.

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35 Im(s)

jzi

Structure

jωi

PPF (1st order)

Re(s)

s = – 1/τ

Figure 1.20 Root locus of the first-order PPF controller (the scale on the real axis has been magnified for the sake of clarity)

unstable for large gains. Proceeding exactly as we did for the second-order controller, one can easily see that the stability condition is that the static loop gain must be lower than 1: gG(0) < 1.

(1.100)

Thus the gain margin is G M = [gG(0)]−1 . The stability condition corresponds to the negative stiffness of the controller overcoming that of the structure. Because of this negative stiffness, the root locus does not leave the open-loop poles orthogonally to the imaginary axis; this is responsible for larger control efforts, as compared to the other strategies considered before. 1.6.4 Integral Force Feedback So far, all the collocated systems that we have considered exhibit alternating poles and zeros, starting with a pole at low frequency (ω1 < z1 < ω2 < z2 < ω3 < . . .). This corresponds to an important class of actuator/sensor pairs, including: force actuator/displacement or velocity sensor; torque actuator/angular or angular velocity sensor; and piezoelectric patches used as actuator and sensor. In this section, we discuss another interlacing

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Active Damping

Figure 1.21 Collocated control system with a first-order PPF controller

pole/zero configuration but starting with a zero (z1 < ω1 < z2 < ω2 < . . .). This situation typically arises in a actuator/sensor pair made of a displacement actuator and a force sensor. Active truss Consider a truss structure controlled with an active strut consisting of a linear displacement actuator collinear with a force sensor (Preumont et al., 1992; Preumont, 2002); see Figure 1.21. In principle, the linear actuator may be of many types (piezoelectric, magnetostrictive, thermal, ball-screw); for the purpose of this discussion, it is enough to assume that the actuator has a constitutive equation = ga u + f /K a

(1.101)

where f is the force in the strut, is the total extension, K a is the strut stiffness, u the control input (e.g. a voltage for a piezoelectric actuator) and ga the input gain (free expansion for a unit input, in meters per volt in the case of a piezoelectric actuator). Equation (1.101) expresses the fact that the total extension is the sum of the controlled expansion ga u and the elastic deformation. The total extension of the active strut is related to the global coordinate system of the truss by = bT x.

(1.102)

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Since the force applied by the strut on the structure is equal and opposite to that applied by the structure to the strut, the virtual work of the conservative forces applied to the truss is δWnc = − f.δ = − f.bT δx.

(1.103)

If the kinetic coenergy and the strain energy of the truss (excluding the active strut) are given respectively by 1 T x K x, 2

(1.104)

[x˙ T Mδ x˙ − xT K δx − f bT δx]dt = 0.

(1.105)

T∗ =

1 T x˙ Mx˙ 2

and

V=

Hamilton’s principle (1.10) leads to

t2 t1

Integrating by parts with respect to time and taking into account that δx(t1 ) = δx(t2 ) = 0, one gets M¨x + K x + b f = 0,

(1.106)

and combining with (1.101) and (1.102), M¨x + (K + bbT K a )x = bK a ga u.

(1.107)

In this formula, K is the stiffness matrix of the truss without the active strut and bbT K a is the contribution of the active strut to the global stiffness matrix. The mass of the active strut can be added to M if necessary; the damping term has been omitted for simplicity. The control force on the right-hand side consists of a pair of self-equilibrating (internal) forces along x, applied by the active strut at the connecting nodes (nodes 3 and 5 in Figure 1.21). The magnitude of the force is the product of the stiffness of the strut K a and the actuator unconstrained expansion ga u (it is interesting to note that this thermal analogy holds for a wide class of actuators).4 4

In thermal problems, the thermal deformations may be computed by applying a set of self-equilibrated thermal loads K a αT, where α is the thermal expansion coefficient and T the temperature difference.

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Active Damping

Open-loop transfer function The output equation follows from the constitutive equations: y = f = K a (bT x − ga u).

(1.108)

If φi and ωi are the mode shapes and the natural frequencies of (1.107) (i.e. of the structure including the strut, but with no control, u = 0), they are solutions of [(K + bbT K a ) − ωi2 M]φi = 0

(1.109)

and satisfy the orthogonality conditions φiT Mφ j = μi δi j , φiT (K + bbT K a )φ j = μi ωi2 δi j .

(1.110) (1.111)

Equation (1.107) can be transformed according to x = z; the modal coordinates satisfy μi zï + μi ωi2 zi = φiT bK a ga u

(1.112)

or, in the Laplace domain, zi =

φiT b K a ga u. μi (s 2 + ωi2 )

(1.113)

From the output equation (1.108), n n y = Ka ( zi bT φi − ga u) = K a [ i=1

i=1

(bT φi )2 K a − 1]ga u. μi (s 2 + ωi2 )

(1.114)

Thus, the open-loop transfer function of the system is n y (bT φi )2 = G(s) = K a ga Ka − 1 . 2 2 u i=1 μi (s + ωi )

(1.115)

Since bT φi is the modal extension of the strut, when the structure vibrates according to mode i, 12 (bT φi )2 K a is the modal strain energy in the strut. On the other hand, from (1.111), 12 μi ωi2 is the total modal strain energy,

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in the whole structure, for the same mode. We define the fraction of modal strain energy: νi =

(bT φi )2 K a = μi ωi2

Strain energy in the strut . Total strain energy mode i

(1.116)

With this definition, the open-loop transfer function can be rewritten n νi −1 . (1.117) G(s) = K a ga 2 2 i=1 s /ωi + 1 The contribution of the high-frequency modes can be truncated and replaced by a quasi-static contribution as we did earlier, leading to m n νi νi − 1 . (1.118) + G(s) K a ga 2 2 i=1 s /ωi + 1 i=m+1 The residues in the modal expansion (1.117) are the fraction of modal strain energy; they are all positive, and this guarantees, once again, interlacing poles and zeros. A typical FRF is represented in Figure 1.22; as compared FRF

ω1

z1

ω2

z2

ω3

z3

ω

Figure 1.22 Open-loop FRF of an active truss. The active strut consists of a piezoelectric actuator and a collocated force sensor

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Active Damping

to Figure 1.5, the important difference is that the zero z1 has a frequency lower than ω1 . Integral force feedback Consider the active truss given by (1.107) and(1.108). Introducing the unconstrained expansion δ = ga u, these respectively become M¨x + (K + bbT K a )x = bK a δ

(1.119)

y = K a (bT x − δ).

(1.120)

and

We now add a positive integral force feedback (IFF) δ=

g y Ka s

(1.121)

(the K a in the denominator is for normalization purposes; y/K a is the elastic extension of the strut). The block diagram is shown in Figure 1.23. The pole/zero pattern of the system is shown in Figure 1.24. It consists of interlacing pole/zero pairs on the imaginary axis (z1 < ω1 < z2 < ω2 < . . .) and the pole at s = 0 from the controller. The root-locus plot is also shown; it consists of the negative real axis and a set of loops going from the openloop poles ± jωi to the open-loop zeros ± j zi . All the loops are entirely contained in left half-plane, so that the closed-loop system is unconditionally stable, for all values of the gain g. Note also that the root-locus plot does not change significantly if the pole at the origin is moved slightly in the left half-plane, to avoid saturation (which is often associated with Unconstrained piezo expansion

Force

gD (s) + −

−g K as

G 0 (s) δ

n

vi − 2 /ω 2 1 1+ s i=1 i

Ka ∑

Figure 1.23 Piezoelectric truss with an IFF control

y

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Im(s)

jω i

Structure

jz i

Re(s) control Figure 1.24 Pole/zero pattern of the active strut and root locus of the IFF

integral control). In fact, piezoelectric force sensors have a built-in highpass filter and cannot measure a d.c. component. Equations (1.120) and (1.121) can be combined, leading to δ=

g bT x s+g

(1.122)

and, substituting into (1.119), one gets the closed-loop characteristic equation

Ms 2 + (K + bbT K a ) − bbT K a

g x = 0. s+g

(1.123)

For g = 0, the eigenvalue problem is identical to (1.109) and the eigenvalues are indeed the open-loop poles, ± jωi . Asymptotically, for g → ∞, the eigenvalue problem becomes [Ms 2 + K ]x = 0,

(1.124)

where K is the stiffness matrix of the structure without the active strut. Thus, the open-loop zeros ± j zi are the natural frequencies of the truss after

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Active Damping

Im(s)

IFF

Re(s)

(a)

(b)

Figure 1.25 (a) IFF root locus of a single mode. (b) Evolution of the root locus as zi moves away from ωi

removing the active strut. This situation should be compared to that discussed earlier (Section 1.5.1) in connection with a displacement sensor. In that case, the open-loop zeros were found to be the natural frequencies of the constrained system where the d.o.f. along which the actuator and sensor operate is blocked. For well=separated modes, the individual loops in the root locus of Figure 1.24 are, to a large extent, independent of each other, and the root locus of a single mode can be drawn from the asymptotic values ± jωi and ± j zi only (Figure 1.25(a)). The corresponding characteristic equation is5 1+g

(s 2 + zi2 ) = 0. s(s 2 + ωi2 )

(1.125)

The actual root locus (Figure 1.24), which includes the influence of the other modes, is only slightly different from that of Figure 1.25(a), with the same asymptotic values at ± jωi and ± j zi . It can be shown that the maximum modal damping for mode i is given by ξimax =

5

ωi − zi 2zi

(zi ≥ ωi /3).

Note the similarity with (1.78) for the lead controller and with (1.92) for the DVF.

(1.126)

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√ It is achieved for g = ωi ωi /zi (Preumont, 2002). This result applies only for ξi ≤ 1, that is, for zi ≥ ωi /3. For zi = ωi /3, the locus actually touches the real axis as indicated in Figure 1.25(b). Equation (1.126) relates clearly the maximum achievable damping and the distance between the pole and the zero. Note, however, that if the system has several modes, there is a single tuning parameter g, and the various loops are traveled at different velocities. As a result, the optimal value of g for one mode will not be optimal for another, and a compromise must be found. Equation (1.123) can be transformed into modal coordinates. Using the orthogonality conditions (1.110)–(1.111) leads to diag(μi )s 2 + diag(μi ωi2 ) − T bbT K a

g = 0. s+g

(1.127)

For small g, the equations are nearly decoupled: s 2 + ωi2 − νi ωi2

g =0 s+g

(1.128)

after using the definition of νi . Since the root-locus plot leaves the openloop pole orthogonally to the imaginary axis, for small gain, one can assume a solution of the form s = ωi (−ξi + j). Substituting into (1.128), one can easily obtain ξi =

gνi . 2ωi

(1.129)

Thus, for small gains, the sensitivity of the closed-loop poles to the gain (i.e. the departure rate from the open-loop poles) is controlled by the fraction of modal strain energy in the active element. This result is very useful for the design of active trusses. The active strut should be located to maximize νi for the critical modes of the structure. Note that νi is readily available from finite element software. In summary, the active strut placement can be made from inspection of the map of modal strain energy in the finite element model. Once the active strut location has been selected, a modal analysis of the truss including the active strut gives the open-loop poles ± jωi , and a modal analysis after removing the active strut gives the open-loop zeros ± j zi . Then, the root-locus plot can be drawn. We will soon examine the case of a truss involving several active struts controlled in a decentralized manner.

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1.6.5 Duality between the Lead and IFF Controllers Root locus of a single mode Figure 1.26 illustrates the duality between the IFF and the lead controllers. If the modes are well separated and if the pole and the zero of the lead controller (1.76) are such that z ωi < zi p, the root-locus plots of every mode turn out to be very similar (Figure 1.26(c)–(d)). The DVF can be regarded as the limit of the lead controller as z → 0 and p → ∞; however, depending on the situation, the individual loops starting from an open-loop pole may go either to a zero with higher frequency (similar to Figure 1.26(d)) or to a zero with lower frequency (similar to Figure 1.26(c)). Open-loop poles and zeros For the IFF, the open-loop poles jωi are the natural frequencies of the structure with the active element working passively (contributing with its own stiffness K a ), while the open-loop zeros, j zi , are the natural frequencies when the force f in the active strut is zero, that is, when the active element is removed. On the other hand, in the control configuration of Figure 1.26(b), the open-loop poles are the natural frequencies when the active element produces no force (as for the zeros in the previous case), and the open-loop zeros are the natural frequencies with the d.o.f. along the actuator blocked ( = 0). Note that these are larger than the open-loop poles in the IFF case, because the stiffness is infinite in this case. Effect of a lag filter Once the pole and the zero are fixed, the size of the loop in the root locus of Figure 1.26(c) is fixed, and with it the maximum damping. However, the loop size can be increased by adding a lag filter in the compensator as shown in Figure 1.27. The departure angle from the open-loop pole and the arrival angle at the zero change as indicated in Figure 1.27. The increments α and β with respect to the previous situation are obtained form the phase diagram of the lag compensator α = φ(ωi ),

β = φ(zi ),

(1.130)

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Lead

f f

-

δ F IF

+

Δ

f

(a)

1+g

s 2 + z 2i

s(s 2 + ω2i)

(b) Im(s)

Im(s)

jωi

jz i

jzi

jωi 1+

=0

2 2 p s + ωi g s(s 2 + z 2) = 0 i

Re(s) IFF (c)

Re(s)

−p

Lead − z (d)

Figure 1.26 Duality between the IFF and the lead (DVF) control configurations. (a) IFF architecture with displacement actuator, force sensor and positive integral force feedback. (b) Force actuator and collocated displacement transducer and (negative) lead controller (DVF is a particular case). (c) IFF control: root locus for a single mode. (d) Lead control: root locus for a single mode z ωi < zi p. The loops of the DVF can be approximated by one configuration or the other, depending on the relative value of ωi and zi

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Active Damping Im(s)

(a)

jωi α

Im(s)

(b) IFF + Lag

jzi

β

ϕ

ψ

}

α

jωi IFF

jzi

β Re(s)

Re(s)

} Lag compensator (c)

φ 0

zi β

ωi α

ω

−90

Figure 1.27 (a),(b) Effect of a lag compensator on the loop size of an IFF control. (c) Phase diagram of the lag compensator

and this tends to widen the loop as shown in Figure 1.27(b). Note, however, that the widening of the loop occurs only if the loop goes from the pole jωi to a zero j zi at a lower frequency; otherwise, the effect is reversed. A lead compensator will have a similar effect on a loop going to a zero at a larger frequency, as in Figure 1.26(d). Note also that the phase margin, originally of 90◦ , is reduced by the phase lag introduced by the lag compensator.

1.7 Decentralized Control with Collocated Pairs 1.7.1 Cross-Talk In this section, we examine the multi-input, multi-output (MIMO) control of a structure with a set of independent control loops using collocated pairs. Consider the shear frame of Figure 1.28 as an example; the control system consists of independent and identical loops, using a force actuator ui and a relative displacement sensor yi . The input–output relationship

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47

x7 x6 x5 x4

(a)

ωk =2Ωsin [π (2k −1) ] 2 (14 +1)

k = 1 ; . . .; 7

(b)

x3 u2 u1

− 1) z l =2Ωsin [ 2π (2l ] (10+1)

l = 1, . . . ,5

x2 y 2 = x2 − x1 x1 y1 = x1

Figure 1.28 (a) Shear frame with two independent control loops (displacement sensor and force actuator). (b) Configuration corresponding to transmission zeros

for this system can be written in compact form as ⎞ ⎡ G 11 y1 ⎝ y2 ⎠ = ⎣ G 21 z G z1 ⎛

G 12 G 22 G z2

⎤⎛ ⎞ u1 G 1w G 2w ⎦ ⎝ u2 ⎠ G zw w

(1.131)

where w is the disturbance and z is the performance metric. The block diagram of the control system with two loops of decentralized control is shown in Figure 1.29. One can see that the output y1 responds to u2 through G 12 and y2 responds to u1 through G 21 , respectively. These terms are called cross-talk, and are responsible for interactions between the two loops. 1.7.2 Transmission Zeros (Case 1) The MIMO system of Figure 1.28 (assumed undamped) is governed by the structure equation M¨x + K x = Bu

(1.132)

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Active Damping

w

z

u2 u1

y2

G21

y1

G12 H1 H2

Figure 1.29 Block diagram of the control system with two loops of decentralized control. The cross-talk terms are indicated by dashed lines

and the output equation y = B T x,

(1.133)

where B defines the topology of the actuator/sensor pairs (the size of the vectors u and y is equal to the number of collocated pairs). The transmission zeros of the system (Franklin et al., 1986; Kailath, 1980) are the values s0 such that an input u = u0 e s0 t (t ≥ 0) applied with appropriate initial conditions x0 produces a system response x = x0 e s0 t and a system output y = 0. From the foregoing equations, (Ms02 + K )x0 = Bu0 , B T x0 = 0 or

Ms02 + K −B 0 BT

x0 u0

(1.134) (1.135)

= 0.

(1.136)

The values s0 for which this system has a non-trivial solution are the transmission zeros of the system. Combining (1.134) and (1.135) gives B T (Ms02 + K )−1 Bu0 = 0;

(1.137)

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s0 is the eigenvalue and u0 is the shape of the control input. Consider the output feedback u = −g H(s)y,

(1.138)

where H(s) is a square matrix and g is a scalar parameter (the discussion is not restricted to decentralized control). The closed-loop eigenvalue problem is obtained by combining equations (1.132), (1.133), and (1.138): [Ms 2 + K + g B H(s)B T ]x = 0.

(1.139)

Theorem 8 of Davison and Wang (1974) extends the results for SISO systems to square (number of inputs = number of outputs) MIMO systems without feedthrough: as g → ∞, the finite eigenvalues of (1.139) coincide with the transmission zeros defined above, for any form of H(s) (not necessarily diagonal). Since the asymptotic solutions of the eigenvalue problem (1.139) do not depend on H(s), they can be computed with H(s) = I . In this case, the transmission zeros are seen as the asymptotic solutions of lim [Ms 2 + K + g B B T ]x = 0.

g→∞

(1.140)

The matrix g B B T is the contribution to the global stiffness matrix of a set of springs of stiffness g connected to all the d.o.f. involved in the control. Asymptotically, when g → ∞, the additional springs act as supports restraining the motion along the controlled d.o.f. Thus, the transmission zeros are the poles (natural frequencies) of the constrained system where the d.o.f. involved in the control are blocked. Since all the matrices involved in (1.140) are symmetrical and positive semi-definite, the transmission zeros are purely imaginary; since blocking the controlled d.o.f. reduces the total number of d.o.f. by the number m of control loops, the number of zeros is 2m less than the number of poles (2n). Example 1

Consider the seven-story shear frame controlled in a decentralized manner with two independent and identical feedback loops shown in Figure 1.28(a). Every actuator ui applies a pair of forces equal and opposite between floor i and floor i − 1, while the sensor yi = xi − xi−1 measures the relative displacement between the same floors. The mass,

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stiffness and B matrices are respectively M = mI7 , ⎛

2 −1 ⎜ −1 2 ⎜ ⎜ 0 −1 K =k⎜ ⎜ ⎜ ⎝ 0 0 0 ...

0 −1 2 ... −1 0

... ... ... 2 −1

0 0 0

⎞

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ −1 ⎠ 1

⎛

⎞ 1 −1 ⎜0 1 ⎟ ⎜ ⎟ ⎜0 0 ⎟ ⎜ ⎟ B=⎜. .. ⎟ . ⎜ .. . ⎟ ⎜ ⎟ ⎝0 0 ⎠ 0 0

(1.141)

The natural frequency of mode l is given by (e.g. Géradin and Rixen, 1997) ωl = 2

π (2l − 1) k sin , m 2 (2n + 1)

l = 1, . . . , n,

(1.142)

where n is the number of stories. The decentralized feedback control law is u = −gh(s) y,

(1.143)

where g is the scalar gain and h(s) is the scalar control law, common to all the loops. According to the foregoing discussion, the transmission zeros are the natural frequencies of the system obtained by constraining (blocking) the first two floors. Equation (1.142) can therefore be used to evaluate the zeros as well, after setting the number of stories to n − 2. Figure 1.30 shows the root locus of a lead compensator h(s) =

1 + Ts 1 + αTs

(α > 1),

(1.144)

while Figure 1.31 shows the root locus for a positive position feedback as in Høgsberg and Krenk (2006), h(s) =

−1 . 1 + τs

(1.145)

In generating Figure 1.30, the zero of (1.144) was chosen very close to the origin and the pole very far on the negative real axis, so that the lead compensator behaves as a DVF. In Figure 1.31, τ is taken such that τ ω1 = 0.5. The two root-locus plots share the same asymptotic values for g = 0

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51

± jωk ± jzl

DVF

Figure 1.30 Root-locus plot for the DVF

± jω k ± jz l

PPF

Stability limit

Figure 1.31 Root-locus plot for the first-order PPF (1.145)

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Active Damping

(open-loop poles) and for g −→ ∞ (transmission zeros); for 0 < g < ∞, the plots depend on the control law. Unlike in the SISO case, it is not possible to draw the entire root-locus plot from the knowledge of the open-loop poles and zeros alone. 1.7.3 Transmission Zeros (Case 2) Another frequent configuration met in active vibration control involves collocated displacement actuator and force sensor pairs (see 1.6.4). Such a system can be represented by the structure equation M¨x + K x = B K a u

(1.146)

y = K a (B T x − u).

(1.147)

and the output equation

Here, K is the global stiffness matrix, including all the active members, B defines the topology of the control system, K a is the diagonal stiffness matrix of the active members and u is the vector of unconstrained expansion of the actuators; these equations apply to a wide variety of actuators such as piezoelectric, magnetostrictive, thermal, SMA, and ball-screw. In the output equation, B T x is the total elongation of the active members and B T x − u is the elastic extension; equation (1.147) simply states that the output force of every sensor is the product of its stiffness and the elastic extension. As compared to the previous case of the force actuator and displacement sensor, the situation is slightly different because of the feedthrough component in (1.147). Proceeding as before, the transmission zeros of the system are the values s0 such that an input u = u0 e s0 t (t ≥ 0) applied with appropriate initial conditions x0 produces a system response x = x0 e s0 t and a system output y = 0. From (1.146)–(1.147), (Ms02 + K )x0 = B K a u0 ,

(1.148)

B x0 − u0 = 0 T

(1.149)

or

Ms02 + K −B K a BT −I

x0 u0

= 0.

(1.150)

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53

Upon eliminating u0 , one gets (Ms02 + K − B K a B T )x0 = 0,

(1.151)

and the condition for a non-trivial solution is det(Ms02 + K − B K a B T ) = 0.

(1.152)

Here, B K a B T is the contribution of the active members to the global stiffness matrix, so that (1.152) is the characteristic equation of the system after removing the contribution of the active members to the global stiffness matrix. Thus, the transmission zeros are the poles (natural frequencies) of the system where the contribution of the active members to the stiffness matrix has been removed; their number is equal to the number of poles. Now let us consider the output feedback u = g H(s) y,

(1.153)

where H(s) is a square matrix and g is a scalar gain (note the positive feedback in this case). Combining (1.153) and (1.147), u = g H(I + gK a H)−1 K a B T x

(1.154)

and substituting in (1.146), one gets [Ms 2 + K − g B K a H(I + gK a H)−1 K a B T ]x = 0.

(1.155)

This is the closed-loop eigenvalue problem. The asymptotic values are, for g = 0, the open-loop poles (natural frequencies of the system including the active members) and, for g → ∞,6 (Ms 2 + K − B K a B T )x = 0

(1.156)

the characteristic equation of which is (1.152). Thus, asymptotically, as g → ∞, the finite eigenvalues coincide with the transmission zeros. This result generalizes earlier ones obtained in the particular case of identical IFF loops (Preumont et al., 2000). An experimental verification with a 6

Because limg→∞ (I + gK a H) ∼ gK a H.

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1) z k = 2 Ωsin[π(2k− 2(14+1) ] k = 1, . . . , 7

(b)

) H(s ) H(s

Figure 1.32 (a) Seven-story shear frame with two independent identical control loops (force sensor and displacement actuator pairs). (b) Configuration corresponding to the transmission zeros ±zk

IFF

Figure 1.33 Root-locus plot for the IFF

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References

55

truss controlled with three decentralized active tendons can be found in (Preumont, 2002; Preumont and Bossens, 2000). Example 2

Consider again the seven-story shear frame of the previous example (Figure 1.32), with active members of stiffness ka acting on the first two floors (ka /k = 5 has been used in the numerical example). Figure 1.33 shows the root-locus plot for the IFF controller h(s) = g/s.

(1.157)

It is worth noting that (i) the zeros of this example are identical to the poles of the previous example; (ii) the poles and zeros alternate on the imaginary axis in this case, but this is not a general rule in the MIMO case (e.g., see Figure 14.14 of Preumont, 2002); and (iii) the root locus of Figure 1.33 is different from that corresponding to a SISO system with the same pole/zero pattern.

References Baz, A., Poh, S., and Fedor, J. (1992) Independent modal space control with positive position feedback. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 114(1), 96–103. Clough, R.W., and Penzien, J. (1975) Dynamics of Structures. McGraw-Hill. Davison, E.J., and Wang, S.H. (1974) Properties and calculation of transmission zeros of linear multivariable systems. Automatica, 10, 643–658. Fanson, J.L., and Caughey, T.K. (1990) Positive position feedback control for large space structures. AIAA Journal, 28(4), 717–724. Franklin, G.F., Powell, J.D., and Emani-Naemi, A. (1986) Feedback Control of Dynamic Systems. Addison-Wesley. Géradin, M., and Rixen, D. (1997) Mechanical Vibrations, 2nd edition. Wiley. Goh, C., and Caughey, T.K. (1985) On the stability problem caused by finite actuator dynamics in the control of large space structures. International Journal of Control, 41(3), 787–802. Høgsberg, J.R., and Krenk, S. (2006) Linear control strategies for damping of flexible structures. Journal of Sound and Vibration, 293, 59–77. Kailath, T. (1980) Linear Systems. Prentice Hall. Martin, G.D. (1978) On the control of flexible mechanical systems. PhD thesis, Stanford University. Miu, D.K. (1993) Mechatronics – Electromechanics and Contromechanics. Springer-Verlag.

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Preumont, A. (1994) Random Vibration and Spectral Analysis. Kluwer Academic. Preumont, A. (2002) Vibration Control of Active Structures: An Introduction, 2nd edition. Kluwer. Preumont, A., Achkire, Y., and Bossens, F. (2000) Active tendon control of large trusses. AIAA Journal, 38(3), 493–498. Preumont, A., and Bossens, F. (2000) Active tendon control of vibration of truss structures: Theory and experiments. Journal of Intelligent Material Systems and Structures, 11(2), 91–99. Preumont, A., Dufour, J.P., Malekian, Ch. (1992) Active damping by local force feedback with piezoelectric actuators. AIAA Journal of Guidance, 15(2), 390–395. Spector, V.A., and Flashner, H. (1989) Sensitivity of structural models for non-collocated control systems. ASME Journal of Dynamic Systems Measurement and Control, 111(4), 646–655.

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2 Active Isolation 2.1 Introduction There are two broad classes of problems in which isolation of vibration is necessary.1 First, operating equipment generates oscillatory forces which can propagate into the supporting structure (Figure 2.1(a)). This situation corresponds to that of an engine in a car, or an attitude control reaction wheel assembly in a spacecraft. Secondly, sensitive equipment may be supported by a structure which vibrates appreciably (Figure 2.1(b)); in this case, it is the support motion which constitutes the source of excitation. This situation corresponds to, for example, a telescope in a spacecraft, a precision machine tool in a workshop, or a passenger seated in a car. The disturbance may be either deterministic, such as a motor imbalance, or random as in a passenger car running on a rough road. For deterministic sources of excitation which can be measured, such as a rotating imbalance, feedforward control can be very effective (see, for example, Fuller et al., 1996, Chapter 7).2 However, the present chapter focuses on the feedback strategies for active isolation; these apply to both deterministic and random disturbances, and they do not need a direct measurement of the disturbance.

1

This chapter was written with the participation of Christophe Collette and Bruno de Marneffe. In feedforward control, it is not necessary to measure directly the disturbance force, but rather a signal which is correlated to it, such as the rotation velocity for the rotating imbalance. 2


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Active Isolation

fd

(a)

(b)

(c) M

M

k

k

c

M

xc

x

k

c xd

xc fa

c xd

fs

Figure 2.1 (a) Operating equipment generating a disturbance force fd. (b) Equipment subjected to a support excitation xd. (c) Active isolation device

Let us begin with the system depicted in Figure 2.1(a), excited by a disturbance force f d . If the support is fixed, the governing equation is Mx¨ + c x˙ + kx = f d .

(2.1)

The force transmitted to the support is given by ˙ f s = kx + c x.

(2.2)

In the Laplace domain, Fd (s) , + 2ξ ωn s + ωn2 ) Fs (s) = M ωn2 + 2ξ ωn s X(s), X(s) =

M(s 2

(2.3) (2.4)

where X(s), Fd (s), and Fs (s) stand for the Laplace transform of x(t), f d (t), and f s (t) respectively, and with the usual notation ωn2 = k/M and 2ξ ωn = c/M. The transmissibility of the support is defined in this case as the transfer function between the disturbance force f d applied to the mass and the force f s transmitted to the support structure; combining the foregoing equations, we get 1 + 2ξ s/ωn Fs (s) = . Fd (s) 1 + 2ξ s/ωn + s 2 /ωn2

(2.5)

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59

Next, consider the second situation illustrated in Figure 2.1(b); the disturbance is the displacement xd of the supporting structure, and the system output is the displacement xc of the sensitive equipment. Proceeding in a similar way, it is easily established that the transmissibility of this isolation system, defined in this case as the transfer function between the support displacement and the absolute displacement of the mass M, is given by Xc (s) 1 + 2ξ s/ωn = , Xd (s) 1 + 2ξ s/ωn + s 2 /ωn2

(2.6)

which is identical to the previous one; the two isolation problems can therefore be treated in parallel. The amplitude of the corresponding frequency response function (FRF), for s = jω, is shown in Figure 2.2 for various values of the damping √ ratio ξ . We observe, firstly, that all the√curves are larger than 1 for ω < √2ωn and become smaller than 1 for ω > 2ωn . Thus the critical frequency 2ωn separates the domains of amplification and attenuation of the isolator. Secondly, when ξ = 0, the high-frequency decay clean body xc

dirty body

dB

xd

xc( jω) xd ( jω)

C

ξ= 0

M

m k

10

0 ξ 2 > ξ1

Objective of the active isolation

−10

ξ1 1/s

−20

1/s² 0.1

1

2

10

ω

ωn

Figure 2.2 Transmissibility of the passive isolator for various values of the damping ratio ξ

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Active Isolation

rate is 1/s 2 , that is −40 dB/decade, while very large amplitudes occur near the corner frequency ωn (the natural frequency of the spring-mass system). Figure 2.2 illustrates the tradeoff in passive isolator design: large damping is desirable at low frequency to reduce the resonant peak, while low damping is needed at high frequency to maximize the isolation. One may already observe that if the disturbance is generated by a rotating imbalance in a motor, there is an obvious benefit in using an damper with variable damping characteristics which can be adjusted according to the √ √ rotation velocity: high when ω < 2ωn and low when ω > 2ωn . Such variable (adaptive) devices will be discussed later. Figure 2.2 also shows the target of an active isolation system which combines a decay rate of −40 dB/decade with no overshoot at resonance. Before discussing active isolation, let us examine how the damping can be reduced at high frequency in a passive way.

2.2 Relaxation Isolator One way to reduce the damping at high frequency is to use a relaxation damper where the viscous damper c is replaced by a Maxwell unit consisting of a damper c and a spring k1 in series (Figure 2.3).3 The governing equations are Mx¨ + k(x − x0 ) + c(x˙ − x˙ 1 ) = 0, c(x˙ − x˙ 1 ) = k1 (x1 − x0 ),

Figure 2.3 Relaxation isolator 3

The relaxation unit is sometimes called Zener or Maxwell generalized.

(2.7) (2.8)

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Relaxation Isolator

61

or, in matrix form using the Laplace variable s, k x −cs Ms 2 + cs + k = x0 . k1 x1 −cs k1 + cs

(2.9)

It follows that the transmissibility is given by x (k1 + cs)k + k1 cs (k1 + cs)k + k1 cs = = . 2 2 2 2 x0 (Ms + cs + k)(k1 + cs) − c s (Ms + k)(k1 + cs) + k1 cs (2.10) One can see that the asymptotic decay rate for large frequencies is ∼ s −2 , that is, −40 dB/decade. Physically, this corresponds to the fact that, at high frequency, the viscous damper tends to be blocked, and the system behaves like an undamped isolator with two springs acting in parallel. Figure 2.4 compares the transmissibility curves for given values

dB

x x0

c=0

c→∞

copt A

0

−20

−40

1

ω/ωn

10

Figure 2.4 Transmissibility of the relaxation isolator for fixed values of k and k1 and various values of c . The first peak corresponds to ω = ωn; the second corresponds to ω = n. All the curves cross each other at A and have an asymptotic decay rate of −40 dB/decade. The curve corresponding to c opt is nearly maximum at A

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Active Isolation

of k and k1 and various values of c. For c = 0, the relaxation isolator behaves like an undamped isolator of natural frequency ωn = (k/M)1/2 . Likewise, for c → ∞, it behaves like an undamped isolator of frequency n = [(k + k1 )/M]1/2 . In between, the poles of the system are solutions of the characteristic equation (Ms 2 + k)(k1 + cs) + k1 cs = (Ms 2 + K )k1 + cs(Ms 2 + k + k1 ) = 0, which can be rewritten in root-locus form 1+

k1 s 2 + ωn2 = 0. c s s 2 + 2n

(2.11)

Comparing with (1.125), one can see that, for fixed values of ωn and n , when the parameter k1 /c is changed,4 the poles of the system move along a root locus similar to that shown in Figure 1.25(a). Using (1.126), the optimum value (producing the system poles with maximum damping ratio) is achieved for 3/2

k1 n = 1/2 c ωn

(2.12)

and the corresponding damper constant is c opt =

k1 n

ωn n

1/2

=

k1 k1 k1 −1/4 k1 −3/4 = . 1+ 1+ n k ωn k

(2.13)

The transmissibility corresponding to c opt is also shown in Figure 2.4; it is nearly maximum at A. 2.2.1 Electromagnetic Realization The principle of the relaxation isolator is simple and it can be realized with viscoelastic materials. However, it is sometimes difficult to integrate into the system, and also to achieve thermal stability. In some circumstances, especially when thermal stability is critical, it may be more convenient to achieve the same effect through an electromechanical converter which consists of a moving coil transducer (to transform the mechanical energy into electrical energy), an inductor L, and a resistor R. 4

k1 /c is the inverse of the relaxation constant; it has the dimension of a frequency.

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63

A moving coil transducer, or voice coil, is an energy converter transforming mechanical energy into electrical energy and vice-versa; its constitutive equations are e = Tv, f = −Ti,

(2.14)

where e is the voltage drop in the coil (in the same direction as the current), v the relative velocity of the two parts, f is the external force required to balance the electromagnetic force, i is the current, and T is the transducer constant, expressed in newtons per ampere or in volt-seconds per meter. Referring to Figure 2.5, if we adopt the electric charge q as our electrical variable, the governing equations of the system are Mx¨ + k(x − x0 ) − T q˙ = 0, L q¨ + T(x˙ − x˙ 0 ) + Rq˙ = 0, or, in matrix form using the Laplace variable, x k −Ts Ms 2 + k = x0 . 2 q Ts Ts Ls + Rs

(2.15) (2.16)

(2.17)

It follows that the transmissibility is given by (Ls + R)k + T 2 s x . = x0 (Ms 2 + k)(Ls + R) + T 2 s

(2.18)

Figure 2.5 Electromagnetic realization of the relaxation isolator

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Active Isolation

Comparing with equation (2.10), one can see that the electromechanical isolator behaves exactly like a relaxation isolator provided that cs + k1 Ls + R = T2 k1 c

(2.19)

or k1 =

T2 , L

c=

T2 . R

(2.20)

These are the two relationships between the three parameters T, L and R so that the transmissibility of the electromechanical system of Figure 2.5 is the same as that of Figure 2.3.

2.3 Sky-hook Damper Consider the single-axis isolator connecting a disturbance source m to a payload M (Figure 2.6). It consists of a soft spring k in parallel with a force actuator f a ; the objective is to isolate the motion xc of the payload M from the motion xd of m due to the disturbance load f d . The governing equations are Mx¨c + k(xc − xd ) = f a , m x¨d + k(xd − xc ) = f d − f a , xd

(2.21) (2.22) xc

accelerometer Fa

dirty body m

clean body M

Fd k

Fa Fk

Figure 2.6 Single-axis active isolator

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Sky-hook Damper

65

or, in matrix form using the Laplace variable s,

Ms 2 + k −k

−k ms 2 + k

Xc Xd

=

Fa F d − Fa

.

(2.23)

Inverting this equation leads to Xc =

s 2 [Mms 2

mFa k Fd + . 2 + (M + m)k] Mms + (M + m)k

(2.24)

The first term of this expression describes the payload response to the disturbance load, while the second term is the payload response to the actuator. If an accelerometer is attached to the payload, measuring the acceleration x¨c , the open-loop transfer function is G(s) =

s 2 Xc ms 2 . = Fa Mms 2 + (M + m)k

(2.25)

We will shortly examine this more closely. But first, consider the closedloop response to a general feedback law based on the absolute velocity x˙ c : Fa = −H(s)s Xc (s).

(2.26)

Introducing this into (2.23) gives

Ms 2 + H(s)s + k −k − H(s)s

−k ms 2 + k

Xc Xd

=

0 Fd

.

(2.27)

From the first line of this equation, the closed-loop transmissibility is k Xc . = 2 Xd Ms + H(s)s + k

(2.28)

This simple equation shows the influence of the feedback control law on the transmissibility; it shows that a simple proportional feedback (Karnopp and Trikha, 1969; Kaplow and Velman, 1980), H(s) = g leads

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66

Active Isolation (b)

(a)

“sky”

Fa = − g s Xc

dirty body m

Fa accelerometer

clean body M

dirty body m

clean body M k

k

Figure 2.7 (a) Isolator with absolute velocity feedback. (b) Equivalent sky-hook damper

to the transmissibility k 1 Xc = = 2 2 , Xd Ms 2 + gs + k s /ωn + gs/k + 1

(2.29)

which complies with the objectives of active isolation stated in Figure 2.2, because the asymptotic decay rate is ∼ s −2 (i.e. −40 dB/decade) and the overshoot at resonance may be controlled by adjusting the gain g of the controller to achieve critical damping. This control law is called the skyhook, because the control force f a = −g x˙ c is identical to that of a viscous damper of constant g attached to the payload and a fixed point in space (the sky); see Figure 2.7. The open-loop transfer function between the input force f a and the output acceleration x¨c is given by (2.25); it has a pair of imaginary poles at

pi = ± j

(M + m)k Mm

(2.30)

and a pair of zeros at the origin. The root locus of the sky-hook is shown in Figure 2.8; it is entirely contained in the left half-plane, which means that the sky-hook damper is unconditionally stable (infinite gain margin).

2.4 Force Feedback We have just seen that the sky-hook damper based on the absolute velocity of the payload is unconditionally stable for a rigid body. However, this is no longer true if the payload cannot be regarded as a rigid body, which

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Force Feedback

67

Im H(s) g s

Fa

G(s) m s2 2 m M s +k(m+M)

s2Xc

Re (2 zeros + 1 pole)

Figure 2.8 Root locus of the sky-hook damper

is a frequent situation, particularly in space applications. Since the absolute acceleration of a rigid body is proportional to the force applied to it, the acceleration feedback of Figure 2.7 may be replaced by a force feedback as shown in Figure 2.9.5 The open-loop transfer function is, in this case, G(s) =

F Mms 2 = Fa Mms 2 + (M + m)k

(2.31)

which has the same pole/zero pattern and the same root locus as Figure 2.8. However, when the payload is flexible, the force applied and the acceleration are no longer proportional and the pole/zero pattern may differ significantly. It can be observed that the feedback based on the acceleration still leads to alternating poles and zeros in the openloop transfer function when the flexible modes are significantly above the suspension mode, but they do not alternate any more when the flexible modes interact with the suspension mode. On the contrary, it has been shown (Preumont et al., 2002) that if two arbitrary undamped flexible bodies are connected by a single-axis isolator with force feedback, the poles and zeros of the open-loop transfer function always alternate on the imaginary axis (Figure 2.10). 5

Besides the advantage of achieving alternating poles and zeros discussed here, a force sensor may be more sensitive than an accelerometer in some low-frequency applications; for example, a force sensor with a sensitivity of 10−3 N is commonplace; for a mass M of 1000 kg (e.g. a space telescope), this corresponds to an acceleration of 10−6 m/s2 ; such a sensitivity is more difficult to achieve. Force sensing is especially attractive in micro-gravity where one does not have to consider the dead loads of a structure.

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Active Isolation

xd

xc

Fa = − (g/s) F

Fa dirty body m

clean body M

F k

Figure 2.9 Sky-hook based on a force sensor

This result is not obvious, because the actuator Fa and the sensor F , if collocated, are not dual as required for alternating poles and zeros (as emphasized in the previous chapter). This can be demonstrated as follows: The system with input Fa and output the relative displacement between the two bodies, X = xc − xd , is collocated and dual; therefore, the FRF (which is purely real in the undamped case) exhibits alternating poles and zeros (solid line in Figure 2.11). On the other hand, the control force Fa , the relative displacement X, and the output (total) force F are related by F = kX − Fa .

(2.32)

It follows that the FRF F/Fa and X/Fa are related by kX F = − 1. Fa Fa

xc

k disturbance source

(2.33)

F

sensitive equipment

Fa xd Figure 2.10 Arbitrary flexible structures connected by a single-axis isolator with force feedback

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69

G(ω)

k ΔFX(ω) a (ω) (ω) F (ω) =k ΔFXa (ω) −1 F a(ω)

1 0

Z1 ω1

ω

Z2 ω2

ω3

1

Figure 2.11 FRFs of the single-axis oscillator connecting two arbitrary flexible structures. The solid line corresponds to kX/F a and the dotted line to F /F a

This equation states that the FRF with force sensor F/Fa can be obtained from that with relative displacement sensor kX/Fa by a simple vertical translation bringing the amplitude to 0 at ω = 0 (from the solid line to the dashed line). This changes the locations of the zeros Zi , but the continuity of the FRF curve between two resonances guarantees that there is a zero between two consecutive poles (natural frequencies): ωi < Zi < ωi+1 .

(2.34)

2.5 Six-Axis Isolator The single-axis isolator considered in the previous section combines an attenuation rate of −40 dB/decade with a tunable overshoot at resonance; the system can be made critically damped (no overshoot) by proper selection of the feedback gain g. If a force feedback implementation is used, the sky-hook damper has guaranteed stability, even if the structures to be isolated are flexible, and if the flexible modes and the suspension modes overlap. To fully isolate two rigid bodies with respect to each other, six

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single-axis isolators must be located judiciously. For earth-based application, the isolator must compensate the gravity loads and the behavior of the isolator is strongly anisotropic. On the other hand, for space applications, the same isolator may be used along every axis. In the present discussion, we assume that the six identical isolators are controlled in a decentralized manner. Two new problems arise: First, the system does not have one, but six suspension modes, with generally different frequencies, and it will not be possible to achieve critical damping simultaneously for all suspension modes with a single gain (the same for all control loops). Second, every single-axis isolator should be mounted on spherical joints, to allow motion orthogonal to its own axis. However, backlash-free spherical joints are difficult to implement and, in precision engineering, they are replaced by elastic joints which have a small rotary stiffness. Even if small, the residual rotary stiffness has a strong effect on the closed-loop performance of the suspension, because it determines the transmission zeros, which are the asymptotic closed-loop poles as g → ∞. An attractive architecture for a generic multi-purpose six-axis isolator is that of a Gough–Stewart platform (Stewart, 1965–6) (Figure 2.12), and several projects have been developed for space applications (Spanos et al., 1995; Rahman et al., 1998; Thayer et al., 1998, 2002; McInroy et al., 1999a,

Figure 2.12 Multi-purpose soft isolator based on a Gough–Stewart platform

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zb

zb 5 4

5 6

4 4 5

7 yb

3 5

8 2

3 6 0

1

xr

1

2

xb

1 xb

3

yr

8+

6

6

4

Payload Plate

Base Plate

3 2

1

yb

2

Figure 2.13 Geometry and coordinate systems for the cubic hexapod isolator. Numbers in bold indicate the struts

1999b; McInroy and Hamann, 2000; McInroy, 2002; Hauge and Campbell, 2004). The system consists of six identical active struts connected to the end plates by spherical joints. Most existing projects, including this one, have opted for a cubic architecture (Geng and Haynes, 1994), where the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube (Figure 2.13). This topology provides a uniform control capability and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other). Figure 2.13 depicts the geometry of the hexapod and the numbering of the nodes and the struts; the base frame {xb , yb , zb } has its origin at node 0; the payload frame {xr , yr , zr } has its origin at the → geometrical center of the hexapod, noted 8, and − zr is perpendicular to − → − → the payload plate; the orientation of xr and yr is shown in Figure 2.13. If one neglects the flexibility of the struts and the bending stiffness of the flexible joints connecting it to the base and payload plates, the equations of motion can be obtained from rigid body dynamics. Assume that the base plate is fixed and denote by B the 6 × 6 projection matrix connecting the force acting along the strut axes and those in the payload plate axes {xr , yr , zr }.6 Then f = B(u − kq), 6

See Preumont (2002, p. 129) for the analytic expression for B.

(2.35)

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where f = ( f x , f y , f z , Mx , My , Mz )T are the forces applied by the legs expressed in the payload axes, u = (u1 , . . . , u6 )T the vector of active control forces in struts 1, . . . , 6, and q = (q 1 , . . . , q 6 )T the vector of leg extensions. In (2.35), k is the stiffness of the suspension spring, assumed the same for all legs. u − kq is the total force in the leg, the sum of the control force u and the elastic restoring force in the spring. From the virtual work principle, the leg extensions and the small displacements and rotations of the payload plate, x = (xr , yr , zr , θx , θ y , θz )T , satisfy q = B T x.

(2.36)

Substituting in (2.35) and writing the dynamic equilibrium of the payload gives M¨x = Bu − kBBT x or M¨x + kBBT x = Bu,

(2.37)

where M is the 6 × 6 mass matrix of the payload M=

mI 0

0 J

.

(2.38)

Here m is the mass and J the inertia tensor of the payload in the payload frame. In (2.38), kBBT is the stiffness matrix of the suspension, resulting exclusively from the axial stiffness of the suspension struts. In practice, however, the spherical joints are responsible for an additional stiffness contribution (unknown at this stage) which we represent by the stiffness matrix K e (in this case, the joints can be viewed as flexible universal joints: low bending stiffness, high axial, shear and torsion stiffness, see below). The total stiffness matrix is kBBT + K e and equation (2.38) becomes M¨x + (kBBT + K e )x = Bu.

(2.39)

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2.5.1 Decentralized Control In order to benefit from the robustness properties discussed earlier, the control strategy consists of a decentralized sky-hook damper based on force feedback (integral force feedback). The isolator is equipped with six force sensors measuring the total axial force in the various legs; the output equation is y = u − kq = u − k B T x,

(2.40)

where y = (y1 , . . . , y6 )T is the vector of the six force sensor outputs (the vector of control forces u and the leg extensions q were defined earlier). Using a decentralized integral force feedback with the same gain g for every loop, the controller equation is u = H(s)y = −

g y s

(2.41)

(g is a scalar in this case). Combining (2.41) and (2.42) gives u=

g k BT x s+g

(2.42)

and, substituting in (2.40), the closed-loop characteristic equation is M¨x + (kBBT + K e )x =

g kBBT x. s+g

(2.43)

First, consider the case of perfect spherical joints, K e = 0. In this case, equation (2.43) becomes

s T (kBB ) x = 0. Ms + s+g 2

(2.44)

The free suspension modes are the solutions for g = 0. Denote by the matrix of the suspension modes, normalized in such a way that T M = I , T (kBBT ) = 2 = diag(i2 ). Equation (2.44) can be transformed into modal coordinates according to x = z. In modal coordinates,

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(a)

(b)

Im jΩ i

Im jΩ i jzi

Re

Re

Figure 2.14 (a) Root locus of the suspension modes of the perfect six-axis isolator (K e = 0) with decentralized integral force feedback. (b) Effect of stiffness of the flexible joints (K e = 0)

the characteristic equation is reduced to a set of uncoupled equations s 2 = 0, s+g i s = 0, 1+ g 2 s + i2 s2 +

(2.45) (2.46)

i = 1, . . . , 6. The corresponding root locus is shown in Figure 2.14(a). It is identical to Figure 2.8 for a single axis isolator; however, unless the six natural frequencies of the suspension modes are identical, a given value of the gain g leads to different pole locations for the various modes, and it is not possible to achieve the same damping for all modes. Better, more balanced performance will be obtained if 1 to 6 are close to each other. It is recommended to locate the payload in such a way that the modal spread 6 / 1 is minimized (Spanos et al., 1995). Let us now investigate the influence of the parasitic stiffness K e introduced by real, flexible joints. The closed-loop characteristic equation in this case becomes Ms 2 + K e +

s (kBBT ) x = 0. s+g

(2.47)

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The asymptotic solutions for high gain (g → ∞) are no longer at the origin s = 0, but satisfy the eigenvalue problem

Ms 2 + K e x = 0.

(2.48)

The solutions are the natural frequencies zi of the system when the axial stiffness of the leg has been removed (k = 0). This shift of the zeros from the origin to finite frequencies (Figure 2.14(b)) has a substantial influence on the practical performance of the isolator, and justifies careful design of the joints. The combined effect of the modal spread and the joint stiffness is illustrated in Figure 2.15; there are only four different curves because of the symmetry of the system. The bullets correspond to the closed-loop poles for a fixed value of g; they illustrate the fact that the various loops travel at different speeds as g increases. How this impacts the transmissibility is examined below. Before closing this section, it is appropriate to mention two additional factors which reduce the closed-loop performances of the isolator. Firstly, the integral controller (2.41) requires the addition of a high-pass filter (to avoid saturation), which negatively impacts the performance of the Im Ω5, Ω 6 Ω4

Ω3 Ω1, Ω2

Re Figure 2.15 Typical root locus of a complete isolation system with real joints. The bullets indicate the location of the closed-loop poles for the adopted value of the gain g

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system. However, the detrimental effect of the high-pass filter can be minimized if its corner frequency is selected far enough below the lowest suspension mode. Secondly, the results of this section have been achieved assuming that the legs are massless and perfectly rigid, except in the axial direction. In practice, however, the legs have their own local dynamics which interfere with that of the isolator and significantly impact the transmissibility in the vicinity of the resonance frequency of the local modes and beyond. Maximizing the natural frequency of the local modes of the legs is a major challenge in the design of a six-axis isolator with broadband isolation capability. The leg design of a six-axis isolator for space applications is briefly discussed in the next subsection (Preumont et al., 2007). 2.5.2 Leg Design This section examines the leg design of a six-axis electromagnetic isolator for use in micro-gravity; it may be skipped by the reader with no interest in the practical design of such an isolator. Two conceptual designs are shown in Figure 2.16. In the first (Figure 2.16(a)), the longitudinal motion and the axial stiffness are achieved with two parallel membranes mounted inside a cylinder which also supports the permanent magnet of the voice coil actuator. The stinger is attached to the center of the membranes; it supports the coil at one end, and the force sensor at the other end; two flexible joints are used to connect the leg, respectively to the base plate and to the payload plate. This design was developed in Abu Hanieh (2003) and tested in parabolic flight in 2002; the isolator works in the frequency band 5–100 Hz with a maximum attenuation of about 20 dB near 50 Hz. However, this leg design suffers two major drawbacks. First, the first lateral vibration mode of the leg occurs near 100 Hz and is responsible for the modest performance of the isolator. This is essentially due to the mass of the magnetic circuit which is contained within the leg. Second, this leg was not able to pass the qualification tests (vibration during launch), because of its large mass and the weakness of the flexible joints. This triggered the second design, with the objective of reducing the weight of the moving part of the leg assembly (Figure 2.16(b)). In the second design, the magnetic circuit, the heaviest moving part in the first design, has been removed from the leg and attached to the base plate. A single membrane acts simultaneously as spring and flexible joint on the base plate side. In this design, the coil axis is allowed to rotate with respect to the magnet axis, which necessitates an increase in the gap in the magnetic circuit. The stinger (made of carbon fiber) is attached

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Flexible joint

Magnetvoice coil

Base Base plate Plate

N

Membranes

Flexible joint Load cell Payload Mobile Plate plate

S

(a)

The membrane is also used as a flexible joint

Base

Base plate Plate

N

Flexible joint Load cell

Mobile Payload plate Plate

S

(b)

Figure 2.16 Two conceptual designs: (a) two membranes, two flexible joints, magnet in the leg; (b) one membrane, one flexible joint, magnet in the base plate

to the center of the membrane; it supports the voice coil at one end, the force sensor at the other end, and is connected to the payload plate by a single flexible joint. This design reduces drastically (by a factor of 8) the weight attached to the membrane and flexible tip. This configuration passed the vibration tests. The natural frequency of the local mode is also raised drastically, leading to a dramatic improvement in performance; the isolator works in the frequency band 5–400 Hz with a maximum attenuation of about 40 dB near 100 Hz (see below). It should emphasized that this considerable improvement was achieved by mechanical design alone, without changing the control law (!). Figure 2.17 shows an exploded view of the leg of the second design; some details of the design follow: The membrane is made of beryllium copper alloy which is a nonmagnetic material with high yield stress (a thin film cover was added to avoid corrosion due to metal–metal interaction with the aluminum support). The membrane geometry was optimized to (i) maximize the

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Active Isolation Magnet Support

Voice-coil

Magnet

Cover

Flexible Joint

Membrane

Stinger

Loadcell

Figure 2.17 Exploded view of the leg

ratio between the radial stiffness and longitudinal stiffness, (ii) achieve a radial stiffness as constant as possible with the longitudinal extension of the leg (the longitudinal extension and the rotation of the stinger axis introduce bending in the membrane), and (iii) reduce the stress concentration to reduce fatigue. Figure 2.18 shows various membrane geometries tested during the course of this project. The membrane is modeled with finite elements and a Guyan reduction is performed to obtain the various spring constants to be integrated in the global model of the platform.

(a)

(b)

(c)

(d)

(e1)

(e2)

Figure 2.18 Various membrane geometries and finite element mesh

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The magnet assembly used in this project is a radial polarity toroid with ferromagnetic core from BEI Kimco. The gap between the voice coil and the magnet is large enough to allow the rotation of the stinger by 1.5◦ . The coil consists of 201 turns on a plastic support (PEEK); this is important in eliminating the eddy currents in a metallic support, which would introduce passive damping in the system, with the detrimental consequences discussed earlier on the asymptotic decay rate. The voice coil maximum force is 2.7 N; the stroke is ±0.7 mm. The stinger consists of a carbon epoxy tube with longitudinal fibers, provided with aluminum connections; it is equipped with mechanical stops, in order to limit damage to the membrane due to excessive stinger axial displacement. The tip of the stinger supports the piezoelectric force sensor and the flexible joint. The influence of the stiffness of the flexible joints on the performance of the platform was stressed earlier (see (2.48)); the behavior of the joint should be as close as possible to that of a spherical joint, that is, it should exhibit high axial and shear stiffness, and low bending and torsional stiffness. On the other hand, the joints play also a key role in the mechanical integrity of the system which would call for strong, and consequently stiff joints.7 The design used in this project is shown in Figure 2.19, manufactured by electro-erosion. The material selected Finite?element mesh (Samcef field)

Figure 2.19 Flexible joint 7

An extensive discussion is available in Abu Hanieh (2003).

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Figure 2.20 View of the isolator

was NiTiNOL alloy, based on its low Young modulus and high yield strength (respectively ∼ 60 GPa and 900–1900 MPa in this case).8 The joint profile was studied numerically with finite elements and a Guyan reduction performed, to obtain the 12 × 12 stiffness matrix of the joint, to be integrated in the global model of the platform. The full isolation platform is shown in Figure 2.20. 2.5.3 Model of the Isolator After Guyan reduction of the individual finite element models of the various components, they can assembled to create a leg model of small size (less than 100 d.o.f., depending on the configuration), with which a model of the whole platform may be generated. Although small in size, this model is accurate in a wide frequency band, up to about 500 Hz, which is appropriate to evaluate the isolation performance of the system. The isolator model can also be combined with a global model of the 8 We also expected to benefit from additional properties of super-elasticity which, we hoped, would add damping to the local transversal modes of the leg, but it did not work out as expected, for unknown reasons (likely, because the strain was not large enough for this application; material data were actually extremely difficult to obtain from the manufacturer). In a later version developed at Micromega Dynamics on behalf of ESA/ESTEC, titanium was used instead of NiTiNOL, because of more reliable material data.

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40

Vertical Transmissibility (dB)

30 20

Without Control

10

With Control

0 –10 –20 –30 –40 –50

0

10

1

10

2

10

Hz

5.10

2

Figure 2.21 Numerical prediction of the transmissibility in the vertical direction

entire spacecraft.9 Next, the dynamic model of the platform can be transformed into state space and coupled with a control model in MATLAB/SIMULINK. Such a coupled model is extensively used to design the components, optimize their shape and size, and tune the controller gain; this model is used to predict the transmissibility matrix. Figure 2.21 shows a numerical simulation of the experiment which is described below. The vertical transmissibility with control exhibits a small overshoot due to the high-pass filter at 0.5 Hz, the residual stiffness of the spherical joints and the modal spread (for the test configuration, 6 /1 = 2.2). The performance of the isolator near 100 Hz is −40 dB and the first local modes occur above 400 Hz.

9

In this application, all finite elements models were developed in SAMCEF.

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2.5.4 Six-axis Transmissibility In this section, we examine the matrix relationship between two sets of six sensors (accelerometers or geophones), attached respectively to the base plate and the payload plate. Let xd = (xd1 , xd2 , xd3 , xd4 , xd5 , xd6 )T and xc = (xc1 , xc2 , xc3 , xc4 , xc5 , xc6 )T be the displacements at the sensor location on the base plate and the payload plate, respectively (xd is the input to the isolator system and xc the output), and let Xd (ω) and Xc (ω) be their Fourier transform. The relationship between the readings at the sensors on the payload plate and the base plate can be expressed using the frequencydependent transmissibility matrix T(ω) as follows: ⎡

Xc (ω) = T(ω)Xd (ω),

T11 ⎢T ⎢ 21 T(ω) = ⎢ ⎣··· T61

T12 T22 ··· T62

··· ··· ··· ···

T16 T26 ··· T66

⎤ ⎥ ⎥ ⎥. ⎦

(2.49)

Ti j (ω) represents the displacement at sensor i of the payload for an imposed displacement of the base Xd (ω) = e j , where e j = (0 0 . . .1 . . . 0)T is the vector with all entries equal to 0 except entry j which is equal to 1. The transmissibility matrix is thus dependent on the choice and orientation of the sensors. Next, assume that the inputs and outputs are transformed into X∗d (ω) = Td Xd (ω),

X∗c (ω) = Tc Xc (ω)

(2.50)

where Td and Tc are constant matrices defining geometric transformations. Equation (2.49) becomes X∗c (ω) = Tc T(ω)Td−1 X∗d (ω);

(2.51)

X∗d (ω), and X∗c (ω) are the Fourier transform of the generalized coordinates. The transmissibility matrix between the generalized coordinates is T ∗ (ω) = Tc T(ω)Td−1 .

(2.52)

A natural choice of generalized coordinates which also leads to a clear physical interpretation is the translations x, y, z and the rotations θx , θ y , θz , of a reference frame located at the geometric center O of

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z Payload plate sensors

Id Shaker Fd x

O y

Shaki ng table Base plate sensors

Figure 2.22 Test setup for the measurement of the transmissibility matrix. The reference frame is located at the geometric center of the platform

the platform (Figure 2.22). Thus, x∗d = (xd , yd , zd , θxd , θ yd , θzd )T and x∗c = (xc , yc , zc , θxc , θ yc , θzc )T are the generalized coordinates of the base plate and the payload plate, respectively, expressed in the same reference frame. In other words, x∗d characterizes the motion of the reference frame when it is attached to the base plate, and x∗c when it is attached to the payload plate. With this choice, column i of the transmissibility matrix T ∗ (ω) corresponds to the amplitude of the harmonic response of the payload to an imposed harmonic generalized displacement of the base plate X∗d (ω) = ei , that is, a pure translation or a pure rotation of the base plate in a given direction x, y or z of the reference frame. Fr¨ obenius norm The transmissibility matrix is a 6 × 6 matrix. To interpret the results and assess the performance of the isolator, it is convenient to define a scalar indicator with a meaning similar to that of the transmissibility of a singleaxis isolator, which provides for every frequency a measure of the isolation

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capability of the isolator. The Frobenius ¨ norm is often used for this purpose (Spanos et al., 1995); it is defined by 1/2 6 6 H 1/2 2 = |Ti j (ω)| ,

T(ω) = trace[T(ω)T(ω) ]

(2.53)

i=1 j=1

where H stands for the Hermitian. This norm can be interpreted as follows: From Parseval’s theorem,

∞ −∞

xcT xc

∞ 1 dt = Xc (ω) H Xc (ω)d ω 2π −∞ ∞ 1 trace Xc (ω)Xc (ω)H d ω = 2π −∞ ∞ 1 trace T (ω)Xd (ω)Xd (ω)H T (ω)H d ω. (2.54) = 2π −∞

By definition of the energy spectral density, Sxx =

1 E[|X(ω)|2 ]. 2π

(2.55)

Assuming that the motion of the base plate is such that the components of Xd (ω) are uncorrelated with unit energy spectral density, 1 E Xd (ω)Xd (ω) H = I 2π

(2.56)

and equation (2.54) becomes

∞

−∞

E xcT xc d t = = =

1 m H trace T (ω) E [Xd (ω)Xd (ω) ] T (ω) d ω 2π −∞ ∞

∞

−∞ ∞ −∞

trace[T (ω)T (ω)H ]d ω

T(ω) 2 d ω.

(2.57)

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Thus, T(ω) 2 represents the frequency distribution of the energy of the payload plate when the six inputs of the base plate are uncorrelated with unity energy spectral density (uniform over all frequencies). In the absence of an isolator, the two plates are rigidly linked, and, with our choice of coordinates, we have x∗c = x∗d . In this particular case, √ the transmissibility matrix T ∗ (ω) is the identity matrix and T ∗ (ω) = 6. Thus, to obtain a performance metric comparable to the transmissibility of a one-axis isolator, one must consider √ (ω) = T ∗ (ω) / 6.

(2.58)

The procedure used to identify the transmissibility matrix experimentally is discussed below. Identification of the transmissibility Consider the experimental setup of Figure 2.22. The base plate is attached to the shaking table to form a rigid body; the payload plate is also a rigid body. In order to identify the transmissibility matrix, it is necessary to input a disturbance force fd to the shaking table; this is achieved with an inertial actuator rigidly attached to it; the input force is controlled through the input current Id of the shaker. The shaker position and orientation can be adjusted to change the disturbance force fd . The current Id is recorded during the experiments, but fd is not measured. The readings from the six sensors on each body (accelerometers in this case) are Xc (ω) and Xd (ω); they can be transformed into the generalized coordinates X∗c (ω) and X∗d (ω) using (2.50). General methodology Since the disturbance force acts on the shaking table which is rigidly linked to the base plate, one can write

X∗c (ω) X∗d (ω)

H11 (ω) = H21 (ω)

0 H12 (ω) , H22 (ω) Fd (ω)

(2.59)

leading to X∗c (ω) = H12 (ω)H22 (ω)−1 X∗d (ω).

(2.60)

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The transmissibility matrix is therefore T ∗ (ω) = H12 (ω)H22 (ω)−1 .

(2.61)

The determination of H12 (ω) and H22 (ω) would require six decoupled excitations in the direction of the six generalized coordinates of the base plate. Because of the practical limitations with respect to the shaker placement in the experiment, it is generally impossible to fulfill this decoupling requirement. However, it is sufficient to use six independent excitations Fdi (ω), i = 1, . . . , 6. In practice, only the current is measured so that we have six different imposed currents Idi (ω), i = 1, . . . , 6 corresponding to six independent shaker orientations. The FRFs between the responses of the payload and the base, and the six imposed currents, are computed and arranged as the columns of a matrix: Hd (ω) = [Hd1 , Hd2 , Hd3 , Hd4 , Hd5 , Hd6 ], Hc (ω) = [Hc1 , Hc2 , Hc3 , Hc4 , Hc5 , Hc6 ],

(2.62)

where Hdi (ω) =

Xdi (ω) Idi (ω)

(2.63)

is the column vector of the six FRFs between the six sensor displacements of the base plate and the ith imposed current (Hci (ω) is defined in the same way for the payload). It follows from (2.49) that Hc (ω) = T(ω)Hd (ω).

(2.64)

If the shaker orientations have been selected in such a way that Hd (ω) is invertible, then T(ω) = Hc (ω)Hd (ω)−1 .

(2.65)

Redundant measurements If more than six excitations are used, the matrices Hd (ω) and Hc (ω) have more than six columns. The transmissibility matrix can still be computed

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from (2.64) using the Moore–Penrose pseudo-inverse: T(ω) = Hc (ω)Hd (ω)+ ,

(2.66)

where the pseudo-inverse of the matrix is defined by −1 Hd+ = HdH Hd HdH

(2.67)

(in practice, a singular value decomposition is used, and singular values smaller than a tolerance are treated as zero and deleted in the calculation of the pseudo-inverse). Once T(ω) has been estimated, the transmissibility in generalized coordinates, T ∗ (ω), can be computed by (2.52). Experimental results Experiments have been conducted in the laboratory and in zero gravity (in parabolic flight). For the ground tests, the gravity was compensated by hanging the payload from three soft elastic springs holding the corners of the upper triangle; a mechanism allowed the spring tension to be adjusted in such a way that the length of the legs was close to their nominal value. The holding mechanism introduces an additional stiffness which increases the corner frequency of the isolator; it also produces additional peaks at the natural frequency of the suspension in the transmissibility curve. During the parabolic flight tests, the zero-gravity environment can be maintained only for 20 s, which reduces the useful part of the signal to about 15 s. This leads to practical difficulties in achieving meaningful results at very low frequency and in eliminating the noise in frequency bands where the transmissibility is low. Fortunately, the flight campaign included 90 parabolas which allowed some averaging to be done. Figure 2.23(a) shows the experimental vertical transmissibility obtained from data measured with the high precision accelerometers during the parabolic flight tests, with and without control; the numerical simulations of Figure 2.21 are also shown as dotted lines in this figure, for comparison purposes; the agreement between the experiment and the simulation is very good. Figure 2.23(b) shows the coherence function measured during the same tests and gives an idea of the quality of the collected data. Single-axis transmissibilities measured along the horizontal axes are very similar to that in the vertical direction. Figure 2.24 shows the Frobenius ¨ norm (ω) as defined by (2.58) and calculated from the measurements of

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Vertical Transmissibility (dB)

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(a) –5 0

10

0

10

1

10

0

10

2

1

10

2

5.10

1 0.8

Coherence

xxx

0.6 0.4 0.2

(b)

0 10

2

2

5.10

Figure 2.23 (a) Experimental transmissibility in the vertical direction, with and without control. (b) Coherence function, with and without control

the 12 regular accelerometers; the simulation results are again shown as dotted lines in the figure; the agreement between the experiments and the simulations is also good, although not as good as in Figure 2.23 (which was obtained with high-precision accelerometers). A detailed examination of the results shows that the main source of discrepancy between the

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40 30

Without Control

20 Transmissibility (dB)

xxx

With Control

10 0 –10 –20 Fröbenius Norm –30 –40 –50 10

0

10

1

10

2

Hz

5.10

2

Figure 2.24 Frobenius ¨ norm (ω) as defined by (2.58), with and without control. The solid lines refer to experiments and the dotted lines to simulation results

values of (ω) computed from the simulation results and the measured data originates from the non-diagonal terms of the transmissibility matrix which are more sensitive to noise. In view of the foregoing experimental results, it seems that the modeling procedure described above allows a reasonably good performance prediction of the isolator up to the first local mode of the legs.

2.6 Vehicle Active Suspension This section gives a short introduction to vehicle active suspensions. The main disturbances acting on a road vehicle are the road roughness, the aerodynamic loads due to wind gusts, and the inertia loadings originating from acceleration, braking and turning. The inertial loads have a limited bandwidth and depend strongly on the driving style. The road profile

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w(t) seen by the vehicle traveling at a speed V can be approximated by the response of a first-order system to a white-noise excitation; the corresponding autocorrelation function is Rw (τ ) = σ 2 e −a V|τ |

(2.68)

where σ is the RMS value of w(t), V the vehicle speed and a is a parameter depending on the road rugosity. The product V|τ | is in fact the distance traveled by the vehicle during the delay τ . The corresponding power spectral density is given by w (ω) =

σ2 aV . 2 π ω + a 2 V2

(2.69)

According to this approximation, the ground input velocity v = w˙ is close to a white noise. Ride comfort requires good vibration isolation, and vehicle handling requires good road holding. Ride comfort is usually measured by the car body acceleration, or sometimes its derivative, called jerk. According to the literature, there is an excellent correlation between the RMS acceleration and the subjective ride rating. Road holding is often measured by the tire deflection, or sometimes by the unsprung mass (wheel) velocity. In addition to the car body acceleration and the tire deflection which define the performance of the suspension (and appear in the performance index), there are design constraints on the suspension travel, that is, the relative displacement between the vehicle body and the wheel. There has been some confusion about the definition of active suspension. A (fully) active suspension is a suspension provided with a wide-band actuator capable of controlling the system in the full bandwidth. There have been very few cars equipped with such a system; the best known is a Lotus Esprit in the mid-1980s. A semi-active (or semi-passive) suspension is a suspension provided with a controllable shock-absorber requiring small external power, but capable of changing the shock-absorber characteristics in wide-band. This requires a fast-responding controllable device. We will call adaptive suspension a suspension equipped with a controllable shock absorber which has only low-frequency capability, allowing the damper characteristics to be adapted to optimize ride comfort and road holding for the current road roughness and driving conditions. Many cars in production are equipped with such a system, with adaption laws of various degrees of sophistication.

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(a)

91

(b)

(c)

Figure 2.25 Quarter-car models of a suspension: (a) one-d.o.f. active suspension; (b) two-d.o.f. passive suspension; (c) two-d.o.f. fully active suspension

In summary, the design of a suspension is concerned with (i) car body acceleration for ride comfort, (ii) tire deflection for road holding and (iii) suspension travel (rattle space) which must remain within fixed limits. 2.6.1 Quarter-car Model Most features of a car suspension can be analyzed with a quarter-car model in which one quarter of the body mass is lumped into the sprung mass, ms . Figure 2.25 summarizes various degrees of sophistication of a quarter-car model; the simplest one neglects the unsprung mass mus (wheel); this model has only 1 d.o.f. and neglects the so-called wheel-hop mode; it constitutes the limit of the best possible performance attainable with a 2 d.o.f. model provided with a single actuator. One-d.o.f. model Consider the one-d.o.f. model of an active suspension (Figure 2.26). The control input u is defined as the force per unit (sprung) mass, u = f /ms . The single-d.o.f. system is fully described by two state variables, x1 , the suspension deflection, and x2 = x˙ s , the absolute velocity of the sprung mass. The road profile is defined by the road velocity v(t) = w(t) ˙ which

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Figure 2.26 One-d.o.f. model of a fully active suspension

can be modeled as a white-noise Gaussian process.10 With this choice of coordinates, the dynamics of the system is given by x˙ 1 = x2 − v, x˙ 2 = u,

(2.70) (2.71)

or, in matrix form,

x˙ 1 x˙ 2

=

0 0

1 0

x1 x2

0 −1 + u+ v. 1 0

(2.72)

A full state feedback has the general form u = −g1 x1 − g2 x2 .

(2.73)

According to the definition of the state variables, the first term, −g1 x1 , corresponds to a spring acting between the road and the sprung mass (Figure 2.27(a)) while the second, −g2 x2 , corresponds to a sky-hook damper (acting on the absolute velocity of the sprung mass). This configuration is true for any full state feedback. Combining equations (2.72) and (2.73), one finds the closed-loop system equation

10

x˙ 1 x˙ 2

=

0 −g1

1 −g2

x1 x2

+

−1 0

v.

It is assumed that the tire is in contact with the road at all times (point-contact follower).

(2.74)

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(b)

(a)

Figure 2.27 Effect of the full state feedback u = −g1 x1 − g 2 x2 on the oned.o.f. model of a fully active suspension. (a) Spring and sky-hook damper. (b) Closed-loop poles

The closed-loop eigenvalues are solutions of the characteristic equation s 2 + g2 s + g1 = 0.

(2.75)

They form a pair of complex conjugate poles which can be assigned anywhere in the complex plane by appropriate choice of g1 and g2 (g1 sets the frequency, g2 the damping). The closed-loop transmissibility between the road velocity and the various state variables is obtained from (2.74): x1 −(s + g2 ) = 2 , v s + g2 s + g1 x2 g1 = 2 . v s + g2 s + g1

(2.76) (2.77)

This equation tells us that the transmissibility between the road velocity v(t) and the sprung mass velocity x2 = x˙ s is a second-order low-pass filter, while the previous one tells that the transmissibility between the road velocity and the relative velocity of the suspension x2 − v −(s 2 + g2 s) sx1 = = 2 v v s + g2 s + g1

(2.78)

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dB

Suspension relative velocity

0

ξ = 0.3 ξ = 0.7

–10

Sprung mass absolute velocity

–20

0.1

ω

1

10

Figure 2.28 Transmissibility between the road velocity and (i) the sprung mass velocity and (ii) the relative velocity of the suspension. The solid line is for ξ = 0.7, the dotted line is for ξ = 0.3

is a high-pass filter. These two signals have widely different frequency content (Figure 2.28). The natural frequency and the damping ratio are respectively ωn = Optimal control

√

g1 ,

ξ=

g2 √ . 2 g1

(2.79)

11

Once again, the foregoing discussion applies to any state feedback with this particular choice of state variables. Since the disturbance v appearing in (2.72) is a white noise, this problem is well suited for a linear quadratic regulator (LQR) formulation. If one uses the suspension deflection (rattle space) as controlled variable, the performance index to be minimized is (2.80) J = E x12 + u2 , 11

This section may be skipped by the reader not familiar with optimal control. The background on optimal control is available in many textbooks, for example Kailath (1980). See also Preumont (2002, Section 7.5) for the symmetric root locus.

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where is a scalar parameter weighting the control effort. According to the optimal control theory, the closed-loop poles are the stable roots (those in the left half-plane) of the characteristic equation 1 + −1 G 0 (s)G 0 (−s) = 0,

(2.81)

where G 0 (s) is the open-loop transfer function between the control input u and the controlled variable x1 appearing in the performance index. From (2.72), G 0 (s) =

x1 1 = 2 u s

(2.82)

and the characteristic equation is s 4 + −1 = 0.

(2.83)

−1/4 s = − √ (1 ± j). 2

(2.84)

The stable solutions are

They lie on straight lines at 45◦ in the complex plane; the distance to the origin depends only on the control weight in the objective function. From Figure 2.27(b), one finds easily that, for this particular objective function, g1 = −1/2 ,

g2 = (2g1 )1/2 . (2.85) √ The corresponding damping ratio is ξ = 2/2 = 0.707; the transmissibility curves are shown in Figure 2.28. Two-d.o.f. model Although the one-d.o.f. model neglects the wheel-hop mode, the state feedback approach used in the previous section has led us to the optimal suspension structure (spring plus sky-hook damper) that is used in most active suspension concepts. This is due to our special choice of state variables (relative displacement and absolute velocity of the sprung mass). It is reasonable to replace the feedback component proportional to the suspension stroke by a passive spring with the same constant; this limits the active control force to the sky-hook damping contribution alone. In this section, we analyze the effect of the wheel-hop mode associated with the unsprung mass. To this end, consider the two-d.o.f. model of Figure 2.29(a), where

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(a)

(b)

Figure 2.29 Quarter-car, two-d.o.f. models. (a) Fully active suspension. (b) Passive suspension with sky-hook damper

two state variables have been added to describe the motion of the wheel. The state variables are selected as follows: x1 = xs − xus is the relative displacement of the sprung mass with respect to the wheel, x2 = x˙ s is the absolute velocity of the car body, x3 = xus − w is the tire deflection, and x4 = x˙ us is the absolute wheel velocity. With this definition of the state variables, the dynamics of the system are given by: ms x˙ 2 mus x˙ 4 x˙ 1 x˙ 3

= f, = − f − kt x3 , = x2 − x4 , = x4 − v.

Defining the force per unit sprung mass u = f /ms , the unsprung mass ratio μ = mus /ms and the tire frequency ωt = (kt /mus )1/2 , the foregoing equations can be rewritten in matrix form: ⎛

⎞ ⎡ 0 x˙ 1 ⎜ x˙ ⎟ ⎢ 0 ⎜ 2⎟ ⎢ ⎜ ⎟=⎢ ⎝ x˙ 3 ⎠ ⎣ 0 x˙ 4

0

1 0 0 0 0 0 0 −ωt2

⎤⎛ ⎞ ⎛ −1 x1 ⎜ ⎥ ⎜ 0 ⎥ ⎜ x2 ⎟ ⎟ ⎜ ⎥⎜ ⎟ + ⎜ 1 ⎦ ⎝ x3 ⎠ ⎝ 0

x4

⎞ ⎛ ⎞ 0 0 ⎜ 0⎟ 1 ⎟ ⎟ ⎜ ⎟ ⎟u + ⎜ ⎟ v. ⎝ −1 ⎠ 0 ⎠ −1/μ 0

(2.86)

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A reasonable performance index for this problem is J = E 1 x12 + 2 x32 + u2 ,

(2.87)

where 1 and 2 are scalar parameters weighting the various contributors to the performance index. The first term is the suspension deflection involved in the rattle space, the second term is the tire deflection involved in the road holding, and the control input u is also the sprung mass acceleration. Because the four state variables appearing in this formulation are not all easy to measure, we will not pursue the full state feedback approach, but rather focus on the simpler architecture of Figure 2.29(b), which consists of a passive suspension supplemented with an ideal force actuator and controlled with a partial state feedback (sky-hook damper). This model, which contains all the features of a passive suspension, is particularly useful for comparing the performance of passive and active suspensions. In this case, using the same state variables, the dynamics of the system are given by ms x˙ 2 mus x˙ 4 x˙ 1 x˙ 3

= f − kx1 + c(x4 − x2 ), = − f − kt x3 + kx1 + c(x2 − x4 ), = x2 − x4 , = x4 − v,

which, upon defining ωn2 = k/ms (ωn is the body resonance) and c/ms = 2ξ ωn , can be rewritten in matrix form as ⎛

⎞

⎡

0 1 0 x˙ 1 ⎢ −ω2 −2ξ ω 0 n ⎜ x˙ ⎟ ⎢ n ⎜ 2⎟ ⎢ ⎜ ⎟=⎢ 0 0 0 ⎝ x˙ 3 ⎠ ⎢ ⎣ ω2 2ξ ωn n x˙ 4 −ωt2 μ μ

⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −1 0 x1 0 ⎥ ⎜ ⎟ 2ξ ωn ⎥ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎥ ⎜ x2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. 1 ⎥⎜ ⎟ + ⎜ 0⎟u + ⎜ ⎥ ⎝ x3 ⎠ ⎜ ⎟ ⎝ −v ⎠ ⎦ ⎝ ⎠ −1 −2ξ ωn x4 0 μ μ (2.88)

Passive suspension Let us first illustrate the tradeoff in the design of a passive suspension with the following example taken from Chalasani (1984): The nominal values of the passive suspension are kt = 160 000 N/m (tire stiffness),

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(a)

(b)

Figure 2.30 Behavior of the passive suspension for various values of the damping constant: c = 200, 980, and 4000 Ns/m. (a) Transmissibility T x¨s v between the road velocity v = w˙ and the body absolute acceleration x˙ 2 = x¨s . (b) Cumulative RMS value of the sprung mass acceleration σx¨s

k = 16 000 N/m (suspension spring stiffness), ms = 240 kg (car body), mus = 36 kg (wheel). The effect of the damping constant is illustrated in Figures 2.30 and 2.31. Figure 2.30(a) shows the transmissibility Tx¨s v between the road velocity v = w˙ and the body absolute acceleration x˙ 2 = x¨s for three values of the damping constant, c = 200, 980, and 4000 Ns/m. For the smallest value of c, one can clearly see the two peaks associated with the body resonance (sprung mass) and the tire resonance (unsprung resonance frequencies are very close to √ mass); the corresponding √ ωn = k/ms and ωt = kt /mus , respectively.12 As the damping increases, the amplitude of the two peaks is reduced; one can clearly see that the passive damping cannot control the body resonance without the isolation at higher frequency deteriorating. The cumulative RMS value of the body

The body resonance is at 7.8 rad/s and the tire resonance is at 69.5 rad/s, while ωn = √ and ωt = kt /mus = 66.7 rad/s.

12

√ k/ms = 8.16 rad/s

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(a)

(b)

(c)

(d)

Figure 2.31 Behavior of the passive suspension for various values of the damping constant: c = 200, 980, and 4000 Ns/m. (a) Transmissibility of the suspension stroke, T x1 v . (b) Cumulative RMS suspension stroke σx 1 . (c) Transmissibility of the tire deflection, T x 3 v . (d) Cumulative RMS tire deflection σx 3

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acceleration is defined by the integral σx¨s (ω) =

0

ω

|Tx¨s v | dν 2

1/2 (2.89)

shown in Figure 2.30(b); since the road velocity is approximately a white noise, σx¨s describes how the various frequencies contribute to the RMS of the body acceleration (in relative terms). When the damping increases, the RMS body acceleration initially decreases, and increases again for larger

(a)

(b)

(c)

Figure 2.32 Influence of the unsprung mass. (a) Transmissibility T x¨s v between the road velocity v = w˙ and the body absolute acceleration x¨s , for mus = 36, 10, and 1 kg. (b) Transmissibility of the suspension stroke, T x1 v . (c) Transmissibility of the tire deflection, T x 3 v

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values of c. Similarly, Figure 2.31 analyzes the effect of the damping coefficient on the relative displacement, x1 (rattle space), and the tire deflection, x3 (road holding). Figure 2.32 analyzes the influence of the unsprung mass; the damping is kept constant (at the lower value c = 200 Ns/m, in order to highlight the resonance peaks). Three values of mus = 36, 10, and 1 kg are considered; reducing mus moves the tire resonance to higher frequency and reduces its amplitude; asymptotically, the system tends towards the one-d.o.f. model of Figure 2.25(a). This confirms that the unsprung mass can only deteriorate the response with respect to the one-d.o.f. model. Active suspension A partial state feedback is added to the passive suspension; it consists of a sky-hook damper, f = −g x˙ s (Figure 2.29(b)). Figure 2.33(a) shows the impact of the control gain on the transmissibility Tx¨s v between the road velocity v = w˙ and the body absolute acceleration x¨s ; the cumulative RMS value of the sprung mass acceleration σx¨s is shown in Figure 2.33(b). One can see that the active control acts very effectively on the body resonance and that the attenuation is achieved without the high-frequency isolation deteriorating. However, the active control is unable to reduce the wheel resonance. The active control produces a significant reduction of the RMS sprung mass acceleration but the reduction of the RMS tire deflection is only modest, because the control system fails to reduce the wheel resonance. Next, we examine how this feature can be improved. Adding a Dynamic Vibration Absorber 13 As we have just seen, the sky-hook feedback has little control over the wheel resonance. Actually, the wheel resonance may be reduced without penalty on the ride comfort by adding a dynamic vibration absorber (DVA) (Den Hartog, 1956) to the unsprung mass, with an absorber mass ma ∼ 5–20% of mus . The situation is depicted in Figure 2.34, with two additional state variables describing the DVA: x5 , the relative displacement of ma with respect to mus ; and x6 , the absolute velocity of the DVA. The other state variables are the same as before. The governing equations are in this 13

According to Hrovat (1997), ‘there has been only one widespread production application of dynamic absorbers with the aim of improving ride comfort. This is the case of the well-known French subcompact Citroen 2 CV, first introduced in 1949. The 2 CV was legendary for having a superior ride for a car of its size.’

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(a)

(b)

(c)

(d)

Figure 2.33 Active suspension for various values of the control gain, g = 0, 1000, and 2000. The damping of the shock absorber is c = 200 Ns/m. (a) and (b) Sprung mass acceleration. (c) and (d) Tire deflection

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Figure 2.34 Quarter-car model: Two d.o.f. plus a DVA added to the wheel. The system is described by 6 state variables

case

ms x˙ 2 mus x˙ 4 x˙ 1 x˙ 3 ma x˙ 6 x˙ 5

= f − kx1 + c(x4 − x2 ), = − f − kt x3 + kx1 + c(x2 − x4 ) + ka x5 + c a (x6 − x4 ), = x2 − x4 , = x4 − v, = −ka x5 − c a (x6 − x4 ), = x6 − x4 .

In matrix form, the state vector x = (x1 , . . . , x6 )T satisfies the equation

x˙ = Ax + Bu + Ev,

(2.90)

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where ⎡

−1 2ξ ωn 1 −2ξ ωn /μ − 2ξa ωa α −1 2ξa ωa

0 1 0 ⎢ −ω2 −2ξ ω 0 n ⎢ n ⎢ ⎢ 0 0 0 A= ⎢ ⎢ ω2 /μ 2ξ ω /μ −ω2 n ⎢ n t ⎢ ⎣ 0 0 0 0 0 0 ⎛

⎞

⎛

0 ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎟ B=⎜ ⎜ −1/μ ⎟ , ⎜ ⎟ ⎜ ⎟ ⎝ 0 ⎠ 0

0 0 0 ωa2 α 0 −ωa2

⎤ 0 ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥, 2ξa ωa α ⎥ ⎥ ⎥ ⎦ 1 −2ξa ωa (2.91)

⎞

0 ⎜ 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −1 ⎟ ⎟ E=⎜ ⎜ 0⎟, ⎜ ⎟ ⎜ ⎟ ⎝ 0⎠ 0

(2.92)

with the same notation as before: u=

f , ms

μ=

mus , ms

ωt2 =

kt , mus

ωn2 =

k , ms

2ξ ωn =

c , ms

(2.93)

supplemented, for the DVA, by α=

ma , mus

ωa2 =

ka , ma

2ξa ωa =

ca . ma

(2.94)

The DVA natural frequency and the damping ratio are tuned on the wheelhop mode. According to the equal peak design (Den Hartog, 1956),14 the optimum DVA is obtained when the frequency ratio and the damping ratio of the DVA are chosen according to 3α 1 ωa and ξa = (2.95) = ωt 1+α 8(1 + α) (the frequency of the wheel-hop mode is approximated by ωt ). Once ma has been selected, these formulas allow ka and c a to be computed. Figure 2.35 shows the response of the active suspension when a DVA is added, 14

The design of the DVA will be reviewed in the next chapter.

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Figure 2.35 Effect of the DVA on the transmissibility of the body acceleration and the tire deflection of the active suspension (mus = 36 kg, c = 200 Ns/m and g = 1000). The three curves correspond to a mass ratio of α = 0 (no DVA), α = 0.05 and 0.2

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tuned on the wheel resonance ωt . The results presented correspond to mus = 36 kg, c = 200 Ns/m, and g = 1000; the two values of α = 0.05 and 0.2 are considered. One can see that the DVA is very successful in reducing the wheel resonance and that it has a very strong influence on the tire deflection, and therefore on the road-holding capabilities of the vehicle. Note that the transmissibility curves do not exhibit exactly equal peaks; this could be further improved by tuning the DVA more precisely on the wheel-hop mode of the suspension.

2.7 Semi-Active Suspension A semi-active suspension consists of a classical suspension provided with a controllable shock absorber, capable of changing its characteristics in real time with a small amount of energy. The device remains essentially passive and can only dissipate energy, that is, produce a force opposing the motion applied to the device. 2.7.1 Semi-Active Devices Two of the most frequently used semi-active devices are illustrated in Figure 2.36, with their respective operating range. Figure 2.36(a) shows a semi-active device consisting of a classical viscous damper with a variable damping coefficient C(u) obtained by controlling the size of the opening of an orifice between the two chambers of the damper (e.g. with an electromagnet). The operating range is the shaded area between two lines corresponding to the minimum and maximum damping coefficients, Cmin < C(u) < Cmax . Figure 2.36(b) shows a device consisting of a magneto-rheological (MR) fluid damper (Carlson and Jolly, 2000); the control variable is the current through the coil, which controls the magnetic field in the active part of the fluid and, as a result, creates the variable yield force in the device. The behavior of the MR device can be represented by a Bingham model consisting of a constant viscous damper in parallel with a variable friction device. The use of controllable shock absorbers in adaptive suspensions is developing quickly; it offers new capabilities to enhance the vehicle dynamics, in connection with the so-called ESP system. Adaptive suspensions involve the whole vehicle dynamics and are outside the scope of this introduction.

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107 (b)

f

f

v f = C(u)v

v

τ (u)

c

f = c v + τ(u) f

f

Cmax

τmax Cmin

v

v

Figure 2.36 Semi-active devices and their operating range. (a) Viscous damper with variable damping coefficient. (b) MR fluid device and its Bingham model

2.7.2 Narrow-band Disturbance Before discussing the semi-active suspension of a quarter-car model, let us revisit the single-axis passive isolation problem of Figure 2.2 equipped with a controllable damper c min < c < c max , in the frequent case where the disturbance force applied to the mass m is harmonic (or nearly so); this situation typically arises when a rotating machine with mass imbalance is accelerated or decelerated. According to the discussion√ at the beginning of this chapter, when the is minimized by setdisturbance frequency is ω < 2ωn , the overshoot √ ting a high damping constant, while above 2ωn the damping should be minimum to enjoy the maximum roll-off rate of −40 dB/decade. This suggests the following control strategy: if if

√ 2ωn √ ω > 2ωn

ω≤

then

c = c max ,

then

c = c min

(2.96)

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If c max is large enough and c min is small enough, the transmissibility achieved in this way fits closely that of the objective of active isolation in Figure 2.2. Thus, the semi-active isolation is optimal in this case. One must be careful, however, that a semi-active isolation device, like any nonlinear system, depends strongly on the excitation. 2.7.3 Quarter-Car Semi-Active Suspension The principle of the semi-active suspension is illustrated in Figure 2.37. The semi-active control unit activates the controllable device to achieve the variable control force f c subject to the constraint imposed by the passivity of the device,15 f c .(x˙ s − x˙ us ) ≤ 0.

(2.97)

As a nonlinear device, the response of a controllable shock absorber depends on the excitation amplitude and on its frequency content, and it has the capability to transfer energy from one frequency to another.

Figure 2.37 Principle of the semi-active suspension. One or several sensors monitor the state of the suspension and a semi-active control unit controls the shock absorber constant c(u) f c = −c(u)(x˙ s − x˙ us ) is the force applied by the shock absorber to the sprung mass. More complex situations may also be considered, in which the spring stiffness is also variable.

15

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109

The semi-active sky-hook consists of trying to emulate the sky-hook control f c −b x˙ s , where b is the targeted damping coefficient of the skyhook damper, with the controllable shock absorber c(u) (Karnopp et al., 1974; Karnopp, 1990; Venhovens, 1994). Because of the passivity constraint (2.97), this is possible only if the sprung mass velocity and the relative velocity have the same sign, x˙ s .(x˙ s − x˙ us ) ≥ 0,

(2.98)

and if the magnitude of the requested control force belongs to the operating range of the controllable shock absorber, c min ≤

|b x˙ s | ≤ c max . |x˙ s − x˙ us |

The damping constant which fits best the requested (sky-hook) control force is ! c(u) = max c min , min

b x˙ s , c max x˙ s − x˙ us

" .

(2.99)

However, the sprung mass velocity x˙ s and the suspension relative velocity x˙ s − x˙ us have widely different frequency contents, as illustrated with a one-d.o.f. model in Figure 2.28, and the foregoing strategy tends to produce a fast switching control force f c , as illustrated below. The control law (2.99) requires a fast, calibrated, proportional valve; an alternative on/off implementation is c(u) = c max c(u) = c min

if if

x˙ s .(x˙ s − x˙ us ) ≥ 0, x˙ s .(x˙ s − x˙ us ) < 0.

(2.100)

Although simpler, this strategy is likely to produce even sharper changes in the control force. The following example illustrates the energy transfer from low frequency to high frequency associated with the semi-active sky-hook control.

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The system of Figure 2.37 is modeled using the same state variables as for the passive suspension (Figure 2.29), x1 = xs − xus , x2 = x˙ s , x3 = xus − w, x4 = x˙ us : ms x˙ 2 mus x˙ 4 x˙ 1 x˙ 3

= −kx1 + c(u)(x4 − x2 ), = −kt x3 + kx1 + c(u)(x2 − x4 ), = x2 − x4 , = x4 − v,

where v = w˙ is the road velocity. Time domain simulations have been conducted with the same numerical data as the passive suspension analyzed earlier: ms = 240 kg, mus = 36 kg, k = 16 000 N/m, kt = 160 000 N/m, b = 2000 Ns/m (gain of the sky-hook control). The shock absorber constant is supposed to vary between c min = 100 Ns/m and c max = 2000 Ns/m. The body resonance and the tire resonance are respectively ωn = (k/ms )1/2 ∼ 8 rad/s and ωt = (kt /mus )1/2 ∼ 70 rad/s. The road velocity v is assumed to be white noise; the control law is (2.99). Figure 2.38 shows various time histories of the quarter-car response: the tire force kt x3 , the body velocity x˙ s = x2 , the relative velocity x˙ 1 = x˙ s − x˙ us , the required force f = −b x˙ s and control force f c = −c(u)(x˙ s − x˙ us ), and finally the damper constant c(u). Note that the relative velocity oscillates much faster (at 70 rad/s) than the body velocity, resulting in sharp changes in the control force f c . Figure 2.39(a) compares the transmissibility between the road velocity and the body acceleration, Tx¨s v of the passive suspension (c = 200 Ns/m), the sky-hook control (c = 200 Ns/m and b = 2000 Ns/m), and the semi-active sky-hook (2.99) with c min = 100 Ns/m and c max = 2000 Ns/m. The first two curves are the same as in Figure 2.33(a). The semi-active control is successful in reducing the body resonance, and the transmissibility of the body acceleration is comparable to that of the active control with b 1000 Ns/m (see Figure 2.33) at low frequency; however, a significant amplification occurs at the wheel resonance, ωt = 70 rad/s, and above ωt the transmissibility rolls off much more slowly than in the previous cases. Besides, one observes peaks at various harmonics of the wheel resonance,16 which are likely to excite flexible modes of the vehicle if nothing is done to attenuate 16

The two peaks at 132 rad/s and 148 rad/s seem to result from the modulation of the second harmonic of the wheel mode (2ωt = 140 rad/s) by the car body mode (ωn = 8 rad/s), producing frequency peaks at 2ωt − ωn and 2ωt + ωn .

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111

(a)

(b)

(c)

(d)

(e)

Figure 2.38 Quarter-car model with continuous semi-active sky-hook control: (a) tire force kt x3 ; (b) body velocity x˙ s = x2 ; (c) relative velocity x˙ 1 = x˙ s − x˙ us ; (d) required force f = −b x˙ s and control force fc = −c(u)(x˙ s − x˙ us ) obtained from (2.99); (e) damper constant c(u)

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Active Isolation

(a)

(b)

Figure 2.39 Quarter-car model with continuous semi-active sky-hook control. (a) Transmissibility between the road velocity and the body acceleration, T x¨s v : passive suspension (c = 200 Ns/m), sky-hook controller (c = 200 Ns/m, b = 2000 Ns/m), semi-active sky-hook (2.99) with c min = 100 Ns/m and c max = 2000 Ns/m. (b) Transmissibility between the road velocity and the tire deflection, T x 3 v

them. Figure 2.39(b) shows the transmissibility between the road velocity and the tire deflection; an amplification at the wheel resonance is also observed, but no spurious high-frequency components appear. The transmissibility diagrams of Figure 2.39 were obtained from timehistories with auto power spectra estimates: Tyx =

yy xx

1/2

! =

E[Y(ω)Y∗ (ω)] E[X(ω)X∗ (ω)]

"1/2 (2.101)

Further evidence of the nonlinear energy transfer from low to high frequencies can be obtained from the coherence function (Figure 2.40)

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References

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Figure 2.40 Quarter-car model with continuous semi-active sky-hook control. Coherence function γx¨2s v between the road velocity and the body acceleration

between the road velocity and the body acceleration, γx¨2s v =

|x¨s v |2 ≤ 1. vv x¨s x¨s

(2.102)

γx¨2s v is equal to 1 for a perfect linear system; it measures the causality of the signal at every frequency; it is a standard tool for detecting the presence of noise and nonlinearities. According to Figure 2.40, the coherence is very good up to the tire mode, and falls rapidly to zero above 100 rad/s, which indicates that at those frequencies the energy content of the body acceleration is not due to the road profile.

References Abu Hanieh, A. (2003) Active isolation and damping of vibrations via Stewart platform. PhD Thesis, ULB-Active Structures Laboratory, Brussels. Carlson, J.D., and Jolly, M.R. (2000) MR fluid, foam and elastomer devices. Mechatronics, 10, 555–569. Chalasani, R.M. (1984) Ride performance potential of active suspension systems. Part 1: Simplified analysis based on a quarter-car model. Paper presented to ASME Symposium on Simulation and Control of Ground Vehicles and Transportation Systems, Anaheim, CA, December. Den Hartog, J.P. (1956) Mechanical Vibrations. McGraw-Hill. Fuller, C.R., Elliott, S.J., and Nelson, P.A. (1996) Active Control of Vibration, Academic Press.

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Geng, Z., and Haynes, L. (1994) Six degree-of-freedom active vibration isolation system using the Stewart platforms. IEEE Transactions on Control Systems Technology, 2(1), 45–53. Hauge, G.S., and Campbell, M.E. (2004) Sensors and control of a spacedbased six-axis vibration isolation system. Journal of Sound and Vibration, 269, 913–931. Hrovat, D. (1997) Survey of advanced suspension developments and related optimal control applications. Automatica, 33(10), 1781–1817. Kailath, T. (1980) Linear Systems. Prentice Hall. Kaplow, C.E., and Velman, J.R. (1980) Active local vibration isolation applied to a flexible telescope. AIAA Journal of Guidance and Control, 3, 227–233. Karnopp, D. (1990) Design principles for vibration control systems using semi-active dampers. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 112, 448–455. Karnopp, D.C., and Trikha, A.K. (1969) Comparative study of optimization techniques for shock and vibration isolation. Transactions of the ASME, Journal of Engineering for Industry, Series B, 91, 1128–1132. Karnopp, D., Crosby, M., and Harwood, R.A. (1974) Vibration control using semi-active suspension control. Transactions of the ASME, Journal of Engineering for Industry, 96, 619–626. McInroy, J.E. (2002) Modeling and design of flexure jointed Stewart platforms for control purposes. IEEE/ASME Transactions on Mechatronics, 7(1), 95–99. McInroy, J.E., and Hamann, J. (2000) Design and control of flexure jointed hexapods, IEEE Transactions on Robotics, 16(4), 372–381. McInroy, J.E., O’Brien, J.F., Neat, G.W. (1999a) Precise, fault-tolerant pointing using a Stewart platform. IEEE/ASME Transactions on Mechatronics, 4(1), 91–95. McInroy, J.E., Neat, G.W., and O’Brien, J.F. (1999b) A robotic approach to fault-tolerant, precision pointing. IEEE Robotics and Automation Magazine (December), 24–37. Preumont, A. (2002) Vibration Control of Active Structures: An Introduction, 2nd edition. Kluwer. Preumont, A., Fran¸cois, A., Bossens, F., and Abu-Hanieh, A. (2002) Force feedback versus acceleration feedback in active vibration isolation. Journal of Sound and Vibration, 257(4), 605–613. Preumont, A., Horodinca, M., Romanescu, I., de Marneffe, B., Avraam, M., Deraemaeker, A., Bossens, F., Abu Hanieh, A. (2007) A six-axis single stage active vibration isolator based on Stewart platform. Journal of Sound and Vibration, 300, 644–661.

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Rahman, Z.H., Spanos, J.T., and Laskin, R.A. (1998) Multi-axis vibration isolation, suppression and steering system for space observational applications. Paper presented to SPIE Symposium on Astronomical Telescopes and Instrumentation, Kona-Hawaii, March 1998. Rivin, E.I. (2003) Passive Vibration Isolation. New York: ASME Press. Spanos, J., Rahman, Z., Blackwood, G. (1995) A soft 6-axis active vibration isolator. In Procedings of the IEEE American Control Conference, 412–416. Stewart, D. (1965–6) A platform with six degrees of freedom, Proceedings of the Institution of Mechanical Engineers, Part 1, 180(15), 371–386. Thayer, D., Vagners, J., von Flotow, A., Hardman, C., Scribner, K. (1998) Six-axis vibration isolation system using soft actuators and multiple sensors (AAS 98-064). Advances in the Astronautical Sciences, 98, 497–506. Thayer, D., Campbell, M., Vagners J., von Flotow, A. (2002) Six-axis vibration isolation system using soft actuators and multiple sensors. AIAA Journal of Spacecraft and Rockets, 39(2), 206–212. Venhovens, P.J.Th. (1994) The development and implementation of adaptive semi-active suspension control. Vehicle System Dynamics, 23, 211– 235.

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3 A Comparison of Passive, Active and Hybrid Control 3.1 Introduction When one examines the huge amount of literature published over the last 20 years on active vibration control, one of the difficulties is to compare the merits of the various techniques which have been proposed and applied on very different and often very limited examples.1 This has led to much confusion amongst the less experienced researchers, and skepticism amongst the more experienced ones. The objective of this chapter is to offer a simple comparison of various techniques on a common example, to highlight their most salient features and to allow a more objective evaluation. The example chosen is a shear frame under seismic excitation. The comparison will consider performance, robustness, and control requirements. A very simple example has deliberately been chosen in order to allow the reader to perform his own independent analysis, conduct parametric studies, and possibly evaluate new control strategies.

1

This chapter was written with the participation of Christophe Collette, Bruno de Marneffe and Mohammed El Ouni.


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In the first part of the discussion, three control techniques working with an inertial mass will be considered: (i) The well-known dynamic vibration absorber (DVA), also called the tuned mass damper, which has been applied very successfully when the targeted frequency is known and sufficiently stable. (ii) The active mass damper (AMD) is also an inertial device which behaves like a single-d.o.f. oscillator coupled with a force actuator; in this case, its natural frequency is chosen well below the first natural frequency of the structure; if the AMD is coupled with an appropriate absolute acceleration, or absolute velocity sensor (geophone), the control system is such that the open-loop transfer function exhibits alternating poles and zeros, which is the key to robustness in active damping. (iii) The next idea is to combine a DVA tuned on the first mode (which is often dominant in the earthquake response) with an actuator (as in the AMD) in an attempt to combine the advantages of the DVA and the AMD. In this way, one can expect to reduce the control effort as compared to the AMD (thanks to the DVA) while improving the performance and extending the bandwidth to higher modes thanks to the active control. However, as always, there is a price to pay: The active control tends to destabilize the lower pole of the dynamic absorber which requires a different tuning of the passive part, and also reduces the stability margin as compared to the active control case. In the second part of the discussion, the control element acts between the ground and the first floor. Two types of control configurations are considered: (i) A force actuator (e.g. hydraulic jack) combined with a relative displacement (or velocity) sensor. This configuration is examined with various compensators: lead, direct velocity feedback (DVF), and firstorder positive position feedback (PPF). (ii) A displacement actuator combined with a collocated force sensor. In this case, the control law is the integral force feedback (IFF). Once again, the various control strategies are compared from the point of view of performance and control effort. The direct calculation of the open-loop pole/zero pattern from classical modal analyses with appropriate boundary conditions is emphasized, and the duality between the DVF and the IFF is highlighted.

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119

3.2 System Description The example used in this comparison is a seven-story plane shear frame excited horizontally by a seismic disturbance (Figure 3.1). The mass and stiffness matrices are respectively M = mI7 and ⎛

2 ⎜ −1 ⎜ ⎜ 0 K =k⎜ ⎜ ⎜ ⎝ 0 0

⎞ −1 0 ... 0 2 −1 . . . 0⎟ ⎟ −1 2 ... 0⎟ ⎟. ⎟ ... ⎟ 0 −1 2 −1 ⎠ ... 0 −1 1

(3.1)

The natural frequency of mode l is given by (e.g. Géradin and Rixen, 1997)

π (2l − 1) k sin , ωl = 2 m 2 (2n + 1)

l = 1, . . . , n,

(3.2)

where n is the number of stories. In the numerical example discussed below, the shear frame has n = 7 floors; m = 10 kg, k = 50 × 106 N/m, and a uniform passive damping of ξi = 0.01 is assumed on all modes. The

Figure 3.1 Seven-story shear frame

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2 2 2

2

1 1

1

1

1

1

1

1

(a)

(b)

Figure 3.2 Notation for the DVA

dynamic response of a structure under single-axis seismic excitation was analyzed in Chapter 1. We will first consider the passive control with a DVA located on the top of the structure. Then, we will examine various active control options and compare them.

3.3 The Dynamic Vibration Absorber We begin our discussion with the dynamic vibration absorber or tuned mass damper,2 which provides a simple and robust solution to the problem when the structural response is dominated by a single mode with a frequency which does not vary significantly with time. The DVA is attributed to Den Hartog (Ormondroyd and Den Hartog, 1928; Den Hartog, 1956). 3.3.1 Single-d.o.f. Oscillator Consider the single-d.o.f. oscillator of Figure 3.2(a), excited by the external force f . The dynamic compliance of the system has the well-known single peak shape of Figure 3.3 (solid line); the dynamic amplification at resonance is 1/2ξ1 . The DVA aims to reduce the maximum amplification by the addition of a small size secondary oscillator (its mass m2 is only a fraction of that of the original system) with appropriate properties, transforming the dynamic compliance curve as indicated in Figure 3.3 2 Various practical realizations of tuned mass dampers and hybrid mass dampers will be discussed in later chapters of this book.

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121

x1 f 2

1 1 k1

With DVA

1 k1

1

Figure 3.3 Dynamic amplification of the single-d.o.f. oscillator and with the addition of a DVA

(dotted line). To analyse this problem, it is customary to introduce the following notation: ω1 = k1 /m1 , ω2 = k2 /m2 , (3.3) c1 c2 , ξ2 = , (3.4) ξ1 = 2m1 ω1 2m2 ω2 μ=

m2 , m1

ν=

ω2 . ω1

(3.5)

The mass ratio μ is fixed by the designer on practical considerations – typically, m2 is a few percent of m1 , and the efficacy of the DVA increases with μ. For a given value of the mass ratio μ and the frequency ratio ν, the dynamic compliance, x1 / f behaves as shown in Figure 3.4 when changing the damping ratio ξ2 of the DVA; all the curves intersect at the two points P and Q. When ξ2 increases, the amplitude increases between P and Q while it tends to decrease outside. For a given mass ratio, the relative magnitude of P and Q depends on the frequency ratio ν. The equal peak

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x1 f

Without DVA

1

1

2 3

1 k1

2

Figure 3.4 Dynamic amplification of the oscillator equipped with a DVA, for various values of the damping ratio ξ2 of the DVA

design, introduced by Den Hartog, consists of selecting the frequency ratio in such a way that P and Q have equal amplitudes. The corresponding frequency ratio is ν=

1 . 1+μ

(3.6)

The next step in the design of the DVA is to minimize the maximum of the dynamic compliance |x1 / f | (sometimes referred to as H∞ optimization). This is achieved when those maxima coincide with P and Q; the corresponding value of the damping is opt ξ2

=

3μ . 8(1 + μ)

(3.7)

The DVA design proceeds as follows: (i) Select the mass ratio μ and compute m2 . (ii) Compute ω2 from the frequency ratio (3.6), and then k2 . (iii) Compute the optimum damping from (3.7), and then c 2 . The equal peak design described above is not the only design procedure. Alternative optimization strategies have been proposed. The H2 optimization (Crandall and Mark, 1963) assumes a white noise excitation f and

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minimizes the variance of the structural response, E[x12 ]. Another alternative consists of maximizing the attenuation rate of the transient vibration by forcing the two pairs of complex conjugate poles as far as possible to the left of the imaginary axis: min[max Re(si )]. A comprehensive review of this subject can be found in Asami et al. (2002). 3.3.2 Multiple-d.o.f. System The DVA may be applied to multiple-d.o.f. systems as well (Figure 3.5); however, one needs to estimate what mass should be considered as m1 in the foregoing design procedure. In fact, it depends on the targeted mode and on the location of the DVA within the system. Consider the dynamic of the multiple-d.o.f. system (defined by its mass and stiffness matrices M and K ) under the force f d coming from the DVA (assuming no damping for simplicity), M¨x + K x = d f d ,

(3.8)

where d is the input vector of the DVA, defining its location in the system. In modal coordinates, x = z, one finds that every mode is governed by μi z¨ i + μi ωi2 zi = φiT d f d = φi (d) f d . DAV tuned

ma

on mode 2

xa ca

ka f xd f

xd With DVA 1

2

Without DVA 3

Figure 3.5 Multiple-d.o.f. system equipped with a DVA

(3.9)

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In this equation, μi is the generalized mass of mode i, ωi its natural frequency and φi (d) = φiT d the modal amplitude at the DVA location. If the DVA is targeted at mode k, it is reasonable to assume that, for frequencies close to ωk , the response is dominated by mode k, and consequently that x ∼ φk zk . This amounts to assuming that the modal amplitude zk and the structural displacement at the DVA location are related by zk ∼

xd . φk (d)

(3.10)

Substituting this into the modal equation for the targeted mode k gives μk

x¨d xd + μk ωk2 = φk (d) f d φk (d) φk (d)

or alternatively μk μk x¨d + 2 ωk2 xd = f d . φk2 (d) φk (d)

(3.11)

Comparing this result with the same equation for a single-d.o.f. oscillator, one can see that the equivalent mass to take into account in the design is m1 =

μk φk2 (d)

(3.12)

(recall that μk is the generalized mass of mode k and φk (d) is the modal amplitude at the location of the DVA). Thus, the design of a DVA for a multiple-d.o.f. system is done as for a single-d.o.f. system of mass m1 given by equation (3.12) and of natural frequency equal to that of the targeted mode, ωk . Observe from the previous equation that a larger modal amplitude will tend to decrease m1 , that is, to increase the mass ratio for a given mass m2 of the DVA. Thus, in order to maximize its efficacy, the DVA should be located where the targeted mode has large modal amplitudes. 3.3.3 Shear Frame Example Consider a DVA attached to the top floor of the shear frame (Figure 3.6(a)); it is tuned to the first mode according to the equal peak design procedure described above. Two values of the DVA mass are considered: ma = 3% of the total mass (mT = 7m in this case) and ma = 1%. Figure 3.7 compares the

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(b)

(a)

Figure 3.6 Shear frame equipped with a vibration control device attached to the top floor: (a) dynamic vibration absorber; (b) active mass damper connected to an absolute velocity sensor (geophone)

transmissibility x¨n /x¨0 between the ground acceleration and the acceleration of the top floor. As expected, the DVA transforms the resonance peak of the first mode into a double peak (note that the amplitudes of the two peaks are not exactly the same because of the simplifying assumptions in the design of the dynamic absorber for the multiple-d.o.f. system), while

1

10

3

Figure 3.7 Transmissibility x¨n/x¨0 between the ground acceleration and the acceleration of the top floor (in decibels)

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2

1

Figure 3.8 Cumulative RMS shear force at the base, f0 = k(x1 − x0 ). A DVA with a mass ma of only 1% of the total mass reduces the RMS response by 50%

the higher modes are not affected by the DVA. The attenuation depends significantly on the absorber mass ma . The shear force at the base, f 0 = k(x1 − x0 ),

(3.13)

is analyzed in Figure 3.8; the figure shows σ f0 =

0

ω

|T f0 w | dν 2

1/2 (3.14)

(according to (1.8), this would represent the cumulative RMS response f 0 to a white-noise ground acceleration of unit spectral amplitude, 0 = 1). One observes that the first mode accounts for 80% of the shear force of the original system and, as a result, the DVA is quite successful in significantly reducing the total response: the RMS response is reduced by about 50% with a dynamic absorber of mass ma only 1% of the total mass. Recall that the structural damping is assumed to be ξi = 0.01 in all modes.

3.4 Active Mass Damper Next, consider the situation of Figure 3.6(b), where an AMD is substituted for the DVA on the top floor. The AMD consists of a spring-mass damper as the DVA, but a force actuator u is added in parallel to the spring. In

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this case, the spring-mass system is selected in such a way that the natural frequency of the AMD is much smaller than that of the first mode of the structure, ωa =

ka ω1 . ma

(3.15)

The damper c a is selected to introduce a significant amount of damping in the actuator system, in the range 0.5 < ξa =

ca < 0.7. 2ma ωa

(3.16)

Under these conditions, the AMD behaves as a near-perfect force generator for ω ≥ ω1 (Preumont, 2002). As far as the sensor is concerned, we assume that an absolute velocity sensor is available on the top floor, y = x˙ n ; this sensor may be a geophone, or an accelerometer combined with an integrator.3 Note that, although both the velocity sensor and the AMD are attached to the same floor, the control system is not collocated, because the system output y is the top floor velocity while the system input u consists of a pair of opposing forces acting respectively on the top floor and on the inertial mass ma (u and y are not dual in the sense of the theorem of virtual work; the dual sensor would be the relative velocity between the inertial mass and the top floor: x˙ a − x˙ n ). However, because the AMD has been designed according to (3.15), the open-loop FRF G = y/u exhibits alternating poles and zeros above ω1 (Figure 3.9). The open-loop pole/zero pattern is shown in Figure 3.10, together with the root locus corresponding to an absolute velocity feedback, u = −g y = −g x˙ n

(3.17)

(if an accelerometer is used, the feedback law includes an integrator, u = −(g/s) y). There is a string of alternating poles and zeros near the imaginary axis, at frequencies jω1 and above, corresponding to the shear 3

To keep things reasonably simple, we will ignore the dynamics of the velocity sensor, although it may be very important in some cases, particularly for a structure with a very low-frequency mode. This is because the sensor itself behaves like a high-pass filter and its corner frequency must be lower than the lowest natural frequency of the building.

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ο ο

ο

Figure 3.9 Open-loop FRF, G = y/u, between the velocity sensor at the top of the shear frame and the force u in the AMD acting on the top floor; there are alternating poles and zeros above ω1

Im(s)

Im(s)

Re(s)

Re(s)

(a)

(b)

Figure 3.10 (a) Pole/zero pattern of the AMD control system with velocity feedback (there are three zeros at the origin). The two poles more heavily damped, at low frequency, correspond to the actuator. The alternating poles and zeros near the imaginary axis correspond the the structure. (b) Detail near the origin

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frame itself.4 The two poles more heavily damped, at low frequency, correspond to the actuator. There are three zeros at the origin, two from the actuator and one from the velocity sensor. Overall, there is one more pole than there are zeros, leading to a single asymptote on the negative real axis. The root locus shows the trajectory of the closed-loop poles when the feedback gain g increases from 0 to ∞. For mode 2 and above, the locus consists of a stable loop going from the pole to the nearby zero. The detail of the root locus near the origin is shown in Figure 3.10(b); as the gain increases, mode 1 moves leftward, eventually reaching the real axis for high gains. At the same time, the actuator poles are moving on loops starting leftward and then rightward to the origin, eventually becoming unstable for large gains. The dots indicate the closed-loop poles for three gains leading respectively to ξ1 = 0.05, 0.1 and 0.15; the corresponding transmissibility curves x¨n /x¨0 between the ground acceleration and the acceleration of the top floor are shown in Figure 3.11(a). An important observation is that the AMD acts on all modes. The transmissibility of the shear force f 0 /x¨0 = T f0 w is shown in Figure 3.11(b); it exhibits alternating resonances and anti-resonances, because x¨0 and f 0 are dual quantities. The cumulative RMS shear force (3.14) is shown in Figure 3.11(c); it decreases when the control gain increases. However, in active control, one also has to consider the control effort which eventually fixes the size of the actuator (and often the investment as well as the running costs). This can be assessed from the cumulative RMS control effort, σu =

ω 0

|Tuw | dν 2

1/2 ,

(3.18)

which is shown in Figure 3.11(d). Comparing the curves for various gains, one can see that the control requirements increase rapidly with g. Contrary to the DVA whose performance depends on the mass ratio μ, and therefore on the absorber mass ma , if the spring-mass system of the AMD is selected according to (3.15), the pole/zero pattern and the control performance are, to a large extent, independent of ma . However, in order to achieve the same control effort u, the actuator stroke (xa − xn ) will be larger for a light actuator (for a given ωa , the stroke is proportional to 1/ma ). 4

The open-loop poles depend on the system dynamics only, while the open-loop zeros depend on the actuator and sensor location.

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Figure 3.11 Response of the structure equipped with an AMD for various values of the control gain: (a) Transmissibility x¨n/x¨0 between the ground acceleration and the acceleration of the top floor. (b) Transmissibility of the shear force, f0 /x¨0 = Tf0 w . (c) Cumulative RMS shear force f0 . (d) Cumulative RMS control effort u

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3.5 Hybrid Control The comparison between the DVA and the AMD may be summarized as follows: on the one hand, the DVA is purely passive, but it operates only on the targeted mode on which it is tuned, leaving the other modes unaffected; on the other hand, the AMD is fully active and operates on all controllable modes, but it requires an adequate sensor and an actuator of appropriate size and a corner frequency satisfying (3.15). The central idea in developing an hybrid control strategy is this: Can we get the best of both worlds, and use a DVA tuned on the first mode (ωa ∼ ω1 instead of ωa ω1 ) as actuator for active control according to the principle of Figure 3.6(b)? Is this useful in order to increase the performance and/or to decrease the control effort? In this case, by (3.6), the actuator is designed in such a way that ωa =

ω1 ka . ma 1+μ

(3.19)

The impact of this change on the pole/zero pattern and the root locus in the vicinity of the origin is shown in Figure 3.12 (the full root locus is shown in Figure 3.10). Figure 3.12(a) reproduces the pole/zero pattern of the AMD when condition (3.15) is fulfilled. The tuning of the actuator in the vicinity of the first mode according to (3.19) results in a pair of neighboring poles as indicated in Figure 3.12(b). The root locus corresponding to the velocity feedback (3.17) is also indicated in the figure; there are three zeros at the origin. The control stabilizes one of the two pairs of poles (that with the higher frequency) while it tends to destabilize the other one (with the lower frequency). This does not sound like a good idea, because the passive design (DVA) was done to achieve similar stability margins in these two poles. However, the situation may be improved if one modifies the tuning of the actuator slightly, in order to increase the initial stability margin of the lower poles (at the expense of lowering the stability margin of the upper one), as indicated in Figure 3.12(c). In this example, this has been achieved by reducing ka by half with respect to the optimum DVA, and multiplying c a by 1.8 to move the open-loop pole leftwards.5 With this pole/zero configuration, the root locus is also shown in the figure and one can operate the controller with moderate gains to produce closed-loop poles with more stability (more to the left) than those of the 5

This is just an example; in practice, some trial and error may be necessary to achieve the appropriate closed-loop behavior.

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A Comparison of Passive, Active and Hybrid Control Im(s)

Re(s)

Im(s)

Re(s)

Im(s)

Re(s)

Figure 3.12 (a) Detail of the root locus near the origin for ωa ω1 . (b) For ωa = ω1 /(1 + μ). (c) After changing the tuning of ωa slightly to increase the stability margin of the lower pole

DVA alone (open-loop poles in Figure 3.12(b)). Besides, the closed-loop poles corresponding to higher-frequency modes (ωi > ω1 ) travel on nice stabilizing loops as in the case of a pure AMD actuator (Figure 3.10), as a result of the interlacing property of poles and zeros. Figure 3.13(a) compares the transmissibility curves x¨n /x¨0 between the ground acceleration and the acceleration of the top floor for the DVA and the hybrid control. One can see that the magnitude near mode 1 is lower with the hybrid controller than with the DVA alone; on the other hand, the hybrid control provides active damping for the higher modes as well, unlike the DVA. Figure 3.13(b) displays the cumulative RMS value of the top acceleration, σx¨n , defined in a way similar to (3.14). Figure 3.14 compares the hybrid control with the purely active control (AMD); in both cases, the gains have been selected to achieve the same magnitude of the transmissibility for mode 1 (Figure 3.14(a)). One observes that the hybrid control has slightly inferior performance on the higher-frequency modes. The cumulative RMS shear force f 0 is shown in Figure 3.14(b) and the cumulative RMS control effort u is shown in

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(a)

(b)

Figure 3.13 (a) Transmissibility x¨n/x¨0 between the ground acceleration and the acceleration of the top floor for the DVA and the hybrid control (in decibels). (b) Cumulative RMS of the top acceleration

Figure 3.14(c); the control effort required by the hybrid control is substantially lower than that of the AMD.

3.6 Shear Control For the remainder of this chapter, we replace the inertial device on the top floor by a device acting between the ground and the first floor (we call it a shear control for want of another name, because the control system tends to modify the shear behavior of the first floor). Two control configurations are considered: (i) a force actuator combined with a relative displacement (or velocity) sensor; and (ii) a displacement actuator combined with a collocated force sensor. These are classical, dual configurations which can

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(a)

(b)

(c)

Figure 3.14 (a) Transmissibility x¨n/x¨0 between the ground acceleration and the acceleration of the top floor for the DVA and the hybrid control (decibels). (b) Cumulative RMS of the shear force f0 . (c) Cumulative RMS control effort u

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be realized with various actuation technologies ranging from electromagnetic to hydraulic for (i), and ball screw or piezoelectric for (ii), depending on the application. The difference between the two architectures stems from the fact that the force actuator brings no stiffness in open loop while the displacement actuator brings the extra stiffness K a of the actuator on the first floor.

3.7 Force Actuator, Displacement Sensor The first control configuration consists of a force actuator attached between the ground and the first floor, with a collocated displacement (or velocity) sensor measuring the relative motion y = x1 − x0 (Figure 3.15(a)). The open-loop poles are the natural frequencies of the system when the actuator is removed. In the case of the shear frame example, pi = ωi = 2

(a)

π 2i − 1 k sin m 2 2n + 1

(i = 1, . . . , n).

(3.20)

(b)

Figure 3.15 (a) Shear frame with a force actuator u and collocated relative displacement sensor y = x1 − x0 . (b) Configuration corresponding to the transmission zeros zi

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According to Chapter 1, the transmission zeros are the natural frequencies of the structure when the sensor output is zero, that is, when the relative motion between the ground and the first floor is blocked (Figure 3.15(b)); these are the natural frequencies of the shear frame with one story less:

π 2i − 1 k sin (i = 1, . . . , n − 1). (3.21) zi = 2 m 2 2n − 1 3.7.1 Direct Velocity Feedback For this actuator/sensor configuration, the most obvious control laws are the lead or the direct velocity feedback. They are very similar and only the DVF will be considered here: u = −gs (x1 − x0 ).

(3.22)

The pole/zero pattern and the root locus are shown in Figure 3.16; the string of interlacing poles and zeros near the imaginary axis are the open-loop features of the structure (a small uniform modal damping of ξi = 0.01 has been assumed), while the zero at the origin belongs to the DVF controller. There is only one pole more than there are zeros, and the root locus has a single asymptote on the negative real axis. The rest Im(s)

Re(s)

Figure 3.16 Shear control with force actuator. Root locus of the DVF, u = −gs (x1 − x0 )

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of the root locus consists of loops going from the open-loop poles to the nearby transmission zeros; the entire root locus is contained in the left half-plane, meaning an infinite gain margin. Note that the loops leave the open-loop poles orthogonal to the imaginary axis as expected for pure viscous damping. We will discuss the results a little later. Note also that the open-loop pole/zero pattern is obtain directly from modal analysis and that the entire root locus can be drawn from the open-loop pole/zero pattern and the control law. This makes the design straightforward. 3.7.2 First-Order Positive Position Feedback An alternative to the DVF is the first-order6 positive position feedback (Baz et al., 1992; Høgsberg and Krenk, 2006), which consists of u=

g (x1 − x0 ). 1 + τs

(3.23)

As its name makes clear, it is a positive feedback; it behaves statically like a negative stiffness of constant g. The root locus is shown in Figure 3.17. The string of interlacing open-loop poles and zeros near the imaginary axis belong to the structure and are the same as in the previous case (DVF), because the two controllers use the same actuator and sensor configuration. The pole on the real axis belongs to the controller; there are three poles more than zeros, leading to three asymptotes, two at ±120◦ and the positive real axis, which means that the controller is conditionally stable. The stability limit is reached when the pole traveling on the real axis reaches the origin. The stability condition is given by (1.100): gG(0) < 1,

(3.24)

where G(s) = y/u is the open-loop transfer function. Since G(0) is in fact the static compliance of the system seen from the actuator, the stability condition simply means that the negative stiffness of the controller should not exceed the static stiffness of the system; this would in fact lead to the static collapse of the first floor. Comparing the two root-locus plots of the DVF and PPF, one observes that the loop corresponding to the low-frequency mode is much wider for the PPF than for the DVF, which might suggest better performance; however, a closer examination reveals 6

PPF controllers were examined in Section 1.6.3; unlike the second-order PPF, it is a wide-band controller which does not need to be tuned on a targeted mode.

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Im(s)

Re(s)

Figure 3.17 Shear control with a force actuator. Root locus of the PPF, g u = 1+τ (x1 − x0 ) s

that only the initial part of the loop is available, because of the stability condition. Note also that in the PPF case, the loops do not leave the openloop poles orthogonal to the imaginary axis as in the DVF case (as a result of the negative stiffness which softens the system), which suggests that the control effort may be larger, as we will shortly see.7 3.7.3 Comparison of the DVF and the PPF A comparison of the DVF and the PPF has been conducted with the shear frame example of Figure 3.15 and is reported in Figure 3.18. The gains of the two controllers have been selected to produce similar performance on the first mode of the transmissibility x¨n /x¨0 between the ground acceleration and that of the top floor (Figure 3.18(a)). Observe that the DVF tends to bring more damping to the higher modes, and that the peaks in the transmissibility for the PPF occur at lower frequencies (softening). The RMS top acceleration is compared in Figure 3.18(b); the results are 7 There is an interesting analogy with the motion of a swing: everybody knows from personal experience that increasing or reducing the amplitude of a swing which vibrates at its natural frequency requires little effort; this amounts to moving the poles a little to the right or left, that is, changing the real part. On the other hand, moving the swing at a frequency different from its natural one requires a lot of effort.

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139

Figure 3.18 Comparison of the DVF and PPF for the shear frame example. (a) Transmissibility x¨n/x¨0 (in decibels); the gains have been selected to achieve similar performance for the first mode. (b) Cumulative RMS top acceleration x¨n. (c) Cumulative RMS shear force at the base f0 . (d) Cumulative RMS control effort u

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extremely close in this problem which is dominated by the first mode. Figure 3.18(c) compares the shear force, f 0 = k(x1 − x0 ). The results obtained with the DVF are substantially better than those of the PPF; this seems to be associated with the negative stiffness inherent to the PPF. This effect is even more apparent in Figure 3.18(d) which compares the control effort (3.18); the control effort of the PPF is more than doubled that of the DVF. This is not surprising, when you think about it, since part of the control is used to soften the structure rather than damping vibration. This will have a direct impact on the actuator design and cost.

3.8 Displacement Actuator, Force Sensor To conclude, consider an active strut consisting of a displacement actuator and a collocated force sensor (Figure 3.19); the control input is the unconstrained expansion u = δ and the actuator stiffness is K a , so that the constitutive equation of the actuator is = u + f /K a ,

(a)

(b)

(3.25)

(c)

Figure 3.19 (a) Shear frame with a displacement actuator u = δ and collocated force sensor y = f , and IFF controller. (b) Configuration corresponding to the open-loop poles pi ; K a is the axial stiffness of the actuator. (c) Configuration corresponding to the transmission zeros zi (actuator removed)

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where is the total expansion of the strut and f is positive in traction.8 This control system being collocated, it has interlacing open-loop poles and zeros. The open-loop poles are obtained by setting u = 0, which means that the actuator will appear as an additional static stiffness K a between the ground and the first floor (Figure 3.19(b)); defining the reduced stiffness of the actuator κa = K a /k, the stiffness matrix of the open-loop system is ⎛ ⎞ 0 ... 0 2 + κa −1 ⎜ −1 2 −1 . . . 0⎟ ⎜ ⎟ ⎜ ⎟ 0 −1 2 . . . 0 ⎟. K =k⎜ ⎜ ⎟ ... ⎜ ⎟ ⎝ 0 0 −1 2 −1 ⎠ 0 ... 0 −1 1

(3.26)

(3.27)

The natural frequencies of this system are the open-loop poles pi of the control system. The transmission zeros are obtained by canceling the sensor output, y = f , which amounts to removing the active strut (Figure 3.19(c)); the natural frequencies of this system are the transmission zeros zi of the control system. Comparing with the dual case discussed in the previous section (DVF), which involves a force actuator and a displacement sensor, we observe that the transmission zeros of this problem are the open-loop poles of the previous problem, given by (3.20), and that the open-loop poles of this problem would become asymptotically identical to the transmission zeros of the previous problem for an infinitely stiff control element, K a → ∞. The control law adopted in this case is the integral force feedback (Preumont et al., 1992) u=δ=

g f . s Ka

(3.28)

It is a positive feedback, based on the force f measured in the strut (K a in the denominator is for normalization purposes: f /K a is the elastic elongation of the strut). The open-loop pole/zero pattern is quite similar to that of the DVF, in reverse order: the string of transmission zeros are at lower frequencies than 8

This equation simply states that the total extension of the active strut is the sum of the unconstrained expansion and the elastic deformation.

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the open-loop poles. The controller consists of a pole at the origin, instead of a zero in the previous case. For well-separated modes, the maximum achievable modal damping is given by (1.126), ξimax =

pi − zi , 2zi

(3.29)

where pi is the open-loop pole and zi is the corresponding transmission zero. This formula may be used to calculate the minimum stiffness of the actuator to achieve a given value of the damping in the targeted mode (mode 1 in this case). With z1 = ω1 given by (3.2) and ξimax fixed by the designer, one gets the minimum value of pi which is the first natural frequency corresponding to the stiffness matrix (3.27); this allows K a to be computed. 3.8.1 Comparison of the IFF and the DVF A comparison of the IFF and the DVF has been carried out on the shear frame example. The active member stiffness has been selected according to κa = K a /k = 5. Figure 3.20 shows the root-locus of the IFF; the control Im(s)

Re(s)

Figure 3.20 Root locus of the IFF u = δ = (g/s)(f/K a ) (the scale on the real axis has been magnified as compared to Figures 3.16 and 3.17)

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Figure 3.21 Comparison of the IFF and the DVF for the shear frame example. (a) Transmissibility x¨n/x¨0 (in decibels); the gains have been selected to achieve similar performance in the first mode. (b) Cumulative RMS value of the top acceleration x¨n. (c) Cumulative RMS value of the force sensor output, y = f , of the IFF, and of the input force u of the DVF. (d) Cumulative MS of the power requirements

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gain has been selected to match the performance of the DVF on the first mode of the transmissibility x¨n /x¨0 . The results of the comparison between the IFF and the DVF are summarized in Figure 3.21. Figure 3.21(a) compares the transmissibility Tx¨n w with that in open loop; as expected, the peaks in the IFF curve have slightly higher frequencies, due to the stiffening from the active member, and the active damping of the higher modes is less pronounced than for the DVF, due to the nature of the control law (the control amplitude decreases with frequency). Figure 3.21(b) compares the cumulative RMS of the top acceleration. Figure 3.21(c) compares the cumulative RMS value of the force sensor output of the IFF, y = f , and the input force u of the DVF.9 Finally, Figure 3.21(d) compares the power requirements of the two control strate˙ for the DVF and E[ f δ˙ ] for the IFF. Taking into account the gies, E[u ] ˙ for the DVF and δ˙ = g f /K a for the IFF respectively, control laws, u = −g the MS power requirements are given by

∞ 2 2 ˙ ˙ = E[u ] = g E[ ] ∼ gω2 |Tw |2 dω, σ DV F 0

∞ g 2 g σ I2F F = E[ f δ˙ ] = E f ∼ |T f w |2 dω Ka K a 0

(3.30) (3.31)

(the seismic excitation is assumed to be white noise). Figure 3.21(d) compares the cumulative MS values for the two control laws; they are very close to each other, and mainly concentrated on the first mode.

References Asami, T., Nishihara, O., and Baz, A. (2002) Analytical solutions to H∞ and H2 optimization of dynamic vibration absorbers attached to damped linear systems. Transactions of the ASME, Journal of Vibration and Acoustics, 124, 284–95. Baz, A., Poh, S., and Fedor, J. (1992) Independent modal space control with positive position feedback. Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, 114(1), 96–103. Crandall, S.H., and Mark, W.D. (1963) Random Vibration in Mechanical Systems. Academic Press. Den Hartog, J.P. (1956) Mechanical Vibrations. McGraw-Hill. 9

Both represent the effort in the control element.

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Géradin, M., and Rixen, D. (1997) Mechanical Vibrations, 2nd edition. Wiley. Høgsberg, J.R., and Krenk, S. (2006) Linear control strategies for damping of flexible structures. Journal of Sound and Vibration, 293, 59–77. Ormondroyd, J., and Den Hartog, J.P. (1928) The theory of the damped vibration absorber. Transactions of the ASME, Journal of Applied Mechanics, 50(7). Preumont, A. (2002) Vibration Control of Active Structures: An Introduction, 2nd edition. Kluwer. Preumont, A., Dufour, J.P., Malekian, Ch. (1992) Active damping by local force feedback with piezoelectric actuators. AIAA Journal of Guidance, 15(2), 390–395.

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4 Vibration Control Methods and Devices 4.1 Introduction Vibration control prevents resonance and unstable vibration, as well as quickly suppressing transient vibration by complementing the shortage of internal damping from the outside or by internally generating a force that cancels external forces. As a conventional vibration control method, passive vibration control does not involve any external energy source. The first active vibration control device introduced in Japan was composed of a sensor and a controller in the form of a servodamper. The control method was proposed in 1970 to prevent the chattering vibration of machine tools (Tominari et al., 1970). While this method proved to have excellent vibration-controlling performance (Tanaka et al., 1976), it was not adopted in practical applications because it was not easily understood and lacked reliability. Active vibration control has recently become a key technology in maintaining and enhancing the capability, performance, and stability properties across a wide range of engineering fields (Seto, 1989, 1991). For example, with industrial robots, lighter weight, energy saving capabilities, higher speeds and greater accuracy are increasingly required. An obstacle to achieving such properties is the lack of availability of light and flexible arms, creating the need to devise a practical method for controlling vibration. Although lighter-weight bodies are needed to decrease consumption in automobiles, it becomes necessary to control the vibration of Active Control of Structures A. Preumont and K. Seto © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-03393-7

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Vibration Control Methods and Devices

these somewhat elastic bodies. Increasingly lighter and taller buildings have been constructed to meet demands for lower costs. Such buildings involve the issue of swaying caused by strong winds. Behind this recent research activity could be the fact that modern control theory has previously been considered hard to understand and thus not likely to be put to practical use, but has recently becomes applicable to control system design because of ongoing developments, advances in computer application techniques, and the appearance of effective software packages such as MATLAB. The dynamic behaviors of systems whose vibration has to be controlled have come to be analyzed in detail, experimentally as well as theoretically (Nagamatu, 1985), and the introduction of fast Fourier transform analyzers has enabled easier examination of vibration control results. These remarkable advances in hardware and software can be listed as follows: (i) the development of software for control systems design; (ii) the availability of high-speed computers and the introduction of digital signal processors; (iii) the development of new active vibration control devices and actuators; (iv) advances in computer-aided vibration analysis methods and experimental modal analyzers; and (v) the appearance of a new control systems design method. This chapter first classifies vibration control methods and controllers, especially active dynamic absorbers that are widely used in mechanical and civil engineering. Then it addresses the construction of active dynamic absorbers, especially for use in civil structures. Finally, it shows that the control system of the active dynamic absorber is designed effectively by employing a control theory such as linear quadratic (LQ) control theory. It is also demonstrated by the use of a simple mechanical vibration system that the LQ control theory is effective in controlling structural vibrations.

4.2 Classification of Vibration Control Methods The main purpose of vibration control is to stabilize an object, and high reliability is required in a vibration control device. Passive vibration control devices have traditionally been used, because they do not require an energy feed and therefore do not run the risk of generating unstable states. However, passive vibration control devices have no sensors and cannot respond to variations in the parameters of the object being controlled or the controlling device. This has resulted in the evolution of a new type of vibration control device, called an active vibration controller, which is equipped with sensors, an actuator, and a controller that requires power.

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149

Figure 4.1 Classification of vibration control methods

Thus, vibration controllers can broadly be classified into passive, semiactive, active, and hybrid types. Since action is followed by reaction, it is necessary to receive a reaction force in some way in order for the actuator to generate a vibration control force. Vibration control methods can also be classified (Seto, 1992) according to the way the reaction force is received: (1) methods using the reaction at a fixed point; (2) methods using the reaction of an auxiliary mass; (3) methods using the reaction of an auxiliary structure. Figure 4.1 gives a schematic overview of vibration control methods. The controlled objects are simplified into a single-d.o.f. system. Of the three methods, method (1) allows for the simplest construction of a control system provided that there is a fixed point near the actuator mounting point. For instance, active vibration isolation systems and active suspension systems for automobiles fall into this category. If there is a fixed point to mount the actuator this method is applicable. Precision

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vibration control techniques to isolate environmental vibrations based on this method will be applicable in many fields, particularly in processing facilities for extra-precision measurement using laser devices, active suspensions of vehicles, micro-gravity environments, etc. However, if a fixed point is not readily available for mounting the actuator then one of the other two methods will be needed to take the reaction force. Method (2) uses the reaction inertia force of an auxiliary mass to produce a vibration control force in an actuator. Several types of dynamic absorbers that use this method are in widespread use in the field of engineering, since it has the advantage of the actuator being mountable at any point. Method (2) is classified, as shown in Figure 4.1, into passive, semiactive, active and hybrid types. In the field of mechanical engineering, the passive type is generally called a dynamic absorber, and can achieve excellent controlling effects by keeping three elements – the mass, the spring and the damper – in optimum adjustment (Ormondroyd and Den Hartog, 1928; Seto and Takita, 1987). In the field of civil engineering, this is called a tuned mass damper. However, since the dynamic absorber does not have its own sensors, it cannot adapt to changes in the characteristics, if any, in the control object or in itself, and it often has difficulty effectively controlling the vibration. This problem is particularly serious where the mass ratio is small. There are two ways to solve this problem. One way is to change the characteristics of the spring or the damper of the passive dynamic absorber, while ensuring that the optimum adjustment condition is maintained. This method is called a semi-active dynamic absorber, although fundamentally it belongs in the passive type category. The other way is to use an active dynamic absorber with a controller, sensors and actuators. In the field of civil engineering, active and hybrid types are called active mass dampers (AMDs) and, hybrid mass dampers (HMDs), respectively. Method (3) generates a control force with an actuator mounted on a structure located parallel to the flexible structure, in order to control the vibration of that structure. An application of this system is shown in Figure 4.2. The control device can be easily assembled, because it only requires insertion of an actuator between the two structures. Figure 4.2(a) illustrates adjacent but independently constructed multi-story buildings. Although this design is architecturally preferable, it is structurally weak because of its susceptibility to strong winds. By placing an active vibration control device in the upper part of the two buildings as shown in Figure 4.2(b), vibrations generated by wind excitation, the Karman vortex excitation, are eliminated. Method (3) – which we call the connected control

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(a)

151

(b)

Figure 4.2 Plan of the method that uses the reaction of an auxiliary structure

method (CCM) – is also classified, as shown in Figure 4.1, into passive, semi-active, active and hybrid types (Mitsuda and Seto, 1992; Seto, 1996). An application of this method is shown later.

4.3 Construction of Active Dynamic Absorber The active dynamic absorbers proposed so far may be roughly classified into hydraulic and electromagnetic types, distinguished by the power source injected into the actuator. Based on the driving method of the auxiliary mass, linear drive and motor drive systems are available. In the former the auxiliary mass is directly moved, and in the latter it is indirectly moved by rotating the ball screw with a servomotor. Moreover, there are three ways to support the auxiliary mass – with a spring support, guide support and pendulum support – and these are shown schematically in Figure 4.3. Since this auxiliary mass is actively controlled, it is also known as an active mass. The active dynamic absorber is formed by substituting the damper of the dynamic absorber with an actuator that is controlled by a sensor and

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Figure 4.3 Active dynamic absorber with electromagnetic actuator

a controller. The hydraulic type active dynamic absorbers are applicable to vibration control of a control object that requires a powerful yet compact vibration controller. To eliminate self-exciting chatter vibration occurring in machine tools, the hydraulic type active dynamic absorber called the servodamper (Figure 4.4) was proposed in the 1960s (Tominari et al., 1970). A typical active dynamic absorber linearly driven by an electromagnetic force and its control system are shown in Figure 4.5. Based on the auxiliary mass driving method, linear drive and motor drive systems are available. The former method directly moves the auxiliary mass and the latter indirectly moves the auxiliary mass by rotating the ball screw with a servomotor (Yoshida et al., 1991). The authors have already proposed two types of spring-supported active dynamic absorbers linearly driven by electromagnetic force (Seto and Furuishi, 1991; Seto et al., 1999), but in the present study, which is general in nature, a standard actuator of active dynamic absorber as shown in Figure 4.5 was composed. It consists of an active mass possessing a ring-shaped magnetic circuit, auxiliary

Figure 4.4 Hydraulic type active dynamic absorber

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Figure 4.5 Electromagnetic type active dynamic absorber

springs to support it from both sides, a moving coil, a main body with a mounting for the control object, and a displacement sensor to detect the relative displacement between the main body and active mass. Since the active mass is supported by the linear guide surface, this active dynamic absorber is of guide-supported electromagnetic linear actuator type. An example of the spring support type of an active dynamic absorber is shown in Figure 4.6. In this example, an active mass is supported by a parallel plate spring and equipped with a linear actuator composed of a pair of permanent magnets and a moving coil. Strain gauges passed on the plate spring are used to measure the relative displacement between the control object and the active mass.

Figure 4.6 Parallel spring type active dynamic absorber

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4.4 Control Devices for Wind Excitation Control in Civil Structures Recently, a number of high-rise buildings and huge bridge towers equipped with AMDs or HMDs have been constructed in Japan (Spencer and Sain, 1997). Although active vibration control techniques have been actively sought as countermeasures against strong winds, some problems remain unresolved in respect of their reliability and energy savings. A practical solution to this problem is the hybrid vibration control method, which combines both passive and active vibration control mechanisms into a single control unit. For example, this hybrid method was employed for the Yokohama Landmark Tower (the 300 m high building shown in Figure 4.7) completed in 1993. An HMD arranges an actuator and a damper in parallel. Since the damper dissipates vibration energy, a certain degree of reliability is ensured, even though failure could be generated by an active system. Two recent applications are presented in the following paragraphs. Figure 4.8 shows an outline of the vibration control device (Abiru et al., 1992) incorporated in the Yokohama Landmark Tower. In this example, the

Figure 4.7 Landmark Tower equipped with two HMDs and an outside view

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155

Figure 4.8 Multi-step pendulum type HMD

supporting method of the auxiliary mass belongs to the pendulum support. The natural frequency of a three-step pendulum for compressing an installed space in the vertical direction is tuned to one of the towers. Figure 4.7 shows the Landmark Tower equipped with two HMDs and an outside view of the HMD realized by Mitsubishi Heavy Industries. Servomotors operating linearly and independently in two horizontal directions drive an auxiliary mass hung by the three-step pendulum. It is possible to tune the natural frequency of the pendulum to that of the building. An important feature of the arrangement is the compact multistep pendulum to tune the natural frequencies of the tower. The counterelectromotive force produced by the rotation of the motor serves as a damper. Figure 4.9 illustrates a system with an active mass, which is formed by cutting the arc of a circle and driven by a servomotor (Tanida et al., 1991). Because the motion of an arc of a circle has constant period, the natural frequency of the mass can be set to any value through proper selection of the radius L. The application of damping in the direction of motion forms a hybrid vibration control device that consists of motor drives and damping forces. The counter-electromotive force of the motor serves as a damper. In addition, passive operation is provided in the event of a power failure. Figure 4.10 shows an improved HMD called a V-shaped HMD developed by Ishikawajima Heavy Industries, because the radius L is selected by changing the angle of the V-shaped rail. Tokyo Park Tower is equipped with three HMDs of this type (Koike et al., 1998).

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Figure 4.9 Roller guide type HMD

4.5 Real Towers Using the Connected Control Method An outline of the Triple Towers located at Harumi Triton Square in dwntown Tokyo is sketched in Figure 4.11 (Asano et al., 2002; Seto and Matsumoto, 2003). The Triple Towers are denoted as X, Y and Z, in order

Figure 4.10 V-shaped HMD and an outside view

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Figure 4.11 Sketch of the Triple Towers located at Harumi Triton Square in downtown Tokyo

of their height, and are arranged at 45◦ angles to Tower Y. Towers X, Y and Z are 195 m, 175 m and 155 m in height, with 48, 43 and 38 stories, respectively. At the design stage, it became clear that a complex vortex excitation from wind was of concern for the first bending mode of the Triple Towers.

Figure 4.12 Outline of the active controlled bridge used in Harumi Triton Square

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Figure 4.13 Schematic view of active dynamic absorber

To control the first bending mode of each tower, it would be necessary to use two AMDs, for each of the two directions; therefore a total of six AMDs would be required if the conventional method was employed. Using our proposed CCM, the minimum requirements are two controlled bridges (devices). The company made a decision to use the CCM and the two controlled bridges. Tower X is connected to Tower Y by one of the controlled bridges at the 39th story. Similarly, Tower Z is connected to Tower Y at the 33rd story. Each tower gains its control force independently and in two directions (x and y). The effectiveness of the CCM for controlling wind-excited vibration of the triple towers will be demonstrated in Chapter 6.

4.6 Application of Active Dynamic Absorber for Controlling Vibration of Single-d.o.f. Systems1 Figure 4.13 shows the structure of an active dynamic absorber with mass m and spring constant k mounted on a control object modeled by a singled.o.f. system with mass M, damping factor C, and spring constant K . This absorber comprises an electromagnetic actuator, two sensors and a controller, as shown in Figure 4.5. The electromagnetic actuator has inductance of moving coil L and resistance Rc and generates a control force f c . The absolute displacement X of the control object and relative displacement xd between an active mass and the control object are ˙ x˙ d are detected by two displacement sensors, and velocity signals X, 1

This section is based on Seto and Furishi (1991).

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produced] by differentiating these displacements. A current-detecting resistance detects also the control current I . These five signals are multiplied by a feedback gain to produce the control value u of a state feedback. 4.6.1 Equations of Motion and State Equation In Figure 4.13, for relative displacement xd = x − X, the equations of motion and the driving current formula at each mass point are expressed as K k Kv Kc C X˙ − X + xd + I+ v, M M M M M K m k 1 C 1 K ˙ X+ X− 1+ xd − − K c I − v, x¨d = M M M m M m M Rc 1 Kb I + u, I˙ = − xd − M L L ¨ =− X

(4.1) (4.2) (4.3)

where K c is the force coefficient, K b is the counter-electromotive coefficient, e b denotes counter-electromotive force, and u and v are respectively the control value and the disturbance force defined by u = ec , fv . v= Kv

(4.4) (4.5)

Then the state vector is given by X = { X˙

x˙ d

X

xd

I } = {X1

X2

X3

X4

X5 }.

(4.6)

Using this state vector, equations (4.1)–(4.3) can be rewritten as the state equation ˙ = AX + bu + dv, X where ⎡

−C/M ⎢ ⎢ C/M ⎢ A= ⎢ ⎢ 1 ⎢ ⎣ 0 0

0 0 0 1 K c /L

(4.7)

⎤ −K /M k/M K c /M ⎥ K /M −(1 + m/M)(k/M) −(1/M + 1/m)K c ⎥ ⎥ ⎥, 0 0 0 ⎥ ⎥ 0 0 0 ⎦ 0 0 1/T (4.8)

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b = [0 0 0 0 1/L]T , d = [K v /M −K v /M 0 0

(4.9) (4.10)

T

0] ,

and the time constant T = L/Rc . The output equation is given by y = cX,

(4.11)

where c = [0

0

1

0

0].

Accordingly, in this active dynamic absorber, the state feedback u = −KX

(4.12)

is applied. Here, the state feedback gain vector K is defined by K = K s [K 1

K2

K3

K4

K 5 ],

(4.13)

where K s is the conversion coefficient of displacement to voltage. As is well known, the value of vector K is determined by employing the LQ control theory, in order to obtain a control system with asymptotic stability. 4.6.2 Representation of a Non-dimensional State Equation However, if the control system is designed directly in the state equation (4.7), the general properties for control system design are not well known. To solve this problem, the state equation is made dimensionless with the following procedure. First, the following non-dimensional quantities are introduced: √

μ = m/M, γ = ωn /n , ζ = C/ 2 MK τ = (1/T) /n , δ = K c K b τ/ (RMn ) ,

(4.14)

√ √ where ωn = k /m, n = K / M are natural frequencies of the dynamic absorber and control object, respectively. Then the following scaled

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conversion factors are introduced: ¨ X˙ = X, ˙ X = X, ¨ = 2n X, X n x¨d = 2n x¨ d , x˙ d = n x˙ d , xd = x d , Rc M 2 M 3˙ M 2 M (4.15) n u, I˙ = n I , I = n I , v = 2n v, Kc Kc Kc K Rc M Rc M 2 Rc n K i (i = 1, 2) , K j = n K j ( j = 3, 4) , K 5 = K 5. Ki = Kc Ks Kc Ks Ks

u=

Using these scale factors, the following non-dimensional state equation is obtained: ˙ = AX + bu + dv, X

(4.16)

where ⎡

−2ζ 0 −1 μγ 2 ⎢ 0 1 − (1 + μ) γ 2 ⎢ 2ζ ⎢ A= ⎢ 0 0 0 ⎢ 1 ⎢ 1 0 0 ⎣ 0 0 δ 0 0 T b= 0 0 0 0 τ , T d = 1 −1 0 0 0 .

⎤ 1 ⎥ − (1 + 1/μ) ⎥ ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ −τ

(4.17)

(4.18) (4.19)

The control value u and state feedback gain vector K are as follows: u = −K X, K = K1 K2

K3

K4

K5 .

(4.20) (4.21)

Meanwhile, one lowers the order of the system when the time constant can be ignored, and the matrix and vector are lowered in order as follows: X = X1

X2

X3

X4

T

,

(4.22)

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⎤ ⎡ −2ζ 0 −1 μγ 2 ⎢ 2ζ 0 1 − (1 + μ) γ 2 ⎥ ⎥ ⎢ A= ⎢ ⎥, ⎦ ⎣ 1 0 0 0 0 1 0 0 T b = 1 − (1 + 1/μ) 0 0 , T d = 1 −1 0 0 , K = K1 K2 K3 K4 .

(4.23)

(4.24) (4.25) (4.26)

4.6.3 Control System Design For the control system design, the LQ control theory is employed. In this case, the design parameters are the weighting matrix Q and the weighting factor r given in the quadratic performance index

∞

J =

XT QX + u2r dt,

(4.27)

0

According to the LQ control theory, the control value u for minimizing this performance index is u = −r −1 bT PX = −KX,

(4.28)

where P is the solution to the Riccati equation PA + AT P − Pbr −1 bT P + Q = 0.

(4.29)

A program for solving the Riccati equation is available, and K can be determined automatically. Even if the control system is unstable, stabilization is guaranteed by this state feedback. In reality, however, the values of Q and r are specified by trial and error since the relation between these and the characteristics of the control system are unknown. Accordingly, in order to solve this problem, it is assumed that Q = diag q 1

q2

q3

q4

q5 ,

(4.30)

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where Q is a non-dimensional weighting matrix of Q. In the nondimensional state equation, all values of r and Q must also be dimensionless. Furthermore, assuming that the time constant T of the actuator is such that 1/T n ,

(4.31)

if the effect of T can be ignored, Q is defined as Q = diag q 1

q2

q3

q4

(4.32)

with the non-dimensional weighting factors q i , where q 1 refers to the velocity of the control object, q 2 to the relative velocity, q 3 to the displacement of the control object, q 4 to the relative displacement, and q 5 to the current of moving coil. 4.6.4 Similarity Law between Dimensional and Non-dimensional System Similarity law between feedback gain vectors According to the scale factors defined in (4.15), the relationship between the non-dimensional state feedback gain vector K and the dimensional gain vector K is given by K=

Rc M 2 K 1 /n Kc Ks n

K 2 /n

K3

K4 .

(4.33)

Therefore, K 1 , . . . , K 4 are determined by solving the non-dimensional Riccati equation and the control system can be designed. Similarity law between weighting factors The non-dimensional Riccati equation corresponding to (4.29) is T

T

P A + A P − Pbr −1 b P + Q = 0.

(4.34)

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If the state vector shown in (4.26) is partitioned into X = [ X11 X22 ]T , where X11 and X22 are the velocity and displacement vectors, respectively, and b = [ b 1 0 ]T , then the corresponding P and P are expressed as P= P=

P11

P12

P21

P22

P11 P21

(4.35)

,

P12 . P22

(4.36)

Therefore, the feedback gain vectors K and K are given by T T K = r −1 b P11 b P12 , K = r −1 bT P11 bT P12 .

(4.37) (4.38)

The relationship between (4.33) and the two equations above leads to P=

Rc M Kc Ks

2 P11 /n r r P21 /2n

P12 /2n . P22 /3n

(4.39)

By comparing the components of Q and Q, after substituting (4.39) into (4.34), the weight matrix Q=

Rc M Kc Ks

is obtained, where Q11 =

2 Q11 /2n r r Q21 /3n

q1 0

0 , q2

Q12 /3n Q22 /4n

Q22 =

q3 0

(4.40)

0 . q4

Therefore, the relation between the weighting matrix and weighting factor is Q = diag q 1 /2n q 2 /2n Kc Ks 2 r= r. Rc M2n

q3

q4 ,

(4.41) (4.42)

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Employing these relations, the result of analysis obtained in the nondimensional system may be directly applied to the dimensional system. 4.6.5 Analysis of Vibration Control Effect In this subsection, first of all, the relation between the characteristics of the control system and q 1 , q 2 , q 3 , q 4 and r is made clear when it is assumed that (4.39) holds and ignoring the effect of T. Then, the case where the effect of T cannot be ignored is discussed. Supposing that the damping of the control object can be ignored and that the mass ratio is not directly related to the vibration control performance, the following values were used in the analysis: ζ = 0, μ = 0.05, δ = 0, and γ = 1. Finally, the effect of τ is also discussed. Effects of q 1 , q 3 on vibrating control performance Root locus The dot-dashed line in Figure 4.14 indicates the root locus when the displacement of the control object is weighted by q 3 = 1, q 1 = q 2 = q 4 = 0, and the dashed line refers to the root locus when the velocity

Figure 4.14 Influence of weighting parameters on vibration control effect indicated by root-locus graph

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is weighted by q 1 = 1, q 2 = q 3 = q 4 = 0. The solid line shows the root locus when both velocity and displacement are weighted by q 1 = q 3 = 1 and q 2 = q 4 = 0. The non-dimensional weighting factor r is varied as a parameter in this figure. From this relation, the following facts are known: 1. The role of q 3 is to move the two roots near the imaginary axis closer to the origin and away from the imaginary axis, and to move the other root away from the origin toward the left in the S-plane. 2. The role of q 1 is to bring the two roots near the imaginary axis into the origin and away from the imaginary axis. Accordingly, the root locus closer to the origin is not moved as far away from the imaginary axis to get active damping. 3. When both q 1 and q 3 are used, the root locus closer to the origin becomes closer to the root locus of q 3 alone, while the root locus remote from the origin becomes closer to that of q 1 . In other words, q 3 largely influences the root locus closer to the origin, while q 1 has more influence on the root locus remote from the origin. Hence, by varying the ratio of q 1 and q 3 , a desired root locus may be obtained in a space enclosed by the root loci of q 1 alone and q 3 alone. As a result, active damping is adjusted by varying the ratio of q 1 and q 3 . Frequency response Figure 4.15 shows the non-dimensional frequency response of the control object when only the displacement of the control

Figure 4.15 Non-dimensional frequency response when only the displacement of the control object is weighted

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Figure 4.16 Non-dimensional frequency response when only the velocity of the control object is weighted

object is weighted by q 3 = 1, q 1 = q 2 = q 4 = 0, and the weighting factor r is changed. Figure 4.16 shows also a non-dimensional frequency response when only the velocity of the control object is weighted by q 1 = 1, q 2 = q 3 = q 4 = 0. As the weighting factor decreases, the resonance peak is suppressed, and the vibration control effect is enhanced, but the lowering of the resonance peak is more obvious in Figure 4.15. In this case, the two peaks tend to be separated on both sides, but they tend to converge into one in Figure 4.16. This means that the weighting velocity term of the control object is effective in suppressing the resonance peak of high frequency rather than one of low frequency, and conversely the weighting displacement term is useful for suppressing the resonance peak of low frequency. This information is useful for controlling multi-mode vibration. Figure 4.17 shows the impulse responses corresponding to Figures 4.15 and 4.16. In both cases, along with the decrease of r , the vibration control effect was improved, but the convergence was slightly faster in the former case. Understanding the correspondence of these responses will be very useful for the design of a control system. Effects of q 2 , q 4 on vibration control performance Root locus Next, the roles of q 2 and q 4 are discussed. Figure 4.18 shows the root loci when the value of r is kept constant and q 1 = q 3 = 1. When q 4 is

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Figure 4.17 Comparison of the vibration control effect by nondimensional impulse response when weighting parameters change

changed, the dot-dashed line indicates the root locus, and a thin solid line shows the root locus when q 2 is changed. The thick solid line corresponds to the root locus in Figure 4.14. From this diagram, the following facts are known: 1. As the value of q 2 is increased, the root locus remote from the origin is brought toward the negative real axis; the root locus closer to the origin is first brought to the negative real axis, and then moves away from the origin and toward the imaginary axis. Therefore, when an optimum

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Figure 4.18 Effects of q 2 and q 4 on vibration control

value is given for q 2 , the resonance peak is suppressed significantly to contribute to the enhancement of the vibration control effect. 2. If the value of q 4 is increased, the root locus closer to the origin moves away from the imaginary axis, but its effect is trifling as compared with that of q 2 . The root locus remote from the origin behaves like the root locus of q 3 in Figure 4.16. Therefore, q 4 does not contribute much to the vibration control effect. It is rather the role of q 4 to suppress the relative displacement xd . As q 4 is increased, the relative displacement decreases, and a proper value may be given to this when the relative displacement is defined due to structural restrictions or suchlike. If an excessive value is given, however, the vibration effect is lost. Frequency response Figure 4.19 indicates the frequency responses when the value of r = 0.1 is kept constant and q 2 is changed, with q 1 = q 3 = 1 and q 4 = 0. As the value of q 2 is increased, the resonance peak is increased, and this reduces the vibration control effect. Therefore, it is better to keep q 2 low, because the motion of an active mass must not be restricted if a large reaction force is desired.

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Figure 4.19 Frequency response for different values of the weighted relative velocity q 2

Figure 4.20 shows frequency responses when q 4 is changed in the conditions r = 0.1, q 1 = q 3 = 1 and q 2 = 0. If the value of q 4 increases, the resonance peak also increases. Therefore, q 4 does not greatly contribute to the vibration control effect; rather, it is the role of q 4 to suppress the relative displacement xd . As q 4 increases, the relative displacement decreases, and a proper value may be given to this when the relative displacement is

Figure 4.20 Frequency response for different values of the weighted relative velocity q 4

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defined due to structural restrictions or suchlike. If an excessive value is given, the vibration control effect is lost. Effects of dynamic characteristics of an actuator Root locus In the analysis so far, assuming equation (4.31), the time constant of the actuator has been ignored, but in fact it cannot always be ignored. For designing the control system with q 3 alone (q 3 = 1, q 1 = q 2 = q 4 = 0), the effects of the two roots of varying the value of the time constant ratio τ are shown by the root loci in Figure 4.21. As is clear from the figure, there is no problem when τ is large, but as its value decreases the root locus remote from the origin approaches the imaginary axis and ends up unstable. When the value of τ is at least 5, there is no particular problem in the control system design based on equation (4.34), but as the value is reduced, the stability of the control system in the high-frequency region is sacrificed. This tendency is more evident when the velocity is weighted.

Figure 4.21 Influence of time constant ratio τ on control stability

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Figure 4.22 Relation between frequency response and the value of τ

Frequency and time responses The effects on the frequency response of varying the value of the time constant ratio τ are shown in Figure 4.22. As is clear from the figure, there is no problem when τ is large, but as its value decreases the frequency response becomes unstable. When the value of τ is 5 or larger, there is no particular problem in the control system design based on (4.31), but as the value is reduced, the stability of the control system is sacrificed, as shown in Figure 4.23. This phenomenon is a kind

Figure 4.23 Relation between impulse response and the valueof τ

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of spillover instability caused by neglecting the actuator dynamics. This tendency is more evident when the velocity is weighted. To avoid this problem, therefore, the time constant of the actuator must be reduced, or all five state feedbacks, including the current feedback, become necessary and the control system must be designed according to equation (4.30). 4.6.6 Experiment Experimental apparatus In the experiment, a control object consisting of parallel plate springs with one end fixed and a mass attached to the other end (Figure 4.24) is used. The active dynamic absorber shown in Figure 4.5 is installed at the mass position. The displacement signal X of the control object is detected by a strain gauge attached to the plate spring, and an inductance conversion type displacement sensor is used to detect the relative displacement xd . The signals are converted into velocity signals through a differentiation circuit, and these four signals are processed by state feedback to obtain the control quantity u. The specifications of this experimental apparatus are as follows: M = 5.34 kg, K = 1.14 × 104 N/m, C = 0 Ns/m, m = 0.34 kg, k = 7.77 × 102 N/m, 1/T = 282 rad/s, R = 0 , K c = 1.22 N/V, and K s = 1 × 103 V/m.

Figure 4.24 Experimental apparatus

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Experimental results The following results were obtained from experiments conducted in order to verify the analysis results of the vibration control effect. According to the above specifications, the dimensionless quantities of the experimental apparatus are as follows: μ = 0.0637,

γ = 1.035,

τ = 6.13,

r = 0.011r .

Using equations (4.41) and (4.42), diemensional quantites are obtained as Q = diag 0.47 × 10−3 × q 1 r = R=

Kc Ks RM2n

2

0.47 × 10−3 × q 2

q3

q4 ,

R = 0.0114 × R.

These numerical values are nearly equal to the non-dimensional quantities used in the theoretical analysis, and the results of experiments obtained from this apparatus can be compared with the analytical results shown in Figures 4.15–4.17. Figure 4.25 shows the experimental results of the frequency response when the weighting factor was changed in three steps, r = R = 0.1, 0.01, 0.001, by applying an independent weight to the displacement of the control object assuming q 3 = 1. This corresponds to

Figure 4.25 Measured frequency response corresponding toFigure 4.15

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Figure 4.26 Measured impulse response corresponding to Figure 4.17(a)

the theoretical analysis shown in Figure 4.15. In this apparatus, since the active mass is supported on the linear guide plane, when r is large and the control quantity is small, the feature of the resonance peak differs slightly due to the effect of friction of the guide plane, but in other conditions a favorable coincidence is observed between the two cases. Similarly, Figure 4.26 is an impulse response result corresponding to Figure 4.17(a). A favorable correspondence is noted, and the vibration control effect shown in the theoretical analysis is known to produce similar results to the experiment. In particular, as the weight r is reduced, it is of note that the initial response is suppressed. In order to clearly demonstrate the control effect, a comparison between the periodic impulse responses of the control object obtained with and without vibration control is presented by Seto and Furuishi (1991). It was also confirmed that a factor of over 5 excellently attenuates the random response. This fact indicates that the dynamic absorber designed by the method discussed in this chapter is suitable for controlling the vibration of high-rise buildings against strong winds or earthquake ground motion and for suppressing the vibration of car bodies induced by a rough road.

4.7 Remarks In the past, vibration control was employed in post-design work and solely as a counter-measure. More recently, vibration control devices have been included in structural designs from the outset. The active control

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device built into the Yokohama Landmark Tower described in this chapter is an important example of this trend. With this concept of designing incorporated vibration control systems, active control can demonstrate its effectiveness; this is considered as a very favorable trend. This chapter has outlined in some detail the general trend toward active control of vibration. It has further considered some recently developed vibration control devices according to the selection of the control system employed. It showed that the control system of the active dynamic absorber was designed by employing the LQ control theory. The control system was represented non-dimensionally in order to lend the analysis results a degree of generality and obtain general reference materials for control system design. A similarity law was presented for mutual conversion of non-dimensional and dimensional systems. Consequently, the result of investigation in the non-dimensional system can be directly utilized in the design of an actual apparatus. In this chapter, although control objects with a single d.o.f. are used, actual structures are more complicated. Therefore, in order to adapt active control devices presented here for real (sophisticated and flexible) structures, a reduced-order physical modeling method is illustrated in the Chapter 5. Following on from this, active vibration control for real structures such as high-rise buildings and flexible bridge towers will be considered in Chapter 6.

References Abiru, H., Fujishiro, M., Matsumoto, T., Yamazaki, S., and Nagata, N. (1992) Tuned active mass damper installed in the Round-Mark Tower. In Proceedings of 1st International Conference on Motion and Vibration Control, pp. 110–152. Asano, M., Yamano, Y., Koike, Y., and Nakagawa, K. (2002) Development of active damping bridges and its application to triple high-rise buildings. In Proceedings of 6th International Conference on Motion and Vibration Control, pp. 7–12. Koike, Y., and Tanida, K. (1998) Application of V-shaped hybrid mass damper to high-rise buildings and verification of damper performance. In Proceedings of Structural Engineers World Congress, San Francisco (CDROM), T198-4. Mitsuta, S., and Seto, K. (1992) Active vibration control of structures arranged in parallel. In Proceedings of 1st International Conference on Motion and Vibration Control, pp. 146–151.

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Nagamatu, A. (1985) Mordal Analysis. Baifukan. Ormondroyd, J., and Den Hartog, J.P. (1928) The theory of the damped vibration absorber. Transactions of the ASME, Journal of Applied Mechanics, 50(7). Seto, K. (1989) Trends in vibration control. Journal of the Society of Instrument and Control Engineers, 28(8), 719–726. Seto, K. (1991) Active vibration control in machinery. Journal of the Acoustical Society of Japan, 12(6), 263–272. Seto, K. (1992) Trends on active vibration control in Japan. Proceedings of 1st International Conference on Motion and Vibration Control, pp. 1–11. Seto, K. (1996) Vibration control method for triple ultra tall buildings. In Proceedings of the First European Conference on Structural Control, pp. 535– 542. World Scientific. Seto, K., and Furuishi, Y. (1991) A study on active dynamic absorber. ASME Conference, DE-Vol. 38, pp. 263–270. Seto, K., and Matsumoto, Y. (2003) Vibration control of multiple connected buildings using active controlled bridges. In Proceedings of the 3rd World Conference on Structural Control, Vol. 3, pp. 253–261. Wiley. Seto, K., and Takita, Y. (1987) Vibration control in multi-degree-of-freedom system by dynamic absorber (Report l). In Proceedings of the 1987 ASME Design Technology Conferences, DE-Vol 4, pp. 169–174. Seto, K., Doi, F., and Ren, M. (1999) Vibration control of bridge towers using a lumped modeling approach. Transactions of ASME, Journal of Vibration and Acoustics, 121, 95–100. Spencer, B.F., and Sain, M.K. (1997) Controlling buildings: A new frontier of feedback. IEEE Control System, 17(6), 19–35. Tanaka, N., Miyashita, M., and Tominari, N. (1976) Output feedback control of servodamper system. Bulletin of the JSME, 19(137), 1278–1284. Tanida, K., Koike, Y., Mutaguchi, M., and Uno, N. (1991) Development of hybrid-type mass damper combining active-type with passive-type. Transactions of the JSME, 57-534C, pp. 485–490. Tominari, N., Seto, K., and Kamiyama, K. (1970) Study on a electrohydraulic servo damper. Preprint of JSME, 704-l I, pp. 23–26. Yoshida, K., Shimogou, T., Suzuki, T., Kageyarna, M., and Hashimoto, J. (1991) Active vibration control for high-rise buildings using a dynamic vibration absorber driven by servomotor. Transactions of the JSME C, 57(534), 472–477.

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5 Reduced-Order Model for Structural Control 5.1 Introduction Most structures are distributed in nature, and cannot be regarded as lumped parameter systems. To design a feedback controller for such a system using linear control theory, it is necessary to obtain a suitable reduced-order model by neglecting high-frequency modes. A feedback controller for this type of infinite-dimensional system is designed based on a reduced-order linear system model expressed by lumped parameter systems. In this context, one major problem is the suppression of spillover caused by the reduction of model order (Balas, 1978). To alleviate such spillover instability, it is necessary to take proper care during the design process. The modeling method presented in this text, called the reducedorder physical model method (also called Seto’s method; see Seto et al., 1998), makes a lumped parameter model of a distributed structure, and designs a controller in a physical state space where the states are directly measurable. It is possible to avoid the spillover by placing actuators or sensors in the node of some neglected higher modes. This prevents the excitation of those particularly neglected modes by means of a feedback control action. The method presented in this chapter involves establishing a lumped parameter model of the distributed structure and designing a controller in a physical state space, where the states are directly measurable. This method differs from the usually employed modal space control method Active Control of Structures A. Preumont and K. Seto © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-03393-7

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Reduced-Order Model for Structural Control

(Merovitch, 1990; Fuller et al., 1996) whereby modal state estimation is necessary, or the so-called direct output feedback control where the sensors are collocated with the actuators and a given actuator force is a function of the sensor output at the same point (Merovitch, 1990). It is not easy to estimate state quantities in modal space accurately if more controlled modes are involved, unless a large number of sensors or distributed sensors are used (Fuller et al., 1996: Bai and Shieh, 1985). The proposed method and skills are fairly easy to manipulate in practice, as verified by experiments, in controlling many bending and twisting modes of flexible structures. Based on this lumped parameter model, a feedback controller is easily designed based on convenient control theories such as LQ control and H-infinity based control law. In Section 5.2, the convectional modeling method of distributed structures is discussed. In Section 5.3, the spillover problem in active control and avoiding or suppressing spillover is described (Seto, 1992). In Section 5.4, the lumped modeling method is explained. Methods for giving physical significance in modal mass and estimating the equivalent masses needed (Seto et al., 1987) for controlling the vibration of a multi-d.o.f. system are explained in Section 5.5. In Sections 5.6–5.8, application examples for making the lumped mass model expressed by the two-d.o.f. model for controlling the first two bending vibration modes of a tower-like structure, the three-d.o.f. model for the first two bending and twisting modes of a plate structure (Kar et al., 2000), and the five-d.o.f. model for a bridge tower structure (Seto et al., 1997) are demonstrated.

5.2 Modeling of Distributed Structures 5.2.1 Equation of Motion for Distributed Structures In general, the physical properties of a structure such as the mass, damping and stiffness are distributed over the structure space. Let these quantities be represented by the functions M(x), C(x), K (x). The structure motion can be described by the following equation: ∂2 ∂ 3 y(x, t) ∂2 ∂ 2 y(x, t) ∂ 2 y(x, t) + 2 C(x) + 2 K (x) = f (x, t). M(x) ∂t 2 ∂x ∂t∂ x 2 ∂x ∂ x2 (5.1) Here, y(x, t) and f (x, t) are displacement and external force at position x, respectively. For example, for a beam with a uniform cross-section (area A,

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Modeling of Distributed Structures

181

Figure 5.1 Cantilever beam with a uniform section

density ρ, moment of inertia I , modulus E, and damping c), the equation of motion becomes Aρ

∂ 5 y(x, t) ∂ 4 y(x, t) ∂ 2 y(x, t) + cI + EI = f (x, t). ∂ x2 ∂t∂ x 4 ∂ x4

(5.2)

Assuming the time dependence of the displacement in the form of y(x, t) = Y(x)e jωt , the equation can be solved by using a variable separation method. For the cantilever case shown in Figure 5.1, the solution at the free end is y(t) =

1 sin βl cosh βl − cos βl sinh βl f (t), EI β cos βl cosh βl + 1

(5.3)

where β = Aρω2 /EI and the damping is ignored. As can be seen, the motion is expressed by super-functions consisting of an infinite order of series. In the control world, only finite orders of a system are of concern because of the limited of numbers of sensors, actuators and the capacity of controllers. In general, a finite-order control yields satisfactory results in most conditions. Therefore, it is necessary to set up a reduced low-order model for application of control theories. 5.2.2 Conventional Modeling of Structures Mode expansion method In vibration control, a structure’s low-order model generally refers to the low modes of the structure. Therefore, it is easier to handle a distributed structure in a modal domain rather than in a physical domain like that shown in equation (5.3). In a modal coordinate system δi (t), the lateral displacement y(x, t) can be expressed as a linear superposition of the

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modal shape functions φi (x), y(x, t) =

∞

δi (t)φi (x).

(5.4)

i=1

By substituting this equation into (5.2) and making use of the orthogonality, one can obtain a decoupled modal motion equation, for the ith mode: δï + 2ζi ωi δ˙i + ωi2 δ = φiT f (t).

(5.5)

Here, ωi , ζi are the natural frequency and damping factors of the ith mode, respectively. Each mode behaves as a one-d.o.f. system. A loworder model can be obtained simply by ignoring the high modes beyond our concern. A controller can then be designed on the basis of the several remaining modal motion equations which, in a control world, are often reformed in the state space. The justification for ‘throwing away’ the higher modes is as follows. The frequency response function of the structure, exciting at point q and monitoring the response at point r , is G(ω) =

∞ φri φq i x(r ) , = 2 2 F (q ) i=1 (ωi − ω ) i

(5.6)

which can be rewritten as G(ω) =

d ∞ φri φq i φri φq i x(r ) = + 2 2 2 2 F (q ) i=1 (ωi − ω ) i i=d+1 (ωi − ω ) i

(5.7)

= G low (ω) + G res (ω). φri , φq i are the mode shape components of the ith mode, ωi is the modal frequency and i is the modal mass. It is then clear that as far as the low frequency is concerned, say, the second term (uncontrolled modes or residual modes) is very small compared with the first term (control modes), therefore G c (ω) =

d φri φq i x(r ) = . 2 2 F (q ) i=1 (ωi − ω ) i

(5.8)

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Spillover

From the point of view of control implementation, a disadvantage of this method is that modal sensors or observers are required to transform between a physical domain and a modal domain. Under general conditions it is not as easy, as this cantilever example shows, to obtain an analytical solution of the modal parameters and shapes. Numerical or experimental methods are therefore required to perform the modal analysis. With the help of commercial software, the finite element method has now become a widely accepted tool for performing this task. Lumped mass model In some (albeit few) cases, the mass of a structure may be concentrated on some specific locations. Such a structure can be directly transferred to a lumped parameter model, whose motion is governed by M¨x + K x = f,

(5.9)

X˙ = AX + BU ,

(5.10)

or in state space

where A=

0 −M−1 K I 0

,B=

M−1 K c 0

x˙ ,X= x

and K c is a force transformation matrix. It is straightforward to design and implement a controller on such a system in a physical domain. The states can be directly measured and no modal sensors or suchlike are required.

5.3 Spillover A problem associated with the reduced-order modeling of a structure in active control is that of spillover. This is a phenomenon of interference between the controlled modes and the omitted, uncontrolled, higher modes. Figure 5.2 illustrates this relationship. The subscript C refers to the controlled modes and the subscript R refers to the uncontrolled residual modes. While injecting control energy into the controlled modes,

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Figure 5.2 Control and observation spillover

the actuator may also excite the higher modes that are not modeled for control. This is called the control spillover. Conversely, the sensors may not only pick up the information related to the controlled modes, but also that of the uncontrolled modes. The spillover thus caused is called the observation spillover. In Figure 5.3, an example of such a spillover is shown by an impulse response and a frequency response experimentally obtained on a flexible structure, where the first two modes are to be controlled and higher modes (including this mode) are not controlled. While the vibration from the first two modes decays quickly under the control, the vibration of the fourth mode grows with time, leading to a spillover. It goes without saying that spillover is a very dangerous phenomenon that may cause local damage or even destroy entire structures. Therefore, avoiding or suppressing spillover is a very important task for a successful control. There are various ways to prevent spillover from occurring. The four primary methods are modal filtering, direct feedback, low-pass filtering, and robust control. Modal filtering was initially proposed by Balas (1982) and is based on separating the information for the controlled modes from uncontrolled modes. This method requires the same number of sensors and actuators as that of the controlled modes. Direct feedback was also proposed by Balas (1979) and requires the placing of sensors and actuators at the same locations (collocation). While the stability of this control system is guaranteed, great control effects could not be expected. The use of low-pass filtering is designed to cut off either the pickup signals or the actuation forces in order to suppress the effects of higher uncontrolled

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185

Figure 5.3 Example of spillover

modes (Hori and Seto, 2000). It is simple but has often failed in controlling multiple modes because controlled higher modes are less suppressed by the low-pass filter. Robust control is fairly new and has seen rapid development in recent years. An improved form of these methods will be introduced and discussed in the next section.

5.4 The Lumped Modeling Method (Seto and Mitsuta, 1992) 5.4.1 A Key Idea for Deriving a Reduced-order Model For a lumped parameter system, it is straightforward to design a controller in a physical state space. Unfortunately, most structures are distributed in nature, and cannot be regarded as lumped parameter systems. A method

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is thus required which can model the main dynamic characteristics of a structure with a lumped parameter model, for the purpose of easy application of control theories. In this section, a method developed by the author is introduced which meets this requirement. The first step of this method is to determine the number and positions of modeling points. This is intimately related to the controllability and observability problem. It is well known that a controller design in state space is only meaningful when the system is observable and controllable. For a distributed system, it is not easy to judge this by any well-established algebraic method. The author has thus proposed a simple method to do so by examining the mode shapes of the structure (Seto, 1991). The cantilever beam shown in Figure 5.1 explains this. Figure 5.4 shows the first four modes of the beam and a graphical depiction of equation (5.8). In this figure, φq i and φri are the ith mode shape components at the position of control force (point q ) and sensor (point r ), respectively. If they are not zero, the system is controllable and observable; if zero, then uncontrollable and unobservable. Obviously, at nodes of a mode, these components are zero. In other words, if the control force is applied at a node of a mode, then the mode is uncontrollable; if a sensor is located at a node, the mode is then unobservable. In Figure 5.4, the first and second modes are controllable and observable, the third mode is observable but uncontrollable, and the fourth mode is controllable but unobservable.

Figure 5.4 Relation between controllability/observability and locations of sensors and actuators

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Therefore, if we want to control the first two modes, two modeling points are needed and can be placed at the nodes of the third mode. In this way, the spillover problem with this mode can be avoided in the modeling phase. The author termed this the ‘structure filter’. The key points of the modeling procedure are as follows: 1. Analyze the vibration mode shapes of the flexible structures to be controlled. 2. Select the nodes of the lowest order non-controlled vibration mode, and place the masses of the lumped parameter system at strategically selected nodes and make an r -d.o.f. system. 3. Determine the values of mass and spring constants using sensitivity analysis. The next section shows how to construct a lumped model with strategically placed masses at specified points on a distributed structure. 5.4.2 Relationship between Physical and Modal Coordinate Systems In physical coordinate systems, the equation of motion of an r -d.o.f. system with negligible damping can be expressed as MX¨ + K X = F,

(5.11)

where M and K are r × r symmetric mass and stiffness matrices, and X and F are displacement and force vectors, respectively. From the solution of the eigenvalue problem given by this equation of motion, the modal matrix is obtained. The relationship between the coordinate system of the physical and modal domains is given by X = δ

(5.12)

and the new equation of motion expressed by the modal domain is given by mδ¨ + kδ = f,

(5.13)

where m and k are r × r diagonal matrices representing the modal mass and stiffness. Similarly, δ and f are the displacement and force vectors.

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The relationship between the mass and stiffness matrices in the physical and modal domains is given by

T M = m,

T K = k.

(5.14) (5.15)

5.4.3 Modification of Normalized Modal Matrix It is reasonable to assign the normalized mode shape values of the real structure at the locations of modeling points, which can be obtained through numerical or experimental modal analysis, to that of the lumped system, that is, √ φri (i) = φri (rr )/ mri . Here, rr refers to the position of the r th modeling point on the structure and mri to the equivalent mass at the r th modeling point and the ith mode. Chapter 6 shows how to estimate the equivalent mass. A new modal matrix normalized by the equivalent mass is expressed as ⎡

√ φ11 /φr 1 / mr 1 √ ⎢ ..

= / m = ⎣ . √ φn1 /φr 1 / mr 1

⎤ √ · · · φ1n /φr n / mr n ⎥ .. .. ⎦. . . √ · · · φnn /φr n / mr n

(5.16)

The normalized modal matrix in (5.14) and (5.15) yields the following set of relationships for the mass M and stiffness K matrices of the physical domain: T −1 , M =

T −1 −1 K =

2 ,

(5.17) (5.18)

where 2 is a diagonal matrix whose non-zero diagonal elements are given by the square of natural frequencies of each mode. Equations (5.17) and (5.18) suggest that if the normalized modal matrix of a lumped parameter system could be obtained, the physical parameters of the system could then be determined. Unfortunately, the directly obtained modal matrix from the finite element model modal analysis data is not guaranteed to

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satisfy (5.17), where both sides should be a diagonal matrix. Since the mode vectors in are sampled from a larger modal matrix of a finite element model based on the structure or vibration mode shapes obtained by the experimental modal analysis, they do not necessarily satisfy (5.17) which is based on a much lower-order lumped parameter model. Substituting into (5.17) results in a matrix, T , whose off-diagonal positions usually contain (albeit small) non-zero values. A sensitivity analysis procedure is proposed here to modify the initial matrix in order to make

T diagonal. Suppose there are p different non-zero components in the off-diagonal positions of the matrix T . We define an error vector ε consisting of these p components, namely ε = ε1

ε2

ε3

...

εp

T

,

(5.19)

in which each error εi is a function of some modal shape components of matrix . Suppose there are, in total, q independent modal shape components involved in (5.19). Then it is reasonable to write εi = f ( φ1 ,

...

φj,

...

φq ) = f (φ),

(5.20)

where φ is a new q × 1 vector consisting of φ j ( j = 1, . . . , q ; here the subscript j is only a sequence number and has nothing to do with the mode order or the position label). The purpose is to modify these modal shape components φ j in order that the errors εi approach zero. It should ¯ if use be pointed out here that in constituting the initial modal matrix , has been made of the symmetry feature of the mode shapes, the number of p and q can be greatly reduced. Define the modification vector of φ as δφ, which is assumed to change the error vector from its initial value ε to 0. Then we can start the correction procedure from the equation

∂ε δφ = 0 − ε = −ε, ∂φ

(5.21)

where [∂ε/∂φ] is a p × q sensitivity matrix whose elements are partially in differentiating the errors, εi (i = 1, . . . , p), with respect to the mode shape

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components, φ j ( j = 1, . . . , q ), namely ⎡

∂ε1 ∂φ2 ∂ε2 ∂φ2 .. . ∂ε p ∂φ2

∂ε1 ⎢ ∂φ1 ⎢ ⎢ ⎢ ∂ε2 ∂ε ⎢ = ⎢ ∂φ1 ⎢ . ∂φ ⎢ .. ⎢ ⎣ ∂ε p ∂φ1

... ... ..

.

...

∂ε1 ∂φq ∂ε2 ∂φq .. . ∂ε p ∂φq

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(5.22)

The modification vector can be obtained directly from (5.21) using the generalized inverse matrix,

∂ε δφ = ∂φ

T

∂ε ∂φ

∂ε ∂φ

T −1 (−ε) ,

(5.23)

and the new vector is calculated by φ + δφ → φ.

(5.24)

Usually, an iterative calculation is necessary in this procedure in order that ε converges to 0. Once a predetermined accuracy has been reached, a final modified modal matrix can be obtained by replacing φ with

which will satisfy the diagonal condition of (5.17). Consequently, M and K can be obtained by introducing into (5.17) and (5.18).

5.5 Method of Equivalent Mass Estimation (Seto et al., 1987) 5.5.1 Meaning of Equivalent Mass Remarkable progress has recently been made in the field of experimental modal analysis. In order to solve vibration problems in mechanical structures, there is a tendency to rectify the cause using mathematical models derived from measured vibration data. In the experimental modal analysis method, a mathematical model is expressed as the sum of the transfer functions of one-d.o.f. systems. Each transfer function consists of modal mass, modal stiffness, modal damping, and natural mode. In order to express the transfer function of an N-d.o.f. system as the sum of the transfer

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Method of Equivalent Mass Estimation

functions of N one-d.o.f. systems, the determination of the modal specific values mentioned above is the key aim of modal analysis methods. Therefore, a transfer function from an experimental modal analysis method is a mathematical model used for explaining physical phenomena. It has been pointed out that the physical significance of the modal specific value is equivocal. This point raises a very important problem from the standpoint of carrying out vibration control in accordance with the results of modal analysis, and the design of a vibration control device cannot be undertaken without understanding the physically derived modal mass of N one-d.o.f. systems. Such a mass is defined as equivalent mass in this chapter. From this point of view, this section shows a method for giving physical significance to the modal mass, and also proposes two methods for estimating the equivalent masses needed for controlling the vibration of a multi-d.o.f. system. One of them is a method derived from the modal analysis method and the other a kind of sensitive analysis method called a mass response method which makes use of the fact that the smaller the equivalent mass at a specific point, the more sensitive the natural frequency becomes when an additional mass is attached at that point. 5.5.2 Eigenvector Method As is well known, in the modal analysis method, modal-specific values such as modal mass, modal stiffness and modal damping are calculated by using eigenvectors. However, since no physical significance is given to the factors of an eigenvector, it has hitherto been indicated that the physical significance of the modal-specific values is equivocal. Therefore, if the eigenvector should attain physical significance, the modal mass at any point on a certain vibration mode obtains a physical quantity. Relating to this, a method for estimating equivalent mass based on this idea is called an eigenvector method here. Let us now prove the theoretical basis of this idea. In the N-d.o.f. system shown in Figure 5.5, it is assumed that the eigenvector Xi of the ith vibration mode is given by Xi = x1

x2

···

xj

···

xN

T

.

(5.25)

The velocities (in meters per second) of each of the mass points in this mode are: ρx1 , ρx2 , . . . , ρx j , . . . , ρxN (where ρ is a constant). From the

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Figure 5.5 N-d.o.f. system moving in ith vibration mode

above, the total kinetic energy Tall of the system moving under this condition yields: Tall =

1 2 m1 ρ 2 x12 + m2 ρ 2 x22 + · · · + m j ρ 2 x 2j + · · · + m N ρ 2 xN . 2

(5.26)

On the other hand, since an N-d.o.f. system in modal coordinates can be disassembled into N single-d.o.f. systems, the N-d.o.f. system of Figure 5.5 can be replaced by N times the single-d.o.f. system shown in Figure 5.6. Then, in order to replace the jth mass point by a single-d.o.f. system, assuming that the jth point in Figure 5.6 is undergoing exactly the same motion as the jth mass point in Figure 5.5, the kinetic energy Tj =

1 Mj (ρx j )2 2

(5.27)

is exhibited, where Mj is the equivalent mass indicated at the jth mass point of the vibration mode under consideration. From the relationship Tall = Tj , the equivalent mass Mj is obtained as M j = m1

x1 xj

2

+ m2

x2 xj

2

+ · · · + mj

xj xj

2

+ · · · + mN

Figure 5.6 Motion at jth point in ith vibration mode

xN xj

2 (5.28)

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or, in matrix form, ⎧ ⎫T ⎡ x1 /x j ⎪ m1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ x /x ⎪ ⎪ 2 j ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎨ .. ⎪ ⎬ ⎢ ⎢ . ⎢ Mj = ⎢ 1 ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ . ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ ⎣ ⎪ ⎪ ⎩ ⎭ xN /x j

0

m2 ..

. mj

0

..

.

⎫ ⎤⎧ x1 /x j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ x2 /x j ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ .. ⎪ ⎬ ⎨ ⎥ . ⎥ ⎥⎪ 1 ⎪. ⎪ ⎥⎪ ⎪ ⎥⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎦⎪ . ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ mN xN /x j

(5.29)

Therefore, the equivalent mass at the jth point can be calculated by premultiplying and postmultiplying the eigenvector, normalized so that the element at the jth point is 1, by the mass matrix. This value is a physical quantity obtained from the equilibrium of kinetic energy. As a result, the modal mass obtained from the equation Mji = XTji MX ji ,

(5.30)

which multiplies the mass matrix M by the eigenvector X ji normalized by the jth component of the eigenvector, is proved to be an equivalent mass with physical significance at the jth point of the ith vibration mode. With equation (5.30), the equivalent mass can be estimated at any point of the natural mode. 5.5.3 Mass Response Method Mass response method using single-d.o.f. system When a vibration system turns into a discrete system, the equivalent mass can easily be detected by the foregoing eigenvector method. However, in order to estimate the equivalent mass under a continuous system, especially during experiments, this method is not suitable since there is no mass matrix. In this case, it is convenient to use the mass response method, which utilizes the fact that the natural frequency of a vibration system changes if an additional mass is placed on it. This method is called the ‘mass response method using a single-d.o.f. system’ because it uses the vibration model of a single-d.o.f. system for estimating the equivalent mass as shown in Figure 5.7. In this case, the equivalent mass Mji at the jth point on the ith vibration mode is expressed

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Δmji Mji

Mji

Kji

Kji Ωji=

Kji

Ωji=

Mji

Kji Mji+Δmji

Figure 5.7 Principle of mass response method using single-d.o.f. system

by the relation Mji =

m ji 2ji i2 − 2ji

(5.31)

where m ji is a known additional mass attached to the jth point on the ith vibration mode, i is the original natural frequency of the ith vibration mode, and ji is the natural frequency of the ith vibration mode after

m ji is attached to the jth point. When applying this method, care must be taken in the selection of the additional mass. That is, a smaller additional mass cause estimation errors, since the change in natural frequency is smaller. Conversely, when the additional mass becomes larger, coupling effects between closed natural modes cause errors of a different type. Therefore, one problem inherent in this method, which remains to be solved, concerns the point that there is no specific guide for the selection of the additional mass. In order to solve this problem, a technique for eliminating such errors is introduced. When three kinds of additional masses are used for estimating equivalent masses, three kinds of equivalent masses are obtained. By plotting additional masses on the horizontal axis and corresponded equivalent masses on the vertical axis, a real equivalent mass is obtained at the vertical axis, as an extrapolation of three points. Mass response method using multi-d.o.f. system The above-mentioned problem can be solved by means of a multid.o.f. system, taking into account the coupling effect. This method is

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provisionally called the ‘mass response method using a multi-d.o.f. system’ and is described in the following paragraphs. The equation for undamped motion of a multi-d.o.f. system due to an excitation force with a frequency ω is "

# K − ω2 M X = F,

(5.32)

where K and M are the stiffness and mass matrices and F and X are the vectors of excitation force and displacement, respectively. By solving the eigenvalue problem of (5.32), eigenvalues and eigenvectors are obtained. To begin with, the eigenvector is normalized so that the largest element at the maximum point is 1. On the assumption that this is an N-d.o.f. system, a modal matrix φ of order N × N, made by N eigenvectors, is formed. Then, provided that an additional mass m j is attached to the jth point of this vibration system, (5.32) can be expanded as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

⎡

⎜ ⎢ ⎜ ⎢ ⎜ ⎢ ⎢ M + K − ω2 ⎜ ⎜ ⎢ ⎪ ⎜ ⎢ ⎪ ⎪ ⎪ ⎝ ⎣ ⎪ ⎪ ⎩

⎤⎞⎫ ⎪ ⎪ ⎥⎟⎪ ⎪ ⎥⎟⎪ ⎪ ⎥⎟⎬ ⎥⎟ X = F . ⎥⎟⎪ ⎥⎟⎪ ⎪ ⎦⎠⎪ ⎪ ⎪ ⎭

0 ..

.

m j ..

.

(5.33)

0

By using the above modal matrix, the displacement vector X in physical coordinates is transformed into a displacement vector δ in modal coordinates: X = φδ.

(5.34)

Therefore, by substituting (5.34) into (5.33), and premultiplying the result by φT , the equation of motion can be written in terms of the coordinates δ as "

# P − ω2 m j Q δ = φT F,

(5.35)

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where ⎡" ⎢ ⎢ ⎢ P=⎢ ⎢ ⎢ ⎣

# 21 − ω2 M1

0 ..

.

"

2ι − ω

# 2

Mi ..

.

# 2N − ω2 MN

"

0 ⎡ ⎢ φ j1 φ j1 ⎢ ⎢ φ j2 φ j1 ⎢ . Q=⎢ ⎢ .. ⎢ ⎢ .. ⎣ . φ j N φ j1

··· .. .

··· φ ji φ ji

···

···

..

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(5.36)

⎤

φ j1 φ j N ⎥ .. ⎥ . ⎥ ⎥ .. ⎥ . . ⎥ ⎥ .. ⎥ .. . . ⎦ · · · φ j Nφ j N .

(5.37)

Hence, Mi is the equivalent mass at the maximum amplitude point of the ith vibration mode, ι is the ith-order natural frequency before the additional mass is attached, and φ ji is the element at the jth point of the ith vibration mode before the additional mass is attached. The equivalent masses of each vibration mode are obtained by solving the characteristic determinants * * det * P − ω2 m j Q* = 0

(5.38)

introduced from (5.35). Equation (5.38) can be rearranged as

A(ω)

N +

Mi +

⎧ ⎪ ⎪ ⎪ N ⎪ ⎨ r =1

i=1

⎪ ⎪ ⎪ ⎪ ⎩

Br (ω)

where A(ω) =

N + "

# i2 − ω2 ,

i=1

N + i=1 i=r

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ Mi

⎪ ⎪ ⎪ ⎪ ⎭

=0

(5.39)

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Modeling of Tower-like Structure N 2 + " 2 # Br (ω) = −ω2 m j φ jr i − ω2 . i=1 i=r

From (5.39), the relationship between the equivalent mass Mi at the maximum amplitude point of the ith vibration mode and A(ω), B(ω) can be arranged in matrix form as ⎡

B1 (ω1 ) ⎢ B1 (ω2 ) ⎢ ⎢ .. ⎢ . ⎢ ⎢ B1 (ωi ) ⎢ ⎢ .. ⎣ . B1 (ω N )

B2 (ω1 ) B2 (ω2 ) .. . B2 (ωi ) .. . B2 (ω N )

··· ···

··· ···

···

···

···

···

⎫ ⎧ ⎫ ⎤⎧ B N (ω1 ) ⎪ A(ω1 ) ⎪ 1/ M1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B N (ω2 ) ⎥ 1 M A(ω ) / ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ . . .. ⎬ ⎨ ⎨ ⎬ . . ⎥ . . . ⎥ = , B N (ωi ) ⎥ A(ωi ) ⎪ ⎪ ⎪ 1/ Mi ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ . ⎪ ⎪ .. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎦⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ ⎭ B N (ω N ) 1/ MN A(ω N ) (5.40)

where ω1 , . . . , ω N are the natural frequencies of order 1, . . . , N after the additional mass m j is attached. Data input to (5.40) are the additional mass m j , the original natural frequencies i and the various natural frequencies ωi (i = 1, . . . , N), and modal components φ ji at the jth point of each vibration mode. When these values are substituted into (5.40), the equivalent mass at the maximum amplitude point of each natural mode can be obtained simultaneously by an inverse matrix calculation. Examples of applications of these estimation methods are given in Seto, et al. (1987).

5.6 Modeling of Tower-like Structure (Seto and Mitsuta, 1992) 5.6.1 Two-d.o.f. Model Although tower-like structures are a typical distributed parameter system with an infinite number of vibration modes, let us consider the first two modes in terms of a two-d.o.f. model. This model is expressed by the equation M−1 =

1/ M1 0

2 2 0 + φ12 φ11 = 1/ M2 φ11 φ21 + φ12 φ22

φ11 φ21 + φ12 φ22 2 2 φ21 + φ22

(5.41)

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based on (5.16). In order to satisfy the condition φ11 φ21 + φ12 φ22 = 0,

(5.42)

an error function of the form φ11 φ21 + φ12 φ22 = ε1

(5.43)

is introduced. Partially differentiating the error function with respect to its elements yields the sensitivity matrix ,

∂ε ∂

-

, =

∂ε1 ∂φ11

∂ε1 ∂φ21

∂ε1 ∂φ12

∂ε1 ∂φ22

(5.44)

The error function in (5.43) thus becomes ,

⎧ ⎫ δφ11 ⎪ ⎪ ⎪ -⎪ ∂ε ⎨ δφ21 ⎬ = −ε1 , ⎪ δφ12 ⎪ ⎪ ∂ ⎪ ⎩ ⎭ δφ22

(5.45)

where δφ11 , δφ21 , δφ12 , and δφ22 are modifying factors for φ11 , φ21 , φ12 , and φ22 . Through modification of φ11 , φ21 , φ12 , and φ22 , a small value of ε1 which converges toward zero can be obtained. The generalized inverse matrix for obtaining convergence is given by ⎫ ⎧ δφ11 ⎪ ⎪ ⎪ ⎪ ⎬ , ∂ε -T , ∂ε - , ∂ε -T −1 ⎨ δφ21 = (−ε1 ) . δφ12 ⎪ ⎪ ∂

∂

∂

⎪ ⎪ ⎭ ⎩ δφ22

(5.46)

5.6.2 Dimension and Dynamic Characteristics of The Tower-like Structure Figure 5.8(a) shows the dimensions of the tower-like structure used in this study, and the virtual lines divided into 15 elements for analysis using the transfer matrix method. The structure is a tube-like flexible cantilever

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199

Figure 5.8 Vibration mode shapes and reduced-order two-d.o.f. model. (a) Tower-like structure. (b) Vibration mode shapes. (c) Lumped model

beam made of steel (1500 mm high, 50 mm × 50 mm outside cross-section, 2.3 mm thick), with a top box (2 kg in mass) prepared for mounting the actuator. The first three vibration mode shapes analyzed using the transfer matrix method are shown by a solid line in Figure 5.8(b). The natural frequencies at each mode lie at 11.7, 114.1, and 355 Hz. Table 5.1 lists the modal values normalized by the maximum valve from points 0 to 15 corresponding to the first three vibration mode shapes. This table shows that the nodes of the third-order mode are located near points 0 and 7. These two points were found to be mass points for the lumped parameter system. Let us now create a two-d.o.f. system at the strategically selected nodes. Using the transfer matrix method and mass response method with a single-d.o.f. system, the equivalent masses of the first two modes at each maximum amplitude point are found to be 5.835 kg at point 0 of the first mode, and 2.430 kg at point 7 of second mode. Modal elements of the first two modes normalized by the above equivalent masses are listed in Table 5.2, where natural frequencies of both modes are f n1 = 11.7[Hz] = 73.5[rad/s], f n2 = 14.1[Hz] = 716.6[rad/s].

2

3

4

5

6

7

8

9

10

11

12

200

1 2 3

14

15

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.414 0.373 0.333 0.293 0.255 0.217 0.182 0.148 0.117 0.088 0.063 0.041 0.024 0.011 0.003 0 0.155 −0.033 −0.211 −0.368 −0.495 −0.586 −0.635 −0.642 −0.607 −0.536 −0.430 −0.324 −0.208 −0.104 −0.029 0 −0.096 0.210 0.455 0.590 0.588 0.451 0.210 0.083 −0.362 −0.567 −0.659 −0.622 −0.476 −0.272 −0.084 0

0

Table 5.2 List of modal elements of the first two modes normalized by equivalent masses

mode

13

1.000 0.902 0.804 0.709 0.615 0.525 0.438 0.357 0.281 0.213 0.152 0.100 0.058 0.026 0.007 0 1.242 −0.052 −0.329 −0.574 −0.772 −0.914 −0.990 −1.000 −0.946 −0.836 −0.683 −0.505 −0.324 −0.161 −0.044 0 −0.146 0.318 0.691 0.896 0.893 0.684 0.318 −0.125 −0.549 −0.862 −1.000 −0.944 −0.722 −0.412 −0.126 0

1

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Table 5.1 List of the modal values normalized by maximum valve from points 0 to 15 corresponding to the first three vibration mode shapes

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5.6.3 Calculation of Parameters of Two-d.o.f. Model A temporarily selected modal matrix normalized by the equivalent mass, which corresponds to equation (5.16), and the diagonal matrix given by natural frequencies of each mode, are given by 0.414 0.155

= , 0.148 −0.642 73.5 0 = . 0 716.6 By introducing these matrices directly into (5.17) and (5.18), the mass and stiffness matrices are calculated as

5.206 0.461 M= , 0.461 2.348 1.62 × 105 −3.71 × 105 . K = −3.71 × 105 1.06 × 106 These results show clearly that the mass matrix does not satisfy the diagonal constraint condition. Therefore, it is necessary to modify the moda1 matrix using (5.23) with the above matrices as the initial condition. After iteration using a computation algorithm indicated in Table 5.3, the modified modal matrix is found to be

0.4216

= 0.1753

0.1165 , −0.6343

leading to

5.197 0 M= , 0 2.329 2.13 × 105 −4.51 × 105 . K = −4.51 × 105 1.11 × 106 Using these matrices the lumped mass model becomes a two-d.o.f. system as shown in Figure 5.8(c). In this model, the following mass and stiffness

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Table 5.3 Program for calculating parameters of two-d.o.f. model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 2DOF Modeling for Tower structure % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clg %%%%%%%%% load eigenvector and eigen value %%%%%%%%%%% ms=zeros(2,2); % Size of Matrix %------------ Setting natural frequency ------------------w1=2*pi*11.7; % Natural frequency of the first mode w2=2*pi*114.1; % Natural frequency of the second mode %------------ Setting initial modal matrix ------v11=1.000 ; % Value of mass point 1 at 1st mode v21=0.357 ; % Value of mass point 2 at 1st mode v12=0.242 ; % Value of mass point 1 at 2nd mode v22=-1.000; % Value of mass point 2 at 2nd mode %---------- Inputting equivalent mass ----------------m1=5.835; % Equivalent mass at mass point 1 of 1st mode m2=2.430; % Equivalent mass at mass point 2 of 2nd mode %%%%%%%%%%%%%% Normalizing by equivalent mass %%%%%%%%% s11=v11/sqrt(m1); % Normalized modal matrix s(1 1) s12=v12/sqrt(m2); % Normalized modal matrix s(1 2) s21=v21/sqrt(m1); % Normalized modal matrix s(2 1) s22=v22/sqrt(m2); % Normalized modal matrix s(2 2) ms(1,1)=s11; ms(1,2)=s12; ms(2,1)=s21; ms(2,2)=s22; ms;

% Normalized modal matrix

ww=[w1ˆ2 0;0 w2ˆ2]; % Square Matrix of natural frequency M2=inv(ms*ms’); K2=inv(ms*inv(ww)*ms’); omomi=[1 0 0 0 % Weighting matrix 0 1 0 0 0 0 1 0 0 0 0 1]; M2 % Initial mass matrix K2 % Initial stiffness matrix %%%%%%%%%%%%%%%%%%%%%%%%%% Modification process %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i=1:8 kando=[ms(2,1) ms(1,1) ms(2,2) ms(1,2)]; % Sensitivity matrix gosa=[ms(1,1)*ms(2,1)+ms(1,2)*ms(2,2)]; % Error function henkou=-omomi*kando’*inv(kando*omomi*kando’)*gosa; % Modification vector ms=[ms(1,1)+henkou(1,1) ms(1,2)+henkou(3,1);ms(2,1)+henkou(2,1) ms(2,2)+henkou(4,1)]; % Modified Modal matrix m2r=inv(ms*ms’); % Modified mass matrix k2r=inv(ms*inv(ww)*ms’); % Modified stiffness matrix end %%%%%%%%%%%%%%%%%%%%%%%%% Modified mass and stiffness matrixes %%%%%%%%%%%%%%% m2r k2r %%%%%%%%%%%%%%%%%%%%%%%%%End %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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Modeling of Plate Structures

matrices are obtained:

M1 0 M= 0 M2 K1 + K2 K = −K 2

−K 2 K2 + K3

By comparing the above two mass and stiffness matrices, the corresponding parameters are determined as M1 = 5.20[kg], M2 = 2.33[kg], K 1 = −2.38 × 105 [N/m], K 2 = 4.51 × 105 [N/m], K 3 = 6.63 × 105 [N/m]. Note that realizing the boundary condition of the free end at the top of a tower-like structure, K 1 takes a negative value. As is well known, motion equation solutions in such distributed structures include the shear force, bending moment, slope angle and deflection. However, the lumped mass model requires only two variables of force and displacement. Therefore, the negative value of spring constant acts as the bending moment. 5.6.4 Comparison between the Distributed Parameter and Two-d.o.f. Systems The vibration mode shapes are plotted as a dotted line in Figure 5.8(b) using the modified modal matrix. At points 0 and 7 the vibration mode shapes of the two-d.o.f. system agree well with those of the distributed parameter system (solid line). Figures 5.9 and 5.10 show the comparison of frequency response calculated at points 0 and 7 with the excitation point 0. The solid line represents the distributed parameter system and the dotted line the two-d.o.f. system. It shows that close agreement between the frequency responses of the distributed parameter system and the twod.o.f. system was obtained within the second-order resonance frequency and that the structure was modeled accurately by the two-d.o.f. system.

5.7 Modeling of Plate Structures (Kar et al., 2000) 5.7.1 Dimensions of a Plate Structure In this sections, the reduced-order model with three d.o.f. for a plate structure is derived. For this purpose, Figure 5.11 shows the schematic diagram

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Figure 5.9 Comparison of frequency responses calculated at the point 0 with the excitation point 0

Figure 5.10 Comparison of frequency responses calculated at point 7 with the excitation point

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205

Figure 5.11 Plate structure

of the plate structure, 1000 mm in height, 320 mm in width and 4 mm in thickness. Figure 5.12 shows the first to fourth vibration modes obtained by the finite element method. In this text, since the first three vibration modes are required to be controlled, it is necessary to place the three mass points at the nodes of the fourth mode and make a lumped mass model

Figure 5.12 Vibration mode shapes of plate structure and modeling points for three-d.o.f. model

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expressed in terms of a three-d.o.f. reduced-order model. This three-d.o.f. model can express the phenomena of the first and second bending and first twisting vibration modes. 5.7.2 Three-d.o.f. Model In order to construct the three-d.o.f. model, the normalized modal matrix

is expressed by the following equation based on (5.16): ⎡

φ11

= ⎣φ21 φ31

φ12 φ22 φ32

⎤ φ13 φ23 ⎦ . φ33

(5.47)

The normalized modal matrix in (5.47) yields the following set of relationships for the mass M: ⎡

M−1

⎤ 1/ M1 0 0 0 ⎦ 1/ M2 =⎣ 0 0 0 1/ M3 ⎡ 2 2 2 φ11 + φ12 + φ13 A 2 2 2 A φ21 + φ22 + φ23 =⎣ B C

⎤ B ⎦ C 2 2 2 φ31 + φ32 + φ33

(5.48)

where the off-diagonal elements are defined as: A = φ11 φ21 + φ12 φ22 + φ13 φ23 , B = φ11 φ31 + φ12 φ32 + φ13 φ33 , C = φ21 φ31 + φ22 φ32 + φ23 φ33 .

(5.49)

In order to satisfy the condition A = B = C = 0,

(5.50)

the error vector ⎫ ⎧ ⎫ ⎧ ⎨ε1 ⎬ ⎨φ11 φ21 + φ12 φ22 + φ13 φ23 ⎬ ε = ε2 = φ11 φ31 + φ12 φ32 + φ13 φ33 ⎭ ⎩ ⎭ ⎩ ε3 φ21 φ31 + φ22 φ32 + φ23 φ33

(5.51)

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is introduced. Partially differentiating the error function with respect to nine elements from φ11 to φ33 yields the sensitivity matrix given by

∂ε ∂ε = ∂φ11 ∂φ

∂ε ∂φ12

∂ε ∂φ13

∂ε ∂φ21

∂ε ∂φ22

∂ε ∂φ23

∂ε ∂φ31

∂ε ∂φ32

∂ε ∂φ33 (5.52)

The error function corresponding to (5.21) thus becomes

∂ε δφ11 δφ12 δφ13 δφ21 δφ22 δφ23 δφ31 δφ32 ∂φ

⎧ ⎫ ⎨ ε1 ⎬ T δφ33 = − ε2 . ⎩ ⎭ ε3 (5.53)

5.7.3 Calculation of Parameters of the Three-d.o.f. Model In this example, an initial normalized modal matrix constructed through vibration mode shapes shown in Figure 5.12 and equivalent masses obtained at each modeling point is given by ⎡

⎤ 0.3039 −0.4719 −0.0523 0.2083 −0.6602⎦ .

= ⎣0.5217 0.5210 0.1316 0.7529 Using the computation algorithm of Table 5.3, the modified modal matrix ⎡

0.2124 ⎣

= 0.5587 0.5658

−0.5025 0.2758 0.1926

⎤ −0.0359 −0.5543⎦ 0.6689

is obtained. Introducing the modified modal matrix into (5.17) and (5.18), the mass matrix M and stiffness matrix K are obtained. Using these matrices the lumped mass model becomes the three-d.o.f. model shown in Figure 5.13. Through the same procedure as that of the example of the two-d.o.f. model, three mass and six stiffness values are determined as shown in Figure 5.13.

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Figure 5.13 Three-d.o.f. model and model parameters of the plate structure

5.7.4 Comparison between Real System and Three-d.o.f. Systems It has been confirmed that close agreement between the frequency responses of the real system shown in Figure 5.11 and the three-d.o.f. system was obtained within the third-order resonance frequency as shown in Figure 5.14 and that the structure was modeled exactly by the three-d.o.f. system. In this calculation of the three-d.o.f. system, a proportional

Figure 5.14 Compared frequency responses

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damping matrix C = α M + β K as well as the modal analysis method were introduced.

5.8 Modeling of a Bridge Tower (Seto et al., 1997) 5.8.1 Dimensions of a Scaled Bridge Tower In the next example a slender but huge bridge tower is introduced, which becomes a controlled subject of many bending and twisting vibration modes. Under strong winds, a vortex excitation vibration may be generated in this bridge tower while under construction. To prevent such vibration, a multimode vibration control technique is required. If this technique is used efficiently, an attempt to drastically reduce the weight of the tower can be made. Figure 5.15 shows a scaled model of a huge bridge tower used for suspending bridges. Modal analysis of the structure is first carried out by a finite element method. Figure 5.16 illustrates the first five vibration modes of the model structure, which are very similar to the mode shapes of the real structure. The first four modes, including bending and twisting modes, are those requiring attention. Choosing four modeling points on the structure, and assuming proper lumped masses and stiffness at and between these points, allows a discrete model to be created. Consequently, the modeling points should be able to reflect the motion of

Figure 5.15 Dimensions of a scaled bridge tower

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Figure 5.16 Vibration mode shapes and their natural frequencies

these four modes as well as possible. In this case, the four modeling points are chosen to be located at the nodes of the fifth mode, as shown in Figure 5.16, with the advantage that the fifth mode will exert no influence on the model. It should be noted that choosing the modeling points is not based on the continuous structure coordinates, but on the finite element model of the structure; that is, only the nodes of the finite element model (here the nodes do not equate to places of zero vibration) are candidates for the modeling points. 5.8.2 Construction of a Four-d.o.f. Model The normalized modal vector components of the structure calculated by the finite element model at the modeling points are defined as the mode vector components of the lumped four-d.o.f. model, as denoted by ⎤ ⎡ φ11 φ12 φ13 φ14 ⎢φ21 φ22 φ23 φ24 ⎥ ⎥ (5.54)

=⎢ ⎣φ31 φ32 φ33 φ34 ⎦ . φ41 φ42 φ43 φ44 In the case of symmetric vibration mode shapes, modal matrix components are expressed as φ11 = φ21 , φ31 = φ41 , φ12 = −φ22 , φ32 = −φ42 , φ13 = φ23 , φ33 = φ43 , φ14 = −φ24 , φ34 = −φ44 .

(5.55)

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Using the above relationship, the normalized modal matrix is replaced by ⎡

φ11 ⎢φ11

=⎢ ⎣φ31 φ31

φ12 −φ12 φ32 −φ32

φ13 φ13 φ33 φ33

⎤ φ14 −φ14 ⎥ ⎥. φ34 ⎦ −φ34

(5.54)∗

Then, the set of relationships between the normalized modal matrix

and the mass M as shown in M−1 = T takes the form ⎡

M−1

1/ M1 ⎢ 0 =⎢ ⎣ 0 0

0 1/ M2 0 0

0 0 1/ M3 0

⎤ ⎡ 0

11 ⎢ 0 ⎥ ⎥=⎢ 0 ⎦ ⎣ 1/ M4

A

22 Sym

B D

33

⎤ C E ⎥ ⎥, F ⎦

44 (5.56)

where 2 2 2 2 − φ12 + φ13 − φ14 , A = φ11 B = E = φ11 φ31 + φ12 φ32 + φ13 φ33 + φ14 φ34 , C = D = φ11 φ31 − φ12 φ32 + φ13 φ33 − φ14 φ34 , 2 2 2 2 − φ32 + φ33 − φ34 . F = φ31

In order to also satisfy the diagonal matrix, the condition A= B = C = D = E = F = 0

(5.57)

is required. For this purpose, the error vector ⎫ ⎧ ⎫ ⎧ 2 2 2 2 − φ12 + φ13 − φ14 φ11 ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ ⎨ ε2 φ11 φ31 + φ12 φ32 + φ13 φ33 + φ14 φ34 = ε= + φ13 φ33 − φ14 φ34 ⎪ ⎪ ⎪ ⎪ ⎪φ11 φ31 −2 φ12 φ32 ⎪ ⎪ε3 ⎪ ⎭ ⎩ ⎭ ⎩ 2 2 2 ε4 φ31 − φ32 + φ33 − φ34

(5.58)

is introduced. Additionally, in this example partially differentiating the error function with respect to nine elements from φ11 to φ44 yields the

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sensitivity matrix ⎡

∂ε1 ∂φ12 ∂ε2 ∂φ12 ∂ε3 ∂φ12 ∂ε4 ∂φ12

∂ε1 ∂φ13 ∂ε2 ∂φ13 ∂ε3 ∂φ13 ∂ε4 ∂φ13

∂ε1 ∂φ14 ∂ε2 ∂φ14 ∂ε3 ∂φ14 ∂ε4 ∂φ14

−2φ12 φ32 −φ32 0

2φ13 φ33 φ33 0

−2φ14 φ34 −φ34 0

∂ε1 ⎢ ∂φ11 ⎢ ⎢ ∂ε ⎢ 2 ⎢ ⎢ ∂φ11 ∂ε ⎢ = ⎢ ∂ε ⎢ 3 ∂φ ⎢ ⎢ ∂φ11 ⎢ ⎢ ∂ε4 ⎣ ∂φ11 ⎡

2φ11 ⎢ φ31 =⎢ ⎣ φ31 0

∂ε1 ∂φ31 ∂ε2 ∂φ31 ∂ε3 ∂φ31 ∂ε4 ∂φ31 0 φ11 φ11 2φ31

∂ε1 ∂φ32 ∂ε2 ∂φ32 ∂ε3 ∂φ32 ∂ε4 ∂φ32

∂ε1 ∂φ33 ∂ε2 ∂φ33 ∂ε3 ∂φ33 ∂ε4 ∂φ33

0 φ12 −φ12 −2φ32

0 φ13 φ13 2φ33

⎤ ∂ε1 ∂φ34 ⎥ ⎥ ∂ε2 ⎥ ⎥ ⎥ ∂φ34 ⎥ ⎥ (5.59) ∂ε3 ⎥ ⎥ ⎥ ∂φ34 ⎥ ⎥ ∂ε4 ⎥ ⎦ ∂φ34 ⎤ 0 φ14 ⎥ ⎥. −φ14 ⎦ −2φ 34

The error function corresponding to (5.21) thus becomes

∂ε δφ11 ∂φ

δφ12

δφ13

δφ14

δφ31

δφ32

δφ33

⎧ ⎫ ε ⎪ ⎪ ⎪ ⎨ 1⎪ ⎬ T ε2 δφ34 = − . ε3 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ε4 (5.60)

5.8.3 Calculation of Parameters of the Four-d.o.f. Model In this example, an initial normalized modal matrix constructed through vibration mode shapes shown in Figure 5.16 and equivalent masses obtained at each modeling point is given by ⎡

0.868 ⎢0.868

=⎢ ⎣0.523 0.523

0.788 −0.788 0.671 −0.671

0.704 0.704 −0.412 −0.412

⎤ −0.917 0.917⎥ ⎥. 0.105⎦ −0.105

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Using the computation algorithm of Table 5.4, the modified modal matrix ⎡

⎤ 0.830 0.612 0.750 −0.936 ⎢0.831 −0.612 0.753 0.940⎥ ⎥

=⎢ ⎣0.411 0.513 −0.456 0.335⎦ 0.413 −0.516 −0.455 −0.336 is obtained. Introducing the modified modal matrix into (5.17) and (5.18), the mass matrix M and stiffness matrix K are obtained. Using these matrices the lumped mass model becomes a three-d.o.f. model shown in Figure 5.13. Through the same procedure as that of the example of the two-d.o.f. model, three mass and six stiffness values are determined as shown in Figure 5.13: ⎡ ⎢ M=⎢ ⎣ ⎡

0

0.400 0.398 1.329 0

⎤

⎡

⎥ ⎢ ⎥=⎢ ⎦ ⎣

0

M1

1.321

M2 M3 0 ⎤

⎤ ⎥ ⎥ , (5.61) ⎦

M4

0.854 −0.684 −0.973 0.647 ⎢−0.684 ⎥ 0.852 0.648 −0.969 ⎥ × 104 K =⎢ (5.62) ⎣−0.973 0.648 1.596 −0.910⎦ 0.647 −0.969 −0.910 1.584 ⎤ ⎛ ⎞ ⎡ −k12 −k13 −k14 K 1 = k10 + k12 + k13 + k14 K1 ⎜ ⎟ ⎢ K 2 −k23 −k24 ⎥ ⎥ , ⎜ K 2 = k12 + k20 + k23 + k24 ⎟ . =⎢ ⎣ K 3 −k34 ⎦ ⎝ K 3 = k13 + k23 + k30 + k34 ⎠ Sym. K4 K 4 = k14 + k24 + k34 + k40 Using the above mass matrix M and stiffness matrix K , the lumped mass model and its parameters form a four-d.o.f. model shown in Figure 5.17. Four mass and ten stiffness values are determined by comparing both sides of (5.61) and (5.62). 5.8.4 Comparison between Real System and Four-d.o.f. Systems Compared frequency responses of acceleration between the real system indicated in Figure 5.15 and the four-d.o.f. system in Figure 5.17 are shown in Figure 5.18. Both systems are excited at the modeling point M1 and measured at four mass points, M1 , M2 , M3 and M4 , respectively. It has been

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Table 5.4 Program for calculating parameters of a four-d.o.f. model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 4DOF Modeling for Tower structure % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clg %%%%%%%%% load eigenvector and eigen value %%%%%%%%%%% ms=zeros(4,4); % Size of Matrix %-------------- Setting natural frequency ------------w1=2*pi*2.512;w2=2*pi*11.66;w3=2*pi*15.98;w4=2*pi*37.21; %-------------- Inputting Normalized modal elements ------------s11=0.868;s12=0.788;s13=0.704;s14=-0.917; s21=0.868;s22=-0.788;s23=0.704;s24=0.917; s31=0.523;s32=0.671;s33=-0.412;s34=0.105; s41=0.523;s42=-0.671;s43=-0.412;s44=-0.105; %---------- Initial modal matrix normalized by equivalent mass ------------ms(1,1)=s11;ms(1,2)=s12;ms(1,3)=s13;ms(1,4)=s14; ms(2,1)=s21;ms(2,2)=s22;ms(2,3)=s23;ms(2,4)=s24; ms(3,1)=s31;ms(3,2)=s32;ms(3,3)=s33;ms(3,4)=s34; ms(4,1)=s41;ms(4,2)=s42;ms(4,3)=s43;ms(4,4)=s44; ww=[w1ˆ2 0 0 0;0 w2ˆ2 0 0;0 0 w3ˆ2 0;0 0 0 w4ˆ2]; % Square matrix of natural frequency M2ini=inv(ms*ms’) % Initial mass matrix K2ini=inv(ms*inv(ww)*ms’) % Initial stiffness matrix omomi=eye(12) % Weighting matrix %%%%%%%%%%%%%%%%%% Modification process %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i=1:5 kando=[2*s11 -2*s12 2*s13 -2*s14 0 0 0 0 0 0 0 0 % Sensitivity matrix s31 s32 s33 s34 0 0 0 0 s11 s12 s13 s14 s31 0 s33 0 0 0 0 0 s11 0 s13 0 0 0 0 0 s31 s32 s33 s34 s21 s22 s23 s24 0 0 0 0 s31 0 s33 0 s21 0 s23 0 0 0 0 0 0 0 0 0 2*s31 -2*s32 2*s33 -2*s34]; gosa=[s11ˆ2-s12ˆ2+s13ˆ2-s14ˆ2; s11*s31+s12*s32+s13*s33+s14*s34; s11*s31+s13*s33; s21*s31+s22*s32+s23*s33+s24*s34; s21*s31+s23*s33; s31ˆ2-s32ˆ2+s33ˆ2-s34ˆ2]; %test=kando*omomi*kando’ %test2=det(kando*omomi*kando’) henkou=-omomi*kando’*inv(kando*omomi*kando’)*gosa; s11=s11+henkou(1,1); s12=s12+henkou(2,1); s13=s13+henkou(3,1); s14=s14+henkou(4,1); s21=s21+henkou(5,1); s22=s22+henkou(6,1); s23=s23+henkou(7,1); s24=s24+henkou(8,1); s31=s31+henkou(9,1); s32=s32+henkou(10,1); s33=s33+henkou(11,1); s34=s34+henkou(12,1); ms(1,1)=s11;ms(1,2)=s12;ms(1,3)=s13;ms(1,4)=s14; ms(2,1)=s11;ms(2,2)=-s12;ms(2,3)=s13;ms(2,4)=-s14;

% Error function

% Modification vector

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ms(3,1)=s31;ms(3,2)=s32;ms(3,3)=s33;ms(3,4)=s34; ms(4,1)=s31;ms(4,2)=-s32;ms(4,3)=s33;ms(4,4)=-s34; ms; % Modified modal matrix m2r=inv(ms*ms’); % Modified mass matrix k2r=inv(ms*inv(ww)*ms’); % Modified stiffness matrix end %%%%%%%%%%%%%%%%%% Masses and stiffness %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ms; m2r k2r MM1=m2r(1,1); MM2=m2r(2,2); MM3=m2r(3,3); MM4=m2r(4,4); KK11=k2r(1,1)-k2r(1,2)-k2r(1,3)-k2r(1,4); KK22=k2r(2,2)-k2r(1,2)-k2r(2,3)-k2r(2,4); KK33=k2r(3,3)-k2r(1,3)-k2r(2,3)-k2r(3,4); KK44=k2r(4,4)-k2r(1,4)-k2r(2,4)-k2r(3,4); KK12=-k2r(1,2); KK13=-k2r(1,3); KK14=-k2r(1,4); KK23=-k2r(2,3); KK24=-k2r(2,4); KK34=-k2r(3,4); %end

confirmed that both frequency responses agree well within four modes. Note that the resonance peak of the fifth mode located at 43.18 Hz does not appear in the frequency responses of the real system, because each modeling point is selected at the node point of the fifth mode. In this way, the fifth mode becomes unobservable so that there will be no spillover

Figure 5.17 Four-d.o.f. model and parameters of the scaled bridge tower

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Figure 5.18 Compared frequency responses of real system with four-d.o.f. model measured at (a) mass point 1; (b) mass point 2; (c) mass point 3: (d) mass point 4


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problem. The fact that the fifth mode is uncontrollable and unobservable allows us to consider a relatively high cut-off frequency for the low-pass filters to eliminate the influence of the fifth (about 43 Hz) and higher modes. Otherwise, an even lower cut-off frequency would be necessary, thus leading to degradation of the control effect on the fourth mode (about 37 Hz).

5.9 Robust Vibration Control for Neglected Higher Modes In actively controlled structures with distributed parameters, the uncontrolled modes are always present, and thus there is always a danger of spillover. To solve this problem, various robust vibration control methods have been developed. Among them, H-infinity control has received a great deal of attention. However, for multi-mode control, the H-infinity controller might become very large. In practical cases, simpler methods are also in demand. Figure 5.19 shows diagrams of three robust control methods. The first, called filtered LQ control (Hori and Seto, 2000), is basically an LQ control which uses low-pass filters to mitigate the effects of uncontrolled higher modes. The low-pass filters can be coded into a program incorporated into a PC-based controller. This method is the simplest one in terms of system implementation. The second method, H-infinity state feedback control (Dorato et al., 1992; Cui et al, 1994), incorporates two kinds of filters (weighting functions) into the dynamic system to form an augmented plant, where the feedback gain is determined by solving one Riccati equation. This method is relatively simple to implement. H-infinity output feedback control, on the other hand, requires solving two Riccati equations to determine the feedback gain, which itself constitutes a dynamic system (compensator). Although this method has the merit of being able to include observers in the system, the controller becomes larger. For example, for a system of four mass points and two AMDs, the controller will be of order 32, making it difficult to implement it on a PC, and a digital signal processor thus becomes necessary.

5.10 Conclusions In order to control the vibration of a distributed structure with an infinite number of vibration modes, it is necessary to reduce the number of modes into an r -order lumped model with r degrees of freedom. Control system design constraints and other factors such as observation requirements

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Figure 5.19 Several robust control methods: (a) filtered LQ control; (b) H∞ control with state feedback; (c) H∞ control with dynamic compensator

make this procedure necessary. This chapter has proposed a method by which a reduced-order lumped model can be determined from information obtained from the vibration mode of a distributed structure. This model makes it easy to design an optimal controller in a physical state space where the states are directly measurable. Spillover problems concerning the omitted higher modes can be suppressed by employing

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the unobservability and uncontrollability characteristics, a kind of ‘structural filter’. Since the modal data of a control object (structure) can be obtained either by experiment or by numerical methods in advance, the modeling method described in this chapter is quite practical and easy to manipulate. The damping effects can also be considered, for example, by using proportional damping such as was applied in the modal analysis. To demonstrate the usefulness of this modeling method for controlling vibration in flexible structures, a tower-like structure with a flexible and distributed parameter system was selected as the control object. Based on the modal analysis approach, physical lumped parameter models of three and four degrees of freedom representing the first three and first four modes of the distributed structure, respectively, were constructed from the finite element model modal data of the structure. The usefulness of the constructed models has been confirmed using a real plate and scaled bridge tower structures. These models will be used for structural vibration control in Chapter 6.

References Bai, M.R., and Shieh, C. (1985) Active noise cancellation by using the linear quadratic Gaussian independent modal space control. Journal of the Acoustical Society of America, 97, 2664–2674. Balas, M.J. (1978) Feedback control of flexible structures. IEEE Transactions on Automatic Control, AC-23, 673–679. Balas, M.J. (1979) Direct velocity feedback control of large space structures. Journal of Guidance, Control and Dynamics, 2(3), 252–253. Balas, M.J. (1982) Theory for distributed parameter systems. In Control and Dynamic Systems 18, pp. 361–421. Academic Press. Cui, W., Nonami, K., and Nishimura, H. (1994) Experimental study on active vibration control of structures by means of H∞ and H2 control. JSME International Journal, Series C, Mechanical Systems, Machine Elements and Manufacturing, 37(3), 462–467. Dorato, P., Fortuna, L., and Muscato, G. (1992) Robust Control for Unstructured Perturbations – An Introduction, Lecture Notes in Control and Information Sciences 168. Berlin: Springer-Verlag. Fuller, C.R., Elliott, S.J., and Nelson, P.A. (1996) Active Control of Vibration. Academic Press. Hori, N., and Seto, K. (2000) Vibration control of flexible space structures based on reduced-order modeling method and filtered LQ control

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theory. JSME International Journal, Series C, Mechanical Systems, Machine Elements and Manufacturing, 43(3), 697–703. Kar, I.N., Doi, F., and Seto, K. (2000) Multimode vibration control of a flexible structure using H∞ based robust control. IEEE Transactions on Mechatronics, 5(1), 23–31. Merovitch, L. (1990) Dynamics and Control of Structures. Wiley. Seto, K. (1991) Active vibration control in machinery. Journal of the Acoustical Society of Japan, 12(6), 263–272. Seto, K. (1992) Trends on active vibration control in Japan. In Proceedings of 1st International Conference on Motion and Vibration Control, pp. 1–11. Seto K. and Mitsuta, S. (1992) A new method for making a reduced order model of flexible structures using unobservability and uncontrollability and its application in vibration control. In Proceedings of 1st International Conference on Motion and Vibration Control, pp. 153–158. Seto, K., Yamashita, S., Ohkuma, M., and Nagamatu, A. (1987) Method of estimating equivalent mass of multi-degree-of-freedom system, JSME International Journal, Series C, Mechanical Systems, Machine Elements and Manufacturing, 30, 1636–1644. Seto, K., Doi, F., and Ren, M. (1997) Modeling and active vibration control for bridge tower structures under construction. In Proceedings of DETC’97, Sacramento, CA, (VIB3808). Seto K., Ren M., and F. Doi, F. (1998) Feedback vibration control of a flexible plate at audio frequencies by using a physical state space approach. Journal of the Acoustical Society of America, 103, pp. 93.4–934.

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6 Active Control of Civil Structures 6.1 Introduction Vibration problems in the construction of large-scale structures can be exemplified through examining large-scale space structures such as those planned by NASA in the 1970s (Kane et al., 1983). As is well known, because space structures are extremely lightweight and large in scale, various oscillation modes exist in the low frequency range. It has been pointed out that when vibration modes, which are a hindrance to accurate communication from Earth, are controlled, vibration modes other than those controlled invite violent excitation. This phenomenon was called ‘spillover instability’ by Balas (1978), and some control system designs were proposed to suppress it, such as the direct feedback and mode control methods (Balas, 1982). This problem of spillover led to a robust control theory (Balas, 1979) in which the uncontrolled modes were regarded as uncertain disturbances, and it became possible to structure a robust system against such disturbances. If issues of vibration are not sufficiently and effectively ameliorated in the construction of large-scale structures such as high-rise buildings and the main towers of long-span suspension bridges, weight increase is unavoidable. However, these days limiting resource and energy expenditure is a necessity, and the problem of weight increase has become an inescapable point in the construction of large-scale structures. Accordingly, although making lightweight large-scale structures is very important, Active Control of Structures A. Preumont and K. Seto © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-03393-7

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Active Control of Civil Structures

flexible structures becoming lightweight tend to be susceptible to more vibration problems, as described above. In this section, we adopt an active vibration control approach to the construction of large-scale structures in which we attempt to successfully balance the task of resource saving without having to compromise the weight reduction of such large-scale structures (Doyle et al., 1989). The purpose of this chapter is to demonstrate the multiple modes of vibration control for civil structures using the lumped modeling approach addressed in Chapter 5 and the H∞ robust control theory in addition to the LQ control theory shown in Chapter 4. First, the classification of structural control for buildings is introduced. Then, two topics are addressed: building vibration control, in particular the connected control method (Seto, 1996, 2004) for multiple high-rise buildings arranged in parallel to each other: and the modeling and control of the multiple vibration modes of flexible towers of long-span suspension bridges (Seto et al., 1999; Kar et al., 2000a).

6.2 Classification of Structural Control for Buildings The classification of structural control methods for buildings based on the excitation level (e.g., wind excitation and small- and large-scale earthquakes) is listed in Table 6.1. The method using inertia masses corresponds to method (2) in Table 4.1, that using mutual action between structures corresponds to method (3), and that utilizing the base isolation corresponds to method (1). Although the method which uses a brace system is peculiar to structural control, it seems to be a combination of methods (1) and (3). Schematic views of each method are illustrated in Figures 6.1– 6.4. The method using inertia mass is effective in controlling the small scale of grand vibration level or wind excitation, because the control force is restricted by the inertia of the mass. The method using mutual action between structures is capable of controlling both low- and high-level excitation Real devices for the AMD, HMD and active bridges for building vibration control have been already reported in Figures 4.8, 4.10 and 4.12, respectively.

6.3 Modeling and Vibration Control for Tower Structures 6.3.1 One-d.o.f. Model In order to control tower-like structures, the most important mode is usually the first one expressed by the single-d.o.f. model. If a one-d.o.f. model

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Table 6.1 Classification of structural vibration control method Excitation level

Method

Type

Passive Wind excitation Traffic excitation

Using mutual action between structures

Structural Control

Using brace system

Active

Active mass damper (AMD)

Hybrid

Hybrid mass damper (HMD)

Passive

Connected damper

Active

Active controlled bridge

Hybrid

Hybrid controlled bridge

Passive

Brace damper

Semi-active

Large-scale earthquakes

Tuned mass damper (TMD) Sloshing damper (LMD)

Using inertia of masses

Small-scale earthquakes

Device

Passive Using base isolation Semi-active

Semi-active brace damper Oil damper, Rubber mount Friction damper, etc. Adjustable oil damper MR damper etc.

Figure 6.1 Schematic view of method using inertia of masses

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Figure 6.2 Schematic view of method using mutual reaction between structures

is suitably obtained, a control system with an active dynamic absorber or AMD is easily designed as indicated in Chapter 4. Figure 6.5 shows the relationship between a modeling point and the corresponding oned.o.f. model, where M1 , M2 and K 1 , K 2 represent the masses and spring constants estimated at two kinds of modeling points, respectively. These values are simply obtained using the mass response method (Seto et al., 1987) of Section 5.5.3. When the modeling point is selected at the top of the tower-like structure, a smaller mass and spring constants are obtained at the first vibration mode. Therefore, the best mounting location of the

Figure 6.3 Schematic view of methods using brace system

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Figure 6.4 Schematic view of method using base isolation

AMD or HMD controller is at the top of the structure for controlling the first vibration mode. 6.3.2 Two-d.o.f. Model for Tower-like Structures and Its LQ Control (Seto et al., 1995) State equation Although high-rise buildings are a kind of tower-like structure with an infinite number of vibration modes, a two-d.o.f. model, which considers the first two bending modes, has been already presented in Figure 5.8. Figure 6.6 shows the two-d.o.f. model with an AMD mounted at the top of the tower-like structure. The AMD is composed of an active mass md ,

Figure 6.5 One-d.o.f. model for tower-like structures

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Figure 6.6 Two-d.o.f. model for tower-like structures with AMD

a supporting spring with constant kd , and an actuator with a control force fd . Suppose that two displacement sensors are mounted at mass points M1 and M2 and a relative displacement sensor is set between mass point M1 and the active mass md . Then the state equation is ˙ = AX + Bu, X where ⎡

⎤ K 10 + K 12 K 12 kd ⎢ ⎥ M1 M1 M1 ⎢ ⎥ K 12 K 20 + K 12 ⎢0 0 0 ⎥ − 0 ⎢ ⎥ M2 M2 ⎢ ⎥ ⎢ K 12 kd ⎥ K 10 + K 12 kd A= ⎢ ⎥, − − + ⎢0 0 0 ⎥ M1 M1 M1 md ⎥ ⎢ ⎢1 0 0 ⎥ 0 0 0 ⎢ ⎥ ⎣0 1 0 ⎦ 0 0 0 0 0 1 0 0 0

T Kf 1 1 1 0 − + 0 0 0 , B= M1 m

Td K s M1 X = x˙ 1 x˙ 2 x˙ 3 x1 x2 x3 . 0

0

0 −

(6.1)

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Here, x1 , x2 and x3 are displacements at the mass points M1 , M2 and the relative displacement xd − x1 , and also K f and Ks are the force factor and a displacement transfer factor expressed by f d = K f u and x = K s u. Using LQ control theory, the state feedback gain matrix K = K1

K2

K3

K4

K5

K6

(6.2)

is determined after setting a weighting matrix Q = diag q 1

q2

q3

q4

q5

q6 ,

(6.3)

where q 1 to q 3 are weighting factors applied to velocities x˙ 1 to x˙ 3 and q 4 to q 6 are weighting factors applied to displacements x1 to x3 , respectively. Vibration control In this simulation, parameters shown in Figure 6.6 are used in addition to K f = 0.77[N/V] and K s = 1 × 10−3 [m/V]. Moreover, a weighting factor R of 10 is selected. Vibration control results are shown by three kinds of frequency responses in Figure 6.7, where the thin solid line indicates the case of weighting velocity factors q 1 = q 2 = 1, q 3 = q 4 = q 5 = q 6 = 0 and the thick solid line indicates the case of weighting velocity factors q 1 = 1 and q 2 = 10, while the dotted line represents no control. When the

Figure 6.7 Vibration control performance dependent on weighting factor

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same weighting velocity factors q 1 = q 2 = 1 are selected, the first resonance peak is well controlled, but the second resonance peak is not controlled well enough on the grounds that the mounting location of the AMD comes close to the node point of the second vibration mode, as shown in Figure 5.8. Adjusting the weighting velocity factor solves this problem; for example, the factor q 2 weighted on velocity x˙ 2 is increased from 1 to 10. As shown in Figure 6.7, the second resonance peak is well suppressed using this method. 6.3.3 Three-d.o.f. Model for Broad Structures and Its H∞ Robust Control (Kar et al., 2000a) Broad structures expressed by three-d.o.f. lumped mass system In this subsection, many modes of vibration control for a broad building structure as shown in Figure 6.8 are considered. Since these types of structures are flexible, it may be necessary to consider the vibration control of twisting modes in addition to bending modes like a plate structure (Figure 5.11). For this purpose, the plate structure is selected as the

Figure 6.8 Broad building structure

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vibration controlled object in order to facilitate the discussion of the vibration control of broad building structures. Figure 5.12 shows the first four vibration modes obtained by the finite element method. When the first three vibration modes need to be controlled, it is necessary to make a three-d.o.f. lumped mass model of the plate structure. Figure 5.13 shows this model and its parameters which could effectively be the phenomena of the first and second bending and the first twisting vibration. State-space equation Figure 6.9 shows the three-d.o.f. model and an actuator. The actuator is connected to mass point 2 of the structure, that is, the actuator is placed at the node of fourth vibration mode. Using this figure, and defining the relative displacement x4 = xd − x2 as a state variable in order to include the actuator dynamics, the following state equation of the control object is obtained: xc = x˙ 1 x˙ 2 x˙ 3 x˙ 4 x1 x˙ c = Ac xc + Bc uc + Bw w,

x2

x3

x4

T

,

Figure 6.9 Three-d.o.f. lumped mass model with actuator

(6.4) (6.5)

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where

Ac = ⎡

A11 A21

A12 , A22

⎤ 0 0 0 0 cd ⎢0 0 0 ⎥ ⎢ ⎥ M2 ⎢ ⎥ A11 = ⎢ 0 0 0 ⎥, 0 ⎢ ⎥ ⎣ cd cd ⎦ 0 0 0 − + M2 md ⎤ ⎡ k10 + k12 + k13 k12 k13 − 0 ⎥ ⎢ M1 M1 M1 ⎥ ⎢ ⎥ ⎢ k23 kd k12 + k20 + k23 k12 ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢ M2 M2 M2 M2 ⎥, A12 = ⎢ ⎥ ⎢ k k k + k + k 13 23 13 23 30 ⎥ ⎢ − 0 ⎥ ⎢ M3 M3 M3 ⎥ ⎢ ⎢ ⎥ ⎣ k12 k23 kd kd ⎦ k12 + k20 + k23 − − − + M2 M2 M2 M2 md A21 = I (4, 4) , A22 = 0 (4, 4) ,

Kc Kc Kc 0 − + 0 0 0 Bc = 0 M2 M2 md T

1 1 1 Bw = 0 0 0 0 0 . M1 M2 M3

T 0

,

In the above K c is the force transmission coefficient between the feedback control and the actual output of the actuator and md is the hybrid mass of the actuator with K c = 1[N/V] and md = 0.15[kg]. The output equation is y c = C c xc where ⎡

1 Cc = 0.75 ⎣ 0 0

0 1 0

0 0 1

0 0 0

4 0 0

0 4 0

0 0 4

⎤ 0 0⎦. 0

H ∞ -based controller design We now turn to a discussion of the H∞ -based robust controller design which is adopted here. The H∞ norm of a stable transfer matrix G( jω) is

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defined as its largest singular value over the entire frequency range, G ( jω)∞ = sup σ¯ {G ( jω)} , ∞

(6.6)

where σ¯ represents the largest singular value of G( jω). In the H∞ optimal controller design, a controller is selected to stabilize the nominal system in such a way that the H∞ norm of a transfer function, which describes certain design objectives, is minimized (or becomes smaller than a specified value). In the present study, the objective is to design a static state feedback controller based on the reduced-order model to control the vibration of the first three modes and suppress the possible instability due to neglected higher modes. These objectives can be achieved by solving a mixed sensitivity minimization problem in the framework of H∞ optimal control (Cui et al., 1994). The block diagram of the H∞ mixed sensitivity problem is shown in Figure 6.10, where Ac , Bc and C c are the parameters of reduced-order nominal model. Here yc is the measurement, uc is the control input, z1 and z2 are the regulated variables. W1 (s) and W2 (s) are the two weighting function matrices to be selected. In the case of additive uncertainty representation, the relation between the actual model and the reduced-order model is given by P (s) = Pr (s) + α (s) ,

Figure 6.10 Block diagram of H∞ -based control system

(6.7)

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where a (s) is the uncertainty and constitutes the dynamics of neglected higher modes. For this case, robust stability will be achieved if the norm inequality α (s) T (s)∞ < 1

(6.8)

holds (Dorato et al., 1992), where T(s) is the transfer function from the disturbance w to control input u. As the exact representation of a (s) is difficult, a weighting function matrix W1 (s) is selected to cover the upper bound of the uncertainty in the entire frequency range, that is,

σ¯ α ( jω) ≤ W1 ( jω) Then equation (6.8) can be simplified as W1 (s) T (s)∞ < 1.

(6.9)

To improve the system performance (to reduce the effects of disturbances on the output), the criterion W2 (s) S (s)∞ < 1

(6.10)

is used, where S(s) is the transfer function between w and output y and W2 (s) is a low-pass weighting filter. The specifications (6.9) and (6.10) are achieved by designing a controller K(s) which satisfies the following mixed sensitivity criteria: W1 (s) T (s) W2 (s) S (s) < 1. ∞

(6.11)

Next the results are exploited to solve a particular class of H∞ optimal control problem to design a static state feedback controller by solving a single Riccati equation. Suppose the augmented system (a nominal plant with two filters in the present investigation) is given by the state space equation xa = Aa x a + Ba 1 w + Ba 2 u, z = C a 1 x + Da u,

(6.12)

where xa is the augmented state vector, u is the control input, w is the disturbance input, and z is the regulated variable. The problem is to design

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a static state feedback control law u = −Kxa such that the following objectives may be achieved: (1) the closed loop matrix (Aa − Ba 2 K ) is stable and (2) Tzw ∞ ≤ 1, where Tzw is the transfer function from w to z. The controller gain is computed by solving the Riccati equation AaT P + PAa + C aT1 C a 1 − θ T −1 θ + PBa 1 BaT1 P = 0,

(6.13)

where Σ = DaT Da ,

θ = DaT C a 1 + BaT2 P,

and the controller gain is given by K = Σ−1 θ.

(6.14)

The details of the derivation of this result are given by Dorato et al. (1992). Selection of weighting filter In this section, special emphasis is given to selecting the weighting filter to represent the upper envelope of the additive uncertainty. In this regard, the placing of the actuator is crucial. In general, the uncertainty a (s) comprises the different peaks that appear at the natural frequency of neglected higher vibration modes. If we can remove some peaks from the uncertainty, then we will say that the uncertainty has been reduced. In the present study, the first three modes are considered to obtain a reducedorder model and the rest of the higher modes are neglected. So, in general the uncertainty will comprise the different peaks that appear at the natural frequency of the fourth and other higher modes. To reduce the uncertainty, the idea of placing the actuator in the node of the fourth vibration mode is employed here. For this purpose, two cases are considered and the corresponding frequency response of the actual system is obtained experimentally as shown in Figure 6.11. In case 1, the frequency response is obtained by exciting a node point of the fourth vibration mode and as a result there is no peak corresponding to the natural frequency of the fourth mode. We call this effect ‘structural filtering’. For case 2, the frequency response is obtained by exciting a point other than the node points of the

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Figure 6.11 Measured frequency responses of two cases

fourth mode and as a result the peak corresponding to the fourth mode appears at 60 Hz. The modeling error is obtained by calculating a remainder between the measured frequency response shown in Figure 6.11 and the frequency response of the three-d.o.f. model shown in Figure 5.14. Figure 6.12 shows the modeling error for case 2, while no peak appears at 60 Hz in case 1. It is observed that in case 1 there is no peak corresponding to the neglected fourth vibration mode. The weighting function is selected to cover the upper bound of the modeling error in the entire frequency range. For these two cases, two weighting functions are selected as shown in Figure 6.12. Based on the error functions for both cases 1 and 2, the corresponding weighting filters are represented by W11 (s) and W12 (s), respectively. The filter selected for case 1 is 2 6.0 × 10−4 1 + 2ζ1n ω1n s + ω1n = C 1 (s I − A1 )B1 + D1 , W11 (s) = 2 1 + 2ζ1d ω1d s + ω1d (6.15) where ζ1n = 0.56, ζ1d = 0.16, ω1n = 40.0 rad/s, ω1d = 90.0 rad/s. The highpass filter selected for case 2 is W12 (s) =

2 18.0 × 10−4 1 + 2ζ1n ω1n s + ω1n 2 1 + 2ζ1d ω1d s + ω1d

= C 1 (s I − A1 )B1 + D1 ,

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Figure 6.12 Frequency responses of two weighting functions and modeling error

where ζ1n = 0.56, ζ1d = 0.16, ω1n = 25.0 rad/s, ω1d = 75.0 rad/s. It is noticed that the cutoff frequency of W12 (s) is less than that of W11 (s) and the gain of W11 (s) is less than that of W12 (s). In other words, we are able to reduce the unmodeled uncertainty by placing the actuator in the node of the fourth mode. It is observed that by using an ideal high-pass filter W1 (s), the controller which satisfies the design specification of equation (6.9) will minimize the influence of neglected higher modes on the overall closed-loop system and thus will avoid the spillover problem. However, in practice, it would be difficult to obtain such results satisfactorily when the controlled modes and neglected (uncontrolled) modes are not sufficiently separated since the filter does not possesses an ideal sharp characteristic that would allow for the cutting off of the uncontrolled modes without influencing the controlled ones. For instance, in order to get rid of the influence of modes higher than the third in this investigation, the cutoff frequency of the filter should be lower than the natural frequency of the third mode, which would in turn unavoidably worsen the control effect on this mode. In this case, the elimination of a sharp peak corresponding to the fourth mode in the error function (hence reducing the system uncertainty) by employing the uncontrollability and/or unobservable characteristics allows us to consider a high-pass filter W11 (s) which has a relatively high cutoff

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frequency. This point is useful for the suitable selection of a filter without the occurrence of much degradation of the control effect on the third mode. This reduction has an effect on the vibration control especially in the case of the third mode which will be clearly seen from the simulation results presented later in this subsection. The weighting matrix W2 (s) is defined as W2 (s) = diag W2 (s) W2 (s) W2 (s) W2 (s) = C 2 (sI − A2 )−1 B2 + D2 , (6.16) where W2 (s) =

2 , with a = 2π × 30.0. s+a

Control system design By combining equations (6.5), (6.15) and (6.16), the state-space equation of the augmented plant is given by x˙ a = Aa x a + Ba 1 w + Ba 2 u, z = C a 1 x a + Da u, y = C a 2 xa ,

(6.17)

where ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ Bc xc Ac 0 0 Bw x a = ⎣ x 1 ⎦ , Aa = ⎣ 0 A1 0 ⎦ , Ba 1 = ⎣ 0 ⎦ , B = ⎣ B1 ⎦ , x2 0 D2 C c 0 A2 0

0 C1 0 D1 Ca1 = , Da = , Ca2 = Cc 0 0 . 0 D2 C c 0 C 2 ⎡

Here, x c , x 1 and x 2 are the state vectors of the plant, high-pass filter W1 (s) and low-pass filter W2 (s), respectively. Figure 6.13 describes the argument system. Using the augmented plant (reduced-order system along with W11 (s) and W2 (s) for case 1), the static state feedback controller is designed by solving a Riccati equation as given in (6.13) and (6.14). This controller gain vector K will be used in experimental studies discussed below.

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Figure 6.13 Block diagram of argument system

Simulation results The simulation results are presented in Figure 6.14. These responses are obtained by exciting mass point 3 and also observing at mass point 3. It should be noted that the first three modes are well suppressed. In order to verify the usefulness of the uncertainty reduction, we also design a controller using the weighting function W12 (s) for case 2. The controlled frequency response for cases 1 and 2 is shown in Figure 6.14. It is observed that the vibration suppression results for case 2 are poorer than

Figure 6.14 Simulated frequency response

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for case 1, especially for the third mode which is situated in the vicinity of the neglected fourth mode. Experimental results The experimental setup for the plate structure shown in Figure 5.11 is used and a personal computer is used as a controller. Three non-contact displacement (gap) sensors are used to measure the displacements of three mass points and a strain gauge is used to obtain the relative displacement between the actuator and the plate. All velocity signals are computed by differentiating the displacement signal, and the state of the filters is also computed on-line within the control algorithm to implement the state feedback control law. The actuator consists of a simple electromagnetic device as shown in Figure 4.6. The feedback control signal changes the current through the coil in a magnetic field and results in the movement of the hybrid mass of the actuator to dissipate the vibration energy. The controller gain vector K, determined using the weighting function W11 (s) for case 1, is used here. The vibration control effects are evaluated by measuring the transfer function (compliance) of the structure under impulse excitation. The experimental frequency response and the time response of the controlled object are shown in Figures 6.15 and 6.16. A hammer is used to provide an impulse input at mass point 3 and the response at the mass point 3 is observed. From both figures, it is evident that the control scheme has

Figure 6.15 Measured frequency response

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Figure 6.16 Measured impulse response

good vibration control effect as all three peaks are well suppressed compared with the non-control frequency response. Also there is no spillover problem that can be clearly discerned from the impulse response. In this case, the experiment was carried out with the same controller gains as the simulation. These experimental results confirm the effectiveness of the proposed control scheme to suppress the vibration of a flexible structure. To verify the robustness of the proposed control scheme against the parameter variations and the lack of accurate knowledge of the natural frequencies, we have deliberately attached two additional masses at each mass point 2 and 3. It was confirmed that several natural frequencies are reduced with good robustness (Kar et al., 2000b). 6.3.4 Four-d.o.f. Model for Bridge Tower and Spillover Suppression Using Filtered LQ Control Bridge tower structure expressed by four-d.o.f. lumped mass system with two active dynamic absorbers In the present subsection, we consider the problem of controlling the first four vibration modes of the bridge tower structure shown in Figure 5.15 with no spillover instability using the filtered LQ control law. The first four vibration modes (two bending and two twisting modes) were illustrated in Figure 5.16. In order to control the vibration, a four lumped mass model was constructed as Figure 5.17. This model is derived using the notion of uncontrollability and unobservability of certain neglected modes by placing the actuators and/or sensors in the nodes of neglected fifth modes. For

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Figure 6.17 Four-d.o.f. lumped mass model of the scaled bridge tower with two active dynamic absorbers

this reason, the resonance peak of the fifth mode located at 43.18 Hz does not appear in the frequency responses of the real system as shown in Figure 5.18. However, the neglected higher modes may invite spillover problems. Figure 6.17 shows the four-d.o.f. lumped mass model that is created by analyzing the first four vibration modes. In this case two active dynamic absorbers are attached at the mass points 3 and 4 coincident with the nodes of the fifth mode. By including the absorber dynamics as an actuator, define the state vector by Xc = x˙ 1

x˙ 2

x˙ 3

x˙ 4

x˙ 5

x˙ 6

x1

x2

x3

x4

x5

x6

T

,

(6.18)

where x5 and x6 refer to the relative displacements: x5 = xd1 − x3 , x6 = xd2 − x4 . Then the system can be described by the state equation ˙ c = Ac Xc + Bc U c , X yc = C c Xc ,

(6.19) (6.20)

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where

Ac =

A11 A21

A12 , A22

with submatrices

a11 A12 = a21

a12 , a22

A21 = I, A22 = 0, where I and 0 are the 6 × 6 unit matrix and null matrix respectively. Here a11 = −M−1 K,

−k13 /M3 −k23 /M3 K 33 /M3 −k34 /M3 0 0 kd1 /M3 , a 12 = a21 = −k14 /M4 −k24 /M4 −k34 /M4 K 44 /M4 0 0 0

0 − (kd1 /md1 + kd1 /M3 ) , a22 = 0 − (kd2 /md2 + kd2 /M4 )

0 kd2 /M4

T ,

the M and K corresponding to (5.61) and (5.62), respectively. In addition, ⎡ ⎢0 0 0 Bc = ⎢ ⎣ K c1 0 0 M3

K c2 M4 0

−K c1

0 1 1 + md1 M3

−K c2

1 1 + md2 M4 0

⎤T 0 0 0 0 0 0⎥ ⎥ , ⎦ 0 0 0 0 0 0

where K c1 , K c2 are the force coefficients of the actuators representing the relationship between the actual control force and the control value in voltage. State equation of low-pass filter Although the influence of the fifth mode (about 43 Hz) is eliminated in the modeling stage, neglected modes higher than the fifth may induce spillover. The influence of the higher modes is reduced by introducing low-pass filters. The state equation of the second-order low-pass filters with the angular cut-off frequency ωn and damping ratio ζ is expressed as ˙ f = Af X f + B f U f , X Y f = C f Xf ,

(6.21) (6.22)

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where the state vector X f with variables of filter output and matrices A f , B f and C f are

T X f = e˙ f 3 e˙ f 4 e f 3 e f 4 , ⎡ 0 −ωn2 −2ζ ωn ⎢ 0 0 −2ζ ωn Af = ⎢ ⎣ 0 0 0 0 0 0

0 0 1 0 Cf = . 0 0 0 1

⎡ 2 ⎤ 0 ωn 2⎥ ⎢ −ωn ⎥ 0 , Bf = ⎢ ⎣ 0 0 ⎦ 0 0

⎤ 0 ωn2 ⎥ ⎥, 0 ⎦ 0

Filtered LQ controller The filter is intended to cut off the control signal; therefore, we have the relationship Y f = Uc .

(6.23)

Then it is easy to obtain an augmented system ˙ = AX + BU, X Y = CX,

(6.24) (6.25)

where

X = Xf

Xc

B = Bf

0

T

T

,

,

Af Bc C f

A=

Cf C= 0

0 , Ac

0 . Cc

This type of built-in software filter will be able to cut off the control signal beyond the frequency range of interest to us. Because the filter characteristics have been taken into account in the system design, there will be no phase lag problem that would hamper control performance as in the case of a hardware filter that is usually difficult to incorporate in the system model. However, in practice, it would be difficult to apply this technique satisfactorily when the controlled and uncontrolled modes are not sufficiently separated, since the filter does not possess an ideally sharp

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characteristic that allows the uncontrolled modes to be cut off without influencing the controlled ones. For instance, in order to get rid of the influences of the modes higher than the fifth in this investigation, the cutoff frequency of the filter should be somewhat lower than the natural frequency of the fourth mode, which would in turn unavoidably worsen the control effect on this mode. On these occasions, it would be quite helpful to make use of the unobservability and/or the uncontrollability characteristics. As already stated, by placing sensors and actuators at the nodes of certain modes, the control system will produce no influence on these modes due to the unobservability and/or the uncontrollability characteristics. It seems that a filter cuts off these modes. We regard this as a structure filter. Usually, it is possible to avoid the spillover problem of one or two modes by means of this structure filter, which will create a favorable frequency allowance for the choice of low-pass filter cutoff frequency. The above consideration is clearly illustrated in Figure 6.18. We refer to the augmented system combining a control object with a filter as the filtered LQ control system. Now, designing an optimal controller using LQ theory becomes a quite standard problem. The control value that minimizes the performance index ∞ T X QX + U T RU dt (6.26) J = 0

is formulated as U = U f = −RBPX = − K f , K c X,

Figure 6.18 Block diagram of filtered LQ control system

(6.27)

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where Q is the state weighting matrix, R is the control weighting matrix and P is the solution of the Riccati equation. K c is the feedback gain matrix from the control object Since the filter is created on the inside of the controller by software, it is very easy to make a state feedback loop with feedback gain matrix K f from the filter. Simulation Figure 6.19 shows the frequency response of the low-pass filter implemented. The cutoff frequency of the low-pass filter is set to 50 Hz, since the natural frequency of the fourth mode is located at 37.2 Hz. The numerical simulations on the discrete model are performed using MATLAB Control System Toolbox routines. In these simulations the values of the components that weight velocity terms for the control object in Q are all taken as unity, while the components weighting displacements are taken as zero, since it has been established by various simulations and experiments that weighting velocity terms is more effective than weighting displacement terms or both. In order to keep the frequency response of the filter, the weighting terms of the filter are taken as zero. Adjusting the weighting matrix changes the feedback gains R. The proper gains examined in simulation will be used in the experiments. Therefore, simulation is not only a test of the model but also an indispensable procedure to prepare the feedback gains for the experiments. The values of R used in this chapter are 0.001. Figures 6.20 and 6.21 show the calculated frequency response (compliance) and the impulse response respectively, when assuming an excitation at mass point 1 and picking up at mass point 2. It is evident that the

Figure 6.19 Frequency response of the low-pass filter used

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Figure 6.20 Calculated frequency response of the model shown in Figure 6.17

first four resonance peaks can be well controlled and large damping is achieved. Experimental setup Experiments are carried out to verify the modeling and control methods mentioned above for controlling the vibration of the bridge tower structure. The experimental setup is shown in Figure 6.22. A personal computer is used as the controller. The displacements at the positions of mass points 1 to 4 in the structure are detected through four absolute displacements sensors and are inputted into the computer via A/D converters. Two active

Figure 6.21 Calculated impulse response of the model shown in Figure 6.17

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Figure 6.22 Experimental setup

dynamic absorbers as shown in Figure 4.6 are installed at the positions of mass points 3 and 4. The displacements of the absorbers are measured by two strain gauges. The six corresponding velocity signals are calculated by software using finite difference approximation. The control values are sent to the D/A converter and then amplified to drive the actuators. The feedback gains obtained from numerical simulation are used to generate the control forces in the experiments. Experimental results without filters In order to demonstrate the effectiveness of filtered LQ control, the measured frequency and impulse responses without filters are first shown in Figures 6.23 and 6.24. In Figure 6.23, although the resonance peak of the fifth mode located at 43 Hz does not appear, because of the structural filtering, the resonance peak located at 142 Hz is increased towards a large amplitude. This is the well-known spillover phenomenon. This phenomenon is illustrated in Figure 6.24, where one observes that large high-frequency oscillations develop after the attenuation of the controlled modes. In spite of setting the control weighting matrix to R = diag[0.005 0.005], this phenomenon is observed; therefore, stronger oscillations will occur when the weighting matrix is chosen as R = diag[0.001 0.001] as in simulations.

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Figure 6.23 Measured frequency response without filter

Experimental results using filtered LQ control Experimental results for filtered LQ control are shown in Figures 6.25 and 6.26, where the weighting matrices Q and R are the same as in the simulation. By exciting at the position of mass point 1 using an impulse hammer and measuring the response at the position of mass point 2, Figure 6.25 shows the experimental frequency response (acceleration) and Figure 6.26 the corresponding impulse response of the structure. It can be seen that the first four vibration modes of the structure are well controlled. No adverse influence of the filters on the control performance has been observed. No spillover occurs for the omitted modes higher than the

Figure 6.24 Measured impulse response without filter

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Figure 6.25 Measured frequency response using filtered LQ control

fifth. In particular, the measured impulse response agrees well with the simulated one without the spillover phenomenon. In Figure 6.25, two kinds of low-pass filters with 50 Hz and 30 Hz as cutoff frequencies are used to eliminate the influence of the seventh (about 68 Hz) and higher modes. When the filter with 50 Hz is used, the resonance peak of the fourth mode is well suppressed in comparison with using 30 Hz. Otherwise, an even lower cutoff frequency leads to degradation of the control effect on the fourth mode (about 37 Hz). Conversely, higher cutoff frequencies may invite spillover. The cutoff frequency of low-pass

Figure 6.26 Measured impulse response using filtered LQ control

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filters is set to 50 Hz in this case, which proves to be a good tradeoff. It should be pointed out that such a high cutoff frequency is possible because the resonance peak of the fifth mode located at 43 Hz is not affected by the structural filter. An advantage of the structural filter is to separate control modes from neglected higher modes. Remarks Spillover problems of omitted higher modes can be suppressed by employing low-pass filters, in combination with use of the unobservability and uncontrollability characteristics, a kind of structural filter. The modeling and the control design have shown by experiment that the first five bending and twisting modes of the structure are very well controlled.

6.4 Active Vibration Control of Multiple Buildings Connected with Active Control Bridges in Response to Large Earthquakes (Seto and Matsumoto, 1999) 6.4.1 Construction of Four Model Buildings Active mass dampers are used mainly in order to reduce the vibration in high-rise buildings caused by small to mid-strength earthquakes or strong wind; however, this equipment has been rendered ineffective in mitigating the vibration caused by large earthquakes or very lowfrequency vibrations which occur in ultra-tall buildings, because of the difficulty of obtaining enough control force to control these movements. In order to solve these problems, the CCM was proposed in previous sections. According to this method, two or more buildings are connected by passages installed with semi-active, active or hybrid control devices, and the buildings are controlled through the interactive force among them. This method has the merit of obtaining sufficient control force under low-frequency conditions. In addition, it is also expected that this method will have further merits such as improving comfort by reducing the wind response of buildings and offering the convenience of being able to move between buildings using connecting bridges as shown in Figure 6.27. In this section, the effectiveness of this method against large earthquakes is demonstrated by the use of four model tower structures which represent four high-rise buildings connected by four controlled bridges shown in Figure 6.27. An outline of the model tower structures is shown in Figure 6.28. The four towers, labeled A, B, C and D, are flexible structures

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Figure 6.27 Schematic view of the four towers with four actuators

with a large number of vibration modes, and they have 9, 10, 11 and 12 stories respectively. Tower A is connected with towers B and C by two actuators on the eighth story. Similarly, tower D is connected with towers B and C by two actuators on the ninth story. Each tower gains the control force in two directions (x and y) independently.

Figure 6.28 Four model tower structures connected with connected with active bridges on assuming Figure 6.27

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Figure 6.29 Vibration mode shapes of the structures

6.4.2 Characteristics of the Tower Structures Vibration characteristics of the four model tower structures are analyzed with the ANSYS finite element model software. The results show that four structures have different natural frequencies but similar vibration mode shapes. Therefore the first three vibration modes of tower A are shown in Figure 6.29 as typical of those of the four tower structures, and the natural frequencies of the four structures are shown in Table 6.2. The bending and twisting modes appear alternately as shown in the figure: the first mode is the first bending mode, the second is the first twisting mode, the third is the second bending mode, and so on.

Table 6.2 Natural frequencies of the structures (Hz)

Tower A Tower B Tower C Tower D

1st mode

2nd mode

3rd mode

10.8 9.6 8.7 7.7

14.8 13.2 12.0 10.7

33.7 30.0 27.0 23.5

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6.4.3 Reduced-order Model of the Four Tower Structures Connected by Four Actuators In this study, the control target modes are the first bending and first twisting modes of each tower. Therefore each tower has to be reduced to a two-d.o.f. lumped mass model. Two modeling points for each tower are selected at the mounting locations of the actuators, as shown in Figure 6.30. The reduced-order models with two degrees of freedom are constructed by the use of the reduced-order modeling method described in Chapter 5. Here it is necessary to verify that this model can express the vibration characteristics of the model tower properly. The comparison between the frequency responses of the model towers and those of the reduced-order lumped model is shown in Figure 6.31. From this figure, the first two resonance peaks of the model towers correspond to those of the reduced-order lumped model. In addition, there are no resonance peaks over the third mode in the frequency responses of the reduced-order lumped model, because the higher modes are neglected in the reduced-order model. From this point it can be said that the

Figure 6.30 Modeling points

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Figure 6.31 Comparison between the model towers and the reduced-order lumped model with 2 degrees of freedom


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reduced-order lumped model can represent well the vibratory characteristics of the model towers. According to the above approach, each of the reduced-order lumped models with 2 d.o.f. and actuators connected with each mass can be created as shown in Figure 6.32. Although proportional damping is usually used instead of structural damping, the damping factors shown in Figure 6.32 are determined in accordance with resonance peaks between measured and calculated frequency responses as shown in Figure 6.31. 6.4.4 Control System Design State-space equation The equation of motion is set up in two directions (x and y) below. Since the towers used in this study have the same characteristics in both directions, the model parameters are the same in both directions as well: ¨ + C Z˙ + KZ = Ff + ha, MZ ¨

(6.28)

where c], K = diag[ k k] ,

M = diag[ m m], C = diag[ c y} , f = { f 1

Z = {x

T

F = [Fx

f2

f3

f 4 },

T

Fy] ,

and stiffness and damping matrices k and c are also obtained separately in the both directions in Figure 6.32. In the above equations, m = diag[ ma 1 x = xa 1 xa 2 y = ya 1 ya 2 ⎡ 0 −1 1 ⎢0 0 0 Fx = ⎢ ⎣0 0 0 0 0 0

ma 2

mb1

xb1

xb2

yb1

yb2

0 0 0 0 0 0 0 −1

0 0 0 0

mb2

mc1

mc2

md1

T xc1 xc2 xd1 xd2 ,

T yc1 yc2 yd1 yd2 , ⎡ ⎤T 0 0 0 0 ⎢ −1 0 0 0⎥ ⎥ , Fy = ⎢ ⎣ 0 0 0 0⎦ 0 1 0 0

md2 ],

0 0 0 0 0 0 0 −1 0 0 0 0

0 1 0 0

0 0 1 0

⎤T 0 0⎥ ⎥ . 0⎦ 0

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Figure 6.32 The reduced-order lumped model and model parameters of the towers


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This equation of the control object is transformed into the state equation as ˙ c = Ac Xc + B1 U c + B2 a, X

(6.29)

where Xc is a state vector, Xc = { x˙

y˙ x

y},

(6.30)

u4 ,

(6.31)

U c is a control value vector, U c = u1

u2

u3

and a is a disturbance vector. Additionally, matrices A, B1 and B2 are given by

−M−1 C Ac = I

−1

−1 −M−1 K M F M h , B1 = K c , B2 = , 0 0 0

where K c is a force factor for connecting f i = K c ui (i= 1, . . . , 4). Filtered LQ control The optimal feedback gain is computed by use of the LQ control theory for the control object represented by (6.32) below. Nevertheless there is some risk that the neglected higher modes may cause spillover instability when a control experiment is done. In this section, low-pass filters are provided in order to prevent spillover instability, and the LQ control theory is applied to the augmented system consisting of the low-pass filter and the control object. When the control variable U f is input to the low-pass filter, the state equation and an output equation concerning the low-pass filters are set as follows: ˙ f = Af X f + B f U f , X Y f = C f Xf .

(6.32)

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Since an input of the control object U c is the output of the filter Y f , the state equation of the augmented system is given by ˙ = AX + BU + B2 a, X Y = CX, where

A=

Af B1 C f

X = {xf

0 Bf Cf ,B= ,C = Ac 0 0

(6.33)

0 , Cc

x c }.

We call such an augmented system combining an LQ control system with a filter a filtered LQ control system. By using the filtered LQ control, it is possible to control the vibration of the structures while spillover is prevented. 6.4.5 Simulated Results of Seismic Response Control Frequency response The simulated results of frequency responses for each tower are shown in Figure 6.33. Thin lines show responses without control and thick lines responses with control. Here the weighting matrices Q and R in the LQ control theory have to be selected suitably. In the case of Q, the weights corresponding to the velocity of each tower are set to 1, and others are set to 0. R is set to R = 5 × 10−4 5 × 10−4 5 × 10−4 5 × 10−4 . From Figure 6.33, it is evident that the peaks of the first (first bending) and second (first twisting) modes are suppressed well. It is obvious that the twisting mode is also controlled effectively. Seismic wave The control effect against the vibration caused by the earthquake is verified through simulation. The conditions of the simulation are as described below. The seismic wave used in the simulation is analogous to that of the 1996 Kobe moving in the east–west direction. This earthquake wave is normalized so that a dominant frequency is brought close to the first

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Figure 6.33 Frequency response of the structures

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Figure 6.34 Seismic wave used in the simulation and its spectrum density

natural frequency of the model structures. The normalized earthquake wave and its spectrum density are shown in Figure 6.34. Earthquake motion response The simulated time responses due to seismic excitation at all towers, which have the frequency responses indicated in Figure 6.33, are shown in Figure 6.35. Again, thin lines show responses without control and thick lines responses with control. This figure shows that the vibration of all towers is reduced quite well. Thus, it is verified by simulation that the control method is effective against earthquakes. 6.4.6 Experiment Experimental setup The experimental setup is shown in Figure 6.36. The sensors measure the relative displacements between the base and each tower. These displacement signals are input through the A/D converter. In the computer the velocity signals are calculated from the displacement signals, and the

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Figure 6.35 Simulated time response due to seismic excitation

control value is calculated by multiplying the feedback gain matrix by the state vector. The control value is output through the D/A converter and the amplifier. The actuators generate the control force and control the vibration of the entire structure. In this experiment, linear actuators consisting of permanent magnets and moving coil are used. Frequency response test The experimental responses measured by fast Fourier transform analyzer under the impulse test are shown in Figure 6.37, under the same conditions

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Figure 6.36 The experimental setup

as the simulation shown in Figure 6.33. From this figure, resonance peaks of the first and second mode of each tower are well controlled. Therefore it is also possible to control the twisting mode effectively by way of the proposed method.

Results of seismic response control The experimental result, which is obtained under the same conditions as the simulation, is shown in Figure 6.38. A thin dotted line shows the response without control, and a thick solid line shows the response with control. This figure shows that the experiment has a favorable effect on the first bending and twisting modes. In order to evaluate the effectiveness of the control statistically, the spectrum density analysis is carried out using the time response data. Figure 6.39 indicates the relationship between the spectrum density with and without control obtained at each tower and frequency. Although each tower withoutcontrol is excited strongly in

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Figure 6.37 Frequency responses of the structures

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Figure 6.38 Measured time response due to seismic excitation

the neighborhood of the first and second natural frequencies, these figures demonstrate that the resonance peaks of the towers are substantially reduced with control. This vibration control method can be applied similarly to any case where structures having different dynamics are arranged in parallel. It is possible to control the vibration of not only high-rise buildings but also the main towers of bridges under construction, space structures and so on.

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Figure 6.39 Spectrum density of time responses

6.5 Vibration Control for Real Triple Towers Using CCM (Seto and Matsumoto, 2003) 6.5.1 Outline of the Triple Towers A building design company has noticed the effectiveness of the CCM, and decided to introduce this method in the Triple Towers. An outline of the Triple Towers located at Harumi Triton Square in downtown Tokyo has been reported in Section 4.5 and sketched in Figure 4.11 Since at the design stage it became clear that a complex vortex excitation from wind was of concern for the first bending mode of the Triple Towers, structural vibration control technology was required. To control the first bending mode of each tower, it was necessary to use two AMDs for each of the two directions. Therefore, in total, six AMDs would be required if the conventional method was employed. Using our proposed CCM, the minimum requirements are two controlled bridges (devices). The company made a decision to use the CCM with two controlled bridges as shown in Figure 4.12.

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6.5.2 Modeling of Towers The vibration characteristics of the towers are analyzed with ANSYS. The results show that the three towers have different natural frequencies, but similar vibration mode shapes. The natural frequencies for the first and second modes of the three towers are shown in Table 6.3. In this study, the control target modes are the first bending of each tower. Therefore each tower has to be reduced to a single-d.o.f. lumped mass model without tower Y. The mass points are selected for the mounting locations of the bridges. Therefore it is necessary to prepare two mass points on tower Y as shown in Figure 6.40. Parameters of the lumped mass model in the x and y directions are determined using the reduced-order modeling method: (x-direction) m21x = 24 098 [tonne] m1x = 38 531 [tonne] k21x = 3.32 × 108 [N/m] k1x = 7.62 × 107 [N/m] 6 c 1x = 1.08 × 10 [Ns/m] c 21x = 4.29 × 106 [Ns/m] m3x = 27 261 [tonne] m22x = 8555 [tonne] 7 k22x = 9.05 × 107 [N/m] k3x = 6.91 × 10 [N/m] 5 c 3x = 8.68 × 10 [Ns/m] c 22x = 1.17 × 106 [Ns/m] (y-direction) m21x = 24 139 [tonne] m1x = 38 715 [tonne] k21x = 3.54 × 108 [N/m] k1x = 7.29 × 107 [N/m] 6 c 1x = 1.06 × 10 [Ns/m] c 21x = 4.62 × 106 [Ns/m] m3x = 26 758 [tonne] m22x = 9073 [tonne] k22x = 8.89 × 107 [N/m] k3x = 7.79 × 107 [N/m] c 3x = 9.13 × 105 [Ns/m] c 22x = 1.16 × 106 [Ns/m] Table 6.3 The natural frequencies of the towers (Hz) x direction

Tower X Tower Y Tower Z

y direction

1st mode

2nd mode

1st mode

2nd mode

0.22375 0.24649 0.25341

0.57726 0.63744 0.67247

0.21842 0.24404 0.27151

0.57437 0.64312 0.71048

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Figure 6.40 Schematic diagram of the Triple Towers

6.5.3 Control System Design The equation of motion is set up in two directions (x and y) as follows: m¨x + c x˙ + kx = f ,

(6.34)

where m = diag m1x m21x m22x m3x m1y m21y m22y m3y , ⎡ 0 0 0 0 0 0 c 1x ⎢ 0 −c 0 0 0 0 c 21x 21x ⎢ ⎢ 0 −c 21x c 21x + c 22x 0 0 0 0 ⎢ ⎢ 0 0 0 0 0 0 c 3x c=⎢ ⎢ 0 0 0 0 0 0 c 1y ⎢ ⎢ 0 −c 0 0 0 0 c 21y 21y ⎢ ⎣ 0 0 0 0 0 −c 21y c 21y + c 22y 0 0 0 0 0 0 0

0 0 0 0 0 0 0 c 3y

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

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⎡

k1x 0 0 0 0 0 ⎢ 0 −k 0 0 0 k 21x 21x ⎢ ⎢ 0 −k21x k21x + k22x 0 0 0 ⎢ ⎢ 0 0 0 0 0 k 3x k=⎢ ⎢ 0 0 0 0 0 k 1y ⎢ ⎢ 0 0 0 0 0 k21y ⎢ ⎣ 0 0 0 0 0 −k21y 0 0 0 0 0 0

T x = x1 x21 x22 x3 y1 y21 y22 y3 , ⎡ ⎤ −kc1 cos 45◦ 0 ⎢ −kc1 cos 45◦ ⎥ 0 ⎢ ⎥ ◦⎥ ⎢ 0 −kc2 cos 45 ⎥ ⎢ ⎢ 0 −kc2 cos 45◦ ⎥ ⎥ u1 . f = kc u = ⎢ ⎢ −kc1 sin 45◦ ⎥ u2 0 ⎢ ⎥ ◦ ⎢ −kc1 sin 45 ⎥ 0 ⎢ ⎥ ⎣ 0 −kc2 sin 45◦ ⎦ 0 −kc2 sin 45◦

267

0 0 0 0 0 −k21y k21y + k22y 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ k3y

This equation is transformed into the state equation as follows: ˙ = AX + Bu + Dw. X

(6.35)

Here X is a state vector and u is a control value vector, X = x˙

x

T

,

u = u1

u2

T

,

(6.36)

and w is a disturbance vector. Additionally, matrices A, B and D are given as follows:

−1

−1 m −m−1 c −m−1 k m kc ,D= . (6.37) A= ,B= 0 0 I 0 The optimal feedback gain is computed for the controlled object represented by means of the LQ control theory. Even so, there is some risk that neglected higher modes may cause spillover instability when a control experiment is undertaken. In this study, low-pass filters were added to prevent spillover instability, and the LQ control theory was applied to the augmented system consisting of the low-pass filter and the controlled object (Seto and Matsumoto, 2003).

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Figure 6.41 Lumped mass model of the Towers

The design target of the control system in this research is to halve the amplitude of each tower against wind excited vibrations. Therefore two kinds of weighting matrices are selected as follows: (Case 1)

Q = diag 1.5 0.1 0.1 1 1.5 0.1 0.1 1 0 0 0 0 0 0 0 0 R = diag 1 × 10−14 1 × 10−14

(Case 2)

Q = diag 1 0.3 0.3 0.7 1 0.3 0.3 0.7 0 0 0 0 0 0 0 0 R = diag 5 × 10−14 5 × 10−14

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269

Figure 6.42 Free vibration responses of Towers Y and Z

6.5.4 Simulation of the Triple Towers Using CCM Free vibration response In order to examine the damping effect of the active controlled bridges, the free vibration responses are calculated after the input of the exciting sinusoidal. Figure 6.42 shows the responses calculated in the x and y directions of Towers Y and Z. It is confirmed that the above design target is realized using case 1. Estimation of control effect against wind excitation The control effects against vibrations caused by wind excitation are verified through simulation. The wind disturbance data obtained by a wind tunnel test is used in the estimation of the control effect. The time

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270


Figure 6.43 Time responses calculated on Towers Y and Z

responses of each tower are calculated, after the data is input to (6.35) as the disturbance vector. As an example, Figure 6.43 shows time responses calculated for Towers Y and Z against the direction of the wind. When the control system is applied, it is confirmed that the displacement amplitude of Towers Y and Z is halved as expected in designing the control system. 6.5.5 Realization of the CCM Active controlled bridge The actual active controlled bridges were designed and developed by Ishikawajima Heavy Industries (Asano et al., 2002). The construction and its capacity are shown schematically in Figure 6.44. The active controlled bridge consists of an outer tube, an inner tube, a slide guide between each

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Total mass

271

80 ton

Control force

± 340 kN

Controlling stroke

± 0.1 m

Movable stroke

± 2.4 m

Figure 6.44 Schematic views of the active controlled bridge and its capacity

tube, a driving mechanism of two ball screws, and two AC servomotors. The AC servomotors, one mounted on the upper side and one on the underside of the inner tube, drive the ball screw to move the outer tube in a forward and backward direction. Therefore both ends of the tube are expanded and contracted by active control of the AC servomotors. Since both ends of the tube are connected to the two towers through ball joints, these towers move smoothly. The outside view of the active controlled bridge connecting Towers Y and Z is shown in Figure 6.45. The first Triple Towers connected by two active controlled bridges using our proposed CCM were constructed in downtown Tokyo in 2001 as shown in Figure 6.46. Results of the damping effect test using the Triple Towers To examine the damping performance of the Triple Towers as a result of the active controlled bridges, the same test as the simulated one shown in Figure 6.42 was carried out using Towers Y and Z. After sinusoidal

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Figure 6.45 Outside view of the active controlled bridge connecting Towers Y and Z

Figure 6.46 Outside view of the Triple Tower connected by active bridges

272

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273

Figure 6.47 Measured frequency responses of Towers Y and Z

excitation of the two towers, the free vibration-damping test was undertaken. The measured results were in good agreement with the calculated ones shown in Figure 6.42. This confirms that the active controlled bridges can be expected to work well in reducing vibrations as shown by measured frequency responses of Towers Y and Z in Figure 6.47. As an example, Figure 6.48 shows the trajectory of acceleration response measured on Tower Y against wind direction 285◦ . It is confirmed that the acceleration amplitude of tower Y is halved as expected in designing the control system. From the above numerical simulation and experimental results, some conclusions can be drawn. According to the authors’ proposed method, the Triple Towers are vibration controlled for wind excitation by using just two active bridges. It is not necessary to use heavy weights like AMDs, therefore by utilizing the proposed method, a lightweight and low-cost control system is realized. Controlled performance was demonstrated using the actual Triple Towers. Although this study used towers with different natural frequencies, it is possible to control towers with the same dynamics. Figure 6.49 shows an outside view of the Triple Towers completed in 2001.

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Figure 6.48 Trajectory of acceleration response measured on Tower Y

6.6 Vibration Control of Bridge Towers Using a Lumped Modeling Approach (Seto et al., 1999) 6.6.1 Vibration Problem of Bridge Towers Under Construction The best-case example where we see the effective weight reduction of a large-scale structure can be found in the construction of the main tower structure of the Kurushima Kaikyo Bridge, which spans the Kurushima Strait in Japan and was completed in 1999. The construction of the Kurushima Kaikyo Bridge started with the construction of the main tower structures as shown in Figure 6.50. Heavy cables were run between the towers and the bridge was suspended from these cables. Consequently, if a strong wind blows while these towers stand independently before being connected with the cables and Karman vortexes occur on the backsides of towers, vortex-induced vibrations occur when the natural vibration frequency of the towers matches that of the frequency of the vortex. Three pairs of main towers were constructed for the Kurushima Kaikyo Bridge. This bridge is very large, with the towers being 179 m, 176 m, 166 m, 143 m, 112 m and 145 m in height, respectively. In constructing the Kurushima Kaikyo Bridge, one of the more important factors was cost reduction. For that purpose, a plan to halve the weight of the bridge was suggested. However, if the weight is halved while keeping the height of the

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Vibration Control of Bridge Towers Using a Lumped Modeling Approach 275

Figure 6.49 Triple Towers completed in 2001

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Figure 6.50 Suspension bridge under construction: (a) towers standing independently; (b) towers connected by a suspension rope

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towers as described above, a new problem of vortex-induced oscillation will naturally arise. The main tower structures had previously only required the consideration of vibration measures of the first-order mode. However, in constructing a lightweight, large-scale structure, the vibrations of higher-order modes require careful consideration. The vibration of the twist mode, in particular, poses a risk of violent vibration accompanied by a pull-in effect, which caused the collapse of the Tacoma Narrows Bridge in Washington State in the past. This problem had been managed by vibration control up to high modes until the safe completion of the construction of the bridges. Each of the six construction companies independently engaged in developing construction plans for the main tower, and various methods including direct velocity feedback, sub-optimal control, H∞ robust control and fuzzy control were applied for vibration control, which ended up becoming a contest of control theories (Spencer and Sain, 1997). The development of active control technology for such large-scale main tower structures is at the forefront of engineering technology. In this section, modeling-based vibration control for the flexible bridge towers with many modes to be controlled is reported.

6.6.2 Controlled Object and Its Dynamic Characteristics Figure 6.51 shows a model bridge tower connected with a model crane tower. The crane tower is connected to the bridge tower during construction. Modal analysis of the structure is first carried out by a finite element method. Figure 6.52 illustrates the first eight vibration modes of the model structure, which are found to be of the same modal shapes as the real structure. The first five modes, including bending and twisting modes, are those requiring attention. In this case, the five modeling points are chosen to be located at the nodes of the sixth mode, as shown in Figure 6.52, with the advantage that the sixth mode will exert no influence on the modeling point. It should be noted that choosing the modeling points is not based on the continuous structure coordinate, but is rather based on the finite element model of the structure; that is, only the nodes of the finite element model are candidates for the modeling points. The normalized modal vector components of the structure calculated by the finite element model at the modeling points are defined as the mode vector components of the lumped 5-d.o.f. model in the same way as shown in equation (5.54).

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278


Figure 6.51 Model structure of a flexible bridge tower with a model crane tower

6.6.3 Five-d.o.f. Modeling of a Scaled Bridge Tower Structure with a Crane Tower The above procedure determines the physical parameters of the lumped parameter system from the available modal data of the distributed parameter system, an inverse modal analysis approach. Figure 6.53 vividly shows the five-d.o.f. lumped parameter model obtained. Two hybrid dynamic absorbers connected to the mass points 3 and 4 are also shown. They are the actuators employed in this chapter (Seto, 1992). This discrete model will make it easy to design a controller in a physical state space where the states are directly measurable. 6.6.4 LQ Control System Design It is not difficult to represent the motion of the lumped system with actuators, as shown in Figure 6.53, in state space. With reference to the symbols shown in Figure 6.53, a state vector is defined as

T X = x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5 x˙ 6 x˙ 7 x1 x2 x3 x4 x5 x6 x7 , (6.38)

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Figure 6.52 Modal shapes of the structure

where x6 and x7 refer to the relative displacements x6 = xd3 − x3 ,

x7 = xd4 − x4 .

(6.39)

Then the system can be described by the state equation ˙ c = Ac Xc + Bc U c , X

(6.40)

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Figure 6.53 The lumped parameter model

yc = C c Xc ,

(6.41)

where

Ac =

A11 A21

A12 A22

with the submatrices defined as ⎡ −K 11 /M1 k12 /M1 k13 /M1 k14 /M1 k15 /M1 −K 22 /M2 k23 /M2 k24 /M2 k25 /M2 ⎢ k12 /M2 ⎢ k23 /M3 −K 33 /M3 k34 /M3 k35 /M3 ⎢ k13 /M3 ⎢ A12 = ⎢ k14 /M4 k24 /M4 k34 /M4 −K 44 /M4 k45 /M4 ⎢ k25 /M5 k35 /M5 k45 /M5 −K 55 /M5 ⎢ k15 /M5 ⎣ −k /M −k23 /M3 K 33 /M3 −k34 /M3 −k35 /M3 13 3 −k14 /M4 −k24 /M4 −k34 /M4 K 44 /M4 −k /M ⎤ 45 4 0 0 0 0 ⎥ ⎥ 0 kd3 /M3 ⎥ ⎥ 0 kd4 /M4 ⎥, ⎥ 0 0 ⎥ ⎦ 0 − (kd3 /md3 + kd3 /M3 ) 0 − (kd4 /md4 + kd4 /M4 )

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with K 11 K 22 K 33 K 44 K 55

= k11 + k12 + k13 + k14 + k15 , = k12 + k22 + k23 + k24 + k25 , = k13 + k23 + k33 + k34 + k35 , = k14 + k24 + k34 + k44 + k45 , = k15 + k25 + k35 + k45 + k55 ,

and A21 = I, A22 = 0, where I and 0 are the 7 × 7 unit and null matrices, respectively. In addition, ⎡ ⎢0 0 0 Bc = ⎢ ⎣ K c3 00 M3

K c4 0 M4 0 0 −K c3

0 1 1 + md3 M3

−K c4

1 1 + md4 M4 0

⎤T 0000000⎥ ⎥ ⎦ 0000000

K c3 , K c4 are the force coefficients of the actuators representing the relationship between the actual control force and the control value in voltage. The first five vibration modes of the structure have been modeled by a lumped parameter system as described above, which forms a standard platform for utilizing LQ control theory to design an optimal controller. In order to handle the remaining high modes properly in avoiding any spillover problems, a low-pass filter and structural filters are introduced. As stated before, by placing sensors and actuators at the nodes of certain modes, the control system will not produce any influence on these modes due to their unobservable characteristics and/or uncontrollability. It seems that a filter cuts off these modes. We regard this as a structural filter. Usually, it is possible to avoid the spillover problem of one or two modes by means of this structure filter, which creates a favorable frequency allowance for the choice of the cutoff frequency of a low-pass filter. The above consideration is clearly illustrated in Figure 6.54. In this figure, a low-pass filter is used to prevent spillover caused by neglected higher modes. Thus, designing an optimal controller using LQ theory becomes a fairly standard problem. Using LQ theory, the optimal feedback gain K = [K f , K c ] is easily designed by an extended system combining the control object denoted by subscript c with a low-pass filter f. It should be noted that due to practical constraints, the states at mass point 5 are not measured; a suboptimal controller (Kosut, 1970) is therefore designed.

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282


Figure 6.54 Block diagram of the filtered optimal controller

Now, designing the optimal controller using LQ theory becomes quite standard. The control values which minimize the performance index ∞ J = (XT QX + U T RU)dt (6.42) are formulated as

0

U = −R−1 BT PX = −KX,

(6.43)

where Q is the state weighting matrix, R is the control weighting matrix, K is the feedback gain matrix and P is the solution of the Riccati equation PA + AT P − PBR−1 BT P + Q = 0.

(6.44)

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It should be mentioned that due to practical constraints, the states at mass point 5 are not measured, a suboptimal controller (Kosut, 1970) is therefore designed based on the sub-state vector Z = x˙ 1

x˙ 2

x˙ 3

x˙ 4

x˙ 6

x˙ 7

x1

x2

x3

x4

x6

x7

T

(6.45)

and the sub-optimal feedback gain reads U = K s Z = KST (SST )−1 SZ,

(6.46)

where S is a transformation matrix between X and Z. 6.6.5 Simulations The numerical simulations on the discrete model are performed using MATLAB Control System Toolbox routines. In these simulations the values of the components that weight velocity terms in Q are all taken as unity, while the components weighting displacements are taken as zero, since it has been established by various simulations and experiments not mentioned in this chapter that weighting velocity terms are more effective than weighting displacement terms, or both. Adjusting the weighting matrix R changes the feedback gains. The proper gains examined in the simulation will be used in the experiments. Therefore, simulation is not only a test of the model but also an indispensable procedure for preparing the feedback gains for the experiments. The values of R used in this chapter are 0.001. In Figures 6.55 and 6.56 the calculated frequency response (compliance) and the impulse response are shown, respectively, under the assumption that an excitation occurs at mass point 1 and picking up at mass point 2. The cut-off frequency of low-pass filters is set to be 30 Hz. It is evident that the five modes can be well controlled. 6.6.6 Experiments Experiments are carried out to verify the modeling and control methods presented in this chapter for controlling the vibration of the bridge tower structure. The experimental setup is shown in Figure 6.57. A personal computer is used as the controller. The displacements at the positions of mass points 1 to 4 in the structure are detected through four absolute displacement sensors, and are input to the computer via A/D converters.

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Figure 6.55 Calculated frequency response

Two hybrid dynamic absorbers (Figure 4.6) are installed at the positions of mass points 3 and 4. The displacements of the absorbers are measured through two strain gauges. The six corresponding velocity signals are calculated by software using finite difference approximation. The control values are sent to the D/A converter and are thereafter amplified to drive the actuators. The feedback gains obtained from numerical simulation are used to generate the control forces in the experiments.

Figure 6.56 Calculated impulse response

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Figure 6.57 Experimental setup

In this experiment, two actuators are placed at the node points of the sixth vibration mode as stated before. Four displacement sensors might also have been placed at the node points of the sixth mode (about 44 Hz) where the modeling points are. However, experimental investigation shows that it is better to place the sensors at the nodal points of the seventh mode (about 65 Hz) which are slightly shifted from the node positions of the sixth mode (see Figure 6.52). In this way, the seventh mode becomes unobservable so that there will be no spillover problem. The sixth and seventh modes, being uncontrollable and unobservable, allow us to consider a relatively high cutoff frequency for the low-pass filters to eliminate the influence of the eighth (about 79 Hz) and higher modes. Otherwise, an even lower cutoff frequency should be necessary, thus leading to degradation of the control effect on the fifth mode (about 37 Hz). The cutoff frequency of the low-pass filters is set as 30 Hz in this research, which proves to be a good tradeoff. It should be pointed out that moving sensors away from the modeling points is only valid when this movement is so small that the detected motions of the controlled modes do not change direction. By exciting at the position of mass point 1 using an impulse hammer and measuring the response at the position of mass point 2, Figure 6.58 shows the experimental frequency response (acceleration) and Figure 6.59

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Figure 6.58 Measured frequency response

the corresponding impulse response of the structure. It can be seen that the first five vibration modes of the structure are well controlled. No adverse influence of the filters on the control performance has been observed. No spillover occurs for the omitted high (e.g. seventh and eighth) modes. 6.6.7 H∞ Robust Control (Kar et al., 2000a) Although the LQ control theory is very useful in designing control systems, H∞ control theory as the dominant robust control theory is used

Figure 6.59 Measured impulse response

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Figure 6.60 Experimental frequency response

for controlling the same bridge tower in order to compare with LQ control. Further discussion will address the effectiveness of this theory as a control system design tool using MATLAB, and DSP. In particular, an interesting experiment has verified its effectiveness in the prevention of spillover. Using the bridge tower under construction shown in Figure 6.51, the effectiveness of H∞ controller with a hybrid dynamic absorber is demonstrated. The experimental frequency and the time responses are shown in Figures 6.60 and 6.61. These responses are obtained using a fast Fourier

Figure 6.61 Experimental impulse response

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transformer analyzer by exciting mass point 1 and observing the response at mass point 2. It is evident that the first four vibration modes are well suppressed and also higher modes are not excited, which implies that there is no spillover instability. The fifth mode (which corresponds to the crane tower) is also well controlled in spite of the unavailability of the state corresponding to the fifth mass point. These measured responses agree well with results of LQ control. In practice, the parameters of a bridge tower change as construction work progresses. To verify the robustness of the proposed control scheme against the parameter variations, an additional mass of 500 g is deliberately attached at the each of the mass points 1 and 2. As a result of the added masses, the natural frequency is reduced and the equivalent mass is increased by about 50% at each of mass points 1 and 2. In this case, the controller gain that is computed at the time of simulation is also used in the control algorithm. The frequency and impulse responses of the controlled structure are shown in Figures 6.62 and 6.63. These results confirm the robustness of the proposed active control scheme against parameter variations. Figure 6.64 shows the outside view of the completed Kurushima Kaikyo Bridge. The development of active control technology for such large-scale main tower structures is at the forefront of engineering technology. It is expected that such a technology will play a primary role in

Figure 6.62 Frequency response with additional mass

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Figure 6.63 Impulse response with additional mass

Figure 6.64 Completed Kurushima Kaikyo Bridge

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290



Figure 6.65 Dream project of large scale suspension bridge

the construction of a dream bridge linking mainland Italy with the island of Sicily, which has been addressed recently. The towers of the planned bridge are to be 370 m in height (Figure 6.65), and naturally, multimode control technology will be required.

6.7 Conclusion In the past, vibration control was employed after design and solely as a countermeasure. More recently, vibration control devices have been included in structural designs from the outset. The active control device built into the Yokohama Landmark Tower described in this chapter is an important example of this trend. With this concept of designing, which incorporates vibration control systems, active control can be demonstrated as being effective, which is considered as a very favorable trend. When high-rise buildings stand together in close proximity, the CCM is effective in reducing the cost of the controller. For example, for controlling three high-rise buildings, a total of six HMDs would normally be required if the conventional method was employed. Using the CCM, the minimum requirements are two controlled bridges (devices). In addition, it is not necessary to use heavy weights like HMDs in the active bridges. Therefore by utilizing the CCM, a lightweight and low-cost control system is realized. The modeling-based vibration control approach is very effective for large-scale continuous structures with distributed parameters, because only vibration mode shapes and their natural frequencies are required to make a lumped mass model. Then it is easy to design a controller using LQ control, robust control theories and so on. This modeling-based vibration control method can similarly be applied to any structures with different dynamics.

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Seto, K. and Matsumoto, Y. (1999) Active vibration control of multiple buildings connected with active control bridges in response to large earthquakes. In Proceedings of the ACC, pp. 1007–1011. Seto, K. and Matsumoto, Y. (2003) Vibration control of multiple connected buildings using active controlled bridges. In Proceedings of the 3rd World Conference on Structural Control, Vol. 3, pp. 253–261. Wiley. Seto, K., Yamashita, S., Ohkuma, M. and Nagamathu, A. (1987) Method of estimating equivalent mass of multi-degree-of-freedom system. JSME International Journal, Series C, Mechanical Systems, Machine Elements and Manufacturing, 30, 1636–1644. Seto, K., Kondo, S. and Ezure, K. (1995) Vibration control method for flexible structures with distributed parameters using a hybrid dynamic absorber. ASME 1995 Design Engineering Technical Conf., DE-Vol. 83-3. Seto, K., Doi, F., and Ren, M. (1999) Vibration control of bridge towers using a lumped modeling approach. Transactions of ASME, Journal of Vibration and Acoustics, 121, 95–100. Spencer, B.F. and Sain, M.K. (1997) Controlling buildings: a new frontier of feedback. IEEE Control System, 17(6), 19–35.

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Index A A/D converters, 283 AC servomotor, 271 active controlled bridge, 157, 223, 269–273 active damping, 1–55, 118, 132, 144, 166 active dynamic absorber, 148, 151–153, 158–175 active mass, 151–153, 155, 158, 169 active mass damper (AMD), 118, 125–130, 150, 223, 249 active truss, 36–37, 40, 43 Active vibration control, 52, 117, 147–148, 150, 154, 176, 222, 249–263 additional mass, 191, 193–197, 239, 288–289 anti-resonance, 16, 19, 129 augmented plant, 217, 236 augmented state vector, 232 B ball screw, 36, 52, 135, 151–152, 271 H∞ -based robust controller, 231–233 bending mode, 157, 225, 228, 251, 264 Bingham model, 106–107 Bode plots, 18 C collocated control system, 15–24 Connected Control Method (CCM), 156–158, 222


constrained system, 16, 18–20, 27, 30, 42, 49 constrained system, 16, 18–20, 27, 30, 42, 49 control input, 3–5, 36, 49, 91, 95, 97, 140, 231–232 control spillover, 184 control value, 159, 161–162, 169, 241, 243, 246, 256, 260, 267, 281–282, 284 control variable, 106, 256 controllability, 25, 186, 219, 235, 239, 243, 249, 281 controlled mode, 180, 183–184, 246, 285 cross-talk, 46–47 cubic architecture, 71 cumulative PSD, 5 cumulative RMS value, 98, 101, 132, 143–144 cut-off frequency of the filter, 217, 283 D D/A converter, 246, 260, 284 decentralized control, 46–55, 73–76 diagonal matrix, 188–189, 201, 211 digital signal processor (DSP), 148, 217 direct output feedback control, 180 direct velocity feedback (DVF), 29–31, 118, 136–137, 277 disturbance input, 232 dominant frequency, 257 dual (pair), 15, 68, 127, 129, 141 DVF–IFF duality, 45, 118

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294 dynamic absorber, 118, 125–126, 148, 150–153, 158–175 dynamic amplification factor, 13 dynamic flexibility matrix, 12–15, 20 dynamic mass, 10–11 dynamic vibration absorber (DVA), 2, 101–106, 118, 120–126, 158 E earthquake, 1, 118, 175, 222–223, 249, 257, 259 effective modal mass, 11 eigenvector, 191–193, 195, 202, 204 energy converter, 2, 63 equal peak design, 104, 122, 124 equivalent mass, 124, 180, 188, 190–197, 199–202, 207, 212, 214 error budget, 4–6 error function, 198, 202, 207, 211–212, 214, 234–235 error vector, 189, 206, 211 experimental modal analyzers, 148 F feedback gain, 16, 69, 129, 159–161, 163–164, 217, 227, 244, 246, 256, 260, 267, 281–284 feedback gain vector, 160–161, 163–164 feedforward, 2, 57 feedthrough, 14, 17, 25, 27, 31, 49, 52 FFT analyzer, 148, 260, 287–288 filtered LQ control, 217–219, 239–249, 256–257 flexible structure, 68–69, 150, 176, 180, 184, 187, 219, 222, 239, 249 force feedback, 28, 35–36, 40–43, 66–69, 73–74, 118, 141 force transmission coefficient, 230 fraction of modal strain energy, 39, 43 Frobenius ¨ norm, 89 G gain stability, 27 Gough–Stewart platform, 70

Index H Hamilton’s principle, 6–7, 37 Hexapod, 71 high authority control (HAC), 25 high-rise building, 154, 175–176, 221–222, 225, 249, 263, 290 H-infinity based control, 180 H-infinity output feedback control, 217 H-infinity state feedback control, 217 hybrid control, 2, 117–144, 223, 249 hybrid mass damper (HMD), 150, 223 I ideal high pass filter, 235 integral force feedback (IFF), 28, 35–36, 40–43, 73–74, 118, 141 J jerk, 90 joints (flexible), 71, 74, 76–77, 79 joints (spherical), 70–73, 81 K Kallman Vortex excitation, 150, 157 Karman vortex, 150, 274 L lag compensator, 44, 46 large earthquake, 249–263 lead compensator, 26–28, 46, 50 linear quadratic (LQ) control, 148 linear quadratic regulator, 94 long-span suspension bridge, 221–222 low authority control (LAC), 25 low-pass filter (s), 93, 185, 217, 236, 241, 243–244, 248–249, 256, 267, 281, 283, 285 LQ control theory, 148, 160, 162, 176, 179, 222, 227, 256, 267, 281, 286 lumped mass model, 180, 183, 201, 203, 205, 207, 213, 229, 239–240, 252, 265

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295

Index M

P

magneto-rheological fluid, 2 mass and stiffness matrixes, 213 mass response method, 191, 193–197, 199, 224 Maxwell unit, 60 membrane, 76–79 mixed sensitivity criteria, 232 mixed sensitivity minimization problem, 231 modal coordinate, 3, 8–10, 12, 29, 38, 43, 73, 123, 181, 187, 192, 195 modal damping, 9, 29–31, 34, 42, 136, 142, 190–191 modal matrix, 187–190, 195, 201, 203, 206–207, 210–213 modal participation factor, 10 modal stiffness, 190–191 modal-space control method, 179 modeling point, 186–188, 205, 207, 209–210, 212–213, 215, 224, 252, 277, 285 modification vector, 189–190 moving coil transducer, 62–63 multiple vibration modes, 222

performance index, 90, 94–95, 97, 162, 243, 282 plate structure, 180, 203–209, 228–229, 238 pole/zero flipping, 23 positive position feedback (PPF), 31–35, 50, 118, 137–140 power requirements, 2, 143–144 power spectral density (PSD), 4, 90 proportional damping, 219, 254 pull-in effect, 277

N neglected (uncontrolled) mode, 235 nodes of a mode, 186–187, 199, 205, 210, 239–240, 243, 277, 281 non-dimensional state equation, 160–162 non-minimum phase, 24 H∞ norm, 230–231 norm inequality, 232 normalized modal matrix, 188, 202, 206–207, 211–212 Nyquist plot, 18 O observability, 186 observation spillover, 184 optimum adjustment, 150 output equation, 38, 48, 52, 73, 160, 230, 256

Q quadratic performance index, 162 quarter-car model, 91, 103, 107, 111–113 R Reduced-Order Physical Modeling method, 179 regulated variable, 231–232 relative displacement, 46, 49, 68–69, 90, 95–96, 101, 118, 133, 135, 153, 158–159, 163, 169–170, 173, 226–227, 229, 238, 240, 259, 279 relaxation isolator, 60–64, residual mode, 3, 14, 17, 182–183 resonance peak, 2, 24, 101, 125, 167, 169–170, 175, 215, 228, 240, 245–246, 248–249, 252, 254, 261, 263 Riccati equation, 162–163, 217, 232–233, 236, 244, 282 road profile, 89, 91, 113 robust control, 184–185, 217–218, 221–222, 228, 230, 277, 286–290 robust vibration control, 217 roll-off, 26–27, 31, 34, 107 Root Locus, 25–28, 30–31, 33–35, 40–45, 50–52, 55, 62, 66–67, 74, 127, 129, 131–132, 136–138, 142, 165–169, 171 Routh–Hurwitz criterion, 33

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296 S seismic excitation, 6–8, 12, 117, 120, 144, 259–260, 263 Seismic Response Control, 257–259, 261–263 seismic response, 6, 257–259, 261–263 semi-active device, 2, 106–107 semi-active sky-hook, 109–113 semi-active suspension, 106–113 sensitivity matrix, 189, 198, 202, 207, 212, 214 servodamper, 147, 152 servomotor, 151–152, 155, 271 shear control, 133–136, 138 shear frame, 46–47, 49, 54–55, 117, 119, 124–126, 128, 135–136, 138–140, 141, 143 six-axis isolator, 69–89 sky-hook damper, 64–66, 67, 69, 73, 92–93, 95–97, 101 spillover instability, 3, 173, 179, 221, 239, 256, 267, 288 spillover, 3, 173, 179-180, 183–185, 187, 215, 217–218, 235, 239–241, 243, 246–248, 256–257, 281, 285–288 state equation, 159–161, 163, 225–227, 229, 240, 241–242, 256–257, 267, 279 state feedback, 92–97, 101, 159–163, 173, 217–218, 227, 231–233, 236, 238, 244 state vector, 103, 159, 164, 232, 236, 240, 242, 256, 260, 267, 278, 283 static state feedback control, 231–233, 236 stiffness and damping matrices, 254 strain gauge, 153, 173, 238, 246, 284 structural filter, 219, 233, 246, 249, 281 structural filtering, 233, 246 suspension (active), 89–106 suspension (adaptive), 90, 106 suspension (semi-active), 106–113 T thermal analogy, 37 time constant, 160–161, 163, 171–173 tower-like structure, 180, 197–203, 219, 222, 225–228

Index transfer function, 4, 16–17, 26–27, 32–34, 38–40, 58–59, 65–67, 95, 111, 118, 137, 190–191, 231–233, 238 transmissibility, 4, 58–59, 61–66, 75–76, 81–83, 85–88, 93–95, 98–101, 105, 108, 110, 112, 125, 129–130, 132–134, 138–139, 143–144 transmission zeros, 14, 17–20, 47–55, 70, 135–137, 140–141 Triple Tower, 156–158, 264–275 tuned mass damper (TMD), 118, 120, 150, 223 twist mode, 277 twisting mode, 180, 209, 228, 239, 249, 251–252, 257, 261, 277 U uncertainty, 231–233, 235, 237 uncontrolled mode, 3, 182, 184, 217, 221, 235, 242–243 unmodeled uncertainty, 235 upper bound of the uncertainty, 232 V vibration isolation, 90, 149 vibration mode shape, 187, 189, 199–200, 203, 205–207, 210, 212, 251, 265, 290 voice coil, 63, 76–79 vortex-induced oscillation, 277 W weight reduction, 222, 274 weighting factor, 162–164, 166–167, 174, 227 weighting filter, 232–234 weighting function, 217, 231–232, 234–235, 237–238 weighting matrix, 162–164, 202, 214, 227, 236, 244, 246, 282–283 wheel-hop mode, 91, 95, 104, 106 Z zeros, 3, 14, 16–27, 31–33, 39–44, 47–55, 66–70, 75, 118, 127–129, 131–132, 135–137, 140–141

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