Vibration Control of Active Structures. Second Edition. An Introduction

Vibration Control of Active Structures. Second Edition. An Introduction

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English

Description

This text is an introduction to the dynamics of active structures and to the feedback control of lightly damped flexible structures; the emphasis is placed on basic issues and simple control strategies that work.
This book consists of 14 chapters. Chapters 2 and 3 are devoted to the dynamics of active structures; the open loop transfer functions are derived from the constitutive equations; the discussion includes active trusses with piezoelectric struts, and beams and shells with embedded laminar piezoelectric actuators and sensors. Chapters 4 and 5 discuss the virtues of collocated actuator/sensor configurations and how they can be exploited to develop active damping with guaranteed stability. Chapter 6 addresses vibration isolation for one and 6 d.o.f.. Chapter 7 discusses optimal control for SISO systems with symmetric root locus. Chapter 8 discusses the design tradeoffs for SISO systems in the frequency domain, including the Bode amplitude/phase relationship. Chapter 9 provides a more general discussion of optimal control using of optimal control using the Riccati equation; spillover is examined. Chapters 10 and 11 review briefly the concepts of controllability, observability and stability. Chapter 12 discusses the semi-active control, including some materials on magneto-rheological fluids. Chapter 13 describes various practical applications to active damping, precision positioning and vibroacoustics, and chapter 14 discusses the active damping of cable- structures.
This book is intended for structural engineers who want to acquire some background in vibration control; it can be used as a textbook for a graduate course on vibration control or active structures. The text is supplemented with 113 problems; a solutions manual is available through the publisher to teachers using this book as a textbook.

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Published 01 January 1983
Reads 3
EAN13 0306484226
License: All rights reserved
Language English

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Contents
1
Preface to the second edition
Preface to the first edition
Introduction 1.1Active versus passive 1.2Smart materials and structures 1.3Control strategies 1.3.1Feedback 1.3.2Feedforward 1.4The various steps of the design 1.5Organization of the book 1.6References 1.7Problems
2Some concepts of structural dynamics 2.1Equation of motion of a discrete system 2.2Vibration modes 2.3Modal decomposition 2.3.1Structure without rigid body modes 2.3.2Structure with rigid body modes 2.3.3Example 2.4Transfer function of collocated systems 2.5Continuous structures 2.6Guyan reduction 2.7References 2.8Problems
3Actuators, piezoelectric materials, and active structures 3.1Introduction 3.2Proofmass actuator 3.3Reaction wheels and gyrostabilizers
vii
xv
xvii
1 1 5 7 7 9 10 12 13 15
17 17 18 20 20 22 25 26 30 31 33 34
37 37 38 40
viii
4
Vibration control of active structures
3.4Piezoelectric actuators 3.4.1Constitutive equations 3.4.2Linear actuator 3.4.3Laminar actuator 3.4.4Laminar sensor 3.4.5Example: Tip displacement of a cantilever beam 3.4.6Spatial modal filters 3.5Passive damping with piezoceramics 3.6Active cantilever beam 3.7Active truss 3.8Piezoelectric shell 3.8.1Twodimensional constitutive equations 3.8.2Kirchhoff shell 3.9Finite element formulation 3.10References 3.11 Problems Collocated versus noncollocated control 4.1Introduction 4.2Polezero flipping 4.3Collocated control 4.4Noncollocated control 4.5Notch filter 4.6Polezero flipping in the structure 4.7Effect on the Bode plots 4.8Relation to the mode shapes 4.9The role of damping 4.10References 4.11Problems
5Active damping with collocated pairs 5.1Introduction 5.2Direct Velocity Feedback 5.2.1Lead compensator 5.3Acceleration feedback 5.3.1Direct Velocity Feedback 5.3.2Second order filter 5.3.3SISO system with many modes 5.3.4Multidimensional case 5.4Positive Position Feedback 5.4.1SISO system 5.4.2Multidimensional case 5.5Integral Force Feedback 5.5.1Modal damping
40 41 46 48 51 52 53 54 55 58 63 64 65 68 70 73
75 75 76 78 80 80 82 84 84 87 88 88
91 91 93 94 96 97 97 98 100 101 101 102 103 104
CONTENTS
6
7
5.6
5.7 5.8
Remarks 5.6.1Controllability, observability 5.6.2Actuator and sensor dynamics References Problems
Active vibration isolation 6.1Introduction 6.2Passive isolator 6.3The “skyhook” damper 6.4Force feedback 6.5Flexible clean body 6.5.1Freefree beam with isolator 6.66 d.o.f. isolator 6.7Decentralized control of the 6 d.o.f. isolator 6.7.1 Remarks 6.8Pointing control 6.9Vehicle suspension 6.10References 6.11 Problems
State space approach 7.1Introduction 7.2State space description 7.2.1Single degree of freedom oscillator 7.2.2Flexible structure 7.2.3Inverted pendulum 7.3System transfer function 7.3.1Poles and zeros 7.4Pole placement by state feedback 7.4.1Example: oscillator 7.5Linear Quadratic Regulator 7.5.1Symmetric root locus 7.5.2Inverted pendulum 7.6Observer design 7.7Kalman Filter 7.7.1Inverted pendulum 7.8Reduced order observer 7.8.1Oscillator 7.8.2Inverted pendulum 7.9Separation principle 7.10Transfer function of the compensator 7.10.1The twomass problem 7.11References
ix
107 107 107 109 110
113 113 114 116 118 119 121 124 127 128 129 131 133 134
137 137 139 139 140 141 142 144 145 146 148 149 149 151 153 154 155 156 156 157 158 159 162
x
8
9
7.12Problems
Vibration control of active structures
Analysis and synthesis in the frequency domain 8.1Gain and phase margins 8.2Nyquist criterion 8.2.1Cauchy’s principle 8.2.2Nyquist stability criterion 8.3Nichols chart 8.4Feedback specification for SISO systems 8.4.1Sensitivity 8.4.2Tracking error 8.4.3Performance specification 8.4.4Unstructured uncertainty 8.4.5Robust performance and robust stability 8.5Bode gainphase relationships 8.6The Bode Ideal Cutoff 8.7Nonminimum phase systems 8.8Usual compensators 8.8.1System type 8.8.2Lead compensator 8.8.3PI compensator 8.8.4Lag compensator 8.8.5PID compensator 8.9References 8.10 Problems
Optimal control 9.1Introduction 9.2Quadratic integral 9.3Deterministic LQR 9.4Stochastic response to a white noise 9.4.1Remark 9.5Stochastic LQR 9.6Asymptotic behaviour of the closedloop 9.7Prescribed degree of stability 9.8Gain and phase margins of the LQR 9.9Full state observer 9.9.1Covariance of the reconstruction error 9.10KalmanBucy Filter (KBF) 9.11Linear Quadratic Gaussian (LQG) 9.12Duality 9.13Spillover 9.13.1Spillover reduction 9.14Loop Transfer Recovery (LTR)
163
165 165 166 166 167 170 171 171 172 173 174 175 178 181 183 185 185 187 187 189 189 189 190
193 193 193 194 196 197 197 198 200 201 202 204 204 205 205 206 209 210
CONTENTS
9.15 Integral control with state feedback 9.16 Frequency shaping 9.16.1 Frequencyshaped cost functionals 9.16.2 Noise model 9.17References 9.18 Problems
10Controllability and Observability 10.1 Introduction 10.1.1Definitions 10.2Controllability and observability matrices 10.3 Examples 10.3.1 A cart with two inverted pendulums 10.3.2Double inverted pendulum 10.3.3 Two d.o.f. oscillator 10.4State transformation 10.4.1 Control canonical form 10.4.2Left and right eigenvectors 10.4.3Diagonal form 10.5PBH test 10.6 Residues 10.7 Example 10.8Sensitivity 10.9 Controllability and observability Gramians 10.10 Relative controllability and observability 10.10.1Internally balanced coordinates 10.11 Model reduction 10.11.1Transfer equivalent realization 10.11.2Internally balanced realization 10.11.3Example 10.12References 10.13 Problems
11Stability 11.1 Introduction 11.1.1Phase portrait 11.2 Linear systems 11.2.1RouthHurwitz criterion 11.3 Liapunov’s direct method 11.3.1Introductory example 11.3.2Stability theorem 11.3.3Asymptotic stability theorem 11.3.4Lasalle’s theorem 11.3.5Geometric interpretation
xi
211 212 212 215 216 217
221 221 222 222 224 224 226 226 227 228 229 230 230 231 232 233 234 235 235 237 237 237 238 240 241
245 245 246 247 248 249 249 249 251 251 252
xii
11.4 11.5 11.6 11.7 11.8 11.9
Vibration control of active structures
11.3.6Instability theorem Liapunov functions for linear systems Liapunov’s indirect method An application to controller design Energy absorbing controls References Problems
12Semiactive control 12.1 Introduction 12.2Magnetorheological(MR) fluids 12.3MR devices 12.4Semiactive control 12.5Openloop control 12.6Feedback control 12.6.1Continuous control 12.6.2Onoff control 12.6.3 Force feedback 12.7References 12.8 Problems
13Applications 13.1 Digital implementation 13.1.1Sampling, aliasing and prefiltering 13.1.2Zeroorder hold, computational delay 13.1.3Quantization 13.1.4Discretization of a continuous controller 13.2 Active damping of a truss structure 13.2.1Actuator placement 13.2.2 Implementation, experimental results 13.3 Active damping generic interface 13.3.1Active damping 13.3.2 Experiment 13.3.3Pointing and position control 13.4 Active damping of a plate 13.4.1Control design 13.5 Active damping of a stiff beam 13.5.1System design 13.6 The HAC/LAC strategy 13.6.1Wideband position control 13.6.2 Compensator design 13.6.3Results 13.7 Volume displacement sensors 13.7.1QWSIS sensor
253 254 255 256 257 259 259
263 263 264 266 267 268 268 268 273 274 275 275
277 277 278 279 280 281 282 282 284 286 287 288 290 290 291 293 294 295 297 299 299 302 303
CONTENTS
13.7.2 Discrete array sensor 13.7.3 Spatial aliasing 13.7.4Distributed sensor 13.8References 13.9 Problems
14Cable StructuresTendon Control of 14.1 Introduction 14.2Tendon control of strings and cables 14.3 Active damping strategy 14.4 Basic Experiment 14.5 Approximate linear theory 14.6 Application to space structures 14.6.1 Guyed truss experiment 14.6.2 JPLMPI testbed 14.6.3 Free floating truss experiment 14.6.4Microvibrations 14.7Application to cablestayed bridges 14.7.1 Laboratory experiment 14.7.2Control of parametric resonance 14.7.3 Large scale experiment 14.8References
Bibliography
Index
xiii
305 308 309 316 319
321 321 323 324 325 327 329 329 331 331 334 335 335 336 338 344
347
361