Vibration control of active structures - Actives Structures Laboratory

500 Pages·2011·7.91 MB·English

Andr´e PREUMONT
Universit´e Libre de Bruxelles
Active Structures Laboratory
Vibration Control of Active
Structures, An Introduction
rd
3 Edition
Springer
Berlin Heidelberg NewYork
HongKong London
Milan Paris Tokyo

“...le travail ´eloigne de nous
trois grands maux:
l’ennui, le vice et le besoin. ”
Voltaire, Candide (XXX)

Contents
Preface to the third edition ...........................X. VII
Preface to the second edition ..........................XIX
Preface to the first edition .............................XXI
1 Introduction ........................................... 1
1.1 Active versus passive ................................. 1
1.2 Vibration suppression................................. 5
1.3 Smart materials and structures ........................ 6
1.4 Control strategies .................................... 8
1.4.1 Feedback ...................................... 9
1.4.2 Feedforward ................................... 10
1.5 The various steps of the design......................... 12
1.6 Plant description, error and control budget .............. 14
1.7 Readership and Organization of the book ............... 16
1.8 References........................................... 17
1.9 Problems............................................ 19
2 Some concepts in structural dynamics.................. 21
2.1 Introduction......................................... 21
2.2 Equation of motion of a discrete system................. 21
2.3 Vibration modes ..................................... 23
2.4 Modal decomposition ................................. 24
2.4.1 Structure without rigid body modes............... 24
2.4.2 Dynamic flexibility matrix ....................... 26
2.4.3 Structure with rigid body modes ................. 28
2.4.4 Example....................................... 31
VIII Contents
2.5 Collocated control system ............................. 33
2.5.1 Transmission zeros and constrained system......... 36
2.6 Continuous structures ................................ 38
2.7 Guyan reduction ..................................... 39
2.8 Craig-Bampton reduction ............................. 41
2.9 References........................................... 43
2.10 Problems............................................ 44
3 Electromagnetic and piezoelectric transducers ......... 47
3.1 Introduction......................................... 47
3.2 Voice coil transducer ................................. 48
3.2.1 Proof-mass actuator ............................ 49
3.2.2 Geophone ..................................... 52
3.3 General electromechanical transducer ................... 53
3.3.1 Constitutive equations .......................... 53
3.3.2 Self-sensing .................................... 54
3.4 Reaction wheels and gyrostabilizers..................... 55
3.5 Smart materials...................................... 56
3.6 Piezoelectric transducer ............................... 57
3.6.1 Constitutive relations of a discrete transducer ...... 58
3.6.2 Interpretation of k2 ............................. 63
3.6.3 Admittance of the piezoelectric transducer ......... 64
3.7 References........................................... 65
3.8 Problems............................................ 66
4 Piezoelectric beam, plate and truss .................... 69
4.1 Piezoelectric material ................................. 69
4.1.1 Constitutive relations ........................... 69
4.1.2 Coenergy density function ....................... 72
4.2 Hamilton’s principle .................................. 74
4.3 Piezoelectric beam actuator ........................... 74
4.3.1 Hamilton’s principle ............................ 75
4.3.2 Piezoelectric loads .............................. 77
4.4 Laminar sensor ...................................... 79
4.4.1 Current and charge amplifiers .................... 79
4.4.2 Distributed sensor output........................ 80
4.4.3 Charge amplifier dynamics....................... 81
4.5 Spatial modal filters .................................. 82
4.5.1 Modal actuator................................. 82
4.5.2 Modal sensor................................... 83
Contents IX
4.6 Active beam with collocated actuator-sensor............. 85
4.6.1 Frequency response function ..................... 85
4.6.2 Pole-zero pattern ............................... 87
4.6.3 Modal truncation ............................... 88
4.7 Admittance of a beam with a piezoelectric patch ......... 89
4.8 Piezoelectric laminate................................. 92
4.8.1 Two dimensional constitutive equations ........... 92
4.8.2 Kirchhoff theory................................ 93
4.8.3 Stiffness matrix of a multi-layer elastic laminate .... 94
4.8.4 Multi-layer laminate with a piezoelectric layer...... 96
4.8.5 Equivalent piezoelectric loads .................... 97
4.8.6 Sensor output .................................. 97
4.8.7 Beam model vs. plate model ..................... 99
4.8.8 Additional remarks ............................. 102
4.9 Active truss ......................................... 102
4.9.1 Open-loop transfer function...................... 106
4.9.2 Admittance function ............................ 107
4.10 Finite element formulation ............................ 108
4.11 References........................................... 109
4.12 Problems............................................ 110
5 Passive damping with piezoelectric transducers ........ 113
5.1 Introduction......................................... 113
5.2 Resistive shunting .................................... 114
5.3 Inductive shunting ................................... 116
5.4 Switched shunt ...................................... 119
5.4.1 Equivalent damping ratio ........................ 123
5.5 References........................................... 124
5.6 Problems............................................ 126
6 Collocated versus non-collocated control ............... 127
6.1 Introduction......................................... 127
6.2 Pole-zero flipping..................................... 128
6.3 The two-mass problem................................ 130
6.3.1 Collocated control .............................. 131
6.3.2 Non-collocated control .......................... 132
6.4 Notch filter.......................................... 133
6.5 Effect of pole-zero flipping on the Bode plots ............ 135
6.6 Nearly collocated control system ....................... 136
6.7 Non-collocated control systems......................... 137
X Contents
6.8 The role of damping .................................. 140
6.9 References........................................... 140
6.10 Problems............................................ 141
7 Active damping with collocated system ................ 143
7.1 Introduction......................................... 143
7.2 Lead control......................................... 145
7.3 Direct velocity feedback (DVF) ........................ 147
7.4 Positive Position Feedback (PPF) ...................... 150
7.5 Integral Force Feedback(IFF) .......................... 153
7.6 Duality between the Lead and the IFF controllers ........ 158
7.6.1 Root-locus of a single mode ...................... 158
7.6.2 Open-loop poles and zeros ....................... 160
7.7 Actuator and sensor dynamics ......................... 160
7.8 Decentralized control with collocated pairs .............. 162
7.8.1 Cross talk ..................................... 162
7.8.2 Force actuator and displacement sensor............ 162
7.8.3 Displacement actuator and force sensor............ 163
7.9 References........................................... 164
7.10 Problems............................................ 165
8 Vibration isolation ..................................... 169
8.1 Introduction......................................... 169
8.2 Relaxation isolator ................................... 172
8.2.1 Electromagnetic realization ...................... 174
8.3 Active isolation ...................................... 175
8.3.1 Sky-hook damper............................... 177
8.3.2 Integral Force Feedback ......................... 177
8.4 Flexible body........................................ 180
8.4.1 Free-free beam with isolator...................... 182
8.5 Payload isolation in spacecraft ......................... 185
8.5.1 Interaction isolator/attitude control............... 186
8.5.2 Gough-Stewart platform......................... 187
8.6 Six-axis isolator...................................... 188
8.6.1 Relaxation isolator.............................. 189
8.6.2 Integral Force Feedback ......................... 190
8.6.3 Spherical joints, modal spread.................... 191
8.7 Active vs. passive .................................... 193
8.8 Car suspension....................................... 195
8.9 References........................................... 200
Contents XI
8.10 Problems............................................ 202
9 State space approach................................... 207
9.1 Introduction......................................... 207
9.2 State space description................................ 209
9.2.1 Single degree of freedom oscillator ................ 209
9.2.2 Flexible structure............................... 210
9.2.3 Inverted pendulum.............................. 212
9.3 System transfer function .............................. 213
9.3.1 Poles and zeros................................. 215
9.4 Pole placement by state feedback....................... 216
9.4.1 Example: oscillator ............................. 218
9.5 Linear Quadratic Regulator ........................... 219
9.5.1 Symmetric root locus ........................... 220
9.5.2 Inverted pendulum.............................. 221
9.6 Observer design...................................... 222
9.7 Kalman Filter ....................................... 225
9.7.1 Inverted pendulum.............................. 226
9.8 Reduced order observer ............................... 228
9.8.1 Oscillator...................................... 228
9.8.2 Inverted pendulum.............................. 229
9.9 Separation principle .................................. 230
9.10 Transfer function of the compensator ................... 231
9.10.1The two-mass problem .......................... 232
9.11 References........................................... 235
9.12 Problems............................................ 235
10 Analysis and synthesis in the frequency domain........ 237
10.1 Gain and phase margins .............................. 237
10.2 Nyquist criterion ..................................... 238
10.2.1Cauchy’s principle .............................. 238
10.2.2Nyquist stability criterion........................ 239
10.3 Nichols chart ........................................ 243
10.4 Feedback specification for SISO systems................. 244
10.4.1Sensitivity ..................................... 245
10.4.2Tracking error.................................. 245
10.4.3Performance specification ........................ 246
10.4.4Unstructured uncertainty ........................ 247
10.4.5Robust performance and robust stability........... 248
10.5 Bode gain-phase relationships.......................... 250
XII Contents
10.6 The Bode Ideal Cutoff ................................ 254
10.7 Non-minimum phase systems .......................... 256
10.8 Usual compensators .................................. 259
10.8.1System type ................................... 259
10.8.2Lead compensator .............................. 260
10.8.3PI compensator ................................ 261
10.8.4Lag compensator ............................... 261
10.8.5PID compensator............................... 263
10.9 Multivariable systems................................. 263
10.9.1Performance specification ........................ 263
10.9.2Small gain theorem ............................. 265
10.9.3Stability robustness tests ........................ 265
10.9.4Residual dynamics .............................. 266
10.10References........................................... 268
10.11Problems............................................ 269
11 Optimal control ........................................ 273
11.1 Introduction......................................... 273
11.2 Quadratic integral.................................... 273
11.3 Deterministic LQR ................................... 274
11.4 Stochastic response to a white noise .................... 276
11.4.1Remark ....................................... 278
11.5 Stochastic LQR ...................................... 278
11.6 Asymptotic behavior of the closed-loop ................. 279
11.7 Prescribed degree of stability .......................... 281
11.8 Gain and phase margins of the LQR.................... 282
11.9 Full state observer.................................... 284
11.9.1Covariance of the reconstruction error ............. 285
11.10Kalman-Bucy Filter (KBF)............................ 285
11.11Linear Quadratic Gaussian (LQG)...................... 286
11.12Duality ............................................. 287
11.13Spillover ............................................ 288
11.13.1Spillover reduction.............................. 291
11.14Loop Transfer Recovery (LTR)......................... 292
11.15Integral control with state feedback..................... 293
11.16Frequency shaping ................................... 294
11.16.1Frequency-shaped cost functionals ................ 294
11.16.2Noise model ................................... 298
11.17References........................................... 299
11.18Problems............................................ 300