Introduction to Structural Motion Control

Cover......Page 1
Part II: Active Control......Page 0
Limitations of conventional structural design......Page 13
Motion based structural design and motion control......Page 15
1.2 Motion versus strength issues for building type structures......Page 16
Example 1.1: Cantilever shear beam......Page 18
Example 1.2: Cantilever bending beam......Page 19
Example 1.3: Quasi-shear beam frame......Page 21
Response for periodic excitation......Page 23
Design criteria......Page 28
Methodology for acceleration controlled design......Page 29
Example 1.4: An illustration of acceleration controlled design......Page 32
Example 1.5: An illustration of displacement controlled design......Page 33
Methodology for force controlled design......Page 35
Example 1.6: Force reduction......Page 37
Response for periodic support motion......Page 38
Example 1.7: Controlling acceleration due to ground motion......Page 39
Example 1.8: Controlling relative motion due to ground motion......Page 40
1.5 Stiffness distribution for a two degree-of-freedom system......Page 42
1.6 Control strategies for motion based design......Page 45
1.7 Scope of text......Page 54
Problem 1.3......Page 57
Problem 1.7......Page 58
Problem 1.10......Page 59
Problem 1.11......Page 60
Problem 1.13......Page 61
2.1 Introduction......Page 63
2.2 Governing equations - transverse bending of planar beams......Page 65
Planar deformation-displacement relations......Page 66
Optimal deformation and displacement profiles......Page 67
Equilibrium equations......Page 69
Force-deformation relations......Page 70
Example 2.1. Composite sandwich beam......Page 71
Example 2.2. Equivalent rigidities for a discrete Truss-beam......Page 72
Governing equations for buildings modelled as pseudo shear beams......Page 76
2.3 Stiffness distribution for a continuous cantilever beam under static loading......Page 81
Example 2.3. Cantilever beam - quasi static seismic loading......Page 83
Example 2.4. Truss-beam revisited......Page 84
2.4 Stiffness distribution for a discrete cantilever shear beam - static loading......Page 87
Example 2.5. 3 DOF shear beam......Page 88
2.5 Stiffness distribution - truss under static loading......Page 89
Example 2.6. Application of least square approaches......Page 93
Example 2.7. Comparison of strength vs. displacement based design......Page 101
Rigidity distribution - undamped response......Page 105
2.7 Stiffness distribution for a discrete shear beam - dynamic response......Page 110
Example 2.8. 3 DOF shear beam......Page 111
Governing equations- fundamental mode response of a discrete shear beam......Page 112
Example 2.9. 3 DOF shear beam revisited......Page 113
Governing equations - fundamental mode response of a continuous beam......Page 117
Example 2.10. One dimensional parameters - continuous beam......Page 120
Stiffness calibration - periodic excitation......Page 121
Example 2.11. 3 DOF shear beam revisited again......Page 123
Example 2.12. Stiffness calibration - continuous beam......Page 125
Example 2.13. Example 2.12 revisited......Page 126
Stiffness calibration - seismic excitation......Page 128
Seismic response spectra......Page 131
Example 2.14. Example 2.12 revisited for seismic excitation......Page 140
Example 2.15. 5 DOF shear beam......Page 141
2.9 Examples of stiffness distribution for buildings......Page 144
Iterative procedure......Page 151
Multiple mode response......Page 152
Building examples - iterated stiffness distribution......Page 155
Problem 2.2......Page 161
Problem 2.4......Page 162
Problem 2.6......Page 163
Problem 2.7......Page 164
Problem 2.10......Page 165
Problem 2.12......Page 166
Problem 2.14......Page 167
Problem 2.16......Page 168
Problem 2.17......Page 169
Problem 2.18......Page 170
Problem 2.20......Page 171
Problem 2.21......Page 172
Problem 2.22......Page 173
Problem 2.25......Page 174
Problem 2.26......Page 175
Problem 2.29......Page 176
3.1 Introduction......Page 179
Viscous damping......Page 183
Example 3.1: Viscous damper......Page 184
Friction damping......Page 186
Hysteretic damping......Page 188
Example 3.2: Stiffness of a rod hysteretic damper......Page 189
Example 3.3: Stiffness of two hysteretic dampers in series......Page 190
3.3 Viscoelastic material damping......Page 191
Example 3.4: Viscoelastic damper......Page 194
3.4 Equivalent viscous damping......Page 196
Example 3.5: Structural and hysteretic damping comparison - seismic excitation......Page 198
Example 3.6: Determining for 3M ISD110 damping material......Page 203
Damping systems......Page 204
Rigid structural members - linear viscous behavior......Page 208
Example 3.7: Example 2.15 revisited......Page 209
Rigid structural members - linear viscoelastic behavior......Page 211
Example 3.8:......Page 212
Example 3.9: Example 3.7 revisited......Page 213
Example 3.10: Viscoelastic damper design......Page 214
Example 3.11: Hysteretic damper design - diagonal element......Page 216
Flexible structural members - linear viscoelastic behavior......Page 217
Example 3.12: Coupled spring-damper model......Page 220
3.6 Damping parameters - truss-beam......Page 221
Linear viscous behavior......Page 223
Linear viscoelastic behavior......Page 224
Multi mode free vibration response......Page 225
Example 3.13: Eigenvalue problem - 2DOF......Page 228
Example 3.14: Modal response for nonproportional damping......Page 235
Stiffness proportional viscous damping......Page 238
Example 3.15: Low rise buildings.......Page 240
Example 3.16: Building #4......Page 248
Problem 3.3......Page 255
Problem 3.5......Page 256
Problem 3.8......Page 257
Problem 3.9......Page 258
Problem 3.10......Page 259
Problem 3.11......Page 260
Problem 3.14......Page 261
Problem 3.16......Page 262
Problem 3.18......Page 263
Problem 3.19......Page 264
Problem 3.20......Page 265
Problem 3.22......Page 267
Problem 3.24......Page 268
Problem 3.25......Page 270
4.1 Introduction......Page 271
4.2 An introductory example......Page 272
Example 4.1: Preliminary design of a TMD for a SDOF system......Page 274
4.3 Examples of existing tuned mass damper systems......Page 276
Undamped structure - undamped TMD......Page 287
Undamped structure - damped TMD......Page 290
Damped structure - damped TMD......Page 303
Example 4.3: Design of a TMD for a damped SDOF system......Page 309
4.5 Case studies - SDOF systems......Page 310
4.6 Tuned mass damper theory for MDOF systems......Page 317
Example 4.4: Design of a TMD for a damped MDOF system......Page 323
Example 4.5: Design of TMD’s for a simply supported beam......Page 325
4.7 Case studies - MDOF systems......Page 332
Problem 4.3......Page 341
Problem 4.4......Page 342
Problem 4.8......Page 343
Problem 4.9......Page 344
Problem 4.11......Page 345
Problem 4.12......Page 346
Problem 4.14......Page 347
Problem 4.16......Page 348
5.1 Introduction......Page 349
SDOF examples......Page 350
Bearing terminology......Page 354
Modified SDOF Model......Page 357
Example 5.1: Stiffness factors for prescribed structure and base motion.......Page 361
Seismic excitation - modified SDOF model......Page 362
Example 5.2: Stiffness parameters - modified SDOF model of Building example #2.......Page 366
Flexibility......Page 367
Rigidity under low level lateral loads......Page 368
Energy dissipation/absorption......Page 369
Modeling of a natural rubber bearing (NRB)......Page 370
Modeling of a lead rubber bearing (LRB)......Page 374
Applicability of base isolation systems......Page 377
5.4 Examples of existing base isolation systems......Page 378
Scaled stiffness distribution......Page 386
Example 5.3: Scaled stiffness for a 4DOF beam with base isolation.......Page 389
Example 5.4: Example 5.3 revisited.......Page 390
Example 5.5: Stiffness calibration for Example 5.4......Page 392
Stiffness distribution - undamped response......Page 396
Fundamental mode equilibrium equation......Page 403
Rigidity calibration - seismic excitation......Page 405
Example 5.6: Stiffness calibration - Example Building #2.......Page 406
Stiffness distribution based on fundamental mode response......Page 407
Stiffness distribution including the contribution of the higher modes......Page 414
Problem 5.1......Page 423
Problem 5.4......Page 424
Problem 5.6......Page 425
Problem 5.7......Page 427
Problem 5.8......Page 428
Problem 5.9......Page 429
Problem 5.10......Page 430
Active versus passive control......Page 433
The role of feedback......Page 436
Computational requirements and models for active control......Page 437
6.2 An introductory example of quasi-static feedback control......Page 438
Example 6.1: Shape control for uniform loading......Page 441
Example 6.2: Discrete displacement data......Page 442
6.3 An introductory example of dynamic feedback control......Page 443
Example 6.3: Illustrative example - influence of velocity feedback......Page 446
Introduction......Page 451
Force application schemes......Page 452
Large scale linear actuators......Page 456
Large scale adaptive configuration based actuators......Page 460
Small scale adaptive material based actuators......Page 461
6.5 Examples of existing large scale active structural control systems......Page 472
Problem 6.1......Page 485
Problem 6.2......Page 487
Problem 6.3......Page 489
7.1 Introduction to control algorithms......Page 491
Passive prestressing......Page 492
Active prestressing......Page 494
Example 7.1: A single force actuator......Page 500
Example 7.2: Two force actuators......Page 504
A general active prestressing methodology......Page 511
Example 7.3: Multiple actuators......Page 515
7.3 Quasi-static displacement control of beams......Page 518
Continuous least square formulation......Page 519
Discrete least square formulation......Page 520
Example 7.4: Cantilever beam - Least square algorithm......Page 522
Extended least square formulation......Page 528
Example 7.5: Example 7.4 revisited......Page 530
Introduction......Page 533
Selection of measures......Page 534
Example 7.6: Illustrative examples of observability......Page 535
Example 7.7: Illustrative examples of controllability......Page 537
Least square control algorithms......Page 539
Example 7.8: Example 7.7 revisited......Page 540
Example 7.9: Example 7.7 revisited with an extended least square algorithm......Page 543
Problem 7.1......Page 547
Problem 7.2......Page 548
Problem 7.3......Page 550
Problem 7.4......Page 551
Problem 7.5......Page 552
Problem 7.7......Page 553
Problem 7.8......Page 554
Problem 7.9......Page 555
8.1 Introduction......Page 557
Governing equations......Page 558
Free vibration uncontrolled response......Page 560
General solution - time invariant systems......Page 562
Example 8.1: Equivalence of equations (8.18) and (8.24)......Page 563
Stability criterion......Page 564
Linear negative feedback......Page 565
Effect of time delay on feedback control......Page 567
Stability analysis for time delay......Page 572
Governing equation......Page 578
Linear negative feedback control......Page 580
Stability analysis for time invariant linear feedback control......Page 581
Example 8.2: Stability analysis - SDOF system with no time delay......Page 587
Example 8.3: Stability analysis -SDOF system with time delay......Page 596
Quadratic performance index......Page 598
An example - linear quadratic regulator control algorithm......Page 600
The continuous-time algebraic Riccati equation......Page 605
The discrete time algebraic Riccati equation......Page 610
Example 8.4: Solution of the discrete time algebraic Riccati equation for a SDOF system......Page 612
Finite interval discrete time algebraic Riccati equation......Page 621
Example 8.5: Example 8.4 revisited......Page 622
Continuous time Riccati differential equation......Page 623
Variational formulation of the continuous time Riccati equation......Page 624
Example 8.6: Application to scalar case......Page 629
Notation and governing equations......Page 635
Free vibration response - time invariant uncontrolled system......Page 637
Example 8.7: Free vibration solution for proportional damping......Page 639
Example 8.8: General uncoupled damping......Page 641
Orthogonality properties of the state eigenvectors......Page 643
Example 8.9: Initial conditions - free vibration response......Page 644
Determination of W and......Page 645
General Solution - time invariant system......Page 646
Modal state space formulation - uncoupled damping......Page 647
Modal state space formulation - arbitrary damping......Page 651
Example 8.10: Modal formulation - undamped case......Page 653
Example 8.11: Modal formulation - uncoupled damping......Page 654
Example 8.12: Modal parameters - 4DOF system......Page 656
Example 8.13: Modal response for example 8.12......Page 663
Example 8.14: Modal response with feedback for example 8.12......Page 666
Stability analysis - discrete modal formulation......Page 671
Example 8.15: Stability analysis for example 8.14......Page 674
Controllability of a particular modal response......Page 691
Example 8.16: Controllability analysis for a 20 DOF Model......Page 692
Observability of a particular modal response......Page 694
Example 8.17: Observability analysis for a 20 DOF model......Page 696
Continuous time modal formulation......Page 698
Discrete time modal formulation......Page 700
Application studies - LQR control......Page 702
Example 8.18: Control force design studies for a 20 DOF shear beam......Page 710
Example 8.19: Alternate choice of response measures......Page 723
Problem 8.5......Page 727
Problem 8.7......Page 728
Problem 8.11......Page 729
Problem 8.16......Page 730
Problem 8.20......Page 731
Problem 8.22......Page 732
Problem 8.23......Page 733
Problem 8.25......Page 734
Problem 8.27......Page 735