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 |  موضوع: كتاب Mechanical Vibration - Analysis, Uncertainties, and Control الأحد 14 أغسطس 2022, 1:43 am | |
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Mechanical Vibration - Analysis, Uncertainties, and Control
FOURTHE DITION
Haym Benaroya • Mark Nagurka • Seon Han
و المحتوى كما يلي :
Contents
1 INTRODUCTION AND BACKGROUND 1
1.1 Challenges and Examples . 2
1.2 Systems and Structures 5
1.3 Basic Concepts of Vibration 5
1.3.1 Modeling for Vibration 6
1.3.2 Idealization and Formulation . 6
1.3.3 Inertia, Stiffness, and Damping 10
1.3.4 Properties of Keyboard Keys . 10
1.3.5 Computational Aspects 11
1.3.6 Is Vibration Good or Bad? 12
1.3.7 Vibration Control . 14
1.4 Types of Vibration . 14
1.4.1 Signal Classification 14
1.4.2 Deterministic Approximations 14
1.4.3 Probability . 15
1.4.4 System Model Uncertainty 15
1.4.5 Random Vibration . 15
1.5 Types of System Models 15
1.5.1 Linear Approximation . 15
1.5.2 Dimensionality . 16
1.5.3 Discrete Models 16
1.5.4 Continuous Models 16
1.5.5 Nonlinear Models . 17
1.6 Basic Dynamics . 17
1.6.1 Statics and Equilibrium 17
1.6.2 The Equations of Motion . 17
1.6.3 Linear Momentum and Impulse . 19
1.6.4 Principles of Work and Energy 19
1.7 Units . 21
1.7.1 Mars Orbiter Loss . 21
1.7.2 U.S. Customary and SI Systems . 21
1.7.3 The Second . 22
1.7.4 Dimensional Analysis . 27
1.8 Concepts Summary . 29
1.9 Quotes 29
2 SINGLE DEGREE-OF-FREEDOM UNDAMPED VIBRATION 31
2.1 Motivating Examples 31
2.1.1 Transport of a Satellite 31
2.1.2 Rocket Propulsion . 32
2.2 Deterministic Modeling . 33
2.2.1 Problem Idealization . 33
2.2.2 Mass, Damping, and Stiffness 34
xvxvi CONTENTS
2.2.3 Deterministic Approximation . 35
2.2.4 Equations of Motion 35
2.2.5 Energy Formulation 43
2.2.6 Representing Harmonic Motion . 46
2.2.7 Solving the Equations of Motion . 50
2.3 Undamped Free Vibration . 50
2.3.1 Alternate Formulation 52
2.3.2 Phase Plane 54
2.4 Harmonically Forced Vibration 55
2.4.1 A Note on Terminology 57
2.4.2 Resonance . 58
2.4.3 Vibration of a Structure in Water 64
2.5 Concepts Summary . 69
2.6 Quotes 69
2.7 Problems 69
3 SINGLE DEGREE-OF-FREEDOM DAMPED VIBRATION 83
3.1 Overview 83
3.2 Introduction to Damping 86
3.3 Damping Models 87
3.3.1 Viscous Damping and Loss Factor 87
3.3.2 Coulomb Damping 89
3.4 Free Vibration with Viscous Damping 91
3.4.1 Critically Damped and Overdamped Systems 92
3.4.2 Some Time Constants . 92
3.4.3 Underdamped Systems 94
3.4.4 Logarithmic Decrement 97
3.4.5 Phase Plane 101
3.5 Free Vibration with Coulomb Damping 101
3.6 Forced Vibration with Viscous Damping . 105
3.7 Forced Harmonic Vibration 106
3.7.1 Response to Harmonic Excitation 106
3.7.2 Harmonic Excitation in Complex Notation . 115
3.7.3 Harmonic Base Excitation 117
3.7.4 Rotating Unbalance 120
3.8 Forced Periodic Vibration . 127
3.8.1 Harmonic/Spectral Analysis . 127
3.8.2 Fourier Series . 127
3.9 Concepts Summary . 129
3.10 Quotes 130
3.11 Problems 130
4 SINGLE DOF VIBRATION: GENERAL LOADING AND ADVANCED TOPICS 137
4.1 Arbitrary Loading: Laplace Transform 137
4.2 Step Loading 142
4.3 Impulsive Excitation 146
4.4 Arbitrary Loading: Convolution Integral . 150
4.5 Introduction to Lagrange’s Equation . 154
4.6 Notions of Randomness 158
4.7 Notions of Control . 159
4.8 The Inverse Problem 159
4.9 A Self-Excited System and Its Stability 160
4.10 Solution Analysis and Design Techniques . 160
4.11 Model of a Bouncing Ball . 166
4.11.1 Time of Contact 167CONTENTS xvii
4.11.2 Stiffness and Damping 167
4.11.3 Natural Frequency & Damping Ratio 168
4.11.4 Approximations 168
4.12 Concepts Summary . 169
4.13 Quotes 169
4.14 Problems 169
5 VARIATIONAL PRINCIPLES AND ANALYTICAL DYNAMICS 175
5.1 Introduction . 175
5.2 Constraints . 176
5.3 Virtual Work 176
5.3.1 Work and Energy . 176
5.3.2 Principle of Virtual Work . 178
5.3.3 D’Alembert’s Principle 182
5.4 Lagrange’s Equation 189
5.4.1 Lagrange’s Equation for Small
Oscillations . 193
5.5 Hamilton’s Principle 196
5.6 Lagrange’s Equation with Damping 197
5.7 Jourdain’s Principle 198
5.7.1 Jourdain’s Principle from d’Alembert’s Principle 198
5.8 Concepts Summary . 199
5.9 Quotes 200
5.10 Problems 200
6 MULTI DEGREE-OF-FREEDOM VIBRATION 209
6.1 Motivating Examples 209
6.1.1 Periodic Structures 209
6.1.2 Inverse Problems . 209
6.1.3 Vehicle Vibration Testing . 210
6.1.4 Scope 211
6.2 The Concepts of Stiffness and Flexibility . 211
6.2.1 Influence Coefficients . 211
6.3 Equations of Motion 216
6.3.1 Mass and Stiffness Matrices . 219
6.4 Undamped Vibration 220
6.4.1 Two Degree-of-Freedom Vibration: Direct Method . 220
6.4.2 Harmonically Forced Vibration: Direct Method 226
6.4.3 Undamped Vibration Absorber 229
6.4.4 Beating Oscillations 232
6.5 Free Vibration with Damping: Direct Method 235
6.6 Modal Analysis . 241
6.6.1 Modal Orthogonality . 241
6.6.2 Modal Analysis with Forcing . 244
6.6.3 Modal Analysis with Proportional Damping 248
6.7 Nonproportional Damping . 251
6.7.1 Phase Synchronization 253
6.8 Real and Complex Modes . 256
6.8.1 Modal Analysis vs. Direct Method 257
6.9 Special Cases 258
6.9.1 Unrestrained Systems . 258
6.9.2 Rigid-Body Mode . 258
6.9.3 Repeated Frequencies . 262
6.10 Eigenvalue Geometry 262
6.11 Periodic Structures . 264xviii CONTENTS
6.11.1 Perfect Lattice Models 264
6.11.2 Effects of Imperfection 265
6.12 Inverse Vibration Problem . 265
6.13 Fluid Sloshing in Container 268
6.14 Stability of Motion . 269
6.15 Rayleigh’s Quotient . 271
6.16 Concepts Summary . 273
6.17 Quotes 273
6.18 Problems 274
7 CONTINUOUS MODELS FOR VIBRATION 283
7.1 Discrete to Continuous . 283
7.2 Vibration of Strings 284
7.2.1 Wave Propagation Solution 285
7.2.2 Wave Equation via Hamilton’s Principle 287
7.2.3 Boundary Value Problem . 288
7.2.4 Modal Solution for Fixed-Fixed Boundary Conditions . 289
7.3 Axial Vibration of Beams . 292
7.3.1 Axial Vibration: Newton’s Approach 292
7.3.2 Axial Vibration: Hamilton’s Approach . 293
7.3.3 Simplified Eigenvalue Problem 294
7.3.4 Eigenfunction Expansion Method 295
7.4 Torsional Vibration of Shafts . 299
7.4.1 Torsion of Shaft with Rigid Disk at One End 300
7.5 Transverse Vibration of Beams 301
7.5.1 Timoshenko Beam . 302
7.5.2 Boundary Conditions . 304
7.5.3 Bernoulli-Euler Beam . 304
7.5.4 Orthogonality of the Modes . 308
7.5.5 Nodes and Antinodes . 314
7.6 Other Transverse Beam Vibration Cases . 314
7.6.1 Beam with Axial Forces 314
7.6.2 Beam with Elastic Restraints . 315
7.6.3 Beam on an Elastic Foundation . 316
7.6.4 Beam with a Moving Support 317
7.6.5 Different Boundary Conditions 318
7.6.6 Beam with Traveling Force 320
7.7 Concepts Summary . 320
7.8 Quotes 321
7.9 Problems 321
8 CONTINUOUS MODELS FOR VIBRATION: ADVANCED MODELS 329
8.1 Vibration of Membranes 329
8.1.1 Rectangular Membranes . 329
8.1.2 Circular Membranes 332
8.2 Vibration of Plates . 336
8.2.1 Rectangular Plates 336
8.2.2 Eigenvalue Problem 338
8.3 Approximate Methods . 341
8.3.1 Rayleigh’s Quotient 341
8.3.2 Rayleigh-Ritz Method . 343
8.3.3 Galerkin Method . 348
8.4 Variables Not Separating 350
8.4.1 Nonharmonic, Time-Dependent Boundary Conditions . 350
8.4.2 Pipe Flow with Constant Tension 355CONTENTS xix
8.5 Concepts Summary . 358
8.6 Quotes 358
8.7 Problems 359
9 RANDOM VIBRATION: PROBABILISTIC FORCES 361
9.1 Introduction . 361
9.2 Motivation 363
9.2.1 Random Vibration . 363
9.2.2 Fatigue Life 364
9.2.3 Ocean Wave Forces 365
9.2.4 Wind Forces 366
9.2.5 Material Properties 366
9.2.6 Statistics and Probability . 367
9.3 Random Variables . 368
9.3.1 Probability Distribution 368
9.3.2 Probability Density Function . 369
9.4 Mathematical Expectation . 370
9.4.1 Variance 371
9.5 Useful Probability Densities 372
9.5.1 Uniform Density 372
9.5.2 Exponential Density 373
9.5.3 Normal (Gaussian) Density 373
9.5.4 Lognormal Density 376
9.5.5 Rayleigh Density 377
9.6 Two Random Variables . 377
9.6.1 Covariance and Correlation 377
9.7 Random Processes . 380
9.7.1 Random Process Descriptors . 380
9.7.2 Ensemble Averaging 380
9.7.3 Stationarity 382
9.7.4 Power Spectrum 383
9.7.5 Units 385
9.7.6 Narrow-Band and Broad-Band Processes 386
9.7.7 White-Noise Process 387
9.8 Random Vibration . 388
9.8.1 Formulation and Preliminaries 388
9.8.2 Mean-Value Response . 389
9.8.3 Response Correlations . 389
9.8.4 Response Spectral Density 390
9.9 Stochastic Response of a Linear MDOF System . 393
9.10 Lunar Seismic Structural Analysis 394
9.11 Random Vibration of Continuous Structures . 398
9.12 Monte Carlo Simulation 400
9.12.1 Random Number Generation . 401
9.12.2 Generating Random Variates . 402
9.12.3 Generating Time History for Random Process . 403
9.13 Inverse Vibration with Uncertain Data 404
9.14 Concepts Summary . 407
9.15 Quotes 407
9.16 Problems 407xx CONTENTS
10 VIBRATION CONTROL 413
10.1 Motivation 413
10.2 Approaches to Controlling Vibration . 415
10.2.1 Why Active Control 416
10.3 Feedback Control 418
10.3.1 Disadvantages of Feedback 421
10.4 Performance of Feedback Control Systems 422
10.4.1 Poles and Zeros of a Second-Order System . 425
10.4.2 System Gain 426
10.4.3 Stability of Response . 427
10.5 Control of Response 427
10.5.1 Control Actions 427
10.5.2 Control of Transient Response 428
10.6 Parameter Sensitivity 430
10.7 State Variable Models . 434
10.7.1 Transfer Function from State Equation . 436
10.7.2 Controllability and Observability . 437
10.7.3 State Variable Feedback . 439
10.8 Multivariable Control 440
10.8.1 State and Output Equations . 441
10.8.2 Controllability and Observability . 441
10.8.3 Closed-loop Feedback of MIMO Systems 441
10.9 Stochastic Control . 442
10.10 Concepts Summary . 443
10.11 Quotes 443
10.12 Problems 444
11 NONLINEAR VIBRATION 447
11.1 Introduction . 447
11.2 Physical Examples . 449
11.2.1 Simple Pendulum: Approximate Solution 451
11.2.2 Simple Pendulum: Exact Solution 451
11.2.3 Duffing and van der Pol Equations 452
11.3 The Phase Plane 452
11.3.1 Stability of Equilibria . 454
11.4 Perturbation or Expansion Methods . 459
11.4.1 Lindstedt-Poincaré Method 462
11.4.2 Forced Oscillations of Quasi-Harmonic Systems 464
11.4.3 Jump Phenomenon 466
11.4.4 Periodic Solutions of Nonautonomous Systems . 466
11.4.5 Subharmonic and Superharmonic Oscillations . 468
11.5 Mathieu Equation . 470
11.6 Van der Pol Equation . 474
11.6.1 Unforced van der Pol Equation 475
11.6.2 Limit Cycles 475
11.6.3 Forced van der Pol Equation . 475
11.7 Motion in the Large 478
11.8 Nonlinear Control 480
11.9 Random Duffing Oscillator . 483
11.10 Nonlinear Pendulum: Galerkin Method 484
11.11 Concept Summary . 485
11.12 Quotes 485
11.13 Problems 486CONTENTS xxi
A MATHEMATICAL CONCEPTS FOR VIBRATION 489
A.1 Complex Numbers . 489
A.1.1 Complex Number Operations . 489
A.1.2 Absolute Value . 490
A.1.3 Equivalent Representation 490
A.2 Matrices . 490
A.2.1 Matrix Operations . 491
A.2.2 Determinant and Matrix Inverse . 491
A.2.3 Eigenvalues and Eigenvectors . 493
A.2.4 Matrix Derivatives and Integrals . 493
A.3 Taylor Series & Linearization . 494
A.4 Ordinary Differential Equations 494
A.4.1 Solution of Linear Equations . 495
A.4.2 Homogeneous Solution 496
A.4.3 Particular Solution 498
A.5 Laplace Transforms . 499
A.5.1 Borel’s Theorem 500
A.5.2 Partial Fraction Expansion 500
A.5.3 Laplace Transform Table . 501
A.5.4 Initial-Value, Final-Value Theorem . 501
A.6 Fourier Series & Transforms 501
A.6.1 Fourier Series . 501
A.6.2 Fourier Transforms 502
A.7 Partial Differential Equations . 502
B VISCOELASTIC DAMPING 505
B.1 Viscoelastic Materials . 505
B.1.1 Work Done Per Cycle . 506
B.2 Viscoelastic Material Models 506
B.2.1 Maxwell Model 506
B.2.2 Voigt Model 509
B.2.3 Maxwell Standard Linear Model . 511
B.2.4 Stress-Strain Equivalent Model 512
B.2.5 Boltzmann Superposition Model . 512
B.2.6 General Nonviscous Damping 513
B.3 Causality Issues in Damping Models . 517
B.4 Concepts Summary . 518
B.5 Quotes 518
C SOLVING VIBRATION PROBLEMS WITH MATLAB 521
C.1 Introduction . 521
C.2 SDOF Undamped System . 521
C.3 SDOF Damped System . 524
C.4 SDOF Overdamped System 526
C.5 SDOF Undamped System with Harmonic Excitation 528
C.6 SDOF Damped System with Harmonic Excitation . 531
C.7 SDOF Damped System with Base Excitation 533
C.8 SDOF Damped System with Rotating Unbalance 537
C.9 SDOF Damped System with Impulse Input . 540
C.10 SDOF Damped System with Step Input . 541
C.11 SDOF Damped System with Square Pulse Input 544
C.12 SDOF Damped System with Ramp Input 548
C.13 SDOF System with Arbitrary Periodic Input 550
C.14 MDOF Undamped System . 552
C.15 MDOF Damped System 555xxii CONTENTS
C.16 General Vibration Solver 558
C.17 Van der Pol Oscillator . 563
C.18 Random Vibration . 565
C.19 Duffing Oscillator with Random Excitation . 567
C.20 Monte Carlo Simulation of a Random System 570
Index 57
Index
accelerometer, 139
admittance, 138
aeroelastic forcing, 67
aerodynamic loads, 363
aircraft, 2
aerodynamic loads, 363
amplitude modulation, 235
anelastic, 86
anelasticity, 83
automobile vibration testing, 62, 210
base-excited motion, 227
beam
axial vibration, 292
time-dependent boundary conditions, 350
Bernoulli-Euler, 301, 304
Galerkin method, 348
random vibration, 398
Rayleigh’s quotient, 341
Rayleigh-Ritz method, 343
space elevator, 296
Timoshenko, 302
transversely vibrating, 301
beam
transversely vibrating
flow in pipe, 355
beating, 60, 232
Bernoulli, 310
Bernoulli-Euler
random vibration, 398
Bernoulli-Euler beam, 301, 304
axial force, 314
boundary conditions, 305
elastic foundation, 316
elastic restraints, 315
equation of motion, 304
Euler buckling load, 315
flow in pipe, 355
harmonic boundary conditions, 318
moving support, 317
non-separable solution
Mindlin-Goodman technique, 350
orthogonality of modes, 308
traveling force, 320
Bessel, 334
Black, 420
Bode, 431
Bode plot, 107
Bode plots, 431
Borel’s theorem, 138, 142
Borgman’s method, 403
bouncing ball, 166
impact, 166
natural frequency and damping, 168
stiffness and damping, 167
time of contact, 167
boundary value problem
string, 289
bridges, 61
damping, 84
London Millenium, 113
Buckingham-Pi Theorem, 28
calculus of variations, 175
characteristic equation, 139, 496
one degree-of-freedom, 91
rectangular membrane, 331
two degrees-of-freedom, 221
characteristic matrix, 226, 233
two degrees of freedom, 221
characteristic polynomial, 226
characteristic time, 93
characteristic values, 221
characteristic vectors, 222
closed-loop transfer function, 419
complex numbers, 46, 489
complex conjugate, 489
Euler’s identity, 489
magnitude, 490
reciprocal, 490
computer keyboards, 10
conservation of energy, 21
conservation of linear momentum, 19
conservative force, 20
constraints
configuration, 176
holonomic, 176
nonholonomic, 176
rheonomic, 176
scleronomic, 176
velocity, 176
continuous system
573574 INDEX
limit of MDOF system, 283
control, 14, 159
active, 416
actuator, 418
block diagram, 418
Bode plots, 431
characteristic time, 425
classical, 417
closed-loop, 418
controllability, 438, 441
feedback, 413
integral, 428
modern, 417
multivariable, 440
negative feedback, 420
nonlinear, 480
observability, 438, 441
open-loop, 418
oscillator, 159
passive, 415
peak overshoot, 423
positive feedback, 419
proportional, 428
proportional plus derivative, 428
proportional plus integral, 428
proportional plus integral plus derivative, 428
settling time, 423
stability, 427
state variable feedback, 439
step response, 422
stochastic, 442
system gain, 426
controllability, 437
convolution integral, 138, 151
Duhamel integral, 150, 249
correlation coefficient, 382
Coulomb, 90
critical damping, 91
critically damped system, 425
d’Alembert, 184
d’Alembert’s principle, 182
for a rigid body, 182
dampers
bridge, 84
damping, 10, 86
Boltzmann superposition model, 512
causality, 517
Coulomb, 86, 89
critical, 91, 92, 425
equivalent viscous damping, 89
general linear nonviscous damping, 513
general nonviscous damping, 513
hysteresis loops, 88
loss factor, 87, 506
material, 86
Maxwell model, 506
Maxwell standard linear model, 511
multi degree-of-freedom systems, 250
overdamped, 91
radiation, 86
standard model, 506
stress-strain equivalent model, 512
structural, 86
underdamped, 91, 94
viscoelastic, 86
viscoelastic materials, 506
viscous, 35, 86
Voigt model, 88, 509
Kelvin-Voigt model, 506
damping
nonproportional, 251
Den Hartog, 110
Den Hartog force vector diagram, 108
design, 160
spacecraft shipping container, 163
water landing space module, 161
differential equations
ordinary, 494
homogeneous solution, 496
non-homogeneous solution, 498
partial, 502
separation of variables, 502
dimensional analysis
similitude theory, 27
Dirac, 148
direct method
free vibration with damping, 235
harmonic forcing, 226
displacement transmissibility, 118
Duffing equation, 452
forced oscillations, 464
jump phenomenon, 466
perturbation method
Lindstedt-Poincaré, 463
random forcing, 483
Duhamel, 153
Duhamel integral, 150
dynamic amplification factor
undamped, 55
dynamics, 1
eigenfunction
axially vibrating beam
orthogonality, 296
string, 290
normalized, 290
eigenvalue problem
axially vibrating beam, 294
characteristic roots, 493INDEX 575
circular membrane, 332
geometric interpretation, 262
rectangular membrane, 330
rectangular plate, 338
repeated frequencies, 262
geometric interpretation, 263
string, 289
eigenvalues, 493
two degrees of freedom, 221
eigenvectors, 493
two degrees of freedom, 222
Einstein, 23
elevator cable system, 224
elliptic integral, 451
energy, 176
engineering, 1
ensemble, 380
ensemble averaging, 380
ergodicity, 382
Euler, 98
Euler buckling load, 315
Euler’s formula, 47, 489
expansion theorem, 242
expected value, 370
fatigue life, 364
fault detection, 159
feedback control, 418
performance, 422
Flügge-Lotz, 482
flexibility coefficient, 211
flexibility matrix, 212
fluid-structure interaction, 64
equivalent mechanical models, 268
flow in pipe, 355
flow-oscillator model, 67
fluid sloshing in containers, 268
Mathieu equation, 470
Pierson-Moskowitz spectrum, 383
Pierson-Moskowitz spectrum generation, 403
van der Pol equation, 474
wave forces
Morrison equation, 365
force transmissibility, 119
force vector diagram, 108
dominated frequency range, 109
forced vibration
direct method
undamped, 226
viscous damping, 105
harmonic, 106
harmonic base excitation, 117
harmonic rotating unbalance, 120
periodic not harmonic, 127
forces
arbitrary, 150
conservative, 177
impulsive, 146
nonconservative, 191
random, 159
step load, 142, 151
wind, 366
Fourier, 125
Fourier series, 501
Fourier series forcing, 127
square wave, 128
Fourier transforms, 502
Fourier transform pairs, 502
free vibration
Coulomb damping, 101
direct method
damped, 235
undamped, 220
undamped, 50
viscous damping, 91
free-body diagram, 34, 35
two degree-of-freedom, 216
frequency
damped, 94
fundamental, 221
inclusion principle, 267
frequency response function, 116, 389
sensitivity, 432
transfer function, 388
Galerkin, 346
Galerkin method
axially vibrating beam, 348
nonlinear pendulum, 484
Galileo, 51
Gauss, 374
generalized coordinate, 189
generalized force, 190
damping, 192
nonconservative, 192
Germain, 340
Gilbreth, 215
Hamilton, 193
Hamilton’s principle, 196
axially vibrating beam, 293
damped case, 218
elastic pendulum, 197
extended principle, 196
physical interpretation, 196
string, 287
Timoshenko beam, 302
two degrees of freedom, 217
harmonic base excitation, 117
displacement transmissibility, 118576 INDEX
force transmissibility, 118
harmonic forcing
complex notation, 115
damped response, 106
direct method, 226
undamped, 55
harmonic motion, 46
rotating vector representation, 46
Hooke’s law, 10, 35, 211
idealization, 6, 33
impedance, 138
impulse response, 147, 389
inclusion principle, 267
inertia, 10, 18
influence coefficients, 211
flexibility, 211
reciprocity, 213
stiffness, 211
inverse problem
deterministic, 266
imperfect system, 404
two degrees-of-freedom, 406
mass and stiffness estimates, 265
single degree-of-freedom system, 159
inverse problems, 209
Jackson, 367
Jeffcott rotor, 123
Jourdain’s principle, 198
principle of virtual power, 198
Kelvin, 513
kinematics, 1
kinetic energy, 43
particle, 20
kinetics, 1
L’Hôpital, 63
Lagrange, 156
Lagrange’s equation, 43, 154, 189, 191
damped case, 198
fluid-conveying pipe, 357
Lagrangian, 191
mass coefficients, 193
simple pendulum, 154
single degree-of-freedom oscillator, 154
small oscillations, 193
stiffness coefficients, 193
two degrees of freedom, 217
Laplace, 143
Laplace transform, 137, 499
Borel’s theorem, 500
impulsive excitation, 147
partial fraction expansion, 500
step loading, 142
table of, 501
Laplacian, 330
Leibnitz, 48
linear momentum, 19
linear superposition, 15, 33, 495
Fourier series solution, 128
lack of for nonlinear problems, 447
linearization, 33
logarithmic decrement, 97
longitudinal vibration
forced beam, 295
loss factor, 87, 506
causality, 88
magnification factor, 106
rotating shaft, 124
mass coefficients, 193
mass matrix
positive definite, 219
mathematical expectation, 370
mathematical modeling, 33
Mathieu equation, 470
matrix, 490
adjoint, 492
determinant, 491
eigenvalues, 493
eigenvectors, 493
inverse, 492
matrix multiplication, 491
matrix transpose, 491
singular, 492
Maxwell, 507
membrane
circular, 332
boundary conditions, 333
eigenvalue problem, 332
modes of vibration, 334
rectangular, 329
boundary conditions, 330
eigenvalue problem, 330
equation of motion, 329
modes of vibration, 331
MEMs, 56
metronome, 38
Miner’s rule, 364
modal analysis, 241
forced, 244
modal analysis vs. direct method, 257
modal coordinates, 242
modal matrix, 242
modal participation factor, 246
proportional damping, 248
random forces, 393
modal participation factor, 246
modal ratios, 222INDEX 577
modes of vibration
circular membrane, 334
degenerate, 333
nodes, 223
real vs. complex, 256
rectangular membrane, 331
rigid-body mode, 258
string, 288
two degrees-of-freedom, 221
Monte Carlo simulation, 400
pseudorandom numbers, 401
random number generation, 401
natural frequency, 36
Newton, 40
Newton’s first law of motion, 17
Newton’s second law of motion, 18, 35
axially vibrating beams, 292
rectangular membrane, 329
string, 285
torsional vibration, 37
unrestrained translation, 261
Newton’s third law of motion, 18
Newton-Euler equation, 36
rectangular plate, 337
unrestrained rotation, 260
nodes and antinodes, 314
nondestructive evaluation, 159
nonlinear vibration
autonomous system, 448
center, 456
combination harmonics, 469
Duffing equation, 466
jump phenomenon, 466
logarithmic spiral, 456
Mathieu equation, 470
stability, 471
motion in the large, 478
node, 456
non-autonomous system, 448
parametric forcing, 447
perturbation method, 459
Lindstedt-Poincaré, 462
saddle point, 456
stability of equilibria, 454
subharmonic oscillation, 469
trajectories, 455
nonproportional damping, 251
nonseparable partial differential equations
nonharmonic boundary conditions, 350
pipe flow, 355
observability, 437
offshore structures, 2, 213
Morrison equation, 365
wave forces, 365
orthogonality of modes
with respect to mass matrix, 242
with respect to stiffness matrix, 242
overdamped system, 91, 425
peak overshoot, 423
pendulum
base excited, 269
coupled pendula, 232
amplitude modulation, 235
elastic, 197
nonlinear
forced, 466
simple, 37, 154, 190
nonlinear, 449
via Lagrange’s equation, 190
periodic structures, 209
periodic systems, 264
imperfect lattice models, 404
imperfect periodicity, 265
perfect lattice models, 264
phase angle
damped system, 94, 95
undamped, 51
viscous damping, 107
phase plane, 54
linear oscillator, 101
nonlinear oscillator, 452
phasing between modes, 237, 252
phase synchronization, 253
plate
rectangular, 336
assumptions, 336
boundary conditions, 339
eigenvalue problem, 338
equation of motion, 338
flexural rigidity, 338
Newton-Euler equation, 337
Poincaré, 461
poles, 139
potential energy, 20, 43, 177
principal coordinates, 242
principle of virtual work, 178, 179
constraint, 178
virtual displacement, 178
principle of work and energy, 177
probability, 362
correlation coefficient, 377
cumulative distribution function, 368
ensemble averaging, 380
excursion frequency, 362
histogram, 362
mathematical expectation, 370
expected value, 370578 INDEX
variance, 371
probability density function, 361, 369
exponential density function, 373
Gaussian or normal, 373
joint density function, 377
lognormal density function, 376
marginal densities, 378
Rayleigh density, 377
standard normal density, 374
uniform density function, 372
probability distribution function
joint distribution function, 377
random process, 363
random variable, 363, 368
statistically independent, 378
proportional damping, 248
random process, 380, see probability
correlation coefficient, 382
correlation function, 381
autocovariance, 382
cross correlation, 381
stationarity, 382
earthquake spectra, 384
ergodicity, 382
narrow- and broad-band, 386
ocean wave height spectra, 383
spectral density, 383
white-noise process, 387
wind velocity spectra, 384
Wiener-Khintchine formulas, 383
random vibration, 363, 388
beams, 398
Borgman’s method, 403
Duffing equation, 483
fundamental theorem, 390
multi degree-of-freedom systems, 393
response mean, 389
response spectral density, 390
response to white noise, 390
response variance, 390
randomness, 158, 361
aleatory, 361
epistemic, 361
Rayleigh, 342
Rayleigh damping, 249
Rayleigh dissipation function, 198
Rayleigh’s quotient
discrete system, 271
three degrees of freedom, 272
two degrees of freedom, 272
transversely vibrating beam, 341
Rayleigh-Ritz method
transversely vibrating beam, 343
resonance, 58, 59
electric motors, 63
Reynolds, 67
rigid-body mode, 258
rigid-body motion, 258
rocket ship, 6, 18, 32, 61, 139
booster vibration, 61
rolling disk, 155
rotating shaft, 123
whirling, 123
rotating unbalance, 120
subcritical operation, 123
supercritical operation, 123
satellites, 31
second, 22
secular term, 60
seismic analysis
lunar structures, 394
self-excited system, 160
parametrically excited system, 160
stability, 160
sensitivity, 430
control, 433
separation of variables, 502
axially vibrating beam, 294
Bernoulli-Euler beam, 305
circular membrane, 332
rectangular membrane, 330
rectangular plate, 338
string, 288
settling time, 423
shaft
frequency dependent boundary condition, 301
natural frequencies, 301
torsional vibration, 36, 299
warping, 299
space elevator, 296
spacecraft
vibration due to shipping, 163
spacecraft shipping container design, 163
spectral density, 383
spring
inertia, 44
series and parallel, 53
stability, 160, 427
state variables, 434
stationarity, 382
stiffness, 10
stiffness coefficients, 193
stiffness matrix, 212
positive semi-definite, 219
singular, 260
strain energy, 43
string
boundary conditions, 289INDEX 579
boundary value problem, 289
equation of motion, 284
Newton’s second law of motion, 285
standing wave, 285, 286
traveling string, 358
wave equation, 285
Hamilton’s principle, 287
wave propagation, 285
synchronicity of motion, 220
system identification, 159, 210
systems, 5
tallest buildings, 7
Taylor, 34
Taylor series, 178, 455, 494
time constants
characteristic time, 92, 425
correlation time, 92
relaxation time, 92
Timoshenko, 312
Timoshenko beam, 302
boundary conditions, 304
deflection equation of motion, 304
equations of motion, 303
shear distortion, 302
torsional vibration
shaft, 299
transfer function, 138, 147
base-excited system, 417
closed-loop, 419
control, 417, 418
poles and zeros, 422, 425
transverse vibration
beam, 301
Timoshenko beam, 302
turbine, 6
underdamped system, 91, 94, 425
units, 21
unrestrained system, 258
constraint matrix, 259
rigid-body mode, 258
rotation, 260
van der Pol, 477
van der Pol equation, 452
flow-oscillator model, 474
forced, 475
frequency entrainment, 476
limit cycle, 475
unforced, 475
variable mass system, 38
variance, 371
variation, 175
variational principles, 175
Vaughan, 291
vibration absorber
pipeline, 232
powered hand tools, 104
Stockbridge, 230
undamped case, 229
vibration control, 14, 413
vibration testing, 387
accelerated testing, 388
automobile, 210
bicycles, 391
viscoelastic materials, 505
Boltzmann superposition model, 512
complex modulus, 506
general nonviscous damping, 513
loss modulus, 506
Maxwell model, 506
standard linear model, 506
storage modulus, 505
Voigt model
Kelvin-Voigt model, 506
viscous damping factor, 36
viscous damping ratio, 36
Voigt, 510
von Kármán, 65
water landing space module, 161
wave equation, 502
beam, 293
rectangular membrane, 330
Weiner, 442
Wiener-Khintchine formulas, 383
wind-induced oscillations, 230
wind-induced vibration, 250
work, 176
done by conservative force, 20
zeros, 139
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