Dynamic Stability of Structures

By: Wei-Chau Xie

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Hardcover | 24 June 2008

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454 Pages

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This 2006 book presents a systematic introduction to the theory of parametric stability of structures under both deterministic and stochastic loadings. A comprehensive range of theories are presented and various application problems are formulated and solved, often using more than one approach. Investigation of an elastic system's dynamic stability frequently leads to the study of dynamic behaviour of the solutions of parametrically excited systems. Parametric instability or resonance is more dangerous than ordinary resonance as it is characterised by exponential growth of the response amplitudes even in the presence of damping. The emphasis in this book is on the applications and various analytical and numerical methods for solving engineering problems. The materials presented are as self-contained as possible, with all of the important steps of analysis provided in order to make the book suitable as a graduate-level textbook and especially for self-study.

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Review of the hardback: 'Writing a monograph intended for engineers on advanced applications of stochastic modelling is a difficult challenge: the subject is extremely mathematical and one runs the risk of either writing for a very small target audience of experts or one has to include entire chapters of background material. Professor Wei-Chau Xie has managed to strike a difficult balance between these two extremes.' Journal of Sound and Vibration

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Non-Fiction Science Physics Classical Mathematics Engineering & Technology Civil Engineering Structural Engineering Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Mechanical Engineering

Technology in General Engineering in General

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Published: 2nd December 2010

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Preface	p. xv
Introduction	p. 1
Concept of Stability	p. 3
Conservative Systems	p. 4
Nonconservative Systems	p. 8
Gyroscopic Systems	p. 18
Motivating Examples	p. 19
Column under Axial Load	p. 20
Beam in Plane Motion	p. 22
Viscoelastic Beam under Dynamic Axial Load	p. 28
Flexural-Torsional Vibration of a Rectangular Beam	p. 33
Vibration of a Rotating Shaft	p. 37
Problems	p. 41
Dynamic Stability of Structures under Deterministic Loadings
Linear Differential Equations with Periodic Coefficients	p. 45
Stability of the Mathieu-Hill Equations	p. 45
Effect of Damping on the Mathieu-Hill Equations	p. 53
Stability of Linear Differential Equations with Peridoic Coefficients	p. 55
Stability of the Mathieu Equations	p. 56
Formulation	p. 56
Boundaries of the First Stability Region	p. 59
Boundaries of the Second Stability Region	p. 60
Boundaries of the Third Stability Region	p. 63
Stability of the Damped Mathieu Equations	p. 64
Periodic Solutions of Period 2T	p. 64
Periodic Solutions of Period T	p. 66
Problems	p. 67
Approximate Methods	p. 68
The Method of Averaging	p. 68
Stability of the Undamped Mathieu Equations	p. 68
Stability of the Damped Mathieu Equations	p. 75
The Method of Multiple Scales	p. 76
Motivating Example	p. 76
Stability of the Undamped Mathieu Equations	p. 79
Multiple Degrees-of-Freedom Non-Gyroscopic Systems	p. 79
Averaged Equations of Motion	p. 79
Parametric Resonances	p. 81
Two Degrees-of-Freedom Gyroscopic Systems	p. 91
Unperturbed System	p. 92
Perturbed System	p. 93
Multiple Degrees-of-Freedom Gyroscopic Systems	p. 101
Problems	p. 106
Nonlinear Systems under Periodic Excitations	p. 108
Pendulum under Support Excitation	p. 108
Column under Harmonic Axial Load	p. 113
Steady-State Solutions	p. 113
Stability of the Steady-State Solutions	p. 116
Non-Trivial Solution	p. 117
Trivial Solution	p. 118
Instability Region	p. 122
Snap-through of a Shallow Arch	p. 125
Equations of Motion	p. 125
Amplitude-Load-Frequency Relation	p. 125
Multiple Degrees-of-Freedom Systems	p. 126
Equations of Motion	p. 126
Resonance Conditions and Averaged Equations	p. 129
Stability of the Steady-State Solutions	p. 130
Examples	p. 138
Column under Harmonic Axial Load	p. 138
Snap-Through of a Shallow Arch	p. 141
Main Resonance	p. 141
Parametric Resonance	p. 143
Problems	p. 151
Dynamic Stability of Structures under Stochastic Loadings
Random Processes and Stochastic Differential Equations	p. 155
Description of Random Processes	p. 156
Processes with Orthogonal Increments	p. 164
Wiener Process or Brownian Process	p. 165
White Noise Process	p. 171
Markov Processes	p. 173
Diffusion Processes	p. 175
Definition	p. 175
Kolmogorov Equations	p. 177
Backward Kolmogorov Equation	p. 177
Forward Kolmogorov Equation (Fokker-Planck Equation)	p. 179
Properties of the Fokker-Planck Equations	p. 181
Stochastic Integrals	p. 187
Stochastic Riemann Integrals	p. 187
Riemann-Stieltjes Integrals	p. 188
Generalized Stochastic Integral	p. 189
The Ito Integral	p. 192
Stochastic Differential Equations	p. 194
General Theory of Stochastic Differential Equations	p. 194
Ornstein-Uhlenbeck Process	p. 196
Bounded Noise Process	p. 199
Ito's Differential Rule (Ito's Lemma)	p. 202
The Stratonovich Integral	p. 204
The Stratonovich Stochastic Differential Equations	p. 206
Approximation of a Physical Process by a Diffusion Process	p. 209
Classification of the Boundaries of a Diffusion Process	p. 210
The Method of Averaging	p. 212
Averaging Method for Stochastic Differential Equations	p. 212
Averaging Method for Integro-Differential Equations	p. 221
Monte Carlo Simulation	p. 222
Generation of Normally Distributed Random Numbers	p. 223
Simulation of the Standard Wiener Process W(t)	p. 224
Simulation of the Stochastic Differential Equations	p. 224
Strong Approximation Schemes	p. 225
Weak Approximation Schemes	p. 227
Example - Duffing-van der Pol Equation	p. 229
Strong Approximation Schemes	p. 230
Weak Approximation Schemes	p. 231
Problems	p. 232
Almost-Sure Stability of Systems under Ergodic Excitations	p. 234
Definitions	p. 234
A.S. Asymptotic Stability of Second-Order Linear Stochastic Systems	p. 236
Basic Equations	p. 237
Systems with Arbitrary Ergodic Coefficients	p. 239
Systems under Random Excitations with Known Probability Density Functions	p. 241
Optimization Model	p. 241
The Complex Method for Constrained Optimization	p. 244
Numerical Solution for Systems with Ergodic Gaussian Coefficients	p. 245
Stability of a Shallow Arch	p. 247
Introduction	p. 247
Static Case	p. 248
Dynamic Case	p. 251
Equations of Motion	p. 251
Two-Dimensional Fokker-Planck Equation	p. 252
Perturbation Equations	p. 253
Sufficient Almost-Sure Asymptotic Stability Boundary	p. 254
Numerical Example	p. 255
Sufficient A.S. Stability Condition Using Schwarz's Inequality	p. 256
Sufficient A.S. Stability Condition Using the Optimization Method	p. 257
Moment Stability of Stochastic Systems	p. 261
Moment Stability of Linear Stochastic Systems	p. 261
Effect of Random Noise on the Stability of a Damped Mathieu Equation	p. 263
Formulation: Averaged Equations	p. 263
Moment Stability	p. 265
Moment Stability of Coupled Linear Systems	p. 268
Introduction	p. 268
Averaged Equations	p. 270
Moment Stability	p. 272
Application: Flexural-Torsional Vibration of a Rectangular Beam	p. 275
Lyapunov Exponents	p. 277
Introduction	p. 277
Simulation of Lyapunov Exponents	p. 280
Lyapunov Exponents of Continuous Stochastic Dynamical Systems	p. 283
Simulation of the Largest Lyapunov Exponent Using Khasminskii's Formulation	p. 289
Systems under Non-White Excitations	p. 289
Systems under White Noise Excitations	p. 291
Lyapunov Exponents of Two-Dimensional Systems	p. 292
Systems Exhibiting Pitchfork Bifurcation	p. 293
Formulation of the Largest Lyapunov Exponent	p. 293
Lyapunov Exponent of the Nilpotent System	p. 295
Lyapunov Exponent by Asymptotic Expansion of Integrals	p. 298
Lyapunov Exponent by Digital Simulation	p. 299
Systems Exhibiting Hopf Bifurcation	p. 301
Lyapunov Exponent by Asymptotic Expansion of Integrals	p. 302
Lyapunov Exponent by Stochastic Averaging	p. 305
Lyapunov Exponent by Digital Simulation	p. 306
Lyapunov Exponent of a Two-Dimensional Viscoelastic System under Bounded Noise Excitation	p. 308
Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems	p. 318
Formulation	p. 318
Non-Singular Case	p. 322
Singular Case	p. 328
White Noise Excitation	p. 330
Generalization to Multiple Degrees-of-Freedom Systems	p. 331
Monte Carlo Simulation	p. 332
Application: Flexural-Torsional Stability of a Rectangular Beam	p. 333
Non-Follower Force Case	p. 334
Follower Force Case	p. 336
White Noise Excitation	p. 336
Moment Lyapunov Exponents	p. 338
Introduction	p. 338
The Concept of Moment Lyapunov Exponent	p. 338
The Partial Differential Eigenvalue Problem for Moment Lyapunov Exponent	p. 342
Monte Carlo Simulation of the Moment Lyapunov Exponents	p. 348
Moment Lyapunov Exponent of an Oscillator under Weak White Noise Excitation	p. 353
Eigenvalue Problem for the Moment Lyapunov Exponents	p. 354
Determination of the Moment Lyapunov Exponents Using a Method of Perturbation	p. 356
Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System	p. 359
Introduction	p. 359
Formulation	p. 361
Lyapunov Exponents	p. 364
Moment Lyapunov Exponents	p. 367
Two-Point Boundary-Value Problems	p. 367
Numerical Solutions of Moment Lyapunov Exponents	p. 369
Moment Lyapunov Exponents of an Oscillator under Real Noise Excitation	p. 373
Formulation	p. 373
Weak Noise Expansion of the Moment Lyapunov Exponent	p. 375
Zeroth-Order Perturbation	p. 376
Solution of L[subscript 0]T([zeta], [phi]) = f([zeta])g([phi])	p. 377
First-Order Perturbation	p. 379
Second-Order Perturbation	p. 380
Higher-Order Perturbation	p. 381
Stability Index	p. 383
Numerical Determination of Moment Lyapunov Exponents	p. 384
Monte Carlo Simulation of Moment Lyapunov Exponents	p. 387
Parametric Resonance of an Oscillator under Bounded Noise Excitation	p. 390
Formulation	p. 390
Weak Noise Expansions of the Moment Lyapunov Exponent Using a Method of Singular Perturbation	p. 394
Perturbation Expansion	p. 394
Zeroth-Order Perturbation	p. 395
First-Order Perturbation	p. 395
Second-Order Perturbation	p. 396
Numerical Determination of the Moment Lyapunov Exponent	p. 401
Concluding Remarks	p. 404
Maple Programs	p. 405
Stability Boundaries (Example 2.1.1, [section]2.1)	p. 405
Stability Boundaries ([section]2.4.2)	p. 405
Method of Averaging ([section]3.1.1)	p. 407
Method of Multiple Scales ([section]3.2.2)	p. 408
Stochastic Averaging (Example 5.16.1, [section]5.16.1])	p. 412
Stochastic Averaging ([section]7.2.1)	p. 414
Method of Regular Perturbation ([section]9.3.2)	p. 415
Lyapunov Exponents ([section]9.4.3)	p. 419
Bibliography	p. 422
Index	p. 429
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Dynamic Stability of Structures

At a Glance

Hardcover

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