Random Vibration and Statistical Linearization

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ISBN: 9780486432403 | 0486432408
Cover: Paperback
Copyright: 12/9/2003

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This self-contained volume explains the general method of statistical, or equivalent, linearization and its use in solving random vibration problems. Subjects include general equations of motion and representation of non-linearities, probability theory and stochastic processes, elements of linear random vibration theory, statistical linearization for simple systems with stationary response, more. 1990 edition.

Preface to the Dover Edition

Preface

xiii

Chapter 1 Introduction

1.1 Random vibration

(2)

1.2 Importance of non-linearities

(1)

1.3 Non-linear random vibration problems

(1)

1.4 Methods of solution

(7)

1.4.1 Statistical linearization

(3)

1.4.2 Moment closure

(1)

1.4.3 Equivalent non-linear equations

(1)

1.4.4 Perturbation and functional series

(1)

1.4.5 Markov methods

(1)

1.4.6 Monte Carlo simulation

(1)

1.5 Role of statistical linearization

(1)

1.6 Scope of book

(1)

1.7 Plan of book

(3)

Chapter 2 General equations of motion and the representation of nonlinearities

2.1 Introduction

(1)

2.2 The general equations of motion

(8)

2.2.1 Small vibrations

(3)

2.2.2 Large vibrations

(1)

2.3 Non-linear conservative forces

(14)

2.3.1 Motion in a gravitational field

(2)

2.3.2 Restoring moments for floating bodies

(1)

2.3.3 Elastic restoring forces

(3)

2.3.4 Non-linear elasticity

(5)

2.3.5 Geometric non-linearities

(2)

2.4 Non-linear dissipative forces

(24)

2.4.1 Internal damping in materials

(7)

2.4.2 Mathematical representation of hysteresis loops

(4)

2.4.3 Interface damping

(4)

2.4.4 Flow induced forces

(6)

Chapter 3 Probability theory and stochastic processes

3.1 Introduction

(1)

3.2 Random events and probability

(1)

3.3 Random variables

(3)

3.3.1 Probability distributions

(1)

3.3.2 Transformation of random variables

(1)

3.4 Expectation of random variables

(3)

3.5 The Gaussian distribution

(5)

3.5.1 Properties of Gaussian random variables

(1)

3.5.2 Expansions of the Gaussian distribution

(3)

3.6 The concept of a stochastic process

(4)

3.6.1 The complete probabilistic specification

(1)

3.6.2 The Gaussian process

(1)

3.6.3 Stationary processes

(1)

3.7 Differentiation of stochastic processes

(1)

3.8 Integration of stochastic processes

(1)

3.9 Ergodicity

(2)

3.10 Spectral decomposition

(4)

3.11 Specification of joint processes

(2)

Chapter 4 Elements of linear random vibration theory

4.1 Introduction

(1)

4.2 General input-output relationships

(2)

4.3 Stochastic input-output relationships

(3)

4.4 Analysis of lumped parameter systems

(8)

4.4.1 Response prediction

(1)

4.4.2 Free undamped motion

(1)

4.4.3 Classical modal analysis

(2)

4.4.4 State variable formulation

(3)

4.4.5 Complex modal analysis

100

(1)

4.5 Stochastic response of linear systems

101

(21)

4.5.1 Single degree of freedom systems

101

(6)

4.5.2 Two degree of freedom systems

107

(4)

4.5.3 Multi-degree of freedom systems

111

(2)

4.5.4 State variable analysis

113

(2)

4.5.5 Analysis using complex modes

115

(7)

Chapter 5 Statistical linearization for simple systems with stationary response

5.1 Introduction

122

(1)

5.2 Non-linear elements without memory

122

(7)

5.2.1 Statistical linearization procedure

123

(2)

5.2.2 Optimum linearization

125

(1)

5.2.3 Examples

126

(3)

5.3 Oscillators with non-linear stiffness

129

(26)

5.3.1 The statistical linearization approximation

131

(2)

5.3.2 Standard deviation of the response

133

(3)

5.3.3 The case of small non-linearity

136

(1)

5.3.4 Power spectrum of the response

137

(1)

5.3.5 Inputs with non-zero means

137

(3)

5.3.6 Asymmetric non-linearities

140

(4)

5.3.7 Systems with a softening restoring characteristic

144

(3)

5.3.8 Systems with multiple static equilibrium positions

147

(4)

5.3.9 Response to narrow-band excitation

151

(4)

5.4 Oscillators with non-linear stiffness and damping

155

(7)

5.4.1 Standard deviation of the response

158

(2)

5.4.2 The case of small non-linearity

160

(1)

5.4.3 Power spectrum of the response

161

(1)

5.4.4 Input and output with non-zero means

161

(1)

5.5 Higher order linearization

162

(2)

5.6 Applications

164

(13)

5.6.1 Friction controlled slip of a structure on a foundation

164

(4)

5.6.2 Ship roll motion in irregular waves

168

(5)

5.6.3 Flow induced vibration of cylindrical structures

173

(4)

Chapter 6 Statistical linearization of multi-degree of freedom systems with stationary response

6.1 Introduction

177

(1)

6.2 The non-linear system

177

(1)

6.3 The equivalent linear system

178

(5)

6.3.1 Formulation

178

(1)

6.3.2 Minimization procedure

179

(1)

6.3.3 Equations for the equivalent linear system parameters

179

(2)

6.3.4 Examination of the minimum

181

(1)

6.3.5 Existence and uniqueness of the equivalent linear system

182

(1)

6.4 Mechanization of the method

183

(1)

6.5 Determination of the elements of the equivalent linear system

184

(3)

6.5.1 Gaussian approximation

184

(1)

6.5.2 Chain-like systems

185

(1)

6.5.3 Treatment of asymmetric non-linearities

186

(1)

6.6 Solution procedures

187

(22)

6.6.1 General remarks

187

(1)

6.6.2 Spectral matrix solution procedure

188

(8)

6.6.3 Modal analysis

196

(6)

6.6.4 State variable solution procedure

202

(3)

6.6.5 Complex modal analysis

205

(4)

6.7 Mode-by-mode linearization

209

(3)

Chapter 7 Non-stationary problems

7.1 Introduction

212

(1)

7.2 General theory

212

(4)

7.3 White noise excitation

216

(9)

7.3.1 Friction controlled slip of a structure on a foundation

217

(5)

7.3.2 Oscillator with asymmetric non-linearity

222

(3)

7.4 Non-white excitation

225

(10)

7.4.1 Decomposition method

226

(1)

7.4.2 Use of pre-filters

227

(3)

7.4.3 An example

230

(5)

Chapter 8 Systems with hysteretic non-linearity

8.1 Introduction

235

(1)

8.2 Averaging method

235

(22)

8.2.1 An alternative approach

239

(2)

8.2.2 Evaluation of the expectations

241

(2)

8.2.3 Application to non-hysteretic oscillators

243

(2)

8.2.4 Inputs with non-zero means

245

(1)

8.2.5 The bilinear oscillator

246

(9)

8.2.6 Allowance for drift motion

255

(2)

8.3 Use of differential models of hysteresis

257

(24)

8.3.1 Oscillators with hysteresis

257

(7)

8.3.2 The bilinear oscillator

264

(7)

8.3.3 The curvilinear model

271

(2)

8.3.4 Inputs with non-zero means

273

(2)

8.3.5 Biaxial hysteretic restoring forces

275

(1)

8.3.6 Multi-degree of freedom systems

276

(5)

8.4 Non-stationary problems

281

(4)

8.4.1 Degrading systems

281

(3)

8.4.2 Non-stationary excitation

284

(1)

Chapter 9 Relaxation of the Gaussian response assumption

9.1 Introduction

285

(1)

9.2 Statistical linearization and Gaussian closure

285

(8)

9.2.1 An example

289

(4)

9.3 Non-Gaussian closure

293

(14)

9.3.1 Moment equations

293

(2)

9.3.2 Closure techniques

295

(2)

9.3.3 An example

297

(10)

9.4 Method of equivalent non-linear equations (ENLE)

307

(17)

9.4.1 Exact solution

308

(3)

9.4.2 Equivalent non-linear equations

311

(3)

9.4.3 Oscillators with linear stiffness and non-linear damping

314

(2)

9.4.4 Oscillators with quadratic damping

316

(2)

9.4.5 Oscillators with linear-plus-cubic damping

318

(3)

9.4.6 An alternative approach

321

(3)

9.5 Reliability estimation

324

(8)

9.5.1 First passage probability

324

(2)

9.5.2 Fatigue life

326

(2)

9.5.3 An example

328

(4)

9.6 Parametric identification

332

(15)

9.6.1 Direct optimization

335

(1)

9.6.2 State variable filters

336

(3)

9.6.3 An example

339

(8)

Chapter 10 Accuracy of statistical linearization

10.1 Introduction

347

(1)

10.2 Exact solutions

347

(5)

10.2.1 Linear damping

348

(1)

10.2.2 Chain-like systems

349

(3)

10.2.3 First-order systems

352

(1)

10.3 Comparison with exact solutions

352

(9)

10.3.1 First-order systems

352

(1)

10.3.2 Oscillators with power-law springs

353

(2)

10.3.3 Duffing oscillators

355

(2)

10.3.4 Oscillators with tangent-law springs

357

(2)

10.3.5 Oscillators with non-linear damping

359

(2)

10.4 Comparison with Monte Carlo simulation results

361

(17)

10.4.1 Simulation technique

361

(2)

10.4.2 Oscillators with non-linear damping

363

(3)

10.4.3 Oscillators with non-linear springs

366

(5)

10.4.4 Oscillators with hysteresis

371

(4)

10.4.5 Multi-degree of freedom systems with hysteresis

375

(1)

10.4.6 Non-stationary response

376

(2)

10.5 Concluding remarks

378

(2)

Appendix A: Evaluation of expectations

380

(2)

Appendix B: A useful integral for random vibration analyses

382

(5)

Addendum to Appendix B

387

(6)

References

393

(12)

Additional References

405

(33)

Author index

438

(4)

Subject index

442

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Random Vibration and Statistical Linearization

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