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9780486432403

Random Vibration and Statistical Linearization

by ;
  • ISBN13:

    9780486432403

  • ISBN10:

    0486432408

  • Format: Paperback
  • Copyright: 2003-12-09
  • Publisher: Dover Publications

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Supplemental Materials

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Summary

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.

Table of Contents

Preface to the Dover Edition xi
Preface xiii
Chapter 1 Introduction
1.1 Random vibration
1(2)
1.2 Importance of non-linearities
3(1)
1.3 Non-linear random vibration problems
4(1)
1.4 Methods of solution
5(7)
1.4.1 Statistical linearization
5(3)
1.4.2 Moment closure
8(1)
1.4.3 Equivalent non-linear equations
9(1)
1.4.4 Perturbation and functional series
9(1)
1.4.5 Markov methods
10(1)
1.4.6 Monte Carlo simulation
11(1)
1.5 Role of statistical linearization
12(1)
1.6 Scope of book
13(1)
1.7 Plan of book
14(3)
Chapter 2 General equations of motion and the representation of nonlinearities
2.1 Introduction
17(1)
2.2 The general equations of motion
17(8)
2.2.1 Small vibrations
21(3)
2.2.2 Large vibrations
24(1)
2.3 Non-linear conservative forces
25(14)
2.3.1 Motion in a gravitational field
26(2)
2.3.2 Restoring moments for floating bodies
28(1)
2.3.3 Elastic restoring forces
29(3)
2.3.4 Non-linear elasticity
32(5)
2.3.5 Geometric non-linearities
37(2)
2.4 Non-linear dissipative forces
39(24)
2.4.1 Internal damping in materials
42(7)
2.4.2 Mathematical representation of hysteresis loops
49(4)
2.4.3 Interface damping
53(4)
2.4.4 Flow induced forces
57(6)
Chapter 3 Probability theory and stochastic processes
3.1 Introduction
63(1)
3.2 Random events and probability
63(1)
3.3 Random variables
64(3)
3.3.1 Probability distributions
65(1)
3.3.2 Transformation of random variables
66(1)
3.4 Expectation of random variables
67(3)
3.5 The Gaussian distribution
70(5)
3.5.1 Properties of Gaussian random variables
72(1)
3.5.2 Expansions of the Gaussian distribution
72(3)
3.6 The concept of a stochastic process
75(4)
3.6.1 The complete probabilistic specification
76(1)
3.6.2 The Gaussian process
77(1)
3.6.3 Stationary processes
78(1)
3.7 Differentiation of stochastic processes
79(1)
3.8 Integration of stochastic processes
80(1)
3.9 Ergodicity
80(2)
3.10 Spectral decomposition
82(4)
3.11 Specification of joint processes
86(2)
Chapter 4 Elements of linear random vibration theory
4.1 Introduction
88(1)
4.2 General input-output relationships
88(2)
4.3 Stochastic input-output relationships
90(3)
4.4 Analysis of lumped parameter systems
93(8)
4.4.1 Response prediction
93(1)
4.4.2 Free undamped motion
94(1)
4.4.3 Classical modal analysis
95(2)
4.4.4 State variable formulation
97(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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