Nonlinear Random Vibration, Analytical Techniques and Applications

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Nonlinear Random Vibration - Analytical Techniques and Applications, Second edition

ISBN-10: 0415898978 ISBN-13: 9780415898973 e-ISBN 9781466512849

Table of contents

Preface to the first edition

Preface to the second edition

Acknowledgements

1 Introduction

2 Markovian and Non-Markovian Solutions of Stochastic Nonlinear Differential Equations

  2.1 Introduction
      2.1.1 Classification based on regularity
      2.1.2 Classification based on memory
      2.1.3 Kinetic equation of stochastic processes
  2.2 Markovian Solution of Stochastic Nonlinear Differential Equations
      2.2.1 Markov and diffusion processes
      2.2.2 It's and Stratonovich integrals
      2.2.3 One-dimensional Fokker-Planck-Kolmogorov equation
      2.2.4 Systems with random parametric excitations
  2.3 Non-Markovian Solution of Stochastic Nonlinear Differential Equations
      2.3.1 One-dimensional problem
      2.3.2 Multi-dimensional problem

3 Exact Solutions of Fokker-Planck-Kolmogorov Equations

  3.1 Introduction
  3.2 Solution of a General Single-Degree-of-Freedom System
  3.3 Applications to Engineering Systems
      3.3.1 Systems with linear damping and nonlinear stiffness
      3.3.2 Systems with nonlinear damping and linear stiffness
      3.3.3 Systems with nonlinear damping and nonlinear stiffness
  3.4 Solution of Multi-Degree-of-Freedom Systems
  3.5 Stochastically Excited Hamiltonian Systems

4 Methods of Statistical Linearization

  4.1 Introduction
  4.2 Statistical Linearization for Single-Degree-of-Freedom Nonlinear Systems
      4.2.1 Stationary solutions of single-degree-of-freedom systems under zero mean Gaussian white noise excitations
      4.2.2 Non-zero mean stationary solution of a single-degree-of-freedom system
      4.2.3 Stationary solution of a single-degree-of-freedom system under narrow-band Excitation
      4.2.4 Stationary solution of a single-degree-of-freedom system under parametric and external random excitations
      4.2.5 Solutions of single-degree-of-freedom systems under nonstationary random excitations
  4.3 Statistical Linearization for Multi-Degree-of-Freedom Systems
  4.4 Applications to Engineering Systems
      4.4.1 Single-degree-of-freedom systems
      4.4.2 Multi-degree-of-freedom systems
  4.5 Uniqueness and Accuracy of Solutions by Statistical Linearization
      4.5.1 Uniqueness of solutions
      4.5.2 Accuracy of solutions
      4.5.3 Remarks

5 Statistical Nonlinearization Techniques

  5.1 Introduction
  5.2 Statistical Nonlinearization Technique Based on Least Mean Square of Deficiency
      5.2.1 Special case
      5.2.2 General case
      5.2.3 Examples
  5.3 Statistical Nonlinearization Technique Based on Equivalent Nonlinear Damping Coefficient
      5.3.1 Derivation of equivalent nonlinear damping coefficient
      5.3.2 Solution of equivalent nonlinear equation of single-degree-of-freedom systems
      5.3.3 Concluding remarks
  5.4 Statistical Nonlinearization Technique for Multi-Degree-of-Freedom Systems
      5.4.1 Equivalent system nonlinear damping coefficient and exact solution
      5.4.2 Applications
  5.5 Improved Statistical Nonlinearization Technique for Multi-Degree-of-Freedom Systems
      5.5.1 Exact solution of multi-degree-of-freedom nonlinear systems
      5.5.2 Improved statistical nonlinearization technique
      5.5.3 Application and comparison
      5.5.4 Concluding remarks
  5.6 Accuracy of Statistical Nonlinearization Techniques

6 Methods of Stochastic Averaging

  6.1 Introduction
  6.2 Classical Stochastic Averaging Method
      6.2.1 Stationary solution of a single-degree-of-freedom system under broad band stationary random excitation
      6.2.2 Stationary solutions of single-degree-of-freedom systems under parametric and external random excitations
      6.2.3 Nonstationary solutions of single-degree-of-freedom systems
      6.2.4 Remarks
  6.3 Stochastic Averaging Methods of Energy Envelope
      6.3.1 General theory
      6.3.2 Examples
      6.3.3 Remarks
  6.4 Other Stochastic Averaging Techniques
  6.5 Accuracy of Stochastic Averaging Techniques
      6.5.1 Smooth stochastic averaging
      6.5.2 Non-smooth stochastic averaging
      6.5.3 Remarks

7 Truncated Hierarchy and Other Techniques

  7.1 Introduction
  7.2 Truncated Hierarchy Techniques
      7.2.1 Gaussian closure schemes
      7.2.2 Non-Gaussian closure schemes
      7.2.3 Examples
      7.2.4 Remarks
  7.3 Perturbation Techniques
      7.3.1 Nonlinear single-degree-of-freedom systems
      7.3.2 Nonlinear multi-degree-of-freedom systems
      7.3.3 Remarks
  7.4 Functional Series Techniques
      7.4.1 Volterra series expansion techniques
      7.4.2 Wiener-Hermite series expansion techniques

References

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