# Nonlinear random vibration : analytical techniques and applications / Cho W.S. To.

- Author:
- To, Cho W. S.
- Published:
- Leiden, the Netherlands : CRC Press/Balkema, [2012]
- Copyright Date:
- ©2012
- Edition:
- 2nd ed.
- Physical Description:
- vii, 295 pages : illustrations ; 26 cm

- Contents:
- Machine generated contents note: 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.Ito'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 -- Appendix Probability, Random Variables and Random Processes -- A.1.Introduction -- A.2.Probability Theory -- A.2.1.Set theory and axioms of probability -- A.2.2.Conditional probability -- A.2.3.Marginal probability and Bayes' theorem -- A.3.Random Variables -- A.3.1.Probability description of single random variable -- A.3.2.Probability description of two random variables -- A.3.3.Expected values, moment generating and characteristic functions -- A.4.Random Processes -- A.4.1.Ensemble and ensemble averages -- A.4.2.Stationary, nonstationary and evolutionary random processes -- A.4.3.Ergodic and Gaussian random processes -- A.4.4.Poisson processes -- References -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Appendix.
- Summary:
- "The book is a systematic treatment of several classes of analytical techniques and applications in nonlinear random vibration. The classes include exact solution of the Fokker-Planck-Komogorov equation, methods of statistical linearization, statistical nonlinearization techniques, methods of stochastic averaging, truncated hierarchy and other techniques. Many examples of the more popular classes, such as methods of statistical linearization, statistical nonllinearization techniques and methods of stochastic averaging, are presented. Many references are quoted. A special feature of the monograph is its incorporation of detailed steps in many examples. Thus, it is suitable for self-study, advanced level graduate students, and research scientists and engineers"--

"A systematic presentation of several classes of analytical techniques in non-linear random vibration. The book also includes a concise treatment of Markovian and non-Markovian solutions of non-linear differential equations"-- - Subject(s):
- ISBN:
- 9780415898973 (hardback)

0415898978 (hardback)

9780203144626 (eBook)

0203144627 (eBook) - Bibliography Note:
- Includes bibliographical references and index.

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