Statistical Linearization Model for the Response Prediction of Nonlinear Stochastic Systems Through Information Closure Method

[+] Author and Article Information
R. J. Chang, S. J. Lin

Department of Mechanical Engineering, National Cheng Kung University, 701 Tainan, Taiwan, R.O.C.

J. Vib. Acoust 126(3), 438-448 (Jul 30, 2004) (11 pages) doi:10.1115/1.1688762 History: Received August 01, 2002; Revised November 01, 2003; Online July 30, 2004
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.


Risken, H., 1989, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed. Springer-Verlag, Berlin.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, New York.
Chang,  R. J., 1993, “Extension in Techniques for Stochastic Dynamic Systems,” Control and Dynamic Systems, 55, pp. 429–470.
Bergman, L. A., Spencer, B. F., Wojtkiewicz, S. F., and Johnson, E. A., 1996, “Robust Numerical Solution of the Fokker-Planck Equation for Second Order Dynamical Systems under Parametric and External White Noise Excitations,” Nonlinear Dynamics and Stochastic Mechanics, Langford, W., Kliemann, W., and Sri Namachchivaya, N., eds., Vol. 9, pp. 23–37, American Mathematical Society, Providence.
Socha,  L., and Soong,  T. T., 1991, “Linearization in Analysis of Nonlinear Stochastic Systems,” Appl. Mech. Rev., 44, pp. 399–422.
Leithead,  W. E., 1990, “A Systematic Approach to Linear Approximation of Nonlinear Stochastic Systems. Part 1: Asymptotic Expansions,” Int. J. Control, 51, pp. 71–91.
Chang,  R. J., 1990, “Model Based Discrete Linear State Estimator for Nonlinearizable Systems with State-Dependent Noise,” ASME J. Dyn. Syst., Meas., Control, 112, pp. 774–781.
Chang,  R. J., 1992, “Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems,” ASME J. Dyn. Syst., Meas., Control, 114, pp. 20–26.
Beaman,  J. J., and Hedrick,  J. K., 1981, “Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems. Part 1: An Extended Statistical Linearization Technique,” ASME J. Dyn. Syst., Meas., Control, 103, pp. 14–21.
Brückner,  A., and Lin,  Y. K., 1987, “Generalization of Equivalent Linearization Method for Nonlinear Random Vibration Problems,” Int. J. Non-Linear Mech., 22, pp. 227–235.
Wojtkiewicz,  S. F., Spencer,  B. F., and Bergman,  L. A., 1996, “On the Cumulant-Neglect Closure Method in Stochastic Dynamics,” Int. J. Non-Linear Mech., 31, pp. 657–684.
Haken, H., 1988, Information and Self-Organization, Springer-Verlag, Berlin.
Chang,  R. J., 1991, “Maximum Entropy Approach for Stationary Response of Nonlinear Stochastic Oscillators,” ASME J. Appl. Mech., 58, pp. 266–271.
Sobczyk,  K., and Trebicki,  J., 1990, “Maximum Entropy Principle in Stochastic Dynamics,” Probab. Eng. Mech., 5, pp. 102–110.
Trebicki,  J., and Sobczyk,  K., 1996, “Maximum Entropy Principle and Non-Stationary Distributions of Stochastic Systems,” Probab. Eng. Mech., 11, pp. 169–178.
Jumarie,  G., 1990, “Solution of the Multivariate Fokker-Planck Equation by Using a Maximum Path Entropy Principle,” J. Math. Phys., 31, pp. 2389–2392.
Phillis,  Y. A., 1982, “Entropy Stability of Continuous Dynamic Systems,” Int. J. Control, 35, pp. 323–340.
Chang,  R. J., and Lin,  S. J., 2002, “Information Closure Method for Dynamic Analysis of Nonlinear Stochastic Systems,” ASME J. Dyn. Syst., Meas., Control, 124, pp. 353–363.
Jaynes,  E. T., 1957, “Information Theory and Statistical Mechanics,” Phys. Rev., 106, pp. 620–630.
Young,  G. E., and Chang,  R. J., 1987, “Prediction of the Response of Nonlinear Oscillators under Stochastic Parametric and External Excitations,” Int. J. Non-Linear Mech., 28, pp. 151–160.
Dimentberg,  M. F., 1982, “An Exact Solution to a Certain Nonlinear Random Vibration Problem,” Int. J. Non-Linear Mech., 17, pp. 231–236.


Grahic Jump Location
Nonstationary moment responses predicted by various Gaussian linearization models and the stationary result derived by exact solution
Grahic Jump Location
Moment responses (m60,l(t)) predicted by different linearization models corresponding to density pl(x1,x2,t),(l=1∼4) and compared with those estimated by density pl(x1,x2,t) and the results of Monte Carlo simulations
Grahic Jump Location
Parametric boundaries of entropy stability predicted by different density modes: 1- p1(x1,x2)=N1,1exp(−λ1,1x12)N2,1exp(−λ2,1x22), 2- p2(x1,x2)=N1,2 exp(−λ1,2x14)N2,2exp(−λ2,2x24), 3- p3(x1,x2)=N1,3 exp(−λ1,3x16)N2,3exp(−λ2,3x26).
Grahic Jump Location
Entropy and second moment responses with varied external excitation intensity (q33) obtained by the improved Gaussian linearization method (IGL), Gaussian linearization method (GL) and exact solution
Grahic Jump Location
A priori information of moment propagation given by Monte Carlo simulation
Grahic Jump Location
Entropy evolution (Hl(t)) estimated by different density pl(x1,x2,t)(l=1∼4)
Grahic Jump Location
Density evolution pl(x1,t)(l=1∼4) compared with the results of Monte Carlo simulations at time instant t=1 and t=8



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In