Abstract
The probability density function (PDF) and the mean up-crossing rate (MCR) of the stationary responses of nonlinear stochastic systems excited by white noise is analyzed based on the assumption that the PDF of the responses is approximated with an exponential function of a polynomial in the state variables. Based upon the approximate PDF, a new technique is developed for the approximate PDF solution of Fokker–Planck–Kolmogorov equation, and consequently, the MCR of the system responses is analyzed. Numerical results showed that the approximate PDFs and MCRs approach to the exact ones as the degree of the polynomial increases.
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References
Booton, R. C., ‘Nonlinear control systems with random inputs’, IRE Transactions on Circuit Theory, CT-1 1, 1954, 9‐19.
Iyengar, R. N. and Dash, P. K., ‘Study of the random vibration of nonlinear systems by the Gaussian closure technique’, ASME Journal of Applied Mechanics 45, 1978, 393‐399.
Baratta, A. and Zuccaro, G., ‘Analysis of nonlinear oscillators under stochastic excitation by the Fokker‐Planck‐Kolmogorov equation’, Nonlinear Dynamics 5, 1994, 255‐271.
Sun, J.-Q. and Hsu, C. S., ‘Cumulant-neglect closure method for nonlinear systems under random excitations’, ASME Journal of Applied Mechanics 54, 1987, 649‐655.
Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, New York, 1963.
Assaf, S. A. and Zirkie, L. D., ‘Approximate analysis of non-linear stochastic systems’, International Journal of Control 23, 1976, 477‐492.
Crandall, S. H., ‘Non-Gaussian closure for random vibration of non-linear oscillators’, International Journal of Non-Linear Mechanics 15, 1980, 303‐313.
Lutes, L. D., ‘Approximate technique for treating random vibration of hysterestic systems’, Journal of the Acoustical Society America 48, 1970, 299‐306.
Cai, G. Q. and Lin, Y. K., ‘A new approximate solution technique for randomly excited non-linear oscillators’, International Journal of Non-Linear Mechanics 23, 1988, 409‐420.
Sobczyk, K. and Trebicki, J., ‘Maximum entropy principle in stochastic dynamics’, Probabilistic Engineering Mechanics 5, 1990, 102‐110.
Pandey, M. D. and Ariaratnam, S. T., ‘Crossing rate analysis of non-Gaussian response of linear systems’, ASCE Journal of Engineering Mechanics 122(6), 1996, 507‐511.
Winterstein, S. R., ‘Nonlinear vibration models for extremes and fatigue’, ASCE Journal of Engineering Mechanics 114(10), 1988, 1772‐1790.
Grigoriu, M., ‘Crossing of nonGaussian translation processes’, ASCE Journal of Engineering Mechanics 110(4), 1984, 610‐620.
Er, G. K., ‘Crossing rate analysis with a non-Gaussian closure method for nonlinear stochastic systems’, Nonlinear Dynamics 14, 1997, 279‐291.
Soong, T. T. and Grigoriu, M., Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, NJ, 1993, p. 145.
Lin, Y. K. and Cai, G. Q., Probabilistic Structural Dynamics, McGraw-Hill, New York, 1995, pp. 185, 331.
Wong, E. and Zakai, M., ‘On the relation between ordinary and stochastic equations’, International Journal of Engineering Science 3, 1965, 213‐229.
Dimentberg, M. F., ‘An exact solution to a certain nonlinear random vibration problem’, International Journal of Non-Linear Mechanics 17, 1982, 231‐236.
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Er, GK. An Improved Closure Method for Analysis of Nonlinear Stochastic Systems. Nonlinear Dynamics 17, 285–297 (1998). https://doi.org/10.1023/A:1008346204836
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DOI: https://doi.org/10.1023/A:1008346204836