Abstract
The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Wang, M. C. and Uhlenbeck, G., ‘On the theory or Brownian motion II’, Reviews of Modern Physics 17, 1945, 323–342. Reprinted in Selected Papers on Noise and Stochastic Processes (N., Wax, ed.), Dover, New York, 1954.
Lin, Y. K., Probabilistic Theory of Structural Dynamics, McGraw Hill, New York, 1967.
Nigam, N. C., Introduction to Random Vibrations, MIT Press, Cambridge, 1983.
Caughey, T. K. and Dienes, J. K., “The behavior of linear systems with white noise input’, Journal of Mathematical Physics 32, 1962, 2476–2479.
Caughey, T. K., ‘Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white noise excitation’, Journal of the Acoustical Society of America 35, 1963, 1683–1692.
Caughey, T. K., ‘Nonlinear theory of random vibrations’, in Advances in Applied Mechanics 11 (Chia-Shun Yih, ed.), Academic Press, 1971, 209–253.
Lin, Y. K. and Cai, G. Q., ‘Exact stationary-response solution for second order nonlinear systems under parametric and external white noise excitation II’, Journal of Applied Mechanics 55, 1988, 702–705.
Soize, C., ‘Steady-state solution of Fokker-Planck equation in higher dimension’, Probabilistic Engineering Mechanics 3, 196–206.
Atkinson, J. D., ‘Eigenfunction expansions for randomly excited non-linear systems’, Journal of Sound and Vibration 30, 1973, 153–172.
Bhandari, R. G. and Sherrer, R. E., ‘Random vibrations in discrete nonlinear dynamic systems’, Journal of Mechanical Engineering Science 10, 1968, 168–174.
Wen, Y. K., ‘Approximate method for nonlinear random vibration’, Journal of the Engineering Mechanics Division, ASCE 101, 1975, 389–401.
Wen, Y. K., ‘Method for random vibration of hysteretic systems’, Journal of the Engineering Mechanics Division, ASCE 102, 1976, 249–263.
Kunert, A., ‘Efficient numerical solution of multidimensional Fokker-Planck equations with chaotic and nonlinear random vibration’, Vibrational Analysis—Analytical and Computational (T. C. Huang, C. S. Hsu, W. Q. Feng, S. C. Sinha, R. A. Ibrahim, and R. L. Engelstad, eds.) 37, 1991.
Langley, R. S., ‘A finite element method for the statistics of non-linear random vibration’, Journal of Sound and Vibration 101, 1985, 41–54.
Langtangen, H. P., ‘A general numerical solution method for Fokker-Planck equations with applications to structural reliability’, Probabilistic Engineering Mechanics 6, 1991, 33–48.
Bolotin, V. V., ‘Statistical aspects in the theory of structural stability’, in Dynamic Stability of Structures (George, Herrmann, ed.), 67–81, Pergamon Press, New York, 1967.
Sun, J.-Q. and Hsu, C. S., ‘First-passage time probability of non-linear stochastic systems by generalized cell mapping method’, Journal of Sound and Vibration 124, 1988, 233–248.
Sun, J.-Q. and Hsu, C. S., ‘The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation’, ASME Journal of Applied Mechanics 57, 1990, 1018–1025.
Naess, A. and Johnsen, J. M., ‘Response statistics of nonlinear dynamic systems by path integration’, Proceedings of IUTAM Symposium on Nonlinear Stochastic Mechanics, Torino, Italy, July 1–5, 1991.
Darling, D. A. and Siegert, A. J. F., ‘The first passage problem for a continuous Markov process’, Annals of Mathematical Statistics 24, 1953, 624–639.
Crandall, S. H., ‘First crossing probabilities of the linear oscillator’, Journal of Sound and Vibration 12, 1970, 285–299.
Yang, J-N. and Shinozuka, M., ‘First passage time problem’, The Acoustical Society of America 47, 1970, 393–394.
Fichera, G., ‘On a unified theory of boundary value problems for elliptic-parobolic equations of second order’, in Boundary Problems in Differential Equations (R. E., Langer, ed.), University of Wisconsin Press, Wisconsin, 1960, 97–102.
Toland, R. H., ‘Random walk approach to first-passage and other random vibration problems’, Ph.D. Thesis, University of Delaware at Newark, Delaware, 1969.
Toland, R. H. and Yang, C. Y., ‘Random walk model for first passage probability’, Journal of Engineering Mechanics Division, ASCE 97, 1971, 791–807.
Bergman, L. A. and Heinrich, J. C., ‘On the moments of time to first passage of the linear oscillator’, Earthquake Engineering and Structural Dynamics 9, 1981, 197–204.
Bergman, L. A. and Spencer, B. F., Jr., ‘Solution of the first passage problem for simple linear and nonlinear oscillators by the finite element method’, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, T & AM Report No. 461, 1983.
Spanos, P.-T. D. and Solomos, G. P., ‘Barrier crossing due to transient excitation’, Journal of Engineering Mechanics, ASCE 110, 1984, 20–36.
Sri Namachchivaya, N., ‘Instability theorem based on the nature of the boundary behavior for one dimensional diffusion’, SM Archives 14, 1989, 131–142.
Chandiramani, K. L., ‘First-passage probabilities of a linear oscillator’, Ph.D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1964.
Crandall, S. H., Chandiramani, K. L., and Cook, R. G., ‘Some first-passage problems in random vibration’, Journal of Applied Mechanics, ASME 33, 1966, 532–538.
Spencer, B. F., Jr., On the Reliability of Nonlinear Hysteretic Structures Subjected to Broadband Random Excitation, Lecture Notes in Engineering (series editors: C. A. Brebbia and S. A. Orszag), 21, 1986, Springer-Verlag.
Bergman, L. A., ‘Numerical solutions of the first passage problem in stochastic structural dynamics’, Computational Mechanics of Probabilistic and Reliability Analysis, Elme Press International, Lausanne, Switzerland, 1989, 479–508.
Spencer, B. F.Jr. and Bergman, L. A., ‘Comments on ‘First-passage time probability of nonlinear stochastic systems by generalized cell mapping method’, Journal of Sound and Vibration 134, 1989, 181–185.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Spencer, B.F., Bergman, L.A. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Nonlinear Dyn 4, 357–372 (1993). https://doi.org/10.1007/BF00120671
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00120671