Summary
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. Several systems are examined, including linear, Duffing, and Van der Pol oscillators, to illustrate the robustness and accuracy of the finite element method.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Atkinson, J.D. Eigenfunction Expansions for Randomly Excited Non-Linear Systems, Journal of Sound and Vibration, Vol. 30, No. 2, pp. 153–172, 1973.
Bergman, L.A. Numerical Solutions of the First Passage Problem in Stochastic Structural Dynamics, in Computational Mechanics of Probabilistic and Reliability Analysis, ElmePress International, Lausanne, Switzerland, pp. 479–508, 1989.
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.
Bhandari, R.G. and Sherrer, R.E. Random Vibrations in Discrete Nonlinear Dynamic Systems, Journal of Mechanical Engineering Science, Vol. 10, No. 2, pp. 168–174, 1968.
Bolotin, V.V. Statistical Aspects in the Theory of Structural Stability, in Dynamic Stability of Structures, (George Herrmann, Ed.), pp. 67–81, Pergamon Press, New York, 1967.
Caughey, T.K. and Dienes, J.K. The Behavior of Linear Systems with White Noise Input, Journal of Mathematical Physics, Vol. 32, pp. 2476–2479, 1962.
Caughey, T.K. Derivation and Application of the Fokker-Planck Equation to Discrete Linear Dynamic Systems Subjected to White Noise Excitation, Journal of the Acoustical Society of America, Vol. 35, No. 11, pp. 1683–1692, 1963.
Caughey, T.K. Nonlinear Theory of Random Vibrations, in Advances in Applied Mechanics Vol. 11 (Chia-Shun Yih, Ed.), Academic Press, pp. 209–253, 1971.
Crandall, S.H., Chandiramani, K.L. and Cook, R.G. Some First-Passage Problems in Random Vibration, Journal of Applied Mechanics, ASME, Vol. 33, pp. 532–538, 1966.
Crandall, S.H. First Crossing Probabilities of the Linear Oscillator, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 285–299, 1970.
Langley, R.S. A Finite Element Method for the Statistics of Non-Linear Random Vibration, Journal of Sound and Vibration, Vol. 101, No. 1, pp. 41–54, 1985.
Langtangen, H.P. A General Numerical Solution Method for Fokker-Planck Equations with Applications to Structural Reliability, Probabilistic Engineering Mechanics, Vol. 6, Nol. 1, pp. 33–48, 1991.
Lin, Y.K. Probabilistic Theory of Structural Dynamics, Mc-Graw Hill, New York, 1967.
Nigam, N.C. Introduction to Random Vibrations, MIT Press, Cambridge, 1983.
Roberts, J.B. and Spanos, P.D. Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems, International Journal of Non-Linear Mechanics, Vol. 21, No. 2, pp. 111–134, 1986.
Roberts, J.B. and Spanos, P.D. Random Vibration and Statistical Linearization, Wiley, Chichester, 1990.
Soize, C. Steady-State Solution of Fokker-Planck Equation in Higher Dimension, Probabilistic Engineering Mechanics, Vol. 3, No. 4, pp. 196–206, 1988.
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), Vol. 21, Springer-Verlag, 1986.
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, Vol. 124, 233–248, 1988.
Sun, J.-Q. and Hsu, C.S. The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-lime Gaussian Approximation, ASME Journal of Applied Mechanics, Vol. 57, pp. 1018–1025, 1990.
Wang, M.C. and Uhlenbeck, G. On the Theory of Brownian Motion H, Reviews of Modern Physics, Vol. 17, No. 2–3, pp. 323–342, 1945.
Reprinted in Selected Papers on Noise and Stochastic Processes (N. Wax, Ed.), Dover, New York, 1954.
Wen, Y.K. Approximate Method for Nonlinear Random Vibration, Journal of the Engineering Mechanics Division, ASCE, Vol. 101, No. EM4, pp. 389–401, 1975.
Wen, Y.K. Method for Random Vibration of Hysteretic Systems, Journal of the Engineering Mechanics Division, ASCE, Vol. 102, No. EM2, pp. 249–263, 1976.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bergman, L.A., Spencer, B.F. (1992). Robust Numerical Solution of the Transient Fokker-Planck Equation for Nonlinear Dynamical Systems. In: Bellomo, N., Casciati, F. (eds) Nonlinear Stochastic Mechanics. IUTAM Symposia. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84789-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-84789-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-84791-2
Online ISBN: 978-3-642-84789-9
eBook Packages: Springer Book Archive