Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Stochastic Mechanics pp 49–60Cite as

Robust Numerical Solution of the Transient Fokker-Planck Equation for Nonlinear Dynamical Systems

  • Conference paper

Part of the book series: IUTAM Symposia ((IUTAM))

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.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atkinson, J.D. Eigenfunction Expansions for Randomly Excited Non-Linear Systems, Journal of Sound and Vibration, Vol. 30, No. 2, pp. 153–172, 1973.

    Article  MATH  ADS  Google Scholar 

  2. 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.

    Google Scholar 

  3. 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.

    Google Scholar 

  4. 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.

    Article  Google Scholar 

  5. 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.

    Google Scholar 

  6. 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.

    MathSciNet  Google Scholar 

  7. 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.

    Article  ADS  MathSciNet  Google Scholar 

  8. 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.

    Google Scholar 

  9. 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.

    Article  ADS  Google Scholar 

  10. Crandall, S.H. First Crossing Probabilities of the Linear Oscillator, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 285–299, 1970.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. 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.

    Article  MATH  ADS  Google Scholar 

  12. 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.

    Article  Google Scholar 

  13. Lin, Y.K. Probabilistic Theory of Structural Dynamics, Mc-Graw Hill, New York, 1967.

    Google Scholar 

  14. Nigam, N.C. Introduction to Random Vibrations, MIT Press, Cambridge, 1983.

    Google Scholar 

  15. 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.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Roberts, J.B. and Spanos, P.D. Random Vibration and Statistical Linearization, Wiley, Chichester, 1990.

    MATH  Google Scholar 

  17. Soize, C. Steady-State Solution of Fokker-Planck Equation in Higher Dimension, Probabilistic Engineering Mechanics, Vol. 3, No. 4, pp. 196–206, 1988.

    Article  Google Scholar 

  18. 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.

    Google Scholar 

  19. 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.

    Article  ADS  MathSciNet  Google Scholar 

  20. 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.

    Article  ADS  MathSciNet  Google Scholar 

  21. 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.

    Google Scholar 

  22. Reprinted in Selected Papers on Noise and Stochastic Processes (N. Wax, Ed.), Dover, New York, 1954.

    MATH  Google Scholar 

  23. Wen, Y.K. Approximate Method for Nonlinear Random Vibration, Journal of the Engineering Mechanics Division, ASCE, Vol. 101, No. EM4, pp. 389–401, 1975.

    Google Scholar 

  24. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Aero. and Astro. Engineering, University of Illinois, Urbana, Illinois, 61801-0236, USA

    L. A. Bergman

  2. Dept. of Civil Engineering, University of Notre Dame, Notre Dame, Indiana, 46556-0767, USA

    B. F. Spencer Jr.

Authors
  1. L. A. Bergman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. F. Spencer Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Torino, Italy

    Nicola Bellomo

  2. Dept. of Structural Mechanics, University of Pavia, Via Abbiategrasso 211, I-27100, Pavia, Italy

    Fabio Casciati

Rights and permissions

Reprints 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

Publish with us

Policies and ethics

Search

Navigation