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Nonlinear Stochastic Dynamic Engineering Systems pp 323–333Cite as

Exact and Approximate Solutions for Response of Nonlinear Systems Under Parametric and External White Noise Excitations

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Summary

By separating the drift coefficients in the Fokker-Planck equation into two parts, one associated with the vanishing probability flow and another with the circulatory probability flow, a method is devised to obtain the exact stationary-state solutions for nonlinear systems under either external or parametric Gaussian white-noise excitations, or both. The conditions under which the method is applicable are less restrictive than the conditions for detailed balance which was used previously to develope a similar method; therefore, the solvable class is broader. Two schemes are then discussed for obtaining approximate solutions which may be either stationary or transient: (1) cumulant closure, and (2) generalized equivalent linearization.

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References

  1. Moyal, J. E., Stochastic processes and statistical physics, J. Roy. Statist. Soc. (London), B11 (1949) 150–162.

    MATH  MathSciNet  Google Scholar 

  2. Wong, E. and Zakai, M., On the relation between ordinary and stochastic equations, Int. J. Eng. Sci., 4 (1965) 213–229.

    MathSciNet  Google Scholar 

  3. Caughey, T. K. and Dienes, J. K., Analysis of a nonlinear first-order system with a white noise input, J. Appl. Physics, 23 (1961) 2476–2479.

    Article  ADS  MathSciNet  Google Scholar 

  4. Caughey, T. K., Nonlinear theory of random vibration, in Advances in Applied Mechanics, 11, (1971) 209–253.

    Google Scholar 

  5. Caughey, T. K. and Ma, F., The exact steady-state solution of a class of nonlinear stochastic systems, Int. J. Nonlinear Mechanics, 17 (1983) 137–142.

    Article  ADS  MathSciNet  Google Scholar 

  6. Lin, Y. K., Probabilistic theory of structural dynamics, McGraw-Hill, New York, 1967, reprinted with revision, Krieger Publishing Co., Melbourne, FL, 1976.

    Google Scholar 

  7. Stratonovich, R. L., Topics in the theory of random noise, Vol. 1, Gordon and Breach, New York, 1963.

    Google Scholar 

  8. Lin, Y. K., Yong, Y. and Cai, G., Exact solutions for nonlinear systems under parametric and external white-noise excitations, Proceedings, US-Austria Joint Seminar on Stochastic Structural Mechanics, Springer-Verlag, 1987.

    Google Scholar 

  9. Graham, R. and Haken, H., Generalized thermo-dynamic potential for Markoff systems in detailed balance and far from thermal equilibrium, Zeitschrift fur Physik, 203, (1971) 289–302.

    Article  ADS  MathSciNet  Google Scholar 

  10. Yong, Y. and Lin, Y. K., Exact stationary response solutions for second order nonlinear systems under parametric and external white-noise excitations, J. of Appl. Mechanics, 54, (1987) 414–418.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Dimentberg, M. F., An exact solution to a certain nonlinear random vibration problem, Int. J. Nonlinear Mechanics, 17 (1982) 231–236.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Wu, W.F. and Lin, Y. K., Cumulant-neglect closure for nonlinear oscillators under random parametric and external excitations, Int. J. Nonlinear Mechanics, 19 (1984) 349–362.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Ibrahim, R. A., Soundararajan, A. and Heo, H., Stochastic response of nonlinear dynamics systems based on a non-Gaussian closure, J. Appl. Mechanics, 52 (1985) 965–970.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Itö, K., On a formula concerning stochastic differentials, Nagoya Mathematical Journal, Japan, 3 (1951) 55–65.

    Google Scholar 

  15. Caughey, T. K., Equivalent Linearization Techniques, J. Acoustical Soc. Am., 35 (1963) 1706–1711.

    Article  ADS  MathSciNet  Google Scholar 

  16. Bruckner, A. and Lin, Y. K., Generalization of the equivalent linearization method for nonlinear random vibration problems, Int. J. Nonlinear Mechanics, to appear.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Center for Applied Stochastics Research, Florida Atlantic University, Boca Raton, FL, 33431, USA

    Y. K. Lin, Y. Yong & G. Q. Cai

  2. Nuclear Research Center, Karlsruhe, Germany

    A. Brückner

Authors
  1. Y. K. Lin
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  2. Y. Yong
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  3. G. Q. Cai
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  4. A. Brückner
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Editor information

Editors and Affiliations

  1. Österreichisches Nationalkomitee für Theoretische und Angewandte Mechanik, Technische Universität Wien, Institut für Allgemeine Mechanik, Karlsplatz 13/201, A-1040, Wien, Österreich

    F. Ziegler

  2. Institut für Mechanik, Universität Innsbruck, Technikerstraße 13, A-6020, Innsbruck, Österreich

    G. I. Schuëller Ph. D.

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© 1988 Springer-Verlag Berlin Heidelberg

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Lin, Y.K., Yong, Y., Cai, G.Q., Brückner, A. (1988). Exact and Approximate Solutions for Response of Nonlinear Systems Under Parametric and External White Noise Excitations. In: Ziegler, F., Schuëller, G.I. (eds) Nonlinear Stochastic Dynamic Engineering Systems. IUTAM Symposium. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83334-2_23

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  • DOI: https://doi.org/10.1007/978-3-642-83334-2_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-83336-6

  • Online ISBN: 978-3-642-83334-2

  • eBook Packages: Springer Book Archive

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