Abstract
This paper proposes a novel method for determining the probabilistic stationary solution of multi-dimensional nonlinear stochastic dynamic systems. In general, the PDF solution is governed by Fokker-Planck equations in multi-dimensions. By dividing the space of the state variables into two subspaces and integrating the Fokker-Planck equation over one of the subspaces, a reduced set of Fokker-Planck equations can be obtained in the state variables of the other subspace. This is achieved by manipulating the integrals and approximating the conditional PDFs resulted from integration. Hence, the reduced set of Fokker-Planck equation will have a smaller number of state variables at choices and can be solved by the exponential polynomial closure method. Examples of the nonlinear stochastic dynamic systems with polynomial nonlinearity are given to show the effectiveness of this novel subspace method. The paper attempts to provide a tool for analyzing the probabilistic solutions of some highly multi-dimensional nonlinear stochastic dynamics systems in various areas of science and engineering.
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
Scheurkogel, A., Elishakoff, I.: Non-linear random vibration of a two-degree-of-freedom system. In: Ziegler, F., Schuëller, G.I. (eds.) Non-Linear Stochastic Engineering Systems, pp. 285–299. Springer, Berlin (1988)
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. International edn. McGraw-Hill, New York (1995)
Caughey, T.K.: Response of a non-linear string to random loading. ASME J. Appl. Mech. 26, 341–344 (1959)
Lin, Y.K.: Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York (1967)
Spanos, P.D.: Stochastic linearization in structural dynamics. Appl. Mech. Review 34, 1–8 (1981)
Stratonovich, R.L.: Topics in the theory of random noise, vol. 1. Gordon and Breach, New York (1963)
Roberts, J.B., Spanos, P.D.: Stochastic averaging: an approximate method of solving random vibration problems. Int. J. Non-Linear Mech. 21, 111–134 (1986)
Er, G.-K.: An improved non-Gaussian closure method for randomly excited nonlinear stochastic systems. Nonlinear Dynamics 17(3), 285–297 (1998)
Er, G.-K., Iu, V.P.: A consistent and effective method for nonlinear random oscillations of MDOF systems. In: Proceedings of the IUTAM Symposium on Recent Developments in Nonlinear Oscillations of Mechanical Systems, March 2-5, pp. 85–94. Kluwer Academic Publishers, Hanoi (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Netherlands
About this paper
Cite this paper
Er, G.K., Iu, V.P. (2011). A New Method for the Probabilistic Solutions of Large-Scale Nonlinear Stochastic Dynamic Systems. In: Zhu, W.Q., Lin, Y.K., Cai, G.Q. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics and Control. IUTAM Bookseries, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0732-0_3
Download citation
DOI: https://doi.org/10.1007/978-94-007-0732-0_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0731-3
Online ISBN: 978-94-007-0732-0
eBook Packages: EngineeringEngineering (R0)