Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.
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January 2010
Research Papers
Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems
Pankaj Kumar,
Pankaj Kumar
Department of Gas Turbine Design,
e-mail: pankajiit1@yahoo.co.in
Bharat Heavy Electricals Limited
, Hyderabad 502032, India
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S. Narayanan
S. Narayanan
Department of Mechanical Engineering,
e-mail: narayans@iitm.ac.in
Indian Institute of Technology, Madras
, Chennai 600036, India
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Pankaj Kumar
Department of Gas Turbine Design,
Bharat Heavy Electricals Limited
, Hyderabad 502032, Indiae-mail: pankajiit1@yahoo.co.in
S. Narayanan
Department of Mechanical Engineering,
Indian Institute of Technology, Madras
, Chennai 600036, Indiae-mail: narayans@iitm.ac.in
J. Comput. Nonlinear Dynam. Jan 2010, 5(1): 011004 (12 pages)
Published Online: November 12, 2009
Article history
Received:
April 30, 2008
Revised:
May 9, 2009
Online:
November 12, 2009
Published:
November 12, 2009
Citation
Kumar, P., and Narayanan, S. (November 12, 2009). "Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems." ASME. J. Comput. Nonlinear Dynam. January 2010; 5(1): 011004. https://doi.org/10.1115/1.4000312
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