Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems
- 12 November 2009
- journal article
- Published by ASME International in Journal of Computational and Nonlinear Dynamics
- Vol. 5 (1), 011004
- https://doi.org/10.1115/1.4000312
Abstract
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.Keywords
This publication has 21 references indexed in Scilit:
- Nonlinear vibration analysis of bladed disks with dry friction dampersJournal of Sound and Vibration, 2006
- Statistics of responses of a mistuned and frictionally damped bladed disk assembly subjected to white noise and narrow band excitationsProbabilistic Engineering Mechanics, 2006
- RANDOM VIBRATION OF A ROTATING BLADE WITH EXTERNAL AND INTERNAL DAMPING BY THE FINITE ELEMENT METHODJournal of Sound and Vibration, 2002
- Efficient path integration methods for nonlinear dynamic systemsProbabilistic Engineering Mechanics, 2000
- On the cumulant-neglect closure method in stochastic dynamicsInternational Journal of Non-Linear Mechanics, 1996
- Response Statistics of Nonlinear Dynamic Systems by Path IntegrationPublished by Springer Science and Business Media LLC ,1992
- Introduction to Random VibrationJournal of Vibration and Acoustics, 1986
- Numerical evaluation of path-integral solutions to Fokker-Planck equationsPhysical Review A, 1983
- A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand SidesTheory of Probability and Its Applications, 1966
- Perturbation Techniques for Random Vibration of Nonlinear SystemsThe Journal of the Acoustical Society of America, 1963