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Research Papers

Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems

[+] Author and Article Information
Pankaj Kumar

Department of Gas Turbine Design, Bharat Heavy Electricals Limited, Hyderabad 502032, Indiapankajiit1@yahoo.co.in

S. Narayanan

Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600036, Indianarayans@iitm.ac.in

J. Comput. Nonlinear Dynam 5(1), 011004 (Nov 12, 2009) (12 pages) doi:10.1115/1.4000312 History: Received April 30, 2008; Revised May 09, 2009; Published November 12, 2009; Online November 12, 2009

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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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

(a) Joint PDF of displacement and velocity. Linear and logarithmic plots of the marginal PDF of (b) displacement and (c) velocity for self-excited oscillator (—) exact results; (○○○○○) PI with non-Gaussian transition PDF; (◻◻◻◻◻) PI with Gaussian transition PDF; ( ⋆⋆⋆⋆) FEM

Grahic Jump Location
Figure 2

Linear and logarithmic plot of the marginal PDF of (a) displacement and (b) velocity for Duffing oscillator; key as in Fig. 1

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Figure 3

Linear and logarithmic plots of the marginal PDF of (a) displacement and (b) velocity for Dimentberg (I) oscillator; key as in Fig. 1

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Figure 4

Linear and logarithmic plots of the marginal PDF of (a) displacement and (b) velocity for Dimentberg (II) oscillator; (—) MCS results; (○○○○○) PI with non-Gaussian transition PDF; (◻◻◻◻◻) PI with Gaussian transition PDF; ( ⋆⋆⋆⋆) FEM

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Figure 5

Joint PDF of Duffing oscillator (β=0.05, λ=0.3, F=0.2, ω=1.2): (a) D=0.002 and (b) D=0.01

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Figure 6

Joint PDF of Duffing oscillator (β=0.05, λ=0.3, F=0.2, D=0.004): (a) ω=1.3 and (b) ω=1.1

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Figure 7

Joint PDF of Duffing oscillator (β=0.05, λ=0.3, ω=1.2, D=0.004): (a) F=0.3 and (b) F=0.15

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Figure 8

Phase plot for Duffing oscillator: (a) S0=0.0025 and (b) S0=0.005

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