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Research Papers

Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems Under Poisson White Noise Excitation

[+] Author and Article Information
Y. Zeng

Department of Mechanics, National Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, People’s Republic of Chinazengyan@zju.edu.cn

W. Q. Zhu1

Department of Mechanics, National Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, People’s Republic of Chinawqzhu@yahoo.com

1

Corresponding author.

J. Appl. Mech 78(2), 021002 (Nov 04, 2010) (11 pages) doi:10.1115/1.4002528 History: Received July 26, 2009; Revised July 25, 2010; Posted September 09, 2010; Published November 04, 2010; Online November 04, 2010

A stochastic averaging method for predicting the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable-Hamiltonian systems with lightly linear and (or) nonlinear dampings subject to weakly external and (or) parametric excitations of Poisson white noises) is proposed. A one-dimensional averaged generalized Fokker–Planck–Kolmogorov equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and two-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The probability density of random excitation ξ of two coupled van der Pol oscillators: ——, Poisson white noise with λ=1.0 and E[Y2]=81.0; - - - -, Gaussian white noise with the same intensity

Grahic Jump Location
Figure 2

The stationary probability density of Hamiltonian H of two coupled van der Pol oscillators for α=0.45, 0.6, 0.75, and 0.9: ——, second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ▲▼●◼, results from simulation

Grahic Jump Location
Figure 3

The stationary marginal probability density of displacement Q2 of two coupled van der Pol oscillators for α=0.45, 0.6, 0.75, and 0.9: ——, theoretical result from second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); ▲▼●◼, results from simulation

Grahic Jump Location
Figure 4

The probability density of random excitation ξ of 2DOF VI system: ——, Poisson white noise with λ=0.1 and E[Y2]=40.0; - - - -, Gaussian white noise with the same intensity

Grahic Jump Location
Figure 5

The probability density of random excitation ξ of 2DOF VI system: ——, Poisson white noise with λ=0.2 and E[Y2]=20.0; - - - -, Gaussian white noise with the same intensity

Grahic Jump Location
Figure 6

The probability density of random excitation ξ of 2DOF VI system: ——, Poisson white noise with λ=0.5 and E[Y2]=8.0; - - - -, Gaussian white noise with the same intensity

Grahic Jump Location
Figure 7

The probability density of random excitation ξ of 2DOF VI system: ——, Poisson white noise with λ=1.0 and E[Y2]=4.0; - - - -, Gaussian white noise with the same intensity

Grahic Jump Location
Figure 8

The stationary probability density of Hamiltonian H of 2DOF VI system for ε=0.1 and λ=0.1, 0.5: ——, second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 9

The stationary marginal probability density of displacement Q2 of 2DOF VI system for ε=0.1 and λ=0.1: ——, theoretical result from second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 10

The stationary probability density of Hamiltonian H of 2DOF VI system for ε=0.2 and λ=0.2, 1.0: ——, second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 11

The stationary marginal probability density of displacement Q2 of 2DOF VI system for ε=0.2 and λ=0.2: ——, theoretical result from second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 12

The stationary probability density of Hamiltonian H of 2DOF VI system for ε=0.3 and λ=0.5, 2.0: ——, second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 13

The stationary marginal probability density of displacement Q2 of 2DOF VI system for ε=0.3 and λ=0.5: ——, theoretical result from second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 14

The stationary probability density of Hamiltonian H of 2DOF VI system for ε=0.4 and λ=1.0, 4.0: ——, second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

Grahic Jump Location
Figure 15

The stationary marginal probability density of displacement Q2 of 2DOF VI system for ε=0.4 and λ=1.0: ——, theoretical result from second-order perturbation solution ρ(h)=ρ0(h)+ερ1(h)+ε2ρ2(h); - - - -, approximate Gaussian solution; ●, results from simulation

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