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Research Papers

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

[+] Author and Article Information
Wantao Jia

Department of Engineering Mechanics,
Northwestern Polytechnical University,
Xi'an 710072, China
e-mail: jiawantao@126.com

Weiqiu Zhu

Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027, China;
Department of Engineering Mechanics,
Northwestern Polytechnical University,
Xi'an 710072, China

Yong Xu

e-mail: robinxuhy@gmail.com

Weiyan Liu

e-mail: liuweiyan@mail.com.edu.cn
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China

1Corresponding author.

Manuscript received June 8, 2013; final manuscript received July 8, 2013; accepted manuscript posted July 29, 2013; published online September 23, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(4), 041009 (Sep 23, 2013) (13 pages) Paper No: JAM-13-1232; doi: 10.1115/1.4025141 History: Received June 08, 2013; Revised July 08, 2013; Accepted July 29, 2013

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

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Figures

Grahic Jump Location
Fig. 1

Stationary marginal probability density p(I1,I2) of the system in Eq. (62). The parameter values in the calculation are α11 = α21 = -0.1, α12 = α22 = 0.04, ω1 = ω2 = 1.0, β1 = β2 = 0.01, 2D1 = 2D2 = 0.015, λ1 = λ2 = 2.5, E[Y12] = E[Y22] = 0.002. (a) From simulation, (b) From proposed averaging method.

Grahic Jump Location
Fig. 2

Stationary marginal probability densities p(I1) and p(I2) of the system in Eq. (62). The parameters are the same as those in Fig. 1 “—”: results from the proposed stochastic averaging method; “▼”: results from digital simulation. (a) Stationary probability densitiy of I1,(b) Stationary probability density of I2.

Grahic Jump Location
Fig. 3

Stationary marginal probability density p(q1,p1) of the system in Eq. (62). The parameters are the same as those in Fig. 1. (a) From simulation, (b) from the proposed averaging method.

Grahic Jump Location
Fig. 4

Marginal stationary probability densities of generalized displacement q1 and generalized momenta p1 of the system in Eq. (62). The parameters are the same as those in Fig. 1 “—”: results from the proposed stochastic averaging method; “▼”: results from digital simulation. (a) Stationary probability density of p1, (b) Stationary probability density of q1.

Grahic Jump Location
Fig. 5

Marginal stationary probability density p(ψ) of the system in Eq. (62) in primary resonant case. The parameters are the same as in Fig. 1 “—”: results from the present stochastic averaging method; “▼”: results from digital simulation.

Grahic Jump Location
Fig. 6

Marginal stationary probability density of generalized displacement q1 for different values of α11. The other parameters are α12 = 0.02, α21 = -0.1, α22 = 0.04, ω1 = ω2 = 1.0, β1 = β2 = 0.01, 2D11 = 2D22 = 0.015, λ1 = λ2 = 2.5, E[Y12] = E[Y22] = 0.002. “—”: results from the proposed stochastic averaging method; “▼♦•”: results from digital simulation.

Grahic Jump Location
Fig. 7

Marginal stationary probability densities p(I1) and p(I2) of stationary response of the system in Eq. (62) obtained from the proposed stochastic averaging method. The parameters are: “—”: 2D11 = 2D22 = 0.011, λ1 = λ2 = 3.0, λ1E[Y12] = λ2E[Y22] = 0.009; “—-”: 2D11 = 2D22 = 0.011, λ1 = λ2 = 60.0, λ1E[Y12] = λ2E[Y22] = 0.009. The rest parameters are: α11 = α21 = -0.1, α12 = α22 = 0.04, ω1 = ω2 = 1.0, β1 = β2 = 0.01. (a) Stationary probability density of I1, (b) stationary probability density of I2.

Grahic Jump Location
Fig. 8

Marginal probability densities p(I1) and p(I2) of stationary response of the system in Eq. (62) obtained from the proposed stochastic averaging method. The parameters are the same as those in Fig. 7 except: “•”: 2D11 = 2D22 = 0.011, λ1 = λ2 = 60.0, λ1E[Y12] = λ2E[Y22] = 0.009; “—”: 2D11 = 2D22 = 0.02, λ1E[Y12] = λ2E[Y22] = 0. (a) Stationary probability density of I1, (b) Stationary probability density of I2.

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