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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April 2014
Research-Article
Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
Wantao Jia,
Wantao Jia
Department of Engineering Mechanics,
e-mail: jiawantao@126.com
Northwestern Polytechnical University
,Xi'an 710072
, China
e-mail: jiawantao@126.com
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Weiqiu Zhu,
Weiqiu Zhu
1
Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027,
China
;Department of Engineering Mechanics,
Xi'an 710072,
Northwestern Polytechnical University
,Xi'an 710072,
China
1Corresponding author.
Search for other works by this author on:
Weiyan Liu
Weiyan Liu
e-mail: liuweiyan@mail.com.edu.cn
Department of Applied Mathematics,
Department of Applied Mathematics,
Northwestern Polytechnical University
,Xi'an 710072
, China
Search for other works by this author on:
Wantao Jia
Department of Engineering Mechanics,
e-mail: jiawantao@126.com
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,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027,
China
;Department of Engineering Mechanics,
Xi'an 710072,
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,
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. Apr 2014, 81(4): 041009 (13 pages)
Published Online: September 23, 2013
Article history
Received:
June 8, 2013
Revision Received:
July 8, 2013
Accepted:
July 29, 2013
Citation
Jia, W., Zhu, W., Xu, Y., and Liu, W. (September 23, 2013). "Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations." ASME. J. Appl. Mech. April 2014; 81(4): 041009. https://doi.org/10.1115/1.4025141
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