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Research Papers

Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

[+] Author and Article Information
Yong Wang

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: yongpi.wang@gmail.com

Xiaoling Jin

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: jinling113@gmail.com

Zhilong Huang

Department of Engineering Mechanics,
Zhejiang University,
#38 Zheda Road,
Hangzhou, Zhejiang 310027, China
e-mail: zlhuang@zju.edu.cn

1Corresponding author.

Manuscript received August 3, 2013; final manuscript received November 5, 2013; accepted manuscript posted November 11, 2013; published online December 10, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(5), 051003 (Dec 10, 2013) (7 pages) Paper No: JAM-13-1325; doi: 10.1115/1.4025954 History: Received August 03, 2013; Revised November 05, 2013; Accepted November 11, 2013

Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

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Figures

Grahic Jump Location
Fig. 1

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on mass disturbance intensity 2Dm. (2De=0.2, b=0.2, c=0.01). —, analytical results; •, ♦, results from MCS.

Grahic Jump Location
Fig. 2

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0on external excitation intensity 2De. (2Dm=0.1, b=0.2, c=0.01). —, analytical results; •, ♦, results from MCS.

Grahic Jump Location
Fig. 3

Stationary probability densities of generalized displacement Q1 and generalized momentum P1 for three cases: proposed analytical results, MCS for original system in Eq. (28), and MCS for approximate system in Eq. (29). (2Dm=0.1, 2De=0.2, b=0.2, c=0.01).

Grahic Jump Location
Fig. 4

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on damping coefficient c. (2Dm=0.1, 2De=0.2, b=0.2).

Grahic Jump Location
Fig. 5

The dependence of mean-square generalized displacement E[Q2] and relative change of mean-square generalized displacement induced by mass disturbance (E[Q2]-E[Q2]|2Dm=0)/E[Q2]|2Dm=0 on nonlinear stiffness coefficient b. (2Dm=0.1, 2De=0.2, c=0.01).

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