Acta Mechanica Sinica

, Volume 29, Issue 4, pp 602–611 | Cite as

Transient stochastic response of quasi integerable Hamiltonian systems

  • Zhong-Hua LiuEmail author
  • Jian-Hua Geng
  • Wei-Qiu Zhu
Research Paper


The approximate transient response of quasi integrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged Itô equations for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independentmotion integrals of the system are derived by applying the stochastic averaging method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of independent motion integrals. Three examples are given to illustrate the application of the proposed procedure. It is shown that the results for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original systems.


Transient response Stochastic averaging method Galerkin method Quasi integrable Hamiltonian system 


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Civil EngineeringXiamen UniversityXiamenChina
  2. 2.Department of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouChina

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