Acta Mechanica Sinica

, Volume 29, Issue 4, pp 602–611

# Transient stochastic response of quasi integerable Hamiltonian systems

Research Paper

## Abstract

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.

## Keywords

Transient response Stochastic averaging method Galerkin method Quasi integrable Hamiltonian system

## References

1. 1.
Caughey, T.K.: Nonlinear theory of random vibration. Advances in Applied Mechanics 11, 209–253 (1971)
2. 2.
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1983)
3. 3.
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. McGraw-Hill, New York (1995)Google Scholar
4. 4.
Sun, J.Q., Hsu, C.S.: The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. Journal of Applied Mechanics 57, 1018–1025 (1990)
5. 5.
Naess, A., Johnsen, J. M.: Response statistics of nonlinear dynamic systems by path integration. In: Bellomo N., Casciati F., eds. Nonlinear Stochastic Mechanics: IUTAM Symposium, 401–414. Springer, Berlin Heidelberg, Berlin (1992)
6. 6.
Yu, J.S., Cai, G.Q., Lin, Y.K.: A new path integration procedure based on Gauss-Legendre scheme. International Journal of Non-linear Mechanics 32, 759–768 (1997)
7. 7.
Spencer, Jr.B.F., Bergman, L.A.: On the Numerical solutions of the Fokker-Planck equations for nonlinear stochastic systems. Nonlinear Dynamics 4, 357–372 (1993)
8. 8.
Pichler, L., Pradlwarter, H.J.: Evolution of probability densities in the phase space for reliability analysis of non-linear structures. Structural Safety 31, 316–324 (2009)
9. 9.
Yue, X.L., Xu, W., Wang, L., et al.: Transient and steadystate responses in a self-sustained oscillator with harmonic and bounded noise excitations. Probabilistic Engineering Mechanics 30, 70–76 (2012)
10. 10.
Atkinson, J.D.: Eigenfunction expansions for randomly excited non-linear systems. Journal of Sound and Vibration 30, 153–172 (1973)
11. 11.
Wen, Y.K.: Approximation method for nonlinear random vibration. Journal of the Engineering Mechanics Division 101, 389–401 (1975)Google Scholar
12. 12.
Spanos, P.D., Sofi, A., Paola, M.Di.: Nonstationary response envelope probability densities of nonlinear oscillators. ASME Journal of Applied Mechanics 74, 315–324 (2007)
13. 13.
Jin, X.L., Huang, Z.L.: Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay. Nonlinear Dynamics 59, 195–206 (2010)
14. 14.
Jin, X.L., Huang, Z.L., Leung, Y.T.: Nonstationary probability densities of system response of strongly nonlinear singledegree-of-freedom system subject to modulated white noise excitation. Applied Mathematics and Mechanics 32, 1389–1398 (2011)
15. 15.
Jin, X.L., Huang, Z.L.: Nonstationary probability densities of nonlinear multi-degree-of-freedom systems under Gaussian white noise excitations. In: Zhu W.Q., Lin Y. K., Cai G.Q., eds. IUTAM Symposium on Nonlinear Stochastic Dynamics and Control 35–44. Springer, Netherlands (2011)
16. 16.
Xu, M., Jin, X.L., Huang Z.L.: First-passage failure of MDOF nonlinear oscillator. Science China Technological Sciences 54, 1999–2006 (2011)
17. 17.
Qi, L.Y., Xu, W., Gu, X.D.: Nonstationary probability densities of a class of nonlinear system excited by external colored noise. Science China Physics, Mechanics & Astronomy 55, 477–482 (2012)
18. 18.
Roberts, J.B., Spanos, P.D.: Stochastic averaging: An approximate method of solving random vibration problems. International Journal of Non-linear Mechanics 21, 111–134 (1986)
19. 19.
Zhu, W.Q., Lin, Y.K.: Stochastic averaging of energy envelope. Journal of Engineering Mechanics 117, 1890–1905 (1991)
20. 20.
Zhu, W.Q., Huang, Z.L., Yang, Y.Q.: Stochastic averaging of quasi-integrable Hamiltonian systems. Journal of Applied Mechanics 64, 975–984 (1997)
21. 21.
Zhu, W.Q., Huang, Z.L.: Lyapunov exponent and stochastic stability of quasi-integrable-Hamiltonian systems. Journal of Applied Mechanics 66, 211–217 (1999)
22. 22.
Zhu, W.Q., Deng. M.L., Huang, Z.L.: First-passage failure of quasi-integrable-Hamiltonian systems. Journal of Applied Mechanics 69, 274–282 (2002)
23. 23.
Gan, C.B.: Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems, Acta Mechanica Sinica 20, 558–566 (2004)
24. 24.
Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equations. International Journal of Engineering Science 3, 213–229 (1965) 