Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

Abstract

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

On the influence of second order non-linearity on random vibrations

“…On the other hand, using the classical SAM [5] one obtain the probability density function of amplitude as follows:…”

“…[1][2][3][4][5]. However, the effect of some non-linear terms cannot be investigated by using the classical first order SAM.…”

The influence of second order narrow-band colored noises on non-linear random vibrations

“…In order to over come this insufficiency, different averaging procedures for obtaining approximate solutions have been developed (see e.g. Mitropolskii et al, 1992; Red-Horse & Spanos, 1992; Sri Namachchivaya & Lin, 1988;Zhu & Lin, 1994;Zhu et al, 1997). Recently, a higher order averaging procedure using Fokker-Planck (FP) equation was developed in (Anh, 1993(Anh, , 1995 and then applied to Van der Pol oscillator under white noise excitation (Anh & Tinh, 1995 ).…”

Stochastic Stability and Bifurcation of A High Speed Rotor-Bearing System with Random Excitation

“…According to the quasi-nonintegrable Hamiltonian system theory, the Hamiltonian function H t converges weakly in probability to a one-dimensional Ito diffusion process [7]. …”