Stochastic Averaging: Some Methods and Applications

Abstract

When studying the dynamics of a structure submitted to some actions of stochastic type, if the excitation is not of too high level, the dynamical geometric properties of the structure will be determinant in the response analysis.

In this contribution, we only deal with dimension one. Amplitude-phase variables, which are inspired by action-angle variables for Hamiltonian systems, are introduced. The main interest is that they are computable in an explicit form, and are well adapted to give results on the maximum of the mechanical variable under consideration, with inequalities which are conservative for reliability analysis. A slow and a fast process then appear. The idea is the same as Lagrange’s variation of constants approach: the slow process, when considered in the unperturbed system, is a first integral of motion. In the stochastic context, the martingale formulation is used, and diffusion approximation results obtained concerning the slow process (which usually is not a diffusion process). These results extend Khasminskii’s results. This is the stochastic averaging method.

The same method was used by [20] to obtain a diffusion approximation of the Hamiltonian itself.

Using this limit averaged diffusion process as an approximation of the slow process, and some results concerning the asymptotic behavior (with respect to level) of level crossings by a one dimensional diffusion process, approximations of the probability distribution of the maximum of the absolute value of the displacement of a strongly nonlinear oscillator on a given time interval are obtained. These formulae, compared with other formulae in the literature, proved to be much better.