Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: theory and computing

Abstract

We study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preserved.

A new stochastic method based on the Taylor expansion to compute response probability densities of uncertain systems

“…An Itô–Taylor expansion‐based numerical integration scheme has been adopted in Reference 8 for a wide class of stochastic nonlinear systems. Recently, the perturbation approach has been applied in terms of the probability density function (PDF) of the response, 9 successively improved by the same research group 10 . Some years ago, it was proposed the so‐called “improved” perturbation approach which considers mean and correlation information on uncertain parameters in computing the mathematical expectation of the response 11 .…”