Unified Second-Order Stochastic Averaging Approach

Abstract

First-order stochastic averaging has proven very useful in predicting the response statistics and stability of dynamic systems with nonlinear damping forces. However, the influence of system stiffness or inertia nonlinearities is lost during the averaging process. These nonlinearities can be recaptured only if one extends the stochastic averaging to second-order analysis. This paper presents a systematic and unified approach of second-order stochastic averaging based on the Stratonovich-Khasminskii limit theorem. Response statistics, stochastic stability, phase transition (known as noise-induced transition), and stabilization by multiplicative noise are examined in one treatment. A MACSYMA symbolic manipulation subroutine has been developed to perform the averaging processes for any type of nonlinearity. The method is implemented to analyze the response statistics of a second-order oscillator with three different types of nonlinearities, excited by both additive and multiplicative random processes. The second averaging results are in good agreement with those estimated by Monte Carlo simulation. For a special nonlinear oscillator, whose exact stationary solution is known, the second-order averaging results are identical to the exact solution up to first-order approximation.