UNIFIED 2ND-ORDER STOCHASTIC AVERAGING APPROACH

First-order stochastic averaging has proven very useful in predicting the response statistics and stability of dynamic systems with nonlinea r damping farces. However, the influence of system stiffness or inerti a nonlinearities is lost during the averaging process. These nonlinear ities can be recaptured only if one extends the stochastic averaging t o second-order analysis. This paper presents a systematic and unified approach of second-order stochastic averaging based on the Stratonovic h-Khasminskii limit theorem. Response statistics, stochastic stability , phase transition (known as noise-induced transition), and stabilizat ion by multiplicative noise are examined in one treatment. A MACSYMA s ymbolic manipulation subroutine has been developed to perform the aver aging processes for any type of nonlinearity. The method is implemente d 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 s pecial nonlinear oscillator, whose exact stationary solution is known, the second-order averaging results are identical to the exact solutio n up to first-order approximation.