Transient responses of stochastic systems under stationary excitations

Stochastic systems to stationary random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. It has been a challenge to obtain exact transient response properties. A novel approximate technique for determining non-stationary probability density function (PDF) of randomly excited nonlinear oscillators is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. Based on the results from statistical linearization, residual error of the FPK equation associated with proposed approximation solution can be treated by weighted residual method. As a result, nonlinear ordinary differential equations are produced. Numerical method is adopted to solve these equations and approximate PDF solutions are then obtained. In order to verify the efficiency of the proposed procedure, four examples of stochastic vibrating systems with additional excitations or/and parametric excitations are considered. It is shown that the results obtained by the proposed procedure agree well with those from Monte Carlo simulation.