Probabilistic response of nonsmooth nonlinear systems under Gaussian white noise excitations

This paper proposes an approach to obtain the steady state probability density function (PDF) of nonsmooth nonlinear dynamical systems driven by the Gaussian white noise. The steady state PDF solution is assumed to contain two probability flows obtained by the detailed balance method. The weighted mean squared residuals of the error of the FPK equation are minimized with respect to the unknown coefficients in the assumed solution. The integrability of the weighted mean squared residuals is found to provide a means to deal with the singularities in the steady state PDF that arise from the nonsmooth dynamics of the nonlinear stochastic system. The accuracy of the proposed approach is estimated by the Monte Carlo simulations via two examples.