Non-stationary response determination of nonlinear systems subjected to combined deterministic and evolutionary stochastic excitations

A semi-analytical method is proposed for determining the response of a lightly damped single-degree-of-freedom nonlinear system subjected to combined deterministic and non-stationary stochastic excitations. This is attained by combining the stochastic averaging and statistical linearization methodologies. Specifically, first, the system response is decomposed into two components, namely the deterministic and the stochastic parts. This leads to a set of coupled differential sub-equations governing, respectively, the deterministic and the stochastic component of the response. Next, aiming at solving the set of differential sub-equations, an additional expression is derived by applying the statistical linearization methodology, followed by the application of a stochastic averaging step to the stochastic sub-equations. Therefore, an equivalent time-varying linear system is defined for the original nonlinear system. The stochastic averaging method is then applied to the linearized system for reducing its order, and thus, its complexity from a solution perspective. In this regard, an additional equation is derived, which connects the deterministic and stochastic components of the response. The latter and the deterministic differential sub-equations are solved simultaneously for determining the system response. A single-degree-of-freedom nonlinear system exhibiting quadratic and cubic nonlinear stiffness is considered for assessing the reliability of the proposed technique. The obtained results are compared with pertinent Monte-Carlo simulation estimates.