A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Abstract

We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics, based on copula functions that has (i) single time statistical structure consistent with the analytical solutions of the Fokker–Planck equation, and (ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to the unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations.