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

TL;DR

This paper introduces a novel moment-equation-closure method using copula-based non-Gaussian PDFs to efficiently approximate the steady-state statistics of nonlinear vibrational systems under correlated noise, capturing complex bimodal behaviors.

Contribution

The proposed method combines moment equations with a copula-based PDF representation, enabling semi-analytical, low-dimensional approximation of non-Gaussian two-time statistics in nonlinear systems.

Findings

Accurately predicts steady-state statistics of bistable energy harvesters.

Shows good agreement with Monte Carlo simulations.

Provides a computationally inexpensive alternative to direct simulation.

Abstract

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and 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 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 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.