Non-stationary response power spectrum determination of linear/non-linear systems endowed with fractional derivative elements via harmonic wavelet

Abstract

This paper presents a wavelet-based method for determining response evolutionary power spectrum density (EPSD) of linear/non-linear systems endowed with fractional derivative elements. Specifically, first, the generalized harmonic wavelets (GHWs) based Galerkin approximation of the stochastic processes are utilized to transform the fractional-order stochastic linear/non-linear differential equations into a set of linear/non-linear algebraic equations with unknown response wavelet coefficients. Next, the linear algebraic equations are solved in a closed-form, while the non-linear ones are treated by the gradient-based standard numerical methods. Further, an analytical relationship between the EPSD of the excitation and of the response for a linear system is derived by considering the wavelet representation of stochastic processes. For a non-linear system, the response EPSD is estimated by repeated solving of sample algebraic equations. Pertinent numerical examples demonstrate the applicability and accuracy of the proposed method.