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

• Published in: Mechanical systems and signal processing 2022-01, Vol.162, p.108024, Article 108024
• Main Authors: Kong, Fan, Zhang, Yixin, Zhang, Yuanjin
• Format: Article
• Language: English
• Subjects: Algebra; Evolutionary power spectrum density; Fractional derivative; Galerkin approximation; Harmonic wavelets; Linear algebra; Nonlinear differential equations; Nonlinear systems; Numerical methods; Stationary response; Stochastic models; Stochastic processes; Wavelet analysis
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2021.108024

•Fractional-order connection coefficients of harmonic wavelets have been derived.•Wavelet-Galerkin formulation of linear/non-linear dynamic systems endowed with fractional elements for deterministic response has been developed.•A wavelet based power spectral density determination method for linear/nonlinear stochastic dynamic systems endowed with fractional elements has been proposed. 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.