Analytical non-stationary response of fractional linear systems endowed with arbitrary rational order fractional derivative elements and subject to evolutionary stochastic excitation

• Published in: Mechanical systems and signal processing Vol. 226; p. 112333
• Main Authors: Xu, Yijian, Kong, Fan, Hong, Xu, Hu, Zhixiang, Ding, Zhaodong
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2025
• Subjects: Eigenvector expansion; Evolutionary stochastic process; Fractional derivatives; Non-stationary response; Non-stationary response; Evolutionary stochastic process; Eigenvector expansion; Fractional derivatives
• ISSN: 0888-3270
• DOI: 10.1016/j.ymssp.2025.112333

This paper presents an analytical solution for the non-stationary response of linear systems endowed with arbitrary rational fractional derivative elements (0<α<2) and subjected to stochastic excitation with separable or non-separable power spectral density. The approach begins with a pseudo-state-space formulation of the fractional equations of motion, leading to a decoupling of the state-space fractional matrix equation via an eigenvector expansion method. The decoupled fractional state-space equations are then solved using the Laplace transform, yielding closed-form expressions for system displacement and its fractional derivative processes under free vibration and impulse excitation. Furthermore, an analytical solution for the stochastic response of fractional linear systems is developed within a frequency domain framework, by introducing the evolutionary frequency response function of the α-order. Numerical examples, including systems with one or two fractional elements subjected to various stochastic excitations, validate the accuracy and applicability of the proposed method through comparisons with Monte Carlo simulations. •Analytical fractional derivative responses of fractional linear systems are derived.•Evolutionary fractional FRF is introduced via the fractional IRF.•Non-stationary second moments of FDR of FLS are developed analytically.