Analytical Nonstationary Response of Linear Stochastic MDOF Systems Endowed with Half-Order Fractional Derivative Elements

Abstract

This paper presents a novel method for obtaining analytical solutions for the nonstationary response of multidegree-of-freedom (MDOF) systems endowed with half fractional derivative elements and subjected to external stochastic excitation. Specifically, first, the proposed technique employs eigenvector expansion of the state-space formulation and Laplace transforms to derive an analytical solution for the impulse/frequency response function (IRF/FRF) of the fractional-order dynamic system. Moreover, by utilizing the Laplace transform method, exact analytical solutions for the second-moment response are obtained in the frequency domain. A comprehensive set of six numerical cases is presented to demonstrate the effectiveness of this novel methodology. These cases include two degenerated scenarios, namely a single-degree-of-freedom (SDOF) system and a two-degree-of-freedom (2-DOF) linear system, both endowed with half-order fractional derivative elements and subjected to stochastic stationary/nonstationary excitations, including white noise, modulated white noise, and modulated colored noise with modified Kanai-Tajimi spectrum. The analytical nonstationary responses derived by the proposed method exhibit exceptional agreement with pertinent Monte Carlo simulations, validating the accuracy and reliability of the proposed approach.