An analytical approach for response power spectral density determination of linear systems using stochastic harmonic function

Abstract

Fractional derivatives have emerged as a powerful tool for characterizing memory-dependent or non-local mechanical behaviors of materials and structures. This paper presents an alternative yet novel method for determining the analytical solution for the non-stationary response of linear dynamic systems with/without fractional derivative elements subjected to stochastic excitation. The technique simplifies the derivation by representing the stochastic excitation as a single harmonic component with random frequency and phase angle, effectively transforming the stochastic dynamic problem into a deterministic one of harmonic response analysis. This significantly reduces the complexity of calculating system response statistics by simply taking mathematical expectations on stochastic harmonic responses. The proposed approach not only offers a new analytical framework for re-deriving conventional Caughey’s solution but also extends readily to linear stochastic systems with fractional derivative elements. Validation through comparison with conventional analytical methods for fractional-order systems developed recently demonstrates the accuracy of the results, providing new insights for further stochastic analysis of fractional-order dynamic systems.