Non-stationary response determination of linear systems/structures with fractional derivative elements

This paper focuses on linear systems/structures endowed with fractional derivative subject to nonstationary excitations with non-separable evolutionary power spectral density. Two efficient approximate analytical solutions of the time-varying response statistics are proposed based on the stochastic averaging technique. Specifically, a convenient approximation of fractional derivatives is employed in the stochastic averaging treatment. This leads to an equivalent natural frequency and damping ratio in the obtained governing equation of response amplitude. The equivalent parameters, which also correspond to an equivalent integer-order linear system, can be determined by solving a quite simple one-dimensional algebraic equation. Further, an approximate analytical determination of time-variant response statistics can be made via two approaches. The first follows the standard stochastic averaging treatment, yielding a concise expression and a corresponding eigenfunction expansion. The second solves directly the equivalent integer-order system subject to the same excitation as the original system, yielding Sconsistently accuracy in general. The accuracy of the two approximate analytical solutions is numerically assessed under various system parameters and excitations, by comparing with pertinent Monte Carlo simulation results.