Stochastic response spectrum determination of nonlinear systems endowed with fractional derivative elements

This paper considers the response to white noise excitation of nonlinear dynamic systems comprising fractional derivative elements. Specifically, it focuses on estimating the stationary power spectral densities of the response displacement and velocity for systems characterized by polynomial nonlinearities, and nonlinearities with memory effects. Various approximations for the spectral response are derived. The first approximation is based on the concept of the conditional spectrum. In doing this, a weighted averaging is performed over a set of surrogate spectral densities. Each surrogate density corresponds to the stationary random response of a linearized system, associated with the dynamics at specific response amplitude levels. These amplitude levels, assumed to vary slowly over time, are treated as constant within individual oscillation cycles. To refine the estimate, an enhanced formulation of the conditional spectrum is pursued. This improved approach introduces a corrective ter...