Closed-form response of a linear fractional visco-elastic oscillator under arbitrary stationary input

Abstract This paper studies the structural response of a single degree-of- freedom structure including a fractional derivative constitutive term. Unlike usual existing models for this kind of structure, the excitation is not necessarily a Markovian process but it is slowly varying in time, so that a timescale separation is used. Following the general formulation of the Multiple Timescale Spectral Analysis [#Denoel2015], the solution is developed as a sum of background and resonant components. Because of the specific shape of the frequency response function of a system equipped with a fractional viscoelastic device, the background component is not simply obtained as the variance of the loading divided by the stiffness of the system. On the contrary the resonant component is expressed as a simple extension of the existing formulation for a viscous system, at least at leading order. As a validation case, the proposed solution is shown to recover similar results (in the white noise excitation case) as former studies based on a stochastic averaging approach [#SPA97, #DIP14]. A better accuracy is however obtained in case of very small fractional exponent. Another example related to the buffeting analysis of a linear fractional viscoelastic system demonstrates the accuracy of the proposed formulation for colored excitation.