Multiple timescale spectral analysis of a linear fractional viscoelastic system under colored excitation

This paper specifies the multiple timescale spectral analysis to the structural analysis 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 also assumed to be colored, in a low-frequency range compared to that of the structural system, but not necessarily as an integer autoregressive filter. This problem further extends the domain of applicability of the multiple timescale spectral analysis. 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. 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. This paper is mostly illustrated with the structural analysis of systems equipped with fractional dampers, but it could be re-interpreted in any of the many other fields of engineering where applications are governed by the same equation.