Time-domain dynamic analysis of structures equipped with fractional viscoelastic solid and fluid dampers via improved pseudo-force approach

A numerical method for the time-domain dynamic analysis of structures with viscoelastic energy dissipation dampers, modeled using fractional derivatives, is presented. Two fractional viscoelastic models are considered: the fractional Kelvin-Voigt model and another one referred to here as the fractional simplified Maxwell model, to distinguish it from the widely used fractional Maxwell model, where two fractional derivatives appear. In the context of the convolution integral method in state variables, the proposed approach, called the improved pseudo-force method involves: i) discretization of fractional derivatives using the Grünwald-Letnikov approximation; ii) piecewise linear interpolation of both the excitation function and the pseudo-force; iii) evaluation of the response in the modal subspace through recursive relations, using operators computed only at the beginning of the procedure; iv) evaluation of the nodal response by the modal superposition method. The accuracy and stability of the method are demonstrated through applications to a Single-Degree-of-Freedom (SDoF) oscillator and a five-story building equipped with viscoelastic dampers.