Finite element analysis of vibrating linear systems with fractional derivative viscoelastic models

Fractional derivative rheological models are known to be very effective in describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. But the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations exceedingly cumbersome. This paper presents a method for reducing the computational effort due to finite element (FE) analysis of vibrating linear systems with a fractional derivative viscoelastic model, namely the Fractional Kelvin–Voigt. The proposed method may be applied also to problems involving other fractional derivative linear models, and it takes under control the computational effort by reducing the main eigenproblem of large dimension to the solution of two standard related eigenproblems of lower size. Numerical examples are provided in order to validate both the accuracy and the efficiency of the proposed methodology.