Finite element analysis of vibrating non-homogeneous beams 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. However the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations too cumbersome for finite element applications. This paper presents a condensation technique based on the computation of two reduced-size eigenproblems. The rheological model adopted is the Fractional Zener or Fractional Standard Linear Solid model, but the same methodology applies to problems involving different fractional derivative linear models.