Homogenised finite element for transient dynamic analysis of unconstrained layer damping beams involving fractional derivative models

This paper presents a homogenised finite element formulation for the transient dynamic analysis of asymmetric and symmetric unconstrained layer damping beams in which the viscoelastic material is characterised by a five-parameter fractional derivative model. This formulation is based on the weighted residual method (Galerkin’s approach) providing a fractional matrix equation of motion. The application of Grünwald-Letnikov’s definition of the fractional derivatives allows to solve numerically the fractional equation by means of two different implicit formulations. Numerical examples for a cantilever beam with viscoelastic treatment are presented comparing the response provided by the proposed homogenised formulation with that of Padovan, based on the principle of virtual work. Different damping levels and load cases are analysed, as well as the influence of the truncation and time-step. From the numerical applications it can be concluded that the presented formulation allows to reduce significantly the degrees of freedom and consequently the computational time and storage needs for the transient dynamic analysis of structural systems in which damping treatments have been applied by means of viscoelastic materials characterised by fractional derivative models.