Transient vibrations of viscoelastic beam systems under arbitrary loading conditions and with fractional derivative damping models

In this paper, an efficient numerical approach called Numerical Assembly Technique is extended to transient vibrations of viscoelastic Euler-Bernoulli beam systems. The viscoelastic behaviour of the material is captured by the fractional Zener model, which uses fractional time derivatives to allow for an efficient representation of a large class of materials. The fractional derivatives pose certain difficulties in classical time-stepping schemes, which are used in e.g. the Finite Element Method. In the present paper, a numerical forward and inverse Laplace transform is applied to solve the transient beam vibration problem, since the fractional derivatives have a simple representation in the transformed domain and initial conditions are easy to include. Several numerical examples are presented, which show the efficiency and accuracy of the proposed
method compared to the Finite Element Method. Furthermore, the effects of the fractional material damping model on the transient vibrations are investigated.