Modal analysis of continuous systems with damping distributions defined according to fractional derivative models

In this paper vibrating continuous linear systems with generalized damping distributions defined according to fractional derivative models are studied in terms of modal analysis, defining and discussing the orthogonality properties of their eigenfunctions. Closed form expressions are presented for the response functions both in the time and in the frequency domain. The attention is focused on the impulse response functions and the receptances, the former enabling the calculation of both transients and responses due to arbitrary distributions of external forces. In particular, the free response problem is analyzed considering different ways of expressing the initial conditions. As a numerical example, the developed theoretical tools are applied to the study of a viscoelastic cantilever Euler-Bernoulli beam.