A new analytical technique for vibration analysis of non-proportionally damped beams

Vibrating linear mechanical systems, in particular continuous systems, are often modelled considering proportional damping distributions only, although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy. In this paper an analytical method is given to take into account the effects of a more general viscous damping model, referred to as non-proportional damping, on a class of vibrating continuous systems. A state-form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form, i.e., complex modes corresponding to non-synchronous motions. Numerical examples are included in order to show the efficiency of the proposed method; non-proportional damping distributions of different type, such as internal and external lumped or distributed viscous damping, are tested on non-homogeneous Euler-Bernoulli beams in bending vibration with different boundary conditions. Finally, a discussion on root locus diagrams behaviour and on modal damping ratio significance for non-proportionally damped systems is presented.