Eigensolutions of non-proportionally damped systems based on continuous damping sensitivity

The viscous damping model has been widely used to represent dissipative forces in structures under mechanical vibrations. In multiple degree of freedom systems, such behavior is mathematically modeled by a damping matrix, which in general presents non-proportionality, that is, it does not become diagonal in the modal space of the undamped problem. Eigensolutions of non-proportional systems are usually estimated assuming that the modal damping matrix is diagonally dominant (neglecting the off-diagonal terms) or, in the general case, using the state-space approach. In this paper, a new closed-form expression for the complex eigenvalues of non-proportionally damped system is proposed. The approach is derived assuming small damping and involves not only the diagonal terms of the modal damping matrix, but also the off-diagonal terms, which appear under higher order. The validity of the proposed approach is illustrated through a numerical example.