On the formation of complex modes in non-proportionally damped systems

Abstract

In the classical studies of non-proportionally damped systems, the resulting complex modal parameters are obtained by solving the generalized eigenvalue problem. In the present study, we propose a unique method to obtain complex modes for discrete and continuous systems. Based on a wave analogy, the difference between a complex mode and a real normal mode is represented by the summation of patterns that propagate from the boundaries. Owing to the spatial non-proportionality of the damping, these patterns undergo changes at a damping intersection. The governing equation for this phenomenon is expressed by Snell's law. We show that, in a similar manner to the refractive index for the medium in which light waves travel, a damping field index can be conceived for individual damping regions, such that they may be scaled against the damping field index of the undamped region, which is assumed to be unity. However, unlike the refractive index, we show that the damping field index is dependent on the spatial distribution of damping. The procedure for obtaining the complex modes is illustrated based on a plate structure with simply supported boundary conditions. The practical applications of the proposed approach and its limitations are discussed based on numerical examples.