A Study of Complex Modes in Nonuniformly Damped Systems with a Wave Based Framework

Abstract

The dissertation discloses a comprehensive study on analysis and interpretation of complex modes that emerge in general nonproportionally damped systems. The key motivation for this study stems from a curious observation of complex modal parameters that are obtained numerically and experimentally. The focus is placed on explaining the underlying reasons for the appearance of complex modes and their differences with respect to their counterpart, real normal modes, in proportionally damped systems.

In the classical studies of viscous and non-viscously damped systems, the resulting modal parameters are obtained by solving the generalized eigenvalue problem. For the nonproportional damping cases, direct and iterative methods are employed to derive the complex eigenvectors. The classical modal analysis is extended to incorporate unique modal characteristics with respect to damping location and magnitude.

Using a wave analogy, a novel method is demonstrated 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 nonproportionality of the damping, these patterns undergo changes at a damping intersection. The governing equation for this phenomenon is expressed by Snell's law. In a similar manner to the refractive index for the medium in which light waves travel, a damping field index is conceived for individual damping regions, such that they are scaled against the damping field index of the undamped region, which is assumed to be unity. However, unlike the refractive index, it is shown 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 method is validated by an experimental procedure on a beam with localized constrained layer damping.

The analytical framework introduced for the nonproportionally damped structure is extended to an acoustic–structure interaction problem. The coupling between a flexible panel and an acoustic enclosure is examined in modal domain. Coupled responses for panel and cavity are formulated using complex modal analysis. Uncoupled modal patterns are used to assess the coupling effectiveness. Comparison with a proportional damping case reveals characteristics that associate complex modes with damping optimality. Such an interrelation is investigated through topology optimization of a damping layer to minimize the acoustic pressure in the cavity.

The findings of this numerical study show a spatial correlation between the imaginary part of the coupling coefficient and the optimum damping layout. Further, it is demonstrated that ratio of damping field indices mapped on the damped panel correlates with the optimum damping layout. The wave-based framework predicts the optimal conditions for a damped coupled system. Thus, the work conducted here has practical implications for vibroacoustic applications and bridges the gap between theoretical and physical studies for nonproportionally damped systems.