Rapid Computation of Resonant Frequencies for Nonproportionally Damped Systems Using Dual Oscillators

Many oscillatory systems of engineering and scientific interest (e.g., mechanical metastructures) exhibit nonproportional damping, wherein the mass-normalized damping and stiffness matrices do not commute. A new modal analysis technique for nonproportionally damped systems, referred to as the "dual-oscillator approach to complex-stiffness damping," was recently proposed as an alternative to the current standard method originally developed by Foss and Traill-Nash. This article presents a critical comparison of the two approaches, with particular emphasis on the time required to compute the resonant frequencies of nonproportionally damped linear systems. It is shown that, for degrees-of-freedom greater than or equal to nine, the dual-oscillator approach is significantly faster (on average) than the conventional approach, and that the relative computation speed actually improves with the system's degree-of-freedom. With 145 degrees-of-freedom, for example, the dual-oscillator approach is about 25% faster than the traditional approach. The difference between the two approaches is statistically significant, with attained significance levels less than machine precision. This suggests that the dual-oscillator approach is the faster of the two algorithms for computing resonant frequencies of nonproportionally damped discrete linear systems with large degrees-of-freedom, at least within the limits of the present study. The approach is illustrated by application to a model system representative of a mechanical metastructure.