Rapid computation of resonant frequencies for non-proportionally damped systems using dual oscillators

Many oscillatory systems of engineering and scientific interest (e.g., mechanical metastructures) exhibit non-proportional damping, wherein the mass-normalized damping and stiffness matrices do not commute. A new modal analysis technique for non-proportionally 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 paper presents a critical comparison of the two approaches, with particular emphasis on the time required to compute the resonant frequencies of non-proportionally 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. To the authors' knowledge, this establishes the dual-oscillator approach as the fastest existing algorithm for computing resonant frequencies of non-proportionally damped linear systems with large degrees of freedom. The approach is illustrated by application to a model system representative of a mechanical metastructure.