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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.

This paper investigates the geometric structure of higher-derivative formulations of classical mechanics. It is shown that every even-order formulation of classical mechanics higher than the second order is intrinsically variational, in the sense that the equations of motion are always derivable from a minimum action principle, even when the system is non-Hamiltonian. Particular emphasis is placed on the fourth-order formulation, as that is shown to be the lowest order for which the governing equations are intrinsically variational. The Noether symmetries and associated conservation laws of the fourth-order formulation, including its Hamiltonian, are derived along with the natural auxiliary conditions. The intrinsic variational structure of higher-derivative formulations makes it possible to treat non-Hamiltonian systems as if they were Hamiltonian, with immediate classical applications. A case study of the classical damped harmonic oscillator is presented for illustration, and an action is formulated for a higher-order Navier-Stokes equation.

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