On the Improved Efficiency of Higher-Order Dynamics for Computing Resonant Frequencies

Abstract It was recently discovered that higher-order dynamics are intrinsically variational, in the sense that higher-derivative versions of the classical equations of motion can always be derived from a minimum-action principle similar to Hamilton’s principle, even when the physical system is non-conservative. This discovery has already led to several applications, including a new and more efficient algorithm for computing a non-proportionally damped system’s resonant frequencies, based on the fourth-order system dynamics. The purpose of this paper is to investigate the source of this improved efficiency in greater detail. We find that the improved efficiency of the new resonant frequency algorithm is due almost entirely to savings in computing the eigenvalues of the system’s stiffness matrix Ω˜. This result is surprising in light of the ostensible complexity of this matrix. Nevertheless, the savings are shown to be statistically significant, with attained significance levels below machine precision. Although a rigorous mathematical explanation remains elusive, empirical results presented here lead us to conjecture that the reason may have to do with the unique block structure of the stiffness matrix, which it inherits from the mathematically Hamiltonian structure of the fourth-order formulation. The present authors believe there may be additional applications of higher-order dynamics waiting to be discovered, and we conclude the paper with a few potential ideas to explore.