Modal evaluation and generalized analysis of the steady-state dynamics of harmonically excited multistable structures

Predictions of multistable structural dynamics are paramount to the development and deployment of air vehicles operating under extreme loading conditions. Although time-stepping numerical techniques may capture the multi-physics interactions that occur in these environments, the generalized insight on parameters that predominantly govern the system behaviors may remain clouded while a large computational expense may be incurred to obtain response predictions. Alternatively, analytical methods may be employed to streamline the prediction process, yet current theoretical approaches do not facilitate such opportunity for multistable structures. Although a recently developed analytical formulation has enabled the prediction of near- and far-from-equilibrium responses for a simplified multistable structure, the preliminary formulation does not illuminate the underlying aspects of modal response and intricate nonlinear coupling manifest in myriad multistable systems. This research rectifies these limitations by a broad expansion of the analytical framework that empowers a new modal perspective of multistable structural dynamics and enables the study of such dynamic systems governed by reduced order models. This new modal analysis indicates that the characteristic frequency response of a single degree-of-freedom Duffing oscillator is preserved in the fundamental equivalent nonlinear mode of a multistable structure. The new analytical formulation is also shown to accurately predict the near- and far-from-equilibrium dynamics of equation systems containing global nonlinear coupling consistent with reduced order models. The advancements achieved in this work contribute to the suite of techniques available to researchers to characterize the near-to- and far-from equilibrium behaviors of nonlinear dynamic systems consisting of many degrees-of-freedom.