Buckling or vibration Galerkin modes for bistable curved beam dynamics: which yields acceptable convergence?

Abstract

Dynamic excitation enables rapid state switching of bistable beams under small forcing amplitudes, yet a controllable switching strategy relies on accurate prediction of beam dynamics. The Galerkin method offers a reliable and efficient approach, but the choice of trial functions varies across studies, leaving the optimal selection for both efficiency and accuracy unresolved. This paper investigates such modal sets for bistable curved beam: straight-beam buckling modes, straight-beam vibration modes, and curved-beam vibration modes. Substituting these trial functions into the governing equation and retaining different numbers of modes, we numerically compute and compare the dynamic behaviour patterns. Low-order truncations yield highly similar patterns among these mode sets, but fail to fully capture the primary resonance and main harmonic regions. The three-mode truncation, however, reveals significant discrepancies in the subharmonic inter-well regions between the buckling modes and the two vibration modes. These discrepancies are traced to inclusion of axial force effects in the buckling problem and their neglect in the vibration problems, a mechanism further substantiated by an axial-to-bending energy ratio analysis. This indicates faster convergence of the buckling modes, which is confirmed by the five-mode truncation results, where all predictions converge and agree qualitatively with the three-mode buckling-mode results. The study demonstrates that the buckling modes exhibit superior convergence and that a five-mode truncation is generally required to capture the full dynamics, providing a clear and reasoned basis for selecting trial functions and truncation order for bistable curved beams.