Modal analysis of post-buckled beams undergoing stable transitions between remote equilibria

Abstract

Post-buckled structures are prone to snap-through instabilities when external loading causes the structure to lose local stability and jump to a remote equilibrium. However, using piezoelectric actuation, structures can undergo entirely stable transitions between remote equilibria, thereby avoiding snap-through. This paper investigates the linearized vibrational properties of post-buckled beams along these stable transition paths. Numerical results are calculated using an elastica model with allowances for piezoelectric actuation. The model is non-dimensionalized to provide a general view of how the natural frequencies and corresponding mode shapes evolve during stable transitions. Results are presented for two types of stable transitions—one in which the transition is accomplished by changing the external load under a constant actuation level, and one in which the actuation levels are changed without external loading. Results show that the first four natural frequencies of the beam undergo complicated changes during a stable transition. Results also indicate that the first two mode shapes tend to be asymmetric and localized during the early and later stages of the transition. Yet, during the middle of the transition, when the static configuration of the beam is an anti-symmetric “S” shape, the first two mode shapes are symmetric and global. The natural frequencies and modes predicted by the numerical model are validated with a series of experiments.