Model development for dynamic energy conversion in post-buckled multi-stable slender columns

Broadband piezoelectric energy harvesting solutions from ambient loading have been extensively studied with the purpose of increasing the efficiency of vibration-based harvesters. Most of the previously developed methods focus on the transducer's properties and configurations, and require vibration input excitations. In contrast, we have previously experimentally shown a mechanical energy concentrator system that exploits the quasi-static input deformations (strains) generated within the structure and induces an amplified amplitude and frequency up-converted response. The tested energy converting devices transform low-amplitude and low-rate service strains into an amplified vibration input to the piezoelectric transducer. The snap-through behavior of bilaterally constrained columns was used as the mechanism for energy concentration. This paper presents a theoretical model, based on energy method, for the post-buckling behavior of a bilaterally constrained slender column under quasi-static axial loadings. The total potential energy of the buckled elastic element is the sum of the potential energies due to bending, compression and external applied force. The transverse deflection is limited by the lateral constraints. Therefore a constrained minimization problem of the total potential energy is solved to determine the equilibrium configurations. Equilibrium transitions are correlated to the changes in the magnitude of the weight coefficients that define the contribution of buckling modes to the deflected shape. Transition states are defined in terms of the axial displacements, axial forces, column shape, and energies stored in the system.