A lumped parameter approach for analysing A metamaterial beam based piezoelectric energy harvester around fundamental resonance

This paper proposes a lumped parameter approach to simplify the modelling of a metamaterial based PEH to predict its energy harvesting performance around the fundamental resonance. The metamaterial based PEH consists of a host beam with a piezoelectric patch bonded at the clamped end. A series of local resonators are attached onto the host beam. In the first case study, the local resonators are modelled as mass-spring systems. By applying Rayleigh’s method and approximating the fundamental mode shape by the static deflection, the host beam is represented by a SDOF system. The equivalent lumped parameters are assumed to concentrate at the tip of the host beam and their explicit expressions are presented. Though the local resonators are identical, they have different influences on the host beam when being attached at different positions. To reflect the interaction degree (i.e., reacting force) between the local resonator and the host beam, a scaling factor that is a function of the attaching position is derived. On the other hand, due to the action of the local resonators, the fundamental mode shape of the host beam is actually changed. Based on the linear superposition principle, the static deflection approximated fundamental mode shape is corrected and the electromechanical coupling coefficient that is sensitive to the slope of the mode shape is updated to improve the accuracy. Based on the derived equivalent lumped parameters and correction factors, a multiple-degree-of-freedom (MDOF) model is constructed to predict the dynamic behavior of the metamaterial based PEH with mass-spring resonators. A corresponding finite element model is built to verify the developed MDOF model. In the second case study, the local resonators are modelled as practical parasitic beams. The parasitic beams are converted into equivalent lumped systems as well. However, the lumped parameters are the effective parameters at the beam tip. For the force interaction at the root of a parasitic beam, a factor is derived to correct the reaction force when a parasitic beam is represented by a SDOF mass-spring system. Using the reaction force correction factor, a MDOF model for the metamaterial based PEH with beam-like resonators is also established and verified by the finite element model.