In recent years, metamaterials have attracted lots of research interests for being used in the design of energy harvesting systems. For a typical cantilevered piezoelectric energy harvester (PEH), there exist various modelling methods. However, metamaterials have relatively more complicated structures than a simple cantilevered beam. In the existing literature, the closed-form solution to the analytical model of a metamaterial based PEH is very cumbersome. Considering that an energy harvester normally delivers the maximum power output around the fundamental resonance, this paper proposes a lumped parameter approach to simplify the modelling of a metamaterial based PEH. The metamaterial based PEH consists of a host beam with a piezoelectric patch bonded at the clamped end. A series of identical resonant beams are periodically mounted onto the host beam perpendicularly. By applying Rayleigh’s method and approximating the fundamental mode shape by the static deflection, the host beam and the resonant beams are represented by independent SDOF models. The equivalent lumped parameters are assumed to concentrate at the tips of the beams and their explicit expressions are derived. The uniformly distributed force onto the beams due to the base excitation are converted into equivalent concentrated forces applied at the tips of the beams. Since the lumped parameters are only effective at the beam tip, for the force interaction at the root of a resonant beam, a factor is derived to correct the reacting force being directly calculated based on the SDOF model of the resonant beam. On the other hand, though the resonant beams are identical, they have different influences on the host beam when being mounted at different positions. To reflect the interaction degree (i.e., reacting force) between a resonant beam and the host beam, a scaling factor that is a function of the mounting position is derived. In addition, due to the action of the resonant beams, 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) system is finally constructed to predict the dynamic behaviour of the metamaterial based PEH. A corresponding finite element model is built to verify the developed MDOF model.


A Lumped Parameter Approach for Analysing a Metamaterial Beam Based Piezoelectric Energy Harvester Around Fundamental Resonance

Division / Sponsors: 
Aerospace Division

Paper Proceedings: https://asmedigitalcollection.asme.org/SMASIS/SMASIS2020/volume/84027

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