Modal analysis of finite-size piezoelectric metamaterial plates

Two-dimensional electromechanical metamaterials composed of thin plates with local piezoelectric resonators can display extreme vibration attenuation characteristics at desired frequencies. The typical bandgap analyses in the literature use the assumption of wave propagation in an infinite elastic structure and do not consider the modal characteristics of the structure. However, for practical implementation and design of finite-size electromechanical metamaterials, modal behaviour of the host structure and piezoelectric elements must be coupled with the dynamics of shunt circuits. To this end, we present a system-level modal analysis framework for finite-size thin plates with a segregated array of piezo-patches connected to resonant shunt circuits. The developed model takes into account the spatially discontinuous flexural rigidity of the metamaterial plate due to discrete placement of piezoelectric resonators on the substrate. Using the developed framework, we show that the electrical quality factor of resonators is critical for transitioning from broadband shunt damping to bandgap formation in piezoelectric plate metamaterials. This enables on-demand tailoring of effective dynamic stiffness of metamaterial plates for the targeted task. Lastly, for a fixed number of discrete resonators, we demonstrate the effect of physical gap size between resonators on the bandgap creation. Overall, the modelling frameworks in this study can be used for predicting the dynamics of piezoelectric plate-type metamaterials for applications in waveguiding, attenuation, filtering, and energy harvesting.