Metamaterials made from flexible structures with piezoelectric laminates connected to resonant shunt circuits can exhibit vibration attenuation properties similar to those of their purely mechanical locally resonant counterparts. Thus, in analogy to purely mechanical metamaterials, electroelastic metamaterials with piezoelectric resonators can exhibit vibration attenuation bandgaps. To enable the effective design of these locally resonant electroelastic metamaterials, the electromechanical behavior of the piezoelectric patches must be reconciled with the modal behavior of the electroelastic structure. To this end, we develop a novel argument for the formation of bandgaps in bimorph piezoelectric beams, relying on modal analysis and the assumption of infinitely many segmented shunted electrodes (unit cells) on continuous piezoelectric laminates bracketing a substrate. As a case study, the frequency limits of the locally resonant bandgap that forms from resonant shunting is derived, and a design guideline is presented to place the bandgap in a desired frequency range. This method can be easily extended to more general circuit impedances, and can be used to design shunt circuits to obtain a desired frequency response in the main structure.

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