Graded multifunctional piezoelectric metastructures for wideband vibration attenuation and energy harvesting

Unlike well-studied locally resonant (LR) metamaterials with a periodic array of identical resonators, 'graded' LR metamaterials consist of an array of resonators with a spatially varying parameter, yielding wideband wave attenuation and mode trapping/localization, among other features. In this work, we explore a graded LR piezoelectric metamaterial-based structure (i.e. metastructure) in which the grading parameter, namely the inductive shunt resonant frequency of the unit cells, follows a predefined variation pattern in space (e.g. first-order, quadratic, or fractional). We investigate the effect of such patterns on (i) the vibration attenuation bandwidth, (ii) the localization of vibration modes, and (iii) the harvested power. To this end, we consider a piezoelectric bimorph cantilever hosting an array of piezoelectric unit cells with spatially varying inductive shunts. Fully coupled electromechanical equations describing the metastructure's linear transverse displacement and unit cell voltages are given with a modal analysis framework and solved using the matrix inversion method. The results show that (i) the first-order grading pattern yields the widest bandgap with 65% increase in the bandwidth compared to the standard uniform LR pattern, (ii) the localization of vibration modes follows in shape the corresponding frequency grading pattern, and (iii) the largest power is harvested for the fractional grading pattern. Furthermore, all of the graded resonator configurations result in wider bandwidth in energy harvesting as compared to the uniform resonators case. Overall, the results unveil the fundamental characteristics of this class of graded piezoelectric metastructures and support the design of such multifunctional piezoelectric metastructures for concurrent vibration attenuation and energy harvesting.