Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications

Energy harvesting at micro- and nanoscales has recently seen a renewed interest that the flexoelectric effect can counter the inability of piezoelectric energy harvesters to generate enough energy at small scales. Almost all small-scale energy harvesters use uniform rectangular geometries, whereas at the macroscale energy harvesters use a wide array of geometries including tapered rectangular geometries. The incorporation of non-uniform effects into a piezoelectric system considering the flexoelectric effect should give insight into how these systems can benefit from different geometries. A non-uniform flexoelectric Euler-Bernoulli cantilever energy harvester is modeled using classical continuum theories and is examined at the microscale. The non-uniformity of the energy harvester is governed by linear and nonlinear tapering effects, with the nonlinearities represented by high-order polynomials. The system is assumed to be linear, only undergoing harmonic base excitation. The varied tapering ratios and powers of the geometric tapering, considering that only the thickness and the width of the beam are tapered, are compared with uniform systems. The results show that non-uniform beams exhibit more harvested power than their uniform counterparts and also increase the range of resonant frequencies where significant power can be generated. Nonlinear tapering increases the amount of power that could be harvested compared to linear tapering; however, the nonlinearity of the tapering effects is limited to cubic and quadratic forms. It is demonstrated that higher-order tapering effects reduce the amount of harvested power compared to the linear taper counterpart. Non-uniform beams prove to be more effective than their rectangular counterparts within a linear system, whereas optimal resistive loads decrease as the tapering effects increase.