Energy harvesting using a clamped–clamped piezoelectric–flexoelectric beam

In this article, the nonlinear vibration problem of a clamped-clamped monolithic piezoelectric flexoelectric beam is investigated when considering the device as an energy harvester. A scalable model of the beam is derived based on Euler-Bernoulli theory and takes into account geometric nonlinearity resulting from relatively large displacements and restrictive boundary conditions. The Galerkin method is used to discretize the nonlinear reduced-order model which is solved numerically using a continuation technique. The analysis of the dynamic response of the device at the micro and nano-scales demonstrate that such structure can be used to harvest energy despite its symmetry. This result was generally ignored in the literature for clamped-clamped single layered structures. The generated energy was demonstrated to be particularly attributed to the introduction of the longitudinal potential variation and the mid-plane stretching into the beam's model. It was also found that the transverse piezoelectric coefficient is mainly responsible for the harvested energy. A 3D finite element model is developed providing the validation of the assumed assumptions including the adopted analytical potential distribution particularly for the piezoelectric case. The nonlinear dynamic behavior of the beam is analyzed by varying the geometric dimensions of the device from the micro to the nano-scale. It is found that a better power density is obtained when larger scaling factors are used. However, a better performance can be achieved for an optimal scale factor at specific narrow frequency bandwidth nearby the fundamental frequency.