Symbolic Modeling of Tapered Beams in Piezoelectric Energy Harvesters

In this paper, we present a novel symbolic MEMS modeling methodology for devices with tapered beam geometries. Such beams are very common in piezoelectric energy harvesters. While a closed-form solution for the device model may be possible in cases without taper, it becomes increasingly difficult to obtain for tapered devices. In the latter cases, we show how a symbolic perturbation expansion with the taper angle as the perturbation parameter can be used to conjure up a closed-form approximation of the modal behavior of the device. A major contribution of our work is the development of a special library of symbolic processing modules in MATHEMATICA to help construct the perturbation expansion, set up a series of symbolic modal solutions of the tapered device, and ultimately, build an equivalent circuit model. Furthermore, a MATLAB script is used for the numerical evaluation of the symbolic model for any given perturbation order. Significant performance improvement is obtained with respect to the finite-element method with almost no loss in accuracy.