# Nonlinear Dynamics of a Piezoelectric Bistable Energy Harvester Using the Finite Element Method
## 2 Piezoelectric Energy Harvester  
A vibration-based energy harvester is employed to convert mechanical into electrical energy. Essentially it is a resonator that is designed to match the ambient vibration source, working under resonant conditions. A typical configuration is a cantilever beam with a tip mass that allows the adjustment of resonant frequency. A piezoelectric energy harvesting device is a bimorph cantilever beam schematically presented in Figure 1. The system consists of a three-layer structure, two of piezoelectric material (bimorph) bonded to a substrate beam. The substrate is assumed to be made of aluminum, and the piezoelectric material is a PZT-5A. The electrical circuit is represented by a resistor connected, in series, to PZT's layers to generate power output. In addition, ambient vibration is modeled as harmonic base excitation. Nonlinear effects are usually exploited to enhance the device's energy harvesting capacity, making it more efficient and with wider bandwidth. Among the possible strategies, magnetic interactions are employed conferring multistable behavior to the structure. On this basis, consider that permanent external magnets generate a homogenous magnetic  
Figure 1: Schematic representation of (**a**) linear energy harvester and (**b**) bimorph beam cross section. field that induces a magnetization per unit of length in the beam, producing a force and a magnetic moment. By assuming a beam displacement \(w\), and that \(w_{L}=w(L,t)\) is the displacement of the tip mass, the nonlinearity is due to the dependence of the magnetic force \(f_{Mz}\) and the beam tip displacement \(w_{L}\), as schematically showed in Figure 2, which considers the horizontal distance \(d_{m}\) and the distance between magnets, \(d_{g}\). A mathematical model is proposed by assuming that the magnets exhibit idealized magnetic behavior with constant properties in time and therefore, magnetic hysteresis and nonlinearities associated with the magnetization and magnetic field relation are neglected. Moreover, magnetic moment and magnetic force in the \(x\)-direction are also neglected due to small tip displacement/rotation. Therefore, the magnetic forces are derived following similar procedure presented by [PERSON] and [PERSON] [45], being expressed as follows:  
\[f_{Mz}=\sum_{k=1}^{2}\frac{\mu_{0}}{4\pi}\mathscr{G}\mathscr{G}^{\prime}\left[ \frac{w_{L}-d_{h}}{\left(d_{p}^{2}+\left(w_{L}+g_{k}\right)^{2}\right)^{3/2}} -\frac{w_{L}-d_{h}}{\left(d_{m}^{2}+\left(w_{L}+g_{k}\right)^{2}\right)^{3/2}}\right] \tag{1}\]  
where \(\mu_{0}\) and \(\mathscr{G}\) are the vacuum permeability and total surface charge, respectively; \(2l_{m}\) is the length of each magnet; \(d_{h}\) is half of the distance between magnets (\(d_{g}\)); on this basis, parameters \(d_{p}\) and \(g_{k}\) are defined as: \(d_{p}=d_{m}+2l_{m}\), \(g_{k}=(-1)^{k}d_{h}\). The mathematical model for the nonlinear energy harvesting device is expressed as follows, assuming a Bernoulli-Euler hypothesis for the beam model and an electric circuit represented by a voltage \(v\), with capacitive (\(C_{p}\)) and a resistive load (\(R\)),  
\[ \begin{split} YI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+m\frac {\partial^{2}w(x,t)}{\partial t^{2}}-\vartheta\ v(t)\left[\frac{d\delta(x)}{ dx}-\frac{d\delta(x-L)}{dx}\right]\\ +c_{a}\frac{\partial w(x,t)}{\partial t}-f_{Mx}\frac{\partial^{2} w}{\partial x^{2}}=-m\frac{\partial^{2}w_{b}(t)}{\partial t^{2}}-c_{a}\frac{ \partial w_{b}(t)}{\partial t}\end{split}  \tag{2}\]  
\[C_{p}\frac{dv(t)}{dt}+\frac{v(t)}{R}+\vartheta\int_{0}^{L}\frac{\partial^{3}w (x,t)}{\partial x^{2}\partial t}dx=0 \tag{3}\]  
where \(YI\) is the equivalent bending stiffness, \(m\) is the mass per unit length and \(\vartheta\) is the piezoelectric coupling. Additionally, \(c_{a}\) is the viscous damping, \(\delta\) is the Dirac function and \(w_{b}(t)\) is a base excitation. For a bimorph beam configuration, these parameters can be obtained from expressions given in Refs.