# Nonlinear Dynamics of a Piezoelectric Bistable Energy Harvester Using the Finite Element Method
## 2 Piezoelectric Energy Harvester
### Discrete Model  
The proper discretization of the continuous energy harvesting model employs the relative transverse vibration of the beam, \(w(x,t)\), which is approximated by a finite and convergent series of orthogonal functions \(\phi_{j}(x)\), satisfying the boundary conditions. A discrete model can be obtained by applying the [PERSON] method, resulting in a system of \(r+1\) ordinary differential equations [46] as follows, where \(r\) is the number of degrees of freedom employed in the approximation,  
\[\ddot{q}_{j}(t)+2\xi_{j}\omega_{j}\dot{q}(t)+\omega_{j}^{2}q_{j}(t)-v(t)\eta_{ j}-\phi_{j}(L)f_{Mz}(w_{L})=f_{j}(t) \tag{6}\]  
\[C_{p}\dot{v}(t)+\frac{1}{R}v(t)+\sum_{j=1}^{r}\eta_{j}\dot{q}_{j}(t)=0 \tag{7}\]  
in which \(q_{j}(t)\) and \(\xi_{j}\) are, respectively, the nodal coordinates and modal damping with \(j=1,2,\ldots,r\). The eigenfunctions \(\phi_{j}(x)\) are obtained by solving the undamped free  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Substrate** & **Piezoceramic** \\ \hline Material & Aluminum & PZT-5A \\ \hline Length (mm) & 30 & 30 \\ Thickness (mm) & 0.5 & 0.5 \\ Mass density (kg/m\({}^{3}\)) & 2700 & 7750 \\ Young Modulus (GPa) & 70 & 61 \\ Poisson’s Ratio & 0.3 & 0.3 \\ Piezoelectric constants (pC/N) & — & -171 \\ Dielectric constants (nF/m) & — & 15.05 \\ \hline Electrical resistance (Ohm) & \(1\times 10^{6}\) & \\ Damping ratio & \(\alpha\) = 2.16 & \(\beta\) = 5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric and material properties of energy device.  
vibration linear problem for fixed-free boundary conditions, under the assumption of a short-circuit connection [8]. The magnetic force \(f_{Mz}\) needs to be properly described as a function of the beam displacement at the end, \(x=L\), and therefore, \(f_{Mz}=f_{Mz}(w_{L})\). By considering just the first vibration mode, the magnetic force can be approximated by a cubic expression,  
\[f_{Mz}=\alpha w_{L}^{3}+\ell w_{L}=\alpha[\phi(L)q(t)]^{3}+\ell\phi(L)q(t) \tag{8}\]  
where the constants \(\alpha\) and \(\ell\) are determined using Equation (1) through least-squares fitting, employing the parameters shown in Table 2.  
On this basis, a Duffing-type nondimensional equation of motion is employed to describe a reduced-order model of the energy harvesting device as follows  
\[\left\{\begin{array}{l}\dot{u}_{1}=u_{2}\\ \dot{u}_{2}=-2\xi\frac{\omega}{\overline{\omega}}u_{2}-\psi u_{1}-\mu u_{1}^{ 3}-\chi u_{3}+F_{j}(t)\\ \dot{u}_{3}=-\gamma\iota_{3}-k\iota_{2}\end{array}\right. \tag{9}\]  
where  
\[\begin{array}{l}\mu=-\omega\phi(L)^{4}\frac{\overline{L}^{2}}{\overline{ \omega}^{2}},\qquad\psi=\frac{1}{\overline{\omega}^{2}}\left[\omega^{2}-\ell \phi(L)^{2}\right]\qquad\text{and}\\ F_{j}=-\frac{m}{\overline{L}\overline{\omega}^{2}}\tilde{w}_{b}\int_{0}^{L} \phi_{j}(x)dx.\end{array} \tag{10}\]  
This Duffing-type equation is characterized by a cubic nonlinearity, which is associated with a double-well potential of the following type,  
\[\Psi_{p}(u_{1})=\frac{1}{2}\psi u_{1}^{2}+\frac{1}{4}\mu\iota_{1}^{4}. \tag{11}\]