# Development of a Nonlinear Piezoelectric Energy Harvester for Alternating Air Load
## 4 The Model of Piezoelectric Patch  
The model of piezoelectric patch presents the relationship between air pressure and output voltage, as shown in Figure 2. The solution used in this article follows the restrictions that there is no shear strain or stress along the thickness direction (\(z\)-direction) and that the deflection is much smaller than the thickness of the patch. In this article, the energy method is used to describe the total energy in a pressure-loaded piezoelectric patch and to ultimately calculate the electrical charge generated from the pressure source.  
As shown in Figure 3, the established model consists of piezoelectric materials and copper substrate, which are marked. The thicknesses of PZT-5H and copper substrate is represented by \(h_{\mathrm{p}}\) and \(h_{\mathrm{s}}\), respectively. The equations governing the transverse deflection, bending, and shearing action of the piezoelectric patch subjected to the uniformly distributed constant pressure load \(P_{x}\) are given by [29]:  
\[\
abla^{4}W_{r}=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial}{\partial r }(\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial W_{r}}{\partial r}) ))=\frac{P_{x}}{D} \tag{9}\]  
\[M_{r}=-D(\frac{\partial^{2}W_{r}}{\partial r^{2}}+\frac{v}{r}\frac{\partial W _{r}}{\partial r}) \tag{10}\]  
\[M_{\Theta}=-D(\frac{1}{r}\frac{\partial W_{r}}{\partial r}+v\frac{\partial^{2} W_{r}}{\partial r^{2}}) \tag{11}\]  
\[Q_{r}=-D\frac{\partial}{\partial r}(\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial W_{r}}{\partial r})) \tag{12}\]where \(W_{r}\) is the deflection of the patch in the z-direction, \(r\) is the distance from the center of the patch to the point of deflection, \(M_{r}\) and \(M_{\Theta}\) are moments in the \(r\) and \(\theta\) directions, respectively, \(\
u\) is Poisson's ratio, \(P_{x}\) is pressure, \(Q_{r}\) is shear force and \(D\) is flexural rigidity of the patch. \(V_{\text{out}}\) is the output voltage. The relationship among \(W_{r}\), \(P_{x}\), \(V_{\text{out}}\) and electric energy \(U\) has been researched in earlier studies [26].  
\[V_{\text{out}}=\frac{3}{2}MR_{s}^{3}h_{s}h_{p}(h_{s}+h_{p})P_{x} \tag{13}\]  
\[U=\frac{9}{8}KR_{s}^{6}R_{p}^{2}h_{s}^{2}h_{p}(h_{s}+h_{p})^{2}{P_{x}}^{2} \tag{14}\]  
According to Equations (2), (7) and (14), \(U\) can be expressed as follow:  
\[U=\frac{9}{8}KR_{s}^{6}R_{p}^{2}h_{s}^{2}h_{p}(h_{s}+h_{p})^{2}(\frac{2}{3} \frac{N}{V_{0}-S_{0}l}\overline{\omega})^{2} \tag{15}\]  
where \(M\) and \(K\) are calculated by the dielectric and piezoelectric parameters.  
Equations (13) and (15) can be used to explore the influence of the relative parameters on energy generation of a piezoelectric harvester. It can be seen that the pressure is proportional to the output voltage. In the pneumatic system, the air condition can be influenced by the piston moving. The compressed air shows a nonlinear characteristic, and as the result, the output voltage of the piezoelectric patch is influenced by the nonlinear compressed air load.  
In the equations, a rough influence trend of the relative parameters on energy generation appears. Although it is difficult or impossible to build an accurate analytical model for accurate calculation the important parameters are shown to guide the simulation and experiments. In this study, the feasibility was validated for the presented harvester to harvest alternating air energy. In the following sections, the influence of relative parameters of the harvester is investigated.