# Dynamic Analytical Solution of a Piezoelectric Stack Utilized in an Actuator and a Generator
## 2 Basic Equations  
Two kinds of piezoelectric stacks are considered in the present paper, the piezoelectric stack under a voltage load (Model A) and the piezoelectric stack under a uniformly distributed load (Model B), as shown in Figure 1. The piezoelectric layers are connected in series mechanically, but in parallel electrically. The number of piezoelectric layers is \(N\). The thickness of piezoelectric layer \(i\) is determined by \(h_{i}-h_{i-1}\). Here, it should be noted that the thickness of the first layer is \(h_{1}-h_{0}\), and its upper and lower surfaces are located at \(z=h_{1}\) and \(z=0\), respectively. To define the thickness of each layer using the above formula, we let \(h_{0}=0\). All the piezoelectric layers are polarized in the \(z\)-direction. For the even layers, the polarization direction is the positive \(z\)-direction, and the piezoelectric coefficient is defined as \(+d_{33}\). For the odd layers, the polarization direction is the negative \(z\)-direction, and the piezoelectric coefficient is defined as \(-d_{33}\). Figure 1: Schematic of the piezoelectric stack: (**a**) under a voltage load (Model A) and (**b**) under a uniformly distributed load (Model B). According to the linear piezo-elasticity theory, in cases without body force and body charge, the basic equations for piezoelectric material in the piezoelectric layer \(i\) can be written as:  
\[\left\{ \begin{array}{l}\rho_{Pi}\frac{\partial^{2}u_{Pi}}{\partial t^{2}}= \frac{\partial v_{Pi}}{\partial z}\\ \sigma_{2 Pi}=C_{33 Pi}\varepsilon_{2 Pi}-e_{33i}E_{zi}\\ \varepsilon_{2 Pi}=\frac{\partial u_{Pi}}{\partial z}\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{l}D_{zi}=e_{33i}\varepsilon_{2 Pi}+\kappa_{33i}^{ \varepsilon}E_{zi}\\ E_{zi}=-\frac{\partial v_{Pi}}{\partial z}\\ \frac{\partial D_{zi}}{\partial z}=0\end{array} \right. \tag{2}\]  
where \(\rho_{Pi}\) is the density of the piezoelectric material; \(u_{Pi}\), \(\sigma_{2 Pi}\), and \(\varepsilon_{2 Pi}\) are the displacement, stress, and strain in the \(z\)-direction, respectively; and \(D_{zi}\), \(E_{zi}\), and \(\phi_{i}\) are the electric displacement, electric field, and electric potential, respectively. \(C_{33 Pi}=\frac{1}{53\mu_{1}}\), \(e_{33i}=\frac{d_{33i}}{53\mu_{1}}=d_{33i}C_{33 Pi}\), \(\kappa_{33i}^{\varepsilon}=\kappa_{33i}^{\varepsilon}-\frac{d_{33i}^{2}}{53 \mu_{3i}}=\kappa_{33i}^{\varepsilon}-d_{33i}^{2}C_{33 Pi}\), where \(S_{33 Pi}\), \(d_{33i}\), and \(\kappa_{33i}^{\varepsilon}\) are the elastic constant, piezoelectric constant, and dielectric constant, respectively. Here, \(e_{33i}\) is taken as a plus or minus value similar to the definition of \(d_{33i}\). That is, when the number of the layer is even, it is defined as \(+e_{33}\); when the number of the layer is odd, it is defined as \(-e_{33}\). In addition, it should be noted that the third condition in Equation (2) adopts a well-known quasi-static approximation in the absence of free charge density, which is the Maxwell-Poisson equation of electrostatics [59; 60; 61]. \[\left\{ \begin{array}{l}V(t)=V_{0}e^{i\omega t}\quad\text{(for Model A)}\\ q(t)=q_{0}e^{i\omega t}\quad\text{(for Model B)}\end{array} \right. \tag{3}\]  
where \(V_{0}\) and \(q_{0}\) are the amplitude of the voltage and distributed load, respectively; \(q_{0}=F/S\), \(F\) is the amplitude of the axial force and \(S\) is the area of the cross-section of the piezoelectric stack; \(j=\sqrt{-1}\); \(\omega=2\pi f\) is the circular frequency; and \(t\) is the time. The displacement \(u_{pi}\) and electric potential \(\phi_{i}\) of the steady vibration can be expressed as:  
\[u_{Pi}=u_{Pi}(z)e^{i\omega t}\quad,\quad\phi_{i}=\phi_{i}(z)e^{i\omega t} \tag{4}\]  
Substituting Equations (2)-(4) into Equation (1) yields:  
\[\frac{\partial^{2}u_{Pi}(z)}{\partial z^{2}}+k_{Pi}^{2}u_{Pi}(z)=0 \tag{5}\]  
where \(k_{Pi}^{2}=\frac{\rho_{Pi}\omega^{2}}{C_{33 Pi}+e_{33i}^{2}/\kappa_{33i}^{ \varepsilon}}\).