# Dynamic Analytical Solution of a Piezoelectric Stack Utilized in an Actuator and a Generator
## 2 Basic Equations
The general solution of Equation (5) has the following form:  
\[u_{Pi}(z)=A_{Pi}\sin k_{Pi}z+B_{Pi}\cos k_{Pi}z \tag{6}\]  
Combining Equations (1), (2), and (6), the stress, electric displacement, and electric potential can be obtained as:  
\[ \begin{array}{l}\sigma_{2 Pi}(z)=p_{Pi}(A_{Pi}\cos k_{Pi}z-B_{Pi}\sin k_{Pi} z)+\frac{\sigma_{33i}^{2}}{\kappa_{33i}^{2}}C_{1i}\\ D_{zi}(z)=-e_{33i}C_{1i}\\ \phi_{i}(z)=\frac{e_{33i}}{\kappa_{33i}^{2}}(A_{Pi}\sin k_{Pi}z+B_{Pi}\cos k_{Pi }z+C_{1i}z+C_{0i})\end{array}  \tag{7}\]  
where \(p_{Pi}=(C_{33 Pi}+\frac{e_{33i}^{2}}{\kappa_{33i}^{2}})k_{Pi}\). The other parameters \(A_{Pi}\), \(B_{Pi}\), \(C_{1i}\), and \(C_{0i}\) are constants to be determined. \[\left\{ \begin{array}{l}a=\frac{e_{33}^{2}}{p_{R}x_{33}^{3}}\\ a_{1}^{1}=-\frac{\sin k_{B}h_{i}-\sin k_{P}h_{i-1}}{h_{i}-h_{i-1}}\\ b_{i}^{1}=-\frac{\cos k_{B}h_{i}-\cos k_{B}h_{i-1}}{h_{i}-h_{i-1}}\\ V_{i}^{1}=\frac{-\frac{\delta k_{33}}{\delta k_{33}}}{|\varepsilon_{33}|(h_{i}-h _{i-1})}\end{array} \right. \tag{18}\]  
\[\left\{ \begin{array}{l}a_{i}^{2}=1+aa_{i+1}^{1}\cos k_{P}h_{i}\\ b_{i}^{2}=ab_{i+1}^{1}\cos k_{P}h_{i}\\ a_{i}^{3}=1+aa_{i}^{1}\cos k_{P}h_{i}\\ b_{i}^{3}=ab_{i}^{1}\cos k_{P}h_{i}\\ V_{i}^{2}=a\cos k_{P}h_{i}(V_{i}^{1}-V_{i+1}^{1})\end{array} \right. \tag{19}\]  
\[\left\{ \begin{array}{l}a_{i}^{4}=-aa_{i+1}^{1}\sin k_{P}h_{i}\\ b_{i}^{4}=1-ab_{i+1}^{1}\sin k_{P}h_{i}\\ a_{i}^{5}=-aa_{i}^{1}\sin k_{P}h_{i}\\ b_{i}^{5}=1-ab_{i}^{1}\sin k_{P}h_{i}\\ V_{i}^{3}=a\sin k_{P}h_{i}(V_{i+1}^{1}-V_{i}^{1})\end{array} \right. \tag{20}\]  
\[\left\{ \begin{array}{l}H_{i}^{1}=a_{i}^{2}b_{i}^{4}-b_{i}^{2}a_{i}^{4}\\ a_{i}^{6}=(a_{i}^{3}b_{i}^{4}-a_{i}^{5}b_{i}^{2})/H_{i}^{1}\\ b_{i}^{6}=(b_{i}^{3}b_{i}^{4}-b_{i}^{5}b_{i}^{2})/H_{i}^{1}\\ V_{i}^{4}=(V_{i}^{2}b_{i}^{4}-V_{i}^{3}b_{i}^{2})/H_{i}^{1}\\ a_{i}^{2}=-(a_{i}^{3}a_{i}^{4}-a_{i}^{5}a_{i}^{2})/H_{i}^{1}\\ b_{i}^{7}=-(b_{i}^{3}a_{i}^{4}-b_{i}^{5}a_{i}^{2})/H_{i}^{1}\\ V_{i}^{5}=-(V_{i}^{2}a_{i}^{4}-V_{i}^{3}a_{i}^{2})/H_{i}^{1}\end{array} \right. \tag{21}\]  
Substituting Equations (6), (7), (14), and (15) into Equation (8), \(A_{P1}\) can be determined as:  
\[A_{P1}=-\frac{d_{0}V_{0}}{d} \tag{22}\]  
where  
\[\left\{ \begin{array}{l}d_{0}=\left(\frac{1}{2}Z_{ext}j\omega\lambda_{N}^{1}- \lambda_{N}^{2}p_{P}\right)\sin k_{P}h_{N}+\left(\frac{1}{5}Z_{ext}j\omega \lambda_{N}^{2}+\lambda_{N}^{1}p_{P}\right)\cos k_{P}h_{N}\\ \quad+\frac{\varepsilon_{33}^{2}}{\varepsilon_{33}^{3}}(a_{N}^{1}\lambda_{N}^{ 1}+b_{N}^{1}\lambda_{N}^{2}+V_{N}^{1})\\ d=\left(\frac{1}{5}Z_{ext}j\omega\delta_{N}^{1}-\delta_{N}^{2}p_{P}\right) \sin k_{P}h_{N}+\left(\frac{1}{5}Z_{ext}j\omega\delta_{N}^{2}+\delta_{N}^{1}p_{ P}\right)\cos k_{P}h_{N}\\ \quad+\frac{\varepsilon_{33}^{2}}{\varepsilon_{33}^{3}}(a_{N}^{1}\delta_{N}^{ 1}+b_{N}^{1}\delta_{N}^{2})\end{array} \right. \tag{23}\]  
For Model A, the electrical admittance \(Y\) can be expressed as:  
\[Y=\frac{1}{Z}=\frac{I_{total}}{V_{0}} \tag{24}\]  
where \(Z\) is the electrical impedance and \(I_{total}=-j\omega S|e_{33}|\sum\limits_{i=1}^{N}C_{1i}\).