# Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack
## 2 Materials and Methods
\]  
Substituting the boundary condition, one can realize that:  
\[V_{h}=-\frac{h}{A\varepsilon(1-r)}\bigg{(}1-r\frac{\tan(ah)}{ah}\bigg{)}q+ \frac{\tan(ah)}{ah}\frac{hd}{eA}F_{h} \tag{15}\]  
where: (\(F_{h}\)) is the load applied to the upper surface, (\(q\)) is electric charge density collected at the upper electrode, (\(E\)) is electric field strength, (\(h\)) denotes the thickness of the plate, and (\(r\)) represents:  
\[r=\frac{d_{33}{}^{2}}{s_{33}\varepsilon_{33}}\]  
The electric charge density gathered at the upper surface as a result of the voltage and force (mostly clamping load) applied along the thickness of the plate is represented by Equation (15). Eventually:  
\[q=\Bigg{(}\frac{1-r}{1-\frac{r\tan(ah)}{ha}}\Bigg{)}\bigg{(}-\frac{V\varepsilon _{33}A}{h}+\frac{\tan(ah)}{ah}d_{33}F_{h}\bigg{)}u_{h}=\frac{1-r}{1-r\frac{ \tan(ah)}{ah}}\frac{\tan(ah)}{ah}\bigg{(}F_{h}\frac{s_{33}h}{A}-d_{33}V\bigg{)} \tag{16}\]  
Equation (16) represents the displacement of the upper surface of the piezoelectric plate and the charge collected with respect to the plate's thickness and the area of electrodes in the consideration of homogeneous mass distribution. Mechanical equivalent electrical circuits have been demonstrated to be useful in the past and appear to be a good technique for modeling electromechanical systems, including smart structures with piezoelectric actuators. The correctness of the differential equations between the physical electromechanical device and the electrical system supports this technique. [PERSON]'s approach is used to develop the equivalent electromechanical model [23; 24]. [PERSON]'s principle states:  
\[ \begin{array}{c}F_{h}=Z_{11}\frac{\partial u}{\partial t}+Z_{12}I\\ V_{h}=Z_{21}\frac{\partial u}{\partial t}+Z_{22}I\end{array}  \tag{17}\]In Equation (18), \(\frac{\partial u}{\partial t}\) denotes the velocity of the upper face, \(I\) denotes the electric current, and \(Z\) represents the impedance. As a recall, the harmonics consists of:  
\[ \begin{array}{c}\frac{\partial u}{\partial t}=j.\omega.u\\ I=j.\omega.q\end{array}  \tag{18}\]  
From Equations (16) and (17), one can find that:  
\(Ks=\frac{A}{s_{3}\hbar}\) the short-circuit device stiffness and \(C=\frac{\epsilon_{3}\omega A}{L}\) device capacity  
\[ \begin{array}{c}F_{h}=\frac{\Delta t}{\tan(ah)}\frac{Ks}{(1-r)}u-\frac{d_{3 }\kappa s}{(1-r)C}q\\ V_{h}=-\frac{d_{3}\kappa s}{(1-r)C}u+\frac{1}{(1-r)C}q\end{array}  \tag{19}\]  
Substituting Equation (19) into [PERSON]'s equation:  
\[\left[ \begin{array}{cc}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{array} \right]=\left[ \begin{array}{cc}\frac{ah}{\tan(ah)} \frac{Ks}{(1-r)}\frac{1}{\
u\omega}&-\frac{d_{3}\kappa s}{(1-r)C}\frac{1}{ \
u\omega}\\ -\frac{d_{3}\kappa s}{(1-r)C}\frac{1}{\
u\omega}&\frac{1}{(1-r)C}\frac{1}{ \
u\omega}\end{array} \right] \tag{20}\]  
In Equations (19) and (20) represents one dimensional analysis by clamping one face and analyzing the deformation and electric displacement on opposite face, the individual terms of the matrix represent the flexibility, admittance, and characteristics of the system. Piezoelectric stack modeling is more complicated than single-plate modeling. The primary purpose of this study was to develop a model composed of layered piezoelectric plates that gives a complicated vibration spectrum that is highly consistent with the experimental data. The model was designed to be a cascade model, with piezoelectric plates connected in series. Experiments were carried out on the composite model, with which the simulation model was compared, and a parameter was identified. The more simplified equation that represents the cascade model of Figure 3 is:  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\left\{ \begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}b_{1}+b_{2}&-b_{2}\\ -b_{2}&b_{2}\end{array} \right]\left\{ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array} \right]\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array}