# Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack
## 2 Materials and Methods
\[S_{3}=s_{33}\sigma+d_{33}E.S_{3}=\frac{\partial u}{\partial x_{3}}\sigma=\frac {1}{s_{33}}S_{3} \tag{3}\]  
The displacement computed in this work is for deformation along the thickness (the item will expand along the thickness) and the parameters \(d_{33}\) and \(s_{33}\) are the piezoelectric coefficient and the compliance along the thickness, respectively. Substituting Equation (3) into Equation (2):  
\[\frac{\partial^{2}u}{\partial t^{2}}=\frac{1}{\rho s_{33}}\frac{\partial^{2}u }{\partial x_{3}{}^{2}}. \tag{4}\]  
The following equation gives the sound velocity (\(v\)) of the piezoceramic based on the elastic property:  
\[v=1/\sqrt{\rho s_{33}} \tag{5}\]  
Substituting Equation (5) into Equation (4):  
\[\frac{\partial^{2}u}{\partial t^{2}}=v^{2}\frac{\partial^{2}u}{\partial x_{3 }{}^{2}}. \tag{6}\]  
The general solution for this equation is given by:  
\[u(x_{3},t)=(c_{1}e^{-j\alpha x_{3}}+c_{2}e^{j\alpha x_{3}})e^{j\omega t} \tag{7}\]  
where:  
\[a=\omega/v \tag{8}\]  
\(c_{1}\) and \(c_{2}\) are the constants determined from the boundary condition by using the linear constituting equation that describes the electromechanical effects. Assuming that the lower face of piezoelectric stack is clamped to the rigid surface and the upper face deforms:  
\[ \begin{array}{c}u(0,t)=0;\\ \sigma(x_{3},t)=\sigma_{h};\end{array}  \tag{9}\]  
Substituting the boundary condition given in Equation (9) into the given solution in Equation (7)  
\[u(0,t)=(c_{1}e^{0}+c_{2}e^{0})e^{j\omega t}=0\]  
The constants are easily determined as follows:  
\[c_{1}=-c_{2}\]  
Then Equation (7) is simplified to:  
\[u(x_{3},t)=2c_{1}\sin(ax_{3})e^{j\omega t} \tag{10}\]  
To obtain \(c_{1}\) from the boundary condition, the simultaneous solution of the linear equation yields:  
\[ \begin{array}{c}+(\frac{\partial u}{\partial x_{3}}=s_{33}\sigma+d_{33}E)\\ -(D=d_{33}\sigma+\varepsilon_{33}E)\frac{d_{33}}{\varepsilon_{33}}\\ \frac{\partial u}{\partial x_{3}}-\frac{d_{33}}{\varepsilon_{33}}D=\sigma(s_{ 33}-\frac{d_{33}{}^{2}}{\varepsilon_{33}}).\end{array} \]Solving for stress gives:  
\[\sigma(x_{3},t)=\frac{1}{\Big{(}s_{33}-\frac{d_{33}}{\varepsilon_{33}}\Big{)}} \bigg{(}\frac{\partial u}{\partial x_{3}}-\frac{d_{33}}{\varepsilon_{33}}D\bigg{)} \tag{11}\]  
\[\frac{\partial u}{\partial x_{3}}=-c_{1}ja\Big{(}e^{-jax_{3}}+e^{jax_{3}}\Big{)} e^{j\omega t}. \tag{12}\]  
Substituting Equation (12) into Equation (11):  
\[ \begin{array}{c}\sigma(x_{3},t)=\frac{1}{\Big{(}s_{33}-\frac{d_{33}}{\varepsilon _{33}}\Big{)}}\Big{(}-c_{1}ja\big{(}e^{-jax_{3}}+e^{jax_{3}}\big{)}e^{j\omega t} -\frac{d_{33}}{\varepsilon_{33}}D\Big{)}.\\ c_{1}=\frac{1}{(j\mu(2\cos(ax_{3})))}\Big{(}-\frac{d_{33}}{\varepsilon_{33}} \frac{D}{e^{j\omega t}}-\sigma_{h}\Big{(}s_{33}-\frac{d_{33}}{\varepsilon_{33 }}\Big{)}\frac{1}{e^{j\omega t}}\Big{)}.\end{array}  \tag{13}\]  
Substituting \(c_{1}\) into Equation (12):  
\[ \begin{array}{c}u(x_{3},t)=2\frac{1}{(a(2\cos(ax_{3})))}\Big{(}\frac{d_{33}} {\varepsilon_{33}}\frac{D}{e^{j\omega t}}+\sigma_{h}\Big{(}s_{33}-\frac{d_{33 }}{\varepsilon_{33}}\Big{)}\frac{1}{e^{j\omega t}}\Big{)}\sin(ax_{3})e^{j \omega t}.\\ u(x_{3},t)=\frac{\tan(ax_{3})}{a}\Big{(}\frac{d_{33}}{\varepsilon_{33}}D+ \sigma_{h}\Big{(}s_{33}-\frac{d_{33}}{\varepsilon_{33}}\Big{)}\Big{)}.\end{array}  \tag{14}\]  
Assuming that the voltage is applied on the upper face grounding the lower face,  
\[ \begin{array}{c}V(0,t)=0\\ V(x_{3},t)=V_{h}\end{array}