# Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack
## 2 Materials and Methods  
The model was developed using identical and different sizes of piezoelectric plates glued together to form a piezoelectric stack. Of the two novel methods, the first method  
Figure 1: The stack composed of two of the same-sized piezos (**b**) and different-sized piezos (**a**). This is a plate representation, in which the actuating voltage is applied separately and assumes non-conductive material between the wafers. involves connecting two piezoelectric plates of the same size (Figure 1b) (30 \(\times\) 30 \(\times\) 10 mm), while the second method involves testing the displacement after connecting piezoelectric plates of different sizes (Figure 1a) (30 \(\times\) 30 \(\times\) 10 and 18 \(\times\) 18 \(\times\) 8 mm). The experiment was carried out in order to compare with the proposed mathematical model. Each plate was controlled by its own circuit, and the plates were separated in nonconductive material, assuming an ideal force transformation from one plate to another. The IEE Ultrasonics, Ferroelectrics, and Frequency Control Society provided the most well acknowledged description of piezoelectric ceramic behavior in 1966 and updated it in 1987. This committee developed linearized constitutive relations characterizing the piezoelectric continuum, which served as the foundation for the piezoelectric behavior model that is now widely used. Piezoelectricity includes two sets of variables: mechanical (\(\sigma\), S) (stress, strain) and electrical (E, D) (electric field, electric induction). The generally recognized mathematical explanation based on the assumption of linearity is provided by IEEE-ANSIS standards [21; 22]. As illustrated in Figure 2, the assumptions made were that plate polarization is along the z direction and that the x-y planes are the electrode planes. The constitutive equation that governs piezoelectric plates is represented by Equation (1), and the equation demonstrates that the electric charge induced in the plate is directly related to the stress applied along the polarized axis. \[ \begin{split} S&=\big{[}s^{E}\big{]}\sigma+[d]^{T}E,\\ D&=[d]\sigma+[e^{\sigma}]E,\end{split}  \tag{1}\]  
where: \(\big{[}s^{E}\big{]}\) is a 6 \(\times\) 6 symmetrical matrix called mechanical matrix of flexibility (compliance tensor), the superscript E in \(\big{[}s^{E}\big{]}\) indicates that either the electrical field is zero or it is a constant, \(\sigma\) is the mechanical stress vector, \(S\) is the strain vector, \([e^{\sigma}]\) is a 3 \(\times\) 3 symmetrical matrix called electric permittivity of the material, the superscript \(\sigma\) in \([e^{\sigma}]\) indicates that either the stress is zero or it is a constant, \([d]\) is a 3 \(\times\) 6 matrix of piezoelectric coefficients, the superscript T refers to matrix transposition, \(E\) is the vector of the applied electric field, and \(D\) is the vector of electric displacement. Since \(D\) is the function of the electric field that depends on electric voltage and plate thickness, we may say that electric displacement is a function of the plate thickness. The dynamic equation that describes the elongation vibration of the plate along the polarization axis (z) as shown in Figure 2 is as follows:  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial\sigma}{\partial x_{3}} \tag{2}\]  
Figure 2: Standard axis nomenclature and piezoelectric plate. The polarization axis determines the axis of vibration. The electrode surface in which the electric field is applied is perpendicular to the poled axis. The linear constitutive equations were calculated on a basis in which the assumed boundary conditions were taken into account. In Equation (2), \(u\) denotes the plate's deformation along the polarization axis, \(x_{3}\) represents the plate's thickness, and \(\rho\) denotes the plate's density. The following equation represents the linear constitutive equation that can be used for boundary condition analysis.