## 3 Calculation Results and Analysis  
To facilitate data processing, each quantity is made dimensionless first  
\[\left\{ \begin{array}{l}\overline{w}=\frac{w}{h}\,\ \overline{\overline{w}}=\frac{ \overline{w}}{h}\,\ \overline{\overline{x}}=\frac{x}{l_{a}}\,\ \overline{y}=\frac{y}{l_{b}}\,\ \lambda=\frac{l_{a}}{l_{b}}\,\ \eta=\frac{l_{a}}{h},\\ \tau=\frac{t}{l_{a}}\sqrt{\frac{A_{10}}{I_{10}}}\,\ \overline{\omega}=\omega l_{a} \sqrt{\frac{I_{10}}{A_{10}}}\,\ \overline{q}=\frac{qH_{A}^{4}}{A_{10}h^{3}}\,\\ \overline{K}=\frac{K_{10}^{4}}{A_{10}h^{2}}\,\ A_{10}=hC_{11(\text{PZT}-4)}\,\ I_{10}=h\rho_{(\text{PZT}-4)}\end{array} \right. \tag{32}\]  
To verify the validity of the model and the accuracy of the calculation, the static bending problem of a piezoelectric thin plate made of a single material was first calculated by referring to the relevant data in the literature [30], and the results were compared with those in the literature [30]. According to the data in Table 1, the numerical results in this paper are in good agreement with those in the literature. Therefore, the validity of the model and the accuracy of the calculation can be verified. Then, based on [PERSON]'s plate theory, the static bending and free vibration behaviors of the four-sided simply supported FGPMs plate are analyzed, and the effects of the functional graded index, length-to-width ratio, length-to-height ratio, foundation stiffness coefficient, temperature change value and external voltage value on the static bending deflection, and fundamental frequency value are discussed. The related physical parameters of the FGMs are shown in Table 2.  
Table 3 shows that the larger the foundation stiffness coefficient is, the smaller the center deflection value is and the larger the fundamental frequency value is. Through changes in data, we can analyze that the increase in elastic coefficient can enhance the equivalent stiffness of the plate, increase the ability of the plate to resist the deformation, improve the fundamental frequency value of the plate and increase the stability of its operation. Table 4 shows that under the same length-to-width ratio, the changes of the gradient index have little influence on the deflection value of the central point. Under the same functional gradient index, the larger the length-to-width ratio is, the smaller the deflection value of the center point is. The increase in the length-to-width ratio increases the equivalent stiffness of the plate to increase its ability to resist deformation. The equivalent stiffness of the plate is not large when the functional gradient index value changes, and therefore, when designing a FGPMs plate, the selection of material components can be flexible. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{\(I_{a}=I_{b}=30h\)} & \(\begin{array}{c}I_{a}=30h\\ I_{b}=50h\end{array}\) & \(\begin{array}{c}I_{a}=I_{b}=50h\end{array}\) \\ \hline \(h=30\) nm & \begin{tabular}{c} This paper \\ literature [30] \\ \end{tabular} & \begin{tabular}{c} 0.3123 \\ 0.3123 \\ \end{tabular} & \begin{tabular}{c} 0.6565 \\ 0.6565 \\ \end{tabular} & \begin{tabular}{c} 0.8203 \\ 0.8203 \\ \end{tabular} \\ \hline \(h=100\) nm & \begin{tabular}{c} This paper \\ literature [30] \\ \end{tabular} & \begin{tabular}{c} 0.8571 \\ 0.8571 \\ \end{tabular} & \
\begin{tabular}{c} 1.6502 \\ 1.6502 \\ \end{tabular} &
\begin{tabular}{c} 8.2253 \\ 8.2253 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of deflection values at the centers of the plates. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Properties** & **PZT-4** & **PZT-5** \\ \hline Elastic constant & \(C_{11}\) & \(81.3\) & \(60.6\) \\ (GPa) & \(C_{12}\) & \(25.6\) & \(23.0\) \\ \(C_{66}\) & \(30.5\) & \(35.5\) \\ Piezoelectric constant & \(e_{31}\) & \(-4.1\) & \(-17.05\) \\ (C/m\({}^{2}  
Table 5 shows that under the same voltage, the greater the temperature change value is, the greater the fundamental frequency value is.