## 3 Results and Discussion  
### Verification and Materials  
To verify the theoretical formula accuracy of this study, two numerical examples are described and discussed in predicting the free vibration behaviors. The first example is a fundamental frequency comparison of the simply supported square functionally gradient plate subjected to different gradient indices (\(p=0,\,0.5,\,1,\,2\)) are presented in Table 1, which was taken from [PERSON] et al. [27] and [PERSON] et al. [28]. Functionally gradient materials (Al/Al\({}_{2}\)O\({}_{3}\) and Al/ZrO\({}_{2}\)) parameters, geometric parameters (\(L_{a}\)/\(L_{b}=1\) and \(h\)/\(L_{a}=0.1\)), and non-dimensional natural frequencies form are given in [29]. Table 1 shows that the results of the present method for the core layer were in good agreement with the existing data on functionally gradient plates in the literature, which certifies the correctness of natural frequencies that was calculated by this study. The second example is vibration analysis of a thermally hybrid composite sandwich plate, which was equipped with SMA fibers in the graphite-epoxy face sheets. The material properties of sheets and core, asymmetric stacking sequence, and geometrical parameters were taken from [PERSON] et al. [30]. In accordance with Table 2, it the correctness has been verified by comparison of non-dimensional natural frequencies for sandwich composite plate-embedded SMA subjected to different SMA volume fractions and pre-strain. Thus, the present procedure has presented acceptable accuracy in the analytical method. In the following, a composite sandwich plate with a length of 0.4 m, width of 0.3 m, and total thicknesses of 0.8 mm is considered, which core thicknesses is 0.64 mm. The plate consisted of 3 layers, of which the two outer layers are SMA/epoxy and the core layer is graphite/epoxy. The SMA wires stacking sequence of the composite face sheets were assumed as (0\({}^{\circ}\)/core/90\({}^{\circ}\)). Moreover, the martensitic phase transition to the austenitic phase is performed during the temperature rise which made the elastic modulus and recovery stress of SMA are shown in Figure 2[31]. In Figure 2, both elastic modulus and recovery stress of the SMA appear to have a strong temperature dependence, while recovery stress also has tensile pre-strain dependent. The material properties of the surface sheets and core are given in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**FGMs**} & \multicolumn{5}{c}{**Gradient Index (\(p\))**} \\  & **0** & **0.5** & **1** & **2** \\ \hline Al/Al\({}_{2}\)O\({}_{3}\) & [PERSON] et al. & 5.7694 & 4.9000 & 4.4166 & 4.0057 \\  & [PERSON] et al. & 5.6763 & 4.8209 & 4.3474 & 3.9474 \\  & Present & 5.7689 & 4.9123 & 4.4328 & 4.0176 \\ Al/ZrO\({}_{2}\) & [PERSON] et al. & 5.7694 & 5.4380 & 5.3113 & 5.2923 \\  & [PERSON] et al. & 5.6763 & 5.1105 & 4.8713 & 4.6977 \\  & Present & 5.7689 & 5.3511 & 5.2696 & 5.1254 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of non-dimensional natural frequencies for functionally gradient plate with [PERSON] et al. and [PERSON] et al. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(V_{s}\)_and_\(\varepsilon_{0}\)} & \multicolumn{5}{c}{**[PERSON] et al.**} & \multicolumn{5}{c}{**Present**} \\  & 25 \({}^{\circ}\)C & 35 \({}^{\circ}\)C & 55 \({}^{\circ}\)C & 65 \({}^{\circ}\)C & 25 \({}^{\circ}\)C & 35 \({}^{\circ}\)C & 55 \({}^{\circ}\)C & 65 \({}^{\circ}\)C \\ \hline Without SMA & 0.862 & 0.696 & 0.485 & 0.259 & 0.877 & 0.712 & 0.493 & 0.261 \\ \(V_{s}=10\%\)\(\varepsilon_{0}=2\%\) & 0.842 & 0.855 & 0.904 & 0.938 & 0.853 & 0.867 & 0.916 & 0.949 \\ \(V_{s}=20\%\)\(\varepsilon_{0}=2\%\) & 0.813 & 0.960 & 1.126 & 1.224 & 0.832 & 0.972 & 1.133 & 1.237 \\ \(V_{s}=20\%\)\(\varepsilon_{0}=1\%\) & 0.811 & 0.925 & 1.084 & 1.171 & 0.826 & 0.932 & 1.091 & 1.180 \\ \(V_{s}=20\%\)\(\varepsilon_{0}=0.5\%\) & 0.803 & 0.888 & 1.005 & 0.952 & 0.811 & 0.895 & 1.011 & 0.963 \\ \hline \end{tabular}
\end{table}