## 3 Results and Discussion
### Validation
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{3}{c}{\(\lambda\)} \\ \cline{2-4} \(C_{e}\) & \(45^{0}/-30^{0}/30^{0}/-45^{0}\) & \(45^{0}/-60^{0}/60^{0}/-45^{0}\) & \(30^{0}/-30^{0}\) \\  & **(KGE/EGE/EGE/KGE)** & **(KGE/EGE/EGE/KGE)** & **(EG/EGE/EGE)** \\ \hline \(-0.2\) & 0.745785 & 0.713475 & 0.490829 \\ \(-0.1\) & 0.742443 & 0.70506 & 0.472436 \\ \(0\) & 0.737531 & 0.695956 & 0.456753 \\ \(0.1\) & 0.732985 & 0.687592 & 0.44392 \\ \(0.2\) & 0.729205 & 0.68026 & 0.433387 \\ \hline \end{tabular}
\end{table} 
Table 3: Relation of exponential thickness variation \(C_{e}\) and the fundamental frequency parameter \(\lambda\). \(n=2\), \(\gamma=0.05\) and \(\beta=0.5\). \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{3}{c}{\(\lambda\)} \\ \cline{2-4} \(\eta\) & \(45^{0}/-30^{0}/30^{0}/-45^{0}\) & \(30^{0}/-30^{0}/-30^{0}\) & \(45^{0}/-60^{0}/60^{0}/-45^{0}\) & \(45^{0}/-45^{0}/45^{0}/-45^{0}\) \\  & **(KGE/EGE/EGE/KGE)** & **(AGE/EGE/EGE/KGE)** & **(KGE/EGE/KGE)** \\ \hline \(0.5\) & 0.71949 & 0.487129 & 0.643041 & 0.84425 \\ \(0.7\) & 0.728127 & 0.503465 & 0.672854 & 0.848355 \\ \(0.9\) & 0.734973 & 0.51629 & 0.690198 & 0.854028 \\ \(1.1\) & 0.739494 & 0.525374 & 0.700368 & 0.855654 \\ \(1.3\) & 0.741874 & 0.531657 & 0.706052 & 0.855741 \\ \(1.5\) & 0.74287 & 0.535933 & 0.709517 & 0.854683 \\ \(1.7\) & 0.742917 & 0.538766 & 0.711673 & 0.852878 \\ \(1.9\) & 0.742298 & 0.540549 & 0.711698 & 0.850596 \\ \(2.1\) & 0.741207 & 0.54156 & 0.710538 & 0.848079 \\ \hline \end{tabular}
\end{table} 
Table 5: Relation of linear thickness variation \(\eta\) and the fundamental frequency parameter \(\lambda\). \(n=2\), \(\gamma=0.05\) and \(\beta=0.5\). Figure 3 shows the variation of the fundamental frequency parameter with respect to the circumferential node number of two-layered annular circular plates for three thickness variations \(C_{s}=0.5\), \(C_{e}=0.2\) and \(\eta=1.5\) under S-S boundary conditions. It can be seen that at circumferential node number \(n=2\) frequency is lowest, which shows that the structure is least rigid and most flexible at this point. As circumferential node number increases, frequency value also increases, which shows that rigidity increases but flexibility decreases. Figure 4 narrates the angular frequency parameter \(\omega\) with respect to radii ratio \(\beta\) for two-layered annular circular plates with different materials. It can be seen that angular frequency increases gradually from \(\beta=0.1\) _to_ 0.5 but strictly increases afterwards. It shows that with the increase in \(\beta\), rigidity increases but flexibility decreases. Figure 5 shows the trend of the angular frequency parameter and radii ratio of four-layered annular plates after fixing thickness variations \(C_{e}=0.2\), \(C_{s}=0.5\) and \(\eta=1.5\). Moreover, a relationship between the angular frequency parameter and the radii ratio of four-layered annular plates after fixing thickness variation \(C_{e}=0.2\) can be seen in Figure 6. However, Figure 7 relates the angular frequency parameter and the radii ratio of four-layered annular plates with different lamination schemes and lamination materials, concluding that both of these factors definitely affect the angular frequency parameter value. The first three mode shapes of vibration of annular circular plates are presented in Figure 8a-c. The shape is not circular because only a small portion of the plate is considered so that the mode shapes can be visualized clearly. Normalization is conducted with respect to the maximum transverse displacement W. As expected, the transverse displacements are most dominant.