## 3 Results and Discussion
### Validation  
Table 1 shows the comparison of a reduced case of current results with [24; 25; 26] for simply supported annular circular plates, which is the validation of the present results with the available results. Tables 2-4 show the influence of taper ratio \(\eta\), exponential coefficient of variation \(C_{e}\) and sinusoidal coefficient of variation \(C_{s}\) on the fundamental frequency parameter \(\lambda\). Two- and four-layered annular circular plates are considered with circumferential node number \(n=2\), ratio of thickness to radius of inner circle \(\gamma=0.05\) and radii ratio \(\beta=0.5\). It can be seen that the value of the frequency parameter increases with the increase in number of layers. Frequency increases as number of layers increases, showing that the structure's rigidity increases and flexibility decreases. Similarly, the frequency parameter value increases with the increase in thickness coefficients, evidently showing that the rigidity of the structure increases and flexibility decreases. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Circumferential** & & & & & \\
**Node Number** & **Nodes** & **Present** & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** (2008)** \\
**\(n\)** & & & & & \\ \hline
1 & 1 & 1583 & 1584.65 & 1583 & 1551 \\  & 2 & 5436 & 5437.61 & 5433 & 5328 \\  & 3 & 11,494 & 11,495.1 & 11,478 & 11,283 \\
2 & 1 & 2307 & 2308.11 & 2306 & 2260 \\  & 2 & 6474 & 6475.13 & 6468 & 6348 \\  & 3 & 12,652 & 12,653.2 & 12,632 & 12,428 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of natural frequency parameter (rad/s) of present study \(\lambda=\omega\ell\sqrt{\frac{t_{0}}{A_{11}}}\) with [24; 25; 26] of simply supported annular circular plates \(\beta=1/6\ and\ \gamma=1/60\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & \\ \cline{2-4} \(\eta\) & \(45^{0}\)/\(-30^{0}\)/\(30^{0}\)/\(-45^{0}\) & \(45^{0}\)/\(-60^{0}\)/\(60^{0}\)/\(-45^{0}\) & \(30^{0}\)/\(-30^{0}\) \\  & **(KGE/EGE/EGE/KGE)** & **(KGE/EGE/EGE/KGE)** & **(EGE/EGE)** \\ \hline
0.5 & 0.71949 & 0.643041 & 0.402384 \\
0.7 & 0.728127 & 0.672854 & 0.421916 \\
0.9 & 0.734973 & 0.690198 & 0.447392 \\
1.1 & 0.739494 & 0.700368 & 0.464501 \\
1.3 & 0.741874 & 0.706052 & 0.476395 \\
1.5 & 0.74287 & 0.709517 & 0.484848 \\
1.7 & 0.742917 & 0.711673 & 0.490948 \\
1.9 & 0.742298 & 0.711698 & 0.495378 \\
2.1 & 0.741207 & 0.710538 & 0.498588 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relation of linear thickness variation \(\eta\) and the fundamental frequency parameter \(\lambda\). \(n=2\), \(\gamma=0.05\) and \(\beta=0.5\). However, Table 5 depicts the comparison of four-layered annular circular plates, concluding that different lamination schemes and material combinations definitely affect the frequency of plates. It is evident that ply orientation \(30^{0}/-30^{0}/30^{0}/-30^{0}\) shows the least frequency, proving that the structure is most flexible using this ply orientation, and \(45^{0}/-45^{0}/45^{0}/-45^{0}\) shows the maximum frequency, which proves that the structure is least flexible using this ply orientation. Table 6 shows the elastic properties of materials used. \begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{3}{c}{\(\lambda\)} \\ \cline{2-4} \(C_{s}\) & \(45^{0}/-30^{0}/30^{0}/-45^{0}\) & \(45^{0}/-60^{0}/60^{0}/-45^{0}\) & \(30^{0}/-30^{0}\) \\  & **(KGE/EGE/EGE/KGE)** & **(KGE/EGE/EGE/KGE)** & **(EG/EGE/EGE)** \\ \hline \(-0.5\) & 0.673905 & 0.637247 & 0.43449 \\ \(-0.3\) & 0.711461 & 0.677643 & 0.452291 \\ \(-0.1\) & 0.731285 & 0.692456 & 0.457029 \\ \(0.1\) & 0.74213 & 0.696611 & 0.455322 \\ \(0.3\) & 0.748382 & 0.693315 & 0.45001 \\ \(0.5\) & 0.751603 & 0.685991 & 0.442494 \\ \hline \end{tabular}
\end{table} 
Table 4: Relation of sinusoidal thickness variation \(C_{s}\) and the fundamental frequency parameter \(\lambda\). \(n=2\), \(\gamma=0.05\) and \(\beta=0.5\).