## 6 Numerical Results and Discussion
For each structure analyzed here (i.e., spherical, hyperbolic paraboloidal, and circular cylindrical shell), we define the dimensionless frequencies as a function of \(\beta\) for different wave numbers along the \(\alpha_{1}\) and \(\alpha_{2}\) directions, as well as for  
According to Figures 4-6, it is worth noting that the dimensionless frequencies increase in value for shells made of pure ceramic (i.e., \(p=0\)). This is due to the fact that ceramic (Al\({}_{2}\)O\({}_{3}\)) features a higher Young's modulus than metal (SUS304). The most important result that can be inferred from Figures 4-6 is that an increased porosity volume fraction leads to an increased dimensionless frequency for a rich ceramic limit. The variation of the dimensionless frequency parameters for mode (1,1) with respect to the power-law index \(p\) for three various values of porosity volume fraction (i.e., \(\beta=0\), \(\beta=0.1\), and \(\beta=0.3\)) in FGM-types 1 and 2, are plotted in Figures 7-9 for a spherical, hyperbolic paraboloidal, and circular cylindrical shell, respectively. As shown in Figures 7-9, the dimensionless frequency increases monotonically with the porosity volume fraction \(\beta\), but decreases for an increasing power-law index. For instance, between \(p=0\) and \(p\approx 1\), the dimensionless frequencies increase with an increasing porosity volume fraction \(\beta\), whereas after \(p=1\) they decrease with the increase in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**FGM-1 Power-Law Distribution**} \\ \hline
**Porosity** & **Mode (\(m,n\))** & \(p=0\) & \(p=0.3\) & \(p=0.6\) & \(p=1\) & \(p=5\) \\ \hline \multirow{4}{*}{(\(\beta=0\))} & (1,1) & 48.4942 & 40.8660 & 37.0253 & 34.1183 & 27.7268 \\  & (2,2) & 53.0808 & 44.3991 & 40.0824 & 36.8778 & 30.1956 \\  & (3,3) & 65.9875 & 54.7970 & 49.3207 & 45.3731 & 37.7164 \\  & (4,4) & 91.8995 & 76.1928 & 68.5752 & 63.1903 & 53.1539 \\ \hline \multirow{4}{*}{(\(\beta=0.1\))} & (1,1) & 50.7297 & 41.7402 & 37.3740 & 34.1238 & 27.0979 \\  & (2,2) & 55.500 & 45.3040 & 40.4061 & 36.8257 & 29.4760 \\  & (3,3) & 68.9586 & 55.8377 & 49.6220 & 45.2055 & 36.7869 \\  & (4,4) & 95.9954 & 77.5601 & 68.8955 & 62.8567 & 51.8369 \\ \hline \multirow{4}{*}{(\(\beta=0.2\))} & (1,1) & 53.8218 & 42.8723 & 37.8124 & 34.1267 & 26.3250 \\  & (2,2) & 58.8601 & 46.4870 & 40.8218 & 36.7640 & 28.5992 \\  & (3,3) & 73.1141 & 57.2240 & 50.0319 & 45.0178 & 35.6715 \\  & (4,4) & 101.7600 & 79.4159 & 69.3675 & 62.4923 & 50.2788 \\ \hline \multicolumn{6}{c}{**FGM-2 Power-law distribution**} \\ \hline
**Porosity** & **Mode (\(m,n\))** & \(p=0\) & \(p=0.3\) & \(p=0.6\) & \(p=1\) & \(p=5\) \\ \hline \multirow{4}{*}{(\(\beta=0\))} & (1,1) & 48.4942 & 41.0805 & 37.3222 & 34.4526 & 27.9614 \\  & (2,2) & 53.0808 & 45.2115 & 41.2050 & 38.1398 & 31.0770 \\  & (3,3) & 65.9875 & 56.2839 & 51.3764 & 47.6810 & 39.3077 \\  & (4,4) & 91.8995 & 78.1084 & 71.2254 & 66.1625 & 55.1808 \\ \hline \multirow{4}{*}{(\(\beta=0.1\))} & (1,1) & 50.7297 & 41.9677 & 37.6866 & 34.4744 & 27.3442 \\  & (2,2) & 55.500 & 46.1629 & 41.5841 & 38.1448 & 30.3976 \\  & (3,3) & 68.9586 & 57.4109 & 51.7815 & 47.6209 & 38.4507 \\  & (4,4) & 95.9954 & 79.5895 & 71.6842 & 65.9730 & 53.9575 \\ \hline \multirow{4}{*}{(\(\beta=0.2\))} & (1,1) & 53.8218 & 43.1169 & 38.1453 & 34.4984 & 26.5867 \\  & (2,2) & 58.8601 & 47.4074 & 42.0719 & 38.1571 & 29.5735 \\ \cline{1-1}  & (3,3) & 73.1141 & 58.9107 & 52.3262 & 47.5720 & 37.4298 \\ \cline{1-1}  & (4,4) & 101.760 & 81.5943 & 72.3353 & 65.7942 & 52.5205 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dimensionless frequency of simply-supported (SSSS) functionally graded porous circular cylindrical shell as functions of the power-law index and porosity volume fraction. Figure 4: _Cont._Figure 4: Dimensionless frequency of FG porous spherical shell against different power-law indices and porosity volume fractions distributions: (**a**) FGM-type 1; (**b**) FGM-type 2.