## 6 Solid Model
### Numerical Verification
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & & \multicolumn{3}{c}{\(\alpha=30^{\circ}\)} & & & & \multicolumn{3}{c}{\(\alpha=60^{\circ}\)} \\  & & \multicolumn{3}{c}{\(n\)} & & & & & \multicolumn{3}{c}{\(n\)} & \\ \cline{2-11}  & & & & & & & & & & & \\ \cline{2-11}  & & & & & & & & & & & \\ \cline{2-11}  & & & & & & & & & & & \\ \cline{2-11}  & & & & & & & & & & & \\ \hline \(f_{1}\)–\(f_{2}\) & 73.76 & 72.43 & 73.38 & 72.01 & 73.23 & 71.65 & 64.23 & 62.47 & 63.98 & 62.16 & 64.69 & 62.64 \\ \(f_{3}\)–\(f_{4}\) & 84.73 & 82.08 & 84.41 & 81.68 & 86.67 & 83.55 & 72.78 & 71.76 & 72.55 & 71.44 & 71.84 & 70.62 \\ \(f_{5}\)–\(f_{6}\) & 104.14 & 103.72 & 103.58 & 103.14 & 101.67 & 101.15 & 77.18 & 74.79 & 76.86 & 74.40 & 79.19 & 76.38 \\ \(f_{7}\) & 124.04 & 120.57 & 123.65 & 120.00 & 127.76 & 123.35 & 82.39 & 81.54 & 82.17 & 81.20 & 80.96 & 79.93 \\ \(f_{8}\) & 124.04 & 120.57 & 123.65 & 120.00 & 127.76 & 123.35 & 105.15 & 102.48 & 104.71 & 101.93 & 108.29 & 104.89 \\ \(f_{9}\) & 151.23 & 147.91 & 150.50 & 147.07 & 149.20 & 147.02 & 105.15 & 102.48 & 104.71 & 101.93 & 108.29 & 104.89 \\ \(f_{10}\) & 151.23 & 147.91 & 150.50 & 147.07 & 151.13 & 147.02 & 119.33 & 115.18 & 118.91 & 114.62 & 119.88 & 114.99 \\ \(f_{11}\) & 152.28 & 151.27 & 151.73 & 150.61 & 151.13 & 148.02 & 121.86 & 117.67 & 121.40 & 117.08 & 122.83 & 117.87 \\ \(f_{12}\) & 156.70 & 152.27 & 156.02 & 151.42 & 158.92 & 153.33 & 121.86 & 117.67 & 121.40 & 117.08 & 122.83 & 117.87 \\ Max. & & & & & & & & & & & \\ Error & 3.22\% & - & 3.34\% & - & 3.74\% & - & 3.61\% & - & 3.74\% & - & 4.25\% & - \\ \hline \end{tabular}
\end{table} 
Table 7: Natural frequencies for conical shells with LVT and CF boundary conditions (FG-II, \(r_{0}/h_{m}=5\), \(h_{m}=3/4\), \(L/h_{m}=10\), \(h_{1}/h_{0}=2\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & & \multicolumn{3}{c}{\(\alpha=30^{\circ}\)} & & & & \multicolumn{3}{c}{\(\alpha=60^{\circ}\)} \\  & & \multicolumn{3}{c}{\(n\)} & & & \multicolumn{3}{c}{\(n\)} & & \\ \cline{2-11}  & & **0.6** & & **1** & & **5** & & **0.6** & & **1** & & **5** \\ \cline{2-11}  & & **7 PL** & **3D** & **7 PL** & **3D** & **7 PL** & **3D** & **7 PL** & **3D** & **7 PL** & **3D** & **7 PL** & **3D** \\ \hline \(f_{1}\)–\(f_{2}\) & 52.69 & 51.04 & 52.43 & 50.75 & 52.89 & 51.00 & 45.53 & 43.8 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(n\)} \\ \cline{2-7}  & \multicolumn{2}{c}{**0.6**} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**5**} \\ \cline{2-7}  & **7 PL** & **3D** & **7 PL** & **3D** & **7 PL** & **3D** \\ \hline \(f_{1}\)–\(f_{2}\) & 64.07 & 63.72 & 63.28 & 62.98 & 62.17 & 61.75 \\ \(f_{3}\) & 123.97 & 121.23 & 122.50 & 120.98 & 119.37 & 121.87 \\ \(f_{4}\) & 129.65 & 121.23 & 129.78 & 120.98 & 134.06 & 124.17 \\ \(f_{5}\) & 129.65 & 126.14 & 129.78 & 124.89 & 134.06 & 124.17 \\ \(f_{6}\)–\(f_{7}\) & 191.89 & 191.01 & 190.34 & 189.49 & 186.81 & 185.65 \\ \(f_{8}\) & 203.99 & 203.77 & 202.46 & 202.23 & 197.49 & 197.18 \\ \(f_{9}\)–\(f_{10}\) & 217.19 & 204.85 & 217.00 & 204.04 & 220.95 & 205.82 \\ \(f_{11}\) & 313.85 & 294.01 & 314.23 & 293.37 & 320.46 & 299.19 \\ \(f_{12}\) & 313.85 & 294.01 & 314.23 & 293.37 & 323.41 & 299.19 \\ Max. Error & 6.95\% & - & 7.28\% & - & 8.10\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Natural frequencies for a cylindrical shell with LVT and CF boundary conditions (FG-II, \(r_{0}/h_{m}=2.5\), \(h_{m}=3/4\), \(L/h_{m}=10\), \(h_{1}/h_{0}=2\)).