## 6 Solid Model
### Numerical Verification
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Model**} & \multicolumn{8}{c}{**Natural Frequency**} \\ \cline{3-10}  & & \(f_{1}\)–\(f_{2}\) & \(f_{3}\)–\(f_{4}\) & \(f_{5}\)–\(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\)–\(f_{12}\)–\(f_{11}\) & \(f_{12}\)–\(f_{13}\) \\ \hline \multirow{4}{*}{0.6} & 7-PL & 115.04 & 162.58 & 162.67 & 190.60 & 194.75 & 194.75 & 205.35 & 218.29 \\  & 3D * & 115.01 & 162.54 & 162.88 & 190.60 & 195.42 & 195.42 & 205.37 & 217.36 \\  & FSDT [23] & 115.40 & 162.85 & 165.07 & 193.28 & 193.28 & 196.66 & 210.82 & - \\ \hline \multirow{4}{*}{1} & 7-PL & 114.15 & 161.55 & 161.83 & 189.18 & 193.88 & 193.88 & 203.94 & 217.37 \\  & 3D * & 114.10 & 161.52 & 161.98 & 189.16 & 194.43 & 194.43 & 203.95 & 216.39 \\  & FSDT [23] & 114.49 & 161.95 & 164.01 & 192.32 & 192.32 & 195.22 & 209.38 & - \\ \hline \multirow{4}{*}{5} & 7-PL & 114.15 & 157.94 & 161.82 & 186.26 & 195.78 & 195.78 & 202.13 & 218.63 \\  & 3D * & 113.98 & 157.87 & 161.79 & 186.15 & 196.02 & 196.02 & 201.99 & 217.16 \\ \cline{1-1}  & FSDT [23] & 114.44 & 160.33 & 161.78 & 194.04 & 194.04 & 194.05 & 207.19 & - \\ \hline \hline \end{tabular}  
* 840,000 SOLID185 elements.  
\end{table} 
Table 3: Comparison for a parabolic shell with UT and SF boundary conditions (FG-I). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Model**} & \multicolumn{8}{c}{**Natural Frequency**} \\ \cline{3-10}  & & \(f_{1}\)–\(f_{2}\) & \(f_{3}\)–\(f_{4}\) & \(f_{5}\)–\(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\)–\(f_{11}\) & \(f_{12}\)–\(f_{13}\) \\ \hline \multirow{4}{*}{0.6} & 7-PL & 115.04 & 162.58 & 162.67 & 190.60 & 194.75 & 194.75 & 205.35 & 218.29 \\  & 3D * & 115.01 & 162.54 & 162.88 & 190.60 & 195.42 & 195.42 & 205.37 & 217.36 \\  & FSDT [23] & 115.40 & 162.85 & 165.07 & 193.28 & 193.28 & 196.66 & 210.82 & - \\ \hline \multirow{4}{*}{1} & 7-PL & 114.15 & 161.55 & 161.83 & 189.18 & 193.88 & 193.88 & 203.94 & 217.37 \\  & 3D * & 114.10 & 161.52 & 161.98 & 189.16 & 194.43 & 194.43 & 203.95 & 216.39 \\  & FSDT [23] & 114.49 & 161.95 & 164.01 & 192.32 & 192.32 & 195.22 & 209.38 & - \\ \hline \multirow{4}{*}{5} & 7-PL & 114.15 & 157.94 & 161.82 & 186.26 & 195.78 & 195.78 & 202.13 & 218.63 \\  & 3D * & 113.98 & 157.87 & 161.79 & 186.15 & 196.02 & 196.02 & 201.99 & 217.16 \\ \cline{1-1}  & FSDT [23] & 114.44 & 160.33 & 161.78 & 194.04 & 194.04 & 194.05 & 207.19 & - \\ \hline \hline \end{tabular}  
* 840,000 SOLID185 elements.  
\end{table} 
Table 4: Vibration mode comparison for a parabolic shell with UT and SF boundary conditions, \(n=5\) (FG-I). From the comparisons presented in Tables 6-8, the maximum values of relative error ranges from 3.22% to 4.34% for conical shells with an angle \(\alpha=30^{\circ}\), and the maximum value is obtained when \(n=5\). Regarding the conical shells with \(\alpha=60^{\circ}\), the maximum errors are found in a range from 2.77% to 5.14%, with the maximum value occurring when \(n=5\). Considering fixed values of the power-law exponent, the natural frequencies decrease as the value of ratio \(r_{0}/h_{m}\) increases, which is an expected behavior since the ratio values reported are related to thick and thin shells. In addition, the angle \(\alpha\) also has an influence on the natural frequencies as it can be seen for fixed ratios of \(r_{0}/h_{m}\), that is, lower values are obtained as the angle \(\alpha\) increases. In Table 9, several vibration modes for a conical shell with LVT, CF boundary conditions, and \(n=0.6\) are compared with those obtained by means of the solid elements (3D model). For both models, the vibration modes are very similar.