## 6 Solid Model
### Numerical Verification
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{4}{c}{**\(n\)**} \\ \cline{2-7}  & \multicolumn{2}{c}{**0.6**} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**5**} \\ \cline{2-7}  & \multicolumn{2}{c}{**7 PL**} & \multicolumn{2}{c}{**3D**} & \multicolumn{2}{c}{**7 PL**} & \multicolumn{2}{c}{**3D**} & \multicolumn{2}{c}{**7 PL**} & \multicolumn{2}{c}{**3D**} \\ \hline \(f_{1}\)–\(f_{2}\) & 119.48 & 115.77 & 118.64 & 114.89 & 118.94 & 116.41 \\ \(f_{3}\)–\(f_{4}\) & 122.05 & 119.91 & 120.77 & 118.66 & 120.81 & 116.84 \\ \(f_{5}\)–\(f_{6}\) & 172.76 & 172.33 & 171.76 & 171.22 & 173.29 & 172.75 \\ \(f_{7}\)–\(f_{8}\) & 208.23 & 210.71  
Lastly, from the natural frequencies of hemispherical shells presented in Tables 18-21, it is noted that the maximum relative errors are above the values obtained in the previous comparisons reported here; they are found in a range between 4.27% and 7.95%. The higher values of maximum relative errors occur when a very thick shell is analyzed, i.e., \(r/h_{m}=2.5\). It must be mentioned that the majority of the maximum relative errors are above 5%. In addition, Table 22 presents comparisons of the vibration modes obtained using both models (7-PL and 3D) for a hemispherical shell with LVT, CF boundary conditions, and \(n=5\). The similarities between models for vibration modes and natural frequencies are noted in these comparisons. In general, the behavior of natural frequencies is similar to that observed when analyzing thin and thick shells, that is, an increase in the frequencies value is observed when the shells become thicker. In addition, the study cases analyzed in this section demonstrate the behavior of the present model. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & & & \(n\) & & \\ \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 \(f_{1}\)–\(f_{2}\) & 213.69 & 204.60 & 212.06 & 203.05 & 213.72 & 203.73 \\ \(f_{3}\)–\(f_{4}\) & 278.82 & 261.85 & 276.89 & 259.97 & 284.29 & 266.47 \\ \(f_{5}\)–\(f_{6}\) & 348.85 & 347.55 & 346.21 & 344.94 & 340.23 & 338.50 \\ \(f_{7}\)–\(f_{8}\) & 428.30 & 410.50 & 425.27 & 407.47 & 435.41 & 416.22 \\ \(f_{9}\) & 478.89 & 465.83 & 474.40 & 461.85 & 471.54 & 457.11 \\ \(f_{10}\)–\(f_{11}\) & 496.81 & 482.31 & 492.13 & 478.18 & 490.25 & 474.15 \\ \(f_{12}\) & 541.01 & 522.79 & 536.21 & 518.42 & 538.24 & 517.91 \\ Max. Error & 6.08\% & - & 6.11\% & - & 6.27\% & - \\ \hline \end{tabular}
\end{table} 
Table 20: Natural frequencies for a hemispherical shell with LVT and CF boundary conditions (FG-I, \(r_{0}/h_{m}=10\), \(r_{0}=2\) m, \(\beta=2\), \(\varphi_{0}=\pi/2\), \(\varphi_{1}=\pi/6\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & & & \(n\) & & \\ \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 \(f_{1}\)–\(f_{2}\) & 166.08 & 156.34 & 165.07 & 155.34 & 165.93 & 159.11 \\ \(f_{3}\)–\(f_{4}\) & 168.36 & 165.52 & 167.19 & 164.37 & 169.40 & 162.67 \\ \(f_{5}\)–\(f_{6}\) & 258.15 & 247.58 & 256.59 & 246.01 & 263.57 & 252.32 \\ \(f_{7}\)–\(f_{8}\) & 330.83 & 331.02 & 328.56 & 328.75 & 321.87 & 321.89 \\ \(f_{9}\)–\(f_{10}\) & 368.54 & 359.67 & 366.25 & 357.34 & 375.09 & 365.31 \\ \(f_{11}\) & 422.31 & 419.84 & 419.00 & 416.72 & 412.14 & 409.15 \\ \(f_{12}\)–\(f_{13}\) & 437.11 & 434.21 & 433.59 & 430.96 & 426.26 & 422.88 \\ Max. Error & 5.87\% & - & 5.89\% & - & 4.27\% & - \\ \hline \end{tabular}
\end{table} 
Table 21: Natural frequencies for a hemispherical shell with LVT and CF boundary conditions (FG-I, \(r_{0}/h_{m}=20\), \(r_{0}=2\) m, \(\beta=2\), \(\varphi_{0}=\pi/2\), \(\varphi_{1}=\pi/6\)).