## 6 Solid Model
### Numerical Verification
\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 \(f_{1}\)–\(f_{2}\) & 58.50 & 56.58 & 58.17 & 56.21 & 58.80 & 56.58 \\ \(f_{3}\)–\(f_{4}\) & 79.10 & 78.91 & 78.44 & 78.29 & 76.87 & 76.63 \\ \(f_{5}\)–\(f_{6}\) & 96.15 & 92.18 & 95.92 & 91.81 & 99.21 & 94.57 \\ \(f_{7}\) & 125.24 & 125.97 & 124.11 & 124.94 & 120.98 & 121.83 \\ \(f_{8}\)–\(f_{9}\) & 157.38 & 153.83 & 156.55 & 152.89 & 156.72 & 152.33 \\ \(f_{10}\)–\(f_{11}\) & 162.94 & 156.82 & 162.61 & 156.20 & 168.25 & 160.75 \\ \(f_{12}\)–\(f_{13}\) & 179.49 & 173.24 & 178.90 & 172.36 & 182.66 & 174.78 \\ Max. Error & 4.31\% & - & 4.48\% & - & 4.91\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Natural frequencies for a cylindrical shell 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} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Vibration Mode**} \\ \cline{2-7}  & \(f_{1}\)–\(f_{2}\) & \(f_{3}\)–\(f_{4}\) & \(f_{5}\)–\(f_{6}\) & \(f_{7}\) & \(f_{8}\)–\(f_{9}\) & \(f_{10}\) & \(f_{11}\)–\(f_{12}\) \\ \hline \multirow{4}{*}{7-PL} & 64.23 Hz & 72.78 Hz & 77.18 Hz & 82.39 Hz & 105.15 Hz & 119.33 Hz & 121.86 Hz \\  & 64.24 Hz & 72.78 Hz & 77.18 Hz & 82.39 Hz & 105.15 Hz & 119.33 Hz & 121.86 Hz \\ \cline{1-1}  & 62.47 Hz & 71.76 Hz & 74.79 Hz & 81.54 Hz & 102.48 Hz & 115.18 Hz & 117.67 Hz \\ \cline{1-1}  & 62.47 Hz & 71.76 Hz & 74.79 Hz & 81.54 Hz & 102.48 Hz & 115.18 Hz & 117.67 Hz \\ \cline{1-1}  & 3D & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 9: Vibration modes for a conical shell with LVT, CF boundary conditions and \(\alpha=60^{\circ}\), \(n=0.6\) (FG-II, \(r_{0}/h_{m}=5\), \(h_{m}=3/4\), \(L/h_{m}=10\), \(h_{1}/h_{0}=2\)). For the natural frequencies presented in Tables 10-12, corresponding to cylindrical shells, the range for the maximum errors is between 4.31% and 8.10%, with the maximum value occurring in the case when \(r_{0}/h_{m}=2.5\) and \(n=5\). Note that the aforementioned radius-to-thickness ratio describes a very thick shell. However, for greater values of the ratio, the value of errors are below 4.91%. For illustration purposes, a comparison of several vibration modes of a cylindrical shell with LVT, CF boundary conditions, and \(n=5\) is presented in Table 13.  
Natural frequencies for parabolic shells with LVT and under SF boundary conditions are presented in Tables 14-16. The volume distribution of the top constituent follows the expression of Equation (12), i.e., FG-I. From the comparisons presented for parabolic shells in Tables 14-16, the maximum values of relative error are within the range of 3.18% to 4.59%. In these comparisons, note that for a thin shell (e.g., \(F/h_{m}=20\)), the maximum error occurs when \(n=0.6\), conversely to minor values of the ratio. For a fixed value of the volume fraction exponent, a decrease in values of the natural frequencies is observed as the ratio \(F/h_{m}\) increases. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{5}{c}{**Vibration Mode**} \\ \cline{2-7}  & \(f_{1}\)–\(f_{2}\) & \(f_{3}\)–\(f_{4}\) & \(f_{5}\)–\(f_{6}\) & \(f_{7}\) & \(f_{8}\)–\(f_{9}\) & \(f_{10}\) \\ \hline \multirow{4}{*}{7-PL} & 58.80 Hz & 76.87 Hz & 99.21 Hz & 120.98 Hz & 156.72 Hz & 168.25 Hz \\  & & & & & & \\ \cline{1-1} \cline{2-7}  & 56.58 Hz & 76.63 Hz & 94.57 Hz & 121.83 Hz & 152.33 Hz & 160.75 Hz \\ \cline{1-1} \cline{2-7}
3D & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Vibration modes for a cylindrical shell with LVT and CF boundary conditions, \(n=5\) (FG-II, \(r_{0}/h_{m}=5\), \(h_{m}=3/4\), \(L/h_{m}=10\), \(h_{1}/h_{0}=2\)).