# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Numeric Validation
#### 6.3.3 Fraction of Cylinder
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Dimensionless** \\ **Frequency** \\ **Parameter** \\ \end{tabular} } & \multicolumn{6}{c}{**r/\(h\)**} \\ \cline{2-10}  & \multicolumn{2}{c}{**100**} & \multicolumn{2}{c}{**50**} & \multicolumn{2}{c}{**20**} & \multicolumn{2}{c}{**10**} & \multicolumn{2}{c}{**5**} \\ \cline{2-10}  & **7-PL** & **3D** & **7-PL** & **3D** & **7-PL** & **3D** & **7-PL** & **3D** \\  & **16** & **645,344** & **16** & **703,360** & **16** & **603,680** & **16** & **639,576** & **16** & **743,400** \\ \hline \multirow{2}{*}{\(\Theta_{1}\)} & 0.0170 & 0.0164 & 0.0240 & 0.0222 & 0.0464 & 0.0445 & 0.0136 & 0.0131 & 0.0186 & 0.0169 \\  & & (4.18) & & (8.17) & & (4.34) & & (4.37) & & (10.02) \\ \hline \multirow{2}{*}{\(\Theta_{2}\)} & 0.0296 & 0.0291 & 0.0416 & 0.0411 & 0.0496 & 0.0457 & 0.0414 & 0.0375 & 0.0530 & 0.0520 \\  & & (1.79) & & (1.34) & & (8.51) & & (10.30) & & (1.94) \\ \hline \multirow{2}{*}{\(\Theta_{3}\)} & 0.0527 & 0.0526 & 0.0742 & 0.0721 & 0.1381 & 0.1344 & 0.0600 & 0.0588 & 0.0864 & 0.0809 \\  & & (0.35) & & (2.97) & & (2.73) & & (2.04) & & (6.79) \\ \hline \multirow{2}{*}{\(\Theta_{4}\)} & 0.0580 & 0.0575 & 0.0832 & 0.0820 & 0.1721 & 0.1663 & 0.0670 & 0.0661 & 0.0901 & 0.0834 \\  & & (0.95) & & (1.40) & & (3.44) & & (1.41) & & (8.04) \\ \hline \multirow{2}{*}{\(\Theta_{5}\)} & 0.0638 & 0.0633 & 0.0988 & 0.0969 & 0.1855 & 0.1834 & 0.1025 & 0.0993 & 0.1877 & 0.1802 \\  & & (0.75) & & (1.89) & & (1.18) & & (3.26) & & (4.15) \\ \hline \multirow{2}{*}{\(\Theta_{6}\)} & 0.0883 & 0.0888 & 0.1646 & 0.1626 & 0.1939 & 0.1937 & 0.1636 & 0.1624 & 0.2112 & 0.2074 \\  & & (0.53) & & (1.23) & & (0.08) & & (0.74) & & (1.85) \\ \hline \multirow{2}{*}{\(\Theta_{7}\)} & 0.1064 & 0.1060 & 0.1709 & 0.1676 & 0.2830 & 0.2789 & 0.1791 & 0.1761 & 0.2493 & 0.2502 \\  & & (0.45) & & (2.01) & & (1.44) & & (1.75) & & (0.36) \\ \hline \multirow{2}{*}{\(\Theta_{8}\)} & 0.1116 & 0.1111 & 0.1711 & 0.1697 & 0.2977 & 0.2933 & 0.2427 & 0.2428 & 0.3068 & 0.3066 \\  & & (0.42) & & (0.79) & & (1.51) & & (0.06) & & (0.05) \\ \hline \multirow{2}{*}{\(\Theta_{9}\)} & 0.1425 & 0.1423 & 0.1946 & 0.1943 & 0.3336 & 0.3295 & 0.2751 & 0.2714 & 0.3272 & 0.3179 \\  & & (0.16) & & (0.17) & & (1.24) & & (1.36) & & (2.91) \\ \hline \multirow{2}{*}{\(\Theta_{10}\)} & 0.1621 & 0.1636 & 0.2266 & 0.2260 & 0.4195 & 0.4119 & 0.2804 & 0.2781 & 0.3761 & 0.3702 \\  & & (0.90) & & (0.28) & & (1.85) & & (0.82) & & (1.60) \\ \hline \multirow{2}{*}{\(\Theta_{11}\)} & 0.1644 & 0.1640 & 0.2446 & 0.2412 & 0.4703 & 0.4642 & 0.2941 & 0.2884 & 0.4884 & 0.4765 \\  & & (0.26) & & (1.41) & & (1.32) & & (1.95) & & (2.51) \\ \hline \multirow{2}{*}{\(\Theta_{12}\)} & 0.1651 & 0.1654 & 0.2535 & 0.2512 & 0.4984 & 0.4924 & 0.3065 & 0.3064 & 0.5238 & 0.5262 \\  & & (0.20) & & (0.92) & & (1.22) & & (0.03) & & (0.45) \\ \hline \multirow{2}{*}{\(\Theta_{13}\)} & 0.1933 & 0.1931 & 0.2626 & 0.2602 & 0.5387 & 0.5348 & 0.3834 & 0.3772 & 0.5518 & 0.5347 \\  & & (0.06) & & (0.93) & & (0.73) & & (1.64) & & (3.20) \\ \hline \multirow{2}{*}{\(\Theta_{14}\)} & 0.2118 & 0.2120 & 0.3215 & 0.3170 & 0.5684 & 0.5610 & 0.3908 & 0.3859 & 0.5821 & 0.5688 \\  & & (0.10) & & (1.40) & & (1.33) & & (1.26) & & (2.33) \\ \hline \multirow{2}{*}{\(\Theta_{15}\)} & 0.2252 & 0.2254 & 0.3467 & 0.3431 & 0.6135 & 0.6133 & 0.4133 & 0.4099 & 0.6773 & 0.6601 \\  & & (0.08) & & (1.05) & & (0.03) & & (0.84) & & (2.59) \\ \hline \hline \end{tabular}
\end{table} 
Table 20: Dimensionless parameter \(\Theta\) for a quarter of cylinder with linear thickness variation. Finally, based on the results presented in Tables 17-21, the following limits for the presented formulation are set:  
Truncated hemisphere: \(20<r/h_{m}\leq 2.5\),  
Cylinder: \(100\leq r/h\leq 5\),  
Fraction of cylinder: \(4<L/r\leq 10\). These parameter ranges demonstrate the capability of the spectral element presented to model thin and moderate thick shells with constant and variable thickness.