# 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** \\ \end{tabular} } & \multicolumn{8}{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** & **7-PL** & **3D** \\  & **25** & **847,800** & **25** & **840,000** & **25** & **628,000** & **16** & **642,560** & **16** & **628,000** \\ \hline \multirow{2}{*}{\(\Theta_{1}\)} & **0.0795** & **0.0798** & **0.1112** & **0.1106** & **0.1803** & **0.1765** & **0.2180** & **0.2131** & **0.3171** & **0.3006** \\  & & (0.37) & & (0.54) & & (2.18) & & (2.32) & & (5.47) \\ \hline \multirow{2}{*}{\(\Theta_{2}\)} & **0.0949** & **0.0960** & 0.1247 & 0.1241 & **0.1804** & **0.1789** & **0.3172** & **0.3068** & **0.3476** & **0.3443** \\  & & (1.15) & & (0.55) & & (0.84) & & (3.38) & & (0.98) \\ \hline \multirow{2}{*}{\(\Theta_{3}\)} & **0.0967** & **0.0966** & 0.1248 & 0.1241 & **0.2784** & **0.2723** & **0.3296** & **0.3286** & 0.5299 & 0.5334 \\  & & (0.07) & & (0.57) & & (2.21) & & (0.31) & & (0.66) \\ \hline \multirow{2}{*}{\(\Theta_{4}\)} & **0.1269** & **0.1290** & **0.1663** & **0.1660** & **0.3221** & **0.3217** & 0.5305 & 0.5167 & **0.5843** & **0.5527** \\  & & (1.61) & & (0.16) & & (0.10) & & (2.66) & & (5.71) \\ \hline \multirow{2}{*}{\(\Theta_{5}\)} & **0.1635** & **0.1634** & **0.1763** & **0.1756** & **0.4212** & **0.4124** & 0.5324 & 0.5167 & **0.7913** & **0.7609** \\  & & (0.06) & & (0.41) & & (2.13) & & (3.03) & & (3.99) \\ \hline \multirow{2}{*}{\(\Theta_{6}\)} & **0.1679** & **0.1707** & 0.2460 & 0.2450 & **0.4327** & **0.4274** & 0.5324 & 0.5314 & 0.8315 & 0.8288 \\  & & (1.61) & & (0.40) & & (1.24) & & (0.20) & & (0.32) \\ \hline \multirow{2}{*}{\(\Theta_{7}\)} & **0.2064** & **0.2057** & 0.2493 & 0.2450 & **0.4774** & **0.4690** & **0.6038** & **0.5943** & **0.8529** & **0.8395** \\  & & (0.32) & & (1.72) & & (1.78) & & (1.59) & & (1.59) \\ \hline \multirow{2}{*}{\(\Theta_{8}\)} & **0.2103** & **0.2096** & **0.2857** & **0.2840** & **0.5210** & **0.5182** & **0.6285** & **0.6125** & **0.9819** & **0.9315** \\  & & (0.34) & & (0.62) & & (0.52) & & (2.61) & & (5.40) \\ \hline \multirow{2}{*}{\(\Theta_{9}\)} & **0.2176** & **0.2190** & **0.2967** & **0.2940** & 0.5306 & 0.5309 & **0.7838** & **0.7790** & 0.9848 & 0.9516 \\  & & (0.62) & & (0.90) & & (0.04) & & (0.62) & & (3.50) \\ \hline \multirow{2}{*}{\(\Theta_{10}\)} & **0.2421** & **0.2416** & **0.3182** & **0.3181** & 0.5957 & 0.5826 & **0.8079** & **0.7851** & 1.0035 & 0.9783 \\  & & (0.22) & & (0.04) & & (2.24) & & (2.90) & & (2.58) \\ \hline \multirow{2}{*}{\(\Theta_{11}\)} & **0.2438** & **0.2426** & **0.3397** & **0.3275** & 0.5999 & 0.5826 & 0.8160 & 0.8030 & 1.0035 & 1.0102 \\  & & (0.49) & & (3.73) & & (2.96) & & (1.62) & & (0.66) \\ \hline \multirow{2}{*}{\(\Theta_{12}\)} & **0.2789** & **0.2737** & **0.3442** & **0.3432** & **0.6187** & **0.6068** & 0.8266 & 0.8030 & 1.0174 & 1.0102 \\  & & (1.93) & & (0.31) & & (1.97) & & (2.94) & & (0.71) \\ \hline \multirow{2}{*}{\(\Theta_{13}\)} & **0.2976** & **0.2936** & 0.3586 & 0.3547 & **0.7530** & **0.7517** & 0.8266 & 0.8148 & 1.0174 & 1.1043 \\  & & (1.38) & & (1.09) & & (0.17) & & (1.45) & & (7.87) \\ \hline \multirow{2}{*}{\(\Theta_{14}\)} & **0.3170** & **0.3169** & 0.3618 & 0.3547 & **0.7619** & **0.7530** & 0.9636 & 0.9608 & 1.1225 & 1.3225 \\  & & (0.02) & & (2.00) & & (1.18) & & (0.29) & & (15.12) \\ \hline \multirow{2}{*}{\(\Theta_{15}\)} & **0.3266** & **0.3263** & 0.4475 & 0.4222 & 0.7721 & 0.7599 & **0.9875** & **0.9866** & 1.3384 & 1.3225 \\  & & (0.10) & & (5.98) & & (1.62) & & (0.09) & & (1.20) \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Dimensionless natural frequencies parameter \(\Theta\) for cylinders with linear thickness variation and CF boundary conditions, \(L/r=2\).