# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Numeric Validation
#### 6.2.2 Shells
A comparison of the natural frequencies for cylinders with linear and linear symmetric thickness variations subjected to CF boundary conditions is presented in Table 15. For the linear thickness variation, the relative errors are smaller than 2.5%, when the results obtained using the present formulation are compared with those reported by [PERSON] & [PERSON] [30], and below 8% when the comparison is made with solid elements. On the other hand, for the cylinder with linear symmetric thickness variation, the results obtained with the present formulation have relative errors lower than 5%, except for the natural frequency \(f_{19}\), when those are compared with the results presented by [PERSON] & [PERSON] [30]. The frequencies obtained with solid elements present errors below 4.4%. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{6}{c}{**Formulation**} \\ \cline{2-9}  & \multicolumn{3}{c}{**Constant Thickness \(\boldsymbol{r/h}=\boldsymbol{5}\)**} & \multicolumn{3}{c}{**Linear Variation Thickness \(\boldsymbol{r/h}_{m}=\boldsymbol{5}\)**} \\ \cline{2-9}  & **Present** & **Kang**[51] & **3D07942** & **Present** & **Kang**[51] & **Bacciocchi**[35] & **3D0 (FSDT)** & **855,000** \\ \hline \multirow{3}{*}{\(\Lambda_{1}\)} & **0.3293** & 0.3241 & **0.3242** & **0.3415** & 0.3485 & 0.3491 & **0.3487** \\  & & (1.60) & (1.58) & & (2.00) & (2.17) & (2.05) \\ \hline \multirow{3}{*}{\(\Lambda_{2}\)} & **0.8507** & 0.8379 & **0.8380** & **0.7442** & 0.7052 & 0.7041 & **0.7054** \\  & & (1.53) & (1.51) & & (5.54) & (5.70) & (5.51) \\ \hline \multirow{3}{*}{\(\Lambda_{3}\)} & **0.8568** & 0.8397 & **0.8396** & **0.9465** & 0.9343 & 0.9364 & **0.9345** \\  & & (2.03) & (2.05) & (1.31) & (1.08) & (1.29) \\ \hline \multirow{3}{*}{\(\Lambda_{4}\)} & 1.5180 & 1.4750 & 1.4753 & 1.5319 & 1.5360 & 1.5298 & 1.5357 \\  & & (2.91) & (2.89) & & (0.26) & (0.14) & (0.25) \\ \hline \multirow{3}{*}{\(\Lambda_{5}\)} & 1.5245 & - & 1.4753 & **1.5688** & 1.5680 & 1.5685 & **1.5681** \\  & & & (3.33) & & (0.05) & (0.02) & (0.05) \\ \hline \multirow{3}{*}{\(\Lambda_{6}\)} & 1.5245 & 1.5160 & 1.5160 & **1.7280** & 1.6600 & 1.6649 & **1.6666** \\  & & (0.56) & (0.56) & & (4.10) & (3.79) & (3.69) \\ \hline \multirow{3}{*}{\(\Lambda_{7}\)} & **1.5669** & 1.5580 & **1.5577** & **1.7346** & 1.6710 & 1.6758 & **1.6714** \\  & & (0.57) & (0.59) & & (3.80) & (3.51) & (3.78) \\ \hline \multirow{3}{*}{\(\Lambda_{8}\)} & **1.9525** & 1.9290 & **1.9288** & **1.8993** & 1.8570 & 1.8622 & **1.8579** \\  & & (1.22) & (1.23) & & (2.28) & (1.99) & (2.23) \\ \hline \multirow{3}{*}{\(\Lambda_{9}\)} & **1.9766** & 1.9300 & **1.9298** & 2.0516 & 2.0900 & 2.1012 & 2.0893 \\  & & (2.42) & (2.43) & & (1.84) & (2.36) & (1.81) \\ \hline \multirow{3}{*}{\(\Lambda_{10}\)} & 2.2298 & 2.205 & 2.2057 & **2.3824** & 2.374 & 2.384 & **2.3743** \\  & & (1.12) & (1.09) & & (0.35) & (0.05) & (0.34) \\ \hline \multirow{3}{*}{\(\Lambda_{11}\)} & 2.3182 & - & 2.2057 & 2.5573 & 2.4230 & 2.4116 & 2.4219 \\  & & (5.10) & & (5.54) & (6.04) & (5.59) \\ \hline \multirow{3}{*}{\(\Lambda_{12}\)} & 2.3182 & 2.2100 & 2.2104 & **2.5816** & 2.4850 & 2.4786 & **2.4844** \\  & & (4.90) & (4.88) & & (3.89) & (4.15) & (3.91) \\ \hline \multirow{3}{*}{\(\Lambda_{13}\)} & 2.4050 & 2.3910 & 2.3914 & **2.6447** & 2.5070 & 2.5150 & **2.5075** \\  & & (0.58) & (0.57) & & (5.49) & (5.16) & (5.47) \\ \hline \multirow{3}{*}{\(\Lambda_{14}\)} & **2.4341** & 2.4180 & **2.4184** & **2.6880** & 2.6430 & 2.6451 & **2.6439** \\  & & (0.67) & (0.65) & & (1.70) & (1.62) & (1.67) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparison of results for a truncated hemisphere with FF boundary conditions.