# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Numeric Validation
#### 6.2.2 Shells
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Formulation**} & \multicolumn{3}{c}{**Formulation**} \\ \cline{2-7}  & \multicolumn{3}{c}{**Linear Variation**} & \multicolumn{2}{c}{**Linear Symmetric Variation**} \\ \cline{2-7}
**Natural Frequency** & **Present** & **Ganesan [30]** & **3D** & **Natural Frequency** & **Present** & **Ganesan [30]** & **1,107,000** \\ \hline \multirow{2}{*}{\(f_{1}\)} & **1132.4** & 1129.6 (0.25) & **1157.0** (2.13) & \multirow{2}{*}{\(f_{1}\)} & **1107.12** & 1112.7 (0.50) & **1136.7** (2.60) \\ \hline \multirow{2}{*}{\(f_{2}\)} & 1160.3 & 1154.2 (0.53) & 1227.0 (5.44) & \multirow{2}{*}{\(f_{2}\)} & **1313.50** (0.20) & 1316.2 (0.51) \\ \hline \multirow{2}{*}{\(f_{3}\)} & 1160.5 & - & 1227.0 (5.42) & \multirow{2}{*}{\(f_{5}\)} & 1376.88 (0.42) & 1435.2 (4.06) \\ \hline \multirow{2}{*}{\(f_{4}\)} & **1462.6** & 1452.2 (0.72) & \multirow{2}{*}{\((6.88)\)} & 1377.28 (6.88) & 1435.2 (4.04) \\ \hline \multirow{2}{*}{\(f_{5}\)} & **1567.1** & 1566.7 (0.02) & **1578.1** (0.70) & \multirow{2}{*}{\(f_{7}\)} & **1866.73** (0.09) & **1950.7** (4.30) \\ \hline \multirow{2}{*}{\(f_{6}\)} & **1905.7** & 1884.3 (1.13) & **2070.4** (7.96) & \multirow{2}{*}{\(f_{9}\)} & **2258.27** (0.05) & **2258.3** (0.001) \\ \hline \multirow{2}{*}{\(f_{7}\)} & **2455.6** & 2396.4 (2.47) & **2652.0** (7.41) & \multirow{2}{*}{\(f_{11}\)} & **2411.96** (0.88) & **2522.6** (4.39) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Comparison of results for a cylinder with constant thickness. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Formulation**} \\ \cline{2-7}
**Natural Frequency** & \multicolumn{3}{c}{**CC**} & \multicolumn{3}{c}{**CF**} \\ \cline{2-7}  & **Present** & **Dai [34]** & **3D** & **Present** & **Dai [34]** & **3D** \\  & **Present** & **Dai [34]** & **750,000** & **Present** & **Dai [34]** & **750,000** \\ \hline \multirow{2}{*}{\(f_{1}\)} & **896.96** & 896.56 (0.04) & **901.05** (0.45) & **315.31** (0.23) & 316.02 (1.06) \\ \hline \multirow{2}{*}{\(f_{2}\)} & **901.15** & 898.18 (0.33) & **904.78** (0.40) & **468.26** (0.20) & 467.34 (0.07) \\ \hline \multirow{2}{*}{\(f_{3}\)} & **1392.09** & 1388.9 (0.23) & **1395.00** (0.21) & **767.59** (0.11) & 768.45 (0.11) & **778.72** (1.43) \\ \hline \multirow{2}{*}{\(f_{4}\)} & 1500.38 & 1501.6 (0.08) & 1518.80 (1.21) & **913.77** (0.15) & 915.13 (0.15) & **921.79** (0.87) \\ \hline \multirow{2}{*}{\(f_{5}\)} & 1500.93 & - & 1518.80 (1.18) & **937.01** (0.08) & 936.25 (0.03) \\  & & (1.18) & & (0.08) & (0.03) \\ \hline \multirow{2}{*}{\(f_{6}\)} & 1675.61 & 1676.00 (0.02) & 1689.40 (0.82) & **1450.34** (0.85) & 1462.80 (0.22) \\ \hline \multirow{2}{*}{\(f_{7}\)} & 1676.11 & - & 1689.40 (0.79) & 1461.22 (0.79) & - & 1482.4 (1.43) \\  & & (0.21) & & (0.06) & & (1.39) \\ \hline \multirow{2}{*}{\(f_{8}\)} & **1884.77** & 1880.9 (0.21) & **2018.70** (0.06) & 1516.70 (0.15) & 1519 (1.22) \\  & & (0.29) & & (0.07) & (0.15) & (1.22) \\ \hline \multirow{2}{*}{\(f_{9}\)} & **2020.03** & 2014.1 (0.29) & **2018.70** (0.07) & 1517.25 (0.15) & - & 1535.6 (1.20) \\  & & (0.29) & & (0.386) & & (1.20) \\ \hline \multirow{2}{*}{\(f_{10}\)} & 2049.35 & - & 2057.30 (0.386) & 1614.93 (0.02) & 1615.2 (0.004) \\  & & (0.37) & & (0.02) & (0.04) \\ \hline \multirow{2}{*}{\(f_{11}\)} & **2182.71** & - & **2182.50** (0.01) & 1709.24 (0.01) & - & 1723.2 (0.81) \\  & & (0.07) & & (1.05) & & (0.81) \\ \hline \multirow{2}{*}{\(f_{13}\)} & **2387.65** & 2386 (0.07) & **2413.00** (1.05) & 1709.72 (0.78) & - & 1723.2 (0.78) \\ \hline \multirow{2}{*}{\(f_{14}\)} & **2474.29** & 2472.6 (0.07) & **2495.90** (0.87) & **2045.12** (0.09) & 2043.3 (0.03) \\  & & (0.07) & & (0.87) & & (0.09) \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Comparison of results for cylinders of ratio \(r/h_{m}=100\), with thickness variation and CF boundary conditions.