# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Numeric Validation
#### 6.2.2 Shells  
Within this section, comparisons between the presented formulation and results available in the literature and those obtained by means of solid elements in ANSYS, for truncated hemispheres and cylinders, are presented. In these comparisons, values of natural frequencies and dimensionless parameters written in bold font represent even frequencies, that is, frequencies with identical value but different vibration mode. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{5}{c}{**Formulation**} \\ \cline{2-6}  & **Present** & **[PERSON] [47] (FSDT)** & **[PERSON] [47] (HSDT)** & **3D** \\ \hline \multirow{2}{*}{\(\lambda_{13}\)} & 14.9194 & 14.6648 & 14.7608 & 14.8173 \\  & & (1.74) & (1.07) & (0.69) \\ \hline \multirow{2}{*}{\(\lambda_{14}\)} & 15.3667 & - & - & 14.8633 \\  & & (3.63) & (3.26) & (3.73) & (2.92) \\ \hline \multirow{2}{*}{\(\lambda_{15}\)} & 16.6314 & - & - & 16.0038 \\  & & & & (3.92) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: _Cont._ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{5}{c}{**Formulation**} \\ \cline{2-5}  & **Present** & **[PERSON] [47] (FSDT)** & **[PERSON] [47] (HSDT)** & **3D** \\ \hline \multirow{2}{*}{\(\lambda_{1}\)} & 2.3533 & 2.2835 & 2.2850 & 2.2818 & 2.2901 \\  & & (3.06) & (2.99) & (3.13) & (2.76) \\ \hline \multirow{2}{*}{\(\lambda_{2}\)} & 4.4315 & 4.3327 & 4.3375 & 4.3298 & 4.3476 \\  & & (2.28) & (2.17) & (2.35) & (1.93) \\ \hline \multirow{2}{*}{\(\lambda_{3}\)} & 4.6585 & 4.5231 & 4.5294 & 4.5207 & 4.5450 \\  & & (2.99) & (2.85) & (3.05) & (2.50) \\ \hline \multirow{2}{*}{\(\lambda_{4}\)} & 6.5960 & 6.4168 & 6.4291 & 6.4107 & 6.4458 \\  & & (2.79) & (2.60) & (2.89) & (2.33) \\ \hline \multirow{2}{*}{\(\lambda_{5}\)} & 7.4785 & 7.2995 & 7.3118 & 7.2945 & 7.3336 \\  & & (2.45) & (2.28) & (2.52) & (1.98) \\ \hline \multirow{2}{*}{\(\lambda_{6}\)} & 7.9704 & 7.7170 & 7.7358 & 7.7157 & 7.7704 \\  & & (3.28) & (3.03) & (3.30) & (2.57) \\ \hline \multirow{2}{*}{\(\lambda_{7}\)} & 9.6596 & 9.3726 & 9.3971 & 9.3631 & 9.4242 \\  & & (3.06) & (2.79) & (3.17) & (2.50) \\ \hline \multirow{2}{*}{\(\lambda_{8}\)} & 9.7117 & 9.3996 & 9.4270 & 9.3922 & 9.4588 \\  & & (3.32) & (3.02) & (3.40) & (2.67) \\ \hline \multirow{2}{*}{\(\lambda_{9}\)} & 11.2627 & - & - & - & 11.0079 \\  & & & & (2.31) \\ \hline \multirow{2}{*}{\(\lambda_{10}\)} & 12.1013 & - & - & - & 11.7525 \\  & & & & & (2.97) \\ \hline \multirow{2}{*}{\(\lambda_{11}\)} & 12.5463 & 12.1072 & 12.1505 & 12.0957 & 12.1901 \\  & & (3.63) & (3.26) & (3.73) & (2.92) \\ \hline \multirow{2}{*}{\(\lambda_{12}\)} & 12.6196 & - & - & - & 12.6188 \\  & & & & (0.01) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison of results for a square plate of ratio \(b/h_{0}=20\), parabolic variation and CCCC boundary conditions. Table 13 presents comparisons for a truncated hemisphere with constant and linear variable thickness. For constant thickness, the relative errors are below 5.10%. For linear thickness variation, frequencies of the present formulation are approximated of those reported in references [35; 51] with maximum errors of 6.04% and 5.70%, respectively. The comparison with the 3D model gives a maximum error of 5.59%. Table 14 presents a comparison for the natural frequencies of a cylinder with constant thickness, subjected to CC and CF boundary conditions. For these boundary conditions, the maximum relative errors are obtained when the present formulation is compared with the three-dimensional models; having values of 1.21% and 1.43%, respectively.