# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 6 Numerical Results
### Numeric Validation
#### 6.2.1 Plates
\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] [44]** & **[PERSON] [45]** & **[PERSON] [46]** & **[PERSON] [46]** & **3D** \\  & **(FOPT)** & **(GJD)** & **(GJP)** & **(HOPT)** & **(HOPT)** & **400,000** \\ \hline \(\lambda_{10}\) & 13.1818 & - & 12.7060 & - & - & 12.7053 \\  & & & (3.74) & & & (3.75) \\ \hline \(\lambda_{11}\) & 14.8717 & - & 14.8700 & - & - & 14.8712 \\  & & & (0.01) & & & (0.003) \\ \hline \(\lambda_{12}\) & 16.1611 & - & - & 15.8491 & 15.9868 & 15.5110 \\  & & & & (1.97) & (1.09) & (4.19) \\ \hline \(\lambda_{13}\) & 16.1611 & - & - & - & - & 15.5110 \\  & & & & & (4.19) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._ \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] [48]** & **[PERSON] [49]** & **[PERSON] [35]** & **3D** \\  & & & (4.70) & (0.09) & (0.09) \\ \hline \(\lambda_{4}\) & 5.4424 & 5.9040 & 5.484 & 5.4838 & 5.2862 \\  & & (7.82) & (0.76) & (0.75) & (2.95) \\ \hline \(\lambda_{5}\) & 6.6003 & 7.0800 & 6.5347 & 6.5345 & 6.4379 \\  & & (6.78) & (1.00) & (1.01) & (2.52) \\ \hline \(\lambda_{6}\) & 6.7904 & 7.3250 & 6.7039 & 6.7038 & 6.5949 \\  & & (7.30) & (1.29) & (1.29) & (2.97) \\ \hline \(\lambda_{7}\) & 8.0989 & - & - & - & 8.0718 \\  & & & & (0.34) \\ \hline \(\lambda_{8}\) & 8.5599 & - & - & - & 8.2568 \\  & & & & (3.67) \\ \hline \(\lambda_{9}\) & 8.5920 & - & 8.5302 & 8.5303 & 8.3459 \\  & & & (0.72) & (0.72) & (2.95) \\ \hline \(\lambda_{10}\) & 8.6431 & - & - & 8.5921 & 8.58 \\  & & & & (0.59) & (0.74) \\ \hline \(\lambda_{11}\) & 8.9568 & 9.5500 & - & - & (8.944) \\  & & (6.21) & & & (0.14) \\ \hline \(\lambda_{12}\) & 9.3862 & 9.6350 & - & - & 9.3532 \\  & & & (2.58) & & (0.35) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of results for a square plate of ratio \(b/h_{0}=20\) and SSSS boundary conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{4}{c}{**Formulation**} \\ \cline{2-5}  & **Present** & **[PERSON]**[50] & **[PERSON] [49] & **3D** \\ \hline \multirow{2}{*}{\(\lambda_{1}\)} & 2.0730 & 2.1460 & 2.0457 & 2.0307 \\  & & (3.40) & (1.33) & (2.08) \\ \hline \multirow{2}{*}{\(\lambda_{2}\)} & 3.8265 & - & 3.7950 & 3.7564 \\  & & & (0.83) & (1.86) \\ \hline \multirow{2}{*}{\(\lambda_{3}\)} & 4.7678 & - & 4.6714 & 4.6667 \\  & & & (3.40) & (2.17) \\ \hline \multirow{2}{*}{\(\lambda_{4}\)} & 5.9229 & - & - & 5.9004 \\  & & & & (0.38) \\ \hline \multirow{2}{*}{\(\lambda_{5}\)} & 6.3816 & - & 6.2870 & 6.2264 \\  & & & (1.50) & (2.49) \\ \hline \multirow{2}{*}{\(\lambda_{6}\)} & 6.8740 & - & 6.7723 & 6.7200 \\  & & & (1.50) & (2.29) \\ \hline \multirow{2}{*}{\(\lambda_{7}\)} & 8.4780 & - & 8.2283 & 8.2623 \\  & & & (3.03) & (2.61) \\ \hline \multirow{2}{*}{\(\lambda_{8}\)} & 9.2778 & - & 9.1022 & 8.9973 \\  & & & (1.93) & (3.12) \\ \hline \multirow{2}{*}{\(\lambda_{9}\)} & 9.9834 & - & - & 9.6869 \\  & & & & (3.06) \\ \hline \multirow{2}{*}{\(\lambda_{10}\)} & 10.5607 & - & - & 10.5382 \\  & & & & (0.213) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of results for a square plate of ratio \(b/h_{0}=20\) and SCSC boundary conditions.