# 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} \hline \hline \multirow{2}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{4}{c}{**Formulation**} \\ \cline{2-5}  & **Present** & **[PERSON] [44]** & **[PERSON] [45]** & **[PERSON] [46]** & **Shufrin [46]** & **3D** \\  & & **(FOPT)** & **(3D)** & **(FOPT)** & **(3D)** \\ \hline \(\lambda_{1}\) & 3.3676 & 3.2921 & 3.3215 & 3.2978 & 3.3046 & 3.3176 \\  & & (2.29) & (1.39) & (2.12) & (1.91) & (1.51) \\ \hline \(\lambda_{2}\) & 6.4928 & 6.2761 & 6.3457 & 6.2877 & 6.3123 & 6.3419 \\  & & (3.45) & (2.32) & (3.26) & (2.86) & (2.38) \\ \hline \(\lambda_{3}\) & 6.4928 & 6.2761 & 6.3457 & 6.2877 & 6.3123 & 6.3419 \\  & & (3.45) & (2.32) & (3.26) & (2.86) & (2.38) \\ \hline \(\lambda_{4}\) & 9.1648 & 8.7934 & 8.9030 & 8.8127 & 8.8594 & 8.8982 \\  & & (4.22) & (2.94) & (4.00) & (3.45) & (3.00) \\ \hline \(\lambda_{5}\) & 10.8503 & 10.3570 & 10.4990 & 10.4299 & 10.4946 & 10.4996 \\  & & (4.76) & (3.35) & (4.03) & (3.39) & (3.34) \\ \hline \(\lambda_{6}\) & 10.9478 & 10.4561 & 10.5980 & 10.4299 & 10.4946 & 10.5988 \\  & & (4.70) & (3.30) & (4.97) & (4.32) & (3.29) \\ \hline \(\lambda_{7}\) & 12.5351 & 12.5246 & 12.5220 & 12.5568 & 12.6471 & 12.5194 \\  & & (0.08) & (0.10) & (0.17) & (0.89) & (0.13) \\ \hline \(\lambda_{8}\) & 12.5351 & 12.5246 & 12.5220 & 12.5568 & 12.6471 & 12.5194 \\  & & (0.08) & (0.10) & (0.17) & (0.89) & (0.13) \\ \hline \(\lambda_{9}\) & 13.1818 & - & 12.7060 & - & - & 12.7053 \\  & & (3.74) & & & (3.75) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of results for a square plate of ratio \(b/h=10\) and CCCC boundary conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Dimensionless Frequency Parameter**} & \multicolumn{4}{c}{**Formulation**} \\ \cline{2-5}  & **Present** & **[PERSON] [44]** & **[PERSON] [45]** & **[PERSON] [46]** & **[PERSON] [46]** & **3D** \\  & & **(FOPT)** & **(3D)** & **(FOPT)** & **(3D)** \\ \hline \(\lambda_{1}\) & 3.3676 & 3.2921 & 3.3215 & 3.2978 & 3.3046 & 3.3176 \\  & & (2.29) & (1.39) & (2.12) & (1.91) & (1.51) \\ \hline \(\lambda_{2}\) & 6.4928 & 6.2761 & 6.3457 & 6.2877 & 6.3123 & 6.3419 \\  & & (3.45) & (2.32) & (3.26) & (2.86) & (2.38) \\ \hline \(\lambda_{3}\) & 6.4928 & 6.2761 & 6.3457 & 6.2877 & 6.3123 & 6.3419 \\  & & (3.45) & (2.32) & (3.26) & (2.86) & (2.38) \\ \hline \(\lambda_{4}\) & 9.1648 & 8.7934 & 8.9030 & 8.8127 & 8.8594 & 8.8982 \\  & & (4.22) & (2.94) & (4.00) & (3.45) & (3.00) \\ \hline \(\lambda_{5}\) & 10.8503 & 10.3570 & 10.4990 & 10.4299 & 10.4946 & 10.4996 \\  & & (4.76) & (3.35) & (4.03) & (3.39) & (3.34) \\ \hline \(\lambda_{6}\) & 10.9478 & 10.4561 & 10.5980 & 10.4299 & 10.4946 & 10.5988 \\  & & (4.70) & (3.30) & (4.97) & (4.32) & (3.29) \\ \hline \(\lambda_{7}\) & 12.5351 & 12.5246 & 12.5220 & 12.5568 & 12.6471 & 12.5194 \\  & & (0.08) & (0.10) & (0.17) & (0.89) & (0.13) \\ \hline \(\lambda_{8}\) & 12.5351 & 12.5246 & 12.5220 & 12.5568 & 12.6471 & 12.5194 \\  & & (0.08) & (0.10) & (0.17) & (0.89) & (0.13) \\ \hline \(\lambda_{9}\) & 13.1818 & - & 12.7060 & - & - & 12.7053 \\  & & (3.74) & & & (3.75) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of results for a square plate of ratio \(b/h=10\) and SCSC boundary conditions. Tables 9-11 present the results for the boundary conditions listed for Cases I, II, and III, applied to square plates with linear thickness variation. The highest relative errors between the presented formulation and solid elements is on the order of 3.92%. However, the highest errors are obtained when the present formulation is compared with the FSDT presented by Shufrin & Eisenberg [47], having a value of 4.11%.