## 5 Results and Discussion
Furthermore, the fact that the present method can predict the first natural frequencies more accurately than the CPT reveals the significant effects of across-thickness shear deformation on the modal characteristics of the plates. It is noted that the value of \(E_{1}/G_{13}=25\) gives the major contribution to the across-thickness shear deformation. Therefore, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{**Normalized Natural Frequency**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline Narita and Leissa [39] & 2.95 & 7.16 & 18.14 & 20.01 & 26.45 \\ \hline Present & 2.95 & 7.16 & 18.14 & 20.01 & 26.45 \\ \hline \({}^{*}\) (error \%) & 0.00\% & 0.00\% & 0.02\% & 0.00\% & 0.00\% \\ \hline ANSYS (Shell99) & 2.96 & 7.18 & 18.48 & 20.35 & 26.93 \\ \hline (error \%) & 0.10\% & 0.18\% & 1.86\% & 1.72\% & 1.82\% \\ \hline \({}^{*}\)error \% = (\(\mid\)Narita and Leissa – Other method \(\mid\)/Narita and Leissa) \(\times\) 100\%. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normalized natural frequencies of thin square glass/epoxy composite cantilever plate with fiber angle equal to \(30^{\circ}\) (\(a/h=150\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{**Natural Frequency (Hz)**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline Experiment [75] & 234.2 & 362 & 728.3 & 1449 & 1503 \\ \hline CPT [75] & 261.9 & 363.5 & 761.8 & 1662 & 1709 \\ \hline \({}^{*}\) (error \%) & 11.83\% & 0.41\% & 4.60\% & 14.70\% & 13.71\% \\ \hline Present & 246.59 & 348.36 & 742.45 & 1535.96 & 1568.75 \\ \hline (error \%) & 5.29\% & 3.77\% & 1.94\% & 6.00\% & 4.37\% \\ \hline \hline \end{tabular} \({}^{*}\)error % = (\(\mid\)Experimental – Theoretical \(\mid\)/Experimental) \(\times\) 100\%.  
\end{table} 
Table 3: Natural frequencies of graphite/epoxy [\(0_{2}\), \(30\), \(-30\)]\({}_{8}\) square cantilever plate (\(a=76.2\) mm, \(a/h=73.2\)). through-thickness shear deformation effects on plate modal characteristics may become too significant to be neglected when the thickness of the cantilever plate or \(E_{1}/G_{13}\) gets larger. The proposed method is then used to predict the natural frequencies of thick cantilever plates. First, consider the free vibration of a thick isotropic cantilever plate with \(a/h=5\). The first four normalized natural frequencies of the plate are listed in Table 6 in comparison with the experimental data available in the literature. It is noted that the results obtained using the present method are acceptable. In particular, for the first mode, the percentage difference between the first theoretical and experimental normalized natural frequencies is less than 1.8% while that of the finite element method (FETM) is larger than 11.24%. Next, the proposed method is used to determine the natural frequencies of the \([0^{\circ}]_{332}\) carbon/epoxy cantilever plate which has been tested. The theoretical natural frequencies predicted using the present method are compared with the experimental ones as tabulated in Table 7. It is noted that for comparison purposes, the 2D Shell99 and 3D Solid186 elements of ANSYS have also been used to determine the natural frequencies. The comparison shows \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Normalized Natural Frequency**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline Experiment [25] & 0.1967 & 0.4392 & 1.0788 & 1.3086 \\ \hline FETM [25] & 0.2188 & 0.4582 & 1.1676 & 1.383 \\ \hline * (error \%) & 11.24\% & 4.33\% & 8.23\% & 5.69\% \\ \hline Present & 0.2002 & 0.4794 & 1.0747 & 1.35 \\ \hline (error \%) & 1.79\% & 9.15\% & 0.38\% & 3.17\% \\ \hline \hline \end{tabular}  
* error % = ( \(|\)Experimental \(-\) Theoretical \(|\)/Experimental) \(\times\) 100%.  
\end{table} 
Table 6: Normalized natural frequencies of an isotropic plate (\(a/h=5\)).