## 5 Results and Discussion  
The proposed method is used to study the free vibration of several thin as well as thick composite cantilever plates the natural frequencies of which are available in the literature. First, a series of convergence tests were performed for thick plates with different length-to-thickness ratios to determine the appropriate number of terms, i.e., \(M_{s}\) and \(N_{s}\), adopted in Equation (18). It is worth pointing out that [PERSON] and [PERSON] used Equation (21) to study the free vibration of thin composite cantilever plates. They found that the use of \(M_{b}=N_{b}=10\) for approximating the thin plate deflection can make the solution converge. Hence, in this study, \(M_{b}=N_{b}=10\) is also used for approximating \(W_{b}\). As for \(W_{s}\), the square cantilever plate with zero fiber angles, \(a=1\) m, \(a/h=5\), and the following information about glass/epoxy are used to perform the convergence test:  
\(E_{1}=60.7\) GPa, \(E_{2}=24.8\) GPa, \(\
u_{12}=0.23\), \(G_{12}=11.99\) GPa, \(G_{23}=11.99\) GPa, \(G_{13}=11.99\) GPa, \(\rho=1000\) kg/m\({}^{3}\), and \(E_{1}/E_{2}=2.45\). The convergence test results listed in Table 1 show that the natural frequencies of the plate converge when \(M_{b}=N_{b}=10\) and \(M_{s}=N_{s}=5\) which are also applicable to other plates with different fiber angles and length-to-thickness ratios. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(M_{s},N_{s}\)} & \multicolumn{5}{c}{**Natural Frequency (Hz)**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline
1 & 235.99 & 332.14 & 672.86 & 1198.49 & 1415.35 \\ \hline
2 & 235.87 & 331.8 & 663.72 & 953.18 & 1199.73 \\ \hline
3 & 235.84 & 331.73 & 663.51 & 911.79 & 1162.57 \\ \hline
4 & 235.84 & 331.73 & 663.51 & 910.89 & 1160.2 \\ \hline
5 & 235.84 & 331.73 & 663.5 & 910.88 & 1160.19 \\ \hline \end{tabular}
\end{table} 
Table 1: Convergence test of thick glass/epoxy composite plate with zero fiber angle and \(a/h=5\) (\(M_{b}=N_{b}=10\)). Figure 3: A typical frequency response function of cantilever composite plate. The proposed method is then used to determine the natural frequencies of a thin square glass/epoxy cantilever plate with a fiber angle equal to \(30^{\circ}\) and aspect ratio \(a/h=150\). The predicted normalized natural frequencies are listed in Table 2 in comparison with those reported in the literature. The small differences between the normalized natural frequencies predicted using the present method and those obtained by [PERSON] and [PERSON] imply that the shear deformation \(W_{s}\) has negligible effects on the natural frequencies of the plate. It is worth pointing out that for this case, the number of terms in \(W_{s}\) also has insignificant effects on the natural frequencies of the plate. For comparison purposes, the 2D Shell99 element of the commercial finite element code ANSYS [74] is also used to determine the natural frequencies of the plate. As shown in Table 2, the natural frequencies predicted using the present method are more accurate than those predicted using Shell99. Next, the proposed method is used to determine the natural frequencies of symmetrically laminated graphite/epoxy cantilever plates with \(a/h=73.2\) and different fiber angles to compare with those obtained using other methods. The properties of the graphite/epoxy material are given as follows:  
\(E_{1}=112\) GPa, \(E_{2}=11\) GPa, \(\
u_{12}=0.25\), \(G_{12}=4.48\) GPa, \(G_{23}=4.48\) GPa, \(G_{13}=4.48\) GPa, \(\rho=1500\) kg/m\({}^{3}\). The natural frequencies of the composite cantilever plates with various lamination arrangements obtained using different methods are listed in Tables 3-5 for comparison. The inspection of the differences between the experimental and theoretical normalized natural frequencies in Tables 3-5 shows that the present method can produce results closely matching the experimental data for all the composite plates under consideration.