## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Natural Frequency (Hz)**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** & **5** \\ \hline Experiment [75] & 196.4 & 418 & 960 & 1215 & 1550 \\ \hline CPT [75] & 224.3 & 421.8 & 1012 & 1426 & 1722 \\ \hline * (error \%) & 14.21\% & 0.91\% & 5.42\% & 17.37\% & 11.10\% \\ \hline SPSDT & 211.6 & 402.78 & 971.41 & 1328.81 & 1620.03 \\ \hline (error \%) & 7.74\% & 3.64\% & 1.19\% & 9.37\% & 4.52\% \\ \hline \hline \end{tabular}  
* error % = ( \(|\)Experimental \(-\) Theoretical \(|\)/Experimental) \(\times\) 100%.  
\end{table} 
Table 4: Natural frequencies of graphite/epoxy \([0,45,\)\(-45,\)\(90]_{\text{s}}\) square cantilever plate (\(a=76.2\) mm, \(a/h=73.2\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Natural Frequency (Hz)**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** & **5** \\ \hline Experiment [75] & 131.2 & 472 & 790.5 & 1168 & 1486 \\ \hline CPT [75] & 138.9 & 499.5 & 805 & 1326 & 1648 \\ \hline * (error \%) & 5.87\% & 5.83\% & 1.83\% & 13.53\% & 10.90\% \\ \hline SFSDT & 133.17 & 475.47 & 771.32 & 1258.3 & 1559.41 \\ \hline (error \%) & 1.50\% & 0.73\% & 2.43\% & 7.73\% & 4.94\% \\ \hline \hline \end{tabular}  
* error % = ( \(|\)Experimental \(-\) Theoretical \(|\)/Experimental) \(\times\) 100.  
\end{table} 
Table 5: Natural frequencies of graphite/epoxy \([45,\)\(-45,\)\(45]_{\text{s}}\) square cantilever plate (\(a=76.2\) mm, \(a/h=73.2\)). that both the present method and ANSYS, no matter whether 2D or 3D elements, are able to produce acceptable results. Compared with the experimental natural frequencies, the absolute percentage differences for 3D Solid186, the present method, and 2D Shell99 are less than or equal to 5.79, 6.36, and 8.68%, respectively. The present method is used to study the effects of length-to-thickness ratio (\(a/h\)) on the natural frequencies of carbon/epoxy plates with different fiber angles. Consider the first six natural frequencies of the composite cantilever plates with different length-to-thickness ratios and lamination arrangements. Herein, the natural frequency \(f\) determined using the present method is normalized with respect to the \(f\)* determined using the CPT. The relations between the normalized natural frequency and length-to-thickness ratio for the \([0]_{8}\), \([0,902,0]_{8}\), and \([45,-45_{2},45]_{8}\) plates are shown in Figures 4-6  
With respect to engineering applications, the natural frequencies predicted using the present method together with the natural frequencies determined experimentally can be used to identify material constants for the quality control of composite plates after fabrication or health assessment of existing cantilever composite plates. Regarding an existing cantilever plate having been operated in a severe environment for some time, the degradation of the constituent material will cause the values of the plate's natural frequencies to decrease. In practice, the current natural frequencies of the plate can be determined via vibration testing. The deviations of the current natural frequencies from the original ones can then be used to identify the current material constants such as \(E_{1}\), \(E_{2}\), and \(G_{12}\) for the health assessment of the plate. It is a common practice to use several measured natural frequencies in an optimization method to identify all the material constants simultaneously. However, for some material constant identification problems, the search for the true solution in the identification process may encounter difficulties in making the solution converge, or the adopted search algorithm, in terms of computation time, is too expensive to be used for practical applications. Herein, for illustration, a procedure  
Figure 5: Relationship between normalized natural frequency and length-to-thickness ratio for the carbon/epoxy [0, 90\({}_{2}\), 0]s square plate.