## 5 Results and Discussions
### Parameter Studies  
To study the effects of different boundary conditions, variations of thickness ratio, modulus ratio, and aspect ratio on the non-dimensional fundamental frequencies of thin orthotropic plates, parametric studies were conducted. The DSM results are also compared with the FEM obtained by ANSYS. For the modeling of the plate, shell element 181 was used in ANSYS. To test the accuracy of FEM, the non-dimensional fundamental frequencies were obtained by DSM based on CPT for a simply supported Levy-type orthotropic plate (\(b/L=1\), \(b/h=20\)) at different modulus ratios, compared to the values obtained by FEM with different mesh sizes, as shown in Table 2. A very good convergence between the FEM and DSM results can be observed. The final choice is a mesh with 20 \(\times\) 20 plate elements, which is a good compromise between the need for accuracy and limiting the computing time. In Tables 3-8, the DSM results are compared against the published results reported by [PERSON] [53] using the differential transform method (DTM) and Taylor collocation method (TCM), and with those reported by authors [15] for different [PERSON]'s modulus ratios (\(E_{1}/E_{2}\)), aspect ratios (\(b/L\)), and thickness ratios (\(b/h\)). It could be observed from Tables 3-8 that the natural frequencies, in general, decrease with an increase in [PERSON]'s modulus, aspect ratio, and thickness ratio for the orthotropic plate. It is observed in Tables 3-7 that with boundary conditions changing from SC to SF, the natural frequencies increase, with the increase in the modulus ratio (\(E_{1}/E_{2}\)) from 3 to 50 because of an increase in the stiffness of the plate. Compared with Ref. [15], the relative errors of DSM are reported in brackets in Tables 3-8. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(b/L\)} & \multirow{2}{*}{\(E_{1}/E_{2}\)} & \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Boundary Conditions**} & \\ \cline{3-8}  & & & **CC** & **SC** & **SS** & **FC** & **FS** & **FF** \\ \hline
2 & 10 & Ref [52] & 25.5184 & 20.5941 & 17.1364 & 12.9377 & 12.2379 & 11.4094 \\  & & Ref [15] & 25.4427 & 20.5543 & 17.1129 & 12.9238 & 12.2259 & 11.3981 \\  & & DSM & 25.4651 & 20.5536 & 17.1046 & 12.9145 & 12.2298 & 11.3998 \\  & & FEM & 25.4958 & 20.5568 & 17.1086 & 12.9324 & 12.2395 & 11.4254 \\  & & Error (\%) & \((-0.0879)\) & \((0.0032)\) & \((0.0483)\) & \((0.0720)\) & \((-0.0319)\) & \((-0.0149)\) \\
25 & & Ref [52] & 35.7303 & 26.8537 & 20.3682 & 13.5562 & 12.2305 & 11.3993 \\  & & Ref [15] & 35.5237 & 26.7659 & 20.3288 & 13.5404 & 12.2186 & 11.3880 \\  & & DSM & 35.7135 & 26.8417 & 20.3596 & 13.5504 & 12.2301 & 11.3895 \\  & & FEM & 35.4925 & 26.7502 & 20.3265 & 13.5433 & 12.2319 & 11.4155 \\  & & Error (\%) & \((-0.5316)\) & \((-0.2822)\) & \((-0.1514)\) & \((-0.0738)\) & \((-0.0940)\) & \((-0.0132)\) \\
40 & & Ref [52] & 43.6154 & 31.9099 & 23.1622 & 14.1271 & 12.2301 & 11.3977 \\  & & Ref [15] & 43.2415 & 31.7630 & 23.1043 & 14.1093 & 12.2182 & 11.3864 \\  & & DSM & 43.6141 & 31.9090 & 23.1616 & 14.1270 & 12.2298 & 11.3944 \\  & & FEM & 43.0347 & 31.6809 & 23.0942 & 14.1072 & 12.2312 & 11.4140 \\  & & Error (\%) & \((-0.8543)\) & \((-0.4576)\) & \((-0.2473)\) & \((-0.1253)\) & \((-0.0949)\) & \((-0.0702)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline _b/L_ & _b/h_ & **Method** & \(\mathbf{E_{1}/E_{2}}\) & & & & \\ \hline \hline \multirow{6}{*}{0.5} & \multirow{6}{*}{20} & \multirow{6}{*}{RPT [15]} & 3 & 10 & 20 & 30 & 40 & 50 \\  & & & 3 & 10 & 20 & 30 & 40 & 50 \\ \hline \multirow{6}{*}{0.5} & \multirow{6}{*}{20} & \multirow