## 5 Results and Discussions
### Comparative Study  
A program in MATLAB was constructed to compute the fundamental natural frequencies of the plate. The natural frequencies obtained by DSM were compared against the results reported by [PERSON] and [PERSON] [52] using CPT and with those reported by [PERSON] and [PERSON] [15] using the variable refined plate theory (RPT), as shown in Table 1; errors incurred by DSM when compared with Ref. [15] are reported in the brackets. The natural frequency changes for different boundary conditions (due to a change in stiffness) were noticed in Table 1. The maximum value of natural frequencies is reported for the CC boundary condition; the minimum value was obtained for the FF boundary condition. In the CC boundary condition, more constraints were introduced at the edges of the plate, which increased the stiffness of the plate, resulting in a higher natural frequency. On the other hand, in the FF boundary condition, no constraint was applied at the edges of the plate, which decreased the stiffness of the plate, and lowered the natural frequency, as shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(b\)/\(L\)} & \multirow{2}{*}{\(E_{1}\)/\(E_{2}\)} & \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**Boundary Conditions**} \\ \cline{3-10}  & & & CC & SC & SS & FC & FS & FF \\ \hline
0.5 & 10 & Ref [52] & 20.6543 & 14.3450 & 9.3421 & 3.5614 & 1.3190 & 0.7124 \\  & & Ref [15] & 20.5603 & 14.3137 & 9.3331 & 3.5600 & 1.3190 & 0.7123 \\  & & DSM & 20.5857 & 14.3116 & 9.3209 & 3.5154 & 1.3188 & 0.7123 \\  & & FEM & 20.6766 & 14.3565 & 9.3455 & 3.5558 & 1.3186 & 0.7141 \\  & & Error (\%) & \((-0.1234)\) & \((0.0144)\) & \((0.1309)\) & \((1.2677)\) & \((0.0152)\) & \((0.0011)\) \\  & & Ref [52] & 32.4390 & 22.4259 & 14.4578 & 5.3051 & 1.3193 & 0.7123 \\  & & Ref [15] & 32.0795 & 22.3069 & 14.4245 & 5.3003 & 1.3192 & 0.7122 \\  & & DSM & 32.4229 & 22.4148 & 14.4507 & 5.3042 & 1.3191 & 0.7122 \\  & & FEM & 32.2347 & 22.3539 & 14.4430 & 5.2925 & 1.3189 & 0.7141 \\  & & Error (\%) & \((-1.0590)\) & \((-0.4813)\) & \((-0.1815)\) & \((-0.0735)\) & \((0.0076)\) & \((0.0000)\) \\
40 & & Ref [52] & 40.9633 & 28.2855 & 18.1876 & 6.6030 & 1.3194 & 0.7123 \\  & & Ref [15] & 40.4278 & 28.0480 & 18.1215 & 6.5937 & 1.3193 & 0.7122 \\  & & DSM & 40.9620 & 28.2846 & 18.1870 & 6.6010 & 1.3190 & 0.7122 \\  & & FEM & 40.4100 & 28.0814 & 18.1426 & 6.5826 & 1.3189 & 0.7141 \\  & & Error (\%) & \((-1.7436)\) & \((-0.8366)\) & \((-0.3603)\) & \((-0.1106)\) & \((0.0227)\) & \((0.0000)\) \\ \hline
1 & 10 & Ref [52] & 21.2889 & 15.2042 & 10.4963 & 5.0586 & 3.6114 & 2.8503 \\  & & Ref [15] & 21.2078 & 15.1747 & 10.4863 & 5.0564 & 3.6105 & 2.8496 \\  & & DSM & 21.2405 & 15.1709 & 10.4750 & 5.0144 & 3.6112 & 2.8501 \\  & & FEM & 21.3090 & 15.2098 & 10.4949 & 5.0538 & 3.6125 & 2.8569 \\  & & Error (\%) & \((-0.1539)\) & \((0.0248)\) & \((0.1080)\) & \((0.8376)\) & \((-0.0194)\) & \((-0.0175)\) \\
25 & & Ref [52] & 32.8464 & 22.9847 & 15.2278 & 6.4146 & 3.6118 & 2.8493 \\  & & Ref [15] & 32.5515 & 22.8835 & 15.1972 & 6.4100 & 3.6110 & 2.8486 \\  & & DSM & 32.8033 & 22.9736 & 15.2207 & 6.4014 & 3.6115 & 2.8490 \\  & & FEM & 32.6351 & 29.9073 & 15.2089 & 6.4025 & 3.6128 & 2.8558 \\  & & Error (\%) & \((-0.8491)\) & \((-0.3921)\) & \((-0.1544)\) & \((0.1343)\) & \((-0.0138)\) & \((-0.0140)\) \\
40 & & Ref [52] & 41.2866 & 28.7305 & 18.8052 & 7.5253 & 3.6121 & 2.8492 \\  & & Ref [15] & 40.7062 & 28.5337 & 18.7477 & 7.5178 & 3.6112 & 2.8485 \\  & & DSM & 41.2853 & 28.7296 & 18.8046 & 7.5045 & 3.6120 & 2.8491 \\  & & FEM & 40.7277 & 28.5213 & 18.7566 & 7.5051 & 3.6130 & 2.8557 \\  & & Error (\%) & \((-1.4027)\) & \((-0.6818)\) & \((-0.3028)\) & \((0.1772)\) & \((-0.0221)\) & \((-0.0211)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of fundamental frequency parameters \(\varpi=\omega\frac{b^{2}}{h}\sqrt{\frac{\rho}{E_{2}}}\) of the Levy-type orthotropic plate (\(b\)/\(h\) = 100).