## 3 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\omega_{i}\) & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{3-10}  & & _L/h = 10_ & _L/h = 12_ & _L/h = 14_ & _L/h = 16_ & _L/h = 18_ & _L/h = 20_ & _L/h = 22_ & _L/h = 24_ \\ \hline
1 & TMM & 5071.0 & 3670.0 & 2806.4 & 2231.2 & 1825.6 & 1526.7 & 1298.8 & 1120.3 \\  & \(C_{f,a}\) & 0.0331 & 0.0542 & 0.0792 & 0.1066 & 0.1351 & 0.1636 & 0.1913 & 0.2179 \\  & \(C_{f,b}\) & 0.7606 & 0.7013 & 0.6488 & 0.6036 & 0.5653 & 0.5332 & 0.5065 & 0.4844 \\  & \(C_{c}\) & 0.2063 & 0.2444 & 0.2720 & 0.2898 & 0.2997 & 0.3032 & 0.3022 & 0.2977 \\  & Equation (36) & 5071.1 & 3670.0 & 2806.4 & 2231.3 & 1825.6 & 1526.7 & 1298.8 & 1120.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Effect of strain energies on the natural frequencies of three-layered beams with the C-F end condition depending on the variation in _L/h_. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{2-9}  & & \(L/h=26\) & \(L/h=28\) & \(L/h=30\) & \(L/h=40\) & \(L/h=50\) & \(L/h=60\) & \(L/h=70\) \\ \hline
1 & TMM & 1180.4 & 1029.4 & 905.78 & 528.24 & 345.30 & 242.96 & 180.05 \\  & C\({}_{f,a}\) & 0.1983 & 0.2191 & 0.2386 & 0.3155 & 0.3658 & 0.3991 & 0.4218 \\  & C\({}_{f,b}\) & 0.6475 & 0.6331 & 0.6206 & 0.5773 & 0.5532 & 0.5386 & 0.5292 \\  & C\({}_{c}\) & 0.1543 & 0.1478 & 0.1409 & 0.1072 & 0.0810 & 0.0623 & 0.0490 \\  & Equation (36) & 1180.5 & 1029.4 & 905.8 & 528.2 & 345.3 & 243.0 & 180.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Effect of strain energies on the natural frequencies of three-layered beams with the C-C end condition depending on the variation in \(L/h\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{2-9}  & & \(L/h=26\) & \(L/h=28\) & \(L/h=30\) & \(L/h=40\) & \(L/h=50\) & \(L/h=60\) & \(L/h=70\) \\ \hline
5 & TMM & 12,360.9 & 10,750.8 & 9447.4 & 5549.2 & 3696.1 & 2657.9 & 2011.8 \\  & C\({}_{f,a}\) & 0.0238 & 0.0286 & 0.0339 & 0.0670 & 0.1066 & 0.1480 & 0.1878 \\  & C\({}_{f,b}\) & 0.8680 & 0.8531 & 0.8384 & 0.7687 & 0.7098 & 0.6621 & 0.6242 \\  & C\({}_{c}\) & 0.1082 & 0.1183 & 0.1278 & 0.1643 & 0.1836 & 0.1900 & 0.1880 \\  & Equation (36) & 12,361.0 & 10,750.9 & 9447.5 & 5549.3 & 3696.1 & 2657.9 & 2011.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{2-9}  & & \(L/h=26\) & \(L/h=28\) & \(L/h=30\) & \(L/h=40\) & \(L/h=50\) & \(L/h=60\) & \(L/h=70\) \\ \hline
1 & TMM & 12,360.9 & 10,750.8 & 9447.4 & 5549.2 & 3696.1 & 2657.9 & 2011.8 \\  & C\({}_{f,a}\) & 0.0238 & 0.0286 & 0.0339 & 0.0670 & 0.1066 & 0.1480 & 0.1878 \\  & C\({}_{f,b}\) & 0.8680 & 0.8531 & 0.8384 & 0.7687 & 0.7098 & 0.6621 & 0.6242 \\  & C\({}_{c}\) & 0.1082 & 0.1183 & 0.1278 & 0.1643 & 0.1836 & 0.1900 & 0.1880 \\  & Equation (36) & 12,361.0 & 10,750.9 & 9447.5 & 5549.3 & 3696.1 & 2657.9 & 2011.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Effect of strain energies on the natural frequencies of three-layered beams with the C-C end condition depending on the variation in \(L/h\).