## 3 Results and Discussions
\begin{table}
\begin{tabular}{c 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 = 26_ & _L/h = 28_ & _L/h = 30_ & _L/h = 40_ & _L/h = 50_ & _L/h = 60_ & _L/h = 70_ \\ \hline
1 & TMM & 206.94 & 178.85 & 156.09 & 88.332 & 56.700 & 39.441 & 29.007 \\  & \(C_{f,a}\) & 0.4484 & 0.4545 & 0.4596 & 0.4757 & 0.4839 & 0.4886 & 0.4915 \\  & \(C_{f,b}\) & 0.5198 & 0.5173 & 0.5152 & 0.5088 & 0.5057 & 0.5040 & 0.5029 \\  & \(C_{c}\) & 0.0318 & 0.0282 & 0.0252 & 0.0155 & 0.0104 & 0.0074 & 0.0056 \\  & Equation (36) & 206.94 & 178.85 & 156.09 & 88.332 & 56.700 & 39.441 & 29.007 \\ \hline
2 & TMM & 1200.3 & 1044.6 & 917.336 & 530.686 & 345.033 & 241.909 & 178.831 \\  & \(C_{f,a}\) & 0.2724 & 0.2904 & 0.3067 & 0.3682 & 0.4061 & 0.4304 & 0.4465 \\  & \(C_{f,b}\) & 0.6060 & 0.5950 & 0.5856 & 0.5537 & 0.5364 & 0.5262 & 0.5197 \\  & \(C_{c}\) & 0.1217 & 0.1146 & 0.1077 & 0.0781 & 0.0575 & 0.0435 & 0.0338 \\  & Equation (36) & 1200.3 & 1044.6 & 917.336 & 530.686 & 345.033 & 241.909 & 178.831 \\ \hline
3 & TMM & 3141.3 & 2740.7 & 2413.5 & 1415.5 & 930.618 & 657.894 & 489.296 \\  & \(C_{f,a}\) & 0.1581 & 0.1753 & 0.1918 & 0.2644 & 0.3189 & 0.3586 & 0.3876 \\  & \(C_{f,b}\) & 0.6906 & 0.6750 & 0.6610 & 0.6091 & 0.5776 & 0.5574 & 0.5440 \\  & \(C_{c}\) & 0.1513 & 0.1497 & 0.1472 & 0.1265 & 0.1036 & 0.0840 & 0.0684 \\  & Equation (36) & 3141.3 & 2740.7 & 2413.5 & 1415.5 & 930.618 & 657.894 & 489.296 \\ \hline
4 & TMM & 5828.8 & 5085.8 & 4480.3 & 2642.1 & 1750.0 & 1245.7 & 931.908 \\  & \(C_{f,a}\) & 0.0853 & 0.0982 & 0.1114 & 0.1770 & 0.2351 & 0.2829 & 0.3210 \\  & \(C_{f,b}\) & 0.7672 & 0.7500 & 0.7339 & 0.6696 & 0.6260 & 0.5962 & 0.5753 \\  & \(C_{c}\) & 0.1475 & 0.1518 & 0.1546 & 0.1535 & 0.1389 & 0.1209 & 0.1037 \\  & Equation (36) & 5828.8 & 5085.8 & 4480.3 & 2642.1 & 1750.0 & 1245.7 & 931.908 \\ \hline
5 & TMM & 9285.2 & 8090.5 & 7119.8 & 4192.8 & 2784.4 & 1990.2 & 1495.2 \\  & \(C_{f,a}\) & 0.0502 & 0.0583 & 0.0671 & 0.1171 & 0.1690 & 0.2168 & 0.2583 \\  & \(C_{f,b}\) & 0.8236 & 0.8079 & 0.7926 & 0.7252 & 0.6746 & 0.6374 & 0.6100 \\  & \(C_{c}\) & 0.1262 & 0.1338 & 0.1403 & 0.1576 & 0.1564 & 0.1458 & 0.1317 \\  & Equation (36) & 9285.2 & 8090.5 & 7119.8 & 4192.8 & 2784.4 & 1990.2 & 1495.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Effect of strain energies on the natural frequencies of three-layered beams with the C-P end condition depending on the variation in _L/h_. \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
5 & TMM & 32,498.4 & 25,214.5 & 19,198.6 & 17,669.0 & 17,189.2 & 15,123.1 & 12,680.9 & 10,779.4 \\  & \(C_{f,a}\) & 0.7235 & 0.0258 & 0.0488 & 0.1008 & 0.0806 & 0.0354 & 0.0372 & 0.0430 \\  & \(C_{f,b}\) & 0.0161 & 0.8850 & 0.6222 & 0.0539 & 0.1269 & 0.8354 & 0.8515 & 0.8390 \\  & \(C_{c}\) & 0.2604 & 0.0892 & 0.3289 & 0.8453 & 0.7926 & 0.1292 & 0.1112 & 0.1181 \\  & Equation (36) & 32,498.4 & 25,214.5 & 19,198.6 & 17,669.0 & 17,189.2 & 15,123.1 & 12,680.9 & 10,779.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._