## 3 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{2-10}  & & _L/h = 26_ & _L/h = 28_ & _L/h = 30_ & _L/h = 40_ & _L/h = 50_ & _L/h = 60_ & _L/h = 70_ \\ \hline
1 & TMM & 818.33 & 727.99 & 652.27 & 407.94 & 279.50 & 203.34 & 154.46 \\  & \(C_{fa}\) & 0.2976 & 0.3223 & 0.3449 & 0.4300 & 0.4813 & 0.5121 & 0.5308 \\  & \(C_{f,b}\) & 0.2755 & 0.2637 & 0.2548 & 0.2370 & 0.2419 & 0.2547 & 0.2698 \\  & \(C_{c}\) & 0.4269 & 0.4140 & 0.4004 & 0.3330 & 0.2769 & 0.2332 & 0.1994 \\  & Equation (36) & 818.34 & 728.00 & 652.28 & 407.95 & 279.51 & 203.35 & 154.47 \\ \hline
2 & TMM & 2242.2 & 1978.9 & 1762.3 & 1087.0 & 744.22 & 543.14 & 414.20 \\  & \(C_{f,a}\) & 0.1347 & 0.1530 & 0.1712 & 0.2561 & 0.3249 & 0.3772 & 0.4159 \\  & \(C_{f,b}\) & 0.5571 & 0.5325 & 0.5105 & 0.4318 & 0.3894 & 0.3679 & 0.3581 \\  & \(C_{c}\) & 0.3081 & 0.3145 & 0.3184 & 0.3121 & 0.2856 & 0.2549 & 0.2259 \\  & Equation (36) & 2242.3 & 1978.9 & 1762.3 & 1087.1 & 744.24 & 543.15 & 414.21 \\ \hline
3 & TMM & 4469.3 & 3918.7 & 3469.7 & 2100.8 & 1427.4 & 1039.6 & 793.22 \\  & \(C_{f,a}\) & 0.0761 & 0.0881 & 0.1005 & 0.1646 & 0.2256 & 0.2792 & 0.3239 \\  & \(C_{f,b}\) & 0.7074 & 0.6841 & 0.6622 & 0.5723 & 0.5106 & 0.4694 & 0.4424 \\  & \(C_{c}\) & 0.2165 & 0.2278 & 0.2373 & 0.2631 & 0.2638 & 0.2515 & 0.2336 \\  & Equation (36) & 4469.3 & 3918.7 & 3469.7 & 2100.8 & 1427.4 & 1039.6 & 793.23 \\ \hline
4 & TMM & 7498.0 & 6547.6 & 5776.0 & 3448.7 & 2325.1 & 1687.2 & 1285.7 \\  & \(C_{f,a}\) & 0.0441 & 0.0522 & 0.0609 & 0.1092 & 0.1600 & 0.2084 & 0.2523 \\  & \(C_{f,b}\) & 0.7946 & 0.7746 & 0.7553 & 0.6701 & 0.6041 & 0.5545 & 0.5178 \\  & \(C_{c}\) & 0.1613 & 0.1731 & 0.1838 & 0.2207 & 0.2359 & 0.2371 & 0.2299 \\  & Equation (36) & 7498.0 & 6547.7 & 5776.0 & 3448.8 & 2325.2 & 1687.2 & 1285.7 \\ \hline
5 & TMM & 11,334.1 & 9869.6 & 8683.4 & 5130.5 & 3436.1 & 2483.8 & 1889.2 \\  & \(C_{f,a}\) & 0.0270 & 0.0323 & 0.0381 & 0.0736 & 0.1147 & 0.1568 & 0.1972 \\  & \(C_{f,b}\) & 0.8493 & 0.8326 & 0.8162 & 0.7394 & 0.6750 & 0.6230 & 0.5819 \\  & \(C_{c}\) & 0.1238 & 0.1351 & 0.1457 & 0.1870 & 0.2103 & 0.2202 & 0.2209 \\  & Equation (36) & 11,334.1 & 9869.6 & 8683.4 & 5130.5 & 3436.1 & 2483.8 & 1889.2 \\ \hline \end{tabular}
\end{table} 
Table 11: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{8}{c}{**Effect of Strain Energy on the Natural Frequency [Hz]**} \\ \cline{2-10}  & & _L/h = 26_ & _L/h = 28_ & _L/h = 30_ & _L/h = 40_ & _L/h = 50_ & _L/h = 60_ & _L/h = 70_ \\ \hline
1 & TMM & 818.33 & 727.99 & 652.27 & 407.94 & 279.50 & 203.34 & 154.46 \\  & \(C_{fa}\) & 0.2976 & 0.3223 & 0.3449 & 0.4300 & 0.4813 & 0.5121 & 0. Figure 4: Variation in the strain energies on the natural frequencies of three-layered beams with the C-F end condition. Figure 5: Variation in the strain energies on the natural frequencies of three-layered beams with the C-P end condition. Figure 6: Variation in the strain energies on the natural frequencies of three-layered beams with the C-C end condition. Figure 7: Variation in the strain energies on the natural frequencies of three-layered beams with the P-P end condition. Moreover, all the strain energies in the fourth and fifth natural frequencies, as shown in Figures 4-7, undergo sudden changes. These phenomena can be analyzed in connection with the third and fourth mode shapes of the three-layered beam with the C-F end condition shown in Figure 3. To analyze the variation in mode shapes in detail, the variation in the mode shapes of three-layered beams with the cantilevered end condition, with respect to the change in _L/h_, was investigated. The mode shapes between _L/h_ = 10 and _L/h_ = 20 exhibited sudden changes in the strain energies.