## 3 Results and Discussions  
To demonstrate the accuracy of the developed method, the natural frequencies computed using the developed transfer matrix method were compared with the natural frequencies discussed in previous works. MATLAB(R2021b) was used to compute the results. The C-F end condition was investigated, and the results are listed in Table 1. The parameters of the sandwich beam used to predict the results shown in Table 1 are as follows [26; 27]: \(E_{f}A_{f}=31,500\) N, \(E_{f}I_{f}=1.362\) Nm\({}^{2}\), \(k_{c}A_{c}G_{c}=1050\) N, \(m_{f}=0.001225\) kg/m, \(h_{f}=0.4527\) mm, \(h_{c}=12.7\) mm, \(L=0.9144\) m. The results of the present study and previous studies showedexcellent agreement. Because there were no results for the C-C, C-P, and P-P end conditions in previous studies, they are listed in Table 2 to analyze the effects of the axial and bending modes, for example, beams with various end conditions. To analyze the dominant frequencies caused by the coupling of the axial and bending displacements for each boundary condition, the first five mode shapes of a sample beam structure with four classical end conditions are illustrated in Figure 2. As shown in Figure 2, the first five modes of this structure for all boundary conditions were dominated by bending frequencies, as expected. Because the length-to-height ratio (_L/h_) of a three-layered beam is very large (67.21), all modes appear to be dominated by the bending frequencies. To investigate the effect of the axial modes, the variation in the first five natural frequencies was examined by changing the length-to-height ratio from 10 to 70 in steps of 10. The results are tabulated in Table 3, and the first five mode shapes of the three-layered beams for _L/h_ = 10 are shown in Figure 3. As shown in Figure 3, the third and fifth mode shapes in the C-P, C-C, and P-P end conditions are clearly shown to dominate the axial frequencies, whereas the third and fourth mode shapes in the C-F end condition appear to be strongly influenced by both the axial and bending modes. This is caused by the coupling of the axial and bending displacements. For a more detailed investigation of this phenomenon, an effective method capable of distinguishing the effects of strain energy is proposed. From Equations (1), (2) and (4), the maximum strain and kinetic energies can be defined as follows:  
\[\mathcal{U}_{max}=\int_{0}^{L}\left\{E_{f}A_{f}\bigg{(}\frac{\mathrm{d}U(x)}{ \mathrm{d}x}\bigg{)}^{2}+E_{f}I_{f}\bigg{(}\frac{\mathrm{d}^{2}W(x)}{\mathrm{ d}x^{2}}\bigg{)}^{2}+\frac{1}{2}k_{c}A_{c}G_{c}\bigg{(}\frac{2}{h_{c}}U(x)+ \bigg{(}1+\frac{h_{f}}{h_{c}}\bigg{)}\frac{\mathrm{d}W(x)}{\mathrm{d}x}\bigg{)} ^{2}\right\}dx \tag{34}\]  
\[\mathcal{T}_{max}=\omega^{2}\int_{0}^{L}m_{f}\Big{\{}\left(U(x)\right)^{2}+ \left(W(x)\right)^{2}\Big{\}}dx \tag{35}\]  
By equating the maximum strain energy (\(\mathcal{U}_{max}\)) and maximum kinetic energy (\(\mathcal{T}_{max}\)), the following equation is obtained:  
\[\omega^{2}=S_{f}\Big{(}=S_{f,a}+S_{f,b}\Big{)}+S_{c} \tag{36}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{3}{c}{**Natural Frequency [Hz]**} \\ \cline{2-4}  & **Present** & **[PERSON] [26]** & **[PERSON] and [PERSON] [27]** \\ \hline
1 & 31.459 & 31.46 & 31.46 \\
2 & 193.71 & 193.7 & 193.7 \\
3 & 529.20 & 529.2 & 529.2 \\
4 & 1006.4 & 1006 & 1006 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the results of previous studies and the present method for the first four natural frequencies of a symmetric three-layered beam with the cantilevered end condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\omega_{i}\)} & \multicolumn{3}{c}{**Natural Frequency [Hz]**} \\ \cline{2-4}  & **C-F** & **C-P** & **C-C** & **P-P** \\ \hline
1 & 31.459 & 179.65 & 194.91 & 166.18 \\
2 & 193.71 & 481.22 & 521.79 & 445.12 \\
3 & 529.2 & 918.52 & 991.81 & 852.16 \\
4 & 1006.4 & 1481.7 & 1590.1 & 1381.5 \\
5 & 1612.8 & 2165.4 & 2308.9 & 2030.7 \\ \hline \hline \end{tabular}
\end{table}