## 2 Theory
The relationship between the two constants (\(H_{j}\) and \(Q_{j}\)) can be defined by substituting Equations (19) and (20) into Equation (12):  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]3  
and  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]4  
Thus, \(W(x)\) can be rewritten as  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]5  
The slope of the bending deformation curve can be obtained by differentiating Equation (23) as:  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]6  
By substituting Equations (23) and (24) into Equations (8)-(10), the expression for the forces can be rewritten as  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]7  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]8  
and  
\[\mathcal{U}=\mathcal{U}_{f}+\mathcal{U}_{c} \tag{3}\]9  
The state vector \(\mathbf{Z}_{x=0}\), expressed in matrix form by substituting \(x=0\) into Equations (19) and (23)-(27), is given by:  
\[\mathcal{T}=\frac{1}{2}\int_{0}^{L}2m_{f}\bigg{(}\frac{\partial u(x,t)}{ \partial t}\bigg{)}^{2}dx+\frac{1}{2}\int_{0}^{L}2m_{f}\bigg{(}\frac{\partial w (x,t)}{\partial t}\bigg{)}^{2}dx, \tag{4}\]0where  
\[\mathbf{Z}_{x=0}=\left\{ \begin{array}{c}U\\ W\\ \Phi\\ F\\ M\\ V\end{array} \right\}_{x=0},\mathbf{C}=\left[ \begin{array}{cccccc}C_{11}&C_{12}&C_{13 }&C_{14}&C_{15}&C_{16}\\ C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\ C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\ C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}\end{array} \right],\mathbf{H}= \left\{ \begin{array}{c}H_{1}\\ H_{2}\\ H_{3}\\ H_{4}\\ H_{5}\\ H_{6}\end{array} \right\}, \tag{29}\]  
and \(C_{1j}=1,\ C_{2j}=\beta_{j},\ C_{3j}=\lambda_{j}\beta_{j},\ C_{4j}=-2E_{f}A_{j}\lambda_{j}\),  
\[C_{5j}=-2E_{f}I_{f}\beta_{j}\lambda_{j}^{2},\ C_{6j}=E_{f}I_{f}\left\{2\lambda_ {j}^{3}\beta_{j}-\frac{a^{2}c}{r^{2}}\big{(}c\lambda_{j}\beta_{j}+2\big{)} \right\}.\]  
From Equation (28), the constant \(\mathbf{H}\) can be obtained as follows:  
\[\mathbf{H}=\mathbf{C}^{-1}\mathbf{Z}_{x=0} \tag{30}\]  
Similarly, the state vector \(\mathbf{Z}_{x=L}\), expressed in matrix form by substituting \(x=L\) into Equations (19) and (23)-(27), is given by:  
\[\mathbf{Z}_{x=L}=\mathbf{GH} \tag{31}\]  
where  
\[\mathbf{z}_{x=L}=\left\{ \begin{array}{c}U\\ W\\ \Phi\\ F\\ M\\ V\end{array} \right\}_{x=L},\mathbf{G}=\left[ \begin{array}{cccccc}G_{11}&G_{12 }&G_{13}&G_{14}&G_{15}&G_{16}\\ G_{21}&G_{22}&G_{23}&G_{24}&G_{25}&G_{26}\\ G_{31}&G_{32}&G_{33}&G_{34}&G_{35}&G_{36}\\ G_{41}&G_{42}&G_{43}&G_{44}&G_{45}&G_{46}\\ G_{51}&G_{52}&G_{53}&G_{54}&G_{55}&G_{56}\\ G_{61}&G_{62}&G_{63}&G_{64}&G_{65}&G_{66}\end{array} \right], \tag{32}\]  
and \(G_{1j}=\mathrm{e}^{\lambda_{j}L},\ G_{2j}=\beta_{j}\mathrm{e}^{\lambda_{j}L}, \ G_{3j}=\lambda_{j}\beta_{j}\mathrm{e}^{\lambda_{j}L},\ G_{4j}=-2E_{f}A_{j} \lambda_{j}\mathrm{e}^{\lambda_{j}L}\),  
\[G_{5j}=-2E_{f}I_{f}\beta_{j}\lambda_{j}^{2}\mathrm{e}^{\lambda_{j}L},\ G_{6j}= E_{f}I_{f}\left\{2\lambda_{j}^{3}\beta_{j}-\frac{a^{2}c}{r^{2}}\big{(}c\lambda_{j} \beta_{j}+2\big{)}\right\}\mathrm{e}^{\lambda_{j}L}.\]  
By substituting Equation (30) into Equation (31), the relationship between state vectors \(\mathbf{Z}_{x=0}\) and \(\mathbf{Z}_{x=L}\) can be expressed as follows:  
\[\mathbf{Z}_{x=L}=\mathbf{TZ}_{x=0}, \tag{33}\]  
where \(\mathbf{T}=\mathbf{GC}^{-1}\), and \(\mathbf{T}\) is the transfer matrix of the three-layered beam element. From Equation (33), the natural frequencies and mode shapes of the three-layered beams with classical boundary conditions can be computed by determining the frequency determinant of the transfer matrix [31]. The end conditions of the displacements and forces used for applying the boundary condition to Equation (33) are \(U,W,\ \Phi=0\) and \(F,\ M,\ V\
eq 0\) at the clamped end and \(U,W,\ \Phi\
eq 0\) and \(F,\ M,\ V=0\) at the free end. In addition, the conditions \(U,\ W,M=0\) and \(\Phi,\ V,\ F\
eq 0\) are applied at the pinned end.