## Solution of CBGCSW Governing Differential Equation  
### Solving Free Vibration Equation of CBGCSW  
On the basis of energy principles and variational principles \(\delta\int_{t_{0}}^{t_{1}}(T-V)dt=0\)[27], the bending vibration differential equation and the natural boundary conditions of a CBGCSW can be obtained.  
\[G_{e}A_{s}(w^{\prime\prime}-\theta^{\prime})-A_{\rho}\ddot{w}=0 \tag{17}\]  
\[E_{c}F_{c}U^{\prime}-E_{c}S_{c}\theta^{\prime\prime}-G_{c}I_{c}U-F_{\rho} \ddot{U}+S_{\rho}\ddot{\theta}=0 \tag{18}\]  
\[G_{e}A_{s}(w^{\prime}-\theta)+E_{c}I_{c}\theta^{\prime\prime}-E_{c}S_{c}U^{ \prime\prime}-I_{\rho}\ddot{\theta}+S_{\rho}\ddot{U}=0 \tag{19}\]  
\[(E_{c}F_{c}U^{\prime}-E_{c}S_{c}\theta^{\prime})\delta U\big{|}_{0}^{L}=0 \tag{20}\]  
\[G_{e}A_{s}(w^{\prime}-\theta)\delta w\big{|}_{0}^{L}=0 \tag{21}\]  
\[(E_{c}I_{c}\theta^{\prime}-E_{c}S_{c}U^{\prime})\delta\theta\big{|}_{0}^{L}=0 \tag{22}\]  
\[I_{\rho}=\rho_{c}\int_{A_{c}}z^{2}dA+\rho_{s}\int_{A_{se}}z^{2}dA \tag{23}\]  
\[F_{\rho}=\rho_{c}\int_{A_{c}}\psi^{2}dA+\rho_{s}\int_{A_{se}}\psi^{2}dA \tag{24}\]  
\[S_{\rho}=\rho_{c}\int_{A_{c}}z\psi dA+\rho_{s}\int_{A_{se}}z\psi dA \tag{25}\]  
\[A_{\rho}=\rho_{c}A_{c}+\rho_{s}A_{se} \tag{26}\]  
\[I_{c}=\int_{A_{c}}z^{2}dA \tag{27}\]  
\[F_{c}=\int_{A_{c}}\psi^{2}dA \tag{28}\]\[S_{c}=\int_{A_{c}}z\psi dA \tag{29}\]  
\[J_{c}=\int_{A_{c}}(\partial\psi/\partial y)^{2}dA \tag{30}\]  
To solve the bending vibration differential equation, assume that  
\[U(x,t)=U_{1}(x)\sin(\omega t+\varphi) \tag{31}\]  
\[w(x,t)=w_{1}(x)\sin(\omega t+\varphi) \tag{32}\]  
\[\theta(x,t)=\theta_{1}(x)\sin(\omega t+\varphi) \tag{33}\]  
\[d^{k}=\frac{\partial^{k}}{\partial x^{k}} \tag{34}\]  
Substituting Equations (31)-(34) into Equations (17)-(19),  
\[\big{(}G_{e}A_{s}d^{2}+A_{\rho}\omega^{2}\big{)}w_{1}-G_{e}A_{s}d\theta_{1}=0 \tag{35}\]  
\[\big{(}E_{c}F_{c}d^{2}-G_{c}I_{c}+F_{\rho}\omega^{2}\big{)}U_{1}-\big{(}E_{c} S_{c}d^{2}+S_{\rho}\omega^{2}\big{)}\theta_{1}=0 \tag{36}\]  
\[\big{(}-E_{c}S_{c}d^{2}-S_{\rho}\omega^{2}\big{)}U_{1}+G_{e}A_{s}dw_{1}+ \big{(}E_{c}I_{c}d^{2}+I_{\rho}\omega^{2}-G_{e}A_{s}\big{)}\theta_{1}=0 \tag{37}\]  
The characteristic equations of Equations (36) and (37) are  
\[\left|\begin{array}{ccc}E_{c}F_{c}d^{2}+F_{\rho}\omega^{2}-G_{c}I_{c}&0&-E_ {c}S_{c}d^{2}-S_{\rho}\omega^{2}\\ 0&G_{e}A_{s}d^{2}+A_{\rho}\omega^{2}&-G_{e}A_{s}d\\ -E_{c}S_{c}d^{2}-S_{\rho}\omega^{2}&G_{e}A_{s}d&E_{c}I_{c}d^{2}+I_{\rho}\omega ^{2}-G_{e}A_{s}\end{array}\right|=0 \tag{38}\]  
where \(|\ |\) is the matrix determinant; the same notation is used hereinafter.  
The solutions of Equations (35)-(37) can be expressed as  
\[U_{1}=\sum_{i=1}^{6}\alpha_{i}\beta_{1i}\exp(d_{i}x) \tag{39}\]  
\[w_{1}=\sum_{i=1}^{6}\alpha_{i}\beta_{2i}\exp(d_{i}x) \tag{40}\]  
\[\theta_{1}=\sum_{i=1}^{6}\alpha_{i}\beta_{3i}\exp(d_{i}x) \tag{41}\]  
\[\beta_{1i}=\big{(}E_{c}S_{c}d_{i}^{2}+S_{\rho}\omega^{2}\big{)}/\big{(}E_{c} F_{c}d_{i}^{2}+F_{\rho}\omega^{2}-G_{c}I_{c}\big{)} \tag{42}\]  
\[\beta_{2i}=(G_{e}A_{s}d_{i})/\big{(}G_{e}A_{s}d_{i}^{2}+A_{\rho}\omega^{2} \big{)} \tag{43}\]  
\[\beta_{3i}=1 \tag{44}\]  
where \(\{\alpha\}=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{6}\}^{T}\) is the integration constant vector; \(d_{i}(i=1,2,\cdots,6)\) is the characteristic root of the characteristic equation Equations (17)-(19); and \(\beta_{i}=\{\beta_{1i},\beta_{2i},\beta_{3i}\}^{T}\) is the characteristic vector corresponding to the characteristic root \(d_{i}\).