## 2 Theoretical Problem
\begin{split}&A_{11}=b\int_{z_{1}}^{z_{1}}\frac{E(z)}{1-\theta^{2} }dz+b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{k}}Q_{11}^{(k)}dz+b\int_{z_{4}}^{z_{5}} \frac{E(z)}{1-\theta^{2}}dz\\ &B_{11}=b\int_{z_{0}}^{z_{1}}\frac{E(z)}{1-\theta^{2}}dz+b\sum_{k=2}^{n=4} \int_{z_{k-1}}^{z_{k}}Q_{11}^{(k)}dz+b\int_{z_{4}}^{z_{5}}\frac{E(z)}{1-\theta^ {2}}dz\\ &D_{11}=b\int_{z_{0}}^{z_{1}}\frac{E(z)}{1-\theta^{2}}z^{2}dz+b\sum_{k=2}^{n=4} \int_{z_{k-1}}^{z_{k-1}}Q_{11}^{(k)}z^{2}dz+b\int_{z_{4}}^{z_{5}}\frac{E(z)}{1- \theta^{2}}z^{2}dz\\ &N_{\rm T}=-b\int_{z_{0}}^{z_{1}}\frac{E(z)}{1-\theta^{2}}\alpha(z)\Delta Tdz -b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{k}}Q_{11}^{(k)}\alpha^{(k)}x\Delta Tdz-b \int_{z_{4}}^{z_{5}}\frac{E(z)}{1-\theta^{2}}\alpha(z)\Delta Tdz\\ &M_{\rm T}=-b\int_{z_{0}}^{z_{1}}\frac{E(z)}{1-\theta^{2}}\alpha(z)z\Delta Tdz -b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{k}}Q_{11}^{(k)}\alpha^{(k)}x\Delta Tdz-b \int_{z_{4}}^{z_{5}}\frac{E(z)}{1-\theta^{2}}\alpha(z)z\Delta Tdz\end{split}  \tag{14}\]  
The first variation of kinetic energy of the composite laminate beam is defined as [34]  
\[\delta K=\int_{0}^{L}\int_{-\frac{b}{2}}^{\frac{b}{2}}\left[\sum_{k=1}^{n}\int_{ Z_{k-1}}^{Z_{k}}\rho^{(k)}\left(\frac{\partial u}{\partial t}\right)^{T}\delta\left( \frac{\partial u}{\partial t}\right)dz\right]dydx \tag{15}\]  
Using the displacement field in Equations (1)-(3),(15), the first variation of kinetic energy over the domain [0,\(T\)] is obtained as \[\int_{0}^{T}\delta kt=\int_{0}^{T}\left\{\underbrace{b_{0}^{\int_{0}^{t} \int_{0}^{t}\int_{0_{0}}^{t}\left[\rho(z)\left(\frac{\partial u}{\partial t} \right)^{T}\delta\left(\frac{\partial u}{\partial t}\right)\right]dx+b_{0}^{ \int_{0}^{t}\left[\sum_{k=2}^{t}\int_{2_{k-1}}^{2_{k}}\rho^{(k)}\left(\frac{ \partial u}{\partial t}\right)^{T}\delta\left(\frac{\partial u}{\partial t} \right)dz\right]dx}}_{\text{Liver loop(Coupling Limit)}}dx+\underbrace{b_{0}^{\int_{0}^{t} \int_{4}^{u}\left[\rho(z)\left(\frac{\partial u}{\partial t}\right)^{T}\delta \left(\frac{\partial u}{\partial t}\right)dz\right]dx}}_{\text{Liver loop(Coupling Limit)}}dx+\right. \tag{16}\]  
where  
\[I_{0}=b\int_{z_{0}}^{z_{1}}\rho(z)dz+b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{k}} \rho^{(k)}dz+b\int_{z_{4}}^{z_{5}}\rho(z)dz \tag{17}\]  
\[I_{1}=b\int_{z_{0}}^{z_{1}}\rho(z)zdz+b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{k}} \rho^{(k)}zdz+b\int_{z_{4}}^{z_{5}}\rho(z)zdz \tag{18}\]  
\[I_{2}=b\int_{z_{0}}^{z_{1}}\rho(z)z^{2}dz+b\sum_{k=2}^{n=4}\int_{z_{k-1}}^{z_{ k}}\rho^{(k)}z^{2}dz+b\int_{z_{4}}^{z_{5}}\rho(z)z^{2}dz \tag{19}\]  
The first variation of the virtual work carried out by the external forces takes the following form,  
\[\delta W^{Ext}=\int_{0}^{L}[F_{u}(x,t)\delta u+F_{w}(x,t)\delta w]dx+\left[N \delta u+\bar{\mathcal{V}}\delta w+\bar{\mathcal{M}}\delta\left(\frac{ \partial w}{\partial x}\right)\right]\left| \begin{matrix}x=L\\ x=0\end{matrix} \right. \tag{20}\]  
where \(F_{u}\) and \(F_{w}\) refer to the distributed external forces along \(x\) and \(z\) directions, respectively. Additionally, \(\bar{N}\), \(\bar{\mathcal{V}}\) and \(\bar{\mathcal{M}}\) stand for the axial force, transverse shear force, and bending moment, respectively, acting on the end sections of the beam. The governing equation of the system can be obtained based on the Hamiltonian principle as [35]  
\[\delta\int_{0}^{T}\left[K-U_{s}+W^{Ext}\right]dt=0 \tag{21}\]  
where \(\delta\) is the variational symbol. Substituting Equations (13), (16) and (20) into Equation (21) leads to the following governing equations  
\[\frac{\partial}{\partial x}\Bigg{[}A_{11}\left(\frac{\partial u}{\partial x}+ \frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)-B_{11}\frac{ \partial^{2}w}{\partial x^{2}}+N_{T}\Bigg{]}+F_{u}=I_{0}\frac{\partial^{2}u}{ \partial t^{2}}-I_{1}\frac{\partial^{3}w}{\partial x\partial t^{2}} \tag{22}\]  
\[