## 2 Theory and Formulation
### Problem Description and Fundamental Equations
\[\delta W_{nc}=\frac{1}{2}\int_{0}^{L}\delta\left(K_{\rm W}(x)\xi^{2}(x,t)+K_{\rm P}(x )\left(\frac{\partial\xi(x,t)}{\partial x}\right)^{2}\right)\mathrm{d}x \tag{17}\]  
which are substituted in the Hamiltonian principle (14) to yield the following equations of motion  
\[u_{2}=\xi(x,t) \tag{10}\]0  
\[u_{2}=\xi(x,t) \tag{10}\]1  
\[u_{2}=\xi(x,t) \tag{10}\]2  
Along with the following boundary conditions  
\[u_{2}=\xi(x,t) \tag{10}\]3  
\[u_{2}=\xi(x,t) \tag{10}\]4  
\[u_{2}=\xi(x,t) \tag{10}\]5  
For symmetry reasons throughout the thickness, the inertial terms \(I_{0}\), \(I_{1}\), \(I_{2}\) become as follows  
\[\left\{ \begin{array}{c}I_{0}(x)\\ I_{2}(x)\end{array} \right\}=(\rho_{m}+(\rho_{\rm CNT}-\rho_{m})V_{\rm CNT}(x) )\bigg{\{} \begin{array}{c}A\\ I\end{array} \right\},\ I_{1}=0 \tag{23}\]  
The stiffness terms \(A\), \(D\) and \(B\) are determined as  
\[\left\{ \begin{array}{c}A(x)\\ D(x)\end{array} \right\}=\int\limits_{-h/2}^{h/2}\bigg{(}\frac{E_{m}+(e_{1}E_{1 1,{\rm CNT}}-E_{m})V_{\rm CNT}(x)}{1-v^{2}(x)}\bigg{)}\bigg{\{} \begin{array}[] {c}1\\ z^{2}\end{array} \bigg{\}}dz,\ B(x)=0 \tag{24}\]  
After a brief mathematical manipulation, the equations of motion and boundary conditions take the following form  
\[I_{0}(x)\frac{\partial^{2}\xi(x,t)}{\partial t^{2}}+K_{\rm W}(x)\xi(x,t)- \frac{\partial}{\partial x}\bigg{(}I_{2}(x)\frac{\partial^{3}\xi(x,t)}{ \partial x\partial t^{2}}+K_{\rm P}(x)\frac{\partial\xi(x,t)}{\partial x}- \frac{\partial D(x)}{\partial x}\frac{\partial^{3}\xi(x,t)}{\partial x^{3}} \bigg{)}=0 \tag{25}\]  
\[I_{0}(x)\frac{\partial\eta^{2}(x,t)}{\partial t^{2}}-\frac{\partial}{\partial x }\bigg{(}A(x)\frac{\partial\eta(x,t)}{\partial x}\bigg{)}=0 \tag{26}\]  
\[\widetilde{V}=(V_{\rm CNT-L}-V_{\rm CNT-R})(k)(1-x)^{(k)} \tag{27}\]  
By substituting Equations (18) and (19) into Equations (20) and (21) and using Equations (4), (5), (11) and (27), the governing equations of the problem become as follows  
\[ \begin{array}{c}\Big{(}\Big{(}1-\Big{(}1-e_{1}\frac{E_{11,{\rm CNT}}}{E_{m}} \Big{)}\Big{(}\frac{\widetilde{V}}{\widetilde{K}}+V_{\rm CNT-R}\Big{)}\Big{)} \frac{\partial^{2}\eta(x,t)}{\partial x^{2}}+\Big{(}1-e_{1}\frac{E_{11,{\rm CNT }}}{E_{m}}\Big{)}\frac{\widetilde{V}}{\Gamma-x}\frac{\partial\eta(x,t)}{ \partial x}\Big{)}\frac{1}{\gamma^{2}(1-v^{2})}+\\ +\Big{(}\Big{(}1-\frac{G_{\rm CNT}}{\rho_{m}}\Big{)}V_{\rm CNT}(x)-1\Big{)} \frac{\partial\eta^{2}(x,t)}{\partial t^{2}}=0\\ -\Big{(}1+\Big{(}\frac{G_{\rm CNT}}{\rho_{m}}-1\Big{)}\Big{(}\frac{\widetilde{ V}}{\widetilde{K}}+V_{\rm CNT-R}\Big{)}\Big{)}\frac{\partial^{2}\xi(x,t)}{ \partial t^{2}}+\Big{(}1+\Big{(}\frac{G_{\rm CNT}}{\rho_{m}}-1\Big{)}\Big{(} \frac{\widetilde{V}}{\widetilde{K}}+V_{\rm CNT-R}\Big{)}\Big{)}\gamma^{2}\frac{ \partial^{2}\xi(x,t)}{\partial x^{2}\partial t^{2}}+\\ -\Big{(}1-\frac{G_{\rm CNT}}{\rho_{m}}\Big{)}\frac{\widetilde{V}}{\Gamma-x} \gamma^{2}\frac{\partial^{2}\xi(x,t)}{\partial x\partial t^{2}}-\Big{(}\Big{(}1 +\Big{(}e_{1}\frac{E_{11,{\rm CNT}}}{E_{m}}-1\Big{)}\Big{(}\frac{\widetilde{V}} {\widetilde{K}}+V_{\rm CNT-R}\Big{)}\Big{)}\frac{\partial^{4}\xi(x,t)}{ \partial x^{4}}+\\ +2\Big{(}1-e_{1}\frac{E_{11,{\rm CNT}}}{E_{m}}\Big{)}\frac{\widetilde{V}}{\Gamma-x }\frac{\partial^{2}\xi(x,t)}{\partial x^{3}}+\Big{(}e_{1}\frac{E_{11,{\rm CNT}}} {E_{m}}-1\Big{)}\frac{(\widetilde{K}-1)\widetilde{V}}{(1-x)^{2}}\frac{\partial ^{2}\xi(x,t)}{\partial x^{2}}\Big{)}\frac{1}{1-v^{2}}+\\ -\Big{(}m\beta x^{m-1}\frac{\partial\xi(x,t)}{\partial x}+K_{\rm W0}(1-x^{m}) \xi(x,t)\Big{)}+K_{\rm P0}(1-\beta x^{m})\frac{\partial^{2}\xi(x,t)}{\partial x ^{2}}=0\end{array}  \tag{29}\]in which \(\gamma=\sqrt{I/(AL^{2})}\). **3.