## 3 Governing Equations of CNTRC Beams
### The Governing Equation for CNTRCs Considering the Thickness Effect  
According to [30], by considering the thickness effect in the composite beam, the total stress can be expressed as:  
\[t_{xx}=\sigma_{xx}+\sigma_{xxx}^{(1)}+\sigma_{xxx}^{(1)} \tag{33}\]  
where \(\sigma_{xx}\) is the nonlocal stress, and \(\sigma_{xxx}^{(1)}\) and \(\sigma_{xxx}^{(1)}\) are higher-order nonlocal stresses. The virtual strain energy is:  
\[\begin{split}&\delta U=\int_{(V)}\ (\sigma_{xx}\delta e_{xx}+\sigma_{xx} \delta\gamma_{xx}+\sigma_{xxx}^{(1)}\sigma_{xx,x}+\sigma_{xxx}^{(1)}\sigma_{ xx}\sigma_{7 xz,z})dV=\int_{0}^{L}(N\delta f+M\delta y+\\ &\quad P\delta h+\mathrm{Q}\Big{(}\frac{\partial w}{\partial x}- \theta\Big{)}\delta h\Big{)}\mathrm{d}x+\Big{[}N^{(1)}\delta f+M^{(1)}\delta g +P^{(1)}\delta h+\mathrm{Q}^{(1)}\delta(w^{\prime}-\theta)\Big{]}_{0}^{L}\\ &\quad\quad\quad\quad\quad(N,M,P)^{T}=\int_{A}\sigma_{xx}(1,z,H(z) )^{T}dA;\ Q=\int_{A}\sigma_{xx}\frac{dH}{dz}dA;\ \Big{(}N^{(1)},M^{(1)},P^{(1)}\Big{)}^{T}=\\ &\quad\  
\[\delta f=\delta u^{\prime}+w^{\prime}\delta w^{\prime};\ \delta g=\left(1-\frac{3}{2}{w^{ \prime}}^{2}\right)\delta w^{\prime\prime}-3w^{\prime}{w^{\prime\prime}}^{2} \delta w^{\prime};\ \delta h=\delta w^{\prime\prime}-\delta\theta^{\prime} \tag{36}\]  
By substituting Equation (36) into Equation (34), after some manipulations, it holds that:  
\[ \begin{array}{c}\delta U=-\int_{0}^{L}\frac{\partial N}{\partial x}dx\delta u +3\int_{0}^{L}\left[\frac{\partial}{\partial x}\left(-Mw^{\prime}w^{\prime \prime}-Nw^{\prime}\Big{(}1-\frac{1}{2}{w^{\prime}}^{2}\Big{)}\right)-\frac{ \partial Q}{\partial x}+\frac{\partial^{2}}{\partial x^{2}}\Big{(}P-M\Big{(}1 -\frac{3}{2}{w^{\prime}}^{2}\Big{)}\Big{)}\right]dx\partial w^{\prime}+\\ \int_{0}^{L}\left(\frac{\partial P}{\partial x}-Q\right)dx\delta\theta+\Big{[} N\delta u+\Big{(}Nw^{\prime}\Big{(}1-\frac{1}{2}{w^{\prime}}^{2}\Big{)}+3 Mw^{ \prime\prime}\Big{)}\delta w+\Big{(}-M\Big{(}-1-\frac{3}{2}{w^{\prime}}^{2} \Big{)}+P\Big{)}\delta w^{\prime}\\ -\frac{\partial}{\partial x}(P-\ M\Big{(}1-\frac{3}{2}{w^{\prime}}^{2}\Big{)} \Big{)}\delta w-P\delta\theta\big{|}|_{0}^{L}\end{array}  \tag{37}\]  
The virtual kinetic energy of CNT is given by:  
\[ \begin{array}{c}\delta K=\int_{A}\int_{0}^{L}\rho(z)\dot{U}_{1}(x,y,z,t) \delta\dot{U}_{1}(x,y,z,t)+\rho(z)\dot{U}_{3}(x,y,z,t)\delta\dot{U}_{3}(x,y,z, t)dAdt\end{array}  \tag{38}\]  
where the dot denotes the derivative with respect to time. Let  
\[(I_{0},I_{1},I_{2},I_{3},I_{4},I_{5})=\int_{A}\rho(z)\Big{(}1,z,z^{2},H(z),zH(z ),H^{2}(z))dA \tag{39}\]  
Taking into account Equations (13), (15), (21), and (39), the virtual kinetic energy becomes:  
\[ \begin{array}{c}\delta K=\int_{0}^{L}\Big{[}I_{0}\dot{u}(x,t)+(I_{3}-I_{1}) \frac{\partial\dot{u}(x)}{\partial x}-I_{3}\dot{\theta}(x,t)\Big{]}dx\dot{u}+ \int_{0}^{L}\Big{[}(I_{3}-I_{1})\dot{u}(x,t)+(I_{5}-2I_{4}+I_{2})\frac{ \partial\dot{w}(x,t)}{\partial x}\Big{]}dx+\\ (I_{4}-I_{5})\dot{\theta}(x,t)]dx\delta\frac{\partial\dot{w}(x,t)}{\partial x} +\int_{0}^{L}\Big{[}-I_{3}\dot{u}(x,t)+(I_{4}-I_{5})\frac{\partial\dot{w}(x,t )}{\partial x}-I_{5}\dot{\theta}(x,t)\Big{]}dx\delta\theta(x,t)+\int_{0}^{L} \dot{w}(x,t)\delta\dot{w}(x,t)dx\end{array}  \tag{40}\]  
The virtual potential energy is given by the electromagnetic actuator, the nonlinear elastic Winkler-Pasternak foundation, the mechanical impact, and the compressive force P:  
\[\overline{q}=\frac{1}{2}\frac{C_{0}V_{x}^{2}}{[d-w]^{2}}-\frac{1}{2}\frac{C_ {0}V_{x}^{2}}{[d+w]^{2}}-K_{1}w-K_{3}w^{3}+F_{0}\delta_{D}\bigg{(}x-\frac{L}{ 2}\bigg{)}+Pw^{\prime} \tag{41}\]  
where \(C_{0}\), \(V_{x}\), \(d\), \(K_{1}\), \(K_{3}\), and \(F_{0}\) are capacitances of the actuator, voltage, the gap width, linear and nonlinear coefficients of the Winkler-Pasternak elastic foundation, and amplitude of the load, and \(\delta_{D}\) is the delta-Dirac function.