## 2 Basic Equations
\begin{array}{c}\int_{0}^{l}H\Big{(}-D\frac{\partial^{2}w}{\partial x^{2}}- \mu D\frac{\partial^{2}w}{\partial x^{2}}\Big{)}\mathrm{d}y-2\int_{0}^{l} \frac{\partial H}{\partial y}\Big{(}-D(1-\mu)\frac{\partial^{3}w}{\partial x^ {2}\partial y}\Big{)}\mathrm{d}y\\ +\int_{0}^{l}\frac{\partial^{2}H}{\partial y^{2}}\Big{(}-\mu D\frac{\partial^{ 2}w}{\partial x^{2}}-D\frac{\partial^{2}w}{\partial y^{2}}\Big{)}\mathrm{d}y -\int_{0}^{l}H\Big{(}1-(e_{0}a)^{2}\
abla^{2}\Big{)}\Big{(}k_{w}w-k_{G}\
abla^ {2}w+c_{t}\frac{\partial w}{\partial t}\Big{)}\mathrm{d}y\\ -\int_{0}^{l}H\Big{(}1-(e_{0}a)^{2}\
abla^{2}\Big{)}\Big{(}N_{Px}\frac{ \partial^{2}w}{\partial x^{2}}+Np_{y}\frac{\partial^{2}w}{\partial y^{2}} \Big{)}\mathrm{d}y\\ -\int_{0}^{l}H\Big{(}1-(e_{0}a)^{2}\
abla^{2}\Big{)}\rho h\Big{(}v^{2}\frac{ \partial^{2}w}{\partial x^{2}}+2v\frac{\partial^{2}w}{\partial t\partial x}+ \frac{\partial^{2}w}{\partial t^{2}}\Big{)}\mathrm{d}y=\mathbf{0}.\end{array}  \tag{14}\]  
For the sake of convenience and generality, the following dimensionless quantities are defined as:  
\[ \begin{array}{c}X=\frac{x}{l_{a}},Y=\frac{y}{l_{b}},W=\frac{w}{h},\lambda= \frac{l_{b}}{l_{a}},\alpha=\frac{\epsilon_{0}a}{l_{a}},T=\frac{t}{l_{b}^{2}} \sqrt{\frac{D}{\rho h}},C=l_{a}v\sqrt{\frac{\rho h}{D}},\\ K_{1}=Np_{x}\frac{l_{a}^{2}}{D},K_{2}=Np_{y}\frac{l_{b}^{2}}{D},K_{w}=\frac{l_{b} ^{2}}{D}k_{w},K_{G}=k_{G}\frac{l_{b}^{2}}{D},L=\frac{l_{b}^{2}}{D},C_{t}= \frac{l_{b}^{2}}{D}\sqrt{\frac{D}{\rho h}}c_{t}.\end{array}  \tag{15}\]  
Using these dimensionless terms, the governing Equation (14) can be rewritten as:  
\[ \begin{array}{c}\int_{0}^{L}\mathbf{H}\Big{(}-\frac{\partial^{4}w}{ \partial x^{2}}-\mu\lambda^{2}\frac{\partial^{4}w}{\partial x^{2}\partial x^{2 }}\Big{)}\mathrm{d}Y-2\int_{0}^{L}\frac{\partial^{2}w}{\partial x^{2}}-(2- \mu)\frac{\partial^{3}w}{\partial x^{2}\partial x^{2}}\Big{)}\mathrm{d}Y+\int_ {0}^{L}\frac{\partial^{2}\mathbf{H}}{\partial\mathbf{H}}\Big{(}-\mu\lambda^{2} \frac{\partial^{2}w}{\partial x^{2}}-\lambda^{4}\frac{\partial^{2}w}{\partial y^ {2}}\Big{)}\mathrm{d}Y\\ -\int_{0}^{L}\mathbf{H}\Big{(}1-\alpha^{2}\Big{(}\frac{\partial^{2}}{\partial x ^{2}}+\lambda^{2}\frac{\partial^{2}}{\partial y^{2}}\Big{)}\Big{)}\Big{(}K_{w} W-K_{G}\Big{(}\frac{\partial^{2}w}{\partial x^{2}}+\lambda^{2}\frac{ \partial^{2}w}{\partial y^{2}}\Big{)}W+C_{t}\frac{\partial w}{\partial y^{2}} \Big{)}\mathrm{d}Y,\\ -\int_{0}^{L}\mathbf{H}\Big{(}1-\alpha^{2}\Big{(}\frac{\partial^{2}}{\partial x ^{2}}+\lambda^{2}\frac{\partial^{2}}{\partial y^{2}}\Big{)}\Big{)}K_{t}\frac{ \partial w}{\partial x^{2}}\mathrm{d}Y+\int_{0}^{L}\mathbf{H}\Big{(}1-\alpha^{2} \Big{(}\frac{\partial^{2}}{\partial x^{2}}+\lambda^{2}\frac{\partial^{2}}{ \partial x^{2}}\Big{)}\Big{)}K_{t}\frac{\partial w}{\partial x^{2}}\mathrm{d}Y +\int_{0}^{L}\mathbf{H}\Big{(}1-\alpha^{2}\Big{(}\frac{\partial^{2}}{\partial x ^{2}}+\lambda^{2}\frac{\partial^{2}}{\partial y^{2}}\Big{)}\Big{)}K_{t}^{2} \lambda^{2}\frac{\partial^{2}w}{\partial t^{2}}\mathrm{d}Y\\ -\int_{0}^{L}\mathbf{H}\Big{(}1-\alpha^{2}\Big{(}\frac{\partial^{2}}{\partial x ^{2}}+\lambda^{2}\frac{\partial^{2}}{\partial y^{2}}\Big{)}\Big{(}C^{2}\frac{ \partial^{2}w}{\partial x^{2}}+2C\frac{\partial^{2}w}{\partial t\partial x}+ \frac{\partial^{2}w}{\partial t^{2}}\Big{)}\mathrm{d}Y=\mathbf{0}.\end{array}  \tag{16}\]  
According to Equation (12), boundary conditions for the axially-moving nanoplate resting on a viscoelastic foundation can be obtained. For simplicity, in the following analysis, we only consider the simply supported boundary constraints for the nanoplate, which are given as follows:  
\[ \begin{array}{c}W(X,Y)\big{|}_{X=0,1}=\left.\frac{\partial W(X,Y)}{\partial X }\right|_{X=0,1}=0,\\ W(X,Y)\big{|}_{Y=0,1}=\left.\frac{\partial W(X,Y)}{\partial Y}\right|_{Y=0,1}=0. \end{array}  \tag{17}\]