## 2 Basic Equations  
A rectangular nanoplate resting on a viscoelastic foundation is shown in Figure 1. The length, width and uniform thickness of the nanoplate are denoted as \(l_{a}\), \(l_{b}\) and \(h\), respectively. The viscoelasticfoundation is represented by a visco-Pasternak foundation model, whose [PERSON]'s modulus parameter is \(k_{w}\), [PERSON]'s modulus parameter is \(k_{G}\) and the damping parameter is \(c_{\rm t}\). The axially-moving velocity of the nanoplate is \(v\). The nanoplate is subjected to a biaxial force \(N_{Px}\) and \(N_{Py}\) along with \(x\) and \(y\) directions according to the rectangular coordinate system \(\alpha xyz\) shown in Figure 1.  
Based on the nonlocal elasticity theory [8], the basic constitutive equations for the nanoplate can be written as:  
\[ \begin{array}{c}\Big{(}1-(\varepsilon_{0}a)^{2}\
abla^{2}\Big{)}\sigma_{ij }=c_{ijkl}\varepsilon_{kl},\\ \sigma_{ij,j}=\rho\ddot{u}_{i},\\ \varepsilon_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,i}\big{)},\end{array}  \tag{1}\]  
where \(u_{i}\), \(\sigma_{ij}\) and \(\varepsilon_{ij}\) are the components of nonlocal displacement, stress and strain vector, respectively. \(c_{ijkl}\) and \(\rho\) are the elastic constants and mass density of the nanoplate, respectively. Moreover, \(\varepsilon_{0}a\) is the nonlocal parameter and \(\
abla^{2}\) is the Laplacian operator, respectively. For simplicity, the displacement fields of the nanoplate can be obtained by neglecting the deformation of the middle surface, given by:  
\[ \begin{cases}&u(x,y,z,t)=-z\frac{\partial uv}{\partial x}\\ &v(x,y,z,t)=-z\frac{\partial v}{\partial y}\\ &w(x,y,z,t)=w(x,y,t)\end{cases} , \tag{2}\]  
where \(u\), \(v\) and \(w\) are, respectively, the displacement components in the \(x\), \(y\) and \(z\) directions. Based on the strain-displacement equations of thin plate theory, the nonzero strain components are given as:  
\[ \begin{cases}&\varepsilon_{xx}=-z\frac{\partial^{2}uv}{\partial x^{2}}\\ &\varepsilon_{yy}=-z\frac{\partial^{2}uv}{\partial y^{2}}\\ &\gamma_{xy}=-2z\frac{\partial^{2}uv}{\partial y^{2}}\end{cases} . \tag{3}\]  
Figure 1: The axially-moving nanoplate resting on a viscoelastic foundation. Based on Equations (1) and (3) and the classical plane stress relation, the nonlocal stress is thus derived as the following form:  
\[\left(1-(e_{0}a)^{2}\
abla^{2}\right)\left\{ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{xy}\end{array} \right\}=\frac{E}{1-\mu^{2}}\left[ \begin{array}{ccc}1 &\mu&0\\ \mu&1&0\\ 0&0&\frac{1+\mu}{2}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{xx} \\ \varepsilon_{yy}\\ \tau_{xy}\end{array} \right\}, \tag{4}\]  
where \(E\) and \(\mu\) denote elastic modulus and Poisson's ratio, respectively. Further, the relation of the bending moment and stress in classical elasticity theory is still valid in the nonlocal stress. Then, the nonlocal bending moment can be presented by corresponding classical counterparts as:  
\[\left(1-(e_{0}a)^{2}\
abla^{2}\right)\left\{ \begin{array}{c}M_{xx}\\ M_{yy}\\ M_{xy}\end{array} \right\}=\left\{ \begin{array}{c}-D\Big{(}\frac{\partial^{2 }w}{\partial x^{2}}+\mu\frac{\partial^{2}w}{\partial y^{2}}\Big{)}\\ -D\Big{(}\frac{\partial^{2}w}{\partial y^{2}}+\mu\frac{\partial^{2}w}{\partial x ^{2}}\Big{)}\\ -D(1-\mu)\frac{\partial^{2}w}{\partial x\partial y}\end{array} \right\}, \tag{5}\]  
where \(D=\frac{Et^{3}}{12(1-\mu^{2})}\). Subsequently, the governing equations and boundary conditions for transverse vibration of the axially-moving nanoplate resting on a viscoelastic foundation can be derived utilizing the variational approach. The strain energy \(\Pi_{U}\) of the nanoplate can be stated as:  
\[\Pi_{U}=\frac{1}{2}\int_{A}\int_{-h/2}^{h/2}\big{(}\sigma_{xx}\varepsilon_{xx} +\sigma_{xy}\tau_{xy}+\sigma_{yy}\varepsilon_{yy}\big{)}\mathrm{d}z\mathrm{d }A, \tag{6}\]  
where \(A\) is the area of the mid-plane of the nanoplate.