## 3 Solution Method  
In this section, the GSDTFM was adopted to investigate the dynamic instability of the axially-moving nanoplate resting on a viscoelastic foundation. Firstly, the nanoplate was divided into a couple of strip elements along the \(y\) direction and there are \(NE\) strip elements and \((NE+1)\) nodal lines for the nanoplate in all, as shown in Figure 2. The length of the \(j\)th strip element is \(l_{a}\) and its width is \(l_{e}\). The transverse displacement \(w(x,y,t)\) of the \(j\)th strip element can be interpolated by:  
\[w(x,y,t)=\mathbf{N}(y)\delta_{e}(x,t), \tag{18}\]  
where \(\mathbf{N}(y)\) is the shape function and \(\delta_{e}(x,t)\) is the unknown displacement of the \(e^{\text{th}}\) strip element. The expression of \(\mathbf{N}(y)\) and \(\delta_{e}(x,t)\) are given in literature [39], which are not herein presented for the sake of brevity. According to the dimensionless quantities defined in Equation (15), Equation (18) can be rewritten as:  
\[W(X,Y,T)=\overline{\mathbf{N}}(Y)\overline{\delta}_{e}(X,T), \tag{19}\]  
where  
\[ \begin{array}{c}\overline{\mathbf{N}}(Y)=\left[\begin{array}{cc}1-3\frac{ \gamma^{2}}{L^{2}}+2\frac{\gamma^{3}}{L^{3}}&l_{b}\Big{(}Y-2\frac{\gamma^{2}}{ L}+\frac{\gamma^{3}}{L^{2}}\Big{)}&3\frac{\gamma^{2}}{L^{2}}-2\frac{\gamma^{3}}{L^{3}}&l_{b} \Big{(}-\frac{\gamma^{2}}{L}+\frac{\gamma^{3}}{L^{2}}\Big{)}\end{array}\right] \\ \overline{\delta}_{e}(X,T)=\left[\begin{array}{cc}W_{j}&\Theta_{j}&W_{j+1}& \Theta_{j+1}\end{array}\right]^{\text{T}},\big{\{}\Theta_{j},\Theta_{j+1} \big{\}}=\frac{1}{h}\{\theta_{j},\theta_{j+1}\},L=\frac{l_{e}}{l_{b}}.\end{array}  \tag{20}\]  
Substituting Equation (19) into Equation (16) with \(\overline{\mathbf{H}}=\overline{\mathbf{N}}^{\text{T}}\) results in the following:  
\[ \begin{array}{c}\overline{\mathbf{K}}_{e}^{4}\frac{\partial^{4}\overline{ \delta}_{e}}{\partial X^{4}}+\overline{\mathbf{K}}_{e}^{2}\frac{\partial^{2} \overline{\delta}_{e}}{\partial X^{2}}+\overline{\mathbf{K}}_{e}^{0}\overline{ \delta}_{e}+\overline{\mathbf{K}}_{e,1}^{1}\frac{\partial^{2}\overline{\delta} _{e}}{\partial X\partial T}+\overline{\mathbf{m}}_{e}^{2}\frac{\partial^{2} \overline{\delta}_{e}}{\partial T^{2}}+\overline{\mathbf{K}}_{e,2}^{2}\frac{ \partial^{4}\overline{\delta}_{e}}{\partial X^{2}\partial T^{2}}+\overline{ \mathbf{K}}_{e,1}^{3}\frac{\partial^{4}\overline{\delta}_{e}}{\partial X^{3} \partial T}=\mathbf{0},\end{array}  \tag{21}\]  
where  
\[\begin{array}{c}\overline{\mathbf{K}}_{e}^{4}=\left[-1-\alpha^{2}K_{1}+ \alpha^{2}C^{2}+\alpha^{2}K_{G}\right]\int_{0}^{L}\overline{\mathbf{N}}^{ \text{T}}\overline{\mathbf{N}}\text{d}Y,\\ \overline{\mathbf{K}}_{e}^{2}=\left[-\mu\lambda^{2}+2\alpha^{2}K_{G}\lambda^{2} -\alpha^{2}\lambda^{2}K_{1}-\alpha^{2}\lambda^{2}K_{2}+\alpha^{2}\lambda^{2}C ^{2}\right]\int_{0}^{L}\overline{\mathbf{N}}^{\text{T}}\frac{\partial^{2} \overline{\mathbf{N}}}{\partial Y^{2}}\text{d}Y\\ +\left[2\lambda^{2}(1-\mu)\right]\int_{0}^{L}\frac{\partial\mathbf{N}^{\text{T }}}{\partial Y}\frac{\partial\mathbf{N}}{\partial Y}\text{d}Y+\left[-K_{G}- \alpha^{2}K_{W}+K_{1}-C^{2}\right]\int_{0}^{L}\overline{\mathbf{N}}^{\text{T }}\overline{\mathbf{N}}\text{d}Y-\mu\lambda^{2}\int_{0}^{L}\frac{\partial^{2} \overline{\mathbf{N}}^{\text{T}}}{\partial Y^{2}}\overline{\mathbf{N}}\text{d }Y,\\ \overline{\mathbf{K}}_{e}^{0}=\left[-\lambda^{2}K_{G}-\lambda^{2}\alpha^{2}K_{ \
u}+\lambda^{2}K_{2}\right]\int_{0}^{L}\overline{\mathbf{N}}^{\text{T}} \frac{\partial^{2}\overline{\mathbf{N}}}{\partial Y^{2}}\text{d}Y+K_{W}\int_{0} ^{L}\overline{\mathbf{N}}^{\text{T}}\text{d}Y \tag{22}\]  
Figure 2: Strip elements for a rectangular nanoplate. where \(\overline{\mathbf{K}}_{e}^{4},\overline{\mathbf{K}}_{e}^{2},\overline{\mathbf{K}}_{e}^ {0},\overline{\mathbf{K}}_{e,1}^{1},\overline{\mathbf{K}}_{e,1}^{3},\overline{ \mathbf{K}}_{e,2}^{2}\) denote stiffness matrices of the strip element and \(\overline{\mathbf{m}}_{e}\) denotes mass matrice of the strip element.