## 4 Numerical Investigation
### Flutter and Divergent Instability
Figure 7 presents the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the [PERSON]'s modulus \(k_{G}=0\), \(k_{G0}\), \(2k_{G0}\) (the [PERSON]'s modulus \(k_{w}=0\) and the damping \(c_{t}=0\)), respectively. Similarly to the [PERSON]'s modulus parameter \(k_{w}\), the instability type of the axially-moving nanoplate with \(k_{G}=0\), \(k_{G0}\), \(2k_{G0}\) still included divergence instability and flutter instability. However, for divergence instability, the instability zeros for \(\omega_{11}\) and \(\omega_{12}\) had a slightly change with the increase of the [PERSON]'s modulus \(k_{G}\), although  
Figure 5: Effect of the [PERSON]’s modulus \(k_{w}\) on dimensionless complex frequencies of the axially-moving nanoplate. Figure 6: Effect of the [PERSON]’s modulus \(k_{w}\) on the flutter and divergence speed of the axially-moving nanoplate. the corresponding divergence speeds \(\mathrm{C_{div1}}\) and \(\mathrm{C_{div2}}\) were still increased. For mode-couple flutter instability, flutter speeds \(\mathrm{C_{flu1}}\) and \(\mathrm{C_{flu2}}\) were also increased with the increase of the [PERSON]'s modulus \(k_{G}\). Also, the dimensionless natural frequencies \(\omega_{11}\) and \(\omega_{12}\) were increased before their divergence instability and \(\omega_{21}\) and \(\omega_{22}\) were also increased before their flutter instability with the increase of the [PERSON]'s modulus \(k_{G}\). The vibrations of divergence speeds \(\mathrm{C_{div1}}\) and \(\mathrm{C_{div2}}\) and flutter speeds \(\mathrm{C_{flu1}}\) and \(\mathrm{C_{flu2}}\) along with the increase of the [PERSON]'s modulus \(k_{G}\) are plotted in Figure 8, illustrating that changes of the flutter speeds are no more dominant than those of the divergence speeds. Figure 9 illustrates the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the damping \(c_{t}=0\), \(c_{t0}\), \(2c_{t0}\)(the [PERSON]'s modulus \(k_{w}=0\) and the Pasternak's modulus \(k_{G}=0\)), respectively. Obviously, the instability type of the axially-moving nanoplate was altered with the appearance of the damping. When the damping \(c_{t}=0\), there was divergence instability in \(\omega_{11}\), \(\omega_{12}\) and flutter instability in \(\omega_{11}\) coupling \(\omega_{21}\) and \(\omega_{12}\) coupling \(\omega_{22}\). However, when the damping \(c_{t}>0\), the flutter instability disappeared and only divergence instability in \(\omega_{11}\) and \(\omega_{12}\) was left. Furthermore, the modes \(\omega_{11}\) /\(\omega_{21}\) and \(\omega_{12}\)/\(\omega_{22}\), which coupled  
Figure 8: Effect of the [PERSON]’s modulus \(k_{G}\) on the flutter and divergence speed of the axially-moving nanoplate. Figure 7: Effect of the [PERSON]’s modulus \(k_{G}\) on dimensionless complex frequencies of the axially-moving nanoplate. when the damping \(c_{t}=0\), separated with the appearance of the damping \(c_{t}\). It was found that the damping \(c_{t}\) had almost no influences on the divergence speeds \(\mathrm{C}_{\mathrm{div1}}/\mathrm{C}_{\mathrm{div2}}\) and the real parts of the first two dimensionless complex frequencies \(\mathrm{Re}(\omega_{11})/\mathrm{Re}(\omega_{12})\) before their divergence instability, but intensively affected the imaginary part of dimensionless complex frequencies. Figure 10 plots the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a viscoelastic foundation with the small-parameter \(\alpha=0.001\), \(0.050\) and \(0.100\), respectively. As the small-parameter \(\alpha\) increased, divergence instability zeros for \(\omega_{11}\) and \(\omega_{12}\) were enlarged but the corresponding divergence speeds \(\mathrm{C}_{\mathrm{div1}}\) and \(\mathrm{C}_{\mathrm{div2}}\) were decreased.