## 4 Numerical Investigation
### Flutter and Divergent Instability
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**C**} & \multicolumn{2}{c}{\(\omega_{\mathbf{11}}\)} & \multicolumn{2}{c}{\(\omega_{\mathbf{12}}\)} \\ \cline{2-5}  & \(\alpha\) = 0.01 & \(\alpha\) = 0.10 & \(\alpha\) = 0.01 & \(\alpha\) = 0.10 \\ \hline
0 & 0.4860 + 0.0297i & 0.4791 + 0.0273i & 0.8345 + 0.0296i & 0.7814 + 0.0219i \\
2 & 0.4761 + 0.0290i & 0.4680 + 0.0265i & 0.8248 + 0.0291i & 0.7692 + 0.0214i \\
4 & 0.4467 + 0.0270i & 0.4359 + 0.0244i & 0.7968 + 0.0277i & 0.7350 + 0.0202i \\
6 & 0.3994 + 0.0240i & 0.3844 + 0.0214i & 0.7532 + 0.0259i & 0.6837 + 0.0186i \\
8 & 0.3345 + 0.0203i & 0.3143 + 0.0177i & 0.6961 + 0.0228i & 0.6179 + 0.0169i \\
10 & 0.2509 + 0.0156i & 0.2226 + 0.0129i & 0.6263 + 0.0214i & 0.5380 + 0.0151i \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensionless natural frequencies for the axially-moving nanoplate resting on a viscoelastic foundation (\(\times 10^{2}\)). instability type of the axially moving nanoplate resting on a viscoelastic foundation was similar to that without a foundation. However, the divergence speeds for \(\omega_{11}\) and \(\omega_{21}\) (C\({}_{\rm div1}\) = 13.3 and C\({}_{\rm div2}\) = 18.3) were larger than those without a foundation (C\({}_{\rm div1}\) = 12.8 and C\({}_{\rm div2}\) = 17.9). This means the divergence speed was more obviously affected by the viscoelastic foundation. Particularly, it is worth pointing out that the phenomenon of mode-couple flutter instabilities of \(\omega_{11}\) coupling \(\omega_{21}\) and \(\omega_{12}\) coupling \(\omega_{22}\) disappeared when the moving nanoplate rested on a viscoelastic foundation. From the Figures 3 and 4, it was found that the viscoelastic foundation had a significant impact on the dynamic instability of the nanoplate. So, the effects of viscoelastic foundation model parameters including the Winkler's modulus \(k_{w}\), the [PERSON]'s modulus \(k_{G}\), and the damping \(c_{t}\) on the dimensionless complex frequencies of the axially-moving nanoplate are respectively investigated in the following text. Figure 5 shows the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Winkler's modulus \(k_{w}=0\), \(k_{w0}\), \(2k_{w0}\) (the [PERSON]'s modulus \(k_{G}\) = 0 and the damping \(c_{t}\) = 0), respectively. Note that the instability type of the axially-moving nanoplate along with the increase of the Winkler's modulus \(k_{w}\) included divergence instability and mode-couple  
Figure 4: Dimensionless complex frequencies vs axially moving speed for the nanoplate resting on a viscoelastic foundation. Figure 3: Dimensionless complex frequencies vs axially moving speed for the nanoplate without a foundation. flutter instability. For divergence instability, as the increase of the Winkler's modulus \(k_{w}\), divergence instability zeros for \(\omega_{11}\) and \(\omega_{12}\) were enlarged and the corresponding divergence speeds \(\mathrm{C_{div1}}\) and \(\mathrm{C_{div2}}\) were increased. For mode-couple flutter instability, flutter speeds \(\mathrm{C_{flu1}}\) and \(\mathrm{C_{flu2}}\) were also increased by raising the \(k_{w}\). This indicates that the divergence instability and flutter instability is delayed when the \(k_{w}\) increases. Besides, the dimensionless natural frequencies \(\omega_{11}\) and \(\omega_{12}\) increased before their divergence instability along with the increase of the \(k_{w}\), whereas \(\omega_{21}\) and \(\omega_{22}\) increased before their flutter instability. Figure 6 gives the vibration of the divergence speeds \(\mathrm{C_{div1}}\) and \(\mathrm{C_{div2}}\) and the flutter speeds \(\mathrm{C_{flu1}}\) and \(\mathrm{C_{flu2}}\) along with the increase of \(k_{w}\). The changes of flutter speeds were dominant over divergence speeds, which indicates that the flutter instability of the nanoplate is more sensitive to the [PERSON]'s modulus \(k_{w}\).