## 4 Numerical Investigation
### Flutter and Divergent Instability  
The flutter instability refers to a dynamic process of structures transiting from stability to instability with a nonvanishing frequency, whereas the divergence instability refers to a structure passing from stability to instability at a zero frequency. In the following text, the mode-couple flutter and divergence instability of an axially-moving nanoplate resting on a viscoelastic foundation with simply supported boundary condition are investigated. The small parameter of the nanoplate was \(\alpha\) = 0.001 and other parameters were the same as Section 4.1. For comparison, the first four dimensionless natural frequencies \(\omega_{11}\), \(\omega_{12}\), \(\omega_{21}\) and \(\omega_{22}\) of the axially-moving nanoplate without a foundation with respect to moving speeds are plotted in Figure 3. As shown in the figure, the dimensionless frequencies were real numbers at the dimensionless speed C = 0. Along with the increase of axially moving speed, the first four dimensionless natural frequencies changed but remained real numbers before C = 12.8. This showed that the nanoplate was stable before C = 12.8. At the speed of C = 12.8, the real part of \(\omega_{11}\) decreased to zero and its imaginary part had two branches. This indicated the nanoplate began divergence instability on its first order mode. C = 12.8 was a threshold critical value of \(\omega_{11}\), denoted as the divergence speed C\({}_{\text{div1}}\). Moreover, \(\omega_{11}\) of the nanoplate was unstable in the interval [12.8, 14.1], but \(\omega_{12}\), \(\omega_{21}\) and \(\omega_{22}\) still kept stable. When the axially moving speed further increased to C = 16.2, \(\omega_{11}\) and \(\omega_{21}\) of the nanoplate coupled with each other into a pair of complex conjugate frequency. This means that the nanoplate exhibited a coupled-mode flutter instability. C = 16.2 was denoted as the flutter speed C\({}_{\text{flu1}}\). When C = 17.9, denoted as the divergence speed C\({}_{\text{div2}}\), the real part of \(\omega_{12}\) decreased to zero with its imaginary part becoming two branches, representing the nanoplate becoming divergent instable again on its second order mode. The nanoplate stayed unstable again during the interval [17.9, 25.5]. At the speed of C = 26.2, denoted as the flutter speed C\({}_{\text{flu2}}\), the nanoplate exhibited coupled-mode flutter instability again with \(\omega_{12}\) and \(\omega_{22}\) merging together. In contrast with Figure 3, Figure 4 presents the first four order dimensionless natural frequencies \(\omega_{11}\), \(\omega_{12}\), \(\omega_{21}\) and \(\omega_{22}\) of the axially-moving nanoplate resting on a viscoelastic foundation with respect to moving speeds. As can be seen, the dimensionless frequencies only had a real component at the speed of C = 0, but the values were slightly larger than those without a foundation. This implies that the rigidity of the nanoplate was enhanced due to the viscoelastic foundation. When the moving speed C > 0, the first four dimensionless frequencies \(\omega_{11}\), \(\omega_{12}\), \(\omega_{21}\) and \(\omega_{22}\) became complex numbers, which were significantly different from vibrations of the nanoplate without a foundation. Also, the divergence