## 4 Numerical Examples and Problem Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Orders**} & \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{\(k=0.1\)} \\ \cline{3-10}  & & \(v=0\) & \(v=0.5\) & \(v=1.0\) & \(v=1.5\) & \(v=2.0\) & \(v=2.5\) & \(v=3.0\) & \(v=4.0\) \\ \hline \multirow{2}{*}{\(\mathrm{Im}(\Omega_{1})\)} & Ref. [16] & 17.60 & 17.51 & 17.261 & 16.83 & 16.23 & 15.43 & 14.44 & 11.69 \\  & Present & 17.592882 & 17.509053 & 17.256451 & 16.831563 & 16.227915 & 15.434911 & 14.435610 & 11.686251 \\  & Ref. [16] & 70.37 & 70.32 & 70.14 & 69.86 & 69.45 & 68.93 & 68.28 & 66.62 \\  & Present & 70.372588 & 70.315273 & 70.143095 & 69.855354 & 69.450866 & 68.927940 & 68.284344 & 66.623139 \\  & Ref. [16] & 158.35 & 158.30 & 158.15 & 157.90 & 157.55 & 157.10 & 156.55 & 155.14 \\  & Present & 158.34866 & 158.29874 & 158.14891 & 157.89904 & 157.54885 & 157.09797 & 156.54593 & 155.13579 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The calculated values of the first three dimensionless natural complex frequencies of the FGM simply supported beam under different axial motion velocities \(v\) (\(k=0.1\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Orders**} & \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{\(k=1.0\)} \\ \cline{3-10}  & & \(v=0\) & \(v=0.5\) & \(v=1.0\) & \(v=1.5\) & \(v=2.0\) & \(v=2.5\) & \(v=3.0\) & \(v=4.0\) \\ \hline \multirow{2}{*}{\(\mathrm{Im}(\Omega_{1})\)} & Ref. [16] & 13.38 & 13.27 & 12.94 & 12.37 & 11.56 & 10.46 & 9.030 & 4.185 \\  & Present & 13.381218 & 13.270916 & 12.937432 & 12.372323 & 11.558739 & 10.465210 & 9.0296415 & 4.1846219 \\ \multirow{2}{*}{\(\mathrm{Im}(\Omega_{2})\)} & Ref. [16] & 53.53 & 53.45 & 53.22 & 52.84 & 52.31 & 51.61 & 50.75 & 48.51 \\  & Present & 53.525680 & 53.450306 & 53.223657 & 52.844129 & 52.308985 & 51.614255 & 50.754565 & 48.510044 \\ \multirow{2}{*}{\(\mathrm{Im}(\Omega_{3})\)} & Ref. [16] & 120.44 & 120.38 & 120.18 & 119.85 & 119.39 & 118.80 & 118.06 & 116.20 \\  & Present & 120.44064 & 120.37500 & 120.17796 & 119.849417 & 119.38802 & 118.79367 & 118.06496 & 116.19830 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calculated values of the first three dimensionless natural complex frequencies of the FGM simply supported beam under different axial motion velocities \(v\) (\(k=1.0\)). supported at both ends (SS), clamp supported at one end and simply supported at the other end (CS), and clamp supported at both ends (CC). When the dimensionless axial velocity is lower than \(v_{\rm D1}\) (\(v_{\rm D1}\) is the first dimensionless divergence velocity), i.e., \(v\leq v_{\rm D1}\), an axially moving FGM beam is stable because all the dimensionless natural frequency values \(\Omega\) are imaginary. However, if the dimensionless axial motion velocity is between \(v_{\rm D1}\) and \(v_{\rm s}\) (\(v_{\rm s}\) is the dimensionless axial motion velocity at the beginning of the second stability zone), that is, when \(v_{\rm D1}\leq v\leq v_{\rm s}\), the first-order natural frequency \(\Omega_{\rm 1}\) is a real number, indicating the occurrence of static instability (i.e., divergence instability) of the beam. Figures 3-5 show that there is a very narrow second stable region between \(v_{\rm s}\) and \(v_{\rm F1}\), that is, when \(v_{\rm s}\leq v\leq v_{\rm F1}\), all natural frequencies are imaginary. If the dimensionless axial velocity of the beam is greater than \(v_{\rm F1}\), that is, when \(v\geq v_{\rm F1}\), then the first and second order dimensionless natural frequencies are a pair of complex conjugates, indicating that the beam has dynamic instability (that is, the first and second order coupled flutter). Therefore, \(v_{\rm F1}\) is the minimum coupled flutter velocity of the beam. In Figures 3-5, \(v_{\rm D2}\) is the second dimensionless divergence velocity, and \(v_{\rm F2}\) is the second dimensionless coupled flutter velocity. When \(v_{\rm D2}\leq v\leq v_{\rm F2}\), it indicates the second occurrence of static instability of the beam (i.e., the second occurrence of divergence instability).