## 4 Numerical Examples and Problem Discussion
When \(v\geq v_{\rm F2}\), it indicates the second occurrence of dynamic instability (i.e., second- and third-order coupled flutter). The values of all critical axial velocities (\(v_{\rm D1}\), \(v_{\rm s}\), \(v_{\rm F1}\), \(v_{\rm D2}\), and \(v_{\rm F2}\)) in Figures 3-5 calculated by the interpolating matrix method (IMM) are listed in Table 4. Tables 5-7 are the IMM calculated values of the dimensionless critical velocity when the gradient index of the functionally graded material is \(k=11\)\(\sim\)15. The calculation results show that with the increase of the material gradient index \(k\), the divergence velocity \(v_{\rm D1}\) and flutter velocity \(v_{\rm F1}\) of the axially moving FGM beam tend to decrease. The main reason is that when the material gradient index \(k\) gradually increases from 0 to, \(\infty\), the stiffness of the beam material gradually decreases. Figure 3: Curve of the imaginary part of the first three complex frequencies of SS beam changing with axial velocity. Figure 4: Curve of the imaginary part of the first three complex frequencies of CS beam changing with axial velocity. Figure 5: Curve of the imaginary part of the first three complex frequencies of CC beam changing with axial velocity. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k\)} & \multirow{2}{*}{
\begin{tabular}{c} **Boundary** \\ **Condition** \\ \end{tabular} } & \multicolumn{4}{c}{**Dimensionless Critical Axial Velocity of SS Beam**} \\ \cline{3-6}  & & \(v_{\mathbf{D1}}\) & \(v_{\mathbf{s}}\) & \(v_{\mathbf{F1}}\) & \(v_{\mathbf{D2}}\) & \(v_{\mathbf{F2}}\) \\ \hline \multirow{3}{*}{10} & SS & 3.45612848 & 6.91230903 & 7.105330000 & 10.31880210 & 11.26140000 \\  & CS & 4.94329742 & 8.49885742 & 8.820195000 & 11.99671700 & 13.10100000 \\ \cline{1-1}  & CC & 6.91230903 & 9.88692206 & 10.76850000 & 13.82628580 & 15.10870000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Critical velocity of dimensionless axial motion of FGM beam with functionally graded materials under different boundary conditions. The modal function of an axially moving Euler-Bernoulli beam has a complex mode. In this paper, the interpolating matrix method (IMM) can not only be used to calculate the natural frequencies of several orders before the transverse free vibration of an axially moving FGM beam but also solve the corresponding modal function. Let the material gradient index \(k=\infty\), and then the FGM beam degenerates into a uniform metal beam. Figure 6 compares the real part of the first-order mode of the uniform metal beam under different axial velocities (\(v\)). The study found that the static SS beam and CC beam with symmetry in the first order mode are distorted due to the influence of the axial velocities \(v\). Therefore, the original symmetry of Euler-Bernoulli beams of uniform metal materials under static SS and CC boundary conditions cannot be applied to the case of axially moving beams, especially for beams with great axial velocities. Moreover, axial motion velocities exert a greater influence on the first-order modal distortion of a beam clamped at both ends (CC) compared to a beam simply supported at both ends (SS). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\(k\)} & \multicolumn{5}{c}{**Dimensionless Critical Axial Velocity**} \\ \cline{2-6}  & \(v_{\text{DI}}\) & \(v_{\text{s}}\) & \(v_{\text{FI}}\) & \(v_{\text{D2}}\) & \(v_{\text{FE2}}\) \\ \hline
11 & 4.91538857 & 8.45087461 & 8.77035300 & 11.9289860 & 13.02699000 \\
12 & 4.89080941 & 8.40861641 & 8.72648000 & 11.8693356 & 12.96194000 \\
13 & 4.86897969 & 8.37108526 & 8.68739900 & 11.8163578 & 12.90400000 \\
14 & 4.84945177 & 8.33751151 & 8.65250000 & 11.7689662 & 12.85200000 \\
15 & 4.83187315 & 8.30728915 & 8.62130000 & 11.7263053 & 12.80540000 \\ \hline \end{tabular}
\end{table} 
Table 6: The effects of gradient parameter \(k\) of material on the dimensionless critical axial velocities of FCM beam (CS).