## 4 Numerical Examples and Problem Discussion  
In order to solve a specific problem, suppose the axially moving FGM beam is made of ceramic and metal, and the elastic modulus \(E(z)\) and density \(\rho(z)\) of the material can be expressed as:  
\[E(z)=(E_{c}-E_{m})\bigg{(}\frac{z}{h}+\frac{1}{2}\bigg{)}^{k}+E_{m} \tag{37a}\] \[\rho(z)=(\rho_{c}-\rho_{m})\bigg{(}\frac{z}{h}+\frac{1}{2}\bigg{)}^{k}+\rho_{m} \tag{37b}\]  
where \(k\) is the material gradient index, the elastic modulus and density of metal materials are \(E_{\text{m}}\) = 210 GPa and \(\rho_{\text{m}}\) = 7800 kg/m\({}^{3}\), respectively, and the elastic modulus and density of ceramic materials are \(E_{\text{c}}\) = 390 GPa and \(\rho_{\text{c}}\) = 3960 kg/m\({}^{3}\), respectively. When the functional gradient index of the beam material \(k\) = 0 and \(k\) = \(\infty\), the functional gradient material beam degenerates into a uniform ceramic material beam and a metal material beam. In order to verify the feasibility and accuracy of the interpolating matrix method (IMM) in solving the natural frequency in free vibration of the FGM beam, the length-height ratio of the FGM beam was set at \(\lambda\) = 10, and the number of discrete elements \(N\) = 6-22. Table 1 shows the value of the dimensionless natural frequency of an axially moving FGM beam calculated by the interpolating matrix method (IMM). The value is compared with the results in the literature [16]. The numerical results show that when the number of discrete beam length elements \(N\) is greater than or equal to 12, the numerical results in this paper gradually converge to the actual value, and with the increase in the number of discrete elements \(N\), the interpolating matrix method (IMM) presents greater computational stability. The number of beam length discrete elements \(N=20\). Tables 2 and 3 illustrate the numerical solutions of the first three-order natural complex frequency using the interpolating matrix method (IMM) for a simply supported beam (SS) with material gradient index \(k=0.1\) and \(1.0\) under different coaxial velocities \(v\), which are completely consistent with the calculated results of the existing literature [16]. The feasibility and accuracy of the interpolating matrix method (IMM) in the analysis of transverse free vibration dynamics of axially moving FGM beams are thus further verified. Figures 3-5 show the change of the dimensionless natural complex frequency value \(\Omega\) of an axially moving FGM beam with axial velocity \(v\) when the FGM beam length-height ratio is \(\lambda=10\), the material gradient index is \(k=10\), and the boundary conditions are simply \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\lambda\) & \(v\) & **Orders** & \(N\) & \(k=0.1\) & \(k=1\) & \(k=10\) \\ \hline \multirow{8}{*}{10} & & & 6 & 17.2640239 & 12.9435352 & 10.3138156 \\  & & & 8 & 17.2612122 & 12.9412079 & 10.3117430 \\  & & & 10 & 17.2590450 & 12.9394837 & 10.3102699 \\  & & & 12 & 17.2579142 & 12.9385884 & 10.3095090 \\  & & & 14 & 17.2573203 & 12.9381188 & 10.3091105 \\  & & & 16 & 17.2569923 & 12.9378596 & 10.3088908 \\  & & & 18 & 17.2568014 & 12.9377088 & 10.3087629 \\  & & & 20 & 17.2566848 & 12.9376167 & 10.3086849 \\  & & & 22 & 17.2564886 & 12.9374618 & 10.3085536 \\  & & & & Ref. [16] & 17.26 & 12.94 & 10.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The change of the natural complex frequency \(\Omega_{1}\) of an axially moving FGM beam with the number of discrete elements \(N\) of the beam Length.