## 5 Conclusions  
Based on the Euler-Bernoulli beam theory and the Hamilton principle, the transverse vibration kinematics control differential equation for an axially moving FGM beam is established. Based on the basic theory of the interpolating matrix method (IMM), the kinematic differential equation with an inherent complex frequency as an eigenvalue is converted into the eigenvalue solution of a standard generalized algebraic equation by using an integral matrix. Finally, the standard generalized algebraic equation set is solved by orthogonal trigonometric decomposition (**QR**), and the natural frequency of transverse vibration of an axially moving FGM beam is obtained. Meanwhile, the corresponding modal function is obtained. The main conclusions are drawn as follows:  
(1) In this paper, the numerical calculation of the dimensionless natural complex frequency of axially moving FGM beams by the interpolating matrix method is in complete agreement with the results obtained in the previous findings, which verifies the calculation accuracy of the interpolating matrix method. At the same time, the numerical results show that, with the increase in the number of discrete elements on the beam length, the algorithm demonstrates strong computational stability.  
(2) With the increase of material gradient index \(k\), the divergence velocity and flutter velocity tend to decrease. When the axial velocity is equal to the lowest divergence velocity, \(v_{\rm D1}\), the first-order natural frequency of the axially moving FGM beam disappears. That is, the first-order bending mode disappears, resulting in static instability. At the same time, there is a very narrow stability region between the first static instability region (divergence) and the first dynamic instability region (first- and second-order coupled chatter).  
(3) In general, the beam clamp supported at both ends (CC) has the strongest constraint, while the beam simply supported at both ends (SS) has the weakest constraint. The calculation results of the interpolating matrix method (IMM) show that the divergence velocity and flutter velocity of an axially moving FGM beam tend to increase with the increase of boundary constraints.  
(4) The axial velocity \(v\) causes symmetry distortion in the real part of the first-order complex mode of the simply supported beam (SS) and clamp supported beam (CC) with uniform metal material. The axial velocity \(v\) has a more significant effect on the real part distortion of the first order mode of the beam clamp supported at both ends (CC) compared to the beam simply supported at both ends (SS).  
Figure 6: Real part of the first mode of FGM beam under different axial velocities.  
**Author Contributions:** Conceptualization, J.-P.W. and Y.T.; methodology, R.-Y.G.; software, R.-Y.G.; validation, R.-Y.G. and Y.T.; formal analysis, J.-P.W.; investigation, J.-P.W.; resources, J.-P.W.; data curation, J.-P.W.; writing--review and editing, J.-P.W. and R.-Y.G.; supervision, Y.T.; project administration, R.-Y.G.; funding acquisition, R.-Y.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Anhui Provincial Natural Science Foundation (Grand No. 1808085 ME147) and the Middle-aged Topnotch Talent Support Program of Anhui Polytechnic University.  
Not applicable.  
The datasets supporting the conclusion of this article are included within the article.  
This work was supported by the Anhui Provincial Natural Science Foundation (Grand No. 1808085 ME147) and the Key Project of the Natural Science Foundation of Anhui Polytechnic University (Grand No. Xjky2022166).  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.