## 6 Conclusions  
This paper analyzes the dynamics of AFG beams resting on inhomogeneous viscoelastic foundations subjected to non-uniformly distributed moving loads. A consistent discrete governing equation that can tackle all the axially varying properties, including viscoelastic foundations, moving distributed loads, functionally graded materials, and cross-sections, is derived based on the Timoshenko beam theory and a Chebyshev spectral method. High-order Chebyshev expansions are employed to approximate the spatial variation of all these variable properties in a unified way and boundary conditions are imposed using a projection matrix method. Two numerical examples are given to evaluate the convergence and accuracy of the proposed method. The results, including natural frequencies and dynamic responses, are compared with those obtained by the FEM or in the literature.  
Figure 16: Effect of material gradient index \(n\) on the mid-point displacement. (**a**) Power material gradient; (**b**) Exponential material gradient.  
Figure 17: Variation of the maximum displacement as the gradient index increases.  
Figure 18: The variation of the maximum displacements of the AFG beams as the load’s moving velocity changes. (**a**) Power material gradient; (**b**) Exponential material gradient.  
Good convergence and excellent accuracy are observed, demonstrating the validity of the proposed method.  
The validated method is then used to investigate the effects of inhomogeneous viscoelastic foundations, AFG materials, and non-uniformly distributed moving loads on the dynamic response of beams. First, a comparison of three moving load models is carried out. They include the realistic non-uniformly distributed moving load model and two idealized equivalent models, i.e., the concentrated moving load model and the uniformly distributed load model. The results show that they are equivalent in most cases, but have some non-negligible differences in some cases. The calculated dynamic response, based on the idealized concentrated load model, including the peak displacements and the maximum displacement when moving loads travel through, will be greater than those based on the non-uniformly distributed moving load model. Conversely, the results obtained by the uniformly distributed load model will be smaller. Second, the effects of inhomogeneous viscoelastic foundations are studied. It can be found that the varying properties of the foundations can alter the maximum displacement and critical speed of beams under moving loads significantly. Finally, the dynamic responses of different AFG beams are compared with those of corresponding homogeneous beams. The results demonstrate that the AFG materials apparently affect vibration amplitudes and periods.  
The proposed numerical method and computational model are available for beams with arbitrary axially varying properties, including AFG materials, inhomogeneous viscoelastic foundations, non-uniform cross-sections, and that are non-uniformly distributed, thus having a wide scope of application. The application of functionally graded materials in high-speed moving loads, such as electromagnetic rail guns and rocket sleds, will make it possible to reduce structural vibration and improve structural stability, thus further improving the limiting speed of the structure. The results of the analysis in this paper provide important support for the design and application of functionally graded materials in complex engineering structures.  
Conceptualization, methodology, writing--review and editing, funding acquisition, Y.H.; Investigation, writing--original draft preparation, software, H.L. Resources, writing--review and editing, Y.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China with Grant Nos. 52205257 and U22B2083.  
Not applicable.  
Data Availability Statement: Data available on request.  
The authors declare no conflict of interest.