## 1 Introduction  
The dynamic analysis of beams on elastic foundations is a fundamental problem of great theoretical and practical significance in many engineering fields, such as railways [1], web processing and laminating lines [2], airport pavements [3], etc. Numerous researchers have carried out various studies on different aspects of this problem, ranging from material, foundation, geometry, and loading. Some of them are interested in beams made of new materials, represented by functionally graded materials (FGMs), and are concerned with the effects of material gradients on the static and dynamic responses of beams. For example, [PERSON] and [PERSON] [4] investigated the bending and free vibration of three-dimensional functionally graded porous beams resting on elastic foundations. [PERSON] et al. [5] studied the free vibration of an axially functionally graded (AFG) curved beam on orthotropic Pasternak. [PERSON] et al. [6] researched the dynamic behaviour of bidirectional functionally graded sandwich beams under a moving mass with partial foundation supporting effect. [PERSON] et al. [7] applied a shifted Chebyshev polynomial-based Rayleigh-Ritz method and [PERSON]'s technique to the analysis of the free vibration of FGM beams embedded in Kerr elastic foundations. [PERSON] et al. [8] proposed an improved first-order beam theory by separating the variables for bending and buckling analysis of thin-walled functionally graded sandwich I-beams resting on a two-parameter elastic foundation. [PERSON] [9] analyzed the free and forced vibrations of AFG Timoshenko beams on a two-parameter foundation using a complementary functions method. [PERSON] and [PERSON] [10] found that the functionally graded distributions of carbon nanotubes through the thickness can increase the fundamental frequencies as well as critical buckling loads of beams on elastic foundations. Another interesting topic in this area is the effect of foundation inhomogeneities and non-uniform cross-sections on beam dynamics. Beam structures resting on viscoelastic foundations are often encountered in engineering practice. Compared with uniform-cross-section beams, beams with variable cross-sections are widely applied in various engineering fields due to their preferable mechanical, structural, functional, and economical characteristics. The dynamic analysis of FGM beams with variable cross-sections resting on a viscoelastic foundation will provide help for the application of functionally graded materials in engineering. [PERSON] et al. [11] proposed an analytical solution for the free vibration of FGM beams with variable cross-sections resting on Pasternak elastic foundations and discussed the effect of variable cross-sections on vibration frequencies and mode shapes. [PERSON] et al. [12] carried out a stability analysis of an exponentially tapered beam on a variable Pasternak foundation. [PERSON] et al. [13] investigated the free and forced vibration of coupled beam systems resting on variable viscoelastic foundations. [PERSON] et al. [14] analyzed the elastic stability of variable cross-section columns by means of some analytical approximation techniques. [PERSON] [15] analysed the dynamic behaviour of axially functionally graded beams resting on variable elastic foundations. [PERSON] et al. [16] developed a FEM program to analyze the dynamic response of simply supported beams on a non-uniform nonlinear foundation subjected to moving loads. [PERSON] et al. [17] gave an optimal control algorithm for suppressing the vibration of beams on elastic foundations under a moving vehicle and random lateral excitations. In addition, the dynamic response of beams under different types of loads, especially moving loads, has also been extensively studied. [PERSON] [18] discussed the dynamic response of an FGM Timoshenko beam on two-parameter elastic foundations subjected to a variable-velocity moving mass. [PERSON] et al. [19] analysed the dynamic response of Euler-Bernoulli beams with discrete viscoelastic supports under sinusoidal loadings.