## 1 Introduction
A wealth of pertinent studies demonstrates that the forces resulting from dynamic vehicle-structure interaction, which leads to the oscillation of the entire system, are primarily triggered by the roughness of the structural surface [23]. Additionally, these interaction forces are influenced by the dynamic deformations of the structure and the vehicle's vibrations [24]. The multilayer medium theory is a method used to analyze and calculate the mechanical behavior of composite structures composed of different material layers. In the field of vehicle-pavement research, this theory is particularly important because pavements are usually made up of multiple layers of different materials, such as asphalt surface layers, base courses, sub-base courses, and the soil beneath. The multilayer medium theory can help engineers understand and predict the interaction between vehicles and pavements, as well as the response of pavement structures under traffic loads [25]. Based on the principles of elasticity mechanics, the theory takes into account the mechanical properties of each layer (such as elastic modulus, Poisson's ratio, etc.) and their interactions. By establishing mathematical models, the theory can describe the transmission and distribution of loads within the multilayer structure [26]. At the interfaces between layers, displacement continuity and stress matching are required to ensure the integrity and functionality of the structure. Utilizing multilayer medium theory allows for the optimization of pavement structure design, selecting appropriate materials and layer thicknesses to enhance the load-bearing capacity and durability of pavements [27]. The advancement of effective dynamic computational techniques, especially those yielding analytical and semi-analytical solutions, is of vital importance for achieving the best possible design [28], ongoing health monitoring [29], and predictive maintenance [30] of the vehicle-pavement systems. In the initial stages, the closed-form solution for the quasi-stationary deflection, also referred to as the steady-state condition in some contemporary studies, of the beam was obtained by adopting a reference frame that moves in conjunction with the source [31]. Subsequently, the models for excitation sources were improved to include a moving mass [32], a moving harmonic line load [33], and a moving distributed mass [34]. [PERSON] employed integral transforms and contour integration to account for transient responses due to non-uniform initial conditions [35]. [PERSON] et al. [36] and [PERSON] et al. [37] used the Adomian decomposition method [38] to analytically solve the dynamic response of a nonlinear foundation beam under the action of a moving load. In conventional road dynamics, the vehicle is regarded as a moving load stimulated by the pavement's surface roughness, with the impact of pavement vibrations often being overlooked. [PERSON] et al. [39] conducted an analytical study to assess the influence of the coupling effect on both pavement displacements and the vertical displacement of the vehicle body. Their findings revealed that the coupling effect only marginally amplifies the pavement response. [PERSON] et al. [40] developed a method that integrates harmonic balance with Fourier transform to extract a semi-analytical solution for the steady-state periodic dynamics of an infinite beam with a vehicle load, considering both the pavement's surface roughness and the nonlinear dynamics of the vehicle-beam interaction. In the analytical study of the aforementioned vehicle-pavement models, there are two issues: First, due to the difficulty in solving, vehicles are often simplified to a moving force or moving mass. However, actual vehicles have complex structures, and simplifying them with moving force or moving mass models can lead to the neglect of the vehicle's inertial characteristics. This may result in the model being unable to accurately reflect the dynamic interaction between the vehicle and the pavement, especially under high-speed or heavy-load conditions.