## 1 Introduction
Moreover, such simplified models cannot consider the coupling force between the vehicle and the pavement, which may lead to less accurate predictions of pavement response. Second, the Winkler foundation model usually assumes a linear relationship, but in practical applications, the behavior of soil is often nonlinear, particularly at higher load levels. In addition, there may be considerable uncertainty in determining the foundation stiffness coefficient in actual engineering, which also affects the accuracy of model predictions. Therefore, it is necessary to analyze the impact of changes in the foundation stiffness coefficient on pavement response within a larger parameter range. Specifically, the issue of pavement response to variations in nonlinear foundation stiffness over a larger parameter range has rarely been addressed in the literature. The nonlinear interaction between the beam and the vehicle, along with the roughness of the beam's surface, are aspects that require further exploration when formulating closed-form analytical solutions. In this paper, we analytically investigate the dynamic response of the vehicle-pavement system, with the vehicle simplified as a two-degree-of-freedom system, the pavement modeled using both the Euler-Bernoulli (E-B) beam and the Timoshenko beam, and the foundation considered as a nonlinear viscous Winkler model. The focus of this paper is on how the nonlinear stiffness of the foundation affects the dynamic response of the pavement. The earliest research by [PERSON] and [PERSON] [41] on wheel/track impact due to rail joints or wheel flats found that the nonlinear stiffness of the foundation cannot be ignored. Further research [20; 36; 42] has confirmed that when the moving force is small, the impact of the foundation's nonlinear stiffness on the deformation of the rail or pavement is not significant. When the moving force is large, the deformation of the rail or pavement increases with the increase in the nonlinear stiffness. Previous studies usually simplify the vehicle as a constant moving force, which cannot reflect the actual dynamic behavior of the vehicle. In this paper, we will use a two-degree-of-freedom vehicle model and consider the impact of pavement roughness. Under this condition, the force of the vehicle on the pavement is not constant. The Fourier transform technique is initially employed to determine the impact force exerted by the vehicle on the pavement. Subsequently, the -Adomian decomposition approach [38] is applied to derive an analytical solution for the pavement's dynamic response. The analytical results are leveraged to underscore a detailed discussion on the influence of nonlinear stiffness on the pavement's maximum deformation. The research finds that when considering a complex vehicle model, the impact of the foundation's nonlinear stiffness on the pavement is quite complex. Changes in nonlinear stiffness can affect the location where the maximum deformation of the pavement occurs. Furthermore, there is a critical value for nonlinear stiffness, and the impact of nonlinear stiffness before and after the critical value has completely different manifestations. Additionally, when the nonlinear stiffness is given, the simulation results of the E-B beam and the Timoshenko beam not only differ in the amount of deformation, but also show significant differences in the distribution of deformation along the length of the beam. Our study's outcomes significantly broaden the insight into the intricate, nonlinear mechanics at play in the dynamics of vehicle-pavement interactions. The remainder of this paper is organized as follows: In Section 2, the vehicle-pavement-foundation system considered in this paper is modeled and solved analytically. In Section 3, the analytical solution of the pavement response is numerically verified and used to analyze the impact of nonlinear stiffness on the pavement response under various parameter changes. The discussion and conclusions are provided in Section 4 and Section 5, respectively.