## 1 Introduction  
Over the past few decades, there has been extensive interest in addressing vehicle-induced vibration issues, including the dynamics of vehicle-pavement, vehicle-bridge, train-track, and track-tunnel-soil interactions [1; 2; 3; 4; 5; 6]. Precisely and swiftly calculating the stresses and deformations in pavements subjected to moving vehicle loads is crucial within the context of the mechanistic-empirical pavement design method [7]. Models for vehicle-pavement interactions and the corresponding dynamic analysis techniques can range from simple to complex. Sophisticated models and analytical methods yield more accurate and realistic outcomes, albeit with increased computational expense. Vehicles are commonly simplified to a moving force or moving mass to reduce computational load [8]. For a more complex approach, they can be simplified into quarter-car [9], half-car [10], and seven-degrees-of-freedom models [11], taking into account the force exerted on the tires due to pavement roughness [12]. Vibration issues in road systems during vehicle travel are a very common type of vehicle-structure interaction problem, with a variety of road models available to choose from. The Timoshenko beam model is a beam theory that takes into account shear deformation and rotational inertia. In pavement simulation, the Timoshenko beam model can be used to analyze the response of different types of pavement structures under traffic loads, including pavement deflection, stress distribution, and displacement [13]. Due to its consideration of shear deformation, the Timoshenko beam model is more effective in analyzing the development of pavement cracks and damage. The Winkler model provides a simple yet effective tool for simulating pavement, treating the foundation as if it is composed of a series of springs, which represent the elastic response of the subgrade [14]. In this model, the pressure intensity at any point on the foundation is proportional to the settlement at that point, but it ignores the shear stress in the foundation. This model is suitable for foundations with mechanical properties similar to water, such as semi-liquid soils (like mud, or soft clay) with very low shear strength or where the plastic zone beneath the foundation is relatively large. The Burmister model regards the foundation as a combination of multiple horizontal elastic layers, each of which can have different stiffness and strength. This model is suitable for situations where the foundation conditions are complex or where it is necessary to consider the stratification of the subgrade [15]. By dividing the foundation into multiple horizontal layers, the Burmister model can better simulate the heterogeneity and nonlinear characteristics of the foundation, thereby providing more accurate analysis results in pavement simulation. The significance of the interaction between vehicles and pavements has been highlighted in numerous scholarly works [16; 17]. [PERSON] et al. explored a three-dimensional vehicle-pavement interaction model [18], which included a comprehensive vehicle model and a simply supported rectangular plate. The plate underwent spatial discretization via the Galerkin method, and the model accounted for the nonlinear coupling between vehicle and pavement [19]. [PERSON] et al. represented the pavement as a Timoshenko beam with nonlinear foundation support, and they utilized the Runge-Kutta method to determine the system's response [20]. [PERSON] and [PERSON] developed an iterative decoupling technique for the dynamic response of the vehicle-pavement system by integrating the Galerkin method, finite difference scheme, and [PERSON]'s method [21]. [PERSON] et al. employed the incremental harmonic balance method to analyze the dynamic instability of a rectangular plate subjected to a moving mass following an irregular trajectory [22].