## 2 Numerical Simulation
### Vehicle-Track-Subgrade Coupled Model
conditions for the finite subgrade structure model. In this study, a viscoelastic boundary was employed for the foundation model because it provides an effective solution for simulating wave propagation into infinite regions, reducing reflections at the boundary, and mimicking the soil's elastic recovery behavior. The viscoelastic boundary simulates wave propagation into an infinite region and the soil's elastic recovery by applying parallel spring-damper systems to the artificially truncated boundary nodes. In ABAQUS, continuously distributed spring-damper systems are applied to the tangential and normal directions of the boundary nodes. The viscoelastic artificial boundary adopts the two-parameter Kelvin solid model, and the expressions for the spring stiffness and damping coefficients are shown in Equation (2). \[\left\{ \begin{aligned} K_{\text{BN}}&=\alpha_{\text{N}} \frac{G}{R}\sum\limits_{i}^{L}A_{i}\\ C_{\text{BN}}&=\rho c_{\text{P}}\sum\limits_{i}^{L}A_{i} \end{aligned} \right.\left\{ \begin{aligned} K_{\text{BT}}& =\alpha_{\text{T}}\frac{G}{R}\sum\limits_{i}^{L}A_{i}\\ C_{\text{BT}}&=\rho c_{\text{S}}\sum\limits_{i}^{L}A_{i} \end{aligned} \right. \tag{2}\]  
where \(K_{\text{BN}}\) and \(C_{\text{BN}}\) represent the normal stiffness and damping coefficient of the spring, respectively; \(K_{\text{BT}}\) and \(C_{\text{BT}}\) represent the tangential stiffness and damping coefficient of the spring, respectively; \(G\) is the shear modulus of the medium; \(\rho\) is the density of the medium; \(R\) is the distance from the vibration source to the artificial boundary point; \(c_{\text{P}}\) and \(c_{\text{s}}\) represent the P-wave velocity and S-wave velocity of the medium, respectively; \(\sum\limits_{i}^{L}A_{i}\) denotes the area of the artificial boundary; the tangential direction \(\alpha_{\text{T}}\) is 2; and the normal direction \(\alpha_{\text{N}}\) is 4. In summary, the finite element model shown in Figure 2 was established. The contact between the wheel and rail primarily considered the normal force and the tangential force. The normal force between the wheel and rail was calculated based on Hertzian contact theory [38], with the formula as follows:  
\[P(t)=\left[\frac{1}{G}\Delta Z(t)\right]^{3/2} \tag{3}\]  
Figure 2: 3D Simulation diagram of the vehicle-track-subgrade. where \(G\) represents the wheel-rail contact constant. For this study, a wear-type wheel tread was used, with \(G=3.68R^{-0.115}\times 10^{-8}\)(m/N\({}^{2/3}\)), \(R\) is the wheel-rail radius, and \(\Delta Z(t)\) represents the elastic compression between the wheel and rail at time \(t\). The tangential force between the wheel and rail mainly includes the tangential creep forces and spin moment. Creep between the wheel and rail generated frictional forces that resist relative sliding at the contact surface. The friction coefficient was the ratio of the vertical force on the wheel-rail to the frictional force, and it varied under different conditions, including static and dynamic friction coefficients. Therefore, the expression for the tangential force was as follows:  
\[F=\mu\times P(t)=\left[\mu_{k}+(\mu_{s}-\mu_{k})e^{-d_{c}\dot{r}_{eq}}\right] \times P(t) \tag{4}\]  
where \(\mu_{k}\) and \(\mu_{s}\) are the dynamic and static friction coefficients, respectively, \(d_{c}\) is the damping coefficient, \(\dot{r}_{eq}\) represents the relative sliding speed at the wheel-rail contact surface, \(P(t)\) is the normal force between the wheel and rail, and the friction coefficient is taken as 0.3.