## 2 Numerical Model
The speed values are approximately minimum and maximum admissible speed in the UK [17]. The boundary conditions enable the vehicle to move in longitudinal and vertical direction; to rotate around lateral (pitch motion) and longitudinal axis (roll motion). Finally yet importantly, properties of the numerical model are presented in Table 1.  
As aforementioned, the coupling between track and vehicle is provided by Hertzian contact theory. According to this theory, it is assumed that contact forces are a product of contact stiffness and elastic deflection, where contact stiffness between two geometries is related to modulus of elasticity (E), Poisson ratio (v), load (Q), and the radii of surfaces of geometries (R). In general, the load is accepted as a static load to facilitate the calculation. Assuming rail and wheel have infinite radius in one direction (then, radii of surfaces for wheel and rail are _Rwheel_,_Rrail_ respectively), an analytical formula for contact stiffness is derived for railway applications [18]. Introducing this contact stiffness into the Hertz formula gives Equation (1). It should be noted that elastic deflection is predicted as an imaginary penetration between nodes in numerical calculations, which equals the distance between wheel (\(\delta_{w}\)) and rail nodes that also contain the information of track surface roughness (\(\delta_{r}+\delta_{ir}\)). \[F=\sqrt[3]{\frac{3E^{2}Q\sqrt{R_{\text{r}wheel}\cdot R_{\text{r} rail}}}{2(1-\
u^{2})^{2}}}\;x\;(\delta_{w}-\delta_{r}-\delta_{ir}) \tag{1}\]  
Track surface roughness in the formula is obtained by field measurements and applied in the simulation to reflect realistic conditions. A roughness profile for 48 m turnout is obtained from [12]. The profile was extracted from 28 measurements for 30-km track length and represents worst track conditions regarding maximum mean and standard deviations. It is noteworthy that rail surfaces have no surface irregularities, perfectly smooth with the exception of a small irregularity at crossing nose representing the geometrical discontinuity (Figure 1). An explicit time integration used for the calculations in this study is well-known for its effectiveness on dynamic problems when large displacements or impact forces are of interest. It inherently provides stability, namely convergence, during solution process. Furthermore, in this study, the solution process is enriched with a dynamic relaxation [19] that ensures the vehicle and track is in an equilibrium before the calculation. Otherwise, the vehicle would bounce on the rails due to gravitational forces caused by small gaps between rails and wheel, which would produce artificial vibrations. Another challenge while using an explicit time integration is to decide the time step for the simulation that depends on the length of an element. Here, the same time step in [12] is preferred to obtain high quality results and acceptable computer clock time for the computation. Model verification is conducted by comparing the current model with the parent model in [12]. It provides flexibility and more importantly, facilitates the verification process, which is quite complex for unsupported bearers due to inherent behavior of the ballast bed that exhibit different characteristic in every case. Therefore, a separate verification with field measurements is omitted. In [12], verification of the model was done based on two different measurements in the literature. Those measurements were based on two different methodologies, such as measuring the acceleration and measuring the contact forces. Acceleration method is simple to apply but insufficient to extract contact forces whereas