## 2 Wheel-Rail Contact Model  
### Wheel-Rail Contact Geometry Calculation  
To solve the wheel-rail contact geometric relationship, two types of methods have been developed currently: the spatial direct search method and the projection method. The spatial direct search method requires searching for wheel rail contact points in the spatial dimension and involves a considerable number of iterative calculations [25; 26; 27]. As a representative of the projection method, the contact trace method [28] uses the idea of constructing spatial traces to transform the three-dimensional search problem into a planar search problem [9], greatly improving computational efficiency and making it widely used in the calculation of wheel rail spatial geometric relationships. Thus, the contact trace method was employed in this study, as illustrated in Figure 1. Two coordinate systems are defined in the system, namely, the absolute coordinate system (\(O\)-\(XYZ\)) and the wheelset coordinate system (\(O_{\text{w}}\)-\(X_{\text{w}}\)\(Y_{\text{w}}\)\(Z_{\text{w}}\)). In the absolute coordinate system, \(O\) is positioned at the center of the track, with the \(OX\) axis pointing in the direction of rolling, the \(OZ\) axis pointing  
In the vertical searching method, the maximum wheel-rail vertical penetration is introduced as the judgement condition when determining the contact point, which is characterized by the length of the green line segment shown in Figure 2. The normal compression amount can be determined by constructing a right triangle, as characterized by the length of the blue line segment shown in Figure 2. The mathematical expression is given as:  
\[\delta_{cv}=\delta_{z}/\cos\theta \tag{4}\]  
where \(\delta_{z}\) is the maximum vertical penetration amount, and \(\theta\) is equal to \(\delta_{w}-\phi_{w}\).  
In the wheel tread penetration zone, as presented in Figure 2a, since \(\theta\) is small, the difference between \(\delta_{ct}\) and \(\delta_{cv}\) is not evident, indicating that the algorithm based on the vertical penetration is applicable in this case. In the wheel flange penetration zone, as shown in Figure 2b, \(\delta_{cv}\) is significantly different from \(\delta_{cn}\) in this case due to the significant impact of \(\theta\). Hence, using the algorithm based on the vertical penetration tends to cause large calculation errors of the normal compression amount in this case.  
In addition, for the consideration of the number of wheel-rail contact points, if only single-point contact is considered, the position corresponding to the maximum penetration in the whole range of wheel tread is generally regarded as the wheel-rail contact point. In contrast, to consider multipoint contact, the segment searching method can be introduced to determine each wheel-rail penetration zone, as illustrated in Figure 3. For each discrete point on the wheel profile, its vertical coordinate is denoted by \(Z_{w}\), and the rail vertical coordinate corresponding to the vertical direction is denoted by \(Z_{r}\). The difference between the two vertical coordinates is denoted by \(\Delta Z\) and can be expressed as:  
\[\Delta Z=Z_{w}-Z_{r} \tag{5}\]  
Thus, each wheel-rail penetration zone can be determined when \(\Delta Z<0\). The contact point position and the normal compression amount can be determined in each penetration zone.  
Figure 3: Wheel-rail segment searching method.  
Figure 2: Determination of wheel-rail contact points and normal compression amounts based on different searching methods: (**a**) wheel-tread contact; (**b**) wheel-flange contact.