# An Innovative and Efficient Method for Reverse Design of Wheel-Rail Profiles
## 2 Geometric Analysis of Wheel-Rail Contact  
### Rolling Radius Difference  
Figure 1 shows the schematic regarding geometric wheel-rail contact, where \(y_{w}\) is the lateral wheelset displacement; \(\theta\) is the rolling angle of the wheelset; \(\Delta Z_{Lmin}\) is the left vertical minimum wheel-rail gap width; \(\Delta Z_{Rmin}\) is the right vertical minimum wheel-rail gap width and \(\Delta Z_{Lmin}=\Delta Z_{Rmin}\); \(r_{L}\) is the rolling radius at the left wheel contact point; \(r_{0}\) is the rolling radius at the wheel contact point when the wheelset is in a central position; \(r_{R}\) is the rolling radius at the right wheel contact point. Rolling radius difference:  
\[\Delta r=\Delta r_{R}+\Delta r_{L} \tag{1}\]  
where,  
\[\left\{ \begin{array}{l}\Delta r_{R}=r_{R}-r_{0}\\ \Delta r_{L}=r_{0}-r_{L}\end{array} \right. \tag{2}\]  
Figure 2 shows the right wheel-rail contact demonstrated using the wheel-rail profile gradient when the lateral wheelset displacement is \(y_{w}\), where the wheel profile is LMa which is widely used in China's high-speed vehicle CRH2, the rail profile shape is CHN60 which is the 60 kg/m standard rail used in China, the rail gauge is 1435 mm and the rail cant is 1/40. As shown in Figure 2, Line 1 indicates the rail gradient and Line 2 indicates the wheel profile gradient when \(y_{w}\
eq 0\) and \(\theta\ =0\), the function of which is expressed by \(zw_{R}(y)\). Line 3 indicates the wheel profile gradient when \(y_{w}\
eq 0\) and \(\theta\ =0\). Line 4 indicates the wheel profile gradient when \(y_{w}\
eq 0\) and \(\theta\ =0\). Line 3 only considers the lateral wheelset displacement, namely moving by \(y_{w}\) to the right based on Line 2. Line 4 considers the rolling angle, namely the wheel profile gradient increase based on Line 3 and a minor lateral displacement to the right. Point a (the intersection point between Line 1 and 2) is the contact point on a wheel when \(y_{w}=0\) and \(\theta\ =0\). Point d (the intersection point between Line 1 and 3) is the right wheel-rail contact point when \(y_{w}\
eq 0\) and \(\theta\ =0\). Point e (the intersection point between Line 1 and 4) is the right wheel-rail contact point when \(y_{w}\
eq 0\) and \(\theta\ \
eq 0\). Point b on Line 1 is the corresponding point of point d. Point c on Line 1 is the corresponding point of point e.  
\[\Delta r_{R1}=r_{R1}-r_{0}=\int_{y_{w}}^{y_{w}}zw_{R}(y)dy=S_{abb^{\prime}a^{ \prime}} \tag{3}\]  
Figure 1: Wheel-rail contact. Figure 2: Right wheel-rail contact demonstrated using the wheel-rail profile gradient. \[\Delta r_{R2}=r_{R2}-r_{0}=\int_{y_{c}}^{y_{a}}zw_{R}(y)dy=S_{acc^{\prime}a^{\prime}} \tag{4}\]  
where \(\Delta r_{R1}\), area \(S_{abb^{\prime}a^{\prime}}\) refer to the \(\Delta r_{R}\) value when \(y_{w}\
eq 0\) and \(\theta\ =\ 0\); \(\Delta r_{R2}\), area \(S_{acc^{\prime}a^{\prime}}\) refer to the \(\Delta r_{R}\) value when \(y_{w}\
eq 0\) and \(\theta\ \
eq 0\); \(r_{R1}\) refers to the rolling radius at the right wheel contact point when \(y_{w}\
eq 0\) and \(\theta\ =\ 0\); \(r_{R2}\) refers to the rolling radius at the right wheel contact point when \(y_{w}\
eq 0\) and \(\theta\ \
eq 0\) and the rolling angle is considered; \(r_{0}\) refers to the rolling radius at the right wheel contact point without lateral displacement and rolling angle. It is obvious that \(\Delta r_{R1}<\Delta r_{R2}\). The contact point at the right wheel when the rolling angle is considered is more biased towards the wheel flange side than when the rolling angle is neglected. The \(\Delta r_{R}\) when the rolling angle is considered is higher than when the rolling angle is neglected. The left wheel-rail contact under wheel-rail profile gradients can be described according to the above method. We can conclude that the contact point at the left wheel when the rolling angle is considered is farther from the flange side than when the rolling angle is neglected. The \(\Delta r_{L}\). when the rolling angle is considered is higher than when the rolling angle is neglected.