# An Innovative and Efficient Method for Reverse Design of Wheel-Rail Profiles
## 2 Geometric Analysis of Wheel-Rail Contact
### Gradient Control  
Two rules about slope control are given. The first rule is: taking the right wheel-rail as an example, when the rail profile does not change and the initial contact point and its right wheel profile gradient (\(>\)0) are given, the closer the left wheel profile gradient of the initial contact point is to the rail profile gradient, the higher the rolling radius difference will become; The second rule is: taking the right wheel-rail as an example, when the wheel profile does not change and the initial contact point and its right rail profile gradient are given, the closer the left rail profile gradient of the initial contact point is to the wheel profile gradient, the higher the rolling radius difference will become. The first rule can be explained based on the following two points (the second rule can be explained in a similar way). (1) Comparison between rolling radii differences, regardless of the rolling angle  
As shown in Figure 3, Line 1 indicates the rail profile gradient, Line 2 indicates a low wheel profile gradient and Line 3 indicates a high wheel profile gradient. Left wheel-rail contact:  
As shown in Figure 3a, the left wheel profile gradient (\(>\)0) of the initial contact point of Line 2 is equal to that of Line 3, so when the wheelset moves right laterally by \(y_{w}\), the left wheel-rail contact points of both wheels are the same. The \(\Delta r_{L}\) values for the left wheel-rail are the same. Figure 3: Wheel-rail contact with different wheel profile gradients, without considering rolling angle. (a) Left wheel-rail; (b) Right wheel-rail. Right wheel-rail contact:  
As shown in Figure 3b, the wheel gradient of Line 3 is higher than that of Line 2. When the wheelset moves right laterally by \(y_{w}\), the wheel contact point of Line 3 is closer to the flange side compared with that of Line 2. The \(\Delta r_{R}\) value of Line 3 is higher than that of Line 2. In summary, the rolling radius difference of corresponding wheels of Line 3 is higher than that of corresponding wheels of Line 2, without considering the rolling angle. (2) Comparison between rolling radii differences, considering the rolling angle  
When the rolling angle is considered, under the same lateral wheelset displacement, the corresponding wheel rolling angle of Line 3 is higher than that of Line 2. Figure 4 shows the geometric contact between the wheel of Line 2 and the rail. The lateral displacement is \(y_{w}\), the rolling angle is \(\theta\), the red intersection point is its wheel-rail contact point. \(\Delta Z_{L\text{min}}\) is the minimum left wheel-rail gap, \(\Delta Z_{R\text{min}}\) is the minimum right wheel-rail gap; at this time, \(\Delta Z_{L\text{min}}=\Delta Z_{R\text{min}}\). The profiles on the right (the side away from the flange) of the initial contact between the wheels of Lines 3 and 2 are the same. When the lateral wheel displacement and rolling angle of Line 3 are equal to those of Line 2, \(\Delta Z_{L\text{min}}=\Delta Z^{\prime}_{L\text{min}}\) (\(\Delta Z^{\prime}_{L\text{min}}\) is the left minimum wheel-rail gap width for the wheel of Line 3). The wheel gradient of Line 3 on the left of the initial contact point is higher than that of Line 2 and \(\Delta Z_{R\text{min}}>\Delta Z^{\prime}_{R}\) (\(\Delta Z^{\prime}_{R}>\Delta Z^{\prime}_{R\text{min}}\), \(\Delta Z^{\prime}_{R\text{min}}\) is the right minimum wheel-rail gap width for the wheel of Line 3), so \(\Delta Z^{\prime}_{L\text{min}}>\Delta Z^{\prime}_{R\text{min}}\). The wheel of Line 3 needs a higher rolling angle in order to allow the vertical minimum wheel-rail gap width on the left to be equal to that on the right. Therefore, the wheel rolling angle of Line 3 is higher than that of Line 2. When the rolling angle is considered, the corresponding wheel rolling angle of Line 3 is higher than that of Line 2. It can be ascertained from the geometric wheel-rail contact performance on gradients that the wheel \(\Delta r_{R}\) and \(\Delta r_{L}\) of Line 3 is higher than that of Line 2.