# An Innovative and Efficient Method for Reverse Design of Wheel-Rail Profiles
## 3 Reverse Profile Design Method
### Method Description  
The left- and right-side wheel profile and rail profile shape are symmetric, only the positive lateral displacement (lateral wheelset movement to the right) must be considered. The lateral displacement can be divided into n parts by a certain step length, which is expressed by \(y_{wk}\) (\(k=1\), \(2\),..., \(n\)). When the lateral displacement varies, the target value of the corresponding rolling radius difference will be \(\Delta R(y_{wk})\).  
Target function:  
\[\left|\frac{\Delta R^{\prime}(y_{wk})-\Delta R(y_{wk})}{\Delta R(y_{wk})} \right|<\varepsilon \tag{5}\]  
where \(\Delta R(y_{wk})\) refers to the target rolling radius difference when the lateral displacement is \(y_{wk}\); \(\Delta R^{\prime}(y_{wk})\) refers to the rolling radius difference of wheel profiles reverse designed when the lateral displacement is \(y_{wk}\).  
As shown in Figure 5, \(W_{y_{wk}}(k=1,2,\ldots,n)\) is the wheel profile gradient section and \(dotk(k=1,2,\ldots,n)\) is the contact point on wheel when the lateral displacement is \(y_{wk}(k=1,2,\ldots,n)\). In order to ensure that the curve of the designed wheel profile is smooth, the wheel profile gradients must be continuous and the wheel profile gradients of the designed sections must decrease monotonously. The gradients between contact points of the wheel profile in this design are straight lines. The profiles of the designed sections are multi-sectional quadratic curves. When the contact section gradients of the wheel profile are higher than 0, along with the wheelset moving right laterally, the contact points on the right wheel approach the flange side and the contact points on the left wheel leave the flange side gradually, enabling design of the wheel sections. As shown in Figure 6, the wheel profile gradient section \(W_{y_{w1}}\) satisfying the target function when the lateral displacement is \(y_{w1}\) is calculated first, then the wheel profile is found that satisfies the requirements by adjusting the wheel profile gradient on the left of the initial contact point \(dot0\) until the target function of the designed profile shape satisfies the requirements. The contact point \(dot1\) on the wheel under this lateral displacement is then chosen and finally the wheel profile gradient section \(W_{y_{w1}}\). In a similar way, the wheel profile gradient section \(W_{y_{wk}}\) is calculated satisfying the target function by adjusting the wheel profile gradient on the left of the contact point \(dot(k-1)\) when the lateral displacement is \(y_{wk}(k=1,2,\ldots,n)\), until the target function satisfies the requirement when the maximum lateral displacement is \(y_{wn}\) and then the completed designed wheel profile is output.  
When the wheel profile gradient on the left of the initial contact point is low, the rolling radius difference corresponding to the wheel profile will be less than the target value. However, when the wheel profile shape gradient on the left of the initial contact point is high, the rolling radius  
Figure 5: Reverse design of wheel profiles.  
Figure 6: Reverse design of wheel profile section \(W_{y_{w1}}\).  
Figure 7: Flowchart.  
difference corresponding to the wheel profile will be higher than the target value; therefore, dichotomy can be adopted to adjust the wheel profile gradient of designed sections. Using dichotomies can quickly converge to the target. Figure 7 shows the program flowchart.