# A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel-Rail Impact Dynamics
## 2 Three-Dimensional Wheel Flat Model
### Worn Flat Model  
According to [8], during the edge-worn stage of a wheel flat, the material between the edges and the center of the flat cannot come into single-point contact with the rail. Thus, the initiation of wear and/or plastification from the edges is geometrically determined. At this stage, the wheel flat geometry is defined as shown in Figure 3. \begin{table}
\begin{tabular}{c c c} \hline
**Notation** & **Meaning** & **Control Parameter or Not** \\ \hline \(L_{0}\) & new flat length & \(\times\) \\ \(L_{w}\) & new flat width & \(\times\) \\ \(d\) & new flat depth & \(\surd\) \\ \(d_{0}\) & nominal depth & \(\times\) \\ \(R_{w}\) & wheel nominal radius & \(\surd\) \\ \(R_{r}\) & curvature at rail head radius & \(\surd\) \\ \(\alpha\) & depth coefficient & \(\surd\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter table for a new flat. From Figure 3, the edge worn flat contains the following two regions: the worn region and the non-worn region. In the nominal rolling section, the endpoint \(A(x_{1},y_{1})\), endpoint \(B(x_{2},y_{2})\) of the worn region are defined. In this case, the depth function of the non-worn region degrades to \(F_{\text{I}}(x)\), the depth function of the worn region is defined as \(g(x)\), and the analytic equation of the rolling circle outside wheel flat region is defined as \(r(x)\). Considering the cubic polynomial approximation for the worn region, the boundary conditions are shown in Equation (6). \[\left\{ \begin{array}{l}r(x_{1})=g(x_{1})\\ r^{\prime}(x_{1})=g^{\prime}(x_{1})\\ F_{\text{I}}(x_{2})=g(x_{2})\\ F^{\prime}_{1}(x_{2})=g(x_{2})\end{array} \right. \tag{6}\]  
Figure 3: Definition of edge worn flat parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Meaning** & **Control Parameter or Not** \\ \hline \(L_{0}\) & new flat length & \(\times\) \\ \(L_{w}\) & new flat width & \(\times\) \\ \(l\) & worn region length & \(\times\) \\ \(L\) & edge worn flat length & \(\times\) \\ \(d\) & edge worn flat depth & \(\surd\) \\ \(d_{0}\) & nominal depth & \(\times\) \\ \(R_{w}\) & wheel nominal rolling circle radius & \(\surd\) \\ \(R_{r}\) & curvature at rail head radius & \(\surd\) \\ \(\alpha\) & depth coefficient & \(\surd\) \\ \(\beta\) & length coefficient & \(\surd\) \\ \(\gamma\) & worn coefficient & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter table for an edge worn flat. In Equation (6), \(r(x)\), \(g(x)\), and \(F(x)\), respectively, are defined as follows:  
\[r(x)=R_{w}-\sqrt{{R_{w}}^{2}-x^{2}} \tag{7}\]  
\[g(x)=a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4} \tag{8}\]  
\[F_{\rm I}(x)=(1-\frac{1}{\alpha})(\sqrt{1-\frac{\alpha^{2}x^{2}}{2\alpha R_{w} d-d^{2}}}+\frac{1}{\alpha})d \tag{9}\]  
For \(x_{1}\), \(x_{2}\), \(x_{1}=-L/2\), \(x_{2}=-L/2+l\), \(\beta=L/L_{0}\), and \(\gamma=l/L_{0}\). The polynomial coefficients of the worn region can be fully determined according to Equation (6). For any section parallel to the rolling circle, the length coefficient of the edge worn flat is replaced by \(\beta^{\prime}\), ignoring the change in the radius of the rolling circle due to the taper of the wheel tread in the wheel flat region. \[\beta^{\prime}=\sqrt{(1-\frac{\Delta y^{2}}{2R_{r}d-d^{2}})}\beta\ \ \ \ (\Delta y \in[0,\frac{L_{w}}{2}]) \tag{10}\]  
The worn coefficient is linearly interpolated based on the section location, as shown in Equation (11). \[\gamma^{\prime}=\frac{L_{w}-2\Delta y}{L_{w}}\gamma \tag{11}\]  
According to Equations (6)-(11), the depth at any position within the worn region of the wheel flat \(g(x,y)\) can be obtained. In summary, the depth function of the edge worn wheel flat \(F_{\rm II}(x,y)\) is shown in Equation (12). \[F_{\rm II}(x,y)= \begin{cases}g(x,y)&x,y\in S_{w}\\ F_{\rm I}(x,y)&x,y\in S_{n}\end{cases}  \tag{12}\]  
where \(S_{w}\) represents the feasible domain of the worn region and \(S_{n}\) represents the feasible domain of the non-worn region.