# A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel-Rail Impact Dynamics
## 1 Introduction
[16] developed a detailed time-domain wheel-rail interaction model with non-Hertzian contact values and used the field-measured roughness of different cross-sections as input. They predicted the vibration impact response and noise caused by the wheel flat using a numerical model. Additionally, the use of a detailed contact model provides significant improvements at high frequencies compared with Hertzian theory. In the existing research, multi-body dynamics (MBD) or finite element methods (FEMs) are primarily used to study the dynamic effects of wheel flats. Regardless of the method used, the accuracy of the flat geometry modeling significantly impacts the final simulation results. This paper summarizes the common methods for describing the geometry of the wheel flat, as shown in Table 1.  
According to Table 1, all existing modeling methods have limitations. Firstly, the most common methods--wheel center trajectory description and radius deviation description--only describe the geometry of the wheel in the longitudinal direction and do not account for the width of the wheel flat. The authors of [17] found that for a fixed flat length, a wider flat increases the wheel-rail vertical impact, and a wide flat results in a jump at the wheel-rail contact point compared with a narrower flat. Secondly, the wear process in each stage of the wheel flat should be continuous. The authors of [8] classified the wear process of the wheel flat in detail and found significant differences in the dynamic response at different wear stages. However, current methods only control the shape of the wheel flat in the complete wear stage by changing the length and do not propose control parameters for the shape in the edge-worn stage. Additionally, reconstruction methods for complex 3D surfaces in flats have not yet been developed for modeling based on measured roughness data in the wheel flat region. The limited number of measured rolling circle sections and simple linear interpolation can lead to geometric discontinuities within the flat region. Therefore, this paper contributes to addressing these issues as follows:  
1. A three-dimensional wheel flat modeling method considering a continuous wear process is proposed. This method uses uniform control parameters and analytical forms to describe wheel flats at different wear stages, with the constraints between the control parameters determined by numerical iteration. 2. A complex 3D wheel flat surface reconstruction method is proposed based on NURBS technology. Three different sizes of wheel flats were machined on the wheel tread, and roughness tests were conducted to demonstrate the effectiveness of the reconstruction method using the measured data. 3. A vehicle-track coupled dynamics model is established to compare the variability in the wheel-rail impact response under different wheel flat morphologies. Based on the flat geometry modeling method proposed in this paper, the pattern of wheel-rail impact under different wear states is investigated, as well as the effect of the wear state on wheel-rail contact geometry. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \multicolumn{2}{c}{**Formula**} & \multicolumn{2}{c}{**2D/3D**} & **Consider Wear** & **Simulation Type** & **Citation** \\ \hline \multirow{3}{*}{Center trajectory} & \multirow{3}{*}{\(z(x)=\left\{ \begin{matrix}\frac{\pi^{2}/2R_{p}}{(1-x)^{2}},\\ (1-x)^{2}/2R_{p}\end{matrix} ,\quad \begin{matrix}0\leq x\leq 1/2\\ (1-x)^{2}/2R_{p}\end{matrix} ,\quad \begin{matrix}1/2<x\leq 1\end{matrix} \right.\) (new flat)} & \multirow{3}{*}{2D} & \multirow{3}{*}{Consider} & \multirow{3}{*}{MD} & \multirow{3}{*}{[5/4]} \\ \cline{2-2} \cline{5-6}  & & \(z(x)=\left\{ \begin{matrix}\frac{\pi^{4}/(1-x)^{2}},\\ \frac{\pi^{4}/(1-x)^{2}}{4},\end{matrix} \right\}/(1-x)<\frac{\pi^{4}/(1-x)}{1/2 <x\leq 1}\) (rounded flat)} & & & \\ \hline \multirow{3}{*}{Rail indentation} & \multicolumn{3}{c}{\(z(x)=\left\{ \begin{matrix}\frac{\pi^{4}/(1-x)^{2}},\\ (1-x)^{2}/2R_{p}\end{matrix} ,\quad \begin{matrix}0\leq x\leq 1\end{matrix}