# A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel-Rail Impact Dynamics
## 6 Conclusions  
This paper focuses on the modeling and simulation methods for the spatially complex morphology of wheel flats and systematically proposes a three-dimensional spatial model of flats using both analytical and data-driven approaches. A wheel-track coupled dynamics model that considers wheel flat geometry is established, revealing the influence of the control parameters of wheel flat geometry on the wheel-rail impact response and the wheel-rail contact geometry. The control parameters of the analytical form of the wheel flat spatial model proposed in this paper are adjustable, making it highly suitable for predicting the impact response at different stages. By combining dynamic indices such as the P1 force, a more accurate safety threshold for the three-dimensional scale of the wheel flat can be established. Furthermore,  
Figure 13: The change in the coordinates of wheel–rail contact points in each iterative step with different worn coefficients. (**a**) Left wheel. (**b**) Right wheel. when spatial sampling points in the wheel flat region are available, the spatial geometric model established through the reconstruction method can be utilized in dynamic simulations to produce high-precision time-frequency response data. This approach reproduces bench test results through numerical computation and offers a new method for evaluating the reliability of key vehicle components under extreme working conditions, such as wheel flats. Overall, the main conclusions are as follows. The analytical modeling method for three-dimensional wheel flats considering multiple worn stages proposed in this paper can effectively characterize each worn morphology during the life cycle of a flat in a parameter-driven approach. By using measured wheel roughness data, an effective reconstruction method for the real morphology of wheel flats is proposed based on the NURBS technique, achieving a minimum reconstruction error of only 2.3%. According to the simulation results, by adjusting the control parameters, the analytical model proposed in this paper can effectively approximate the existing haversine wheel flat modeling method, with an average error of only 0.927% under different depth conditions. This modeling method, which considers multiple worn stages, allows for a detailed investigation of the effects of different worn states on the wheel-rail impact response and contact geometry through numerical simulation. Based on the method, the following patterns were found: In the subcritical regime, the wear degree of the wheel flat predominantly affects the wheel-rail impact force, while in the transcritical regime, both wear degree and velocity determine the magnitude of the wheel-rail impact force. As the degree of wear increases, the wheel lateral jump occurs earlier, with the maximum wear degree increasing the jump duration by 9.7% compared with the minimum wear degree. The spatial modeling method of flats proposed in this paper is highly flexible and provides a new technical approach to simulating the dynamic behavior of wheel-rail contact accurately when a flat is present. Conceptualization, G.Z. and C.C. ; methodology, G.Z. ; software, G.Z. ; validation, G.Z., C.C. and Y.S. ; formal analysis, Y.S. ; investigation, G.Z. ; writing--original draft preparation, G.Z. ; writing--review and editing, G.Z. and N.L. ; visualization, G.Z. and Y.S. ; supervision, N.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (No. 6227001, 61877002) and the China Academy of Railway Sciences Fund Project (No. 2022 YJ145). Not applicable. The example scripts and data in this investigation can be downloaded through the following link: [[https://github.com/Legend19960404/wheel-flat](https://github.com/Legend19960404/wheel-flat)]([https://github.com/Legend19960404/wheel-flat](https://github.com/Legend19960404/wheel-flat)) accessed on 10 May 2024.