# A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel-Rail Impact Dynamics
## 1 Introduction  
A wheel flat is a common wheel defect that typically occurs when a wheel slides along a rail during braking, as illustrated in Figure 1.  
This sliding causes severe wear on the wheel tread where it contacts the rail, forming a nearly circular or oval damage area. Once a wheel flat develops, it generates significant impact forces on both the vehicle and the rail, as well as periodic impact noise. These effects pose substantial challenges to vehicle operation safety and riding comfort. Figure 1: Wheel flat. The effect of wheel flats on vehicles and track systems has attracted considerable attention from researchers. [PERSON] et al. [1] investigated the dynamic contact loads at the wheel-rail contact points and found that the nonlinear rail pad and ballast model could better predict the wheel-rail impact force. On this basis, they developed a railway vehicle-track model containing multiple flats [2] and analyzed in detail the effect of multiple flats on the peak acceleration of wheels of different sizes and relative positions. To further simulate the high-frequency response during impacts involving wheel flats, a rigid-flexible coupling model was considered. [PERSON] et al. [3] established a rigid-flexible vehicle-turnout dynamic coupling model and investigated the most unfavorable phase of flats on the wheels of a vehicle crossing a turnout. The studies [4; 5] established a joint simulation model considering the combination of an electric drive subsystem and a coupled locomotive-track dynamics model to analyze the dynamic response of a locomotive under the impact of flats of different sizes. [PERSON] et al. [6] developed a numerical model for predicting radiated noise from wheel tracks in the presence of flats. [PERSON] et al. [7] conducted a field test in which unevenness was introduced at the rail head to simulate a series of vehicle flats, monitoring the loads and rail stresses. [PERSON] et al. [8] classified different stages of wheel flats and concluded that the circumferential curvature of the wheel tread is the key parameter controlling the dynamic interaction between the wheel and the rail when the wheel flat is worn. [PERSON] et al. [9] developed a vehicle-track finite element model and investigated the factors affecting the impact loads on wheel flats. [PERSON] et al. [10] performed a three-dimensional elasto-plastic finite element analysis of undamaged wheels and wheels containing flats, considering the effect of impact loads on the parameters of the contact area that influence stresses. [PERSON] et al. [11] developed a three-dimensional finite element model to analyze the effect of flat impact on railroad track performance. The authors of [12] analyzed the dynamic effects of wheel flats on axle box bearings using rigid-flexible coupling modeling. To accurately predict the impact response of a wheel flat, most complex programs use the Hertzian contact model to evaluate the elastic properties of the wheel-rail contact. However, when the wheel-rail contact occurs in the flat region, the basic assumptions of the Hertzian contact model are not satisfied, leading to deviations between the model's calculations and field test results. Consequently, detailed studies have been conducted on wheel-rail contact modeling in the presence of flats. [PERSON] et al. [13] compared the Hertzian contact model with a non-Hertzian contact model in simulating wheel flat impacts and found that the peak contact force calculated using the non-Hertzian contact model was significantly lower. [PERSON] et al. [14] analyzed the effect of contact modeling on simulated wheel flat impacts and investigated wheel-rail dynamic interactions using a three-dimensional non-Hertzian contact time-domain model based on [PERSON]'s variational method [15]. They determined the errors introduced by the simplified method. Their results showed that accurately modeling the longitudinal geometry of the wheel flat actually has a greater impact on the maximum impact force than the choice of the contact model. [PERSON] et al.