# Research on Wheel Flat Recognition Based on Wayside Wheel-Rail Force
## 4 Wheel Flat Shock Effect and Its Mapping Relationship
By extracting all peak forces, the force-speed relationship plot in Figure 13 is generated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Units** & **Values** \\ \hline Carbody mass & kg & 90,297 \\ Center of gravity position of carbody (from rail surface) & m & 2.232 \\ Carbody inertia about \(x\)-axis & kg\(\cdot\)m\({}^{2}\) & 2.163 \(\times\) 10\({}^{5}\) \\ Carbody inertia about \(y\)-axis & kg\(\cdot\)m\({}^{2}\) & 9.963 \(\times\) 10\({}^{5}\) \\ Carbody inertia about \(z\)-axis & kg\(\cdot\)m\({}^{2}\) & 9.839 \(\times\) 10\({}^{5}\) \\ Center of gravity position of the pillow (from the rail surface) & m & 0.506 \\ Lateral stiffness of the second suspension (end of bolster) & MN/m & 3.127 \\ Vertical stiffness of the second suspension (end of bolster) & MN/m & 3.127 \\ Longitudinal stiffness of the second suspension (end of bolster) & MN/m & 4.235 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Changed values of parameters involved in the heavy load dynamic model. Figure 11: Mapping the relationship between vehicle speed, wheel–rail force, and wheel flat length in an empty load situation. Under heavy-load conditions, wheel-rail force peaks exhibit a similar trend to the empty-load model: an initial increase with speed, followed by a decrease. Notably, force growth with speed is less steep in loaded conditions, reaching a maximum at approximately 50 km/h before declining. As with the empty-load case, a near-linear relationship persists between impact force magnitude and wheel flat length. To improve the applicability of practical engineering applications, it is necessary to obtain the mapping relationship between fault responses and fault characteristics. This mapping relationship can be derived by fitting the curves of wheel-rail impact force changes under different fault conditions. In real-world detection, vehicle speed can be accurately captured by speed sensors. Balancing computational constraints and curve-fitting precision typical in engineering applications, a two-step fitting approach was adopted: first, wheel-rail force was curve-fitted against wheel flat length; then, the coefficients of the initial fitting function were further regressed with vehicle speed. This iterative process yielded a comprehensive fitting function relating wheel-rail force to both flat length and vehicle speed. Simulations were conducted on empty and loaded vehicles, each featuring a flat on the right front wheel of the leading bogie, with a fixed speed of \(v=80\) km/h. Considering  
Figure 12: Fifty mm flat, 30 km/h, wheel/rail force time-domain figure. Figure 13: Mapping the relationship between vehicle speed, wheel–rail force, and wheel flat length in a loaded situation. the trade-off between simulation accuracy and available computational resources, a second-order polynomial was employed for curve fitting of empty vehicle data, while a third-order polynomial was applied to loaded vehicle data. The fitting result is shown in Figure 14, where both empty load state and loaded state show a good mapping result. Figure 15 displays the fitted contour curves for empty and heavy vehicles. Across the speed range \(v=\) 10-80 km/h, the vertical wheel-rail force of empty vehicles monotonically increases with wheel flat length. Under heavy-load conditions, at low speeds, the vertical force first decreases and then increases with flat length, whereas at high speeds, it increases monotonically with flat length. Within the flat length range from 10 mm to 70 mm, the vertical force for both vehicle types exhibits a non-monotonic speed dependence, first increasing, then decreasing, and subsequently increasing again as speed changes. Notably, heavy vehicles show more pronounced vertical force variations compared to empty vehicles. The figure demonstrates excellent curve-fitting performance, with randomly selected wheel-rail impact forces for both vehicle types falling within a 5% error margin of the fitted  
Figure 14: Wheel–rail impact force fitting result (\(v=\) 80 km/h).