# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 5 Conclusions  
Through mathematical modeling, the physical problem of the wear of reduced-diameter railway wheels was tackled. The algorithm which was constructed allowed emphasizing the importance of diameter in the wear problem. The algorithm links several calculation models and methods, balancing accuracy with computational efficiency. It accounts for key factors affecting wheel wear, including both vehicle-related and superstructure-related elements. By incorporating additional boundary conditions and assumptions, the algorithm can compute wear for wheels of various diameters. In this instance, wheel wear calculations were employed to generate diameter-mileage curves for various scenarios: 920-mm diameter wheels on the Y-25 bogie, 680-mm diameter wheels on the Saas-z 702 bogie, and 355-mm diameter wheels on the Graz Pauker 702 bogie. In light of these results, when analyzing the evolution of one particular wheel (920, 680, or 355 mm), the wheel degradation worsens as the diameter diminishes, so the reprofiling span shortens as a consequence. This result agrees with the initial assumption of the paper: \"presumably, reduced-diameter wheels do not undergo the same degradation as the ordinary-diameter wheels due to its greater angular contact with the rail (number of revolutions)\". The results also prove that smaller wheels can travel for shorter mileages than bigger wheels, as expected. Notwithstanding, the trend observed when extrapolating the results of 680-mm and 355-mm wheels as though they could be reprofilied as 980-mm exhibits non-linear behavior, i.e., halving the diameter does not imply that the lifespan will halve as well. The comprehension of wheel, wheelset and bogie kinematics, and dynamics (which this work has enabled), allows finding the root causes responsible for this behavior:  
1. Contemplating kinematics, reduced-diameter wheels navigate curves more smoothly than standard-diameter wheels due to their lower uncentering, which results in less frequent flange contact with the rails and a lower threshold radius. 2. Contemplating dynamics, the contact between the flange and the rail is gentler with reduced-diameter wheels. When the flanges of these wheels make contact with the rails, the interaction is less forceful due to the reduced uncentering forces. Additionally, because bogies using reduced-diameter wheels carry less load, the force exerted by the rails on the flange is lower. As a result, the balance of forces and torques leads to reduced rail-flange forces. As a continuation of this research study, many actions can be implemented besides those commented upon in reference [1]:  
* Repetition of the results for other track gauges present around the world. Figure 18: (**a**) Partial uncentering for different \(r_{o}\) and \(R\) values; and (**b**) total uncentering for the same values. * Computation of the speed effects through finite elements, proving more accurate results by obtaining the elastic distortions for every different speed. * Computation of the exact load distribution between the tread and the flange in the event of simultaneous contacts. Finite elements would allow for knowing the real deformations, strains, stresses, and forces at both areas. * Inclusion of more superstructure factors modifying wheel (and rail) wear, such as warp, rail deflection, joints, irregularities and cant excess and deficiency under low static friction conditions. * Inclusion of dynamic loads, especially those appearing as the wheels traverse a turnout/switch. * Inclusion of impacts between the wheels and the superstructure, especially those of the wheels with the switch frogs and track devices. * Inclusion of other types of wheel damage shortening wheel life, such as cracks, flats and spalling. * Extension of the algorithm to cover any other bogies belonging to the wagon (wagons have at least one more bogie). Conceptualization, D.S.P. and E.L.; methodology, D.S.P. ; software, D.S.P. ; validation, D.S.P. and E.L.; formal analysis, D.S.P. ; investigation, D.S.P. ; resources, D.S.P.