# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Hypotheses  
The ensuing hypotheses were considered, in addition to the application assumptions of [PERSON]'s solution, [PERSON]'s method, and the wear calculation:  
1. It is quasi-static, i.e., it does neither regard dynamic loads nor the vibrations derived from them, based on other quasi-static analyses [17; 18]. 2. The method relies on global calculations for the contact patch without dividing it into finite elements. 3. It is stationary, meaning it does not account for changes in variables over time. In transition curves, where these variations are more significant, average values are calculated. 4. It ignores any rail wear and the prior wear of the wheels, meaning it does not update the contact parameters as the profile changes, since this profile is consistently renewed. 5. It is applied to all the bogie wheels. For each wheel, the parameters and wear calculations are stored separately, as the wear varies among the different bogie wheels [18]. 6. It is applied to a single bogie of a wagon. Typically, a wagon comprises two bogies, which can generally rotate independently of one another. 7. It overlooks the tractive and compressive forces that certain wagons transfer to one another through couplings while navigating curves, due to the inherent coupling slacks [19]. 8. It considers that the track bed stiffness is infinite and its damping is null, so that its deformation does not affect the contact patch shape, as is usually the case in these analyses [17; 18]. 9. It disregards non-holonomic bonds/ties, which is usually the case in these analyses [17; 18]. 10. It can account for up to two contact patches on the same wheel: one on the tread and the other on the flange [17]. 11. In [PERSON]'s and [PERSON]'s equations, the spin is assumed to be positive when it is clockwise, as it must comply with the sign convention applied for creepage. This spin is later passed on to the energy transfer model employed. 12. Creepage is derived from the kinematic analyses of the wheelset rather than from the non-dimensional slips, which involve partial derivatives typically not used in global calculations. 13. Throughout this study, the radial deformation \(\delta_{o}\) is considered negligible compared to the wheel radius \(r_{o}\) (\(\delta_{o}\ll\ r_{o}\)). Additionally, since \(\Phi\ll\ \gamma_{o}\) (specifically, \(\Phi\ll\ \psi\ \ll\ \gamma_{o}\), based on the values obtained in Ref. [17]), the influence of \(\Phi\) on \(\gamma_{o}\) can also be disregarded. 14. However, the influence of \(\dot{\Phi}\ (=\mathrm{d}\Phi/dt)\) on the wear occurring at transition curves is taken into account, as it contributes to a slight increase in wear. 15. The displacements from the bogie suspensions and anti-yaw mechanisms are not considered in the kinematic analysis. 16. The variation in wheel and rail curvatures at transition curves is ignored because although the location of the rail-wheel contact shifts across the widths of the wheel and rail (affecting their curvatures), these changes are typically minimal. When the variations are significant, the most unfavorable values are used directly (for example,the curvatures at the flange-rail contact when such contact is expected to occur at a specific transition curve). (q) Defects like cracks, spalling, squats, or flats are disregarded, and only adhesive and abrasive wear are considered [20; 21]. (r) RCF is only anticipated without calculating the extent of the damage caused, often involving subsurface cracks [20]. (s) The bogie wheels are assumed to be non-powered, which means \(F_{t}=0\) at the contact patches. (t) The bogie wheels are assumed to have disk brakes that do not cause wear on the wheels [17]. (u) The railway vehicle is expected to navigate curves (both circular and transitional) at a constant speed, applying brakes (if necessary) before entering the curve, resulting in \(F_{f}=0\) during the curve. An exception to this occurs when the vehicle is traveling downhill, as outlined in the next hypothesis.