# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Input Data  
As illustrated in Figure 3, the algorithm requires input related to several factors: wheelset and bogie geometry, wheel and rail geometries, material properties in contact, load characteristics, railway line geometry, and vehicle speed. The vehicle speed can be aligned with the railway line's specifications if the vehicle operates at the maximum allowable speed for that infrastructure.  
For the three scenarios, the wheel profile follows the 1/40 standard and is made from ER8 steel, while the rail profile adheres to the 60E1 standard and is made from R260 steel [47]. Most wheelset and bogie characteristics, sourced from the bogie comparison in Ref. [17], are consistent across all scenarios. Similarly, parameters for friction variation with speed, based on [PERSON]'s method (which incorporates variable friction under dry conditions), are the same. These common input parameters are in Table A3 (Appendix B).  
Regarding railway line parameters, calculations for the three scenarios use data from a hypothetical railway line. Railway line design parameters are referenced from Refs. [26; 48; 49; 50], though not all are used for wear calculations. In Ref. [17], a railway line is defined segment by segment with the following parameters:  
* Initial and final metric points (\(Q_{in}\) and \(Q_{f}\), respectively).
* Stretch type: RECTA (straight), CIR (circular curve), CLO (clothoid), PARACUAD (quadratic parabola), or PARACUB (cubic parabola).
* Curve direction: NING (the stretch is straight), IZDA (curve to the left), or DCHA (curve to the right).
* Bogie position on the curve: NING (the stretch is straight), ENT (the bogie is entering the curve), or SAL (the bogie is exiting the curve).
* Curve radius (\(R\)), cant (\(h_{r}\)), and inclination (\(i\)).
* Initial and final maximum allowed speeds (\(V_{in}\) and \(V_{fn}\), respectively).  
Constant values, such as the track gauge (1.668 m), remain uniform across all segments. Gauge widening is defined as a piecewise function imported from Ref. [26], with the parameter \(\xi\) varying by curve radius (\(R\)). For instance, \(\xi\) is zero for curves with R greater than 300 m and 20 mm for curves with \(R\) between 100 and 150 m.  
Additional parameters, such as those for transition curves, are pre-defined, while others are derived from the provided values. For example, the distance between two metric points is simply the difference between them.  
The 333 defined stretches from Ref. [17] are available in the Supplementary Materials. The curve radii range from a minimum of 265 m (where the ratio \(e/R_{min}\) is less than 0.01, indicating restricted movement) to a maximum approaching infinity in straight sections (\(\infty\) is not accepted in EES, so it is approximated as 5\(\times\)10\({}^{7}\)), with 200-800 m radii being the most common. For added realism, fictional names are used: station 1 is Albarque, station 2 is Zacarin, and there is an intermediate station named Milbello.  
Finally, the Supplementary Materials includes details on incorporating [PERSON]'s and [PERSON]'s coefficients through polynomials and the 150 equations not displayed in this document due to space constraints but are part of the algorithm [51].