# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 4 Discussion  
An algorithm capable of calculating wheel wear was designed in this work. As seen in Figure 16, it was found that a 920-mm wheel installed on a Y-25 bogie can travel for  
Figure 16: (**a**) Diameter–mileage curve for the 920 mm scenario; (**b**) diameter–mileage curve for the 680 mm scenario; (**c**) diameter–mileage curve for the 355 mm scenario; and (**d**) representation of a worn-out wheel profile after a random distance traveled. 124,275 km, a 680-mm wheel mounted on a Saas-z 702 bogie can travel for 75,648 km, while a 355-mm wheel assembled on a Graz Pauker can travel for 26,985 km. Should the last two wheels undergo the same number of reproffiling cycles as the first one then their results would be greater than that of the 920-mm wheel: 118,683 and 101,433 km, respectively. To fully understand these results, it is essential to examine specific aspects uncovered during the overall analysis of this work. This includes a detailed review of the underlying equation blocks that ultimately produce the diameter-mileage curves:  
* Comparing the real life ends (124,275; 75,648; and 26,985 km) in percentual terms with respect to the first value, it is obtained that the 680-mm wheels' life is 30.13% shorter and the 355-mm wheels' life is 78.29% shorter. * Due to the elevated life shortening of 355-mm wheels, operators prefer using bigger wheels. For example, in Ref. [17], 380-mm wheels, which are mounted on the Saad-kms690 bogie, are presented, which can be reprofiled until reaching 335 mm and the difference between both values (45 mm) is 25 mm higher than for 355-mm wheels (20 mm). Escalating the life of 355-mm wheels heuristically with the ratio 45/20, the result is 43,176 km, only 65.26% shorter than 920-mm wheels' life. This is very advantageous despite the elevation in 25 mm of the loading plane height, so replacing 355-mm wheels by 380-mm wheels will ultimately depend on the application (semi-trailers' heights and tunnels and bridges' loading gauges). * The distance difference between reprofiling (the reprofiling span) is very variable when reprofiling a same wheel and, obviously, when moving across wheels, so adopting arithmetic mean values is required. The mean value is 4,603 km for 920-mm wheels, 4396 km for 680-mm ones, and 3757 km for 355-mm ones. * Should the wagons perform \(n\) routes Albarque-Zacarin-Albarque (75.272 km) a week, then reprofiling the periodicity should be \(Reprofiling\ span\cdot(7/(75.272n))\). Using the average value of 4250 km, the approximate result obtained is \(56\cdot(7/n)\). * If all of the wheels were reprofiled the same number of cycles (always eliminating 80 mm in diameter), then the wheels' life (fictional, as eliminating 80 mm would be against the manufacturers and operators' regulations) would be: 118,683 km for 680-mm wheels and 101,433 km for 355-mm wheels. The former value is 4.50% lower than that of 920-mm wheels (124,275 km), while the latter value is 18.38% lower. * These trends are summarized in Figure 17, where it can be seen that neither the behavior of the real life end nor that of the fictional life end are linear:
* This non-linear behavior responds to the different kinematic response of reduced-diameter wheels when negotiating curves. As demonstrated by [PERSON]'s formula, uncentering is proportional to wheel radius (to wheel diameter in turn, as the radius is half), so not only do reduced-diameter wheels uncenter less than ordinary-diameter ones, but also their flanges will push against the rails less intensely. Moreover, the bogies where reduced-diameter wheels are mounted are less loaded, which will further reduce the force exerted by the rail on the flange (coming from force and torque balances).