# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Calculation Scenarios  
The objective is to calculate the wear as a function of the nominal diameter for various wheels and compare the results, for which the calculation scenarios and input data must be previously set. In Ref. [17], a review of various types of bogies revealed that bogies equipped with reduced-diameter wheels require more wheels to support the same load. This is because smaller wheels can handle lower axle loads compared to standard-sized wheels, due to their reduced material strength. Consequently, more wheels are needed to distribute the same load across the bogie. Additionally, the minimum diameter after reprofiling cycles is more restrictive for reduced-diameter wheels to ensure operational safety. For comparison, these commercial bogies, used or proposed on rail motorways, are examined:  
* Y--25. This bogie has four wheels (two wheelsets) and supports a total load of 45 tons (22.5 tons per axle) at a maximum speed of 120 km/h. The total wheelbase (\(e\)) is variable, and the wheels are typically braked by brake shoes. The nominal wheel diameter (\(D\)) ranges from 920 mm (original maximum) to 840 mm (operational minimum). * Saas-z 703. This bogie also features four wheels (two wheelsets) and supports a total load of 32 tons (16 tons per axle) at a maximum speed of 100 km/h. The wheelbase (\(e\)) is variable, and the wheels are braked by brake disks. The nominal wheel diameter (\(D\)) ranges from 680 mm to 630 mm. * Graz Pauker 702. This bogie has eight wheels (four wheelsets) and supports a total load of 20 tons (5 tons per axle) at a maximum speed of 100 km/h. The wheelbase (\(e\)) is variable, and the nominal wheel diameter (\(D\)) ranges from 355 mm to 335 mm. While these bogies differ significantly, comparisons should be made under consistent conditions, focusing solely on the parameter being assessed--in this case, the nominal diameter (\(D\)). However, achieving identical conditions for comparison is challenging [17]:  
1. Axle load (\(\lambda_{eje}\)). Maintaining a constant axle load across all scenarios would result in some wheels being overloaded and others underloaded. For instance, axle loads of 22.5 tons would be unrealistic for 680 mm and 355 mm wheels, whereas loads as low as 5 tons would be feasible but would place extreme stress on the smallest wheels. To ensure comparable conditions, the axle load that generates a normal pressure of 1235 MPa is chosen. This value is common since maximum axle loads usually induce between 1100 and 1300 MPa on the wheel, with 1235 MPa being a mean value, even if it exceeds the manufacturer's limit for the smallest wheels. 2. Flange radius (\(r_{p}\)). This is the sum of the nominal rolling radius (\(r_{o}\), which is half of \(D\)) and a constant. Therefore, \(r_{p}\) decreases in proportion to \(D\). As a result, some data will vary depending on the wheel in question. These scenario-dependent data are detailed in Ref. [17] and are summarized in Table 1:  
The remaining conditions, such as the wheelbase, are kept consistent and are detailed in Section 2.7. Only realistic, feasible, and plausible values are used, and variations in geometry and friction, including the effect of speed on dry friction, are considered. Given these parameters, three scenarios are defined: wheels with diameters of 920 mm, 680 mm, and 355 mm. For each scenario, the input data are initially entered into the system. The program then executes the algorithm for each segment of the railway line, reversing \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \begin{tabular}{c} **Value for 920-mm** \\ **Wheels Scenario** \\ \end{tabular} & \
\begin{tabular}{c} **Value for 680-mm** \\ **Wheels Scenario** \\ \end{tabular} &
\begin{tabular}{c} **Value for 355-mm** \\ **Wheels Scenario** \\ \end{tabular} \\ \hline \(D\) (m) & 0.920 & 0.680 & 0.355 \\ \(n_{eis}\) (\(\varnothing\)) & 4 & 4 & 8 \\ \(r_{p}\) (m) & 0.467–0.475 & 0.347–0.355 & 0.185–0.193 \\ \(\lambda_{eje}\) (kg) & 18,784 & 15,325 & 6996 \\ \hline \hline \end{tabular}
\end{table}