# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Calculation Model
#### 2.4.8 Characterization of Flange-Rail Contact  
Flange-rail contact is an aggressive interaction that occurs on tight or narrow curves where gauge widening is insufficient for smooth negotiation. In this scenario, the wheel flange presses laterally against the rail, which in turn exerts a reaction force on the wheel, increasing pressure on the flange, a region with small radii. Even when flange-rail contact is present, the typical tread-rail contact does not disappear (at least under the assumptions considered here), making it crucial to understand the load on each contact point.  
To determine the reaction force exerted by the rail on the flange, Ref. [28] introduces the center of friction model. This model posits that for each bogie negotiating a curve, there exists a point where, if a wheel were mounted, it would rotate ideally without slipping. This point is known as the center of friction, and identifying it allows for the calculation of forces acting on the flange-rail contact through force and torque balances.  
According to this model, there can be one flange-rail contact (free motion) or two flange-rail contacts (restricted motion). The latter situation occurs in the tightest curves when both wheels of a diagonal pair touch the rails.  
Regarding load distribution at each contact, Ref. [39] explains the Sauvage model. This model states that the total indentation (\(\delta_{o}\)) is the sum of the indentation at the tread-rail contact (\(\delta_{br}\)) and at the flange-rail contact (\(\delta_{pe}\)). Since these indents depend on the normal force, it is possible to derive the normal force at the tread contact (\(N_{br}\)) and the flange  
Figure 12: (**a**) Evolution of slip and stick regions as the creepage, purely longitudinal here, grows; (**b**) traction curve for the case of pure longitudinal creepage; and (**c**) tangential forces and torques for a four-wheel bogie. Source: own elaboration and [1].  
contact (\(N_{p}\)). However, this method is heuristic unless supported by a robust finite-element analysis to determine the precise values of \(\delta_{br}\) and \(\delta_{pe}\).  
Ref. [17] compiles both models and simplifies the Sauvage model by introducing the load distribution coefficient (\(\alpha_{fn}\)), which ranges from 0.5 (indicating equal normal load for both contacts) to 1 (where the tread contact would be unloaded). Typical values for this coefficient, derived from the Sauvage model results, fall between 0.7 and 0.8.  
Figure 13 illustrates the parts of the center of friction model. It can be noted that the motion is restricted when both the wheels \(W_{1}\) and \(W_{4}\) touch the rail, receiving the reactions \(\zeta_{h.1}\) and \(\zeta_{h.4}\) from the rails, respectively. Otherwise, when the motion is free, only the wheel \(W_{1}\) touches the rail and \(\zeta_{h.4}\) is null. Below Figure 13, the main equations of this model and the normal force distribution by means of \(\alpha_{fn}\) are presented:  
\[\zeta_{v}=\alpha_{fn}N_{e} \tag{46}\] \[N_{p}=\zeta_{v}\ cos\gamma_{o}+\zeta_{h}\ sen\gamma_{o}\] (47) \[\left(-F_{t}\right|+F_{f}\right)=-\sum_{i=1}^{i=Z_{w}+2}F_{x,i}^{\prime}\] (48) \[\zeta_{h,1}-\zeta_{h,Z_{w}}=\sum_{i=1}^{i=Z_{w}+2}F_{y,i}^{\prime}\] (49) \[\zeta_{h,1}u_{fl,1}-\zeta_{h,Z_{w}}u_{fl,A}=\sum_{i=1}^{i=Z_{w}+2} \left(F_{y,i}^{\prime}\ \omega_{f,i}\right)+\sum_{i=1}^{i=Z_{w}+2}\left(F_{x,i}^{ \prime}\ \omega_{f,i}\right) \tag{50}\]