## 2 Materials and Methods
### Description of a Wheel Movement with a Traditional Wheel Design
The angle \(\chi\) between the projection \(\vec{V}_{1 XZ}^{B}\) of the slippage velocity vector \(\vec{V}_{1}^{B}\) in the center of the flange contact \(B_{1}\) on the plane \(xoz\) in the coordinate system associated with the wheel and the horizontal axis can be determined using the following relation (Figure 1):  
\[\chi\ =\ \arcsin(\frac{x_{G}}{|CB_{1}|}) \tag{3}\]  
The relative wheel slippage on the rail at the center of the main contact point \(A_{1}\) will be  
\[\varepsilon\ =\ \frac{V_{1}^{A}}{V_{W}}. \tag{4}\]  
From the geometry shown in the calculating scheme depicted in Figure 1 we can write:  
\[|OC|\ =\ r_{1}^{A}-|A_{1}C|,\ |DC|\ =\ h_{F}+|A_{1}C|,\ |CB_{1}|\ =\ \sqrt{|DC|^{2}+X_{F}^{2}}, \tag{5}\]  
\[\dot{\phi}_{W}\ =\ V_{1}^{A}/|A_{1}C|\ =\ V_{W}/|OC|\Rightarrow\frac{V_{1}^{A}}{V _{W}}\ =\ \varepsilon\ =\ \frac{|A_{1}C|}{r_{1}^{A}-|A_{1}C|} \tag{6}\]  
After transformations of expression (6), we obtain  
\[|A_{1}C|\ =\ r_{1}^{A}\cdot\frac{\varepsilon}{1+\varepsilon}, \tag{7}\]  
\[\dot{\phi}_{W}\ =\ V_{W}/|OC|\ =\ V_{1 XZ}^{B}/|CB_{1}|\Rightarrow\]  
Figure 2:0  
Respectively,  
Figure 2:1  
Analyzing expressions (3) and (6) we can see, that the value of the angle \(\chi\) and the projection modulus \(V_{1 XZ}^{B}\) slip velocity vector in a flange contact \(\vec{V}_{1}^{B}\) is unambiguously determined by the geometric characteristics of the contact between the wheel and the rail, the parameters of their profiles, as well as the value of slippage in the center of the main contact. Figure 3 shows the projections of the slip velocity vector \(\overset{\rightarrow}{V}_{1}^{B}\) in the flange contact in the axis of the coordinate system related with the wheel. Modulus of the corresponding projections of the slip velocity vector \(\overset{\rightarrow}{V}_{1}^{B}\) in the flange contact in the coordinate axis are determined as follows:  
Figure 2:2  
Then  
\[ \begin{array}{rcl}V_{1}^{B}&=&\sqrt{\left(V_{1X}^{B}\right)^{2}+\left(V_{1Y}^ {B}\right)^{2}+\left(V_{1Z}^{B}\right)^{2}}\ =\ V_{1 XZ}^{B}\cdot\sqrt{\left(\cos\chi\right)^{2}+\left(\frac{\sin\chi}{\tan \beta}\right)+\left(\sin\chi\right)^{2}}=\\ &=&V_{W}\cdot\frac{\sqrt{\left(h_{F}+\Gamma_{1}^{A}\frac{\varepsilon}{1+ \varepsilon}\right)^{2}+X_{F}^{2}}}{r_{1}^{A}\left(1-\frac{\varepsilon}{1+ \varepsilon}\right)}\cdot\sqrt{1+\left(\frac{\sin\chi}{\tan\beta}\right)^{2} }\end{array}  \tag{11}\]  
It is obvious, that in this case, the modulus of the vector \(\overset{\rightarrow}{V}_{1}^{B}\) is also unequivocally determined by the geometry of the contact of the wheel and the rail by the longitudinal velocity of the wheel and the level of slip in the main contact. The influence of the level of relative slippage \(\varepsilon\) in the main contact of the traditional wheel design on the value of the slip velocity in its flange contact for various motion conditions are shown by the graphs depicted in Figure 4. The calculations were carried out for the following values of the variables: \(V_{W}=20\) m/s, \(r_{1}^{A}=0.475\) m, \(h_{F}=0.01\) m.  
Analyzing the graph shows a significant effect of the level of relative slippage \(\varepsilon\) in the main contact on the value of the slip velocity \(V_{1}^{B}\) in the flange contact of the traditional wheel design. This effect is especially noticeable when the wheel moves with small angles to attack on the rail [27; 28]. The graph in Figure 4 clearly shows the minimum value \(V_{1}^{B}\), which is reached when the value \(\varepsilon\approx 0.02\). This effect is obviously related with the peculiarities of the spatial arrangement of the contact points between the wheel and the rail under the given geometric characteristics and running conditions. For \(\psi=0.0\) rad, the center of the flange contact coincides with the instantaneous c  
Figure 3: A calculation scheme—front view, the traditional wheel design.