## 2 Materials and Methods
### Description of the Wheel Movement with a Perspective Wheel Design
 \tag{16}\]  
The modulus of the velocity \(\overset{\rightarrow}{V}_{1 XZ}\) will be determined when considering the calculation scheme in Figure 5 as follows:  
* From one side: \[\dot{\phi}_{F}\ =\ \frac{V_{1 XZ}^{BA}}{|EB_{1}|},\] (17)
* From the opposite side: \[\dot{\phi}_{F}\ =\ \frac{V_{W}}{|EO|}.\] (18) where \[|EO|\ =\ r_{1}^{A}+h_{F}+|ED|,\] (19) Then we can write: \[\frac{V_{1 XZ}^{BA}}{|EB_{1}|}\ =\ \frac{V_{W}}{|EO|},\] (20)After simple transformations we get:  
\[V_{1 XZ}^{BA}\ =\ V_{W}\cdot\frac{\sqrt{\left|ED\right|^{2}+X_{F}^{2}}}{r_{1}^{A}+h_ {F}+\left|ED\right|}, \tag{21}\]  
By considering (1) and (15) we get:  
\[ \begin{array}{ll}V_{1 XZ}^{BA}&=\ V_{W}\cdot\frac{\sqrt{\left[ \left(\hat{\varphi}_{F}\cdot\frac{\hat{\varphi}_{W}}{\hat{\varphi}_{F}\cdot(1+ \varepsilon)}-1\right)r_{1}^{A}-h_{F}\right]^{2}+X_{F}^{2}}}{\frac{\hat{ \varphi}_{W}}{\hat{\varphi}_{F}\cdot(1+\varepsilon)}\cdot r_{1}^{A}}=\\ &=\ \hat{\varphi}_{F}\cdot\sqrt{\left[\left(\frac{\hat{\varphi}_{W}}{\hat{ \varphi}_{F}\cdot(1+\varepsilon)}-1\right)\cdot r_{1}^{A}-h_{F}\right]^{2}+ \left[\left(r_{1}^{A}+h_{F}\right)\cdot\tan\psi\cdot\tan\beta\right]^{2}}. \end{array}  \tag{22}\]  
Using expressions similar to expressions (10) we can write:  
\[ \begin{array}{ll}V_{1}^{BA}&=\sqrt{\left(V_{1X}^{BA}\right)^{2}+\left(V_{1Y} ^{BA}\right)^{2}+\left(V_{1Z}^{BA}\right)^{2}}\ =\ V_{1 XZ}^{BA}\cdot\sqrt{\left(\cos\chi^{ \prime}\right)^{2}+\left(\frac{\sin\chi^{\prime}}{\tan\beta}\right)^{2}+\left( \sin\chi^{\prime}\right)^{2}}=\\ &=\ \hat{\varphi}_{F}\cdot\sqrt{\left[\left(\frac{\hat{\varphi}_{W}}{\hat{ \varphi}_{F}\cdot(1+\varepsilon)}-1\right)\cdot r_{1}^{A}-h_{F}\right]^{2}+ \left[\left(r_{1}^{A}+h_{F}\right)\cdot\tan\psi\cdot\tan\beta\right]^{2}} \cdot\sqrt{1+\left(\frac{\sin\chi^{\prime}}{\tan\beta}\right)^{2}}.\end{array}  \tag{23}\]  
Obviously, if there is a constructive possibility of turning of the flange relative to the wheel tread surface, then the angle \(\chi^{\prime}\) of inclination to the horizontal plane of the projection \(\overset{\to BA}{V_{1 XZ}}\) of slip velocity vector of the flange on the rail and the modulus of this vector will also depend on the ratio of the angular velocity of the wheel rotation and the flange. Given the values \(\hat{\varphi}_{F}\) at a fixed value \(\hat{\varphi}_{W}\) and \(\varepsilon\), one can obtain the corresponding values of the angle \(\chi^{\prime}\) (Figure 6) and the modulus of the velocity vector \(\overset{\to BA}{V_{1}}\) (Figure 7). When we analyze the graph shown in Figure 6, we can see that the value of the angle \(\chi^{\prime}\) is relatively small, essentially in the entire considered range of values of angular velocity \(\dot{\varphi}_{F}\) and only at values \(\dot{\varphi}_{F}\rightarrow\dot{\varphi}_{W}\cdot\frac{r_{1}^{A}}{r_{1}^{B}}\) at the given longitudinal velocity of the wheel, the value of the angle \(\chi^{\prime}\rightarrow\pi/2\). In this case, the projection of the slip velocity vector \(\overset{-BA}{V}_{1 XZ}\) of the flange along to the side edge of the rail head onto the rolling plane of the wheel directs perpendicularly to the horizontal direction and the modulus of this vector reaches the minimum value. For example, for the angle to attack \(\psi=0.015\), \(\varepsilon=0.01\), and \(r_{1}^{A}=475\) mm, the ratio \(r_{1}^{A}/r_{1}^{B}\) is of 0.979. With the indicated ratio of the angular velocities of the tread surface and the flange of the wheel, the linear longitudinal velocity of movement of the main centers and the flange contacts can be considered approximately equal, when the wheel is rolling at a constant linear velocity \(V_{W}=20\) m/s. From the graph depicted in Figure 7 it can be seen, that at \(\chi^{\prime}=\pi/2\) for the given running conditions, the value of the slip velocity in the flange contact \(V_{1}^{BA}\) is small, but it does not equal to zero. Analyzing the expression (19), it is possible to determine the possible modes of the wheel motion, in which \(V_{1}^{BA}\to 0\).