# Development of a New Test-Rig for Wheel-Rail Contact Experiments under Large Slip Conditions
## 2 Overall Design
### Wheel-Rail Contact and Cercling Diameter  
The adhesion recovery under large slip conditions as well as the increase in available braking or traction force caused by it is a vehicle longitudinal dynamics issue that highlights the significance of longitudinal creepage and creep force on the test-rig similar to the actual vehicle. Looking at the wheel-rail contact interface in Figure 3, the wheel-rail system on the test-rig has no yaw angle, contact angle or lateral displacement geometrically. As a result, three directional creepages on the test-rig can be calculated by their respective definitions [33] as follows:  
\[\xi_{x}=\frac{v-\omega_{1}R}{v}=\frac{\omega_{2}\rho-\omega_{1}R}{\omega_{2}\rho} \tag{8}\]  
\[\xi_{y}=0 \tag{9}\]  
\[\xi_{ep}=\frac{\omega_{2}}{v}=\frac{1}{\rho} \tag{10}\]  
where \(\xi_{x}\) is longitudinal creepage; \(\xi_{y}\) is lateral creepage; \(\xi_{ep}\) is spin creepage; \(v\) is longitudinal speed of the wheel; \(\omega_{1}\) is rotation speed of the wheel; \(\omega_{2}\) is circling speed of the test-rig; \(R\) is for wheel radius while \(\rho\) is for circling radius of the test-rig.  
However, the test-rig circling leads to a different distribution of longitudinal creepage on the contact patch. Coincidentally, a similar distribution of longitudinal creepage occurs to the actual vehicle contact patch owing to tread equivalent conicity of the wheel set. Figure 4 presents the contact patch of the test-rig (right) and actual vehicle (left) when the wheel rolls. The tread equivalent conicity of the wheel set brings about the contact angle \(\delta\). As we can see, the contact patch can be divided into a slip zone and adhesive zone. Longitudinal creepage on the left side of the slip zone is negative while longitudinal creepage on the right side is positive on the contact patch of the actual vehicle. The contact patch of the test-rig shows high similarity, only with the opposite sign of creepage.  
Figure 3: The wheel on test-rig with the diameter of 210 mm and the cross-section radius of 75 mm.  
The maximum longitudinal creepages on the contact patch of the actual vehicle and test-rig are calculated by Equations (11) and (12).  
\[\xi_{xmax}=\frac{v-\omega_{1}(R_{0}+b_{0}\text{tan}\,\delta)}{v}=\xi_{x}-(1-\xi_ {x})\frac{b_{0}\text{tan}\,\delta}{R_{0}} \tag{11}\]  
\[\xi_{xmax}=\frac{v-\omega_{1}R(\rho+b)/\rho}{v}=\xi_{x}-(1-\xi_{x})\frac{b}{\rho} \tag{12}\]  
In order to make the longitudinal creepage on the test-rig similar to that of the actual vehicle, Equations (11) and (12) should be equal as follows:  
\[\frac{b_{0}\text{tan}\,\delta}{R_{0}}=\frac{b}{\rho} \tag{13}\]  
As the wheel radius of the actual vehicle is 420 mm, the ratio of the contact patch is 1:4 because of the 1:4 scaled test-rig and the common tread equivalent conicity is 0.1 to 0.4, the circling radius of the test-rig shall range from 525 mm to 2100 mm according to Equation (13). Finally, the circling diameter of the test-rig is rounded to 2 m for engineering convenience.  
To verify the correctness of the wheel-rail contact relationship of the test-rig, the longitudinal creep forces of the actual vehicle and test-rig with the circling diameter of 2 m under diverse longitudinal creepage are calculated by FASTSIM, as shown in Figure 5. It indicates that the contact patch of the test-rig is approximately a quarter of the actual vehicle contact patch in length. The creep force increases rapidly with small creepage, following the description of Figure 4, and then the shear stress remains at the maximum value of about 350 MPa in both the actual vehicle and test-rig. In summary, the evolution of creep force of the test-rig is consistent with that of the actual vehicle along with the variation of creepage, guaranteeing the accuracy of vehicle longitudinal dynamics on the test-rig.  
Figure 4: The distribution of longitudinal creepage on the contact patch of test-rig (**right**) and actual vehicle (**left**) with the contact angle \(\delta\).