# Development of a New Test-Rig for Wheel-Rail Contact Experiments under Large Slip Conditions
## 3 Detailed Description
### Dynamics Analysis  
The test-rig generates the wheel-rail creepage through the gear transmission system by two motors, as shown in Figure 8. One motor (Motor V) drives the test-rig circling to simulate the vehicle velocity while the other (Motor W) produces a velocity difference between vehicle and wheel. In this instance, it is possible to achieve decoupling of two motors by selecting an appropriate gear ratio of the transmission system.  
In Figure 8, \(R\) is wheel radius, \(\rho\) is circling radius of the test-rig. \(\omega_{1}\), \(M_{1}\) are angular velocity and torque of the wheel. \(\omega_{2}\), \(M_{2}\) are angular velocity and torque of the test-rig frame driven by Motor V (PHASE, U31340F) while \(\omega_{3}\), \(M_{3}\) are angular velocity and torque of the central level gear driven by Motor W (PHASE, U31320F). \(r_{1}\) is the pitch circle radius of the bevel gear in the wheel shaft while \(r_{2}\) is the pitch circle radius of the central level gear, \(B\) is the brake force between wheel and rail, \(F\) is gear meshing force of the bevel gears.  
Figure 8: Sketch of the gear transmission system of the test-rig: (1) Motor V, (5) Motor W, (2) (6) clutches, (3–4) (7) (10–12) bevel gears, (8) wheel and (9) test-rig frame.  
Figure 7: Details of one-wheel assembly of the test-rig. (**a**) Front view of the wheel assembly with multiple sensors; (**b**) modification of single-wheel assembly for track grinding.  
According to the bevel gear system, the composition of velocities is as follows:  
\[(\omega_{2}-\omega_{3})r_{2}=\omega_{1}r_{1} \tag{14}\]  
The creepage \(s\) in the test-rig is calculated by the definition of longitudinal creepage of wheel-rail contact, as shown in Equation (15).  
\[s=\xi_{x}=1-\frac{\omega_{1}R}{\omega_{2}\rho} \tag{15}\]  
According to force balance, the dynamics analysis of the wheel shaft is as follows, regardless of any resistance:  
\[\left\{\begin{aligned} M_{1}=BR,M_{3}=Fr_{2}\\ Fr_{1}-BR=0\\ Fr_{2}-B\rho+M_{2}=0\end{aligned}\right. \tag{16}\]  
The adhesion coefficient \(\mu\) in the test-rig is calculated by Equation (17).  
\[\mu=\frac{B}{N}=\frac{M_{1}}{NR} \tag{17}\]  
Let \(R/\rho\) be equal to \(r_{1}/r_{2}\), then the torques produced by Motor V and Motor W are calculated as follows:  
\[M_{2}=BR\bigg{(}\frac{\rho}{R}-\frac{r_{2}}{r_{1}}\bigg{)}=0,M_{3}=B\rho \tag{18}\]  
Meanwhile, the creepage \(s\) can be rewritten as follows:  
\[s=1-\bigg{(}1-\frac{\omega_{3}}{\omega_{2}}\bigg{)}\frac{Fr_{2}}{\rho r_{1}}= \frac{\omega_{3}}{\omega_{2}} \tag{19}\]  
Equation (18) indicates that, regardless of any resistance, the torque on Motor V is 0, and the brake force between wheel and rail is completely borne by Motor W, making these two motors decoupled. Therefore, when the gear ratio \(r_{1}/r_{2}\) is equal to \(R/\rho\), the change in creep force between wheel and rail will not affect the vehicle speed, keeping the creepage stable. Moreover, it can be seen in Equation (19) that the creepage is the specific value of two motor speeds, which brings convenience to slip control. Angular velocities \(\omega_{2}\) and \(\omega_{3}\) can be measured by the torque meters (SANIING, JN338-AF) separately. Equation (17) shows how to calculate the adhesion coefficient and longitudinal torque \(M_{1}\) and vertical force \(N\) of wheel-rail contact can be measured by the torque meter (SANIING, JN338-AF) and the load sensor (ZHEN-DAN, GH-4L).