# Development of a New Test-Rig for Wheel-Rail Contact Experiments under Large Slip Conditions
## 2 Overall Design
### Scale and Power  
The scale of the test-rig refers to the ratio of wheel diameter of the test-rig to the actual wheel diameter. So, the scale of the test-rig determines the wheel diameter of the test-rig and the vertical force applied to the wheel of the test-rig as well as the power required by the test-rig. With the limitation of total power in the laboratory, the downscale wheel-on-ring test-rig is developed.  
Figure 2: On a wheel-on-roller test-rig, the creep force of wheel set is \(F_{1}\), opposing the creep force of roller \(F_{2}\). Consequently, the direction of the rotation \(\omega_{1}\) and torque \(M_{1}\) of the motor driving the wheel set is the opposite, making the motor operate in a generator condition.  
The research on the scaling strategies is complete and these methods have been widely adopted and used in many scaled test-rigs [32]. The fundamental theory to be satisfied is the Hertz equivalent relationship:  
1. The ratio of the longitudinal semi-axis length \(a\) to the transversal semi-axis length \(b\) of the elliptical contact patch is the same in both test-rig and reality;
2. The maximum contact pressure \(q\) of contact patch is the same in both test-rig and reality. The equations for \(q\), \(a\) and \(b\) are as follows: \[q=\frac{3N}{2\pi ab}\] (1)  
\[a=m\sqrt[3]{\frac{3N}{4A}\bigg{(}\frac{1-\
u_{w}^{2}}{E_{w}}+\frac{1-\
u_{r}^{ 2}}{E_{r}}\bigg{)}} \tag{2}\]  
\[b=n\sqrt[3]{\frac{3N}{4A}\bigg{(}\frac{1-\
u_{w}^{2}}{E_{w}}+\frac{1-\
u_{r}^{ 2}}{E_{r}}\bigg{)}} \tag{3}\]  
where \(N\) is the vertical force applied to the wheel; \(m\), \(n\) are non-dimensional Hertz coefficients; \(E_{w}\), \(E_{r}\) are the elastic moduli of wheel and rail material; \(\
u_{w}\), \(\
u_{r}\) are [PERSON]'s ratios of wheel and rail material; \(A\) and \(B\) are equivalent radii of wheel and rail. The equations for \(A\) and \(B\) are as follows:  
\[A=\frac{1}{2}\bigg{(}\frac{1}{R_{w1}}+\frac{1}{R_{w2}}+\frac{1}{R_{r1}}+\frac {1}{R_{r2}}\bigg{)} \tag{4}\]  
\[B=\frac{1}{2}\bigg{(}\frac{1}{R_{w1}}-\frac{1}{R_{w2}}+\frac{1}{R_{r1}}-\frac {1}{R_{r2}}\bigg{)} \tag{5}\]  
where \(R_{w1}\), \(R_{r1}\) are the rolling circle radii of wheel and rail; \(R_{w2}\), \(R_{r2}\) are the cross-section radii of wheel tread and rail. \(R_{w1}\) = 420 mm, \(R_{w2}\) = \(\infty\), \(R_{r1}\) = \(\infty\), \(R_{r2}\) = 300 mm in actual wheel-rail contact. The ratio between \(A\) and \(B\) is the same in both test-rig and reality.  
From Equations (1)-(3), the ratio between vertical force in test-rig \(N_{1}\) and reality \(N_{2}\) is the square of the ratio between the equivalent radius of wheel in reality \(A_{2}\) and test-rig \(A_{1}\):  
\[\frac{N_{1}}{N_{2}}=\bigg{(}\frac{A_{2}}{A_{1}}\bigg{)}^{2} \tag{6}\]  
The required power of the test-rig \(P\) can be calculated by the following equation:  
\[P=\frac{\mu_{max}\cdot N_{max}\cdot\
u_{max}\cdot S_{max}}{\eta} \tag{7}\]  
where \(\mu_{\max}\) is the maximum adhesion coefficient, taking 0.6; \(N_{\max}\) is the maximum vertical force of the wheel; \(\
u_{\max}\) is the maximum running speed, taking 160 km/h; \(s_{\max}\) is the maximum creepage, taking 0.3; \(\eta\) is the mechanical efficiency of the test-rig, taking 0.9. The required power and other parameters of the test-rig in different scales are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scale** & **1:1** & **1:2** & **1:4** \\ \hline \(R_{w1}\) (mm) & 420 & 210 & 105 \\ \(R_{w2}\) (mm) & 300 & 150 & 75 \\ \(N_{\max}\) * (kN) & 85 & 21.3 & 5.3 \\ \(P\) (kW) & 756 & 189 & 47 \\ \hline \hline \end{tabular}  
* The maximum vertical load is 17 t per wheel set in actual vehicle in this research.  
\end{table}
Table 2: The parameters of the test-rig in different scales.  
As the total power of the laboratory is no more than 150 kW, a 1:4 scaled test-rig is adopted and the cross-section of the wheel on the test-rig is shown in Figure 3, according to the parameters given in Table 2.