# Formation Mechanism of Metro Rail Corrugation Based on Wheel-Rail Stick-Slip Behaviors
## 3 Theoretical Analysis on the Cause of Rail Corrugation  
When the vehicle is running along the line, in order to avoid the frequent contact between the wheel flange and the rail side and facilitate the vehicle to pass through the curve, the outside distance between the left and right wheel flanges is less than the track gauge, so the wheelset can make lateral displacement and yaw angle relative to the track. Under conditions of different lateral displacements and yaw angles, the contact points between the left and right wheels and rails have different positions, so the contact parameters between wheel and rail also change accordingly, that is, the geometric parameters of wheel-rail contact are actually functions of lateral displacement and yaw angle of the wheelset relative to the track. Assuming the wheel has LM wear-type tread and the rail has standard CHN60 profile, when the center of the wheelset is consistent with the center line of the track, the contact angles between the left and right wheels and rails are all \(\delta_{0}\), and the rolling circle radii of wheels are all \(r_{0}\). When the lateral displacement \(y_{\text{nz}}\) occurs to the right of the wheelset, the diagram of the wheel-rail motion relationship is shown in Figure 3, and the contact angles of the left and right wheels and rails are as follows:  
\[\left\{ \begin{array}{l}\delta_{L}=\delta_{0}-\frac{\epsilon y_{\text{nz}}}{ \delta_{\text{R}}}\\ \delta_{\text{R}}=\delta_{0}+\frac{\epsilon y_{\text{nz}}}{\delta_{\text{R}} }\end{array} \right. \tag{1}\]  
Figure 1: Corrugation pictures on-site. (**a**) Short pitch corrugation on the inner rail. (**b**) Long wavelength corrugation on the inner rail. (**c**) Side corrugation on the outer rail. (**d**) Rail corrugation on the straight line. Figure 2: Amplitude distribution diagram of rail corrugation. where, \(\delta_{L}\) is the left wheel-rail contact angle; \(\delta_{R}\) is the right wheel-rail contact angle; \(s\) is the half of rolling circle span of left and right wheels; \(\varepsilon\) is the contact angle parameter, representing the variation rate of the slope of contact surface with respect to the lateral displacement of the wheelset, which can be expressed as follows:  
\[\varepsilon=\frac{s}{(R-R\prime)}\bigg{(}\frac{s+R\delta_{0}}{s-r_{0}\delta_ {0}}\bigg{)}\approx\frac{s}{(R-R\prime)} \tag{2}\]  
where, \(R\) is the curvature radius of the wheel tread; \(R\prime\) is the curvature radius of the rail top surface. When the lateral displacement of the wheelset is \(y_{w}\), the distances between the wheelset center line and the wheel-rail contact points are as follows:  
\[\left\{ \begin{array}{l}s_{L}=s+\xi y_{w}\\ s_{R}=s-\xi y_{w}\end{array} \right. \tag{3}\]  
where, \(s_{L}\) is the distance between the wheelset center line and the left wheel-rail contact point; \(s_{R}\) is the distance between the wheelset center line and the right wheel-rail contact point; \(\xi\) is the variation rate of the distance between the wheelset center line and the wheel-rail contact point with respect to the lateral displacement of the wheelset, which can be expressed as follows:  
\[\xi=\frac{R}{(R-R\prime)}\bigg{(}\frac{s+R\prime\delta_{0}}{s-r_{0}\delta_{0} }\bigg{)}\approx\frac{R}{(R-R\prime)} \tag{4}\]  
When the wheelset passes the circular curve line with a radian \(\alpha\) and a radius \(R_{T}\), the driving distances of the left and right wheels can be expressed as:  
\[\left\{ \begin{array}{l}D_{1L}=\alpha(R_{T}-s_{L})=\alpha(R_{T}-s-\xi y_{w}) \\ D_{1R}=\alpha(R_{T}+s_{R})=\alpha(R_{T}+s-\xi y_{w})\end{array} \right. \tag{5}\]  
Figure 3: Diagram of the wheel–rail motion relationship. where, \(D_{1L}\) is the driving distance of the left wheel; \(D_{1R}\) is the driving distance of the right wheel.