# Formation Mechanism of Metro Rail Corrugation Based on Wheel-Rail Stick-Slip Behaviors
## 3 Theoretical Analysis on the Cause of Rail Corrugation
Furthermore, the driving distance difference \(\Delta D_{1}\) between the left and right wheels is obtained as follows:  
\[\Delta D_{1}=D_{1R}-D_{1L}=2\alpha s \tag{6}\]  
Because the wheelset can be approximately regarded as a rigid body, when the wheelset rotates around its center line, the rotation speed of each part is the same, and the larger the radius of the wheel rolling circle, the longer the running distance at the same angle. The greater the radius difference between the left and right wheels of the same wheelset, the greater the running distance difference between the left and right wheels. When the wheelset lateral displacement is \(y_{w}\), the rolling circle radii of the left and right wheels are:  
\[\left\{ \begin{array}{l}r_{L}=r_{0}-\lambda y_{w}\\ r_{R}=r_{0}+\lambda y_{w}\end{array} \right. \tag{7}\]  
where, \(r_{L}\) is the rolling circle radius of the left wheel; \(r_{R}\) is the rolling circle radius of the right wheel; \(\lambda\) is the equivalent slope, representing the variation rate of the rolling circle radius difference of the left and right wheels on the lateral displacement of the wheelset, and its expression is:  
\[\lambda=\frac{R\delta_{0}}{(R-R^{\prime})}\bigg{(}\frac{s+R\prime\delta_{0}}{ s-r_{0}\delta_{0}}\bigg{)}\approx\frac{R\delta_{0}}{(R-R^{\prime})} \tag{8}\]  
Assuming that when the wheelset passes through a circular curve with a radian \(\alpha\), the rotation angle of wheelset around its center line is \(\beta\), then the running distances of the left and right wheels are obtained as follows:  
\[\left\{ \begin{array}{l}D_{2L}=\beta r_{L}=\beta(r_{0}-\lambda y_{w})\\ D_{2R}=\beta r_{R}=\beta(r_{0}+\lambda y_{w})\end{array} \right. \tag{9}\]  
where, \(D_{2L}\) is the running distance of the left wheel; \(D_{2R}\) is the running distance of the right wheel. Similarly, it can be further obtained that the running distance difference \(\Delta D_{2}\) when the left and right wheels are rolling is:  
\[\Delta D_{2}=D_{2R}-D_{2L}=2\beta\lambda y_{w} \tag{10}\]  
Thus, it can be acquired that the driving distance difference \(\Delta D\) between the left and right wheels on the rails when the rolling circle radius difference of the wheels is considered is:  
\[\Delta D=\Delta D_{1}-\Delta D_{2}=2(\alpha s-\beta\lambda y_{w}) \tag{11}\]  
When \(\Delta D\) is greater than 0, it means that the running distance difference between the left and right wheels is not enough to offset the driving distance difference between the left and right wheels on the rails, so that the wheelset lateral displacement is further expanded to make up for the driving distance difference between the left and right wheels. When \(\Delta D\) is equal to 0, it means that the running distance difference between the left and right wheels is equal to the driving distance difference between the left and right wheels on the rails, and the wheelset is in a pure rolling state. When \(\Delta D\) is less than 0, it means that the running distance difference between the left and right wheels exceeds the driving distance difference between the left and right wheels on the rails, so that the wheelset lateral displacement tends to decrease, so as to realize the steady rolling of the wheelset through the curve. Through the above analysis, it can be seen that when the wheelset passes through the curve, the contact motion relationship between the inner and outer wheels (in order to distinguish from the straight line, we used the inner and outer wheels instead of the left and right wheels) and rails has been in a continuously changing process. When the running distance difference between the inner and outer wheels is not equal to the driving distance difference between the inner and outer wheels on rails, the inner and outer wheelswill receive the lateral force at wheel-rail contact points, and the rails will receive the reaction force at same positions. The existence of the above force creates the wheel and rail bear torque.