# Formation Mechanism of Metro Rail Corrugation Based on Wheel-Rail Stick-Slip Behaviors
## 4 Analysis of Wheel-Rail Stick-Slip Characteristics
### Wheel-Rail Stick-Slip Characteristics of Small Radius Curve  
Referring to the measured line Section 1, the line-type composition was set as (50 m straight line + 100 m transition curve + 300 m circular curve + 100 m transition curve), as shown in Figure 7. The circular curve radius was 300 m, the track superelevation was 100 mm, the track gauge widening was 5 mm, the wheel-rail static coefficient of friction was 0.3, the rail surface had no initial irregularity, and the running speed of the vehicle on the circular curve was 55 km/h. Based on the vehicle-track space rigid-flexible coupling model, the wheel-rail longitudinal and lateral creep forces/creepages, wheel-rail normal force, and other parameters can be obtained by calculation. Meantime, the wheel-rail adhesion coefficient can be obtained by dividing the wheel-rail creep force by the normal force, as shown in Equation (12). This section mainly analyzes the relationship between the wheel-rail adhesion coefficient and creepage to study the cause of rail corrugation in a small radius curve. \[\left\{ \begin{array}{l}\mu_{1}=F_{1}/p_{3}\\ \mu_{2}=F_{2}/p_{3}\end{array} \right. \tag{12}\]  
where, \(\mu_{1}\) and \(\mu_{2}\) are the longitudinal and lateral adhesion coefficients; \(F_{1}\) and \(F_{2}\) are the longitudinal and lateral creep forces; \(p_{3}\) is the normal force. As the vehicle passes through the curve, the movement form and stress condition of the guiding wheelset and driven wheelset were different. Therefore, two wheelsets of the front bogie of a single vehicle (namely, the guiding wheelset and the driven wheelset, a total of four wheels) were selected for analysis, and the corresponding wheel-rail stick-slip relationship curves are shown in Figures 8 and 9 (the red solid line denotes the fitting curve, similar in the following sections). It can be seen from Figure 8 that with the increase of the lateral creepage, the lateral adhesion coefficient of the inner rail-inner wheel of the guiding wheelset first decreased and then increased, and a negative slope phenomenon appeared in the wheel-rail stick-slip curve, that is, with the increase of the lateral velocity difference between the inner wheel and inner rail, the lateral adhesion coefficient of the wheel-rail interface decreased, which illustrates that the lateral relative slip between the inner wheel and inner rail occurred. The alternation of wheel-rail lateral adhesion and slip states leads to the wheel-rail lateral torsional stick-slip vibration, and the alternation of the two states is cyclic, so as to achieve the smooth passage of the vehicle through the curve. When the inner wheel and inner rail stick together, the rail wear is small; when the inner wheel slips on the inner rail, the rail wear is large, so with the increase of the number of vehicles running, the inner rail surface eventually forms a wavy wear. The longitudinal adhesion coefficient of the inner rail-inner wheel and the longitudinal and lateral adhesion coefficients of the outer rail-outer wheel are all positively correlated with the corresponding creepages, and there is no negative slope phenomenon, indicating that there will be no microscopic slip at the corresponding wheel-rail interface, so it is not easy to produce rail corrugation. Figure 7: Schematic diagram of line-type composition. According to Figure 9, it can be found that the lateral stick-slip curve of the outer rail-outer wheel of the driven wheelset had a negative slope, and the other wheel-rail stick-slip curves had no obvious negative slope phenomenon, which shows that the outer rail-outer wheel of the driven wheelset will also produce lateral torsional stick-slip vibration, which causes the outer rail to form corrugation. However, the adhesion coefficient of 0.069 required for the lateral torsional stick-slip vibration of the outer rail-outer wheel of the driven wheelset was much greater than the adhesion coefficient of 0.0084 required for the lateral torsional stick-slip vibration of inner rail-inner wheel of the guiding wheelset.