# The Impact of Hollow Wear on the Stability of High-Speed Railway Vehicles
## 2 The Influence of Tread Hollow Wear on the Wheel-Rail Contact Relationship
### Nonlinear Characteristics and Revised Equivalent Conicity for worn Wheels  
The nominal equivalent conicity is commonly used to characterize the wheel-rail contact geometry, and this method is only applicable for uniform wear but not for hollow wear. In fact, due to hollow wear on the wheel tread, the equivalent conicity decreases from the outer side towards the flange and then gradually increases. The equivalent conicity values under varying lateral displacements exhibit nonlinear characteristics. As a result, the nominal equivalent conicity cannot accurately reflect the wheel-rail geometric contact relationship of the tread.  
Figure 5: Wheel–rail contact geometry evolutions.  
The equivalent conicity values on multiple wheelsets of a certain type of high-speed train on the Ha-Da high-speed railway line were measured, and four wheels were selected and marked as A, B, C, D. The equivalent conicity curves are shown in Figure 6, characterized by the same equivalent conicity for 3 mm amplitude, but different values for other wheelset amplitudes.  
In response to this issue, [PERSON] [20] considered the nonlinear characteristics of the equivalent conicity curve and expressed the equivalent conicity at point N on the wheel tread as follows:  
\[\lambda_{N}=\frac{\lambda_{\text{N+1}}-\lambda_{\text{N-1}}}{2} \tag{1}\]  
According to the UIC 518 [21], when evaluating conicity values for wheelsets, it is essential to analyze the conicity values associated with amplitudes of 2 mm and 4 mm alongside the conicity value for an amplitude of 3 mm. By examining these three specific conicity values collectively, it becomes possible to establish a more comprehensive understanding of the conicity function. This approach allows for the calculation of a nonlinear parameter that characterizes how the conicity function changes in response to an increase in wheelset amplitude of 1 mm, specifically within the range of amplitudes from 2 mm to 4 mm. However, the lateral displacement of the wheelset increases under the poor railway line conditions, making it essential to broaden the equivalent conicity range. On this condition, the equivalent conicity can be revised as follows:  
1 By calculating the equivalent conicity values corresponding to the integer lateral displacements of the wheelset within the 1-6 mm range, the standard deviation is expressed as Equation (2)  
\[K=\sqrt{\frac{1}{6}\overset{6}{\underset{\text{i=1}}{\sum}}\overset{2}{( \lambda_{\text{i}}-\mu)}} \tag{2}\]  
where \(\lambda_{\text{i}}\) represents the equivalent conicity values for i = 1, 2,..., 6, and \(\mu\) represents the average of these equivalent conicity values. A small standard deviation \(K\) indicates that the equivalent conicity curve changes less.  
2 A nonlinear factor for the equivalent conicity curve was proposed, and it was defined as  
\[\Delta\lambda_{N}=a*K=\frac{\lambda_{4}-\lambda_{2}}{|\lambda_{4}-\lambda_{2 }|}*\sqrt{\frac{1}{6}\overset{6}{\underset{\text{i=1}}{\sum}}\overset{2}{( \lambda_{\text{i}}-\mu)}} \tag{3}\]  
Figure 6: Different equivalent conicity curves.  
3 The nonlinear factor was subtracted from the original nominal equivalent conicity value, after which the nonlinear equivalent conicity \(\lambda_{\text{G}}\) was revised.  
\[\lambda_{\text{G}}=\lambda_{3}-\Delta\lambda_{N} \tag{4}\]  
To elucidate the impact of hollow wear on equivalent conicity, five types of wheels were matched with standard profile rail. The equivalent conicity curves, generated under various lateral displacements of the wheelset, are presented in Figure 7. The simulation results also showed that as the train travel distance increased, the average equivalent conicity value increased, and the equivalent conicity curve gradually showed a negative slope and nonlinear characteristics, further proving that the nonlinear equivalent conicity can more accurately reflect the differences in the transformation of the wheel-rail contact relationship than the nominal equivalent conicity.