# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 2 Materials and Methods  
### Theories of Friction in Rolling Contacts  
Building upon [PERSON]'s foundational work, numerous wheel-rail contact theories have emerged over the past few decades. These theories strive to establish a mathematical relationship between the forces acting at the contact point, the spin moment experienced by the wheelset, and both the linear and spin creepage values [28]. However, each theory has limitations, leading to various approaches, including linear theory, strip theory, empirical theory, simplified theory, and the exact three-dimensional rolling contact theory. The linear theory [13; 14; 15] is suitable for scenarios with minimal spin and creepage values. However, this assumption disregards the pressure distribution at the contact area'sedges. Consequently, it fails to enforce the boundary conditions at the trailing edge, where the contact between wheel and rail particles terminates. The strip theory suffers from similar drawbacks. While it delivers highly accurate predictions regarding the contact area's shape, it exhibits limitations in accurately predicting the tangential forces. The empirical theory [23] defines the relationship between longitudinal and lateral creepage, as well as the total creep force experienced at the contact point. Experimental studies have shown that the predicted total force aligns well with measured values for low creepage conditions. However, as the creepage value increases, the expected force values deviate from the measured data, often exceeding the actual measurements. The simplified theory [25] adopts the assumption of an elliptical contact area. This simplification introduces an error estimated to be around 15% [29]. However, this error may be acceptable in practice because contamination can significantly reduce the friction coefficient, leading to variations in the creep forces' saturation level. Experimental validation has shown that extremely high accuracy in calculating creep forces is not essential [30]. Therefore, the simplified traction-displacement relationship can be a sufficient tool for railway vehicle design. The three-dimensional rolling contact theory [29] offers the most comprehensive analysis, but its high computational demands make it less suitable for complex multibody models. Similarly, alternative theories for conformal contact (where the wheel and rail surfaces perfectly match) also necessitate intricate computational procedures. Early time-domain solution methods for the friction-creep relationship modeled saturation as an evolution of linear expressions [31]. Various mathematical formulations have been proposed to describe this saturation stage, including [PERSON]'s law, [PERSON]'s empirical proposition, and exponential, hyperbolic, or arctangent laws [31; 32]. Studies on locomotives operating under saturated adhesion conditions often employ simplified linear adhesion laws [32]. These laws typically feature a steep slope in the increasing portion (adhesion) and a mild negative slope in the saturated stage (slip) [33; 34]. A recurring characteristic of these friction laws is that they are derived by fitting curves to measured data rather than representing a physical model of the contact mechanics. Another critical aspect is the significant difference in magnitude between the creep values in the initial stages and those observed in the saturated stage. The previous paragraphs summarize the key takeaways regarding rolling friction models for railway vehicles, which are as follows:  
1. Real-world variations: Friction laws vary significantly due to the influence of rail conditions (contamination, wetness, etc.). 2. Balancing accuracy and efficiency: Experimental results suggest that high precision in calculating creep forces is not crucial [30]. This allows for simplified traction-displacement relationships, which are computationally efficient and suitable for railway vehicle design. 3. Disparity in creep values: The range of creep values in the unsaturated friction regime is much smaller than that in the saturated regime. 4.