# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 2 Materials and Methods
### A Rolling Friction Model with Saturation  
The dependence of tangential force on creep velocity in rolling contacts exhibits a characteristic maximum value, referred to as friction saturation. As creep increases from zero, the tangential force rises steadily until it reaches saturation. Beyond this point, further increases in creep lead to a decrease in tangential force. Figure 1 depicts a typical measured relationship between creep force and creep velocity. It is important to note that the shape of this curve, particularly the initial slope, varies significantly depending on whether the rail surface is wet, dry, or contaminated.  
In the drivetrain dynamics, the focus is typically on the longitudinal direction of motion. When considering only the longitudinal forces (and neglecting spin), the saturation mechanism is represented by three distinct stages, as illustrated in Figure 1. The curve's initial linear portion (I) corresponds to a condition of full adhesion between the wheel and rail. As creep increases, the second stage commences (II), characterized by the onset of partial slip within the contact area. Finally, the third stage signifies the achievement of fully saturated slip (III).  
This work proposes a solution that addresses these challenges by applying a non-smooth rolling friction model, as presented in Equation (1). Non-smooth models have been demonstrated to provide efficient and accurate solutions for dynamic simulations of rigid multibody systems [36]. Examples include [37], which explores the application of a switch model, and [38], a comprehensive study on non-smooth train dynamics. Recognizing the specific characteristics of the phenomena under investigation, a set-valued friction law was employed to describe the unsaturated friction regime. This approach closely resembles the set-valued models used for static friction.  
\[F_{f}\in\left\{\begin{array}{c}F_{s}(t,\ v),\ \text{if}|v|>0,\ (\text{slip})\\ F_{a}\in[0\ F_{sMax}]\ \text{if}\ v=0,\ (\text{stick})\end{array}\right. \tag{1}\]  
The key distinguishing feature of this friction law lies in treating the two friction regimes. While the saturated friction force \(F_{s}(v)\) is described by a classical constitutive law similar to those mentioned previously, the adhesion force \(F_{a}\) takes any value within the  
Figure 1: Friction force in rolling contacts.  
set [0, \(F_{sMax}\)] as long as the relative velocity, \(\
u\), remains zero. This set-valued nature of \(F_{a}\) necessitates the formulation of the system dynamics as a differential inclusion [39].  
Stick-slip motion inherently involves distinct mechanisms, leading to discontinuities in the differential equations that govern its behavior. While such a model offers a more realistic representation compared to stiff differential systems, it also presents greater complexity for analysis [36; 39; 40].  
The non-smooth approach necessitates working with separate, non-overlapping time intervals during the computation. Within each interval, the system exhibits a specific configuration, characterized by either stick or slip phases between the contacting bodies, essentially representing the system's mode of operation.  
This solution strategy requires specialized techniques to identify the switching moments (events) that mark transitions between these modes. A core challenge lies in determining the specific value of the adhesion force within the set [0, \(F_{sMax}\)]. Another complication arises from the non-zero value of the upper bound of the stick phase, \(\eta\), as will be explained in the following sections. While this value was derived primarily through numerical considerations, physical reasoning was incorporated [35]. These aspects are elaborated upon in detail in the following sections.