# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 1 Introduction  
This present paper proposes a non-smooth model for the study of railway vehicles' drive wheels' stick-slip oscillations. The slip of the wheel relative to the contact point leads to wear, and the wheel rotational oscillations generate overloads in the entire drivetrain. Both wear and overloads represent hazards for vehicle safety, increase build and operation costs, and decrease the vehicle lifetime. Frequent slippage is explicitly mentioned [1, 2, 3] as the main cause of driving wheelsets breakage. This phenomenon has been documented in numerous safety warnings and accident reports issued by railway authorities (starting from the steam locomotive era). The operation of powered rail vehicles necessitates sophisticated control strategies to ensure optimal wheel-rail adhesion. Driving wheelsets may encounter torsional vibrations when initiating train movement, leading to compromised adhesion and potentially causing wheelset damage. These vibrations are also considered in studies on rail corrugation [4, 5]. Adhesion saturation and a negative friction coefficient slope are the primary triggers for self-sustained axial vibrations [6]. Due to its importance in the railway domain, stick-slip has been the focus of vehicle dynamics researchers. An important domain of traction control study is the fast detectionof the stick-slip, e.g., [7; 8]. The phenomenon is evaluated and treated by the traction control unit which implements a mathematical model and proposes a solution for solving a particular case, as in [9; 10]. Previous studies of wheel-rail tangential force describe the stick phase of the contact employing a \"physical model\". Physical models aim to remove discontinuity or indefiniteness due to body contact using elasticity and defining constitutional friction laws [11; 12]. While this approach exhibits limit cycles, it is not able to feature the stick phase, and it is numerically inefficient. These shortcomings are solved in the present work through a non-smooth approach to the phenomenon. This paper proposes a new method for analyzing railway vehicle drive wheels that offers significant improvements over the existing approaches documented in scientific publications. The method proves to be an order of magnitude more numerically efficient than the classic approach. A brief literature review is presented in the sequel. Pioneering work by [13] established the relationship between the torsional vibrations of driving wheelsets and the velocity dependence of the friction coefficient. Subsequent theoretical advancements have focused on refining the model parameters proposed by [13]. These refinements encompass the torsional stiffness and inertia of transmission components and specific wheel-rail contact characteristics. Researchers have employed various test setups, including twin-disk rigs [14; 15] and full-scale roller rigs [16], to analyze wheel-rail adhesion under diverse operating conditions, such as varying speeds, axle loads, and contamination levels. This research aims to gain a deeper understanding of several key issues. The focus areas include the analysis of drivetrain system vibration amplitudes and stability through theoretical, numerical, and experimental approaches as evidenced by studies such as [17; 18]. Additionally, investigations have explored the potential role of torsional vibrations in rail corrugation formation, aside from the well-established risk of axle and transmission damage [19]. A significant area of research involves the development of dynamic models that integrate mechanical and electrical aspects. These models aim to understand the influence of torsional vibrations on stick-slip phenomena in driving wheelsets and inform the development of control strategies, as exemplified by the work presented in [20; 21; 22]. Control strategies must overcome the inherent limitation of slow response times compared to the rapid dynamics of wheel-rail contact phenomena.