# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 2 Materials and Methods
### Non-Smooth Model for Wheelset Stick-Slip Vibrations  
The simplified model proposed below focuses on the core elements governing the system's dynamics, specifically the non-smooth rolling friction law. This approach aims to accurately represent the stick-slip behavior of the driving wheelsets while excluding extraneous details. The model comprises the wheel (the wheelset) and the actuation system. The motor torque is delivered to the wheels through a drivetrain. The drive force acting upon the wheel results in a wheel/rail friction force, which enables the train to move (Figure 3). The essential elements for the study of the powered wheelsets' stick-slip are the inertia of the drivetrain (including the motor) and wheelset and the interactions between motor/wheelset and wheelset/rails. The motor/wheelset interaction occurs through the transmission and is considered elastic in nature, while the wheelset/rail tangential interaction is through friction. In order to emphasize the stick-slip phases between wheel and rail masses, movements are represented as translations, as shown in Figure 4. This representation has the advantage that displacement and non-zero velocities of the wheel body clearly indicate a slip phase, while for a model with rotating masses, the wheel moves evenly without slipping on the rail. The motor and drive torque are also  
Figure 3: Motor vehicle power train. The wheel/rail friction force is denoted by \(F_{f}\), the driving force by \(F_{el}\), the wheel inertia by m\({}_{1}\), and the motor inertia by \(m_{2}\). The parameter values are defined (and their meaning is again indicated) in the simulation chapter. The system's states are \(x_{1}\), the position of the wheel relative to the rail, and \(x_{2}\), the position of the drivetrain relative to the wheel. As stated, angular displacements are equivalated with translational ones. The equivalent translational model is illustrated in Figure 5. The dynamics of the system are defined by Equation (4), where \(m_{1}\) is the wheelset inertia, and \(m_{2}\) the actuation system inertia:  
\[ \begin{array}{c}m_{1}\tilde{x}_{1}=F_{el}-F_{f}\\ m_{2}\tilde{x}_{2}=F_{m}-F_{el}\end{array}  \tag{4}\]  
As indicated by Equations (1)-(3), the friction force during the adhesion phase is set-valued. This implies it takes on any value within a specific set. The ambiguity arises because, in the stick phase, the friction force is identical to the traction force. The piecewise structure of the friction law, as presented in Equation (3), transforms the system defined by Equation (4) into a switch model [37], as described in Equations (5)-(7). \[ \begin{array}{c}\text{if }\big{|}\dot{x}_{1}\big{|}>\eta\\ \text{(slip phase)}\\ m_{1}\tilde{x}_{1}=F_{el}-F_{f}\\ m_{2}\tilde{x}_{2}=F_{m}-F_{el}\end{array}  \tag{5}\]  
Equation (5) defines the slip phase, i.e., there is relative displacement between the bodies in contact, and the slip friction force \(F_{f}\) (i.e., the saturated friction force) is described by a classical constitutive law. \[ \begin{array}{c}\text{if }\big{|}\dot{x}_{1}\big{|}\leq\eta\text{ and }|F_{el}|\leq F_{sMax}\\ \text{(stick phase)}\\ m_{1}\tilde{x}_{1}=0\\ m_{2}\tilde{x}_{2}=F_{m}-F_{el}\end{array}  \tag{6}\]  
Equation (6) describes the system in the stick phase. The logical conditions for the switch to this mode are relative velocity below the threshold \(\eta\) and external force below  
Figure 4: A simplified system for the study of wheelset stick-slip vibrations. Figure 5: An equivalent model with masses in translation. saturated friction force \(F_{sMax}\). A striking feature of this mode is the collapse of the degrees of freedom from two to one, as the acceleration of the wheel is set to null. \[ \begin{array}{l}\text{if}\big{|}\tilde{x}_{1}\big{|}\leq\eta\text{ and}|F_{el}|>F_{sMax}\\ (\text{stick to slip transition})\\ m_{1}\tilde{x}_{1}=F_{el}-F_{sMax}\\ m_{2}\tilde{x}_{2}=F_{m}-F_{el}\end{array}