# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 3 Numerical Simulation Results
### Stick-Slip Simulation Using the Switch Model  
The following simulations demonstrate the potential and advantages of using a non-smooth approach to analyze wheelset stick-slip torsional vibrations, which can occur during locomotive running or can be used to perform traction control tasks. The model incorporates only the most critical forces and their essential characteristics to achieve this. The specific parameters used in the model are provided in Table 1. The friction-related values have been approximated compiling data available in the paper references. The magnitude of the friction-related data is consistent with rail vehicles specifications. Inertia and rigidity are approximated from scale models available in the National University of Science and Technology POLITEHNICA Bucharest. The saturation creep limit, \(\eta\), is also chosen according to theoretical and experimental research, e.g., [15; 25]. The saturated friction law is defined using a negative friction coefficient slope in order, Equation (6), and the saturated stage in Figure 6 (abs(\(v\)) \(>\) \(\eta\)) to provide the conditions for the stick-slip occurrence. The function is commonly used for modeling of the saturated friction in locomotives, e.g., [47]. \[F_{f}\!\in\!\left\{ \begin{array}{cc}F_{sMax}((1-B)\mathrm{e}^{-A|\hat{x}_{1}|}+ \mathrm{B})\mathrm{sign}(\dot{x}_{1}),&\mathrm{if}\ |v|>\eta\\ F_{d},&\mathrm{if}\ |v|\leq\eta\ \mathrm{and}\ |\Sigma|\leq F_{sMax}\\ F_{sMax}\ \mathrm{sign}(F_{d}),&\mathrm{if}\ |v|\leq\eta\ \mathrm{and}\ |\Sigma|>F_{sMax}\end{array} \right. \tag{8}\]  
When using a constant force as input from the motor, only unstable oscillations were obtained. As proposed in Table 1, a negative slope characteristic of the motor force/motor speed is used to simulate stable stick-slip limit cycles. The proposed motor force function with a negative slope represents constant power output. It is based on a running speed of 10 m/s, and \(\dot{x}_{2}\) are the motor oscillations relative to the running speed. The plot of the numerical solution presented in Figures 6-8 features the characteristics described in the previous chapters:  
1. Alternative stick and slip phases;
2. Discontinuities at the acceleration level;
3. The system is overloaded due to drivetrain force variation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Symbol** & **Value** & **Unit** \\ \hline Saturated friction maximum & \(F_{sMax}\) & 10 & kN \\ Wheel inertia & \(m_{1}\) & 200 & kg \\ Drivertain and motor inertia & \(m_{2}\) & 2000 & kg \\ Drivertain rigidity & \(k\) & \(10^{6}\) & N/m \\ Drive force & \(F_{d}\) & \(-k(x_{1}-x_{2})\) & N \\ Saturation creep limit & \(\eta\) & \(5\times 10^{-3}\) & m/s \\ Friction parameters & \(A\); \(B\) & 5; 0.9 & s/m \\ Kinetic friction force & \(F_{f}\) & \(F_{sMax}\Big{(}(1-B)\mathrm{e}^{-A|\hat{x}_{1}|}+B\Big{)}\mathrm{sign}(\dot{x }_{1})\) & N \\ Force produced by the motor & \(F_{m}\) & \(\frac{9.5}{10+x_{2}}F_{sMax}\) & N \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters values (suitable for a locomotive). Figure 6: Wheel/rail relative velocity. Values are (almost) zero during stick-slip phases. The distinctive feature of the wheel-rail relative movement plot is the (almost) zero-velocity interval specific to the stick phases (see Figure 6). The relative velocity is not exactly zero due to the fact that the wheel acceleration becomes null when the absolute relative velocity is smaller than the saturation creep limit \(\eta\) according to the principles related to zero-crossings detection. It might be noteworthy that the algorithm is refined to set the relative speed to zero in the stick phase, as mentioned in Section 3, but this is not relevant within the scope of this research. The motor force of the system \(F_{m}\) has rather small oscillations at about 9 kN. Nevertheless, in spite of this almost constant value of the motor force, the reactions induced in the system feature spectacular variations.