# Efficient Wheel-Rail Stick-Slip Numerical Modeling for Railway Traction Vehicles
## 3 Numerical Simulation Results
### Main Differences in the Simulation with the Classical Model
The steep variation of the friction force in this phase determines the stiffness of the classical models. It should also be noted that if the stage of the onset of partial slip within the contact area is considered, the linear variation takes place in an interval even smaller than \(\eta\), and the model becomes even more stiff. As opposed to the constitutive friction law, the switch model uses Equation (6) in the stick phase. The value of the friction force equals the external force, i.e., \(F_{e}\); hence, there is no dramatic change in the value of the system derivative depending on the relative velocity. In fact, in the stick phase, due to the nature of the set-valued friction values, there is no variation of the relative velocity. As seen in Figure 10, in the stick phase, the numerical integration of the classic model requires 5 times more steps because the equations are stiff. The discontinuous model required 13 steps, and the classical model 64, in the conditions described in Table 1.  
Another notable aspect is that the longer the stick phase is compared to the slip phase, the bigger is the ratio of the integration steps for the two approaches. This is a consequence of the fact that the derivative is very small for the switch model in the stick phase, and integration steps are increased. The ratio of 5 was obtained because at transitions from stick to slip and vice versa the steps become again smaller. If the steps ratio is examined in the middle of the stick phase, in the interval 0.028-0.034, the value raises to 25. In fact, in traction control problems, the system should not slip at all, and the stick has to be maintained at the adhesion limit to provide powerful traction force and acceleration. Hence, the switch model would be by far the best option. It should be further noted that if the constitutive friction law definition considers the onset of partial slip within the contact area stage (II) described in Figure 1, the equations of the classic approach would become even stiffer because the linear initial portion (I), corresponding to a condition of full adhesion, varies in an interval smaller than \(\eta\). More relevant details given in Figure 10 are the values of the relative velocities for both models in the stick phase. While the velocity of the wheel in the non-smooth approach remains constant (at a value very close to \(\eta\), the wheel velocity is unnecessarily computed with considerable effort in the classic approach). Numerical experiments were further performed with the aim of a better assessment of the results' validity. The motor force was varied and its influence upon stick-slip oscillations and numerical efficiency was studied. Both the switch model and the classical model were analyzed in the same conditions. The resulting values and behavior were again almost identical. The described differences of the numerical efficiency were preserved for all the considered values of the motor force. Stable stick-slip oscillations occur if the motor force \(F_{m}\) numerator (Table 1) is in the interval \([9.4F_{SMax},\,10.1F_{SMax}]\). Regardless of the motor force value, the described structure and number of integration steps show almost no difference from the situation described in Figure 10. If the value of the motor force drops below \(9.4F_{SMax}\), the stick phase is stable. Conversely, if the motor force value is above \(10.1F_{SMax}\), the slip phase is stable. These results indicate that the observed numerical improvement achieved by the switch model is valid and consistent in all situations when stick-slip oscillations occur.