## 4 Verification
### Friction Coefficient  
Due to the approach used, the friction forces are directly proportional to the normal force, thus even small variations in the friction coefficient have a significant influence. The effect is particularly clear in connection with the secondary torque \(\vec{T}_{2 XY}\) in the joint, which is defined according to the authors of [4; 26] as  
\[\vec{T}_{2 XY}=\vec{T}_{L}\cdot\tan\biggl{(}\frac{\beta_{ba}}{2}\biggr{)},\text{ for }\mu=0\quad, \tag{25}\]  
with the load torque \(\vec{T}_{L}\) and the deflection angle \(\beta_{ba}\) (see Figure 6). This results on the one hand from the joint geometry as soon as the joint is loaded with a torque \(M_{t}\) in the deflected state. On the other hand, previous studies [2; 4] have already shown that friction in the tracks affects the magnitude of the resulting secondary torque, which can be explained by the friction forces acting tangentially. With the model presented here, this relationship can be proven. The functional correlation between friction coefficient and secondary torque for 2000 Nm under 2 \({}^{\circ}\) deflection angle is shown in Figure 11. For the friction-free consideration, the secondary torque can be verified from Equation (25), while an approximate quadratic dependence for the consideration of friction between the coefficient of sliding friction \(\mu_{r}\) and secondary torque is obtained for the case of \(T_{L}=2000\) Nm and a deflection angle of \(2^{\circ}\). The quadratic regression function for low friction coefficients (\(\mu_{r}=0.00\)-\(0.05\), red line) is given in the legend of Figure 11.  
Considering the other data points with higher friction coefficients, a clear deviation from this regression function is shown. Moreover, the negative coefficient of the linear part of the regression function for all data points (blue line) cannot be physically motivated. A possible explanation for this behaviour is the superposition of the high contact stiffness with the simple Coulomb's law, which leads to very high friction forces and thus to a significant increase in the secondary moment. At this point, the need for a more detailed friction model becomes obvious. One possible approach would be to calculate the frictional forces using the Pade approximation [27]. In this case, it is possible to consider the real track geometry and still obtain the computationally efficient performance of the analytical contact determination. Figure 10: Forces acting on the ball due to the inclination angles \(\alpha_{incl}\) and \(\beta_{incl}\) and the cage. The effect of friction can also be seen when evaluating the contact forces in the tracks and in the cage window. Figure 12 shows the force curves in the track over the angle of rotation of the inner race for different friction coefficients. The contact force on the track flank in the load direction is shown as a solid line, while the contact force on the opposite flank is shown as a dashed line. If the frictional forces are neglected (see red lines in Figure 12), a force curve is established, which, in addition to a relatively constant component, causes a force drop starting at a rotation angle of approximately \(-70\)\({}^{\circ}\) in the curve of the flank in load direction (solid red line), resulting in a slight increase of the contact force on the counter flank (dashed red line). If the joint is as a consequence operated under bending, a swelling load is generated in the tracks. Taking friction into account (see blue and green lines in Figure 12), this effect is even more pronounced, so that the maximum contact force on both track flanks increases as the coefficient of friction increases. In addition, the maximum contact force in the load flank is shifted in the direction of rotation of the inner race. The increase in the friction coefficient also has a significant influence on the ball kinematics.