## 4 Verification  
### Contact Stiffness  
To guarantee the functionality of the simulation model, it is necessary to verify it first. Therefore, the influence of the numerical contact parameters on the joint kinematics is investigated. These parameters must be chosen in such a way that the model behaviour corresponds as closely as possible to the real joint behaviour. This includes, above all, that the homokinetic condition [12] be fulfilled, which is characterised by the position of the balls in the bisecting plane between the inner and outer race (see Figure 6). However, due to the chosen approach, which allows penetrations between the contact bodies, this cannot be completely fulfilled. For this reason, a compromise must be found for the contact stiffness, under which a sufficient fulfillment is guaranteed. In a previous work [25], it is shown that the use of the contact stiffnesses listed in the literature, which were calculated according to the method of [PERSON] [22] for the Hertzian contact theory, results in a too soft system behaviour for our application case. This means that the deviation from the homokinetic condition becomes too large, so that the physical behaviour of the CVJ is no longer represented sufficiently, which also leads to a high number of iterations during the numerical integration and can ultimately cause the termination of the simulation. Therefore, the equivalent stiffness \(K_{eqv}\) was increased in such a way that the homokinetic condition is sufficiently fulfilled over the entire load range, which is shown exemplarily for the cage deflection angle in Figure 7. The cage is used for positioning the balls and must therefore also be located in the plane bisecting the angle between the inner and outer race. The diagram shows the applied deflection angle for the inner race (\(\beta_{ba,IR}\), black) and the resulting cage deflection angles for different load moments and equivalent contact stiffnesses (\(\beta_{ba,CA}\), coloured) over time. According to theory [12], the cage flexion angle should be set exactly at \(\beta_{ba,CA}=\frac{\beta_{ba,IR}}{2}\). Figure 7 shows that this is sufficiently fulfilled for contact stiffness of \(K_{eqv}=6e12\frac{N}{m^{15}}\), while there is an unacceptable deviation for contact stiffness of \(K_{eqv}=3e11\frac{N}{m^{15}}\). However, an exact fulfillment only applies to ideally rigid contact bodies, so that a deviation with increasing external load and thus elastic deformation is plausible. The divergence from the original Hertzian contact stiffness can be explained by the assumptions of Hertzian contact theory. On the one hand, this was initially derived for friction-free contact zones, which cannot be assumed in the application under consideration. On the other hand, it can be presumed that the contact area between the ball and the track is no longer sufficiently small compared to the ball surface. Finally, an ideal point contact was also assumed for the model in order to achieve the fastest possible computing times. However, based on the real wear behaviour of such joints, it can be expected that there will be flat contact between the ball and the track, which will lead to a further deviation from the assumptions made. Figure 5: Coulomb friction coefficient for different functional relationships \(\mu=f(v_{t})\). In addition to the homokinetic condition, the model is verified with regard to the contact forces occurring in the tracks and cage windows. For this purpose, a comparison with analytical calculation results for the statically undeflected joint condition is made. The force curves determined are shown in Figure 8. In a first approximation, the small inclination angles of the tracks can be neglected so that the analytical contact forces can be calculated using Equation (23) for the track contact forces  
\[F_{0}=\frac{2\cdot T_{L}}{m\cdot PCD\cdot\sin(\alpha_{ca})}\quad, \tag{23}\]  
Figure 6: Homokinetic condition: The balls and the cage are in the bisecting plane between outer race and inner race for each deflection angle \(\beta_{ba}\).