## 4 Numerical Analysis of the Contact Characteristics
### Numerical Analysis of Stress Distribution in Contact Area
Comparing the values of points \(A\), \(B\), and \(C\), which are \(-1.3\), \(-1.2\), and \(-1.0\), respectively, it can be seen that the stress mutation is not proportional to the \(k\); the larger \(k\) is, the larger the stress mutation value is; the larger the stress mutation rate is, the easier it is to fail at the contact point. When the friction coefficient \(\mu\) is constant, the faster the rolling element moves, the larger the transmission force is. High speed causes \(k\) to decrease and the tangential force to increase. The contact area of the local slip region becomes larger and can lead to slip. Moreover, under the condition of a periodic load, the wear and fatigue are more likely to occur at the variable diameter raceway, resulting in the failure of variable diameter discrete action. Figure 8: Stress \(\sigma_{x}/p_{0}\) for different values of \(k\). Figure 7: Distribution of dimensionless stress \(\tau_{xz}/p_{0}\) for \(k=0.8\) and \(\mu=0.15\). Figure 9 shows that the variation law of stress components \(\sigma_{x}/p_{0}\) for \(k\) is constant and there are varying values of \(\mu\). The results in Figure 9 show that the friction coefficient \(\mu\) has a slight effect on the stress. The change in friction coefficient \(\mu\) within the 0.1-0.4 range causes the change in dimensionless stress \(\sigma_{x}/p_{0}\) for no more than 18.07%, which in the stick region is not more than 20.1%, and in the slip region is not more than 18.05%. The tangential stress in the slip region is less than the tangential stress in the stick region. When the friction coefficient \(\sigma_{x}/p_{0}\) is larger, the relative difference between compressive stress and tensile stress is larger, which leads to the stress mutation in the contact area more seriously. The stress mutation occurs at the junction of the stick region and the slip region (points \(A\), \(B\) and \(C\)). Equation (6) shows that the composition of the contact stress in the stick region and the slip region is different. The friction resistance in the slip region is opposite to the direction of motion, for this reason the stress in the slip region is less than that in the stick region. The mechanical properties in the contact area conform to the Amontons-Coulomb friction law; hence with the increase in \(\mu\), the tangential friction force increases and the tangential stress component along the moving direction of the rolling element is larger. Figures 10 and 11 show the effects of different friction coefficients \(\mu\) and stick-slip coefficients \(k\) on stress components; when \(\mu=0.15\) and \(y=0\) and varying values of \(k\). The variation law of stress components \(\tau_{xz}=p_{0}\) is shown in Figure 10.  
When \(k=1\) the whole contact area is a stick region, there is no slip phenomenon or stress mutation in the contact ellipse. When \(k<1\), as \(k\) increases, the width of the stick region gradually increases, the contact stress gradually decreases. According to the position of the three points (stress mutation points \(A\), \(B\), and \(C\)), it can be analyzed that as \(k\) increases, the position of the stress mutation moves away from the rolling direction. At this time, the stick contact area increases, which proves that the stress mutation occurs at the stick-slip junction. The reason for this is as \(k\) increases, the creep phenomenon in the contact area decreases; therefore, the slip at the contact point decreases, and the stress mutation value decreases. In the slip region, the rate of change in the tangential stress component is independent of \(k\). This is because in the slip region, the tangential friction is constant because the friction coefficient is constant, which provides the external tangential load of the tangential stress component. Figure 11 shows that when \(k=0.8\), the variation of \(\tau_{xz}=p_{0}\) for different \(\mu\). When the friction coefficient \(\mu\) decreases, the stress of contact area decreases.