## 4 Numerical Analysis of the Contact Characteristics
### Numerical Analysis of Stress Distribution in Contact Area  
From the analysis, it is observed that the stress state in the contact area affect the contact slipping and rolling states of the rolling element and the raceway. The friction coefficient \(\mu\) and the ratio \(k\) of the stick region and the slip region are the main factors. The effects of different values of \(\mu\) and \(k\) on the stress components are shown in Figure 5.  
As shown in Figure 5a, the rolling elements roll in the negative direction of the \(x\)-axis, which offsets the stress of the stick region in the rolling direction. The normal stress is compressive in the rolling direction and tensile in the opposite direction and the amplitude of the compressive stress is larger than that of the tensile stress as shown in Figure 5b. The existence of tangential friction during the contact between the rolling element and the raceway causes the velocity difference of the relative movement of the contact area, which results in the creep phenomenon in the micro-contact area. The maximum compressive stress is shifted to the rolling direction, while the maximum tensile stress is at the trailing edge of the contact area. The stress mutation occurs at the junction of the stick region and the slip region of the adopted Carter theory. The non-continuous distribution of stress in the contact area leads to uneven stress on the material. In actual contact, there is a large stress gradient at the junction of the stick region and the slip region. Figure 4: Relationship between contact stress and position angle for different speed. Figure 6 illustrates that the variation of \(\sigma_{z}/p_{0}\) in the contact area. The stress value reaches the maximum when \(x/a=0\), and it is distributed in a hemispherical shape in the contact area. There is no stress mutation in the normal direction. Normal stress has no effect on rolling and slipping in the contact area. Figure 7 shows the variation of \(\tau_{xz}/p_{0}\) for \(k=0.8\) and \(\mu=0.15\). It can be clearly observed that in the slip region, the stress is larger. Stress mutation occurs at the junction of the stick region and the slip region, moreover, the maximum stress value appears in the tangential slipping direction. Figures 5-7 show the maximum contact stress between the rolling element and the variable diameter raceway in the contact area, there is no stress mutation in the normal direction, and the tangential stress mutation is related to the existence of the slip region and the stick region. The tangential stress is related to the friction coefficient \(\mu\) and the stick-slip coefficient \(k\). Therefore, the influence of different friction coefficient \(\mu\) and stick-slip coefficient \(k\) on the dimensionless tangential stress \(\sigma_{x}/p_{0}\), \(\tau_{xz}/p_{0}\) is described below. Figure 5: Distribution of dimensionless stress \(\sigma_{x}/p_{0}\) for \(k=0.8\) and \(\mu=0.15\). Figure 6: Distribution of dimensionless stress \(\sigma_{z}/p_{0}\) for \(k=0.8\) and \(\mu=0.15\). Figure 8 is the variation law of stress components \(\sigma_{x}/p_{0}\) for fixed values of \(\mu=0.15\) and \(y/b=0\) and varying values of \(k\). Points \(A\), \(B\), and \(C\) are the stress mutation points under different \(k\). When \(k=1\), the whole contact area is a stick region, there is no slip phenomenon or stress mutation in the contact ellipse which is an ideal situation without slip. When \(k<1\), the smaller \(k\) is, the smaller the stick region is, and the location of the stress mutation is closer to the rolling direction of the rolling elements. Moreover the stress in the stick region increases with the decrease in \(k\), and the compressive stress in the slip region increases with the \(k\) increases. The faster the tensile stress disappears with the increase in \(k\).