## 3 Analysis of Contact Stress Distribution between Rolling Element and Variable Diameter Raceway
\(a\) and \(b\) are calculated by Equations (3)-(5):  
\[a=a^{*}\left(\frac{3Q_{i}}{2E^{*}\overline{\sum}\rho}\right)^{1/3},\ b=b^{*} \left(\frac{3Q_{i}}{2E^{*}\overline{\sum}\rho}\right)^{1/3}\]  
The contact load between the rolling elements and the variable diameter raceway is an alternating load. Friction and sliding occur with the change in the radius of curvature, which makes the contact stress of variable diameter raceway alternating stress. There will be a spatially varying shear stress field in the contact area due to the alternating stress. From a macroscopic point of view, the judgment of the rolling state between the rolling element and the variable diameter raceway is directly affected by the shear stress field. From a microscopic point of view, the impact failure in the contact area is due to abrupt changes within the spatial stress field. To ensure that slipping and stress concentration do not occur when the rolling elements pass through the variable diameter raceway, it is necessary to analyze the contact state and stress distribution in the contact area. When the rolling elements enter the variable diameter raceway, the normal force in the contact area is only provided by the centrifugal force and the gravity of the rolling element, the tangential stress at the boundary of the contact area tends to infinity. Hence, the contact condition cannot be satisfied and results in relative slipping. In the transition region, the normal load is provided by the radial load and the centrifugal force of the rolling elements of the bearing. As the position angle of the rolling element gradually increases, the tangential friction force gradually decreases. Therefore, under the combined action of the tangential stress and the normal stress, the stick-slip phenomenon occurs in the contact area when the rolling elements pass through the variable diameter raceway. When stick-slip occurs between the rolling elements and the variable diameter raceway, one side of the contact area along the rolling direction is compressed and the other side is stretched. The contact area moves along the direction of the tangential stress, the offset distances of all the points are equal. The shape of the contact area is shown in Figure 2. The semi-axes \(c\) and \(d\) of the stick region (ellipse 2) coincide with the \(x^{\prime}\) and \(y^{\prime}\)-axis. The stick region is offset along the rolling body direction under the action of tangential stress, the shape and boundary of the contact area remains unchanged in ellipse three during the displacement process. The front part of the contact area is the stick region, the rear part is the slip region, and the slipping offset distance is s.  
According to [PERSON]'s rolling contact theory, the distribution of the tangential stress in the three-dimensional rolling contact area is accomplished by the superposition principle, it is calculated as follows:  
\[p=\left\{ \begin{array}{cl}p^{\prime}(x,y)&\text{slip region}\\ p^{\prime}(x,y)+q^{\prime}(x,y)&\text{stick region}\end{array} \right. \tag{6}\]  
where \(p^{\prime}(x,y)\) and \(p^{\prime}(x,y)\pm q^{\prime}(x,y)\) denote the distributions of the tangential stress in the slip and stick region, respectively,\(p^{\prime}(x,y)=-\frac{3\mu Q}{2\pi ab}\Big{(}1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2 }}\Big{)}^{1/2}\quad(2c-a<x_{1}<a)\), \(q^{\prime}(x,y)=-\frac{3\mu Q}{2\pi a^{2}b}\Big{(}1-\big{(}\frac{x+s}{c}\big{)} ^{2}-\big{(}\frac{y}{d}\big{)}^{2}\Big{)}^{1/2}\quad(-a\leq x_{1}\leq 2c-a))\quad\). A negative sign indicates that the tangential stress is in the opposite direction of the motion. Since the width of the ellipse contact area changes with the \(y\)-axis, \(a\) is much smaller than \(b\). Contact stress \(p_{1}\) is very small near \(x_{1}=a\). The elliptical contact area was divided into numerous narrow bands, and the corresponding contact width is \(2x\), the normal stress in the contact area can be calculated as: \(p_{1}=\frac{3Q}{2\pi ab}\sqrt{1-\big{(}\frac{x_{1}}{a}\big{)}^{2}-\big{(}\frac {y_{1}}{b}\big{)}^{2}}\).