## 3 Analysis of Contact Stress Distribution between Rolling Element and Variable Diameter Raceway  
Based on the rolling contact theory, a three-dimensional tangential stress distribution model is obtained under the combined effect of the variable diameter raceway structure and load conditions at the rolling contact area. In rolling contact between the rolling elements and variable diameter raceway, the contact area is under compression. A slight deformation occurs in the contact area (\(x^{\prime}\)-\(y^{\prime}\) plane), where \(o^{\prime}\) is the initial contact point, the elliptical contact area is defined, as shown in Figure 2.  
Figure 1: Structural model of cageless bearing with variable diameter raceway. Figure 2: State of contact area. The \(x^{\prime}\)-axis is the direction of motion, and the contact normal load is the direction of the \(z^{\prime}\)-axis. \(a\) and \(b\) denote the minor and major semi-axes of the ellipse, respectively. Semi-axis \(a\) is along the \(x^{\prime}\)-axis, and semi-axis \(b\) is along the \(y^{\prime}\)-axis. According to the Hertz contact theory, the equation of the contact deformation ellipse boundary between the rolling element and the variable diameter raceway can be expressed as:  
\[\frac{{x^{\prime}}^{2}}{a^{2}}+\frac{{y^{\prime}}^{2}}{b^{2}}=1 \tag{1}\]  
The distribution of the normal stress \(p\) (\(x^{\prime},y^{\prime}\)) in the contact area can be expressed as:  
\[p\big{(}x^{\prime},y^{\prime}\big{)}=p_{0}\sqrt{1-\frac{{x^{\prime}}^{2}}{a^{2 }}-\frac{{y^{\prime}}^{2}}{b^{2}}} \tag{2}\]  
where \(p_{0}\) is maximum surface stress at the center point \(o^{\prime}\) of contact area, and is the maximum contact stress in the ellipse contact area, \(p_{0}\) can be obtained as:  
\[p_{0}=\frac{3Q_{i}}{2\pi ab}=\left[\frac{6 QE^{z^{2}}}{\pi^{3}{R^{*}}^{2}} \right]^{\frac{1}{3}} \tag{3}\]  
where the normal variable load of the rolling element enters and exits at both ends of the variable diameter raceway is: \(Q_{i}=(d_{i}+d_{o})m\omega_{cj}^{2}\cot\gamma/2-mR_{1}\dot{\alpha}\dot{\alpha }_{cj}\).The variable diameter raceway is: \(Q_{i}=\frac{(R_{1}+d_{i})2m\omega_{cj}^{2}+C\sin(\mathrm{mod}(\varphi_{i},2\pi) )}{2\cos a}\). \(d_{o}\) is outer raceway diameter, \({d_{o}}^{\prime}\) is diameter of the contact trajectory within variable diameter raceway. \(E^{z}\) is the equivalent elastic modulus, which is calculated by Equation (4):  
\[\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}} \tag{4}\]  
where \(v_{1}\), \(v_{2}\), \(E_{1}\) and \(E_{2}\) are the Poisson's ratio, elastic modulus of the materials of the rolling elements and the variable diameter raceway. \(R^{*}\) is the equivalent radius of curvature at the contact point in Equation (3). According to the geometrical characteristics of the variable diameter raceway, the equivalent radius can be obtained from Equation (5):  
\[R^{*}=(R_{1}R_{1})^{\frac{1}{2}}=\left(\frac{R_{11}R_{i}R_{11}r_{i}}{(R_{11}+R _{i})(R_{11}+r_{i})}\right)^{\frac{1}{2}} \tag{5}\]  
where \(R_{1}\) and \(R_{11}\) are the relative radii of the curvatures at the contact point. \(R_{11}\) and \(R_{11I}\) are the main radii of the curvatures of the rolling element and the variable diameter raceway at the initial contact point, respectively (as shown in Figure 1). \(R_{1}\) is the radius of the rolling element. When the contact area is in the transition region of both ends of the variable diameter raceway, \(R_{i}=r_{o}\); \(r_{i}=r\). When the contact area is in the non-transition region of both ends of the variable diameter raceway, \(R_{i}=r\); \(r_{i}=\frac{d_{o}}{2}\cos\frac{\theta_{i}}{2}\). \(r\) is the transition radius of the variable diameter raceway. \({d_{o}}^{\prime}\) is the diameter of the circle formed by the contact point,\({d_{o}}^{\prime}=d_{o}-2\left(R_{1}-\left(R_{1}^{2}-\left(r_{o}\sin\left(\frac{ \theta_{2}}{2}\right)\right)^{2}\right)^{0.5}\right)\).