## 2 Quadratic Contact Surface Analysis and Force Modeling
\[f(x,y)=z=f(0,0)+x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y }+\frac{1}{2}(x^{2}\frac{\partial^{2}z}{\partial x^{2}}+2 xy\frac{\partial^{2 }z}{\partial x\partial y}+y^{2}\frac{\partial^{2}z}{\partial y^{2}}) \tag{7}\]  
\[z=\frac{1}{2}(-\frac{a}{w}x^{2}-\frac{2 hxy}{w}-\frac{b}{w}y^{2}) \tag{8}\]  
Eccentricity is a function depicting the angular and spatial relationship between the axes of two cylindrical bodies. A higher eccentricity results in a broader contact surface between the two cylinders. The contact surface on the XY plane can be characterized using a quadratic surface, as described by Equation (8). Within this equation, parameters '\(a\)' and '\(b\)' govern the curvature of the contact surface in the X and Y directions, respectively, reflecting the width of the contact area, while '\(h\)' determines the rotation angle of the contact surface. Figure 1: Two cylinders meshing. When the axes of the two cylinders are not perfectly parallel, an eccentricity distinct from zero emerges, implying that neither '_a_' nor '_b_' are zero. Consequently, the shape of the contact surface manifests as an ellipsoidal paraboloid. This suggests that the pressure distribution on the contact surface might be non-uniform, exhibiting peak values at the center and diminishing towards the edges. Upon an in-depth examination of the contact surface geometry in conjunction with the actual operational conditions, it becomes evident that the clearance during the meshing process between the pinion teeth and the cycloid gear leads to the misalignment of the axes of the two contacting bodies. This misalignment culminates in an ellipsoidal paraboloid-shaped contact region. Moreover, the oscillation or 'runout' of the pinion teeth undeniably has repercussions on the stress distribution across the face of the cycloid gear, as shown in Figure 2.  
In scenarios considering pinion oscillation, this analysis operates on the presumption of contact between infinitesimally small convex bodies. The pressure distribution on the contact surface, under these circumstances, still conforms to the Hertzian stress distribution model. Notably, we disregarded any contact effects beyond the direct meshing of the pinion with the cycloid gear. This means that during the 'runout' phase of the pinion, there are no other collisions to account for. To sum up, when the pinion engages with the cycloid gear, the situation can be approximated as the contact problem of two cylindrical bodies with misaligned axes. The deformation at the contact site manifests as an ellipsoidal paraboloid. Hence, when computing the contact forces between the cycloid and the pinion, the impact of a conical contact surface on the force dynamics is incorporated into the analysis. Assuming that the angle between the axes of two cylinders is \(\theta\), where \(\theta\in\left(0^{\circ},90^{\circ}\right]\), it should be noted that when \(\theta\) is \(0^{\circ}\), the relationship between the contact force and deformation between the cylinders is linear, and a different contact force model is used; hence, this value is not included in the range. In addition, the radii of the two cylinders are \(R_{1}\) and \(R_{2}\), respectively. It is possible to calculate the heights of the two surfaces near the contact point, as shown in Equation (9). \[h=z_{1}+z_{2}=\frac{x^{2}}{2R_{1}}+\frac{\left(x\cos\theta-y\sin\theta\right) ^{2}}{2R_{2}} \tag{9}\]  
where \(R_{1}\) and \(R_{2}\) denote the radii of the two cylinders, respectively. By forming a quadratic form through the surface height \(h\), the principal curvatures can be obtained, and consequently, the results for the equivalent radius are derived. This allows for the substitution of the equivalent rigid sphere and elastic half-space body contact force model (which is identical to the computational model used when the angle between the axes is \(90^{\circ}\)), as shown in Equation (10).