## 4 Calculation of Contact Forces and Normal Deformation of the Cycloidal Gear and Pinwheels
### Calculation of the Meshing Tooth Number and Normal Deformation of the Cycloidal Gear  
The position of the maximum contact force of the pinwheel is at the point with the maximum arm of force \(l_{\text{max}}=r_{c}^{\prime}\). By excluding small nonlinear deviation caused by the equivalent curvature, the contact force acting on the \(i\)-th pinwheel is  
\[F_{i}=\frac{l_{i}}{r_{c}^{\prime}}F_{\text{max}}. \tag{28}\]  
Assume that pinwheels are fixed in the coordinate system in Figure 3; the torque \(T_{c}\) is exerted on the cycloidal gear. Due to normal deformation between the cycloidal gear and pinwheels, the cycloidal gear rotates an angle \(\beta\). The torque \(T_{c}\) is  
\[T_{c}=\sum_{i=m}^{n}F_{i}l_{i}=\frac{F_{\text{max}}}{r_{c}^{\prime}}\sum_{i=m} ^{n}l_{i}^{2}=F_{\text{max}}r_{c}^{\prime}z_{p}\left(\frac{\sum\limits_{i=m}^{ n}l_{i}^{2}}{r_{c}^{\prime 2}z_{p}}\right)=\frac{1}{4}F_{\text{max}}r_{c}^{\prime}z_{p}. \tag{29}\]  
Based on Equation (29), the maximum contact force between the cycloidal gear and pinwheels is  
\[F_{\text{max}}=\frac{4T_{c}}{z_{p}r_{c}^{\prime}}=\frac{4T_{c}}{K_{1}z_{c}r_{p }^{\prime}}. \tag{30}\]  
Figure 3: Meshing diagram of the cycloidal gear and pinwheels. Due to the manufacturing error, the torque \(T\) transmitted by the RV speed reducer is not equally divided on two cycloidal gears. Equation (29) can be simplified as \(T_{c}=0.55T\). Equation (30) can also be written as  
\[F_{\text{max}}=\frac{2.2T}{K_{1}z_{c}r_{p}^{\prime}}. \tag{31}\]  
There is no simultaneous meshing of multiple pairs of cycloidal gears and pinwheels after tooth profile modification if normal deformation is not considered. Tooth profile modification leads to an initial normal clearance \(\Delta(\varphi)_{i}\) in the normal direction between a cycloidal gear and a pinwheel, which can be represented as  
\[\Delta(\varphi)_{i}=\Delta r_{rp}(1-\frac{\sin\varphi_{i}}{\sqrt{1+{K_{1}}^{2 }-2K_{1}\cos\varphi_{i}}})-\frac{\Delta r_{rp}(1-K_{1}\cos\varphi_{i}-\sqrt{1-{ K_{1}}^{2}}\sin\varphi_{i})}{\sqrt{1+{K_{1}}^{2}-2K_{1}\cos\varphi_{i}}}, \tag{32}\]  
where \(\varphi_{i}\) is the angle between the \(i\)-th pinwheel and arm of force. Let \(\Delta(\varphi)_{i}=0\); one has  
\[\cos\varphi_{i}=K_{1}, \tag{33}\]  
\[\varphi_{i}=\varphi_{0}=\arccos K_{1}. \tag{34}\]  
The implication of this result is that the initial normal clearance is zero at this angle. The cycloidal gear and pinwheels begin to mesh at this angle. When the cycloidal gear transfers the torque \(T_{c}\), contact deformation of the cycloidal gear \(w\) and that of the pinwheel \(f\) occur. The maximum contact stress of cycloidal gears and pinwheels is obtained by using Abaqus [24; 31], as shown in Figure 4. This is a quasi-static simulation. The hexahedral mesh is adopted to divide the model. Pinwheels are fixed and only two cycloidal gears can rotate around the central axis. The torque of the cycloidal gear transmission acts at the geometric center of cycloidal gears. Finite element meshes of the cycloidal gear transmission are shown in Figure 5. The simulation result shows that the maximum magnitude of the contact stress of the cycloidal gear is \(1.102\times 10^{4}\) Pa, while that of the pinwheel is \(4.89\times 10^{-26}\) Pa. The contact deformation of the pinwheel \(f\) is produced by its contact stress. Since the maximum magnitude of contact stress of the cycloidal gear is \(10^{30}\) times that of pinwheels, contact deformation of the pinwheel \(f\) is neglected here.