## 4 Calculation of Contact Forces and Normal Deformation of the Cycloidal Gear and Pinwheels
### Calculation of the Number of Meshing Gears and the Contact Force  
Since initial clearances \(\Delta(\varphi)_{i}\) are different after modification of the cycloidal gear, the relationship between \(F_{i}\) and \(\delta_{i}=l_{i}\beta\) is nonlinear. However, since changes in \(F_{i}\) and \(\delta_{i}-\Delta(\varphi)_{i}\) are small, the relationship between \(F_{i}\) and \(\delta_{i}-\Delta(\varphi)_{i}\) is assumed to be linear. Based on the above analysis, pinwheels in the simultaneous meshing state are classified as pinwheels from the \(m\)-th to the \(n\)-th pinwheel. The contact force of the \(i\)-th pinwheel is  
\[F_{i}=\frac{\delta_{i}-\Delta(\varphi)_{i}}{\delta_{\max}}F_{\max}, \tag{41}\]  
where \(F_{\max}\) is the maximum contact force at the position where the pinwheel is at or close to the position \(\varphi_{i}=\varphi_{0}=\arccos K_{1}\). Based on the moment equilibrium equation, \(T_{c}\) can be represented as  
\[T_{c}=\sum_{i=m}^{n}F_{i}l_{i}, \tag{42}\]  
where \(\delta_{i}=l_{i}\beta\) and \(\delta_{\max}=r_{c}^{\prime}\beta\). Submitting Equation (41) into Equation (42) yields  
\[T_{c}=F_{\max}\sum_{i=m}^{n}\big{(}\frac{l_{i}}{r_{c}^{\prime}}-\frac{\Delta( \varphi)_{i}}{\delta_{\max}}\big{)}l_{i}. \tag{43}\]  
Therefore, the maximum contact force \(F_{\max}\) on the pinwheel with tooth profile modification can be derived as  
\[F_{\max}=\frac{T_{c}}{\sum\limits_{i=m}^{n}\big{(}\frac{l_{i}}{r_{c}^{\prime}} -\frac{\Delta(\varphi)_{i}}{\delta_{\max}}\big{)}l_{i}}=\frac{0.55T}{\sum \limits_{i=m}^{n}\big{(}\frac{l_{i}}{r_{c}^{\prime}}-\frac{\Delta(\varphi)_{i }}{\delta_{\max}}\big{)}l_{i}}. \tag{44}\]  
One can calculate \(F_{\max}\) by giving an initial value of \(F_{\max}\) that is \(F_{\max_{0}}\), which can be represented as  
\[F_{\max 0}=\frac{1}{2}(F_{1}-F_{2}). \tag{45}\]  
According to Equation (31), \(F_{1}\) can be represented as  
\[F_{1}=\frac{2.2T}{z_{c}r_{p}^{\prime}}. \tag{46}\]  
By assuming that there is only one pair of cycloidal gear and pinwheel meshing at the position \(\varphi_{i}=\varphi_{0}=\arccos K_{1}\) with tooth profile modification, \(F_{2}\) can be represented as  
\[F_{2}=\frac{0.55T}{r_{p}^{\prime}}. \tag{47}\]  
By substituting \(F_{\max_{0}}\) into Equations (37)-(44), the initial value of \(\delta_{\max}\) can be obtained as \(\delta_{\max_{0}}\). Substituting \(F_{\max_{0}}\) and \(\delta_{\max_{0}}\) into Equations (37)-(44), one can calculate the number of meshed pinwheels that are from the \(m\)-th to the \(n\)-th pinwheel. The maximum contact force calculated in each iteration is denoted as \(F_{\max_{k}}\). The value of \(F_{\max_{1}}\) at the first iteration is obtained by substituting \(\delta_{\max_{0}}\) into Equation (44). By comparing \(F_{\max_{0}}\) with \(F_{\max_{1}}\), if the absolute value of the difference between \(F_{\max_{0}}\) and \(F_{\max_{1}}\) is greater than 0.1% of \(F_{\max_{1}}\), it is necessary to calculate \(\delta_{\max_{1}}\) by substituting \(F_{\max_{1}}\) into Equations (37)-(44). The second iteration result of \(F_{\max_{2}}\) can be obtained by substituting \(\delta_{\max_{1}}\) into Equation (44). The iteration is implemented many times until the result of the \(k\)-th iteration \(F_{\max_{k}}\) satisfies the above condition. Finally, the exact \(F_{\max}\) can be obtained by  
\[F_{\max}=\frac{1}{2}(F_{\max_{k}}+F_{\max_{k-1}}). \tag{48}\]A flowchart of the tooth contact analysis method for cycloidal gear transmissions is shown in Figure 6.