# Modification and Optimization of Cycloidal Gear Tooth Profile Based on Machining Error Compensation
## 4 Modifying the Parameter Optimization
### Optimization Model
According to the RV transmission characteristics, the deformation of the two parts is produced simultaneously under the same meshing force, the contact between the cycloidal gear and the pin wheel satisfies the Hertzian contact, and the maximum deformation \(\delta_{\text{max}}\) can be obtained from the Hertzian contact formula [37] as:  
\[\delta_{\text{max}}=\frac{2F_{\text{max}}}{\pi b}\bigg{[}\frac{1-\mu_{1}{}^{2} }{E_{1}}\bigg{(}\frac{1}{3}+\ln\frac{4\rho}{c}\bigg{)}+\frac{1-\mu_{2}{}^{2}} {E_{2}}\bigg{(}\frac{1}{3}+\ln\frac{4r_{rp}}{c}\bigg{)}\bigg{]} \tag{21}\]  
where \(E\) is the modulus of elasticity, \(\mu\) is Poisson's ratio, and \(\rho\) is the radius of curvature. In the loaded state, the cycloidal gear and the pin teeth produce multi-teeth meshing due to elastic deformation. In this state, there will be multiple meshing points. The deformation amount is different for pin teeth at different positions of the meshing point, and the deformation amount of the meshing point can be obtained according to Formula (22). When the load deformation between the cycloidal gear and the pin teeth is greater than their initial meshing clearance, the pairing of the cycloidal gear and the pin teeth will enter meshing; otherwise, they will not participate in meshing. According to this principle, the evaluation of the number of meshing teeth of the cycloidal gear and the pin teeth can be performed simultaneously. \[\delta_{i}=l_{i}\beta=l_{i}\frac{\delta_{\text{max}}}{l_{\text{max}}}\big{(}i= 1,2,\cdots,z_{p}/2\big{)} \tag{22}\]  
where \(l\) is the distance from the common normal of the meshing point to the center of the cycloidal gear; under the action of load, the cycloidal gear and the pin wheel will produce contact deformation, and the cycloidal gear will turn through the \(\beta\) angle; assuming that the maximum deformation of the pair of cycloidal gears and the pin teeth under the largestload is \(\delta_{\max}\), the distance between the common normal of the meshing point and the center of the cycloidal gear is \(l_{\max}\), and obviously, \(\beta=\delta_{\max}/l_{\max}\). For the cycloidal gears and pin teeth, (m-n) refers to all the teeth between the m-th tooth and the n-th tooth that are in a simultaneous meshing state; because the number of meshing teeth of the modified cycloidal gear cannot reach half the number of teeth at the same time, and a certain initial gap between the modified cycloidal gear and the pin teeth in the normal direction of the meshing point \(\Delta(\varphi)_{i}\) exists, the force \(F_{i}\) of the i-th tooth is proportional to the difference between the deformation and the initial gap [38]; that is:  
\[F_{i}=\frac{\delta_{i}-\Delta(\varphi)_{i}}{\delta_{\max}}F_{\max} \tag{23}\]  
The torque \(T_{c}\) on the cycloidal gear is transmitted by all the cycloidal gear teeth simultaneously participating in the meshing. According to the torque balance condition, we have:  
\[T_{c}=\sum\limits_{i=m}^{i=n}F_{i}l_{i}\Rightarrow F_{\max}=\frac{T_{c}}{ \sum\limits_{i=m}^{i=n}\Big{(}\frac{l_{i}}{r_{c}^{2}}-\frac{\Delta(\varphi)_{ i}}{\delta_{\max}}\Big{)}l_{i}} \tag{24}\]  
The above solution process shows that \(\delta_{\max}\) is required to solve \(F_{\max}\), and \(F_{\max}\) should be determined to solve \(\delta_{\max}\). Thus, the iterative calculation method is utilized to solve the problem. An initial value \(F_{\max0}\) of \(F_{\max}\) is given, and the initial value \(\delta_{\max0}\) of \(\delta_{\max}\) can be solved. Finally, the exact value of \(F_{\max}\) can be solved by Formula \(F_{\max}=\Big{(}F_{\max}+F_{\max(k-1)}\Big{)}/2\). In the RV transmission, the smaller the maximum normal contact force \(F_{\max}\), the greater the carrying capacity of the RV reducer, while considering the other parameters unchanged.