# Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load
## 2 Analytical Model of RV Reducer with Tooth Profile Modifications  
### Contact Force Calculation  
#### 2.1.1 Contact Force Model  
A traditional cycloid drive is the basis used to analyze an RV reducer, and Figure 1 shows a traditional cycloid drive coordinate system. Here, \(xoy\) and \(x_{c}o_{c}y_{c}\) represent the static coordinates of a pin wheel and cycloid gear, \(O\) and \(O_{c}\) denote the center of the pin wheel and cycloid gear, and \(r_{p}\) and \(r_{c}\) denote the pitch circle radius of a pin wheel and cycloid gear, respectively. Pitch point \(P\) is the instant velocity center of a cycloid drive and \(OO_{c}\) is the position of the crank. When the crank rotates\(\theta_{cr}\), the cycloid gear will rotate \(\theta_{cy}\) according to the transmission ratio. The rotation of a cycloid gear is output using a parallelogram mechanism. The track of the meshing point of a cycloid-pin in a static coordinate system is part of the circular arc of the pin. According to the meshing principle of a cycloid drive, the tooth profile of a cycloid gear can be obtained as follows [20]:  
\[\left\{ \begin{array}{l}x=\left(R_{p}+\Delta R_{p}\right)\!\cos\!\left[\left(1 -i^{H}\right)\theta\right]-a\cos\!\left(i^{H}\theta\right)+\left(r_{rp}+\Delta r _{rp}\right)\!S^{-\frac{1}{2}}\!\left\{Kcos\!\left(i^{H}\theta\right)-cos\! \left[\left(1-i^{H}\right)\theta\right]\right\}\\ y=\left(R_{p}+\Delta R_{p}\right)\!sin\!\left[\left(1-i^{H}\right)\theta \right]+a\sin\!\left(i^{H}\theta\right)-\left(r_{rp}+\Delta r_{rp}\right)\!S^ {-\frac{1}{2}}\!\left\{Ksin\!\left(i^{H}\theta\right)+sin\!\left[\left(1-i^{H }\right)\theta\right]\right\}\end{array} \right., \tag{1}\]  
where  
\[i^{H}=\frac{Z_{p}}{Z_{c}}, \tag{2}\]  
\[K=\frac{aZ_{p}}{R_{p}+\Delta R_{p}}, \tag{3}\]  
\[S=1+K^{2}-2 Kcos\,\theta. \tag{4}\]  
Here, \(\theta\) represents the angle of the pin position vector; \(\Delta R_{p}\) and \(\Delta r_{rp}\) refer to the modification coefficient of the moved distance and equidistance, respectively; \(Z_{c}\) and \(Z_{p}\) denote the cycloid disc teeth number and pin number; \(r_{rp}\) and \(R_{p}\) represent the pin radius and pin center circle radius, respectively; and \(a\) represents the eccentricity. The pitch circle radius of a pin wheel and cycloid gear can be calculated using the following equations:  
\[\left\{ \begin{array}{l}r_{p}=aZ_{p}\\ r_{c}=aZ_{c}\end{array} \right. \tag{5}\]  
Figure 2 shows the mechanism of an RV reducer. Because the machining precision of the crank, cycloid gear, and planetary carrier is extremely high, these are generally known as the three key parts of an RV reducer. Two cycloid gears are supported in the planet carrier by three cranks. The purpose of these two cycloid gears is to balance the radial force and improve the load capacity. An RV reducer contains three freedoms of rotation, namely, the rotations of the input gear shaft, pin wheel, and planet carrier. Any fixed parts can form a mechanism for increasing or decreasing the speed. Because an  
Figure 1: Traditional cycloid drive coordinates. involute planet drive is applied to the high-speed stage, its TE is significantly attenuated by the transmission chain [5; 6]. Therefore, to focus on the main factors affecting the transmission performance of an RV reducer, an involute gear transmission is not considered in the present study. The torque is applied to the three cranks as an input of the involute planetary drive. The RV reducer parameters used in this paper are listed in Table 1.  
A geometric model of an RV reducer is shown in Figure 3. Here, \(O\) and \(O_{1}\) represent the center of a pin wheel and cycloid gear 1, respectively. In addition, pitch point \(P\) represents the instant velocity center of a cycloid gear 1, \(O_{pi}\) denotes the center of the _ith_ pin, \(c_{i}\) refers to the meshing point of the _ith_ pin with cycloid gear 1, and \(l_{i}\) represents the normal arm length.