# 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
Moreover, \(\Delta OPO_{1}\) is defined as the meshing triangle, where \(\phi_{i}\), \(\psi_{i}\), and \(\theta_{i}\) denote the contact angle, normal angle, and azimuth of the _ith_ pin. Because three cranks are supported in the planet carrier, as a difference between the RV reducer and a traditional cycloid drive, the cranks have an extra rotation relative to the pin wheel. The angle between the center line \(\overline{OO_{1}}\) and the \(X\) axis is \(\varphi=\theta_{cr}/i^{H}\), where \(\theta_{cr}\) is the crank rotation angle. The rotation angle of the planet carrier is \(\theta_{pl}=\theta_{cr}/Z_{p}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Transmission ratio & 81 \\ Pin center circle radius \(R_{p}\) (mm) & 114.5 \\ Crank center circle radius \(R_{z}\) (mm) & 70 \\ Pin radius \(r_{rp}\) (mm) & 5 \\ Eccentricity \(a\) (mm) & 2.2 \\ Pin number \(Z_{p}\) & 40 \\ Number of cycloid disc teeth \(Z_{c}\) & 39 \\ Width of cycloid disc \(L\) (mm) & 18 \\ Rated output torque \(T_{o}\) (N-m) & 784 \\ Elasticity modulus of pin \(E_{p}\) (MPa) & \(2.19\times 10^{5}\) \\ Poisson’s ratio of pin \(\upsilon_{p}\) & 0.3 \\ Elasticity modulus of cycloid disc \(E_{c}\) (MPa) & \(2.11\times 10^{5}\) \\ Poisson’s ratio of cycloid disc \(v_{c}\) & 0.292 \\ Modification parameter \(\Delta r_{rp}\) (mm) & 0.05 \\ Modification parameter \(\Delta R_{p}\) (mm) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of RV reducer. Figure 2: Construction of RV reducer. The contact angle can be obtained based on the geometric relationship of \(\Delta OPO_{pi}\), which can be expressed as Equation (6) [21]:  
\[\phi_{i}=tan^{-1}\left[\frac{aZ_{p}\,sin\,\alpha_{i}}{R_{p}-aZ_{p}\,cos\,\alpha _{i}}\right]=tan^{-1}\left[\frac{sin\,\alpha_{i}}{R_{p}/(aZ_{p})-cos\,\alpha_ {i}}\right], \tag{6}\]  
where \(\alpha_{i}=\varphi-\frac{2\pi}{\beta t}\cdot i\), and \(i\) is the pin number, i.e., \(i=1,2,3\ldots Z_{p}\). The _ith_ pin azimuth may be obtained using Equation (7). \[\theta_{i}=mod(\frac{2\pi}{Z_{p}}\cdot i-\varphi,\pi) \tag{7}\]  
Furthermore, the normal angle \(\psi_{i}\) can be obtained from the inner angle relationship of \(\Delta OPO_{pi}\). \[\psi_{i}=\pi-\theta_{i}-\phi_{i} \tag{8}\]  
The static coordinate value of the theoretical meshing point \(c_{i}\) may be obtained through a coordinate transformation as follows:  
\[\left[ \begin{array}{c}x_{pi}\\ y_{pi}\\ 1\end{array} \right]=\left[ \begin{array}{ccc}cos\,\theta_{pl}&-sin\,\theta_{ pl}&a\,cos(\theta_{\sigma}/t^{H})\\ sin\,\theta_{pl}&cos\,\theta_{pl}&a\,sin(\theta_{\sigma}/t^{H})\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}x_{i}\\ y_{i}\\ 1\end{array} \right], \tag{9}\]  
where \(x_{i}\) and \(y_{i}\) are the values in Equation (1), which may be calculated by substituting \(\alpha_{i}\) into \(\theta\). The contact deformation caused by torque load \(T_{o}\) between the cycloid gear 1 and the _ith_ pin is \(\delta_{i}\) in the enlarged view of Figure 3. As a result of a contact deformation, the cycloid gear 1 will rotate a small angle \(\Delta\beta\) around \(O_{1}\) in the direction of the torque load. Figure 3: Geometric model of RV reducer. The contact deformation \(\delta_{i}\) may be expressed as follows:  
\[\delta_{i}=l_{i}\Delta\beta-E_{i} \tag{10}\]  
Here, \(E_{i}\) is the initial tooth gap along the normal line caused by a tooth profile modification (TPM), and is calculated as follows [20]:  
\[E_{i}(\theta_{i})=\Delta r_{rp}\Bigg{(}1-\frac{sin\,\theta_{i}}{\sqrt{1+K^{2}- 2K\cos\,\theta_{i}}}\Bigg{)}-\Delta R_{p}\cdot\frac{1-K\cos\,\theta_{i}-\sqrt{ 1-K^{2}sin\,\theta_{i}}}{\sqrt{1+K^{2}-2K^{2}cos\,\theta_{i}}} \tag{11}\]  
The normal arm length \(l_{i}\) at meshing point \(c_{i}\) may be calculated using Equation (12). \[l_{i}=r_{c}\sin\psi_{i} \tag{12}\]  
If \(\delta_{i}\geq 0\), the pin will mesh with cycloid gear 1; otherwise, the pin will not participate in the transmission. This condition is used to judge the number of pins that are simultaneously engaged.