# Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load
## 3 Effects of Modification Coefficients and Load  
### Tooth Profile Modification Method  
We applied three modification methods in this study. The first two methods are commonly used in engineering when forming a modification profile by changing the modification coefficients \(\Delta r_{rp}\) and \(\Delta R_{p}\). The cycloid profile of these two methods can be obtained using Equation (1). Case 1: Modification by fixing coefficient \(\Delta r_{rp}\)  
The modification coefficient \(\Delta r_{rp}\) is a constant in the case 1 method, which focuses on the impact of the modification coefficient \(\Delta R_{p}\) on the RV reducer. Case 2: Modification by changing coefficients \(\Delta r_{rp}\) and \(\Delta R_{p}\) with the same value  
In the case 2 method, the modification coefficients \(\Delta r_{rp}\) and \(\Delta R_{p}\) are changed simultaneously to the same value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Harmonic** & **Amplitudes** & **Harmonic** & **Amplitudes** & **Harmonic** & **Amplitudes** \\
**Components** & **(Arcsec)** & **Components** & **(Arcsec)** & **Components** & **(Arcsec)** \\ \hline
1 & 4.81 & 3n & 2.44 & 6n & 1.42 \\ n & 3.28 & 4n & 1.56 & 8n & 0.68 \\
2n & 2.68 & 5n & 0.64 & 39n & 0.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Main harmonic components of the test TE. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Analytical Method** & **Test Results** & **Difference (\%)** \\ \hline Backlash (arcsec) & 28.61 & 31.03 & 7.79\% \\ TE (arcsec) & 0.49 & 0.47 & 4.08\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Percentage differences of analytical method relative to test results of backlash and TE caused by tooth profile modification (TPM) and load. Case 3: Modification based on traditional contact force formula  
The case 3 method is based on the characteristics of the contact force distribution along the cycloid profile, which uses the contact force curves as the modification curves. The case 3 method is defined as the reverse modification method in this study. The inverse modification curve is obtained using the traditional contact force formula of a cycloid gear [29]. The contact force of a cycloid profile may be calculated as follows:  
\[P_{i}=\frac{2 MS^{-\frac{1}{2}}sin\,\theta_{i}}{KZ_{c}R_{p}}, \tag{33}\]  
where \(M\) is the load torque on the output shaft in the formula, but is a dimensionless coefficient used to obtain different modification curves in this study. In a cycloid gear coordinate system, the modification amount is as follows:  
\[\left\{ \begin{array}{l}x_{si}=P_{i}\cos(\psi_{i}-\varphi)\\ y_{si}=P_{i}\sin(\psi_{i}-\varphi)\end{array} \right. \tag{34}\]  
Then, the reverse modification cycloid profile is obtained as:  
\[\left\{ \begin{array}{l}x_{fi}=x_{i}-x_{si}\\ y_{fi}=y_{i}-y_{si}\end{array} \right., \tag{35}\]  
where, \(x_{i}\) and \(y_{i}\) are the stand cycloid profile coordinates in Equation (1). Then, the initial tooth gap caused by a TPM using the case 3 method may be calculated as follows:  
\[E_{i}(\theta_{i})=\sqrt{x_{si}^{2}+y_{si}^{2}} \tag{36}\]  
The modification parameters of the three methods are listed in Table 6 based on the same maximum clearance of 0.012 to 0.105 mm. In addition, the TPM profiles with the same maximum modification gap are shown in Figure 11. To investigate the load effect, the transmission performances of an RV reducer under three load conditions were calculated respectively, including one to three times rated output torque. The transmission performance of an RV reducer includes three main aspects. The first is the transmission accuracy, which can be measured based on the amplitude and mean value of the \begin{table}
\begin{tabular}{c c c c} \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline \(\Delta r_{rp}\) & \(\Delta R_{p}\) & \(\Delta R_{p}=\Delta r_{rp}\) & \(M\) \\
0.01 mm & 0.02 to 0.1 mm & 0.01 to 0.225 mm & 41 to 161 \\ \hline \end{tabular}
\end{table} 
Table 6: Modification conditions. Figure 11: Modification curves. transmission error.