# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 5 Results and Discussion
### Test Results of 65 BX-121-RVE Reducers  
(1) Transmission error test results  
The TE test results for the four 65 BX-121-RVE prototypes are shown in Figure 10. The TE for prototype 5 is 42\({}^{\prime\prime}\), for prototype 6 is 50\({}^{\prime\prime}\), for prototype 7 is 73\({}^{\prime\prime}\), and for prototype 8 is 83\({}^{\prime\prime}\). The time-domain distribution of the TE for the 65 BX-121-RVE reducer, as shown in Figure 10, also displays periodicity and complex beat frequency characteristics. (2) Transmission error spectrum analysis results  
The TE spectrum analysis results for the four 65 BX-121-RVE prototypes are shown in Figure 11. It can be seen that the spectral energy of the TE for all four tested prototypes is also concentrated in the low-frequency range (not exceeding 1 Hz), with negligible amplitude responses in frequencies above 1 Hz. Apart from the trend term (0 Hz), the high-amplitude response frequencies for the tested prototypes are also approximately evenly spaced at intervals of 0.1612 Hz. Figure 10: Test results of transmission error for the 65 BX-121-RVE reducer prototypes: (**a**) prototype 5; (**b**) prototype 6; (**c**) prototype 7; and (**d**) prototype 8. (3) Characteristic frequency comparison and verification results  
In the spectrum results, the characteristic frequencies were identified, and the identification results are presented in Table 8. A visual representation of Table 8 is shown in Figure 12. It is evident that the analysis results for the 65 BX-121-RVE model also clearly respond to all characteristic frequencies. Similar to the results for the 190 BX-121-RVE model, the four tested 65 BX-121-RVE prototypes indicate that the TE response amplitude at the sun gear characteristic frequencies is also relatively small, suggesting that the sun gear has a minimal impact on the overall TE of the reducer. In prototypes 5-8, the response amplitude at the characteristic frequency of 0.1653 Hz is also typically the highest, further indicating the significant influence of the cycloidal gear's tangential composite error on the overall TE. For prototype 5, the response amplitudes at characteristic frequencies 0.1612 Hz, 0.3223 Hz, and 0.004132 Hz are all around 3'', suggesting good quality in the planetary gears, crankshaft, flange, and pin gear. This also indicates good positional accuracy of the crankshaft hole on the cycloidal gear. While the  
Figure 11: Frequency spectrum analysis results of transmission error for the 65 BX-121-RVE reducer prototypes: (**a**) prototype 5; (**b**) prototype 6; (**c**) prototype 7; and (**d**) prototype 8. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & \multicolumn{4}{c}{**Transmission Error Amplitude (′′)**} \\ \cline{3-6}
**Component** & **Frequency (Hz)** & **Prototype 5** & **Prototype 6** & **Prototype 7** & **Prototype 8** \\ \hline Gear shaft & 0.5 & 0.08813 & 0.9439 & 1.0777 & 0.8153 \\ Gear shaft & 1.4876 & 0.04335 & 0.0609 & 0.5880 & 0.2287 \\ Planet gear, crankshaft, Flange & 0.1612 & 3.7702 & 3.7851 & 3.7575 & 1.9491 \\ Crankshaft, Flange, Pin gear & 0.3223 & 2.7016 & 6.7005 & 1.1094 & 1.1582 \\ Flange, Cycloidal gear & 0.004132 & 3.0597 & 3.8212 & 1.7252 & 7.9031 \\ Cycloidal gear & 0.1653 & 10.6515 & 6.5387 & 23.9579 & 23.6949 \\ \hline \end{tabular}
\end{table} 
Table 8: Characteristic frequency response of transmission error for 65 BX-121-RVE reducers. transmission precision is already high, there remains room for improvement by refining the tangential composite error of the cycloidal gear. A comprehensive comparison of the results at the three characteristic frequencies (0.1612 Hz, 0.3223 Hz, and 0.004132 Hz) shows that in prototype 6, the pin gear is another component, besides the cycloidal gear, that warrants quality improvement. For prototype 7, the crankshaft, flange, and pin gear demonstrate good quality and positional accuracy, with the planetary gear exhibiting relatively notable composite errors.