# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 5 Results and Discussion
### Test Results of 190 BX-121-RVE Reducers  
(1) Transmission error test results  
The time-domain TE test results for the four 190 BX-121-RVE prototypes are depicted in Figure 7. The TE of the reducer is defined as the difference between the maximum and minimum values of the dynamic TE curve. From Figure 7, the TEs for prototypes 1, 2, 3, and 4 are 41\({}^{\prime\prime}\), 54\({}^{\prime\prime}\), 69\({}^{\prime\prime}\), and 75\({}^{\prime\prime}\), respectively. Observing the time-domain distribution of the TE shown in Figure 6, it is evident that the TE of the 190 BX-121-RVE reducer exhibits periodicity and a complex beat frequency characteristic. However, the source and characteristics of the error cannot be directly identified from the time-domain graph, necessitating a frequency-domain analysis for further insights. Figure 6: Flowchart of the analysis method. 2 Transmission error spectrum analysis results  
The TE spectrum analysis results for the four 190 BX-121-RVE prototypes are illustrated in Figure 8. To accurately identify all the characteristic frequencies, particularly the ultra-low frequency \(f_{\rm cyc}=0.004132\) Hz, the total sampling point count for the spectral analysis was set to N = 4840. In this setting, the Fourier analysis frequency resolution is \(f_{0}=f_{s}/N=0.002067\) Hz, approximately half of \(f_{\rm cyc}\), thus allowing for the precise identification of all characteristic frequencies. From Figure 8, it can be seen that most of the TE spectral energy for the four test prototypes is concentrated in the low-frequency range (no greater than 1 Hz), with multiple high-amplitude responses within this range. Beyond the 1 Hz range, the amplitude response is minimal and can be ignored. Excluding the trend term (0 Hz), the high-amplitude response frequencies of the test prototypes are distributed at approximately equal intervals, with an interval around \(f_{\rm pla}=0.1612\) Hz. Figure 8: Frequency spectrum analysis results of transmission error for the 190 BX-121-RVE reducer prototypes: (**a**) prototype 1; (**b**) prototype 2; (**c**) prototype 3; and (**d**) prototype 4. Figure 7: Test results of transmission error for the 190 BX-121-RVE reducer prototypes: (**a**) prototype 1; (**b**) prototype 2; (**c**) prototype 3; and (**d**) prototype 4. Characteristic frequency validation results  
Characteristic frequencies were identified in the spectrum results to validate the accuracy of the characteristic frequency analysis. The identification results are presented in Table 7. It is evident that all the characteristic frequencies exhibit a clear TE amplitude response, which confirms the accuracy of the previously discussed kinematic analysis and characteristic frequency analysis for the components of the RV reducer. A visual representation of Table 7 is shown in Figure 9.  
As observed in Table 7 and Figure 9, there is a distinct TE response at the two characteristic frequencies corresponding to the sun gear, 0.5 Hz and 0.9917 Hz, but their amplitude is noticeably smaller than at other characteristic frequencies. This suggests that the sun gear error is relatively minor and may have a limited impact on the overall TE of the reducer. Additionally, the characteristic frequency of 0.1653 Hz is primarily contributed by the cycloidal gear, indicating the impact of the cycloidal gear's error on the meshing between the cycloidal gear and the pin gear. The results for this frequency across the four prototypes indicate that the tangential composite error of the cycloidal gear has a significant impact on the cycloid-to-pin gear meshing, with the largest error amplitude among all characteristic frequencies. Thus, ensuring high-quality cycloidal gear is essential for maintaining high transmission accuracy in RV reducers. The frequencies of 0.1612 Hz, 0.3223 Hz, and 0.004132 Hz do not have a single source, complicating their analysis. These frequencies require a comprehensive analysis in conjunction with other characteristic frequency results.