# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 3 Transmission Error Frequency Analysis
### Characteristic Frequencies of Transmission Error
#### 3.3.1 Condition 1 with Fixed Flange  
(1) Gear shaft  
The primary error forms for the gear shaft include overall coaxial deviation and gear manufacturing errors. These errors complete one full cycle over a single rotation of the shaft and engage with \(n_{\rm pla}\) (the number of planet gears and crankshafts) planet gears, completing \(n_{\rm pla}\) cycles of influence on the cycloidal-to-pin gear engagement. Therefore, the characteristic frequencies for the gear shaft are \(f_{\rm sun}\) and \(n_{\rm pla}f_{\rm sun}\). (2) Planet gear  
Similar to the gear shaft, the primary errors for the planet gear are coaxial deviations and gear manufacturing errors. These errors complete one full meshing cycle relative to the gear shaft over one rotation. Thus, the characteristic frequency for the planet gear is \(f_{\rm pla}\). Since each planet gear engages only with one gear shaft and has no direct interaction with other planet gears, the number of planet gears does not affect this frequency. Crankshaft  
The main errors in the crankshaft are coaxial errors and dimensional errors (primarily positional errors in the crank portion). Over one full rotation, these errors complete an influence cycle on the meshing between the gear shaft and planet gear, with a characteristic frequency of \(f_{\rm pla}\). As the crankshaft is double-cranked and linked to two cycloidal gears, it completes two cycles of influence on the cycloidal-to-pin gear engagement, giving it an additional characteristic frequency of \(2f_{\rm pla}\). Flange  
Although fixed, the positioning precision of the crankshaft mounting holes on the flange affects TE. The impact of the hole-positioning errors is similar to that of the crankshaft errors, with characteristic frequencies of \(f_{\rm pla}\) and \(2f_{\rm pla}\). The primary errors in the cycloidal gear include geometric errors of the gear teeth, tangential composite errors, and positional errors of the crank hole. With a fixed flange, the cycloidal gear does not self-rotate. These errors complete one full influence cycle on the cycloidal-to-pin gear engagement over a single orbit. Since the orbital motion of the cycloidal gear synchronizes with the crankshaft's rotation, the characteristic frequency for these errors is \(f_{\rm pla}\). The pin gear consists of cylindrical pins mounted in the pin housing, where manufacturing quality is challenging to maintain. The primary errors include pin alignment and position accuracy. Analogous to gear errors, the primary errors in the pin gear are geometric errors and tangential composite errors. These errors affect the engagement in two ways: first, completing one full cycle over a single rotation, influencing the cycloidal-to-pin engagement with a characteristic frequency of \(f_{\rm pin}\); and second, completing two influence cycles when all pins complete one engagement cycle with two cycloidal gears. Given that \(\omega_{\rm pla}\) and \(\omega_{\rm pin}\) rotate in the same direction (as shown in Figure 3), the characteristic frequency is \(2(f_{\rm pla}-f_{\rm pin})\). Crankshaft  
The error impact for the planet gear is primarily considered in terms of its self-rotation. Therefore, its characteristic frequency is similar to Condition 1, that is, \(f_{\rm pla}\). Crankshaft  
The crankshaft moves with the planet gear, therefore its considerations are also similar to Condition 1, so its characteristic frequencies are \(f_{\rm pla}\), and \(2f_{\rm pla}\). (4) Flange  
When not fixed, errors in the flange contribute to TE through its self-rotation. Its geometric and coaxial errors complete one cycle per rotation, with a characteristic frequency of \(f_{\rm cyc}\). Positioning errors in the crank holes also affect TE through the crankshaft, with the same characteristic frequencies \(f_{\rm pla}\) and \(2f_{\rm pla}\). (5) Cycloidal gear  
The error influence in the cycloidal gear exhibits two characteristic frequencies.