# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 3 Transmission Error Frequency Analysis
### Calculation of Fundamental Rotational Speeds  
We commence our analysis with Condition 2, wherein the RV reducer operates as a fixed-axis gear system. Under this condition, the rotational speeds of the planetary reduction stage and the small-tooth-difference cycloidal pin gear stage are related as follows:  
\[\left\{\begin{array}{l}\frac{\omega_{\rm pla}}{\omega_{\rm sun}}=-\frac{z_{ \rm sun}}{z_{\rm pla}}\\ \frac{\omega_{\rm pin}}{\omega_{\rm pla}}=\frac{z_{\rm pin}-z_{\rm cyc}}{z_{\rm pin }}\end{array}\right., \tag{3}\]  
where \(z_{\rm sun}\) is the teeth number of the gear shaft, \(z_{\rm pla}\) is the teeth number of the planet gear, \(z_{\rm cyc}\) is the teeth number of the cycloidal gear, and \(z_{\rm pin}\) is the number of pins mounted on the pin housing.  
According to Equation (3), the rotational speed of the planetary gear is  
\[\omega_{\rm pla}=-\frac{z_{\rm sun}}{z_{\rm pla}}\omega_{\rm sun} \tag{4}\]  
Based on Equation (3), the self-rotation speed of the planet gear is as follows:  
\[\omega_{\rm pin}=\frac{z_{\rm pin}-z_{\rm cyc}}{z_{\rm pin}}\omega_{\rm pla}= -\frac{z_{\rm sun}}{z_{\rm pla}}\frac{z_{\rm pin}-z_{\rm cyc}}{z_{\rm pin}} \omega_{\rm sun} \tag{5}\]  
In Equations (3)-(5), the negative sign (\"\(-\)\") immediately following the equals sign indicates that the rotation direction is opposite.  
For Condition 1, when the pin gear is fixed, the cycloidal gear drives the flange, forming an epicyclic gear system. Analyzing the speed relationships in this configuration is more complex. By applying an opposite rotation of equal magnitude to the cycloidal gear across the entire structure, the system can be treated as a fixed-axis gear system, with the rotational speeds related as follows:  
\[\frac{\omega_{\rm pla}-\omega_{\rm cyc}}{\omega_{\rm sun}-\omega_{\rm cyc}}=- \frac{z_{\rm sun}}{z_{\rm pla}}, \tag{6}\]  
\[\frac{\omega_{\rm pin}-\omega_{\rm cyc}}{\omega_{\rm pla}-\omega_{\rm cyc}}= \frac{z_{\rm pin}-z_{\rm cyc}}{z_{\rm pin}} \tag{7}\]  
Substituting \(\omega_{\rm pin}\) = 0 into Equation (7),  
\[\omega_{\rm pla}=-\frac{z_{\rm cyc}}{z_{\rm pin}-z_{\rm cyc}}\omega_{\rm cyc} \tag{8}\]  
Substituting Equation (8) into Equation (6),  
\[\omega_{\rm cyc}=\frac{(z_{\rm pin}-z_{\rm cyc})z_{\rm sun}}{(z_{\rm pin}-z_{ \rm cyc})z_{\rm sun}+z_{\rm pin}z_{\rm pla}}\omega_{\rm sun} \tag{9}\]  
Substituting into Equation (8),  
\[\omega_{\rm pla}=-\frac{z_{\rm cyc}z_{\rm sun}}{(z_{\rm pin}-z_{\rm cyc})z_{ \rm sun}+z_{\rm pin}z_{\rm pla}}\omega_{\rm sun} \tag{10}\]  
In Equations (6)-(10), the negative sign \"\(-\)\" immediately following \"=\" also indicates an opposite rotation direction. Thus, for Condition 1, given the rotational speeds of the gear shaft, \(\omega_{\rm sun}\), and the pin gear, \(\omega_{\rm pin}\) = 0, the self-rotation speeds of the planet gear \(\omega_{\rm pla}\) and the cycloidal gear \(\omega_{\rm cyc}\) can be determined using Equations (10) and (9). For Condition 2:With the gear shaft speed \(\omega_{\rm sun}\) and the cycloidal gear speed \(\omega_{\rm cyc}=0\) known, the speeds of the planet gear \(\omega_{\rm pla}\) and the pin gear \(\omega_{\rm pin}\) can be calculated using Equations (4) and (5).