## 3 Fault Diagnosis Method of Planet Gear Based on CVS and MED
### Meshing Regularity of Planet Gear Tooth  
As shown in Figure 2, planet gear of interests passes the sensor once in a carrier cycle. According to analysis in Section 2.1, when the planet gear meshes with ring gear at the sensor position, sensor captured signals will be dominated by the very planet meshing. Additionally, signal of the planet gear is less influenced by planetary movements-caused modulation effects at the time. If we extract a segment of signal from the monitoring signal that corresponds to the moment when the planet gear meshes at the sensor position, the extracted signal segment mainly contains dynamic responses of the planet gear that meshes at the sensor, moreover, the extracted signal is less influenced by the modulation effect. When the planet gear meshes at the sensor again after a carrier cycle, the tooth aligns with the sensor will be different. Comparing Figure 2a,f, the 1 st tooth of the planet gear aligns with the sensor when \(\theta_{\text{c}}=0\) while does not aligns with the sensor when \(\theta_{\text{c}}=2\pi\). It can be easily concluded that planet gear tooth number aligns with the sensor changes periodically as planetary gearbox runs, but the detailed meshing regularity should be investigated in order to build mapping relationship between extracted signal segments and planet gear teeth. Without losing generality, assuming planet gear tooth of P\({}_{n}\) that aligns with the sensor in \(k\)-th carrier cycle will be \(N_{n,k}^{\star\star}\), accordingly, after a full carrier cycle, when planet gear meshes at the sensor again, planet gear will mesh with ring gear \(N_{r}\) times. The planet gear tooth that aligns with the sensor will be  
\[N_{n,k+1}^{\star\star}=\overline{\text{mod}}(N_{n,k}^{\star\star}+N_{r},N_{p}), \tag{8}\]  
where  
\[\overline{\text{mod}}(x,y)= \begin{cases}\text{mod}(x,y)&if\,\text{mod}(x,y) \
eq 0\\ y&if\,\text{mod}(x,y)=0\end{cases} , \tag{9}\]  
where mod(\(x\), \(y\)) is modulo function. Equation (8) is a recurrence formula, it is equivalent to the following equation  
\[N_{n,k}^{\star\star}=\overline{\text{mod}}(N_{n,1}^{\star\star}+(k-1)N_{r},N_{ p}). \tag{10}\]  
Equation (10) reveals how planet gear tooth number, which aligns with the sensor, changes as planetary gearbox runs. Tooth number is neither monotone increasing from 1 to \(N_{p}\) nor monotone deceasing from \(N_{p}\) to 1, but it changes as addition of modulo \(N_{p}\). Accordingly, period of tooth number can be derived as well. The period is equal to Planet Gear Hunting Tooth Ratio (P\({}_{\rm HTR}\)). P\({}_{\rm HTR}\) is the minimum rotation of carrier to reset the planet gear to its initial state [44]. \[{\rm P}_{\rm HTR}=\frac{\text{LCM}(N_{r},N_{p})}{N_{r}}, \tag{11}\]  
where LCM(\(x,y\)) is the least common multiple of \(x\) and \(y\). All planet gear will be reset to its own initial condition when one of them is reset to initial condition since movements of planet gears are identical in a planetary gearbox during operation. \(N_{n,1}^{**}\) is an important parameter in Equation (10) since tooth numbers in following carrier cycles are evaluated based on it. However, it is difficult to define first tooth of planet gear since planet gear has rotational symmetric. In Refs. [31; 32], first tooth of planet gear is defined as the gear tooth that meshes with ring gear under initial condition. In this way, gear tooth number that meshes at the sensor in first carrier cycle is evaluated according to the initial phase differences \(\psi_{n}\) (\(n=1,2,\ldots,NP\)) between planet gear and sensor by the following equation  
\[N_{n,1}^{**}=\overline{\text{mod}}\bigg{(}\text{round}(\frac{\psi_{n}N_{r}}{2 \pi}),N_{p}\bigg{)}. \tag{12}\]  
Given initial phase difference, tooth number of ring gear and planet gear, tooth index of planet gears in each carrier cycle could be evaluated accordingly. For example, a planetary gear that contains 4 planet gears, its ring gear tooth number is 96, sun gear number is 28, and planet gear number is 34.