## 4 Planetary Gearbox Digital Twin Model Simulation Analysis
### Simulation Analysis of Normal Working Condition of Sun Gear and Planetary Gear  
The simulation model was set up according to the above parameters, and the sensor measurement point was set up at the point directly above the ring gear, where the vibration velocity signal in the Y direction of the ring gear was used, and then the simulation data were processed in MATLAB with a sampling frequency of 5000 Hz, and the Hilbert transform was performed on the time domain spectrum, and the final obtained time domain waveform and frequency spectrum are shown in Figure 7 below. The spectrum was taken between 450 Hz and 550 Hz.  
Figure 7 shows the time domain waveform and the Fourier spectrum of the planetary gearbox vibration signal under normal conditions. The time domain waveform is uniformly distributed as a whole, without any obvious evidence of shock. 1 and 2 in the figure are \(f_{m}-7f_{c}\) and \(f_{m}+7f_{c}\), respectively. It can be seen from the Fourier spectrum that shocks with frequencies of \(f_{m}\pm nf_{c}\) (\(n=1\),\(2\),... ) appear in the sideband of the meshing cycle,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Meshing** \\ **Frequency** \\ \(f_{m}\) \\ \end{tabular} } & \multicolumn{2}{c}{**Absolute Rotational Frequency**} & \multicolumn{2}{c}{**Local Fault**} \\ \cline{2-5}  & & & **Characteristic Frequency** \\ \cline{2-5}  & **Sun Gear** & **Carrier** & **Sun Gear** & **Planet Gear** \\ \cline{2-5}  & \(f_{s}^{\prime}\) & \(f_{c}\) & \(f_{s}\) & \(f_{p}\) \\ \hline
500.1 Hz & 30.0 Hz & 6.2 Hz & 71.4 Hz & 16.7 Hz \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation of planetary gearbox characteristic frequency.  
and the sideband is spaced at the planetary frame rotation frequency. These phenomena indicate that the passing effect of the planetary gear plays a major role in amplitude modulation of the meshing vibration, which is in line with the theoretical expectation of the normal planetary gearbox vibration signal, and also basically in line with the results of the subsequent experimental verification.  
Simulation Analysis of the Combined Fault Conditions of Sun Gear Missing Teeth and Planetary Gear Pitting  
The simulation model was also set up according to the above parameters, the sampling frequency was kept constant, the simulation data were imported into MATLAB for processing, and then the Hilbert transform was performed on the time domain spectrum, and the final time domain waveform and spectrum are shown in Figure 8 below.  
Figure 8 shows the time domain waveform and Fourier frequencies of the combined fault vibration signal of the sun gear missing teeth and planetary gear pitting. In the time domain waveform, more obvious evidence of shock appears, and more modulation frequencies appear in the Fourier spectrum. For easy identification, only the meshing frequency and the modulation frequency associated with the fault are marked in the figure.  
Figure 7: Normal simulation vibration signal of sun gear and planetary gear: (**a**) Time domain waveform; (**b**) Fourier spectrum.  
Due to the modulation effect caused by the planetary frame and sun gear self-propagation, the sideband interval \(f_{c}\) and its multiplicative correlation modulation sideband appear on both sides of \(f_{m}\). For example, at point 5 is \(f_{m}-f_{c}\). Moreover, due to the presence of three planetary gears, it can be seen in the figure that peaks occur at the positions \(f_{m}\pm nf_{s}/3\pm f_{c}\) of the meshing frequency, and the sideband is spaced as \(nf_{s}/3\) times the characteristic frequency of the sun gear fault, where \(n\) is an integer, 1 is \(f_{m}-f_{s}/3-f_{c}\). Peaks appear at the positions \(f_{m}\pm nf_{p}\pm nf_{c}\), related to the characteristic frequency \(f_{p}\) associated with the planetary gear, where \(n\) is an integer, such that 2, 3, and 4 would be \(f_{m}-f_{p}\), \(f_{m}+f_{p}\), and \(f_{m}+2f_{p}+f_{c}\), respectively, so the comprehensive diagnosis is that both the sun gear and the planetary gear are faulty.