## 4 Simulation and Experimental Analysis
### Simulation Analysis  
The fourth-order Runge-Kutta method is used to numerically simulate the sun planetary vibration signal of the planetary gear in this section. Before obtaining the dynamic response, first to calculate the corresponding fault characteristic information under the set working conditions to verify the correctness of the solution stiffness and the establishment of the model. The following is the calculation process of fault characteristic information:  
Sun gear rotation frequency:  
\[f_{\rm s}=n_{t}/60 \tag{57}\]  
Planetary carrier rotation frequency:  
\[f_{H}=\frac{Z_{1}}{Z_{1}+Z_{3}}f_{\rm S} \tag{58}\]  
Planetary gear rotation frequency:  
\[f_{p}=f_{H}-\frac{Z_{1}}{Z_{2}}(f_{\rm s}-f_{H}) \tag{59}\]  
Gear mesh frequency:  
\[f_{m}=f_{H}Z_{3}=(f_{\rm s}-f_{H})Z_{1}=\frac{Z_{1}Z_{3}}{60(Z_{1}+Z_{3})}n_{t} \tag{60}\]  
Among them, \(Z_{1}\) is the number of sun gear teeth, \(Z_{2}\) is the number of planetary gear teeth, \(Z_{3}\) is the number of ring gear teeth, and \(n_{t}\) is the input speed, that is, the sun gear speed. When \(n_{t}\) = 1200 r/min, the calculated sun gear rotation frequency \(f_{\rm s}\) = 20 Hz, the rotation frequency of planet carrier is \(f_{H}\) = 4.38 Hz, the rotation frequency of planet gear is \(f_{p}\) = 7.78 Hz, planetary gear train mesh frequency \(f_{m}\) = 437.5 Hz. When a component of the planetary gear system fails, the carrier frequency of vibration response is the planetary gear system mesh frequency, and the corresponding fault characteristic frequencies of the sun gear and planetary gear are \(N(f_{\rm s}-f_{H})\) and \(N(f_{p}+f_{H})\) (\(N\) = 1, 2, 3,... ).  
The relevant parameters of planetary gear system dynamics modeling and simulation are shown in the following Table 1.  
The main purpose of this paper is to verify the effectiveness of the TVMS solution method and the establishment of the dynamic model. To verify the correctness of the model, the vibration response of the planetary gear and sun gear under the healthy state and the cracked state is solved. The time-domain waveforms of the TVMS of the sun gear in healthy and faulty states are obtained, as shown in the following Figure 12.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Sun Gear** & **Planet Gear** & **Ring Gear** & **Planet Carrier** \\ \hline Number of teeth & 28 & 36 & 100 & - \\ Modulus (mm) & 1 & 1 & 1 & - \\ Young’s modulus (Pa) & \(2.068\times 10^{11}\) & \(2.068\times 10^{11}\) & \(2.068\times 10^{11}\) & - \\ Poisson’s ratio & 0.3 & 0.3 & 0.3 & - \\ Tooth width (mm) & 10 & 10 & 10 & - \\ Mass (kg) & 0.7 & 1.822 & 2.58 & 10 \\ Pressure angle & \(20^{\circ}\) & \(20^{\circ}\) & \(20^{\circ}\) & - \\ Rotation frequency (Hz) & \(f_{\rm s}\) = 20 & \(f_{p}\) = 7.78 & - & \(f_{H}\) = 4.38 \\ Fault characteristic frequency (Hz) & \(N(f_{\rm s}-f_{H})\)=\(15.63\times N\) & \(N(f_{p}+f_{H})=12.16\times N\) & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dynamic simulation parameters of the planetary gear train.  
Figure 16 shows the envelope spectrum in a crack condition, from which the mesh frequency \(f_{m}=437.5\) Hz and its multiplication can be found clearly. Simultaneously, a large number of sidebands appear near the mesh frequency in the vibration signal spectrum. In order to facilitate the analysis, the low-frequency band was amplified, and it was found that the sun gear fault characteristic frequency (15.63 Hz) and its multiplication frequency existed (31.26 Hz and 46.9 Hz). The spectrum structure obtained by simulation analysis is also consistent with the theoretical analysis results.  
Figure 16: Simulation signal spectrum in crack condition.  
Figure 15: Simulation signal spectrum in healthy condition.