## 3 Experimental Study
### Introduction to the Experiment
The smaller the entropy is, the less the coupling degree between the two systems is, and the richer the fault information contained in the fault system is. The Renyi entropy with the parameter value \(\alpha=4\) between the IMF component of each broken tooth state of the planetary gearbox and the healthy signal is calculated, respectively, and the calculated results are shown in Figure 8. In Figure 8a, IMF\({}_{1}\) \(>\) IMF\({}_{4}\) \(>\) IMF\({}_{2}\) \(>\) IMF\({}_{6}\) \(>\) IMF\({}_{3}\) \(>\) IMF\({}_{5}\); In Figure 8b, IMF\({}_{1}\) \(>\) IMF\({}_{4}\) \(>\) IMF\({}_{2}\) \(>\) IMF\({}_{6}\) \(>\) IMF\({}_{3}\) \(>\) IMF\({}_{5}\). As shown in the figure, the Renyi entropy between the IMF\({}_{1}\) component and healthy signal is the largest under the condition of planetary gear teeth broken, indicating that the coupling relationship between the IMF\({}_{1}\) information feature and normal signal is the largest, and the fault feature information is relatively the least. The Renyi entropy corresponding to IMF\({}_{3}\), IMF\({}_{5}\) and IMF\({}_{6}\) is small, indicating that there is a strong nonlinear relationship between the broken tooth IMF component and the healthy signal, and the fault information is more abundant. The components of IMF\({}_{3}\), IMF\({}_{5}\) and IMF\({}_{6}\) are in the high-frequency domain. After data fusion, the influence of noise can be reduced and the workload of data calculation can also be reduced. So, IMF\({}_{3}\), IMF\({}_{5}\) and IMF\({}_{6}\) are selected to extract and process the follow-up fault information, which provides the basis for the follow-up fault diagnosis. Multi-channel homologous information gathering realizes the superposition and complementarity of the information of each degree of freedom, and fully reflects the integrity of the original information. Figure 9 shows the spectrum diagram of the broken tooth state of the planetary gearbox and the local magnification of part of the meshing frequency obtained by multi-channel acquisition and full-vector spectrum processing. As shown in Figure 9a, in the spectrum diagram, the sun gear in the broken tooth state will have a peak value at its engagement frequency and its frequency doubling, and will generate periodic impact signal (\(f_{s}=20.84\) Hz). Due to the influence of manufacturing error, there is a certain difference in the impact characteristics between the solar wheel and the three planetary gears. The local fault characteristic frequency of the solar wheel will become \(f_{s}\)/3 = 6.947 Hz and there will be a peak value at its frequency doubling. At the same time, based on the modulation effect of the transmission effect of the system on the vibration signal, the peak value will appear at the absolute rotation frequency \(f_{c}\) and its frequency doubling of the planetary frame. Based on the above factors, the peak value will appear at the frequency of \(kf_{m}\pm mf_{c}\pm(m\pm n)f_{s}\)/3 Hz in the frequency spectrum of the multi-stage planetary gearbox when the gear is broken. The side band is formed around the meshing frequency and its doubling frequency, and its amplitude and width are larger than the healthy signal [70]. As shown in Figure 9b, peak values of \(1.422\times 10^{-3}\), \(2.396\times 10^{-3}\), \(8.519\times 10^{-4}\), \(1.950\times 10^{-3}\) and \(3.453\times 10^{-3}\) appeared at positions \(9f_{m}+7f_{c}-5f_{s}=1211\) Hz, \(9f_{m}-3f_{c}-3f_{s}=1237\) Hz, \(9f_{m}+2f_{c}+f_{s}\)/3 = 1314 Hz, \(9f_{m}+f_{c}+5f_{s}\)/3 = 1340 Hz, \(9f_{m}+3f_{c}+4=1357\) Hz and so on, respectively. In Figure 9c, peak values of \(8.331\times 10^{-4}\), \(9.986\times 10^{-4}\), \(1.856\times 10^{-4}\), \(4.919\times 10^{-4}\), \(1.280\times 10^{-3}\) appeared at positions \(11f_{m}+3f_{c}-2f_{s}=1557\) Hz, \(11f_{m}-2f_{c}-2f_{s}\)/3 = 1577 Hz, \(11f_{m}+f_{c}=1597\) Hz, \(11f_{m}+\(5f_{s}3=1630\) Hz, \(11f_{m}+f_{c}+2f_{s}=1637\) Hz and so on, respectively.