## 3 The Proposed Multi-Scale Demodulation Method
### Results and Discussion  
The time waveforms and the corresponding spectra of the planetary gearbox under 100 rpm, 300 rpm and 500 rpm rotating speeds are shown in Figure 11. It can be found that several obvious impulsive shocks were caused by the spalling fault, as shown in Figure 11a-c, and the impact frequency and the amplitude of impulsive shocks became larger when the rotating speed increased from 100 rpm to 500 rpm. Besides, there were also some changes of the spectrum, as shown in Figure 11d-f. Under a 100 rpm working condition, the dominant frequencies of the spectrum were 224.1 Hz and 1087 Hz. Under a 300 rpm working condition, the dominant frequencies of the spectrum were 213 Hz and 1208 Hz. Under a 500 rpm working condition, the dominant frequencies of the spectrum were 448 Hz and 1345 Hz, and more obvious sidebands could be found. Figure 12a-c illustrates the TFR of the vibration signal after the denoising process for 100 rpm, 300 rpm and 500 rpm conditions, respectively. It can be found that impulsive shocks can be observed clearly in the TFRs of vibration signals, and the impulsive shocks became more intensive as the rotating speed increased from 100 rpm to 500 rpm. The frequency range of these impulsive shocks also became wider as the rotating speed increased; under the 100 rpm working condition, the impulsive shocks were present in a frequency range below 4000 Hz, while under the 500 rpm working condition, the impulsive shocks were present over almost the entire frequency range. According to the proposed method, both the time-frequency impulse envelope waveforms and the corresponding demodulation spectra could be calculated based on the TFR of the vibration signals, as shown in Figure 13. The impulse envelope waveforms of the three datasets are shown in Figure 13a-c. The corresponding demodulation spectra of the three datasets are shown in Figure 13d-f. Under the 100 rpm working condition, five impulsive shocks caused by the spalling fault were found, as shown in Figure 13a. The sun gear fault characteristic frequency of 5.625 Hz and its harmonics could be observed clearly in the demodulation spectrum, as shown in Figure 13d. Under a 300 rpm working condition, 17 impulsive shocks caused by the spalling fault were found, as shown in Figure 13b. The sun gear fault characteristic frequency of 16.88 Hz and its harmonics could be observed clearly in the demodulation spectrum, as shown in Figure 13e. Similarly, 28 impulsive shocks caused by the spalling fault were found in the 500 rpm working condition, as shown in Figure 13c. The sun gear fault characteristic frequency of 28.13 Hz and its harmonics could be observed clearly in the demodulation spectrum, as shown in Figure 13f. The conclusion can be drawn that the proposed method can accurately extract the modulation components caused by a spalling fault without the selection of variable optimal modulation sub-bands under different rotating speed conditions. The original fast Kurtogram proposed by [PERSON] (1991), the enhanced Kurtogram proposed by [PERSON] et al. [PERSON] et al. (2019) and the traditional envelope analysis method were applied to analyze the same datasets for comparison. The demodulation results based on the fast Kurtogram method are shown in Figure 14. The fast Kurtograms of the three datasets are shown in Figure 14a-c. The corresponding demodulation spectra of the three datasets are shown in Figure 14d-f. Under the 100 rpm working condition, the optimal demodulation scale chosen by the fast Kurtogram was at level 4, and the frequency ranged from 1920 to 2560 Hz, as shown in Figure 14a. The corresponding demodulation spectrum is shown in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Speed (rpm)** & **fsun (Hz)** & **fc (Hz)** & **fp (Hz)** & **fm (Hz)** & **fds (Hz)** \\ \hline
100 & 1.667 & 0.267 & 0.412 & 22.4 & 5.6 \\
300 & 5 & 0.8 & 1.237 & 67.2 & 16.8 \\
500 & 8.333 & 1.333 & 2.061 & 112 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristic frequencies of the planetary gearbox.