## 3 The Proposed Multi-Scale Demodulation Method
### Results and Discussion  
The vibration signal waveform of the gearbox while increasing and decreasing speed is shown in Figure 21. It could be found that the amplitude of the gearbox vibration had the same change trend as the increase and decrease of speed. We selected 500 rpm data in the initial stage and 700 rpm data in the intermediate stage to verify whether the algorithm could extract the fault characteristic vibration adaptively. The data interception time was 2 s, assuming that the vibration data in 2 s was quasi-static. The time-domain waveforms and spectra of vibration signals at 500 rpm and 700 rpm operating conditions are shown in Figure 22. Through analysis, it could be found that the spectrum structure changed to a certain extent as the operating condition increased from 500 rpm to 700 rpm. At 500 rpm, the energy mainly concentrated near 383.4 Hz, 725.6 Hz and 1745 Hz. At 700 rpm, the energy mainly concentrated around 536.6 Hz, and the spectrum energy distribution drifted. Based on the decomposition and reconstruction of the weighted-EMD, the time-frequency distribution of the vibration signals of gearbox under 500 rpm and 700 rpm operating conditions was calculated as shown in Figure 23a,b. It could be found that the impact components of the time-frequency vibration caused by the root crack fault were more disordered than those caused by the sun wheel flaking fault test, but impact characteristics of comparative regularity could still be found. When the working condition increased from 500 rpm to 700 rpm, the energy of the impact component increased and the time interval became shorter. The time-frequency shock envelope characteristic signals and their demodulation spectra under two conditions were extracted and calculated as shown in Figure 24. At 500 rpm, the extracted time-frequency impulse characteristic signal was as shown in Figure 24a. Five impulse components were clearly visible within one second, and the corresponding impulse demodulation spectra were clearly visible at 5 Hz and 10 Hz, which corresponded to the local fault characteristic frequency of the solar wheel and its 2\(\times\) harmonic frequency, respectively, as shown in Figure 24b. At 700 rpm,  
Figure 21: Vibration waveform of the gearbox under working conditions of increasing and decreasing speed. Figure 20: The seeded tooth root crack fault. the extracted time-frequency impulse characteristic signal was as shown in Figure 24c. Seven impulse components were clearly visible in one second, and the corresponding impulse demodulation spectra were 6.875 Hz, 13.75 Hz and 20.63 Hz, which corresponded to the local fault characteristic frequency and second-order and third-order harmonic frequencies of the solar wheel, respectively, as shown in Figure 24d. Figure 23: Comparisons of the time–frequency distribution of the gearbox vibration signal between the two working conditions: (**a**) 500 rpm, (**b**) 700 rpm. Figure 22: Vibration waveform and spectrum of the gearbox in two working conditions: (**a**) time-domain waveform under the 500 rpm condition, (**b**) frequency spectrum under the 500 rpm condition, (**c**) time domain waveform under the 700 rpm condition, (**d**) frequency spectrum under the 700 rpm condition. Based on the fast Kurtogram method, the performance of the demodulation spectrum under the two conditions was analyzed as shown in Figure 25. Under the 500 rpm operating condition, the optimum demodulation scale selected by the fast Kurtogram method was in three layers, as shown in Figure 25a, and the frequency range was 2240-2560 Hz. Envelope demodulation analysis was carried out to obtain the corresponding envelope spectrum. As shown in Figure 25b, it was found that the characteristic frequency of the gear crack fault was 5 Hz, but compared with the impulse demodulation spectrum, the envelope spectrum contained a large amount of noise. At 700 rpm, the optimum demodulation scale selected by the fast Kurtogram method was also in three layers, as shown in Figure 25c, and the frequency range was 2240-2560 Hz.