## 3 The Proposed Multi-Scale Demodulation Method
### Results and Discussion
Then, the corresponding envelope spectrum was obtained by the envelope demodulation analysis of the scale, as shown in Figure 25d As shown in the diagram, the characteristic frequency of the gear crack fault was 6.875 Hz, but compared with the impulse demodulation spectrum, the noise interference was more obvious, and no higher-order harmonic components of the characteristic frequency of the gear crack were found. Based on the enhanced Kurtogram method, the performance of the demodulation spectrum under two conditions was analyzed. As shown in Figure 26, the optimum demodulation scale selected by the enhanced Kurtogram method under both conditions was at (4,8) nodes of the wavelet decomposition, and the frequency range was 1280-1440 Hz, as shown in Figure 26a,c respectively. The corresponding demodulation spectra under the two conditions are shown in Figure 26b,d. At 500 rpm, the characteristic frequency of the gear crack fault was found to be 5 Hz. However, compared with the impulse demodulation spectra, the envelope spectra contained a large amount of noise. At 700 rpm, the characteristic frequency of the gear crack fault was 6.875 Hz, but compared with the impulse demodulation spectrum, the envelope spectrum also contained a large number of noise interference components. The demodulation spectrum calculated based on the traditional envelope demodulation analysis method is shown in Figure 27. Under the 500 rpm operating condition, a frequency band near six times the meshing frequency of the gearbox was selected for filtering and modulation analysis according to experience. Specifically, the dominant frequency of the spectrum was 680 Hz to 780 Hz. The characteristic frequency of the tooth root crack could not be found in the envelope demodulation spectrum, which contained a large amount of noise, as shown in Figure 27a. Under the 700 rpm operating condition, a frequency band near three times the meshing frequency of the gearbox was selected for filtering and demodulation analysis according to experience. Specifically, the dominant frequency of the spectrum was 470 Hz to 570 Hz. In the envelope demodulation spectrum, frequencies of two times and three times the characteristic frequency of the tooth root crack fault are found, but a  
Figure 25: Fast Kurtogram under(**a**) 500 rpm, (**c**) 700 rpm working conditions and the corresponding demodulation spectra under (**b**) 500 rpm, (**d**) 700 rpm working conditions. large amount of noise was included compared with the impact demodulation spectrum of the proposed method, as shown in Figure 27b. The \(MPR\) values calculated based on the four demodulation methods varied with speed, as shown in Figure 22.0 It can be seen that the time-frequency multi-scale envelope demodulation always performed well and was superior to the other three methods. The \(MPR\) value of the method proposed in this chapter was 19.86 dB higher than that of the fast spectral kurtosis plot method, 16.78 dB higher than that of the enhanced spectral kurtosis plot method and 17.00 dB higher than that of the traditional envelope demodulation method. In conclusion, the time-frequency multi-scale envelope demodulation algorithm proposed in this work can accurately extract the modulation components caused by faults during the increase and decrease of speed of a fixed-axle gearbox. The fault feature modulation components extracted by the fast spectral kurtosis method at 500 rpm and 700 rpm are heavily contaminated by noise. The modulation components of fault features extracted by the enhanced spectral kurtosis method are more obvious than those obtained by the fast spectral kurtosis method, but they are heavily contaminated by noise compared with the time-frequency multi-scale envelope demodulation  
Figure 22.1 \(MPR\) values of the four methods with different rotating speeds. Figure 22.2 Enhanced Kurtogam under (**a**) 500 rpm, (**c**) 700 rpm working conditions and the corresponding demodulation spectra under (**b**) 500 rpm, (**d**) 700 rpm working conditions.