## 6 Lab Experimental Validation
### Signal Analysis
#### 6.2.1 Sun Gear Fault  
Figure 6 shows the sun gear fault signal, its Fourier spectrum and envelope spectrum. Figure 7 displays the five IMFs output from VMD and their respective instantaneous frequency waveform. Among them, IMF3 has a center frequency of 1136 Hz corresponding to four times the gear meshing frequency, and its instantaneous frequency fluctuates around 1136 Hz, as shown in Figure 7f. According to the sensitive IMF selection criterion, we choose IMF3 for further amplitude and frequency demodulation analysis.  
Figure 6: Sun gear fault signal: (**a**) Waveform; (**b**) Fourier spectrum; and (**c**) Envelope spectrum.  
Figure 7: _Cont._Figure 8 shows the envelope spectrum and the Fourier spectrum of instantaneous frequency of IMF3. The envelope spectrum (Figure 8a) presents a similar structure to that of raw signal (Figure 6c). The sun gear fault characteristic frequency \(f_{\rm s}\) appears with a prominent magnitude, indicating the sun gear fault. In addition, the fractional harmonics of sun gear fault characteristic frequency, \(1/3f_{\rm s}\) and \(2/3f_{\rm s}\), also emerge with pronounced magnitudes, further confirming the occurrence of sun gear fault. For a real planetary gearbox, it is impossible for all the planet gears to be perfectly identical. The inevitable difference between planet gears will generate three different impulse trains when the faulty sun gear meshes with the three planet gears, thus leading to presence of the fractional harmonics. Meanwhile, the planet carrier rotating frequency harmonics (1-3)\(f_{\rm c}\), as well as their sum and difference combinations with the fractional harmonics of sun gear fault characteristic frequency \(m/3f_{\rm s}\pm nf_{\rm c}\) also have high magnitudes. This is because the sun gear fault will result in an uneven load distribution among planet gears, and this uneven load distribution will magnify the modulation effect of planet carrier rotation on gear meshing vibrations.  
The Fourier spectrum of instantaneous frequency (Figure 8b) shows a simpler structure than both the Fourier and envelope spectra (Figure 6b,c). Only four prominent peaks appear at the sun gear fault characteristic frequency \(f_{\rm s}\), as well as its sum combination with the planet carrier rotating frequency \(f_{\rm s}+f_{\rm c}\), twice the sun gear rotating frequency \(2f_{\rm s}({\rm r})\) and the third harmonic of planet carrier rotating frequency \(3f_{\rm c}\). This implies existence of the sun gear fault, consistent with the actual experimental setting.  
Figure 7: VMD result: (**a**) IMF1 waveform; (**b**) IMF1 instantaneous frequency; (**c**) IMF2 waveform; (**d**) IMF2 instantaneous frequency; (**e**) IMF3 waveform; (**f**) IMF3 instantaneous frequency; (**g**) IMF4 waveform; (**h**) IMF4 instantaneous frequency; (**i**) IMF5 waveform; and (**j**) IMF5 instantaneous frequency.  
A comparison study again demonstrates the outperformance of proposed joint amplitude and frequency demodulation analysis method over conventional Fourier and envelope spectra. The proposed amplitude demodulation analysis of sensitive IMF avoids the difficulty due to intricate sidebands in Fourier spectrum, and is equally effective as the envelope spectrum of raw signal (see Figure 8a vs. Figure 6b,c). The proposed frequency demodulation analysis of sensitive IMF is free from intricate sidebands, further suppresses the irrelevant transfer path effect, and highlights gear fault features (see Figure 8b vs. Figure 6b,c).