## 4 Procedure of Demodulation Analysis  
Planetary gearbox vibration signals have strong AM-FM features. Gear faults are manifested by the gear fault characteristic frequency that modulates the gear meshing vibration in terms of both amplitude and frequency. Amplitude and frequency demodulation analysis is effective to identify the modulating frequency of both the AM and the FM parts, thus providing an approach to extraction of gear fault features. Amplitude demodulation needs properly selecting a constituent component sensitive to the gear fault, and frequency demodulation requires fine mono-component decomposition for accurate estimation of instantaneous frequency. However, planetary gearbox vibration signals are usually composed of complex multiple AM-FM components. It is difficult to reveal the modulation characteristics from raw signals, due to their multi-component nature. VMD is effective in decomposing complex signals into constituent AM-FM components, and empirical AM-FM decomposition is capable to separate the carrier signal from any AM-FM component. A fusion of these two methods provides an effective solution to estimation of instantaneous frequency and amplitude envelope, and thereby offering insight into the modulation nature of planetary gearbox vibration signals. In our proposed method, we exploit the advantages of VMD and empirical AM-FM decomposition to jointly amplitude and frequency demodulate AM-FM signals for planetary gearbox fault diagnosis. The procedure is detailed below. 1. Decompose the signal into some IMFs via VMD. The number of IMFs to be separated \(K\) is a key parameter for VMD in real applications. We propose a criterion to determine the number according to the AM-FM nature and the spectral characteristics of planetary gearbox vibration signals. Planetary gearbox vibration signals can be modeled as a sum of some AM-FM signals considering their modulation nature (see Appendix A). Each of these AM-FM signals has a specific center frequency equal to one of the gear meshing frequency harmonics. Conversely, each gear meshing frequency harmonic corresponds to an AM-FM component (an IMF equivalently). Therefore, the gear meshing frequency harmonic orders covered in the effective frequency band of a signal, i.e. 0 through half the sampling frequency, roughly amounts to the number of IMF center frequencies. Accordingly, we can estimate the number of IMFs \(K=\left\lfloor\frac{F_{\text{s}}}{2f_{\text{m}}}\right\rfloor\), where \(\left\lfloor\cdot\right\rfloor\) is the floor operator, \(F_{\text{s}}\) is the sampling frequency, and \(f_{\text{m}}\) the gear meshing frequency. 2. Separate the amplitude envelope \(a(t)\) and the carrier signal \(c(t)\) of each IMF via empirical AM-FM decomposition. For each IMF, calculate its instantaneous frequency based on the local phase derivative of carrier signal \(c(t)\) using Hilbert transform  
\[f(t)=\frac{1}{2\pi}\frac{d}{dt}\text{arctan}\left\{\frac{H[c(t)]}{c(t)}\right\}, \tag{13}\]  
where \(H(\cdot)\) stands for Hilbert transform. This instantaneous frequency estimation approach is called normalized Hilbert transform (NHT). 3. Choose the IMF with an instantaneous frequency fluctuating around the gear meshing frequency or its harmonics as a sensitive mono-component for further demodulation analysis, because gear fault information are mainly carried by the meshing vibration or its harmonics. If more than one IMF satisfies this condition, choose the IMF with a higher center frequency, since gear fault induced vibration signals feature high frequency impulses. 4. Fourier transform the amplitude envelope and the instantaneous frequency of the selected sensitive IMF, thus obtaining the amplitude and frequency demodulated spectra, i.e. envelope spectrum and Fourier spectrum of instantaneous frequency. 5. Diagnose the gear fault by matching peaks in the demodulated spectra with the gear fault characteristic frequencies. The envelope spectrum and the Fourier spectrum of instantaneous frequency are free from complex sidebands.