## 5 Numerical Simulation Evaluation  
In this section, we illustrate the proposed demodulation analysis with a numerical simulated vibration signal of planetary gearboxes. Without loss of generality, we consider a signal synthetic of three AM-FM components centered at the first three gear meshing frequency harmonics respectively. According to the planetary gearbox fault vibration signal model in [39], the simulated signal is generated by  
\[ \begin{array}{ll}x(t)&=[1+0.5\cos(2\pi f_{\mathrm{d}}t)][1+0.5\cos(2\pi f_{ \mathrm{d}}t)]cos[2\pi f_{\mathrm{m}}t+0.5 sin(2\pi f_{\mathrm{d}}t)]\\ &+0.8[1+0.5\cos(2\pi f_{\mathrm{f}}t)][1+0.5\cos(2\pi f_{\mathrm{d}}t)]cos[4\pi f _{\mathrm{m}}t+0.5 sin(2\pi f_{\mathrm{d}}t)]\\ &+0.5[1+0.5\cos(2\pi f_{\mathrm{f}}t)][1+0.5\cos(2\pi f_{\mathrm{d}}t)]cos[6\pi f _{\mathrm{m}}t+0.5 sin(2\pi f_{\mathrm{d}}t)]+n(t),\end{array}  \tag{14}\]  
where \(f_{\mathrm{f}}\), \(f_{\mathrm{d}}\) and \(f_{\mathrm{m}}\) are the vibration transfer path varying frequency, the gear fault characteristic frequency and the gear meshing frequency respectively, and \(n(t)\) is a white Gaussian noise at a signal-to-noise-ratio of 0 dB mimicking the background interferences. Table 1 lists the parameter values. Figure 1 shows the signal waveform, its Fourier spectrum and envelope spectrum. The effective frequency band of this simulated signal is 0-650 Hz (\(F_{\mathrm{s}}/2=650\) Hz), and covers up to the third harmonics of gear meshing frequency \(f_{\mathrm{m}}=182\) Hz. According to the proposed criterion for selecting the number of IMFs to be separated, we set the number \(K=3\) in the VMD. In the Fourier spectrum, (Figure 1b), the gear meshing frequency, its harmonics up to third order, and their associated sidebands dominate the signal. In the envelope spectrum, which is obtained by \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Signal Length** & \begin{tabular}{c} **Sampling** \\ **Frequency \(F_{\mathrm{s}}\)** \\ \end{tabular} & \begin{tabular}{c} **Meshing** \\ **Frequency \(f_{\mathrm{m}}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Gear Fault** \\ **Frequency \(f_{\mathrm{d}}\)** \\ \end{tabular} &
\begin{tabular}{c} **Transfer Path** \\ **Frequency \(f_{\mathrm{t}}\)** \\ \end{tabular} \\ \hline
10 s & 1300 Hz & 182 Hz & 42 Hz & 16 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in simulated signal. Figure 1: Simulated signal: (**a**) Waveform; (**b**) Fourier spectrum; and (**c**) Envelope spectrum. Fourier transforming the amplitude envelope of raw signal (Figure 1c), the gear fault frequency, transfer path frequency, and their sum and difference combinations show up in a pronounced magnitude. Figure 2 shows the three IMFs that are obtained by VMD and their instantaneous frequencies. For each IMF, its instantaneous frequency is calculated via both NHT based on empirical AM-FM decomposition and conventional Hilbert transform (HT) approaches. As shown in Figure 2b,d,f, negative values exist with HT results. This phenomenon does not comply with the physical meaning of frequency. On the contrary, NHT results are free from the presence of misleading negative values. This demonstrates that empirical AM-FM decomposition can improve instantaneous frequency estimation, thus being helpful for further frequency demodulation analysis. For each IMF, its center frequency increases with the order of IMFs, and its instantaneous frequency fluctuates around the gear meshing frequency or its harmonic. These characteristics are consistent with both the numerical simulation settings and the findings from the Fourier spectrum. They also match well with the wavelet packet like property of VMD. Based on these findings, the three IMFs contain the major information about gear fault.