## 1 Introduction
Therefore, planetary gearbox vibration signal spectra exhibit a very intricate sideband structure. Planetary gearbox fault diagnosis often relies on effective detection of the gear fault characteristic frequency and monitoring its magnitude change. Through conventional Fourier spectrum analysis, one has to identify the sideband spacing first, and then conduct fault diagnosis by matching it with the gear fault characteristic frequency. However, it is not easy to pinpoint the gear fault characteristic frequency via sideband analysis. Because of multiple modulations and their coupling effects, planetary gearbox vibration signals feature a highly complex spectral structure: sideband clusters around the carrier frequency (meshing frequency or its harmonics) and even sidebands within a sideband cluster. The multiple modulation sources may result in different sideband spacing values. The spacing between two adjacent sideband peaks may not exactly equal the gear fault characteristic frequency, instead may equal the sum or difference of the gear fault characteristic frequency and the frequency of time-varying vibration transfer paths. This leads to difficulty in planetary gearbox fault diagnosis through conventional Fourier spectrum analysis. The modulating frequency of both the AM and the FM parts links to the gear fault characteristic frequency. If we can separate the AM and the FM parts, the Fourier spectrum of AM part or FM instantaneous frequency alone can directly reveal the gear fault characteristic frequency, thus avoiding the complex sidebands inherent in Fourier spectrum of the whole AM-FM signal. This inspires the idea of fault characteristic frequency identification through amplitude and frequency demodulation analysis. However, demodulation analysis needs proper selection of a sensitive AM-FM component containing gear fault information. Most importantly, frequency demodulation analysis requires effective mono-component decomposition for accurate instantaneous frequency estimation. Recently, [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) proposed the variational mode decomposition (VMD). This method can non-recursively decompose a complex multi-component signal into constituent AM-FM components, and it is robust to noise. [PERSON] and his colleagues [PERSON] and his colleagues [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2019); [PERSON] and [PERSON] (2020) discovered the wavelet packet like frequency band decomposition property of VMD based on fractal Gaussian noise simulations, extended VMD from real-valued to complex-valued signals, and applied them to extract the fundamental, sub-harmonics, super-harmonics, and impacts of rotor-stator rubbing fault, as well as the impulsive components of rolling bearing fault. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) utilized VMD to analyze the nonstationary pressure fluctuation signal of a hydraulic turbine draft tube. They showed that VMD is better than EMD in suppressing mode mixing and improving time-frequency readability. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) developed a denoising method based on VMD and approximate entropy, they demonstrated its better performance than wavelet transform, and applied it to hydropower unit vibration analysis. [PERSON] et al. [PERSON] et al. (2018) proposed to decompose signals into multiple components, and then use independent component analysis to solve the underdetermined blind source separation problem. They detected the compound fault of rolling bearings with the proposed method. [PERSON] et al. [PERSON] et al. (2018) extracted features from the VMD based time-frequency representation, and identified rolling bearing faults. [PERSON] et al. [PERSON] et al. (2018) improved the robustness of VMD to sampling and noise via particle swarm optimization, and utilized the proposed method to detect rolling bearing faults. [PERSON] et al. [PERSON] et al. (2018) determined the number of modes in VMD following a correlation criterion, and presented a time-frequency analysis method based on VMD and energy separation.