## 4 Conclusions  
We propose a new method for planetary gearbox fault detection based on compressed sensing, 2D-VMD and full-vector spectrum theory. The conclusions are as follows:  
1. In view of the complexity and low signal-to-noise ratio of the vibration signal extracted from the multi-stage planetary gearbox, compressed sensing is used to denoise the signal, which proves that compressed sensing has strong robustness and noise reduction ability for noise signals.  
2. In order to realize the synchronous noise reduction of multi-degree-of-freedom signals, we propose a new idea of multi-sensor data fusion, which extends one-dimensional variational mode decomposition to multi-dimensional. The 2D-VMD method with non-recursive mode and strong mathematical theory is used to process the planetary gearbox fault signal, and Renyi entropy is used to be the optimization index of 2D-VMD performance by measuring the coupling degree of information, so as to extract the fault features contained in the effective target IMF component, and the feasibility of the method is proved by analyzing and showing the fault characteristic frequency of the solar wheel in the planetary gearbox. The proposed method realizes the synchronous processing of multivariate  
Figure 10: Results of demodulation of the broken tooth state signal of the planetary gearbox in the \(x\)-direction by the energy operator.  
vibration signals and can efficiently and comprehensively extract the effective spectrum characteristics of planetary gearbox fault state signals.  
Using the full-vector spectrum technique, the homologous information collected by multiple sensors is data layer fused in the spatial domain and the time domain to increase the comprehensiveness and certainty of the fault information.  
When the multi-stage planetary gearbox is in the state of broken teeth, there are peak values at the meshing frequency, at its doubling frequency, and at \(kf_{m}\pm mf_{c}\pm(m\pm n)f_{s}/3\), and the side frequency band is formed at the same time, while the width and amplitude are greater than the healthy signal. Potential low-frequency features need to be demodulated from the high-frequency domain using energy operators, and the energy operator has higher resolution than envelope demodulation. When the multi-stage planetary gearbox is in local fault, the peak values will appear at the frequency multiplication (\(nf_{s}/3\)) of the sun wheel's local fault characteristic frequency, at the frequency multiplication (\(mf_{c}\)) of the planet carrier's absolute rotation frequency, and at the frequency multiplication (\(kf_{a}\)) of the sun wheel's absolute rotation frequency; the peak values appear at the combined position of the three and form the side band at the same time.  
Conceptualization, Z.W.; Data curation, Q.Z.; Formal analysis, S.T.; Software, S.T.; Writing--original draft, Z.W.; Writing--review & editing, Q.Z. and L.C.  
This research was funded by The National Natural Science Foundation of China [51705130, 51706058], Hebei University Science and Technology Research Youth Fund Project [QN2018168], and China National Key Laboratory of Vehicle Transmission Fund Project [9140C340101101].  
The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.  
The authors declare no conflict of interest.