Coupling Fault Feature Extraction Method Based on Bivariate Empirical Mode Decomposition and Full Spectrum for Rotating Machinery

To accurately extract the fault characteristics of vibration signals of rotating machinery is of great significance to the unit online monitoring and evaluation. However, because the current feature extraction methods are mainly for single channel, the results of feature extraction are often inaccurate. To this end, a coupling fault feature extraction method based on bivariate empirical mode decomposition (BEMD) and full spectrum is proposed for rotating machinery. Firstly, the two-dimensional orthogonal signal obtained by orthogonal sampling technique is decomposed by bivariate empirical mode decomposition to obtain the intrinsic mode function with phase information. In order to obtain the sensitive modal components, the sensitivity coefficients are constructed on the basis of mutual information. Then, calculate the sensitivity coefficient of each intrinsic mode function, and the intrinsic mode function with the larger sensitive coefficient is selected as the sensitive component. Finally, the full spectrum of the sensitive component is obtained by using the full vector envelope technique, so as to get a comprehensive and accurate characteristic component. The results of simulations experiment and an application example show that this method can extract the fault characteristic component of the rotating machinery comprehensively and accurately. It is of great significance to realize the accurate diagnosis of coupling faults of rotating machinery.


Introduction
Rotating machinery is one of the key parts of large mechanical equipment, such as hydraulic turbine, steam turbine and wind turbine, which have found wide application in various industrial fields [1]. Due to the influence of load, damping, friction and other factors, the rotating machinery usually exhibits complex dynamic behavior, which is related to non-stationarity and nonlinearity [2]. And the vibration signals of rotating machinery often show coupling. In order to ensure the safe and reliable operation of the equipment, the equipment condition monitoring and fault diagnosis technology needs to have higher precision. Whether it can accurately extract the characteristic signal of coupling fault is the key to the condition monitoring and fault diagnosis of rotating machinery [3,4].
At present, many scholars have done some research on the feature extraction of rotating machinery fault signal, especially in the matter of signal processing methods. The main methods are wavelet transform (WT) [5,6], Hilbert-Huang transform, empirical mode decomposition (EMD) [7,8] and variational mode decomposition (VMD) [9,10] and so on, and the characteristic of different fault feature extraction method as shown in Tab.1. Wavelet transform is the improvement of the Fourier transform, has a certain ability to deal with nonlinear signals. But its essence is the inner product principle characteristic waveform signal decomposition based on basis function, exists the choice problem for basis functions, and the different vibration will show the characteristics of different waveforms [11,12]. Therefore, the adaptive ability of wavelet transform is poor.
EMD has strong adaptability and good local analysis ability, and has been widely used in the field of fault diagnosis. However, the EMD method is lack of rigorous theoretical basis, and there is a serious phenomenon of modal aliasing in the decomposition process [13,14]. Variational mode decomposition (VMD) is a new adaptive signal decomposition method, which has a solid theoretical foundation, and can solve the problem of mode mixing in EMD, but there are parameter optimization problems in VMD [15].
However, due to the complex mechanism of the fault of rotating machinery, and the vibration signal is often nonlinear, the vibration signals in different directions may represent different fault characteristic information [16]. Through single channel signal feature extraction and fault diagnosis, it often leads to misjudgment and omission [17,18]. Therefore, it is more accurate to collect the signal of multiple channels, and through the effective information fusion. Therefore, taking into account the rotating machinery generally has two sensors in horizontal and vertical directions, this paper utilize bivariate empirical mode decomposition (BEMD) to decompose the signal collected from two directions.
In 2007, bivariate empirical mode decomposition (BEMD) was firstly introduced by Gabriel Rilling et al [19]. It is not limited to single direction processing of real valued signals, and it can analyze complex signals consisting of two orthogonal directions contained phase information and synchronization [20,21].  [27,28].
In this paper, we proposed a coupling fault Conclusions come in Section 7.

Bivariate Empirical Mode Decomposition
The essence of bivariate empirical mode decomposition (BEMD) is to decompose a two-dimensional signal adaptively into intrinsic mode components with physical meaning. The modal component obtained from the decomposition is a series of single component signals from high frequency to low frequency in two directions. This paper takes the algorithm II in [19] to perform BEMD. For a two-dimensional signal ) (t x , the basic decomposition process is as follows.
Step 1: Determine the direction of Step 2: The two-dimensional signal Step 3: Extract the corresponding moment for Step 4: Calculate the mean of the maximum Step 5: Similar to the EMD decomposition process, the residual component is calculated: Step 6: Record the resulting IMF and remove it from the original signal. And get IMF1 as: , residual component as: Step 7: Repeat the above steps until you get all the IMFs. The original signal can be expressed as: Where, K represents the total number of IMFs.

Mutual information
Mutual information can measure the degree of interdependence between the two variables, which means that the information content is shared between the two variables [29,30]  ) max( 3) Calculate the sensitivity coefficient of each modal component IMFs as follows:

Full spectrum theory
There are usually two vertical sensors in the The length of the long axis of the ellipse ai R is defined as the main vibration vector.
The length of the short axis of the ellipse bi R is defined as vice vibration vector.
i  is the angle between the main vibration vector and the X axis.
Re Re Im Im tan (14) Where, Step 2: The orthogonal fault vibration signal is decomposed by BEMD to obtain K intrinsic Step 3: Calculate the sensitivity of each modal component IMFs.
Step 4: Select the modal component IMFs with the larger sensitive coefficient as sensitive modal components IMFs.
Step 5: Full spectrum analysis of the selected modal components IMFs is carried out to detect the characteristic frequency of coupling fault signals.

Simulation experiment
There are two common phenomena that a change in amplitude at the typical frequency [35] and a new frequency appearance [36] when rotating machinery fault. In this paper, the vibration signal of the hydropower unit bearing is simulated. Suppose that the rotating frequency of hydropower unit is f 0 , It is considered that the vibration fault signal of the actual hydroelectric generating set often contains f 0 , 2f 0 , 3f 0 , 50Hz, 100Hz and some common characteristic signals and noise. In order to make the simulation result more fitting with the actual situation and verify the effectiveness of the proposed method, this paper in the processing of two-dimensional signals, simulate the normal and fault signals of the hydropower unit to verify. The specific simulation vibration signal is as follows.  Fig.2(b). And Fig.2 The IMF1 and IMF3, whose sensitivity coefficient is greater than 1, are selected as sensitive IMFs. Utilize the full vector envelope technique to fuse the sensitive IMFs are selected, and obtain full spectrum of the sensitive IMFs. As shown in Fig. 5.
Based on the full spectrum of the sensitive IMFs in Fig.5, this method proposed in the paper can fuse the orthogonal signals in two directions.
And it can accurately detect the new frequency appearance at a frequency of 5f 0 and the change in amplitude at the typical frequency of 50Hz.
Obviously, the method has strong compatibility and effectiveness. In order to illustrate the advantages of BEMD, the real and imaginary parts of the fault signal are separately decomposed by EMD, the results are shown in Fig.9

conclusions
This paper proposed a method to extract a feature from orthogonal signals in two directions sensors for the condition monitoring of rotating machinery based on bivariate empirical mode decomposition (BEMD) and full spectrum.
BEMD is employed to decompose signals from two orthogonal sources together, thus a complicated rotation can be represented by a set of simpler rotation components. BEMD is proved to outperform standard EMD for two orthogonal signals, because the intrinsic mode functions