Compound Fault Diagnosis of Rolling Bearing Based on Singular Negentropy Difference Spectrum and Integrated Fast Spectral Correlation

Compound fault diagnosis is challenging due to the complexity, diversity and non-stationary characteristics of mechanical complex faults. In this paper, a novel compound fault separation method based on singular negentropy difference spectrum (SNDS) and integrated fast spectral correlation (IFSC) is proposed. Firstly, the original signal was de-noised by SNDS which improved the noise reduction effect of singular difference spectrum by introducing negative entropy. Secondly, the de-noised signal was analyzed by fast spectral correlation. Finally, IFSC took the fourth-order energy as the index to determine the resonance band and separate the fault features of different single fault. The proposed method is applied to analyze the simulated compound signals and the experimental vibration signals, the results show that the proposed method has excellent performance in the separation of rolling bearing composite faults.


Introduction
Rotating machinery is widely used in modern industries, such as helicopters, airplanes, machining centers, track loaders, mining tracks and wind turbines [1], as shown in Figure 1. Rolling bearing is one of the most common components in rotating machines. Whether the rolling bearing runs normally is directly related to the running state of the whole rotating machinery [2,3]. Due to the bad working environment and long-term working conditions, several key parts of the rolling bearing are easy to be damaged at the same time, resulting in composite faults [4]. Composite faults are more harmful to machinery than the single faults, so it is important to diagnose early complex faults. However, all kinds of fault features are closely coupled and interfere with each other, which makes the separation of rolling bearing composite faults more challenging [5].
x(N − n + 1) x(N − n + 2) · · · x(N) where N is the length of the signal X, 1 < n < N. Let m = N-n+1, A∈R m×n which is the reconstructed attractor orbital matrix, and then perform singular value decomposition on it. Perform singular value decomposition (SVD) on the matrix A, then the following equation can be obtained: where U = [u 1 , u 2 , . . . , u m ]∈R m×m , V = [v 1 , v 2 , . . . , v n ]∈R n×n and i = 1, 2, . . . , k, k = min(m, n). S = diag(λ 1 , λ 2 ,..., λ k ) is a diagonal matrix arranged in descending order, and its diagonal element is the singular value of matrix A. A i is the submatrix corresponding to singular value λ i obtained via SVD decomposition. The submatrix A i is inversely transformed to obtain the component signal P i . The result of linear superposition of k component signals obtained in this way is the original discrete signal X, that is:

Singular Negentropy Difference Spectrum
Shannon Entropy is an effective index to quantitatively evaluate the uncertainties of signal or system state [25]. When the impact component appears, the Shannon entropy becomes smaller. The negentropy was defined, which is the negative value of the Shannon entropy, in order to keep the two changing regularities.
The definition of negentropy is as follows: where y = (y 1 , y 2 , . . . , y n ) denotes a random variable, p i is the probability of y i . In order to effectively preserve useful fault features and minimize the impact of noise, the singular negentropy difference spectrum (SNDS) based on SVD was introduced in this paper. The definition of SNDS is as follows: where E y (i) means the negentropy of the first i-order reconstructed signal, i denotes the number of singular values. The schematic diagram of SNDS is shown in Figure 2. When the negentropy of two reconstructed signals are quite different, a significant peak will appear in the difference spectrum. There will inevitably be a maximum peak in the whole difference spectrum, that is, the position SNDS q of the maximum mutation of the peak degree. The maximum break point SNDS q not only shows that there are abundant fault impact characteristics in the reconstructed signal, but also shows the boundary between the useful signal and the noise component. Thus, the first q-order reconstructed signal is selected.

Fast Spectral Correlation
For a non-stationary random signal x(t n ), its spectral correlation is given by [24]: where F s denotes the sampling frequency, t n means the sampling time, t n = n /F s , R x (t n , τ) denotes the cyclic autocorrelation function of x(t n ), τ denotes time delay, α means the cyclic frequency, f means frequency. Fast spectral correlation (FSC) is an improved version of spectral correlation, which shorten the calculation time and improve the efficiency by short-time Fourier transform (STFT) [24]. STFT of signal x(t n ) is as follows: The phase-corrected STFT might be presented as follows: where N w is the window length of STFT, R denotes the block shift, w[n] means the function of time index n, f k denotes the k-th discrete frequency and f k = k∆f (∆f is the frequency resolution and it is expressed as ∆f = F s /N w ), L is the length of signal x(t n ).
Assume f = f k = k∆f and α = p∆f + δ, then f -α = f k -α ≈ f k-p , hence α ≈ p∆f. Substitute these results into Equation (8) Substituting Equation (9) into Equation (6), the following equation is described as: If p = 0, the band [f k -∆f /2, f k +∆f /2] is flowed by the energy. Otherwise, the energy will flow between bands Therefore, the expression of fast spectral correlation is given by [24]: where R w (α) = Fs represents the kernel function, and when α=0 the expression of kernel function is as R w (0) = w 2 .

Integrated Fast Spectral Correlation
In order to effectively separate the fault features, the integrated fast spectral correlation (IFSC) theory was proposed in this paper. The spectral correlation matrix S obtained by fast spectral correlation is an I×J matrix, whose dimension is expressed as frequencies × cyclic frequencies. After the spectral correlation matrix is obtained, the fourth-order energy at all the cyclic frequencies is added together, and the resonance band can be identified by observing the distribution of the accumulated energy along the frequency axis. In general, different resonance bands in complex faults represent different characteristics of faults. By observing the resonance band, the frequency range [f 1 , f 2 ] of the resonance band is determined, and the spectral correlation matrix is integrated to obtain the integrated fast spectral correlation results. The expression of fourth-order energy is given by:

The Proposed Method
Since the actual collected rolling bearing signal is usually covered by many noise components which have an interference effect on the extraction of fault features. In order to effectively improve signal-to-noise ratio and separate composite faults, the following steps are taken: Step 1: Collect the composite fault signal.
Step 2: Singular negentropy difference spectrum is applied to separate the trend component which contains the fault feature information from the interference component.
Step 3: Fast spectral correlation is utilized to obtain the fast spectral correlation spectrum, the fourth-order energy of whose frequency axis is added together. As the result, the resonance band can be identified.
Step 4: Fast spectral correlation is used to separate different faults and obtain the enhanced envelope spectrum of each fault frequency band.
The flow chart of the method presented in this paper is shown in Figure 3.

Simulation Analysis
In order to verify the effectiveness of the proposed method, a simulated composite fault signal of inner and outer ring is performed.
where x 1 (t) is the simulation signal of rolling bearing with inner ring fault whose inner ring fault characteristic frequency is expressed as f i = 130 Hz and natural frequency f 1 is expressed as f 1 = 3000 Hz. x 2 (t) is the simulation signal of rolling bearing with outer ring fault whose outer ring fault characteristic frequency is expressed as f o = 90 Hz and natural frequency f 2 is expressed as f 2 = 1000 Hz. n(t) is white noise. In the simulation, the sampling frequency of the example signal is f s = 8192 Hz and the sampling number is N = 4096. Figure 4a shows the time domain waveform of the simulation signal. It can be seen from Figure 4a that the simulation signals x 1 (t), x 2 (t) and x 3 (t) all contain periodic impulse components, while the periodic impulse components in x(t) have been submerged by noise. Figure 4b shows that the envelope spectrum of x(t) has no prominent frequency components. The Hankel matrix is constructed from the original signal and processed by SVD. The negentropy of first 50 points and singular negentropy difference spectrum are shown in Figure 5. It can be found that the 42th point is the maximum mutation points of the difference spectrum, retaining the first 42 singular values obtained by SVD processing, and the other singular values are all set to 0. The reconstruction signal shown in Figure 6 is obtained by singular value reconstruction. Compared with Figures 6 and 4a, the periodic impulse components are more obvious in the time domain waveform after SVD reconstruction.   Then, the fast spectral correlation is performed on the reconstructed signal, and the fast spectral correlation spectrum shown in Figure 7 is obtained. Two resonance bands are found in Figure 7, among which the red box area means the resonance band of the inner ring fault and the green box area is the resonance band of the outer ring fault. Next, the fourth-order energy of the ordinate in Figure 7 is carried out to obtain the resonance band screening diagram shown in Figure 8. According to Figure 8, the two resonance bands are 512-1408 Hz and 2304-3712 Hz, respectively. Fast spectral correlation is conducted for the frequency ranges of the two resonance bands respectively, the outer and inner ring faults after separation are finally obtained as shown in Figures 9 and 10. Figure 9 describes the fast spectral correlation and enhanced envelope spectrum of outer ring fault, from which f o , 2f o , 3f o , 4f o , 5f o , 6f o and 7f o can be recognized obviously. Figure 10 describes the fast spectral correlation and enhanced envelope spectrum of inner ring fault. f i and its 2X, 3X, 4X, 5X frequency doubling can be clearly identified from Figure 10.    In order to show the superiority of the proposed method, the integrated fast spectral correlation based on singular difference spectrum and spectral kurtosis based on wavelet [26] are used to analyze the simulation signal. Figure 11 shows the singular difference spectrum of the simulation signal. As shown in Figure 11, i = 44 is the mutation point. The singular values after i= 44 are set to 0, and the time domain diagram of the reconstructed signal is shown in Figure 12. While the periodic impulse in the reconstructed signal is more obvious than that in the simulation signal, it still contains some noise components. The fast spectral correlation spectrum is shown in Figure 13. Compared with Figure 7, the resonance bands of outer and inner ring faults cannot be clearly identified.    Figure 14 shows the separation of resonance bands, from which, the inner fault resonance band can be accurately identified, while the outer fault resonance band cannot be clearly identified. Figure 15 shows the fast spectral correlation and enhanced envelope spectrum of outer ring fault. From Figure 15a,b, only f o , 2f o , 3f o , 6f o and 7f o can be recognized. It can be also found from Figure 15b that there are many interference components. The fast spectral correlation and enhanced envelope spectrum of inner ring fault is shown in Figure 16. f i and its 2X, 3X, 4X, 5X frequency doubling can be clearly identified from Figure 16. Compared with Figures 9 and 10, the separation effect of the proposed method is better than that of the integrated fast spectral correlation based on singular difference spectrum.   As shown in Figure 17, periodic impact components cannot be noticed in the time-domain waveform after wavelet denoising. Kurtogram of the denoised signal is shown in Figure 18, and two frequency bands was selected according to it. It can be seen from Figure 19a,b, the envelope spectrums contain the characteristic frequency of inner ring and outer ring, but the composite fault is not separated into single ones. The results show that the proposed method has a better separation performance than the method combing wavelet transform and spectral kurtosis.

Introduction of Experiment
In order to prove the effectiveness of the proposed method in practical application, QPZZ model test-bed is applied to simulate the composite fault of inner and outer rings of rolling bearings. Figure 20a, b and c show the experimental devices, sensor distribution and composite faults of inner and outer rings, respectively. The test bench is mainly composed of a motor, a shaft support, coupling, bearings, block and accelerometers. The geometric parameters of rolling bearings are shown in Table 1. The speed of motor is 1470 r/min during the experiment. The sampling frequency f s is 12,800 Hz, the sampling point N is 6400 and the rotating frequency f r is 24.5 Hz. The characteristic frequency of inner ring fault f i is 132 Hz and the outer ring fault characteristic frequency f o is 88 Hz.   Figure 21a,b are the time domain waveform and envelope spectrogram of the original signal, respectively. From the time-domain waveform of the original signal, we can see that it contains many noise components, which cannot be identified from the envelope spectrum. What is more, the fault type cannot be determined from Figure 21b. Figure 22 shows the negentropy and singular negentropy difference spectrum of the signal reconstructed from the first i-th singular values after SVD decomposition. From Figure 22, we can see that i = 47 is the mutation position of negentropy value. Hence, the negentropy values after i= 47 are set to 0. The signal is reconstructed, then the time domain waveform of Figure 23 is obtained.   The fast spectral correlation analysis of the reconstructed signal is carried out, and the fast spectral correlation spectrum of Figure 24 is obtained. From Figure 24, two resonance bands are found, among which the green rectangular frame denotes the resonance band of the outer ring fault and the red rectangular frame means the resonance band of the inner ring fault. By calculating the fourth-order energy of the frequency of Figure 24, the resonance bands of the outer and inner ring faults can be separated. It can be seen from Figure      The result of integrated fast spectral correlation method based on singular difference spectrum is compared to show the superiority of SNDS. Figure 28 shows the singular difference spectrum of the reconstructed signals which is composed of the first i-th singular values. As shown in Figure 28, i=44 is the mutation point. The singular values after i = 44 are set to 0, and the time domain diagram of reconstructed signal is shown in Figure 29. Although the periodic pulse in reconstructed signal is more obvious than that in simulated signal, it still contains some noise components. The fast spectral correlation spectrum is shown in Figure 30. Compared with Figure 24, the resonance bands of outer and inner ring faults cannot be clearly identified. Figure 31 shows the separation of the resonance band. It can be seen from Figure 31 that the internal fault resonance band can be accurately identified, while the external fault resonance band cannot be clearly identified. Figure 32 shows the fast spectral correlation and enhanced envelope spectrum of the outer ring fault. From Figure 32a,b, we can only identify f o , 2f o , 3f o and 4f o . Figure 33 describes the fast spectral correlation and enhanced envelope spectra of inner ring faults. f i and its 2X, 3X, 4X and 5X multiples can be clearly identified in Figure 33. Compared with Figures 26 and 27, the separation effect of this method is better than that of the integrated fast spectral correlation method based on singular difference spectrum.      Similar to simulation analysis, the method based on wavelet transform and spectral kurtosis was performed to compare with the proposed method. It can be seen from Figure 34 that the period impact components are more obvious after wavelet denoising. The two frequency bands with the strongest impact were selected from Kurtogram shown in Figure 35. The fault with weak impact is easy to be submerged by the fault with strong impact. Hence, as shown in Figure 36, only the outer ring fault is separated. The frequency of inner ring fault cannot be identified from both Figure 36a

Conclusions
In this paper, a compound fault feature separation method based on singular negentropy difference spectrum and integrated fast spectral correlation was proposed. Singular negentropy difference spectrum was applied to de-noise and then integrated fast spectral correlation was performed. The results show that the composite fault signal can be separated effectively and accurately. The following conclusions can be drawn: (1) Singular negentropy difference spectrum (SNDS) can adaptively determine the effective singular value, effectively remove the noise components and retain useful fault information. What is more, the comparison between SNDS and the singular difference spectrum shows that SNDS has better denoising performance. (2) The fourth-order energy was used as the index by integrated fast spectrum correlation (IFSC) to select different resonance bands, so as to realize the separation of different faults. The method combining wavelet transform and spectral kurtosis was used to compare with the proposed method in this paper, the results show that the proposed method can separate the composite faults better. (3) Limited to the experimental conditions, the composite fault diagnosis of rolling bearing is only discussed in this paper. There are many other composite faults of rotating machines, such as compound fault of gear, compound fault of gear and rolling bearing. In the future, we will continue to study the difficult problem of fault diagnosis of other fault modes.
Author Contributions: T.T. conceived the method, performed the experiment and completed the paper writing; G.T. provided guidance. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.