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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

#### 2.1. Singular Negentropy Difference Spectrum

#### 2.1.1. SVD

**A**is constructed as follows [19]:

**A**∈

**R**

^{m}

^{×n}which is the reconstructed attractor orbital matrix, and then perform singular value decomposition on it.

**A**, then the following equation can be obtained:

**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.

_{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:

#### 2.1.2. Singular Negentropy Difference Spectrum

_{1}, y

_{2}, …, y

_{n}) denotes a random variable, p

_{i}is the probability of y

_{i}.

_{y}(i) means the negentropy of the first i-order reconstructed signal, i denotes the number of singular values.

_{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.

#### 2.2. Integrated Fast Spectral Correlation

#### 2.2.1. Fast Spectral Correlation

_{n}), its spectral correlation is given by [24]:

_{s}denotes the sampling frequency, t

_{n}means the sampling time, t

_{n}= n /F

_{s}, ${R}_{x}({t}_{n},\tau )$ denotes the cyclic autocorrelation function of x(t

_{n}), $\tau $ denotes time delay, α means the cyclic frequency, f means frequency.

_{n}) is as follows:

_{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}= kΔf (Δf is the frequency resolution and it is expressed as Δf = F

_{s}/N

_{w}), L is the length of signal x(t

_{n}).

_{k}= kΔf and α = pΔf + δ, then f-α = f

_{k}-α ≈ f

_{k}

_{-p}, hence α ≈ pΔf. Substitute these results into Equation (8)

_{k}-Δf/2, f

_{k}+Δf/2] is flowed by the energy. Otherwise, the energy will flow between bands [f

_{k}-Δf/2, f

_{k}+Δf/2] and [f

_{k}

_{-p}-Δf/2, f

_{k}

_{-p}+Δf/2].

#### 2.2.2. Integrated Fast Spectral Correlation

**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:

## 3. The Proposed Method

## 4. Simulation Analysis

_{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.

_{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 Figure 6 and Figure 4a, the periodic impulse components are more obvious in the time domain waveform after SVD reconstruction.

_{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.

_{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 Figure 9 and Figure 10, the separation effect of the proposed method is better than that of the integrated fast spectral correlation based on singular difference spectrum.

## 5. Experimental Analysis

#### 5.1. Introduction of Experiment

_{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.

#### 5.2. Compound Fault Diagnosis

_{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 Figure 26 and Figure 27, the separation effect of this method is better than that of the integrated fast spectral correlation method based on singular difference spectrum.

## 6. Conclusions

- (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

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Some application examples of rotating machinery [1].

**Figure 9.**Outer ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 10.**Inner ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 15.**Outer ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 16.**Inner ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 19.**The results of separation: (

**a**) the envelope spectrum of band 1; (

**b**) the envelope spectrum of band 2.

**Figure 20.**(

**a**) Experiment stand; (

**b**) sensors distribution and (

**c**) rolling bearing with inner ring fault and outer ring fault.

**Figure 21.**(

**a**) Time domain waveform of the original signal; (

**b**) the envelope spectrum of the original signal.

**Figure 26.**Outer ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 27.**Inner ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 32.**Outer ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 33.**Inner ring fault after separation: (

**a**) the fast spectral correlation spectrum; (

**b**) the enhanced envelope spectrum.

**Figure 36.**The results of separation: (

**a**) the envelope spectrum of band 1; (

**b**) the envelope spectrum of band 2.

Type | Diameter of Balls, d (mm) | Pith Diameter, D (mm) | Number of Balls, z | Contact Angle, α (°) | Damage Size of Inner Ring, (mm) | Damage Size of Outer Ring, (mm) |
---|---|---|---|---|---|---|

SKF6205 | 7.5 | 38.5 | 9 | 0 | 0.008 | 0.059 |

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**MDPI and ACS Style**

Tang, G.; Tian, T.
Compound Fault Diagnosis of Rolling Bearing Based on Singular Negentropy Difference Spectrum and Integrated Fast Spectral Correlation. *Entropy* **2020**, *22*, 367.
https://doi.org/10.3390/e22030367

**AMA Style**

Tang G, Tian T.
Compound Fault Diagnosis of Rolling Bearing Based on Singular Negentropy Difference Spectrum and Integrated Fast Spectral Correlation. *Entropy*. 2020; 22(3):367.
https://doi.org/10.3390/e22030367

**Chicago/Turabian Style**

Tang, Guiji, and Tian Tian.
2020. "Compound Fault Diagnosis of Rolling Bearing Based on Singular Negentropy Difference Spectrum and Integrated Fast Spectral Correlation" *Entropy* 22, no. 3: 367.
https://doi.org/10.3390/e22030367