# Rolling Bearing Fault Diagnosis Based on Optimal Notch Filter and Enhanced Singular Value Decomposition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory Background

#### 2.1. Optimal Notch Filter Based on Teager Energy Entropy Index

#### 2.1.1. Notch Filter

_{0}, the frequency- amplitude curve of the filter should be zero at ω

_{0}and almost be constant at the other frequencies [26].

_{0}. In order to increase the amplitude to a constant instantly once the frequencies are not ±ω

_{0}, two extreme points are set near the zero points. The extreme point is described as: $z=r{\mathrm{e}}^{\pm \mathrm{j}{\omega}_{0}}$, and the transfer function of the notch filter is shown in Equation (1):

_{0 }denotes the gain factor. The frequency band of notch filter at −3 dB is defined as the bandwidth (Bw). The bandwidth Bw and the gain coefficient b

_{0}can be expressed as:

_{s}represents the sampling frequency. When normalized frequency is used, the sampling frequency f

_{s}is 1 and the bandwidth Bw equals to 1 − r. The frequency-amplitude curve of notch filter varies with Bw.

_{0}varies severely when Bw changes. Once Bw decreases, the depth of notch will decrease as well. The frequency-amplitude curve of the notch filter demonstrates that the selection of Bw has a remarkable influence on the analysis results of notch filter. A proper value of Bw should be selected to achieve the best result when applying the notch filter to the vibration signals of bearings.

#### 2.1.2. Teager Energy Entropy Index

_{1}(t) is used to simulate the periodic impulse signal of the outer race fault, which is shown in Equation (6) [35]:

_{o}is the interval of the impulsive pulses, ξ is the structural damping factor, f

_{n }represents the natural frequency of the system, ${f}_{d}={f}_{n}\sqrt{1-{\xi}^{2}}$ is the damping natural frequency, A and θ represent the amplitude and initial phase respectively.

_{o }= 0.01 s, A = 2, ξ = 0.06, f

_{n }= 2000 Hz and θ = 0.

_{2}(t), which is used to simulate the fundamental frequency signal:

_{1}= 15 Hz is the fundamental frequency.

_{3}(t) is used to simulate the noise, and its value can be generated in MATLAB using A*randn(1,N), where A = 1.5 and N = 2048.

_{4}(t) can be described as: x

_{4}(t) = x

_{1}(t) + x

_{3}(t). The fifth simulated signal x

_{5}(t) was obtained by adding a single impulse to x

_{3}(t).

_{1}(t) is 14.4072, which is the highest and the TEE value of x

_{1}(t) is 5.9126, which is the smallest. x

_{2}(t) and x

_{3}(t) have small kurtosis values and high TEE values. Both the kurtosis and TEE indexes measure the impulsive characteristics of x

_{1}(t), x

_{2}(t) and x

_{3}(t) effectively. Whereas, the kurtosis value of x

_{5}(t) is higher than that of x

_{4}(t), which demonstrates that the accidental pulse results in an erroneous evaluation when using the kurtosis index. The TEE value of x

_{5}(t) is higher than that of x

_{4}(t), which demonstrates that the TEE index can be also effective when an accidental pulse appears. And it should be noticed that a smaller TEE value means that the periodic pulse characteristics of the signal are more prominent.

#### 2.1.3. Optimal Bandwidth Selection Based on Teager Energy Entropy Index

- (1)
- Measure the vibration signal of the defective bearing.
- (2)
- Set the fundamental frequency as the center frequency of the notch filter and perform the notch filter analysis with varying Bws (Bw = [0.01f
_{s}, 0.99f_{s}], the step length is 0.01f_{s}) to achieve a series of notch filter signals. - (3)
- Calculate the TEE value of each notch filter signal, determine the optimal bandwidth with the smallest TEE value and select the corresponding notch filter signal as the optimal notch filter signal.

_{r}is the fundamental frequency, A(f

_{r}), A(2f

_{r}), A(3f

_{r}) represent the amplitude of f

_{r}, 2f

_{r}, 3f

_{r}respectively in the envelope spectrum of the original signal, and W represents the total energy of the local envelope spectrum in the frequency band [0, 3f

_{r}].

#### 2.2. Enhanced Singular Value Decomposition

**U**ϵ

**R**

^{m}

^{×m}and

**V**ϵ

**R**

^{n}

^{×n}are orthogonal matrices,

**D**ϵ

**R**

^{m}

^{×n}is the obtained diagonal matrix. The expressions of the three matrices are as follows:

_{i}(i = 1, 2, …, q) denotes the singular values of the matrix

**H**, q is the number of the nonzero singular values. With a comprehensive consideration of Equations (10)–(13),

**H**can be described as Equation (14):

**H**

_{i}= σ

_{i}

**u**

_{i}

**v**

_{i}

^{T}(i = 1, 2, …, q), denotes the corresponding matrix component of the singular value σ

_{i}. Therefore, the matrix

**H**can be expressed as:

**H**, it is necessary to select the effective singular values and corresponding matrix components to reconstruct the matrix. A series of criterions have been proposed by scholars to select the proper singular components [38], and the difference spectrum of singular values (DSSV) [37] is the most widely used one. The number of the effective singular values can be determined by observing the difference spectrum sequence. The DSSV is achieved by backward reduction of singular values:

_{i}sets {b

_{1}, b

_{2}, …, b

_{q}

_{−1}}, is called the DSSV. As pointed out in [38], the maximum peak point b

_{k }is the cut-off point of effective and useless singular values. Then the matrix

**H**is reconstructed using the former k singular value components:

_{i}. If RAR(j) is smaller than a threshold, for j > m, j = m + 1, m + 2, …, q − 1, the last maximum peak of DSSV can be identified as b

_{m}. When the threshold is set to a smaller value, more singular values are selected for reconstruction, that means more information and more noise of the raw signal will be retained. On the contrary, when the threshold is set to a greater value, the reconstructed signal will contains less noise and some useful information may be also lost. The threshold was set to 10% in this paper based on many tests.

#### 2.3. The Presented Method

## 3. Simulated Analysis

_{1}(t) represents the simulated outer race (OR) fault signal and the OR characteristic frequency f

_{o}= 1/T

_{o}= 100 Hz. x

_{2}(t) and x

_{3}(t) are used for simulating the fundamental frequency signal and noise, respectively. The simulated OR fault signal, fundamental frequency signal and noise component were respectively shown in Figure 2a–c in above analysis. A sampling frequency of 8192 Hz was used and the signal length of S(t) is 2048.

_{2}(t), that is f

_{1}, could be found from the envelope spectrum. Then, the simulated signal was processed by notch filters with f

_{1 }as the center frequency, and with varying Bws (from 0.01f

_{s}to 0.99f

_{s}and the step length was 0.01f

_{s}). The TEE values of the notch filter signals under different Bws were exhibited in Figure 7a. From Figure 7a, it can be found that the TEE value is minimum when Bw = 0.43f

_{s}. So the optimal Bw is 0.43f

_{s}and the corresponding optimal notch filter signal was depicted in Figure 7b. The optimal notch filter signal reflects the same impulsive feature with x

_{1}(t). However, the interferences are very obvious. Figure 7c shows the envelope spectrum of the optimal notch filter signal, some apparent peaks can be visible at the frequencies of f

_{o}, 2f

_{o}, 3f

_{o}, 4f

_{o}, 5f

_{o}and 6f

_{o}, but the noise interferences are also visible.

_{o}and some amplitudes of noise interferences can be found from Figure 9b. Figure 9c shows the ESVD de-noised signal of Figure 9a,d displays its envelope spectrum. Some peaks can be found at the frequencies of f

_{o}and its harmonics, but less harmonics can be detected from Figure 9d compared with Figure 8d. Figure 10a displays the Kurtogram of S(t), the optimal narrowband can be identified at level 4 with center frequency of 2048 Hz. A designed band-pass filter based on the information of the optimal narrowband was conducted on the raw vibration signal and the filtered signal is plotted in Figure 10b. The filtered signal is also contaminated by noise. Figure 10c shows envelope spectrum of Figure 10b. From Figure 10c, the first five harmonics of f

_{o}and some amplitudes of noise interference can be visible. Figure 10d depicts the ESVD de-noised signal of Figure 10b. Figure 10e shows the envelope spectrum of Figure 10d, which displays less harmonics of f

_{o}compared with Figure 8d. Figure 11 reflects the analysis results obtained using the Infogram method. We can find that the proposed method shows a better performance than the Infogram method in enhancing the impulsive features and extracting more harmonics of f

_{o}.

## 4. Experimental Analysis

#### 4.1. Experiment 1

_{r}) was kept at 24 Hz. The vibration signals were collected by eddy current sensors as Figure 12 reflects. The sampling frequency is 12,800 Hz and the number of sampling points is 6400. The theoretical characteristic frequency of IR fault (f

_{i}) is 172 Hz.

_{s}based on the results of Figure 15a. Figure 15b shows the corresponding optimal notch signal. Prominent impact features can be detected in Figure 15b. However, the noise is also visible. From the envelope spectrum of optimal notch filter signal shown in Figure 15c, there are two peak points at the frequencies of 172 Hz and 344 Hz, corresponding to f

_{i}and 2f

_{i}, respectively. Figure 15d reflects the signal obtained by performing ESVD de-noising on the optimal notch filter signal. Compared with the optimal notch filter signal, Figure 15d presents more obvious and cleaner impact features. Figure 15e illustrates the envelope spectrum of Figure 15d. Obvious spectral lines can be detected at the frequencies of f

_{i}, 2f

_{i}, 3f

_{i}, 4f

_{i}and 5f

_{i}. Moreover, the sidebands with the interval of f

_{r}are also very clear. After the ESVD de-nosing analysis, the noise interferences of the optimal notch signal was suppressed and more harmonic frequencies of can be detected.

_{i}and 2f

_{i}can be detected from Figure 16b, some peaks that reflect the noise can also be visible. Figure 16c displays the signal obtained by performing ESVD de-noising on the MED filtered signal and Figure 16d shows its envelope spectrum. Only the first three harmonics of f

_{i}can be detected from Figure 16d. Figure 17a,b respectively illustrate the Kurtogram and the filtered signal obtained based on Figure 17a. Figure 17c shows the envelope spectrum of Figure 17b, which exhibits the first four harmonic frequencies of f

_{i}. Figure 17d displays the ESVD de-noised signal of Figure 17b,e displays its envelope spectrum. The first five harmonics of f

_{i}can be identified in Figure 17e. But the fourth and fifth harmonics of f

_{i}and their sidebands shown in Figure 17e are less prominent than that shown in Figure 15d obtained using the proposed method. Figure 18 shows the process results obtained via the Infogram method. By comparing Figure 18 and Figure 15, we can find that the Infogram method is not as effective as the proposed method. The comparison results demonstrate the advantages of the proposed method.

#### 4.2. Experiment 2

#### 4.2.1. Case 1: Detection of Rolling Element Defect

_{b}) is about 114.97 Hz.

_{r}) and its harmonics can be detected in the envelope spectrum. It is suggested that the fundament signal plays a role as an interference signal. Then the raw vibration data was analyzed by the proposed method and the analysis results are shown in Figure 21. As Figure 21a reflects, the TEE index has the minimum value when Bw = 0.85f

_{s}, so we set the optimal bandwidth as 0.85f

_{s}. Figure 21b shows the waveform of the optimal notch filter signal corresponding to the optimal bandwidth, and its envelope spectrum is plotted in Figure 21c. From Figure 21c, the fault characteristic frequency of rolling element f

_{b}can be visible, but we can find some interference peaks which make it hard to identify f

_{b}. Figure 21d displays the signal obtained by performing the ESVD de-noising on the optimal notch filter signal. From Figure 21d, obvious shock features can be visible. And from the envelope spectrum of the ESVD de-noised signal, the spectral lines corresponding to the frequencies of f

_{b}, 2f

_{b}, 3f

_{b}, 4f

_{b}and 5f

_{b }are apparent. Moreover, no interferences can be visible from Figure 21e.

#### 4.2.2. Case 2: Detection of Outer Race Defect

_{o}) as 90.17 Hz.

_{s}based on Figure 26a. Figure 26b displays the optimal notch filter signal corresponding to the optimal bandwidth and Figure 26c shows its envelope spectrum. Two characteristic frequencies, f

_{o}and 2f

_{o}, can be detected from Figure 26c, whereas the noise interferences are very obvious. Figure 26d reflects the de-nosed signal by performing ESVD on the optimal notch filter signal and Figure 26e shows its envelope spectrum. Some prominent peaks corresponding to the frequencies of f

_{o}, 2f

_{o}, 3f

_{o}, 4f

_{o}, 5f

_{o}and 6f

_{o}can be visible from Figure 26e. The proposed method detects the OR fault effectively.

## 5. Conclusions

- (1)
- To adaptively determine the optimal bandwidth of the notch filter and implement the optimal notch filter analysis, a new indicator for evaluating the periodic impulsive features named Teager energy entropy index was presented. The Teager energy entropy index performs better in overcoming the accidental shocks than the kurtosis index.
- (2)
- The optimal notch filter analysis based on Teager energy entropy index (with the fundamental frequency as the center frequency) shows its ability in inhibiting the interference of the fundamental frequency signal and enhancing the shock feature signal.
- (3)
- An enhanced singular value decomposition de-noising method was proposed to improve the noise reduction of singular value decomposition.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The Five simulation signals: (

**a**) x

_{1}(t); (

**b**) x

_{2}(t); (

**c**) x

_{3}(t); (

**d**) x

_{4}(t); (

**e**) x

_{5}(t).

**Figure 3.**The evaluation results of the simulated signals with signal-to-noise ratios (SNRs) via Teager energy entropy (TEE) index: (

**a**) 5 dB; (

**b**) 10 dB; (

**c**) 15 dB; (

**d**) 20 dB; (

**e**) the cure of the TEE index with different SNRs.

**Figure 7.**The analysis results of S(t) using the optimal notch filter analysis: (

**a**) the curve of the TEE index under different bandwidths; (

**b**) the optimal notch filter signal; (

**c**) the envelope spectrum of (

**b**).

**Figure 8.**The ESVD de-noising results of the optimal notch signal: (

**a**) the DSSV; (

**b**) the SVD de-noised signal; (

**c**) the ESVD de-noised signal; (

**d**) the envelope spectrum of (

**c**).

**Figure 9.**The analysis results of S(t) using MED: (

**a**) the filtered signal; (

**b**) its envelope spectrum; (

**c**) the ESVD de-noised signal of (

**a**); (

**d**) the envelope spectrum of (

**c**).

**Figure 10.**The analysis results of S(t) using Kurtogram: (

**a**) the Kurtogram; (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 11.**The analysis results of S(t) using Infogram: (

**a**) the average infogram ∆

**I**

_{1/2}(

**f**; ∆

**f**); (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 15.**The analysis results of inner race fault signal using the proposed method: (

**a**) the curve of the TEE index under different bandwidths; (

**b**) the optimal notch filter signal; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 16.**The analysis results of inner race fault signal using MED: (

**a**) the filtered signal; (

**b**) the envelope spectrum of (

**a**); (

**c**) the ESVD de-noised signal of (

**a**); (

**d**) the envelope spectrum of (

**c**).

**Figure 17.**The analysis results of inner race fault signal using Kurtogram: (

**a**) the Kurtogram; (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 18.**The analysis results of inner race fault signal using Infogram: (

**a**) the average infogram ∆

**I**

_{1/2}(

**f**; ∆

**f**); (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 21.**The analysis results of rolling element fault signal using the proposed method: (

**a**) the curve of the TEE index under different bandwidths; (

**b**) the optimal notch filter signal; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 22.**The analysis results of rolling element fault signal using MED: (

**a**) the filtered signal; (

**b**) the envelope spectrum of (

**a**); (

**c**) the ESVD de-noised signal of (

**a**); (

**d**) the envelope spectrum of (

**c**).

**Figure 23.**The analysis results of rolling element fault signal using Kurtogram: (

**a**) the Kurtogram; (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 24.**The analysis results of rolling element fault signal using Infogram: (

**a**) the average infogram ∆

**I**

_{1/2}(

**f**; ∆

**f**); (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 26.**The analysis results of outer race fault signal using the proposed method: (

**a**) the curve of the TEE index under different bandwidths; (

**b**) the optimal notch filter signal; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 27.**The analysis results of outer race fault signal using MED: (

**a**) the filtered signal; (

**b**) the envelope spectrum of (

**a**); (

**c**) the ESVD de-noised signal of (

**a**); (

**d**) the envelope spectrum of (

**c**).

**Figure 28.**The analysis results of outer race fault signal using Kurtogarm: (

**a**) the Kurtogram; (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

**Figure 29.**The analysis results of outer race fault signal using Infogram: (

**a**) the average infogram ∆

**I**

_{1/2}(

**f**; ∆

**f**); (

**b**) the filtered signal of the optimal narrowband; (

**c**) the envelope spectrum of (

**b**); (

**d**) the ESVD de-noised signal of (

**b**); (

**e**) the envelope spectrum of (

**d**).

Roller Diameter (mm) | Pith Diameter (mm) | Number of Rollers | Contact Angle (°) |
---|---|---|---|

7.5 | 38.5 | 12 | 0° |

Ball Diameter (mm) | Pith Diameter (mm) | Number of Balls | Contact Angle (°) |
---|---|---|---|

6.75 | 28.5 | 8 | 0° |

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## Share and Cite

**MDPI and ACS Style**

Pang, B.; He, Y.; Tang, G.; Zhou, C.; Tian, T.
Rolling Bearing Fault Diagnosis Based on Optimal Notch Filter and Enhanced Singular Value Decomposition. *Entropy* **2018**, *20*, 482.
https://doi.org/10.3390/e20070482

**AMA Style**

Pang B, He Y, Tang G, Zhou C, Tian T.
Rolling Bearing Fault Diagnosis Based on Optimal Notch Filter and Enhanced Singular Value Decomposition. *Entropy*. 2018; 20(7):482.
https://doi.org/10.3390/e20070482

**Chicago/Turabian Style**

Pang, Bin, Yuling He, Guiji Tang, Chong Zhou, and Tian Tian.
2018. "Rolling Bearing Fault Diagnosis Based on Optimal Notch Filter and Enhanced Singular Value Decomposition" *Entropy* 20, no. 7: 482.
https://doi.org/10.3390/e20070482