# Rolling Bearing Fault Diagnosis Based on EWT Sub-Modal Hypothesis Test and Ambiguity Correlation Classification

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## Abstract

**:**

## 1. Introduction

## 2. Denoising Method Based on EWT and Hypothesis Test

#### 2.1. Basic Principles of EWT

#### 2.2. Filtering Method Based on EWT Sub-Modal Hypothesis Test

#### 2.3. Simulation Experiment

## 3. Ambiguity Correlation Classifier

#### 3.1. Ambiguity Correlation Theory

- (1)
- Calculate correlation function of the ambiguity function of the two signals x(t) and y(t)$${\rho}_{xy}(\tau ,\theta )=\underset{{\tau}_{0},{\theta}_{0}}{\mathrm{max}}\left|{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}{A}_{x}(\tau ,\theta ){A}_{y}(\tau -{\tau}_{0},\theta -{\theta}_{0}){d}_{\tau}d\theta}}\right|$$
- (2)
- Calculate the normalized correlation coefficient using correlation function, with mathematical expression shown as follows:$${\rho}_{xy}(\tau ,\theta )=\frac{\underset{{\tau}_{0},{\theta}_{0}}{\mathrm{max}}\left|{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}{A}_{x}(\tau ,\theta ){A}_{y}(\tau -{\tau}_{0},\theta -{\theta}_{0}){d}_{\tau}d\theta}}\right|}{{[{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{x}^{2}(\tau ,\theta ){d}_{\tau}{d}_{\theta}{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{y}^{2}}}}}(\tau ,\theta ){d}_{\tau}{d}_{\theta}]}^{\frac{1}{2}}}$$
- (3)
- Take the correlation coefficient when τ = 0 or θ = 0$${\rho}_{xy}(0,\theta )=\frac{\underset{{\tau}_{0},{\theta}_{0}}{\mathrm{max}}\left|{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}{A}_{x}(0,\theta ){A}_{y}(0-{\tau}_{0},\theta -{\theta}_{0}){d}_{\tau}d\theta}}\right|}{{[{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{x}^{2}(0,\theta ){d}_{\tau}{d}_{\theta}{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{y}^{2}}}}}(0,\theta ){d}_{\tau}{d}_{\theta}]}^{\frac{1}{2}}}$$$${\rho}_{xy}(\tau ,0)=\frac{\underset{{\tau}_{0},\theta 0}{\mathrm{max}}\left|{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}{A}_{x}(\tau ,0){A}_{y}(\tau -{\tau}_{0},0-{\theta}_{0}){d}_{\tau}d\theta}}\right|}{{[{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{x}^{2}(\tau ,0){d}_{\tau}{d}_{\theta}{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+}{A}_{y}^{2}}}}}(\tau ,0){d}_{\tau}{d}_{\theta}]}^{\frac{1}{2}}}$$
- (4)
- Calculate the ambiguity correlation coefficient$$\overline{\rho}=\sqrt{\frac{{\rho}_{xy}^{2}(0,\theta )+{\rho}_{xy}^{2}(\tau ,0)}{2}}$$

#### 3.2. Basic Principles of Classifiers

## 4. Experimental Research

#### Experimental Data Collection

## 5. Conclusions

- (1)
- Using EWT to decompose the vibration signal, the exact component can be obtained, and the mode aliasing phenomenon can be eliminated compared with EMD decomposition.
- (2)
- The sub-modes of EWT are tested by Gaussian distribution hypothesis to identify Gaussian noise. This method does not have parameter selection and has good adaptability.
- (3)
- Aiming at the shortcomings of traditional BP and SVM classifiers, such as too many parameters and slow convergence speed, the proposed classifier does not need parameter settings, and the calculation is simple. The experimental results show that the classifier can monitor different working conditions of rolling bearings, and the recognition rate is higher than BP and SVM.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**The collected vibration signals: (

**a**) normal signal; (

**b**) outer race fault signal; and (

**c**) inner race fault signal.

Modal | Sub-Modal | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

F1 | 1 | 1 | 1 | 0 | ||||

F2 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | |

F3 | 0 | 0 | 1 | 0 | ||||

F4 | 1 | 0 | 0 | 1 | 0 | 1 | ||

F5 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | |

F6 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 |

F7 | 0 | 0 | 1 | 0 | ||||

F8 | 1 | 0 | 1 | 0 | 1 | |||

F9 | 0 | 1 | 0 | 0 | ||||

F10 | 0 | 0 | 0 | 1 | 0 | |||

F11 | 0 | 0 | 0 | 0 | ||||

F12 | 0 | 1 | 0 | 0 | 1 | 0 |

Original SNR | SNR after Filtering | ||||||
---|---|---|---|---|---|---|---|

Median Filter 6 Order | Median Filter 3 Order | Moving Average Filter 5 Point | Moving Average Filter 2 Point | Wavelet Filter Soft Threshold | Wavelet Filter Hard Threshold | EWT Sub-Modal Hypothesis Test | |

13.5392 | 3.3115 | 9.3115 | 3.3679 | 13.5392 | 5.9103 | 14.713 | 14.0713 |

5.5804 | 1.8441 | 5.6641 | 2.5437 | 5.5804 | 0.3794 | 5.5804 | 7.9106 |

0.4700 | 0.4619 | 1.8384 | 1.0831 | 0.4750 | −0.3066 | 0.4750 | 6.1843 |

−5.5456 | −2.7814 | −3.4708 | −2.4105 | −5.5456 | −2.4269 | −5.5456 | −2.2995 |

−14.065 | −9.9563 | −11.6723 | −3.2688 | −14.0650 | −8.5990 | −14.0650 | −8.2256 |

Different Groups | 1# | 2# | 3# | |
---|---|---|---|---|

Different Working Conditions | Normal State | Outer Race Fault | Inner Race Fault | |

Normal state | mean value standard deviation | 0.5074 0.0630 | 0.4664 0.0719 | 0.4042 0.0853 |

Outer race fault | mean value standard deviation | 0.4664 0.0719 | 0.2415 0.0540 | 0.3650 0.0710 |

Inner race fault | mean value standard deviation | 0.4042 0.0853 | 0.3650 0.0710 | 0.2789 0.0552 |

Different Groups | 1# | 2# | 3# | |
---|---|---|---|---|

Different Working Conditions | Normal State | Outer Race Fault | Inner Race Fault | |

Normal state | mean value | 0.7563 | 0.2934 | 0.0925 |

standard deviation | 0.0235 | 0.1064 | 0.0909 | |

Outer race fault | mean value | 0.2934 | 0.6407 | 0.0585 |

standard deviation | 0.1064 | 0.0205 | 0.0493 | |

Inner race fault | mean value | 0.1025 | 0.0685 | 0.4353 |

standard deviation | 0.0869 | 0.0494 | 0.0372 |

Different Working Conditions | Different Methods | Different Groups | ||||
---|---|---|---|---|---|---|

1# | 2# | 3# | 4# | 5# | ||

Normal state | The proposed method | 91.5% | 93.6% | 100% | 98.3% | 100% |

BP | 88.1% | 82.8% | 89.5% | 80.0% | 80.0% | |

SVM | 87.2% | 83.2% | 88.5% | 80% | 77.8% | |

Outer race fault | The proposed method | 96.2% | 96.2% | 100% | 98.1% | 100% |

BP | 88.9% | 82.9% | 82.0% | 84.6% | 80.0% | |

SVM | 84.5% | 76.0% | 86.1% | 84.6% | 78.2% | |

Inner race fault | The proposed method | 91.5% | 95.5% | 100% | 98.0% | 100% |

BP | 77.5% | 82.5% | 87.0% | 84.0% | 85.0% | |

SVM | 87.5% | 87.0% | 82.2% | 77.5% | 82.0% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ge, M.; Wang, J.; Xu, Y.; Zhang, F.; Bai, K.; Ren, X.
Rolling Bearing Fault Diagnosis Based on EWT Sub-Modal Hypothesis Test and Ambiguity Correlation Classification. *Symmetry* **2018**, *10*, 730.
https://doi.org/10.3390/sym10120730

**AMA Style**

Ge M, Wang J, Xu Y, Zhang F, Bai K, Ren X.
Rolling Bearing Fault Diagnosis Based on EWT Sub-Modal Hypothesis Test and Ambiguity Correlation Classification. *Symmetry*. 2018; 10(12):730.
https://doi.org/10.3390/sym10120730

**Chicago/Turabian Style**

Ge, Mingtao, Jie Wang, Yicun Xu, Fangfang Zhang, Ke Bai, and Xiangyang Ren.
2018. "Rolling Bearing Fault Diagnosis Based on EWT Sub-Modal Hypothesis Test and Ambiguity Correlation Classification" *Symmetry* 10, no. 12: 730.
https://doi.org/10.3390/sym10120730