# Fault Diagnosis for Rolling Element Bearings Based on Feature Space Reconstruction and Multiscale Permutation Entropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. EEMD-Based Feature Space Reconstruction

#### 2.1. A Brief Overview of EMD and EEMD

_{1}can be obtained as the highest frequency of the time series.

_{i}(t) denotes the i-th IMF, and r

_{i}(t) denotes the residue of the signal.

_{i}(t) to the original signal x(t) (repeated M realizations in this work), where n

_{i}(t) represents the i-th added zero-mean noise. Then, x

_{i}(t) = x(t) + n

_{i}(t).

_{i}(t) into K IMFs c

_{ij}(t) (i = 1, 2, …, K) by EMD, where c

_{ij}(t) presents the j-th IMF in the i-th realization.

#### 2.2. Feature Space Reconstruction Based on EEMD (FSRE)

_{j}), when E(c

_{j})/E(c

_{j}

_{+ 1}) > § and $\sum _{j=1}^{j}{c}_{j}}/{\displaystyle \sum _{j=1}^{n-1}{c}_{j}}>0.95$, where j = 1, 2, 3, …, n − 1, E(c

_{1}) > E(c

_{2}) >…> E(c

_{n}

_{− 1}), and § is the closeness measure of energy between adjacent components, which is usually set as 10.

#### 2.3. Analysis of Simulating Bearing Fault Signals

_{o}is the characteristic frequency of the outer race, f

_{r}is the modulation frequency, and N(t) is the white noise. Here, we set A = 1, f

_{o}= 105 Hz, f

_{r}= 400 Hz, and the signal-to-noise ratio (SNR) of added white noise to −9.5 dB.

## 3. Permutation Entropy and Multiscale Permutation Entropy

#### 3.1. Permutation Entropy

_{1}− 1)r) = x(i + (j

_{2}− 1)r), their original positions can be sorted according to the index j*. Then, any time series X(i) can be transformed into a collection of symbols as

_{1}, P

_{2}, …, P

_{l}, consequently, the PE can be defined as the Shannon entropy,

_{p}≤ 1. When P

_{j}= 1/m!, H

_{p}(m) reaches the maximum value ln(m!).

_{p}is small, the time series tends to be regular. Conversely, the time series tends to be random.

#### 3.2. Multiscale Permutation Entropy

_{j}

^{(s)}represents the coarse-grained time series. When scale s = 1, y

_{j}

^{(1)}is simply the original time series.

## 4. The Proposed Method

## 5. Experimental Results

#### 5.1. Experimental Data Description

#### 5.2. Results and Analysis

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) The decomposed results of S(t) using ensemble empirical mode decomposition (EEMD); (

**b**) envelope spectrum of intrinsic mode functions (IMFs).

**Figure 8.**Multiscale permutation entropy (MPE) values over 10 scales of feature space reconstruction.

**Figure 9.**Classification results using the proposed approach: (

**a**) Group 1 with four working conditions; (

**b**) Group 2 with seven working conditions; (

**c**) Group 3 with ten working conditions.

Working Conditions | Defect Size (inches) | Number of Training Data Points | Number of Testing Data Points | Label of Classification |
---|---|---|---|---|

Normal | 0 | 80 | 30 | 0 |

Ball 1 | 0.007 | 80 | 30 | 1 |

Ball 2 | 0.014 | 80 | 30 | 2 |

Ball 3 | 0.021 | 80 | 30 | 3 |

Inner race 1 | 0.007 | 80 | 30 | 4 |

Inner race 2 | 0.014 | 80 | 30 | 5 |

Inner race 3 | 0.021 | 80 | 30 | 6 |

Outer race 1 | 0.007 | 80 | 30 | 7 |

Outer race 2 | 0.014 | 80 | 30 | 8 |

Outer race 3 | 0.021 | 80 | 30 | 9 |

Group | Fault Label | Label of Classification | Number of Training Data Points | Number of Testing Data Points |
---|---|---|---|---|

1 | Normal | 0 | 80 | 30 |

B007 | 1 | |||

IR007 | 2 | |||

OR007 | 3 | |||

2 | Normal | 0 | 80 | 30 |

B007 | 1 | |||

B021 | 2 | |||

IR007 | 3 | |||

IR021 | 4 | |||

OR007 | 5 | |||

OR021 | 6 | |||

3 | Normal | 0 | 80 | 30 |

B007 | 1 | |||

B014 | 2 | |||

B021 | 3 | |||

IR007 | 4 | |||

IR014 | 5 | |||

IR021 | 6 | |||

OR007 | 7 | |||

OR014 | 8 | |||

OR021 | 9 |

**Table 3.**Comparison results of different approaches. PE—permutation entropy; MPE—multiscale permutation entropy; MSE—multiscale sample entropy; IMPE—intrinsic mode permutation entropy.

Approach | Group 1 | Group 2 | Group 3 |
---|---|---|---|

PE | 92.50% | 85.88% | 73.25% |

MPE | 98.33% | 94.14% | 87.33% |

MSE | 97.17% | 92.46% | 84.46% |

IMPE | 100% | 95.85% | 91.90% |

Proposed method | 100% | 98.5% | 94.7% |

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**MDPI and ACS Style**

Zhang, W.; Zhou, J.
Fault Diagnosis for Rolling Element Bearings Based on Feature Space Reconstruction and Multiscale Permutation Entropy. *Entropy* **2019**, *21*, 519.
https://doi.org/10.3390/e21050519

**AMA Style**

Zhang W, Zhou J.
Fault Diagnosis for Rolling Element Bearings Based on Feature Space Reconstruction and Multiscale Permutation Entropy. *Entropy*. 2019; 21(5):519.
https://doi.org/10.3390/e21050519

**Chicago/Turabian Style**

Zhang, Weibo, and Jianzhong Zhou.
2019. "Fault Diagnosis for Rolling Element Bearings Based on Feature Space Reconstruction and Multiscale Permutation Entropy" *Entropy* 21, no. 5: 519.
https://doi.org/10.3390/e21050519