# Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

## 2. The Theory of the Proposed Method

#### 2.1. The Theory of Adaptive Local Iterative Filtering

Algorithm 1 Adaptive local iterative filtering algorithm |

ALIF Algorithm IMF = ALIF(f) |

IMF = { }While the number of extrema of $f\ge 2$ do${f}_{1}=f$ While the stopping criterion is not satisfied doCompute the filter length ${l}_{n}(x)$ for ${f}_{n}(x)$ ${f}_{n+1}(x)={f}_{n}(x)-{\displaystyle {\int}_{-{l}_{n}(x)}^{{l}_{n}(x)}{f}_{n}(x+t){w}_{n}(x,t)dt}$ $n=n+1$ End while$\mathrm{I}\mathrm{M}\mathrm{F}=\mathrm{I}\mathrm{M}\mathrm{F}\cup \left\{{\mathrm{f}}_{\mathrm{n}}\right\}$ End while$\mathrm{I}\mathrm{M}\mathrm{F}=\mathrm{I}\mathrm{M}\mathrm{F}\cup \left\{\mathrm{f}\right\}$ |

- (1)
- $g(a)=g(b)=0$ and $g(t)>0$ for $x\in (a,b)$
- (2)
- $h(a)<0<h(b)$

- (1)
- $p(x)>0$, for any $x\in (a,b)$
- (2)
- $p(x)=0$, for any $x\notin (a,b)$

#### 2.2. Adaptive Local Iterative Filtering Based on Particle Swarm Optimization

#### 2.3. The Desired Component Selection Base on PE

## 3. Numerical Simulation Analysis

## 4. Experimental Study

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**The best intrinsic modal function (IMF) of the original signal decomposed by empirical mode decomposition (EMD) in time-domain.

**Figure 13.**The Intelligent Maintenance System of Cincinnati: (

**a**) Bearing test rig; (

**b**) sensor placement illustration.

Inner Race Frequency ${\mathit{f}}_{\mathit{i}}$ (Hz) | Outer Race Frequency ${\mathit{f}}_{\mathit{o}}$ (Hz) | Rolling Element Frequency ${\mathit{f}}_{\mathit{o}}$ (Hz) |
---|---|---|

180 | 135 | 80 |

${\mathbf{A}}_{0}$ | ${\mathit{f}}_{\mathit{r}}$ (Hz) | ${\mathit{f}}_{\mathit{m}}$ (Hz) | ${\mathit{f}}_{\mathit{n}}$ (Hz) | $\mathit{C}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{C}}_{\mathit{A}}$ | ${\mathit{\phi}}_{\mathit{A}}$ | ${\mathit{\phi}}_{\mathit{W}}$ |
---|---|---|---|---|---|---|---|---|

3 | 20 | 20 | 2048 | 800 | 0.01 | 1 | 0 | 0 |

IMF Order | 1 | 2 | 3 | 4 |

PE Value | 0.8887 | 0.7072 | 0.5184 | 0.3754 |

Bearing Type | Outside Diameter d^{2}/mm | Ball Diameter/mm | Ball Number n | Contact Angle α |
---|---|---|---|---|

ZA2115 | 71.5 | 8.4 | 16 | 15.17 |

Group | Sample Channels | Fault Description |
---|---|---|

Group 1 | 8 | Inner race defect that occurred in bearing 3 and roller element defect occurred in bearing 4. |

Group 2 | 4 | Outer race failure that occurred in bearing 1. |

Group 3 | 4 | Outer race failure that occurred in bearing 3. |

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**MDPI and ACS Style**

Lv, Y.; Zhang, Y.; Yi, C.
Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis. *Entropy* **2018**, *20*, 920.
https://doi.org/10.3390/e20120920

**AMA Style**

Lv Y, Zhang Y, Yi C.
Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis. *Entropy*. 2018; 20(12):920.
https://doi.org/10.3390/e20120920

**Chicago/Turabian Style**

Lv, Yong, Yi Zhang, and Cancan Yi.
2018. "Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis" *Entropy* 20, no. 12: 920.
https://doi.org/10.3390/e20120920