# Intelligent Ball Bearing Fault Diagnosis Using Fractional Lorenz Chaos Extension Detection

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chua’s Circuit

_{RL}is defined as Equation (2).

_{a}and G

_{b}are the slopes, and E

_{a}is the breakpoint.

_{a}and V

_{b}waveform characteristics. which are introduced into chaos as an extension matter-element model, using the extension algorithm.

## 3. Experiment System

## 4. Chaos Theory

- $0.00<\left|\alpha \right|\le 0.20$: For arithmetic quantification as well as proportional application
- $0.20<\left|\alpha \right|\le 1.00$: For classification and control of non-arithmetical values

_{A}and V

_{B}and e

_{1}, e

_{2}, and e

_{3}, to make trace diagrams of the phase domain. The important characteristics are the four bearing state signals: Normal signal, bearing fault, inner ring fault and outer ring fault. The dynamic errors are:

_{A}and V

_{B}signals.

_{A}and V

_{B}test signals into the master synchronization (MS) system. The e

_{2}and e

_{3}dynamic error signals were used for plotting the trace diagrams. The trajectories existing for each state were used to establish a matter-element model. The signals emanating from the output of the monitoring system were identified using an extension theory.

## 5. Extension Theory

#### 5.1. Matter-Element Theory

#### 5.2. Extension Sets

#### 5.3. An Extension Theory Matter-Element Model

## 6. Experiment Results

_{1}, e

_{2}and e

_{3}have been made from observations. The different states provided by the experimental database include normal, outer ring, roll ring and inner ring faults. Extension matter-element models were designed for distinguishing intelligent monitoring output systems which work on a 48 k (Hz) base sampling rate. One second data volumes are formed from observation of important characteristics of the different state dynamic errors, e

_{2}and e

_{3}.

## 7. Conclusions

`(`EtherCAT), this might helpful in several ways. For example, the data could be uploaded to the cloud and immediately be available for other users anywhere. Such a database would be extremely useful as a check for review and evaluation, or the improvement of existing methods. It would certainly help the development of Industry 4.0.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**The order 1 fault dynamic error trace diagrams for all fault diameters. (

**a**) Normal state; (

**b**) fault diameter = 0.007; (

**c**) fault diameter = 0.014; (

**d**) fault diameter = 0.021.

**Figure 6.**The order 0.9 fault dynamic error trace diagrams for all fault diameters. (

**a**) Normal state; (

**b**) fault diameter = 0.007; (

**c**) fault diameter = 0.014; (

**d**) fault diameter = 0.021.

**Figure 7.**The order 0.7 fault dynamic error trace diagrams for all fault diameters. (

**a**) Normal state; (

**b**) fault diameter = 0.007; (

**c**) fault diameter = 0.014; (

**d**) fault diameter = 0.021.

**Figure 8.**The order 0.5 fault dynamic error trace diagrams for all fault diameters. (

**a**) Normal state; (

**b**) fault diameter = 0.007; (

**c**) fault diameter = 0.014; (

**d**) fault diameter = 0.021.

**Figure 9.**The order 0.3 fault dynamic error trace diagrams for all fault diameters. (

**a**) Normal state; (

**b**) fault diameter = 0.007; (

**c**) fault diameter = 0.014; (

**d**) fault diameter = 0.021.

**Figure 10.**Fuzzy and extension sets. (

**a**) Relationship of a fuzzy set to an extension set; (

**b**) extension set diagram.

Sampling Frequency (Hz) | Motor Load (HP) | Fault Single Point Diameter (inches) | Fault Single Point Depth (inches) | Fault Condition |
---|---|---|---|---|

12 k 48 k | 0 1 2 3 | 0.007 0.014 0.021 | 0.011 | Normal ball bearing fault inner ring fault outer ring fault |

Notation | Definition | Notation | Definition |
---|---|---|---|

X | The system states of the master system | Г( ) | Gamma function |

Y | The system states of the slave system | a’, b’, c’ | System parameters of fractional–order system |

f | Non-linear function | Ф_{i} | Dynamic error equation |

U | Control input | g, h | The upper and lower limits of the classical domain |

A | System parameter vector | r, s | The upper and lower limits of the joint domain |

a, b, c | System parameters of the Chen-Lee Chaos System | О | The name of a matter-element |

e | System error state vector | ε | The characteristics of the matter-element |

D | Differential operator | μ | The values corresponding to the characteristics |

α | The value of differential order | Ω | The universe of discourse |

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**MDPI and ACS Style**

Tian, A.-H.; Fu, C.-B.; Li, Y.-C.; Yau, H.-T. Intelligent Ball Bearing Fault Diagnosis Using Fractional Lorenz Chaos Extension Detection. *Sensors* **2018**, *18*, 3069.
https://doi.org/10.3390/s18093069

**AMA Style**

Tian A-H, Fu C-B, Li Y-C, Yau H-T. Intelligent Ball Bearing Fault Diagnosis Using Fractional Lorenz Chaos Extension Detection. *Sensors*. 2018; 18(9):3069.
https://doi.org/10.3390/s18093069

**Chicago/Turabian Style**

Tian, An-Hong, Cheng-Biao Fu, Yu-Chung Li, and Her-Terng Yau. 2018. "Intelligent Ball Bearing Fault Diagnosis Using Fractional Lorenz Chaos Extension Detection" *Sensors* 18, no. 9: 3069.
https://doi.org/10.3390/s18093069