# Ball Bearing Fault Diagnosis Using Recurrence Analysis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Fault Bearing Diagnosis

#### 1.2. Motivation and Aim

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Recurrence Method

#### 2.3. Fault Modeling

## 3. Results and Discussion

#### 3.1. Measured Time Series

#### 3.2. Recurrence Plot Analysis

- The recurrence plot is covered by a small overlap window of size w that slides with steps s,
- The time signal is divided into overlapping segments, from which the RP diagrams and RQA indicators are calculated.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Laboratory rig for dynamic tests of bearings. The laboratory system is located at the Polish Bearings Factory in Krasnik.

**Figure 3.**Images of rolling bearing no. 6208C3 (

**a**), artificial fault on the ball (

**b**), fault on the inner ring (

**c**) and fault in the outer ring (

**d**).

**Figure 5.**Measured time series for the tested rolling bearings: bearing without fault (

**a**), bearing with ball fault (

**b**), bearing with inner ring fault (

**c**), and bearing with outer ring fault (

**d**).

**Figure 6.**Results of FNN (

**a**) and AMI (

**b**) methods for estimating embedding parameters m and d. The points represent the estimated optimal lag values.

**Figure 7.**Recurrence plots calculated for the bearing: without defect (

**a**), with ball defect (

**b**), with inner ring defect (

**c**) and outer ring defect (

**d**).

**Figure 8.**Recurrence quantifications versus shifting time window: DET(

**a**), ENT (

**b**), LAM (

**c**), TT (

**d**), ${L}_{MAX}$ (

**e**), ${V}_{MAX}$ (

**f**), L (

**g**), T1 (

**h**), T2 (

**i**), RTE (

**j**), Trans (

**k**) and Clust (

**l**).

**Table 1.**Definition of the most useful recurrence quantifications [28,30,31,32,33]. $P\left(l\right)$ and $P\left(v\right)$ denote the distribution of the length of diagonal and vertical lines. ${N}_{l}$ and ${N}_{v}$ are the numbers of diagonal and vertical lines. ${R}_{i}$ is the recurrence point that belongs to the state $\overline{{x}_{i}}$, and $Hv\left(v\right)$ is the distribution.

Quantification | Equation | Description |
---|---|---|

Recurrence Rate $RR$ | $\frac{1}{{N}^{2}}$${\sum}_{i,j=1}^{N}R{P}_{i,j}$ | Recurrence point density. |

Determinism $DET$ | $\frac{{\sum}_{l={l}_{min}}^{N}lP\left(l\right)}{{\sum}_{i,j=1}^{N}R{P}_{i,j}}$ | Portion of recurrence points forming diagonal lines. |

Entropy $ENT$ | $-{\sum}_{l={l}_{min}}^{N}P\left(l\right)ln\left(P\left(l\right)\right)$ | Entropy of the frequency distribution of the diagonal lines. |

Laminarity $LAM$ | $\frac{{\sum}_{v={v}_{min}}^{N}vP\left(v\right)}{{\sum}_{v=1}^{N}vP\left(v\right)}$ | Amount of recurrence points that form vertical lines. |

Trapping Time $TT$ | $\frac{{\sum}_{v={v}_{min}}^{N}vP\left(v\right)}{{\sum}_{v={v}_{min}}^{N}P\left(v\right)}$ | Average length of vertical lines. |

Longest diagonal line ${L}_{max}$ | $max(\{{l}_{i};i=1,...,{N}_{l}\}$ | Maximal line length in the diagonal direction. |

Longest vertical line ${V}_{max}$ | $max(\{{v}_{i};i=1,...,{N}_{v}\}$ | Maximal length of the vertical structures. |

Averaged diagonal line L | $\frac{{\sum}_{l={l}_{min}}^{N}lP\left(l\right)}{{\sum}_{l={l}_{min}}^{N}lP\left(l\right)}$ | Average diagonal line length. |

Recurrences time $T1$ | $|\{i,j:\phantom{\rule{3.33333pt}{0ex}}\overline{{x}_{i}},\overline{{x}_{j}}\}\in {R}_{i}\left\}\right|$ | Recurrence time of the 1st Poincare recurrence. |

Recurrences time $T2$ | $|\{i,j:\phantom{\rule{3.33333pt}{0ex}}\overline{{x}_{i}},\overline{{x}_{j}}\}\in {R}_{i},\overline{{x}_{j}}\notin {R}_{i}\}|$ | Recurrence time of the 2nd Poincare recurrence. |

Recurrence time entropy $RTE$ | $-\frac{1}{ln{V}_{max}}{\sum}_{v=1}^{{V}_{max}}{H}_{v}\left(v\right)ln{H}_{v}\left(v\right)$ | Shannon entropy of the recurrence times. |

Transitivity $Trans$ | $\frac{{\sum}_{i,j,k=1}^{N}R{P}_{i,j}R{P}_{j,k}R{P}_{k,i}}{{\sum}_{i,j,k=1}^{N}R{P}_{i,j}R{P}_{k,i}}$ | Local recurrence rate. |

Clustering coefficient $Clust$ | ${\sum}_{i=1}^{N}\frac{{\sum}_{j,k=1}^{N}R{P}_{i,j}R{P}_{j,k}R{P}_{k,i}}{R{R}_{i}}$ | The probability that two recurrence states are neighbors. |

Location of Defect | Embedding Dimension, m | Lag, d | Recurrence Rate, RR | Threshold, $\mathit{\u03f5}$ |
---|---|---|---|---|

No defect | 6 | 8 | 0.02 | 0.88 |

Ball | 6 | 4 | 0.02 | 0.80 |

Outer ring | 6 | 5 | 0.02 | 0.61 |

Inner ring | 6 | 8 | 0.02 | 0.88 |

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**MDPI and ACS Style**

Kecik, K.; Smagala, A.; Lyubitska, K.
Ball Bearing Fault Diagnosis Using Recurrence Analysis. *Materials* **2022**, *15*, 5940.
https://doi.org/10.3390/ma15175940

**AMA Style**

Kecik K, Smagala A, Lyubitska K.
Ball Bearing Fault Diagnosis Using Recurrence Analysis. *Materials*. 2022; 15(17):5940.
https://doi.org/10.3390/ma15175940

**Chicago/Turabian Style**

Kecik, Krzysztof, Arkadiusz Smagala, and Kateryna Lyubitska.
2022. "Ball Bearing Fault Diagnosis Using Recurrence Analysis" *Materials* 15, no. 17: 5940.
https://doi.org/10.3390/ma15175940