# Time-Frequency Fusion Features-Based GSWOA-KELM Model for Gear Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

- (1)
- This study proposes the GSWOA-KELM model for the first time. In the new model, the GSWOA is used to find the optimal parameters of the KELM, and the results show that compared with the existing model, the proposed GSWOA-KELM model has higher diagnostic accuracy, faster convergence speed, and stronger global search capability;
- (2)
- The time-domain and frequency-domain features are extracted and fused in this study, which overcomes the limitations of single-domain features and improves the fault diagnosis ability of the model. Meanwhile, the superiority of multi-domain features in representing information ability is examined in this study, which provides a reference basis for the application of feature extraction work in other aspects.

## 2. Time-Frequency Features Extraction

#### 2.1. Time-Domain Features

#### 2.2. Frequency-Domain Features

#### 2.3. Fusion Features

**T**and the frequency-domain feature vector matrix

**F**, respectively. Assuming that the total number of samples is n, then:

**T**and frequency-domain feature vector matrix

**F**are fused to form the fused feature vector

**TF**, then:

## 3. GSWOA-KELM Fault Diagnosis Model

#### 3.1. Kernel Extreme Learning Machine

#### 3.2. Whale Optimization Algorithm

#### 3.3. Global Search Whale Optimization Algorithm

#### 3.4. Kernel Extreme Learning Machine Optimized Using the Global Search Whale Optimization Algorithm

## 4. Experimental Verification and Result Analysis

#### 4.1. Data Acquisition and Preprocessing

#### 4.2. Time-Frequency Features Extraction

#### 4.3. Fault Diagnosis and Result Analysis

#### 4.3.1. Fault Diagnosis and Result Analysis without Feature Fusion

#### 4.3.2. Fault Diagnosis and Result Analysis with Feature Fusion

## 5. Conclusions

- (1)
- Compared with KELM, SSA-KELM, and WOA-KELM, the GSWOA-KELM has faster convergence speed, stronger global search capability, and higher recognition accuracy;
- (2)
- When constructing a GSWOA-KELM model for gear fault diagnosis, the GSWOA-KELM performance can be improved by considering the fusion features rather than the single time-domain or frequency-domain features;
- (3)
- Compared to KELM, SSA-KELM, and WOA-KELM, the GSWOA-KELM model proposed in this study improved the fault diagnosis accuracy by 11.33%, 8.67%, and 1.33%, respectively.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**GSWOA-KELM fault diagnosis results for time-domain features as input. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 8.**GSWOA-KELM fault diagnosis results for frequency-domain features as input. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 9.**GSWOA-KELM fault diagnosis results for fusion-domain features as input. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 11.**The fault diagnosis results of KELM. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 12.**The fault diagnosis results of SSA-KELM. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 13.**The fault diagnosis results of WOA-KELM. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

**Figure 14.**The fault diagnosis results of GSWOA-KELM. (

**a**) Predictive classification results; (

**b**) confusion matrix for the test set.

Dimensional | Formula | Dimensionless | Formula |
---|---|---|---|

Mean Value | $\overline{\mathrm{x}}=\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}}\mathrm{x}\left(\mathrm{n}\right)$ | Pulse Factor | $\mathrm{I}=\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left|\mathrm{x}\left(\mathrm{n}\right)\right|}{\overline{\mathrm{x}}}$ |

Standard Deviation | ${\mathsf{\sigma}}_{\mathrm{x}}=\sqrt{\frac{1}{\mathrm{N}-1}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}}{\left[\mathrm{x}\left(\mathrm{n}\right)-\overline{\mathrm{x}}\right]}^{2}}$ | Margin Factor | $\mathrm{L}=\frac{\mathrm{m}\mathrm{a}\mathrm{x}\left|\mathrm{x}\left(\mathrm{n}\right)\right|}{{\left(\frac{1}{\mathrm{N}}\sum _{\mathrm{n}=1}^{\mathrm{N}}\sqrt{\left|\mathrm{x}\left(\mathrm{n}\right)\right|}\right)}^{2}}$ |

Root-Mean-Square Value | ${\mathrm{x}}_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}}{\mathrm{x}}^{2}\left(\mathrm{n}\right)}$ | Waveform Factor | $\mathrm{W}=\frac{{\mathrm{x}}_{\mathrm{r}\mathrm{m}\mathrm{s}}}{\overline{\mathrm{x}}}$ |

Maximum Value | ${\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{x}}_{\mathrm{n}}\right)$ | Kurtosis | $\mathrm{K}=\frac{{\sum _{\mathrm{n}=1}^{\mathrm{N}}\left[\mathrm{x}\left(\mathrm{n}\right)-\overline{\mathrm{x}}\right]}^{4}}{(\mathrm{N}-1){\mathsf{\sigma}}_{\mathrm{x}}^{4}}$ |

Minimum Value | ${\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\mathrm{x}}_{\mathrm{n}}\right)$ | Skewness | $\mathrm{S}=\frac{{\sum _{\mathrm{n}=1}^{\mathrm{N}}\left[\mathrm{x}\left(\mathrm{n}\right)-\overline{\mathrm{x}}\right]}^{3}}{(\mathrm{N}-1){\mathsf{\sigma}}_{\mathrm{x}}^{3}}$ |

Peak-peak Value | ${\mathrm{x}}_{\mathrm{p}\mathrm{p}}={\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ | Amplitude Factor | $\mathrm{A}=\frac{{\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{x}}_{\mathrm{r}\mathrm{m}\mathrm{s}}}$ |

Energy | $\mathrm{E}={\sum}_{\mathrm{n}=1}^{\mathrm{N}}{\mathrm{x}\left(\mathrm{n}\right)}^{2}$ |

Frequency Domain Characteristic Parameters | Formula |
---|---|

Amplitude Mean | $\mathrm{A}\mathrm{M}=\frac{1}{\mathrm{K}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{K}}}\mathrm{s}\left(\mathrm{k}\right)$ |

Center Frequency | $\mathrm{C}\mathrm{F}=\frac{\sum _{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{f}}_{\mathrm{k}\xb7}\mathrm{s}\left(\mathrm{k}\right)}{\sum _{\mathrm{k}=1}^{\mathrm{K}}\mathrm{s}\left(\mathrm{k}\right)}$ |

Mean Square Frequency | $\mathrm{M}\mathrm{S}\mathrm{F}=\frac{\sum _{\mathrm{k}=1}^{\mathrm{K}}{{\mathrm{f}}_{\mathrm{k}}}^{2}\mathrm{s}\left(\mathrm{k}\right)}{\sum _{\mathrm{k}=1}^{\mathrm{K}}\mathrm{s}\left(\mathrm{k}\right)}$ |

Root-Mean-Square Frequency | $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{F}=\sqrt{\frac{\sum _{\mathrm{k}=1}^{\mathrm{K}}{{\mathrm{f}}_{\mathrm{k}}}^{2}\mathrm{s}\left(\mathrm{k}\right)}{\sum _{\mathrm{k}=1}^{\mathrm{K}}\mathrm{s}\left(\mathrm{k}\right)}}$ |

Frequency Variance | $\mathrm{F}\mathrm{V}\mathrm{A}\mathrm{R}=\frac{\sum _{\mathrm{k}=1}^{\mathrm{K}}{{(\mathrm{f}}_{\mathrm{k}}-\mathrm{C}\mathrm{F})}^{2}\xb7\mathrm{s}\left(\mathrm{k}\right)}{\sum _{\mathrm{k}=1}^{\mathrm{K}}\mathrm{s}\left(\mathrm{k}\right)}$ |

Fault Type | Fault Description | Classification Label | Sample Number | Total Sample Number | |
---|---|---|---|---|---|

Train Set | Test Set | ||||

Health | Healthy gear. | 1 | 70 | 30 | 100 |

Chipped | The gear is cracked or even broken. | 2 | 70 | 30 | 100 |

Miss | Gear defect. | 3 | 70 | 30 | 100 |

Root | There is a crack at the root of the gear. | 4 | 70 | 30 | 100 |

Surface | Gear surface wear. | 5 | 70 | 30 | 100 |

Input | Accuracy Rate |
---|---|

T | 86.67% |

F | 85.33% |

TF | 100% |

Fault Diagnosis Model | Fault Type | Accuracy Rate | Overall Accuracy |
---|---|---|---|

KELM | Health | 100% | 88.67% |

Chipped | 100% | ||

Miss | 88.0% | ||

Root | 93.3% | ||

Surface | 65.7% | ||

SSA-KELM | Health | 100% | 91.33% |

Chipped | 100% | ||

Miss | 82.6% | ||

Root | 83.5% | ||

Surface | 79.4% | ||

WOA-KELM | Health | 100% | 98.67% |

Chipped | 100% | ||

Miss | 100% | ||

Root | 100% | ||

Surface | 96.8% | ||

GSWOA-KELM | Health | 100% | 100% |

Chipped | 100% | ||

Miss | 100% | ||

Root | 100% | ||

Surface | 100% |

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## Share and Cite

**MDPI and ACS Style**

Hu, Q.; Zhou, H.; Wang, C.; Zhu, C.; Shen, J.; He, P.
Time-Frequency Fusion Features-Based GSWOA-KELM Model for Gear Fault Diagnosis. *Lubricants* **2024**, *12*, 10.
https://doi.org/10.3390/lubricants12010010

**AMA Style**

Hu Q, Zhou H, Wang C, Zhu C, Shen J, He P.
Time-Frequency Fusion Features-Based GSWOA-KELM Model for Gear Fault Diagnosis. *Lubricants*. 2024; 12(1):10.
https://doi.org/10.3390/lubricants12010010

**Chicago/Turabian Style**

Hu, Qin, Haiting Zhou, Chengcheng Wang, Chenxi Zhu, Jiaping Shen, and Peng He.
2024. "Time-Frequency Fusion Features-Based GSWOA-KELM Model for Gear Fault Diagnosis" *Lubricants* 12, no. 1: 10.
https://doi.org/10.3390/lubricants12010010