# Evaluation of Rolling Bearing Performance Degradation Based on Comprehensive Index Reduction and SVDD

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Feature Extraction and Dimensionality Reduction

#### 2.1. Improved Variational Mode Decomposition

- (1)
- Initialize $X(t)$ and ${U}_{l}(t)$. Assign the original signal to $X(t)$, and assign the low-frequency mode ${U}_{l}(t)$ to 0.
- (2)
- Remove low-frequency modes. remove the low-frequency mode ${U}_{l}(t)$ from $X(t)$ at each run as follows:$$X(t)=X(t)-{U}_{l}(t).$$
- (3)
- Perform VMD decomposition. The number of fixed decompositions K is set to 2, and the penalty factor parameter alpha is set to 2000. The relevant literature showed that the penalty factor demonstrates strong applicability when the penalty factor is set to 2000 and VMD is suitable for extracting low-frequency modal components when the penalty factor is beyond 2000. Therefore, the VMD was run one at a time to obtain both high- and low-frequency modes.
- (4)
- Iteration stop judgment. Stop the iteration when the posting progress of central frequencies of the two modes obtained from the decomposition is less than the set threshold to obtain the final decomposed mode. The posting progress is defined as follows:$${f}_{d}=({f}_{h}-{f}_{l})/{f}_{l},$$

- (5)
- Take the kurtosis. Mode ${U}_{i}(t)$ is represented by ${u}_{k}$, and the kurtosis is calculated for all modes ${u}_{k}$. The maximum kurtosis value is denoted as $K1$.
- (6)
- Initial mode classification. Fast Fourier transform (FFT) is performed on mode ${u}_{k}$ to obtain the corresponding spectrum of each mode as follows:$${U}_{k}=FFT[{u}_{k}],k=1,2\cdots ,K,$$$${\widehat{U}}_{k}=\frac{{U}_{k}}{\sqrt{{\displaystyle \sum _{k=0}^{N-1}{U}_{k}^{2}}}},k=1,2,,K.$$

- (7)
- Modal reorganization determination. On the basis of the kurtosis, the reorganization of the class with more than one mode is determined and the maximum kurtosis S of modes in the class is compared with the value of kurtosis $K2$. When $K2<K1$, the modal classification in the class is considered to be dominated by noise, the modal reorganization in the class. When $K2<K1$, all modalities in the class are combined with $K2$ corresponding modalities $m0$, respectively, and, based on the above principle, the decision is made whether to reorganize or not. Finally, all modalities $U$ are outputted after regrouping.

#### 2.2. Mode Selection Based on KCI Features

#### 2.3. Defect Frequency Amplitude Ratio

#### 2.4. Comprehensive Indicator Downscaling

## 3. Support Vector Data Description

## 4. Performance Degradation Assessment Based on Combined Metric Downscaling and SVDD

## 5. Experimental Analysis and Validation

#### 5.1. Presentation of Experimental Data

#### 5.2. Effectiveness of VMD Improvements

#### 5.3. Comparative Analysis of the Effect of LLE Downscaling

#### 5.4. Assessment of Performance Degradation of Full-Life Rolling Bearings

#### 5.5. Fault Verification in All Stages of Bearing Degradation

## 6. Conclusions

- (1)
- The original vibration signal is decomposed using the improved VMD method, and modalities selected by the weighted kurtosis index are demodulated via enveloping with evident filtering effect. This effect is beneficial for separating early fault characteristics from the disturbance noise and conducive to the extraction of feature indicators.
- (2)
- Defect frequency amplitude ratio indicators, which are sensitive to early faults and more sensitive to early fault onset than traditional RMS indicators, are extracted to address the problem of strong sensitivity of effective feature indicators to the initial stage.
- (3)
- LLE dimensionality reduction is carried out on extracted comprehensive feature indicators to extract the main features, and the SVDD degradation assessment model combined with the relative distance indicator is used in the degradation assessment of the whole-life bearing. The proposed method is important for online health monitoring and early warning of bearings in practical production.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**The experimental equipment of the state degradation of rolling bearings; (

**a**) bearing test rig illustration; (

**b**) sensor placement in the test rig [34].

**Figure 6.**Time-domain waveform and envelope spectrum of the original signal; (

**a**) time-domain waveform; (

**b**) spectral envelope.

**Figure 11.**Comparison of dimensionality reduction effects of different methods; (

**a**) LLE feature dimension reduction results; (

**b**) PCA feature dimension reduction results.

**Figure 20.**Envelope demodulation of key sample points; (

**a**) 532nd data point; (

**b**) 533rd data point; (

**c**) 705th data point; (

**d**) 850th data point; (

**e**) 972nd data point.

**Figure 23.**Envelope pectrum of key sampling points in bearing 3; (

**a**) 854th data point; (

**b**) 872nd data point; (

**c**) 902nd data point.

Method | VMD | Improvement of VMD |
---|---|---|

PSNR | 19.627 | 22.495 |

Sample point | 533rd | 705th | 850th | 972nd |

Amplitude | 0.0087 | 0.1215 | 0.135 | 0.2787 |

Bearing condition | Early failure | Medium failure | Severe failure | Very severe failure |

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**MDPI and ACS Style**

Xin, H.; Zhang, H.; Yang, Y.; Wang, J.
Evaluation of Rolling Bearing Performance Degradation Based on Comprehensive Index Reduction and SVDD. *Machines* **2022**, *10*, 677.
https://doi.org/10.3390/machines10080677

**AMA Style**

Xin H, Zhang H, Yang Y, Wang J.
Evaluation of Rolling Bearing Performance Degradation Based on Comprehensive Index Reduction and SVDD. *Machines*. 2022; 10(8):677.
https://doi.org/10.3390/machines10080677

**Chicago/Turabian Style**

Xin, Hongwei, Haidong Zhang, Yanjun Yang, and Jianguo Wang.
2022. "Evaluation of Rolling Bearing Performance Degradation Based on Comprehensive Index Reduction and SVDD" *Machines* 10, no. 8: 677.
https://doi.org/10.3390/machines10080677