# Recognition of Crack-Rubbing Coupling Fault of Bearing under High Water Pressure Based on Polar Symmetry Mode Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (1)
- Dimensionality reduction was carried out for the system to improve the efficiency and the accuracy of the fault recognition and support the fault detection.
- (2)
- The internal characteristics of the signal were extracted.
- (3)
- According to the extraction results of characteristic parameters, the probability neural network was introduced to achieve the final recognition of bearing crack-rub coupling fault under high water pressure.
- (4)
- The validity of the fault recognition method was verified via experiment and discussion.
- (5)
- Summary was proposed, and prospects were put forward.

#### 2.1. Dimensionality Reduction of System

_{n}, ${P}_{z}$ represents the rectangular projection of ${X}_{z}$. ${Q}_{z}=I-{P}_{z}$ is the projection operator. The Equation (1) can be broken down as follows via linear variation:

#### 2.2. Feature Parameters Extraction Based on Polar Symmetry Mode Decomposition

- (1)
- All the extreme points of time series X’ are marked and expressed via ${E}_{i}\left(1\le i<n\right)$.
- (2)
- The midpoint of the line among the extreme points is marked as ${{F}^{\prime}}_{i}\left(1\le i<n-1\right)$.
- (3)
- Boundary points ${{F}^{\prime}}_{0}$ and ${{F}^{\prime}}_{n}$ are added via linear interpolation method.
- (4)
- According to ${{F}^{\prime}}_{i}\left(1\le i<n-1\right)$, curves ${L}_{1},\cdots ,{L}_{p}\left(p\ge 1\right)$ are obtained via direct interpolation algorithm. The ${L}^{\ast}$ is calculated.$${L}^{\ast}=\frac{\left({L}_{1}+\cdots +{L}_{p}\right)}{p}$$
- (5)
- ${X}^{\prime}-{L}^{\ast}$. The above four steps are repeated until $\left|{L}^{\ast}\right|\le \epsilon $ is set up, where the $\epsilon $ represents the allowable error.In another situation, when the number iterations achieves the set value ${K}^{\prime}$, the modal function ${{M}^{\prime}}_{1}$ is obtained.
- (6)
- As shown in Equation (10), for ${{R}^{\prime}}_{1}$, the ${{M}^{\prime}}_{1}$ is solved repeatedly to obtain ${{M}^{\prime}}_{2},{{M}^{\prime}}_{3},\cdots $, respectively, until the end condition is achieved.$${{R}^{\prime}}_{1}={X}^{\prime}-{{M}^{\prime}}_{1}$$
- (7)
- The value of maximum screen number is changed in section $\left[{{K}^{\prime}}_{\mathrm{min}},{{K}^{\prime}}_{\mathrm{max}}\right]$. The decomposition method is repeated to obtain a series of results.
- (8)
- The ${{K}^{\prime}}_{0}$ making the $\sigma /{\sigma}_{0}$ minimum value in the section $\left[{{K}^{\prime}}_{\mathrm{min}},{{K}^{\prime}}_{\mathrm{max}}\right]$. Then, the ${{K}^{\prime}}_{0}$ is substituted. The IMF is obtained via ESMD decomposition. The obtained R’ is the optimal AGM curve.

#### 2.3. Fault Recognition

- (1)
- ESMD decomposition was carried out for the vibration signal.
- (2)
- The correlation index of the obtained IMF components and the original signal were calculated and ranked respectively. The former n IMF components were taken to characterize the vibration signal (the value of n depended on the specific vibration signal decomposition).
- (3)
- In order to facilitate PNN training and classification of the data, the obtained IMF component with high correlation was calculated in terms of energy. We created a vector and normalize the obtained vector as a whole.
- (4)
- ${{T}^{\u2034}}^{*}$ was input into the PNN as the eigenvector to train until the optimal solution was output. The final classification results of the fault were obtained.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Dimensionality reductions effect of the recognition method of crack-rubbing coupling fault of bearing under high water pressure based on polar symmetry mode decomposition.

Model | SKF6203 | SKF6205 |
---|---|---|

Rolling element diameter | 6.746 | 15.001 |

Outer ring diameter | 39.99 | 51.9989 |

Inner diameter | 17.0002 | 25.001 |

Pitch diameter | 28.50 | 39.0398 |

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**MDPI and ACS Style**

Huang, J.; Hua, W.; Xie, T.; Yao, Y.; Dong, S.
Recognition of Crack-Rubbing Coupling Fault of Bearing under High Water Pressure Based on Polar Symmetry Mode Decomposition. *Symmetry* **2021**, *13*, 59.
https://doi.org/10.3390/sym13010059

**AMA Style**

Huang J, Hua W, Xie T, Yao Y, Dong S.
Recognition of Crack-Rubbing Coupling Fault of Bearing under High Water Pressure Based on Polar Symmetry Mode Decomposition. *Symmetry*. 2021; 13(1):59.
https://doi.org/10.3390/sym13010059

**Chicago/Turabian Style**

Huang, Jiuzhou, Wen Hua, Tianzhou Xie, Yanchao Yao, and Shiming Dong.
2021. "Recognition of Crack-Rubbing Coupling Fault of Bearing under High Water Pressure Based on Polar Symmetry Mode Decomposition" *Symmetry* 13, no. 1: 59.
https://doi.org/10.3390/sym13010059