# A Technique for Bearing Fault Diagnosis Using Novel Wavelet Packet Transform-Based Signal Representation and Informative Factor LDA

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## Abstract

**:**

## 1. Introduction

- (1)
- A new WPT-based signal representation is introduced for the extraction of bearing fault-related components.
- (2)
- A variant of LDA, IF-LDA, is introduced to increase the discriminancy of the feature space based on the informative factor.

- (1)
- WPT is used with a novel R-value criterion for mother wavelet selection in analyzing bearing signals. The R-value criterion considers the energy-to-entropy ratio of the signal power spectrum to select the mother wavelet that provides the most uneven energy distribution in a specific WPT node while preserving high signal energy.
- (2)
- The proposed method constructs the final signal representation node by node, based on the R-value of each node’s reconstruction. As nodes are selected from WPT trees decomposed using different mother wavelets, the method is referred to as a novel WPT-based signal representation.
- (3)
- The introduction of a novel feature engineering approach that greatly benefits linear discriminant analysis. This approach ensures minimal scatteredness among features within the same class and maximizes between-class separation, leading to improved accuracy in model predictions and easier generalization.

## 2. Testbeds, Experiments, and Collected Data

#### 2.1. Paderborn University Bearing Data with Artificial Damage (PUA Dataset)

#### 2.2. Paderborn University Bearing Data with Real Damage (PUR Dataset)

#### 2.3. Case Western Reserve University Bearing Data (CWRU Dataset)

## 3. Technical Background

#### 3.1. Wavelet Packet Transform

#### 3.2. Approaches for Mother Wavelet Selection

^{th}coefficient at the s level, then ${p}_{i}$ is defined in the following manner:

#### 3.3. Linear Discriminant Analysis

## 4. Proposed Method

#### 4.1. Vibration Signal Processing

#### 4.2. Vibration Signal Processing

_{i}is the probability of the signal power being in the frequency bin, which is defined as follows:

_{i}is the power in the i-th frequency bin.

#### 4.3. Feature Extraction and Feature Pool Configuration

#### 4.4. Feature Extraction and Feature Pool Configuration

#### 4.5. Bearing Fault Classification

## 5. Experimental Results and Discussion

^{®}Core™ i7-9700K CPU and 16 GB of RAM.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Correction Statement

## References

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**Figure 10.**True positive rates for each bearing fault acquired as a result of testing the proposed and comparison methods on three datasets. The columns in black, orange, grey, and yellow stand for the bearing fault class. Each set of columns has a caption that indicates the comparison method to which it belongs.

**Figure 11.**Averaged performance metrics for the proposed and comparison methods obtained from testing on three datasets. The columns in black, orange, grey, and yellow stand for the bearing fault class. Each set of columns has a caption that indicates the comparison method to which it belongs.

No. | Rot. Speed (rpm) | Load Torque (Nm) | Radial Force (N) |
---|---|---|---|

0 | 1500 | 0.7 | 1000 |

1 | 900 | 0.7 | 1000 |

2 | 1500 | 0.1 | 1000 |

3 | 1500 | 0.7 | 400 |

Bearing Type | Bearing Code |
---|---|

Healthy | K: 001, 002, 003, 004, 005, 006 |

Outer ring damage | KA: 01, 03, 05, 06, 07, 08, 09 |

Inner ring damage | KI: 0, 03, 05, 07, 08 |

Bearing Type | Bearing Code |
---|---|

Healthy | K: 001, 002, 003, 004, 005, 006 |

Outer ring damage | KA: 04, 15, 16, 22, 30 |

Inner ring damage | KI: 04, 14, 16, 17, 18, 21 |

Outer + inner ring fault | KB: 23, 24, 27 |

Bearing Type | Bearing Code |
---|---|

Healthy | 97–100 |

Outer ring damage | 130–133, 144–147, 156–160, 197–200, 234–237, 246–249, 258–261 |

Inner ring damage | 056–059, 105–108, 169–172, 209–212 |

Ball damage | 048–051, 118–121, 185–188, 222–225 |

Feature Name | Equation | Feature Name | Equation |
---|---|---|---|

Peak value | ${X}_{p}=\underset{i}{\mathrm{max}}\left|{x}_{i}\right|$ | Entropy | $H(x)=-{\displaystyle \sum _{i=1}^{N}P\left({x}_{i}\right)}\cdot {\mathrm{log}}_{2}P\left({x}_{i}\right)$ |

RMS | ${X}_{RMS}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{{\displaystyle x}}_{i}^{2}}}$ | Mean | $\mu =\frac{1}{N}{\displaystyle \sum _{i-1}^{N}{x}_{i}}$ |

Kurtosis | ${X}_{kurtosis}=\frac{1}{N}\left(\frac{{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{4}}}{{\sigma}^{4}}\right)$ | Skewness | ${X}_{kurtosis}=\frac{1}{N}\left(\frac{{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{3}}}{{\sigma}^{3}}\right)$ |

Crest factor | ${C}_{f}=\frac{{X}_{p}}{{X}_{RMS}}$ | Shape factor RMS | $S{F}_{RMS}=\frac{{X}_{RMS}}{\mu}$ |

Clearance factor | $L=\frac{{X}_{p}}{{\left(\left(1/N\right){\displaystyle \sum _{i=1}^{N}\sqrt{\left|{x}_{i}\right|}}\right)}^{2}}$ | Peak-to-peak value | ${x}_{ptp}=\mathrm{max}\left|x\right|-\mathrm{min}\left|x\right|$ |

Impulse factor | $L=\frac{\mathrm{max}\left\{\left|{x}_{i}\right|\right\}}{\left(\left(1/N\right){\displaystyle \sum _{i=1}^{N}\left|{x}_{i}\right|}\right)}$ | Energy of signal | $e={\displaystyle \sum _{i=1}^{N}{x}_{i}^{2}}$ |

Root variance frequency | $RVF=\sqrt{\frac{{\displaystyle \underset{0}{\overset{\infty}{\int}}{({f}_{i}-FC)}^{2}s\left({f}_{i}\right)df}}{{\displaystyle \underset{0}{\overset{\infty}{\int}}s\left({f}_{i}\right)}df}}$ | RMS frequency | $RMSF=\sqrt{\frac{{\displaystyle \underset{0}{\overset{\infty}{\int}}{f}_{i}^{2}s\left({f}_{i}\right)df}}{{\displaystyle \underset{0}{\overset{\infty}{\int}}s\left({f}_{i}\right)}df}}$ |

Square mean root | ${X}_{SMR}={\left(\frac{{\displaystyle \sum _{i=1}^{N}\sqrt{{x}_{i}}}}{N}\right)}^{2}$ | Frequency center | $FC=\frac{{\displaystyle \underset{0}{\overset{\infty}{\int}}fs\left(f\right)df}}{{\displaystyle \underset{0}{\overset{\infty}{\int}}s\left(f\right)}df}$ |

5th normalized moment | $HOMn5=\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{5}}}{{\left(\sqrt{\frac{1}{N-1}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{2}}}\right)}^{5}}$ | 6th normalized moment | $HOMn6=\frac{\frac{1}{n}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{6}}}{{\left(\sqrt{\frac{1}{N-1}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\mu \right)}^{2}}}\right)}^{6}}$ |

Shape factor SMR | $S{F}_{SMR}=\frac{{X}_{SMR}}{\mu}$ |

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**MDPI and ACS Style**

Maliuk, A.S.; Ahmad, Z.; Kim, J.-M.
A Technique for Bearing Fault Diagnosis Using Novel Wavelet Packet Transform-Based Signal Representation and Informative Factor LDA. *Machines* **2023**, *11*, 1080.
https://doi.org/10.3390/machines11121080

**AMA Style**

Maliuk AS, Ahmad Z, Kim J-M.
A Technique for Bearing Fault Diagnosis Using Novel Wavelet Packet Transform-Based Signal Representation and Informative Factor LDA. *Machines*. 2023; 11(12):1080.
https://doi.org/10.3390/machines11121080

**Chicago/Turabian Style**

Maliuk, Andrei S., Zahoor Ahmad, and Jong-Myon Kim.
2023. "A Technique for Bearing Fault Diagnosis Using Novel Wavelet Packet Transform-Based Signal Representation and Informative Factor LDA" *Machines* 11, no. 12: 1080.
https://doi.org/10.3390/machines11121080