# Condition Monitoring of Bearing Faults Using the Stator Current and Shrinkage Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fault Signatures

_{1}is the main supply frequency, n is an entire number, and f

_{v}is the vibration characteristic frequency. f

_{v}depends on the type of bearing fault (outer or inner race, balls and train defect), with expressions that are a function of the geometry and composition of the bearing [7].

## 3. Diagnosis

_{i}be the m predictors (or fault signatures in the context of condition monitoring) and y

_{i}the response for the n cases of the problem. A linear model tries to estimate the m + 1 coefficients (b

_{0}, …, b

_{m}). Using a least squares fitting approach, b

_{i}are selected to minimize (a).

_{0}, since this term is just an estimation of the mean when the predictors are zero [42].

_{2}penalty for an l

_{1}one [42]. The use of an l

_{1}norm has the inconvenience of turning the function to minimize into one that is nondifferentiable, although there are available methods to proceed with the minimization, such as proximal gradient ones [43]. This way of considering the penalty gives rise to the method known as Lasso. As opposed to Ridge Regression, with Lasso, some variables are canceled, so perform as variable selection, depending the number of the variables to be selected on the value of λ (as λ grows, less variables are selected).

## 4. Results

#### 4.1. Test Bench

#### 4.2. Classification with 968 Fault Signatures

#### 4.3. Classification with 968 Fault Signatures by Applying Shrinkage

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Evolution of the condition states of the bearing (see Table 3): (

**a**) C1; (

**b**) C2; (

**c**) C3; (

**d**) C4; (

**e**) C5.

**Figure 3.**Performance of Lasso classifier, depending on the value of the regularization parameter for different supplies (S1–S6) and loads (HL: high load; LL: low load).

Traditional Approach | Proposed Approach | |
---|---|---|

Bearing fault frequencies | BPFO ^{1}, BPFI ^{2}, FTF ^{3}; BSF ^{4} | BPFO ^{1}, BPFI ^{2}, FTF ^{3}; BSF ^{4} |

Harmonics considered | 1 | 1, 2, …, 11 |

Sidebands around harmonic | 1 | 1, 2, …, 11 |

Number of fault signatures | 8 | 968 |

^{1}Ball Pass Frequency of the Outer race;

^{2}Ball Pass Frequency of the Inner race;

^{3}Fundamental Train Frequency;

^{4}Ball Spin Frequency.

Supply Identification | Power Source | Operating Frequency | Switching Frequency |
---|---|---|---|

S1 | utility | 50 Hz | - |

S2 | Power converter | 50 Hz | 4 kHz |

S3 | Power converter | 25 Hz | 4 kHz |

S4 | Power converter | 50 Hz | 5 kHz |

Condition | Evolution of the Fault | Number of Tests Per Supply |
---|---|---|

C1 | healthy state | 20 |

C2 | incipient fault | 15 |

C3 | intermediate fault | 15 |

C4 | developed fault | 10 |

C5 | complete breakdown | 10 |

**Table 4.**Comparison between the traditional (eight fault signatures) and proposed approach (968 fault signatures) with the algorithms included in the MATLAB 2019a Classification learner App.

Supply Identification | Load | Best Accuracy 968 Fault Signatures | Best Algorithm 968 Fault Signatures | Best Accuracy 8 Fault Signatures | Best Algorithm 8 Fault Signatures |
---|---|---|---|---|---|

S1 | Low | 95.7% | KNN ^{1} | 40% | SVM ^{2} |

High | 92.9% | SVM ^{2} | 40% | SVM ^{2} | |

S2 | Low | 88.6% | SVM ^{2} | 32.9% | BT ^{5} |

High | 98.6% | SVM ^{2} | 44.3% | BT ^{5} | |

S3 | Low | 81.4% | BT ^{5} | 41.4% | SVM ^{2} |

High | 97.1% | LD ^{4} | 60% | KNN ^{1} | |

S4 | Low | 85.7% | LD ^{4} | 41.4% | SVM ^{2} |

High | 98.6% | BT ^{5} | 42.9% | SVM ^{2} |

^{1}Fine k-Nearest Neighbor;

^{2}Quadratic Support Vector Machines;

^{3}Gaussian Naive Bayes;

^{4}Linear Discriminant;

^{5}Bagged Tress.

Supply Identification | Load | Lasso | Elastic Nets | Ridge Regression |
---|---|---|---|---|

S1 | Low | 100 | 100 | 100 |

High | 90.48 | 90.48 | 85.71 | |

S2 | Low | 95.24 | 90.48 | 80.95 |

High | 95.24 | 100 | 100 | |

S3 | Low | 80.95 | 76.19 | 76.19 |

High | 90.48 | 95.24 | 95.24 | |

S4 | Low | 90.48 | 80.95 | 85.71 |

High | 100 | 95.24 | 85.71 |

Supply | Load | Regularization Parameter |
---|---|---|

S1 | High | 0.02 |

Low | 0.05 | |

S2 | High | 0.0003 |

Low | 0.02 | |

S3 | High | 0.01 |

Low | 0.01 | |

S4 | High | 0.005 |

Low | 0.005 |

Low Load | High Load | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Predicted Class | Predicted Class | ||||||||||

True class | C1 | C2 | C3 | C4 | C5 | C1 | C2 | C3 | C4 | C5 | |

C1 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | |

C2 | 0 | 5 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | |

C3 | 0 | 0 | 4 | 0 | 0 | 0 | 1 | 4 | 0 | 0 | |

C4 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | |

C5 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 |

Low Load | High Load | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Predicted Class | Predicted Class | ||||||||||

True class | C1 | C2 | C3 | C4 | C5 | C1 | C2 | C3 | C4 | C5 | |

C1 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | |

C2 | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | |

C3 | 0 | 0 | 5 | 0 | 0 | 0 | 1 | 4 | 0 | 0 | |

C4 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | |

C5 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 |

Low Load | High Load | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Predicted Class | Predicted Class | ||||||||||

True class | C1 | C2 | C3 | C4 | C5 | C1 | C2 | C3 | C4 | C5 | |

C1 | 5 | 1 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | |

C2 | 0 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | |

C3 | 0 | 0 | 3 | 1 | 1 | 1 | 1 | 3 | 0 | 0 | |

C4 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | |

C5 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 3 |

Low Load | High Load | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Predicted Class | Predicted Class | ||||||||||

True class | C1 | C2 | C3 | C4 | C5 | C1 | C2 | C3 | C4 | C5 | |

C1 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | |

C2 | 0 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | |

C3 | 0 | 0 | 4 | 0 | 1 | 0 | 0 | 5 | 0 | 0 | |

C4 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 3 | 0 | |

C5 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Duque-Perez, O.; Del Pozo-Gallego, C.; Morinigo-Sotelo, D.; Fontes Godoy, W. Condition Monitoring of Bearing Faults Using the Stator Current and Shrinkage Methods. *Energies* **2019**, *12*, 3392.
https://doi.org/10.3390/en12173392

**AMA Style**

Duque-Perez O, Del Pozo-Gallego C, Morinigo-Sotelo D, Fontes Godoy W. Condition Monitoring of Bearing Faults Using the Stator Current and Shrinkage Methods. *Energies*. 2019; 12(17):3392.
https://doi.org/10.3390/en12173392

**Chicago/Turabian Style**

Duque-Perez, Oscar, Carlos Del Pozo-Gallego, Daniel Morinigo-Sotelo, and Wagner Fontes Godoy. 2019. "Condition Monitoring of Bearing Faults Using the Stator Current and Shrinkage Methods" *Energies* 12, no. 17: 3392.
https://doi.org/10.3390/en12173392