# Classification Framework of the Bearing Faults of an Induction Motor Using Wavelet Scattering Transform-Based Features

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## Abstract

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## 1. Introduction

- Investigate the applicability of the WST technique for extracting fault features to classify bearing states with ensemble ML algorithms and ANN.
- The classification performance exhibits that the resulting coefficients can directly be used as features, thus no additional feature calculation step from the coefficients is required.
- Resolve the feature extraction complexity of current signal-based bearing classification approaches due to their poor SNR and indirect measurement.

## 2. Theoretical Background

#### 2.1. Bearing Fault Frequencies

_{b}is the number of rolling components (balls), D

_{b}is the diameter of the ball, D

_{c}is the diameter of the cage, β is the load angle from the radial plane, and f

_{m}is the frequency of rotation.

_{k}is equivalent to $\frac{2\mathsf{\pi}{\mathit{f}}_{\mathit{bearing}}}{\mathit{p}}$.

_{bearing}is the harmonic frequency, which can be written as $\left|{f}_{s}\pm m{f}_{v}\right|$, and p denotes the operating machine’s pole pair number. Furthermore, m and f

_{s}denote the harmonic index and supply frequency, respectively. However, f

_{v}can be expressed as either f

_{inner}or f

_{outer}.

#### 2.2. Wavelet Scattering Transform (WST)

- 1.
- At first, x is convolved with the dilated mother wavelet ψ, which has the center frequency of λ, to calculate the WST. This operation can be expressed as $x*{\psi}_{\lambda}$. Here, the average of the convolved signal, which oscillates at a scale of 2j, is zero.
- 2.
- After that, a nonlinear operator, such as a modulus, is applied to the convolved signal to eliminate these oscillations (i.e., $\left|x*{\psi}_{\lambda}\right|$). This procedure is used to make up for the information lost due to down sampling by doubling the frequency of the given signal.
- 3.
- Finally, a low-pass filter φ is applied to the resultant absolute convolved signal, which is equivalent to $\left|x*{\psi}_{\lambda}\right|*\phi $

_{0x},S

_{1x},and S

_{2x}.

_{0x}represents the zero-order scattering coefficients, which evaluate the local translation invariance of the given input signal. The high-frequency components of the convolved signal are lost during each stage’s averaging operation, but they can be recovered in the following stage’s convolution operation with the wavelet. The WST method possesses the stability of time warp deformation, conversion in energy, and contraction, which makes the overall system robust in a noisy environment and appropriate for many classification tasks [30].

_{x}inherit properties of wavelet transforms, which make them stable against local deformations. This also allows the scattering decomposition to detect subtle changes in bearing signals’ amplitudes under different conditions and makes the classification task easier. Therefore, the wavelet scattering network can be used as an effective way to create robust representations of different bearing conditions that minimize the differences under the same condition and maintain enough discriminability to distinguish among different bearing conditions.

_{1x}and S

_{2x}. Through the cascaded wavelet decomposition, the WST can extract detailed feature information, and the local averaging technique can lessen the impact of noise. For these reasons, the WST can be considered a useful technique for extracting features in order to identify fault features in signals.

#### 2.3. Feature Extraction Mechanism

_{λ}), the Q factor, and the layer number of the scattering transform (m). Researchers found that, as long as the wavelet is complex, the outcome of the scattering transform is independent of the wavelet selection [46]. In the case of choosing the mother wavelet, Morlet (Gabor) wavelets were applied in this study. This wavelet can be expressed by Equation (9).

_{σ}and c

_{σ}represent the admissibility criterion and normalization constant, respectively.

#### 2.4. Classification with Ensemble Classifiers

#### 2.4.1. Random Forest (RF)

#### 2.4.2. Extreme Gradient Boosting (XGBoost)

_{i}, the predicted output ŷ

_{i}for the GBDT can be written as shown in Equation (10) [53]:

_{k}that fit the data extremely well while training and identifying the regions accordingly. Therefore, XGB adds the regularization factor Ω(t

_{k}) to reflect the complexity of the tree, and it uses Equation (11) to define the objective function of the optimization in the training model.

#### 2.5. Classification with Artificial Neural Network

- Step 1: Initialize weights and bias and perform forward propagation

^{l}, as shown by the Equation (13).

- Step 2: Estimating error values

- Step 3: Performing backpropagation

- Step 4: Update Parameters

## 3. Experimental Testbed and Data Description

## 4. Proposed Method

_{1}= 8 and Q

_{2}= 1 are presented in Figure 9. This architecture preserved the most signal information for classification, as compared to other settings, for the invariance scale and wavelet octave resolution.

#### Fault Classification Performance Evaluation Parameters

## 5. Results and Discussion

^{2}metric was used to optimize model performance. The ranges of parameters considered, along with the optimum values, are listed in Table 4. All the associated programs are executed in a desktop computer equipped with an Intel(R) Core (TM) i7-9700 CPU @3.6 GHz, and 16 GB RAM.

#### Comparison with Other Works

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Time series bearing data, 0th and 1st order scattering coefficients of the inner race faulty conditions.

**Figure 9.**(

**a**) The Morlet wavelet and its low−pass filter with a scaling function. (

**b**) Frequency response of the first and second filter banks with eight and one wavelets per octave, respectively.

**Figure 12.**The boxplot represents the accuracy matrix of over 100 experiments for the SVM, KNN, RF, XGB, and ANN models.

Layer Type | Shape of the Output | Numbers of Parameters |
---|---|---|

dense_1 | (None, 256) | 128,000 |

dense_2 | (None, 128) | 32,896 |

dense_3 | (None, 32) | 8256 |

dense_4 | (None, 3) | 195 |

Total params: 169,347 | Trainable params: 169,347 | Non-trainable params: 0 |

Serial No | Rotating Speed (S) [rpm] | Radial Force (F) [N] | Load Torque (M) [Nm] |
---|---|---|---|

1 | 1500 | 1000 | 0.7 |

2 | 1500 | 1000 | 0.1 |

3 | 900 | 1000 | 0.7 |

4 | 1500 | 400 | 0.7 |

Bearing Conditions | Bearing Code | Class Label |
---|---|---|

Normal Bearing | K001, K002, K003, K004, K005, K006 | 0 |

Outer Ring | KA04, KA15, KA16, KA22, KA30 | 1 |

Inner Ring | KI04, KI14, KI16, KI17, KI18, KI21 | 2 |

RF | XGB | ANN | ||||||
---|---|---|---|---|---|---|---|---|

Model Parameters | Considered Range | Optimum Value | Model Parameters | Considered Range | Optimum Value | Model Parameters | Considered Range | Optimum Value |

Minimum sample leaf | (1, 2, 3) | 3 | Maximum depth | 1 to 20 | 15 | Number of epochs | (20, 50, 100, 200, 250] | 200 |

Minimum sample split | (2, 4, 8, 16) | 8 | Gamma | 0.1 to 1 | 1 | Batch size | (32, 64, 128, 256) | 32 |

Number of estimator | (20, 30, 50, 100, 150, 200, 250) | 150 | Number of estimator | 50 to 1000 | 500 | Learning rate | (0.001, 0.01, 0.1, 0.2, 0.3) | 0.2 |

Maximum features | (3, 5, 7, 9) | 3 | Learning rate | 0.1 to 1 | 0.1 | Momentum | (0.0, 0.2, 0.4, 0.6, 0.8, 0.9) | 0.9 |

Precision | Recall | F1_score | Accuracy (%) | |
---|---|---|---|---|

SVM | 0.92 | 0.90 | 0.92 | 92.88 |

KNN | 0.89 | 0.86 | 0.88 | 89.89 |

RF | 0.99 | 0.99 | 1.00 | 99.26 |

XGB | 0.99 | 1.00 | 0.99 | 99.54 |

ANN | 0.99 | 0.99 | 0.99 | 99.13 |

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**MDPI and ACS Style**

Toma, R.N.; Gao, Y.; Piltan, F.; Im, K.; Shon, D.; Yoon, T.H.; Yoo, D.-S.; Kim, J.-M.
Classification Framework of the Bearing Faults of an Induction Motor Using Wavelet Scattering Transform-Based Features. *Sensors* **2022**, *22*, 8958.
https://doi.org/10.3390/s22228958

**AMA Style**

Toma RN, Gao Y, Piltan F, Im K, Shon D, Yoon TH, Yoo D-S, Kim J-M.
Classification Framework of the Bearing Faults of an Induction Motor Using Wavelet Scattering Transform-Based Features. *Sensors*. 2022; 22(22):8958.
https://doi.org/10.3390/s22228958

**Chicago/Turabian Style**

Toma, Rafia Nishat, Yangde Gao, Farzin Piltan, Kichang Im, Dongkoo Shon, Tae Hyun Yoon, Dae-Seung Yoo, and Jong-Myon Kim.
2022. "Classification Framework of the Bearing Faults of an Induction Motor Using Wavelet Scattering Transform-Based Features" *Sensors* 22, no. 22: 8958.
https://doi.org/10.3390/s22228958