# A Bearing Fault Diagnosis Method under Small Sample Conditions Based on the Fractional Order Siamese Deep Residual Shrinkage Network

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## Abstract

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## 1. Introduction

- (1)
- The one-dimensional vibration signals are converted into two-dimensional time series feature maps, which is convenient for the neural network model to extract the feature of the signal. The combination of the DRSN and Siamese network is conducive to improving the feature extraction ability of fault signals under small sample conditions.
- (2)
- In the parameter updating process of neural network backpropagation, momentum and fractional order calculus are applied to the gradient descent optimizer to make it converge to the optimal solution, thus improving the accuracy of fault diagnosis in the case of limited training data.
- (3)
- In order to simulate the limited data conditions in engineering applications, four sets of small sample training data were selected from the CWRU dataset to analyze and verify the FO-SDRSN method, which provides a possibility for its further application in bearing fault diagnosis with small sample data.

## 2. Proposed Method

#### 2.1. Data Processing and Feature Extraction

_{i}, x

_{j}), where i, j = 1, 2…, N and i ≠ j. If x

_{i}and x

_{j}are fault sample pairs of the same type, Y = 1 is assigned; otherwise, Y = 0 is assigned, and p(⋅) denotes the labeling process for the sample pairs.

_{i}is brought into F

_{1}for feature extraction as follows:

_{1}operations.

_{i}is the output after passing through the sigmoid.

_{i}is the output after the threshold calculation.

_{i}using FO-SDRSN, it is expressed as follows:

_{i}after feature extraction from one side of the Siamese deep residual shrinkage network. Similarly, the output feature of x

_{j}after feature extraction from the other side of the Siamese Network can be defined as

_{i}) and H(x

_{j}) represent the fault features extracted from x

_{i}and x

_{j}, respectively, after applying the FO-SDRSN method. The value d is used to determine whether the input pair is of the same fault type: a smaller value of d indicates that the sample pair is similar, whereas larger values of d suggest that the sample pair is dissimilar. The contrastive loss function L

_{1}is expressed as follows:

#### 2.2. Fault Diagnosis and Parameter Update

_{i}) and H(x

_{j}) were fed to a SoftMax classifier. The output y

_{n}of the SoftMax classifier is defined by

_{n}is the predicted probability that the sample belongs to the n-th class, and N

_{class}is the number of categories, such that $\sum {y}_{n}}=1$. In the above formula, N

_{class}= 10. After the probability of the sample belonging to 10 categories is predicted, respectively, the category with the largest prediction probability value is selected as the result of this fault diagnosis. The cross-entropy loss function is usually used as the objective function of multi-classification problems. In this paper, this function is the loss function of the fault diagnosis, which is expressed as

_{n}indicates the actual fault type. When the sample belongs to the n-th class, t

_{n}= 1; otherwise, t

_{n}= 0. L

_{1}and L

_{2}are combined as total losses as follows:

_{0}is the initial value, and $\mathsf{\Gamma}(\alpha )={\displaystyle \underset{0}{\overset{\infty}{\int}}{x}^{\alpha -1}{e}^{-x}}dx$ is the Gamma function. The above formula can be derived as follows:

^{−8}.

## 3. Experiments and Evaluations

#### 3.1. Data Acquisition

#### 3.2. Experiments

## 4. Discussion

## 5. Conclusions

- (1)
- The FO-SDRSN method can be used to diagnose bearing fault types under small sample conditions. This method can further reduce the loss during the repeated iterative updating of the network parameters, and the results are constantly close to the optimal solution, thus improving the accuracy of bearing fault diagnosis under small sample conditions.
- (2)
- The experiments indicated that the FO-SDRSN method was more accurate and stable than other progressive methods under the given four small sample datasets. When the number of samples for each fault was 15, the average fault diagnostic accuracy was 2.27% higher than that of the progressive Siamese–DRSN method. The Discussion Section shows that the fault diagnosis performance of the FO-SDRSN method under different orders was associated with the quantity of small sample data.
- (3)
- In cases where there are limited data, the improvement in the accuracy of bearing fault diagnosis is crucial for the subsequent rapid and targeted maintenance and enhancement of the working efficiency of rotating machinery. The improvements demonstrated in this study also provide a new approach for the fault diagnosis of bearings equipment under actual industrial operation and maintenance conditions. This study was validated with publicly available datasets, so the robustness and applicability of the proposed method will be further verified in different engineering scenarios.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Time-domain waveform graphs of signals: (

**a**–

**j**) represent signals corresponding to labels 0–9 respectively.

**Figure 3.**Two-dimensional time characteristic sequence graphs of signals: (

**a**–

**j**) represent signal corresponding to labels 0–9 respectively.

**Figure 4.**Box plot of fault diagnostic accuracy distribution of different models under the four types of small sample datasets: (

**a**–

**d**) represent small sample datasets with 10, 15, 20, and 30 training samples for each type of fault, respectively.

**Figure 5.**Test accuracy curve of each model changes with epoch on four small sample datasets: (

**a**–

**d**) represent small sample datasets with 10, 15, 20, and 30 training samples for each type, respectively.

**Figure 6.**Confusion matrix for various models when the number of training samples of each type of fault is 15. (

**a**) SVM model; (

**b**) CNN model; (

**c**) MSFACNN model; (

**d**) DRSN model; (

**e**) Siamese–DRSN model; (

**f**) FO-SDRSN model.

**Figure 7.**Feature visualization using various models when the number of training samples of each type of fault is 15. (

**a**) Feature visualization of data distribution before fault diagnosis; (

**b**) CNN model; (

**c**) MSFACNN model; (

**d**) DRSN model; (

**e**) Siamese–DRSN model; (

**f**) FO-SDRSN model.

**Figure 8.**Average accuracy of different fractional orders on four small sample datasets: (

**a**–

**d**) represent small sample datasets with 10, 15, 20, and 30 training samples for each type of fault, respectively.

**Figure 9.**Box plot of 5 repeated experiments under different fractional orders on four small sample datasets: (

**a**–

**d**) represent small sample datasets with 10, 15, 20, and 30 training samples for each type of fault, respectively.

**Figure 10.**Precision curve of different fractional orders with epoch on four small sample datasets: (

**a**–

**d**) represent small sample datasets with 10, 15, 20, and 30 training samples for each type of fault, respectively.

Label | Fault Size (Inch) | Fault Location | Number of Training Samples in Four Small Sample Datasets |
---|---|---|---|

0 | 0.007 | Roller | 10, 15, 20, 30 |

1 | 0.014 | Roller | 10, 15, 20, 30 |

2 | 0.021 | Roller | 10, 15, 20, 30 |

3 | 0.007 | Inner race | 10, 15, 20, 30 |

4 | 0.014 | Inner race | 10, 15, 20, 30 |

5 | 0.021 | Inner race | 10, 15, 20, 30 |

6 | 0.007 | Outer race | 10, 15, 20, 30 |

7 | 0.014 | Outer race | 10, 15, 20, 30 |

8 | 0.021 | Outer race | 10, 15, 20, 30 |

9 | - | Health | 10, 15, 20, 30 |

Model | Number of Training Samples for Each Type of Fault Is 10 | Number of Training Samples for Each Type of Fault Is 15 | Number of Training Samples for Each Type of Fault Is 20 | Number of Training Samples for Each Type of Fault Is 30 | ||||
---|---|---|---|---|---|---|---|---|

Average Accuracy (%) | Standard Deviation | Average Accuracy (%) | Standard Deviation | Average Accuracy (%) | Standard Deviation | Average Accuracy (%) | Standard Deviation | |

FO-SDRSN | 82.6091 | 0.985746 | 91.46106 | 0.603937 | 92.6948 | 0.761442 | 96.20128 | 1.296661 |

MSFACNN | 74.62782 | 5.53897 | 83.49514 | 1.033596 | 90.84142 | 0.952347 | 90.08976 | 0.724449 |

CNN | 54.56308 | 3.814642 | 67.99352 | 2.788438 | 67.37864 | 2.33054 | 78.73786 | 3.899128 |

DRSN | 61.51612 | 4.303248 | 81.70968 | 4.418985 | 91.93034 | 1.348297 | 95 | 1.145057 |

Siamese–DRSN | 81.42856 | 2.994431 | 89.18832 | 2.465018 | 90.35718 | 1.108664 | 94.22076 | 0.392317 |

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## Share and Cite

**MDPI and ACS Style**

Li, T.; Wu, X.; Luo, Z.; Chen, Y.; He, C.; Ding, R.; Zhang, C.; Yang, J.
A Bearing Fault Diagnosis Method under Small Sample Conditions Based on the Fractional Order Siamese Deep Residual Shrinkage Network. *Fractal Fract.* **2024**, *8*, 134.
https://doi.org/10.3390/fractalfract8030134

**AMA Style**

Li T, Wu X, Luo Z, Chen Y, He C, Ding R, Zhang C, Yang J.
A Bearing Fault Diagnosis Method under Small Sample Conditions Based on the Fractional Order Siamese Deep Residual Shrinkage Network. *Fractal and Fractional*. 2024; 8(3):134.
https://doi.org/10.3390/fractalfract8030134

**Chicago/Turabian Style**

Li, Tao, Xiaoting Wu, Zhuhui Luo, Yanan Chen, Caichun He, Rongjun Ding, Changfan Zhang, and Jun Yang.
2024. "A Bearing Fault Diagnosis Method under Small Sample Conditions Based on the Fractional Order Siamese Deep Residual Shrinkage Network" *Fractal and Fractional* 8, no. 3: 134.
https://doi.org/10.3390/fractalfract8030134