# A Non-Contact Fault Diagnosis Method for Bearings and Gears Based on Generalized Matrix Norm Sparse Filtering

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## Abstract

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## 1. Introduction

## 2. Proposed Method

#### 2.1. GMNSF

**W**is a filter applied to each sample. Concretely, let ${f}_{j}^{i}$ denote the jth feature value (row) for the ith example (column). $g(.)$ is used as activation function. The feature calculation is

#### 2.2. Intelligent Fault Diagnosis Framework

- Model training: The GMNSF model is trained through the original acoustic dataset. Set the training sample set as ${\left\{{x}^{i},{y}^{i}\right\}}_{i=1}^{M}$where $M$ is the number of samples, ${x}^{i}\in {\Re}^{N\times 1}$ denotes the $N$ input variables in the $i$ sample, and ${y}^{i}$ is the label of the sample ${x}^{i}$. The sample is randomly divided into ${N}_{s}$ segments and each segment includes ${N}_{in}$ input variables. Thus, a training set ${\left\{{s}^{j}\right\}}_{j=1}^{M{N}_{s}}$ contains $M{N}_{s}$ segments, where ${s}^{j}\in {\Re}^{{N}_{in}\times 1}$ is the jth segment of the training dataset. Noticeably, the segments are obtained through overlapping. The following are the specific steps of model training. Then, the obtained ${\left\{{s}^{j}\right\}}_{j=1}^{M{N}_{s}}$ is directly inputted to GMNSF to train the weight matrix
**W**. - Feature extraction: The discriminant features can be obtained by the optimal weight matrix $W$. The training sample ${x}_{i}$ is separated into $K$ sections, and the $K$ is an integer and equal to $N/{N}_{in}$. The dataset ${\left\{{x}_{k}^{i}\right\}}_{k=1}^{K}$ is composed of $K$ segments and ${x}_{k}^{i}\in {\Re}^{{N}_{in}\times 1}$. Then, the trained sparse filtering model is used to calculate the local feature ${\stackrel{\u2323}{f}}_{k}^{i}\in {\Re}^{{N}_{out}\times 1}$ of each sample. Learned features are obtained by combining local features using the average method, leading to$${\stackrel{\u2323}{f}}^{i}=\frac{1}{K}{\displaystyle \sum _{k=1}^{K}{\stackrel{\u2323}{f}}_{k}^{i}}$$
- Fault recognition: The learned features of all samples are combined with the labels and then trained through the softmax classifier. First, Z-Score is normalized to activate training and test data. The calculation method is as follows:$$\mathit{F}=\frac{\mathit{f}-\overline{\mathit{f}}}{\sigma}$$Then, the softmax regression model is trained by the learned characteristic set ${\left\{{\stackrel{\u2323}{f}}^{i}\right\}}_{i=1}^{M}$ and the healthy condition label set ${\left\{{y}^{i}\right\}}_{i=1}^{M}$, ${y}^{i}\in \left\{1,2,\dots ,D\right\}$. The softmax regression output probability $\rho \left({y}^{i}=d/{\stackrel{\u2323}{f}}^{i}\right)$ that ${\stackrel{\u2323}{f}}^{i}$ is the label of the feature vector. The softmax regression model of weight matrix $\theta $ is acquired by minimizing the nest cost function.$$H(\theta )=\frac{\lambda}{2}{\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{{\rm N}}_{o\upsilon \tau}}{({\theta}_{i}^{j})}^{2}}}-\frac{1}{{\rm M}}\left[{\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{d=1}^{D}\{{y}^{i}=d\}}}\mathrm{log}\frac{{e}^{{\theta}_{d}}{\stackrel{\u2323}{\mathit{f}}}^{i}}{{\displaystyle \sum _{c=1}^{D}{e}^{{\theta}_{c}}{\stackrel{\u2323}{\mathit{f}}}^{i}}}\right]$$

## 3. Experimental Validation

#### 3.1. Rolling Bearing Data Verification and Analysis

#### 3.2. Gear Data Verification and Analysis

## 4. Discuss Weight Matrix

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Method | Description | Training Samples (%) | Testing Accuracy (%) |
---|---|---|---|

1 | SF | 50 | 92.96 ± 1.22% |

2 | ICF | 50 | 87.35 ± 1.58% |

4 | CSF | 50 | 73.35 ± 3.78% |

5 | GMNSF | 50 | 97.93 ± 0.96% |

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**MDPI and ACS Style**

Bao, H.; Shi, Z.; Wang, J.; Zhang, Z.; Zhang, G.
A Non-Contact Fault Diagnosis Method for Bearings and Gears Based on Generalized Matrix Norm Sparse Filtering. *Entropy* **2021**, *23*, 1075.
https://doi.org/10.3390/e23081075

**AMA Style**

Bao H, Shi Z, Wang J, Zhang Z, Zhang G.
A Non-Contact Fault Diagnosis Method for Bearings and Gears Based on Generalized Matrix Norm Sparse Filtering. *Entropy*. 2021; 23(8):1075.
https://doi.org/10.3390/e23081075

**Chicago/Turabian Style**

Bao, Huaiqian, Zhaoting Shi, Jinrui Wang, Zongzhen Zhang, and Guowei Zhang.
2021. "A Non-Contact Fault Diagnosis Method for Bearings and Gears Based on Generalized Matrix Norm Sparse Filtering" *Entropy* 23, no. 8: 1075.
https://doi.org/10.3390/e23081075