# Rolling Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and SOA-SVM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Method

#### 2.1. WOA-VMD

_{i}in automation mode.

_{1},…u

_{i}} is a series of decomposed intrinsic mode functions, and {ω

_{k}} = {ω

_{1},…ω

_{k}} is the center frequency corresponding to each intrinsic mode function. In order to arrive at the best solution in Equation (2), the Lagrange penalty factor L and secondary penalty factor α are introduced.

- (1)
- The initialization of parameters such as whale individual population, location, and iteration times. The i-th individual location is:$${\mathrm{X}}_{\mathrm{i}}=\mathrm{r}\cdot \left(\mathrm{ub}-\mathrm{lb}\right)+\mathrm{lb}$$
- (2)
- When p < 0.5 and |A| < 1, shrink and surround according to the best search agent, as shown in Equation (5):$$\{\begin{array}{l}\overrightarrow{\mathrm{X}}\left(\mathrm{i}+1\right)=\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{i}\right)-\overrightarrow{\mathrm{A}}\cdot \overrightarrow{\mathrm{D}}\hfill \\ \overrightarrow{\mathrm{D}}=\left|\overrightarrow{\mathrm{C}}\cdot \overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{i}\right)-\overrightarrow{\mathrm{X}}\left(\mathrm{i}\right)\right|\hfill \\ \overrightarrow{\mathrm{A}}=2\overrightarrow{\mathrm{a}}\cdot {\overrightarrow{\mathrm{r}}}_{1}-\overrightarrow{\mathrm{a}}\hfill \\ \overrightarrow{\mathrm{C}}=2\cdot {\overrightarrow{\mathrm{r}}}_{2}\hfill \\ \overrightarrow{\mathrm{a}}=2-2\left(\mathrm{i}/{\mathrm{i}}_{\mathrm{max}}\right)\hfill \end{array}$$

- (3)
- Check if the termination requirements have been satisfied or if the maximum number of repetitions has been reached. If not, return to step (2). If yes, output the best search agent.

#### 2.2. Multiscale Permutation Entropy and Its Parameter Setting

#### 2.3. SOA-SVM

_{i}is the ith input value of the sample feature space. In the linearly separable state, the optimal hyperplane solved by the support vector classifier can be transformed into the following constraint problem:

_{i}, x

_{j}) must meet the positive definite matrix condition, that is,

_{i},x

_{j}), the nonlinear samples can be linearized and classified. After the relaxation variable ξ

_{i}is introduced, the expression of the original classification hyperplane is

_{i}, x

_{j}) is the radial basis function, the debugging of penalty factor C and kernel width g is the major focus of SVM parameter adjustment.

- (1)
- Avoid collision. In order to prevent the occurrence of a collision between adjacent seagulls, add a new variable a. The formula is:$$\mathrm{c}\left(\mathrm{t}\right)=\mathrm{A}\times \mathrm{p}\left(\mathrm{t}\right)$$$$\mathrm{A}=\mathrm{f}-\mathrm{t}\times \left(\mathrm{f}/{\mathrm{n}}_{\mathrm{max}}\right)$$
- (2)
- The direction of the best position. On the premise of not colliding with other individuals, seagulls will move in the direction of the best position. The formula is$$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{B}\times [\mathrm{Zbest}\xb7\left(\mathrm{t}\right)-\mathrm{p}\left(\mathrm{t}\right)]$$$$\mathrm{B}=2\times \mathrm{A}\times \mathrm{A}\times {\mathrm{r}}_{\mathrm{d}}$$
- (3)
- Approaching the best position. The seagull will soar in the route of the best position to achieve a new one after landing in a safe location away from other seagulls.$$\mathrm{d}\left(\mathrm{t}\right)=\mid \mathrm{c}\left(\mathrm{t}\right)+\mathrm{m}\left(\mathrm{t}\right)$$

_{best}(t) indicates the best seagull position. The steps of optimizing support vector machine with seagull algorithm are as follows:

- (1)
- Initialize the population parameters of seagull optimization algorithm, the number of iterations, and the value range of C and g.
- (2)
- Determine the fitness function of seagull optimization algorithm and evaluate the adaptability of seagull individuals on the basis of the value of fitness function. According to the principle of seagull optimization algorithm, find the optimal fitness value and the optimal position obtained by seagull.
- (3)
- According to the best individual position of seagull, the optimal values of parameters C and g are obtained.
- (4)
- The optimal parameters C and g are assigned to the support vector machine for training, and the optimized support vector machine classification model is obtained.
- (5)
- Input the test samples, then the optimized SVM classification model will output the predicted labels of the test samples and compare the predicted labels with the actual labels to obtain the classification accuracy.

## 3. Experiment and Results

#### 3.1. Experimental System

#### 3.2. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

K | decomposition levels |

α | secondary penalty factor |

${\mathrm{X}}_{\mathrm{i}}$ | whale location vector |

r | random number |

$\mathrm{lb}$ | lower boundary |

ub | upper boundary |

p | random number |

$\overrightarrow{\mathrm{A}}$, $\overrightarrow{\mathrm{C}}$ | coefficient vectors |

$\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{i}\right)$ | best solution obtained so far |

$\cdot $ | element-by-element multiplication |

${\mathrm{Q}}_{\mathrm{j}}$ | probability distribution |

${\mathrm{E}}_{\mathrm{e}}$ | envelope entropy |

s | scale factor |

N | time-series length |

m | encapsulation dimension |

$\mathsf{\tau}$ | time delay |

n | number of samples |

x_{i} | the ith input value |

$\mathsf{\omega}$ | the normal vector of the hyperplane |

m | offset |

Λ | high-dimensional space |

Ψ | nonlinear mapping |

ξ_{i} | relaxation variable |

k(x_{i},x_{j}) | kernel function |

C | penalty factor |

g | kernel width |

$\mathrm{c}\left(\mathrm{t}\right)$ | position of seagulls |

$\mathrm{t}$ | number of iteration |

$\mathrm{p}\left(\mathrm{t}\right)$ | initial position of the seagull |

$\mathrm{m}\left(\mathrm{t}\right)$ | the direction of the best position |

$\mathrm{d}\left(\mathrm{t}\right)$ | new position where the seagull meets three conditions |

u,v | correlation constant |

θ | angle |

${\mathrm{Z}}_{\mathrm{best}}\left(\mathrm{t}\right)$ | the best seagull position |

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**Figure 4.**The different defects of rolling bearings: (

**a**) rolling balls, (

**b**) inner race, and (

**c**) outer race.

Type | Fault Size/mm | Category Label |
---|---|---|

Normal | —— | 1 |

Inner Race Fault | 0.1778 | 2 |

Outer Race Fault | 0.1778 | 3 |

Rolling Element fault | 0.1778 | 4 |

Normal | Inner Race | Outer Race | Rolling Element | |
---|---|---|---|---|

K | 6 | 10 | 10 | 10 |

α | 1108 | 1219 | 1469 | 2000 |

Starting Point | Imf1 | Imf2 | Imf3 | Imf4 | Imf5 | Imf6 | Imf7 | Imf8 | Imf9 | Imf10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 3.110 | 3.699 | 4.460 | 2.159 | 3.583 | 2.249 | 2.695 | 2.687 | 4.751 | 5.099 |

2049 | 2.975 | 2.649 | 4.488 | 2.281 | 3.570 | 2.358 | 2.897 | 2.915 | 5.241 | 5.067 |

4097 | 3.177 | 2.410 | 4.561 | 2.205 | 3.620 | 2.374 | 2.763 | 2.740 | 5.163 | 5.429 |

Starting Point | Imf1 | Imf2 | Imf3 | Imf4 | Imf5 | Imf6 | Imf7 | Imf8 | Imf9 | Imf10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2.859 | 2.128 | 3.565 | 2.534 | 2.951 | 3.150 | 2.997 | 2.109 | 3.106 | 3.625 |

2049 | 2.972 | 2.020 | 3.545 | 2.480 | 2.782 | 3.138 | 2.573 | 1.930 | 2.712 | 3.218 |

4097 | 2.699 | 2.017 | 3.417 | 2.475 | 2.920 | 3.149 | 2.619 | 2.335 | 2.452 | 3.500 |

Starting Point | Imf1 | Imf2 | Imf3 | Imf4 | Imf5 | Imf6 | Imf7 | Imf8 | Imf9 | Imf10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2.493 | 2.676 | 3.403 | 2.990 | 2.654 | 2.363 | 2.714 | 2.482 | 2.699 | 3.018 |

2049 | 2.885 | 2.867 | 3.324 | 3.400 | 3.324 | 2.974 | 2.630 | 2.723 | 2.941 | 3.340 |

4097 | 2.450 | 2.891 | 3.367 | 3.013 | 2.992 | 2.925 | 3.032 | 2.514 | 2.689 | 3.333 |

Starting Point | Imf1 | Imf2 | Imf3 | Imf4 | Imf5 | Imf6 |
---|---|---|---|---|---|---|

1 | 2.825 | 3.110 | 1.545 | 2.559 | 3.956 | 3.014 |

2049 | 2.970 | 3.261 | 1.583 | 2.706 | 2.946 | 2.854 |

4097 | 2.329 | 3.030 | 1.600 | 2.996 | 2.911 | 3.316 |

Starting Point | Imf1 | Imf2 | Imf3 | Imf4 | Imf5 | Imf6 |
---|---|---|---|---|---|---|

1 | 0.649 | 0.644 | 0.304 | 0.382 | 0.090 | 0.048 |

2049 | 0.631 | 0.658 | 0.300 | 0.383 | 0.097 | 0.049 |

4097 | 0.632 | 0.654 | 0.289 | 0.384 | 0.092 | 0.047 |

Feature Value 1 | Feature Value 2 | Feature Value 3 | Label |
---|---|---|---|

0.597 | 0.742 | 0.482 | 1 |

0.682 | 0.621 | 0.832 | 2 |

0.902 | 0.834 | 0.750 | 3 |

0.802 | 0.846 | 0.811 | 4 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Wang, H.; Ren, M.; He, M.; Jin, L.
Rolling Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and SOA-SVM. *Machines* **2022**, *10*, 485.
https://doi.org/10.3390/machines10060485

**AMA Style**

Zhang X, Wang H, Ren M, He M, Jin L.
Rolling Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and SOA-SVM. *Machines*. 2022; 10(6):485.
https://doi.org/10.3390/machines10060485

**Chicago/Turabian Style**

Zhang, Xi, Hongju Wang, Mingming Ren, Mengyun He, and Lei Jin.
2022. "Rolling Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and SOA-SVM" *Machines* 10, no. 6: 485.
https://doi.org/10.3390/machines10060485