# A Comprehensive Diagnosis Method of Rolling Bearing Fault Based on CEEMDAN-DFA-Improved Wavelet Threshold Function and QPSO-MPE-SVM

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A CEEMDAN-DFA-improved wavelet thresholding denoising method was proposed. The method uses the CEEMDAN algorithm to decompose the vibration signal, performs DFA on the obtained IMF, calculates the scalar function value of each IMF component, selects the noise-dominated IMF component, and applies an improved wavelet threshold function to denoise it.
- (2)
- Combining QPSO-MPE-SVM into an effective fault diagnosis method can accurately extract fault features and improve the identification accuracy of bearing faults.
- (3)
- Experimental cases were used to illustrate the effectiveness of the proposed method in bearing vibration signal denoising, fault feature extraction, and fault identification.

## 2. Denoising Algorithm of the CEEMDAN-DFA-Improved Wavelet Threshold Function

#### 2.1. Basic Algorithm Related to CEEMDAN-DFA-Improved Wavelet Threshold Function

#### 2.1.1. CEEMDAN Algorithm

- (1)
- The jth IMF component generated by the signal decomposition by the EMD is defined as ${E}_{j}\left(\cdot \right)$. The jth IMF by the CEEMDAN is defined as $IM{F}_{j}{}^{\prime}$. ${n}^{i}\left(t\right)$ is for the Gaussian white noise. The CEEMDAN performs I EMD decomposition on the noisy signal $x\left(t\right)+{\epsilon}_{0}\cdot {n}^{i}\left(t\right)$ formed by the combination of the original signal and the white noise. Then the first IMF component decomposed by CEEMDAN can be expressed as:$$IM{F}_{1}^{\prime}\left(t\right)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}IM{F}_{1}^{i}\left(t\right)}$$
- (2)
- First residual sequence of the first stage (j = 1) is expressed as:$${r}_{1}\left(t\right)=x\left(t\right)-IM{F}_{1}^{\prime}\left(t\right)$$
- (3)
- The ${r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({n}^{i}\left(t\right)\right)\left(i=1,2,\cdot \cdot \cdot \right)$ is processed several times using the EMD algorithm until the first IMF component is generated. The second IMF component is expressed as:$$IM{{F}^{\prime}}_{2}\left(t\right)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}{E}_{1}\left({r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({n}^{i}\left(t\right)\right)\right)}$$
- (4)
- Perform step (3) above for the other remaining stages ($j=2,3,\cdot \cdot \cdot J$), then the $j+1$ IMF component is expressed as:$${r}_{j}\left(t\right)={r}_{j-1}-IM{F}_{j}\left(t\right)$$$$IM{{F}^{\prime}}_{j+1}\left(t\right)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}{E}_{1}\left({r}_{j}\left(t\right)+{\epsilon}_{j}{E}_{j}\left({n}^{i}\left(t\right)\right)\right)}$$
- (5)
- Add 1 for j and repeat step (4) until the residual sequence cannot be processed. The number of IMF components is J. The final calculated residual sequence is expressed as:$$r\left(t\right)=x\left(t\right)-{\displaystyle \sum _{j=1}^{J}IM{F}_{j}{}^{\prime}\left(t\right)}$$
- (6)
- The original signal $x(t)$ represented by the IMF component and the residual component is expressed as:$$x\left(t\right)={\displaystyle \sum _{j=1}^{J}IM{F}_{j}^{\prime}\left(t\right)+r\left(t\right)}$$

#### 2.1.2. DFA Algorithm

- (1)
- $\overline{x}(i)$ is defined as the average of the time series $x(i)$ in the time intervals [1,N], and denoted as:$$\overline{x}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}x\left(i\right)}$$
- (2)
- The time series y(k) is segmented into segments of length n. It is denoted as:$$y\left(k\right)={\displaystyle \sum _{i=1}^{k}\left[x\left(i\right)-\overline{x}\right]},k=1,2,3,\cdot \cdot \cdot $$
- (3)
- The trend ${y}_{s}\left(i\right)$ of each series segment is calculated as:$${y}_{s}\left(i\right)={\displaystyle \sum _{n=0}^{k}{a}_{n}{i}^{n}}$$
- (4)
- After removing the uncertain trend in each series segment, the second-order fluctuation coefficient of the segment series is expressed as:$${F}^{2}\left(n,s\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left(y\left[\left(s-1\right)n+i\right]-{y}_{s}\left(i\right)\right)}^{2}}$$$${F}_{q}\left(n\right)={\left[\frac{1}{{N}_{n}}{\displaystyle \sum _{s=1}^{{N}_{s}}{F}^{2}\left(n,s\right)}\right]}^{1/2}$$
- (5)
- Change the segment length n in step (1), and repeat steps (2) and (3) to obtain the change in the fluctuation function of the time series. The correlation of the time series represented by the Hurst function is expressed as:$$H=\frac{{\mathrm{log}}_{2}{F}_{q}\left(n\right)}{{\mathrm{log}}_{2}\left(n\right)}=\frac{{\mathrm{log}}_{2}\left[\frac{1}{{N}_{n}}{\displaystyle \sum _{s=1}^{{N}_{n}}{F}^{2}\left(n,s\right)}\right]}{{\mathrm{log}}_{2}\left(n\right)}$$
- (6)
- The relationship between the scalar function α and the time series fluctuation function F(n) is expressed as:$$F\left(n\right)\propto {n}^{\alpha}$$

#### 2.1.3. Improved Wavelet Threshold Function

- (1)
- A good continuity is maintained at the set threshold;
- (2)
- The threshold function has monotonicity and continuity when the wavelet coefficient is greater than the set threshold;
- (3)
- The threshold function should have an asymptote, and the curve $y\left(x\right)=x$ can overlap with the asymptote.

- (1)
- When ${\psi}_{j,k}\to \lambda $, ${e}^{-\left(\left|{\psi}_{j,k}\right|-\lambda \right)/k}\to 1$ and ${\stackrel{\u2322}{\psi}}_{j,k}\to 0$, the improved wavelet threshold function is continuous at the threshold $\lambda $. When ${\psi}_{j,k}\to \infty $, ${e}^{-\left(\left|{\psi}_{j,k}\right|-\lambda \right)/k}\to 0$ and ${\stackrel{\u2322}{\psi}}_{j,k}\to {\psi}_{j,k}$, the improved wavelet threshold function is an asymptote, which makes the reconstructed signal closer to the actual value.
- (2)
- When $k\to 0$, the properties of the improved wavelet threshold function are close to the hard threshold function. When $k\to \infty $, the properties of the improved wavelet threshold function are close to the soft threshold function.
- (3)
- When $\left|{\psi}_{j,k}\right|\le \lambda $, a very large part of the wavelet coefficient is noise, and the a value is adjusted to be as small as possible to remove the noise interference.

#### 2.2. The Validation of CEEMDAN-DFA-Improved Wavelet Threshold Function Denoising Algorithm

## 3. Fault Feature Extraction and Identification Algorithm of the QPSO-MPE-SVM

#### 3.1. Basic Algorithm Related to the QPSO-MPE-SVM

#### 3.1.1. MPE Algorithm

- (1)
- The initial time series $x\left(i\right)$ is coarsely granularized to obtain the coarse-grained series ${y}_{j}^{\tau}$, which is calculated as follow:$${y}_{j}^{\tau}=\frac{1}{\tau}{\displaystyle \sum _{i=(j-1)+1}^{\tau}{x}_{i}}\hspace{1em}1\le j\le \frac{N}{\tau}$$
- (2)
- Calculate multi-scale alignment entropy based on sequence ${y}_{j}^{\tau}$.$$MPE(x,\tau ,m,\lambda )=PE({y}_{j}^{\tau},m,\lambda )$$

#### 3.1.2. QPSO Algorithm

_{best}is particle optimum average. M is race number. P

_{g}is the global optimal solution for the particle. $\varsigma $ is compression expansion factor.

#### 3.1.3. SVM Algorithm

#### 3.2. The Validation of the QPSO-MPE-SVM Algorithm

- (1)
- The denoised vibration signal is again disintegrated by CEEMDAN, and the IMFs are selected according to the correlation coefficient and Kurtosis values of the IMF for signal reconstruction;
- (2)
- The initial parameters of the MPE are optimized using the QPSO to obtain the better MPE parameters;
- (3)
- The MPE values of the reconstructed signals are calculated using the optimized MPE parameters, and the MPE values with obvious differentiation are selected to construct the bearing fault feature set;
- (4)
- The obtained MPE fault feature set is input to the SVM for fault identification.

#### 3.2.1. Optimize MPE Values Using QPSO

#### 3.2.2. Fault Feature Extraction and Identification Using the QPSO-MPE-SVM

## 4. Fault Diagnosis of Rolling Bearing of Sine Roller Screen Based on Vibration Signal

#### 4.1. Rolling Bearing Feature Frequency Calculation

- (1)
- Journal rotation frequency is expressed as:

- (2)
- Inner-ring fault characteristic frequency is expressed as:

- (3)
- Outer-ring fault characteristic frequency is expressed as:

- (4)
- Rolling-body fault characteristic frequency is expressed as:

#### 4.2. Rolling Bearing Fault Signal Denoising and Feature Extraction

#### 4.3. Rolling Bearing Failure Identification and Control Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Plots of IMF component correlation coefficients and Kurtosis values. (

**a**) The correlation coefficients of IMF; (

**b**) The Kurtosis values of IMF.

**Figure 6.**The HHT envelope spectrum of the inner-ring fault signal. (

**a**) The envelope spectrum analysis of the original signal; (

**b**) The envelope spectrum analysis of the reconstructed signal.

**Figure 16.**The MPE values of the four running signals. (

**a**) The MPE values using the initial parameters; (

**b**) The MPE values using the optimized parameters.

**Figure 17.**Four types of test sets identify results. (

**a**) The results of MPE-SVM fault identification on the original signal; (

**b**) the results of the QPSO-MPE-SVM fault identification on the original signal; (

**c**) the results of MPE-SVM fault identification on the denoised signal; (

**d**) the results of QPSO-MPE-SVM fault identification on the denoised signal.

IMF | 1 | 2 | 3 | 4 | 5 | 6 |

α | 0.4639 | 0.4214 | 0.3651 | 0.3124 | 0.5154 | 0.5712 |

IMF | 7 | 8 | 9 | 10 | 11 | 12 |

α | 0.6812 | 0.7948 | 0.8106 | 0.8637 | 0.8961 | 0.8942 |

Method | SNR | RMSE |
---|---|---|

CEEMDAN-DFA | 4.9134 | 0.12143 |

CEEMDAN-DFA-wavelet hard threshold function | 9.7542 | 0.08719 |

CEEMDAN-DFA-wavelet soft threshold function | 13.6718 | 0.06024 |

CEEMDAN-DFA-improved wavelet threshold function | 15.2324 | 0.05047 |

Signal | $\mathit{N}$ | $\mathit{\lambda}$ | $\mathit{m}$ | $\mathit{\tau}$ |
---|---|---|---|---|

Normal signal | 1182 | 1 | 5 | 12 |

Inner-ring fault signal | 1467 | 2 | 6 | 14 |

Outer-ring fault signal | 1384 | 3 | 7 | 13 |

Rolling-body fault signal | 953 | 1 | 6 | 12 |

$\mathbf{Inner}\text{}\mathbf{Ring}\text{}\mathbf{Diameter}\text{}{\mathit{D}}_{1}$ | $\mathbf{Outer}\text{}\mathbf{Ring}\text{}\mathbf{Diameter}\text{}{\mathit{D}}_{2}$ | $\mathbf{Rolling}\text{}\mathbf{Body}\text{}\mathbf{Diameter}\text{}\mathit{d}$ | $\mathbf{Bearing}\text{}\mathbf{Mid}\text{}\mathbf{Diameter}\text{}\mathit{D}$ | $\mathbf{Number}\text{}\mathbf{of}\text{}\mathbf{Rolling}\text{}\mathbf{Bodies}\text{}\mathit{n}$ | $\mathbf{Contact}\text{}\mathbf{Angle}\text{}\mathit{\theta}$ |
---|---|---|---|---|---|

70 mm | 130 mm | 20.43 mm | 100 mm | 8 | 0° |

Rotational Speed | Inner Ring Fault Frequency | Outer Ring Fault Frequency | Rolling Body Fault Frequency |
---|---|---|---|

500 r/min | 40.143 Hz | 26.523 Hz | 19.545 Hz |

IMF | IMF_{1} | IMF_{2} | IMF_{3} | IMF_{4} | IMF_{5} | IMF_{6} | IMF_{7} | IMF_{8} |
---|---|---|---|---|---|---|---|---|

$\alpha $ value | 0.4138 | 0.4096 | 0.2961 | 0.4537 | 0.5068 | 0.5564 | 0.6224 | 0.6743 |

IMF | IMF_{1} | IMF_{2} | IMF_{3} | IMF_{4} | IMF_{5} | IMF_{6} | IMF_{7} | IMF_{8} |
---|---|---|---|---|---|---|---|---|

Correlation coefficient | 0.9214 | 0.4327 | 0.3961 | 0.2715 | 0.2049 | 0.0621 | 0.0592 | 0.0357 |

Kurtosis value | 4.5217 | 4.9371 | 4.1964 | 3.4922 | 3.2614 | 2.5147 | 2.2291 | 2.0634 |

Signal | N | λ | m | τ |
---|---|---|---|---|

Normal Signal | 1285 | 1 | 5 | 12 |

Inner-ring fault signal | 1836 | 2 | 7 | 14 |

Outer-ring fault signal | 1587 | 1 | 6 | 14 |

Rolling-body fault signal | 1054 | 1 | 5 | 13 |

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**MDPI and ACS Style**

Wang, Y.; Xu, C.; Wang, Y.; Cheng, X.
A Comprehensive Diagnosis Method of Rolling Bearing Fault Based on CEEMDAN-DFA-Improved Wavelet Threshold Function and QPSO-MPE-SVM. *Entropy* **2021**, *23*, 1142.
https://doi.org/10.3390/e23091142

**AMA Style**

Wang Y, Xu C, Wang Y, Cheng X.
A Comprehensive Diagnosis Method of Rolling Bearing Fault Based on CEEMDAN-DFA-Improved Wavelet Threshold Function and QPSO-MPE-SVM. *Entropy*. 2021; 23(9):1142.
https://doi.org/10.3390/e23091142

**Chicago/Turabian Style**

Wang, Yi, Chuannuo Xu, Yu Wang, and Xuezhen Cheng.
2021. "A Comprehensive Diagnosis Method of Rolling Bearing Fault Based on CEEMDAN-DFA-Improved Wavelet Threshold Function and QPSO-MPE-SVM" *Entropy* 23, no. 9: 1142.
https://doi.org/10.3390/e23091142