# Self-Adaptive Fault Feature Extraction of Rolling Bearings Based on Enhancing Mode Characteristic of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. CEEMDAN Algorithm

- (1)
- Apply EMD to decompose signal ($x\left(t\right)+{w}_{0}{\epsilon}^{i}\left(t\right)$) to extract the first IMF;$${c}_{1}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{c}_{1}^{i}\left(t\right)}\hspace{1em}i\in \left\{1,\dots ,N\right\}$$Here, $\epsilon \left(t\right)$ is white noise with unit variance, and ${w}_{0}$ is the corresponding amplitude.
- (2)
- Obtain the differential signal by Equation (2);$${r}_{1}\left(t\right)=x\left(t\right)-{c}_{1}\left(t\right)$$
- (3)
- Extract the first mode by decompose ${r}_{1}\left(t\right)+{w}_{1}{E}_{1}\left({\epsilon}^{i}\left(t\right)\right)$ and appoint the second IMF using Equation (3);$${c}_{2}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{E}_{1}}\left({r}_{1}\left(t\right)+{w}_{1}{E}_{1}\left({\epsilon}^{i}\left(t\right)\right)\right)$$
- (4)
- Decompose the $k$-th residue and extract the first mode, and the $\left(k+1\right)$-th mode can be obtained using the following equation ($k=2,\dots ,K$):$${c}_{k+1}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{E}_{1}}\left({r}_{k}\left(t\right)+{w}_{k}{E}_{k}\left({\epsilon}^{i}\left(t\right)\right)\right)$$Here, ${E}_{k}(\cdot )$ indicates the symbol function of the k-th IMF.
- (5)
- Obtain last mode by Equation (5) when the residue has fewer than (or equal to) two extrema by repeating step,$$R\left(t\right)=x\left(t\right)-{\displaystyle \sum _{k=1}^{K}{c}_{k}\left(t\right)}$$Signal $x\left(t\right)$ is decomposed using equation,$$x\left(t\right)={\displaystyle \sum _{k=1}^{K}{c}_{k}\left(t\right)}+R\left(t\right)$$

#### 2.2. De-Trended Fluctuation Algorithm (DFA)

- (1)
- Divide the following series into integrated time:$$y\left(k\right)={\displaystyle \sum _{i=1}^{k}\left[x\left(i\right)-\overline{x}\right]}\hspace{1em}k=1,2,3,\dots ,N$$Here, $\overline{x}$ indicates the mean of the series $x\left(i\right)$.
- (2)
- Slice $y\left(k\right)$ into n length sub-section;
- (3)
- Apply least square method fitting to obtain the local trend ${y}_{n}\left(k\right)$;
- (4)
- Extract ${y}_{n}\left(k\right)$ from the $y\left(k\right)$ to obtain fluctuation function $F\left(n\right)$;$$F\left(n\right)={\left(\frac{1}{N}{\displaystyle \sum _{k=1}^{N}{\left[y\left(k\right)-{y}_{n}\left(k\right)\right]}^{2}}\right)}^{\frac{1}{2}}$$
- (5)
- Obtain different $F\left(n\right)$ by different length segments;
- (6)
- Calculate the slope $\alpha $ (fractal scaling index) between $\mathrm{log}\left(F\left(n\right)\right)$ and $\mathrm{log}\left(k\right)$; the bigger slope $\alpha $ of signal, the smoother time series (as shown in Equation (9)):$$F\left(n\right)\propto {n}^{\alpha}$$

#### 2.3. Modified Hausdorff Distance (MHD)

## 3. Proposed Works

#### 3.1. Identifying of IMF with Minimum Number and Physical Meaning

_{2}and CIMF

_{3}, CIMF

_{5}and CIMF

_{6}are not similar. CIMF

_{2}combines IMF

_{1}and IMF

_{2}, and CIMF

_{3}combines IMF

_{1}, IMF

_{2}and IMF

_{3}; this illustrates that IMF

_{3}is not similar to IMF

_{1}and IMF

_{2}. Likewise, IMFs after IMF

_{6}are not similar to IMF

_{3}, IMF

_{4}and IMF

_{5}; this means that the IMF

_{3}, IMF

_{4}and IMF

_{5}can been organized into a class. Figure 7 shows a relational graph of CIMFs (‘Y’ means that there is a similarity among adjacent CIMFs, while ‘N’ means non-similarity). Thus, the CIMFs are obtained using following equation:

_{2}and FIMF

_{3}are almost in accordance with theoretic frequencies in Equation (15) (75 Hz and 10 Hz). The FIMF

_{1}and R are the noisy IMF and residue, respectively; this proves that the proposed method can be used to distinguish the frequency components of a complicated signal and avoid the interference of spurious mode.

#### 3.2. Selection of Intrinsic Information Mode

- (1)
- Decompose the vibration signal into sets of IMFs by CEEMDAN;
- (2)
- Combine the adjacent IMFs into CIMF;
- (3)
- Obtain the FC by calculating the FFT of each CIMF;
- (4)
- Get PFC by estimating the PDF of each FFT;
- (5)
- Obtain the FIMF by calculating the MHD of adjacent PDF;
- (6)
- Obtain the minimum amount of mode information with physical meanings;
- (7)
- Identify the IIM by comprehensive evaluate index;
- (8)
- Identify the feature frequency of rolling bearings using the IIM envelope.

## 4. Results and Discussions

#### 4.1. Diagnose Inner Raceway Fault of Rolling Bearing

_{10}). To extract the IIM of the vibration signal, the CIMFs were first obtained using Equation (16), and then FFT and the corresponding local zoom (main information) were conducted for each CIMF (as shown in Figure 13). It was found by comparing FC1 with FC2 that there were additional frequency bands. Similarly, the frequency component in FC6 is different from that of FC7 (the detail difference is shown Figure 14). That in FC8 is also different from that in FC9; this means that the feature difference is easy to distinguished among IMFs using the proposed method.

_{1}is not similar to CIMF

_{2}, CIMF

_{3}is not similar to CIMF

_{4}, CIMF

_{6}is not similar to CIMF

_{7}, CIMF

_{8}is not similar with CIMF

_{9}, and the residues are similar.

_{2}; this illustrates that FIMF

_{2}should be used to extract fault information from the vibration signal. Meanwhile, to compare performance of proposed method, the original CEEMDAN, combined with comprehensive evaluated index, is also used to extract fault information for rolling bearings (as shown in Figure 20). It was found that the second point was the maximum point; this means that the IMF

_{2}contains the main fault features of vibration signal. Figure 21 shows the identified IMF2 and FIMF2.

_{2}and FIMF

_{2}(as shown in Figure 22a,b). It was found that $2{f}_{r}$,${f}_{i}$,$2{f}_{i}$,$3{f}_{i}$ are all identified for IMF

_{2}and FIMF

_{2}. The envelope spectrum of FIMF

_{2}does not contain any interference frequency. However, that of IMF

_{2}abundantly contains the interference frequency; this means that the proposed method can effectively identify fault information of rolling bearings. Meanwhile, the original signal was also used for the envelope spectrum (as shown in Figure 23). It was found that there is a greater interference component compared to IMF2. In addition, the computational load of the proposed method is compared with CEEMDAN; they are 20.24 s and 19.26 s, respectively (the amplitude of added white noise and ensemble trails are 0.1 and 100, respectively). There was only a difference of 0.98 s (the used laptop has Core i5-3230M CPU, 2.60 Ghz, RAM is 6 GB). However, the decomposed effectiveness for proposed method outperformed that of CEEMDAN. Thus, the proposed method is more suitable for fault diagnoses of rolling bearings.

#### 4.2. Diagnose Outer Raceway Fault of Rolling Bearing

## 5. Conclusions

- (1)
- Our research introduces a method which combines adjacent IMFs into a novel mode function (CIMF) to enhance difference characteristics among IMFs (as shown in step 2).
- (2)
- It proposes a method of combing FFT, PDF and MHD to obtain final intrinsic information mode (FIMF) of minimum number and with physical meanings (as shown in step 3).
- (3)
- It introduces a comprehensive evaluate index (DFA and Kurtosis) to identify intrinsic information mode (IIM) to extract the feature frequencies of rolling bearings (as shown in step 4).

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

CEEMDAN | complete ensemble empirical mode decomposition with adaptive noise | IMFs | intrinsic mode functions |

CIMFs | Combined mode functions | FFT | Fast Fourier Transform |

probability density function | MHD | modified Hausdorff distance | |

FIMFs | final intrinsic mode functions | DFA | de-trended fluctuation analysis |

EMD | empirical mode decomposition | EEMD | ensemble EMD |

CEEMD | Complementary EEMD | ELMD | ensemble local mean decomposition |

CELMDAN | complete ensemble local mean decomposition with adaptive noise | K-L | Kullback-Leibler |

LMD | local mean decomposition | HD | Hausdorff distance |

FFT of CIMF | FC | PDF of FC | PFC |

IIM | intrinsic information mode | AE | approximate entropy |

SE | Sample entropy | FE | Fuzzy entropy |

VMD | variational mode decomposition |

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**Figure 10.**Experiment rig of rolling bearings [52].

**Figure 29.**(

**a**) Identifying of IIM with kurtosis method for IMF; (

**b**) Identifying of IIM with kurtosis method for PF.

Bearing Type | $\mathbf{Ball}\text{}\mathbf{Number}\mathit{n}$ | $\mathbf{Pitch}\text{}\mathbf{Diameter}\mathit{D}$ (mm) | $\mathbf{Ball}\text{}\mathbf{Diameter}\mathit{d}$ (mm) | $\mathbf{Contact}\text{}\mathbf{Angle}\mathit{\alpha}$ (°) |
---|---|---|---|---|

6205-2RS | 9 | 52 | 8 | 0 |

$\mathbf{Ball}\text{}\mathbf{Number}\mathit{n}$ | $\mathbf{Pitch}\text{}\mathbf{Diameter}\mathit{D}$ (mm) | $\mathbf{Ball}\text{}\mathbf{Diameter}\mathit{d}$ (mm) | $\mathbf{Contact}\text{}\mathbf{Angle}\mathit{\alpha}$ (°) |
---|---|---|---|

10 | 46 | 7.9 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Ma, F.; Zhan, L.; Li, C.; Li, Z.; Wang, T.
Self-Adaptive Fault Feature Extraction of Rolling Bearings Based on Enhancing Mode Characteristic of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise. *Symmetry* **2019**, *11*, 513.
https://doi.org/10.3390/sym11040513

**AMA Style**

Ma F, Zhan L, Li C, Li Z, Wang T.
Self-Adaptive Fault Feature Extraction of Rolling Bearings Based on Enhancing Mode Characteristic of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise. *Symmetry*. 2019; 11(4):513.
https://doi.org/10.3390/sym11040513

**Chicago/Turabian Style**

Ma, Fang, Liwei Zhan, Chengwei Li, Zhenghui Li, and Tingjian Wang.
2019. "Self-Adaptive Fault Feature Extraction of Rolling Bearings Based on Enhancing Mode Characteristic of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise" *Symmetry* 11, no. 4: 513.
https://doi.org/10.3390/sym11040513