# Research of Planetary Gear Fault Diagnosis Based on Permutation Entropy of CEEMDAN and ANFIS

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## Abstract

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## 1. Introduction

## 2. Building Model

#### 2.1. CEEMDAN Signal Decomposition Method

- (1)
- The white noise $X(t)+{\epsilon}_{0}{\omega}^{i}(t)$ is added to the original signal, and Ith EMD decomposition is performed. Then complete the average operation on the result to get ${\mathrm{IMF}}_{1}$.$${\mathrm{IMF}}_{1}=\frac{1}{I}{\displaystyle \sum _{i=1}^{1}{E}_{1}(X(t)+{\epsilon}_{0}{\omega}^{i}(t))}$$
- (2)
- The first stage residual component can be calculated.$${r}_{1}(t)=X(t)-{\mathrm{IMF}}_{1}$$The white noise ${r}_{1}(t)+{\epsilon}_{1}{E}_{1}({\omega}^{i}(t))$, $i=1,2,\dots ,I$ is added to the first stage residual component, and the EMD is performed. Then ${\mathrm{IMF}}_{2}$ can be calculated with the mean value of the first IMF.$${\mathrm{IMF}}_{2}=\frac{1}{I}{\displaystyle \sum _{i=1}^{1}{E}_{1}}({r}_{1}(t)+{\epsilon}_{1}{E}_{1}({\omega}^{i}(t)))$$For k = 1, 2, …, K, the Kth residual component can be calculated.$${r}_{k}\left(t\right)={r}_{k-1}\left(t\right)-{\mathrm{IMF}}_{k}$$
- (3)
- Adding white noise ${r}_{1}(t)+{\epsilon}_{1}{E}_{1}({\omega}^{i}(t))$, $i=1,2,\dots ,I$ to the kth residual component and performing EMD decomposition. Then ${\mathrm{IMF}}_{k+1}$ can be calculated with the mean value of the first IMF.$${\mathrm{IMF}}_{k+1}=\frac{1}{I}{\displaystyle \sum _{i=1}^{1}{E}_{1}({r}_{k}(t)+{\epsilon}_{k}{E}_{k}({\omega}^{i}(t))})$$
- (4)
- Repeat Step (4) and Step (5) until the value of residual component is less than two extremes, then the decomposition stops. Eventually the residual variable is obtained.$$r(t)=X(t)-{\displaystyle \sum _{k=1}^{K}{\mathrm{IMF}}_{k}}$$$$X(t)=r(t)+{\displaystyle \sum _{k=1}^{K}{\mathrm{IMF}}_{k}}$$

#### 2.2. Permutation Entropy

#### 2.3. Adaptive Neuro-Fuzzy Inference System

- (1)
- Layer 1 is input layer which is composed of square nodes, and the membership degree of the output fuzzy set corresponding to each input is calculated by blurring the input quantity. The transfer function transmitted from the first layer nodes to the second layer nodes can be expressed as follows:$$\{\begin{array}{l}{O}_{i,j}={\mu}_{{A}_{i}}({x}_{1}),i=1,2\\ {O}_{i,j}={\mu}_{{B}_{i-2}}({x}_{2}),i=3,4\end{array}$$
- (2)
- Layer 2 is rule operation layer which is composed of round nodes. Each node represents one rule. The fitness of each rule is obtained by performing product operation, which is expressed as follows:$${O}_{2,j}={w}_{i}={\mu}_{{A}_{i}}({x}_{1}){\mu}_{{B}_{i}}({x}_{2})\text{}i=1,2$$
- (3)
- Layer 3 is normalized layer which is composed of circular nodes, whose function is to the normalize fitness fuzzy rules.$${O}_{3,j}={\overline{w}}_{i}=\frac{{w}_{i}}{{w}_{1}+{w}_{2}}\text{}i=1,2$$
- (4)
- Layer 4 is rule output layer which is composed of square nodes. Each node’s transfer function is a linear function, whose role is to calculate the output of all fuzzy rules, expressed as follows:$${O}_{4,j}={\overline{w}}_{i}{f}_{i}={\overline{w}}_{i}({p}_{i}{x}_{1}+{q}_{i}{x}_{2}+{r}_{i})\text{}i=1,2$$
- (5)
- Layer 5 is output layer which is composed of round nodes, whose role is to calculate the sum of all outputs. It can be expressed as follows:$${O}_{5,i}={\displaystyle \sum _{i}\overline{{w}_{i}}{f}_{i}}=\frac{{\displaystyle {\sum}_{i}{w}_{i}{f}_{i}}}{{\displaystyle {\sum}_{i}{w}_{i}}}$$The essence of fault diagnosis research using ANFIS is to adjust the premise parameter and conclusion parameter of the model constantly. The correction method generally includes BP algorithm and hybrid algorithm. However, in the practical application of the fault diagnosis, both hybrid algorithm and BP algorithm have slow training speed, which can make algorithms easily fall into the local minimum. Therefore, this study overcomes slow convergence which BP algorithm usually has by using numerical optimization technique and Levenberg-Marquart algorithm.

## 3. Experimental Equipment and Data Acquisition

## 4. Experimental Analysis

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Lei, Y.G.; Tang, W.; Kong, D.T.; Lin, J. Vibration signal simulation and fault diagnosis of planetary gearboxes based on transmission mechanism analysis. J. Mech. Eng.
**2014**, 50, 61–68. [Google Scholar] [CrossRef] - Djeziri, M.A.; Benmoussa, S.; Sanchez, R. Hybrid method for remaining useful life prediction in wind turbine systems. Renew. Energy
**2017**, 116, 173–187. [Google Scholar] [CrossRef] - Praveenkumar, T.; Sabhrish, B.; Saimurugan, M.; Ramachandran, K.I. Pattern recognition based on-line vibration monitoring system for fault diagnosis of automobile gearbox. Measurement
**2018**, 114, 233–242. [Google Scholar] [CrossRef] - Prusa, Z.; Balazs, P.; Sondergaard, P.L. A noniterative method for reconstruction of phase from STFT magnitude. IEEE/ACM Trans. Audio Speech Lang. Process.
**2017**, 25, 1154–1164. [Google Scholar] [CrossRef] - Wigner, E.P. On the quantum correction for thermodynamic equilibrium. Phys. Rev.
**1932**, 40, 749. [Google Scholar] [CrossRef] - Ananou, B.; Ouladsine, M.; Pinaton, P.; Nguyen, L.; Djeziri, M.; Djohor, F. Fault prognosis for batch production based on percentile measure and gamma process. J. Process Control
**2016**, 48, 72–80. [Google Scholar] - Cheng, G.; Cheng, Y.L.; Shen, L.H.; Qiu, J.B.; Zhang, S. Gear fault identification based on Hilbert-Huang transform and SOM neural network. Measurement
**2013**, 46, 1137–1146. [Google Scholar] [CrossRef] - Liu, D.; Zeng, H.T.; Xiao, Z.H.; Peng, L.H.; Malik, O.P. Fault diagnosis of rotor using EMD thresholding-based de-noising combined with probabilistic neural network. J. Vibroeng.
**2017**, 19, 5920–5931. [Google Scholar] - Guo, T.; Deng, Z.M. An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing. Appl. Acoust.
**2017**, 127, 46–62. [Google Scholar] [CrossRef] - Wu, Z.H.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal.
**2005**, 1, 1–21. [Google Scholar] [CrossRef] - Choudhary, V.; Dhami, S.S.; Pabla, B.S. Non-contact incipient fault diagnosis method of fixed-axis gearbox based on CEEMDAN. R. Soc. Open Sci.
**2017**, 4, 170616. [Google Scholar] - Torres, M.E.; Colominas, M.A.; Schlotthauer, G.; Flandrin, P. A complete ensemble empirical mode decomposition with adaptive noise. Brain Res. Bull.
**2011**, 125, 4144–4147. [Google Scholar] - Li, C.W.; Zhan, L.W.; Shen, L.Q. Friction signal denoising using complete ensemble EMD with adaptive noise and mutual information. Entropy
**2015**, 17, 5965–5979. [Google Scholar] [CrossRef] - Humeau-Heurtier, A.; Mahe, G.; Abraham, P. Multi-dimensional complete ensemble empirical mode decomposition with adaptive noise applied to laser speckle contrast images. IEEE Trans. Med. Imaging
**2015**, 34, 2103–2117. [Google Scholar] [CrossRef] [PubMed] - Reljin, N.; Reyes, B.A.; Chon, K.H. Tidal volume estimation using the blanket fractal dimension of the tracheal sounds acquired by smartphone. Sensors
**2015**, 15, 9773–9790. [Google Scholar] [CrossRef] [PubMed] - Chen, J.Y.; Zhou, D.; Lyu, C.; Lu, C. An integrated method based on CEEMD-SampEn and the correlation analysis algorithm for the fault diagnosis of a gearbox under different working conditions. Mech. Syst. Signal Process.
**2017**. [Google Scholar] [CrossRef] - Shi, Z.L.; Song, W.Q.; Taheri, S. Improved LMD, permutation entropy and optimized K-Means to fault diagnosis for roller bearings. Entropy
**2016**, 18, 70. [Google Scholar] [CrossRef] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.Y.; Liang, Y.T.; Zhou, J.Z.; Zang, Y. A novel bearing fault diagnosis model integrated permutation entropy, ensemble empirical mode decomposition and optimized SVM. Measurement
**2015**, 69, 164–179. [Google Scholar] [CrossRef] - Zhen, J.D.; Cheng, J.S.; Yang, Y. Multiscale permutation entropy based rolling bearing fault diagnosis. Shock Vib.
**2014**. [Google Scholar] [CrossRef] - Jiang, P.; Hu, Z.X.; Liu, J.; Yu, S.N.; Wu, F. Fault diagnosis based on chemical sensor data with an active deep neural network. Sensors
**2016**, 16, 1695. [Google Scholar] [CrossRef] [PubMed] - Jang, J.S.R. ANFIS: Adaptive-network-based fuzzy inference systems. IEEE Trans. Syst. Man Cybern.
**1993**, 23, 665–685. [Google Scholar] [CrossRef] - Chilbani, A.; Chadli, M.; Shi, P.; Braiek, N.B. Fuzzy Fault Detection Filter Design for T-S Fuzzy Systems in Finite Frequency Domain. IEEE Trans. Fuzzy Syst.
**2017**, 25, 1051–1061. [Google Scholar] [CrossRef] - Youssef, T.; Chadli, M.; Karimi, H.R.; Wang, R. Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model. J. Frankl. Inst.
**2017**, 354, 2524–2542. [Google Scholar] [CrossRef] - Pradhan, B. A comparative study on the predictive ability of the decision tree, support vector machine and neuro-fuzzy models in landslide susceptibility mapping using GIS. Comput. Geosci.
**2013**, 51, 350–365. [Google Scholar] [CrossRef][Green Version] - Uğuz, H. Adaptive neuro-fuzzy inference system for diagnosis of the heart valve diseases using wavelet transform with entropy. Neural Comput. Appl.
**2012**, 21, 1617–1628. [Google Scholar] [CrossRef] - Cheng, G.; Chen, X.H.; Li, H.Y. Study on planetary gear fault diagnosis based on entropy feature fusion of ensemble empirical mode decomposition. Measurement
**2016**, 91, 140–154. [Google Scholar] [CrossRef] - Yan, R.Q.; Liu, Y.B.; Gao, R.X. Permutation entropy: A nonlinear statistical measure for status characterization of rotary machines. Mech. Syst. Signal Process.
**2012**, 29, 474–484. [Google Scholar] [CrossRef]

**Figure 3.**Four types of gears, (

**a**) normal gear, (

**b**) broken gear, (

**c**) gear with one missing tooth, (

**d**) gear with a tooth root crack.

**Figure 4.**Layout mode of acceleration sensor (

**a**) location of acceleration sensor, (

**b**) type of acceleration sensor.

**Figure 7.**Decomposition of broken gear, (

**a**) EEMD decomposition of broken gear, (

**b**) CEEMDAN decomposition of broken gear.

**Figure 10.**Changes of the training RMSE and the training step size during the training process: (

**a**) the changes of training RMSE; (

**b**) the changes of the training step size.

Parameter | Transmission Ratio | Pressure Angle | Material | Module | Number of Sun Gear | Number of Teeth on the Sun Gear | Number of Inner Ring Gear |
---|---|---|---|---|---|---|---|

Value | 4.57 | 20° | S45C | 1 | 28 | 36 | 100 |

Type | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 |
---|---|---|---|---|---|---|

Normal gear | 0.9422 | 0.7792 | 0.9012 | 0.8525 | 0.6132 | 0.4595 |

0.9527 | 0.7924 | 0.9119 | 0.8679 | 0.6170 | 0.4684 | |

0.9513 | 0.7930 | 0.9029 | 0.8603 | 0.6320 | 0.4837 | |

Broken gear | 0.9500 | 0.8356 | 0.9151 | 0.8431 | 0.6277 | 0.4497 |

0.9517 | 0.8199 | 0.9109 | 0.8541 | 0.6045 | 0.4583 | |

0.9517 | 0.8276 | 0.9132 | 0.8507 | 0.6069 | 0.4748 | |

Gear with one missing tooth | 0.9687 | 0.8366 | 0.8859 | 0.8774 | 0.6258 | 0.4968 |

0.9688 | 0.8379 | 0.8882 | 0.8784 | 0.6233 | 0.4889 | |

0.9694 | 0.8319 | 0.8850 | 0.8855 | 0.6130 | 0.4942 | |

Gear with a tooth root crack | 0.9286 | 0.8156 | 0.8954 | 0.8734 | 0.6245 | 0.4794 |

0.9379 | 0.8109 | 0.9009 | 0.8767 | 0.6411 | 0.4885 | |

0.9328 | 0.8161 | 0.8972 | 0.8707 | 0.6170 | 0.4776 |

**Table 3.**Recognition rate of permutation entropy of CEEMDAN and BP and permutation entropy of CEEMDAN and ANFIS.

Fault Modes | Normal Gear | Broken Gear | Gear with One Missing Tooth | Gear with a Tooth Root | Overall |
---|---|---|---|---|---|

The number of trained samples | 30 | 30 | 30 | 30 | 120 |

The number of test samples | 20 | 20 | 20 | 20 | 80 |

Permutation entropy of CEEMDAN and BP Accuracy (%) | 85 | 95 | 100 | 60 | 85 |

Permutation entropy of CEEMDAN and ANFIS Accuracy (%) | 80 | 95 | 100 | 85 | 90 |

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**MDPI and ACS Style**

Kuai, M.; Cheng, G.; Pang, Y.; Li, Y. Research of Planetary Gear Fault Diagnosis Based on Permutation Entropy of CEEMDAN and ANFIS. *Sensors* **2018**, *18*, 782.
https://doi.org/10.3390/s18030782

**AMA Style**

Kuai M, Cheng G, Pang Y, Li Y. Research of Planetary Gear Fault Diagnosis Based on Permutation Entropy of CEEMDAN and ANFIS. *Sensors*. 2018; 18(3):782.
https://doi.org/10.3390/s18030782

**Chicago/Turabian Style**

Kuai, Moshen, Gang Cheng, Yusong Pang, and Yong Li. 2018. "Research of Planetary Gear Fault Diagnosis Based on Permutation Entropy of CEEMDAN and ANFIS" *Sensors* 18, no. 3: 782.
https://doi.org/10.3390/s18030782