# Research and Development of a Chaotic Signal Synchronization Error Dynamics-Based Ball Bearing Fault Diagnostor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Ball Bearing Test System

## 3. Chaos-Fractal-Extenics Theory

#### 3.1. Data Preprocessing

#### 3.2. The Chen-Lee Chaos Synchronization System

_{1}, y

_{1}and z

_{1}of the Master System a zero voltage signal.

_{2}, and the vibration signal can be expressed as x[1667].The chaotic system is a three-dimensional system. This paper uses the concept of phase space reconstruction to reconstruct the one-dimensional signal y to be measured as a three-dimensional signal. Equation (4) is obtained, corresponding to the vibration signal used in this paper to obtain Equation (5).

_{1}= x

_{1}− x

_{2}, e

_{2}= y

_{1}− y

_{2}, e

_{3}= z

_{1}− z

_{2}, so as to obtain the dynamic error equation of the Chen-Lee system (Equation (4)). The small signal amplified by chaos theory can be obtained.

_{1}, ė

_{2}and ė

_{3}generated by the chaos synchronization system in Figure 4. The differences among various states can be observed from two of them. For example, Figure 5 uses the signals of ė

_{1}and ė

_{2}as the two-dimensional signal of normal bearing in two-dimensional motion trajectory. The master system bears a normal zero voltage signal; the slave system bears 1665 data of ball bearing signals to be measured. In the slave system, y

_{1}bears Sth data to [(s + 1664) − 2]th data; y

_{2}bears (s + 1)th data to [(s + 1664) − 1]th data; y

_{3}bears (s + 2)th data to [(s + 1664)]th data.

#### 3.3. Fractal Theory

_{1}, e

_{2}and e

_{3}of chaos theory into a 10 × 10 three-dimensional characteristic matrix. Then it is magnified to 64 × 64 by nearest-neighbor interpolation, so as to amplify the error signal. Matlab displays the matrix in graph form. Figure 6 shows significant differences. Figure 7 shows the process of building the characteristic matrix. The detailed information of the computation of the matrix can be found in [25].

#### 3.3.1. Fractal Dimension

_{s})]. The fractal box dimension can be estimated from s = L/M. Where L can be a positive integer, s is the calculated fractal dimension value and M is the magnification of the object. Figure 8 shows the condition of L = 6.

#### 3.3.2. Lacunarity

#### 3.4. Extenics Theory

_{0}= <a, b> is in the real domain; Equation (7) can be regarded as the “distance” between x and x

_{0}.

## 4. Results and Discussion

#### 4.1. Fourier Analysis

#### 4.2. Wavelet Packet Analysis

#### 4.3. Chaos Synchronization

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Bianchini, C.; Immovilli, F.; Cocconcelli, M.; Rubini, R.; Bellini, A. Fault detection of linear bearings in brushless ac linear motors by vibration analysis. IEEE Trans. Ind. Electron
**2011**, 58, 1684–1694. [Google Scholar] - Immovilli, F.; Bellini, A.; Rubini, R.; Tassoni, C. Diagnosis of bearing faults in induction machines by vibration or current signals: A critical comparison. IEEE Trans. Ind. Appl
**2010**, 46, 1350–1359. [Google Scholar] - Immovilli, F.; Cocconcelli, M.; Bellini, A.; Rubini, R. Detection of generalized-roughness bearing fault by spectral-kurtosis energy of vibration or current signals. IEEE Trans. Ind. Electron
**2009**, 56, 4710–4717. [Google Scholar] - Frosini, L.; Bassi, E. Stator current and motor efficiency as indicators for different types of bearing faults in induction motors. IEEE Trans. Ind. Electron
**2010**, 57, 244–251. [Google Scholar] - Lau, E.C.C.; Ngan, H.W. Detection of motor bearing outer raceway defect by wavelet packet transformed motor current signature analysis. IEEE Trans. Instrum. Meas
**2010**, 59, 2683–2690. [Google Scholar] - Zhou, W.; Lu, B.; Habetler, T.G.; Harley, R.G. Incipient bearing fault detection via motor stator current noise cancellation using Wiener filter. IEEE Trans. Ind. Appl
**2009**, 45, 1309–1317. [Google Scholar] - Zhou, W.; Habetler, T.G.; Harley, R.G. Bearing condition monitoring methods for electric machines: A general review. Proceedings of IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives (SDEMPED 2007), Cracow, Poland, 6–8 September 2007.
- Frosini, L.; Bassi, E. Bearing fault diagnosis for direct-drive wind turbines via current-demodulated signals. IEEE Trans. Ind. Electron
**2013**, 60, 3419–3428. [Google Scholar] - Amarnath, M.; Praveen Krishna, I.R. Empirical mode decomposition of acoustic signals for diagnosis of faults in gears and rolling element bearings. IET Sci. Meas. Technol
**2012**, 6, 279–287. [Google Scholar] - Tavakkoli, F.; Teshnehlab, M. A ball bearing fault diagnosis method based on wavelet and EMD energy entropy mean. Proceedings of International Conference on Intelligent and Advanced Systems (ICIAS 2007), Kuala Lumpur, Malaysia, 25–28 November 2007.
- Seryasat, O.R.; Shoorehdeli, M.A.; Honarvar, F.; Rahmani, A. Multi-fault diagnosis of ball bearing using FFT, wavelet energy entropy mean and root mean square (RMS). Proceedings of IEEE International Conference on Systems Man and Cybernetics (SMC 2010), Istanbul, Turkey, 10–13 October 2010.
- Chen, K.; Li, X.; Wang, F.; Wang, T.; Wu, C. Bearing fault diagnosis using wavelet analysis. Proceedings of International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (ICQR2MSE 2012), Chengdu, China, 15–18 June 2012.
- Fang, S.; Wei, Z. Rolling bearing fault diagnosis based on wavelet packet and RBF neural network. Proceedings of Chinese Control Conference (CCC 2007), Zhangjiajie, China, 26–31 July 2007.
- Prieto, M.D.; Cirrincione, G.; Cocconcelli, M.; Espinosa, A.G.; Ortega, J.A. Bearing fault detection by a novel condition-monitoring scheme based on statistical-time features and neural networks. IEEE Trans. Ind. Electron
**2013**, 60, 3398–3407. [Google Scholar] - Liu, J.; Wang, W.; Golnaraghi, F. An enhanced diagnostic scheme for bearing condition monitoring. IEEE Trans. Instrum. Meas
**2010**, 59, 309–321. [Google Scholar] - Yang, Y.; Tang, W. Study of remote bearing fault diagnosis based on BP neural network combination. Proceedings of the Seventh International Conference on Natural Computation (ICNC 2011), Shanghai, China, 26–28 July 2011.
- Meng, X.; Ni, J.; Zhu, Y. Research on vibration signal filtering based on wavelet multi-resolution analysis. Proceedings of International Conference on Artificial Intelligence and Computational Intelligence (AICI 2010), Sanya, China, 23–24 October 2010.
- Liu, D.; Michel, A.N. Sparsely interconnected neural networks for associative memories with applications to cellular neural networks. IEEE Trans. Circuits Syst. II
**1994**, 41, 295–347. [Google Scholar] - Li, F.; Gong, W.; Li, Y.; Liang, Y.; Wang, X. Research of fractal dimension calculation algorithm based on Mobile Box-Counting Method. Proceedings of the Seventh International Conference on Natural Computation (ICNC 2011), Shanghai, China, 26–28 July 2011.
- Zheng, H.; Wang, J.; Liu, D.; Peng, G. Dimension estimation of subdivision fractal and its application. Proceedings of the 5th International Conference on Computer Science and Education (ICCSE 2010), Hefei, China, 24–27 August 2010.
- Cheng, Y.; Sun, J.; Hua, J.; Li, X. A method of calculating image fractal dimension based on fractal Brownian mode. Proceedings of the International Forum on Information Technology and Applications (IFITA 2010), Kunming, China, 16–18 July 2010.
- Case Western Reserve University Bearing Data Center Website. Available online: http://csegroups.case.edu/bearingdatacenter/home accessed on 30 August 2014.
- Chen, H.; Li, C. Anti-control of chaos in rigid body motion. Chaos Solitons Fractals
**2004**, 21, 957–965. [Google Scholar] - Yau, H.T.; Kuo, C.L.; Yan, J.J. Fuzzy sliding mode control for a class of chaos synchronization with uncertainties. Int. J. Nonlinear Sci. Numer. Simul
**2006**, 7, 333–338. [Google Scholar] - Chen, H.-C.; Yau, H.-T.; Chen, P.-Y. Chaos synchronization error technique-based defect pattern recognition for GIS through partial discharge signal analysis. Entropy
**2014**, 16, 4566–4582. [Google Scholar] - Poirier, J.R.; Aubert, H.; Jaggard, D.L. Lacunarity of rough surfaces from the wavelet analysis of scattering data. IEEE Trans. Antennas Propag
**2009**, 57, 2130–2136. [Google Scholar] - Lu, M. The study of fault diagnosis algorithm based on extension neural network. Proceedings of the 2nd IEEE International Conference on Information and Financial Engineering (ICIFE 2010), Chongqing, China, 17–19 September 2010.
- Tao, R.; Li, Y.L.; Wang, Y. Short-time fractional Fourier transform and its applications. IEEE Trans. Signal Process
**2009**, 58, 2568–2580. [Google Scholar] - Qiu, H.; Lee, J.; Lin, J.; Yu, G. Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics. J. Sound Vib
**2006**, 289, 1066–1090. [Google Scholar]

**Figure 4.**Three-dimensional diagram of chaotic signals of one revolution in various bearing states in the case of 21 mil, Hp = 0.

**Figure 5.**Two-dimensional diagram of chaotic signals of one revolution in various bearing states in the case of 21 mil, Hp = 0.

**Figure 6.**Schematic diagram of the characteristic matrix of the bearing in various states in the case of 21 mil, Hp = 0.

**Figure 13.**Wavelet packet signal of the fundamental frequency band in various bearing states in the case of 21 mil, Hp = 0.

**Figure 14.**Schematic diagram of the wavelet packet signal characteristic matrix image of the fundamental frequency band in various bearing states in the case of 21 mil, Hp = 0.

**Figure 16.**Extracted characteristics of the bearing in various states in the case of 21 mil, Hp = 0.

System quantities | Values/Conditions |
---|---|

Sampling frequency | 12 K/48 K (Hz) |

Motor load | 0/1/2/3 (Hp) |

Fault diameter | 7/14/21 (mil) |

Fault depth | 11 (mil) |

Fault state | Normal/inner ring fault/outer ring fault/ball fault |

System quantities | Values | |||
---|---|---|---|---|

Load | 0 Hp | 1 Hp | 2 Hp | 3 Hp |

Motor speed (rpm) | 1797 | 1772 | 1750 | 1730 |

Motor speed (rps) | 29.95 | 29.53333333 | 29.16666667 | 28.83333333 |

Period (s) | 0.033388982 | 0.033860045 | 0.034285714 | 0.034682081 |

Sampled points per revolution | 1602 | 1625 | 1645 | 1664 |

Method | Discrete Fourier | Wavelet and fractal theory | CS and fractal theory | ||||||
---|---|---|---|---|---|---|---|---|---|

Fault diameter (mil) | 7 | 14 | 21 | 7 | 14 | 21 | 7 | 14 | 21 |

Normal state | 100% | 100% | 98.75% | 98.75% | 100% | 97.50% | 100% | 100% | 100% |

Inner ring fault | 91.25% | 91.25% | 95% | 96.25% | 100% | 98.75% | 100% | 100% | 100% |

Ball fault | 91.25% | 93.75% | 93.75% | 96.25% | 98.75% | 95% | 100% | 100% | 100% |

Outer ring fault | 93.75% | 95% | 97.50% | 95% | 95% | 98.75% | 100% | 100% | 100% |

Diagnostic rate | 95.10% | 97.50% | 100% |

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**MDPI and ACS Style**

Kuo, Y.-C.; Hsieh, C.-T.; Yau, H.-T.; Li, Y.-C. Research and Development of a Chaotic Signal Synchronization Error Dynamics-Based Ball Bearing Fault Diagnostor. *Entropy* **2014**, *16*, 5358-5376.
https://doi.org/10.3390/e16105358

**AMA Style**

Kuo Y-C, Hsieh C-T, Yau H-T, Li Y-C. Research and Development of a Chaotic Signal Synchronization Error Dynamics-Based Ball Bearing Fault Diagnostor. *Entropy*. 2014; 16(10):5358-5376.
https://doi.org/10.3390/e16105358

**Chicago/Turabian Style**

Kuo, Ying-Che, Chin-Tsung Hsieh, Her-Terng Yau, and Yu-Chung Li. 2014. "Research and Development of a Chaotic Signal Synchronization Error Dynamics-Based Ball Bearing Fault Diagnostor" *Entropy* 16, no. 10: 5358-5376.
https://doi.org/10.3390/e16105358