# Performance Degradation Assessment of Rolling Element Bearings Based on an Index Combining SVD and Information Exergy

^{1}

^{2}

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^{*}

## Abstract

**:**

**PACS Codes:**65.40.gd; 33.20.Tp; 07.05.Rm; 43.40.At; 91.55.Ax

## 1. Introduction

## 2. Information Exergy

#### 2.1. Generalized Information Entropy

_{i}, limited to A

_{i}∩A

_{j}= Φ (i≠ j) and C = ∪A

_{i}, is a division of the symbol sequence information C generated by the discrete system. The function μ(.) is the measurement defined on the dividing space, much like a probability mass function.

**Y**={y

_{1}, y

_{2}, …, y

_{L}}. Following Equation (1), the same normalized form of generalized information entropy can be obtained as:

#### 2.2. Information Exergy

**E**= [E(1), E(2),…, E(M − 1)]

^{T}(M is the total number of instants and T is the transposition operator) can be obtained by assigning all sequentially evaluated information exergy in a vector form. The information exergy vector integrates multiple pieces of condition monitoring information in sequential instants or possible operating conditions and process information is obtained instead of instant information, thus uncertainties of operational condition variations and temporal variabilities can be reduced using the information exergy vector.

**F**can be accordingly constructed by arranging information exergy vector

**E**of each individual sensing point in different columns respectively, as:

**F**for errors of one or more sensors can be compromised by the remaining sensors, thus robust machine condition monitoring can be expected based on information exergy matrix.

## 3. Information Exergy Index for Degradation Assessment

#### 3.1. Information Exergy Index Combining SVD and Information Exergy

**F**of size (M−1)×K, its SVD is as:

**U**is a (M−1)×(M−1) orthogonal matrix and

**V**is a K×K orthogonal matrix, while

**Λ**is a (M−1)×K diagonal matrix as:

_{i}(for i = 1, 2,…, rank(

**F**)) are called singular values of the matrix

**F**and they are arranged in a descending order, i.e., σ

_{1}≥ σ

_{2}≥ … ≥ σ

_{rank(}

_{F}_{)}.

**F**can be reconstructed from the first Q eigenvectors as

**F**) by another matrix (

**F̂**) of smaller rank in a least square sense, so that SVD is equivalent to principal component analysis (PCA) [41].

**F**and only a few of them are needed to preserve major information of the matrix

**F**. Therefore, truncated singular values [σ

_{1}, σ

_{2},…, σ

_{N}]

^{T}are defined as information exergy index here to facilitate the application of information exergy. For determination of the preserved singular values number (N), cumulative contribution limit (or explained variance) method is used:

#### 3.2. Degradation Assessment Based on the Proposed Information Exergy Index

## 4. Experimental Validation of Information Exergy Index

#### 4.1. Experiment Description

#### 4.2. Experimental Results and Analysis

#### 4.2.1. Inner Raceway Degradation Assessment

#### 4.2.2. Rolling Element Degradation Assessment

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zio, E.; Gola, G. A neuro-fuzzy technique for fault diagnosis and its application to rotating machinery. Reliab. Eng. Syst. Saf
**2009**, 94, 78–88. [Google Scholar] - Jardine, A.K.S.; Lin, D.M.; Banjevic, D. A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mech. Syst. Signal Process
**2006**, 20, 1483–1510. [Google Scholar] - Chen, B.; Yan, Z.L.; Chen, W. Defect detection for wheel-bearings with time-spectral kurtosis and entropy. Entropy
**2014**, 16, 607–626. [Google Scholar] - Randall, R.B.; Antoni, J. Rolling element bearing diagnostics—A tutorial. Mech. Syst. Signal Process
**2011**, 25, 485–520. [Google Scholar] - Liao, L.X.; Lee, J. A novel method for machine performance degradation assessment based on fixed cycle features test. J. Sound Vib
**2009**, 326, 894–908. [Google Scholar] - Niu, G.; Yang, B.S.; Pecht, M. Development of an optimized condition-based maintenance system by data fusion and reliability-centered maintenance. Reliab. Eng. Syst. Saf
**2010**, 95, 786–796. [Google Scholar] - Yan, R.Q.; Gao, R.X.; Wang, C.T. Experimental evaluation of a unified time-scale-frequency technique for bearing defect feature extraction. J. Vib. Acoust
**2009**. [Google Scholar] - Tao, B.; Zhu, L.M.; Ding, H.; Xiong, Y.L. An alternative time-domain index for condition monitoring of rolling element bearings—A comparison study. Reliab. Eng. Syst. Saf
**2007**, 92, 660–670. [Google Scholar] - Feng, Z.P.; Zuo, M.J.; Hao, R.J.; Chu, F.L.; Badaoui, M.E. Gear damage assessment based on cyclic spectral analysis. IEEE Trans. Reliab
**2011**, 61, 21–32. [Google Scholar] - Pan, Y.N.; Chen, J.; Li, X.L. Spectral entropy: A complementary index for rolling element bearing performance degradation assessment. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci
**2009**, 223, 1223–1231. [Google Scholar] - Hong, H.B.; Liang, M. Fault severity assessment for rolling element bearings using the Lempel-Ziv complexity and continuous wavelet transform. J. Sound Vib
**2009**, 320, 452–468. [Google Scholar] - Yu, J.B. Local and nonlocal preserving projection for bearing defect classification and performance assessment. IEEE Trans. Ind. Electron
**2012**, 59, 2363–2376. [Google Scholar] - Liu, W.B.; Zhong, X.; Lee, J.; Liao, L.X.; Zhou, M. Application of a novel method for machine performance degradation assessment based on Gaussian mixture model and Logistic regression. Chin. J. Mech. Eng
**2011**, 24, 879–884. [Google Scholar] - Shen, Z.J.; He, Z.J.; Chen, X.F.; Sun, C.; Liu, Z.W. A monotonic degradation assessment index of rolling bearings using fuzzy support vector data description and running time. Sensors
**2012**, 12, 10109–10135. [Google Scholar] - Hong, S.; Zhou, Z.; Zio, E.; Hong, K. Condition assessment for the performance degradation of bearing based on a combinatorial feature extraction method. Digit. Signal Process
**2014**, 27, 159–166. [Google Scholar] - Xu, R.Z.; Xie, L.; Zhang, M.C.; Li, C.X. Machine degradation analysis using fuzzy CMAC neural network approach. Int. J. Adv. Manuf. Technol
**2008**, 36, 765–772. [Google Scholar] - Moghaddass, R.; Zuo, M.J. A parameter estimation method for a condition-monitored device under multi-state deterioration. Reliab. Eng. Syst. Saf
**2012**, 106, 94–103. [Google Scholar] - Lee, S.; Li, L.; Ni, J. Online degradation assessment and adaptive fault detection using modified hidden Markov model. J. Manuf. Sci. Eng
**2010**. [Google Scholar] - Hu, J.Q.; Zhang, L.B.; Liang, W. Dynamic degradation observer for bearing fault by MTS-SOM system. Mech. Syst. Signal Process
**2013**, 36, 385–400. [Google Scholar] - Zhang, Y.; Zuo, H.F.; Bai, F. Classification of fault location and performance degradation of a roller bearing. Measurement
**2013**, 46, 1178–1189. [Google Scholar] - Tamilselvan, P.; Wang, P.F. Failure diagnosis using deep belief learning based health state classification. Reliab. Eng. Syst. Saf
**2013**, 115, 124–135. [Google Scholar] - Widodo, A.; Yang, B.S. Application of relevance vector machine and survival probability to machine degradation assessment. Expert Syst. Appl
**2011**, 38, 2592–2599. [Google Scholar] - Safizadeh, M.S.; Latifi, S.K. Using multi-sensor data fusion for vibration fault diagnosis of rolling element bearings by accelerometer and load cell. Inf. Fusion
**2014**, 18, 1–8. [Google Scholar] - Xiao, W.B.; Chen, J.; Dong, G.M.; Zhou, Y.; Wang, Z.Y. A multichannel fusion approach based on coupled hidden Markov models for rolling element bearing fault diagnosis. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci
**2012**, 226, 202–216. [Google Scholar] - Lin, Y.F.; Chen, M.Y.; Zhou, D.H. Online probabilistic operational safety assessment of multi-mode engineering systems using Bayesian methods. Reliab. Eng. Syst. Saf
**2013**, 119, 150–157. [Google Scholar] - Li, Z.X.; Yan, X.P. Study on data fusion of multi-dimensional sensors for health monitoring of rolling bearings. Insight-Non-Destr. Test. Cond. Monit
**2013**, 55, 147–151. [Google Scholar] - Pan, Y.N.; Chen, J.; Dong, G.M. A hybrid model for bearing performance degradation assessment based on support vector data description and fuzzy c-means. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci
**2009**, 223, 2687–2695. [Google Scholar] - Lybeck, N.; Marble, S.; Morton, B. Validating prognostic algorithms: A case study using comprehensive bearing fault data. Proceedings of the 2007 IEEE Aerospace Conference, Big Sky, MT, USA, 3–10March 2007; pp. 1–9.
- Jiang, L.L.; Liu, Y.L.; Li, X.J.; Chen, A.H. Degradation assessment and fault diagnosis for roller bearing based on AR model and fuzzy cluster analysis. Shock Vib
**2011**, 18, 127–137. [Google Scholar] - Zhang, B.; Sconyers, C.; Byington, C.; Patrick, R.; Orchard, M.; Vachtsevanos, G. A probabilistic fault detection approach: application to bearing fault detection. IEEE Trans. Ind. Electron
**2011**, 58, 2011–2018. [Google Scholar] - Sawalhi, N.; Randall, R.B. Simulating gear and bearing interactions in the presence of faults: Part I. The combined gear bearing dynamic model and the simulation of localised bearing faults. Mech. Syst. Signal Process
**2008**, 22, 1924–1951. [Google Scholar] - Shannon, C.E. A mathematical theory of communication. Mob. Comput. Commun. Rev
**2001**, 5, 3–55. [Google Scholar] - Wu, B.F.; Wang, K.C. Robust endpoint detection algorithm based on the adaptive band-partitioning spectral entropy in adverse environments. IEEE Trans. Speech Audio Process
**2005**, 13, 762–775. [Google Scholar] - Yu, D.J.; Yang, Y.; Cheng, J.S. Application of time-frequency entropy method based on Hilbert-Huang transform to gear fault diagnosis. Measurement
**2007**, 40, 823–830. [Google Scholar] - Wu, S.D.; Wu, C.W.; Wu, T.Y.; Wang, C.C. Bearing fault diagnosis based on multiscale permutation entropy and support vector machine. Entropy
**2012**, 14, 1343–1356. [Google Scholar] - Ren, W.X.; Sun, Z.S. Structural damage identification by using wavelet entropy. Eng. Struct
**2008**, 30, 2840–2849. [Google Scholar] - Sciubba, E.; Wall, G. A brief commented history of exergy from the beginnings to 2004. Int. J. Thermodyn
**2007**, 10, 1–26. [Google Scholar] - Chen, F.; Huang, S.H.; Yang, T.; Gao, W.; He, G.Q. Information exergy diagnosis method of vibration faults of rotating machinery. J. Mech. Eng
**2009**, 45, 65–71. [Google Scholar] - Zhang, B.; Zhang, L.J.; Xu, J.W.; Liu, J. Information exergy-based method for structural damage diagnosis. J. Vibroeng
**2013**, 15, 1606–1618. [Google Scholar] - Rajwade, A.; Rangarajan, A.; Banerjee, A. Image denoising using the higher order singular value decomposition. IEEE Trans. Pattern Anal. Mach. Intell
**2013**, 35, 49–862. [Google Scholar] - Shirali, G.A.; Mohammadfam, I.; Ebrahimipour, V. A new method for quantitative assessment of resilience engineering by PCA and NT approach: A case study in a process industry. Reliab. Eng. Syst. Saf
**2013**, 119, 88–94. [Google Scholar] - Tamura, M.; Tsujita, S. A study on the number of principal components and sensitivity of fault detection using PCA. Comput. Chem. Eng
**2007**, 31, 1035–1046. [Google Scholar] - Ocak, H.; Loparo, K.A. Estimation of the running speed and bearing defect frequencies of an induction motor from vibration data. Mech. Syst. Signal Process
**2004**, 18, 515–533. [Google Scholar] - Zhang, L.J.; Xu, J.W.; Yang, J.H.; Yang, D.B.; Wang, D.D. Multiscale morphology analysis and its application to fault diagnosis. Mech. Syst. Signal Process
**2008**, 22, 597–610. [Google Scholar] - Yu, J.B. Bearing performance degradation assessment using locality preserving projections. Expert Syst. Appl
**2011**, 38, 7440–7450. [Google Scholar]

**Figure 4.**RMS errorbar graph of inner race degradation vibration signals under four operating conditions: (

**a**) drive end and (

**b**) fan end.

**Figure 5.**SEn errorbar graph of inner race degradation vibration signals under four operating conditions: (

**a**) drive end and (

**b**) fan end.

**Figure 6.**Information exergy indices errorbar graph of inner race degradation vibration signals: (

**a**) SEx; (

**b**) Mv; (

**c**) Std and (

**d**) Pr.

**Figure 7.**RMS errorbar graph of rolling element degradation vibration signals under four operating conditions: (

**a**) drive end and (

**b**) fan end.

**Figure 8.**SEn errorbar graph of rolling element degradation vibration signals under four operating conditions: (

**a**) drive end and (

**b**) fan end.

**Figure 9.**Information exergy indices errorbar graph of rolling element degradation vibration signals: (

**a**) SEx; (

**b**) Mv; (

**c**) Std and (

**d**) Pr.

Step 1 | Condition monitoring signals of multiple instants and multiple sensors are preprocessed for calculation of generalized information entropy, such as fast Fourier transform (FFT). |

Step 2 | Specific generalized information entropy such as spectral information entropy of multi-instant and multi-sensor condition monitoring signals is calculated. |

Step 3 | Information exergy matrix is constructed using the specific generalized information entropies of multiple instants and multiple sensors. |

Step 4 | Information exergy index is extracted by SVD of the above information exergy matrix and singular value truncating. |

Step 5 | Empirical relationship between information exergy index and degradation severity is established by clustering information exergy indices with known degradation severities. |

Step 6 | Degradation severity is assessed by the established empirical relationship when new information exergy index is input. |

Index | Normal | 7 Mils Fault | 14 Mils Fault | 21 Mils Fault |
---|---|---|---|---|

SEx | 2.9817(0.0060) * | 3.6606(0.0137) * | 4.0467(0.0034) * | 3.9153(0.0080) * |

Mv | 0 | 0.2536(0.0845) | 0.3987(0.0208) * | 0.3432(0.0361) * |

Std | 0 | 0.0170(0.1480) * | 0.0392(0.0487) * | 0.0810(0.0772) * |

Pr | 0 | 0.3607(0.0812) * | 0.5659(0.0207) * | 0.5598(0.0259) * |

Index | Normal | 7 Mils Fault | 14 Mils Fault | 21 Mils Fault |
---|---|---|---|---|

SEx | 2.9817(0.0040) * | 3.5170(0.0047) * | 3.5826(0.0094) * | 3.4769(0.0112) * |

Mv | 0 | 0.1901(0.0569) * | 0.2147(0.0981) * | 0.1838(0.1247) * |

Std | 0 | 0.0161(0.0749) * | 0.0221(0.1448) * | 0.0113(0.3443) * |

Pr | 0 | 0.2697(0.0555) * | 0.3200(0.0770) * | 0.2648(0.1249) * |

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**MDPI and ACS Style**

Zhang, B.; Zhang, L.; Xu, J.; Wang, P.
Performance Degradation Assessment of Rolling Element Bearings Based on an Index Combining SVD and Information Exergy. *Entropy* **2014**, *16*, 5400-5415.
https://doi.org/10.3390/e16105400

**AMA Style**

Zhang B, Zhang L, Xu J, Wang P.
Performance Degradation Assessment of Rolling Element Bearings Based on an Index Combining SVD and Information Exergy. *Entropy*. 2014; 16(10):5400-5415.
https://doi.org/10.3390/e16105400

**Chicago/Turabian Style**

Zhang, Bin, Lijun Zhang, Jinwu Xu, and Pingfeng Wang.
2014. "Performance Degradation Assessment of Rolling Element Bearings Based on an Index Combining SVD and Information Exergy" *Entropy* 16, no. 10: 5400-5415.
https://doi.org/10.3390/e16105400