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

^{1}

^{2}

^{3}

^{*}

## 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

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**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