# Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Incipient Fault Prediction for Bearings

#### 2.1. Minimum Entropy Deconvolution

#### 2.2. Extracting Fault Feature in Frequency Domain

_{r}= 29.95 Hz, the inner ring fault’s characteristic frequency f

_{i}= 162 Hz, the outer race’s characteristic frequency f

_{o}= 107 Hz, the rolling element’s characteristic frequency f

_{rs}= 141 Hz. Then, the MED maximum cycle number is set to 30, the error is 0.01, FIR filter points is 40, so that we can verify the weak fault diagnostic effect by the MED method. Figure 2 shows the time-domain vibration signal and filtered signal with MED. Combined with the envelope spectrum, the filtered vibration signal envelope spectrum is shown in Figure 3, from which it can be seen that the envelope spectrum exists at the rotation frequency f

_{r}= 29.95 Hz, outer race fault’s characteristic frequency f

_{o}= 107 Hz, and its twice frequency, triple frequency, four times frequency components. After filtering, the fault characteristic frequency of the signal is more obvious, the rest of the frequency components are almost entirely eliminated, which can accurately realize the fault feature extraction of the outer race.

#### 2.3. Experimental Confirmation

#### 2.4. Distinguishing Equipment Weak Faults

## 3. The Fault Trend Prediction of Mechanical Equipment by FBM

#### 3.1. Calculate the Hurst Index of Vibration Intensity by R/S Method

#### 3.2. FBM Model

#### 3.2.1. Characteristic Analysis of FBM

_{0}be an arbitrary real number, Let B

_{H}(t) be fractional Brownian motion (FBM), so B

_{H}(t) is defined by [29]:

_{H}(t) is a filtered B(t) under the operation of Equation (6) and Brownian motion can be regarded as a special case of FBM.

#### 3.2.2. Stochastic Differential Equations (SDE) Based on FBM

_{H}(t) is FBM. Therefore, the methods and ideas based on FBM in the financial field with the long memory are applied to the time series with LRD characteristics and can predict the time series changes in future time.

#### 3.3. Generating FBM Series

^{H}is used to simulate the increment of FBM.

_{j}(j = 0, 1, 2, …, N), the increment of the FBM is discretized:

_{1}(t), w

_{2}(t) are independent and are standard normal distributions.

#### 3.4. Parameters Estimation of FBM

_{0}, y

_{Δt}, …, y

_{N}

_{Δt}), time vector is t = (0, Δt, …, NΔt), fractional Brown motion vector is B

_{H}(t) = [B

_{H}(0), B

_{H}(Δt), …, B

_{H}(NΔt)], so the parameters μ and σ by the maximum likelihood estimation are determined as follows [37,38]:

## 4. Predicting Fault Trends

#### 4.1. Property of Vibration Intensity

#### 4.2. Experiment Result Validation

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**Envelope spectrum by MED on different days. (

**a**) Envelope spectrum by MED on first day; (

**b**) Envelope spectrum by MED on second day; (

**c**) Envelope spectrum by MED on third day; (

**d**) Envelope spectrum by MED on fourth day; (

**e**) Envelope spectrum by MED on fifth day; (

**f**) Envelope spectrum by MED on sixth day; (

**g**) Envelope spectrum by MED on seventh day; (

**h**) Envelope spectrum by MED on eighth day.

**Figure 13.**The intensity prediction by FBM model in different time periods. (

**a**) The prediction results by FBM in level A; (

**b**) The prediction results by FBM in level B; (

**c**) The prediction results by FBM in level C; (

**d**) The prediction results by FBM in level D; (

**e**) The prediction results by FBM in level E.

Method | Aggregate Variance Estimation | Absolute Value Estimation | Curve Fitting Estimation | R/S Estimation | Periodogram Estimation | Wavelet Estimation |
---|---|---|---|---|---|---|

Hurst index | 0.672 | 0.9578 | 0.914 | 0.7967 | 0.8009 | 0.8652 |

**Table 2.**The Hurst value corresponding to different sections of the vibration intensity sequence V1.

V1 | (1:640) | (10:20) | (10:30) | (10:40) | (10:70) | (20:240) | (20:50) |

H | 0.6393 | 0.4725 | 0.4773 | 0.5275 | 0.60 | 0.4495 | 0.5456 |

V1 | (50:100) | (100:140) | (120:140) | (120:180) | (150:200) | (240:300) | (300:350) |

H | 0.5543 | 0.5509 | 0.4621 | 0.64 | 0.54 | 0.5336 | 0.5198 |

V1 | (380:420) | (400:450) | (420:450) | (450:550) | (480:580) | (500:600) | (550:640) |

H | 0.5988 | 0.563 | 0.6139 | 0.727 | 0.638 | 0.621 | 0.779 |

V1 | (1:640) | (10:20) | (10:30) | (10:40) | (10:70) | (20:240) | (20:50) |

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**MDPI and ACS Style**

Song, W.; Li, M.; Liang, J.-K.
Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution. *Entropy* **2016**, *18*, 418.
https://doi.org/10.3390/e18110418

**AMA Style**

Song W, Li M, Liang J-K.
Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution. *Entropy*. 2016; 18(11):418.
https://doi.org/10.3390/e18110418

**Chicago/Turabian Style**

Song, Wanqing, Ming Li, and Jian-Kai Liang.
2016. "Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution" *Entropy* 18, no. 11: 418.
https://doi.org/10.3390/e18110418