# Evaluation of Rolling Bearing Performance Degradation Using Wavelet Packet Energy Entropy and RBF Neural Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Backgrounds

#### 2.1. Wavelet Packet Energy Entropy

#### 2.1.1. Wavelet Packet Decomposition Layers and Selection of Wavelet Basis

#### 2.1.2. Wavelet Packet Energy Entropy Feature Extraction

#### 2.2. RBF Neural Network Model

#### 2.3. Adaptive Threshold Setting

## 3. Establishment of Performance Degradation Assessment Model

- Step 1:
- Perform wavelet packet decomposition on the rolling bearing vibration signal $X(\mathrm{t})$, and obtain all sub-band decomposition coefficients, a total of 8;
- Step 2:
- Step 2: Reconstruct the wavelet packet decomposition coefficients:$${a}_{j,k}(i)={\displaystyle \sum _{n}{a}_{j-1,k}(n){p}_{i-2n}}+{\displaystyle \sum _{n}{b}_{j+1,k}(n){q}_{i-2n}}$$
- Step 3:
- The energy value of the last wavelet packet reconstruction coefficient ${a}_{3,k}(i)$ was obtained. Calculate the total energy of wavelet decomposition. Finally, the energy ratios of each wavelet packet node are obtained:$${g}_{i}={e}_{i}/{\displaystyle \sum _{i=1}^{8}{e}_{i}}$$

- Step 4:
- Extracting the WPEE as the input eigenvector, and the RBF neural network model is established by using the early faultless samples and the failed samples of similar bearings. The model classifies all samples by using Euclidean distance, and obtains the cluster centers of the faultless samples and the failed samples, respectively.
- Step 5:
- Keep the model unchanged, and input the WPEE feature of the full-life bearing test data into the trained model through iterative method to obtain the model output value. According to the theory of the model, the output value of the model is the performance degradation evaluation index.
- Step 6:
- Calculate adaptive threshold curves, identify early failure points, and perform quantitative assessments.

## 4. Experiment and Result Analysis

#### 4.1. Bearing Discrete Data Verification

#### 4.2. Bearing Full Life Data Experimental Verification

#### 4.2.1. Test Bench Introduction

#### 4.2.2. RBF Neural Network Model Evaluation Results

## 5. Envelope Spectrum Analysis

## 6. Conclusions

- —
- The WPEE is used to deal with the non-stationary and nonlinear characteristics of the vibration signal.
- —
- The RBF neural network has the advantages of fast convergence speed, good approximation performance, and simple structure, which improves the accuracy and real-time performance of bearing performance degradation evaluation.
- —
- The effects of three different radial basis functions on performance degradation evaluation results are compared, and the superiority of the RBF neural network based on the Gaussian basis function is highlighted.
- —
- The box plot is used in the performance degradation assessment curve, and it is used as the alarm threshold method for bearing early fault determination. The box plot uses a quartile of certain robustness to calculate the actual appearance of the data. The results are objective and reliable, and can overcome the shortcomings of the previous failure thresholds that need to meet certain conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\psi $ | A basic wavelet |

${S}_{i}$ | Energy entropy of i-th wavelet packet |

${g}_{i}$ | Energy radio of the i-th wavelet packet node |

${h}_{j}$ | Output of the j-th RBF node |

${c}_{j}$ | Center value |

${r}_{j}$ | Width of the j-th RBF node |

${y}_{k}$ | The k-th output of the network |

$m$ | Number of hidden nodes |

${w}_{kj}$ | Connection weight of the k-th output node and the j-th hidden node |

${b}_{k}$ | The offset |

$K$ | Outlier |

${L}_{1}$ | Upper quartile |

${L}_{3}$ | Lower quartile |

$p\left(n\right)$ | Impulse response of the conjugate image filter |

$q\left(n\right)$ | Impulse response of the conjugate image filter |

${a}_{j,k}$ | Low frequency decomposition coefficients |

${b}_{j,k}$ | High frequency decomposition coefficients |

${e}_{i}$ | Energy of the i-th wavelet packet node |

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**Figure 6.**Time domain and spectrum diagram of normal sample and fault sample: (

**a**) The normal sample; (

**b**) The fault sample of 0.05 mm; (

**c**) The fault sample of 1.00 mm; (

**d**) The fault sample of 1.50 mm.

**Figure 9.**Rolling bearing accelerated fatigue life test bench: (

**a**) Test bench schematic; (

**b**) Partial picture.

**Figure 10.**Performance degradation evaluation results of RBF neural network based on Gaussian function.

**Figure 11.**Performance degradation evaluation results of RBF neural networks with two different basis functions: (

**a**) Reflected Sigmoidal Function; (

**b**) Inverse Multiquadrics Function.

**Figure 14.**Envelope demodulation diagrams of early fault and faultless samples: (

**a**) The 533 sample envelope demodulation diagram; (

**b**) The 532 sample envelope demodulation diagram.

Type | Parameter |
---|---|

data acquisition card | NI-USB4431 |

sensor | DH107 piezoelectric sensor |

motor speed | 1218 rpm |

load | 80 kg |

sample frequency | 12,000 Hz |

sample length | 1024 |

bearing designation | N205EM |

bearing bore diameter | 25 mm |

bearing outer diameter | 52 mm |

number of rolls | 9 |

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**MDPI and ACS Style**

Zhou, J.; Wang, F.; Zhang, C.; Zhang, L.; Li, P.
Evaluation of Rolling Bearing Performance Degradation Using Wavelet Packet Energy Entropy and RBF Neural Network. *Symmetry* **2019**, *11*, 1064.
https://doi.org/10.3390/sym11081064

**AMA Style**

Zhou J, Wang F, Zhang C, Zhang L, Li P.
Evaluation of Rolling Bearing Performance Degradation Using Wavelet Packet Energy Entropy and RBF Neural Network. *Symmetry*. 2019; 11(8):1064.
https://doi.org/10.3390/sym11081064

**Chicago/Turabian Style**

Zhou, Jianmin, Faling Wang, Chenchen Zhang, Long Zhang, and Peng Li.
2019. "Evaluation of Rolling Bearing Performance Degradation Using Wavelet Packet Energy Entropy and RBF Neural Network" *Symmetry* 11, no. 8: 1064.
https://doi.org/10.3390/sym11081064