# Fault Diagnosis for Rotating Machinery Using Vibration Measurement Deep Statistical Feature Learning

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodologies

#### 2.1. Statistical Features of the Vibration Sensor Signals

^{j}− 1; j = 0, …, J, which is the number of the decomposition levels), a WP function ${T}_{j,k}^{n}(t)$ is defined by:

^{j}packets with the order n = 1, 2, …, 2

^{j}. For simplicity, we index the WP node as (j, n) whose coefficients are given by ${P}_{j,k}^{n}$.

_{0}is the length of x(t).

_{1,p}, …, F

_{9,p}stand for kurtosis, skewness factor, crest factor, clearance factor, shape factor, impulse indicator, variance, denominator of clearance factor (the square of the averaged square roots of absolute amplitude), and mean of absolute amplitude values of the p-th vector of M(p,q), respectively [32]. Note that there are nine statistical features for the time domain representation M(p,q) = x(t), 9 for the frequency domain representation M(p,q) = X(f), and 9(2

^{j}

^{+}

^{1}− 1) for the time-frequency domain representation M(p,q) = $[{P}_{1,k}^{1},{P}_{1,k}^{2},\mathrm{...},{P}_{j,k}^{{2}^{j}}]$. The feature set F is therefore given by:

#### 2.2. Statistical Feature Representation by Unsupervised Boltzmann Machines

_{1}is 0, but that for F

_{5}will be a negative number. This means that the conventional RBM is difficult to cope with our statistical features for the fault diagnosis.

**v**,

**h**) is given by:

**v**and

**h**denote the visible and the hidden neurons, b

_{i}and

**c**

_{i}stand for the offsets of the visible layers, w

_{ij}represents the weights for the connection matrix, σ

_{i}is the standard deviation associated with a Gaussian visible neuron v

_{i}, and θ is the Gaussian parameter [35]. The traditional gradient-based training of the GRBM has difficulty learning σ

_{i}, which is constrained to be positive. Hence, some algorithms fix σ

_{i}as unity. With the improved energy function, Cho et al. [35] proposed conditional probabilities for the visible and the hidden neurons as follows:

^{2}, and S(.) is a sigmoid function. The upgraded gradients with respect to the GRBM parameters are given by:

_{d}and <.>

_{m}represent the expectation computed over the data and the model distributions, respectively.

**v**and the GRBM results GR(F) =

**h**. In this way, the n

_{v}statistical features are represented by n

_{h}neurons [36]. For condition monitoring and fault type classification, GRBM representations can be input to a classifier such as a support vector machine (SVM), decision tree, or random forest.

#### 2.3. Deep Statistical Feature Learning and Classification

_{1}, GRBM

_{2}, and GRBM

_{3}). Each GRBM consists of one visible layer and one hidden layer, and the hidden layer of the previous GRBM is just the visible layer of the next GRBM. In this way, the first layer (data layer) and the second layer (hidden layer 1) forms the GRBM

_{1}, the second layer and the third layer (hidden layer 2) forms the GRBM

_{2}, the third layer and the last layer (output layer) forms the GRBM

_{3}, and the three GRBMs are stacked together to form the GDBM.

_{v}= 2 corresponds to the duplication of the visible layer. Similarly, the energy for the topmost GRBM

_{L}during the pretraining is given by:

**w**can be discriminatively fine-tuned using a back-propagation (BP) algorithm [41]. The supervised BP method uses labeled data as an extra MLP layer of variables to train the GDBM model for the classification. Unlike the unsupervised training process considering one GRBM at a time, the BP training considers all the layers in a GDBM simultaneously, which is in the same way as for the standard feed forward neural networks [42]. In this way, the GDBM can be regarded as an improvement of the MLP, or neural networks. It is capable of dealing with the classification for nonlinear, abnormal (non-Gaussian) data using a “deeper” fashion [43]. Of course, this “deeper” learning is much more time-consuming than the conventional ones.

- Step 1.
- Collect the vibration signals x(t), define the fault patterns and the diagnosis problems;
- Step 2.
- Calculate the statistical feature set F according to Equation (8);
- Step 3.
- Step 4.
- Pretrain the GDBM model and its constituting GRBMs using the layer-by-layer unsupervised learning algorithm from the training dataset;
- Step 5.
- Fine-tune the GDBM weights using the BP algorithm from the training dataset; and
- Step 6.
- Diagnose the rotating machinery condition using the trained GDBM model.

## 3. Data Collection Experiments for the Fault Diagnosis

#### 3.1. Experimental Procedure for Gearbox Fault Diagnosis

#### 3.2. Experimental Procedure for Bearing Fault Diagnosis

## 4. Results and Discussion

#### 4.1. Gearbox Fault Diagnosis Results

^{®}. One may note that in this work we have not employed more “shallow” learning models such as the decision tree, the random forest, and the neural network. The reason is that the SVM has been proven the prominent representative which outperforms most of the “shallow” learning members.

#### 4.2. Bearing Fault Diagnosis Results

#### 4.3. Remarks

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the network connections with a GRBM. Note the GRBM exhibits same structure compared to its RBM counterpart.

**Figure 2.**Schematic of the three-layer GDBM: (

**a**) network structure; and (

**b**) pretraining and composition of the GDBM.

**Figure 3.**Flowchart of the deep statistical feature learning technique for the fault diagnosis of the rotating machinery.

**Figure 4.**Gearbox fault diagnosis configurations: (

**a**) experimental set-up; and (

**b**) three different faulty gears and five different faulty pinions.

**Figure 5.**Fault diagnosis configurations for the rolling element bearings: (

**a**) experimental set-up; and (

**b**) 3 different faulty bearings with an inner race fault (

**left**), an outer race fault (

**middle**) and a ball fault (

**right**), respectively.

**Figure 6.**Time domain features for the gearbox fault diagnosis: (

**a**) time domain waveform of the first signal; (

**b**) time domain statistical features of the first signal; (

**c**) time domain waveforms of the 3600 collected signals; and (

**d**) time domain statistical features of the 3600 collected signals.

**Figure 7.**Frequency domain features for the gearbox fault diagnosis: (

**a**) frequency domain representation of the first signal; (

**b**) frequency domain statistical features of the first signal; (

**c**) frequency domain representations of all the collected 3600 signals; and (

**d**) frequency domain statistical features of all the collected 3600 signals.

**Figure 8.**Time-frequency domain features for the gearbox fault diagnosis: (

**a**) WPT representation of the first signal; (

**b**) time-frequency domain statistical features of the first signal; (

**c**) time-frequency domain representations of all the collected 3600 signals; and (

**d**) time-frequency domain statistical features of all the collected 3600 signals.

**Figure 9.**Bearing fault diagnosis experiments: (

**a**) the time domain signals; (

**b**) the time domain statistical features; (

**c**) the frequency domain representations; (

**d**) the frequency domain statistical features; (

**e**) the WPT results; and (

**f**) the time-frequency domain statistical features.

**Figure 10.**Relationship between the classification rate and the number of the modeling epochs: (

**a**) classification rates v.s. pretraining epochs; and (

**b**) classification rates vs. fine-tuning epochs of the time-frequency domain GDBM models.

**Figure 11.**Bearing fault diagnosis results in different domains: (

**a**) the time domain; (

**b**) the frequency domain; and (

**c**) the time-frequency domain.

Experimental Setup | Pattern Label | Component 1 | Component 2 | Load |
---|---|---|---|---|

Gearbox (component 1-pinion; component 2-gear) | A | Normal | Normal | zero, small, great |

B | Chaffing tooth | Normal | zero, small, great | |

C | Worn tooth | Normal | zero, small, great | |

D | Chipped tooth 25% | Normal | zero, small, great | |

E | Chipped tooth 50% | Normal | zero, small, great | |

F | Missing tooth | Normal | zero, small, great | |

G | Normal | Chipped tooth 25% | zero, small, great | |

H | Normal | Chipped tooth 50% | zero, small, great | |

I | Normal | Missing tooth | zero, small, great | |

J | Chipped tooth 25% | Chipped tooth 25% | zero, small, great | |

Bearing (component 1-bearing 1; component 2-bearing 2) | 1 | Normal | Normal | Zero, 1, 2 flywheel(s) |

2 | Normal | Inner race fault | Zero, 1, 2 flywheel(s) | |

3 | Normal | Outer race fault | Zero, 1, 2 flywheel(s) | |

4 | Normal | Ball fault | Zero, 1, 2 flywheel(s) | |

5 | Outer race fault | Inner race fault | Zero, 1, 2 flywheel(s) | |

6 | Ball fault | Inner race fault | Zero, 1, 2 flywheel(s) | |

7 | Ball fault | Outer race fault | Zero, 1, 2 flywheel(s) |

**Table 2.**Fault classification rates for the testing dataset (%), where N represents the device, d denotes the domain of the feature.

Device (N) | Domain (d) | Fault Diagnosis Model | ||||
---|---|---|---|---|---|---|

#1 Peer | GDBM | #2 Peer | #3 Peer | Average ^{a} | ||

Gearbox | Time domain | 26.08 | 62.58 | 60.83 | 35.83 | 46.33 |

Frequency domain | 52.67 | 91.75 | 52.83 | 79.42 | 69.17 | |

Time-frequency domain | 45.25 | 95.17 | 69.50 | 78.42 | 72.09 | |

Average ^{b} | 41.33 | 83.17 | 61.05 | 64.56 | 62.53 | |

Bearing | Time domain | 18.52 | 60.63 | 59.58 | 41.96 | 45.17 |

Frequency domain | 39.95 | 87.57 | 80.74 | 82.91 | 72.79 | |

Time-frequency domain | 58.84 | 91.75 | 81.53 | 82.70 | 78.71 | |

Average ^{b} | 39.10 | 79.98 | 73.95 | 69.19 | 65.508 |

^{a}the average value of the left four models;

^{b}the average value of the above three domains.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, C.; Sánchez, R.-V.; Zurita, G.; Cerrada, M.; Cabrera, D.
Fault Diagnosis for Rotating Machinery Using Vibration Measurement Deep Statistical Feature Learning. *Sensors* **2016**, *16*, 895.
https://doi.org/10.3390/s16060895

**AMA Style**

Li C, Sánchez R-V, Zurita G, Cerrada M, Cabrera D.
Fault Diagnosis for Rotating Machinery Using Vibration Measurement Deep Statistical Feature Learning. *Sensors*. 2016; 16(6):895.
https://doi.org/10.3390/s16060895

**Chicago/Turabian Style**

Li, Chuan, René-Vinicio Sánchez, Grover Zurita, Mariela Cerrada, and Diego Cabrera.
2016. "Fault Diagnosis for Rotating Machinery Using Vibration Measurement Deep Statistical Feature Learning" *Sensors* 16, no. 6: 895.
https://doi.org/10.3390/s16060895