A Hybrid Gearbox Fault Diagnosis Method Based on GWO-VMD and DE-KELM
Abstract
:Featured Application
Abstract
1. Introduction
1.1. Motivations
1.2. State of the Art of Gearbox Fault Diagnosis
1.2.1. Feature Signal Adaptive Processing Aspect
1.2.2. Fault Identification Aspect
1.3. Organization of This Article
2. Common Faults of Gearboxes and Causes Analysis
3. Vibration Signal Preprocessing and Fault Feature Extraction
3.1. GWO-VMD-Based Vibration Signal Decomposition
3.1.1. Basic Principle of VMD
3.1.2. Deficiencies of VMD
3.1.3. GWO-Based [K, α] Optimization for VMD
3.2. Correlation-Based Gearbox Vibration Signal Reconstruction
3.3. Fault Feature Extraction
3.4. Method Verification Experiments
4. DE-KELM-Based Gearbox Fault Classification
4.1. Basic Principle of KELM
4.2. DE-KELM-Based Fault Classification
4.3. Method Verification Experiments
5. Experimental Validation and Result Analysis
5.1. Gearbox Fault Diagnosis Experimental Data
5.2. Contrast Experiment I—Gearbox Fault Diagnosis with Contrasting Vibration Signal Decomposition Methods: GWO-VMD and VMD
5.2.1. Vibration Signals Decomposition through GWO-VMD and VMD, Respectively
5.2.2. Fault Classification Results on the Two Sets of Feature Vectors Obtained through GWO-VMD and VMD, Respectively
5.3. Contrast Experiment II—Fault Classification by DE-KELM and KELM, Respectively, on the Same Fault Feature Vectors
6. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Acronyms or Variables | Connotations |
VMD | Variational Mode Decomposition |
GWO | Wolf Grey Optimizer |
KELM | Kernel Extreme Learning Machine |
DE | Differential Evolutionary |
IMF | Intrinsic Mode Function |
TFA | Time-Frequency Analysis |
STFT | Short-Time Fourier Transform |
WVD | Wigner-Ville Distribution |
DWT | Discrete Wavelet Transform |
EMD | Empirical Mode Decomposition |
PSO | Particle Swarm Optimization |
EEMD | Ensemble Empirical Mode Decomposition |
CNN | Convolution Neural Network |
DRN | Deep Residual Network |
ELM | Extreme Learning Machine |
TL | Transfer Learning |
SVM | Support Vector Machine |
NN | Neural Network |
DL | Deep Learning |
SNR | Signal-Noise Ratio |
PE | Permutation Entropy |
GA | Genetic Algorithm |
v(t) | Gearbox vibration signal |
uk(t) | The kth IMF of v(t) |
K | Number of imfs |
ωk | Frequency centers of the kth IMF (Hz) |
α | Penalty factor in VMD algorithm |
λ | Lagrange multiplier in VMD |
fn | Fitness function of GWO |
kt | Kurtosis in GWO fitness function |
Hp | Permutation entropy in GWO fitness function |
μ | Mean value for Kurtosis calculation |
σ | Standard deviation for Kurtosis calculation |
m1, q | Dimensional parameters of the reconstruction matrix for Hp calculation |
pi | Probability of each row of the reconstruction matrix |
alpha, beta, delta, omega | Grey wolf names in their social hierarchy |
Corr | Correlation coefficient |
Corrth | Correlation coefficient threshold for signal reconstruction |
v′(t) | Reconstructed gearbox vibration signal |
vmax | Maximum amplitude of v′(t) |
vmin | Minimum amplitude of v′(t) |
vp | Peakedness of v′(t) |
vpp | Peak-to-peak value of v′(t) |
vm | Mean value of v′(t) |
vs | Slant of v′(t) |
vrms | Root mean square value of v′(t) |
vr | Root square amplitude of v′(t) |
vma | Mean of absolute value of v′(t) |
vσ | Variance of v′(t) |
vsf | Shape factor of v′(t) |
vsn | Skewness of v′(t) |
vif | Impulse factor of v′(t) |
vcf | Crest factor of v′(t) |
vclf | Clearance factor of v′(t) |
vkf | Kurtosis factor of v′(t) |
vE | Energy of v′(t) |
vpe | Permutation entropy of v′(t) |
Nv | Length of v′(t) |
T | Fault feature vector |
x(t) | Constructed experimental signal with noise |
s(t) | Periodic impulsive signal for experiments |
xn(t) | Gauss white noise for experiments |
n | KELM input node number |
L | KELM hidden layer node number |
m | KELM output node number |
N | Number of KELM training sample pairs |
xi | Input vector of the ith KELM sample |
ti | Target output of xi |
Y | N × m output matrix of KELM |
H | N × L output matrix of KELM hidden layer |
ω | KELM input weight matrix |
b | Bias vector of KELM hidden layer |
x | KELM sample input matrix |
β | L × m output weight matrix of KELM |
C | Regularization factor of KELM |
ξi | Training error of KELM |
k(x1, x2) | Kernel function of KELM |
f | Output function of KELM |
g | Kernel parameter of RBF kernel |
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Elements | Common Faults | Causes |
---|---|---|
Gears | Broken Teeth |
|
Pitting |
| |
Cracks |
| |
Bearings | Inner Ring Faults | Mainly include two types: wearing and pitting.
|
Outer Ring Faults | Mainly include three types: wear, pitting and fracture.
| |
Roller Faults |
|
No. | Parameter Names | Notations | Mathematical Expressions |
---|---|---|---|
1 | Maximum amplitude | vmax | |
2 | Minimum amplitude | vmin | |
3 | Peakedness 1 | vp | |
4 | Peak-to-peak value | vpp | |
5 | Mean value | vm | |
6 | Slant | vs | |
7 | Root mean square value | vrms | |
8 | Root square amplitude | vr | |
9 | Mean of absolute value | vma | |
10 | Variance | vσ | |
11 | Shape factor | vsf | |
12 | Skewness | vsn | |
13 | Impulse factor | vif | |
14 | Crest factor | vcf | |
15 | Clearance factor | vclf | |
16 | Kurtosis factor | vkf | |
17 | Energy | vE | |
18 | Permutation entropy | vpe |
Parameters | Parameter Values or Ranges |
---|---|
Population number | 15 |
Maximum iterations number | 20 |
Range of K | [2, 8] |
Range of α | [100, 3000] |
IMF 1 | IMF 2 | IMF 3 | IMF 4 | IMF 5 | IMF 6 | IMF 7 | IMF 8 | |
---|---|---|---|---|---|---|---|---|
CORR | 0.550 | 0.744 | 0.326 | 0.266 | 0.249 | 0.243 | 0.240 | 0.229 |
Parameters | s(t) | x(t) | Reconstructed x(t) through GWO-VMD | Reconstructed x(t) through VMD |
---|---|---|---|---|
Kurtosis | 7.7708 | 5.3719 | 5.4677 | 4.7667 |
Signal–noise ratio | / | 5.0021 | 8.7756 | 8.1462 |
RMSE | / | 0.1019 | 0.0660 | 0.0710 |
Permutation entropy | 0.4545 | 0.9320 | 0.6145 | 0.8036 |
Parameters | Parameters Values or Ranges |
---|---|
Population size | 9 |
Maximum iteration number | 30 |
Mutation operator | 0.7 |
Crossover operator | 0.6 |
Regularization coefficient C | [0.01, 100] |
Kernel parameter g | [0.01, 10] |
Recognized as → | Healthy | Inner Ring Fault | Outer Ring Fault | Roller Fault | Accuracy |
---|---|---|---|---|---|
Healthy | 10 | 0 | 0 | 0 | 100% |
Inner ring fault | 0 | 10 | 0 | 0 | 100% |
Outer ring fault | 0 | 0 | 10 | 0 | 100% |
Roller fault | 0 | 0 | 1 | 9 | 90% |
Recognized as → | Healthy | Inner Ring Fault | Outer Ring Fault | Roller Fault | Accuracy |
---|---|---|---|---|---|
Healthy | 10 | 0 | 0 | 0 | 100% |
Inner ring fault | 0 | 10 | 0 | 0 | 100% |
Outer ring fault | 0 | 0 | 10 | 0 | 100% |
Roller fault | 0 | 1 | 3 | 6 | 60% |
Gearbox Elements | Classification Labels | Operating Conditions | Sample Numbers |
---|---|---|---|
Bearing | 1 | healthy bearing | 150 |
2 | inner ring fault | 150 | |
3 | outer ring fault | 150 | |
4 | roller fault | 150 | |
Gear | 5 | healthy gear | 150 |
6 | pitting fault | 150 | |
7 | broken teeth | 150 | |
8 | tooth root cracks | 150 |
Parameters | Parameter Values or Ranges |
---|---|
Population number | 9 |
Maximum iterations number | 100 |
Mode Number | [2, 15] |
Penalty factor range | [100, 6000] |
Gearbox Elements | Working Conditions | [K, α] |
---|---|---|
Bearing | healthy bearing | [11, 4018] |
inner ring fault | [15, 139] | |
outer ring fault | [5, 2755] | |
roller fault | [5, 2678] | |
Gear | healthy gear | [5, 2330] |
pitting fault | [15, 100] | |
broken teeth | [5, 3652] | |
tooth root cracks | [5, 3019] |
Gearbox Elements | Working Conditions | [K, α] |
---|---|---|
Bearing | healthy bearing | [3, 2000] |
inner ring fault | [4, 2000] | |
outer ring fault | [4, 2000] | |
roller fault | [5, 2000] | |
Gear | healthy gear | [4, 2000] |
pitting fault | [5, 2000] | |
broken teeth | [4, 2000] | |
tooth root cracks | [4, 2000] |
Signal Decomposition Method | Classification Accuracy of the Training Set (by DE-KELM) | Classification Accuracy of the Testing Set (by DE-KELM) |
---|---|---|
GWO-VMD | 100% | 98.125% |
VMD | 100% | 91.875% |
Fault Classification Methods | Classification Accuracy of the Training Set | Classification Accuracy of the Testing Set |
---|---|---|
DE-KELM | 100% | 98.125% |
KELM | 97.9808% | 97.5% |
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Yao, G.; Wang, Y.; Benbouzid, M.; Ait-Ahmed, M. A Hybrid Gearbox Fault Diagnosis Method Based on GWO-VMD and DE-KELM. Appl. Sci. 2021, 11, 4996. https://doi.org/10.3390/app11114996
Yao G, Wang Y, Benbouzid M, Ait-Ahmed M. A Hybrid Gearbox Fault Diagnosis Method Based on GWO-VMD and DE-KELM. Applied Sciences. 2021; 11(11):4996. https://doi.org/10.3390/app11114996
Chicago/Turabian StyleYao, Gang, Yunce Wang, Mohamed Benbouzid, and Mourad Ait-Ahmed. 2021. "A Hybrid Gearbox Fault Diagnosis Method Based on GWO-VMD and DE-KELM" Applied Sciences 11, no. 11: 4996. https://doi.org/10.3390/app11114996