The fault diagnosis process is essentially a class discrimination problem. However, traditional class discrimination methods such as SVM and ANN fail to capitalize the interactions among the feature variables. Variable predictive model-based class discrimination (VPMCD) can adequately use the interactions. But the feature extraction and selection will greatly affect the accuracy and stability of VPMCD classifier. Aiming at the nonstationary characteristics of vibration signal from rotating machinery with local fault, singular value decomposition (SVD) technique based local characteristic-scale decomposition (LCD) was developed to extract the feature variables. Subsequently, combining artificial neural net (ANN) and mean impact value (MIV), ANN-MIV as a kind of feature selection approach was proposed to select more suitable feature variables as input vector of VPMCD classifier. In the end of this paper, a novel fault diagnosis model based on LCD-SVD-ANN-MIV and VPMCD is proposed and proved by an experimental application for roller bearing fault diagnosis. The results show that the proposed method is effective and noise tolerant. And the comparative results demonstrate that the proposed method is superior to the other methods in diagnosis speed, diagnosis success rate, and diagnosis stability.
Fault diagnosis is essentially considered as a class discrimination problem. Various methods have been applied to build classifiers to fulfill fault diagnosis [
It is known that there are some interactions among feature variables and VPMCD method can adequately use these interactions. However, in the application to fault diagnosis of rotating machinery, we found that the feature extraction and selection have a great influence on the performance of VPMCD classifier. As a new time-frequency signal processing method, local characteristic-scale decomposition (LCD) method can decompose a nonstationary signal into several intrinsic scale components (ISCs). Many applications show that LCD is superior to empirical mode decomposition (EMD) [
After feature extraction, we need to answer the following questions: which feature variables cause interrelationship that can describe the system’s dynamic characteristics more effectively? How to select more representative feature variables to improve the performance of the VPMCD classifier? In many practical applications, operators often have not a clear professional theory as guidance, so they cannot select better input features to design better VPMCD classifier. In this case, the accuracy of VPMCD classifier will decrease and seriously affect the accuracy for fault diagnosis. In other words, feature selection is fairly critical to design VPMCD classifier with better performance. Mean impact value can sensitively capture the interaction between the independent variable and dependent variable [
The rest of this paper is organized as follows. VPMCD method is introduced in Section
It is known that different system behaviors are always quantified by measurable features and interactions among them. For mechanical fault diagnosis, there exist linear or nonlinear associations among the features extracted from the vibration signals in different work conditions. In VPMCD, variable predictive models (VPMs) are defined to distinguish linear/nonlinear and direct/indirect quantitative relationships among the features using one of the mathematical equations in the form of the following formulas:
Suppose that there are
If there are
Taking a fault diagnosis problem, for example, VPMCD algorithm includes two steps. The first step is to train VPMs of each class; the second step is to repredict feature variables by mapping on each of these VPMs and then to establish classifier. The detailed procedure is given as follows.
(1) Collect
(2) Extract feature vector
(3) For any predicted variable
(4) For the classification problem with
(1) For unknown sample, extract feature vector
(2) Repredict each feature variable
(3) Calculate the sum of squared prediction errors
(4) The unknown sample is classified into class
A trajectory matrix
It is known that reconstruction parameters, such as lag time and embedding dimension, would have effect on the result of SVD method. It is difficult to determine reconstruction parameters. In order to solve this problem, LCD-SVD technique is presented. We introduced LCD method as follows.
LCD method can decompose a complex multicomponent signal into series of intrinsic scale components (ISCs), in which each ISC is a monocomponent signal whose instantaneous frequency has specific physical meaning.
That is, the original signal is decomposed into
MIV is the evaluation index showing how much the independent variables influence the dependent variable. Its absolute value represents the relative importance degree of the independent variables. In combination with ANN, we use MIV to rank the independent variables to select more representative feature. The ANN-MIV algorithm is described as follows.
To elaborate the algorithm,
For a special class
The
The pair of new samples are, respectively, tested, and a pair of simulation outputs, noted as
For the
The process from Step 2 to Step 5 is repeated for the other
The value of
A novel fault diagnosis model based on LCD-SVD-ANN-MIV and VPMCD for rotating machinery was proposed in this paper. Firstly, LCD-SVD technique was introduced for the fault feature extraction. Subsequently, more suitable features were selected by ANN-MIV approach to form feature vector. Lastly, VPMCD method was utilized to design the classifier to identify the work condition. The flow chart of the proposed fault diagnosis model is given in Figure
Fault diagnosis model for rotating machinery based on LCD-SVD-ANN-MIV and VPMCD.
All datasets and system investigations of the roller bearing were downloaded from the website of the Case Western Reserve University. The vibration signals were acquired by the accelerometer, which had been mounted on the bearing housing at the driver end of the motor. The bearing type at drive end is 6205-2RS SKF, whose parameters are given in Table
Parameters of 6205-RS JEM SKF.
Inside diameter | Outside diameter | Ball diameter | Pitch diameter |
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0.9843 inches | 2.0472 inches | 0.3126 inches | 1.537 inches |
Time domain waveforms for seven running conditions of roller bearing. (a) Normal condition, (b) inner-race fault with fault diameter of 0.007 inches, (c) outer-race fault with fault diameter of 0.007 inches, (d) ball fault with fault diameter of 0.007 inches, (e) inner-race fault with fault diameter of 0.021 inches, (f) outer-race fault with fault diameter of 0.021 inches, and (g) ball fault with fault diameter of 0.021 inches.
Using LCD method mentioned above, the vibration signal of the roller bearing was decomposed into about 10 ISCs, whose frequency bands ranged from high frequency to low frequency. Here, we give the ISCs of the vibration signals with bearing fault with fault diameter of 0.021 inches in Figure
The ISCs of the vibration signals with the bearings fault with bearing fault in fault diameter of 0.021 inches.
From Figure
Cross-correlation coefficient between the
ISC | ISC1 | ISC2 | ISC3 | ISC4 | ISC5 | ISC6 | ISC7 | ISC8 | ISC9 | ISC10 | ISC11 |
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0.6224 | 0.5631 | 0.3427 | 0.2759 | 0.2173 | 0.1063 | 0.0891 | 0.0832 | 0.0146 | 0.0089 | 0.0042 |
Note:
Feature selection results based on ANN-MIV.
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Selected times | 6 | 5 | 5 | 3 | 2 | 2 | 5 | 1 |
Feature variables of seven conditions of vibration signals.
Different running condition | Feature variables | |||
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Normal condition | 1.5508 | 1.3899 | 0.7705 | 0.6698 |
Inner-race fault with fault diameter of 0.007 inches | 11.3170 | 5.2801 | 3.9301 | 2.3817 |
Outer-race fault with fault diameter of 0.007 inches | 24.0114 | 5.7474 | 3.0605 | 1.9921 |
Ball fault with fault diameter of 0.007 inches | 6.3502 | 1.2586 | 1.0221 | 0.8439 |
Inner-race fault with fault diameter of 0.021 inches | 20.8725 | 3.2570 | 2.7492 | 2.0920 |
Outer-race fault with fault diameter of 0.021 inches | 23.6590 | 5.9063 | 4.4414 | 3.6721 |
Ball fault with fault diameter of 0.021 inches | 4.3274 | 1.3387 | 1.1446 | 0.9301 |
In this application, ten VPMs in total have been used to perform the initial tests for seven work conditions by combining all four models and different predictor order
The test results of different classifier based VPMCD methods.
Different model and different predictor order | Accuracy (%) | Cost time (s) | Least number for acquired training samples |
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Linear VPM with |
94.16 | 0.1063 | 2 |
Linear VPM with |
95.34 | 0.1156 | 3 |
Linear VPM with |
96.65 | 0.1280 | 4 |
Linear + interaction VPM with |
99.74 | 0.1656 | 4 |
Linear + interaction VPM with |
100 | 0.1653 | 7 |
Quadratic + interaction VPM with |
99.48 | 0.1518 | 6 |
Quadratic + interaction VPM with |
100 | 0.1281 | 10 |
Pure quadratic VPM with |
98.74 | 0.1391 | 3 |
Pure quadratic VPM with |
100 | 0.1218 | 5 |
Pure quadratic VPM with |
99.74 | 0.1012 | 7 |
After the detailed comparative analysis in Table
In order to avoid the occasionality of the testing accuracy, Monte Carlo test approach is applied. In this experiment, all samples under each running condition were divided into three groups for training, validating, and testing randomly. Using the mentioned VPMCD algorithm in Section
The mathematical expression of part
Running condition |
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Predictor variables |
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Inner-race fault with 7 mil |
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Outer-race fault with 7 mil |
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Since the experiment was performed with no masking source element such as gear vibration, Gaussian noise was added to the original vibration signals. The noisy signals with SNR
Classification results of the proposed methods for the noisy signals.
Noisy signals | Average accuracy (%) | Standard deviation (%) | Cost time (s) |
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SNR +10 dB | 98.41 | 1.42 | 0.2655 |
SNR −10 dB | 91.17 | 1.12 | 0.2657 |
SNR −20 dB | 88.78 | 1.37 | 0.2514 |
Comparison between different diagnosis methods.
Method | Sample size | Feature variables | Average accuracy | Cost time |
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SSA and BP-ANN [ |
Training: 336 |
4 singular values | 96.53% | — |
3 energy features | 95.14% | — | ||
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Multiscale entropy and SVM [ |
Training: 525 |
6 entropies at different time scale | 97.42% | — |
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Envelope spectra and SVM [ |
Training: 60 |
3 fault characteristic frequencies in the envelope spectra | 100% | 76.68 s |
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LCD-SVD and LSSVM | Training: |
8 singular values of ISCs | 95.23% | 146.13 s |
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LCD-SVD and VPMCD | Training: |
8 singular values of ISCs | 96.19% | 12.84 s |
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LCD-SVD-ANN-MIV and LSSVM | Training: |
4 singular values of ISCs | 96.67% | 67.42 s |
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LCD-SVD-ANN-MIV and VPMCD | Training: |
4 singular values of ISCs | 100% | 0.1028 s |
To prove the proposed model, Table
In the literature [
In the literature [
In the literature [
As seen in Table
A novel fault diagnosis model was presented in this paper. Firstly, a new singular value decomposition technique based on local scale decomposition, called LCD-SVD technique, was introduced for roller bearing fault feature extraction. The LCD-SVD technique avoids the difficulty of selecting the parameters, which affects the accuracy of traditional SVD technique. Secondly, feature selection approach based on ANN-MIV was proposed to choose more suitable feature variables as input features for VPMCD classifier. Thirdly, a fault diagnosis model based on LCD-SVD-ANN-MIV and VPMCD was proposed. Lastly, the proposed model was applied to roller bearing fault diagnosis. At the same time, the effect of noise on classification performance was studied and the comparison has been made. The investigation shows that the proposed model performs well for the signal with a low SNR. The comparative results demonstrate that the proposed model is superior to the other methods in diagnosis speed, diagnosis success rate, and stability.
The authors declare that they have no competing interests.
The authors would like to acknowledge the support from Chinese National Science Foundation Grant (no. 51375152), Cooperative Innovation Center for the Construction & Development of Dongting Lake Ecological Economic Zone (XJT