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As a crucial and widely used component in industrial fields with great complexity, the health condition of rotating machinery is directly related to production efficiency and safety. Consequently, recognizing and diagnosing rotating machine faults remain to be one of the main concerns in preventing failures of mechanical systems, which can enhance the reliability and efficiency of mechanical systems. In this paper, a novel approach based on blind parameter identification of MAR model and mutation hybrid GWO-SCA optimization is proposed to diagnose faults for rotating machinery. Signals collected from different types of faults were firstly split into sets of intrinsic mode functions (IMFs) by variational mode decomposition (VMD), the decomposing mode number

Rotating machinery is a crucial part in modern industrial manufacture, and its failure may result in serious safety accidents and economic losses. Therefore, one of the major topics to be investigated in preventing failures of mechanical systems is recognizing and diagnosing rotating machine faults [

Owing to rich information carried by vibration signals, most of fault diagnosis methods for rolling bearings rely on analyzing vibration signals [

The essence of fault diagnosis for rolling bearings is a pattern recognition problem; that is, the fault type is determined by fault features of samples. In engineering application, many alternative methods are available for addressing this issue. For instance, Bayesian decision has strong recognition ability by considering prior probability as well as class conditional probability, and its good accuracy requires assumption of a suitable prior model [

Although SVM has superior ability in pattern recognition, its performance is affected by parameters. In view of this, various optimization algorithms are developed and applied to search the best parameters, such as genetic algorithm (GA) [

The paper is organized as follows: Section

Variational Mode Decomposition (VMD) is a new adaptive signal preprocessing method [

For the above variational problem, quadratic penalty function term and Lagrange multiplier are employed to transform it into an unconstrained problem. Then problem (

Then alternate direction method is utilized to obtain the saddle point of Lagrange multiplier, that is, solving (_{k},_{k}, and _{k} and_{k} are formulated as (

Solving problems (

The Lagrange multipliers can be iterated with

The main steps of VMD can be summarized as follows:

Initialize

Execute loop,

Update_{k} and_{k} based on (

Update

If

Autoregressive (AR) model deduces the regressive variables with itself, namely, using a linear combination of white noise at time^{2}, and

When multivariate modeling is required, AR model must be analyzed and established multiple times. For this reason, the model order determined by different variables may be inconsistent. Compared with AR model, multivariate autoregressive (MAR) model only models once and can reflect the internal relationship among different variables. Let

Support vector machine (SVM) is a machine learning model developed by Vapnik [

Taking binary classification issue as an example, to classify samples correctly with sufficient classification interval, all samples are required to meet the following constraint.

In the sequel, classification interval can be calculated as

When mapping samples to high-dimensional space, different inner product kernel functions will form different algorithms. Among them, radial basis function is widely employed in application of pattern recognition, which can be described as

The solution of constrained optimization problem (

With the optimal value of_{i} obtained from above dual problem, the optimal classification discriminant function can be ascertained as follows.

Multivariate autoregressive (MAR) model is the multivariate version of autoregressive (AR) model. By synchronous autoregressive modeling and parameter identification of multiple variables, it possesses advantages of high efficiency and consideration of variable correlation. Parameter identification is a critical part of MAR analysis and appropriate parameters will contribute to a better fitting effect, during which process the order selection of MAR model not only relates to the extraction of essential features implicated in the sequence, but also has a great impact on recognition efficiency. In this research, Schwartz Bayes Criterion (SBC) is adopted to estimate the order of MAR model. The SBC function is as follows [_{p}=_{22} is the fourth part of the upper triangular matrix obtained by QR decomposition of K matrix of the model (see [

The specific method of order selection is that, within the given range of model order,

Considering the inner relation among different variable sequences, there will be a mass of parameters in MAR model. Generally, minimum variance criterion is used to estimate model parameters directly [

Sine cosine algorithm (SCA) is a newly swarm optimization algorithm, which is simple in structure and easy to realize. Using only sine and cosine functions, SCA achieves good optimization effect. Exploration and exploitation are the two key phases when SCA processes optimization problem [_{2} is a random number in the scope of _{3} within _{4} is a random number in the range of

Grey wolf optimizer is a swarm intelligence algorithm based on strict organization system of grey wolves and its sophisticated cooperative hunting method. GWO algorithm simulates the hierarchy and hunting behavior of grey wolves in the wild. Wolves are divided into four subgroups, i.e., _{1},_{2}, and_{3} are random numbers in _{1}=2_{2}=2_{3}=2

In the updating strategy of GWO, the position of _{1},_{2}, and_{4} are the same as in SCA, and the meaning of _{1},

Analysis of GWO shows that diversity of individuals is gradually losing with iteration and it is likely to result in the local optimum in the later stage of iteration, which will affect the optimization effect. To solve this problem, mutation operators are introduced to enrich diversity of individuals [

The main steps of proposed mutation hybrid GWO-SCA (MHGWOSCA) are shown below:

Initialize the population randomly in the given range of variables and set mutation parameters.

Calculate the fitness value of each individual.

Choose and save the best three individuals as

Update other individuals (

Update parameters

If the maximum number of iterations is not met, go to Step 2.

Output the position of individual

In this section, to promote the fault diagnosis accuracy, a fault diagnosis method based on blind parameter identification of MAR model and mutation hybrid GWO-SCA optimized SVM is proposed. The specific steps are detailed as follows:

Collect the vibration signals.

Determine the mode number

Decompose the samples into sets of IMFs with VMD.

Establish the MAR model of all IMFs and apply SBC to determine the model order; then MAR parameter vectors of different fault samples can be identified by QR decomposition.

Extract principal features from parameter vectors based on PCA; thus the feature vectors of different fault types are obtained.

Optimize the parameters

Train the SVM model with optimal parameters

Apply the optimal SVM model to classify different fault samples and evaluate the performance of the model.

The flowchart of the proposed fault diagnosis model is shown in Figure

The flowchart of the proposed fault diagnosis method.

Vibration signals with different fault locations and sizes gathered from Bearings Data Center of Case Western Reserve University [

Description of the experimental data.

Position of fault | Defect size (inches) | Label of classes | Number of samples |
---|---|---|---|

Inner race | 0.007 | L1 | 59 |

Inner race | 0.014 | L2 | 59 |

Inner race | 0.021 | L3 | 59 |

Ball | 0.007 | L4 | 59 |

Ball | 0.014 | L5 | 59 |

Ball | 0.021 | L6 | 59 |

Outer race | 0.007 | L7 | 59 |

Outer race | 0.014 | L8 | 59 |

Outer race | 0.021 | L9 | 59 |

Experiment stand and its measurement sensors in bearing data center.

In this application, adjusted Rand index (ARI), normalized mutual information (NMI), F-measure (F), and accuracy (ACC) as four widely used evaluation measures are employed to evaluate the diagnosis results [

Let _{11} is the number of sample pairs with the same label in both_{00} is the number of sample pairs with different labels in

NMI is one of the common external evaluation metrics that can measure the degree of agreement between two data distributions [

F and ACC are derivations from the confusion matrix whose structure is reflected in Table

The confusion matrix.

Actual label | |||
---|---|---|---|

Positive | Negative | ||

Classification result | Positive | True positive (TP) | False positive (FP) |

Negative | False negative (FN) | True negative (TN) |

To demonstrate the superiority of the proposed VMD-MAR-MHGWOSCA-SVM method, the experiment was carried out in a comparative form at each stage, as follows: EMD was applied for comparison in the signal processing stage; fuzzy entropy (FE) [

In the proposed method, the first step is to decompose signal of each fault type into a set of IMFs. The mode number

Normalized central frequencies with different

Number of modes | Normalized central frequencies | ||||||||
---|---|---|---|---|---|---|---|---|---|

2 | 0.2256 | 0.0907 | |||||||

3 | 0.2978 | 0.2275 | 0.0999 | ||||||

| | | | | |||||

5 | 0.3066 | 0.2834 | 0.2271 | 0.1148 | 0.0525 | ||||

6 | 0.3072 | 0.2854 | 0.2364 | 0.2142 | 0.1139 | 0.0519 | |||

7 | 0.3170 | 0.2981 | 0.2757 | 0.2270 | 0.1242 | 0.0982 | 0.0493 | ||

8 | 0.3178 | 0.2990 | 0.2777 | 0.2363 | 0.2147 | 0.1231 | 0.0975 | 0.0492 | |

9 | 0.4022 | 0.3133 | 0.2970 | 0.2768 | 0.2363 | 0.2146 | 0.1230 | 0.0975 | 0.0492 |

Time domain waveforms of signals with different faults: (a) inner race 0.007 inches, (b) inner race 0.014 inches, (c) inner race 0.021 inches, (d) ball 0.007 inches, (e) ball 0.014 inches, (f) ball 0.021 inches, (g) outer race 0.007 inches, (h) outer race 0.014 inches, (i) outer race 0.021 inches.

The VMD decomposition results of 0.007 inches fault signals: (a) inner race, (b) ball, (c) outer race.

When decomposed components of a sample were obtained, a multivariate autoregressive (MAR) model was established for all IMFs. The values of SBC value

Principal components of feature vectors.

Fault label | Principal components | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PC1 | PC2 | PC3 | PC4 | PC5 | PC6 | PC7 | PC8 | PC9 | PC10 | PC11 | PC12 | PC13 | |

L1 | -0.465 | 0.498 | -0.055 | -0.516 | 0.095 | 0.008 | -0.101 | 0.334 | 0.151 | 0.100 | -0.040 | 0.078 | -0.097 |

-0.459 | 0.510 | -0.076 | -0.515 | 0.116 | 0.003 | -0.082 | 0.297 | 0.137 | 0.083 | -0.032 | 0.076 | -0.095 | |

| |||||||||||||

L2 | -0.362 | -0.252 | 0.257 | -0.034 | 0.268 | -0.861 | -0.699 | -0.181 | -0.131 | -0.702 | 0.053 | 0.071 | -0.185 |

0.396 | -0.547 | -0.194 | -0.347 | 0.161 | -0.432 | -0.415 | 0.160 | 0.059 | -0.438 | 0.510 | -0.070 | -0.393 | |

| |||||||||||||

L3 | 0.883 | 0.374 | 1.127 | 0.456 | 0.168 | 0.145 | -0.120 | 0.229 | -0.036 | 0.061 | 0.054 | -0.031 | 0.020 |

0.903 | 0.393 | 1.113 | 0.462 | 0.140 | 0.162 | -0.158 | 0.221 | -0.054 | 0.065 | 0.081 | -0.054 | 0.030 | |

| |||||||||||||

L4 | -0.724 | -1.063 | 0.770 | 0.037 | -0.593 | 0.167 | 0.025 | -0.271 | -0.099 | 0.045 | 0.053 | 0.124 | -0.244 |

0.277 | -1.649 | 0.483 | -0.168 | 0.614 | 1.125 | 0.160 | -0.182 | 0.285 | -0.320 | 0.017 | 0.495 | 0.519 | |

| |||||||||||||

L5 | -0.387 | 0.075 | -0.232 | -0.161 | 0.232 | 0.566 | 0.057 | -0.033 | -0.077 | -0.069 | 0.124 | 0.013 | 0.072 |

-0.334 | -0.002 | -0.292 | -0.107 | 0.242 | 0.600 | 0.082 | 0.003 | -0.120 | -0.102 | 0.127 | 0.039 | 0.082 | |

| |||||||||||||

L6 | 0.350 | -0.174 | 0.128 | -0.874 | -0.099 | -0.441 | 0.413 | 0.533 | -0.149 | 0.306 | -0.065 | 0.204 | 0.002 |

-1.335 | 0.256 | -0.407 | 0.647 | -0.262 | 0.114 | 0.339 | 0.128 | -0.068 | 0.374 | -0.047 | -0.265 | -0.284 | |

| |||||||||||||

L7 | 1.847 | 0.174 | 0.321 | 0.410 | 0.250 | -0.255 | -0.026 | -0.182 | -0.008 | 0.198 | -0.283 | 0.083 | -0.011 |

1.995 | 0.217 | 0.045 | 0.276 | 0.170 | -0.364 | -0.070 | -0.297 | -0.048 | 0.130 | -0.086 | 0.061 | 0.005 | |

| |||||||||||||

L8 | -1.110 | 0.912 | 0.024 | 1.937 | -1.912 | -0.797 | 0.348 | 0.720 | -0.036 | -0.714 | 0.657 | 0.399 | 0.280 |

-1.337 | 0.116 | -0.700 | 0.553 | 0.894 | -0.020 | 0.247 | -0.459 | -0.077 | 0.289 | -0.036 | 0.308 | 0.335 | |

| |||||||||||||

L9 | -0.845 | 0.547 | -0.570 | 0.382 | -0.046 | 0.345 | 0.295 | 0.237 | 0.015 | -0.127 | -0.100 | -0.218 | 0.005 |

0.753 | 0.129 | -0.638 | 0.636 | 0.267 | 0.380 | 0.208 | -0.340 | -0.274 | -0.473 | -0.328 | -0.078 | -0.162 |

Selection of MAR model order

Principal component distribution of feature vectors.

3D projection of feature vectors for different faults.

During optimization stage, the feature vectors of different fault types were divided randomly into two parts with 30 and 29 vectors, respectively, and the 30 ones were used for training while the 29 ones for testing. The proposed MHGWOSCA approach was applied to optimize the penalty factor

The average fitness value of individuals during iterations.

Iteration number | Optimization methods | |||
---|---|---|---|---|

MHGWOSCA | HGWOSCA | GWO | SCA | |

81 | 88.12 | 97.25 | 97.15 | 95.95 |

82 | 97.32 | 97.25 | 97.17 | 96.06 |

83 | 97.32 | 97.25 | 97.17 | 95.87 |

84 | 97.32 | 97.25 | 97.17 | 96.03 |

85 | 97.33 | 97.26 | 97.17 | 96.13 |

86 | 89.70 | 97.25 | 97.17 | 96.16 |

87 | 97.32 | 97.25 | 97.17 | 96.04 |

88 | 97.33 | 97.25 | 97.18 | 96.02 |

89 | 97.32 | 97.25 | 97.18 | 96.08 |

90 | 97.32 | 97.25 | 97.17 | 96.09 |

91 | 88.87 | 97.26 | 97.19 | 96.25 |

92 | 97.32 | 97.25 | 97.18 | 96.33 |

93 | 97.32 | 97.25 | 97.18 | 96.34 |

94 | 97.32 | 97.25 | 97.19 | 96.29 |

95 | 97.32 | 97.26 | 97.19 | 96.43 |

96 | 88.99 | 97.26 | 97.19 | 96.37 |

97 | 97.32 | 97.26 | 97.19 | 96.40 |

98 | 97.32 | 97.26 | 97.18 | 96.43 |

99 | 97.32 | 97.26 | 97.19 | 96.46 |

100 | 97.32 | 97.26 | 97.19 | 96.46 |

Convergence procedure of proposed optimization approach: (a) overall trend, (b) partial enlarged detail.

Comparison of different optimization methods: (a) overall trend, (b) partial enlarged detail.

With the optimal parameters

Fault diagnosis results for different methods are shown in Table

Fault diagnosis results with different methods.

Processing methods | | | Result evaluation | |||
---|---|---|---|---|---|---|

Adjusted Rand index (ARI) | Normalized mutual information (NMI) | F-measure (F) | Accuracy (ACC) | |||

VMD-FE-HGWOSCA-SVM | 184.0764 | 2.2262 | 0.9001 | 0.9046 | 0.9519 | 0.9525 |

[-0.034, 0.034] | [-0.032, 0.034] | [-0.019, 0.017] | [-0.018, 0.017] | |||

VMD-PE-HGWOSCA-SVM | 2.5155 | 414.0703 | 0.9216 | 0.9221 | 0.9635 | 0.9636 |

[-0.037, 0.053] | [-0.043, 0.048] | [-0.017, 0.025] | [-0.017, 0.025] | |||

EMD-MAR-GWO-SVM | 172.6222 | 0.0010 | 0.9284 | 0.9244 | 0.9662 | 0.9667 |

[-0.031, 0.038] | [-0.033, 0.039] | [-0.017, 0.018] | [-0.016, 0.018] | |||

EMD-MAR-MHGWOSCA-SVM | 254.3134 | 0.0339 | 0.9296 | 0.9236 | 0.9661 | 0.9667 |

[-0.072, 0.053] | [-0.074, 0.055] | [-0.041, 0.026] | [-0.039, 0.026] | |||

VMD-MAR-GWO-SVM | 515.8658 | 0.1028 | 0.9325 | 0.9311 | 0.9691 | 0.9693 |

[-0.048, 0.033] | [-0.054, 0.039] | [-0.024, 0.016] | [-0.023, 0.015] | |||

VMD-MAR-HGWOSCA-SVM | 324.7595 | 0.0925 | 0.9380 | 0.9366 | 0.9719 | 0.9720 |

[-0.039, 0.037] | [-0.053, 0.037] | [-0.018, 0.017] | [-0.018, 0.016] | |||

VMD-MAR-SCA-SVM | 212.6435 | 0.3115 | 0.9431 | 0.9401 | 0.9741 | 0.9743 |

[-0.028, 0.032] | [-0.041, 0.032] | [-0.012, 0.014] | [-0.013, 0.014] | |||

| | | | | | |

| | | |

Comparison of evaluation values for different methods.

Boxplots of evaluation values for different methods; the x-axis tick labels correspond to 1: VMD-FE-HGWOSCA-SVM, 2: VMD-PE-HGWOSCA-SVM, 3: EMD-MAR-GWO-SVM, 4: EMD-MAR-MHGWOSCA-SVM, 5: VMD-MAR-GWO-SVM, 6 VMD-MAR-HGWOSCA-SVM, 7: VMD-MAR-SCA-SVM, and 8: VMD-MAR-MHGWOSCA-SVM.

For rolling bearing fault diagnosis, a new method based on blind parameter identification of MAR model and mutation hybrid GWO-SCA optimized SVM is proposed in this study. Due to the nonstationarity of original signal, signals collected from different types of faults were firstly split into a set of IMFs by VMD, the decomposing mode number

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (nos. 51741907 and 51409095), the Fundamental Research Project for Application Supported by Yichang Science and Technology Bureau (no. A17-302-a12), the Open Fund of Hubei Provincial Key Laboratory for Operation and Control of Cascaded Hydropower Station (no. 2017KJX06), and the Research Fund for Excellent Dissertation of China Three Gorges University (no. 2019SSPY070).