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The classification frameworks for fault diagnosis of rolling element bearings in rotating machinery are mostly based on analysis in a single time-frequency domain, where sensitive features are not completely extracted. To solve this problem, a new fault diagnosis technique is proposed in the mixed domain, based on the crossover-mutation chaotic particle swarm optimization support vector machine. Firstly, fault features are generated using techniques in the time domain, the frequency domain, and the time-frequency domain. Secondly, the weighted maximum relevance minimum redundancy (WMRMR) algorithm is adopted to reduce the dimension of the feature set and to establish the representative feature set. Thirdly, a new crossover-mutation strategy is suggested to reduce the local minima in optimization, and an optimization disturbance is added. Finally, the support vector machine is optimized using the improved chaotic particle swarm to improve fault classification diagnosis. The effectiveness of the proposed new bearing fault diagnostic technique is verified by experimental tests under different bearing conditions. Test results showed that the bearing fault classification accuracy of CMCPSO-SVM in the mixed domain was much higher than those in a single feature domain.

The rolling bearings in hydraulic pumps operate in harsh environments with high pressure and high temperature, leading to operation degradation or even complete shutdown of the entire mechanical system [

In recent years, support vector machine (SVM) and artificial neural networks have been widely concerned in the classification and recognition field. Although neural network has the characteristics of self-learning and nonlinear processing, it has some clear disadvantages, such as the “black box” effect, expensive computing costs, and time-consuming data specimen training [

To solve these problems of SVM algorithm, optimization algorithms such as genetic algorithm (GA) and particle swarm optimization algorithm (PSO) are used by some scholars to optimize the penalty parameter and kernel function parameter. Ravasan et al. [

The above fault diagnosis techniques mainly focused on time-frequency characteristics of the decomposed signal, and the characteristics of the original signal in the time domain and the frequency domain were usually ignored. Thus, the fault representative features and deep features may not be recognized completely and accurately. Scholars have done a lot of research to solve the abovementioned problems. Jia et al. [

To solve the aforementioned problems, a new bearing fault diagnosis technique based on crossover-mutation chaotic particle swarm in the mixed domain is proposed in this paper. Firstly, the extracted features in the time domain, frequency domain, and time-frequency domain are combined to establish a mixed domain feature set. Secondly, the WMRMR technique is used to select the representative feature subset by sorting the features. Finally, the new crossover-mutation strategy and an optimization disturbance are added to the traditional chaotic particle swarm to optimize SVM, which can improve the accuracy of fault classification.

The time domain feature is a statistical feature that can intuitively discover important feature information through observation [_{1}–_{8} include maximum value, minimum value, peak value, effective value, square root amplitude, absolute mean, mean square value, and kurtosis. Dimensionless parameters _{9}–_{12} are kurtosis parameter, pulse parameter, margin parameter, and peak parameter.

The fault features represented in the frequency domain of vibration signals can be described according to the distribution of spectral information [_{13}–_{15}, respectively.

Bearing fault related features are usually transient and time-varying. Therefore, rolling bearing faults cannot be found intuitively and reliably by the simple time-frequency domain features [

Several PF_{1} components are obtained by LMD decomposition of the measurement signal

The combined reconstruction and noise reduction of _{1} (

In order to improve the noise reduction effect and the accuracy of feature extraction, a new signal processing method of LMD decomposition—noise reduction—signal recombination—LMD decomposition is adopted to reduce the vector dimension and the sparsity of the SVM from the source.

The total energy of each component

The characteristic matrix _{N} is composed of elements

The larger value of the total energy _{N} is normalized by

_{j} is the probability of symbol sequence occurrence,

In general, bearing fault feature information is mainly concentrated in the first five PF components, so the first five PF component permutation entropies will be extracted as the time-frequency domain set, which is written as

The maximum relevance minimum redundancy (MRMR) algorithm can select features by mutual information, correlation or distance similarity scores. Similar to the maximum dependent feature selection algorithm [

The mutual information between random variables

The MRMR evaluation index is determined by_{N} is the number of features; _{i}; _{i} and _{i}; _{j}) is the mutual information between the samples _{i} and _{j}. _{i};

In order to minimize the redundancy between features and maximize the relevance, the MRMR evaluation criterion is defined as

The traditional MRMR method cannot select optimal features of complex feature sets with high relevance and redundancy. Therefore, the weighting coefficient of the evaluation criterion will be optimized. The revised evaluation criterion is expressed as

The SVM is a linear classifier that can maximize the distance between categories according to certain criteria. SVM can search for the optimal hyperplane

The PSO has the advantages of convenient operation, fast convergence, and good optimization effect [

Assuming that a swarm contains M-dimensional particles and the size of the particle swarm is

In each optimization iteration of the particle swarm, the position and velocity need to be tracked dynamically. The velocity _{1} and _{2} are the self-learning weight coefficient and the social learning weight coefficient, respectively; and _{ij} is the _{gj} is the

However, due to the weak local search ability and lack of information communication, the PSO algorithm is easy to fall into a cycle of local minima for complex sample sets. Some methods were proposed by scholars to solve this problem, such as chaotic motion, which has advantages of sensitive initial conditions and good randomness to optimize the PSO algorithm [

The CPSO algorithm can optimize the inertial weight _{1}, and the social learning weight coefficient _{2} by equation (_{i} is the fitness of the

However, the CPSO algorithm is likely to fall into the local minima for samples with complex interference. In order to solve this shortcoming of the CPSO algorithm, a crossover-mutation chaos particle swarm optimization algorithm (CMCPSO) is proposed.

Based on the crossover-mutation process of genetic algorithm, a new adaptive crossover-mutation strategy [

In order to increase the communication between the particles, a certain number of particles are randomly selected to perform the cross operation according to the cross probability

To prevent local minima of the CPSO algorithm, an adaptive mutation strategy is suggested in this work. The adaptive mutation probability

When the globally optimal and locally optimal particles are found, better particles in their neighborhood may exist. Thus, an optimization perturbation will be added into the global and local optimal solutions to improve the global optimization ability of the algorithm. The particle velocity formula is expressed as

By simulation, it is found that

The key part of SVM is the optimization of the penalty factor _{max} is the maximum number of iterations.

Initialize chaotic particle swarm parameters, including inertial weight _{1} and _{2}, maximum number of iteration _{max}

Initialize the position and velocity of the particles. The chaotic swarm is generated by logistic mapping, the optimal position of particles is determined, and the fitness is calculated.

Update the optimal position and velocity of particles. The update position, velocity, and the fitness variance

Crossover-mutation operations are performed based on _{m} and _{c} to enrich the diversity of particles and form the new swarm. Furthermore, the individual and global optimal fitness values are updated.

Optimize

The process of SVM optimized by CMCPSO.

The proposed CMCPSO-SVM technique will be implemented for rolling bearing fault diagnosis in this section. Its effectiveness will be examined by the use of the comparison with other related techniques. The overall flow chart of rolling bearing fault diagnosis is shown in Figure

The flow chart of rolling bearing fault diagnosis in the mixed domain.

The first step is the vibration signal acquisition using an accelerator and a data acquisition model. The second step is mixed domain feature extraction by the statistical analysis, spectral analysis, and dual-LMD methods, and a mixed domain feature set is established.

The third step is the selection of representative features. The weight factor

The fourth step is to apply the proposed CMCPSO-SVM technique for bearing fault diagnosis. The selected representative features are divided into training samples and test samples, and they are input into the CMCPSO-SVM classification model to diagnose rolling bearing faults.

Figure

The bearing fault experiment platform.

Experimental parameters.

Rotating speed | Bearing | Parameter (mm) |
---|---|---|

940r/min | Outer diameter | 52 |

Inner diameter | 25 | |

Pitch circle diameter | 39 | |

Basic rated dynamic load | 12.6 | |

Basic rated static load | 3.3 |

Tested bearings have simulated defect on the inner ring, outer ring, and rolling element of the normal bearings, tested under load and no-load conditions, respectively. The faults of the inner ring, outer ring, and ball failure under the load condition are recorded as LIR, LOR, and LBR, respectively, and the faults of inner ring, outer ring, and ball failure under no-load condition are denoted as IR, OR, and BR, respectively, as shown in Figure

Fault signals under different working conditions. (a)∼(f) are fault signals of BR, LBR, OR, LOR, IR, and LIR, respectively.

The bearing representative features under six operating conditions are extracted in the time domain, frequency domain, and time-frequency domain, respectively. The results of dual-LMD extraction in time-frequency domain are shown in Figure _{,} respectively. The average frequency, frequency variance, and centroid frequency of frequency domain features are marked as

The double LMD extraction results. (a)∼(f) are the double LMD extraction results of BR, LBR, OR, LOR, IR, and LIR, respectively.

Energy normalization results of dual-LMD decomposition.

Operating status | Fault type | E1 | E2 | E3 | E4 | E5 | E6 | E7 | E8 | E9 |
---|---|---|---|---|---|---|---|---|---|---|

No-load | Outer ring failure | 0.5004 | 0.1578 | 0.1302 | 0.1270 | 0.0921 | 0.0651 | 0.0406 | 0.0427 | 0.0241 |

Ball failure | 0.5589 | 0.1731 | 0.1187 | 0.0754 | 0.0393 | 0.0224 | 0.0065 | 0.0031 | 0.0026 | |

Inner ring failure | 0.4514 | 0.2081 | 0.1524 | 0.1055 | 0.0652 | 0.0434 | 0.0079 | 0.0065 | 0.0050 | |

With load | Outer ring failure | 0.3425 | 0.3047 | 0.1484 | 0.1337 | 0.0343 | 0.0253 | 0.0071 | 0.0027 | 0.0013 |

Ball failure | 0.3435 | 0.1998 | 0.1453 | 0.1428 | 0.0702 | 0.0391 | 0.0045 | 0.0023 | 0.0014 | |

Inner ring failure | 0.4001 | 0.2835 | 0.1318 | 0.0925 | 0.0732 | 0.0165 | 0.0048 | 0.0029 | 0.0016 |

The weight factor _{i} corresponding to the weight factor

Feature ranking.

Feature order | |
---|---|

0.1 | TF_{19}, _{4}, F_{13}, F_{14}, F_{15}, TF_{17}, _{12}, TF_{18}, TF_{20}, TF_{25}, _{5}, _{1}, |

_{2}, TF_{24}, TF_{23}, TF_{22}, TF_{21}, TF_{16}, _{6}, _{8}, _{9}, _{3}, _{7}, _{10}, _{11} | |

0.2 | TF_{19}, _{4}, F_{13}, F_{14}, F_{15}, TF_{17}, _{12}, TF_{18}, _{8}, _{9}, _{3}, _{10}, _{11}, |

TF_{20}, TF_{25}, _{5}, _{7}, TF_{23}, TF_{24}, _{1}, _{2}, TF_{22}, TF_{21}, TF_{16}, _{6} | |

0.3 | TF_{19}, _{4}, F_{13}, F_{14}, F_{15}, TF_{22}, TF_{17}, _{3}, _{12}, _{2}, _{8}, _{9}, TF_{18}, |

TF_{21}, TF_{24}, _{7}, TF_{16}, TF_{20}, TF_{25}, _{1}, _{5}, _{6}, TF_{23}, _{10}, _{11} | |

0.4 | TF_{19}, _{4}, F_{13}, F_{14}, _{11}, TF_{17}, TF_{24}, TF_{21}, _{2}, _{8}, _{9}, TF_{22}, |

TF_{23}, F_{15}, TF_{18}, _{1}, _{5}, TF_{16}, TF_{20}, TF_{25}, _{3}, _{6}, _{7}, _{10}, _{12} | |

0.5 | TF_{19}, _{4}, F_{13}, _{11}, TF_{16}, TF_{17}, _{5}, _{6}, TF_{21}, TF_{22}, TF_{24}, F_{13}, |

TF_{20}, _{2}, _{3}, F_{15}, _{8}, F_{14}, _{7}, _{9}, _{10}, _{1}, _{12}, TF_{23}, TF_{18}, TF_{25} | |

0.6 | TF_{19}, TF_{22}, _{9}, _{10}, TF_{23}, _{4}, _{12}, TF_{21,}_{8}, TF_{20}, TF_{24}, _{3}, |

_{7}, _{8}, TF_{16}, _{1}, F_{13}, _{5}, _{2}, F_{14}, F_{15}, _{6}, _{11}, TF_{17}, TF_{18}, TF_{25} | |

0.7 | TF_{19}, _{11}, TF_{17}, _{12}, _{10}, TF_{22}, _{3}, TF_{21}, TF_{24}, _{7}, _{1}, TF_{23}, |

_{6}, _{2}, F_{13}, F_{14}, F_{15}, TF_{16}, TF_{20}, _{4}, _{5}, _{8}, _{9}, TF_{18}, TF_{25} | |

0.8 | TF_{19}, _{7}, _{11}, _{12}, TF_{17}, _{10}, TF_{21}, _{3}, TF_{22}, _{1}, TF_{23}, TF_{24}, |

_{6}, _{2}, F_{13}, F_{14}, F_{15}, TF_{16}, TF_{20}, _{4}, _{5}, _{8}, _{9}, TF_{18}, TF_{25} | |

0.9 | TF_{19}, _{7}, _{11}, _{12}, TF_{17}, TF_{21}, _{10}, _{3}, TF_{24}, _{1}, TF_{22}, TF_{23} |

_{6}, _{2}, F_{13}, F_{14}, F_{15}, TF_{16}, TF_{20}, _{4}, _{5}, _{8}, _{9}, TF_{18}, TF_{25} | |

1 | TF_{19}, _{7}, _{11}, TF_{17}, _{12}, TF_{21}, _{10}, _{3}, _{1}, TF_{24}, TF_{22}, TF_{23,} |

_{6}, _{2}, F_{13}, F_{14}, F_{15}, TF_{16}, TF_{20}, _{4}, _{5}, _{8}, _{9}, TF_{18}, TF_{25} |

The selected low-dimensional feature subset

The accuracy of classification results with different weight factors

As can be seen from Figure

To verify the effectiveness of the CMCPSO-SVM classification model, the low-dimensional optimal feature subset

It can be seen from Figure

Since the iteration number of the classifier is also an important factor which can affect the classification performance, the fault diagnosis accuracy and time under different iteration number working conditions are listed in Table

There are many types of classifiers, such as neural network classifier and Bayesian classifier. Therefore, experimental comparison between SVM classifiers is not enough to illustrate the superiority of CMCPSO-SVM classifier. To further examine the effectiveness of the proposed CMCPSO-SVM classifier, the radial basis function neural network (RBFNN), extreme learning machine (ELM), K nearest neighbor algorithm (KNN), BP neural network algorithm (BPNN), and CMCPSO-SVM classifier are selected to compare the accuracy of fault diagnosis. Figure

Different feature subset classification accuracy rate curves corresponding to different

: The classification results and the change law of fitness value. (a)∼(c) are the classification results and the change law of fitness value of PSO-SVM, CPSO-SVM, and CMCPSO-SVM classification models, respectively.

Classification accuracy of the three diagnosis models.

Fault type | Bearing fault label | Number of training samples | Number of test samples | Classifier type | ||
---|---|---|---|---|---|---|

PSO-SVM (%) | CPSO-SVM (%) | CMCPSO-SVM (%) | ||||

BR | 1 | 25 | 25 | 88 | 100 | 100 |

LBR | 2 | 25 | 25 | 92 | 88 | 96 |

OR | 3 | 25 | 25 | 92 | 88 | 100 |

LOR | 4 | 25 | 25 | 84 | 92 | 100 |

IR | 5 | 25 | 25 | 80 | 80 | 100 |

LIR | 6 | 25 | 25 | 92 | 96 | 96 |

Total/average | - | 150 | 150 | 88 | 90.51 | 98.67 |

Fault diagnosis accuracy and time under different iteration number working conditions.

Fault type | Number of iterations | |||||||
---|---|---|---|---|---|---|---|---|

50 | 100 | 150 | 200 | |||||

Accuracy | Time | Accuracy | Time | Accuracy | Time | Accuracy (%) | Time (s) | |

BR | 97.26 | 30.25 | 100 | 55.37 | 97.47 | 75.35 | 95.25 | 105.62 |

LBR | 97.01 | 33.51 | 96 | 52.96 | 96.29 | 77.64 | 96.32 | 112.31 |

OR | 96.59 | 37.65 | 100 | 52.13 | 97.02 | 79.26 | 96.54 | 99.86 |

LOR | 98.37 | 38.01 | 100 | 50.64 | 97.64 | 70.66 | 95.91 | 102.45 |

IR | 97.05 | 34.55 | 100 | 54.75 | 96.65 | 80.51 | 96.56 | 103.34 |

LIR | 96.34 | 33.19 | 96 | 52.12 | 96.11 | 78.96 | 97.79 | 105.75 |

Average | 97.10 | 34.53 | 98.67 | 52.99 | 96.86 | 77.06 | 96.40 | 104.89 |

Average classification accuracy of different classification algorithms in different domains.

In order to solve the problem of sensitive features incomplete extraction in time-frequency domain, a fault diagnosis method based on crossover-mutation chaotic particle swarm optimization support vector machine in the mixed domain is proposed in this paper. The innovation points of this paper are as follows:

The sensitive characteristic values of the fault signals are selected in mixed domain, and the WMRMR algorithm is used to reduce the dimension of the sensitive feature set to obtain the optimal low-dimensional feature subset.

A new CMCPSO-SVM classifier is proposed. The new crossover-mutation strategy and an optimization disturbance are added to the CPSO algorithm to avoid the local optimization and improve the classification accuracy.

Different algorithms are used to classify bearing faults under different working conditions. Results showed that the average accuracy of CMCPSO-SVM was 98.67%, which was higher than that of the PSO-SVM and CPSO-SVM. In addition, the algorithm proposed in this paper, and the other four commonly algorithms were used for fault classification in different domains. From experimental analysis, it can be seen that the fault diagnosis accuracy of the CMCPSO-SVM algorithm in the mixed domain is the highest.

In this paper, CMCPSO-SVM was used to diagnose single faults of bearings under different working conditions, and the ideal results could be achieved. And for the complicated working environment with variable bearing faults, the multiclass SVM classification strategy based on the CMCPSO-SVM might be a good choice to diagnose the bearing compound faults. As for future work, the combining multiple binary SVM classifiers could be utilized for bearing compound faults diagnosis.

All data are provided in full in the numerical simulation and discussion section of this article.

The authors declare that they have no conflicts of interest.

All authors contributed equally and all of them read and approved the final manuscript.

This research was supported by the National Natural Science Foundation of China (grant nos. 51805299 and 51465009).