The vibration signals resulting from rolling bearings are nonlinear and nonstationary, and an approach for the fault diagnosis of rolling bearings using the quantile permutation entropy and EMD (empirical mode decomposition) is proposed. Firstly, the EMD is used to decompose the rolling bearings vibration signal, and several IMFs (intrinsic mode functions) spanning different scales are obtained. Secondly, aiming at the shortcomings of the permutation entropy algorithm, a new permutation entropy algorithm based on sample quantile is proposed, and the quantile permutation entropy of the first few IMFs, which contain the main fault information, is calculated. The quantile permutation entropies are accordingly seen as the characteristic vector and then input to the particle swarm optimization and support vector machine. Finally, the proposed method is applied to the experimental data. The analysis results show that the proposed approach can effectively achieve fault diagnosis of rolling bearings.

Rolling bearings are the most frequently used component in a rotary machines, which play an essential role in the modern industry [

The application of entropy theory provides a new idea for rolling bearing fault diagnosis. Pincus proposed the concept of approximate entropy (ApE) to measure the probability of new patterns in signals according to the characteristics of time series [

However, due to the complexity of mechanical systems, the randomicity and dynamic changes of the vibration signal behave on different scales, making it necessary to analyze the vibration signal with PE in a multiscale way [

Compared with the traditional feature extraction method, this paper solves the problem of embedding dimension selection in the PE calculation process by SQPE entropy calculation and improves the feature extraction ability of PE at the same scale. The proposed method is applied to the bearing experimental data. The results show that the method can effectively distinguish the fault type of rolling bearings and is an effective fault diagnosis method.

The remainder of this paper is organized as follows: Section

In order to realize the feature extraction of different vibration signals of rolling bearings, the multiscale decomposition method is first used to decompose the signals into different time scales. Then, using the SQPE proposed in this paper, the SQPE value of the IMF component containing the main fault information is calculated to deeply mine the rolling bearing fault information.

EMD was put forward by Huang N. E. in 1998, which seemed much more promising and had appealed a lot of attention [

The EMD algorithm can decompose a nonlinear and nonstationary signal self-adaptively into some IMFs. Each IMF contains fault information of different frequency bands and different scales of the original vibration signal [

The SQPE proposed in this paper is an improved algorithm based on PE, which mainly solves the problem of PE in embedding dimension selection and improves the feature extraction ability of PE for complex signals through the quantile principle. The definition of SQPE is based on PE, and PE is described as follows.

Consider a time series,

It is noticed that

Obviously, the values of

There are two parameters to be set in the calculation of PE: the embedding dimension

In order to improve the adaptability of the selection with embedding dimension and improve the feature extraction ability of PE. On the basis of PE, this paper introduces the concept of sample quantile and defines SQPE. Sample quantile divides the probability distribution range of a random variable into equal numerical points, without considering the time dimension characteristics of the time series, and directly and effectively characterizes the characteristics of the fluctuation of the signal time series data [

Given a time series

The sample quantile has the advantages of simple calculation and small calculation. And the sample quantile is a nonparametric statistic, which can accurately reflect the aggregation characteristics of the data at a certain quantile in the absence of the overall distribution prior information. The SQPE is defined as follows:

Given a time series

The 0.75 sample quantile of

SQPE is used instead of PE as the feature information for fault diagnosis

The proposed SQPE algorithm can be described briefly as in Figure

Flow chart of SQPE.

In the calculation of SQPE, the problem of embedding dimension selection is not considered, but the entropy value is solved by calculating the sample quantile of all embedding dimensions. This avoids the effect of embedding dimension on feature extraction to some extent.

In addition to EMD-SQPE, the proposed fault diagnosis also adopts the PSO-SVM algorithm. The SVM was originally a deterministic algorithm for finding the linear separating hyperplane of a binary labeled dataset. Similar with the neural network, it has machine learning features. The core algorithm of SVM is to change the original data space to a high-dimensional feature space through a nonlinear mapping [

Since the introduction of the PSO algorithm in 1995, researchers have put much effort to improve the original version of PSO. Shi and Eberhart proposed the idea of inertia weight in order to balance the local and global search during the optimization process, in which the inertia weight is linearly decreasing over iterations [

Based on the advantages of EMD, SQPE, and PSO-SVM, the proposed rolling bearing fault diagnosis algorithm is described as follows.

Decomposing the initial rolling bearing vibration signals into several IMF components using the EMD method.

SQPE of the first several IMF components which contain the main failure information is calculated and taken as the feature vector. According to [

where

The feature vectors are seen as the inputs to the SVM classifier. Here, the feature information in (

PSO is applied into adaptive selection of the best penalty parameter and kernel function parameter.

The output results of testing samples achieve to discriminate the fault categories automatically.

The proposed EMD-SQPE-PSOSVM fault diagnosis algorithm can be described briefly as in Figure

Flow chart of the proposed method.

In order to validate the capability of the EMD-SQPE-PSOSVM method, experimental analyses on rolling bearing faults were conducted. The 6205-2RS JEM SKF deep groove ball bearing was used in the experimental, and single point faults were introduced to the test rolling bearings using electrodischarge machining with fault diameters of 0.3556 mm [

In the experiments, the feature vectors are seen as the inputs to the PSOSVM model, and the output of the PSOSVM model is the label value corresponding to each fault state, set to NORM (label1), IRF (label2), REF (label3), and ORF (label4).

PE of the data sets is calculated and taken as the feature vector. The embedding dimension and the delay of PE were set to

The output results and the desired outputs of testing sets with feature vector consisting of the original data’s PE: (a) SVM diagnosis results; (b) PSOSVM diagnosis results.

The accuracy of 160 testing sets with the feature vector consisting of PE by the SVM classifier is 84.375% (135/160), and the accuracy of the PSOSVM classifier is 87.5% (140/160). The best penalty parameter is 1.8232, and the best kernel function parameter is 645.2294 after PSO optimized.

In order to visually explain the fault diagnosis results of PE and SQPE, SQPE of the data sets is calculated and taken as the feature vector. The accuracy of 160 testing sets with the feature vector consisting of SQPE by the SVM classifier is 98.125% (157/160), and the accuracy of the PSOSVM classifier is 98.75% (158/160). The best penalty parameter is 0.1, and the best kernel function parameter is 2796.8326 after PSO optimized.

As shown in Figure

The output results and the desired outputs of testing sets with feature vector consisting of the EMD-PE: (a) SVM diagnosis results; (b) PSOSVM diagnosis results.

The accuracy of 160 testing sets with the feature vector consisting of EMD-PE by the SVM classifier is 95% (152/160), and the accuracy of the PSOSVM classifier is 96.875% (155/160). The best penalty parameter is 13.2931, and the best kernel function parameter is 1960.7666 after PSO optimized.

As shown in Figure

The output results and the desired outputs of testing sets with the feature vector consisting of the EMD-SQPE: (a) SVM diagnosis results; (b) PSOSVM diagnosis results.

The accuracy of 160 testing sets with the feature vector consisting of EMD-SQPE by the SVM classifier is 98.75% (158/160), and the accuracy of the PSOSVM classifier is 100% (160/160). The best penalty parameter is 0.1, and the best kernel function parameter is 1307.5593 after PSO optimized.

In order to visually explain the fault diagnosis results on the vibration signal of the rolling bearing, the results obtained by each method are shown in Table

Accuracy rate comparison of three algorithms.

Fault diagnosis method | NORM | IRF | REF | ORF | Accuracy (%) |
---|---|---|---|---|---|

PE-SVM | 40 | 39 | 35 | 21 | 84.375 |

PE-PSOSVM | 40 | 37 | 38 | 25 | 87.5 |

EMD-PE-SVM | 40 | 38 | 40 | 34 | 95 |

EMD-PE-PSOSVM | 40 | 39 | 40 | 36 | 96.875 |

EMD-SQPE-SVM | 40 | 40 | 40 | 38 | 98.75 |

EMD-SQPE-PSOSVM | 40 | 40 | 40 | 40 | 100 |

As shown in Table

To illustrate the influence of the number of training sets, the experiments were designed by different training set scales (10%, 20%, 30%, and 40% of total data sets), and the remaining sets are used for test; the accuracies of the EMD-SQPE-PSOSVM and EMD-PE-PSOSVM for different percentages of the samples used for training are presented in Table

Accuracies with different training set scales.

Fault diagnosis method | Percentage of the samples used for training | ||||
---|---|---|---|---|---|

10% | 20% | 30% | 40% | 50% | |

EMD-PE-PSOSVM | 97.9167% | 96.875% | 97.7679% | 97.3958% | 96.875% |

EMD-SQPE-PSOSVM | 100% | 100% | 100% | 100% | 100% |

As shown in Table

In this paper, a new rolling bearing fault diagnosis method has been introduced based on the vibration signal analysis using EMD-SQPE and PSOSVM. The test data analysis conclusions illustrate that the proposed algorithm can not only analyze among different fault categories but also identify the level of fault severity. The experimental results show the following:

SQPE is a valid method to measure the complexity of nonlinear and nonstationary vibration signals. Compared with PE, SQPE can extract the features with high distinguishability.

SQPE can avoid the selection of embedding dimension, and it is not necessary to determine the specific embedding dimension in the solution process, which has better adaptability.

EMD has a good application prospect in analyzing the nonlinear signal. After the decomposition of the original signal by EMD, entropy is applied to measure the feature of the IMFs obviously.

PSO can select the best penalty parameter and kernel function parameter adaptively, which improved the performance of SVM.

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

The authors declare that they have no conflicts of interest.

This research was funded by the Shandong Natural Science Foundation of China (grant number ZR2017MF036) and Defense Science and Technology Project Foundation of China (grant number F062102009).