Refined composite multiscale slope entropy and its application in rolling bearing fault diagnosis.

Rolling bearing is the key component of rotating machinery, and its vibration signal usually exhibits nonlinear and nonstationary characteristics when failure occurs. Multiscale permutation entropy (MPE) is an effective nonlinear dynamics analysis tool, which has been successfully applied to rolling bearing fault diagnosis in recent years. However, MPE ignores the deep amplitude information when measuring the complexity of the time series and the original multiscale coarse-graining is insufficient, which requires further research and improvement. In order to protect the integrity of information structure, a novel nonlinear dynamic analysis method termed refined composite multiscale slope entropy (RCMSlE) is proposed in this paper, which introduced the concept of refined composite to further boost the performance of MPE in nonlinear dynamical complexity analysis. Furthermore, RCMSlE utilizes a novel symbolic representation that takes full account of mode and amplitude information, which overcomes the weaknesses in describing the complexity and regularity of bearing signals. Based on this, a GWO-SVM multi-classifier is introduced to fulfill mode recognition, and then a new intelligent fault diagnosis method for rolling bearing based on RCMSlE and GWO-SVM is proposed. The experimental results show that the proposed method can not only accurately identify different fault types and degrees of rolling bearing, but also has a short computation time and better performance than other comparative methods.

Order-frequency Holo-Hilbert spectral analysis for machinery fault diagnosis under time-varying operating conditions.

Holo-Hilbert spectral analysis (HHSA) has been demonstrated to be an effective instantaneous feature demodulation tool for revealing the coupling relationship between the frequency-modulated (FM) carriers and amplitude-modulated (AM) characteristics within nonlinear and non-stationary mechanical vibration signals. However, it is unable to acquire the time varying AM characteristics from the vibration signals of the equipment operates under variable speed conditions. To decode such signals, inspired by HHSA, a novel angle-time double-layer decomposition structure termed order-frequency HHSA (OFHHSA) is established to demodulate the fault information from the time varying vibration signals in this paper. The corresponding spectrogram, namely, order-frequency Holo-Hilbert spectrum (OFHHS) is acquired for describing the interaction relationship between time and angle domains. Besides, the order AM-marginal spectrum is derived from the OFHHS via integrating the carrier variable to exhibit the fault characteristic-related orders. Moreover, the differences between OFHHSA and angle-time cyclo-stationary framework-based order-frequency spectral correlation (OFSC) are analyzed for time varying machinery fault diagnosis. Finally, from the analyses of simulated and tested data of mechanical equipment, the OFHHSA method has avoided the limitations of the two OFSC estimators on periodic assumption and the maximum cut-off order, and the proposed method obtained a more accurate rate in fault identification and more robust ability of anti-noise.

Spectral envelope-based adaptive empirical Fourier decomposition method and its application to rolling bearing fault diagnosis.

Adaptive empirical Fourier decomposition (AEFD) is a recently developed approach of nonstationary signal mode separation. However, it requires to set the spectrum segmentation boundary relying on the users' professional experience ahead of time. In this paper, a novel spectral envelope-based adaptive empirical Fourier decomposition (SEAEFD) method is proposed to improve the performance of AEFD for rolling bearing vibration signal analysis. In the proposed SEAEFD approach, fast Fourier transform (FFT) of the raw signal is calculated to obtain the frequency spectrum at first. Then, the spectral envelope processing is implemented on the spectrum signal obtained by FFT to achieve an adaptive segmentation. In the traditional segmentation method, generally, the minima and midpoints between adjacent extreme points are taken as the spectrum segmentation boundary, in which the obtained frequency band contains more interference components. To achieve the effect of denoising and restrain the noise that existed in the collected vibration signal, SEAEFD is proposed to optimize the spectrum segmentation boundary so that the obtained frequency band contains the least noise components. Lastly, the inverse FFT is used to reconstruct the component signal within each frequency band and the gained signals are termed as Fourier intrinsic mode functions (FIMFs). Therefore, SEAEFD enables a nonstationary signal to be decomposed into several single-component signals with instantaneous frequencies of physical significance. The proposed SEAEFD method is compared with recently developed methods, including EAEFD, AEFD, EWT, VMD and EMD methods, by analyzing the simulation signals and the measured data of rolling bearing. The results indicate that SEAEFD is valid in diagnosing rolling bearing faults and gets a better diagnosis performance than the compared methods.

Composite Multivariate Multi-Scale Permutation Entropy and Laplacian Score Based Fault Diagnosis of Rolling Bearing.

As a powerful tool for measuring complexity and randomness, multivariate multi-scale permutation entropy (MMPE) has been widely applied to the feature representation and extraction of multi-channel signals. However, MMPE still has some intrinsic shortcomings that exist in the coarse-grained procedure, and it lacks the precise estimation of entropy value. To address these issues, in this paper a novel non-linear dynamic method named composite multivariate multi-scale permutation entropy (CMMPE) is proposed, for optimizing insufficient coarse-grained process in MMPE, and thus to avoid the loss of information. The simulated signals are used to verify the validity of CMMPE by comparing it with the often-used MMPE method. An intelligent fault diagnosis method is then put forward on the basis of CMMPE, Laplacian score (LS), and bat optimization algorithm-based support vector machine (BA-SVM). Finally, the proposed fault diagnosis method is utilized to analyze the test data of rolling bearings and is then compared with the MMPE, multivariate multi-scale multiscale entropy (MMFE), and multi-scale permutation entropy (MPE) based fault diagnosis methods. The results indicate that the proposed fault diagnosis method of rolling bearing can achieve effective identification of fault categories and is superior to comparative methods.

Generalized refined composite multiscale fuzzy entropy and multi-cluster feature selection based intelligent fault diagnosis of rolling bearing.

Extracting the failure related information from vibration signals is a very important aspect of vibration-based fault detection for rolling bearing Multiscale entropy and its improvement, multiscale fuzzy entropy (MFE), are significant complexity measure tools of time series. They have been successfully applied to extract vibration failure features for rolling bearing condition monitoring . However, MFE over different scales will fluctuate with increase of scale factor. A new nonlinear dynamic parameter termed generalized refined composite multiscale fuzzy entropy (GRCMFE) is firstly developed to enhance the performance of MSE and MFE in data complexity measurement. Then three algorithms are developed and compared with MSE and MFE, as well as two algorithms of generalized MFE to verify the availability and superiority by analyzing two kinds of noise signals. In addition, based on three algorithms of GRCMFE, a novel fault diagnosis approach for rolling bearing is proposed with linking multi-cluster feature selection for supervised learning and the gravitational search algorithm optimized support vector machine for failure pattern recognition. Last, the proposed fault diagnostic approach was utilized to analyze two kinds of bearing test data sets. Analysis results indicate that our proposed fault diagnosis approach could effectively extract nonlinear dynamic complexity information and gets the highest identifying rate and the best performance among the comparative approaches.

Enhanced periodic mode decomposition and its application to composite fault diagnosis of rolling bearings.

The impulse components of different periods in the composite fault signal of rolling bearing are extracted difficultly due to the background noise and the coupling of composite faults, which greatly affects the accuracy of composite fault diagnosis. To accurately extract the periodic impulse components from the composite fault signals, we introduce the theory of Ramanujan sum to generate the precise periodic components (PPCs). In order to comprehensively extract major periods in composite fault signals, the SOSO-maximum autocorrelation impulse harmonic to noise deconvolution (SOSO-MAIHND) method is proposed to reduce noise and enhance the relatively weak periodic impulses. Based on this, an enhanced periodic mode decomposition (EPMD) method is proposed. The experimental results indicate that the EPMD is an effective method for composite fault diagnosis of rolling bearings.

Mean-optimized mode decomposition: An improved EMD approach for non-stationary signal processing.

As an effective signal separation method of non-stationary signal, empirical mode decomposition (EMD) has been widely used in the data or time series analysis of many engineering fields. However, the decomposing result of EMD often is affected by the fitting in mean curve construction and the sifting process. In this paper, the mean-optimized mode decomposition (MOMD) procedure is proposed to enhance the performance of the original EMD in mean curve construction. Also, the proposed MOMD algorithm is compared with original EMD through analyzing two artificial signals and the analysis results demonstrate that MOMD has much more significantly improvement in decomposition performance and precision than the original EMD. Last, MOMD is introduced to the signal processing stemming from the faulty rolling bearing and the rotor system with failure. Also, the comparison of the proposed MOMD method with EMD was made and the analysis results show that MOMD obtains much more accurate IMFs and fault diagnostic effect than the original EMD method.

Composite Interpolation-Based Multiscale Fuzzy Entropy and Its Application to Fault Diagnosis of Rolling Bearing.

Multiscale fuzzy entropy (MFE), as an enhanced multiscale sample entropy (MSE) method, is an effective nonlinear method for measuring the complexity of time series. In this paper, an improved MFE algorithm termed composite interpolation-based multiscale fuzzy entropy (CIMFE) is proposed by using cubic spline interpolation of the time series over different scales to overcome the drawbacks of the coarse-grained MFE process. The proposed CIMFE method is compared with MSE and MFE by analyzing simulation signals and the result indicates that CIMFE is more robust than MSE and MFE in analyzing short time series. Taking this into account, a new fault diagnosis method for rolling bearing is presented by combining CIMFE for feature extraction with Laplacian support vector machine for fault feature classification. Finally, the proposed fault diagnosis method is applied to the experiment data of rolling bearing by comparing with the MSE, MFE and other existing methods, and the recognition rate of the proposed method is 98.71%, 98.71%, 98.71%, 98.71% and 100% under different training samples (5, 10, 15, 20 and 25), which is higher than that of the existing methods.

Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing.

Multi-scale permutation entropy (MPE) is an effective nonlinear dynamic approach for complexity measurement of time series and it has been widely applied to fault feature representation of rolling bearing. However, the coarse-grained time series in MPE becomes shorter and shorter with the increase of the scale factor, which causes an imprecise estimation of permutation entropy. In addition, the different amplitudes of the same patterns are not considered by the permutation entropy used in MPE. To solve these issues, the time-shift multi-scale weighted permutation entropy (TSMWPE) approach is proposed in this paper. The inadequate process of coarse-grained time series in MPE was optimized by using a time shift time series and the process of probability calculation that cannot fully consider the symbol mode is solved by introducing a weighting operation. The parameter selections of TSMWPE were studied by analyzing two different noise signals. The stability and robustness were also studied by comparing TSMWPE with TSMPE and MPE. Based on the advantages of TSMWPE, an intelligent fault diagnosis method for rolling bearing is proposed by combining it with gray wolf optimized support vector machine for fault classification. The proposed fault diagnostic method was applied to two cases of experimental data analysis of rolling bearing and the results show that it can diagnose the fault category and severity of rolling bearing accurately and the corresponding recognition rate is higher than the rate provided by the existing comparison methods.

Multiscale Distribution Entropy and t-Distributed Stochastic Neighbor Embedding-Based Fault Diagnosis of Rolling Bearings.

As a nonlinear dynamic method for complexity measurement of time series, multiscale entropy (MSE) has been successfully applied to fault diagnosis of rolling bearings. However, the MSE algorithm is sensitive to the predetermined parameters and depends heavily on the length of the time series and MSE may yield an inaccurate estimation of entropy or undefined entropy when the length of time series is too short. To improve the robustness of complexity measurement for short time series, a novel nonlinear parameter named multiscale distribution entropy (MDE) was proposed and employed to extract the nonlinear complexity features from vibration signals of rolling bearing in this paper. Combining with t-distributed stochastic neighbor embedding (t-SNE) for feature dimension reduction and Kriging-variable predictive models based class discrimination (KVPMCD) for automatic identification, a new intelligent fault diagnosis method for rolling bearings was proposed. Finally, the proposed approach was applied to analyze the experimental data of rolling bearings and the results indicated that the proposed method could distinguish the different fault categories of rolling bearings effectively.

Time-Shift Multiscale Fuzzy Entropy and Laplacian Support Vector Machine Based Rolling Bearing Fault Diagnosis.

Multiscale entropy (MSE), as a complexity measurement method of time series, has been widely used to extract the fault information hidden in machinery vibration signals. However, the insufficient coarse graining in MSE will result in fault pattern information missing and the sample entropy used in MSE at larger factors will fluctuate heavily. Combining fractal theory and fuzzy entropy, the time shift multiscale fuzzy entropy (TSMFE) is put forward and applied to the complexity analysis of time series for enhancing the performance of MSE. Then TSMFE is used to extract the nonlinear fault features from vibration signals of rolling bearing. By combining TSMFE with the Laplacian support vector machine (LapSVM), which only needs very few marked samples for classification training, a new intelligent fault diagnosis method for rolling bearing is proposed. Also the proposed method is applied to the experiment data analysis of rolling bearing by comparing with the existing methods and the analysis results show that the proposed fault diagnosis method can effectively identify different states of rolling bearing and get the highest recognition rate among the existing methods.