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On karst steep slopes, subsurface flow frequently emerges at the rock-soil interface, thereby exacerbating soil erosion. However, there have been few studies that quantitatively assess the impact of subsurface flow on soil erosion in this region. We conducted indoor artificial rainfall experiments to observe runoff and sediment yield under two conditions: with subsurface flow (SF) and without subsurface flow input (WSF). We considered various rainfall intensities (30 mm h−1, 60 mm h−1, 90 mm h−1) and slope gradients (30°, 35°, 40°). The results demonstrated that the presence of subsurface flow significantly amplified both surface runoff and sediment yield, while also intensifying the influence of rainfall intensity and slope gradient on runoff and sediment yield. Compared to WSF, under SF conditions, surface runoff increased by 1.55 to 4.43 times, and sediment yield increased by 4.60 to 8.72 times. Slope gradient played a substantial role in soil loss, while rainfall intensity primarily impacted surface runoff. Under WSF conditions, soil loss only occurred at a 40° slope gradient or under a rainfall intensity of 90 mm h−1, whereas under SF conditions, soil loss could occur at a 30° slope gradient or under a rainfall intensity of 30 mm h−1. This research provides a scientific foundation for a more profound understanding of the mechanisms driving soil erosion on karst slopes, particularly in the context of subsurface flow.

Multi-channel signal has more abundant and accurate state characteristic information than single channel signal. How to separate fault characteristic information from the multi-channel signal is the key of fault diagnosis. As two typical multi-channel signal decomposition methods, multivariate empirical mode decomposition (MEMD) and multivariate variational mode decomposition (MVMD) are widely used in multi-channel signal analysis. However, MEMD and MVMD use cyclic iteration to complete the analysis of multi-channel signals, and it is difficult to overcome their inherent defects. In view of this, based on nonlinear sparse mode decomposition (NSMD), this paper proposes a multivariate nonlinear sparse mode decomposition (MNSMD) by constraining singular local linear operators to separate the natural oscillation modes in multi-channel signal. By constraining singular local linear operators into signal decomposition, MNSMD has obvious advantages in restraining mode aliasing and robustness. In addition, the local narrow-band component is used as the basis function for iteration, and the component signal is obtained by approaching the original signal. Through the simulation signal and gear fault signal analysis, the results show that, compared with MEMD and MVMD methods, MNSMD method can effectively complete gear fault diagnosis.

The vibration signals collected by the sensor often have non-stationary and non-linear characteristics owing to the complexity of working environment of rolling bearing, so it is difficult to obtain useful and stable vibration information for diagnosis. Empirical Wavelet Transform (EWT) can effectively decompose non-stationary and nonlinear signals, but it is not suitable for signal analysis of bearing with a complicated spectrum. In this paper, an improved EWT (IEWT) method is proposed by developing a new segmentation approach. Meanwhile, the IEWT is compared with empirical mode decomposition (EMD) and EWT to verify the superiority of IEWT in decomposition accuracy. By combining with the refined composite multiscale dispersion entropy (RCMDE), which is a powerful nonlinear tool for irregularity measurement of vibration signals, a new diagnosis method based on IEWT, RCMDE, multi-cluster feature selection and support vector machine is proposed. Then the method is applied to analysis of bearing in this paper and the results show that the new method has higher identifying rate and better performance than that of the methods of RCMDE combining with EMD or EWT. Also, the superiority of RCMDE to dispersion entropy and multiscale dispersion entropy is investigated, together with the superiority of MCFS for feature selection.

To solve the problem that the limited time-frequency features cannot fully represent the deep-seated state information of rolling bearing, the time-frequency analysis method, whale optimization algorithm (WOA) and support matrix machine (SMM) are combined, and a fault diagnosis model based on multisynchrosqueezing transform (MSST) and WOA-SMM is proposed. First, the time-frequency trait of the original signal is extracted by MSST. Then, using the time-frequency spectrum processed by MSST as the input of SMM, MSST solves the problem of state information loss when constructing a characteristic matrix. Finally, the parameters of the SMM are optimized by WOA and the ideal parameters can be obtained adaptively, and the problem of setting parameters subjectively is solved. The experimental analysis of the two datasets shows that WOA-SMM is superior not only to other classifiers in classification performance, but also has higher in convergence accuracy and speed for rolling bearing fault diagnosis.

Many nonlinear dynamic and statistic methods, including multiscale sample entropy (MSE) and multiscale fuzzy entropy (MFE), have been widely studied and employed to fault diagnosis of the rolling bearing. Multiscale dispersion entropy (MDE) is a powerful tool for complexity measure of time series, and compared with MSE and MFE, it gets much better performance and costs less time for computation. Since single-channel time series analysis will cause information missing, inspired by multivariate multiscale sample entropy (MMSE) and multivariate multiscale fuzzy entropy (MMFE), refined composite multivariate multiscale dispersion entropy (RCMMDE) was proposed in this paper. After that, RCMMDE was compared with MDE, MMSE, and MMFE by analyzing synthetic signals and the results show that the RCMMDE has certain advantages in terms of robustness. A hybrid fault diagnostics approach is proposed for rolling bearing with a combination of RCMMDE, multi-cluster feature selection, and support vector machine. Also, the proposed method is compared with MDE, MMSE, and MMFE, as well as multivariate multiscale dispersion entropy-based fault diagnosis methods by analyzing the experimental data of rolling bearing, and the result shows that the proposed method gets a higher identification rate than the existing other fault diagnosis methods.

Incipient damages of wind turbine rolling bearing are very difficult to be detected because of the interference of multi-frequency components and strong ambient noise. To solve this problem, this paper proposes a new detected method named VMD-AMCKD, combining complementary advantages of variational mode decomposition (VMD) and adaptive maximum correlated kurtosis deconvolution (AMCKD). A novel index is proposed to screen out the most sensitive mode containing fault information after VMD decomposition. The mode also can determine a suspectable range for the fault frequency, based on which the optimized range of devolution period T in MCKD can be pre-determined. The Grasshopper optimization algorithm (GOA) is adapted to adaptively select the key parameters in MCKD. The proposed method can successfully diagnose the simulated signal mixed with strong white Gaussian noise. Its robustness is further proven by the diagnosis for three different types of experimental signal from CWRU bearing data center. Finally, the VMD-AMCKD is applied to detect incipient damages of rolling bearings in a laboratory wind turbine.

The squared envelope spectrum (SES) is one of the most effective methods in rolling element bearing fault diagnosis. Being a technique based on demodulation, the success of SES depends highly on the extraction of information caused by the bearings fault. Therefore, a preprocessing step of choosing an optimal frequency band (OFB), before the implementation of the SES, is absolutely needed and should always be taken prior. Fast kurtogram (FK), as the most commonly used method targeting resonance frequency band which is excited by the bearing fault, is applied as the benchmark tool for the selection of the OFB. However, recent theoretical works show that the kurtosis used as the criterion for OFB selection was both sensitive to the impulsiveness and cyclostationarity, which means irrelevant impulsive and cyclostationary components are able to mislead the kurtosis based OFB selection techniques and subsequently bring great difficulties in bearing fault diagnosis. To tackle this problem, in this paper, a novel statistical model parameter-based method, called Distcsgram, is proposed as a substitution to the traditional FK technique so that an OFB can be selected with the least influences from the irrelevant cyclostationary and impulsive interferences. Together with the SES, the Distcsgram is explained and validated in both simulation and experimental studies. Furthermore, through the comparisons with the traditional methods, the superiority of the proposed method over those traditional methods for the diagnosis of rolling element bearing is proved.

As an effective tool for nonlinear and non-stationary signal separation method, empirical mode decomposition (EMD) has attracted a lot of attention of many scholars and has been successfully applied to many engineering areas. Since the kernel of EMD is to define a baseline and then implement the sift process, it is important to select an appropriate baseline to improve the performance of EMD for an accurate decomposition. However, it is difficult to choose the most suitable method for dealing with a given arbitrary signal. Besides, the baseline constructed through the existing methods generally does not equal the expected one. To address these problems, we propose a new generalized framework for adaptive mode decomposition (GF-AMD) based on EMD, in which the optimal baseline is firstly chosen from different ones and a weighted factor is introduced to adjust the baseline for getting the optimal one. Then the optimal baseline is implemented to the sift process of EMD. Two simulation signals are used to verify the effectiveness and superiority to EMD. Finally, the proposed GF-AMD method is applied to the vibration signal analysis of faulty rotary machinery by comparing with EMD method and the analysis results indicate its effect and superiority.

Effective condition monitoring provides some benefits such as improving safety and reliability. Roller bearing is the key component of rotating machinery, and a novel roller bearing condition monitoring method based on rational Hermite interpolation-local characteristic-scale decomposition (RHLCD) and fusion variable predictive model-based class discriminate method (FVPMCD) is proposed in this paper. RHLCD can adaptively decompose any complex signal into a sum of rational intrinsic scale components (RISCs), whose instantaneous frequency has physical meaning. In addition, targeting the limitation of variable predictive model-based class discriminate method (VPMCD), FVPMCD is presented. First, four kinds of common models are used to recognize a sample. Then, the recognition results of each model are satisfied, and the recognition probability of each state is calculated. Finally, the largest recognition probability of the state is chosen to recognize categories. The analytical results of experimental signals indicate that the proposed condition monitoring approach can identify the states of roller bearing effectively.

To enhance the performance of deep auto-encoder (AE) under complex working conditions, a novel deep auto-encoder network method for rolling bearing fault diagnosis is proposed in this paper. First, multiscale analysis is adopted to extract the multiscale features from the raw vibration signals of rolling bearing. Second, the sparse penalty term and contractive penalty term are used simultaneously to regularize the loss function of auto-encoder to enhance the feature learning ability of networks. Finally, the cuckoo search algorithm (CS) is used to find the optimal hyperparameters automatically. The proposed method is applied to the experimental data analysis. The results indicate that the proposed method could more effectively distinguish fault categories and severities of rolling bearings under different working conditions than other methods.

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.

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 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.

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.

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.

The vibration signals of rolling bearings are often nonlinear and non-stationary. Multiscale entropy (MSE) has been widely applied to measure the complexity of nonlinear mechanical vibration signals, however, at present only the single channel vibration signals are used for fault diagnosis by many scholars. In this paper multiscale entropy in multivariate framework, i.e., multivariate multiscale entropy (MMSE) is introduced to machinery fault diagnosis to improve the efficiency of fault identification as much as possible through using multi-channel vibration information. MMSE evaluates the multivariate complexity of synchronous multi-channel data and is an effective method for measuring complexity and mutual nonlinear dynamic relationship, but its statistical stability is poor. Refined composite multivariate multiscale fuzzy entropy (RCMMFE) was developed to overcome the problems existing in MMSE and was compared with MSE, multiscale fuzzy entropy, MMSE and multivariate multiscale fuzzy entropy by analyzing simulation data. Finally, a new fault diagnosis method for rolling bearing was proposed based on RCMMFE for fault feature extraction, Laplacian score and particle swarm optimization support vector machine (PSO-SVM) for automatic fault mode identification. The proposed method was compared with the existing methods by analyzing experimental data analysis and the results indicate its effectiveness and superiority.

A new rolling bearing fault diagnosis approach based on multiscale permutation entropy (MPE), Laplacian score (LS), and support vector machines (SVMs) is proposed in this paper. Permutation entropy (PE) was recently proposed and defined to measure the randomicity and detect dynamical changes of time series. However, for the complexity of mechanical systems, the randomicity and dynamic changes of the vibration signal will exist in different scales. Thus, the definition of MPE is introduced and employed to extract the nonlinear fault characteristics from the bearing vibration signal in different scales. Besides, the SVM is utilized to accomplish the fault feature classification to fulfill diagnostic procedure automatically. Meanwhile, in order to avoid a high dimension of features, the Laplacian score (LS) is used to refine the feature vector by ranking the features according to their importance and correlations with the main fault information. Finally, the rolling bearing fault diagnosis method based on MPE, LS, and SVM is proposed and applied to the experimental data. The experimental data analysis results indicate that the proposed method could identify the fault categories effectively.