Application of Auto-Regulative Sparse Variational Mode Decomposition in Mechanical Fault Diagnosis

This paper proposes a novel method to improve the variational mode decomposition (VMD) method and to automatically acquire the sensitive intrinsic mode function (IMF). First, since fault signals are impulsive and periodic, a weighted autocorrelative function maximum (AFM) indicator is constructed based on the Gini index and autocorrelation function to serve as the optimization objective function. The mode number K and the penalty parameter α of VMD are automatically obtained through an optimal parameter searching process underpinned by the improved particle swarm optimization (PSO) algorithm with a variety of inertia weights. This improvement solves one of the major drawbacks of the conventional VMD method, that is, the need to manually set parameters. Then, an optimal IMF automatic selecting process is performed for single-failure faults and compound faults, according to the principles of the maximum weighted AFM indicator and maximum spectrum peak ratio (SPR), respectively. The sensitive IMFs are then subjected to an envelope demodulation analysis to obtain the fault characteristic frequency. The results of simulations and experiments show that the proposed method can effectively identify fault characteristics early, especially compound faults, demonstrating great potential for real-world applications.

The variational mode decomposition (VMD) method for signal decomposition is severely affected by the number of components of the VMD method. In order to determine the decomposition modal number, K, in the VMD method, a new center frequency method of the multi-threshold is proposed in this paper. Then, an improved VMD (MTCFVMD) algorithm based on the center frequency method of the multi-threshold is obtained to decompose the vibration signal into a series of intrinsic modal functions (IMFs). The Hilbert transformation is used to calculate the envelope signal of each IMF component, and the maximum frequency value of the power spectral density is obtained in order to effectively and accurately extract the fault characteristic frequency and realize the fault diagnosis. The rolling element vibration data of the motor bearing is used to test the effectiveness of proposed methods. The experiment results show that the center frequency method of the multi-threshold can effectively determine the number, K, of decomposed modes. The proposed fault diagnosis method based on MTCFVMD and Hilbert transformation can effectively and accurately extract the fault characteristic frequency, rotation frequency, and frequency doubling, and can obtain higher diagnostic accuracy.

A motor bearing system is a nonlinear dynamics system with nonlinear support stiffness. It is an asymmetry system, which plays an extremely important role in rotating machinery. In this paper, a center frequency method of double thresholds is proposed to improve the variational mode decomposition (VMD) method, then an adaptive VMD (called DTCFVMD) method is obtained to extract the fault feature. In the DTCFVMD method, a center frequency method of double thresholds is a symmetry method, which is used to determine the decomposed mode number of VMD according to the power spectrum of the signal. The proposed DTCFVMD method is used to decompose the nonlinear and non-stationary vibration signals of motor bearing in order to obtain a series of intrinsic mode functions (IMFs) under different scales. Then, the Hilbert transform is used to analyze the envelope of each mode component and calculate the power spectrum of each mode component. Finally, the power spectrum is used to extract the fault feature frequency for determining the fault type of the motor bearing. To test and verify the effectiveness of the DTCFVMD method, the actual fault vibration signal of the motor bearing is selected in here. The experimental results show that the center frequency method of double thresholds can effectively determine the mode number of the VMD method, and the proposed DTCFVMD method can accurately extract the clear time frequency characteristics of each mode component, and obtain the fault characteristics of characteristics; frequency, rotating frequency, and frequency doubling and so on.

Aiming at the problems of difficulty in extracting vibration signal features and low recognition accuracy of bearing fault, a new fault diagnosis method based on ISSA-VMD and IMSE was proposed. This method leverages an improved sparrow search algorithm (ISSA) to optimize the variational mode decomposition (VMD) method and employs an improved multiscale sample entropy (IMSE) for fault feature extraction. Initially, the ISSA algorithm optimizes two critical parameters of the VMD method: the number of modes K and the penalty factor α, to obtain the optimal parameter combination [K, α]. The optimized VMD method is then used to decompose the bearing vibration signals for signal reconstruction analysis. Subsequently, the IMSE entropy algorithm is applied to the reconstructed signals to extract fault features, resulting in the required fault feature vector set. Finally, this extracted fault feature vector set is input into a multi-kernel extreme learning machine model for fault classification and diagnosis. The results show that the identification accuracy of the fault diagnosis of reciprocating compressor plain bearings and automobile rolling bearings is significantly improved, and the method has better fault feature extraction effect.

Recently, magnetocardiography (MCG) has attracted increasing attention as a non-invasive and non-contact technique for detecting electrocardioelectric functions. However, the severe background noise makes it difficult to extract information. Variational Mode Decomposition (VMD), which is an entirely non-recursive model, is used to decompose the non-stationary signal into the intrinsic mode functions (IMFs). Traditional VMD algorithms cannot control the bandwidth of each IMF, whose quadratic penalty lacks adaptivity. As a result, baseline drift noise is still present or medical information is lost. In this paper, to overcome the unadaptable quadratic penalty problem, an improved VMD model via correlation coefficient and new update formulas are proposed to decompose MCG signals. To improve the denoising precision, this algorithm is combined with the interval threshold algorithm. First, the correlation coefficient is calculated, to determine quadratic penalty, in order to extract the first IMF made up of baseline drift. Then, the new update formulas derived from the variance that describes the noise level are used, to perform decomposition on the rest signal. Finally, the Interval thresholding algorithm is performed on each IMF. Theoretical analysis and experimental results show that this algorithm can effectively improve the output signal-to-noise ratio and has superior performance.

In this study, a new approach is proposed to detection of Parkinson’s disease by using Electroencephalography (EEG) signals and three different subband signals generated using Variational Mode Decomposition (VMD) method. In the proposed method, EEG signals and subband signals are applied separately to the generated 1D CNN model and the classification results are compared. The classification results showed that the VMD-sub band signals obtained from EEG signals were successful in diagnosing Parkinson’s. The highest classifier accuracy was obtained from second VMD subband data by 98.10%.

Amyotrophic lateral sclerosis (ALS), also known as motor neuron disease, is a neurological disease that occurs as a result of damage to the nerves in the brain and restriction of muscle movements. Electroencephalography (EEG) is the most common method used in brain imaging to study neurological disorders. Diagnosis of neurological disorders such as ALS, Parkinson's, attention deficit hyperactivity disorder is important in biomedical studies. In recent years, deep learning (DL) models have been started to be applied in the literature for the diagnosis of these diseases. In this study, event-related potentials (ERPs) were obtained from EEG signals obtained as a result of visual stimuli from ALS patients and healthy controls. As a new method, variational mode decomposition (VMD) is applied to the produced ERP signals and the signals are decomposed into subbands. In addition, empirical mode decomposition (EMD), one of the popular decomposition methods in the literature, was also analyzed, and ERP signals were divided into subbands and compared with the VMD method. Subband signals were classified in two stages with the one-dimensional convolutional neural network (1D CNN) model, which is one of the DL techniques proposed in the study. Accuracy, sensitivity, specificity, and F1-Score measurements were obtained using 5- and 10-fold cross-validation to evaluate classifier performance. In the first stage of classification, only VMD and EMD subband signals were used and 92.95% classification accuracy was obtained by the VMD method. In the second stage, VMD, EMD subband signals, and original ERP signals were all classified together with the VMD+ERP model achieving the maximum classification accuracy rate of 90.42%. It is thought that the results of the study will contribute to the diagnosis of similar neurological disorders such as ALS, attention studies based on visual stimuli, and the development of brain-computer interface (BCI) systems using the method applied to the proposed ERP signals.

This paper aims to present a non-destructive, optimized variational mode decomposition (VMD)-based ground-penetrating radar (GPR) method developed for identifying void defects in reinforced concrete (RC) structures. This study also presents an enhanced framework for defect detection in RC by integrating advanced spectrum analysis with deep learning techniques. A GPR investigation was conducted on an RC bridge deck with known structural defects to generate a representative dataset reflecting both intact and void-defective conditions. In addition to conventional spectral techniques such as fast Fourier transform (FFT), spectrogram, and scalogram, an optimized variational mode decomposition (VMD) method was implemented. The VMD approach decomposes GPR signals into intrinsic mode functions, enabling refined feature extraction beyond traditional spectral methods and allowing clear differentiation between intact and defective signals. The limited availability and quality of GPR small datasets have restricted the application of a functional 1D-CNN which generally requires at least several hundred datasets. To address this challenge, a data augmentation strategy is adopted. FFT-based features were successfully utilized to train a one-dimensional convolutional neural network (1D-CNN) for automated defect identification. The results demonstrate that both the advanced spectrum-based approach and the hybrid framework combining spectral analysis with deep learning significantly improve defect detection performance. Overall, the proposed methodology provides an effective and intelligent solution to support timely, data-driven decision-making for maintenance and safety assurance of bridge infrastructure.

As the gear is one of the most critical components of the high-speed rotating machinery, the fault identification of the gear has attracted increasing attention recently. However, the nonlinear and non-stationary vibration signals with mode mixing phenomenon and heavy background noise make it difficult to excavate the hidden weak fault feature. Variational Mode Decomposition (VMD), which can decompose the non-linear and non-stationary signals into a couple of Intrinsic Mode Functions (IMF) adaptively and non-recursively, bring a feasible tool. While the heavy background noise seriously affects the setting of the model number, which may lead to information loss or over decomposition problem. In this paper, a new improved VMD method, namely, Generalized Variational Mode Decomposition (GVMD) method is proposed to extract the fault characteristic information buried in the vibration signals adaptively. For the gear fault quantitative diagnosis, the corresponding relationship between the characteristic parameters and the working conditions is established under a specific condition, and it is no longer applicable to other conditions. Based on the dynamic model of the gear transmission system, the characteristic parameters that are sensitive and correlative to the gear tooth crack level, but not sensitive to the working conditions, can be extracted from the dynamic responses. Then the Normalized Feature Vector (NFV) can be formed with the extracted parameters. Based on the methods mentioned above, a novel intelligent method that combines the GVMD with NFV extraction to realize the quantitative diagnosis of the gear fault under variable working conditions is proposed. The NFV can be extracted from the experimental vibration signals preprocessed by the GVMD, and establish the relationship between the NFV and the gear tooth crack level. Then the quantitative diagnosis of the gear fault can be realized by the Support Vector Machine (SVM). The results show that the intelligent quantitative diagnosis method proposed in this paper is a potential quantitative diagnosis tool for the gear fault.

A distributed optical fiber acoustic sensing system (DAS) based on phase-sensitive optical time domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR) can detect the vibration information along an optical fiber and has been widely used for various engineering detections; however, the high-coherence detection light source used in a DAS will cause coherent fading in the fiber, leading to system fading noise and affecting the monitoring of the target vibration events. In order to solve the problem of the fading noise in the DAS, this study considered a method of suppressing the fading noise based on the genetic algorithm (GA) and variational mode decomposition (VMD). Furthermore, the parameter selection problem in the VMD method is improved by introducing the GA. By introducing sample entropy as a fitness function and combining it with the GA, this method adaptively searches for the optimal decomposition parameters by iterating the minimum value of the fitness function according to the type of input signal. Then, it decomposes the noisy DAS data into multiple intrinsic mode functions (IMFs), and the permutation entropy and time-frequency domain information of each component were used to extract the effective IMF components for signal reconstruction to obtain the denoised signal. Experimental results show that compared with empirical mode decomposition (EMD), ensemble EMD (EEMD), and complete EEMD (CEEMD) methods, the proposed method can eliminate the mode aliasing problem, effectively suppress the fading noise of the DAS, improve the signal-to-noise ratio, and reduce the root-mean-square error. It is, thus, possible to more accurately monitor vibration events, improving the practical application value of the DAS.

Machines learning models have recently been proposed for predicting rivers water temperature (Tw) using only air temperature (Ta). The proposed models relied on a nonlinear relationship between the Tw and Ta and they have proven to be robust modelling tools. The main motivation for this study was to evaluate how the variational mode decomposition (VMD) contributed to the improvement of machines learning performances for river Tw modelling. Measured data collected at five stations located in Poland from 1987 to 2014 were acquired and used for the analysis. Six machines learning models were used and compared namely, K-nearest neighbor's regression (KNNR), least square support vector machine (LSSVM), generalized regression neural network (GRNN), cascade correlation artificial neural networks (CCNN), relevance vector machine (RVM), and locally weighted polynomials regression (LWPR). The six models were developed according to three scenarios. First, the models were calibrated using only the Ta as input and obtained results show that the models were able to predict consistently water temperature, showing a high determination coefficient (R2) and Nash-Sutcliffe efficiency (NSE) with values near or above 0.910 and 0.915, respectively, and in overall the six models worked equally without clear superiority of one above another. Second, the air temperature was combined with the periodicity (i.e., day, month and year number) as input variable and a significant improvement was achieved. Both models show their ability to accurately predict river Tw with an overall accuracy of 0.956 for R2 and 0.955 for NSE values, but the LSSVM2 have some advantages such as a small errors metrics, and high fitting capabilities and it slightly surpasses the others models. Thirdly, air temperature was decomposed into several intrinsic mode functions (IMF) using the VMD method and the performances of the models were evaluated. The VMD parameters appeared to cause much influence on the prediction accuracy, exhibiting an improvement of about 40.50% and 39.12% in terms of RMSE and MAE between the first and the third scenarios, however, some models, i.e., GRNN and KNNR have not benefited from the VMD. This research has demonstrated the high capability of the VMD algorithm as a preprocessing approach in improving the accuracies of the machine learning models for river water temperature prediction.

The research about online monitoring and leakage automatic location of water distribution networks (WDN) has a wide range of applications that include water resource protection, monitoring, and allocation. Variational mode decomposition (VMD) and cross-correlation (CC) based leakage location is a popular and effective method in WDN. However, the value of K intrinsic mode functions (IMFs) based on VMD decomposition needs to be determined artificially, which affects the separation effect of signal frequency band characteristics directly. Hence, this work proposes an adaptive method to determine the parameter K of leakage vibration signal’s IMFs, which will be applied to automatic leakage location in WDN. Firstly, the number of saddle points in the frequency domain envelope of the sampled signal in different step sizes is calculated. The parameter K is determined according to the curvature change of the number of saddle points and the sampled signal. Finally, the selective IMFs are reconstituted into a new signal, which can determine a leak position using CC based time-delay estimation (TDE). To verify the effectiveness of the proposed algorithm, the different methods based on EMD and Fast ICA are compared. The experimental results demonstrate that the proposed parameter K value adaptive VMD (KVA-VMD) decomposition method is more suitable for leakage location in WDN.

Variational mode decomposition has been widely applied to machinery fault diagnosis during these years. However, it remains difficult to set proper hyperparameters for the variational mode decomposition, including number of decomposed modes, initial center frequencies, and balance parameter. Moreover, the low efficiency of the existing variational mode decomposition methods hinders their applications to practical diagnostic task. This article proposes an adaptive and efficient variational mode decomposition method after thoroughly investigating its convergence property characteristic. A convergent tendency phenomenon is discovered and is explained mathematically for the first time. Motivated by the convergent tendency phenomenon, the proposed method rapidly and adaptively determines the number and the optimal initial center frequencies of signal latent modes with the guidance of the convergent tendencies of the initial center frequencies changing from small to large. In the proposed method, the number of decomposed modes and initial center frequencies are not hyperparameters that require to be set in advance any more, but are parameters learned from the analyzed signals. The determined parameters enable efficient extraction of the main latent modes contained in the analyzed signals. Therefore, the proposed variational mode decomposition method represents a major improvement in parameter adaption and decomposition efficiency over the existing variational mode decomposition methods. In the application for bearing fault diagnosis, the faulty modes are selected adaptively and the corresponding balance parameters are further optimized efficiently. Two experimental cases validate the proposed method and its superiority over the existing variational mode decomposition methods and the classical fast spectral kurtosis in bearing fault diagnosis.

The performance of the traditional variational modal decomposition (VMD) method is greatly affected by the number of modalities decomposed by artificial settings when the signal is modally decomposed, and the traditional learning method has a slow training speed. It is easy to trap in the local minimum and is sensitive to the selection of the learning rate. In view of the above problems, this paper proposes a method for fault diagnosis of rotating bearings based on an improved VMD algorithm combined with extreme learning machine (ELM). First, the vibration signal is decomposed by using VMD according to the number of different modes, and the information entropy of each mode obtained after each decomposition. At the same time, the minimum value of each information entropy is selected, and Compared with the minimum values of the information entropy under different modes, the minimum information entropy is selected, the modal number corresponding to the minimum information entropy is selected as the best modal number, and the information of each intrinsic mode function (IMF) corresponding to the optimal modal number is finally selected. Information entropy is sent to ELM as a feature for modal recognition. Experimental results show that this method can classify more than 93% of faults in rotating bearings. Key Words: variational modal decomposition, center frequency method, information entropy, ELM

Inner Mongolia is one of the most sensitive regions to global change with vegetation significantly affected by global warming and human activities. Therefore, exploring the dynamic change of vegetation and its impact mechanism is of great significance for guiding the protection ecosystem and policy formulation. Based on the 1982–2015 growing season and the interannual normalized difference vegetation index (NDVI), simultaneous temperature and precipitation data sets, we extracted the characteristic components of the NDVI and climatic factors by the Variational Mode Decomposition (VMD) methods. And the periodic characteristics of the NDVI in Inner Mongolia and its relationship with climate over a period of 34 years were analyzed. The results show that: 1) The VMD algorithm overcomes the “mode aliasing” problem of Empirical Mode Decomposition (EMD) and can more accurately extract the periodic information of long time series vegetation NDVI dataset containing complex frequencies.; 2) the NDVI in the growing season of Inner Mongolia has main change cycles of one growing season (7 months) and half a growing season, and temperature and NDVI have the same double-cycle relationship, whereas precipitation has a periodic change of one growing season; 3) the NDVI cycle change of Inner Mongolia vegetation from 1982 to 2015 shows multiple time scales, showing 16.95a, 6.8a, 4.85a, 4.25a, 3.4a and 2.83a change cycles; 4) vegetation NDVI in Inner Mongolia from 1982 to 2015 has a good correlation with climate factors, and the partial correlation coefficients are different in Inner Mongolia and three sampling areas. NDVI and precipitation have relatively good periodic consistency, and there is a significant positive correlation between vegetation NDVI and temperature and precipitation before and after removing the interference of periodic variables. In general, temperature and precipitation jointly affect the NDVI cyclical changes of Inner Mongolia. The VMD algorithm has significant effects on determining the cyclic dynamic changes of NDVI and climatic factors.