Damage Identification of Curtain Wall Cables based on Improved VMD and Hilbert Transform

Signal analysis is a fundamental field in engineering and data science, focused on the study of signal representation, transformation, and manipulation. The accurate estimation of harmonic vibration components and their associated parameters in vibrating mechanical systems presents significant challenges in the presence of very similar frequencies and mode mixing. In this context, a hybrid strategy to estimate harmonic vibration modes in weakly damped, multi-degree-of-freedom vibrating mechanical systems by combining Empirical Mode Decomposition and Variational Mode Decomposition is described. In this way, this hybrid approach leverages the detection of mode mixing based on the analysis of intrinsic mode functions through Empirical Mode Decomposition to determine the number of components to be estimated and thus provide greater information for Variational Mode Decomposition. The computational time and dependency on a predefined number of modes are significantly reduced by providing crucial information about the approximate number of vibratory components, enabling a more precise estimation with Variational Mode Decomposition. This hybrid strategy is employed to compute unknown natural frequencies of vibrating systems using output measurement signals. The algorithm for this hybrid strategy is presented, along with a comparison to conventional techniques such as Empirical Mode Decomposition, Variational Mode Decomposition, and the Fast Fourier Transform. Through several case studies involving multi-degree-of-freedom vibrating systems, the superior and satisfactory performance of the hybrid method is demonstrated. Additionally, the advantages of the hybrid approach in terms of computational efficiency and accuracy in signal decomposition are highlighted.

The bearing fault diagnosis is of vital significance in maintaining the safety of rotation machine. Among various fault detection techniques, the diagnosis based on vibration signal is widely applied in monitoring the condition of rotation machine. Variational mode decomposition (VMD) is a novel signal analysis method, which can decompose a multi-component signal into a certain number of band-limited intrinsic mode functions (BLIMFs) nonrecursively. VMD could overcome some problems such as mode mixing, the inference of noise, the determination of wavelet base, which exist in empirical mode decomposition, ensemble empirical mode decomposition, wavelet transform, respectively. However, the empirical selection of the parameters for VMD would affect the result of the decomposition. This paper presents an adaptive VMD method with parameter optimization for detecting the localized faults of rolling bearing. Kurtosis, sensitive to transient impulsive components, is employed as optimization index to evaluate the performance of the VMD. Two parameters in the VMD, namely the number of decomposition modes and data-fidelity constraint, are optimized synchronously based on the kurtosis index through artificial fish swarm algorithm. Executing VMD with the acquired parameters, the optimal BLIMF is obtained. The spectrum analysis of the optimal BLIMF could identify the characteristic frequency caused by the localized crack effectually. The validity of the proposed method is proved by means of a cyclic transient impulse response signal and two experiments with practical vibration signals of rolling bearings. Compared to several existing methods, the proposed method demonstrates reinforced results.

The Impact Echo (IE) method is a widely used method for the detection of concrete structures. The signal has non-stationary characteristics. Empirical mode decomposition (EMD) is one of the typical ways of dealing with non-stationary signals. Due to EMD having mode overlap and a breakpoint effect, Variational Mode Decomposition (VMD) has been proposed to overcome these problems. In this study, VMD is introduced into the analysis of the concrete component detection signals, and the impact echo analysis method based on VMD and marginal spectrum is established. The proposed method to solve the marginal spectrum is used in order to analyze the numerical simulation model’s IE signal. This method is able to correctly obtain the main frequency. When compared with the traditional Fourier spectrum, the marginal spectrum that is based on the VMD method shows that the band width that is present in the main frequency is narrow. Even in the presence of a highly noisy background, the method is able to correctly obtain the main frequency. At last, through the detection signal analysis of the non-defective concrete specimen and the concrete specimen with the circular hole, the results show that the proposed method can distinguish the signal in the different defective states of the concrete structures, which has obvious advantages compared with the Fourier spectrum.

Random noise suppression is of great importance in seismic processing and interpretation, and time–frequency peak filtering (TFPF) is a classic denoising approach. In TFPF, pseudo Wigner–Ville distribution (PWVD) is used to linearise the given signal for an unbiased estimation of the instantaneous frequency. However, window length is a trade-off parameter for preserving valid signals and attenuating random noise. A long window length may cause loss of the desired signal, whereas a short window length may be inadequate to suppress noise. To ensure a good trade-off between signal preservation and noise reduction, empirical mode decomposition (EMD) has been introduced into the TFPF method. Although the EMD-TFPF method can achieve good results, the mode mixing problem in EMD is non-negligible. In this article, we introduce variational mode decomposition (VMD) to overcome the mode mixing problem. VMD decomposes a signal into an ensemble of modes that own their respective centre frequencies. Thus, the modes obtained by VMD contain less noise, which simplifies selection of the window width of TFPF. Therefore, we propose the VMD-based TFPF (VMD-TFPF) method to suppress random noise. Synthetic and field seismic data examples are employed to illustrate the superior performance of the proposed method in attenuating random noise and preserving the desired signal.

The vehicle scanning method (VSM) is widely used for bridge damage identification (BDI) because it relies solely on vehicle dynamic responses. The recently introduced contact point response, which is derived from vehicle dynamics but devoid of vehicle-related natural frequencies, shows great potential for application in the vehicle scanning method. However, its application in bridge damage detection remains understudied. The aim of this paper is to propose a new bridge damage identification method based on the contact point response. The method uses variational modal decomposition (VMD) to solve the problem of mode mixing and spurious frequencies in the signal. The continuous wavelet transform (CWT) is then utilized for damage identification. The introduction of variational modal decomposition makes the extracted signal more accurate, thus enabling more accurate damage identification. Numerical simulations validate the method’s robustness under varying conditions, including the vehicle speed, wavelet scale factors, the number of bridge spans, and pavement roughness. The results demonstrate that variational modal decomposition eliminates signal artifacts, producing smooth variational modal decomposition–continuous wavelet transform curves for accurate damage detection. In this study, we offer a robust and practical solution for bridge health monitoring using the vehicle scanning method.

Optical time domain reflectometry (OTDR) plays an important role in optical fiber communications. To improve the performance of OTDR, we propose a method based on the Variational Mode Decomposition (VMD) and Hilbert transform (HT) for fiber events detection. Firstly, the variational mode decomposition is applied to decompose OTDR data into some intrinsic mode functions (imfs). To determine the decomposition mode number in VMD, an adaptive estimation method is introduced. Secondly, the Hilbert transform is utilized to obtain the instantaneous amplitude of the imf for events localization. Finally, the Dynamic Time Warping (DTW) is used for identifying the type of event. Experimental results show that the proposed method can locate events accurately. Compared with the Short-Time Fourier Transform (STFT) method, the VMD-HT method presents a higher accuracy in events localization, which indicates that the method is effective and applicable.

Soldiers rely heavily on their vision, which is crucial not only for daily activities but also for the effective operation of defense systems, weaponry, and other military applications. However, various eye disorders, such as those related to increased intra-ocular pressure, can lead to irreversible vision loss, severely impacting a soldier’s operational capabilities. While extensive research has been conducted on detecting such ocular conditions, there remains a critical need for more accurate diagnostic methods to ensure early detection and treatment. In this study, we propose a novel approach combining Empirical Mode Decomposition (EMD) and Variational Mode Decomposition (VMD) for enhanced detection of eye disorders from retinal fundus images. The proposed method includes a comprehensive preprocessing phase, followed by decomposition using EMD and VMD techniques. The decomposed images undergo feature extraction through feature combination, with subsequent normalization and selection using z-score and the Relief method, respectively. Classification is performed using Support Vector Machines (SVM) with various kernels, including cubic, Gaussian, linear, and quadratic. The results demonstrate that the proposed method achieves high accuracy, with SVM kernel functions yielding accuracies of 98.30 %, 96.59 %, 96.59 %, and 97.87 % for 10-fold cross-validation, respectively. Additionally, the evaluation metrics, including sensitivity and specificity, indicate superior performance compared to state-of-the-art methods for similar datasets. This advanced diagnostic approach offers significant improvements in detecting eye disorders, which could be crucial in defense applications. Early and accurate diagnosis by military ophthalmologists can lead to better decision-making and timely interventions, ultimately preserving the vision and effectiveness of soldiers in the war.

We have proposed a new denoising method for the simultaneous noise reduction and preservation of seismic signals based on variational mode decomposition (VMD). VMD is a recently developed adaptive signal decomposition method and an advance in non-stationary signal analysis. It solves the mode-mixing and non-optimal reconstruction performance problems of empirical mode decomposition that have existed for a long time. By using VMD, a multi-component signal can be non-recursively decomposed into a series of quasi-orthogonal intrinsic mode functions (IMFs), each of which has a relatively local frequency range. Meanwhile, the signal will focus on a smaller number of obtained IMFs after decomposition, and thus the denoised result is able to be obtained by reconstructing these signal-dominant IMFs. Synthetic examples are given to demonstrate the effectiveness of the proposed approach and comparison is made with the complete ensemble empirical mode decomposition, which demonstrates that the VMD algorithm has lower computational cost and better random noise elimination performance. The application of on field seismic data further illustrates the superior performance of our method in both random noise attenuation and the recovery of seismic events.

This study presents a comparative framework for seismic signal classification based on adaptive decomposition and Hilbert spectral analysis. A self-recorded vibration dataset containing approximately 4,500 one-second segments of pedestrian, bicycle, and vehicle activities was acquired at a 1 kHz sampling rate using a buried geophone sensor. The raw signals were preprocessed through direct-current removal, fourth-order Butterworth band-pass filtering (1–50 Hz), and min–max normalization to mitigate environmental and instrumental noise. Three decomposition methods—Variational Mode Decomposition (VMD), Empirical Mode Decomposition (EMD), and Matching Pursuit Decomposition (MPD) were independently applied to obtain narrow-band oscillatory components. The Hilbert Transform (HT) was subsequently employed to derive marginal-spectral features, including spectral energy, spectral entropy, and dominant frequency, preserving a clear physical correspondence to source–soil coupling characteristics. These features were used to train multiple classifiers, including Support Vector Machine, Logistic Regression, k-Nearest Neighbors, Gradient Boosting, Random Forest, and Extreme Gradient Boosting (XGBoost), under group-aware cross-validation to avoid trial-level data leakage. Experimental evaluation demonstrates that VMD-HT features provide the highest discriminative performance, achieving a test accuracy of 91.4% and a macro-F1 score of 0.89 using the Random Forest classifier, followed by XGBoost with 90.3% accuracy. The EMD-HT framework attained 87.5% accuracy (macro-F1 = 0.84), while MPD-HT achieved 84.2% accuracy (macro-F1 = 0.81). Ensemble classifiers consistently outperformed linear models, confirming the advantage of nonlinear decision boundaries for decomposed spectral representations. The observed results indicate that VMD yields superior mode separation and spectral compactness compared with EMD and MPD, leading to improved feature stability and reduced overlap across classes. The proposed framework thus offers a robust and physically interpretable approach for ground-vibration classification, with potential applications in intrusion detection, structural health monitoring, and environmental sensing.

Application of tentative variational mode decomposition in fault feature detection of rolling element bearing

Variational mode decomposition (VMD), which is an alternative to empirical mode decomposition (EMD), has been widely used to extract the feature components of nonstationary signals. However, as a parameterized method, the performance of VMD is heavily influenced by its parameters. Meanwhile, it cannot efficiently extract weak feature components submerged in powerful ones. To address these problems, a novel method based on an optimized VMD is developed for precisely extracting the weak feature of viscoelastic sandwich cylindrical structures (VSCSs). In this method, a parameter optimization algorithm first is proposed to simultaneously select the crucial parameters in the VMD, and to reveal the characteristics of the influence of these parameters on the decomposition performance. Then, the weak feature components submerged in low-frequency strong ones are extracted twice by using the VMD with the optimized parameters. The effectiveness of the proposed method is verified by simulation signals and the experiment vibration signal collected from the VSCS. Its robustness to noise is also discussed. The results indicate that the parameter optimization algorithm can adaptively obtain optimal parameters, and compared with the optimized complementary ensemble EMD (CEEMD) and the original VMD, the proposed method can precisely extract the weak feature components submerged in strong ones.

A new denoising algorithm and feature extraction algorithm that combine a new kind of permutation entropy (NPE) and variational mode decomposition (VMD) are put forward in this paper. VMD is a new self-adaptive signal processing algorithm, which is more robust to sampling and noise, and also can overcome the problem of mode mixing in empirical mode decomposition (EMD) and ensemble EMD (EEMD). Permutation entropy (PE), as a nonlinear dynamics parameter, is a powerful tool that can describe the complexity of a time series. NPE, a new version of PE, is interpreted as distance to white noise, which shows a reverse trend to PE and has better stability than PE. In this paper, three kinds of ship-radiated noise (SN) signal are decomposed by VMD algorithm, and a series of intrinsic mode functions (IMF) are obtained. The NPEs of all the IMFs are calculated, the noise IMFs are screened out according to the value of NPE, and the process of denoising can be realized by reconstructing the rest of IMFs. Then the reconstructed SN signal is decomposed by VMD algorithm again, and one IMF containing the most dominant information is chosen to represent the original SN signal. Finally, NPE of the chosen IMF is calculated as a new complexity feature, which constitutes the input of the support vector machine (SVM) for pattern recognition of SN. Compared with the existing denoising algorithms and feature extraction algorithms, the effectiveness of proposed algorithms is validated using the numerical simulation signal and the different kinds of SN signal.

Variational mode decomposition is a new signal decomposition method, which can process non-linear and non-stationary signals. It can overcome the problems of mode mixing and compensate for the shortcomings in empirical mode decomposition. Permutation entropy is a method which can detect the randomness and kinetic mutation behavior of a time series. It can be considered for use in fault diagnosis. The complexity of wind power generation systems means that the randomness and kinetic mutation behavior of their vibration signals are displayed at different scales. Multi-scale permutation entropy analysis is therefore needed for such vibration signals. This research investigated a method based on variational mode decomposition and permutation entropy for the fault diagnosis of a wind turbine roller bearing. Variational mode decomposition was adopted to decompose the bearing vibration signal into its constituent components. The components containing key fault information were selected for the extraction of their permutation entropy. This entropy was used as a bearing fault characteristic value. The nearest neighbor algorithm was employed as a classifier to identify faults in a roller bearing. The experimental data showed that the proposed method can be applied to wind turbine roller bearing fault diagnosis.

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.

Diesel engines are widely used in railway systems, particularly in freight trains. Despite their high efficiency in energy conversion, they usually generate high levels of acoustics pollution during operation. In order to mitigate this problem, a series of active/passive acoustics control methods are used to reduce noise. Most of these methods are only effective if the prior knowledge of sources is given. In other words, it is essential to recognize the acoustics source. Variational mode decomposition (VMD) is a signal processing method that enhances the signal corrupted by background noise. However, the decomposed results of VMD depend on their mode parameter and penalty parameter. Therefore, an evaluation method based on system modal parameters (natural frequency and damping ratio) is proposed to select the mode parameter, and the penalty parameter can be selected from the power spectra of signals. In order to increase the accuracy of decomposition for diesel engines and find out the sources of acoustics, a method combining VMD, fast independent component analysis, and Hilbert transformation (VMD‐FastICA‐HT) is proposed for the separation and identification of different sources for diesel engines. The optimization results indicate that when the penalty parameter value is 1.5 to 16 times the maximum signal amplitude, better decomposition results can be achieved. Therefore, the separated independent acoustics are more accurate in source identification. Furthermore, both simulation data and in situ operational data of diesel engines for vehicles are used to validate the effectiveness of the proposed method.