Vibration

25 pages, 14530 KB

Open AccessArticle

Symplectic Geometry Matrix Machine Controlled by the Whale Optimization Algorithm and Its Application in Bearing Fault Diagnosis

by Yonghua Jiang, Zhiqiang He, Zhilin Dong, Jianjie Zhang, Hongkui Jiang, Chao Tang, Jianfeng Sun, Xiaohao Chen and Weidong Jiao

Vibration 2026, 9(2), 34; https://doi.org/10.3390/vibration9020034 - 13 May 2026

Abstract

In the field of industrial equipment condition monitoring, accurate rolling bearing fault diagnosis is critical yet challenging due to high-dimensional vibration signals and complex operating conditions. Traditional machine learning methods often struggle with insufficient feature separability and sensitivity to model parameters, leading to [...] Read more.

In the field of industrial equipment condition monitoring, accurate rolling bearing fault diagnosis is critical yet challenging due to high-dimensional vibration signals and complex operating conditions. Traditional machine learning methods often struggle with insufficient feature separability and sensitivity to model parameters, leading to fluctuating diagnostic accuracy. To address these challenges, this study introduces the whale optimization algorithm-guided symplectic geometry matrix machine (WOA-SGMM) and proposes the application of the whale optimization algorithm (WOA) to optimize the symplectic geometry matrix machine (SGMM), forming a WOA-SGMM diagnostic framework. (1) The symplectic geometry spectral transformation (SGST) effectively converts high-dimensional vibration signals into low-dimensional feature matrices while preserving intrinsic geometric and topological structures, enhancing noise robustness. (2) Leveraging WOA, we adaptively search for the optimal hyperparameters of the proposed SGMM, specifically addressing the limitations of traditional SMM, to mitigate the risk of overfitting. (3) Experimental validation on three benchmark datasets demonstrates that WOA-SGMM achieves superior multi-class fault diagnosis accuracy (up to 100%) under varying operating conditions. Compared to traditional methods, the proposed WOA-SGMM demonstrates improved classification accuracy and enhanced robustness against noise interference in the tested experimental scenarios, highlighting its potential for real-world industrial applications. Full article

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26 pages, 24612 KB

Open AccessArticle

Effects of Mixed Air on the Performance and Stiffness of a Viscous Fluid Damper

by Junwen Wei, Yurong Wang, Yi Wang and Qiangsheng Luo

Vibration 2026, 9(2), 33; https://doi.org/10.3390/vibration9020033 - 8 May 2026

Abstract

Viscous fluid dampers are widely used for mechanical vibration reduction to ensure the stability and safety of structures and systems. However, when a small amount of air (less than 10%) is mixed into the fluid, the compressibility of the fluid increases, leading to [...] Read more.

Viscous fluid dampers are widely used for mechanical vibration reduction to ensure the stability and safety of structures and systems. However, when a small amount of air (less than 10%) is mixed into the fluid, the compressibility of the fluid increases, leading to a decrease in the physical series stiffness of the damper. Consequently, under dynamic excitation, the proportion of elastic force in the total output force rises, resulting in an increase in the equivalent parallel additional stiffness—a concept often conflated with the series stiffness in the literature. This paper aims to demonstrate these two aspects of stiffness change by investigating the dynamic characteristics of air-mixed viscous fluid dampers through nonlinear modeling, finite element simulation, and experimental validation. Starting from a nonlinear series model comprising nonlinear damping and a nonlinear fluid spring (series stiffness), the energy dissipation and physical series stiffness under different air mixtures are simulated using a finite element model. To further explore the influence of air, an equivalent linear parallel model is established based on the equal energy principle, yielding an equivalent parallel additional stiffness. The results reveal that the energy dissipation effectiveness and the dynamic stiffness of viscous fluid dampers decrease as the air mixture increases. Nevertheless, the additional stiffness is increased with the air content. When the amount of air mixing is the same, the energy dissipation characteristics of the viscous fluid damper under different excitation frequencies vary. Both the damper efficiency and the additional stiffness are increased with the increase of the excitation frequency. The proposed equivalent linear model effectively captures the coupled effects of air mixture and excitation conditions on damper performance. Full article

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35 pages, 12550 KB

Open AccessArticle

Comparative Study on the Interaction Between Underwater Explosion Bubbles and Elastic Plates with Vertical and Horizontal Orientations

by Kexin Chen, Lin Lu, Changan Xu, Luyue Xi and Xianghong Huang

Vibration 2026, 9(2), 32; https://doi.org/10.3390/vibration9020032 - 8 May 2026

Abstract

Underwater explosion bubbles generate intense pressure pulses and high-speed re-entrant jets during their expansion and collapse processes, posing significant threats to ships and submerged structures. In practical engineering, plate-like structures with different orientations are widely encountered; therefore, investigating the influence of boundary orientation [...] Read more.

Underwater explosion bubbles generate intense pressure pulses and high-speed re-entrant jets during their expansion and collapse processes, posing significant threats to ships and submerged structures. In practical engineering, plate-like structures with different orientations are widely encountered; therefore, investigating the influence of boundary orientation on bubble dynamics is of great importance. In this study, underwater electrical explosion experiments were conducted using a capacitor discharge voltage of 300 V, with stand-off distances ranging from 1 mm to 30 mm. Two typical boundary configurations were established, namely a vertical plate and a horizontal plate. High-speed imaging was employed to capture the complete bubble evolution process, while coupled Eulerian–Lagrangian (CEL) simulations were performed to analyze bubble dynamics and structural response. The results indicate that, under the vertical plate condition, the maximum bubble diameter decreases monotonically with increasing stand-off distance, whereas the oscillation period exhibits a non-monotonic variation. At a stand-off distance of 5 mm, the maximum bubble diameter in the vertical plate configuration is 40.3% larger than that in the horizontal plate configuration. The reflected shock wave from the elastic boundary modifies the surrounding pressure field, thereby influencing the evolution of the bubble interface. In the presence of a vertical elastic plate, the bubble exhibits a centroid displacement during the expansion phase, and a re-entrant jet directed toward the boundary forms during collapse. In contrast, under the horizontal elastic plate condition, the bubble maintains a nearly axisymmetric evolution, and the re-entrant jet develops along the vertical direction. As the standoff distance between the plate and the charge center increases, the boundary effect gradually weakens, and the bubble morphology approaches that under free-field conditions. This study provides experimental evidence for understanding bubble–structure interaction (BSI) between underwater explosion bubbles and ship plate structures, and offers valuable insights for blast-resistant design of naval structures and the evaluation of underwater explosion loads. Full article

(This article belongs to the Special Issue Free Vibration and Dynamic Characteristics of Microheterogeneous Materials and Structures)

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26 pages, 16610 KB

Open AccessArticle

Experimental and Numerical Investigation of Stiffness-Optimized Vibration Isolators on Vibration Transmission in Cylindrical Shell Structures: A Comparative Land-Based and Underwater Study

by Quansheng Hu, Sheng Liu, Kun Zhang, Chaoying Wang, Qichao Xue, Guangping Zou, Yonghui Wang, Mingtao Chen and Deshui Xu

Vibration 2026, 9(2), 31; https://doi.org/10.3390/vibration9020031 - 29 Apr 2026

Abstract

Optimizing isolator stiffness is essential for controlling vibration transmission in cylindrical shell structures operating in cross-environment conditions. This study investigates the influence of isolator stiffness on vibration transmission and fluid-coupled response through coordinated land-based experiments, water-immersed experiments, and ABAQUS simulations. Two damped spring [...] Read more.

Optimizing isolator stiffness is essential for controlling vibration transmission in cylindrical shell structures operating in cross-environment conditions. This study investigates the influence of isolator stiffness on vibration transmission and fluid-coupled response through coordinated land-based experiments, water-immersed experiments, and ABAQUS simulations. Two damped spring isolators with stiffness values of 290 N/mm and 970 N/mm were tested under representative excitations of 25 Hz and 40 Hz. The results show that the lower-stiffness isolator provides consistently stronger vibration attenuation and produces higher vibration level differences than the higher-stiffness isolator. The measured vibration level differences between land-based and water-immersed conditions remain generally within 3 dB, indicating good cross-environment consistency. The numerical results agree well with the experimental trends, with deviations generally below 5 dB in the main low-frequency range. Mechanism analysis indicates that reducing isolator stiffness weakens the transmission of excitation energy from the raft frame to the base and shell, thereby reducing near-field fluid-coupled response around the excitation region. These findings support the use of lower-stiffness isolators and provide a practical framework for vibration assessment and parameter selection in cylindrical shell structures working under coupled air–water conditions. Full article

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31 pages, 2300 KB

Open AccessArticle

MDCAD-Net: A Multi-Dilated Convolution Attention Denoising Network for Bearing Fault Diagnosis

by Ran Duan, Ruopeng Yan and Guangyin Jin

Vibration 2026, 9(2), 30; https://doi.org/10.3390/vibration9020030 - 24 Apr 2026

Abstract

Bearing fault diagnosis is an important task for condition monitoring and predictive maintenance of rotating machinery. Nevertheless, many existing deep learning-based methods have difficulty in jointly modeling multi-scale fault characteristics, adaptively highlighting informative features, and maintaining robustness under noisy measurement conditions. To address [...] Read more.

Bearing fault diagnosis is an important task for condition monitoring and predictive maintenance of rotating machinery. Nevertheless, many existing deep learning-based methods have difficulty in jointly modeling multi-scale fault characteristics, adaptively highlighting informative features, and maintaining robustness under noisy measurement conditions. To address these issues, this study presents MDCAD-Net, a multi-dilated convolution attention denoising network that integrates multi-scale temporal feature extraction, attention-based feature refinement, and explicit noise suppression within an end-to-end learning framework. Parallel dilated convolutions with different dilation rates are employed to capture short-duration transient impulses as well as long-range periodic patterns in vibration signals. Channel-wise feature recalibration using squeeze-and-excitation networks and spatial-temporal attention via a convolutional block attention module are combined to enhance informative representations. In addition, a denoising block with gated attention and residual connections is introduced to reduce noise interference while retaining fault-related signal components. Experiments conducted on the Case Western Reserve University bearing dataset show that the proposed method achieves a classification accuracy of 98.93% and yields competitive performance compared with several commonly used deep learning models. Ablation studies and feature visualization results further illustrate the contributions of the individual components and the separability of the learned feature representations under noisy conditions. The results indicate the potential of the proposed framework for practical bearing fault diagnosis under noisy operating conditions. Full article

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29 pages, 3706 KB

Open AccessArticle

Reliability Analysis of Tuned Mass Damper-Equipped Structures Under Stochastic Excitation

by Lun Shao, Alexandre Saidi, Abdel-Malek Zine and Mohamed Ichchou

Vibration 2026, 9(2), 29; https://doi.org/10.3390/vibration9020029 - 20 Apr 2026

Abstract

Tuned mass dampers (TMDs) are commonly used to reduce excessive vibrations in engineering structures. Although their vibration control performance has been widely studied, the reliability of TMD-equipped structures under stochastic excitations has not been sufficiently investigated. In practical applications, random loads and system [...] Read more.

Tuned mass dampers (TMDs) are commonly used to reduce excessive vibrations in engineering structures. Although their vibration control performance has been widely studied, the reliability of TMD-equipped structures under stochastic excitations has not been sufficiently investigated. In practical applications, random loads and system uncertainties may significantly affect structural safety, and an efficient evaluation of failure probability remains a challenging task. Thus, the applications of these methods are greatly limited in vibration control. In this work, the structural reliability of systems equipped with TMDs is analyzed by adopting the first-passage time (FPT) as the failure criterion. Numerical investigations are performed on continuous beam models with TMDs under different types of stochastic excitation. In addition, an experimental study on a two-story steel frame structure is conducted to further examine the reliability performance of TMD-controlled systems. To reduce the computational cost associated with Monte Carlo simulation, a data-driven classification method is employed to approximate the failure domain based on a limited number of samples. The results indicate that the proposed approach enables accurate reliability estimation with a substantial reduction in computational cost, making it suitable for large-scale reliability analysis of vibration-controlled structures under stochastic excitation. The experimental results further demonstrate the applicability of the proposed reliability assessment method for practical vibration control problems. Full article

(This article belongs to the Special Issue New Trends in Experimental and Numerical Vibroacoustic Techniques—Physics Guided and Datas Guided Approaches)

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28 pages, 5786 KB

Open AccessArticle

Multi-Wavelet Fusion Transformer with Token-to-Spectrum Traceback for Physically Interpretable Bearing Fault Diagnosis

by Hongzhi Fan, Chao Zhang, Mingyu Sun, Kexi Xu, Wenyang Zhang and Ximing Zhang

Vibration 2026, 9(2), 28; https://doi.org/10.3390/vibration9020028 - 15 Apr 2026

Abstract

Rolling bearing fault diagnosis under complex and noisy operating conditions requires not only high diagnostic accuracy but also interpretability that can be quantitatively verified against physically meaningful excitation structures. However, many existing deep learning approaches rely on a single time–frequency (TF) representation and [...] Read more.

Rolling bearing fault diagnosis under complex and noisy operating conditions requires not only high diagnostic accuracy but also interpretability that can be quantitatively verified against physically meaningful excitation structures. However, many existing deep learning approaches rely on a single time–frequency (TF) representation and provide limited, non-verifiable links between model decisions and the original vibration patterns. To address this issue, we propose MBT-XAI, a multi-wavelet TF fusion network with a Token-to-Spectrum Traceback (TST) mechanism for structure-preserving, physics-consistent interpretability. Three complementary wavelets, namely Morlet, Mexican Hat, and Complex Morlet, are used to construct multi-view TF representations, which are encoded into RGB channels and adaptively fused via cross-channel attention within a Transformer backbone. TST maps patch-token attributions back to the TF domain, enabling quantitative evaluation of physics consistency through overlap-based metrics. Experiments on the public CWRU dataset and an industrial IMUST dataset show that MBT-XAI achieves 98.13 ± 0.24% and 96.23 ± 0.31% accuracy at SNR = 0 dB, outperforming the strongest baseline by 2.83% and 2.43%, respectively. Under AWGN contamination, MBT-XAI maintains 95.44 ± 0.38%/93.45 ± 0.47% accuracy on CWRU and 95.80 ± 0.33%/92.91 ± 0.51% accuracy on IMUST at SNR = −2/−4 dB. Under colored-noise contamination, the proposed method also preserves robust performance under pink and brown noise at the same SNR levels. Quantitative interpretability evaluation further indicates high alignment between salient frequency regions and theoretical fault-characteristic bands, with IoU = 80.21 ± 0.86% and Coverage = 91.70 ± 0.63%. In addition, MBT-XAI requires 10.393 M parameters and 10.678 GFLOPs, with an inference latency of 14.7 ms per sample (batch size = 1) on an NVIDIA GeForce RTX 3060 GPU. These results suggest that multi-wavelet TF modeling with attention-based fusion and TF-level traceback provides an accurate, robust, and physics-consistent framework for intelligent bearing fault diagnosis. Full article

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21 pages, 3293 KB

Open AccessArticle

Numerical and Experimental Study of Structural Parameter Identification for Jacket-Type Offshore Wind Turbines

by Xu Han, Chen Zhang, Zhaoyang Guo, Wenhua Wang, Qiang Liu and Xin Li

Vibration 2026, 9(2), 27; https://doi.org/10.3390/vibration9020027 - 14 Apr 2026

Abstract

Offshore wind energy has developed rapidly in recent years as a crucial component of renewable energy. However, offshore wind turbines (OWTs) face significant challenges in operations under complex marine environmental conditions, such as multimodal nonlinear vibrations, reliable structural monitoring, efficient maintenance, and sustainable [...] Read more.

Offshore wind energy has developed rapidly in recent years as a crucial component of renewable energy. However, offshore wind turbines (OWTs) face significant challenges in operations under complex marine environmental conditions, such as multimodal nonlinear vibrations, reliable structural monitoring, efficient maintenance, and sustainable long-term operations. The model-updating-based parameter identification takes advantages of structural vibration measurements, assisting in structural health monitoring. However, the traditional methods have not fully accounted for the parameter uncertainties and the need for real-time state updating, making them insufficient to meet the long-term online monitoring requirements for OWTs. This study introduces an innovative structural parameter identification framework that integrates modal parameter identification with Bayesian recursive updating. The proposed framework enables more efficient updates and uncertainty quantification of critical physical parameters for OWTs. It combines the covariance-driven stochastic subspace identification (COV-SSI) method for automatic modal parameter identification with the unscented Kalman filter (UKF) for parameter estimation. A 10 MW jacket-type offshore wind turbine was used as a case study. First, the numerical simulations were conducted to generate synthetic measurements for method validation and demonstration, enabling stepwise updating of the tower material’s elastic modulus across different sea conditions. A comparison of update speed and the convergence rate with the traditional time-step-based UKF method demonstrated the superiority of the proposed sea-condition-based approach in terms of computational efficiency and stability. Finally, the proposed framework was systematically validated using scaled model experimental data of a jacket-type OWT with a 4.2% identification error, confirming its engineering applicability. This research provides reliable technical support for the safety assessment of offshore wind turbine structures. Full article

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25 pages, 8275 KB

Open AccessArticle

Optimization of a Ship-Based Three-Magnet Energy Harvester Using Wave Excitation via the Flower Pollination and Simulated Annealing Algorithms

by Ho-Chih Cheng, Min-Chie Chiu and Ming-Guo Her

Vibration 2026, 9(2), 26; https://doi.org/10.3390/vibration9020026 - 10 Apr 2026

Abstract

In response to the urgent requirement for sustainable power supply for deep-sea or offshore underwater sensing equipment, this work investigates autonomous power generation aboard marine vessels. The vertical vibrations induced by wave excitation at the bottom of the vessel are utilized to drive [...] Read more.

In response to the urgent requirement for sustainable power supply for deep-sea or offshore underwater sensing equipment, this work investigates autonomous power generation aboard marine vessels. The vertical vibrations induced by wave excitation at the bottom of the vessel are utilized to drive the vibration energy harvesters on the deck for power generation. In a scenario involving automatic steering, a multiplicity of magnetoelectric harvesters mounted on the deck would move vertically in response to surface wave motion, enabling continuous conversion of wave energy into electrical power. The key feature of this study is that the ship-based self-power generation system is simple to install and safe, with the vibration energy harvesters mounted above the sea surface to avoid the unpredictable underwater sea conditions. This study presents a numerical case analysis of a three-magnet energy harvester designed to generate induced electrical power under wave conditions characterized by a speed of V = 3.0 m/s, amplitude of Z_o = 0.4 m, and wavelength of λ = 2.0 m. Prior to optimizing the ship-based energy harvester, the mathematical model of a three-magnet vibration system was validated against experimental data to ensure accuracy. Subsequently, a sensitivity study was performed to evaluate the influence of wave parameters (e.g., amplitude and wavelength) and the harvester’s geometric parameters on the electrical power output. To maximize power generation, the flower pollination algorithm—an efficient bio-inspired optimization method known for its robustness in global search—was integrated with the objective function defined as the root-mean-square electrical power. Simulation results indicate that the optimized harvester is capable of producing up to 0.1943 W. These findings highlight the potential of ship-based energy harvesters as a sustainable and reliable source of electrical power. Full article

(This article belongs to the Special Issue Vibration Control and Energy Harvesting Towards Autonomous Structural Systems)

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15 pages, 8492 KB

Open AccessArticle

Posture Prediction of Individuals Using Agricultural Machinery Under Whole-Body Vibration in a Lab Environment

by Brian Fiegel, Yash Kumar Dhabi, Salam Rahmatalla, Geb Thomas, Tyler Guzowski, Elizabeth Ritchie, David Wilder and Nathan B. Fethke

Vibration 2026, 9(2), 25; https://doi.org/10.3390/vibration9020025 - 9 Apr 2026

Abstract

Low back pain associated with exposure to whole-body vibration (WBV) is common among agricultural workers, and seated posture significantly affects health outcomes from WBV exposure. Current posture assessment methods rely on manual observation or body-worn sensors, which are labor-intensive and impractical for continuous [...] Read more.

Low back pain associated with exposure to whole-body vibration (WBV) is common among agricultural workers, and seated posture significantly affects health outcomes from WBV exposure. Current posture assessment methods rely on manual observation or body-worn sensors, which are labor-intensive and impractical for continuous monitoring. We developed a machine learning approach to classify seated posture using force sensors and accelerometers integrated into a vibration sensing seat pad for use in agricultural machinery, avoiding the need for body-worn sensors. Twenty-four participants were exposed to WBV in different upper body postures while seat pad force and acceleration data were recorded. We compared four machine learning architectures: Logistic Regression, Random Forest, Support Vector Machine, and Recurrent Neural Network with Gated Recurrent Unit (GRU). The GRU architecture substantially outperformed baseline models, achieving 89% accuracy (weighted F1 = 0.89) in classifying forward and backward leaning postures. To our knowledge, this study demonstrates the first application of machine learning to classify seated postures from seat pad force measurements during WBV exposure. Temporal modeling with an 18 s window proved essential for accurate classification, enabling non-invasive, continuous posture monitoring. Full article

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15 pages, 3134 KB

Open AccessArticle

Impact of Lateral Hollow Wear Depth on 400 km/h Wheel–Rail Contact and Noise Radiation

by Mandie Tu, Laixian Peng, Xinbiao Xiao, Jian Han and Peng Wang

Vibration 2026, 9(2), 24; https://doi.org/10.3390/vibration9020024 - 5 Apr 2026

Abstract

Lateral wear inevitably develops on the wheel treads of high-speed trains after a period of operation. Extensive research has been dedicated to circumferential wear (e.g., wheel polygonization), whereas studies on lateral tread wear and its impact on wheel-rail noise remain limited. This study [...] Read more.

Lateral wear inevitably develops on the wheel treads of high-speed trains after a period of operation. Extensive research has been dedicated to circumferential wear (e.g., wheel polygonization), whereas studies on lateral tread wear and its impact on wheel-rail noise remain limited. This study investigates this issue through a combined approach of field measurements and numerical simulation. First, lateral wear profiles are measured on in-service high-speed train wheels, and their patterns are systematically analyzed. Subsequently, a three-dimensional transient wheel-rail rolling contact model is developed using the explicit finite element method. This model is employed to analyze the effects of the lateral hollow wear depth on the contact patch position and wheel-rail forces at 400 km/h. Finally, these calculated forces are imported into a coupled wheel-rail vibration and acoustic radiation model to predict noise characteristics at different wear depths. This study clarifies the coupling of lateral tread hollow wear with wheel-rail contact characteristics at 400 km/h and quantifies its mechanical influence on high-frequency wheel-rail noise via contact patch evolution and structural receptance variation. The results demonstrate that lateral wear manifests as hollow wear, with a maximum depth of approximately 1 mm within a reprofiling cycle. It has been found that as the hollow wear depth increases, the contact patch center shifts toward the wheel flange, and its major axis elongates. Consequently, wheel-rail noise increases significantly with greater wear depth. Specifically, a wear depth increase of 0.78 mm leads to increments of 2.3 dB in wheel noise, 0.9 dB in rail noise, and 1.0 dB in total wheel-rail noise. These findings underscore that tread hollow wear is a significant contributor to high-speed wheel-rail noise, highlighting the need for its consideration in maintenance and noise control strategies. Full article

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19 pages, 5614 KB

Open AccessArticle

CNN-BiLSTM-CA Model with Visualized Bayesian Optimization for Structural Vibration Prediction During Flood Discharge

by Guojiang Yin and Shuo Wang

Vibration 2026, 9(2), 23; https://doi.org/10.3390/vibration9020023 - 30 Mar 2026

Abstract

Accurate prediction of vibration responses in hydraulic structures during flood discharge is essential for ensuring safe and stable operation. This study develops a hybrid deep learning model that combines Convolutional Neural Networks (CNN), Bidirectional Long Short-Term Memory (BiLSTM), and a Channel Attention (CA) [...] Read more.

Accurate prediction of vibration responses in hydraulic structures during flood discharge is essential for ensuring safe and stable operation. This study develops a hybrid deep learning model that combines Convolutional Neural Networks (CNN), Bidirectional Long Short-Term Memory (BiLSTM), and a Channel Attention (CA) mechanism, optimized through Bayesian Optimization (BO), to predict dam gantry crane beam displacements. Time-lagged Pearson correlation and Maximum Information Coefficient (MIC) are applied to select the informative input features. The CNN-BiLSTM-CA model captures both spatial patterns and temporal dependencies in vibration signals. BO tunes model hyperparameters, while Partial Dependence (PD) analysis provides insight into how these parameters affect prediction accuracy. The model is validated using vibration data from an arch dam in Southwest China during flood discharge. Results show that CNN parameters have a greater impact on prediction accuracy than BiLSTM parameters, underscoring the importance of spatial feature extraction. Ablation studies confirm each component’s contribution. Compared with existing methods, the proposed model achieves superior accuracy with a Root Mean Square Error (RMSE) of 5.49, Mean Absolute Error (MAE) of 4.34, and correlation coefficient (R) of 99.42%. This framework provides a reliable and interpretable tool for predicting structural vibrations in hydraulic engineering under complex discharge conditions. Full article

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16 pages, 1313 KB

Open AccessArticle

New Accurate Local-Buckling Analysis of Equal-Leg Angle Steels in Transmission Towers

by Dongrui Song, Xiaocheng Tang, Zhiwei Sun, Dong Han, Xiaozhuo Guan and Huashun Li

Vibration 2026, 9(1), 22; https://doi.org/10.3390/vibration9010022 - 22 Mar 2026

Abstract

This study presents a specific analytical solution procedure to the local-buckling problem in angle steels using a two-dimensional improved Fourier-series method (2D-IFSM). The effect of coupling between the sub-plates of an angle steel on its local-buckling behaviour is studied by incorporating rotational spring [...] Read more.

This study presents a specific analytical solution procedure to the local-buckling problem in angle steels using a two-dimensional improved Fourier-series method (2D-IFSM). The effect of coupling between the sub-plates of an angle steel on its local-buckling behaviour is studied by incorporating rotational spring constraints between them. The proposed solution procedure enables one to convert the local-buckling problem of angle steels into solving sets of linear algebraic equations, thereby effectively simplifying its solution process. The critical load and related buckling-mode results obtained in this study are in good agreement with the existing analytical solutions and finite-element-method numerical data, verifying the effectiveness of the proposed method. Based on the derived solutions, a quantitative analysis is conducted to investigate the influences of aspect ratio, width–thickness ratio, and rotational constraint degree on the local-buckling behaviour of angle steels. Full article

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20 pages, 24152 KB

Open AccessArticle

Excitation and Transmission of Train-Induced Ground and Building Vibrations—Measurements, Analysis, and Prediction

by Lutz Auersch, Samir Said and Werner Rücker

Vibration 2026, 9(1), 21; https://doi.org/10.3390/vibration9010021 - 18 Mar 2026

Abstract

Measurement results of train-induced vibrations are evaluated for characteristic frequencies, amplitudes and spectra, leading to a prediction which is based on transfer functions of the vehicle–track–soil system, the soil, and the building–soil system. The characteristic frequencies of train-induced vibrations are discussed following the [...] Read more.

Measurement results of train-induced vibrations are evaluated for characteristic frequencies, amplitudes and spectra, leading to a prediction which is based on transfer functions of the vehicle–track–soil system, the soil, and the building–soil system. The characteristic frequencies of train-induced vibrations are discussed following the propagation of vibrations from the source to the receiver: out-of-roundness frequencies of the wheels, the sleeper passage frequency, the vehicle–track eigenfrequency, the car-length frequency and multiples, axle-distance frequencies, bridge eigenfrequencies, the building–soil eigenfrequency, and floor eigenfrequencies. Amplitudes and spectra are compared for different train and track types, for different train speeds, and for different soft and stiff soils, where high frequencies are typically found for stiff soil and low frequencies for soft soil. The ground vibration is between the cut-on frequency due to the layering and the cut-off frequency due to the material damping of the soil, but the dominant frequency range also changes with distance from the track. The frequency band of the axle impulses due to the passing static loads obtains a signature from the axle sequence. The high amplitudes between the zeros of the axle-sequence spectrum are measured at the track, the bridge, and also in the ground vibrations, which are even dominant in the far field. A prediction software is presented, which includes all three parts: the excitation by the vehicle–track interaction, the wave transmission through the soil, and the transfer into a building. Full article

(This article belongs to the Special Issue Railway Dynamics and Ground-Borne Vibrations)

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15 pages, 3229 KB

Open AccessArticle

Nonlinear Characterisation of Wind Turbine Gearbox Vibration Dynamics Driven by Inhomogeneous Helical Gear Wear

by Khaldoon F. Brethee, Ghalib R. Ibrahim and Al-Hussein Albarbar

Vibration 2026, 9(1), 20; https://doi.org/10.3390/vibration9010020 - 16 Mar 2026

Abstract

Helical gear transmissions in wind turbine gearboxes operate under high torque, variable speed, and complex rolling–sliding contact conditions, where friction-induced wear evolves in a spatially non-uniform manner. However, most existing dynamic models assume uniform or mild wear and therefore fail to capture the [...] Read more.

Helical gear transmissions in wind turbine gearboxes operate under high torque, variable speed, and complex rolling–sliding contact conditions, where friction-induced wear evolves in a spatially non-uniform manner. However, most existing dynamic models assume uniform or mild wear and therefore fail to capture the nonlinear coupling between localised tooth surface degradation, gear mesh dynamics, and vibration response. In this work, a nonlinear dynamic model of a helical gear pair is formulated by incorporating time-varying mesh stiffness, elasto-hydrodynamic lubrication (EHL)-based friction forces, and wear-dependent contact geometry. The governing equations of motion are derived to explicitly account for the influence of inhomogeneous tooth wear on the contact load distribution and frictional excitation during meshing. Wear evolution is represented as a spatially varying modification of tooth surface topology, enabling the progressive coupling between wear depth, mesh stiffness perturbations, and dynamic transmission error. The model is employed to analyse the effects of non-uniform wear on system stability, vibration spectra, and dynamic response under wind turbine operating conditions. Numerical results reveal that uneven wear introduces nonlinear modulation of gear mesh forces and generates characteristic sidebands and amplitude variations in the vibration signal that are absent in conventional mild-wear formulations. These wear-induced dynamic features provide mathematically traceable indicators for the onset and progression of uneven tooth degradation. The proposed framework establishes a physics-based link between wear evolution and measurable vibration responses, providing a rigorous foundation for advanced vibration-based diagnostics and model-driven condition monitoring of wind turbine gearboxes. Full article

(This article belongs to the Special Issue Free Vibration and Dynamic Characteristics of Microheterogeneous Materials and Structures)

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24 pages, 11796 KB

Open AccessArticle

Visual Servoing Sliding Mode Control with Vibration Model Compensation for Trajectory Tracking in a 2-DOF Ball Balancer System

by Mohammed Abdeldjalil Djehaf, Ahmed Hamet Sidi and Youcef Islam Djilani Kobibi

Vibration 2026, 9(1), 19; https://doi.org/10.3390/vibration9010019 - 11 Mar 2026

Abstract

Ball balancers are nonlinear, electromechanical, multivariable, open-loop unstable systems widely used in research laboratories, aerospace, military, and automotive industries to evaluate control mechanism effectiveness. The inherent difficulty in precisely managing ball position, combined with actuator saturation and system sensitivity to disturbances, makes trajectory [...] Read more.

Ball balancers are nonlinear, electromechanical, multivariable, open-loop unstable systems widely used in research laboratories, aerospace, military, and automotive industries to evaluate control mechanism effectiveness. The inherent difficulty in precisely managing ball position, combined with actuator saturation and system sensitivity to disturbances, makes trajectory tracking a persistent challenge. Conventional controllers often exhibit oscillatory responses with steady-state errors exceeding acceptable limits. Sliding mode control (SMC) offers robustness against model uncertainties; however, chattering finite-frequency, finite-amplitude oscillations near the sliding surface caused by switching imperfections, time delays, and actuator dynamics remain a significant limitation. This study addresses chattering through explicit vibration model compensation integrated into the SMC design for a 2-DOF ball balancer system using a visual servoing approach. A double-loop control architecture is implemented, where the inner loop handles servo angular position control and the outer loop manages ball position tracking through visual servoing feedback. The sliding mode controller is designed with a power rate reaching law, synthesizing two control laws: one with explicit vibration model compensation incorporating damping and stiffness terms, and one without. Experimental validation confirmed that SMC with compensation achieved significantly reduced steady-state error (0.034 mm vs. 0.386 mm) and lower overshoot (3.95% vs. 13.81%) compared to the uncompensated variant, with chattering amplitude reduced by approximately 72%. Full article

(This article belongs to the Special Issue Vibration Damping)

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10 pages, 1487 KB

Open AccessArticle

The Influence of Boundary Conditions on Trapped Modes in Semi-Infinite Elastic Waveguides

by Marcus Dykes, Julius Kaplunov and Danila Prikazchikov

Vibration 2026, 9(1), 18; https://doi.org/10.3390/vibration9010018 - 10 Mar 2026

Cited by 1

Abstract

This work investigates trapped modes induced by localized inhomogeneities in semi-infinite elastic waveguides in the form of a point mass or a meta-spring attached to the edge. Explicit relations linking the parameters of the meta-spring and the mass are presented with a string [...] Read more.

This work investigates trapped modes induced by localized inhomogeneities in semi-infinite elastic waveguides in the form of a point mass or a meta-spring attached to the edge. Explicit relations linking the parameters of the meta-spring and the mass are presented with a string or beam resting on a Winkler foundation. Asymptotic expansions are derived to describe the limiting behavior of the obtained solutions, including small- and large-mass regimes. Special emphasis is placed on the less-studied trapped modes in an elastically supported beam, providing new insights into the peculiarities of wave localization phenomena, e.g., the analysis of the associated frequency equation. Full article

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17 pages, 842 KB

Open AccessArticle

In-Plane Vibration Analysis of Annular Plates Considering All Combinations of Edge Conditions

by Yoshihiro Narita

Vibration 2026, 9(1), 17; https://doi.org/10.3390/vibration9010017 - 9 Mar 2026

Abstract

The Ritz method is applied to an in-plane vibration analysis to obtain accurate frequencies of isotropic annular plates. The method is formulated in a manner that allows all combinations of free boundary conditions, two types of supported (constraining only either radial or circumferential [...] Read more.

The Ritz method is applied to an in-plane vibration analysis to obtain accurate frequencies of isotropic annular plates. The method is formulated in a manner that allows all combinations of free boundary conditions, two types of supported (constraining only either radial or circumferential displacement) boundary conditions, and clamped boundary conditions. Admissible functions for the two displacement components are chosen as products of trigonometric functions in the circumferential coordinate and special algebraic polynomials in the radial coordinate, enabling all possible boundary-condition combinations to be satisfied. In the numerical study, after the solution’s accuracy is verified through convergence and comparison tests, extensive and accurate frequency parameters are presented to cover all combinations of the four in-plane boundary conditions along the outer and inner edges of the annular plates. Full article

(This article belongs to the Special Issue Structural Vibration: Modeling, Analysis, Optimization and Engineering Applications)

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34 pages, 3198 KB

Open AccessArticle

The Energy-Dispersion Index (EDI) and Cross-Domain Archetypes: Towards Fully Automated VMD Decomposition for Robust Fault Detection

by Ikram Bagri, Achraf Touil, Rachid Oucheikh, Ahmed Mousrij, Aziz Hraiba and Karim Tahiry

Vibration 2026, 9(1), 16; https://doi.org/10.3390/vibration9010016 - 2 Mar 2026

Abstract

Variational Mode Decomposition (VMD) is a powerful formalism for the time-scale analysis of vibration signals from rotating machinery. However, its performance is often compromised by complex parameter configuration, where subjective manual tuning leads to mode mixing or information loss. In this study, we [...] Read more.

Variational Mode Decomposition (VMD) is a powerful formalism for the time-scale analysis of vibration signals from rotating machinery. However, its performance is often compromised by complex parameter configuration, where subjective manual tuning leads to mode mixing or information loss. In this study, we present a physics-guided framework that generalizes VMD optimization across diverse operating conditions. We utilized a meta-dataset combining three distinct sources (CWRU, HUST, UO) to validate the approach. Through a shaft-normalized segmentation strategy and K-Means++ clustering, we identified six distinct signal archetypes based on spectral morphology. Central to this framework is the Energy-Dispersion Index (EDI), a novel physically interpretable metric designed to differentiate between structured fault transients and stochastic noise. Extensive validation via a full-factorial Design of Experiments (8640 trials) confirmed the statistical superiority of EDI over benchmarks like kurtosis and envelope entropy, yielding an 8.3% improvement in modal fidelity. Furthermore, a rigorous ablation study demonstrated that the proposed archetype-based parameterization is highly efficient. This strategy achieved a

392 \times

speedup over online optimization while maintaining statistically equivalent diagnostic accuracy. Additionally, by generalizing parameters from high-quality archetype representatives, the framework reduced spectral leakage (Orthogonality Index) by 51.4% compared to instance-wise optimization. The resulting framework provides a mathematically rigorous, real-time solution for automated vibration signal decomposition. Full article

(This article belongs to the Special Issue New Trends in Experimental and Numerical Vibroacoustic Techniques—Physics Guided and Datas Guided Approaches)

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