Vibration doi: 10.3390/vibration9030056

Authors: Gianmarco Battista Marcello Vanali

Condition monitoring of automatic elevator doors is essential for ensuring operational reliability, passenger safety, and optimized maintenance scheduling. Although scientific literature extensively covers fault diagnostics for some types of machinery (e.g., rotating machinery), complex electromechanical door systems also deserve close attention. This paper proposes an intuitive and easily deployable monitoring methodology designed for broad industrial implementation. Feature extraction is performed using Autoregressive (AR) models on time-series signals. A condition monitoring index is formulated by combining Mahalanobis Distance with Box–Cox transformation, thereby normalizing the baseline data and enabling straightforward and reliable threshold setting. The methodology can be applied to both individual components and the whole system, within the same theoretical framework. Fault classification is effectively achieved using Linear Discriminant Analysis (LDA). A diagnostic interpretation tool, termed Pairwise Inter-Class Mahalanobis Distance Analysis (PICMDA), is introduced to facilitate the visual assessment of class separation. Validation on a full-scale elevator door system demonstrates high sensitivity to incipient faults and excellent diagnostic capabilities. The framework provides an effective tool for condition monitoring and fault diagnosis, with a flexible architecture that is not strictly restricted to elevator doors but can be readily extended to various other mechanical systems.

Vibration doi: 10.3390/vibration9030055

Authors: Elke Ochsmann Alexandra Corominas Cishek Uwe Kaulbars

Background: Single-shock exposure to the hand–arm system is a distinct form of occupational mechanical exposure that is rarely assessed separately from continuous vibration. Due to methodological challenges and the lack of standardized definitions, epidemiological evidence on its prevalence and health relevance remains limited. Objectives: This study aimed to explore the occurrence of occupational single-shock exposure in a general working population and to examine its association with self-reported hand–arm health problems. A secondary objective was to assess the usefulness of an exploratory ISCO-08-based job–exposure matrix (JEM). Methods: A cross-sectional survey was conducted among workers from diverse occupational backgrounds. Participants reported the use of predefined hand-held tools emitting single shocks, as well as additional tools identified via free text. Job titles were coded according to ISCO-08 and combined with expert-based exposure assessments to develop an exploratory JEM. Associations between exposure and health outcomes were analyzed descriptively and analytically in three exposure regression models. Results: Overall, 35% of participants (N = 162) reported occupational exposure to single shocks across a wide range of occupations, including technical and mixed-task jobs, indicating broader exposure than expected from ISCO-based classifications. Although exposure was generally of low to moderate intensity, it was associated with reported hand–arm health problems, particularly in distal regions (fingers, hand, and wrist (FHW); OR 4.32, 95% CI 1.38–13.44), where high forces seemed to be the main risk aspect (OR 6.26, 95% CI 1.67–23.50). Conclusions: Single-shock exposure appears to be common and may be associated with adverse hand–arm outcomes even at relatively low levels. The exploratory JEM was useful for identifying underestimated occupations and heterogeneity but cannot replace detailed task- or measurement-based assessment. Improved exposure definitions, assessment methods, and longitudinal studies in larger cohorts are needed to clarify health risks.

Vibration doi: 10.3390/vibration9030054

Authors: Friedrich Beyer Tom Wühle M. Ercan Altinsoy

Vibrations occurring when playing a musical instrument can affect how the quality of the instrument is perceived by the player. Previous studies on violins have investigated how vibrotactile feedback influences certain perceptual attributes and the perceived quality or preference of the instruments. These studies have found that, in addition to auditory feedback, vibrotactile feedback had an important influence on the corresponding ratings. In these studies, however, to control the vibrotactile feedback independently from the auditory feedback was only possible to a limited extent, or not at all. To overcome this limitation, the present study introduces a concept for the creation of a violin-like instrument on which both vibrotactile and auditory feedback can be controlled individually. To apply this concept, the feedback that the instrument exposes by itself to the player is required to be as minimal as possible. This keeps interference with the intended feedback to a minimum. It was decided to use an electric violin as the basis, as the usual absence of a hollow body being the most important airborne excitation source meets this requirement in the auditory domain already quite well. In the vibrotactile domain, however, meeting this requirement is more challenging, as most of the structure-borne excitation sources persist. Therefore, the present study investigated the vibrotactile feedback of electric violins. Two setups for measuring the vibrotactile feedback exposed to the player at the neck of a violin are presented. One setup was intended to ensure high reproducibility while the other one was intended to resemble real playing scenarios more closely. Using these two setups, measurements were carried out for a set of six electric violins. The results were discussed regarding physical and perceptual aspects. Finally, one electric violin was selected for the creation of a violin-like instrument with independently controllable auditory and vibrotactile feedback.

Vibration doi: 10.3390/vibration9030053

Authors: Zengjun Lu Xinlong Zhu Rongjiang Tang Zhengxiong Chen Kefang Cai

A unified modeling framework is presented in this work to predict the free vibration and loss factor characteristics of damping sandwich rotational plates. The formulation starts from the first-order shear deformation theory, where the zigzag hypothesis and interlayer displacement continuity are combined to couple the displacement fields of the individual plies. An artificial spring scheme is adopted to enforce the layer–layer compatibility and the external boundary restraints, which leads to a Lagrangian functional composed of the kinetic energy, the strain energy, and the potential energies contributed by the boundary and coupling springs. The displacement unknowns are discretized with Chebyshev polynomials of the first kind, and the natural frequencies and damping loss factors are extracted by solving the resulting eigenvalue problem with the Rayleigh–Ritz method. Convergence tests are conducted, and the reliability of the model is validated against finite element results. Finally, a series of numerical examples is presented to systematically investigate the effects of key model parameters on the vibration characteristics of the structure. The results indicate that increasing the thicknesses of the inner and outer layers of the damping sandwich rotational plate structure can significantly raise the natural frequencies. Increasing the inner diameter helps to reduce the area of the low-frequency region, where the difference between the two sides exceeds 40 Hz, caused by the close thicknesses of the inner and outer layers. When only the outer boundary is clamped, the natural frequencies of the annular plate are more than twice those of the solid rotational plate, although the solid rotational plate yields a larger loss factor. When only the outer circular edge is fixed, increasing the total thickness of the structure can effectively raise the natural frequencies, with a maximum increase exceeding 110 Hz, while the loss factor decreases significantly.

Vibration doi: 10.3390/vibration9030052

Authors: Fengguo Li Peng Yao Yao Hou Xinyu Mao Zhonglei Zhang Wei Wu Jiarong Bai Jubin Zhang Tonghui Hu Hongyi Sun Jiaofeng Ma Yang Yu Wenxiu Yu

The stable and reliable operation of precision motion stages constitutes a core prerequisite for accurate machining and operation in high-end equipment. Aiming at solving the problems of insufficient dynamic modeling accuracy, difficulty in achieving the precise prediction of guide rail parameters, and inadequate system performance optimization, this paper takes a four-axis precision motion stage as the research object and conducts a systematic study on dynamic modeling, guide rail parameter prediction, and system performance optimization. Verification results show that the maximum errors of natural frequencies in the x, y, z, and β directions are 1.3%, 4%, 5%, and 1%, respectively. The predicted results of the model are in good agreement with the experimental data, based on the dynamic model of the precision motion stage and the prediction model of guide rail dynamic parameters. On this basis, the guide rail-related parameters were optimized. After optimization, compared with the original state, the performance in the z and β directions was significantly improved: the natural frequency in the z direction was increased from 57 Hz to 70 Hz, and the natural frequency in the β direction was increased from 110 Hz to 154 Hz. This improvement enhances the overall anti-vibration capability at low frequencies and effectively ensures the stability and reliability of the precision motion stage during operation. The method used in this study has generality and transferability, which can provide a theoretical basis and technical support for the dynamic design and parameter optimization of various precision motion systems.

Vibration doi: 10.3390/vibration9030051

Authors: Aires Colaço Pedro Alves Costa Mohammed F. M. Hussein Ahmed M. Abouelmaty

The continued development and modernisation of railway networks, together with increasingly demanding expectations regarding human comfort, require careful assessment and mitigation of the effects of railway infrastructure on nearby buildings, their occupants, and vibration-sensitive equipment. Using a recently completed railway line in Portugal as a case study, this paper provides insights into predicting railway-induced ground-borne noise and vibration (GBN&V) during the environmental impact assessment stage. The initial predictions are compared with experimental measurements obtained during the exploration of the railway infrastructure, revealing a good match between predicted and observed ground-borne vibration levels. However, the direct application of the empirical FTA vibration-to-noise conversion factors underestimates the measured ground-borne noise levels. This discrepancy highlights the importance of the building-specific vibro-acoustic response and indicates that simplified conversion factors should be applied with caution.

Vibration doi: 10.3390/vibration9030050

Authors: Oleksandr Pashchenko Yevhenii Koroviaka Volodymyr Khomenko Oleksandr Kamyshatskyi Valerii Rastsvietaiev Serhii Shypunov

Downhole vibrations are a primary cause of premature wear and failure of rock-cutting elements (RCEs) during well drilling. This study develops an integrated finite element (FEM) model that couples axial and torsional vibrations with the evolution of the wear flat, friction, and temperature. The model is validated against laboratory experiments on a drilling stand using MEMS accelerometers. Two types of tungsten-cobalt (WC-Co) inserts were compared: uncoated and coated with a 3–5 nm titanium nitride (TiN) layer. Thirty tests were performed on granite and sandstone under varying single-RCE weight on bit (WOB = 1.0–2.2 kN) and rotation speed (RPM = 80–120). The TiN coating, by providing a low-friction running-in surface, reduced axial RMS acceleration by 18%, torsional amplitude by 24%, and the steady-state wear rate by 27% (from 0.154 to 0.112 mm/h) in granite. Frequency spectra revealed a resonant torsional peak at 55 Hz when RPM exceeded 120, with torque fluctuations increasing by 240%. A safe operating chart was constructed, defining green (WOB 1.0–1.6 kN, RPM 80–110), yellow, and red zones. The recommended regime (WOB = 1.7 kN, RPM = 105) gives 94% of maximum rate of penetration while reducing predicted wear by 35% compared to the red zone. The model prediction errors are 8–12% for axial and 10–15% for torsional vibrations. This work demonstrates that the proposed laboratory framework, combining nanoscale TiN-coated inserts with low-cost MEMS sensors, enables improved characterization of drilling vibrations and wear and supports the development of practical operating charts for drilling optimization.

Vibration doi: 10.3390/vibration9030049

Authors: Aleksandar Pavic

When Vibration was launched in 2018, its Aims and Scope reflected the journal we wanted to build from scratch [...]

Vibration doi: 10.3390/vibration9030048

Authors: Hilal Koç Mertol Tüfekci Ekrem Tüfekci

This study investigates the forced transverse vibration of a thin rotating annular disc subjected to time-varying point forces that are fixed in space and act perpendicular to the disc surface. Earlier analytical treatments of this problem have almost always been restricted to a single support condition, most often the clamped–free disc of a hard-disk drive. The boundary conditions of the disc have not been treated as a design variable of the forced response. The contribution of this work is to remove that restriction: the same generalised Galerkin formulation is applied to clamped–clamped, clamped–free and free–clamped rotating annular discs, and the sensitivity of the forced response to the excitation parameters is compared across all three. The governing differential equation, which includes gyroscopic coupling and the membrane stresses induced by rotation, is nondimensionalised and solved by the Galerkin method with polynomial radial trial functions. The modal equations are then integrated in state-space form with light modal damping. The physical transverse response at a fixed observation point is characterised by power spectral density diagrams assembled into waterfall plots, and the corresponding steady-state harmonic responses are obtained in closed form from the frequency-domain resolvent of the same state-space model. The formulation is verified against an independent radial finite-element model for all three boundary conditions and against published rotating clamped–free natural frequencies. The central finding concerns how the excitation parameters act on the response. The excitation frequency, the radial position of a force, and the angular separation and phase of a pair of forces act as largely independent levers. The quantitative sensitivity to each lever, however, is set by the boundary conditions, as is the force placement that minimises a chosen travelling-wave family. The radial position that minimises the excitation of a chosen radial family is governed by the interior node of that mode, which lies at r≈52, 68 and 44 mm for the clamped–clamped, clamped–free and free–clamped discs, respectively, and does not in general coincide with a free edge. Angular separation, by contrast, suppresses a nodal-diameter family in a manner that is essentially boundary-condition independent. The parameter dependences are shown to be steady-state properties: the driven spectral line of the finite-duration records reproduces the resolvent solution to within 0.09 dB.

Vibration doi: 10.3390/vibration9030047

Authors: El Mahdi Rhiate Khawla Gaouzi Farah Abdoun Lahcen Azrar

This paper develops a reduced-order electromechanical model for piezoelectric energy harvesting from a beam traversed by a moving mass. The beam is described by the Euler–Bernoulli theory, and the coupled equations of motion are derived through modal expansion combined with the linear piezoelectric constitutive relations. Unlike most existing formulations, the model accounts for non-uniform transit by including moving-mass acceleration, accommodates an arbitrary number of piezoelectric patches distributed along the span, and incorporates von Kármán strain–displacement relations. So, moderately large deflections and mid-plane stretching as well as various boundary conditions may be investigated within the same framework. The resulting coupled nonlinear ordinary differential equations are integrated in time using a numerical solver. On the other hand, predictions of midpoint deflection, output voltage, and harvested power are validated against the COMSOL Multiphysics Finite element model. The experimental setup has been established, and a dedicated laboratory experiment provides additional verification under controlled conditions. Parametric analyses investigating the individual and combined effects of the mass ratio, velocity ratio, acceleration profile, patch length, patch position, number of patches, and external load resistance are elaborated. Distributed multi-patch configurations are shown to recover more energy than a single-centered patch once higher modes contribute appreciably to the response. Design charts relating the governing parameters to the harvested power are constructed for each set of support conditions. These results are intended to assist the preliminary sizing and placement of piezoelectric transducers on some practical energy harvesting applications.

Vibration doi: 10.3390/vibration9030046

Authors: Oleksii Vodka Mariia Shapovalova Vitalii Ovcharenko Olena Avdieieva

This paper presents a linear mathematical model of bass guitar string vibration with experimentally identified, frequency- and fret-dependent modal damping, aimed at high-fidelity generative sound synthesis. To identify the string damping parameters across various frets and configurations, an experimental framework was developed to benchmark four structural identification methods: half-power bandwidth, I. Yoshida’s method, Discrete Fourier Transform Interpolation, and Hilbert-transform envelope approximation. Experiments were systematically conducted on Cort C4H, Ibanez RB 630, and Yamaha bass guitars. Based on the extracted parameter space, two audio generation strategies are formulated: a spectrum-driven harmonic reconstruction method (Method 1) and a physical modeling approach utilizing spatial wave equations (Method 2). The proposed linear approximation framework effectively captures the inverse relationship between the damping factor and fret numbers specifically on the E-string, while mapping linear increases on the G and D-strings. Quantitative verification using Sobolev norm differences demonstrates good agreement between the synthesized and original signals for the spectrum-driven method (Q = 0.031–0.057) and moderate agreement for the physics-based wave equation method (Q = 0.058–0.153). This reflects a trade-off in which the former achieves tighter spectral convergence, while the latter better preserves the physical, time-domain waveform structure. As both synthesis strategies are closed-form and computationally lightweight, the model is suitable for real-time implementation and the dynamic control of playing techniques (e.g., plucking location and, in principle, slap-type excitation), without relying on heavy, multi-gigabyte audio sample libraries.

Vibration doi: 10.3390/vibration9030045

Authors: Longjie Wu Tianhang Jiang Hanjie Yuan Yuxiang Zhu Jie Yang Yana Wang Zhen Li Yisheng Zhang Yilin Wang

The micro-vibration of overhead transmission lines often leads to conductor fatigue and damage to hardware, and the reliability of the connection of vibration dampers is of vital importance. To prevent loosening and detachment during operation, this study investigated the slippage mechanism of vibration dampers’ wire clamps under dynamic loads. The static friction coefficient was measured through the pull-off force experiment, and the dynamic sliding characteristics of the two types of clamp covers (pressure block type and hinge type) under different vibration conditions were systematically tested. The experiments showed that vibration significantly reduces the dynamic friction force, resulting in a “friction reduction effect”. The pressure block type structure is prone to slip under low tightening torque, while the hinge type structure has excellent anti-loosening performance due to its lever amplification design. The study clarified that the tightening torque, vibration parameters, and structural form are the key influencing factors, providing a basis for the optimization design and installation of anti-vibration dampers.

Vibration doi: 10.3390/vibration9030044

Authors: Aires Colaço Hassan Liravi Pedro Alves Costa

Rapid urbanisation, the demand for efficient mobility, and the need to mitigate climate change are among the major societal challenges of our time [...]

Vibration doi: 10.3390/vibration9030043

Authors: Ruixia Ma Chenchen Xie Chong Xu Kai Wu Wei Wang Xiwei Xu

Rescue-induced vibrations easily trigger dynamic instability and secondary collapse of post-disaster metastable structures, threatening rescue safety and efficiency. This paper comprehensively reviews research on vibration-induced instability and dynamic responses and mitigation strategies of these discontinuous structural systems. We analyze vibration propagation, energy concentration and progressive collapse mechanisms, and summarize parameterized modeling, physical tests and mainstream numerical methods including FEM, DEM and F-DEM, with their pros and cons compared. Typical vibration-mitigation technologies such as passive support, damping reinforcement, and semi-active and active control are classified and discussed, and nonlinear energy sinks as well as anti-phase control are elaborated on. Validation studies in rescue-training bases are also presented. Finally, the study is synthesized to clarify the interconnections among dynamic monitoring, structural modeling, and vibration mitigation. This review-derived synthesis identifies current knowledge gaps and outlines future research directions for rescue-oriented dynamic safety assessment. This review provides theoretical and engineering references for safe disaster rescue.

Vibration doi: 10.3390/vibration9020042

Authors: Ziyu Tao James A. Moore Masoud Sanayei Said Bolourchi

Train-induced ground vibrations can propagate into pile foundations, potentially causing undesirable vibration in nearby buildings, laboratories housing vibration-sensitive equipment, and manufacturing facilities for high-precision processes. This paper presents an innovative method for predicting building vibration from free-field ground vibration measurements at locations away from the tracks during train pass-bys. The proposed method accounts for site-specific soil profiles and train-soil-pile-structure interactions and is implemented in four steps. In Step 1, train-induced vibration transmission into the ground is estimated using an axisymmetric finite element model that simulates wave propagation through layered soils from the tracks to free-field ground locations. Step 2 estimates free pile head vibration using a three-dimensional finite-element model that captures the ground-borne transmission of track inputs through soil layers to the pile. Step 3 estimates vibration at the junction of the pile head and depot column base using a finite-element model to estimate the pile head impedance and an analytical impedance model for the depot structures supported by the pile. In Step 4, estimates of column-base vibration that transmits into over-track buildings are compared to measured column-base vibration levels obtained during train pass-bys. The method was applied at a metro depot in China, where tracks were in close proximity to columns supporting over-track buildings. Ground and column base vibration levels were measured during multiple train pass-bys. The estimated vibration levels at the base of depot columns closely agreed with the measured vibration levels at the columns during six-car train pass-bys. It demonstrated the potential effectiveness of this hybrid method for assessing vibration transmission into structures atop existing railway tracks. By integrating field measurements, finite element simulations, and analytical impedance models, the proposed hybrid method provides a framework for evaluating the transmission of the train-induced vibration to nearby building structures.

Vibration doi: 10.3390/vibration9020041

Authors: Qiang Wei Heng Zhong Chao Fan Yanqiang Bi Hongye Zhang

Spacecraft are evolving toward larger scales and higher performance, enabling widespread application of sophisticated space structures such as space antennas and flexible solar arrays. Such structures may experience thermally induced vibration (TIV) due to the influence of sudden solar radiation heat flows when it enters and leaves the Earth’s shadow in orbit. This paper focuses on a space thin-walled tube structure as the test specimen, and conducts ground-based TIV experiments in a vacuum environment, comparing the results with numerical simulations. The numerical simulation results for various key parameters show good agreement with the experimental data. The relative errors of average temperature, quasi-static displacement, and vibration frequency are approximately 5%, while the relative error of vibration amplitude is around 10%. Leveraging the validated numerical model, this study further investigates the influence of gravity on the TIV of large space structures. The results indicate that the TIV response amplitude under orbital conditions is significantly larger than that obtained from ground-based experiments.

Vibration doi: 10.3390/vibration9020040

Authors: Tamer A. El-Sayed Moustafa S. Taima Fady E. Shoukry Mohamed M. Z. Ahmed

Pipes conveying fluids are important fluid–structure interaction systems encountered in aerospace, energy, marine, and industrial applications. Their dynamic behavior is strongly influenced by the interaction between structural motion and internal or external flow, leading to complex phenomena such as divergence, flutter, and flow-induced vibration. This review presents a comprehensive assessment of the dynamics and stability of pipes conveying fluids by integrating classical theories with recent developments in modeling, computation, materials, and control. The review covers mathematical formulations based on Euler–Bernoulli, Rayleigh, Timoshenko, and shell theories, together with analytical and numerical solution methods used for stability and vibration analysis. The effects of geometry, boundary conditions, flow configuration, damping, and material properties on dynamic response and instability thresholds are discussed. Special attention is given to composite, viscoelastic, functionally graded, and smart materials, as well as micro- and nanoscale pipe systems. Recent advances in vibration suppression, reduced-order modeling, machine learning, and physics-informed computational approaches are also reviewed. Finally, the paper identifies current challenges and future research directions, including multiphysics coupling, experimental validation, digital twins, and AI-assisted predictive modeling for fluid-conveying pipe systems.

Vibration doi: 10.3390/vibration9020039

Authors: Shivan Ramnarace Jacqueline Bridge Kefu Liu

Hysteretic nonlinear vibration systems can exhibit jumps, coexisting attractors, and strong dependence on the initial state, particularly when material hysteresis is coupled with geometric nonlinearity. This paper investigates the global nonlinear dynamics of a harmonically forced single-degree-of-freedom oscillator incorporating pseudoelastic shape memory alloy (SMA) wires in a perpendicular geometric configuration. Cyclic force–displacement tests on pseudoelastic SMA wires are used to calibrate the constitutive response, after which steady-state dynamics are analyzed using time integration, numerical continuation (COCO), and basin-of-attraction computations over representative excitation frequencies, pre-tension levels, and the number of wires. The calibrated model predicts rich response regimes including jump phenomena, coexisting stable solutions, multistability, asymmetric periodic responses, and the pronounced dependence of the achieved steady response on initial conditions and internal state. Basin computations reveal sensitive partitioning of the state space between competing attractors, highlighting the influence of the initial and internal state in oscillators that combine pseudoelastic hysteresis with geometric stiffening. Additional numerical exploration of a negative pre-tension extension indicates transitions to more complex responses, including quasi-periodic and chaotic behaviour, but these are presented as secondary results outside the directly validated tension-wire regime. The results clarify how calibrated SMA hysteresis and geometric nonlinearity jointly shape multistability and basin structure in pseudoelastic oscillators.

Vibration doi: 10.3390/vibration9020038

Authors: Junbiao Xie Yuying Jiang Chen Wang Jingcheng Dai Yiming Yu Chenglong Pan

To address the vibration reduction problem of the six-degrees of freedom(6-DOF) half-vehicle model and to improve ride comfort and handling stability, a piezoelectric stack actuator based on the inverse piezoelectric effect was introduced. A 6-DOF half-vehicle dynamic model coupling the cab, body, and wheels was established based on the Lagrange equation. Based on this model, a vertical-pitch dual sliding surface RBF neural network sliding mode control strategy was proposed, with two independent RBF neural networks designed to separately approximate, online, the comprehensive uncertainties in the vertical and pitch channels associated with unmodeled dynamics, external disturbances, and modeling simplifications. The variable-speed reaching law (dsat) function was used to design the sliding mode reaching law, balancing sliding surface convergence speed and vibration suppression. Six indicators, including vertical acceleration of the cab and vertical acceleration of the vehicle body, were selected as performance evaluation metrics to establish the fitness function. Combined with a genetic algorithm, the dual sliding surface coefficients, RBF network parameters, adaptive update rates, and variable-speed reaching law parameters were globally optimized. The vibration reduction effects of four schemes—passive control, traditional sliding mode control, RBF sliding mode control, and genetic algorithm optimized RBF dual-sliding-mode control—were compared and analyzed. Simulation results show that the genetic algorithm optimized RBF dual-sliding-mode control achieves improved vibration suppression in several key ride-comfort-related indices and provides better overall coordination among ride comfort, suspension working space, and tire dynamic deflection. The research results validate the effectiveness of this method and provide a new solution for addressing vehicle vibration reduction problems.

Vibration doi: 10.3390/vibration9020037

Authors: Yanbin Shen Lijia Yang Diming Guo Luyue Xi

This study introduces the dynamic characteristics of stepped functionally graded (FG) cylindrical shells under general boundary conditions using the Legendre–Ritz method. The calculated model is established based on the first-order shear deformation theory and the domain decomposition method, and the artificial spring is introduced to simulate the boundary conditions and ensure segment continuity. The Legendre polynomials and the Fourier series are used to form the admissible displacement function. The Rayleigh–Ritz method is employed to determine the free and forced vibration characteristics of stepped FG cylindrical shells. Results are presented for various boundary conditions, material parameters and geometric dimensions, and comparisons with published studies are performed. The method demonstrates good accuracy, providing a basis for analyzing the vibration behavior of stepped FG cylindrical shells.

Vibration doi: 10.3390/vibration9020036

Authors: Shasha Chen Bernard F. Rolfe James Griffin Arnaldo Delli Carri Michael P. Pereira

Ground engaging tool (GET) wear monitoring is important for mining excavator maintenance, but progressive multi-tooth wear estimation remains insufficiently explored. This study presents a vibration-based framework for GET wear estimation during operations using modal analysis, finite element (FE) modelling, and machine learning as a supporting evaluation tool. A laboratory-scale mining bucket surrogate with detachable attached masses was used to represent progressive tooth wear through controlled mass-loss conditions. Experimental impact hammer tests under approximately free-free boundary conditions were conducted to validate the FE modal model through natural-frequency comparison and qualitative mode correspondence. The validated FE model was then used to generate a broader dataset of multi-tooth wear scenarios, from which the first ten natural frequencies were extracted as modal features. Linear Regression (LR) was adopted as a simple and interpretable baseline to evaluate both overall wear estimation and individual tooth wear estimation. High accuracy was obtained for overall wear estimation for both the non-symmetric and symmetry-augmented datasets, with R2 values of 0.9983 and 0.9976, respectively. In contrast, individual tooth prediction was more challenging, and the symmetry-augmented results showed that mirrored tooth locations can produce non-unique frequency-based signatures. An additional asymmetric FE sensitivity study further confirmed that structural symmetry can limit local wear identifiability when only global natural frequencies are used. These findings demonstrate the potential of FE-derived modal frequency features for laboratory-scale GET wear assessment, while also highlighting the limitations of frequency-only features for unique local wear localisation in symmetric structures. This is a promising approach for wear estimation during mining operations.

Vibration doi: 10.3390/vibration9020035

Authors: Guo Li Qingyun Ye Xuze Wu Muyang Wu Wen Liu Hang Wang

Brake-induced steering wheel shimmy is a critical nonlinear dynamic phenomenon that severely compromises vehicle handling stability and driving safety. While clearances in steering mechanism kinematic pairs are widely recognized as a primary cause of shimmy instability, the coupling effect of multiple concurrent clearances remains poorly characterized, particularly under transient braking conditions. In this work, a 5-degree-of-freedom non-autonomous dynamic model of brake-induced shimmy is developed using Lagrange’s equations. The model comprehensively incorporates the non-smooth contact behavior of multiple clearance joints, transient braking axle load transfer, and the longitudinal–lateral coupling nonlinearity of tires. The nonlinear dynamic evolution of the system is investigated through phase portraits, Poincaré sections, and continuous wavelet transform analysis. Numerical results demonstrate that multi-clearance coupling increases the peak shimmy angle by more than 40% compared to the single-clearance case. As the clearance magnitude increases from 0.05 mm to 0.40 mm, the system undergoes a transition from stable periodic motion to high-dimensional chaos, accompanied by a 67% reduction in vibration energy concentration at the 0.4 mm clearance level. This study elucidates the nonlinear mechanism underlying clearance-induced brake shimmy, providing a robust theoretical foundation for steering system parameter optimization and shimmy mitigation strategies.

Vibration doi: 10.3390/vibration9020034

Authors: Yonghua Jiang Zhiqiang He Zhilin Dong Jianjie Zhang Hongkui Jiang Chao Tang Jianfeng Sun Xiaohao Chen Weidong Jiao

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.

Vibration doi: 10.3390/vibration9020033

Authors: Junwen Wei Yurong Wang Yi Wang Qiangsheng Luo

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.

Vibration doi: 10.3390/vibration9020032

Authors: Kexin Chen Lin Lu Changan Xu Luyue Xi Xianghong Huang

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.

Vibration doi: 10.3390/vibration9020031

Authors: Quansheng Hu Sheng Liu Kun Zhang Chaoying Wang Qichao Xue Guangping Zou Yonghui Wang Mingtao Chen Deshui Xu

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.

Vibration doi: 10.3390/vibration9020030

Authors: Ran Duan Ruopeng Yan Guangyin Jin

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.

Vibration doi: 10.3390/vibration9020029

Authors: Lun Shao Alexandre Saidi Abdel-Malek Zine Mohamed Ichchou

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.

Vibration doi: 10.3390/vibration9020028

Authors: Hongzhi Fan Chao Zhang Mingyu Sun Kexi Xu Wenyang Zhang Ximing Zhang

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.

Vibration doi: 10.3390/vibration9020027

Authors: Xu Han Chen Zhang Zhaoyang Guo Wenhua Wang Qiang Liu Xin Li

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.

Vibration doi: 10.3390/vibration9020026

Authors: Ho-Chih Cheng Min-Chie Chiu Ming-Guo Her

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 Zo = 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.

Vibration doi: 10.3390/vibration9020025

Authors: Brian Fiegel Yash Kumar Dhabi Salam Rahmatalla Geb Thomas Tyler Guzowski Elizabeth Ritchie David Wilder Nathan B. Fethke

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.

Vibration doi: 10.3390/vibration9020024

Authors: Mandie Tu Laixian Peng Xinbiao Xiao Jian Han Peng Wang

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.

Vibration doi: 10.3390/vibration9020023

Authors: Guojiang Yin Shuo Wang

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.

Vibration doi: 10.3390/vibration9010022

Authors: Dongrui Song Xiaocheng Tang Zhiwei Sun Dong Han Xiaozhuo Guan Huashun Li

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.

Vibration doi: 10.3390/vibration9010021

Authors: Lutz Auersch Samir Said Werner Rücker

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.

Vibration doi: 10.3390/vibration9010020

Authors: Khaldoon F. Brethee Ghalib R. Ibrahim Al-Hussein Albarbar

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.

Vibration doi: 10.3390/vibration9010019

Authors: Mohammed Abdeldjalil Djehaf Ahmed Hamet Sidi Youcef Islam Djilani Kobibi

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%.

Vibration doi: 10.3390/vibration9010018

Authors: Marcus Dykes Julius Kaplunov Danila Prikazchikov

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.

Vibration doi: 10.3390/vibration9010017

Authors: Yoshihiro Narita

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.

Vibration doi: 10.3390/vibration9010016

Authors: Ikram Bagri Achraf Touil Rachid Oucheikh Ahmed Mousrij Aziz Hraiba Karim Tahiry

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× 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.

Vibration doi: 10.3390/vibration9010015

Authors: Yousef Sardahi Asad Salem Isaac W. Wait Gang S. Chen Kirk McCormick Killian Blake Tanner Samples Luke Lanham

Impact-echo/impact response testing is widely used to detect cracks, voids, and delamination, but transient signals and crowded spectra can complicate diagnosis. This study presents a nonlinear, harmonic-based framework that characterizes delamination using higher-order harmonics in the impact-free response, instead of the amplitude-dependent resonance–frequency shift. The delaminated region is formulated as a locally vibrating nonlinear plate/oscillator with polynomial material and geometric nonlinearities, predicting harmonic components whose levels depend on impact intensity and nonlinearity parameters. The approach is validated on a concrete slab containing an artificial delamination, excited by repeatable impacts, and measured with an accelerometer. Frequency-domain analysis shows that intact regions exhibit a distinct spectral pattern, whereas the delaminated region produces a clear fundamental component and, with modestly increased impacts, a strong second harmonic that serves as a defect signature; time series metrics corroborate nonlinearity. The results demonstrate a nondestructive technique that can localize and characterize delamination without driving the specimen into damaging strain. Looking ahead, the same harmonic signature principle can be extended to vibroacoustic/impact monitoring of lithium-ion batteries to flag mechanically induced internal defects (e.g., separator/electrode delamination) that can precipitate internal short circuits and elevate thermal runaway risk, improving quality control and in-service safety.

Vibration doi: 10.3390/vibration9010014

Authors: Yilang Dong Xuewu Dai Dongliang Cui Dong Zhou

Rolling bearings are critical components in rotating machinery, and their failures may lead to significant economic losses and safety hazards. However, early fault signals are often weak and masked by strong background noise, making accurate fault diagnosis extremely challenging. To address this issue, this paper proposes a fault diagnosis method for rolling bearings based on improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), an autoregressive (AR) model, and observer-based eigenvalue extraction, combined with a particle swarm optimization-based kernel extreme learning machine (PSO-KELM). Targeting rotating machinery with rolling bearings, the approach begins by applying ICEEMDAN as a preprocessing step to decompose non-stationary vibration signals into multiple intrinsic mode functions (IMFs), from which all essential fault-related information is extracted. The preprocessed vibration signal is then reconstructed. Subsequently, an AR model is used to establish a state-space representation for the observer, which processes the reconstructed signal and generates a residual output by comparing it with the actual mechanical signal. Features are then extracted from the residual signal, including its mean, variance, maximum and minimum values, kurtosis, waveform factor, pulse factor, and clearance factor. These features serve as inputs to the PSO-KELM classifier for fault diagnosis. To validate the method, real vibration data from electric motor bearings were employed in a case study, covering normal conditions and three typical fault types: outer race fault, inner race fault, and rolling element fault. The results demonstrate that the proposed method effectively enables fault feature extraction and accurate identification of bearing conditions.

Vibration doi: 10.3390/vibration9010013

Authors: Lin Lu Zhe Zhao Ting Li Cong Gao Jiajun Zheng

This study develops a semi-analytical method for free vibration analysis of joined conical–cylindrical shell with axially stepped thickness. The computational framework is built through the domain decomposition method, artificial spring technology and shear deformation shell theory. Kinematic admissible functions are constructed via superposition of Chebyshev orthogonal polynomials and trigonometric series. Subsequently, the Rayleigh–Ritz method is employed to solve for the system’s characteristic frequencies. The accuracy of the method is further verified by the excellent agreement between the current results and those from published studies and finite element simulations. Ultimately, the influence of boundary conditions, structural parameters and stepped thickness distribution on the free vibration characteristics of conical–cylindrical shells are systematically discussed. These findings reveal the critical methodological constraints in free vibration modeling of stepped thickness shell systems, thereby advancing vibration design optimization for the stepped thickness structures.

Vibration doi: 10.3390/vibration9010012

Authors: Rui Yang Wenyou Zha Ruixiang Zhang Yongtao Yao Yanju Liu

Honeycomb structures are widely utilized in engineering due to their light weight, high strength, high stiffness, excellent energy absorption, and outstanding vibration isolation performance. In this study, we propose a novel tuning fork–honeycomb megastructure, which demonstrates excellent tunable vibration isolation capabilities. The geometric configuration of the structure before and after shape memory–induced deformation is described, and a theoretical model for the natural frequency of the initial configuration is established. The vibration isolation performance of the structure is validated through simulations and experiments, and three strategies for tuning its vibrational behavior are proposed. First, by exploiting variable stiffness, shape memory materials are used to achieve a linear shift in the bandgap position. At 75 °C, the starting frequency of the bandgap decreases to 95% of its value at room temperature. Second, based on shape memory programming, the deformed structure exhibits a 20% reduction in the center frequency of the first bandgap and a 47% reduction in the center frequency of the second bandgap compared to the undeformed configuration. Then, by altering the geometry of the tuning fork structure, in–plane deformation is shown to provide superior low–frequency vibration isolation performance compared to out–of–plane deformation. Finally, the design method of programmable mechanical pixel metamaterials is introduced. This method achieves tunable full–band vibration isolation through shape–memory–induced deformation and temperature–induced stiffness variation. It enhances the structural diversity, modularity, and reconfigurability. Moreover, a shape memory tuning fork structure could be combined with any type of cellular structure with excellent vibration isolation performance. It offers a new paradigm for designing structures with adjustable wide–frequency vibration isolation performance.

Vibration doi: 10.3390/vibration9010011

Authors: Eckardt Johanning

In a sentinel health event investigation of a back disorder claim, the vibration exposure and ergonomic function of a modified suspension seat were assessed. Background: In a forensic occupational injury investigation, an aftermarket-altered operator seat in a railroad rail-track tamper machine was evaluated. Methods: Detailed whole-body vibration (WBV) exposure measurements were conducted according to current applicable technical standards and guidelines (i.e., ISO 2631-1:1997) on a 09-16 DYNACAT Continuous Action Tamper with Stabilizer during routine track repair services. The modified Grammer Mfg. suspension operator seat was evaluated for performance and ergonomic features (i.e., adjustability, posture, and suspension quality). Results: The tested seat appeared to underperform and was overloaded with the aftermarket control devices, attachments and modifications. The suspension system’s end-stopper was damaged. The seat system had excessive play and wobbles; it was not firmly braced and attached. The vector sum (av) results ranged from 0.26 m/s2 (no tamping) to a maximal 0.55 m/s2 (tamping). The seat transfer (SEAT) analysis showed magnification of vibration input and variable performance of the suspension depending on operational tasks. Conclusions: The modified suspension seat underperformed and seemed to magnify and worsen the vibration, jolts and shock exposures of the seated operator. The heavy and bulky seat modifications likely limited the suspension function. The malfunctioning seat was more likely than not a contributing factor in the pathogenesis of the spinal disorders of the injured machine operator.

Vibration doi: 10.3390/vibration9010010

Authors: Francesco Pellicano Yuri V. Mikhlin Konstantin V. Avramov Antonio Zippo

Nonlinear vibration phenomena play a central role in modern engineering, spanning applications from large-scale civil infrastructure to microscale and nanoscale systems [...]

Vibration doi: 10.3390/vibration9010009

Authors: Nicolae Lobontiu

A new lumped-parameter matrix method is proposed to model the decoupled, in-plane longitudinal and transverse free undamped vibrations of a collinear system with fixed ends and formed of two end flexible and prismatic members linked by a middle rigid connector. The method calculates the natural frequencies associated with the system’s three degrees of freedom by solving a linear algebraic characteristic equation related to the dynamic matrix, which is obtained from the system compliance and mass matrices. The linear, small-displacement model characterizes either long or short beams by adequately formulating the compliance and mass matrices. The lumped-parameter model is comprehensively validated by two separate distributed-parameter models, which determine the system’s longitudinal-vibration and long-beam, bending-vibration natural frequencies. Numerical simulations are performed with the lumped-parameter model to identify the sensitivity of the natural frequencies to system parameters variations and model variants. The system’s matrices are also utilized to perform frequency-domain analysis of the three-member system in a displacement/acceleration sensing application. The method can be adapted and expanded to describe more complex configurations with multiple, non-collinear, and non-prismatic members.

Vibration doi: 10.3390/vibration9010008

Authors: Ru Li Bowen Xu Weiqi Yang

With the development of aerospace technology, hypersonic flight vehicles are evolving towards larger size, lighter weight, and higher performance. Their cross-domain maneuverability and extreme flight environment led to the rigid–flexible coupling effect and became the core bottleneck restricting performance improvement, seriously affecting flight stability and control accuracy. This paper systematically reviews the research status in the field of control for high-speed rigid–flexible coupling aircraft and conducts a review focusing on two core aspects: dynamic modeling and control strategies. In terms of modeling, the modeling framework based on the average shafting, the nondeformed aircraft fixed-coordinate system, and the transient coordinate system is summarized. In addition, the dedicated modeling methods for key issues, such as elastic mode coupling and liquid sloshing in the fuel tank, are also presented. The research progress and challenges of multi-physical field (thermal–structure–control, fluid–structure–control) coupling modeling are analyzed. In terms of control strategies, the development and application of linear control, nonlinear control (robust control, sliding mode variable structure control), and intelligent control (model predictive control, neural network control, prescribed performance control) are elaborated. Meanwhile, it is pointed out that the current research has limitations, such as insufficient characterization of multi-physical field coupling, neglect of the closed-loop coupling characteristics of elastic vibration, and lack of adaptability to special working conditions. Finally, the relevant research directions are prospected according to the priority of “near-term engineering requirements–long-term frontier exploration”, providing Refs. for the breakthrough of the rigid–flexible coupling control technology of the new-generation high-speed aircraft.

Vibration doi: 10.3390/vibration9010007

Authors: Biljana Mladenović Andrija Zorić Dragan Zlatkov Danilo Ristic Jelena Ristic Katarina Slavković Bojan Milošević

Construction of precast reinforced concrete (PRC) industrial halls in seismically active areas has been increasing in recent decades. As connections are one of the most sensitive and vulnerable zones of PRC structures, there is a need to pay special attention to their investigation and modeling in seismic analysis. Knowing that each PRC system is specific and unique, this study aims to evaluate the actual seismic performances of PRC industrial halls built in the AMONT system, which represent a significant portion of the existing industrial building stock in Italy, the Balkans, and Turkey. As there is a lack of published research data on its specific joints, the results of the quasi-static full-scale experiments carried out up to failure on the models of four characteristic connections are presented. Since the implementation of nonlinear dynamic analysis in everyday engineering practice can be demanding, a simplified model of the structure considering the effects of the connections’ stiffness is proposed in this paper. The differences in the roof top displacements between the proposed model and the model with the rigid joints of the analyzed frames are in the range from 16.53% to 66.93%. The values of inter-story drift ratios are larger by 10–100% when the real stiffness of connections is considered, which is above the limit value provided by standard EN 1998-1. These results confirm the necessity of considering the nonlinear behavior and stiffness of connections in precast frame structures when determining displacements, which is particularly important for the verification of the serviceability limit state of structures in seismic regions.

Vibration doi: 10.3390/vibration9010006

Authors: Jiaqi Zhao Mingxin Liu Xulong Jin Youlong Du Ye Zhuang

Off-road vehicle control designs often neglect the complex tire–soil interactions inherent to soft terrain. This paper proposes a Variable Impedance Control (VIC) strategy integrated with a high-fidelity terramechanics model. First, a real-time sinkage estimation algorithm is derived using experimentally identified Bekker parameters and the quasi-rigid wheel assumption to capture the nonlinear feedback between soil deformation and vehicle dynamics. Building on this, the VIC strategy adaptively regulates virtual stiffness, damping, and inertia parameters based on real-time suspension states. Comparative simulations on an ISO Class-C soft soil profile demonstrate that this framework effectively balances ride comfort and safety constraints. Specifically, the VIC strategy reduces the root-mean-square of vertical body acceleration by 46.9% compared to the passive baseline, significantly outperforming the Linear Quadratic Regulator (LQR). Furthermore, it achieves a 48.6% reduction in average power relative to LQR while maintaining suspension deflection strictly within the safe range. Moreover, unlike LQR, the VIC strategy improves tire deflection performance, ensuring superior ground adhesion. These results validate the method’s robustness and energy efficiency for off-road applications.

Vibration doi: 10.3390/vibration9010005

Authors: Yueyin Ma Zhenhua Chen Wanhua Chen Bin Ma Xinyu Gao Xutao Nie Daokui Li

The free vibration of stiffened plates analyzed using classical plate–beam theoretical theory (PBM) simplified the vibrations of stiffeners parallel to the plane of the stiffened plate as the first-order torsional vibration of the stiffener cross-section. This simplification introduces errors in both the natural frequencies and mode shapes of the structure for stiffened plates with relatively tall stiffeners. To mitigate the issue previously described, this paper proposes an enhanced plate–beam theoretical model (EPBM). The EBPM decouples stiffener deformation into two components: (1) bending deformation along the transverse direction of the stiffened plate, governed by Euler–Bernoulli beam theory, and (2) transverse deformation of the stiffeners, modeled using thin plate theory. Virtual torsional springs are introduced at the stiffener–plate and stiffener–stiffener interfaces via penalty function method to enforce rotational continuity. These constraints are transformed into energy functionals and integrated into the system’s total energy. Displacement trial functions constructed from Chebyshev polynomials of the first kind are solved using the Ritz method. Numerical validation demonstrates that the EBPM significantly improves accuracy over the BPM: errors in free-vibration frequency decrease from 2.42% to 0.63% for the first mode and from 9.79% to 1.34% for the second mode. For constrained vibration, the second-mode error is reduced from 4.22% to 0.03%. This approach provides an effective theoretical framework for the vibration analysis of structures with high stiffeners.

Vibration doi: 10.3390/vibration9010004

Authors: Pratik Sarker M. Shafiqur Rahman Uttam K. Chakravarty

Helicopter vibration is an inherent characteristic of rotorcraft operations, arising from transmission dynamics and unsteady aerodynamic loading, posing challenges to flight control and longevity of structural components. Excessive vibration elevates pilot workload and accelerates fatigue damage in critical components. Leveraging advances in optimal control and microelectronics, the active vibration control methods offer superior adaptability compared to the passive techniques, which are limited by added weight and narrow bandwidth. In this study, a comprehensive vibration analysis and optimal control framework are developed for the Bo 105 helicopter rotor blade exhibiting flapping, lead-lag, and torsional (triply coupled) motions, where a Linear Quadratic Regulator (LQR) is employed to suppress vibratory responses. An analytical formulation is constructed to estimate the blade’s sectional properties, used to compute the coupled natural frequencies of vibration by the modified Galerkin method. An orthogonality condition for the coupled flap–lag–torsion dynamics is established to derive the corresponding state-space equations for both hovering and forward-flight conditions. The LQR controller is tuned through systematic variation of the weighting parameter Q, revealing an optimal range of 102–104 that balances vibration attenuation and control responsiveness. The predicted frequencies of the vibrating rotor blade are compared with the finite element modeling results and published experimental data. The proposed framework captures the triply coupled rotor blade dynamics with optimal control, achieves modal vibration reductions of approximately 60–90%, and provides a clear theoretical benchmark for future actuator-integrated computational and experimental studies.

Vibration doi: 10.3390/vibration9010003

Authors: Joerg Bienert Simon Regnet

Safety is the most important requirement in flight operations. This also affects the application for an extreme lightweight glider in this paper. Essential properties are the target weight below 120 kg and the electric propulsion. The unsymmetric inertia from the two-blade propeller at the rear in combination with the light and flexible aluminium tube support makes it necessary to investigate the risk of mechanical instability. Starting from the equations of motion, the time-variant system matrices are set up. The simulation of Floquet multiplier and Hill’s hyper-eigenvalue problem provide the necessary information about the system stability. The conclusion is that the potential instability due to structural damping in the observed system can be avoided in the range of operation. The damping, experimentally determined by approximately 2%, is sufficient.

Vibration doi: 10.3390/vibration9010002

Authors: Lei Chen Longxin Cui Dongliang Zou Yakun Wang Peiquan Wang Wenxuan Shi

Enterprises in industries such as coking and metallurgy possess extensive industrial equipment requiring real-time monitoring and timely fault detection. Transmitting all monitoring data to servers or cloud platforms for processing presents challenges, including substantial data volumes, high latency, and significant bandwidth consumption, thereby compromising the monitoring system’s real-time performance and stability. This paper proposes a cloud–edge collaborative approach for edge feature extraction in equipment monitoring. A three-tier collaborative architecture is established: “edge pre-processing-cloud optimization-edge iteration”. At the edge, lightweight time-domain and frequency-domain feature extraction modules are employed based on equipment structure and failure mechanisms to rapidly pre-process and extract features from monitoring data (e.g., equipment vibration), substantially reducing uploaded data volume. The cloud node constructs a diagnostic feature library through threshold self-learning and data-driven model training, then disseminates optimized feature extraction parameters to the edge node via this threshold learning mechanism. The edge node dynamically iterates its feature extraction capabilities based on updated parameters, enhancing the capture accuracy of critical fault features under complex operating conditions. Verification and demonstration applications were conducted using an enterprise’s online equipment monitoring system as the experimental scenario. The results indicate that the proposed method reduces data transmission volume by 98.21% and required bandwidth by 98.25% compared to pure cloud-based solutions, while effectively enhancing the monitoring system’s real-time performance. This approach significantly improves equipment monitoring responsiveness, reduces demands on network bandwidth and data transmission, and provides an effective technical solution for equipment health management within industrial IoT environments.

Vibration doi: 10.3390/vibration9010001

Authors: Robbert van der Kruk

This article explores whether jerk, the derivative of acceleration, should be limited in trajectory planning for position-controlled mechanical systems or in the controller. The excess jerk excites structural resonances and increases actuator wear, motivating the use of a limited jerk. However, we question the necessity of incorporating the jerk directly in trajectory planning by comparing third-order jerk-limited trajectories with second-order trajectories with reduced controller bandwidth that regulate torque gradients. We demonstrate by a typical practical application that reducing controller bandwidth can achieve comparable or superior jerk reduction without extending overall motion time for point-to-point trajectories. As a result, second-order parabolic trajectory profiles simplify on-line implementation. This investigation relies on a detailed sensitivity analysis of a one-dimensional model, incorporating crucial elements such as signal and sensor quantisation, sampling, and modes of structural resonances. The study shows that smooth trajectories reduce resonant vibrations and wear, but the jerk limitation may be addressed more effectively within the controller rather than within the trajectory generator. We conclude that although the limitation of the jerk in the trajectories is valuable, feedback controllers can reduce the jerk more effectively by bandwidth reduction, allowing simpler point-to-point trajectory designs without compromising performance.

Vibration doi: 10.3390/vibration8040081

Authors: Océane Topenot Gaël Chevallier Scott Cogan Christophe Oulerich

Physics-based simulations are now widely employed in mechanical engineering. Flexible Multibody dynamic Simulations (FMBSs) have proven to be effective in representing the behavior of complex structures with local damping and stiffness nonlinearities. However, due to the broad range of component flexibilities as well as contact behavior between structural elements, time integration analyses can result in high computational burden. The challenge addressed in this article concerns the implementation of an efficient model reduction procedure in order to provide an acceptable tradeoff between calculation time and loss of accuracy in the prediction of system responses and dynamic loads. In most FMBS commercial software, the behavior of linear elastodynamic components is taken into account via imported Craig–Bampton superelements. In this context, dynamic mode selection techniques have been shown to provide a better order reduction than the standard low-frequency truncation. This article provides a review of dynamic mode selection methods that can be found in the literature, followed by a comparison based on simulations of an aircraft engine stator integrated in the full industrial engine model and tested on a speed ramp-up with unbalance.

Vibration doi: 10.3390/vibration8040080

Authors: Mihai Gingarasu Daniel Ganea Elena Mereuta

The steering linkage represents a key subsystem of any automobile, playing a direct role in vehicle handling, driving safety, and overall comfort. Within this mechanism, the tie rod and tie rod end are crucial for transmitting steering forces from the gear to the wheel hub. A typical issue that gradually develops in these components is the clearance appearing in the spherical joint, caused by wear, corrosion, and repeated operational stresses. Even small clearances can noticeably reduce stiffness and natural frequencies, making the system more sensitive to vibration and premature failure. In this work, the effect of spherical joint clearance on the dynamic behavior of the tie rod-tie rod end assembly was analyzed through numerical simulation combined with experimental observation. Three-dimensional CAD models were meshed with tetrahedral elements and subjected to modal analysis under several clearance conditions, while boundary constraints were set to replicate real operating conditions. Experimental measurements on a dedicated test rig were used to assess joint clearance and wear in service parts. The results indicate a strong nonlinear relationship between clearance magnitude and modal response, with PTFE bushing degradation identified as the main source of clearance. These findings link the evolution of clearance to the change in vibration characteristics, providing useful insight for diagnostic approaches and predictive maintenance aimed at improving steering reliability and vehicle safety.

Vibration doi: 10.3390/vibration8040079

Authors: Xiaohan Phrain Gu Anbin Wang Hongdong Huang

This paper reviews, analyses, and suggests practical mitigation techniques at source for reducing vibration-induced annoyance to occupants in building structures that are built on top of significant railway infrastructure. The dynamic characteristics of vibration caused by wheel-rail interaction at metro train depots are different from those on main-lines and conventional studies. Ground-borne vibration in a building directly above a double-deck railway depot was investigated, focusing on vibration attenuation through rail track design, which is more effective and economic compared to treatments at receivers or along prorogation paths. A 2.5-Dimensional finite element model was established to simulate vibration transmission using different combinations of track-forms. Source contribution under different train running conditions has been evaluated by computing vibration levels along the main transmission path. Vibration levels at representative positions in the building rooms have been predicted using the numerical model and have been compared against site measurements at the corresponding locations after the completion of the construction of the depot and buildings. It was found that the 2.5D FE model enables a reasonable prediction of ground-borne vibration from the metro depot, and that by appropriate design of the track-form, a good level of vibration attenuation can be achieved in an economical way.

Vibration doi: 10.3390/vibration8040078

Authors: Ihab Abu Ajamieh Vincent Iacobellis Ali Radhi

Vibration band gap structures are advanced materials for vibration wave mitigation from metamaterials to phononic crystals from simple geometrical manipulations. Here, we present geometrical structures, made from platonic solids, that are capable of providing multi-passband frequency ranges with face symmetry in each unit cell. We fabricated the metamaterial structures using stereolithography, after which we experimentally characterized band gaps through impulse vibration testing. Experimental results have shown that the band gaps can be changed for different types of platonic structures along with the loading direction. This provided a comparison between axial and two bending direction band gaps, revealing ranges where the structures behave in either a “fluid-like” or an “optical-like” manner. Dodecahedron unit cells have exhibited the most promising results, when compared with reduced relative densities and a number of stacking unit cells. We utilized the coherence function during signal processing analysis, which provided strong predictions for the band gap frequency ranges.

Vibration doi: 10.3390/vibration8040077

Authors: Chao Zhang Chang Sun Wanbin Ren Yuchen Liao Ming Li Jian Liu

The environmental adaptability of outdoor power connectors exerts a crucial influence on the reliability of electrical systems. In this work, the current-carrying performance degradation of commercial power connectors under forced mechanical vibration conditions is investigated comprehensively. The variations in the instantaneous electrical contact resistance (ECR) of power connectors are accurately recorded in real time, and then effects of vibration amplitude, frequency, and load current on the ECR are interpreted explicitly. Furthermore, multi-cycle swept-sine vibration tests are carried out, and the open circuit failure of power connectors is reproduced. The continuous carrying of a heavy current combined with the mechanical fretting between socket and plug results in surface coating wear, debris melting, and the formation of copper oxide. The observed surface morphology and element contents support the presented failure mechanisms of power connectors under external vibrations.

Vibration doi: 10.3390/vibration8040076

Authors: Kaijian Hu Xiaojing Sun Ruoteng Yang Rui Han Meng Ma

Elevated stations are essential auxiliary structures within the high-speed rail (HSR) network. The newly constructed integrated elevated station for bridge building possesses a distinctive construction and intricate force transmission pathways, complicating the assessment of the dynamic coupling of train vibrations. Consequently, it is essential to examine the dynamic reaction of trains at such stations. This study utilises numerical simulation and field measurement techniques to examine the dynamic features of the newly constructed integrated elevated station for bridge building. Initially, vibration tests were performed on existing integrated elevated stations for bridge construction to assess their dynamic properties. The collected data were utilised to validate the modelling approach and parameter selection for the numerical model of existing stations, yielding a numerical solution method appropriate for bridge-station integrated stations. Secondly, utilising this technology, a numerical model of the newly integrated elevated station for bridge construction was developed to examine its dynamic features. Moreover, the impact of spatial configuration, train velocity, and operational organisation on the dynamic characteristics was analysed in greater depth. The vibration response level in the waiting hall was assessed. Research results indicate that structural joints alter the transmission path of train vibration energy, thereby significantly affecting the vibration characteristics of the station. The vibration response under double-track operation is notably greater than that under single-track operation. When two trains pass simultaneously at a speed of 200 km/h or higher, or a single train passes at 350 km/h, the maximum Z-vibration level of the waiting hall floor exceeds 75 dB, which goes beyond the specification limit.

Vibration doi: 10.3390/vibration8040075

Authors: Botao Li Yinhui Bao Guoqing Hu Xun Zhang

Due to its effective noise reduction, the semi-enclosed noise barrier is increasingly being applied in the construction of high-speed railways. However, there is still a lack of systematic research on the wind load distribution characteristics under natural crosswind, especially for the complex aerodynamic behavior of the intersection section of multi-line bridges. Therefore, the wind load distribution characteristics on the surface of the sound barrier under crosswind conditions are explored within the engineering context of a semi-enclosed acoustic barrier at the junction of a single-track bridge and a three-track bridge, using a combination of wind tunnel testing and numerical simulation. A rigid-body model with a geometric scale of 1:10 is established for the wind tunnel test. The wind load distribution characteristics of the two acoustic barriers are analyzed from the perspectives of mean wind pressure, pulsating wind pressure, and extreme wind pressure, respectively. FLUENT 2022 software is utilized to model the flow field characteristics of the sound barrier under two working conditions: windward and leeward. The results show that under the action of crosswind, the surface wind load of the sound barrier at the junction of the single/three-line bridge is very prominent, the maximum negative pressure shape coefficient is −4.516, and its distribution is dominated by negative pressure; that is, the sound barrier mainly bears suction. Compared with the semi-closed sound barrier on the single-track bridge, the extreme wind pressure at the semi-closed sound barrier on the three-track bridge and the junction of the two is more significant, which shows that this kind of area needs special attention in wind-resistant design.

Vibration doi: 10.3390/vibration8040074

Authors: Stylianos Vasileios Kontomaris Gamal M. Ismail Anna Malamou Andreas Stylianou

This paper examines the generic case of nonlinear mechanical oscillation under the influence of Hertzian-type restoring forces, a model relevant to phenomena involving elastic contact. The study addresses the complexity of strongly nonlinear systems by focusing on the differential equation governing the oscillation of a rigid sphere interacting with an elastic half-space, which includes a full series expansion to account for large deformations. Since no closed-form solution exists for the amplitude-dependent oscillation period, a new approximate analytical approach is introduced. This method preserves the system’s dominant Hertzian scaling while incorporating higher-order corrections through an averaged factor. For amplitudes where the deformation is less than or equal to the sphere’s radius, this approximation is nearly identical to the numerical solution. For larger amplitudes, the accuracy is further enhanced by introducing a semi-empirical linear adjustment to the relative error. This framework provides a reliable analytical description of the system’s behavior, offering a useful tool for theoretical studies and comparison with numerical results.

Vibration doi: 10.3390/vibration8040073

Authors: Aires Colaço Hassan Liravi Paulo J. Soares Jelena Ninić Pedro Alves Costa

Ground-borne vibrations caused by railway traffic represent a significant environmental concern, particularly in densely populated or vibration-sensitive urban areas. These phenomena can lead to discomfort and annoyance among residents, interfere with the operation of sensitive equipment, and even threaten the integrity of heritage sites or structurally vulnerable buildings and infrastructures. Building on these concerns, this paper presents a comprehensive review of the current state of knowledge on the subject. It begins by examining the impacts of ground-borne vibrations on both people and structures, followed by an overview of the regulatory frameworks implemented in different countries to manage these effects, with a focus on four examples from Europe and North America. The review then systematically explores the key factors associated with the generation and propagation of ground-borne noise and vibrations. Furthermore, prediction methodologies are categorised into four groups—analytical and semi-analytical, numerical, empirical and AI-based models—and critically assessed. Finally, the paper reviews mitigation strategies applied at the source, along the propagation path, and at the receiver, assessing their effectiveness in reducing the identified impacts.

Vibration doi: 10.3390/vibration8040072

Authors: Rade Vignjevic Nenad Djordjevic Javier de Caceres Prieto Nenad Filipovic Milos Jovicic Gordana Jovicic

The natural frequencies and damping characterisation of a new aerospace grade composite material were investigated using a modified impulse method combined with the half power bandwidth method, which is applicable to the structures with a low damping. The composite material of interest was unidirectional carbon fibre reinforced plastic. The tests were carried out with three identical square 4.6 mm thick plates consisting of 24 plies. The composite plates were clamped along one edge in a SignalForce shaker, which applied a sinusoidal signal generated by the signal conditioner exiting the bending modes of the plates. Laser vibrometer measurements were taken at three points on the free end so that different vibrational modes could be obtained: one measurement was taken on the longitudinal symmetry plane with the other two 35 mm on either side of the symmetry plane. The acceleration of the clamp was also recorded and integrated twice to calculate its displacement, which was then subtracted from the free end displacement. Two material orientations were tested, and the first four natural frequencies were obtained in the test. Damping was determined by the half-power bandwidth method. A linear relationship between the loss factors and frequency was observed for the first two modes but not for the other two modes, which may be related to the coupling of the modes of the plate and the shaker. The experiment was also modelled by using the Finite Element Method (FEM) and implicit solver of LS Dyna, where the simulation results for the first two modes were within 15% of the experimental results. The novelty of this paper lies in the presentation of new experimental data for the natural frequencies and damping coefficients of a newly developed composite material intended for the vibration analysis of rotating components.

Vibration doi: 10.3390/vibration8040071

Authors: Pedro Galvín Antonio Romero Mario Solís Emma Moliner María Dolores Martínez-Rodrigo

In this work, a time–frequency analysis of two railway bridges included in the InBridge4EU project database is presented. The study focuses on the identification of modal parameters from free responses after train passages and their comparison with estimations obtained from ambient vibration data. The wavelet transform is introduced as a valuable tool for detecting both free and forced bridge responses due to different train passages, as well as for conducting time–frequency analysis. This approach is particularly relevant for the identification of structural damping, given its dependence on vibration amplitude, as it enables the estimation of realistic values representative of bridge behavior under operational conditions. Additionally, the paper examines the complementary use of free vibrations for identifying natural frequencies and comparing them with results from ambient vibration tests. Wavelet analysis further reveals the predominant frequencies in the structural response before, during, and after train crossings, thereby capturing the influence of the moving vehicle on bridge dynamics.

Vibration doi: 10.3390/vibration8040070

Authors: Reza Rafiee-Dehkharghani Kamran Esmaeili Meysam Najari

This paper proposes a sequential bidirectional gated recurrent unit (BGRU) model to predict construction-induced ground vibrations. The ground vibration time histories for twelve real construction projects in Toronto, Canada, are collected and used to develop the BGRU model. A single time-step method is used to predict the vibrations, and the time window is swept continuously over the whole training data. In addition to the BGRU method, and for comparison, two other methods, autoregressive integrated moving average (ARIMA) and random forest (RF), are used to predict the ground vibrations. The results show that the BGRU method performs much better than ARIMA and RF methods in forecasting construction-induced ground vibrations. The BGRU method captures the construction-induced and background vibrations very well, and this method remains accurate when the training data includes both background and construction vibrations. Therefore, this method can be used to predict ground vibrations in real projects where there is always a potential for missing some parts of the ground vibration data due to the malfunction of the vibration recording units.

Vibration doi: 10.3390/vibration8040069

Authors: Liang Huang Kang Li Jinke Li Panjie Li Can Cui Pengfei Zheng

The high cost of traditional structural health monitoring systems limits their application to only a few major bridges, leaving most structures unmonitored between manual inspections. To address this issue, this study proposes a UAV mobile detection device (UMD) system that integrates a Raspberry Pi, data acquisition module, and accelerometer for rapid, contact-based vibration measurement. A vibration transmission model between the UMD and the bridge deck is developed to guide hardware design and quantify the influence of isolator stiffness and damping. The UMD’s performance is validated through both laboratory floor tests and field bridge experiments, demonstrating reliable identification of modal frequencies in the range of 0.00–51.95 Hz with a maximum acceleration error below 0.01 g and a relative modal frequency deviation within 3.4%. The analysis further determines that an accelerometer resolution of 0.02×10−1 g is required for accurate frequency domain measurement. These findings establish the UMD as a fast, low-cost, and accurate tool for rapid bridge vibration assessment and lay the groundwork for future multi-UAV synchronized monitoring.

Vibration doi: 10.3390/vibration8040068

Authors: Rajesh Shah Vikram Mittal Michael Lotwin

Vibration-based predictive maintenance is an essential element of reliability engineering for modern automotive powertrains including internal combustion engines, hybrids, and battery-electric platforms. This review synthesizes advances in sensing, signal processing, and artificial intelligence that convert raw vibration into diagnostics and prognostics. It characterizes vibration signatures unique to engines, transmissions, e-axles, and power electronics, emphasizing order analysis, demodulation, and time–frequency methods that extract weak, non-stationary fault content under real driving conditions. It surveys data acquisition, piezoelectric and MEMS accelerometry, edge-resident preprocessing, and fleet telemetry, and details feature engineering pipelines with classical machine learning and deep architectures for fault detection and remaining useful life prediction. In contrast to earlier reviews focused mainly on stationary industrial systems, this review unifies vibration analysis across combustion, hybrid, and electric vehicles and connects physics-based preprocessing to scalable edge and cloud implementations. Case studies show that this integrated perspective enables practical deployment, where physics-guided preprocessing with lightweight models supports robust on-vehicle inference, while cloud-based learning provides cross-fleet generalization and model governance. Open challenges include disentangling overlapping sources in compact e-axles, coping with domain and concept drift from duty cycles, software updates, and aging, addressing data scarcity through augmentation, transfer, and few-shot learning, integrating digital twins and multimodal fusion of vibration, current, thermal, and acoustic data, and deploying scalable cloud and edge AI with transparent governance. By emphasizing inverter-aware analysis, drift management, and benchmark standardization, this review uniquely positions vibration-based predictive maintenance as a foundation for next-generation vehicle reliability.

Vibration doi: 10.3390/vibration8040067

Authors: Ali F. Abdulla Soroush Arghavan Jihyun Cho Ibrahim F. Gebrel Mohamed Bognash Samuel F. Asokanthan

One of the challenges in inertia sensor applications is the need for a class of devices that operate at one of the ring resonant frequencies to achieve large amplitudes of vibration. However, large amplitudes tend to produce undesirable nonlinear effects due to geometrical nonlinearities. Hence, a rigorous experimental dynamic analysis of rotating thin circular ring-type structures is considered important to gain a deeper understanding of the device’s nonlinear behavior as well as the potential performance improvements. This study aims to experimentally investigate the nonlinear dynamic behavior of rotating thin circular rings and the effects of angular rate as well as mass mismatch variations on the system natural frequency. A prototype made of a macroscale thin cylindrical structure is employed to study the nonlinear dynamic behavior of rotating thin circular rings. Using a precision rate table equipped with a slip ring as well as non-contact sensors/actuators, experiments that closely represent the actual physical operating conditions of angular rate sensors are developed. Natural frequency variations due to the input angular rate changes are measured in time and frequency domains. Useful experimental observations on the frequency split and mass mismatch effects have been performed. Typical nonlinear behavior, such as jump phenomena of a rotating thin circular cylinder, is noted. The nonlinear dynamic behavior of a ring-type resonator system, which is subjected to external excitations, is experimentally investigated. Results from the present experimental study on the mechanics of the ring structure are expected to provide further insight into the design and operation of ring-type resonators for angular rate sensing applications.

Vibration doi: 10.3390/vibration8040066

Authors: Denyue Sun Yousef Sardahi Gang S. Chen Wael Zatar Hien Nghiem Zhaohui (Joey) Yang

This paper presents a study on the nonlinear vibrations in the impact-echo (IE) method for void flaw detection of solid structures. Linear theory has historically served as the foundational framework for non-destructive methods, including the IE method, particularly for estimating flaws in solids. This paper gives a comprehensive analysis of the nonlinear theory behind the IE method for detection of voids in solids such as concrete structures. The general equation of motion is presented for the flexural vibration of a void-defected solid with general nonlinear constitutive material properties, and then the simplified solutions for polynomial nonlinearity and hysteresis nonlinearity are derived comprehensively. The solutions of principal frequency and sub- and super-harmonics as well as the frequency of combined modes are elaborated, and the theoretical formula of resonant frequency shift with amplitude is derived. As conventional nonlinear IE methods have been conducted by only using a phenomenological model of linear shift in resonant frequency with amplitude, the proposed new frame of nonlinear vibration theory can be used to implement the IE method more comprehensively and accurately for void detection in solids.

Vibration doi: 10.3390/vibration8040065

Authors: Seyed Hamed Seyed Hosseini Seyedhossein Hajzargarbashi Gabriel Côté Zhaoheng Liu

Robots are used more and more in manufacturing, especially in tasks like robotic machining, where understanding their vibration behavior is very important. However, robot vibrations vary with posture, and evaluating all representative postures requires significant time and cost. This study proposes a deep learning (DL) based transfer learning (TL) approach to predict robot vibration behavior using fewer experiments. A large dataset was collected from a KUKA KR300 robot (Robot A) by testing nearly 250 postures. This dataset was then used to train a model to predict modal parameters such as natural frequencies (ω_n), damping ratios (ξ), and modal stiffness (k) within the workspace. TL was then used to apply the knowledge from Robot A to two other robots: a Comau NJ 650-2.7 (Robot B, high-payload) and an ABB IRB 4400 (Robot C, low-payload). Only a small number of postures were tested for Robots B and C. They were chosen carefully to cover different workspace areas and avoid collisions. Hammer tests were performed, and a four-step process was used to identify the real vibration modes. Stabilization diagrams were applied to confirm valid modes and remove noise. The results show that TL can accurately predict modal parameters for both Robot B and Robot C, even with limited data. These predictions were also used to estimate frequency response functions (FRFs), which matched well with experimental results. The main novelties of this work are: achieving accurate prediction of posture-dependent dynamics using minimal experimental data, demonstrating generalization across robots with different payload capacities, and revealing that data coverage across the workspace is more critical than dataset size.

Vibration doi: 10.3390/vibration8040064

Authors: Ioannis Kalyvas Dimitrios Dimogianopoulos

A contactless passive magnetoelastic sensing setup, recently proposed for detecting pest/substance accumulation in confined spaces (labs, museum reserves), is optimized for enhanced low-frequency performance. The setup uses a short flexible polymer slab, clamped at one end. There, a short Metglas® 2826MB magnetoelastic ribbon is fixed upon the slab’s surface. The opposite end receives excitation by a remotely controlled module of ultra-low amplitude vibration. When vibrating (with the slab), the ribbon generates magnetic flux, which depends on (and reflects) the slab’s dynamics. This changes when loads accumulate on its surface. The flux induces voltage in a contactless manner in a low-cost pick-up coil suspended above the ribbon. Voltage monitoring allows for evaluation of the vibrating slab’s real-time dynamics and, consequently, the detection of load-induced changes. This work innovates by introducing a low-cost passive circuit for real-time voltage processing, thus achieving an accurate representation of the low-frequency dynamics of the magnetic flux. Furthermore, it introduces an algorithm, which statistically detects load-induced changes using the voltage’s low-frequency power characteristics. Both additions enable load detection at relatively low frequencies, thus addressing a principal issue of passive contactless sensing setups. Extensive testing at different occasions demonstrates promising load detection performance under various conditions, especially given its cost-efficient hardware and operation.

Vibration doi: 10.3390/vibration8040063

Authors: Eckardt Johanning Alice Turcot

This systematic review examined the health risk assessment methods of studies of whole-body vibration exposure from occupational vehicles or machines utilizing the International Standard ISO 2631-1 (1997) and/or the European Machine Directive 2002/44. This review found inconsistent reporting of measurement parameters in studies on whole-body vibration (WBV) exposure. Although many authors treat the ISO 2631-1 HGCZ as a medical health standard with defined threshold levels, the epidemiological evidence for these limits is unclear. Similarly, the EU Directive offers more comprehensive risk management guidance, but the numeric limits are equal without supporting scientific evidence. Both guidelines likely represent the prevailing societal and interdisciplinary consensus at the time. Authors note discrepancies between international and national standards and adverse WBV exposure outcomes are reported below given boundaries. Future publications should report all relevant parameters from ISO 2631-1 and clearly state study limitations, exercising caution when applying ISO 2631-1 HGCZ in health and safety assessments and considering different susceptibility of diverse populations. We advise reducing WBV exposure to the lowest technically feasible limits wherever possible and applying the precautionary principle with attention to individual differences, instead of depending solely on numeric limits.

Vibration doi: 10.3390/vibration8040062

Authors: Javad Sadeghi Alireza Toloukian Sogand Mehravar

The application of dynamic vibration absorbers (DVAs) is a countermeasure to suppress vibrations induced by railway traffic. A key advantage of the DVA application is that it does not require any changes to the path of vibration propagation or the receiver of vibration. A review of the literature reveals the necessity of deriving the optimum properties of DVA to mitigate railway vibrations. To this end, the optimum DVA properties were investigated through the development of a two-dimensional finite element model of the track-tunnel-soil system. The model was validated using the results of a field test. A parametric study was made to obtain the optimum properties of DVA for different soils surrounding the tunnel. The results of the model analysis indicate that the DVA has better vibration reduction for metro tunnels built in soft soils as compared to those surrounded by medium and stiff soils. Also, the results disclose that the DVA reduces vibration radiated on the ground surface when the DVA natural frequency is tuned to a low frequency. Using the results of the parametric study, graphs are suggested to select the optimum properties of the DVA as a function of the soil around the tunnel.

Vibration doi: 10.3390/vibration8040061

Authors: Kenton Hummel Jay Hix Edna Cárdenas

An accelerometer mounting technique has large implications on the frequency range and accuracy of the measurement, with stiffness and the mass relative to the monitored structure as the primary concerns. The International Organization for Standardization (ISO) gives an extensive list in 5348:2021, detailing mounting methods, and provides recommendations for testing mounts that are not specifically defined. In the nuclear industry on the laboratory scale, there is a need for vibration measurements for predictive maintenance and process monitoring that are nondestructive and capable of working in high-temperature environments. Commercial adhesive products with easy application and removal were tested as nondestructive methods, while quick mounts to a commonly used aluminum frame were tested as nondestructive and have potential applicability in high-temperature environments. The sinusoidal excitation method was used, measuring frequencies from 50 Hz to 10 kHz in one-third octave band intervals, utilizing three accelerometers and comparing the results to those obtained with the stud-mounting method. Using the lowest ±3 dB threshold across each accelerometer, foam dots and poster strips were not successful, and foam tapes were accurate up to 2000 Hz, hose clamps and zip ties up to 800 Hz, and a custom 3D printed mount up to 1000 Hz. Knowing the limitations of each mounting technique allows for accurate measurements within the appropriate range.

Vibration doi: 10.3390/vibration8040060

Authors: Andrii Velychkovych Vasyl Mykhailiuk Andriy Andrusyak

Vibration loads during deep drilling are one of the main causes of reduced service life of drilling tools and emergency failure of downhole motors. This work investigates the adaptive operation of an original elastic element based on an open cylindrical shell used as part of a drilling shock absorber. The vibration protection device contains an adjustable radial clearance between the load-bearing shell and the rigid housing, which provides the effect of structural nonlinearity. This allows effective combination of two operating modes of the drilling shock absorber: normal mode, when the clearance does not close and the elastic element operates with increased compliance; and emergency mode, when the clearance closes and gradual load redistribution and increase in device stiffness occur. A nonconservative problem concerning the contact interaction of an elastic filler with a coaxially installed shaft and an open shell is formulated, and as the load increases, contact between the shell and the housing, installed with a radial clearance, is taken into account. Numerical finite element modeling is performed considering dry friction in contact pairs. The distributions of radial displacements, contact stresses, and equivalent stresses are examined, and deformation diagrams are presented for two loading modes. The influence of different cycle asymmetry coefficients on the formation of hysteresis loops and energy dissipation is analyzed. It is shown that with increasing load, clearance closure begins from local sectors and gradually covers almost the entire outer surface of the shell. This results in deconcentration of contact pressure between the shell and housing and reduction of peak concentrations of equivalent stresses in the open shell. The results confirm the effectiveness of the adaptive approach to designing shell shock absorbers capable of reliably withstanding emergency overloads, which is important for deep drilling where the exact range of external impacts is difficult to predict.

Vibration doi: 10.3390/vibration8040059

Authors: Yubo Lyu Yu Guo Jiangbo Li Haipeng Wang

This study proposes a novel framework to enhance inner race fault features in servo motor bearings by acquiring rotary encoder-derived instantaneous angular speed (IAS) signals, which are obtained from a servo motor encoder without requiring additional external sensors. However, such signals are often obscured by strong periodic interferences from motor pole-pair and shaft rotation order components. To address this issue, three key improvements are introduced within the cyclic blind deconvolution (CYCBD) framework: (1) a comb-notch filtering strategy based on rotation domain synchronous averaging (RDA) to suppress dominant periodic interferences; (2) an adaptive fault order estimation method using the autocorrelation of the squared envelope spectrum (SES) for robust localization of the true fault modulation order; and (3) an improved envelope harmonic product (IEHP), based on the geometric mean of harmonics, which optimizes the deconvolution filter length. These combined enhancements enable the proposed improved CYCBD (ICYCBD) method to accurately extract weak fault-induced cyclic impulses under complex interference conditions. Experimental validation on a test rig demonstrates the effectiveness of the approach in enhancing and extracting the fault-related features associated with the inner race defect.

Vibration doi: 10.3390/vibration8040058

Authors: Amir Mohamad Kamalirad Reza Fotouhi

This research aims to actively suppress vibrations at the end-effector of a flexible manipulator. When configured in a locked state, the system behaves as a two-link manipulator subjected to disturbances on the first link. To analyze its behavior, Finite Element Analysis (FEA) is employed to extract the natural frequencies (eigenvalues) and corresponding mode shapes (eigenvectors) of a two-link, two-joint flexible manipulator (2L2JM). The obtained eigenvectors are transformed into uncoupled state-space equations using balanced realization and the Match-DC-Gain model reduction algorithm. An H-infinity controller is then designed and applied to both the full-order and reduced-order models of the manipulator. The objective of this study is to validate an analytical framework through FEA, demonstrating its applicability to complex manipulators with multiple joints and flexible links. Given that the full state-space representation typically results in high-dimensional matrices, model reduction enables effective vibration control with a minimal number of states. The derivation of the 2L2JM state space, its model reduction, and a subsequent control strategy have not been previously addressed in this manner. Simulation results showcasing vibration suppression of a cantilever beam are presented and benchmarked against two alternative modeling approaches.

Vibration doi: 10.3390/vibration8040057

Authors: Lin Sun Xudong Li Xiaopei Liu

For the first time, the influence of random parametric errors (RPEs) on the nonlinear dynamic behaviors of a laminated composite cantilever beam (LCCB) is studied. A nonlinear dynamic model for the LCCB is first established based on Hamilton’s principle. In a numerical simulation, four different cases are presented to analyze the dynamic behavior of the studied LCCB. This study reveals that varying RPE levels cause significant changes in the dynamic response of the LCCB. The results indicate that RPE not only induces a transition from a periodic to a chaotic behavior but may also alter the maximum amplitude of chaotic vibrations, providing a critical theoretical basis for incorporating uncertainty factors in engineering design.

Vibration doi: 10.3390/vibration8040056

Authors: Limin Ma Zhangpeng Li Shengrong Yang Jinqing Wang

This paper presents a systematic review of vibration sensors and their application in industrial-monitoring systems, aiming to provide a comprehensive reference for both academic research and practical applications in this field. Through the classification of measured parameters and sensing principles, this work endeavors to establish a structured understanding of vibration sensor’s working mechanism and deliver an in-depth analysis of their recent research achievements. By integrating practical cases from typical domains, this manuscript comprehensively demonstrates the practical value and application potential of vibration sensors in equipment-monitoring systems, illustrating how these sensors are utilized to detect mechanical failures and enhance the performance and safety of industrial systems, such as wind turbine, tunnel boring machine, and aerospace engine. Looking forward, with the rapid advancement of the Internet of Things (IoT) and artificial intelligence (AI) technologies, vibration sensors are anticipated to evolve towards multifunctionalization, miniaturization and intelligentization, thereby forming a comprehensive monitoring network that improves overall efficiency and reliability of the mechanical systems.

Vibration doi: 10.3390/vibration8030055

Authors: Claudia Marin-Artieda

Wire-rope isolators (WRIs) are widely used in vibration and seismic protection due to their multidirectional flexibility and amplitude-dependent hysteretic damping. However, their complex nonlinear behavior, especially under inclined and combined-mode loading, poses challenges for predictive modeling. This study presents a simplified finite-element modeling framework using constant-property Timoshenko beam elements with tuned Rayleigh damping to simulate WRI behavior across various configurations. Benchmark validation against analytical ring deformation confirmed the model’s ability to capture geometric nonlinearities. The framework was extended to five WRI types, with effective cross-sectional properties calibrated against vendor-supplied quasi-static data. Dynamic simulations under sinusoidal excitation demonstrated strong agreement with experimental force-displacement loops in pure modes and showed moderate accuracy (within 29%) in inclined configurations. System-level validation using a rocking-control platform with four inclined WRIs showed that the model reliably predicts global stiffness and energy dissipation under base accelerations. While the model does not capture localized nonlinearities such as pinched hysteresis due to interstrand friction, it offers a computationally efficient tool for engineering design. The proposed method enables rapid evaluation of WRI performance in complex scenarios, supporting broader integration into performance-based seismic mitigation strategies.

Vibration doi: 10.3390/vibration8030054

Authors: Marcel Ilie

The reduction in carbon footprint and scarcity of energy resources have increased the demand for renewable and sustainable energy resources, and thus, significant efforts have been concentrated on harnessing renewable and sustainable energy resources. The oscillating water column (OWC) wave energy converter has proven to be the most promising approach for harnessing wave energy. The OWC offers the benefits of a long operating time span and low maintenance, as air serves as the driving fluid. The hydrodynamic efficiency of OWC depends on the wave motion and its interaction with the OWC structure. Therefore, the present research concerns the impact of the incident wave signature on the OWC’s efficiency voltage output, and it is carried out experimentally using a laboratory-scale wave tank. Four different waves, of different amplitudes and frequencies, and their impact on the OWC voltage output are experimentally investigated. This study shows that the four waves exhibit different characteristics, such as crests and troughs of different slopes and amplitudes. However, although the wave crests exhibit relatively similar amplitudes, the wave troughs exhibit significantly different characteristics. This study also reveals that the OWC voltage output exhibits a nonlinear behavior due to the nonlinear nature of the incident waves and compressible air inside the OWC chamber. The maximum voltage output is obtained for a maximum air compressibility factor. However, lower voltage outputs are obtained for both compression and decompression of the air inside the OWC chamber.

Vibration doi: 10.3390/vibration8030053

Authors: Ali Mohammad Mohammadi Atefeh Soleymani Hashem Jahangir Mohsen Khatibinia José Viriato Araújo dos Santos Hernâni Miguel Lopes

This paper presents a novel method for damage identification in aluminum beams using the contourlet transform. Four aluminum beams were used in the study: one was undamaged, while the other three had different damage scenarios. The damage included middle and side slots with depth-to-thickness ratios of 7% and 28%. Damage is identified using the proposed index of contourlet transform of the modal rotations and modal curvatures of the beams for the free-free condition. The beam’s first three modal rotations are directly measured with digital shearography, and the corresponding modal curvatures are obtained through their numerical differentiation. The results indicated that to detect the exact locations and identify damage severities using the proposed damage indices, instead of modal rotations, the modal curvatures should be introduced as the input. Moreover, they revealed that the proposed damage indices need modal data of the undamaged state as a baseline to identify smaller damage. In addition, comparing the proposed contourlet-based damage indices with previously suggested wavelet-based damage detection methods revealed that, although the wavelet-based damage index is more sensitive to damage severity, it also exhibits higher noise levels in undamaged locations. The Tukey windowing process was introduced to address the boundary effect problem.

Vibration doi: 10.3390/vibration8030052

Authors: Mahardika Rizki Pynasti Chang-Yong Song

Ship stiffened panels, typically flat plates reinforced with various types of stiffeners, form a substantial part of a ship’s structure and are susceptible to resonance, especially in areas such as the after peak structure, engine room, and accommodation compartments. These vibrations are primarily excited by main engine and propeller forces. Conventional methods such as finite element analysis (FEA) and plate theory are widely used to estimate vibration frequencies, but they are time-consuming and computationally intensive when applied to numerous stiffened panels. This study proposes a machine learning approach using an artificial neural network (ANN) algorithm to efficiently predict the vibration frequencies of ship stiffened panels. A crude oil tanker is chosen as the case study, and FEA is conducted to generate the vibration frequency and mass data for panels across critical regions. The input layer features for the ANN include panel area, thickness, number and area of stiffeners, fluid density, number of fluid contact sides, and overall structural stiffness. The ANN model predicts two outputs: the fundamental vibration frequency and the mass of the panel structure. To evaluate the model performance, hyperparameters such as the number of hidden neurons are optimized. The results indicate that the ANN achieves accurate predictions while significantly reducing the time and resources required compared with conventional methods. This approach offers a promising tool for accelerating the local vibration analysis process in ship structural design.

Vibration doi: 10.3390/vibration8030051

Authors: Lianhua Wang Guowen Yao Xuanbo He

The half-through arch bridge short suspender is more prone to damage due to its high linear stiffness and special force characteristics. To analyze the vehicle-induced vibration characteristics of the short suspender during service, a half-through arch bridge finite element model and a three-axis vehicle model were established to realize the coupled vibration of the suspender axle under bridge deck unevenness excitation. The suspender was simulated using LINK element and BEAM element and separated along its axial and radial directions, and its tension-bending response characteristics was studied. The study found that the short suspender’s amplitude and frequency are higher than those of the long suspender as vehicle critical duration increases. Influenced by the tensile bending effect, the vibration, cross-section equivalent force amplitude, and impact coefficient at the anchorage end are larger than those at the center, and the lower anchorage end’s cross-section peak stress is biased towards the direction of the side column. The internal force of the short suspender is consistent with the deformation trend; its internal force coincides with the deformation trend; and its axial alternating load is generated by the axial relative deformation between the arch rib and the bridge deck, while the bending alternating load originates from the rotational deformation of the short suspender.

Vibration doi: 10.3390/vibration8030050

Authors: Yan Zhang Qi Li Haodong Sun Kaiming Sun Yuanjing Mou Jie Wan

This study addresses the performance optimization of piezoelectric cantilever beam energy harvesters by proposing a design method based on surface arrayed groove modulation. Through systematic investigation of the effects of single grooves (upper surface, lower surface, and double-sided grooves) and arrayed grooves on the power generation performance of piezoelectric cantilever beams, the coupling mechanism of stiffness modulation, Local resonance instability phenomenon, and energy conversion in groove design is revealed. The results show that while single grooves can improve the output voltage by altering the neutral axis position, groove widths exceeding 20 mm induce Local resonance instability phenomenon, leading to energy dissipation. In contrast, arrayed grooves effectively suppress Local resonance instability phenomenon by uniformly distributing the grooves, significantly enhancing energy conversion efficiency. The optimized arrayed groove configuration (groove width: 4 mm, depth: 1 mm, number: 7) achieves a peak voltage of 549.525 mV, representing a 17.3% improvement over the ungrooved structure, without inducing narrow-bandwidth effects. Additionally, this design exhibits excellent process compatibility and can be fabricated using conventional machining methods, reducing costs by 30–45% compared to additive manufacturing. This study provides important optimization directions and technical references for the design of piezoelectric cantilever beam energy harvesters.

Vibration doi: 10.3390/vibration8030049

Authors: Wenxu Zhang Chaoyong Ma Gehao Feng Yanping Zhu Kun Zhang Yonggang Xu

The Fourier decomposition technique has notable advantages in filtering vibration acceleration signals and enhances the feasibility of frequency-domain mode decomposition. To improve the accuracy of mode extraction, this paper proposed a novel Fourier decomposition technique based on spectral clustering. The methodology comprises three key steps. First, spectral clustering is performed using feature vectors derived from the spectrum envelope, specifically the frequency and amplitude of its maximum value, along with the average amplitude of local spectral peaks. Subsequently, the spectrum is adaptively segmented based on clustering feedback to determine spectral segmentation boundaries. Followed by this, a filter bank is constructed via Fourier decomposition for signal reconstruction. Finally, a harmonic correlation index is computed for all decomposed components to identify fault-sensitive modes exhibiting the highest diagnostic relevance. These selected modes are subsequently subjected to demodulation for fault diagnosis. The effectiveness of the proposed method is validated through both simulated signals and experimental datasets, demonstrating its improved ability to capture critical fault information.

Vibration doi: 10.3390/vibration8030048

Authors: Minoru Sasaki Mizuki Takeda Joseph Muguro Waweru Njeri

This study explores trajectory control in flexible manipulators using neural-network-based forward and inverse modeling. Unlike traditional approaches that enhance precision by increasing structural rigidity—often at the cost of added weight and energy consumption—this work focuses on lightweight flexible manipulators, which are more suitable for aerospace and other weight-sensitive applications but introduce control complexities due to elastic deformations. To address these challenges, neural-network-based models are proposed for a two-link, three-degree-of-freedom (3-DOF) flexible manipulator. Simulation and experimental results show that incorporating system delay compensation into the training data significantly improves tracking accuracy. Nonetheless, difficulties remain in achieving smooth trajectory generation. The findings highlight the potential of neural networks in adaptive control and point to future opportunities for refining input–output modeling to better align theoretical developments with practical implementation.

Vibration doi: 10.3390/vibration8030047

Authors: Alexandre Castanheira-Pinto Luís Godinho Pedro Alves Costa Aires Colaço

Train-induced vibrations negatively impact residents in nearby buildings and are increasingly recognized as a public health concern. To address this issue, both effective mitigation measures and simplified design procedures are essential. This study investigates the mitigation pattern induced by an array of stiff inclusions employing a modal dispersive analysis. However, applying this type of analysis to a half-space medium presents challenges. To overcome this limitation, a wave-scattering methodology is proposed. This approach enables the computation of the mitigation pattern in a specific direction and at a particular location. It also highlights the conditioning energy content, thereby identifying the key frequency target for attenuation.

Vibration doi: 10.3390/vibration8030046

Authors: Jesús M. Barraza-Contreras Manuel R. Piña-Monárrez María M. Hernández-Ramos Secundino Ramos-Lozano

Communication via optical fiber is increasingly being used in harsh applications where environmental vibration is present. This study involves a Weibull reliability analysis focused on the performance of fiber optic connectors when they are subjected to mechanical random vibration stress to simulate real-world operating conditions, and the insertion loss (IL) degradation is measurable. By analyzing the testing times and stress levels, the Weibull shape (β) and scale (η) parameters are estimated directly from the maximal and minimal principal IL stresses (σ1, σ2), enabling the prediction of the connector’s reliability with efficiency. The sample size n is derived from the desired reliability (R(t)), and the GR-326 mechanical vibration test (2.306 Grms for six hours) is performed on optical SC angled physical contact (PC) polish fiber endface connectors that are monitored during testing to evaluate the IL transient change in the optical transmission. The method is verified by an experiment performed with σ1=0.3960 and σ2=0.1910 where the IL measurements are captured with an Agilent N7745A source-detector optical equipment, and the Weibull statistical results provide a connector’s reliability R(t) = 0.8474, with a characteristic value of η = 0.2750 dB and β = 3. Finally, the connector’s reliability is as worthy of attention as the telecommunication sign conditions.

Vibration doi: 10.3390/vibration8030045

Authors: Chein-Shan Liu Chia-Cheng Tsai Chih-Wen Chang

The development of simple and yet accurate formulations of frequency–amplitude relationships for non-conservative nonlinear oscillators is an important issue. The present paper is concerned with integral-type frequency–amplitude formulas in the dimensionless time domain and time domain to accurately determine vibrational frequencies of nonlinear oscillators. The novel formulation is a balance of kinetic energy and the work during motion of the nonlinear oscillator within one period; its generalized formulation permits a weight function to appear in the integral formula. The exact values of frequencies can be obtained when exact solutions are inserted into the formulas. In general, the exact solution is not available; hence, low-order periodic functions as trial solutions are inserted into the formulas to obtain approximate values of true frequencies. For conservative nonlinear oscillators, a powerful technique is developed in terms of a weighted integral formula in the spatial domain, which is directly derived from the governing ordinary differential equation (ODE) multiplied by a weight function, and integrating the resulting equation after inserting a general trial ODE to acquire accurate frequency. The free parameter is involved in the frequency–amplitude formula, whose optimal value is achieved by minimizing the absolute error to fulfill the periodicity conditions. Several examples involving two typical non-conservative nonlinear oscillators are explored to display the effectiveness and accuracy of the proposed integral-type formulations.

Vibration doi: 10.3390/vibration8030044

Authors: Yihao Wang Wei Li Drazan Kozak

This study proposes an integrated framework that combines nonlinear stochastic vibration analysis with reliability assessment to address the safety issues of cable systems under damage conditions. First of all, a mathematical model of the damaged cable is established by introducing damage parameters, and its static configuration is determined. Using the Pearl River Huangpu Bridge as a case study, the accuracy of the analytical solution for the cable’s sag displacement is validated through the finite difference method (FDM). Furthermore, a quantitative relationship between the damage parameters and structural response under stochastic excitation is developed, and the nonlinear stochastic dynamic equations governing the in-plane and out-of-plane motions of the damaged cable are derived. Subsequently, a Gaussian Radial Basis Function Neural Network (GRBFNN) method is employed to solve for the steady-state probability density function of the system response, enabling a detailed analysis of how various damage parameters affect structural behavior. Finally, the First-Order and Second-Order Reliability Method (FORM/SORM) are used to compute the reliability index and failure probability, which are further validated using Monte Carlo simulation (MCS). Results show that the severity parameter η shows the highest sensitivity in influencing the failure probability among the damage parameters. For the system of the Pearl River Huangpu bridge, an increase in the damage extent δ from 0.1 to 0.4 can reduce the reliability-based service life of by approximately 40% under fixed values of the damage severity and location, and failure risk is highest when the damage is located at the midspan of the cable. This study provides a theoretical framework from the point of stochastic vibration for evaluating the response and associated reliability of mechanical systems; the results can be applied in practice with guidance for the engineering design and avoid potential damages of suspended cables.

Vibration doi: 10.3390/vibration8030043

Authors: Tomasz Więcek Zygmunt L. Warsza

The paper describes the vibration method of measuring the dynamic Young’s modulus for a ferromagnetic steel element. The parameters of vibrations at the resonant frequency induced by an external magnetic field are studied for an unmagnetized and magnetized steel element. A fiber optic reflective sensor is used to study the vibration parameters of this element. The dynamic Young’s modulus is determined from these studies. A theory describing the amplitude of vibrations of the tested sample induced by the interaction of a magnetic field is developed and used. The conclusions resulting from the studies using this method on the experimental stand are discussed and the scope of its further studies are proposed.

Vibration doi: 10.3390/vibration8030042

Authors: Desejo Filipeson Sozinando Bernard Xavier Tchomeni Alfayo Anyika Alugongo

A concentric double-flange butterfly valve (DN-500, PN-10) was analyzed to examine its dynamic behavior and internal fluid flow across multiple opening angles. Finite Element Analysis (FEA) was employed to determine natural frequencies, mode shapes, and effective mass participation factors (EMPFs) for valve positions at 30°, 60°, and 90°. The valve geometry was discretized using a curvature-based mesh with linear elastic isotropic properties for 1023 carbon steel. Lower-order vibration modes produced global deformations primarily along the valve disk, while higher-order modes showed localized displacement near the shaft–bearing interface, indicating coupled torsional and translational dynamics. The highest EMPF in the X-direction occurred at 1153.1 Hz with 0.2631 kg, while the Y-direction showed moderate contributions peaking at 0.1239 kg at 392.06 Hz. The Z-direction demonstrated lower influence, with a maximum EMPF of 0.1218 kg. Modes 3 and 4 were critical for potential resonance zones due to significant mass contributions and directional sensitivity. Computational Fluid Dynamics (CFD) simulation analyzed flow behavior, pressure drops, and turbulence under varying valve openings. At a lower opening angle, significant flow separation, recirculation zones, and high turbulence were observed. At 90°, the flow became more streamlined, resulting in a reduction in pressure losses and stabilizing velocity profiles.

Vibration doi: 10.3390/vibration8030041

Authors: Jingshun Zuo Jingchao Guan Wei Zhao Keisuke Minagawa Xilu Zhao

Ensuring the safety of large spherical water storage tanks in seismic environments is critical. Therefore, this study proposed a vibration control device applicable to general spherical water tanks. By utilizing the upper interior space of a spherical tank, a novel tuned mass damper (TMD) system composed of a mass block and four elastic springs was proposed. To enable practical implementation, the vibration control mechanism and tuning principle of the proposed TMD were examined. Subsequently, an experimental setup, including the spherical water tank and the TMD, was developed. Subsequently, shaking experiments were conducted using two types of spherical tanks with different leg stiffness values under various seismic waves and excitation directions. Shaking tests using actual El Centro NS and Taft NW earthquake waves demonstrated vibration reduction effects of 34.87% and 43.38%, respectively. Additional shaking experiments were conducted under challenging conditions, where the natural frequency of the spherical tank was adjusted to align closely with the dominant frequency of the earthquake waves, yielding vibration reduction effects of 18.74% and 22.42%, respectively. To investigate the influence of the excitation direction on the vibration control performance, shaking tests were conducted at 15-degree intervals. These experiments confirmed that an average vibration reduction of more than 15% was achieved, thereby verifying the validity and practicality of the proposed TMD vibration control system for spherical water tanks.

Vibration doi: 10.3390/vibration8030040

Authors: Linn Ye Yiqing Yang Wenshuo Ma Wenjing Wu

Vibration control constitutes a critical consideration in structural design, as excessive oscillations may precipitate fatigue damage, operational instability, and catastrophic failures. Dynamic vibration absorbers (DVAs), serving as passive control devices, demonstrate remarkable efficacy in mitigating structural vibrations across engineering applications. This study systematically investigates the design of DVAs for vibration suppression of a cantilevered plate through integrated theoretical modeling, parameter optimization, structural implementation, and experimental validation. Key methodologies encompass receptance coupling substructure analysis (RCSA) for system dynamics characterization and H∞ optimization for absorber parameter identification. Experimental results reveal 74.2–85.7% vibration amplitude reduction in target mode, validating the proposed design framework. Challenges pertaining to boundary condition uncertainties and manufacturing tolerances are critically discussed, providing insights for practical implementations.

Vibration doi: 10.3390/vibration8030039

Authors: Elie Al Shami Xu Wang

This paper introduces a novel approach by employing a Monte Carlo simulation to investigate the impact of various design parameters on the performance of two-body wave energy converters. The study uses a simplified analytical model that eliminates the need for complex simulations such as boundary elements or computational fluid dynamics methods. Instead, this model offers an efficient means of predicting and calculating converter performance output. Rigorous validation has been conducted through ANSYS AQWA simulations, affirming the accuracy of the proposed analytical model. The parametric investigation reveals new insights into design optimization. These findings serve as a valuable guide for optimizing the design of two-body point absorbers based on specific performance requirements and prevailing sea state conditions. The results show that in the early design stages, device dimensions and hydrodynamics affect performance more than the PTO’s stiffness and damping. Furthermore, for lower frequencies, adjustments to the buoy’s height emerge as a favorable strategy, whereas augmenting the buoy radius proves more advantageous for enhancing performance at higher frequencies.

Vibration doi: 10.3390/vibration8030038

Authors: Milad Asadi Farhad S. Samani Antonio Zippo Moslem Molaie

Spiral bevel gears are used in a wide range of industries, such as automotive and aerospace, to transfer power between intersecting axes. However, a certain level of vibration is always present in the systems, primarily due to the complex dynamic forces generated during the meshing of the gear teeth affected by the tooth profile. To address these challenges, this research developed a comprehensive dynamic model with eight degrees of freedom, capturing both translational and rotational movements of the system’s components. The study focused on evaluating the effects of two different tooth profile modifications, namely topology and flank modifications, on the vibration characteristics of the system. The system comprised a spiral bevel gear pair with mesh stiffness in forward rotation. The results highlighted that optimizing the tooth profile and minimizing tooth surface deviation significantly reduce vibration amplitudes and improve dynamic stability. These findings not only enhance the performance and lifespan of spiral bevel gears but also provide a robust foundation for the design and optimization of advanced gear systems in industrial applications, ensuring higher efficiency and reliability. In this paper, it was observed that some modifications led to a 68% reduction in vibration levels. Additionally, three modifications helped improve the vibrational behavior of the system, preventing chaotic behavior, which can lead to system failure, and transforming the system’s behavior into periodic motion.