This article systematically investigates the isolation performance of a double-beam system interconnected by a nonlinear X-shaped connector. Based on Hamilton's principle and geometric analysis, a dynamic model of the system is derived. This model is used to explore the sensitivity of vibration suppression performance to key parameters, including the tuning ratio and connector design. The analytical findings are subsequently validated through finite-element simulations, ensuring consistency between theoretical predictions and numerical results. The theoretical findings demonstrate that: (a) compared to linear connectors, the nonlinear X-shaped connector enables the double-beam system more effectively isolate vibration transmission between flexible bodies and perform better under large-amplitude excitations by utilizing the geometric nonlinearity of the connector; (b) their adjustable parameters allow flexible tuning to meet vibration reduction needs in various environments; (c) an unequal-length design of the upper and lower beams improves vibration control; (d) adapting the system's tuning ratio according to excitation frequency yields superior vibration attenuation. This research proposes a potential method for the vibration isolation of continuous systems utilizing bio-inspired nonlinear characteristics.
This study addresses the challenging issue of analytical modeling of forced vibration of rectangular plates in thermal environments, which involves mathematical difficulties in treating complex boundary value problems in higher-order partial differential equations. An effective symplectic superposition method is extended for the present issue, focusing on non-L & eacute;vy-type boundary conditions that were not accurately analyzed by conventional analytical methods. To be specific, an original problem is decomposed into three subproblems, which are solved rigorously through separation of variables followed by symplectic eigen expansion, and the original problem's solution is determined by superposing the subproblems' solutions. Various forced vibration results under different thermal environments and different harmonic load scenarios are presented, showing good agreement with finite element numerical simulation results. Furthermore, the effects of temperature variation, harmonic frequency, simple harmonic load amplitude, and boundary conditions, among others, on the thermal vibration characteristics are explored. The findings delve into the significant impact of thermal environments on the forced vibration performance of rectangular plates, offering a theoretical basis for related structural designs.
Total focusing method (TFM) imaging is widely adopted in nondestructive evaluation (NDE) using ultrasonic phased arrays (UPA) due to its high imaging resolution in detecting and identifying subsurface defects. Plane-wave TFM (PWI-TFM) combines plane-wave transmission with full-aperture reception, which can enhance acoustic energy in the region of interest and improve the signal-to-noise ratio (SNR) of defect echoes. However, the use of a finite steering range and discrete angular steps in PWI transmission often leads to nonuniform acoustic energy coverage, which results in spatially varying gains and elevated sidelobes. To address these limitations, this article proposes a spatio-angular coherence factor minimum variance distortionless response (SACF-MVDR) adaptive weighting method for PWI-TFM imaging. The proposed method combines coherence information from the receive-aperture and steering-angle domains with diagonally loaded MVDR beamforming to improve robustness against noise and clutter. Simulations for near-surface defect imaging in railway wheels indicate that the proposed method provides better background suppression and defect separation than conventional TFM, Hanning-windowed TFM, phase coherence factor (PCF), sign coherence factor (SCF), and MVDR. Among the compared methods, SACF-MVDR achieves the lowest image entropy while maintaining high lateral resolution.
Cascade cylindrical piezoelectric transducers (CCPTs) are widely used in ultrasonic engineering, and accurate vibration analysis is essential for understanding their operating mechanisms. Traditional one-dimensional (1D) vibration theory assumes the transducer's length significantly exceeds its diameter, neglecting radial vibrations. However, in practical applications, radial dimensions influence the overall vibration behavior and must be considered. In this article, the equivalent elastic method is applied to analyze the coupled longitudinal-radial vibration of CCPTs by introducing a mechanical coupling coefficient. An electromechanical equivalent circuit is formulated, from which resonance frequency equations and the effective electromechanical coupling coefficient are derived. Numerical and experimental results demonstrate that the method agrees well with finite element method results and predicts the vibration behavior more accurately than the 1D theory. In addition, the equivalent elastic method reveals the influence of geometric dimensions and material parameters on the coupled vibration behavior. The results indicate that when the longitudinal and radial dimensions approach each other, the mechanical coupling becomes strong and the radial vibration must be considered in vibration analysis.
Abstract This study presents a complete analysis of novel length–mass-insensitive shapers designed for nonzero initial conditions, followed by experimental verification of their effectiveness. Input shaping, a preconditioning technique that modifies the reference input to reduce or eliminate residual vibrations, is applied to overhead cranes modeled as a double-pendulum system. Modal analysis is employed to characterize system performance and governing equations. The proposed shapers are designed to suppress oscillations arising from nonzero initial conditions, including the system's inherent initial energy, which reflects realistic scenarios such as abrupt stops or external disturbances. With an arbitrarily selected maneuvering time, the proposed shapers effectively eliminate residual vibrations and robustly compensate for variations in system masses and cable lengths. Numerical simulations and experimental results confirm the effectiveness of the proposed shapers in eliminating residual vibrations while maintaining feasible input accelerations. Sensitivity analysis highlights the vulnerability of the zero-vibration shaper and the robustness of the zero-vibration and length derivative and zero-vibration and mass-derivative shapers, which effectively mitigate length and mass uncertainties, respectively. Additionally, feasible and infeasible solution regions are identified, emphasizing the importance of proper shaper selection. The results demonstrate that shaped inputs significantly enhance system stability, minimizing oscillations even under parameter variations. Moreover, the findings reveal the influence of system parameters on trolley velocity behavior, including sign alternations, emphasizing the necessity of adaptive shaping strategies. The study highlights the critical role of robust input shaping for real-world crane applications, where variations in payload mass and cable length are inevitable.
Abstract An equality-based weighted residual (E-WR) formulation is proposed to analyze the periodic responses of continuous systems subject to discrete frictional occurrences. A cantilever beam with an attached friction damper is considered to evaluate the ability of friction to mitigate response amplitudes. The system is a common, simple model of a bladed-disk sector with an attached platform wedge or ring damper. Friction, governed by Coulomb’s classical law, is expressed as a nonsmooth equality condition that augments the equations of motion into a mixed displacement-contact force formulation, and periodic solutions are obtained via a Ritz-Galerkin procedure using Fourier basis functions. The formulation employs an exact equality representation of Coulomb’s law for interfaces with mass and operates entirely in the frequency domain, thereby eliminating the need for time-domain calculations of contact forces and avoiding common assumptions like regularization, penalization, or massless interfaces. Coulomb’s law and intricate stick-slip behavior at low-frequency subresonances are captured accurately, as validated by time integration results. The effectiveness of the friction damper at mitigating the beam’s vibration response and the effect of damper mass are assessed. Results confirm that the damper is optimal for an intermediate range of excitation amplitudes, at which response behavior is strongly nonlinear. The effect of damper mass on the response is found to be nonmonotonic and can be significant for a range of damper-to-beam mass ratios. Overall, the approach proves to be compact, robust, computationally efficient, and free of convergence issues up to large numbers of harmonics.
The health condition of rolling bearings is crucial for the safe and stable operation of rotating machinery. However, bearing vibration signals acquired in industrial settings typically exhibit nonstationarity, strong nonlinearity, and extremely weak early fault characteristics, which significantly limit the accuracy and practicality of traditional diagnostic methods. To address these core challenges, this study proposes a novel end-to-end fault diagnosis model named LNFormer. It enhances the modeling of raw vibration signal sequences through an input transposition operation, eliminating the need for additional positional encoding. Furthermore, by removing the Softmax function from the self-attention mechanism, the model's computational complexity is substantially reduced, improving processing efficiency for long, high-sampling-rate vibration signals common in industry and supporting real-time or near-real-time monitoring. An innovative multi-head layer normalization (MHLN) structure is also designed as a nonlinear operator to effectively capture complex fault patterns while suppressing overfitting, thereby enhancing the model's robustness and generalization capability under noisy and varying operational conditions. Extensive experiments on eight public bearing datasets demonstrate that LNFormer achieves superior diagnostic accuracy and F1 score compared to existing advanced methods, along with excellent generalization and computational efficiency. A practical case study further confirms its significant potential for direct industrial application without complex data preprocessing.
Accurate prediction of vibrational dissipation in aerospace structures is essential for the design of these structures, which undergo extreme stresses during launches. These complex structures include a main load-bearing structure and various nonstructural elements that ensure the system's proper functioning. Recent studies have revealed that nonstructural elements in spacecraft can significantly dampen the overall structure. This explains the differences observed between numerical models and experimental results for large amplitude vibrations. Vibrational energy dissipation is often modeled by modal damping, which does not represent the actual physical phenomena involved. Aerospace structures incorporate numerous nonstructural elements such as electrical harnesses, thermal protection, fluid lines, and various equipment. The mass of these elements, often representing 10-30% of the total mass of spacecraft, is significant. However, their dynamic behavior is not accounted for in global models, with only a distributed static mass being considered. In this article, to improve predictive models of aerospace structures, a specific identification method with noncollocated measures is necessary to develop a predictive dynamic model. The objective is to propose an inverse approach to identify the dynamic behavior of nonstructural elements based on a thorough understanding of the main structure and vibrational tests on the coupled structure. Experimental tests on the coupled structure, that is, the main structure with the nonstructural elements attached, are also conducted to establish an inverse identification method for the dynamic behavior of the nonstructural elements.
Abstract Several damping mechanisms contribute to the damping of mechanical structures, with every damping mechanism being interpretable as a relaxation process. While vibration decay in beams and rods is generally associated with exponential envelopes, a power law often dominates decay curves observed in many applications. Here, we derive a linear stress-strain relation that accounts for several relaxation processes in material damping without assuming an a-priori rheological topology. The solution of the wave equation and measurements of vibrations in steel beams and timber rods show the distinct decay behaviour in these structural systems. Flexural vibrations of an impacted cantilever beam decay by a power law. Using variational mode decomposition, vibration decay can be decomposed into several empirical modes, each exhibiting distinct characteristics. We show that persistent vibration energy within a particular mode is an indicator of energy efficiency. This persistence can then be analysed in greater detail through the Hilbert–Huang spectrum, which highlights the mode's time scales and the physical phenomena involved. The methods can be used to identify damping in the time-domain if non-exponential decay is observed, which is of particular significance to determine damping in timber structures, although we found that the mode identification in timber rods is less statistically robust than in the steel beam.
This work examines the multistable response of a nonlinear energy sink (NES) incorporating piecewise linear stiffness. The introduction of piecewise linear stiffness modifies the system's stiffness characteristics, broadens the resonance conditions, allows the existing damping elements to dissipate energy more efficiently, and limits the displacement of the nonlinear energy sink. The governing equations, derived from Newton's second law, are solved using the incremental harmonic balance (IHB) method to compute the steady-state responses. Stability analysis reveals the coexistence of two periodic responses, two quasi-periodic oscillations, and chaotic states within these regions. The energy dissipation efficiency of the NES is evaluated, and the energy distribution across the different response types is analyzed; specifically, within the multistable response region in the frequency interval of 38.260-44.280 rad/s, the optimal response branch achieves nearly 100% maximum energy absorption, and the average energy absorption efficiency reaches 80%. Numerical simulations confirm the IHB results, showing excellent agreement in predicting the complex nonlinear dynamics.
Rolling bearings undergo progressive degradation during service, where localized raceway defects evolve from initial pitting to extended wear, often exhibiting asymmetric defect-edge geometries and shoulder formation. Such geometric evolution plays a critical role in failure development but is inadequately represented in conventional dynamic models based on rectangular or idealized defect assumptions, limiting their ability to explain failure-induced vibration responses observed in practice. This study develops a physics-based dynamic model to investigate the failure mechanisms associated with asymmetric edge-wear evolution of raceway defects. The model explicitly incorporates evolving edge profiles and shoulder geometries through piecewise displacement excitation functions, enabling a mechanistic description of rolling-element motion and transient contact interactions across different defect regions. A direct relationship is thereby established between defect morphology, transient contact forces, and vibration responses. The proposed model is validated using finite element simulations of contact forces and experimental vibration measurements under defective conditions. Results show that neglecting defect-edge evolution leads to systematic overestimation of impact severity in rectangular defect models, whereas edge steepness and shoulder height dominate transient impact intensity and vibration persistence. These findings explain why defects of identical length can produce markedly different vibration amplitudes. By clarifying the role of defect geometry in failure-related dynamics, this work provides a mechanism-oriented interpretation of bearing vibration behavior and offers quantitative parameters for vibration-based fault diagnosis, defect localization, and prognosis, contributing to improved bearing health monitoring and reliability assessment.
Vibration and control problems often lead to solving intricate transcendental univariates. However, finding the entire solution set to such functions is an open problem. Existing methods employ (i) approximations to convert the problem into a simpler, or generally a proxy, function amenable to the entire solution set, and then (ii) refinements to remove the approximation error. However, the former could be hampered by the degree of the nonlinearity of the problem, and the latter could miss solutions if converged to the same solution for distinct approximate ones. In this article, a deterministic algorithm is proposed for finding the entire solution set to any finitely smooth univariate for a given interval. The algorithm, utilizing no proxy functions, sweeps the solutions to the univariate through the application of the proposed Principle of Next Solution and a localized solver, the restrained Newton's method. The next solution is the endpoint of the solutions to the cascade of higher order derivatives of the function. This is useful for parametric studies and eigenvalue problems in vibrations and control, where solutions within particular ranges are of interest. The algorithm also solves the global optimization problem for transcendental univariates. It finds the entire extrema of the problem, among which it extracts the global optimizer. The algorithm is naturally parallelizable, allowing the independent processing of subdivisions of the input interval, thus speeding up the computations. Examples from nonlinear eigenvalue problems to a suite of univariate global optimization test problems are presented to demonstrate the superb performance of the proposed algorithm.
Abstract In this work, as a representative of continuous systems impacting with a body, we consider the problem of a ball bouncing on a beam, focusing our attention on the infinite dimensionality of the system. A modal expansion of the equation of motion is used, and the solution during the contact phase is obtained in closed form. It is shown that the time duration of the impact can be shrunk to zero, i.e. the impact can be considered instantaneous in time. Then, the distribution of the impact contact force in space is addressed. We prove that it cannot be considered as a Dirac delta function, i.e. localized in space, since this leads to an inconsistency, caused by the divergence of the series involved in the problem and entailing an unphysical situation, that has not been previously highlighted (to the best of authors' knowledge). The conclusion is that in continuous bodies the impact cannot be simultaneously istantaneous and localized. This marks a strong difference with the finite dimension (discrete) counterpart, in which an impulsive contact force in both space and time is possible and is commonly used. A solution is proposed to overcome this apparent paradox and a simple impact model is developed and illustrated with some examples.
This article presents a two-dimensional composite phononic crystal with a graphene-inspired twisted bilayer configuration. The structure consists of tungsten-rubber core-shell scatterers embedded in an elastic matrix and arranged in a honeycomb lattice. A minimal unit cell was constructed, and finite-element calculations were performed to investigate the low-frequency bandgap characteristics of the structure. By systematically varying the geometric parameter q, the independent dimensions of the core and coating, and the twist angle theta, the effects of geometric scaling and twisting on the first two low-frequency bandgaps were analyzed. The results show that the low-frequency bandgap characteristics exhibit a pronounced nonmonotonic dependence on q, while independent scaling of the core and coating dictates the bandgap frequency position and width, respectively. Under the representative geometric condition of q = 2.5, twisting significantly modifies the bandgap structure by broadening the low-frequency bandgap range, particularly the second bandgap, and shifting the first bandgap toward lower frequencies. In addition, the twist angle provides an additional geometric degree-of-freedom for tuning both bandgap width and frequency position. Finite-structure wave propagation simulations and frequency-response analysis further confirm that elastic-wave transmission is effectively suppressed within the predicted bandgap ranges, in good agreement with the band structure calculations. These results show that the proposed twisted bilayer model provides an effective strategy for designing low-frequency phononic crystals and regulating low-frequency elastic-wave propagation.
Abstract Accurately predicting critical speeds and shaft dynamic responses to imbalance is essential in rotordynamic analysis. Comprehensive simulations must be performed prior to manufacturing gas turbines or jet engines to avoid costly post-production design modifications. While 3D solid finite element models provide high accuracy, they demand substantial computational resources for both preprocessing and post-processing. Consequently, industries continue to rely on 1D beam models due to their efficiency, despite their inherent limitations in terms of accuracy. This article presents a novel approach for optimizing 1D rotor models using a genetic algorithm (GA). Correction parameters for the mass, transverse moment of inertia, and Young's modulus are introduced into the 1D beam model and optimized using a GA with the objective of minimizing the discrepancies in amplitude, phase angle, and gravity-induced displacement compared with a 3D solid model. Transient analyses are conducted for three cases: (1) a conventional 1D beam model, (2) a 3D solid model, and (3) a GA-optimized 1D beam model. The results demonstrate that the GA-optimized 1D beam model closely replicates the behavior of the 3D solid model, whereas the conventional 1D beam model exhibits significant deviations, particularly near critical speeds. Additionally, the results reveal that variations in disk size substantially affect critical speeds, leading to changes in imbalance amplitude and overall dynamic response.
Maximizing the longitudinal fundamental frequency of rod-like structures is essential for broadening the operational bandwidth of high-frequency transmission systems. This article derives a closed-form analytical solution for the optimal cross-sectional profile of a rod subject to prescribed length and area constraints (A(min )and A(max)). Using a variational formulation based on Rayleigh's principle, the optimal profile is identified as a multisegment configuration comprising an exponential transition bridged by uniform segments at the geometric limits. A key finding is that the maximum fundamental frequency scales linearly with the logarithm of the area ratio (kappa), implying theoretically unbounded frequency enhancement as the area ratio increases without bound. The analysis is further extended to complex boundary conditions, revealing the existence of critical tip-mass thresholds that discretely simplify the optimal topology into two-segment or uniform profiles. To validate engineering applicability, this theory is implemented in the design of an electrodynamic shaker's moving coil. Experimental harmonic response tests confirm that the optimized geometry achieves a 42.4% increase in the first natural frequency compared to traditional designs. This work provides a generalized theoretical framework and a rigorous design methodology for optimizing the dynamic performance of variable cross section components.
Abstract We study the free vibrations of two hanging strings that remain in mutual contact over a time-varying span and separated elsewhere. The contact is enforced by reversible Johnson–Kendall–Roberts (JKR) type adhesion, so the contact length is an unknown, time-dependent quantity. Using the variational approach, we derive the governing equations, boundary conditions, and a transversality condition that determines the contact length. As exact solutions are unavailable, we apply asymptotic analysis to obtain displacement fields and the motion of the contact point. Static equilibria appear above a critical adhesive strength and undergo a saddle-node bifurcation; one branch is unstable due to divergence instability, and the bifurcation point is independent of material, geometric, and adhesive properties. Two families of normal modes (NMs) emerge: one with fixed contact length (single string behavior) and one with contact-point motion. The associated eigenvalue problems (EVPs) are self-adjoint and the computed NMs satisfy orthonormality properties. Impulsive responses are examined based on the modal expansion. We have explored approximate solutions based on Ritz method which incorporates the moving contact, without invoking the transversality condition. This numerical framework validates the asymptotic predictions for equilibria, frequency spectrum, and contact point dynamics. We identify a counterintuitive threshold of the contact length for sustained oscillations and show that beyond these limits the strings separate irreversibly as t ∞ 8.
Abstract The theoretical basis for modelling sound propagation in marine waveguides essentially leads to analysis for the boundary value problem of the Helmholtz equation and the ocean bottom is generally assumed to be an uneven interface composed of different media. This paper presents an analytical solution to the classical acoustic problem of sound propagation by a point source in a waveguide which comprises a cylindrical seamount or a cavity. The velocity potential is constructed for each part of the waveguide as a series of normal modes. Conditions of continuity of acoustic field have led to an infinite system of linear algebraic equations in terms of the unknown coefficients of normal modes. It is shown for the first time that the derived infinite system is quasi-regular and has a unique bounded solution. Asymptotical behavior of unknowns in the system is established with the help of the law of singularity of particle velocity at the edge of the waveguide. The derived asymptotic solution of the unknowns permits the use of the method of improved reduction for the determination of the coefficients of normal modes. Examples of numerical implementation of the proposed theory are presented by varying significant parameters of geophysical waveguides.
Abstract In rotating machinery, large amplitude vibrations can result in increased wear and reduced system life. These vibrations can be a result of response localization, which can result in localized vibrations. It is of interest to explore means to reduce such vibrations in these systems. As a step towards attenuating vibrations, the use of noise to steer the system away from a large amplitude response state is explored in this work. With potential application in turbomachinery in mind, the influence of different noise perturbations is considered. The noise models considered include Gaussian white noise, one based on forcing amplitude modulation, and another based on turbine engine spectral data. An experimental test rig with a rotating cantilever and an arrangement of magnets is considered. This cantilever system, which has softening behavior, has multiple response states. Noise perturbation-induced movement from a large amplitude response state to a low amplitude response state of this rotating cantilever is examined. To explain experimental observations and support them, numerical studies are pursued. The experimental and numerical results for escape times are compared and discussed. The findings from this work are expected to be relevant to noise-induced transitions in other physical systems.
Abstract The rectangular thin plate is one of the common components in engineering structures. Diverse constraints and connections result in various dynamical characteristics of the plate, which in turn have significance on the major structure. For the rectangular thin plate with a specific length constraint rather than the entire edge being restricted, the traditional polynomial method has limitations in dynamical modeling. This paper proposes the power series polynomial to express such rigid physical interval boundary conditions of the thin plate. Based on the Rayleigh–Ritz method, the power series polynomial is applied directly to the structural energy expression in the form of a constraining functional. The unknown coefficients of the polynomial can be obtained simultaneously with the weight coefficients of the plate displacement expression. Comparisons of natural frequencies and modal shapes obtained by the presented method and the traditional orthogonal polynomial for the classical boundary conditions demonstrate the correctness of the power series polynomial. For the plate with a certain length constraint on the boundary, the natural characteristics are obtained by the modal experiment. Results show that the relative errors of natural frequencies are slight enough and the proposed method is valid for the interval constraint on the boundary of the plate. The computational efficiency of the proposed method also has significant advantages. Therefore, the proposed method can provide a theoretical foundation for component dynamical modeling with sophisticated boundary conditions.
