Euler-Bernoulli Beam Research Articles

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Influence of Elastic Properties and Nodes on the Flexural Behaviour of Bamboo Culms

This study investigates the influence of elastic properties and nodes on the flexural behaviour of bamboo culms by comparing different characterization techniques and theoretical approaches. The most representative parts of the bamboo culm were selected using microscopic images of bamboo cross-sections. These were sliced from the bottom, middle, and top parts of a single culm and were analyzed with Digital Image Processing. Four-point bending tests were conducted on twelve culms of Phyllostachys aurea, subdivided into groups of untreated (UN) and heat-treated (HT) samples. The axial modulus of elasticity (Eb) and the shear modulus (G) were determined experimentally using four different mathematical models: (i) a global deflection model using the Euler-Bernoulli beam theory according to the ISO Standard; (ii) a global deflection model using the Timoshenko beam theory; (iii) a global deflection model based on the Timoshenko beam theory but accounting for the presence of nodes; and finally, (iv) a local model using extensometry. The dominant failure modes for UN and HT samples are described and discussed, and were influenced by the moisture content (MC). Approaches (i) and (ii) showed good agreement, giving reliable parameters to assess Eb. The third approach (iii) indicated that the nodes significantly influence the flexural behaviour of the culms. Approach (iv) was appropriate for determining G, but resulted in higher values of Eb, typically not representative of the material.

Dynamics of a compressed Euler–Bernoulli beam on an elastic foundation with a partly prescribed discrete spectrum

Dynamics of a compressed Euler–Bernoulli beam on an elastic foundation with a partly prescribed discrete spectrum

Transient Shallow Water Wave Interactions with a Partially Fragmented Ice Shelf

This work investigates the interaction between water waves and multiple ice shelf fragments in front of a semi-infinite ice sheet. The hydrodynamics are modelled using shallow water wave theory and the ice shelf vibration is modelled using Euler–Bernoulli beam theory. The ensuing multiple scattering problem is solved in the frequency domain using the transfer matrix method. The appropriate conservation of energy identity is derived in order to validate our numerical calculations. The transient scattering problem for incident wave packets is constructed from the frequency domain solutions. By incorporating multiple scattering, this paper extends previous models that have only considered a continuous semi-infinite ice shelf. This paper serves as a fundamental step towards developing a comprehensive model to simulate the breakup of ice shelves.

Modeling of visco-electro-elastic responses of PZT-based functionally graded beam benders

The modeling of the visco-electro-elastic behavior of functionally graded beam benders with PZT constituents is accomplished via a hierarchical framework that is based on the homogenization technique for composite layers and the laminate theory for a composite laminate. The representation of a bulk PZT constituent is based on linear visco-electro-elastic constitutive equations. The resulting bending displacements of PZT-graded bimorph and multimorph are obtained under the assumption of the Euler–Bernoulli beam. The experimental data of the bending displacements versus applied voltage are compared with the predictions for a bimorph and a multimorph, resulting in a good agreement. The responses of a bender to a complete cycle of applied voltage are shown in order to reveal the critical hysteretic actuation due to the presence of a visco-electro-elastic PZT material in a functionally graded piezoelectric beam bender which is made by functionally graded piezoelectric materials.

Vibration suppression of flexible hose for autonomous aerial refueling based on adaptive boundary compound controller

The suppression of vibrations in flexible hoses during autonomous aerial refueling (AAR) is crucial for enhancing mission success rates and safety, advancing AAR technology. In this paper, an adaptive boundary compound control (ABCC) strategy is proposed to address the vibration suppression problem of the flexible hose of a tanker subject to the bow wave effect (BWE) of the receiver, state output constraint, and partial actuator failure. Unlike previous studies, this research takes into account the impact of the receiver's BWE on these vibrations. The flexible hose is modeled as a three-dimensional Euler-Bernoulli beam (TDEBB), utilizing partial differential equations (PDEs) to provide a more accurate description of its dynamic characteristics. Furthermore, the proposed ABCC strategy accurately detects all disturbances, ensures that the system state output remains within specified limits, and maintains stability even in the face of actuator failures. Last but not least, numerical comparisons and simulations demonstrate that the ABCC method significantly suppresses hose vibrations and maintains end displacement within a narrow range.

A Fast Numerical Approach for Investigating Adhesion Strength in Fibrillar Structures: Impact of Buckling and Roughness

This study presents a numerical investigation into the adhesion strength of micro fibrillar structures, incorporating statistical analysis and the effects of excessive pre–load leading to fibril buckling. Fibrils are modeled as soft cylinders using the Euler–Bernoulli beam theory, with buckling conditions described across three distinct states, each affecting the adhesive properties of the fibrils. Iterative simulations analyze how adhesion strength varies with pre–load, roughness, number of fibrils, and the work of adhesion. Roughness is modeled both in fibril heights and in the texture of a rigid counter surface, following a normal distribution with a single variance parameter. Results indicate that roughness and pre–load significantly influence adhesion strength, with excessive pre–load causing substantial buckling and a dramatic reduction in adhesion. This study also finds that adhesion strength decreases exponentially with increasing roughness, in line with theoretical expectations. The findings highlight the importance of buckling and roughness parameters in determining adhesion strength. This study offers valuable insights into the complex adhesive interactions of fibrillar structures, offering a scalable solution for rapid assessment of adhesion in various rough surface and loading scenarios.

Analytical solution for the kinematic response of end-bearing piles subjected to SH waves in a homogeneous layer by an improved beam-foundation model

This paper presents an analytical solution for evaluating kinematic response in piles subjected to shear waves. The proposed model treats the pile as a Timoshenko beam supported by a two parameter Winkler foundation, capturing both transverse and rotational ground reactions. Validation against both analytical and numerical solutions from the literature, particularly 3D finite element models, shows very good agreement. The proposed model generalizes different solutions currently applied in practice, covering various pile slenderness ratios and ground conditions. This approach addresses limitations in existing Euler-Bernoulli beam resting on Winkler or Vlasov-Pasternak foundation models, by incorporating shear distortions in the structure and ground friction at the interface. Shear stresses induced by the rotational springs are found to significantly influence the kinematic factors, as well as bending and, particularly, shear forces. Finally, simplified expressions for kinematic interaction factors and forces are provided for expedited analyses of practical engineering problems.

Modelling Human-Structure Interaction in Pedestrian Bridges Using a Three-Dimensional Biomechanical Approach

Pedestrian bridges, which are essential in urban and rural infrastructures, are vulnerable to vibrations induced by pedestrian traffic owing to their low mass, stiffness, and damping. This paper presents a novel predictive model of Human-Structure Interaction (HSI) that integrates a three-dimensional biomechanical model of the human body, and a pedestrian bridge represented as a simply supported Euler-Bernoulli beam. Using inverse dynamics, the human model accurately captures three-dimensional gait and its interaction with structural vibrations. The results show that this approach provides precise estimates of human gait kinematics and kinetics, as well as the bridge response under pedestrian loads. The incorporation of a three-dimensional human gait model reflects the changes induced by bridge vibrations, providing a robust tool for evaluating and improving the effect of structural vibrations on the properties and gait patterns.

Modeling and Estimation of Continuous Flexible Structure Using Theory of Functional Connections

This paper presents a novel method for modeling and estimating the dynamics of a continuous structure based on a limited number of noisy measurements. The goal is reached using a Kalman filter in synergy with the recently developed mathematical framework known as the Theory of Functional Connections (TFC). The TFC allows deriving a functional expression capable of representing the entire space of the functions that satisfy a given set of linear and, in some cases, nonlinear constraints. The proposed approach exploits the possibilities offered by the TFC to derive an approximated dynamical model for the flexible system using the Lagrangian mechanics. The result is a representation of the structural dynamics using a finite number of states, in contrast to the infinite-dimensional model that would be obtained by application of the traditional continuum mechanics models that are based on sets of partial differential equations. The limited number of states enables the application of the well-known Kalman filter framework to improve the estimation of the displacements and displacement velocities. In addition, the continuous displacement field of the structure can be reconstructed with high fidelity. The theoretical development of the method is presented in relation to the case of an Euler–Bernoulli beam. Finally, the obtained model is used to carry out a simulation campaign aimed at assessing the accuracy, efficiency, and robustness of the proposed method.

Travelling waves in beam-like structures submerged in water

There are many examples in nature of travelling waves used for propulsion purposes, e.g., micro-organisms and sea creatures. Structural travelling waves can be used to induce momentum in a surrounding media creating a net propelling force. Recent research has tried to capture this interaction in engineering devices. Nonetheless, some challenges remain to fully exploit this phenomenon so that travelling-waves propelled devices can be optimally designed. One such challenge is that the interaction between the structure and the surrounding fluid heavily influences the amplitude of the waves and how they travel through the structure. This paper proposes a systematic qualitative and quantitative analysis of travelling waves in a slender cantilever beam submerged in water. The novelty of this work is demonstrated through two key aspects: The application of the Euler-Bernoulli beam equation combined with the Galerkin approximation, enabling a deeper understanding of how travelling waves form at resonant frequencies rather than non-resonant ones; and An analytical approach using a Galerkin approximation to characterise the nonlinear fluid-structure interaction, followed by linearisation for a comprehensive parametric study of the problem. In this investigation, the contributions of the first five vibration modes are considered in relation to the travelling waves observed near the resonant peaks. Experimental tests validate the analytical results and assess the accuracy of the proposed models. The results demonstrate that the model presented effectively characterises the travelling waves, making a suitable tool for the design of travelling-wave propelled devices.

Chebyshev–Ritz and Navier's methods for hygro‐magneto vibration of Euler–Bernoulli nanobeam resting on Winkler–Pasternak elastic foundation

AbstractThis research employs Chebyshev–Ritz method along with Navier's method to investigate the vibration characteristics of a nanobeam subject to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is characterized by a Winkler–Pasternak elastic foundation and follows the nonlocal Euler–Bernoulli beam theory. The governing equation of motion is derived using Hamilton's principle, and non‐dimensional frequency parameters are computed for Simply Supported‐Simply Supported (SS), Clamped‐Clamped (CC), and Clamped‐Free (CF) boundary conditions. The motivation behind this study is to provide a comprehensive and efficient analytical framework for understanding the dynamic behavior of nanobeams in complex environments. By investigating the influence of magnetic and hygroscopic factors on the vibration characteristics of nanobeams, this research aims to offer valuable insights for the design and optimization of nanoscale structures. Employing shifted Chebyshev polynomials as shape functions in Chebyshev–Ritz method offers several advantages in the proposed model. Firstly, these polynomials possess orthogonal properties, which can significantly enhance computational efficiency. The orthogonality of shifted Chebyshev polynomials allow for simpler and more streamlined numerical computations compared to non‐orthogonal basis functions. Additionally, the orthogonality ensures that the resulting system of equations is well‐conditioned, even for higher‐order polynomial approximations. A closed‐form solution for SS boundary condition is obtained through Navier's method. Convergence analysis is performed to validate the accuracy and effectiveness of the proposed model against existing models. The non‐dimensional frequency parameters obtained using both Navier's method and Chebyshev–Ritz method demonstrate strong agreement, further validating the proposed nanobeam model. Additionally, a comprehensive parametric study evaluates the impact of various characteristics, including the small‐scale parameter, Winkler modulus, shear modulus, magnetic parameter, and hygroscopic parameter. The findings contribute to a nuanced understanding of nanobeam vibrations under the influence of a magnetic field and hygroscopic environment, providing valuable insights for the design and optimization of nanoscale structures in practical applications.

Scaling law for a buckled elastic filament in a shear flow.

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equationin the shear plane, found earlier by Liu etal. [Phys. Rev. Fluids 9, 014101 (2024)2469-990X10.1103/PhysRevFluids.9.014101] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ[over ̃]=-ηsin2ϕsin^{2}θ, where η is the elastoviscous number and the filament orientation is determined by the zenith angle θ with respect to the vorticity direction and the azimuthal angle ϕ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise hydromultipole numerical codes.

Theoretical and experimental research on multi-modal vibration control of flexible beams via the multiple-input multiple-output decoupling control strategy

In this study, a multiple-input multiple-output decoupling control (MIMO-DC) strategy is proposed based on the linear active disturbance rejection control (LADRC) algorithm to suppress the multi-modal vibration of flexible beams. Firstly, the dynamic equation of flexible beams is established using the Euler-Bernoulli beam theory. Then, a virtual control vector is introduced to decouple the MIMO system. The effectiveness of the flexible beam model is verified by comparing it with the experimental frequency response. Finally, experimental studies are carried out to analyze the vibration suppression performance of three control forms in suppressing the vibration of flexible beams under first-order modal excitation, second-order modal excitation, multi-modal excitation, variable multi-modal excitation, and multi-modal excitation with random disturbance, respectively. The robustness of the MIMO-DC strategy against parameter perturbations is also studied. The results show that the proposed MIMO-DC strategy not only exhibits excellent control performance for various types of modal excitations but also has strong adaptability to variable multi-modal excitation and strong anti-disturbance ability. Furthermore, the strategy exhibits strong robustness against parameter perturbations.

Dynamic Modeling and Analysis of a Temperature- Dependent Axially Translating Functionally Graded Cantilever Beam

In this paper, the dynamics of an axially translating functionally graded cantilever beam (FGCB) with time-varying length are studied under environmental temperature variations. Firstly, the governing partial differential equation for the FGCB under different temperatures is established based on the Euler–Bernoulli beam theory. Secondly, the Galerkin discretization and the assumed mode method (AMM) are employed to derive the vibration equations for each mode. Then, the coupling effects of the axial translation motion and the bending deformation on the vibration response of the FGCB are analyzed through numerical simulation. The results showed that different initial lengths, velocities and accelerations can influence the dynamic response greatly. Moreover, the increase in temperature will increase the response amplitude and decrease the frequency of FGCB, but the increase in functionally gradient parameter will decrease both the amplitude and the frequency of the FGCB. Finally, the wavelet transform (WT) is utilized to perform a time–frequency analysis. It is found that the time-dependent frequency obtained by using WT is consistent with the first-order static frequency obtained theoretically. The variation of frequency with time can be obtained quickly and accurately by using WT. The results of this research are of great significance for the application of composite axially translating beams in practical engineering.

End jitter suppression using FONFTSMC for rigid-flexible coupling systems of PMSpM based on NDO

A permanent magnet spherical motor (PMSpM) with three degrees of freedom rotation characteristics in the rigid rotor has broad application prospects. But the adding flexible material will inevitably produce end jitter issue in the process of the rigid-flexible coupling system (RFCs) movement. To solve the above problem, a fractional order non-singular fast terminal sliding mode control method with a disturbance observer was proposed for the rigid-flexible coupling system of a permanent magnet spherical motor (RFCs-PMSpM). Firstly, a dynamic model of RFCs-PMSpM was established by using the Hamilton principle and Euler–Bernoulli beam theory. Secondly, the influences on the end jitter were discussed from the perspectives of load mass, material parameters, and driving torque of the flexible shaft. The sensitivity analysis is carried out, and the key factors affecting the jitter are obtained. Then, a fractional order non-singular fast terminal sliding mode controller based on a nonlinear disturbance observer (NDO-FONFTSMC) is proposed. The stability of the closed-loop control system was proved by the Lyapunov method. Finally, the effectiveness of the proposed jitter suppression strategy was validated and compared with existing approaches, which also provides an important reference for the future application of RFCs-PMSpM in high-precision industry.

Dynamic Analysis of a Uniform Microbeam Resting on a Nonlinear Foundation Considering Its Curvature Subjected to a Mechanical Impact and Electromagnetic Actuation.

This study proposes an investigation into the nonlinear vibration of a simply supported, flexible, uniform microbeam associated with its curvature considering the mechanical impact, the electromagnetic actuation, the nonlinear Winkler-Pasternak foundation, and the longitudinal magnetic field. The governing differential equations and the boundary conditions are modeled within the framework of a Euler-Bernoulli beam considering an element of the length of the beam at rest and using the second-order approximation of the deflected beam and the Galerkin-Bubnov procedure. In this work, we present a novel characterization of the microbeam and a novel method to solve the nonlinear vibration of the microactuator. The resulting equation of this complex problem is studied using the Optimal Homotopy Asymptotic Method, employing some auxiliary functions derived from the terms that appear in the equation of motion. An explicit closed-form analytical solution is proposed, proving that our procedure is a powerful tool for solving a nonlinear problem without the presence of small or large parameters. The presence of some convergence-control parameters assures the rapid convergence of the solutions. These parameters are evaluated using some rigorous mathematical procedures. The present approach is very accurate and easy to implement, even for complicated nonlinear problems. The local stability near the primary resonance is studied.

Elastica-plastica theory of Euler-Bernoulli beams subjected to concentrated loads

Elastica theory plays an important role in the study of large displacements of slender rods. However, the majority of the studies are still limited to elastic material only. In this paper, the traditional elastica theory is extended to an elastica-plastica theory.For the Euler-Bernoulli beams subjected to concentrated loads, the Legendre elliptic integral solution to elastic rods and the plastica solution to elastic-perfectly plastic rods are used. The deformations of elastic-perfectly plastic rods are classified into several cases, and the classification criteria are given. Furthermore, general solving methods that are applicable to various loading conditions are proposed. For an initial value problem, the highly efficient and accurate solution is completely analytical, while for a boundary value problem, the solution adopts a combination of the shooting method and the quasi-Newton method. The accuracy of the current elastica-plastica theory is verified by the corresponding two-dimensional FE simulations.

On the positive invertibility of a differential operator on a graph

In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.

Dynamics of a cross beam with a straight crack in a mining linear vibrating screen

Cross beam fracture is one of the common failures of vibrating screens, and crack is the early manifestation of fracture, which is hard to detect. In order to meet the screening requirement of the vibrating screen and improve the service life of the cracked beam, the cracked Euler-Bernoulli beam model is established to investigate the dynamics of the cross beam with a straight crack under different weights of eccentric block, processing capacities, and Rayleigh damping coefficients based on the finite element method in this paper. The local flexibility coefficients are derived from the principles of fracture mechanics and strain release energy and solved by the adaptive five-point Gaussian Legende algorithm. The stiffness matrix of the cracked beam element is calculated through the inverse method of total flexibility. The four order Runge-Kutta algorithm and MATLAB tools are used to solve the dynamic equation of the cracked cross beam. The relationship between the vibration amplitude of the cracked cross beam and the weight of the eccentric block is studied by fitting formulas using the least squares method. The influence of different weights of eccentric block, processing capacities, and Rayleigh damping coefficients on the vibration amplitude and service life of the cracked beam are discussed. The results show that the greater the weight of the eccentric block, the shorter service life of the beam. When the damping is greater, the service life of the cracked beam is longer.