Euler-Bernoulli Beam Research Articles

Dynamics of the Euler — Bernoully beam with distributed hysteresis properties

In this paper, we present a new mathematical approach to the analysis of a beam with distributed hysteresis properties. These hysteresis characteristics are described by two methods: phenomenological (Bouc — Wen model) and constructive (Prandtl — Ishlinskii model). The equations for beam are developed using the well-known Hamilton method. We investigate the dynamic response of a hysteresis beam under various external loads, including impulse, periodic and seismic loads. The results of numerical simulations show that the hysteresis beam exhibits differently to external influences as compared to the classical Euler-Bernoulli beam. In particular, under the same external loads, the vibration amplitude and energy characteristics of the hysteresis beam are lower than those of the classical one. These findings can be useful for buildings developers in the design of external load resistant buildings and structures

Constraint Realization‐Based Hamel Field Integrator for Geometrically Exact Planar Euler–Bernoulli Beam Dynamics

ABSTRACTIn this article, we first introduce a Hamel field integrator designed for a geometrically exact Euler–Bernoulli beam with infinite‐dimensional holonomic constraints, constructed using a Lagrange multiplier. This method addresses the complexities introduced by constraints, but the additional multiplier introduces a new degree of freedom and hence results in a system with mixed‐type partial differential equations. To address this issue, we further propose a constraint realization method based on perturbation theory for infinite‐dimensional mechanical systems within the framework of Hamel's formalism. This method circumvents the use of additional Lagrange multiplier, significantly reducing the computational complexity of modeling problems. Building on this, we construct a perturbed Hamel field integrator optimized for parallel computing and incorporate artificial viscosity to accelerate constraint convergence. While applicable to three dimensions, our method is demonstrated in a simplified context using planar Euler–Bernoulli beam examples to illustrate the effectiveness of the unified mathematical framework.

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

Aeroelasticity of an aircraft wing with nonlinear energy sink

In this paper, the effectiveness of nonlinear energy sinks on enhancing aeroelastic stability and post-instability response of aircraft wings is investigated. The wing has two degrees of freedom in bending and torsion, and is modelled using an extended Euler-Bernoulli beam theory with hardening nonlinearity. A Nonlinear Energy Sink (NES) absorber is embedded inside the wing distributed along the wing span. The wing attached unsteady aerodynamic loads are simulated using the Wagner's indicial lift model. The structural dynamics of the wing are derived using Extended Hamilton's Principle and it is discretised using Galerkin's method. The NES mass is connected to the wing spar through a linear damper and nonlinear spring with a cubic stiffness nonlinearity. The coupled aeroelastic equations are then transformed to state space. Then, integrated numerically to resolve the bending and torsional response of the wing to study the impact of spanwise and chordwise positions of the embedded NES on flutter suppression and instability response enhancement. The results demonstrate that the NES is most efficient and is most sensitive to changes in the stiffness when placed at the wingtip. For a given chordwise location, it is found that there is a range of flow speeds over which the NES is most effective and reducing the chordwise offset lowers the speed of the peak efficiency range and moves it closer to the flutter speed. In addition, increasing the stiffness coefficient of the NES improves the efficiency of the device in the immediate post-flutter region. Two near optimum NES devices are proposed with a mass ratios of 1% (located at the wingtip) and 2.5% (located at 75% span). Both of these improve the flutter speed by 5%, and reduce the post-flutter response by 64.5% and 59.2%, respectively.

Instability analysis of Li-ion battery electrode particles induced by chemomechanical growth using incremental deformation theory

Volumetric swelling due to lithiation into active battery materials is a well-known phenomenon. A chemo-mechanical framework can suitably explain the deformations of the electrode and the generation of diffusion-induced stresses (DIS). Various experimental works have shown that the effects of DIS further cause the active material to lose its structural stability. However, relatively few frameworks exist that can investigate such phenomena in a comprehensive manner. These are mainly based on the Euler–Bernoulli beam theory and the von-Karman plate theory. In the present work we present a generalized three-dimensional instability framework incorporating the theory of incremental deformation. We use this theory on a widely used chemo-mechanical framework. Notably, similar existing instability investigations on negative electrode materials are based only on incompressible material behaviour. As opposed to it, in this work we have considered a compressible material. Also, we account for the spatial and temporal nature of growth, which is a significant departure from the existing instability frameworks. Therefore, this current work brings the investigation of structural instability of electrode material closer to physical reality. In our study, we use the compound matrix method to find the critical lithiation parameter. Then we consider three sets of problems, each with a particular combination of boundary conditions applied on the inner and outer cylindrical surfaces of hollow cylindrical anode particles. These are free-free, fixed-free, and free-fixed, where each combination represents a particular type of interaction between an electrode particle and its surroundings. Most importantly, we find that the free-free and fixed-free conditions show only a pure axial and mixed mode of instability, and do not show the pure circumferential mode. The free-fixed conditions, however, shows all three modes of instability. Additionally, the presence of mode crossover makes the instability pattern of such electrode particles significant.

Long-term effect of soil settlement on lateral dynamic responses of end-bearing friction pile

This paper presents a closed-form solution to evaluate the lateral dynamic responses of pile foundation adjacent to superficial surcharge. The equilibrium equation of soil is analyzed using continuum solutions, and the Euler-Bernoulli beam theory is employed to model the pile. The consolidation settlement of surrounding soil induced by the superficial surcharge is determined by Terzaghi's consolidation theory. The lateral soil resistance is formulated using potential functions, and the dynamic responses of the end-bearing friction pile are subsequently obtained by considering the vertical load and skin friction. Our results are compared with analytical approaches from previous literature to validate the effectiveness of the present solution. The evolutions of lateral dynamic responses of pile foundation are systematically analyzed. It is suggested that the superficial surcharge influences the distribution of skin friction of the pile foundation, thereby altering its lateral dynamic responses. This phenomenon does not manifest immediately but demonstrates significant long-term effects during the consolidation settlement of surrounding soil.

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

Spatial and frequency identification of the dynamic properties of thin plates with the Frequency-Adapted Virtual Fields Method

Vibroacoustic inverse methods use the measured response of a vibrating structure to identify a structural parameter or a dynamic load. Two inverse methods are considered, the Force Analysis Technique (FAT) and the Virtual Fields Method (VFM). The Corrected Force Analysis Technique (CFAT) is a variant of FAT that corrects its singularity. This correction allows the method to be applied in the high-frequency domain, when the number of measurement points per flexural wavelength becomes small. In this study, the proposed novelty is the development of a Frequency-Adapted VFM (FA VFM) to the case of a Love–Kirchhoff plate. Thanks to this method, the VFM can now be applied to identify the equivalent bending stiffness and structural damping of a thin plate when the number of measurement points per wavelength is small. The method has previously been developed for an Euler–Bernoulli beam. An experimental identification of the complex bending stiffness of an locally damped aluminium plate using Laser Doppler Velocimetry (LDV) data and the developed method is performed. The experimental study shows for the first time that the FA VFM can be used to map the equivalent bending stiffness and structural damping as a function of position on a plate and identify these parameters as a function of frequency over a large frequency band. The results of the Frequency-Adapted VFM are compared with those of CFAT and the classical VFM approach. FA VFM results are more accurate than those of classical VFM and similar to those of CFAT.

Natural frequencies of a Timoshenko beam with cracks

A problem of calculating of natural frequencies of a Timoshenko beam weakened by a finite number of transverse open cracks is considered. The problem is solved for both known crack models. In one model every crack is simulated by a single, massless rotational spring. In the other model every crack is simulated by two massless springs (one extensional and another one rotational). An effective method for calculation of the natural frequencies of a vibrating beam with cracks, which was previously successfully applied to Euler-Bernoulli beams, is extended to the case of Timoshenko beam. The developed method makes it possible to significantly reduce the order of the determinant, the zeros of which are natural frequencies. Numerical examples are considered. The results are compared with the known where it is possible.

A Novel Approach of the Viscoelasticity of Axially Functional Graded Bar and Application of Harmonic Vibration Analysis of an Isotropic Beam as Support

The use of smart materials and passive controllers in modern technologies has stimulated the study of vibration in elastic systems with viscoelastic damping. It is also possible to create components with precise material distribution coefficients and distinct properties, such as Functionally Graded Materials. This work investigates the resonant frequency characteristics of a beam supported at its ends by Axially Functionally Graded (AFG) viscoelastic bars using the finite element method. The set of equations governing motion is obtained by assuming Euler–Bernoulli beam theory for the beam and bar theory for the bars using Lagrange’s equations. The material properties of the functionally graded bar is assumed to vary through the length according to the power law distribution. The longitudinal loss factor values are used to define the internal damping coefficient, which is also dependent on the Young’s modulus value varying along the bar. The effects of the length-varying material properties and internal damping of the FG support bars on the force transmission TR and frequency parameters λ are examined in detail. No study has been found in the literature on the vibration of viscoelastic FG bar-supported beams subjected to a harmonic force at the centre point. It is shown that using bars formed with combinations of different materials considering material damping will be useful to keep the vibration level and force transmission at a certain value and control the frequency parameters.

Physics-informed neural networks for localized assessment in Euler-Bernoulli beams

This study employs the Physics-Informed Neural Networks (PINNs) to address an inverse problem, specifically evaluating local changes in material properties within an Euler-Bernoulli beam through numerical simulations. In our numerical model, a viscoelastic material is strategically introduced to a localized zone of the beam, inducing non-constant and complex alterations in the storage modulus. The training process incorporates the partial differential equation into the neural network's loss function, coupled with essential stability conditions for the machine learning model, enhancing the efficiency of the training process. Notably, this methodology excels in characterizing structural behavior influenced by these intricate local changes in material properties, bypassing the need to identify the entire structure and consequently to deal with uncertainties related to boundary conditions. The effectiveness of the approach is validated through numerical simulations, underscoring its potential for applications in efficiently assessing and understanding local material alterations within complex structural systems.

The vibration response of five beam theories under eccentric moving harmonic loads

The shear rotary inertia caused by the shear deformation affects the natural frequencies and the vibration characteristics of bridges. To guarantee the safety of bridges, it is necessary to investigate the shear rotary inertia as a crucial content. Unlike previous studies, the motivation of this paper is to improve the Timoshenko beam model and further explore the influence of the shear rotary inertia on flexural–torsional natural frequencies and flexural–torsional vibration properties of bridges. Based on the Timoshenko beam theory, a modified dynamic equation of the Timoshenko beam is established. Its natural frequencies are obtained, and its analytical solutions under moving harmonic loads are derived using the integral transform method. In order to further understand the shear rotary inertia, the other four beams (namely, the Euler–Bernoulli beam, the shear beam, the Rayleigh beam, and the Timoshenko beam) are introduced to make comparisons with the modified Timoshenko beam. Subsequently, in the parameter analysis, the natural frequencies and vibration responses of five different beams with various lengths are analyzed numerically. It is discovered that the shear deformation and the shear rotary inertia have great effect on the high-mode vertical and bridge flexural–torsional frequencies, and it can improve the accuracy of calculating the natural frequencies of beams by about 6.00%. Furthermore, the shear deformation effect on vertical dynamic responses under harmonic moving loads intensifies with the beam spans increasing. The shear beam theory and the Timoshenko beam theory are appropriate for the vibration analyses of short beams, and the Rayleigh beam theory and the Timoshenko beam theory are suitable for the vibration analyses of long beams, while the modified Timoshenko beam theory, being more generally, is suitable for both beams.

Stability analysis of a vertical cantilever pipe with lumped masses conveying two-phase flow

This study investigates the vibration behavior of a vertical cantilever pipe conveying gas-liquid two-phase flow, focusing on the influence of lumped masses attached to the vertical cantilevered pipe. The governing motion equation based on small deflection Euler–Bernoulli beam theory is solved by using the generalized integral transforms technique. The proposed solution approach was first validated against available numerical and experimental results in the literature. The effects of the mass ratios, number and position of lumped masses on the stability of the pipe are investigated. Numerical results show that the parameters of the lumped masses affect significantly the stability of the pipe conveying two-phase flow, by altering the fluid–structure interaction dynamics and impacting natural frequencies and vibration modes of the pipe. Specifically, as the position of a single lumped mass moves downward from the fixed end to the free end, the critical flow velocity initially increases and subsequently decreases, thereby reducing the stability of pipe. Moreover, increasing the number of lumped masses significantly impacts the critical flow velocity due to the mass ratios and locations. Notably, modal “jumping” phenomena are observed, which demonstrate continuous shifts between equilibrium and non-equilibrium states in the cantilever pipes. These findings are crucial for ensuring the safe operation of pipes with discrete masses across various engineering applications.

Quantized Time-Varying Fault-Tolerant Control for a Three-Dimensional Euler–Bernoulli Beam With Unknown Control Directions

Quantized Time-Varying Fault-Tolerant Control for a Three-Dimensional Euler–Bernoulli Beam With Unknown Control Directions

Exact closed-form solution for buckling and free vibration of pipes conveying fluid with intermediate elastic supports

In this paper, the exact closed-form solution is given to investigate the influence of the intermediate elastic support on the buckling and free vibration of an elastically supported pipe. According to the Euler–Bernoulli beam theory, the mechanical model of the pipe is established. The exact equilibrium configuration is derived using the generalised function method without enforcing continuity conditions. A simple solution to the eigenvalue problem is formulated using the methods of complex mode superposition and Laplace transformation. The comparative study shows the differences in the supercritical vibration characteristics and highlights the limitations of previous studies. Parametric studies are carried out to investigate the influence of elastic support and intermediate support conditions on the equilibrium configuration, critical flow velocity, and natural frequency. The results demonstrate that the proposed closed-form solution can determine the support conditions that lead to the maximum critical flow velocity and natural frequency of a pipe with multiple intermediate supports. The maximum values are required to adjust the support conditions leading to the nodes of higher-order equilibrium configurations and complex modes. Furthermore, the natural frequencies of the pipe conveying supercritical fluid no longer satisfy the monotonicity for the support stiffness, the symmetry for the support position, and the ‘zero-point’ property for the support number.

A modified neural network method for computing the Lyapunov exponent spectrum in the nonlinear analysis of dynamical systems

This study presents a modified neural network method for computing the Lyapunov exponent spectrum in non-linear dynamical systems. Its mathematical description is an introduction. The proposed modified neural network method allows the addition of bias and constant neurons to the neural network topology and the use of different numbers of activation functions to suit different cases. Various algorithms for computing Lyapunov exponents, such as the Benettin method, the Wolf method, the Rosenstein method, the Kantz method, the Synchronisation method, the Sano-Sawada algorithm and the proposed modification of the neural network method, are used for classical problems in nonlinear dynamics. These problems include the generalised Hénon map, the chaotic attractor of the Baier-Klein map, and the vibrations of mechanical systems such as the flexible Bernoulli-Euler beam and the flexible functionally graded porous closed cylindrical shell under alternating load. The comparative analyses presented in this study are aimed at validating the accuracy and effectiveness of the methods, and at identifying the most relevant approaches for different types of systems and classes of problems. The proposed method is demonstrated to be superior to existing methods based on time-series evaluation in terms of sample size and accuracy. Furthermore, it does not require the initial system equations.

A frequency domain analysis method for the dynamic response of a submerged floating tunnel under wave action

To investigate the dynamic response of a submerged floating tunnel (SFT) under the action of waves, a modal superposition method in the frequency domain was established, and the amplitudes of the modals were determined using the generalized wave loads and hydrodynamic coefficients of the structural modals. In this model, the SFT was simplified as a constant-cross-section Euler–Bernoulli beam with multiple anchor cables, and its structural modals were resolved using the finite element method (FEM). The generalized wave loads and generalized hydrodynamic coefficients were obtained using the Fourier expansion strategy of the modal function, by which the hydrodynamic problem can be converted into 2D problems with the modified Helmholtz controlling equation and solved using the matched higher-order boundary element method (HOBEM) and eigenfunction expansion. The effects of the diameter of the anchor cables and length of the SFT on the stiffness of the structure and the dynamic response of an SFT with anchor cables under wave action were investigated. Furthermore, the effects of different added mass calculation methods on the structural response were studied. In addition, the wave exciting forces were calculated using potential flow theory or linearized Morison equations according to the wave scale. Finally, the effects of the incident wave frequencies and incident angles were investigated in detail to reveal the combined resonance mechanism.

Analysis of dynamic modeling and power flow of ABH laminated composite thin beam with elastic boundaries

To investigate the energy distribution characteristics of a laminated structure with an acoustic black hole (ABH), this study introduces the dynamic modeling and power flow analysis of an ABH laminated thin beam (ABH laminated beam) with elastic boundary conditions for the first time. ABH profile is defined by a power function in the dynamic modeling process. Utilizing the Euler-Bernoulli beam theory, the constitutive equation of ABH laminated beams is formulated by using the isogeometric method. Three sets of artificial springs are employed to replicate the elastic boundary conditions, incorporating the potential energy of the springs. Subsequently, the dynamic model of the ABH laminated beam is developed by solving the Lagrange equation concerning the total energy within the uniform region, ABH region, and boundary. Through numerical examples, the convergence and accuracy of the proposed modeling approach are validated by comparing the results with those obtained from COMSOL Multiphysics and experimental data. The analysis of power flow and structural intensity elucidates the energy propagation behavior in ABH laminated beams and the underlying mechanism of the ABH effect. The results demonstrate that the ABH laminated beam displays remarkable absorption and dissipation characteristics, introducing a new perspective for the design and application of ABH structures.