Euler Bernoulli Research Articles

A comprehensive study of elastically constrained beams under compression and tensile forces employing analytical and numerical approaches

This research investigates elastically constrained beams, including Timoshenko, shear, Rayleigh, and Euler–Bernoulli beams, positioned on diverse foundations such as Pasternak, Hetényi, and Filonenko–Borodich, and subjected to compression and tensile forces. The significance of this study lies in providing a critical dynamics response to the behavior of various beam types under varying foundation conditions, elucidating the influences of important factors such as shear modulus, flexural rigidity, foundation stiffness, rotatory effect, and shear deformation effects. The determination of the dynamical behavior of the problem is particularly challenging, dealing simultaneously with four engineering theories of beams positioned at different foundations under elastic constraints. The problem has been analytically solved by extending the separation of variables approach, involving coupled eigenvalues in the dispersion relation and leading to the development of the spatial matrix for slopes and displacements, along with their derivatives. The complexity of the problem necessitates a rigorous procedure for coupling eigenvalues, deviating from the standard approach. Consequently, analytical handling proves challenging and relatively sensitive to variations. Therefore, a numerical scheme using Galerkin finite element method (GFEM) has been obtained, recognized for its accuracy compared to other techniques like Differential Transform Technique and Rayleigh Ritz technique in finite element methods. The GFEM approach employed here offers a relatively simple and convenient solution, demonstrating effective convergence to exact solutions without shear locking. This further opens a pathway for delving into more intricate and challenging problems in beam dynamics within the scope of investigation. The obtained results indicate that beam eigenfrequencies of different models are higher under tensile forces and decrease with reduced compression forces. The study provides valuable insights for ideal design under specific conditions, highlighting the importance of incorporating foundation characteristics, compression, and tensile forces in beam dynamics.

Motion Characteristics of Self-Sensing Piezoelectric Actuator for Yarn Micro-Gripper

In order to solve the problem of low response frequency and poor consistency of conventional yarn grippers in weft accumulators, in this study, a piezoelectric yarn gripper is used instead of conventional yarn grippers and the motion characteristics of its actuator are studied. This gripper uses a bimorph piezoelectric bending actuator with a low-cost, well integrated self-sensing method based on charge measurement. The modeling of the piezoelectric micromanipulator is based on the piezoelectric and Euler–Bernoulli beam equations. The static and dynamic characteristics of the piezoelectric actuator as well as the self-sensing capability were experimentally tested. The experimental results show that the maximum output displacement at the end of the piezoelectric actuator is 834 μm, and the maximum output force is 388 μN at 150 V driving voltage. The stability and consistency of its response are also very good, with a response speed of 24 ms. The self-sensing test of the output force also proved the feasibility of the self-sensing method used, with an error of 0.74%. The piezoelectric yarn gripper studied in this paper is promising for practical clamping applications.

Displacement Reconstruction Based on Physics-Informed DeepONet Regularizing Geometric Differential Equations of Beam or Plate

Physics-informed DeepONet (PI_DeepONet) is utilized for the reconstruction task of structural displacement based on measured strain. For beam and plate structures, the PI_DeepONet is built by regularizing the strain–displacement relation and boundary conditions, referred to as geometric differential equations (GDEs) in this paper, and the training datasets are constructed by modeling strain functions with mean-zero Gaussian random fields. For the GDEs with more than one Neumann boundary condition, an algorithm is proposed to balance the interplay between different loss terms. The algorithm updates the weight of each loss term adaptively using the back-propagated gradient statistics during the training process. The trained network essentially serves as a solution operator of GDEs, which directly maps the strain function to the displacement function. We demonstrate the application of the proposed method in the displacement reconstruction of Euler–Bernoulli beams and Kirchhoff plates, without any paired strain–displacement observations. The PI_DeepONet exhibits remarkable precision in the displacement reconstruction, with the reconstructed results achieving a close proximity, surpassing 99%, to the finite element calculations.

Dynamic Amplification of Railway Bridges under Varying Wagon Pass Frequencies

Train configurations give rise to a primary wagon pass forcing frequency and their multiples. When any one of these frequencies coincides with the natural frequency of vibration of the bridge, a resonant response can occur. This condition can amplify the dynamic response of the bridge, leading to increased levels of displacement, stresses and acceleration. Increased stress levels on critical bridge structural elements increases the rate at which fatigue damage accumulates. Increased bridge acceleration levels can affect passenger comfort, noise levels, and can also compromise train safety. For older bridges the effects of fatigue, and being able to predict the remaining life, has become a primary concern for bridge engineers. Better understanding of the sensitivity of fatigue damage to the characteristics of the passing train will lead to more accurate remaining life predictions and can also help to identify optimal train speeds for a given train–bridge configuration. In this paper, a mathematical model which enables the dynamic response of railway bridges to be assessed for different train configurations is presented. The model is based on the well established closed from solution of the Euler–Bernoulli Beam (EBB) model, for a series of moving loads, using the inverse Laplace–Carson transform. In this work the methodology is adapted to allow different train configurations to be easily implemented into the formulation in a generalised form. A generalised equation, which captures the primary wagon pass frequency for any train configuration, is developed and verified by presenting the results of the bridge response in the frequency domain. The model, and the accuracy of the equation for predicting the primary wagon pass frequency, is verified using independently obtained measured field train–bridge response data. The main emphasis of this work is to enable the practicing engineer, railway operators and bridge asset owners, to easily and efficiently make an initial assessment of dynamic amplification, and the optimal train speeds, for a given bridge and train configuration. This is visually presented in this work using a Campbell diagram, which shows dynamic amplification and compares this with those calculated based on the design code, across a range of train speeds. The diagram is able to identify train speeds at which a resonance response can occur, and the wagon pass frequency, or its multiples, which are causing the increased dynamic amplification. The model is implemented in Matlab and demonstrated by analysing a range of short- to medium-single span simply supported plate girder railway bridges, typically found on the UK railway network, using the standard BS-5400 train configurations. The model does not consider the effects of the train mass and suspension system as this would require a non-closed form numerical solution of the problem which is not practical for the purposes of an initial assessment of the train–bridge interaction problem.

Frequency Design of a Vertical Cantilever Beam Carrying Tip Mass

Previous research has revealed that vertical cantilever beam structures, after considering the effects of gravity, can exhibit a 1:3 internal resonance between two transverse modes. However, whether a similar phenomenon would still occur in the system when a proof body is attached to the free end of cantilever beams, and how to perform parameter design to deliberately induce internal resonance are still open issues. The present study conducts research on the natural frequencies of a vertical cantilever beam with a tip mass. Within the framework of the Euler–Bernoulli beam theory, Hamilton’s principle is utilized to derive the integral-partial differential equation of free vibration of the system. The corresponding frequency equation is then solved using the Chebyshev pseudospectral method, with an algebraic equation of the natural frequency obtained. Moreover, an explicit analytical expression for the fundamental frequency is obtained using parameter fitting, which is then used to reveal the relationship among parameters under the critical state of buckling and evaluate the extent of gravity’s impact on the fundamental frequency. The study finally investigates the ratio of the first two natural frequencies. A parameter design method is proposed, which derives a parametric curve in terms of the fundamental frequency, ensuring a 1:3 ratio between the first two natural frequencies. The free vibration time history at the midpoint of the beam is acquired by directly solving the system’s partial differential equation through the finite element method. Fourier transformation of this time history verifies that the fundamental frequency aligns with a predetermined value, and the ratio between the first two natural frequencies is 1:3, thus confirming the efficacy of the proposed method. It is found that the addition of a tip mass provides more flexibility in adjusting parameters to achieve internal resonance, which facilitates the miniaturization of actual structures.

Thermal effects on nonlinear vibration of nonlocal nanobeam embedded in nonlinear elastic medium

Nanotechnology has an impact on our lives in a many ways, from better medical treatments and more efficient energy sources to stronger and lighter materials and advanced electronics and this article presents the implementation of a perturbation method for the vibration analysis of simply supported and clamped–clamped Euler–Bernoulli nanobeams resting on nonlinear elastic foundations in thermal environment using nonlocal elasticity theory. Hamilton's principle is used to construct the differential equation of motion of a nanobeam in conjunction with appropriate boundary conditions. The equations of motion of the Euler–Bernoulli nanobeam are determined using nonlocal elasticity theory. It is shown how thermal loadings affect the vibration of the Euler–Bernoulli nanobeam. The multiple scale method, which is one of the perturbation method, is used to get an approximated solution for the presented system. The effects of temperature, Winkler, Pasternak and nonlinear foundation parameters on the vibration analysis of simply supported and clamped–clamped nanobeams are determined and results are given in tables and graphs.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

Stability of a Multi-Span Current-Carrying Beam Resting on Periodic Supports and Exposed to an External Magnetic Field

Abstract The stability of a current-carrying beam in an external magnetic field is studied in this paper. The beam rests on periodic supports and is modeled as an Euler–Bernoulli elastic beam. Based on the linearized stability equation and propagator transfer matrix approach for a multi-span finite-length beam, the stability equation and solutions of the several boundary value problems are obtained. While the Floquet–Bloch theory is widely used in the dynamic problem of phononic and photonic structure conditions, its application to the solution of infinite length beam static stability problem is novel. The stability values of the periodic infinite beam model are in very good agreement with one of the finite-length multi-span beams.

Free vibrations of functionally graded porous hanging and standing cantilever beams

The free oscillations of a functionally graded (FG) porous vertical cantilever beam in the frame work of Euler–Bernoulli beam theory is investigated. The beam is subjected to the gravity-load and the properties of the FG material such as the modulus of elasticity and the density are supposed to change through the thickness of the beam according to power-law relations. The equation of motion is derived using Newton’s second law. The Numerical Chebyshev collocation method is utilized to determine the transverse frequencies of the FG porous hanging and standing cantilever Euler–Bernoulli beams. A parametric study is conducted to determine the effects of various factors such as the transverse functionally graded index, the porosity factor, and the elastic and the mass density ratios on the natural frequencies and the mode shapes of the FG porous vertical hanging and standing cantilever thin beams under their self-weight. The accuracy of the proposed numerical method is evaluated through comparisons of the frequencies obtained from the present approach with those available in previous literature. In general, it was observed that the elastic ratio has a softening impact on the frequencies except for the fundamental frequency which remains constant as the elastic ratio increases. Moreover, the porosity parameter and the power-law index may have a softening or hardening impact on the frequencies, and the behavior of these frequencies depends on the values of the elastic and the mass density ratios.

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Linear active disturbance rejection control algorithm for active vibration control of piezo-actuated beams: Theoretical and experimental studies

The active vibration control (AVC) of lightweight thin-walled structures on spacecraft has been a proven solution for vibration suppression. In this study, a linear active disturbance rejection control (LADRC) algorithm is presented for vibration suppression of piezo-actuated beams. Firstly, the governing equation of a piezo-actuated beam is obtained by using the Euler–Bernoulli beam theory and the Lagrange equation. The LADRC algorithm is designed for the piezo-actuated beam based on the governing equation. Then, the accuracy of the piezo-actuated beam model is confirmed by comparison with ANSYS and experiments. Finally, numerical simulations and experiments are carried out on the active vibration suppression of the piezo-actuated beam under the fixed harmonic excitation, variable harmonic excitation, and fixed harmonic excitation with measuring noise, respectively. The results show that the LADRC algorithm has an excellent control effect on structural vibration suppression, strong adaptability to various external excitations, and anti-noise ability.

Nonlinear vibrations of a cantilevered conical beam with axially functionally graded material carrying a longitudinally loaded mass

In this article, the free and forced vibrations of a cantilevered conical beam with axially functionally graded material 1 , carrying a concentrated mass and subjected to a distributed load, are investigated. The beam has a rectangular cross-section with a variable thickness and properties along the length. The governing equation of the system is derived using the Euler–Bernoulli beam theory based on the Hamilton principle. The response of the system is determined under the assumption of the first mode dominance using the method of multiple scales 2 . The validity of the results is confirmed by comparing them with the results of the numerical solution. This investigation breaks new ground by examining the nonlinear effects of curvature and inertia, which contribute to the occurrence of hardening and softening behavior in the frequency response. Additionally, this article provides new insights into the profound impact of material distribution on the system’s frequency response at the initial resonance state. Therefore, the motivation of this study lies in the convergence of FGM, nonlinear behavior, and resonance of a beam.

Adaptive Distributed Boundary Vibration Control of Multiagent Euler–Bernoulli Beams via Cooperative Disturbance Observer Network

Adaptive Distributed Boundary Vibration Control of Multiagent Euler–Bernoulli Beams via Cooperative Disturbance Observer Network

Optimal multiple tuned mass dampers for monopile supported offshore wind turbines using Genetic Algorithm

This study investigates the parametric optimization of multiple-tuned mass dampers (MTMDs) deployed along the length of a tower and monopile structure subjected to stochastic wind and wave forces. The structural components, comprising the tower and monopile, are conceptualized as Euler–Bernoulli beams. The interaction with the soil is modeled through viscoelastic springs distributed along the monopile. The rotor nacelle assembly (RNA) and blades are abstracted into a concentrated mass at the tower’s apex. The MTMDs are characterized as spring–mass–dashpot systems strategically positioned at intervals along the tower and monopile. The investigation involves a random vibration analysis under wind and wave loads, employing the Kaimal spectrum for wind and the JONSWAP spectrum for waves. The study focuses on optimizing the TMD parameters using H2 optimization techniques, with the optimization criteria being the minimization of the standard deviation of the tower’s top displacement response. Additionally, meta-heuristic algorithms like the genetic algorithm (GA) are utilized to refine the determination of these optimal parameters. A key aspect of the research includes a parametric study that reveals variations in optimal parameters as the number of TMDs is altered, maintaining a consistent total mass across the MTMDs.

Nonlinear Dynamic Analysis of FG Fluid Conveying Micropipes with Initial Imperfections

This paper investigates the nonlinear dynamics of FG-FCMPs with initial imperfections. Based on the MCST, the nonlinear equations of motion and the corresponding boundary conditions are established by applying Hamilton’s principle, the Euler–Bernoulli beam theory, and von-Kármán geometric nonlinearity. To describe the initial geometric imperfection of the FG-FCMP, the first-order vibrational mode is employed. Subsequently, in cases of primary parametric resonance, 1:2 subharmonic resonance for the first-order mode, as well as primary resonance for the second-order mode, Galerkin’s method, and the multiple scale method are utilized to analyze the amplitude–frequency responses of the imperfect FG-FCMP. The numerical simulations test the influence of flow velocity, micro-scale parameter, geometrical imperfections, and external loads on the nonlinear characteristics of a coupled system with two DOFs. It is found that, as the increase of flow velocity, micro-scale effect, and external load, the amplitude of the first two modes can be increased. The hardening characteristics are converted into the softening characteristics due to the imperfect effect. Furthermore, numerical results provide a more comprehensive understanding of the nonlinear dynamics of FCMPs for both periodic and chaotic motions.

Event-triggered robust fault-tolerant control of a class of Euler–Bernoulli beam equations via sliding mode control

Event-triggered robust fault-tolerant control of a class of Euler–Bernoulli beam equations via sliding mode control

Time history nonlinear analysis for 2D modelization of an existing building using flexibility and displacement-based formulation

The object of the research is the response of a structure in time between formulations, namely force-based (flexibility-based) and displacement-based. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility based) and displacement based. This research aimed to compare between two formulations namely displacement based and force-based (flexibility-based), in order to show the accuracy and the reliability of the second one. In the case of nonlinear time history response of a residential two-storey reinforced concrete building, which is representative of typical residential building construction in Italy in the 1970’s and 1980’s with the aim of taking advantages of flexibility-based formulation if the comparison proves conclusive. In the course of the research, the former approach is a classical two-node, displacement-based, Euler-Bernoulli frame element with the disadvantage of discretizing the structure thus increasing the number of degrees of freedom and equation, in order to achieve the required precision. The latter is a two-node, force-based, Euler Bernoulli frame element. The main advantage of the second element is that it is «exact» within the relevant frame element theory. This implies that one element per frame member (beam or column) is used in preparing the frame mesh, thus leading to a reduction of the global number of degrees of freedom. Therefore, a reduction in number of equations to solve, resulting in a saving of time with increased precision. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility-based) and displacement-based. As a result of the research, it is shown that with flexibility-based approach no discretization error occurs and all governing equations are satisfied exactly, making a possibility of simpler and faster numerical modeling. In the proposed approach will be of great use in the field of numerical modeling due to its ease and accuracy. This elegant formulation is promising for future work thanks to its advantages.

Dynamic analysis of short-fiber reinforced composite nanobeams based on nonlocal strain gradient theory

This study deals with the dynamic behavior of short-fiber reinforced composite nanobeams. It is assumed that short-fibers are aligned or randomly distributed in the composite nanobeams. Nonlocal strain gradient theory is applied to composite nanobeam mechanics including Euler–Bernoulli and Timoshenko beam models. The transverse vibration of these composite nanobeams is investigated for various boundary conditions. Approximate Ritz method is used for obtaining the natural frequencies of short-fiber reinforced composite nanobeams. In addition to vibration analysis, wave propagation in short-fiber reinforced composite nanobeams is investigated and wave dispersion relations are analytically obtained for both Euler-Bernoulli and Timoshenko beam models. The vibration and wave dispersion results of short-fiber reinforced composite nanobeams are obtained for aligned and randomly distributed cases. The results obtained from this paper showed that there is no significant difference between the aligned and randomly oriented short-fiber composite nanobeams. This provides great convenience to designers where it is not possible to orient the reinforcement material in composites. The present study may be useful for the mechanical analysis and design of micro/nano-electromechanical systems (MEMS/NEMS), nanoprobes, nanosensors, nanoactuators, and atomic force microscopes.

Linear matrix inequality-based vibration control of three-dimensional Euler–Bernoulli beam with external disturbance

In this paper, we address the LMI-based vibration control problem of a three-dimensional Euler–Bernoulli beam subject to external disturbance. First, the dynamics of the system is characterized by several partial differential equations (PDEs). Second, in order to realize the disturbance rejection, we propose the disturbance observers. Based on these, the boundary control protocols are designed via linear matrix inequality (LMI) method for suppressing elastic vibration. A significant feature of LMI is that the control gains can be efficiently solved by this method. Finally, some simulation examples illustrate the effectiveness of the proposed theoretical results.

Static bending analysis of a transversely cracked strip tapered footing on a two-parameter soil using a new beam finite element

In this paper, a new beam Euler–Bernoulli finite element for the transverse static bending analysis of cracked slender strip tapered footings on an elastic two-parameter soil is presented. Standard Hermitian cubic interpolation functions are selected to derive the closed-form expressions of complete stiffness matrix and the load vector. The efficiency of the proposed finite element is verified on an example with several width tapering variations of a simple cracked footing with the results of governing differential equation. Another novelty of this study is improved bending moment functions with included discontinuity conditions at the crack location. These functions now accurately describe the bending moments in the vicinity of the crack of the finite element.