Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.

A State-Based Peridynamic Formulation for Functionally Graded Euler-Bernoulli Beams

In this study,a new state-based peridynamic formulation is developed for functionally graded Euler-Bernoulli beams.The equation of motion is developed by using Lagrange’s equation and Taylor series.Both axial and transverse displacements are taken into account as degrees of freedom.Four different boundary conditions are considered including pinned support-roller support,pinned support-pinned support,clamped-clamped and clamped-free.Peridynamic results are compared against finite element analysis results for transverse and axial deformations and a very good agreement is observed for all different types of boundary conditions.

A Meshless Method for Retrieving Nonlinear Large External Forces on Euler-Bernoulli Beams

We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data,comprising two-end fixed,cantilevered,clamped-hinged,and simply supported conditions in this study.Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams;however,an effective numerical algorithm to solve these inverse problems is still not available.We cope with the homogeneous boundary conditions,initial data,and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions.The unknown nonlinear large external force can be recuperated via back-substitution of the solution into the nonlinear Euler-Bernoulli beam equation when we acquire the solution by utilizing the boundary shape function scheme and deal with a smallscale linear system to gratify an additional right-side boundary data.For the robustness and accuracy,we reveal that the current schemes are substantiated by comparing the recuperated numerical results of four instances to the exact forces,even though a large level of noise up to 50%is burdened with the overspecified conditions.The current method can be employed in the online real-time computation of unknown force functions in space-time for varied boundary supports of the vibrating nonlinear beam.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

Stability analysis for an Euler-Bernoulli beam under local internal control and boundary observation

An Euler-Bernoulli beam system under the local internal distributed control and boundary point observation is studied. An infinite-dimensional observer for the open-loop system is designed. The closed-loop system that is non- dissipative is obtained by the estimated state feedback. By a detailed spectral analysis, it is shown that there is a set of generalized eigenfunctions, which forms a Riesz basis for the state space. Consequently, both the spectrum-determined growth condition and exponential stability are concluded.

The Response of Viscously Damped Euler-Bernoulli Beam to Uniform Partially Distributed Moving Loads

The paper investigates the response of non-initially stressed Euler-Bernoulli beam to uniform partially distributed moving loads. The governing partial differential equations were analyzed for both moving force and moving mass problem in order to determine the behaviour of the system under consideration. The analytical method in terms of series solution and numerical method were used for the governing equation. The effect of various beam observed that the response amplitude due to the moving force is greater than that due to moving mass. It was also found that the response amplitude of the moving force problem with non-initial stress increase as mass of the mass of the load M increases.

Exact geometry based quasi-conforming analysis for Euler-Bernoulli beam

The problem of quick analysis using exact geometry data was proposed by Hughes et al. and the isogeometric analysis framework was introduced as a solution. In this letter, the exact geometry concept is combined into the quasi-conforming framework and a novel method, i.e., the exact geometry based quasi-conforming analysis is proposed. In present method the geometry is exactly described by non-uniform rational B-spline bases, while the solution space by traditional polynomial bases. Present method combines the merits of both isogeometric analysis and quasi-conforming finite element method. In this letter Euler-Bernoulli beam problem is solved as an example and the results show that the present method is effective and promising.

Inverse Problems for an Euler-Bernoulli Beam: Identification of Bending Rigidity and External Loads

We present a method for identifying the flexural rigidity and external loads acting on a beam using the finite-element method. We used mixed beam elements possessing transverse deflection and the bending moment as the primary degrees of freedom. The first step is to determine the bending moment from the transverse deflection and boundary conditions. The second step is to substitute the bending moment into the final equations with respect to the unknown parameters (flexural rigidity or external load). The final step solves the resulting system of equations. We apply this method to some inverse beam problems and provide an accurate estimation. Several numerical examples are performed and show that present method gives excellent results for identifying bending stiffness and distributed load of beam.

Vibration response of Euler-Bernoulli-damped beam with appendages subjected to a moving mass

This paper addresses the problem of a viscoelastic Euler-Bernoulli beam under the influence of a constant velocity moving mass and different types of appendages.Four types of boundary conditions are considered:pinned-pinned,fixed-pinned,fixed-free(or cantilever),and fixed-fixed.Appendages considered include lumped masses,dampers,and springs.The modal decomposition method is employed to derive the equation of motion of the beam,for which an analytical closed-form expression of the dynamic vibration response is generated.The proposed method enables the study of the effect of a single appendage or a combination of the three types of appendages on the non-dimensional dynamic response of the beam.Numerical examples are presented to illustrate the effects of these appendages and compare them to the reference cases of a beam with no appendages.The results demonstrate the importance of considering these parameters in the design of structures.The proposed method is compared to other techniques in the literature and found to be advantageous due to its direct approach.The method also offers a versatile tool for investigating various configurations,aiding in engineering design and structural analysis for which establishing a precise prediction of beam vibrations is crucial.

A novel solution treatment and aging for powder bed fusion-laser beam Ti-6Al-2Sn-4Zr-6Mo alloy:Microstructural and mechanical characterization

Ti-6Al-4Zr-2Sn-6Mo alloy is one of the most recent titanium alloys processed using powder bed fusion-laser beam(PBF-LB)technology.This alloy has the potential to replace Ti-6Al-4V in automotive and aerospace applications,given its superior mechanical properties,which are approximately 10%higher in terms of ultimate tensile strength(UTS)and yield strength after appropriate heat treatment.In as-built conditions,the alloy is characterized by the presence of soft orthorhombicα″martensite,necessitating a postprocessing heat treatment to decompose this phase and enhance the mechanical properties of the alloy.Usually,PBFed Ti6246 components undergo an annealing process that transforms theα″martensite into anα-βlamellar microstructure.The primary objective of this research was to develop a solution treatment and aging(STA)heat treatment tailored to the unique microstructure produced by the additive manufacturing process to achieve an ultrafine bilamellar microstructure reinforced by precipitation hardening.This study investigated the effects of various solution temperatures in theα-βfield(ranging from 800 to 875℃),cooling media(air and water),and aging time to determine the optimal heat treatment parameters for achieving the desired bilamellar microstructure.For each heat treatment condition,differentα-βmicrostructures were found,varying in terms of theα/βratio and the size of the primaryα-phase lamellae.Particular attention was given to how these factors were influenced by increases in solution temperature and how microhardness correlated with the percentage of the metastableβphase present after quenching.Tensile tests were performed on samples subjected to the most promising heat treatment parameters.A comparison with literature data revealed that the optimized STA treatment enhanced hardness and UTS by13%and 23%,respectively,compared with those of the annealed alloy.Fracture surface analyses were conducted to investigate fracture mechanisms.

Low-profile,low sidelobe array antenna with ultrawide beam coverage for UAV communication

This paper presents a design method to implement an antenna array characterized by ultra-wide beam coverage,low profile,and low Sidelobe Level(SLL)for the application of Unmanned Aerial Vehicle(UAV)air-to-ground communication.The array consists of ten broadside-radiating,ultrawide-beamwidth elements that are cascaded by a central-symmetry series-fed network with tapered currents following Dolph-Chebyshev distribution to provide low SLL.First,an innovative design of end-fire Huygens source antenna that is compatible with metal ground is presented.A low-profile,half-mode Microstrip Patch Antenna(MPA)is utilized to serve as the magnetic dipole and a monopole is utilized to serves as the electric dipole,constructing the compact,end-fire,grounded Huygens source antenna.Then,two opposite-oriented end-fire Huygens source antennas are seamlessly integrated into a single antenna element in the form of monopole-loaded MPA to accomplish the ultrawide,broadside-radiating beam.Particular consideration has been applied into the design of series-fed network as well as antenna element to compensate the adverse coupling effects between elements on the radiation performance.Experiment indicates an ultrawide Half-Power Beamwidth(HPBW)of 161°and a low SLL of-25 dB with a high gain of 12 d Bi under a single-layer configuration.The concurrent ultrawide beamwidth and low SLL make it particularly attractive for applications of UAV air-to-ground communication.

具有边界记忆阻尼的Euler-Bernoulli梁模型的指数稳定性

该文研究了具有混合边界条件的Euler-Bernoulli梁模型,其中仅一端边界条件具有黏弹性记忆阻尼,而内部无其他阻尼影响。通过选择合适的乘子构造Lyapunov辅助函数,并利用能量估计方法,证明了此模型在其记忆核函数满足适当的假设条件时具有指数形式的衰减率。该结论表明,对于混合边界条件欧拉-伯努利梁模型,一个部分边界黏弹性记忆阻尼也足以产生一个整体的耗散机制,并且确保与全局内部摩擦/记忆效应的情况同样具有指数型衰减率。

Plates, Beams and Shells Reinforced by CNTs or GPLs: A Review on Their Structural Behavior and Computational Methods

Since the initial observation of carbon nanotubes(CNTs)and graphene platelets(GPLs)in the 1990 and 2000s,the demand for high-performance structural applications and multifunctional materials has driven significant interest in composite structures reinforced with GPLs and CNTs.Incorporating these nanofillers into matrix materials markedly enhances the mechanical properties of the structures.To further improve efficiency and functionality,func-tionally graded(FG)distributions of CNTs and GPLs have been proposed.This study presents an extensive review of computational approaches developed to predict the global behavior of composite structural components enhanced with CNT and GPL nanofillers.The analysis focuses on key structural elements,such as plate-type configurations,cylindrical and curved shells,and beams,emphasizing the computational techniques utilized to simulate their mechanical behavior.The utilization of three-dimensional elasticity theories and equivalent single-layer(ESL)frameworks,which are widely employed in the modeling and analysis of these composites,is comprehensively discussed.Additionally,the paper examines various mechanical performance aspects,including static,buckling,post-buckling,vibrational,and dynamic responses for the mentioned structures.The unique features of hybrid nanocomposites,combining CNTs and GPLs,are also analyzed.Furthermore,the study delves into the fabrication and processing techniques of these materials,with a particular focus on strategies to mitigate nanofiller agglomeration.The review extends to cover thermal and electrical properties,durability under environmental exposure,fatigue resistance,and vibration-damping characteristics.In conclusion,the paper underscores the necessity for ongoing advancements in computational modeling to facilitate improved design,analysis,and optimization of nanocomposite structures.Future research opportunities in this rapidly advancing domain are also outlined.

Exponential Decay of Laminated Beam with Nonlinear Time-varying Delay and Microtemperature Effect

This research paper addresses a topic of interest to many researchers and engineers due to its effective applications in various industrial areas.It focuses on the thermoelastic laminated beam model with nonlinear structural damping,nonlinear time-varying delay,and microtemperature effects.Our primary goal is to establish the stability of the solution.To achieve this,and under suitable hypotheses,we demonstrate energy decay and construct a Lyapunov functional that leads to our results.

Natural frequencies analysis of a composite beam consisting of Euler-Bernoulli and Timoshenko beam segments alternately

Present investigation is concerned with the free vibration property of a beam with periodically variable cross-sections.For the special geometry characteristic,the beam was modelled as the combination of long equal-length uniform Euler-Bernoulli beam segments and short equal-length uniform Timoshenko beam segments alternately.By using continuity conditions,the hybrid beam unit(ETE-B) consisting of Euler-Bernoulli beam,Timoshenko beam and Euler-Bernoulli beam in sequence was developed.Classical boundary conditions of pinned-pinned,clamped-clamped and clamped-free were considered to obtain the natural frequencies.Numerical examples of the equal-length composite beam with 1,2 and 3 ETE-B units were presented and compared with the equal-length and equal-cross-section Euler-Bernoulli beam,respectively.The work demonstrates that natural frequencies of the composite beam are larger than those of the Euler-Bernoulli beam,which in practice,is the interpretation that the inner-welded plate can strengthen a hollow beam.In this work,comparisons with the finite element calculation were presented to validate the ETE-B model.

Stabilization of an Euler-Bernoulli Beam with a Tip Mass Under the Unknown Boundary External Disturbances

This paper studies the stabilization problem of an Euler-Bernoulli beam with a tip mass,which undergoes unknown but uniform bounded disturbance at tip mass. Here the nonlinear feedback control law is used to cancel the effects of the external disturbances. For the controlled nonlinear system,the authors prove the well-posedness by the maximal monotone operator theory and the variational principle. Further the authors prove that the controlled nonlinear system is exponential stable by constructing a suitable Lyapunov function. Finally, some numerical simulations are given to support these results.

STABILIZATION OF EULER-BERNOULLI BEAM WITH A NONLINEAR LOCALLY DISTRIBUTED FEEDBACK CONTROL

This paper studies the stabilization problem of uniform Euler-Bernoulli beam with a nonlinear locally distributed feedback control. By virtue of nonlinear semigroup theory, energy-perturbed approach and polynomial multiplier skill, the authors show that, corresponding to the different values of the parameters involved in the nonlinear locally distributed feedback control, the energy of the beam under the proposed feedback decays exponentially or in negative power of time t as t →∞.

Stability analysis of the Euler-Bernoulli beam with multi-delay controller

In this paper,we investigate the stabilization of an Euler-Bernoulli beam with time delays in the boundary controller.The boundary velocity feedback law is applied to obtain the closed-loop system.It is shown that this system generates a C_(0-)semigroup of linear operators.Moreover,the stability of the closed-loop system is discussed for different values of the controller constants and time delays via using spectral analysis and a suitable Lyapunov function.