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.

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.

ENERGY DECAY ESTIMATES FOR AN EULER-BERNOULLI BEAM WITH A TIP MASS

In this paper,we study the energy decay estimates for an Euler-Bernoulli beam with a tip mass,which is clamped at one end and attached a tip mass to the free end.

A PINNED NETWORK OF EULER-BERNOULLI BEAMS UNDER FEEDBACK CONTROLS

In this paper, the authors design boundary feedback controllers at the interior node to stabilize a star-shaped network of Euler-Bernoulli beams. The beams are pinned each other, that is, the displacements of the structure are continuous but the rotations of the beams are not continuous. The weil-posed-ness of the closed loop system is proved by the semigroup theory. The authors show that the system is asymptotically stable if the authors impose a bending moment control on each edge. Finally, the authors derive the exponential stability of the system.

Stabilization of Nonuniform Euler-Bernoulli Beam with Locally Distributed Feedbacks

In this article, we study the stabilization problem of a nonuniform Euler-Bernoulli beam with locally distributed feedbacks. Firstly, using the semi-group theory, we establish the well-posedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.

Generalized stiffness and effective mass coefficients for power-law Euler-Bernoulli beams

We extend the well-known concept and results for lumped parameters used in the spring-like models for linear materials to Hollomon’s power-law materials.We provide the generalized stiffness and effective mass coefficients for the power-law Euler-Bernoulli beams under standard geometric and load conditions.In particular,our mass-spring lumped parameter models reduce to the classical models when Hollomon’s law reduces to Hooke’s law.Since there are no known solutions to the dynamic power-law beam equations,solutions to our mass lumped models are compared to the low-order Galerkin approximations in the case of cantilever beams with circular and rectangular cross-sections.

基于蒙特卡洛法的Euler-Bernoulli梁基频和振型求解方法

将Rayleigh法和蒙特卡洛法相结合,在Euler-Bernoulli梁理论假设下求解了均匀梁、变截面梁和附带集中质量的变截面梁自由振动问题.对原本连续的梁结构模型进行离散化处理,利用蒙特卡洛法给出梁结构的假设振型.将假设得到的梁结构振型函数代入Rayleigh法,多次计算过程中,将历次基频所得值与计算所得最小值进行比较,根据其相对误差判断是否满足收敛条件,进而求得基频及对应的振型.结果表明,不同计算模型中基频最大误差不超过10%,能够满足工程需求,且精度和时间的控制参数调整灵活,使用者可根据自身需要自行调节.该方法理论简明,适用范围广泛,能够快速准确地求解诸多类型的梁结构基频和振型.

A semi-infinite beam theoretical model on predicting rock slope subsidence induced by underground mining

When the mining goaf is close to the cliff,rock slope subsidence induced by underground mining is significantly affected by its boundary conditions.In this study,an analytical method is proposed by considering the key strata as a semi-infinite Euler-Bernoulli beam rested on a Winkler foundation with a local subsidence area.The analytical solutions of deflection are derived by analyzing the boundary and continuity conditions of the cliff.Then,the analytical solutions are verified by the results from experimental tests,FEM and InSAR,respectively.After that,the influence of changing parameters on deflections is studied with sensitivity analysis.The results show that the distance between goaf and cliff significantly affects the deflection of semi-infinite beam.The response of semi-infinite beam is obviously determined by the length of goaf and the bending stiffness of beam.The comparisons between semi-infinite beam and infinite beam illustrate the ascendancy of the improved model in such problems.