AN EXISTENCE THEOREM OF POSITIVE SOLUTIONS FOR ELASTIC BEAM EQUATION WITH BOTH FIXED END-POINTS

By using the degree theory on cone an existence theorem of positive solution for a class of fourth-order two-point BVP's is obtained. This class of BVP's usually describes the deformation of the elastic beam with both fixed end-points.

Positive solutions of nonlinear elastic beam equations with a fixed end and a movable end

The existence of n positive solutions is studied for a class of fourth-order elastic beam equations where one end is fixed and other end is movable. Here, n is an arbitrary natural number. Our results show that the class of equations may have n positive solutions provided the “heights” of the nonlinear term are appropriate on some bounded sets.

Existence and multiplicity of positive solutions for an elastic beam equation

This paper investigates the boundary value problem for elastic beam equation of the formu″″（t） q（t）f（t, u（t）,u′（t）,u″（t）,u′″（t））, 0〈t〈1,with the boundary conditionsu=（0）=u′（1）=u″（0）=u′″（1）=0.The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. By using Leray-Schauder nonlinear alternate, Leray-Schauder fixed point theorem and a fixed point theorem due to Avery and Peterson, we establish some results on the existence and multiplicity of positive solutions to the boundary value problem. Our results extend and improve some recent work in the literature.

SOME MULTIPLICITY RESULTS FOR AN ELASTIC BEAM EQUATION AT RESONANCE

This paper deals with multiplicity results for nonlinear elastic equations of the typewheregi[0,1] X R R satisfies Caratheodory condition L2[0,1]. The solvability of this problem has been studied by several authors, but there isn't any multiplicity result until now to the author's knowledge. By combining the Lyapunov-Schmidt procedure with the technique of connected set, we establish several multiplicity results under suitable condition.

On Reducibility of Beam Equation with Quasi-periodic Forcing Potential

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation utt ＋ △^2u ＋ αu ＋ εφ（t）（u ＋ u^3） = 0 , α 〉 0 in the dimension one is considered, where u（t,x） and φ（t） are analytic quasi-periodic functions in t, and e is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

Sound field prediction of ultrasonic lithotripsy in water with spheroidal beam equations

With converged shock wave, extracorporeal shock wave lithotripsy（ESWL） has become a preferable way to crush human calculi because of its advantages of efficiency and non-intrusion. Nonlinear spheroidal beam equations（SBE） are employed to illustrate the acoustic wave propagation for transducers with a wide aperture angle. To predict the acoustic field distribution precisely, boundary conditions are obtained for the SBE model of the monochromatic wave when the source is located on the focus of an ESWL transducer. Numerical results of the monochromatic wave propagation in water are analyzed and the influences of half-angle, fundamental frequency, and initial pressure are investigated. According to our results, with optimization of these factors, the pressure focal gain of ESWL can be enhanced and the effectiveness of treatment can be improved.

Quasi-periodic Solutions of the General Nonlinear Beam Equations

In this paper, one-dimensional （1D） nonlinear beam equations of the form utt - uxx ＋ uxxxx ＋ mu = f （u） with Dirichlet boundary conditions are considered, where the nonlinearity f is an analytic, odd function and f（u） = O（u3）. It is proved that for all m ∈ （0, M＊] R （M＊ is a fixed large number）, but a set of small Lebesgue measure, the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.

On Uniform Decay of Solutions for Extensible Beam Equation with Strong Damping

This paper investigates the existence and uniform decay of global solutions to the initial and boundary value problem with clamped boundary conditions for a nonlinear beam equation with a strong damping.

Stochastic Nonlinear Beam Equations with Lévy Jump

In this paper, we study stochastic nonlinear beam equations with Levy jump, and use Lyapunov functions to prove existence of global mild solutions and asymptotic stability of the zero solution.

Existence of Positive Solutions for Eigenvalue Problems of Fourth-order Elastic Beam Equations

In this paper, we investigate the positive solutions of fourth-order elastic beam equations with both end-points simply supported. By using the approximation theorem of completely continuous operators and the global bifurcation techniques, we obtain the existence of positive solutions of elastic beam equations under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, when the nonlinear term is non-singular or singular, and allowed to change sign.

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.

TRAVELING WAVE SOLUTIONS TO BEAM EQUATION WITH FAST-INCREASING NONLINEAR RESTORING FORCES

On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vibrating beam supported by cables,which are treated as a spring with a one\|sided restoring force.The existence of a traveling wave solution to the above piece\|wise linear equation has been proved by solving the equation explicitly (McKenna & Walter in 1990).Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u\++-1+g(u) (Chen & McKenna,1997).However,the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u-1 for u >1.Since an interesting “multiton” behavior,that is ,two traveling waves appear to emerge intact after interacting nonlinearly with each other,has been observed in numerical experiments for a fast\|increasing nonlinearity f(u)=e u-1 -1 ,it hints that the conclusion of the existence of a traveling wave solution with fast\|increasing nonlinearities shall be valid as well.\;In this paper,the restoring force function of the form f(u)=u·h(u)-1 is considered.It is shown that a traveling wave solution exists when h(u)≥1 for u≥1 (with other assumptions which will be detailed in the paper),and hence allows f to grow faster than u-1 .It is shown that a solution can be obtained as a saddle point in a variational formulation.It is also easy to construct such fast\|increasing f(u) 's for more numerical tests.

A Random Attractor Family of the High Order Beam Equations with White Noise

In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.

Attractor for the Extensible Beam Equation with Nonlocal Weak Damping on Time–Dependent Space

In this article,we consider the long-time behavior of extensible beams with nonlocal weak damping:ε(t)u_(tt)+Δ^(2)u-m(‖▽u‖^(2))Δu+‖u_(t)‖^(p_(u_(t)))+f(u)=h,whereε(t)is a decreasing function vanishing at infinity.Within the theory of process on time-dependent spaces,we investigate the existence of the time-dependent attractor by using the Condition(C_(t))method and more detailed estimates.The results obtained essentially improve and complete some previous works.

Global Attractors of Strong Solutions for the Beam Equation of Memory Type

The model with linear memory arise in the case of a generalized Kirchhoff viscoelastic bar, where a bending-moment relation with memory was considered. In this paper, the exponential decay is proved if the memory kernal satisfies the condition of the exponential decay. Furthermore, we show that the existence of strong global attractor by verifying the condition （C） introduced in [3].

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt+△2u+σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.

ON THE NONLINEARTIMOSHENKO-KIRCHHOFF BEAM EQUATION

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D’Alembert equation. Later on, WoinowskyKrieger (Nash & Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends.Here a new equation for the small transverse vibrations of a simply supported beam is proposed. Such equation takes into account Kirchhoff’s correction, as well as the correction for rotary inertia of the cross section Of the beam and the influence of shearing strains, already present in the Timoshenko beam equation (of the Mindlin-Timoshenko equation for the plate).The model is inspired by a remark of Rayleigh, and by a joint paper with Panizzi & Paoli. It looks more complicated than the one proposed by Sapir & Reiss, but as a matter of fact it is easier to study if a suitable change of variables is performed.The author proves the local well-posedness of the initial-boundary value problem in Sobolev spaces of order ≥2.5. The technique is abstract, i.e. the equation is rewritten as a fourth order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for the plate).

BOUNDARY FEEDBACK CONTROL OF ELASTIC BEAM EQUATION WITH STRUCTURAL DAMPING AND STABILITY

In this paper, we consider the partial differential equation of an elastic beam with structuraldamping by boundary feedback control. First, we prove this closed system is well--posed; then weestablish tbe exponential stability for this elastic system by using a theorem whichbelongs to F. L.Huang; finally, we discuss the distribution and multiplicity of the spectrum of this system. Theseresults are very important and useful in practical applications.

On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations

This paper discusses the mathematical modeling for the mechanics of solid using the distribution theory of Schwartz to the beam bending differential Equations. This problem is solved by the use of generalized functions, among which is the well known Dirac delta function. The governing differential Equation is Euler-Bernoulli beams with jump discontinuities on displacements and rotations. Also, the governing differential Equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness are obtained in the space of generalized functions. The operator of one of the governing differential Equations changes so that for both Equations the Dirac Delta function and its first distributional derivative appear in the new force terms as we present the same in a Euler-Bernoulli beam. Examples are provided to illustrate the abstract theory. This research is useful to Mechanical Engineering, Ocean Engineering, Civil Engineering, and Aerospace Engineering.

GENERAL DYNAMIC EQUATION AND DYNAMICAL CHARACTERISTICS OF VISCOELASTIC TIMOSHENKO BEAMS

In this paper, a governing differential equation of viscoelastic Timoshenko beam including both extension and shear viscosity is developed in the time domain by direct method. To measure the complex moduli and three parameters of standard linear solid, the forced vibration technique of beam is successfully used for PCL and PMMA specimens. The dynamical characteristics of viscoelastic Timoshenko beams, especially the damping properties, are derived from a considerable number of numerical computations. The analyses show that the viscosity of materials has great influence on dynamical characteristics of structures, especially on damping, and the standard linear solid model is the better one for describing the dynamic behavior of high viscous materials.