Nonlinear Beam Equation Research Articles

Decay Estimates for Fourth Order Wave Equations

may be thought of as a nonlinear beam equation. In this paper we obtain both L-L estimates and space-time integrability estimates on solutions to the linear equation. We also use these estimates to study the local existence and asymptotic behavior of solutions to the nonlinear equation, for nonlinear terms which grow like a certain power of u. The main L-L estimate (Theorem 2.1) states that solutions of the linear equation with initial data (u(0), ut(0)) in W 2,q(Rn) ⊕ Lq(Rn) are bounded in L(R) ⊕ W−2,q(Rn) for 2 ≤ q ≤ 2∗∗ for all time and that their norm in this space decays at the (optimal) rate t n 2q−4 . Here and throughout 2∗∗ = { 2n n−4 for n ≥ 5 ∞ for 1 ≤ n ≤ 4 (1.2)

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.

A Nonlinear Beam Equation Arising in the Theory of Elastic Bodies

We study the global solvability of a nonlinear Cauchy problem, which arises in the theory of oscillations in elastic bodies. We show that the linearized problem defines a contraction semigroup, which is then used to transform the Cauchy problem into an integral equation. Finally, it is shown that the corresponding integral operator has a unique fixed point, which gives rise to a global solution of the original nonlinear problem.

Post-buckling Analysis of Multiply Delaminated Composite Plates

This paper presents an elastic post-buckling analysis of an axially loaded beam-plate with two central across-the-width delaminations located at arbitrary depths. The analysis is based on the nonlinear beam equations, combined with the appropriate kinematic continuity and equilibrium conditions. A perturbation technique is employed, which transforms the nonlinear equations into a sequence of linear equations. An asymptotic solution of the post-buckling behavior of the plate is thus obtained. It is shown that with two delaminations, both the maximum deflection and the internal load of the first buckled (top) subplate increase as the external load increases. Of particular interest is the redistribution of load among subplates, which keeps the increase rate of internal load of the top buckled subplate much less than that of the external load. In other words, the load of the buckled subplate is close to the critical value even though the externally applied load is much larger than the critical load. In addition to the two-delamination configuration, a single delamination case is studied based on the present approach in order to verify the accuracy of the method. Also, a comparison with available finite element results is performed.

Inertial manifolds and stabilization of nonlinear beam equationswith Balakrishnan‐Taylor damping

In this paper we study a hinged, extensible, and elastic nonlinear beam equation with structural damping and Balakrishnan‐Taylor damping with the full exponent 2(n + β) + 1. This strongly nonlinear equation, initially proposed by Balakrishnan and Taylor in 1989, is a very general and useful model for large aerospace structures. In this work, the existence of global solutions and the existence of absorbing sets in the energy space are proved. For this equation, the feature is that the exponential rate of the absorbing property is not a global constant, but which is uniform for the family of trajectories starting from any given bounded set in the state space. Then it is proved that there exists an inertial manifold whose exponentially attracting rate is accordingly non‐uniform. Finally, the spillover problem with respect to the stabilization of this equation is solved by constructing a linear state feedback control involving only finitely many modes. The obtained results are robust in regard to the uncertainty of the structural parameters.

Global attractor and inertial set for the beam equation

This paper contains some results about the asymptotic behaviour of the solutions of an n–dimensional nonlinear beam equation with strong damping. The dynamical system associated to this equation is shown to possess a global attractor and a finite dimensional exponentially attracting set.

Remarks on the existence and decay of the nonlinear beam equation

We will consider a class of nonlinear beam equation and we will prove the existence and decay weak solution

Computation of solutions of nonlinear boundary value problems

We develop an explicit method allowing efficient computation of solutions of nonlinear boundary value problems with nonlinear boundary conditions. We apply our results to the nonlinear beam equation, and to second order problems in the case when the growth of nonlinearities in the first derivative of the solution is not restricted, for which there are very few theoretical results.

Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation

A mathematical model for the transverse deflection of an extensible beam of length L whose ends are held at fixed distance apart is equation ∂ 2 u/∂t 2 +α∂ 4 u/∂x 4 +(β+∫ 0 L uξ 2 (ξ,t)dξ)(−∂ 2 u/∂x 2 )=0, which has been proposed by Woinowsky and Krieger [18], where α is a positive constant, β is a constant, not necessarily positive, and the nonlinear term represents the change in the tension of the beam due to its extensibility

Properties of bounded solutions of linear and nonlinear evolution equations: Homoclinics of a beam equation

The objective of this work is to discuss the existence, bifurcation, and regularity, with respect to time and parameters, of bounded solutions of infinite dimensional equations. The authors present an application of their results to the study of homoclinic solutions of a nonlinear forced beam equation. They use the approach of alternative method and semigroup theory.

On generalized periodic solutions of some nonlinear telegraph and beam equations

On generalized periodic solutions of some nonlinear telegraph and beam equations

Periodic solutions of a nonlinear beam equation

Periodic solutions of a nonlinear beam equation

On periodic solution of a nonlinear beam equation

the existence of an $\omega$-periodic solution of the equation $\frac {\partial^2u}{\partial t^2} + \alpha \frac {\partial^4u} {\partial x^4} + \gamma \frac {\partial^5u}{\partial x^4\partial t} - \tilde{\gamma} \frac {\partial^3u}{\partial x^2\partial t} + \delta \frac {\partial u}{\partial t} - \left[\beta + \aleph\int^n_0{\left(\frac {\partial u}{\partial x}\right)}^2 (\cdot,\xi)d\xi + \sigma \int^n_0 \frac {\partial^2u}{\partial x \partial t} (\cdot,\xi) \frac {\partial u}{\partial x}(\cdot,\xi)d \xi \right] \frac {\partial^2u}{\partial x^2}=f$ sarisfying the boundary conditions $u(t,0)=u(t,\pi)=\frac{\partial^2u}{\partial x^2}\left(t,0\right)=\frac{\partial^2u}{\partial x^2}\left(t,\pi\right)=0$ is proved for every $\omega$-periodic function $f\in C\left(\left[0,\omega\right],L_2\right)$.

Nonlinear noncoercive equations and applications

This paper deals with the periodic solvability of the nonlinear beam equation \beta u_t + u_{tt} + u_{xxxx} - \lambda u + \varphi (u) = f, which depends on non-linear \varphi : \mathbb R \to \mathbb R . This paper continues the subject of the paper by S. Fucik [6]. We present some new methods and results which are not included in [6].

Nontrivial periodic solutions of a nonlinear beam equation

AbstractWe consider the existence of a nontrivial solution of the following equation: where g is a nondecreasing function defined on R1, satisfies g(O) = O, and some other additional conditions.Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]‐[8]: .

Initial-boundary value problems for an extensible beam

In this paper we discuss certain initial-boundary value problems for the nonlinear beam equation

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.

Large Deflections of Curved Cantilever Springs of Trapezoidal Profile

The large deflections of a circular-arc cantilever spring of trapezoidal profile subjected to a concentrated force at the free end in the lateral direction have been studied. A solution of the nonlinear beam equation is obtained in the form of power series. However, the power series used yields poor convergency as the shape of spring approaches the triangular profile and it diverges lastly. An approximate solution is also derived from the nonlinear beam equation after approximated appropriately. In comparison between the results calculated from the series-form and approximate solutions, respectively, it is seen that the approximate one gives practically satisfactory results. Numerical results according to both the series-form and approximate solutions are also presented.

Large Deflections of Curved Cantilever Springs of Trapezoidal Profile

The large deflections of a circular-arc cantilever spring of trapezoidal profile subjected to a concentrated force at the free end in the lateral direction have been studied. A solution of the nonlinear beam equation is obtained in the form of power series. However, the power series used yields poor convergency as the shape of spring approaches the triangular profile and it diverges lastly. An approximate solution is also derived from the nonlinear beam equation after approximated appropriately. In comparison between results calculated from the seriesform and approximate solutions respectively, it has been seen that the approximate one gives practically satisfactory results. Numerical results due to both the series-form and approximate solution are also presented.

On the Nonlinear Differential Equation for Beam Deflection: EI d 2 y / dx 2 = M ( x ) [ 1 + ( dy / dx ) 2 ] 3 ⁄ 2

Abstract A general solution of the nonlinear beam equation is given for all problems in which the moment can be expressed as a function of the independent variable alone.