Timoshenko's beam equation as limit of a nonlinear one-dimensional von Kármán system

Abstract

We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko's model may vanish.

Stabilization of Berger–Timoshenko's equation as limit of the uniform stabilization of the von Kármán system of beams and plates

“…Indeed, the analysis in [17,18] allows to get the convergence of solutions in bounded intervals of time. However, the decay properties we have in mind require the analysis of convergence as time goes to infinity as well.…”

“…The connections between the various available models for a given mechanical problem may be often described precisely in mathematical terms by means of the analysis of the underlying singular perturbation problem. At this respect, in addition to the works [17,18] discussed above on the beam models, we also refer to the monograph by Ciarlet [2] in which various plate models are derived as singular limits from the equations of 3 − d elasticity in thin domains, and to [3] and [22] for the asymptotic analysis of beam models.…”

“…In order to make more precise the problem we have in mind let us recall the essentials of [17,18]. In these works the following von Kármán system for the vibrations of a beam occupying the inverval (0, L) was considered subject to various boundary conditions.…”

“…In [17,18] it was proved that, as ε → 0, and for appropriate boundary conditions, the solution of (1.1) is such that w ε converges (in an appropriate topology) towards the solution of the Berger-Timoshenko model for the transversal vibrations of the beam:…”

“…Recently, it was proved that the Berger-Timoshenko beam model may be derived as a singular limit of the von Kármán beam model. This was done in [17,18] in the conservative case and under various boundary conditions. Here the same analysis is developed for the corresponding damped systems.…”