Asymptotics and stabilization for dynamic models of nonlinear beams

Abstract

We prove that the von Kármán model for vibrating beams can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth-order dispersive operator is added. We also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k, when suitable damping terms are added. As k → ∞, one deduces the uniform exponential decay of the energy of the von Kármán model.

Exponential stabilization of the full von Kármán beam by a thermal effect and a frictional damping

“…Most of these devices are internal and/or boundary dampers. For instance, two frictional dampings, one acting on the longitudinal component and one acting on the transversal component, have been used by Perla Menzala et al [18] and Araruna et al [1] to stabilize the system exponentially. There is also an "extra" strong damping in the transversal component when the rotational inertia of the beam is taken into account.…”

Well-Posedness and Exponential Stability of the Von Kármán Beam With Infinite Memory

“…A substantial range of literature use this type of model, where they addressing the problems of existence, uniqueness and asymptotic behavior in time when some damping effects are considered , (See Refs. [4,17,25] and the references therein for more information).…”