Stability of an $N$-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation

We consider the transmission problem of a Timoshenko's beam composed by N components, each of them being either purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional damping mechanism. Our main result is that the rate of decay depends on the...

Full description

Saved in:
Bibliographic Details
Published inElectronic journal of differential equations Vol. 2018; no. 136; pp. 1 - 18
Main Authors Tita K. Maryati, Jaime E. Munoz Rivera, Amelie Rambaud, Octavio Vera
Format Journal Article
LanguageEnglish
Published Texas State University 01.07.2018
Subjects
beam equation
exponential and polynomial stability
lack of exponential stability
localized dissipation
Timoshenko's model
viscoelaticity
Online AccessGet full text

Cover

More Information
Summary:We consider the transmission problem of a Timoshenko's beam composed by N components, each of them being either purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional damping mechanism. Our main result is that the rate of decay depends on the position of each component. More precisely, we prove that the Timoshenko's model is exponentially stable if and only if all the elastic components are connected with one component with frictional damping. Otherwise, there is no exponential stability, but a polynomial decay of the energy as $1/t^2$. We introduce a new criterion to show the lack of exponential stability, Theorem 1.2. We also consider the semilinear problem.
ISSN:1072-6691