Polynomial Decay and Computational Numerical Modeling of Timoshenko beam with partial dissipation
DOI:
https://doi.org/10.17268/sel.mat.2018.02.04Keywords:
Differential partial equations, beam, semigroup, polynomial stabilityAbstract
We studied the uniform stabilization of a class of Timoshenko systems with partial dissipation of the beam. Our main result is to prove that the semigroup associated to this model has polynomial decay. Moreover, we prove that the semigroup decays polynomially to zero. The system decays polynomially with rate depending
on the coeficients of the problem. We also show the computational modeling of the system showing the results obtained theoretically.
References
Timoshenko, S. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine. 1921; 41:744-746.
Kim, J.U. and Renardy, Y. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 1987; 25:1417-1429.
Raposo, C.A., Ferreira, J., Santos, M.L., and Castro, N.N.O. Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Letters. 2005; 18:535-541.
Soufyane, A. andWehbe, A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron J. Diff. Equations. 2003; 29:1-14.
Kafini, M. General energy decay in a Timoshenko-type system of thermo-elasticity of type III with a viscoelastic damping. J. Math. Anal. Appl. 2011; 375:523-537.
Messaoudi, S.A., and Said-Houari, B. Energy decay in a Timoshenko-type system of thermo-elasticity of type III. J.Math. Anal. Appl. 2008; 348:298-307.
Messaoudi, S.A. and Mustafa, M.I. On the internal and boundary stabilization of Timoshenko beams. NoDEA Nonlinear Differential Equations Appl. 2008; 15:655-671.
Muñoz Rivera, J.E. and Avila, A.I. Rates of decay to non homogeneous timoshenko model with tip body. J. Diff. Equations. 2015; 258:3468-3490.
Muñoz Rivera, J.E. Estabilizacao de Semigrupos e Aplicacoes. LNCC-UFRJ. Brasil.2009, 1 edicaoo, 120 p.
Acasiete, F. Modelagem Computacional da viga de Timoshenko submetida a cargas pontuais. Dissertacao de Mestrado-LNCC. 2016.
Borichev, A. and Tomilov, Y. Optimal polynomial decay of functions and operator semigroups. Math. Ann. 2009; 347:455-478.
Strikwerda, J.C. Finite difference schemes and partial differential equations. SIAM. Philadelphia, 2004, 435 p.
Strauss, W., and A Vasquez, L. Numerical solution of a nonlinear klein-gordon equation. Journal of Computational Physics. 1978; 28:271-278.
Negreanu, M. and Zuazua, E. Uniform boundary controllability of a discrete 1-d wave equation. System and Control Letters. 2003; 48:261-280.
Published
How to Cite
Issue
Section
License
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.