Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

Pages: 1447 - 1462, Volume 10, Issue 5, September 2011

doi:10.3934/cpaa.2011.10.1447       Abstract        References        Full Text (597.1K)       Related Articles

Emine Kaya - Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042, United States (email)
Eugenio Aulisa - Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042, United States (email)
Akif Ibragimov - Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042, United States (email)
Padmanabhan Seshaiyer - Department of Mathematical Sciences, George Mason University, Fairfax VA, 22030, United States (email)

Abstract: In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different models of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.

Keywords:  Stability analysis, non-linear partial differential equations, Euler-Bernoulli beam
Mathematics Subject Classification:  Primary: 49K20, 70K20; Secondary: 74K10.

Received: May 2009;      Revised: November 2010;      Published: April 2011.

 References