An Asymptotic Theory for a Weakly Nonlinear Beam Equation with a Quadratic Perturbation

Abstract

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a quadratic nonlinearity will be studied. The initial-boundary value problem can be regarded as a simple model describing free oscillations of flexible structures like suspension bridges. Using a two-timescales perturbation method an approximation for the solution of this initial-boundary problem will be constructed. For a class of initial-boundary value problems the existence and uniqueness of solutions and the asymptotic validity of approximations on a large timescale are shown. It will be shown that for specific values of the beam parameters complicated internal resonances occur.