Nonlinear Vibrations of Beams Including Shear Deformations and Rotatory Inertia

Abstract

The free large-amplitude oscillation of a beam with whatever boundary conditions are investigated on the basis of a polar continuum model in which a priori ordering assumptions are avoided. The dynamic configuration is described through three displacement variables which allow to account for the shear deformation and rotatory inertia. By using an assumed mode technique and Hamilton’s principle, two ordinary differential equations with nonlinearities up to order three are obtained. They are solved by Lindstedt-Poincare perturbation method to study the frequency-amplitude relationship of a transverse mode of shear-deformable beam, taking into account the coupling with the longitudinal displacement. Numerical investigation has been performed for simply-supported beams with movable and immovable ends and the influence of shear deformations and rotatory inertia has been evidenced.