Influence of axial loads on the nonplanar vibrations of cantilever beams

In this paper an inextensible cantilever beam subject to a concentrated axial load and a lateral harmonic excitation is investigated. Special attention is given to the effect of the axial load on the frequency-amplitude relation, bifurcations and instabilities of the beam. To this aim, the nonlinear integro-differential equations describing the flexural-flexural-torsional coupling of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. Due to symmetries of the beam cross section, the beam exhibits a 1:1 internal resonance which has an important role on the nonlinear oscillations and bifurcation scenario. The results show that the axial load influences the stiffness of the beam changing its nonlinear behavior from hardening to softening. A detailed parametric analysis using several tools of nonlinear dynamics unveils the complex dynamic behavior of the beam in the parametric and external resonance regions. Bifurcations leading to multiple coexisting solutions are observed.