Influence of axial loads on the nonplanar vibrations of cantilever beams

Abstract

The three-dimensional motions of cantilever beams have been extensively studied in the past. This structural element can be found in several applications, including MEMS and NEMS. In many applications the beam is subjected to axial loads which can play an important role in the dynamics of very slender beams. In this paper a cantilever inextensible beam subject to a concentrated axial load and a lateral harmonic excitation is investigated. Special attention is given to the effect of axial load on the frequency-amplitude relation, bifurcations and instabilities of the beam, a problem not tackled in the previous literature on this subject. To this aim, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings 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 numerical integration 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 dynamics of the beam in the parametric or external resonance regions. Bifurcations leading to multiple coexisting solutions are observed.