Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes

Abstract

In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.