Dynamics of variable length geometrically exact beams in three-dimensions

Abstract

We investigate the three-dimensional dynamics of a variable-length, geometrically-exact beam. The beam is released / retracted from a guide. Geometrically exact beam theory, like regular beam theory, is phrased in terms of the deformation of a neutral axis and the rotations of the cross-sections normal to the neutral axis. In contrast to beams in linear elasticity, geometrically exact beams allow large rotations of the beam’s cross-section and large deformations of the line of centroids. We derive the necessary governing equations for the system which, in three dimensions, contain non-trivial contributions from three-dimensional rotations coupled to the varying length of the cable. These equations are then solved through a Galerkin finite element method, after first mapping the physical domain to a fixed computational domain. Large rotations are incorporated into the finite element procedure by utilizing the exponential representation of the rotation tensor. Finally, as an important application of our computational formulation, we investigate the three-dimensional dynamics of a spinning, vibrating, variable-length, flexible beam through a sequence of increasingly complex examples.