Dynamics of Beams Using a Geometrically Exact Elastic Rod Approach

Abstract

The dynamics of a long slender beam, intrinsically straight, is addressed systematically for 3-D problems using the Cosserat rod theory. The model developed allows for bending, extension/compression and torsion, thus enabling the study of the dynamics of various types of elastic deformations. In this work a linear constitutive relation is used, also, the Bernoulli hypothesis is considered and the shear deformations are neglected. The fundamental problem when using any finite element (FE) formulation is the choice of the displacement functions. When using Cosserat rod theory this problem is handled using approximate solutions of the nonlinear equations of motion (in quasi-static sense). These nonlinear displacement functions are functions of generic nodal displacements and rotations. Based on the Lagrangian approach formed by the kinetic and strain energy expressions, the principle of virtual work is used to derive the nonlinear ordinary differential equations of motion that are solved numerically. As an application, a curved rod, formed by many straight elements is investigated numerically. When using the Cosserat rod approach, that take into account all the geometric nonlinearities in the rod, the higher accuracy of the dynamic responses is achieved by dividing the system into a few elements which is much less than the traditional FE methods, this is the main advantage when using this approach. Overall, the Cosserat model provides an accurate way of modelling long slender beams and simulation times are greatly reduced through this approach.