Equivalence of the floating frame of reference approach and finite element formulations

Abstract

The floating frame of reference formulation which is widely used in flexible multibody simulations leads to a highly non-linear inertia matrix. This matrix which exhibits a strong inertia coupling between the reference motion and the elastic deformation can be expressed in terms of a unique set of inertia shape integrals that depend on the assumed displacement field. In this paper, we demonstrate that the floating frame of reference formulation leads to the same dynamic relationship obtained using large deformation finite element formulations. This result clearly demonstrates that the floating frame of reference formulation can also be used in the large deformation analysis of constrained flexible bodies provided that no infinitesimal or finite rotations are used as nodal coordinates; instead, displacements and slopes are used to describe the element configuration. The relationship between the local and the global slopes is defined and used to establish a coordinate transformation which is used to demonstrate the equivalence of the dynamic relationships and the inertia forces obtained using two different finite element formulations. Using this equivalence relationship, a simple and systematic procedure for evaluating all the inertia shape integrals that appear in the floating frame of reference formulation from the constant mass matrix that appears in linear structural dynamics is developed. In order to develop this procedure, the concepts of the local and global shape functions are introduced. The local shape function, which is used in the floating frame of reference formulation, does not have any rigid body modes and is defined using an appropriate set of reference conditions. The global shape function, on the other hand, has a complete set of rigid body modes that can describe an arbitrary rigid body displacement. The analysis presented in this paper demonstrates that the floating frame of reference formulation does not lead to a separation between the rigid body motion and the elastic deformation, and in such a formulation, the reference motion cannot be considered as the rigid body motion of the deformable body.