Exact Modeling of the Spatial Rigid Body Inertia Using the Finite Element Method

In the classical finite element literature beams and plates are not considered as isoparametric elements since infinitesimal rotations are used as nodal coordinates. As a consequence, exact modeling of an arbitrary rigid body displacement cannot be obtained, and rigid body motion does not lead to zero strain. In order to circumvent this problem in flexible multibody simulations, an intermediate element coordinate system, which has an origin rigidly attached to the origin of the deformable body coordinate system and has axes which are parallel to the axes of the element coordinate system in the undeformed configuration was introduced. Using this intermediate element coordinate system and the fact that conventional beam and plate shape functions can describe an arbitrary rigid body translation, an exact modeling of the rigid body inertia can be obtained. The large rigid body translation and rotational displacements can be described using a set of reference coordinates that define the location of the origin and the orientation of the deformable body coordinate system. On the other hand, as demonstrated in this investigation, the incremental finite element formulations do not lead to exact modeling of the spatial rigid body mass moments and products of inertia when the structures move as rigid bodies, and such formulations do not lead to the correct rigid body equations of motion. The correct equations of motion, however, can be obtained if the coordinates are defined in terms of global slopes. Using this new definition of the element coordinates, an absolute nodal coordinate formulation that leads to a constant mass matrix for the element can be developed. Using this formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, beam and plate elements can be treated as isoparametric elements.