Definition of the Elastic Forces in the Finite Element Formulations

The equivalence of the finite element formulations used in flexible multibody dynamics is the focus of this investigation. This equivalence will be used to address several fundamental issues related to the deformations, flexible body coordinate systems, and the geometric centrifugal stiffening effect. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used in this investigation. The first one is the absolute nodal coordinate formulation, in which beams and plates can be treated as isoparametric elements. This formulation leads to a constant and symmetric mass matrix and highly nonlinear elastic forces. It is demonstrated in this study that different element coordinate systems which are used for the convenience of describing the element deformations lead to similar results as the element size is reduced. In particular, two element frames are used in this study the pinned and the tangent frames. The pinned frame has one of its axes passes through two nodes of the element, while the tangent frame is rigidly attached to one of the ends of the element. Numerical results obtained in this investigation using these two different frames are found to be in a good agreement as the element size decreases. The relationship between the coordinates used in the absolute nodal coordinate formulation and the floating frame of reference formulation is presented. This relationship is used to obtain the highly nonlinear expression of the strain energy used in the absolute nodal coordinate formulation from the simple energy expression used in the floating frame of reference formulation. It is shown in this paper that the source of the nonlinearity is due to the finite rotation of the element.