An Absolute Nodal Coordinate Formulation for the Large Rotation and deformation Analysis of Flexible Bodies

There are three basic finite element formulations which are often used in flexible multibody dynamics. These are the floating frame of reference approach, the incremental method, and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulations in the large reference displacement analysis does not lead to the exact rigid body inertia when the structures rotate as rigid bodies. In this paper, a new and simple procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformations is avoided, but also a simple expression for the inertia forces is obtained. Furthermore, the resulting mass matrix is constant and it.is the same mass matrix that appears in linear structural dynamics. It is demonstrated in this paper, that this coordinate description preserves the exactness of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function of time even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation-of finite rotations. Furthermore, no redundant rigid body modes are used to describe the motion of the element, and as such there is no need for the use of a system of differential and algebraic equations for the finite element. Using the procedure developed in this paper, it can also be shown that the current floating reference formulations can be, with minor modifications, used for the large deformation analysis of constrained mechanical systems. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three-dimensional case is straightforward. Using this new procedure, which represents a departure from the classical finite element approach used in the transient dynamic analysis since not all the coordinates in the new procedure have a physical meaning, beams and plates can be treated as isoparametric elements.