Dynamics of Multibody Systems

In the classical finite-element (FE) formulation for beams, plates, and shells infinitesimal rotations are used as nodal coordinates. As a result, beams, plates, and shells are not considered as isoparametric elements. Rigid body motion of these nonisoparametric elements does not result in zero strains and exact modeling of the rigid body inertia using these elements cannot be obtained. In this chapter, a formulation for the large reference displacement and small deformation analysis of deformable bodies using nonisoparametric finite elements is presented. This formulation, in which infinitesimal rotations are used as nodal coordinates, leads to exact modeling of the rigid body dynamics and results in zero strains under an arbitrary rigid body motion. It is crucial in this formulation that the assumed displacement field of the element can describe an arbitrary rigid body translation. Using this property and an intermediate element coordinate system, a concept similar to the parallel axis theorem used in rigid body dynamics can be applied to obtain an exact modeling of the rigid body inertia for deformable bodies that have complex geometrical shapes. More discussion on the use of the parallel axis theorem in modeling the inertia of rigid bodies with complex geometry is presented in Chapter 8 of this book. It is recommended that the reader reviews the basic materials presented in Chapter 8 in order to recognize that the coordinate systems used to develop the large displacement FE/FFR formulation presented in this chapter are the same as the coordinate systems used to model the complex geometry in the case of rigid body dynamics.