Finite Element Incremental Approach and Exact Rigid Body Inertia

Abstract In the dynamics of multibody systems that consist of interconnected rigid and deformable bodies, it is desirable to have a formulation that preserves the exactness of the rigid body inertia. As demonstrated in this paper, the incremental finite element approach, which is often used to solve large rotation problems, does not lead to the exact inertia of simple structures when they rotate as rigid bodies because the physical nodal coordinates can not be used to describe large rotations in the case of beams and plates. Nonetheless, the exact inertia properties, such as the mass moments of inertia and the moments of mass, of the rigid bodies can be obtained using the finite element shape functions that describe large rigid body translations by introducing an intermediate element coordinate system. The results of application of the parallel axis theorem can be obtained using the finite element shape functions by simply changing the element nodal coordinates. A simple rigid body rotation, however, can cause a significant error if the element shape function and the nodal coordinates are used to evaluate the inertia properties of bodies that undergo large rigid body rotations. As demonstrated in this investigation, the exact rigid body inertia properties in case of rigid body rotations can be obtained using the shape function if the nodal coordinates are defined using trigonometric functions that lack a physical meaning. Linearization of the nodal coordinate vector can lead to different results when different methods are used to define the rigid body inertia. For example, the calculation of the mass moment of inertia using position coordinates only leads to results which are different from those obtained using energy expressions or the laws of motion.