INTEGRATING FINITE ROTATIONS

We study the integration of problems of evolution in the rotation grou p. Instead of attacking the problem in the nonlinear differential mani fold SO(3) (pure rotational dynamics), as is usually done, we derive e quations for the complete problem of motion (translational and rotatio nal dynamics) on an extended manifold. We develop a generalization of Runge-Kutta methods that, by design, ensures that the solution will re main on the manifold for any choice of the tableau. This is obtained t hrough configuration updates performed via the exponential map. We sho w how certain terms can be approximated, while retaining the order of accuracy of the scheme, and how the method conserves the total momentu m of the system. Within this framework, we develop two nonlinearly unc onditionally stable time integration schemes, that are associated with discrete laws of conservation/dissipation of the total energy. The di ssipating algorithm generalizes to the nonlinear case the high frequen cy damping characteristics provided by some well-known conventional me thods. We present numerical results to support our analysis, and we de velop a complete application of this methodology to the nonlinear dyna mics of three-dimensional rods undergoing large displacements and fini te rotations, under the assumption of small strains. (C) 1998 Elsevier Science S.A. All rights reserved.