Explicit Momentum Conserving Algorithms for Rigid Body Dynamics

Two new explicit time integration algorithms are presented for solving the equations of motion of rigid body dynamics that identically preserve angular momentum in the absence of applied torques. This is achieved by expressing the equations of motion in conservation form. Both algorithms also eliminate the need for computing the angular acceleration. The first algorithm employs a one-pass predictor-corrector scheme while the second algorithm is based upon the staggered time integration approach of Park. Numerical results are presented comparing the new algorithms to the algorithms of Simo and Wong and Park et al. The predictor~orr~tor algorithm is shown to suffer weak instabilities while the staggered conserving algorithm exhibits improved performance compared to the staggered algorithm of Park et al. 1. INTRODu~ION The development of time integration schemes for solving equations associated with finite rotations has received considerable attention during the past several years. These research efforts have been motivated by the development of structural models in which proper orthogonal matrices are used to describe the motion; see, e.g., [l-5] as well as the references contained therein. The work of Simo and Wong is particularly interesting in that they present an implicit, single-step, time integration algorithm that identically preserves angular momentum and energy in the absence of applied torques. The basic idea of their approach is to write the equations of motion in conservation form. Integrating the equations over a time step ensures conservation of angular momentum for torque-free motions provided the configuration and angular velocity updates are computed properly. The algorithm obviates the need to compute the angular acceleration; if needed. the acceleration may be obtained by post-processing. An explicit version of the algorithm is presented in their paper that preserves angular momentum but requires the angular acceleration to be computed. Within the context of linear structural dynamics, Tamma and Namburu developed an explicit algorithm in which the acceleration need not be computed [6]. An extension of their approach to rigid/flexible body dynamics is given in [8]. However, this extended algorithm loses the desirable momentum conservation property achieved by the original version for linear structura1 dynamics. One purpose of this paper is to present an algorithm that conserves angular momentum using the Tamma-Namburu approach. As described in Sec. 4.3, this requires a one pass predictor-corrector strategy. Park et al. developed an explicit-implicit time integration procedure for multibody dynamics that uses a staggered form of the central difference algorithm coupled with a midpoint update of the configuration[9]. One advantage of their procedure is the implicit treatment of constraint equations; compared to standard central difference methods, twice the number of equations needs to be solved per time step, Their algorithm is not cast in conservation form and thus does not conserve angular momentum. In this paper, an alternate staggered algorithm is presented that inherits the momentum conservation property. An outline of the paper follows. Since the time integration algorithms for the rotation group may be considered as extensions of algorithms developed for structural dynamics, a review of the classical and staggered central difference algorithms and the Tamma-Namburu algorithm is given in Sec. 2 for linear structural dynamics. Section 3 briefly presents the notation and fo~ulation of rigid body kinematics and dynamics. In Sec. 4, parametrization of the rotation group is discussed, the Simo-Wong algorithm and the proposed momentum conserving predictor-corrector and staggered time integration algorithms are presented and different choices for evaluating configuration-de~ndent moments are given. Results from numerical simulations are provided in Sec. 5. Conclusions are drawn in Sec. 6. 2. EXPLICIT SECOND-ORDER ACCURATE ALGORITHMS FOR LiNEAR STRUCTURAL DYNAMICS To provide a basis for presenting explicit algorithms for rigid body rotations, we review two explicit, second-order accurate algorithms for solving the equations of linear structural dynamics: classical central differences and the algorithm of Tamma and 1291