Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum

We show that, for rigid body dynamics, the mid-point rule formulated in body co-ordinates exactly conserves energy and the norm of the angular momentum for incremental force-free motions, but fails to conserve the direction of the angular momentum vector. Further, we show that the mid-point rule formulated in the spatial representation is, in general, physically and geometrically meaningless. An alternative algorithm is developed which exactly preserves energy, and the total spatial angular momentum in incremental force-free motions. The implicit version of this algorithm is unconditionally stable and second order accurate. The explicit version conserves exactly angular momentum in incremental force-free motions. Numerical simulations are presented which illustrate the excellent performance of the proposed procedure, even for incremental rotations over 65 degrees. The procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells.

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