Explicit Newmark/Verlet algorithm for Time Integration of the Rotational Dynamics of Rigid Bodies

We consider the problem of integration of the initial value problem of the rotational rigid body dynamics. We are motivated by a very practical problem: time integration of the equations of motion of drill bits as they cut through rock. The drill bit geometry is fully three-dimensional, as is the surface of the rock to be cut, and the force laws of cutting, scraping, impacts, and persistent contacts are heuristic, but computationally intensive anyway. Consequently, evaluation of the forces acting at various points on the bit is very expensive. (Similar observations may be made in many molecular dynamics simulations, where the evaluation of the forcing may take as much as 90% or more of the CPU time, or in finite element dynamics calculations.) Therefore it makes eminent sense to try to limit the number of evaluations of the external torque, and we are thus led to consider methods that are explicit in the torque calculation, i.e. the torque is evaluated just once per time step, in preference to implicit methods (torque evaluated many times per time step). We are generally looking for an integrator that is robust, accurate, and efficient. Conservation properties are often essential for the preservation of key qualitative features of the motion. Equally important are often the dissipative and forcing characteristics [19]. In addition, three-dimensional rotations pose a special challenge in that the configuration set is a curved manifold, not a vector space. The integrator should maintain the configuration on the manifold. Methods for this type of problem have been studied by many researchers, both in the computational mechanics community, and in the applied mathematics community [5, 27, 1, 24, 20, 32, 21, 29, 25, 42, 23, 8, 39, 2, 28, 10, 40, 22, 17, 33, 15, 41, 31, 26, 13, 6].