Second-order explicit integrator via composition for coupled rotating rigid bodies applied to roller cone drill bits

Abstract

We present a derivation of the equations of motion of a roller cone bit as an example of coupled rotating rigid bodies and apply a state-of-the-art numerical integrator to produce an algorithm for use in a bit dynamics software application. The equations are derived using the virtual power method, which naturally handles the constraint between the bit body and the cones. These equations are fully three-dimensional (three degrees of freedom for the body, plus one degree of freedom for each cone) and nicely parallel to the equations of motion of a single rigid body. We apply the composition of adjoint first-order integrators (reminiscent of the approach used earlier to derive an explicit midpoint Lie method (Int. J. Numer. Meth. Eng. 2005; 63:2171–2193) to produce an algorithm that maintains the properties of the original three degree-of-freedom integrator: second-order convergence, symplecticness, remarkable accuracy, and momentum conservation. This algorithm can be applied to other applications where one or more rigid bodies with a single rotational degree of freedom are attached to another rotating rigid body. Copyright © 2007 John Wiley & Sons, Ltd.