The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.