The dynamics of a liquid drop which is supported on a solid rod that is forced to undergo large-amplitude, time-periodic oscillations along its axis is studied using a computational approach based on the Galerkin/finite element method and an adaptive mesh generation technique which enables discretization of overturning interfaces and analysis of drop breakup. When the forcing amplitude is small, the drop deformations are small and the drop remains intact as it undergoes shape oscillations. Larger forcing amplitudes result in the formation of a liquid thread, or neck, which connects two fluid masses: the fluid adjacent to the rod and a nearly globular fluid mass. If the drop deforms such that its length is sufficiently large, the thread ruptures and the globular fluid mass-a so-called primary drop-is ejected from the fluid remaining on the rod. The critical forcing amplitude A c necessary to attain this length and hence drop breakup, the interface shape at breakup, and the volume of the ejected primary drop are determined computationally as functions of the Reynolds number, forcing frequency, and drop size. Over a wide range of values of the forcing amplitude above A c , ejection occurs as the drop recedes from its maximum length during its second period of oscillation. These results show that A c increases as Reynolds number and/or drop size decreases. The maximum length the drop reaches prior to ejection and the position and velocity of the rod for times approaching breakup are shown to profoundly affect the dynamics of drop breakup and the resulting interface shapes. These results show that the forcing amplitude and/or frequency can be chosen so as to prevent the formation of long liquid necks, which typically favor the formation of satellite droplets after breakup. Because drop ejection does not rely on external forces other than those due to rod motion, this method of drop formation holds promise for microgravity applications as well as terrestrial drop-on-demand technologies in which gravitational force is negligible compared to surface tension force.