Drillstring vibrations and in particular stick-slip and bit-bounce are detrimental to oil-well drilling operations. Controlling these vibrations is essential because they may cause equipment failures and damage to the oil-well. A simple model that adequately captures the dynamics is used to simulate the effects of varying operating conditions on stick-slip and bit-bounce interactions. It is shown that the conditions at the bit/formation interface, such as bit speed and formation stiffness, are major factors in shaping the dynamic response. Due to the varying and uncertain nature of these conditions, simple operational guidelines or active rotary table control strategies are not sufficient to eliminate both stick-slip and bit-bounce. It is demonstrated that an additional active controller for the axial motion can be effective in suppressing both stick-slip and bit-bounce. It is anticipated that if the proposed approach is implemented, smooth drilling will be possible for a wide range of conditions. Copyright © 2006 by ASME.