Limit cycles occurring in oil well drillstrings result from the interaction between the drill bit and the rock during drilling operations. In this paper, we propose to use the weight on the bit (WoB) force as an additional control variable to extinguish limit cycles when they occur. An approximate analysis based on the bias describing function and completed with some simulations, provides good evidence that the rotational dynamics of the oil well drillstring displays such a behavior. In particular, we propose an adaptation law for the WoB named D-OSKIL mechanisms, which results from a variant of the oscillation killer (OSKIL) mechanism studied in detail in [6]. In opposition to the heuristic control structure proposed in [7], we show that the new WoB (W _oB ) control law results in a globally asymptotically stable closed-loop system. Simulations applying the D-OSKIL mechanism show that the stick-slip oscillations can be eliminated without requiring a redesign of the velocity rotary table control.