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The mechanisms leading to torsional vibrations in drilling systems are considered in this paper. Thereto, a drill string model of the axial and torsional dynamics is proposed, where coupling is provided by a rate-independent bit-rock interaction law. Analysis of this model shows that the fast axial dynamics exhibit an axial stick-slip limit cycle. This generates an apparent velocity-weakening effect in the torque-on-bit, explaining the onset of torsional vibrations. Based on this analysis, a strategy for control design aiming at the suppression of torsional vibrations is presented.