The interest in using delayed design is emphasized in recent research on the stabilization of finite/infinite dimensional dynamical systems. In particular a property called multiplicity-induced-dominancy (or MID for short) allowing for a reduced-complexity delayed- controller design showed its efficiency in fast damping of harmful oscillations. This contribution is concerned with oil-well drilling systems torsional vibrations, which constitute an important source of economic losses; drill bit wear, pipes disconnection, borehole disruption and prolonged drilling time, among other consequences. Such torsional vibrations are assumed to be governed by a wave equation with weak damping term. The MID-based design is further exploited to quench the torsional vibrations along the rotary drilling system. The proposed control law guarantees the existence of robustness margins with respect to delays and parameters uncertainties. Numerical investigation of the performance as well as the robustness of the corresponding control law will be presented.