The latter round bids of the pre-salt for exploration and production of oil and natural gas in Brazil indicate the drilling operations will become more intense in coming years. The rotational drilling process is largely used to reach the oil reservoirs and because of diameter-to-length ratio of the drilling system, torsional vibration mode is present in most all drilling processes and may reach an undesired severe stage: the stick-slip phenomenon. In order to address this problem, the torsional vibration mode is isolated and the stick-slip is observed in a fully instrumented drill-string experimental set-up in this work. During this phenomenon, another torque may be applied on an intermediate position of the test bench. The lumped parameter mathematical model is obtained and it is compared to experimental data to validate whether the mathematical model represents the experimental apparatus. A stability analysis is performed using the validated mathematical model in order to identify stable solutions of the system. Therewith, one observed that there is a range of the bifurcation parameter in which stable equilibrium and periodic solutions may coexist. For a given stick-slip situation in bi-stability range, two mitigation strategies of torsional vibration were considered which consisted of imposing perturbations in the system via torques on the intermediate position of the test bench: (i) torques applied only against the direction of motion of the system, and (ii) torques applied in both directions. The strategies were tested numerically and presented eficiency so that the stickslip was completely mitigated: the energies of the system and the work created by the intermediate torque were compared in order evaluate the feasibility and reasonableness of the strategy. Experimentally, the system continued to oscillate, however it presented a significant reduction of stick phase even with limitations of torque applications.