The torsional vibration of drill-strings is one of the most destructive types of vibration present in drilling operations and if lateral and axial vibrations are small, a pure torsional model is able to represent the main features of the nonlinear drill-string dynamics [30]. The stick-slip phenomenon is considered the worst case in torsional vibration [1, 20, 42] and is commonly associated to the non-linear relationship between friction torque and the angular speed at the bit [17]. Given its origin, the stick-slip only occurs when the bit is in contact with the rock [1, 20, 42]. A variety of models have been developed to study the torsional vibration of drill-strings. In [2, 4, 7, 13, 39], the torsional vibration is modeled using a distributed parameter approach. On the other hand, the drill-string was modeled as a torsional pendulum in a lumped parameter approach in [14, 24, 29], with usually one or two degrees of freedom (DOF). Recently, the torsional pendulum model was used in [9, 18, 36, 43] and is the strategy adopted in this article. The top rotational system is responsible for imposing the rotational speed of the drill-string and is sometimes modeled as a inertia with a control system [5, 23, 25] or, as in this paper, using the boundary condition of constant top rotational speed [17, 28, 29, 36, 42, 43]. The bit-rock interaction also plays an important role in every model in the literature. The uncoupled torsional vibration analysis usually use a friction model to include the bit-rock interaction into the model. Some authors use the dry friction model [17, 36, 43], while others include a decaying function in the slip phase [25, 37, 38]. The approach with the decaying function is observed in experimental results given in [5, 30].