Axial Stick-slip Limit Cycling in Drill-string Dynamics with Delay

In a novel approach to model stick-slip vibrations occurring when drilling with drag bits, the axial and torsional dynamics are coupled through the boundary conditions via a state-dependent delay. Moreover, friction is modelled by a rate-independent discontinuous term. A regime characterized by a low amplitude of the torsional vibrations and a high drilling efficiency is numerically observed for some sets of parameters. In this regime, the axial fast vibrations have a stabilizing effect on the torsional equilibrium. To understand this stabilizing mechanism, we are studying the decoupled axial equation obtained by freezing the delay. This approximation reflects the small variations of the delay when the bit experiences small torsional vibrations. Axial periodic solutions may be analysed independently. Particularities of this equation lie in the presence of a delayed term and a non-smooth non linearity. In this paper, we apply different well-known methods to study the periodic orbits of the axial dynamics. The results and limitations of semi-analytical (Describing Functions Method) and numerical procedures (Finite Difference Method, ShootingMethod) are exposed here. We use these numerical techniques to investigate some particular properties of the system, such as the dependency of period time with the delay.