Nonlinear instabilities and control of drill-string stick-slip vibrations with consideration of state-dependent delay

Abstract

In this paper, a reduced-order drill-string model with coupled axial and torsion dynamics is studied. Nonlinear effects associated with dry friction, loss of contact, and the state-dependent delay, which all arise from cutting mechanics, are considered. The linearized system equations are derived, and the local stability analysis is carried out analytically by using the D-subdivision method. The obtained results are presented in terms of stability crossing curves, which are parameterized by using the rotation speed and the cutting depth. A numerical continuation method is developed and used to follow periodic orbits of systems with friction, loss of contact, and state-dependent delay. Bifurcation diagrams are constructed to capture the possible routes from either a nominal stable operational state or a stable limit-cycle motion without stick slip to a limit-cycle motion with stick slip. It is shown that the system can experience subcritical Hopf bifurcations of equilibrium solutions and cyclic fold bifurcations. Based on this work, an observer based control scheme is proposed by using a continuous pole placement method for time-delay systems. The effectiveness of the controller in suppressing stick-slip behavior is shown through simulations.