Influence of PDC bit cutter layout on stick–slip vibrations of deep drilling systems

This paper extends the class of two degrees-of-freedom (2 DOF) models that simulate the self-excited vibrations of a rotary drilling system, by considering PDC bits equipped with round cutters and a realistic cutter layout. The extension requires an elaborate bit/rock interaction model that involves bit characterization, cutter projection and correlating the bit/rock engagement with the bit motion history. The governing equations are state-dependent delay differential inclusions with two major sources of nonlinearities: the regenerative effect in both cutting and frictional components of the drilling force and the unilateral nature of the frictional force. To solve for the state variables, the technical challenges involve a search of many delays and a fast and accurate geometric computation, which are respectively resolved with a modified inverse algorithm and the polygon clipping technique. The linear stability analysis is formulated based on the geometric variations of the bit/rock engagement surfaces, which result in critical rotation speeds that separate various stability regions of the decoupled axial dynamics. Determined by the bit design, the resulting stability boundaries are defined in terms of the rotation speed and the nominal bit penetration per revolution. With the time simulation, it is verified that the stability map, in conjunction with the exact location of unstable axial poles, can be used to predict the occurrence of axial stick–slip oscillations in the coupled dynamics, which eventually lead to sustained torsional stick–slip oscillations.