Multiple Mode Analysis of the Self-Excited Vibrations of Rotary Drilling Systems

This paper extends the approach proposed by Richard et al (2007) to analyze the axial and torsional vibrations of drilling systems that are excited by the particular boundary conditions at the drag bit-rock interface, by basing the formulation of the model on a continuum representation of the drillstring rather than on a characterization of the drilling structure by a two degree of freedom system. These boundary conditions account for both cutting and frictional contact at the interface. The cutting process combined with the quasi-helicoidal motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring can be detected. In particular, stickslip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters.