Bifurcations in the axial-torsional state-dependent delay model of rotary drilling

We present a detailed bifurcation analysis of the state-dependent delay model of rotary drilling considering only the axial and torsional modes. This analysis is presented for the general case of independent natural frequencies of these two modes. The regenerative effect accompanying axial vibrations gives rise to a delayed model with the delay determined by the torsional oscillations. It is observed that steady drilling loses stability through a Hopf bifurcation. The nature of bifurcation is ascertained by the method of multiple scales for the general values of system parameters. Analytical results suggest that both supercritical and subcritical bifurcations exist for different choices of operating and system parameters. These analytical findings are further confirmed by numerical simulations. Possible unfoldings of the dynamics near the codimension-2 point, guided by numerical simulations and analytical results for the codimension-1 Hopf branches, are also presented. We find two different scenarios at the primary codimension-2 point viz. both Hopf branches having supercritical bifurcation, and one branch being supercritical while the other being subcritical. Our numerical simulations suggest that the dynamics near the codimension-2 point is dominated by the low-frequency limit cycles in both the scenarios.