This paper describes an alternative formulation of the nonlinear equations governing the coupled axial and torsional dynamics of a discrete model of a rotary drilling system equipped with a drag bit. This model assumes a rate-independent bit/rock interface law that accounts for both cutting and frictional contact processes. The regenerative effect associated with the cutting introduces a term with delay, and thus a feedback, in the equations of motion, while the frictional contact results in set-valued contact forces and discontinuities in the boundary conditions, which are responsible for the occurrence of axial and torsional stick-slip oscillations. Inspired by P. Wahi and A. Chatterjee (Self-interrupted regenerative metal cutting in turning, International Journal of Non-Linear Mechanics 43:2, 111–123, 2008), the regenerative effect is captured by a bit trajectory function, whose evolution is governed by a first order partial differential equation. With this approach, the original state-dependent delay differential equations (SDDDEs) governing the dynamics of the discrete model are replaced by the partial differential equation of the bit trajectory function together with the axial and angular equations of motion of the two degrees-of-freedom model. The equation governing the evolution of the bit trajectory function is further approximated by a system of first order ordinary differential equations through application of the Galerkin method. The introduction of the bit trajectory function makes it possible to consider a bit with two state-dependent delays (and in principle multiple delays) without reducing the computational efficiency. The accuracy of this approximation is validated using published results, and the convergence and robustness properties of the proposed method are analyzed using three cases corresponding to different regimes of instability. Finally, we extend the discrete model to a more generic lumped-parameter model with multiple degrees-of-freedom, which captures the propagation of axial and torsional vibrations along the drillstring.
- Generic lumped-parameter model
- Multiple state-dependent delays
- Nonlinear drillstring oscillations
- Rate-independent bit-rock interaction