Self-excited oscillations of a rotary drilling system induced by bit-rock interaction with two state-dependent delays

This paper describes a generic lumped-parameter model with multiple degrees-of-freedom (referred to as the MDOF model) to study the coupled axial and torsional vibrations of the drilling system with a drag bit consisting of full and partial blades. The interaction between such a bit with the rock is composed of a cutting process and a frictional contact process. The cutting process introduces a feedback with state-dependent delays into the equations of motion. The feedback is related to the depth of cut whose value depends on the current and a prior axial positions of the bit - the axial regenerative effect. The friction contact process is associated with nonlinear boundary conditions due to the occurrence of stick events in both axial and torsional vibrations. In this work, the axial regenerative effect is captured by a bit trajectory function, which makes it more efficient to calculate the depth of cut of a drag bit with full and partial blades than the approaches used in previous studies. The evolution of the bit trajectory function is governed by a partial differential equation (PDE), which is inherently coupled with the ordinary differential equations (ODEs) that govern the axial and torsional dynamics of the MDOF model. Through the application of Galerkin method, the coupled system of PDE-ODEs is further approximated by a finite set of ODEs, which is integrated to obtain the dynamic responses of the MDOF model. The characteristics of axial and torsional vibrations along the drillstring are captured with the MDOF model in conjunction with drag bits equipped with full and partial blades. The dynamic responses between a bit with full and partial blades and a bit with identical full blades are also compared in this paper.