LAUSR

Abstract A common strategy for analyzing the stability of coupled axial and torsional dynamics in drilling is by using a two-degree-of-freedom mass-spring-damper model derived as a second-order ordinary differential equation. This implies using a single point mass to represent the inertia of the bottom-hole assembly and thus the characteristics of an otherwise distributed system. For the bit-rock model developed in Richard et al. [ 33 ], the stability characteristics of the drilling system are in many previous works derived by assuming that the axial dynamics can be decoupled from the slower torsional dynamics. Using this bit-rock model, the friction forces and torques in the system are dependent on a time-delay term, dictating the stability of the system. Consequently, a set of critical drilling parameters, i.e., rotation speed, drill string stiffness, and fluid damping, can be investigated to identify the properties of the equilibrium for the nominal solution. In this paper, we have generalized the drill string dynamics by modeling the drill string as a series of alternating springs and point masses. By making the same assumptions as in previous works, the dynamics are derived as a lumped-multi-element second-order model including a time-delay dependent on the system states. The model is then defined as a state-dependent delay differential equation. The stability analysis of the decoupled generalized system indicates that the stable region for the chosen drilling parameters reduces and is trending towards zero when the number of model elements increases. The implication of this is that the nonlinearity of the coupled axial and torsional dynamics is required for a thorough understanding of the stability of the drilling system. Furthermore, we have performed a numerical analysis of the coupled dynamics by simulating relevant drilling scenarios.