The drill string is subjected to complex coupled dynamics. The first step for system analysis and control is having a good dynamic model that can represent the physical behavior of real drill strings. The most widely used dynamic models for such systems are the lumped element models which neglect the system distributed feature. In this paper, a dynamic model called neutral-type time delay model is modified to investigate the coupled axial-torsional vibrations in drill strings. Note that this model is derived directly from the distributed parameter model and described by delay differential equations which are infinite dimensional. Coupling of axial and torsional vibration modes occurs in the bit-rock interface. For the first time, the neutral-type time delay model is combined with a bit-rock interaction model that regards cutting process in addition to frictional contact. Moreover, mistakes made in some previous studies are corrected. The model consists of two constant delays which arise from the dynamic model and a state-dependent delay, introduced by the bit-rock interaction model. The simulation results demonstrate bit-bounce and stick-slip phenomena. The bit-rock interaction forces and torques are also presented.

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