State-dependent delay influenced drill string dynamics and stability analysis

In this paper, the authors present a discrete system model to study the coupled axial-torsional dynamics of a drill string. In this model, nonlinearities such as dry friction, loss of contact, and state-dependent time delay are taken into consideration. Simulations are carried out by using a 32-segment model with 128 states. Bit bounce is observed through time histories of axial vibrations, while stick-slip phenomenon is noted in the torsion response. The normal strain contours of this spatialtemporal system demonstrate the existence of strain wave propagation along the drill string. The shear strain wave shows the wave node and wave loop along the drill string, which indicates that the torsion motion has the properties of a standing wave. By varying the penetration rate, qualitative changes are observed in the system response, which includes chaotic and hyperchaotic behavior. Stability analysis shows a stable region for the degenerated one-segment model, while the stable region becomes infinitesimally small, as the resolution of spatial discretization is increased. This finding suggests the drill string motions are always likely to be self-exited in practical drilling operations.