Nonlinear Dynamics of Torsional Waves in a Drill-string Model with Spatial Extent

In this paper we investigate the dynamics and bifurcations of an oil-well drill-string model that takes the form of a neutral delay differential equation. We consider the torsional mode of the drill-string and investigate the associated stick-slip motion. To analyze the model we develop numerical continuation routines based on Fourier methods since existing routines based on polynomial approximations are unable to cope with the presence of arbitrarily weakly damped modes. We find "resonance peaks" in the dynamics where a high-frequency mode is superimposed on the underlying periodic behavior, causing large torsional waves in the drill-string. We show that the resonance peaks are robust to small perturbations in the friction parameters but disappear if static friction forces are neglected completely.