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 (NDDE). We consider the torsional mode of the drill-string and investigate the associated stick-slip motion. To analyse 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 behaviour 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