Nonlinear dynamic stability analysis of the coupled axial-torsional motion of the rotary drilling considering the effect of axial rigid-body dynamics

Dynamic stability of the coupled axial-torsional motion of a drill string is studied in this paper. Considering the drill string as a continuous system, governing equations of axial and torsional motions are derived. The drill string motion is divided into the rigid-body motion and elastic vibrations. Dynamic behavior of axial rigid-body motion is included through an independent equation. Natural vibration mode shapes of a free-free bar and a fixed-free shaft is introduced as trial functions for discretizing axial and torsional equations of elastic vibration, respectively. Due to the presence of state-dependent delay in equations of motion, semi-discretization method is implemented for stability analysis of the delay differential equations. The effect of nominal rotational speed of the drill string (n) and nominal weight on bit (Wo) are investigated on drilling stability. Stability charts are presented and maximum allowable value of Wo is depicted with respect to Q. Comparison of the results reveals that considering a constant axial velocity for the top of the drill string which is a common simplifying assumption in the previous researches, affects stability analysis of the drill string dynamics. It is observed that drilling operation is unstable for speeds lower than a threshold value. Based on the stability analysis, the maximum allowable value of Wo shows an increasing and then decreasing behavior with respect to the nominal rotary speed of the drill string. However, by increasing the damping of the system, its behavior changes into a fully decreasing profile with respect to Q. It can be recognized from the results that both velocity-weakening friction and regenerative cutting have great effect on stability of the system and should be considered in dynamic modeling of the drill string. In this paper effects of several parameters of rock formation and drill string on the stable region are also presented. Finally, variation of steady-state axial speed is depicted with respect to the rotary table angular speed and nominal WOB which can be useful for choosing optimal and safe parameters for drilling.