Investigation on bit stick-slip vibration with random friction coefficients

Torsional vibration of a drillstring, bit stick-slip in particular, is investigated with a simplified single degree of freedom (DOF) model in this paper. The friction in the bit-rock interaction is modeled as random, and the motions in stick and slip stages are treated separately. In the stick stage, it is treated as in static equilibrium. A single point or a range of points, at which the stick stage is completed and the slip stage is started, are determined depending on deterministic or random static friction coefficient assumed respectively. In the slip stage, the response at a time instant becomes a spreading area centered around the means due to the randomness of the kinetic friction. By assuming the random kinetic friction as White noise, the probabilistic distribution of the responses with displacement and velocity is calculated with the numerical path integration (PI) technique. For verification purpose, Monte Carlo (MC) simulation is also conducted. Also due to the randomness in friction, the time instants leaving the slip stage, or entering the stick stage, become random. Comparison between PI and MC shows that results from the two methods are in good agreement. In addition, parametric studies on damping, rotary speed, weight on bit, drillstring length and different combination of pipe and collar are conducted for both deterministic and random cases.