Whirl Interaction of a Drill Bit with the Bore-Hole Bottom

Nabil W Musa,V I Gulyayev,L V Shevchuk,Hasan Aldabas

doi:10.4236/mme.2015.53005

Abstract

This paper deals with the theoretic simulation of a drill bit whirling under conditions of its contact interaction with the bore-hole bottom rock plane. The bit is considered to be an absolutely rigid ellipsoidal body with uneven surface. It is attached to the lower end of a rotating elastic drill string. In the perturbed state, the bit can roll without sliding on the bore-hole bottom, performing whirling vibrations (the model of dynamic equilibrium with pure rolling when maximum cohesive force does not exceed the ultimate Coulombic friction). To describe these motions, a nonholonomic dynamic model is proposed, constitutive partial differential equations are deduced. With their use, the whirling vibrations of oblong and oblate ellipsoidal bits are analyzed, the functions of cohesive (frictional) forces are calculated. It is shown that the system of elastic drill string and ellipsoidal bit can acquire stable or unstable whirl modes with approaching critical Eulerian values by the parameters of axial force, torque and angular velocity. The analogy of the found modes of motions with ones of the Celtic stones is established. It is shown that the ellipsoidal bits can stop their whirling vibrations and change directions of their circumferential motions in the same manner as the ellipsoidal Celtic stones do. As this takes place, the trajectories of the oblate ellipsoidal bits are characterized by more complicated paths and irregularities.

Highlights

Late in the 19th century, one of the most fascinating dynamic effects of the nonholonomic mechanics was found
To study the phenomenon of a bit rolling without sliding on a bore-hole bottom surface, it is necessary to state the problem about elastic bending vibration of the drill string (DS) taking into account the constraints fulfilling the roles of boundary conditions for the dynamic equations of the DS
The elaborated techniques were used for computer simulation of whirling vibrations of DSs with elongated and oblate ellipsoidal bits

Summary

Introduction

Late in the 19th century, one of the most fascinating dynamic effects of the nonholonomic mechanics was found. To study the phenomenon of a bit rolling without sliding on a bore-hole bottom surface, it is necessary to state the problem about elastic bending vibration of the DS taking into account the constraints fulfilling the roles of boundary conditions for the dynamic equations of the DS. Consider that it rotates with constant angular velocity ω. This part of analysis is performed on the basis of the Cauchy problem statement

Statement of Boundary Conditions for the Constitutive Equations

Statement of the Cauchy Problem for the Bottom Assembly Dynamics

Simulation of Whirl Vibrations of Oblong and Oblate Ellipsoidal Bits

Conclusions