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