Frequency Analysis of Drill Bit Whirlings on Uneven Bottoms of Deep Bore-Holes

Abstract

An incipient stage of a rotating drill string bit whirl motion proceeding on a well bottom surface is studied on the basis of kinematic (nonholonomic) models of mechanic interaction between the contacting uneven bodies. The bit is considered to be an absolutely rigid spherical or ellipsoidal body, the well bottom surface can be plane or spherical. It is assumed that the system coaxiality is disturbed through small initial curvature of the drill string, imperfections of the bit and bore-well geometry or the system mass debalance. Linearized equations of the bit whirling are deducted, the frequencies of periodic motions are calculated, and their modes are constructed for different geometric parameters of the spherical and ellipsoidal bits. It is demonstrated that, depending on the system properties, the bit motion can acquire the regimes of forward and backward whirlings or to transit to the state of stationary spinning relative to an immovable center of velocities. The most unfavorable and atypical whirling modes are characteristic for oblate ellipsoidal bits and curvilinear surfaces of the well bottom.