Kinematics of the Cone Bit

Abstract

Abstract The bits' and bit cones' angular velocities are variable with time. All published papers on cone bit kinematics, however, are based on the supposition that the motions of cones are uniform. This supposition, therefore, has been the main obstacle to determining the objective law of the working cone bit. The apparatus, methods, and tests results for measuring the bit and cone instantaneous angular velocities are described. A new theoretical analysis of cone bit motion is proposed and some interesting new conclusions are obtained that will help designers and users of cone bits. Introduction Although cone bits have been used for more than 70 years, their kinematics and dynamics have not been investigated thoroughly. Unfortunately, many questions, such as the variation of velocity of a bit tooth during its impact against the hole bottom and the distance or velocity of slip between the bit tooth and the hole bottom, both of which concern the designers and users of cone bits, have not been answered explicitly. Many papers on kinematics or dynamics of cone bits proposed that the rotations of bit and cones were proposed that the rotations of bit and cones were uniform. Some papers treated the motion of the cone as a rigid body motion with a fixed point"; others looked on the motion of a tooth-row as a linear rolling. None of these concepts has reflected the actual behavior accurately. In recent years, a few authors have noticed the problem of the nonuniform rotations of cones, but no detailed problem of the nonuniform rotations of cones, but no detailed study has been carried out. This paper discusses the Southwestern Petroleum Inst. Rock Bit Research Laboratory's experimental equipment for measuring the motion of the frill-scale rock bit. and the single-tooth-row roller. Some test results are shown. The relational equations among the kinematic and geometric parameters of the single-tooth-row roller sheet with respect to double frames of reference have been derived by using polar coordinates. These equations are used to explain the test data. Experimental Equipment A square hole is made in the thrust button in each cone of the frill-scale bit. A round hole is made through each journal along its axis. In each journal, a small shaft is inserted in the round hole. The square end of the small shaft is fixed into the square hole in the thrust button and the other end is joined to the shaft of a sensitive DC tachometer generator. A magneto-electric oscillograph is used for recording the output voltage of the tachometer generator. The graph of the instantaneous angular velocities of cones are recorded on film. A drilling machine is converted into testing equipment for studying the kinematics of a single-tooth-row roller. It is shown schematically in Fig. 1. In this figure, (1) denotes the DC motor with stepless speed variation; (2) represents the gear box; (3) shows some large iron disks used to simulate the bit weight and the moment of the upper part of the drillstring; (4) is a sensitive DC tachometer generator joined to (5) and used for measuring the angular speed of the imaginary upper drillstring; (5) is a vertical slender shaft; and (6) denotes a special experimental bit body. (One-, two-, or three-cone assemblies may be fixed on this bit body and their offset may be controlled.) The (7) represents the roller fixed to (8); (8) denotes a shaft that is supported on the bit body by two roller bearings; (9) shows the second sensitive DC tachometer generator used for measuring the angular velocity of the roller; and (10) represents a displacement transducer with strain gauges, which is used for measuring the vertical displacement of the bit body. The information produced by Instruments (4), (9), and (10) shown in Fig. 1 are recorded by an oscillograph on film. Brief Description of Test Results The full-scale bit test results indicated that the angular velocities of cones varied randomly over time and produced large-amplitude oscillations. Fig. 2 gives the produced large-amplitude oscillations. Fig. 2 gives the angular velocities vs. time graphs of the three cones of the rock bit XHP-215Z, taken while drilling in medium-hard sandstone. Careful observation of these graphs shows that the number of angular speed large-amplitude oscillations in one cycle of the cone is equal to or approximately equal to the tooth number in a certain tooth-row on that cone. This tooth-row usually is found in the second or third row of the cone and not in the first (gauge) row, as mentioned in some papers. The total number of oscillations of the cone per cycle usually approximates the total number of teeth on that cone. The tooth-rows are the "cells" of cones and a specific tooth-row plays an important role in the motion of cones. Therefore, a more detailed study of the motion of a single-tooth-row roller is necessary to explain the motion of the rock bits. Valuable results have been obtained from the tests of various rollers under different test conditions. These results generally illustrated that the bit cones' rotation has never been uniform. SPEJ p. 321