Nonholonomic dynamics of drill string bit whirling in a deep bore-hole

This article studies the problem of wirl vibrations of a rotating drill string bit under conditions of its friction interaction with the bore-hole bottom surface at the wandering contact point. The mechanism of the bit spinning and rolling without sliding on the rigid surface has been analyzed. To study the wirl vibrations, the methods of nonholonomic mechanics are used. As an example, spherical bit whirling on spherical bore-hole bottom is considered. The kinematic inducement of the rotating bit motion without sliding is shown to be the main cause of its stability loss. The detailed study of the bit whirling revealed three types of its stable and unstable motions associated with direct and inverse rolling as well as pure spinning. It is found that the most detectable influence on dynamics of the system and its stability is exerted by the overall stiffness of the drill string tube, which depends on parameters of compressive axial force and torque and diminishes as their critical values are approached.