Shevchuk L. Simulation of drill string whirling on the basis of friction and nonholonomic models

The dissertation is devoted to statement and solving the new urgent problem on elastic bending non-linear vibrations of rotating drill string structures in cavities of deep bore-holes in the regimes of selfexcitated whirling of their bits on the bore-hole bottom surface. For the first time, the mathematic models of rotating elastic drill string vibrations induced by rolling of their spherical or ellipsoidal (oblong or oblate) bits on the bore-hole bottom. The constitutive differential equations of the drill string bending are deduced with allowance made for frictional and nonholonomic mechanisms of the bit rolling. The techniques for the integrating of the constructed partial differential equations are elaborated. With their use, the computer analysis of elastic vibrations of the drill strings and the bits is performed for different values of their geometric and mechanic parameters. The whirling modes of the bit whirling in fixed and rotating coordinate systems are constructed. It is demonstrated that the calculated trajectories of the bit motion can have shapes of expanding (unstable motion) or contracting (stable motion) spirals, as well as multilobed figures and multibeam stars, which are observed in drilling practice. They are characterized by large values of accelerations and so represent danger for the system. These regimes can be represented as the regimes of frictional or kinematic selfgeneration of the system vibrations via dynamic actions caused by cohesion forces between the rotating bit and the rock surface.