Quantized attractors in wave models of torsion vibrations of deep-hole drill strings

We pose the problem of self-excitation of elastic wave torsional vibrations of a rotating drill string, which arise as a result of frictional interaction of the drill bit with the rock at the bottom of the deep hole. We use d'Alembert's solution of the wave equation to construct a mathematical model of the wave torsion pendulum in the form of a nonlinear ordinary differential equation with retarded argument. We show that there exists a range of variation in the angular velocity of the drill string rotation, where, along with the unstable stationary solution characterized by the absence of vibrations, there are oscillatory solutions in the form of a stable limit cycle (attractor). The self-excitation of these vibrations is soft, and the self-oscillations themselves belong to the class of relaxation vibrations, because their period can be divided into several separated intervals corresponding to slow and fast variations in the state of the system. The velocities of the drill bit elastic motions on each of these time intervals remain constant, and the durations of all of them are the same and equal to the time interval (quantum) of the twist mode propagation from the drill bit to the drill string top and conversely.