A wave model of torsional vibrations of rotating drill strings is set up. The ranges of rotational speed in which self-excited vibrations occur are found. Andronov-Hopf bifurcations occur at the limits of these ranges. The conditions for the occurrence and development of self-excited oscillatory processes are established Keywords: deep drill string, self-excited torsional vibrations, wave model, Andronov-Hopf bifurcation Introduction. Drilling of superdeep oil and gas wells has recently become an important field of mining. The most popular deep drilling technology is the rotary method in which the well is drilled by a rotating bit fastened to the downhole end of the drill string suspended from its upper end. The bit is rotated by applying a torque to the upper end of the drill string.A complex dynamic effect that may cause accidents during deep well drilling is the self-excited torsional vibrations of the rotating drill string (DS). In this case, the DS can be modeled by a torsion pendulum. At its lower end, the friction between the bit and broken rock causes the stick-slip effect, which disturbs the stationary energy flow from the driving mechanism to the environment, leading to self-excited torsional vibrations of the bit.Since the bit is fixedly connected to the DS, its stick-slip torsional vibrations generate elastic (generally nonharmonic) torsional waves that run from the downhole end of the DS to its upper end and, mirroring from it, return to the bit, affecting significantly its vibration mode. Therefore, the motion of such a system can only be described using a torsion pendulum as a mathematical model, with the bit playing the role of a flywheel. Such models were used in [7,8,10,11] to study the dynamics of a bit, with traveling torsional waves represented by harmonic functions.We will demonstrate that if there is no wave dispersion and the solution of the wave equation can be represented in the d'Alembert form, then the wave equation with boundary conditions can be reduced to one ordinary differential equation of the second order with time t as a delayed argument and angular speed w as a variable parameter. Depending on the value of this parameter, the equation can have either a stationary solution j( ) t =const or an oscillatory solution j( ) t . The change of the former solution to the latter one is called Andronov-Hopf bifurcation, and the associated value of w is called bifurcational [2][3][4][5][6]9]. In what follows, we will use a mathematical model describing the torsional vibrations of a DS to find the bifurcational values of w for different DS lengths and to establish some patterns of self-excited vibrations.1. Wave Torsion Pendulum as a Mathematical Model. We will model the DS by a torsion pendulum with a flywheel (modeling the bit) attached to its lower end. The upper end of the DS rotates with constant speed w. We choose an inertial frame of reference OXYZ with the origin at the center of mass of the bit and the OZ-axis aligned with the DS axis. We also choose a frame of reference Oxyz rotat...
The non-linear wave model of torsion pendulum is elaborated with the aim to simulate bifurcations of auto-oscillations of superdeep drill strings (DSs). The constitutive differential equation with delay argument and small parameter before the superior derivative is deduced. The results of its analysis testify that the self-excited torsional oscillations of the DS are of the relaxation type with fast and slow motions and proceed in the manner of quantized time. In doing so, the rotation speed _ ' changes by portions and continues to be constant during time segments Á, which equaled the duration of the torsion wave up and down propagation through the DS length.The states of limit cycle birth and loss in superdeep DSs are found, and the DS torsion modes are constructed for the critical situations. The dependence of the auto-oscillation generation states on the friction (cutting) moment parameters is analysed.
539.3: 622.24 and O. V. GlushakovaWe address a problem of self-excitation of elastic torsional vibrations of a rotating drill string due to frictional interaction between the drill bit and rock at the deep well bottom. Using the d'Alembert solution to a wave equation, a mathematical model is constructed for a wave torsional pendulum in the form of a nonlinear ordinary differential equation with delayed argument. Special features of generation of self-excited vibrations of drill strings have been determined through computer simulation.Introduction. Rotary drilling is the most widely accepted method for producing oil and gas wells; it involves rock destruction at the well bottom by means of a drill bit attached to the lower end of the drill string (DS) to which rotation is imparted through the torque applied to the DS top end. In the course of drilling, a drill string experiences a number of mechanical actions the most significant of which are the longitudinally nonuniform DS tension force, torque, centrifugal and compound centrifugal forces induced by the drilling fluid flow inside DS, forces of frictional interaction between the DS and the formation, etc. The factors listed above initiate longitudinal, torsional, and bending vibrations in the string and contribute to its flexural buckling. This may cause a stuck DS, wellbore wall falling in, and overall instability of the system [1-5]. One of the dynamic phenomena that may eventually lead to an emergency occurrence in drilling is the self-excitation of torsional vibration of a rotating DS. Since a DS is a torsional pendulum, where an outflow of energy from the drive to environment occurs at the DS bottom part due to energy-dissipative interaction between the drill bit and the formation, the string may pass from the steady equilibrium rotation to torsional self-excited vibration when the conditions of this energy outflow are upset. To study this phenomena we will use here a wave model of a torsional pendulum, which allows for the effects of the finite-velocity propagation of torsional strains along the DS.Special Features of Processes of Self-Excitation of Torsional Vibration in Long Drill Strings. The effects of self-excitation vibration fundamentally differ from other types of vibration processes in dissipative systems in that no periodic external action is needed for their excitation [6,7]. If a mechanical system's transition from some initial state to self-excited vibration conditions occurs without any addition "push," this is a soft self-excitation. If a vibration starts increasing spontaneously only from some limiting amplitude, this is called the hard self-excitation. To periodic self-excited vibration there corresponds a closed loop path in the phase space, which all neighbor paths tend to; this path has been named the stable limit cycle or the attractor [8].With regard to the phenomena that accompany the DS rotation, a study of generation of the DS self-excited torsional vibration enables one to answer three important questions: (i) at what values of the...
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