Spiral waves in rotating twisted elastic pipes

Abstract

The problem of propagation of bending waves in rotating pipes prestressed by longitudinal force and torque is stated and solved. Such waves are shown to be spiral ones. It is established that four waves exist for every wavelength, two of which are right-handed spirals and the other two are left-handed. These waves propagate with different velocities in different directions Introduction. How the perturbation in a wave deflects from the propagation direction in a continuum is determined by polarization. If perturbation oscillates in one direction all the time, this is the elementary case of a linearly polarized wave. In media transmitting transverse waves, other, more complex types of polarization are possible. For example, if the extreme point of the perturbation vector describes a circle or an ellipse in the plane of oscillation, then the polarization is circular or elliptic. Perturbations with such polarization can be excited as displacements and velocities in elastic waves or intensity of electric and magnetic fields in electromagnetic waves [2,5]. If two harmonic transverse elastic waves of equal length propagate in perpendicular planes of an elastic rod with a phase shift of p/2, the resulting traveling wave will have the form of a spiral. However, the situation will be essentially different if the rod is prestressed by a torque. It will be shown below that no waves other than spiral ones are possible in such a rod. It will be established that for every wavelength there are four such waves, two of which are left-handed spirals, while the other two are right-handed spirals. A dispersion analysis of these waves shows that they are dispersing. The velocities of the right-and left-handed spiral waves are different, and the velocities of each of these waves in positive and negative directions of the rod axis are different too.These results can be used to analyze the dynamic behavior of drillstrings in the oil and gas industries. While rotating, drillstrings, usually several thousand meters long, are subject to torque and longitudinal force. Quite specific dynamic effects are caused by the inertial forces induced by the fluids moving inside them. Therefore, dynamic processes excited in such columns are very complex and are yet to be understood. Some issues of the dynamics of such systems are addressed in [8][9][10][11][12][13].1. Problem Formulation. Consider an infinite elastic pipe subjected to a longitudinal force T and a torque M z . The pipe spins with constant angular velocity r w. A fluid of density r fl moves with velocity V inside the pipe. Let us derive the

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Stability of drillstrings in ultradeep wells: an integrated design model

“…Considering the importance of such technologies for the power industry of Ukraine and the absence of a scientific theory of oil and gas extraction from ultradeep wells, we may conclude that the mathematical simulation of deep-well drillstrings of various configurations is an important scientific and applied problem. Similar problems of the dynamics of thin-walled rods and beams are addressed in [8,9,19].During well drilling, the drillstring is subjected to gravity, torque, and inertial forces exerted by the rotating drill pipe and internal fluid. These factors induce longitudinal, torsional, and flexural vibrations and flexural buckling of the drillstring.…”