On dynamic stability of drill strings in a supersonic gas flow

Abstract

In this work stability of the drill string nonlinear dynamics, complicated by the effect of an external
axial load, initial curvature of the drill string, geometric nonlinearity and the influence of a
supersonic gas flow as a circulating medium is studied. The drill string is modelled as a rotating
elastic isotropic rod with constant cross-section. Pressure of the gas flow used to clean the borehole
from drill cuttings and to transport them from the bottom to the surface is determined by the
nonlinear dependences of the piston theory in the third approximation. Utilization of the Galerkin
method allows to reduce the drill string mathematical model to an ordinary differential equation
for the generalized time function, containing an asymmetric nonlinear characteristic, which is further
eliminated by introducing the corresponding substitution. Considering a small perturbation
to the system and applying the harmonic balance method, characteristic determinants are constructed.
Equations describing boundaries of instability zones of basic resonance, which allow to
determine the range of dangerous frequency regimes and to increase safety of the drilling process,
are obtained.