Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 113

4.1.2.3 Torsional vibrations.
Torsional or rotational oscillations are exactly what they imply: twists and
turns. They are (somewhat) elementary to model, because like axial vibrations,
both satisfy the classical wave equation. While the partial differential equations
governing axial and torsional vibrations do not couple, dynamic coupling is
achieved downhole at the bit through boundary conditions. For example, the
“dynamic weight-on-bit” due to longitudinal oscillations will affect the
“dynamic torque-on-bit” due to friction at the rock interface, and likewise,
drillbit bouncing implies simultaneous axial and torsional free ends. As will be
demonstrated later, transfer of energy between mean and dynamic components
of the torsional strain field can also lead to “stick-slip” vibrations, nonuniform
rpm' s and “torque reversals,” their time scales being determined by the dynamic
weight-on-bit. Axial and torsional vibrations, through interactions at the bit,
mutually affect each other' s evolution in time when nonlinear back-interaction
through lateral coupling is allowed. As we demonstrated in our eraser
experiment, torsion couples both static bending modes, and likewise, it must
couple both lateral vibration modes. Thus, the single beam theory studied in
elementary elasticity must be replaced by coupled beam equations, which
generally contain variable torques and axial forces. All of the vibrations modes
discussed will be coupled and transient.
4.1.2.4 Whirling vibrations.
Whirling motions are often confused with torsional vibrations. The
confusion arises because whirling vibrations, like torsional ones, cannot exist
without established drillstring rotation. By contrast, a drillstring will deform
axially and laterally even without rotation, e.g., a case in point is the non-
rotating BHA for a turbodrilled well. Whirling motions are a subset of lateral
vibrations; these represent rotating displacements about the equilibrium axis of
the drillstring. The motion of the drill collars may vary from simple whirling
motion, like that of an unbalanced centrifuge, to the highly irregular motions
induced by nonlinear effects such as fluid forces, stabilizer clearance and
borehole-wall contact. More specialized discussions on whirl appear in
Vandiver, Nicholson, and Shyu (1989) and Jansen (1992), who respectively
consider axial-lateral wave coupling, and the effects of eccentricity of the drill
collar's center of mass, fluid damping, stabilizer clearance, plus friction, on
forward and backward whirl.
4.1.2.5 Coupled axial, torsional and lateral vibrations.
Beginning students often ask, “How are drillstring vibrations different
from those of conventional rotating machine shafts? ” Let us restate the above
observations with slightly different emphasis. First, machine shafts are not
loaded by axial stress fields that change signs. Second, the boundary condition