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have constrained nodes at the ends of the shaft, as assumed, and the n th mode
also has (n-1) internal nodes, equally spaced along its length.
Example 4-7. Generalized whirl. We have given, in the above analysis, a
self-contained derivation for whirling oscillations that is independent of the
lateral vibration equations to be covered. However, we emphasize that circular
whirl can be viewed as the resultant of two perpendicular lateral vibrations v(x,t)
and w(x,t), which combine to form a local radial displacement {v 2 (x,t) +
w 2 (x,t)} from an equilibrium x = 0 state. Thus, the results obtained here, and
their extensions under more practical boundary conditions, should emerge as
subsets of the general coupled model to be presented later. A combined lateral
displacement variable similar to this is used by Dunayevsky and Judzis (1983),
Dunayevsky, Judzis and Mills (1984), and Dunayevsky, Judzis, Mills and
Griffin (1985) in their studies of precessional motions. Thus, it is not necessary
to consider special differential equations for whirling motions; if they exist for
specific non-simple sets of boundary conditions, the transient coupled axial,
torsional and lateral equations derived later will model them.
4.4.2.8 Causes of whirling motions.
In drillstring vibrations, whirling represents the centrifugally induced
bowing of drill collars resulting from rotation. If the center of gravity of a drill
collar is not located precisely along the center line of the hole, then the collar
rotates, and a centrifugal force acts at the center of gravity causing the collar to
bend. Eccentricity may arise, for example, from initially bent drill collars, or
from collar sag due to gravity and high compressive loads at the bit. Motions of
drill collars may vary from simple whirling motion, like that of an unbalanced
centrifuge, to highly irregular motion caused by nonlinear effects of fluid forces,
stabilizer clearance and borehole wall contact. While whirling does not
generally cause precipitous drill collar failures, because lateral amplitudes
limited by wall contact, constant wall contact does result in high levels of
abrasion and damaging wear.
Practical questions such as stabilizer modeling arise in the course of
vibration simulation. Stabilizers have larger diameters than drillpipes, but small
length. Thus, they can be viewed as simple lumped masses on beams, at least in
the first approximation; where stabilizer friction due to rotation and borehole
contact with the formation are important, nonzero right-side “q” terms or special
elastic responses may be invoked and defined as necessary. An excellent study
on whirling appears in Vandiver, Nicholson, and Shyu (1989), providing
qualitative descriptions of the vibration process based on extensive high-data-
rate field measurements. Jansen (1992) investigated the effects of eccentricity
of the center of mass of the drill collar, fluid damping, stabilizer clearance and
friction on forward and backward whirl.
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