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also reduces the number of unknown variables). Several boundary condition
models were allowed, namely, (1) simple supports at the surface (“top-drive
condition”), or at the bottom (e.g., “bit with slick assembly”), with v = v xx = 0 at
x = 0 and L, and (2) clamped at top (e.g., “kelly rotary table”), or at the bottom
(“packed assembly condition”), with v = v x = 0 at x = 0 and L. Importantly, the
authors noted that the effects due to drillstring rotary motion and twisting
vibrations are ignored in the precession model. Thus, the model does not
include torque, and hence, it cannot model stick-slip oscillations and their
interaction with bending. Of course, the objective of the authors' model was to
examine lateral vibrations as induced by axial excitation; in that regard, the
work represents a significant contribution to the literature.
4.5.4.10 Direct simulation of bit precession.
We indicate that the formulation of Dunayevsky will produce stability
envelopes only: it will define parameter ranges for which dangerous motions are
anticipated, but it will not determine forces, moments, displacements, velocities
or accelerations. But direct transient analysis will satisfy both objectives. Since
actual amplitude outputs are obtained from the fast difference scheme given
earlier, exact magnitudes are available, where the results are to be interpreted
along with the user's (subjective) assessment of stability.
The formulation given here will allow prescribed circular motions at the
bit, and in fact, the computed results given above assumed just such a model.
The Fortran listing which implemented this motion is given in Figure 4.5.9,
where the first source block reads 1 x v(1) + 0 x v(2) + 0 x v(3) =
PRERAD*SIN(WMGPRE*T) , or v(1) = PRERAD*SIN(WMGPRE*T) . Similarly, the
second block reads w(1) = PRERAD*COS(WMGPRE*T) . Here, PRERAD denotes a
prescribed precession radius R prec , while WMGPRE is a precession frequency prec .
Since sin 2 prec t + cos 2 prec t = 1, it follows that v(1) 2 + w(1) 2 = R prec 2 applies
at the bit index i = 1. Of course, we could have parameterized v(1) and w(1) to
any other functions of time, in order to model the effects of bit motions other
than perfect circles. Prescribing circular motions at the bit with a radius equal to
that of the hole, in our analytical model, in no way constrains the entire
drillstring to “wrap around” the borehole wall - unless, of course, that is the
outcome of the calculations. Thus, the approach is more general, because it
allows a dynamical system to seek out its own stable equilibria, rather than
imposing their existence on an a priori basis.
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