Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 206

be ascertained from real-time, downhole MWD measurements and transmitted
from time to time to the surface, more controlled surface monitoring is possible.
This lateral-to-axial mode of energy transfer was first noted in Chin (1988a,b).
It was later suggested independently by Rewcastle and Burgess (1992), who
stated explicitly that bending vibration can induce axial vibration at twice the
frequency. However, these authors did not publish details related to this
statement. The literature provides little evidence regarding this speculation.
Monitoring surface axial vibrations for downhole lateral instability, needless to
say, is neither obvious nor ordinary.
An interesting clue is offered by Paslay, Jan, Kingman and Macpherson
(1992), who monitored the surface axial vibration spectrum due to a drillstring
in a vertical well. Their experiments were conducted with a drill bit rotation rate
in the 2.3 Hz (or 138 rpm) range, and its corresponding spectral line prominently
appears in their measurements. For example, their Figure 3 clearly shows a
strong 2.3 Hz line, while their Figure 5 similarly shows a well-defined 2.1 Hz
line. Also noted on these respective figures, though, are unexpectedly strong 4.6
and 4.2 Hz lines. An examination of their Figure 1 shows that the lateral
resonant frequency happens to be 2.2 Hz, which is extremely close to their 2.1
and 2.3 Hz rotation rates. If BHA lateral resonance was, in fact, ongoing at the
time axial data was collected, the 4 Hz axial signals may well represent twice
the resonant lateral frequency and not so much a confusing twice-rotation-rate.

4.3.6 Linear boundary value problem formulation.
So far we have discussed fundamental physical ideas only: in particular,
why lateral vibrations tend to be violent downhole and undetectable at the
surface, plus the possibility of detecting their downhole occurrence by
monitoring the axial wave spectrum at the surface. Here we formulate the linear
boundary value problem, to compute solutions satisfying prescribed conditions.
By linear , we refer to lateral displacement solutions when the axial strain field is
specified, without determining the back-interaction of lateral displacements on
axial stresses. We also give a stable numerical finite difference scheme that
leads to easily inverted algebraic equations requiring efficient banded matrix
solvers only.

4.3.6.1 General linear equation.
Our beam equation for the transverse displacement v(x,t), Equation 4.3.1,
or EI 4 v/ x 4 + (N v/ x)/ x + A 2 v/ t 2 = 0, included only the essential
terms needed to model wave focusing and trapping. In general, the complete
equation takes the form
EI

4 v/ x 4 + (N v/ x)/ x
+ kv +

v/ t + A

2 v/ t 2 = q (v) (x,t)

(4.3.40)