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We are not suggesting fault with existing approaches, e.g., conventional
modal theories, nonlinear models, buckling simulations or parametric
instabilities. In fact, they most likely will apply to other aspects of the vibration
process; however, the observed failures occurring at the neutral point can only
be described by bending models that specifically allow for axial load variations.
Importantly, we have provided a physically plausible explanation without
recourse to nonlinear models. The model is consistent with observed
phenomena, and supports the view points expressed by recent authors. We
emphasize that
the frequencies associated with large vibration amplitudes are
not necessarily the natural frequencies of the system.
Linear system resonances
always lead to instabilities. However, they are not the only source of violent
fluctuations, and in the problem at hand, they do not represent the most
dangerous form of downhole vibration. It is also interesting that our general
breakdown criterion does not depend on the details (that is, the boundary or
initial conditions) of the bottomhole assembly or drillstring configuration used.
Thus, our wave trapping instability model supports the speculation behind
Rewcastle and Burgess (1992) that shock behavior is more stable than a purely
resonant response. Of course, the details will be important to the extent that
they
do
determine the exact conditions under which dangerous waves are
produced and the resultant magnitudes; in the final analysis, detailed boundary
value problem solutions are required. The above results lead us to speculate that
impending failures might be avoided by substantially increasing or decreasing
weight-on-bit (WOB), at least temporarily, so that the drillstring is completely in
compression or in tension, in order to remove the neutral point singularity. This
would relieve the intense, localized, dynamic stress concentrations formed as a
result of wave trapping. But the possibility that another more conventional type
of instability will appear in response to this change in WOB must be explored
for the bottomhole assembly of interest.
4.3.4.4 Designing safe drill collars.
It is clear that the equations derived above can be used to design
(un)tapered drillstrings (with continuously varying A, E, I, and ) that are less
susceptible to the undesirable effects associated with wave trapping and energy
accumulation. For a given target frequency
0
, the wavenumber solution k =
[{N+
(N
2
+4 EI A
0
2
)}/(2EI)]
1/2
shown in Equation 4.3.13 can be used to
simplify the group velocity formula C
g
= (2EIk
3
-Nk)/ A
0
given by Equation
4.3.6. The resulting expression
C
g
= (2EI [{N+ (N
2
+4 EI
0
2
)}/(2EI)]
3/2
A
0
2
)}/(2EI)]
1/2
)/ A
- N[{N+ (N
2
+4 EI
A
(4.3.33)
0
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