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extent that there may exist some form of coupling through rock-bit boundary
conditions. If we revisit the more complete system of coupled governing
equations that apply before the linearizing assumption behind Equation 4.3.37
was made, it turns out that
A 2 u/ t 2 - EA 2 u/ x 2 = EI {( 3 v(x,t)/ x 3 ) ( v/ x)}/ x (4.3.38)
applies at the next hierarchical level of approximation, where v(x,t) represents
the lateral bending displacement (Graff, 1975).
The right side term describes the nonlinear coupling between longitudinal
and bending modes, one that will in general be weak; it is usually neglected
because it is quadratic in a small dimensionless lateral displacement, thus
leading to classical wave equation for u(x,t), which is independent of axial
contraction and expansion due to bending. When lateral vibrations are large,
though, the right side of Equation 4.3.38 will not be small. But the simplicity
behind the classical wave operator can be retained: insofar as the wave-like left
side of Equation 4.3.38 is concerned, the right side plays the role of an external
forcing function , literally providing a source of distributed axial force.
This topic-keeping is not mere accounting: it is consistent with actual
observation. Since up and down-going axial waves do exist whether or not
downhole lateral vibrations are small, our interpretation of Equation 4.3.38 as a
type of almost linear axial wave equation is physically justified. For large
lateral vibrations, we can therefore view the left side of Equation 4.3.38 as we
did in Chapter 1, but interpret the right side of Equation 4.3.38 as a forced
excitation. Additional simplifications are possible. Since the bending
singularity is only sizable near the neutral point, its localized nature allows us to
view this “compact source” of axial stress as a concentrated load, particularly
amenable to delta function modeling as described in Chapter 1.
4.3.5.3 Detecting lateral vibrations from the surface.
We will accept this physical interpretation. Thus, if b is the dominant
frequency characterizing downhole lateral excitation, then the quadratic nature
of the right-side source term indicates that the axial vibration frequency
spectrum measured at the surface should contain a spectral line having
frequency 2 b - this exists only if downhole lateral vibrations are significant!
In other words, a drillstring that is undergoing large-amplitude trapped
transverse vibrations downhole at
b will create longitudinal oscillations near
the neutral point having frequency 2
b , which will, in turn, be freely transmitted
to the surface.
Continuous surface monitoring of the axial spectrum for sudden
unexpected changes, therefore, should provide a clue as to the likelihood of
impending danger. Furthermore, if drill collar lateral vibration information can
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