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drillstring. However, they express cautious optimism, suggesting that it may be
possible to see evidence of downhole bending vibration in the surface axial and
torsional vibration signals although the relationships are not understood well
enough to be used diagnostically.
Direct downhole observation remains a possibility, for example, by real-
time monitoring of the timewise density of transverse shocks exceeding a pre-
programmed threshold amplitude, and transmitting the result to the surface using
MWD transmissions. Such a technique, in fact, was originally reported in
Rewcastle and Burgess (1992) and is now commonplace. However, the low
data rates characteristic of existing MWD systems mean that any such
transmissions are likely to provide no more than highly compressed information
of limited value, e.g., a rms value indicative of dynamic stresses at some
arbitrary point in the bottom hole assembly without bearing to nodal or anti-
nodal properties is typically measured. Such MWD vibrations measurements
are necessarily taken at the expense of directional and formation data.
Paslay, Jan, Kingman and Macpherson (1992) attempt to detect large
downhole lateral vibrations from surface longitudinal and torsional
measurements. Their analysis, based on simplified geometrical beam deflection
arguments, predicted motions at top of string due to BHA forward synchronous
whirl. We will discuss their field and calculated results in more detail later.
Here, we will develop a different conceptual framework, and draw upon formal
results in nonlinear elasticity. From a theory of nonlinear vibrations perspective,
we view bending motions as distributed sources of axial disturbances, which
are detectable at the surface as longitudinal oscillations, and possibly,
additionally through their effect on torsional oscillations. This effect
mathematically manifests itself as an external forcing function to the linear
longitudinal wave operator; again, the modeling of forcing functions is
discussed in Chapter 1, and in more specialized contexts, in our treatment of
axial vibrations. Thus, the spectral content obtained by monitoring surface
axial vibrations should change noticeably and predictably when large bending
motions are encountered prior to impending drillstring failure.
4.3.5.2 Nonlinear axial equation.
To illustrate the basic ideas, we will ignore the effects of static and
dynamic torsion for now, concentrating instead on longitudinal and lateral mode
interactions only. Earlier we indicated that undamped longitudinal oscillations
for the axial displacement variable u(x,t) satisfy the classical wave equation
2 u/ t 2 - E 2 u/ x 2 = 0 (4.3.37)
where is the mass density per unit volume and E is Young's modulus.
Equation 4.3.37 holds when all other vibration modes are weak in amplitude; it
shows that u(x,t) acts independently of other types of vibration, except to the
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