Geology Reference
In-Depth Information
4.3.9 Nonlinear interaction between axial and lateral bending
vibrations.
Our closed form solution shows that the static axial stress field imposed
along a drillstring can lead to the trapping of lateral waves, and hence, localized
high cycle fatigue. But as lateral amplitudes become singular, they can alter the
axial stress state that induced the severe bending, and theoretically anyway, act
as to remove the cause of the instability. But this is probably unlikely. Severe
lateral vibrations will remain an issue, as unfortunate field experiences
constantly remind us; thus, the axial-lateral mode coupling to be discussed is
likely to provide only minor changes to the linear model discussed so far. This
statement represents, of course, opinion and not fact.
For formulation purposes, it is worthwhile to return to two previous
equations, namely,
EI
4
v/ x
4
- {(AE u/ x) v/ x}/ x
+ kv +
2
v/ t
2
= q
(v)
v/ t + A
(4.3.42)
A
2
u/ t
2
- EA
2
u/ x
2
= EI {(
3
v(x,t)/ x
3
) ( v/ x)}/ x
(4.3.38)
Again, note that the second term in Equation 4.3.42 represents the effects of
variable axial loading, with the force AE u
x
< 0 for compression and AE u
x
> 0
for tension; in terms of our earlier notation, we can write N(x,t) = -EA u/ x.
In elementary beam theory, Equation 4.3.42 is solved with all coefficients
uniquely defined; for example, A, E, I, k, and are typically constants, and
u/ x and q
(v)
are prescribed functions of x. While complicated, the beam
formulation is still a linear one. Equation 4.3.38, if the right-side is set to zero,
is no more than the classical wave equation discussed at length previously. This
model is a linear one. However, when lateral vibrations are significant
downhole, v(x,t) and its spatial derivatives may no longer be small, and the
right-side of Equation 4.3.38 cannot in general be neglected. Thus, Equations
4.3.42 and 4.3.38 couple nonlinearly, and together describe a
nonlinear model
in
which axial and lateral bending modes continuously exchange energy in time.
The coupled system in general requires numerical solution methods, which at
first sight may appear overwhelming. However, the only visible structural
change is the nonzero right-side of Equation 4.3.38. This suggests that all of the
recipes we have developed so far still apply, provided we update our axial
displacement results at each time step using latest available bending solutions.
The nonlinear boundary value problem in this manner is rendered amenable to
numerical solution.
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