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In our analysis, we modeled harmonic bending disturbances as they
propagated on a time-stationary axial stress field, with N(x) depending only on x
and not t. The resulting dynamically steady problem, greatly simplified, then
yielded to closed form analytical solutions that indicated strong instabilities due
to energy accumulation and wave trapping. This new mechanism is consistent
with Wolf, Zacksenhouse and Arian (1985). In their evaluation of downhole
lateral and longitudinal interactions, the authors observed that weight-on-bit
“WOB” fluctuations were relatively low, or were in the process of decaying,
when high bending moments developed. When high WOB fluctuations
developed, interestingly, bending moments turned out to be low. These limited
observations suggest that the root cause of large bending fluctuations may have
to do with static distributions of N(x) rather than fluctuating ones.
N* k* N* k*
-5.0
0.31467
0.0
1.0000
-4.8
0.32103
0.2
1.1044
-4.6
0.32778
0.4
1.2153
-4.4
0.33497
0.6
1.3290
-4.2
0.34265
0.8
1.4424
-4.0 0.35086 1.0 1.5538
-3.8 0.35969 1.2 1.6619
-3.6 0.36920 1.4 1.7665
-3.4 0.37949 1.6 1.8673
-3.2 0.39065 1.8 1.9645
-3.0 0.40284 2.0 2.0582
-2.8 0.41619 2.2 2.1486
-2.6 0.43090 2.4 2.2361
-2.4 0.44721 2.6 2.3207
-2.2 0.46541 2.8 2.4028
-2.0 0.48587 3.0 2.4824
-1.8 0.50904 3.2 2.5598
-1.6 0.53553 3.4 2.6351
-1.4 0.56610 3.6 2.7086
-1.2 0.60171 3.8 2.7802
-1.0 0.64359 4.0 2.8501
-0.8 0.69327 4.2 2.9185
-0.6 0.75246 4.4 2.9853
-0.4 0.82282 4.6 3.0508
-0.2 0.90543 4.8 3.1150
Table 4.3.2.
Wavenumber versus axial force.
4.3.4.3 Bending amplitude distribution in space.
In this section, we will expand upon the physical arguments offered above,
and provide more rigorous and exact results. It can be shown (Whitham, 1974;
Chin, 1976, 1980a) that the “wave momentum density” M and the “wave energy
density” E satisfy
M/ t +
(C
g
M) / x= M{2
i
-
r,x
(k,x,t)/k}
(4.3.19)
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