Geology Reference
In-Depth Information
Deflected spring:
2
v(0,t)/ x
2
= 0
(4.3.51)
3
v(0,t)/ x
3
= - k
d
v(0,t)
EI
(4.3.52)
Torsional spring:
v(0,t) = 0
(4.3.53)
EI
2
v(0,t)/ x
2
= k
T
v(0,t)/ x
(4.3.54)
Mass loaded:
2
v(0,t)/ x
2
= 0
(4.3.55)
- EI
3
v(0,t)/ x
3
= m
2
v(0,t)/ t
2
(4.3.56)
Dashpot dampened:
2
v(0,t)/ x
2
= 0
(4.3.57)
3
v(0,t)/ x
3
=
EI
v(0,t)/ t
(4.3.58)
Although we have chosen to emphasize the rock-bit contact point x = 0, the
above conditions also apply at the surface x = L as necessary. In the foregoing
boundary conditions, m, k
T
, k
d
, and are suitable constants.
Finally, suitable
initial conditions
must be used to start any analytical or
numerical integrations. If we assume that the initial drillstring is straight and
starts from rest, we can write
v(x,0) = 0 (4.3.59)
v(x,0)/ t = 0 (4.3.60)
We are now in a position to simulate lateral bending vibrations. We emphasize
that while numerous analytical solutions are available in the mechanical
engineering literature, they do not contain variable N(x,t) functions that hold
particular relevance to drillstring vibrations. Thus, we will refrain from
presenting these, preferring instead to consider more powerful computational
methods that can be ultimately extended to handle general coupled vibrations.
4.3.7 Finite difference modeling.
Because lateral vibrations play significant roles in downhole mechanical
failure and borehole instability, the ability to model and simulate their behavior
holds practical importance. Spanos and Payne (1992), for example, have
developed a numerical “finite element” method based on “Euler-Bernoulli beam
theory” to simulate
harmonic
events occurring downhole. We have already
identified the predominant physical mechanisms controlling downhole
instability, using kinematic wave theory as applied to individual Fourier wave
components, so we will now concentrate on fully transient theory capable of
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