Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 229

and lateral vibrations are all present and strongly coupled. Bit bounce, stick-
slip, forward and backward whirl, linear and parametric coupling between axial
and lateral vibration all occur during the drilling process.
4.5.3 Notes on the coupled model.
Before presenting mathematical details, we wish to emphasize those
aspects of the physics that are incorporated in our formulation. Importantly, the
analytical solutions previously given show that a bending equation that takes
account of the static axial stress field will successfully predict catastrophic
lateral vibrations at the neutral point, and simultaneously explain the
disappearance of violent bending motions uphole. Thus, the coupled equations
presented below contain two important physical characteristics. First, torque is
included to couple both lateral modes; and second, axial forces are included in
both bending equations to account for extensions of the wave trapping model
given earlier.
The axial formulation, even by itself, is nontrivial. Recall that this model
was designed to simulate bit bounce, rate of penetration and rock-bit interaction.
Likewise, our torsional wave model was designed to use information available
from axial calculations, in order to model torque reversal and stick-slip motions.
How all of these models worked in concert was the main research objective of
this chapter. Drillstring vibration analyses can support field operations in
several ways. They can identify rotation rates that are likely to induce fatigue,
and define safe operating envelopes for damage avoidance. Also, properly
designed simulators can help drillers penetrate more efficiently, by harnessing
the vibration energy available within the drillstring. The model developed
below, hopefully, represents a first step in this direction.
4.5.4 Coupled axial, torsional and bending vibrations.
We now present the partial differential equations for a drillstring vibrations
model that embodies coupled axial, torsional and (two) lateral bending modes.
The coupled equations in their entirety have not been given explicitly, either in
the petroleum industry, or in related mechanical and aerospace applications.
However, their general form can be inferred from Timoshenko and Goodier
(1934), Love (1944), Den Hartog (1952) and Clough and Penzien (1975) on
beams and shafts. In addition, Nordgren (1974) formulated a three-dimensional
nonlinear, large-amplitude, transient problem, and solved it by finite differences;
Garrett (1982) considered the nonlinear dynamics of slender rods. Recent clues
related to the form of the coupled equations can be found in Dunayevsky and
Judzis (1983), Dunayevsky, Judzis, and Mills (1984), Dunayevsky, Judzis,
Mills, and Griffin (1985), and Dunayevsky, Judzis, and Abbassian (1989),
dealing with drillstring buckling and precessional motions.