Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 112

will partially reflect and transmit, following well-defined conservation laws that
are specific to the application. Other elements that control the vibrations are rig
details at the surface, the kinematics of the drillbit used, and rock-bit interaction
occurring at the formation interface. Rock-bit interactions are identical in form
to “mixed boundary conditions” used by mathematicians and discussed in this
topic; this means that advances in vibration modeling are possible with modest
levels of technology transfer.
4.1.2.2 Transverse vibrations.
Axial vibrations are to be contrasted with transverse vibrations, e.g., the
oscillations seen on jump ropes, violin and guitar strings. Such “strings” are
one-dimensional and support tension only; without tension, they collapse and do
not admit wave propagation. “Drillstrings” are not “strings” in this sense: they
are beams that support transverse vibrations even in compression because they
possess stiffness. For more details, refer to the beam references in Chapter 3.
Transverse vibrations are also known as lateral or bending vibrations.
The subtleties behind lateral bending are easily visualized in a simple must
experiment using the long flexible rubber erasers found in early draftsmen' s
tools. Apply an axial load and watch the eraser bend in-plane . Next, twist the
eraser somewhat and reapply the same load: now the eraser bends in two planes;
when the eraser is long, the two bending deflections occur in mutually
orthogonal (or, perpendicular) planes. Thus two coupled modes of lateral
deflection are the rule in practice. As if the physics were not complicated
enough, understand that the million-pound, mile-long drillstring never
completely rests on the rock bottom; if it did, the formation would crack, and the
bit would never rotate with noticeable speed.
Because the drillstring is lifted from the surface, it is in tension there; since
it must contact the formation in order to drill, it must be in compression at the
bit. Hence, the axial load in general changes from tension at the surface to
compression at the bit, the exact point where the sign of the normal stress
changes being the “neutral point” (the neutral point is located in the drill collar).
Other neutral point definitions are available, but ours suffices for this topic' s
objectives. Students with strength of materials experience understand that beam
deflections satisfy a fourth-order differential equation. This contains a second-
derivative term related to the superimposed axial load. At the neutral point, its
sign changes, leading to what mathematicians call a “turning point problem.”
This turning point, due to sign reversals in axial stress, is primarily responsible
for violent lateral vibrations near the neutral point plus their failure to be
observed and detected from the surface. We will provide an unequivocal proof
later in this chapter. Understanding the physics, therefore, requires a firm
appreciation of the mathematics; both, consequently, are necessary before
computer models can be properly formulated or developed.