Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 129

4.2.2.5 More on AC/DC interactions.
Static DC axial displacements (which are, again, affected by AC solutions)
are important in our analysis for two reasons. For one, the end result of
successful drilling is decreased drillbit elevation, a DC effect if the bit is to
advance on the average. This philosophy contrasts with the usual specification
“u(0,t) = u 0 sin t” in which the drillbit executes purely periodic motions but
does not drill. Second, later results demonstrate, consistently with limited
observations, that the static component of the axial displacement field is
responsible for violent bending oscillations near the neutral point. Treating the
total AC/DC displacement allows us to model this effect naturally.
Similar arguments are given for studying the complete dynamic and static
torsional displacement rather than each separately. But the reason is quite
different. In order to model “stick-slip” oscillations, the cyclic transformation of
mean potential strain energy to disturbance kinetic energy, and vice-versa,
suggests that it is simpler and more natural to deal with a single entity. Thus, in
this topic, unless specifically noted, all axial, torsional and lateral disturbances
are modeled in their totality without book-keeping them into distinct AC and
DC components. But in the next section, we will separate AC and DC solutions,
in order to develop conventional ideas on mechanical vibrations and to
demonstrate possible pitfalls incurred by this separation.

4.2.3 Conventional separation of AC/DC solutions.
To see how the usual ideas about static and dynamic formulations follow
from Equation 4.2.1, which describes the complete problem, we write the total
longitudinal displacement u(x,t) in the form
u(x,t) = u s (x) + u d (x,t) (4.2.2)
where s and d denote static and dynamic . Substitution in Equation 4.2.1 yields
2 {u s (x) + u d (x,t)}/ t 2 + {u s (x) + u d (x,t)}/ t
- E 2 {u s (x) + u d (x,t)}/ x 2 + g = F e

(4.2.3)

or

2 u d (x,t)/ t 2 + u d (x,t)/ t
- E 2 {u s (x) + u d (x,t)}/ x 2 + g = F e

(4.2.4)

Now, we choose the function u s (x) so that it satisfies the ordinary
differential equation
d 2 u s (x)/dx 2 = g/E (4.2.5)
The accounting implicit in Equation 4.2.5 assumes that static wall friction, if
any, is not book-kept in u s (x); the appearance of frictional effects is
conspicuously absent. This is not to say that friction does not exist, only that