Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 134

For one, he noted that the right excitation source does not appear all too often.
And second, when it does, the frequency bandwidth for drill string resonance is
narrow - thus, delicately maintaining a constant rotary speed that is just right is
essential (the width of the frequency band reduces with drillstring length).
Finally, the irregularities which create the excitation source are often drilled
away before the resonant vibration mode has time to develop. The possibility
that other transient vibration modes exist, though, was raised in his paper. We
might note that if these exist, the actual drilling process itself would have to be
modeled in order to simulate reality, while somehow retaining the periodic,
kinematic character of the rotating drillbit.
Dareing (1972) importantly observed that harmonic motions may not be
the most general downhole, and that they may not be the predominant form of
vibration while drilling. He noted that large dynamic forces can occur in the
drillstring and at the bit, at non-resonant frequencies - in fact, the bit forces
measured by the downhole recorder were at non-resonant frequencies. Dareing
noted that vibration modes do not develop instantaneously, because time is
required to establish normal modes; but while acknowledging that general
transient motions may well predominate, he did not propose any other physical
mechanism that may be responsible for dangerous transient motions. In
summary, fixed displacement boundary conditions only describe drilling in hard
rock, or possibly, drilling over short time scales, or both. Any resonances
predicted by the model are likely to have disappeared by the time required for
the harmonically steady solutions to establish themselves, since the conditions
needed to create them will have been drilled away. In formulating a realistic
model, the boundary conditions must be generalized, and transient, rather than
harmonic calculations should be performed.
We have agreed that the differential equation drillstring model in Equation
4.2.1, and the surface boundary condition model in Equation 4.2.17, are likely to
apply to real situations. But Equation 4.2.19 is not acceptable, since predicted
resonances do not exist physically, not to mention its complete failure in
modeling forward drilling. We are left with a dilemma posed by two simple
wishes: how can we model both drilling ahead and drillbit kinematics? Since
Equation 4.2.19 does not model forward drilling, penetration rate optimization
cannot be studied with it; nor can we model bit-bounce, which clearly satisfies
stress-free boundary conditions, when it does occur. Before developing a
unified model, further discussion of existing models is useful.
4.2.4.3 Modeling rock-bit interactions.
We address the former question first, but we need to return to
fundamentals, examining available options. In Chapter 1, we studied models of
the form u(0,t) = 0 and u x (0,t) = 0; the first describes rigid ends, whereas the
second simulates stress free ends. These models are extreme limits representing