Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 221

The arguments behind the energy transfer mechanisms described are
quantified by considering the “energy density” 1/2 { J ( / t) 2 + C ( / x) 2 }.
The first term represents the system kinetic energy, and applies to the dynamic
solution only; the second represents the potential strain energy due to torsion,
and applies to both static and dynamic displacement modes. If we now integrate
this energy density over the entire length of the drillstring, the net system energy
can be described by the energy integral E(t), where

E(t) = 1/2 { J ( / t) 2 + C ( / x) 2 } dx (4.4.7)

In Equation 4.4.7, we have omitted the limits of integration x = 0 to L for
brevity. Now, let us resolve the angle of twist into steady and time-dependent
parts, taking (x,t) = s (x) + d (x,t). We follow an integration by parts
procedure similar to that in our treatment of axial vibrations to show that
changes to E(t) are due to internal dissipation (e.g., arising from the / t term
in Equation 4.4.6) and power input at x = 0 and x = L. The power contribution
depends on the product ( / t)( / x) or ( d / t)( s / x + d / x) at upper and
lower ends. This contribution is zero, positive or negative, depending on the
complete dynamics of the system, which in turn depends on the if-then boundary
condition at the bit, as conceptualized in our axial vibrations analysis.
As we have indicated that, when the bit bounces, the boundary condition
downhole is the torsional free end / x = 0. But when it drills ahead and
makes hole, the appropriate prescription is an indicator of total bit torque, say
T(0,t) = GJ (0,t)/ x = f AE { u(0,t)/ x}, where f is an empirical “Coulomb
friction” factor. In this physically oriented model, the instantaneous value of the
axial strain u(0,t)/ x is obtained from the complementary axial vibrations
problem. These ideas are consistent with the results of Clayer, Vandiver and
Lee (1990). They found that the natural frequencies of bottomhole assemblies
in torsion are best modeled by free stiffness boundary conditions at the bit, in
particular, descriptions involving (0,t)/ x in conjunction with large bit
damping. The e external excitation shown in Equation 4.4.6 plays a role
analogous to the F e we encountered in modeling axial vibrations. If borehole
wall friction or substantial borehole contact is significant, e cannot be
neglected along the (contact) length of the drillstring. This will also affect the
energy balance, as indicated in our axial vibrations treatment.

4.4.2.2 Static torque effects on bending.
Earlier we emphasized the need to study a single static plus transient
torque, rather than separate static and transient combinations. This is all the
more important for lateral vibrations: the existence of the slightest torque will
dynamically couple both orthogonal lateral bending modes, as discussed in our
prior “eraser experiment” and also later on. Thus, even when dynamic torsional