Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 192

as variable coefficients. The loading, compressive at the bit and tensile at the
surface, vanishes at the neutral point (this is the definition of “neutral point”).
We will show that the group velocities of dangerous waves do, in fact, vanish
near the neutral point, thus trapping the waves there, leading to large,
undetectable local accumulations of vibration energy.
General downhole vibrations consist of waves with many component
frequencies. Whitham (1974) describes, as we will below in greater detail, how
group velocity can be calculated for monochromatic waves of fixed frequency.
Thus, it is possible to identify those waves and frequency ranges most likely to
inflict damage. Exact integrals for steady wave motion can be obtained, and
general qualitative and quantitative formulas will be derived. Detailed
numerical results for all pertinent vibration parameters, based on the general
dimensionless equations, will be tabulated and plotted in order to provide
qualitative measures of danger at various points along the drillstring. Typical
applications, for example, may include bottomhole assembly design, the optimal
placement and design of downhole MWD tools, and the location of sensors
within MWD subs. We will also discuss means to detect violent bending
vibrations from the surface. Finally, the use of tapered drillstrings with varying
material and geometrical properties (e.g., constructed by using different drill
collar and pipe sections with the required variations in cross-sectional
properties) is discussed.

4.3.4.1 Beam equation analysis.
Let represent the mass density per unit volume, A the cross-sectional
area, E the Young' s modulus, I the moment of inertia, k the wave number, 0
the vibration frequency and N the variable axial force distributed along the
drillstring. In the absence of torsion, the dual bending modes v and w that
generally coexist will decouple, so that it suffices to consider each of the lateral
modes individually. Here, we let v(x,t) denote transverse displacements
satisfying
EI 4 v/ x 4 + (N v/ x)/ x + A 2 v/ t 2 = 0 (4.3.1)
The axial force distribution N(x,t) in Equation 4.3.1 satisfies N < 0 for those
portions in tension, and N > 0 for those portions in compression. The transition
N = 0 defines the neutral point of the drillstring. Once the solution for v(x,t) is
available, the “bending moment” is known from M = -EI 2 v/ x 2 , while the
“shear force” satisfies V = M/ x = -EI 3 v/ x 3 . If the “stiffness” EI vanishes
identically, and a constant axial force N < 0 remains in tension, Equation 4.3.1
reduces to the classical wave equation. In this limit, the resultant equation
2 v/ t 2 - c 2 2 v/ x 2 = 0 would describe the transverse vibrations of a string, and
the speed of sound c would satisfy c 2 = - N/ A; and since N = - EA < 0, it

follows that c 2 = E/

> 0, as is well known from classical wave theory.