Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 193

Equation 4.3.1, applicable to straight beams, is discussed in Graff (1975).
The (N v/ x)/ x term demonstrates the coupling between N and v(x,t). The
assertion in Vandiver, Nicholson, and Shyu (1989) that linear coupling will not
occur on a perfectly straight beam excited by an axial load which is less that the
critical buckling load is therefore incorrect. However, the coupling mechanism
that they investigate, namely that due to initial BHA curvature, represents a
valid physical problem. In Equation 4.3.1, we have not included viscous
dissipation, although the extension is trivial; we will discuss the qualitative
effects of internal viscous damping later. But an effective damping, however, is
introduced by the effects of heterogeneity alone, as we will show; in the case of
lateral vibrations, it is completely due to axial variations in the longitudinal
force N(x,t).

4.3.4.2 Kinematic wave modeling.
Let us consider, as we had in our example for the transverse vibrations of a
string, a sinusoidal Fourier wave component having a wavenumber k and a
frequency . Since Equation 4.3.1 is linear, this involves no further
assumptions than those underlying elastic small displacement theory. Thus,
v(x,t) = e i(kx- t) (4.3.2)
Substitution of Equation 4.3.2 in Equation 4.3.1 leads to the “complex
dispersion relation”
A 2 = EI k 4 - Nk 2 + i kN x (4.3.3)
where i = -1. Since Equation 4.3.3 involves terms with and without “i,” thre
frequency must necessarily be complex . In order to obtain useful results, we
rewrite the complex frequency

explicitly as the sum of a real part

r and an

imaginary part i i , where both r and i are real. Thus, we assert that
= r + i i (4.3.4)
Equations 4.3.2 and 4.3.4 allow us to express v(x,t) as the product of two
quantities, a propagating sinusoidal "exp{i(kx-

r t)}" wave, and an exponential

“exp( i t)” damping, that is,
v(x,t) = exp( i t)exp{i(kx- r t)} (4.3.5)
We now consider only those waves that are not critically damped, that is, only
propagating waves with weakly damped oscillations satisfying | r | >> | i |. In
this limit, substitution of Equation 4.3.4 in Equation 4.3.3, and equating real and
imaginary parts leads to
A

r 2 = EI k 4 - Nk 2

(4.3.6)

and

i = kN x /(2 r A)

(4.3.7)