Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 207

where we have introduced a general loading q (v) (x,t) to account for additional
sources of external excitation, and a resistive force kv + v/ t consisting of
elastic (kv) and dissipative ( v/ t) contributions. In Equation 4.3.40, N(x,t)
represents the axial force, positive for compression, and negative for tension; in
the limit of vanishing stiffness EI and dissipation , we recover the classical
wave equation for transverse vibrations. The reader should perform a global
energy analysis similar to that for axial vibrations, constructing energy
inequalities, in order to understand the exact physical meaning behind each term
in Equation 4.3.40.
Now recall from our axial vibrations work how our sign convention
assumed that the axial force satisfies AE u x < 0 for compression, while we have
AE u x > 0 for tension. Thus, we consistently set
N(x,t) = -AE u/ x

(4.3.41)

so that the bending equation takes the explicit form
EI 4 v/ x 4 - {(AE u/ x) v/ x}/ x
+ kv + v/ t + A 2 v/ t 2 = q (v) (4.3.42)
The following review is not intended to replace a complete study of strength of
materials and elasticity. For the transverse deflection v(x,t), the slope is simply
v(x,t)/ x. The moment M(x,t) and the shear V(x,t) satisfy
M = - EI 2 v(x,t)/ x 2

(4.3.43)

and

3 v(0,t)/ x 3

V = - EI (4.3.44)

even when axial forces are present.

4.3.6.2 Auxiliary conditions.
Equation 4.3.42, a partial differential equation for v(x,t), must be solved
with auxiliary boundary and initial conditions. Let us first review the
fundamental boundary conditions applicable to beam bending formulations. In
short, we have,

Pinned end:
v(0,t) = 0 (4.3.45)

2 v(0,t)/ x 2 = 0

(4.3.46)

Fixed end:
v(0,t) = 0

(4.3.47)

v(0,t)/ x = 0 (4.3.48)

Free end:
2 v(0,t)/ x 2 = 0

(4.3.49)

3 v(0,t)/ x 3 = 0

(4.3.50)