Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 124

4.2.1 Pioneering axial vibration studies.
With several decades of hindsight, it is correct to say that many early
findings on vibration control are over-simplified. Nonetheless, several studies
remain of lasting value for their key physical insight. One such work is the
paper by Bailey and Finnie (1960). The authors modeled axial vibrations, and
their natural frequencies and resonances, using the linear partial differential
equation governing the undamped longitudinal oscillations of a bar. At the
surface, a mass-spring (but damper less ) boundary condition was postulated,
where the mass of the swivel, traveling block and kelly was viewed as a
concentrated mass M. The cables and derrick were treated as a spring, and
simple experiments were suggested to determine its stiffness. The spring
constant was inferred by observing the free oscillations of the system once it had
been set into motion by various means. Displacements were assumed to be zero
at the drillbit. The longitudinal bar was formed by the drillpipe and collar. At
their interface, continuity of displacement and force were assumed in order to
connect the sinusoidal wave solutions obtained for each side. These, taking an
unwieldy “(A sin x/c + B cos x/c)(D sin t + E cos t)” form, were further
related by a transcendental equation which required graphical solution. A
closely related torsional vibration analysis, which satisfied similar governing
equations, was also undertaken by the authors. They suggested that the surface
boundary condition might be taken as a fixed end, treating the bottom as more or
less free. At the drill pipe and collar interface, the authors invoked continuity of
angular displacement and torque.
Finnie and Bailey (1960) described experimental techniques developed for
measuring time-dependent force, torque, axial and rotational motions at the top
of a drillstring. Some observations agreed with theoretical predictions, but
many results did not. Paslay and Bogy (1963) studied the axial vibration nature
of intermittent bit teeth and bottomhole contact, showing that appreciable bit
load variations may be detected by surface instruments; they also demonstrated
that downhole excitations may induce oscillating bit forces with amplitudes of
the same order as those of the static bit load. Bailey and Finnie (1960) assumed
an undamped drillstring to simplify the mathematics.
Later, Angona (1965) studied wave attenuation in drillstrings by
measuring the decay of normal stress pulses propagating along them. These
were created by falling weights which struck the top end of the string. Strain
gauges measured pulse amplitudes as they executed multiple reflections.
Angona obtained the relationship between attenuation and frequency by Fourier
analyzing the pulse signal. Typically, three round trips were possible before
complete attenuation, indicating that the system was lengthy enough to be
regarded as wave-like, as opposed to being a lumped mass. The local decay,
taking an “e -

x ” character, was weakly exponential with space; this