Geology Reference
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4.4.2.7 Whirling motions.
Initially straight machine shafts, carrying a torque between bearings, with
or without pulleys or similar loads, will at certain rotation speeds “whirl” or
rotate in a bent configuration about the original axis (Bickley and Talbot, 1961).
When the shaft rotation rate equals the natural frequency of the shaft in bending,
highly destructive motions are possible in rotating machinery. The mechanical
engineering literature deals with the vibration and balancing of flexible rotating
shafts and a considerable body of literature is available (e.g., see Bishop (1959),
or Bishop and Gladwell (1959)). Whirling motions are solutions of the dynamic
beam equation governing lateral vibrations. However, they are introduced in
this chapter on torsional vibrations, if only because such motions cannot exist
unless rotation is present to initiate the whirling.
From Chapter 3, if E is Young' s modulus, I is the moment of inertia, A is
the cross-sectional area and is the mass per unit volume, the elementary
bending equation is given by
EI
4
v/ x
4
= - A
2
v/ t
2
(4.4.8)
If v(x,t) is the shaft centerline deflection from its equilibrium straight position at
any point x, the equivalent elastic restoring force must equal that necessary to
produce the centripetal acceleration. The exact form of the centripetal
acceleration term can be obtained by examining the cross-plane perpendicular to
the shaft, where the linear coordinate s of the circular motion and the angle are
related by s = R , with R being the radius. The linear speed is therefore ds/dt =
R d /dt. From particle dynamics, the centripetal acceleration per unit length of
shaft is A(ds/dt)
2
/R or A(d /dt)
2
R. If denotes the constant angular velocity
d /dt of the rotating shaft and v(x) is the local radius R, we have the expression
A
2
v. Thus, whirling motions satisfy
2
v
EI
4
v/ x
4
= A
(4.4.9)
where the explicit dependence on time disappears in this “dynamically steady”
limit (in real drilling, “transient whirling” is typically the rule in the presence of
arbitrary axial and torsional static and transient fields).
Example 4-6. Machine shaft example.
Let us rewrite Equation 4.4.9 as
d
4
v/dx
4
=
4
v where
4
= A
2
/EI. The general solution to this fourth-order
ordinary differential equation is v(x) = A cosh x + B sinh x + C cos x + D
sin x. For the simple “pinned-pinned” machine shaft problem defined on 0 < x
< L, v(x) and the moment EI d
2
v/dx
2
both vanish at x = 0 and x = L. The
sinusoidal mode shape v(x) = sin n x/L is obtained, where n = 1, 2, 3, ... with
the corresponding natural frequencies
n
satisfying
n
2
= n
4 4
EI/ AL
4
. Note
that the frequencies do not form a complete harmonic sequence, although they
are all integral multiples of the fundamental. All of the mode configurations
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