Page 76 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.4 Whirling of Rotating Shafts 63

directions then become
—kr —cr = ml r —rd^ eo)^ cos (o)i - 0)]

- crd = m\rO 2r0 - eco^ sin (cot - 0)j
which can be rearranged to
r -h + ~ = eo)^ cos (cot - 0) (3.4-3)

r 0 - f ( ^ r - h 2 r ) ^ = eco^ sin {c ot - 6 ) (3.4-4)
The general case of whirl as described by the foregoing equations comes
under the classification of self-excited motion, where the exciting forces inducing
the motion are controlled by the motion itself. Because the variables in these
equations are r and 6, the problem is that of 2 DOF. However, in the steady-state
synchronous whirl, where 6 ^ co and 6 = r = r = 0, the problem reduces to that of
1 DOF.

Synchronous whirl. For the synchronous whirl, the whirling speed 6 is
equal to the rotation speed o>, which we have assumed to be constant. Thus, we
have

6 —0)
and on integrating we obtain
6 = cot — (f)
where cf) is the phase angle between e and r, which is now a constant, as shown in
Fig. 3.4-1. With 0 = r = r = 0, Eqs. (3.4-3) and (3.4-4) reduce to
k
r = eco cos cf)
(3.4-5)
— cor = eco^ sin 6
m ^
Dividing, we obtain the following equation for the phase angle:
c
0)^
m u —
tan (3.4-6)
m 1 -

where oo^ = yjk/m is the critical speed, and ^ = c/c^. Noting from the vector
triangle of Fig. 3.4-2 that
k 7
m
cos (f)