Handbook of Turbomachinery Second Edition Revised - Ventech!

Figure 18 Whirl speed map: Laval–Jeffcott rotor rotatory motion.
The backward synchronous critical speed is
nb ¼
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kr
Id þ Ip
ð21bÞ
For the choice when Ip ¼ 0:80Id, these two critical speeds are nf ¼
21; 540 rpm and nb ¼ 7; 181 rpm.
Note that Eq. (21a) is singular when Ip ¼ Id and that this equation
yields an extraneous solution when Ip > Id. WhenIp ¼ Id, the disk is said to
be ‘‘inertially spherical.’’ As illustrated in Fig. 18, when Ip is larger than Id,
an intersection of the synchronous excitation line and the forward natural
frequency line does not exist. In general, forward natural whirl speeds
increase in value due to a ‘‘gyroscopic stiffening’’ of the shaft as the rotor
spin-speed increases, and backward natural whirl speeds decrease due to a
‘‘gyroscopic softening.’’ For the special case when Ip > Id, the Laval–
Jeffcott rotor system ‘‘stiffens’’ such that the forward natural frequency is
always larger than the rotor spin-speed. Thus, a forward critical speed does
not exist for this situation. For additional information on the effects of
rotatory inertia and gyroscopic moments, refer to the vibrations book by
Den Hartog [3].
Copyright © 2003 Marcel Dekker, Inc.