Handbook of Turbomachinery Second Edition Revised - Ventech!

support stiffness and the spin speed. For the ‘‘squeeze film-mounted’’
configuration, the radial stiffness k and transverse viscous damping
coefficient c of the squeeze film may be calculated from closed-form
equations, e.g., Vance [21], based on short bearing theory and a centered,
circular orbit assumption:
Where
k ¼ 2RL3 moe
C 3 ð1 e 2 Þ 2 c ¼
pRL3m 2C3ð1 e2Þ 3=2
R ¼ radius.
L ¼ length.
C ¼ radial clearance.
m ¼ oil viscosity.
o ¼ whirl speed.
e ¼ eccentricity ratio (orbit magnitude/clearance).
Note that both the stiffness and damping coefficients become very large as
the eccentricity ratio approaches 1.0.
Figure 39 illustrates typical response characteristics for in-phase and
out-of-phase unbalance of the two disks. For the in-phase unbalance and
‘‘hard-mounted’’ bearings, the ‘‘cylindrical’’ mode responds strongly with a
dynamic magnification slightly greater than 10 (0.014-in. (356 mm) whirl
radius/0.001-in. (25.4 mm) mass eccentricity) while the ‘‘conical’’ mode does
not respond significantly. With the out-of-phase unbalance condition, the
response is larger at the ‘‘conical’’ mode. The maximum response for a given
mode occurs when the unbalance distribution is aligned with the mode
shape, thus producing the maximum modal force. Although not shown in
this example, the response at the either the third or fourth critical speed
could be very large if the unbalance distribution resulted in a significant
modal excitation force. The beneficial external damping from the bearing
support structure and/or squeeze film dampers would be largely ineffective
due to the low percent strain energy at the bearings (see Fig. 38). Figure 39
also illustrates the dramatic reduction in response at both of the ‘‘rigidbody’’
critical speeds with squeeze film dampers at both bearings. Response
at the first and second critical speeds is reduced by factors of five and two,
respectively. The axial length or radial clearance of the squeeze film dampers
could be changed to better balance the response at the critical speeds,
recognizing that their stiffness increases linearly with the whirl frequency.
Some high-speed machines have been designed that operate between
the first and second bending modes. These types of machines are usually
referred to as ‘‘supercritical rotors.’’ Since most of the strain energy is in the
Copyright © 2003 Marcel Dekker, Inc.