Handbook of Turbomachinery Second Edition Revised - Ventech!

on the shaft is zero. Thus, the steady-state unbalance response is not
influenced by the internal damping mechanism. The normalized amplitude
and the phase of the unbalance response are graphed in Bode and polar plot
format in Fig. 12 versus the normalized spin-speed.
When the rotor spin-speed equals the undamped natural frequency,
ot, the phase lag is 908. This condition is referred to as the phase resonance
condition of the rotor. The peak unbalance response for the rotor, however,
occurs at a slightly higher spin-speed of ot= 1 2z 2 1=2
e . This condition is
referred to as the amplitude resonance condition of the rotor. For small
values of damping, the difference between the two conditions is negligible in
a practical sense.
SeveralconfigurationsofthesteadyresponsearealsoshowninFig.13.
These configurations display the location of the disk geometric center o and
mass center c for the cases associated when with the shaft spin-speed is less
than, equal to, and greater than the undamped natural frequency ot. The
geometric center lags the excitation associated with the mass center c as the
result of a transverse viscous damping force that is opposite in direction to
the motion of the rotor geometric center. This phase lag ranges from near
zero at low spin-speeds to 908 (the phase resonance) when the spin-speed
equals the undamped natural frequency, to nearly 1808 at very high spinspeeds.
Inspection of Eqs. (10) and (11) and the associated response plots in
Fig. 12 reveal that the radial position of the mass center c approaches zero
as the spin-speed approaches infinity. Thus, the rotor tends to spin about its
mass center at high spin-speeds, and the geometric center o then approaches
a circular orbit with amplitude equal to the mass eccentricity, e, of the disk.
Damped Free Response—External Damping Only. It is instructive to
initially consider the situation when only external damping is present. For
this case, the homogeneous form of the rotor equations of motion relative to
ðx, y, zÞ, Eq. (9), reduces to
m 0
0 m
€u
€v þ ce 0
0 ce
_u
_v
þ k 0
0 k
u
v
¼ 0
0
ð12Þ
For lightly (under) damped systems, i.e., 0 < z e < 1, the solutions from
Eq. (12), for an arbitrary set of initial conditions,
uð0Þ ¼uo
vð0Þ ¼vo
and
Copyright © 2003 Marcel Dekker, Inc.
_uð0Þ ¼_uo
_vð0Þ ¼_vo
ð13Þ