Handbook of Turbomachinery Second Edition Revised - Ventech!

are 1808 out of phase (i.e., the bearing center, mass center, and rotor center
form a straight line). As the rotor speed approaches infinity, the amplitude
of the response approaches the cg eccentricity, e, of the rotor. Thus, at very
high speeds, the rotor tends to spin about its mass center, providing an orbit
radius equal to the cg eccentricity.
Undamped Free Response. The free response, or homogeneous solution, is
obtained from Eq. (4) with the excitation forces excluded, i.e.,
m 0
0 m
€u
€v þ kt 0
0 kt
For an arbitrary set of initial conditions,
uð0Þ ¼uo
vð0Þ ¼vo
and
the solution of Eq. (6) yields
_uð0Þ ¼ _uo
_vð0Þ ¼ _vo
u
v
¼ 0
0
uðtÞ ¼uo cos ott þ _uo
sin ott ð8aÞ
ot
vðtÞ ¼vo cos ott þ _vo
ot
ð6Þ
ð7Þ
sin ott ð8bÞ
This free-vibration response is clearly harmonic with the same natural
undamped frequency ot ¼ ffiffiffiffiffiffiffiffiffiffi p
kt=m in both the x- and y-directions of motion.
Atypicalfree-vibrationorbit,relativetoanobserverinðx,y,zÞ,isillustrated
in Fig. 8. The orbit may precess either in aforward or backward direction
depending on the initial conditions. The precession direction is defined as
forward if it is in the same direction as the rotor spin-speed and backward if
in the opposite direction.
The orbit is generally a standard ellipse, however, a few special
degenerate cases may exist. First, if the initial conditions for the x- and
y-components of displacement and velocity are proportionally related, i.e.,
vð0Þ ¼uð0Þ tan y and _vð0Þ ¼ _uð0Þ tan y, the orbit reduces to an oscillatory
straight-line motion at the angle y relative to the x-axis, i.e., an ellipse with
zero semi-minor axis. A second special case exists if the initial conditions
satisfy the relations _uð0Þ ¼ vð0Þot and _vð0Þ ¼uð0Þot. For this case, the
resulting orbit is circular, i.e., an ellipse with zero eccentricity. These two
special cases are illustrated in Fig. 9.
It is common practice in rotordynamics to graph the natural
frequencies of the rotor system as a function of the rotor spin-speed. These
graphs are typically referred to as ‘‘whirl speed maps’’ or ‘‘Campbell
diagrams for whirl speeds.’’ For the case of the classic Laval–Jeffcott rotor,
Copyright © 2003 Marcel Dekker, Inc.