Handbook of Turbomachinery Second Edition Revised - Ventech!

this graph is quite simple; however, it is presented here to introduce the
concept as a forerunner of whirl speed maps for more complicated systems.
For the case of a Laval–Jeffcott rotor in translation only, two natural
frequencies of whirl exist. The first is a forward whirl (denoted as 1f) with
natural whirl frequency, ot, and the second is a backward whirl (denoted as
1b) at the frequency, ot. These natural frequencies are not a function of
the rotor spin-speed since the natural motion of the rigid disk is pure
translation. Thus, gyroscopic moments are not present for this simple
translational model.
One option is to graph the positive and negative natural whirl
frequencies as a function of the spin-speed including the entire range of
positive and negative spin-speeds (i.e., a four-quadrant map). This graph is
illustrated in Fig. 10(a) for the Laval–Jeffcott rotor. When the rotor is
defined as having a positive spin-speed, a positive whirl frequency
corresponds to a forward mode and a negative whirl frequency corresponds
to a backward mode. When the rotor is defined as having a negative spinspeed,
a positive whirl frequency corresponds to a backward mode and a
negative whirl frequency corresponds to a forward mode. Recall: a forward
mode is simply defined as a whirling motion in the same sense as the spin
while a backward mode is in the opposite sense.
Because of the polar symmetry of the four-quadrant graph, it is
common practice to present a whirl speed map as a single-quadrant graph
associated with a positive spin-speed and include both the positive and
negative whirl frequencies in the first quadrant. Quite often, different line
types are used to distinguish the positive (e.g., solid) and negative (e.g.,
dashed) whirl frequencies. A single-quadrant whirl speed map for the Laval–
Jeffcott rotor is illustrated in Fig. 10(b). For this simple system, the positive
and negative whirl frequencies have the same absolute value and are
therefore coincident on the graph.
It is common practice to include an excitation frequency line, e.g.,
oexc ¼ rO, on whirl speed maps. This line indicates an excitation frequency
that is a multiple, r, of the rotor spin-speed, O. The line shown in Fig. 10(a)
and (b) corresponds to r ¼ 1 and indicates an excitation frequency that is
synchronous with the rotor spin-speed. This excitation is generally of
interest because rotating unbalance is always present in a rotor system, and
the associated excitation is synchronous with the rotor spin-speed, i.e.,
r ¼ 1. An intersection of the excitation frequency line with a whirl speed line
identifies a ‘‘critical speed’’ or rotor spin-speed, n, associated with a
resonance condition. For the Laval–Jeffcott rotor, only one forward critical
speed exists for synchronous spin-speed excitation, with a value of nf ¼ ot.
A second type of graph that is often used by designers to assist in
making design parameter decisions is called a ‘‘critical speed map.’’ This
Copyright © 2003 Marcel Dekker, Inc.