Handbook of Turbomachinery Second Edition Revised - Ventech!

vibration (either forward or backward) with a frequency equal to a
submultiple of o (i.e., o=n). This situation may exist when the rotor tends to
respond at one of the natural frequencies of whirl and is simultaneously
excited by a mechanism at the rotor spin-speed or by a self-excitation
mechanism. When n is an integer, the period of the superposed motion is
nð2p=oÞ.
TwoexamplesofsuperposedellipticvibrationsareillustratedinFig.5.
The first case [Fig. 5(a)] is a superposition of two forward circular orbits.
Both orbits have an amplitude of 1.0, one with a whirl frequency o, and the
other with a frequency of o=2. The second case [Fig. 5(b)] is a superposition
of a forward circular orbit with amplitude 1.0 and frequency o, and a
backward circular orbit with an amplitude 1.0 and frequency o=2. More
discussion on the superposition of elliptic vibrations is included in Tondl [19].
SINGLE-DISK ROTOR SYSTEMS
The Laval–Jeffcott Rotor: Translational Motion
An elementary rotor system model, frequently used by designers and
analysts to study and simulate basic rotordynamic system characteristics, is
a single rigid disk centrally located on a uniform flexible shaft as illustrated
inFig.6.Theshaftisconsideredtohavenegligiblemassandissupportedon
bearings that are considered to be infinitely stiff. A form of this single-disk
rotor model was first presented by A. Fo¨ppl [8] in Germany in 1895. Fo¨ppl
named this elementary model the Laval rotor in recognition of the
contributions made in the area of turbomachinery by the Swedish engineer
Gustav Patric De Laval. In 1919, the English engineer, H. H. Jeffcott [11]
subsequently presented a study using the same elementary model. As the
result of references made to these two pioneering works over the past several
decades, this single-disk rotor model is referred to as the Laval rotor in some
sections of the world and as the Jeffcott rotor in others. In order to
recognize the early work of both Fo¨ppl and Jeffcott, the name Laval–
Jeffcott rotor is adopted here.
The rigid disk is of negligible axial extent and has a mass, m, with mass
center located a distance, e, in the x-direction from the geometric disk
center, o. The diametral and polar mass moments of inertia for the disk are
Id and Ip, respectively. The shaft has a translational stiffness, kt (associated
with shaft bending), which depends in magnitude on the shaft geometrical
and material properties. For a simply supported shaft with modulus of
elasticity, E, and diametral area moment of inertia, I, the translational
stiffness at the center of the beam is kt ¼ 48EI=L3 . Only the undamped
translational motion of the disk in the ðx, yÞ-plane is included for this classic
Copyright © 2003 Marcel Dekker, Inc.