ROTOR
DYNAMIC
EQUATION
IMPROVEMENTS
Earlier this year, to assist one of our customers, Don Bently authored a short synopsis of work done by himself and
Dr. Agnes Muszynska related to the defining equations for rotor dynamics. We felt the summary would be valuable
for a broader audience and provide it here for the benefit of all our ORBIT readers.
ry,
or Dynamic Theo
ot
R
al
ic
ss
la
C
in
to
ral Deficiencies
rch Corporation
ea
es
R
ic
Summar y of Seve
am
yn
D
Bently Rotor
and Work Done by
iciencies
Correct These Def
15
tute (EPRI) about
wer Research Insti
Po
c
tri
ec
El
e
th
ed by
mic studies publish
In some rotor dyna
om was used.
ed
fre
gle degree of
sin
a
is impossible to
ly
on
o,
ag
s
year
freedom is that it
of
ee
gr
de
e
gl
sin
on a
y directions must
a study based up
tics – both x and
ris
th
te
wi
ac
ar
lem
ch
ob
ss
pr
ne
e
is a serious
Th
amic Stiff
forward and reverse
n
from reverse Dyn
ee
d
ar
tw
rw
be
fo
sh
sh
ui
ui
ng
ng
an forward
disti
to disti
different effects th
ally, the inability
er
tly
en
nc
G
sti
.
di
ad
s
ste
ce
in
du
be used
stiffness intro
ckward (reverse)
een the two.
mistake because ba
differentiate betw
to
l
ta
vi
both lateral
is
it
d
an
stiffness,
excite a rotor in
at
th
es
di
stu
ic
m
dyna
paratus for
relies upon rotor
this, we employ ap
e,
or
do
ef
To
er
.
th
ns
a,
io
ad
ct
re
ev
eral di
vice such
Bently N
response in both lat
a unidirectional de
its
an
th
re
su
er
ea
th
m
ra
,
d
ce
an
vi
,
ce de
ckward
directions
a rotating unbalan
teristic from the ba
as
ac
ar
ch
ch
su
s
n,
es
io
ffn
at
sti
rb
d
circular pertu
separate the forwar
studies allow us to
as a shaker. Such
ty and guesswork.
ka and
eliminate uncertain
d
an
,
tic
ris
te
ac
Dr. Agnes Muszyns
,
ar
ch
om
ed
fre
of
s
ee
gr
ents using two de
equations.
nce upon experim
tal rotor dynamic
en
am
nd
fu
e
In addition to relia
th
to
s
ge
an
ch
r
three majo
I have introduced
represented as
tl y
By D on al d E. B en
Term
1. Cross Stiffness
lled “cross
erature uses so-ca
Much existing lit
the cross
er,
ture. Furth
stiffness” nomencla
ated as
tre
are
and K yx
stiffness terms K xy
and I
ka
ns
zy
us
bles. Dr. M
independent varia
ent
nd
pe
de
in
t
terms are no
showed that these
cterized
d are properly chara
of one another an
ping, λ
m
da
is
, where D
by the term DλΩ
average
l
tia
en
fer
id circum
(lambda) is the flu
speed.
ive
tat
ro
d Ω is shaft
velocity ratio, an
tial
gen
tan
e
th
as
m, DλΩ,
We refer to this ter
the K xy
. It is equivalent to
or quadrature term
enjoys
re
tu
cla
en
t our nom
and K yx terms, bu
it de fin es
ad va nt ag e th at
th e sig ni fic an t
r dynamic
to
ro
of the basic
stiffness in terms
eed, and
mping, rotative sp
parameters of da
l model
es it a more usefu
lambda. This mak
rstanding.
for diagnostic unde
34
ORBIT
3Q01
is sometimes
“Cross stiffness”
a rotor.
t and right below
springs at 45Þ lef
ess is
ffn
sti
correct. Cross
This is totally in
, or
ial
nt
ge
s effect – a tan
actually a sideway
to
lar
cu
di
that acts perpen
quadrature, term
t.
en
em
lac
sp
of ro to r di
th e di re cti on
e
es
th
to
re fer
pe op le
So m eti m es ,
of
sed
po
m
ms as being co
“perpendicular” ter
wever,
y” components; ho
ar
“real” and “imagin
ture.
cla
en
m
using such no
I have never liked
th e
t
ou
ab
g “im ag in ar y”
Th er e is no th in
ays
alw
da
va
and Bently Ne
tangential term,
of
ad
ste
in
“q ua dr at ur e”
us es th e wo rd
a
in
d
ve
ser
can be easily ob
“imaginar y.” It
wn
do
sh
pu
u
m. When yo
rotor dynamic syste
t also to
not only down, bu
es
on a rotor, it mov
the side.
ROTOR
2. Direct Stiffness Term
Even the Direct Stiffness term has
been badly
misunderstood over the years beca
use it is
often modeled in terms of rotative spee
d under
the assumption that only the circu
mferential
profile of the supporting pressure wed
ge is of
imp orta nce.
Our wor k with exte rnal ly
pres suri zed bear ings show s that
Dire ct
Stiffness is the result of the conv
erging /
diverging pressure wedge that is axia
l (along
the roto r) in orie ntat ion, rath
er than
circu mferentia l (aro und the roto
r). This
clari fies a conc ept that has
been
misunderstood and misapplied for
more than
a century.
3. Fluid Iner tia Term
Our rotor dynamic model introduce
d a term
to quantify the fluid iner tia effect.
This effect
can become very large and exhibit
influence
on the rotor that is quite significant,
adding to
the iner tial effects of the rotor mass
itself.
However, although it behaves like iner
tia, and
our model characterizes the roto
r behavior
quite accurately, it is incorrect to thin
k of our
term as a literal mass-based iner tia.
This can
be observed very easily by introduci
ng an air
bubble into a cylindrical bearing syste
m. The
fluid iner tia effect disappears insta
ntly, and
the system reverts to normal spring,
mass, and
dam ping char acte risti cs. The refo
re, it is
perfectly clear that it is neither a Cori
olis nor
gyroscopic 1 effect. This fluid iner tia
effect acts
very muc h like a nega tive spri
ng and
introduces a negative stiffness effec
t.
DYNAMIC
EQUATION
IMPROVEMENTS
The befo re-m enti oned imp rove
men ts to the
defin ing equa tion s are nece ssary
for the mos t
accurate modeling and diagnostics
of modern rotor
dynamic systems. They also more
clearly describe
both simple and complex rotor syste
ms. Finally, in
addition to using the most up-to-da
te equations as
the basis for calculations, we strongly
advocate the
use of a graphical analysis method
known as RootLocus in studying the stability of
rotor dynamic
systems. It has been very widely appl
ied to study
the stability of electronic control syste
ms since its
introduction by Walter Evans in 1954
. However, it
is less commonly applied to rotor dyna
mic systems,
which is a shame since it is such a pow
erful analysis
tool, vastly superior to older meth
ods such as
Cam pbel l diag ram s or loga rithm
ic-d ecre men t
plots. Root-Locus is addressed very
capably in the
following excellent texts:
1. Evans, W. R., Control-System Dyn
amics,
McGraw-Hill Book Company, Inc.
, New York,
New York, 1954.
2. Nise, Norman S., Control Systems
Engineering
with MATLAB, Third Edition, John
Wiley &
Sons, 2000.
3. Ogata, Katsuhiko, Modern Control
Engineering,
Third Edition, Prentice-Hall, Inc.,
1997.
If you would like further assistance
with any of the
concepts or methods covered in this
summar y, I
would be pleased to help.
1 It should be
pointed out, however, that while fluid
inertia is clearly not a gyroscopic effec
gyroscopic behavior are correctly done
t, the existing descriptions of
. Therefore, gyroscopics continue to
be a valuable part of properly chara
rotor’s behavior.
cterizing a
3Q01
ORBIT
35