-
International Journal of Rotating Machinery1999, Vol. 5, No. 3,
pp. 193-201Reprints available directly from the
publisherPhotocopying permitted by license only
(C) 1999 OPA (Overseas Publishers Association) N.V.Published by
license under
the Gordon and Breach SciencePublishers imprint.
Printed in Malaysia.
Revisiting Rotor Rigid Body Modes:Parametric Study of
Stability
NATHAN B. LITTRELL, AGNES MUSZYNSKA* and PAUL GOLDMAN
Bently Rotor Dynamics Research Corporation, P.O. Box 2529,
Minden, NV 89423-2529, USA
(Received 16 April 1998," In finalform 20 June 1998)
This paper presents an analytical model of a rigid rotor
supported in two fluid film bearingswith an emphasis on predicting
the instability threshold speed. The factors contributing to
thestability of the rotor are discussed and presented graphically
using root locus plots. Theparametric study of the stability starts
from the discussion of the rotor/bearing system with"mirror
symmetry". Three basic cases are considered:
(i) Rotor with relatively small gyroscopic effect (small polar
moment of inertia) andrelatively high transverse moment of inertia.
It is found that the pivotal modeinstability exists, but the
lateral mode controls stability.
(ii) Highly gyroscopic rotor (relatively large polar moment of
inertia) with also rela-tively low transverse moment of inertia. It
is found that the pivotal mode is infinitelystable and the lateral
mode controls stability.
(iii) Highly gyroscopic rotor with relatively high transverse
moment of inertia. It isfound that the pivotal mode exists and
controls stability. The lateral mode alwaysexists.
Both asymmetry in rotor geometry (location of center of mass
with respect to thebearings) and fluid bearing parameters
(stiffness, damping) are considered. It is shownthat, for a given
bearing asymmetry parameter, the maximum stability is achieved
whenthe geometric asymmetry parameter is of equal value. The
recommendations on theoptimal design from the stability standpoint
are given.
Keywords." Gyroscopic, Fluid bearing, Optimal design, Stability,
Root locus
INTRODUCTION
The gyroscopic effect, as related to rotordy-namics, has been
researched in many papers,starting from pioneering works by
Yamamoto
(1954), Dimentberg (1961) and Crandall (1982,1961) considering
various ways of describingthe rotor lateral and angular motion. A
descrip-tion of gyroscopic effects together with more com-plete
lists of references can be found in books by
Corresponding author. Tel.: 702 782-3611. Fax: 702 782-9236.
E-mail: [email protected].
193
-
194 N.B. LITTRELL et al.
Ehrich (1992) and Vance (1988). An experimentalwork dealing with
parameter identification for therotor system with large gyroscopic
influence isreported by Bently et al. (1986). The
interactionbetween the stabilizing effect of gyroscopics
anddestabilizing effect of fluid-induced tangentialforces has been
investigated by Muijderman (1986).An earlier paper by Hatch and
Bently (1995)
presented a model and stability criteria for a rigidrotor with
significant gyroscopic effects supportedin one fluid film bearing
and one roller elementbearing. The latter was assumed to have
infinitelateral stiffness. The significant finding ofthis paperwas
that it is possible to make a rotor pivotal modeabsolutely
resistant to fluid induced instabilities byusing the gyroscopic
effect to counteract the forcesdriving the fluid instability.
Experimentation veri-fied the conclusions drawn in this paper,
leading tothe feasibility of employing the stability criteria
onreal machines to correct fluid instability problems.One of the
questions that came up was, "what if a
machine has two fluid film bearings?" By addinganother fluid
film bearing, it is expected that therewill be a translational mode
as well as a pivotalmode. The second fluid film bearing introduces
anew, strictly lateral, degree of freedom, and anothermode.
Investigating the stability criteria for thismore complex model was
the motivation for thisstudy.
ROTOR MODEL
A diagram of the rotor modeled is shown in Fig. 1.The model
consists of a rotor supported in two fluidlubricated bearings with
fluid film radial stiffnessKb, and rotating damping Db, subscripted
with aor 2 to indicate which bearing is referred to. Therotor
itself has parameters of mass M, polar massmoment of inertia Ip,
and transverse mass momentsof inertia Ix, Iy.The coordinate system
for describing the rotor
motion is shown in Figs. 2 and 3. It is a combinationof a
Cartesian and a spherical system. The basis ofthe absolute
stationary coordinate system xyz has
Bearing Bearing 2
FIGURE Diagram of rotor system used in model.
FIGURE 2 Coordinate systems for the rigid rotor.
its origin at the point O which is coincident with therotor mass
center, Or, when the rotor is centered inthe bearing clearances.
The translational motion ofthe rotor is assumed planar, and is
described bylateral displacements x and y. Additionally, aspherical
coordinate system is introduced withorigin at the rotor mass center
Or, and, and asangles of yaw and pitch respectively (Fig. 3).
Notethat for the rotor as shown in Fig. 1, distance z isnegative.
Of course, generally a rotor may have itsmass center outside of the
bearing span in whichcase zl and z2 are both of the same sign. Such
rotorsare usually called "overhung" rotors.Any lateral displacement
of the rotor relative to
its center of mass, Or, can be described by the twoangles plus
the distance z from Or. The combinationof the (x,y) and (X, )
coordinate systems providethe independent coordinates used for a
Lagrangian
-
ROTOR RIGID BODY MODES 195
FIGURE 3 Angles X, b of yaw and pitch.
derivation of the equations of motion (notpresented in this
paper).
Assumptions
Bearing fluid film stiffness and damping proper-ties are
considered laterally isotropic. This istrue for the case of lightly
loaded bearings/seals.The rotor is assumed rigid.It is assumed that
the angular displacement oftherotor is small, consequently the
distance from theorigin along the axis is maintained as simply
z.Additionally, gyroscopic effects of second orderor higher are
neglected.There is no axial motion of the rotor.The fluid film
damping in the bearings is directlyproportional to the fluid film
stiffness.The fluid circumferential average velocity ratio,
A(Muszynska, 1988), is considered the same inboth bearings.The
rotative speed f, of the rotor is assumedconstant. There is no
torsional vibration. Allcalculations consider variation of rotor
speeddiscretely.
Equations of Motion
The four degrees of freedom x, y, X and can bereduced to two by
introduction of the complexcoordinates r=x+jy, and similarly, tI,
=X +jb.Using these identities, the equations for the free
response of the system are as follows:
M/: + (Dbl + Db2)? + [(Kbl + Kb2)jAf(Dbl + Dbz)]r. + (Db2z2 +
DblZl)
+ [Kb2z2 + Kblz --j(Db2z2 + Obz)AQ]t 0,(1)
It + [(Db,z2 + Db2z)--jlpf] + [(Kb,z2 + Kb2z22)--jAf(Db,Zl2 +
Db2z22)] + (Db2z2 + Db,z,+ (Kbzz2 + Kbz jAf(Db2z2 + Dblz))r O,
(2)where A is the fluid circumferential average velocityratio as
shown by Muszynska (1988). In order toreduce the complexity of the
equation set andpossibly gain some insight into the
physicalbehavior of the rotor system, nondimensionalforms of Eqs.
(1) and (2) are generated using therelations given in Table I:
h" + 2th + (1 2jtA)h+ bto(O -java0) 4- atoO O, (3)
I,"+ 2/(0 (P jka2) + T]2J
bto ato+ --(h jcoh) + --h 0. (4)
STABILITY OF THE MIRROR SYMMETRICSYSTEM-UNCOUPLED MODES
The rotor system (Fig. 1) is considered "mirrorsymmetric" if the
distance from the center ofmass to either bearing is identical
(]zl]and the bearing characteristics are the same (DblOb2 Kbl-
Kb2). In that case the cross-couplingfactors, ato and bto, vanish
and translational andpivotal modes, Eqs. (3) and (4) become
uncoupled.
(1) For translational mode:
h" + 2th + (1 2jtAco)h 0. (5)
-
196 N.B. LITTRELL et al.
TABLE Nomenclature of factors used to nondimensionalize
equations of motion
Radius of gyration, p
Nondimensional displacement, h
Ratio of transverse to polar moment of inertia,
Lateral natural frequency, ut
Angular natural frequency, "u0Lateral damping factor, tAngular
damping factor, 0Natural frequency ratio,
Stiffness cross coupling factor, ato
Nondimensional rotor speed, co
Nondimensional time, r
Damping cross coupling factor, bto
(2) For pivotal mode:
0"-j0 + 2r/0(0- jAco0) + r/Z0 0. (6)The instability threshold
speed for the translationalmode is as follows:
(t) /t v/Kbl -[- Kb2From Eq. (7) it can be seen that the
stability ofthe
translational mode can be maximized three ways:Increasing direct
stiffness values, decreasing themodal mass, M, or decreasing the
bearing fluidcircumferential average velocity ratio, A. All
willraise the speed at which the translational modebecomes
unstable. There is no unconditionalstability for the translational
mode. The instabilitythreshold speed for the pivotal mode is as
follows:
(o) uo /KblZ -I- Kb2zth
The interesting result from Eq. (8) is that anunconditional
stability criterion can be derivedusing finite and achievable
parameters as shown
by Hatch and Bently (1995). By setting the term inthe
denominator under the radical less than orequal to zero, the
following unconditional stabilitycriterion is reached:
-
ROTOR RIGID BODY MODES 197
lOOO
800
600
400
200
-200-400
-500 -400 -300 -200 -100 O 100 200Decay Rate (rad/s)
SYSTEM PARAME;rERS
M 2 kgKbl 28e3 N/mKb2 28e3 N/mDbl 262.5 N s/mDb2 262.5 N s/mZl
=-0.102 mz2 0.102 m
0.45Ip 0.022 kg mIt Ip / A kg rn
FIGURE 4 Root locus plot of "mirror symmetric" rotor with ratio
= 1/A. Plot was generated using the parameters givenabove and
varying rotative speed, f2. The Os indicate the beginning of the
locus at ft 0. The xs indicate the position of theroots at the
point where the system becomes unstable at f- 3581 rpm.
500
400
3O0
200
lOO
-100
o lOO
FIGURE 5 Root locus plot of symmetric rotor with
systemparameters as in Figure 4, except for the parameter
It=2Ip/Akgm2. The Os indicate the beginning of the locus atf 0. The
xs indicate the position of the roots at the pointwhere the system
first becomes unstable at f 2435 rpm.
100
150 O0 -50 0 50Decay Rate (rad/s)
FIGURE 6 Root locus plot of symmetric rotor with
systemparameters as in Figure 4, except for the parameter It=
1.1Ip/Akgm2. The Os indicate the beginning of the locus atf-0. The
xs indicate the position of the roots at the pointwhere the system
first becomes unstable at f 3581 rpm.
Figure 6 illustrates a case where the criterion (9) isnot met,
yet the lateral mode still controls stability.
ROTOR AXIAL ASYMMETRY: COUPLINGOF THE MODES
The introduction of asymmetry in either geometricor bearing
parameters causes a coupling of thelateral and pivotal modes. The
purpose of this
investigation is to determine if there is ever animprovement in
the rotor stability from asym-metry. The deviations of the rotor
system fromthe mirror symmetric case can be expressed by
twocoefficients:
Coefficient of geometric asymmetry a
The distances from the center of mass to thebearings, Z and Z2,
can be expressed in terms of
-
198 N.B. LITTRELL et al.
TABLE II Transformation of nondimensionalization parameters by
asymmetry factors
Lateral natural frequency, u,Natural frequency ratio, r/Lateral
damping factor, tAngular damping factor, 0Damping cross coupling
factor, bwstiffness cross coupling factor, ato
rl L/(2pv/-)v/i + a 2ab, D/(2x/--)
0=,b,o= 2,a,o
ato L/(2p)(a- b)
the total distance between the bearings, L, and anasymmetry
factor, a, as follows:
L Lz1 -(a- 1)--, z2 -(1 / a)-. (10)
If a=0, the rotor system is geometricallysymmetric, if al >
then the rotor has overhungdesign (i.e. the rotor center of mass
lies outside ofthe two bearing supports).
Coefficient of stiffness asymmetry b
Similarly, the fluid film direct stiffness Kbl and Kb2can be
parametrized in terms of the total stiffnessK-Kb + Kb2, and an
asymmetry parameter, b, asfollows:
K KKbl (1 / b)-, Kb2 (1 b)-. (11)
The same parameter, b, can be used to describethe damping
asymmetry based on the assumptionthat the fluid film damping is
proportional tostiffness. D represents the total damping Dbl /
Db2.
D D (12)+ Z) 2
The parameter b can range from 0 to 1. Asymmetric system
corresponds to b =0. The non-dimensional parameters listed in Table
I are nowtransformed in terms of the asymmetry parameters,as
follows in Table II:Taking the relations in Table II into account,
the
characteristic equation for Eqs. (3) and (4) canbe presented in
the following nondimensional
format:
[$2/ 2IS / (1- 2jt/CO)] [$2/ (2]t --j)S / ]22jtr/ACO a2to[1 /
2t(S-- jAco)] 2 0, (13)
where s is an eigenvalue. This is a fourth-orderequation and
consequently has four roots, two ofwhich can have positive real
parts. The followingexpression represents the relation for the
instabilitythreshold of the system:
/,t q ]2 /( _)2 a2toth----" /
-1-" 1-- /--t
(14)
where a is related to the stability criterion given by(9) as
follows"
ip.It
If cr > 0, the system has a finite pivotal modeinstability
threshold, cr < 0 corresponds to infinitestability of the
pivotal mode. Note that expression(14) for ato-0 turns into
instability thresholds (7)and (8) for the mirror symmetric case.
Dependingon the parameters, there can be instability thresh-olds
corresponding to either the lateral or pivotalmodes. The effects of
asymmetry on the instabilitythreshold can now be investigated.
Figure 7 shows a family of curves, each repre-senting a constant
value ofthe fluid film asymmetryparameter, b. The stability
criterion for the pivotalmode (9) is not met here, consequently the
pivotalmode is the stability controlling factor. In this case
-
ROTOR RIGID BODY MODES
700
600
500
400300_-- 200
100
0-2
.
--
Tr=nshttional mode.,.,.
-1 0 2Geometric Asymmetry Factor
SYSTEM PARAMETERS
M 2 kgK 56e3 N/m
lip 0.204 rn0.453.7e-2 kgmI 2Ip/2 kgm
(r>0)
FIGURE 7 Instability threshold versus geometric parameter
asymmetry.
199
0 2Geometric Asymmetry Factor
sYstem parameters asin Figure 7, exceptrotor length
isincreased.L .684 rn
(o>0)
FIGURE 8 Instability threshold versus geometric parameter
asymmetry.
it can be seen that asymmetry in either a or b canhave only
detrimental effects on the instabilitythreshold.The next case
investigated is shown in Fig. 8. All
the parameters are identical with Fig. 7, except thatrotor
length, L, is increased. The important insightgained from this case
is that in all cases the insta-bility threshold is higher than with
a short bearingspan. Additionally, note that the peak
instabilitythreshold at b a =0.6 is the same as with a sym-metric
system b a 0. Peak stability is reduced atgreater values of fluid
film asymmetry, b 0.9, yetis still improved over the short rotor
case.
Figure 9 shows the case where the stabilitycriterion (9) is
satisfied. The rotor length, L, is
returned to the original smaller value used in Fig. 7.Remarkable
about this case is the fact that the peakinstability thresholds are
equal regardless of b andthat the peak value of stability is
reached at a b.The translational mode is now entirely
responsiblefor determining stability.
Figure 10 is similar to Fig. 9 with the exceptionthat rotor
length, L, has been increased. At firstglance, the results appear
identical because the peakvalues are the same, but closer
inspection will showthat the curves take slightly different paths
to arriveat the peak values. From this it can be concludedthat
lengthening the rotor does not have the samebenefits when the
gyroscopic mode is not thecontrolling factor for stability.
-
200 N.B. LITTRELL et al.
400.
350
:oo
2sI200
150 .
00
50
-0.5 0 0.5 1.5Geometric Asymmetry Factor
System prameters asin Figure 7 excepttransverse moment
ofinertia
It .51p/2 kgm:.(o
-
ROTOR RIGID BODY MODES 201
is achieved when the rotor center of mass is shiftedaway from
the stiffer bearing on the geometricasymmetry parameter of the same
value. Theinstability threshold maximum corresponds to thesymmetric
rotor.
This implies that stability may be maximized inthis case by
distributing the mass of the rotor insuch a way as to minimize the
ratio of transversemoment of inertia to polar moment of inertia,
.
In the case where the translational mode iscontrolling
stability, The instability threshold canonly be managed by
manipulation of the bearingparameters or total rotor mass.
NOMENCLATURES
a
atobbtoDbl, Db2
D
Kbl, Kb2
KZ1, Z2LMOOrs
geometric asymmetry parameterstiffness cross-coupling
parameterstiffness asymmetry parameterdamping cross-coupling
parameterbearing and 2 fluid film damping,respectivelytotal lateral
fluid film dampingtransverse and polar moments ofinertia,
respectivelybearing and 2 direct fluid filmstiffnesstotal lateral
fluid film stiffnessbearing and 2 location, respectivelytotal
distance between the bearingsrotor system massabsolute coordinate
system originrotor center of masseigenvalue
r-x+jy
-x+J
th
lateral displacement of rotor centerof mass in the stationary
system ofcoordinatesfluid bearing circumferential averagevelocity
ratioangle of yaw and pitch, respectivelycomplex angular
displacementrotative speedinstability thresholds for translationand
pivotal modes
ReferencesBently, D.E. et al. (1986). Identification of the
modal parameters
by perturbation testing of a rotor with strong gyroscopiceffect,
Proc. of International Conference on Rotordynamics,Tokyo,
Japan.
Brosens, P.J. and Crandall, S.H. (1982). Whirling of
unsymme-trical rotors, Journal ofAppliedMechanics, Paper 61-APM-
10.
Crandall, S.H. and Karnopp, D.C. (1961). Dynamics ofMechan-ical
and Electromechanical Systems, Krieger Publishing Co.,Malabar,
F1.
Dimentberg, F.M. (1961). Flexural Vibrations ofRotating
Shafts,Butterworths, London.
Ehrich, F.E. (1992). Handbook ofRotordynamics,
McGraw-Hill.Hatch, C.T. and Bently, D.E. (1995). Moment
equation
representation and stability analysis of a 1-CDOF overhungrotor
model with fluid bearing and gyroscopic effects,BRDRC Report No.
8.
Muijderman, E.A. (1986). Algebraic formulas for the thresholdand
mode of instability and the first critical speed of a
simpleflexibly supported (overhung) rotor-bearing system, Proc.
ofthe International Conference on Rotordynamics, Tokyo, Japan,p.
201.
Muszynska, A. (1988). Improvements in lightly loaded
rotor/bearing and rotor/seal models, Trans. ASME Journal
ofVibration and Acoustics, 110(2), 129-136.
Muszynska, A. (1995). Modal testing of rotors with
fluidinteraction, International Journal of Rotating Machinery,1(2),
83-116.
Vance, J.M. (1988). Rotordynamics ofRotating Machinery,
JohnWiley & Sons, New York.
Yamamoto, T. (1954). On the critical speeds of a shaft,
Memoirsof the Faculty ofEngineering, Nagoya University.
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