Revisiting Rotor Rigid Body Modes

10
International Journal of Rotating Machinery 1999, Vol. 5, No. 3, pp. 193-201 Reprints available directly from the publisher Photocopying permitted by license only (C) 1999 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers 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 final form 20 June 1998) This paper presents an analytical model of a rigid rotor supported in two fluid film bearings with an emphasis on predicting the instability threshold speed. The factors contributing to the stability of the rotor are discussed and presented graphically using root locus plots. The parametric 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) and relatively high transverse moment of inertia. It is found that the pivotal mode instability 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 infinitely stable and the lateral mode controls stability. (iii) Highly gyroscopic rotor with relatively high transverse moment of inertia. It is found that the pivotal mode exists and controls stability. The lateral mode always exists. Both asymmetry in rotor geometry (location of center of mass with respect to the bearings) and fluid bearing parameters (stiffness, damping) are considered. It is shown that, for a given bearing asymmetry parameter, the maximum stability is achieved when the geometric asymmetry parameter is of equal value. The recommendations on the optimal 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 describing the 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

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Rotor Rigid Body Modes

Transcript of Revisiting Rotor Rigid Body Modes

  • 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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