Revisiting Rotor Rigid Body Modes : Parametric Study of Stability

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:


INTRODUCTION
The gyroscopic effect, as related to rotordynamics, 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: agnes@brdrc.com.Ehrich (1992)  and  Vance (1988).An experimental work dealing with parameter identification for the rotor system with large gyroscopic influence is reported by Bently et al. (1986).The interaction between the stabilizing effect of gyroscopics and destabilizing effect of fluid-induced tangential forces has been investigated by Muijderman (1986).
An earlier paper by Hatch and Bently (1995) presented a model and stability criteria for a rigid rotor with significant gyroscopic effects supported in one fluid film bearing and one roller element bearing.The latter was assumed to have infinite lateral stiffness.The significant finding of this paper was that it is possible to make a rotor pivotal mode absolutely resistant to fluid induced instabilities by using the gyroscopic effect to counteract the forces driving the fluid instability.Experimentation veri- fied the conclusions drawn in this paper, leading to the feasibility of employing the stability criteria on real machines to correct fluid instability problems.One of the questions that came up was, "what if a machine has two fluid film bearings?"By adding another fluid film bearing, it is expected that there will be a translational mode as well as a pivotal mode.The second fluid film bearing introduces a new, strictly lateral, degree of freedom, and another mode.Investigating the stability criteria for this more complex model was the motivation for this study.

ROTOR MODEL
A diagram of the rotor modeled is shown in Fig. 1.
The model consists of a rotor supported in two fluid lubricated bearings with fluid film radial stiffness Kb, and rotating damping Db, subscripted with a or 2 to indicate which bearing is referred to.The rotor itself has parameters of mass M, polar mass moment of inertia Ip, and transverse mass moments of inertia Ix, Iy.
The coordinate system for describing the rotor motion is shown in Figs. 2 and 3.It is a combination of a Cartesian and a spherical system.The basis of the absolute stationary coordinate system xyz has  its origin at the point O which is coincident with the rotor mass center, Or, when the rotor is centered in the bearing clearances.The translational motion of the rotor is assumed planar, and is described by lateral displacements x and y.Additionally, a spherical coordinate system is introduced with origin at the rotor mass center Or, and, and as angles of yaw and pitch respectively (Fig. 3).Note that for the rotor as shown in Fig. 1, distance z is negative.Of course, generally a rotor may have its mass center outside of the bearing span in which case zl and z2 are both of the same sign.Such rotors are usually called "overhung" rotors.
Any lateral displacement of the rotor relative to its center of mass, Or, can be described by the two angles plus the distance z from Or.The combination of the (x,y) and (X, ) coordinate systems provide the independent coordinates used for a Lagrangian derivation of the equations of motion (not presented in this paper).

Assumptions
Bearing fluid film stiffness and damping properties are considered laterally isotropic.This is true for the case of lightly loaded bearings/ seals.The rotor is assumed rigid.It is assumed that the angular displacement of the rotor is small, consequently the distance from the origin along the axis is maintained as simply z.Additionally, gyroscopic effects of second order or higher are neglected.There is no axial motion of the rotor.The fluid film damping in the bearings is directly proportional to the fluid film stiffness.
The fluid circumferential average velocity ratio, A (Muszynska, 1988), is considered the same in both bearings.The rotative speed f, of the rotor is assumed constant.There is no torsional vibration.All calculations consider variation of rotor speed discretely.

Equations of Motion
The four degrees of freedom x, y, X and can be reduced to two by introduction of the complex coordinates 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,z where A is the fluid circumferential average velocity ratio as shown by Muszynska (1988).In order to reduce the complexity of the equation set and possibly gain some insight into the physical behavior of the rotor system, nondimensional forms of Eqs.(1) and (2) are generated using the relations given in Table I: I," ' + 2/(0 (P' jka2) + T]2 J bto ato + --(h' jcoh) + --h 0. (4) STABILITY OF THE MIRROR SYMMETRIC SYSTEM-UNCOUPLED MODES The rotor system (Fig. 1) is considered "mirror symmetric" if the distance from the center of mass to either bearing is identical (]zl] and the bearing characteristics are the same (Dbl Ob2 Kbl-Kb2).In that case the cross-coupling factors, ato and bto, vanish and translational and pivotal modes, Eqs. ( 3) and (4) become uncoupled.
(5) (2) For pivotal mode: 0"-j0' + 2r/0(0'-jAco0) + r/Z0 0. (6) The instability threshold speed for the translational mode is as follows: From Eq. ( 7) it can be seen that the stability of the translational mode can be maximized three ways: Increasing direct stiffness values, decreasing the modal mass, M, or decreasing the bearing fluid circumferential average velocity ratio, A. All will raise the speed at which the translational mode becomes unstable.There is no unconditional stability for the translational mode.The instability threshold speed for the pivotal mode is as follows: (o) uo /KblZ -I-Kb2z th The interesting result from Eq. ( 8) is that an unconditional stability criterion can be derived using finite and achievable parameters as shown by Hatch and Bently (1995).By setting the term in the denominator under the radical less than or equal to zero, the following unconditional stability criterion is reached: <A.
(9) p- Figure 4 shows a root locus plot of the rotor system with the criterion of Eq. ( 9) satisfied.By examining the roots of each of the decoupled equations ( 5) and ( 6), it can be seen that the lateral mode is the only source of the instability.A quick check of this is provided by observing that the roots generated by the gyroscopic term have a branch that is asymptotically approaching the zero real axis.
In Fig. 5 it can be seen that the lateral roots are unchanged, but now the gyroscopic root crosses the zero axis at lower rotative speed than the lateral root, thus controlling stability.This is due to the criterion set in (9) not being met.It should be noted that the stability of the gyroscopic mode does not necessarily have to be absolute, just better than the lateral mode, as the lowest threshold dominates.Root locus plot of "mirror symmetric" rotor with ratio = 1/A.Plot was generated using the parameters given above and varying rotative speed, f2.The O's indicate the beginning of the locus at ft 0. The x's indicate the position of the roots at the point where the system becomes unstable at f-3581 rpm. Figure 6 illustrates a case where the criterion (9) is not met, yet the lateral mode still controls stability.

ROTOR AXIAL ASYMMETRY: COUPLING OF THE MODES
The introduction of asymmetry in either geometric or bearing parameters causes a coupling of the lateral and pivotal modes.The purpose of this investigation is to determine if there is ever an improvement in the rotor stability from asym- metry.The deviations of the rotor system from the mirror symmetric case can be expressed by two coefficients: Coefficient of geometric asymmetry a The distances from the center of mass to the bearings, Z and Z2, can be expressed in terms of the total distance between the bearings, L, and an asymmetry factor, a, as follows: L L z1 -(a-1)--, z 2 -(1 / a)-. (10)  If a=0, the rotor system is geometrically symmetric, if al > then the rotor has overhung design (i.e. the rotor center of mass lies outside of the two bearing supports).

Coefficient of stiffness asymmetry b
Similarly, the fluid film direct stiffness' Kbl and Kb2 can be parametrized in terms of the total stiffness K-Kb + Kb2, and an asymmetry parameter, b, as follows: K K Kbl (1 / b)-, Kb2 (1 b)-. (11)  The same parameter, b, can be used to describe the damping asymmetry based on the assumption that the fluid film damping is proportional to stiffness.D represents the total damping Dbl / Db2.
The parameter b can range from 0 to 1.A symmetric system corresponds to b =0.The non- dimensional parameters listed in Table I are now transformed 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) can be presented in the following nondimensional format: where s is an eigenvalue.This is a fourth-order equation and consequently has four roots, two of which can have positive real parts.The following expression represents the relation for the instability threshold of the system: /,'t q ]2 /( _)2 a2to th----" / ' -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 mode instability threshold, cr < 0 corresponds to infinite stability of the pivotal mode.Note that expression (14) for ato-0 turns into instability thresholds (7)   and (8) for the mirror symmetric case.Depending on the parameters, there can be instability thresh- olds corresponding to either the lateral or pivotal modes.The effects of asymmetry on the instability threshold can now be investigated.
Figure 7 shows a family of curves, each repre- senting a constant value of the fluid film asymmetry parameter, b.The stability criterion for the pivotal mode (9) is not met here, consequently the pivotal mode is the stability controlling factor.In this case it can be seen that asymmetry in either a or b can have only detrimental effects on the instability threshold.
The next case investigated is shown in Fig. 8.All the parameters are identical with Fig. 7, except that rotor length, L, is increased.The important insight gained from this case is that in all cases the insta- bility threshold is higher than with a short bearing span.Additionally, note that the peak instability threshold at b a =0.6 is the same as with a symmetric system b a 0. Peak stability is reduced at greater values of fluid film asymmetry, b 0.9, yet is still improved over the short rotor case.
Figure 9 shows the case where the stability criterion ( 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 peak instability thresholds are equal regardless of b and that the peak value of stability is reached at a b.The translational mode is now entirely responsible for determining stability.
Figure 10 is similar to Fig. 9 with the exception that rotor length, L, has been increased.At first glance, the results appear identical because the peak values are the same, but closer inspection will show that the curves take slightly different paths to arrive at the peak values.From this it can be concluded that lengthening the rotor does not have the same benefits when the gyroscopic mode is not the controlling factor for stability.

CONCLUSIONS
The rotor system considered is a rigid rotor supported in two fluid film bearings.The latter are assumed to have isotropic stiffness and rotating damping characteristics as by Muszynska (1988).
This model, based on the strength of circumfer- ential flow, describes the "cross-coupled stiffness coefficient" as a product of damping, rotative speed, and A. This model was identified by extensive modal testing of rotor with fluid interaction by Muszynska (1995).The application of such a model allows a complete parametric stability analysis of the system.Consideration of stability of the steady state regime in the rotor system with "mirror symmetry" allows for three basic cases: (i) Highly gyroscopic rotor with high transverse moment of inertia.There are two instability thresholds: one for both translational and pivotal modes.The latter as the lowest controls the rotor stability.
(ii) The rotor with relatively low gyroscopic effect and high transverse moment of inertia: Again there are two instability thresholds, but in this case the translational mode threshold is the lowest, thus controlling overall stability.
(iii) The case of a rotor with relatively low trans- verse moment ofinertia.In this case the pivotal mode is always stable and the rotor stability is determined by the translational mode insta- bility threshold.
The parametric study of the asymmetric rotor system based on the results of the symmetric system stability analysis yields that for the first case the pivotal mode controls the stability for any set of asymmetry parameters.For a given stiffness asym- metry parameter the maximum instability threshold is achieved when the rotor center of mass is shifted away from the stiffer bearing on the geometric asymmetry parameter of the same value.The instability threshold maximum corresponds to the symmetric rotor.
This implies that stability may be maximized in this case by distributing the mass of the rotor in such a way as to minimize the ratio of transverse moment of inertia to polar moment of inertia, .
In the case where the translational mode is controlling stability, The instability threshold can only be managed by manipulation of the bearing parameters or total rotor mass.and 2 direct fluid film stiffness total lateral fluid film stiffness bearing and 2 location, respectively total distance between the bearings rotor system mass absolute coordinate system origin rotor center of mass eigenvalue r-x+jy -x+J th lateral displacement of rotor center of mass in the stationary system of coordinates fluid bearing circumferential average velocity ratio angle of yaw and pitch, respectively complex angular displacement rotative speed instability thresholds for translation and pivotal modes

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FIGUREDiagram of rotor system used in model.

FIGURE 2
FIGURE 2 Coordinate systems for the rigid rotor.

FIGURE 3
FIGURE 3 Angles X, b of yaw and pitch.
FIGURE 4 Root locus plot of "mirror symmetric" rotor with ratio = 1/A.Plot was generated using the parameters given

FIGURE 5 FIGURE 6
FIGURE 5 Root locus plot of symmetric rotor with system parameters as in Figure4, except for the parameter It=2Ip/Akgm2.The O's indicate the beginning of the locus at f 0. The x's indicate the position of the roots at the point where the system first becomes unstable at f 2435 rpm.

FIGURE 8
FIGURE 7 Instability threshold versus geometric parameter asymmetry.

FIGURE 10
FIGURE 9 Instability threshold versus geometric parameter asymmetry.

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TABLE
Nomenclature of factors used to nondimensionalize equations of motion Natural frequency ratio, Stiffness cross coupling factor, ato Nondimensional rotor speed, co Nondimensional time, r Damping cross coupling factor, bto