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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St, New York, N.Y SThe Society shall not be responsible for stateents or opinions advanced in papers or discussion at eetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published 95-GT-217 in an ASME Journal. Authorization to photocopy aterial for internal or personal use under circustance not falling within the fair use provisions of the Copyright Act is granted byasme to libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service provided that the base fee of $0.30 per page is paid directly to the CCC, 27 Congress Street, Sale MA Requests for special perission or bulk reproduction should be addressed to the ASME Technical Publishing Departent Copyright 1995 by ASME All Rights Reserved Printed in U.S.A. OPTIMIZATION OF BEARING LOCATIONS FOR ROTOR SYSTEMS WITH MAGNETIC BEARINGS Srira Srinivasan United Technologies Research Center East Hartford, Connecticut Eric H. Maslen and Lloyd E. Barrett Departent of Mechanical, Aerospace and Nuclear Engineering University of Virginia Charlottesville, Virginia ABSTRACT This paper presents a ethod for quickly evaluating the effect of changes in bearing location on bearing design for stability of rotating achinery. This ethod is intended for use by rotating achinery designers to select the "best" bearing locations prior to the bearing design process. The purpose of the ethod is to iprove the design process by separating the proble of deterining the "best" bearing locations fro that of deterining the actual bearing design. The ethod is independent of the type of bearing eployed. For each candidate bearing configuration, the ethod provides a scalar easure of the relative ability of bearings to eet stability specifications. Within certain liits, the stability specifications are defined by the designer. The scalar easure is used to rank the candidate bearing locations and thereby select the best one. The scalar easure is copared to a practical easure of agnetic bearing design such as the infinity nor of the controller for an exaple design of a ulti-stage centrifugal copressor. INTRODUCTION In the design of hydrodynaic or agnetic bearings for rotor systes, ost of the effort has been directed towards bearing design ( eg.: Barrett, 1983; Maslen, 1991a; Dhar, 1991; and Larsonneur, 1988) while relatively little attention has been given to bearing locations. Generally the bearing designer assues that the bearing locations are given and the design process starts fro there. In any instances, poorly chosen bearing locations can result in unrealistic or unattainable bearing designs to achieve desired rotor stability or unbalance response characteristics. Thus, a proper selection of the bearing locations should be part of the overall rotor design process, preferably siultaneous with design for aero or therodynaic perforance, in order to obtain the best syste perforance. In the present work, the ter bearing configuration eans the set of selected locations for the rn. bearings in the syste and does not directly iply anything about the bearing properties. The nuber of bearings, rn, is typically 2, but could be higher. The key syste properties involved in selecting bearing configurations are controllability and observability (Kailath, 1982) which are dual properties. For collocated bearings, it turns out that the controllability and observability properties are equal and thus, in the succeeding discussion only the controllability property is discussed. Controllability is a yes-no property which indicates whether all of the syste eigenvalues can be directly affected by the bearing dynaic copliances at the given locations. Rotor systes are controllable by virtually all bearing configurations and therefore a relative easure of controllability is needed to copare potential configurations. Measures of controllability have been defined in several ways: energy approach (Moore, 1981), size of recovery region (Vishwanathan, 1984), eigenvalue obility (Maslen, 1991b), the distance between certain subspaces (Hadan, 1989) and the nor of the odal residues of the transfer atrix (Tarokh, 1990). These easures have liited practical application to free free rotors for the following reasons: All of these easures ephasize the least controllable ode; for rotor systes, the lower odes are of priary interest. This subset ight not include the least controllable ode. Moore's (1981) energy approach is undefined for undaped systes and very sensitive to uncertain structural daping for lightly daped rotors. The easures of Hadan (1989) and Tarokh (1990) are based upon only the eigenvectors while ignoring the associated eigenvalues. This can lead to the easure being decided by a higher ode. Presented at the International Gas Turbine and Aeroengine Congress & Exposition Houston, Texas - June 5-8,1995 This paper has been accepted for publication in the Transactions of the ASME Discussion of it will be accepted at ASME Headquarters until Septeber 30, 1995

2 NOMENCLATURE s Laplace variable in original y p-vector of sensor outputs K feedback atrix in s-plane coplex plane a', a Low-ode daping paraeters fc feedback atrix in.-plane s Laplace variable in transfored specifying acceptable region in scalar easure A coplex plane Q Cutoff frequency paraeter II II' Frobenius nor of atrix: n x n syste atrix specifying acceptable region E M, 2 B n x actuator location atrix y Real constant specifying ^^ (^ Infinity nor of transfer C p x n sensor location atrix bilinear transforation function: ax, A, B, C Syste atrices in s-plane S scale factor to select y v Real part of eigenvalue x n-vector of syste states a, b, Real constants of bilinear w Iag. part of eigenvalue X Laplace transfor of x c, d transforation,, eigenvalue of syste u -vector of bearing forces J Objective function for bearing U Laplace transfor of u design The ethod of Tarokh (1990) is a foralization of the widely eployed technique of coparing rotor freefree odeshape aplitudes at potential bearing locations. When such a criterion is applied only to selected odeshapes, it can provide a qualitative coparison, but provides no echanis to decide what relative influence each of the selected odeshapes should have on the final selection. When each of the odeshapes individually indicate the sae selection, then the ranking of bearing configurations is unabiguous and probably results in the sae selection as would be ade by the technique presented here. Several approaches to the actuator/sensor location proble in flexible structures have been proposed (Ki, 1991; Li, 1992; Chiang, 1992; Choe, 1992). The controllability easures obtained by these approaches are based upon the controllability Graians of Moore (1981) or they are obtained by a non-convex nonlinear optiization proble which can be coputationally expensive. In this paper, a non-iterative, nuerically robust ethod is presented for selecting bearing configurations in undaped free-free rotor systes. This is iportant because any design ethod producing designs realizable in practice, will probably use tie consuing optiization ethods. The basis for coparison is a easure of the dynaic stiffness of bearings required to produce syste eigenvalues which be in a specified region of the coplex plane. As a result, the iplicit objective is to achieve a certain level of daping, or logarithic decreent, rather than directly approaching the forced response proble. Although the ethod can be shown to be applicable to placeent of hydrodynaic bearings, the present work restricts its attention to agnetic bearings. The proposed ethod is described in detail in the second section. An exaple is presented in the third section where three candidate bearing configurations are ranked by this technique. Magnetic bearing designs for the candidate configurations are also obtained. The infinity nor of each controller is used as a easure of "goodness". This nor is copared to the ranking predicted by the proposed ethod. The last section concludes the paper. METHOD The proposed ethod is predicated on the following assuptions: 1. the priary goal of the bearing design is to achieve a general stability objective: the rotor/bearing syste eigenvalues ust lie outside of a specific region in the coplex plane 2. this goal should be et with the sallest bearings possible where sall is, conceptually, a cobined easure of bearing stiffness and daping Given such a stability objective and a specification of the bearing locations along the rotor, it is possible to copute a bound for the required bearing size with a single noniterative calculation. Several candidate bearing locations can then be copared on the basis of this bearing size easure. The dynaic character of the undaped rotor syste without bearings is represented with a state space for of the odel equations (see e.g. Ki, 1991): i = A(S2)x+Bu+B ef (1) y = Cx where A is a n x n atrix which contains the dynaics of the undaped free-free rotor at a given rotational speed 52, B is a n x in atrix which deterines the bearing locations, C is a p x n atrix which deterines the sensor locations, x is an n-vector of the generalized displaceents and velocities of the rotor, u is a -vector of bearing forces, y is a p-vector of sensor outputs, and f represents other external forces acting on the rotor such as ass ibalance or aerodynaic loads. There are no specific liitations on the diensions n,, and p except that the nuerical conditioning and coputational speed of the proble will suffer if they are too large. Nuerical probles will be an issue if n gets larger than about 100, but nuerical ethods exist for handling the required coputations even with uch larger values of n. An alternative approach is to liit the rotor odel size through odal truncation to contain only twenty or thirty states. This protects the nuerical conditioning and preserves all of the rotor eigenvalues which are likely to have any physical significance.

3 A set of bearings is a dynaic feedback echanis which generates control forces on the rotor in response to otion of the rotor: Zc = Axc + By (2) U = Cx,+Ky so that the rotor/bearing cobination is described by dt x } [A(Q) + BKC BC, x I + I o ] jl f L J (3) y=cx where the dynaic properties of the bearing ("stiffness" and "daping") are deterined by A, B, C., and K. Clearly, the stability (closed loop eigenvalues) of the rotor/bearing syste depends upon the interaction of all of these atrices. In this work, we take advantage of this description of bearings to perit a fairly autoatic assessent of how large they ust be in order to satisfy certain stability objectives for the closed loop syste. Specifically, we adopt a sipler odel for the bearings, referred to as state feedback, which takes the for u = Kx (4) to produce the closed loop rotor/bearing cobination.i = [A(fI)+ BK]x+Bu (5) y = Cx The choice of state feedback rather than output feedback (u = Ky) is less restrictive and therefore can be expected to produce a lower bound on bearing capacity. The syste stability objectives considered (Dhar, 1991), are to provide acceptable log decreents for lower odes of interest while not shifting the higher odes above a cutoff frequency. In other words, the dynaic properties of the bearings ust be chosen so that the eigenvalues of the rotor/bearing syste be within a certain acceptable region of the coplex plane without uch caring about the exact locations. The acceptable region that is considered by this ethod is represented by the unshaded region of the coplex plane shown in Figure 1. This particular choice of acceptable region is a atheatical convenience. However, it does satisfy coon rotor stability objectives: the lower odes are required to be ore heavily daped than the higher odes and the highest odes are not required to be daped at all. The acceptable region is defined by the low-ode daping paraeter a and the cutoff frequency 3 which are to be specified by the designer. As a is ade ore negative, the lower odes are required to be ore heavily daped; as ^3 is ade larger, the higher odes are required to be ore heavily daped. As will be deonstrated, the agnitude of a ust be less than that of 3. The basic idea of the proposed ethod is to find an unique state feedback K for which the closed loop syste eigenvalues lie in the acceptable region. This atrix is coputed Figure 1: ACCEPTABLE REGION IN THE S-PLANE: o SYM- BOLS REPRESENT ROTOR FREE FREE EIGENVALUES by solving a Linear Quadratic Regulator (LQR) proble (Molinari, 1977). Solution of the LQR is non-iterative and nuerically robust (Laub, 1979). In addition, the LQR approach lends itself to the stability objective described above and can be tailored to iniize K subject to this objective (treated as a constraint). The Frobenius nor of the state feedback atrix is used as the required scalar easure for the candidate actuator configuration: A, B, a, 3 LQR K. = IIKIIF (6) In soe sense, the Frobenius nor easures the "effort" needed to place the syste eigenvalues in the acceptable region. The choice of the Frobenius nor is soewhat arbitrary and without direct physical otivation. However, the exaples in the next section deonstrate a strong correlation between this easure and soe properties of bearings which are designed to place the daped eigenvalues of the syste in the acceptable region. The LQR proble is defined by two cost function atrices, Q and R, and produces the Algebraic Ricatti equation PA + A'P + Q + PBR ' B'P = 0 (7) with the solution K = R ' B'P. The atrices Q and R, can be freely chosen as long as they are syetric, Q is positive sei definite, and R is positive definite (soe other restrictions also apply: see Molinari, 1977). The specific choice of these atrices, in conjunction with the syste description {A, B}, uniquely deterines the location of the closed loop eigenvalues. It is not at all apparent that the right choice of Q and R can be ade which will place the closed loop eigenvalues in the acceptable region while iniizing the Frobenius nor of K. However, if Q is ade arbitrarily sall, it can be shown that the resulting K atrix will stabilize the closed loop syste with a inial control effort, which is essentially equivalent to a iniized Frobenius nor for K. Such a choice of weighting atrices fors what is called the infinite cost of control LQR (Molinari, 1977). Using the infinite cost of control LQR (ICCLQR) eliinates the need to select Q and R and also tends to iniize

4 Let K be the state feedback atrix obtained by infinite cost of control LQR in the s-plane: Urn PA+A'P+Q+PBB'P=0 k= - 'p (12) Q-+O The equivalent state feedback atrix K in the s-plane is obtained by (Srinivasan, 1993): K = (I - ck(ai - ca) -'B) -1 K (13) Figure 2: ACCEPTABLE REGION IN S-PLANE: o SYMBOLS INDICATE FREE-FREE ROTOR EIGENVALUES IIKIIF but does not, in itself, provide a solution which satisfies the stability objective. This proble is overcoe with a siple trick. It is observed that the ICCLQR proble reflects any unstable open-loop syste eigenvalues across the iaginary axis into the stable half of the coplex plane. If a transforation can be found which distorts the coplex plane so that, in the transfored syste, any eigenvalues which originally lay outside the unacceptable portion of the coplex plane now be in the right half plane, then the LQR proble applied to the transfored syste will reflect the unacceptable eigenvalues into the acceptable region. This distortion is indicated in Figure 2. The resulting state feedback can then be transfored back to the original proble space and a iniu Frobenius nor state feedback atrix K will be obtained which satisfies the stability objective. Noting that the Laplace transfor of the syste differential equation is slx(s) = Ax(s) + Bu(s) (8) we introduce the bilinear transforation (Churchill, 1974) as+b s= csi + d (9) where a, b, c and d are all real nubers. After a little arithetic (Srinivasan, 1993), the syste {A, B} in the s-plane is siix(si) = Ax(i) + Bu(s) (10) where A = (ai - ca)-'(da- bi) B = (ca+di)(ai -ca) -1 B (11) This is an invertible transforation which does not change the diension of the syste {A, B} when transfored into {A, B}. Further, the atrices A and B are real. Note that the transforation fro A to A is independent of B. This coupled with the fact that ICCLQR is used iplies that the locations in the acceptable region to which the free-free eigenvalues are shifted will be the soe for all the candidate actuator configurations. However, the state feedback atrix which achieves this will be different for each configuration. Equivalence eans that the eigenvalues of A = A - BK in the s"-plane are apped to the eigenvalues of A = A - BK in the s-plane according to the bilinear transforation (9). The constants a, b, c and d of (9) are related to the specification of the acceptable region through a=-c z b=ya a<0 c=1 d=y ^i,y>0 (14) The nuerical conditioning of the LQR proble can be protected by selecting y as Y = /32 /lal (15) This choice akes the iaginary parts of the eigenvalues of A be approxiately the sae order of agnitude as those of A. The sae value should be used for all the candidate bearing locations being ranked. Fro the geoetry of Figure 2, it is clear that the freefree eigenvalues on the boundary of the circle in the right half plane are reflected to locations within the acceptable region as long as the center of the circle, -2(1 - ') < 0. This iplies that al </9 Another technical condition that ust be satisfied arises due to the fact that {A, B} ust be controllable in order for LQR to work. It can be shown (Srinivasan, 1993) that if {A, B} is controllable and A has no eigenvalue equal to -/3 2/a then {A, B} is controllable. This condition will always be satisfied for undaped free-free rotors. The ethod, as ipleented above, will place the closed loop eigenvalues below the cutoff frequency 3 to within but not on the boundary of the acceptable region. When bearings are actually designed (Dhar, 1991), the syste eigenvalues tend to be on or close to the boundary. In view of this, a better correlation with the bearing design ethod will result if the eigenvalues are placed on the boundary. A slight odification results in a state feedback which will place all the syste eigenvalues below the cutoff frequency,(3 directly on the boundary of the acceptable region. This is achieved with the auxiliary circle shown in Figure 3 which has the sae cutoff frequency paraeter /3 as before but a different daping paraeter a': (/3 2 + /9/92 + 2)/ (16) The entire ethod described above is used with {a',,3} instead of {a,/3}. 4

5 Table 1: SCALAR MEASURES M FOR REGIONS A & B pecified circle circle Brg. Config. Acceptable region A a=-75, /3=5500 Acceptable region B a=-117, Q=5500 ax ax() E E E E E E Figure 3: ACCEPTABLE REGION IN S-PLANE , ^^ Figure 4: SCHEMATIC OF COMPRESSOR WITH THE CAN- DIDATE BEARING CONFIGURATIONS: ALL DIMENSIONS ARE IN CM A scalar easure for the actuator configuration {A, B} has been obtained. Assuing perfectly collocated bearings, the scalar easure of the sensor configuration will also be by duality. The ethod can be suarized as follows: given a free-free rotor syste dynaic atrix A and atrices B; each corresponding to a candidate bearing configuration: 1. Specify the acceptable region by selecting the the lowode daping paraeter a and the frequency cutoff paraeter j3 with al <i3. 2. Choose a' according to (16) and -y according to (15). 3. For the syste {A, B}, obtain {A, B} fro (11). 4. Use ICCLQR on {A, B} to copute the state feedback atrix K. 5. Obtain K fro (13) and K. 6. Copute the scalar easure = IIKIIF 7. Rank the candidate bearing configurations according to the agnitude of the easures. EXAMPLE This exaple copares the ranking of candidate bearing configurations to the optiized perforance of actual agnetic bearing designs for the sae configurations. The perforance of these designs are assessed by the infinity nor of the controller transfer functions. A ulti-stage centrifugal copressor is depicted in Figure 4 which shows three candidate bearing configurations, each with two bearings. The 220 c long rotor has an average diaeter of 12.7 c and is odeled as a bea of 41 discrete ass stations. Nine disks are odeled as luped asses as shown. Gyroscopic effects are sall for this rotor and are neglected, peritting a planar odel. The state space odel {A, B, C}, was obtained by odal truncation, retaining two rigid body odes and bending odes at frequencies (in rad/sec) of 733.2, , , , and The frequency of the highest retained ode (at rads/sec) is greater than the cutoff frequency /3 = 5500 and is already in the acceptable region. Two acceptable regions are considered. Both regions have frequency cut off paraeters of ^3 = 5500; region A has a low-ode daping paraeter a = 75 while region B has a low-ode daping paraeter a = 117. Physically, region B provides ore log-decreent to the lower odes than region A. Table shows the scalar easures obtained by the proposed ethod for the three candidate bearing configurations. Configuration 3 requires the least "effort" while configuration 1 requires the ost for both regions A and B. In designing optiized agnetic bearings for coparison, the following assuptions are ade: A linear odel of the agnetic bearing is considered. The agnetic bearings are collocated and the sensor at each bearing feeds back only to the sae bearing. The sensors and actuators are infinite bandwidth. The open loop stiffness of the agnetic bearings is zero. The controller transfer functions have the for: as + b C s = 10 ( ) = s 10 k (s Z) (17) 2 + cs + d (s PI)(s P2) where a, b, c, d > 0 which ensures that the poles and zeros of C(s) are always in the left half plane. The agnitudes of the poles are liited by requiring c<2.0x104 andd<1.5 x108 The bearing design proble is to deterine the paraeters of the controller transfer functions such that the daped eigenvalues of the syste are within the acceptable region. There ay be no or ultiple solutions (Johnson, 1970). If there are ultiple solutions, it is desirable fro a

6 Table 2: MAGNETIC BEARING DESIGN FOR ACCEPTABLE REGION A (a = 75, ^3 = 5500) Brg. controller gain, zero and poles Config. Jn,i n brg k z P1 P left right left right left right Table 3: MAGNETIC BEARING DESIGN FOR ACCEPTABLE REGION B (a = 117, 3 = 5500) Brg. controller gain, zero and poles Config. Jin brg k z P1 I P left ± 1031i right ±2070i left right left right ± 2337i practical viewpoint to choose a solution which has the lowest infinity nor which is the peak value of the gain of the transfer function and is denoted by ICI.. The bearing design proble can be forulated as a nonlinear constrained optiization proble. The cost function J = 10 6 C 0 is iniized by choice of the transfer function paraeters such that the daped eigenvalues of the syste are within the acceptable region. (Srinivasan, 1993) Results are suarized in Tables 2 and 3. The ranking of the bearing configurations based on the easure J = of the controller designs is the sae as was predicted by the proposed ethod where a large value of J i is undesirable. CONCLUSIONS The locations of bearings in a rotor-bearing syste deterine the controllability/observability characteristics of the syste which in turn are iportant in bearing design for stability. A non-iterative, nuerically robust ethod for the purpose of ranking candidate bearing configurations has been presented. This ethod can be used by the agnetic bearing designer to deterine the "best" locations before the actual bearing design. A design exaple on a ultistage centrifugal copressor has been presented wherein, agnetic bearings with iniu infinity nor controllers are designed to eet the stability specifications. The ranking of the bearing designs for the candidate bearing configurations in ters of the infinity nors are well correlated with the ranking obtained by the proposed ethod. This indicates the utility of the new ethod as a design tool. REFERENCES Barrett, L. E., 1983, "The Optiization of Hydrodynaic Journal Bearings for Aerodynaically Excited Flexible Rotor Bearing Systes," ASME Journal of Mechanical Design, VOL. 105, No. 2, pp Chiang, H. D.,Thorp, J.S., Wang, J.C., Lu, J. and Aubert, B., 1992, " Optial Controller Placeents in Large-Scale Linear Systes," IEE Proceedings-D, VOL 139, pp Choe, K. and Baruh, H., 1992, "Actuator Placeent in Structural Control," AIAA Journal of Guidance and Control, VOL 15, pp Churchill, R.V., Brown, J.W. and Verhey, R.F., 1974, Coplex Variables and Applications, McGraw-Hill. Dhar, D., Barrett, L.E. and Knospe, C.R., 1991, "Design of Bearings for Rotor Systes Based on Stability," Proc. of the Int'l. Syp. on Magnetic Bearing Suspension Tech., NASA Langley Research Ctr., Hapton VA, August Hadan, A.M.A. and Nayfeh, A.H., 1989, "Measures of odal controllability and observability," AIAA Journal of Guidance and Control, VOL 12, pp Johnson, T.L., Athans, M., 1970, "On the design of optial constrained dynaic copensators for linear constant systes," IEEE Trans. Autoat. Contr. VOL AC-16, pp Kailath, T., 1982, Linear Systes, Prentice Hall, Englewood Cliffs, N.J. Ki, Y. and Junkins, J.L., 1991, "Measure of Controllability for Actuator Placeent," AIAA Journal of Guidance and Control VOL 14, pp Larsonneur, R. and Herzog, R., 1988, "Optial Design of Structure Predefined Discrete Control for Rotors in Magnetic Bearings," Proc., 1st Int'l. Syp. on Magnetic Bearings, Springer-Verlag, pp Laub, A.J., 1979, "A Schur Method for Solving Algebraic Ricatti Equations," IEEE Trans. Autoat. Contr. VOL AC-24, pp Li, K.B., 1992, "Method for Optial Actuator and Sensor Placeent for Large Flexible Structures," AIAA Journal of Guidance and Control, VOL 15, pp Maslen, E.H., 1991a, "Magnetic bearing synthesis for Rotating Machinery," Ph.D. dissertation, University of Virginia, Charlottesville, VA. Maslen, E.H., 1991b, "Controllability of Rotors," Proc., ROMAG `91 Conference on Magnetic Bearings and Dry Gas Seals, Alexandria, VA. Molinari, B.P., 1977, "The tie invariant Linear Quadratic optial control proble," Autoatica, VOL 13, pp Moore, B.C., 1981, "Principal Coponent analysis in Linear systes: Controllability, Observability and Model Reduction," IEEE Trans. Autoat. Contr. VOL AC-29, pp Srinivasan, S., 1993, "A Stability Based Method for Selecting Bearing Locations in Rotor Bearing Systes," Ph.D. dissertation, Univ. of Virginia, Charlottesville, VA. Tarokh, M., 1990, "Measures for Controllability and Eigenvalue Mobility," Proc., 29 1h Conference on Decision and Control, pp Vishwanathan, C.N., Longan, R.W. and Likins, P.W., 1984, "A degree of controllability definition: Fundaental concepts and application to odal systes," AIAA Journal of Guidance and Control, VOL 7, pp