modeling of tilting-pad journal bearings with transfer...

International Journal of Rotating Machinery2001, Vol. 7, No. 1, pp. 1-10Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in Malaysia.

Modeling of Tilting-Pad Journal Bearings withTransfer Functions

J.A. VAZQUEZ* and L.E. BARRETT

Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science,University of Virginia, Charlottesville, VA 22903, USA

(Received in finalform 6 July 1999)

Tilting-pad journal bearings are widely used to promote stability in modern rotating machi-nery. However, the dynamics associated with pad motion alters this stabilizing capacitydepending on the operating speed of the machine and the bearing geometric parameters,particularly the bearing preload. In modeling the dynamics of the entire rotor-bearing sys-tem, the rotor is augmented with a model of the bearings. This model may explicitly includethe pad degrees offreedom or may implicitly include them by using dynamic matrix reductionmethods. The dynamic reduction models may be represented as a set of polynomials in theeigenvalues of the system used to determine stability. All tilting-pad bearings can then berepresented by a fixed size matrix with polynomial elements interacting with the rotor. Thispaper presents a procedure to calculate the coefficients of polynomials for implicit bearingmodels. The order of the polynomials changes to reflect the number of pads in the bearings.This results in a very compact and computationally efficient method for fully including thedynamics of tilting-pad bearings or other multiple degrees of freedom components thatinteract with rotors. The fixed size of the dynamic reduction matrices permits the method tobe easily incorporated into rotor dynamic stability codes. A recursive algorithm is developedand presented for calculating the coefficients of the polynomials. The method is applied tostability calculations for a model of a typical industrial compressor.

Keywords." Rotating machine dynamics, Tilting-pad bearings, Dynamic reduction,Stability calculations, Transfer functions

INTRODUCTION

Barrett et al. (1988) and Brockett and Barrett (1993)showed the importance of including all dynamiccoefficients of the tilting-pad bearing for stability

analysis. In general a tilting-pad bearing, like thatshown in Fig. 1, with n pads has 2(5n + 4) coeffi-cients. The variable number of coefficients couldpresent a problem in some analysis programs sincethe number of equations of motion completely

Corresponding author. E-mail: [email protected]: [email protected].

J.A. V,ZQUEZ AND L.E. BARRETT

representing the bearings will change with the num-ber of pads. Brockett and Barrett (1993) showedthat by using dynamic condensation equivalentstiffness and damping coefficients can be obtainedwhich are functions of the complex frequency s.

Thus the number ofequations representing a tilting-pad bearing remains fixed although the bearingcoefficients are variable throughout an analysis.The dynamic condensation method used byBrockett and Barrett (1993) follows that developedby Leung (1978, 1989) for structural analysis. Con-versely, if a single input single output transferfunction is known as a ratio of polynomials withknown coefficients, an equivalent mass-stiffness-damping matrix model can be obtained (Maslenand Bielk, 1992).The work presented here shows another ap-

proach. Instead of reducing the pad degrees offreedom for each complex frequency s, complexdynamic coefficients are calculated as a function ofthe complex variable s. The complex dynamic coeffi-cients are represented by a ratio of polynomials ortransfer functions. The advantage of this approachis that the resultant dynamic coefficient matrix isalways 2 x 2. A different number of pads will onlychange the order of the polynomials but the generalsize of the matrix remains the same. This approachfacilitates the inclusion of tilting-pad bearings in

analysis tools because it requires few modificationsfrom the standard eight coefficient hydrodynamicbearings. If the analysis tools already include mag-netic bearing control transfer function models, thismethod can be used directly and no modificationsare required (Brockett and Barrett, 1995). The trans-fer function representation of tilting-pad bearingsalso facilitates understanding the behavior of thebearing as a function of the vibration frequency.

PROCEDURE

Tilting-pad bearing models can be expressed by

+_K6, K t 0

Dynamically reducing the pad degrees of freedomfrom Eq. (1) we get

(2)

C b: R R

P3"e](:)d (ib//C

FIGURE Schematic of a tilting-pad journal bearing.

TILTING-PAD JOURNAL BEARINGS

or

,’[M]{,} + [K,q(,)] {,} {}.

where

[eq(’)] { [,C.,. + ..]- [,C., +

X [S2Ip -t- sC6 -nL Kaa] -1 [sCu -t- K&] }.(4)

For an n pad bearing

Changing variables

Ad2i-1 ai,

Ad2i ai,

Bdzi-1 hi,

Bdzi cl,

Dd2i-I el,

Dd2 f,.,

i--1...n.(9)

sCxx + Kxx[Keq (s)] sCyx -3r- Syx

(SCx5i + Kx(Si)(sC(siX + KSiX (SCx(5"i + Kx(Si)(sCSiy + K’iysCxy + Kxy i=1 S2Ipi + sCe,e, + K-i,i i=1 S2Ipi + sCe,e, + Keie,

SCyy @ <y @ (SCy, + <i)(sC,x @ Kaix) @ (sCyi @ Ky6i)(sCiy @ Ke,y)S2 S21p. + sCe,e, + Ke.e,

(5)

The 1, m equivalent complex bearing coefficientscan be written as

Keq (s) SClm -- Klm (6)S21pi at- sC(si6i at- K5.(5i

Using partial-fraction expansion the summationterm can be written as

where

Oi

Ei

i= S2Ipi @ sCt,6i qL Ke,e,

(sai + bi sa(+___di

Cl6 C6irn V/Ip ei Di + D2i -El,

ClsiK6i -Jr- K, C6imV

I/Ip,

fi--D,- D2i-E,,

Kl(si K8

C,<b

C- de2Ip,’ "K(5ihi Aip

ai ci 2

j[Bi ai(ei +fi)] Ci

(7)

Then Eq. (7) becomes

i=S2Ipi 7-Sbi 7 g(sisi

2n sAdi + Bdi77+ Ddi

(10)i=1

or

(c, +i=1

S2Ip -AI- sC(s (S -Jr-

f2nS2n -+-f2n-1S2n-1 @ -qLf S -+-fog2nS2n @ g2n- s2n- _+_ at_ g S at- go

After some manipulation it can be shown that

g2n 1,

g2n-1 Ddi,i=1

2n- 2n

g2n-2--Z Ddi Ddj,i=1 j=l+l

2n-2 2n- 2n

g2n-3- Z Ddi Ddj Ddk,i=1 j=i+l k=j+l

(11)

J.A. V/ZQUEZ AND L.E. BARRETT

2n-3 2n-2 2n- 2n

g2n-4- Ddi Z 04 Z odk Z Od,,i=1 j=i+l k=j+l l:k+l

2 4

gl--Z Ddi Z D4 Z Ddk...i=1 j--i+l k--j+l

2n

x E Ddr2,,_,,r2n- r2n- @

2 2n

go- ZDdi Z D4" Z Ddk"" Zi-- j--i/ k=-j+ rzn =-rz,,- +2n

I-[D&,i=1

Ddrzn

(12)

2n 2n

fo ’ai 1-Ii=1 j--1

j#i

Therefore

f,,# s2n+l ,t 2n -]-f sKeql,m(S 2n+l -JnS +"" +fo

g2nS2n -+- g2n-1 s2n-1 nt_ 7 I-S 2r- gO

where

fd Klmgo fo,

f( Clmgo -+- Klmgl fl,

(14)

the numerator in Eq. (11) follows a similar pat-tern as

2n

i=1

2n 2n

f2n-I- Z gdi+AdiZDdji=1 .1=1

2n 2n 2n-- 2n

f2n-2-- Z BdiZD4-t--AdiZ Ddi Z Di=1 j=l j=l k=j+l

jfi ji ki

2n 2n- 2n

f2n-3 Z BdiZD4 Z Ddki:1 j=l k:j+l

j#i ki

2n-2 2n- 2n

@Adi D4 Z adk E Ddlk=j+ l=k+

,ji ki li

2n 2

fl- Z BdiZ04 Z Ddk""i= j= k=j+

ji

2n

/AdiUD"j=l

r2n- =r2n-2-r2n-li

Ddr2n_

(is)fn- Clmg2n-2 @ Klmg2n-1 f2n-1,

fn Clmg2n-12r- Klmg2n f2nfn+ Clmg2n

Numerically, the complex stiffness coefficientscan be expressed as equivalent real stiffness and realdamping coefficients for any oscillatory complexfrequency, s, from the relationships:

keqz.m (s) Re(Keq,... (s)) Re(s) Im(Kemm (s))Im(s)(16)

Ceq,,m (S) Im(s---- Im(Ke%m (S)). (17)

NUMERICAL EXAMPLE

The rotor used for this example, shown in Fig. 2,is an eight stage centrifugal compressor used fornatural gas re-injection at an offshore drilling site,running at 5626 rpm. The rotor is approximately2.8 m long and 954 kg with the mass center near themid-span. Table I lists the model for the rotor. It is


Tilting Pad, Beating, AerodynarmcBTfleta’gagPal A Cross-coupling

FIGURE 2 Rotor model.

TABLE Rotor model

Station Section Outer Transverse Mass Stationno. length (m) diameter (m) moment of (kg) no.

inertia (kgm2)

Section Outer Transverse Mass (kg)length (m) diameter (m) moment of

inertia (kgme)

0.0345 0.0699 0.0002 72.60 192 0.1712 0.1107 0.0209 10.20 203 0.0381 0.1270 0.0229 12.92 214 0.0434 0.1270 0.0046 6.57 225 0.0881 0.1651 0.0199 13.92 236 0.0544 0.1681 0.0269 17.68 247 0.0897 0.1580 0.0249 17.28 258 0.0864 0.1824 0.0392 23.22 269 0.0625 0.1702 0.0357 21.08 2710 0.0787 0.1702 0.1186 45.89 2811 0.0625 0.1702 0.0334 20.36 2912 0.0787 0.1702 0.1186 45.89 3013 0.0625 0.1702 0.0334 20.36 3114 0.0787 0.1702 0.1186 45.89 3215 0.0625 0.1702 0.0334 20.36 3316 0.1600 0.1702 0.1585 56.50 3417 0.1600 0.1702 0.1124 41.72 3518 0.0721 0.1702 0.1610 57.59

0.0909 0.1702 0.0426 24.030.0721 0.1702 0.1251 48.840.0909 0.1702 0.0426 24.030.0721 0.1702 0.1251 48.840.0909 0.1702 0.0426 24.030.1273 0.1702 0.1464 56.000.0734 0.1702 0.0860 35.010.0587 0.1778 0.0277 17.870.0902 0.1681 0.0321 19.500.0544 0.1681 0.0287 18.270.0876 0.1651 0.0267 17.590.0434 0.1270 0.0197 13.520.0381 0.1270 0.0046 6.030.1445 0.1189 0.0186 11.740.0475 0.0950 0.0175 15.010.1400 0.0935 0.0092 14.510.0000 0.0935 0.0082 8.48

composed of 35 mass stations with identical tilting-pad bearings acting at nodes 4 and 31. The completeset of tilting-pad bearing coefficients is shown inTable II. The stiffness and damping coefficients forthese bearings can be calculated by using severalmethods (Shapiro and Colsher, 1977; Branagan,1988). The geometry of the bearings in this analysisare given in Table III. Aerodynamic cross-couplingwas assumed to act at station 16 with a value of2.539 x 106N/m. A stability analysis is performedfor this rotor using synchronously reduced coeffi-cients for the bearings assuming that s=icos inEq. (5), where COs is the shaft spin frequency; amethod often employed in rotor stability calcula-tions. The analysis is repeated using the transfer

function representation of the bearings calculatedusing the algorithm presented in this paper. Thesynchronously reduced bearing coefficients areshown in Table IV. Table V shows the coefficientsof the four transfer functions obtained from thecoefficients of Table II using the method describedin this paper (Eq. (12) and (15)).

DISCUSSION OF RESULTS

The eigenvalues obtained using the frequencydependent transfer function representation pro-posed in this paper are shown in Table VI. The

J.A. VZQUEZ AND L.E. BARRETT

TABLE II Tilting-pad bearing coefficients

Translational coefficients

Kx.; (N/m)Kxy (N/m)Kyx (N/m)Kyy (N/m)Pad coefficientsPad number

Kp.,a (N)Kpex (N)Kp.,,t (N)Kp:y (n)Kp (Y m)Cpx. (N)cpex (N)Cp,. (N)Cpe.,Cpe (N m)

6.08E+077.33E+07

-1.83E+082.19E/08

1512-4168954617272781.2015.5797.58313.6080.221

2

-8610-29294387-3432278

-6.839-11.2163.4859.5090.221

Cxx (Ns/m)Cxy (N s/m)Cx (N s/m)Cyy (N s/m)

3

-2.35E+061.03E+05-9.06E/05-1.09E+05

8595-132.338-132.338--321.587-321.58716.825

2.65E+05-3.10E+04-3.07E+046.32E+05

4

1.28E+061.28E+06-1.10E+078.45E+048.16E+041242.361242.36-2315.94-2315.9478.890

5

2.44E+061.95E+04

-6.49E+051.49E+05

8596296.110296.110182.384182.38416.825

TABLE III Bearing characteristics

Number of pads 5Bearing load 4680 Nm (preload) 0.0Load direction On padRadius 63.5 mmRadial clearance 0.1016 mmLength 63.5 mmRotor speed 5626 rpmOffset factor 0.5Pad arc length 60Pad moment 1.13 x 10.-4 kg/m

difference in the eigenvalues obtained using fulltilting-pad models and those obtained withsynchronously reduced bearing coefficients can bedetermined by comparing Tables VI and VII. Thereal part of the eigenvalues in Table VI are morenegative than the real part of the eigenvalues inTable VII. In this model, therefore, synchronouslyreduced bearing coefficients tends to over-predictstability. This is consistent with the findings byBarrett et al. (1988) and Brockett and Barrett (1993,1995). In particular, synchronously reducedbearing coefficients predicted the system to bestable (all eigenvalues have negative real part) whilethe full tilting-pad models found the rotor to beunstable.

TABLE IV Synchronously reducedcoefficients

Kxx 2.23 107N/mKxy -1.50 105N/mK,.. 1.01 105N/mKyy 1.61 x 10 N/mC.. 1.69 105 N s/mC.y 319.78N s/mCy. -119.55 N s/mCyy 3.56 x 105 N s/m

When using the synchronously reduced bearingcoefficients it is not possible to locate some of theeigenvalues in Table VI. The reason for this is thatsome of the dynamics of the system are neglectedwhen using the synchronously reduced bearingcoefficients. This changes the dynamic system beinganalyzed and different results are expected. Inparticular, eigenvalues 9-12 in Table VI are directlyrelated to the natural frequencies of the bearingpads, while eigenvalues 3-6 in Table VI are theresult of the dependency of the bearing coefficientson the eigenvalues. Figure 3 shows the effect of thereduction frequency on the equivalent coefficients.The coefficients are plotted as a ratio of thesynchronously reduced coefficients. It is shownthat the reduction frequency has a great effect on


TABLE V Transfer functions representing the tilting-pad bearings

Power of s Num.,. Denx. Numx. Denxy Num.,x Den,x Numy Den,

11 2.650E/05 -3.096E/04 -3.072E+0410 2.504E/ 11 1.000E/00 -6.268E/09 1.000E/00 -6.29 E+09 1.000E/009 5.734E+ 16 1.000E+06 2.264E+ 14 1.000E+06 -2.940E+ 14 1.000E+068 4.010E+21 2.349E+11 3.441E+18 2.349E/11 -5.704E+18 2.349E+17 2.271E+25 1.674E+16 2.811E+22 1.674E+16 -2.867E+22 1.674E+166 6.774E+28 9.598E/19 9.560E/25 9.598E/19 -7.730E/25 9.598E/195 1.233E+32 2.866E+23 1.776E/29 2.866E/23 -1.221E+29 2.866E+234 1.460E+35 5.185E+26 1.949E+32 5.185E+26 -1.167E+32 5.185E+263 1.106E+38 6.001E+29 1.133E/35 6.001E+29 -5.846E+34 6.001E+292 5.066E+40 4.280E/32 2.504E/37 4.280E/32 -1.141E+37 4.280E/32

1.377E/43 1.645E+35 -5.776E+37 1.645E/35 -5.776E+37 1.645E/350 1.888E+45 2.542E+37 3.956E+38 2.542E+37 3.956E+38 2.542E+37

6 324E+055 839E+11332E+ 17

9 322E+215 282E+25600E/29

2.984E/323.690E/352.981E+381.524E+414.508E+435.853E/45

1.000E+001.000E+062.349E+111.674E+169.598E+192.866E/235.185E+266.001E/294.280E+321.645E/352.542E/37

TABLE VI Eigenvalues using the transfer function represen-tation of the tilting-pad bearings

Eigenvalue no. Damping exp. (s-l) Frequency (rpm)

-11.74 20082 2.76 21013 -124.8 32344 -156.6 37075 -393.5 48896 -338.8 50057 -13.55 79948 -17.50 8359 -981.8 1169010 -979.5 1170011 -977.7 1171012 -978.1 11 71013 -275.1 13 55014 -132.0 1542015 -198.8 16430

the value of the equivalent coefficients. Thereforethe full dynamics of the bearing should be used forstability calculations.

CONCLUSIONS

It is possible to model a tilting-pad bearing withtransfer functions using the algorithm presentedhere. The stability analysis shows that synchro-nously reduced coefficients can predict eigenvaluessignificantly different than those predicted using a

TABLE VII Eigenvalues using the synchronously reducedcoefficients

Eigenvalue no. Damping exp. (s-1) Frequency (rpm)

-44.31 19962 -1.23 21223 -14.44 79774 -20.67 81475 --583.7 114306 -104.7 156407 -190.3 15850

complete model of the tilting-pad bearings. It isshown that using the synchronously reduced bear-ing coefficients tends to over-predict the stability ofthe system while some eigenvalues are eliminatedcompletely from the model. A case is presentedwhere the synchronously reduced coefficients pre-dicted the rotor to be stable while a complete modelof the tilting-pad bearing predicted instability. Theevidence shown in this work supports the require-ment of using all dynamics of the bearing whenperforming stability analysis. The transfer func-tion representation is a simple way to include alldynamics of the bearing in existing computercodes.The algorithm presented in this paper is just one

of many ways to calculate the coefficients of thetransfer functions representing a tilting-pad bear-ing. This one was chosen partly because it is pos-sible to write each coefficient independently.

(a) 3.5

2.5

2

1.5

0.5

o Kxx/KSxx_-- Kxy/KSxy

-- 4- ......--.._

0.25 0.5Frequency Ratio (o/o

0.75

(b) 1.2 L1.1

0.9

0.8

. 0.7

tw 0.6

.=_" 0.5

I10.4

0.3

0.2

0.1

-0.1o 0.25 0.5 O.75

Frequency Ratio o/o

FIGURE 3 (a) Effect of the reduction frequency on the equivalent bearing stiffness coefficients. (b) Effect of the reductionfrequency on the equivalent bearing damping coefficients.


NOMENCLATURE

ai bi, cidi, ei,f

Ai, Bi, Ci,Oi, Ei

Ad, Bd, Dd

[Coq(S)]

Cij

[keq(S)]

[Keq(S)]

Coefficients for the partial-fraction expansionCoefficients for the partial-fraction expansionChange of variables to obtain thetransfer function coefficientsEquivalent bearing dampingcoefficient matrix as a functionof the complex frequency s, N s/mThe i,j equivalent dampingcoefficients, reduced at a

frequency different thansynchronous(used in Fig. 3), N s/mThe i,j synchronously reduceddamping coefficients

(used in Fig. 3), N s/mDamping coefficient matrixfor the shaft degrees offreedom, N s/mCross-coupled dampingcoefficient matrices betweenthe shaft and pad degreesof freedom, N s

Damping coefficient matrixfor the pad degrees offreedom, N-m-sCoefficients in the transferfunctionPad inertia matrix, kgm2

Equivalent bearing stiffnesscoefficient matrix as a functionof the complex frequency s, N/mEquivalent complex bearingcoefficient matrix as a functionof the complex frequency s, N/mThe l, m equivalent bearingcoefficients, where or mcould be x or yThe i,j equivalent stiffnesscoefficients, reduced at a

frequency different thansynchronous (used in Fig. 3), N/m

Ksij

m

x,y

The i,j synchronously reducedstiffness coefficients

(used in Fig. 3), N/mStiffness coefficient matrix forthe shaft degrees of freedom, N/mCross-coupled stiffness coefficientmatrices between the shaft andpad degrees of freedom, NStiffness coefficient matrix for thepad degrees of freedom, N-mPad preload, dim.Mass matrix for the journal, kgNumber of padsComplex frequency, 1,/sDisplacement vector for the shaftdegrees of freedom {x y}T, mDisplacement coordinates of theshaft in the horizontal and verticaldirections, mPad offset factor, dim.Rotation coordinates of pad1,2,...,n, dim.Rotation vector for the padsdegrees of freedom{61(2 6.}T, dim.Pad arc length, dim.Pivot arc, dim.Reduction frequency for thebearing coefficients(used in Fig. 3), rad/sSpin speed of the rotor, rad/s

ReferencesBarrett, L.E., Allaire, P.E. and Wilson, B.W., 1988, "The

eigenvalue dependence of reduced tilting pad bearing stiffnessand damping coefficients," Tribology Transactions, 31(4),411-419.

Branagan, L.A., 1988, "Thermal analysis of fixed and tiltingpad journal bearings including cross-film viscosity varia-tions and deformations," Ph.D. Dissertation, University ofVirginia.

Brockett, T.S. and Barrett, L.E., 1993, "Exact dynamic reduc-tion of tilting-pad bearing models for stability analyses,"STLE Tribology Transactions, 36(4), 581-588.

Brockett, T.S. and Barrett, L.E., 1995, "Magnetic bearingmodels with flexible supports in rotor stability analyses,"

10 J.A. V/kZQUEZ AND L.E. BARRETT

Proceedings of the 1995 Design Engineering Technical Con-ferences, Vol. 3, Part B, pp. 1063-1072, Paper DE-Vol. 84-2.

Leung, A.Y.-T., 1978, "An accurate method of dynamiccondensation in structural analysis," International Journal.for Numerical Methods in Engineering, 12, 1705-1715.

Leung, A.Y.-T., 1989, "Multilevel dynamic substructures,"International Journal for Numerical Methods in Engineering,28, 181-191.

Lund, J.W., 1964, "Spring and damping coefficients for thetilting-pad journal bearing," ASLE Transactions, 7, 342-352.

Maslen, E.H. and Bielk, J.R., 1992, "A stability model forflexible rotors with magnetic bearings," Journal of DynamicSystems, Measurement, and Control, 114, 172-175.

Shapiro, W. and Colsher, R., 1977, "Dynamic characteristics offluid film bearings," Proceedings of the 6th TurbomachinerySymposium, Texas A&M University, pp. 39-54.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


modeling of tilting-pad journal bearings with transfer...

Documents