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Proc. of 21st

IMAC, Paper no. 230, Feb. 2003

MULTI PLANE BALANCING OF A ROTATING MACHINE

USING RUN-DOWN DATA

A. W. Lees, J. K. Sinha and M. I. Friswell*

Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK(a.w.lees@swansea.ac.uk, mesinhjk@swansea.ac.uk)

*Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK(m.i.friswell@bristol.ac.uk)

ABSTRACT

Earlier studies have suggested a reliable estimation of the

state of the rotor unbalance (both amplitude and phase) atmulti plane of a flexibly supported machine from measuredvibration data on a machine during a single machine run-down. Now a method has been proposed that can reliablyestimate both the rotor unbalance and misalignment frommeasured vibration during a single machine run-down. Forthis identification, it has been assumed that the source ofmisalignment in the rotor is the coupling of the multi-rotorsystem, and that will generate constant forces andmoments at the couplings depending upon the extent of theoff-set between the two rotors irrespective of the machinerotating speed. The theoretical concept, and the completecomputational implementation used are presented in thispaper. The method is demonstrated using experimentaldata from a machine with 2 journal bearings and a flexible

coupling to the motor.

1. INTRODUCTION

Power station turbogenerators may be considered toconsist of three major parts; the rotor, the fluid journalbearings and the foundations. In many modern plants,these foundation structures are flexible and have asubstantial influence on the dynamic behaviour of themachine. These machines have a high capital cost andhence the development of condition monitoring techniquesfor such rotating machines is important. Perhaps thevibration-based identification of faults such as rotorunbalance, rotor bent, crack, rub, misalignment, fluidinduced instability is well-developed [1] and widely used in

practice, however the quantification parts the extent ofidentified faults and their locations - have been active areasof research for many years. Over the past thirty years,theoretical models have played an increasing role in therapid resolution of problems in rotating machinery. Doeblinget al. [2] gave an extensive survey on the crack detectionmethods. Parkinson [3] and Foiles et al. [4] give somecomprehensive reviews on the rotor balancing. Muszynska[5] gave a thorough review of the rubbing phenomenon.Edwards et al. [6] gave a brief review of the wider field offault diagnosis. However the study on the rotor

misalignment is limited to understanding this phenomenon[7-13] but none of them have suggested any method for thedirect quantification of misalignment.

In spite of the success of model based estimation of faults,the construction of a reliable mathematical model is stillimpossible. Often a good finite element (FE) model of therotor and an adequate model of the fluid bearing [14] maybe constructed. Indeed several FE based softwarepackages are available for such modelling. However areliable FE model for the foundation is difficult, if notimpossible, to construct due to a number of practicaldifficulties [15]. Inclusion of the foundation model is veryimportant as it is observed that the dynamics of the flexiblefoundation also contributes significantly to the dynamics ofthe complete machine. Many research studies [16-22] havebeen carried out to derive the foundation models directlyfrom the measured machine responses but more researchis needed on their practical application. Hence a completemathematical model of a machine is still not available forcondition monitoring in many cases.

Considering the above limitations, an alternate method hasbeen suggested by Lees and Friswell [23] for the reliableestimation of the multi plane rotor unbalance using a priorirotor and bearing models along with the measuredresponse at bearing pedestals from a single machine run-down. They estimated the foundation parameters as a by-product to account for the foundation dynamics. Lees andFriswell [23] demonstrated the method on a simplesimulated example. The method has been further validatedon a small simple experimental rig [24]. In both the cases

the number of modes of the system were less than themeasured DoF in the run-down frequency range. Howeverfor systems like TG sets the number of modes excited maybe more than the measured DoF in the run-down frequencyrange. Thus the estimated unbalance may not account forall the critical speeds. Hence the method has been modifiedfurther to reliably estimate the rotor unbalance by splittingthe whole frequency range into bands for the foundation sothat the band dependent foundation models account for allcritical speeds. The advantages of the suggested approach

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have been demonstrated on a complex simulated exampleand on an experimental rig [25-27].

However in a multi-rotor system there is always thepossibility of rotor misalignment and it may influence themachine response and hence the unbalance estimation.The above method has been further modified to estimateboth the rotor unbalance and misalignment. The generalperception and observation is that the misalignment in

multi-coupled rotors generates a 2X (twice the rotatingspeed) component in the response of the machine [8-9] andthe effect on the 1X component is assumed to be small.These features are usually used for the detection of thepresence of rotor misalignment [28]. Many simple analyticalsimulations have been carried out to understand thisphenomena [7, 10]. However in the present paper theidentification of rotor unbalance and misalignment has beencarried out using the 1X response at the bearing pedestalsof the machine from a single run-down, even though thedirect influence on the 1X response due to misalignmentmay be small.

For the current identification, it has been assumed that thesource of misalignment in the rotor is the coupling of themulti-rotor system. Such a misalignment will generateconstant forces and moments at the couplings dependingupon the extent of the off-set between the two rotors andirrespective of the machine rotating speed [7]. The forcevector in the equation of motion was assumed to consist ofboth the unbalance forces and the constant forces andmoments at the couplings, and these parameters wereestimated along with the foundation model. The theoreticalconcept, and the complete computational implementationused are presented in this paper. The method isdemonstrated using experimental data from a machine withtwo bearings and a flexible coupling to the motor.

Figure 1. The abstract representation of a turbogeneratorsystem

2. THEORY

Figure 1 shows the abstract representation of aturbogenerator, where a rotor is connected to a flexiblefoundation via oil-film journal bearings. The equations ofmotion of the system may be written [1, 26] as

0

0

f

r

r

r

ZZZ0

ZZZZ

0ZZ u

bF

bR

iR

FBB

BBbbRbiR

ibRiiR

,

,

,

,,

,, (1)

where Z is the dynamic stiffness matrix, the subscripts band i refer to internal and bearing (connection) degrees offreedom respectively, and the subscripts F, R, and B referto the foundation, the rotor and the bearings. r are the

responses and uf are the force vectors, which are

assumed to be applied only at the rotor internal degrees offreedom. The dynamic stiffness matrix of the foundation,

FZ , is defined only at the degrees of freedom connecting

the bearings and the foundation. In practice this will be areduced order model, where the internal foundation degreesof freedom have been eliminated [1]. The dynamic stiffness

matrix of the bearings is given by BZ . It has been assumed

that the inertia effects within the bearings are negligible,although these could be included if required.

Solving equation (1) to eliminate the unknown response ofthe rotor gives,

1 1, ,

1,

F F b B R bi R,ii u

B B F b

Z r Z P Z Z f

Z P Z I r (2)

where1

, , ,R bb B R bi R,ii R ib P Z Z Z Z Z . It is assumed

that good models for the rotor and the bearings, RZ and

BZ , are known a priori and ,F br is measured. Thus, the

only unknown quantities in equation (2) are the foundation

model, FZ , and the force vectors, uf .

2.1. PARAMETER ESTIMATION

The force vectors can be defined as

munu fff , (3)

where unf is the vectors of the unbalance forces and mf is

the vectors of forces and moments at the couplings.

Unbalance Forces: Although the unbalance will bedistributed throughout the rotor, this is equivalent to adiscrete distribution of unbalance, provided there are asmany balance planes as active modes. Suppose the

unbalance planes are located at nodes 1 2, , , pn n n ,

where pis the number of planes. The associated amplitudeof unbalance (defined as the unbalance mass multiplied by

distance between the mass and geometric centres) and

phase angles are1 2

T[ , , , ]pn n n

u u u and

1 2

T[ , , , ]pn n n

respectively. These amplitudes and

phase angles can be expressed, for the ith balance plane,

as the complex quantity , ,exp j ji i i in n r n i nu e e .

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Hence, the unbalance forces, unf , in the horizontal and

vertical directions, can be written as [26]

Tef 2un , (4)

where1 2 1 2

T

, , , , , ,p pr n r n r n i n i n i ne e e e e e

e and

Tis a selection matrix indicating the location of the balanceplanes.

Figure 2 Schematic of rotor with misalignment at a coupling

Misalignment Forces and Moments:It has been assumedthat the misalignment in the rotor exists at the couplingsbetween the multi-rotors. The nature of the rotormisalignment could be parallel, angular or combined asshown in Figure 2, but all of them would generate forcesand moments. Let us assumed that there are c couplings

in the rotor located at nodes 1m , 2m , , cm . The

associated amplitude of the forces and moments are

1 1 1 1 2 2, , , , , ,, , , , , ,m z m y m y m z m z m y mf f M M f f e

, , , ,, , ,c c c c

T

z m y m y m z mf f M M , where the subscripts

yand zrepresent the horizontal and vertical directions and fand M are forces and moments respectively. Hence, the

misalignment force, mf , can be written as [1]

mmm eTf , (5)

where mT is the transformation matrix indicating the

location of the couplings. Substituting equations (4) and (5)into equation (2) produces,

bFBBm

miiRbiRBbFF

,

,,,

rIZPZ

e

eTTZZPZrZ

1

211

(6)

To identify the foundation parameters and forces in a leastsquares sense, the foundation parameters are grouped intoa vector v. We will assume that the foundation dynamic

stiffness matrix, FZ , is written in terms of mass, damping

and stiffness matrices. If there are nmeasured degrees offreedom at the foundation-bearing interface, then vwill takethe form,

,11 ,12 , ,11 ,12

T

, ,11 ,12 ,

F F F nn F F

F nn F F F nn

k k k c c

c m m m

v

(7)

where the elements in v are individual elements of thestructural matrices. With this definition of v, there is a lineartransformation such that

,F F b Z r W v , (8)

where W contains the response terms at each measuredfrequency [1]. For the qth measured frequency

0 1 2q q q q W W W W , (9)

where, if all elements of the foundation mass, damping andstiffness matrices are identified,

T,

T,

T,

( )

( )( ) j

( )

F b q

k F b qk q q

F b q

r 0 0

0 r 0W

0 0 r

, (10)

for 0, 1, 2k . Equation (6) then becomes

qm

qmqq

Q

e

e

v

RRW , (11)

where the form of R, mR and Q may be obtained by

comparing equations (6), (10) and (11), as

2 1 1,( ) ( ) ( ) ( )q q B q R bi q R,ii q R Z P Z Z T (12)

1,( ) ( ) ( ) ( ) ( )q B q q B q F b q

Q Z P Z I r (13)

mqiiRqbiRqqBqm T)(Z)(Z)(P)(Z)(R ,, 11

(14)

Clearly there is an equation of the form of (11) at everyfrequency. The equations generated may be solved in aleast squares sense directly, although the solution via thesingular value decomposition (SVD) is more robust. Suchan equation error approach does not optimise the error inthe response directly, and thus the accuracy of thepredicted response is not assured. The great advantage isthat the equations are linear in the parameters. However anon-linear optimisation (output error) may be performed,starting with linear estimated parameters, if a more accurateprediction of the response is required [1, 22]. In the present

z

Parallel Misalignment

y

Angular Misalignment

z

x

Combined Misalignment

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paper, only the equation error approach has beenconsidered in order to concentrate on the influence offrequency range subdivision. Furthermore, the unbalanceand misalignment seems to be estimated robustly by theequation error approach, even if the foundation is relativelyinaccurate [24].

2.2. SPLITTING THE FREQUENCY RANGE

Suppose that the frequencies at which the response ismeasured are q , 1, ,q N . Let us assume that the

run-down frequency range is split into b frequency bands.The vectors of the foundation parameters are identified in

each frequency band, and are denoted 1v , 2v ,, bv .

Hence combining the frequency band dependent foundationmodels and the global unbalance and misalignment similarto the unbalance estimation [26], gives from equation (11),

b

m

b

1

bmbb

m

m

band_

band_2

1band_

2

band,bandband_

2band,2bandband_2

1band,1bandband_1

Q

Q

Q

e

e

v

v

v

RRW00

RR0W0

RR00W

__

__

__

(15)

Let us assume that the stiffness of an ith coupling is ic,K

then the linear misalignment, iy & iz

, and the angular

misalignment, iy, & iz, at the ith coupling in the

horizontal and the vertical directions can be calculated as[7]

iz

iy

iy

iz

ic

iz

iy

i

i

y

z

,

,

,

,

,

,

,

]K[

M

M

f

f

1 (16)

2.3. REGULARISATION

Equation (15) is a least-squares problem, and its solution islikely to be ill-conditioned [21]. Generally two types of

scaling, namely row scaling and column scaling, may beapplied to least squares problems [29]. Column scaling isnecessary because of the different magnitudes of the

elements of the FM , FC and FK matrices, and the

scaling factors used here were 1, and 2 respectively,where is the mean value of the frequency range. Thescaling of the columns of R and mR depend upon

engineering judgement based on the unbalance and

misalignment magnitudes expected. Truncated SVD wasused to solve the equations [30].

Other physically based constraints may be applied to thefoundation model to improve the conditioning. For example,the mass, damping and stiffness matrices of the foundationmay be assumed to be symmetric, therefore reducing thenumber of unknown foundation parameters. Otherconstraints could be introduced, such as a diagonal mass or

damping matrix, or block diagonal matrices if bearingpedestals do not interact dynamically.

3. THE EXPERIMENTAL EXAMPLE

Figure 3 shows a photograph of the rig. Each foundation ofthis rig, shown in Figure 3, consists of a horizontal beam(500mm x 25.5mm x 6.4 mm) and a vertical beam (322mmx 25.5mm x 6.4mm) made of steel. The horizontal beam isbolted to the base plate and the vertical beam to thebearing assembly as seen in the photograph. A layer ofacrylic foam (315mm x 19mm x 1mm) was bonded betweenthe vertical beam and thin layers of metal sheet (315mm

x19mm x 350m) to increase the damping. In fact themodal experiment confirms that the damping of the

foundation increased to 1.4% from a value of 0.6% at thefirst lateral mode. A 12mm OD (d) steel shaft of 980mmlength is connected to these flexible supports and iscoupled to the motor through a flexible coupler. One flexiblefoundation is connected at 15 mm and other at 765 mmfrom the right end of the shaft through self-lubricating ballbearings. The shaft also carries two identical balancingdisks of 75mm OD and 15mm thickness and placed at 140mm and 640 mm from the right end of the shaft. Disk A isnear to the Motor.

An FE model was created for the rotor using two-nodedTimoshenko beam elements, each with two translationaland two rotational degrees of freedom. Sinha [1] givesdetails of the dynamic characterization of the rig by modaltests and FE analysis.

Figure 3 Photograph of the rig in Swansea (UK)

Different run-down experiments were performed with therotor speed reducing from 2500 RPM to 300 RPM fordifferent combinations of added masses to the balancedisks A and B listed in Table 1. Runs 1 and 4 were the

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residual runs-down i.e., without any added mass to thedisks. The order tracking was performed such that each setof the run-down data consisted of the 1X component of thedisplacement responses in the frequency range from5.094Hz to 40.969Hz in steps of 0.125Hz.

Four critical speeds (two in the horizontal and one each inthe vertical and axial directions) of the machine werepresent in the run-down frequency range of 5.094Hz to

40.969Hz. The unbalance and misalignment estimation wascarried out by the suggested method for individual runsassuming misalignment forces and moments at the couplingof the rotor with the motor. The frequency range was splitinto three bands; 5.094Hz to 12.094Hz, 12.094Hz to27.469Hz, and 27.469Hz to 40.969Hz based on theobservation that the estimated responses were a close fit tothe measured responses. The estimated results are listed inTable 1 and Figure 4 compares typical measured andestimated responses.

Figure 4 shows that the fit of the estimated response isclose to the measured. Table 1 also shows that theestimated unbalance is excellent and close to the actualvalues and the estimated misalignment in the rotor at thecoupling is quite consistent for each run. The order of theestimated misalignment (approximately 0.1 and 0.2 mm inthe horizontal and vertical directions and their relatedangular misalignment is 0.4 and 0.2 degrees respectively) isvery small and such a misalignment is quite possible duringthe rig assembly. The small deviation in the estimation maybe because of noise in the measurements, however theestimation seems to be quite robust. Hence thisexperimental example confirms that an accurate estimationof both the rotor unbalance and misalignment is possibleusing measured responses from a single run-down of amachine.

5 10 15 20 25 30 35 4010

8

106

104

102

Frequency, Hz

HorizontalDispl.,m

ExperimentalEst. found., unb. & misalignment

5 10 15 20 25 30 35 4010

7

106

105

104

103

Frequency, Hz

VerticalDispl.,m

ExperimentalEst. found., unb. & misalignment

Figure 4 Measured and estimated responses at bearing A,

for run 2 for the experimental rig

4. CONCLUSION

An estimation method for both the state of rotor unbalance(amplitude and phase) and the misalignment of a rotor-bearing-foundation system has been presented. Theestimation uses a priori rotor and bearing models along withmeasured vibration data at the bearing pedestals from a

single run-down or run-up of the machine. The method alsoestimates the frequency band dependent foundationparameters to account for the dynamics of the foundation.The suggested method has been applied to a smallexperimental rig and the estimated results were excellent.Hence the suggested method seems to be reliable for theestimation of both rotor unbalance and misalignment andneeds to be tested on real machines, such as a TG set, tofurther enhance the confidence level in the approach.

5. ACKNOWLEDGEMENTS

The authors acknowledge the support of EPSRC throughgrant number GR/M52939. Jyoti K. Sinha acknowledgesMr R. K. Sinha, Associate Director, RD & DG of his parentorganization B.A.R.C., India for consistent support andencouragement.

6. REFERENCES

1. SINHA, J. K., Health Monitoring Techniques forRotating Machinery, PhD Thesis, University of WalesSwansea, 2002.

2. DOEBLING, S. W., FARRAR, C. R. and PRIME, M. B.,

A Summary Review of Vibration-based DamageIdentification Methods, Shock and Vibration Digest30(2), 91-105, 1998.

3. PARKINSON, A. G., Balancing of Rotating Machinery,IMechE Proceedings C Journal of MechanicalEngineering Science, 205, 53-66, 1991.

4. FOILES, W. C., ALLAIRE, P. E., and GUNTER, E. J.,Review: Rotor Balancing, Shock and Vibration 5(5-6),325-336, 1998.

5. MUSZYNSKA, A., Rotor to Stationary Element Rub-Related Vibration Phenomena in Rotating MachineryLiterature Survey, Shock and Vibration Digest 21(3), 3-11, 1989.

6. EDWARDS, S., LEES, A.W. and FRISWELL, M.I.,Fault diagnosis of rotating machinery, Shock andVibration Digest, 30(1), 4-13, 1998.

7. GIBBONS, C. B., Coupling Misalignment Forces,Proceedings of the 15

thTurbomachinery Symposium,

Gas Turbine Laboratory, 111-116, Texas, 1976.

8. DEWELL, D. L., and MITCHELL, L. D., Detection of aMisaligned Disk Coupling using Spectrum Analysis,Transactions of the ASME - Journal of VibrationAcoustics Stress and Reliability in Design 106(1), 9-16,1984.

9. EHRICH, F. F. (Editor), Handbook of Rotordynamics.NY: McGraw-Hill, 1992.

10. SEKHAR, A. S., and PRABHU, B. S., Effects of

Coupling Misalignment on Vibrations of RotatingMachinery, Journal of Sound and Vibration 185(4),655-671, 1995.

11. SIMON, G., Prediction of Vibration Behaviour of largeTurbo-machinery on Elastic Foundations due toUnbalance and Coupling Misalignment, Proceedings ofthe Institution of Mechanical Engineers, Part C Journal of Mechanical Engineering Science 206(C1),29-39, 1992.

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12. XU, M., and MARANGONI, R. D., Vibration Analysis ofa Motor-Flexible Coupling-Rotor System subjected toMisalignment and Unbalance, Part I: Theoretical Modeland Analysis, Journal of Sound and Vibration 176(5),663-679, 1994.

13. XU, M., and MARANGONI, R. D., Vibration Analysis ofa Motor-Flexible Coupling-Rotor System subjected toMisalignment and Unbalance, Part II: ExperimentalValidation, Journal of Sound and Vibration 176(5), 681-691, 1994.

14. HAMROCK, B.J., Fundamentals of Fluid FilmLubrication, McGraw-Hill, Inc., NJ, 1994.

15. LEES, A.W. and SIMPSON, I.C., The dynamics ofturbo-alternator foundations, IMechE ConferencePaper C6/83, 37-44, 1983.

16. LEES, A.W., The Least Squares Method Applied toIdentified Rotor/Foundation Parameters, IMechEConference on Vibrations in Rotating Machinery,Paper C306/88, 209-216, 1988.

17. ZANETTA, G. A., Identification Methods in theDynamics of Turbogenerator Rotors, IMechEConference on Vibrations in Rotating Machinery, Bath

Paper C432/092, 173-181, 1992.18. FENG, N. S. and HAHN, E. J., Including Foundation

Effects on the Vibration Behaviour of RotatingMachinery, Mechanical Systems and SignalProcessing 9, 243-256, 1995.

19. VANIA, A., Estimating Turbo-Generator FoundationParameters, Politecnico di Milano, Dipartimento diMeccanica, Internal Report 9-96, 1996.

20. PROVASI, R., ZANETTA, G. A., and Vania, A., TheExtended Kalman Filter in the Frequency Domain forthe Identification of Mechanical Structures excited bySinusoidal Multiple Inputs, Mechanical Systems andSignal Processing 14(3), 327-341, 2000.

21. SMART, M.G., FRISWELL, M.I. and LEES, A.W.,Estimating Turbogenerator Foundation Parameters

Model Selection and Regularisation, Proceedings ofthe Royal Society A456, 1583-1607, 2000.

22. SINHA, J. K., LEES, A. W., FRISWELL, M. I., andSINHA, R. K., The Estimation of Foundation Models ofFlexible Machines, Proceedings of 3

rdInternational

Conference Identification in Engineering Systems,300-309, Swansea, UK, April 15-17, 2002.

23. LEES, A.W. and FRISWELL, M.I., The Evaluation of

Rotor Unbalance in Flexibly Mounted Machines,Journal of Sound and Vibration,208, 671-683, 1997.

24. EDWARDS, S., LEES, A.W. and FRISWELL, M.I.,Experimental Identification of Excitation and SupportParameters of a Flexible Rotor-Bearing-FoundationSystem from a Single Run-Down, Journal of Soundand Vibration, 232(5), 963-992, 2000.

25. SINHA, J. K., LEES, A. W., and FRISWELL, M. I.,Estimating the Unbalance of a Rotating Machine froma Single Run Down, Proceedings of 19

thInternational

Modal Analysis Conference, Florida, 109-115, 2001.

26. SINHA, J. K., FRISWELL, M. I., and LEES, A. W., TheIdentification of the Unbalance and the FoundationModel of a Flexible Rotating Machine from a Single

Run Down, Mechanical Systems and SignalProcessing 16(2-3), 255-271, 2002.

27. LEES, A. W., SINHA, J. K., and FRISWELL, M. I., TheIdentification of the Unbalance of a Flexible RotatingMachine from a Single Run-Down, Proceedings ofASME Turbo Expo Conference, ASME paper GT-2002-30420,Amsterdam, Holland, 3-6 June, 2002.

28. JORDAN, M. A., What are Orbit plots, anyway?, Orbit,8-15, Bently Nevada Corporation, USA, Dec., 1993.

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rdedition, 1996.

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Table 1 Estimation of both the rotor unbalance and misalignment from the experimental run-down data

Estimated Misalignment Added Unbalance (g@ deg

Run Disk

ActualUnbalance(g @ deg.)

EstimatedUnbalance(g @ deg.)

Hori., y

(mm)

Vert., z (mm)

Hori. Angle

z (deg.)Ver. Angle

y (deg.)With respect

to Run 1With respe

to Run 4

1A

B

Residual

Residual

1.84 @ 130

1.46 @ 350

0.10 0.21 0.40 0.69 ----- -----

4A

B

Residual

Residual

1.99 @ 127

1.44 @ 341

0.09 0.18 0.32 0.20 ----- -----

2A

B

0.76 @ 180

0.76 @ 0

2.48 @ 144

2.33 @ 354

0.14 0.20 0.75 0.45 0.82 @ 181

0.88 @ -4

0.82 @ 19

0.98 @ 14

3A

B

1.52 @ 180

1.52 @ 0

2.75 @ 158

2.96 @ 1

0.14 0.18 0.42 0.16 1.41 @ 191

1.55 @ 11

1.46 @ 20

1.67 @ 18

5A

B

0.76 @ 45

0.76 @ 225

2.54 @ 114

1.34 @ 323

0.08 0.21 0.24 0.17 0.95 @ 79

0.66 @ 236

0.76 @ 76

0.45 @ 22