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Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and AirGT2009

June 8-12, 2009, Orlando, Florida, USA

GT2009-59256

PREDICTED ROTORDYNAMIC BEHAVIOR OF A LABYRINTH SEAL AS ROTORSURFACE SPEED APPROACHES MACH 1

Manish R. ThoratResearch Assistant

Turbomachinery Laboratory, Texas A&MUniversity

Dara W. ChildsLeland T. Jordan Professor of MechanicalEngineering, Turbomachinery Laboratory,

Texas A&M University, College Station,TX-77843

ABSTRACTPrior one-control-volume (1CV) models for rotor-fluid

interaction in labyrinth seals produce synchronously-reduced

(at running-speed), frequency-independent stiffness and

damping coefficients. The 1CV model, consisting of a leakage

equation, a continuity equation, and a circumferential-momentum equation (for each cavity) was stated to be invalid

for rotor surface speeds approaching the speed of sound.

However, the present results show that, while the 1CV fluid-

mechanic model continues to be valid, the calculated

rotordynamic coefficients become strongly frequencydependent.

A solution is developed for the reaction-force components

for a range of precession frequencies, producing frequency-dependent stiffness and damping coefficients. They can be

used to define a Laplace-domain transfer-function model for the

reaction-force/rotor-motion components. Calculated

rotordynamic results are presented for a simple Jeffcott rotor

acted on by a labyrinth seal. The seal radius Rs and runningspeed cause the rotor surface velocity Rs to equal the speed

of sound c0 at =58 krpm.

Calculated synchronous-response results due to imbalance

coincide for the synchronously-reduced and the frequency-

dependent models.

For an inlet preswirl ratio of 0.5, both models predict thesame log decs out to 14.5 krpm. The synchronously-reduced

model predicts an onset speed of instability (OSI) at 15 krpm,but a return to stability at 45 krpm, with subsequent increases inlog dec out to 65 krpm. The frequency-dependent model

predicts an OSI of 65 krpm. The frequency-dependent models

predict small changes in the rotors damped natural frequencies.

The synchronously-reduced model predicts large changes.

The stability-analysis results show that a frequency-dependent labyrinth seal model should be used if the rotor

surface speed approaches a significant fraction of the speed of

sound. For the present example, observable discrepancies arose

when Rs = 0.26 c0.

INTRODUCTIONLabyrinth seals are used widely in turbomachinery to

reduce leakage. However, they are known to cause latera

rotordynamic instabilities.

A tooth-on-stator/smooth-rotor labyrinth is considered hereSeveral models have been developed to predict the

rotordynamic coefficients for this seal. Kurohashi et al. [1] and

Iwatsubo [2] developed a 1CV analysis. Childs and Scharrer [3

modified the 1CV model to include angular area derivative inthe continuity and momentum equations. Wyssman et al. [4]

developed the first two-control-volume (2CV) model using abox-in-box model to account for the through flow and vortex

flow in the labyrinth cavity. An alternative 2CV model wasdeveloped by Scharrer [5]. However, experimental results by

Picardo [6] showed that the 1CV model by Childs and Scharre

yielded better predictions than Scharrers 2CV approach

Analysis using these models produced synchronously-reducedfrequency-independent results based on the assumption that the

lowest acoustic (circumferential direction) natural frequency in

the seal cavity is much higher than the rotor speed. Picardos

measured results produced frequency-independent coefficients.If the Rs is comparable to c0, frequency dependency of the

coefficients is expected. The present analysis uses the model o

Childs-Scharrer but adopts the analysis procedure by Thielekeand Stetter [7].

NOMENCLATURE

A cross-sectional area of controlvolume

[L2]

B Tooth height [L]

C, c direct and cross coupled damping [F t/L]

1 Copyright 2009 by ASME


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c0 speed of sound [L/t]

FX, FY seal reaction forces [F]

fr ,f radial and circumferential

dynamic stiffness coefficients

[F/L]

H seal Clearance [L]

j 1 K,k direct and cross coupled stiffness [F/L]kr

Jeffcott rotor stiffness [F/L]

L pitch of seal strip [L]

m&

leakage flow rate [M/L t]

P pressure [F/L2]

R Gas constant [L2/Tt2]

Rs radius of seal [L]

s complex variable for Laplace

transform

[1/t]

T Temperature [T]

t time [t]

U circumferential flow velocity [L/t]

u0(0) U(0)/Rs

, Preswirl ratio circumferential coordinate

density [M/L3]

shear stress [F/L2]

kinematic viscosity [L2/t]

rotor speed [1/t]

Excitation frequency [1/t]

SUBSCRIPTS

0,1 zeroth and first order

i Seal cavity index

r, radial and circumferential

R,S rotor and stator

THEORETICAL MODEL

The labyrinth seal model is based on the control volumeshown in Figs. 1 and 2.

Fig. 1 Axial View of Labyrinth Seal Cavity and ControlVolume

Fig. 2 Radial View of Labyrinth Seal Cavity andControl Volume

Continuity and Circumferential momentum equations are

derived for this control volume based on the analysis by Childs

and Scharrer [3]. The gas is ideal and temperature is constant.

Continuity Equation:

1( ) ( )i i i

i i i is

AUA m m

R t

+

0+ + =

& & (1)

Circumferential Momentum Equation:

2

1 1( ) ( )

( )

i i ii i i i i i i

s

i i Ri Ri Si Si is

AUAU m U m U

R t

A Pa a LR

+

+ +

= +

& &

(2)

where aR and aS are dimensionless lengths upon which shear

stress acts.Rotor and stator shear stresses are modeled using a Blasius

shear-stress model. To account for the curvature effects, the

shear stress terms are modified according to Martinez-Sanchez

et al. [8].

0.25

0.25

| |(1.0 0.075( ) )

2

| |(1.0 0.075( ) )

2

s i i iRi Ri

s

i i iSi Si

s

UR Dh D

R

Dh Dh

R

U

+

=

=

+

h

where, Dhi is the hydraulic diameter defined by

2( )

( )

i i i

i

i i i

H B LDh

H B L

+=

+ +

Leakage Model:

2 21

1 2i i

i i i

P Pm H

RT

=& (3)

1im +&

Control VolumePi

Ui

im&

ROTOR

STATOR

Pi-1Ui-1

Pi+1

Ui+1



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where, 1i is the Chaplygins flow coefficient [9], 2 is the

Neumanns kinetic energy carryover coefficient [10].

PERTURBATION ANALYSIS

A perturbation analysis is used to solve the governing

equations. The eccentricity ratio is the perturbation parameterfor small motion about a centered position. The zeroth-order

equations define the leakage flow rate, plus steady-statecircumferential velocities and cavity pressures at each cavity.

The first-order equations define the perturbations in pressure

and circumferential velocity of flow due to rotor motion.

First-Order Perturbation Equations:

Continuity Equation

1 1 0 1 1 0 11

2 014 1 1 5 1 1 1 2

16

i i i i i i ii

s s

i ii i i i i i

s

P G U P G P U G G

t

G U

3 1i iP

H H

G P G P G H

R R

RG t

+

+ + +

+ + =

(4)

Circumferential Momentum Equation

1 1 0 1 0 11

2 1 0 1 1 3 1 4 1 1 5 1

i i i i i ii

s S

i i i i i i i i

U X U U A P X

t

X

R R

U m U X P X P X H

+ +

+ + + =&

(5)

Here, , are the perturbed circumferential velocities and

pressures in the ith cavity, and

1iU 1iP

1H is the first order clearance

perturbation. Due to insufficient space, the coefficients G1i and

X1i are not given in this paper; however they can be referred to

in ref. [3].

Elliptic-Orbit Solution

For small motion about a centered position, labyrinth seal

reaction forces are typically modeled as

X

Y

F K k X C c X

F k C Y c C Y

= +

&

&

i

(6)

Starting with Iwatsubo, the solution for the four unknownsk, K, c and C, was obtained by assuming an elliptical orbit of

the form:

(7), = s= cos int Y bX a tWhere the rotor is assumed to be precessing at the running

speed . The clearance function associated with this solution is:

1 cos sin sos ncH t b t a = (8)

A separation-of-variables approach is used for Eqs.(4-5) for

this clearance excitation. Substituting Eq.(7) into Eq. (6) yields:

)cos ( )sin

) cos ( ) sin

(

(

X

Y

t kb Ca t F Ka c

F t Kb caCb t

b

ka

Because of the separate sine and cosine terms, and the two

independent parameters a and b in these two equations, two

independent equations are developed from each equation to

solve for the four unknowns K, k, C, c.

The solution from this approach is assumed to be valid ifthe lowest acoustic natural frequency in the cavity is much

higher than the rotor speed. However if the rotor speed

approaches the lowest cavity natural frequency, the frequency-independent solution breaks down. The natural frequency

considered here is that of a standing wave in the circular

annulus of a labyrinth seal. To determine the acoustic

frequency, the circular annulus can be considered to be an

open-ended pipe. If the flow velocity within the pipe isnegligible compared to the sonic velocity, and continuity is

enforced at the two ends of the pipe, the natural frequencies are

0 0 02( ) ( ) ; 1,2,.2

ni nis s

i i icpipe cavit

Ry i

L

c

R

c

= = = = ..

Hence, the running speed that equals the 1st acousticresonance is:

01n s

s

c RR

c = 0= = (10)

Note that the limiting condition 1n = is the same as the

rotor surface velocity approaching the acoustic velocity

i.e., 0sR c = . Isothermal sonic velocity, 0c R= T

in

is

considered in the present analysis, as the 1CV model is

isothermal.

Circular Orbit Solution

Using an approach developed by Childs and Kim [11] forliquid annular seals, Thieleke and Stetter [7] developed a

solution for Eqs.(4-5) assuming a circular orbit solution of the

form:(11),= = scost Y aX a t

Substituting this solution into Eq.(6) gives

(12)cos sin

ns so ic

X

Y

t ca t F Ka

k t taF Ca

+

=

=

Resolving these components into radial term that is parallel to

the rotating vector a and to a circumferential term that isperpendicular to a nets:

( )r rF

f K ca

f k Ca

F

= = +

= =

(13)

Equation (13) provides two equations for the fourunknowns; however, using two precession frequencies provides

enough equations. Providing more than two frequencies creates

a circumstance where ( ), )(rf f

1n

are curve fitted to solve for

the unknowns. Thieleke and Stetter did not consider

circumstances where ; i.e. ,where the frequency

independent model of Eq.(6) would become invalid. The

= +

= +

(9)



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example in the following section, explains the inadequacy of

using the frequency independent model when 1n

Test Case for a Long Labyrinth Seal

The test conditions, seal geometry and operating conditions

used in this analysis are obtained from the tests conducted by

Picardo [6]. The input data are shown in Table 1. Air is used inPicardos tests. The mass flow rate for the given operating

conditions at rotor speed of 20200 RPM and preswirl ratio of0.578 is 0.373 kg/sec [6].The predicted exit axial flow Mach

number is 0.29

Table 1 Geometric, Operating, and Input DataReservoir Pressure 70.149 bar

Sump Pressure 36.694 bar

Temperature 288.559 K

Radial Clearance 0.198 mm

Seal Radius 57.340 mm

Tooth Pitch 4.293 mm

Tooth Height 4.293 mmRotor Friction Constant 0.079

Rotor Friction Exponent -0.250

Stator Friction Constant 0.079

Stator Friction Exponent -0.250

Compressibility Factor 1.000

Ratio of Specific Heats 1.400

Kinematic Viscosity 0.00001510 m2/s

Gas Constant 286.900 J/kg K

Number of Teeth 20

Tooth Location Stator

Speed Influence on Acoustic Resonance Location

Figures 3a and 3b show fr and f versus non-dimensionalexcitation frequencies for =15.2 krpm. The excitation

frequency is normalized with respect to the rotor speed .The plots show the resonant peaks of the 1st acoustic mode of

the labyrinth cavity. As increases, the dynamic-stiffness

coefficients at the rotor speed are strongly influenced by

resonance. Thus, the fr and f coefficients can no longer be

defined by linear expressions of Eq.(13).

6.0E+07

5.0E+07

4.0E+07

3.0E+07

2.0E+07

1.0E+07

0.0E+00

1.0E+07

2.0E+07

3.0E+07

0.0 1.0 2.0 3.0 4.0 5.

fr

,RadialDyna

micStiffness

Coefficie

nt(N/m)

Non-Dimensional Frequency

Fig. 3a Radial Dynamic Stiffness Coefficient VsExcitation frequency, (Preswirl ratio=0)

8.0E+07

7.0E+07

6.0E+07

5.0E+07

4.0E+07

3.0E+07

2.0E+07

1.0E+07

0.0E+00

0.0 1.0 2.0 3.0 4.0 5

f

,CircumferentialDynamic

StiffnessCoefficient(N/m)

Non-Dimensional Frequency

Fig. 3b Circumferential Dynamic Stiffness CoefficientVs Excitation frequency, (Preswirl ratio=0)

Figure 4, shows the predicted change in the acoustic

resonant frequency versus . For lower rotor speeds, the

resonant frequency closely matches the predicted isothermaacoustic frequency of 5022 rad/sec from Eq. (10). As the 1CV

model is isothermal, acoustic frequency is evaluated using

isothermal sonic speed, 0c R= T . For higher rotor speeds

the resonant frequency increases with increasing . This

dependency can be attributed to (i) the circumferential flowvelocity in the annulus becoming an appreciable fraction of c0and (ii) convective acceleration terms that are retained here but

neglected in conventional acoustics.



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0

1000

2000

3000

4000

5000

6000

7000

8000

0 10000 20000 30000 40000 50000 60000 70000

AcousticFrequency(rad/sec)

Rotor S eed RPM

Fig.4 Predicted 1st

acoustic damped naturalfrequency versus running speed. (Preswirl ratio=0)

Transfer-Function Models for Reaction Force Components

If Rs approaches c0, fr and f cannot be modeled by the

frequency-dependent model of Eq. (6). A similar situation

arises in honeycomb/hole-pattern stator seals where the

apparent acoustic velocity for flow within the seal can be

reduced due to the effect of gas compressibility within theholes/cells, dropping the lowest acoustic frequency within the

operating region. Kleynhans and Childs [12] present solutions

that produce frequency-dependent rotordynamic coefficients forthese types of seals, following the approach of Bolleter et al.

[13]. Their transfer-function model is

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

X

Y

F s s s x s

F s s s y

=

D E

E D s

]

(14)

with D and Edefined as follows:

) Re[ ( )] Im[ ( )]

) Re[ ( )] Im[ ( )]

) Re[ ( )] Im[ ( )

) Re[ ( )] Im[ ( )

(

(

]

(

(

r

rf

f

f

f

+

+

=

= +

= +

= + +

D j E j

D j E j

E j D j

E j D j

(15)

Here, the + power indicates positive excitation frequenciesand - power indicates negative excitation frequencies. The

complex functions D(j) and E(j) are obtained by adding and

subtracting terms in Eq.(15).Analytical expressions are obtained forD and Eby curve-

fitting to standard polynomial forms. Figures 5 and 6 providerepresentative results for = 15200 rpm and zero-preswirl.

101

102

103

104100

120

140

160

GaindB

Freq. [rad/sec]

Data and Curve Fit; 5 poles, 4 zeros D(s)

Curve Fit

Data

101

102

103

104

-400

-300

-200

-100

Phase[Deg]

Freq. [rad/sec]

Fig. 5 Magnitude and Phase Plot of Calculated andCurve-fitted D transfer function (Zero preswirl, RotorSpeed 15200 RPM)

101

102

103

104

110

120

130

140

150

GaindB

Freq. [rad/sec]

Data and Curve Fit; 5 poles, 4 zeros E(s)

Curve Fit

Data

101

102

103

104

-400

-300

-200

-100

0

Phase[Deg]

Freq. [rad/sec]

Fig. 6 Magnitude and Phase Plot of Calculated andCurve-fitted E transfer function (zero pre-swirl, RotorSpeed 15200 RPM)

The following D(s) and E(s) transfer functions were obtained

by curve fitting frand f for 0-8000 rad/sec:

4 3 2

5 4 3 2

8 4 13 3 17 2 19 2

12 16 20 23

5 8 13 16

3

3

5 4 7 3 11 2

4.7 10 1.1 10 1.3 10 1.3 10 2.8 10( )

2.7 10 6.5 10 1.4 10 1.7 10 1.7 10

1.24 10 2.4 10 3.6 10 3.6 10 2.3 10

3.2 10 5.4 10 1.0 1( )

0

s s s sD s

s s s s s

s s s s

s s sE s

s

+ =

+ +

+ + + =

+

25

20

14 176.8 10 4.8 10s +

(16)

In Eq. (16), D(s) has unstable poles that are discounted in the

stability analysis.



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JEFFCOTT ROTOR MODEL-IMBALANCE RESPONSE

AND STABILITY ANALYSIS

Fig. 7 Simple Jeffcott Rotor

The Jeffcott rotor model of fig. 7, acted on by labyrinth seal

forces is used to consider the effect of frequency dependency

on synchronous response and stability analysis.

The rotor parameters are:

Mass (m) 100 kg

Natural frequency , rk

m 795.87 rad/sec = 7.6 krpm

This natural frequency is half the rotor speed of 15.2 krpm.

This choice was made to amplify any possible impact of the

labyrinth on rotor stability. The labyrinth seal parameters usedhere are shown in Table 1. Labyrinth seal forces are the only

source of viscous damping in the model.

Imbalance-Response calculations

The governing equation for the Jeffcott rotor model of figure 7

with conventional labyrinth seal forces is:

2

2

0

0

cos

sin

r

r

K k km x C c x

k K km y c C y y

me t

m te

+ + + +

=

&& &

&& &

x

(17)

where kr is the rotor stiffness and e is the rotor imbalance.

Comparisons were made for the speed-dependent

(frequency-independent) model and the frequency and speeddependent model for a range of inlet swirl ratios and rotor

speeds. Figure 8 shows the (same) calculated amplitude results

for both models. Although not shown, the phase plots also

coincide. This outcome applies because, for response toimbalance, the rotordynamic coefficients are calculated for

forward precession at for both model types.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

10 100 1000 10000 100000

Magnitu

deRatio(X/e)

Rotor Running Speed (RPM)

transfer function model

synchronously reducedcoefficient model

Labyrinth seal

Bearing

Fig. 8 Magnitude Plot for frequency-independenmodel (synchronously reduced coefficients) andfrequency dependent model (Transfer FunctionModel) foru0(0) = 0.5

STABILITY CALCULATIONS

Speed-Dependent, Frequency-Independent Model

Approach

The frequency-independent model is implemented

considering synchronously-reduced rotordynamic coefficients

evaluated using elliptical precessional orbit at the rotor runningspeed. The homogeneous version of Eq.(6) applies.

Frequency-Dependent Models

Frequency dependency can be accounted by using Eq.(6)

with frequency dependent rotordynamic coefficients, with the

corresponding precession frequencies equal to the rotorsdamped natural frequencies. This approach is regularly used to

account for calculated frequency dependency of tilting-padbearings. Thus, the rotordynamic coefficients for forward and

backward damped natural frequencies can be extracted from theD and Etransfer functions via:

Re[ ( )]

Re[ ( )]

Im[ ( )]

Im[ ( )]

nr

nr

r

n

nr

n

r

r

n

K

k

C

c

+

+

+

+

+

+

+

+

++

=

=

=

=

D j

E j

D j

E j

Re[ ( )]

Re[ ( )]

Im[ ( )]

Im[ ( )]

nr

nr

r

n

nr

n

r

r

n

K

k

C

c

=

=

=

=

D j

E j

D j

E j

(18)

Where, nr is the damped natural frequency of the rotor, and

superscripts + and - indicate rotors forward and backward

modes.

For hole-pattern stator seals, seal forces can significantlychange the damped natural frequencies of the rotor. In such

cases, the rotordynamic coefficients are re-evaluated at the

calculated damped natural frequencies, and the procedure is



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repeated until there is convergence between the assumed and

calculated natural frequency. Thus, the stability analysis

becomes iterative in nature. However, Labyrinth seal forces do

not significantly change the rotors damped natural frequencies,

and in the present example the rotors critical speed closelyapproximates the first forward and backward damped natural

frequencies. The stability analysis in this example is non-

iterative with the evaluation of rotordynamic coefficients at therotors critical speed.

Re[ ( )]

Re[ ( )]

Im[ ( )]

Im[ ( )]

cr

cr

cr

cr

cr

cr

K

k

C

c

=

=

=

=

D j

E j

D j

E j

(19)

Where, cris the rotor critical speed.The second frequency-dependent approach involves directly

implementing the D and E transfer-function results into the

rotor model using a state-space format.Figures 9-11 present calculated log-dec results for the

model versus the inlet preswirl ratio u0(0), which is varied from0 to 0.8. Zero corresponds to a highly-effective swirl brake,

and 0.8 corresponds to a high preswirl value as might be

expected for a balance-piston seal with no swirl brake. Resultswere obtained using the following three approaches:

1. Speed dependent, frequency-independent model.2. Frequency-dependent stiffness and damping

coefficients

3. D and Etransfer-function model.Results are presented for both forward and backwards-

precessing roots.

TEST CASE 1, = 15.2 krpm; Rs = 0.26 c0

Figure 9 shows the following outcomes for forward precession:

The synchronously reduced model predicts instability forforward precession at a pre-swirl ratio of ~0.57.

The frequency-dependent model predicts instability at apre-swirl ratio of ~0.74

Transfer Function Model predicts instability at a preswirlratio of ~0.76.

Although not presented, the improved models predict that the

backward-precessing mode is more stable than predicted by the

synchronously-reduced model.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8

LogarithmicDecrement

Preswirl Ratio

synchronously reduced coefficients

non-synchronously reduced coefficients


Fig. 9 Log-Dec versus u0(0) for Forward Critical Speed

TEST CASE 2, = 40 krpm; Rs = 0.7c0Figure 10 shows the following outcomes:

The synchronously-reduced model predicts instability forforward precession at all preswirl ratios up to 0.8.

The frequency-dependent and transfer-function modelspredicts instability at a preswirl ratio greater than ~ 0.66

Although not illustrated, the improved models predict reduced

stability for the backward-precessing mode as compared to

predictions from the synchronously-reduced model.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8

LogarithmicDecrement

Preswirl Ratio




Fig. 10 Log-Dec Vs Pre-swirl Ratio for ForwardCritical Speed

TEST CASE 3 = 70 krpm; Rs = 1.2c0Figures 11a and 11b illustrate the following outcomes:

For the forward precessing mode, the synchronously-reduced model predicts stability for 0 u0(0) 0.8



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The frequency-dependent model predicts instability at u0(0) 0.43

The transfer-function model predicts instability at u0(0) 0.51

For the backwards precessing mode, the improved modelspredict stability for 0 u0(0) 0.8; the synchronously

reduced model predict instability for 0 u0(0) 0.8.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6

L

ogarithmicDecrement

Pr

0.8

eswirl Ratio


non-synchronously reduced coeffic ients


Fig. 11a Log-dec versus u0(0) for forward whirlingmode

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8

LogarithmicDec

rement

Preswirl Ratio




Fig. 11b Log-Dec versus u0(0) for backward whirling

mode

Evaluation of onset speed of Instability:

The onset speeds of instability for forward and backward

precessing modes are evaluated for u0(0)=0.5. Figure 12aillustrates the predicted log dec for the forward precessional

mode, showing that stability calculations are about the same for

synchronously-reduced and frequency-dependent solution up to

~14 krpm, but diverge from this speed onwards. The frequency-

dependent model predicts an onset speed of instability of 65krpm.

Fig. 12a Log-dec Vs rotor speed for forward whirling

modeFor the backwards-precessing mode, figure 12b also shows

about the same calculated log dec for speeds out to ~14 krpmThe synchronously-reduced model predicts an onset speed of

instability for the backward mode at ~50 krpm. The frequency

dependent model predicts a continuing increase in the log dec

as the running speed increases. Hence, the rotor backward

precessional mode is predicted to remain stable for the

complete speed range.

Fig. 12b Log-dec Vs rotor speed for backwardwhirling mode

The historical experience with labyrinth seals is that they

produce low values of direct stiffness and have a minimalimpact on the rotor natural frequencies. Figure 13a shows the

calculated damped natural frequency for the rotors forward

precession mode. The synchronously-reduced model shows ifirst dropping and then increasing sharply as the increases

The frequency-dependent results show a modest drop in the



9/11

natural frequency with increasing . A comparison of figures

13a and 12a suggests that differences between the log-dec

predictions for the models arise mainly due to erroneous

predictions of the damped natural frequency for the

synchronously-reduced model.

Fig. 13a Damped natural frequencies Vs rotor speedfor forward whirling mode

Fig. 13b Damped natural frequencies Vs rotor speedfor backward whirling mode

Figure 13b presents calculated results for backward-precessing natural frequency. The frequency-dependent model

shows a slow drop for out to about 48 krpm, followed by a

gradual increase. The synchronously-reduced model shows aninitial sharp drop ending at about 27 krpm, followed by a sharprise as increases further.

SUMMARY, DISCUSSION, AND CONCLUSIONS

The 1CV model for a see-through labyrinth seal continuesto be used; however, the solution procedure is modified to

account for frequency-dependency as the rotors surface

velocity approaches the speed of sound. Calculated results are

presented for a Jeffcott model acted on by a labyrinth seal

using: (i) A synchronously-reduced (traditional) model, and (ii)frequency-dependent models. Frequency dependency can be

accounted for by considering frequency dependent stiffness and

damping coefficients or by using transfer-function models. Thetransfer-function models can be integrated into a conventiona

rotordynamic code by converting them into a state-space form

The traditional speed-dependent (but frequency independent)

stiffness and damping coefficients continue to be valid for

synchronous imbalance response.

Rotordynamic calculations were performed for a Jeffcottmodel acted on by a labyrinth seal. The rotor diameter at the

seal produces a predicted surface velocity equal to the speed osound at =58 krpm. Log-dec predictions from the two mode

types coincide for speeds out to about 15k rpm. The frequency

dependent models predict an onset speed of instability (OSI)

around 65 krpm. The frequency-independent model predicts an

initial OSI at ~15k rpm followed by a return to predicted

stability around 45 krpm. The stability results show that thetraditional model is not valid for rotor surface velocities

approaching a significant fraction of the speed of sound.The new frequency dependent model needs to be validated

by experiments. Test data by Picardo [5], provides dynamic

stiffness coefficients for a frequency range of excitation of 10

to 150 Hz. However, to verify the predictions in this analysis

test data are needed from rotors running at high rotor surface

velocities (approaching the speed of sound), or the excitation

frequency range must be expanded significantly beyond therotor running speed.

ACKNOWLEDGMENTSThe work reported here was supported by the

Turbomachinery Research Consortium of the Texas A&MUniversity Turbomachinery Laboratory.

REFERENCES[1] M. Kurohashi, et al., 1980, Spring and Damping

Coefficients of the Labyrinth Seals, Paper No. C283/80

Proceedings of the 2nd International Conference on Vibrations

in Rotating Machinery (Institution of Mechanical Engineers)held at Churchill College, Cambridge University, pp. 215-222.

[2] Iwatsubo, T., 1980, Evaluation of Instability Forces of

Labyrinth Seals in Turbines or Compressors, NASA CP 2133Proceedings of a workshop at Texas A&M University 12-14

May 1980, entitled Rotordynamic Instability Problems in High

Performance Turbomachinery, pp.205-222.[3] Childs, D., Scharrer, J., 1986, An Iwatsubo-Based Solution

for Labyrinth Seals: Comparison to Experimental ResultsJournal of Engineering for Gas Turbines and Power, Vol.108

pp.325-331.

[4] Wyssman, H., Pham, T., and Jenny, R. (1984), "Prediction

of Stiffness and Coefficients for Centrifugal CompressorLabyrinth Seals," ASME J. ofEngineering for Gas Turbines andPower, 106, 920-926.



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[5] Scharrer, J., 1987, A Comparison of Experimental and

Theoretical Results for Labyrinth Gas Seals, Ph.D.

dissertation, Texas A&M University.

[6] Picardo, A., 2003, High Pressure Testing of See-Through

Labyrinth Seals, M.S. thesis, Texas A&M University.[7] Thieleke, G., Stetter, H., 1990, Experimental Investigations

of Exciting Forces caused by Flow in Labyrinth Seals, TAMU

Instability Workshop.[8]Martinez-Sanchez, M., Lee, O.W.K., Czajkowski, E., 1984,

The prediction of Force Coefficients for Labyrinth Seals,

Nasa CP 2338,Rotordynamic Instability Problems in High

Performance Turbomachinery, proceedings of a work held at

Texas A&M University, pp. 235-256.[9]Childs, D., Chang-Ho Kim,1984, Analysis and testing of

Rotordynamic Coefficients of Turbulent Annular Seals with

Different, Directionally Homogeneous Surface-RoughnessTreatment for Rotor and Stator Elements, Nasa CP

2338,Rotordynamic Instability Problems in High Performance

Turbomachinery, proceedings of a work held at Texas A&M

University, pp. 313-340.

[10]Gurevich, M.I., 1966, The Theory of Jets in an IdealFluid, Pergamon Press, pp. 115-122.[11]Neumann, K., 1964, Zur Frage der Verwendung von

Durchblickdichtungen im Dampfturbinenbau,

Maschinentechnik, Vol.13, No.4, pp.188-195.[12]Kleynhans, G., and Childs, D., 1997, The Acoustic

Influence of Cell Depth on the Rotordynamic Characteristics of

Smooth-Rotor/Honeycomb-Stator Annular Gas Seals, ASME

Trans., J. of Engineering for Gas Turbines and Power, Vol.

119, No. 4, pp. 949-957.

[13] Bolleter, U., Leibundgut, E., Strchler, R., and McCloskey,T., 1989, Hydraulic Interaction and Excitation Forces of High

Head Pump Impellers, in: PumpingMachinery1989,

Proceedings of the Third Joint ASCE/ASME MechanicalConference, La Jolla, CA, pp. 187-194.

APPENDIX A

Circular Orbit Solution:

The seal clearance function can be defined as:

1 ( ) cos ( ) sinH yx t t =

The pressure and velocity perturbations can be expressed

as:

1 1 1

1 1 1

cos sin

cos sin

i ic is

i ic is

P

U

P P

UU

+

+

=

=

Substituting the above variable definitions in the first order

continuity and momentum equations eliminates the theta

dependency. The time dependency is eliminated by considering

circular whirl orbit:

$

1 0

1 1

1 1

( ) ( ) j t

j ti i

j ti i

H x t jy t r e

P p e

U u e

= + =

=

=

uur

ur

ur

Where is the radius of the circular whirl orbit. and

are complex amplitudes:

0r

1 1

1i

p

$1iu

$

1

1

i

i 1 1

ic is

ic is

p P jP

u U jU

= +

= +

The separation of variables approach yields algebraic

equations of the form:

[ ]{ } [ ]{ } { } {1 1 2 1 3 1 21 1i i i }A x A x A x x ++ + =

Where,

{ } { }{ }

[ ]

1 1 1 11

2 0 2 6 52 0 0 0

4

41

4 0

4 0

0

0 0 0

0 0 0

0 0

0 0

T

ic is ic isi

T

i i i i i

i

i

i

i

s

i

x P P U U

G U G G X x r r r

R

G

GA

X m

X m

=

0r

= +

=

&

&

[ ]

1 0 1 01 3

1 0 1 03 1

2

0 1 03 1 2

0 13 2

0

0

i i i ii i

s s

i i i ii i

s s

i i ii i is s

i ii i

s s

U G PG G

R

U GG G

RA

A UX X XR R

A UX X

G

R

G

R

X

XR R

X 01

i

P

=

+

5

53

0 0

0 0 0

0 0 0 0

0 0 0 0

i

i

G

GA

0 =

The above set of equations is for a single cavity. The cavity

equations are assembled to form a 4NC 4NC matrix equation

(NC= number of cavities). Pressure & velocity perturbations a

the entry and exit are zero. These form the boundary condition

for the problem.



11/11

The solution is of the form:

01 1

01 1

ic ic

is is

rP P

rP P

=

=

The radial and circumferential forces, and)(rF

( )F ,

on the seal are evaluated from the pressure perturbations in the

labyrinth seal cavities.

0 1

1

( )

NC

r s

i

F rR

=

= icp

Similarly,

0 1

1

( )

NC

s is

i

F R r

=

p=

The radial and circumferential dynamic stiffness

coefficients are obtained from these forces:

0

0

((

( )(

))

)

rrf

f

F

rF

r

=

=
