1 xii congreso y exposición latinoamericana de turbomaquinaria, queretaro, mexico, february 24 2011...

1

XII Congreso y Exposición Latinoamericana de Turbomaquinaria, Queretaro, Mexico, February 24 2011

Dr. Luis San AndresMast-Childs Tribology Professor

ASME Fellow, STLE Fellow

[email protected]

Identification of Force Coefficients in Mechanical Components:

Bearings and SealsA guide to a frequency domain technique

Turbomachinery Laboratory, Mechanical Engineering DepartmentTexas A&M University

2

Turbomachinery

A turbomachinery is a rotating structure where the load or the driver handles a process fluid from which power is extracted or delivered to.

Fluid film bearings (typically oil lubricated) support rotating machinery, providing stiffness and damping for vibration control and stability. In a pump, neck ring seals and inter stage seals and balance pistons also react with dynamic forces. Pump impellers also act to impose static and dynamic hydraulic forces.

3

Turbomachinery

Acceptable rotordynamic operation of turbomachinery

Ability to tolerate normal (even abnormal transient) vibrations levels without affecting TM overall performance (reliability and efficiency)

4

Model structure (shaft and disks) and find free-free mode natural frequencies

Model bearings and seals: predict or IDENTIFY mechanical impedances (stiffness, damping and inertia force coefficients)

Eigenvalue analysis: find damped natural frequencies and damping ratios for various (rigid & elastic) modes of vibration as rotor speed increases (typically 2 x operating speed)

Synchronous response analysis: predict amplitude of 1X motion, verify safe passage through critical speeds and estimate bearing loads

Certify reliable performance as per engineering criteria (API 610 qualification) and give recommendations to improve system performance

Rotordynamics primer (2)

5

The need for parameter identification

• to predict, at the design stage, the dynamic response of a rotor-bearing-seal system (RBS);

• to reproduce rotordynamic performance when troubleshooting RBS malfunctions or searching for instability sources, &

• to validate (and calibrate) predictive tools for bearing and seal analyses.

The ultimate goal is to collect a reliable data base giving confidence on bearings and/or seals operation under both normal design conditions and extreme environments due to unforeseen events

Experimental identification of force coefficients is important

6

The physical modelFor lateral rotor motions (x, y), a bearing or seal reaction force vector f is modeled as

( )

( )

( ) ( ) ( )

( ) ( ) ( )

t

t

X t t tXX XY XX XY XX XY

t t tYX YY YX YY YX YYY

f x x xK K C C M M

y y yK K C C M Mf

( ) ( )witht t

x

y

X

Y

ff = = - K z +C z+Mz z

f

K,C,M are matrices of stiffness, damping, and inertia force coefficients (4+4+4 = 16 parameters) representing a linear physical system.

The (K, C, M) coefficients are determined from measurements in a test system or element undergoing small amplitude motions about an equilibrium condition.

X

Y

Z

Lateral displacements (X,Y)

7

Bearings: dynamic reaction forces

Stiffness coefficients

Damping coefficients

Typical of oil-lubricated bearings: No fluid inertia coefficients accounted for.

Force coefficients are independent of excitation frequency for incompressible fluids (oil). Functions of speed & applied load

X XX XY XX XY

Y YX YY YX YYB B

f K K C Cx x

f K K C Cy y

X

Y

Z

Lateral displacements (X,Y)

8

Liquid seals:

Stiffness coefficients

Inertia coefficients

Damping coefficients

Typically: frequency dependent force coefficients

X XX XY XX XY XX XY

Y YX YY YX YY YX YYS S S

f K K C C M Mx x x

f K K C C M My y y

Gas seals

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

XX XY XX XYX

YX YY YX YYY S S

K K C Cf x x

K K C Cf y y

Seals: dynamic reaction forces

9

Strictly valid for small amplitude motions.

Derived from SEPThe “physical” idealization of force coefficients in lubricated bearings and seals

;j

iij X

FK

j

iij X

FC

Stiffness:

Damping:

Inertia:;

j

iij X

FM

i,j = X,Y

Kxx, Cxx

journal

bearing

X

Y

Kxy, Cxy

Kyx, Cyx

Kyy Cyy

The concept of force coefficients

10

Modern parameter identification

Modern techniques rely on frequency domain procedures, where force coefficients are estimated from transfer functions of measured displacements (or velocities or accelerations) due to external loads of a prescribed time varying structure.

Frequency domain methods take advantage of high speed computing and digital signal processors, thus producing estimates of system parameters in real time and at a fraction of the cost (and effort) than with antiquated and cumbersome time domain algorithms.

11

A test system example

Kh,Ch: support stiffness

and damping

Mh : effective mass

X

YKYY, CYY

KXY, CXY

force, fY

Bearing or seal

Journal

KYX, CYX

KXX, CXX

Ω

force, fX

KhX, ChX

KhY, ChY

SoftSupport structure

Consider a test bearing or seal element as a point mass undergoing forced vibrations induced by external forcing functions

(K,C,M): test element stiffness, damping & inertia

force coefficients

12

Kh,Ch: structure

stiffness and damping

Mh : effective mass

(K,C,M): test element force coefficients

For small amplitudes about an equilibrium position, the EOMs of a linear mechanical system are

h h hM +M z + C +C z + K +K z = f

, X

Y

fx

fy

z fwhere

Note: The system structural stiffness and damping coefficients, {Kh,Ch}i=X,Y, are obtained from prior shake tests results under dry conditions, i.e. without lubricant in the test element

X

YKYY, CYY

KXY, CXY

force, FY

Bearing or seal

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KhX, ChX

KhY, ChY


Equations of motion (EOMs)

13

Identification model (1)

Apply two independent force excitations on the test element

X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


How to apply the forces?Use impact hammers, mass imbalances,

shakers(impulse, periodic-single frequency, sine-swept,

random, etc)

Step (1) Apply 1

1 ( )

x

y t

f

f

and measure ( )

( )

1

1

t

t

x

y

and measureStep (2) Apply2

2 ( )

x

y t

f

f

( )

( )

2

2

t

t

x

y

14

Excitations with shakers

X Y

15


Obtain the discrete Fourier transform (DFT) of the applied forces and displacements, i.e.,

X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


and use the property

1( ) 1( ) 1( )

1( ) 1( ) 1( )

1( )

1( )

; ;t

t

X x t

tY Y

F f X xDFT DFT

yF f Y

2( )2( ) 2( )

22( ) 2( ) ( )

2( )

2( )

;t

t

X x t

tY Y

XF f xDFT DFT

yYF f

2( ) ( ) ( ) ( );t ti X DFT x X DFT x

where, 1i

16


The DFT operator transforms the EOMS from the time domain into the frequency domain

For the assumed physical model, the EOMS become algebraic

2 i h h hK +K M +M C +C Z = F

, X

Y

FX

FY

Z F

17


Define the complex impedance matrix

2 i h h hH K +K M +M C +C

0 200 400 600 800 10005 10

6

0

5 106

1 107

Re(H)Im(H)

Ideal impedance

frequency (rad/s)

Rea

nd &

Im

agin

ary

Impe

danc

e

xx

The impedances are functions of the excitation frequency ().

REAL PART = dynamic stiffness,

IMAGINARY PART = (quadrature stiffness), proportional to viscous damping

K - 2 M

C

XX XY

YX YY

H H

H H

H

18


With the complex impedance

The EOMS become, for the first & second tests

( ) ( ) 1

( ) ( ) 1

1

1

XX XY X

YX YY Y

H H FX

YH H F

( ) ( ) 2

( ) ( ) 2

2

2

XX XY X

YX YY Y

H H FX

YH H F

Add these two eqns. and reorganize them as

1 2

1 2

1 2

1 2

X XXX YX

XY YY Y Y

F FH H X X

H H Y Y F F

At each frequency (ωk=1,2,…n), the eqn. above denotes four independent equations with four unknowns, (HXX, HYY , HXY , HYX)

2 i h h hH K +K M +M C +C

19


Find H

Then

where

The need for linear independence of the test forces (and ensuing motions) is obvious

1 2

1 2

1

1 2

1 2

X XXX YX

XY YY Y Y

F FH H X X

H H Y YF F

1(1) (2) (1) (2) H= F F Z Z

1 2

1 2

1 2(1) (1) (2) (2)

1 2

& , &X X

Y Y

F FX X

Y YF F

F Z F Z

since

20

Condition numberIn the identification process, linear independence is MOST important to obtain reliable and repeatable results.

In practice, measured displacements may not appear similar to each other albeit producing an identification matrix that is ill conditioned, i.e., the determinant of

In this case, the condition number of the identification matrix tell us whether the identified coefficients are any good.

(1) (2) ~ 0 Z Z

X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


Test elements that are ~isotropic or that are excited by periodic (single frequency) loads producing circular orbits usually determine an ill conditioned system

21

The estimated parameters

Estimates of the system parameters

{M, K, C},j=X,Y

are determined by curve fitting of the test derived discrete set of impedances

(HXX, HYY , HXY , HYX ) k=1,2….,

one set for each frequency ωk,

to the analytical formulas over a pre-selected frequency range.

For example:

X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


2 RealXX hX h XXK K M H

ImaXX hX XXC C H

22

Meaning of the curve fit

Analytical curve fitting of any data gives a correlation

coefficient (r2) representing the goodness of the fit.

A low r2 << 1, does not mean the test data or the obtained impedance are incorrect, but rather that the physical model (analytical function) chosen to represent the test system does not actually reproduce the measurements.

On the other hand, a high r2 ~ 1 demonstrates that the physical model with stiffness, damping and

inertia giving K-ω2M and ωC, DOES model well the system response with accuracy.

X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


23

Transfer functions=flexibilities

Transfer functions (displacement/force) are the

system flexibilities G derived fromX

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY


X

YKYY, CYY

KXY, CXY

force, FY

Bearing

Journal

KYX, CYX

KXX, CXX

Ω

force, FX

KSX, CSX

KSY, CSY

SoftSupport structureG=H-1

1 2

1 2

( ) ; ( )

( ) ; ( )

YY XYXX XY

YX XXYX YY

H HG TF X G TF X

H HG TF Y G TF Y

where H H H HXX YY XY YX

24

The instrumental variable filter method

Fritzen (1985) introduced the IVFM as an extension of a least-squares estimation method to simultaneously curve fit all four transfer functions from measured displacements due to two sets of (linearly independent) applied loads.

The IVFM has the advantage of eliminating bias typically seen in an estimator due to measurement noise

GH = I

In the experiments there are many more data sets (one at each

frequency) than parameters (4 K, 4 C, 4 M=16).

Recall that 1 0

0 1

I

25

The IVFM (1)

However, in any measurement process there is always some

noise. Introduce the error matrix (e) and set

GH = I

Since

The product1 0

0 1

I

G=H-1

2 i G H G K M C I + e

Above G is the measured flexibility matrix while H represents the (to be) estimated test system impedance matrix

26

The IVFM (2)It is more accurate to minimize the approximation errors (e) rather than directly curve fitting the impedances.

2( ) i

M

H I I I C

K

1 0

0 1

I

2k kk ki

M

G I I I C I + e

K

Hence G H I + e

2k kk ki A G I I I

Letk k

M

A C I +e

K

Let

27

The IVFM (3)

Stack all the equations, one for each frequency k= 1,2…,n , to

obtain the set

where

M

A C Ι e

K

1 1

2 2,n n

A e

A A e e

A e

0 1 0 1 0 1 .. .. .. .. 0 1

1 0 1 0 1 0 .. .. .. .. 1 0T

I

A contains the stack of measured flexibility functions at discrete

frequencies k=1,2…,n. Eqs. make an over determined set, i.e. there

are more equations than unknowns.

Hence, use least-squares to minimize the Euclidean norm of e

28

The IVFM (4)

The minimization leads to the normal equations

A first set of force coefficients (M,C,K) is determined

In the IVFM, the weight function A is replaced by a new matrix

function W created from the analytical flexibilities resulting from the (initial) least-squares curve fit.

W is free of measurement noise and contains peaks only at the

resonant frequencies as determined from the first estimates of K, C, M coefficients

1

T T

M

C = A A A I

K

29

The IVFM (5)

At step m,

where

1

1

m

T Tm m

M

C W A W I

K

21 1( )1

2( )

m

m

mn nn

i

F I i I I

W

F I I I

1

2( )

m

mi

M

F I I I C

K

when m=1 use W1=A = least-squares solution. Continue iteratively until a given convergence criterion or tolerance is satisfied

30

The IVFM (6)

At step m, 1

1

m

T Tm m

M

C W A W I

K

Substituting W for the discrete measured flexibility A (which also contains noise) improves the prediction of parameters.

Note that the product ATA amplifies the noisy components and adds them. Therefore, even if the noise has a zero mean value, the addition of its squares becomes positive resulting in a bias error.

On the other hand, W does not have components correlated to the

measurement noise. That is, no bias error is kept in WTA. Hence, the approximation to the system parameters improves.

31

The IVFM (7)

In the IVFM, the flexibility coefficients (G) work as weight functions of the errors in the minimization procedure.

Whenever the flexibility coefficients are large, the error is also large. Hence, the minimization procedure is best in the neighborhood of the system resonances (natural frequencies) where the dynamic flexibilities

are maxima (i.e., null dynamic stiffness, K-2M=0)

External forcing functions exciting the test system resonances are more reliable because at those frequencies the system is more sensitive, and the measurements are accomplished with larger signal to noise ratios

32

An example of parameter identification

33

Sponsor: Pratt & Whitney Engines

Luis San Andrés Sanjeev SeshagiriPaola Mahecha

Research Assistants

SFD EXPERIMENTAL TESTING & ANALYTICAL METHODS DEVELOPMENT

Identification of force coefficients in a SFD

Texas A&M UniversityMechanical Engineering Dept. – Turbomachinery Laboratory

34

P&W SFD test rig

Static loader

Shaker assembly (Y direction)

Shaker assembly (X direction)

Static loader

Shaker in X direction

Shaker in Y direction

SFD test bearing

35

Test rig description

shaker Xshaker Y

Static loader

SFD

basesupport rods

Static loader

X

Y

shaker Xshaker Y

Static loader

SFD

basesupport rods

Static loader

X

Y

36

P & W SFD Test Rig – Cut Section

in

Test rig main features

Journal diameter: 5.0 inch

Film clearance: 5.1 mil

Film length: 2 x 0.5 inch

Support stiffness: 22 klbf/in

Bearing Cartridge

Test Journal

Main support rod (4)

Journal BasePedestal

Piston ring seal

(location)

Flexural Rod (4, 8, 12)

Circumferential groove

Supply orifices (3)

37

Lubricant flow pathOil inlet

in

38

Objective & task

Evaluate dynamic load performance of SFD with a central groove.

Dynamic load measurements: circular orbits (centered and off centered) and identification of test system and SFD force coefficients

39

Circular orbit tests• Frequency range: 5-85 Hz

• Centered and off-centered, eS/c = 0.20, 0.40, 0.60• Orbit amplitude r/c = 0.05 – 0.50

ISO VG 2 OilViscosity at 73.4 oF [cPoise] 2.95

Density [kg/m3] 784

Inlet pressure [psig] 7.5

Outlet pressure [psig] 0

Radial Clearance [mil] c

Journal Diameter [inch] 5.0

Central groove length [inch] L

Land length, L [inch] L

Total Length [inch] 3L

Oil out, Qb

BaseSupportrod

Bearing Cartridge

Journal (D) Oil out, Qt

Oil in, Qin

Central groove

L

L

L

End groove

End groove

Oil outOil collector

c

40

Typical circular orbit tests

• Frequency range: 5-85 Hz

• Centered eS=0• Orbit amplitude r/c=0.66

Dmax 5.1 Lmax 160

5.1 2.55 0 2.55 5.1

5.1

2.55

2.55

5.1

5 Hz15 Hz25 Hz35 Hz45 Hz55 Hz65 Hz75 Hz85 Hz95 Hztrace 11

X Displacement [mil]

Y D

ispl

acem

ent [

mil]

160 80 0 80 160

160

80

80

160

X Load [lbf]

Y L

oad

[lbf

]

Forces (fy vs. fx)motion (y vs. x)

41

Typical circular orbit tests

• Frequency: 85 Hz

• Off-centered at eS/c= 0.31• Orbit amplitude r=0.05 – 0.5

Dmax 5.1 Lmax 80 f 9

80 40 0 40 80

80

40

40

80

X Load [lbf]

Y L

oad

[lbf

]

5.1 2.55 0 2.55 5.1

5.1

2.55

2.55

5.1

0.26 mil0.32 mil0.60mil1.04 mil0.64 mil1.31 mil2.62 miltrace 8trace 9

X Displacement [mil]

Y D

ispl

acem

ent [

mil]

Forces (fy vs. fx)motion (y vs. x)

42

Typ system direct impedances

HXX

0 50 1000

5 103

1 104

1.5 104

2 104

From IVFFrom test data

Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.948

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104

2 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.996

ryyImd0.966

0 50 1000

5 103

1 104

1.5 104

2 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.948

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104

2 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.996

ryyImd0.966

HYY

r/c= 0.66, centered es=0

0 50 1000

5 103

1 104

1.5 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1002 10

4

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.932

0 50 1001 10

4

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.99

ryyImd0.978

Imaginary partReal part

43

Typ. system direct impedances

HXX

0 50 1000

5 103

1 104

1.5 104

2 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.948

0 50 1003 10

4

2 104

1 104

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104

2 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.996

ryyImd0.966


0 50 1000

5 103

1 104

1.5 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1002 10

4

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.932

0 50 1001 10

4

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.99

ryyImd0.978

Excellent correlation between test data and physical model

REAL PART = dynamic stiffness

IMAGINARY PART proportional to viscous damping

K - 2 M C

44

Test cross-coupled impedances

HXY

HYX

Cross Coupled terms

0 50 1003 10

3

2 103

1 103

0

1 103


Re (Hxy)

Frequency [Hz]

Re(

Hxy

) [

lbf

/ in]

0 50 1000

1 103

2 103

3 103


Im (Hxy)

Frequency [Hz]

Im(H

xy)

[lb

f / i

n]

rxyred0.82

rxyImd0.73

0 50 1002 10

3

1.5 103

1 103

500

0

500


Re (Hyx)

Frequency [Hz]

Re(

yx)

[lb

f / i

n]

0 50 1000

1 103

2 103

3 103


Im (Hyx)

Frequency [Hz]

Im(H

yx)

[lb

f / i

n]ryxred

0.866

ryxImd0.629

Cross Coupled terms

0 50 1003 10

3

2 103

1 103

0

1 103


Re (Hxy)

Frequency [Hz]

Re(

Hxy

) [

lbf

/ in]

0 50 1000

1 103

2 103

3 103


Im (Hxy)

Frequency [Hz]

Im(H

xy)

[lb

f / i

n]

rxyred0.82

rxyImd0.73

0 50 1002 10

3

1.5 103

1 103

500

0

500


Re (Hyx)

Frequency [Hz]

Re(

yx)

[lb

f / i

n]

0 50 1000

1 103

2 103

3 103


Im (Hyx)

Frequency [Hz]

Im(H

yx)

[lb

f / i

n]

ryxred0.866

ryxImd0.629

0 50 1000

5 103

1 104

1.5 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1002 10

4

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.932

0 50 1001 10

4

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.99

ryyImd0.978

0 50 1000

5 103

1 104

1.5 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1002 10

4

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.932

0 50 1001 10

4

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.99

ryyImd0.978

0 50 1000

5 103

1 104

1.5 104


Im (Hxx)

Frequency [Hz]

Im(H

xx)

[lb

f / i

n]

0 50 1002 10

4

1 104

0

1 104

2 104

3 104


Real (Hxx)

Frequency [Hz]

Re(

Hxx

) [

lbf

/ in]

rxxred0.999

rxxImd0.932

0 50 1001 10

4

0

1 104

2 104

3 104


Real (Hyy)

Frequency [Hz]

Re(

Hyy

) [

lbf

/ in]

0 50 1000

5 103

1 104

1.5 104


Im (Hyy)

Frequency [Hz]

Im(H

yy)

[lb

f / i

n]

ryyred0.99

ryyImd0.978

One order of magnitude lesser than

direct impedances = Negligible

cross- coupling

effects


Imaginary partReal part

45

SFD force coefficients

SFDKs = 21 klbf/in

Ms = 40 lbCs= 7 lbf-s/in

Nat freq = 73-75 HzDamping ratio

= 0.04

DRY system parameters

CSFD=Clubricated - Cs

MSFD=Mlubricated - Ms

KSFD=Klubricated - Ksh

Difference between lubricated system and dry system (baseline) coefficients

46

SFD damping coefficients

CXX

0

5

10

15

20

25

30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Orbit radius (r) + static eccentricity (es)(mil)

Da

mp

ing

co

eff

icie

nts

(lb

f-s

/in)

C XX SFD

e s = 0 mil e s = 1.56 mil

e s = 2.4 mil

Damping increases mildly as

static eccentricity

increases

CYY ~ CXX for circular orbits, independent of

static eccentricity

47

SFD mass coefficients

MXX

0

5

10

15

20

25

30

0.0 1.0 2.0 3.0 4.0 5.0

Orbit radius (r) + static eccentricity (es)(mil)

Mas

s C

oe

ffic

ien

ts (

lbm

)

M XX SFD

e s = 0 mil e s = 1.56 mil e s = 2.4 mil

MXX ~ MYY decreases with orbit radius (r) for

centered motions. Typical nonlinearity

48

Conclusions

• SFD test rig: completed measurements of dynamic loads inducing small and large amplitude orbits, centered and off-centered.

• Identified SFD damping and inertia coefficients behave well. IVFM delivers reliable and accurate parameters.

• Comparison to predictions are a must to certify the confidence of numerical models.

49

Acknowledgments

• Thanks to Pratt & Whitney Engines• Turbomachinery Research Consortium

Learn more

http:/rotorlab.tamu.edu

Questions (?)

50

Fritzen, C. P., 1985, “Identification of Mass, Damping, and Stiffness Matrices of Mechanical Systems,” ASME Paper 85-DET-91.

Massmann, H., and R. Nordmann, 1985, “Some New Results Concerning the Dynamic Behavior of Annular Turbulent Seals,” Rotordynamic Instability Problems of High Performance Turbomachinery, Proceedings of a workshop held at Texas A&M University, Dec, pp. 179-194.

Diaz, S., and L. San Andrés, 1999, "A Method for Identification of Bearing Force Coefficients and its Application to a Squeeze Film Damper with a Bubbly Lubricant,” STLE Tribology Transactions, Vol. 42, 4, pp. 739-746.

L. San Andrés, 2010, “identification of Squeeze Film Damper Force Coefficients for Jet Engines,” TAMU Internal Report to Sponsor (proprietary)

References

1 xii congreso y exposición latinoamericana de turbomaquinaria, queretaro, mexico, february 24 2011...

Documents

damping coefficients

concept of force coefficients

c xy force

seals stiffness

fluid inertia coefficients

coefficients typical

test element stiffness

structure stiffness