1
XII Congreso y Exposición Latinoamericana de Turbomaquinaria, Queretaro, Mexico, February 24 2011
Dr. Luis San AndresMast-Childs Tribology Professor
ASME Fellow, STLE Fellow
Identification of Force Coefficients in Mechanical Components:
Bearings and SealsA guide to a frequency domain technique
Turbomachinery Laboratory, Mechanical Engineering DepartmentTexas A&M University
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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.
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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)
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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
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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)
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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)
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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
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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
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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.
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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
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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
SoftSupport structure
Equations of motion (EOMs)
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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
SoftSupport structure
X
YKYY, CYY
KXY, CXY
force, FY
Bearing
Journal
KYX, CYX
KXX, CXX
Ω
force, FX
KSX, CSX
KSY, CSY
SoftSupport structure
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
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Excitations with shakers
X Y
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Identification model (2)
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
SoftSupport structure
X
YKYY, CYY
KXY, CXY
force, FY
Bearing
Journal
KYX, CYX
KXX, CXX
Ω
force, FX
KSX, CSX
KSY, CSY
SoftSupport structure
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
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Identification model (3)
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
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Identification model (4)
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
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Identification model (5)
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
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Identification model (6)
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
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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
SoftSupport structure
X
YKYY, CYY
KXY, CXY
force, FY
Bearing
Journal
KYX, CYX
KXX, CXX
Ω
force, FX
KSX, CSX
KSY, CSY
SoftSupport structure
Test elements that are ~isotropic or that are excited by periodic (single frequency) loads producing circular orbits usually determine an ill conditioned system
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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
SoftSupport structure
X
YKYY, CYY
KXY, CXY
force, FY
Bearing
Journal
KYX, CYX
KXX, CXX
Ω
force, FX
KSX, CSX
KSY, CSY
SoftSupport structure
2 RealXX hX h XXK K M H
ImaXX hX XXC C H
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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
SoftSupport structure
X
YKYY, CYY
KXY, CXY
force, FY
Bearing
Journal
KYX, CYX
KXX, CXX
Ω
force, FX
KSX, CSX
KSY, CSY
SoftSupport structure
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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
SoftSupport structure
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
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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
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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
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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
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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
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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
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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.
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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
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An example of parameter identification
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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
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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
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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
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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)
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Lubricant flow pathOil inlet
in
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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
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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)
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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)
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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
From IVFFrom test data
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
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
2 104
From IVFFrom test data
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
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
2 104
From IVFFrom test data
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hxx)
Frequency [Hz]
Re(
Hxx
) [
lbf
/ in]
rxxred0.999
rxxImd0.932
0 50 1001 10
4
0
1 104
2 104
3 104
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
2 104
From IVFFrom test data
Im (Hyy)
Frequency [Hz]
Im(H
yy)
[lb
f / i
n]
ryyred0.996
ryyImd0.966
r/c= 0.66, centered es=0
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
From IVFFrom test data
Real (Hxx)
Frequency [Hz]
Re(
Hxx
) [
lbf
/ in]
rxxred0.999
rxxImd0.932
0 50 1001 10
4
0
1 104
2 104
3 104
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
From IVFFrom test data
Re (Hxy)
Frequency [Hz]
Re(
Hxy
) [
lbf
/ in]
0 50 1000
1 103
2 103
3 103
From IVFFrom test data
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
From IVFFrom test data
Re (Hyx)
Frequency [Hz]
Re(
yx)
[lb
f / i
n]
0 50 1000
1 103
2 103
3 103
From IVFFrom test data
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
From IVFFrom test data
Re (Hxy)
Frequency [Hz]
Re(
Hxy
) [
lbf
/ in]
0 50 1000
1 103
2 103
3 103
From IVFFrom test data
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
From IVFFrom test data
Re (Hyx)
Frequency [Hz]
Re(
yx)
[lb
f / i
n]
0 50 1000
1 103
2 103
3 103
From IVFFrom test data
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hxx)
Frequency [Hz]
Re(
Hxx
) [
lbf
/ in]
rxxred0.999
rxxImd0.932
0 50 1001 10
4
0
1 104
2 104
3 104
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hxx)
Frequency [Hz]
Re(
Hxx
) [
lbf
/ in]
rxxred0.999
rxxImd0.932
0 50 1001 10
4
0
1 104
2 104
3 104
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
From IVFFrom test data
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
From IVFFrom test data
Real (Hxx)
Frequency [Hz]
Re(
Hxx
) [
lbf
/ in]
rxxred0.999
rxxImd0.932
0 50 1001 10
4
0
1 104
2 104
3 104
From IVFFrom test data
Real (Hyy)
Frequency [Hz]
Re(
Hyy
) [
lbf
/ in]
0 50 1000
5 103
1 104
1.5 104
From IVFFrom test data
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
r/c= 0.66, centered es=0
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