Transcript Document

Turbomachinery Laboratory, Mechanical Engineering Department
Texas A&M University
Identification of Force Coefficients in
Mechanical Components:
Bearings and Seals
A guide to a frequency domain technique
Dr. Luis San Andres
Mast-Childs Tribology Professor
ASME Fellow, STLE Fellow
[email protected]
XII Congreso y Exposición Latinoamericana de Turbomaquinaria,
Queretaro, Mexico, February 24 2011
1
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.
2
Turbomachinery
Acceptable rotordynamic
operation of turbomachinery
Ability to tolerate normal (even
abnormal transient) vibrations
levels without affecting TM
overall performance (reliability
and efficiency)
3
Rotordynamics primer (2)
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
4
The need for parameter identification
Experimental identification of force coefficients is important
•
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
5
The physical model
For lateral rotor motions (x, y), a bearing or seal
reaction force vector f is modeled as
f 
 x
f(t ) =  X  = - K z + C z + M z  with z (t )   
 y
f Y 
 f X(t ) 


 fY( t ) 
 K XX
K
 YX
K XY   x(t )  C XX




KYY   y(t )   CYX
Y
Z
X
Lateral displacements (X,Y)
C XY   x(t )   M XX




CYY   y(t )   M YX
M XY   x(t ) 
M YY   y(t ) 
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.
6
Bearings: dynamic reaction forces
Y
Z
 fX 
 K XX
 f K
 Y
 YX
K XY   x  C XX
  

KYY  B  y   CYX
C XY   x 
 
CYY  B  y 
X
Lateral displacements (X,Y)
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
7
Seals: dynamic reaction forces
Liquid seals:
 fX 
 K XX
 f K
 Y
 YX
K XY   x  C XX
  

KYY  S  y   CYX
Stiffness
coefficients
C XY   x   M XX
  

CYY  S  y   M YX
M XY   x 
 

M YY  S  y 
Inertia
coefficients
Damping
coefficients
Gas seals
 K XX ( )
 fX 
 f K
 Y
 YX ( )
K XY ( )   x  C XX ( )
  

KYY ( )   y   CYX ( )
S
C XY ( )   x 
 
CYY ( )   y 
S
Typically: frequency dependent force
coefficients
8
The concept of force coefficients
Kxx, Cxx
Stiffness:
Kxy, Cxy
Kyy
Cyy
Damping:
journal
Fi
K ij  
X j
;
Fi
Cij  
X j
Y
Inertia:
Kyx,
Cyx
bearing
M ij  
 Fi
;


Xj
i,j = X,Y
X
The “physical” idealization of force
coefficients in lubricated bearings and
seals
Strictly valid for small
amplitude motions.
Derived from SEP
9
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.
10
A test system example
Consider a test bearing or seal element as a point mass
undergoing forced vibrations induced by external forcing
functions
force, fY
KYY, CYY
Kh,Ch: support
Y
KXY, CXY
KXX, CXX
stiffness and damping
Mh : effective mass
Bearing or
seal
Ω
Journal
X
(K,C,M): test element
force, fX
stiffness, damping & inertia
force coefficients
KhY, ChY
Soft
Support
structure
KYX, CYX
KhX, ChX
11
Equations of motion (EOMs)
For small amplitudes about an equilibrium position, the
EOMs of a linear mechanical system are
Mh + M z + Ch +C z + Kh + K  z = f
force, FY
KYY, CYY
Y
KXY, CXY
KXX, CXX
 fX 
 x
z   , f   
 y
 fY 
Bearing or
seal
Ω
Journal
X
KhY, ChY
Soft
Support
structure
KYX, CYX
force, FX
where
Kh,Ch: structure
stiffness and damping
Mh : effective mass
(K,C,M): test element
force coefficients
KhX, ChX
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
12
Identification model (1)
Apply two independent force excitations on the test element
 f x1 
Step (1) Apply  
 f y1  ( t )
 f x2 
Step (2) Apply  f 
 y2  ( t )
and measure
and measure
 x1( t ) 


y
 1( t ) 
 x2( t ) 


y
 2( t ) 
force, FY
KYY, CYY
Y
KXY, CXY
KXX, CXX
Bearing
Ω
Journal
X
KSY, CSY
Soft
Support
structure
KYX, CYX
force, FX
How to apply the forces?
Use impact hammers, mass imbalances,
shakers
(impulse, periodic-single frequency, sine-swept,
random, etc)
KSX, CSX
13
Excitations with shakers
X
Y
14
Identification model (2)
Obtain the discrete Fourier transform (DFT) of the applied
forces and displacements, i.e.,
force, FY
KYY, CYY
 FX1( ) 
 f x1( t ) 

  DFT 
;
 FY1( ) 
 fY1( t ) 
 X 1( ) 
 x1(t ) 
;

  DFT 

 Y1( ) 
 y1(t ) 
 FX 2( ) 
 f x2( t ) 

  DFT 
;
 FY2( ) 
 fY2( t ) 
 X 2( ) 
 x2(t ) 

  DFT 

y
Y
 2( ) 
 2(t ) 
Y
KXY, CXY
KXX, CXX
Bearing
Ω
Journal
X
force, FX
and use the property
i  X ( )  DFT  x( t )  ;  2 X ( )  DFT  x( t ) 
KSY, CSY
Soft
Support
structure
KYX, CYX
KSX, CSX
where, i  1
15
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
 K h + K    2  M h + M   i   Ch + C   Z = F
F 
X
Z  , F X 
 Y   
 FY  
16
Identification model (4)
 H XX
H
 H YX
H XY 
H YY 
Define the complex impedance matrix
2

H   K h + K     M h + M   i   Ch + C  
Reand & Imaginary Impedance
The impedances are
functions of the excitation
frequency ().
Ideal imp ed ance
7
1 10
6
5 10
xx
0
6
5 10
0
200
400
600
frequency (rad/s)
Re(H)
Im(H )
800
1000
K
REAL PART =
dynamic stiffness,
- 2 M
IMAGINARY PART = C
(quadrature stiffness),
proportional to viscous
damping
17
Identification model (5)
With the complex impedance H   K h + K    2  M h + M   i   Ch + C  


The EOMS become, for the first & second tests
 H XX ( )

 HYX ( )
H XY(  )   X  
 FX1 

1
    
HYY(  )   Y1  
 FY1 

 H XX ( )

 HYX ( )
H XY(  )   X  
 FX 2 

2
    
HYY( )   Y2  
 FY2 

Add these two eqns. and reorganize them as
 H XX
H
 XY
H YX   X 1
H YY   Y1
X 2   FX1


Y2   FY1
FX 2 

FY2 
At each frequency (ωk=1,2,…n), the eqn. above denotes four
independent equations with four unknowns, (HXX, HYY , HXY , HYX)
18
Identification model (6)
Find H
 H XX
H
 XY
since
HYX   FX1


H YY   FY1
H = F

Then
where
F
(1)
(1)
 FX1 
 X1 
(1)

 &Z  ,
 FY1 
 Y1 
F
FX 2   X 1

FY2   Y1
(2)  

F
(2)
Z
(1)
Z
X2
Y2 
(2) 
1
1

 FX 2 
 X2 
(2)

 &Z   
 FY2 
 Y2 
The need for linear independence of the test forces (and ensuing motions)
is obvious
19
Condition number
In the identification process, linear independence is
MOST important to obtain reliable and repeatable
results.
force, FY
KYY, CYY
Y
KXY, CXY
KXX, CXX
Bearing
Ω
Journal
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
(1)
(2)
Z

Z
X
KSY, CSY
Soft
Support
structure
KYX, CYX
KSX, CSX
 ~0

In this case, the condition number of the
identification matrix tell us whether the identified
coefficients are any good.
Test elements that are ~isotropic or that are
excited by periodic (single frequency) loads
producing circular orbits usually determine an ill
conditioned system
20
force, FX
The estimated parameters
Estimates of the system parameters
force, FY
KYY, CYY
Y
KXY, CXY
KXX, CXX
Bearing
{M, K, C},j=X,Y
Ω
Journal
X
KSY, CSY
are determined by curve fitting of the test
derived discrete set of impedances
Soft
Support
structure
KYX, CYX
KSX, CSX
(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:
 K XX  KhX   
2
M h  Real H XX 
  CXX  ChX   Ima H XX 
21
force, FX
Meaning of the curve fit
force, FY
Analytical curve fitting of any data gives a correlation
coefficient (r2) representing the goodness of the fit.
KYY, CYY
Y
KXY, CXY
KXX, CXX
Bearing
Ω
Journal
X
KSY, CSY
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.
Soft
Support
structure
KYX, CYX
KSX, CSX
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.
22
force, FX
Transfer functions=flexibilities
force, FY
Transfer
functions
(displacement/force)
are
the
KYY, CYY
Y
KXY, CXY
KXX, CXX
system flexibilities G derived from
Bearing
Ω
Journal
X
KSY, CSY
-1
G=H
Soft
Support
structure
KYX, CYX
KSX, CSX
H YY
 H XY
; GXY  TF ( X 2 ) 


 H YX
H XX
 TF (Y1 ) 
; GYY  TF (Y2 ) 


GXX  TF ( X 1 ) 
GYX
where
  H XX HYY  H XY HYX
23
force, FX
The instrumental variable filter method
In the experiments there are many more data sets (one at each
frequency) than parameters (4 K, 4 C, 4 M=16).
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
Recall that
GH = I
1 0 
I

0 1 
24
The IVFM (1)
Since
G=H-1
The product
GH = I
1 0 
I

0 1 
However, in any measurement process there is always some
noise. Introduce the error matrix (e) and set
G  H  G K   M  i  C   I + e
2
Above G is the measured flexibility matrix while H
represents the (to be) estimated test system
impedance matrix
25
The IVFM (2)
It is more accurate to minimize the approximation errors (e) rather than
directly curve fitting the impedances.
Let
Hence
Let
H( )
M 
 
2


  I i  I I  C 


K 
 
GH  I +e
A G
k
k
k2 I

1 0 
I

0 1 
M 
 
G k k2 I i k I I   C   I + ek


K 
 
i k I I 

M 
k  
A  C   I + ek
K 
 
26
The IVFM (3)
Stack all the equations, one for each frequency
obtain the set
k= 1,2…,n , to
M 
 
A C  Ι e
K 
 
where
 A1 
 e1 
 
 
2
A   A  , e  e 2 
 n
 n
 A 
e 
0 1 0 1 0 1 .. .. .. .. 0 1 
I 

1
0
1
0
1
0
..
..
..
..
1
0


T
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
27
The IVFM (4)
The minimization leads to the normal equations
M 
1
 
T
T
C
=
A
A
A
I


 
K 
 
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
28
The IVFM (5)
At step m,
M 
 
C
K 
 
m 1


  W
A



m T
1
m T
W
I


where
 F m   2I i  I
1
  (1)  1
W m  
 m  2
 F(n )  n I i n I
I   
 



I   

m
F ( )
m

M  

  
2
    I 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
29
1
The IVFM (6)
At step m,
M 
 
C
K 
 
m 1


  W
A



m T
1
m T
W
I


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.
30
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
31
An example of parameter
identification
32
Texas A&M University
Mechanical Engineering Dept. – Turbomachinery Laboratory
Identification of force
coefficients in a SFD
Luis San Andrés
Sanjeev Seshagiri
Paola Mahecha
Research Assistants
Sponsor: Pratt & Whitney Engines
SFD EXPERIMENTAL TESTING & ANALYTICAL METHODS DEVELOPMENT
33
P&W SFD test rig
Static loader
Shaker assembly (Y
direction)
Shaker assembly
(X direction)
Static loader
Shaker in Y
direction
Shaker in X
direction
SFD test
bearing
34
Test rig description
Static loader
shaker Y
shaker X
SFD
Y
Static loader
support rods
base
X
35
P & W SFD Test Rig – Cut Section
Test rig main features
Piston ring seal
Test Journal
(location)
Bearing Cartridge
Supply orifices (3)
Circumferential groove
Flexural Rod
(4, 8, 12)
Journal diameter: 5.0 inch
Film clearance: 5.1 mil
Film length: 2 x 0.5 inch
Support stiffness: 22 klbf/in
Main support
rod (4)
Journal Base
Pedestal
in
36
Lubricant flow path
Oil
inlet
in
37
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
38
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
Oil in, Qin
ISO VG 2 Oil
Viscosity at 73.4 oF [cPoise]
Density
[kg/m3]
Inlet pressure [psig]
Journal (D)
Oil out, Qt
2.95
784
7.5
Outlet pressure [psig]
0
Radial Clearance [mil]
c
End groove
Central
groove
c
L
L
L
Bearing
Cartridge
End groove
5.0
Oil out, Qb
Central groove length [inch]
L
Oil collector
Land length, L [inch]
L
Total Length [inch]
3L
Journal Diameter [inch]
Oil out
Base
Support
rod
39
Typical circular orbit tests
• Frequency range: 5-85 Hz
• Centered eS=0
• Orbit amplitude r/c=0.66
 5.1
Lmax  160
5.1
160
2.55
80
 2.55
0
2.55
5.1
 2.55
 5.1
 160
 80
0
80
160
 80
 160
X Displacement [mil]
5 Hz
15 Hz
motion
25 Hz
35 Hz
45 Hz
Y Load [lbf]
Y Displacement [mil]
Dmax  5.1
(y vs. x)
X Load [lbf]
Forces (fy vs. fx)
40
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
2.55
 5.1
 2.55
0
2.55
 2.55
 5.1
40
 80
 40
0
40
80
 40
 80
X Displacement [mil]
0.26 mil
0.32 mil(y vs.
motion
0.60mil
1.04 mil
0.64 mil
5.1
Y Load [lbf]
Y Displacement [mil]
5.1
x)
X Load [lbf]
Forces (fy vs. fx)
41
Typ system direct impedances
Real (Hxx)
4
 3104
 210 0
0
4
 110
4
100
210
50
0
 310
0
Frequency
[Hz]
50
4
ryy
Real (Hyy)red
4
2104
310
04
110
4
 110
0
4
 2104
 110
4
 3104
 210 0
50
Frequency
[Hz]
50
From
IVF
Frequency
[Hz]
From test data
From IVF
From test data
Real part
5100
0
50
50
r/c= 0.66,
rxxIm  0.932
centered
es=0
3
510
d
100
0
0
100
Frequency
[Hz]
50
100
4
210
4
110
0
100
0
ryyre  0.99
d
Im (Hyy)
4
2104
1.510
4
1.5104
110
Im (Hyy)
4
1.510
4
110
3
0
50
100
0
100
0
0
Frequency [Hz]
From IVF
From test data
50
ryyIm  0.978
d
3
510
ryyIm  0.966
d
510
0
100
From IVF
From test data
ryyIm  0.966
d
4
1103
510
100
50
Frequency [Hz]
From
IVF
Frequency
[Hz]
From test data
From IVF
From Im
test(Hyy)
data
4
4
0
3
rxxIm  0.948
d
Real (Hyy)
210
4
310
 110
4
 310
rxxIm  0.948
d
4
From IVF
From test data
 0.996
4
110
3
1
5
10
10
Frequency [Hz]
ryyre  0.996
d
4
1104
210
110
0
100
Im (Hxx)
4
1.5104
0
From
IVF
Frequency
[Hz]
From test data
From IVF
(Hyy)
FromReal
test data
310
Re(Hyy)
Re(Hyy)
[lbf /[lbf
in] / in]
110
rxxre  0.999
d
4
4
1.52
10
10
1.510
Im(Hxx) [lbf / in]
4
 2104
 110
4
4
Im(Hyy) [lbf / in]
4
 110
0
210
rxxre  0.999
d
Im(Hyy)
Im(Hyy)
[lbf /[lbf
in] / in]
04
110
4
HYY
Re(Hxx) [lbf / in]
4
1104
210
4
Im(Hxx)
Im(Hxx)
[lbf /[lbf
in] / in]
310
Real (Hxx)rxxred  0.999
Re(Hyy) [lbf / in]
HXX
Re(Hxx)
Re(Hxx)
[lbf /[lbf
in] / in]
310
210
4
310
Im (Hxx)
4
4
4
2104
Im (Hxx)
Real (Hxx)
Frequency
[Hz]
50
From
IVF
Frequency
[Hz]
From test data
From IVF
From test data
0
50
100
100
Frequency [Hz]
From IVF
Imaginary part
From test data
42
Typ. system direct impedances
Im (Hxx)
Real (Hxx)
4
4
4
210
4
110
0
4
 110
210
rxxre  0.999
d
4
4
110
0
4
 110
4
 210
 210
0
Im (Hxx)
4
1.510
4
110
r/c= 0.66,
centered es=0
4
110
3
510
rxxIm  0.932
d
rxxIm  0.948
d
3
510
0
50
0
100
50
100
0
4
 310
50
0
100
Frequency [Hz]
K- M
Real (Hyy)
Re(Hyy) [lbf / in]
4
310
4
310
210
4
110
0
4
 110
4
4
0
4
 110
4
0
0
50
4
1.510
Excellent correlation between
test data and physical model
110
 310
Im (Hyy)
Im (Hyy)
4
4
1.510
4
110
4
110
3
510
50
0
100
0
0
100
0
ryyIm  0.978
d
3
510
ryyIm  0.966
d
REAL PART = dynamic stiffness
 210
From IVF
From test data
C
210
4
210
Frequency [Hz]
From IVF
From test data
ryyre  0.99
d
ryyre  0.996
d
4
100
Frequency [Hz]
From IVF
From test data
From
2 IVF
From test data
Real (Hyy)
50
Frequency [Hz]
Im(Hyy) [lbf / in]
0
Re(Hyy) [lbf / in]
rxxre  0.999
d
4
210
4
Im(Hyy) [lbf / in]
HXX
Re(Hxx) [lbf / in]
310
310
Im(Hxx) [lbf / in]
Re(Hxx) [lbf / in]
Real (Hxx)
1.510
Im(Hxx) [lbf / in]
4
50
50
100
100
Frequency [Hz]
Frequency [Hz]
Frequencyproportional
[Hz]
Frequency [Hz]damping
IMAGINARY PART
to
viscous
From IVF
From IVF
From test data
From test data
From IVF
From test data
From IVF
From test data
43
Test cross-coupled
impedances
Re (Hxy)
Im (Hxy)
Cross Coupled terms
3
310
Coupled terms
rxyre  0.82 310
d
4
3
 110
0
3
 210
3
 110
3
 310
30
50
 210
4
210
4
110
0
100
Frequency [Hz]
3
4
 110
4
 210
 310
50
From IVF
From
test data[Hz]
Frequency
3
3
 1.5500
10
3
3

1
2
10
10
4
0
4
 110
4
 210
0
0
rxxre  0.999
4
2d10
ryxre  0.866
d
110
50
50
3
 1.510
4
110
0
100
100
4
 110
0
Frequency [Hz]
Frequency
[Hz]
From IVF
3
 210
0
50
100
4
Real
part
From IVF
ryyre  Im
0.99(Hxx)
d
4
Im (Hyx)
4
1.510
3
3
2
310
4
110
33
103
1
2510
rxxIm  0.932
d
0
0
3
1100
50
1.510
4
110
ryxIm  0.629
d
0
50
100
50
100
Frequency
[Hz] ryx
100
Imd  0.629
[Hz]
From IVF
From
IVF
Im (Hyy)
From test data From
test data[Hz]
Frequency
0
ryyre  0.99
d
4
From test data
4
110
0
4
 110
50
From IVF
From
test data[Hz]
Frequency
100
One order of
magnitude
Frequency [Hz]
From IVF lesser than
From test data
direct
Im (Hyy)
impedances
= Negligible
crosscoupling
effects
50
100
ryyIm  0.978
d
3
510
0
0
50
100
Frequency [Hz]
0
210
0
50
From IVF
Frequency [Hz]
Frequency
From test
data
From IVFFrom test data
Real (Hyy)
From
test data[Hz]
Frequency
310
0
From test data
3
Real310(Hyy)
Im(Hxx) [lbf / in]
3
 1100
4
210
0
100
From IVF
From test data From IVF
Im(Hyx)
Im(Hyx)
[lbf / in][lbf / in]
Re(Hxx) [lbf / in]
500
 500
rxxIm  0.932
d
rxyIm  0.73 100
d
Frequency [Hz]
50 0
100
4
1.510
Im(Hyy) [lbf / in]
4
50
Im (Hyx)
Re(Hyy) [lbf / in]
Re (Hyx)
0
3
510
Frequency [Hz]
ryxre  0.866
4
310
310
4
110
rxyIm  0.73
d
0
Real d(Hxx)
0
Re(Hyy) [lbf / in]
Re(yx) Re(yx)
[lbf / in][lbf / in]
HYX
3
3
1
210
100
From IVF
Re (Hyx)
From test data
500
4
1.510
rxxre  0.999
d
0
0
Im (Hxx)
2
310
1100
r/c= 0.66,
centered es=0
Im (Hxy)
Real 3(Hxx)
Im(Hyy) [lbf / in]
3
Im(Hxx) [lbf / in]
Re (Hxy)
0
110
Re(Hxx) [lbf / in]
Re(Hxy)
Re(Hxy)
[lbf / in][lbf / in]
HXY
rxyre  0.82
Cross
d
Im(Hxy)
Im(Hxy)
[lbf / in][lbf / in]
3
110
4
110
3
510
0
50
100
From IVF
From test data
Imaginary part
From IVF
From test data
ryyIm  0.978
d
44
SFD force coefficients
SFD
Difference between lubricated
system and dry system
(baseline) coefficients
CSFD=Clubricated - Cs
MSFD=Mlubricated - Ms
DRY system parameters
Ks = 21 klbf/in
Ms = 40 lb
Cs= 7 lbf-s/in
Nat freq = 73-75 Hz
Damping ratio
= 0.04
KSFD=Klubricated - Ksh
45
CXX
Damping coefficients (lbf-s/in)
SFD damping coefficients
30
Damping
increases
mildly as
static
eccentricity
increases
25
e s = 2.4 mil
20
e s = 0 mil
15
e s = 1.56 mil
10
5
C XX SFD
0
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)
CYY ~ CXX for circular orbits, independent of
static eccentricity
46
SFD mass coefficients
MXX
Mass Coefficients (lbm)
30
25
20
e s = 0 mil
15
e s = 1.56 mil
e s = 2.4 mil
10
5
M XX SFD
0
0.0
1.0
2.0
3.0
4.0
5.0
Orbit radius (r) + static eccentricity (es)(mil)
MXX ~ MYY decreases with orbit radius (r) for
centered motions. Typical nonlinearity
47
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.
48
Acknowledgments
• Thanks to Pratt & Whitney Engines
• Turbomachinery Research Consortium
Learn more
http:/rotorlab.tamu.edu
Questions (?)
49
References
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)
50