A Linear Fluid Inertia Model for Improved Prediction of Force Coefficients in Grooved Squeeze Film Dampers and Grooved Oil Seal Rings
Modern Lubrication – Appendix to Notes 11 & 13
Luis San Andres
Adolfo Delgado
Texas A&M University © 2009
Bearing & seal dynamic reaction forcesBearing & seal dynamic reaction forces
Stiffness coefficients
Damping coefficients
Typically: Small clearances (c/R < 0.001)Re*<<1No fluid inertia forces are included
YX
CCCC
YX
KKKK
FF
BYYYX
XYXX
BYYYX
XYXX
Y
X
2* cRe
Viscous fluid force
Unsteady flow fluid inertia forceSqueeze film Reynolds
number
rotor
X
Z
Bearing coordinate system
Y
X
Fluid inertia Fluid inertia –– When is it important?When is it important?
Added mass coefficients:- comparable to journal mass (i.e. can shift system natural frequency)- larger in grooves (i.e. large clearances)
-Large clearances and/or groove depths -Long axial length-High frequencies
Grooved Oil Seals & Squeeze Film Dampers.
Dynamic reaction forces:
Stiffness coefficients Damping coefficients
YX
MMMM
YX
CCCC
YX
KKKK
FF
SYYYX
XYXX
SYYYX
XYXX
SYYYX
XYXX
Y
X
Inertia coefficients
Oil seal:
SFD-CCOcentered:
0 0 00 0 0
X XX XX XX
Y YY YY YYS S S
F K C MX X XF K C MY Y Y
from centering spring
2
12* cRe
Fluid mass inside film is just a few grams, but…. since added mass is proportional to Diameter and 1/clearance MXX >> mass of solid steel journal [1]
M fluid D L c M steel steel D2
2 L
M XX D2
3
Lc 1
tanhLD
L
D
SHORT SEALM XX 2.91 kg0.2
M fluid 2.76 10 3 kg M XX
M steel0.67
M steel 4.34 kg
M XX 42.03 kg0.5
M fluid 6.9 10 3 kg M XX
M steel3.88
M steel 10.84 kg
LONG SEAL
Fluid inertia Fluid inertia –– How large is it?How large is it?
[1] Reinhardt, F., and Lund, J. W., 1975, “The Influence of Fluid Inertia on the Dynamic Properties of Journal Bearings,”ASME J. Lubr. Technol., 97(1), pp. 154-167.
L/D=0.2 L/D=0.5
Squeeze film damperSqueeze film damper
Known issue: Poor predictions of fluid film added masses for grooved SFDs
Common configurations include grooves and recesses to supply oil and to prevent oil starvation in squeeze film lands.
In aircraft gas turbinesand compressors,
squeeze film dampers aidto attenuate rotor
vibrations and providemechanical isolation.
Feed
groove
oil inlet
seal
housing
ournal
lubricant film
shaft
nti-rotation pin
ball bearing Discharge
groove
Lubricant
film
Added mass coefficient Added mass coefficient -- Literature reviewLiterature review
Grooved SFDsSan Andrés, 1992 (Predictions)
- SFD with a shallow groove behaves at low frequencies as a single land damper
- Groove influences dynamic force coefficients
Arauz and San Andrés, 1994 (Prediction and experiments)
- Dynamic pressure levels at the groove (cg/c < 10) and film land are of the same order of magnitude- Model underestimates the radial force and overestimates the tangential (damping) force at the damper film land.
Qingchang et al., 1998 (Predictions and experiments)- Experimental results show that radial (inertial) force is underpredicted by a factor of three.
San Andres and Delgado, 2006 (experiments)Added mass coefficients are underpredicted by classical model (5 times smaller)
Predictions of the inertia coefficient correlate well experimental data while damping coefficients are underestimated (two-stage liquid seal)
Kim and Lee, 2005 (Predictions and experiments)
Current models do not properly predict both damping and added mass coefficients in grooved SFDs
Lund et al., 2003 (Predictions and experiments)Fluid inertia force coefficient is overpredicted (up to 70 %) for increasing groove volumes
Added mass coefficient Added mass coefficient -- Literature review (cont.)Literature review (cont.)
Grooved SFDs
Oil seal ringsOil seal rings
Typical oil seal multi-ring assembly
- Locked oil seal rings can induce instability in compressors.
- Seals are grooved to reduce cross-coupled stiffness and lower lock-up forces.
In centrifugal compressorsoil seals are commonly used to prevent leakage of process gas into ambient.
Shaft
Outer seal
Inner seal
Oil supply (PS+P)
Process Gas (PS)Pa
Outer seal land
Inner seal land
Anti-rotation pinSeal loading spring
Shaft
Outer seal
Inner seal
Oil supply (PS+P)
Process Gas (PS)Pa
Outer seal land
Inner seal land
Anti-rotation pinSeal loading spring
[1] Graviss, M., 2005, “The Influence of a Central Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., College Station, TX.
Parallel oil seal configuration [1]
17 mm
76c
c
25 mm
136c
Oil supplySeal lengt
Journal
Parallel seal configuration (balance thrust force due to pressure drop across the seals)
Includes ‘deep’ inlet (central) groove to feed seals
Parameter identification: FSEAL= 1/2 FTest conf.
Predictions do not consider groove or fluid inertia effects (Zirkelback and San Andrés 1996)
Smooth oil sealsSmooth oil seals-- Experimental resultsExperimental results
Childs et al., (2006, 2007)
Results Force coefficients are well predicted (C,K) except added mass coefficients
Large added mass coefficients (~15 kg)
Added mass predictions using Classical model (Reinhardt & Lund –1975) (single land- i.e. not including inlet groove) (2.84 kg)
2L
cJournal
Constant pressure
Compare smooth and grooved seals of short lengthCompare smooth and grooved seals of short lengthDamping, cross-coupled stiffness & added mass
3 3
3
18 ; ; 84 2 20XX XY XX XX
D L D LC K C Mc c
2-land seal: (deep groove divides lands)
3 3
3
1; ;4 2 20XX XY XX XX
D L D LC K C Mc c
L
c
30c
5c
3 3
3
12 ; ; 24 2 20XX XY XX XX
D L D LC K C Mc c
Coeffs are ¼ of original seal
flow
Grooved oil sealsGrooved oil seals-- Literature reviewLiterature review
- Grooves should reduce force coefficients by a factor of four, i.e.
Semanate and San Andrés, (1993)
Kxy (1 land)= 4 Kxy(2 lands)Cxx (1 land)= 4 Cxx(2 lands)
-Fluid inertia effects not predicted (considered negligible)
Baheti and Kirk, (1995)
Predictive models
- Reynolds and energy equation (Finite element solution)- Grooves should effectively isolate seal lands - Cross-coupled stiffness and damping coefficients are reduced by ~60 % for grooved configurations
- Bulk flow equation model
Childs et al., (2006)Single inner land groove and multiple groove oil seal (single clearance)
Large added mass coefficients (~30 kg)-
Higher than for ‘smooth’ seal
Childs et al., (2007)
Grooved oil sealsGrooved oil seals-- Experimental resultsExperimental results
One inner land groove with groove depths (5c,10c,15c)
Inner land groove does not effectively separate seal lands
Results Force coefficients are underpredicted (grooved seal)
[1] Graviss, M., 2005, “The Influence of a Central Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., College Station, TX.
17 mm
76c
c
25 mm
136c
Oil supplySeal length
Journal
Parallel oil seal configuration [1]
Inner land groove
Test oil sealTest oil seal
Predictions Experiments≠
Inner land groove should reduce crossed-coupled stiffness and direct damping coefficients by a factor of four
Groove does not effectively separate seal lands
Kxy (1 land)= 4 Kxy(2 lands)Cxx (1 land)= 4 Cxx(2 lands)
≠Null (neglected) added mass coefficients
Large added mass coefficients, increasing with increasing groove depth
Need for better predictive models
Inlet groove not considered (null dynamic pressure)- Null added mass coefficients
Large added mass coefficients
≠At most:
Kxy (1 land)= 2 Kxy(2 lands)Cxx (1 land)= 2 Cxx(2 lands)
Ps- Pd >0
Pd :discharge pressure
yz
feed plenum groovemid-land groove
oil supply, Ps
Streamlines in axially symmetric grooved annular cavity.
PdPd
Fluid flow predictive modelFluid flow predictive model• Bulk flow, centered operation, incompressible fluid
• Qualitative observations of laminar flow field
• Boundary Conditions
• Characteristic groove depth
Groove effective depthGroove effective depth
Test seal
CFD simulations show: streamline separating flow regions IS a physical boundary delimiting the domain for squeeze film flow due to journal radial motions.
Inner land groove close up (CFD -Pressure driven flow)
Laminar flow
Ps= supply pressurePa= ambient pressure
10c
15c
Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.
PaPs
, ,...I II N
Linear fluid inertia modelLinear fluid inertia model
3 3 12 6P Ph h h R hx x z z t x
n+1
Oil supply
zI zIIIzII zIV
1 2 3 4
zN
nIV NIIIIII
No fluid inertia advection
In each flow region:
2
22h h
t
with temporal fluid inertiaReynolds equation
0i t
X X Y YP = P + e P + e Pe
, ,...I II N
0
0
X Y
X Y
i tx x X x Y x
i tz z X z Y z
q = q + e q + e qe
q = q + e q + e qe
Perturbation flow analysisPerturbation flow analysis
Pressure field
Flow field
Axial
Circumferential
x= R
Y
X
e
h
R cos sini t
X Yh = c + e ( ) + e ( )e
,X Ye e < < c
Film thickness
For each individual flow region
View of rotating and whirling journal
Centered operation
n+1
Oil supply
zI zIIIzII zIV
1 2 3 4
zN
nIV NIIIIII
Boundary conditionsBoundary conditions
No generation of dynamic pressure
First-order pressures and axial flow rates must be equal
Null axial flow rate(geometrical symmetry)
PaPa
Ps= supply pressurePa= ambient pressure
Laminar flowPs
Journal centered operation
;
220
221 10 0
0
( )( ) co s( )i
2 2 - + i ( ) s in ( )
( )
L
XL R2 N NXX X X X X X
LYX YX YX X
X
R f z d zf z - + CK M d x d z
C g zK MR g z d z
( ) ( ) cos( ) ( ) sin( ) X X XP z f z g z
2
*3
2
3
( ) cosh sinh 12 1 Re
6( ) cosh sinh
X f f
X g g
Rz zf z c s i iR R c
Rz zg z c sR R c
Pressure field (Pressure field (PPXX))
Zeroth order (Static Pressure) 0P a s z
First order (Dynamic Pressure)
( , , , )f g f gc c s s
Obtained from boundary conditions
2
*Re12
c
Modified local squeeze film Reynolds number
Force Coefficients
First order solution
2 in (50.8 mm)
Shaft
Oil inlet
Discharge groove
Journal
Oil inlet plenum
Inlet groove
Squeeze film land
Test Squeeze film damperTest Squeeze film damper
Clearance= 127 m
Inlet groove
Film land
Dischargegroove
c
78 c
31 c
6 mm
4 mm
25 mm
I
II
III
y
z
Flow in
Sealed-end SFD assembly cut view.
Journal diameter: 127 mm
Mas
s co
effic
ient
[kg]
Dam
ping
coe
ffici
ent [
kN.s
/m]
Inlet groove-to-clearance ratio (cI/c)
201551
20151051
10
20
50
100
200
0
150
3010
Good correlation with experimental results for
both damping and added mass coefficients
Dischargegroove
Inlet groove
Film landc
78 c
31 c
6 mm
4 mm
25 mm
I
II
III
y
z Inlet groove
Film landc
78 c
31 c
6 mm
4 mm
25 mm
I
II
III
y
z
c
78 c
31 c
6 mm
4 mm
25 mm
I
II
III
y
z
Grooved SFDGrooved SFD
DynamicDynamic
pressurepressureExperiments
z
x=R
Inlet
Test grooved oil sealTest grooved oil seal
Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.
Clearance= 86 m
Journal diameter: 117 mm
Parallel oil seals Configuration [Childs et al]
17 mm
76c
c(0-15) c
25 mm
136c
Oil supplySeal length
Discharge plenum
Buffer sealJournal
2 mmx=R
Damping decreasesrapidly as groove effective
depth increases (~1/c
I II c
Mas
s co
effic
ient
[kg]
Dam
ping
coe
ffici
ent [
kN.s
/m]
Inlet groove-to-clearance ratio (cI/c)
20151051
20151051
Mass coefficient lesssensitive to groove
effective depth (~1/c)20
40
60
0
50
100
200
0
150
Smooth seal (inlet groove)Smooth seal (inlet groove)
Classical model prediction
Experiments
Experiments
Best predictions
Best predictions
136c
151051
151051
Mas
s co
effic
ient
[kg]
Dam
ping
coe
ffici
ent [
kN.s
/m]
Mid-land groove-to-clearance ratio (cIII/c)
0
20
40
60
0
50
100
200
150
I II III IV c
c12cI
c7cI
c12cI
c7cI
Grooved seal (Grooved seal (mid or inner land groovemid or inner land groove))
Best correlation withexperimental results for
groove effective depthat ~ 50% of groove
physical depth
Best predictions
Best predictions
cIIIc
cIII= 16c
Experiments
cIIIccIII= 16c
Experiments
4000 6000 8000 100000
20
40
60
Rotor speed [RPM]
I II III IV c
Oil seal (Oil seal (grooved and grooved and ungroovedungrooved))
Cross-coupled stiffness
Cro
ss-c
oupl
ed s
tiffn
ess
[kN
/m]
Model effectively predicts reduction of cross-coupled stiffness due to inner land groove.
(16c groove)
Kmodel ( =11 , =7I III
c c c c Kxy(model)
(no inner land groove)Kmodel ( =11 , =
I IIIc c c c Kxy(model)
kxy -kyx
cIII= cII (exp.) kxy -kyx
cIII= cII (exp.)
cIII=cII +15cII (exp.) kxy -kyx
Finite element solutionFinite element solutionOff-centered journal operation
x= R
Y
X
e
h
R e
Reynolds eqn. with temporal fluid inertia
0 cos sini tX Yh = h + e ( ) + e ( )e
Film thickness
3 3
22
212 6
P Ph hx x z z
h R h h ht x t
22
2h ht
z
x=R
Inlet
Test grooved oil sealTest grooved oil seal
Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.
Clearance= 86 mJournal diameter= 117 mm
Parallel oil seals Configuration [Childs et al]
17 mm
76c
c(0-15) c
25 mm
136c
Oil supplySeal length
Discharge plenum
Buffer sealJournal
2 mm
12
34
e
e
0h
Outlet plane
z
Inlet plane0 2
Grooves
x=R
Seal operating conditionsSeal operating conditions
-1.0
-0.8
-0.5
-0.3
0.0
0.3
0.5
0.8
1.0
-1.0 -0.8 -0.5 -0.3 0.0 0.3 0.5 0.8 1.0
x
y
smooth seal
Grooved seal
Load
Smooth seal
Grooved seal
Journal center locusindicates seal
operates with oilcavitation at the
largest testeccentricities
Journal locus
Shaft speed: 10,000 rpmStatic eccentricity ratio:
0, 0.3, 0.5, 0.7
Supply pressure: 70 bar
Test dataY
X
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6
Static journal eccentricity ratio (e/c)
Leak
age
[kg/
s]Seal LeakageSeal Leakage
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6
Static journal eccentricity ratio (e/c)
Leak
age
[kg/
s]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6
Static journal eccentricity ratio (e/c)
Leak
age
[kg/
s]
Smooth seal- Predictions
Smooth seal- Experiments
Grooved seal- Predictions
Grooved seal- ExperimentsPredicted leakage
correlates well withexperiments for
both smooth landand grooved seal
Smooth SealGrooved Seal
(cg= 16c)
(c = 7c)
10,000 rpm, 70 bar
Smooth seal- Predictions
Smooth seal- Experiments
Grooved seal- Predictions
Grooved seal- Experiments
Direct dampingDirect damping
0
100
200
300
0 0.2 0.4 0.6
Static journal eccentricity ratio (e/c)
Dam
ping
[kN
.s/m
]
0
100
200
300
0 0.2 0.4 0.6
Static journal eccentricity ratio (e/c )
Dam
ping
[kN
.s/m
]
0
100
200
300
0 0.2 0.4 0.6
Static journal eccentricity ratio (e/c)
Dam
ping
[kN
.s/m
]
0
100
200
300
0 0.2 0.4 0.6
Static journal eccentricity ratio (e/c )
Dam
ping
[kN
.s/m
]
0
0
0
0 0.2 0.4
Static journal eccentricity ratio (e/c)0 0.2 0.4 0
Static journal eccentricity ratio (e/c)
Model predictsaccurately reduction
in direct damping dueto inner land groove.
Smooth Seal
Grooved Seal
Smooth Seal
Grooved Seal
Cyy
Cxx
10,000 rpm, 70 bar
(cg= 16c)
(c = 7c)
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
Smooth seal- Predictions
Smooth seal- Experiments
Grooved seal- Predictions
Grooved seal- Experiments
CrossCross--coupled stiffnesscoupled stiffness
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000
Rotor speed (RPM)
Stiff
ness
[MN
/m]
Model effectivelypredicts reduction
in cross-coupledstiffness due to mid-
land groove.
Smooth SealGrooved Seal
Kxy
Kxy
Eccentricity ratio=0.3
Eccentricity ratio=0
= 0, 0.3
Smooth SealGrooved Seal
70 bar
(cg= 16c)
(c = 7c)
0
5
10
15
20
25
30
35
40
0 0.1 0.2 0.3 0.4 0.5 0.6
Static journal eccentricity ratio (e/c)
Add
ed M
ass
[kg]
Added massAdded mass
5
10
15
20
25
30
35
40
Experimental data shows relatively large added mass coefficients. Predictions correlate well with experimental results.
Added mass coefficients are larger for grooved seal
Classical theory [1] predicts ~ 1/10 of test
value
[1] Reinhardt, F., and Lund, J. W., 1975, “The Influence of Fluid Inertia on the Dynamic Properties of Journal Bearings,” ASME J. Lubr. Technol., 97(1),
pp. 154-167.
Mxx
Smooth seal- Predictions
Smooth seal- Experiments
Grooved seal- Predictions
Grooved seal- Experiments
10,000 rpm, 70 bar
Smooth Seal
Grooved Seal
(cg= 16c)
(c = 7c)
• Damping and cross-coupled stiffness decrease rapidly as the effective groove depth increases(~1/c3).Added mass coefficients less sensitive to effective depth (~1/c).
• Predicted force coefficients (K,C,M) correlate well with experimental data for a narrow range of groove effective depths
• For shallow and short mid-land groove (oil seal) predicted (K,C,M) correlate best with test data when using 50% of actual groove depth.
• In oil seals, an inner land groove does not uncouple adjacent film lands!!
Conclusions:Conclusions:
The dynamic pressure field in deep grooves may be difficult to measure for most practical excitation frequency ranges. (fluid inertia pressures are proportional to 2).
Deep grooves do generate dynamic pressures of mainly inertial nature, which lead to large added mass coefficients.
There is a specific groove depth where the added mass coefficient peaks. This groove depth value for the studied configurations is < 10c.
Force coefficients in test configurations, like SFDs and oil seals, are a function of the ancillary geometries like feeding/discharge arrangements.
Conclusions (cont):Conclusions (cont):