proceedings of asme turbo expo 2015: turbine …rotorlab.tamu.edu/tribgroup/2015 trc san andres/2015...
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Luis San AndrésMast-Childs Chair Professor, Fellow ASMETexas A&M University
Orbit-Model Force Coefficients for Fluid Film Bearings: a Step beyond Linearization
Sung-Hwa JeungGraduate Research Assistant, Student Member ASME Texas A&M University
Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, June 15-19, 2015, Montreal, Canada
Paper GT2015-43487
Accepted for journal
publication
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BackgroundRotordynamics models bearings and seals reactionforces F={FX, FY}T with linearized force coefficients(stiffness K, damping C, and inertia M).
( )t eF =F -K z -Cz -Mz : rotor (journal) displacements about static position (e).
z={x,y}T
, , ,; ;
e e e
X Y XXY YX XX
x y x y x y
F F FK C M
y x x
Linearized force coefficients denote changes in reaction force toinfinitesimally small amplitude motions about an equilibriumposition:
0
lim eY Y
x
F F
x
( ) 0lim
( )e
e
X X
y y e
F F
y y
0lim eX X
x
F F
x
Y
X
3
A concern:
Linearized force coefficients (K, C,M) are often inadequate to produceaccurate reaction forces for rotormotions of a sizeable magnitude.Squeeze film dampers (SFDs) are acase in point.
In practice rotor-bearing systems do not show infinitesimally smallmotions, hence
The ever present questions are:• How good are the linearized force coefficients?• Can they be used for rotor motions of large amplitude, like when
crossing a critical speed? • How can one obtain better (more accurate) coefficients? Should
one use higher order terms?
Y
X
4
Choy et al. (1991) Calculate higher-order force coefficients derived from a Taylor-series expansion.
( )
2 32 3
2 3
2 32 3
2 3
2
2 32 3
2 3
1 1 .....2! 3!
1 1 .....2! 3!
.....
1 1 .....2! 3!
t e
X X XX X
e e e
X X X
e e e
X
e
X X X
e e e
F F FF F x x xx x x
F F Fy y yy y y
F x yx y
F F Fx x xx x x
2 32 3
2 3
2
1 1 .....2! 3!
..... ....
X X X
e e e
X
e
F F Fy y yy y y
F x yx y
How many nonlinear terms are needed?
Past literature
Issues:What is the threshold
between small and large amplitude motions?
How to do it fast (in a computer)?
Past literature El-Shafei and Eranki (1994)Characterize a bearing (nonlinear) reaction force response to deliver best
estimations for the bearing stiffness and damping force coefficients.
Sawicki and Rao (2001)
Müller-Karger and Granados (1997)Nonlinearity of force coefficients depends on the size and shape of the orbital path.
Czolczynski (1999)Estimates force coefficients
from orbits (harmonic) motions.
Non-linear analyses
None of the methods benchmarked (to date) to experimental data.
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6
In an analysis, how are the true linearized force coefficients obtained?
Y
X
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Extended Reynolds eqn. for laminar thin film flows
3 2 2
21 2 2 1 2h P h h h hP
t t
c : clearance
eX, eY: journal eccentricity (varies with time)
P : film pressure
h : film thickness
: viscosity
Wo X
Y
eXo
eY
eo
X
clearancecircle
YStatic load
Journal center
Wo X
Y
eXo
eY
eo
X
clearancecircle
YStatic load
Journal center
o
Wo X
Y
eXo
eY
eo
X
clearancecircle
YStatic load
Journal center
Wo X
Y
eXo
eY
eo
X
clearancecircle
YStatic load
Journal center
o
Film thickness: 0 cos sin i th h X Y e
Pressure:
Let journal move with small (~0) amplitude displacements(∆X, ∆Y) with whirl frequency ωabout an equilibrium position:
cos sinX Yh c e e
0 + + .. i tX YX YP P P X P X P Y P Y e
0 + + ...e e e e
P P P PP P X X Y Y
X X Y Y
,...X Ye e
P PP P
X Y
i t
i t
X X e
Y Y e
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Substitute h and P into modified Reynolds equation to obtain
30 0
0 0( )12 2h hP P
L
20
0( ) 1 Re2 4s
h hP i i h h P
L
Zeroth-order:
First-order:
Pressure fields & forces
00
cossin
o
o
LX
Y
FP Rd dz
F
Static (equilibrium) forces:
0 0
0 0
cos cosRe ; Im
sin sin
cos cosRe ; Im
sin sin
L LXX XX
X XXY XY
L LXY XY
Y YYY YX
K CP Rd dz P Rd dz
K C
K CP Rd dz P Rd dz
K C
True Linearized Force coefficients:
Limitation: journal motion of infinitesimally small amplitude about an equilibrium position.
F0+W=0balance of static forces
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Orbit analysis
Method follows an experimental identification procedure.
Estimates (numerically) force coefficients accurate over a wide frequency range. In particular for rotor whirl motions of large amplitude and statically off-centered.
Y
X
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Journal center kinetics and forcesJournal motion (X vsY) bearing reaction forces (FX vs FY).
Dots denote discrete points along the orbital path during a full period of whirl motion
Journal motion is neither circular nor small in amplitude.
Forces (N)
X
YJournal center orbit (µm)
X
Y
ω
Specify X(t),Y (t)
Solve unsteady Reynolds equation find pressure P( X,Y,dX/dt, dY/dt)
Integrate pressure field on journal surface
find ForcesFX, FY
Continue to complete whole orbit path.
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The orbit analysis methodFor given whirl orbit with given shape and over specified frequency range:
• Calculate bearing reaction forces.
• Conduct Fourier analysis.• Estimate force coefficients.
0
0
( ) ( ) ( )
( ) ( ) ( )
cos( ) sin( )
sin( ) cos( )X t X X t Y t
Y t Y X t Y t
e e a a
e e a a
( ) cos( )X t Xa r t ( ) sin( )Y t Ya r t
,X Yr r : amplitudes of motion along X,Y axes.
Journal motion with frequency (ω)
X/c
Y/c Clearance (c)
4
0.3Xr c
0.4Yr c
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Bearing reaction forces
The bearing reaction force: F = Fstatic + Fdyn(t).
The dynamic portion of the fluid film reaction force will be modeled in a linearized form as
dynF K z Cz+Mz
where z is a vector of dynamic displacements and are matrices of stiffness, viscous damping and inertia force coefficients
cos( ) sin( )X Y
Y X
i t i tr i r
i r re e
1z z
In the frequency domain, the journal dynamic motion is:
( , , )K C M
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Fourier analysisThe time varying part of the reaction force is periodic with fundamental period T=2/.Using Fourier series decomposition,
2 31 ....i t i t i te e e dyn II IIIF F F F
To first order effects (fundamental frequency)
1i te dynF F 1 1F H z
, i.e. reaction force is proportional to motion with fundamental frequency.
is a matrix of complex stiffness
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Procedure to obtain multiple solutions
Forward () and backward (-) whirl orbits ensure linear independence of two forces needed for identification. z1 F1 z2 F2
X
Y
+ωX
Y
-ω
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Y
X
FY
FX
1z1 1F11
FY
FX
2z1 2F12
3, 4,….……N-1,
ω
Forward whirl motions
2 22 21i te dynF F
2 21 1F H z
FY
FX
2z1 2F1
Fourier analysis:
FX
Nz1 NF1N
FY
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Y
X
FY
FX
1z2 1F21
FY
FX
2z2 2F22
-ω
Backward whirl motions
2 22 22i te dynF F
2 22 2F H z
FY
FX
2z2 2F2
Fourier analysis:
3, 4,….……N-1,
FX
Nz2 NF2N
FY
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Procedure to obtain complex stiffnesses
To find :
1 2k k k k
k 1 2F F H z z
Solve algebraic equations at each frequency (k):
XX XYk
YX YY k
H H=H H
H
1,.....k N H H H z1 F1 z2 F2
X
Y
+ωX
Y
-ω
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Stack Hk for a set of frequencies (k= 1,2,….N). A simple linear curve fit delivers:
2Re( )
Im( )k
k
k
k
H K - M
H C
Derived K,C,M coefficients
The orbit analysis procedure numerically replicates experimental procedures to identify force coefficients.
, ,K C M : stiffness, damping and mass coefficientsvalid over a frequency range and for specified whirl amplitude (and shape) sets.
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Bearing forces - compare Damper B (cB = 127 μm)
Orbit model reproduces best periodic force at highest frequency.
(Numerical) nonlinear force: integration of pressure field (solution of Reynolds eqn.)
Orbit analysis:
First Fourier component: 1i te F
21 i 1F K M C z
50% off-centered orbit (20% c)
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Comparisons of experimental results to predictions from models (orbit and linearized)
Orbit model
TrueLinear
Note in the following slides:
( , , )K C M
Y
X
( , , )K C M
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SFD Test Rig
Static loader
Shaker in X direction
Shaker in Y direction
Top view
SFD test bearing
2 electro magnetic-shakers (2 kN ~ 550 lbf)Static loader at 45°Customizable SFD test bearing.
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Example: Squeeze Film Damper
San Andrés, L., and Jeung, S.-H., 2015, “Experimental Performance of an Open Ends, Centrally Grooved, Squeeze Film Damper Operating with Large Amplitude Orbital Motions,” ASME J. Eng. Gas Turbines Power, 137(3).
L/D=0.4, 2 x 25.4 mm lands
L/D=0.2, 25.4 mm lands
Damper A Damper BJournal diameter, D 127 mm 127 mmRadial clearance, cA, cB 0.251 mm 0.129 mmFilm land length, L 25.4 mm 25.4 mmCentral Groove axial length, LG 12.7 mm none
depth, dG 9.5 mmLubricant ISO VG 2
Density, r 785 kg/m3 799 kg/m3
Viscosity m at TS 0.0029 Pa.s 0.0025 Pa.s
Frequency range (Hz) 10-100Whirl amplitude, r (μm) 20 - 178 6.- 76Static eccentricity, es (μm) 0 - 191 0 - 63
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Evaluate SFD force coefficients fromMax. clearance (c)
X Displacement [μm]
YD
ispl
acem
ent [μm
]Tests conducted Excitation frequency 10 – 100 Hz
circular orbits: amplitude (r) grows.Y
X
with offset or static eccentricity (es) – 45o from
X-Y axes
Operating condition Damper A0.251 mm
Damper B0.129 mm
Whirl amplitude r (μm) 20 – 178 6 – 76
Static eccentricity es (μm) 0 - 191 0 – 63
Max. squeeze film Reynolds No. (Res)
10.5 3.2
2
ScRe Squeeze film
Reynolds #
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Complex stiffness vs. frequency
Ima(H)Real(H)
Model
force coefficients
Goodness of Fit (R2)
MXX [kg] CXX [kN.s/m] Re (XX) Im (XX)
Fit to test data 6.5 5.8 0.99 0.99
Linear model 7.0 6.0 0.95 0.99
Orbit model 7.2 6.5 0.93 0.94
Damper B (cB = 127 μm)
Centered orbit (30% c)
Orbit model True Linear
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Complex stiffness vs. frequency
Ima(H)Real(H)
Model
force coefficients
Goodness of Fit (R2)
MXX [kg] CXX [kN.s/m] Re (XX) Im (XX)
Fit to test data 6.1 6.1 0.99 0.99
Linear model 7.0 6.0 0.98 0.99
Orbit model 6.1 6.1 0.98 0.99
Damper B (cB = 127 μm)
Centered orbit (50% c)
Orbit model True Linear
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Complex stiffness vs. frequency
Ima(H)Real(H)
Model
force coefficients
Goodness of Fit (R2)
MXX [kg] CXX [kN.s/m] Re (XX) Im (XX)
Fit to test data 6.4 7.8 0.93 0.97
Linear model 8.2 9.2 0.73 0.85
Orbit model 5.8 8.6 0.92 0.93
Damper B (cB = 127 μm)
40% off-centered orbit (30% c)
Orbit model True Linear
orbit model ~ test data
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SFD effective force coefficients
2X eff XX XY XXK K C M
XYX eff XX XY
KC C M
For circular orbits (only), dynamic forces reduce to
radial eff
tangential eff
F K r
F C r
r
-Fradial
-Ftangential
r
X
Y
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Orbit modelpredictions agree
well with experimental
results.
(es/cB=0)
Effective force coefficients Damper B (cB = 127 μm)
Circular centered orbits
Large amplitude motions.
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Orbit modelpredictions agree
well with experimental results.
SFD effective force coefficients Damper B (cB = 127 μm)
(es/cB=0.4)Circular off-centered orbits
Large amplitude eccentric motions.
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Coefficients from elliptical orbits
Y
X
Y
X
Y
X
Dynamic load measurements:Elliptical orbit amplitude (r) grows.rX : rY = 5 : 1 at static eccentricity eS/cB=0.1.
(rX, rY)=(0.1, 0.02) cB
(rX, rY)=(0.35, 0.07) cB
(rX, rY)=(0.6, 0.12) cB
Damper B (cB = 127 μm)
Note: True force coefficients can’t be found large motions.
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Coefficients from elliptical orbits Damper B (cB = 127 μm)
Predicted and experimental results show CXX ≠ CYY and MXX ≠ MYY.Orbit-based force coefficients follow trend of test coefficients;but over predict MYY by ~42% at rX/cB=0.6.
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SFD forces and energy dissipation
( )E dt TSFDz F
BCM SFD S S SF L M z C z K z a
SFD SFD SFD SFDF M z C z K z
Mechanical work
Experimentally derived force
Orbit model derived force
Over a full period of motion
: Applied Dynamic load
: Mass of bearing cartridge
: Rig structure force coefficients, ,S S SM C K LBCM
= Dissipated energyA way to characterize the goodness of the force coefficients.
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Evaluation of dynamic forcesω=100 Hz
Y
X(a)
(c)
(b)
For small amplitude motions, experimental
force and that constructed from orbit
model are similar.
For a large amplitude orbit, the differences
are significant.
Damper A (cA = 253 μm)
small to large size orbits:
34Energy dissipation increases with increasing orbit amplitude.
E < 0 = energy dissipated by SFD
Mechanical energy dissipation ( )E dt TSFDz F
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Energy difference (%)100 Hz
For all cases the difference is less than ~25%.
experimental
orbit_model
1diff
EE
E
Test damper A Test damper B
The measure gives validity to the force coefficients derived from an orbit analysis.
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ConclusionThe orbit analysis numerically replicates an experimentalprocedure, it calculates forces as a function of (any)motion amplitude and identifies force coefficients (K,C,M)over a frequency range.
Comparison to test results from two distinct SFDsFor large amplitude whirl orbits (r/c>0.3),
• True linearized force coefficients correlate poorly with test data.
• Orbit model force coefficients predict well the experimentallyforce coefficients, reproduce measured forces, and dissipatesame amount of energy as that in the experiments.
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Observation
One step beyond the conventional….
Orbit analysis model proposes a methodology to escape the limitations of linearized force coefficients while
avoiding time-consuming (transient response) numerical integration.
(*) In the near future, with ubiquitous super fast & super cheap computer power, every engineering process will be modeled with the greatest fidelity possible (number crunching rules! ).
for the time being (*)
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Thanks to Pratt & Whitney Engines
Learn more at http://rotorlab.tamu.edu
Questions (?)
TAMU Turbomachinery Research Consortium
Acknowledgements
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Other slidesY
X
Y
X
Y
X
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Types of journal motionx= R
Y
X
e
h
R
e: amplitude of motion
whirl frequency
eXo
eYo
whirlingjournal
Film thickness
x= R
Y
X
2rX
rX, rY : amplitudes of motion
whirl frequency
eo
2rY
K, C, M (force coefficients)RBS stability analysis
Applications:
FX, FY (reaction forces)RBS imbalance response& transient load effects
(a) Small amplitude journal motions (b) Large amplitude journal motions