Computational Model for Tilting Pad Journal Bearings Yujiao Tao Research Assistant Dr. Luis San...
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Transcript of Computational Model for Tilting Pad Journal Bearings Yujiao Tao Research Assistant Dr. Luis San...
Computational Model for Tilting Pad Journal
BearingsYujiao Tao
Research AssistantDr. Luis San AndresMast-Childs Professor
TRC project 2010-2011 TRC 32514/15196B/ME
START DATE: September 1, 2010
Justification and Objective
Journal Pad
Pivot
e
Rotor speed
Fluid film
W, static load
The accurate prediction of tilting pad journal bearing (TPJB) static and dynamic forced performance is vital to the successful design and operation of high-speed rotating machinery.
Pivot flexibility reduces bearing force coefficients for operation with heavy loads.XLTRC2 TFPBRG code shows poor predictions for dynamic force coefficients when compared to test data.
Research objective: To develop an advanced computational program, benchmarked by test data, to predict the static and dynamic forced performance of modern TPJBs accounting for thermal effects and the (nonlinear) effects of pivot flexibility.
Work to date
(a) Reviewed literature on TPJBs
(b) Developed analysis for effect of pivot flexibility on TPJBs load response.
(c) Took XLPRESSDAM® code and began modifications
(d) Obtained initial predictions for a near-rigid TPJB
Comprehensive table summing
46 papers
Literature review
• Reviewed 46 papers on TPJBs (1964-2011) and prepared a table that includes analysis methods, test methods and force coefficient identification, lubricant feeding arrangements, etc.
• Reviewed oil feed arrangements and other conditions to improve TPJBs’ performance.
Views of leading edge groove in TPJB (Ball, J. H., and Byrne, T. R., 1998)
Single externally adjustable pad fluid film bearing (Shenoy B. S. and Pai R.2009)
Literature review 46 papers on TPJBs (1964-2011)
Literature review 46 papers on TPJBs (1964-2011)
Work to date
(a) Reviewed literature on TPJBs
(b) Developed analysis for effect of pivot flexibility on force coefficients of TPJBs.
(c) Took XLPRESSDAM® code and began modifications
(d) Obtained initial predictions for near-rigid TPJB
Physical model and equations
follow
Major assumptions:• Laminar flow• Includes temporal fluid inertia
effects• Average viscosity across the
film
3 3 2 2
2 2
1
12 12 2 12J
h P h P h h h h
R z z t t
On kth pad
h : fluid film thickness P : hydrodynamic pressure
μ : lubricant viscosity : journal speed
RJ : journal radius
Journal Pad
Pivot
e
Rotor speed
Fluid film
W, static load
Reynolds equation for thin film bearing
Thermal energy transport in thin film flows
Nomenclature
T: film temperature
h : film thickness
U,W: circ. & axial flow velocities
Cv: viscosity & density, specific
heat
hB, hJ : heat convection coefficients
TB, TJ : bearing and journal temperatures
: journal speed
Major assumptions:
Neglect temperature variations across-
film. Use bulk-flow velocities and
temperature
22 2212
12 2
v B B J JC U h T W h T h T T h T TR z
R RW U
h
CONVECTION + DIFFUSION= DISSIPATION
(Energy Disposed) = (Energy Generated)
hhh
dyTh
TdyWh
WdyUh
U000
;~1
;~1
;~1
cos sin cos sinp X Y piv p p piv d p ph C e e r R
Film thickness in a pad
cos
sin
p piv p p
piv d p p
h C r e
R e
Cp : Pad radial clearance
Rd = Rp+t : pad thickness
rp : pad dimensional preload
p : pad tilt angle
pivpivpivot radial and
transverse deflections
Y
θp
h
e
WX
Pivot
Fluid film
Journal
OB
RP
WY
OP’
θ
X
P’
OP
piv
piv
p
X
Y
P
Pad
Journal static equilibrium in a TPJB
0 0 0
0 0 01
padX
Y
kNX X
kkY Y
FW F
W F F
sin cosd X p Y p dM R F F R F
Fluid film moment on pad
0
0
2
0
2
cos
sin
kt
kl
L kkpX k
Jk kLY p
FP R d dz
F
k=1,…Npad
jF
jFF
F
M
Journal
X
Y
Pad
WY
WX
’
p
pivFpivF
P
P’
p
X
Y
Op
p piv
pad piv piv
piv piv
M M
F F
F F
M
Pad equations of motion about pivot point P
is pad mass matrixpad M
Perturbation analysis
• Consider small journal motion perturbations with frequency () about the equilibrium position , the journal displacements are:
0
0
( )
( )
XX X i t
Y YY
ee t ee
e t ee
• Journal motions induce changes in the rotation of the kth pad and its pivot displacements with the same frequency ()
0
0
0
( )
( )
( )
pp pi t
piv piv piv
piv pivpiv
t
t e
t
• And, journal and pad motions induce changes in the film thickness and pressure fields
0
0
( )
( )
piv piv
piv piv
i tX X Y Y p piv piv
i tX X Y Y p piv piv
h t h h e h e h h h e
P t P P e P e P P P e
Reduced force coefficients
• 25 force impedances for the kth pad
12R R
R R
XX XY
R XY a s pad c bYX YY
Z Z
Z Z
Z Z Z Z M Z Z
XX XYXY
YX YY
Z Z
Z Z
ZX X X
aY Y Y
Z Z Z
Z Z Z
ZX Y
b X Y
X Y
Z Z
Z Z
Z Z
Z
c
Z Z Z
Z Z Z
Z Z Z
Z
/ 2
/ 2
l
l
L
L
Z P h Rd dz
X, Y,
• The reduced force impedances are
s s s
s s s s
s s s
Z Z Z
Z Z Z
Z Z Z
Z
Reduced force coefficients (in pad coordinates)
Alternatively, reduced impedances (ZR) are also obtained in pad local coordinates.
s s s
s s s s
s s s
Z Z Z
Z Z Z
Z Z Z
Z
12R JP s pad P PJ
Z Z Z Z M Z Z
, ,
2d d d
P d c
d
Z R Z R Z R Z Z Z
Z R Z Z Z Z Z
Z R Z Z Z Z Z
Z Zd d
PJ
Z R Z R
Z Z
Z Z
Z
d
JPd
Z R Z Z
Z R Z Z
ZZ Z
Z Z
Z
According to the perturbation analysis, the reduced impedances obtained by two methods are identical:
X
Y
R T
RZ A Z A
Work to date
(a) Reviewed literature on TPJBs
(b) Developed analysis for effect of pivot flexibility on force coefficients of TPJBs.
(c) Took XLPRESSDAM® code and began modifications
(d) Obtained initial predictions for near-rigid TPJB
Fortran program and
Excel GUI
Modified Fortran program and Excel GUI
• Uses finite element method to solve Reynolds equation (hydrodynamic pressure)
• Uses control volume method to solve energy transport equation
• Program updated for ideal TPJB with pivot flexibility. At this time, it works only for a near-rigid pivot (Difficulties in convergence).
Work to date
(a) Reviewed literature on TPJBs
(b) Developed analysis for effect of pivot flexibility on force coefficients of TPJBs.
(c) Took XLPRESSDAM® code and began modifications
(d) Obtained initial predictions for near-rigid TPJB
Comparison with other
predictions and some
experimental results
Predictions for a (near rigid) TPJB bearing
*Someya, T., 1988, Journal-Bearing Databook, Springer-Verlag, Berlin , pp. 227-229.
Number of Pads, N 5
Configuration Load on Pad
L/D 0.5
Dimensionless Preload , rp 0.5
Pad Arc Angle, p 60º
Rotor Diameter, D 0.06 m (2.36 inch)
Bearing Axial Length, L 0.03 m (1.18 inch)
Pad radial Clearance, Cp 120 μm (0.004724 inch)
Lubricant Viscosity, 0.028 Pa.s
Rotor Speed 6000 rpm
Offset 0.5
(Someya*) Five pad, tilting pad bearing (LOP)
• Isothermal flow, isoviscous
• Synchronous speed reduced force coefficients
1 RIGID pivot (Someya’s data)
2 RIGID pivot (My code)
3 Pivot stiffness Kp =3 GN/m (almost rigid)
Comparison of results for
W Y
X
0
0.1
0.2
0.3
0.4
0.5
0.01 0.1 1 10Sommerfeld number
Ec
ce
ntr
icit
y
d d
Someya's
rigid pivot
near rigid pivot
Predictions for static load versus journal eccentricity
p
e
C
2
p
LD RS
W C
TPJB model with flexible pivot predicts a larger eccentricity than that with rigid pivot, especially at heavy loads (small S).
W Y
X
Near rigid pivot
Rigid pivot
Predicted stiffness coefficients
10
100
1000
10000
0.01 0.1 1 10Sommerfeld number
Stif
fnes
s K
xx (
MN
/m)
cc
c
Someya's
rigid pivot
near rigid pivot
Near rigid pivot
Rigid pivotKXX
KYY
W Y
X
1
10
100
0.01 0.1 1 10Sommerfeld number
Stif
fnes
s K
yy (
MN
/m)
c
c
c
Kyy (Someya's)
Kyy (rigid pivot)
Kyy (Near rigid pivot)
Rigid pivotNear rigid pivotPivot flexibility lowers the
direct stiffness coefficient KXX (along load direction), in particular for large loads.
KP
10
100
1000
10000
0.01 0.1 1 10Sommerfeld number
Da
mp
ing
Cxx
(kN
.s/m
)
cc
c
Cxx (Someya's)
Cxx (rigid pivot)
Cxx (Near rigid pivot)
Near rigid pivot
Rigid pivot
10
100
1000
10000
0.01 0.1 1 10Sommerfeld number
Da
mp
ing
Cyy
(kN
.s/m
)
cc
c
Cyy (Someya's)
Cyy (rigid pivot)
Cyy (Near rigid pivot)Rigid pivot
Near rigid pivot
Predicted damping coefficients
CXX
CYY
W Y
X
Pivot flexibility lowers the direct damping coefficient CXX (along load direction),
in particular for large loads.
Comparison with recent test data
Number of Pads, N 5
Load Configuration Load on Pad
Pad Arc Angle, P 60º
Offset 0.5
Rotor Diameter, D 101.59mm (4.0 in)
Bearing Axial Length, L 55.88 mm (2.20 in)
Pad Radial Clearance, CP 120.65 μm (4.75 mil)
Bearing Radial Clearance, Cb 68 μm (2.67mil)
Bearing Preload, 0.44
Pad Mass, mp 0.44kg (0.97 lb)
Pad Inertia, IG 2.49 kg-cm2 ( 0.851 lb-in2)
Pad thickness, t 19.05mm (3.228inch)
Bearing pivot stiffness, Kp nonlinear, ~0.5GN/m
Bearing Lubricant DTE 797, ISO VG-32
Wilkes* five pad, rocker-back pivot, tilting pad bearing (LOP)
*Proceedings of ASME Turbo Expo 2011, Paper No. GT2011-46510
pr
Operating condition
Journal speed : 4,400 rpm
Unit load: 1566 kPa (227 psi)
Lubricant supply temperature :25 oC
Used pivot stiffness:
Pivot radial stiffness: 2 GN/m
W
Y
X
0
200
400
600
800
1000
1200
0 100 200 300
Excitation frequency (Hz)
Rea
l par
t of
th
e im
ped
ance
s
(M
N/m
)c
Re (Zxx)-prediction
Re (Zyy)-prediction
Re (Zxx)-measurement
Re (Zyy)-measurement
MN
/m d
Predicted & Test impedances versus frequency
Measured Predicted
X 0.009 0.006
Y -0.381 -0.306
Dimensionless
Eccentricity
K-C model:
Z=K + iωC
Stiffness: K=Re (Z)
Damping:C=Im (Z)/ ω
W
Y
X
Real part of impedances
Re (ZYY)-prediction
Re (ZYY)-measurement
Re (ZXX)-measurementRe (ZXX)-prediction
Dynamic stiffness KYY over predicted
Predicted & Test impedances versus frequency
Imaginary part of impedances
W
Y
X
Im (ZYY)-measurement
Im (ZYY)-prediction
Im (ZXX)-measurement
Im (ZXX)-prediction
Both damping coefficients are underpredicted.
Conclusions
• Updated XLTRC2 XLPRESSDAM code works for TPJBs with a near rigid pivot stiffness
• Predictions agree with published predictions for ideal, rigid pivot, TPJB.
• Comparisons with recent TPJB impedance data vs frequency, show damping coefficients are largely
underpredicted while the off-load stiffness coefficients is over predicted. Test results at odds with prior
test data. Current code used pivot stiffness ~ 4 times magnitude of that in test bearing.
Proposed work for 2nd year1.Complete analysis of reduced frequency force coefficients for TPJBs
for NONLINEAR pivot stiffness depending on the type of contact.
2. Derivation of iterative search scheme to update the pad radial and transverse deformations and ensure reliable convergence to an equilibrium solution.
3. Implementation of various oil feed arrangements in the FE model to model TPJBs with leading edge groove supply systems and scrapers.
4. Comparison of predictions from the enhanced TPJB code to test data for various bearing geometries tested by Childs and students and preparation of a technical report (MS. Thesis).
TRC Budget
Year II
Support for graduate student (20 h/week) x $ 1,800 x 12 months $ 21,600
Fringe benefits (0.6%) and medical insurance ($191/month) $ 2,419
Travel to (US) technical conference $ 1,200
Tuition three semesters ($3,802 x 9 ch) $ 10,132
Office (PC & HD storage) $ 200
(2011-12) Year II $ 35,558
(2010-11) Year I $ 34,863
End product (code) will enable TRC members to model modern TPB configurations and to improve predictions of dynamic forced response (K-C-M model)
Code for Tilting Pad Bearings
Questions (?)