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Transcript of 1 A Novel Computational Model for Tilting Pad Journal Bearings with Soft Pivot Stiffness Yujiao Tao...
1
A Novel Computational Model for Tilting Pad Journal Bearings
with Soft Pivot Stiffness
Yujiao TaoResearch Assistant
Dr. Luis San AndrésMast-Childs Professor
32nd Turbomachinery Research Consortium Meeting
TRC 32514/15196B
Year II
May 2012
22
JustificationPivot flexibility reduces the force coefficients in heavily loaded tilting pad journal bearings (TPJBs).
XLTRC2 TFPBRG code shows poor predictions for TPJB force coefficients.
W, static load
Y
X
Housing
Pad
Pivot
Fluid film
Journal
, Journal speed
, Pad tilt angle
Research objective: To develop a code, benchmarked by test data, to predict
the K-C-M coefficients of TPJBs. Code accounts for thermal energy transport and the (nonlinear) effects of pivot flexibility.
33
Tasks completed• Completed derivation of reduced frequency
force coefficients for TPJBs
• Developed iterative search scheme to update pad radial and transverse displacements
• Constructed GUI for new TPJB code as per XLTRC2 standards
• Compared predictions from the TPJB code to test data
44
• Completed derivation of reduced frequency force coefficients for TPJBs
Includes pivot NL deformations
• Developed sound iterative search scheme to update the pad radial and transverse displacements
• Constructed GUI for new TPJB code as per XLTRC2 standards
• Compared predictions from the enhanced TPJB code to test data
Tasks completed
55
cos sin
cos sin
p X Y
piv p p piv d p p
h C e e
r R
Film thickness in a padCp : Pad radial clearance
CB = Cp-rp Bearing assembled clearance
Rd= Rp+t : Pad radius and thickness
rp : Pad dimensional preload
p : Pad tilt angle
pivpivPivot radial and transverse
deflections
Unloaded Pad
Journal
P
X
Y
p
p
h
e
RJ
WX
RB
piv
OB
RP
WY
OP
piv
P’
p
t
L
Bearing CenterPad CenterFluid film
Loaded Pad
Film thickness:
66
• Laminar flow• Includes temporal fluid inertia
effects• Average viscosity across 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
Reynolds equation for thin film bearing
Y
X
Housing
Pad
Pivot
Fluid film
Journal
, Journal speed
W, static load
77
Thermal energy transport in thin film flows
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
Neglects 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)
8
R
Upstream pad
Journal
8
Pad inlet thermal mixing coefficientInlet thermal mixing coefficient (0<<1) is empirical parameter.
in s hF F F
in in s s h hF T F T F T
Downstream pad
Fh
Th
Fs Ts
Fin
Tin
Fs , Fh , Fin Volumetric flow rates
Ts , Th , Tin Fluid flow temperatures
~0.6-0.9 for conventional lubricant feed arrangements with deep grooves and wide holes.
small (<< 1) for TPJBs with LEG feed arrangements and scrappers.
Hot oil Mixing oil
Cold oil
Is constant for all conditions?
99
Nonlinear pivot deflection & stiffnessPivot deformation is typically nonlinear depending on the load (Fpiv), area of contact, hardness of the materials, surface conditions.
• Sphere on a sphere
Pivot and housing:Ep , Eh Elastic modulus p ,h Poisson’s ratiosDp , Dh Diameter of the curvature L Contact length
22 2
23
1 11.04 P H H P
piv pivP H P H
D DF
E E D D
2
2
2 (1 ) 4 ( )2ln
3 2.15piv P H H P
pivpiv
F LED D D D
LE F
2 22 2
331 1
0.52 1pivH P P H
pivH P P H
FD D
D D E E
1 piv
piv pivK F
2 3 4
0 1 2 3 4piv piv piv piv pivF a a a a a piv
pivpiv
FK
• Sphere on a cylinder
•Cylinder on a cylinder
• Load-deflection function (empirical)*Kirk, R.G., and Reedy, S.W., 1988, J. Vib. Acoust. Stress. Reliab. Des., 110(2), pp. 165-171.
1010
Tasks completed• Completed derivation of reduced frequency
force coefficients for TPJBs
• Developed sound iterative search scheme to update the pad radial and transverse displacements
• Constructed GUI for new TPJB code as per XLTRC2 standards
• Compared predictions from the enhanced TPJB code to test data
Convergence to the pad and journal equilibrium positions
1111
Iterative scheme to find pad equilibrium position
Journal displacement, pivot radial and transverse displacements
converge to equilibrium solution for TPJB with flexible pivots
Flow chart of iterative scheme
Set initial journal center displacements
Check convergence on loads (W0)
Find tilt angle for kth pad
Estimate the initial pivot radial displacement for kth pad
Find pivot radial and transverse displacements for
kth pad
Find tilt angle for kth pad
Update e
Check convergence on kth pad tilt angle and pivot
displacements
End of procedure
Check convergence on loads
(W0)
Update e Take TPJB with flexible pivots, find pads tilt angles, radial and transverse displacements and journal eccentricity
Estimate the pivot radial displacement and journal eccentricity
Take TPJB with rigid pivots, find pads tilt angles and journal eccentricity
1212
Tasks completed• Completed derivation of reduced frequency
force coefficients for TPJBs
• Developed sound iterative search scheme to update the pad radial and transverse displacements
• Constructed GUI for new TPJB code as per XLTRC2 standards
• Compared predictions from the enhanced TPJB code to test data
Fortran program and Excel GUI
1313
Excel GUI and Fortran code• Modifications/enhancements to XLPRESSDAM® code • FEM to solve Reynolds equation (hydrodynamic pressure)• Uses control volume method to solve energy transport eqn
Excel GUI
Parameters of pivot: type, radii of contact & material properties (E
Different pads: geometry parameters
•Sphere on a sphere•Cylinder on a cylinder•Sphere on a cylinder•Rigid pivot•Load-deflection function
1414
Tasks completed• Completed derivation of reduced frequency
force coefficients for TPJBs
• Developed sound iterative search scheme to update the pad radial and transverse displacements
• Constructed GUI for new TPJB code as per XLTRC2 standards
• Compared predictions from TPJB code to test data
Bearings tested by Childs and students(TurboLab)
1515
Predictions for a four-pad TPJB(Childs and Harris*) Four pad, tilting pad bearing (LBP)
*Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
Number of pads, Npad 4
Configuration LBPRotor diameter, D 101.6 mm (4 inch)Pad axial length, L 101.6 mm (4 inch)
Pad arc angle, P 73o
Pivot offset 65%Pad preload, 0.37, 0.58
Nominal bearing clearance, CB 95.3 m (3.75 mil)
Measured bearing clearance, CB
54.6 m (2.15 mil)99.6 m (3.92 mil)
Pad inertia, IP 7.91kg.cm2 (2.70lb.in2)
Oil inlet temperature ~40 oC (104 oF)Lubricant type ISO VG32, DTE 797
Oil supply viscosity, 0 0.032 Pa.s
Pr
Specific load, W/LD 0 kPa-1,896 kPa (275 psi)Journal speed, 4 krpm-12 krpm
Cold conditions
16
CB=55 m
16
Predictions for a four-pad TPJB
*Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX.
Specific load, W/LD 1.9 MPa (275 psi)Journal speed, 4 krpm-12 krpm
Lubricant arrangements: Spray bar blocker, By pass cooling
Pivot type: Ball-in-socket pivot
Measured pivot stiffness: 350 MN/m
Nominal cold bearing clearance
CB=100 m
CB=55 m
CB=100 m
Measured cold bearing clearances on Pads #1 and #3 are ~40% smaller than the nominal cold clearance.
X
Y
2.45o
p=73o
W
Measured cold bearing clearance
CB=95 m
Pad 1
Pad 4
Pad 2
Pad 3
X
Y
W
Pad 1
Pad 4
Pad 2
Pad 3
1717
0
20
40
60
80
100
0 400 800 1200 1600 2000Specific Load (kPa)
Jo
urn
al D
isp
lac
em
en
t (
m)
Journal eccentricity vs. static load
-eY
eX
Rotor speed =10 krpm
Rotor speed =6 krpm
Predicted journal eccentricity correlates well with measurements.
X
Y
W X
Y
W
Symbols: test dataLines: prediction
*Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
0
20
40
60
80
100
0 400 800 1200 1600 2000
Specific Load (kPa)
Jo
urn
al D
isp
lac
em
en
t (
m)
-eY
eX
Max. 275 psi
1818
0
10
20
30
40
0 400 800 1200 1600 2000Specific Load (kPa)
Te
mp
era
ture
Ris
e a
t P
ad
Tra
ilin
g E
dg
e (
oC
)Film temperature rise vs. static loadmeasured pad sub-surface temperature rise at pad trailing edge
Rotor speed =6 krpm
X
Y
W
Pad 3
Pad 2
Pad 4
Pad 1
Trailing edges
Input: inlet thermal mixing coefficient =0.5
At 6 krpm, film temperature rises at pad trailing edges are
considerable even with no load applied
Pad 1Pad 2Pad 3
Pad 4
*Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX.
Symbols: test dataLines: prediction
Max. 275 psiOil Inlet temperature ~40oC
1919
0
10
20
30
40
0 400 800 1200 1600 2000Specific Load (kPa)
Te
mp
era
ture
Ris
e a
t P
ad
Tra
ilin
g E
dg
e (
oC
)
Film temperature rise vs static loadmeasured pad sub-surface temperature rise at pad trailing edge
X
Y
W
Pad 3
Pad 2
Pad 4
Pad 1
Trailing edges
Input: inlet thermal mixing coefficient =0.95
Film temperatures underpredicted at 10 krpm.
Film heats little with load
Rotor speed =10 krpm
Pad 1Pad 2Pad 3
Pad 4
*Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX.
Symbols: test dataLines: prediction
Max. 275 psi
Effectiveness of spray bar blocker diminishes
Oil Inlet temperature ~40oC
20
200
400
600
800
1000
0 400 800 1200 1600 2000
Specific Load (kPa)
TP
JB
Sti
ffn
es
s C
oe
ffic
ien
ts
(MN
/m)
200
400
600
800
1000
0 400 800 1200 1600 2000
Specific Load (kPa)
TP
JB
Sti
ffn
es
s C
oe
ffic
ien
ts
(MN
/m)
20
Stiffness coefficients
Rotor speed =10 krpm
Rotor speed =6 krpm
=0.5
=0.95
KYY
KXX =KYY, TFBBRG
Kpiv=350 MN/m
Kpiv=350 MN/m
Very soft pivot produces
~ constant K’s, invariant with
load and speed.
X
Y
W X
Y
W
Symbols: test dataLines: prediction
*Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
TFBBRG code in XLTRC2-TPJB model with rigid pivotKXX
Prediction KXX=KYY
KYY
KXX =KYY, TFBBRG
KXX
Max. 275 psi
2121
Rotor speed =10 krpm
Rotor speed =6 krpm
=0.5
=0.95Soft pivot renders
nearly constant damping coefficients. Good correlation with
test data
Damping coefficientsSymbols: test dataLines: prediction
*Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
CYY
CXX =CYY, TFBBRG
CXX
CYY
CXX =CYY, TFBBRG
CXX
X
Y
W X
Y
W
Synchronous speed coefficients
Prediction CXX=CYY
0
200
400
600
800
1000
0 400 800 1200 1600 2000
Specific Load (kPa)
TP
JB
Da
mp
ing
Co
eff
icie
nts
(k
N.s
/m)
0
200
400
600
800
0 400 800 1200 1600 2000
Specific Load (kPa)
TP
JB
Da
mp
ing
Co
eff
icie
nts
(k
N.s
/m)
Max. 275 psi
2222
Predictions for a five-pad TPJB(Wilkes and Childs*) Five pad, tilting pad bearing (LOP)
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Pr
Specific load W/LD: 3,132 kPa (454 psi)Journal speed 4.4 krpm-13.1 krpm
X
Y
Pad
W
Journal
Fluid film
2323
0
200
400
600
800
1000
0 5 10 15 20
Pivot Radial Deflection (m)
Piv
ot
Ra
dia
l Sti
ffn
es
s
(MN
/m)
0
2
4
6
8
10
12
14
0 5 10 15 20
Pivot Radial Deflection (m)
Piv
ot R
ad
ial F
orc
e (
kN)
Pivot stiffness & hot bearing clearance
Pivot load-deflection function
Pivot stiffness
Hot bearing clearance
Rocker back pivotPivot stiffness-deflection function
CB,cold-CB,hot=(Thot-Tcold)Bearing clearance decreases due to thermal expansion of the rotor and pad surfaces.
=0.396 m/oC
Pivot radial force
Pivot radial stiffness
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
EXPERIMENTAL
Hot bearing clearance CB: 48 m~58 m
Nominal cold CB=68 m
Empirical
2424
Pad bending stiffness
Pad bending stiffness*
kpad =5.4644×104Mp+1.1559×105 (N.m/rad)
MP ,Pad bending moment from fluid film
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
/2 /22 2
/2 /2
sin sinP T
L P
L L
p p p
L L
M P R d dZ P R d dZ
Equivalent pad-pivot stiffness: series pivot + pad bending
21 1
eq piv pad
l
k k k
12 P Pl R
Derived from experiments
Pad ½ length
RP
MPMP
OP
Used in code TPJB
25
0
20
40
60
80
0 500 1000 1500 2000 2500 3000 3500
Specific Load (kPa)
Jo
urn
al D
isp
lac
em
en
t (
m)
0
20
40
60
80
0 500 1000 1500 2000 2500 3000 3500
Jo
urn
al D
isp
lac
em
en
t (
m)
25
Journal eccentricity slightly under/over predicted at the
low/high rotor speeds.
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Journal eccentricity vs. static load
Symbols: test dataLines: prediction
-eY
-eY
Max. 454 psi
X
Y
W
Rotor speed =13.1 krpm
Rotor speed =4.4 krpm
cB= 57m<-eY
cB= 49m~-eY
=0.8
=0.9
cold CB=68 m
26
Rotor speed =13.1 krpm
0
200
400
600
800
0 500 1000 1500 2000 2500 3000 3500Specific Load (kPa)
TP
JB S
tiffn
ess
Co
effic
ien
ts
(MN
/m)
0
200
400
600
800
0 500 1000 1500 2000 2500 3000 3500
TP
JB S
tiffn
ess
Co
effic
ien
ts
(MN
/m)
26
Static stiffness coefficients over
predicted at large specific loads
KYY
KXX
KYY
KXX
K-C-M model
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Static stiffness coefficientsSymbols: test dataLines: prediction TPJBDashed: Wilkes preds.
Max. 454 psi
X
Y
W
Rotor speed =4.4 krpm=0.8
=0.9
27
0
100
200
300
400
0 500 1000 1500 2000 2500 3000 3500Specific Load (kPa)
TP
JB
Da
mp
ing
Co
eff
icie
nts
(k
N.s
/m)
0
100
200
300
400
0 500 1000 1500 2000 2500 3000 3500Specific Load (kPa)
TP
JB
Da
mp
ing
Co
eff
icie
nts
(k
N.s
/m)
27
Damping coefficients vary little with static load.
CYY
CXX
CYY
CXX
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Damping coefficients
X
Y
W
Max. 454 psi
=0.8
=0.9Rotor speed =13.1 krpm
Rotor speed =4.4 krpm
Synchronous speed coefficients
Symbols: test dataLines: prediction TPJBDashed: Wilkes preds.
28
-150
-100
-50
0
50
100
0 500 1000 1500 2000 2500 3000 3500
Vir
tua
l Ma
ss
Co
eff
icie
nts
(k
g)
-150
-100
-50
0
50
100
0 500 1000 1500 2000 2500 3000 3500
Specific Load (kPa)
Vir
tua
l Ma
ss
Co
eff
icie
nts
(k
g)
28
Large negative virtual masses at 4.4 krpm.
Dynamic stiffness increases with frequency.
MYY
MXX
MYY
MXX
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Virtual mass coefficients
X
Y
W
Rotor speed =13.1 krpm
Rotor speed =4.4 krpm=0.8
=0.9
Symbols: test dataLines: prediction TPJBDashed: Wilkes pred.
29
0
100
200
300
400
500
600
700
800
0 50 100 150 200 250 300 350Excitation Frequency (Hz)
Re
al p
art
of
imp
ed
an
ce
c
oe
ffic
ien
ts R
e(Z
) (M
N/m
)
0
100
200
300
400
500
600
700
800
0 50 100 150 200 250 300 350Excitation Frequency (Hz)
Re
al p
art
of
imp
ed
an
ce
c
oe
ffic
ien
ts R
e(Z
) (
MN
/m)
29
Real part
At 4.4 krpm, dynamic stiffness HARDENs at frequencies 2.
Re( ZXX)
Re( ZYY)
Synchronous frequency
Re( ZXX)
Re( ZYY)Synchronous frequency
Re(Z)=K-M2
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Impedances Specific load = 227psi
X
Y
W
Rotor speed =13.1 krpm
Rotor speed =4.4 krpm
Symbols: test dataLines: prediction TPJBDashed: Wilkes pred.
30
0
100
200
300
400
500
0 50 100 150 200 250 300 350Excitation Frequency (Hz)
Ima
gin
ary
pa
rt o
f im
pe
da
nc
e
co
eff
icie
nts
Im (
ZX
X)
(MN
/m)
0
100
200
300
400
500
0 50 100 150 200 250 300 350Excitation Frequency (Hz)
Ima
gin
ary
pa
rt o
f im
pe
da
nc
e
co
eff
icie
nts
Im(Z
) (
MN
/m)
30
At 4.4 krpm, predicted damping is frequency dependent for >2
Im( ZXX)
Im(ZYY)
Synchronous frequency
Im( ZXX)
Im( ZYY)Synchronous frequency
Ima(Z)=C
*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Impedances
Rotor speed =13.1 krpm
Rotor speed =4.4 krpm
Imaginary part
Specific load = 227psi
X
Y
W
Symbols: test dataLines: prediction TPJBDashed: Wilkes pred.
3131
Predictions for other test TPJBs
Kulhanek and Childs, 2012, ASME, J. Eng. Gas Turbines Power, 134, 052505 1-11.• TPJB code delivers good predictions for stiffness and damping by estimating the (actual) hot bearing clearances and using a constant pivot radial stiffness.
In the works. Will be part of forthcoming Y. Tao M.S. Thesis (TRC report 2012)
Delgado et al., 2010, ASME Paper GT2010-23802• TPJB code takes TPJB pivots as rigid and uses hot bearing clearances.• Predicted stiffness and damping correlate well with test data (45 psi).
Tschoepe and Childs, 2012, not yet published• TPJB code uses measured pivot load-deflection function and hot bearing clearances.• Predicted stiffness and damping are in agreement with test TPJB data.
3232
Conclusions
• For TPJBs with very soft pivots (Kpiv<<Kfilm), pivot stiffness determines bearing stiffness.
• Film temperatures at no load condition are high. At high rotor speeds (> 10 krpm), LEG and spray bar blockers have less effectiveness in cooling a bearing.
• Bearing & pad clearances change due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea.
•A-priori knowledge of pivot stiffness and bearing & pad clearances is required to obtain accurate predictions of TPJB performance.
• Bearing & pad clearances change a lot due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea.
3333
2012 Proposal to TRC (2 years)
Hydrodynamic pressure P Pad surface elastic & thermal deformations change bearing & pad clearances
K = P + C∆TK, Pad stiffness matrix
P, Fluid film pressure vector
C, Mechanical-thermal stiffness matrix
, Pad displacement vector
Objective: Enhance TPJB code to accurately predict pad surface deformations
Hot oil flow
Pivot constraint
FE pad structural analysis by Yingkun Li
3434
Proposed work 2012-2013
• Build a 3-D FE model of commercial pads (ANSYS® or
SolidWorks®) to obtain pad stiffness matrix. Reduce model with active DOFs, perform structural modal analysis for easy off-line evaluation of pad surface deformations and pivot deflections.
• Implement oil feed arrangements (LEG, spray bar blockers etc.) in the FE model
• Construct new Excel GUI and Fortran code for XLTRC2
• Digest more test data and continue to update predictions using enhanced code.
35
TRC Budget
Year III
Support for graduate student (20 h/week) x $ 2,200 x 12 months $ 26,400
Fringe benefits (0.6%) and medical insurance ($197/month) $ 2,522
Travel to (US) technical conference $ 1,200
Tuition & fees three semesters ($227/credit hour) $ 9,262
Other (PC+software+storage supplies) $ 1,600
2012-2013 Year III $ 40,984
Enhanced TPJB code will model current (commercial) TPJBs and improve predictions of force coefficients with minimum User expertise for specification of empirical parameters.
2012-2013 Year III
3636
Questions (?)