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1
A FLOW STARVATION MODEL FOR TPJBS
AND EVALUATION OF FRFS: A CONTRIBUTION
TOWARDS UNDERSTANDING THE ONSET OF
LOW FREQUENCY SHAFT MOTIONS
Luis San AndrésMast-Childs Chair Professor
Texas A&M University
Makoto HemmiSenior Researcher
Hitachi, Ltd. R&D Group
Bonjin KooGraduate Research Assistant
Texas A&M University
Paper GT2017-64822
Proceedings of ASME Turbo Expo 2017: Turbomachinery Technical Conference and
Exposition, June 26-30, 2017, Charlotte, NC, USA
Funded by Hitachi, Ltd. R&D Group Accepted for journal publication
2
Tilting Pad Journal Bearings (TPJBs)
X
Y
Shaft rotation
speed, Ω
Static load, W
Bearing
housing
Pivot
Fluid film
Tilting pad journal bearings (TPJB) support rotors
with minimal destabilizing forces.
X
Y
Shaft rotation
speed, Ω
Static load, W
Pad
Pivot
Load between Pads (LBP) Load on a Pad (LOP)
Loaded
pads
Unoaded
pads
3
Leopard (1970), Mikula (1983-1988), Huart (1984): quantify improvement of direct lubrication methods - relative to flooded lubrication:
Power loss reduces up to 35%.
Bearing max. temperature reduces up to 25%.
Haragonzo (1990), Brockwell and DeCamillo (1994): Direct lubricated bearings without end seals (evacuated) operate with less drag and
have lower film & pad temperatures than flooded bearings.
Dmochowski and Blair (2006): Bearing power loss and temperature reduce as a result of the TPJB flow evacuation. Reduced oil flow has no effect
on min. film thickness but reduces bearing stiffness and damping.
Direct lubrication & bearing without end seals
Lower supply flow and faster oil evacuation
Churning loss ↓ Power loss and temperature↓
Direct lubrication methods
A too low oil flowrate (starvation) causes rotor sub
synchronous vibrations (SSV)
SSV
1X
4
SSV Issues due to lubricant starvation
DeCamillo et al. (2008) experimentally find.
• For low flow, evacuated TPJB shows less SSV than sealed ends bearing.
• At intermediate oil flow, SSV is pervasive in evacuated - direct lube TPJBs
• LOP configuration produces larger amplitude SSV than LBP.
• Shaft SSV correlates with vibrations of one of the pads. Unloaded pads
have the largest SSV but rarely correlate with shaft SSV.
• Adding side grooves suppresses SSV.
DeCamillo et al. (2008)
A model for flow starved
bearings
Reynolds and thermal energy transport
On kth pad,
3 2
2
212 2 12
k kk k k
kh hh h h
Pt t
2 2
212
12 2
k k k
v s
Jk k J
k
C h T Q
R RV U
h
V
Reynolds equation:
Thermal energy transport eqn.:
cos sin
cos sin
k
p X Y
k k k k k
p p p p J p
h C e e
r R
Film thickness
+ pivot and pad surface (mechanical and thermal) deformation
equations.
7
Effective arc length model for a lubricated pad
In a lubricant starved pad, fluid filling the gap between a pad and
the journal starts at angle θf. Pad has a lesser effective arc
length (and pivot arc).
Ω
sQ
hQ
Flow from
upstream padf Effective arc length
Starved film
*Concept introduced by He et al. (2005)
Flow distribution into each bearing pad
X
Y
Ω
W
es=0 (no load) supply flow
distributes evenly as
totali ss
pad
N
1
sQ
2
sQ
3
sQ
4
sQ
1
sQ2
sQ3
sQ4
sQ
total
sQ
Operation with load (W) journal eccentricity:• Unloaded pads (low pressure) → demand more lubricant
With a reduced supply flow, flow into each pad decreases while the flow
fraction si=Qi/Qtotal is constant; as per Dmochowski & Blair (2006) when noting that a reduction in flow has no effect on minimum film thickness.
Numerical model for flow starved
bearings
Solve flow equations numerically using FE
methods. Iterative search to satisfy inlet flow
and update effective arc length.
Read more:
San Andres & Tao [2013], San Andres & Li [2015]
ASME J. Eng. Gas Turbines Power
Equations of motion for rotor
supported on TPJBs
EOM for rotor-bearing system
In the frequency domain:
Degrees of freedom
lateral rotor displacements
Rotor-TPJB, including pad tilt angles
DOFs and equations of motion
Let
G(w) is a system complex stiffness matrix
,T
x yz
1, , , , padT
Nx y z
0
i te w Mz Cz Kz F
2
0 0 0i
ww w K M C z G z F
0
i te wz z
12
(four-pad TPJB example)
2 1 2 3 4
2 1 2 3 4
1 1 1 2 1
2 2 2 2 2
3 3 3 2 3
4 4 4 2 4
0 0 0
0 0 0
0 0 0
0 0 0
pad pad
pad pad
N N
i i
xx J xy x x x x
i i
N N
i i
yx yy J y y y y
i i
x y pad
x y pad
x y pad
x y pad
g M g g g g g
g g M g g g g
g g g I
g g g I
g g g I
g g g I
w
w
w
w
w
w
w
G
2 2
2 2
;
;
i i i i
xx xx xx xx x x x x
x x x x
g K i C M g K i C M
g K i C M g K i C M
w w w w
w w w w
G matrix for whole system
(Ki, Ci, Mi) are pad stiffness, damping and added mass coefficients [3 DOF (x, y, i) each]. G size = (2+Npad)
2
Note: computational model includes also pivot deformation and
pad transverse displacement (+ 2 DOF x Npad)
1, , , , padT
Nx y z
13
2
( )
XX J XYXX XX XY XY
YX YY JYX YX YY YY
M M MK i C K i C
M M MK i C K i Cw
w ww
w w
G
with KCM coefficients
with frequency reduced force coefficients (as per Lund):
2
( )
1 0
0 1
XX XX XY XY
J
YX YX YY YY
K i C K i CM
K i C K i C
w w w w
w w w w
w
w ww
w w
G
2-DOF rotor-bearing system
MJ = rotor mass and (K,C) are 2×2 bearing stiffness & damping coefficients.
2Re
Im
K i C K M
K i C i C
w w
w w
w w
w w
,T
x yz
Frequency reduced KCM model (Childs)
Frequency response functions (FRF) for
rotor-bearing system
FRFs show natural frequencies and damping ratios
for various modes of operation evidence sensitivity of a journal or pads to external (periodic) forces
15
Simple rotor-TPJB system
1, , , , padT
Nx y z
1
0 0w
z G F
MJ
KXX,CXX
KYY,CYYMpad, IpadK, C
Pad #1
Mpad, IpadK, C
Pad #2
Mpad, IpadK, C
Pad #3
Mpad, IpadK, C
Pad #4
2
0
0 0
i
w
w w
K M C z
G z F EOM
,T
x yz
1
0 0w
z G F 10 0
z G F
Full coefficient
Freq. reduced KCM
16
Effect of lubricant starvation
on bearing performance
(1) Brockwell et al. (1994)
X
Y
W5-pads LBP TPJB
TEST
Flow reduced to 40% nominal.
Measured pad temperatures.
Five-pad TPJB (LBP):Brockwell et al. (1994)
*Nominal flow rate=28 LPM
Diameter, D 98 mm
Diametrical clearance 304 µm
Length, L 38 mm
Pad arc length, Qp 56°
Pad preload 0.0
Pivot offset 0.6
Lubricant viscosity, 21 mPa-s
density, 856 kg/m3
supply temperature 49 oC
Rotor speed, 16,500 rpm
Applied load, W 5,338 N
Specific load, W/(LD) 1.4 MPa
X
Y
W
Pad 3
Pad 2
Pad 1
Pad 5
Pad 4
18
Flow starvation: temperature & pressure1X= 16.5 krpm, W/LD=1.4 MPa
50
55
60
65
70
75
80
85
90
95
100
0 90 180 270 360
100% (Measurement)100% Flow (Prediction)40% Flow (Measurement)40% Flow (Prediction)
Pad 1 Pad 2 Pad 3 Pad 4 Pad 5
Angular location (°)
Te
mp
era
ture
(°C
)
0
10
20
30
40
50
60
70
80
0 90 180 270 360
100% Flow
40% Flow
Pad 1 Pad 2 Pad 3 Pad 4 Pad 5
Angular location (°)
Pre
ssure
(b
ar)
X
Y
W
Pad 3
Pad 2
Pad 1
Pad 5
Pad 4As flow rate decreases, so does the
effective pad length and film peak
pressure.
Pad temperature increases.
Measured & predicted temperatures
agree.
fEffective pad length Effective pad length
19
Flow starvation: temperature & power loss
1X= 16.5 krpm, W/(LD)=1.4 MPa
As flowrate decreases,
Pad temperature increases
Power loss decreases
0
2
4
6
8
10
12
14
40 60 80 100Supply flow (%)
Pow
er
loss (
kW
)
30
35
40
45
50
40 60 80 100
Supply flow (%)
Max.
tem
p. rise (
°C)
Oil temp.
risePower loss
20
Effect of lubricant starvation
on bearing performance
X
Y
W
(2) Dimond et al. (2010)
4-pads LBP TPJB
Small journal size
Preloaded pads (0.3)
Center pivot
Four-pad TPJB (LBP): Dimond et al. (2010)
X
Y
W
*Nominal flow rate=45LPM
½ Rotor Mass MJ = 850 kg
* Bearing geometry adapted from Dimond(2009) and set MJ= ½ W/g.
Diameter, D 127 mm
Diametral clearance, CD=2CB 305 µm
Length, L 95.3 mm
Pad arc length, Qp 75°
Pad preload 0.3
Pivot offset 0.5
Lubricant viscosity, 33 mPa-s
density, 850 kg/m3
supply temperature 32 oC
Pad inertia, Ipad 716 kg-mm2
Rotor speed, 5,000 rpm
(83.3 Hz)
Applied load, W 8.3 kN
Specific load, W/(LD) 689 kPa
0
5
10
15
20
25
30
35
40
45
0 90 180 270 360
Cente
rlin
e P
ressure
(bar)
Angle (o)
22
Flow starvation: Pressure
Pad #3 Pad #4 Pad #2 Pad #1
41% of nominal flowrate
71% of nominal flowrate
Nominal flowrate
Flowrate ↓ Unloaded pads (Pads #1&2) do not generate
pressure Peak pressure: 71% flow < 41% flow
1X= 5 kpm (83.3 Hz)
23
How does the flow reduction affect the system natural frequency and damping ratio?
Flow starvation: force coefficients
Flow
(LPM) MN/m kN.s/m kg
45 (100%) 328 861 299
32 (71%) 243 356 358
18 (41%) 415 282 401
XX YYK K
XX YYC C
XX YYM M
KCM force coefficients
Bearing stiffness increases for lowest flow rate (41%) while damping decreases quickly with more flow starvation.
1X= 5 kpm (83.3 Hz)
24
Flow starvation: Natural frequency and damping ratio
1X= 5,000 rpm (83.3 Hz)
Natural frequency (Hz)
Modes
Flow
rate1,2 3,4 KCM
Frequency
Reduced
100% 66 66 85 80
71% 65 65 71 68
41% 86 86 92 88
Damping ratio
Modes
Flow
rate1,2 3,4 KCM
Frequency
Reduced
100% 0.48 0.48 0.70 0.65
71% 0.19 0.19 0.33 0.23
41% 0.15 0.15 0.20 0.17
With flow
starvation
system natural
frequency
increases and
damping ratio
quickly
decreases.
KCM model
overpredicts
damping and
nat frequency.
0
2
4
6
8
10
12
0 50 100 150 200
Frequency (Hz)
Am
plit
ud
eo
f re
sp
on
se (
nm
/N)
0
2
4
6
8
10
12
0 50 100 150 200Frequency (Hz)
Am
plit
ud
eo
f re
sp
on
se (
nm
/N)
0
2
4
6
8
10
12
0 50 100 150 200
Frequency (Hz)
Am
plit
ud
eo
f re
sp
on
se (
nm
/N)
FRF: amplitude/force |y /FY|
25
1X 1X
1X
KCM Frequency reduced (K,C)
All force coefficients model
As flow decreases,
damping ratio decreases
and natural frequency
increases (crosses 1X).
KCM model predicts the most
damping.
100% Flow
71% Flow
41% Flow71% Flow
71% Flow
100% Flow
100% Flow
41% Flow
41% Flow
26
Effect of lubricant starvation
on bearing performance
X
Y
W
(3) Commercial
turbo machine
SSV observed
under flow starvation.
5-pads LOP TPJB
Center pivot (0.5)
No preload pads
Heavy rotor (12t)
½ Rotor Mass MJ =11,980 kg
X
Y
Pad 3
Pad 2
Pad 4
W
Journal
Fluid film
Pad 5
Pad 1
Turbo machine: Rotor-TPJB (LOP)
Diameter, D 432 mm
Length, L 254 mm
Diametral clearance, 874 µm
Pad arc length, Qp 56°
preload, rp 0.0
Pivot offset 0.5
Lubricant temp. at supply 45 oC
Viscosity, 18 cPoise
Density, 860 kg/m3
Rotor speed, 3,600 rpm
Pad mass, Mpad 40 kg
inertia, Ipad 0.395 kg-m2
External radial load, W 117 kN
Specific load, W/(LD) 1.070 MPa
28
Known operation (pads w/o preload)
Top pads are unloaded. If fully
lubricated there is an increase in
drag power loss.
The large amount of flow pads 4&5
require (Qs) is a waste.
0
10
20
30
40
50
60
-10 80 170 260 350
Ce
nte
rlin
e P
ressu
re (
bar)
Angle (degrees)
Pad #1 Pad #2 Pad #3 Pad #4 Pad #5
No pressure
X
Y
WPad #1
Pad #2
Pad #3
Pad #4Pad #5
29
Known operation of bearing
In actual operation: flow supplied is lesser than one predicted (~ 50%)
The top (unloaded) pads take less flow, i.e. not fully flooded.
Delivered flow is enough to fill the three loaded pads: (3-
pads model require 0.5Qs)
X
Y
WPad #1
Pad #2
Pad #3
Pad #4Pad #5
X
Y
WPad #1
Pad #2
Pad #3
Pad #4
Predictions: 5-pad vs 3-pad (flooded and starved)
0
200
400
600
800
-10 80 170 260 350
Film
thic
kness (
m
)
Angle (degrees)
Pad 1 Pad 2 Pad 3 Pad 4 Pad 5
Film thickness
0
10
20
30
40
50
60
-10 80 170 260 350Angle (degrees)
Cente
rlin
e p
ressure
(bar)
Pad 1 Pad 2 Pad 3
Pad 4 Pad 5
Pressure
0
5
10
15
20
25
30
-10 80 170 260 350
Tem
p. ra
ise (
oC
)
Angle (degrees)
Pad 1 Pad 2 Pad 3 Pad 4 Pad 5
Temperature
•Starved (3-pads, 88% flow)
•Flooded (3-pads)
•Flooded (5-pads)
•Starved (3-pads, 88% flow)
•Flooded (3-pads)
•Flooded (5-pads)
•Starved (3-pads, 88% flow)
•Flooded (3-pads)
•Flooded (5-pads)
1X= 3.6 krpm, W/(LD)=1.1 MPa
5-pad and 3-pad models show
similar results.
Starved 3-pad model predicts ~0
pressure for pads #1 & #3.
Pad #2 is one generating
pressure, identical for three cases.
30
0
5
10
15
20
25
30
0 1 2 3 4
Flooded (5-Pads)
Flooded (3-Pads)
Starved (3-Pads, 88% flow)
Da
mp
ing
(M
N-s
/m)
Speed (krpm)
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 1 2 3 4
Flooded (5-Pads)
Flooded (3-Pads)
Starved (3-Pads, 88% flow)
Stiff
ne
ss (
MN
/m)
Speed (krpm)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4
Flooded (5-Pads)
Flooded (3-Pads)
Starved (3-Pads, 88% flow)D
am
pin
g (
MN
-s/m
)
Speed (krpm)
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4
Flooded (5-Pads)
Flooded (3-Pads)
Starved (3-Pads, 88% flow)Stiff
ne
ss (
MN
/m)
Speed (krpm)
31
K,C coefficients: 5-pad vs 3-pad (flooded and starved)
YYK
YYC
XXCXXK
Pad #2 has a full film albeit the other pads starve Constant KYY & CYYKYY >>KXX, CYY>>CXX. For 88% flow, PPads #1,3,4,5~0 bar (KXX,CXX)~0
0.6
0.7
0.8
0.9
50 60 70 80 90 100
Ecce
ntr
icity r
atio
(e
/Cp)
% of nominal flowrate
0
0.5
1
1.5
2
2.5
3
3.5
4
50 60 70 80 90 100
Da
mp
ing
(M
N-s
/m)
% of nominal flowrate
0
500
1000
1500
2000
2500
3000
3500
50 60 70 80 90 100
Stiff
ne
ss (
MN
/m)
% of nominal flowrate
32
e/Cp
KYY >> KXX ,CYY >>CXX
Predictions: 1 pad (starves) vs 3 & 5 pads
YYC
YYK
3-pads
5-pads
Flowrate ↓
reduction in Qs increases e
and KYY but decreases CYY.
For 1-pad, 5 & 3-pad
models: e, KYY, and CYY remain constant as Pad #2 is
flooded although other pads
are flow starved.
3-pads
5-pads
3-pads
5-pads
For fully flooded, 5-pad:Qs, 3-pad:⅟2Qs, 1-pad:⅟4Qs
33
Natural frequency (Hz)
Modes
1,2 3,4 5,6 7,8 9,10 11,12 13,14
5 Pads 9 9 17 60 114 117 180
3 Pads 17 61 111 116 181
1 Pad 60 179
Damping ratio
Modes
1,2 3,4 5,6 7,8 9,10 11,12 13,14
5 Pads 0.03 0.03 0.11 0.31 0.88 0.87 0.99
3 Pads 0.11 0.31 0.86 0.85 0.99
1 Pad 0.32 0.99
Top pads tilting
5-pad model 3-pad model 1-pad model
Natural frequencies and damping ratios for 5-pad, 3-pad and 1-pad
models – fully flooded condition (pads effective arc=actual arc)
X
Y
W
Journal#2
X
Y
W
#1#2
#3X
Y
W
#1
#2
#3
#4#5
Only 5-pads model
predicts top pads’ tilting
mode.
Low damping ratio
1-pad model predicts two
pairs of N.F.s for {y, 2}.
For same mode, predicted
N.F.s are ~same,
regardless of model.
34
Amplitude FRF: pads’ rotation | /FX|
5-pad model 3-pad model
5-pad model gives lowest natural frequency (9 Hz) with small
damping ratio (0.03) tilting of unloaded pads (4&5).
5-pad and 3-pad models identical amplitude for 2nd mode at
17 Hz tilting of loaded pads 1-3.
3-pad vs. 5-pad modelsFully flooded
1
10
100
1000
1 10 100Excitation frequency (Hz)
Am
plit
ude
of re
sp
on
se
(×
10
-9ra
d/N
)
Excitation frequency (Hz)Excitation frequency (Hz)Excitation frequency (Hz)
1X
Pad4
Pad5
Pad1
Pad2
Pad31
10
100
1000
1 10 100Excitation frequency (Hz)
Am
plit
ude
of re
sp
on
se
(×
10
-9ra
d/N
)
Excitation frequency (Hz)Excitation frequency (Hz)Excitation frequency (Hz)
1X
Pad1
Pad2
Pad3
Unloaded pads
Loaded pads Loaded pads
X
Y
W
#1
#2
#3
#4#5
35
Fully flooded
Pad resonance frequency (~ 9 Hz)
does not excite rotor.
DeCamillo (2008) noted pad flutter does not excite test rotor.
Predicted nat. frequency at 17 Hz is
slightly lower than measured SSV
frequency at ~20 Hz.
FRF: rotor displacements |x/FX| & |y/FY|3-pad vs. 5-pad models
0.1
1
10
1 10 100Excitation frequency (Hz)
Am
plit
ude
of re
sp
on
se
(nm
/N)
Excitation frequency (Hz)
5-Pad
1X3-Pad
0.00001
0.0001
0.001
0.01
0.1
1 10 100Excitation frequency (Hz)
Am
plit
ude
of re
sp
on
se
(nm
/N)
Excitation frequency (Hz)
5-Pad1X
3-Pad
5-pad
model
3-pad
model
X
Y
W
#1#2
#3X
Y
W
#1
#2
#3
#4#5
(x /FX) (y /FX)
Pads flutter
Pads flutter
Measured resonance
Measured resonance
1
10
100
1000
1 10 100Excitation frequency (Hz)
Am
plit
ud
e o
f re
sp
on
se (
nm
/N)
1
10
100
1000
1 10 100Excitation frequency (Hz)
Am
plit
ud
e o
f re
sp
on
se (
nm
/N)
36
|x /FX| |2 /FX|
FRF: pad 2 |x /FX| & rotation |2 /FX| 3-pad model
88% flow
90% flow100% flow
95% flow
95% flow
88% flow
90% flow
100% flow
Flow reduction
Flow reduction
X
Y
W
#1#2
#3
At 88% of nominal flow, the peak
amplitudes are unbounded as the
damping ratio is negative (z
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
85 90 95 100
Da
mp
ing
ra
tio
% of nominal flowrate
Damping ratio
0
2
4
6
8
10
12
14
16
18
85 90 95 100
Na
tura
l fr
eq
ue
ncy (
Hz)
% of nominal flowrate
Nat. Freq.
X
Y
Pad 3
Pad 2
W
Journal
Fluid film
Pad 1
Reducing flow to 90%
decreases nat. frequency and
damping ratio
(17→ 11 Hz, z 0.11→0.01).
As flow decreases (to 88%),
upstream and downstream
pads (1&3) starve
natural freq. drops to 5 Hz, Damping turns negative
SSV Hash
37
Natural frequencies and damping ratios for 3-
pad model – STARVED FLOW
Unstable
38
ConclusionA FLOW STARVATION MODEL FOR TPJBS AND
EVALUATION OF FRFS: A CONTRIBUTION
TOWARDS UNDERSTANDING THE ONSET OF
LOW FREQUENCY SHAFT MOTIONS
39
A model predicts the performance of TPJBs
operating with an oil supply flowrate well below its
design condition.
A dynamic force coefficients model constructs FRF
for a rotor supported on TPJBs
Example 1 (Brockwell) Predicted pads’ temperature
agrees well with measurements in a flow starved
bearing.
Conclusion Paper GT2017-64822
40
As supplied flowrate decreases:
(a) Pad temperature increases and power loss
decreases.
(b) The force coefficients rapidly change, and the
system N.F. and damping ratios drastically
decreases.
(c) Flutter of unloaded pads does not excite a rotor.
(d) LOP configuration is more sensitive to the
decrease in the flow rate than LBP TPJBs.
(e) Frequency reduced coefficients cannot predict
pads’ modes of vibration or onset of SSV from
flow starvation.
Conclusion Paper GT2017-64822
41
Thanks to Hitachi, Ltd. R&D Group
Questions (?)
Acknowledgment
Learn more: http://rotorlab.tamu.edu