paper gt2017-64822 a f starvation model for s … san andres trc... · 2020. 12. 12. · paper...

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

QQ

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


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



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 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


(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



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


preload, rp 0.0

Pivot offset 0.5

Lubricant temp. at supply 45 oC

Viscosity, 18 cPoise

Density, 860 kg/m3


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)


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)


Temperature

•Starved (3-pads, 88% flow)

•Flooded (3-pads)

•Flooded (5-pads)


•Flooded (3-pads)

•Flooded (5-pads)


•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)


0

500

1000

1500

2000

2500

3000

3500

50 60 70 80 90 100

Stiff

ne

ss (

MN

/m)


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


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


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


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


Am

plit

ud

e o

f re

sp

on

se (

nm

/N)

1

10

100

1000


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


Damping ratio

0

2

4

6

8

10

12

14

16

18

85 90 95 100

Na

tura

l fr

eq

ue

ncy (

Hz)


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

paper gt2017-64822 a f starvation model for s … san andres trc... · 2020. 12. 12. · paper...

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