1

TRC 2008

The Effect of (Nonlinear) Pivot Stiffness on Tilting Pad

Bearing Dynamic Force Coefficients – Analysis

Jared GoldsmithResearch Assistant

Dr. Luis San AndresMast-Childs Professor

TRC Project 32513/1519 T3

2

XLTRC2

Project Goals

Enhance tilting pad bearing model by including nonlinear pivot flexibility for rocker, spherical, and flexure type pivots

3

Tilting Pad Journal Bearing Pivot Types

Tilting Pad Bearings

Y

X

PAD ROTATION

Y

X

PAD ROTATION

Y

X

PAD ROTATION

FLEXURE WEB

4

pivot

P

l

Y

X

e

journalpad

t

pivot

P

l

Y

X

e

journalpad

t

Film Thickness

Tilting pad journal bearing & coordinates

pivot

pad

pivot

pad

rotational stiffness

radial stiffness

PK

PK

)sin()()cos()()sin()cos( pkk

ppk

YXpk RreeCh

XoW

YoW

journal speed

film thickness

Film thickness:

Pad clearance ( ) and preload ( ) and journal eccentricity ( )pC pr YX ee ,

Pad angular rotation ( ), radial ( ) and transverse displacement ( ) fork k k padNk ,..1

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Perturbation Analysis

Small amplitude journal motions about an equilibrium position

YoXo WW ,

YoXo ee ,

koP kohthk k

oko

ko ,,

tiXXoX eeete )( ti

YYoY eeete )(

tikko

k et )(

tikko

k et )(

tikko

k et )(

padNk ,..1

Applying an external static load with components ( ) to the journal determines its static equilibrium position ( ) with fluid

static pressure field , film thickness , and corresponding

equilibrium pad rotation and deflections ( )

Consider small amplitude journal center motions ( ) of

frequency about the static equilibrium point. Hence

YX ee ,

Consider small amplitude journal and pad motions about static equilibrium position (SEP)

and

for

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Load and Pad EOMs

Bearing forces and Pad Equations of Motion

k

k

k

kP

kP

kP

k

k

k

kpad

F

F

M

F

F

M

M

padN

k

kX

tiXXoX FeWWW

1

padN

k

kY

tiYYoY FeWWW

1

The sum of the pad fluid film forces balance the external load applied on the journal, i.e.,

kkk

kkk

kkkkkP

kpad

mcm

mbm

cmbmI

M

0

0

Force and Moment EOMs for pad:thk

Matrix representing pad inertia and mass

Y

X

Journal Rotation

XoW

YoW

Bearing

Fluid film

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Nonlinear Pad Pivot

Typical Nonlinear Pivot

Radial Force

PK

PF

oPK oPF

Po

PF

P

Radial Deflection

)( fFP

PoPoPP KFF

)( PooP fF

PoPK

The assumption of small amplitude motions about an equilibrium position allows the pivot reaction radial force to be expressed as

where

is the static load on the pivot and

is the force due to radial displacement

Consider a typical nonlinear force ( ) versus pivot radial deflection ( ) in a bearing pivot

PF

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Pad Forces and Moment

Forces and moments acting on a pad

Pad Fluid Film Forces = integration of hydrodynamic pressure fields on pad

R

L

kkl

kl

L

L

kk

kY

kX dzRdPF

F

sin

cos

ti

k

k

kY

X

YX

YYYYXYY

XXXXYXX

ko

kYo

kXo

k

kY

kX

e

e

e

ZZZZZ

ZZZZZ

ZZZZZ

M

F

F

M

F

F

kpP

kXP

kY

k FRFFtRM ]sincos)[(

Moment on Pad:

Substitution of zeroth and first order pressure fields gives

Fluid film impedances: },{ kkk CiKZ ,,,,, YX

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Reduced Force Coefficients

Frequency reduced force coefficients for tilting pad bearing

pad

RR

RR

N

k

kb

kfP

ka

kXYRR

YYYX

XYXXR ZZZZCiK

ZZ

ZZZ

1

1 ][]][[][

where

and

Assuming the pads move with the same excitation frequency ω as the journal whirl frequency, the frequency reduced coefficients are

k

YYYX

XYXXkXY ZZ

ZZZ

k

YYY

XXXka ZZZ

ZZZZ

k

YX

YX

YX

b

ZZ

ZZ

ZZ

Z

k

kc

ZZZ

ZZZ

ZZZ

Z

][][][][][ 2 k

masskc

kpivot

kpivot

kfP MZCiKZ

Matrices representing pivot stiffness and damping coefficients

k

PP

P

PPkpivot

KK

K

KK

K

0

00

0

][

k

PP

P

PPkpivot

CC

C

CC

C

0

00

0

][

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Progress

Tilting pad bearing pivot

Modified tilting pad bearing model now accounts for spherical and rocker nonlinear pivot stiffness

0

100

200

300

400

500

0 2 4 6 8 10

Load (kN)

Piv

ot S

tiffn

ess

(MN

)

Spherical pivot stiffness versus load

Pivot

Pivot housing

PF Assumptions:

•Spherical pivot – point contact

•Rocker pivot – line contact

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Test Bearing

Test bearing description

Y

X

PAD ROTATION

Carter and Childs* five pad, rocker pivot, tilting pad bearing (LBP) and (LOP)

2m

Bearing Parameters Values

Rotor diameter 101.59 mm

Pad axial length 60.33 mm

Pivot offset 60%

Pad number (arc length) 5 (57.87)

Radial pad clearance .1105 mm

Pad inertia 2.49E-4 kg-

Preload 0.282

Radial bearing clearance .0792 mm

Mobile DTE ISO 32

31 cSt

5.5 cSt

Viscosity @ 40° C

Viscosity @ 100° C

Density @ 15°C

Specific heat

850 kg/m3

1951 J/(kg-K)

Fluid Properties

* ASME Paper No. GT2008-5069

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(LBP) Rotordynamic Force Coefficients

Experimental and predicted direct stiffness and damping coefficients

Carter and Childs measured and predicted nonsynchronous direct stiffness

Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP)

0

100

200

300

400

500

600

700

800

0 5 10 15

Load [kN]

Sti

ffn

es

s [

MN

/m]

Kxx

Kxx Th

Kyy

Kyy Th

The TPB model (rigid pivot) generally over predicts stiffness and damping coefficients

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Static Results

Direct static stiffnesses versus load

Original XLTRC2 (rigid pivot) and modified XLTRC2 (flexible pivot) predicted direct static stiffnesses versus static load

0

50

100

150

200

250

300

0 2 4 6 8

Load [kN]

Dir

ect

Sti

ffn

ess [

MN

/m]

Kxx - Flexible Pivot

Kxx - Rigid Pivot

Kyy - Flexible Pivot

Kyy - Rigid Pivot

Flexible pivot

Rigid pivot

Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP)

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-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 1 2 3 4 5 6 7

Load [kN]

Ey/

Cp

Flexible Pivot

Rigid Pivot

Static Results

Journal eccentricity versus load

Original XLTRC2 (rigid pivot) and modified XLTRC2 (flexible pivot) predicted journal eccentricity versus static load

Flexible pivot

Rigid pivot

Rotor speed = 4000 RPM Bearing loaded in –Y direction (LBP)

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Future work

Future work

•Perform extensive comparisons between predictions and Childs et al. experimental TPB stiffness and damping coefficients

•Include pivot friction for spherical pivots

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Statement of work

Future work

1. Account for pad clearance variations due to thermal and mechanical deformation effects

2. Improve I/O operations for Excel interface

3. Implement informed eccentricity “guess” for starting calculations