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 (?)