2011 Model Tilting Pad Bearing 15196B 3

8/23/2019 2011 Model Tilting Pad Bearing 15196B 3

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Computational Model for Tilting Pad Journal

Bearings Yujiao Tao

Research Assistant

Dr. Luis San AndresMast-Childs Professor

TRC project 2010-2011 TRC 32514/15196B/ME

START DATE: September 1, 2010



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 beganmodifications

(d) Obtained initial predictions for a near-rigid

TPJB

Comprehensivetable summing

46 papers



Literature review

• Reviewed 46 papers on TPJBs (1964-2011) andprepared 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 padfluid film bearing (Shenoy B. S. andPai R.2009)



Literature review 46 papers on TPJBs (1964-2011)



Work to date


(b) Developed analysis for effect of pivotflexibility on force coefficients of TPJBs.

(c) Took XLPRESSDAM® code and began

modifications

(d) Obtained initial predictions for near-rigid

TPJB

Physicalmodel and

equationsfollow



Major assumptions:• Laminar flow• Includes temporal fluid inertia

effects• Average viscosity across the

film

3 3 2 2

2 2

1

12 12 2 12 J

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

R J : journal radius

JournalPad

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

, , C v : viscosity & density, specific heat

h B , h J : heat convection coefficients

T B , T

J : bearing and journal temperatures

: journal speed

Major assumptions:

Neglect temperaturevariations across-

film. Use bulk-flowvelocities and

temperature

22 2212

12 2

v B B J J C U h T W h T h T T h T T

R z

R RW U

h

CONVECTION + DIFFUSION= DISSIPATION(Energy Disposed) = (Energy Generated)

hhh

dyT h

T dyW h

W dyU h

U 000

;~1

;~1

;~1



h

cos sin cos sin p X Y piv p p piv d p ph C e e r R x h

Film thickness in a pad

cos

sin

p piv p p

piv d p p

h C r e

R e

x

h

x

h

C p : Pad radial clearance

Rd = R p+t : pad thickness

r p : pad dimensional preload

p : pad tilt angle

x piv , h piv : pivot radial and

transverse deflections

Y

θ p

h

e

W X

Pivot

Fluid film

Journal

O B

R P

W Y

O P ’

θ

X

P’

O P

xpiv

hpiv

p

X

Y

h

P

Pad



Journal static equilibrium in a TPJB

0 0 0

0 0 01

pad X

Y

k N X X

k k Y Y

F W F

W F F

sin cosd X p Y p d M R F F R F h

Fluid film moment on pad

0

0

2

0

2

cos

sin

k t

k l

L k k p X k

J k k LY p

F P R d dz

F

k =1,… Npad

j F x

j F h

F h

F x

M

Journal

X

Y

Pad

W Y

W X

h

x ’

p

piv F x piv F h

P

P ’

p

X

Y

h O p p piv

pad piv piv

piv piv

M M F F

F F

x x

h h

x

h

M

Pad equations of motionabout pivot point P

is pad mass matrix pad M



Perturbation analysis

• Consider small journal motion perturbationswith frequency (w) about the equilibrium

position , the journal displacements are:

0

0

( )

( )

X X X i t

Y Y Y

ee t e

ee t ee

w

• Journal motions induce changes in therotation of the k th pad and its pivotdisplacements z,h with the samefrequency (w)

0

0

0

( )

( )

( )

p p p

i t

piv piv piv

piv piv piv

t

t e

t

w

x x x

h h h

• And, journal and pad motions inducechanges in the film thickness and pressurefields

0

0

( )

( )

piv piv

piv piv

i t

X X Y Y p piv piv

i t

X 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

w

x h

w

x h

x h

x h



Reduced force coefficients

• 25 force impedances for the k th pad

12 R R

R R

XX XY R XY a s pad c b

YX YY

Z Z

Z Z w

Z Z Z Z M Z Z

XX XY

XY

YX YY

Z Z

Z Z

Z

X X X

a

Y Y Y

Z Z Z

Z Z Z

x h

x h

Z

X Y

b X Y

X Y

Z Z

Z Z

Z Z

x x

h h

Z

c

Z Z Z

Z Z Z

Z Z Z

x h

x xx xh

h hx hh

Z

/ 2

/ 2

l

l

L

L

Z P h Rd dz

, X, Y, x, h,

• The reduced force impedances are

s s s

s s s s

s s s

Z Z Z

Z Z Z

Z Z Z

x h

x xx xh

h hx hh

Z



Reduced force coefficients (in pad coordinates)

Alternatively, reduced impedances (ZR) are alsoobtained in pad local coordinates.

s s s

s s s s

s s s

Z Z Z

Z Z Z Z Z Z

x h

x xx xh

h hx hh

Z

12

R JP s pad P PJ w

Z Z Z Z M Z Z, , x h

2

d 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

hh hx hh x h

xh xx xh x xx xh

hh hx hh h hx hh

Z Z

d d

PJ

Z R Z R

Z Z

Z Z

hx hh

xx xh

hx hh

Z

d

JP d

Z R Z Z

Z R Z Z

xh xx xh

hh hx hh

Z

Z Z

Z Z

xx xh

hx hh

Z

According to the perturbation analysis, the reducedimpedances obtained by two methods are identical:

X

Y

h

R T

R Z A Z A



Work to date



flexibility on force coefficients of TPJBs.

(c) Took XLPRESSDAM ® code and beganmodifications

(d) Obtained initial predictions for near-rigid

TPJB

Fortranprogram 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 transportequation

• Program updated for ideal TPJB with pivot flexibility. Atthis time, it works only for a near-rigid pivot (Difficultiesin convergence).



Work to date



flexibility on force coefficients of TPJBs.

(c) Took XLPRESSDAM® code and beganmodifications

(d) Obtained initial predictions for near-

rigid TPJB

Comparisonwith other

predictionsand 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 , r p 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, C p 120 μm (0.004724 inch)

Lubricant Viscosity, 0 0.028 Pa.s

Rotor Speed 6000 rpm

Offset 0.5

(Someya*) Five pad, tilting pad bearing (LOP)

• Isothermal flow, isoviscous

• Synchronous speed reduced forcecoefficients

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 10

Sommerfeld number

E c

c e n t r i c i t y

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 rigidpivot, especially at heavy loads (small S ).

W Y

X

Near rigid pivot

Rigid pivot

P di t d tiff ffi i t



Predicted stiffness coefficients

10

100

1000

10000

0.01 0.1 1 10

Sommerfeld number

S t i f f n e s s K x x ( M N / m )

Someya'srigid pivot

near rigid pivot

Near rigid pivot

Rigid pivotKXX

K YY

W Y

X

1

10

100

0.01 0.1 1 10Sommerfeld number

S t i f f n e s s K y y ( M N / m )

Kyy (Someya's)

Kyy (rigid pivot)

Kyy (Near rigid pivot)

Rigid pivot

Near rigid pivotPivot flexibility lowers thedirect stiffness coefficientKXX (along load direction),in particular for largeloads.

K P



10

100

1000

10000

0.01 0.1 1 10

Sommerfeld number

D a m p i n g C x x ( k

N . s

/ m )

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

D a m p i n g C y y ( k N . s

/ m )

Cyy (Someya's)

Cyy (rigid pivot)

Cyy (Near rigid pivot)Rigid pivot

Near rigid pivot

Predicted damping coefficients

CXX

C YY

W Y

X

Pivot flexibility lowers thedirect damping coefficientCXX (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, C P 120.65 μm (4.75 mil)

Bearing Radial Clearance, C b 68 μm (2.67mil)

Bearing Preload, 0.44

Pad Mass, m p 0.44kg (0.97 lb)

Pad Inertia, I G 2.49 kg-cm2 ( 0.851 lb-in2)

Pad thickness, t 19.05mm (3.228inch)

Bearing pivot stiffness, K p 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

o

CUsed 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)

R e a l p a r t o f t h e i m p e d a n c e s

Re (Zxx)-prediction

Re (Zyy)-prediction

Re (Zxx)-measurementRe (Zyy)-measurement

M N / m

Predicted & Test impedances versus frequency

Measured Predicted

X 0.009 0.006

Y -0.381 -0.306

Dimensionless

Eccentricity

K-C model:

Z=K + i ωC

Stiffness: K= Re (Z)

Damping:C= Im (Z)/ ω

W

Y

X

Real part of impedances

Re (Z YY)-prediction

Re (Z YY)-measurement

Re (ZXX)-measurementRe (ZXX)-prediction

Dynamicstiffness K YY over predicted

P di t d & T t i d f



-100

0

100

200

300

400

500

600

0 100 200 300

I m a g i n a r y p a r t o f t h e i m

p e d a n c e s M N / m

Excitation frequency (Hz)

Im (Zxx)-prediction

Im (Zyy)-prediction

Im (Zxx)-measurement

Im (Zyy)-measurement

Predicted & Test impedances versus frequency

Imaginary part of impedances

W

Y

X

Im (Z YY)-measurement

Im (Z YY)-prediction

Im (ZXX)-measurement

Im (ZXX)-prediction

Both dampingcoefficients areunderpredicted.

C l i



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 vsfrequency, 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.

P d k f 2nd



Proposed work for 2nd year 1.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 andtransverse deformations and ensure reliable convergence to anequilibrium 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 datafor various bearing geometries tested by Childs and students andpreparation 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 modelmodern TPB configurations and to improve predictions of dynamic forced response (K-C-M model)

Code for Tilting Pad Bearings



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2011 Model Tilting Pad Bearing 15196B 3

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