A Novel Computational Model for Tilting Pad Journal...

1

A Novel Computational Model for Tilting Pad Journal Bearings

with Soft Pivot Stiffness

Yujiao TaoResearch Assistant

Dr. Luis San AndrésMast-Childs Professor

32nd Turbomachinery Research Consortium Meeting

TRC 32514/15196B

Year II

May 2012

22

JustificationPivot flexibility reduces the force coefficients in heavily loaded tilting pad journal bearings (TPJBs).

XLTRC2 TFPBRG code shows poor predictions for TPJB force coefficients.

W, static load

Y

X

Housing

Pad

Pivot

Fluid film

Journal

Ω , Journal speed

δ, Pad tilt angleξ

η

Research objective:To develop a code, benchmarked by test data, to predict

the K-C-M coefficients of TPJBs. Code accounts for thermal energy transport and the (nonlinear) effects of pivot flexibility.

33

Tasks completed• Completed derivation of reduced

frequency force coefficients for TPJBs

• Developed iterative search scheme to update pad radial and transverse displacements

• Constructed GUI for new TPJB code as per XLTRC2 standards

• Compared predictions from the TPJB codeto test data

44

• Completed derivation of reduced frequency force coefficients for TPJBs

Includes pivot NL deformations

• Developed sound iterative search scheme to update the pad radial and transverse displacements


• Compared predictions from the enhanced TPJB code to test data

Tasks completed

55( ) ( ) ( ) ( )

cos sin

cos sinp X Y

piv p p piv d p p

h C e e

r R

θ θ

ξ θ θ η δ θ θ

= + + +

− − + − −

Film thickness in a padCp : Pad radial clearance

CB = Cp-rp Bearing assembled clearance

Rd= Rp+t : Pad radius and thicknessrp : Pad dimensional preload

δp : Pad tilt angleξpiv, ηpiv : Pivot radial and transverse

deflections

Unloaded Pad

Journal

P

Ω

X

Y

δp

η

ξ

θp

θh

eRJ

WX

RB

ξpiv

OB

RP

WY

OP

ηpiv

P’

Θp

t

θL

Bearing CenterPad CenterFluid film

Loaded Pad Film thickness:

66

• Laminar flow• Includes temporal fluid inertia

effects• Average viscosity across film

3 3 2 2

2 2

112 12 2 12J

h P h P h h h hR z z t t

ρθ μ θ μ θ μ

⎧ ⎫ ⎧ ⎫∂ ∂ ∂ ∂ ∂ Ω ∂ ∂+ = + +⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂ ∂ ∂ ∂⎩ ⎭ ⎩ ⎭

On kth pad

h : fluid film thickness P : hydrodynamic pressure

μ : lubricant viscosity Ω : journal speed

RJ : journal radius

Reynolds equation for thin film bearingY

X

Housing

Pad

Pivot

Fluid film

Journal

Ω , Journal speed

W, static load

77

Thermal energy transport in thin film flowsT: film temperature

h : film thickness

U,W: circ. & axial flow velocities

μ, ρ, Cv : viscosity & density, specific heat

hB, hJ : heat convection coefficients

TB, TJ : bearing and journal temperatures

Ω : journal speed

Neglects 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 Uh

ρ

μ

⎡ ⎤∂ ∂+ + − + −⎢ ⎥∂ Θ ∂⎣ ⎦

⎛ ⎞Ω Ω⎡ ⎤= + + + −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠CONVECTION + DIFFUSION= DISSIPATION(Energy Disposed) = (Energy Generated)

8

ΩR

Upstream pad

Journal

8

Pad inlet thermal mixing coefficientInlet thermal mixing coefficient λ (0< λ <1) is empirical parameter.

in s hF F Fλ= +

in in s s h hF T FT F Tλ= +

Downstream pad

FhTh

FsTs

FinTin

Fs , Fh , Fin Volumetric flow ratesTs , Th , Tin Fluid flow temperatures

λ ~0.6-0.9 for conventional lubricant feed arrangements with deep grooves and wide holes.

λ small (<< 1) for TPJBs with LEG feed arrangements and scrappers.

Hot oil Mixing oil

Cold oil

Is λ constant for all conditions?

99

Nonlinear pivot deflection & stiffnessPivot deformation is typically nonlinear depending on the load (Fpiv), area of contact, hardness of the materials, surface conditions.

• Sphere on a sphere

Pivot and housing:Ep , Eh Elastic modulus νp ,νh Poisson’s ratiosDp , Dh Diameter of the curvature L Contact length

( )22 22

31 11.04 P H H P

piv pivP H P H

D DFE E D D

υ υξ⎛ ⎞− − −

= +⎜ ⎟⎝ ⎠

2

2

2 (1 ) 4 ( )2 ln3 2.15

piv P H H Ppiv

piv

F LED D D DLE F

υξ

π⎛ ⎞− −

= +⎜ ⎟⎜ ⎟⎝ ⎠

( )2 22 233

1 10.52 1 pivH P P Hpiv

H P P H

FD DD D E E

υ υξ⎛ ⎞ ⎛ ⎞− − −

= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

1 piv

piv pivK Fξ∂

=∂

( ) ( ) ( )2 3 4

0 1 2 3 4piv piv piv piv pivF a a a a aξ ξ ξ ξ= + + + +piv

pivpiv

FK

ξ∂

=∂

• Sphere on a cylinder

•Cylinder on a cylinder

• Load-deflection function(empirical)

*Kirk, R.G., and Reedy, S.W., 1988, J. Vib. Acoust. Stress. Reliab. Des., 110(2), pp. 165-171.

1010


frequency force coefficients for TPJBs• Developed sound iterative search scheme to

update the pad radial and transverse displacements



Convergence to the pad and journal equilibrium positions

1111

Iterative scheme to find pad equilibrium positionJournal displacement, pivot radial and transverse displacements converge to equilibrium solution for TPJB with flexible pivots

Flow chart of iterative scheme

Set initial journal center displacements

Check convergence on loads (W0)

Find tilt angle for kth pad

Estimate the initial pivot radial displacement for kth pad

Find pivot radial and transverse displacements for kth pad

Find tilt angle for kth pad

Update e

Check convergence on kth pad tilt angle and pivot displacements

End of procedure

Check convergence on loads (W0)

Update e Take TPJB with flexible pivots, find pads tilt angles, radial and transverse displacements and journal eccentricity

Estimate the pivot radial displacement and journal eccentricity

Take TPJB with rigid pivots, find pads tilt angles and journal eccentricity

1212






Fortran program and Excel GUI

1313

Excel GUI and Fortran code• Modifications/enhancements to XLPRESSDAM® code • FEM to solve Reynolds equation (hydrodynamic pressure)• Uses control volume method to solve energy transport eqn

Excel GUI

Parameters of pivot: type, radii of contact & material properties (E,ν)

Different pads: geometry parameters

•Sphere on a sphere•Cylinder on a cylinder•Sphere on a cylinder•Rigid pivot•Load-deflection function

1414





• Compared predictions from TPJB code to test data

Bearings tested by Childs and students(TurboLab)

1515

Predictions for a four-pad TPJB(Childs and Harris*) Four pad, tilting pad bearing (LBP)

*Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502

Number of pads, Npad 4Configuration LBPRotor diameter, D 101.6 mm (4 inch)Pad axial length, L 101.6 mm (4 inch)Pad arc angle, ΘP 73o

Pivot offset 65%Pad preload, 0.37, 0.58Nominal bearing clearance, CB 95.3 μm (3.75 mil)

Measured bearing clearance, CB54.6 μm (2.15 mil)99.6 μm (3.92 mil)

Pad inertia, IP 7.91kg.cm2 (2.70lb.in2)Oil inlet temperature ~40 oC (104 oF)Lubricant type ISO VG32, DTE 797Oil supply viscosity, μ0 0.032 Pa.s

Pr

Specific load, W/LD 0 kPa-1,896 kPa (275 psi)Journal speed, Ω 4 krpm-12 krpm

Cold conditions

16

CB=55 μm

16

Predictions for a four-pad TPJB

*Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX.

Specific load, W/LD 1.9 MPa (275 psi)Journal speed, Ω 4 krpm-12 krpm

Lubricant arrangements: Spray bar blocker, By pass cooling

Pivot type: Ball-in-socket pivot

Measured pivot stiffness: 350 MN/m

Nominal cold bearing clearanceCB=100 μm

CB=55 μmCB=100 μm

Measured cold bearing clearances on Pads #1 and #3 are ~40% smaller than the nominal cold clearance.

X

Y

2.45o

Θp=73o

W

Measured cold bearing clearance

CB=95 μm

Pad 1

Pad 4

Pad 2

Pad 3

X

Y

W

Pad 1

Pad 4

Pad 2

Pad 3

1717

0

20

40

60

80

100

0 400 800 1200 1600 2000Specific Load (kPa)

Jour

nal D

ispl

acem

ent

( μm

)Journal eccentricity vs. static load

-eY

eX

Rotor speed Ω =10 krpm

Rotor speed Ω =6 krpm

Predicted journal eccentricity correlates well with measurements.

XY

W XY

W

Symbols: test dataLines: prediction


0

20

40

60

80

100

0 400 800 1200 1600 2000

Specific Load (kPa)

Jour

nal D

ispl

acem

ent

( μm

)

-eY

eX

Max. 275 psi

1818

0

10

20

30

40


Tem

pera

ture

Ris

e at

Pad

Tr

ailin

g Ed

ge (o C

)Film temperature rise vs. static loadmeasured pad sub-surface temperature rise at pad trailing edge

Rotor speed Ω=6 krpm

X

Y

WPad 3

Pad 2

Pad 4

Pad 1

Trailing edges

Input: inlet thermal mixing coefficient λ=0.5

At 6 krpm, film temperature rises at pad trailing edges are

considerable even with no load applied

Pad 1Pad 2Pad 3

Pad 4



Max. 275 psiOil Inlet temperature ~40oC

1919

0

10

20

30

40


Tem

pera

ture

Ris

e at

Pad

Tr

ailin

g Ed

ge (o C

)

Film temperature rise vs static loadmeasured pad sub-surface temperature rise at pad trailing edge

X

Y

WPad 3

Pad 2

Pad 4

Pad 1

Trailing edges

Input: inlet thermal mixing coefficient λ=0.95

Film temperatures underpredicted at 10 krpm.

Film heats little with load


Pad 1Pad 2Pad 3

Pad 4



Max. 275 psi

Effectiveness of spray bar blocker diminishes

Oil Inlet temperature ~40oC

20

200

400

600

800

1000

0 400 800 1200 1600 2000

Specific Load (kPa)

TPJB

Stif

fnes

s C

oeffi

cien

ts

(MN

/m)

200

400

600

800

1000

0 400 800 1200 1600 2000

Specific Load (kPa)

TPJB

Stif

fnes

s C

oeffi

cien

ts

(MN

/m)

20

Stiffness coefficients



λ=0.5

λ=0.95

KYY

KXX =KYY, TFBBRG

Kpiv=350 MN/m

Kpiv=350 MN/m

Very soft pivot produces

~ constant K’s, invariant with

load and speed.

XY

W XY

W



TFBBRG code in XLTRC2-TPJB model with rigid pivotKXX

Prediction KXX=KYY

KYY

KXX =KYY, TFBBRG

KXX

Max. 275 psi

2121



λ=0.5

λ=0.95Soft pivot renders

nearly constant damping coefficients. Good correlation with

test data

Damping coefficients Symbols: test dataLines: prediction


CYY

CXX =CYY, TFBBRG

CXX

CYY

CXX =CYY, TFBBRG

CXX

XY

W XY

W

Synchronous speed coefficients

Prediction CXX=CYY

0

200

400

600

800

1000


TPJB

Dam

ping

Coe

ffici

ents

(k

N.s

/m)

0

200

400

600

800


TPJB

Dam

ping

Coe

ffici

ents

(k

N.s

/m)

Max. 275 psi

2222

Predictions for a five-pad TPJB(Wilkes and Childs*) Five pad, tilting pad bearing (LOP)

*Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.

Pr

Specific load W/LD: 3,132 kPa (454 psi)Journal speed Ω: 4.4 krpm-13.1 krpm

X

Y

Pad

W

Journal

Fluid film

2323

0

200

400

600

800

1000

0 5 10 15 20

Pivot Radial Deflection (μm)

Pivo

t Rad

ial S

tiffn

ess

(MN

/m)

0246

8101214

0 5 10 15 20

Pivot Radial Deflection (μm)

Piv

ot R

adia

l For

ce (k

N)

Pivot stiffness & hot bearing clearance

Pivot load-deflection function

Pivot stiffness

Hot bearing clearance

Rocker back pivotPivot stiffness-deflection function

CB,cold-CB,hot=α(Thot-Tcold)Bearing clearance decreases due to thermal expansion of the rotor and pad surfaces.

α=0.396 μm/oC

Pivot radial force

Pivot radial stiffness


EXPERIMENTAL

Hot bearing clearance CB: 48 μm~58 μmNominal cold CB=68 μm

Empirical

2424

Pad bending stiffnessPad bending stiffness*

kpad =5.4644×104Mp+1.1559×105 (N.m/rad)

MP ,Pad bending moment from fluid film


/2 /22 2

/2 /2

sin sinP T

L P

L L

p p pL L

M P R d dZ P R d dZβ β β βΘ Θ

− Θ − Θ

= − =∫ ∫ ∫ ∫

Equivalent pad-pivot stiffness: series pivot + pad bending21 1

eq piv pad

lk k k

= +

12 P Pl R= Θ

Derived from experiments

Pad ½ length

β

RP

MPMP

OP

Used in code TPJB

25

0

20

40

60

80

0 500 1000 1500 2000 2500 3000 3500Specific Load (kPa)

Jour

nal D

ispl

acem

ent (

μm

)

0

20

40

60

80

0 500 1000 1500 2000 2500 3000 3500

Jour

nal D

ispl

acem

ent (

μm

)

25

Journal eccentricity slightly under/over predicted at the

low/high rotor speeds.


Journal eccentricity vs. static load


-eY

-eY

Max. 454 psi

X

Y

W

Rotor speed Ω =13.1 krpm


cB= 57μm<-eY

cB= 49μm~-eY

λ=0.8

λ=0.9

cold CB=68 μm

26


0

200

400

600

800


TPJB

Stif

fnes

s C

oeffi

cien

ts

(MN

/m)

0

200

400

600

800

0 500 1000 1500 2000 2500 3000 3500

TPJB

Stif

fnes

s C

oeffi

cien

ts

(MN

/m)

26

Static stiffness coefficients over

predicted at large specific loads

KYY

KXX

KYY

KXX

K-C-M model


Static stiffness coefficients Symbols: test dataLines: prediction TPJBDashed: Wilkes preds.

Max. 454 psi

X

Y

W

Rotor speed Ω =4.4 krpmλ=0.8

λ=0.9

27

0

100

200

300

400


TPJB

Dam

ping

Coe

ffici

ents

(k

N.s

/m)

0

100

200

300

400


TPJB

Dam

ping

Coe

ffici

ents

(k

N.s

/m)

27

Damping coefficients vary little with static load.

CYY

CXX

CYY

CXX


Damping coefficients

X

Y

W

Max. 454 psi

λ=0.8

λ=0.9Rotor speed Ω =13.1 krpm


Synchronous speed coefficients

Symbols: test dataLines: prediction TPJBDashed: Wilkes preds.

28

-150

-100

-50

0

50

100

0 500 1000 1500 2000 2500 3000 3500

Virtu

al M

ass

Coe

ffici

ents

(kg)

-150

-100

-50

0

50

100


Virtu

al M

ass

Coe

ffici

ents

(kg)

28

Large negative virtual masses at 4.4 krpm.

Dynamic stiffness increases with frequency.

MYY

MXX

MYY

MXX


Virtual mass coefficients

X

Y

W


Rotor speed Ω =4.4 krpmλ=0.8

λ=0.9

Symbols: test dataLines: prediction TPJBDashed: Wilkes pred.

29

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250 300 350Excitation Frequency (Hz)

Rea

l par

t of i

mpe

danc

e co

effic

ient

s R

e(Z

) (M

N/m

)

0

100

200

300

400

500

600

700

800


Rea

l par

t of i

mpe

danc

e co

effic

ient

s R

e(Z

) (M

N/m

)

29

Real part

At 4.4 krpm, dynamic stiffness HARDENs at frequencies ω > 2Ω.

Re( ZXX)

Re( ZYY)

Synchronous frequency

Re( ZXX)

Re( ZYY)Synchronous frequency

Re(Z)=K-Mω2


Impedances Specific load = 227psi

X

Y

W




30

0

100

200

300

400

500


Imag

inar

y pa

rt of

impe

danc

e co

effic

ient

s Im

(ZXX

) (M

N/m

)

0

100

200

300

400

500


Imag

inar

y pa

rt of

impe

danc

e co

effic

ient

s Im

(Z)

(MN

/m)

30

At 4.4 krpm, predicted damping is frequency dependent for ω>2Ω

Im( ZXX)

Im(ZYY)

Synchronous frequency

Im( ZXX)

Im( ZYY)Synchronous frequency

Ima(Z)=Cω


Impedances



Imaginary part

Specific load = 227psi

X

Y

W


3131

Predictions for other test TPJBsKulhanek and Childs, 2012, ASME, J. Eng. Gas Turbines Power, 134, 052505 1-11.• TPJB code delivers good predictions for stiffness and damping by estimating the (actual) hot bearing clearances and using a constant pivot radial stiffness.

In the works. Will be part of forthcoming Y. Tao M.S. Thesis (TRC report 2012)

Delgado et al., 2010, ASME Paper GT2010-23802• TPJB code takes TPJB pivots as rigid and uses hot bearing clearances.• Predicted stiffness and damping correlate well with test data (45 psi).

Tschoepe and Childs, 2012, not yet published• TPJB code uses measured pivot load-deflection function and hot bearing clearances.• Predicted stiffness and damping are in agreement with test TPJB data.

3232

Conclusions• For TPJBs with very soft pivots (Kpiv<<Kfilm), pivot stiffness determines bearing stiffness.

• Film temperatures at no load condition are high. At high rotor speeds (> 10 krpm), LEG and spray bar blockers have less effectiveness in cooling a bearing.

• Bearing & pad clearances change due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea.

•A-priori knowledge of pivot stiffness and bearing & pad clearances is required to obtain accurate predictions of TPJB performance.

• Bearing & pad clearances change a lot due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea.

3333

2012 Proposal to TRC (2 years)

Hydrodynamic pressure P Pad surface elastic & thermal deformations change bearing & pad clearances

Kδ = P + C∆TK, Pad stiffness matrix

P, Fluid film pressure vectorC, Mechanical-thermal stiffness

matrixδ, Pad displacement vector

Objective: Enhance TPJB code to accurately predict pad surface deformations

Hot oil flow

Pivot constraint

FE pad structural analysis by Yingkun Li

3434

Proposed work 2012-2013• Build a 3-D FE model of commercial pads (ANSYS® or

SolidWorks®) to obtain pad stiffness matrix. Reduce model with active DOFs, perform structural modal analysis for easy off-line evaluation of pad surface deformations and pivot deflections.

• Implement oil feed arrangements (LEG, spray bar blockers etc.) in the FE model

• Construct new Excel GUI and Fortran code for XLTRC2

• Digest more test data and continue to update predictions using enhanced code.

35

TRC BudgetYear III

Support for graduate student (20 h/week) x $ 2,200 x 12 months $ 26,400Fringe benefits (0.6%) and medical insurance ($197/month) $ 2,522Travel to (US) technical conference $ 1,200Tuition & fees three semesters ($227/credit hour) $ 9,262Other (PC+software+storage supplies) $ 1,600

2012-2013 Year III $ 40,984

Enhanced TPJB code will model current (commercial) TPJBsand improve predictions of force coefficients with minimum User expertise for specification of empirical parameters.

2012-2013 Year III

3636

Questions (?)

A Novel Computational Model for Tilting Pad Journal...

Documents

Transcript of A Novel Computational Model for Tilting Pad Journal...