MEASURED AND PREDICTED ROTOR-PAD TRANSFER …

MEASURED AND PREDICTED ROTOR-PAD TRANSFER FUNCTIONS FOR A ROCKER-PIVOT TILTING-PAD JOURNAL BEARING

Presented By: Jason Wilkes

PhD: Final Examination

Turbomachinery LaboratoryMechanical Engineering Department

Texas A&M UniversityCollege Station, Texas

T L

Dedication:To Monica – who has worked tirelessly at home while I put in my time at the lab.

Overview:Introduction

Previous Work

Objective of the ResearchTheoretical Model

Geometry, Kinematics, and Pad DynamicsRotor-Pad Transfer functionBearing Impedances

HardwareTest Rig DescriptionBearing Description/Instrumentation

ResultsPad and Pivot StiffnessBearing Clearance and Static MeasurementsMeasured vs. Predicted Rotor-Pad Transfer functionsMeasured vs. Predicted Bearing Impedances

Summary and Outlook

bO( )X 0i

( )Y 0jbmjm

byfbxf

jOx

y

Introduction: BearingsPrimary Functions of a bearing

Permit relative rotationTransmit reaction forces

Types of bearingsRolling elementMechanical contact (sliding bearings)Magnetic levitationFluid-film bearings

Reaction force modelsStiffness (kij) and damping (cij) (KC) model

Stiffness, damping, and virtual-mass (mij) (KCM) model

Direct Stiffness Cross-Coupled Stiffness

xx xy xx xybxyx yyyx yyby

f k k c cx xy c c yf k k

⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎧ ⎫⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦⎢ ⎥⎣ ⎦⎩ ⎭

− = +

xx xy xx xy xx xybxyx yy yx yyyx yyby

f k k c c m mx x xy c c m my yf k k

⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎧ ⎫⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦⎩ ⎭

− = + +

Introduction: Tilting Pad Journal Bearings (TPJB)Composed of multiple pads, or shoes, that are free to tiltThis feature results in a reaction force that is directed through each pad’s pivot.Inherently stable (reduced/eliminated destabilizing )

Fluid Film Pressure Profile

Pad

Journal

jy

reactionf

ω

Journal

yf

( )Load on Pad LOP

yf

( )Load Between Pad LBP

Bearingxyk

Introduction: TPJB Terminology

: lc lcctlc lt

Offset β ββ β β= =+

p p jc r r= − 1 bp

cPreload c= −

Bearing

bc

lcβ ctβ

PadFulcrum, cO

ltβ

ω

LeadingEdge

Trailing Edge

Journal

jrbr

pr

Center of Pad's Surface Arc, pO

( )( )

Center of Bearing and Journal

b

j

OO

Rocker-Pivot

Sliding-Pivot

Flexure-Pivot

BearingPad

Pivot configurations

Introduction: TPJB TheoryLund’s (1964) Pad Assembly Method

Solve for static equilibriumWrite perturbed equations of motion (EOMs) for a pad and the journal about equilibriumAssume harmonic rotor motion (initially assumed synchronous)

Eliminate pad motion using a harmonic reduction (solve for pad motion as a function of the assumed rotor motion)Calculate reduced direct and cross-coupled stiffness and damping coefficients for the padSum impedances for all pads to determine bearing impedance

( ) ( ), , j tj j j jx y x y e Ω=

TPJB Theory (Improvements)Pad Flexibility: Nilsson (1978)

Approximates pad deformation using curved beam theory such that fluid film pressures result in a change in pad radiusAsserts that pad compliance having a small impact on static characteristics can dramatically affect dynamic characteristics (stiffness and damping)Shows a 90°arc pad to have 40% less damping than a rigid pad when heavily loaded

Undeflected PadSurface Arc

por

poO

pcM

pt

Undeflected PadSurface Arc por

poO

pO

pr

prδ

pp po rr r δ= +

pcM

Deflected PadSurface Arc

TPJB Theory (Improvements)Pad and Pivot Flexibility: Lund and Pederson (1987)

Include pad and pivot complianceAnalytically perturbed Reynolds equation to obtain stiffness and damping coefficients of the oil filmShows a significant reduction in bearing damping with increasing pivot flexibilityAssert that stability calculations should be performed using the systems damped eigenvalue, not the synchronous frequency.

,c kc ξ,c kk ξ, (Radial Pad Motion)c kξ

Bearing Housing

Introduction: Predicted DampingDmochowski (2005, 2007)

Damping was reduced at higher excitation frequencies by as much as 75% for spherical pivots and 25% for rocker pivots.Obtained moderately good agreement between experiments and predictions using a model having pivot flexibility (though his data showed significant scatter)

Carter (2007) and Kulhanek (2010)Measured damping was independent of frequency, speed, and load.Principal damping was significantly over-predicted by codes, especially at low speeds and high loads.

Objective:Determine the underlying sources responsible for the discrepancy between measured and predicted tilting-pad bearing damping.Approach

Reevaluate the fundamental assumptions governing theoretical and experimental practicesFocus on comparisons between predicted and measured pad motion (an adequate pad perturbation model should produce accurate bearing coefficients)

Model: The Reference State

crbr

jrαpo

br

r−

cpt

por

gobη

poO ( )X 0i

( )Y 0j

( )kη ki

( )kξ kj

ctβcfrom Xθ

bc

oO

gobξ

lcβ

coOpt

ω

ltβ

bm

jm

pm

,jo boO O

( ) ( )( ) ( )

cos sinsin cos

k k

k k

α αα α

⎡ ⎤⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥− ⎩ ⎭⎩ ⎭ ⎣ ⎦⎧ ⎫= ⎨ ⎬⎩ ⎭

kQ

η,k X

Yξ,k

X

Y

b bbb

bb

Coordinate Transformation

Arbitrary Pad Centerof Gravity (C.G.)Attention to Physical Dimensions

Rigid Body Pad DOFs, Pivot Reaction Forces

Angular Reaction Force

Radial Reaction Force

Transverse Reaction Force

, 0, 1,

, ,0, 0, 0, 1, 1,

1, 1,,

cz k cz k cz k

cz k cz kcz k c k k k k

k k

M M MM M

M ξ φ φ φφ φ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= +

∂ ∂= + +∂ ∂

, 0, 1,

, ,0, 0, 1, 1,

1, 1,0 0

c k c k c k

c k c kc k c k c k c k

c k c k

f f f

f ff

ξ ξ ξ

ξ ξξ ξ ξ ξξ ξ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= +

∂ ∂= + +∂ ∂

, 0, 1,

, ,0, 0, 0, 1, 1,

1, 1,0 0

,

c k c k c k

c k c kc k c k c k c k c k

c k c k

f f f

f ff

η η η

η ηη ξ η η ηη η

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= +

∂ ∂= + +∂ ∂

, (Radial Pad Motion)c kξ,(Transverse Pad Motion) c kη

(Tilt)kφ

Padthk

( )kη ki

( )kξ kj

,c kf ξ,c kf η

(Tilt)φ

( )kη ki

( )kξ kj

,cz kM

cO

cO

Equilibrium Pivot Reaction Forces

Angular Stiffness and Damping

Radial Stiffness and Damping

Transverse Stiffness and Damping

, ,, ,

1, 1,0 0

,c k c kc k c k

c k c k

f fk cη ηη ηη η

∂ ∂= =∂ ∂

, ,, ,

1, 1,0 0

,c k c kc k c k

c k c k

f fk cξ ξξ ξξ ξ

∂ ∂= =∂ ∂

, ,, ,

1, 1,0 0

,cz k cz kcz k cz k

k k

M Mk cφ φ

∂ ∂= =∂ ∂

,c kc ξ,c kk ξ

cO,c kk η

,c kc η

,cz kc,cz kk

Bearing Housing

Bending Moment in a Pad

Pressure induced bending moment

Pivot Discontinuity

/2

,/2

, /2

,/2

0

0

sin ,

sin ,

p

p lcp p ct

p

L

p n pL

c n L

p n pL

p r r d dZM

p r r d dZ

β

ββ

β

β β β β

β

β β β β

⎧⎪

⎛ ⎞ ⎛ ⎞⎪ ⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪⎪⎛ ⎞⎨⎜ ⎟

⎝ ⎠ ⎪⎛ ⎞ ⎛ ⎞⎪⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪

⎪⎩

−−

−

<

>

−

=

−

∫ ∫

∫ ∫

, ,0 0p pc n cz c nM M M⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− ++ =

( ), 0pc nM −

( ), 0pc nM +

Trailing SegmentLeading Segment

Fluid Film Pressure

czM

( )Fluid Film Pressure p β

pO

β

Neutral Axis

pr,p nr

Approach taken by Nilsson, & Lund and Pederson

Average Bending Moment (to be applied as an end moment on a curved beam)

The deflected pad radius given by

where

and is the pad’s bending

stiffness taken from curved

beam theory.

psck

, ,1 t

p pl

c n c nlt

M M dβ

ββ ββ⎛ ⎞⎜ ⎟⎝ ⎠

−= ∫

,0 1

pp

p

c nr p p

sc

Mr r

kδ = + =

pp po rr r δ= +jO

pcM

pt

Undeflected PadSurface Arc por

poO

pO

pr

jr

pocpc pcδ

pcM


p pc rδ δ=

Pad deflection in the Current Work

Nonlinearity presentin the current workrequires the developmentof a slightly different pad bending model.

We will assume that changes in pad clearance are given by

is the applied fluid film moment at “A”

Finite element analysis (FEA) is used to determine the nonlinear

bending stiffness

12p p pc c cM M Mβ β⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

− += +

pp

p

cc

sc

Mkδ =

pp

p

csc

c

Mk δ=

Pivot Insert

gap

Pad

Contact

pcM

A

Theoretical Model: DOFJournal, Oj:

Bearing, Ob:

Pad pivot location, Oc:

Center of pad surface arc, Op

Relative Rotor-Pad Motion

coO

cξ

φ

αoO

gη

jOpO

bObξ

jξpξ

bη

pηjη

cηcO

gξ

( )X 0i

( )Y 0j

( )kη ki

( )kξ kj

ψ

bmjm

pm

bξbη

, ,j k j kη ξ= +j,k k ke i j

, ,b k b kη ξ= +b,k k ke i j

, , , ,b k c k b k c kη η ξ ξ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= + + +c,k k ke i j

, ,

, , ,

, ,

p k p k

b k c k cp k k

pb k c k b

r

c c

η ξ

η η φ

ξ ξ

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

= +

= + −

+ + + −

p,k k k

k

k

e i j

i

j

, ,

, , , , , , ,

pj k pj k

pj k b k c k cp k k j k b k c k br c c

η ξ

η η η φ ξ ξ ξ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= − = +

= − − + + − − − +

pj,k j,k p,k k k

k k

e e e i j

i j

Theoretical Model: Fluid FilmReynolds Equation (variable viscosity)

where hk is the fluid film height

Pressure perturbation

Transverse and radial fluid-film reaction forces

Bending moment applied by the zeroth and first order fluid-film pressures

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1,p pk k k k k k c k p k k k k c k p kkp p p p p c p p p cη ξ η ξ

η ξ η ξ= + + + + + +

0, 1, , ,cosk k k k k k p k k pj kh h h cψ ψ ψ ψ ψ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠= + = − −pj,ke

3 31: 12 12 2k k k k

k kj jk k k

h h h hp pr r z z tω

ψ μ ψ μ ψ

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎡ ⎤ ⎡ ⎤⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

∂ ∂∂ ∂ ∂ ∂ℜ = + = +∂ ∂ ∂ ∂ ∂ ∂

, ,

,

/2, 0, 1,

0,, 0, 1, 0

cos2

sin

p k t k

l k

Lkk k k

k p k kk k k k

f f fp r d dZf f f

ψη η η

ψξ ξ ξ

ψψ

ψ

⎧ ⎫⎛ ⎞⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝ ⎠⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬

⎛ ⎞⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎜ ⎟⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎪ ⎪⎝ ⎠⎩ ⎭

= + =− ∫ ∫

, ,

1,

/2

, 0, 1, 0, 0,0

cosp k c k

p p pl k

L

c k c k c k p k k k p k kM M M r p r d dZψ

ψψ ψ⎛ ⎞⎜ ⎟⎝ ⎠

= + =− ∫ ∫

Equilibrium Fluid Film Reaction Forces/Moments

Static Equilibrium: Net body forces and moments on the journal, bearing, and each pad are zero.

Fluid Film Stiffness and Damping

αoO

cO

jO

pO

bO

,i kk ξ,i kc ξ

,i kk η

,i kcη

,c kk ξ

,c kc ξ

,c kk η

,c kc η

,cz kk ,cz kc

( )X 0i

( )Y 0j

( )kη ki

( )kξ kj

bmjm

pm,, , ,, ,

, , , , , ,

, , ,, , ,

, , , , ,p p

p p

p p p pp p p p

c kk k c kk k

k k c k k k c k

c k c k c c kc k c k c c k

kk k cc ck k k c c c

c c ck k k

ηηη ηξ ηηη ηξ

ξη ξξ ξ ξη ξξ ξ

η ξη ξ

⎧ ⎫⎧ ⎫ ⎧ ⎫ ⎧⎧ ⎫ ⎧ ⎫⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭ ⎩ ⎭

⎧ ⎫⎫⎪ ⎪⎪⎪ ⎪

⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎬⎪ ⎪ ⎪⎪⎪ ⎪ ⎪⎪⎪ ⎪ ⎪⎪⎩ ⎭⎩ ⎭

, ,

,

/2

1, 1, 1, 1, 1, 0,1,0

0,

2cos

, , , , , 2sin

cos

p k t k

p pl k

kL

k k k k k k p k kk

p k k

c cp p p p p p r d dz

r

ψ

η ξ η ξψ

ψ

ψ ψ

ψ

⎧ ⎫⎪ ⎪⎛ ⎞

⎜ ⎟⎪ ⎪⎝ ⎠⎪ ⎪⎧ ⎫⎪ ⎪⎪ ⎪⎛ ⎞⎨ ⎬⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎪ ⎪⎩ ⎭⎪ ⎪⎛ ⎞⎪ ⎪⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

= ∫ ∫

Journal/Bearing/Pad Perturbed EOMsFor a single pad, the reaction force components arising from the kth pad are given for the

Journal Bearing

And for the kth pad

In matrix notation

, , , , ,

, , , , ,

, , , , , ,

, , 1, 1, , 1,p p p p

g k p k g k k c k

g k p k g k k c k

zc k c k k p k cp k k cz k

c k c k p k c k sc k p k

F m f fF m f f

M I m r f MM m c M k c

η η η

ξ ξ ξ

η

ηξ

φ ⎛ ⎞⎜ ⎟⎝ ⎠

= =− +

= =− +

= + × = +

= = −

∑∑∑∑

cgo,k c,kb e

, , ,

, , ,

jj k j k k

jj k j k k

F m fF m fη η

ξ ξ

ηξ

= =

= =∑∑

, , ,

, , ,

b k b b k c k

b k b b k c k

F m fF m fη η

ξ ξ

ηξ

= =−

= =−∑∑

⎧ ⎫ ⎧ ⎫⎡ ⎤⎡ ⎤ ⎪ ⎪ ⎪ ⎪⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎣ ⎦ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎣ ⎦⎩ ⎭ ⎩ ⎭

+ +j1,k j1,kjj jj,k jp,k jb,k jj,k jp,k jb,k

pp,k pb,k pp,k pb,k pp,k pp1,k pj,k p1,k pj,k

bb bp,k bb,kbj,kb1,k b1,k

U UM 0 0 C C C K K K0 M M U C C C U K K K0 0 M C C CU U

⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎩ ⎭ ⎩ ⎭⎣ ⎦

=j1,k j0,k

b,k p1,k p0,k

bp,k bb,kbj,k b1,k b0,k

U FU F

K K K U F

1,

1,

1,

1,

1,1,

1,

1,

j k

j k

k

c k

c kp k

b k

b k

c

ηξφηξ

ηξ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪

⎧ ⎫ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

= =j1,k

1,k p1,k

b1,k

UU U

U

Options for Including Pad in a SystemFull/Unreduced Bearing Model: Explicitly include all pad degrees of freedom in the structural model

BenefitsMore AccurateFrequency Independent

DrawbacksComplexity (Requires 4np additional degrees of)

Reduced Bearing Model: Eliminate pad degrees of freedom using a harmonic reduction to produce 2×2 stiffness, damping, (and possibly virtual-mass) coefficients

BenefitsSimplicityReadily Identifiable

DrawbacksFrequency Dependence

Full/Unreduced Bearing ModelFor the journal and bearing, the sum of reaction force components on the journal and bearing in the X/Y directions is

and for the kth pad,

1

1

1

p

p

p

n

kn

kn

k

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

=

=

=

+

∑

∑

∑

Tk k kjj j1 jj,k j1 jp,k p1,k jb,k b1

Tk k kjj,k j1 jp,k p1,k jb,k b1 j1

Tbb k k bp,k bb,k kb1 bj,k j1 p1,k b1

Tk k bp,k bb,bj,k j1 p1,k

M U + Q C Q U +C U +C Q U

Q K Q U +K U +K Q U =F

M U + Q C Q U +C U +C Q U

+ Q K Q U +K U +K1

pn

k

⎛ ⎞⎜ ⎟⎝ ⎠=

∑ k k b1 b1Q U =F

, 1 pk n

+

+ =

pp,k pb,k k k pp,k pb,k kp1,k b1 pj,k j1 p1,k b1

k pp,k pb,k kpj,k j1 p1,k b1

M U M Q U +C Q U +C U +C Q U

K Q U +K U +K Q U =0 …

Full/Unreduced Bearing ModelIn matrix notation

1 1

1 1

p p

p p

n n

k k

n n

k k

⎡⎢⎧ ⎫⎡ ⎤ ⎢⎪ ⎪⎢ ⎥ ⎢⎪ ⎪⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥

⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦ ⎪ ⎪⎩ ⎭

⎣

= =

= =

+

∑ ∑

∑ ∑

T T Tk k k k kjj,k jp,k jb,kj1jj

k pp,k pb,k kpj,kpp,k pb,k k p1,k

T T Tbb b1 k k k bp,k k bb,k kbj,k

Q C Q Q C Q C QUM 0 0C Q C C Q0 M M Q U

0 0 M U Q C Q Q C Q C Q

1 1

1 1

p p

p p

n n

k k

n n

k k

⎤⎥ ⎧ ⎫⎥ ⎪ ⎪⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥ ⎨ ⎬⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭⎢ ⎥

⎢ ⎥⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= =

= =

+

∑ ∑

∑ ∑

j1

p1,k

b1

T T Tk k k k kjj,k jp,k jb,k

j1

k pp,k pb,k kpj,k p1,k

T T Tk k k bp,k k bb,k kbj,k

UU

U

Q K Q Q K Q K Q UK Q K K Q U

Q K Q Q K Q K Q

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

=j1

b1b1

F0

FU

1

1

1

1,11,1

1,11,1

11

j

j

c

cp

bb

xy

c

xy

φηξ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪

⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎪ ⎪

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

= =

j1

p1,k=1

b1

UUU

U

Reduced Bearing ModelAssume that where such that

where

Solve for pad motion using the second equation yields

where is the pad-journal or pad-rotor transfer-function matrix

and is similarly defined as the pad-bearing transfer-function matrix

( ), , , j tsti i i i i ie e λξ η ξ η ξ η⎛ ⎞ ⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

− + Ω= =⎡ ⎤⎡ ⎤ ⎧ ⎫⎢ ⎥⎢ ⎥ ⎪ ⎪⎢ ⎥⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎢ ⎥⎢ ⎥ ⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎢ ⎥⎢ ⎥ ⎪ ⎪⎣ ⎦ ⎩ ⎭ ⎣ ⎦

=jj,k jp,k jb,kjj,k jp,k jb,k j1,k j1,k

2 2pp,k pp,k pb,k pb,k pp,k pb,kpj,k p1,k pj,k

bp,k bb,kbj,k b1,k bp,k bb,kbj,k

A A AI I I U UI M s +I M s +I U A A AI I I U A A A

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪

⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭

−=−

2jj j1,k

p1,k2

bb b1,kb1,k

M s UU 0

M s UU

s jλ= + Ω

ij,k ij,k ij,kI =C s+K

=− − =-1 -1pp,k pp,k pb,k pb,kp1,k pj,k j1,k b1,k pj,k j1,k b1,kU A A U A A U Γ U +Γ U

, ,

, ,

, ,

1, 1,

11, 1,

1, 1,

1, 1,

j j j jk kj j j jc k c k

j j j jc k c k

j jj jp k p k

k k

jc k c k

c k c k

p k p kc c c c

η ξ η ξφ φη ξ η ξη ηη ξ η ξξ ξ

η ξη ξ

φ φ

ηη η

ξ ξ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

Γ Γ

Γ Γ= =− ≡Γ Γ

Γ Γ

-1pp,kpj,k pj,kΓ A A

1,

11,

00k

j kξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−

−

p1,kU

pj,kΓ

pb,kΓ

Reduced Bearing ModelSubstituting back into the previous set of equations yields

where the elements of are commonly referred to as impedances, or complex dynamic stiffnesses, where

Rotating into the X/Y coordinate system and summing impedances across all pads nets the journal/bearing impedances

p1,kU

⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎧ ⎫⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦⎣ ⎦

−

−

2pb,kjj,k jp,k pj,k jb,k jp,k jj,k jb,kj1,k j1,k jj j1,k

2bb,kbj,kbp,k bb,k bp,k pb,k bbbj,k pj,k b1,k b1,k

A +A Γ A +A Γ H HU U M s U= =H HA +A Γ A +A Γ U U M s U

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭b1,k

( ) ( ) ( ), ,

, , ,

Im, Re , ,k k

k k ij k

H HH Hηη ηξ

ξη ξξ

⎧ ⎫⎡ ⎤ ⎨ ⎬⎢ ⎥ ⎧ ⎫ ⎩ ⎭⎢ ⎥ ⎨ ⎬⎢ ⎥ ⎩ ⎭⎢ ⎥⎣ ⎦

Ω = Ω = Ω = Ωij,k

ij,k ij,k ij,k ij,k

HH K H C

ij,kH

ij,kH

1 1

1 1

p p

p p

n n

k kn n

k k

⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

= =

= =

=∑ ∑

∑ ∑

T Tk k k kjj,k jb,kjj jb

T Tbbbjk k k bb,k kbj,k

Q H Q Q H QH HH H Q H Q Q H Q

Test Rig: Overview

Test Bearing

X Y

X-Stinge

rY-Stinger

Radial Pad-Stator Probes

Loaded Shaft Rot.Pad

Pad-StatorProbes

Tangential Pad-Stator Probes

Static Loader

Prev. Pad-StatorProbe Location

Properties of the bearing at room temp. (24 °C).

Operating conditionsSpeed: 4400-13100 rpmUnit Load: 0-3134 kPa (0-454 psi)

Number of Pads 5

Loading Configuration Load on pad (LOP)

Pad Arc Length (βlt) 58.9°

Rotor Diameter 101.587 mm (3.9995 in)

Pad Axial Length 55.88 mm (2.200 in)

Cold Bearing Radial Clearance1 68 μm (2.67 mils)

Cold Pad Radial Clearance1 120.65 μm (4.75 mils)

Cold Bearing Preload1 0.44

Offset 0.50

Pad Mass (mp) 0.385 kg (0.849 lb)

Pad Inertia about Oc (Ic,k) 1.807e-4 kg-m2 (0.851 lb-in2)

Pad C.G (bηgo,bξgo) (0,0.0127) m, (0,0.5) in

Bearing Lubricant DTE 797, ISO VG-32

Pad Motion Measurement

T L

,c kξReferencePad

0Axis , ,k k Zη ξ

PerturbedPad

,c kη

kφ

0Z

kξ

kη

1,kη

1, 1, 1,Axis , ,k k kZη ξ

1,kξ

, (yaw)c kξφ

, (pitch)c kηφ

(tilt) Motion Probe Arrangement

Pad Degrees of Freedom

Pad Strain MeasurementStrain gages were applied to the side of the loaded pad

Differential Wheatstone Bridge Configuration

Changes in pad clearance were determined using

where will be determined by correlating differential strains to changes in pad radius using FEA

12dε1ε2ε

pcM

pcMUndeflected Pad

Surface Arc


( )12 1 2 12ooutv v kεε ε ε= − = −

12 12p pc ck εδ ε=

12pck ε

Test Rig: Data AnalysisWriting an EOM for the stator and taking an FFT nets

Given orthogonal excitations, we can solve for using

and likewise for the pad-rotor transfer functions using

where the x/y superscripts denote data recorded during orthogonal X/Y stator excitations.

ex xx xybx bx x

yey yx yyby by

F m A H H UUF m A H H

⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎩ ⎭⎣ ⎦⎪ ⎪⎩ ⎭

−=

−

1y yx x yxex ex xx xybx bx bx bx x xyy y xx x

y yey ey yx yyby by by by

F m A F m A H HU UU UF m A F m A H H

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

−− −

=− −

, ,

,,

, 1,

1, 1,

1, 1,

1, 1,

1, 1,

j jj j

k kj j j jc k c k

j j j jp kc k

j jj jp k c k

k k

yxc k c k x xyx

y yc c k c k

c p k p k

U UU U

c cξ

η ξ η ξφ φη ξ η ξη ηη ξ η ξξ

η ξη ξφ

φ φ

η η

ξ ξ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

Γ Γ

Γ Γ= =Γ Γ

Γ Γ

kpj,kΓ Q1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−

ijH

pj,kΓ

Results: Pivot Load-vs-Deflection

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-5

0

2

4

6

8

10

12

Deflection, δ (m)

Rad

ial P

ivot

For

ce (k

N)

Measured and Predicted Pivot Load vs. Deflection (δ)

FMeas. (δPad-Stator)

FMeas. (δRotor-Stator)

FPred.

DecreasingPivot Stiffness

Predicted Hertzian Contact Force

δ Measured with Rotor-Stator Probes

δ Measured with Pad-Stator Probes

Results: Pivot Stiffness

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-5

0

200

400

600

800

1000

1200

Deflection, δ (m)

Stif

fnes

s (M

N/m

)Measured and Predicted Pivot Stiffness vs. Deflection (δ)

KMeas. (δPad-Stator)

KMeas. (δRotor-Stator)

KPred.

DecreasingPivot Stiffness

Predicted Hertzian Contact Stiffness

δ Measured with Rotor-Stator Probes

δ Measured with Pad-Stator Probes

Results: Pivot Stiffness

21 4 17 3 13 2 87.321 10 -6.653 10 +3.339 10 +1.773 10 -64.44cf ξ δ δ δ δ= × × × ×

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-5

0

200

400

600

800

1000

1200

Deflection, δ (m)

Stif

fnes

s (M

N/m

)Accuracy of Quadratic Approximation of Pivot Load-vs-Deflection Curve

KMeas. (δPad-Stator)

KFit (δPad-Stator)=1.73×1013×δ+2.93×108

Overestimation of Pivot Stiffnessfor Small and Large Deflections

Dashed Line Obtained from a Quadratic Fit ofthe Pad-Stator Load-Versus-Deflection Curve

Results: Pad Bending StiffnessPivot insert design

Nonlinear bending stiffnessIncreases with increasing bending moment

12dε1ε2ε

pressf

pressd

2pressd

2pressd

4p

press pressc

f dM =

Press Experiment

Pivot Insert

gap

Pad

Contact

pcM

Region 1(light/no bending moments)

Region 2(heavy bending moments)

Results: FEA Validation

0 20 40 60 80 100 120 1400

100

200

300

400

500

600

700

Moment Mcp (N.m)

Stra

in ( μ

ε)Pad Strain (με) vs. Bending Moment (Mc

p)

ε12,Meas.

ε12,Fit,Region 1 =9.984×Mcp

-1.319

ε12,Fit,Region 2 =4.625×Mcp

+91.931

ε12,FEA,Composite

Composite FEA Prediction

Region

Decreasing PadStiffness

21

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Angular Location (°)

Nor

mal

ized

Rad

ial D

efle

ctio

n ( μ

m/N

.m)

Pad Deflection (μm) vs. Angular Location

cp,exp (Region 1)=2.09×10-6, kscp

=4.78×105

cp,exp (Region 2)=1.15×10-6, kscp

=8.69×105

cp,punif

(Region 1)=1.85×10-6, kscp

=5.39×105

cp,punif

(Region 2)=1.08×10-6, kscp

=9.23×105

Results: FEA Pad Bending Stiffness

Loadingkcpε12

(μm/με)

Experiment (Region 1) 0.2787

Experiment (Region 2) 0.3397

Uniform Pressure (Region 1) 0.2560

Uniform Pressure (Region 2) 0.2782

120.2671 /pck mε μ με=

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7x 106

Bending Moment (N.m)

Pad

Ben

ding

Stif

fnes

s (N

.m/m

)Pad Bending Stiffness vs Bending Moment

4400 RPM7300 RPM10200 RPM13100 RPMLinear Fit

Results: Pad Bending Stiffness

4 40=7.548 10 +9.983 10p psc ck M× × ×

Operating Bearing Clearance

T L

Large reduction in bearing clearance during experiments.Measured after shutdown with slow stator precession

Squashed clearances at large loadDecrease in clearance results in increased stiffness and damping predictionsClearance inversely proportional to average pivot surface temperatureCharacteristic thermal length

( ),0.396 / 34.7mm 1.37in11.4 /

bmc C

T charmat C

lμ

μαα

°

°= = =

-60 -40 -20 0 20 40 60 80 100

-60

-40

-20

0

20

40

60

80

100

X (μm)

Y ( μ

m)

Clearance Measurement

Cb = 69.89 μm at 23.5°C

Cb = 66.39 μm at 24.0°C

Cb = 53.40 μm at 61.4°C

Cb = 50.82 μm at 70.2°C

Cb = 48.69 μm at 74.6°C

20 30 40 50 60 70 8040

45

50

55

60

65

70

Rad

ial C

lear

ance

( μm

)

Clearance vs Average Pad Surface Temperature at Pivot

Average Pad Surface Temperature at the Pivot Location (°C)

cb(Tref) - αcb(Tavg-Tref), cb(24.4)=68.6, αcb=0.396 [μm/°C

]

Experimental Measurements

Static Eccentricity Measurement

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20-40

-20

0

20

40

60

80

X [μm]

Y [ μ

m]

Static Eccentricity Measurement

4441 rpm 7407 rpm10271 rpm13240 rpm

Cold

Bea

ring

Clea

ranc

eHo

t Bea

ring

Cle

aran

ce

IncreasingLoad

Static Pad Radial Displacement

0 500 1000 1500 2000 2500 3000 3500-40

-35

-30

-25

-20

-15

-10

-5

Unit Load (kPa)

Rad

ial P

ad D

ispl

acem

ent ξ

c [ μm

]Radial Pad Displacement vs Static Load

4441 rpm 7407 rpm10271 rpm13240 rpm

IncreasingSpeed

Static Pad Radial Displacement

0 500 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

220

Unit Load (kPa)

Pad

Cle

aran

ce c

p [ μm

]Measured Operating Pad Clearance vs Static Load

4441 rpm 7407 rpm10271 rpm13240 rpm

IncreasingSpeed

Installed Pad Clearance (cpo)

Installed Bearing Clearance (cbo)

0 50 100 150 200 250 300 350-2

0

2

4

6

8x 108

Re(

Hij) (

N/m

)

A) Real Part of Impedance Coefficients

0 50 100 150 200 250 300 350

0

1

2

3x 108

Frequency (Hz)

Im(H

ij) (N

/m)

B) Imaginary Part of Impedance Coefficients

Noticable Difference with 10× Heavier Pads

Journal vs. Bearing Impedances

Impedances for the test bearing Impedances for the test bearing having 10x heavier pads

0 50 100 150 200 250 300 3500

2

4

6

8x 108

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 350-1

0

1

2

3x 108

Frequency (Hz)

Im(H

ij) (N

/m)


Hxxjj

HxyjjHyxjj

Hyyjj

Hxxbb

HxybbHyxbb

Hyybb

Slight Difference at High Frequencies (Ω)

No Difference in Bearing vs. JournalImpedances at Low Frequencies Ω

0 50 100 150 200 250 300 350-1

-0.5

0

0.5

1x 109

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 350-3

-2

-1

0

1

2

3x 108

Frequency (Hz)

Im(H

ij) (N

/m)


Hxxjj

HxyjjHyxjj

Hyyjj

Hxxjb

HxyjbHyxjb

Hyyjb

Journal vs. Bearing Cross-ImpedancesFor the test bearing

Reaction forces result from relative rotor-stator motions.Previous comparisons are valid!

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

−

−jj jb jj jj

bbbj jj jj

H H H HH H H H

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Am

plitu

de ( μ

m/ μ

m)

(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|

Γφ, Meas. 1566kPa

Γξc

, Meas. 1566kPa

Γηc

, Meas. 1566kPa

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|

Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Γφη, Ratio of Normalized Pad Tiltto Transverse Rotor Motion

Depiction of TransverseRotor Motion

Γξc

η , Γηc

η , Ratio of Radial and Transverse Pad

Motions to Transverse Rotor Motion

Γηc

ξ , Ratio of Transverse Pad Motion

to Radial Rotor Motion

Depiction of RadialRotor Motion

Γφξ, Ratio of Normalized Pad Tilt

to Radial Rotor Motion

Γξc

ξ , Ratio of Radial Pad Motion ξc

to Radial Shaft Motion

Anatomy of the Pad-Rotor TF

Normalized Pad Tilt

25.4mmφ φ= ×

Pad-Rotor TF (4400 RPM, 10 Hz)

T L



Pad-Rotor TF vs unit load at 4400 rpm

Pad-Rotor Tracking AnomalyRatio of radial pad motion to radial shaft motion increases with increasing unit load.Ratio of pad tilt to radial shaft motion decreases with increasing unit load.Ratio of transverse pad motion to radial and transverse shaft motions is minimal

jφ

ηΓ

0 500

0.1

0.2

0.3

0.4

0.5

Am

plitu

de ( μ

m/ μ

m)

Γφ, Meas. 0kPa

Γξc

, Meas. 0kPa

Γηc

, Meas. 0kPa

Γφ, Meas. 1566kPa

Γξc

, Meas. 1566kPa

Γηc

, Meas. 1566kPa

Γφ, Meas. 3132kPa

Γξc

, Meas. 3132kPa

Γηc

, Meas. 3132kPa0 50 100 150 200 250 300 350

0

0.1

0.2

0.3

0.4

0.5

0.6


Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Am

plitu

de ( μ

m/ μ

m)


Γφη, Pad Tilt TF Anomaly onlyObserved at Zero Unit Load

Transverse Rotor Motion Induces SmallTransverse and Radial Pad Motions

Radial Rotor Motion InducesSmall Transverse Pad Motions

Γφξ, Pad Tilt TF Decreases

with Increasing Unit Load

Γξc

ξ , Radial Pad TF Increases



Pad-Rotor Tracking AnomalyRatio of radial pad motion to radial shaft motion increases with increasing unit load.Ratio of pad tilt to radial shaft motion increaseswith increasing unit loadRatio of transverse pad motion to radial and transverse shaft motions is minimalLess frequency dependence

0 500

0.1

0.2

0.3

0.4

0.5

Am

plitu

de ( μ

m/ μ

m)

Γφ, Meas. 0kPa

Γξc

, Meas. 0kPa

Γηc

, Meas. 0kPa

Γφ, Meas. 1566kPa

Γξc

, Meas. 1566kPa

Γηc

, Meas. 1566kPa

Γφ, Meas. 3132kPa

Γξc

, Meas. 3132kPa

Γηc

, Meas. 3132kPa

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Am

plitu

de ( μ

m/ μ

m)


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4


Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Γφξ, Pad Tilt TF Increases


Transverse Rotor Motion Induces SmallTransverse and Radial Pad Motion

Γξc

ξ , Radial Pad TF Increases with

Increasing Unit Load

Pad Tilt TF Anomaly onlyObserved at Zero Unit Load


Ratio of pad clearance change to radial shaft motion decreases with increasing unit load.The pad is much stiffer at high unit loads

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

Am

plitu

de ( μ

m/ μ

m)


Γcp

, Meas. 0kPa

Γcp, Meas. 783kPa

Γcp, Meas. 1566kPa

Γcp, Meas. 2350kPa

Γcp, Meas. 3132kPa

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|

Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Transverse Rotor Motion Induces SmallChanges in pad Clearance

Γcp

ξ , Ratio of in Pad Clearance Change to

Radial Rotor Motion Decreases withIncreasing Unit Load

TF AmplitudeMeas. vs. Pred.: 4400 rpm, 0 kPa

Tilt tracking anomaly not predicted.Radial pad motion predicted wellPad tilt due to radial rotor motion slightly underpredictedPredicted transverse pad motions are minimal

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Am

plitu

de ( μ

m/ μ

m)


Γφ, Meas. 0kPa

Γξc

, Meas. 0kPa

Γηc

, Meas. 0kPa

Γφ, Pred. 0kPa

Γξc

, Pred. 0kPa

Γηc

, Pred. 0kPa

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|

Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Radial and Transverse Pad Motionsdue to Transverse Rotor Motion

Pad Tilt Tracking Anomaly Not Predicted

Γφξ, Pad Tilt due to Radial Rotor

Motion is Underpredicted

Γξc

ξ , Radial Pad Motion due to Radial

Rotor Motion is Predicted Well

TF AmplitudeMeas. vs. Pred.: 4400 rpm, 3132 kPa

Pad tracking slightly overpredictedRadial pad motion predicted wellPad tilt due to radial rotor motion moderately underpredictedPredicted transverse pad motions are minimal

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Am

plitu

de ( μ

m/ μ

m)


Γφ, Meas. 3132kPa

Γξc

, Meas. 3132kPa

Γηc

, Meas. 3132kPa

Γφ, Pred. 3132kPa

Γξc

, Pred. 3132kPa

Γηc

, Pred. 3132kPa

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|

Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Γφη, Pad Tilt Tracking TransverseRotor Motion is Overpredicted

Γξc





Γηc

ξ , Predicted Transverse Pad Motion is Minimal


TF Amplitude Meas. vs. Pred.: 10200 rpm, 3132 kPa

Pad tracking still slightly overpredictedRadial pad motion predicted very wellPad tilt due to radial rotor motion still underpredictedPredicted transverse pad motions are minimal

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Am

plitu

de ( μ

m/ μ

m)


Γφ, Meas. 3132kPa

Γξc

, Meas. 3132kPa

Γηc

, Meas. 3132kPa

Γφ, Pred. 3132kPa

Γξc

, Pred. 3132kPa

Γηc

, Pred. 3132kPa

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4


Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)


Γηc

ξ , Predicted Transverse Pad Motion is Minimal



Γξc



Γφη, Pad Tilt Tracking Transverse

Rotor Motion Moderately Overpredicted

TF AmplitudeMeas. vs. Pred.:4400 rpm, 3132 kPa

Predicted change in pad clearance due to transverse rotor motion is minimalChange in pad clearance due to radial rotor motion predicted well at low frequencies, and slightly underpredicted at higher frequencies.

0 50 100 150 200 250 300 3500

0.05

0.1

0.15

0.2

Am

plitu

de ( μ

m/ μ

m)


Γcp

, Meas. 3132kPa

Γcp, Pred. 3132kPa

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Am

plitu

de ( μ

m/ μ

m)

Frequency (Hz)

Predicted Change in Pad Clearance due toTransverse Rotor Motion is Minimal

Γcp

ξ , Change in Pad Clearance due to Radial Rotor

Motion is Slightly Underpredicted at High Frequencies

TF Phase Meas. vs. Pred.: 4400 rpm, 1566 kPa

Sharp phase shift indicates that transverse pad motion is lightly damped (resonance not seen in amplitude)Phase of pad tilt and radial pad motion relative to radial rotor motion predicted well

0 50 100 150 200 250 300 350-200

-150

-100

-50

0

50

100

150

200

Pha

se ( °

)

(A) Phase of Pad Transfer Function Due to Transverse Rotor Motion ∠Γη

0 50 100 150 200 250 300 350-200

-150

-100

-50

0

50

100

150

200

Frequency (Hz)

Pha

se ( °

)

(B) Phase of Pad Transfer Function Due to Radial Rotor Motion ∠Γξ

Γφ, Meas. 1566kPa

Γξc

, Meas. 1566kPa

Γηc

, Meas. 1566kPa

Γφ, Pred. 1566kPa

Γξc

, Pred. 1566kPa

Γηc

, Pred. 1566kPa

Phase of Pad Tilt and Radial Motions due toRadial Rotor Motion are Predicted well

Sharp Phase Shift is Predictedat Lower Frequency than Measured

Sharp Phase Shift Suggests TransversePad Motion is Lightly Damped

Meas. vs. Pred. Brg. Impedances: 4400 rpm, 3132 kPa

Real part of direct impedance coefficients slightly overpredictedat high frequenciesImaginary part of direct impedance coefficients underpredicted at high frequencies

0 50 100 150 200 250 300 350

0

1

2

3

4

5

6

7x 108

Re(

Hij) (

N/m

)

(A) Real Part of Impedance Coefficients

0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 108

Frequency (Hz)

Im(H

ij) (N

/m)

(B) Imaginary Part of Impedance Coefficients

0-0.5

0

0.5

1

1.5

Im(H

ij

Hxx - Meas.Hyy - Meas.Hxy - Meas.Hyx - Meas.Hxx - Pred.Hyy - Pred.Hxy - Pred.Hyx - Pred.Speed, ω

0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 108

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 108

Frequency (Hz)

Im(H

ij) (N

/m)



Real and imaginary impedances are predicted quite wellSlight difference in the frequency dependence of the real part of Hij at high frequencies

0-0.5

0

0.5

1

1.5

Im(H

ij



Real and imaginary impedances are predicted quite well

0 50 100 150 200 250 300 350

0

1

2

3

4

5

6x 108

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 108

Frequency (Hz)

Im(H

ij) (N

/m)


0-0.5

0

0.5

1

1.5

Im(H

ij


0 50 100 150 200 250 300 3500

2

4

6

8

10

12

14

16

18x 108

Re(

Hij) (

N/m

)

A) Real Part of Impedance Coefficients in the Loaded Direction

0 50 100 150 200 250 300 3500

2

4

6

8

10

12

14x 108

Frequency (Hz)

Im(H

ij) (N

/m)

B) Imaginary Part of Impedance Coefficients in the Loaded Direction

Rigpad,pivot

Rigpad,pivot

Flexpad,Rigpivot

Rigpad,Flexpivot

Flexpad,pivot

Flexpad,Rigpivot

Rigpad,Flexpivot

Flexpad,pivot

Impact of Pad and Pivot Flexibility : 4400 rpm, 3132 kPa

Relative error in predictionPad Model Type kyy (%) cyy (%)

Flexpad,pivot 0.95 -14.10

Rigidpad,Flexpivot 35.53 77.93

Flexpad, Rigidpivot 88.88 136.33

Rigidpad,pivot 201.61 810.78

0 500

2

4

I

Hyy-Flexpad,piv ot

Hyy-Rigpad

,Flexpiv ot

Hyy-Flexpad

,Rigpiv ot

Hyy-Rigpad,piv ot

Hyy-Meas.Speed, ω

0 50 100 150 200 250 300 3500

2

4

6

8

10

12

14

16x 108

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 3500

1

2

3

4

5

6x 108

Frequency (Hz)

Im(H

ij) (N

/m)


Rigpad,pivot

Flexpad,Rigpivot

Rigpad,Flexpivot

Flexpad,pivot

Rigpad,pivot Flexpad,Rigpivot

Flexpad,pivot

Rigpad,Flexpivot

Impact of Pad and Pivot Flexibility : 10200 rpm, 3132 kPa


Flexpad,pivot 10.63 -2.15




0 500

2

4

I

Hyy-Flexpad,piv ot

Hyy-Rigpad

,Flexpiv ot

Hyy-Flexpad

,Rigpiv ot

Hyy-Rigpad,piv ot

Hyy-Meas.Speed, ω

0 50 100 150 200 250 300 3500

1

2

3

4

5

6x 108

Re(

Hij) (

N/m

)


0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

3.5

4x 108

Frequency (Hz)

Im(H

ij) (N

/m)


Rigpad,pivot

Flexpad,Rigpivot

Rigpad,Flexpivot Flexpad,pivot

Rigpad,pivot Flexpad,Rigpivot

Rigpad,Flexpivot

Flexpad,pivot

Impact of Pad and Pivot Flexibility: 10200 rpm, 783 kPa


Flexpad,pivot 4.59 5.41




0 500

2

4

I

Hyy-Flexpad,piv ot

Hyy-Rigpad

,Flexpiv ot

Hyy-Flexpad

,Rigpiv ot

Hyy-Rigpad,piv ot

Hyy-Meas.Speed, ω

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

Unit Load (kPa)

Per

cent

Rel

ativ

e E

rror i

n K

xyD

esta

biliz

ing

Relative Error in Magnitude of Destabilizing Cross-Coupled Stiffness Causing System Stability

4400 rpm - Subsynchronously Reduced4400 rpm - Synchronously Reduced10200 rpm - Subsynchronously Reduced10200 rpm - Synchronously Reduced

Error in Stability Prediction usingSynchronously Reduced Model

Error in Stability Prediction usingSubsynchronously Reduced Model

Subsynchronously reduced model results in 1% errorSynchronously reduced model results in 25% error. Error is worse at low loads.

.DestabK .BrgC.BrgK.BrgC.BrgK

0.25L 0.25L0.5L

( )( )( )

Speed rpm 4400 10200m 1.61 1.15

m 0.3 0.25mid

LR

Bearing model for Stability Calculation?

Synchronously reduced stiffness is 1%-6% lowerSynchronously reduced damping is 10%-15% higher

Frequency Dependent Damping is Predicted

0 500 1000 1500 2000 2500 3000 3500-6

-5

-4

-3

-2

-1

0

1

100 ×

[kij( ω

)-kij( Ω

)]/k ij( Ω

)

(A) Relative Error in Synchronously Reduced Stiffness

4400 rpm - kxx

4400 rpm - ky y

10200 rpm - kxx

10200 rpm - ky y

0 500 1000 1500 2000 2500 3000 35008

9

10

11

12

13

14

15

Unit Load (kPa)

100 ×

[cij( ω

)-cij( Ω

)]/c ij( Ω

)

(B) Relative Error in Synchronously Reduced Damping

4400 rpm - cxx

4400 rpm - cy y

10200 rpm - cxx

10200 rpm - cy y

0 50 100 150 200 250 300 350

0

0.5

1

1.5

2

2.5

3

x 105

Frequency (Hz)

Dam

ping

(Cij) (

N-s

/m)

Frequency Dependent Damping Coefficients

Cxx - Meas.Cyy - Meas.Cxy - Meas.Cyx - Meas.Cxx - Pred.Cyy - Pred.Cxy - Pred.Cyx - Pred.Speed, ω

Increasing Damping withIncreasing Excitation Frequency

Falloff in DampingPredicted

At high loads, yes!

Was frequency dependent damping measured?

Frequency dependent damping at 10200 rpm, 3132 kPa unit load

0 50 100 150 200 250 300 350

0

0.5

1

1.5

2

2.5

3

x 105

Frequency (Hz)

Dam

ping

(Cij) (

N-s

/m)

Frequency Dependent Damping Coefficients

Cxx - Meas.Cyy - Meas.Cxy - Meas.Cyx - Meas.Cxx - Pred.Cyy - Pred.Cxy - Pred.Cyx - Pred.Speed, ω

At lower loads, No!

Was frequency dependent damping measured?

Frequency dependent damping at 10200 rpm, 783 kPa unit load

Both the full bearing predictions and synchronously measured impedances underestimate system stability.

Predicted system stability with measured KC?

500 1000 1500 2000 2500 3000 35000.7

0.8

0.9

1

1.1

1.2

1.3

1.4x 107

Unit Load (kPa)

Kxy

Des

tabi

lizin

g (N/m

)

Magnitude of Destabilizing Cross-Coupled Stiffness Causing System Stability

10200 rpm - Subsynchronously Reduced Measured Data10200 rpm - Synchronously Reduced Measured Data10200 rpm - Full Bearing Model Predictions

Summary and ConclusionsOriginal Contributions:

Including perturbations of both the journal and bearing in TPJB analysisPerturbing pad tilt, radial and transverse pad motion, and changes in pad clearance (all previously perturbed in literature, but not in the same analysis).Though allowing for an arbitrary pad center of gravity was included in the analysis, this feature was insignificant for the bearing tested. Though previous researchers solved for pad motion as a function of rotor motion while reducing bearing impedances, this is the first work to define the rotor-pad transfer functions, and to suggest that comparisons between measured and predicted rotor-pad transfer functions may be used to rectify modeling deficiencies.Measuring static and dynamic pad rotations (tilt, pitch, yaw), pad translations (radial and transverse pivot motion), and changes in pad clearance.Converting measured pad motions into rotor-pad transfer functions, and comparing them to predicted rotor-pad transfer functions.Measuring bearing clearances using slow frequency circular excitation, and correlating these measured clearances to changes to pad surface temperatures.

Summary and ConclusionsStatic Measurements

Measured hot bearing clearances after shutdown are up to 30% smaller than the cold bearing clearance.Measured hot bearing clearance is inversely proportional to average surface temperature at each pad’s pivot location.Approximating reductions in clearance based on the expansion of a pad at elevated temperatures accounts for only ½ of the reduction in measured bearing clearance (note that this observation pertains to bearing clearances measured on a floating bearing test rig)Measured operating pad clearances were 60% larger than the installed pad clearance (this effect will tend to reduce the frequency dependence of measured bearing impedances)

Journal vs. bearing perturbations

For the test bearing, journal perturbed impedances were nearly the same as bearing perturbed impedances (slight differences between the two were only noted at higher frequencies).For the test bearing having 10×heavier pads, differences in journal and bearing perturbed impedances were significant at high frequencies.Comparing journal perturbed impedance predictions to impedances measured on a floating bearing test rig appears to be valid for bearings similar to the test bearing, but this may not necessarily be the case for larger bearings having heavier pads, or possibly gas bearings with significantly smaller impedances.

Summary and ConclusionsRotor-pad transfer functions

The rotor and pad have the same frequencies of motion. These motions occur at 1x, 2x, etc. in which the harmonics have an amplitude of 5-10% of the fundamental frequency.Pad tilt due to transverse rotor motion showed that the pad tracked the rotor, thus the pivot allowed the pad to rotate freely.The current work shows that pivot compliance allows for significant radial pad motion. Neglecting this degree of freedom produces large errors in predicted bearing impedances.A similar result was shown for pad flexibility; however, for the test bearing, pad flexibility appears to be less significant than radial pivot flexibility

Rotor-pad transfer functionsTransverse pad pivot motion was predicted and observed; however, this motion appears to be lightly damped, which suggests that it is caused by transverse compliance of the pivot, not slipping.In general, the rotor-pad transfer functions were predicted well; however, pad tilt due to radial rotor motion had a tendency to be underpredicted.

Bearing ImpedancesThe current work shows that when the rotor-pad transfer functions are predicted well, this resulted in a decent prediction of bearing impedances.The accuracy of the predicted bearing impedances was moderately good at 4400 rpm, and very good at 10200 rpm.

Summary and ConclusionsFull versus Reduced bearing models

Stability predictions using a subsynchronously reduced bearing model were within 1% of a full bearing model (explicitly containing all pad degrees of freedom).Stability predictions using a synchronously reduced model overestimated stability by as much as 25% compared to the full bearing model.This error in stability calculation resulted from an increase in predicted damping at synchronous frequencies relative to subsynchronously reduced damping (frequency dependent damping).

Frequency Dependent Damping

An increase in direct damping in the loaded direction was measured only for highly loaded operation, while this trend was predicted for all loads.Calculating stability using the full bearing model and synchronously measured coefficients were more conservative than stability calculations using subsynchronously measured bearing impedances

Unresolved IssuesWhy does predicted damping increase with frequency while measurements show the opposite?

AcknowledgementsTo my family for supporting me throughout my tenure as a student.To the Turbomachinery Research Consortium (TRC) for sponsoring this research. To my committee for their advice on this subject and the manner in which it is presented.To Chris Kulhanek and Gustavo Vignolo, for helping me to get the test rig in its current conditionTo Eddie Denk and all of the A&M TurbomachineryLaboratory staff.

T L

Questions?

T L Jason Wilkes (PhD?)

Thank You

Static Eccentricity Prediction

T L

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

60

70

80

Unit Load (kPa)

Y [ μ

m]Journal Eccentricity vs Static Load

Meas - 4441 rpmPred - 4400 rpmMeas - 7407 rpmPred - 7197 rpmMeas - 10271 rpmPred - 10182 rpmMeas - 13240 rpmPred - 13211 rpm

IncreasingSpeed

Power Loss

T L

Pad Tilt Angle

T L

0 500 1000 1500 2000 2500 3000 3500

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Unit Load (kPa)

φ [m

Rad

]Tilt Angle φ vs Static Load

4441 rpm 7407 rpm10271 rpm13240 rpm

IncreasingSpeed

Determination of Offset for Pad Tilt

T L

-60 -50 -40 -30 -20 -10 0 10 20 30-0.2

0

0.2

0.4

0.6

0.8

1

1.2φ versus ηj

ηj (μm)

φ (m

illira

dian

)

φMeas.

φFit =1.52×10-2×ηj+7.30×10-1

Offset = φMeas. at the Reference Position

Upper Left Corner of Pad Clearance Pentagon

Midpoint of the Loaded Pad(Midpoint of Top Side)

Upper Right Corner of Pad Clearance Pentagon

Traversing the Loaded (Top) Pad from Left to Right

050

100150

200250

300350

0

100

200

300

4000

0.1

0.2

0.3

0.4

0.5

Response Frequency (Hz)Excitation Frequency (Hz)

Nor

mal

ized

Pad

Tilt

, Γφ ( μ

m/ μ

m) Ratio of Normalized Pad Tilt to

Radial Rotor Motion

2× and 3×Excitation Response

1×, 2×, and 3×Synchronous Response

Tilt Waterfall Plot

T L

Additional Moment term due to Rolling Without Slipping

T L

, 0, ,c cf k c k f kM fη ξ η=−

, ,sin sinc c

rf k rh f k rh krrh

rr r r rη θ φ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

= = −

,cr

f k krrh

rr rθ φ= −

1 , 0, , 1,cosc c

r rhf k c k f k krrh

r rM f r rη ξ θ φ⎛ ⎞⎜ ⎟⎝ ⎠

=− −

Change in offset creates a moment

where the moment arm

Additional reaction moment

Pad atthk

(Tilt) kφ

Reference State

, , Generalcf k

O

ReferenceState

Contact Location

,cf kθ

,cf kη

Pad Tiltedthkan Angle kφ

rhrrr

,Contact Location, cf o k

O

, , Reference Statecf o k

OContact Location

,c kf ξ

,c kf ξ

Documents

MEASURED AND PREDICTED ROTOR-PAD TRANSFER …