MEASURED AND PREDICTED ROTOR-PAD TRANSFER …
Transcript of 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
Deflected PadSurface Arc
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
Deflected PadSurface 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
)
A) Real Part of Impedance Coefficients
0 50 100 150 200 250 300 350-1
0
1
2
3x 108
Frequency (Hz)
Im(H
ij) (N
/m)
B) Imaginary Part of Impedance Coefficients
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
)
A) Real Part of Impedance Coefficients
0 50 100 150 200 250 300 350-3
-2
-1
0
1
2
3x 108
Frequency (Hz)
Im(H
ij) (N
/m)
B) Imaginary Part of Impedance Coefficients
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 (4400 RPM, 166 Hz)
Pad-Rotor TF (4400 RPM, 342 Hz)
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
0.7(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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
with Increasing Unit Load
Pad-Rotor TF vs unit load at 10200 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 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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|
Am
plitu
de ( μ
m/ μ
m)
Frequency (Hz)
Γφξ, Pad Tilt TF Increases
with Increasing Unit Load
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
Pad-Rotor TF vs unit load at 4400 rpm
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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
ξ , Radial Pad Motion due to Radial
Rotor Motion is Predicted Well
Γφξ, Pad Tilt due to Radial Rotor
Motion is Underpredicted
Γηc
ξ , Predicted Transverse Pad Motion is Minimal
Radial and Transverse Pad Motionsdue to Transverse Rotor Motion
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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(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
Γηc
ξ , Predicted Transverse Pad Motion is Minimal
Γφξ, Pad Tilt due to Radial Rotor
Motion is Underpredicted
Γξc
ξ , Radial Pad Motion due to Radial
Rotor Motion is Predicted Well
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)
(A) Magnitude of Pad Transfer Function Due to Transverse Rotor Motion |Γη|
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
0.8(B) Magnitude of Pad Transfer Function Due to Radial Rotor Motion |Γξ|
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
)
(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
Meas. vs. Pred. Brg. Impedances: 10200 rpm, 783 kPa
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
Hxx - Meas.Hyy - Meas.Hxy - Meas.Hyx - Meas.Hxx - Pred.Hyy - Pred.Hxy - Pred.Hyx - Pred.Speed, ω
Meas. vs. Pred. Brg. Impedances: 10200 rpm, 3132 kPa
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
)
(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
3.5x 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 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
)
A) Real Part of Impedance Coefficients in the Loaded Direction
0 50 100 150 200 250 300 3500
1
2
3
4
5
6x 108
Frequency (Hz)
Im(H
ij) (N
/m)
B) Imaginary Part of Impedance Coefficients in the Loaded Direction
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
Relative error in predictionPad Model Type kyy (%) cyy (%)
Flexpad,pivot 10.63 -2.15
Rigidpad,Flexpivot 41.77 56.90
Flexpad, Rigidpivot 99.84 140.29
Rigidpad,pivot 176.65 512.99
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
)
A) Real Part of Impedance Coefficients in the Loaded Direction
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)
B) Imaginary Part of Impedance Coefficients in the Loaded Direction
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
Relative error in predictionPad Model Type kyy (%) cyy (%)
Flexpad,pivot 4.59 5.41
Rigidpad,Flexpivot 8.49 42.27
Flexpad, Rigidpivot 64.34 88.09
Rigidpad,pivot 51.16 181.93
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