2.76 multi-scale system design & manufacturing · 2021. 4. 28. · 2.76 multi-scale system...
TRANSCRIPT
2.76 Multi-scale System Design & Manufacturing
2.76 / 2.760 Lecture 12: Interfaces
2.76 Multi-scale System Design & Manufacturing
AnnouncementsSchedule TA Meetings
Thursday design review (Oct. 21 not Nov. 21)FRs & DPsConcepts & approachInitial modeling (HTMs, systematic, random, actuation, flexures)Sensitivity and prioritizationScheduling & risk/mitigation
2.76 Multi-scale System Design & Manufacturing
Homogeneous Transformation MatricesRotations:
For small θ, cos(θ)~1 & sin(θ)~θ
Order of multiplication is important for large θ
( )( ) ( )( ) ( )
1000
0cossin0
0sincos0
0001
xx
xxxB
AHθθ
θθθ
−=
( )
( ) ( )
( ) ( )1000
0cos0sin
0010
0sin0cos
yy
yy
yBAH
θθ
θθ
θ−
=
( )
( ) ( )( ) ( )
1000
0100
00cossin
00sincos
zz
zz
zBAH
θθ
θθ
θ
−
=
x
y
z
2.76 Multi-scale System Design & Manufacturing
Purpose of todayModeling of mechanical interfaces:
“Rigid”FlexibleRigid-flexible
Examples:“Rigid”: Kinematic couplings“Rigid”-Flexible: Flexure-couplings“Rigid”-Flexible: Quasi-kinematic coupling
Thursday: Manufacturing & assemblyCross-scale interfacesBolted joints
2.76 Multi-scale System Design & Manufacturing
Constraintconceptreview
2.76 Multi-scale System Design & Manufacturing
Exact constraintAt some scale, everything is a mechanism
Everything is compliant
What we’ve done: KinematicsArranging constraints in optimum topologyIdeal constraints & small motionsConstraints = lines (ideal)
Now: Kinematics and mechanicsModel & calculate stiffness/stress/motion
Figure: Layton Hales PhD Thesis, MIT.
2.76 Multi-scale System Design & Manufacturing
Constraint fundamentalsRigid bodies have 6 DOF
DOC = # of linearly independent constraints
DOF = 6 - DOC
A linear displacement can be visualized as a rotation about a point which is “far” away
Figure: Layton Hales PhD Thesis, MIT.
2.76 Multi-scale System Design & Manufacturing
Points on a constraint line move perpendicular to the constraint line
Constraints along thisline are equivalent
Statements
Figure: Layton Hales PhD Thesis, MIT.
Diagram removed for copyright reasons.Source: Blanding, D. L. Exact Constraint:
Machine Design using Kinematic Principles. New York: ASME Press, 1999.
2.76 Multi-scale System Design & Manufacturing
StatementsIntersecting, same-plane constraints are
equivalent to other same-plane intersecting constraints
Instant centers are powerful tool for visualization, diagnosis, & synthesis
2.76 Multi-scale System Design & Manufacturing
Abbe errorError due to magnified moment arm
r
2.76 Multi-scale System Design & Manufacturing
StatementsConstraints remove rotational degree of
freedom
Length of moment arm determines the quality of the rotational constraint
2.76 Multi-scale System Design & Manufacturing
StatementsParallel constraints may be visualized/treated
as intersecting at infinity
2.76 Multi-scale System Design & Manufacturing
Past / nextWe know how to lay out constraints to
achieve a given behavior
Need to understand how to model and optimize them
Reality checks and useful numbers for decision making
2.76 Multi-scale System Design & Manufacturing
“Rigid” Constraints
2.76 Multi-scale System Design & Manufacturing
Generic mechanical interfaceGeometry & topology
Relative formSurface finish
MaterialModulus, Poisson’s ratio, stressOxide & contaminationSurface energy
FlowsMass (wear)Momentum (interface force)Energy (elastic, thermal, etc…)
A
B
2.76 Multi-scale System Design & Manufacturing
Local geometry
Waviness can be due to:Clamping forcesLow frequency vibrationsEquipment set upBearing errors
Roughness can be due to:High frequency vibrationsTool geometry and rubbingVariations in materialChip contact
Shape / form
Waviness
Surfaceroughness
2.76 Multi-scale System Design & Manufacturing
Exact constraint couplingsExact constraint:
Constraint points = DOF to be constrainedDeterministic saves $Balls (inexpensive)Grooves (more difficult to make)
Kinematic design the issues are:KNOW what is happening in the systemMANAGE forces, deflections, stresses and friction
Balls
V groove
Tetrahedralgroove
Figures: Layton Hales PhD Thesis, MIT.
2.76 Multi-scale System Design & Manufacturing
Passive kinematic couplingsFabricate and forget: no active elementsGoal: Repeatable location
(1/4 micron is the norm)
What is important?Contact point & contact force directionContact stressStiffnessFriction & hysteresisOrder of engagement
Wear in period
Trial #
Cen
troi
d D
ispl
acem
ent [
µm
]
Radial (x2 +y2)0.5 repeatability
0.5
1.0
1.5
2.0
0 10 20 30
Trial #
Cen
troi
d D
ispl
acem
ent [
µm
]
Radial (x2 +y2)0.5 repeatability
0.5
1.0
1.5
2.0
0 10 20 30
Y error motion
-0.1
0.0
0.1
0.2
0.3
0 200 400 600 800 1000Cycle
Y [ µ
m]
No flexures With flexures
Figure: Layton Hales PhD Thesis, MIT.
2.76 Multi-scale System Design & Manufacturing
Common mechanical interfaces
Elastic averagingElastic averaging Passive kinematicPassive kinematicCompliant kinematicCompliant kinematic QuasiQuasi--kinematickinematic Active kinematicActive kinematic
AccuracyAccuracy RepeatabilityRepeatability
Desired Position
Elastic BElastic B
Passive KCPassive KC Active KCActive KC
Elastic CElastic CElastic AElastic A
QuasiQuasi--KCKC
ErrorError
Accuracy & repeatabilityAccuracy & repeatability
ErrorError
2.76 Multi-scale System Design & Manufacturing
Minimize variation
System
EnvironmentchangesPeople
Desired output
Materialproperties Stress
Actual output
Forces Vibration Procedure Friction
Measured output
PerceivedError
Realerror
2.76 Multi-scale System Design & Manufacturing
Instant center can help you identify how to best constrain or free up a mechanism
*Pictures from Precision Machine Design
Stability & balanced stiffness
PoorInstant center ifball 1 is removedGood
1||
|| <<⋅→⋅ ⊥
⊥δδ
δ CMCMKK
k
Diagram removed for copyright reasons.Source: Alex Slocum, Precision Machine Design.
2.76 Multi-scale System Design & Manufacturing
Perspective on couplingsIdeal: Couplings are staticReality: Deflections = motion
How/why they moveFriction and mating hysteresisContact deflections
Source of error motion
BA
λB
A
λ
2.76 Multi-scale System Design & Manufacturing
Distance of approach (δn)
Max shear stress occurs below surface, in the member with largest R
δr
δzδn
δl
Point AInitial Position ofBall’s Far Field Point
Final Position ofBall’s Far Field Point
r
z
l
n
Point BFinal True Groove’sFar Field Point
Initial Position ofGroove’s Far Field Point
Initial Contact Point
Final Contact Point
Max τ
Contact Cone
θc
2.76 Multi-scale System Design & Manufacturing
Model ball-groove contacts as 6 balls on flats
Important relationships for ball-flat contact
Solving for coupling errors
Contact pressure Stiffness[normal to groove surface]
2
3 1.5 for metals2 tensile
Fqa
σπ
= ⋅ < ⋅1
32
2
916n
e
FR E
δ⎛ ⎞
= ⋅⎜ ⎟⋅⎝ ⎠
1 12 22n ek R Eδ= ⋅ ⋅ ⋅
F
Distance of approach
F
F
31
43
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅=
e
en
ERFa
2.76 Multi-scale System Design & Manufacturing
System of six contact points
Ideal in-plane constraints Real in-plane constraints
2.76 Multi-scale System Design & Manufacturing
Modeling round interfaces
31
2
2
169
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅=
ee
nn ER
Fδ
( ) ( ) 5.05.02 neenn ERk δδ ⋅⋅⋅=
Important scaling law
( ) ( ) 31
32
31
Constant neenn FERFk ⋅⋅⋅=
ormajorormajor
e
RRRR
R
min22min11
11111
+++=
2
22
1
21 11
1
EE
Ee ηη −+
−=
Contact stiffness
0
05
10
20
30
0 250 500 750 1000Fn [N]
k [N
/mic
ron]
Poisson’s ratio
Young’s modulus
Equivalent modulus
Equivalent radius
2.76 Multi-scale System Design & Manufacturing
Preload keeps the fixture parts together
Increase preload:increase ball-groove in contactincrease contact stiffness
Stiffness scales as:
Preload should have high repeatability
Importance of preload
In-plane stiffness
0
20
40
60
80
0 250 500 750 1000Preload [N]
k [N
/mic
ron]
31
2
2
169
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅=
ee
nn ER
Fδ
( ) ( ) 5.05.02 neenn ERk δδ ⋅⋅⋅=
( )n
nn ddFkδ
δ =Important scaling law
( ) ( ) 3126
1Constant neenn FERFk ⋅⋅⋅=
31
43
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅=
e
en
ERFa
2.76 Multi-scale System Design & Manufacturing
Force balance (3 equations)
Moment balance (3 equations)
Given geometry, materials,preload force, error force,solve for local distance of approach
Load balance: Ford and moment
( ) ( )6_5_4_3_2_1_0 BallBallBallBallBallBallErrorpreloadrelative FFFFFFFFF +++++++==Σ
Ball far-field point
Groove far-field points
Constraint
( ) ( )6_6_5_5_4_4_3_3_2_2_1_1_ BallBallBallBallBallBallBallBallBallBallBallBallErrorerrorpreloadpreloadrelative FrFrFrFrFrFrFrFrM ×+×+×+×+×+×+×+×=Σ
( ) ( ) ( ) ( ) iBalliBallErrorerrorpreloadpreloaderrorpreloadiBallirelative FrFrFrMMMM ___16 ×Σ+×+×=++Σ= =
A B
2.76 Multi-scale System Design & Manufacturing
Ball motions: Displacements
iABall
iABallee
niABalln n
ER
F_
_
31
2
2
__ ˆ169
⋅⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅⋅−=Σδ
Ball far-field point
Groove far-field points
Constraint
iBBalliABalliBall ___ δδδ +=Σ
iBBall
iBBallee
niBBalln n
ER
F_
_
31
2
2
__ ˆ169
⋅⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅⋅−=Σδ
B1
B3
B2
A B
2.76 Multi-scale System Design & Manufacturing
Coupling motionsGiven 3 points (e.g. ball centers)
(1) Define centroid of the points & attach CS(2) Define plane equation & normal vector at CS
Compliance errors cause plane/normal to shift
Differences in coefficients give:Centroid (δx, δy, θz)Normal vector (θx, θy, δz)
For small θiεz ~ θisin(θi)~θicos (θi) ~1
2.76 Multi-scale System Design & Manufacturing
Step 1: Analyze ball center displacementsStep 2: Obtain centroid displacements
Translations: δx, δy, δz Rotations: εx, εy , εzStep 3: Calculating error at any point
Solving for coupling errors
B1
B3
B2
Error at i Centroid errors Abbe arm
∆ x
∆ y
∆ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
1
ε z
ε y−
0
ε z−
1
ε x
0
ε y
ε x−
1
0
δ x
δ y
δ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
⋅
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
−
i
i
i
i
i
i
i
i
i
F
Fixture Loading Deflection Errors
2.76 Multi-scale System Design & Manufacturing
Solving for coupling errors
B1
B3
B2
∆ x
∆ y
∆ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
1
ε z
ε y−
0
ε z−
1
ε x
0
ε y
ε x−
1
0
δ x
δ y
δ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
⋅
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
−
i
i
i
i
i
i
i
i
i
Error HTMAmplification
ArmDistance from
centroid
2.76 Multi-scale System Design & Manufacturing
Environmental: TemperatureThermal expansion
Which is better with respect to in-plane errors if:∆Ttop > ∆Tbottom∆Ttop = ∆Tbottom∆Ttop < ∆Tbottom
xTx ⋅∆⋅=∆ αα
µm/ o C/m µ inch/oF/inchCast iron 13 7Carbon steel [AISI 1005] 13 7Stainless steel [AISI 303] 18 10Aluminum 6061 T6 24 13Yellow brass 20 11Delrin (polymer) 85 47
Balls
V groove
Tetrahedralgroove
Figure: Layton Hales PhD Thesis, MIT.
2.76 Multi-scale System Design & Manufacturing
Example: Surface finish trace & metric
∫⋅=L
a dxyL
R0
1
y
x
Mea
n
2.76 Multi-scale System Design & Manufacturing
Surface finishProducing fine surface is expensive/time
consuming
Reference chart:
2000 1000 250500 125 63 32 16 8 4 2 1 ½
Sawing
Drilling
Milling
Turning
Grinding
Polishing
Process
Common surface roughness (Ra in micro-inches)
2.76 Multi-scale System Design & Manufacturing
Contact problemsSurface topology (finish):
50 cycle repeatability ~ 1/3 µm RaFriction depends on surface finish!Finish should be a design specSurface may be brinelled if possible
Wear and Fretting:High stress + sliding = wearMetallic surfaces = frettingUse ceramics if possible (low µ and high strength)Dissimilar metals avoids “snowballing”
Friction:Friction = Stored energy, over constraintFlexures can help (see right)Lubrication (high pressure grease) helpsBeware of settling time
BA
λB
A
λ
Mate n Mate n + 1
Wea
r on
Gro
ove
Image removed for copyright reasons. See Figure 4, “Ball in a V Groove…” in
Schouten, et. al., “Design of a kinematic coupling for precision applications.”Precision Engineering, vol. 20, 1997.
2.76 Multi-scale System Design & Manufacturing
Repeatability with & w/o lubricationThe trend of
the data is important
Magnitude depends on coupling design and test conditions
Number of Trials
Radial Repeatability (Unlubricated)2
0D
ispl
acem
ent, µm
600
Number of Trials
Radial Repeatability (Lubricated)
600
Dis
plac
emen
t, µm
2
0
Slocum, A. H., Precision Engineering, 1988: Kinematic couplings for precision fixturing–Experimental determination of repeatability and stiffness
Graph removed for copyright reasons. See Slocum, A. H. “Kinematic couplings for precision fixturing:
Experimental determination of repeatability and stiffness.” Precision Engineering, 1988.
Graph removed for copyright reasons. See Slocum, A. H. “Kinematic couplings for precision fixturing:
Experimental determination of repeatability and stiffness.” Precision Engineering, 1988.
2.76 Multi-scale System Design & Manufacturing
Centroid εy error motion[ Stability test ]
-500
-400
-300
-200
-100
0
100
200
300
400
500
0 500Cycle
Dis
plac
emen
t [µ
radi
an]
Centroid δy error motion[ Stability test ]
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
80.00
0 100 200 300 400 500 600 700Cycle
Dis
plac
emen
t [m
icro
ns]
Effect of Hard CoatingsWhat should be the affect of adding TiN
coating?TiN = low frictionTiN = high surface energy
Results of non-lubricated, TiN coated tests
2.76 Multi-scale System Design & Manufacturing
Flexible constraints
2.76 Multi-scale System Design & Manufacturing
Why? Want less than 6 DOF constraintE.g.: Planar alignment ∆z for sealing/stiffness
Add compliance to permit z motionThis is equivalent to adding a DOF
Care must be taken to make sureCompliant direction is not in a sensitive directionParasitic errors in sensitive directions are acceptable
Adding and taking away constraints
Compliant ↕ Stiff ↔
Compliant ↕ Stiff ↔
Compliant ↕ Stiff ↔
2.76 Multi-scale System Design & Manufacturing
Out-of-plane use of flexuresCompliant ↕ Stiff ↔
Compliant ↕ Stiff ↔
Compliant ↕ Stiff ↔
Compliant ↕ Stiff ↔ SL
Fnest
Ferror
z z
Compliant ↕ Stiff ↔
L - S
Fnest
Ferror
Rigid PassiveCompliance
2.76 Multi-scale System Design & Manufacturing
System of six contact points
Ideal in-plane constraints Real in-plane constraints
2.76 Multi-scale System Design & Manufacturing
CharacteristicsStroke ≤ 0.25 inchesRepeatability 5 -10 micronsBall movement in non-sens. direction
F
t
L
w
δ
Cost$ 10 - 200
Design Issues (flexure)1. Kr ~
2. Tolerances affect Kr
w2
t2
Applications/Processes1. Assembly2. Casting
Example: Cantilevered balls
U.S. Patent 5, 678, 944, Slocum, Muller, Braunstein
Courtesy of Prof. Alex Slocum. Used with permission.
2.76 Multi-scale System Design & Manufacturing
Characteristics1. Repeatability (2.5 micron)2. Stroke ~ 0.5 inches
Cost$ 2000
Design Issues (flexures)1. Kr =
2. Press fit tolerances
KguideKspring
Applications/Processes1. Assembly2. Casting3. Fixtures
Axial bushings with spring preloadU
.S. P
aten
t 5, 6
78, 9
44,
Slo
cum
, Mul
ler,
Bra
unst
ein
Courtesy of Prof. Alex Slocum. Used with permission.
2.76 Multi-scale System Design & Manufacturing
System of six contact points
Ideal in-plane constraints Real in-plane constraints
Flexure arms
2.76 Multi-scale System Design & Manufacturing
Flexure arms
Flexure grooves reduce friction effect
1
23
Y displacements
-50
-25
0
25
50
-50 -25 0 25 50Command [microns]
Mea
sure
d [m
icro
ns]
Y error motion
-0.1
0.0
0.1
0.2
0.3
0 200 400 600 800 1000Cycle
Y [ µ
m]
+ =
yx
z
2.76 Multi-scale System Design & Manufacturing
“Rigid”-flexible constraints
2.76 Multi-scale System Design & Manufacturing
Exact & near kinematic constraint
KinematicContact: Point / small areaRepeatability: 80 nmCost: $10s to $1000sSealing: With flexuresTime: > 60 minutes
Quasi-kinematicArc250 nm$ 0.75Integrated< 20 s
Ball
V groovesBall
Axi symmetricgroove
Groove Seat
Side Reliefs
[US Pat. 6 193 430]
2.76 Multi-scale System Design & Manufacturing
x
y
Contact Point
Ball
Groove SurfacePeg Surface
Relief
Cone Seat
QKC methods vs kinematic method
Relief
Components and Definitions
Force Diagrams
2.76 Multi-scale System Design & Manufacturing
Contact angle and degree of over constraint
QKC constraint compared to “ideal” coupling
x
y
Constraint in QKCs
θcontact θcontact
QKC Ideal KC
Bisector
Bisector||
planes bisecting tolar perpendicuStiffnessplanes bisecting toparallelStiffness
⊥==k
kCMk
2.76 Multi-scale System Design & Manufacturing
Step 1: Analyze ball center displacementsStep 2: Obtain centroid displacements
Translations: δx, δy, δz Rotations: εx, εy , εzStep 3: Calculating error at any point
Solving for coupling errors
B1
B3
B2
Error at i Centroid errors Abbe arm
∆ x
∆ y
∆ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
1
ε z
ε y−
0
ε z−
1
ε x
0
ε y
ε x−
1
0
δ x
δ y
δ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
⋅
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
−
i
i
i
i
i
i
i
i
i
F
Fixture Loading Deflection Errors
2.76 Multi-scale System Design & Manufacturing
ResultantForces[ni & Fi]
AppliedLoads
[Fp & Mp]
Deflectionsδ -> ∆r
RelativeError
GeometryMaterial
New model for QKC stiffness
QKC ModelGeometry
Material
Displacements
Contact Stiffnessfn(δn)
Force/Torque
Stiffness
2.76 Multi-scale System Design & Manufacturing
Force per unit of line contact
0
100 000
200 000
300 000
400 000
500 000
0 50 100 150 200Distance of Approach [microns]
Con
tact
Uni
t For
ce [N
/m]
Initial Loading
Re-
load
Unl
oad
Contact mechanics( ) ( )[ ]brnrn Kf θδθ =k
n l
i
^
Rc
^
^
^
nl
s
^^
Rotating CS
FEA ProfileHertz Profile
Surface Contact Pressure - Elastic Hertz
0
50
100
150
200
-0.02 -0.01 0 0.01 0.02Distance in L, in
Con
tact
Pre
ssur
e, k
psi
2.76 Multi-scale System Design & Manufacturing
QKC Performance
Probe 2Probe 3
Probe 1
x
y
1st Component
2nd Component
QKC In-plane Repeatability
0.0
0.5
1.0
1.5
2.0
0 5 10 15 20 25 30-8
-6
-4
-2
0
2
4
6
8Centroid DisplacementCentroid Rotation
Rad
ial C
entr
oid
Dis
plac
emen
t [ µ
m ] C
entroid Rotation (θz) [ µ
radians]
Trial
2.76 Multi-scale System Design & Manufacturing
Characteristics:Ford V6 300,000 / year Cycle time: < 30 s
Coupling+ others
Process
Practical use: DuratecTM assembly
Repeatability0.01 µm 0.10 µm 1.0 µm 10 µm
Elastic averagingCompliant kinematicQuasi-kinematicActive kinematicPassive kinematic
Rotation
Oil
Sensitive to misalignment
10 µm
5 µm
0 µm
Photos by Prof. Martin Culpepper, courtesy of Ford Motor Company. Used with permission.
2.76 Multi-scale System Design & Manufacturing
Components
Pinned joint engine QKC equipped engine
Block
Bedplate
Assembly Bolts
Goal: 5 micron assembly
Block Bedplate
Coupling+ others
Process
10 µm
5 µm
Photos by Prof. Martin Culpepper, courtesy of Ford Motor Company. Used with permission.
2.76 Multi-scale System Design & Manufacturing
DuratecTM QKC in detail
Photos by Prof. Martin Culpepper, courtesy of Ford Motor Company. Used with permission.
2.76 Multi-scale System Design & Manufacturing
Kinematic elements Manufacturing
Constraint diagrams Metrics
+
+
OR
Low-cost couplings
Example QKC Joint Metrics
0.2
0.4
0.6
0.8
1.0
0 30 60 90 120 150 180
θcontact
CM
50
100
150
200
250
Kr m
ax [ N/µm ]
CM Kr max
Diagrams removed for copyright reasons.“Cast + Form Tool = Finished”
2.76 Multi-scale System Design & Manufacturing
Solving for coupling errors
B1
B3
B2
Error at i Centroid errors Abbe arm
∆ x
∆ y
∆ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
1
ε z
ε y−
0
ε z−
1
ε x
0
ε y
ε x−
1
0
δ x
δ y
δ z
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
⋅
X xc−
Y y c−
Z z c−
1
⎛⎜⎜⎜⎜⎝
⎞
⎟⎟
⎠
−
i
i
i
i
i
i
i
i
i
Error HTMAmplificationArm
Centroiddisplacement
2.76 Multi-scale System Design & Manufacturing
Load sources: AlignmentForces/loads can varyResult is variation in position
yxJournal 4
Tf2
Tf1 Tf3
δy at Journal 4
0
1000
2000
-0.515 -0.455 -0.395 -0.335 -0.275
microns
Freq
uenc
y
Photos by Prof. Martin Culpepper, courtesy of Ford Motor Company. Used with permission.
2.76 Multi-scale System Design & Manufacturing
Form-in-place process
Groove surface after burnishing processElastic recovery restores gap
Assemble Mate Pre-load Recovery
microns
mic
rons