2.76 Multi-scale System Design & Manufacturing · 2021. 4. 28. · 2.76 Multi-scale System Design & Manufacturing Announcements Schedule TA Meetings Thursday design review (Oct. 21

2.76 Multi-scale System Design & Manufacturing

2.76 / 2.760 Lecture 12: Interfaces


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


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


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


Constraintconceptreview


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.


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



Points on a constraint line move perpendicular to the constraint line

Constraints along thisline are equivalent

Statements


Diagram removed for copyright reasons.Source: Blanding, D. L. Exact Constraint:

Machine Design using Kinematic Principles. New York: ASME Press, 1999.


StatementsIntersecting, same-plane constraints are

equivalent to other same-plane intersecting constraints

Instant centers are powerful tool for visualization, diagnosis, & synthesis


Abbe errorError due to magnified moment arm

r


StatementsConstraints remove rotational degree of

freedom

Length of moment arm determines the quality of the rotational constraint


StatementsParallel constraints may be visualized/treated

as intersecting at infinity


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


“Rigid” Constraints


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


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


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.


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



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


Minimize variation

System

EnvironmentchangesPeople

Desired output

Materialproperties Stress

Actual output

Forces Vibration Procedure Friction

Measured output

PerceivedError

Realerror


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.


Perspective on couplingsIdeal: Couplings are staticReality: Deflections = motion

How/why they moveFriction and mating hysteresisContact deflections

Source of error motion

BA

λB

A

λ


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


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


System of six contact points

Ideal in-plane constraints Real in-plane constraints


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


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


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


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


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


Step 1: Analyze ball center displacementsStep 2: Obtain centroid displacements

Translations: δx, δy, δz Rotations: εx, εy , εzStep 3: Calculating error at any point


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



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


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



Example: Surface finish trace & metric

∫⋅=L

a dxyL

R0

1

y

x

Mea

n


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)


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.


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.


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


Flexible constraints


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 ↔




Out-of-plane use of flexuresCompliant ↕ Stiff ↔



Compliant ↕ Stiff ↔ SL

Fnest

Ferror

z z


L - S

Fnest

Ferror

Rigid PassiveCompliance





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.


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.




Flexure arms


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


“Rigid”-flexible constraints


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]


x

y

Contact Point

Ball

Groove SurfacePeg Surface

Relief

Cone Seat

QKC methods vs kinematic method

Relief

Components and Definitions

Force Diagrams


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


Step 1: Analyze ball center displacementsStep 2: Obtain centroid displacements

Translations: δx, δy, δz Rotations: εx, εy , εzStep 3: Calculating error at any point


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

F

Fixture Loading Deflection Errors


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


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


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


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.


Components

Pinned joint engine QKC equipped engine

Block

Bedplate

Assembly Bolts

Goal: 5 micron assembly

Block Bedplate

Coupling+ others

Process

10 µm

5 µm



DuratecTM QKC in detail



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”



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 HTMAmplificationArm

Centroiddisplacement


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



Form-in-place process

Groove surface after burnishing processElastic recovery restores gap

Assemble Mate Pre-load Recovery

microns

mic

rons

Documents

2.76 Multi-scale System Design & Manufacturing · 2021. 4. 28. · 2.76 Multi-scale System Design & Manufacturing Announcements Schedule TA Meetings Thursday design review (Oct. 21