2.76 multi-scale system design & manufacturing · 2021. 4. 28. · 2.76 multi-scale system...

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

( )( ) ( )( ) ( )

0cossin0

0sincos0

AHθθ

θθθ

( ) ( )

( ) ( )1000

0cos0sin

0sin0cos

( ) ( )( ) ( )

00cossin

00sincos

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

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…)

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

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 #

Radial (x2 +y2)0.5 repeatability

0 10 20 30

Trial #

Radial (x2 +y2)0.5 repeatability

0 10 20 30

Y error motion

0 200 400 600 800 1000Cycle

Y [ µ

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

|| <<⋅→⋅ ⊥

⊥δδ

δ CMCMKK

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

Distance of approach (δn)

Max shear stress occurs below surface, in the member with largest R

δzδn

Point AInitial Position ofBall’s Far Field Point

Final Position ofBall’s Far Field Point

Point BFinal True Groove’sFar Field Point

Initial Position ofGroove’s Far Field Point

Initial Contact Point

Final Contact Point

Max τ

Contact Cone

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]

3 1.5 for metals2 tensile

= ⋅ < ⋅1

δ⎛ ⎞

= ⋅⎜ ⎟⋅⎝ ⎠

1 12 22n ek R Eδ= ⋅ ⋅ ⋅

Distance of approach

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅=

System of six contact points

Ideal in-plane constraints Real in-plane constraints

Modeling round interfaces

⎟⎟⎠

⎞⎜⎜⎝

⋅⋅=

( ) ( ) 5.05.02 neenn ERk δδ ⋅⋅⋅=

Important scaling law

( ) ( ) 31

Constant neenn FERFk ⋅⋅⋅=

ormajorormajor

min22min11

Ee ηη −+

Contact stiffness

0 250 500 750 1000Fn [N]

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

0 250 500 750 1000Preload [N]

⎟⎟⎠

⎞⎜⎜⎝

⋅⋅=

( ) ( ) 5.05.02 neenn ERk δδ ⋅⋅⋅=

nn ddFkδ

δ =Important scaling law

( ) ( ) 3126

1Constant neenn FERFk ⋅⋅⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅=

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= =

Ball motions: Displacements

iABall

iABallee

niABalln n

__ ˆ169

⋅⎟⎟⎟

⎜⎜⎜

⋅⋅−=Σδ

Ball far-field point

Groove far-field points

Constraint

iBBalliABalliBall ___ δδδ +=Σ

iBBall

iBBallee

niBBalln n

__ ˆ169

⋅⎟⎟⎟

⎜⎜⎜

⋅⋅−=Σδ

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

Error at i Centroid errors Abbe arm

⎛⎜⎜⎜⎜⎝

⎟⎟

ε y−

ε z−

ε x−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

Fixture Loading Deflection Errors

⎛⎜⎜⎜⎜⎝

⎟⎟

ε y−

ε z−

ε x−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

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

V groove

Tetrahedralgroove

Example: Surface finish trace & metric

∫⋅=L

a dxyL

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

Mate n Mate n + 1

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

ent, µm

Number of Trials

Radial Repeatability (Lubricated)

t, µm

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 ]

0 500Cycle

Centroid δy error motion[ Stability test ]

-80.00

-60.00

-40.00

-20.00

0 100 200 300 400 500 600 700Cycle

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

Ferror

Rigid PassiveCompliance

CharacteristicsStroke ≤ 0.25 inchesRepeatability 5 -10 micronsBall movement in non-sens. direction

Cost$ 10 - 200

Design Issues (flexure)1. Kr ~

2. Tolerances affect Kr

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

t 5, 6

Courtesy of Prof. Alex Slocum. Used with permission.

Flexure arms

Flexure grooves reduce friction effect

Y displacements

-50 -25 0 25 50Command [microns]

Y error motion

0 200 400 600 800 1000Cycle

Y [ µ

“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

V groovesBall

Axi symmetricgroove

Groove Seat

Side Reliefs

[US Pat. 6 193 430]

Contact Point

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

Constraint in QKCs

θcontact θcontact

QKC Ideal KC

Bisector

Bisector||

planes bisecting tolar perpendicuStiffnessplanes bisecting toparallelStiffness

⊥==k

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

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

⎛⎜⎜⎜⎜⎝

⎟⎟

ε y−

ε z−

ε x−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

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

100 000

200 000

300 000

400 000

500 000

0 50 100 150 200Distance of Approach [microns]

Initial Loading

Contact mechanics( ) ( )[ ]brnrn Kf θδθ =k

Rotating CS

FEA ProfileHertz Profile

Surface Contact Pressure - Elastic Hertz

-0.02 -0.01 0 0.01 0.02Distance in L, in

QKC Performance

Probe 2Probe 3

Probe 1

1st Component

2nd Component

QKC In-plane Repeatability

0 5 10 15 20 25 30-8

8Centroid DisplacementCentroid Rotation

t [ µ

entroid Rotation (θz) [ µ

radians]

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

Sensitive to misalignment

10 µm

Photos by Prof. Martin Culpepper, courtesy of Ford Motor Company. Used with permission.

Components

Pinned joint engine QKC equipped engine

Bedplate

Assembly Bolts

Goal: 5 micron assembly

Block Bedplate

Coupling+ others

Process

10 µm

DuratecTM QKC in detail

Kinematic elements Manufacturing

Constraint diagrams Metrics

Low-cost couplings

Example QKC Joint Metrics

0 30 60 90 120 150 180

θcontact

ax [ N/µm ]

CM Kr max

Diagrams removed for copyright reasons.“Cast + Form Tool = Finished”

⎛⎜⎜⎜⎜⎝

⎟⎟

ε y−

ε z−

ε x−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

X xc−

Y y c−

Z z c−

⎛⎜⎜⎜⎜⎝

⎟⎟

Error HTMAmplificationArm

Centroiddisplacement

Load sources: AlignmentForces/loads can varyResult is variation in position

yxJournal 4

Tf1 Tf3

δy at Journal 4

-0.515 -0.455 -0.395 -0.335 -0.275

microns

Form-in-place process

Groove surface after burnishing processElastic recovery restores gap

Assemble Mate Pre-load Recovery

microns

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