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2.72 Elements of Mechanical Design Spring 2009

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2.72 Elements of

Mechanical Design

Lecture 09: Alignment

Schedule and reading assignment Quiz

� Thursday: Hale 6.1

� Soon: Bolted joint qualifying quiz

Topics � Lab notebooks � Alignment methods � Kinematic coupling grade bump = ½ grade for use/design

Reading assignment • Read: 8.2 • Examples: All in 8.2

© Martin Culpepper, All rights reserved 2

Lab notebooks Technical quality/quantity

� Appropriate equations, codes � Units � Important results highlighted/boxed/noted/explained

Graphical quality/quantity � Appropriate sketches/pictures

� Pasted CAD/etc…

Archival quality � Can this be copied?

� Understood by others?

Best practices � Dating and number of pages

� Permanent pen

� No blank spaces (X out)


Ideal alignment interface Repeatable

Accuracy

Stiffness (sensitive?)

Load capacity

“Perfect” constraint

Lowest energy state

6 DOF

Metal molds

High natural frequency


Common alignment methods

Elastic averagElastic averagiingng Compliant kinCompliant kineematicmatic QuasiQuasi--kinematickinematic Active kinematicActive kinematic Passive kinematicPassive kinematic

AccuracyAccuracy RepeatabilityRepeatability Accuracy & repeatabilityAccuracy & repeatability

Desired Position

Elastic BElastic B

Passive KCPassive KC Active KCActive KC

Elastic CElastic CElastic AElastic A

QuasiQuasi--KCKC

ErrorErrorErrorError


Pin-hole

6 DOF

Metal molds


3 – 2 – 1 Alignment schemes


Exact constraint couplings Exact constraint (EC):

� Constraints = DOF to be constrained � Deterministic saves $ � Balls (inexpensive) & grooves (more difficult to make)

In KC design the issues are:� KNOW what is happening in the system (coupling) � MANAGE forces, deflections, stresses and friction

There are many types of EC couplings, our time limits us to a semi-focused study on kinematic couplings

Balls

Tetrahedral groove

Maxwell V groove Kelvin


Passive kinematic couplings Fabricate and forget

¼ micrometer with best practices, 10s of nm recently

What is important?� Contact forces � Contact stress � Stiffness vs. geometry � Stiffness vs. preload � Friction & settling � Thermal loading � Preload repeatability

Preload (nesting load) is the force applied to keep the coupling components engaged and prevent tipping

Preload


Ball motions: Displacements

iABall

iABall ee

n iABalln n

ER

F _

_

31

2

2

__ ˆ 16

9 ⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅ ⋅−=Σδ

v iBBall

iBBall ee

n iBBalln n

ER

F _

_

31

2

2

__ ˆ 16

9 ⋅

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⋅ ⋅−=Σδ

v

Hertz 1857-1894 vvv

ΣδBall _ i = δBall _ iA +δBall _ iB

This assumes that the ball-ball stiffness is > ~10x ball-groove stiffness

Ball far-field point

B1

B3

B2

A B

iABall _ δ v

iBBall _ δ v

iBall _ δ v

Groove far-field points© Martin Culpepper, All rights reserved 10

Load balance: Force and moment Preload

Force balance (3 equations)Error

Contact

ΣF v

relative = 0 = ( preloadFv

+ ErrorFv )+ (

Moment balance (3 equations) M r

relative = ( iBalli M _ 1 6

r = )Σ + ( preloadM

r )+ ( errorM r )= ( preloadpreload Fr

vv × + Errorerror Fr vv × )+ Σ iBalliBall Fr __

vv ×

Goal: 1. Solve 6 equations for contact forces 2. Solve normal displacements 3. Solve relative displacements/rotations

Ball far-field point

Given geometry, materials,preload force, error force,solve for local distance of approach

A B

Groove far-field points

)6_5_4_3_2_1_ BallBallBallBallBallBall FFFFFF vvvvvv

+++++


Modeling round interfaces Equivalent radius 1R = e 1 1 1 1

+ + +R1major R1min or R2major R2min or

Equivalent modulus 1Ee = 2 2 Poisson’s ratio1−η1 1−η2+

E1 E2

Young’s modulus

1

⎛ 9 F 2 ⎞ 3nδ = ⎜ ⋅ ⎟ n ⎜

⎝16 Re ⋅Ee 2 ⎟⎠

Important scaling law

Scaling with Mat’l properties and geometry

Contact stiffness

0

05

10

20

30

0 250 500 750 1000 Fn [N]

k [N

/mic

ron]

Preload should be repeatable in magnitude & direction

Degree of nonlinearity is reduced as preload is increased

3 ⋅E 3 3k δ = 2 ⋅R ⋅E ⋅δ( ) ( 0.5 ) 0.5 k ( )F = Constant ⋅ (R 1 2 )⋅F

1

n n e e n n n e e n


Friction and lubrication The trend of

the data is important

Wear in vs. “snow balling”

Magnitude depends on coupling design and test conditions

Slocum, A. H., Precision Engineering, 1988: Kinematic couplings for precision fixturing– Experimental determination of repeatability and stiffness

Number of Trials

Radial Repeatability (Unlubricated) 2

0 D

ispl

acem

ent, μm

Radial Repeatability (Lubricated)

Dis

plac

emen

t, μm

2

0 Number of Trials

© Martin Culpepper, All rights reserved Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.

13

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