03 KinematicsHertzFlexures Sullivan

Precision Machine DesignME 250

Kinematic Design 2Hertz Contact Stresses

Flexures

Mark Sullivan

September 18, 2008

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Precision Machine Design

Kinematic DesignHertz Contact

StressesFlexures

SullivanSep 18, 2008

• Kinematic Couplings – History, Design Guidelines, Load Capacity, Examples

• Quasi Kinematic Couplings• Constraints• Hertz Contact Stresses• Flexures• References

Agenda

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Acknowledgements

• Text and figures in these lecture notes are taken from the following sources:– Blanding, D., Exact Constraint: Machine Design Using Kinematic

Principles, ASME Press, New York, 1999.

– Hale, L. C., “Precision Engineering Principles,” ASPE Tutorial, Monterey, 2006.

– Smith, S. T., Chetwynd, D. G., Foundations of Ultraprecision Mechanism Design, Taylor & Francis, 1994.

– Hale, L. C., “Principles and Techniques for Designing Precision Machines,” UCRL-LR-133066, Lawrence Livermore National Laboratory, 1999. (http://www.llnl.gov/tid/lof/documents/pdf/235415.pdf)

– Slocum, A. H., Precision Machine Design, SME, 1992.

– Slocum, A. H., FUNdaMENTALs of Design, MIT, 2008.

– Precision Engineering Research Group, MIT• http://pergatory.mit.edu/• http://pergatory.mit.edu/kinematiccouplings/

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Kinematic Couplingsand Exact Constraint

Chart from “FUNdaMENTALs of Design,” Slocum

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Kinematic Couplings

• Two common forms of the kinematic coupling.

“2 – 2 – 2”(Maxwell)

“3 – 2 – 1”(Kelvin Clamp)

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Kinematic Couplings:Some History

Chart from “Mechanics of Designing Precision Machines,” Slocum

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Kinematic Couplings:Three-groove Design Guidelines

Chart from FUNdaMENTALs of Design, Slocum

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Kinematic Couplings:Load Capacity of Couplings


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Kinematic CouplingExample: Bal-Tec Components

• Bal-Tec Kinematic Coupling Components– High repeatability– Low cost– Limited load– Limited stiffness

Cone Vee Flat

http://www.precisionballs.com/index.html

Ball

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Kinematic CouplingExample: Bal-Tec Clamp

http://www.precisionballs.com/index.html

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Example of Kinematic Coupling: Adjustable Mirror Mount

• A common example of a kinematic coupling is the adjustable mirror mount found in most optics labs.

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Quasi-Kinematic Couplings


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Quasi-Kinematic CouplingExample: Ford Engine Assembly


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Quasi-Kinematic Couplings:Engine Assembly Performance


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• Always a good idea to keep in mind the size of things– The thickness of this paper =

– The diameter of a human hair =

– Computer hard drive track spacing =

– Diameter of a fiber optic =

– Visible light wavelength (mid-spectrum) =

– Size of a typical virus =

– Atomic diameter =

Recall: The Size of Things

100 μm

20 – 180 μm

1 μm

4 or 62.5 μm core, 125 μm cladding

550 nm

0.1 – 0.6 nm (He to Cs)

10 – 400 nm

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Quasi-Kinematic Couplings (2)

• Quasi-Kinematic Couplings (QKCs) approximate Kinematic Couplings– Reduced repeatability– Low cost– Increased load– Increased stiffness

Designs based on line contact offer a significant increase in load capability and stiffness (Hale).

Three cones with radial flexures

Three tooth coupling

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Quasi-Kinematic CouplingExample: Spherolinder

• Spherolinder Quasi-Kinematic Coupling– Reduced repeatability– Increased cost– Increased load (~100X higher)– Increased stiffness

Spherolinder Vee Cone

Retainer

http://www.g2-engineering.com/index.html

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Pop Quiz: Constraints

• How many degrees of freedom does this coupling have?• What are they?

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Idealized KinematicConstraint Configurations

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Review: Constraints

Constraints Configuration

1 Ball on flat

2Link on flat

Ball in groove

3

Ball in trihedral

Link with one ball in groove and other on flat

Three linked balls on flat

4

Link with one ball in trihedral and other ball on flat

Link with 2 balls both in a vee groove

Link of 3 balls with 2 on a flat & one in a groove

Link of 4 balls on 2 inclined flats

5

Link of 2 balls with one in a trihedral hole & the other in a vee groove

Link of 3 balls with 2 in vee grooves & one on a flat

Link of 4 balls with one in a vee groove & 3 on a flat

Link of 5 balls on 2 inclined flats

6

Link of 3 balls in 3 vees (“2-2-2” kinematic mount)

Link of 3 balls with one in a trihedral, one in a vee & one on a flat

(“3-2-1” kinematic mount)

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Constraint Model of Kinematic Couplings

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Rotational DOF

• Statement 9: A constraint (C) properly applied to a body (i.e., without overconstraint) has the effect of removing one of the body’s rotational degrees of freedom (R). The R removed is the one about which the constraint exerts a moment. A body constrained by n constraints will have 6 – n rotational degrees of freedom, each positioned such that no constraint exerts a moment about it. In other words, each R will intersect all Cs.

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Exercise 1

Pick one corner that is not on a constraint line and add one orthogonal constraint at a time until the block is fully constrained. How many constraints must you add? Identify each constraint added with the rotational axis that it constrains (Hint: Use Statement 9).

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Equivalent Pairs of Rotational DOF

• Statement 10: Any pair of intersecting rotational degrees of freedom (R) is equivalent to any other pair intersecting at the same point and lying in the same plane. This holds true for small motions.

• Statement 11: Two parallel Rs are equivalent to any two parallel Rs parallel to the first pair and lying on the same plane. They are also equivalent to a single R parallel to the first pair and lying in the same plane; and a T perpendicular to that plane.

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Series and Parallel Connections

• Statement 12: When parts are connected in series (cascaded), add the degrees of freedom. When the connections occur in parallel, add constraints.


Hertz Contact Stresses

Mark Sullivan

September 18, 2008

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Hertz Contact Stresses


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Hertz Contact Stresses (2)

Equations from FUNdaMENTALs of Design, Slocum

This is the “general case.”

For solved cases, see Roark or MathCAD

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Shear

Radial

Compressive

Graph and equations from FUNdaMENTALs of Design, Slocum

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• To reduce Hertz Contact Stresses:– Decrease force

– Increase “ball” radius

– Decrease E

• To reduce deflection:– Decrease force

– Increase “ball” radius

– Increase E

• To reduce contact area:– Decrease force

– Decrease “ball” radius

– Increase E

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Kinematic Coupling Analysis

• Also, MathCAD


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Pop Quiz: Contact Stress

• Which 3 DOF mount has lower Hertz contact stresses? Why?• How could you make the stresses even lower?

3-Ball Nest

Tetrahedron


Flexures

(Adapted from ME 119 / ME 324 material by D. DeBra, Stanford University)

Mark SullivanSeptember 18, 2008

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Conceptual Basis for Flexure Design

• Kinematic Design– A rigid body has 6 DOF with respect to a reference frame (or

another rigid body)– With exactly 6 constraints suitably arranged, no relative motion.– If more than 6 constraints are applied to the body, it is

overconstrained and can be strained if its support base strains– If less than 6 constraints are applied, movement is made possible

(e.g., bearings):• 1 rotation free - spindle, rotary bearing• 1 translation free - carriage on ways

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From Hale, 1999

Is this mount over-constrained?

Kinematic and Semi-Kinematic Constraint

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Constraintsand Strain Attenuation

• A constraint is (relatively) stiff along its line of constraint– Can substitute suitable arranged flexible elements to provide

functionally equivalent constraint• Ex. Your stick models

• Strain attenuation is important– Frictional forces from contacts can transmit unwanted strain

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Basic Building Blocks of Flexures

• Rod– Which DOF are Stiff

Which are Flexible?

• Bellows– Which DOF are Stiff

Which are Flexible?

• Ex. of combining rods and bellows to achieve flexural elements.

x

z

y

x

z

y Rx

Rz

Ry

Rx

Rz

Ry

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Basic Building Blocks of Flexures (2)

• Sheets or plates– Which DOF are Stiff

Which are Flexible?

– bh determines strengthL influences buckling strength

• Ex. of combining sheet flexures to reduceconstraints.

• Ex. of combining sheet flexures to increaseconstraints.

x

z

y

h

b

L

Rz

Rx Ry

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Blade Flexures

• Rigid constraint in its own plane (x, y, & θz)

• Three degrees of freedom: z, θx, & θy.

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Parallel-Blade Flexure

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Cross-Blade Flexure

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Commercial Flexures

http://www.c-flex.com/home.html

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Rowland’s Ruling Engine

Henry Augustus Rowland III [1848-1901] - American physicist

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Anti-Distortion Mountings

Jones, 1961

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Other Flexures

Jones, 1962

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Other Flexures, cont.

Jones, 1962

Why have 2 sets of cantilevered blade flexures?

(At least 2 reasons)

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Other Flexures, cont.

Jones, 1962

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Series and ParallelConnections of Springs

• Rule 1: The equivalent compliance of springs connected in series is the sum of their individual compliances.

• Rule 2: The equivalent stiffness of springs connected in parallel is the sum of their individual stiffnesses.

cseries

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Series and ParallelConnections of Springs (2)

• Corollary: When springs are connected in series, add stiffnesses in parallel. When springs are connected in parallel, add stiffnesses in series.

k1 k2 k3

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References

• Blanding, D., Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York, 1999.

• DeBra, D. ME 119 Lecture Notes on Flexures, Stanford University, 1987.

• Jones, R. V., “Anti-distortion Mountings for Instruments and Apparatus,” J. of Sci. Instr., vol. 38, October 1961, pp. 408-409.

• Jones, R. V., “Some Uses of Elasticity in Instrument Design, J. of Sci. Instr., vol. 39, 1962, pp. 193-203.

• Hale, L. C., “Principles and Techniques for Designing Precision Machines,” UCRL-LR-133066, Lawrence Livermore National Laboratory, 1999. (http://www.llnl.gov/tid/lof/documents/pdf/235415.pdf)

• Smith, S. T., Chetwynd, D. G., Foundations of Ultraprecision Mechanism Design, Gordon and Breach Science Publishers, Switzerland, 1992.

Documents

03 KinematicsHertzFlexures Sullivan