KINEMATIC COUPLINGS - Massachusetts Institute of …pergatory.mit.edu/kinematiccouplings/.../kinematic_couplings.pdfContact pressure 12,162 Stress ratio (must be less than 1) 1.22

© 2001 MIT PSDAM AND PERG LABS

KINEMATIC COUPLINGS

The good, the bad, the ugly….


Defining constraint

Penalties for over constraintClever use of constraint


Exact constraint (Kinematic) designExact Constraint: Number of constraint points = DOF to be constrained

These constraints must be independent!!!

Assuming couplings have rigid bodies, equations can be written to describe movements

Design is deterministic, saves design and development $

KCs provide repeatability on the order of parts’ surface finish¼ micron repeatability is commonManaging contact stresses are the key to success


Making life easier“Kinematic Design”, “Exact Constraint Design”…..the issues are:

KNOW what is happening in the systemManage forces and deflectionsMinimize stored energy in the couplingKnow when “Kinematic Design” should be usedKnow when “Elastic Averaging” should be used (next week)

Boyes ClampKelvin Clamp

Picture from Precision Machine Design, Slocum, A. H.


Kinematic couplingsKinematic Couplings:

Deterministic Coupling# POC = # DOFDo Not Allow Sealing ContactExcellent Repeatability

Performance

Power of the KC

Accuracy Repeatability Accuracy & Repeatability


Modeling Kinematic coupling error motions

InterfaceForces

[Fi]

AppliedLoads

[Fp & Mp]

Deflectionsδ -> ∆r

RelativeError

GeometryMaterialni

Σ Fi = FP Σ Mi = MP

6 Unknown Forces & 6 Equilibrium Equations

Hertzian Point Contact for Local Displacementsδi = f(EB, EG, νB, νG, RB, RG)

Kinematic Coupling Groove

Mating Spherical Element

Contact Force

Coupling Centroid

Angle Bisectors

Coupling Triangle


KC error motion analysisNeed δx, δy, δz, εx, εy , εz to predict effect of non-repeatability

Hertz deflections -> displacements of ball centers

Three ball centers form a plane

Analyze relative position of “before” and “after” planes for error motions

B1

B3

B2

Original Positions Final Positions


Kinematic couplings and distance of approachHow do we characterize motions of the ball centers?

δn = distance of approach

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

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


Contact mechanics – Hertzian contactHeinrich Hertz – 1st analytic solution for “near” point contact

KC contacts are modeled as Hertz Contacts

Enables us to determine stress and distance of approach, δn

Ronemaj 1.00E+06Ronemin 0.06250Rtwomaj 0.25000Rtwomin -0.06500Applied load F 13Phi (degrees) 0Max contact stress 10,000Elastic modulus Eone 3.00E+07Elastic modulus Etwo 4.40E+05Poisson's ratio vone 0.3Poisson's ratio vtwo 0.3Equivalent modulus Ee 4.77E+05Equivalent radius Re 0.2167Contact pressure 12,162Stress ratio (must be less than 1) 1.22Deflection at the one contact interface Deflection (µunits) 829

F

F

∆ = 2 δn

Radii

Load

Modulus

ν Ratio

Stress

Deflection


Key Hertzian physical relationsEquivalent radius and modulus:

cos(θ) function (φ is the angle between the planes of principal curvature of the two bodies)

Solution to elliptic integrals estimated with curve fits

cos = R e 1 R 1 major

- 1 R 1 minor

2 + 1

R 2 major - 1

R 2 minor

2

+ 2 R 1 major

- R 1 minor

R 2 major

- R 2 minor

cos2 1/2

R e = 1 1

R 1 major + 1

R 1 minor + 1

R 2 major + 1

R 2 minor

E e = 1

1 - 1 2

E 1 + 1 - 2

2

E 2

= 1.939 e -5.26 + 1.78 e -1.09 + 0.723/ + 0.221

= 35.228 e -0.98 - 32.424 e -1.0475 + 1.486 - 2.634

= -0.214 e -4.95 - 0.179 2 + 0.555 + 0.319

q = 3F2 cd

1.5 tensile for metals = 2 F 2

3 R e E e 2

1/3

Minor Contact Axis

Major Contact Axis

Distance of Approach

Contact Pressure Minor Contact Axis

Major Contact Axis

Distance of Approach

Contact Pressure

β 3FR e 2Ee

1/3 β 3FR e

2Ee

1/3 c = α 3FR e

2Ee

1/3 c = α 3FR e

2Ee

1/3

1 1 1 1


KEY Hertzian relation scaling lawsContact Pressure is proportional to:

Force to the 1/3rd powerRadius to the –2/3rd powerModulus to the 2/3rd power

Distance of approach is proportional to:Force to the 2/3rd powerRadius to the –1/3rd powerModulus to the –2/3rd power

Contact ellipse diameter is proportional to:Force to the 1/3rd powerRadius to the 1/3rd powerModulus to the –1/3rd power

DO NOT ALLOW THE CONTACT ELLIPSE TO BE WITHIN ONE DIAMETER OF THE EDGE OF A SURFACE!


Calculating error motions in kinematic couplingsMotion of ball centers -> Centroid motion in 6 DOF -> ∆x, ∆y, ∆z at X, Y, Z

Coupling Centroid Translation Errors

Rotations

Error At X, Y, Z (includes translation and sine errors)

c 1

L 1c

2

L 2c

3

L 3c

L 1c L 2c L 3c

3

x z1

L 1 23 cos 23

z2

L 2 31 cos 31

z3

L 3 12 cos 12

y z1

L 1 23 sin 23

z2

L 2 31 sin 31

z3

L 3 12 sin 12

z1 B1 1 B2 2 2

B1 1 B2 2 2

x1 xc 2 y 1 y c 2 SIGN B1 1 B2 2 z

z1 z2 z3

3

x

y

z

1

1

z

y

0

z

1

x

0

y

x

1

0

x

y

z

1

X xc

Y y c

Z z c

1

X xc

Y y c

Z z c

1


Kinematic coupling centroid displacementPicture from Precision Machine

Design, Slocum, A. H.


General design guidelines

1. Location of the coupling plane is important to avoid sine errors

2. For good stability, normals to planes containing contact for vectors should bisect angles of coupling triangle

3. Coupling triangle centroid lies at center circle that coincides with the three ball centers

4. Coupling centroid is at intersection of angle bisectors

5. These are only coincident for equilateral triangles

6. Mounting the balls at different radii makes crash-proof

7. Non-symmetric grooves make coupling idiot-proof


Kinematic coupling stability theory, instant centers

Poor Design

Good Design

Picture from Precision Machine Design, Slocum, A. H.

*Pictures courtesy Alex Slocum

Precision Machine Design


Sources of errors in kinematic couplings

ThermalErrorsSurface

FinishMaterialProperty

Disturbance

GeometryDisturbanceDisplacement

Disturbance

CouplingSystem

Force Disturbance

Desired Outputs•Desired Location

Error

Kinematics MaterialInputs•Force•Displacement Geometry Others

Actual Outputs•Actual LocationError Loads

Preload Variation


Problems with physical contact (and solutions)Surface 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 = Hysteresis, stored energy, over constraintFlexures can help (see right)Lubrication (high pressure grease) helps

- Beware settling time and particlesTapping can help if you have the “magic touch”

BA

λB

A

λ

Ball in V-Groove with Elastic Hinges

Mate n Mate n + 1

Wea

r on

Gro

ove

Picture from:Schouten, et. al., “Design of a kinematic coupling for precision applications”, Precision Engineering, vol. 20, 1997.


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

Number of Trials

Radial Repeatability (Unlubricated)2

0

Dis

plac

emen

t, µm

600

Number of Trials

Radial Repeatability (Lubricated)

600

Dis

plac

emen

t, µm

2

0


Practical design of kinematic couplingsDesign

Specify surface finish or brinell on contacting surfacesNormal to contact forces bisect angles of coupling triangle!!!

Manufacturing & PerformanceRepeatability = f (friction, surface, error loads, preload variation, stiffness)Accuracy = f (assembly) unless using and ARKC

Precision Balls (ubiquitous, easy to buy)Baltec sells hardened, polished kinematic coupling balls or…..

Grooves (more difficult to make than balls)May be integral or inserts. Inserts should be potted with thin layer of epoxy

MaterialsCeramics = low friction, high stiffness, and small contact pointsIf using metals, hardenUse dissimilar materials for ball and groove

Preparation and AssemblyClean with oil mistLubricate grooves if needed


Example: Servo-controlled kinematic couplingsLocation & automatic leveling of precision electronic test equipment

Teradyne has shipped over 500 systems

Ph.D. Thesis: Michael Chiu


Example: Canoe-Ball kinematic interface elementThe “Canoe Ball” shape is the secret to a highly repeatable design

It acts like a ball 1 meter in diameter

It has 100 times the stiffness and load capacity of a normal 1” ball

Large, shallow Hertzian zone is very (i.e. < 0.1 microns) repeatable

M.S. Thesis, Bernhard Muellerheld


Canoe-Ball repeatability measurementsTest Setup

-0.20

-0.10

0.00

0.10

0 5 10 15 20 25 30 35 40 45 50

erro

r [ µ

m ]

0.1 µm

-0.2 µm

Coupling

-0.10

0.00

0.10

0 5 10 15 20 25 30 35 40 45 50

erro

r [

µm

] 0.1 µm

-0.1 µm

Meas. system


There are MANY uses for Kinematic Couplings….The Kinematic Sheet Coupling was created

for the PCB industry

It provides 10x greater repeatability than traditional 4 pins-in-4-slots method

Documents

KINEMATIC COUPLINGS - Massachusetts Institute of …pergatory.mit.edu/kinematiccouplings/.../kinematic_couplings.pdfContact pressure 12,162 Stress ratio (must be less than 1) 1.22