© 2001 Martin Culpepper
Design of Constraints in Precision Systems
Background
History
Reasons
Requirements
Problems
Classes of Constraint
Kinematic
Quasi-Kinematic
Variable Geometry
Partial/Compliance
Hardware/discussion time
Elastic Averaging will be done next lecture
© 2001 Martin Culpepper
Common Coupling Methods
Elastic Averaging
Pinned Joints
Kinematic Couplings
Flexural Kin. Couplings
Non-Deterministic
No Unique Position
Kinematic Constraint
Kinematic Constraint
Quasi-Kinematic Couplings
Near Kinematic Constraint
© 2001 Martin Culpepper
Perspective: What the coupling designer faces…
APPLICATION
SYSTEM SIZE
REQ’D PRECISION
Fiber Optics
Meso
Nano
Optical Resonators
Meso
Nano
Large array telescopes
60 ft diam.
Angstrom
Automotive
3 ft
1 micron
Problems due to strain affects
Thermal Affects
Gravity
Stress Relief
Loads
Air, hands, sunlight
Sagging
Time variable assemblies
Stiffness
Problems due to sub-optimal designers
Competing cost vs performance
Automotive (no temperature control, large parts, resistance to change)
© 2001 Martin Culpepper
General Service Requirements & Applications
Ideal couplings:
Inexpensive
Accurate & Repeatable
High Stiffness
Handle Load Capacity
Sealing Interfaces
Well Damping
Example Applications:
Grinding
Optic Mounts
Robotics
Automotive
Sensitivity
What are the sensitive directions?!?!?!?!?
© 2001 Martin Culpepper
Couplings Are Designed as Systems
You must know what is going on (loads, environment, thermal)!
Shoot for determinism or it will “suck to be you”
Displacement
Disturbance
Inputs
•Force
•Displacement
Force
Disturbance
Material
Property
Disturbance
Geometry
Disturbance
Kinematics
Geometry
Coupling
System
Material
Desired Outputs
•Desired Location
Others
Error
Actual Outputs
•Actual Location
© 2001 Martin Culpepper
The good, the bad, the ugly….
© 2001 Martin Culpepper
Defining Constraint
Clever use of constraint
© 2001 Martin Culpepper
Penalties for over constraint
Exact Constraint (Kinematic) Design
Exact 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 finish
¼ micron repeatability is common
Managing contact stresses are the key to success
© 2001 Martin Culpepper
Making Life Easier
“Kinematic Design”, “Exact Constraint Design”…..the issues are:
KNOW what is happening in the system
Manage forces and deflections
Minimize stored energy in the coupling
Know when “Kinematic Design” should be used
Know when “Elastic Averaging” should be used (next week)
© 2001 Martin Culpepper
Kelvin Clamp
Boyes Clamp
Kinematic couplings
Kinematic Couplings:
Deterministic Coupling
# POC = # DOF
Do Not Allow Sealing Contact
Excellent Repeatability
Performance
Power of the KC
© 2001 Martin Culpepper
Accuracy
Repeatability
Accuracy &
Repeatability
Modeling Kinematic Coupling Error Motions
Geometry
Material
ni
Applied
Loads
[Fp & Mp]
Interface
Forces
[Fi]
Deflections
δ -> ∆r
Relative
Error
6 Unknown Forces & 6 Equilibrium Equations
Σ Mi = MP
Σ Fi = FP
Hertzian Point Contact for Local Displacements
δi = f(EB, EG, νB, νG, RB, RG)
Kinematic Coupling Groove
Mating Spherical Element
Contact Force
Coupling Centroid
Angle Bisectors
Coupling Triangle
© 2001 Martin Culpepper
KC Error Motion Analysis
Need 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
© 2001 Martin Culpepper
Final Positions
Kinematic Couplings and Distance of Approach
How do we characterize motions of the ball centers?
n
l
δn = distance of approach
Initial Position of
Ball’s Far Field Point
δl
δr
δn
Final Position of
Ball’s Far Field Point
Initial Contact Point
δz
Initial Position of
Groove’s Far Field Point
z
Final True Groove’s
Far Field Point
r
Max τ
Final Contact Point
© 2001 Martin Culpepper
Max shear stress occurs below surface, in the member with larges R
Contact Mechanics – Hertz Contact
Heinrich 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
Radii
Load
Modulus
ν Ratio
Stress
Deflection
© 2001 Martin Culpepper
Ronemaj
Ronemin
Rtwomaj
Rtwomin
Applied load F
Phi (degrees)
Max contact stress
Elastic modulus Eone
Elastic modulus Etwo
Poisson's ratio vone
Poisson's ratio vtwo
Equivalent modulus Ee
Equivalent radius Re
Contact pressure
Stress ratio (must be less than 1)
Deflection at the one contact interface
Deflection (µunits)
1.00E+06
0.06250
0.25000
-0.06500
13
0
10,000
3.00E+07
4.40E+05
0.3
0.3
4.77E+05
0.2167
12,162
1.22
829
F
∆ = 2 δn
F
Key Hertz Physical Relations
Equivalent radius and modulus:
Re =
1
R1major
+
1
R1minor
1
+
1
R2major
+
Ee =
1
R2minor
1
1 - η 21 1 - η 22
+
E1
E2
cos(θ)
θ function (φφ is the angle between the planes of principal curvature of the two bodies)
cosθ = Re
+2
1
1
-
2
1
R1major R1minor
-
1
+
1
-
1
-
2
1
R2major R2minor
1
R1major R1minor R2major R2minor
cos2φ
1/2
Solution to elliptic integrals estimated with curve fits
α = 1.939e-5.26θ + 1.78e-1.09θ + 0.723/ θ + 0.221
β = 35.228e-0.98θ - 32.424e-1.0475θ + 1.486θ - 2.634
λ = -0.214e-4.95θ - 0.179θ 2 + 0.555θ + 0.319
Contact Pressure
q = 3F ≈ 1.5σ tensile for metals
2π cd
© 2001 Martin Culpepper
Distance of
Approach
δ=λ
2F2
3ReE2e
1/3
Major Contact
Axis
3FRe 1/3
c= α
2E e
Minor Contact
Axis
1/3
β 3FR e
2E e
KEY Hertz Relations
Contact Pressure is proportional to:
Force to the 1/3rd power
Radius to the –2/3rd power
Modulus to the 2/3rd power
Distance of approach is proportional to:
Force to the 2/3rd power
Radius to the –1/3rd power
Modulus to the –2/3rd power
Contact ellipse diameter is proportional to:
Force to the 1/3rd power
Radius to the 1/3rd power
Modulus to the –1/3rd power
DO NOT ALLOW THE CONTACT ELLIPSE TO BE WITHIN ONE DIAMETER OF
THE EDGE OF A SURFACE!
© 2001 Martin Culpepper
Calculating Errors Motions in Kinematic Couplings
Motion of ball centers -> Centroid motion in 6 DOF -> ∆
∆x, ∆
∆y, ∆
∆z at X, Y, Z
Coupling Centroid Translation Errors
δ ζc
⎛ δ 1ζ δ 2ζ δ 3ζ ⎞ L 1c + L 2c + L 3c
+
+
⎜
⎟⋅
3
⎝ L 1c L 2c L 3c ⎠
Rotations
εx
εy
δ z1
L 1 , 23
δ z1
L 1 , 23
ε z1
(
)
(
)
⋅ cos θ 23 +
⋅ sin θ 23 +
δ z2
L 2 , 31
δ z2
L 2 , 31
(
δ z3
)
⋅ cos θ 31 +
(
)
⋅ sin θ 31 +
L 3 , 12
δ z3
L 3 , 12
(
⋅ cos θ 12
(
)
⋅ sin θ 12
(α B1⋅ δ 1 + α B2⋅ δ 2)2 + (β B1⋅ δ 1 + β B2⋅ δ 2) 2
(x 1 − x c)2 + (y 1 − y c)2
)
(
⋅ SIGN α B1⋅ δ 1 − α B2⋅ δ 2
Error At X, Y, Z (includes translation and sine errors)
⎛∆ x⎞
⎜
⎟
⎜∆ y ⎟
⎜∆ z ⎟
⎜
⎟
⎝ 1 ⎠
© 2001 Martin Culpepper
⎛ 1 −ε z ε y
⎜
⎜ ε z 1 −ε x
⎜ −ε y ε x 1
⎜
0
0
⎝ 0
δx⎞
⎛ X − xc ⎞
⎟⎜
⎟
δy ⎟ ⎜Y − y c⎟
⋅
δz ⎟ ⎜ Z − zc ⎟
⎟⎜
⎟
1 ⎠⎝ 1 ⎠
⎛ X − xc ⎞
⎜
⎟
Y − yc⎟
⎜
−
⎜ Z − zc ⎟
⎜
⎟
⎝ 1 ⎠
)
εz
ε z1 + ε z2 + ε z3
3
Kinematic Coupling Centroid Displacement
*Pictures courtesy Alex Slocum
Precision Machine Design
Centroid Displacement:
© 2001 Martin Culpepper
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
© 2001 Martin Culpepper
Kinematic Coupling Stability Theory
Poor Design
Good Design
*Pictures courtesy Alex Slocum
Precision Machine Design
© 2001 Martin Culpepper
Sources of Errors in KCs
Thermal
Errors
Surface
Finish
Displacement
Disturbance
Kinematics
Inputs
•Force
•Displacement
Geometry
Force
Disturbance
Error Loads
Preload Variation
© 2001 Martin Culpepper
Material
Property
Disturbance
Geometry
Disturbance
Coupling
System
Material
Desired Outputs
•Desired Location
Others
Error
Actual Outputs
•Actual Location
Problems With Physical Contact (and solutions)
Surface topology (finish):
50 cycle repeatability ~ 1/3 µm
µ Ra
Friction depends on surface finish!
Finish should be a design spec
Surface may be brinelled if possible
A
A
B
λ
B
λ
Wear and Fretting:
High stress + sliding = wear
Metallic surfaces = fretting
Use ceramics if possible (low µ and high strength)
Dissimilar metals avoids “snowballing”
Mate n + 1
Wear on Groove
Mate n
Friction:
Friction = Hysteresis, stored energy, overconstraint
Flexures can help (see right)
Lubrication (high pressure grease) helps
- Beware settling time and particles
Tapping can help if you have the “magic touch”
© 2001 Martin Culpepper
Ball in V-Groove with Elastic Hinges
Experimental Results –Repeatability & Lubrication
Displacement, µm
Radial Repeatability (Unlubricated)
2
0
Number of Trials
60
0
Displacement, µm
Radial Repeatability (Lubricated)
2
0
© 2001 Martin Culpepper
Number of Trials
60
0
Practical Design of Kinematic Couplings
Design
Specify surface finish or brinell on contacting surfaces
Normal to contact forces bisect angles of coupling triangle!!!
Manufacturing & Performance
Repeatability = 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
Materials
Ceramics = low friction, high stiffness, and small contact points
If using metals, harden
Use dissimilar materials for ball and groove
Preparation and Assembly
Clean with oil mist
Lubricate grooves if needed
© 2001 Martin Culpepper
Example: Servo-Controlled Kinematic Couplings
Location & automatic leveling of precision electronic test equipment
Teradyne has shipped over 500 systems
© 2001 Martin Culpepper
Example: Canoe-Ball Kinematic Interface Element
The “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
© 2001 Martin Culpepper
Canoe Ball Repeatability Measurements
Coupling
error [ µ
µm ]
Test Setup
0.1 µ
µm
0.10
0.00
0
5
10
15
20
25
30
35
40
45
50
Meas. system
error [ µ m ]
-0.10
-0.20
-0.2 µ
µm
0.10
0.1 µ
µm
0.00
0
-0.10
© 2001 Martin Culpepper
5
10
15
20
25
30
35
40
45
50
-0.1 µ
µm
Why do it the easy way when you can do it the lazy way?
© 2001 Martin Culpepper
Quasi-Kinematic (QKC) Alignment
QKC characteristics:
Spherical Protrusion
Arc contact
Submicron repeatability
Stiff, sealing contact
Less expensive than KCs
Easier to make than KCs
Groove Seat
Feature Height, mils
Side Reliefs
QKC Function:
Ball & groove comply
Burnish surface irregularities
Elastic recovery restores gap
Distance along Cone Face, [inches]
F or ∆
δinitial
© 2001 Martin Culpepper
δ=0
δfinal
Details of QKC Element Geometry
PAIRS OF QKC ELEMENTS
+
+
OR
TYPE 2 GROOVE MFG.
ASSEMBLED JOINT
BOLT
+
CAST
© 2001 Martin Culpepper
BLOCK
=
FORM TOOL
FINISHED
BEDPLATE
PEG
QKC Methods vs Kinematic Method
Components and Definitions
y
x
Relief
Relief
Cone Seat
Ball
Groove Surface
Peg Surface
Force Diagrams
© 2001 Martin Culpepper
Contact Point
Modeling QKC Stiffness
Geometry
Material
Applied
Loads
[Fp & Mp]
Resultant
Forces
[ni & Fi]
Deflections
δ -> ∆r
Relative
Error
24 Unknowns & 7 Eqxns.
Material
Contact Stiffness
fn(δ
δn)
QKC Model
Force/Torque
Geometry
Displacements
Stiffness
© 2001 Martin Culpepper
^l
Contact Mechanics
^
n
MECHANICS:
^
s
• Use Rotating Coordinate System
• Assume Sinusoidal Normal Distance of Approach
• Obtain Contact Stress Profile as Function of Above
Rotating CS
• Integrate Stress Profile in Rotating CS thru contact
Surface Contact Pressure - Elastic Hertz
^
k
^
n
^l
Rc
^
i
Contact Pressure, psi
200000
FEA Profile
150000
Hertz Profile
100000
50000
0
-0.0200
-0.0100
0.0000
0.0100
Distance in L, in
© 2001 Martin Culpepper
0.0200
Example: Duratec Assembly
Characteristics:
• Ford 2.5 & 3.0 L V6
10 µm
µ
• > 300,000 Units / Year
• Cycle Time: < 30 s
Coupling + Others
Process
5µ
µm
0 µ
µm
Rough Error Budget
© 2001 Martin Culpepper
Example: Assembly of Duratec Block and Bedplate
COMPONENTS
Block
Bedplate
ERROR
ASSEMBLY
δe MAX = 5 microns
Assembly Bolts
δe
Bedplate
C B Halves
JL
Block
JR
r
Block Bore CL
a
© 2001 Martin Culpepper
Bedplate Bore CL
Bearing Assemblies in Engines
Block
Crank Shaft Journal
Block Main
Bearing Half
Piston
Crank Shaft
Bedplate
Main Bearing
Halves
© 2001 Martin Culpepper
Bedplate Main
Bearing Half
Results of Duratec QKC Research
MANUFACTURING:
Engine Manufacturing Process With Pinned Joint
Op. #10
• Mill Joint Face
• Drill/Bore 16 Holes
• Drill Bolt Holes
Op. #30
• Drill Bolt Holes
Op. #50
• Press in 8 Dowels
• Assemble
• Load Bolts
• Torque Bolts
Op. #100
• Semi-finish crank bores
• Finish crank bores
Modified Engine Manufacturing Process Using Kinni-Mate Coupling
Op. #10
• Mill Joint Face
• Drill/Bore 3 Peg Holes
• Drill Bolt Holes & Form
3 Conical Grooves
DESIGN:
© 2001 Martin Culpepper
Op. #30
• Drill Bolt Holes
Op. #50
• Press 3 Pegs in BP
• Assemble
• Load Bolts
• Torque Bolts
Op. #100
• Semi-finish crank bores
• Finish crank bores
Engine Assembly Performance
Axial
JL Cap Probe
Sensitive
1st Block Fixture
CMM Head
Bedplate Fixture
Bedplate
JR Cap Probe
Axial Cap Probe
2nd Block Fixture
QKC Error in Axial Direction
QKC Error in Sensitive Direction
2.0
δc, microns
JR
1.0
0.5
0.0
-0.5 0
1
2
3
4
5
6
-1.0
-1.5
-2.0
Trial #
(Range/2)|AVG = 0.65 µm
© 2001 Martin Culpepper
7
8
δa, microns
JL
1.5
2.0
1.5
1.0
0.5
0.0
-0.5 0
-1.0
-1.5
-2.0
Max x Dislacement
1
2
3
4
5
Trial #
(Range/2) = 1.35 µm
6
7
8