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## Kinematic Design

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**1. **Kinematic Design Mark Sullivan
September 11, 2008

**2. **Review
Introduction
Kinematic Constraint / Exact Constraint
Constraint Line
Instantaneous Center of Rotation
Axes of Rotation
Constraining Rotational Motion
Equivalent Sets of Constraints
Table of Orthogonal Constraints
Blade Flexures
Wire Flexures
Kinematic Coupling
Summary
References Agenda

**3. **Review: Scale

**4. **Always a good idea to have a physical feel for 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 = Pop Quiz: The Size of Things

**5. **Machines and instruments are made up of elements that are suitably arranged and many of which that are movably connected.
Two parts that are in contact and move relative to one another are called kinematic “pairs” or can be thought of as being kinematically coupled.
Sliding, rotating, or helical motion (screwing) are “lower” pairs
Other combinations of the above are “higher” pairs
A connection between two pairs is a “link”
Any “rigid body” in free space has six degrees of freedom.
3 translations
3 rotations
Can resolve any motion of the rigid body into translations parallel to coordinate axes and rotations around these axes.
Non-rigidity adds extra degrees of freedom Introduction

**6. **The number of contact points between any two rigid bodies is equal to the number of their mutual constraints.
Examples:
Sphere on a plane
One contact point
Motion with respect to plane is constrained in z
Sphere in a trihedral hole
Three points in contact
Three translations are constrained
Others:
See Smith & Chetwynd, p. 49
How should I constrain a body to have zero degrees of freedom?
Provide six contact points
Ex: Smith & Chetwynd, p. 52
Bear in mind that we can come up with geometric singularities that degenerate and allow degrees of freedom. Introduction (2)

**7. **Introduction (3) Sphere on a Flat Plate:

**8. **Kinematic Constraint

**9. **Exact Constraint

**10. **Kinematic Design in Precision Engineering Mechanism designers routinely use the principles of kinematics because overconstrained and underconstrained devices simply will not function.
Note to Precision Engineers: At some scale, everything is a mechanism.
The component that must remain stable to nanometers will not if it is overconstrained to a structure that deforms by micrometers.
(? Isolate structural loop from metrology loop!)

**11. **The Constraint Line& Equivalent Constraints Statement 1: Points on the object along the constraint line can move only at right angles to the constraint line, not along it.
Statement 2: Any constraint along a given constraint line is functionally equivalent to any other constraint along the same constraint line (for small motions).

**12. **Instantaneous Center of Rotation(Instant Center) Statement 3: Any pair of constraints whose constraint lines interact at a given point, is functionally equivalent to any other pair in the same plane whose constraint lines intersect at the same point. This is true for small motions and where the two constraints lie on distinctly different constraint lines.

**13. **Exercise 1 A ladder is constrained by a vertical wall and a horizontal floor. On the sketch, show the instant center. Also show the path of the instant center for the ladder between the extreme positions, that is, vertical and horizontal. What geometric shape is the path?

**14. **Axes of rotation intersectall constraints Statement 4: The axes of a body’s rotational degrees of freedom will each intersect all constraints applied to the body.

**15. **Constraining Rotational Motion Statement 5: A constraint applied to a body removes that rotational degree of freedom about which it exerts a moment.

**16. **Equivalent Sets of Constraints Statement 6: Any set of constraints whose constraint lines intersect a complete and independent set of rotational axes, is functionally equivalent to any other set of constraints whose constraint lines intersect the same or equivalent set of rotational axes. This is true for small motions and when each set contains the same number of independent constraints.

**17. **Exercise 2 The three constraints in the figure allow rotation about three non-intersecting axes. Draw in the three axes (hint: use Statement 4).

**18. **Table of Orthogonal Constraints

**19. **Blade Flexures Statement 7: An ideal blade flexure imposes absolutely rigid constraint in its own plane (x, y, & ?z), but it allows three degrees of freedom: z, ?x, & ?y.

**20. **Wire Flexures Statement 8: An ideal wire flexure imposes absolutely rigid constraint along its axis (x), but it allows fived degrees of freedom: y, z, ?x, ?y, & ?z.

**21. **Kinematic Coupling We will most often be interested in couplings that allow zero DOF (also called clamps or mounts) and 1 DOF (slides, ways, bearings).
What happens if more constraints are introduced than are necessary?
Overconstraint ? subject to internal stress, hence strain
Redundant constraint ? 4-legged stool vs. 3-legged stool
Four key problems:
Non-repeatable relative motions between elements and/or repositioning capability (ex. 3 vs. 4-legged stool)
Transmission of distortion
Inability to accommodate relative thermal dimensional changes without causing internal stresses and strains
Generally higher accuracy of construction and assembly required, hence higher cost to achieve comparable levels of performance.

**22. **Kinematic Coupling (2) Why?
Hold two bodies without motion and avoid problems of overconstraint or redundant constraint.
Allow bodies to be separated and rejoined with high degree of repeatability.
Optical mounts
Kelvin clamp (cone, vee, flat) insensitive to thermal changes
Alternate with 3 vees
Additional way to make trihedral hole
3-ball nest
machined version
Note: There are degenerate cases
3 vees tangent to a circle
cone-vee-slot, where cone lies on normal to slot
See Braddick, p. 66
Constraining surface should be put normal to displacement vector that could take place at that point without violating the other constraints if the surface were removed.

**23. **Kinematic Coupling (3) Kinematic coupling provide a rigid, repeatable connection between two objects.

**24. **Kinematic Coupling (4) Variations on the basic three-vee configuration may be considered to better suit the application.

**25. **Kinematic Coupling (5) Modified 3-Vee Coupling

**26. **Kinematic Coupling (6)

**27. **Idealized Kinematic Constraints

**28. **Kinematic Design Summary Degrees of Freedom and constraints
The number of contact points between determines the number of constraints
Be careful with degenerate configurations of contacts
Ex: Degenerate Kelvin clamp
Couplings of zero DOF ? 6 contact points
Couplings of one DOF ? 5 contact points
Kinematic design
Design mating components so that they impose only the necessary and sufficient constraints for the desired effect.
Why?
To minimize non-repeatable relative motions between elements or non-repeatable assembly
To minimize transmission of distortion
To minimize effects of thermal dimensional changes
To minimize required accuracy of parts and construction to achieve performance goals.

**29. **Kinematic Design Summary (2) Semi-Kinematic Design
Apply principles of kinematic design
Theoretical point contacts are expanded into lines and/or surfaces
Needed when loads are large and point contacts would result in too high stresses
Ex. Slideway design where point supports have been replaced by pad supports.

**30. **References 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)
Furse, J. E., “Kinematic Design of Fine Mechanisms in Instruments,” J. Phys. E: Sci. Instrum., vol. 14, 1981, p. 264-272.
Braddick, H. J. J., The Physics of Experimental Method, Chapman and Hall, London, 1966.