Kinematic Design Principles

Transcription

1 Kinematic Design Principles BJ Furman 24SEP97 Introduction 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 1

2 Introduction, cont. 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 C&S p. 49 How should I constrain a body to have zero degrees of freedom? Provide six contact points Ex: C&S p. 52 Bear in mind that we can come up with geometric singularities that degenerate and allow degrees of freedom. Exact Constraint Design Blanding (1999) has developed a comprehensive approach to kinematic design. The basic concepts are: 1. Points on the object along the constraint line can move only at right angles to the constraint line, not along it. The constraint (link) is rigid, so it cannot stretch or shorten. The component of motion must be perpendicular to the constraint. An unconstrained 2-D object has 3 DOF (two translations, one rotation). One constraint will eliminate 1 degree of translational freedom. As a result, the object will have two independent degrees of freedom: a rotational degree of freedom that intersects the constraint line (anywhere), and a translational degree of freedom perpendicular to the constraint line. 2. Try to avoid overconstraint! Parts will not fit properly Assembly will either be too loose or too tight Internal stresses will buildup or will be transferred Warping, bowing, non-repeatable behavior due to stick-slip, premature failure 2

3 Exact Constraint Design, cont. 3. Any constraint along a given constraint line is functionally equivalent to any other constraint along the same constraint line (for small motions). For small motions, it doesn t matter how long the link is or where it connects to the body along the constraint line. 4. Any pair of constraints whose constraint lines intersect 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. This describes an instantaneous center of rotation (instant center). Try to avoid shallow angles (approaching 0 or 180 deg) 5. The axes of a body s rotational degrees of freedom will each intersect all constraints applied to the body. If there is an axis that intersects all the applied constraint lines, then none of those constraints can exert a moment about that axis (lever arm is zero), hence the body is free to rotate about that axis. Exact Constraint Design, cont. 6. A constraint applied to a body removes that rotational degree of freedom about which it exerts a moment. From Hale, 1999 The length of the lever arm is a measure of effectiveness of rotational constraint. For planar problems, in the absence of a good reason to do otherwise, try to balance the design by choosing equal lever arm lengths. 3

4 Exact Constraint Design, cont. From Hale, 1999 Exact Constraint Design, cont. 7. 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. From Hale, 1999 Blanding s Chart 4

5 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 --> 3 legged stool vs. 4 Four key problems: Non-repeatable relative motions between elements and/or repositioning capability (ex. 3 legged stool vs. 4) 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. Kinematic Mounts (Zero Degrees of Freedom) 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 Braddick, p vees tangent to a circle cone-vee-slot, where cone lies on normal to slot 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. 5

6 Kinematic Mounts (Zero Degrees of Freedom), cont. Wiffle tree Support of large mirrors or heavy plane objects These couplings need force closure. Gravity may suffice. Stewart platform Kinematic Couplings for 1 DOF motion Mallock s Vibrometer (1880) Vee and flat (S&C, p. 54) Alternate construction using balls and rods (Furse, p. 267, Braddick) 6

7 Kinematic Design Principles (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 nonrepeatable reassembleability To minimize transmission of distortion To minimize effects of thermal dimensional changes To minimize required accuracy of parts and construction to achieve performance goals. Kinematic Design Principles (summary), cont. 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. 7

8 Kinematic Couplings for Rotary Motion Example from Furse and Braddick Other Couplings and Drives Screw Couplings Axially preloaded nut Radially preloaded nut 8

9 References Blanding, D., Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York, Furse, J. E., Kinematic Design of Fine Mechanisms in Instruments, J. Phys. E: Sci. Instrum., vol. 14, 1981, p Hale, L. C., Principles and Techniques for Designing Precision Machines, UCRL-LR , Lawrence Livermore National Laboratory, ( Smith, S. T., Chetwynd, D. G., Foundations of Ultraprecision Mechanism Design, Gordon and Breach Science Publishers, Switzerland, Braddick, H. J. J., The Physics of Experimental Method, Chapman and Hall, London,