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TECHNICAL PAPERS

Equivalence Classes for Shape Synthesis of Moving Mechanical Parts

[+] Author and Article Information
Horea T. Ilieş

Ford Motor Company*e-mail: hilies@ford.com

Vadim Shapiro

University of Wisconsin-Madison, Spatial Automation Laboratory, Department of Mechanical Engineering, 1513 University Avenue, University of Wisconsin-Madison, Madison, WI 53706 USAe-mail: vshapiro@engr.wisc.edu

J. Comput. Inf. Sci. Eng 4(1), 20-27 (Mar 23, 2004) (8 pages) doi:10.1115/1.1641794 History: Received July 01, 2003; Revised November 01, 2003; Online March 23, 2004
Copyright © 2004 by ASME
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References

Suh, N. P., 1990, “The Principles of Design,” Oxford Series on Advanced Manufacturing, Oxford University Press, New York, Chap. 3.
Pahl, G., and Beitz, W., 1996, “Engineering Design: a Systematic Approach,” Springer, London, Chap. 6.
Gupta,  R., and Jakiela,  M., 1994, “Kinematic Simulation and Shape Synthesis via Small Scale Interference Detection,” Res. Eng. Des., 6, pp. 103–123.
Stahovich, T. F., 2001, “Artificial Intelligence for Design,” In E. K. Antonsson and J. Cagan, eds., In Formal Engineering Design Synthesis, Cambridge University Press, pp. 228–269.
Lee, C.-Y., Ma, L., and Antonsson, E. K., 2001, “Evolutionary and Adaptive Synthesis Methods,” In E. K. Antonsson and J. Cagan, eds., Formal Engineering Design Synthesis, Cambridge University Press, pp. 270–320.
Caine, M. E., 1993, “The Design of Shape From Motion Constraints,” PhD thesis, Massachusetts Institute of Technology, Cambridge.
Joskowicz, L., and Addanki, S., 1988, “Innovative Shape Design: A Configuration Space Approach,” Technical Report 356, Courant Institute of Mathematical Sciences, New York University, NY.
Joskowicz,  L., and Sacks,  E., 1999, “Computer-Aided Mechanical Design Using Configuration Spaces,” Comput. Sci. Eng., 1(6), pp. 14–21.
Stahovich,  T. F., Davis,  R., and Shrobe,  H. E., 2000, “Qualitative Rigid-Body Mechanics,” Artif. Intel., 119(1–2), pp. 19–60.
Ilieş, H., 2000, “On Shaping Moving Mechanical Parts,” PhD thesis, University of Wisconsin, Madison.
Ilieş, H., and Shapiro, V., 1996, “An Approach to Systematic Part Design,” In Proceedings of the 5th IFIP WG5.2 Workshop on Geometric Modeling in Computer Aided Design, pp. 383–392.
Ilieş,  H., and Shapiro,  V., 1999, “The Dual of Sweep,” Comput.-Aided Des., 31(3), pp. 185–201.
Ilieş, H., and Shapiro, V., 2003, “On the Synthesis of Functionally Equivalent Mechanical Designs,” In Computational Synthesis: From Basic Building Blocks to High Level Functionality, AAAI Spring Symposium Series, AAAI Press.
Ilieş,  H., and Shapiro,  V., 2002, “A Class of Forms From Function: the Case of Parts Moving in Contact,” Res. Eng. Des., (13), pp. 157–166. A preliminary version of this paper appeared in the Proceedings of the ASME 2001 DETC, 13th International Design Theory and Methodology Conference.
Horsch,  T., and Jüttler,  B., 1998, “Cartesian Spline Interpolation for Industrial Robots,” Comput.-Aided Des., 30(3), pp. 207–224.
Greenwood, D. L., 1988, Principles of Dynamics, Prentice-Hall, Englewood Cliffs, NJ, Chap. 1.

Figures

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A secondary hood latch must engage and retain a vertically moving striker
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The largest shape (set of points) remaining inside a containing set E during a known motion M(t) is given by unsweep (E,M). The motion is a clockwise rotation with an angle of 21°.
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Applied force (a) and torque (b) with components opposite to the normal contact force Fn at a contact point P
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A conjugate triplet, which requires continuous geometric contact, does not always exist for given object and motion (a). If one conjugate triplet exists (b), there are infinitely many conjugate triplets one of which is shown in (c).
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The largest connected latch seen here from three different directions, which include all other positionally equivalent shapes moving in contact with the given striker according to a known motion. It corresponds to set BmaxA in equation (4).
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A cylinder rolling on a plane under the influence of an externally applied force. Set B1 in (a) shows all contact points between the cylinder and the plane. Set B2 shows a subset of B1 that may be sufficient to carry the applied force.
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A cylindrical follower A maintains contact with B while moving under the influence of externally applied loads. For loading condition shown in (a) only the points on the outer profile of B are carrying the loads, but different loading conditions may redistribute these load-bearing points as shown in (b) and (c).
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The discrete relative configurations of the striker A relative to the latch to be designed: the motion of the striker was separated into its “downward” motion (a) and its “upward” motion (b) because they correspond to different functions of the latch. Figure (c) shows the top view of the BmaxA forming the maximal triplet 〈AmaxA,BmaxA,M〉 according to equations (4).
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Figure (a) illustrates the largest shape satisfying the containment constraints. Figures (b–c) show the largest unbounded connected shape, which includes all other statically equivalent shapes satisfying the given contact function. The load-bearing boundary points have been computed according to the functional classification detailed in appendix A and are shown in lighter color. Figure (d) displays the largest latch satisfying both contact and containment constraints obtained through an intersection of the two corresponding maximal shapes. Figures (e–f) show two other functional latches that are statically equivalent with the maximal latch shown in (d).
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Any functional part must concurrently satisfy all imposed functional requirements, and thus must be an element of all three classes of equivalence

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