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Mathematical Modeling of Assembly

This article explains the mathematical modeling of assembly using coordinate frames and composite transforms. It covers topics such as nominal mating of parts, part location variation, error accumulation, and precision versus accuracy.

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Mathematical Modeling of Assembly

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  1. Mathematical Modeling of Assembly • Coordinate frames • each part has a base coordinate frame • Relationships between parts are expressed as 4x4 matrix transforms

  2. Matrix Math • 4x4 matrices relate adjacent frames • Matrix contains rotational part and translational part • Translation occurs first, so rotation does not change position of new frame

  3. Basic Translation and Rotation

  4. Watch Transform Ordering!

  5. Composite Transforms

  6. Nominal Mating of Parts

  7. Example - Pin & Hole Mating(pin translated)

  8. Example - Pin & Hole Mating(pin rotated)

  9. Example - Pin & Hole Mating(feature on first part)

  10. Example - Pin & Hole Mating(feature on second part)

  11. Example - Pin & Hole Mating(Assembling two parts)

  12. Part Location Variation • Varied location of Part B calculated from nominal location of Part A • Uses same math as nominal model!

  13. Chaining together parts

  14. Variation Analysis • Error types: • Change in process average (mean shift) • Process variation around average (variance)

  15. Precision vs. Accuracy

  16. Desired Distribution • LSL = lower specification limit • USL = upper specification limit

  17. Error Accumulation • Worst case tolerancing: • assume all errors at extremes • errors accumulate linearly w/ # of parts • deterministic, not statistical • Statistical tolerancing • assume errors distributed randomly between limits • errors accumulate as sqrt of # of parts • if mean is equal to nominal dimension!

  18. Error Accumulation • Sums of zero-mean errors accumulate as sqrt(N), because + and - errors cancel • Sums of non-zero-mean errors accumulate as N, because there are no cancellations

  19. How do Non-zero-mean errors occur? • Example: • operator stops material removal as soon as part enters tolerance zone

  20. Example - Stapler Variations

  21. (small) Error Transform

  22. Using Error Transform

  23. Using Error Transform

  24. Using Error Transform - Part 2

  25. Using Error Transform - Part 2

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