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CATS 1-D XL

CATS 1-D XL. One-Dimensional Tolerance Stackup Spreadsheet. by Ken Chase and Jonathan Wittwer Brigham Young University. Complex Assemblies. 3s. Capability. Process Variation. Mean. Standard Deviation. -1s. +1s. Rejects. LL. UL. -3s. +3s. A frequency plot shows how a

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CATS 1-D XL

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  1. CATS 1-D XL One-Dimensional Tolerance Stackup Spreadsheet by Ken Chase and Jonathan Wittwer Brigham Young University

  2. Complex Assemblies

  3. 3s Capability Process Variation Mean Standard Deviation -1s +1s Rejects LL UL -3s +3s A frequency plot shows how a process varies from the mean

  4. Models for Predicting Assembly Tolerance Stackup • Worst Case (WC) • Statistical (RSS) • Six Sigma (6s) • Measured Data (Meas)

  5. Models for Predicting Assembly Tolerance Stackup

  6. Centering the Mean Center Rejects Xmean LL UL Centering the process in the LL/UL window minimizes rejects

  7. 3s Capability Decrease Variation Decrease Standard Deviation Reduce rejects LL UL -3s +3s

  8. LL UL -6s -5s -4s -3s -2s -1s 0 +1s +2s +3s +4s +5s +6s ±3s ±4.5s ±6s Higher s Values Mean Higher Quality

  9. Controlling Variation Leads to Higher Yields and Fewer Defects LimitsYield Defects ±3s 99.73% 2700 / Million ±4.5s 99.99966% 3.4 / Million ±6s 99.9999998% 2 / Billion

  10. Controlling Variation Leads to Higher Yields and Fewer Defects LimitsYield Defects ±3s 99.73% 2700 / Million ±4.5s 99.99966% 3.4 / Million ±6s 99.9999998% 2 / Billion

  11. Mean Shifts Happen Mean Shift Mean variation is time-dependent due to tool wear, temperature, etc.

  12. Accounting for Mean Shifts 1.5s 4.5s LL UL The Six Sigma Model allows the engineer to model assembly variation due to mean shifts

  13. Measures of Process Capability UL LL Process Capability Index: 3 ± s Cp adjusted for k: k UL LL Cpk= Cp(1-k) 6 ± s

  14. UL LL 3 ± s k UL LL 6 ± s 6 Variation Defined RSS: 6s: where Cpk= Cp(1-k) and

  15. 3s Capability Six SigmaDynamic Mean Shift 99.73% (Short Term) 95.45% (Long Term) LSL USL -3s +3s Comparing Short and Long Term Variation Distribution of parts tool wear Distribution of parts when tool is new. when tool is old. - tolerance + tolerance min. max. Figure 7. Scenario 3. As the tool wears, the mean of the most recent parts shifts from the left to the right

  16. Requirements for High Quality 99.9999998% (Short Term) LL UL 99.99932% (Long Term) -6s +6s 6s Capability A goal of 4.5s long term requires a 6s process in the short term

  17. Excel Statistical Functions NORMDIST(x,mean,stand_dev,T/Fflag) =area under the Normal distrib at point x NORMSDIST(z) =area under the Standard Normal distrib STANDARDIZE(x,mean,stand_dev) =(x - mean)/stand_dev NORMSINV(Probability) =z corresp to a given probability(for z <5) SUMSQ(G1:G25)^0.5 =square root of the sum of squares of si

  18. a Yield (1- a) Rejects Z Yield Fraction a Source 3s 6s 15 Place Math Tables 0.998650101968370000 0.999999999013000000 6th Order Polynomial Fit 0.998650187471804000 0.999999998751939000 Excel 0.998650032776765000 0.999999999009878000 Reject Fraction (1-a) Source 3s 6s 15 Place Math Tables * 0.001349898031629990 0.000000000987000000 6th Order Polynomial Fit 0.001349812528196100 0.000000001248061000 Excel 0.001349967223235000 0.000000000990122000 StatisticalFunction Accuracy * the 6s value is from an alternate source, accurate to 12 places.

  19. CATS 1-D Spreadsheet

  20. CATS 1-D Inputs

  21. Mean and Variance Comparison

  22. Yield and Rejects Calculated in s units, %, or parts-per-million

  23. X = 0 Standard Normal Distribution Transformation ZL ZU Mean Standard Deviation sx = 1.0 Z (s units) Used to determine % rejects from standard tables. Does not show mean shift. Transformed data all looks alike.

  24. Modified Normal Distribution LL UL Shows the mean shifts Shows the quality levels Allows all 3 curves to be plotted for comparison

  25. Plot Controls

  26. What can CATS 1-D users expect? Probably won’t see: An end to poverty and misery All men treating each other as brothers World peace But, you might notice: An increased understanding of the role of statistics in design Fewer problems on the factory floor Engineering and production talking to each other without shouting.

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