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17

  1. 1. Treating simulation data with mathematical functions using chi-square technique **…formulas achieved by using Chi - square based Java applets on http://csbsju.edu *…relationship btw outputs ( filling,spread ) and inputs ( thickness red.,friction,…)
  2. 2. Thickness reduction ( r ) variation Rolling model Filling factor = H 1 – t (H 1 = central height)
  3. 3. Fixed condition with: friction 0.15; sample width 40mm, thickness 5mm, groove 10mm and 20 degrees. 0.00 40.00 5.00 0 1.22 42.36 4.22 40 1.07 41.52 4.32 35 0.95 41.27 4.45 30 0.76 41.06 4.51 25 0.59 41.03 4.59 20 0.24 40.33 4.74 10 Filling factor (as drawn) S (mm) spread H r (mm) (central) Thickness reduction,%
  4. 4. Other fixed conditions, relationship btw Hr and r (thickness red.) Drawn with Origin 7.0 from simulation results The central (max) height
  5. 5. Linear model : y = a + bx ( http://www.physics.csbsju.edu/stats/WAPP.html ) Hr = 4.992 – 0.019r Test with r = 40  Hr = 4.992 – 0.019 x 40 = 4.232 Measured H r (40%) = 4.22 mm  error (relative) = (4.232 – 4.22)/4.22 = 0.28% Test with r = 10  Hr = 4.992 – 0.019 x 10 = 4.80. error = (4.8 – 4.74)/4.74 = 1.25%
  6. 6. Graph showing the regression line and the data points Regression line Data point
  7. 7. Filling factor and thickness reduction
  8. 8. Rolled width vs thickness reduction Plotted with Origin
  9. 9. Quadratic model : y = a + bx + cx 2 ( http://www.physics.csbsju.edu/cgi-bin/stats/WAPP ) Sr = 40.03 + 0.01862r + 0.0009461r 2 Test with r = 35  Sr = 40.03 + 0.01862x35 + 0.0009461x35 2 = 41.84 Measured S r (35%) = 4.1.52 mm  error (relative) = (41.84 – 41.52)/41.52 = 0.77%
  10. 10. Graph showing the regression curve and the data points Regression curve Data point
  11. 12. Friction factor ( m ) variation
  12. 13. Friction factor variation data from DEFORM 4.22/4.76 4.47/40.97 4.57/40.68 0.60 4.27/41.73 4.44/41.02 4.50/40.65 0.40 4.27/41.70 4.48/41.05 4.54/40.76 0.35 4.28/41.75 4.51/41.04 4.53/40.79 0.30 4.25/41.90 4.48/41.17 4.53/40.82 0.25 4.18/42.28 4.47/41.24 4.57/40.89 0.20 4.22/42.36 4.45/41.27 4.59/40.96 0.15 40% 30% 20%
  13. 14. Not significant effect on central height
  14. 15.  Try parabolic model for (1) (1) On width… Half – quadratic curve
  15. 16. Sr = 43.61 – 10.48m + 14.36m 2 <ul><li>Test with m = 0.20 </li></ul><ul><li>Sr = 43.61 – 10.48x0.2 + 14.36x0.04) </li></ul><ul><li>= 42.09 </li></ul><ul><li>Error = (42.28 – 42.09)/42.09 </li></ul><ul><li>= 0.45% </li></ul>
  16. 17. Quadratic graph
  17. 18. <ul><li>Central height & filling vs thickness reduction relationship can be treated with linear function </li></ul><ul><li>Spreading vs thickness reduction can be well treated with quadratic relationship </li></ul><ul><li>Friction does not affect central height </li></ul><ul><li>Spread vs friction can be described by a quadratic function </li></ul><ul><li>Upcoming: thickness, width, vacancy, inclined angle ,… </li></ul>Conclusions
  18. 19. Appendix
  19. 20. Appendix

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