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• 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. Thickness reduction ( r ) variation Rolling model Filling factor = H 1 – t (H 1 = central height)
• 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. Other fixed conditions, relationship btw Hr and r (thickness red.) Drawn with Origin 7.0 from simulation results The central (max) height
• 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. Graph showing the regression line and the data points Regression line Data point
• 7. Filling factor and thickness reduction
• 8. Rolled width vs thickness reduction Plotted with Origin
• 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. Graph showing the regression curve and the data points Regression curve Data point
• 11.
• 12. Friction factor ( m ) variation
• 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%
• 14. Not significant effect on central height
• 15.  Try parabolic model for (1) (1) On width… Half – quadratic curve
• 16. Sr = 43.61 – 10.48m + 14.36m 2
• Test with m = 0.20
• Sr = 43.61 – 10.48x0.2 + 14.36x0.04)
• = 42.09
• Error = (42.28 – 42.09)/42.09
• = 0.45%