lecture13 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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MIT OpenCourseWare ____________ http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .
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2.830J/6.780J/ESD.63J 1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #13 Modeling Testing and Fractional Factorial Designs April 1, 2008
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2.830J/6.780J/ESD.63J 2 M anufacturing Outline • Full Factorial Models – Contrasts – Extension to 2 k – Model Term Significance: ANOVA – Checking Adequacy of Model Form • Tests for higher order fits (curvature) • Experimental Design – Blocks and Confounding – Single Replicate Designs – Fractional Factorial Designs NB: Read Montgomery Chapter 12
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2.830J/6.780J/ESD.63J 3 M anufacturing 2 2 Model Based on Contrasts This defines a 3-D “ruled surface” y - + - + ˆ y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 (Regression model) x 2 x 1 Two factor, two level experiments:
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2.830J/6.780J/ESD.63J 4 M anufacturing General Form for Contrasts Trial A B AB (1) --+ a +-- b -+- ab +++ Contrast A = Trial Column · A Contrast B = Trial Column · B Contrast AB = Trial Column · AB A :[ a + ab b (1)] B b + ab a AB ab + (1) a b ]
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2.830J/6.780J/ESD.63J 5 M anufacturing Extension to 2 k Consider 2 3 (3 factors, 2 levels each factor): Factor Levels Run Number Treatment Combination A B C 1 (1) -1 -1 -1 2a 1 -1 -1 3b -1 1 -1 4a b 1 1 -1 5c -1 -1 1 6a c 1 -1 1 7b c -1 11 8 abc 1 1 1 x 1 x 2 x 3 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8
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2.830J/6.780J/ESD.63J 6 M anufacturing Generalization A C B + - + + - - k 2 number of factors number of levels Courtesy of Dan Frey. Used with permission.
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2.830J/6.780J/ESD.63J 7 M anufacturing Contrasts as “Surface” Average Differences A B C + - + + - - [] ) 1 ( ) ( ) ( ) ( 4 1 ) ( ) ( ) ( ) ( 4 1 + + + + + + = bc c b a ac ab abc A (1) ( a ) ( b ) ( c ) ( ab ) ( abc ) ( bc ) ( ac ) Courtesy of Dan Frey. Used with permission.
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2.830J/6.780J/ESD.63J 8 M anufacturing A B C + - + + - - (1) ( a ) ( b ) ( c ) ( ab ) ( abc ) ( bc ) ( ac ) [] ) 1 ( ) ( ) ( ) ( 4 1 ) ( ) ( ) ( ) ( 4 1 + + + + + + = ac c a b bc ab abc B Contrasts for Main Effect Courtesy of Dan Frey. Used with permission.
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2.830J/6.780J/ESD.63J 9 M anufacturing A B C + - + + - - (1) ( a ) ( b ) ( c ) ( ab ) ( abc ) ( bc ) ( ac ) [] ) ( ) ( ) ( ) ( 4 1 ) ( ) ( ) ( ) 1 ( 4 1 bc ac b a abc c ab AB + + + + + + = Contrasts for Interaction Effect Courtesy of Dan Frey. Used with permission.
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2.830J/6.780J/ESD.63J 10 M anufacturing Contrasts for 2 3 Contrast A :[ a + ab + ac + abc b c bc (1)] Contrast ABC a + b + c + abc ab ac bc Effect = Contrast n 2 k 1 A = 1 4 n [ a + ab + ac + abc b c bc Factorial Combination Treament Combination I A B AB C AC BC ABC (1) 1 -1 -1 1 -1 1 1 -1 a 1 1 - 1- 1 1 1 b 1 -1 1 -1 -1 1 -1 1 ab 1 1 1 1 -1 -1 -1 -1 c 1 -1 -1 1 1 -1 -1 1 ac 1 1 -1 -1 1 1 -1 -1 bc 1 -1 1 -1 1 -1 1 -1 a b c 11111111 where n is the number of replicates at each treatment combination
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2.830J/6.780J/ESD.63J 11 M anufacturing Factorial Combinations Note: this is the scaled X matrix in the regression model Factorial Combination Treament Combination I A B AB C AC BC ABC (1) 1 -1 -1 1 -1 1 1 -1 a 1 1 - 1- 1 1 1 b 1 -1 1 -1 -1 1 -1 1 ab 1 1 1 1 -1 -1 -1 -1 c 1 -1 -1 1 1 -1 -1 1 ac 1 1 -1 -1 1 1 -1 -1 bc 1 -1 1 -1 1 -1 1 -1 a b c 11111111
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2.830J/6.780J/ESD.63J 12 M anufacturing Relationship to Regression Model A is the Effect of input 1 averaged over all other input changes (-1 to +1 or a total range of 2) B is the Effect of input 2 averaged over all other input changes, …….
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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lecture13 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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