Influence of the Processes Parameters on the Properties of The ...

Chapter 3.
Analytical Methods and Designs <strong>of</strong> Experiments
Y =a 0 + a 1 X 1 + a 2 X 2 + a 12 X 1 X 2 + a 11 X 1
2
+ a 22 X 2
2
(3.29)
where a 0 is <strong>the</strong> average value <strong>of</strong> <strong>the</strong> centre <strong>of</strong> <strong>the</strong> domain, a 1 and a 2 are <strong>the</strong> effects <strong>of</strong> <strong>the</strong> two
factors, a 12 represents <strong>the</strong> interaction between <strong>the</strong> two factors and a 11 and a 22 are <strong>the</strong> quadratic effects <strong>of</strong> both
variables. The five levels <strong>of</strong> factor 1 correspond to lines 2 to 8 in Table 3.2.
Figure 3.27: Distribution <strong>of</strong> experimental points for a Doehlert’s design <strong>of</strong> 2-variables.
Table 3.2: Matrix <strong>of</strong> experiments X.
I X 1 X 2 X 1 X 2
2
X 1
2
X 2
1 1.000 0.000 0.000 0.000 0.000
1 0.500 0.866 0.433 0.250 0.750
1 -0.500 0.866 -0.433 0.250 0.750
1 -1.000 0.000 0.000 1.000 0.000
1 -0.500 -0.866 0.433 0.250 0.750
1 0.500 -0.866 -0.433 0.250 0.750
1 0.000 0.000 0.000 0.000 0.000
The multi-linear regression analysis <strong>of</strong> <strong>the</strong> experimental points controls <strong>the</strong>se five coefficients
minimizing <strong>the</strong> error adjustment <strong>of</strong> <strong>the</strong> ma<strong>the</strong>matical model. The relationship matrix that correlates <strong>the</strong>se
factors toge<strong>the</strong>r in <strong>the</strong> vector <strong>of</strong> coefficients â to <strong>the</strong> vector <strong>of</strong> response Y is given by
â = (X t . X) -1 . X t .Y (3.30)
where X is <strong>the</strong> matrix <strong>of</strong> experiments defined in Table 3.2.
The numerical values <strong>of</strong> <strong>the</strong> coefficients <strong>of</strong> <strong>the</strong> vector â, determine what factors and interactions
are <strong>the</strong> more influent. To clarify whe<strong>the</strong>r coefficients are significant or not, we calculated <strong>the</strong> experimental
standard deviation S from three tests at <strong>the</strong> centre <strong>of</strong> <strong>the</strong> experimental domain. The standard deviation on
<strong>the</strong> various coefficients can be determined from <strong>the</strong> relationship:
â= S (diagonal <strong>of</strong> <strong>the</strong> dispersion matrix) 1/2 (3.31)
in which <strong>the</strong> dispersion matrix (X t .X) -1 represents <strong>the</strong> inverse matrix <strong>of</strong> <strong>the</strong> product <strong>of</strong> <strong>the</strong> transposed matrix
<strong>of</strong> X by X. Each coefficient a i can be considered significant, if it has a numeric value greater than three times
its uncertainty a i .
7.2 Screening Plans: Taguchi’ Design
The method, created by Taguchi [Roy, 1990], aims to simplify <strong>the</strong> implementation <strong>of</strong> experimental
designs. It <strong>of</strong>fers a collection <strong>of</strong> tables and tools to help to choose <strong>the</strong> most appropriate table. Collection <strong>of</strong>
Taguchi’ tables are actually three types <strong>of</strong> information related to each o<strong>the</strong>r:
Taguchi’ Tables: <strong>the</strong>y specify <strong>the</strong> content <strong>of</strong> <strong>the</strong> matrix <strong>of</strong> experience, and were chosen based on <strong>the</strong>
number <strong>of</strong> terms, factors, interactions.
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