Issue 44

G. G. Bordonaro et alii, Frattura ed Integrità Strutturale, 44 (2018) 1-15; DOI: 10.3221/IGF-ESIS.44.01

S TEPS FOR D ESIGN OF C OMPUTER E XPERIMENT

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he goal of this analysis is to determine effects and possible interaction effects of a set of different process variables on significant process responses [24]. Based on experience from rolls designers, four factors are thought to affect the rolling product characteristics (billet temperature, billet size, rolls angular velocity, draught). The mathematical model, containing one constant, four linear terms, six interactions and four quadratic terms, is the following: Y = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + b 12 X 1 X 2 + b 13 X 1 X 3 + b 14 X 1 X 4 + (2) + b 23 X 2 X 3 + b 24 X 2 X 4 + b 34 X 3 X 4 + b 1 2 X 1 2 + b 2 2 X 2 2 + b 3 2 X 3 2 + b 4 2 X 4 2 where terms X i are the process variables:  X 1 : workpiece temperature [°C]  X 2 : rolls angular velocity [RPM]  X 3 : workpiece diameter [mm]  X 4 : draught in terms of diameter reduction [%] In this design each variable has three levels, respectively coded as -1, 0 and +1:  X 1 : 800, 1000, 1200 [°C]  X 2 : 5, 17.5, 30 [RPM]  X 3 : 20, 40, 60 [mm]  X 4 : 20%, 40%, 60% [%] The following responses, measured after reaching a steady-state condition, are taken into account, with Y 1 and Y 2 to be maximized and Y 3 and Y 4 to be minimized:  Y 1 : spread [mm]  Y 2 : effective stress in the contact region [MPa]  Y 3 : effective stress in the non-contact region [mm]  Y 4 : rolls reaction force (average) [kN] A standard experimental design such as the Central Composite Design with just one center point would require a total of 25 experiments (i.e., 2 4 + 2·4 + 1). In order to further reduce the experimental effort, a D-Optimal design has instead been performed. This statistical tool allows to identify, among all the possible experiments (in our case 81, i.e., 3 4 ), the subset leading to the best possible compromise between experimental effort and quality of information by maximizing the normalized determinant of the information matrix (det( X ' X }/n p ), where X is the model matrix, n is the number of experiments and p is the number of parameters in the model. Figure 8 shows the variation of the normalized determinant versus the number of experiments. The solution with 22 experiments has been selected.

Figure 8 : Normalized determinant of the information matrix is represented versus the number of experiments.

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