# Optimizing Roller Design to Improve Web Strain Uniformity in Roll-to-Roll Process
## 3 Roller Design Optimization for Minimizing Strain Deviation
### Design of Experiments (DOE) Considering Interactions  
Since we confirmed that interactions between design factors occur, our DOE analysis considered these interactions. Table 3 presents an orthogonal array for the case in which the interactions between the design factors are considered. Factors A, B, and C are in columns 1, 2, and 5, and their interactions are indicated by AB, BC, and AC, respectively. In the DOE-based FEA, only columns 1, 2, and 5 with factors A, B, and C were considered. Then, in the process of selecting the optimum condition, the factors of the remaining columns were considered. The ABC factor, which is a higher-order interaction, was not considered and was pooled in the error term. Subsequently, a sensitivity analysis was conducted to determine whether changes in design factor values affected the strain deviation in the MD.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & & & \\ \cline{3-14} \multirow{2}{*}{**No.**} & **A** & **B** & **AB** & **AB** & **C** & **AC** & **AC** & **BC** & **ABC** & **ABC** & **BC** & **ABC** & **ABC** \\ \cline{2-14}  & **A** & **B** & **AB** & **AB** & **C**  
The initial factor levels in the sensitivity analysis were the factor levels in Table 2, and the results are shown in Figure 9. The objective function to minimize the strain deviation of MD was selected so that the difference between the maximum and minimum values of the strain distribution was to be the smallest. The same was applied to the DOE-based FEA and verification experiments. Accordingly, the strain deviation in the MD was minimal when the thickness gradient (A) of the skin was approximately 6 mm and the total diameter (B) of the roller was 40-60 mm. The lower the Young's modulus (C) of the rubber, the greater the reduction of the MD strain deviation.  
Thus, the range of factor A was further sub-divided from 6 to 9 mm and the range of factor B was set to 40-60 mm for the FEA. The levels of the modified factors are shown in Table 4, and Table 3 shows the orthogonal array. Moreover, Figure 10 shows the results of the sensitivity analysis based on the modified factor levels in Table 4. For a skin thickness gradient (A) of 6 mm, roller diameter (B) of 60 mm, and low Young's modulus (C) of the skin, the strain deviation in the MD was minimized. In addition, each design factor influenced the reduction of the strain deviation in the MD.  
Subsequently, we performed a quantitative analysis of variance (ANOVA) to select the optimal design. The ANOVA represents the variation in the characteristic values obtained as a result of the DOE as the total sum of squares [15]. This value is expressed as the total sum of squares for each factor, and the remainder is interpreted as an error variation. The variance of each factor is then compared to the variance of the error to determine how much influence each factor has on the characteristic values. The larger the F value in the analysis of variance, the greater the effect on the objective function. Table 5 shows the results of the ANOVA including the interactions between the factors; SS is the sum of squares, DF is the degrees of freedom, MS is the mean square, E is the error term, and T is the total  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **A (mm)** & **B (mm)** & **C (MPa)** \\ \hline Level 1 & 6 & 40 & 2.85 \\ Level 2 & 7 & 50 & 7.85 \\ Level 3 & 8 & 60 & 12.85 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Modified factor levels for determining the global minimum.  
Figure 10: Sensitivity analysis considering interactions based on factor levels in Table 4.  
Figure 9: Sensitivity analysis considering interactions based on factor levels in Table 2.  
deviations in the MD of the conventional and barrel rollers. The total deformation of the web is shown in Figure 12. The proposed design is effective according to these results.