## 4 Mathematical Modeling of Forming Pressure  
Based on the data in Table 3, a mathematical model describing the effect of the input parameters on the forming pressure was built. Equation (1) is rewritten as Equations (4) and (5). \[\mathrm{P_{max}}=\mathrm{f(Q,S_{0},\,\sigma)} \tag{4}\]  
i.e \(\mathrm{P_{max}}=\mathrm{Yi}=\mathrm{f}\left(\mathrm{x_{1},\,x_{2},\,x_{3}}\right)\) (5)  
where the physical variable is converted into a coded variable in Equation (6). The values of the coding variables are given in Table 4.  
\[\mathrm{x_{1}}=\frac{\mathrm{Q}-97.5}{17.5}\ ;\ \mathrm{x_{2}}=\frac{\mathrm{S_{0} }-1}{0.2}\ ;\ \mathrm{x_{3}}=\frac{\sigma-1.9}{0.6} \tag{6}\]  
where \(\mathrm{x_{1},\,x_{2},\,and\,x_{3}}\) are coded variables of the binder pressure, the thickness of the workpiece, and the ratio of stress, respectively. According to orthogonal second-order design theory [30], the formula of forming pressure is represented by Equation (7):  
\[\mathrm{Y}=\mathrm{b_{0}}+\sum_{j=1}^{2}\mathrm{b_{j}}\mathrm{x_{j}}+\sum_{ \mathrm{i}j=1\
eq j}^{2}\mathrm{b_{ij}}\mathrm{x_{i}}\mathrm{x_{j}}+\sum_{j=1 }^{2}\mathrm{b_{jj}}\mathrm{x_{j}^{2}} \tag{7}\]  
where \(\mathrm{b_{0},\,b_{j},\,b_{ji},\,and\,b_{jj}\) are the coefficients of the regression function and are determined by Equations (8)-(11). \[\mathrm{b_{0}}=\frac{1}{n}\sum_{u=1}^{n}\mathrm{Y_{u}} \tag{8}\]  
\[\mathrm{b_{j}}=\frac{\sum_{u=1}^{n}\mathrm{x_{uj}}\mathrm{Y_{u}}}{\sum_{u=1}^ {n}\mathrm{x_{uj}^{2}}} \tag{9}\]  
\[\mathrm{b_{ij}}=\frac{\sum_{u=1}^{n}(\mathrm{x_{ui}\,x_{uj}})\mathrm{Y_{u}}}{ \sum_{u=1}^{n}\left(\mathrm{x_{ui}\,x_{uj}}\right)^{2}} \tag{10}\]  
\[\mathrm{b_{jj}}=\frac{\sum_{u=1}^{n}\mathrm{x_{uj}^{\prime}\,x_{u}}}{\sum_{u=1 }^{n}\left(\mathrm{x_{ui}}\right)^{2}} \tag{11}\]  
The coefficients of Equation (7), including independent and dependent coefficients, were calculated and checked by the Student criterion [30]. After calculating, a variance \(\mathrm{s^{2}}\) of 72.3 was calculated, the average \(\mathrm{t}\) was determined to be 517.29, and the standard deviation \(\mathrm{s}\) was computed as 8.5. Based on the experimental data in Tables 2 and 3, the mathematical equation of forming pressure is shown in Equation (12). \[\mathrm{Y}=510.22-16.8\ \mathrm{x_{1}}^{2}+22.8\ \mathrm{x_{2}}^{2}+84.3\ \mathrm{x_{1}}+25.58\ \mathrm{x_{2}}+31.58\ \mathrm{x_{3}}+The suitability of Equation (15), determined by the Fisher criterion, is represented by Equation (13). It shows that the function is suitable.  
\[\mathrm{F}_{\alpha}=5.56\leq\mathrm{F}_{\mathrm{t}}=19.33 \tag{13}\]  
where \(\mathrm{F}_{\mathrm{t}}\) is tabulated according to the Fisher criterion and \(\mathrm{F}_{\alpha}\) is the adequacy according to the Fisher criterion [30]. In order to determine the influence of each input parameter on the maximum forming pressure value, Equations (14)-(16) are rewritten from Equation (12) as follows:  
\[\mathrm{Y}_{1}(\mathrm{x}_{1},\,1,\,1)=569.68+121.8\mathrm{x}_{1}-16.8\, \mathrm{x}_{1}{}^{2} \tag{14}\]  
Table 4.0  
Table 4.1  
Based on the coefficients of Equations (14)-(16), it can be seen that the binder pressure (\(\mathrm{x}_{1}\)) is the parameter that has the greatest impact on the forming pressure. Next, the second most influential parameter is the workpiece thickness. The ratio of stress is the parameter that has the least influence on the value of the maximum forming pressure. A comparison of the value of the experiment with the value obtained from Equation (12) is shown in Table 4.2 The error of the mathematical model and the experimental result is 2.4%. It can be seen that the difference between the experimental results and results calculated from the mathematical models is not significant. The analysis results from the MATLAB software for Equation (12) show the influence of each factor on the forming pressure.