## 4 Modern Dimensional Analysis Involved in Additive Manufacturing
Based on the mentioned unique protocol of Szirtes, the demanded elements of _ML_ were obtained as follows:  
\[\mathrm{v}\cdot\gamma^{0.2}\cdot\left(E\cdot I_{z}\right)^{-0.2}=1\ \Rightarrow\ S_{\mathrm{v}}=\left(\frac{S_{EI_{z}}}{S_{\gamma}}\right)^{0.2}= \frac{\mathrm{v}_{2}}{\mathrm{v}_{1}}, \tag{3}\]  
\[a*\cdot\gamma^{0.2}\cdot\left(E\cdot I_{z}\right)^{-0.2}=1\ \Rightarrow\ S_{**}=\left(\frac{S_{EI_{z}}}{S_{\gamma}}\right)^{0.2}= \frac{a_{2}^{*}}{a_{1}^{*}}, \tag{4}\]  
\[b*\cdot\gamma^{0.2}\cdot\left(E\cdot I_{z}\right)^{-0.2}=1\ \Rightarrow\ S_{bs}=\left(\frac{S_{EI_{z}}}{S_{\gamma}}\right)^{0.2}= \frac{b_{2}^{*}}{b_{1}^{*}}, \tag{5}\]  
\[c*\cdot\gamma^{0.2}\cdot\left(E\cdot I_{z}\right)^{-0.2}=1\ \Rightarrow\ S_{c*}=\left(\frac{S_{EI_{z}}}{S_{ \gamma}}\right)^{0.2}=\frac{c_{2}^{*}}{c_{1}^{*}}, \tag{6}\]  
\[A_{1}\cdot\gamma^{0.4}\cdot\left(E\cdot I_{z}\right)^{-0.4}=1\ \Rightarrow\ S_{v}=\left(\frac{S_{EI_{z}}}{S_{ \gamma}}\right)^{0.4}=\frac{A_{1,2}}{A_{1,1}}, \tag{7}\]  
\[F\cdot\gamma^{-0.4}\cdot\left(E\cdot I_{z}\right)^{-0.6}=1\ \Rightarrow\ S_{F}=\left(S_{ \gamma}\right)^{0.4}\cdot\left(S_{EI_{z}}\right)^{0.6}=\frac{F_{2}}{F_{1}}, \tag{8}\]  
\[L\cdot\gamma^{0.2}\cdot\left(E\cdot I_{z}\right)^{-0.2}=1\ \Rightarrow\ S_{L}=\left(\frac{S_{EI_{z}}}{S_{ \gamma}}\right)^{0.2}=\frac{L_{2}}{L_{1}}, \tag{9}\]  
By a priori selection, the prototype had:  
60% filling percentage, L\({}_{1}=0.5\) m, b\({}_{1}=0.03\) m, a\({}_{1}=0.018\) m, F\({}_{1}=2.943\) N, E\({}_{1}\cdot\)I\({}_{2,1}=116.5\) N\(\cdot\)m\({}^{2}\), \(\gamma_{1}=7806\) N/m\({}^{3}\). In the same manner, the corresponding model had:  
30% filling percentage, \(E_{2}\cdot I_{z,2}=14.0\) N\(\cdot\)m\({}^{2}\); \(\gamma_{2}=5986\) N/m\({}^{3}\). Resulting:  
\[S_{E\cdot I_{z}}=\frac{E_{2}\cdot I_{z,2}}{E_{1}\cdot I_{z,1}}=0.12018\ \mathrm{and}\ S_{\gamma}=\frac{\gamma_{2}}{\gamma_{1}}=0.76682.\]  
From the mentioned _ML_ result:  
Figure 7:0  
Figure 7:1  
Figure 7:2  
By high-accuracy measurements on model, \(\mathrm{v}_{2,meas}=0.97\) mm was obtained, resulting from _ML_ for the prototype the predictable displacement in: \(\mathrm{v}_{1 MDA}=\frac{\mathrm{v}_{2,meas}}{\mathrm{S}_{\mathrm{v}}}=1.405\) mm. In order to validate the _ML_, experimental investigations were similarly performed on the prototype, obtaining \(\mathrm{v}_{1,meas}=1.42\) mm, a deviation of 1.056 %. In this case, by adapting \(E\cdot I_{z}\) as independent variable, there is no restriction in choosing different materials for the prototype and model, as well as different cross-sectional shapes. In addition, by means of \(\gamma\), one can choose different filling degrees for prototype and model, which was happened in this case. The second proposed and validated Dimensional Set, detailed in Table 4, had as independent variables \((b,\ L,\ F,\ E)), with \(a\) becoming a dependent variable. In similar manner, the matrixes A, B, C, and D can be identified. Figure 7:3  
Figure 7:4Figure 7:5  
Figure 7:6  
Figure 7:7  
Figure 7:8  
For the prototype, by a priori selection, one has: 60% filling percentage, \(b_{1}=0.03\) m, \(L_{1}=0.45\) m, \(F_{1}=1.962\) N, \(E_{1}=2.31\cdot 10^{9}\)\(\frac{\text{N}}{\text{m}^{2}}\), respectively \(a_{1}=0.018\) m.  
The corresponding model, by a priori selection, has:  
30% filling percentage, \(\text{b}_{2}=0.02\) m, \(\text{L}_{2}=0.25\) m, \(\text{F}_{2}=0.981\) N, \(\text{E}_{2}=\text{E}_{1}=2.31\cdot 10^{9}\)\(\frac{\text{N}}{\text{m}^{2}}\). By unitary calculi, \(S_{b}=0.66667\); \(S_{L}=0.55556\); \(S_{F}=0.5\); and \(S_{a}=0.75\) are obtained. By applying the correspondent element of \(ML\), i.e., (15), we obtain \(a_{2}=0.0135\) m. Performing accurate measuring on the model, we obtain \(\text{v}_{2,meas}=0.49\) mm, resulting from \(ML\) for the prototype the predictable displacement in: \(\text{v}_{1 MDA}=\frac{\text{v}_{2,meas}}{\text{S}_{\text{v}}}=0.7417\) mm. In order to validate the \(ML\), experimental investigations on the prototype were similarly performed, obtaining \(\text{v}_{1,meas}=0.73\) mm, a deviation of 1.603 %.