## 3 Results and Discussion
### Flat Window and Accelerated Printing Platform  
Following the procedure proposed in [25], an expression for the separation force in case of arbitrary elevation speed, \(V(t)\), can be obtained. Here we consider a case with the gap varying as \(h_{a}=h_{0}+V_{0}t+\frac{at^{2}}{2}\) and corresponding elevation speed, \(V(t)=V_{0}+at\). Then the separation force reads as:  
\[F_{s,a}=\frac{(V_{0}+at)\pi R^{4}}{4\beta h_{a}^{3}(1+\alpha_{a})}, \tag{7}\]  
where \(\alpha_{a}=24\sum_{n=1}^{\infty}\frac{((-1)^{n}-1)^{2}}{\tau^{4}n^{4}}e^{-( \frac{\pi n}{2})^{2}vt}\) and \(a\) is the acceleration of the printing platform. The behavior of the parameter \(\alpha_{a}\) with respect to initial elevation speed \(V_{0}\) and time \(t\) are given in Figure 2b,d where the acceleration is fixed at \(a=0.01\) mm/s\({}^{2}\). Interestingly, even for such a relatively low value of acceleration, the behavior of the parameter \(\alpha_{a}\) changes significantly compared to \(\alpha\). In particular, the maximal value of \(\alpha_{a}\) is achieved approximately two times faster compared to \(\alpha\). Moreover, the value of the initial elevation speed for which the value of the parameter \(\alpha_{a}\) becomes non-negligible, decreases as well. To better illustrate the difference between \(\alpha\) and \(\alpha_{a}\), we introduced \(\alpha\)-reduction parameter, \(\eta_{\alpha}\), given by:  
\[\eta_{\alpha}=1-\frac{\alpha}{\alpha_{a}} \tag{8}\]  
Figure 2: The dependence of transient parameter (**a**) \(\alpha\) from the time for the different values of the initial elevation speed; (**b**) \(\alpha_{a}\) from the time for the different values of the initial elevation speed; (**c**) \(\alpha\) from the elevation speed for the different moments of time; (**d**) \(\alpha_{a}\) from the elevation speed for the different moments of time. The initial gap \(h_{0}=10\)um. The comparison of \(\eta_{a}\) dependence on the initial elevation speed for different values of the acceleration is given in Figure 3.  
The value of the \(\alpha\)-reduction parameter remains constant (\(\eta_{a}\to 1\)) for relatively low initial elevation speeds and drops dramatically with time. As the acceleration increases, the point where \(\eta_{a}\) starts dropping shifts to the higher values of the initial elevation speeds, thus demonstrating the positive role of acceleration in decreasing parameter \(\alpha_{a}\) and consequent reduction of the separation force, as outlined below. As the behavior of the transient parameter was defined, we proceeded with the optimization of the separation force. Let us introduce the non-dimensional force reduction parameter, \(\chi\), given by:  
\[\chi=1-\frac{F_{s,a}}{F_{s}} \tag{9}\]  
The time dependence of the parameter \(\chi\) for different values of elevation speed varying from 0.1 mm/s to 0.4 mm/s is given in Figure 4. In top (a-d) and bottom (e-h) rows the cases of 1 mm and 10 um initial gaps are compared. It was observed that for the relatively large initial gap the acceleration of the moving platform led to the resonant increase of the separation force. The effect, however reversed after several seconds and the separation force started to drop. In contrast, for a relatively small initial gap this effect was almost negligible, as can be seen from the bottom row of Figure 4e-h. The ranges of elevation speed and acceleration correspond to previous experimental studies [13]. The minimal value of the separation force reduction, \(\chi_{min}\), with respect to time, \(t\), derived from \(\frac{\partial\chi}{\partial t}=0\) and given by:  
\[t_{min}=\frac{A^{2}-(V_{0}^{2}+5h_{0}a)A-B}{6V_{0}aA},\chi_{min}=\chi|_{t=t_{ min}}, \tag{10}\]  
where \(A\) and \(B\) stand for:  
\[A=\bigg{(}3V_{0}^{4}h_{0}a+123h_{0}^{2}V_{0}^{2}a^{2}-V_{0}^{6}- 125h_{0}^{3}a^{3}+6h_{0}V_{0}a\sqrt{36V_{0}^{4}h_{0}a-9V_{0}^{6}+339h_{0}^{2} V_{0}^{2}a^{2}-750h_{0}^{3}a^{3}}\bigg{)}^{1/3},\\ B=2V_{0}^{2}h_{0}a+V_{0}^{4}+25h_{0}^{2}a^{2} \tag{11}\]  
We can also obtain the values of \(t\), where \(\chi=0\).