## 3 Mathematical Model Development
### Behavioral Analysis of Forces Exerted on DUT
Figure 6: (a) Analysis for SOIC package moving down in the air, (b) air–dielectric fluid interface, and (c) SOIC package moving up in a dielectric fluid. Moving on to Figure 6b, which represents a constant velocity, zero acceleration, and downward displacement in a dielectric fluid, we obtained Equation (27):  
\[-F_{s}+W_{SOIC}-B=0 \tag{27}\]  
Here, the buoyant force (\(B\)) is equal to the dielectric fluid density (\(\rho_{oil}\)) multiplied by gravity acceleration (\(g\)) and by the volume of the SOIC package base (\(\gamma_{SOIC}\)). The drag force (\(F_{d}\)) is proportional to the buoyant force and the friction coefficient (\(\mu_{oil}\)). The maximum velocity (\(V_{max}\)) for downward displacement in the dielectric fluid can be determined using Equation (28):  
\[|V_{max_{Oil_{D}}}|=\sqrt{\frac{2(F_{s}+B-W_{SOIC})}{\mu\cdot\rho_{oil}\cdot a _{SOIC}}} \tag{28}\]  
In the context of upward displacement in the dielectric fluid, as depicted in Figure 6c, the drag force (\(f_{d}\)) exhibits an inverse direction about the flotation force, acting only over the upper areas of the SOIC package (excluding the area under the nozzle \(a_{n_{cyl}}\)). This results in Equation (29):  
\[-F_{s}+W_{SOIC}-B+Fd=0 \tag{29}\]  
By substituting the equivalent values for the drag force, the maximum speed for upward displacement in the dielectric fluid can be expressed using Equation (30):  
\[V_{max_{Oil_{L}}}=\sqrt{\frac{2(F_{s}+B-W_{SOIC})}{\mu\cdot\rho\cdot(a_{SOIC}- a_{n_{cyl}})}} \tag{30}\]  
Behavioral Analysis of Section Force  
The minimum suction force (\(F_{s}\)) required to securely hold the SOIC package during vertical movement, as described in Equation (31), is directly linked to several factors. These include the mass of the SOIC package (\(m_{SOIC}\)), the acceleration due to gravity (\(g\)), the linear machinery acceleration (\(a_{L}\)), a minimum safety factor (\(S_{f}\)) typically set at 1.5 or 2.0 (which accounts for the secure holding of rigid surfaces with minimal porosity), the nozzle diameter of 3.607 mm, and the reciprocal of the drag coefficient (\(\mu\)) of the dielectric fluid [37]. The formula for calculating the minimum suction force is given as follows:  
\[F_{s}=m_{SOIC}\cdot(g+\frac{a_{L}}{\mu})\cdot S_{f} \tag{31}\]  
The maximum linear machinery acceleration obtained was 40.77 m/s\({}^{2}\). For the experimentation and analysis, a minimum safety factor of 1.5 was selected. Table 8 presents the values of the pneumatic line's minimal pressure and the corresponding suction force necessary to ensure the secure attachment of the SOIC package to the nozzle. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dielectric** & **SOIC** & **Minimum Suction Force** & **Vacuum Pressure** \\
**Medium** & **Package** & **N** & **N/m\({}^{2}\)** \\ \hline  & SOP-16 & 0.0138 & 458.78 \\ AIR & SOP-20 & 0.0471 & 1560.70 \\  & SOP-24 & 0.0613 & 2031.89 \\ \hline  & SOP-16 & 0.0239 & 791.59 \\ FR3 & SOP-20 & 0.0790 & 2617.87 \\  & SOP-24 & 0.0999 & 3309.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Minimum suction force and vacuum pressure for each SOIC package. After determining the suction force values, we substituted them into Equations (26), (28) and (30) to calculate the maximum speed of continuous displacement along the Z-axis during testing. The results of these calculations are presented in Table 9, which provides the maximum speed values required for transporting the DUT through two different media, namely air and oil. These speed values account for constant acceleration and are essential for the testing process. To calculate the maximum speed (\(V_{ZIF}\)) needed to close the ZIF socket, we applied a force to the spring-loaded mechanism until the spring shifted to a position that corresponds to the total distance \((X_{D}+X_{E})\), as shown in Figure 5. We can determine the magnitude of this force using Equation (32), where \((K)\) denotes the spring's elasticity constant, following [PERSON]'s law [38].