## 3 Mathematical Model Development
### Behavioral Analysis of Forces Exerted on DUT  
Analyzing the forces acting on the SOIC package, especially near its contour, the complexity increases due to the variety of forces encountered in different fluid environments. To understand a device's behavior in different fluids, we must deduce the forces acting during its linear displacement. Understanding these forces helps determine the SOIC package's maximum permissible speed while maintaining its alignment [35]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stage** & **Accel-Decel** & **Stage** & **Accel-Decel** & **Stage** & **Accel-Decel** \\ \hline
**I and XX** & \(a_{A}=\frac{V_{max}}{\Delta t_{A}}\) & **III and XVIII** & \(a_{C}=\frac{V_{max}-V_{ZIF}}{\Delta t_{C}}\) & **V and XVI** & \(a_{E}=\frac{V_{ZIF}}{\Delta t_{E}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Equations of acceleration or deceleration on trapezoid pattern. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stage** & \
\begin{tabular}{c} **Downward Period** \\ **Distance Covered** \\ \end{tabular} & **Stage** &
\begin{tabular}{c} **Upward Period** \\ **Distance Covered** \\ \end{tabular} \\ \hline
**I** & \(X_{A}=\frac{1}{2}a_{A}{\Delta t_{A}}^{2}\) & **XVI** & \(X_{E}=\frac{1}{2}a_{E}{\Delta t_{E}}^{2}\) \\
**II** & \(X_{B}=\hat{V}_{max}\Delta t_{B}\) & **XVII** & \(X_{D}=\hat{V}_{ZIF}\Delta t_{D}\) \\
**III** & \(X_{C}=V_{max}\Delta t_{C}-\frac{1}{2}a_{C}{\Delta t_{C}}^{2}\) & **XVIII** & \(X_{C}=V_{ZIF}\Delta t_{C}+\frac{1}{2}a_{C}{\Delta t_{C}}^{2}\) \\
**IV** & \(X_{D}=V_{ZIF}\Delta t_{D}\) & **XIX** & \(X_{B}=V_{max}\Delta t_{B}\) \\
**V** & \(X_{E}=V_{ZIF}\Delta t_{E}-\frac{1}{2}a_{E}{\Delta t_{E}}^{2}\) & **XX** & \(X_{A}=V_{max}\Delta t_{A}-\frac{1}{2}a_{A}{\Delta t_{A}}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Equations of the intervals covered. near-zero instantaneous acceleration concerning the displacement of the SOIC package, we analyzed the resulting forces during its trajectory and the associated energy losses for each density, including transitions between different fluids. Figure 6 shows the force models we developed for each medium, detailing how we moved the SOIC package and facilitating the analysis of static and dynamic force behaviors. The force analysis depicted in Figure 6 reveals a model for the initial moment without vertical movement in Figure 6a. In this scenario, the machine force (\(F_{m}\)), drag force (\(F_{d_{air}}\)), and normal force (\(N\)) are all zero. Therefore, the initial suction force (\(F_{S}\)) must be equal to or greater than the weight of the SOIC package (\(W_{SOIC}\)). The resulting force, considering the machine force (\(F_{m}\)), acceleration, and an initial velocity of zero, is given by Equation (24):  
\[-F_{s}+W_{SOIC}-N-Fd=0 \tag{24}\]  
Here, the normal force (\(N\)) results from multiplying the air density (\(\rho_{air}\)), gravity acceleration (\(g\)), and the volume of the SOIC package base (\(\gamma_{SOIC}\)). The drag force (\(f_{d}\)) depends on the normal force, and we multiply it by the friction coefficient (\(\mu_{air}\)). To determine the maximum velocities for each medium, we use [PERSON]'s second law for force analysis and apply the principle of conservation of energy to symmetric temporal translation [36]. In this case, the Z-axis moves vertically, holding a SOIC package under suction pressure. The drag force exerted by the fluid medium depends on the square of the velocity and the area of the SOIC package (\(a_{SOIC}\)) in contact with the dielectric fluid, given by Equation (25):  
\[F_{d}=\frac{V_{max}^{2}*\rho*\mu*a_{SOIC}}{2} \tag{25}\]  
By substituting the drag force into Equation (24), we can derive the maximum speed for downward displacement in the air, as shown in Equation (26):  
\[|V_{max_{Air_{D}}}|=\sqrt{\frac{2(F_{s}+N-W_{SOIC})}{\mu\cdot\rho_{air}\cdot a _{SOIC}}} \tag{26}\]  
We considered the velocity as an absolute value and denote its direction with the suffix \(D=downward\) and \(U=upward\).