## 2 Materials and Methods
### Experimental Station and Measurements  
The primary objective of this study was to evaluate the effectiveness of magnetic abrasive finishing with planetary kinematics in enhancing surface roughness following vibratory processing. This investigation aimed to determine whether the magnetically assisted method, characterized by its controlled abrasive action under the influence of a magnetic field, could serve as a complementary or subsequent finishing technique capable of further surface-texture refining beyond what is achievable through conventional vibratory machining. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample No.**} & \multicolumn{5}{c}{**Roughness [\(\mu\)m]**} \\ \cline{2-6}  & **Ra** & **Rq** & **Ry** & **Rz** & **Sm** \\ \hline
1 & 1.15 & 1.47 & 4.11 & 6.86 & 248.3 \\
2 & 1.34 & 1.83 & 4.21 & 7.71 & 227.7 \\
3 & 1.13 & 1.48 & 4.17 & 7.06 & 233.9 \\ Average & 1.21 & 1.59 & 4.16 & 7.21 & 236.6 \\ SD & 0.09 & 0.17 & 0.04 & 0.36 & 8.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Change in profile roughness heights after vibratory machining of Inconel 939. Figure 2: Diagram of the custom-designed sample holder assembly. (**a**) Head assembly components: 1—main body of the head; 2—detachable jaw; 3—bar holder; 4—bar on which the sample mount is fixed; 5—jaw fastening screws. (**b**) Sample clamp components: 1—main clamp mounted on the rotating shaft; 2—retaining plate; 3—M3x10 stainless steel hex-head screw. (**c**) Fully assembled and mounted sample holder, ready for installation in the head. The experiment was conducted using a custom-built experimental station equipped with pairs of neodymium magnets (50 mm in diameter and 20 mm in height) positioned at the top and bottom of the setup. A specialized sample holder (Figure 2) was employed, designed to enable simultaneous dual-axis rotation. The primary rotational movement occurred along the axis of the head, while the secondary rotation was induced along the axis of the sample, which was mounted eccentrically relative to the main rotational axis. This configuration allowed for the generation of complex planetary kinematics during the MAF. During the experiment, the sample was consistently immersed in the abrasive medium to the same depth (Figure 3). The machining gap, defined as the distance between the magnets, was set to 45 mm. Courtesy of POLIMAG [24], the study employed solid ferromagnetic Fe-Si abrasive grains with a particle size range of 160/80 \(\upmu\)m (Figure 4). The selection of Fe-Si abrasive grains with a size range of 160/80 \(\upmu\)m was motivated by the requirement to achieve a balance between effective material removal and surface refinement while maintaining sufficient adaptability to treat complex geometries typical of AM components. Additionally, this grain size was chosen to enable comparison with previous studies involving the MAF of AISI 304 stainless steel, where similar abrasive characteristics were reported to yield favorable finishing performance. The research was conducted on an Olympus BX53M optical microscope (Olympus Corporation, Tokyo, Japan) and Joel 7000 SEM with chemical analysis equipment (JEOL Ltd., Tokyo, Japan). The study was conducted using a planned three-factor, three-level experimental design. The investigated parameters included eccentricity value a [mm], machining time t [min], and rotational speed n [rpm] (Tables 4 and 5). A total of 13 samples were processed, with 1 sample assigned to each unique parameter combination defined by the experimental matrix. The surface roughness of each sample was measured on the front, rear, and lateral faces. Each of these surfaces was measured twice, and the average value of the measurements was subsequently calculated for each sample. Figure 3: Sample mounted in the holder prior to immersion in the flexible magnetic abrasive tool on the experimental station. It should be acknowledged that for each set of process parameters, only a single sample (n = 1) was used in the experimental procedure.