## 3 Results
### The Measurement and Modeling of Zero Bead Geometry
The laser profilometer precisely measures the distance between the laser beam and the bead surface at a series of discrete points, thus capturing a detailed representation of the bead's transversal profile. The center of mass of the function formed by the measured profile points is calculated (Figure 3a). To do this in two dimensions, it can be found by first dividing the domain of the function into a regular grid of points. Then, the weighted average of the coordinates of these points can be calculated, where the weights are the values of the function at these points. Equation (3) for \(x_{cm}\) and \(z_{cm}\) represents the calculation of the center of mass in the \(x\) and \(z\) directions, as shown in Figure 3a, respectively, where \(x_{i}\) and \(z_{i}\) are the point clouds of the profile, and \(\text{m}_{\text{i}}\) represents the mass associated with each distance component. Equation (3) shows the formula for the center of mass (\(x_{cm},z_{cm}\)). \[ \begin{split} x_{cm}&=\frac{\sum_{i}x_{i}\cdot m_{i }}{\sum_{i}m_{i}}\\ z_{cm}&=\frac{\sum_{i}x_{i}\cdot m_{i}}{\sum_{i}m_{ i}}\\ \end{split}  \tag{3}\]  
These equations give you the coordinates of the center of mass for a two-dimensional object with either continuous or discrete mass distribution. Adjust the integrals or sums based on the specific form of your mass distribution. The reference plane or dimension 0 is defined using the profilometer, where the initial and final sections of the bead are detrended and adjusted to the zero reference of the plane. This adjustment ensures that the zero reference of the plane aligns with the bead's starting and ending points, therefore providing a standardized reference point for the measurements. The data can then be used to fit a parabolic equation, representing the idealized parabolic shape of the zero bead (Figure 3b). The fitted parabola provides valuable insights into the bead's geometry, including its height and width. Across the 46 profiles extracted from the initial tested bead, the R-squared values vary between 0.807 and 0.948, averaging at 0.887. On average, this indicates that the model accounts for 88.7% of the data variability. To further enhance the accuracy of the bead geometry representation, 3D modeling techniques can be employed, as can be seen in Figure 4. In this representation, the fitted parabola is obtained from the 46 measured profiles of the bead in the middle section at a rate of 1 profile per mm in order to develop a full 3D model. By combining the fitted parabola with the measured bead dimensions, a comprehensive 3D model can be  
Figure 3: Zero bead morphology: (**a**) point cloud and center of masses of the zero bead and (**b**) the modelled parabola for the profile. developed. This model provides a holistic view of the bead's shape, size, and distribution of material. The comparison of the 3D model to the theoretical models of zero beads provides valuable insights into the influence of arc-DED parameters on bead geometry. By understanding the relationship between the process parameters and bead shapes, engineers can optimize arc-DED settings, allowing them to achieve the desired bead geometry and material properties. The measuring and modeling of the zero bead geometry with arc-DED is a critical step in understanding and optimizing the process. The insights gained from these measurements can be used to ensure the consistent quality of the parts, enhance material properties, and optimize the design of arc-DED applications. Table 3 summarizes the results of the bead height (h) and bead width (w) for the manufacturing conditions. Three measurements are taken at locations along the trajectory of tests 1 to 9, denoted as w1, w2, and w3, and h1, h2, and h3. These measurements correspond to the widths (w1, w2, w3) and heights (h1, h2, h3) along the initial, middle, and final sections, respectively, of the 46 profiles extracted for each bead. The data indicates that there is a significant relationship between the welding parameters and the resulting bead dimensions.