## 4 Analysis of the Consecutive Discharge Occurrence  
Using the detection method described in the previous section, the locations of all the discharges were identified. A histogram of the distances between the \(i\)th discharge (\(d_{i}\)) and the \(i-1\)th discharge (\(d_{i-1}\)), \(i-5\)th discharge (\(d_{i-5}\)), and \(i-20\)th discharge (\(d_{i-20}\)) are shown, respectively, in Figure 9a, Figure 9b, and Figure 9c. In addition, a random distribution model was created in MATLAB as a reference. The random distribution model was designed under the assumption that the discharge locations are determined completely at random, without influence from previous discharges or external factors. Maintaining the same number of discharges as in the experiment, 713 discharge positions within a range  
Figure 8: Discharge location identified by the grayscale analysis. Figure 7: Recorded signals: gap voltage, discharge current, and framing signal. of 10 mm, corresponding to the workpiece thickness, were randomly generated using the random function in MATLAB, ensuring no correlation between consecutive discharges. Through analyzing these random discharge locations, as shown in Figure 9d, the histogram of the distance between two consecutive discharges was obtained. Figure 9: Histogram of the distance between discharges: (**a**) between \(d_{i}\) and \(d_{i-1}\); (**b**) between \(d_{i}\) and \(d_{i-5}\); (**c**) between \(d_{i}\) and \(d_{i-20}\); and (**d**) between two consecutive discharges in the random model. In Figure 9a, it is clear that 24% of the distances between \(d_{i}\) and \(d_{i-1}\) are shorter than 0.5 mm, which is 14.1% higher than that of the random model shown in Figure 9d. Thus, comparing Figure 9a-c suggests that with more discharges between two discharges, there is a lower percentage of distances below 0.5 mm. Moreover, if there are a sufficiently large number of discharges between two discharges, the distribution tends to be random. This means that the discharge \(d_{i-1}\) exerts the greatest influence on the location of \(d_{i}\). Given that [PERSON] and [PERSON] [10] found that most of the debris particles accumulate at the boundary of the bubble generated by the discharge, the influence of \(d_{i-1}\) on the occurrence of \(d_{i}\) is considered to be closely related to the bubble generated by \(d_{i-1}\). To verify this assumption, the set of discharges \(d_{i}\) that occurred within 0.5 mm from the location of \(d_{i-1}\) was selected for further analysis. The change in bubble radius throughout the entire process from formation to dissipation was measured using the image processing software ImageJ (1.54g). These measurements, displayed in Figure 10, show that the size of the bubble is almost uniform and, thus, the oscillation period of the bubble is similar. The first cycle of bubble oscillation is always around 200 \(\upmu\)s, and the bubbles disappear after approximately 500 \(\upmu\)s due to the reaction between hydrogen and oxygen in the gap [19]. Since it is known that more debris particles are generated with greater discharge, this makes the image too dark to accurately locate the discharge and the bubble. Thus, to obtain a clearly observable image, sufficient flushing flow was essential for cleaning the machining area. However, due to the existence of flushing flow, the bubble generated by \(d_{i-1}\) was convective before \(d_{i}\) occurred, suggesting a slight movement of the bubble. Therefore, the distance (\(  
Furthermore, for understanding the influence of the bubble on discharge occurrence, the relationship between the discharge delay time and the discharge location is presented in Figure 13. In this figure, all the discharges are divided into two groups. Group 1 includes the discharges \(d_{i}\) which occurred within 0.5 mm from \(d_{i-1}\), while Group 2 includes the discharges \(d_{i}\) which occurred at a distance greater than 0.5 mm from \(d_{i-1}\).