# Study of the Critical Speed of Ultra-High-Speed Diamond Drilling Technology in Marble
## 3 Simulation and Discussion
### Effect of Varied Rotational Speed on Energy
In ABAQUS, the internal energy (ALLIE) encompasses several aspects, including the recoverable elastic strain energy (ALLSE), energy dissipation associated with inelastic processes (ALLPD), viscoelasticity, and pseudo-strain energy (ALLAE). It is important to note that pseudo-strain energy accounts for energy stored in hourglass resistance and shear within the shell and beam units and is not part of the core system itself. However, in the simulation results of the rock-crushing process by the drill teeth, the contribution of viscoelastic energy dissipation is negligible. Figure 17 illustrates the percentage of internal energy distribution during each phase of the rock crushing process by the drill teeth at different speeds. Notably, as the rotational speed increases, the dissipation of plastic strain energy decreases significantly. Specifically, during stages I (2 m/s-7 m/s)  
Figure 16: Relationship between cutting force and time: (**a**) 1.1 s–1.130 s; (**b**) 1.1 s–1.105 s.  
and III, plastic energy dissipation decreases markedly and stabilizes. These results further underscore how an increase in rotational speed alters the rock-crushing mechanism, leading to a notable reduction in energy consumption during diamond rock crushing. Based on existing research into the forces and energy involved in rock crushing by drill bits, it is commonly recognized that the forces and work performed by the drill bit can be broadly categorized into two components. These components encompass the work required to compress and crush the rock via the front face of the bit and the work needed for the frictional interaction between the back face of the bit and the rock. In this study, we further break down the energy dissipation during rock crushing by diamond into two distinct parts: the energy dissipated through friction (ALLFD) and the energy dissipated as a result of rock crushing and crack damage (ALLDMD). Figure 18 provides insights into the ratio of frictional energy dissipation to energy dissipation from rock damage at various speeds. During stages I and II, the trend of decreasing proportions between these two energy components closely aligns with the increase in linear velocity. This reduction in the ratio of energy spent on rock damage versus energy spent on friction suggests that, at higher rotational speeds, a larger share of energy is directed toward raising the temperature of the rock. This observation underscores that conventional speeds do not result in improved rock-breaking efficiency by the diamond. However, when the rotational speed surpasses 9 m/s, as observed in stage III, the proportion of energy allocated to rock crushing and friction increases. This shift signifies that, at ultra-high speeds, more energy is being utilized for the purpose of rock crushing. By reducing the portion of unused, ineffective work, a greater amount of energy is effectively channeled toward rock fragmentation, thereby significantly enhancing drilling efficiency. Considering both Figures 18 and 19, it becomes evident that the primary contributor to energy consumption during rock crushing is the energy expended in the process of rock damage, while the fraction associated with friction energy consumption remains relatively minor. At 5 m/s, friction energy consumption makes up 32% of the total energy consumption. However, within the speed range of 7-9 m/s (corresponding to stage II), friction energy consumption is notably reduced when compared to stage I. Subsequently, after reaching 9 m/s, friction energy consumption gradually increases. These observations lead us to conclude that the optimal rotational speed interval lies between 7-9 m/s. Figure 17: Internal energy distribution during diamond crushing.