## 4 Cluster Design of the Edge Buttons of the Bit and the Rotation Speed Matching  
The process of DTH drilling is a transient dynamic crushing of brittle materials, and the cracks extend along the grain or joint of the rock. The edge buttons cluster arrangement is adopted, and the three ball buttons are as close as possible to meet the installation requirements. The rock between the ball buttons in the cluster is broken through crack propagation [31; 32; 33]. The equivalent diameter is  
\[d_{e}\geq 2d \tag{1}\]  
where \(d_{e}\) is the button-equivalent diameter, mm; and \(d\) is the diameter of a single tooth, mm.  
The equivalent diameter of the spherical button is at least two times larger than that of a single spherical button. According to the \"optimal angle\" relation of drilling, the edge buttons of the DTH hammer bit are arranged in the cluster as shown in Figure 6b, making it equivalent to a large-diameter bit edge button. It meets the requirement of rock button spacing (\(\Delta\)S\({}_{2}\) < \(\Delta\)S\({}_{1}\)) for two impacts at the bottom of the hole near the drill button; reduces the residual rock ridge; eliminates the tangential reaction force on the bit button; and can deduce the relationship between the impact frequency, bit diameter, ball button diameter, and bit speed.  
\[n=\frac{k}{\pi}\cdot\frac{d_{e}}{D}f=0.255\frac{d_{e}}{D}f \tag{2}\]  
where \(n\) is bit rotation speed, rps; \(k\) is the rock fracture coefficient, here \(0.8;f\) is the impact frequency, Hz; and \(D\) is the drill bit diameter, mm.  
The reasonable rotation speed is proportional to the equivalent diameter of the cluster buttons and the impact frequency and inversely proportional to the bit diameter. It is beneficial to prevent the side button from the fractures by matching the reasonable rotary speed of the bit.  
When the impact frequency is 25-28 Hz, the equivalent diameter of the three spherical buttons with a spherical button diameter of 16 mm is 32 mm, and the bit diameter \(D\) is 311 mm; then, the upper rotation speed of the bit can be estimated as 44 rpm according to Equation (2). Most of the rigs with rotary table or top drive head can drill a 311 mm diameter hole, increasing the limit of the excessive rotation speed in the lowest gear position of the rig.  
Figure 6: (**a**) Clustering design of edge button of DTH hammer drilling hammer bit in large-diameter and high-strength rock formation. (**b**) Cluster of edge buttons of the bit and the equivalent diameter.  
Table 2 shows the reasonable rotation speed of the new large-diameter bit under high impact frequency. It should be noted that if the drill with larger diameter and the impactor with lower impact frequency are used, the corresponding upper rotation speed should not be lower than the minimum rotation speed of the bit. In special cases, a larger-diameter bit button can be selected to obtain a greater upper rotation speed.  
It is worth noting that when using a larger-diameter bit or an impactor with a lower impact frequency, the corresponding upper rotation speed should not be lower than the minimum rotation speed of the rig. A larger-diameter bit button can be selected for a greater upper rotation speed.  
By expanding the equivalent diameter of the edge buttons (see Figure 6a), the edge buttons cluster allows for drilling at a higher rotation speed, ensuring that two adjacent fracture holes are connected at normal rotation speed, and the ridge between them is clipped, improving the life of the bit in drilling high-strength formations. According to the idea of cluster button arrangement, the bit (diameter is 311 mm) with a drilling hammer was innovatively designed, as shown in Figure 6. Meanwhile, the bit button density was appropriately increased for hard strata. The area of chute should be increased to meet the requirements of chute removal [34; 35].