STUDY OF THE EFFECT OF THE RAKE ANGLE OF CUTTERS OF CARBIDE-TIPPED BITS ON THE PENETRATION DEPTH AND ROCK CUTTING FORCES

V. I. Zvarygin, S. S. Sulakshin

(Presented by the scientific seminar of the Department of Exploration Engineering)

The effectiveness of carbide-tipped bits is ultimately determined by the penetration depth and the durability of the cutters. These indicators depend on many factors, including the magnitude of the rake angle of the cutters used. In this connection, determining the optimal value of the rake angle is of great importance for the development of rational designs of carbide-tipped bits. At present there are two views on the dependence of the cutting force on the magnitude of the cutter rake angle.

I. A. Ostroushko [1], referring to the experiments of V. D. Romanov, F. A. Shamshev, B. E. Sveshnikov, and others, believes that in cutting rocks, rake angles having values of less than 90° do not exert a noticeable influence on the cutting forces. At cutting angles above 90°, the conditions for chip removal deteriorate, friction and additional crushing increase, as a result of which the forces increase sharply.

Yu. N. Kazak [2], on the basis of experiments in cutting anthracite, found that increasing the cutting angle from 30 to 60° practically does not lead to an increase in the cutting forces. When the cutting angle is increased from 60 to 90°, the cutting forces increase by 40–90%, and especially intense growth is observed when the angle is increased above 90°. According to the data of V. N. Getopanov [2], with an increase in the cutting angle from 30 to 60°, the forces change, in comparison with the forces for a vertically mounted cutter, by 65–70%, and from 60 to 90° by 70–100%; with an increase in the angle from 90 to 120°, the forces increase by 170%.

In order to determine the influence of the rake angle on cutting forces and cutter penetration depth, and also to establish the relationship between these parameters, we carried out investigations over a wide range of variation in cutter rake angles, from −60 to +60°. The experiments were conducted on a special installation while cutting fine-grained and medium-grained sandstones and siltstones of Kuzbass.

The laboratory installation on which the investigations were carried out is a hydraulic machine with a horizontally arranged cylinder (Fig. 1). To measure the forces acting on the cutter, a vertical spindle dynamometer was developed, in which the physical quantities (cutting forces and cutter penetration depth) were converted into electrical quantities and recorded on the film of an N-700 oscillograph.

The tested rock specimens were fastened on a special table, along which a rod moved with a spindle dynamometer attached to its end by means of a coupling.

Fig. 1. a. Diagram of the laboratory setup for investigating cutters: 1 — frame, 2 — lower bed, 3 — upper bed, 4 — hydraulic cylinder, 5 — hydraulic-control instrument, 6 — oil tank, 7 — oil line, 8 — lever, 9 — table, 10 — electric motor, 11 — oil pump, 12 — channel bar, 13 — upright, 14 — coupling, 15 — spindle dynamometer, 16 — recess for inserts, 17 — pin, 18 — platform, 19, 20 — pressure gauges, 21 — DS-02 dynamometer, 22, a — sensor coil, 22, b — armature, 23 — groove, 24 — bridge, 25 — spring, 26 — sensor coil, 27 — core; 28 — plate, 29 — rock specimen, 30 — tray;
b) diagram of the electrical system of the laboratory setup;
c) bridge circuit of the laboratory setup.

The cutters were mounted in dies, which, in turn, were fastened in a socket in the lower part of the dynamometer. Depending on the required inclination of the cutter, the corresponding dies were selected.

On the basis of the oscillograms obtained, graphs were plotted showing the dependence of the penetration depth and cutting forces on the magnitude of the rake angle. Each curve was constructed from the results of 15–20 cuts under constant test conditions.

The graph (Fig. 2, a) shows the dependence of the axial cutting forces \(P\) on the penetration depth for various values of the cutter rake angle \(\gamma\). Figure 2, b gives the graph of the dependence of the horizontal forces \(F\) on the penetration depth for different values of the cutter rake angle

Fig. 2.
a) graph of the dependence of the axial force on the penetration depth of the cutter for various values of the rake angle;
b) graph of the dependence of the horizontal forces on the penetration depth of the cutter for various values of the rake angle;
c) graph of the dependence of the cutting forces on the rake angle of the cutter

cutter \(\gamma\). Figure 2, b gives the graph of the dependence of the horizontal forces \(F\) on the penetration depth for different values of the rake angle

cutter. Finally, Fig. 2 shows the dependence of the cutting forces \((P, F)\) on the magnitude of the rake angle. From the graphs obtained the following conclusions follow:

1. The dependences \(F=f(h)\); \(P=f(h)\) are linear in character. A. I. Beron, A. S. Kazanskii, and E. Z. Pozin [3] arrived at the same conclusion in cutting anthracite.

2. With an increase in the rake angle, at the same depth of penetration of the cutter, the axial forces increase faster than the horizontal ones.

3. For positive and negative angles, the increase in cutting forces proceeds according to different laws:

a) when the rake angle is varied within the range \(+60 \div 0^\circ\), the axial forces fluctuate only slightly;

b) when the rake angle is varied from \(0\) to \(-60^\circ\), the horizontal and axial forces increase more intensively than for \(\gamma = 0 \div +60^\circ\).

The experimentally obtained dependences make it possible to substantiate the choice of the optimum rake angle of the cutters, proceeding from the conditions of their most effective operation and sufficient resistance to fracture (breakage).

For the case of cutting rock with carbide cutters, we shall use a simplified scheme of cutter operation (Fig. 3). As a result of the small magnitude of the contact between the cutter and the rock, the horizontal and axial forces are transferred to the cutting edge of the cutter.

Fig. 3.

It should be noted that, when cutting rock, in addition to the horizontal shearing forces, there is also a frictional force of the cutter against the rock. Therefore the cutting process should be considered as a whole, as the resistance of the rock to the displacement of the cutter. From Fig. 3 it is seen that

\[ P = k \cdot F = F \cdot \operatorname{tg}\Psi, \tag{1} \]

where \(k = \operatorname{tg}\Psi\) is the coefficient of resistance of the rock to displacement of the cutter; \(\Psi\) is the angle of resistance to displacement.

The magnitude of the angle \(\Psi\) also changes when the rake angle of the cutter is changed; therefore, before determining the function \(P=f(F,\gamma)\), it is necessary to find the function \(\Psi=f(\gamma)\). To establish this dependence, we shall compile a table based on the materials of laboratory investigations of the dependence of \(\Psi\) and \(\gamma\) (Table 1). The angle \(\Psi\) is calculated by formula (1).

From this table it is seen that, for cutters with positive rake angles, the value of \(\psi\) remains approximately constant, \(26^\circ 30'\). Therefore,

\[ P = F \cdot \tg \psi = F \cdot \tg 26^\circ 30' = 0.5F . \tag{2} \]

Table 1

For cutters with negative rake angles, as is seen from the table, the angle \(\psi\) will be approximately equal to

\[ \psi = 26^\circ - \frac{\gamma}{4} \tag{3} \]

and then

\[ P = F \cdot \tg \psi = F \cdot \tg \left(26^\circ - \frac{\gamma}{4}\right). \tag{4} \]

In order to determine the dependence of the cutting forces on the cutter rake angle, we shall use the formula derived earlier by us

\[ \sigma = -\frac{k_1 \cdot F \cdot \cos \alpha}{h(b + l)}, \tag{5} \]

where \(\sigma\) is the critical stress arising in the rock at the moments when a rock element is separated;
\(\alpha\) is the angle of chipping of the element;
\(h\) is the depth of penetration of the cutter,
\(b\) is the width of the cutter;
\(l\) is a constant quantity;
\(k_1\) is a coefficient depending on cutting conditions.

When cutting along a free surface \(k_1 = 2.1\); when cutting under bottom-hole conditions \(k_1 = 2.1\); during repeated cutting (cutting along the edge of a groove) \(k_1 = 9\).

It was established earlier by us that

\[ \alpha = \frac{\alpha_0}{2}, \tag{6} \]

where \(\alpha_0\) is the angle between the vertical and the direction of the resultant \(R\)—the force acting normally to the rake face of the cutter \(F_0\), and the axial force \(P\) (Fig. 3).

As the rake angle \(\gamma\) changes, the direction of the resultant will change and, consequently, the angle \(\alpha\) will change.

From Fig. 3 it is seen that

\[ \alpha_0 = 90^\circ + \varphi, \tag{7} \]

here \(\varphi\) is the angle between the direction of the resultant \(R\) and the horizontal plane. As is seen from the figure,

\[ \varphi=\varphi_0-\gamma . \]

Substituting this value into formulas (7) and (6), we obtain

\[ \alpha=\frac{90+\varphi_0-\gamma}{2}. \tag{8} \]

Let us assume that \(\varphi_0\) is a constant quantity (below we shall see that this assumption proves to be correct), and then its magnitude can easily be determined.

When cutting with a cutter having a rake angle \(\gamma=0\),

\[ \varphi=\psi=\varphi_0=26^\circ . \]

Then formula (8) will take the form

\[ \alpha=\frac{90+26-\gamma}{2}=58-\frac{\gamma}{2}. \tag{9} \]

Substituting the value of \(\alpha\) into formula (5), we obtain

\[ \sigma=-\frac{k_1 F\cdot \cos\left(58-\frac{\gamma}{2}\right)}{h(b+l)} . \tag{10} \]

or

\[ F=-\frac{\sigma\cdot h(b+l)}{k_1\cdot \cos\left(58-\frac{\gamma}{2}\right)} , \tag{11} \]

i.e., the horizontal forces in cutting rocks are inversely proportional to

\[ \cos\left(58^\circ-\frac{\gamma}{2}\right). \]

To verify the derived formula, let us use the experiments obtained by us and by other researchers (Table 2). For ease of comparison, the horizontal forces \(F\) for cutters with \(\gamma=0\) are taken as \(100\%\); the remaining values are equal to

\[ \frac{F\cdot\gamma}{F_1}\cdot 100. \]

As is seen from Table 2, the formula obtained agrees well with the actual data.

From all that has been said above, it may be concluded that it is most rational to install cutters in the bit at an angle \(\gamma>0\). Moreover, the greater the rake angle, the smaller the forces acting on the cutter at the same effect.

However, while taking into account the reduction of cutting forces with an increase in rake angle, it is also necessary to take into account the strength of the cutters. With an increase in \(\gamma\), the included angle of the cutter \(\beta_0\), and consequently the cross section of the cutter, will decrease. On the other hand, the force causing bending of the cutter (the breaking force) will decrease, while the compressive force of the cutter will increase. As a result, a cutter with a large rake angle, with the same cutter cross section, will be able to withstand greater loads. Table 3 gives the critical values of the forces at which fracture of the cutters occurs.

Table 2

It is evident from the table that the greatest resistance of the cutter to fracture occurs at a rake angle equal to \(30 \div 45^\circ\). In addition, as was shown earlier, with a decrease in the rake angle to \((30 \div 40^\circ)\), the penetration depth of the cutters as they move along the face increases substantially.

Table 3

Consequently, the investigations carried out make it possible to recommend, for a given type of rock, the most optimal values of the rake angle within the range \(+(30 \div 45^\circ)\).

REFERENCES

1. I. A. Ostroushko. Destruction of rocks during drilling. Gostgeolizdat, 1952.

2. Scientific Works of the Moscow Mining Institute, collections Nos. 19 and 21, 1957.

3. A. I. Beron, A. S. Kazanskii, B. M. Leibov, E. Z. Pozin. Coal cutting. Gosgortekhizdat, 1962.

4. Studies of the destruction of rocks by means of cutters (review of foreign literature). TsNIITEIneftegaz, 1963.