ON THE RATIONAL CUTTING ANGLE OF THE TEETH OF EARTHMOVING MACHINES

I. G. Basov, Yu. N. Luzgin

(Presented by the Department of Mining Machines, Mine Transport, and Mining Mechanics)

The principal factors determining the magnitude of the rational cutting angle of the teeth of earthmoving machines may be: a) the cutting force; b) the average value of the volume of the main element of the separated chip; c) the area of contact of the front face of the tooth that is in contact with the abrasive medium (frozen ground); d) the strength of the tooth.

It has been established experimentally that the elementary volumes of the chip separated by the cutter periodically vary within certain limits, thereby causing a stepwise change in the cutting force \(P_p\). The maximum cutting force corresponds to the shearing of the principal, largest element of the chip. Its magnitude depends on the cutting angle \(\alpha\) (Fig. 1). Hence follows the need to consider the question of the separation by the tooth from the mass of frozen ground of the maximum possible chip element at the minimum possible force \(P_p\), and the dependence of these quantities on \(\alpha\). The chip-shear element in the cutting plane is bounded, on one side, by the front face of the tooth and, on the other, by the shear line (Fig. 1). As experiments have shown, for \(30^\circ \le \alpha \le 60^\circ\) the shear line passes approximately perpendicular to the cutter, while for \(60^\circ \le \alpha \le 90^\circ\) the shear line in the cutting plane passes at an angle to the horizon approximately equal to the angle of internal friction of frozen ground (about \(30^\circ\)).

With constant tooth width \(b\) and cutting depth \(h\), the volumes of the separated chip elements will be proportional to the area \(F\) (Fig. 1) bounded by the front face of the cutter and the shear line. The indicated area, for constant \(h\) and variation of the cutting angle within \(30^\circ \le \alpha \le 60^\circ\), can be found from the expression

\[ F=\frac{h^2}{2}\int_{\alpha}^{\alpha+90^\circ}\frac{d\alpha}{\sin^2\alpha} =\frac{h^2}{2}(\operatorname{tg}\alpha+\operatorname{ctg}\alpha). \tag{1} \]

When the cutting angle varies within \(60^\circ \le \alpha \le 90^\circ\), the area \(F\) (Fig. 1) decreases as \(\alpha\) increases. Consequently, the maximum possible area \(F\) will correspond to cutting angles varying within \(30^\circ \le \alpha \le 60^\circ\). To find the extremal value of the desired area, we set the first derivative of equation (1) equal to zero:

\[ \frac{dF}{d\alpha} = \frac{h^2}{2} \left( \frac{1}{\cos^2\alpha} - \frac{1}{\sin^2\alpha} \right) =0. \]

Fig. 1. Change in cutting force \(P_p\)—(1), area \(S\)—(2) of contact of the front face of the cutter with the ground, volume \(V\)—(3) of the principal element of the chip, expressed in relative units, as a function of the cutting angle \(\alpha\)

This condition is satisfied only at \(\alpha = 45^\circ\).

The second derivative of the equation under consideration (1) is

\[ \frac{d^2 F}{d\alpha^2} = 8h^2\frac{\sin^4\alpha+\cos^4\alpha}{\cos^3 2\alpha} >0, \]

and this means that at the angle \(\alpha = 45^\circ\) the minimum area \(F\) is obtained.

To determine the maximum area \(F\), we substitute into equation (1) the minimum and maximum values of the angle \(\alpha\) and find that

\[ F_{30^\circ}=F_{60^\circ}=1.155h^2. \]

Thus, the maximum volume of the separated chip element will correspond to the area \(F = 1.55h^2\) and is determined by cutting angles equal to \(30^\circ\) and \(60^\circ\). As the angle \(\alpha\) changes from \(30^\circ\) to \(90^\circ\), the cutting force \(P_p\), expressed in relative units, increases (Fig. 1). The minimum of the function \(P_p=f(\alpha)\), determined by the point of tangency of the curve with a line drawn from the origin, corresponds to the angle \(\alpha = 60^\circ\).

The next factor influencing the choice of cutting angle is the area \(S\) of contact of the tooth with the medium being destroyed. This area can be represented as a function of the cutting angle

\[ S=\frac{b\cdot h}{\sin\alpha}. \tag{2} \]

It follows from equation (2): the smaller the cutting angle, the greater the value of \(S\). As the angle \(\alpha\) decreases from \(30^\circ\) to \(60^\circ\), the area \(S\) decreases by approximately 70% (Fig. 1). A further increase of the angle \(\alpha\) to \(90^\circ\) only slightly decreases the area \(S\).

Frozen soil is a highly abrasive medium [1], and this must be taken into account when selecting the cutting angle of the tooth. The value of \(\alpha\) will determine the area of contact between the soil and the front face of the tooth and the angle at which abrasive particles meet the metal of the cutter. Consequently, the friction force on the front face of the tooth and the intensity of wear depend on the angle.

It has been established [2] that, in wear in a stream of abrasive particles, the character of the wear depends on the angle at which the particles meet the metal surface. Under a normal impact of the particles (angle of attack \(90^\circ\)), the surface layer of the metal wears as a result of repeated deformation. At an impact angle of less than \(90^\circ\), wear occurs as a result of the scratching action of the particles and is considerably more intense than under a normal impact of the particles. V. D. Kuznetsov [3] showed that wear by fixed particles does not differ in principle from wear by free particles. It follows from this that, in order to reduce the intensity of wear of cutting teeth, the angle \(\alpha\) should be taken as large as possible, since in this case both the area and the intensity of abrasion of the tooth decrease.

In addition, it must be taken into account that the smaller the initial angle \(\alpha\), the faster it increases as a result of intensive wear of the tooth. At the same time the forces resisting cutting increase sharply. Teeth with a large angle \(\alpha\), other conditions being equal, do not lead to an abrupt change in the load on the earth-cutting machine. It should also be noted that the smaller the cutting angle, the less strong the tooth design will be, additionally weakened by the necessity of having a rear cutting angle \(\gamma\) within the range \(8^\circ\)—\(10^\circ\).

Thus, with an increase in \(\alpha\) beyond \(60^\circ\), the cutting force \(P_p\) increases considerably, while the area \(S\) decreases only slightly. At \(\alpha < 60^\circ\), the force decreases insignificantly, but the area of contact with the frozen soil increases rapidly; the angle at which the front face of the cutter meets the abrasive particles decreases, and, consequently, the intensity of wear will increase.

In this connection, the rational cutting angle for the teeth of earth-cutting machines should be considered an angle close to \(60^\circ\).

REFERENCES

1. N. I. Goryushkin. Study of steel wear as applied to the operation of moving parts of earth-cutting machines in frozen soils. Candidate dissertation. Library of TPI, Tomsk, 1965.

2. V. N. Kashcheev, V. M. Glazkov. Wear in a stream of moving abrasive particles. Proceedings of the conference “Methods of Testing for Wear.” Publishing House of the Academy of Sciences of the USSR, 1962.

3. V. D. Kuznetsov. Physics of the Solid State, vol. IV. TSU Publishing House, Tomsk, 1947.