ON THE EFFICIENCY OF IMPACT IN DRILLING HAMMERS

P. M. ALABUZHEV, O. D. ALIMOV, and A. G. TSUKANOV

The question of the efficiency (e.p.) of the impact of striking and drilling hammers is considered in works [3; 5; 7; 14; 24]. In most cases, the transfer of mechanical energy during the impact of the striker on the drill or pick and the action of the latter on the rock being broken are considered from the standpoint of the classical theory of impact, set forth in [12].

When considering the question of the e.p. of the impact of striking and drilling hammers, it is usually assumed that, in terms of the transfer and utilization of impact energy, these two cases are analogous.

Corresponding Member of the Academy of Sciences of the Ukrainian SSR P. S. Kucherov [14], investigating pneumatic striking hammers, considers that all the kinetic energy of the striker is imparted to the pick, less the energy going into inelastic deformations during impact; denoting the ratio of the mass of the pick—\(m_2\)—to the mass of the striker—\(m_1\)—by \(k\), we obtain, according to the classical theory of impact, that in this case the e.p. of impact is

\[ \eta = 1 - (1 - c^2)\frac{k}{1+k}. \]

Taking the coefficient of restitution for impact on steel as \(c = 5{,}9\), we obtain

\[ \eta = 1 - 0{,}7\frac{k}{1+k}. \tag{1} \]

Yu. M. Malakhov [16], considering the transfer of kinetic energy from the striker to the drill, proposes, in order to take into account the influence of the rock on the collision of the striker and the drill, that in calculating the e.p. of impact one adopt a “fictive mass of the drill”—\(m_2\), not equal to its actual mass; however, Yu. M. Malakhov does not give practical data for determining the fictive mass of the drill and in fact considers, in the example given in the work, the collision of the striker and the drill with their actual masses according to the classical theory, as the impact of not wholly elastic bodies, conventionally dividing all losses into losses from elastic and plastic deformation.

In connection with the need to drill deep blastholes and boreholes, the importance of the question of the e.p. of the transfer of impact to the rock being destroyed, for various tool sizes, has increased. The solution of this question makes it possible not only to correctly standardize drilling productivity, but also to outline ways of reconstructing drilling tools and drilling machines. Works [4; 23] are devoted to questions of the drilling speed of deep blastholes. Analyzing the reason for the decrease in drilling speed with increasing blasthole depth, I. M. Shulgin and N. V. Mishchenko [23] believe that the main losses of kinetic energy occur during the collision of the striker with the drill rod; assuming, during collision, the velocity of displacement in various sections of the rod

is not the same, it is not difficult to come to the conclusion that the impact efficiency can be determined by the formula

\[ \eta=\frac{1}{1+\dfrac{k}{3}} . \tag{2} \]

Here, as before,

\[ k=\frac{Q_2}{Q_1}, \]

where \(Q_1\) is the weight of the striker,
\(Q_2=ql\) is the weight of the drill, of length \(l\).

The authors [23] believe that the drill, acquiring kinetic energy from the striker upon impact, then expends it equally on “shooting” into the rock and on “recoil” (reflection and crushing of the striker). It should be noted that such an “equal” distribution of the drill energy would be possible only if both the collision of the striker with the drill and that of the drill with the rock occurred at the same time, and if the resistances during deformation of the striker and of the rock were equal. The absence of rebound of the striker from the drill during drilling, from the point of view of the authors [23], can be explained only by considerable plastic deformation of the striker; however, as experience shows, nothing of the kind occurs in the operation of drilling hammers.

Prof. B. I. Bokii [7], considering manual drilling of blast-holes, holds that the impact efficiency must be determined as the product of two coefficients:

\[ \eta=\eta_1\cdot\eta_2, \tag{3} \]

\[ \eta_1=(1+c)^2\cdot\frac{Q_1Q_2}{(Q_1+Q_2)^2} \]

is the coefficient of transfer of impact energy from the hammer to the drill. Here \(c\) is the coefficient of restitution in impact \((0<c<1)\); \(Q_1\) and \(Q_2\) are, respectively, the weight of the striker and the weight of the drill; \(\eta_2\) is the coefficient of utilization of the energy directly for the destruction of the rock, which varies within the limits \(0.5<\eta_2<0.7\). Prof. B. I. Bokii believes that, taking into account other losses, the overall impact efficiency in drilling cannot be considered to exceed \(0.05\text{--}0.07\).

I. A. Afanas’ev [3] also defines the efficiency in drilling as consisting of two coefficients: the efficiency of impact on the drill and the efficiency of the drill’s impact on the rock. The overall efficiency is

\[ \eta=\frac{m_1m_2(1+c_1)^2(1-c_2^2)}{(m_1+m_2)^2}, \tag{4} \]

where \(m_1\) is the mass of the striker,
\(m_2\) is the mass of the drill,
\(c_1\) is the coefficient of restitution for the impact of the striker on the rod,
\(c_2\) is the coefficient of restitution for the impact of the drill on the rock.

Formula (4) for determining the impact efficiency for drilling hammers is given in the reference book on mining [26], where, in addition, a table of values of restitution coefficients is given according to the data of engineer A. V. Varzin. It follows from these tables that the coefficient of restitution depends on the impact velocity and the hardness of the steel, while \(c_2\) depends only on the rock. However, as shown in the work of engineer O. D. Alimov [2], the coefficient of restitution in the impact of the drill against the rock is not a constant quantity and depends to a considerable extent on the mechanism of failure during penetration, the impact energy, and the dimensions of the penetrating tool.

The results of experimental and theoretical works [5; 17] cast doubt on the correctness of determining the impact efficiency from the condition that the greatest losses occur in the collision of the striker and the drill. Thus, for example,

N. M. Batuev [5], on the basis of processing the data of a number of experiments and observations of the operation of pneumatic hammers, believes that when the striker impacts the shank, elastic impacts practically occur, and therefore the coefficient of restitution must be taken as close to unity. To determine the efficiency of impact of pneumatic hammers, N. M. Batuev recommends using the formula

\[ \eta = 4\alpha \frac{Q_1 Q_2}{(Q_1+\alpha Q_2)^2}, \tag{5} \]

where, for soft rocks, the coefficient \(\alpha\), which takes into account the influence of the rock being broken, may be taken equal to unity; i.e., in that case formula (5) coincides with formulas (4) and (3).

V. M. Mostkov [17], proceeding from the fact that when the striker impacts the drill, practically no appreciable heating of the latter nor residual deformations are observed, believes that energy losses during the transmission of impact occur only between the drill and the rock. On the basis of the work of N. M. Gersevanov [8] on the conditions of elastic impact of piles, V. M. Mostkov believes that the impact between the striker and the drill may be regarded as perfectly elastic if the condition is satisfied:

\[ E \cdot \frac{v_0}{a} < \sigma_{\text{el}}^{1)}. \]

Considering the transmission of impact through the drill on the basis of wave theory, V. M. Mostkov comes to the conclusion that “the efficiency of impact does not depend on the velocity of the piston at the instant of impact and is equal to \(\eta_2=\frac{A_y}{A}\)” [17, 103].

Here, according to V. M. Mostkov,

\[ A=\frac{m v_0^2}{2} \]

is the kinetic energy of the striker (brought to the drill) before impact;

\[ A_y=\gamma \cdot A_0 \]

is the useful work performed by the drill during its penetration into the rock, after impact by the piston (striker),

where

\[ A_0=\frac{m_b v_0^2}{2} \]

is the kinetic energy of the drill if it were to acquire the velocity of the striker at the instant of impact—\(v_0\);

\[ \gamma=\frac{k_*}{\varepsilon}\,[2l^{-\mu}(1-l^{-\varepsilon})]^2 \]

is the “work coefficient”;

\(k_*\) is a correction index depending on the number of the phase in which the penetration of the drill reaches its maximum;

\[ \mu=\frac{Q_b}{2Q_n} \]

is the ratio of the weight of the drill to twice the weight of the striker (piston);

\[ \varepsilon=\frac{k_v\omega_0 l}{2E\omega_b}, \]

where, in turn, the following are denoted:

\(k_v\) — coefficient of penetration,
\(l\) — length of the drill,
\(\omega_0\) — effective area of the drill bit head,
\(\omega_b\) — cross-sectional area of the drill,
\(E\) — modulus of elasticity of the drill material.

\(^{1)}\) With the modulus of elasticity of steel \(E=21\cdot10^6\ \text{t}/\text{m}^2\), the speed of sound in steel \(a=5.15\cdot10^3\ \text{m}/\text{s}\), and the elastic limit for carbon steel \(\sigma_{\text{el}}=3.5\cdot10^4\ \text{t}/\text{m}^2\), the value of the piston velocity for satisfying this condition is \(V_0 \leq 9\ \text{m}/\text{s}\) [17, 31].

Thus, the efficiency of impact, according to V. M. Mostkov, is

\[ \eta_2=\frac{A_y}{A}= \frac{\dfrac{m_{\delta}v_0^2}{2}\cdot\gamma}{\dfrac{mv_0^2}{2}} = \frac{m_{\delta}}{m}\cdot\frac{k_*}{\varepsilon} \left[2l^{-\mu}\cdot(1-l^{\varepsilon})\right]^2 . \tag{6} \]

It is proposed to calculate the overall efficiency of drilling as the product of the efficiency of the drilling machine proper and the efficiency of the drill impact, i.e. \(\eta=\eta_1\cdot\eta_2\),

where \(\eta_1=\dfrac{N}{N_n}\) is the efficiency up to impact, equal to the ratio of the useful power of the drilling machine before penetration of the drill into the rock to the power supplied to the drilling machine.

However, in formulating the differential equations and boundary conditions for determining the transfer of energy through the drill for the destruction of rock, V. M. Mostkov proceeds from the premise that the blow of the piston on the drill occurs when the latter rests against an elastic medium (rock), and penetration takes place according to a linear law. In reality, as shown in [2], the blow of the striker on the drill occurs when the latter is at some distance from the bottom, and the destruction of the rock during penetration is discontinuous in character [2, 25]. Therefore, as indicated below, when experimental data are compared with data calculated according to V. M. Mostkov’s theory, a considerable discrepancy is observed; the better agreement of the theoretical data with the experimental verification carried out by the author can be explained by the fact that, “in conducting the experiments, in order to avoid distortions that might arise on account of torsional oscillations of the drill (reaming of the walls of the blasthole), the dogs holding the helical rod of the drilling machine against rotation were removed… the thrust force of the drilling machine on the bottom was, as far as possible, kept constant, and so large that recoil did not affect the additional oscillations of the drilling machine” [17, 131].

Thus, V. M. Mostkov’s experimental data recorded the wave character of energy transfer through the drill for a mode of operation uncharacteristic of a drilling hammer—without rotation of the drill and without a clearance between the drill bit and the bottom of the blasthole, which provides the possibility of rotating the drill by means of the hammer’s turning device.

Nor can one agree with the proposed method for estimating the efficiency of drilling, since the energy of impact and the efficiency of the drilling machine proper,

\[ \eta_1=\frac{N}{N_n} \]

depend on the properties of the rock being destroyed. The overall efficiency of the machine and of its individual links must be studied concretely under definite conditions of interaction of the operating drilling machine with the rock being destroyed.

K. N. Shmargunov [24] points out that “for a hammer the efficiency of impact will depend not only on the ratio of the masses of the striker and the tool, but also on what material we are deforming: whether there will be rebound phenomena or not, whether the rebound of the striker will be in resonance with the motion of the mechanism or not” [24, 111].

As is evident from the quoted statement, K. N. Shmargunov gives the concept of impact efficiency a very broad content, in fact including the parameters of the impact machine, the parameters of the medium being destroyed, and the operating mode of the machine. Unfortunately, K. N. Shmargunov does not give a quantitative expression for the impact efficiency and, considering N. M. Batuev’s approximation

completely incorrect, recommends using the following formula for determining the impact efficiency:

\[ \eta = 1 - 0.7 \frac{k}{1+k}. \tag{1} \]

The overall efficiency of the hammer, according to K. N. Shmargunov, is equal to:

\[ \eta = \eta_{\mathrm{el}} \cdot \eta_{\mathrm{mech}} \cdot \eta_{\mathrm{imp}}, \]

where \(\eta_{\mathrm{el}}\) is the efficiency of the electric motor,
\(\eta_{\mathrm{mech}}\) is the efficiency of mechanical transmission,
\(\eta_{\mathrm{imp}}\) is the impact efficiency.

Thus, consideration of the state of the question of the transfer of energy from the striker to the tool of a drilling hammer and of the impact efficiency shows that at present there is no general opinion on this question. This is explained by the complexity of the processes occurring over short intervals of time in the colliding bodies during impact, and also by the complexity of the study of impact processes in general [9; 10; 11; 15] and of the impact action of the drilling-hammer tool on the rock in particular.

V. P. Shubin [22], considering the question of certain characteristics of metals under impact, writes: “In the study of the phenomenon of impact, two theories are most widespread. One of them takes into account the vibrations of the colliding bodies under the influence of waves of pressure and rarefaction; the other, the local phenomena in the impact zone. Experimental verification of these theories, Prof. S. O. Dobrogurskii [10] points out, has shown that each of them agrees with experimental results under its own strictly defined conditions: the vibration theory is valid for rods of considerable length and low rigidity, for example those having the form of springs; the compression theory (Hertz’s) is valid for the opposite case.” Proceeding from this, it seems most correct to us to consider the transfer of energy from the striker to the drill on the basis of wave theory, taking into account elastoplastic deformations in the colliding bodies, and the action of the drill on the rock on the basis of compression theory.

The first problem—the collision of the striker and the drill—can be solved most accurately on the basis of the work of Soviet scientists on the propagation of elastoplastic waves in colliding parts [8; 15; 18; 21], taking into account the mechanical characteristics of metals under impact [9; 13; 22].

In the present work we shall dwell chiefly on the questions of the second problem—the determination of the impact efficiency when the drill strikes the rock.

On the basis of observations carried out of the operation of drilling hammers, we consider that, in solving the question of impact efficiency, it is necessary to take into account:

1) The conversion of the striker’s energy into work occurs in two separate collisions—the striker against the drill and the drill against the rock—since, under the most characteristic operating conditions, at the moment when the striker strikes the drill the latter, owing to the rebound from the rock that almost always occurs, is at some distance from the face. After the drill has been imparted a certain kinetic energy, the drill moves toward the face and strikes the rock; the striker moves together with the drill for some time and then stops; rebound of the striker from the drill is observed in the case when the path of motion of the drill is insufficient, i.e., when there is an excessively large feed force (in this case there is incomplete transfer of energy from the striker to the drill).

2) The destruction of the rock under the impact of the drill occurs mainly at the expense of elastic deformation, in a discontinuous manner, when the potential energy of deformation of the volume of rock being destroyed exceeds a certain definite value [2, 25]. The magnitude of the potential energy of deforma-

of the rock before fracture depends on the mechanical properties of the rock and on the magnitude of the volume being deformed. After the first jump-like fracture, subsequent jump-like fractures are possible as long as the reserve of kinetic energy of the drill is sufficient for elastic deformation both of the drill itself and of the rock being broken up to its critical stress. If, during the last compression of the drill and the rock, no fracture follows, then the potential energy of compression is partially restored in the form of the kinetic energy of rebound of the drill from the bottom hole. The coefficient of rebound of the drill is not constant and depends mainly on the mechanism of rock fracture (on the magnitude of the impact energy, the shape and dimensions of the tool, and the area of contact of the tool with the rock [2]).

3) As the usefully expended part of the supplied (initial) energy of the striker

\[ A_0=\frac{m v_0^2}{2} \]

during impact, it is most expedient to take the work going (almost entirely) into the elastic deformation of the fractured volume of rock—\(A_n\); and as the efficiency of impact, in rotary-percussive drilling, the ratio of the useful work obtained in breaking the rock—\(A_n\)—to the energy absorbed by the drill during the impact of the striker and equal to

\[ A_y=\frac{m v_0^2}{2}(1-c^2)=A_0(1-c^2), \]

where

\[ c=\frac{v_{om}}{v_0} \]

is the coefficient of restitution (coefficient of rebound).

The measure of absorption by the drill of the energy supplied during the impact of the striker against the drill is the coefficient

\[ f=\frac{A_y}{A_0}=1-c^2. \]

When the drill strikes the rock, the kinetic energy of the drill \(T_\delta=A_u\) is transformed into the potential energy of elastic compression of the drill rod \(A_\delta\) and of the deformed volume of rock \(A_n\), with

\[ A_n+A_\delta=A_y. \]

Thus, the overall efficiency of impact in rotary-percussive drilling is:

\[ \eta=\frac{A_n}{A_y}=\frac{A_n}{A_u}\cdot\frac{A_u}{A_y}=\eta_2\cdot\eta_1, \]

where the efficiencies are

\[ \eta_1=\frac{A_u}{A_y} \]

—accounts for unproductive energy expenditures during the impact of the striker on the drill (heating, plastic deformations, sound);

\[ \eta_2=\frac{A_n}{A_u} \]

—accounts for unproductive expenditures during the impact of the drill on the rock (heating of the bit, plastic deformations).

The chief difficulty in finding the efficiency of impact consists in determining the work \(A_y=A_n+A_\delta\), which goes into the elastic deformation of the fractured volume of rock—\(A_n\), and also into the elastic deformation of the drill rod—\(A_\delta\).

Neglecting energy losses for compaction of the rock when the drill head comes into contact with it, we shall assume that all the kinetic energy of the drill rod passes into the potential energy of elastic deformation of the drill rod and of the fractured volume of rock—\(V\). Suppose that the magnitude of the area \(F_z\) of contact of the drill with the rock precisely ensures the complete expenditure of the drill’s kinetic energy in one “jump” of pressure and fracture;

Moreover, the maximum stress required for the destruction of the deformable volume of rock coincides in time with the maximum deformation of the entire rod, and the distribution of stresses along the length of the rod under impact corresponds to the distribution of weight [6, 664]. Since, with sufficient potential energy, the destruction of rock occurs abruptly, the compressed drill rod at that moment as it were instantaneously loses its support (the external forces deforming the drill instantaneously disappear). As a consequence of this, the potential energy going into the elastic deformation of the drill cannot be transformed into relative motion and is lost as sound and heat. The potential energy of elastic deformation of the volume of rock being destroyed, upon dispersion of the latter, is transformed into the energy of the newly formed surface, into sound, and into heat [13].

In determining the potential energy of elastic deformation of rock, we shall conditionally assume that the deformed volume is equal to the destroyed volume, and that the latter, prior to destruction, is in a volumetrically stressed state. Such an assumption is quite permissible as a first approximation, since, owing to the considerable rate of deformation under impact, the region of maximum elastic deformations is localized to a significant degree by the place of contact.

To determine the influence of deformation of the blade of the drill bit on the efficiency of impact, let us consider the impact on the rock of a drill with an end in the form of a triangular prism (Fig. 1). We introduce the following notation: \(l\) is the length of the drill rod; \(d\) is the length of the drill blade; \(\beta\) is the angle of sharpening of the drill; \(H\) is the length of the sharpened part of the drill (the height of the bit); \(\varphi\) is the chipping angle, depending on the coefficient of internal friction [19].

Elastic deformations of the rock and the drill occur after the drill compacts the semi-destroyed surface of the blasthole and comes into contact with it over a certain area \(F_{\mathrm{cor}} = d \cdot b\), which may be taken as the blunting of the drill tip, where \(b\) is the “width” of the blunting area of the bit tip.

Let us denote by \(k\) the distance of the blunting area \(F_{\mathrm{cor}}\) from the point of intersection of the faces, \(O\) (which we take as the origin of coordinates).

We denote by \(\xi\) the ratio of the depth of penetration of the drill (from the bottom surface of the blasthole) during compaction, \(k\), to the height of the bit, \(H\) (i.e., \(k = \xi H\)).

The maximum tangential stresses under elastic compression of two bodies, as was first indicated by A. N. Dinnik [11], arise at a certain depth, \(h\), from the pressure surface, thereby creating at this place more favorable conditions for the onset of destruction. In turn, the depth \(h\) depends on the blunting of the drill. It may be assumed that \(h = \varepsilon \cdot b\), where \(\varepsilon = 0.5 \div 1.0\).

The work expended on elastic deformation under impact of a drill rod of length \(l\), with modulus of elasticity of the material \(E\) and cross-sectional area of the rod \(F_{\mathrm{st}}\), may be determined by the formula:

\[ A_{\mathrm{st}} = \frac{\sigma_c^2 \cdot F_{\mathrm{st}} l}{6E} \quad [6, 665], \]

where \(\sigma_c\) is the stress in section \(C D\) (Fig. 1):

To determine the energy \(A_k\) expended on elastic deformation of the sharpened part of the drill (the bit), we first determine the stress in any section \(y\) only from the sharpened part of the drill, developed under impact by the \(c\) plane.

The elementary volumetric weight of the volume shaded in Fig. 1 is

\[ dQ_k = \gamma \, dv = 2d \cdot \gamma \cdot x \, dy, \quad \text{but} \quad x = y \tan \beta . \]

Consequently, the weight of the pointed part lying above the section \(y\)

\[ Q_y=\int_y^H 2d\gamma \tg\beta \cdot y\,dy =\gamma d \tg\beta\,(H^2-y^2), \]

Fig. 1

stress

\[ \sigma'_y=\frac{Q_y}{2xd} =\frac{\gamma}{2}\left(\frac{H^2}{y}-y\right), \]

when \(y=k\),

\[ \sigma'_k=\frac{\gamma}{2}\left(\frac{H^2}{k}-k\right). \]

Then

\[ \sigma'_y = \sigma'_k\, \frac{\left(\dfrac{H^2}{y}-y\right)} {\left(\dfrac{H^2}{k}-k\right)}. \]

Assuming that, in a free impact, the stresses in the impacting body (in any cross section) are proportional to the distribution of pressure from its own weight [6], we obtain:

\[ \sigma_c:\sigma'_k=\frac{Q_{\mathrm{st}}}{F_{\mathrm{st}}}:\frac{Q_{\mathrm{cor}}}{F_{\mathrm{cor}}} \quad \text{and} \quad \frac{\sigma_k}{\sigma'_k}=\frac{Q_{\mathrm{st}}+Q_{\mathrm{cor}}}{Q_{\mathrm{cor}}}; \]

whence

\[ \sigma_c=\sigma'_k \cdot \frac{Q_{\mathrm{st}}\cdot F_{\mathrm{cor}}}{Q_{\mathrm{cor}}\cdot F_{\mathrm{st}}} = \sigma_k \cdot \frac{Q_{\mathrm{st}}\cdot F_{\mathrm{cor}}}{(Q_{\mathrm{st}}+Q_{\mathrm{cor}})\cdot F_{\mathrm{st}}}. \]

Here \(\sigma_k\) is the resultant stress on the area \(F_{\mathrm{cor}}\) from the drill rod and the bit.

We assume that the stress \(\sigma'_c\) in any cross section of the sharpened part of the drill, located from the origin of coordinates at a distance \(y\) (only from the rod), is determined from the relation:

\[ \frac{\sigma'_c}{\sigma_c}=\frac{H}{y}, \quad \text{i.e.} \quad \sigma'_c=\frac{H}{y}\sigma_c. \]

By the law of independence of the action of forces, the stress under impact in any cross section of the sharpened part of the drill will be equal to the sum of the stress due directly to the rod and the stress due to the sharpened part of the drill lying above the section considered, i.e.

\[ \sigma_y=\sigma'_c+\sigma'_y = \frac{H}{y}\sigma_c + \sigma_k\cdot \frac{\dfrac{H^2}{y}-y}{\dfrac{H^2}{k}-k} = \]

\[ = \sigma_c \left[ \frac{H}{y} + \frac{Q_{\mathrm{cor}}\cdot F_{\mathrm{st}}\left(\dfrac{H^2}{y}-y\right)} {Q_{\mathrm{st}}\cdot F_{\mathrm{cor}}\left(\dfrac{H^2}{k}-k\right)} \right]. \]

The amount of energy expended on the deformation of the elementary volume \(dV\) is

\[ dA_k= \frac{\sigma_y^2\cdot F_y\cdot dy}{2E} = \frac{\sigma_y^2\cdot F_{\mathrm{st}}\cdot y\cdot dy}{2EH}. \]

\[ \left( \text{since } \frac{F_y}{F_{\mathrm{st}}}=\frac{y}{H} \right). \]

Replacing \(\sigma_y\) by its expression in terms of \(\sigma_c\) and integrating, we obtain the work expended on the elastic deformation of the bit

\[ A_k= \int_k^H \frac{\sigma_y^2\cdot F_{\mathrm{st}}\cdot y\cdot dy}{2EH} = \frac{\sigma_c^2\cdot F_{\mathrm{st}}\cdot H}{2E}\cdot \psi, \]

where the dimensionless coefficient \(\psi\) has the value:

\[ \psi = H\delta\left(2\ln\frac{1}{\xi}+\xi^2-1\right) + \delta^2H^2 \left( \ln\frac{1}{\xi}+\xi^2-\frac{\xi^4}{4}-\frac{3}{4} \right) + \ln\frac{1}{\xi}, \]

\[ \delta = \frac{Q_{\mathrm{cor}}\cdot F_{\mathrm{st}}} {Q_{\mathrm{st}}\cdot F_{\mathrm{cor}}\left(\dfrac{H^2}{k}-k\right)} = \frac{1}{2l(1-\xi^2)}; \qquad [\delta]=\frac{1}{\mathrm{mm}}, \]

since

\[ k=\xi H;\qquad F_{\mathrm{cor}}=\frac{k}{H}\cdot F_{\mathrm{st}};\qquad Q_{\mathrm{st}}=F_{\mathrm{st}}l\gamma \quad\text{and}\quad Q_{\mathrm{cor}}=\frac{H\cdot F_{\mathrm{st}}\cdot\gamma}{2}. \]

The total work expended on the elastic deformation of the drill is

\[ A_{\delta}=A_{\mathrm{st}}+A_{\mathrm{k}} = \frac{\sigma_c^2\cdot F_{\mathrm{st}}\cdot l}{6E} + \frac{\sigma_c^2\cdot F_{\mathrm{st}}\cdot H}{2E}\cdot\psi = \]

\[ = \frac{\sigma_c^2\cdot F_{\mathrm{st}}}{2E} \left(\frac{l}{3}+H\psi\right); \]

or, replacing \(\sigma_c\) by \(\sigma_k\) (the stress at the end of the drill resting against the rock), we obtain:

\[ A_{\delta}= \frac{\sigma_k^2 F_{\mathrm{st}} l^2 \xi^2} {2E\left(l+\dfrac{H}{2}\right)^2} \cdot \left(\frac{l}{3}+H\psi\right). \]

The complete work of deformation of a unit volume under a complex stressed state [6, 145]

\[ a=\frac{1}{2E}\cdot \left[ \sigma_1^2+\sigma_2^2+\sigma_3^2 - 2\mu(\sigma_1\sigma_2+\sigma_1\sigma_3+\sigma_2\sigma_3) \right]. \]

Let us assume that \(\sigma_1=\sigma_2=\sigma_3=\sigma_n=\sigma_k\), i.e., that the volume being destroyed is in a uniform volumetric stressed state; then

\[ a=\frac{3\sigma_n^2}{2E_n}\cdot(1-2\mu). \]

In two-sided shearing of the rock, the destroyed volume (Fig. 1) may be found from the formula:

\[ V=h^2\cdot d\cdot \operatorname{tg}\frac{\varphi}{2}. \]

The work expended on the elastic deformation of the destroyed volume of rock is:

\[ A_n=V\cdot a\cdot\lambda = \frac{3\sigma_n^2}{2E_n}\cdot(1-2\mu)\cdot h^2d\cdot \operatorname{tg}\frac{\varphi}{2}\cdot\lambda \]

Here \(\lambda\) is a coefficient that takes into account the losses of energy for deformation of the surrounding volume and the inaccuracies in determining the volumetric work of deformation solely by allowing for a uniform volumetric stressed state.

Since

\[ h=\varepsilon\cdot b;\qquad b=2k\cdot \operatorname{tg}\frac{\beta}{2};\qquad F_{\mathrm{cor}}=bd=\xi\cdot F_{\mathrm{st}}, \]

then after substitution we obtain:

\[ A_n= \frac{3\sigma_n^2}{E_n}\cdot \lambda\cdot\xi\cdot\varepsilon^2\cdot k\cdot F_{\mathrm{st}}\cdot (1-2\mu)\cdot \operatorname{tg}\frac{\varphi}{2}\cdot \operatorname{tg}\frac{\beta}{2}. \]

The total work of elastic deformation of the drill and the rock.

\[ A_y=A_\delta+A_{\mathrm{p}} = \frac{\sigma_{\mathrm{p}}^{2}\cdot F_{cm}\cdot l^{2}\cdot \xi^{2}} {2E\left(l+\dfrac{H}{2}\right)^2} \cdot \left(\frac{l}{3}+H\psi\right) + \]

\[ +\frac{3\sigma_{\mathrm{p}}^{2}}{E_{\mathrm{p}}}\cdot \lambda\,\xi^{2}k\,F_{cm}(1-2\mu)\cdot \operatorname{tg}\frac{\varphi}{2}\cdot \operatorname{tg}\frac{\beta}{2} \]

efficiency of the impact of the drill on the rock

\[ \eta_2= \frac{A_{\mathrm{p}}}{A_{\mathrm{p}}+A_\delta} = \frac{1}{1+\dfrac{A_\delta}{A_{\mathrm{p}}}} = \]

\[ = \frac{1} {1+ \dfrac{ E_{\mathrm{p}}\cdot\left(\dfrac{l}{3}+H\psi\right) }{ 6E\cdot\left(1+\dfrac{H}{2l}\right)^2\cdot \lambda\cdot\varepsilon^2\cdot H(1-2\mu)\cdot \operatorname{tg}\dfrac{\varphi}{2}\cdot \operatorname{tg}\dfrac{\beta}{2} }} \tag{7} \]

The sharpening angle \(\beta\) may be expressed through the area of the rod \(F_{cm}\), the length of the blade \(d\), and the height of the bit; from Fig. 1 it is seen that

\[ \operatorname{tg}\frac{\beta}{2} = \frac{F_{cm}}{2dH}; \]

after substitution we obtain:

\[ \eta_2= \frac{1} {1+ \dfrac{ E_{\mathrm{p}}\cdot d\cdot\left(\dfrac{l}{3}+H\psi\right) }{ 3E\left(1+\dfrac{H}{2l}\right)^2\cdot \lambda\cdot\varepsilon^2(1-2\mu)\cdot F_{cm}\cdot \operatorname{tg}\dfrac{\varphi}{2} }} \tag{7} \]

As a special case, for \(H=0\), we have:

\[ \eta_2= \frac{1} {1+ \dfrac{ E_{\mathrm{p}}d\cdot l }{ 9E\lambda\varepsilon^2(1-2\mu)\cdot F_{cm}\cdot \operatorname{tg}\dfrac{\varphi}{2} }} \]

It follows from this that the efficiency of the impact of the drill on the rock increases with a decrease in the ratio \(\dfrac{A_\delta}{A_{\mathrm{p}}}\), i.e., \(\eta_2\) increases: with a decrease in the length of the drill \(l\), the height of the bit \(H\), and the modulus of elasticity of the rock \(E_{\mathrm{p}}\); with an increase in the modulus of elasticity of the entire drill (and especially of its sharpened part—the blade); and with an increase in the sharpening angle of the drill \(\beta\).

All this agrees well with practical data and explains the reason for the increase in drilling speed when using blades reinforced with hard alloy, as well as the relatively poor drillability by the impact method of “viscous” rocks, which, while having low compressive strength, possess a high modulus of elasticity, such as, for example, certain types of marble.

Thus, the formula for \(\eta_2\) relates the losses during impact penetration of the drill bit into the rock to the geometry of the tool and to the principal mechanical characteristics of the colliding materials (the modulus of elasticity and Poisson’s ratio). However, we have considered the limiting case in which the destruction of the rock by impact occurs in a single pressure jump, and we have not taken into account losses due to friction and possible plastic deformations. In reality, the kinetic energy of the drill bit is expended over several chips, between which the blade of the bit displaces and compacts already destroyed or semi-destroyed rock, spending part of the energy on this; moreover, the last chip, as a rule, does not make full use of the kinetic energy of the drill bit, and therefore some part of it is restored in the form of rebound of the bit from the bottom. Consequently, the value of the efficiency \(\eta_2\) will in fact be somewhat smaller than that calculated from the formula we have derived; however, it may be assumed that the influence of all additional factors can be taken into account by a coefficient \(\lambda\), determined experimentally.

As is clear from the preceding, the use of all the impact energy for a single chip is most expedient, but this is possible only when the magnitude of the contact area corresponds exactly to the given impact energy. With an increase in the impact energy, the magnitude of this area must also increase. On this basis one can explain the fact, seemingly “paradoxical” from the point of view of existing theories [25], that “the minimum volumetric work for a blunted bit, whose initial contact area is large, is less than for a sharp bit” [25]. In the existing literature this fact is explained only by the influence of the scale factor [25].

The efficiency of the blow of the striker on the drill bit can be calculated by the formula:

\[ \eta_1=\frac{A_u}{A_y} = \frac{ \sigma_n^2 F_{cm}\xi^2 \left[ \dfrac{\dfrac{l}{3}+H\psi}{2E\left(1+\dfrac{H}{2l}\right)^2} + \dfrac{3\lambda \varepsilon^2 H}{E_n}(1-2\mu)\,\tg\frac{\varphi}{2}\,\tg^3\frac{\beta}{2} \right] }{ A_0(1-c^2) }. \]

Here a certain additional difficulty may arise in determining the value of \(\sigma_n\). In addition, knowing the magnitude of the overall efficiency of impact

\[ \eta=\eta_1\cdot\eta_2 \]

for known values of \(\eta_2\) and \(\eta\), we determine:

\[ \eta_1=\frac{\eta}{\eta_2}. \]

Comparing the values of the impact efficiency determined by the formula (7) proposed by us with experimental data known in the literature on the drilling of deep boreholes, as well as with the data of our investigations, we came to the conclusion that, for calculations with sufficient accuracy, the efficiency of the transfer of kinetic energy from the striker to the drill bit can be determined by the formula:

\[ \eta_1=1-(1-c_1^2)\cdot\frac{k}{1+k}, \]

i.e., in fact by formula (1), if the value of the impact coefficient is taken as \(c_1 \simeq 0.80 \div 0.85\). This is also confirmed by the experimental investigations of engineer A. V. Varzin [26].

The coefficient of restitution (of impact) \(c_1\) should not be confused with the coefficient \(c\), which characterizes rebound from the drill bit during the drilling process (the characteristics of the system: striker—drill bit—bit crown—rock), the value of which can also be determined experimentally from graphs of the motion of the striker [1], or by studying the process of impact and destruction on copra [2]. As our experiments showed, when drilling in the optimal regime with a clearance between the blade of the bit and the bottom of the shothole, at the moment of impact of the striker on the drill bit, with a drill length of more

meter, there was practically no rebound of the striker from the drill (i.e., \(c = 0\)); therefore, under the optimum drilling regime the energy losses \((A_0 c^2)\) due to rebound of the striker from the drill may be neglected.

In this case, apparently, a phenomenon takes place analogous to the exchange of velocities in an elastic impact \((c_1 \ne 0;\ v_{\text{striker}} = 0\) after the impact; \(v_{\text{drill}} > 0)\).

Special investigations must be carried out to determine the value of the coefficient of restitution \(c_1\) as a function of the factors affecting its magnitude. One of the interesting investigations in this direction is the work of V. P. Shubin [22], in which a formula is given for determining the coefficient of restitution when a rod strikes hard coal, expressed in terms of the elastic-plastic deformation.

Fig. 2

Finally, from the standpoint of unproductive losses in drilling, it is necessary to take into account the presence of friction of the drill against the rock and the comminution of fragments; the coefficient accounting for losses due to friction during compaction and displacement of the destroyed rock particles by the drill over several chips will be denoted by \(\eta_3\)¹); then the overall impact efficiency is

\[ \eta_0 = \eta_1 \cdot \eta_2 \eta_3 . \tag{7″} \]

In Fig. 2 a comparison is given of the results of calculating the impact efficiency according to formulas (1)—(7) for the case of drilling deep blastholes. The case of drilling deep blastholes was chosen because, depending on the depth of the blasthole (bore—

¹) The value of \(\eta_3\) varies from zero to unity and depends on the ratio of impact energy to tool size, the mechanical properties of the rock, and the method of removing the drill cuttings.

changes the ratio of the masses and dimensions of the colliding parts of the hammer and the drill, i.e., the main factors influencing the value of the efficiency of the impact. Moreover, for this case the literature contains the largest amount of experimental data. As an empirical formula for determining the change in the drilling speed of deep boreholes, we use the formula recommended by L. I. Baron [4].

\[ v=\frac{a}{l}, \tag{8} \]

where \(v\) is the drilling speed; \(l\) is the borehole depth; \(a\) is a constant for the given mining-geological and technical conditions.

As can be seen from the graphs in Fig. 2, the values of the impact efficiency calculated by different formulas for one and the same case differ considerably both in magnitude and in the character of their variation, depending on the drill length.

Fig. 3

The greatest value of the impact efficiency, as was to be expected, is obtained from formulas (1) and (5), which do not take into account the loss of energy when the drill acts on the rock. According to formula (6), for a drill length of more than 4 meters, all destruction of the rock by impact should practically have ceased.

Figure 3 gives a comparison of the data on the change in drilling speed, calculated as a function of the impact efficiency on the basis of formulas (1)—(8), with the experimental data obtained from [4].

In constructing the calculated curves, indicated in Figs. 2 and 3 by the numbers of the corresponding formulas, the initial drilling speeds were taken—

are taken equal to the experimental ones, according to [4]. In all cases the agreement between the experimental values and those calculated by the formula obtained by us \((7'')\) proved to be sufficiently close.

Unfortunately, in the literature we have examined we did not find data on the influence on the drilling rate of such mechanical characteristics of rocks as the modulus of elasticity and the coefficient of internal friction, which would make it possible to check the correctness of taking these factors into account in determining the efficiency of impact by formula \((7'')\).

Consideration of the theoretical and experimental data on the efficiency of impact in drilling hammers shows that rod drilling should expediently be used only for hole depths not exceeding \(4\text{–}5\) m, and quite evidently emphasizes the need to carry out work on the creation and mastering, for drilling deep blast-holes and boreholes, of drilling hammers with an impact node directly entering the blast-hole or borehole¹.

REFERENCES

1. Alabuzhev P. M., Yudin I. P. Experimental investigation of an electromagnetic hammer. Proceedings of the Mining and Geological Institute, Western Siberian Branch of the Academy of Sciences, issue 8, Novosibirsk, 1950.

2. Alimov O. D. On the mechanism of rock destruction in percussive-rotary drilling. Proceedings of TPI, vol. 75, 1954.

3. Afanas’ev I. A. Electrohydraulic drilling machine. Gornyi zhurnal, No. 7–8, 1937.

4. Baron L. I. The use of deep boreholes for underground ore mining. Metallurgizdat, 1951.

5. Batuev N. M. Increasing the productivity of electric demolition hammers and reducing their weight. Journal Mechanization of Heavy and Labor-Intensive Work, No. 5, 1948.

6. Belyaev N. M. Strength of Materials. Gostekhizdat, 1939.

7. Bokii B. I. Practical Course in Mining, vol. II, GNTI, Moscow–Leningrad, 1931.

8. Gersevanov N. M. Theory of a longitudinal elastic impact and its application to determining the resistance of piles. Collected Works, vol. I, Stroiizdat, 1948.

9. Davidenkov N. N. The problem of impact in metal science. ONTI, 1938.

10. Dobrogurskii S. O. On stresses and forces under impact. Problems in the calculation and design of machine parts. Institute of Machine Science, Academy of Sciences of the USSR, 1942.

11. Dynnik A. N. Selected Works, vol. I, Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1952.

12. Zhukovskii N. E. Theoretical Mechanics. Oborongiz, 1949.

13. Kuznetsov V. D. Physics of the Solid State, vol. V. Tomsk, 1949.

14. Kucherov P. S. “Coal,” No. 93, 1933.

15. Kil’chevskii N. A. Theory of the Collision of Solid Bodies. Gostekhizdat, 1949.

16. Malakhov Yu. M. Theory of operation of the pneumatic hammer. Gornyi zhurnal, No. 2, 1934.

17. Mostkov V. M. Fundamentals of the theory of pneumatic drilling. Ugletekhizdat, 1952.

18. Mochalov S. D. A graphical method for investigating the longitudinal elastic-plastic impact of rods. Scientific Notes of Tomsk State University, No. 17, Tomsk, 1952.

19. Ostroushko A. On the theory of drilling of I. S. Pokrovskii. Gornyi zhurnal, No. 7, 1950.

20. Rakhmatulin Kh. A. On the propagation of unloading waves. PMM, vol. IX, issue 1, 1945.

21. Sokolovskii V. V. Propagation of elastic-viscous-plastic waves in rods. PMM, vol. XII, issue 3, 1948.

22. Shubin V. P. On certain characteristics of metals under impact. Abstract. Tomsk, 1953.

23. Shul’gin I. M. and Mishchenko. The influence of blast-hole depth on the productivity of drilling. Gornyi zhurnal, No. 7, 1937.

24. Shmargunov K. N. Electric hammers. Mashgiz, 1950.

25. Shreiner L. A. Physical foundations of the mechanics of rocks. Gostoptekhizdat, 1950.

26. Handbook of Mining Engineering, vol. I. Metallurgizdat, 1952.

1. Since the completion of the work and its submission for publication, we have encountered the book by engineer N. M. Batuev, The Energetics of Electric Hammers and Ways to Increase Their Productivity, Proceedings of VNIISDORMASH, No. 6, 1953; the work indicates the possibility of increasing the efficiency of impact by using a drill consisting of a number of identical successively colliding bodies equal in weight to the striker. ↩