On the Methodology for Studying Electropneumatic Hammers

P. M. Alabuzhev, E. K. Medlina, E. N. Nevstruev

The present work was carried out in connection with the processing of indicator diagrams of electropneumatic hammers with a one-sided air connection (Fig. 1), obtained on an experimental test stand.

The hammer operates as follows: the electric motor, by means of crank 1 and connecting rod 2, sets piston 3 into reciprocating motion. When the piston moves to the left, a rarefaction is created between it and striker 5, and, under excess air pressure, the striker (ram) moves to the left. During the piston stroke to the right, the air is compressed; the ram changes the direction of its motion and, at the end of its stroke, having acquired a certain velocity \(v_b\), delivers a blow to the tool. To maintain stable operation of the hammer at the moment of impact, the air contained in the hammer cylinder comes into contact with atmospheric air through special compensation openings 4.

Fig. 1

An experimental study of an electropneumatic hammer of this scheme was performed in the Laboratory of Mining Machines of the ZSFAN by P. M. Alabuzhev and I. P. Yudin [1]. In this study, the motion curves of the piston and striker were recorded simultaneously by a mechanical method in the coordinates: path—\(s\) and time—\(t\). From the recorded path curve of the ram, its velocity and acceleration were determined graphically; from the latter value, the air pressure in the hammer cylinder was found. On the basis of the constructed indicator diagram in the coordinates: pressure—\(P\) and volume \(v\), the efficiency of the air cushion was determined:

\[ \left( \text{eff.} = \eta = \frac{L_{\text{exp}}}{L_{\text{comp}}} = 0.86 \right) \]

and the mean value of the polytropic exponent over the cycle \((m_{\text{av}} \approx 1.23)\).

Experiments to determine the efficiency of the air connection and the value of the polytropic exponent were repeated at the Tomsk Polytechnic Institute (on a stand specially built by the Tomsk Electromechanical Plant) for various machine parameters. The path curves of the ram and piston were recorded by an optical method; the procedure for carrying out the expe-

…of the experiment is set forth in the work of P. M. Alabuzhev and O. D. Alimov [2]. In addition to the indicated curves, the curves of the air pressure \(P\) and the angular velocity \(\omega\), as well as a mark of the angle of rotation of the crankshaft, were simultaneously observed and recorded on a cathode-ray oscillograph.

The experimental curves for one of the operating conditions are presented in Fig. 2. It should be noted that, when processing the diagrams, when superposing the curves on one graph, in a number of cases difficulties arise, and there are certain discrepancies with the initial data obtained in carrying out the experiment. In one

Fig. 2

of the tests the calibration curve of the capacitive pressure transducer was lost; the question arose—how to calibrate the recorded pressure curve \(P\)^[1], having the diagrams of the path of the striker \(s_b\), the path of the piston \(s_n\), as well as the value of the weight of the striker \(Q_b\) and the period of the cycle \(T\)?

First we shall show how, with a known approximation, one can check the correctness of the calibration of the pressure diagram. After superposing the curves on one graph, when processing the experimental results, it is necessary to divide the period of the cycle by the straight line \(AA\) into two parts, corresponding to the idle stroke of the striker during the time \(t_x\) and to the working stroke during the time \(t_p\), ending with the blow against the tool. The position of the vertical \(AA\) may

^[1] Under the condition of linearity of the calibration of the pressure diagram.

it is sufficiently accurate to draw through the point corresponding to the maximum stroke of the striker (i.e., when \(v_b = 0\)). Knowing the scale of the striker-path curve, \(l_c\), and the time scale, \(t_c\), from the formula

\[ v=\frac{l_c}{t_c}\cdot \operatorname{tg}\alpha \]

we can find the velocity at any instant of time (here \(\alpha\) is the angle of inclination of the tangent to the graph \(s_b\) at the corresponding point).

We determine the velocity of the striker at the moment of impact, \(v_{ud}\), and upon rebound, \(v_{ot}\). The ratio of the rebound velocity, \(v_{ot}\), to the velocity at the moment of impact, \(v_{ud}\) (which is the initial velocity), is called the coefficient of restitution in impact; obviously, this ratio

\[ k=\frac{v_{ot}}{v_{ud}} \tag{1} \]

is, in the present case, a characteristic of the system “striker—tool—material.”

In addition, the velocity of the striker at any instant of time, as well as the mean pressure over any part of the cycle \(T\), can be determined on the basis of the impulse theorem [3].

We draw the straight line \(BB\) parallel to the zero pressure line \(P=0\), corresponding to the excess air pressure in the hammer cylinder above atmospheric pressure, which we take approximately equal to: \(P_0 = 1 \dfrac{\mathrm{kg}}{\mathrm{cm}^2}\).

Fig. 3

Then, as is evident from Fig. 3, the area of the air-pressure impulse is divided by the straight lines \(AA\) and \(BB\) into a number of elements, the number of which depends on the operating regime of the machine, its parameters, and the interaction with the energy-absorbing medium during impact.

The impulse area lying to the left of the straight line \(AA\) corresponds to the idle stroke of the striker; denote it by \(S_x\) (from Fig. 3 it is obvious:

\[ S_x=S'_x-S''_x+S'''_x \]

).

The area of the impulse of forces located to the right of the straight line \(AA\), corresponding to the working stroke of the striker, we denote by \(S_p\) (from Fig. 4 it is seen:

\[ S_p=S'_p-S''_p+S'''_p \]

). \(C_x\) and \(C_p\) are the centers of gravity of the impulse areas \(S_x\) and \(S_p\).

According to the impulse theorem:

\[ mv-(-mv_{om})=S_t, \tag{2} \]

where

\[ m=\frac{Q_b}{g}; \]

therefore, the velocity of the striker \(v\) at any instant of time \(t\) can be determined from relation (2), by the formula

\[ v=-v_{om}+\frac{S_t}{m}. \tag{2a} \]

At the end of the idle stroke, i.e., when \(t=t_x\), \(v_b=0\); consequently,

\[ v_{om}=\frac{S_x}{m}. \]

At the moment of impact, when \(t=t_x+t_p=T\),

\[ v_{yd}=-v_{om}+\frac{S_T}{m} =-v_{om}+\frac{S_x+S_p}{m}= \]

\[ =-v_{om}+\frac{S_x}{m}+\frac{S_p}{m}=\frac{S_p}{m}. \]

Thus, the velocity of the striker at the moment of impact is equal to the ratio of the impulse of the force acting on the striker during the working part of the stroke \(t_p\) to the mass of the striker.

Let us find the average values of the air-pressure forces over the times \(t_x\), \(t_p\), and \(T\), denoting them respectively by \(P_x\), \(P_p\), \(P_T\).

\[ \left. \begin{aligned} S_x&=mv_{om}=\frac{Q_b}{g}\cdot v_{om} =P_x\cdot\frac{\pi D^2}{4}\cdot t_x, &&\text{whence } P_x=\frac{q\cdot v_{om}}{g\cdot t_x},\\[6pt] S_p&=mv_{yd}=\frac{Q_b}{g}\cdot v_{yd} =P_p\cdot\frac{\pi D^2}{4}\,t_p, && P_p=\frac{q\cdot v_{yd}}{g\cdot t_p},\\[6pt] S_T&=S_x+S_p=\frac{Q_b}{g}(v_{om}+v_{yd}) =P_T\frac{\pi D^2}{4}\cdot T, && P_T=\frac{q(v_{om}+v_{yd})}{gT}. \end{aligned} \right\} \tag{3} \]

Here

\[ q=\frac{Q_b}{\dfrac{\pi D^2}{4}} \]

is the weight of the striker per unit area, and \(D\) is the diameter of the cylinder (or striker).

The value of the average air pressure acting on the striker can be related to the parameters of the hammer and to its operating mode:

\[ P_T=\frac{q\,(v_{om}+v_{yd})}{gT} =\frac{q\cdot Z(1+k)v_{yd}}{60\cdot g} =\frac{Q_b\cdot Z(1+k)\cdot v_{yd}}{15\pi D^2 g}, \]

i.e., the average pressure over the hammer operating cycle is proportional to the weight of the striker \(Q_b\), the number of blows per minute \(Z\), and the impact velocity \(v_b\); it is inversely proportional to the square of the cylinder diameter \(D\), and depends on the coefficient of restitution in impact \(k\).

For a definite type of design, on the basis of experimental data, under various operating conditions, one can find the relation between

with the maximum air pressure \(P_{max}\) and the mean pressure \(P_T\) over the period \(T=\dfrac{60}{Z}\), i.e., to determine the dependence \(P_{max}=f(P_T)\), as well as the most probable value of the ratio \(\dfrac{t_p}{t_x}\), and thus to construct theoretically the indicator diagram (analogously one could express \(P_{max}\) through \(P_x\) and \(P_p\)). Knowing the maximum pressure \(P_{max}\), on the basis of the gas equation of state \(Pv=RT^\circ\), or the known relation

\[ \left(\frac{P_2}{P_1}\right)^{\frac{m-1}{m}}=\left(\frac{T_2^\circ}{T_1^\circ}\right), \]

one can, in the first approximation, determine the maximum temperature of the air in the hammer cylinder \(T^\circ\) and thereby know the heating temperature of the lubricant and the degree of heating of the hammer.

Between the force impulses \(S_x\), \(S_p\), and \(S_T\), on the basis of (3), the following relations can be composed:

\[ \left. \begin{aligned} \frac{S_x}{S_p} &= \frac{m v_{om}}{m v_{ud}}=\frac{P_x \cdot t_x}{P_p \cdot t_p}=k; \qquad \frac{S_x}{S_T}=\frac{S_x}{S_x+S_p}=\frac{k}{k+1};\\ \frac{S_p}{S_T} &= \frac{S_p}{S_x+S_p}=\frac{1}{k+1}. \end{aligned} \right\} \tag{4} \]

For an inelastic impact \(k=0\), and consequently \(S_x=S_x'-S_x''+S_x'''=0\); for a perfectly elastic impact \(k=1\), then \(S_x=S_p\); for steel \(k \simeq 5/9\), hence:

\[ \frac{S_x}{S_p}\simeq \frac{1}{2}; \qquad \frac{S_x}{S_T}\simeq \frac{1}{3}; \qquad \frac{S_p}{S_T}\simeq \frac{2}{3}. \]

It can be shown that between the mean values of the air pressure there exist relations expressed through the coefficient of restitution \(k\) and the parts of the cycle time \(t_x\) and \(t_p\); from (3) it follows that

\[ \frac{P_x}{P_p}=k\cdot\frac{t_p}{t_x}; \qquad \frac{P_x}{P_T}=\frac{k}{k+1}\cdot\frac{T}{t_x}; \qquad \frac{P_p}{P_T}=\frac{1}{k+1}\cdot\frac{T}{t_p}. \tag{5} \]

If qualitative curves of the path of the striker \(s_b\) and of the air pressure \(P\) in the hammer cylinder, obtained from experiment with scales marked on them, are available, it is not difficult by planimetry to verify that the areas of the impulses \(S_x\) and \(S_p\) satisfy relations (4), and, by formulas (3), to determine the mean pressure values \(P_x\), \(P_p\), \(P_T\) and check the scale of the calibration obtained on the basis of the experiment.

As an example, let us consider the graphs presented for one of the investigated operating regimes in Fig. 2.

In this regime

\[ Q_b=2{,}400\ \text{kg}, \qquad D=68{,}0\ \text{mm}, \qquad q=\frac{Q_b}{\dfrac{\pi D^2}{4}}=0{,}066\ \frac{\text{kg}}{\text{cm}^2}, \]

\[ Z=845\ \text{blows/min}, \qquad r=40{,}0\ \text{mm}, \qquad l_0=45{,}0\ \text{mm} \]

(impact on a metal rod resting on granite).

The cycle period

\[ T=\frac{60}{Z}=0{,}0710\ \text{sec}=146\ \text{mm} \]

on the drawing (with a measurement error of \(0{,}5\ \text{mm}\); tachometer reading error \(1{,}5\%\)).

Idle-stroke time \(t_x = 0.0578\) sec \(= 119\) mm on the drawing.

Working-stroke time \(t_p = 0.0132\) sec \(= 27\) mm on the drawing.

\[ \frac{t_p}{t_x}=0.227;\qquad \frac{t_p}{T}=0.18 \]

—the coefficient characterizing the utilization of the machine’s working time.

Time scale

\[ t_c=\frac{0.071}{146}=0.000485\ \frac{\text{sec}}{\text{mm on the drawing}}. \]

Linear scale

\[ l_c=2.11\ \frac{\text{mm}}{\text{mm on the drawing}}; \]

angle of inclination of the tangent to curve \(s_b\) at the moment of impact \(\alpha_y=53^\circ\),
angle of inclination of the tangent to curve \(s_b\) at the moment of rebound \(\alpha_{om}=13^\circ\).

Velocity at the moment of impact

\[ v_b=\frac{l_c\cdot \operatorname{tg} 53^\circ}{t_c} =\frac{2.11\cdot 1.33}{0.000485} =580\ \frac{\text{cm}}{\text{sec}}. \]

Velocity at the moment of rebound

\[ v_{om}=\frac{l_c\cdot \operatorname{tg} 13^\circ}{t_c} =\frac{2.11\cdot 0.23}{0.000485} =100\ \frac{\text{cm}}{\text{sec}}. \]

Coefficient of restitution in impact

\[ k=\frac{v_{om}}{v_{yb}}=\frac{1.00}{5.80}=0.17. \]

Impact energy at the striker

\[ A_b=\frac{Q_b\cdot v_b^{2}}{2g} =\frac{2.4\cdot 5.8^{2}}{2\cdot 9.8} =4.12\ \text{kg}\cdot\text{m}. \]

Rebound energy

\[ A_{om}=\frac{Q_b\cdot v_{om}^{2}}{2g} =\frac{2.4\cdot 1.0^{2}}{2\cdot 9.8} =0.12\ \text{kg}\cdot\text{m}. \]

Since the rebound energy of the striker is not lost energy for the machine, the measure of energy transfer to the tool is the impact-transfer coefficient

\[ \eta_{\text{per}}=\frac{A_{yb}-A_{om}}{A_{yb}}=1-k^2\approx 0.97. \]

This characteristic is not the value of the impact efficiency, since it does not take into account useless energy losses during impact; the efficiency in the collision of bodies \((\eta_{yb})\) depends on the ratio of their masses, as well as of the masses attached to them, on the physical and geometrical properties of the colliding materials and their state, and varies within very wide limits (from zero to unity).

Let us find the impulses of the pressure forces per unit area:

\[ \left. \begin{aligned} S_x&=\frac{q}{g}\cdot v_{om} =\frac{0.066}{981}\cdot 100 =0.00675\ \frac{\text{kg}^2\cdot\text{sec}}{\text{cm}^2}; &\frac{S_x}{S_p}&=k=0.17,\\[6pt] S_p&=\frac{q}{g}\,v_{yb} =\frac{0.066}{981}\cdot 580 =0.03915, &\frac{S_x}{S_T}&=\frac{k}{k+1}=0.15,\\[6pt] S_T&=S_x+S_p=0.04585, &\frac{S_p}{S_T}&=\frac{1}{k+1}=0.85 \end{aligned} \right\} \tag{4'} \]

The measured areas of the unit impulses \(S_x = 122\ \mathrm{mm}^2\) and \(S_p = 710\ \mathrm{mm}^2\) satisfy relations (4′), consequently:

\[ P_x=\frac{S_x}{t_x}=\frac{0.00675}{0.0578}=0.117\ \frac{\mathrm{kg}}{\mathrm{cm}^2} =\frac{122\ \mathrm{mm}^2}{119\ \mathrm{mm}}=1.05\ \mathrm{mm} \quad \text{on the drawing} \]

\[ P_p=\frac{S_p}{t_p}=\frac{0.0391}{0.0132}=2.970\ \frac{\mathrm{kg}}{\mathrm{cm}^2} =\frac{710\ \mathrm{mm}^2}{27\ \mathrm{mm}}=26.1\ \mathrm{mm} \quad " \]

\[ P=\frac{S_{\mathrm{T}}}{T}=\frac{0.04585}{0.071}=0.645\ \frac{\mathrm{kg}}{\mathrm{cm}^2} =\frac{832\ \mathrm{mm}^2}{146\ \mathrm{mm}}=5.7\ \mathrm{mm} \quad " \]

whence the calibration scale is:

\[ 1\ \frac{\mathrm{kg}}{\mathrm{cm}^2} =\frac{1.05}{0.117} =\frac{26.1}{2.97} =\frac{5.7}{0.645} =8.85\ \mathrm{mm} \quad " \]

According to the pressure scale and the graph, we determine

\[ P_{max}=1.0+\frac{60.0}{8.85}=7.8\ \frac{\mathrm{kg}}{\mathrm{cm}^2}; \qquad P_{min}=\frac{4.5}{8.85}=0.51\ \frac{\mathrm{kg}}{\mathrm{cm}^2}, \]

\[ P_{max}=C\cdot P_{\mathrm{T}},\quad \text{whence}\quad C=\frac{P_{max}}{P_{\mathrm{T}}}=\frac{7.8}{0.645}=12.5. \]

On the basis of processing a number of operating regimes of a hammer of this type with variable parameters, the mean value of the coefficient may be taken as \(C \approx 10\), with a maximum deviation of \(\pm 25\%\).

The error in determining the calibration scale by the computational method depends on the degree of accuracy of the initial data, and also on the error with which the individual quantities are determined during processing; in the example considered, the errors of the scale \(P_c\)

\[ P_c=\left|1\ \frac{\mathrm{kg}}{\mathrm{cm}^2}\right| = \frac{\dfrac{S_{\mathrm{T}}\ \mathrm{mm}^2}{T\ \mathrm{mm}}} {\dfrac{q(1+k)\cdot v_{ud}}{T}}; \]

the error is

\[ |\Delta P_c| = |\Delta S_{\mathrm{T}}\ \mathrm{mm}^2| + |\Delta T\ \mathrm{mm}| + |\Delta q| + |\Delta v_{ud}| + |\Delta v_{ot}| + |\Delta T\ \mathrm{sec}| = \]

\[ =(2.5+0.5+0.5+2.0+2.0+1.5)\%=9\%. \]

The error in measuring the air pressure \(P\) with a capacitive sensor by observing the beam deflection on the screen of a cathode oscilloscope is determined from the ratio of the thickness of the electron-beam line \(\Delta h\) in mm to the mean value of the beam deflection from the zero line \(h\) in mm and the error permitted by the experimental method of investigation.

Figure 4 shows the pressure curve \(P\), the graph of the calibration curve taken before and after the experiment; on the right are drawn the lines on the oscilloscope screen and the corresponding pressures: the lower line corresponds to a pressure of \(1\ \mathrm{kg}/\mathrm{cm}^2\), the upper to \(5\ \mathrm{kg}/\mathrm{cm}^2\).

The discrepancy in the value of the pressure obtained from the experiment in this case is equal to

\[ \frac{\Delta}{P_{max}}\cdot 100 = \frac{3.6}{60}\cdot 100 = 6\%. \]

Thus, the error in determining the pressure scale \(P_c\) from the displacement graph is comparable with the error obtained directly when measuring the observed quantity on the oscilloscope screen (exceeding it by 1.5–2 times).

From the example considered it is clear how, when a graph of the striker displacement \(S_b\) is available, one can calibrate the recorded experimental curve of the air pressure \(P\); this is important not only in those cases when, for example, the calibration graph has been lost or, for some reason during operation, the zero line has shifted, or the scale of the measured quantity \(P\) has accidentally changed (for example, the oscilloscope knob was turned!). Having recorded on the photograph only the shape of the pressure curve \(P\), one can subsequently process this curve by the method indicated above. This calibration method assumes especially great importance in studying the phenomenon of impact between the striker and the tool.

Fig. 4

Procedure for performing the work

1. We superpose the curves \(S_b\) and \(P\) on one graph at the same time scale; then we draw the most probable positions of the coordinate axes \(AA\) and \(BB\) (the position of the vertical \(AA\) can be drawn with sufficient accuracy from \(S_{b\max}\)); we determine

\[ k=\frac{v_{om}}{v_{yd}}. \]

1. By planimetering the areas of the impulses \(S_x\) and \(S_p\) in two or three trials, we check the validity of relation (1), i.e.

\[ k=\frac{S_x}{S_p} \]

and establish the final position of the line \(BB\), which can subsequently be taken as the “zero” reference line, i.e. allowing for the counterpressure of the air on the other side of the striker; in most cases, within the limits of experimental error (i.e. 8–9%), such a construction is sufficiently accurate.

The position of the line \(BB\) can be found in two ways:

1) by specifying the position of an arbitrary line \(B_1B_1\) and determining, by planimetering, the values of the areas \(S'_x\) and \(S'_p\);

2) then we find the distance \(y\) from the relation:

\[ \frac{S_x}{S_p}=\frac{S'_x-y t_x}{S'_p-y t_p}=k \]

and establish the final position of the line \(BB\), as well as the values of the impulse areas \(S_x\) and \(S_p\) (Fig. 3).

1. On the basis of relations (3) we determine the mean values of the pressures \(P_x\), \(P_p\), \(P_m\) and the calibration scale \(P_c\), and consequently also \(P_{\max}\) and \(P_{\min}\).

The computations are simplified by the use of the nomogram (Fig. 5). If a calibrated pressure curve \(P=f(t)\) is available, this procedure, with a known approximation, can be used to estimate the error in calibration of the sensor and, consequently, to carry out a check. It is appropriate to note here that

Fig. 5

the pressure curve can be reproduced from the graph of the motion of the striker \(S_b\)—by twice performing graphical differentiation, i.e., by finding the acceleration curve, see [1], and, from it and the known value of the unit weight of the striker \(q\), computing the pressure \(P\). According to our verification calculations, the error here too is of the order of 9–10% (the greatest error being at the pressure “peak” \(P_{\max}\)), but the amount of computational work by this method increases many times over.

1. Having the curves \(s_b\) and \(s_p\), as well as the expression \(P=f(t)\), it is not difficult to construct indicator diagrams:

a) for the piston stroke — \(P=f_1(s_p)\);
b) for the striker stroke — \(P=f_2(s_b)\);
c) for the change in the volume of the air cushion — \(P=f_3(l)\).

The indicated diagrams are presented in Fig. 6; they show the action of the air coupling on the piston \(A_p\), the striker \(A_b\), and give the magnitude of the compressor losses (efficiency of the air coupling on the order of \(\simeq 71\%\)); from the last diagram one can, in a first approximation, determine the average values of the polytropic exponents: for compression \(m_{\text{comp}}=1.05\), for expansion \(m_{\text{exp}}=0.92\).

Fig. 6.

From Fig. 6 it is evident that the curves are not closed; consequently, the course of the process in the air linkage, strictly speaking, is not polytropic, since the weight quantity of air between the striker and the piston is not constant, owing to gas leaks through the compensating openings and through the clearances between the cylinder and the striker, as well as between the cylinder and the piston.

1. Knowing the efficiency of the machine $\eta$ and the coefficient of utilization of its time

\[ \eta_{vr}=\frac{t_p}{T}, \]

according to the literature data on the study of forging hammers [6; 23], we determine the power coefficient

\[ R=\eta\cdot\frac{t_p}{T}; \]

this coefficient characterizes the effectiveness of the machine’s operation. In our example,

\[ \frac{t_p}{T}=0.18 \]

(which is close to the value

\[ k=\frac{v_{om}}{v_{ud}}=0.17 \]

).

The consumed electric-current power is $2.53\ \text{kW}$ and the efficiency is

\[ \eta=\frac{A_b\cdot z/60\cdot 9.8}{N}=0.22, \]

therefore:

\[ R=\eta\cdot\frac{t_p}{T}=0.22\cdot 0.18=0.04=4\%. \]

1. The variation of a number of quantities over the cycle period is represented in dimensionless coordinates in Fig. 7. The study of a family of similar curves will make it possible to determine the limits of the most probable values of the variation of the quantities of interest to us over fractions of the cycle.

Fig. 7

The indicated relations are valid for electropneumatic hammers with free motion of the striker, both for the case of one-sided and two-sided air linkage; they are also valid for pneumatic hammers. The two straight lines $AA$ and $BB$ are the principal coordinate axes for the force diagram.

In the case where there is only one curve of the change in pressure (or force) as a function of time, and the value of the coefficient of restitution is also known,

\[ k=\frac{v_{om}}{v_{y\partial}}, \]

the position of the axes \(AA\) and \(BB\) can be determined on the basis of two relations:

\[ S_x = k \cdot S_p, \tag{4} \]

and

\[ S_x \cdot tc_x = S_p \cdot tc_p = m \cdot S_{b\,max}, \tag{6} \]

where \(tc_x\) and \(tc_p\) are the distances of the centers of gravity of the impulse areas (see Fig. 3) from the beginning and the end of the cycle.

Relation (6) is a consequence of the theorem on the displacement of a mass during the action of a force, proved by B. V. Sudnishnikov [4] and applied by the author, together with N. N. Esin, to the study of pneumatic hammers [5]. On the basis of relation (6), it is possible, from a graphically specified force impulse, to calculate directly the maximum stroke of the striker \(S_{b\,max}\), as well as the approach of the colliding bodies during the impact.

LITERATURE

1. P. M. Alabuzhev and I. P. Yudin. Experimental study of an electropneumatic hammer. Proceedings of the Mining-Geological Institute, issue 8, WSB AS USSR, Novosibirsk, 1950.

2. P. M. Alabuzhev and O. D. Alimov. On a method for determining the energy of an impact on the striker. Published in the present volume of Izvestiya TPI.

3. L. G. Loitsyanskii and A. I. Lurie. Course of Theoretical Mechanics, part II, Gostekhizdat, Moscow, 1948.

4. B. V. Sudnishnikov. The theorem on the displacement of a mass during the action of a force. Proceedings of the Mining-Geological Institute, issue 8, WSB AS USSR, Novosibirsk, 1950.

5. B. V. Sudnishnikov and N. N. Esin. On the methodology for studying pneumatic hammers. Proceedings of the Mining-Geological Institute, issue 8, WSB AS USSR, Novosibirsk, 1950.

6. V. F. Shcheglov. Study of forge hammers with valve distribution. Izvestiya of the Tomsk Industrial Institute named after S. M. Kirov, vol. 57, issue III, 1939.