Proceedings of the Tomsk Technological Institute of Emperor Nicholas II.
Vol. XXVII, 1912, No. 3.

Study of Pneumatic Hammers.

(After Möller).

S. K. Konyukhov.

(With 1 table).

In recent times, small pneumatic tools—hammers, chisels, riveters, chippers, etc.—have become widely used in the technology of plants and factories. Comparatively light and convenient to handle, they successfully perform many tasks, in the sense of handiness and economy in the expenditure of concentrated energy, that are often done by hand; and therefore it is not surprising that interest in them is growing with every day. One may say without exaggeration that not a month passes without a new type appearing in technical journals and patent books, intended to eliminate the shortcomings inherent in tools that appeared earlier and are in use.

The new pneumatic tools have, in their basic idea, much in common with perforators, which are successfully used in mining and construction. From the mechanical point of view these tools are interesting in that in them the piston moves under the influence not of a rigid body (at the moment of change of stroke), such as, for example, the connecting rod in piston machines, but of an elastic medium (in particular, compressed air). It is true that the use of an elastic medium as an intermediate link in the kinematic chain is not a great novelty, since steam, air, and gas hammers are built precisely on this principle; nevertheless, small pneumatic tools have their own specific features, which compel us to separate them into a special group. First, these tools are of a more delicate construction; secondly, they very often make up to 1500 strokes per minute; thirdly, they are intended for the most part to operate upon bodies of definite resistance, such as iron, steel, copper, and upon areas of a definite size. To apply to them the method of calculation for steam and air hammers is already unsuitable for more than one reason.

because forging tools (hammers) are calculated too roughly. Thus, for example, the piston area in steam hammers of small weight sometimes exceeds the theoretical one by a factor of 5, and exceeding it by two or three times is a completely ordinary occurrence. There is no particular need to go into the causes of such imperfect calculation of hammers in detail; in brief, it all comes down to the following: 1) the elasticity of steam in boilers is not a constant quantity, 2) it may become necessary to forge objects of somewhat larger size in form and weight than was assumed in calculating the hammer, 3) it is often necessary to increase the number of blows per unit time, etc. In small pneumatic tools such deviations can almost not be made, and therefore their calculation must be carried out more accurately, so that the tools operate economically and justify their purpose.

The Germans and the French have worked most on the investigation of pneumatic tools. As to whom the right of priority belongs, the question remains open. The point is that Möller’s doctoral dissertation and the work of the engineer Baril appeared almost simultaneously. It is true that Möller completed his work, which concerned the investigation of only two or three types of pneumatic hammers, while Baril, having taken upon himself the more tedious task of investigating and describing pneumatic tools in their entirety, continues to publish his works in the Revue de mécanique to this day. Their methods of investigation are different, but their conclusions are in many respects similar. In the present article we shall touch only on Möller’s work, and, in view of its extensiveness and unfinished state, we shall leave Baril’s work aside.

The best and most graphic way to obtain a picture of the process of air operation is, of course, by means of an indicator diagram. But here we encounter the following obstacle: even the optical indicator of O. Schulze, designed for 1500 strokes, cannot give reliable results for the following reasons. The volume of air operating during one stroke is not large, whereas the volume of air in the supply hoses—or, more precisely, in the hoses themselves—is rather considerable. The presence of these hoses, at present directly unavoidable, significantly increases the harmful space. Movable, flexible hoses leading from the hammer to the stationary indicator are especially difficult to eliminate.

It was precisely these considerations that led P. Möller to choose the “graphical-numerical method” for the investigation of pneumatic hammers.

Strictly speaking, this is not one method of investigation, but two: the graphical and the numerical, the method of graphical analysis and the experimental method. The work is based on the investigation of the following questions: 1) the study of the nature of the change of pressure inside pneumatic tools, 2) the determination of the piston velocities, and 3) the determination of the work delivered during the stroke, in kilogram-meters. The numerical data obtained from experiment serve to verify the accuracy of the calculations by the graphical method. Thus, for example, on the basis of the diagram one can find the air consumption, the number of strokes or blows.

The same data can also be obtained experimentally. If there is sufficiently good agreement of the results, it means that the graphical method of investigation has been chosen successfully. It is necessary to resort to control comparisons of this kind fairly often; moreover, it is very important that the investigations be conducted independently and, if possible, by different persons; otherwise, enthusiasm for the conclusions obtained by one method of investigation may influence the nature of the conclusions of the other method, tendentiously directing the experimenter along one course in order to obtain already known conclusions. Professor Jouguet, one of the most talented, in my opinion, and most original thermodynamicists of our time, made excellent use of the method of control in the investigation of the balance of a steam boiler.

One of the first tasks in the study of pneumatic hammers is to clarify the old and incorrect assumption that, at the beginning of admission and the beginning of exhaust of the working medium (gas, air, steam) into the working cylinder, an instantaneous equalization of pressure occurs. Thus, for example, in calculating steam hammers it is assumed, according to Tauer, that after the upper or lower part of the cylinder, filled with steam, is connected with the atmosphere through buffer or any other exhaust passages, atmospheric pressure is established instantaneously in the steam space of the hammer. When the question concerns rough, approximate calculations, perhaps there is no point in a detailed study of the circumstances of the motion and of the phenomena accompanying them; but in the delicate tools of our time, an exact accounting of phenomena in design and calculation is the best guarantee of the success of the machine-tool in operation.

Let us draw attention to the fact that in pneumatic hammers, operating at 6–7 atmospheres of pressure in the supply hoses, the air velocities are very high, and the openings for the admission of the working…

of air and for the exit of the exhaust from the hammer are very small. It is these inlet and outlet velocities that determine the degree of equalization of pressure. It is therefore impossible to be satisfied with drawing on the diagram a vertical line indicating instantaneous equalization of pressure: the equalization of pressure proceeds not by a jump, but along a curve. Thus, the immediate problem takes the following form: it is necessary to introduce into the study of the process of work in pneumatic hammers the velocities of the incoming and discharged air. Further, one may already speak of the direction of the pressure curve when admission and exhaust are coupled with compression and expansion. There is no particular need to explain that here we shall throughout assume operations on dry air, as a permanent gas.

Let us suppose that we have a cylinder with a piston, on one side of which there is air of volume \(V\) at pressure \(p\). If the piston moves to the right into an infinitely close position, the volume of the air will change by \(dV\). Since the left-hand part of the cylinder is in communication, during the motion of the piston, with the air reservoir feeding the tool, then from the latter, owing to the fall of pressure in the cylinder, a new volume of air at pressure \(p'\) will pass into the cylinder. Thus, when the piston is displaced to the right by \(dS\), there will occur the change

\[ V \text{ into } V + dV \]

\[ p \text{ into } p + dp \]

When the question concerns the outflow of air into a volume \(dV'\) at pressure \(p'\), then \(dV'\) must be regarded as a negative quantity. However, we shall direct our attention not to volumes, but to weights. Using the general equation of thermodynamics \(pV = RT\), we write

\[ \tag{1} \frac{(V+dV)(p+dp)}{R(T+dT)}=\frac{Vp}{RT}+\frac{p'dV'}{RT} \]

But in the presence of a change of state of the air along an isotherm, \(dT=0\). Then equation (1) is rewritten in the simpler form

\[ \tag{2} (V+dV)(p+dp)=Vp+p'dV' \]

Or, neglecting infinitesimals of the second order,

\[ \tag{3} p\,dV+V\,dp=p'dV' \]

\[ \tag{4} \text{whence}\qquad dp=\frac{p'dV'}{V}-\frac{p\,dV}{V} \]

Graphically this derivation may be represented as follows: it is known that in an orthogonal system the geometric locus of one of the

from the vertices of equal-area rectangles is the rectangular hyperbola

\[ XY=C \]

Thus we shall also transform each of the terms of equation (4) into a rectangle, one side of which is \(V\). The method of transformation, well known from elementary geometry, consists in the following: suppose that we need to transform the rectangle \(a \times b\) into the rectangle \(c \times d = a \times b\).

\[ AD=a;\quad AB=b;\quad AH=d. \]

Lay off from \(A\) to the right (Fig. 2), along the line \(AD\), the segment \(AH\) equal to \(d\), with \(AB=b,\ AD=a\); then \(AG\) will be equal to \(c\).

In our case we proceed in this way. We draw the orthogonal system of coordinates \(XOY\) (Fig. 3) and lay off, to some scale, \(p-p'-dv\) and \(dV\). Then

\[ 0-3=1-9=p;\quad 0-4=2-9=p';\quad 3-1=0-9=V; \]

\[ 0-5=dV;\quad 0-6=dV'. \]

Fig. 1.

Fig. 2.

Fig. 3.

It is not difficult to see that

\[ 8-9:dV=p:V \quad \text{or} \quad 8-9=\frac{p\,dV}{V} \quad \text{and} \quad 7-9=\frac{p' dV'}{V} \]

Hence the absolute magnitude \(dp\) will be represented graphically by the segment \(7-8\),

\[ 7-8=dp=\frac{p' dV'}{V}-\frac{p\,dV}{V}. \]

Further, it is of interest to determine the magnitude \(dV'\) of the inflowing air. A priori one may say that this quantity is directly proportional to the cross-section \(f\), to the infinitely small interval of time \(dt\), during which the piston traverses the path \(dS=dV:F\) (\(F\) is the area of the piston), and to the velocity \(U\) in the inlet orifice. Thus,

\[ dV'=f\cdot U\cdot dt \tag{5} \]

For a certain completeness and clarity of the phenomenon under study, it is necessary to remember that the volume \(dV'\) is measured at the pressure \(p_a\) in the receiver. With a small difference in the pressures between \(p_e\) (the supply space) and \(p_a\), one may put \(p_a=p_o\) (the pressure in the plane of the smallest cross-section of the jet). When, however, the difference between \(p_e\) and \(p_a\) is large, then

\[ p_o=p_e\left(\frac{2}{n+1}\right)^{\frac{n}{n-1}} \quad \text{or} \quad p_e=p_o\left(\frac{n+1}{2}\right)^{\frac{n}{n-1}} \tag{6} \]

The last two expressions represent the assumption, or hypothesis, of Saint-Venant and Wantzel, which gives a basis for the following conclusions: 1) the velocity of the air jet (in our case) in the plane of the smallest cross-section is equal to the velocity of sound in a medium whose state is identical with the state of the elastic fluid in this plane \((p_o V_o T_o)\); 2) the quantity of gas-like fluid flowing out through the orifice does not depend on the pressure \(p_a\) in the receiver, but is determined only by the pressure \(p_e\) in the source or supply vessel; 3) the pressure \(p_o\) has the value at which the function determining the quantity of fluid flowing out through the orifice passes through a maximum (see Rerikh, Theoretical Justification of the Hypothesis of Saint-Venant and Wantzel). The experimental justification of the hypothesis of Saint-Venant and Wantzel is a long-known fact. There is a deficiency in the theoretical justifications as well, but Rerikh performed the task of theoretically substantiating the hypothesis that interests us better than others and more successfully in the sense of depth of analysis.

Study of Pneumatic Hammers

hypotheses of K. Rerikh. (Proceedings of the St. Petersburg Polytechnic Institute, 1904).

If the ratio \(p_e : p_o\) does not exceed a certain critical value, then the quantity of air passing through the nozzle per unit time, at pressure \(p_o\) or \(p_a\), will have one and the same maximum value, even if the pressure ratio continues to fall. The law of continuity of outflow, expressed by the equality

\[ dV' = f \cdot u \cdot dt \tag{7} \]

continues to hold, and only \(dV\) and \(U\) change; moreover, \(dV\) is measured at the pressure in the receiver. Thus one may write

\[ dV'' = f u' \cdot dt. \tag{8} \]

The quantity \(U'\) has a certain constant value, independent of elasticity; namely, this will be the velocity of the air leaving the feeder into the intermediate pipe, whereas \(U\) was the velocity of the air leaving the intermediate pipe into the receiver. Using Zeuner’s formula, one can find the velocity of outflow

\[ U = 44{,}4 \sqrt{T\left[1 - \left(\frac{p_e}{p_o}\right)^{\frac{n-1}{n}}\right]} \tag{9} \]

Putting \(t = 16^\circ\), we obtain

\[ U = 754{,}8 \sqrt{1 - \left(\frac{p_e}{p_o}\right)^{\frac{n-1}{n}}} \tag{10} \]

Here \(n\) denotes the exponent of outflow, whose value is given by experiments. If the pipe nozzles have the form of short cylinders, then \(n = 1{,}286\). A somewhat different value is given by Weisbach, according to whom \(n = 1{,}243\text{–}1{,}252\). Thus, as the mean value for \(n\), one may take

\[ n_{\mathrm{av}} = \frac{1{,}286 + 1{,}243 + 1{,}252}{3} = 1{,}260 \tag{11} \]

Further. If one takes into account a certain resistance to the outflow on the part of the nozzle itself, then the outflow exponent must be reduced somewhat, \(n = 1{,}25\). For this value the critical pressure ratio will be

\[ p_e : p_o = \left(\frac{2}{n + 1}\right)^{\frac{n}{n-1}} = 0{,}555 \tag{12} \]

In the case when \(\dfrac{p_o}{p_e} \leqslant 0.555\), the velocity \(U\) assumes the maximum value

\[ U_{\max}=754.8\sqrt{\frac{n-1}{n+1}}=252\ \mathrm{mtr/sec}. \]

For the critical pressure it is characteristic that the weight of the outflowing air, whether we calculate it by the formula

\[ dV'=f u\,dt \]

or by the formula

\[ dV''=f\cdot u'\cdot dt, \]

will be one and the same. Thus, for an isothermal change we shall have

\[ \tag{13} p_e\,dV'=p_o\,dV'' \]

or

\[ \tag{14} U_{\max}p_e=U'\cdot p_o, \]

whence

\[ U'=U_{\max}\frac{p_o}{p_e}=252\cdot 0.555=139.86\ \mathrm{mtr/sec}. \]

If the passages through which the flow of air takes place have bends, then it is better to take smaller values for the velocities, namely, to introduce a coefficient of proportionality \(\alpha=0.9\).

Let us denote the velocity of the piston at a given instant by \(V\), and the path by \(ds\); then

\[ \tag{15} dt=\frac{ds}{V} \]

Let the mass of the piston be \(m\), its area \(F\), the mean pressure \(p_m\), and \(S\) the path traversed. The theorem of live forces gives us the right to write

\[ \tag{16} \frac{mV^2}{2}=p_mF\cdot S;\qquad V=\sqrt{\frac{2p_mFS}{m}} \]

Since the mean value \(p_m\) can be obtained from the diagram, the magnitude of the velocity at a given instant is determined without difficulty. Here the friction of the piston against the cylinder wall has not yet been taken into account. Usually po-

act as follows: decrease or increase the corresponding pressure, taking the coefficient of friction as \(0.15\).

Subsequently it is necessary to calculate the velocity for some position of the piston and to find the time \(dt\), on the assumption that over the path \(ds\) the velocity is constant. After this, the quantity of air flowing out will be found, and the pressure \(p+dp\) for the piston position \(S+dS\) will be determined graphically. Thus, on the diagram there will be found the point \(S+ds\) \((V+dV)\), for which the velocity is

\[ V=\sqrt{\,2p_m\frac{F(S+ds)}{m}\,} \tag{17} \]

The stroke intervals are taken as \(0.5\)—\(1\)—\(2\) millimeters, depending on the character of the change in velocity.

Attempts were made to pass from small intervals to differentials, in order to represent the pressure curve analytically, but the differential equation thereby obtained proved not to admit of solution. It was necessary to determine the time of rise graphically, \(\Sigma dt=t\).

P. Möller’s investigations concerned primarily pneumatic hammers without a distributing valve, of the Collet & Engelhardt system. As can be seen from drawing 4, the piston has shoulders and a neck in the middle. The space between the inner surface of the cylinder and the surface of the neck is the distributing chamber. The compressed air entering this chamber exerts pressure on the annular area and continuously pushes the piston toward the handle. As soon as the upper edge of the recess (the neck) opens the connecting channel during the motion, the compressed air rushes into the rear part of the cylinder and exerts pressure on the entire area of the piston. As a result, a downward stroke is observed.

The drawing of the diagram of the working process of a pneumatic hammer is preceded by establishing the magnitude of the pressure of the compressed air at entry into the distributing chamber. Compressed air is usually supplied to the hammer through a flexible hose of small internal diameter. It is therefore clear that measuring the pressure in the hose is irrational: the losses due to friction against the walls and resistance in the bends are large. On the other hand, it is directly impossible to determine, before the hammer, the losses of elasticity of the air at each given moment, because of the large number of oscillations. It is therefore simpler to stop at determining experimentally the greatest and the least loss of elasticity. In order to

To carry out this task, flexible hoses of various diameters and lengths are taken and connected to the hammer. The manometric readings at the hammer itself and before entry into the hammer give quite clear indications in this respect. But as soon as the hammer begins to work, the pointer of the manometer at the hammer immediately begins to oscillate. Although these oscillations of the pointer over the dial are very rapid, nevertheless the extreme positions of the indicator can still be distinguished with the naked eye. (Here it would have been advisable to make use of cinematographic photographs.) The whole difficulty, however, is that these readings cannot be given great confidence. Owing to the rapid motion of the pointer, motion by inertia plays a large role here. It is often observed that the pressure readings by the pointer exceed the prevailing pressure in the hose. This excess of pressure may be taken, with a sufficient degree of accuracy, as the average measure of the error in the manometer readings. Thus, if the highest pressure according to the manometer is \(p_{max}\), and in the air main \(p\), then \(p_{max} - p\) will be the error in the direction of increased pressure. But the same deviation will also be in the direction of the lowest pressure. In the experiments it was found, for example, that \(p_{max} = 6.3\), while \(p = 6\). Hence \(0.3\) also represents the excess of pressure. The minimum pressure is \(p_{min} = 4.8\), and therefore actually \(5.1\).

For the first experiments a pneumatic hammer was taken, the short hose to which (and, together with it, the hammer nipple) had an internal diameter of 10 millimeters and a length of 400 millimeters. At the free end a manometer was placed; from the latter the test hose went to the air main. The results of the experiments are concentrated in the following table.

Table 1.

New test hoses. Diameter, in millim.	New test hoses. Length, in meters	Pressure in the air main	Deflection of manometer readings. Highest, atm.	Deflection of manometer readings. Lowest, atm.	Mean pressure before the hammer
22	6	6	6.30	5.2	5.75
13	4	5.9	6.30	4.8	5.55
10	5.2	5.8	6.00	4.3	5.15
10	10.2	5.8	5.50	3.7	4.60

From the experiments it is clear that the drop of pressure from hoses of different length and diameter is quite considerable. Therefore one should not

one must use supply hoses from the air line to the hammer that are neither too wide nor too narrow. The most suitable proved to be a hose with a diameter of 13 millimeters and a length of 6 meters. Of course, in each particular case these quantities must always be determined experimentally, so as not to waste a store of expensive energy needlessly. Nor should one place much confidence in the particularly advertising character of the firms’ guarantees. One’s own experiments are more reliable and more faithful than any guarantees, and finding, by means of them, the most advantageous conditions for operation will always be rewarded with a large surplus.

The decrease in pressure in the elasticity of the air amounted, before the hammer, to 12%–15%. The mean working pressure was \(6 \times 0.85 = 5.1\). Here the pressure fluctuations immediately before the hammer have not been taken into account. For the pressure falling on the annular area in the hammer piston, this figure will be low, owing to the fact that, with the inlet closed, the air in the hoses is almost at rest. As the mean pressure for the annular area, the arithmetic mean \(6 + 5.1 : 2 = 5.55\) is taken. In the diagram (Fig. 5), the inlet channels \((E)\) and the exhaust channels \((A)\) are shown by circles at natural size. The distance of the origin of coordinates is found by dividing the volume of the cylinder by its cross-section. The pressure on the annular area of the piston, taken as constant, is indicated in the drawing by a thick line. The thin horizontal lines give the critical pressures. The piston diameters taken into account were 30–32 millimeters, and the weight, determined by weighing, was 0.58 kilogram. Thus the mass of the piston is

\[ m = 0.58 : 9.81 = 0.0592 = 2 \cdot 0.0296 \frac{\text{kil. sec}^2}{\text{m}} . \]

The friction and weight of the piston are taken into account here, assuming the coefficient of friction to be 0.15. The hammer itself must have an unchanging direction at an angle of 30° to the surface of the object being worked. At the beginning of the experiment the end of the chisel touches the surface being worked, and from this position begins to move upward. As soon as the inlet channel at the hammer handle is opened, the pressure is distributed over the annular area, and already at the very beginning of the motion the pressure may be considered full. On the opposite side of the piston, at this time, the exhaust valve is open, so that atmospheric pressure is established there. The longer and less completely the exhaust openings are closed, the more noticeable are the phenomena simultaneous with exhaust and compression (the part of curve I from \(a\) to \(b\)).

Pure compression (between point $b$ and point $c$) lasts a shorter time, since admission begins immediately, and then there follows a period of simultaneous admission and compression (from $c$ to $d$). The piston changes its direction of motion when the area of the diagram on its upper side becomes approximately equal to the rectangle equal in magnitude to the annular area. On this return path (between $d$ and $e$), admission and expansion predominate. In order to construct on the diagram the rightward-falling part of the curve from point $d$ to $e$, one adopts, for an arbitrary path at one’s discretion, some fall of pressure and computes the subsequent course of the curve by the method set forth on p. 7. In general a zigzag line is obtained, which can easily be replaced by a straight line. This method is quite sufficiently accurate for practical purposes, and it is simpler than, for example, Blass’s method (see Z. d. V. d. Ing. 1905). As soon as admission is closed, a short pure expansion begins (between points $e$ and $f$), lasting until the exhaust passage is opened. Then follows a period in which expansion and exhaust predominate (between points $f$ and $d$), and this continues until the moment when the piston meets the tool. At this moment a drop in pressure occurs (the time of impact). To take this circumstance into account is difficult, owing to the short duration of the impact, and the drop in pressure itself is small. Hence it is clear that the bases for calculating the duration of the impact are insufficient.

In parallel with the calculation of the individual points of the pressure curve, the piston velocities are found; in the diagram (Fig. 5) this is represented by curve I. As soon as the piston has touched the blunt end of the tool, it rebounds backward, but with a lower velocity. During the return stroke of the piston there is observed a period of compression and exhaust of the compressed air remaining in the cylinder (curve II). Then follows a short period of compression, then admission and compression simultaneously, and so on.

For the velocities, curve II (Fig. 5) has been plotted; it happens to be partly overlapped by curves I. The process can, of course, be traced further as well; however, it is clear that after a certain interval of time there is a repetition of the phenomena, so that the subsequent curves do not differ from the preceding ones (the lower part of Fig. 5).

Concerning the velocity of the piston on the return stroke immediately after the blow, the following must be said: this velocity depends entirely on the shape and material of the bodies colliding, on the velocity of impact, and on the resistance of the air. The more elastic the colliding bodies and the backing, the greater the velocity, which follows directly from the theory

impact. In view of the fact that the material being worked has such a great influence on the rebound, it is urgently necessary to construct differently chisels, riveting hammers, and sand stamps (dies). This circumstance has not escaped the manufacturers engaged in the production of stamping tools and instruments.

The influence of rebound in power hammers, the so-called bouncing of the ram, has often been observed, but no conclusions were drawn from it regarding the actual processes inside the hammers. Thus, for example, Professor Lindner (Z. d. V. d. Ing. 1902, p. 37) gives the diagram of a steam hammer and, from his observations in connection with experimental data of other persons, concludes: with strong blows the ram, it seems, quickly rebounds backward, after which, over a path of about 150 millimeters, a slowing of the motion is observed, and then the velocity increases. If the segment in the ascending line (representing the path of the piston) is not the result of vibrations of the experimental apparatus, then it indicates that, when the ram strikes the piece of wood placed beneath it, it rebounds with a greater velocity than that imparted to it by the steam pressure, and only after this does it move solely under the steam pressure. Something similar is also found in the work of Lüdicke (Z. d. V. d. Ing. 1900, p. 1787), where a diagram is given of an air-spring hammer. Here the suction vacuum, extending over two thirds of the stroke, is so small that the piston cannot move under the influence of this cause alone; the ram must therefore, upon striking a hard steel hoop, rebound so strongly backward that it needs lifting force only in the last third of its stroke.”

On the basis of these considerations it was quite natural to suppose that rebound plays a large role also in pneumatic hammers. Thus there arose a new problem: to obtain an idea of the magnitude of the rebound. Möller undertook the study of this question on cutting hammers of two firms, Collet & Engelhardt and Pokorny & Wittekind. The experiments had to be conducted by imitating the operation of pneumatic hammers in reality. In particular this concerned the guides of the cutter in the bushing and the position of the cutter itself with respect to the object being worked. The piston was made to fall freely, taking as the initial point that height of fall for which the velocity followed from the first line, not subjected to the influence of rebound. (Curve I, Fig. 5).

The experiments at Pokorny & Wittekind were carried out in the following manner: a heavy piece of old boiler iron was taken (Fig. 6) and fastened to the wall at an angle of 60°, which corresponded

most commonly used in practice for cutting hammers. Under the channel iron there was placed a cast-iron pad in the form of a verb, to which the beam being worked was attached by bolts, so that it formed, as it were, a single whole. On one flange of the channel iron (7 millimeters thick) a notch was made, so that the removal of the chip by the chisel began precisely from this place. The chisel was guided by a firmly fastened bushing, as is done in hammers. Although it had only just been sharpened, it was nevertheless considered quite suitable for the work. The blade was arc-shaped, 20 millimeters wide. The upper surface of the chisel was set with a spirit level. As for the piston, it was raised so that it hung vertically above the chisel. Between the chisel and the piston a piece of cardboard with a round hole was fixed, so that the piston could conveniently fall through this hole. However, when rebounding it never entered exactly into the hole, but invariably struck the cardboard. It was then determined experimentally at what height the cardboard had to be placed above the chisel so that the piston, upon rebounding, only just touched the cardboard. Light bodies were placed on the latter, the slight oscillation of which also indicated contact. In order that the piston should fall as nearly vertically as possible, the thread on which it was suspended was burned through each time.

From the experiments it was possible to calculate the velocity with which the piston rebounded, on the basis of the consideration that the velocities of falling are related to one another as the square roots of the heights. If, therefore, the height of fall was \(a\), and the height of rise \(b\), then the rebound velocity is obtained by multiplying the velocity of impact by \(\sqrt{\frac{b}{a}}\).

The following table gives the results of the experiments.

Table II.

Height of lifting \(a\), in meters.	Height of rebound \(b\), in meters.	Reciprocal value of the coefficient of rebound \(\sqrt{\frac{a}{b}}\)
1.80	0.110	4.05
1.80	0.100	4.05
2.00	0.125	4.00
2.00	0.125	4.00
2.25	0.120	4.34

The last line reveals a certain deviation from the previously obtained values of the coefficients of restitution, which led to

…thought to check the upper face of the cutter by means of a level. It turned out that it had been brought out of the horizontal position. The face was checked, and the experiments continued. Their results are seen from the following Table III.

Height of fall \(a\).	Height of rebound \(b\)	\(\sqrt{\dfrac{a}{b}}\)
2,00	0,130	3,94	From this moment the cutter was checked after each blow.
2,25	0,135	4,08	From this moment the cutter was checked after each blow.
1,50	0,116	3,60	From this moment the cutter was checked after each blow.
1,80	0,130	3,72	From this moment the cutter was checked after each blow.
2,00	0,138	3,81	From this moment the cutter was checked after each blow.

The following experiments were carried out at Collet & Engelhardt with a cast-iron block weighing 8 kilograms and placed on the concrete floor of the workshop. The guide bushing for the cutter was fastened into the groove of a cast-iron beam supported by two iron bars. Wooden wedges served for shifting the bars. A roller was fixed on the beam, over which a cord was passed, and the piston was suspended from the cord. When the free end of the cord was released, the piston fell directly onto the cutter. The height of rebound was set, as in the previous experiments, only in order to save time; it was not measured each time at what height the cartridge should stand, but they confined themselves to observing whether the piston touched the paper, or whether it remained below or above the latter. As before, the cutter was taken from the work and was of medium sharpness.

Table IV.

Height of fall \(a\), in meters.	Set rebound height \(b\), in meters.	Lifting height of the piston.	Reciprocal value of the coefficient of rebound \(\sqrt{\dfrac{a}{b}}\)
2,00	0,150	more guaranteed by 3–5 millim.	3,65
2,00	0,155	\(>\)	3,59
2,00	0,160	\(>\)	3,54
2,00	0,160	less by a millim.	3,54
2,00	0,155	\(>\)	3,59
1,80	0,160	a little more.	3,35
1,80	0,165	a little more.	3,30
1,80	0,170	more.	3,26
1,80	0,180	more.	3,16

Since from these experiments it seemed to follow that the height to which the piston rebounds is directly proportional to the depth of penetration of the cutter or chisel, a series of experiments was carried out to determine how many blows must be made in order for the piston to rebound to a definite height. In these experiments the chisel was driven in with a hand hammer. With one fall of the piston from a height of 1.5 meters, it rebounded to a height greater than 0.190 meter. However, upon inspection it was found that the chisel had a broken edge. Here are the data of the experiment.

Table V.

Height of fall \(a\), in meters.	Established height of rebound \(b\), in meters.	Number of blows.
1.80	0.175	6
1.80	0.190	6+3
1.50	0.190	6+3+1

The following experiments were carried out at Collet & Engelhardt; in them the arrangement was adopted in the same way as before. For the test, strip iron 8 millimeters thick was taken, which was clamped between two iron bars. An iron beam placed on trestles served as the backing. For the test two chisels were taken: a sharp one and a blunt one. The test bar was again placed obliquely to the blade of the chisel. The results of the experiments are set out in the following two tables. From these experiments it follows that the rebound is greater the more deeply the cutter penetrates the metal, the blunter the edge is, and the less accurately the lower surface of the piston strikes the upper surface of the cutter, and hence the smaller the work of cutting.

A) Blunt cutter. Table VI.

Height of fall \(a\), in meters.	Established height of rebound \(b\).	Height of rise of the piston.	Reciprocal value of the rebound coefficient \(\sqrt{\dfrac{a}{b}}\)
1.80	0.125	Much higher.	—
1.80	0.140	Higher by 3–5 millim.	3.59
1.80	0.150	Somewhat higher < 3 millim.	3.47
1.80	0.160	Contact.	3.35
1.80	0.165	Somewhat higher.	3.31
1.80	0.180	Somewhat higher.	3.16
1.80	0.200	Lower by exactly 10 millim.	3.00
1.80	0.190	Contact.	3.08

of a theoretical consideration to experiment II. Möller had no possibility, nor did he have the figures obtained. However, the reliability of the conclusions obtained is attested by the excellent agreement of the figures with the experimental data advertised by the firm itself. Thus the firm gives the number of blows as 1400 per minute, and the air consumption as 0.25–0.50 cubic meter.

Turning then to the diagram in Fig. 15, one may observe that here it has also been investigated what smallest length the cylinder must have in order to ensure reliable operation. Here the most unfavorable case may occur when there is no rebound, and after the spool has changed its position, the inlet valve at the handle of the hammer is closed. With such a state of affairs, the air above the piston will be compressed, while below the piston it will at the same time expand and escape. From the external, indicative side, the process may be described in the following manner: the piston, touching the head of the cutter, occupies its lower position until such time as the pressure above the piston, owing to the discharge of air, falls to a pressure equal to the increasing pressure below the piston. As soon as such an equalization of pressures occurs, the piston begins to move. From this moment the pressure below the piston continues to rise, while above the piston it gradually approaches atmospheric pressure, as is also seen in Fig. 15 in the form of a dotted line. In this drawing, the dotted line rising from the left-hand side, which in the upper diagram represents the compression line for the upper side of the piston. In the lower diagram the dotted lines indicate velocities. It is not difficult to observe that the actual length of the cylinder \(l\) is not quite sufficient for the most unfavorable case, so that the piston may touch the cylinder cover, though with a small velocity.

This may conclude the exposition of the study of pneumatic hammers according to the method of P. Möller. There is no doubt whatever that this method may be applied with great benefit in the investigation and calculation of steam high-speed hammers, air hammers, and gas hammers, since it has an entirely general character. At the present time, for obtaining exact diagrams of high-speed hammers, the cinematograph is being used with extraordinary success. I do not intend to touch in detail here upon this new question, since the analysis of the application of the cinematograph to the study

the velocity gradually approaches zero. If, for this finite segment of the velocity curve, we assume that the pressure curve is a straight line (Fig. 7), then the time \(t\) is obtained from the relation

\[ \frac{mV^2}{2}=pFC \quad \text{and} \quad t=\int \frac{ds}{V} \]

\[ t=\frac{mS}{(p_0-p_1)F}\int_{p_1}^{p_0}\frac{dp}{\sqrt{p_0^2-p_1^2}} =\sqrt{\frac{mS}{(p_0-p_1)F}}\cdot\left(\frac{\pi}{2}-\operatorname{arc.\ sin}\frac{p_1}{p_0}\right) \tag{18} \]

When the pressure \(p=\mathrm{const.}\), then the pressure curve is parallel to the axis of abscissas and

\[ t=\frac{mV}{pF}; \qquad d\frac{mV^2}{2}=(l\cdot p)FS=pFV\,dt; \]

\[ m\,dV=pF\,dt; \qquad \int_0^{V'} m\,dV=\int_0^t pF\,dt; \qquad mV'=pFt; \qquad t=\frac{mV'}{pF}; \tag{19} \]

Thus, the dotted curves on the diagram (Fig. 5) have been obtained. They show that the piston requires \(27.54\sigma\) for the lift (where \(\sigma=0.001\) second); \(13.05\sigma\) for the working stroke from top to bottom, and in all \(40.59\sigma\) for the complete swing. Hence the required number of blows per minute is readily obtained:

\[ n=\frac{60.1000}{40.59}=1480 \]

Finally, the diagram gives the length of the piston stroke as 44 millimeters.

These quantities, obtained by the graphical-numerical method, had to be verified experimentally under normal working conditions, which was done at the Collet & Engelhardt works. It proved rather difficult to measure directly the work of the piston in its action. It was easier to determine experimentally the air consumption, the number of blows, and, with a known degree of accuracy, the magnitude of the lift.

The air consumption could best be measured in the following way: fill the largest possible air receiver with compressed air, then, having disconnected this receiver from the supply pipes, use the air for the needs of the hammer’s operation until the pressure has fallen to a definite value. P. Möller used for this purpose the ordinary Glover gas meters. Since instruments of this kind indicate the quantity of air at a temporary pressure of \(p\) absolute atmospheres, the calcul—

it should be noted that this assumption is arbitrary, but this arbitrariness is of the same character as the assumption that friction in a state of rest passes directly and suddenly into friction in a state of motion. Now the question naturally arises: how is this inaccuracy of the assumption, or rather this arbitrary assumption, reflected in the results of the investigation? It turns out that this apprehension is unnecessary. If an error has been made here in the calculations in the sense of increasing the pressure, then in the further course it disappears, because for the second, third, etc. lifts the use of formulas 20 and 21 is superfluous, since the piston, owing to the rebound, already has at the very beginning of the lift a certain velocity.

The matter is somewhat different with a piston valve, when the same formulas 20 and 21 are applied to it in order to find at what point of the piston stroke the change of position of the valve occurs. Since each rise or stroke of the piston here is independent of the following one, any admitted inaccuracy remains fixed in the process. In addition, other influences arise here which are difficult to take into account, such as, for example, the effect of leakage (with the narrowness of the closing surface of the piston valve this circumstance plays a more significant role than in the working piston), the formation under the lower surface of the valve of an air cushion, which results in the appearance under the valve now of compression, now of expansion, and finally the pressure acting on the hammer, which, although partly absorbed by the muscles of the hands, nevertheless causes vibration of the whole hammer. All these phenomena, difficult to take into account, compel one to be content with estimating by eye the time required for the piston valve to change its position. This operation is carried out as follows: let us suppose that, as soon as the distributing edges of the working piston have crossed the openings in the cylinder wall, the equalization of pressure for the distributing valve occurs in such a way that in one case full pressure suddenly comes above the valve, and in the other case likewise full pressure, prevailing in the working cylinder, suddenly comes. Then, from equation 19, one may calculate the time sufficient for the displacement of the valve. Multiplying this quantity by the average velocity of the working piston, we obtain the distance traveled by the piston up to the moment at which the change of position of the valve takes place. For simplicity it is assumed that the change of position of the valve occurred suddenly at the moment when the distributing valve half-opened the ca—

Table VIII.

1. Chisel formerly in use.

Hammer No.	Pressure in the air main \(p'\), atm.	Duration of the experiment, min.	Measured air consumption during the experiment, cu. meter	Measured air consumption, cu. m/min.	Calculated air consumption, cu. m/min.	Notes
861	5,90	1	0,350	0,350	0,402	2 readings, one after the other.
861	5,80	1	0,325	0,325	0,369	2 readings, one after the other.
664	5,75	1	0,325	0,325	0,366	2 readings, one after the other.
664	5,75	1,5	0,560	0,375	0,420
664	5,90—5,80	1	0,380	0,380	0,434
				average	0,398

Table IX.

2. Well-sharpened chisel.

Hammer No.	Pressure in the air main \(p'\), atm.	Duration of the experiment, min.	Measured air consumption during the experiment, cu. meter	Measured air consumption, cu. m/min.	Calculated air consumption, cu. m/min.	Notes
664	5,80	0,5	0,18	0,36	0,408	three readings, one after the other.
664	5,80	1	0,33	0,33	0,374	three readings, one after the other.
664	5,80	1,5	0,48	0,32	0,364	three readings, one after the other.
				average	0,382

Table X.

Hammer No.	Pressure in the mains, atmospheres	Number of blows per minute	Quality of the chisel
664	5,80	1496	Blunt.
664	5,80	1440	Sharp.
861	5,80	1284	Blunt.
861	5,80	1280	Sharp.

From the preceding, one readily observes good agreement between the experimental data and the results of the calculations, and this recommends the computational method favorably.

It is not difficult to obtain the air consumption, in cubic meters, per one kilogram-meter of work, reducing the quantity of air to atmospheric pressure:

\[ q=\frac{mV^2}{2}=0{,}0002703:1{,}11=0{,}000243\ \text{cu. meter}. \]

when the piston begins to move, and 2) what the pressure will be when the piston has traversed the path \(S\)? The first of these questions is answered without particular difficulty. The motion of the piston begins when the pressure \(p_0\) acting upon it balances its entire weight together with the force of friction (in the state of rest). Assuming further that the hammer is held at an angle of \(30^\circ\) to the object being worked, and that the coefficient of friction at rest is twice the coefficient of friction in motion \((2 \times 0.15)\), we obtain for the expression of \(p_0\) the following relation:

\[ p_0=\frac{0.277\ \mathrm{kg}}{0.25\pi \cdot 2.6^2\ \mathrm{sq.\ cm}}=0.0522\ \mathrm{atm}. \]

As soon as the hammer is set in motion, the friction decreases, and the pressure producing motion increases by the difference between the friction at rest and the friction in motion, \(p_0-p_r=0.0089\ \mathrm{atm}\).

In order to solve the second problem posed, concerning the magnitude of the pressure at the end of the path \(S\) traversed by the piston, it is necessary to adopt some probable assumption representing the dependence between the increase in pressure \(p\) and the path traversed \(S\).

Over a small interval, as the simplest such assumption, one may choose the linear dependence \(p:S=\mathrm{const}\). This assumption makes it possible subsequently to evaluate the integral

\[ \int \frac{dS}{V_x}. \]

If we now turn our attention (Fig. 15) to the path of the piston \(x\), after traversing which the piston acquires the velocity \(V_x\), then, using the familiar notation, we obtain

\[ V_x^2=\left(\frac{\Delta p\,x}{2S}+p_r\right)\frac{F_x}{m/2} \]

and from this we find the time for the path \(x\)

\[ t=\int \frac{dx}{V_x} = \sqrt{\frac{m}{F}} \int \frac{dx}{\sqrt{2p_r x+\Delta p\,\frac{x^2}{S}}}. \]

Extending the integration over the limits from \(0\) to \(S\), we shall have

\[ t= \sqrt{\frac{mS}{F\Delta p}} \left\{ \lg_n\left(p_r+\Delta p+\sqrt{\Delta p}\sqrt{2p_r+\Delta p}\right) - \lg_n p_r \right\} \tag{20} \]

becomes smaller. On the contrary, with small openings the lift increases, and the piston strikes against the cylinder cover.

The size of the exhaust ports also has its own peculiarities. Thus, when the ports \(A\) are large, the line of simultaneous exhaust and expansion \((f—g)\) falls more steeply. The same must be said with respect to the line of simultaneous exhaust and compression \((g—b)\). When the exhaust openings are very large, the area of the diagram is reduced, and with it the striking force of the hammer. With very small exhaust openings the air does not escape quickly enough, and therefore the lift becomes shorter.

It is interesting to trace what influence recoil—rebounding at the blow—has on the operation of the hammer. Observations show that the weaker this rebound is, the smaller the number of blows. The explanation of this is easy to find if one turns to the forces acting on the hammer during operation. Indeed, immediately after the blow, recoil and the pressure on the annular area act on the piston from below upward, while from above downward acts the air pressure on the full surface of the piston, since the air has not yet had time to escape. If, under the influence of these opposing forces, the piston comes to rest before the pressure on the annular area exceeds the pressure on the full area on the opposite side, then the piston remains in one place, or returns back to the cutter. Thus the number of useful strokes of the piston will be considerably reduced in the presence of such circumstances. In the case where the recoil is equal to zero, the piston does not strike against anything, and this is easily perceived by the ear. An analogous phenomenon is observed in pneumatic rammers, when an insignificant rebound is due to the quality of the molding sand. Here the piston, after passing a small distance, stops for an instant and then continues its path. The diagrams in Figs. 8 and 9, drawn for the same conditions as the diagram in Fig. 5, give a sufficiently clear idea of the influence of the magnitude of recoil and the actual limits of this influence, with only the difference that in the diagram of Fig. 8 the recoil velocity is taken at \(15\%\) of the stroke, and in the diagram of Fig. 9 at \(35\%\) of the stroke magnitude.

From the examination of these diagrams it becomes clear that in all three cases curve 1, not yet subject to the influence of recoil, coincides with the others.

The following table presents a grouping of the numerical results, and Fig. 10 their graphical representation.

The table just presented, together with Fig. 13, contains a summary of the results from these experiments. As was to be expected, the number of blows, their force, and the air consumption increase with an increase in pressure. In contrast, the specific power attains its highest value at 5.1 atmospheres. It follows from this that, although the absolute power does increase with a rise in pressure above that for which the hammer was designed, in the end this increase in pressure proves uneconomical.

The hammer just investigated, made by the firm Collet-Engelhard, belongs to the type of hammers with automatic distribution, that is, those in which the admission and exhaust are controlled by the piston. However, most current hammer designs belonging to the type of valve hammers have an auxiliary valve, or more properly a piston slide-valve, which controls the regulation of the air, whereas the slide-valve piston itself functions from the working piston.

Hammers without valves are distinguished by the relative simplicity of their construction, but since their pistons are longer, the cylinders also become longer, which causes an increase in weight. But the increase in weight deprives them of such a valuable quality as convenience in operation, which is why these hammers are not used, for example, in riveting work.

The method of investigation applied to the hammer with automatic distribution is quite suitable also for valve hammers. For this purpose let us take a hammer of the firm Pokorny-Wittekind, Fig. 14. In this design, now already abandoned by the firm because of its shortcomings, the piston slide-valve is made in the form of a differential piston, the annular surface of which is always under full pressure, so that the slide-valve tends to move upward. But if air is made to press on the upper end surface, the piston moves downward and is held there. Compressed air obtains access to this end surface only when the openings \(a\) and \(b\), owing to the rotation of the piston, come into communication. If channel \(c\) is not covered by the piston, the compressed air can escape into the space between the upper surface of the cutting tool and the lower surface of the piston. In addition to the openings \(a\), \(b\), and \(c\), there is one more opening \(d\) in the wall of the cylinder, independent of the slide-valve and intended for blowing out the air.

When examining hammers with automatic distribution, it could be observed that only during the working stroke of the piston in the downward direction—

In hammers, such parts as the cylinder and the piston are not separated from one another, as happens with a core or a shot charge. The worker’s hand perceives only that rebound with which the working end of the tool, the chisel or die, flies back from the piece being worked. The tool is usually guided by a bushing, fastened by being screwed into the cylinder. If the chisel has cut into the object being worked, it separates from the bushing. To eliminate this phenomenon, it is necessary that the cylinder also take part in the forward motion, i.e., the worker must press on the handle of the hammer with a certain force, so that there is no play between the linkage and the bushing; then the rebound, or more precisely the recoil, should disappear. This is in fact observed in experiment. However, there remains the tension of the muscles of the arm, which cannot be eliminated, and it is transmitted in the form of pressure to the upper cover of the cylinder. During the entire ascent of the hammer this pressure undergoes oscillations from a certain minimum value to a maximum. It is this oscillation of pressure, perceived by the worker’s hand, that greatly fatigues the arm. It is therefore necessary to take care that these oscillations do not proceed in jumps, i.e., that the curve \(c-d\) not rise too steeply. How is this to be achieved? There are several means here. One may, for example, keep the inlet openings small, as was already indicated earlier, or else increase the harmful space. In the hammer tested by P. Möller (Fig. 4), the second path was chosen, for which purpose an air chamber was attached to the cylinder. It must, however, be noted that concern for convenience of work is purchased at no small cost, since in this case the consumption of compressed air increases considerably.

Finally, there are also cases in which the jolt is caused by an imperfection in the design of the hammer. Here the chief attention must be paid to the length of the cylinder. Thus the question arises: what length must the piston have so that, even under the most unfavorable conditions, no jolt occurs? Investigations show that such a jolt occurs under the following circumstances: if the hammer is in operation, then at the moment when the admission of air above the piston is to begin, the inlet valve at the hammer handle is closed. The air contained in the channels and in the recesses of the hammer (piston) then rushes to the upper side of the piston, and if at this moment the piston comes to rest, then the pressures on both its sides are compensated. If one investigates, as shown in drawing 9, the processes occurring in this case—

in hammers such parts as the cylinder and the piston do not separate from one another, as happens with a cannonball or a shotgun charge. The worker’s hand perceives only the rebound with which the working end of the tool, the cutter or punch, flies away from the piece being worked. For guiding the tool there is usually a bushing, fastened by being screwed into the cylinder. If the cutter has bitten into the object being worked, then it separates from the bushing. To eliminate this phenomenon it is necessary that the cylinder also take part in the progressive motion, i.e. the worker must, with a certain force, press on the handle of the hammer, so that there is no play between the link and the bushing; and then the rebound, or rather the recoil, should disappear. This is in fact observed in practice. However, there remains the tension of the muscles of the arm, which cannot be eliminated, and it is transmitted in the form of pressure to the upper cover of the cylinder. During the whole rise of the hammer this pressure undergoes oscillations from some minimum value to a maximum one. It is this oscillation of pressure, perceived by the worker’s hand, that greatly tires the arm. It is therefore necessary to see to it that these oscillations do not proceed by jumps, i.e. that the curve \(c-l\) does not rise too steeply. How is this to be achieved? There are several means here. One may, for example, keep the inlet openings small, as was already indicated earlier, or else increase the harmful space. In the hammer tested by P. Möller (Fig. 4) the second method was chosen, for which purpose an air chamber was attached to the cylinder. It must, however, be stipulated that concern for convenience in work is purchased not cheaply, since the consumption of compressed air is thereby considerably increased.

Finally, there are also cases when the shock is caused by imperfection in the design of the hammer. Here the principal attention must be directed to the length of the cylinder. Thus the question arises: what length must the piston have so that, even under the most unfavorable conditions, no shock occurs? Investigations show that such a shock occurs under the following circumstances: if the hammer is in operation, then at the moment when the admission of air above the piston is to begin, the inlet valve at the handle of the hammer is closed. The air contained in the channels and in the recesses of the hammer (piston) then rushes to the upper side of the piston, and if at this time the piston comes to rest, this means that the pressures on both its sides are balanced. If one investigates, as shown in drawing 9, the processes occurring in this case—

The table just given, together with Fig. 13, contains a summary of the results from these experiments. As was to be expected, the number of blows, their force, and the air consumption increase with the rise in pressure. On the other hand, the specific power reaches its highest value at 5.1 atmospheres. It follows from this that, although the absolute power does increase with the pressure rising above that for which the hammer is designed, in the end this increase in pressure proves uneconomical.

The hammer just investigated, made by the firm Collet-Engelhardt, belongs to the type of hammers with automatic distribution, that is, those in which the piston controls the distribution of admission and exhaust. However, the majority of current designs of hammers belonging to the type of valve hammers have an auxiliary valve or, more properly, a piston slide valve, which controls the regulation of the air, while the piston slide valve itself is actuated by the working piston.

Hammers without valves are distinguished by the comparative simplicity of their construction, but since their pistons are longer, the cylinders also become longer, which causes an increase in weight. This increase in weight deprives them of so valuable a quality as convenience in operation, which is why these hammers are not used, for example, in riveting work.

The method of investigation applied to the hammer with automatic distribution is also fully suitable for valve hammers. For this purpose let us take the hammer of the firm Picornie-Wittekind, Fig. 14. In this design, now already abandoned by the firm because of its shortcomings, the piston slide valve is made in the form of a differential piston, the annular surface of which is always under full pressure, so that the slide valve has a tendency to move upward. But if the air is made to press on the upper end area, the piston moves downward and is held there. Compressed air obtains access to this end area only when the openings $a$ and $b$, as a result of the rotation of the piston, come into communication. If the channel $c$ is not covered by the piston, then the compressed air can pass into the space between the upper face of the cutting tool and the lower face of the piston. Besides the openings $a$, $b$, and $c$, there is in the wall of the cylinder one more opening, $d$, independent of the slide valve and intended for blowing out the air.

In considering hammers with automatic distribution, it could be observed that only during the working stroke of the piston along the direction of—

becomes smaller. Conversely, with small openings the lift increases, and the piston strikes the cylinder cover.

The size of the exhaust windows also has its own peculiarities. Thus, when the windows \(A\) are large, the line of simultaneous exhaust and expansion \((f-g)\) falls more steeply. The same must be said with respect to the line of simultaneous exhaust and compression \((g-b)\). When the exhaust openings are very large, the area of the diagram decreases, and together with it also the striking force of the hammer. With very small exhaust openings the air does not escape quickly enough, and therefore the lift becomes shorter.

It is of interest to trace what influence rebound—bouncing back at the blow—has on the operation of the hammer. Observations show that the weaker this bouncing back, the smaller the number of blows. The explanation of this is easy to find if one considers the forces acting on the hammer in operation. Indeed, immediately after the blow, rebound and pressure on the annular area act on the piston from below upward, while from above downward there acts the pressure of the air on the full surface of the piston, since the air has not yet had time to escape outward. If, under the influence of these opposing forces, the piston comes to rest before the pressure on the annular area has exceeded the pressure on the full area from the opposite side, then the piston remains in one place, or returns back toward the cutter. Thus, the number of useful strokes of the piston is considerably reduced in the presence of such circumstances. In the case when the rebound is equal to zero, the piston does not strike anything, and this is easily perceived by the ear. An analogous phenomenon is observed in pneumatic tampers, when an insignificant recoil is due to the quality of the molding sand. Here the piston, after passing a small distance, stops for an instant and then continues its path. Diagrams in Figs. 8 and 9, drawn for the same conditions as the diagram in Fig. 5, give a sufficiently clear idea of the influence of the magnitude of the rebound and of the actual limits of this influence, with the only difference that in the diagram of Fig. 8 the rebound velocity is taken at \(15\%\) of the stroke, and in the diagram of Fig. 9 at \(35\%\) of the magnitude of the stroke.

From consideration of these diagrams it becomes clear that in all three cases curve 1, not yet having come under the influence of rebound, coincides with one another.

The following table presents a grouping of the numerical results, and Fig. 10 a graphical representation.

when the piston begins to move, and 2) what the pressure will be when the piston has traversed the distance \(S'\)? The first of these questions is answered without particular difficulty. The motion of the piston begins when the pressure \(p_0\) acting upon it balances its weight together with the frictional force (in the state of rest). Assuming further that the hammer is held at an angle of \(30^\circ\) to the object being worked, and that the coefficient of friction at rest is twice the coefficient of friction in motion \((2 \times 0.15)\), we obtain for the expression \(p_0\) the following relation:

\[ p_0=\frac{0.277\ \mathrm{kg.}}{0.25\pi\cdot 2.6^2\ \mathrm{sq.\ cm.}}=0.0522\ \mathrm{atm.} \]

As soon as the hammer is set in motion, the friction decreases and the driving pressure increases by the difference between the friction at rest and the friction in motion, \(p_0-p_c=0.0089\ \mathrm{atm.}\)

In order to solve the second problem posed, concerning the magnitude of the pressure at the end of the path \(S\) traversed by the piston, it is necessary to stop at some probable assumption representing the dependence between the increase in pressure \(p\) and the distance traversed \(S'\).

Over a small interval, the simplest such assumption may be to choose a linear dependence, \(p:S=\mathrm{const.}\) This assumption makes it possible subsequently to solve the integral

\[ \int \frac{dS}{V_x}. \]

If we now consider (Fig. 15) the path of the piston \(x\), in passing through which the piston acquires the velocity \(V_x\), then, applying familiar notations, we obtain

\[ V_x^2=\left(\frac{\Delta p\,x}{2S}+p_c\right)\frac{F_x}{m/2} \]

and hence we find the time for the path \(x\)

\[ t=\int \frac{dx}{V_x} =\sqrt{\frac{m}{F}}\int \frac{dx}{\sqrt{2p_c x+\Delta p\,\frac{x^2}{S}}}. \]

Extending the integration over the limits from \(0\) to \(S\), we shall have

\[ t=\sqrt{\frac{mS}{F\Delta p}}\{lg_n(p_c+\Delta p+\sqrt{\Delta p}\sqrt{2p_c+\Delta p})-lg_n p_c\} \tag{20} \]

Table VIII.

1. A previously used chisel.

Hammer No.	Pressure in the air pipeline, \(p'\) atm.	Duration of the experiment, min.	Measured air consumption during the experiment, cu. m	Measured air consumption, cu. m/min.	Calculated air consumption, cu. m/min.	Notes
861	5.90	1	0.350	0.350	0.402	2 readings, one after the other.
861	5.80	1	0.325	0.325	0.369	2 readings, one after the other.
664	5.75	1	0.325	0.325	0.366	2 readings, one after the other.
664	5.75	1.5	0.560	0.373	0.420	2 readings, one after the other.
664	5.90—5.80	1	0.380	0.380	0.434	2 readings, one after the other.
				average	0.398

Table IX.

2. A well-sharpened chisel.

Hammer No.	Pressure in the air pipeline, \(p'\) atm.	Duration of the experiment, min.	Measured air consumption during the experiment, cu. m	Measured air consumption, cu. m/min.	Calculated air consumption, cu. m/min.	Notes
664	5.80	0.5	0.18	0.36	0.408	three readings, one after another.
664	5.80	1	0.33	0.33	0.374	three readings, one after another.
664	5.80	1.5	0.48	0.32	0.364	three readings, one after another.
				average	0.382

Table X.

Hammer No.	Pressure in the pipes, atm.	Number of blows per minute	Quality of the chisel.
664	5.80	1496	Blunt.
664	5.80	1440	Sharp.
861	5.80	1284	Blunt.
861	5.80	1280	Sharp.

From the foregoing, a good agreement between the experimental data and the results of the calculations is readily seen, and this recommends the computational method favorably.

It is not difficult to obtain the air consumption in cubic meters per kilogram-meter of work, by referring the quantity of air to atmospheric pressure

\[ q=\frac{m V^{2}}{2}=0.0002703:1.11=0.000243\ \text{cu. m}. \]

make the reservation that this assumption is arbitrary, but this arbitrariness is of the same character as the assumption that friction at rest passes directly and suddenly into friction in motion. Now the question naturally arises: how is this inaccuracy of the assumption—or, more correctly, the arbitrary assumption—reflected in the results of the investigation? It turns out that this apprehension is superfluous. If an error has been made here in the calculation in the sense of increasing the pressure, then in the subsequent course it disappears, because for the second, third, etc., rise the use of formulas 20 and 21 is unnecessary, since, as a result of the rebound, the piston already has a certain velocity at the very beginning of the rise.

The matter stands somewhat differently with the piston valve, to which the same formulas 20 and 21 are applied in order to find at what point of the piston stroke the change in the position of the valve takes place. Since each rise or stroke of the piston here is independent of the following one, any admitted inaccuracy remains fixed in the process. In addition, other influences also arise here which are difficult to take into account, such as, for example, the influence of leakage (owing to the narrowness of the sealing surface of the piston valve this circumstance plays a more significant role than in the working piston), the formation of an air cushion under the lower surface of the valve, which results in the appearance under the valve of either compression or expansion, and finally, the pressure acting on the hammer, which, although partly absorbed by the muscles of the hands, nevertheless causes vibration of the whole hammer. All these phenomena, which are difficult to take into account, compel one to be satisfied with an estimate by eye of the time required for the piston valve to change its position. This operation is performed in the following way: let us suppose that as soon as the distributing edges of the working piston have passed over the openings in the wall of the cylinder, equalization of pressure for the distributing valve takes place in such a way that in one case full pressure suddenly appears above the valve, while in another case the same full pressure, prevailing in the working cylinder, likewise suddenly appears. Then, by equation 19, one can calculate the time sufficient for the displacement of the valve. Multiplying this quantity by the average velocity of the working piston, we obtain the path traversed by the piston up to the moment at which the change in the position of the valve occurs. For the sake of simplicity it is assumed that the change in the position of the valve occurred suddenly at the moment when the distributing valve had half-opened the ca-

the velocity gradually approaches zero. If, for this final segment of the velocity curve, we take the pressure curve to be a straight line (Fig. 7), then the time \(t\) is obtained from the relation

\[ \frac{mV^2}{2}=pFC \quad \text{and} \quad t=\int \frac{ds}{V} \]

\[ t=\frac{mS}{(p_0-p_1)F}\int_{p_1}^{p_0}\frac{dp}{\sqrt{p_0^2-p_1^2}} =\sqrt{\frac{mS}{(p_0-p_1)F}}\cdot\left(\frac{\pi}{2}-\operatorname{arc.\ sin}\frac{p_1}{p_0}\right) \tag{18} \]

When the pressure \(p=\mathrm{const}\), then the pressure curve is parallel to the axis of abscissas, and

\[ t=\frac{mV}{pF}; \qquad \frac{dmV^2}{2}=dpFS=pFVdt; \]

\[ mdV=pFdt; \qquad \int_0^{V'} m\,dV=\int_0^t pF\,dt; \qquad mV'=pFt, \quad t=\frac{mV'}{pF}; \tag{19} \]

In this way, the dotted curves were obtained on the diagram (Fig. 5). They show that the piston requires \(27.54\sigma\) for the lift (where \(\sigma=0.001\) seconds); for the working stroke from top to bottom, \(13.05\sigma\), and in all for the full swing \(40.59\sigma\). Hence one readily obtains the number of blows per minute

\[ n=\frac{60.1000}{40.59}=1480 \]

Finally, the diagram gives the length of the piston stroke \(=44\) millimeters.

These quantities, obtained by the graphical-numerical method, had to be checked experimentally, under normal working conditions, which was done at the Collet & Engelhardt works. To measure directly the work of the piston in the process proved rather difficult. It turned out to be simpler to determine experimentally the air consumption, the number of blows, and, with a known degree of accuracy, the magnitude of the lift.

The air consumption could best be measured in the following way: fill as large an air receiver as possible with compressed air, then disconnect this receiver from the supply pipes, and use the air for the needs of the hammer’s operation until the pressure falls to a specified degree. P. Möller used for this purpose ordinary Glover gas meters. Since devices of this kind show the quantity of air at temporary pressure in \(p\) absolute atmospheres, the count-

theoretical consideration with P. Möller’s experiment was not possible, nor was comparison of the numbers obtained. However, the excellent agreement of the figures with the experimental data advertised by the firm speaks for the reliability of the conclusions obtained. Thus the firm gives the number of blows as 1400 per minute, and the air consumption as 0.25–0.50 cubic meter.

Turning then to the diagrams in Fig. 15, it may be observed that here it has also been investigated what least length the cylinder must have in order to ensure reliable operation. Here the most unfavorable case may occur when there is no rebound, and after the spool has changed its position the exhaust valve at the handle of the hammer is closed. In such a state of affairs, the air above the piston will be compressed, while under the piston it will at the same time expand and escape. From the external, indicated side, the process may be described as follows: the piston, touching the head of the cutter, remains in the lower position until the pressure above the piston has fallen, owing to the discharge of air, to a pressure equal to the increasing pressure under the piston. As soon as such an equality of pressures occurs, the piston begins to move. From this moment the pressure under the piston continues to rise, while above the piston it gradually approaches atmospheric pressure, as is also seen in Fig. 15 in the form of a dotted line. In this drawing the dotted line rising on the left-hand side, in the upper diagram, represents the compression line for the upper side of the piston. In the lower diagram the dotted lines indicate velocities. It is not difficult to observe that the actual length of the cylinder \(l\) is not quite sufficient for the most unfavorable case, so that the piston may touch the cylinder cover, although only with a small velocity.

With this one may conclude the exposition of the investigation of pneumatic hammers by P. Möller’s method. There is no doubt that this method may be applied with great benefit in the investigation and calculation of steam high-speed hammers, air hammers, and gas hammers, since it is of a completely general character. At present, for obtaining accurate diagrams of high-speed hammers, the cinematograph is being used with unusual success. I do not intend to deal in detail here with this new question, since the analysis of the application of the cinematograph to the investigation

I shall devote a separate article to high-speed hammers. For now I will confine myself to the observation that a successful combination of both methods gives designers a reliable way to study more precisely the most delicate part of hammers, namely the distribution of energy by slide valves, valves, cocks, etc. The former arbitrariness in the calculation of hammers, under the friendly pressure of the fruitful ideas of the new age, must rapidly depart into the realm of tradition and take a modest place in the field of the history of technology.

NOTES.

To p. 4. In the cylinder, on one side of the piston, there is air in volume \(V\) at pressure \(p\). When the piston moves through the path \(dS\), the volume \(V\) increases by \(dV\). Into the new volume \(V+dV\) there passes from the air receiver a new quantity of air \(dV'\) at pressure \(p'\). If the weight of the former air was \(G_1\), and that of the entering air \(G_2\), then the final weight of air \(G\) will be

\[ G = G_1 + G_2. \]

The well-known equation of thermodynamics gives

\[ GRT = pV;\quad G = pV : RT \]

Hence

\[ G_1 = \frac{pV}{RT};\quad G_2 = \frac{p'dV'}{RT};\quad G = \frac{pV}{TR} + \frac{p'dV'}{RT}. \]

Here it is assumed that the absolute temperature of the entering air is the same as the absolute temperature of the air already present in the cylinder.

If the air in the volume \(V\) were to expand, its temperature would fall. But the air flowing into the volume \(dV'\) produces compression of the air, accompanied by an increase in temperature. Thus, without risking a large error, it may be admitted that over the path \(dS\) the temperature remains unchanged.

For the weight \(G\) one may write

\[ G = \frac{(p+dp)(V+dV)}{R(T+dT)} = \frac{pV}{RT} + \frac{p'dV'}{RT}. \]

But since \(dT=0\), then

\[ pV + Vdp + pdV + dp\,dV = pV + p'dV'. \]

Performing the cancellation and discarding the term \(dp\,dV\) as a quantity of the second order, we obtain

\[ dp = \frac{p'dV' - pdV}{V}. \]

To pp. 5–7. Möller regards the volume of air \(dV'\) as the exiting volume, although this volume flows into the hammer cylinder. The apparent misunderstanding is explained by the fact that this air leaves the supply hose and enters the cylinder. (See Fig. 19.) Thus, into the cylinder there flows

\[ dV' = U \cdot f \cdot dt. \]

From the reservoir during this time there issues \(dV''\), or, by weight,

\[ dG=\frac{p_a dV''}{RT}. \]

These volumes may also be unequal if, upon entering the cylinder, the air expands. However, until the moment when the critical pressure ratio \(p_c : p_a\) is reached, the weights of the air will be equal. Thus,

\[ \frac{p_c dV'}{RT}=\frac{p_a dV''}{RT};\quad p_c dV'=p_a dV''. \]

Or, substituting for \(dV'\) and \(dV''\) the values obtained earlier, we shall have

\[ f \cdot U \cdot dt \cdot p_c = f \cdot U' dt \cdot p_a;\quad Up_c=U'p_a. \]

At the moment when the critical pressure ratio is reached, we shall have

\[ U'=U_{\max}\frac{p_c}{p_a}=U_{\max}\beta. \]

To p. 8. As regards the corrections to the manometer readings, Möller is hardly right. The point is that if one impulse is quickly followed by another in the opposite direction, the pointer will not have sufficient time for a full swing. Therefore the readings of a manometer located not far from the hammer will be correct, for these readings are conditioned by a constant increase of pressure in a column of moving elastic medium, if the pressure drops instantaneously. A good illustration of this proposition is furnished, for example, by hydraulic shock in pumps. The same thing is also observed in valve chests. The simple explanation of the instantaneous increase in pressure is that the elastic medium continues to move by inertia.

On pp. 9–10. In constructing the diagram for the hammer, the pressure is taken equal to the arithmetic mean of the manometer readings

\[ p_a=\frac{p_{\max}+p_{\min}}{2} \]

As for the return stroke, for it the arithmetic mean of the working pressure \(p_l\) and the maximum is taken,

\[ p'_a=\frac{p_l+p_a}{2} \]

Thus,

\[ p_a=5.1+6:2=5.55. \]

Further. The piston has diameter \(d_2=32\) and the neck \(d_1=30\). Consequently, the annular area is

\[ f=\frac{d_2^2-d_1^2}{4}\pi=97.39\ \text{sq. mm.} \]

And the total pressure \(P\) on this area will be

\[ P=5.55\cdot 97.39=5.401\ \text{kilograms}. \]

Per one centimeter of area \(F\) there will be

\[ p=5.401:F=5.401:\frac{\pi d_2^2}{4}=0.663\ \text{kilograms}. \]

This pressure is indicated on the diagram by a thick line. The critical pressure for admission is obtained from the ratio \(p_v:p_a=0.555=p_v:6.1\). Hence \(p_e=3.3855\).

For the period of exhaust, the back pressure is assumed to be \(p_r=1\) atmosphere (absolute), and then \(p_a=1.8\) abs. atm.

The weight and friction figuring in the question under consideration are taken into account in the following way: if the hammer is inclined to the vertical at an angle of \(30^\circ\), then the component of the weight in the direction of motion will be

\[ p_v=0.58\sin 30^\circ=0.29\ \text{kilograms}. \]

Or, referring this figure to the square centimeter of piston area \(F\),

\[ 0.29:\frac{\pi d_2^2}{4}=0.0356\ \text{kilograms}, \]

During the working stroke this force will accelerate the motion, and during the return stroke upward it will retard it.

The frictional force is also found without any difficulty:

\[ p_{m\varphi}=0.58\cos 30^\circ\cdot 0.15=0.0753\ \text{kilogram} \]

or, per one sq. centimeter,

\[ 0.0753:8.1384=0.0094\ \text{kilogram}. \]

The frictional force is always a harmful resistance.

Construction of the curve \(d-c\). Regarding the construction of this curvilinear segment, Möller speaks somewhat obscurely. The following considerations, explained along the way by calculation, should remove the apparent ambiguity. Suppose that the course of the curve is sought from the point at which the pressure \(p_1=6.1\) abs. atm. At this moment, behind the piston there will be enclosed a volume of air

\[ V_1=5.88\cdot 8.134=47.854\ \text{cu. cent.} \]

The distance of the piston from the turning (dead) point is equal to \(0.1\) cent. With further motion of the piston over the path \(dS_1=0.2\) centim., the pressure will fall by \(0.05\) atmosphere. Thus, the absolute pressure \(p_2\) will be equal to \(6.05\) atmospheres. The volume, however, will increase by \(dV_1=0.2\cdot 8.1348=\)\(=1.628\) cu. cent.

Now it is necessary to determine the pressure for the further motion of the piston over the path \(dS_2=0.2\) cent., which must correspond to the new increase in volume by \(dV_2=1.628\) cu. cent.

The velocity of the piston at the second point will be

\[ V=\sqrt{\frac{p_m\cdot F\cdot S\cdot 2}{m}}. \]

Here \(p_m=5\) atm. represents the mean pressure on the upper side of the piston. From this it is necessary to subtract the mean pressure of \(0.667\) atmosphere on the lower side of the piston and the friction \(p_{m\varphi}=0.0094\) atm., and to add the force of weight \(p_b=0.0356\).

Thus,

\[ p_m=5-0.667-0.0094+0.0356=4.409\ \text{atmospheres}. \]

Further, \(F=8.1384;\ S=0.12+0.20=0.32=0.0032\) meter, and the mass \(m\) is equal to \(0.58:0.81=0.0592\).

Thus,

\[ V=\sqrt{\frac{4{,}409\cdot 8{,}1384\cdot 0{,}0032}{0{,}0296}}=2{,}05 \text{ meters}. \]

\[ dt=\frac{dS}{V}=\frac{0{,}002}{2{,}05}\simeq 0{,}001 \text{ seconds}. \]

Now let us determine the volume \(dV'\). In view of the fact that the air enters through two openings with cross-section \(f\),

\[ dV'=2\cdot f\cdot u\cdot dt. \]

But

\[ 2f=\frac{2\pi S^2}{4}=0{,}3925 \text{ sq. cm} \]

Hence

\[ U=0{,}9\cdot 755\sqrt{1-\left(\frac{6{,}05}{6{,}10}\right)^{\frac{1}{5}}}=2759 \text{ cm}. \]

Therefore,

\[ dV'=0{,}3925\cdot 2759\cdot 0{,}001=1{,}12 \text{ cubic cm}. \]

The pressure drop \(dp\) is determined from the formula given earlier

\[ dp=\frac{p'dV'-p_2dV_2}{V_3} \]

Since here \(p'=p_2\), and \(V_3=V_1+dV_1+dV_2=51{,}11\) cubic cm,

then

\[ dp=\frac{(1{,}12-1{,}628)6{,}05}{51{,}11}=-0{,}06. \]

Ultimately,

\[ p_3=p_2+dp=6{,}05-0{,}06\simeq 6 \text{ atm.} \]

If \(p_2=6\) absolute atmospheres. Then

\[ U=0{,}9\cdot 755\sqrt{1-\left(\frac{6}{6{,}1}\right)^{\frac{1}{5}}}=39{,}10 \text{ m}, \]

\[ dV'=2\cdot 0{,}1964\cdot 3910\cdot 0{,}001=1{,}584 \text{ cubic cm}. \]

\[ dp=\frac{(1{,}584-1{,}628)6}{51{,}11}=-0{,}004 \]

\[ p=6-0{,}004\simeq 6 \text{ atm.} \]

This calculation shows that, under both assumptions, the final pressure will be approximately the same.

Observing the chronological sequence, we must note that after the admission period there begins pure expansion between points \(e\) and \(f\) of the diagram. The curve \(e—f\) is constructed from the zero point of the system. Between points \(f—g\), when expansion and exhaust occur, the curve is constructed in the same way as the curve between points \(d—e\), with the sole difference that here \(p_a\) denotes the pressure in the hammer, and \(p_c\) the atmospheric pressure.

Finally, at the moment of impact the pressure stress falls. On meeting the chisel, the piston rebounds, but with a lower velocity than it had during the forward stroke. After the impact the return stroke begins, characterized by the occurrence of combined compression and exhaust (curve II), etc.

To p. 13. If “\(a\)” in meters is the height of fall, and “\(b\)” the height of rebound, then the velocity of impact is \(V_1 = \sqrt{2ga}\), and the velocity of rebound is \(V_2 = \sqrt{2gb}\).

Consequently,

\[ V_2 : V_1 = \sqrt{b} : \sqrt{a}. \]

To p. 15. In pointing out that not all the work of the blow is utilized in the hammer, Möller merely states the fact, but gives no idea of the magnitude of this loss. Meanwhile, it is not at all difficult to calculate it. Indeed, if the piston strikes with velocity \(V\) and rebounds with velocity \(V_1\), then the useful work of the blow will be approximately

\[ \frac{mV^2 - mV_1^2}{2}. \]

Assuming, for example, that \(V_1 = V : 4\), we obtain for the useful work the value

\[ \frac{15}{16}\,\frac{mV^2}{2}. \]

Other losses, such as, for example, air resistance, are not taken into account at all by Möller; yet at the high velocities possessed by the piston, and with it the chisel or die (see the study of pneumatic hammers according to Barillo, Proceedings of the Ekaterinoslav Branch of the Imperial Russian Technical Society, 1910), these losses are not so insignificant that they can be neglected in delicate tools, especially since it is not difficult to take them into account.

We shall simply use the generally accepted formula for air resistance (see Wellner, Flying Machines, p. 9)

\[ W = a F \frac{\gamma}{g} V^{2} = 0.125 F V^{2} \]

Then, of course, one must not lose sight of the worker’s ability to handle the tool. But to take account of the individual traits of the worker by means of analysis is a very difficult task.

To p. 19. The theoretical coefficient of useful action is determined by the ratio

\[ \eta_i = \frac{1.11}{2.63} = 0.422. \]

In reality it will be somewhat smaller, since from the foregoing it is clear that part of the energy of the moving piston will go into rebound. Thus, it will be more correct to calculate the coefficient of useful action as

\[ \eta_i = \frac{1.11 - \frac{1}{16}\cdot 1.11}{2.63} = 0.40. \]

The same must also be said regarding the coefficient of useful action of the entire installation:

\[ \eta = \frac{1.11}{7.60} = 0.146 \quad \text{(according to Möller)};\quad \eta = \frac{1.04}{7.60} = 0.137 \quad \text{(more correctly)}. \]

Of Möller’s whole work, this point is the most vulnerable. The difference between 0.422 and 0.146 is already too large, or, with the correction, between 0.4 and 0.137. It is hardly possible to be satisfied with the explanation that the large losses are due to leaks in the joints, to the friction of the air against the walls of the air conduit, to losses in the compressor, to cooling of the air, etc., since the newest compressor installations are not so very poor. It is most likely that the error here lies in the inaccuracy of the application of Zeuner’s formula, since here between the feeder and the receiver there is an intermediate conduit, a connecting link. Here attention must be paid to the drop in the elastic pressure of the air during admission. It cannot be assumed that there is constant pressure in the hose along its whole length.

I shall try to explain this situation with examples. The pressure in a gas flow is connected with the square of the velocity by the following dependence: if the square of the velocity increases, the pressure decreases, and conversely. But in a channel, hose, or pipe, under steady motion, the velocity פּר.

...the moving particles varies depending on the magnitude of the cross-section of the gas pipeline. And this cross-section is never invariable.

Furthermore. With a light gas it is more correct to suppose, in accordance with the hypothesis of Professor V. E. Grum-Grzhimailo, beautifully substantiated by I. Esman, that this motion may be likened not only to the motion of droplet liquids in pipelines under pressure, but also to the motion by gravity in open channels. In the latter case the motion may be likened to motion in a river turned upside down, with its bottom upward. Then, as soon as the inlet ports open, expansion will immediately begin, and consequently a fall in pressure (until air arrives from the receiver). The very short path of transition from the end of the sleeve through the ports into the cylinder itself fully allows for such a possibility. It is hardly a correct assumption that the mean inlet pressure according to the manometer is free from error. The manometer is located at a distance of 40 centimeters from the hammer, and this circumstance plays a large role. It is very probable that the pressure in front of the hammer is much lower than that assumed by Möller. As for the maximum pressure in front of the hammer, it will be close to the manometer readings.

Furthermore, the pressure, or elasticity, of the air must be further reduced by the bends in the air conduit. The introduction of a coefficient of 0.9 for the velocity, as an allowance for losses at bends, elbows, transitions, etc., can in no case be regarded as sufficient.

On the basis of these considerations, when constructing the diagram one must bear in mind that the admission curve will lie lower than in Möller’s, but by how much—this question can be resolved in a positive sense only by experiment, and separately in each particular case.

Then one must bear in mind that part of the heat of the air, while passing through the passages, is transferred to the hammer itself and is partly better dissipated.

This phenomenon belongs to the category of irreversible processes. For completeness of the investigation it would have been necessary to determine this loss also, and if it were significant, it would have been necessary to develop a new principle for the construction of delicate tools similar to pneumatic ones, in which the idea of the smallest surface for heat dissipation would be carried through, or else the principle of the Dewar vessel.

Explanations for the diagrams in the table:

1. Diagram of the hammer of the firm “Collet Engelhardt” (Fig. 5). Piston diameters 30–32 millimeters; piston weight 0.58 kilogram; mean admission pressure 5.1 atmospheres; rebound coefficient 0.25. Scales: for lift, 12.5 millimeters = 10 millimeters; for pressure, 12.5 millimeters = 1 atmosphere; for velocities, 12.5 millimeters = 1 meter per second.

2. Diagram of the hammer of the firm “Collet Engelhardt” (Fig. 8). Piston diameters 30–32 millimeters; piston weight 0.58 kilogram; mean admission pressure 5.1 atmospheres; rebound coefficient 0.15. Scales: for lift, 12.5 millimeters = 10 millimeters; for pressure, 12.5 mm = 1 atmosphere; for velocities, 12.5 mm = 1 meter per second.

3. Diagram of the hammer of the firm “Collet Engelhardt” (Fig. 9). Piston diameters 30–32 millimeters; piston weight 0.58 kilogram; mean admission pressure 5.1 atmospheres; rebound coefficient 0.35; scales: for lift, 12.5 mm = 10 millimeters; for pressure, 12.5 millimeters = 1 atmosphere; for velocities, 12.5 millimeters = 1 meter per second.

4. Diagram of the hammer of the firm “Collet Engelhardt” (Fig. 11). Piston diameters 30–32 millimeters; piston weight 0.58 kilogram; mean admission pressure 4.5 atmospheres; rebound coefficient 0.25; scales: for lift, 12.5 mm = 10 millimeters; for pressure, 12.5 millimeters = 1 atmosphere; for velocities, 12.5 millimeters = 1 meter per second.

5. Diagram of the hammer of the firm “Collet Engelhardt” (Fig. 12). Piston diameters 30–32; piston weight 0.58; mean admission pressure 5.7 atmospheres; rebound coefficient 0.25; scales: for lift, 12.5 mm = 10 mm; for pressure, 12.5 mm = 1 atmosphere; for velocities, 12.5 millimeters = 1 meter per second.

6. Diagram of the hammer of the firm “Pokorny—Wittekind” (Fig. 15). Piston diameter 26 mm; piston weight 0.365 kg; mean admission pressure 5.1 atmospheres; rebound coefficient 0.25; scales: for lift, 12.5 mm = 10 mm; for pressure, 12.5 mm = 1 atmosphere; for velocities, 12.5 millimeters = 1 meter per second.

To the article by S. K. Konyukhov, “Study of Pneumatic Hammers” (after P. Möller).

Fig. 5

Atmospheres

Curve of pressure \(E\).

Curve of pressure \(A\).

Theoretical diagram

Indicator diagram

Visible curve and point labels: \(I\), \(II\), \(A\), \(E\), \(4{,}5\), \(5\), \(7{,}546\), \(13{,}656\).

Fig. 8

Atmospheres

Curve of pressure \(E\).

Curve of pressure \(A\).

Theoretical diagram

Indicator diagram

Visible curve and point labels: \(I\), \(II\), \(A\), \(E\), \(4{,}5\), \(5\), \(35{,}656\), \(12{,}966\).

Drawing 12.

Visible labels:

• atmosphere
• pressure curve \(E\)
• pressure curve \(A\)
• pressure curve \(E\)
• pressure curve \(A\)
• \(A\)
• \(E\)
• \(I\)
• \(II\)
• \(12{,}306\)
• \(21{,}326\)
• \(12{,}306\)
• \(p_0\)

Drawing 13.

Visible labels:

• \(v_2\)
• \(n\)
• \(q\)
• atmosphere

Drawing 2.

Visible labels:

• \(p_0\)
• \(p\)
• \(v_1\)

Drawing 4.

Drawing 10.

Visible labels:

• \(p_0\)
• \(p\)
• \(v\)

Drawing 16.

Visible labels:

• \(\Delta p\)
• \(p_a\)
• \(p_b\)
• \(s\)
• \(x\)
• \(x_1\)

Drawing 14.

Drawing 5.

Fig. 9.

Labels visible in the figure:

• atmosphere
• center of striker
• center of piston
• critical velocity \(E\)
• critical velocity \(A\)
• atmosphere 6
• \(A\)
• \(E\)
• \(L_1\)
• \(I\), \(II\)
• \(150\)
• \(15/76\)
• \(27/86\)

Fig. 11.

Labels visible in the figure:

• atmosphere
• center of striker
• center of piston
• critical pressure \(E\)
• critical pressure \(A\)
• \(A\)
• \(E\)
• \(I\), \(II\)
• \(15/76\)
• \(27/86\)