APPLICATION OF THE THEORY OF SIMILARITY AND DIMENSIONS TO THE STUDY (MODELING) OF IMPACT MACHINES

P. M. ALABUZHEV

Introduction

As a result of the development of socio-historical practice and the need to build similar types of machine and structure designs, modeling arose, the basis of which is the similarity of phenomena according to a number of characteristics. In constructing new objects, without operating experience and without a correctly built model, in most cases the “experiment” may turn out to be blind, risky, and costly; therefore, wishing to study a phenomenon or process of interest to us and to foresee the fate of a structure, it is necessary first to study the phenomenon on a model and then to make a forecast concerning its fate over some interval of time. In the practice of the great construction projects of communism, modeling finds application as a method of cognition, investigation, and calculation in solving problems of the construction of hydraulic-engineering structures (the Small Stalingrad Hydroelectric Station, the model of the Great Don, the study of soil properties in a centrifuge, etc.), high-rise buildings, bridges, ship and aircraft structures, and a number of new machines, the serial production of which requires preliminary study of samples previously created by practice (which are prototypes) and experimentation with new samples.

Experiment is one of the cornerstones of cognition, a method of investigation by means of which the validity of hypotheses, conjectures, and models is tested and the values of a number of coefficients and indices are established. The theory of similarity is, to a certain extent, the “grammar” of experiment. Acad. M. V. Kirpichev, in his works [24—27], showed that the theory of similarity is the theory of experiment and modeling, that it indicates how an experiment should be set up, how experimental data should be processed, and how the results obtained should be generalized and extended to other objects.

The formulation and task of an experiment on the basis of the theory of dimensions and the theory of similarity is simplified and facilitated when there is a functional relationship between entire complexes of quantities determining the phenomenon, since in a number of cases there is no need to study the influence on the process of each factor separately and, in addition, there is the possibility (within known limits) of extending the results of a single experiment to similar systems.

Modeling may be physical or conditional (by analogy). In physical modeling there is a change of scale while the nature of the phenomenon is preserved. The qualitative and quantitative relations of similar phenomena are established in the form of criterial relations. Analysis of natural phenomena in the form of generalized (criterial) dependences makes it possible to single out the most general and essential, and gives the possibility of seeing that “the unity of nature is revealed in a ‘striking analogicity’”

differential equations pertaining to different regions of phenomena” (V. I. Lenin, Materialism and Empirio-Criticism, OGIZ, GIPL, 1948, p. 272).

One may arrive in various ways at establishing the connection between similarity criteria: the most reliable solution and the obtaining of generalizing conclusions can be achieved by applying the theory of similarity on the basis of the differential equations describing the process and the conditions of uniqueness, by which the limits of applicability of a single experiment are established. Having a well-functioning specimen of a successfully designed machine, in which the experience of universal human practice has already to a certain extent been embodied, it is comparatively easy and simple (and sufficiently reliable), on the basis of the theory of similarity, to produce, with a known approximation, a series of machines of the same type in accordance with the required indicators.

From the historical point of view, as L. Civita and U. Amaldi note [70, p. 361], Galileo had already posed the question: “Why does a model in miniature function perfectly, whereas the machine built thereafter in full size does not give the expected results?” The theorem of mechanical similarity was first formulated by Newton in his book Mathematical Principles of Natural Philosophy, translated into Russian by Academician A. N. Krylov [49].

On the basis of the theorem of similarity, Newton derived the law of resistance of a fluid to a solid body moving in it. The theory of similarity received further development, varied in its applications and original in character, in the works of Russian and Soviet scientists. Academician I. I. Artobolevsky points out that Euler checked the calculations of the model of an arch bridge across the Neva River, built by the brilliant Russian inventor-mechanic I. P. Kulibin. All the calculations proved to be correct. Euler set forth his considerations and computations in the article “A Simple Rule by Which, from the Model of a Wooden Bridge or Another Similar Machine Whose Strength Must Be Known, It Is Possible to Determine Whether the Same Can Be Done on a Larger Scale Than in the Model,” published by him in the Mesiatseslov with instructions for the year 1776 [4;16].

The author of classical works on structural mechanics and the theory of elasticity, the organizer of higher technical education in Russia, Prof. V. L. Kirpichev, who trained several generations of Russian engineers, in his book Conversations on Mechanics [23], first published in 1907 and which has not lost its significance even at the present time (an excellent supplement to courses in theoretical mechanics), set forth, in a lively and engaging form, the theorem on similarity in mechanics and dynamic models. V. L. Kirpichev, in 1874, was the first to formulate and prove the third theorem of similarity for elastic phenomena. Here we shall also note that his authorship of the discovery of the law of equal resistance was incorrectly attributed to the foreign scientists Barba and Kick.

Academician A. N. Krylov, while working in the Admiralty experimental basin, further and more deeply developed the method for calculating ship resistance proposed by Froude. In his book My Recollections he writes that, working in the experimental basin, he carried out “a consistent systematic testing of models for the selection of such a combination of elements as would ensure the proper seagoing qualities of ships of the projected type” [33, p. 141]. By A. N. Krylov, the role of ship modeling was raised to a tremendous height. The years of his work in the experimental basin were years of great searches and of the creation of the theory of unsinkable ships, the correctness of whose conclusions was demonstrated in practice. At the suggestion of A. N. Krylov, Prof. E. B. Lunts carried out the work “Determination of Critical Speeds of Shafts by the Method of Dynamic Similarity” [40], considering the case of shafts with variable cross-section, not amenable to the usual calculation methods.

In the works of Academician V. P. Goryachkin [13] and Professor N. D. Luchinsky [41], the application of the principle of similarity is shown clearly and convincingly for

1) constructing series of various agricultural machines;
2) determining the magnitude of various forces;
3) clarifying the dependence of resistance on dimensions.

In the twentieth century, the fundamental theorem of the theory of similarity and dimensions was proved, concerning the possibility of expressing physical laws in the form of a dependence between dimensionless numbers characterizing a phenomenon—the so-called Π-theorem. K. D. Voskresensky writes: “The American physicist E. Buckingham, who in 1914 published a work devoted to this theorem, is unjustly called the author of the Π-theorem. In reality the Π-theorem was first formulated and proved, and also found practical applications, in Russia. In 1911 there appeared a work by A. Federman, a lecturer at the Petersburg Polytechnic Institute, who proved a more general theorem, from which the Π-theorem follows as a consequence. It has recently been established that the Π-theorem was already being used in 1909 in the works of the Kuchino Aerodynamic Institute, organized by N. E. Zhukovsky and working under his direction” [10, p. 32].

Let us add that the proof of this theorem was published in the proceedings of the Kuchino Institute in 1912; the borrowing of results, with reference to these works, is contained in the book by L. Civita and U. Amaldi [70, p. 370].

Further development and reflection of the theory of similarity as applied to the mechanics of continuous media can be found in works on hydro- and aeromechanics by N. E. Kochin [30], L. G. Loitsyansky [38;39], L. Prandtl [54], L. I. Sedov [57], and others.

It should be noted that hydrodynamic modeling had a substantial influence on the development of thermal modeling. In connection with the modeling of thermal devices, Soviet scientists Academician M. V. Kirpichev and M. A. Mikheev developed a general theory of the existence of similarity [26]. The mathematical foundations of the theory of similarity and dimensional analysis were considered by Academician M. V. Kirpichev and P. K. Konakov [25; 28], as well as by P. V. Bridgman [6] and M. F. Malikov [43]. The application of the theory of similarity to the modeling of ventilation systems was considered by E. V. Kudryavtsev [35] and S. I. Pitkin [69].

The extension of the theory of similarity to the case of a medium with variable physical properties was carried out by A. A. Gukhman [15]. The application of the theory of similarity in geological modeling was shown by B. L. Shneerson [71].

Electrical modeling found its fullest development and reflection in the works of A. A. Kharkevich [67], V. V. Furduev [66], L. I. Guttenmacher [14], and V. A. Venikov [9]. Modeling in the broad sense of the word was considered by Professor L. S. Eigenson [73].

The literature on the theory of similarity and dimensions is extensive. Materials on the theory of similarity and modeling have entered the literature of an instructional [7; 23;50;70] and reference character [44; 51; 55].

However, the possibility of applying the theory of similarity and dimensions to the modeling of impact machines has been insufficiently covered in the literature.

B. V. Sudnishnikov, on the basis of V. L. Kirpichev’s work [23], was the first to apply the method of dynamic similarity to the calculation of electric hammers and perforators with spring coupling [59]. The development of B. V. Sudnishnikov’s work as applied to electropneumatic hammers of the ZERT type was shown by N. N. Esin [17].

Without claiming completeness or finality, the present work illuminates certain results of our investigations concerning the application of the theory of similarity and dimensions to the modeling of impact machines:

a) the work shows that similarity with respect to a number of dynamic characteristics also exists for forging hammers;

b) the found values of a number of similarity criteria for electro-pneumatic hammers and drills are given, by which, on the basis of experimental data, a series of parameters of impact machines can be selected for the optimal operating regime;

c) the direction of further work and a rational form for processing experimental data in drilling rocks have been clarified;

d) the role of essential quantities and the degree of complexity of the question in studying the operation of the air connection have been clarified;

e) the possibility of applying electrical modeling to study the working process of electro-pneumatic impact machines has been shown.

Some information from the theory of similarity and its application to piston machines

1. The literature on the theory of similarity is striking in its unusual variety and diversity of notation. We shall agree to denote similarity coefficients by the same letter as that by which the given quantity is denoted [16], but with the index \(c\). The basic specimen (prototype), adopted as the model, will be denoted by the value of the corresponding quantity (in necessary cases with the addition of the index \(m\)—model). Each subsequent specimen of the series, similar to the prototype (model), will be marked by the same letter, but with the index 1 (or \(n\)—full-scale).

Then, in similarity:

geometric (similarity of shape), the ratio of linear scales is

\[ l_c=\frac{l_m}{l_n}=\frac{l}{l_1}=\frac{x}{x_1}=\frac{y}{y_1}=\frac{z}{z_1}; \tag{1} \]

kinematic (similarity of motions and processes occurring in time),

\[ t_c=\frac{t_m}{t_n}=\frac{t}{t_1}; \tag{2} \]

material (similarity of masses),

\[ m_c=\frac{m_m}{m_n}=\frac{m}{m_1}=\frac{\rho l^3}{\rho_1 l_1^3}=\rho_c l_c^3; \tag{3} \]

dynamic (similarity of forces, assuming that the forces are identically oriented at corresponding points),

\[ F_c=\frac{F_m}{F_n}=\frac{F}{F_1}=m_c\frac{l_c}{t_c^2}. \tag{4} \]

Consequently, the ratio of velocities

\[ v_c=\frac{v_m}{v_n}=\frac{v}{v_1}=\frac{l_c}{t_c}, \tag{5} \]

“ accelerations

\[ a_c=\frac{a_m}{a_n}=\frac{a}{a_1}=\frac{l}{t_c^2}, \tag{6} \]

“ numbers of revolutions

\[ n_c=\frac{n_m}{n_n}=\frac{n}{n_1}=\frac{\omega}{\omega_1}=\omega_c=t_c^{-1}, \tag{7} \]

“ works

\[ A_c=\frac{A_m}{A_n}=\frac{A}{A_1}=m_c\frac{l_c^2}{t_c^2}=m_cv_c^2, \tag{8} \]

“ powers

\[ N_c=\frac{N_m}{N_n}=\frac{N}{N_1}=m_c\frac{l_c^2}{t_c^3}. \]

From (9) and (4) it follows that

\[ N_c=\frac{\overline{l_c}\cdot \sqrt{F_c^3}}{\sqrt{m_c}}=\frac{\sqrt{F_c^3}}{l_c\sqrt{\rho_c}}, \tag{10} \]

for \(\rho_c=1\) (equal densities)
and \(l_c=1\) (the same linear scale)

\[ N_c=\sqrt{F_c^3}=F_c^{3/2}. \tag{11} \]

The last expression (11) is of interest for one and the same machine operating in different regimes, if the resistance forces vary in the same way as the inertial forces developed by the machine.

Under the condition \(\rho=1\) and \(l_c=1\), taking into account relations (4) and (7), we obtain:

\[ F_c=n_c^2 \quad \text{and} \quad N_c=n_c^3. \]

The power of a piston machine is:

\[ N=p_{cp}\frac{\pi d^2}{4}\cdot l\cdot \frac{n}{60\cdot 75}\ \text{h.p.}, \]

where \(p_{cp}\) is the mean gas pressure,
\(l\) is the stroke length,
\(d\) is the diameter;

then

\[ N_c=\frac{N}{N_1}=p_c l_c^3 n_c. \]

Usually \(p_c=1\) (equal compression ratios).

Consequently, for \(p_c=1\)

\[ \frac{l_c^3}{N_c}=\frac{1}{n_c} \]

or

\[ \frac{Q}{N}=\frac{c}{n}, \]

where \(c=\text{const}\),

the weight \(Q=\gamma V\), that is, the ratio of the weight or (volume) of the machine to its power is inversely proportional to the number of revolutions of the machine.

For electric motors (at the same current density) it is not difficult to show that

\[ \frac{Q_c}{N_c}=\frac{1}{n_c l_c} \]

or

\[ \frac{Q}{N}=\frac{c}{nl}, \]

or the relative weight of an electrical machine (in relation to its power) is inversely proportional to the number of revolutions and to the linear scale, that is, as Prof. N. D. Luchinskii notes, “in the very nature of electrical machines there are conditions for concentrating their power” [41, p. 46].

The problem of the hand-held electric chipping hammer is ultimately a problem of weight (a problem of the strength of parts at reduced weight). It is necessary to bear in mind the existence of two contradictory tendencies in reducing the relative weight of a hammer with an electric motor built into it, depending on the linear scale and the number of revolutions: when the linear scale is reduced, the relative weight increases; this suggests the need to increase the number of revolutions of the electric motor, that is, to use high-frequency motors.

Let us find the ratio of the powers of two similar pneumatic hammers. Here

\[ F_c=m_c\frac{l_c}{t_c^2}=p_c l_c^2, \]

whence

\[ t_c=\sqrt{\frac{m_c}{p_c l_c}}, \]

therefore,

\[ N_c=\frac{N}{N_1}=m_c\frac{l_c^2}{t_c^3} =m_c\frac{l_c^2}{\left(\sqrt{\frac{m_c}{p_c l_c}}\right)^3} =\frac{l_c^2\sqrt{l_c}}{\sqrt{m_c}}\,p_c\sqrt{p_c} =\frac{l_c}{\sqrt{\rho_c}}\,p_c\sqrt{p_c}. \]

or, for one and the same hammer design, to within a constant \(C\), one can reconstruct the structure of the formula for determining the power of a pneumatic piston machine

\[ N = C d^3 \sqrt{l}\cdot \frac{p\sqrt{p}}{\sqrt{Q_b}}, \]

where
\(Q_b\) — weight of the ram,
\(d\) — diameter,
\(l\) — ram stroke.

We encountered this formula, derived from other considerations, when studying the work of Engineer Yu. M. Malakhov [42].

For similarity of electropneumatic hammers it is necessary that the degrees of compression be the same, that is, \(p_c = 1\); then, with \(p_c = 1\), \(N_c = l_c^2\).

1. We became interested in the question—in what measure is the law of dynamic similarity valid as applied to various types of forging hammers? It turned out that in most cases, for designs of one and the same type, the maximum deviations from the law of similarity for individual indices lie within up to 30% relative to one of the hammers of the given series, taken as the standard.

As an example, let us consider some data from Table 50 of the handbook Mechanical Engineering, vol. 8 [45, p. 386].

Table 1

Main parameters	I (standard)	II	III	IV	V	VI
Hammer ram stroke, in mm	355	430	535	686	850	1065
Linear scale \(l_c\)	1	1,22	1,51	1,94	2,41	3,0
Weight of falling parts, in kg	100	150	250	500	1000	2000
Scale for masses \(m_c\)	1	1,5	2,5	5	10	20
Number of blows per minute	200	170	140	110	90	80
Time scale \(t_c\)	1	1,17	1,43	1,82	2,22	2,5
Electric-motor power, in h.p.	12	15	25	45	80	150
Scale for power \(N_c\)	1	1,25	2,08	3,74	6,55	12,5
Impact energy, in kg·m, approximately	200	350	750	1500	3500	6000
Scale for work \(A_c\)	1	1,75	3,74	7,5	17,4	30
According to the theory of similarity \(A_c = m_c \dfrac{l_c^2}{t_c^2}\)	1	1,62	2,8	5,7	11,8	28,8
Error \(\Delta A_c\)	—	−0,13	−0,94	−1,8	−5,6	−1,2
Error, in %	—	−7,5	−25,1	−24,0	−32,2	−4,0
\(N_c = m_c \dfrac{l_c^2}{t_c^3}\)	1	1,38	1,96	3,13	5,3	11,5
Error \(\Delta N_c\)	—	+0,13	−0,12	−0,61	1,35	−1,0
Error, in %	—	+10,4	−5,80	−16,3	−20,3	−8,0

The greatest discrepancy is found in determining the impact energy (the greatest error \(\approx 32.2\%\); the average value of the error \(\approx 18\)), which re-

partly explained by the approximate data on the energy of a single blow (?!). These data are somewhat overstated, which is also confirmed by the fact that the efficiency of the hammer, although not given in the table, turns out to be very high; for example, for variant V

\[ \text{efficiency}=\eta=\frac{A_{\delta}Z}{60\cdot 75} =\frac{3500\cdot 90}{60\cdot 75\cdot 80}=88\%. \]

According to the data of V. A. Keldyshchev [22] and N. E. Glikin [12], the efficiency of pneumatic hammers with an individual drive is on the order of 60–65%.

The power of the electric motor was measured more accurately: the discrepancy between the values of the actual power of the machine and that calculated according to the theory of similarity, if one of the hammers of the series is taken as the prototype, lies within 8–20% (average error \(\approx 12\%\)).

In the last four machines (variants), in comparison with the first two, the electric motor is somewhat overloaded, which is quite permissible for short-time operation.

From the example given it is clear that dynamic similarity exists among impact machines for one and the same series of machines having identical conditions of uniqueness!

The greatest discrepancy in power (power requirement \(N_c\)) found by us, on the order of \(\approx 30\%\), is in Table 48 of the handbook Mechanical Engineering [45, p. 385].

Similarity of electropneumatic impact machines

1. Let us consider the similarity of electropneumatic impact machines using as an example a hammer of the ZERT scheme (for the electrification of mining labor); from the fundamental point of view, no special difficulties will be encountered in considering other schemes and designs; it is only necessary to know the equations of motion describing the working process of the machine, and in most cases the similarity criteria have the same form if reduced lengths, areas, volumes, etc., are used.

Fig. 1

The hammer operates as follows: the electric motor, through a reduction gear (Fig. 1), by means of crank 1 and connecting rod 2, sets piston 3 into reciprocating motion. When the piston moves to the left, a vacuum is created between the piston and striker 5, and by the excess air pressure the striker is displaced to the left. During the stroke of the piston to the right, compression of the air begins; the striker changes the direction of its motion and, at the end of its stroke, having acquired a certain velocity, delivers a blow to the tool. To preserve stable operation of the hammer, at the moment of impact the air in the hammer cylinder comes into contact with atmospheric or crankcase air through special compensation ports 4, which are a distinctive feature of the design.

Here, in the machine, the air coupling plays the role of an elastic element that does not satisfy Hooke’s law.

Theoretical and experimental studies of individual questions concerning the operation of hammers of this scheme are partially covered in works [1; 2; 5; 17].

In the simplest case, neglecting resistances, the working process of the hammer is described by two nonlinear second-order differential equations with variable coefficients [1].

The equation of motion of the tup is

\[ \frac{Q_b}{g}\cdot a = p_0\left[1-\left(\frac{l_0}{l}\right)^{\alpha}\right]\frac{\pi d^2}{4}, \tag{1} \]

the equation of motion of the motor is

\[ I\varepsilon = M_k + M_p - \frac{\omega^2}{2}\frac{dI}{d\varphi}. \tag{2} \]

Here

\(Q_b\)—weight of the tup, in kg;
\(g = 9.8\ \text{m}/\text{sec}^2\)—acceleration of free fall;
\(a\)—acceleration of the tup;
\(p_0\)—atmospheric pressure, kg/cm²;
\(l_0\)—initial length of the air coupling, cm;
\(l\)—length of the air coupling at the given instant, cm;
\(\alpha\)—polytropic exponent (mean value over the cycle \(\alpha = 1.32\));
\(d\)—diameter of the hammer cylinder, cm;
\(I\)—reduced moment of inertia of the moving masses referred to the crankshaft, kg·m·sec²;
\(M_k\)—torque on the shaft of the electric motor, determined by its characteristic: \(M_k = f(n)\), where \(n\)—number of revolutions per minute;

\[ \varepsilon = \frac{d\omega}{dt} = \frac{d^2\varphi}{dt^2} \]

—angular acceleration of the crankshaft, \(1/\text{sec}^2\),

where \(\varphi\)—angle of rotation of the crankshaft,

\[ \omega = \frac{d\varphi}{dt} = \frac{\pi n}{30} \]

—angular velocity of the crankshaft, which is not constant during one revolution;

\[ M_p = r(p-p_0)\frac{\pi d^2}{4}\sin\varphi + \frac{\lambda}{2}(\sin 2\varphi) \]

—moment of the air pressure acting on the crankshaft, where \(\lambda = \frac{r}{L}\), and \(r\) is the crank radius and

\(L\)—length of the connecting rod.

The solution of equations (1) and (2) can be carried out by approximate methods with any degree of accuracy [1], but the solution is rather cumbersome and labor-intensive and can be performed only for specified machine parameters.

Having a hammer that has been calculated and verified in operation, subsequently, on the basis of the theory of similarity, one can compile a series of machine variants for the given design.

To achieve dynamic similarity, it is necessary to keep in mind the conditions of uniqueness, which are:

1) Geometric conditions, arising from the design of the machine; in the present case the quantities determining the operation of the impact unit are \(r, l, d\), the compensation openings, and the shape of the striker.

2) Physical conditions: the weight of the striker \(Q_{\sigma}\), or the weight of the striker per unit area,

\[ q=-\frac{Q_{\sigma}}{\dfrac{\pi d^2}{4}}; \]

the air pressure in the hammer cylinder is assumed to vary according to the polytropic law

\[ p=p_0\left(\frac{l_0}{l}\right)^{\alpha}, \]

where \(\alpha=1.32\).

3) Initial conditions at \(t_0=0\): \(p=p_0\); \(l=l_0\), \(\omega=\omega_0\), \(v_0=v'_0=0\); \(\varphi=0\) are not essential, since the working process is considered in the established operating regime of the machine, whereas the transient process is of the order of 5–6 revolutions of the crankshaft.

4) The boundary condition for the striker is the change in its velocity at impact, characterized by the value of the coefficient of restitution:

\[ k=\frac{v}{v_0}. \]

Thus, the determining criteria will be those composed of the quantities entering into the conditions of uniqueness, or their ratios; in the present case \(l_c, m_c, k, \alpha\).

Here the boundary condition

\[ k=\frac{v}{v_0} \]

is not connected with the equations of motion and can only be taken into account in an approximate numerical solution of the problem; one of the determining quantities does not enter into the similarity criteria obtained from consideration of equations (1) and (2). Consequently, when considering the operation of similar machines, it is necessary to preserve the same boundary conditions, i.e.,

\[ k=\frac{v}{v_0}=\mathrm{const}. \]

2. For a similar system, with identical values of \(\alpha\) and \(k\), the equation of motion of the striker has the following form:

\[ -\frac{Q_{15}}{g_1}\,a_1 = p_{01}\left[1-\left(\frac{l_0}{l}\right)^{\alpha}\right]\frac{\pi d_1^2}{4}, \tag{1'} \]

since

\[ Q_c=\frac{Q_{\sigma}}{Q_{1\sigma}}=m_c;\qquad g_c=\frac{g}{g_1};\qquad a_c=\frac{a}{a_1}=\frac{l_c}{t_c^2}=l_c\omega_c^2; \]

\[ p_{0c}=\frac{p_0}{p_{01}};\qquad l_c=\frac{l_0}{l_{01}}=\frac{l}{l_1}=\frac{d}{d_1}, \]

then, after substitution, equation \((1')\) takes the form:

\[ \frac{Q_{\sigma}\cdot a}{g}\cdot \frac{g_c}{Q_c l_c\omega_c^2} = \frac{p_0}{p_{0c}} \left[1-\left(\frac{l_0}{l}\right)^{\alpha}\right] \frac{\pi}{4}\cdot \frac{d^2}{d_c^2}. \tag{1''} \]

From comparison of equations \((1')\) and \((1'')\) it follows that, for two machines similar in the dynamic sense, there must be equality of the expressions

\[ \frac{g_c}{Q_c l_c\omega_c^2} = \frac{1}{p_{0c}d_c^2}, \quad\text{or}\quad \frac{p_{0c}g_c d_c^2}{Q_c l_c\omega_c^2} = \frac{p_{0c}g_c}{q_c l_c\omega_c^2} = 1, \]

where

\[ q_c=\frac{q}{q_1}=-\frac{Q_c}{d_c^2}. \]

Whence it follows that, for similar systems, one and the same value must be possessed by the expression (similarity criterion \(\Pi\)):

\[ \Pi=\frac{p_0 g d^2}{\omega^2 Q_0 l_0} =\frac{p_{01} g_1 d_1^2}{\omega_1^2 Q_{16} l_{10}} =\mathrm{idem} \]

(one and the same: a constant dimensionless number, identical for the given group of machines that are similar in the dynamic sense).

Since the pressure in the hammer cylinder varies according to the law of an exponential function \(p=p_0\left(\dfrac{l}{l_0}\right)^\alpha\), consequently, for similar hammers the expressions for the arguments must be identical, that is,

\[ \frac{l_{01}}{l_1}=\frac{l_0}{l}=\mathrm{const}, \]

which is equivalent to the condition of preserving geometric similarity \(l_c=\mathrm{const}\), or, in other words: the condition of similarity of electro-pneumatic machines is the necessity of having identical compression ratios. This condition was first stated by B. V. Sudnishnikov and noted in works [1 and 17].

The condition of identical compression ratios can be written as follows: \(p_c=1\), or

\[ p_c=p_{0c}=1=\frac{Q_c l_c \omega_c^2}{g_c d_c^2} =m_c\,\frac{l_c}{t_c^2}\left(\frac{d_1}{d}\right)^2 \]

(the Sudnishnikov criterion).

The last relation is the condition of force similarity and gives the possibility of departing from complete geometric similarity with respect to the diameter.

Here we shall also note that Eng. E. P. Unksov [64] proposed a method for solving the problem of the motion of the hammer ram by composing an approximate equation describing the working process, and an exact solution of the equation obtained. Linearizing the problem and considering the forced motion of the hammer ram in the presence of a disturbing force from the electric motor, transmitted through the piston and the air coupling, and proceeding from the assumption that at resonance an optimum regime takes place, E. P. Unksov obtains a condition relating the machine parameters under the resonant (“optimum”) regime. In our case, for a hammer of the ZERT scheme, it is not difficult to show that this condition has the form:

\[ \omega^2=\frac{\pi d^2 g p_0 \alpha}{4l_0Q_0} =\frac{1000d^2}{l_0Q_0}, \]

or

\[ \frac{p_0 g d^2 \frac{\pi\alpha}{4}}{\omega^2 Q_0 l_0}=1. \]

Taking the value \(\dfrac{\pi\alpha}{4}=1\). Then the resonant (optimum) regime, characterized by a complex of quantities satisfying the relation

\[ \frac{p_0 g d^2}{\omega^2 Q l_0}=1, \]

is one of a series of regimes of the group of impact machines similar in the dynamic sense and satisfying the condition:

\[ \Pi=\frac{p_0 g d^2}{\omega^2 Q l_0}=\mathrm{idem}, \]

where \(\mathrm{idem}\), within known limits, may have any value (at resonance \(\Pi_{\mathrm{rez}}=1\)).

If, however, we write the condition of similarity in the form of a relation between similarity coefficients, then the characteristic number will in this case be the same as the corresponding criterion under the resonant regime; indeed:

\[ \frac{p_{0c} g_c d_c^2}{Q_c l_c \omega_c^2}=1 . \]

The same also holds for hammers with a spring connection, for which the similarity condition is the relation: \(\frac{Q_c \omega_c^2}{g_c C_c}=1\), while the condition of the resonant regime is expressed by \(\frac{Q\omega^2}{gC}=1\), where \(C\) is the stiffness of the spring.

The similarity criterion for electropneumatic impact machines of the ZERT scheme may be represented in the following form

\[ \Pi=\frac{p_0 g d^2}{Q l_0 \omega^2} \approx \frac{p_0 g \frac{\pi d^2}{4}\cdot \alpha}{Q l_0 \omega^2} = \frac{\alpha p_0}{\frac{Q}{\frac{\pi d^2}{4}}\cdot \frac{l_0\omega^2}{g}} = \frac{\frac{\alpha p_0}{q}}{\frac{l_0\omega^2}{g}} =\tg\alpha=\mathrm{idem}. \]

In the numerator is the ratio of atmospheric pressure to the weight of the striker per unit area; in the denominator is the ratio of the inertial acceleration developed by the machine to the acceleration of free fall.

The value of a series of similarity criteria \(\Pi\) can be represented as a family of straight lines having a common origin in dimensionless coordinates: \(\frac{\alpha p_0}{q}\) and \(\frac{l_0\omega^2}{g}\). Each point of the plane relates six quantities, four of which are determined by the design of the machine. The straight line \(\Pi_{\mathrm{res}}=1\), which is the bisector of a right angle, corresponds to the resonant (optimal) regime.

The table below gives the values of the similarity criterion \(\Pi\) for hammers built according to the ZERT scheme.

Table 2

Hammer name	\(d\), cm	\(l_0\), cm	\(Q_b\), kg	\(\omega=\frac{\pi n}{30}\), \(1/\mathrm{sec}\) (crank)	\(\Pi=\frac{p_b g d_i^2}{\omega^2 Q l_0}\)
ZERT-2	5.5	4.73	0.94	98	0.70
KM-1	6.8	5.8	1.40	89	0.70
OEM-1	7.8	6.8	1.16	73.2	1.42
Hammer ZSFAN	5.47	6	0.8	115	0.46
ZERT-10	3.5	5.8	0.44	125.5	0.30

As is seen from Table 2, for the hammers designed according to the ZERT scheme there exists a definite relation between the parameters, and the first two designs are similar in the dynamic respect \((\Pi=0.7=\mathrm{idem})\).

It should be noted that there is a deviation from the conditions of the optimal “resonance” mode. Here, apparently, either the parameters determining the machine’s working process have not been chosen quite rationally, or the condition for the optimal choice of the parameters of impact machines requires additional investigation. It is possible that the given condition:

\[ \Pi_{rez}=\frac{p_0 g d^2}{\omega^2 Q l_0}=1, \]

which is valid for the case of forced linear oscillations of the striker, must, in the presence of an abrupt connection (impact of the striker and rebound!), have a somewhat different expression with a correction for impact; moreover, the magnitude of the correction coefficient \(C\), apparently, can be determined only experimentally, for a particular case, proceeding from the best operating regime (at maximum efficiency); that is, for an equation characterizing nonuniform motion, with a nonlinear connection between the striker and the piston (in the presence of impact, which increases this nonlinearity), if the same structure of the formula is retained, we shall have:

\[ \Pi_{rez}=C\frac{p_0 g d^2}{\omega^2 Q l_0}=1, \]

where \(C\) is found from experiment.

Having the value of \(C\), we very simply determine, for example, the length of the air cushion

\[ l_0 \approx \frac{100 C d^2}{\omega^2 Q} \]

in the optimal operating regime of the machine.

For a selected value of the diameter \(d\), finding the relation between \(l, Q\), and \(\omega\) is simplified.

The established criterion of the optimal regime is very useful, since it gives the designer the possibility, from both the qualitative and quantitative sides, to relate and evaluate the role of the parameters determining the machine’s working process; in addition, new design schemes can, in preliminary investigation, be related to already known designs.

1. Let us consider the dynamic similarity of the electric motors of hammers with a pneumatic connection between the striker and the piston on the basis of equation (2)

\[ J\varepsilon = M_k + M_p - \frac{\omega^2}{2}\cdot \frac{dJ}{d\varphi}, \]

\[ J_1\varepsilon_1 = M_{1k}+M_{1p}-\frac{\omega_1^2}{2}\cdot \frac{dJ_1}{d\varphi_1}. \]

Denoting

\[ \varepsilon_c=\frac{\varepsilon}{\varepsilon_1},\quad J_c=\frac{J}{J_1};\quad \omega_c=\frac{\omega}{\omega_1};\quad M_{ck}=\frac{M_k}{M_{1k}}=\frac{M_{max}}{M_{1max}}=\frac{A}{A_1}=m_c\frac{l_c^3}{t_c^2} \]

(with identical slip)

\[ M_{cp}=\frac{M_p}{M_{1p}}=p_c l_c^3;\quad M_{ck}= \frac{\dfrac{\omega^2}{2}\dfrac{dJ}{d\varphi}} {\dfrac{\omega_1^2}{2}\dfrac{dJ_1}{d\varphi_1}} = \omega_c^2\cdot \frac{Q_c}{g_c}l_c = J_c\omega_c^2, \]

the equation of the first machine after substitution will have the following expression:

\[ J_c\varepsilon_c J_1\varepsilon_1 = M_{ck}M_{1k}+M_{cp}M_{1p} - M_{cu}\frac{\omega_1^2}{2}\cdot \frac{dJ_1}{d\varphi_1}. \]

For similarity to be fulfilled, it is necessary that

\[ J_c\varepsilon_c=M_{ck}=M_{cp}=M_{cu}=M_c. \]

Consequently, between the moments of the forces acting in similar systems described by equations (2), there exists the relation:

\[ M_c=\varepsilon_c J_c=m_c\frac{l_c^2}{t_c^2}=p_{oc}l_c^3=\omega_c^2\frac{Q}{g_c}l_c^2=\omega_c^2J_c. \tag{2′} \]

Expression (2′) can be used to determine the characteristics of similar electric motors. Let us note that from the relation \(M_c=\varepsilon_cJ_c\) it follows that, for machines similar in the dynamic sense, under conditions of nonuniform motion and variable moments of inertia, there exists between the scale coefficients the same relation which in its form resembles the equation of rotational motion of a rigid body about a fixed axis \((\varepsilon J=M,\ \text{where } J=\mathrm{const})\).

From the relation \(m_c\dfrac{l_c}{t_c^2}=p_{oc}l_c^2\) it follows that, for equal degrees of compression,

\[ p_{oc}=1=m_c\frac{l_c}{t_c^2}\cdot\frac{1}{l_c^2} =m_c\frac{l_c}{t_c^2}\left(\frac{d_1}{d}\right)^2 \quad\text{(the condition of B. V. Sudnishnikov).} \]

Dividing expression (2′) term by term by \(\omega_c^2J_c\), we obtain a series of characteristic numbers:

\[ \frac{M_c}{\omega_c^2J_c} = \frac{\varepsilon_c}{\omega_c^2} = \frac{m_cl_c^2}{J_c\omega_c^2t_c^2} = \frac{p_{oc}l_c^3}{J_c\omega_c^2} = \frac{Q_cl_c^2}{g_cJ_c} =1 \]

and the corresponding series of similarity criteria.

Dividing each term of expression (2′) by \(Q_cl_c\), we obtain:

\[ \frac{M_c}{Q_cl_c} = \frac{J_c\varepsilon_c}{Q_cl_c} = \frac{l_c}{g_ct_c^2} = \frac{p_c}{q_c} = \frac{\omega_c^2l_c}{g_c} = \frac{J_c\omega_c^2}{Q_cl_c}. \]

Here the three last expressions are of greatest interest:

\[ \frac{p_c}{q_c} \]
— the ratio of the forces of air pressure and unit weight;

\[ \frac{\omega_c^2l_c}{g_c} \]
— the ratio of the inertia forces developed by the machine to the force of gravity;

\[ \frac{J_c\omega_c^2}{Q_cl_c} \]
— the ratio between the energy of the machine and the energy of the striker in free fall.

A graphical representation of the underlined relations between the similarity coefficients is given in Fig. 2. From this same relation it is not difficult to obtain the already known characteristic number

\[ \frac{p_{oc}g_c}{q_c\omega_c^2l_c} = \frac{p_{oc}g_cd_c^2}{Q_c\omega_c^2l_c} =1, \]

obtained earlier from the equation of motion of the striker (1).

1. The calculation of a hammer by the method of dynamic similarity is based on a manufactured and tested hammer specimen whose parameters are known. Let us note that in the design of the rebound part of a coal-cutting pick with an “active edge,” by Stalin Prize laureate P. M. Emelyanov, hammer samples KM—1 were adopted as the initial models.

and ZERT-2. The KM-1 spike-driving hammer, constructed according to the ZERT scheme, has the following parameters: \(A_5 = 3.2\ \text{kg·m}\); \(n = z = 850\ \text{blows/min}\); \(r = 32\ \text{mm}\); \(l_0 = 58\ \text{mm}\); \(d = 68\ \text{mm}\); \(Q_b = 1.4\ \text{kg}\); \(N_{\mathrm{eff}} = 800\ \text{W}\) at the motor shaft.

\[ \frac{p_c}{q_c}=\frac{\omega_c^2 l_c}{g_c} =\frac{J_c\omega_c^2}{p_c Q_c} \]

Fig. 2

According to the technical specifications, for one of the principal variants it was necessary to have:

\[ n_1 = 760\ \text{blows/min}; \qquad Q_{1b} = 2.47\ \text{kg} \quad \text{and} \quad d'_1 = 85\ \text{mm}. \]

Then

\[ t_c=\frac{t}{t_1} = \frac{\dfrac{1}{850}}{\dfrac{1}{760}} = \frac{760}{850} =0.895, \]

\[ m_c=\frac{Q_b}{Q_{1b}} = \frac{1.4}{2.47} =0.567 \qquad (\text{since } g_c=1), \]

\[ F_c=\frac{F}{F_1} =p_0 l_c^2 =1\left(\frac{d}{d_1}\right)^2 = \left(\frac{68}{85}\right)^2 =0.64. \]

We determine \(l_c, A_c, N_c\)

\[ l_c=\frac{F_c\cdot t_c^2}{m_c}=\frac{0,64\cdot 0,895^2}{0,569}=0,9, \]

\[ \frac{d}{d_1}=\frac{68}{85}=0,8\ne l_c, \]

\[ A_c=m_c\frac{l_c^2}{t_c^2}=0,567\cdot\frac{0,9^2}{0,895^2}=0,576, \]

\[ N_c=m_c\frac{l_c^2}{t_c^3}=0,567\cdot\frac{0,9^2}{0,895^3}=0,65. \]

We check fulfillment of the condition of equality of compression ratios:

\[ p_c-1=m_c\frac{l_c}{t_c^2}\left(\frac{d_1}{d}\right)^2 = \frac{0,565\cdot 0,9\cdot\left(\frac{85}{68}\right)^2}{0,895^2} =1, \]

and also the value of the similarity criterion

\[ \Pi=\frac{p_0 g d^2}{Q\omega^2 l_0} = \frac{p_0 g d_1^2}{Q_1\omega_1^2 l_{01}} =\mathrm{idem}=0,7. \]

The results of the calculation are given in Table 3.

Table 3

Name of quantity	Dimension	Designation	Initial data	Scale coefficient: designation	Scale coefficient: value	Parameters of the new machine
Work per striker	kgm	\(A_b\)	3,2	\(A_c\)	0,575	\(A_{1b}=3,2:0,577=5,6\)
Number of blows	blows/min	\(z\)	850	\(t_c\)	0,895	\(z_1=850\cdot0,895=760\)
Radius	cm	\(r\)	32	\(l_c\)	0,9	\(r_1=32:0,9=36\)
Length of air cushion	cm	\(l_0\)	58	\(l_c\)	0,9	\(l_{10}=58:0,9=65\)
Weight of striker	kg	\(Q_b\)	1,4	\(m_c\)	0,567	\(Q_{1b}=1,4:0,567=2,47\)
Diameter	mm	\(d\)	68	\(\sqrt{F_c}\)	0,8	\(d_1=68:0,8=85\)
Power	watt	\(N\)	800	\(N_c\)	0,65	\(N_1=800:0,65=1230\)

Because of the absence of geometric similarity (with respect to the diameter \(d\)), the ratio of the densities of the strikers is

\[ \rho_c=\frac{m_c}{l_c^3}=\frac{0,567}{0,73}=0,79. \]

Consequently, the new striker must be longer in order to preserve the weight, or be made of a denser material, which is also corrected by filling the striker with lead.

The task of studying the working process of electropneumatic machines of impact action is not only a dynamic problem. To a certain extent it is also a thermodynamic problem.

The operation of the machine is accompanied by energy losses in all links and units of the machine; energy losses ultimately turn into heat. The overall efficiency of electropneumatic hammers and perforators is, at best, on the order of 40–50%; the efficiency of the air connection is on the order of 70–80% [2].

The application of the theory of similarity in locomotive heat engineering is very extensively considered in the work of P. K. Konakov [29]; the methodology developed there can be applied in its entirety to the processing of test results for pneumatic hammers, rock drills, and forging hammers, where air acts as the energy carrier instead of steam. A specific feature of the operation of an electropneumatic hammer is that air performs the role of the medium transmitting energy from the electric motor to the striker; that is, it acts as an elastic coupling and replaces the steel spring in the machine. From a fundamental point of view, the energy of the electric motor could be replaced by the energy of a wind motor, a hydraulic motor, an internal-combustion engine, etc.

Fig. 3

Since we are interested in the operation of air as an elastic element in an impact machine, let us note the principal parameters of air that determine its state. These are: the weight of the air \(G\) in kg; the volume \(V\) in m\(^3\); the pressure \(p\) in kg/cm\(^2\); the temperature \(T_{\mathrm{abs}} = 273^\circ + t^\circ\)C.

When selecting the parameters of a machine, it is necessary to take into account the heating of the air, if only to prevent coking of the lubricant (one must know the flash temperature of the oil!). The change in the temperature of compressed air in an adiabatic process is shown in Fig. 3 [22, p. 34].

Let us use the thermodynamic relation, known for the work of air,

\[ L = \frac{R}{\alpha - 1}(T_0 - T) \]

[37, p. 86], which in our case, for a hammer of the ZERT scheme, has the form:

\[ A_b = \frac{Q_b V_b}{2g} = \frac{R}{\alpha - 1}(T_0 - T)G, \]

where \(R = 29.27\ \text{kg·m/kg·deg.}\) is the gas constant of air;

\[ G = l_0 \frac{\pi d^2}{4}\,\gamma \cdot 10^{-6}\ \text{kg} \]

is the weight of the air in the hammer cylinder between the piston and the striker;

\(T\) is the absolute temperature of the air at the moment of maximum compression;

\(T^\circ\) is the absolute initial temperature of the air cushion.

In addition, for the stability of the hammer’s operation, the role of compensation of the air in the hammer cylinder by atmospheric air is of extremely great importance.

For the selection of the parameters of impact machines by the similarity method, it is useful to present a number of relations in the form of tables, graphs, and nomograms.

Application of Dimensional Theory to the Study of Impact Machines

1. Dimensional analysis, as a method for the preliminary study of natural phenomena, is often used in the study of sections of physics, mechanics, hydrodynamics, aerodynamics, and has become firmly established in educational and scientific literature.

literature [6; 7; 56; 57; 58; 70]. Without dwelling on the variety of viewpoints concerning the doctrine of dimension, and sufficiently fully covered in the work of Prof. M. F. Malikov [43], let us note that, on the basis of dimensional analysis, it is possible to establish “the structure of functional relationships between physical quantities” [57], even in the case when there is no differential equation describing the process; moreover, it is possible to establish relationships not only between the physical quantities characterizing the phenomenon, but also between their dimensionless combinations, that is, relationships between similarity criteria. In this way, the formulation of the experiment, its direction, and the processing and systematization of the experimental data obtained are facilitated.

Prof. K. M. Polivanov writes: “The methods of the theory of similarity and of dimensional analysis are extremely close, and the difference between them can be discerned only in the fact that dimensional analysis does not consider the basic equations of precisely the given particular system, the processes in which can be determined only by part of the general equations underlying the chosen system of units” [51, p. 233], and somewhat further: “…However, even this difference is erased in those cases when, for greater definiteness of the solution, in the analysis one uses a system of units specially constructed so as to be most convenient for analyzing the processes of precisely the given system” [51, p. 234].

On the basis of dimensional analysis it is easy to verify the correctness of equations. The principle of similarity is closely connected with the principle of homogeneity. However varied in character the acting forces may be, their homogeneity is beyond doubt.

The method of zero dimensions is based on the theorem of homogeneity of formulas of physical quantities. On the basis of this method, which is a variety of dimensional analysis, one can establish a relationship between dimensionless quantities—similarity criteria, sometimes with accuracy up to a constant determined from experiment.

If there exists a functional relation between a number of dimensional \((u)\) and dimensionless \((c_i)\) quantities of the form: \(f(u_1,u_2,\ldots,u_n;c_1,c_2,\ldots,c_m)=0\), then from this equation any quantity can be isolated and represented as a function of the remaining quantities, for example:

\[ u_1=f_1(u_2,u_3,\ldots,u_n;c_1,c_2,\ldots,c_m). \tag{1} \]

Let us choose from among the arguments any three mutually independent quantities, for example \(u_2,u_3,u_4\), and take them as the basic units of our system; then the remaining quantities \(u_1,u_5,\ldots,u_n\) will be derived quantities in this new system of units and can be expressed through the basic units \(u_2,u_3,u_4\). The dimension of \(u_1\) in the new system will be \([u_1]=u_2^p u_3^q u_4^r\); equation (1) must possess triple homogeneity with respect to \(u_2,u_3,u_4\).

The quantities \(u_2,u_3\), and \(u_4\) are mutually independent if the determinant composed of the exponents of their dimensions is not equal to zero, i.e.

\[ \Delta= \begin{vmatrix} p_1 & q_1 & r_1\\ p_2 & q_2 & r_2\\ p_3 & q_3 & r_3 \end{vmatrix} \ne 0. \]

Replacing in (1) the values \(u_2, u_3, u_4,\ldots,u_n\) by the ratios of these quantities to their dimensions in the new system \((u_2,u_3,u_4)\), we obtain:

\[ \frac{u_1}{u_2^{p}u_3^{q}u_4^{r}} = \Phi\left(1,1,1,\frac{u_5}{[u_5]_{2,3,4}},\ldots,\frac{u_n}{[u_n]_{2,3,4}};\ c_1,c_2,\ldots,c_m\right). \]

Here

\[ \frac{u_i}{[u_i]_{2,3,4}} \]

denotes the quantity \(\Pi_i\) (similarity criterion), equal to the ratio of the quantity \(u_i\) to its dimension in the system of units \((u_2,u_3,u_4)\), and it is obvious that

\[ \frac{u_2}{[u_2]_{2,3,4}}=1;\quad \frac{u_3}{[u_3]_{2,3,4}}=1;\quad \frac{u_4}{[u_4]_{2,3,4}}=1. \]

There is a dependence between the dimensionless expressions:

\[ \Pi_1=\Phi(\Pi_5,\Pi_6,\ldots,\Pi_n), \]

which is precisely the expression of the \(\Pi\)-theorem.

Or

\[ u_1=u_2^{p}u_3^{q}u_4^{r}\,\Phi(\Pi_5,\Pi_6,\ldots,\Pi_n,c_1,c_2,\ldots,c_m). \]

If the number of physical parameters is equal to \(n\), and the number of fundamental units is equal to \(k\) (usually \(k=3\)), then the number of independent dimensionless expressions will be \(n-k\).

Obviously, the smaller the difference \(n-k\), the more easily and confidently one can find the form of the function \(\Phi\) relating the dimensionless quantities.

1. Let us find the similarity criteria for an electropneumatic hammer of the ZERT system. We shall do this for comparison of the results and for greater confidence in the correctness of the conclusions, and for those cases when there is no equation describing the physical process, for example, in drilling rock, etc.

The working process of an electropneumatic hammer, in a first approximation, is determined by a number of quantities between which there exists a functional relation, written in general form as:

\[ \Phi(\omega,r,l,d,Q,g,J,p,\alpha,k)=0. \]

Let us choose, as the fundamental units of our system, any three quantities independent of one another, for example: \(\omega,l,Q\) (it is not difficult to show that the determinant \(\Delta\ne0\)).

Then, in dimensionless form, the function \(\Phi\) will have the expression:

\[ \Phi\left(\frac{r}{l},\frac{d}{l},\frac{g}{\omega^2l},\frac{J\omega^2}{Ql},\frac{pl^2}{Q},\alpha,k\right)=0. \]

For given values of the parameters \(r,d,\alpha\) and \(k\), we have

\[ \Phi\left(\frac{g}{l\omega^2},\frac{J\omega^2}{Ql},\frac{pl^2}{Q}\right)=0. \]

Denoting

\[ \Pi_1=\frac{g}{l\omega^2} \]

—the criterion indicating how many times the forces of gravity are smaller than the inertial forces developed in the machine;

\[ \Pi_2=\frac{J\omega^2}{Ql} \]

—the criterion indicating the relation between the kinetic energy of the machine and the work of the striker during free fall from a height equal to \(l\);

\[ \Pi_3=\frac{pt^2}{Q}=\frac{p}{\dfrac{Q}{l^2}}=\frac{p}{q} \]

—a criterion giving the ratio between the forces of air pressure and the weight of the striker per unit cross-sectional area.

The form of the function \(\Phi(\Pi_1,\Pi_2,\Pi_3)=0\) is determined from experiment. The result of the experimental data may be represented by a nomogram, for example, in the coordinates \(\Pi_1\) and \(\Pi_2\) with the parameter \(\Pi_3\) (abacus), or spatially in the form of a surface in dimensionless coordinates: \(\Pi_1,\Pi_2\), and \(\Pi_3\).

Comparison of the solutions obtained on the basis of the theory of similarity and the theory of dimensions makes it possible to conclude that the structure of the similarity criteria is one and the same, but finding the relationship between the similarity criteria and extending the results obtained in the latter case is more difficult.

The working process of a forging hammer with an individual drive is characterized by the following quantities: the weight of the hammer ram \(Q\), as well as the weight of the hammer itself \(Q_m\), the ram stroke \(s\), the number of blows \(z\), the work of the blow \(A_b\), and the power of the machine \(N\), which may be written briefly as:

\[ \Phi(Q,Q_m,s,z,A,N)=0. \]

If, as the basic independent quantities, we take \(Q,s,z\), then the functional relationship between the dimensionless arguments, whose number is reduced by three, will have the form:

\[ \Phi\left(\frac{Q_m}{Q},\frac{A}{Qs},\frac{N}{Qsz}\right)=0. \]

It is not difficult to see that the efficiency is

\[ \text{efficiency}=\eta=\frac{Az}{N}=\frac{A}{\dfrac{N}{z}}=\frac{\dfrac{A}{Qs}}{\dfrac{N}{Qsz}}=\tg \varphi . \]

The function \(\Phi\) is found on the basis of experimental data. As an illustration we shall use the data already known to us from Table 50 of the handbook Machine Building [45].

Figure 4 shows the characteristics of forging hammers, constructed by us in dimensionless coordinates. Each point of the plane relates all the indicated quantities to one another, although the parameter region of the constructed hammer machines is comparatively small. On the basis of a nomogram constructed from a comparatively small number of points for a given series of machines, by interpolation one can easily select the parameters of a new machine.

Figure 5 gives the characteristics of hammers according to the data of V. A. Keldysh [22]; since the table lacks the value of the piston stroke \(s\), the distance between the lower striker and the lower edge of the ram guide has been taken as the characteristic length. Here the ratio of the hammer weight to the ram weight lies within the limits of the relation:

\[ \frac{Q_m}{Q}=30\div 40. \]

The nomograms presented for forging hammers indicate the direction of experimental investigations, of material processing, and of the selection of parameters of electropneumatic hammers and perforators.

The work on the striker depends on the power of the electric motor \(N\), the angular velocity of the crank \(\omega\), the weight of the striker \(Q\), the feed force \(P\), the length of the air cushion \(l\), the cylinder diameter \(d\), and the material on which the blow is performed, with coefficient of restitution \(k\).

or

\[ A_{\sigma}=\Phi(N,\omega,Q,P,l,d,r,k), \]

if, as the basic independent quantities, we take \(\omega, l, Q\), then

\[ \frac{A_{\sigma}}{Ql}=\Phi\left(\frac{N}{\omega Ql},\frac{P}{Q},\frac{d}{l},\frac{r}{l},k\right). \]

For a selected value of the diameter \(d\) and a definite value of \(k\),

\[ \frac{A_{\sigma}}{Ql}=\Phi\left(\frac{N}{\omega Ql},\frac{P}{Q},\frac{r}{l}\right). \]

Fig. 4. Characteristics of forging hammers in dimensionless coordinates (according to the data of table No. 50, Handbook of Mechanical Engineering, vol. 8).

Designations:

\[ \circ\;-\;\frac{75\cdot 60\cdot N}{Q\cdot S\cdot Z} =f\left(\frac{Q_{m}}{Q}\right); \qquad \bullet\;-\;\frac{75\cdot 60\cdot N}{Q\cdot S\cdot Z} =f\left(\frac{A}{Q\cdot S},\eta\right). \]

Example. For point “\(K\)” we find:

\[ \frac{Q_{m}}{Q}=25;\qquad \frac{A}{Q\cdot S}=4.15;\qquad \frac{75\cdot 60\cdot N}{Q\cdot S\cdot Z}=4.7;\qquad \eta=\operatorname{tg}\varphi=48\%. \]

The form of the function \(\Phi\) is determined from experiment; moreover, for certain invariable quantities the number of arguments decreases, and thereby the determination of the functional relation between them is facilitated; for example, at an unchanged value of

\[ \frac{P}{Q} \]

\[ \frac{A_{\sigma}}{Ql}=\Phi\left(\frac{N}{\omega Ql},\frac{r}{l}\right). \]

The latter relation, on the basis of experimental data, is not difficult to represent graphically in dimensionless coordinates.

Further, for the selected operating mode of the machine, with its given parameters, a special experimental study is needed to establish the relation between the work on the ram and the material on which the blow is delivered, i.e., to find \(A_b = Ql\Phi(k)\).

Of very great importance is the question of choosing the parameters \(l_0, d, r\), which determine the operation of the impact unit for a prescribed value of the work of a unit blow \(A_b\) and the number of blows \(z\), or the value

\[ \omega=\frac{\pi n}{30}=\frac{\pi z}{30}. \]

Fig. 5. Characteristics of forging hammers in dimensionless coordinates
(according to data of V. A. Keldyushev [22]).

Designations:

\[ \cdot\!-\ \frac{75\cdot 60\cdot N}{Q\cdot l\cdot Z} = f\!\left(\frac{Q_m}{Q}\right); \qquad \bullet\!-\ \frac{75\cdot 60\cdot N}{Q\cdot l\cdot Z} = f\!\left(\frac{A}{Q\cdot l},\,\eta\right) \]

Example. For point “\(K\)” we find:

\[ \frac{Q_m}{Q}=0.35;\qquad \frac{A}{Q\cdot l}=5.5;\qquad \frac{75\cdot 60\cdot N}{Q\cdot l\cdot Z}=8;\qquad \eta=\tg\gamma=69\%. \]

The velocity at the moment of impact depends on the following quantities

\[ v=f(\omega,r,l,Q_b d,\rho_0,P,k). \]

Choosing \(\omega\), \(r\), and \(Q_b\) as the fundamental independent units of the system, we obtain:

\[ \frac{v}{r\omega} = f\!\left( \frac{d}{r},\, \frac{\rho_0 r^2}{Q},\, \frac{l}{r},\, \frac{P}{Q},\, k \right). \]

Here it is useful to determine the dependence on the feed force and the coefficient of restitution at impact

\[ \frac{v}{r\omega} = f_1\!\left( \frac{P}{Q},\,k \right). \tag{1} \]

with the remaining data unchanged, and also

\[ \frac{v}{r\omega}=f_2\left(\frac{p_0}{q},\,\frac{l}{r}\right). \tag{2} \]

In addition, it is necessary to find

\[ \frac{\omega^2 r}{g}=\Phi\left(\frac{p_0}{q}\right). \tag{3} \]

Then, for given \(A_\delta\) and \(z\), we choose \(Q\) and \(q\).

Consequently, the values are determined as

\[ v=\sqrt{\frac{2A_\delta}{Q}} \quad \text{and} \quad d=\sqrt{\frac{4Q}{\pi q}};\quad \omega=\frac{\pi n}{30}. \]

From (3) we determine \(r\), and from (2), \(l\).

As a check condition we use the thermodynamic expression:

\[ A_\delta=\frac{Q_\delta v_\delta^2}{2g} =\frac{R}{a-1}\left(T_0-T\right)\frac{\pi d^2}{4}\gamma l_0\,10^{-6}. \]

The dependences (1), (2), and (3), found from experimental data, should be presented in the form of graphs and nomograms.

It would be very interesting to have graphs of relations (2) and (3) over the period of one cycle, knowing from experience the changes in \(l\), \(\omega\), and \(v\) during one revolution of the crankshaft.

1. Let us see what indications the theory of dimensions gives with respect to the study of the destruction of rocks. Let us consider this question from the standpoint of penetration rate and power consumption.

According to the assertion of Prof. L. A. Schreiner that the drilling rate is inversely proportional to hardness \([72]\), it follows that for hard rocks the degree of their crushing (dispersion \(\beta\)) must also be increased. Further, if we take into account the considerations of Prof. P. M. Tsimbarevich \([68]\) on the theory of drillability, then the drilling rate is (to a first approximation) a function of the following quantities:

\[ v=\Phi(N,D,d,\beta), \]

where

\[ \begin{aligned} v &\text{— drilling rate, } \mathrm{cm/sec};\\ N &\text{— power, } \mathrm{kg\,m/sec};\\ D &\text{— drillability, } \mathrm{cm^3/kg\cdot m};\\ d &\text{— bit diameter, } \mathrm{cm};\\ \beta &\text{— dispersion, } 1/\mathrm{mm}. \end{aligned} \]

If we take \(N\), \(D\), and \(d\) as the basic independent units (it is not difficult to show that \(\Delta \ne 0\)), then we obtain:

\[ v=\frac{ND}{d^2}\cdot \Phi(d\beta), \]

the form of the function \(\Phi\) being determined from experiment.

Since the crushing (dispersion) of the rock is inversely proportional to its hardness \(R_z\), \(\mathrm{kg}/\mathrm{cm^2}\), and if we set \(\Phi(d\beta)=\dfrac{k}{R_z}\), where \(k\)—

a dimensional constant determined from experiment, the drilling formula will take the form

\[ v = k \frac{ND}{d^3 R_z} = c \frac{N}{d^2} \]

(the Medvedko formula [46]).

Here the coefficient \(c = \frac{k}{R_z}\). \(D\) must be found from experiment.

Under the indicated assumptions, the drilling rate is proportional to the power and inversely proportional to the square of the diameter. Incidentally, we note that the work of fracture of brittle bodies is inversely proportional to the dispersity of the material being broken (Rittinger’s law), and, according to the law of similarity (V. L. Kirpichev), is proportional to the volumes of the deformable bodies. As L. A. Schreiner notes, when the linear dimensions of a body are reduced, its strength increases: “The law of similarity with a correction for the scale factor must be consistent with practical data. Nothing else can be expected, since physically it is fully justified, as is Rittinger’s law” [72, p. 161]. A. I. Ter-Grigoryan notes: “None of the proposed formulas determining the drillability of rocks can claim completeness. The complex problem of establishing the drillability of rocks may be considered solved only when an equation is given that relates the mechanical drilling rate to the physical properties of the rock and the medium, the elements of the drilling regime, the parameters of the bit and its variable wear” [61; 36].

On the basis of familiarity with the literature on the fracture of rocks [8; 65; 74], as well as with the solid-state physics of Corresponding Member of the Academy of Sciences of the USSR V. D. Kuznetsov [36] and the mechanics of rocks of Prof. P. M. Tsimbarevich [68], the author attempts to consider the question of the drilling formula from the standpoint of dimensional theory.

The drilling rate \(v\) depends on the quality of the tool, the drilling regime, the properties of the rock being broken, the development system, and the organization of the work. We shall consider the physicotechnical side of the question; then the drilling rate is

\[ v = f(Q_b, d, l, \beta, e,\ \Theta,\ R_z, \mu, \lambda,\ N, M_{kr}, A_b, P, \omega, v_b, z), \]

where

\[ \begin{aligned} Q_b &\text{— weight of the bit (drill), kg;}\\ d &\text{— bit diameter, cm;}\\ l &\text{— bit length, cm;}\\ \beta &\text{— bit sharpening angle, }[\beta]=1;\\ e &\text{— number of bit wings, }[e]=1;\\ \Theta &\text{— wear of the bit due to friction forces, kg/cm}^2;\\ R_z &\text{— absolute hardness of the rock, kg/cm}^2;\\ \mu &\text{— coefficient of friction between the medium and the bit, }[\mu]=1;\\ \lambda &\text{— coefficient characterizing cleaning of the blast-hole, }[\lambda]=1;\\ N &\text{— power consumed by the machine, hp (kW);}\\ M_{kr} &\text{— torque on the bit, kg/m;}\\ A_b &\text{— work of one blow on the striker, in percussion-rotary drilling, kg/m;}\\ P &\text{— feed force, kg;}\\ \omega &\text{— angular velocity of rotation of the bit, }1/\text{s};\\ v_b &\text{— velocity of the striker at the moment of impact, cm/s;}\\ z &\text{— number of striker blows per min.} \end{aligned} \]

Here we have thirteen dimensional quantities and four dimensionless ones. We take as the basic independent quantities: \(P,\ d,\ \omega\) \((\Delta \ne 0)\).

Thus, in the given formulation of the problem we have ten dimensionless expressions \((n-k=13-3=10)\), interconnected by some function \(\Phi\)

\[ \frac{v}{d\omega} = \Phi\left( \frac{Q_b}{P},\, \frac{l}{d},\, \frac{\Theta}{P d^{-2}},\, \frac{R_z}{P d^{-2}},\, \frac{N}{P d\omega},\, \frac{M_{kr}}{P d},\, \frac{A_b}{P d},\, \frac{v_b}{d\omega},\, \frac{z}{\omega} \right). \]

In the general case, finding the function \(\Phi\) is very difficult. From the dimensionality of the arguments, the idea arises of a connection between

1) bit wear and rock hardness, i.e. \(\Theta=\psi(R_z)\);

2) work on the bit and torque

\[ M_{kr}=\psi_2(A_b); \]

3) machine power, bit speed, and feed force, that is \(N=\psi_3(P,\ v_b)\).

In order to reveal the basic connections and relationships, let us introduce certain simplifying assumptions.

In rotary drilling the role of \(A_b\) is not significant; then, for the same tool parameters and the same operating mode of the drilling machine, neglecting wear, we have:

\[ v=\omega d \cdot \Phi\left(\frac{R_z}{P d^{-2}}\right), \]

where the form of the function \(\Phi\) is found from experiment.

If we assume that

\[ \Phi\left(\frac{R_z}{P d^{-2}}\right)=\frac{P d^{-2}}{R_z}, \]

the guess, or guiding idea, is the obvious condition that the rate of advance decreases as the hardness of the rock increases; then the drilling speed will be inversely proportional to the hardness of the rock, i.e.

\[ v=\omega d \cdot \frac{P}{R_z d^2} = \frac{\omega P}{R_z d} = K\frac{P}{R_z}\,^{1)}. \]

If, at one and the same rock hardness, we are interested in the rate of advance as a function of power, then it is not difficult to obtain the structure of the formula

\[ v=d\omega \Phi\left(\frac{N}{P d\omega}\right), \]

if we set

\[ \Phi\left(\frac{N}{P d\omega}\right)=K\frac{N}{P d\omega}, \]

where \(K=\mathrm{const}\), determined from experiment,

\(^{1)}\) According to the data of Prof. L. A. Schreiner, the volumetric rate of destruction is

\[ v=\alpha \cdot \frac{\omega}{p} P = K' \frac{P}{p}. \]

Here \(\alpha\) is a dimensional coefficient; \(\omega\) is the displacement velocity; \(p\)-\(R_z\) is the indentation hardness; \(P\) is the load; \(K'\) is a dimensional coefficient [74, pp. 149--151].

According to L. A. Schreiner, when developing rational drilling regimes one should proceed from the hardness of rocks.

then

\[ v = K \frac{N}{P}, \]

i.e., at one and the same power, the drilling rate is inversely proportional to the feed force (with excessive pressure, the friction forces between the bit and the rock are large). Under the optimum regime, the magnitude of the axial pressure must correspond to the power of the drilling machine [8]. For one and the same impact energy, for one and the same medium, the drilling rate is, within known limits, proportional to the number of impacts

\[ v = d \omega \cdot \Phi\left(\frac{z}{\omega}\right), \]

if we set

\[ \Phi\left(\frac{z}{\omega}\right)=K\frac{z}{\omega}, \quad \text{then} \quad v=Kzd=K'z. \]

Preservation of the linearity of the rate of advance with respect to the number of impacts or revolutions over broader limits is, obviously, accompanied by a simultaneous increase in the pressure and power of the drilling machine; in fact:

\[ v = K \frac{N}{P}=K'z. \]

With the combined action of a number of factors on the drilling process, finding the function \(\Phi\), which characterizes the drilling rate \(v\), is complicated and, evidently, the principal regularities will most likely be revealed experimentally, with rational processing of the experimental data.

From the structure of the formula for the function \(\Phi\), the idea arises of setting up experiments to reveal relations of the form:

\[ v = \omega d \cdot \Phi_1\left(\frac{Q_b}{P},\ \frac{v_b}{\omega d},\ \frac{A_b}{Pd}\right), \tag{1} \]

\[ v = \omega d \cdot \Phi_2\left(\frac{N}{\omega Pd},\ \frac{M_{kr}}{Pd},\ \frac{z}{\omega}\right), \tag{2} \]

\[ v = \omega d \cdot \Phi_3\left(\frac{R_z}{Pd^{-2}},\ \frac{\Theta}{Pd^{-2}},\ \mu,\ \lambda\right), \tag{3} \]

\[ v = \omega d \cdot \Phi_4\left(\frac{l}{d},\ \beta,\ e,\ \frac{Q_b}{P}\right). \tag{4} \]

According to formula (1), the drilling rate is found for percussion and percussion-rotary drilling and, as a special case, only for rotary drilling, with \(A_b=0\), as a function of the ratio of the colliding masses, the impact energy, and the deformation rate; according to formula (2), the drilling rate is determined as a function of the relation between dimensionless quantities that include, as the defining ones, power, torque, and number of revolutions; according to formula (3), the rate of destruction is determined as a function of tool wear and the properties of the medium; formula (4) gives an idea of the drilling rate as a function of the bit parameters and axial pressure; finally, in order to link these formulas with one another, one can set up a series of additional experiments and, accordingly, reveal a number of additional dependences, for example

\[ v = d\omega \cdot \Phi_5\left(\frac{Q_b}{P},\ \frac{N}{\omega dP},\ \frac{R_z}{Pd^{-2}}\right), \tag{5} \]

and others.

On the basis of the experimental material obtained, one can construct a series of tables, graphs, and nomograms which, as a first approximation, reflect the complex process of drilling in parts and from various aspects. For example, one can determine the influence, on the magnitude of the power of the machine performing the drilling, of the following criteria:

\[ N = P_{\omega} d \cdot f_1 \left( \frac{R_z}{P d^{-2}}, \frac{M_{kp}}{P d}, \frac{A_b}{P d} \right); \]

\[ N = P_{\omega} d \cdot f_2 \left( \frac{z}{\omega}, \psi, \lambda \right); \]

\[ N = P_{\omega} d \cdot f_3 \left( \frac{l}{d}, \frac{Q_b}{P}, \beta, e \right). \]

Undoubtedly, it would be more desirable to have a single formula relating all the above-mentioned quantities; but, as is noted in the literature, there is as yet no satisfactory formula, and at the present stage of development of the science of drilling theory it is hardly possible in the near future to obtain a single universal formula comprehensively encompassing the complex (many-sided) process of drilling—a formula sufficiently simple and convenient for practical applications.

It is not useless to recall the statement of the major Russian scientist—theoretician, practical engineer, and founder of mine-plant mechanics I. A. Time—that “... the magnitude of the errors is proportional to the number of coefficients, and therefore simpler (approximate) formulas, corrected by a smaller number of coefficients, may often prove in practice to be more accurate than more complex ones” [63, p. 5].

The relationships established on the basis of dimensional theory in the form of generalized dependencies make it possible to single out the most general and essential: the general structure of formulas indicating the direction of experimental investigations and of the processing of the materials obtained, on the basis of which the existing regularities between the medium being destroyed and the tool are revealed, or, more broadly, between the rock being extracted and the machine producing the destruction.

On the basis of dimensional theory and similarity theory it is possible to create a general methodology for conducting experimental work on drilling, as well as to systematize and generalize the available materials.

The scientific foundations of drilling regimes by individual stages have been most fully elucidated by Prof. V. S. Fedorov [65]. Unfortunately, the book does not describe drilling regimes with all the data characterizing the drilling process.

Before proceeding to consideration of the question of the physical quantities determining the operation of an air cushion, and to finding the functional relationship between them, as an example of applying the method of zero dimensions, let us find the equation of the thermodynamic state of an ideal gas, if it is known that it contains four quantities—\(R, T, v, p\), whose dimensions in the technical system of units \((\kappa—l—t)\) are as follows:

gas constant \(R\), kg·m/kg·deg, or

\[ [R] = \frac{1}{\kappa}, \qquad \text{whence } \kappa = \frac{1}{R}; \]

temperature \(T\), kg·m, or

\[ [T] = \kappa l, \qquad \text{whence } l = \frac{T}{\kappa} = TR; \]

specific volume \(v\), m\(^3\)/kg, or

\[ [v] = \frac{l^3}{\kappa}, \qquad \text{whence } \kappa = \frac{l^3}{v} = \frac{T^3 R^3}{v}; \]

air pressure \(p\), kg·m\(^{-2}\), or

\[ [p]=\frac{K}{l^2}=\frac{T R}{v T^2 R^2}=\frac{T R}{v}; \]

the equation relating the quantities, in general form, is written as follows:

\[ f(p,\ v,\ R,\ T)=0 \quad \text{or} \quad p=f_1(v,\ R,\ T). \]

Here the number of physical parameters is four, and the number of basic independent units is three; consequently, there must be (according to the \(\Pi\)-theorem) one dimensionless expression.

Taking \(v, R, T\) as the basic independent units, the dimensions of all quantities in this new system of units will evidently be:

\[ [v]=1;\quad [R]=1;\quad [T]=1;\quad [p]=\frac{T R}{v}. \]

Replacing the quantities \(p, v, R, T\) by their ratios to their dimensions in the new system, we obtain:

\[ \frac{p}{\dfrac{T R}{v}}=f(1,1,1)=\mathrm{const}=C. \]

From experiment we determine \(C=1\).

Consequently, the equation of state of an ideal gas is: \(pv=RT\), which can be represented graphically as a thermodynamic surface in the coordinates \(p, v, T\).

The general form of the equation of state of real gases, considered by M. P. Vukalovich and N. N. Novikov [11, pp. 43–58], is the expression:

\[ pv=RT\left[1-\sum \frac{k}{k+1}\frac{B_k}{v^k}\right], \]

where

\(B_k\) are functions of temperature, for a known law of interaction of the gas molecules with one another.

The cited work contains a compilation of various equations of state of real gases (150 in number), which, apart from its historical value, brilliantly illustrates the fundamental proposition of dialectical materialism concerning the relation between relative and absolute truths.

The physical content of the processes occurring in an air cushion is very complex: within a limited and relatively small volume there occurs a fairly rapidly time-varying interaction of the piston with the striker through a deformable elastic medium. The quality of the air coupling depends on the design of the machine, taking into account compensation for clearances and the conditions of its operation (cooling, lubrication, etc.).

To study the processes in the air cushion (air buffer), it is necessary to apply various methods developed by human practice:

1) theoretical investigations using the differential equations of motion of a continuous medium, the equation of state, continuity, heat transfer, etc.;

2) hydrodynamic and thermal modeling, relying on experimental data in this direction, and the establishment of relations between the quantities characterizing the working process of the air buffer; over the period of machine operation it is necessary to determine experimentally the deformations of the air volume, the change in air pressure and temperature; to determine, over a cycle, the mean value of the polytropic exponent and the efficiency of the air coupling for the given type of machine.

Finally, proceeding from the work of impact \(A\), for the optimum operating regime of the machine, determine the required quantity of air between the ram and the piston and the volume ratio for an impact-action machine with a two-sided air coupling.

On the basis of the method of zero dimensions, let us attempt to clarify the degree of difficulty of the problem of establishing the relationship between the quantities that determine the operation of the air coupling. Essential for the working process of the hammer, of the ZERT scheme, are the following quantities: air pressure \(p\), \(\text{kg}/\text{cm}^2\); unit weight of the ram \(q\), \(\text{kg}/\text{cm}^2\); angular velocity of the crank \(\omega\), \(1/\text{sec}\); length of the air cushion \(l\), \(\text{cm}\); crank radius \(r\), \(\text{cm}\); air density \(\rho\), \(\text{kg}\cdot\text{sec}^2/\text{cm}^4\); air viscosity, characterized by the coefficient of dynamic viscosity \(\mu\), \(\text{kg}\cdot\text{sec}/\text{cm}^2\text{sec}\) (or by the kinematic viscosity

\[ \nu=\frac{\mu}{\rho},\ \text{cm}^2/\text{sec}). \]

With viscosity are associated the dissipation of energy and its transformation into heat, as well as the exchange of momentum between neighboring layers of air; acceleration of gravity \(g\), \(\text{cm}/\text{sec}^2\); polytropic exponent \(\alpha\), which, generally speaking, changes with change in temperature, \([\alpha]=1\); coefficient of thermal diffusivity of air \(a\), \(\text{cm}^2/\text{sec}\); \(v\)—the velocity of displacement of the deformation of the air cushion. The state of the deforming air column is determined by the relationship among eleven quantities:

\[ f(p,\ q,\ \omega,\ l,\ \rho,\ \mu,\ g,\ a,\ r,\ \alpha,\ v)=0. \]

Taking \(\omega, l, \rho\) as the basic independent units, then between the eight dimensionless expressions there exists the dependence

\[ \Phi\left( \frac{p}{\rho l^2\omega^2},\ \frac{q}{l^2\omega^2\rho},\ \frac{\mu}{\rho l^2\omega},\ \frac{g}{l\omega^2},\ \frac{a}{l^2\omega},\ \frac{r}{l},\ \frac{v}{\omega l},\ \alpha \right)=0. \]

Hence the polytropic exponent

\[ \alpha=\Phi_1(\Pi_1,\ \Pi_2,\ \Pi_3,\ \Pi_4,\ \Pi_5,\ \Pi_6,\ \Pi_7), \]

where

\[ \Pi_1=\frac{p}{\rho l^2\omega^2}=\frac{p}{\rho v^2} \]

is the Euler criterion, which may be transformed as follows. Taking into account that the adiabatic speed of sound is equal to

\[ c=\sqrt{k\frac{p}{\rho}}, \qquad [38,\ \text{p. }159], \]

where

\[ k=\frac{c_p}{c_v}=1.4 \]

(for air

\[ c=\sqrt{k g R T}=20.1\sqrt{T}\ \text{m/sec}), \]

we obtain:

\[ \frac{p}{\rho v^2} = \frac{kp}{k\rho v^2} = \frac{c^2}{kv^2} = \frac{1}{k\left(\dfrac{v}{c}\right)^2} = \frac{1}{k\cdot M^2}, \]

where the number

\[ M=Ma=\frac{v}{c} \]

is the similarity invariant of N. V. Maievskii \([29,\ \text{p. }51]\).

characterizing the influence of elastic and compressibility forces and equal to the ratio of the velocity of motion of the medium (or in the medium) to the velocity of sound.

Criterion

\[ \Pi_2=\frac{q}{\rho l^2\omega^2} =\frac{q}{\rho v^2} =\frac{q}{\frac{\gamma}{g}\,v^2} =\frac{g\cdot q}{v^2\gamma} =\frac{g}{v^2}\cdot H, \]

where

\[ H=\frac{q}{\gamma} \]

is the reduced height to the length of the air column.

Obviously,

\[ \Pi_1=f_2(\Pi_2), \]

\[ \Pi_3=\frac{\mu}{\rho l^2\omega} =\frac{\mu}{\rho lv} =\frac{\nu}{lv} =\frac{1}{Re}; \]

\(Re\) is the Reynolds number, characterizing the ratio of inertial forces to viscous forces.

Criterion

\[ \Pi_4=\frac{g}{l\omega^2} =\frac{gl}{v^2} =\frac{1}{Fr}, \]

where

\[ Fr=\frac{v^2}{gl} \]

is the Froude number, characterizing the ratio between inertial forces and gravity forces.

Criterion

\[ \Pi_5=\frac{a}{l^2\omega} =\frac{a}{lv} =\frac{1}{Pe}, \]

where

\[ Pe=\frac{vl}{a} \]

is the Péclet criterion, characterizing the motion of a medium with a given thermal state.

The ratio of the Péclet criterion to the Reynolds criterion gives a new criterion, the Prandtl criterion,

\[ Pr=\frac{Pe}{Re} =\frac{\nu}{a} =\frac{c_p\mu}{\lambda}, \]

characterizing the ratio between the molecular exchange of heat and the molecular exchange of momentum between the air and the surface of a solid body.

Here, \(\lambda\) is the coefficient of thermal conductivity,

\[ [\lambda]=\text{kcal}/\text{cm}\cdot\text{sec}\cdot\text{deg}. \]

The physical constants for air, with indication of a number of criteria, are given in the literature [47 and 44].

\[ \Pi_6=\frac{r}{l} \]

is a simplex characterizing the ratio of the crank radius to the length of the air column.

\[ \Pi_7=\frac{v}{\omega l}=H_0 \]

is the homochronicity criterion, characterizing the uniformity of processes in time (or, in other words, the periodicity of the phenomenon).

The obtained criteria could have been derived from consideration of the equations of hydrodynamics and thermokinetics. As a supplement, let us find

criterion characterizing the intensity of heat exchange between a moving medium and the bounding surface.

According to Newton’s law, the process of heat exchange between a moving medium and a surface is expressed by the relation: \(-\lambda \operatorname{grad} T = \alpha \Delta T\), where \(\alpha\) is the heat-transfer coefficient, and \(\Delta T\) is the temperature difference between the moving medium and the solid wall.

Obviously, for such a system

\[ -\frac{\lambda_c T_c}{l_c}\operatorname{grad} T = \alpha_c T_c \alpha \Delta T, \]

whence

\[ \frac{\lambda_c T_c}{l_c}=\alpha_c T_c, \]

or

\[ \frac{\lambda_c}{\alpha_c l_c}=1. \]

The similarity invariant

\[ \frac{\alpha_1 l_1}{\lambda_1}=\frac{\alpha l}{\lambda}=\mathrm{idem}=Nu \quad \text{(Nusselt criterion).} \]

As an example, we shall note that experimental studies of heat transfer during the motion of a heat carrier in a pipe have established the existence of a relationship among the similarity criteria \(Nu\), \(Re\), \(Pr\),

\[ Nu = 0.023\,(Re)^{0.8}(Pr)^{0.4} \qquad [47,\ \text{p. }70]. \]

In our case, the solution of the problem posed proved to be too complex, which in turn is explained by the complexity of the processes occurring in the air cushion and by the large number of factors affecting the process; moreover, as already noted above, one must take into account the presence of air compensation and the influence of the clearances between the striker and the cylinder, which also enter into the conditions of uniqueness.

In the air connection (buffer) we have a medium with variable physical properties; extending similarity theory to this case encounters a number of difficulties and, obviously, can be carried out by narrowing the intervals of the phenomenon under study and by using averaged values of a number of quantities.

The application of the similarity method to the turbulent motion of a homogeneous incompressible viscous fluid was shown by L. G. Loitsyanskii [39], with an exposition of the foundations of the theory of approximate “internal” similarity of a unit phenomenon, by dividing the regions in which the process takes place into a number of separate regions (sections) and determining the conditions of similarity of the phenomena in these intervals.

The method of “internal” similarity of a unit phenomenon, it seems to us, can also be applied when considering the working process of single-type electropneumatic impact machines. From the nature of the change in a number of parameters over the course of one cycle, and even within individual intervals, it is possible to determine the conditions of similarity of the processes within these intervals.

The calculation method proposed by us [1] must be supplemented with tables (graphs) reflecting the similarity of phenomena within selected intervals according to the angle of rotation \(\varphi\) of the crankshaft (or according to time). It should be noted here that the calculation is cumbersome; thus, in production, carrying out all computations for one cycle requires 10–12 hours of work. Along with the development of the propo—

... approximate (numerical) methods of calculation, in recent years Soviet scientists have very fruitfully developed and applied electrical modeling. This method, as an effective method of calculation, could be applied to the investigation of operating regimes of impact-action machines, to finding solutions of the differential equations describing the working process of the machine, and to finding electrical analogs—similarity criteria for impact-action machines and the relations between them.

Method of Electromechanical Analogies

(as applied to impact-action machines)

1. In a number of branches of scientific knowledge (physics, hydromechanics), a working hypothesis is often constructed concerning the nature of one phenomenon or another. A model of the phenomenon is constructed which does not claim to be an exact representation of the phenomenon itself, but constitutes only a scheme facilitating the analysis of the phenomenon from a mathematical (or physical) point of view. As our knowledge expands and deepens, the model of the phenomenon is improved or replaced by a new model, which serves as a milestone (a step) on the path of approaching the knowledge of absolute truth.

Acad. A. F. Ioffe writes: “A successfully constructed model simplifies the conclusions drawn from known facts and makes it possible to pose new experiments that advance science. A physical model is not an exact image of a phenomenon, but a simplified schematic picture based on analogy... Within the limits in which the analogy actually holds, a successful physical model makes it possible to predict the results of experiments, to seek new manifestations of the processes being studied and, on their basis, to refine the model. Often over a long period of time the model serves as a guiding thread of scientific research; ...but most often the model is only a companion up to one of the turns where the paths of the phenomenon under study and of its model diverge... The same task as the model is often better and more fully solved by mathematical theory... Its significance is determined by the field of experimental facts that it encompasses” [19, p. 348].

Inference by analogy is based on the similarity of phenomena according to several features and asserts the possibility of similarity also according to a number of other features; that is, such an inference, proceeding from the particular to the particular, has a certain degree of probability. The probability of an inference by analogy, in turn, depends on the extent of our information about the phenomena under consideration; moreover, it is important to know not only the similarities but also the differences of the objects being studied.

The linking of unresolved problems with problems solved by analogy is a preliminary construction indicating the direction of the investigator’s further work, whose task is to discover the essential regularities hidden beneath the surface of external similarities and differences, and to create on this basis a new theory (hypothesis), on the basis of which one can predict the course of the process under real conditions. The scientific value of analogies, as conjectures, as a basis for constructing hypotheses, can be sufficiently high; Prof. P. E. Krasnushkin writes: “A vivid example of the generality of wave regularities and of their use is the discovery by P. N. Lebedev of light pressure, who in his experiments was guided by the existence of an analogous phenomenon for acoustic and hydrodynamic waves” [31, p. 89].

An example of the generality of oscillatory regularities is the use of methods of celestial mechanics in nonlinear radio engineering (the method of limit cycles) [3 and 60].

Acad. A. N. Krylov [34] writes very vividly: “Many such analogies can be cited among problems from entirely different fields, yet leading to identical differential equations. It would seem, what can there be in common between calculating the motion of celestial bodies under the action of attraction to the sun and the self- and rolling of a ship in waves, or between determining the so-called secular inequalities in the motion of celestial bodies and the torsional oscillations of the shaft of a multi-cylinder diesel engine working on a ship’s propeller or on an electric generator? Meanwhile, if one writes only the formula or the equations, without words, then it is impossible to distinguish which of these problems is being solved: the equations are the same.”

The question of dynamic models, with an analysis of a number of examples, is set forth most simply and accessibly by V. L. Kirpichev in his conversations on mechanics: “It often happens that problems of dynamics or of mathematical physics, differing from one another in essence, lead to equations that are completely identical in form. The analytical form of the equations proves to be the same for two or more problems, although the letters entering into the terms of the equations represent, in these problems, entirely different, often heterogeneous quantities. Such formal similarity makes it possible to apply the same mathematical techniques for integrating and solving equations: we use the solution obtained for one problem and apply it to others represented by the same equations. One problem serves as a model, or prototype, for several others; we can simply copy the already prepared solution, finding it entirely superfluous to repeat all the previous calculations and derivations anew” [23, p. 320].

Prof. L. S. Eigenson writes: “The method of modeling (in the broad interpretation of this term) makes it possible to replace the study of a phenomenon of one class by the study of a phenomenon of another class” [73, p. 3].

The method of modeling in the broad sense of the word is based on the “...remarkable analogousness” of differential equations, noted with genius by V. I. Lenin, “relating to different domains of phenomena” (see Materialism and Empirio-Criticism). Dynamic analogies are treated in detail in the literature [18; 21; 50; 66; 67].

Prof. A. I. Gutenmakher notes: “The principle of mathematical analogy makes it possible experimentally to find the solution of a differential equation on a model in that domain where the experiment is carried out most easily and accurately.

It is difficult to find a field of technology more suitable for this purpose than electrical engineering” [14, p. 11].

The method of electromechanical analogies is based on the formal similarity of the mathematical apparatus describing dynamic processes in mechanics and in electrical engineering.

Three systems of electromechanical analogies are known:

1) in the first system, the equation of the balance of forces of a dynamic system corresponds to the equation of the balance of voltages of a definite closed circuit of the scheme;

2) in the second scheme, the equation of the balance of forces of a dynamic system corresponds to the equation of the balance of currents of a definite node of the scheme;

3) in the third scheme of electromechanical analogies, all the sought elements of the mechanical system can be represented only by the magnitude of the voltage at the corresponding points of the electric circuit.

The transition from the parameters of dynamic systems (translational or rotational) to electrical quantities will be carried out in accordance with two developed systems of electromechanical analogies, presented below in Table 4.

Table 4

No.	Mechanical quantities: Translational system	Designation	Mechanical quantities: Rotational system	Designation	Electrical quantities: 1st system of analogies	Designation	Electrical quantities: 2nd system of analogies	Designation
1	Mass	\(m\)	Moment of inertia	\(I\)	Self-inductance	\(L\)	Capacitance	\(C\)
2	Linear displacement	\(s\)	Angular displacement	\(\varphi\)	Electric charge	\(q\)	Magnetic flux linkage	\(\psi\)
3	Linear velocity	\(v\)	Angular velocity	\(\omega\)	Current	\(i\)	Voltage	\(u\)
4	Force	\(F\)	Torque	\(M\)	Voltage	\(u\)	Current	\(i\)
5	Loss resistance	\(k\)	Rotational resistance	\(\kappa_v\)	Ohmic resistance	\(R\)	Ohmic conductance	\(\dfrac{1}{R}\)
6	Compliance (reciprocal of stiffness)	\(e\)	Compliance (elasticity)	\(c\)	Capacitance	\(C\)	Self-inductance (mutual inductance)	\(L\) \((M)\)

From the equation of motion of the electric motor we obtained the expression:

\[ M_c=\varepsilon_c I_c=m_c\frac{l_c^2}{t_c^2}=p_c l_c^3=\omega_c^2\frac{Q_c}{g_c}\,l_c^2=\omega_c^2\cdot c. \]

The corresponding electrical analogs (according to the first system of electrical analogies) will be the following:

\[ u_c=\frac{i_c}{t_c}L_c=L_c\frac{q_c^2}{t_c^2}=\frac{u_c}{q_c^2}\,q_c^2=\omega_c^2 L_c q_c^2=\omega_c^2 L_c. \]

To the criterion of the optimal regime of the mechanical system,

\[ \Pi_{\mathrm{res}}=\frac{p_0 g d_2}{Q_0\omega^2 l}=1, \]

for the electrical circuit there corresponds the expression:

\[ \frac{u}{Lq\omega^2}=1. \]

To the condition

\[ p_c=1=m_c=\frac{l_c}{t_c^2}\cdot\left(\frac{d_1}{d}\right)^2 \]

in the electrical circuit there corresponds the expression:

\[ \frac{L}{t^2q}=1. \]

Let us find the ratio of the times (or frequencies) of the corresponding processes in the mechanical system and in the electrical circuit, which is its analog:

\[ t_{c.\mathrm{mech}}= \sqrt{\frac{m_c}{p_c l_c}} = \sqrt{\frac{m_c l_c^2}{I_c\omega_c^2}} = \frac{1}{\omega_{c.\mathrm{mech}}}, \]

\[ t_{c.\mathrm{el}}= \sqrt{\frac{L_c q_c^3}{u_c}} = \sqrt{\frac{i_c L_c}{u_c}} = \sqrt{\frac{L_c q_c^2}{\omega_c^2 L_c}} = \frac{q_c}{\omega_{c.\mathrm{el}}}, \]

whence

\[ \frac{t_{c.\mathrm{mech}}}{t_{c.\mathrm{el}}} = \frac{\omega_{c.\mathrm{el}}}{\omega_c q_{c.\mathrm{mech}}}. \]

or

\[ \frac{\omega_{c.\mathrm{el}}}{\omega_{c.\mathrm{mech}}} = q_c \frac{t_{c.\mathrm{mech}}}{t_{c.\mathrm{el}}} = q_c \frac{ \sqrt{\dfrac{m_c}{p l_c}} }{ \sqrt{\dfrac{L_c q_c^2}{u_c}} } = \frac{ \sqrt{\dfrac{m_c}{l_c}} }{ \sqrt{\dfrac{L_c}{u_c}} }. \]

Consequently,

\[ \frac{\omega_{\mathrm{el}}}{\omega_{\mathrm{mech}}} = C_1 \frac{ \sqrt{\dfrac{m}{l}} }{ \sqrt{\dfrac{L}{u}} }, \quad \text{where } C_1=\mathrm{const}. \]

Thus, equality of the frequencies in the mechanical system and the electrical model is not necessarily required.

The similarity condition for spring hammers in the resonant regime is the relation

\[ \frac{Q\omega^2}{g C_2} = m\omega^2 l = 1, \quad \text{where } l=\frac{1}{c_2},\ c_2 \text{ is the stiffness of the spring}, \]

whence

\[ \omega=\frac{1}{\sqrt{me}} \quad \text{or} \quad t_{\mathrm{mech}}=\frac{2\pi}{\omega}=2\pi\sqrt{me}. \]

For the electrical circuit that is its analog,

\[ t_{\mathrm{el}}=2\pi\sqrt{LC}, \quad \text{where } C \text{ is the capacitance}. \]

Consequently, the ratio of the times of the occurring processes is

\[ \frac{t_{\mathrm{mech}}}{t_{\mathrm{el}}} = \frac{\sqrt{me}}{\sqrt{LC}} = \frac{\omega_{\mathrm{el}}}{\omega_{\mathrm{mech}}}. \]

Further investigation of the problem can be carried out in two directions: a) by composing electrical circuits that are analogs of mechanical systems and studying the processes occurring in them (with variable values of the corresponding parameters); b) by studying the nature of the forces acting on the ram over the period of one cycle, that is, by determining the dependence \(F=f(t)\) or \(F=f_1(\varphi)\), and then, with the aid of an electrointegrator, finding the law of motion of the ram.

When composing electrical circuits that are analogs of mechanical systems, each of the elements of the mechanical system is represented in the form of a two-terminal network, from which the mechanical circuit is composed.

On the basis of the rules set forth in the literature [18; 62; 66; 67], which provide the theoretical and practical foundations for constructing electrical analogs of systems 1 and 2, let us represent the system of an electrical hammer (Fig. 6a), with an elastic (spring) connection between the ram and the piston, in the form of a mechanical circuit (Fig. 6b) consisting of two-terminal networks that include the piston, spring, ram (with the resistance accompanying its motion), and the tool (the work absorber); in Fig. 6c the action of the piston is schematically replaced by the force \(F\); the compliance of the elastic element is denoted by \(e\); the mass of the ram is denoted by \(m\); the frictional resistance against the wall

cylinder—by \(K\); the absorber of work is the tool (pick), through which energy is transmitted to the medium being destroyed and is partially returned back when the striker rebounds.

Figure 6d shows equivalent electrical circuits compiled according to systems 1 and 2 of electrical analogies. Here \(R_1\) is the energy absorber, a nonlinear element, and \(D\) is a detector (interrupter).

Fig. 6

Modeling an electro-pneumatic hammer is more complicated, since the air coupling does not satisfy Hooke’s law.

In the mechanical system there is a sudden coupling (impact), sharply changing the motion of the striker, while in the equivalent electrical circuit there is a detector that operates at a definite voltage; in both systems, mechanical and electrical, there are elements whose properties depend substantially on the applied forces arising during impact and on voltages (or currents flowing in the electrical circuit), that is,

nonlinear elements. In general, each of the nonlinear circuits constitutes a self-oscillating system [3; 60].

The consideration of impact machines as self-oscillating systems deserves more detailed study.

Let us consider the question of applying the corresponding voltage to an electrical circuit. With a sufficiently powerful electric motor, the applied voltage corresponds to the exciting oscillations of the piston, which in a first approximation may be taken as harmonic; we note that, from the electrical point of view, it presents no special difficulty to take into account the length of the connecting rod as well.

To obtain undamped oscillations in the circuit of the electrical model, the latter must be supplied periodically with portions of energy compensating for its losses. The form of the oscillations is determined mainly by the properties of the circuit; and since the damping of the circuit must be large, the external source of energy must feed it during a considerable part of the period.

The device feeding the circuit must deliver energy to it at quite definite moments of time, determined by the elements of the circuit \(R, L, C\); consequently, this device too must “wait” until the voltage on it or the current through it reaches a certain value; when it operates, it will bring the system into some other state. The role of such a device is performed by various elements: a neon lamp, a thyratron, etc.

By changing the parameters of the electrical model \(L, C, R\) and varying (altering) the frequency, magnitude, and form of the applied pulses, it thus appears possible, on the electrical model, to study certain questions connected with the operation of an impact machine.

1. Still more attractive is the finding of effective methods for solving the system of differential equations describing the motion of the striker and of the electric motor. For the existing electropneumatic hammers, or on a stand specially constructed for these purposes, it is necessary to study the curves of air pressure as a function of the parameters and operating regime of the machine. Knowing the forces acting on the striker as functions of time \(p(t)\), or of the angle of rotation \(p(\varphi)\), and the limits within which the curve varies, and specifying the initial conditions of integration, we shall find the curve of force impulses and the velocity of the striker; by repeated integration we shall determine the law of motion of the striker, the integration being possible with the aid of electrical circuits. In the absence of an electronic integrator, it is necessary to develop apparatus of the multichannel cathode-oscillograph type, with the possibility of observing and studying several processes simultaneously, and also with differentiating and integrating circuits in it.

One may go still further in this direction: knowing the change in air pressure over the period \(p(t)\) and the characteristic of the electric motor \(M_k = f(\omega)\), it is evidently possible to know the resultant torque on the crankshaft, and consequently the kinetic energy of the motor and its power as functions of time. Then, for a known value of the moment of inertia \(J(\varphi)\), one may know the law of motion \(\varphi = \varphi(t)\) and \(\omega = \omega(t)\).

Having transformed the air link from a passive two-terminal network into an active one (after the analogy of an internal-combustion engine\(^1\)), by applying to the instrument voltage or current pulses corresponding to the pressure\(^1\), we have the possibility of obtaining almost instantaneously the law of motion of both the striker and the motor—

\(^1\) It is desirable, on the basis of experimental investigations, to record the pressure curve \(p(t)\) on magnetic tape or motion-picture film; then, with the aid of photocells, voltage (or current) pulses of the corresponding form should be applied to a multichannel oscillograph having integrating circuits.

...body (not necessarily even an electrical one!). In this direction, modeling is essentially mathematical, and it has already found reflection in the machine-building literature [44; 62]. The theory of electrical models and mathematical machines is treated quite fully in the cited works [14; 21; 48; 75].

It is possible that, in mathematical modeling, it will be necessary to turn to the study of the theory of electrical circuits, in particular to the use of the method considered by G. E. Pukhov in the work “Theory of the Subcircuit Method,” based on decomposing a complex circuit into the simplest parts—subcircuits [76].

Conclusion

1. The formulation of the topic “Study of the working process of electropneumatic impact machines,” and the multiplicity of the questions touched upon from adjacent fields of knowledge, require the application of various methods and techniques of investigation to individual stages and sections of the work. Effective methods for solving the problem may be the methods of the theory of similarity and dimensions, which constitute the basis of modeling. Modeling, as one of the methods for cognizing the regularities of the world around us, has enormous practical and general educational significance: the difficulties of calculating and constructing structures which in a number of cases require considerable material expenditures and power compel the research engineer to study the phenomenon on a simplified model at a somewhat different scale, sometimes schematizing the process and identifying the principal determining factors. In the study of new machine designs, the correctness of the theory, assumptions, allowances, and calculation is first checked on a model; on a model it is possible to study various operating regimes of the machine, to permit the experimenter’s active intervention in the observed process, to control the new process, to reveal the influence of individual elements, to eliminate shortcomings, and to improve the design of the machine as a whole. Modeling is the reproduction of a phenomenon similar to a specimen (prototype), when there is a need for serial production of similar structures at a somewhat different scale.

Depending on the nature of the motion and the forces manifested in it, various scale time–space relations between real objects are possible, which to a certain extent are also disclosed by the theory of similarity and dimensions. For example, considering special cases of modeling, if the main force in the phenomenon under study is the force of gravity, then it is not difficult to show that the transition number between the time and space scales, for one and the same place \((g_c = 1)\), is the expression:

\[ t_c = l_c^{1/2}) \]

Likewise, for the case of a central (Newtonian) force, according to Kepler’s law, it is not difficult to obtain the relation:

\[ t_c^2 = l_c^3. \]

For some systematization of the presentation of the question, we give a table (Table 5) of time–space scale relations, for cases—

cases when, in modeling, the predominant forces are: the force of gravity, viscosity, elasticity, centrifugal force, and central (Newtonian) force.

Table 5

Characteristic	Designation	Gravity	Viscosity	Elasticity	Centrifugal force	Central force \(F=f\dfrac{m_1m_2}{r^2}\)	Note
Similarity criteria . . . . .	\(\Pi\)	\(\dfrac{v^2}{lg}\)	\(\dfrac{v}{l\gamma}\)	\(\dfrac{v}{\sqrt{E_y/\rho}}\)	\(\dfrac{g}{\omega^2 R}\)	\(\dfrac{v^2}{f\rho l^2}\)
Characteristic numbers . . .	\(\Lambda\)	\(\dfrac{v_c^2}{l_c g_c}\)	\(\dfrac{v}{l_c\gamma_c}\)	\(\dfrac{v_c}{\sqrt{E_{yc}/\rho_c}}\)	\(\dfrac{g_c}{\omega_c^2 R_c}\)	\(\dfrac{v_c^2 l_c}{f_c m_c}\)
Transition number (scale) in time . . .	\(t_c\)	\(l_c^{1/2}\)	\(l_c^2\)	\(l_c\)	\(l_c^x\)	\(l_c^{3/2}\)	For identical materials
Transition number for forces . . .	\(F_c\)	\(l_c^3\)	\(1\)	\(l_c^2\)	\(l_c^2\)	\(l_c^{-2}\)	For identical materials
Ratio of stresses (pressures) . . . . .	\(\sigma_c\)	\(l_c\)	\(l_c^{-2}\)	\(1\)	\(1\)	\(l_c^{-4}\)	For identical materials
Ratio of velocities . . . . .	\(v_c\)	\(l_c^{1/2}\)	\(l_c^{-1}\)	\(1\)	\(l_c^{1-x}\)	\(l_c^{-1/2}\)	For identical materials
Ratio of works . . . . .	\(A_c\)	\(l_c^4\)	\(l_c\)	\(l_c^3\)	\(l_c^3\)	\(l_c^{-1}\)	For identical materials
Ratio of powers . . . . .	\(N_c\)	\(l_c^{7/2}\)	\(l_c^{-1}\)	\(l_c^2\)	\(l_c^{3-x}\)	\(l_c^{-5/2}\)	For identical materials

Notes to the table:

1. The transition numbers, expressed in the general case by the dependence \(t_c=f(\rho_c l_c)\), combine into a single whole the spatial and temporal characteristics of the moving matter (the table gives the simplest cases; in reality there is a great variety of forms in which this relationship manifests itself).

2. In the case when several forces act simultaneously, the problem becomes more complicated, and approximate modeling then takes place.

3. The similarity criteria can be obtained in various ways. Let us find some of them, proceeding from the general structure of the formula \(E=mc^2\), which gives the relation between mass and energy [19 and 56].

Dimension

\[ [E]=[m][v^2] \]

or

\[ [me^2t^{-2}]=[\rho l^3 v^2]. \]

Dividing both parts of the last relation by the dimension of volume \([l^3]\), we obtain the dimension of the so-called “density” of energy, or energy per unit volume,

\[ [me^{-1}t^{-2}] = [\rho v^2]. \]

But the dimension of the expression on the left-hand side is nothing other than the dimension of pressure \([p]\) or stress \([\sigma]\).

Consequently, \([p] = [\sigma] = [\rho v^2]\), whence it is not difficult to obtain the structure of the formula \(p = \rho v^2\) and, for the velocity of propagation of a wave process in an elastic medium:

\[ v = \sqrt{\frac{p}{\rho}} \quad \text{or} \quad v = \sqrt{\frac{\sigma}{\rho}} . \]

In the case when \(\sigma = E_y\), (where \(E_y\) is the modulus of elasticity), then

\[ v = \sqrt{\frac{E_y}{\rho}}, \]

therefore,

\[ v_c = \sqrt{\frac{E_{yc}}{\rho_c}}, \]

whence

\[ \frac{v_c}{\sqrt{\dfrac{E_{yc}}{\rho_c}}} = 1, \]

or

\[ Ca = \frac{v}{\sqrt{\dfrac{E_y}{\rho}}} = \mathrm{idem} \quad \text{(Cauchy criterion).} \]

It would have been possible to carry out the entire reasoning in the reverse order, starting from the formula

\[ v = \sqrt{\frac{p}{\rho}}, \]

to obtain the structure of the formula \([E] = [mv^2]\).

Let us find the value of the energy per unit weight

\[ [G] = [V \cdot \gamma] = [e^3\gamma], \]

where \(\gamma\) is the specific weight, kg/cm\(^3\).

Dividing the value of the energy by the weight of the body, we obtain:

\[ \left[\frac{E}{G}\right] = \left[\frac{p}{\gamma}\right] = \left[\frac{\rho v^2}{\gamma}\right] = \left[\frac{v^2}{g}\right] = [l] = [pv] = [RT]. \]

Here

\[ v = \frac{V}{G} = \frac{1}{\gamma} \]

is the specific volume.

Consequently, the dimensionless expression will be the ratio of the dimension of the energy per unit weight to the dimension of length

\[ Fr = \frac{\left[\dfrac{v^2}{g}\right]}{[l]} = \left[\frac{v^2}{gl}\right]. \]

or

\[ \frac{v^{2}}{gl}=\mathrm{idem}. \]

1. If, in the expression for the similarity criterion for the force of gravity, we take the characteristic linear dimension \(l=R\) and replace \(v=\omega R\), we obtain:

\[ \frac{v^{2}}{lg}=\frac{\omega^{2}R^{2}}{Rg}=\frac{\omega^{2}R}{g}. \]

The identity of the similarity criteria naturally suggests the identity of the forces of inertia and gravitation, which to a certain extent is also substantiated by the theory of relativity.

Centrifugal modeling is widely applied in solving various problems of structural mechanics, soil science, and rock pressure \([32; 52; 53; 20]\).

When modeling in a centrifuge (Fig. 7), the centrifugal forces developed at each point of the material under investigation, as a result of the rotation of the centrifuge, are a model for the field of gravity. With sufficiently small model dimensions and a large radius of rotation, the centrifugal forces of inertia may be considered parallel.

a) Schematic drawing of a centrifuge for testing models
b) Section of a box with a model of a canal slope

Fig. 7

The acceleration on the model is

\[ a=\sqrt{g^{2}+\omega^{4}R^{2}} \approx \omega^{2}R, \]

then the “weight” on the model is

\[ G_{m}=m\omega^{2}R. \tag{1} \]

In the case when the essential force is weight, the stress ratio is determined by the relation

\[ \sigma_{c}=\frac{\sigma_{m}}{\sigma_{n}}=\frac{\sigma}{\sigma_{1}} = \frac{\dfrac{\rho l^{3}}{l^{2}}}{\dfrac{\rho_{1}l_{1}^{3}}{l_{1}^{2}}} = \rho_{c}l_{c}. \]

If it is desired that the stresses be the same both in the model and in the prototype, i.e. \(\sigma_{c}=1\), then \(\rho_{c}l_{c}=1\). If \(\rho_{c}=1\) (the material is the same), then \(l_{c}=1\), i.e. the model must coincide with the prototype or must be made of a denser material, but with elastic moduli and allowable stress identical to those of the prototype, which is impossible.

The solution is to increase the gravity (weight) on the model by the same factor by which the linear dimensions are reduced. Consequently, the weight of the body \(G = mg\), located in the carriage during rotation of the centrifuge, must be equal to

\[ G_m = \frac{mg}{l_c}. \tag{2} \]

Equating expressions (1) and (2), we obtain:

\[ m\omega^2 R = \frac{mg}{l_c}, \]

whence

\[ \omega = \sqrt{\frac{g}{R l_c}} \]

and

\[ l_c = \frac{g}{\omega^2 R}. \]

Determination of the time scale, which makes it possible to predict the development over time of the processes being studied [20], is of especially great practical importance; here \(t_c = l_c^x\).

In view of the diversity of the properties of rocks, the value of the exponent \(x\) depends on the material being tested, is determined from experience, and has different values within the limits: \(1 < x < 2\); when \(x = 1\), all characteristics in centrifugal modeling correspond to the characteristics of elastic force; when \(x = 2\), the characteristics correspond to viscous forces except for \(A_c\) and \(N_c\), since the excess energy on the model is obtained from the motor rotating the centrifuge, that is, from outside, which also ensures, under the additionally loaded state, the accelerated course of the process on the model.

1. The theory of similarity is based on three theorems: the first theorem of similarity formulates the properties of similar systems, asserting that similar phenomena have identical similarity criteria.

Similarity criteria can be found in various ways: either from the identity conditions of the equations describing the processes, or from dimensional analysis, a variety of which is the method of zero dimensions. The difference lies only in the methods of solving the problem; the result, in the final analysis, is one and the same.

The advantage of dimensional analysis consists in applying the method to complex systems for which we cannot write equations characterizing the process, but it is necessary to know all \(n\) physical quantities participating in the process; it is necessary to know the number of fundamental units \(k\) underlying the system and, finally, to know the law on the basis of which the system of units is constructed.

However, “finding similarity criteria from the solution of specific equations, under known uniqueness conditions, proves to be more definite and reliable” [9, p. 28]. Sharing this point of view, we note that often the effectiveness, simplicity, and elegance of solving a problem depend to a considerable degree on the nature and content of the problem, on the choice of the method of solution, and, finally, on the orientation of the researcher’s mind and skills.

The proof of the possibility of reducing an equation to criterial form constitutes the content of the second theorem of similarity. However, the application of the dimensional-analysis method does not raise the question of sufficient conditions for the existence of similarity, which may lead to the danger of excessively broad generalizations.

Clarity in the question of the limits of the regular extension of an individual experiment is indicated by the third theorem of the theory of similarity, sometimes called the modeling theorem, which is briefly formulated as follows: “Similar phenomena are those which have similar conditions of uniqueness and identical determining criteria” [9; 24; 35].

The determining criteria are composed of mutually independent quantities that enter into the conditions of uniqueness (geometric relationships, physical parameters, boundary conditions: initial and limiting).

The theory of similarity and dimensions is the connecting link between theory and experiment. “A specific point of view on an experiment as on modeling enables the physicist to discern from the experiment general laws that go beyond the limits of the specific physical nature of the phenomenon” [31, p. 90].

Experiment with models, carried out on the basis of similarity theory, frees us from the necessity of an analytical solution of the problem, which is not always possible.

The greatest effect is possible with the coordinated development of the methods of physical and mathematical modeling, with their mutual supplementation of one another; moreover, the advantage, in the sense of the reliability of conclusions, will belong to physical modeling, since the nature of the phenomenon remains one and the same, although here too there are shortcomings:

a) the manufacture of models often takes too much time and resources;

b) in connection with a change in scale, distortion of the “nature” of the phenomenon is possible.

One may approach the formulation of an experiment in various ways. Acad. M. V. Kirpichev, in the work “Similarity Theory as the Basis of Experiment” [24], writes: “... The theorems of similarity establish a close connection between theory and experience. At a certain stage in the development of science it becomes possible to derive mathematical relationships for the phenomenon of nature under consideration. In this, as a rule, the mathematical relationship can be established only in the form of a differential equation. The compilation of equations and the finding of their conditions of uniqueness, consisting in the selection of boundary conditions and the proof of uniqueness of the solution, from which the theoretician begins the integration of the equations, are, as we see, equally necessary for the experimenter as well. Their paths diverge only when the first takes up the pencil, and the second arms himself with a measuring instrument. This connection between theory and experiment becomes still clearer if, in the differential equations, one passes from absolute to relative units of measurement and thereby makes the equations invariant for similar phenomena.”

Methods for processing experimental results on the basis of similarity theory may be found in manuals [26 and 47].

Stalin Prize laureate Prof. M. A. Mikheev writes: “... particular dependences are attractive by their simplicity. However, simple formulas can be used only in the case when, in the apparatus being designed, the conditions of the course of the process correspond exactly to those that existed when carrying out the experiments on the basis of which these formulas were obtained... If the conditions that existed in the experiment and in the apparatus being designed are different, then in calculations one should use such formulas as would take into account a larger number of variables determining the course of the process. Only generalized dependences in criterion form satisfy this requirement. Therefore, when choosing a calculation formula, they should be given unconditional preference” [47, p. 70].

The relations established by the theory of dimensionality and similarity are often only a preliminary stage of investigation (a first approximation...), but these relations are not accidental; for the basis of the theory of dimensions and the theory of similarity is the socio-historical productive practice of humanity as a whole. Observation and experiment have a subordinate significance in relation to developing productive practice, to the requirements of socio-historical practice as a whole.

It must be noted that the theory of similarity (and dimensionality), while revealing a number of connections and relations and indicating a rational form for processing experimental data, can give no more than what is contained in the equations describing the phenomenon, and is not a universal theory suitable for every case, still less for the various speculative constructions in the spirit of bourgeois scholars, for example Tolman’s “On the Miniature Universe” [70, p. 374].

“...The theory of similarity makes it possible, without integrating differential equations, to obtain from them similarity criteria and to establish criterial dependences that are valid for all mutually similar processes. However, it should be remembered that such generalized dependences are limited by the conditions of similarity, and conclusions extending beyond these limitations cannot be drawn from them. The theory of similarity does not provide a general solution: it merely makes it possible to generalize experimental data within a region bounded by the conditions of similarity. When using the similarity method, these limitations must always be kept in mind” [47; 64].

Along with establishing the relations given by the theory of similarity and dimensionality (which are not complete), it is necessary to study more deeply the specificity of each phenomenon, of each process under concrete conditions, for the task of science is to study not only the relations between objects of the real world but, chiefly and above all, the properties of the objects themselves; otherwise, form may become detached from content, and consequently the danger of idealism arises.

1. On the basis of the methods of the theory of similarity and the theory of dimensionality considered in this work, it seems to us necessary to set up experimental investigations on the study and modeling of impact-action machines in the following directions:

1). Since similarity occurs among forging hammers, it is evidently necessary to study more deeply the operation of the best types of hammers with individual drive; at the same time, it is necessary to design a universal test stand for testing electropneumatic hammers and perforators; to investigate the operation of various designs (of the impact unit) under variable parameters and different operating regimes of the machine. To verify the validity of the condition for the optimal regime of the machine; to determine the influence of rebound of the striker on the working process of the machine; to determine the values of certain coefficients and indicators (for example, the polytropic exponent, etc.). To study more deeply the specific character of the work and efficiency of each separate unit of the machine (engine, transmissions, feed, impact unit, drill-rotation mechanism). To study the role and influence on the working process of the machine of the following factors: lubrication, cooling, compensation, and the magnitude of the clearances between striker and cylinder. To pay attention to the strength and reliability of the operation of individual elements of the machine, and to study failures of parts.

2). To clarify the interaction of the operating machine with the medium being destroyed and the quantities characterizing the process of rock destruction under specified operating conditions. Along with the task of creating a reliable and more effective tool for drilling, it is necessary to investigate new drilling regimes and the application of a number of factors that facilitate rock destruction (hardness reducers, removal of drilling fines from the borehole, etc.). It is necessary to set up a more refined experiment

in studying the mechanism of rock destruction. It is obvious that the question of the destruction of rocks and of the interaction of the rock being broken with the machine that performs the breaking (and extraction) cannot be solved in isolation from other fields of science and technology.

3). Rationally process the results of experimental data in the form of tables, graphs, and nomograms; select the best operating regimes as models for designing new impact-action machines. Systematize the existing and developed methods of calculation for electro-pneumatic impact-action machines.

Further develop work in the direction of modeling—electrical modeling, modeling of the air cushion, etc. Devise a methodology for carrying out experimental work and create the necessary apparatus for observing and recording the processes under study.

4). Develop a number of questions of a prospective nature, such as:

a) the hammer as a self-oscillating system;

b) selection of the parameters of an impact-action machine for the optimal regime, as a variational problem;

c) investigation of transient processes and selection of the corresponding characteristic of the motor;

d) questions of efficiency in impact; transmission of energy in impact through a series of links;

e) extension of the developed methods of investigation and calculation to other machines (solenoid hammer, the possibility of using a hydraulic motor, etc.).

The indicated scope of work can be carried out quickly and effectively only by a collective of workers, with a combination of theory and practice, on the basis of creative cooperation between workers of science and production, and on the basis of the developed methods of investigation and calculation. A scientific substantiation of processes in the field of impact-action machines will make it possible to master this field from the standpoint of controlling the phenomenon, and from the standpoint of creating new high-productivity machines and mechanisms.

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1. We illustrate with a well-known example: the period of oscillation of a mathematical pendulum for small angles of deviation from the vertical is determined by the formula \(t = 2\pi\sqrt{\frac{e}{g}}\); then for one and the same place \(g_c = 1\), and the ratio of the periods of oscillation of two pendulums of different lengths will be \(t_c = \sqrt{l_c}\). ↩