Study of the Working Process of an Electric Jackhammer with an Elastic (Air) Coupling

P. M. Alabuzhev

Docent, Candidate of Physical and Mathematical Sciences

§ 1. Introduction

In this work, carried out at the suggestion of the design bureau of the Tomsk Electromechanical Plant (TEMZ), a study is made of the working process of an electropneumatic hammer for given parameters determining its operation. Methods are indicated for the simultaneous solution of the equations of motion of the striker and the motor, when the elastic coupling between the striker and the piston is air. As an example, a numerical solution is given in which the entire working process of the hammer over the period of its operation is considered, and the role and significance of the initial data are also clarified.

Here the author considers it his duty to express gratitude to Prof. V. K. Nechaev for reviewing the work and for the advice he gave, and also to senior engineer of TEMZ N. M. Batuev for assistance in preparing the drawings and providing the necessary materials and manuals.

The notation adopted in the work:

$r$—radius of the crank;
$L$—length of the connecting rod;
$\lambda$—ratio of the crank radius to the connecting-rod length;
$d$—diameter of the piston (and striker);
$B$—area of the piston (and striker);
$l_0$—initial distance between the piston and the striker;
$l$—distance between the piston and the striker;
$s$—distance traveled by the piston from the initial position;
$x$—distance traveled by the striker from the initial position;
$u=x-s$—relative distance between the striker and the piston;
$v_p=\dot{s}$—velocity of the piston;
$v_b=\dot{x}$—velocity of the striker;
$\dot{u}=\dot{x}-\dot{s}$—relative velocity between the striker and the piston;
$W_p=\ddot{s}$—acceleration of the piston;
$W_b=\ddot{x}$—acceleration of the striker;
$\varphi$—angle of rotation of the crank;
$\omega=\dot{\varphi}$—angular velocity of the crank;
$\varepsilon=\ddot{\varphi}$—angular acceleration of the crank;
$t$—time;
$h$—time step (interval) adopted in the calculations;
$T$—period of the working cycle;

$A_b$—work on the striker;

$N$—motor power;

$n$—number of revolutions per minute;

$i$—gear ratio between the motor shaft and the crank shaft;

$E$—kinetic energy;

$M_p$—moment of the air-pressure forces with respect to the crank shaft;

$M_q$—driving moment determined by the motor characteristic.

$M$—total torque on the shaft;

$J$—moment of inertia of the mass;

$Q_b$—weight of the striker;

$Q_p$—weight of the piston;

$Q_{\text{sh}}$—weight of the connecting rod;

$Q_m$—weight of the hammer body;

$g = 9.81\ \mathrm{m/sec^2}$—acceleration due to gravity;

$z$—number of blows of the striker per minute;

$y_m$—travel of the hammer body;

$v_m = \dot y_m$—velocity of the hammer body;

$\alpha = 1.32$—polytropic exponent;

\[ \beta=\frac{\dot x}{\dot x_0}=\frac{v_b\ \text{rebound}}{v_b\ \text{impact}} \]
—coefficient of restitution at impact;

$p$—air pressure between the striker and the piston;

$p_0$—atmospheric pressure (taken hereafter as $=1\ \mathrm{kg/cm^2}$);

$P=pB$—resultant force of air pressure;

$F$—inertial force;

$\Delta$—difference;

$f$—designation of a function;

\[ \left. \begin{aligned} \eta &= h\omega\\ \xi &= h^2\ddot x\\ \psi &= h\varphi \end{aligned} \right\} \]
auxiliary notations in calculation by the Adams–Störmer method.

$*$—mean value of a quantity;

$k$—variable index $k=0,\ 1,\ 2\ldots$

$b$—index referring to the striker;

$p$—same, referring to the piston;

$m$—same, referring to the hammer.

The present study of the working process of an electric jackhammer with an elastic (air) coupling has been carried out with allowance for the nonuniform motion of the motor, for any prescribed motor characteristic. In addition, a sufficiently accurate and simple method of calculation is indicated, making it possible relatively quickly to obtain an idea of the working process inside the hammer, as well as the principal indicators characterizing its operation. A preliminary theoretical study of the design, even before it is manufactured, makes it possible to eliminate observed shortcomings in good time. Of course, one may also proceed purely experimentally, but this path is more difficult and expensive. The best combination is the interrelation and mutual supplementation of theoretical and experimental methods of investigation.

The problem of the jackhammer has not yet been fully solved: industry needs an economical, reliable, lightweight hammer with reduced recoil. Pneumatic hammers are reliable and productive, but too expensive (taken together with the entire “pneumatic system”). In addition, a major shortcoming of theirs is the considerable recoil. Electromechanical hammers, from the standpoint of energy utilization, are more economical than pneuma-

tic, but they in turn also have a number of significant shortcomings, such as: low impact work (on the order of 1 kgm); great weight (up to 15 kg), rapid wear of the springs, and considerable heating.

In the mining industry there is an acute need for a satisfactory design of an electric jackhammer. Breakage and rapid wear of steel springs lead designers to attempts to use air as the elastic material.

G. I. Maleev [11] gave the design and calculation of an electropneumatic hammer with a two-sided air connection (Fig. 1). This hammer has the work of one blow \(A_b = 3.0\ \text{kgm}\) with piston diameter \(d = 3.2\ \text{cm}\) and lengths of the “air springs” \(l_1 = 3.6\ \text{cm}\) and \(l_2 = 5.8\ \text{cm}\). But the entire calculation of the hammer was made on the assumption that the angular velocity is constant, which, of course, cannot be the case, since the “elastic springs” \(l_1\) and \(l_2\) will substantially affect the piston, and consequently also the motion of the motor.

Fig. 1

Senior engineer of the design bureau of TEMZ N. M. Batuev proposed a design of a gearless electropneumatic hammer, for which a detailed study of the dynamics and working process was carried out by a team of workers of the Tomsk Polytechnic Institute (TPI), consisting of Prof. Kh. F. Ketov, Docent P. M. Alabuzhev, and Assistant N. G. Shlenkina, under the direction of Prof. Kh. F. Ketov, in accordance with a contract with TEMZ in 1942 [7].

In the report submitted to the plant, a numerical solution of the problem was set out under the assumption of constancy of the moment of resistance forces and of a linear law for the motor characteristic; the working process and the basic design elements determining its operation were also clarified, and the theory and calculation of the inertial switch were developed.

It should be noted that the method of numerical solution proposed there by us was too cumbersome. In addition, the phenomenon of rebound of the striker and recoil of the hammer was not touched upon at all in the work; nevertheless, this investigation was the first step toward further work on bringing the design to a prototype and toward studying the working process in other mechanisms with an elastic (air) coupling.

We shall give here only the technical characteristics of the designed hammer (its description will be given separately): average power on the motor shaft \(N = 350\ \text{W}\); average number of revolutions per minute \(n = 2800\), number of blows per minute \(z = 700\), work of a blow on the striker \(A_b = 2\ \text{kgm}\); weight of the striker \(Q_b = 0.6\ \text{kg}\); weight of the hammer (without the pick) \(Q_m = 11.5\ \text{kg}\); piston diameter \(d = 4.5\ \text{cm}\). When pressure on the pick ceases, the hammer goes into idle running.

Let us dwell further on the design of the OEM-1 electropneumatic hammer (Fig. 2), developed according to the scheme of Fig. 3 (proposed earlier by engineers E. A. Berner and N. I. Zhadaev) by TEMZ designers N. I. Shokhor and M. F. Klimova under the general direction of chief designer B. V. Sudnishnikov. When the piston moves from the extreme right position, a rarefaction is created between the piston and the striker, and by the excess pressure of the atmospheric air the striker moves to the left. When the piston moves to the right, compression of the air begins,

the striker changes the direction of motion and, by the end of its stroke, acquires a certain impact velocity.

From the standpoint of simplicity and strength, the design raises no objections. The question of reliability in operation (stability, cycle period, the role of initial conditions, etc.) and of sufficient efficiency required preliminary investigation—theoretical and experimental.

Fig. 2

The former was carried out by us before the prototype was manufactured. The technical characteristics of the OEM-1 hammer according to the design are:
\(N = 390 w;\ n = 2700\) rpm; \(z = 700\) blows/min; \(A_b = 2.5\) kgm; \(Q_b = 1.16\) kg; \(Q_m = 13\) kg; \(d = 5.5\) cm. The hammer is switched off (passes to idle running) when the pressure on the pick is stopped.

As a calculation example in this work we have taken the electropneumatic hammer OEM-1. In principle, however, by the method we propose one can calculate any electropneumatic hammer (or perforator).

Fig. 3

§ 2. Equations of motion of the striker and crank

A hammer with a pneumatic coupling (Fig. 3) is a mechanism with two degrees of freedom; therefore, an understanding of the working process of the hammer can be obtained by solving jointly two differential equations: the equation of motion of the striker and the equation of motion of the motor (or piston).

We shall formulate the equations of motion on the basis of the following assumptions and simplifications:

1) the hammer is arranged horizontally;
2) frictional forces in the mechanism are absent;
3) the quantity of air between the piston and the striker is constant (there are no leaks);
4) the change in air pressure in the working volume occurs according to the law \(p = p_0 \left(\dfrac{l_0}{l}\right)^a\), where \(a\) is the polytropic exponent.

We consider the motor characteristic to be given.

Note. In view of the specific operating conditions of the mechanism, in the calculation it is necessary to use not only the rectilinear (or straightened) part of the characteristic; therefore, in the absence of a graph or table of the dependence of torque on slip-

... for a given slip (or rotational speed \(n\)), having catalog data, one may use Kloss’s first formula, whose error, however, sometimes reaches 17–20%. Kloss’s second formula, which takes into account the resistance of the primary winding of the motor, has a more complicated form and requires additional data. Eng. L. V. Geyler [4] proposed for the torque of an asynchronous motor a formula which, in accuracy, is not inferior to Kloss’s second formula, but is more convenient for calculation and, according to the author of the formula, suitable for a broader class of motors. According to our calculations, Geyler’s formula gives an error of up to 6% relative to the characteristic taken from the motor.

Under the stated assumptions and according to Fig. 4, as well as the adopted notation, the equation of motion of the striker has the form:

\[ \frac{Q_b}{g}\,W_b=-B\,(p-p_0). \tag{1} \]

The equation of motion of the crank can be written in the form

\[ J\cdot \varepsilon=M_q+M_p, \tag{2} \]

where

\[ J=J_0+\frac{Q_p+\frac{1}{3}Q_{\sh}}{g}\cdot r^2\cdot \left(\sin\varphi+\frac{\lambda}{2}\cdot\sin 2\varphi\right)^2 \]

is the moment of inertia of the moving masses reduced to the crank axis;

\(J_0=\mathrm{Const}\) is the moment of inertia of the motor, crank, and \(2/3\) of the connecting-rod mass;

\(M_q=f(\omega)\) is the driving torque determined by the specified motor characteristic.

\[ M_p=r\cdot B\cdot(p-p_0)\cdot \left(\sin\varphi+\frac{\lambda}{2}\sin 2\varphi\right) \]

is the moment of the air-pressure forces on the piston with respect to the crank axis. The quantity \(M_p\) may be positive or negative, depending on the pressure \(p\) in the cylinder and the crank angle \(\varphi\).

Note. When accounting for resistance forces, it is necessary to suppose in Eq. (1), in addition to Coulomb friction, air-resistance forces proportional to the velocity or to the square of the striker velocity (depending on the magnitude of \(v_b\)); and in Eq. (2) to add to the right-hand side the moment of the resistance forces, expressed as a percentage of \(M_p\). In addition, for comparatively large values of the quantities \(r,\omega,Q_p,Q_{\sh}\), it is necessary also to take into account on the shaft the moment due to the inertia forces of the piston and connecting rod.

Fig. 4

We shall determine the paths of the piston and striker by their distances from the initial position (in Fig. 4a, the extreme right position), and we shall call the difference \(x-s=u\) the relative distance between them.

At \(t=0;\ \varphi=0,\ u_0=0\).

From Fig. 4 it is seen that \(l_0+s=l+x\); consequently: \(u=x-s=l_0-l\),

whence \(l=l_0-u\), and the pressure

\[ p=p_0\left(\frac{l_0}{l_0-u}\right)^\chi . \]

The displacement of the piston \(s\) can be expressed as a function of the angle of rotation \(\varphi\) by the formula:

\[ s=r\left[\left(1+\frac{\lambda}{4}\right)-\left(\cos\varphi+\frac{\lambda}{4}\cdot\cos 2\varphi\right)\right]. \]

Two simultaneous differential equations (1) and (2) describe the motion of the striker and the piston, which are connected to one another by an elastic (air) coupling. Since this coupling does not obey Hooke’s law, the equations of motion are nonlinear differential equations, as can also be seen directly by presenting them in the following general form:

\[ W_{\sigma}=\ddot{x}=a[x+b(\varphi)]^{-\alpha}+c=f[x(t),\varphi(t)], \tag{1} \]

\[ \varepsilon=\ddot{\varphi}=\frac{M_q(\omega)+b\cdot(\varphi)\cdot\ddot{x}}{J(\varphi)} =f_1[x(t),\varphi(t),\dot{\varphi}(t)]. \tag{2} \]

An approximate solution of the system (1) and (2) of nonlinear differential equations with variable coefficients may be carried out by various methods, both graphically and numerically; in the latter case, with any desired degree of accuracy.

§ 3. Approximate integration of the equations of motion (1) and (2)

We shall set forth the essence of the methods and the procedure for calculating the system of equations (1) and (2) without deriving the formulas and without rigorous justifications, since the latter can be found in the literature cited in this section, as well as at the end of the work. We dwell on this question because the long-standing problem of calculating an electropneumatic hammer had for a relatively long time remained unsolved. In addition, the results we have obtained make it possible to judge that this same calculation method can be used on a broader scale in studying the working process of various mechanisms and machines.

The question of the choice of method and degree of accuracy is determined to a considerable extent by the amount of computational work, and also by the purpose of the calculation. Here it is useful to recall the statement of Academician A. N. Krylov that “the accuracy of computations must correspond to the accuracy of the data and to the practical need for which it is carried out” [9, 486].

In selecting the principal parameters \((l_0,\lambda,Q_{\sigma},M_q)\), which must provide the required cycle, the calculation may be made by Euler’s first method, assuming the pressure \(p\) and the torque on the shaft within each interval (step) \(\Delta t\) to be constant and equal to their values at the beginning of the step; in other words, the computations are to be carried out by the formulas of uniformly accelerated motion.

The computer himself must choose the interval \(\Delta t\) according to the character of variation of the quantities entering into the calculation (pressure \(p\), etc.).

The advantage of this calculation method is its simplicity and the possibility, as needed, of immediately changing the step \(\Delta t\); its disadvantage is the accumulation of errors and a comparatively large error in relation to calculations by more accurate methods. In our work, with the same values of the step \(\Delta t\), the error amounted to as much as 25% for various indicators.

A good and simple method of solution, giving sufficiently accurate results, is indicated by Prof. P. S. Timoshenko [22] and Prof. E. L. Nikolai [17] (the difference in the calculation in the indicated manuals is insignificant). Prof. V. P. Vetchinkin [3, 305] classifies the method of integri-

... Timoshenko—Nikolai integration among the “unforgettable” ones for the computer.

Let us explain the order of the computations in applying to our problem the joint solution of the equations of motion of the striker and the crank:

\[ W_b=\ddot{x}=f(x,\varphi)\ldots \tag{1} \qquad \text{and} \qquad \varepsilon=\ddot{\varphi}=f_1(\varphi,\dot{\varphi},x)\ldots \tag{2}, \]

where \(x=x(t)\), \(\varphi=\varphi(t)\), and \(\dot{\varphi}=\dot{\varphi}(t)\).

Let, at \(t=t_k\), where \(k=0,1,2,3,\ldots\), the values \(x_k,\ \varphi_k,\ \dot{x}_k,\ \dot{\varphi}_k\) be known, and consequently,

\[ \ddot{x}_k=f(x_k,\varphi_k) \qquad \text{and} \qquad \ddot{\varphi}_k=f_1(\varphi_k,\dot{\varphi}_k,x_k). \]

In passing from the value \((k)\) to the next \((k+1)\), over the interval \(\Delta t=t_{k+1}-t_k\), we find, in the first approximation for the time \(t_{k+1}\), the values:

\[ \begin{aligned} (\dot{x}_{k+1})&=\dot{x}_k+\ddot{x}_k\Delta t,\\ (x_{k+1})&=x_k+\frac{\dot{x}_k+(\dot{x}_{k+1})}{2}\cdot\Delta t, \end{aligned} \qquad \left| \begin{aligned} (\dot{\varphi}_{k+1})&=\dot{\varphi}_k+\ddot{\varphi}_k\Delta t,\\ (\varphi_{k+1})&=\varphi_k+\frac{\dot{\varphi}_k+(\dot{\varphi}_{k+1})}{2}\cdot\Delta t. \end{aligned} \right. \]

Next, from Eqs. (1) and (2), we determine, in the first approximation,

\[ (\ddot{x}_{k+1})=f[(x_{k+1}),(\varphi_{k+1})] \qquad \text{and} \qquad (\ddot{\varphi}_{k+1})=f_1[(\varphi_{k+1}),(\dot{\varphi}_{k+1}),(x_{k+1})]. \]

Finally, more accurate values for the time \(t_{k+1}\) in the second approximation are obtained:

\[ \begin{aligned} \dot{x}_{k+1}&=\dot{x}_k+\frac{\ddot{x}_k+(\ddot{x}_{k+1})}{2}\cdot\Delta t,\\ x_{k+1}&=x_k+\frac{\dot{x}_k+\dot{x}_{k+1}}{2}\cdot\Delta t, \end{aligned} \qquad \left| \begin{aligned} \dot{\varphi}_{k+1}&=\dot{\varphi}_k+\frac{\ddot{\varphi}_k+(\ddot{\varphi}_{k+1})}{2}\cdot\Delta t,\\ \varphi_{k+1}&=\varphi_k+\frac{\dot{\varphi}_k+\dot{\varphi}_{k+1}}{2}\cdot\Delta t. \end{aligned} \right. \]

Thereafter, in the same order, we carry out the computations for the next interval, and so on. All the computational material is arranged in tabular form (two lines for each point).

“Taking the intervals \(\Delta t\) sufficiently small and carrying out the computations for each value of \(t\) twice, as indicated above, in order to obtain the second degree of approximation, we can, by this method of numerical integration, always obtain an accuracy sufficient for practical applications” [22, 85].

The error of computation by this method, in comparison with computations by more exact methods (approximately over the same intervals), amounted in our work, for individual indicators, to \(2.8\%\). (It is not useless to recall that the degree of accuracy of any method is determined to a significant extent by the patience of the computer and can be considerably increased by reducing the interval.)

In order to reduce the computational work, Prof. Vetchinkin proposes the “method of mean ordinates,” which gives almost the same accuracy as the Timoshenko—Nikolai method.

“The drawback of the Timoshenko—Nikolai method is the necessity of computing the function \(f\) twice in order to obtain each point. Meanwhile, for continuing the process of integration it is possible to obtain almost the same accuracy by means of a single computation of the func-”

…of the function \(f\) at each point. For this purpose one should compute the area under the curve by means of the ordinate drawn in the middle of the interval of integration, which is extended to a length \(2h''\) [2, 88].

In applying this to the solution of our problem, having the equations

\[ \ddot{x}=f[x,\varphi]\ldots (1) \qquad \text{and} \qquad \ddot{\varphi}=f_1[\varphi,\dot{\varphi},x]\ldots (2), \]

and also the initial values of the quantities \(t_0, x_0, \varphi_0, \dot{x}_0, \dot{\varphi}_0\), we compute, by the Timoshenko—Nikolai method, for the next point \(t_1\) the values: \(x_1, \dot{x}_1, \varphi_1, \dot{\varphi}_1\), and in subsequent computations pass to the “method of mean ordinates.”

For \(t=t_2\),

\[ \begin{aligned} x_2 &= x_0 + 2h\cdot \dot{x}_1,\\ \dot{x}_2 &= \dot{x}_0 + 2h\cdot \ddot{x}_1, \end{aligned} \qquad \begin{aligned} \varphi_2 &= \varphi_0 + 2h\cdot \dot{\varphi}_1,\\ \dot{\varphi}_2 &= \dot{\varphi}_0 + 2h\cdot \ddot{\varphi}_1. \end{aligned} \]

Knowing the values \(x_2,\dot{x}_2,\varphi_2,\dot{\varphi}_2\), we compute from equations (1) and (2) the values \(\ddot{x}_2\) and \(\ddot{\varphi}_2\), and pass to the next point \(t_3\).

\[ \begin{aligned} x_3 &= x_1 + 2h\cdot \dot{x}_2,\\ \dot{x}_3 &= \dot{x}_1 + 2h\cdot \ddot{x}_2, \end{aligned} \qquad \begin{aligned} \varphi_3 &= \varphi_1 + 2h\cdot \dot{\varphi}_2,\\ \dot{\varphi}_3 &= \dot{\varphi}_1 + 2h\cdot \ddot{\varphi}_2. \end{aligned} \]

From equations (1) and (2) we find \(\ddot{x}_3\) and \(\ddot{\varphi}_3\), and so on.

The chief drawback of this method of calculation is the limited range of its application, since, having the initial values of the functions, the next value must be computed by another method (which is significant when the step \(h\) is changed, especially near peak positions).

In order to obtain a general picture of the working process of the mechanism without a significant expenditure of computational labor, it is first necessary to make the calculation by the “combined method”—by the Timoshenko—Nikolai method at the first point of the interval taken and near the peak positions, while at the remaining points using the “method of mean ordinates” and even the first Euler method, if the air-pressure curve changes slowly.

To speed up the calculation and eliminate motley variation in the computations, it is expedient to take, as the independent variable instead of time \(t\), the angle \(\varphi\) of rotation of the crank. Then it becomes possible to prepare in advance a table of values of \(s\) and \(J\) as functions of the angle \(\varphi\). (Incidentally, such tables for various \(\lambda\) already exist in the literature on internal-combustion engines.) In addition, it is very useful to prepare two more tables (which remains valid also with respect to other methods of computation):

1) from the engine characteristic—the dependence of the driving torque on the crankshaft on its angular velocity, i.e. \(M_d=f(\omega)\), computed approximately through \(\Delta\omega=0.5\ 1/\text{sec}\);

2) a table of pressure values as a function of the relative distance between the striker and the piston, i.e. \(p=f(u)\).

The interval in compiling the tables should be chosen so that eighth parts of the second-order difference may be neglected, i.e. the change of the quantity over the given interval may be considered linear.

In view of the uniformity of the work, the compilation of the tables does not require much labor, and the time spent on compiling them is fully repaid in the subsequent calculation; moreover, it is possible quickly to make a new calculation by changing some parameter (for example, \(Q_b\), etc.). It is also useful to plot the computed values immediately on a graph in one operation, thanks to which unnoticed accidental errors in computation can be detected in good time.

Taking the crank angle \(\varphi\) as the independent variable, we write the equations of motion of the striker (1) and of the crank (2) in the form

\[ W_{\delta}=\frac{p_{0}B g}{Q_{\delta}}\cdot \left[1-\left(\frac{l_{0}}{l}\right)^{\alpha}\right] \]

and

\[ \Delta E=\Delta\left[J\frac{\omega^{2}}{2}\right]= \]

\[ =(M_{q}+M_{p})\Delta\varphi, \]

after which the calculation is carried out on one sheet according to the following universal scheme, represented by Table 1.

Having the initial values \(\varphi_{0}, s_{0}, x_{0}, \dot{x}_{0}, \ddot{x}_{0}, p_{0}, \omega_{0}\) indicated in the table, we shall show the procedure for filling in the remaining columns and for passing to the next row, for which these quantities will be the new initial ones. Columns (3), (5), (14), (17), (22) are filled in, as needed, from tables compiled earlier. In the first approximation (denoted by round brackets), when passing from the initial position to the next point, i.e. from \(k=0\) to \(k=(1)\), we shall perform the calculation by Euler’s first method.

First of all we find the magnitude of the total moment \(M_{0}\) (column 18), and the increment of kinetic energy \((\Delta E_{0})=M_{0}\Delta\varphi\) (column 20); then for the next row in column (21) we determine the kinetic energy \((E_{1})=E_{0}+(\Delta E_{0})\). Knowing (from the table) the value of the reduced moment \(J_{1}\), we find the angular velocity \((\omega_{1})\) (column 23); in column (24) we write the increment of angular velocity \((\Delta\omega_{0})\), and in column (25) we enter the mean angular velocity over the interval \(\Delta\varphi\), equal to

\[ (\omega^{*}_{01})=\omega_{0}+\frac{(\Delta\omega_{0})}{2}; \]

in column (26) we determine the time interval

\[ (\Delta t_{01})=\frac{\Delta\varphi}{\omega^{*}_{01}} \]

of the motion of the crank. In column

Table 1

No.	Quantity / formula	0	(1)	1	2	3	4
27	\(\tau\)
26	\(\Delta t\)	\((\Delta t_{01})\)	\(\Delta t_{01}\)	\(\Delta t_{12}\)	\(\Delta t_{23}\)	\(\Delta t_{34}\)
25	\(\omega^{*}\)	\((\omega^{*}_{01})\)	\(\omega^{*}_{01}\)	\(\omega^{*}_{12}\)	\(\omega^{*}_{23}\)	\(\omega^{*}_{34}\)
24	\(\Delta\omega\)	\((\Delta\omega_{0})\)	\(\Delta\omega_{0}\)	\(\Delta\omega_{1}\)	\(\Delta\omega_{2}\)	\(\Delta\omega_{3}\)
23	\(\omega=\sqrt{\frac{2E}{J}}\)	\(\omega_{0}\)	\((\omega_{1})\)	\(\omega_{1}\)	\(\omega_{2}\)	\(\omega_{3}\)
22	\(J\)	\(J_{0}\)	\((J_{1})\)	\(J_{1}\)	\(J_{2}\)	\(J_{3}\)	\(J_{4}\)
21	\(E\)	\(E_{0}\)	\((E_{1})\)	\(E_{1}\)	\(E_{2}\)	\(E_{3}\)	\(E_{4}\)
20	\(\delta\varphi\cdot M=\Delta E\)	\((\Delta E_{0})\)	\(\Delta E_{0}\)	\(\Delta E_{1}\)	\(\Delta E_{2}\)	\(\Delta E_{3}\)	\(\Delta E_{4}\)
19	\(M\)	\(M_{0}\)
18	\(M=M_{q}+M_{p}\)	\(M_{0}\)	\((M_{1})\)	\(M_{1}\)	\(M_{2}\)	\(M_{3}\)	\(M_{4}\)
17	\(M_{p}\)
16	\(M_{q}\)
15	\(r-Q\)
14	\(Q\)	\(Q_{0}\)	\((Q_{1})\)	\(Q_{1}\)	\(Q_{2}\)	\(Q_{3}\)	\(Q_{4}\)
13	\(s-x-n\)	\(U_{0}\)	\((U_{1})\)	\(U_{1}\)	\(U_{2}\)	\(U_{3}\)	\(U_{4}\)
12	\(x=\dot{x}_{m}\)	\(x_{0}\)
11	\(x=\dot{x}_{m}\)	\(\dot{x}_{0}\)	\((\dot{x}_{1})\)	\(\dot{x}_{1}\)	\(\dot{x}_{2}\)	\(\dot{x}_{3}\)	\(\dot{x}_{4}\)
10	\(x=\ddot{x}_{m}\)	\((\ddot{x}_{0})\)	\(\ddot{x}_{0}\)
9	\(\dot{x}\,v=\dot{x}_{1}v\)	\((\Delta\dot{x}_{0})\)	\(\Delta\dot{x}_{0}\)	\(\Delta\dot{x}_{02}\)	\(\Delta\dot{x}_{13}\)	\(\Delta\dot{x}_{24}\)
8	\(\dot{x}=\dot{x}_{1}\)	\(\dot{x}_{0}\)	\((\dot{x}_{1})\)	\(\dot{x}_{1}\)	\(\dot{x}_{2}\)	\(\dot{x}_{3}\)	\(\dot{x}_{4}\)
7	\(xv\)	\((\Delta x_{0})\)	\(\Delta x_{0}\)	\(\Delta x_{02}\)	\(\Delta x_{13}\)	\(\Delta x_{24}\)
6	\(x\)	\(x_{0}\)	\((x_{1})\)	\(x_{1}\)	\(x_{2}\)	\(x_{3}\)	\(x_{4}\)
5	\(s\)	\(s_{0}\)	\(s_{1}\)	\(s_{1}\)	\(s_{2}\)	\(s_{3}\)	\(s_{4}\)
4	\(l\)
3	\(6.3^{m}15\frac{2}{4}+6.4^{m}5\)
2	\(\delta\)	\(\varphi_{0}\)	\((\varphi_{1})\)	\(\varphi_{1}\)	\(\varphi_{2}\)	\(\varphi_{3}\)	\(\varphi_{4}\)
1	\(\nu\)	0	(1)	1	2	3	4

(9)—the increment of the striker velocity: \(\Delta \dot{x}_0=\ddot{x}_0(\Delta t_{01})\), and in column (8)—the striker velocity: \((\dot{x}_1)=\dot{x}_0+(\Delta \dot{x}_0)\). In column (7) we enter the increment of the striker path over the interval \(\Delta \varphi\):

\[ (\Delta x_0)=\dot{x}_0\cdot(\Delta t_{01})+\frac{\ddot{x}_0(\Delta t_{01})^2}{2} = \left[\dot{x}_0+\frac{(\Delta \dot{x}_0)}{2}\right]\cdot(\Delta t_{01}). \]

Then in column (6) we write: \((x_1)=x_0+(\Delta x_0)\). In column (5) we enter (from the tables) the piston path \(s\), and then fill in (13), (14), (15), (16) for the value of the angle \((\varphi_1)\). In column (4), for checking (and for comparison with \(v_b\)), we place the piston velocity, determined by the formula

\[ v_n=\omega\cdot r\cdot\left[\sin\varphi+\frac{\lambda}{2}\sin2\varphi\right]. \]

The control point in the calculation is the equality: \(v_b=v_n\) at extreme values of the pressure \(p\). In the same sequence the calculation is carried out when filling the third row, etc.

Timoshenko–Nikolai Method

If the values obtained in the second row are taken as the first approximation, denoted by parentheses \((\,)\), then, from the mean values of the velocities \(v_b^*\) (column 10) and accelerations \(W_b^*\) (column 12), in the second approximation one can find the velocity

\[ \dot{x}_1=\dot{x}_0+\frac{\ddot{x}_0+(\ddot{x}_1)}{2}\Delta t_{01} \]

and the striker path

\[ x_1=x_0+\frac{\dot{x}_0+\dot{x}_1}{2}\Delta t_{01} \]

for the first point. In this case the time

\[ \Delta t_{01}=\frac{\Delta\varphi}{\omega_{01}^*} = \frac{\Delta\varphi}{\dfrac{\omega_0+\omega_1}{2}} \]

is determined from column (26), after in \(\Delta E_0\) (column 20), for the multiplier at \(\Delta\varphi\), we insert the mean value of the total moment

\[ M_{01}^*=\frac{M_0+(M_1)}{2} \]

and determine: \(E_1=E_0+\Delta E_0\) (column 21), and \(\omega_1\) (column 28).

The further calculation process is performed in the same sequence.

Thus, taking the angle \(\varphi\) as the independent variable, here we have the possibility of carrying out the calculation in the second approximation without changing the calculation scheme, merely repeating the computations for each point twice.

Method of “Mean” Ordinates

Having the initial and first rows completed, we fill in the second row \((k=2)\).

From column (21): \(E_2=E_0+\Delta E_{02}=E_0+M_1\cdot2\Delta\varphi\); from column 23 we determine \(\omega_2\) and the increment of angular velocity \(\Delta\omega_1=\omega_2-\omega_1\), the mean angular velocity of the crank

\[ \omega_{12}^*=\frac{\omega_1+\omega_2}{2} = \omega_1+\frac{\Delta\omega_1}{2}, \]

(column 25), and the time

\[ \Delta t_{12}=\frac{\Delta\varphi}{\omega_{12}^*} \]

from column (26). The striker velocity (column 8):

\[ \dot{x}_2=\dot{x}_0+\ddot{x}_1(t_2-t_1) = \dot{x}_0+\ddot{x}_1(\Delta t_{01}+\Delta t_{12}) = \dot{x}_0+\Delta\dot{x}_{02}; \]

the striker path (column 6):

\[ x_2=x_0+\Delta x_{02} = x_0+\dot{x}_1(\Delta t_{01}+\Delta t_{12}); \]

we compute: \(x_2\) (column 11), and \(M_2\) (column 18). Thus we fill the second row. For the third row we proceed analogously: \(E_3=E_1+M_2\cdot2\Delta\varphi\). Hence \(\Delta t_{23}=\dfrac{\Delta\varphi}{\omega_{23}^*}\) and for the striker:

\[ \dot{x}_3=\dot{x}_1+\ddot{x}_2(\Delta t_{12}+\Delta t_{23}), \]

\[ x_3=x_1+\dot{x}_2(\Delta t_{12}+\Delta t_{23}). \]

We compute \(\ddot{x}\) and \(M_3\), etc. Here the “mean” values of the ordinates of the velocity \(\dot{x}_3\) and acceleration \(\ddot{x}_3\) correspond to the middle of the interval in \(\varphi\) and, owing to the nonuniformity of the crank motion, do not coincide with the midpoint of the time interval; but in the calculation this difference is insignificant.

In order to determine the degree of accuracy of the methods considered, we carried out, for one of the variants, a comparison of the results with computations by the Adams–Störmer method. This method is most widespread among engineers thanks to the works of Academician A. N. Krylov. According to Prof. Vetchinkin, Acad. Krylov presented the method “with exhaustive clarity and completeness, with detailed instructions for starting and continuing the integration process and with numerous examples of application” [2, 6].

Prof. Markovich [12] points to the application of this same method to the study of the operation of a pneumatic hammer (although here we have only one equation of motion of the hammer ram).

To give an idea of the volume of work by the Adams–Störmer method, as well as of the order in which it is performed when applied to our problem, we shall show how to continue the tables, carrying out the computations with second-order differences. Having the equations of motion

\[ W_\sigma=\ddot{x}=f[x,\varphi]\ldots(1)\quad \text{and}\quad \varepsilon=\ddot{\varphi}=f_1[\varphi,\omega,x]\ldots\quad (2), \]

we compile the principal Table II, which is the final table for each point and whose filling requires no special explanation, and the auxiliary Table III, in which we shall assume all columns up to and including row \((k)\) to be filled. Here the following notation has been introduced:

\[ \xi_k=h^2\cdot w_{\sigma k}=h^2\cdot f[x_k,\varphi_k]; \]

\[ \psi_k=h\cdot \varepsilon_k=h\cdot f_1[\varphi_k,\omega_k,x_k]; \]

\[ \eta_k=h\cdot \omega_k, \]

where \(h=\Delta t;\ x=x(t);\ \varphi=\varphi(t);\ \omega=\omega(t)\).

Then, according to the formulas given below and in the corresponding order, we continue filling Table III, i.e., we jointly solve the system of equations (1), (2).

\[ \begin{aligned} 1)\quad &\Delta\omega_k=\psi_k+\frac{1}{2}\Delta\psi_{k-1}+\frac{5}{12}\Delta^2\psi_{k-2} &&\text{column }(12)\\ 2)\quad &\omega_{k+1}=\omega_k+\Delta\omega_k &&\text{” }(11)\\ 3)\quad &\Delta\varphi_k=\eta_k+\frac{1}{2}\Delta\eta_{k-1}+\frac{5}{12}\Delta^2\eta_{k-2} &&\text{” }(17)\\ 4)\quad &\varphi_{k+1}=\varphi_k+\Delta\varphi_k &&\text{” }(16)\\ 5)\quad &\eta_{k+1}=h\cdot \omega_{k+1} &&\text{” }(18)\\ 6)\quad &\Delta\eta_k=\eta_{k+1}-\eta_k &&\text{” }(19)\\ 7)\quad &\Delta^2\eta_{k-1}=\Delta\eta_k-\Delta\eta_{k-1} &&\text{” }(20)\\ 8)\quad &\Delta^2 x_k=\xi_k+\frac{1}{12}\Delta^2\xi_{k-2} &&\text{” }(7)\\ 9)\quad &\Delta x_k=\Delta x_{k-1}+\Delta^2 x_{k-1} &&\text{” }(6) \end{aligned} \]

\[ \begin{aligned} 10)\quad & x_{k+1}=x_k+\Delta x_k && (5)\\ 11)\quad & \Delta \dot{x}=\frac{1}{h}\left(\xi_k+\frac{1}{2}\Delta \xi_{k-1}+\frac{5}{12}\Delta^2 \xi_{k-2}\right) && (4)\\ 12)\quad & v_b=\dot{x}_{k+1}=\dot{x}_k+\Delta \dot{x}_k && (3)\\ 13)\quad & \xi_{k+1}=h^2\cdot W_{k+1}=h^2\cdot f[x_{k+1},\varphi_{k+1}] && (8)\\ 14)\quad & \Delta \xi_k=\xi_{k+1}-\xi_k && (9)\\ 15)\quad & \Delta^2\xi_{k-1}=\Delta \xi_k-\Delta \xi_{k-1} && (10)\\ 16)\quad & \psi_{k+1}=h\cdot \varepsilon_{k+1}=h\cdot f[\varphi_{k+1},\omega_{k+1},x_{k+1}] && (13)\\ 17)\quad & \Delta\psi_k=\psi_{k+1}-\psi_k && (14)\\ 18)\quad & \Delta^2\psi_{k-1}=\Delta\psi_k-\Delta\psi_{k-1} && (15) \end{aligned} \]

Next we fill in the \((k+1)\)-th row of Table II to the end and, in the same order, carry out the calculation for the \((k+2)\)-th row, beginning with \(\Delta \omega_{k+1}\), etc. Here, for each point, the calculation is performed once. All the computational material can be placed in a single table, i.e., Tables II and III can be combined.

It should be noted that, in order to make it possible to apply the Adams–Störmer method, the calculations at the beginning of the tables must first be carried out by the method of successive approximations (up to the third approximation inclusive, if one confines oneself to second-order differences).

Table II

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
\(k\)	\(\varphi\)	\(\sin^2 \frac{\varphi}{2}\)	\(r\)	\(s\)	\(x\)	\(a=x-s\)	\(p\)	\(p-1\)	\(W_b\)	\(N_b\)	\(M_b\)	\(n\)	\(\xi\)	\(\omega\)	\(V_b\)	\(v_b\)	\(u'=v_b-v_n\)	\(t_0\)
0
1

Further filling of the tables is carried out by the Adams–Störmer method (Table III). However, when the air pressure in the cylinder changes rapidly, a number of quantities \((v_b,\omega,x)\) also change very rapidly; the second differences are no longer constant, and therefore differences of higher orders have to be introduced, which complicates the calculations. If, on the other hand, the interval is reduced, it is necessary to set up the beginning of the tables anew (by successive approximations), which again makes the calculations more complicated. It is more advantageous to take a smaller interval at first and, in the course of the work, if necessary, to double the interval (the corresponding scheme and example of calculation were considered by Academician Krylov [9]).

The question of the accuracy of the calculation is resolved as follows. In an electric jackhammer the velocity of the striker changes very rapidly; it is therefore necessary to make sure that the second differences \(\Delta^2 \xi\) for the striker are constant and correspond to the selected accuracy of the calculation. Suppose, for example, that from preliminary calculations it is known that the stroke of the striker is \(x=10\) cm. Consequently, in one revolution of the crank the path traversed by the striker may be taken approximately as \(\simeq 20\) cm. If an error of \(0.1\%\) is allowed, the greatest error must be \(0.2\) mm. Taking the interval corresponding to a crank rotation \(\Delta\varphi=15^\circ\), we find that the error per calculation must not numerically exceed

\[ \frac{0.2}{24}=0.008\ \text{mm}. \]

(10)	(6)	(8)	(4)	(3?)	(5?)	(7)	(2)	[[unclear]]	[[unclear]]	(3)	(6)	(8)	(7)	(9)	(5)	(4)	(2)	[[unclear]]	[[unclear]]
\(y_2\Delta\)	\(y_1\Delta\)	\(C\)	\(y\Delta\)	\(C\)	\(y_2\Delta\)	\(y_1\Delta\)	\(\lambda\)	\(\Delta\omega\)	\(\omega\)	\(y_2\Delta\)	\(y_1\Delta\)	\(\delta\)	\(\Delta x_2\)	\(\Delta x\)	\(x\)	\(\Delta x\)	\(x\)	\(t\)	\(x\)
•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•	•
\(K-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y\)	\(z-y\)	\(z-y_1\)	\(z-y\)
\(b?\)	\(y_2\)	\(y\)	\(y_1\)	\(y\)	\(\lambda y_2\)	\(\lambda y_1\)	\(\lambda\)	\(\omega\)	\(\omega\)	\(\delta y_2\)	\(\delta y_1\)	\(\delta\)	\(x_2\)	\(x_1\)	\(x\)	\(x\)	\(x\)	\(t\)	\(x\)
\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y_1\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y_2\)	\(z-y_1\)	\(z-y\)	\(z-y\)	\(z-y\)	\(z-y_1\)	\(z-y\)

Periods

In the formula

\[ \Delta^{2}x_{k-1}=\xi_k+\frac{1}{12}\Delta^{2}\xi_{k-2}+\frac{1}{12}\Delta^{3}\xi_{k-3} \]

the magnitude of the discarded first term must not exceed \(0.008 \ll \frac{1}{12}\Delta^{3}\xi_{k-3}\). If this rule is satisfied up to the end of the calculation, then the error of the computations over a cycle does not exceed \(0.1\%\). With an allowable error of \(0.1\%\), as a result of the computations the number of correct significant figures must be equal to three [1, 18].

Of the methods of graphical calculation known to us, the simplest and most suitable, when applied to the solution of the present problem, is the method of Eng. B. V. Sudnishnikov [20], but here too, besides graphical work, a number of computations accompanying the construction of the graphs is unavoidable; moreover, there is no criterion by which to judge the degree of accuracy of the calculation.

Let us also point out an additional and very effective method of calculation—the method of dynamic similarity, developed as applied to hammers by Eng. Sudnishnikov [19] and making it possible, when an exact calculation for a given type of hammer is available, to carry out a series of calculations for identical degrees of compression of other hammers similar in dynamic respect.

§ 4. “Combined method” of calculation, using the OEM-1 hammer as an example

Initial data for calculating the electropneumatic hammer OEM-1:

\[ r=3.8\ \text{cm};\quad L=9.5\ \text{cm};\quad l_0=6.8\ \text{cm};\quad d=5.5\ \text{cm};\quad Q_b=1.16\ \text{kg};\quad Q_p=0.19\ \text{kg}; \]

\[ Q_{\sh}=0.12\ \text{kg};\quad Q_m=13\ \text{kg};\quad J_b=8.7\cdot 10^{-4}\ \text{kg}\cdot\text{m}\cdot\text{sec}^{2};\quad \alpha=1.32;\quad i=\frac{10}{38}=0.263; \]

the energy on rebound is not more than \(10\%\) of the impact energy (here the value \(\beta=\sqrt{0.1}=0.316\) is adopted for the coefficient of restitution); the duration of the blow is \(\Delta t_0=0.001\ \text{sec}\). The motor characteristic is shown in Fig. 5. The position of the hammer is horizontal (Fig. 3); the transition from idle running to working operation is accomplished by pressing the lance. We take, at \(t_0=0\):

Fig. 5

\[ \varphi_0=0;\quad x_0=s_0=0;\quad l-l_0;\quad p_0\simeq 1\ \frac{\text{kg}}{\text{cm}^{2}};\quad v_0=\dot{s}_0=0;\quad \omega_0=82.5\ \text{sec}^{-1}. \]

At the beginning, the calculations from \(\varphi=0\) to \(\varphi=180^\circ\) were carried out with an interval \(\Delta\varphi=15^\circ\); near the peak position \(\Delta\varphi=2.5^\circ\). In steady motion, out of 44 points for one revolution of the crank, twelve were calculated by the Timoshenko—Nikolai method, while the remaining 32 were calculated by the “method of mean ordinates.” The time spent calculating one revolution of the crank was approximately 10 working hours. The results of the calculation for 6 revolutions, obtained by the method described, are combined in Table IV.

TABLE IV

Revolution	\(x^{\min}\) (cm)	\(v_b^{\max}\) (cm/sec)	\(v_b^{\min}\) (cm/sec)	\(v_b^{\mathrm{ud}}\) (cm/sec)	\(p^{\max}\) (kg/cm²)	\(p^{\min}\) (kg/cm²)	\(\varphi_{\mathrm{ud}}\) (°)	\(\omega^{\max}\) (1/sec)	\(\omega^{\min}\) (1/sec)	Revolution time (sec.)
1	9,67	+362	−798	−781	9,83	0,42	302,5	82,5	37,0	0,0921
2	10,44	+345	−741	−722	7,62	0,52	301,7	80,5	50,4	0,0889
3	6,73	+298	−749	−749	8,78	0,34	340,0	83,2	44,0	0,0853
4	8,52	+255	−721	−717	6,22	0,55	320,6	82,0	47,8	0,0885
5	7,82	+248	−751	−750	9,60	0,45	322,8	82,5	44,4	0,0887
6	7,76	+243	−739	−739	9,34	0,46	323,5	82,5	43,7	0,0892

Table V presents the values of \(x, p, v_b, \omega\) for each revolution for \(\varphi=180^\circ\) and \(\varphi=360^\circ\).

TABLE V

Revolution	\(\varphi=180^\circ,\ S=7.6\) cm: \(x\)	\(\varphi=180^\circ,\ S=7.6\) cm: \(p\)	\(\varphi=180^\circ,\ S=7.6\) cm: \(v_b\)	\(\varphi=180^\circ,\ S=7.6\) cm: \(\omega\)	\(\varphi=360^\circ,\ S=0\): \(x\)	\(\varphi=360^\circ,\ S=0\): \(p\)	\(\varphi=360^\circ,\ S=0\): \(v_b\)	\(\varphi=360^\circ,\ S=0\): \(\omega\)
1	4,23	0,56	+230	80,4	3,59	3,58	−91	59,1
2	6,50	0,82	+339	79,9	3,16	2,81	+76	71,2
3	0,62	0,35	+135	80,0	1,54	1,50	+208	55,2
4	4,98	0,61	+180	81,0	2,23	1,90	+167	64,8
5	3,32	0,48	+161	80,3	2,44	2,06	+162	62,5
6	3,23	0,50	+158	80,5	2,37	1,98	+160	59,6

For clarity, the results obtained are presented in the form of graphs.

Characteristic of the Working Process on the Basis of Tables IV–V and the Graphs

Let us dwell on some principal features of the transient regime, represented in Fig. 6 by graphs of the variation of the striker travel, the piston travel, and the angular velocity of the crank as functions of the angle \(\varphi\) of rotation of the crank.

In terms of force, the first revolution is the most difficult for the mechanism, since the initial velocity of the striker is zero; but, owing to the prolonged vacuum, the striker, when moving to the left, acquires a considerable velocity, \(\max v_b = 362\ \text{cm/sec}\), which it must lose during the subsequent compression of the air; here the greatest compression of the air and a decrease in the angular velocity of the mechanism take place. After the blow at \(\varphi = 302.5^\circ\) we have opposite motion of the striker and the piston, as a result of which there is a considerable change in the initial conditions for the second revolution.

As is seen from Table V, at the end of the first revolution the pressure is \(p = -3.58\ \text{kg/cm}^2\), and the striker changes the direction of its motion, having a velocity \(v_b = -91\ \text{cm/sec}\), and at \(\varphi = 55^\circ\) a secondary (weaker accompanying) blow occurs; after it the striker has an initial velocity \(v_b = +95\ \text{cm/sec}\) and moves in the same direction as the piston; since the relative distance between them is smaller than in the first case, the vacuum is also smaller, and the striker is accelerated to the left to the velocity \(\max v_b = +345\ \text{cm/sec}\). As a result, during compression of the air cushion, \(p_{\max} = 7.62\ \text{kg/cm}^2\) is obtained. Thus the striker, having an initial velocity in the presence of rarefaction, and then a comparatively small compression, traverses a distance \(x_{\max} = 10.44\ \text{cm}\). At the second main blow, owing to the lower pressure (and vacuum), the striker has a lower impact velocity than at the first blow \((v_{b,\text{impact}} = -722\ \text{cm/sec})\), and consequently also a lower rebound velocity. During the opposing motion of the striker and the piston from \(\varphi = 301.7^\circ\) to \(360^\circ\), the distance traversed by the striker up to the beginning of the third revolution is \(x = 3.16\ \text{cm}\), which corresponds to a pressure \(p = 2.81\ \text{kg/cm}^2\); there is a delay in the change in the direction of the striker velocity, and in the third revolution of the crank the second accompanying blow occurs at \(\varphi = 135^\circ\), with a rebound velocity equal to \(+4.1\ \text{cm/sec}\); owing to this, by \(180^\circ\) of the third revolution the striker travel is \(x = 0.62\ \text{cm}\). Here we have the lowest pressure in the working part of the cylinder, \(p_{\min} = 0.35\ \text{kg/cm}^2\), and a rapid increase in the striker velocity; but since the time of motion of the striker from \(\varphi = 135^\circ\) is considerably less than in the preceding revolution, and, moreover, the initial velocity after rebound is small, the distance traversed by the striker to the left is \(x_{\max} = 6.73\ \text{cm}\). The increased pressure \(p_{\max} = 8.78\ \text{kg/cm}^2\) is explained by the fact that, for this position of the striker, the displacements of the piston from the initial position are comparatively small; therefore \(u = x - s\) is large. In the third revolution, for the reasons indicated, the impact point is displaced \((\varphi_{\text{impact}} = 340^\circ)\), as a result of which, from the instant of impact to the end of the revolution (i.e., from \(340^\circ\) to \(360^\circ\)), the time of motion of the striker is shorter than in the preceding case, and by the end of the third revolution the striker traverses a distance \(x = 1.5\ \text{cm}\). The pressure in the cylinder is insufficient to change the direction of the striker’s velocity; thus, the secondary accompanying blow is already absent here. In the fourth revolution the relative distances between the striker and the piston are comparatively small; therefore the degree of compression is also not large \((p_{\max} = 6.22\ \text{kg/cm}^2)\); the smallest striker velocity is \(v_b = -721\ \text{cm/sec}\); but in the presence of vacuum (since

the distance between the striker and the piston at this moment is greater than the initial one), a loss of velocity occurs, and by the moment of impact, at \(\varphi = 320.6^\circ\), the impact velocity is

\[ V_b = -717.0 \ \text{cm/sec}. \]

By the beginning of the fifth revolution of the crank, the striker, from the moment of impact, travels a distance \(x = 2.23\) cm (since the time of motion is greater than in the preceding case), to which corresponds a pressure \(p = 1.9\) kg/cm\(^2\); the resistance of the air pressure is sufficient to change the direction of the striker’s velocity, but owing to the small velocity in the reverse direction, and also to the occurrence of a vacuum at \(\varphi = 60^\circ\), in the curve representing the motion of the striker we have a slight downward dip (a trough).

The small oscillations of the quantities \(x\), \(p\), \(v_b\), \(\omega\), corresponding to the end of the fourth and fifth revolutions of the crank (Table V), show that the steady regime begins with the fifth revolution, 0.375 seconds after the mechanism is put into operation. The period \(T\) of the established motion corresponds to one revolution of the crankshaft, which is also confirmed by the check calculation for the sixth revolution. The discrepancy in the principal quantities lies entirely within the limits of the accuracy of the calculation.

Let us dwell in somewhat more detail on the consideration of the established motion of the mechanism over the period of its operation.

In Fig. 7 are shown, as functions of the angle \(\varphi\) of rotation of the crank, the path of the piston \(s\) and of the striker \(x\). The difference \(x - s = u\) gives the relative distance between the striker and the piston. \(x_{\max} = 7.82\) cm corresponds to \(\varphi = 285^\circ\); when \(u = 0 = x - s\), the air is in the initial state: \(p = p_0 \approx 1\) kg/cm\(^2\).

Fig. 7. Steady motion.
Axes and labels visible in the figure: \(\varphi^\circ\); paths \(x, s\) in cm; velocity \(v_b\) in cm/sec.

\(u > 0\) corresponds to compression of the air in the working part of the cylinder; here the striker gives up energy through the air cushion to the crank-and-connecting-rod mechanism and the motor.

\(u<0\) corresponds to a vacuum; here the striker receives energy through the air cushion from the motor.

In Fig. 8 are shown the piston velocity \(v_n\), the striker velocity \(v_b\), the angular velocity of the crank \(\omega\), and the air pressure \(p\) in the working part of the cylinder over one period. The difference \(v_b-v_n=\dot u\) gives the velocity of the relative motion between the piston and the striker. \(\dot u=0\), i.e. \(v_b=v_n\), corresponds to the extreme values of the pressure, which is a good check on the calculations in the process of carrying out the work (4 points are shown on the graph). In particular, the greatest compression is \(p_{\max}=9.6\ \mathrm{kg/cm^2}\), when \(v_b=v_n=-224.3\ \mathrm{cm/sec}\), at \(\varphi=290^\circ\).

The air-pressure curve, moreover, can be represented as a function:

Fig. 7

Fig. 8

1) \(p=f(x)\)—indicator diagram with respect to the stroke of the striker (gives the action of the air on the striker);
2) \(p=f(s)\)—indicator diagram with respect to the stroke of the piston;
3) \(p=f(u)\)—graph of the polytrope.

These curves make it possible to isolate that part of the striker’s energy which it receives as a result of the expansion of the air between the piston and the striker.

If the connection between the striker and the piston were rigid, and the striker could separate from the piston at its maximum velocity (which corresponds to \(\varphi \approx 75^\circ\), when the sign of the piston acceleration changes, \(W_n=0\)),

\[ {}_{\max}v_n=3.16\ \mathrm{m/sec}, \]

the striker would have the energy

\[ E=\frac{1}{2}\frac{Q_b}{g}\,{}_{\max}v_n^2=0.59\ \mathrm{kgm}. \]

In our case the striker velocity is \({}_{\max}v_{y0}=7.5\ \mathrm{m/sec}\); consequently, the impact energy in the striker is

\[ E_1=\frac{1}{2}\frac{Q_b}{g}\,v_{b\max}^2=3.33\ \mathrm{kgm}. \]

Comparison of the quantities

\[ \frac{E_1}{E}=\left(\frac{{}_{\max}v_{\text{impact}}}{{}_{\max}v_n}\right)^2 =\left(\frac{7.50}{3.16}\right)^2=5.65 \]

shows,

which can be “squeezed” out of the motor by introducing air as an elastic coupling between the piston and the striker. The overwhelming part of the striker’s impact energy is accumulated through the work of expansion of the compressed air.

But such “drawing” of energy from the motor has a very substantial effect on the graph of angular velocity \(\omega\), on the overall nonuniformity of the motion (coefficient of nonuniformity \(\simeq 0.6\)).

Figure 9 shows the acceleration graphs of the striker and the piston. The force acting on the striker is the air pressure; therefore the striker-acceleration curve is the air-pressure curve on the corresponding scale.

The piston-acceleration curve is connected with the change in the velocity of the latter, and since

\[ v_n = r \cdot \omega \cdot \left( \sin \varphi + \frac{\lambda}{2}\sin 2\varphi \right), \]

the acceleration is

\[ W_n = \frac{dv_n}{dt} = r\omega^2(\cos\varphi + \lambda \cos 2\varphi) + r\varepsilon \left( \sin\varphi + \frac{\lambda}{2}\sin 2\varphi \right). \]

In addition, one may be satisfied with the mean values of the acceleration, determined by the formula \(W_n = \dfrac{\Delta v_n}{\Delta t}\).

Figure 10 presents: the kinetic energy of the mechanism

\[ E = J \frac{\omega^2}{2}, \]

the driving moment \(M_q\), determined by the motor characteristic, and the moment of the forces of air pressure \(M_p\) as functions of the angle \(\varphi\) of rotation of the crank. The moment of air pressure \(M_p\) is large not only because the pressures in the cylinder are great, but also because these high pressures coincide with the maximum values of the factor

\[ \left( \sin \varphi + \frac{\lambda}{2}\sin 2\varphi \right). \]

Fig. 9

Fig. 10

In the hammer, if the greatest pressures are shifted in \(\varphi\) toward the end of the revolution, then smaller values of \(M_p\) are obtained and, consequently, smaller losses in the kinetic energy \(E\).

We define the recoil of the hammer as the motion of the hammer body under the action of all the forces applied to it. Since the external forces are not specified (the pressure force of the worker, and the force of the ground resistance during penetration), we shall determine the recoil of the hammer in its horizontal position under the action only of the internal forces: 1) the excess force of the air pressure in the cylinder and 2) the inertia force of the crank-and-connecting-rod mechanism. In Fig. 11 these forces are shown separately, as are also the velocity of the body and its displacement, under the assumption that \((v_M)_0 = 0\) and \((y_M)_0 = 0\).

As can be seen from the graph, in one revolution of the crankshaft, i.e., in the time \(\Delta t = 0.08875\ \text{sec}\), the velocity of the hammer body will attain the value \({}^{\max} v_M = -86.6\ \text{cm/sec}\), and its displacement during the same time will be \(y_M = -2.21\ \text{cm}\).

In the presence of an additional (resultant) external force \(R\), acting on the body in the direction from the handle toward the bit and equal to its weight, \(R = Q_M = 13\ \text{kg}\), after one revolution of the crankshaft the velocity of the hammer body is \(v_M = +0.4\ \text{cm/sec}\), and its displacement is \(y_M = +1.64\ \text{cm}\), reckoned from the initial position in the direction of action of the additional force.

As can be seen from the graphs and the calculation, here we have a mechanism with reduced recoil, which is explained by the comparatively large weight of the hammer, as well as by the presence of rarefaction in the cylinder, the short duration of the increased pressure, and the complete utilization of the energy of the compressed air in imparting velocity to the striker.

The Role of the Initial Data in the Working Process of the OEM-1 Hammer

It is, of course, impossible to obtain the desired working process for arbitrary values of the specified parameters. Only with a successful combination and selection of them is it possible to “squeeze” everything possible out of the design, in accordance with the requirements imposed upon it.

1. Since it is desirable to obtain the principal part of the impact energy on the striker through the work of the air, the values of the quantities \(r\), \(l_0\), \(d\), and \(\alpha\) must be chosen with particular care. The quantities \(r\) and \(l_0\) jointly determine the degrees of rarefaction and compression, as well as the position of the impact point (\(\varphi\) impact), which in general it is desirable to have near \(360^\circ\). With very small values of \(l_0\), after the rebound of the striker during its counter-motion with the piston, large pressures develop, “stopping” the motor; with large values of \(l_0\) in comparison with \(r\), there may be no motion of the striker at all if \(v_0 = 0\) and the friction forces are taken into account; in the presence of an initial velocity of the striker (for example, from a push), there may occur the phenomenon of a missed blow owing to turning of the crankshaft.

The magnitude of the pressure force \(P = pB\), acting on the striker and the hammer body, as well as the total weight of the mechanism, depends on the size of the diameter \(d\).

The adopted value of the polytropic exponent \(\alpha\) has a very strong influence on the character of the working process. (In our calculation the value \(\alpha = 1.32\) was taken on the basis of previous work carried out at the plant.)

1. In the working process of the hammer, a very substantial role is played by the motor and its characteristic.

In our example the motor operates for a comparatively long time on the “unstable” part of the characteristic (approximately \(1/6\) of a revolution of the crankshaft, or \(1/4\) of the period in time), which must lead to considerable losses in heating. Sometimes the motor operates as a dynamo, owing to the vacuum, having an angular velocity greater than the synchronous one. With unchanged

of the remaining data, a motor of special design with a stiffer characteristic is required.

The initial angular velocity \(\omega_0\) is approximately equal to the synchronous angular velocity, since the mechanism has a structural device in the form of a spring catch, Fig. 2, owing to which the transition from idling to working operation is effected by pressing the hammer body against the pick. The purpose of this device is to eliminate the possibility of contact between the striker and the slider at the initial instant for any operating position of the mechanism. In our calculation

\[ \omega_0=\omega_{\mathrm{syn}}=\frac{\pi n}{30}\, i = 82.5\ \mathrm{s}^{-1} \]

on the crankshaft. If the idling speed is taken as \(\omega_0=76\ \mathrm{s}^{-1}\) (with the same characteristic), then, as the calculation showed,

Fig. 11

the mechanism already at \(\varphi=90^\circ\) passes to the same value of angular velocity as if the motion had begun with \(\omega_0=82.5\ \mathrm{s}^{-1}\).

1. In the OEM-1 hammer, when pressed against the pick, switching to working operation may be effected at various values of the angle \(\varphi\), which corresponds to a variable value of the length of the air cushion within the limits from \(l_0\) to \(l_0+2r\), and consequently to different initial operating regimes.

For a stationary installation of the mechanism and in the absence of impacts from the pick shank, the initial velocity of the striker is \(v_0=0\); when the hammer body is pressed, approaching the pick, it releases the striker from the catch. The striker in this case does not completely cover the openings in the sleeve between the inner working part of the cylinder and the outer part, placed, in order to avoid clogging with dust, in the casing of the mechanism (the width of the remaining

openings \(\simeq 1.5\) mm). Here, when switching on in the interval from \(180^\circ\) to \(360^\circ\), the air from the sleeve will be pushed out by the piston through the uncovered openings into the space surrounding the sleeve and the motor; with further motion of the piston to the left, a rarefaction is created in the cylinder, and under the excess pressure of the atmospheric air the striker will move to the left. Fig. 12 gives the general picture of the working process for \(\varphi_0 = 90^\circ\) (“incomplete blow”); the other possible cases of switching on in the interval from \(\varphi_0 = 0\) to \(\varphi_0 = 90^\circ\) will be located between these limiting cases. At angles close to \(180^\circ\), the rarefaction in the cylinder is insufficient to impart motion to the striker (especially in the presence of frictional force).

At the moment of striking the pick (with the hammer housing stationary), the striker receives an initial velocity, the magnitude of which depends on the magnitude of the transmitted impulse. An initial working stroke is possible at any \(\varphi\); consequently, in the OEM-1 there are many possible initial switchings-on of the striker, but, with the presence of openings in the sleeve and also of rebound of the striker, the mechanism very quickly reaches its normal working state. Specially conducted tests of the prototype only for switching on showed in all cases stable operation of the hammer (by ear—with approximately the same impact force) without any interruptions; evidently, the most probable initial switching-on angle is the angle \(\varphi_0\) between \(0^\circ\) and \(90^\circ\). The question of transient regimes in connection with switching on the striker at arbitrary angles \(\varphi\) and initial striker velocities \(v_0\) still requires special investigation.

1. The position of the mechanism affects the character of the working process: with a vertical or inclined position of the hammer it is necessary in equation (1) to take into account the force of gravity of the striker, but, as calculation and experimental tests have shown, the influence of the position of the mechanism on the character of the hammer’s working process is small and is important mainly for the initial moment.

Fig. 12

1. The coefficient of restitution \(\beta\) has a large influence on the character of the working process and on the duration of the transient regime.

Fig. 13

In our calculation it was assumed that the rebound energy of the striker \(= 10\%\) of the impact energy; consequently, \(\beta = \sqrt{0.1} = 0.316\). The corresponding curve \(x = f(\varphi)\) in Fig. 13 after the first blow (from \(\varphi = 302.5^\circ\) to \(\varphi = 180^\circ\) of the next revolution) is designated by the number 1. Shown here as well is the path of the striker after impact for

became equal to \(\beta = 5/9\) (curve 2 in Fig. 13). In this case, during the accompanying blow, greater pressures develop than in the first case, and after the second rebound the striker, having a considerable velocity, overtakes the piston, giving it energy. Owing to the small value of \(u = x - s\), the compression ratios must be smaller and, consequently, the impact velocity of the striker also decreases.

If the blow is inelastic, i.e., \(\beta = 0\), the striker gives up all its energy; the accompanying blow is absent. The curve corresponding to this case is denoted by the number 3.

§ 5. Conclusion

By means of preliminary theoretical investigation and calculation, it is possible to obtain diagrams of the working process of the hammer being designed and, by studying them, to eliminate the observed shortcomings before producing a prototype. In numerical integration by the methods described, it is possible to carry out the calculation taking into account resistance forces, the position of the hammer, the motion of its body, heating, and air discharge. The “combined” method can be applied to the study of the operation of other mechanisms, and by comparison one can also study the operation of a given motor under various kinds of loads.

As applied to electropneumatic hammers, it is necessary in further work to study and determine more carefully such quantities as the polytropic exponent, friction forces, the moment of resistance forces, the coefficient of restitution, resistance forces during the passage of the hammer, heating, air leakage, etc.

It is necessary to develop a more rigorous methodology for laboratory tests, with the establishment of the corresponding indicators and standards (at present almost every inventor proposes his own methodology). In the theoretical part of the investigation, an attempt should be made to find the general solution of the simultaneous equations (1) and (2), and also to clarify the role of the initial conditions in the general case. If a general solution is not obtained, or if it proves to be very complicated, then it is necessary, with a known degree of accuracy, taking into account laboratory tests of prototypes, to compute a number of tables and construct nomograms by which the calculation can already be made comparatively simply. Having a number of accurate calculations, it will then be possible, on the basis of similarity theory, to carry out (without great computational labor) a series of calculations for dynamically similar mechanisms.

LIST OF LITERATURE AND HANDBOOKS USED

1. Bezikovich N. and Friedman A.—Approximate Calculations. Gosizdat, Leningrad, 1925.

2. Prof. Vetyankin V. P.—Manual on Approximate Calculations. TsAGI Publishing House, Moscow, 1935.

3. Prof. Vetyankin V. P.—Methods of Approximate and Numerical Integration of Ordinary Differential Equations (Issue III). Publishing House of the Air Force Academy of the RKKA, Moscow, 1935.

4. Geiler, L. B.—Formula for the Torque of an Asynchronous Motor. Bulletin of the Electrical Industry, 1928, No. 12.

5. Gnichenko V. A.—Pick Hammers and Their Application in the Mechanization of Coal Mining in Steeply Dipping Seams. Soyuzugol Publishing House, 1930.

6. Dobrogursky O.—Problems of the Theory of Impact. Academy of Sciences of the USSR, Institute of Mechanical Engineering. Moscow—Leningrad, 1942.

7. Prof. Ketov Kh. F., Assoc. Prof. Alyabuzhev P. M., Assist. Shlepkina N. G.—Study of the Dynamics and Working Process of a Gearless Electropneumatic Hammer. Report on work carried out under contract with TEMZ, 1942 (manuscript).

8. Prof. Ketov Kh. F. and Prof. Kolchin P. I.—Theory of Mechanisms and Machines. Mashgiz, Moscow—Leningrad, 1939.

9. Krylov A. N.—Lectures on Approximate Calculations. Academy of Sciences of the USSR, Leningrad, 1933.

1. Kucherov P. S.—On the question of the study of pneumatic jackhammers. Coal, 1933, No. 33.

2. Maleev G. I.—Electropneumatic hammer. Mining Machine Building, 1939, Nos. 4–5.

3. Prof. Markovich—Study of the pneumatic driving hammer of the Besche and Grose type. Publishing House, Moscow, 1932.

4. Melentiev P. V.—Several new methods and techniques of approximate calculations. ONTI, Moscow–Leningrad, 1937.

5. Moskvitin A. I.—Electropneumatic or solenoid hammer. Electricity, 1934, Nos. 5–6.

6. Moskvitin A. I.—The problem of the electric jackhammer. History of Technology, issue II, ONTI, 1934, pp. 161–182.

7. Pokrovsky N. M.—Designing hammers of a new construction. Mining Journal, 1933, No. 1.

8. Prof. Nikolai E. L.—Lectures on theoretical mechanics, Part II. ONTI, Moscow–Leningrad, 1934.

9. Scarborough D. S.—Numerical methods of mathematical analysis, GTTI, Moscow–Leningrad, 1934.

10. Sudnishnikov B. V.—Some questions in the theory of impact machines. Dissertation defended at TII in 1944.

11. Sudnishnikov B. V.—An elementary graphical solution of certain dynamic problems. Bulletin of Engineers and Technicians, 1940, No. 2.

12. Acad. Terpigorev A. M., Assoc. Prof. Demidov P. N., Assoc. Prof. Protodiakonov M. M. et al. Mining machines for the extraction of minerals (chapter “Jackhammers,” pp. 482–509), Moscow, 1940.

13. Timoshenko S. P.—Theory of vibrations in engineering. GNTI, Moscow–Leningrad, 1931.

14. Unskov E. P.—Theoretical calculation of a spring hammer with a flat leaf spring (from the works of TsNIIMASH), Bulletin of the Metal Industry, 1936, No. 7.

15. Filippov M. F.—On the equation of the characteristic for a three-phase asynchronous motor. Proceedings of TII, vol. 59, II, 1941.

16. Shargunov K. N.—Electric jackhammers. Design, calculations, and testing. Moscow–Sverdlovsk, 1937.