On the Oscillatory Motion of the Housing of an Electric Demolition Hammer with a Crank-and-Connecting-Rod Mechanism

P. M. Alabuzhev

§ 1. “Electromechanical demolition hammers are of extremely great practical interest not only for the mining industry, but also for a whole range of other branches of the national economy. This determines the need for their comprehensive theoretical investigation, to which due attention has not been paid in the literature,” writes Corresponding Member of the Academy of Sciences of the Ukrainian SSR P. S. Kucherov [8].

The present work is an attempt to clarify certain questions connected with the recoil in electromechanical and electropneumatic demolition hammers with a crank-and-connecting-rod mechanism, which is driven by an electric motor built into the hammer housing. The unpleasant phenomena inherent in machines with reciprocating moving parts are well known in practice (oscillations of foundations, vibrations of crankshafts, etc.) and have been treated quite fully in the literature. For an electric demolition hammer this question is especially important, and one of the conditions imposed on a demolition hammer is the requirement of reduced recoil at minimum weight. Sometimes recoil is understood as the oscillatory motion of the housing of an impact machine during its operation; in addition, it is also customary to call recoil the negative effect of the operating impact machine on the operator (worker). It is more correct to define recoil in the second sense, i.e., as the reactive action of the oscillating hammer on the operator; moreover, this reactive action (recoil) can be significantly reduced only by introducing an elastic element between the hammer and the operator.

Reducing recoil means improving working conditions and increasing labor productivity. Reducing the oscillations of the housing and of individual parts of the machine means striving for the durability and reliability of the machine itself. A radical reduction of recoil can be achieved only by rational design of the impact machine (the proper choice of the parameters determining its working process, balancing the forces that cause oscillations, and introducing elastic elements). Without dwelling on the description of the various schemes and designs of electric hammers with a crank-and-connecting-rod mechanism, we shall note that the choice of the principal parameters is the most important and responsible part of the work in design engineering. For the most part, the designer proceeds by feel on the basis of existing experience, sometimes applying the principle of dynamic similarity.

A theoretical investigation, with an indication of a method for calculating the parameters of electromechanical spring hammers, was carried out by P. S. Kucherov [8]; in this investigation the influence of resistances is not taken into account, nor is the rebound of the striker after impact, but nevertheless it is a valuable contribution to the design of “compromised” (by operating experience) spring hammers.

The author of the present work has carried out a study of the working process of an electro-pneumatic hammer [1] for given values of its parameters; on the basis of a number of calculations by the indicated method, and of experiments, it is possible to obtain the desired results, i.e., to select the appropriate values of the parameters determining the operation of the impact unit, taking into account the nonuniformity of rotation of the motor, the rebound of the striker, and various kinds of resistance. The essence of the calculation consists in the numerical solution of two simultaneous nonlinear differential equations with variable coefficients: the equation of motion of the striker

\[ m_b W_b = F \tag{1} \]

and the equation of motion of the motor (piston)

\[ \frac{d}{d\varphi}\left(I\frac{\omega^2}{2}\right)=M, \tag{2} \]

where

\[ \begin{aligned} m_b &\text{ is the mass of the striker;}\\ W_b &\text{ is the acceleration of the striker;}\\ F &\text{ is the resultant of all forces applied to the striker (depending on the design, the components may be: the excess force } F_p \text{ of the pressure of the air enclosed between the piston and the striker,}\\ &\quad \text{the spring force } -F_{pr}, \text{ the resistance force } -F_c, \text{ as well as the force transmitted through the rigid connection between the piston and the striker } -F_{cb}, \text{ the force of gravity } -F_g);\\ I &\text{ is the moment of inertia of the moving masses, reduced to the crankshaft;}\\ \omega &\text{ is the angular velocity of the crank;}\\ M &\text{ is the resultant moment of forces on the crankshaft (depending on the design, the components may be: } M_k \text{—the torque, reduced to the crankshaft and determined by the motor characteristic;}\\ &\quad M_Q \text{—the moment of the inertia forces of the reciprocating masses; } M_p \text{—the moment of the excess air pressure acting on the piston; } M_{pr} \text{—the moment of spring pressure; } M_c \text{—the moment of friction forces).} \end{aligned} \]

From the solution of equations (1) and (2) one can find, as a function of the angle \(\varphi\) of rotation of the crank (or of the time \(t\)), the law of motion of the striker \(x=f(t)\) and its velocity \(v_b\), the angular velocity of the crank \(\omega\), the air pressure \(p\), etc. For clarity, the numerical calculations obtained may be presented in the form of graphs, which give the designer a complete picture of the working process of the machine. Figure 1 shows a diagram of the hammer.

Labels in Fig. 1: operator; link; striker; pick; hammer housing.

Fig. 1

with a crank-and-connecting-rod transmission between the motor and the striker, moreover between the striker and the piston there is some connection (rigid or elastic, which at certain moments of time may also be absent). Let us choose the origin of coordinates \(O\) on the axis of the crankshaft, directing the \(x\)-axis along the axis of the hammer; we shall agree to call the plane \(xOy\) the transverse plane with respect to the axis of the hammer and the axis of the shaft. As is seen from Fig. 1, the distance of the piston from the shaft axis is determined by the relation:

\[ S=R\cos\varphi+L\cos\beta,\qquad \text{besides } R\sin\varphi=L\sin\beta, \]

where

\[ \left. \begin{aligned} R&\text{— crank radius,}\\ L&\text{— length of the connecting rod} \end{aligned} \right\} \quad \text{we denote by} \quad \lambda=\frac{R}{L}\quad \left(\text{usually } \lambda=\frac{1}{3}\right) \]

\[ \cos\beta=\sqrt{1-\sin^2\beta} =(1-\lambda^2\sin^2\varphi)^{1/2} \simeq 1-\frac{\lambda^2}{2}\left(\frac{1-\cos 2\varphi}{2}\right) \]

In view of the smallness of \(\lambda\), we restrict ourselves to the first two terms of the series, whence, after substitution, we obtain an expression sufficiently accurate for practical purposes:

\[ s=L\left(1-\frac{\lambda^2}{4}\right) +R\left(\cos\varphi+\frac{\lambda}{4}\cos 2\varphi\right), \]

the piston velocity

\[ v_n=\frac{ds}{dt} =-R\omega\left(\sin\varphi+\frac{\lambda}{2}\sin 2\varphi\right), \]

where

\[ \omega=\frac{d\varphi}{dt} \]

and the piston acceleration

\[ W_n=\frac{d^2s}{dt^2} =-R\omega^2(\cos\varphi+\lambda\cos 2\varphi) -R\varepsilon\left(\sin\varphi+\frac{\lambda}{2}\sin 2\varphi\right), \]

where

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

In further considering the question of balancing the inertia forces developed by the connecting-rod-and-crank mechanism, we reduce the moving masses to two points \(A\) and \(B\) with the reduced masses:

\[ m_A=m_1\cdot\frac{e}{R}+m_3\frac{b}{L}, \]

\[ m_B=m_2+m_1\frac{a}{L}, \]

where

\[ a+b=L;\qquad a=AC_3,\qquad \text{and}\qquad b=BC_3, \]

\[ \begin{aligned} m_1&\text{— mass of the crank, concentrated at its center of gravity }—G_1,\\ m_2&\text{— mass of the piston} \hfill —G_2,\\ m_3&\text{— mass of the connecting rod} \hfill —G_3. \end{aligned} \]

Then the inertia force of the translationally moving mass is

\[ Q=-m_B\cdot W_n. \]

The inertia force of the rotating mass has two components: the normal

\[ Q_{nA}=-m_A\cdot R\omega^2 \]

(centrifugal force) and the tangential

\[ Q_{tA}=-m_A R\varepsilon \]

(tangential inertia force).

With a sufficient degree of accuracy, the kinetic energy of the system motor—crank—connecting rod—piston can be represented in the form

\[ E = I_0 \cdot \frac{\omega^2}{2} + m_B \cdot \frac{v_n^2}{2} = I_0 \cdot \frac{\omega^2}{2} + \frac{m_B R^2 \omega^2}{2} \left( \sin \varphi + \frac{\lambda}{2}\sin 2\varphi \right)^2, \]

where \(I_0\) is the moment of inertia of the electric-motor rotor and of the reduced mass \(m_A\). Consequently, the moment of inertia reduced to the crankshaft is

\[ I = I_0 + m_B R^2 \left( \sin \varphi + \frac{\lambda}{2}\sin 2\varphi \right)^2 . \]

The kinetic energy of the system \(E = I \dfrac{\omega^2}{2}\) will be expressed as a function of the angle \(\varphi\).

§ 2. Forces Acting on the Hammer Housing

External forces are applied to the hammer housing: from the worker and the reactive force of the tool (or of the material being worked). These forces balance the action of the internal forces and their moments during operation of the machine. It is known that “an engine is considered balanced if, during the steady-state regime of its operation, the rigid supports of the engine experience certain constant pressures” [4].

Having the equations of motion of the striker (1) and piston (2), for any angle \(\varphi\) of crank rotation we can find these internal forces and their moments. The reactive moment \(M_k\), applied to the hammer housing, is equal in magnitude to the torque on the shaft \(M_k\) (determined by the characteristic of the motor), and is directed opposite to the rotation of the crank; this can be verified from consideration of Fig. 2. According to the known laws of mechanics, if the moment of the resultant couple \(M = \operatorname{mom}(PP)\), then, resolving the force \(P\), applied at point \(O\), into the components \(P_1\) and \(P_2\), transferring the force \(P\), applied in the crank pin, to point \(B\) and resolving it into the components \(P'_1\) and \(P'_2\), we obtain a balanced system of forces \((P_2\) and \(P'_2)\) and a couple of forces \((P_1P'_1)\), whose moment is equal to the moment of the original couple \((PP)\).¹

Fig. 2

To balance this couple, it is necessary (in accordance with Newton’s third law) to apply to the housing an external reactive couple \((R'R'_1)\).

Changes in the angular velocity \(\omega\), i.e., the nonuniformity of rotation of the motor (or crankshaft), cause a change in the moment acting on the hammer housing, as a result of which the latter will perform oscillatory motion in the transverse plane (in the plane \(xOy\)). If a mechanism with \(M_{kr} = \mathrm{const}\) could be built, then the oscillations of the housing would be minimal. An indicator of the relative merit of different con—

of constructions, from the standpoint of the magnitude of the reactive moment, is the ratio

\[ k=\frac{\max M_{\mathrm{cr}}}{\mathrm{mean}\, M_{\mathrm{cr}}}. \tag{3} \]

We shall assume that the rotating masses are dynamically balanced, i.e.: 1) the center of gravity of the rotating mass coincides with the axis of rotation (by introducing counterweights, balancing, etc.); 2) the axis of rotation is the principal axis of inertia for the given mass.

With a dynamically unbalanced rotating mass (for example, because of inhomogeneity of the material or inaccuracy of manufacture), centrifugal forces and moments develop, rocking the housing. Suppose, for example, that in a gearless MB hammer the center of gravity is displaced relative to the shaft axis by 1 mm; then at 3000 revolutions per minute, per 1 kg of rotating mass there will act a centrifugal force equal to 10 kg, changing its direction 50 times per second.

On the hammer housing, in the direction of its axis (Fig. 1), there acts the force \(F=m_b W_b\) (the force of constraint) and the inertia force, developed by the crank-and-connecting-rod mechanism, of the translating mass \(m_B\), equal to \(Q_B=m_B w_B\), since \(W_p=W_B\).

Let us denote the resultant of these forces by \(R\)

\[ R=F+Q. \]

The force \(R\) is internal for the machine; therefore the center of gravity of the whole system must be immobile, which in turn requires, when there are reciprocating masses inside the hammer (\(m_b\) and \(m_B\)), an oscillatory motion of the hammer housing in the direction of its axis.

The greatest value of the inertia force from the translating parts of the machine will be at \(\varphi=0\), i.e.

\[ \max Q_B=m_B\cdot R\omega^2(1+\lambda) \]

As an example (Table 1), let us estimate \(\max Q_B\) for the designs of the hammers KNSh-2, OM-1, MB-3 (at no load).

Table 1

These forces \(Q_B\) will cause shaking of the housing and will be very perceptible to the operator (especially in gearless hammers, in which at no load the number of piston strokes is equal to the number of revolutions of the electric motor).

Complete balancing of the first-order inertia forces (i.e., the term \(m_B R\omega^2\cos\varphi\)) of a single-cylinder engine can be achieved by means of a system of counterweights mounted on two auxiliary axes and driven by gear transmissions. But this device—

… because of its bulkiness, it can be recommended only for stationary installations. But in the case of a stationary installation (of a mobile impact machine), it is more expedient to design an impact machine equivalent to a multi-cylinder engine.

Knowing the weight of the hammer \(G_m\), its moment of inertia relative to the axis passing through the crankshaft axis \(I_m\) (or through its center of gravity), one can write the equations of motion of the hammer housing under the action only of internal forces

\[ -\frac{G_m}{g}\cdot w_m = R \]

\[ I_m \cdot \varepsilon_m = M_k \tag{3} \]

Integrating these equations, one can determine, over a period (cycle), for the given design the maximum amplitude of the linear and angular oscillations of the housing, \(\max X\) and \(\max \varphi\), as well as the values of \(v_m\) and \(\omega_m\) for any angle \(\varphi\).

By introducing additional external forces \(F^{e}\) and moments \(M^{e}\), it is possible to set the oscillations of the housing within “quite permissible limits.” Having the values of the quantities \(\omega_m = f(t)\) and \(\varphi_m = f_1(t)\) in the second approximation, one can calculate and estimate the influence of the gyroscopic moment of the rotating motor, the magnitude and direction of which are determined by the theorem of moments of momentum:

\[ \frac{\overline{dK}}{dt}=\overline{M}_{\mathrm{gyr}}, \]

where we take the moment of momentum to be \(K=I_0\omega\).

Here the following cases, characteristic from the design standpoint, are possible:

1) when the motor axis is perpendicular to the hammer axis (as, for example, in MB and OM-1), then the oscillations of the hammer housing will occur about an axis passing through the vector \(\overline{K}\) (in the transverse plane); the velocity of its end will be equal to zero, and there will be no reactive gyroscopic moment in this case (in the absence of external forces and rotations of the hammer in any direction);

2) when the motor axis is directed along the hammer axis or parallel to it (for example, in the designs of the Siemens-Schuckert, KNSH, and EM hammers), here oscillations of the hammer axis in the transverse plane relative to the axis passing through the crankshaft will cause the appearance of a velocity of the vector \(\overline{K}\), this velocity being equal to \(|U_k|=K\omega_m=I_0\omega\omega_m=M_{\mathrm{gyr}}\).

The gyroscopic reactive moment \(M_{\mathrm{gyr}}\) will cause motion of the hammer housing relative to a new axis, perpendicular to the plane passing through the hammer axis and the axis of rotation of the crankshaft, which in turn will generate a new motion, opposite in direction to the initial motion (in some direction) in the transverse plane to the hammer axis. Consequently, designs with the same axial arrangement of the electric motor and the hammer will be “more resistant” to oscillatory motions of the housing caused by changes in torque on the crankshaft.

It is also possible to calculate and estimate the Coriolis inertia forces developed by the reciprocating masses (\(m_b\) and \(m_B\)) in their relative motion with respect to the hammer housing at a known angular velocity \(\omega_m\) of the latter. To find the direction of the Coriolis acceleration \(W_c\), one must project the velocities of the piston \(v_p\) and the striker \(v_b\) onto a plane perpendicular to the axis of rotation of the hammer housing and rotate them by \(90^\circ\) in the direction of rotation (i.e., \(\omega_m\)). The magnitude of the Coriolis inertia forces

is calculated separately for the striker, \(Q_{c\delta}=2m_{\delta}v_{\delta}\omega_m\), and for the translationally moving mass reduced to the piston, \(=2m_{B}v_n\omega_m\) (if the hammer rotates in the transverse plane).

The moment due to the Coriolis inertia forces will be

\[ (M_{c\delta}=Q_{c\delta}\cdot x \quad \text{and} \quad M_{cB}=Q_{cB}S) \]

in the transverse plane to the axis of the hammer and causes additional oscillations of the housing; moreover, these oscillations are intensified when the masses \(m_{\delta}\) and \(m_{в}\) move from the tool toward the handle (during the return stroke) and are reduced during the working stroke, i.e., when these masses move in the direction toward the tool.

The magnitude of these forces (and moments) can be reduced only by reducing \(\omega_m\), i.e., the initial oscillations of the hammer housing.

For the OM-1 hammer:

\[ M_{cp}=0.6\ \text{kg}\cdot\text{m}; \qquad M_{\text{gir}}\simeq 0.06\omega; \qquad M_{\text{cor}}\simeq 0.75\omega\ \text{kg}\cdot\text{m}. \]

§ 3. Expression of the disturbing forces in the form of a Fourier series and determination of the natural frequencies of oscillation of the hammer elements

The period of the hammer’s working process depends on its design and operating mode. For a hammer of the OM-1 and KNSh types the period is equal to one revolution of the crankshaft; for the MB hammer at idle stroke the period is equal to one revolution of the crankshaft; under steady operating conditions the period is equal to a multiple number of revolutions of the electric motor, depending on the selected rate of missed blows. Both the period \(T\) and a number of quantities characterizing the working process of the hammer (angular velocity \(\omega\), torque on the shaft \(M_k\), striker travel \(x_{\delta}\), the resulting motion of the constraint forces on the striker and the hammer housing, etc.), for its selected parameters, can be found from the joint solution of equations (1) and (2), or experimentally. It is very useful to represent some of these quantities not only in the form of a graph, but also to have an analytical expression. This applies especially to periodically acting disturbing forces causing forced oscillations, which can be represented as components of a Fourier series:

\[ f(\varphi)=A_0+\sum_{h=1}^{\infty}(A_h\sin h\varphi+B_h\cos h\varphi), \]

where \(h=1,2,3\ldots\)

“Every investigation of forced oscillations should begin with an expansion into a Fourier series,” Prof. E. L. Nikolai [14] points out. In view of the practical importance of the problem of the harmonic expansion of a periodic function, the literature contains several procedures that make it possible to compute the coefficients \(A_h\) and \(B_h\), such as: analytical and graphical methods, tabular procedures, and mechanical techniques.

In the literature on the dynamics of internal-combustion engines there are usually tables facilitating calculations [10; 13].

In computation practice, some finite number of terms of this series is usually sufficient, depending on the form of the function being expanded; for example, in expanding \(R\) for the OM-1 design, 8 terms of the series were required.

When the frequency of the natural oscillations of any of the hammer parts coincides with the frequency of one of the harmonics of the disturbing force (resonance), oscillations of large amplitude may arise, as a result of which the distur—

possible overstresses and breakages of individual elements of the hammer (the hammer housing, the crankshaft, the shaft of the electric-motor rotor, springs, etc.). When the frequency of oscillation of the hammer housing coincides with one of the frequencies of the disturbing force or moment, the largest vibrations of the hammer housing and of the electric-motor stator are possible. In the general case this question is quite complicated, and finding the natural frequencies of oscillation of the hammer housing can be considered on the basis of Rayleigh’s approximate energy method, the essence and description of which may be found in the literature cited at the end of the work [5]; [18]; [21].

Determining the magnitudes of the displacements of various points of the hammer housing during its longitudinal elastic oscillations, as well as the stresses arising from them, is not part of our task.

The frequencies of the natural longitudinal oscillations of the hammer housing can, to a first approximation, be found as the oscillation of a prismatic rod by the following formulas. For a rod with both ends free or with both ends fixed,

\[ \omega_h=\frac{\pi h}{l}\sqrt{\frac{E g}{\gamma}} . \]

For a rod with one free and the other fixed end,

\[ \omega_h=\frac{(2h-1)\pi}{l}\sqrt{\frac{E g}{\gamma}}, \]

where

\[ \begin{aligned} h&\text{ is an integer corresponding to the order of oscillation,}\\ g&\text{ is the acceleration of gravity,}\\ E&\text{ is the modulus of elasticity of the rod material in tension,}\\ \gamma&\text{ is the specific weight of the rod material,}\\ l&\text{ is the length of the rod.} \end{aligned} \]

These formulas are given in the cited literature, for example [18]. Their derivation may be found in [5; 6; 7; 21].

Let us consider, for example, according to the first formula, the frequency of free oscillations of the fundamental tone, i.e. \(h=1\), for \(l=50\ \text{cm}\), \(E=2.2\cdot10^6\ \text{kg}/\text{cm}^2\), \(\gamma=7.8\ \text{g}/\text{cm}^3\):

\[ \omega_1=\frac{3.14}{50}\sqrt{\frac{2.2\cdot10^6\cdot980}{7.8\cdot10^{-3}}}\simeq 3.3\cdot10^4\ \frac{1}{\text{sec}} . \]

For a hammer without a reduction gear the synchronous angular velocity is \(\omega=\pi\cdot10^2\); consequently, there is no basis for fearing the occurrence of resonance.

The determination of the frequencies of the natural longitudinal oscillations of coiled springs is set forth with exhaustive completeness by S. D. Ponomarev [16], and the necessary formulas for calculation with the corresponding nomogram are given by S. V. Serensen [17].

Here the author considers it appropriate to note that it is hardly expedient to orient designers toward tuning, in spring hammers, the oscillating impact part to resonance with the disturbing force, since 1) owing to changes in the line voltage, the number of revolutions of the electric motor is not constant, and 2) as a result of the elastic impact, there is a sharp change in the initial conditions in the established resonant oscillations of the striker.

Since a reactive torque acts on the hammer housing and, in addition, the hammer is used as a lever when breaking out rock, it is obvious that the strength of the housing must also be checked for transverse bending and for transverse oscillations. The question of torsional oscillations for existing designs of electric demolition hammers is not urgent, since the phenomenon of resonance in torsional oscilla-

... vibrations in the shaft with the crank are practically absent, owing to the small reduced length of the reduction shaft \(l_{red.}\) and the small mean moment of inertia of the reciprocating masses (over one revolution) in comparison with the moment of inertia \(J_0\) of the rotating masses (of the electric motor and reduction gear) reduced to the crankshaft.

For an approximate calculation of the lowest natural frequency of torsional vibrations, we may regard the electric motor and the crank-and-connecting-rod mechanism as a two-mass system, for which the period of oscillations is determined by the formula:

\[ T = 2\pi \sqrt{\frac{I_1 J_0 l_{red.}}{g I_p (I_1 + I_0)}} , \]

cm, for example [10], where

\(g\)—modulus of elasticity in shear,
\(I_p\)—polar moment of inertia of the cross section of the reduction shaft.

In Table 2 below, for the designs KNP-2, OM-1, and MB-3, a comparison is shown between the quantities: the number of natural oscillations per minute of the two-mass system \(n_1\) and the number of revolutions of the electric motor per minute \(n\).

Table 2

In designs having a fan, it is possible (by the corresponding formulas) to determine the principal frequencies of oscillation of a three-mass system—the rotor of the electric motor—the fan and the crank-and-connecting-rod mechanism, reduced to the shaft of the electric motor (if the fan is mounted on it). Vibrational oscillations in the stator of the electric motor can be eliminated by designing the rotor with the appropriate “squirrel-cage” winding [5]. Finally, when calculating the dynamic strength of the housing (and also the recoil), one must take into account the presence of impact when the hammer housing seats on the collar of the tool. Here, as a first approximation, one may make use of the problem of longitudinal impact of prismatic rods considered by S. P. Timoshenko [21].

§ 4. Possibilities for Reducing Recoil

As is evident from the preceding section, for an electro-pneumatic (and electromechanical) hammer the oscillatory motions of the hammer housing, regarded as a rigid body, are of primary importance (and, to a lesser extent, the elastic vibrations of individual elements are important).

In order to reduce oscillations of the housing, it is necessary:

1) to carry out thorough dynamic balancing of the rotating masses (especially in gearless systems);
2) to equalize the magnitude of the torque;
3) to use dynamic dampers;
4) to introduce elastic elements into the system.

Equalization of the torque of the motor can be achieved with the aid of a pneumatic accumulator proposed by Prof. L. K. Martens [11]; rational design and constructive imple-

the design of which will require considerable labor and time. Moreover, as Prof. E. D. Lvov notes, the accumulator of surplus work “should be designed for the engine load and the number of revolutions most frequently occurring in operation of the engine being designed” [10]. The use of Martens accumulators in an electric demolition hammer would complicate the design and would cause an undesirable increase in dimensions and weight. This device may be kept in mind as a reserve when designing a column-mounted rock drill.

Of essential importance for increasing the efficiency of the machine and evening out the magnitude of the torque are the motor and its characteristic. In the present case a special motor is desirable (corresponding to the operation of the percussion unit), with a small variation of torque with the number of revolutions.

To a certain extent, a means that evens out the torques on the shaft is a system of pendulum dampers, which develop inertia forces directed in the opposite direction with respect to the external forces rocking the system. Pendulum dampers find their principal application in changing torsional oscillations in a crankshaft. With dynamic damping there is no dissipation of energy in the system. Successful applications of pendulum dampers are designs in which movable counterweights are used as dampers [5].

It is seen from Fig. 3 that, with retarded rotation of the crank \((\varepsilon < 0)\), the pendulum counterweight is deflected in the direction of rotation and develops an additional moment from the inertia force \(Q\) in the direction of rotation; for \(\varepsilon > 0\) the pendulum will act as a brake.

Besides the reactive moment, a periodically varying force \(R\) acts on the hammer housing in the direction of its axis; this force is the resultant of the inertia forces developed by the reciprocating masses and of the constraint force acting from the side of the striker. The designer has the possibility of weakening this force by arranging a dynamic vibration absorber tuned to the corresponding harmonic of the disturbing force.

Fig. 3

Den Hartog (and also Olson) describes a device for an electric hair clipper with a dynamic vibration absorber, which is used to eliminate oscillations of the housing that are extremely unpleasant for the barber.

“A mechanism considered as a free body in space, deprived of the action of external forces, must have a fixed center of gravity and principal axes of inertia” [5]. Here there is an analogy with the problem considered by us: in the hammer, the striker, piston, and connecting rod are in motion under the action of internal forces, and from the condition that the center of gravity of the system is fixed there follows the motion of the hammer housing in the opposite direction.

The question of a dynamic vibration absorber deserves attention, and the undertaking of computational and experimental work in application to percussion machines is necessary.

Structurally, for example, a dynamic vibration absorber can be made in the form of a mass \(m\), attached by flat steel springs (or rubber) to the hammer housing and even in its handle (Fig. 4). It is necessary that the mass of the absorber lie on the line of action of the disturbing force and be able to move in this direction.

Finally, one must consider the possibility of designing a dynamic absorber in the direction of the barrel, by boring long channels of small diameter in it, Fig. 4, so that an absorber of small mass can execute large strokes. Academician A. N. Krylov gives a numerical example in which “the damper reduces the amplitude of oscillations of the mass \(m\) by a factor of 6, although its own mass is only \(1/1000\) of the mass \(m\), but the amplitude of its oscillations will be 50 times greater than the amplitude of oscillations of the mass \(m\)” [7]. The same example is also considered by L. G. Loitsyanskii and A. I. Lur’e [9]. If the weight of the hammer is of the order of 15 kg, then the weight of the vibration absorber is sufficient (according to this example) at \(\simeq 15\) g. In addition, resistances may be introduced into the absorber (for example, viscous oil, which changes its properties little with heating). These resistances, while damping the oscillations of the vibration absorber, will at the same time also damp the oscillations of the main system.

Fig. 4

The question of the influence of an elastic support on the magnitudes of the support reaction was considered by De Villiers [4]. A chapter in Den Hartog [5] is devoted to the question of vibration isolation as applied to electric machines and the automobile; it turns out that the question of vibration isolation is closely connected with the ratio of the frequency of the natural oscillations \(\omega_0\) of the mass \(m\) on an elastic connection to the frequency of the disturbing force \(\omega\).

The graph of the variation of the amplification factor \(\beta\) (the resonance factor) as a function of the frequency ratio \(\nu = \dfrac{\omega}{\omega_0}\) is shown in Fig. 5. The relation between the maximum dynamic and static deformations is expressed as:

\[ x_{\max}=\beta x_{cm}, \quad \text{where } x_{cm}=\frac{R_1}{c}, \]

\(c\) — the stiffness of the spring (elastic connection) between the hammer housing and the handle (Fig. 6),

\(R_1\) — the maximum value of the component of the disturbing force;

\(R_x=R_1\sin\omega t\) — one of the harmonic components of the disturbing force,

\(P\) — the force transmitted to the operator, with \(P_{\max}=c x_{\max}\).

Let us define the force-transmission coefficient as the ratio of the maximum value of the transmitted force to the maximum value of the disturbing force

\[ \alpha=\frac{P_{\max}}{R_1}=\frac{c x_{\max}}{R_1}=\frac{x_{\max}}{x_{\mathrm{st}}}=\beta, \]

therefore, the force-transmission coefficient of the elastic connection coincides with the value of the resonance multiplier. In the absence of resistance,

Fig. 5

the dynamic coefficient (or coefficient of growth of oscillation) is determined by the well-known formula:

\[ \beta=\frac{1}{1-\left(\frac{\omega}{\omega_0}\right)^2}, \]

in our case \(\omega\) is the frequency of the disturbing force,

Fig. 6

\[ \omega_0=\sqrt{\frac{c}{m}} \]

is the frequency of free oscillations of the hammer housing on the spring; the weight of the hammer is \(G_m=gm\).

As is seen from Fig. 5, in order that the transmitted force be small, it is necessary for the motor to operate in the “beyond-resonance” region (i.e., for the disturbing force to be opposite in phase to the position of the oscillating mass of the hammer).

Knowing the mass of the hammer and the frequency of the disturbing force, it is easy to calculate also the stiffness of the elastic suspension. Let, for example:

\[ \nu=\frac{\omega}{\omega_0}=5,\quad \text{then } \beta=\alpha=\frac{1}{1-5^2}=\left|-\frac{1}{24}\right| \]

and

\[ P_{\max}=\frac{R_1}{24}, \]

the transmitted force is reduced by a factor of 24.

\[ \omega_0=\frac{\omega}{5}=\sqrt{\frac{c}{m}},\quad \text{whence } c=\frac{\omega^2}{25}\cdot m. \]

For the MB hammer,

\[ \omega \simeq 300\ \frac{1}{\mathrm{sec}}, \quad G_M=gm=10\ \mathrm{kg} \]

and the spring stiffness is

\[ c=36\ \mathrm{kg}/\mathrm{cm} \]

at

\[ \nu=\frac{\omega}{\omega_0}=10,\quad \alpha=\frac{1}{99}\quad \text{and } c=9\ \mathrm{kg}/\mathrm{cm}. \]

The selected spring must provide a pressing force from the operator’s side of 20–30 kg.

When rubber is used instead of steel springs for damping purposes, it is necessary to take the damping forces into account.

In this case the force-transmission coefficient is:

\[ \alpha=\sqrt{\frac{1+4\mu^2\nu^2}{(1-\nu^2)^2+4\mu^2\nu^2}}, \]

where

\[ \nu=\frac{\omega}{\omega_0}\quad \text{and}\quad \mu=\frac{b}{b_{\mathrm{cr}}}, \]

\(b\) is the coefficient of velocity damping (the damping factor at velocity),

\[ b_{\mathrm{cr}}=2\sqrt{cm}=2m\omega_0 \]

is the critical value of the damping factor, which separates the oscillatory type of motion from the decaying one in the case of free vibrations.

At \(\mu=0\) we have the case shown in Fig. 5. The presence of some damping \((\mu\ne 0)\) has an advantage over a wider range of variation of angular velocity (during starting and passage through resonance), although it increases the value of \(\alpha\) in the region of effectiveness of the elastic suspension.

Without going into the various forms of constructive implementation, we shall point out the possibility of using rubber, which works well in shear, Figs. 6–6.

It is desirable to introduce an elastic connection between the electric motor and the crank, which will cut off the “peaks” in the torque (especially important in electropneumatic hammers).

Finally, an elastic connection in the form of a shock absorber that softens impact is important when seating the barrel on the shank of the tool.

§ 5. Conclusion

An electric demolition hammer is a machine subjected to the action of various forces that vary with time. When the masses are unbalanced and measures are not taken to equalize the resultant forces and moments, extremely unpleasant shaking of the housing arises, rapid deterioration of the moving parts occurs, and labor productivity is reduced (recoil, noise).

The following may be regarded as indicators of the oscillations of the housing (recoil):

1) given a known law of variation of the force \(R\) and the moment \(M\), what is characteristic is the very variation of these quantities with time (or angle), or, in other words, as B. V. Sudnishnikov [20] notes, the change in the form of the impulse of forces and moments. Since recoil depends on the operating regime, its characteristics may be the angular velocity \(\omega\) and the number of blows per minute \(z\). In addition, indicative quantities are the ratios of the maximum values of the quantities \(R\) and \(M\) to their mean values:

\[ K_1=\frac{R_{\max}}{R_{\mathrm{mean}}}\quad \text{and}\quad K_2=\frac{M_{\max}}{M_{\mathrm{mean}}}; \]

2) the amplitudes of oscillation of the housing \(x_{\max}\) and \(\psi_{\max}\), both under the action only of internal forces and moments and under various kinds of applied forces (for the operator, in the sense of negative effect, the amplitudes of oscillations are of greater importance).

In order to reduce recoil it is necessary to select the parameters of hammers rationally, which is equivalent to the proper choice of the force diagram.

It is also necessary to formulate and carry out computational and experimental studies on prototypes of the corresponding hammer designs, using various possible means that reduce recoil; to develop a methodology for experimental investigation (taking into account the existing experience in the vibration departments of design bureaus), and to determine and evaluate the mutual influence of the operator and the impact machine. Objective coefficients characterizing the degree of recoil should be established.

There is no doubt that the cost of the initial expenditures in the manufacture of a good impact machine will increase in connection with preliminary studies and experiments, the search for special wear-resistant materials for its manufacture, and the care required in manufacture and assembly itself; but, on the other hand, reliability and long service life will increase, and consequently so will economy.

In the hand-held demolition hammer, as in a focus, there is concentrated a series of the most difficult questions for design, since what is needed is an economical hammer, reliable in operation, light in weight, and with reduced recoil.

From the fundamental point of view, the effectiveness of the blow itself in demolition makes it possible to think that not only hand-held demolition hammers are possible, but also mobile stationary combined impact machines, where questions of the action of the blow and of recoil on the foundation (the machine bed) still do not lose their relevance.

References Used

1. Alabuzhev P. M. Investigation of the working process of an electric demolition hammer with an elastic (air) connection. Izvestiya TPI, vol. 61, no. 1, 1947.

2. Batuev N. M. Electric impact drills, journal Mechanization of Labor-Intensive and Heavy Work, Moscow, 1947, No. 4.

3. Wilson V. K. Balancing of Internal-Combustion Engines, ONTI, Moscow, 1935.

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4. Krylov, A. N. On Certain Differential Equations of Mathematical Physics Having Applications to Technical Problems, Leningrad, 1932.

5. Kucherov, V. S. Theoretical investigation and method for calculating the parameters of electromechanical demolition hammers with a crank-and-connecting-rod drive. Transactions of the Institute of Mining Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev, 1947.

6. Loitsyanskii, L. G., and Lurie, A. I. Theoretical Mechanics, vol. III, Leningrad, 1943.

7. Lvov, E. D. Dynamics of Piston Engines, ONTI, Moscow, 1936.

8. Martens, L. K. Dynamics of Piston Engines, ONTI, Moscow, 1932.

9. Mechanical Engineering. Encyclopedic Handbook, vol. 1, issue 2. Moscow, 1947.

10. Neiman, I. Sh. Dynamics of Aircraft Engines. Moscow, 1940.

11. Nikolai, E. L. Theoretical Mechanics, part III, GONTI, Leningrad, 1939.

12. Nikolai, E. L. Oscillatory motions and resonance phenomena in the foundation of a sawmill frame. Journal “Lesnoe khozyaistvo i lesnaya promyshlennost’,” Leningrad, 1929.

13. Olson, H. Dynamical Analogies, Moscow, 1947.

14. Ponomarev, S. D. Calculation and Design of Helical Springs, Moscow, 1938.

15. Serensen, S. V.; Tetelbaum, I. M.; Prigorovskii, N. I. Dynamic Strength in Mechanical Engineering, Mashgiz, 1945.

16. Smirnov, V. I. A Course of Higher Mathematics for Physicists and Engineers, vol. 2, Moscow, 1931.

17. Sudnishnikov, B. V. Certain Questions in the Theory of Impact Machines, Novosibirsk, 1949.

18. Timoshenko, S. P. Theory of Vibrations in Engineering, ONTI, Moscow, 1931.

19. Shmargunov, K. N. Electric Demolition Hammers—Designs, Calculations, and Tests, Moscow, 1937.

1. Since \(\operatorname{mom}(PP) = 2 S_{\triangle OKB}\) and \(\operatorname{mom}(P_1P'_1) = 2 S_{\triangle ODB}\); but \(S_{\triangle OKB} = S_{\triangle ODB}\) (\(OB\) is common and \(DK \parallel OB\), hence the heights are equal). ↩