ON THE DYNAMICS OF FORCES ACTING ON THE BODY OF A PNEUMATIC HAMMER

V. F. Gorbunov, V. I. Baburov

(Presented by the Department of Mining Machines and Mine Transport)

The vibration of hand-held pneumatic hammers leads to workers contracting vibration disease. Therefore, the study of vibration, carried out with the aim of creating hand-held machines that are safe with respect to vibration, is of great practical importance. At present, many scientists and inventors have proposed many different methods for reducing the effect of vibration on a person.

In our article [1] it is shown that the problem of vibration isolation and vibration damping can be solved only through the combined application of various methods. This conclusion follows, in particular, from the fact that during operation of the hammer various forces act on it, which vary both in magnitude and in duration of action. As a result, the amplitude–force characteristic of hammers is qualitatively complex, with bends in the form of maxima and minima [1—3]. In article [2], depending on the magnitude of the pressing force, we identified a number of the most characteristic operating regimes of the hammer—the first transitional, floating, second transitional, limiting basic, third transitional, and the regime in which vibration of the body occurs only from the rebound impacts of the tool. The external feature adopted for these regimes is the geometric form of the machine’s oscillations.

The present work is devoted to an analysis of the forces acting on the body of hand-held pneumatic hammers under various regimes of their operation. Figure 1 shows the geometric forms of oscillation of the body of the KE-16 hammer at an air pressure in the supply network of 5 atm and during cutting of a steel billet.

a) First transitional regime

The first transitional regime is the most unstable operating regime of the hammer.

The pressing force in this regime is equal to the minimum permissible value and is so small that in each cycle the hammer body moves farther and farther away from the workpiece being machined. Its backward displacement continues until the recoil becomes equal to the value of the limitation of the striker stroke by the shank of the working tool. Consequently, the greatest recoil of the hammer is determined by its design. The geometric form of oscillation of the body corresponding to the first transitional regime is shown in Fig. 1, a. The oscillations of the hammer are accom-

are effected under the action of the force of impact of the striker against the body \(F\), the pressing force \(R_{\mathrm{н}}\), and the pressure of the compressed air \(P_t\), with the amplitude of oscillations \(A\) being maximal (Fig. 1, a).

It is known that the compressed air entering the front chamber exerts pressure simultaneously on the striker and on the limiting bridge of the machine. Moreover, the impulse from the air pressure on the striker from the side of the front chamber (we shall call this impulse the initial one) is expended on compressing the air in the rear chamber and, thus, is also

Fig. 1. Geometric forms of oscillations of the body of a pneumatic hammer

transmitted to the body. Consequently, the resultant action of the initial impulse on the body is equal to

\[ I_{\mathrm{п}} = F_y \int_{0}^{t_{\mathrm{п}}} P_{\mathrm{п}}\,dt - (F_y - f_{\mathrm{ш}})\int_{0}^{t_{\mathrm{п}}} P_{\mathrm{п}}\,dt = f_{\mathrm{ш}}\int_{0}^{t_{\mathrm{п}}} P_{\mathrm{п}}\,dt, \quad \text{kg}\cdot\text{s}, \tag{1} \]

where \(I_{\mathrm{п}}\) is the magnitude of the initial impulse acting on the hammer body, \(\text{kg}\cdot\text{s}\);

\(P_{\mathrm{п}}\) is the air pressure in the front chamber, variable in magnitude and time, \(\text{kg}/\text{cm}^2\);

\(t_{\mathrm{п}}\) is the duration of action of the air pressure in the front chamber, s;

\(F_y\) — cross-sectional area of the striker, cm\(^2\);

\(f_u\) — cross-sectional area of the shank of the working tool, cm\(^2\).

The authors obtained empirical expressions for determining the impulse \(I_n\) (1) in chipping and riveting hammers under the conditions of their operation on steel.

For riveting hammers,

\[ I_n=\frac{f_u S P_0}{870P_0+3600},\ \text{kg·s}, \tag{2} \]

where \(S\) is the distance from the end of the tool shank to the center of the exhaust ports, cm;

\(P_0\) — air pressure in the network, kg/cm\(^2\).

For chipping hammers,

\[ I_n=\frac{f_u S P_0}{7000},\ \text{kg·s}. \tag{3} \]

Thus, the value of \(I_n\) depends on the length of the front cavity of the hammer, the air pressure in the network, and the area of the tool shank.

Compression of the air in the rear cavity occurs not only as a result of the initial impulse, but also due to the impulse of rebound of the striker from the tool. The rebound impulse is found from the expression

\[ I_0=\sqrt{2A_0m},\ \text{kg·s}, \tag{4} \]

where \(I_0\) is the rebound impulse of the striker, kg·s;

\(A_0\) — rebound energy, kg·m;

\(m\) — mass of the striker, kg·s\(^2\)/m.

During the working stroke of the striker, an impulse of force acts on the hammer body, the magnitude of which is determined by the kinetic energy of the striker [4]:

\[ I_p=F_y\int_0^{t_p} P_p\,dt=\sqrt{2A_y m},\ \text{kg·s}, \tag{5} \]

where \(I_p\) is the impulse of compressed-air pressure on the hammer body during the working stroke of the striker, kg·s;

\(t_p\) — time of the working stroke of the striker, s;

\(P_p\) — variable air pressure on the body (striker) over the time interval \(t_p\) of the cycle, kg/cm\(^2\);

\(A_y\) — impact energy of the striker, kg·m.

The magnitude of the impulse of the force \(F\) acting on the hammer at the moment when the striker strikes the bridge can be calculated by equating it to the change in momentum of the machine body, i.e.,

\[ I_F=\int_0^t F\,dt=m_k V_k''-(-m_k V_k'),\ \text{kg·s}, \tag{6} \]

where \(I_F\) is the impulse of the force \(F\) from the collision of the striker with the body, kg·s;

\(t\) — time of collision of the striker with the body, s; \(V_k'\), \(V_k''\) — velocities of the body before and after the collision, m/s.

It is practically impossible to determine \(V_k'\) and \(V_k''\) analytically, since, in addition to the impact force, the pressing force, the inertia force, and the air pressure act on the hammer body. Graphical differentiation of the vibrogram for specific operating conditions of the hammer shows—

indicates that the velocity \(V'_k\) is proportional to the velocity of the striker before impact, the proportionality coefficient being determined by the ratio of the masses of the body and the striker. According to differentiation data, the velocity of the body \(V'_k\) of the KE-22 hammer in the first transient mode at \(5\) atm is \(0.5\text{--}0.65\ \mathrm{m/sec}\). The striker velocity in this case is \(10.0\text{--}10.7\ \mathrm{m/sec}\), i.e., the body velocity is smaller than the striker velocity by almost as many times as the mass of the former is greater than the mass of the latter.

Thus,

\[ V'_k = V_y \frac{m}{m_k}. \tag{7} \]

From the law of conservation of impulse it follows that

\[ R_n T + I_F - I_p - I_0 - I_n = 0. \tag{8} \]

Substituting the values of the impulses into expression (8) and combining the impulses \(I_p\) and \(I_0\), we obtain a formula for determining the pressing force for the first transient mode:

\[ R_n = \frac{n}{60} \left[ \sqrt{2 A_y m}\,(\kappa_0 + 1) + f_u \int_0^{t_p} P_p\,dt \right] - \frac{n'}{60} \int_0^t F\,dt,\ \mathrm{kg}, \tag{9} \]

where \(R_n\) is the reduced pressing force (taking into account the weight of the hammer), kg;
\(\kappa_0\) is the coefficient of rebound of the striker;
\(n\) is the frequency of strokes of the striker per minute;
\(n'\) is the frequency of impacts of the striker against the hammer body, impacts/min.

Collisions of the striker and the body do not occur in every cycle. The frequency of collisions depends on the pressing force. Let us find the magnitude of the pressing force at which impacts on the bridge will occur in every operating cycle of the machine.

If all the kinetic energy of the striker is transmitted to the body, i.e., collisions of the striker and the body occur in every cycle, then the equality

\[ I_p = I_F \quad \text{or} \quad \sqrt{2 A_y m} = \int_0^t F\,dt \tag{10} \]

will be valid.

The algebraic sum of the second and third terms of expression (8) becomes zero. Consequently, the pressing force for the indicated mode of operation of the hammer is equal to

\[ R_n = \frac{n}{60} \left( \sqrt{2 A_0 m} + f_u \int_0^{t_p} P_p\,dt \right),\ \mathrm{kg}. \tag{11} \]

Substituting the values of the impulse \(I_n\) from formulas (2) and (3) into expression (11), we find the value of the limiting minimum pressing force in the first transient mode. For riveting hammers of the KE type this force at \(5\) atm is \(9\text{--}12\ \mathrm{kg}\), and for chipping hammers of the MR type it is \(4.5\text{--}5\ \mathrm{kg}\).

b) Floating mode

With an increase in the pressing force in the case under consideration from \(18\) to \(21\ \mathrm{kg}\), the hammer passes into a floating mode of operation. The oscillations of the body occur smoothly. The geometric form of the oscillations is close to sinusoidal (Fig. 1, b). The amplitude of vibration in the floating mode is significantly smaller than in the first transient mode [2]. During oscillation

the body does not reach the collar of the tool. The magnitude of the pressing force ensuring operation of the hammer in the floating mode is

\[ R_{\mathrm{н}}=\frac{n}{60}\left[\sqrt{2A_y m(\kappa_0+1)}+I_n\right]. \tag{12} \]

In this case the pressing force compensates for the action on the body only of the internal disturbing force. Hygienists consider the floating mode the most favorable from the sanitary-hygienic point of view, since in it there are no abrupt changes in the velocity of motion of the body, and the force acting on it changes smoothly.

The studies of A. M. Petreev [5, 6] have established that the pressing force required to maintain the floating mode is entirely independent of the shape of the indicator diagram and is determined only by the magnitude of the total impulse of the compressed-air pressure force acting on the body of the machine during a cycle. At the same time, the phase shift between the diagram of the resultant air pressure and the curve of body motion is always such that the dimensionless parameter derived by A. M. Petreev has a constant value equal to \(\frac{1}{2}\).

Despite the fact that the floating mode is the most favorable from the hygienic standpoint, its practical use is very difficult because the worker must constantly catch the required phase shift between the pressure diagrams and the vibrogram. However, the study of this mode is of practical importance. To obtain a sinusoidal form of oscillation of the handle, one must strive to ensure that all forces acting on it vary smoothly.

c) Second transient mode

A slight increase in the pressing force takes the machine out of the floating mode. The hammer body performs complex oscillatory motions characterized by a gradual departure from the limiting collar of the tool and subsequent approach to it (Fig. 1, c). The geometric form of the body oscillations in some striker-stroke cycles is close to the floating mode; in others it is similar to the form of oscillations of the limiting main mode (considered below). The alternation of such forms in this transient mode occurs periodically. This is easily established when studying vibrograms.

The amplitude of vibration of the machine body in the second transient mode increases as the pressing force increases [2]. The increase in amplitude, in our opinion, is explained by the fact that in this mode of hammer operation new forces begin to act on its body, arising upon impacts with the limiting collar of the tool. From the action of the forces \(N\) during seating, the body receives an additional impulse \(I_N\), the magnitude of which depends on the mass of the body and the parameters of the machine and is equal to the change in the quantity of motion of the body [4].

During seating, the velocity of the body changes abruptly (the moments of seating are marked by the letters \(A\) in Fig. 1, c). The duration of the striker-stroke cycle in the second transient mode varies within wide limits, since in each cycle, as a result of the body’s departure, the stroke length of the striker changes its value.

The value of the pressing force, determined by us through the principal parameters of the hammer, for the second transitional mode is equal to

\[ R_{\mathrm{н}}=\frac{n}{60}\left[\sqrt{2A_y m}\,(\kappa_0+1)+I_n\right] +\frac{n''}{60}m_{\mathrm{k}}V_{\mathrm{п}}(\kappa_{\mathrm{k}}+1),\ \text{kg}, \tag{13} \]

where \(n\) is the frequency of blows of the striker per minute;
\(n''\) is the frequency of collisions of the body with the shoulder of the tool, blows/min;
\(V_{\mathrm{п}}\) is the seating velocity of the body, m/sec;
\(m_{\mathrm{k}}\) is the mass of the body, kg·sec\(^2\)/m;
\(\kappa_{\mathrm{k}}\) is the coefficient of rebound of the body.

To determine the pressing force by formula (13), in addition to the principal parameters of the machine, it is necessary to know the velocity of the body before and after seating, as well as their frequency. These characteristics are found from vibrograms.

d) Limiting principal mode

With a further increase in the pressing force, seatings of the body onto the limiting shoulder become more frequent. In the limiting principal mode they occur in every cycle of the striker stroke. The vibration amplitude reaches a maximum [2].

The geometric form of the oscillations of the body in the limiting principal mode is qualitatively similar to the diagram of the striker path.

At the beginning of the mode the vibration amplitude differs in different cycles (Figs. 1, 2); subsequently the oscillations are performed with the same amplitude (Fig. 1, d). The pressing force that must be applied to the handle in order to obtain the limiting principal mode is calculated by the formula

\[ R_{\mathrm{н}}=\frac{n}{60}\left[\sqrt{2A_y m}(\kappa_0+1)+I_n +m_{\mathrm{k}}V_{\mathrm{п}}(\kappa_{\mathrm{k}}+1)\right],\ \text{kg}. \tag{14} \]

The seating velocity of the body onto the shoulder \(V_{\mathrm{п}}\) and the coefficient \(\kappa_{\mathrm{k}}\) are determined from vibrograms recorded on film.

In addition, \(V_{\mathrm{п}}\) can also be determined by the formula proposed by B. V. Sudnishnikov [7]:

\[ V_{\mathrm{п}}=\sqrt{\frac{2A}{m_{\mathrm{k}}}}\,R_{\mathrm{н}},\ \text{m/sec}, \tag{15} \]

where \(A\) is the vibration amplitude in the limiting principal operating mode of the hammer, m.

Analysis of vibrograms shows that, in the limiting principal mode of hammer operation, the motion of its body toward the handle always begins at the moment when the force of air pressure on the body becomes greater than the pressing force. When these forces are equal, the path of the body is zero. The time of motion of the body in the limiting principal mode is equal to the duration of the striker-stroke cycle. With an increase in the pressing force on the hammer handle, the time of motion of the body in the mode under consideration decreases. Similar geometric forms of oscillations were studied by B. V. Sudnishnikov [7] and N. A. Klushin [8]. For chipping and riveting hammers, when the pressing force is increased, the geometric form of the oscillations changes qualitatively (Fig. 1, e). This mode of hammer operation has been singled out by us separately and called the third transitional mode (considered below).

Thus, despite the large pressing force, the vibration amplitude of the hammer in the limiting principal mode is greater than in the floating mode. In the works known to us by Soviet and foreign authors devot-

devoted to the study of the vibration of hand-held pneumatic hammers (in most cases chipping hammers), it is indicated that the vibration amplitude gradually decreases as the pressing force increases, i.e., the amplitude–force characteristic changes smoothly, without inflections. In our opinion, for chipping hammers the change in vibration amplitude as a function of the pressing force apparently occurs qualitatively in the same way as for the hammers we have studied. In the work of N. A. Klushin [8] the following example is given: the oscillation range of the body of an OMSP-5 hammer in the limiting basic regime \((R_{\mathrm{n}} = 30\ \text{kg})\) is \(7\ \text{mm}\), while in the floating regime \((R_{\mathrm{n}} = 15\ \text{kg})\) it is \(4\ \text{mm}\), that is, 1.7 times smaller, which confirms our assumptions.

d) Third transitional regime

After the limiting basic regime, as the pressing force is increased, the vibration amplitude decreases [2]. The form of the vibrogram also changes (Fig. 1, e, zh, z). The hammer body undergoes a complex motion, first under the action of the pressure of compressed air and then as a result of the rebound impact of the tool. Accordingly, two maxima appear on the vibrogram (indicated by the numbers 1 and 2, Fig. 1, e, zh, z). Moreover, with a gradual increase in the pressing force, the ordinate of the first maximum decreases, while that of the second increases (in comparison with the first). In [3] it is shown that in this case the amplitude–force characteristic may have a second maximum. In the third transitional regime, the hammer body is acted upon by forces of collision with the collar of two kinds. The nature of some of the collision forces has already been considered by us. These are the forces acting on the body during landings occurring during the return stroke of the striker, when the tool is at rest and, as a rule, at some distance from the material being worked.

The other forces are the rebound impacts of the tool. Their magnitude depends primarily on the properties of the materials being worked, on the ratio and properties of the masses of the colliding bodies, and also on the impact energy. Rebound impacts of the tool cause a sharp change in the velocity of motion of the body, which, as a rule, leads to an increase in the vibration amplitude [3]. From the point of view of hygienists, this operating regime of hammers is dangerous for workers.

The magnitude of the pressing force for the third transitional regime is determined from the expression

\[ R_{\mathrm{n}} = \frac{n}{60} \left[ \sqrt{2A_{\mathrm{y}}m}\,(\kappa_{0}+1) + I_{\mathrm{n}} + m_{\mathrm{k}}V_{\mathrm{n}}(\kappa_{\mathrm{k}}+1) + \int_{0}^{t} F_{\mathrm{o.y}}\,dt \right], \tag{16} \]

where \(F_{\mathrm{o.y}}\) is the force of the rebound impact of the tool on the hammer body, kg;
\(t\) is the time of the rebound impact, sec.

Judging from the forms of oscillation of the hammer body, the rebound-impact force depends on the distance at which the body is from the workpiece at the moment of collision. After the striker impacts, the tool, under the action of the impact forces and the energy transmitted to it, penetrates into the material being worked, rebounds from it, and moves toward the body. The stopping of the tool in the extreme rear position occurs as a result of collision with the body and of air pressure on the end of the shank, or as a result only of the latter factor (in all regimes except the third transitional one).

Under the action of air pressure, the tool is displaced a second time until it collides with the material being worked and rebounds from it, depending on the properties of the material, another 1–2 times.

The magnitude of the air pressure in the front cavity, in turn, also depends on the recoil of the body. The pressure is lower when the recoil is smaller. Thus, the force of the rebound impact will be maximal when there is contact between the body and the collar before the striker impacts the shank of the tool (Figs. 1, 3). In [3] it is calculated that the force of the rebound impact on the body during operation of the KE-22 hammer on steel is equal to 246 kg, and during operation on wood, 76 kg (at 5 atm).

e) Operating regime in which oscillations of the body are produced by rebound impacts of the tool

When the force pressing on the hammer handle is increased above 32 kg, oscillations of the body are produced only by rebound impacts of the tool. The geometric form of the oscillations is shown in Figs. 1, i, k. In this operating regime of the hammer, the pressing force compensates the action of the following forces:

\[ R_{\mathrm{p}} = \frac{n}{60} \left[ \sqrt{2A_y m}\,(k_0 + 1) + I_n + \int_0^t F_{\mathrm{oy}}\,dt \right] + R_k,\ \text{kg}, \tag{17} \]

where \(R_k\) is the contact force of the hammer body with the limiting collar of the tool, kg.

In the formulas given for determining the pressing force for each operating regime of the hammer, the friction forces in the guide bushing were not taken into account, nor were the weight of the hose and the reactive action of the exhaust jet of compressed air. Despite these assumptions, in most cases the calculated values of the pressing force coincide with the experimental ones. The interrelation between the feed force and the main parameters of a pneumatic impact machine was first established by O. D. Alimov [4]. On the basis of the theorem on quantity of motion, he came to the conclusion that the pressing force compensates the action on the machine of a number of forces. In accordance with this, O. D. Alimov introduced new concepts, the so-called components of the feed force.

The first component \(R_1\) compensates the action on the hammer body of the impulse from the collision of the piston with the drill. The second component \(R_2\) compensates the forces of collision of the body with the collar during seating. Subsequently, developing the propositions on the components of the feed force, V. F. Gorbunov [9] introduced a third component, the magnitude of which is determined by expression (1).

In the floating operating regime of the machine, the pressing force may conventionally be divided into two components—the first and the third; in the second transient and limiting basic regimes—into three: the first, second, and third. In the third transient regime, in addition to the indicated forces, the pressing force compensates the action on the body of the machine of rebound impacts of the tool. Consequently, one more component of the pressing force appears—the fourth. The part of the pressing force that compensates the contact forces may be separated out as a fifth component.

As is evident from the mathematical expressions for determining the components of the pressing force, some of them can be found through the main parameters of the machine. To determine other components, experimental data are necessary. It is practically impossible to determine them analytically.

Dividing the magnitude of the pressing force into components makes it possible to analyze the interrelations of the main parameters of the machine under various operating regimes with the magnitude of the pressing force and to outline ways of reducing it. The pressing force can be significantly reduced

by changing the shape of the resultant-force diagram according to the method of B. V. Sudnishnikov, by reducing the diameter of the tool shank to rational limits determined by strength and by the energy-transfer coefficient, by introducing vibration-damping devices into the design of the machines, etc.

Conclusions

1. The amplitude–force characteristic of hammers, showing the nature of the change in vibration amplitude as a function of the pressing force, is complex. The presence of bends in the amplitude–force relationship is explained by the fact that, under certain operating conditions, additional forces act on the hammer body which, by their nature, are impact forces. In the limiting main mode, such a force arises when the body collides with the limiting collar of the tool (during seating of the body). In the third transient mode, the impact forces are the reverse blows of the tool against the body.

2. The collision forces of the body with the tool collar, in their absolute magnitudes, considerably exceed the pressing force (for certain modes); therefore, the increase in vibration amplitude occurs precisely at those moments when these forces arise.

Literature

1. V. I. Baburov, V. F. Gorbunov, A. V. Trikhanov. Causes of vibration of hand-held pneumatic hammers and methods of combating it. Article in the present collection.

2. V. F. Gorbunov, V. I. Baburov, Yu. A. Oparin, L. S. Redutinsky. Evaluation of the vibration characteristics of chipping-and-riveting hammers according to sanitary standards. Proceedings of TPI, vol. 129. Publishing House of Tomsk University, 1965.

3. V. F. Gorbunov, V. I. Baburov, L. S. Redutinsky, A. V. Trikhanov, G. G. Pestov. Investigation of the vibration of a hand-held pneumatic hammer when working on various materials. Mining Machines and Automation, TsNIITEIugol, No. 56, Nedra, 1964.

4. O. D. Alimov. Interrelation of the feed force with the main parameters of a drilling hammer. Proceedings of TPI, vol. 108, Metallurgizdat, 1959.

5. A. M. Petreev. On the theory of the working cycle of pneumatic percussion machines. Collection Mathematical Methods in Mining (conference proceedings), part 1, Novosibirsk, 1963.

6. A. M. Petreev. On reducing the recoil of hand-held pneumatic percussion machines by improving the working cycle. Proceedings of the Siberian Branch of the Academy of Sciences of the USSR, Technical Sciences Series, No. 6, issue 2, 1963.

7. B. V. Sudnishnikov. On the theory of recoil of hand-held percussion machines. Novosibirsk, 1953.

8. N. A. Klushin. Pneumatic hammers with a new cycle that reduces recoil. Candidate dissertation. Novosibirsk, 1958.

9. V. F. Gorbunov. On the calculation of minimum pressing forces when working with hand-held pneumatic hammers. Proceedings of TPI, vol. 129. Publishing House of Tomsk University, 1964.