ANALYSIS OF PRESSING AND ENERGY-ABSORBING ELEMENTS OF SETUPS FOR STUDYING HAND-HELD IMPACT MACHINES

A. I. PASHNIN, V. F. GORBUNOV, V. I. BABUROV, A. F. KOZLOV

(Presented by the scientific seminar of the Department of Mining Machines, Mine Transport, and Mining Mechanics)

Many works have been devoted to the study of vibrations of pneumatic impact machines. However, there is as yet no generally accepted methodology for testing and experimentally studying these machines. The variety of methods and research apparatus employed makes it difficult to analyze and generalize the results obtained by researchers, even for machines of one and the same type. The purpose of the present work is to develop theoretical foundations for creating a unified methodology for studying the vibration properties of hand-held impact machines.

Fig. 1. Calculation scheme.

\(m_k\) — mass of the hammer body; \(m_b\) — mass of the striker; \(x_k\) — its vibration displacement;
\(p_z, p_p\) — air pressure in the rear and front cavities of the hammer, respectively;
\(m_p\) — mass of the impact-power absorber; \(C\) — stiffness of the absorber;
\(Q\) — pressing force: a) pressing by a spring; b) manual pressing; c) pressing by a pneumatic feeder.

The vibration parameters, for example, of pneumatic hammers depend on many factors (air pressure in the network, pressing force, position of the hammer, design of the working tool, properties of the materials being worked, etc.). But even under the same conditions

and, with identical measuring equipment, the test results may be different. They depend on the type of pressing devices and impact-energy absorbers used.

In bench tests, pressure on the pneumatic hammer is applied in several ways: by a precompressed spring, by a pneumatic feeder, by a weight, or by hand. The impact-power absorbers for hammers are designed as a pneumatic buffer, a friction rock drill, a spring damper, a special tool embedded in lead or clamped in wood. The influence of the designs of the pressing devices and impact-power absorbers on the vibration of pneumatic hammers can be evaluated experimentally or by calculation.

In the general case, the following act on the body of an operating hammer: the inertial force; the pressing force; friction forces; forces arising from the collision of the body and the tool; and the exciting force.

The computational scheme adopted for the purposes of this work is presented in Fig. 1.

In the calculation the following assumptions are adopted:

a) the elastic forces arising from the collision of the body and the tool, as well as the forces of dry and viscous friction, are not taken into account;

b) the exciting force \(S\delta(P_3-P_{\mathrm{p}})\), as a function of time, described by a complex polyharmonic law, is replaced by a cosine wave shifted along the ordinate axis by the amount \(\Delta F\) (Fig. 2);

Fig. 2. Graph of the disturbing force acting on the hammer.
\(1\) — actual curve, \(2\) — adopted for calculation

c) in calculations of vibrations of the hammer body with various pressing devices, the impact-energy absorber is assumed to be absolutely rigid.

Case 1. A spring-type pressing device with spring stiffness coefficient \(C_{\mathrm{p}}\).

The pressing force for this case (Fig. 1,a) is:

\[ Q = C_{\mathrm{p}} x_{\mathrm{k}} + Q_0,\ \text{kg}, \]

where \(Q_0\) is the magnitude of the preliminary compression of the spring, kg. Equa-

the motion of the hammer housing is described by a linear differential equation of the second order

\[ m_{\mathrm{k}}\frac{d^2 x_{\mathrm{k}}}{dt^2}+C_{\mathrm{p}}x_{\mathrm{k}}=A\cos\omega t+F, \tag{1} \]

where \(\omega=\dfrac{2\pi}{T_{\mathrm{c}}}\)

\(T_{\mathrm{c}}\) is the time of one cycle,

\(F=\Delta F-Q_0\).

In solving this equation the restriction \(x_{\mathrm{k}}>0\) is imposed.
The characteristic equation has the form:

\[ \kappa^2+\frac{C_{\mathrm{p}}}{m_{\mathrm{k}}}=0. \tag{2} \]

The roots of equation (2) are

\[ \kappa_1,\ \kappa_2=\pm i\sqrt{\frac{C_{\mathrm{p}}}{m_{\mathrm{k}}}}=if, \]

where \(f\) is the natural frequency of the system: housing mass—spring, and the general solution of the linear homogeneous equation is

\[ \bar{x}=C_1\cos ft+C_2\sin ft. \]

We seek a particular solution in the form

\[ \bar{x}=B\cos\omega t+D\sin\omega t+\varepsilon, \]

and after differentiating and substituting we find

\[ B=\frac{A}{C_{\mathrm{p}}(1-4\pi^2)};\quad D=0;\quad \varepsilon=\frac{F}{C_{\mathrm{p}}}. \]

Thus, the solution of equation (1) will be:

\[ \left\{ \begin{aligned} x_{\mathrm{k}}&=C_1\cos ft+C_2\sin ft+\frac{A}{C_{\mathrm{p}}(1-4\pi^2)}\cdot\cos\omega t+\frac{F}{C_{\mathrm{p}}},\\ x_{\mathrm{k}}&>0 \end{aligned} \right. \]

The constants \(C_1\) and \(C_2\) are found from the initial conditions:

\[ \begin{matrix} x_{\mathrm{k}}=0\\ \dot{x}_{\mathrm{k}}=\xi \end{matrix} \Bigg|\, t=0 \]

Proceeding from the solution obtained, the vibratory displacement of the housing will depend on the stiffness of the spring; specifically, the larger the stiffness coefficient \(C_{\mathrm{p}}\), the smaller the vibratory displacement.

Case 2. Manual pressure (Fig. 1, б).
The pressing force in this case is determined through the impedance of the human hand \(l\), which, according to Shlobakh’s data, is equal to \(20\ \text{kg}\cdot\text{s}/\text{m}\)

\[ Q=l\frac{dx_{\mathrm{k}}}{dt}+Q_0,\ \text{kg}, \]

and the equation of motion of the hammer housing will have the form

\[ m_{\mathrm{k}}\frac{d^2 x_{\mathrm{k}}}{dt^2}+l\frac{dx_{\mathrm{k}}}{dt}=A\cos\omega t+F \tag{3} \]

(the notation is the same).

Solving equation (3) in an analogous manner, we obtain:

\[ \begin{cases} x_k=C_1 e^{-\frac{lt}{m_k}}+ \dfrac{AT_{\mathrm{c}}^{3}l}{(4\pi^2 m_k+l^2T_{\mathrm{c}}^2)2\pi}\cdot \sin \omega t -\dfrac{Am_kT_{\mathrm{c}}^2}{4\pi^2m_k^2+l^2T_{\mathrm{c}}^2}\cdot \cos \omega t,\\ x_k>0 \end{cases} \]

\(C_1\) is found from the initial conditions.

Case 3. Pressing by a pneumatic feeder (Fig. 1, c).
The pressing force for this case (determined experimentally) is

\[ Q=p_0S_{\mathrm{p}}. \]

The equation of oscillations of the body is written as:

\[ m_k\frac{d^2x_k}{dt^2}=A\cos \omega t+N, \tag{4} \]

where \(N=\Delta F-p_0\cdot S_{\mathrm{p}}\).

The solution of equation (4) is found by double integration:

\[ \begin{cases} x_k=-\dfrac{A}{\omega^2m_k}\cdot \cos \omega t+ \dfrac{Fp^0S_{\mathrm{p}}}{2m_k}\cdot t,\\ x_k>0 \end{cases} \]

As can be seen from the obtained solutions of equations (1), (3), and (4), the oscillatory motion of the pneumatic hammer body depends substantially on the design of the pressing device used.

Fig. 3. Amplitude–force relationships of the BSZ-135 hammer under various pressing devices.
\(1—C_{\mathrm{p}}=1.8\ \text{kg/mm},\ 2—C_{\mathrm{p}}=1.3\ \text{kg/mm},\ 3—C_{\mathrm{p}}=0.5\ \text{kg/mm},\)
\(4\)—manual pressing, \(5—C_{\mathrm{p}}=0.39\ \text{kg/mm},\ 6\)—pressing by a pneumatic feeder.

Substituting into the solutions of equations (1), (3), and (4) the values of the constants for a particular pneumatic hammer and assigning various values of the preliminary compression, one can construct the dependence of vibrational displacement on the pressing force.

Figure 3 shows experimental and theoretical dependences of vibration on the pressing force for the BSZ-135 chipping hammer. From

it is clear from the plots that the character of the experimental (solid) curves agrees satisfactorily with the theoretical (dashed) curves. The difference is explained by the assumptions made:

a) the theoretical curves have no extremal values corresponding to the floating and limiting basic operating modes of the hammer, since the elastic forces arising when the hammer body seats on the shoulder of the tool were not taken into account;

b) the theoretical curves lie somewhat above the experimental ones, since friction forces were not taken into account in solving the differential equations.

An exact solution of the equations of oscillation of the hammer body can be obtained on a computer, taking into account all acting forces.

As can be seen from Fig. 3, curve 4, corresponding to the vibration displacement of the hammer body under manual pressure, lies between curves 3 and 6. Curve 3 corresponds to the vibration displacement under pressure by a spring with \(C_p = 0.5\ \mathrm{kg/mm}\), while curve 6 corresponds to pressure by a pneumatic feeder. The assessment of pneumatic-hammer vibration in laboratories must correspond to production conditions, that is, to pressure by hand. Therefore, in laboratory tests of hand-held pneumatic percussion machines, these requirements are satisfied by spring-type pressing devices with a stiffness coefficient of \(0.3\text{--}0.5\ \mathrm{kg/mm}\), or by pressure using a pneumatic feeder.

Effect of impact-energy absorbers

We shall assume that, for different absorbers, pressure is applied only by a spring. Under spring pressure \(Q = C_p x_k\), and the elastic force acting through the tool on the hammer body is

\[ R = C_m \cdot x_k,\quad \text{for } x_k < 0, \]

where \(C_m\) is the stiffness of the impact-energy absorber, \(\mathrm{kg/cm}\).

This is valid for the case when the tool is pressed tightly against the body. Then the equation of motion of the body will be written in the form

\[ m_k \frac{d^2 x_k}{dt^2} + (C_p - C_m)x_k = A \cos \omega t + F . \tag{5} \]

Let us consider some of the existing absorbers:

a) an energy absorber consisting of a metal block (G. S. Wilner’s test bench).

The settlement of the material being processed can be found from Hooke’s law:

\[ \Delta l = \frac{\Delta P l}{E S}, \]

where \(l\) is the thickness of the material;

\(S\) is the contact area;

\(E\) is the modulus of elasticity.

Assuming \(l\) and \(S\) to be constant, then

\[ C_m = \frac{\Delta P}{\Delta l} = \frac{E S}{l}, \]

and the solution of equation (5) will be analogous to the solution of equation (1):

\[ x_k = C_1 \cos \sqrt{\frac{l C_p - E S}{m_k l}} \cdot t + C_2 \sin \sqrt{\frac{l C_p - E S}{m_k l}} \cdot t + \]

\[ + \frac{A}{\left(C_p - \dfrac{E S}{l}\right)(1 - 4\pi^2)} \cdot \cos \omega t + \frac{F}{C_p - \dfrac{E S}{l}} . \]

b) the energy absorber is an air shock absorber (the test stand of S. I. Dobroborskii)

\[ C_m=\frac{P S_{\mathrm p}}{x_k}, \]

where \(S_{\mathrm p}\) is the piston area.

Since the process is isothermal,

\[ P=-\frac{p_0 V_0}{V_0-S_{\mathrm p}x_k} \]

and

\[ C_m=\frac{p_0 V_0 S_{\mathrm p}}{x_k\left(V_0-S_{\mathrm p}x_k\right)}, \]

since \(x_k=1\),

\[ C_m=\frac{p_0 V_0 S_{\mathrm p}}{V_0-S_{\mathrm p}}. \]

The solution of equation (5) will be:

\[ x_k=C_1\cos \sqrt{\frac{C_{\mathrm p}}{m_k}-\frac{p_0V_0S_{\mathrm p}}{(V_0-S_{\mathrm p})m_k}}\cdot t+ \]

\[ {}+C_2\sin \sqrt{\frac{C_{\mathrm p}}{m_k}-\frac{p_0V_0S_{\mathrm p}}{(V_0-S_{\mathrm p})m_k}}\cdot t+ \]

\[ {}+\frac{A}{\left(C_{\mathrm p}-\dfrac{p_0V_0S_{\mathrm p}}{V_0-S_{\mathrm p}}\right)}\cdot \cos \omega t+\frac{F}{C_{\mathrm p}-C_m}. \]

As can be seen from the solutions obtained, the vibration displacement of the housing will be greater the larger the modulus of elasticity, or the lower the air pressure under the piston.

Fig. 4. Effect of the pressing force and of the properties of the energy absorber on the magnitude of vibration of the KE-16 hammer. 1, 2, 5—vibration acceleration; 3, 4, 6—vibration displacement during operation of the hammer, respectively, on the ERD-3 stand and in cutting cast iron and aluminum.

To check these assumptions, studies were carried out to measure the vibration displacement of the KE-16 hammer. The hammer

was mounted on the ERD-3 stand; the tool used was a chisel with a spherical tip. Cast-iron and aluminum blanks and a piston pneumatic device were taken as impact-energy absorbers.

The test results are presented in Fig. 4; they coincide with the results of solving equation (5). Since impact-energy absorbers exert a significant influence on the magnitude of the vibrational displacement, and the results of laboratory tests must coincide with production results, it is necessary to conduct studies with impact-energy absorbers made of the materials on which cutting or riveting is performed by pneumatic hammers. In this case a situation may arise in which the measuring apparatus, owing to low sensitivity, a small linearity range, or small oscillation amplitudes, does not make it possible to measure the vibration parameters (displacement, velocity, acceleration). In this case it is recommended to use materials with a large modulus of elasticity (hardened steel \(E = 2.2 \cdot 10^6\ \text{kg}/\text{cm}^2\)) or with lower elasticity (aluminum \(E = 0.7 \cdot 10^6\ \text{kg}/\text{cm}^2\), lead \(E = 0.18 \cdot 10^6\ \text{kg}/\text{cm}^2\)) or another energy absorber, and then, using the relationship obtained, to recalculate to the original material, i.e., to apply the method of physical modeling.

It has been established experimentally that, as the pressing force increases, the vibrational acceleration increases. This is evident from the differential equation of motion of the hammer body, since with an increase in the static pressing force, and also in the elasticity of the material being processed, the vibrational acceleration must increase,

\[ \frac{d^2 x_k}{dt^2}. \]

Thus, the vibration of pneumatic hammers can be assessed not only by vibrational displacement, but also by the magnitude of vibrational acceleration, which is solved comparatively easily by using standard instruments of the PIU-1 and PIU-2 type.

Conclusions

1. In laboratory studies of the vibration of impact-action machines, it is necessary that the pressing device provide the same vibrational displacements of the hammer as are observed under production conditions when pressed by hand. Cylindrical helical springs with a stiffness coefficient of \(0.3\text{–}0.5\ \text{kg}/\text{mm}\) and a pneumatic feeder meet these requirements.

2. In order for the results of laboratory tests to coincide with production results, it is necessary to use impact-energy absorbers made of the materials on which the technological process is carried out (cutting, riveting). If the measuring apparatus does not make it possible to measure the vibration parameters (displacement, velocity, acceleration), then it is recommended to take a material with a lower modulus of elasticity (aluminum, lead) and recalculate the obtained data to the original material.

3. It has been established experimentally that, as the pressing force and the elasticity of the energy absorber increase, the vibrational acceleration of the hammer handle increases. Thus, the vibration of hand-held impact machines can be assessed by the magnitude of the vibrational acceleration.