STUDY OF VIBRATION ISOLATION OF HANDLES OF HANDHELD PNEUMATIC HAMMERS BY MATHEMATICAL MODELING

V. F. Gorbunov, A. V. Trikhanov

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

An inherent drawback of pneumatic hammers is the presence of vibration of their body under the action of variable air pressure in the cavities of the hammer cylinder and of impact processes of energy transmission to the tool. As a result of prolonged work with pneumatic hammers possessing significant vibration, pathological changes arise in the bodies of workers, leading in a number of cases to disability [3].

The theory of vibration isolation as applied to handheld machines of impact action has not yet been developed sufficiently fully, owing to the complexity of the analytical solution of the equations describing the behavior of such systems [1, 2, 10].

Electronic computing machines of continuous action—mathematical models—possess great capabilities in solving theoretical and design problems of vibration isolation [6, 7]. The model of a pneumatic hammer with a vibration-isolating device in the form of a spring represents a mechanical oscillatory system with four degrees of freedom. The behavior of the mechanical model (Fig. 1) is described by a system of four nonlinear differential equations of the second order.

\[ m_{\mathrm{u}} \frac{d^2 x_{\mathrm{u}}}{dt^2} = -\eta S_{\mathrm{b}} p_{\mathrm{u}} - r_{\mathrm{u}} \frac{d x_{\mathrm{u}}}{dt} + \xi_{\mathrm{k}} C_{\mathrm{k}}(x_{\mathrm{k}}-x_{\mathrm{u}}) - \xi_{\mathrm{m}} C_{\mathrm{m}} x_{\mathrm{u}}, \tag{1} \]

\[ m_{\mathrm{b}} \frac{d^2 x_{\mathrm{b}}}{dt^2} = -S_{\mathrm{b}} p_{3} + S_{\mathrm{b}} p_{\mathrm{u}} - r_{\mathrm{b}} \frac{d x_{\mathrm{b}}}{dt} - \xi_{\mathrm{b.k}} C_{\mathrm{b.k}}(x_{\mathrm{b}}-x_{\mathrm{k}}), \tag{2} \]

\[ m_{\mathrm{k}} \frac{d^2 x_{\mathrm{k}}}{dt^2} = + S_{\mathrm{b}} p_{3} - S_{\mathrm{b}} p_{\mathrm{u}} - r_{\mathrm{k}} \frac{d x_{\mathrm{k}}}{dt} - \xi_{\mathrm{k}} C_{\mathrm{k}}(x_{\mathrm{k}}-x_{\mathrm{u}}) + \xi_{\mathrm{b.k}} C_{\mathrm{b.k}}(x_{\mathrm{b}}-x_{\mathrm{k}}) + \]
\[ + C_{\mathrm{r}}(x_{\mathrm{r}}-x_{\mathrm{k}}) - Q, \tag{3} \]

\[ m_{\mathrm{r}} \frac{d^2 x_{\mathrm{r}}}{dt^2} = - r_{\mathrm{r}} \frac{d x_{\mathrm{r}}}{dt} - C_{\mathrm{r}}(x_{\mathrm{r}}-x_{\mathrm{k}}), \tag{4} \]

where \(m \frac{d^2 x}{dt^2}\) are the inertial forces;

\[ Sp \]
are the disturbing forces (air pressure);

\[ Cx \]
are the elastic forces of the springs modeling the elasticity of the material of the colliding parts;

\(\xi\) are coefficients determining the occurrence of elastic spring forces. For example,

\[ \xi_k = \begin{cases} 0, & x_i \leq x_k,\\ 1, & x_i > x_k. \end{cases} \]

The principal notation adopted in describing the system is given in Fig. 1.

Mathematical modeling provides for reducing the equations to machine form by replacing physical variables with voltages, with the introduction of the corresponding scale factors. As the object for calculating real initial data, we have adopted the KE-22 hand-held pneumatic hammer, manufactured serially by the Tomsk Electromechanical Plant. The scale factors and coefficients of equations (5–8) are given in Tables 1 and 2.

Fig. 1. Calculation diagram of a pneumatic hammer with an elastic handle

Table 1

Reducing the equations to machine form, we obtain:

\[ \frac{d^2 U_i}{d\tau^2} = - B_1 U_{1,2} - B_1' \frac{dU_i}{d\tau} + \xi_k D_1 (U_k - U_i) - \xi_m \Gamma_1 U_i, \tag{5} \]

\[ \frac{d^{2}U_{\mathrm{b}}}{d\tau^{2}} = - A_{2}U_{3,4}+B_{2}U_{1,2} - V_{2}\frac{dU_{\mathrm{b}}}{d\tau} -\xi_{\mathrm{b.k}}G_{2}(AU_{\mathrm{b}}-U_{\mathrm{k}}), \tag{6} \]

\[ \frac{d^{2}U_{\mathrm{k}}}{d\tau^{2}} = A_{3}U_{3,4}-B_{3}U_{1,2} - V_{3}\frac{dU_{\mathrm{k}}}{d\tau} -\xi_{\mathrm{k}}D_{3}(U_{\mathrm{k}}-U_{\mathrm{h}}) + \tag{7} \]

\[ +\xi_{\mathrm{b.k}}G_{3}(AU_{\mathrm{b}}-U_{\mathrm{k}}) +Zh_{3}(BU_{\mathrm{r}}-U_{\mathrm{k}}) -T_{3}U_{q}, \]

\[ \frac{d^{2}U_{\mathrm{r}}}{d\tau^{2}} = - V_{4}\frac{dU_{\mathrm{r}}}{d\tau} - Zh_{4}(BU_{\mathrm{r}}-U_{\mathrm{k}}). \tag{8} \]

In accordance with the description, the mathematical model of the pneumatic hammer was assembled on the basis of the IPT-5 electronic modeling installation.

Table 2

The structural diagram of the model (Fig. 2) consists of four interconnected chains of computing blocks corresponding to equations (5–8). Thus, for example, equation (5) is solved by a set of blocks 1–7, of which integrators 2 and 3 are the principal ones; block 5 compares the coordinates of the body and the tool; block 4 models the limiter of the tool on the side of the material being worked, $C_{\mathrm{m}}$; block 6 specifies the velocity of the tool after its collision with the striker.

The resultant force from the air pressure on the hammer body, striker, and tool $(p_{3}, p_{\mathrm{p}})$ was specified with the aid of nonlinear blocks BN-1, 2, 3, 4 on the basis of results from indicator diagrams of various pneumatic-hammer designs in the laboratory of mining machines of Tomsk Polytechnic Institute [4]. The impact processes were implemented by electronic-relay blocks specially developed for this purpose [7]. The model operated at a 230-fold slowed rate (with a time scale of $4.34\cdot10^{-3}$). Thus it was possible, visually on the oscilloscope screen, to analyze each cycle of oscillation of the body, handle, and tool and to photograph the most characteristic regimes.

Fig. 2. Structural diagram of the model

The amplitude of oscillation of the handle \(A_p\) decreases as the pressing force increases (Fig. 3). With a handle weight of \(2\ \mathrm{kg}\) and a spring stiffness of \(400\ \mathrm{kg/m}\), the amplitude of the handle is 5–10 times smaller than the amplitude of the body. It is also interesting to note that, as the pressure of the air supplied to the hammer increases, the vibration-isolation effect improves. Thus, when the limiting basic operating mode of the hammer is maintained [11] and the air pressure is increased from 5 to 6 atm, the vibration amplitude of the body practically does not increase, while the amplitude of the handle decreases from 0.43 to 0.32 mm (the pressing forces are, respectively, 25 and 30 kg).

If, however, the comparison is made at equal pressing forces, then as the air pressure increases the vibration amplitude of the handle increases (Fig. 3), and the ratio by which the handle amplitude is reduced in comparison with the body amplitude decreases.

The character of the change in the velocity of the handle under any operating conditions of the hammer remains very smooth, close to sinusoidal.

Fig. 3. Amplitude–force relationships of the vibration of the body \(A_k\) and the handle \(A_p\) of the hammer

Summing up the analysis of the dependence of the vibration amplitude of the handle on the air pressure in the hammer and on the pressing force, we note that qualitatively this dependence is analogous to the corresponding characteristics of the first experimental models of hammers with elastic handles [5], while quantitative values for hammers of the KE type have been obtained for the first time.

As studies of a two-mass oscillatory system on an electronic model [6] have shown, the vibration amplitude of the handle depends very strongly on the stiffness of the elastic link. To reveal this dependence as applied to KE-type hammers, we investigated the system (Fig. 1) while varying \(C_p\) from 200 to \(1600\ \mathrm{kg/m}\), with a handle weight of \(2\ \mathrm{kg}\). The weight of the hammer body was taken as unchanged for this series of experiments, \(G_k = 8.75\ \mathrm{kg}\). The air pressure (the diagram of resultant forces) was also taken as unchanged, corresponding to \(p = 5\) atm.

The general character of the change in the amplitude of oscillation of the handle as a function of the stiffness coefficient of the spring was shown by us in previous work on modeling [6]. Qualitatively, this dependence is also predicted by other studies on vibration damping [1, 2, 9, 10].

The data of the present investigations characterize the quantitative interrelations of these indices for KE-type hammers. A certain increase in the amplitude of oscillation of the handle when passing from the floating mode

mode to the limiting basic one (Fig. 4), evidently, naturally as a result of a similar change in the amplitude of oscillation of the body. For a handle weighing 2 kg and hammer operation in the limiting basic mode, the vibration amplitude of the handle increases with an increase in the spring stiffness coefficient from 0.2 mm (at \(C_p = 200\ \text{kg/m}\)) to 1.6 mm (at \(C_p = 1600\ \text{kg/m}\)). In other modes of hammer operation this proportionality is not observed. With an increase in spring stiffness, the amplitude of the handle in the zone of large pressing forces increases to a lesser degree, since under these conditions the body oscillates for a relatively short time and its action on the handle through the spring approaches an impact action. Evidently, when an impact impulse is transmitted, the stiffness of the spring has less influence on the effect of vibration isolation than under a smoother, harmonic oscillation of the hammer body.

In the zone of small pressing forces, the amplitude of oscillation of the handle is comparatively greater because of the increased vibration of the body, as well as because of the presence of cycles with significantly different amplitudes. This leads to a change not only in the vibration amplitude of the handle itself, but at the same time also to a change in the position of its dynamic equilibrium.

Fig. 4. Dependence of the vibration amplitude of the handle on its weight and pressing force

Qualitatively, the dependence of the handle oscillation amplitude on its weight is expressed by a formulation encountered in many literary sources: “With an increase in the weight of the elastic handle, the vibration-isolation effect improves.” The validity of this is obvious. A quantitative expression of this dependence for hand-held pneumatic hammers, to our knowledge, has been obtained for the first time.

As follows from the modeling data, a change in the weight of the handle has a substantial effect on the amplitude of its oscillation and practically does not affect the vibration amplitude of the hammer body. As the weight of the handle increases, its amplitude decreases proportionally.

The main regularities of vibration damping as applied to pneumatic hammers must be considered in interrelation with the magnitude of the principal oscillating mass—the hammer body. Naturally, the greater the mass of the body, the smaller the amplitude of its oscillation should be and, consequently, the vibration amplitude of the handle will decrease correspondingly. In our experiments, reducing the mass of the body from 0.9 to \(0.6\ \text{kg}\cdot\text{sec}^2/\text{m}\) led to an increase in the amplitude of its oscillation from 2.7–2.8 to 3.2 mm (at \(p = 5\ \text{ati}\) and a pressing force ensuring the limiting basic operating mode).

Quantitative data on the dependence of the oscillation amplitudes of the handle and the body when their masses are varied are given in Fig. 5.

The study of oscillatory systems analogous to hammers with elastic handles is usually carried out using dimensionless dependences. In particular, the vibration-damping effect, determined by the rela-

with a decrease in the amplitudes of the handle and of the body (or conversely), is invariably associated with the relative frequency of oscillations

\[ \beta = \frac{\omega}{\omega_0}, \]

where \(\omega\) is the frequency of action of the exciting force (air pressure), and \(\omega_0\) is the natural (partial) frequency of the free vibrations of the handle on the spring \(C_p\).

The expressions for the frequencies are generally known:

\[ \omega = 2\pi n, \qquad \omega_0 = \sqrt{\frac{C_p}{m_p}}, \]

where \(n\) is the hammer blow frequency per second.

Fig. 5. Dependence of the vibration amplitude of the handle on its weight, for hammer-body weights of 5.9; 6.87; 7.85 and 8.75 kg

The dependence of the relative vibration amplitude of the handle on the ratio of the forced and natural vibration frequencies, constructed from the data of modeling the system (Fig. 1), is shown in Fig. 6.

A similar graphical dependence was obtained by Yu. P. Tsivinskii [2] in studying a mechanical two-mass oscillatory system with a limiter. Despite the different approach to solving the problem and the different initial parameters of the system, the curves in dimensionless coordinates coincide.

Fig. 6. Amplitude–frequency characteristic of a model of a hammer with an elastic handle: \(\dfrac{A_p}{A_k}\) is the ratio of the vibration amplitudes of the handle and the body; \(\dfrac{\omega}{\omega_0}\) is the dimensionless frequency

Incidentally, we note that the indicated dependence, constructed by Yu. P. Tsivinskii [2] from the data of our earlier studies [6], also lies in the region of real amplitude–frequency characteristics

stics of two-mass systems. All this proves the objectivity of the data obtained and their importance from the standpoint of their use for the development and improvement of methods for calculating and designing hammers with elastic handles.

The analytical expression for the vibration-damping effect through the frequency ratio, derived in [2], reflects the behavior of the system over a wide range of frequencies of the disturbing force in the presence of friction

\[ \frac{A_{\mathrm{p}}}{A_{\mathrm{k}}} = \frac{1}{ \sqrt{ \left(1-\frac{\omega^{2}}{\omega_{0}^{2}}\right) + \frac{\gamma \omega^{2}}{C_{\mathrm{p}}^{2}} } }, \tag{9} \]

where \(\gamma\) is a coefficient taking friction into account.

Restricting the variation of the frequencies \(\omega\) and \(\omega_{0}\) to the real limits of the parameters of pneumatic hammers at \(p = 5\) ati, formula (9) can be simplified and, taking into account the modeling results, written as follows:

\[ \frac{A_{\mathrm{p}}}{A_{\mathrm{k}}} = \frac{0.035 \cdot \omega_{0}}{\omega} = \frac{0.056 \sqrt{C_{\mathrm{p}}}}{n \sqrt{m_{\mathrm{p}}}}. \tag{10} \]

The empirical coefficient of formula (10) was obtained in processing the modeling data for the system shown in Fig. 1.

The amplitude of vibration of the handle can also be determined analytically, using the results of processing the graphs (Figs. 5, 6):

\[ A_{\mathrm{p}} = \kappa_{1} \sqrt{\frac{C_{\mathrm{p}}}{m_{\mathrm{k}} m_{\mathrm{p}}}} = \kappa_{1}\omega \sqrt{\frac{1}{m_{\mathrm{k}}}}, \quad \text{mm}, \tag{11} \]

where \(\kappa_{1}\) is an empirical coefficient.

For KE-type hammers with elastic handles, when operating in the limiting basic mode and at an air pressure of 5 ati, \(\kappa_{1} = 8.5 \cdot 10^{-3}\).

The formulas (10), (11) presented are recommended for an approximate determination of the vibration-damping effect achieved by applying elastic handles to KE- and MR-type hammers. They do not claim to provide an exhaustive treatment of the problem of vibration isolation of all pneumatic hammers and will evidently be refined after the study of a number of experimental designs of hammers with vibration-isolating devices. It should be noted, incidentally, that in the modern literature it is recognized that “the methods and techniques of regulating oscillations and damping them are equally objects of both art and science” [9]. In this connection, the success of the work depends not only on a correctly chosen method of vibration damping and an objective calculation of vibration-damping devices, but is also determined to a large extent by design solutions and the accuracy of their execution.

The data from modeling the dynamics of operation of the KE-22 pneumatic hammer with an elastic handle show that limiting the harmful effect of vibration of pneumatic hammers on the worker can be accomplished very effectively by introducing an elastic element (a spring) into the design of the hammers. In this case the amplitude of vibration of the handle is reduced by a factor of 5–10 relative to the amplitude of the housing, which guarantees safe operation with the hammers from the sanitary-hygienic standpoint.

The quantitative dependences obtained in the present study of handle vibration on hammer parameters and on the stiffness of the elastic element can be used in designing experimental hammer constructions. The expected vibration-damping effect can be determined by

given in the present work by analytical and graphical dependences.

REFERENCES

1. P. M. Alabuzhev, V. I. Kopytov. On the question of the possibility of applying a dynamic vibration absorber to impact-action machines. Izvestiya TPI, vol. 108, Metallurgizdat, 1959.

2. P. M. Alabuzhev, Yu. P. Tsivinsky. Amplitude-frequency characteristic of the oscillations of a two-mass system with impact against a limiter of one of the masses. Mechanisms and machines of impact, vibratory, and vibratory-impact action. Interuniversity collection of papers No. 1, Novosibirsk, 1963.

3. E. Ts. Andreeva-Galanina, B. G. Artamonova, E. A. Drogichina. Vibration disease. Medgiz, 1961.

4. O. D. Alimov, I. G. Basov, V. F. Gorbunov, D. N. Malikov. Drilling machines. Gosgortekhizdat, 1960.

5. V. F. Gorbunov, V. I. Kopytov, I. F. Vysotsky. Some results of the study of an experimental model of a jackhammer with an elastic handle. Izvestiya vuzov, Mashinostroenie, No. 10, 1963.

6. V. F. Gorbunov, V. M. Razin, A. V. Trikhanov. Some results of modeling the oscillations of pneumatic hammers with vibration-isolating devices. Proceedings of the conference “Mathematical Methods in Mining.” Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, 1963.

7. V. F. Gorbunov, V. M. Razin, A. V. Trikhanov. Electronic model of elastic impact. GOSINTI publication, collection, 1964.

8. V. F. Gorbunov, V. I. Kopytov, M. P. Shumsky. Method for calculating the vibration-isolating element of pneumatic hammers with elastic handles. Izvestiya TPI, vol. 123, TTU Publishing House, 1963.

9. Kim N. Tong. Theory of mechanical vibrations. Mashgiz, 1963.

10. A. E. Kobrinsky. On the theory of impact vibration damping. Izvestiya AN SSSR, No. 5, 1957.

11. A. M. Petreev. On the theory of the working cycle of pneumatic impact-action machines. Proceedings of the conference “Mathematical Methods in Mining.” Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, 1963.