On the Modeling of the Dynamics of Pneumatic Hammers

V. F. Gorbunov

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

Experimental and theoretical studies of pneumatic impact machines, carried out at the Tomsk Polytechnic Institute under the direction and with the participation of Prof. P. M. Alabuzhev and O. D. Alimov, made it possible to formulate in a new way the requirements for the working process and design of pneumatic hammers. Studies of the mechanism of rock destruction in percussive-rotary drilling and of the transmission of energy, etc., as well as of the influence of hammer operating conditions on the nature of the internal processes, showed that an isolated description of the thermodynamic processes in the hammer without taking into account the motion of its body, the tool, and the acting feed forces does not give a correct solution of the working cycle.

On the basis of the results of studying the interrelation of the feed force with the principal parameters of a pneumatic hammer [1], of our own studies of the working process and vibration of pneumatic hammers, and of the results of solving the basic equations of the working cycle on electronic models [2, 3], the description of the dynamics of a pneumatic hammer appears to us as follows.

The equations describing the internal process of a pneumatic hammer may be compiled by introducing certain assumptions and simplifications which, to a certain extent, simplify the mathematical formulation of individual terms of the equations. Such assumptions include the following:

1) the thermodynamic processes taking place in the hammer cylinder are assumed to be equilibrium (quasi-static);

2) compressed air obeys the laws of an ideal gas;

3) external heat exchange is absent;

4) the air pressure at the inlet to the machine is constant;

5) air leakage through piston clearances is neglected;

6) the periods during which the cross section of the inlet ports changes are excluded from consideration;

7) compressed air cannot enter both cavities of the cylinder simultaneously.

Such assumptions may be accepted without distorting the physical meaning of the phenomena under study, although the accuracy of the solution will naturally decrease. However, without these simplifications, solving the equations would be altogether impossible.

Without presenting the entire derivation of the equations of state of the air in the cavities, which is set out in detail in [2, 3, 4] and elsewhere, we write them as functions of time:

\[ \frac{dp_1}{dt} = \frac{\kappa}{F_1 x} \left[ nRT_c \mu_1 f_1 \sqrt{\frac{2g\kappa}{\kappa-1}}\cdot \sqrt{\frac{p_c}{W_1}}\, \Phi\!\left(\frac{p_1}{p_c}\right) - F_1 p_1 \frac{dx}{dt} \right], \tag{1} \]

\[ \frac{dp_2}{dt} = \frac{\kappa}{aF_1(L_c-x)} \left[ -\mu_2 f_2 \sqrt{2g\frac{\kappa}{\kappa-1}}\cdot \sqrt{RT_2}\cdot p_0^{\frac{\kappa-1}{2\kappa}}\cdot p_2^{\frac{3\kappa-1}{2\kappa}} \Phi\!\left(\frac{p_a}{p_2}\right) + aF_1p_2\frac{dx}{dt} \right], \tag{2} \]

where

\[ a=\frac{F_2}{F_1};\qquad pV=RT;\qquad W_1=F_1x, \]

\[ \Phi\!\left(\frac{p_a}{p_2}\right) = \sqrt{ \left(\frac{p_a}{p_2}\right)^{\frac{2}{\kappa}} - \left(\frac{p_a}{p_2}\right)^{\frac{\kappa+1}{\kappa}} } \quad \text{for subcritical outflow} \]

\[ \left( \beta \geq \beta_{\mathrm{cr}}=\frac{p_a}{p_2}=0.53 \right); \]

\[ \Phi\!\left(\frac{p_a}{p_2}\right) = \sqrt{ \left(\frac{2}{\kappa+1}\right)^{\frac{2}{\kappa-1}} - \frac{\kappa+1}{\kappa-1} } = \mathrm{const} \quad \text{for supercritical outflow }(\beta<\beta_{\mathrm{cr}}); \]

\[ \Phi\!\left(\frac{p_1}{p_c}\right) = \sqrt{ \left(\frac{p_1}{p_c}\right)^{\frac{2}{\kappa}} - \left(\frac{p_1}{p_c}\right)^{\frac{\kappa+1}{\kappa}} } \quad \text{for subcritical outflow;} \]

\[ \Phi\!\left(\frac{p_1}{p_c}\right) = \sqrt{ \left(\frac{2}{\kappa+1}\right)^{\frac{2}{\kappa-1}} - \left(\frac{2}{\kappa+1}\right)^{\frac{\kappa-1}{\kappa+1}} } = \mathrm{const} \quad \text{for supercritical outflow;} \]

\[ \mu_1 f_1 \sqrt{2g\frac{\kappa}{\kappa-1}}\cdot \sqrt{\frac{p_a}{W_1}}\, \Phi\!\left(\frac{p_1}{p_c}\right) = G \quad \text{— weight flow rate of the air entering the cylinder;} \]

\[ \mu_2 f_2 \sqrt{2g\frac{\kappa}{\kappa-1}}\cdot \sqrt{\frac{p_2}{W_2}}\, \Phi\!\left(\frac{p_a}{p_2}\right) = G_p \quad \text{— weight flow rate of the air leaving the cylinder.} \]

\[ \kappa=1.4 \]
— process exponent;

\[ f_1 \text{ and } f_2 \]
— cross-sectional areas of the intake and exhaust channels;

\[ \mu_1;\ \mu_2 \]
— discharge coefficients of the intake and exhaust openings;

\[ F_1 \text{ and } F_2 \]
— piston areas on the sides of the working-stroke and return-stroke cavities;

\[ L_c \]
— reduced length of the cylinder;

\[ x \]
— piston coordinate;

\[ g \]
— acceleration of free fall;

\[ \beta_{\mathrm{cr}}=\frac{p_{a,1}}{p_{c,2}}=0.53 \]
— index of the characteristic of the outflow process.

The equation of motion of the piston in the hammer cylinder can be derived from the assumption that the following forces act on the piston:

\[ p_1 F_1 \]
—the force due to the air pressure in the rear cavity;

\[ p_2 F_2 \]
—the counterpressure force from the front cavity;

\[ p_a (F_1 - F_2) \]
—the force arising from the action of atmospheric pressure;

\[ m \frac{d^2 x}{dt^2} \]
—the inertia force;

\[ r \frac{dx}{dt}, \quad R_{\mathrm{tr}}\operatorname{sign}\frac{dx}{dt} \]
—the force of viscous or dry friction;

\[ \xi C_i \Delta \]
—the force arising when the piston strikes the tool.

Then

\[ m \frac{d^2 x}{dt^2} = p_1 F_1 - p_2 F_2 - p_a(F_1 - F_2) - r \frac{dx}{dt} + \xi C_i \Delta \]

or

\[ \frac{d^2 x}{dt^2} = \frac{F_1}{m}\left[p_1 - \alpha p_2 - p_a(1-\alpha)\right] - \frac{r}{m}\frac{dx}{dt} + \xi \frac{C_i \Delta}{m}. \tag{3} \]

Equations (1), (2), and (3) are compiled for the condition in which the body of the hammer and the tool are immobile. Therefore, in order to reveal the overall dynamics of hammer operation, it is necessary to supplement the system of equations with two more second-order differential equations [5]:

\[ m_k \frac{d^2 x_k}{dt^2} = \varphi_1(p_1) - \varphi_2(p_2) - \frac{r_k dx_k}{dt} + \xi_k C_k(x_k - x_i) - \]

\[ - \xi_{b.k} C_{b.k}(x_b - x_k) - Q; \tag{4} \]

\[ m_i \frac{d^2 x_i}{dt^2} = \varphi_3(p_2) - \frac{r_i dx_i}{dt} - \xi_m C_m x_i - \xi C_i \Delta - \xi_k C_k(x_k - x_i), \tag{5} \]

where

\[ m_k \text{ and } m_i \]
—are the masses of the hammer body and the tool;

\[ x_k,\ x_i \]
—are the coordinates of \(m_k\) and \(m_i\);

\[ \varphi_1(p_1);\ \varphi_2(p_2);\ \varphi_3(p_3) \]
—are the forces of air pressure on the hammer elements, specified by equations (1), (2);

\[ \xi C x \]
—are elastic forces arising in the mutual impact of the hammer elements;

\[ Q \]
—is the pressing force applied to the hammer.

An analytical solution of the system of differential equations (1–5) with accuracy sufficient for practice does not appear possible. The only acceptable method of solution, in our opinion, is the use of mathematical machines, which make it possible not only to resolve the nonlinear functions of the thermodynamic process, but also to implement nonlinearities of the elastic-impact type. Electronic modeling installations have great capabilities in this respect; their use makes it possible to overcome a number of formal mathematical difficulties in solving nonlinear problems.

Equations (1–3) could have been solved by dividing the entire cycle into a number of intervals, as some authors have done [4, 6]. However, in all these cases the calculated data differ significantly from the experimental data. Solving the system of all five equations is still more difficult because the literature lacks an established

opinion on the values of the coefficient of restitution of piston velocity when it strikes the tool. Experimental and theoretical studies by a number of authors give highly contradictory results, of little use for determining the real value of this coefficient. All studies of the coefficient of restitution of velocity in the collision of two free bodies (including in the presence of a third constraint—the rock or the clamping of the tool), in our opinion, do not fully reflect the physics of the collision process of the piston of a pneumatic hammer with the tool, owing to the presence, in actual designs, of air pressure on the piston at the moment of impact.

Thus, the exact solution of the system of equations describing the working process of a pneumatic machine depends on the accuracy with which the initial data are specified and on the resolving power of the chosen solution method. The former can be accomplished by accumulating a sufficient amount of experimental material obtained with the use of the most advanced measuring instruments. As a universal means for solving systems of differential equations, mathematical machines of continuous or discrete action are now generally recognized.

Preparation for modeling the compiled system of differential equations, as is known [7], consists in choosing the ranges and scale relations of the quantities, transforming the quantities under study into voltages, drawing up the structural diagram of the model, and setting individual functions on variable-coefficient blocks and nonlinearity blocks.

This programming of the problem is, in essence, easy to carry out if the available electronic model has high resolving power. However, the models of the IPT-5, MN-7, and other types that are widespread at nonspecialized enterprises have a very limited number of operational blocks. In this connection the problem has to be solved in parts, depending on the requirements imposed. Thus, for example, when studying the internal process of the PR-24 l drilling hammer on the IPT-5 model, we were forced to introduce the assumptions that the hammer body and the tool are immovable, system of equations (1—3). And after obtaining diagrams of the internal process, the problem of the oscillations of the hammer body during its operation was solved separately (system of equations (3—5)). Such an approach to the solution of complex problems is acceptable, but the accuracy of the solution suffers somewhat [8].

In formulating the problem of studying vibration and methods of limiting it, we confined ourselves to solving only those equations that describe the displacement of the elements of the system, that is, equations (3—5). In this case the disturbing forces (air pressure) were introduced into the model in the form of functions of the displacement of the piston of the pneumatic hammer and were set on nonlinearity blocks [9].

As experience with modeling shows, solving problems of the dynamics of a pneumatic hammer in parts is acceptable not only when small models are available, but also from the point of view of speeding up the programming of the problem and increasing the accuracy of the solution. The presence of a large number of functional blocks also leads to a reduction in the reliability of the model and, as a consequence, to large expenditures of time in finding faults in the course of solving the problem.

Modeling installations make it possible to obtain the results of the solution in the form of an output voltage delivered to the control panel. There it is recorded by a pointer instrument and can be fed to an electronic indicator. For recording the signal it is possible to use photographic devices or visual readout. However, when solving

In many problems there arises a need to carry out sequential processing of the output signals. In particular, in studying the vibration of pneumatic hammers it is highly desirable to obtain arithmetic mean and root-mean-square quantities. Because of the comparatively low power of the signal delivered by the model control panel, it is practically impossible to process the signal on electromechanical and electromagnetic counters. Therefore there arises a need to develop special electronic units for these purposes. Obviously, such units should be manufactured and supplied together with the model.

Together with staff members of the Department of Mathematical and Computing Instruments and Devices of TPI, we developed one such unit [10], based on the ÉTsPV-1 electronic printing voltmeter. The use of the unit for automatic calculation of the amplitude of hammer vibration makes it possible to speed up the conduct of investigations considerably.

Conclusions

1. The solution of the system of differential equations describing the dynamics of pneumatic hammers on low-power modeling installations is possible when the method of sequential modeling of the individual parts of the system is used.

2. An expedient variant for modeling the system of equations describing the thermodynamic processes in the hammer and the motion of its parts is to divide the system according to the equation of motion of the piston, which must enter both the first and the second part of the problem being solved.

3. For automatic processing of the output signal of the model, it is advisable to use special electronic units, the manufacture of which should be carried out together with the model itself.

LITERATURE

1. O. D. Alimov. The interrelation of feed force with the basic parameters of a drilling hammer. Izvestiya TPI, vol. 108, Metallurgizdat, 1959.

2. G. A. Terekhov, A. D. Shkolnikov. Electronic modeling of the working cycle of pneumatic breaker hammers. Izvestiya vuzov, Gornyi zhurnal, No. 4, 1963.

3. E. V. Gerts, G. V. Kreinin. Theory and Calculation of Pneumatic Devices. Publishing House of the Academy of Sciences of the USSR, 1960.

4. V. M. Mostkov. Fundamentals of the Theory of Pneumatic Drilling. Ugletekhizdat, 1952.

5. V. F. Gorbunov. Investigation of working processes and vibration of pneumatic hammers. Doctoral dissertation, Tomsk, 1964.

6. A. P. German. The Use of Compressed Air in Mining. Gorno-geol.-neft. Publishing House, Moscow–Leningrad, 1933.

7. I. M. Tetelbaum. Electrical Modeling. Fizmatgiz, 1951.

8. A. V. Baranov, I. A. Ordanovich. Expansion of the capabilities of modeling devices. Report at the IV Interuniversity Conference on the Application of Physical and Mathematical Modeling, Collection No. 3, Moscow, 1962.

9. V. F. Gorbunov, A. V. Trikhanov, I. G. Smyshlyaeva. An electronic model for studying vibration processes in pneumatic hammers. Izv. TPI, vol. 141, 1965.

10. V. F. Gorbunov, V. M. Razin, N. I. Sablin, I. G. Smyshlyaeva, A. V. Trikhanov. A device for automatic calculation of vibration amplitude in the study of vibration characteristics of hand-held impact machines on electronic modeling installations. Abstracts of a report at the XX All-Union Scientific Session devoted to Radio Day, Moscow, 1964.