On the Theory of Calculation of Electropneumatic Impact Machines

P. M. Alabuzhev

This article considers certain questions in the theory and calculation of electropneumatic impact machines, using as an example a hammer with a one-way air connection (Fig. 1).

The operating process may be characterized by the graphs presented in Figs. 2, 3, 4, and 5. In Fig. 2, over the period of a cycle \(T\), graphs are shown of the variation of the angular velocity \(\omega\), the piston travel \(s_p\), the striker travel \(s_b\), the striker velocity \(v_b\), the air pressure \(p\) between the striker and the piston, and the value of the striker energy \(E\); Fig. 3 shows the energy balance of an electropneumatic hammer in one of the best regimes during its testing on a stand [5]. The characteristics of the air connection are shown in Fig. 5. Fig. 4a shows the variation, over a cycle, of the striker velocity as a function of the striker travel. The so-called limiting cycle in the phase plane \(s_b, v_b\) is established after 4--5 blows of the striker. The limiting cycle in dimensionless coordinates \(\dfrac{v_0}{R\omega}\) and \(\dfrac{s_b}{l_b}\) is shown in Fig. 4b.

Each point of the phase plane corresponds to a definite state of the oscillating system. The “phase portrait” of the behavior of the system makes it possible at once to encompass all possible states of the system [7].

The dependence \(v_b = f(s_b)\) determines, in the phase plane, all the features of the motion of the system, since on the basis of the limiting cycle one can obtain:

a) the character of the transient and steady-state regime of the machine;

b) the period of the cycle of established operation;

c) the coefficient of restitution in impact
\[ e=\frac{v_{ot}}{v_{yd}}; \]

d) the air pressure
\[ p=p_0-\frac{v_b}{k}\operatorname{tg}\psi, \]

where
\[ k=\frac{\pi D^2}{4}\cdot\frac{g}{Q_b}. \]

It is also of interest to study the dependence \(v_p=f(s_p)\) both during idling of the machine and during working operation. The curves \(v_b=f(s_b)\) and \(v_p=f(s_p)\) are most simply obtained experimentally, through the use of electrical sensors and differentiating circuits.

That pneumatic rebound hammers are self-oscillating systems is stated in [9].

The functional diagram of the system: feeder--hammer--rock is shown in Fig. 6.

From the point of view of self-oscillating systems, electropneumatic impact machines (in contrast to pneumatic machines) have the following features:

1) these machines are energy converters with pronounced elements of self-oscillation (the presence of a falling characteristic in the curve \(p=f(v_b)\), a limiting cycle, and independence of the working cycle from the initial conditions). But in most designs the frequency of oscillation of the striker is determined externally, since electropneumatic hammers have a comparatively powerful system of rotating parts, with respect to which the oscillatory motion of the striker has a subordinate significance;

Fig. 1. Data:
\(R=36\) cm; \(d=5.8\) cm; \(l=76\) cm; \(Q_b=10\) kg; \(z=1200\) blows/min;
\(S_b+l=S_n+l_0\).

Fig. 2. Pressure, displacement, velocity, and energy diagrams of the working cycle.

Fig. 3. Energy balance.
Visible labels include: “input energy”; “supplied energy”; “losses of energy for air leakage”; “compression”; “at rebound of striker”; “for tool displacement”; “impact energy.”

Fig. 4. Characteristics of the air cushion:
(a) approximately linear characteristic;
(b) region with pressure drop;
(c) limiting cycle;
(d) triangular characteristic with “compression” and “expansion.”

Fig. 5. Functional diagram of the system.
Visible labels include: “power source”; “transducer”; “feedback”; “compressor”; “actuating mechanism”; “class 2”; “tool”; “energy absorber (threshold)”; “model of the hammer—oscillatory system.”

Fig. 6. Functional diagram of the system.
Visible labels include: “power source”; “transducer”; “feedback”; “actuating mechanism”; “class 2”; “tool”; “energy absorber (threshold)”; “model of the hammer—oscillatory system.”

2) the feedback (the action of the vacuum, compensation for air leakage, rebound of the striker) is predominantly of the reactive type;

3) valve 1 is “open” when the length of the air cushion has the value \(l=l_0\); valve 2 is “open” at the moment of impact;

4) the system contains a rigid excitation, since the hammer is put into operation (or stopped) by means of a switch.

Assuming the angular velocity of the electric motor to be constant, and neglecting frictional resistances when the hammer is in the horizontal position, we can

write the equation of motion of the striker for an impact-action machine with a one-sided air connection in the following form:

\[ \frac{Q_{\sigma}}{g}\cdot \frac{dv_{\sigma}}{dt} = -\frac{\pi D^{2}}{4}(p-p_{0})+F_{y}, \]

where \(F_{y}\) is the reaction force of the tool, acting only at the moment of impact.

Applying the energy method [7] to the solution, one can determine the maximum stroke of the striker

\[ s_{\sigma \max} = \frac{1}{p_{0}}\cdot \int p_{1}\,ds_{\sigma} - \frac{q}{p_{0}}\cdot \frac{v_{om}^{2}}{2g}, \]

where \(p_{1}\) is the pressure read from the air-expansion curve, Fig. 5b.

Here \(q\) is the weight of the striker per unit area,

\[ q=\frac{4Q_{\sigma}}{\pi D^{2}}. \]

For the working stroke,

\[ s_{\sigma \max} = \frac{1}{p_{0}}\cdot \int p_{2}\,ds_{\sigma} - \frac{q}{p_{0}}\cdot \frac{v_{y0}^{2}}{2g}, \]

where \(p_{2}\) is the pressure read from the air-compression curve, Fig. 5b.

It is not difficult to show that, in the steady operating regime of the machine, the work \(A_{p}\) absorbed by the tool is equal to the energy of the striker \(A_{\sigma}\), accumulated by it from the air cushion during the cycle time \(T\), and

\[ A_{p}=A_{\sigma}=\frac{\pi D^{2}}{4}\cdot \int (p_{2}-p_{1})\,ds_{\sigma}. \]

Applying the impulse theorem to the solution of the equation of motion of the striker, and bearing in mind that the maximum stroke of the striker has one and the same value both in idle running and in the working stroke, it is not difficult to establish the following relation:

\[ e=\frac{v_{om}}{v_{00}}=\frac{t_{p}}{t_{x}}. \]

A more accurate value in the distribution of the cycle time \(T=t_{x}+t_{p}\) into the parts corresponding to the idle-stroke time \(t_{x}\) and the working-stroke time \(t_{p}\) can be obtained by taking into account the influence of the shape of the impulse on the motion of the striker mass [8].

Experimental data [3,5] confirm the existence of the noted relation

\[ e=\frac{t_{p}}{t_{x}}, \]

with an error of the order of \(\approx 10\%\). Hence the influence is clear of the material absorbing energy during impact on the working process of impact-action machines and on the distribution of the cycle time \(T\).

From the experimental investigations we conducted [5], it was established that, in the impact-load regime, the dependence of the air pressure \(p\) on the degree of compression

\[ \frac{l_{0}}{l} \]

is almost linear; moreover, the average value of the polytropic exponent over the cycle may be taken as \(m_{cp}=1.25\).

Linearizing the polytropic equation, Fig. 5a, we obtain:

\[ p=p_{0}\left[k\left(\frac{l_{0}}{l}-1\right)+1\right], \]

where \(k\) is determined, according to [6], from the condition

\[ \frac{d}{dk}\int_{y_1}^{y_2}\left[y^m-k(y-1)-1\right]^2\,dy=0. \]

Here: \(y=\dfrac{l_0}{l}\); for \(m=1.25\) and \(y=0.5\div5.0\); \(k=1.54\).

From Fig. 1 it is seen that \(s_{\delta}+l=l_0+s_{\mathrm{p}}\); for \(\lambda=\dfrac{R}{L}=0\), \(s_{\mathrm{p}}=R(1+\cos\varphi)\),

where \(\varphi=\omega t\);
then

\[ \frac{ds_{\delta}}{dt}+\frac{dl}{dt} = \frac{ds_{\mathrm{p}}}{dt} = -R\omega\sin\varphi; \]

\[ \frac{d^2s_{\delta}}{dt^2}+\frac{d^2l}{dt^2} = \frac{d^2s_{\mathrm{p}}}{dt^2} = -R\omega^2\cos\varphi. \]

The equation of motion of the striker will be:

\[ \frac{Q_{\delta}}{g}\cdot \frac{d^2s_{\delta}}{dt^2} = -\frac{\pi D^2}{4}(p-p_0) = -\frac{\pi D^2}{4}\cdot p_0 k\left(\frac{l_0}{l}-1\right). \]

Denoting

\[ \frac{\pi D^2}{4}\cdot \frac{gkp_0}{Q_{\delta}}=h, \]

we obtain:

\[ \frac{d^2l}{dt^2} - h\left(\frac{l_0}{l}-1\right) = -R\omega^2\cdot \cos\omega t \]

—a nonlinear and nonhomogeneous equation of the air coupling of the striker with the piston. Assuming the solution \(l=l_0+r\cos\nu\), where \(\nu=\omega t-\delta\) (here \(r\) and \(\delta\) are functions varying slowly with time), in the first approximation we shall have:

\[ \frac{dl}{dt}\simeq -r\omega\sin\nu \quad \text{and} \quad \frac{d^2l}{dt^2}\simeq -r\omega^2\cos\nu. \]

After substitution we obtain:

\[ -r\omega^2\cdot \cos\nu + h\cdot \frac{\cos\nu}{l_0+r\cos\nu} \cdot \frac{\sin\nu}{\sin\nu} = \]

\[ = -R\omega^2\cdot(\cos\nu\cos\delta-\sin\nu\sin\delta), \]

whence

\[ \sin\delta=\frac{hr\operatorname{ctg}\nu}{R\omega^2 l}; \qquad \cos\delta=\frac{r}{R}; \qquad \operatorname{tg}\delta=\frac{h\cdot \operatorname{ctg}\nu}{\omega^2\cdot l}. \]

From the condition \(\sin^2\nu+\cos^2\nu=1\), we obtain

\[ r^4+ar^3+br^2+cr+d=0, \]

where

\[ a=\frac{2l_0}{\cos\nu}; \qquad b=\frac{l^2}{\cos\nu} + \frac{h^2}{\omega^4\sin^2\nu} - R^2; \]

\[ c=\frac{2R^2\cdot l_0}{\cos\nu}; \qquad d=-\frac{l_0^2\cdot R^2}{\cos^2\nu}. \]

Procedure for solving the problem by the algebraic method:

1) for known parameters \(R, l_0, Q_b, k, D, \omega\), we assign \(v\) and find the coefficients \(a, b, c, d\);

2) solving the fourth-degree equation, we determine \(r\);

3) we find the phase-shift angle, \(\delta\);

4) we determine the crank angle \(\varphi = \omega t = v + \delta\);

5) we compute the piston travel \(s_p = R(1 + \cos \varphi)\);

6) we determine the striker travel \(s_b = l_0 + s_p - l\);

7) we find, for the given position of the mechanism, the air pressure

\[ p = p_0 \left[ k \left( \frac{l_0}{l} - 1 \right) + 1 \right]. \]

Note: \(\varphi_{ud} = v + \delta = \pi\) and \(s_b = s_p = 0\), if \(\operatorname{tg}^2 v = \dfrac{h}{\omega^2 l_0} = 1\), that is, the complex of quantities \(\dfrac{\pi D^2 g k p_0}{4 Q_b \omega^2 l_0}\) is equal to unity.

For small values of \(l_0, Q_b\), and \(\omega\), we have oscillations of the striker on an elastic (air) coupling under the action of the air cushion, perturbed by the piston. There may also be no impact of the striker on the tool if the excess mean air pressure in the air cushion over the hammer working cycle is equal to zero.

In [10] a numerical example of solving the problem by an algebraic method is given. Despite the cumbersome nature of the solution and the limited scope of application, this method opens up the possibility of an analytical solution of the problem, of selecting optimal parameters for an impact machine, and of investigating the stability of the motion.

In [1,4] we have shown the possibility of applying numerical and graphical methods of calculation to the investigation of the working process of a hammer.

In conclusion, let us show the possibility of selecting the parameters of an impact machine on the basis of the theory of similitude [2].

Let the well-working machine be a hammer with the parameters: \(R = 39\) mm, \(l_0 = 58\) mm, \(Q = 1.65\) kg, \(n = z = 1400\) strokes/min, \(N = 4.95\) kW, \(D = 68\) mm, and \(A_b = 7\) kg m. Suppose that we are interested in a new machine with the number of blows \(Z_1 = 1100\) strokes/min, with striker weight \(Q_1 = 3\) kg and work on the striker \(A_1 = 10\) kg m.

Then, on the basis of the theory of dynamic similitude, we determine the scale factors: the ratio of the times of the processes taking place

\[ t_c = \frac{t}{t_1} = \frac{z_1}{z} = \frac{1100}{1400} = 0.785; \]

the ratio of masses

\[ m_c = \frac{m}{m_1} = \frac{Q}{Q_1} = \frac{1.65}{3} = 0.55; \]

the ratio of works

\[ A_c = \frac{A}{A_1} = \frac{7}{10} = m_c \cdot v_c^2, \]

Then the linear scale is

\[ l_c = v_c t_c = 1.13 \cdot 0.785 = 0.885. \]

Since \(l_c = \dfrac{l}{l_1} = \dfrac{R}{R_1} = \dfrac{D}{D_1}\), the new length of the air cushion will be

\[ l_1=\frac{l}{l_c}=\frac{58}{0.885}=65.5\ \text{mm}, \]
the new crank radius
\[ R_1=\frac{R}{l_c}=\frac{39}{0.885}=44.3\ \text{mm}; \]
the new diameter
\[ D_1=\frac{D}{l_c}=\frac{68}{0.885}=77\ \text{mm}. \]

Power ratio
\[ N_c=\frac{N}{N_1}=m_c\,\frac{l_c^2}{t_c^3} =0.55\,\frac{0.885^2}{0.785^3}=0.89. \]

Whence the power of the new machine is
\[ N_1=\frac{N}{N_c}=\frac{4.95}{0.89}=5.55\ \text{kW}. \]

For electropneumatic machines with a one-sided air connection, the similarity criterion is the expression, valid under identical operating conditions of the machine (rebound of the striker, air compensation, clearances and lubrication),

\[ \Pi=\frac{p_c g D^2}{Q\omega^2 l_0}=\textit{idem}\ \text{(one and the same)}\ [2]. \]

In the present case
\[ \Pi=\frac{1\cdot 980\cdot 6.8^2}{1.65\cdot 146.5^2\cdot 5.8} = \frac{1\cdot 980\cdot 7.7^2}{3\cdot 116^2\cdot 6.55} =0.22. \]

If, for the machine under consideration, we change only the number of revolutions, assuming the harmful resistances to be as before, that is, if we take a new regime in time while preserving the linear dimensions as before, then \(l_c=1\); \(m_c=1\).

Force ratio
\[ F_c=m_c\,\frac{l_c}{t_c^2}=1\cdot \frac{1}{t_c^2}=n_c^2. \]

Work ratio
\[ A_c=F_c\cdot l_c=n_c^2. \]

Power ratio
\[ N_c=m_c\cdot \frac{l_c^2}{t_c^3}=1\cdot \frac{1}{t_c^3}=n_c^3. \]

Let, for example, \(n_1=z_1=700\ \text{rpm}\); then
\[ n_c=\frac{n}{n_1}=\frac{1400}{700}=2;\qquad F_c=n_c^2=2^2=4; \]

\[ A_c=\frac{A}{A_1}=n_c^2=2^2=4 \quad \text{and} \quad A_1=\frac{A}{A_c}=\frac{7}{4}=1.75\ \text{kgm}. \]

\[ N_c=\frac{N}{N_1}=n_c^3=2^3 \]
and the power of the new machine is
\[ N_1=\frac{N}{2^3}=\frac{4.95}{8}=0.64\ \text{kW}. \]

From the equation of motion of the crankshaft [2]

\[ I\varepsilon=M_k+M_p-\frac{\omega^2}{2}\cdot \frac{dI}{d\varphi} \]

it follows that between the coefficients for the moments of forces, \(-M\), acting in similar systems, the following relations hold:

\[ M_c=\frac{M}{M_1}=I_c\varepsilon_c=m_c\frac{l_c^2}{t_c^2}=p_{oc}l_c^3=\omega_c^2\frac{Q_c}{g_c}\,l_c^2=\omega_c^2 J_c, \]

which may be used to determine the characteristics of similar electric motors operating under impact-load conditions.

REFERENCES

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

2. Alabuzhev, P. M. Application of the theory of similarity and dimensions to the investigation (modeling) of impact-action machines. Izvestiya TPI, vol. 73, 1952.

3. Alabuzhev, P. M., Medlina, E. K., Novostruev, E. N. On a method for investigating indicator diagrams of electropneumatic impact-action machines. Izvestiya TPI, vol. 76, 1953.

4. Alabuzhev, P. M. Application of a grapho-analytical calculation method to the investigation of the working process of an electropneumatic hammer. Izvestiya TPI, vol. 75, 1954.

5. Alabuzhev, P. M. Some results of an experimental investigation of electropneumatic impact-action machines. Izvestiya TPI, vol. 78, 1954.

6. Panovko, Ya. G. The action of periodic impacts on a nonlinear elastic system with one degree of freedom. Proceedings of the Institute of Physics, vol. V, Academy of Sciences of the USSR, Riga.

7. Strelkov, S. B. Introduction to the Theory of Oscillations. GITTL, Moscow–Leningrad, 1950.

8. Sudnishnikov, B. V. Some problems in the theory of impact-action machines. ZSFAN, Novosibirsk, 1949.

9. Kharkevich, A. A. Self-Oscillations. GITTL, Moscow, 1953.

10. Alabuzhev, P. M. Investigation of the working process of electropneumatic impact-action machines. (Dissertation), TPI, 1954.