Application of the Graphical-Analytical Calculation Method to the Study of the Working Process of Electropneumatic Impact Machines¹

P. M. ALABUZHEV

The working process of an electropneumatic hammer is described by two nonlinear differential equations of the second order with variable coefficients: by the equation of motion of the striker

\[ \frac{Q_b}{g}\cdot \frac{d^2 x}{dt^2} = \frac{\pi D^2}{4}\cdot (p-p_0), \tag{1} \]

and by the equation of motion of the electric motor

\[ d\left(J\frac{\omega^2}{2}\right)=(M_k+M_p)\cdot d\varphi=dE, \tag{2} \]

where the following notation is used:

\(Q_b\) — weight of the striker;
\(g\) — acceleration of free fall;
\(x\) — displacement of the striker;
\(D\) — diameter of the hammer cylinder;
\(p\) — air pressure between the striker and the piston, which is assumed to vary according to the law \(p=p_0\left(\dfrac{l_0}{l}\right)^m\), where \(m\) is the polytropic exponent, and \(l\) is the distance between the striker and the piston,

\[ J=J_0+m_n r^2\left(\sin\varphi+\frac{\lambda}{2}\cdot \sin 2\varphi\right) \]

— the value of the moment of inertia reduced to the crankshaft of radius \(r\) (here \(J_0=\mathrm{const}\) and \(\lambda=\dfrac{r}{L}\), where \(L\) is the length of the connecting rod);

\[ \omega=\frac{d\varphi}{dt} \]

— angular velocity of the crankshaft;

\[ M_k=f(\omega) \]

— the specified graphical characteristic of the electric motor;

\[ M_p=(p-p_0)\cdot r\cdot \frac{\pi D^2}{4} \left(\sin\varphi+\frac{\lambda}{2}\cdot \sin 2\varphi\right) \]

— the moment of the air-pressure forces (\(M_p\) may have a positive value, a negative value, or zero);

\(E\) — kinetic energy of the system: electric motor, reducer, crank-and-connecting-rod mechanism, piston.

Equations (1) and (2) were compiled under the assumption that the position of the hammer is horizontal, the quantity of air between the striker and the piston is constant, and the resistance forces are negligible.

The solution of equations (1) and (2) is given in [1] by a numerical-integration method, which makes it possible to carry out the calculation with any prescribed degree of accuracy. Among the shortcomings of this method one should include its considerable cumbersomeness and the need to spend a comparatively large amount of time on carrying out the calculation; in addition, it is necessary to construct a series of graphs giving the designer an idea of the working process of the machine, since “the regularity of any process is best revealed in a graphical representation” [5, 133].

An experimental investigation of the working process of an electropneumatic hammer, carried out by us [2], confirmed the correctness of the conclusions drawn in [1] and gave a clearer picture of the operation of the air cushion (in particular, the average value of the polytropic exponent over the cycle was obtained experimentally: \(m_{cp}=1.23\), and the efficiency of the air cushion is of the order of \(\simeq 86\%\)).

An elementary graphical solution of certain dynamic problems as applied to the calculation of electric hammers was given in [9] for the case when \(\omega=\mathrm{const}\). Works [10; 7; 4; 3] to a known extent gave an impetus to, and directed attention toward, the application of the graphical-analytical calculation method to the study of the working process of impact machines. Unfortunately, the graphical-analytical methods developed in [3, 4] cannot be applied to the solution of our problem, since the variation of the air pressure

\[ p=p_0\left(\frac{l_0}{l}\right)^m \]

with time is not known in advance. An attempt to prescribe the value of the functional relation \(p=p(t)\) leads to verification of a number of preliminary assumptions that are not always feasible in actual implementation. The form of the function \(p=p(t)\) depends on the type of hammer design, its parameters, and its operating regime, and is found in the course of solving the specific problem.

In [3; 4] a very important question is considered concerning the error of the graphical-analytical method, and it is shown that the step error is of order \((\Delta t)^3\), while the resulting error is of order \((\Delta t)^2\), where \(\Delta t\) is the integration step (in our case the integration step with respect to the angle of rotation of the crank is denoted by \(\Delta\varphi\)). In [3] it is shown that, for one and the same integration step, the step error by the method of mean ordinates is the smallest in comparison with the methods of the tangent and the secant¹.

In the calculation we apply the method of mean values, which is essentially the method of numerical solution proposed in [10]. “Taking the intervals \(\Delta t\) sufficiently small and performing the computations for each value of \(t\) twice, in order to obtain the second degree of approximation, we can by this method of numerical integration always obtain an accuracy sufficient for practical applications” [10, 85].

At present there are no methods for obtaining a general solution of equations of the type (1) and (2); therefore it is first necessary to select the parameters of the impact machine. The initial parameters (depending on the purpose of the machine) primarily include the impact energy \(A_b\), the number of blows \(z\), the feed force \(P\), and the angle \(\alpha\) of rotation of the drill between blows of the striker. From these initial data, in the first approximation, the parameters of the impact unit \((l_0, D, Q_b, r)\) are selected. Knowing the motor power and its characteristic \(M_k=f(\omega)\), one can, by the graphical-analytical method, carry out a refining

calculation taking into account the interaction of the striker and the electric motor over the machine’s operating cycle.

Given a well-thought-out arrangement of the material, auxiliary tables (graphs), and templates, one can rather quickly obtain an idea of the working process of the machine and its performance indicators. The advantages of the graphical-analytical method include:

1) the clarity of the solution; in effect, alongside the solution one immediately obtains an image (graph) of the change in the working process of the machine and of its characteristics;

2) the presence of good control, since calculation errors are immediately revealed in the course of the work;

3) the effectiveness of the method with an accuracy sufficient for practice; application of the method is not limited by the number of terms of the equation or by the nature of the variation of individual quantities during the machine’s operating cycle (the possibility of taking into account the variation of the angular velocity \(\omega\), the polytropic exponent \(m\), air leakage, the phenomena of impact and rebound of the striker, etc.);

4) the relative ease of changing the integration step depending on the nature of the variation of the quantities \((p, \omega, x)\);

5) carrying out the calculation by the graphical-analytical method does not require a highly qualified operator and is accessible to a wide circle of engineering and technical workers and inventors.

Let us explain the procedure for carrying out the calculation using as an example the hammer of the EZRT scheme, known in the literature and described in sufficient detail in works [1; 2]. The calculation method set forth below has also been used by us to study other designs of hammers and perforators.

First of all, on one sheet of millimeter paper we construct graphs of the following functions:

1) the piston-displacement graph

\[ s = r \cdot \left[\left(1 + \frac{\lambda}{4}\right) - \left(\cos \varphi + \frac{\lambda}{4}\cos 2\varphi\right)\right] \]

and the graph of piston velocity

\[ v_n = \frac{ds}{dt} = r \cdot \omega \cdot \left(\sin \varphi + \frac{\lambda}{2}\sin 2\varphi\right) \]

as a function of the crankshaft rotation angle \(\varphi\);

2) the value of the moment of inertia \(J\);

3) the coefficient

\[ K = \frac{\pi D^2}{4} \cdot r \cdot \left(\sin \varphi + \frac{\lambda}{2}\cdot \sin 2\varphi\right); \]

4) the characteristic of the electric motor \(M_k = f(\omega)\);

5) the graph of the change in air pressure (polytrope)

\[ p = p_0 \left(\frac{l_0}{l}\right)^m . \]

The approximate arrangement of the graphs and the calculation scheme is shown in Fig. 1. In the lower left-hand corner of the calculation sheet we place the hammer diagram and the necessary formulas. The integration step \(\Delta\varphi = \varphi_2 - \varphi_1\) is chosen in accordance with the nature of the variation of the quantities that determine the calculation (on the order of \(\Delta\varphi \simeq 5^\circ \div 15^\circ\)).

Let us show how, from the crankshaft rotation angle \(\varphi_1\), to which correspond the values of the quantities \(p_1, M_1, E_1, \omega_1, x_1\), one passes to the angle \(\varphi_2\), finding the values of the same quantities—\(p_2, M_2, E_2, \omega_2, x_2\)—while already having constructed graphs of \(s, v_n, J, K, M_k, p\).

\[ \left(\frac{p_0}{\ell_1}\right) \]

1. \[ p_1=p_0\left(\frac{\ell_0}{\ell_1}\right)^m \]

2. \[ K_1=\frac{\pi D^2}{4}R\left(\sin\psi_1+\frac{\lambda}{2}\sin 2\psi_1\right) \]

3. \[ M_{1p}=(p_1-p_0)K_1 \]

4. \[ M_1=M_{1k}+M_{1p} \]

5. \[ \Delta E=M_1\Delta\psi \]

6. \[ E_2'=E_1+\Delta E_1 \]

7. \[ \omega_2=\sqrt{\frac{2E_2}{J_2}} \]

8. \[ \omega_{\mathrm{cp}}=\frac{\omega_1+\omega_2}{2} \]

9. \[ \Delta U_b'= \frac{\pi D^2(p_1-p_0)\Delta\psi}{4Q\,\gamma/g\,\omega_{\mathrm{cp}}} \]

10. \[ U_{2b}'=U_{1b}+\Delta U_b' \]

11. \[ \Delta X'=\left(U_{1b}+\frac{\Delta U_b'}{2}\right)\frac{\Delta\psi}{\omega_{\mathrm{cp}}} \]

12. \[ X_2=X_1+\Delta X' \]

13. \[ \ell_2=\ell_0+S_2-X_2 \]

14. \[ p_2'=p_0\left(\frac{\ell_0}{\ell_2}\right)^m \]

\[ \Delta\psi=15^\circ=0.262 \]

\[ \Delta\psi=10^\circ=0.175 \]

\[ \Delta\psi=5^\circ=0.087 \]

\[ S=R\left[\left(1+\frac{\lambda}{4}\right)-\left(\cos\psi+\frac{\lambda}{4}\cos2\psi\right)\right] \]

\[ U_n=R\omega\left(\sin\psi+\frac{\lambda}{4}\sin2\psi\right) \]

\[ \ell_0+S=\ell+x \]

Fig. 1

Calculation procedure:

a) first approximation

(method of tangents)

\[ \text{1) }\quad p_1=p_0\left(\frac{l_0}{l_1}\right)^m \]

\[ \text{2) }\quad K_1=\frac{\pi D^2}{4}\cdot r\left(\sin\varphi_1+\frac{\lambda}{2}\sin2\varphi_1\right) \]

\[ \text{3) }\quad M_{1p}=(p_1-p_0)\cdot K_1 \]

\[ \text{4) }\quad M_1=M_{1k}+M_{1p} \]

\[ \text{5) }\quad \Delta E_1=M_1\cdot\Delta\varphi \]

\[ \text{6) }\quad E^1_2=E_1+\Delta E \]

\[ \text{7) }\quad \omega^1_2=\sqrt{\frac{2E^1_2}{J_2}} \]

\[ \text{8) }\quad \omega^1_c=\frac{\omega_1+\omega^1_2}{2} \]

\[ \text{9) }\quad \Delta v^1_b=\frac{\pi D^2}{4}\cdot\frac{p_1-p_0}{Q_b/g}\cdot\frac{\Delta\varphi}{\omega^1_c} \]

\[ \text{10) }\quad v^1_b=v_{1b}+\Delta v^1_b \]

\[ \text{11) }\quad \Delta^1 x=\left(v_{1b}+\frac{\Delta v^1_b}{2}\right)\cdot\frac{\Delta\varphi}{\omega^1_c} \]

\[ \text{12) }\quad x^1_2=x_1+\Delta x^1 \]

\[ \text{13) }\quad l^1_2=l_0+s_2-x^1_2 \]

\[ \text{14) }\quad p^1_2=p_0\left(\frac{l_0}{l^1_2}\right)^m \]

b) second approximation

(the subscript “c” denotes mean values of the quantities in the interval)

\[ \text{15) }\quad p_c=\frac{p_1+p^1_2}{2} \]

\[ \text{16) }\quad K_c=\frac{\pi D^2}{4}\cdot r\left(\sin\varphi_c+\frac{\lambda}{2}\sin2\varphi_c\right) \]

\[ \text{17) }\quad (M_c)_p=(p_c-p_0)\cdot K_c \]

\[ \text{18) }\quad M_c=(M_c)_p+(M_c)_k \]

\[ \text{19) }\quad \Delta E_c=M_c\cdot\Delta\varphi \]

\[ \text{20) }\quad E_2=E_1+\Delta E_c \]

\[ \text{21) }\quad \omega_2=\sqrt{\frac{2E_2}{J_2}} \]

\[ \text{22) }\quad \omega_c=\frac{\omega_1+\omega_2}{2} \]

\[ \text{23) }\quad (\Delta v_b)_c=\frac{\pi D^2}{4}\cdot\frac{p^1_2-p_0}{Q_b/g}\cdot\frac{\Delta\varphi}{\omega_c} \]

\[ \text{24) }\quad v_{2b}=v_{1b}+(\Delta v_b)_c \]

\[ \text{25) }\quad \Delta x=\left(v_{1b}+\frac{\Delta v_{bc}}{2}\right)\cdot\frac{\Delta\varphi}{\omega_c} \]

\[ \text{26) }\quad x_2=x_1+\Delta x \]

\[ \text{27) }\quad l_2=l_0+s_2-x_2 \]

\[ \text{28) }\quad p_2=p_0\left(\frac{l_0}{l_2}\right)^m. \]

The entire calculation is performed with the aid of a slide rule, a compass, and a pencil, with simultaneous plotting of the corresponding values on the graph. A comparison of the calculation performed by the graphical-analytical method with more accurate methods of numerical integration (for example, the Adams-Störmer method developed in [8]) shows that the resulting error for individual indices, with the same step \(\Delta\varphi=15^\circ\), lies within the range of \(2\)–\(6\%\); this error can be reduced by decreasing the step \((\Delta\varphi=5\div10^\circ)\), by using a slide rule \(50\ \text{cm}\) long, and also by increasing the scale of the graphical construction.

The following characteristic points may serve as a good check on the graphical-analytical method during the work:

1) when \(l=l_0\), then \(p=p_0\) (rarefaction is replaced by compression);

2) when \(x=x_{\max}\), in this case \(v_b=0\); the pressure curve \(p\) is divided into two parts with impulses corresponding to: a) the working stroke of the striker, over the time \(t_p\), and equal to

\[ S_p=\frac{Q_b}{g}\cdot v_{ud}, \]

b) the idle stroke of the striker, over the time \(t_x\), and equal to

\[ S_x=\frac{Q_b}{g}\cdot v_{ot}, \]

where the ratio

\[ \frac{S_x}{S_p}=\frac{v_{ot}}{v_{ud}}=k \]

may be called

... by the coefficient of restitution in the impact and rebound of the striker from the tool (in the system of three bodies: striker—tool—material). The value of the coefficient \(k\) must be known in the calculation;

3) at \(p = p_{max}\), the length of the air connection \(l = l_{min}\);

4) furthermore, the extreme values of the air pressure \(p\) correspond to equality of the velocities of the striker and the piston. Indeed, from the polytropic equation

\[ \frac{dp}{dt}=-p_0m\left(\frac{l_0}{l}\right)^{m-1}\cdot\frac{\dot l}{l^2}\cdot\frac{dp}{dt}=0,\quad \text{if } \dot l=\frac{dl}{dt}=0; \]

but \(l=l_0+s-x\); consequently, the deformation rate of the air connection is \(\dot l=\dot S-\dot x=v_n-v_b=0\), if \(v_b=v_n\).

From the value of the velocity at impact \(v_{ud}\) and the weight of the striker \(Q_b\), we determine and check the value of the impact energy

\[ A_b=\frac{Q_b\cdot v_{ud}^2}{2g}=\frac{S_p\cdot v_{ud}}{2}, \]

and also the mean air pressure over the cycle

\[ p_{sr}=\frac{S_p+S_x}{t_p+t_x} =\frac{Q_b(v_{ot}+v_{ud})}{g\cdot T} =\frac{Q_b\cdot z\cdot v_{ud}\cdot(1+k)}{60\cdot g}. \]

On the basis of the pressure diagram \(p=p(\varphi)\), it is not difficult to perform the force calculation of the machine, and also to make a verification calculation of the flywheel masses in the presence of a periodically recurring short-duration peak load and of the electric-motor power by the methods set forth in the relevant literature [6, 7].

By applying the graphical-analytical calculation method in combination with experimental research methods, it is possible to study more deeply and fully questions of the dynamics of impact machines (the influence of the choice of parameters on the working process of the machine, questions of the transient regime; the influence of air leaks and of clearances between the striker and the cylinder on the working process, etc.).

Knowing the distribution of losses in the individual links of the machine, it is possible to estimate the overall efficiency of the machine.

REFERENCES

1. Alabuzhev P. M. Study of the working process of an electric chipping hammer with an elastic (air) connection. Izvestiya TPI, No. 61, issue 1, 1947.

2. Alabuzhev P. M. and Yudin I. P. Experimental study of an electropneumatic hammer. Trudy ZSPhAN, issue 8, 1950.

3. Bashkirov D. A. Graphical-analytical method for constructing transient processes in automatic-control systems. LKVIA, Moscow, 1952.

4. Bey-Su-Si et al. Graphical analysis of nonlinear systems. Journal Applied Mathematics and Machine Building, No. 6, Foreign Literature Publishing House, Moscow, 1952.

5. Goryachkin V. P. Approximate calculations and graphical representations. Theory, design, and manufacture of agricultural machines, vol. 1, Trudy VISKhOM, Selkhozgiz, Moscow–Leningrad, 1935.

6. Zinoviev V. A. Calculation of flywheel masses with a driving torque dependent on velocity. Proceedings of the Seminar on the Theory of Machines and Mechanisms, vol. XII, issue 46, Publishing House of the Academy of Sciences of the USSR, Moscow, 1952.

7. Iroshnikov A. N. On a graphical method for the dynamic investigation of a multilink mechanism when the torque varies as a function of velocity. Journal Vestnik Mashinostroeniya, No. 8, 1949.

8. Krylov A. N. Lectures on approximate calculations. Collected Works, vol. III, Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1949.

9. Sudnishnikov B. V. Some questions in the theory of impact machines. ZSPhAN, 1949.

10. Timoshenko S. P. Theory of vibrations in engineering. GITL, Moscow–Leningrad, 1931.

1. The method of mean ordinates, for the same \(\Delta t\), is twice as accurate as the secant method, and the tangent methods are 12 times as accurate [3]. ↩