Electronic Model for Studying Vibration Processes in Pneumatic Hammers
Abstract Generated abstract
This paper presents an electronic analog model for investigating vibrational processes in pneumatic hammers, based on previously derived differential equations for the coupled motion of the tool, striker, and housing. It describes the conversion of physical variables into machine voltages, gives representative scale factors and coefficients for the KE-22 pneumatic chipping and riveting hammer, and outlines the block diagram implementing nonlinear air-pressure characteristics, impacts, contact forces, switching, and post-impact velocities. The model permits slowed-time simulation of rapid impact cycles and is intended to study vibration amplitudes and frequencies under varied operating conditions. The authors argue that such modeling can help identify optimal regimes, improve productivity, reduce harmful vibration exposure, and guide design variants for impact and vibrational machines.
Full Text
Electronic Model for Studying Vibrational Processes in Pneumatic Hammers
V. F. Gorbunov, V. M. Razin, I. G. Smyshlyaeva, A. V. Trikhanov
(Presented by the seminar of the Faculty of Automation and Computer Engineering)
In our previous articles [1, 2] on electronic modeling of processes of vibration isolation and vibration damping, the initial assumptions for modeling, the computational scheme, and the mathematical description of a pneumatic hammer were given.
To solve the system of differential equations indicated in [2] on mathematical modeling installations, it is necessary to replace physical quantities by voltages, with the introduction of the corresponding scale factors. It is not difficult to show that the equations in machine variables, convenient for drawing up the block diagram of the model, will have the following form:
\[ \frac{d^2 U_i}{dt^2} = - B_1 U_{1(2)} - V_1 \frac{dU_i}{dt} + \xi_k D_1 (U_k - U_i) - \xi_m G_1 U_i; \tag{1} \]
\[ \frac{d^2 U_b}{dt^2} = - A_2 U_{3(4)} + B_2 U_{1(2)} - V_2 \frac{dU_b}{dt} - \xi_{bk} G_2 (U_b A - U_k); \tag{2} \]
\[ \frac{d^2 U_k}{dt^2} = A_3 U_{3(4)} - B_3 U_{1(2)} - T_3 U_q - V_3 \frac{dU_b}{dt} - \xi_k D_3 (U_k - U_i) + \]
\[ \qquad\qquad \xi_{bk} G_3 (A U_b - U_k), \tag{3} \]
where
\[ \xi_k = \begin{cases} 0, & U_k > U_i;\\ 1, & U_k < U_i; \end{cases} \qquad \xi_m = \begin{cases} 0, & U_i > 0;\\ 1, & U_i < 0; \end{cases} \]
\[ \xi_{bk} = \begin{cases} 0, & A U_b > U_k;\\ 1, & A U_b < U_k. \end{cases} \]
\(A_1, A_2, A_3, A, B_1, B_2, B_3, V_1, V_2, V_3, G_1, G_2, G_3, D_1, D_3\), and \(T_q\) are machine coefficients, the values of which for studying, for example, the KE-22 pneumatic chipping-and-riveting hammer are given in Table 1.
The block diagram for connecting the blocks is shown in Fig. 1. Nonlinearity blocks BN1, BA2, computing blocks 1, 2, 3, 4, 5, 6, 7, 10, 13, and switching device PU1 are used to solve the differential equation of motion of the tool.
BN1 specifies the nonlinear dependence of the change in air pressure in the front cavity of the hammer during the forward stroke of the piston; BN2, during the return stroke.
Table 1
| Name | Mathematical expression | Dimension | Values |
|---|---|---|---|
| Scale of striker displacement | $a_{\mathrm{b}}$ | m/V | $1.61\cdot 10^{-3}$ |
| Scale of displacement of the body and tool | $a_{\mathrm{k}}, a_{\mathrm{i}}$ | m/V | $2.15\cdot 10^{-4}$ |
| Time scale | $a_t$ | $4.34\cdot 10^{-3}$ | |
| Scale of axial force | $a_q$ | kg/V | 1 |
| Scale of disturbing force | $a_f$ | kg/V | 1 |
| Mass of the hammer body | $m_{\mathrm{k}}$ | $\dfrac{\mathrm{kg}\cdot \mathrm{sec}^2}{\mathrm{m}}$ | 0.894 |
| Mass of the striker | $m_{\mathrm{b}}$ | " | 0.056 |
| Mass of the tool | $m_{\mathrm{i}}$ | " | 0.094 |
| Coefficients of equations 1, 2, 3 | $B_1=\eta\dfrac{a_t^2 a_f}{m_{\mathrm{i}}a_{\mathrm{i}}}$ | 0.771 | |
| $B_1=\dfrac{a_t\cdot a_f\cdot r_{\mathrm{i}}}{m_{\mathrm{i}}}$ | m/sec | 0.83 | |
| $D_1=\dfrac{C_{\mathrm{k}}a_t^2}{m_{\mathrm{i}}}$ | 1/sec | 59.5 | |
| $\Gamma_1=\dfrac{C_{\mathrm{i}}a_t^2}{m_{\mathrm{i}}}$ | 1/sec | 59.5 | |
| $E_1=K_{\mathrm{ii}}$ | — | 0 | |
| $F_1=K_{\mathrm{ib}}K_{\mathrm{i}}\dfrac{a_{\mathrm{b}}}{a_{\mathrm{i}}}$ | 0.217—0.876 | ||
| $A_2=\dfrac{a_t^2 a_f}{m_{\mathrm{b}}a_{\mathrm{b}}}$ | 0.173 | ||
| $B_2=A_2$ | — | 0.173 | |
| $B_2=\dfrac{r_{\mathrm{b}}a_t}{m_{\mathrm{b}}}$ | m/sec | 0.050 | |
| $\Gamma_2=\dfrac{C_{\mathrm{bk}}a_t^2}{m_{\mathrm{b}}}$ | 1/sec | 61.0 | |
| $E_2=K_{\mathrm{bi}}\dfrac{a_{\mathrm{i}}}{a_{\mathrm{b}}}$ | 0 | ||
| $F_2=K_{\mathrm{bb}}$ | 0.5—0.35 | ||
| $A_3=\dfrac{a_t^2 a_f}{m_{\mathrm{k}}a_{\mathrm{k}}}$ | 0.081 | ||
| $B_3=\dfrac{r_{\mathrm{k}}a_t}{m_{\mathrm{k}}}$ | m/sec | 0.240 | |
| $\Gamma_3=\dfrac{C_{\mathrm{bk}}a_t^2}{m_{\mathrm{k}}}$ | 1/sec | 6.25 | |
| $D_3=\dfrac{C_{\mathrm{k}}a_t^2}{m_{\mathrm{k}}}$ | " | 6.25 | |
| $T_3=\dfrac{a_t^2 a_q}{m_{\mathrm{k}}a_{\mathrm{k}}}$ | 0.097 | ||
| $E_3=K_{\mathrm{ki}}$ | 0.019 | ||
| $F_3=K_{\mathrm{kk}}$ | 0 |
The output of one or another BH unit is connected to the input of the operational amplifier depending on the position of the polarized relay R3, which is connected to the output of decision unit No. 16. The input of the latter is supplied with the velocity of the striker with a negative sign, \(-\dot U_b\). During the forward stroke of the piston, the middle contact of relay R3 is in the upper position. When the striker impacts the tool or the front wall of the housing, the sign of the striker velocity changes to the opposite. After this, the reverse stroke of the striker begins; the middle contact R3 assumes the lower position.
Fig. 1. Block diagram of the model.
Unit No. 4 realizes a positive impact force on the tool when the latter strikes the medium being worked. In other cases, at the output of this unit, owing to limitation at the level \(0\), the voltage is equal to zero.
Unit No. 5 realizes a negative impact force on the tool when the latter collides with the housing. When the housing moves away from the tool, a positive voltage is obtained at the output of this unit, which is limited at the zero level by means of a diode limiter.
The post-impact velocity of the tool when it is struck by the striker is set by means of units 2 and 6, which are included in the model of elastic impact [3]. When the condition for the indicated impact is fulfilled, relay R1 operates, switching...
of the switching device PU1. The triples of contacts of this relay, 1P1, 2P1, 3P1, 4P1, interchange the capacitors, one of which was connected into the negative-feedback circuit of the first computing block, at whose output the velocity of the tool is obtained with negative sign, while the other is at the output of the sixth computing block, where the after-impact velocity of the tool is reproduced.
When the indicated capacitors are interchanged, the voltage at the output of block 2 is instantaneously replaced by the voltage that had been established before switching at the output of block 6. This occurs each time the condition of impact of the striker on the tool is fulfilled.
To determine this condition \((x_b \geq x_i + l)\), blocks 7, 10, and 13 are used.
The circuit for solving the differential equation of motion of the striker includes the nonlinear blocks BN1, BN2, BN3, BN4, and the computing amplifiers 7, 8, 9, 10, 12, 13, 17.
Blocks BN3 and BN4 reproduce the air pressure in the rear cavity of the casing during the forward and return strokes of the striker, respectively. The outputs of these blocks are switched to the input of block 8 by means of the second triple of contacts of relay P3.
Assignment of the after-impact velocity for the striker when the latter strikes the tool is implemented by means of blocks 8 and 12, similarly to the way the after-impact velocity was assigned for the tool when struck by the striker.
The capacitors \(C_3\) and \(C_4\) in these blocks are interchanged by means of four triples of contacts 1P2, 2P2, 3P2, 4P2 of relay 2, located in the switching block PU1.
The positive impact force for the striker (and the casing) when it strikes the wall of the casing is implemented by means of block 17, which has diode limiting at the level 0 at its output.
The nonlinear blocks BN1, BN2, BN3, BN4—the last two when studying the KE22 hammer should not be used—and the computing amplifiers 5, 14, 15, 17, 19 are used to solve the differential equation of motion of the casing.
Blocks 14 and 19 serve to assign the after-impact velocity of the casing when it is struck by the tool after the latter rebounds from the rock being worked.
The capacitors \(C_5\) and \(C_6\) of these blocks are interchanged by means of four triples of contacts 1P4, 2P4, 3P4, 4P4 of relay P4, located in the switching device PU2.
This device switches relay P4 each time the voltage at the output of block 5 passes through zero in the negative direction; that is, when the condition of impact between the casing and the tool is fulfilled \((x_k \leq x_i)\).
Between the tool and the casing, besides impacts, prolonged contacts occur; this is modeled by means of block 17, at whose output, when \(x_k \leq x_i\), a positive voltage appears corresponding to the contact force.
When assigning the after-impact velocities of the striker and the casing, instead of the negative velocity of the tool it is necessary to use the positive velocity of the tool. For this purpose inverting amplifier 18 is used.
Block 11 is intended to feed to the input of the nonlinearity blocks the quantity \(-(A U_b - U_k)\).
Multiplication by constant coefficients, whose value varies in each experiment, is carried out by means of constant-coefficient blocks of the SB-4A type, indicated in the diagram by circles.
The initial condition \(x_6(0)=x_{6\max}\) was set in the circuit from the model control panel, while the quantities of interest at the output were recorded by means of an I-4M electronic indicator and a voltmeter located on the IPT-5 panel.
For the study of rapidly occurring processes, an important advantage of electronic simulation installations is the possibility of conducting investigations on a slowed time scale. The choice of the time scale is determined by the resolving power of the recording equipment or by the desire to observe visually the processes under study. For investigating machines of impact and vibrational action with a cycle frequency of 15–50 Hz, it is convenient to choose the time scale within the limits \(10^{-3}\)—\(10^{-4}\).
The proposed electronic model makes it possible to study the dynamics of hammers during metal cutting, the manufacture of blanks, the destruction of rocks and frozen soils, etc. Particularly important in such investigations is the study of the frequency and amplitude of vibrations of the hammer body, by which the quality of hammers is assessed from the sanitary-hygienic point of view [4].
Investigation of the vibration characteristics of machines of impact and vibrational action by the method of mathematical modeling makes it possible to determine their behavior under wide variation of the initial data (characteristics, operating conditions and regimes, etc.). On the basis of the modeling results, zones of optimal operating regimes of existing machines are determined, and reserves for increasing their productivity while simultaneously reducing the vibrational effect on the worker’s organism are revealed. In addition, modeling makes it possible to predict a number of expedient design variants of new machines of better quality.
This model was developed at the Department of Mathematical and Computing Instruments and Devices of the Tomsk Polytechnic Institute and was tested in work during the investigation of pneumatic chipping and riveting hammers of the Tomsk Electromechanical Plant and electromagnetic hammers of TPI.
REFERENCES
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V. I. Baburov, V. F. Gorbunov, V. M. Razin, I. G. Smyshlyaeva, A. V. Trikhanov. Initial prerequisites for electronic modeling of vibration isolation and vibration damping of pneumatic hammers. (To be printed in the present collection).
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V. F. Gorbunov, V. M. Razin, I. G. Smyshlyaeva, A. V. Trikhanov. Calculation scheme and mathematical description of a pneumatic hammer for electronic modeling. (To be printed in the present collection).
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V. F. Gorbunov, V. M. Razin, A. V. Trikhanov. Some investigations of pneumatic hammers with vibration-isolating devices on an electronic model. Mathematical Methods in Mining. Collection of reports. Mining Machines and Automation. Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1963.
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Collection of the Most Important Official Materials on Questions of Occupational Hygiene and Industrial Sanitation, I–II, Medgiz, 1962.