Calculation Scheme and Mathematical Description of a Pneumatic Hammer for Electronic Modeling
Abstract Generated abstract
The paper develops a computational scheme and mathematical description for electronic modeling of a pneumatic hammer. It represents the hammer as a three-mass nonlinear oscillatory system consisting of body, striker, and tool, with elastic impacts, viscous friction, material stiffness, pressing force, and time-varying air pressures in the front and rear cavities. The authors formulate differential equations with switching conditions for contact and impact, discuss the selection of post-impact velocity coefficients from experiments and literature, and propose representing pressure diagrams as functions of striker displacement relative to the housing. The resulting model is intended to support electronic simulation of hammer dynamics under varying working materials, air pressures, and operating conditions.
Full Text
COMPUTATIONAL SCHEME AND MATHEMATICAL DESCRIPTION OF A PNEUMATIC HAMMER FOR ELECTRONIC MODELING
V. I. Baburov, 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 order to become a real object of study, a process must have a sufficiently accurate mathematical description [1]. Such a description must be an abstract-formal characterization of the process, that is, its mathematical model. The development of a mathematical model is, as a rule, preceded by the construction of a computational scheme for the object under study.
In any theoretical investigation of a real physical system, we are always compelled, to a greater or lesser degree, to idealize the properties of the object being described. Some idealization of the problem is inevitable, since even if we were able to take into account a significant part, or even all, of the factors affecting the behavior of the system, the problem would become so complicated that its solution would be extremely difficult or altogether impossible. In this connection the question arises: to what extent may the properties of an object be idealized and still yield satisfactory results? The answer to this is ultimately given only by experience, by experiment. By comparing the data of an idealized consideration of a given question with experimental results, we can judge how legitimate a particular idealization is.
Fig. 1. Computational scheme of a pneumatic hammer.
In [2], the basic requirements for an electronic model of a pneumatic hammer were formulated. These requirements are more fully satisfied by the computational scheme of a pneumatic hammer presented in Fig. 1.
This computational scheme has been proposed on the basis of an analysis of experimental studies of pneumatic hammers carried out by the staff of the Department of Mining Machines of TPI, and of a number of published works [3, 4, 5, 6, 7]. Electromagnetic hammers with free striker rebound have an analogous computational scheme [8, 9].
In the diagram (Fig. 1) and in the subsequent exposition the following notation is adopted:
\(m_k, m_b, m_i\) — the masses, respectively, of the hammer body, striker, and tool;
\(p_z\) — air pressure in the rear cavity of the hammer;
\(p_p\) — air pressure in the front cavity of the hammer;
\(S_b\) — cross-sectional area of the striker;
\(Q\) — constant force acting on the body (the worker’s pressing force);
\(d\) — thickness of the front wall of the body;
\(l\) — length of the tool shank;
\(l_1\) — length of the tool without the shank;
\(C_{b.k}\) — spring simulating the elasticity of the material of the striker and the body during their collision;
\(C_k\) — spring simulating the elasticity of the material of the body and the tool during their contact;
\(C_m\) — spring simulating the elasticity of the material being worked;
\(x_k, x_b, x_i\) — displacement of the body, striker, and tool relative to the uncompressed spring \(C_m\);
\(m_y\) — mass of the absolutely inelastic limiter (\(m_y = 0\));
\(r_b, r_k, r_i\) — viscous friction of the striker, body, and tool;
\(\eta\) — ratio of the cross-sectional area of the tool shank to the cross-sectional area of the striker.
In what follows we assume that \(d = 0\).
In this case, instead of \(l\) one must consider \(l_2 - d\), which may be denoted by \(l\).
The introduced assumption is valid if oscillations in the tool and in the wall are absent.
Under the action of air pressure, all component parts of the hammer perform oscillatory motions, and elastic impacts occur between them.
The differential equations describing the operating dynamics of the three-mass nonlinear oscillatory system under consideration may be written in the form:
\[ m_i \frac{d^2 x_i}{dt^2} = -\eta S_b p_p - r_i \frac{dx_i}{dt} + \xi_k C_k (x_k - x_i) - \xi_m C_m x_i; \tag{1} \]
\[ m_b \frac{d^2 x_b}{dt^2} = - S_b p_z + S_b p_p - r_b \frac{dx_b}{dt} - \xi_{b.k} C_{b.k} (x_b - x_k); \tag{2} \]
\[ m_k \frac{d^2 x_k}{dt^2} = - Q - S_b p_p + S_b p_z - r_k \frac{dx_k}{dt} - \xi_k C_k (x_k - x_i) + \]
\[ + \xi_{b.k} C_{b.k} (x_b - x_k), \tag{3} \]
where
\[ \xi_m = \begin{cases} 0, & x_i > 0;\\ 1, & x_i < 0; \end{cases} \qquad \xi_k = \begin{cases} 0, & x_k > x_i;\\ 1, & x_k < x_i; \end{cases} \tag{4} \]
\[ \xi_{b.k} = \begin{cases} 0, & x_b \gg x_k;\\ 1, & x_b < x_k. \end{cases} \]
The specification of the post-impact velocities of the striker, the tool, and the body—at the instants of impact of the striker against the tool, of the tool against the striker, and of the tool against the body, respectively—is carried out by means of the expressions:
\[ \dot{x}_b = \kappa_{b.b}\dot{x}_{b.u} + \kappa_{b.i}\dot{x}_{i.u}; \tag{5} \]
\[ \dot{x}_i = \kappa_i\left(\kappa_{i.i}\dot{x}_{i.u} + \kappa_{i.b}\dot{x}_{b.u}\right); \tag{6} \]
\[ \dot{x}_k = \kappa_{k.k}\dot{x}_{k.u} + \kappa_{k.i}\dot{x}_{i.u}. \tag{7} \]
where the indicated coefficients take into account the following:
\(\kappa_{b.b}\) — the influence of the striker velocity before impact on its post-impact velocity;
\(\kappa_{b.i}\) — the influence of the tool velocity before impact on the post-impact velocity of the striker;
\(\kappa_{i.i}\) — the influence of the tool velocity before impact on its impact velocity;
\(\kappa_{i.b}\) — the influence of the striker velocity before impact on the post-impact velocity of the tool;
\(\kappa_{k.k}\) — the influence of the body velocity before impact on its post-impact velocity;
\(\kappa_{k.i}\) — the influence of the tool velocity before impact on the post-impact velocity of the body;
\(\kappa_i\) — the coefficient of restitution of the tool velocity.
A considerable difficulty in modeling is encountered in the selection of objectively justified velocity coefficients for the collision of the striker with the tool, of the tool with the material being worked, etc. The coefficients adopted by us (Table 1) are based on investigations of actual hammers under laboratory conditions and on literature data [3, 4, 6, 10].
Table 1
| Symbol | Name of coefficient | Steel | Granite | Limestone | Aluminum | Wood | Data sources |
|---|---|---|---|---|---|---|---|
| \(\kappa_{b.b}\) | Coefficient of rebound of the striker when impacting the shank of the tool | 0.35 | 0.30 | 0.2 | 0.1 | 0.05 | [6, 10] and own investigations |
| \(\kappa_i\) | Coefficient of restitution of the tool velocity when impacting the material being worked | 0.3 | 0.25 | 0.17 | 0.1 | 0.05 | [4, 6] |
| \(\kappa_{i.i}\) | 0 | 0 | 0 | 0 | 0 | ||
| \(\kappa_{b.i}\) | 0 | 0 | 0 | 0 | 0 | ||
| \(\kappa_{i.b}\) | 0.39 | 0.42 | 0.48 | 0.54 | 0.57 | ||
| \(\kappa_{k.k}\) | 0 | 0 | 0 | 0 | 0 | ||
| \(\kappa_{k.i}\) | 0.19 | 0.19 | 0.19 | 0.19 | 0.19 |
Certain difficulties are presented by the question of introducing disturbing forces into the mathematical description.
All the component parts of the system constituting the pneumatic hammer move under the action of disturbing forces—the air pressures in both cavities of the hammer cylinder, which vary with time owing to the valve device in accordance with the displacement of the striker relative to the body.
The forms of the diagrams of air pressure in both cavities of the hammer may be different.
During operation of the hammer, depending on the air pressure in the line, the hardness of the rock being worked, and the pressing force, the air-pressure diagrams in both cavities of the hammer change mainly in quantitative relation. The changes are especially substantial for the front cavity, whose volume may change as a result of the tool shank leaving the body cavity.
Fig. 2. Diagrams of the change in air pressure in the KE-22 pneumatic hammer: \(p_1\)—air pressure in the rear cavity of the hammer; \(p_2\)—air pressure in the front cavity of the hammer; \(1\)—\(4\)—change of pressures during the direct (working) stroke of the striker; \(2\)—\(3\)—change of pressures during the return stroke of the striker.
For example, for the KE-22 hammer, when the pressure in both cavities of the hammer is increased, the blow frequency increases.
As the hardness of the rock being worked increases, the operating frequency of the hammer increases.
A decrease in the pressing force leads to an increase in the pressure in the front cavity, which causes an increase in the operating frequency.
Apparently, something similar should also occur for other disturbing forces. These circumstances must be borne in mind when considering the question of implementing disturbing forces.
In our opinion, it is most expedient to proceed from specifying the forms of the diagrams as functions of the displacement of the striker relative to the housing. Thus, for example, (see Fig. 2a) the dependencies of these air pressures in both cavities of the hammer on the displacement of the striker relative to the housing look like this for ordinary diagrams (Fig. 2b).
Such an approach to specifying air-pressure diagrams makes it possible:
a) to conduct investigations for different forms of disturbing forces;
b) to take into account the influence of the hammer dynamics on the diagrams of the disturbing forces.
When implementing pressure diagrams in both cavities, it will obviously, as a rule, be necessary to use four nonlinearity blocks. Two blocks are used to reproduce the diagrams during the return stroke, and two during the forward stroke.
The air-pressure diagram in one cavity is therefore obtained with the aid of two blocks, the output of which is used alternately depending on the stroke of the striker.
When reproducing diagrams that have steep fronts not at the points where the strokes change, in our opinion it is most expedient to use relay switching of the outputs of the nonlinear blocks at these points.
The coefficients of viscous friction \(r_b, r_k, r_i\) may be chosen equal to 0.65, 50, and 18, respectively.
The stiffness coefficients of the springs modeling the elasticity of the materials of the elements of the circuit and of the material being worked may be chosen as follows:
\[ C_k = C_{b.k} = 3 \cdot 10^5 \text{ kg/m}, \]
\(C_m\)—the stiffness of the spring simulating the material being worked—\(= 15 \cdot 10^5 \text{ kg/m}\).
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