Equation of Motion of a Bar-Type Earth-Cutting Machine
Abstract Generated abstract
This paper formulates the traction balance and equation of motion for a bar-type earth-cutting machine operating on inclined ground while cutting frozen soil. It decomposes rolling resistance, grade resistance, inertial resistance, and soil-cutting resistance, then derives the cutting and feed forces on an individual bar tooth from soil reactions, friction, tooth geometry, chip thickness, and bar inclination. By relating the resistance of all engaged teeth to machine displacement, the authors obtain a nonlinear second-order differential equation for rectilinear motion, in which acceleration decreases with the square of displacement. The resulting model is proposed as a basis for calculating realistic velocities and accelerations, particularly by finite-difference solution, for use in the design of earth-cutting machines.
Full Text
Equation of Motion of a Bar-Type Earth-Cutting Machine
I. G. Basov, B. L. Stepanov
(Presented by the Department of Mining Machines, Mine Transport, and Mining Mechanics)
During operation, a bar-type earth-cutting machine is under the influence of external forces—the forces of resistance to the motion of the installation, resistance to the cutting of frozen ground, and also the inertia forces of the machine.
Fig. 1. Diagram of the operation of a bar-type earth-cutting machine
The traction-balance equation of an earth-cutting machine for the general case (the machine is operating on a section inclined at an angle \(\alpha_1\), with the bar inclined at an angle \(\beta\), Fig. 1) can be written in the following form:
\[ P_{\mathrm{k}} = P_f + P_{\alpha} + P_j + R. \tag{1} \]
Let us consider the individual components of equation (1). The driving force causing the displacement of the earth-cutting machine is the tangential tractive force \(P_{\mathrm{k}}\), directed parallel to the path of motion of the machine.
The rolling-resistance force \(P_f\) is equal to the product of the rolling-resistance coefficient of the machine \(f\) and the component of its weight \(G\):
\[ P_f = fG\cos\alpha_1. \tag{2} \]
Since the machine operates on a section inclined at an angle \(\alpha_1\), there appears one more component of the weight \(G\), which offers resistance to ascent,
\[ P_a = G\sin\alpha_1. \tag{3} \]
When the machine operates on a horizontal section this resistance is equal to zero; when it operates on a downgrade it will have a minus sign.
The resistance of the inertia forces can be found from the equation
\[ P_j=\delta_{\mathrm{rot}}\frac{G}{g}\cdot j, \tag{4} \]
where \(\delta_{\mathrm{rot}}\) is a coefficient taking into account the influence of the inertia of the rotating masses of the earth-cutting machine;
\(g\) is the acceleration of gravity;
\(j\) is the acceleration of the rectilinear translational motion of the machine.
The last term of the traction-balance equation is the component, parallel to the path of motion of the machine, of the resistance force to cutting frozen soil, \(R\). To determine it, it is necessary to know the forces acting on one cutter tooth of the bar. Let us consider these forces, shown in Fig. 2, where the following notation is adopted:
\(P_r\) — cutting effort;
\(N_r\) — the reaction force of the frozen soil on the tooth, arising as a result of the application of the cutting force;
\(N_r\tan\rho\) — the friction force on the front face of the tooth;
\(N_r\) — the resultant cutting-resistance force;
\(\rho\) — the angle of friction of the tooth against the frozen soil;
\(P_p\) — feed effort;
\(N_p\) — the reaction force of the frozen soil on the tooth, arising as a result of the application of the feed effort;
\(N_p\tan\rho\) — the friction force on the blunting surface of the tooth;
\(N_p\) — the resultant resistance force to feeding;
\(\alpha\) — cutting angle;
\(h\) — depth of the chip of frozen soil removed by the tooth;
\(\beta\) — angle of inclination of the bar.
In the process of cutting frozen soil, the tooth is subjected to active forces, which include the cutting effort \(P_r\) and the feed effort \(P_p\), as well as resistance forces, which include the reactions of the frozen soil due to the application of the cutting and feed efforts, \(N_r\) and \(N_p\), and also the friction forces on the front face of the tooth, \(N_r\tan\rho\), and on the blunting surface, \(N_p\tan\rho\).
The effort \(P_r\) acts along the cutting line, and the effort \(P_p\) acts in a direction parallel to the path of motion of the machine, i.e., the \(x\)-axis. In this connection, in order to determine them, the resistance forces must be resolved into components parallel and perpendicular to the cutting line and parallel to the \(x\) and \(y\) axes (Fig. 2). The analytical expressions for these components will have the following form:
\[ N_{r1}=N_r\frac{\sin(\alpha+\rho)}{\cos\rho}, \tag{5} \]
\[ N_{r2}=N_r\frac{\cos(\alpha+\rho)}{\cos\rho}, \tag{6} \]
\[ N_{\rho x}=-N_{\rho}\frac{\cos(\beta+\alpha+\rho)}{\cos\rho}, \tag{7} \]
\[ N_{\rho y}=N_{\rho}\frac{\sin(\beta+\alpha+\rho)}{\cos\rho}, \tag{8} \]
\[ N_{\mathrm{n}x}=N_{\mathrm{n}}\frac{\cos(\beta-\rho)}{\cos\rho}, \tag{9} \]
\[ N_{\mathrm{n}y}=N_{\mathrm{n}}\frac{\sin(\beta-\rho)}{\cos\rho}. \tag{10} \]
The cutting force is the sum of the resistance forces acting parallel to the cutting line, and can be found from the expression
\[ P_{\rho}=N_{\rho 1}+N_{\mathrm{n}}\tan\rho. \]
Fig. 2. Diagram of the forces acting on the cutting tooth.
After replacing \(N_{\rho 1}\) by the value from (5) and carrying out the corresponding transformations, the last equation assumes the form
\[ P_{\rho}=\frac{N_{\rho}\sin(\alpha+\rho)+N_{\mathrm{n}}\sin\rho}{\cos\rho}. \tag{11} \]
The feed force is equal to the sum of the forces acting parallel to the \(x\)-axis (Fig. 2):
\[ P_{\mathrm{n}}=N_{\mathrm{n}x}+N_{\rho x}. \]
Substituting the values (9) and (7) into the last expression, we obtain
\[ P_{\mathrm{n}}=\frac{N_{\mathrm{n}}\cos(\beta-\rho)-N_{\rho}\cos(\beta+\alpha+\rho)}{\cos\rho}. \tag{12} \]
The force acting on the tooth, parallel to the \(y\)-axis, is equal to the sum of the components of the cutting-resistance force \(N_{py}\) and of the feed force \(N_{ny}\) (Fig. 2)
\[ N_{py}-N_{ny}=\frac{N_p\sin(\beta+\alpha+\rho)-N_n\sin(\beta-\rho)}{\cos\rho}. \tag{13} \]
In equations (11–13) the forces \(P_p\) and \(P_n\), as well as the force \(N_{py}-N_{ny}\), depend on the reaction forces of the frozen soil \(N_p\) and \(N_n\). In order to relate these forces to the displacement of the earth-cutting machine during operation, let us express them through the thickness \(h\) of the chip removed by the tooth.
The reaction forces of the frozen soil are
\[ N_p=\sigma_{vr}F_p; \tag{14} \]
\[ N_n=\sigma_{vr}F_n; \tag{15} \]
where \(\sigma_{vr}\) is the averaged temporary resistance of the frozen soil to compression over the depth of the slot cut by the bar;
\(F_p\) is the area of the front face of the tooth on which the force \(N_p\) acts;
\(F_n\) is the blunting area of the tooth on which the force \(N_n\) acts.
In its shape, the tooth of the bar of the earth-cutting machine is a pyramid, and its front face and blunting area are trapezoids. If the half-sum of the bases of the trapezoid is represented as the product of its height by a certain proportionality coefficient depending on the shape and dimensions of the trapezoid, then the area of the latter will be equal to the product of the square of the height by the proportionality coefficient.
In that case the area (Fig. 2) is
\[ F_p=\kappa_1\cdot AB^2=\kappa_1\frac{h^2}{\sin^2\alpha}, \]
where \(\kappa_1\) is a proportionality coefficient depending on the shape of the tooth;
\(AB\) is the height of the trapezoid of the front face of the tooth.
Let us express the blunting area through \(F_p\) and a certain proportionality coefficient \(\kappa_2\), depending on the shape of the tooth; then
\[ F_n=\kappa_2F_p\cos\alpha=\frac{\kappa_1\cdot\kappa_2\cos\alpha}{\sin^2\alpha}h^2. \]
Substituting the values found for \(F_p\) and \(F_n\) into equations (14) and (15), we obtain
\[ N_p=\frac{\sigma_{vr}\kappa_1}{\sin^2\alpha}h^2, \tag{16} \]
\[ N_n=\frac{\sigma_{vr}\kappa_1\kappa_2\cos\alpha\cdot h^2}{\sin^2\alpha}. \tag{17} \]
Substituting the values (16) and (17) into equations (11) and (12), we obtain
\[ P_p=\frac{\sigma_{vr}\kappa_1}{\sin^2\alpha} \left[ \frac{\sin(\alpha+\rho)+\kappa_2\cos\alpha\cdot\sin\rho}{\cos\rho} \right]h^2, \tag{18} \]
\[ P_n=\frac{\sigma_{vr}\kappa_1}{\sin^2\alpha} \left[ \frac{\kappa_2\cos\alpha\cdot\cos(\beta-\rho)-\cos(\beta+\alpha+\rho)}{\cos\rho} \right]h^2. \tag{19} \]
To simplify equation (19), denote
\[ q=\frac{\sigma_{vr}\kappa_1}{\sin^2\alpha} \left[ \frac{\kappa_2\cos\alpha\cdot\cos(\beta-\rho)-\cos(\beta+\alpha+\rho)}{\cos\rho} \right], \]
then
\[ P_{\mathrm{p}} = qh^2. \tag{20} \]
The resistance force of the frozen soil acting on the bar in a direction parallel to the path of motion of the earth-cutting machine is equal to the product of the resistance forces acting on an individual tooth of the cutting chain in the same direction and the number of teeth \(n\) in contact with the soil, i.e.,
\[ R = P_{\mathrm{p}}\cdot n = qn\cdot h^2. \tag{21} \]
The component of the cutting-resistance force perpendicular to the path of motion of the machine (Fig. 1) is
\[ N = (N_{\mathrm{ru}} - N_{\mathrm{pu}})\,n. \]
Substituting into this expression the values (13), (16), (17), and, after the corresponding transformations, we obtain
\[ N = \frac{\sigma_{\mathrm{vr}}\cdot k_1}{\sin^2 \alpha}\, nh^2\, \frac{\sin(\beta+\alpha+\rho)-k_2\cos\alpha\cdot\sin(\beta-\rho)} {\cos\rho}. \tag{22} \]
This force does not enter into the traction-balance equation of the earth-cutting machine, but it must be taken into account in calculations of the stability of these machines and of their adhesion forces with the soil.
Since all components of the traction-balance equation are now known, substituting the values (2), (3), (4), and (21) into equation (1), we obtain
\[ P_k = fG\cos\alpha + G\sin\alpha + \delta_{\mathrm{vr}}\frac{G}{g}j + qnh^2. \]
The magnitude of the chip thickness of frozen soil \(h\), removed by a bar tooth, is equal to the path \(x\) traveled by the machine during the time \(t\).
But since the bar is at an angle \(\beta\), then
\[ h=\frac{x}{\cos\beta}. \]
From the traction-balance equation we determine the acceleration of the rectilinear translational motion of the machine, first replacing \(h\) by \(x\):
\[ j = \frac{g\left(P_k-G\sin\alpha_1-fG\cos\alpha_1\right)} {\delta_{\mathrm{vr}}G} - \frac{qgn}{\delta_{\mathrm{vr}}G\cos^2\beta}\,x^2. \tag{23} \]
For simplification, let us denote
\[ \frac{g\left(P_k-G\sin\alpha_1-fG\cos\alpha_1\right)} {\delta_{\mathrm{vr}}G} = A^2, \]
\[ \frac{qg\cdot n}{\delta_{\mathrm{vr}}\cdot G\cos^2\beta} = B^2. \]
Then \(j=A^2-B^2x^2\).
Since the acceleration of the rectilinear translational motion of the machine is the second derivative with respect to the path \(x\), the equation of motion of the bar-type earth-cutting machine can be written in differential form as
\[ x'' = A^2 - B^2x^2. \tag{24} \]
Equation (24) is a nonlinear second-order differential equation. Solving it by the finite-difference method, one can obtain the true values of the velocities and accelerations that must be known when designing earth-cutting machines.
REFERENCES
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- M. P. Sitnikov. Control of the motion of a cutting machine. Scientific Works of the Moscow Mining Institute. Issue 8. Ugletekhizdat, Moscow–Leningrad, 1950.
- D. A. Chudakov. Fundamentals of the Theory of the Tractor and the Automobile. Selkhozgiz, Moscow, 1962.