ON THE THEORY OF THE INTERACTION OF A TRENCHING MACHINE WITH SOIL

I. G. Basov

(Submitted by the Department of Mining Machines, Mine Transport, and Mining Mechanics)

During operation of a trenching earth-moving machine, its propulsor moves along the ground surface, while the working member moves in the slot-trench cut in the frozen layer of soil. Therefore the loads on the working member will be determined not only by the resistance of the soil to destruction, but also by the character of motion of the earth-moving machine, which depends on the microrelief of the ground surface, the relative position of the working member and the propulsor, and the control of the machine.

Fig. 1

1. In the general case, when the machine moves along a laid-out route, there may be different profiles of the ground surface under its right and left tracks (wheels). As a result, the body of the machine, and with it the trenching actuating working member located in the slot, will perform a complex motion in space. A rapid change in the coordinates of displacement in the longitudinal and especially in the transverse directions (Fig. 1) can lead to significant overloads of the trenching frame and other units of its suspension.

The load on the bar frame will depend on the ratio of the speed \(v_{\mathrm{p}}\) of cutting the slot (the machine feed) to the speed \(v_{\mathrm{rot}}\) of change of the current coordinates of the points of rigid attachment of the left \((y_{\mathrm{l}})\) and right \((y_{\mathrm{r}})\) tracks (wheels) to the machine body.

In order to prevent excessive lateral loads from arising on the working member, it is necessary that only the teeth of the cutting chain be in contact with the soil. In other words, the condition must be satisfied (Fig. 1)

\[ v_z = v_{\mathrm{p}} \frac{(B_{\mathrm{sl}} - T_1)\cos\beta}{2D_1}, \tag{1} \]

where \(v_z\) is the linear speed of rotation of the end of the bar in the \(yz\) plane;
\(B_{\mathrm{sl}}\) is the slot width;
\(T_1\) is the thickness of the bar measured over the cams;
\(D_1\) is the width of the bar measured over the cams;
\(\beta\) is the angle of inclination of the bar to the vertical.

On the other hand, from Fig. 1 it follows that

\[ v_z = \frac{v_{\mathrm{rot}}(H_{\mathrm{sl}}^2 + C^2)}{B}, \tag{2} \]

where \(v_{\mathrm{rot}}\) is the linear speed of rotation of one track relative to the other in the \(yz\) plane;
\(C\) is the distance from the point of rotation to the bar;
\(B\) is the machine gauge.

Equating the right-hand sides of equations (1) and (2), we find the permissible speed

\[ v_{\mathrm{rot}} = \frac{v_{\mathrm{p}} B(B_{\mathrm{sl}} - T_1)\cos\beta} {2D_1(H_{\mathrm{sl}}^2 + C^2)} . \tag{3} \]

The speed \(v_{\mathrm{rot}}\) depends on the change in the relative slope \(i\) along the machine path,

\[ v_{\mathrm{rot}} = v_{\mathrm{p}}\tan(\alpha_1 - \alpha_2) = v_{\mathrm{p}}\cdot i, \tag{4} \]

where \(\alpha_1\) and \(\alpha_2\) are the angles of ascent or descent of the right and left tracks (wheels).

Solving equations (3) and (4) jointly, we obtain an expression for determining the magnitude of the maximum permissible slope

\[ i = \frac{B(B_{\mathrm{sl}} - T_1)\cos\beta} {2D_1(H_{\mathrm{sl}}^2 + C^2)} . \tag{5} \]

When calculating \(i\) for machines with two and a larger number of bars, and also when one bar is arranged off-axis (Fig. 1), the value \(C\) must be taken equal to the minimum distance from the bar working member to the track (or wheel).

Thus, from expression (5) it follows that machines with a wider base, a larger cutter overhang, a smaller bar width, and also a greater cutting depth will react less to changes in the microrelief of the route.

It should also be noted that the influence of the microrelief of the route on the occurrence of a lateral load on the bar working member depends to a considerable degree on the type of machine suspension. With a rigid suspension [1], the probability of a lateral load arising on the bar will be considerably greater than with a semi-rigid or equalizing suspension.

1. The relative position of the bar-type working member and the tracks (or driving wheels) of the machine determines the stability of its motion in the specified direction.

An asymmetric arrangement of the bar-type working member relative to the longitudinal axis of the machine causes it to turn during slot cutting. The magnitude of the turning moment \(M_{\mathrm{p}}\) can be found from the equation of moments with respect to point \(O_2\) (Fig. 1).

\[ \sum M_{O_2}=M_{\mathrm{p}}=0.5X_{\mathrm{k}}\left(\frac{B}{2}+C_1\right)+0.5X_{\mathrm{p}}\left(\frac{B}{2}-C_1\right)- \]

\[ -0.5X_{\mathrm{p}}\left(\frac{B}{2}+C_1\right)-0.5X_{\mathrm{k}}\left(\frac{B}{2}-C_1\right). \]

After transformations the equation takes the form

\[ M_{\mathrm{p}}=C_1(X_{\mathrm{k}}-X_{\mathrm{p}}). \tag{6} \]

It may be assumed that the pushing force is

\[ X_{\mathrm{k}}=P_{\mathrm{p}}+X_{\mathrm{p}}, \tag{7} \]

where \(P_{\mathrm{p}}\) is the feed force of the working member;
\(X_{\mathrm{p}}\) is the resistance force to rolling of the tracks (wheels).

Substituting the value of \(X_{\mathrm{k}}\) from (7) into (6), we obtain

\[ M_{\mathrm{p}}=C_1P_{\mathrm{p}}. \]

Thus, the magnitude of the moment causing the machine to turn when the working member is arranged off-axis is directly proportional to the magnitude of the bar displacement relative to the longitudinal axis of the machine and to the feed force.

1. The bar-type working member must allow a certain deviation from rectilinear motion of the machine when cutting a slot, and no additional loads should arise in doing so. In this case the minimum permissible turning radius of the recessed bar (Fig. 1) must be equal to the radius of the circle inscribed [2] in a trapezoid with height

\[ h=\frac{B_{\mathrm{sh}}-T}{2} \]

and a base equal to the horizontal projection of the recessed part of the bar

\[ L=D+H_{\mathrm{sh}}\operatorname{tg}\beta, \tag{8} \]

where \(T\) is the thickness of the bar frame;
\(D\) is the width of the bar at the teeth.

The value of the turning radius of the bar in this case can be found from the formula

\[ r=\frac{L^2+(B_{\mathrm{sh}}-T)^2}{4(B_{\mathrm{sh}}-T)}. \tag{9} \]

The minimum permissible turn of an earth-cutting machine with a recessed bar can be calculated from the expression

\[ R=\frac{2(L_{\mathrm{b}}\sin\beta+A)r}{L}, \tag{10} \]

where \(L_{\mathrm{b}}\) is the length of the bar;

\(A\) is the distance from the axis of rotation of the bar to the line connecting the centers of the tracks (for a tracked prime mover) or to the axis of the rear wheels (for a wheeled prime mover).

Substituting the values of \(L\) and \(r\) from (8) and (9) into (10), we obtain

\[ R=\frac{(L_{\mathrm{b}}\sin\beta+A)\left[(D+H_{\mathrm{sh}}\operatorname{tg}\beta)^2+(B_{\mathrm{sh}}-T)^2\right]}{2(D+H_{\mathrm{sh}}\operatorname{tg}\beta)(B_{\mathrm{sh}}-T)}. \]

It follows from the last formula that the minimum permissible turning radius of the machine will be the smaller, the smaller the values of \(D\), \(H_{\mathrm{sh}}\), \(\beta\), and \(A\). The minimum permissible turning radius of wheeled machines will always be smaller than that of tracked machines, because for them the quantity \(A\) is smaller, all other conditions being equal.

Thus, when designing earth-moving machines, it is desirable to bring the bar-type working member closer to the center, and during operation to deepen it to a vertical position.

REFERENCES

1. G. P. Lyzo, A. P. Lyzo, V. A. Lomovsky. Tractors, automobiles, engines. Vysshaya shkola, Moscow, 1962.

2. I. N. Bronshtein, K. A. Semendyaev. Handbook of Mathematics for Engineers and Students of Higher Technical Schools. Gostekhizdat, Moscow, 1953.