Proceedings
of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order of the Red Banner of Labor
named after S. M. Kirov

Vol. 225　　　　　　　　　　　　　　　　　　　　　　　　　1972

On the Calculation of the Specific Cutting Resistance of a Frozen Soil Layer by a Chain Working Organ

I. G. Basov, A. N. Shchipunov

(Presented by the Department of Mining Machines)

The specific resistance to cutting of a frozen soil layer can be found from experimental data obtained when cutting soil both with a single cutter and with a group cutting tool (a bar, disk cutter, rotor) of existing earthmoving machines. In the latter case, to determine \(k_p\), the dependence of the force \(P_c\) on the cutting tool on the travel speed of the machine \(v_p\) may be used. For an earth-cutting machine with a bar-type working organ, this dependence is rectilinear (Fig. 1) and in general form can be described by the equation

\[ P_c = A + Bv_p. \tag{1} \]

In equation (1), the coefficient \(A\) is numerically equal to the segment cut off by the straight line \(P_c = f(v_p)\) on the ordinate axis, and represents the pulling force of the cutting chain at \(v_p = 0\), but in the presence of a certain feed force \(P_p\), that is, the friction force

\[ P_{\mathrm{fr}} = P_p(\mu + \omega), \tag{2} \]

where \(\mu\) is the coefficient of friction of the teeth against the soil,
\(\omega\) is the coefficient of resistance to motion of the chain in the bar slot.

Fig. 1. Dependence of the cutting force \(P_c\) on the feed speed \(v_p\) at cutting speeds:
1 — 0.76 m/s; 2 — 1.48 m/s; 3 — 1.8 m/s;
4 — 2.6 m/s; 5 — 4.6 m/s; 6 — 5.4 m/s.
Soil — loam; \(H_p = 2\) m, \(H_c = 1.3\) m;
chain — “Ural-33,” \(z_{\mathrm{av}} = 16.8\) teeth.

Experiments show that the force \(P_p\) at \(v_p = 0\) depends only on the physicomechanical properties of the soil being destroyed and on the degree of wear of the cutting teeth, and does not depend on the speed of motion of the cutting

of the chain \(v_r\). The coefficients of friction of the cutters against the soil and of resistance to the motion of the chain in the bar slot depend to a considerable degree on \(v_r\).

The total values \((\mu+\omega)\) can be found from oscillograms recording the torque \(M_r\) on the shaft of the working member. Since, at \(v_r=\mathrm{const}\) and \(v_p=\mathrm{const}\), the recording curves of \(M_r\) and \(P_p\) on the oscillogram are usually equidistant, it may be assumed that the increment \(\Delta P_c\) of the pulling force of the cutting chain is proportional to the increment \(\Delta P_p\) of the feed force, and consequently,

\[ \frac{\Delta P_c}{\Delta P_p}=(\mu+\omega). \]

Processing of the experimental results established that, when cutting loam with a relative moisture content of 14.5%, a temperature along the depth of the slot of \(-1.5 \div -6^\circ\mathrm{C}\), a DorNII impact-hammer index \(C=60\text{--}130\), and \(v_r=1.1\text{--}3.5\ \mathrm{m/sec}\), the value \((\mu+\omega)\) lies within the range 1.2–1.9. The second component of the pulling force of the cutting chain in equation (1) is, in turn, the sum of the forces expended on breaking up and transporting the soil from the slot.

The cutting force of frozen soil may be found from the expression

\[ P_r=\kappa_r \frac{B_{sh}h z_{cp}}{z_e},\ kg, \tag{3} \]

where \(\kappa_r\) is the specific cutting resistance, \(kg/cm^2\);

\[ h=\frac{n_l t_c z_k v_p}{3600\,zv_r}\cos\beta \]

is the average thickness of the cut chip, cm;

\[ z_{cp}=\frac{H_{sh}z}{t_c z_k} \]

is the average number of cutters in contact with the soil;

\(z_e\) is the average number of cutters in the repeating element of their arrangement scheme (for example, a “herringbone”);

\(n_l\) is the number of cutting lines;

\(t_c\) is the chain pitch, cm;

\(z_k\) and \(z\) are the number of cams and cutters in the chain;

\(B_{sh}\) and \(H_{sh}\) are the width and depth of the slot cut by the bar, cm;

\(\beta\) is the angle of inclination of the bar to the vertical, degrees.

After substituting the values of \(h\) and \(z_{cp}\) into equation (3), we obtain

\[ P_r=\kappa_r \frac{B_{sh}H_{sh}n_l v_p \cos\beta}{3600\,z_e v_r},\ kg. \tag{4} \]

The force expended on transporting the soil may be determined by the formula

\[ P_t=\frac{H_{sh}B_{sh}\cos\beta\,v_p}{3600\,v_r} \left[H_{sh}\gamma(1+\tg\beta\,\mu_t)\right],\ kg, \tag{5} \]

where \(\gamma\) is the unit weight of the soil in the virgin state, \(g/cm^3\);

\(\mu_t\) is the coefficient of friction of loosened soil against the wall of the slot.

From equations (4) and (5) we find that the sum of the forces of cutting and transporting soil from the slot by the cutting chain is

\[ P_r+P_t=(\kappa_r+\kappa_t) \frac{B_{sh}H_{sh}n_l v_p\cos\beta}{3600\,z_e v_r},\ kg, \tag{6} \]

where

\[ \kappa_t=\frac{H_{sh}z_e\gamma}{n_l}(1+\tg\beta\mu_t) \]

is the specific force of transporting loosened soil, \(kg/cm^2\).

Fig. 2. Dependence of the specific cutting force \(k_{rt}\) on the feed rate \(v_p\) at cutting speeds:
1 — 2.6 m/s; 2 — 1.8 m/s;
3 — 1.48 m/s; 4 — 0.76 m/s; 5 — 4.6 m/s;
6 — 5.4 m/s.

Fig. 3. Dependence of the specific cutting force \(k_{rt}\) on the cutting speed \(v_r\) for various soils
(1 — sandy loam; 2 — sand; 3 — loam; 4 — ice)
with KMP chains (solid lines) and “Ural-33” chains (dashed lines), with different numbers of teeth \(z_{sr}\) in contact with the soil.

The circumferential force on the drive sprocket of the cutting chain is

\[ P_{\mathrm{c}} = P_{\mathrm{tr}} + P_{\mathrm{r}} + P_{\mathrm{t}}, \]

or, substituting the values of the individual terms from (2) and (6),

\[ P_{\mathrm{c}} = P_{\mathrm{p}}(\mu+\omega) + \kappa_{\mathrm{rt}}\, \frac{B_{\mathrm{c}}H_{\mathrm{c}}n_{\mathrm{l}}v_{\mathrm{p}}\cos\beta} {3600\,z_{\mathrm{e}}v_{\mathrm{r}}}, \ \text{kg}, \tag{7} \]

where \(\kappa_{\mathrm{rt}}=\kappa_{\mathrm{r}}+\kappa_{\mathrm{t}}\) is the specific force of cutting and transporting soil from the slot, kg/cm\(^2\).

Under operating conditions that preclude clogging of the cutting chain, the value of \(\kappa_{\mathrm{t}}\) is very small in comparison with \(\kappa_{\mathrm{r}}\). Therefore, in such cases one may in practice take \(\kappa_{\mathrm{rt}}=\kappa_{\mathrm{r}}\).

It follows from equation (7) that the specific cutting force of the frozen soil layer can be calculated by the formula

\[ \kappa_{\mathrm{r}} = \frac{3600\,z_{\mathrm{e}}v_{\mathrm{r}}\left[P_{\mathrm{c}}-P_{\mathrm{p}}(\mu+\omega)\right]} {B_{\mathrm{c}}H_{\mathrm{c}}n_{\mathrm{l}}v_{\mathrm{p}}\cos\beta}, \ \text{kg/cm}^2. \tag{8} \]

It follows from formula (8) that the coefficient \(\kappa_{\mathrm{r}}\) depends on the physicomechanical properties of the frozen soil, the speed \(v_{\mathrm{r}}\), and the number of cutters in contact with the soil. The coefficient \(\kappa_{\mathrm{r}}\) is practically independent of the speed \(v_{\mathrm{p}}\), the dimensions of the working member, and the angle \(\beta\), since, when these parameters change, the forces \(P_{\mathrm{c}}\) and \(P_{\mathrm{p}}\) change proportionally. This is confirmed by experimental data (Figs. 2, 3), obtained under operating conditions ensuring \(h=0.5\text{–}2.5\) cm.

The minimum values of \(\kappa_{\mathrm{r}}\) for the soils investigated were obtained at \(v_{\mathrm{r}}=2.0\text{–}3.5\) m/sec.