Dynamics of a corn stripping cutter
Abstract Generated abstract
This paper analyzes the cutting dynamics and power requirements of a corn stripping cutter, whose helical chip breaker geometry divides the removed layer into thin and thick chip portions. It derives expressions for chip geometry, specific cutting work, and mean milling power under several depth-to-diameter regimes, treating the cutter as operating by constrained, non-free cutting and relating its constants to those used for face milling. Experimental torque measurements on steel and cast iron, compared with cylindrical cutters, are used to test the formulas over a broad range of chip thicknesses. The results support the proposed power equations and indicate that the corn cutter can reduce cutting work relative to an ordinary cylindrical cutter, especially at small feeds and thin chips.
Full Text
Dynamics of a Corn Stripping Cutter
A. M. Rozenberg
Professor, Doctor of Technical Sciences
1. Geometry of the Chip
The stripping corn cutter shown in Fig. 1 is a cutter with a distinctive form of chip breakers, which constitute a wide helical groove. Owing to this form of chip breakers, the cutting of the chip, besides the main cutting edge located on the cylinder of the cutter, is performed predominantly by one lateral side of the chip breaker, which has a more favorable geometry from the standpoint of the cutting process. In some part of its width the chip being removed has a considerably increased thickness, which leads to a substantial reduction in energy consumption during operation of the corn cutter in comparison with an ordinary cylindrical cutter. The rear faces of the chip breakers are formed on a lathe-relieving machine, whereby the necessary relief angles are provided.
On Fig. 2 is shown the development of the cylindrical surface of the corn cutter. From Fig. 2 one can clarify the principle of operation of the corn cutter and see the form of its cutting edges.
Fig. 1
On Fig. 3 is shown the successive arrangement of the cutting edges for a corn cutter having 6 teeth around the circumference \((z = 6)\).
For an understanding of the subsequent exposition it is necessary to clarify that, under the chip, there pass simultaneously first the teeth 4, 4, 4, then the teeth 3, 3, 3, then 2, 2, 2, then 1, 1, 1. In reality the teeth 1, 1, 1 or 2, 2, 2, etc., are nothing other than one and the same tooth of a cylindrical cutter, divided into separate cutting elements by the helical groove of the chip breaker. Therefore, instead of an integral chip removed over the entire width of the milled surface by the continuous cutting edge of an ordinary cylindrical cutter, a tooth of the corn cutter removes a series of separate chips, each of which is removed by a cutting element located between two adjacent helical grooves of the chip breakers.
In Fig. 4 is shown the section of the cutting surface by a plane passing through the axis of the cutter (heavy outline), and also by the shaded ...
by sections are shown the cross sections of shavings simultaneously removed by individual cutting elements of one and the same tooth of the corn cutter. Here, as in all subsequent derivations, we shall conventionally assume that the cutting edge of the tooth of the corn cutter is located not along a helical spiral, as is actually the case, but along the generatrix of a cylinder, i.e., we shall take the angle of inclination of the spiral to be \(0^\circ\),
Fig. 2
the helix angle to be \(90^\circ\). Our aim is to derive an equation for the mean power expended during operation of the corn cutter. Therefore the indicated assumption will not introduce any distortions into the result of the derivation,
Fig. 3
since the helix angle within the range from \(40^\circ\) to \(90^\circ\) does not affect the magnitude of the mean power [1], [2].
Fig. 4
In Fig. 5 the outline of the cutting edge of a corn cutter is shown, with designations for its individual elements. The parts of the cutting edge \(fg\) and \(mn\) are located on the cylinder of the cutter and perform the principal cutting work; the parts \(ef\), \(gk\), \(lm\), and \(no\) work only with small sections adjacent to the corners \(f\), \(g\), \(m\), \(n\), and play the role of auxiliary blades. The quantity \(h\) represents the pitch of the helical spiral of the chip-breaker groove.
Fig. 5
In Fig. 6 the form of the cross section of the shaving removed by one cutting element of a tooth of the corn cutter is shown. The shaving is cut off
along the perimeter 1234; moreover, along sides 12 and 34 the chip is cut off by the auxiliary blades of the cutting element, and along side 23 by the main blade. The chip consists of two parts: a thin one, \(q_1\), having thickness \(S_{e1}\), and a thick one, \(q_2\), having some average thickness \(S_{e2}\). Since, for identical values of feed, width, and milling depth, a corn
Fig. 6
cutter removes the same volume of chip as an ordinary cylindrical cutter, the area of the transverse section of the chip removed by one cutting element of the corn cutter will be
\[ q = q_1 + q_2 = S_{e1}\cdot h. \]
We determine the separate parts \(q_1\) and \(q_2\) approximately as follows:
\[ q_1 = S_{e1}\cdot\left(r-\frac{h}{z}\right), \tag{1} \]
\[ q_2 = q - q_1 = S_{e1}\cdot\left(h+\frac{h}{z}-r\right). \tag{2} \]
According to equations (1) and (2), \(q_1\) will have a value somewhat less than the actual one, since in determining it the magnitude of triangle 125 has not been taken into account; correspondingly, \(q_2\) will have a value somewhat greater than the actual one. The relative error in this case will be very small, and there is no sense in complicating the equations in order to eliminate it. The thickness of the thin part of the chip, \(S_{e1}\), will be the same as for an ordinary cylindrical cutter [2]:
\[ S_{e1}=S_z\cdot\sin\psi, \]
where \(S_z\) is the feed per tooth in mm,
\(\psi\) is the angle at which the cutting edge is located on the arc of contact at the given moment.
The thickness of the thick part of the chip, as is evident from Fig. 6, is not constant over the width, and its average value may be obtained in the following way:
\[ S_{e2}=\frac{q_2}{\dfrac{h}{z}} = S_{e1}\cdot\left(z-\frac{rz}{h}+1\right). \tag{3} \]
Figure 6 shows that the principal cutting work is performed along side 23 of the chip by a cutting edge located on the cylinder of the cutter and having length \(r\). Side 12 of the perimeter of the transverse section of the chip is insignificant in magnitude and is cut off by one side cutting edge of the cutting element of the cutter. Side 34 is relatively large...
is more significant and is cut off by the other lateral cutting edge of the cutting element. Thus, here there occurs a process of constrained cutting, since, in addition to the main cutting edge 23, the auxiliary cutting edges 12 and 34 take part in the cutting work. The cutting edge 34, which performs a noticeable share of the cutting work, has a favorable geometry from the standpoint of the cutting process, possessing an increased rake angle. If, for simplification, we assume that the front face of the cutter tooth in the immediate vicinity of the cutting edges is an element of a plane, and denote the rake angle in the plane perpendicular to the axis of the cutter by \(\gamma\), the helix angle by \(\sigma\), the angle in the plan of the lateral cutting edge by \(\Theta\), and its inclination angle by \(\psi\) (see Fig. 7), then the rake angle in the plane perpendicular to the lateral cutting edge, \(\gamma_R\), can be determined in the same way as it is determined on the blade of a lathe tool [3]:
Fig. 7
\[ \tg \gamma_R = (\tg \gamma \cdot \cos \Theta + \tg \sigma \cdot \sin \Theta)\cdot \cos \psi; \]
\[ \cos \psi = \frac{1}{\sqrt{1+(\tg \sigma \cdot \cos \Theta-\tg \gamma \cdot \sin \Theta)^2}}; \]
\[ \tg \gamma_R = \frac{\tg \gamma \cdot \cos \Theta+\tg \sigma \cdot \sin \Theta} {\sqrt{1+(\tg \sigma \cdot \cos \Theta-\tg \gamma \cdot \sin \Theta)^2}}. \]
2. Power Expended During Operation of a Corn Cutter
Previously we proved, both theoretically and experimentally, that the average power during milling is expressed by the equation [4]:
\[ N_e=\frac{A_w \cdot t \cdot b \cdot s}{1000\cdot 4500}, \]
where
\(A_w\) — specific work in \(kgmm/mm^3\),
\(t\) — milling depth in \(mm\),
\(b\) — milling width in \(mm\),
\(s\) — feed per minute in \(mm/min\).
Thus, here the power is expressed through the specific work. We shall use this to derive the equation for the average power during operation of the corn cutter.
Since, during operation of the corn cutter, a process of constrained cutting takes place, the specific work will depend on the thickness and width of the chip and will have the general expression
\[ A_w=\frac{\tau^{k+1}}{k+2}\cdot \lambda \cdot S_m^\kappa \cdot U^l, \tag{4} \]
Here:
\(\lambda\) — a constant depending mainly on the material being machined,
\(S_m\) — chip thickness at the middle of the contact angle (mean thickness),
\(U\) — chip width,
\(\kappa, l\) — negative and fractional exponents depending on the material being machined, the degree of dulling of the cutting edge, and the presence of lubrication and cooling.
Equation (4) is suitable for determining the specific work in the case of milling with such ratios $\frac{t}{D}$, when the contact angle does not exceed $60^\circ$ $\left(\frac{t}{D}\leq 0.25\right)$.
In the case when the milling depth is large and the contact angle exceeds $60^\circ$, for the specific work we shall have a more exact expression [5]:
$$ A_w=\lambda\cdot S_m^k\cdot U^l. \tag{5} $$
It may be assumed (this will subsequently be confirmed experimentally) that, in the operation of a corn cutter, the quantities $\lambda$, $k$, $l$ must have the same numerical values as in the operation of face mills, since in both cases a process of non-free cutting takes place.
Equations (4) and (5) show that it is not possible to calculate the specific work and power for the entire cross section of the chip on the cutting element of a corn cutter; they must be determined separately for the thick and thin parts of the chip. Therefore we must represent the equation for power in the following form:
$$ N_e=N_{e1}+N_{e2}= \frac{A_{w1}\cdot t\cdot b\cdot S}{1000\cdot 4500}\cdot\frac{\omega_1}{\omega} + \frac{A_{w2}\cdot t\cdot b\cdot S}{1000\cdot 4500}\cdot\frac{\omega_2}{\omega}. \tag{6} $$
Here $A_{w1}$ and $N_{e1}$ are the specific work and power corresponding to the thin part of the chip; $A_{w2}$ and $N_{e2}$ are the same for the thick part of the chip cross section.
$\omega_1$ is the volume of metal removed by the thin parts of the chip.
$\omega_2$ is the volume of metal removed by the thick parts of the chip.
$\omega$ is the total volume of chip removed by the cutter, where
$$ \omega=\omega_1+\omega_2. $$
The ratio of the volumes may be expressed by the ratio of the areas of the chip cross section
$$ \frac{\omega_1}{\omega}=\frac{q_1}{q}=\frac{r-\frac{h}{z}}{h}; $$
$$ \frac{\omega_1}{\omega}=\frac{r}{h}-\frac{1}{z}. \tag{6} $$
$$ \frac{\omega_2}{\omega}=\frac{q_2}{q}=\frac{h+\frac{h}{z}-r}{h}; $$
$$ \frac{\omega_2}{\omega}=1+\frac{1}{z}-\frac{r}{h}. \tag{7} $$
Using relations (6) and (7), we obtain:
$$ N_e= \frac{A_{w1}\cdot t\cdot b\cdot S}{1000\cdot 4500} \cdot \left\{ \left(\frac{r}{h}-\frac{1}{z}\right) + \frac{A_{w2}}{A_{w1}} \left(1+\frac{1}{z}-\frac{r}{h}\right) \right\}. \tag{8} $$
In equation (8)
$$ A_{w1}=\frac{2^{k+1}}{k+2}\lambda\cdot S_{m1}^k\cdot U_1^l; $$
$$ A_{w2}=\frac{2^{k+1}}{k+2}\lambda\cdot S_{m2}^k\cdot U_2^l. $$
Here \(S_{m1}\) and \(S_{m2}\) are the mean thicknesses for the thin and thick parts of the chip, respectively, and \(U_1\) and \(U_2\) are the widths of the thin and thick parts of the chip, with
\[ U_1 = r - \frac{h}{z}; \qquad U_2=\frac{h}{z}. \]
Therefore
\[ N_e = \frac{2^{k+1}}{k+2}\,\lambda\,S_{m1}^{k}\, \frac{t\cdot b\cdot S}{1000\cdot 4500}\cdot \left(r-\frac{h}{z}\right)^l \cdot \left\{ \left(\frac{r}{h}-\frac{1}{z}\right) + \left(\frac{S_{m2}}{S_{m1}}\right) \cdot \left(\frac{h}{rz-h}\right)^l \cdot \left(1+\frac{1}{z}-\frac{r}{h}\right) \right\} \tag{9} \]
Here:
\[ \frac{S_{m2}}{S_{m1}}=\frac{S_{e2}}{S_{e1}} = z-\frac{rz}{h}+1; \]
\[ S_{m1}=S_z\cdot \sqrt{\frac{t}{D}}; \]
\(D\) is the cutter diameter in mm.
After substitutions and transformations we obtain:
\[ N_e = \frac{2^{k+1}}{k+2}\cdot \lambda\cdot S_z^{k+1}\cdot \frac{z\cdot n\cdot b}{1000\cdot 4500}\cdot t^{\frac{k+2}{2}}\cdot D^{-\frac{k}{2}}\cdot \left[ \frac{h^l}{z^{l+1}} \left\{ \left(\frac{r}{h}\cdot z-1\right)^{l+1} + \left(z-\frac{r}{h}\cdot z+1\right)^{k+1} \right\} \right], \tag{10} \]
Equation (10) represents the dependence of the mean power on the milling regime and on the structural dimensions and elements of the corn cutter. The part of equation (10) outside the square brackets coincides completely in form with the power equation for a normal cylindrical cutter, with the special feature that in equation (10) one must substitute the values of \(\lambda\) and \(k\) obtained for face cutters, since during operation of the corn cutter a process of non-free cutting takes place.
Equation (10) has been derived for the case in which the corn cutter operates as a cylindrical cutter, at contact angles not greater than \(60^\circ\), i.e., at a ratio of milling depth to diameter
\[ \frac{t}{D}\leq 0.25. \]
In the case in which the corn cutter operates with \(\frac{t}{D}=1\), for the derivation one should use the method applied by us in deriving the power equation for face cutters [5], and in this case we would obtain
\[ N_e = \frac{2^{-\frac{k}{2}}\cdot \lambda\cdot S_z^{k+1}\cdot b\cdot z\cdot n\cdot D} {1000\cdot 4500} \cdot \left[ \frac{h^l}{z^{l+1}} \left\{ \left(\frac{r}{h}\cdot z-1\right)^{l+1} + \left(z-\frac{r}{h}\cdot z+1\right)^{k+1} \right\} \right]. \tag{11} \]
In the case in which the corn cutter operates with \(1>\frac{t}{D}>0.5\), the following equation will be obtained:
\[ N_e = \frac{2^{-\frac{k}{2}}\cdot \lambda\cdot S_z^{k+1}\cdot b\cdot z\cdot n\cdot D} {1000\cdot 4500} \left\{ 1-2^{\frac{k}{2}}\cdot \left(1-\frac{t}{D}\right)^{\frac{k+2}{2}} \right\} \times \]
\[ \times\left[\frac{h^l}{z^{l+1}}\cdot\left\{\left(\frac{r}{h}\cdot z-1\right)^{l+1}+\left(z-\frac{r}{h}\cdot z+1\right)^{k+1}\right\}\right]. \tag{12} \]
Finally, in the case where \(0.5 \ge \dfrac{t}{D} > 0.25\),
\[ N_e= \frac{\lambda\cdot S_z^{k+1}\cdot b\cdot z\cdot n\cdot t^{\frac{k+2}{2}}\cdot D^{-\frac{k}{2}}}{1000\cdot 4500} \cdot \left[ \frac{h^l}{z^{l+1}}\cdot \left\{ \left(\frac{r}{h}\cdot z-1\right)^{l+1} + \left(z-\frac{r}{h}\cdot z+1\right)^{k+1} \right\} \right] \tag{13} \]
In equations (11), (12), and (13), \(b\) denotes the width of the milled surface, measured parallel to the axis of the cutter. The quantities \(h\) and \(r\) must also be measured parallel to the axis of the cutter.
3. Experimental Verification of the Conclusions
To verify the correctness of the equations obtained, experiments were carried out on a milling machine with a dynamometer built into the gearbox for measuring and recording the torques on the cutter. The design of the dynamometer and its calibration have been described in detail by us [4], and therefore we do not give that description here.
For the experiments, both corn cutters and normal cylindrical cutters were taken for comparison. The dimensions and design elements of both cutters are presented in Table 1.
Table 1
| Name of cutter | Diameter \(D\) | Number of teeth \(z\) | Front angle \(\gamma\) | Back angle \(\alpha\) | Helix angle | \(h\) | \(r\) | \(\dfrac{h}{z}\) |
|---|---|---|---|---|---|---|---|---|
| Corn | 75 | 8 | 12° | 10° | 65° | 12 | 4 | 1.5 |
| Cylindrical | 75 | 8 | 12° | 10° | 60° | — | — | — |
As Table 1 shows, the one cutter and the other differ somewhat in the helix angle. This difference cannot have any significance, since we established that a helix angle within the range of 40°–90° has no effect on the average power. The front angle \(\gamma = 12^\circ\) is given in a plane perpendicular to the axis of the cutter. The cylindrical cutters had a pointed tooth, while the corn cutters had a relieved tooth; moreover, as follows from the table, both had the same back angle.
The experiments were carried out on two materials—Steel 35 and soft cast iron. Their strength characteristics are given in Table 2. This choice of materials for the experiments is explained by our desire to verify the applicability of our conclusions both to the case of machining ductile metals and to the case of machining brittle metals.
Table 2
| Name of metal | Brinell hardness | Rockwell hardness, scale \(B\) | Temporary tensile strength \(\sigma_B\), kg/mm² | Elongation at rupture, % | Transverse contraction, % |
|---|---|---|---|---|---|
| Steel 35, normalized | 163—170 | 84—88 | 64.6 | 21.6 | 51.0 |
| Cast iron | 120 | 66—73 | — | — | — |
According to cast iron, the experiments were carried out with both corn and cylindrical cutters. For the steel for the cylindrical cutters, we used data from one of our earlier works [6], which had been carried out not only on the same grade of steel, but even on the very same blank on which the test of the corn cutter was also conducted in the present study.
Fig. 8.
(Graph annotations visible in the figure: “Steel 3”; legend values \(t = 1.05\) mm, \(t = 2.05\) mm, \(t = 3.0\) mm, \(t = 4.2\) mm, \(t = 5.5\) mm; curves labeled “cylindrical cutter” and “corn cutter.”)
The procedure of the experiments was as follows. With a constant number of revolutions \((n = 32)\) and with a constant milling width, experiments were conducted at different milling depths, with a wide range of feeds at each milling depth.
Fig. 9.
(Graph annotations visible in the figure: “Cast iron HB = 120”; \(\Omega = 25\) mm, \(b = 8\); legend values \(t = 1.0\) mm, \(t = 3.0\) mm, \(t = 5.1\) mm; curves labeled “cylindrical cutter” and “corn cutter.”)
In each experiment the torque on the cutter \((M)\) was recorded by a dynamometer; from its value the specific work \(A_w\) was then calculated according to the expression:
\[ A_w=\frac{6.28\,M}{b\cdot t\cdot S_z\cdot z}. \]
The specific work was made dependent on the average chip thickness \((S_m)\), as a result of which a graph was obtained on which the results of experiments carried out at different \(t\) and \(S_z\) could be expressed by a single line.
In Figs. 8 and 9, the results of experiments on steel and cast iron with corn-type and cylindrical cutters are presented in logarithmic coordinates \((S_m - A_w)\).
First of all, it is noteworthy that the specific work in milling with corn-type cutters is considerably lower than in milling with cylindrical cutters. This is the result of the fact that a portion of the cutting element of the corn-type cutter removes a much thicker chip than the cylindrical cutter does (see Fig. 6). The ratio of the chip thicknesses for those corn-type cutters that were used in our experiments is (Eq. 3)
\[ \frac{S_{e2}}{S_{e1}} = z - \frac{r \cdot z}{h} + 1 = 8 - \frac{4 \cdot 8}{12} + 1 = 6.34. \]
In addition, it is evident from the figures that, as the average thickness \(S_m\) increases, the reduction in specific work for the corn-type cutter in comparison with the cylindrical cutter becomes less and less significant. If at \(S_m < 0.01\ \mathrm{mm}\) it amounts to tens of percent, which in practice is extremely substantial, then for very thick chips (which we did not reach in the experiments) it disappears completely. This is explained by the fact that the corn-type cutter operates according to the principle of non-free cutting, and as the chip thickness increases, the work of the auxiliary blades increases.
To assess the correctness of the results obtained from the quantitative side, let us calculate the ratio of the specific works for the corn-type and cylindrical cutters.
We have already noted that, when a corn-type cutter is operating, we deal with two values of the specific work—one for the thin part of the chip, \(S_{e1}\), and the other for the thick part, \(S_{e2}\). The specific work calculated from the experiments and represented in Figs. 8 and 9 in the form of experimental points is a certain average specific work \(A_{w\,avg}\), which can be obtained from equations (10), (11), (12), and (13). Since our experiments were carried out at
\[ \frac{t}{D} < 0.25, \]
then, to obtain \(A_{w\,avg}\), we shall use equation (10).
The average specific work in milling with a corn-type cutter is obtained as
\[ A_{w\,avg} = \frac{1000 \cdot 4500 \cdot N_e}{t \cdot b \cdot s}. \]
Substituting into this equation, in place of \(N_e\), its expression from equation (10), we obtain:
\[ A_{w\,avg} = \frac{2^{k+1}}{k+2}\,\lambda\,S_m^k \cdot \left[ \frac{h^l}{z^{l+1}} \cdot \left\{ \left(\frac{r}{h}\cdot z - 1\right)^{l+1} + \left(z - \frac{r}{h}z + 1\right)^{k+1} \right\} \right]. \tag{14} \]
For the cylindrical cutter, the equation for the specific work will be:
\[ A_w = \frac{2^{k+1}}{k+2}\,\lambda\,S_m^k. \tag{15} \]
In equation (14), the quantities \(\lambda\), \(k\), \(l\) must have the same values as when end cutters operate, that is, values different from those in equation (15). Let us find
ratio of the specific works for the corn and cylindrical cutters; for this purpose we divide equation (14) by equation (15).
\[ \frac{A_{w\mathrm{cp}}}{A_w} = C \cdot S_m^{m} \cdot \left[ \frac{h^l}{z^{l+1}} \cdot \left\{ \left(\frac{r}{h}\cdot z - 1\right)^{l+1} + \left(z-\frac{r}{h}\cdot z+1\right)^{k+1} \right\} \right]. \tag{16} \]
In equation (16) we have denoted by \(C\) the ratio of the quantities \(\dfrac{2k+1}{k+2}\lambda\), where \(m\) is the difference between the values of \(k\) for the corn and cylindrical cutters.
In equation (16), the expression in square brackets is, for a given corn cutter, a constant quantity. Therefore equation (16) shows that, with an increase in \(S_m\), the ratio \(\dfrac{A_{w\mathrm{cp}}}{A_w}\) increases, which is also confirmed by the results of the experiments presented in Figs. 8 and 9.
Using equation (16), we can verify the correctness of the ratios of specific works obtained by us for the corn and cylindrical cutters. For this purpose let us take the values of the quantities \(\lambda\), \(k\), \(l\) for steel 35 (normalized) and cast iron \(H_B=120\) from our previous work [6] (see Table 3).
Table 3
| Cutter | Steel 35 (normalized), \(\lambda\) | Steel 35 (normalized), \(k\) | Steel 35 (normalized), \(l\) | Cast iron No. 1; \(H_B=120\), \(\lambda\) | Cast iron No. 1; \(H_B=120\), \(k\) | Cast iron No. 1; \(H_B=120\), \(l\) |
|---|---|---|---|---|---|---|
| Cylindrical | 136 | \(-0.29\) | — | 46.5 | \(-0.38\) | — |
| Face | 190 | \(-0.23\) | \(-0.09\) | 72 | \(-0.32\) | \(-0.07\) |
After substituting into equation (16) the values of \(\lambda\), \(k\), \(l\) from Table 3 and the values of \(r\), \(h\), \(z\) from Table 1, we obtain:
steel 35:
\[ \frac{A_{w\mathrm{cp}}}{A_w}=0.98\cdot S_m^{0.06}, \tag{17} \]
cast iron:
\[ \frac{A_{w\mathrm{cp}}}{A_w}=0.97\cdot S_m^{0.06}. \tag{18} \]
In Figs. 8 and 9 the straight lines for the cylindrical cutter have been drawn so that they most closely fit the arrangement of the experimental points; the straight lines for the corn cutter have been drawn relative to the straight lines for the cylindrical cutter so that between the former and the latter the ratio of specific works according to equation (17) for steel and according to equation (18) for cast iron is maintained. Thus, in Figs. 8 and 9, for the corn cutter the points were obtained experimentally, while the straight lines were drawn according to equations (10) and (16), which we derived theoretically with allowance for the shape of the cutting edge and the geometry of the cutting process of the corn cutter. Therefore the agreement in Figs. 8 and 9 of the straight lines with the experimental points for the corn cutter is a criterion of the correctness of all our theoretical conclusions, both qualitatively and quantitatively. Figs. 8 and 9 show a satisfactory arrangement of the straight lines relative to the experimental points. A certain deviation for cast iron, not exceeding 5%, may be explained by the nonuni-
by the homogeneity of the workpiece on which the experiments were carried out, and is not substantial. Figs. 8 and 9 make it possible to draw the following conclusions.
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Experiments with corn cutters were carried out over wide ranges of variation in chip thickness, from \(0.0006\) mm to \(0.175\) mm for steel and from \(0.0024\) mm to \(0.3\) mm for cast iron, thereby fully covering the limits of cutting conditions encountered in practice.
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All the experimental points obtained at four depths of cut for steel and three depths of cut for cast iron were arranged, without significant deviations, rectilinearly in logarithmic coordinates \(S_m - A^m\), which confirms the correctness of the structure of equations (4), (10), and (14), and shows that for a corn cutter of constant geometry the specific work depends on the mean chip thickness.
-
The satisfactory agreement of the experimental points with the straight lines drawn according to the theoretically calculated equations (17) and (18) confirms the correctness of the conclusions we have drawn and indicates that, for corn cutters operating by the method of constrained cutting, the same values of the constants \(\lambda\), \(k\), \(l\) should be used in the equations as for end mills.
In Table 4 we give the values of \(\lambda\), \(k\), \(l\) for various materials, obtained by us when working with end mills (for the front angle \(\gamma = 10^\circ\)).
Table 4
| Material being machined | \(\sigma_B\) | \(H_B\) | \(\lambda\) | \(k\) | \(l\) |
|---|---|---|---|---|---|
| Mild steel | 34 | 90 | 185 | -0,29 | 0,00 |
| Steel 10, rolled | 40,1 | 110 | 200 | -0,27 | -0,02 |
| Steel 20, rolled | 44,6 | 120 | 185 | -0,24 | -0,04 |
| Steel 20, normalized | 46,3 | 122 | 195 | -0,26 | -0,02 |
| Steel 35, rolled | 62,1 | 165 | 187 | -0,21 | -0,05 |
| Steel 35, normalized | 64,6 | 167 | 190 | -0,23 | -0,09 |
| Steel 40, rolled | 64,4 | 172 | 203 | -0,2 | -0,09 |
| Steel 40, annealed | 62,8 | 162 | 198 | -0,21 | -0,05 |
| Steel 40, normalized | 65,8 | 178 | 197 | -0,2 | -0,05 |
| Steel 45, improved by quenching in oil | 69 | 206 | 213 | -0,2 | -0,09 |
| Steel 40, improved by quenching in water | 84 | 215 | 215 | -0,2 | -0,1 |
| Steel 45, improved by quenching in water | 94,5 | 260 | 220 | -0,22 | -0,06 |
| Steel 51, annealed | 78 | 217 | 190 | -0,19 | -0,04 |
| Steel 20Kh, rolled | 50,3 | 140 | 188 | -0,23 | -0,09 |
| Steel 20Kh, normalized | 50,3 | 139 | 193 | -0,25 | -0,1 |
| Steel 40Kh, rolled | 71,5 | 201 | 190 | -0,16 | -0,03 |
| Steel 40Kh, annealed | 71,3 | 200 | 196 | -0,2 | -0,06 |
| Steel 40Kh, normalized | 81,1 | 215 | 200 | -0,175 | -0,04 |
| Steel 45Kh, rolled | 74 | 210 | 193 | -0,2 | -0,04 |
| Steel 20KhN, normalized | 54,4 | 143 | 203 | -0,245 | -0,1 |
| Steel 40KhN, rolled | 80,3 | 225 | 187 | -0,2 | -0,1 |
| Steel Kh4N, annealed | 91 | 265 | 210 | -0,2 | -0,06 |
| Steel Kh1N, improved | 106 | 311 | 225 | -0,19 | -0,03 |
| Gray cast iron | -- | 95 | 70 | -0,27 | -0,05 |
| Cast iron | -- | 118 | 72 | -0,32 | -0,07 |
| Cast iron | -- | 150 | 93 | -0,31 | -0,09 |
| Cast iron | -- | 195 | 110 | -0,25 | -0,08 |
| Pearlite cast iron | 25 | 217 | 142 | -0,26 | -0,11 |
| Chromium-nickel cast iron | 7,65 | 145 | 90 | -0,31 | -0,11 |
| Cast iron | 18,5 | 198 | 110 | -0,28 | -0,1 |
| Malleable cast iron | 40 | 145 | 123 | -0,26 | -0,12 |
| Bronze | -- | 81 | 67 | -0,34 | -0,21 |
4. Conclusion
The theoretical derivations and experimental investigation carried out make it possible to draw the following conclusion.
-
The corn cutter operates by the method of non-free cutting, and for power calculations one must use those values of the quantities \(\lambda\), \(k\), \(l\) that were obtained for the operation of face mills (Table 4).
-
In view of the fact that a thickened chip is removed by part of the cutting edges of the corn cutter, a reduction in cutting work and power occurs here in comparison with a cylindrical cutter; this reduction is the more considerable, the smaller the feed per tooth and the thinner the chip. From this point of view, in the case when an increase in productivity is limited by the power of the machine tool, the corn cutter will give a higher productivity than an ordinary cylindrical cutter.
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For calculating the average power in milling with a corn cutter, equations (10), (11), (12), and (13) should be used.
The work was carried out with the participation of student of the Mechanical Faculty of the Tomsk Polytechnic Institute, Comrade Shokun.
LIST OF CITED LITERATURE
-
Rozenberg A. M.—Elementary mechanics of the process of free cutting of metals. Loniomash. Materials for the conference on metal cutting. Leningrad, 1940.
Prof. Rozenberg A. M.—Dynamics of milling. Publishing House Sovetskaya Nauka, 1945.
Assoc. Prof. Rozenberg A. M. Elementary mechanics of the process of free cutting of metals. Proceedings of the Tomsk Industrial Institute, vol. 59, issue 1. -
Assoc. Prof. Rozenberg A. M. and Sudnishnikov B. V.—Theory of the operation of a cylindrical cutter with a helical tooth. Vestnik Metallopromyshlennosti, 1933, No. 4.
-
Prof. Semenchenko—Cutting Tool, Part 1, p. 77.
-
Rozenberg A. M.—Operation of a cylindrical cutter. Organinformmash, 1935, No. 11.
-
Rozenberg A. M.—Dynamics of face milling. Proceedings of the Tomsk Industrial Institute, 1937, vol. 56, issue 1.
Rozenberg A. M.—Operation of a face mill. Organinformmash, 1936, No. 7. -
Rozenberg A. M.—Dynamic tests of cylindrical and face mills. Proceedings of the Tomsk Industrial Institute, vol. 59, issue 1.
Noted misprints
| Page | Line | Printed | Should read |
|---|---|---|---|
| 71 | Formula 8 | $$m=\frac{\lg\left\{\dfrac{\tau_n-1+\dfrac{1}{e}}{\tau_n-\Theta}\right\}}{\lg C}$$ | $$m=\frac{\lg\left\{\dfrac{\tau_n-1}{\tau_n-\Theta}+\dfrac{1}{e}\right\}}{\lg C}$$ |
| 82 | Formulas 17 and 18 | \(A w\, er\) | \(A w\, sr\) |
| 83 | 2 from bottom | carried-out years years | reduced years |
| 93 | 14 from top | \(c\ st\) | \(\mathrm{const}\) |
| 99 | 5 from top | quantities \(H\) can | the quantity \(H\) and can |
| 99 | 6 from top | \(2.7(91\) | \(2{,}7(91\,S)^{0.535-0.006V}\) |
| 102 | 2 from bottom | property | meaning |
| 104 | 7 from top | action | influence on |
| 107 | 19 from top | in physical properties | in the physical sense |
| 114 | 10 from top | of curved surfaces. | of curves. |
| 119 | 9 from top | [[unclear: no entry visible in the “Printed” column]] | [[unclear: no entry visible in the “Should read” column]] |
Proceedings of TPI, volume 61, issue 3.