THE INFLUENCE OF THE PRINCIPAL ANGLE IN PLAN ON THE MAIN COMPONENT OF THE CUTTING FORCE

Yu. A. Rozenberg

(Submitted by Prof. Dr. Tech. Sci. A. M. Rozenberg)

Many scientists have engaged in the study and theoretical analysis of the influence of the principal angle in plan on the main component of the cutting force \(P_z\). N. N. Zorev, in his works [1, 2, and 3], analyzing the greater part of these investigations, indicates that all these studies may be divided into two groups. One group of researchers, to whom Zorev assigns Schlesinger, Chelyustkin, Boston, A. M. Rozenberg, and himself, obtained, in cutting steels, results indicating a uniform decrease in the cutting force as the principal angle in plan increases. Another group of researchers (Gobbart, Savvin, Nicholson, Zverev, and Glebov) obtained results indicating the presence of a minimum on the curve \(P_z = f(\varphi)\).

The same division, in the sense of evaluating the influence of \(\varphi\) on \(P_z\), is also found in the cutting of cast iron. Here Chelyustkin, Polikarpov, and Besprozvannyi belong to the first group of researchers, and Gobbart, Fogler, and Glebov to the second.

In the greater part of the textbook literature on the theory of metal cutting (Krivoukhov [4], Klushin [5], Danielyan [6], Arshinov and Alekseev [7], Granovskii, Grudov, Krivoukhov, Larin, and Malkin [8], etc.), the question of the influence of \(\varphi\) on \(P_z\) is treated one-sidedly. It is considered that, in cutting steels, the curve \(P_z = f(\varphi)\) has a minimum, while in cutting cast iron there is a uniform decrease in the cutting force as the principal angle in plan increases.

All these authors explain the presence of a minimum on the curve \(P_z = f(\varphi)\) in cutting steel mainly by the fact that, as the angle \(\varphi\) increases, both the chip thickness and the length of the curvilinear portion of the cutting edge increase. The increase in chip thickness supposedly leads to a decrease in the deformations characterized by chip shrinkage and, consequently, to a decrease in the force \(P_z\). With an increase in the principal angle in plan, the relative importance of non-free cutting (the curvilinear part of the cutting edges) and of secondary cutting increases, with a simultaneous decrease in the rake angle on the auxiliary cutting edge, which leads to an increase in shrinkage at angles \(\varphi = 50 \div 90^\circ\). These explanations are confirmed by a graph of the dependence of shrinkage on \(\varphi\). The authors explain the uniform decrease in the cutting force with an increase in the principal angle in plan when cutting cast iron by the fact that cast iron has little capacity for plastic deformation, and an increase in the length of the curvilinear portion in the range \(\varphi = 60 \div 90^\circ\) will have practically no effect.

The Commission on Metal Cutting, apparently adhering to these same propositions, recommends taking into account the influence of the angle in plan \(\varphi\) on \(P_z\) by the formulas:

for steel

\[ P_z = \frac{\mathrm{const}}{\varphi^{0.18}} \quad \text{for } \varphi < 55^\circ \tag{1} \]

\[ P_z = \mathrm{const.}\,\varphi^{0.22} \quad \text{for } \varphi > 55^\circ, \]

for cast iron

\[ P_z=\frac{\mathrm{const}}{\varphi^{0.18}}. \tag{2} \]

Zorev, criticizing the conclusions set forth above in his first works devoted to this question, believes that the data indicating the presence of a minimum on the curve \(P_z=f(\varphi)\) were obtained as a result of methodological errors in the setting up of the experiments. Zorev assumes that such a character of the curve should in fact occur if the angle in plan is changed by regrinding along the main flank without regrinding the rake face, as a result of which, with a change in \(\varphi\), the cutting angle measured in a plane perpendicular to the cutting edge of the tool changes. In this connection he refers to the investigations of A. M. Rozenberg, who carried out special experiments both with regrinding only on the flank and with regrinding on both the flank and the rake face, i.e., with preservation of a constant cutting angle [9]. In the first case A. M. Rozenberg obtained a curve \(P_z=f(\varphi)\) having a minimum, while in the second case he obtained a uniform decrease in the cutting force with an increase in the main angle in plan.

In one of his recent works [3] Zorev arrives at the important conclusion that the influence of the angle in plan on the cutting force should be determined not for a constant cutting speed, but for a constant temperature of contact between the chip and the rake face. In this case, with an increase in the angle in plan, the main component of the cutting force will decrease. Zorev explains this decrease in \(P_z\) by a reduction in the length of the contact strip of the flank and the cutting surface, which leads to a decrease in the magnitude of the forces on the flank, with unchanged forces on the rake face.

Recently, in the metal-cutting laboratory of the S. M. Kirov Tomsk Polytechnic Institute, a number of investigations have been carried out in which the question of the influence of the main angle in plan on the cutting forces was also considered. These include the author’s work on investigating the cutting process of gray cast iron [10] and A. I. Promitov’s work on investigating the dynamics of milling steels [11]. Since up to the present time there is no unified view in the literature on this question, and erroneous dependences are presented in textbooks, we consider it necessary to clarify this question on the basis of the experimental materials available to us, which confirm the correctness of Zorev’s conclusions.

All our conclusions are based on the following propositions, advanced and proved in a number of recent works [12], [13], [14], [10], [11], [15] of the Department of Machine Tools and Metal Cutting of the Tomsk Polytechnic Institute.

1. Cutting forces, coefficients of friction, built-up edge geometry, shear angle, relative shear, and specific works in the cutting of metals vary along curves whose character—and consequently the course of the metal-cutting process and the phenomena accompanying it—are determined only by the cutting temperature.

2. The method of determining the influence of chip thickness and width, the angle in plan, the rake angle of the tool, and other cutting-regime parameters on cutting forces at a constant cutting speed is incorrect and leads to gross errors. The only correct method is to determine the influence of these quantities on the cutting forces at a constant temperature.

3. In the cutting process, two systems of forces should mainly be distinguished: 1) forces located on the rake face of the tool; 2) forces located on the flank of the tool.

4. At a constant temperature, the forces on the rake face of the tool are directly proportional to the area of the chip cross section, while the forces

on the back face of the tool are directly proportional to the width of cut and do not depend on the thickness of cut.

At constant values of depth of cut and feed, a change in the principal cutting-edge angle changes the shape of the chip cross section. A decrease in the principal cutting-edge angle reduces the thickness of cut and increases the width of cut, while an increase in the cutting-edge angle leads to the opposite changes in the thickness and width of cut. It would seem that, if it is considered established that a change in the thickness of cut leads to a change in the specific cutting force, while a change in the width of cut does not affect the specific cutting force (or, more precisely, in oblique cutting affects it only quite insignificantly), then one may conclude that at constant depth of cut and feed, with a decrease in the principal cutting-edge angle the cutting force should increase continuously, while with an increase in the cutting-edge angle, on the contrary, it should decrease and reach its smallest values at $\varphi = 90^\circ$.

However, a number of experimental works carried out to determine the influence of the cutting-edge angle on cutting forces have shown that these simple relations between the cutting-edge angle and the force $P_z$ are not justified. Thus, Glebov [16], having analyzed all the studies available on this question, came to the conclusion that the force $P_z$ and the pressure have the smallest value at a principal cutting-edge angle of $45^\circ$ when cutting steel and $60^\circ$ when cutting cast iron; when the cutting-edge angle is changed from these values in either direction, the force $P_z$ and the pressure increase. This proposition, as we indicated above, has been reflected in many research and scientific works. The authors try to explain such a deviation of the experimental results from simple logical conclusions by various circumstances connected with the peculiarities of the process of oblique cutting. In our opinion, the discrepancy between the experimental results and the logical conclusions given above is a consequence of the fact that the authors did not take into account the influence of cutting temperature on the cutting force, and all experiments with different cutting-edge angles, which were intended to reveal the dependence of the cutting force on the cutting-edge angle, were carried out at a constant cutting speed.

Naturally, changing the cutting-edge angle while maintaining a constant cutting speed, by changing the heat-removal conditions, immediately changed the temperature, which in itself already affected the cutting force and thus distorted the direct influence of the principal cutting-edge angle.

Fig. 1

As our experiments have shown, decreasing the cutting-edge angle at constant values of cutting speed, depth of cut, and feed leads, owing to an increase in the width and a decrease in the thickness of cut, to an improvement in heat removal from the cutting edge and to a reduction in the cutting temperature. On the other hand, if, while changing the cutting-edge angle, the depth of cut and feed are changed at the same time in such a way that the thickness of cut “$a$” and the width of cut “$b$” remain unchanged, the condition of heat removal will not change, and the cutting-edge angle should not influence the cutting temperature. This is well confirmed by Fig. 1, constructed from the results of our experiments.

Thus, one may come to the conclusion that the influence of the actual plan angle can be revealed only if the experiments are carried out at constant temperature. In this case, the logical conclusions that we made earlier must be confirmed. As our investigations [10] have shown, in cutting gray cast iron, just as in cutting steels, when the cutting temperature is kept constant, the coefficients of friction between the chip and the tool and between the tool and the machined surface remain constant. In this case, the force on the rake face of the tool will be directly proportional to the thickness and width of the cut, while the force on the flank face is directly proportional to the width of the cut:

\[ P_{\text{front face}} = A_1ab = A_1s \cdot t, \tag{3} \]

\[ P_{\text{flank face}} = K_1b = K_1 \frac{t}{\sin\varphi}, \tag{4} \]

\[ P_z = P_{\text{front face}} + P_{\text{flank face}} = A_1s \cdot t + K_1 \frac{t}{\sin\varphi} = t\left(A_1s + \frac{K_1}{\sin\varphi}\right) \tag{5} \]

or

\[ P_z = A_1ab + K_1b = b(A_1a + K_1). \tag{6} \]

As the experiments have shown, the coefficients \(A_1\) and \(K_1\) are constant at constant cutting temperature.

These simple expressions, which have a physical meaning, show that at constant depth of cut and feed, and under conditions of maintaining a constant temperature on the rake face, the force \(P_z\), with an increase in the plan angle \(\varphi\), must decrease monotonically; moreover, this decrease must be the less significant the less noticeable the forces on the flank face are in comparison with the forces on the rake face, that is, the greater the feed.

Fig. 2, which presents the results of our experiments, fully confirms this. It shows, for a small feed, an entirely distinct increase in \(P_z\) as the plan angle decreases from 90 to 45° (that is, precisely in that range of plan angles in which many investigators obtained a decrease in the force with decreasing plan angle). For large feeds, the influence of the plan angle becomes hardly noticeable, which also confirms the correctness of our considerations.

Fig. 2

If, when changing the plan angle, the depth of cut and feed are kept constant, and the width “\(b\)” and thickness “\(a\)” of the cut are not kept constant, then, under constant-temperature conditions, the plan angle should not affect the force \(P_z\), because in this case the forces on the flank face, proportional to the wi-

cutting plane will remain constant. This is also indicated by equation (6), where the angle in the plan has not been reflected.

Figure 3 presents the results of an experimental proof of this proposition. The experiments were carried out by us with such values of the depth of cut and feed that, for different angles in the plan, the width of cut \(b = 3\) mm and the thickness of cut \(a = 0.26\) mm remained constant.

Fig. 3

As Fig. 3 shows, under these conditions the angle in the plan has no effect on the force \(P_z\). Fig. 3 also shows that under these conditions the force \(R_{x.y.}\), which is the resultant in the horizontal plane, does not depend on the angle in the plan, which is a consequence of the same propositions.

It is interesting to note the fact that if, from the same experimental data, the dependence \(P_z = f(\varphi)\) is plotted at constant cutting speed, then, as Fig. 4 shows, at a cutting speed \(V = 75\) m/min, with an increase of the angle in the plan from 45 to 90°, the force \(P_z\) will increase, whereas at a speed \(V = 125\) m/min, with the same change of the angle in the plan, the force \(P_z\) will decrease. Fig. 4 clearly proves the erroneousness of the method for determining the influence of the principal angle in the plan on the cutting forces at constant cutting speed, since in this case contradictory results may be obtained.

Fig. 4

The question of the influence of the angles in the plan on the components \(P_x\) and \(P_y\) was discussed by us earlier [17], and therefore in the present work we shall not dwell on it.

A. I. Promtov also arrived at similar conclusions on the question of the influence of the principal angle in the plan on the cutting forces, while studying the dynamics of high-speed milling of steels with face mills [11]. In his work Promtov found that a change in the principal angle in the plan is not reflected in the magnitude of the circumferential force applied to the rake face, because, as he indicates, at constant cutting temperature and chip shrinkage it does not depend on the ratio between the thickness and the width.

of the cut, but is determined only by the area of the cross section of the cut, on the magnitude of which the principal angle in plan has no effect.

Thus, on the basis of investigations carried out in the metal-cutting laboratory of the Tomsk Polytechnic Institute, the following conclusions may be drawn:

1. Both in cutting steels and in cutting gray cast iron, under conditions of constant temperature (this condition corresponds to constancy of tool life and therefore is of interest for production), with an increase in the angle in plan the force \(P_z\) decreases. This decrease is the smaller, the larger the feed, and at finishing and roughing feeds it is so insignificant that it may be disregarded.

2. The reason for the incorrect assessment by other investigators of the influence of the angle in plan \(\varphi\) on the force \(P_z\) was the fact that the influence of temperature on the force \(P_z\) was not taken into account and all experiments were carried out at a constant cutting speed.

3. The only correct method is that of determining the influence of the principal angle in plan on the cutting forces at constant temperature.

LITERATURE

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2. Zorev N. N. The main cutting force and a new method of calculating it. Collection of TsNIITMASH. New investigations in the field of metal cutting. Mashgiz. 1948.

3. Zorev N. N. Investigation of elements of the mechanics of the cutting process. Mashgiz. 1952.

4. Krivoukhov V. A. Metal Cutting. Mashgiz. 1938.

5. Klushin M. I. Metal Cutting. Mashgiz. 1953.

6. Danielyan A. M. Metal Cutting and the Tool. Mashgiz. 1950.

7. Arshinov V. A. and Alekseev G. A. Metal Cutting. Mashgiz. 1953.

8. Granovskii G. I., Grudov P. P., Krivoukhov V. A., Larin M. N., Malkin A. Ya. Metal Cutting. Mashgiz. 1954.

9. Rozenberg A. M. Dynamics of Milling. Sovetskaya Nauka. 1945.

10. Rozenberg Yu. A. Investigation of the process of cutting gray cast iron. Dissertation. Tomsk. 1952.

11. Promptov A. I. Dynamics of high-speed milling of steels with face mills. Dissertation. Tomsk. 1954.

12. Eremin A. N. The Physical Essence of Phenomena in Cutting Steel. Mashgiz. 1951.

13. Zimin Yu. P. Investigation of the process of high-speed cutting of steels. Dissertation. Tomsk. 1948.

14. Rozenberg A. M. and Eremin A. N. Toward a theory of the cutting process. “Stanki i instrument,” No. 10, 1949.

15. Khvorostukhin L. A. Calculation of forces in high-speed cutting on the basis of the physicomechanical characteristics of metals. Dissertation, Tomsk. 1953.

16. Glebov S. F. Theory of the Most Advantageous Cutting of Metals. Gosmashmetizdat. 1933.

17. Rozenberg A. M. and Rozenberg Yu. A. Equations of force in cutting fragile metal (cast iron). Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov. Vol. 75. 1954.