Elementary Mechanics of the Process of Free Metal Cutting

Assoc. Prof. A. M. Rozenberg

A process of free cutting is understood to be a process in which only one cutting edge participates in the cutting. Such a case may occur when turning a bar or disk on a lathe with transverse feed of the tool, the length of the tool blade exceeding the thickness of the bar or disk (the width of the chip). Such a case may occur when planing ridges on a planing machine, or when an end countersink works on the end face of a ring.

The same case is encountered when a cylindrical milling cutter is operating, when, as usual, the width of the surface being milled is less than the length of the cutter.

If the cutting edge of the tool in the process of free cutting is located perpendicular to the direction of the cutting motion, then the chip being separated forms an Archimedean spiral, and there is no doubt that both the chip as a whole and the individual metal particles in it move in a plane normal to the cutting blade. In this plane is also located the effective cutting angle.

Fig. 1.

If, however, the cutting blade does not form a right angle with the direction of its motion, then the chip curls into a helical spiral with a direction and pitch depending on the inclination of the cutting edge to the direction of motion. This phenomenon is well illustrated by Fig. 1, borrowed from Hippler’s book¹). The impression is created that in this case the chip comes off not in a plane normal to the cutting edge, and therefore the cutting angle must be measured not in the plane normal to the cutting edge, but in another plane, including the direction of motion of the chip along the front face of the tool. In general, by the cutting angle we shall mean the angle between the tangent to the direction of motion of a given point of the blade and the direction of motion of the metal particles of the chip, passing through the given point of the blade, along the front face of the tool. This angle determines the deformation of the metal in the formation of the chip. In Fig. 2 a wedge-tool is shown. The plane ABCD is the cutting plane. If the tool moves in the direction DA, and the metal particles of the chip move along the front face of the tool in the direction AE, the cutting angle is EAD. If the direction of motion of the tool is AH, and the direction of motion of the metal particles along the front face of the tool is Ag, the cutting angle is gAH. If one could imagine a case in which, during the motion of the tool in

Fig. 2.

in the direction \(DA\), the chip particles would move in the direction \(Ag\); the cutting angle would have to be taken as \(gAD\).

In 1896 Sellergren²) proposed that, when the blade is inclined to the direction of motion, the cutting angle should be measured in a plane deviating from the normal to the blade by the same angle \(\gamma\) by which the blade deviates from the perpendicular to the direction of motion. According to Sellergren, the cutting angle must be measured not in the plane \(NNN\), but in the plane \(NTT\) (Fig. 3), and the cutting angle \(\delta_T\) is determined from the expression

\[ \tan \delta_T = \tan \delta_N \cdot \cos \gamma . \]

Fig. 3.

Prof. Glebov called this proposition “Sellergren’s law,” and, although Sellergren in his work does not mention the direction of chip motion along the front face of the cutter, Prof. Glebov gave the following formulation of “Sellergren’s law”³): “the angle of deviation of the chip from the perpendicular to the blade is equal to the angle of deviation of the blade from the perpendicular to the direction of motion.”

In addition, Prof. Glebov believes that “the helical motion of the chip when the blade is inclined to the direction of motion is explained by the fact that in this case the chip deviates to the side from its normal direction, perpendicular to the blade.”

From all these propositions it follows that the inclined position of the blade favors the process of chip separation, since the “effective” cutting angle \(\delta_T\) in this case is smaller than the sum of the sharpening angle and the clearance angle, and at the same time, without worsening the conditions for heat removal from the cutting edge, it is possible to reduce the load on the cutter.

The same working conditions can be created by imparting to the cutter, in addition to the principal motion in the direction \(V_N\), an additional motion in the direction \(V_S\), parallel to the blade (Fig. 4); in this case the cutting blade will act as if it were inclined to the direction of motion.

Fig. 4.

A classic example is the knife which, when drawn along the line of its blade, penetrates more easily into the body being cut. On this matter Prof. Glebov writes:

“Everyone knows from everyday life the reduction of the effective cutting angle by inclining the blade to the direction of motion: when, for example, we cut bread with an ordinary knife, we never move the knife with its blade perpendicular to the cut, but always move it like a saw, forward or backward; as a result, the direction of motion of the knife makes a very small angle, close to zero, with the direction of the knife blade; this reduction of the angle of inclination leads to a reduction of the cutting angle, and we in effect cut with a knife considerably sharper than the ordinary sharpening of the blade. The indicated reduction of the cutting angle is especially important because it gives all the advantages of reducing pressure due to the smaller angle \(\alpha\) (of cutting) while preserving the sharpening angle; thus, by reducing the pressure, we lose neither in the durability of the cutter nor in its ability to conduct away heat.”⁴)

Prof. Pankin illustrates this same phenomenon with exceptional persuasiveness in his article “Kinematic Sharpening of Cutting Blades.”⁵)

Calling kinematic sharpening the reduction of the cutting angle (or wedge angle) under any motion of the cutter (or knife) composed of two motions, one of which is normal to the cutting edge and the other parallel to it, Prof. Pankin notes that “we encounter the practical application of this kinematic sharpening in remote epochs in the use of edged weapons. When cutting a vine, the saber is drawn along its axis; Turkish yataghans had a peculiar curved shape that ensured a reduction of the angle without drawing the weapon.”

Thus, it is considered firmly established that any inclined position of the cutting blade relative to the direction of its motion causes the chip to curl into a helical spiral, to deviate during its motion along the rake face of the cutter from the direction perpendicular to the cutting edge, and to leave the cutter toward the side with the smaller cutting angle.

Despite the persuasiveness and complete unanimity with which the propositions stated above are presented in the technical literature, they are nothing more than a general misconception based on external observation of the chip-separation process. A detailed consideration of this question, as well as experimental verification, leads to entirely different conclusions.

Fig. 5.

Figure 5 shows the inclined position of the cutting edge \(AB\) of a planing tool (\(\beta < 90^\circ\)). Let the edge move from position \(AB\) to position \(A'B'\), moving in the direction of the arrow \(V\), i.e., perpendicular to its position. With such motion, each particle of the chip will move in a plane normal to the blade, as is not difficult to verify if one imagines the surface from which the chip is removed divided into a series of strips perpendicular to the blade \(AB\). Each of the strips, in forming the chip, will curl into an Archimedean spiral, while the whole chip will take the form of a helical spiral leaving the cutter in the direction from \(A\) to \(B\) and creating the complete impression that the chip runs off toward the side of the reduced cutting angle. If the chip is unrolled into a ribbon, placed in front of the blade \(A'B'\), and it is assumed that there is no chip shrinkage, then point \(D\) will be located at point \(D_1\), and \(C\) at \(C_1\), with \(DD_2 = D_2D_1\), \(CC_2 = C_2C_1\). The inclination angle of the chip spiral will then be equal to the angle \(\beta\). Thus, a helical spiral chip will be obtained, running off along the rake face of the cutter toward the side of the smaller cutting angle, but the formation of this chip occurs by the process of displacement of individual metal particles in a plane normal to the cutting blade. For illustration, such a chip is shown on the right in Fig. 5.

If one takes into account the shrinkage of the chip, which always accompanies the cutting process, an even more deceptive impression is obtained.

Chip shrinkage, i.e., its shortening, will occur in the direction of motion of the individual metal particles, i.e., in the plane normal to the cutting blade. Figure 6 shows two positions of the blade \(AB\) and \(A'B'\) and the chip unrolled in front of the blade \(A'B'\), which, during its formation, has undergone a certain shrinkage expressed by a certain shrinkage coefficient

\[ K = \frac{1}{l_0}, \]

where \(l_0\) is the distance traveled by the cutter, and \(l\) is the length of the chip obtained from this distance. The surface of the workpiece being machined, traversed by the cut-

by the cutter as it moves from position \(AB\) to position \(A'B'\), schematically divided into a number of strips parallel to the cutting edge. These same strips are shown on the chip, but here, owing to the shrinkage that occurs in the plane normal to the cutting blade, these strips have become narrower.

From Fig. 6 it follows that:

\[ CE = l_0; \]

\[ EC_1 = l = K l_0 \]

\[ EC_1 \cos \beta' = CE \cos \beta; \]

\[ K l_0 \cos \beta' = l_0 \cos \beta; \]

\[ \cos \beta' = \frac{\cos \beta}{K}. \]

Since the shrinkage coefficient “\(K\)” is always less than unity and, for thin chips, may reach values \(K = 0.2\text{—}0.3^{6}\), it follows that the angle \(\beta'\), in the presence of shrinkage, will always turn out to be considerably smaller than the angle \(\beta\), and an even stronger impression will be created that the chip is running off along the cutter toward the side of the smaller cutting angle, whereas in reality each metal particle, during chip formation, will move in a plane normal to the cutting edge of the cutter.

Fig. 6.

Fig. 7.

In both cases considered, the cutting edge moved perpendicular to its own orientation. Let us now examine the case when the cutting edge is inclined to the direction of motion. In Fig. 7, \(\beta < 90^\circ\), and the motion of the cutter takes place in the direction of the velocity \(V\). Such motion may be represented as consisting of two motions: one in the direction \(V_N\), normal to the cutting edge, and the other in the direction \(V_S\), parallel to the cutting edge.

If we move the blade from position \(AB\) to position \(A'B'\), first by motion in the direction \(V_N\), then we must repeat everything that was said above with respect to Fig. 6. The chip will then occupy the position \(EC_1D_1F\), and its particles will move in planes normal to the cutting edge. Then we move the cutter in the direction \(V_S\). If, in doing so, we assume that the friction force between the cutter and the chip is completely absent, then no action of the cutter on the chip arises from the displacement in the direction \(V_S\); in other words, in the absence of friction between the chip and the cutter, the chip particles, for any orientation of the blade relative to the direction of its motion, always move in planes normal to the blade.

In the presence of a friction force between the chip and the cutter, the displacement of the cutter in the direction \(V_s\) will be associated with additional deformations in the chip; namely, the chip will be compressed in the direction from \(B'\) to \(A'\) and will take the position \(EC_3D_3F\). Such additional compression of the chip hinders its formation and increases the shrinkage (reduces the shrinkage coefficient).

From the photographs of chips shown by us in Figs. 8, 9, and 10 it is clearly seen that, when the blade is positioned obliquely to the direction of motion, the chip turn is shorter (more significant shrinkage) than when the blade is positioned perpendicular to the blade. The same follows from Table 1, where the results of measurements of such chips are presented. The chips were taken with small thickness, since the investigation of this question interested us as applied to the process of milling with a cylindrical milling cutter, where the chip thickness is usually measured in hundredths of a millimeter.

Table 1.

Motion of the cutter	Chip thickness, mm	Length traversed by the cutter \(l_0\)	Length of the chip obtained \(l\)	Shrinkage coefficient \(k = \dfrac{l}{l_0}\)	Note
According to the scheme of Fig. 6	0.05	62	33	0.53	Without lubricant
According to the scheme of Fig. 6	0.05	63	38	0.6	With lubricant
According to the scheme of Fig. 6	0.1	62	34	0.55	Without lubricant
According to the scheme of Fig. 6	0.1	63	39	0.62	With lubricant
According to the scheme of Fig. 7	0.05	61	16	0.26	Without lubricant
According to the scheme of Fig. 7	0.05	63	23	0.36	With lubricant
According to the scheme of Fig. 7	0.1	61	19.5	0.32	Without lubricant
According to the scheme of Fig. 7	0.1	63	23	0.36	With lubricant

It follows from Table 1 that, with an obliquely positioned blade, the shrinkage of the chip increases considerably, which is the result of additional deformation under the influence of the relative displacement of the cutter in the direction \(V_s\) and of the friction force acting on the chip in this direction.

From all that has been set forth above, one may conclude that, if the cutting edge is positioned obliquely to the direction of motion (\(\beta < 90^\circ\)), the chip particles deviate from the plane normal to the cutting edge, which is the result of the relative displacement of the cutter in the direction \(V_s\). In Fig. 11, \(QO\) is the direction of motion of the cutter, whose blade \(AB\) is positioned obliquely to the direction of motion. The rake face of the cutter is \(ABCD\). According to Sellergren, Glebov, and Pankin, the chip in this case should move along the rake face of the cutter in the direction \(OS\), separating at the cutting angle \(SOQ\).

The analysis carried out by us shows that such a displacement of the chip as a whole would be possible in the absence of chip shrinkage and in the absence of a friction force between the chip and the cutter. But even then, in the direction \(OS\) only the chip as a whole would move; the individual metal particles, during formation of this chip, would move in the direction \(OR\), normal to the blade.

If, however, we take the real cutting process with the presence of chip shrinkage and of a friction force between the chip and the rake face of the cutter, then in this case the chip will be formed by the displacement of metal particles in a direction deviating from the normal to the blade toward \(B\) to \(A\). This direction in Fig. 11 is shown conventionally (it cannot be indicated exactly) by the letters \(OP\). The cutting angle in this case will be \(QOP\). As we see, it

Fig. 8.

\(\delta = 0.05\ \mathrm{mm}\)

With lubricant.

Fig. 9.

\(\delta = 0.1\ \mathrm{mm}\)

With lubricant.

Fig. 10.

will be greater than the angle \(QOS\) and, in exactly the same way, greater than the angle \(ROF\), measured in the plane normal to the blade.

Thus, a detailed analysis of the question, supported by experiment, leads us to the conclusion that inclining the blade toward the direction of motion does not decrease the cutting angle, but increases it.

Naturally the question arises: how, then, can one explain the easier penetration of a knife-wedge into the body being cut when, in addition to the motion normal to the blade, the knife is also given an additional motion of drawing along the line of the blade? To answer this question let us turn to Fig. 12, which shows a wedge moving in a direction perpendicular to its blade. \(P\) are the forces acting normally to its side

Fig. 11.

Fig. 12.

faces from the body being cut. \(f\) is the coefficient of friction on the side faces. \(fP\) is the friction force. \(Q_1\), the force that must be applied to the wedge for its penetration, in this case will be

\[ Q_1 = 2P \left[ \sin \frac{\alpha}{2} + f . \cos \frac{\alpha}{2} \right] \tag{1} \]

For small angles \(\alpha\), the second term in the brackets, i.e. the term depending on the friction force, will play the prevailing role in the force \(Q_1\).

Fig. 13.

Let the wedge be given, in addition to the principal motion normal to the blade with velocity \(V_N\), an additional displacement parallel to the blade with velocity \(V_S\) (Fig. 13). The resultant velocity \(V\) and the direction of motion are determined from the rectangle \(V_S\)—\(V_N\). The cutting edge of the wedge is situated with respect to the direction of motion at an angle \(\beta < 90^\circ\), where

\[ \operatorname{tng}\beta = \frac{V_N}{V_S}. \]

The velocity \(V_N\) will give, on the side face of the wedge, a component \(V_N . \cos \dfrac{\alpha}{2}\), which, together with the velocity \(V_S\), will give on the side face of the wedge a resultant \(V_1\), directed at an angle \(\beta_1\) to the blade of the wedge, where

\[ \operatorname{tng}\beta_1 = \operatorname{tng}\beta \cdot \cos \frac{\alpha}{2} = \frac{V_N}{V_S} \cdot \cos \frac{\alpha}{2}. \]

If the normal pressure on the lateral side of the wedge is \(P\), then the frictional force \(fP\) will act in the direction opposite to the velocity \(V_1\); in the direction normal to the cutting edge there will act only the part of the frictional force

\[ fP\sin\beta_1 . \]

This part of the frictional force will offer resistance to the penetration of the wedge into the body being separated, and the force \(Q_2\), which must be applied to the wedge in the direction normal to its blade, is determined as:

\[ Q_2=2P\left[\sin\frac{\alpha}{2}+f\sin\beta_1\cos\frac{\alpha}{2}\right] \tag{2} \]

In this expression

\[ \sin\beta_1= \frac{\sin\beta\cos\frac{\alpha}{2}} {\sqrt{\cos^2\beta+\sin^2\beta\cos\frac{\alpha}{2}}}. \]

That is,

\[ Q_2=2P\left[ \sin\frac{\alpha}{2} + f\frac{\sin\beta\cos^2\frac{\alpha}{2}} {\sqrt{\cos^2\beta+\sin^2\beta\cos\frac{\alpha}{2}}} \right]. \tag{3} \]

It is not difficult to see that equation 1 is a special case of equations 2 and 3 in the case when \(\beta=\beta_1=90^\circ\) and \(V_S=0\), i.e., when the cutting edge moves normally to its position. Thus, equations 2 and 3 are general equations for determining the force \(Q\), regardless of the position of the wedge edge relative to the direction of its displacement.

From equations 2 and 3 it is evident that, with decreasing \(\beta\), i.e., with increasing

\[ \frac{V_S}{V_N}, \]

the force \(Q\) decreases; moreover, the cause of this decrease is the decrease of the second term in the brackets, that is, the decrease in the counteraction of the frictional force, and in no case a decrease in the cutting angle, as follows from the explanations of Glebov—Pankin.

Let us imagine that the frictional force is absent \((f=0)\); in this case equations 1, 2, and 3 will give one and the same magnitude of the force, and no amount of drawing along the blade can change the force \(Q\).

Since a knife or wedge always has a small angle \(\alpha\), judging from equations 2 and 3, a decrease of the second term in the brackets will especially strongly reduce the force \(Q\).

In the case of a real wedge or knife, the reduction of the force in the presence of drawing may, in addition to all that has been stated above, be partly due to the inevitable presence on every real blade of the finest irregularities or notches, which, during drawing, will act like a saw, producing tears, thereby separating particles of the material and facilitating the penetration into it of the wedge blade.

Forces on an Inclined Blade

It is customary to think that, in the process of free cutting, no matter how the cutting edge is positioned relative to the direction of motion, the resultant of all forces on the cutter is located in the plane normal to the cutting ...

blade. Such a conclusion can be reached on the basis of the force-equilibrium diagram given in the works of Semenchenko ^7), Reznikov ^8), Danielian ^9), Savin ^10), Rudnik ^11), Krivoukhov ^12), Currein ^13), Schlesinger ^14), and others. If this proposition is accepted and the forces acting in the cutting plane are considered, we obtain the following geometrical relation between them (Fig. 14).

\[ \frac{P_N}{P}=\frac{1}{\sin \beta} \tag{4} \]

\[ \frac{P_a}{P}=\frac{1}{\operatorname{tng}\beta} \tag{5} \]

In Fig. 14, \(AB\) is the cutting edge, located at an angle \(\beta\) to the direction of motion; \(b\) is the width of the surface from which the chip is removed.

Fig. 14.

\(P_N\) is the force in the plane normal to the blade, \(P\) is the force in the direction of the resultant velocity \(V\), and \(P_a\) is the transverse force. We obtain the same picture if we imagine the developed surface of cutting of a cylindrical milling cutter with a helical tooth having helix angle \(\beta\). In this case \(P\) is the circumferential (tangential) force, and \(P_a\) is the axial force on the tooth of the milling cutter.

We know of no experiments in which the force \(P_N\) has been measured during free cutting with an inclined blade; the ratio \(\frac{P_a}{P}\), however, can be obtained from experiments with milling cutters carried out by various authors. In Table 2 we give a summary of the calculations we performed from the works of various authors; it should be noted that all authors, with the exception of Boston, Gilbert, and Kaiser, measured not the circumferential force, but the feed force during milling, \(P_h\). From the feed force we calculated the circumferential force by means of the relation

\[ P=0.9P_h\cdot S_m^{0.03} \]

Here \(S_m\) is the chip thickness at the middle of the contact arc. This relation was obtained by us as a result of special experimental work and gives an error of not more than \(\pm 5\%\) (for \(\frac{t}{D}\leq 0.15\)).

According to all authors, with the exception of Aizel and Boston, as \(S_m\) increases the ratio \(\frac{P_a}{P}\) increases. The limits of variation are given in the table.

In Aizel, the fluctuations of \(\frac{P_a}{P}\) are completely irregular.

Figure 15 presents a comparison of all these data. They are arranged in such a way that a certain average curve can be drawn through them; this curve indicates that the actual ratio \(\frac{P_a}{P}\), determined from experiments, is approximately three times smaller than according to equation 5.

Table 2.

Front angle $\gamma$	Spiral lead angle $\beta$	$\dfrac{P_a}{P}$	Material being machined	Source
15°	80°	0.066	Steel 50.11	Stöwer 15)
15°	70°	0.125	Steel 50.11	Stöwer 15)
15°	60°	0.194	Steel 50.11	Stöwer 15)
15°	50°	0.256	Steel 50.11	Stöwer 15)
15°	41°	0.2–0.32	Steel 50.11	Stöwer 15)
15°	40°	0.326	Steel 50.11	Stöwer 15)
15°	30°	0.435	Steel 50.11	Stöwer 15)
15°	25°	0.51	Steel 50.11	Stöwer 15)
15°	20°	0.3–0.6	Steel 50.11	Stöwer 15)
27°	58°	0.11–0.15	Steel $\sigma_{\mathrm{в}} = 36$ kg/mm²	Schlesinger 14)
10°	48°	0.2–0.22	Steel $\sigma_{\mathrm{в}} = 6?--70$	Kurrein 13)
10°	65°	0.1–0.28	Steel $\sigma_{\mathrm{в}} = 70$	Aizel 16)
8°	52°	0.134–0.333	Steel VCN—35	Kienzebush and Pohl 17)
5°–20°	65°	0.12–0.16	Steel 3150	Boston, Gilbert and Kaiser 18)
15°	45°	0.35	Steel 3150	Boston, Gilbert and Kaiser 18)

We find an attempt to explain this phenomenon only in Stöwer, who believes that part of the circumferential force $P$ is a friction force on the clearance face, not producing an axial force. But this explanation is unsatisfactory in view of the threefold discrepancy between the value of $\dfrac{P_a}{P}$ measured in the experiments and that calculated from equation 5, since in this case it would have to be assumed that the predominant part of the force $P$ is a friction force on the clearance face, which is quite improbable.

The reason for such a discrepancy between the equilibrium picture and the experimental results, in our opinion, lies in the incorrect prevailing conception of the forces acting on the blade. Since, with an inclined position of the blade ($\beta < 90^\circ$), the resultant velocity can be resolved into a component normal to the blade $V_N$ and a component parallel to the blade $V_s$ (Fig. 7), and the latter of these, as we showed earlier, produces an additional upsetting of the chip, then in the direction parallel to the blade there must act from the cutter on the chip a force producing this upsetting. This force $P_s$, together with $P_N$, gives the resultant $R$, which will no longer lie in the plane normal to the blade (Fig. 16).

Fig. 15.

Comparison of Figs. 14 and 16, in which the force $P$ is plotted with the same magnitude, shows that according to the scheme of Fig. 16 the forces $P_a$ and $P_N$ are obtained as considerably smaller than according to Fig. 14, i.e., according to equations 4 and 5. In the case of work of the—

With a lathe or planer tool we very rarely encounter the process of free cutting; moreover, in these cases the angle \(\beta\) is usually very close to \(90^\circ\), and therefore, for these cases, the indicated change in the scheme of action of the forces is of no substantial significance. In milling with a cylindrical cutter, on the contrary, it is important to be able to calculate the actual magnitude of the axial force \(P_a\), which here, owing to the presence of angles \(\beta\) that differ considerably from \(90^\circ\), may have a substantial magnitude and will exert a noticeable effect on the spindle bearings, the fixtures, load the cross-feed screw, the guides, etc.

From Fig. 16 we obtain

\[ P = P_N \cdot \sin \beta + P_s \cdot \cos \beta \tag{6} \]

\[ P_a = P_N \cos \beta - P_s \cdot \sin \beta \tag{7} \]

Whence it is not difficult to obtain:

\[ \frac{P_a}{P} = \frac{\dfrac{P_N}{P} - \sin \beta}{\cos \beta} \tag{8} \]

\[ \frac{P_s}{P} = \frac{1 - \dfrac{P_N}{P}\sin \beta}{\cos \beta} \tag{9} \]

\[ \frac{P_s}{P} = \cos \beta - \frac{P_a}{P}\sin \beta \tag{10} \]

We set ourselves the aim of determining experimentally the relation of the forces on an inclined blade for different values of \(\beta\). This was necessary in order to evaluate the significance of the force \(P_s\), and also to decide whether the specific cutting pressure is a constant independent of the angle \(\beta\), and in what direction this specific pressure should be taken when calculating forces, torques, and powers. Up to now it had been taken as constant, independent of \(\beta\), in the direction normal to the cutting edge, and this, according to the scheme of Fig. 16, should have led to an increase of the force \(P\) as \(\beta\) decreases, with \(b=\mathrm{const}\), owing to the influence of the force \(P_s\) on \(P\).

Fig. 16.

From equations 8, 9, 10 and Fig. 16 it is evident that, in order to determine all the forces, it is sufficient to determine two of them. But we considered that, for greater reliability, it was desirable to carry out a direct measurement of all the forces.

In addition, we were interested in comparing the characteristics of the cutting process in planing and milling in order to solve the question of the possibility of transferring the fundamental dependencies of one to the other.

We first determined the dependence of the force \(P\) on the angle \(\beta\) in planing with different chip thicknesses. For this purpose the tool was fastened in a special holder on the traverse of a universal milling machine. On the machine table (see Fig. 17) a measuring table was installed for measuring the horizontal feed force during milling. In this table were fastened

...was a metal bar to be planed, and in this way we could measure the principal cutting pressure \(P\) on the planing tool, which, in its fixture, could be set at any angle \(\beta\). The measuring table, manufactured in the metal-cutting laboratory of the Tomsk Industrial Institute, had high precision, since it was mounted on balls, was equipped with a hydraulic load gauge and a self-recording manometer. Before operation it was calibrated; moreover, during calibration it was found that the lateral and vertical forces inevitably acting on the measuring table in the cutting process had no influence on its readings, owing to the special features of its design.

Fig. 17.

Figures 18 and 19 present the results of measuring the force \(P\) in planing, with different angles \(\beta\), steel \(\sigma_{в} = 60\ \text{kg}/\text{mm}^2\), which show that, with a constant planing width \(b\), the force \(P\) (in the direction of motion of the cutter) for angles \(\beta = 90^\circ\text{--}30^\circ\) does not depend on the angle \(\beta\), despite the fact that when \(\beta\) is decreased the chip width \(l\) increases, where

\[ l = \frac{b}{\sin \beta} \]

At \(\beta < 30^\circ\), the force \(P\) begins to increase.

A similar phenomenon was obtained in experiments with cast iron \(H_b=150\), with the difference that the increase in the force \(P\) here begins at angles \(\beta<40^\circ\).

All these experiments were carried out with a cutter having a rake angle \(\gamma=10^\circ\), at a cutting speed \(V=200\text{–}760\ \mathrm{mm/min}\).

In parallel with this, milling experiments were carried out on the same materials with cylindrical cutters having helix angles \(\beta=80^\circ\), \(\beta=64.5^\circ\), \(\beta=45^\circ\), and \(\beta=25^\circ\). In the experiments the torque on the cutter was measured by means of a special dynamometer of the author’s design \(^{19}\). The cutters had a rake angle \(\gamma=0^\circ\). Figs. 20, 21, and 22 present the results of the cutter experiments. The graphs are plotted in coordinates \(A_w\)—specific work in \(\mathrm{kg\,mm/mm^3}\)—and \(S_m\)—chip thickness at the middle of the contact arc in microns. The ratio of the values of \(A_w\) at the same \(S_m\) gives the ratio of the average circumferential forces in milling.

Thus, Figs. 20, 21, and 22 show that in milling we have the same dependence of the force \(P\) on the angle \(\beta\) as in planing. Fig. 23 gives a comparison of these and other results, the points for milling with \(\beta=90^\circ\) and \(\beta=75^\circ\) being plotted from our experiments with a pendulum dynamometer \(^{19}\).

Next we carried out experiments to measure the normal force \(P_N\) on a planing cutter with \(\beta<90^\circ\). To measure the normal force on an inclined blade, we proceeded as follows. The cutter was fixed in the above-described device on the cross-slide of a milling machine so that its blade was positioned perpendicular to the motion of the longitudinal feed; during the experiments, simultaneously with the longitudinal feed, we imparted to the table of the milling machine a transverse feed parallel to the cutter blade. As a result of these two motions, the workpiece fastened on the table moved relative to the cutter in a direction making an angle \(\beta\) with its cutting edge; if the speed of the longitudinal feed is \(V_N\), and the speed of the transverse feed is \(V_s\), then

\[ \tg \beta=\frac{V_N}{V_s} \]

Thus, by creating different ratios of \(V_s\) and \(V_N\) (for which purpose the motion was transmitted to the transverse-feed screw at different speeds from an auxiliary drive), we could obtain cutting with different \(\beta\) and record the force \(P_N\) with the same measuring table by which, in the above-described series of experiments, the force \(P\) was recorded. The magnitude of the angle \(\beta\) was recorded in the experiments by means of a pencil fastened to the cross-slide, which drew a straight line on a strip of paper fastened to the table. This straight line is nothing other than the direction of the resultant displacement of the workpiece (the strip of paper) relative to the cutter (the pencil).

As a result of this arrangement of the experiments, for the first time in the history of the study of cutting processes we directly measured (and did not obtain as the result of recalculation by any formulas) the force \(P_N\) on an inclined blade.

The results of the experiments are presented in Figs. 24 and 25. The experiments were carried out with a constant length of cutting edge \(l\) located under the chip; therefore along the ordinate axis in Figs. 24 and 25 the quantity \(P_N\sin\beta\) is plotted, i.e., the normal force referred to a constant length of cutting edge \((l=\mathrm{const})^{1}\).

\(^{1}\) In view of the special features of the experimental arrangement, in the experiments we have \(l'=b\), independently of the angle \(\beta\). In Figs. 24 and 25 the quantity \(b\) is indicated.

Fig. 18.
In-figure labels: \(P\); \(\delta_{\mathrm{mm}}\).
Annotation: steel [[unclear: grade]] 60; planing; \(\beta = 90^\circ\), \(\gamma = 10^\circ\); \(b = 6\ \mathrm{mm}\).

Fig. 19.
In-figure labels: \(P\); \(\delta_{\mathrm{mm}}\).
Annotation: planing; steel [[unclear: grade]] 60; \(\gamma = 10^\circ\); \(b = 6\ \mathrm{mm}\).
Legend: \(\beta = 75^\circ\); \(\beta = 60^\circ\); \(\beta = 47^\circ\); \(\beta = 30^\circ\); \(\beta = 20^\circ\).

Fig. 20.
In-figure labels: \(P_H\); \(S_{\mathrm{m}}\).
Left annotation: \(D = 74\ \mathrm{mm}\); \(z = 18\); \(\beta = 64.5^\circ\); \(\gamma = 0^\circ\).
Legend: \(t = 1.05\ \mathrm{mm}\); \(t = 3\ \mathrm{mm}\); \(t = 5\ \mathrm{mm}\).
Right annotation: steel [[unclear: grade]] 60; \(D = 62\ \mathrm{mm}\); \(z = 18\); \(\beta = 80^\circ\); \(\gamma = 0^\circ\).
Legend: \(t = 1.5\ \mathrm{mm}\); \(t = 2.5\ \mathrm{mm}\).

Fig. 21.
In-figure labels: \(P_H\); \(S_{\mathrm{m}}\).
Left annotation: \(D = 44\ \mathrm{mm}\); \(z = 8\); \(\beta = 45^\circ\); \(\gamma = 0^\circ\).
Legend: \(t = 1.0\ \mathrm{mm}\); \(t = 2\ \mathrm{mm}\); \(t = 4\ \mathrm{mm}\); \(t = 6.2\ \mathrm{mm}\).
Right annotation: steel [[unclear: grade]] 60; \(D = 62\ \mathrm{mm}\); \(z = 3\); \(\beta = 25^\circ\); \(\gamma = 0^\circ\).
Legend: \(t = 1\ \mathrm{mm}\); \(t = 4.1\ \mathrm{mm}\).

From Figs. 24 and 25 one may conclude that, with constant chip thickness and width, i.e., with a constant chip cross-section, the force \(P_N\) decreases considerably as the angle \(\beta\) decreases. In our opinion, this phenomenon is explained as follows. First, the smaller the angle \(\beta\), the more the direction of motion of the chip particles along the rake face deviates from the normal to the blade; this is connected with a decrease in the component of the friction force on the rake face in the direction normal to the blade, which undoubtedly affects the magnitude of the force \(P_N\). Second, as the angle \(\beta\) decreases, the effect of undercutting the chip by the blade becomes more significant, since with decreasing \(\beta\) the ratio

\[ \frac{V_s}{V_N}, \]

increases, which evidently facilitates the cutting process. Third, as the angle \(\beta\) decreases, the force \(P_S\), directed parallel to the blade, increases. This force (or a part of it applied on the rake face), being an external force acting on the chip, deflects it to the side, which apparently causes the destruction of metal particles in front of the cutter blade and thereby helps reduce the force \(P_N\).

Fig. 22.

As the chip thickness increases, the decrease in the force \(P_N\) with decreasing \(\beta\) becomes less significant, which is especially clearly seen from Fig. 24, since the circumstances indicated above have a relatively smaller specific weight for a thick chip than for a thin one.

From Figs. 24 and 25 one may conclude that the specific cutting pressure \((K_s)\) cannot be considered, in the direction normal to the blade, independent of the angle \(\beta\), as was assumed by many authors as applied to the milling process.

Fig. 23.

Fig. 24.

Further, we carried out a series of experiments to determine the axial force \(P_a\) in milling with cylindrical cutters having various helix angles \(\beta\), and, measuring simultaneously the axial force on the cutter and the circumferential force \(P\), obtained the possibility of determining the ratio

\(\dfrac{P_a}{P}\). All these experiments were carried out with a uniform milling width, which ensured smooth operation of the milling cutters and accurate records from the recording instruments. The circumferential force \(P\) was determined from the torque recorded by the dynamometer mentioned above\(^{19}\). The axial force on the cutter, \(P_a\), was determined by a special measuring device, shown schematically in Fig. 26. In the body of the milling machine (1), a spindle (2) with an internal bore is mounted on bearings with tapered rollers. Into the spindle is inserted the taper of the cutter arbor 3, which carries cutter 13. The taper of the arbor does not touch the internal taper of the spindle, but rests on a ball bearing whose inner ring is firmly fastened to the taper of the arbor, while the outer ring is fitted into part 4, fastened at the front end of the spindle. The inner and outer rings of the ball bearing, between which the balls are enclosed, are made cylindrical and without grooves, owing to which they can move relative to one another in the axial direction. Between special projections of part 4 and the shoulder of arbor 3 are placed pin 6 and tension spring 7, which forcibly rotate the arbor with the cutter when the spindle rotates, but do not prevent its axial displacement under the influence of the axial force on the cutter. The arbor rests in cylindrical sliding bearing 5. Into the tapered shank of the arbor is screwed rod 8, which, passing freely through the spindle bore, bears through a thrust ball bearing against measuring diaphragm 9. The preliminary tension of the whole system is produced by spring 10, which is tightened by bolt 11, screwed into bracket 12, and through the thrust ball bearing presses on arbor 3, rod 8, and diaphragm 9.

Fig. 25.

The axial forces arising during operation of the cutter will be transmitted to diaphragm 9 and recorded by it.

When considering the scheme of Fig. 26, the supposition may arise that harmful resistances in bearing 5 will distort the value of the axial force recorded by diaphragm 9. In actuality, however, these harmful resistances will be very small for the following reasons. The maximum displacement of the piston of the diaphragm, and hence also the axial displacement of arbor 3 and rod 8, when the axial load changes from zero to the maximum, was only \(0.10\) mm.

Fig. 26.

The friction force \(F\) in bearing 5 in the direction of the circumferential velocity of the journal is

\[ F = fN, \]

where \(f\) is the coefficient of friction,
\(N\) is the reaction.

Let us take \(2N\) equal to the geometric sum of the forces on the cutter tooth:

\[ 2N=\sqrt{P^2+P_r^2} \]

\(P\) is the circumferential force on the cutter tooth,
\(P_r\) is the radial force on the cutter tooth.

Let us take \(P_r=0.50P\); then

\[ N=0.56P \]

Let us take \(f=0.1\) (in reality it will be lower). Then

\[ F=0.056P \]

Let us take the minimum ratio between the measured axial force on the cutter \(P_a\) and the circumferential force \(P\) to be

\[ \frac{P_a}{P}=0.1 \]

Then

\[ F=0.56P_a. \]

This is the magnitude of the friction force on the journal in the direction of its circumferential velocity.

To determine the friction force in the axial direction, let us assume that the load on the cutter changes from zero to maximum \(2Z\) times per revolution, where \(Z\) is the number of teeth of the cutter.

Then the friction force in the axial direction \(F_a\) will be smaller than the friction force \(F\) by the same factor by which the axial displacement of the journal is smaller than the circumferential displacement:

\[ \frac{F_a}{F}=\frac{2Z\cdot 0.1}{\pi d} \]

The journal diameter is \(d=50\) mm.
The number of cutter teeth is \(Z_{\max}=18\).

\[ F_{a\max}=\frac{0.56\cdot 2\cdot 18\cdot 0.1P_a}{\pi\cdot 50} \]

\[ F_{a\max}=0.01280\,P_a \]

In reality the magnitude \(F_a\) will be still smaller, since our experiments were conducted with a uniform width of milling, and the load fluctuations on the cutter were always very small; this was accompanied by axial displacements of the journal in bearing 5 that were even less noticeable than those assumed in the calculation. The device described was designed and manufactured in the metal-cutting laboratory of the Tomsk Industrial Institute.

The results of experiments measuring the axial forces \(P_a\) on cutters are presented in Figs. 27 and 28 in coordinates \(S_m\) (mean chip thickness) as abscissa and \(\dfrac{P_a}{P}\) as ordinate.

In Fig. 27, moreover, values calculated from equation 8 on the basis of experiments with a planing cutter are plotted with crosses. The good agreement indicates the complete identity of the processes occurring in planing and milling.

Fig. 27.

Fig. 28.

Fig. 29.                Fig. 30.

In Fig. 29 the dependence

\[ \frac{P_a}{P}=f(\beta) \]

is presented; moreover, the data relating to milling are plotted directly from Figs. 27 and 28, while the data for planing were obtained by calculation from equation 8. Fig. 29 shows an exceptionally good agreement between the results of the planing and milling experiments and, in addition, allows one to assume the same dependence for cast iron and steel.

Fig. 29 is drawn for a certain mean value of the chip thickness,

\[ \delta=S_m=0.1\ \text{mm}, \]

but in general, as Figs. 27 and 28 show,

\[ \frac{P_a}{P} \]

depends, though only to a very slight degree, on the chip thickness.

According to equation 9 for the planing experiments and according to equation 10 for the milling experiments, we calculated the quantities

\[ \frac{P_s}{P} \]

and plotted them on the graph of Fig. 30 for \(\delta=S_m=0.1\ \text{mm}\). From Fig. 30 it may be concluded that the force \(P_s\), directed parallel to the cutting edge, reaches a large magnitude at small values of the angle \(\beta\), and it can by no means be neglected. This also explains why equations 4 and 5, hitherto accepted as correct, are not confirmed by experiment, but in reality are based on a grossly erroneous conception of the relation of the forces on an inclined blade.

Since, by our conclusions, we deny views that have existed for a long time and put forward entirely new propositions for calculating the forces on a helical milling cutter, we considered ourselves obliged to carry out a direct measurement of the force \(P_s\), in order to show that Fig. 30, obtained by calculation, gives its correct magnitude, and also to show that the picture of the acting forces proposed by us (Fig. 16) is exhaustively complete and that in the plane of cutting there are no forces other than those accounted for by us.

Fig. 31.

For direct measurement of the force \(P_s\) we carried out a series of experiments on a lathe (DIP—30) with a three-component measuring carriage of the firm Losenhausen.

For carrying out the direct measurement of the force \(P_s\) we proceeded as follows. A cylindrical blank was mounted between the centers of the machine; on this blank there had previously been made a projecting ribbon helical thread with one or another helix angle of interest to us (Fig. 31). In the lathe measuring carriage a tool of claw-like form was clamped, with a blade parallel to the axis of the blank. The blade of the tool had a length exceeding the width of the thread; the tool had a rake angle \(\gamma=10^\circ\). Bringing the tool to the surface of the helical thread and setting it to the desired chip thickness, we gave the tool a feed equal to the pitch of the previously made ribbon thread, so that it cut from the surface of the thread a continuous chip of the specified thickness and of the width of the thread.

Fig. 32.

The principal cutting pressure in this case was nothing other than the force \(P_N\), and the feed pressure the force \(P_s\).

In Fig. 32 the results of the experiments are given; since in these experiments we directly measured \(P_N\) and \(P_s\), while we had no possibility of measuring the force \(P\) for different \(\beta\), the graph in Fig. 32 is given in the coordinates

\[ \frac{P_s}{P_N} - \delta \]

(chip thickness). It would have been preferable to have not

\[ \frac{P_s}{P_N}, \]

but

\[ \frac{P_s}{P} \]

for comparison with Fig. 30; however, below we shall show that \(P_N\) differs only very slightly from \(P\).

Fig. 32 gives results fully corresponding to Fig. 27. Indeed, if we assume that with increasing chip thickness at \(\beta = \mathrm{const}\) the ratio

\[ \frac{P_a}{P} \]

increases, then from equation 10 it follows that the ratio

\[ \frac{P_s}{P} \]

must decrease, which is what we observe in Fig. 32.

The decrease of the ratio

\[ \frac{P_s}{P} \]

with increasing chip thickness can be explained as follows. The force \(P_s\) is applied partly on the back face and partly on the front face of the cutter. The latter part is nothing other than the projection of the friction force between the chip and the cutter onto the direction parallel to the cutting edge. The resultant friction force is directed opposite to the motion of the chip along the front face. This direction, as we showed above, is deflected from the normal to the cutting edge. The deflection from the normal is associated with an additional deformation of the chip; moreover, a thin chip offers comparatively less resistance to deformation, is deflected comparatively more from the normal to the cutting edge, and for a thin chip the projection of the friction force onto the direction parallel to the cutting edge has a relatively greater value than for a thick chip.

In Fig. 33, based on Fig. 32, the ratio

\[ \frac{P_s}{P_N} \]

is given as a function of \(\beta\); here we have complete agreement with Fig. 30, which convinces us of the correctness of our conclusions.

In Figs. 34 and 35 we have given the ratios

\[ \frac{P_N}{P}, \]

and for the planing cutter they were obtained directly from the experiments; for milling they were calculated from equation 8, into which we substituted the experimentally obtained

\[ \frac{P_a}{P}. \]

Figs. 34 and 35 show that the force \(P_n\) is approximately equal to and differs only slightly from the force \(P\), whereas according to the previously existing notions, for \(\beta < 90^\circ\), \(P_N\) always exceeds the force \(P\) to a more or less considerable degree (equation 4).

In Fig. 36 we give a comparison of the force ratios obtained by us experimentally (solid lines) with the ratios according to the views that existed until now, from equations 4 and 5 (dashed curves). Fig. 36 requires no explanation and characterizes quite vividly the gross erroneousness of equations 4 and 5.

Fig. 33.

Visible graph labels:

• \(P_z/P\)
• \(\beta^\circ\)
• Steel
  turning
  \(\gamma = 10^\circ\)
  \(\delta = 0.1\ \mathrm{mm}\)

Fig. 34.

Visible graph labels:

• \(P_N/P\)
• Steel \(6_{\mathrm{B}} = 60\)
• ○ — milling
• × — planing
• \(\delta : S_m = 0.2\ \mathrm{mm}\)
• \(\delta : S_m = 0.1\ \mathrm{mm}\)
• \(\delta : S_m = 0.05\ \mathrm{mm}\)
• \(\beta^\circ\)

Fig. 35.

Visible graph labels:

• \(P_N/P\)
• Cast iron
• ○ — milling
• × — planing
• \(\delta : S_m = 0.2\ \mathrm{mm}\)
• \(\delta : S_m = 0.1\ \mathrm{mm}\)
• \(\delta : S_m = 0.05\ \mathrm{mm}\)
• \(\beta^\circ\)

Fig. 36.

Visible graph labels:

• \(\dfrac{P_z}{P}\)
• \(\dfrac{P_a}{P}\)
• \(\dfrac{P_s}{P}\)
• \(\beta^\circ\)

Conclusions

On the basis of the experiments carried out, it is possible to establish the correctness of the equations proposed by us (6, 7, 8, 9, 10) for the process of free cutting of metals, both in planing and lathe machining and in milling.

The ratio of the forces on the cutting blade according to Figs. 29 and 30 can be expressed with sufficient accuracy by the relations

\[ \frac{P_a}{P}=\frac{0.28}{\operatorname{tg}\beta} \tag{11} \]

\[ \frac{P_S}{P}=0.72\cos\beta \tag{12} \]

From this one may obtain the relation

\[ \frac{P_N}{P}=\frac{0.28}{\sin\beta}+0.72\sin\beta \tag{13} \]

which agrees well with the curves of Figs. 34 and 35.

For the force \(R\) we obtain the following:

\[ \frac{R}{P}=\sqrt{\frac{0.0784}{\sin^2\beta}+0.921} \tag{14} \]

Expressions 11, 12, 13, 14 are equally suitable both for cast iron and for steel within the range of angles \(\beta\) from \(20^\circ\) to \(90^\circ\); they give the greatest accuracy for a chip thickness \(\delta(S_M)=0.1\) mm, but for other values of chip thickness the deviations will be so insignificant that there is no sense in complicating them, especially since for the angles \(\beta>45^\circ\), most commonly encountered in milling, these deviations are particularly small.

The work carried out by us makes it possible to establish the correct form of the dependence of the total and specific cutting pressure on the cutting factors, and also, no less importantly, the fundamentally correct relation between the total and specific cutting pressure. To clarify this, we point out that up to the present time it has not been sufficiently clear how, from the specific cutting pressure, to pass to the total force on the cutting edge in the case of an inclined cutting edge \((\beta<90^\circ)\) in the process of free cutting.

Prof. Savvin, taking for milling the specific pressure

\[ K_s=C\delta_{\max}^{k} \]

considers that this expression makes it possible to calculate the normal force \(P_N\), by summing the specific pressure over the entire cross section of the chip, and obtains the force \(P\) (in the direction of the resultant cutting speed) by multiplying \(P_N\) by the sine of the helix angle \(\beta\). Many other authors follow the same basic path, using for the specific pressure the expression of Oxford and Airey

\[ K_s=CS_e^{k} \]

This is done by Prof. Reznikov \(^{8}\), Prof. Rudnik \(^{11}\), Prof. Krivoukhov \(^{12}\), Eng. Dembinskii \(^{20}\), and others. We did the same in our “theory of the operation of a cylindrical milling cutter with a helical tooth” \(^{21}\).

Other authors obtain the force \(P\) by distributing the specific cutting pressure not over the cross section of the chip, but over its projection onto a plane simultaneously normal to the cutting surface and to the direction of cutting motion. This is usually done in calculating the force on a lathe tool; the same method is followed in determining the circumferential force on a milling cutter by Pohl \(^{22}\), Klein \(^{26}\), and Neustadt \(^{24}\).

According to the results of our experiments, taking Figs. 18, 19, and 23 as a guide, the expression for the force \(P\) may be given in the following form:

\[ P=C_{\beta}\lambda\cdot b\cdot \delta^{x} \tag{15} \]

Here \(C_{\beta}\) is a coefficient taking into account the effect of the blade inclination,
\(\lambda\) is a coefficient depending on the material being machined and on the cutting angle.

Hence, for the specific pressure we obtain the expression

\[ K_S=\frac{P}{b\cdot\delta}=C_{\beta}\lambda\delta^{k} \tag{16} \]

where this will be the specific force referred to a unit area of the projection of the chip cross-section onto a plane normal to the direction of the cutting motion.

On the basis of the experimental material presented, which confirms the correctness of equation 6, one may conclude that by simply projecting the actual value of the normal force \(P_N\) onto the direction of the resultant velocity it is not possible to obtain the actual value of the force \(P\), since the force \(P\) also includes the projection of the force \(P_S\).

In expressions 15 and 16 the coefficient \(C_{\beta}\) has the following value.

\(\beta\)	\(C_{\beta}\) Cast iron	\(C_{\beta}\) Steel
\(90^\circ\)—\(40^\circ\)	1	1
\(35^\circ\)	1.05	1
\(30^\circ\)	1.12	1.03
\(25^\circ\)	1.2	1.14
\(20^\circ\)	1.32	1.28

In application to the milling process, the circumferential force \(P\) on the cutting edge of a helical-tooth milling cutter is obtained, taking equation 16 as a guide, in the following way.

Fig. 37.

Specific pressure

\[ K_S=C_{\beta}\lambda S_e^{k} \]

Here \(S_e\) is the variable chip thickness.

Let us isolate on the edge of the tooth a segment of differentially small magnitude \(db\) (Fig. 37). The force on this segment is

\[ dP=K_S\cdot S_e\cdot db; \]

\[ db=\frac{D}{2}\operatorname{tng}\beta\,d\psi; \]

\[ dP=C_{\beta}\lambda\frac{D}{2}\operatorname{tng}\beta\,S_z^{k+1}\sin^{k+1}\psi\,d\psi \]

Here \(S_z\) is the feed per 1 tooth of the cutter.
\(\psi\) is the current angle determining the position of the isolated segment of the tooth edge along the arc of contact.

\[ P=C_{\beta}\lambda\frac{D}{2}\operatorname{tng}\beta\,S_z^{k+1} \int_{\psi_1}^{\psi_2}\sin^{k+1}\psi\,d\psi \]

Solving the integral by Salomon’s method, we obtain

\[ P = D C_{\beta}\lambda S_z^{\kappa+1}\operatorname{tg}\beta\, \frac{2^{\kappa+1}}{\kappa+2} \left\{ \sin^{\kappa+2}\left(\frac{\psi_2}{2}\right) - \sin^{\kappa+2}\left(\frac{\psi_1}{2}\right) \right\} \tag{17} \]

Equation 17 differs from the equation for the force on a milling-cutter tooth previously derived by us¹ by the presence of the coefficient \(C_{\beta}\).

Thus, the quantity \(C_{\beta}\) must also be introduced into the equation for power in milling. In our previous milling experiments this quantity could not be detected, since we used cutters with angles \(\beta \ge 45^\circ\); in this case, as we established above, \(C_{\beta}=1\).

Equations 15 and 17, together with equations 11, 12, 13, and 14, make it possible to determine any of the forces on an inclined blade in the plane of cutting.

The author’s associate B. V. Sudnishnikov took part in carrying out the present work.

Literature Cited

1. Gippler.—Turning Practice and Its Tools.

2. Sellergren.—Das Messen des Widerstandes der Metalle bei Anwendung von Schneidestählen. Z. d. Österreichischen Ingenieur und Architekten-Vereines. 1896, No. 2.

3. Glebov.—Theory of the Most Advantageous Cutting of Metals. 1933, p. 52.

4. Ibid., p. 75.

5. Pankin.—Kinematic sharpening of cutting blades. Machines and Tools. 1936, No. 1.

6. Rosenberg.—Experimental study of the process of formation of a metallic chip. Proceedings of the Siberian Technological Institute. 1929, vol. 51, issue IV.

7. Semenchenko.—The Cutting Tool, vol. II, 1938, p. 23.

8. Reznikov.—Theory of Metal Cutting. 1934, p. 229.

9. Danielyan.—Dynamics of Milling. 1936, p. 32.

10. Sawin.—Mechanical Engineering, 1926; Genie Civil, 1925.

11. Rudnik.—Theory of Metal Cutting. 1932, p. 198.

12. Krivoukhov.—Machining of Metals by Cutting. 1932, p. 269.

13. Kurrein.—Fräser und Fräsmaschine. Werkstattstechnik, 1929, No. 9.

14. Schlesinger.—Z. d. VDY, 1929.

15. Stuck.—Fräser-Handbuch. 1933.

16. Eisele.—Dynamische Untersuchungen des Fräsvorganges. 1931.

17. Kiekebusch und Pohl.—Kräfte und Kraftschwankungen beim Fräsen. Maschinenbau, 1935, No. 9/10.

18. Boston, Gilbert and Kaiser.—Power and Forces in Milling. Trans. ASME, 1937.

19. Rosenberg.—Operation of a cylindrical milling cutter. Orgainformatsiya, 1935, No. 11.

20. Dembinsky.—Theoretical foundations of the operation of cylindrical milling cutters. Machines and Tools. 1936, No. 9.

21. Rosenberg and Sudnishnikov.—Theory of the operation of a cylindrical milling cutter with a helical tooth. Bulletin of the Metal Industry. 1933, No. 4.

22. Schröder.—Die Bedeutung der Spandicke bei Walzenfräsern. Maschinenbau, 1934, No. 19/20.

23. Klein.—Die Zeitliche Verlauf der Umfangskraft beim einem Walzenfräser. Ingenieur-Archiv, 1937, Dezember.

24. Neishtadt.—Elements of chip cross-section and the required power in the operation of a cylindrical milling cutter with a helical tooth. Bulletin of the Metal Industry. 1938, No. 1.

1. Reference number as printed in the source text. ↩