Associate Professor A. M. Rozenberg.

Operation of a Cylindrical Milling Cutter

(An Experimental Study)

Despite the considerable number of works devoted to the theoretical and experimental study of the milling process, the question of the influence of cutting factors on the power expended cannot be regarded as finally clarified. Some investigators¹ attempted to transfer to the milling process the relationships obtained for the turning process; in most cases, however, the question was subjected to experimental study.

The general tendency in the influence of the principal cutting factors (feed, depth of cut, number of teeth) on the power expended in milling was established already by the works of de Leeuw² and Reindl³, after which the works of Eirey-Oxford⁴, Sawin⁵, Beckh⁶, Salomon⁷, Bahlecke⁸, Eisele⁹, Schlesinger¹⁰, Stoewer¹¹, Besprozvannyi¹², Boston and Kraus¹³, Brödner¹⁴, Schröder¹⁵, the University of Michigan¹⁶, and others confirmed and, to a greater or lesser degree, refined the dependence of power in milling on cutting factors, without altering the general tendency—revealed by the first experiments of de Leeuw and Reidl—toward increased productivity with a transition to larger chips. In contradiction to these numerous and substantial works stands only Guvernor¹⁷, who in his theory of “rational milling,” incidentally strongly recommended in the preface by Professor Pankin, takes us back 35 years to Fischer’s¹⁸ views, asserting a direct proportionality between power, on the one hand, and feed and depth of milling, on the other; this, of course, can be regarded only as an unfortunate misunderstanding.

We still do not have an exact expression for the influence of all cutting factors on the power consumed, and the results of all the works listed above show considerable discrepancies.

The least clarified question is the influence of the helix angle of the milling-cutter tooth on milling power, and here we encounter completely opposite views. Bahlecke⁸, Eisele⁹, Schlesinger¹⁰, and Boston and Kraus¹³ do not touch upon this question; Brödner¹⁴, the experiments of the University of Michigan¹⁶, and Safonov and Novikov²³ indicate the absence of any influence of the tooth helix angle on the power consumed; Stoewer¹¹ points to an increase in the power consumed with a decrease in the helix angle (with an increase in the angle of inclination) of the spiral; Beckh⁶, Sawin⁵, Sokolov²⁰, Reznikov¹, Krivoukhov²¹, and Isaev²² assert the opposite influence.

The author of the present work published in 1933 a theory of the operation of a cylindrical milling cutter with a helical tooth²⁴, in which, in general form, the relationship between all cutting factors in milling was given; in addition, the error of a whole series of investigators in calculating the chip thickness on the helical tooth of a milling cutter was revealed—an error that led to an incorrect estimate of the influence of the helix angle on the chip cross-section, the circumferential force, and the power consumed in milling. The theory of the operation of a milling cutter with a helical tooth was given in elementary form without taking into account

some secondary factors (such as: frictional forces on the front face of the spiral tooth; the phenomenon of chip filing by the cutting edge of the tooth, as a result of the presence of a velocity component in the direction of the tooth edge; changes in the actual cutting angle), since these factors could not be taken into account theoretically, and their influence can be determined only by experiment. The proposed work was intended to verify experimentally the theory published by the author, and also to find the characteristics of various materials being machined that enter into the power equation in milling.

Method of Investigation and Apparatus Used

In investigating the milling process, a considerable number of studies have been carried out using measuring tables that record the feed force $P_h$, the vertical pressure force $P_v$, and the axial force $P_a$, which

Fig. 1

nos. of “Transactions of the TII,” 9.

does not make it possible to proceed to the expenditure of power, since to this day we do not know the exact relationship between the circumferential force on the cutter and the above-mentioned quantities. Work has been carried out by this method by Eisele ^9), Brödner ^14), Beckh ^6), Besprozvannyi ^12); the last work was accompanied by determination of the efficiency of the machine tool by braking, in order to be able to compute the effective power at the spindle from the power consumed by the motor, which can hardly give accurate results because of the variability of losses in the machine tool. Bahlecke ^8) used only the braking method, and did not even take into account additional resistances under the load of the machine tool. Boston and Kraus ^13) used a special pendulum dynamometer determining the work expended in cutting a single chip; unfortunately, nothing is known about the calibration of this dynamometer. Sawin, Salomon ^25) used special dynamometers to record the torques on the spindle of the machine tool, and Salomon’s dynamometer was practically inertia-free, which made it possible to record the change in forces and moments occurring over short intervals of time. In most cases the measuring apparatus was based on the hydraulic principle, or on the principle of elastic deformations of a standard, which introduced its own distortions into the results of the experiments owing to the inertia of the recording instruments.

A detailed description of the apparatus is given by Eisele ^26), Kurrein ^27), Schlesinger ^28).

We attach exceptional importance to the experimental procedure and to the apparatus used, since this to a considerable degree determines the reliability of the results obtained. Unfortunately, in many investigations there is either no description at all of the apparatus used, or in most cases nothing is said about its calibration, which makes it impossible to judge the accuracy of the results obtained.

We, in carrying out experiments in the metal-cutting laboratory of the Tomsk Industrial Institute, used two methods of measuring the quantities of interest to us in the milling process.

A special pendulum dynamometer was made, by means of which it was possible, with a high degree of accuracy, to determine the work expended in cutting one chip by a cutter tooth. The diagram and photograph of the dynamometer are presented in Figs. 1 and 2. On the spindle of the dynamometer, mounted in its housing on ball bearings, a cutter with one tooth (1) was fastened. For reliability of fastening, the neck of the spindle and the holes of the single-tooth cutters were made conical. On the cone the cutter was tightened with a nut and was kept from turning by a key. Under the cutter, on the guides of the instrument, there was placed a table in which the specimen was fastened by means of a special wedge clamp. The table was fastened to the guides with two bolts, after it had been fed by means of special lead screws (4), with reading by the indicator (3), to an accuracy of up to

Fig. 2

0.01 mm. After the feed had been made and the table clamped, the lever with the weight was raised to a definite height, fixed by a special pawl, and through a ratchet was coupled to the spindle and the cutter. Being released from the pawl, the lever with the weight fell, rotated the cutter, cut off one chip, and then rose, owing to the remaining reserve of energy, to a certain height, which was recorded automatically by a pointer on the scale (2). The difference between the initial reserve of work of the pendulum and its remainder was expended on cutting off one chip of definite, previously known dimensions and could easily be calculated.

The calibration of the pendulum was carried out as follows. In Fig. 3 a diagram of the operation of the pendulum is shown. The initial position of the center of gravity of the lever and weight is \(h_0\), and the reserve of work in this case is \(Qh_0\). The final point to which the center of gravity rises is \(h_1\), and the remaining work in this case is \(Qh_1\).

The expenditure of work for cutting off one chip, or, what is the same thing, the work of one tooth, which we shall hereafter denote by \(A_z\), is:

\[ A_z = Q\,(h_0 - h_1). \]

Fig. 3

If the distance from the axis of rotation to the center of gravity of the rotating system is denoted by \(R\), then

\[ A_z = Q \cdot R \,[\sin \alpha_0 - \sin(\alpha_1 - 90^\circ)], \quad \text{for } \alpha_1 > 90^\circ \]

and

\[ A_z = Q \cdot R \cdot [\sin \alpha_0 + \cos \alpha_1] \quad \text{for } \alpha_1 < 90^\circ. \]

Thus, in order to find \(A_z\), it is necessary to know \(QR\) and to record, during the experiments, the angles \(\alpha_0\) and \(\alpha_1\).

\(QR\) is nothing other than the moment of the weight and lever with respect to the axis of rotation, and was determined by us with the aid of a decimal balance.

The angle \(\alpha_0\) was determined by us as the angle through which the pendulum rose during its free fall without cutting a chip; in this way the harmful resistances of the idle stroke were eliminated of themselves. The angle \(\alpha_1\) was automatically recorded by a pointer on the scale after the chip had been cut off. During \(1\frac{1}{2}\) years of work with the apparatus, the angle \(\alpha_0\) fluctuated by no more than \(\pm 0.15^\circ\). It was determined regularly before each series of experiments.

During cutting, additional forces arose on the spindle of the apparatus: in the case of a straight-tooth cutter, radial forces; in the case of a helical cutter, in addition, axial forces as well, causing additional harmful resistances in the bearings. These additional harmful resistances were determined by us by means of a special calibration.

To determine the additional harmful resistances arising under the influence of radial forces, a ball bearing (3) was mounted on the spindle of the apparatus in place of the cutter; through this bearing various loads were applied to the spindle, and the corresponding change in the height of rise of the pendulum was deter-

the work expended in overcoming the harmful resistances in the three bearings (see Fig. 4). Subsequently it was assumed that the work of the harmful resistances of the instrument is equal to \(^{2}/_{3}\) of the work determined in this way (although, in calibration, it could have been set equal to \(^{1}/_{2}\) of the work of the harmful resistances, since the force acting on bearing 3 is equal to the sum of the forces acting on bearings 1 and 2, being their resultant, and the harmful resistances in bearing 3 must constitute more than \(^{1}/_{3}\) of the harmful resistances in all three bearings). From the experimental results the diagram in Fig. 4 was constructed, where along the ordinate axis is plotted the work of the harmful resistances for one swing of the pendulum, \(A_0\), in kg·mm, and along the abscissa axis the circumferential force on the cutter, i.e. the cutting force \(P\). For the transition from the force \(R^*)\) to the force \(P\) it was assumed that the radial component in milling is equal to one half of the circumferential cutting force, i.e.

Fig. 4

\[ P_r = 0.5 \cdot P \]

although Kurrain gives \(P_r = (0.3—0.4)\cdot P^{27})\).

Thus:

\[ R=\sqrt{P^2+P_r^2}=\sqrt{P^2+(0.5P)^2}=1.12P. \]

\[ P=0.893R. \]

From Fig. 4 one can find:

\[ A_0=0.225P. \]

If we assign the work of the harmful resistances to the circumference of the cutter, then we obtain an additional force on the cutter tooth, \(P_{\text{add.}}\), which distorts the instrument readings in the experiments as a result of the presence of harmful resistances in the instrument. We may write:

\[ A_0=P_{\text{add.}}\cdot \frac{D}{2}(a_0+a_1) \]

where: \(D\) is the cutter diameter.

\[ \begin{aligned} a_0&\text{— the angle of rise of the pendulum before its swing,}\\ a_1&\text{— the angle of rise of the pendulum after the swing.} \end{aligned} \]

\(a_0+a_1\) is the total angle traversed by the pendulum in one swing. In our case \(a_0=145^\circ=\text{const}\); \(a_1\) during calibration decreased with increasing force \(R\); at \(R=0\), \(a_1=144^\circ\); at \(R=1250\) kg, \(a_1=142^\circ\).

\[ \text{*) Here } R \text{ is the resultant force on the spindle.} \]

For the calculations it was assumed that

\[ (\alpha_0+\alpha_1)=145+142=287^\circ. \]

Thus:

\[ A_0=P_{\text{add.}}\cdot \frac{D}{2}(\alpha_0+\alpha_1) =P_{\text{add.}}\cdot \frac{D}{2}\cdot \frac{2\pi}{360}\cdot 287 \]

Whence

\[ P_{\text{add.}}=0.4\,\frac{A_0}{D}. \]

But according to Fig. 4

\[ A_0=0.225\cdot P. \]

Substituting, we obtain

\[ P_{\text{add.}}=0.09\,\frac{P}{D}, \]

i.e., the additional force on the cutter (as a result of the harmful resistances of the apparatus) will be greatest at the minimum cutter diameter. In our experiments with a pendulum diameter \(D_{\min}=60\) mm, and for this case

\[ P_{\text{add.}}=0.09\,\frac{P}{60}=0.0015\cdot P. \]

Such is also the ratio between the work of cutting and the work of harmful resistances. In other words, the harmful resistances of the apparatus distort its readings by only \(0.15\%\), which, of course, may be completely disregarded.

In the same manner, the harmful resistances of the apparatus arising under the influence of axial forces were determined (see Fig. 5).

Bearing 3 in this case was mounted on the end of the spindle, and through it an axial load was applied to the spindle during calibration.

Fig. 5

In this case, at \(R=900\) kg, \(\alpha_1=138^\circ\), we obtained

\[ A_0=P'_{\text{add.}}\cdot \frac{D}{2}\cdot \frac{2\pi}{360}\cdot 283 \]

\[ P'_{\text{add.}}=0.405\cdot \frac{A_0}{D}. \]

According to Fig. 5,

\[ A_0 = 0.83 R. \]

Moreover,

\[ R=\frac{P}{\operatorname{tng}\beta}, \]

where \(\beta\) is the helix angle of the cutter tooth.

Hence

\[ P'_{\text{add.}}=0.405\cdot 0.83\cdot \frac{P}{D\operatorname{tng}\beta} =\frac{0.336P}{D\operatorname{tng}\beta}. \]

In the experiments \(\beta_{\min}=45^\circ;\ D_{\min}=60\) mm, and the maximum value of \(P'_{\text{add.}}\) will be:

\[ P'_{\text{add.}}=\frac{0.336}{60}\cdot P=0.0056P. \]

That is, the force and work of the detrimental resistances distort the experimental readings by only \(0.56\%\), which likewise need not be taken into account.

Thus we have satisfied ourselves as to the sufficient accuracy of our instrument. On the pendulum dynamometer we could use cutters with diameters from 55 mm to 100 mm. The milling width that could be used in the experiments was up to 25 mm and was determined by the width of the specimen clamped in the table. After the specimen had been fastened in the table, the table was mounted on the milling-machine table, where, by milling, the specimen was given a strictly horizontal surface and, at the same time, a definite milling depth \(t\) was produced for work on the dynamometer. The length of the specimen was 65 mm, which made it possible, at one milling depth \(t\) and with various feeds, to obtain 150–300 experiments without resetting the specimen. The milling depth was determined accurately with an indicator.

The pendulum of the dynamometer had a work reserve of 38.57 kilogram-meters, and this made it possible to remove chips of considerable size. During the experiments we brought the feed per tooth up to 1 mm. In operation the pendulum dynamometer showed good qualities, especially in cast-iron experiments, as can be seen from the sample record given below (see Table 1).

In working on steel, we had somewhat less constancy of readings (with \(S_z=\mathrm{const};\ t=\mathrm{const};\ b=\mathrm{const}\)), evidently because, when removing chips from a ductile metal, irregularities and tears were formed at the place where the tooth emerged from under the chip, affecting the expenditure of work in cutting the next chip.

Protocol No. 54, 29/VI-34.

Table 1.

Machined material: cast iron \(H=134\).

Cutter of carbon steel. Sharpening 2/VI-34.

\[ D=60.5\text{ mm},\quad \gamma=0^\circ,\quad \beta=45^\circ;\quad \delta=5^\circ \]

Milling width \(b=25\) mm. Milling depth \(t=7\) mm. \(\alpha_0=144^\circ\).

Experiment Nos.	\(S_z\), mm	\(\alpha_0\)	\(A_z\), kg mm	\(A_z\), average	Note
1223	0.05	134.8	2222	2400	\(S_z\) — feed per 1 tooth
1224	0.05	134.9	2196	2400	\(A_z\) — work for cutting one chip.
1225	0.05	133.6	2500	2400	\(A_z\) — work for cutting one chip.
1226	0.05	133.6	2500	2400	\(A_z\) — work for cutting one chip.
1227	0.05	133.5	2565	2400	\(A_z\) — work for cutting one chip.

Experiment Nos.	$S_z$, mm	$\alpha_0$	$A_z$, kg mm	Mean $A_z$	Note
1228	0.1	128.5	3970	3960
1229	0.1	128.6	3940	3960
1230	0.1	128.9	3860	3960
1231	0.1	128.5	3970	3960
1232	0.1	128.0	4120	3960
1233	0.1	128.7	3920	3960
1234	0.2	121.0	6260	6250
1235	0.2	121.2	6200	6250
1236	0.2	120.7	6350	6250
1237	0.2	121.2	6200	6250
1238	0.2	121.1	6230	6250
1239	0.3	114.5	8400	8180
1240	0.3	115.0	8230	8180
1241	0.3	115.6	8030	8180
1242	0.3	115.0	8230	8180
1243	0.3	115.7	8000	8180
1244	0.4	110.5	9780	9810
1245	0.4	110.6	9740	9810
1246	0.4	110.5	9780	9810
1247	0.4	110.0	9950	9810
1248	0.5	105.3	11620	11710
1249	0.5	105.0	11720	11710
1250	0.5	104.0	11750	11710
1251	0.5	105.0	11720	11710
1252	0.6	100.6	13320	13230
1253	0.6	100.9	13210	13230
1254	0.6	101.0	13170	13230
1255	0.8	92.0	16500	16463	Control: at $S_z = 0.1$ mm gave 128.8° = 3890 kg mm
1256	0.8	92.2	16425	16463	Control: at $S_z = 0.1$ mm gave 128.8° = 3890 kg mm
1257	0.8	92.1	16463	16463	Control: at $S_z = 0.1$ mm gave 128.8° = 3890 kg mm

When cutting cast iron, the line along which the tooth emerged from under the chip was completely even. Therefore, in machining cast iron it was considered necessary, under constant conditions, to carry out not fewer than three tests, and in machining steel not fewer than five.

The single-tooth milling cutters used in work with the dynamometer had the dimensions and tooth profile given in Table 2.

Table No. 2

No.	Cutter diameter $D$, mm	Cutter material	Helix angle $\beta$	Rake angle $\gamma$	Clearance angle $\delta$
1	60	Carbon steel	90°	0°	5–7°
2	60	Carbon steel	75°	0°	5–7°
3	60	Carbon steel	65°	0°	5–7°
4	60	Carbon steel	45°	0°	5–7°
5	79	High-speed steel	90°	0°—5°—10°—15°	19°—14°—9°—−4°

Cutter 5, shown in Fig. 6, was made with an inserted tooth in the middle; moreover, the four slots in the body of the cutter were made with different inclinations to the radial plane, so that by transferring the tooth from one slot of the cutter to another we thereby changed the rake angle. This made it possible for us to carry out experiments with different rake angles, from

Fig. 6

\(\gamma = 0^\circ\) to \(\gamma = 15^\circ\), with one and the same tooth, having one and the same condition of the cutting edge, and thus to determine precisely the influence of the rake angle on the work expenditure. We considered that the condition of the edge practically did not change over the course of the 200–250 experiments required for the investigation, at one milling depth and various feeds with four rake angles; this was also confirmed by special check experiments.

When the tooth was transferred to another slot, the clearance angle inevitably changed within the limits from \(\delta = 19^\circ\) (at \(\gamma = 0\)) to \(\delta = 4^\circ\) (at \(\gamma = 15^\circ\)). By a specially conducted series of experiments it was established that a change in the clearance angle \(\delta\) within the limits from \(4^\circ\) to \(19^\circ\), with all other conditions constant, has absolutely no effect on the work expenditure.

The use of a pendulum dynamometer enabled us, with a small expenditure of material, to carry out a large number of experiments of high accuracy. In addition, when working with the dynamometer we were entirely free from the influence on the results of the experiments of undesirable factors inevitably accompanying work on a milling machine with normal cutters, namely cutter runout and varying degrees of sharpness of the cutter teeth.

In parallel with the experiments on the pendulum dynamometer, we carried out a large number of experiments on a milling machine of the Tula plant, having 12 spindle speeds and 12 feeds from 20 to 700 mm per minute. The machine was equipped with a special self-recording hydraulic dynamometer of the author’s design, manufactured in the metal-cutting laboratory of the Tomsk Industrial Institute and built into the speed gearbox of the machine. The dynamometer recorded the torques on the machine spindle during operation. The principle of operation of the dynamometer is clear from Fig. 7.

On the shaft that drives the spindle of the milling machine, gear 3 is mounted on a key; rotating together with the shaft, through gear 4 it rotates gear 5, which is connected (by a special coupling not shown in the diagram) with the spindle; the intermediate gear 4 rotates on ball bearings in two brackets 6, which in turn are mounted on ball bearings on the shaft on which gear 3 is keyed.

The earrings are firmly connected to one another. When torque is transmitted to the spindle through gears 3, 4, 5, the circumferential forces arising on gear 4 tend to turn it together with earrings 6 clockwise (if the spindle also rotates clockwise). But the earrings bear, through a special pin, against hydraulic capsule 7, fastened to the side wall of the machine. In Fig. 8, on the right wall of the machine, the fastened capsule is visible (shown by an arrow). Since the liquid in the capsule is practically incompressible, earrings 6 move hardly at all; at the same time, the pressure on the piston of the capsule will be the greater, the greater the torque on the spindle, which can thus be recorded by a self-recording pressure gauge on a moving paper tape.

Fig. 7

A special spring is provided to draw earrings 6 toward capsule 7, which creates an initial pressure and eliminates all backlash and vibration.

Calibration of the dynamometer was carried out with a “Prony” brake. The calibration curves are shown in Fig. 9. The same figure shows the points of direct (independent) calibration of one capsule 7 outside the dynamometer. The points of this calibration fell exactly on the calibration curve of the dynamometer in the machine, which indicates imperceptibly small losses in the spindle bearings; this is quite natural, since the machine spindle is mounted on bearings with tapered rollers. Fig. 10 shows the displacement of the capsule piston under various loads. It can be seen that, at the maximum spindle torque of 40,000 kg mm, the displacement of the capsule piston is only 0.16 mm, while the displacement of gear 4 of the dynamometer is only 0.1 mm. If one takes into account that there is always an initial tension provided by the spring, and that during operation with accurately ground milling cutters the load ...

never falls from the maximum to zero, one can understand why the dynamometer gives good results in operation. We carried out the calibration several times at more or less long intervals of time, and we always obtained one and the same calibration curve. The zero of the dynamometer, set before work and approximately equal in our case to 25% of the maximum load, never drifted during operation, which also testifies to the high qualities of the dynamometer.

4) In order to take account of the additional parasitic resistances in the bearings of the arbor and of the spindle during operation, arising under the influence of the forces on the cutter, an additional calibration was performed, the results of which, with respect to the radial forces, are presented in Fig. 11. The axial forces did not produce any additional resistances in the spindle bearings, which was also tested. Therefore it may be assumed that the additional moments \(M_{\text{add.}}\) according to Fig. 11 arise not in the spindle bearings (tapered roller bearings), but in the plain bearings of the outer end of the cutter arbor. Thus, even with the latest types of electrical dynamometers, it does not appear possible to free oneself from these additional resistances. The \(M_{\text{add.}}\) were subtracted by us from the moments obtained in the experiments, thanks to which we obtained the pure cutting moment.

Fig. 8

5) Through the dynamometer we could obtain 6 different spindle speeds. With the aid of a special locking device it was possible to disengage the dynamometer and thereby obtain 12 different spindle speeds. Thus the machine was always ready for operation without the installation of any additional attachments, which created favorable conditions for carrying out the experiments.

Our dynamometer, of course, owing to its “inertia,” could not register with absolute precision the changes in torque on the spindle occurring over very short intervals of time; therefore, in the work, we always chose such a milling width \(b\) that was connected

with the cutter diameter \(D\), the helix angle \(\beta\), the lead \(H\) of the cutter tooth helix, and the number of teeth \(Z\) by the relation:

\[ b = c \cdot \frac{H}{Z} = c \cdot \frac{\pi \cdot D \cdot \operatorname{tg}\beta}{Z}, \]

where \(c\) is any integer.

Fig. 9

For this milling width, the total cross-section of the chip on all simultaneously working teeth of the cutter is constant, both in magnitude and in shape; the total circumferential force and the torque on the cutter are likewise constant quantities under the prescribed operating conditions; and if there are any oscillations due to the unavoidable runout of the cutter, these oscillations occur once per revolution of the cutter and can be accurately recorded by a dynamometer \(^{24}\). In Fig. 12 two dynamometer records are shown, one for \(b = \dfrac{H}{Z}\), the other for \(b < \dfrac{H}{Z}\). In the second, the marks are clearly visible—

… corresponding to each tooth of the milling cutter, which also indicates the sensitivity of the dynamometer.

The cutters used in the experiments on the machine are shown in Fig. 13. They

Fig. 10.

were sharpened by us without being removed from the arbors; for this purpose the arbor with the cutter was mounted, by means of a special fixture, between the centers of a grinding

Fig. 11.

machine. In this way we were able to sharpen the cutters more accurately and to keep the maximum eccentricity (runout) within 0.02 mm.

Methodology for Processing the Results of the Experiments

Thus, our experiments were carried out by two entirely different methods: when using the pendulum dynamometer, the expenditure of work for cutting one chip was determined; when working on the milling machine, the hydraulic dynamometer recorded the torque on the cutter. In the event of sufficiently close agreement between the results of experiments conducted by these two different methods, with different measuring apparatus, we had a complete guarantee of the accuracy of the results obtained.

According to the equation for power in milling that we derived earlier \(^{24}\):

\[ N_e = \frac{2^k+1}{K+2}\cdot \frac{\lambda \cdot n^{-k}}{1000\cdot 4500}\cdot b\cdot S^{k+1}\cdot t^{\frac{k+2}{2}}\cdot Z^{-k}\cdot D^{-\frac{k}{2}} \ldots\ldots\ldots 1 \]

Power depends on:

\(n\)—the number of revolutions of the cutter per minute.
\(b\)—the width of milling in mm.
\(S\)—the feed per minute in millimeters.
\(t\)—the depth of milling in mm.
\(Z\)—the number of teeth of the cutter.
\(D\)—the diameter of the cutter in mm.
\(k\)—the characteristic of the material being machined.
\(\lambda\)—the characteristic of the material being machined and of the tooth profile of the cutter.

In addition, the feed per 1 tooth is

\[ S_z=\frac{S}{n\cdot z}. \]

Equation 1 was derived by us, taking as its basis the equation for specific pressure in milling according to the data of Airey—Oxford:

\[ K=\lambda \cdot S_e^k \ldots\ldots\ldots\ldots\ldots\ldots\ldots 2 \]

where: \(S_e\) is the chip thickness at the given point of the tooth at the given moment.

Since the main purpose of our work was the experimental verification of the correctness of Eq. 1 and the determination, for various materials being machined, of the values \(K\) and \(\lambda\) for equation 1, the logical course of the work should have been as follows:

1) Having cutters of one and the same diameter, with one and the same number of teeth and tooth profile, but with different helix angles, to investigate the influence of the helix angle on the power expended under various regimes.

2) Having cutters of one and the same diameter, with one and the same helix inclination and tooth profile, but with different numbers of teeth, to investigate the influence of the number of teeth under various regimes.

3) In the same way, to vary the diameter while keeping the other conditions constant, and to investigate the influence of diameter.

4) To vary the tooth profile, and so on.

For the purpose of simplifying the work without loss of its practical accuracy, we chose another method, which will become clear from certain transformations of equation 1 and equation 3, which characterizes the expenditure of work on one tooth in one revolution, i.e. on the removal of one chip \(^{24}\):

\[ A_z= \frac{2^{k+1}}{K+2}\lambda\cdot b\cdot S_z^{k+1}\cdot t^{\frac{k+2}{2}}\cdot D^{-\frac{k}{2}} \ldots\ldots\ldots 3 \]

Let us divide the right- and left-hand sides of the equation by the volume of the chip removed

On these materials we carried out 3500 experiments. On the machine tool, experiments were carried out on all the indicated materials; on the pendulum dynamometer, of these materials there were tested: cast iron \(H = 140\); steel of medium hardness and chromium steel. The working conditions in the experiments were varied

Fig. 12.

Fig. 13.

within the following extreme limits: milling depth from 0.5 mm to 9.6 mm; feed per 1 tooth from 0.018 mm to 1.05 mm.

\[ \frac{A_z}{t \cdot b \cdot S_z} = \frac{1000 \cdot 4500\,N_e}{t \cdot b \cdot S_z \cdot Z \cdot n} = \frac{1000 \cdot 4500 \cdot N_e}{t \cdot b \cdot S} \]

\[ \frac{1000 \cdot 4500 \cdot N_e}{t \cdot b \cdot S} = \frac{2^{k+1}}{K+2}\cdot \lambda \cdot n^{-k}\cdot S^k \cdot Z^{-k}\cdot t^{\frac{k}{2}}D^{\frac{k}{2}} \]

\[ \frac{1000 \cdot 60 \cdot 75\,N_e}{t \cdot b \cdot S} = \frac{2^{k+1}}{K+2}\lambda \left(S_z \cdot \sqrt{\frac{t}{D}}\right)^k \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \quad 8 \]

In the right-hand side of equation 8 we have obtained the same thing as in the right-hand side of equation 4. In the left-hand side of equation 8 we have in the numerator the power \(N_e\), converted by multiplication by \(1000 \cdot 60 \cdot 75\) into work in kilogram-millimeters per minute, and in the denominator the volume of chips removed per minute. That is, the left-hand side of equation 8 represents the specific work \(A_w\), and we obtain equation 7.

\[ A_w = C \cdot S_{\mathrm{m}}^K \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \quad 7 \]

This equation tells us:

If equation 1 has been correctly derived by us and expresses the actual relation between the power expended and the cutting factors, then in that case the specific consumption of work in machining the given material depends only on the mean chip thickness. The mean chip thickness is the result of the quantities \(S\), \(n\), \(Z\), \(S_z\), \(t\), \(D\), and whatever feeds, numbers of teeth, numbers of revolutions, milling depths, and cutter diameters we may work with, if in combining these various quantities in different cases of work we have one and the same mean chip thickness, the specific consumption of work must be one and the same. This will serve as proof of the correctness of equation 1.

The same conclusion, as applied to a straight-tooth cutter, only with certain assumptions and by a graphical method, was reached by Zinke, Germar, and Ehrenreich¹. We see that here there are no assumptions, and relation 7 follows in a strictly mathematical manner from relations 1 and 3.

If relation 7 is represented graphically in logarithmic coordinates \(S_{\mathrm{m}}\) (abscissa) and \(A_w\) (ordinate), we obtain a straight line with angle of inclination \(\alpha\) to the axis of abscissas, where \(\operatorname{tg}\alpha = K\); this straight line, at \(S_{\mathrm{m}} = 1\) mm, gives the value \(A_w = C\), and from this, knowing \(K\), it is not difficult to determine the quantity \(\lambda\)

\[ \lambda = \frac{K+2}{2^{k+1}}\cdot C. \]

Thus, if the results of a series of experiments (on the given machined material with the given tooth profile) with different feeds, milling depths, milling widths, cutter diameters, numbers of teeth, and helix leads are expressed through the specific work \(A_w\), and if from the experimental data the graph \(A_w = f(S_{\mathrm{m}})\) is constructed, then on this graph the points of all experiments must lie on one straight line. This will serve, on the one hand, as proof of the correctness of equation 1, and on the other hand will make it possible to determine, from a large number of experiments, the values of \(K\) and \(\lambda\). This method of processing experimental results was adopted by us.

Milling cast iron

First of all, the question of the influence of the helix angle of the tooth \(\beta\) on the consumption of work in cutting cast iron \(H = 140\) was investigated. For this purpose, on a pendu-

by one tooth, i.e., we shall refer the work to a unit volume of chip. In this case we obtain

\[ \frac{A_z}{t \cdot b \cdot S_z} = \frac{2^{k+1}}{K+2}\,\lambda \cdot S_z \cdot \frac{t^{\frac{k}{2}}}{D^{\frac{k}{2}}}, \]

or

\[ \frac{A_z}{t \cdot b \cdot S_z} = \frac{2^{k+1}}{K+2}\cdot \lambda \cdot \left[ S_z \cdot \left(\sqrt{\frac{t}{D}}\right)^k \right] \qquad \ldots\ldots\ldots\ldots 4 \]

But

\[ \sqrt{\frac{t}{D}}=\sin\frac{\varphi}{2}, \]

where: \(\varphi\) is the contact angle over which the chip is removed by the tooth of the milling cutter (see Fig. 14).

And hence

\[ S_z \cdot \sqrt{\frac{t}{D}} = S_z \cdot \sin\frac{\varphi}{2} = S_m \qquad \ldots\ldots 5 \]

\(S_m\) is the chip thickness at the middle of the contact angle, or, in other words, the “mean chip thickness.” That \(S_m\) is indeed the chip thickness at the middle of the angle \(\varphi\) is easy to see by comparing equation 5 with the equation for the chip thickness at any current angle \(\psi\) ²⁴)

\[ S_e = S_z \cdot \sin \psi \]

In the present case \(\psi=\frac{\varphi}{2}\), and \(S_e=S_m\).

Substituting equation 5 into 4, we obtain

\[ \frac{A_z}{t \cdot b \cdot S_z} = \frac{2^{k+1}}{K+2}\,\lambda \cdot S_m^k \qquad \ldots\ldots\ldots\ldots 6 \]

Fig. 14.

The work \(A_z\), referred to the chip volume \(t \cdot b \cdot S_z\), will be called the specific work and denoted by \(A_w\). The quantities \(\frac{2^{k+1}}{K+2}\cdot\lambda\) depend only on the material being machined and on the profile of the cutter and may be denoted by the constant \(C\). Then we have:

\[ A_w = C \cdot S_m^k \qquad \ldots\ldots\ldots\ldots 7 \]

Equation 7 can also be obtained from the equation for the power consumption 1.

\[ N_e=\frac{A_z \cdot Z \cdot n}{1000 \cdot 4500}. \]

\[ A_z=\frac{1000 \cdot 4500}{Z \cdot n}\cdot N_e \]

The indicated experiments with different helix angles were carried out with a sharp cutting edge. After each series of experiments, the conditions of the first experiment of the series were repeated, in order to determine, by comparing the results, whether any dulling had occurred. The straight lines located lower in Figs. 15 and 16 refer to sharp cutters.

In the same figures, under the designation “dulled tooth,” the straight lines located higher have been plotted; these will be discussed below.

Each point is the result of no fewer than 3–5 experiments. From Figs. 15 and 16, for a sharp tooth, it may be concluded that the helix angle \(\beta\) has absolutely no effect on the work consumption. It does not appear possible to state that any one of the cutters everywhere gave points located higher or everywhere lower than the other cutters. Some very slight scattering of the points may, admittedly, be explained by the nonuniform hardness of the cast iron.

For clarity, we give the log of experiments with two extreme values of the angle \(\beta = 90^\circ\) and \(45^\circ\) (Table 5).

Table No. 5

Cast iron \(H = 130—150\); \(b = 25\) mm; \(t = 2.6\) mm
\(D = 61\) mm; \(\gamma = 0^\circ\)

\(S_z\) mm	\(A_z\) at \(\beta = 90^\circ\), kg mm	\(A_z\) at \(\beta = 45^\circ\), kg mm	Remark
0.1	1250	1180	Each value is the average of 3–5 experiments.
0.2	2590	2430	Each value is the average of 3–5 experiments.
0.3	3310	3310	Each value is the average of 3–5 experiments.
0.4	4030	4120	Each value is the average of 3–5 experiments.
0.5	4560	4900	Each value is the average of 3–5 experiments.
0.6	5360	5450	Each value is the average of 3–5 experiments.
0.8	6480	6420	Each value is the average of 3–5 experiments.
0.1	1240	1250	Controls.

After the experiments with sharp cutters, these single-tooth cutters were mounted on the arbor of the milling machine and put to work until all of them had acquired the same dulling, characterized by an increase in the work consumption for cutting one chip at \(b = 25\) mm, \(S_z = 0.3\) mm, \(t = 4.5\) mm by 35% in comparison with sharp cutters. This dulling is not yet limiting, and we selected it only in order to check whether a dulled tooth would give a different regularity than a sharp one. From the upper straight lines in Figs. 15 and 16 we see that, for a dulled tooth, the helix angle \(\beta\) has no effect whatever on the work consumption. These same cutters, dulled to the same degree, were subsequently also used in experiments on steel.

In parallel with these experiments, experiments were carried out on a milling machine with cast irons \(H = 100—105\), \(H = 140\), and \(H = 170\). The results of the experiments are presented in Figs. 17, 18, 19. Here again the results are given for work with a sharp and with a dulled cutter. The dulled cutter was far from destruction of the cutting edges and gave a completely normal machined surface. To produce dulling, the cutter worked for 30 min at a speed \(v = 24\) m/min, with a feed \(S = 100\) mm/min, and with a depth \(t = 2\) mm on rail steel. The degree of dulling of the single-tooth cutters was taken to be considerably greater than that of a normal multi-tooth cutter.

From Figs. 17, 18, 19 it may be concluded that experiments carried out with different \(D, S, t, Z, b, \beta\) fall exactly on one straight line, which

Experiments were carried out on a Martens dynamometer with cutters \(D = 61\) mm with an undercut tooth, having helix angles \(\beta = 90^\circ\) (straight-tooth cutter); \(\beta = 75^\circ\); \(\beta = 65^\circ\), and \(\beta = 45^\circ\). The experiments were conducted with a milling width \(b = 25\) mm, milling depths \(t = 7\) mm and \(t = 2.5\) mm. The feed per tooth was varied from \(S_z = 0.05\) mm to \(S_z = 0.8\) mm. The results

Fig. 15.

Fig. 16.

of the experiments for \(\beta = 45^\circ, 65^\circ, 75^\circ\) are presented in Fig. 15; for \(\beta = 90^\circ\), in Fig. 16. In Fig. 16, points are also plotted that relate to the cutter \(D = 79\) mm, at \(t = 3.0\) mm and \(t = 5\) mm.

a slight discrepancy, which can be explained by nonuniformity of the material, by the different degree of sharpness of the milling-cutter teeth, by the unavoidable

Fig. 19.

runout of the cutter, etc. That these discrepancies are insignificant may be judged by comparing these figures with Fig. 20, which presents, for comparison, the results of Bahlecke’s experiments\(^{8}\) on cast iron \(H = 190\), processed by us. Here we see incomparably larger discrepancies even within the limits of experiments carried out by one method. From the graphs shown it is not difficult to obtain the values of \(K\) and \(\lambda\) for equation 1. But first let us point out that, from these same graphs, it is evident that the value of \(K\), as the tangent of the angle of inclination of the straight line to the axis of abscissas, will be greater for a dulled tooth than for a sharp one.

Fig. 20.

We investigated the character of the change in the value \(K\) with increasing dulling of the tooth. A carbon-steel cutter with one tooth was put to work on cast iron at \(t = 3\) mm, \(S_z = 0.025\) mm, and \(v = 55\) m/min. After certain intervals of time the cutter was removed from the machine and placed on a pendulum dynamometer, where the necessary experiments were carried out. The results are presented in Fig. 21, where three curves are plotted giving the change in work consumption at three different feeds as a function of the time of operation of the cutter on the machine, and the curve \(K = f(T)\), giving the character of the change in the value \(K\) with the operating time of the cutter, i.e., with the degree of dulling of the tooth.

From Fig. 21 it can be seen that one and the same dulling is characterized, for thin chips, by a more significant increase in work consumption than for thick ones. According to Fig. 21, at \(T = 28\) minutes, for \(S_z = 0.5\) mm we have an increase in work consumption of only 10%, while for \(S_z = 0.1\) mm the increase reaches 60%. This is quite understandable, since a thin chip bears against the very edge of the cutter, the condition of which is the principal factor determining work consumption. A thick chip, bypassing the edge, bears against the front face of the tooth. At \(T = 43\) min. destruction of the tooth was observed, manifested by chipping of the cutting edge from the flank face.

categorically confirms the accuracy of equation 1 derived by us for the power consumption in milling, and also the accuracy of equation 2 for the specific cutting pressure, which we adopted as the starting point.

The helix angle \(\beta\) has no influence whatever on the rate of work consumption, and therefore on the average power in milling cast iron, both with a sharp and with a blunted cutter tooth.

Our experiments on cast iron were carried out over wide ranges of feeds and depths. The maximum feed per 1 tooth for a cutter with \(D = 46\) mm and \(Z = 8\)

Fig. 17.

reached the value \(S_z = 1.05\) mm, which, with the normal value of cutting speed for a high-speed-steel cutter on cast iron of medium hardness, \(v = 20\) m/min.\(^{29)}\), gives a feed in millimeters per minute of \(S = 1100\) mm/min. Thus we have fully covered, and believe we have even exceeded,

Fig. 18.

the limits of feeds used in roughing work. This serves as a sufficiently good indication of the applicability of our conclusions to practice.

Comparing Fig. 15 (the results of experiments with the pendulum dynamometer) with Fig. 18 (experiments on the same cast iron on the machine), one may note some-

…given in Ivanov’s work², can hardly be recommended for so expensive a tool as a milling cutter. From Fig. 21 it is evident that, from the very beginning of operation, the value \(K\) (the quantity entering into the exponents of the variables of equation 1) quickly begins to increase and varies within wide limits, from \(K=-0.31\) for a freshly sharpened cutter to \(K=-0.62\) for a dulled one. This must be taken into account when establishing the coefficients and exponents in equation 1, since if we give them for a freshly sharpened cutter, then after a short interval of cutter operation this equation will give an energy consumption lower than the actual one.

Fig. 22.

Figure 22 presents the dependence of the work expenditure at various clearance angles, from \(\gamma=0^\circ\) to \(\gamma=15^\circ\). The experiments were performed on a pendulum dynamometer with one and the same tooth, reset into different slots of the holder (see Fig. 6).

The influence of the clearance angle on the work expenditure in machining cast iron may be expressed by coefficients (Table 6).

Table No. 6

Clearance angle \(\gamma\)	Coefficient of work expenditure	Note
\(0^\circ\)	1
\(5^\circ\)	0.96
\(10^\circ\)	0.92
\(15^\circ\)	0.87
\(20^\circ\)	0.82	Presumably.

On the basis of experiments carried out with cast iron, we have the following values of the constants in the equation of power and specific pressure (Table 7).

Table No. 7

No.	Material machined	\(\lambda\)	\(K^\*\), sharp cutter	\(K^\*\), dulled cutter	Note
1	Cast iron \(H=100\)	50	\(-0.4\)	\(-0.51\)	Fig. 17
2	Cast iron \(H=140\)	60	\(-0.39\)	\(-0.52\)	Fig. 18
3	Cast iron \(H=170\)	75	\(-0.34\)	\(-0.46\)	Fig. 19
4	Cast iron \(H=190\)	95–115	\(-0.29\)	—	Fig. 20 (after Bahlecke)

edge of the tooth. Between \(T = 30\) minutes and \(T = 40\) minutes we see in Fig. 21 a more rapid increase in the expenditure of work than before this interval of time. At \(T = 43\) minutes, when the tooth may be considered sufficiently dulled and in need of regrinding, the increase in the expenditure of work at \(S_z = 0.1\) mm was \(112\%\), while at \(S_z = 0.5\) mm it was only \(29\%\). The figures for the percentage increase in the expenditure of work are marked on the curves in Fig. 21. Experiments carried out in the same way on the gradual dulling of a normal (multi-tooth) high-speed-steel cutter when working on steel showed exactly the same phenomenon. From the moment the cutter begins working, the expenditure of work immediately begins to increase. The thicker the chip (the greater the feed \(S_z\) or the depth \(t\)), the smaller the percentage increase in the expenditure of work characterizes the same dulling of the cutter.

Fig. 21.

Thus, when a milling cutter is operating, dulling of the cutting edge appears and is revealed through the expenditure of work differently than when a turning tool is operating; for the latter, the cutting pressure increases significantly only just before the very moment of dulling of the tool, remaining approximately constant throughout the whole experiment, which applies especially to high-speed-steel tools. Moreover, with small chips a carbon-steel tool gives a higher cutting pressure than a high-speed-steel one\({}^{30}\); in our investigation, however, we obtained no difference in the operation of carbon-steel and high-speed-steel cutters, as can be seen from the graphs presented.

On the basis of everything said concerning the dulling of the cutter, one may conclude that the criterion of cutter dulling cannot be taken to be an increase in the energy expenditure during cutting by a definite percentage, since this objective indicator will not, under different cutting conditions, characterize one and the same state of the edge. From this point of view, Safonov’s experiments\({}^{31}\) raise serious doubts; in them, according to his statement, with thin chips he was unable to obtain a significant increase in feed pressure even for a very blunt cutter, whereas with thick chips a significant increase was present. Evidently the apparatus used by him in the experiments was insufficiently reliable, since from Fig. 21, as well as from all the curves we have presented earlier for cast iron and those presented below for various steels, from a comparison of the curves for a sharp cutter and a dulled one, on the basis of hundreds of experiments, one can draw the completely opposite conclusion. Evidently, as a criterion of dulling, one must adopt either the surface finish of the machined surface, objectively measured, or the increase in energy expenditure, different for different cutting conditions, determined by preliminary experiments and characterizing one and the same dulling of the edge. Wear of the cutting edge, adopted as a criterion of dulling—

… in Ivanov’s work² can hardly be recommended for so expensive a tool as a milling cutter. From Fig. 21 it is evident that, from the very initial moment of operation, the value \(K\) (a quantity entering into the exponents of the variables of equation 1) rapidly begins to increase and varies within wide limits, from \(K=-0.31\) for a newly sharpened cutter to \(K=-0.62\) for a dulled one. This must be taken into account when establishing the coefficients and exponents in equation 1, since if we give them for a newly sharpened cutter only, then after a short interval of cutter operation this equation will give an energy consumption lower than the actual one.

Fig. 22.

Figure 22 presents the dependence of the work consumption at various angles of undergrinding from \(\gamma=0^\circ\) to \(\gamma=15^\circ\). The experiments were carried out on a pendulum dynamometer with one and the same tooth, reset into various slots of the arbor (see Fig. 6).

The influence of the angle of undergrinding on the work consumption in machining cast iron may be expressed by coefficients (Table 6).

Table No. 6

Angle of undergrinding \(\gamma\)	Coefficient of work consumption	Note
\(0^\circ\)	1
\(5^\circ\)	0.96
\(10^\circ\)	0.92
\(15^\circ\)	0.87
\(20^\circ\)	0.82	Presumably.

On the basis of experiments carried out with cast iron, we have the following values of the constants in the equation of power and of specific pressure (Table 7).

Table No. 7

No.	Material machined	\(\lambda\)	\(-K^*\), sharp cutter	\(-K^*\), dulled cutter	Note
1	Cast iron \(H=100\) . . . .	50	\(-0.4\)	\(-0.51\)	Fig. 17
2	Cast iron \(H=140\) . . . .	60	\(-0.39\)	\(-0.52\)	Fig. 18
3	Cast iron \(H=170\) . . . .	75	\(-0.34\)	\(-0.46\)	Fig. 19
4	Cast iron \(H=190\) . . . .	95—115	\(-0.29\)	—	Fig. 20 (according to Bahlecke)

From the graphs in Figs. 15, 16, 17, 18, 19, one can see that \(\lambda\) for a dulled tooth has the same value as for a sharp tooth, and the increase in energy consumption is obtained through a change in the value of \(K\).

Milling of steel.

In Fig. 23 are presented the results of experiments on steel of medium hardness 1, \(K_z = 54\) kg/mm, on a pendulum dynamometer with various angles of helix rise \(\beta\), from \(90^\circ\) to \(45^\circ\), with \(t = 5\) mm. Here too we see the absence of any influence of the angle of helix rise of the tooth on the work consumption, and hence also on the average power. Fig. 23 gives the work of sharp cutters.

Fig. 23.

In Figs. 24 and 25 are presented experiments on steel of medium hardness on a milling machine with various \(Z, D, S, S_z, t, b, \beta\), and they give the same regularities of which we spoke in the appendix to cast iron. In these two

Fig. 24.

figures we have cutter helix-rise angles \(\beta = 45^\circ\), \(\beta = 64.5^\circ\), and \(\beta = 80^\circ\), and we see that, in the experiments on the machine as well, all these cutters, with so considerable a difference in helix rise, give one and the same work consumption. For all these cutters, the milling width was likewise selected so as to ensure constancy of the torque on the cutter.

In Fig. 24, in addition, are given the results of operation with a dulled cutter, which before the experiments had worked for 30 min at \(v = 24\) m/min, \(t = 2\) mm, \(S = 100\) mm/min on rail steel (see above).

We see the same phenomenon of the effect of dulling on the work consumption in milling steel as in milling cast iron.

Fig. 25.

In Fig. 26 a comparison is given for different degrees of cutter dulling. A sharp cutter, just sharpened, gives \(K=-0.25\).

A cutter after 1500 revolutions at a cutting speed \(v=6\) m/min., which for a high-speed-steel cutter on steel, \(K_z=54\), is much lower than normal, gives \(K=-0.28\).

The uppermost straight line in Fig. 26 gives the work consumption of cutters with different helix angles \(\beta\). Here there is still more considerable dulling (but the same for all cutters with different \(\beta\)) (as was discussed above), and the value is \(K=-0.49\). We see how significantly the value of \(K\) changes with a change in the condition of the cutting edge of the tooth; moreover, it should be noted that we did not bring the edge to failure. In that case, evidently, the value of \(K\) would have increased still more.

From the same Fig. 26 it is seen that dulled cutters do not give a difference in work consumption when milling with different helix angles \(\beta\), i.e., in application to steel we have the very same thing as in machining cast iron.

Fig. 26.

Fig. 27 gives the results of milling mild steel with \(K_z = 34\ \text{kg/mm}^2\). It is interesting to note that, when milling mild steel, we had exactly the same expenditure of work as when milling medium-hard steel,

Fig. 27.

as is evident from a comparison of Figs. 27 and 24, although the tensile strengths of these two steels differ considerably. This is apparently explained by the high toughness of the mild steel, which had a relative elongation at fracture of 39%, as against 24% for the medium-hard steel (see Table 4).

Fig. 28 presents the results of machining hard steel, \(K_z = 76\ \text{kg/mm}^2\), likewise confirming everything stated above. The chromium steel (see Table 4)

Fig. 28.

was tested both on the machine tool and on the pendulum dynamometer. Figs. 29, 30, 31, and 32 make it possible to judge the influence of the helix lead angle on the expenditure of work. Here we see some influence of the angle \(\beta\). At \(\beta = 45^\circ\) the expenditure of work is 6% lower than at \(\beta = 90^\circ\). Such a small difference cannot serve as a basis for introducing the influence of the helix lead angle into the power equation.

Figs. 33 and 34 give the results of experiments on the same chromium steel on a milling machine. The points from the experiments on the pendulum dynamometer are also plotted here, and they agree very well with the results of the experiments on the machine tool.

When machining steel on a machine tool with \(K_z = 54\ \text{kg/mm}^2\) and chromium steel, the influence of the undercutting angle on the expenditure of work was also determined. These re—

Fig. 29.

Fig. 30.

Fig. 31.

Fig. 32.

results are expressed by the coefficients in Table 8, into which the coefficients for cast iron from Table 6 have also been entered.

Table 8

Clearance angle \(\gamma\)	Work-expenditure coefficient — Steel	Work-expenditure coefficient — Cast iron
\(0^\circ\)	1	1
\(5^\circ\)	0.94	0.96
\(10^\circ\)	0.87	0.92
\(15^\circ\)	0.80	0.87

Fig. 33.

The change in the expenditure of work with change in the clearance angle, according to our experiments, comes close to the data of Beckh \(^{5}\), Oxford \(^{4}\), and to Glebov’s data, developed on the basis of the experiments of Nicholson, Stanton and Hyde, the Manchester Committee, and Boston. Sokolov’s data \(^{33}\), indicating a decrease in the expenditure of work by 5% for each \(1^\circ\) of the tooth clearance angle, are wholly improbable.

Fig. 34.

The limiting maximum feed per tooth that we reached in the experiments on steel was \(S_z = 1.05\) mm; thus here also we fully covered the limits of roughing work. The maximum depth of cut \(t = 7.6\) mm likewise appears to us to be quite sufficient.

From all the graphs presented for steel it may be concluded that the experimental points are scattered somewhat more than in the experiments on cast iron, but the mean straight lines drawn satisfy all the experiments, carried out over wide ranges of variation of the variables, quite well, and must be recognized as reliably expressing the dependence of the work on the cutting factors.

In Fig. 35 are given the results of experiments on aluminum, which, having been carried out over still wider ranges of cutting depths, confirm everything stated above.

Thus an ever greater number of experiments, carried out by us by two entirely different methods, with different apparatus, indicate the correctness of the equation theoretically derived by us for the expenditure of power when a cylindrical milling cutter with straight and helical teeth is operating ^24). The angle of rise of the helix has no influence on the expenditure of work and on the mean power in milling. We think that the experimental results presented are the best proof of the proposition which, contrary to the general opinion, was advanced by us in our previous work: that the thickness of the chip on the helical tooth of a milling cutter, just as on a straight one, depends only on the feed per tooth and on the instantaneous position of the given point of the tooth, and is entirely independent of the angle of rise of the tooth helix, \(\beta\), since only in this case can there be no influence of the angle \(\beta\) on the expenditure of work. This proposition, in essence simple and sufficiently clear to us, analytically, and then also graphically, proved by engineer Eremin ^34), nevertheless, despite its simplicity, raises doubts among many, as we judge from the technical literature ^21), ^22), ^35), ^36), which appeared both before and after our work, and from a whole series of letters received by us. Engineer Tishin gives entirely inadmissible settings, based on the data of Prof. Savvin, in the collection of guiding data of Gipromash on milling conditions ^37), where he gives the following table of coefficients for conversion from the power of a straight-tooth cutter to the power of a helical cutter.

Fig. 35.

\(90^\circ-\beta\)	\(5^\circ\)	\(10^\circ\)	\(15^\circ\)	\(20^\circ\)	\(25^\circ\)	\(30^\circ\)	\(35^\circ\)	\(40^\circ\)	\(45^\circ\)
Coefficient . . . . . . .	0.73	0.57	0.47	0.42	0.39	0.37	0.35	0.34	0.33

In other words, according to the “guiding materials,” when a cutter with \(\beta=45^\circ\) is operating, the expenditure of power amounts to 33% of that for a straight tooth with \(\beta=90^\circ\).

We think that such a colossal difference can be noticed without any instruments, and our apparatus undoubtedly should have recorded it. On the basis of the material presented, the following values of \(\lambda\) and \(K\) can be given (Table 9).

Table 9

No.	Material being machined	\(\lambda\)	\(K\), sharp cutter	\(K\), dull cutter
1	Soft steel . . . . . . . . . . . . \(K_z = 34\)	142	\(-0.3\)	\(-0.36\)
2	Medium-hard steel . . . . . . . . \(K_z = 54\)	142	\(-0.3\)	\(-0.36\)
3	Hard steel . . . . . . . . . . . . \(K_z = 76\)	180	\(-0.26\)	\(-0.33\)
4	Chromium steel . . . . . . . . . . \(K_z = 74\)	205	\(-0.23\)	\(-0.28\)
5	Cast aluminum . . . . . . . . . . \(H = 60\)	53	\(-0.22\)	—

The values of \(K\) given in Tables 7 and 9 may be used within the limits up to \(S_m \leq 1\) mm, which is quite sufficient for the heaviest cutting conditions. For \(S_m > 1\) mm, if we used the values of \(K\) given in the tables, we would obtain, when working with a dull cutter, a lower power than when working with a sharp one, which is, of course, absurd. On examining the graphs presented, it can be seen that, for a dull cutter, we do not have an exactly straight line, but rather certain curves that asymptotically approach the straight lines for a sharp cutter in the region of large chips. Thus, for a dull cutter, \(K\) is not a constant quantity, but gradually, as \(S_m\) increases, decreases, approaching the value of \(K\) for a sharp cutter, and from some value of \(S_m\) onward the value of \(K\) for a dull cutter will be equal to the value of \(K\) for a sharp one.

Not wishing to introduce unnecessary complication, we have also taken \(K = \mathrm{const}\) for the dull cutter, which, within the range of values \(S_m < 1\) mm applicable to milling, agrees sufficiently well with the location of the experimental points.

In view of the fact that the blunting of a milling-cutter tooth proceeds differently than that of a turning tool, beginning from the very first moment of its operation, which entails a gradual change in the value of \(K\), it would be erroneous to recommend, for practical calculations, the value of \(K\) for a freshly sharpened cutter alone. We believe that here it is necessary to orient oneself toward a medium-dull cutter in working condition and to take, for this purpose, a value of \(K\) intermediate between that for a sharp and that for a dull cutter. Such values, suitable for practical calculations, are given by us in Table 10.

Table 10

No.	Material being machined	\(\lambda\)	\(K\)	Note
1	Cast iron . . . . . . . . . . . . \(H = 100\)	50	\(-0.45\)	According to Bahlecke
2	Cast iron . . . . . . . . . . . . \(H = 140\)	60	\(-0.45\)	According to Bahlecke
3	Cast iron . . . . . . . . . . . . \(H = 170\)	75	\(-0.40\)	According to Bahlecke
4	Cast iron . . . . . . . . . . . . \(H = 190\)	100	\(-0.35\)	According to Bahlecke
5	Steel . . . . . . . . . . . . . . \(K_z = 34\)	142	\(-0.33\)	According to Bahlecke
6	Steel . . . . . . . . . . . . . . \(K_z = 54\)	142	\(-0.33\)	According to Bahlecke
7	Steel . . . . . . . . . . . . . . \(K_z = 76\)	180	\(-0.29\)	According to Bahlecke
8	Chromium steel . . . . . . . . . \(K_z = 74\)	205	\(-0.26\)	According to Bahlecke
9	Aluminum . . . . . . . . . . . . \(H = 60\)	53	\(-0.24\)	According to Bahlecke

We have included here cast iron \(H = 190\) according to Bahlecke’s experiments, in view of the fact that the experiments on medium-hard steel carried out by us gave exceptionally close agreement with similar experiments carried out by Bahlecke; this makes it possible to suppose that his data for cast iron are also accurate.

Table 10 is represented graphically in Fig. 36 in the coordinates \(S_{\mu}—A_w\). Here we note first of all that the stronger the material, the higher the straight line characterizing the expenditure of work in milling it is located.

Fig. 36.

In addition, as a result of the different inclination of the straight lines (different values of \(K\)), the straight lines in the region of small chips converge, which gives one and the same expenditure of work when milling materials of different strength. This phenomenon is quite natural, since Schlesinger\(^{38}\) already noted that in grinding, when especially thin chips occur, the circumferential force in machining even such different materials as cast iron and steel shows no difference.

Here it must also be noted that the ratios of the coefficients \(\lambda\) for steels of different hardness by no means confirm Prof. Savin’s indication that this coefficient in the equation for the specific cutting pressure for steel should be equal to the ultimate tensile strength. In that case, for steel with \(K_z = 76\) we should have obtained \(\lambda = 304\), whereas we obtained \(\lambda = 180\).

In addition, it should be pointed out that the mechanical transfer of dependencies and coefficients found for turning to the milling process, as Prof. Reznikov does, cannot give satisfactory results, since in that case we should have had to obtain one and the same value of \(K\) for materials of different hardness. From Tables 7, 9, and 10 it is evident that, with an increase in the hardness of the material being machined, the value of \(K\) decreases, which agrees well with the experiments of Boston and Kraus\(^{13}\).

We can give the following equations for determining the useful power in milling (in horsepower) (see Table 11).

The equations are rather complicated, and it is simpler to use their graphical solution. The construction of a nomogram for this purpose presents no difficulties. As an example, in Fig. 37 we give a nomogram for solving the power equation when machining cast iron \(H = 170\) (according to Fig. 36). Similar nomograms can easily be constructed for other materials being machined. In this nomogram, as in the equations given, the influence of the rake angle is not taken into account; i.e., they are given for a tooth without undercutting. If undercutting is present, the results obtained must be multiplied by the corresponding coefficients of Table 8. In doing so, it is necessary to orient—

Table 11

Work material	Power equation for a sharp cutter	Power equation for a medium-dull cutter
1. Cast iron \(H=100\text{–}105\)	\[\begin{aligned} N_e &= \frac{2^{0,6}}{1,6}\cdot \frac{50}{4500\cdot 1000}\cdot n^{0,4}\cdot b\cdot S^{0,6}\cdot l^{0,8}\cdot Z^{0,4}\cdot D^{0,2} \\ \text{or:}\quad N_e &= \frac{1}{951\cdot 100}\cdot n^{0,4}\cdot b\cdot S^{0,6}\cdot l^{0,8}\cdot Z^{0,4}\cdot D^{0,2} \end{aligned}\]	\[\begin{aligned} N_e &= \frac{2^{0,55}}{1,55}\cdot \frac{50}{4500\cdot 1000}\cdot n^{0,45}\cdot b\cdot S^{0,55}\cdot l^{0,775}\cdot Z^{0,45}\cdot D^{0,225} \\ \text{or:}\quad N_e &= \frac{1}{951\cdot 100}\cdot n^{0,45}\cdot b\cdot S^{0,55}\cdot l^{0,775}\cdot Z^{0,45}\cdot D^{0,225} \end{aligned}\]
2. Cast iron \(H=140\)	\[\begin{aligned} N_e &= \frac{2^{0,61}}{1,61}\cdot \frac{60}{4500\cdot 1000}\cdot n^{0,39}\cdot b\cdot S^{0,61}\cdot l^{0,805}\cdot Z^{0,39}\cdot D^{0,195} \\ \text{or:}\quad N_e &= \frac{1}{792\cdot 100}\cdot n^{0,39}\cdot b\cdot S^{0,61}\cdot l^{0,805}\cdot Z^{0,39}\cdot D^{0,195} \end{aligned}\]	\[\begin{aligned} N_e &= \frac{2^{0,55}}{1,55}\cdot \frac{60}{4500\cdot 1000}\cdot n^{0,45}\cdot b\cdot S^{0,55}\cdot l^{0,775}\cdot Z^{0,45}\cdot D^{0,225} \\ \text{or:}\quad N_e &= \frac{1}{794\cdot 100}\cdot n^{0,45}\cdot b\cdot S^{0,55}\cdot l^{0,775}\cdot Z^{0,45}\cdot D^{0,225} \end{aligned}\]
3. Cast iron \(H=170\)	\[\begin{aligned} N_e &= \frac{2^{0,66}}{1,66}\cdot \frac{75}{4500\cdot 1000}\cdot n^{0,34}\cdot b\cdot S^{0,66}\cdot l^{0,83}\cdot Z^{0,34}\cdot D^{0,17} \\ \text{or:}\quad N_e &= \frac{1}{630\cdot 100}\cdot n^{0,34}\cdot b\cdot S^{0,66}\cdot l^{0,83}\cdot Z^{0,34}\cdot D^{0,17} \end{aligned}\]	\[\begin{aligned} N_e &= \frac{2^{0,6}}{1,6}\cdot \frac{75}{4500\cdot 1000}\cdot n^{0,4}\cdot b\cdot S^{0,6}\cdot l^{0,8}\cdot Z^{0,4}\cdot D^{0,2} \\ \text{or:}\quad N_e &= \frac{1}{634\cdot 100}\cdot n^{0,4}\cdot b\cdot S^{0,6}\cdot l^{0,8}\cdot Z^{0,4}\cdot D^{0,2} \end{aligned}\]
4. Cast iron \(H=190\)	\[\begin{aligned} N_e &= \frac{2^{0,71}}{1,71}\cdot \frac{100}{4500\cdot 1000}\cdot n^{0,29}\cdot b\cdot S^{0,71}\cdot l^{0,855}\cdot Z^{0,29}\cdot D^{0,145} \\ \text{or:}\quad N_e &= \frac{1}{471\cdot 100}\cdot n^{0,29}\cdot b\cdot S^{0,71}\cdot l^{0,855}\cdot Z^{0,29}\cdot D^{0,145} \end{aligned}\]	\[\begin{aligned} N_e &= \frac{2^{0,65}}{1,65}\cdot \frac{100}{4500\cdot 1000}\cdot n^{0,35}\cdot b\cdot S^{0,65}\cdot l^{0,825}\cdot Z^{0,35}\cdot D^{0,175} \\ \text{or:}\quad N_e &= \frac{1}{474\cdot 100}\cdot n^{0,35}\cdot b\cdot S^{0,65}\cdot l^{0,825}\cdot Z^{0,35}\cdot D^{0,175} \end{aligned}\]

Machined material	Power equations for a sharp cutter	Power equations for a moderately blunted cutter
5. Mild steel \(K_z=34\), and medium-hard steel \(K_z=76\)	$$N_e=\frac{2^{0.7}}{1.7}\cdot\frac{142}{4500\cdot1000}\cdot t^{0.3}\cdot n^{0.3}\cdot b\cdot S^{0.7}\cdot \rho^{0.85}\cdot Z^{0.3}\cdot D^{0.15}$$ or: $$N_e=\frac{1}{332\cdot100}\cdot n^{0.3}\cdot b\cdot S^{0.7}\cdot \rho^{0.85}\cdot Z^{0.3}\cdot D^{0.15}$$	$$N_e=\frac{2^{0.67}}{1.67}\cdot\frac{142}{4500\cdot1000}\cdot n^{0.33}\cdot b\cdot S^{0.67}\cdot \rho^{0.835}\cdot Z^{0.33}\cdot D^{0.165}$$ or: $$N_e=\frac{1}{332\cdot100}\cdot n^{0.33}\cdot b\cdot S^{0.67}\cdot \rho^{0.835}\cdot Z^{0.33}\cdot D^{0.165}$$
6. Hard steel \(K_z=76\)	$$N_e=\frac{2^{0.74}}{1.74}\cdot\frac{180}{4500\cdot1000}\cdot n^{0.26}\cdot b\cdot S^{0.74}\cdot \rho^{0.87}\cdot Z^{0.26}\cdot D^{0.13}$$ or: $$N_e=\frac{1}{260\cdot100}\cdot n^{0.26}\cdot b\cdot S^{0.74}\cdot \rho^{0.87}\cdot Z^{0.26}\cdot D^{0.13}$$	$$N_e=\frac{2^{0.71}}{1.71}\cdot\frac{180}{4500\cdot1000}\cdot n^{0.29}\cdot b\cdot S^{0.71}\cdot \rho^{0.855}\cdot Z^{0.29}\cdot D^{0.145}$$ or: $$N_e=\frac{1}{261\cdot100}\cdot n^{0.29}\cdot b\cdot S^{0.71}\cdot \rho^{0.855}\cdot Z^{0.29}\cdot D^{0.145}$$
7. Chromium steel \(K_z=74\)	$$N_e=\frac{2^{0.77}}{1.77}\cdot\frac{205}{4500\cdot1000}\cdot n^{0.23}\cdot b\cdot S^{0.77}\cdot \rho^{0.885}\cdot Z^{0.23}\cdot D^{0.115}$$ or: $$N_e=\frac{1}{228\cdot100}\cdot n^{0.23}\cdot b\cdot S^{0.77}\cdot \rho^{0.855}\cdot Z^{0.23}\cdot D^{0.115}$$	$$N_e=\frac{2^{0.74}}{1.74}\cdot\frac{205}{4500\cdot1000}\cdot n^{0.26}\cdot b\cdot S^{0.74}\cdot \rho^{0.87}\cdot Z^{0.26}\cdot D^{0.13}$$ or: $$N_e=\frac{1}{228\cdot100}\cdot n^{0.26}\cdot b\cdot S^{0.74}\cdot \rho^{0.87}\cdot Z^{0.26}\cdot D^{0.13}$$
8. Aluminum alloy \(H=60\)	$$N_e=\frac{2^{0.78}}{1.78}\cdot\frac{53}{4500\cdot1000}\cdot n^{0.22}\cdot b\cdot S^{0.78}\cdot \rho^{0.89}\cdot Z^{0.22}\cdot D^{0.11}$$ or: $$N_e=\frac{1}{881\cdot100}\cdot n^{0.22}\cdot b\cdot S^{0.78}\cdot \rho^{0.89}\cdot Z^{0.22}\cdot D^{0.11}$$	$$N_e=\frac{2^{0.76}}{1.76}\cdot\frac{53}{4500\cdot1000}\cdot n^{0.24}\cdot b\cdot S^{0.76}\cdot \rho^{0.88}\cdot Z^{0.24}\cdot D^{0.12}$$ or: $$N_e=\frac{1}{883\cdot100}\cdot n^{0.24}\cdot b\cdot S^{0.76}\cdot \rho^{0.88}\cdot Z^{0.24}\cdot D^{0.12}$$

…be oriented to the relief angle, in the plane normal to the cutting edge of the tooth. It may be assumed that the influence of relief on a helical tooth will be somewhat more significant than that determined by us, owing to the fact that the chip, in its motion, deviating from the direction normal to the edge of the tooth, will thereby change the actual cutting angle; however, judging from Glebov’s data 39), this change will be so small that it will not produce any noticeable reduction in power.

Fig. 37.

To calculate the power for a specified cutting condition, one may also use Fig. 36. For this purpose, knowing \(S\), \(n\), \(Z\), \(t\), and \(D\), it is necessary to determine \(S_m\); then, for the obtained \(S_m\), find from Fig. 36 the corresponding value of \(A_w\) for the specified material being machined, multiply this value by \(b . S . t\), and divide by \(4500 . 1000\). In the case of orientation to a sharp cutter, for this purpose it is necessary to use the corresponding straight lines from Figs. 17, 18, 19, 20, 24, 27, 28, 33, and 35.

Conclusion.

On the basis of the work carried out, we can draw the following conclusions:

1. All the theoretical derivations and propositions put forward by us in the “Theory of the Operation of a Cylindrical Milling Cutter with a Helical Tooth” 24) are fully confirmed by experiment.

2. The power at the cutter is determined by equation 1 and Table 11.

3. The chip thickness on a straight and a helical tooth is expressed by the relation:

\[ S_e = S_z \cdot \sin \psi, \]

The helix angle \(\beta\) has no effect on the chip thickness.

1. The chip cross-section on a helical tooth is expressed by the relation

\[ q=\frac{D}{2}\cdot\frac{S_z}{\cos \beta}(\cos \psi_1-\cos \psi_2). \]

1. The circumferential force on a helical tooth of the milling cutter is expressed by the relation

\[ F=D\cdot \lambda \cdot S_z^{k+1}\cdot \operatorname{tng}\beta \frac{2^{k+1}}{K+2}\cdot \left[\sin^{k+2}\left(\frac{\psi_2}{2}\right)-\sin^{k+2}\left(\frac{\psi_1}{2}\right)\right] \]

1. The helix angle \(\beta\) has no effect on the expenditure of work or on the mean power in milling.

2. Prof. Savvin’s conclusions are not confirmed by experiment and lead to considerable errors, amounting in some cases to hundreds of percent of deviation.

We were unable to make a broader comparison of the results of our work with existing experimental data, owing to the considerable length of the present article.

The work was carried out in the metal-cutting laboratory of the Tomsk Industrial Institute at the expense of credits from the GUUZ NKTP.

Tomsk, May 1935.

Literature cited in the article:

1. Engel. Über Schnittgeschwindigkeit und Schnittdruck beim Fräsen. Z. d. V. D. I. 1925 S.—819. Reznikov. Theory of Metal Cutting. 1934, p. 208 ff.
2. Schütte-Blätter für den Betrieb. 1910. Die de Leeuwchen Versuche an Cincinnati-Hochleistungsfräsmaschinen.
3. Reindl. Vergleichende Fräsversuche mit Walzenfräsern. Schütte-Blätter für den Betrieb 1911. N 1.
4. Airey-Oxford. On the Art of Milling. Mech. Eng. 1921 p. 783.
5. Sawin. Theorie of Milling Cutters. Mach. Eng. 1926 p.—1203. Sawin. Genie Civil 1925.
6. Beckh. Die Metallbearbeitung mittels Walzenfräsern. Maschinenbau 1926. S—497.
7. Salomon. Die Fräsarbeit. Werkstattstechnik 1926, S—469. Salomon. Zur Theorie des Fräsvorganges. Z. d. V. D. I. 1928.
8. Bahlecke. Richtwerte für das Fräsen mit Walzenfräsern. Maschinenbau 1930. S—437.
9. Eisele. Dynamische Untersuchungen des Fräsvorganges. Berichte über betriebswissenschaftliche Arbeiten. N 7. 1931.
10. Schlesinger. Werkstattstechnik 1931 No. 17.
11. Stoewer. Maschinenbau 1932 No. 23.
12. Bessprozvannyi. Influence of the operating regime and the design of a cylindrical milling cutter on force and energy consumption. Org.-Informatsiya 1933 No. 5.
13. Boston and Kraus. The Elements of Milling. A. S. M. E. Transactions. Vol. 54. 1932.
14. Brödner. Neuere Untersuchungen des Fräsvorganges. Z. d. V. D. I. 1933. S—1113.
15. Schröder. Die Bedeutung der Spandicke bei Walzenfräsern. Maschinenbau 1934. S—541.
16. ENIMS. Issue No. 9. 1934. Milling cutters and milling.
17. Guverner. Milling practice. 1934. Pp. 267—303.
18. Fischer. Die Werkzeugmaschine. 1900.
19. Stoewer. Richtlinien für Wirtschaftliches Fräsen. Maschinenbau 1934. S—11.
20. Sokolov. Tool Engineering. P. 146. 1933.
21. Krivoukhov. Machining of Metals by Cutting.
22. Isaev. Cold Working of Metals.
23. Safonov and Novikov. Technological Standardization 1935. No. 2.
24. Rozenberg and Sudnishnikov. Theory of operation of a cylindrical milling cutter with a helical tooth. Bulletin of the Metal Industry. 1933. No. 4.
25. Salomon. Trägheitslöse Zerspanungsmessungen. Loewe-Notizen Bd. 14. 1929. S—118.
26. Eisele. Beschreibung und Kritik der bisher veröffentlichten Schnittdruckmesseinrichtungen für Fräsmaschinen. Maschinenbau 1931.
27. Kurrein. Fräser und Fräsmaschine. Werkstattstechnik 1929 No. 9. Werkstattstechnik 1928. S—33.

1. Schlesinger. Wechselwirkung zwischen Werkstoff und Werkzeug. Werkstattstechnik 1927. S—605.

Schlesinger. Messungen bei Zerspanungsversuche an Fräsmaschine. Werkstattstechnik 1930. S—102.

1. Avrutin. Cutting conditions and the design of milling cutters. P. 16.

2. Okochia Okoshi. Scientific paper of the Inst of Physical and Chemical Research N 84. Maschinenbau 1929. S—318.

Glebov. Theory of the most advantageous cutting. P. 197.

1. Safonov and Novikov. On milling speeds. Tekhnormirovanie 1933 No. 9.

2. Ivanov. Milling operations. Vestnik Metallopromyshlennosti 1934 No. 11.

3. Sokolov. Tool engineering. P. 155.

4. Eremin. Chip thickness on the tooth of a cylindrical milling cutter. Tekhnormirovanie 1934 No. 2.

5. Danilenko. Tekhnormirovanie 1935 No. 2.

6. Reznikov. Theory of metal cutting. P. 222.

7. Collection of guidance materials of Gipromash. Cutting conditions on milling machines. Series II. Issue 34. 1934. P. 17, table 5.

8. Schlesinger. Versuche über die Leistung von Schleifscheiben. Mitteilungen über Forschungsarbeiten 1907. No. 5.

9. Kenen. Elements of machine tools. P. 18, table 5.

Dzhigit. Grinding practice. P. 19, table 18.

1. Glebov. Theory of the most advantageous cutting. P. 75, table 10.

SUMMARY.

In spite of a considerable amount of work devoted to the investigation of the process of milling, there is no sufficient clarity in the question of the influence of various factors on the expenditure of work during milling.

The least explained is the question of the helix angle of the cutter’s tooth affecting the energy of milling. Bahlecke \(^{8}\), Eisele \(^{9}\), Schlesinger \(^{10}\), Brödner \(^{14}\), experiments of the University in Michigan \(^{16}\), Saffonov and Novikov \(^{23}\) indicate the absence of the influence of the angle of helix lifting upon the expenditure of work; Stoewer \(^{11)19)}\) indicates the increase of the expenditure of work with the diminution of the helix lifting; Beckh \(^{6}\), Sawin \(^{5}\), Sokolov \(^{20}\), Reznikov \(^{1}\), Krivoukhov \(^{21}\), Issaev \(^{22}\) assert the opposite.

The author of this work published in 1933 the theory of work of a plain milling cutter with a helical tooth \(^{24}\); in this work a connection among all factors of cutting during the milling was given, and it was proved theoretically that the expenditure of work during the milling with a plain milling cutter does not depend on the helix angle of the tooth.

The purpose of the present work was to verify experimentally the theory suggested by the author, as well as to determine characteristics of different materials in milling, entering into the equation of the milling energy.

This whole work was performed in parallel using two experimental methods. By means of a special pendulum-actuated milling dynamometer (see Figs. 1 and 2) the expenditure of work on the removal of one chip with one tooth of the cutter was determined; and, besides that, by means of a dynamometer installed into the construction of the milling machine (see Figs. 7 and 8), the turning moments on the cutter of the work with different working regimes were recorded. The exact coincidence of both methods gave a complete assurance in the reliability of the conclusions.

The whole apparatus was carefully examined, possible errors were determined and were taken into consideration in working out the experimental data.

The cutters used during the work with the pendulum-actuated dynamometer are represented in Table 2 and in Fig. 6. The cutters used during the work on the milling machine are represented in Table 3 and in Fig. 13.

The materials which were used in the tests are represented in Table 4. 3500 tests were performed and they confirmed fully the author’s theory, showed a mutual connection between different factors of cutting and the expenditure of work, as well as measured the influence of the rake angle and the influence of dulling of the cutter upon the expenditure of work.

Tables 7 and 9 afford the values of quantities entering into the equation of energy (1).

Figs. 15—20 are related to Table 7.

Figs. 23—35 to Table 9.

Table 8 shows the influence of the rake angle.

Table 11 affords the equations of the expenditure of energy in the case of work with different materials in milling with a sharp cutter and with a slightly dull one.

The work was performed in the Laboratory of Cutting of Metals of the Industrial Institute in Tomsk and was finished in May 1935.

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