Dynamic Testing of Cylindrical and Face Milling Cutters

Associate Professor A. M. Rozenberg.

The work was carried out on assignment from the Technical Department of Narkommash. The purpose of the work was to determine the data necessary for calculating powers and torques during the operation of cylindrical and face milling cutters under various cutting conditions for steels and cast irons.

The experiments were carried out on the following materials being machined (Table 1).

Table 1.

No.	Name of material	Tensile strength, kg/mm², $\sigma_B$	Elongation, $\delta_{10}\%$	Reduction of area, $\psi\%$	Brinell hardness, $H_B$	Rockwell hardness, $R_B$	Impact toughness, kg·m/cm²
1	Steel 1010-p	40,1	29,6–31,6	68,4–70	109–122	55–58	16
2	Steel 1020-p	44,6	26,4–30	61–64	118–124	66–70	14,6
3	Steel 1020-n	46,3	29–31	67,0	118–126	70	17,3
4	Steel 1035-p	62,1	20–21,5	43,7–46,5	163–166	78–83	4,37
5	Steel 1035-n	64,6	21,6	51,0	163–170	84–88	7,05
6	Steel 1040-p	64,4	20,0–20,6	46,0	170–174	86,5–89,5	2,8
7	Steel 1040-o	62,8	22,4–23,4	47,0	159–165	82,5–84,5	5,1
8	Steel 1040-n	65,8	21,3–22	53–55	174–183	87,5–88	5,85
9	Steel 1040-u	69,0	20	64,7	196–217	95–97	7,45
10	Steel 1050-o	78,0	12,5	24	212–223	95,5–96	1,19
11	Steel 5120-p	50,3	26,0–27,9	61,5–62,9	137–143	77–78	—
12	Steel 5120-n	50,3	27,5–27,9	69,7–70,4	134–143	74–78	13,4
13	Steel 5140-p	71,5	15,3–15,7	30–31	196–207	91,5–93	2,0
14	Steel 5140-o	71,3	16,3–19,3	46–47	192–207	93	3,13
15	Steel 5140-n	81,1	16–16,4	60,8	212–217	98,5–100,5	2,95
16	Steel 5145-p	74,0	16,8	29,3	210	92–93	4,1
17	Steel 3120-n	54,4	26,0	66,6	140–146	79–80	15,8
18	Steel 3140-p	80,3	14,7–15,7	39,5–41,6	223–228	96–97	5,75
19	Steel 3340-o	91,0	9,5	51,0	265	101–102	4,5
20	Steel 3340-u	106,0	11,5	57,7	311	106–109	5,62
21	Cast iron No. 1 (gray)	4,5–9,3	—	—	109–124	66–73	2,44
22	Cast iron No. 2 (pearlitic)	25,0	—	—	212–223	95–96	1,94
23	Cast iron No. 3 (alloyed)	18,5	—	—	183–212	93–95	0,725
24	Cast iron No. 4 (alloyed)	7,65	—	—	137–156	80–85	0,75
25	Cast iron No. 5 (malleable)	36,6–42,0	12,8	14,8	134–156	76–78	2,25
26	Bronze	—	—	—	81	60–62	1,19

In Table 1 the steels are designated according to the American standard. The indices following the steel number have the following meanings: p — rolled stock, o — annealing, n — normalizing, u — heat treatment.

Steel 3340, before annealing and heat treatment, was rolled from a section of 100 × 100 mm to a section of 50 × 100 mm. The mechanical properties were determined:

1. The temporary tensile strength, \(\sigma_B\), by rupturing two specimens of each material. The dimensions of the specimens for the tensile test were: diameter \(10\) mm, gauge length \(100\) mm. Rupture was carried out on a 10/50-ton Amsler-Laffon machine in the institute’s materials-strength laboratory.

2. Relative elongation was determined in the usual manner.

3. Reduction of area was calculated as the ratio of the difference between the cross-sectional areas of the specimen before and after rupture to the cross-section before rupture.

The tensile specimens were cut from those parts of the blanks that had been subjected to milling.

1. Brinell hardness was determined on the blanks at the places where milling was carried out; moreover, hardness was determined both before milling and after milling. Table 1 gives the extreme limits of hardness variation. Hardness was determined with a ball \(d = 10\) mm under a load of \(3000\) kg on an “Alpha” Brinell press in the institute’s materials-strength laboratory.

Fig. 1.

1. Rockwell hardness was determined on a Rockwell durometer in the metal-cutting laboratory. Hardness was determined on the B scale (steel ball, load \(100\) kg), in no fewer than 5 places. Table 1 shows the limits of hardness variation.

2. Impact toughness was determined on \(10 \times 10 \times 65\) mm specimens with a notch \(2\) mm deep, with the inclined sides of the notch at \(45^\circ\) to each other, and with a notch-root radius of \(0.5\) mm. This notch shape was chosen as the simplest of those in use, since the “general methods” did not provide any specimen shape for fracture at all.

The specimen was cut directly from the milled surface of the blank and was tested on a 15 kg Sharpy pendulum in the materials-strength laboratory of the Tomsk Institute of Railway Transport Engineers. In the event that the specimen did not break under a single blow, a second blow was applied to it, the work of the second blow being added to the first.

Figure 1 gives a comparison of the strength characteristics of various steels. The heat treatment of the blanks (annealing, normalizing, hardening and tempering) was carried out in the institute’s metallographic laboratory, partly in a Heraeus muffle electric furnace and partly in an oil furnace. Temperature control was maintained by means of thermocouples; in addition, the electric furnace was equipped with automatic temperature regulation. The heat-treatment schedule is given in Table 2.

The dimensions of the steel blanks for milling were \(50 \times 80 \times 300\) mm. The duration of heating during heat treatment was chosen in accordance with literature data, depending on the smallest dimension of the blank (50 mm) (Podoprigora. Heat Treater’s Handbook. p. 179. Signi-

Table 2.

No.	Material being machined	Heat treatment	Heating temperature	Cooling
1	Steel 1020—n	Normalizing	880°	In air
2	” 1035—n	same	880°—900°	In air
3	” 1040—o	Annealing	760—780°	With the furnace
4	” 1040—n	Normalizing	880°	In air
5	” 1040—u	Hardening and tempering	820°; tempering after quenching 530°	In oil; in air
6	” 1050—o	Annealing	Carried out at the Kuznetsk Metallurgical Plant. Regime not known
7	” 5120—n	Normalizing	860°	In air
8	” 5140—o	Annealing	860°	With the furnace
9	” 5140—n	Normalizing	855—860°	In air
10	” 3120—n	Normalizing	880°—900°	In air
11	” 3340—o	Annealing	810°—820°	With the furnace
12	” 3340—u	Hardening and tempering	800—810°; tempering after quenching 580°	In oil; in oil

menskii. Metalworker’s Handbook. p. 348. Chernyshev. Testing, Heat Treatment, and Application of Steel. p. 73).

The heating time for reaching the required temperature was taken as 1 hr. 30 min. Holding at constant temperature: 1 hr. 30 min. The quenching oil was machine oil; the oil temperature was 15—20°C. Cooling with the furnace during annealing continued for 18—24 hours, after which the blanks, at a temperature of 100—120°C, were taken out into the air. The chemical composition of the materials being machined is given in Table 3.

The data of chemical analysis for cast irons No. 1 and No. 2 are completely absent, since the cast irons were sent by Gipromash without chemical composition, and no chemical analysis on site was provided for. The heat treatment of malleable cast iron (No. 5) was carried out in the malleable-cast-iron foundry of the Stalin Automobile Plant according to the following regime.

1. Heating to a temperature of 900°C—11—12 hours.
2. Holding at a temperature of 900—980°C—35 hours.
3. Lowering the temperature from 900° to 750°C—28 hours.
4. Lowering the temperature from 750° to 600°C—5 hours.
5. Cooling in air.

Total duration of the cycle (1—4)—80 hours.

All steels were received in blanks of section 100 × 100 mm.

Cast iron No. 1 and No. 2: blanks 100 × 100 × 500 mm.
Cast iron No. 3: 33 × 76 × 315 mm.
Cast iron No. 4: 49 × 80 × 420 mm.
Cast iron No. 5.

In Figs. 2—27 microphotographs of the materials being machined are given. For the microphotographs we provide the conclusion of the institute’s metallographic laboratory on the structural constituents.

Steel 1010—p (Fig. 2). Ferrite grains with isolated inclusions of pearlite islands.

Steel 1020—p (Fig. 3). Ferrite grains with isolated inclusions of pearlite islands.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Fig. 16.

Fig. 17.

Fig. 18.

Fig. 19.

Fig. 20.

Fig. 21.

Fig. 22.

Fig. 23.

Fig. 24.

Fig. 25.

Table 3

No.	Name of material	Source from which material was obtained	C	Mn	Si	P	S	Ni	Cr	Source from which chemical analysis was obtained
	Steel 1010	Krasny Oktyabr Plant	0,13	0,44	0,17	0,015	0,018	—	—	Certificate of the Krasny Oktyabr Plant
	1020	Chelyabinsk Tractor Plant	0,15	0,39	0,24	0,017	0,029	—	—	Chemical laboratory, Tomsk Industrial Institute
	1035	Krasny Oktyabr Plant	0,35	0,62	0,3	0,023	0,023	—	—	Certificate of the Krasny Oktyabr Plant
	1040	Ch.T.Z.	0,35—0,45	0,5—0,8	0,17—0,37	\<0,045	\<0,045	—	\<0,2	OST 7123
	1050	Kuznetsk Metallurgical Plant	0,54	0,64	0,22	0,01	0,2	—	—	Chemical laboratory, K. M. Plant
	5120	Ch.T.Z.	0,16	0,39	0,29	0,02	0,035	—	0,68	Chemical laboratory, Tomsk Industrial Institute
	5140	Ch.T.Z.	0,35	0,61	0,31	0,021	0,045	—	0,89	Chemical laboratory, Tomsk Industrial Institute; OST 7124
	5145	Ch.T.Z.	0,4—0,5	0,5—0,8	0,17—0,37	\<0,01	\<0,04	0,3	0,8—1,1	OST 7124
	3120	Krasny Oktyabr Plant	0,18	0,35	0,28	0,013	0,016	1,26	0,57	Certificate of the Krasny Oktyabr Plant
	3140	Ch.T.Z.	0,35—0,45	0,5—0,8	0,17—0,37	\<0,04	\<0,04	1,0—1,5	0,45—0,75	OST 7124
	3340 (E—10)	Elektrostal Plant	0,39	0,38	0,34	0,023	0,018	3,44	1,33	Certificate No. 412, OST, Elektrostal Plant
	Cast iron No. 3 (alloyed)	Automobile Plant named after Stalin	3,30	0,65	2,45	0,2	0,1	0,36	0,28	Data of the laboratory of the Automobile Plant named after Stalin
	Cast iron No. 4	Automobile Plant named after Stalin	3,41	0,61	2,36	0,179	0,131	0,24	0,23	Data of the laboratory of the Automobile Plant named after Stalin
	Cast iron No. 5 (malleable)	Automobile Plant named after Stalin	3,30	0,43	1,35	0,106	0,122	—	—	Data of the laboratory of the Automobile Plant named after Stalin

Steel 1020—n (Fig. 4). Ferritic field with isolated inclusions of pearlite islands.

Steel 1035—p (Fig. 5). Ferrite plus pearlite.

Steel 1035—n (Fig. 6). Refined pearlite plus ferrite.

Steel 1040—p (Fig. 7). Pearlite plus ferrite. Sorbite-like pearlite.

Steel 1040—o (Fig. 8). Pearlite plus ferrite.

Steel 1040—n (Fig. 9). Pearlite grains surrounded by a ferrite network. The ferrite is distributed unevenly over the section of the polished specimen. There are individual light areas in the form of accumulations of ferrite.

Steel 1040—u (Fig. 10). Fine pearlite grains surrounded by a ferrite network.

Steel 1050—o (Fig. 11). The main field consists of fine-lamellar pearlite surrounded by a thin, locally broken ferrite network.

Steel 5120—p (Fig. 12). Coarse grains of pearlite and ferrite.

Steel 5120—n (Fig. 13). Refined pearlite surrounded by ferrite.

Steel 5140—p (Fig. 14). Coarse pearlite grains surrounded by a thin broken ferrite network. Fine-lamellar pearlite.

Fig. 26.

Fig. 27.

Steel 5140—o (Fig. 15). Pearlite grains surrounded by a ferrite network. Sorbite-like pearlite.

Steel 5140—n (Fig. 16). Pearlite grains surrounded by a ferrite network. The pearlite grains are refined. Elongated slag inclusions occur in large quantity.

Steel 5145—p (Fig. 17). Pearlite grains surrounded by a thin ferrite network, not everywhere uniform in thickness. Sorbite-like pearlite, locally fine-lamellar.

Steel 3120—n (Fig. 18). Ferrite grains and isolated pearlite islands. The distribution of pearlite and ferrite over the section of the polished specimen is not uniform. Noticeable rolling traces.

Steel 3140—p (Fig. 19). Fine pearlite grains surrounded by a ferrite network. In places a banded structure is retained. Fine-lamellar pearlite.

Steel 3340—o (Fig. 20). Fine grains of sorbite-like pearlite surrounded by an indistinct ferrite network.

Steel 3340—u (Fig. 21). Finely dispersed sorbite.

Cast iron No. 1 (Fig. 22). Pearlite-ferrite structure with graphite inclusions in the form of elongated strands and isolated islands of phosphide eutectic.

Cast iron No. 2 (Fig. 23). Pearlitic cast iron. The principal field consists of finely lamellar pearlite with graphite inclusions in the form of elongated filaments of uneven length.

Cast iron No. 3 (Fig. 24). Alloyed (chromium-nickel) cast iron. The principal field consists of finely lamellar pearlite, with graphite inclusions in the form of curved filaments of uneven thickness. In places the graphite occurs in the form of small globules. The distribution of graphite over the section of the specimen is uniform.

Cast iron No. 4 (Fig. 25). Alloyed (chromium-nickel) cast iron. The principal field consists of finely lamellar pearlite, with graphite inclusions in the form of curved filaments of uneven thickness. In places the graphite inclusions occur in the form of globules.

Cast iron No. 5 (Fig. 26). Malleable cast iron. The principal field is pure ferrite with graphite inclusions in the form of globules with jagged outlines. In places there are separate islands of finely lamellar pearlite.

Bronze (Fig. 27). The structure is a dendritic structure of a molten solid solution O,C.

All the microsections examined were taken from the metal along the rolling direction. Preliminary investigations showed that there is no appreciable difference between the structures along and across the rolling. Therefore, for uniformity and comparison of all the steels studied, the microsections were examined along the rolling direction.

Etching of the steel and cast-iron specimens was carried out with a 2% solution of $HNO_3$ in ethyl alcohol. The bronze was etched with the following composition: $20\ \mathrm{cm}^3$ $HCl$—10 grams of ferric chloride, diluted in $60\ \mathrm{cm}^3$ of ethyl alcohol.

Microinvestigations were carried out with a Reichert bench metallurgical microscope at magnifications of 100, 200, 300, 500, 750, and 1000. Microphotographs were taken with a large Reichert metallurgical microscope and partly with a Zeiss metallurgical microscope. The magnifications used in photographing the microsections were as follows:

Steels 3340-o and 3340-u: magnification 320
The remaining steels: 120
Cast irons and bronze: 60.

Microinvestigations showed that the metal structures after heat treatment (annealing, normalizing, quenching and tempering) correspond to the standard microstructures after the corresponding treatment.

The metallographic investigations were carried out by Engineer V. Strokopytov.

Method of Work.

A. Cylindrical Cutter.

We derived theoretically¹) and then confirmed by an extensively conducted experiment²) equations characterizing the dependence of the circumferential force and power during operation of a cylindrical cutter on the factors of the cutting process. These equations are as follows:

Specific cutting pressure

$$ K_s = \lambda S_e^k \tag{1} $$

Circumferential force on the cutter for the given instantaneous position of its teeth under the chip:

$$ P = D\lambda S_z^{\kappa+1}\operatorname{tng}\beta \frac{2^{\kappa+1}}{k+2} \sum \left\{ \sin^{\kappa+2}\left(\frac{\psi_2}{2}\right) - \sin^{\kappa+2}\left(\frac{\psi_1}{2}\right) \right\} \tag{2} $$

Average Power When a Cylindrical Milling Cutter Is Operating

\[ N_e=\frac{2^{\kappa+1}}{\kappa+2}\, \frac{\lambda n^{-\kappa}}{1000\cdot 4500}\cdot b\cdot s^{\kappa+1}\, t^{\frac{\kappa+2}{2}}\, z^{-\kappa}\cdot D^{-\frac{\kappa}{2}} \tag{3} \]

These relationships were confirmed by us experimentally, both by means of tests on a milling machine and by tests on a special pendulum dynamometer when working on various metals (steel, cast iron, aluminum), and therefore may be considered entirely reliable.

Here:

\(\lambda\) — a constant characterizing the material being machined and the tooth profile of the cutter (the presence or absence of a rake angle).

\(S_e\) — the chip thickness at the given moment at the given point of the cutter tooth.

\(k\) — a constant depending on the material being machined and on the sharpness of the cutter tooth. The quantity is negative and fractional.

\(D\) — cutter diameter in mm.

\(S_z\) — feed per tooth in mm.

\(\beta\) — angle of rise of the tooth helix.

\(\psi_1, \psi_2\) — end angles characterizing the given instantaneous position of the tooth under the chip.

\(n\) — number of revolutions of the cutter per minute.

\(b\) — milling width in mm.

\(s\) — feed in millimeters per minute.

\(t\) — milling depth (thickness of the layer being removed).

\(z\) — number of cutter teeth.

According to equation 3, in order to determine the expenditure of power in milling a given work material, it is sufficient to know \(\lambda\) and \(k\), whose determination must be carried out experimentally for the various materials machined.

The simplest experimental method, requiring a minimum number of experiments and not inferior in accuracy to any other, was applied by us in our previous experimental work. It will become clear from the following derivations.

From equation 3 one can obtain an expression for the work expended by one cutter tooth in removing one chip per 1 revolution:

\[ A_z=\frac{1000\cdot 4500\,N_e}{zn}\ \text{kgmm}. \]

Substituting, in place of \(N_e\), the right-hand side of equation 3, we obtain

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

The work \(A_z\) is expended on the removal of a chip volume

\[ W=b\cdot t\cdot S_z\ \text{mm}^3 \]

Dividing \(A_z\) by \(W\), we obtain the specific work \(A_w\), expended on the removal of \(1\ \text{mm}^3\) of chip.

\[ A_w=\frac{A_z}{b\cdot t\cdot S_z} =\frac{2^{\kappa+1}}{\kappa+2}\lambda \left\{S_z\sqrt{\frac{t}{D}}\right\}^{\kappa} \ \text{kgmm/mm}^3 \]

Here

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

where \(\varphi\) is the angle of contact over which the chip is removed by the cutter tooth.

Hence

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

\(S_m\) is nothing other than the chip thickness at the middle of the contact angle, or, otherwise, the “mean chip thickness.” Finally we have

\[ A_w=\frac{2\kappa+1}{\kappa+2}\,\lambda S_m^{\kappa} \tag{4} \]

Equation 4 shows that, whatever the values of \(D, z, n, b, t, s, s_z\) at which the milling operation is performed, the specific work, for a constant work material and tooth profile \((\lambda, \kappa=\mathrm{const})\), depends only on the mean chip thickness. Thus, if the results of measuring the useful power \(N_e\), expended in work with various diameters, numbers of teeth, rotational speeds, milling widths, depths of milling, and feeds, are recalculated into the specific work \(A_w\), and the results of all these experiments are plotted on a graph with logarithmic coordinates \(S_m\) (abscissa) and \(A_w\) (ordinate), then on this graph we must obtain, for all tests on the given work material, only one straight line. The inclination of this straight line to the abscissa axis \((S_m)\) will give \(\tan \alpha=\kappa\); the value of \(A_w\) at \(S_m=1\) will give \(\frac{2\kappa+1}{\kappa+2}\lambda\), i.e., the quantities \(k\) and \(\lambda\) can readily be determined, and this is all that we must obtain for power equation 3.

Everything just set forth was fully confirmed by our previous experimental work.

Thus, in order to determine the dependence of useful power on the cutting factors in milling, it would be sufficient, taking a cutter with any diameter and number of teeth, to carry out a series of experiments with one depth of milling and with a range of feeds, and from this it would already be possible to obtain the values of \(k\) and \(\lambda\) required for equation 3. But, for greater reliability of the procedure presented by us and approved by the Technical Department of the People’s Commissariat of Heavy Machine Building, it was provided that tests be carried out with not fewer than two depths of milling for each material being machined.

For the tests the following depths were selected:

\[ t=1\ \text{mm}\quad \text{(finishing)} \]

\[ t=4\ \text{mm}\quad \text{(roughing)} \]

Since

\[ N_e=\frac{Mn}{716200}, \]

where \(M\) is the torque in \(\mathrm{kg\,mm}\), the specific work can be expressed as follows:

\[ A_w=\frac{1000\cdot 4500\,N_e}{z\cdot n\cdot b\cdot t\cdot s_z} \]

\[ A_w=6{,}28\,\frac{M}{b\cdot t\cdot s_z\cdot z} \tag{5} \]

Thus, in the experiments, in order to determine \(A_w\), it was necessary to measure the quantities \(b, t, s_z, z, M\); in addition, in order to determine \(S_m\), it was necessary to measure \(D\). The method of measuring the torque on the cutter will be discussed below in the section on the experimental apparatus; the remaining quantities were measured as follows:

The diameter of the cutter \(D\) was measured with a micrometer to an accuracy of \(0.05\) mm. Greater accuracy was not required here, since, judging from equation 3, the influence of the diameter on power consumption is very small.

The milling width “\(b\)” was measured with a micrometer to an accuracy of \(0.01\) mm.

The milling depth \(t\) was measured with an indicator and a depth micrometer to an accuracy of \(0.01\) mm.

The feed was measured with an indicator to an accuracy of \(0.005\) mm.

B. Face cutter.

For the operation of the face cutter we derived and experimentally verified the following dependences\(^{3}\):

Specific cutting pressure

\[ K_s=\lambda \cdot S_e^{\kappa}\cdot t^l \tag{6} \]

Since the tooth of a face cutter operates according to the principle of non-free cutting and cuts a chip both in width and in thickness, the specific pressure is affected, in addition to the chip thickness \(S_e\), also by its width, which in the given case is the thickness of the layer removed, the milling depth \(t\). The quantity \(l\) characterizes the influence of \(t\) on \(K_s\) (Fig. 28).

Circumferential force on the cutter.

\[ P=\lambda \cdot S_z^{\kappa+1} t^{l+1}\sum \sin^{\kappa+1}\psi \tag{7} \]

Fig. 28.

Useful power in milling in the case when the milling width is equal to the diameter \((b=D)\).

\[ N_e=\frac{2^{-\frac{\kappa}{2}}}{1000\cdot 4500}\lambda \cdot n^{\,l-\kappa}\cdot t^{l+1}\cdot s^{\kappa+1}\cdot z^{-\kappa}\cdot D \tag{8} \]

Useful power in the case when \(b<D\) and the cutter is positioned symmetrically relative to the milled surface:

\[ N''_e=N_e\left[1-\left(1-\frac{b}{D}\right)^{\frac{\kappa+2}{2}}\right] \tag{9} \]

Useful power in the case when \(b>\dfrac{D}{2}\) and the cutter, being located on one side of the machined surface (side cutting), removes a chip similarly to a cylindrical cutter, beginning from zero thickness.

\[ N'_e=N_e\,2^{\frac{\kappa}{2}}\left(\frac{b}{2}\right)^{\frac{\kappa+2}{2}} \tag{10} \]

In expressions 9 and 10, \(N_e\) is the power for the case when \(b=D\), i.e., according to equation 8. The milling width \(b\) in all cases is measured in the plane normal to the axis of the cutter and normal to the feed direction (Fig. 28). All these equations were also confirmed in our previous work\(^{3}\).

In the present work the face cutter always operated with a milling width equal to the diameter \((b=D)\), since this produced the smoothest operation of the cutter. For this case the work of one tooth

\[ A_z=2^{-\frac{\kappa}{2}}\lambda \cdot S_z^{\kappa+1}\cdot t^{l+1}\cdot D \]

Specific work

\[ A_w=\frac{A_z}{t\cdot D\cdot S_z}=2^{-\frac{k}{2}}\lambda\cdot S_z^k\cdot t^l \]

\[ A_w=\lambda\cdot(0.707\,S_z)^k\cdot t^l \]

Here

\[ 0.707=\sin 45^\circ=\sin\frac{\varphi}{2} \]

\[ 0.707\,S_z=S_z\sin\frac{\varphi}{2}=S_m \]

whence

\[ A_w=\lambda S_m^k\cdot t^l \tag{11} \]

If equation 11 is expressed graphically in logarithmic coordinates \(S_m—A_w\), then we obtain as many straight lines as there were different milling depths used in the experiments (provided that \(l\ne 0\)). The slope of these straight lines makes it possible to determine the value of \(k\). With \(S_m=1\) and \(t=1\) we obtain the value \(\lambda\). From the ratios of \(A_w\) at a constant (any) value of \(S_m\), but at different \(t\), we obtain the quantity \(l\). Thus, in order to determine the values \(\lambda\), \(k\), \(l\), needed for the power equations, it is sufficient to carry out, for each material being machined, a series of experiments with any cutter diameter and with any number of teeth, at several different milling depths and with a range of feeds at each milling depth.

For the reliability of the conclusions, the procedure provided for carrying out experiments at four milling depths.

\[ t=1\ \text{mm};\ 2\ \text{mm};\ 4\ \text{mm and }6\ \text{mm}. \]

Cutting tool.

From the above it follows that, in order to determine the power characteristics, it is sufficient to carry out experiments with only one cutter size. Therefore, one cutter each was taken for cylindrical and face milling.

Cylindrical cutter (Fig. 29):

Diameter \(\sim 59\) mm.
Number of teeth 8.
Helix lead angle to the end face \(\beta=60^\circ\).
Helix inclination angle to the axis \(30^\circ\).
Rake angle in the plane of the end face \(\gamma=10^\circ\).
Relief angle \(6—8^\circ\).
Cutter material: steel RF—1.

Fig. 29.

The cutter, throughout the entire operation, was not removed from the arbor and, in order to obtain minimum eccentricity of the teeth, was sharpened on the same arbor.

The eccentricity of the cutter teeth after sharpening was checked with an indicator, with the cutter arbor fixed in the working position in the spin-

on the body of the milling machine. One of such measurements, carried out at three different positions of the cutter (along its length), is given in Table 4.

Table 4

Tooth No.	Indicator readings in hundredths of a millimeter	Indicator readings in hundredths of a millimeter	Indicator readings in hundredths of a millimeter
Tooth No.	Section 1	Section 2	Section 3
1	0	0	1
2	1	0	1
3	1	1	1
4	1	1	0
5	2	1	1
6	2	2	1
7	2	2	1
8	1	1

The table shows that the eccentricity does not exceed 0.02 mm.

Face milling cutter (Fig. 30).

Diameter 64 mm.
Number of teeth 10.
Rake angle on the blade of the cylindrical part 8°.
Rake angle on the face blade 15°.
Relief angle on the cylindrical part 8–10°.
Relief angle on the face blade 6–8°.
Inclination of the face blade to the end-face plane 1½–2°.
Radius of rounding at the junction of the face and cylindrical blades 0.5 mm.
Cutter material—steel RF-1.

Fig. 30.

The cutter was likewise sharpened on an arbor, and the results of measuring the eccentricity of the teeth were obtained qualitatively the same as for the cylindrical cutter.

For the tests, cutters were used only with a sharp tooth. The cutters were sent for resharpening at the moment when a land 0.1 mm wide had formed on the relief face. Check tests showed that such dulling does not have a noticeable effect on the power consumption. Sharpening was usually carried out on the relief face, and only in the case when a notch visible to the eye appeared on the rake face was the rake face touched up.

After sharpening, the cutters were run for several passes without recording the loads. This was necessary in order to remove from the blade the burr produced during sharpening and to bring it into a certain stable working condition. Otherwise, the results of tests with a freshly sharpened cutter deviated somewhat from the results of all the other tests.

Measurement of the angles on the cutters was carried out both with a protractor and by the coordinate method; in all cases we obtained good agreement of the results.

Equipment and Experimental Apparatus

All tests were carried out on a universal milling machine of the Tula plant, having 12 spindle speeds and 12 feeds from 20 to 700 mm per minute. When working with the face milling cutter, the blank was fastened to an auxiliary—

...on the test blank in such a way that the surface being machined faced the spindle of the machine, in which the face-milling cutter was mounted. In all cases of operation, after the necessary setting for depth of cut, the vertical- and cross-feed guides were firmly clamped in order to protect the system from the play and vibrations inevitable in these guides.

The face-milling cutter was given such a direction of rotation that the blank being machined was pressed against the table by the acting cutting forces, rather than being torn away from it.

During the experiments it was necessary to record the torques on the milling cutter (without parasitic resistances). For this purpose a dynamometer of the author’s design was used, built into the speed box of the machine. The principle of operation of the dynamometer is clear from Fig. 31. 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

Fig. 31.

5, connected 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 brackets 6 are rigidly connected to each other. 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 brackets 6, clockwise (if the spindle also rotates clockwise). But the brackets bear, through a special pin, against hydraulic load cell 7, fastened to the side wall of the machine. In Fig. 32, on the right wall of the machine, the arrow indicates the load cell, from which tubes run to indicating and recording manometers fastened to the wall to the left of the machine. Since the liquid in the load cell is practically incompressible, brackets 6 move almost not at all. The pressure on the piston of the load cell will be the greater, the greater the torque on the spindle. The torque can thus be recorded by a self-recording manometer on a moving paper tape.

A special spring is provided to press the earrings 6 against the dynamometer 7, which creates an initial preload and eliminates play. In practice, during operation the dynamometer registers not only the net cutting torque, but also harmful resistances in the spindle bearings, in the bearings of gear 4, in the gear pair 4–5, and, when a cylindrical cutter is used, also in the outer bearing of the cutter arbor. The machine spindle is mounted on tapered roller bearings, gear 4 on ball bearings, and the losses here are practically negligible. The losses in pair 4–5 and in the outer bearing of the arbor were taken into account by means of special calibration and were then excluded from the total torque registered by the dynamometer.

Fig. 32.

The main calibration of the dynamometer was carried out with a “Prony” brake. The calibration graph is shown in Fig. 33. Fig. 34 shows the displacement of the piston of the dynamometer chamber measured during calibration. The maximum displacement of the chamber piston is only 0.16 mm, while the displacement of gear 4 of the dynamometer in this case does not exceed 0.1 mm. If one takes into account that there is an initial preload approximately equal to 25% of the maximum load calculated for the dynamometer, and that during operation with accurately ground cutters the load never fluctuates

from the maximum to zero, one can imagine how stable the entire system is in operation, and the dynamometer operates quite flawlessly. The dynamometer has been in operation for more than 4 years already, and all the dozens of calibrations carried out during this time, as well as repeated check calibrations carried out in the course of the present work, in all cases gave one and the same calibration curve.

To take account of the additional losses arising under the influence of the forces acting on the cutter, a special calibration was carried out separately for the case of operation of a cylindrical cutter (the presence of an outer bearing for the cutter arbor), the results of which are presented in Fig. 35, and separately for the case of operation of a face cutter (absence of an outer bearing); the results are presented in Fig. 36. The magnitudes of the additional torques, as functions of the forces acting on the cutter, are very small, but nevertheless we subtracted them from the torques obtained in the tests and in this way obtained the pure cutting torque.

Fig. 33.

The axial loads on the cutter, as a specially conducted test showed, had absolutely no effect on the readings of the dynamometer.

The dynamometer, owing to its “inertia,” could not register with absolute accuracy changes in torque occurring over very short intervals of time; therefore, when operating with a cylindrical cutter, such a milling width \(b\) was chosen that it was related to the cutter diameter \(D\), the helix angle \(\beta\), the helix lead of the cutter tooth \(H\), and the number of teeth \(Z\) by the relation

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

In this case, as is known, the total cross-section on all simultaneously working teeth is constant both in magnitude and in form; the total circumferential force and torque are likewise constant for a given operating condition, and if they do have certain oscillations owing to the inevitable runout of the cutter, these oscillations occur once per revolution of the cutter and can be accurately recorded by the dynamometer.

Fig. 34.

The speed of the paper tape on which the torque diagram was recorded by the recording manometer was such that one revolution of the milling cutter occupied 40 mm on the diagram, and all load fluctuations were clearly represented.

Fig. 35.

All diagrams without exception were planimetered, in such a way that the planimeter covered the portion of the graph necessarily corresponding to an integral number of revolutions of the milling cutter (2–3); from the area the mean height was found, and from this, using the calibration graph, the magnitude of the torque was determined.

In the course of the tests it became clear that, when working with a face milling cutter at a milling depth of 6 mm, on steel, with the existing feeds of the machine, it is possible, within the load permissible for the dynamometer, to use only the 3–4 smallest feeds and thus, at this milling depth, to obtain only 3–4 experimental points. Some of the tests were carried out in this way. Then a transmission was built into the feed mechanism of the machine, reducing the feed magnitudes by a factor of 60. Thanks—

Fig. 36.

to this, within the limits of the load permissible for the dynamometer, it became possible to use only the 3–4 smallest feeds and to obtain, at this milling depth, only 3–4 experimental points. Some of the tests were carried out in this way. Then a transmission was built into the feed mechanism of the machine, reducing the feed magnitudes by a factor of 60. Thanks—

As a result, this machine tool already had not 12 but 24 feeds, and it seemed possible to obtain a larger number of experimental points at any depth of milling.

All the tests were carried out at a spindle speed of 34 rpm. The following feeds per one revolution of the cutter and per one cutter tooth were used (Table 5).

Table 5

No.	Feed in mm/rev \(S_n\)	Feed per 1 tooth \((S_z)\), mm — cylindrical cutter \(Z = 8\)	Feed per 1 tooth \((S_z)\), mm — face cutter \(Z = 10\)
1	0.073	0.0094	0.0073
2	0.1	0.0125	0.01
3	0.134	0.0167	0.0134
4	0.2	0.025	0.02
5	0.272	0.034	0.0272
6	0.37	0.0463	0.037
7	0.585	0.0732	0.0585
8	0.805	0.1006	0.0805
9	1.10	0.137	0.11
10	1.59	0.1987	0.159
11	2.21	0.276	0.221
12	3.0	0.375	0.3
13	4.46	0.557	0.446
14	6.21	0.776	0.621
15	8.4	1.05	0.84
16	12.1	1.51	1.21
17	16.85	2.106	1.685

It is evident from the table that the feeds used extend far beyond, on the one hand, the finest finishing feeds and, on the other, the largest roughing feeds. All tests were conducted without cooling or lubrication. The influence of lubricant on power consumption has been established by us in another work and will be published in the near future.

Tests with a Cylindrical Cutter

Before proceeding to the main tests, whose purpose was to determine the values of \(k\) and \(\lambda\) for various metals, we carried out a series of tests to determine the influence of cutting speed on torque, with all other conditions held constant \((t, b, D, Z, S_z = \mathrm{const}.)\). Although this was not provided for by the research program, it was extremely necessary, since according to Stoewer’s data\(^{4}\) cutting speed has a substantial influence on torque when milling steel of medium hardness.

According to Stoewer, this influence is such that, when the cutting speed is increased from 5 to 30 meters per minute, the torque, with a constant feed per revolution, falls by 10%. All other researchers of the milling process, without exception, paid no attention to this dependence, evidently believing that the specific pressure in milling does not depend on the cutting speed. Stoewer’s data, for their part, required verification, since in his investigations there was no special device for recording the torque, and the torque was determined from the power consumed and the efficiency of the machine by the Bahlecke braking method, which cannot be regarded as correct in view of the variability of losses in the individual links of the machine when the load and speed change. We performed this experiment using our dynamometer, which for this purpose was calibrated at various spindle speeds. Fig. 37, constructed from the results of our experiments, shows the complete absence of any influence of cutting speed on torque.

Fig. 37.

The experiments were conducted at a constant milling depth equal to 2.4 mm and a width of 40 mm. We were unable to achieve a speed above 27 m/min with our dynamometer.

On the basis of Fig. 37 it may be concluded that experiments can be carried out at any of the speeds up to 30 m/min.

For the purpose of obtaining calmer operation, better preservation of the cutting edge during work, and lower consumption of the material being machined, in the subsequent work we used a speed of 34 revolutions per minute, which for a cylindrical milling cutter corresponded to a cutting speed of 5.3 m/min.

Below, in Tables 6–9, we give the magnitudes of the torques obtained in experiments on the materials tested. The torque values from these tables were recalculated by us into specific work \(A_w\), and graphs were constructed in logarithmic coordinates \(S_m — A_w\) (Figs. 38–63). In the graphs, the mean chip thickness is plotted in microns.

In all the graphs without exception we see how satisfactory the results obtained are. There is no doubt about the rectilinearity of the dependence \(A_w = f(S_m)\) in logarithmic coordinates. In some cases (Figs. 50, 51, 62), with the thinnest chips (in the left-hand part of the graphs), the experimental points deviate downward from the straight line drawn. This is explained by the following circumstance. The indicated deviation of the points is observed in Figs. 50, 51, 52 at a mean chip thickness of less than 5–7 microns. With such thin chips, even with very careful sharpening of the cutter teeth, the chip is not removed by every successive tooth of the cutter; rather, some teeth, differing in height from the neighboring ones, either along the entire length of the cutting edge or along part of it, do not remove chips, leaving a thicker chip for the next tooth. A thicker chip requires less specific work, which is reflected in a lowering of the point on the graph.

Some series of experiments were repeated by us under completely identical conditions, and this repetition was carried out after a certain considerable interval of time. Such experiments were conducted on steels 1040—u and 5140—n.

Table 6

Torques, in kilogram-meters, when machining with a cylindrical milling cutter with the number of teeth \(Z = 8\) and helix angle \(\beta = 60^\circ\)

| Machined material | \multicolumn{2}{c}{Steel 1010—p
\(D=58.5\) mm
\(b=39\) mm} | \multicolumn{2}{c}{Steel 1020—n
\(D=58.8\) mm
\(b=40.3\) mm} | \multicolumn{2}{c}{Steel 1020—p
\(D=58.8\) mm
\(b=40.3\) mm} | \multicolumn{3}{c}{Steel 1035—p
\(D=58.5\) mm
\(b=39\) mm} | \multicolumn{2}{c}{Steel 1035—n
\(D=58.5\) mm
\(b=40.3\) mm} | \multicolumn{3}{c}{Steel 1040—p
\(D=58.8\) mm
\(b=40.3\) mm} | \multicolumn{2}{c}{Steel 1040—o
\(D=58.8\) mm
\(b=40.3\) mm} | \multicolumn{2}{c}{Steel 1040—n
\(D=58.8\) mm
\(b=40.3\) mm} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| \(S_z\), feed per 1 tooth | \(t=1.03\) mm | \(t=4.14\) mm | \(t=1.04\) mm | \(t=4.0\) mm | \(t=1.0\) mm | \(t=4.1\) mm | \(t=1.02\) mm | \(t=1.8\) mm | \(t=4.0\) mm | \(t=1.05\) mm | \(t=4.2\) mm | \(t=1.02\) mm | \(t=2.0\) mm | \(t=4.15\) mm | \(t=1.0\) mm | \(t=4.1\) mm | \(t=1.0\) mm | \(t=4.1\) mm |
| 0.0091 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — |
| 0.0125 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — |
| 0.01674 | — | 2896 | — | — | — | 980 | 2203 | — | — | — | — | — | — | — | — | — | — | — |
| 0.0250 | — | 3485 | — | — | — | 1276 | 2787 | — | — | 2970 | — | — | — | — | — | — | — | — |
| 0.0340 | — | 4368 | — | — | — | 1620 | 2994 | — | — | 3925 | — | — | — | — | — | — | — | — |
| 0.0463 | — | 5497 | — | — | — | 2110 | 3554 | — | — | 4760 | — | — | — | — | — | — | — | — |
| 0.0732 | 2012 | 7286 | 2445 | 7838 | 2395 | 2896 | 5565 | 2061 | 6325 | 1973 | — | 3337 | 5879 | 1915 | 6575 | 1944 | 6623 | — |
| 0.1006 | 2676 | 9240 | 3043 | 9261 | 2894 | 3535 | 6830 | 2615 | 8165 | 2208 | 4352 | 7190 | 2641 | 8311 | 2115 | 7895 | — | — |
| 0.1374 | 3313 | 10921 | 3828 | 11263 | 3857 | 4363 | 8580 | 3410 | 9630 | 3092 | 5506 | 9000 | 3140 | 10240 | 3170 | 9898 | — | — |
| 0.1987 | 4515 | 13766 | 5005 | 14323 | 4466 | 6085 | 11477 | 4220 | 12552 | 3907 | 7064 | 11574 | 4344 | 13402 | 4319 | 12528 | — | — |
| 0.2765 | 5423 | 17075 | 6035 | 17654 | 5830 | 7308 | 14747 | 5638 | 16747 | 4908 | 8960 | 14854 | 5496 | 16235 | 5488 | 15145 | — | — |
| 0.3750 | 6477 | 19513 | 7455 | 21280 | 7073 | 9532 | 18578 | 6775 | 19385 | 6300 | 10970 | 18540 | 7004 | 20525 | 6075 | 19233 | — | — |
| 0.5570 | 8213 | 24917 | 9390 | 30250 | 8854 | 12829 | 24620 | 9115 | 26655 | 8431 | 14660 | 25645 | 9224 | 27595 | 8995 | 25276 | — | — |
| 0.7760 | 9948 | 31160 | 11244 | — | 10922 | 15915 | 32620 | 11210 | 31525 | 10776 | 18497 | 30202 | 11660 | — | 10980 | — | — | — |
| 1.0500 | 12285 | — | 13461 | — | 12917 | 19931 | — | 13862 | — | 12965 | 23007 | — | 13669 | — | 13645 | — | — | — |
| 1.5120 | 14864 | — | 16593 | — | 16255 | — | — | 17965 | — | 16641 | — | — | 17108 | — | 17365 | — | — | — |
| 2.1060 | — | — | 20335 | — | 19193 | — | — | 22330 | — | — | — | — | 21101 | — | 21795 | — | — | — |

Table 7

Torques in kilogram-millimeters during operation of a cylindrical milling cutter with number of teeth \(Z = 8\) and helix angle \(\beta = 60^\circ\)

Workpiece material	Steel 1040—U \(D=58.0\) mm, \(b=40.3\) mm \(S_z=1.0\)	Steel 1040—U \(D=58.0\) mm, \(b=40.3\) mm \(S_z=1.04\)	Steel 1040—U \(D=58.0\) mm, \(b=40.3\) mm \(S_z=3.85\)	Steel 1050—O \(D=59.7\) mm, \(b=39.7\) mm \(S_z=1.0\)	Steel 1050—O \(D=59.7\) mm, \(b=39.7\) mm \(S_z=3.0\)	Steel 3120—H \(D=58.5\) mm, \(b=40.3\) mm \(S_z=1.05\)	Steel 3120—H \(D=58.5\) mm, \(b=40.3\) mm \(S_z=4.25\)	Steel 3140—P \(D=58.0\) mm, \(b=39.7\) mm \(S_z=1.12\)	Steel 3140—P \(D=58.0\) mm, \(b=39.7\) mm \(S_z=4.07\)	Steel 3340—O \(D=58.0\) mm, \(b=39.75\) mm \(S_z=1.05\)	Steel 3340—O \(D=58.0\) mm, \(b=39.75\) mm \(S_z=4.0\)	Steel 3340—U \(D=58.0\) mm, \(b=39.9\) mm \(S_z=1.1\)	Steel 3340—U \(D=58.0\) mm, \(b=39.9\) mm \(S_z=4.1\)	Steel 5120—P \(D=58.0\) mm, \(b=40.3\) mm \(S_z=1.0\)	Steel 5120—P \(D=58.0\) mm, \(b=40.3\) mm \(S_z=4.0\)	Steel 5120—H \(D=58.0\) mm, \(b=40.3\) mm \(S_z=1.02\)	Steel 5120—H \(D=58.0\) mm, \(b=40.3\) mm \(S_z=4.0\)
0.0091	—	—	—	—	—	—	—	1423	—	1375	—	1225	—	—	—	—	—
0.0125	—	—	—	—	—	—	—	1765	—	1815	—	1670	—	—	—	—	—
0.01674	—	—	—	—	—	—	—	2160	—	2260	—	2305	—	—	—	—	—
0.0250	—	—	—	—	—	3690	—	2748	—	2945	—	3140	—	—	—	—	—
0.0340	—	—	—	—	—	4770	—	3580	—	3975	—	3875	—	—	—	—	—
0.0463	—	—	—	—	—	5888	—	4023	—	4820	—	5005	—	—	—	—	—
0.0732	2258	2356	6281	1775	4953	2420	—	2208	6183	2500	6920	2355	6870	1964	—	2223	7162
0.1006	2879	2896	8335	2173	6223	3215	—	2748	7210	2845	8630	2995	8725	2876	8150	3249	9012
0.1374	3436	3475	10863	2845	7495	4025	3690	2940	7800	3730	10875	3780	10970	3524	10298	3828	10776
0.1987	4858	4859	13014	3726	9985	5203	4770	4220	11555	5155	11320	4955	14420	4466	13606	4948	13848
0.2765	5938	6173	16544	4610	12145	5638	5888	5055	14275	6330	17945	6820	18280	5627	16515	6207	17076
0.3750	7455	7494	20421	5785	15032	7950	7944	6428	18149	8215	22450	8335	23240	7171	19955	7455	20790
0.5570	9679	9567	27825	8247	20475	10020	9918	9118	24800	10580	28875	10260	32225	9031	26003	9390	26977
0.7760	11672	12158	—	10910	26290	12065	12065	11215	—	12965	—	14225	—	10873	—	11920	—
1.0500	14119	14767	—	13803	—	14168	—	13395	—	15725	—	17220	—	12878	—	13781	—
1.5120	18728	18637	—	—	—	17967	—	18620	—	20381	—	21800	—	16400	—	17663	—
2.1060	—	22699	—	—	—	21865	—	23795	—	—	—	—	—	20430	—	20324	—

Table 8

Torques, in kilogram-millimeters, during operation of a cylindrical milling cutter with number of teeth \(Z = 8\) and helix angle \(\beta = 60^\circ\)

Feed \(S_z\), per tooth	Steel 5140-П D=58.8 b=40.3 t=1.0	Steel 5140-П D=58.8 b=40.3 t=1.0	Steel 5140-О D=58.8 b=40.3 t=1.0	Steel 5140-О D=58.8 b=40.3 t=4.1	Steel 5140-Н D=58.4 b=39 t=1.0	Steel 5140-Н D=58.4 b=39 t=3.15	Steel 5140-Н D=58.4 b=39 t=4.15	Steel 5140-Н D=58.8 b=40.3 t=1.02	Steel 5140-Н D=58.8 b=40.3 t=3.0	Steel 5145-П D=58.4 b=39 t=1.05	Steel 5145-П D=58.4 b=39 t=4.05	Steel 5145-П D=59.7 b=39.75 t=3.1	Cutter No. 1 D=58.5 b=39 t=1.0	Cutter No. 1 D=58.5 b=39 t=4.04	Cutter No. 2 D=58.0 b=40.2 t=1.03	Cutter No. 2 D=58.0 b=40.2 t=3.55
0.0091	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
0.0125	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
0.01671	—	—	—	2060	—	—	—	—	—	—	—	—	—	1030	—	—
0.0250	—	—	—	2600	—	—	—	—	—	—	—	—	—	1473	—	2305
0.0340	—	—	—	3240	—	—	—	—	—	—	—	—	—	1865	—	2945
0.0463	—	—	—	3875	—	—	—	—	—	—	—	—	—	2405	—	3530
0.0732	1963	6501	1816	6173	1765	1865	4375	1925	5228	2258	6575	5138	835	3632	1665	4365
0.1006	2464	8327	2430	7700	2455	2258	5693	2223	6550	2945	8583	6753	1130	4025	2110	5250
0.1374	2729	9825	2896	10240	2745	2600	6965	2552	7749	3335	10095	8665	1464	4885	2455	6180
0.1987	3361	11808	3927	12771	3827	3335	9360	3435	9752	4515	13550	11050	1895	6735	3140	8775
0.2765	4941	14698	5300	15120	5200	4660	11555	4427	11564	5840	16980	13808	2208	7308	3975	10680
0.3750	6354	18578	6281	19002	5985	5840	14515	5815	14283	7260	20810	17105	2750	8798	4710	13170
0.5570	8238	25682	8825	26360	8090	7600	—	7828	19040	9850	28985	23277	2994	9860	6230	17075
0.7760	10752	29875	10912	30522	10140	10140	25625	10182	25170	12335	—	—	3750	12844	7845	21800
1.0500	12917	—	13340	—	12285	—	—	12949	32425	15000	—	—	4415	14525	9945	—
1.5120	17437	—	17141	—	16260	—	—	16785	—	19860	—	—	6085	19717	13160	—
2.1060	20950	—	21935	—	22080	—	—	21702	—	—	—	—	—	—	—	—

Table 9

Torques in kilogram-millimeters when working with a cylindrical milling cutter with number of teeth \(Z=8\) and helix angle \(\beta=60^\circ\)

Machined material	Cast iron No. 3	Cast iron No. 3	Cast iron No. 4	Cast iron No. 4	Cast iron No. 5	Cast iron No. 5	Bronze	Bronze	Bronze
\(D\), mm	58.0	58.0	58.0	58.0	58.0	58.0	58.0	58.0	58.0
\(b\), mm	40	40	40.2	40.2	40.2	40.2	40.3	40.3	40.3
\(t\), mm	1.33	4.0	1.0	4.0	1.0	4.0	1.0	2.45	6.15
\(S_z\), feed per 1 tooth
0.0091	-	1080	-	835	-	-	-	-	-
0.0125	-	1225	-	1130	-	1130	-	-	-
0.01574	-	1470	-	1325	-	1325	-	-	-
0.0250	-	2060	-	1865	-	1865	-	-	-
0.0340	-	2505	-	2060	-	2305	-	-	-
0.0463	-	3040	-	2600	-	2945	-	-	-
0.0732	1470	3975	1325	3090	1425	4220	932	1816	4025
0.1006	1960	4710	1375	4170	1815	5155	1227	2160	5025
0.1374	2160	5500	1375	5005	2110	5985	1423	2749	5624
0.197	2845	7015	1865	6575	2845	7845	1758	3338	7162
0.2765	4070	8480	2355	8725	3335	9410	1855	3877	8726
0.3750	4910	10435	2945	9165	3825	11700	2181	4956	10434
0.5570	6330	15245	3675	12720	5055	15485	3219	6525	13790
0.7760	7895	19335	4910	15870	6280	20195	3769	7788	16632
1.0500	9410	23655	6185	19770	8285	25400	5025	9752	20620
1.5120	12770	-	8335	25810	10630	-	6379	12820	-
2.1060	-	-	10580	-	-	-	8238	-	-

Table 10

Cylindrical milling cutter. Clearance angle \(10^\circ\)

No.	Machined material	\(\lambda\)	\(K\)
1	Steel 1010-P	116	-0.37
2	Steel 1020-P	118	-0.37
3	Steel 1020-N	123	-0.365
4	Steel 1035-P	131	-0.3
5	Steel 1035-N	136	-0.29
6	Steel 1040-P	131	-0.29
7	Steel 1040-O	138	-0.31
8	Steel 1040-N	138	-0.3
9	Steel 1040-U	151	-0.3
10	Steel 1050-O	154	-0.26
11	Steel 5120-P	128	-0.34
12	Steel 5120-N	128	-0.36
13	Steel 5140-P	138	-0.295
14	Steel 5140-O	142	-0.285
15	Steel 5140-N	146	-0.26
16	Steel 5145-P	162	-0.28
17	Steel 3120-N	139	-0.34
18	Steel 3140-P	144	-0.28
19	Steel 3340-O	176	-0.26
20	Steel 3340-U	188	-0.25
21	Cast iron No. 1	46.5	-0.38
22	Cast iron No. 2, pearlitic	95	-0.33
23	Cast iron No. 3, alloyed	79	-0.33
24	Cast iron No. 4, alloyed	62	-0.35
25	Cast iron No. 5, malleable	81	-0.33
26	Bronze	42	-0.4

Fig. 38.

Fig. 39.

Fig. 40.

Fig. 41.

Fig. 42.

Fig. 43.

Fig. 44.

Fig. 45.

![Figure 46]

Fig. 46.

Annotations in the figure:

• \(A_w\)
• \(S_m\)
• Steel 1040-U
  \(D = 58.0;\ Z = 8\)
  \(\beta = 60^\circ;\ \gamma = 10^\circ\)
  ○ \(t = 1.04\ \text{mm}\)
  × \(t = 3.85\ \text{mm}\)

![Figure 47]

Fig. 47.

Annotations in the figure:

• \(A_w\)
• \(S_m\)
• Steel 1050-O
  \(D = 59.7;\ Z = 8\)
  \(\beta = 60^\circ;\ \gamma = 10^\circ\)
  ○—\(t = 1.0\ \text{mm}\)
  ×—\(t = 3.0\ \text{mm}\)

![Figure 48]

Fig. 48.

Annotations in the figure:

• \(A_w\)
• \(S_m\)
• Steel 3120-[[unclear: final letter]]
  \(D = 58.5;\ Z = 8\)
  \(\beta = 60^\circ;\ \gamma = 10^\circ\)
  ○ \(t = 1.05\ \text{mm}\)
  × \(t = 4.25\ \text{mm}\)

![Figure 49]

Fig. 49.

Annotations in the figure:

• \(A_w\)
• \(S_m\)
• Steel 3140-P
  \(D = 58.4;\ Z = 8\)
  \(\beta = 60^\circ;\ \gamma = 10^\circ\)
  ○ \(t = 1.12\ \text{mm}\)
  × \(t = 4.07\ \text{mm}\)

Fig. 50.

Fig. 51.

Fig. 52.

Fig. 53.

Fig. 54.

Fig. 55.

Fig. 56.

Fig. 57.

Fig. 58.

Fig. 59.

Fig. 60.

Fig. 61.

The satisfactory agreement of the results of tests carried out at different times serves as a good check on all the apparatus and as an indicator of the reliability of the results obtained. The main tests on steel 1040-u were carried out on 27.2.37, and the repeated tests on 25.3.37. The main tests on steel 5140-n were carried out on 25.2.37, and the repeated tests on 21.3.38.

Fig. 62.

As a result of processing the experimental data in the form of the graphs in Figs. 38-63, we obtained the values of the quantities \(a\) and \(k\) for the operation of a cylindrical milling cutter, presented in Table 10.

Fig. 63.

The values of \(a\) and \(k\) show good agreement with the results of work previously carried out by us, and if their graphical comparison were performed, we would obtain a superposition of one set of results on the other.

The obtained values of \(a\) and \(k\) make expressions 1, 2, and 3 quite definite.

In addition, by transforming expression 3 one can obtain the following relationship:

\[ N_e = C_N \cdot n \cdot b \cdot S_z^{q_N} \cdot t^{y_N} \cdot Z D^{x_N}\ \text{horsepower}. \tag{12} \]

From this, the expression for the mean circumferential force on the milling cutter can also be obtained:

\[ P = \frac{4500\,N_e}{V} = \frac{4500\,N_e \cdot 1000}{\pi D n} \]

\[ P = C_p \cdot b \cdot S_z^{q_p} \cdot t^{y_p} \cdot Z D^{x_p} \tag{13} \]

In these expressions

\[ C_N=\frac{2^{k+1}}{k+2}\cdot \frac{\lambda}{1000\cdot 4500}; \]

\[ C_p=\frac{2^{k+1}}{k+2}\cdot \frac{\lambda}{k}; \]

\[ q_N=q_p=k+1; \]

\[ y_N=y_p=\frac{k+2}{2}; \]

\[ x_N=-\frac{k}{2}; \]

\[ x_p=-\frac{k}{2}-1 \]

The values of these quantities are given in Table 11.

Table 11

No.	Material being machined	\(C_N\)	\(q_N\)	\(y_N\)	\(x_N\)	\(C_p\)	\(X_p\)
1	Steel 1010—P	0,0000245	0,63	0,815	0,185	35	−0,815
2	Steel 1020—P	0,0000249	0,63	0,815	0,185	35,7	−0,815
3	Steel 1020—N	0,000026	0,635	0,817	0,182	37,2	−0,817
4	Steel 1035—P	0,0000261	0,7	0,85	0,15	37,4	−0,85
5	Steel 1035—N	0,000029	0,71	0,855	0,145	41,4	−0,855
6	Steel 1040—P	0,0000278	0,71	0,855	0,145	39,8	−0,855
7	Steel 1040—O	0,0000293	0,69	0,845	0,155	42	−0,845
8	Steel 1040—N	0,0000293	0,7	0,85	0,15	42	−0,85
9	Steel 1040—U	0,0000321	0,7	0,85	0,15	46	−0,85
10	Steel 1050—O	0,0000327	0,74	0,87	0,13	46,8	−0,87
11	Steel 5120—P	0,0000272	0,66	0,83	0,17	39	−0,83
12	Steel 5120—N	0,0000272	0,64	0,82	0,18	39	−0,82
13	Steel 5140—P	0,0000293	0,705	0,852	0,147	42	−0,852
14	Steel 5140—O	0,0000302	0,715	0,857	0,142	43,2	−0,857
15	Steel 5140—N	0,000031	0,74	0,87	0,13	44,4	−0,87
16	Steel 5145—P	0,0000344	0,72	0,86	0,14	49,3	−0,86
17	Steel 3120—N	0,0000295	0,66	0,83	0,17	42,3	−0,83
18	Steel 3140—P	0,0000306	0,72	0,86	0,14	43,8	−0,86
19	Steel 3340—O	0,0000374	0,74	0,87	0,13	53,5	−0,87
20	Steel 3340—U	0,0000401	0,75	0,875	0,125	57,5	−0,875
21	Cast iron No. 1	0,0000099	0,62	0,81	0,19	14,1	−0,81
22	Cast iron No. 2, pearlitic	0,0000202	0,67	0,835	0,165	28,9	−0,835
23	Cast iron No. 3, alloyed	0,0000168	0,67	0,835	0,165	24,0	−0,835
24	Cast iron No. 4, alloyed	0,0000132	0,65	0,825	0,175	18,9	−0,825
25	Cast iron No. 5, malleable	0,0000172	0,67	0,835	0,165	24,6	−0,835
26	Bronze	0,0000089	0,6	0,8	0,2	12,8	−0,8

Since materials with a wide range of hardnesses figured in our experiments, we attempted to put the constants \(\lambda\) and \(k\) in relation to Brinell hardness. In Fig. 64, in ordinary coordinates, the dependence \(\lambda=f(H_B)\) is presented for steels. This same dependence is given in logarithmic coordinates in Fig. 65. The arrangement of the points relative to the mean straight line drawn is such that the greatest deviation of the points from the line does not exceed 10%. In Fig. 66 the dependence for steels—\(K=f(H_B)\)—is given in logarithmic coordinates.

The same graphs for cast irons are given in Figs. 67 and 68. Here the points with tails are taken from our previous works. One may note a good-

Fig. 64.

Fig. 65.

Fig. 66.

Fig. 67.

...a better agreement of them with the data of the present study. In addition, it may be concluded that, judging from graphs 67 and 68, cast irons give a more accurate dependence of the constants \(\lambda\) and \(k\) on \(H_B\). Malleable cast iron has not been placed on these graphs by us, since, possessing the properties of toughness, it has a considerably greater resistance to cutting than ordinary cast iron having the same hardness. Therefore it cannot be included in the general dependence.

Fig. 68.

On the basis of Figs. 64–68, the following analytical dependences may be given.

For steels:

\[ \lambda = 13.5\,H_B^{0.455} \tag{14} \]

\[ \lambda = 74.6 + 0.37\,H_B \tag{15} \]

\[ K = -\frac{2.5}{H_B^{0.403}} \tag{16} \]

\[ C_N = 0.00000287\,H_B^{0.455} \tag{17} \]

\[ C_p = 4.11\,H_B^{0.455} \tag{18} \]

For cast irons.

\[ \lambda = 0.42\,H_B \tag{19} \]

\[ K = -\frac{1.34}{H_B^{0.263}} \tag{20} \]

\[ C_N = 0.0000000895\,H_B \tag{21} \]

\[ C_p = 0.128\,H_B \tag{22} \]

Using these dependences, knowing the hardness of the material being machined, one can calculate its dynamic constants for the milling process. It should be borne in mind that such a calculation for steels may give deviations of up to 10% from the actual values. For cast irons these dependences may be considered practically sufficiently accurate.

In conclusion to the section on the cylindrical milling cutter, let us recall that, using the graphs of specific work \(A_w\) (Figs. 38–63), one can simply calculate the expenditure of power if, for a given cutting regime, the average chip thickness is determined from the expression:

\[ S_m = S_z \sqrt{\frac{t}{D}}, \]

for the obtained value of \(S_m\), from the graph for the corresponding material being machined find the corresponding value of \(A_w\) and calculate the power by the expression:

\[ N_e=\frac{A_w\cdot b\cdot t\cdot S}{1000\cdot 4500}\ \text{hp}. \]

This method of calculation is simpler, since it does not require raising variables to fractional powers; however, it makes it possible to determine the power for a specified regime, but not the regime utilizing a given power.

Tests with a Face Milling Cutter

Here, just as in work with a cylindrical milling cutter, we first of all turned to determining the dependence of the torque on the cutting speed, with all other conditions constant \((b,t,S_z,D=\mathrm{const}.)\). Fig. 69, presenting the results of this series of tests, shows that cutting speeds in the range from 6.8 to 30 m/min have no influence on the cutting resistance. Therefore, we conducted all subsequent tests with a face milling cutter at the lowest cutting speed, which corresponded to a cutter speed per minute

Fig. 69.

\(n=34\). Tables 12–18 given below contain the values of the torques obtained when working with a face milling cutter on various materials being machined. In all cases the work was carried out with a width equal to the cutter diameter. The data in the torque tables were recalculated into specific work, and graphs were plotted (Figs. 70–95).

When milling steels 1010—p, 1020—n, 5120—p, we did not perform a cutting depth of 6 mm, since, owing to the very considerable toughness of these steels, at this cutting depth chips became wedged between the teeth, which made the tests impossible and threatened breakage of the cutter.

A number of test series with completely identical conditions, here as well as when working with the cylindrical milling cutter, were repeated by us at different times for the purpose of checking the results obtained.

![Figure 70]

Visible annotations: \(R_{\mathrm{и}}\); face mill; steel 1010-P; \(D=64\); \(z=10\); \(t=1.0\ \mathrm{mm}\); \(t=2.0\ \mathrm{mm}\); \(t=4.05\ \mathrm{mm}\); \(S_m\).

Fig. 70.

![Figure 71]

Visible annotations: \(R_{\mathrm{и}}\); face mill; \(D=64\), \(z=10\); steel 1020-P; \(t=1.0\ \mathrm{mm}\); \(t=2.1\ \mathrm{mm}\); \(t=3.05\ \mathrm{mm}\); \(t=4.4\ \mathrm{mm}\); \(t=6.2\ \mathrm{mm}\); \(S_m\).

Fig. 71.

![Figure 72]

Visible annotations: \(R_{\mathrm{и}}\); face mill; \(D=64\); \(z=10\); steel 1020-N; \(t=1.0\ \mathrm{mm}\); \(t=2.0\ \mathrm{mm}\); \(t=3.0\ \mathrm{mm}\); \(t=4.0\ \mathrm{mm}\); \(S_m\).

Fig. 72.

![Figure 75]

Visible annotations: \(R_{\mathrm{и}}\); face mill; \(D=64\), \(z=10\); steel 1050-P; \(t=1.0\ \mathrm{mm}\); \(t=2.0\ \mathrm{mm}\); \(t=3.0\ \mathrm{mm}\); \(t=4.0\ \mathrm{mm}\); \(t=6.0\ \mathrm{mm}\); \(S_m\).

Fig. 75.

Fig. 74.

Fig. 73.

Fig. 76.

Fig. 77.

Fig. 78.

Fig. 79.

Fig. 80.

Fig. 81.

Fig. 82.

Fig. 83.

Visible inscriptions in the figure:

• \(A_w\)
• Face mill
  \(D = 64\), \(z = 10\)
  Steel [[unclear: grade appears as “35Х10-У”]]
• \(t = 0.95\ \text{mm}\)
• \(t = 2.0\ \text{mm}\)
• \(t = 4.0\ \text{mm}\)
• \(t = 6.0\ \text{mm}\)
• \(S_m\)

Fig. 84.

Visible inscriptions in the figure:

• \(A_w\)
• Face mill
  \(D = 64\), \(z = 10\)
  Steel \(5Г20\text{-}П\)
• \(t = 0.5\ \text{mm}\)
• \(t = 1.0\ \text{mm}\)
• \(t = 2.0\ \text{mm}\)
• \(t = 4.0\ \text{mm}\)
• \(S_m\)

Fig. 85.

Fig. 88.

Fig. 86.

Fig. 87.

Fig. 89.

Fig. 90.

Fig. 91.

Fig. 92.

Fig. 93.

Fig. 94.

Fig. 95.

Table 12

Torque moments when milling in the operation of a cutter. Cutter diameter \(D=64\); number of teeth \(Z=10\); milling width \(b=D=64\) mm.

Material being machined, \(t\), mm	Steel 1010-p, \(S_z=1.0\)	Steel 1010-p, \(S_z=1.0\)	Steel 1010-p, \(S_z=2.0\)	Steel 1010-p, \(S_z=2.0\)	Steel 1010-p, \(S_z=4.05\)	Steel 1020-u, \(S_z=1.0\)	Steel 1020-u, \(S_z=2.0\)	Steel 1020-u, \(S_z=2.10\)	Steel 1020-u, \(S_z=3.05\)	Steel 1020-u, \(S_z=4.40\)	Steel 1020-u, \(S_z=4.40\)	Steel 1020-u, \(S_z=6.20\)	Steel 1020-v, \(S_z=1.0\)	Steel 1020-v, \(S_z=2.0\)	Steel 1020-v, \(S_z=2.0\)	Steel 1020-v, \(S_z=2.1\)	Steel 1020-v, \(S_z=3.0\)	Steel 1020-v, \(S_z=4.0\)	Steel 1020-v, \(S_z=4.1\)
0.0073	—	—	—	—	2348	—	—	—	—	—	—	—	—	—	—	—	1400	2200	—
0.0100	—	—	—	—	2857	—	—	—	—	—	—	—	—	—	—	—	1850	2900	—
0.0134	—	—	—	1788	3485	—	—	—	—	—	—	—	—	—	—	—	2650	3695	—
0.0200	—	—	—	2408	4865	—	—	—	—	—	—	—	—	—	—	—	3445	5195	—
0.0272	—	—	—	3096	6143	—	—	—	—	—	—	—	—	—	—	—	4795	6590	—
0.0370	—	—	—	3920	6942	—	—	—	—	—	—	—	—	—	—	—	5415	8090	—
0.0585	2248	2824	5545	5694	10635	2548	4805	5369	7715	10709	10784	14331	2400	4650	5100	4874	8190	11085	10824
0.0805	2496	3595	7340	7450	14147	2896	6144	6759	9878	13982	13674	17757	3350	6750	7050	6942	10335	14475	13634
0.1100	4445	4715	8938	9208	17360	3695	7990	8390	12279	15965	16538	22075	4000	8600	8600	8969	13030	18605	16364
0.1590	5893	6123	11880	11705	22371	5194	10285	11456	16105	20448	—	31192	5800	11280	11482	11540	17015	—	22750
0.2210	7640	7790	14770	14370	30167	6543	12678	15107	20200	27136	—	—	7250	13225	—	14620	21335	—	32618
0.3000	9488	9313	17710	17350	—	8140	15856	17360	25664	—	—	—	9400	17660	—	17608	28550	—	—
0.4460	12530	12193	23260	22757	—	11488	20738	23212	—	—	—	—	12230	22520	—	22668	—	—	—
0.6210	15420	15442	28800	28648	—	14025	24805	—	—	—	—	—	15270	29085	—	—	—	—	—
0.8400	18800	19135	—	—	—	17760	—	36733	—	—	—	—	19000	—	—	—	—	—	—
1.2100	—	25620	—	—	—	24395	—	—	—	—	—	—	25325	—	—	—	—	—	—
1.6850	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 13

Torques, in kilogram-millimeters, during operation of a face mill. Cutter diameter \(D = 64\); number of teeth \(Z = 10\); milling width \(b = D = 64\) mm

Workpiece material	\(S_z\), feed per tooth	Steel 1035-p, \(t=0.97\) mm	Steel 1035-p, \(t=2.05\) mm	Steel 1035-p, \(t=4.05\) mm	Steel 1035-p, \(t=6.0\) mm	Steel 1035-n, \(t=0.98\) mm	Steel 1035-n, \(t=1.0\) mm	Steel 1035-n, \(t=1.95\) mm	Steel 1035-n, \(t=2.0\) mm	Steel 1035-n, \(t=4.0\) mm	Steel 1035-n, \(t=6.0\) mm	Steel 1050-o, \(t=1.0\) mm	Steel 1050-o, \(t=2.0\) mm	Steel 1050-o, \(t=3.0\) mm	Steel 1050-o, \(t=4.0\) mm	Steel 1050-o, \(t=6.0\) mm
	0.0073	—	—	1050	1799	—	—	—	—	—	1948	—	—	—	—	—
	0.0100	—	—	1609	2298	—	—	—	—	—	3016	—	—	—	—	—
	0.0134	—	1200	2348	3246	—	—	—	—	—	3645	—	—	—	—	—
	0.0200	—	1624	3215	4345	—	—	1998	—	—	5221	—	—	—	—	—
	0.0272	—	2118	4185	5794	—	—	—	—	—	6942	—	—	—	—	—
	0.0370	—	2896	5275	7815	—	—	3122	—	—	8440	—	—	—	—	—
	0.0585	2048	4495	7940	11685	2498	2473	4620	3197	8490	12349	2000	3500	5350	7700	11250
	0.0805	2798	5895	10665	15442	3445	3096	5744	5745	11182	15845	2650	4850	7700	10200	15000
	0.1100	3596	7540	13863	20090	3794	4035	7416	7490	13875	20586	3430	6400	9200	13275	19350
	0.1590	3895	10225	19000	30093	5094	5420	10005	9987	19172	29475	5000	9150	13700	17860	28400
	0.2210	6193	13225	24935	—	6343	6693	12528	12772	24786	—	6400	11800	17400	23060	—
	0.3000	7590	16290	31663	—	8590	8565	15915	15370	30925	—	9200	15000	22330	29000	—
	0.4460	10535	22915	—	—	11075	10982	21580	20688	—	—	11100	20550	30200	—	—
	0.6210	13325	29205	—	—	14073	13973	27770	26500	—	—	14275	26300	—	—	—
	0.8400	16165	—	—	—	17260	17335	33580	—	—	—	17000	31100	—	—	—
	1.2100	22620	—	—	—	23705	23605	—	—	—	—	21125	—	—	—	—
	1.6850	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 14

Twisting moments in kilogram-millimeters during operation of a face mill. Cutter diameter \(D=64\); number of teeth \(Z=10\); milling width \(b=D=64\) mm

Workpiece material	\(t\), mm	Feed per 1 tooth \(S_z\)	Steel 1040-п 1.0	Steel 1040-п 2.05	Steel 1040-п 3.10	Steel 1040-п 4.05	Steel 1040-п 6.0	Steel 1040-о 1.04	Steel 1040-о 2.10	Steel 1040-о 3.10	Steel 1040-о 4.05	Steel 1040-о 6.0	Steel 1040-н 1.0	Steel 1040-н 2.0	Steel 1040-н 3.1	Steel 1040-н 4.05	Steel 1040-н 6.0	Steel 1040-у 1.0	Steel 1040-у 2.06	Steel 1040-у 4.1	Steel 1040-у 6.0
		0.0073	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0100	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0134	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0200	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0272	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0370	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
		0.0585	2198	4071	5894	7790	10435	2400	4145	6213	8290	11989	1998	3827	5844	7566	11082	2300	4300	8165	4615
		0.0805	2897	5254	7591	10136	14004	3173	5568	7990	10843	15667	2747	5444	7467	10011	14490	3150	5850	10958	6293
		0.1100	3956	6667	10136	12961	18313	4145	7117	10834	14082	21133	3696	7072	10061	13340	19148	3990	7467	13883	8123
		0.1590	5594	9508	13674	18055	25548	5793	9987	15218	18899	28720	5369	10038	14475	18432	28162	5410	10186	20140	11033
		0.2210	7272	12777	17658	23779	—	7092	12428	20041	25366	—	7052	12677	18552	24738	—	7350	12400	25647	12059
		0.3000	9163	16155	21993	31603	—	8989	16522	25042	32531	—	8690	15995	25120	32641	—	8250	16298	31410	16120
		0.4460	12353	21860	—	—	—	12278	23384	—	—	—	11680	22104	34652	—	—	12100	22915	—	19445
		0.6210	15586	28900	—	—	—	15243	30259	—	—	—	14590	29708	—	—	—	15200	29409	—	27577
		0.8400	18949	36473	—	—	—	17808	—	—	—	—	18924	—	—	—	—	18750	—	—	—
		1.2100	25689	—	—	—	—	24688	—	—	—	—	24905	—	—	—	—	25866	—	—	—
		1.6850	32723	—	—	—	—	33052	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 15.

Torque moments, in kilogram-millimeters, during operation of a face milling cutter.
Cutter diameter \(D = 64\); number of teeth \(Z = 10\); milling width \(b = D = 64\ \mathrm{mm}\)

Workpiece material	\(t\), mm	\(S_z\), feed per tooth	Steel 3120-н, 0.95	Steel 3120-н, 2.00	Steel 3120-н, 4.00	Steel 3120-н, 6.00	Steel 3140-п, 0.97	Steel 3140-п, 0.97	Steel 3140-п, 1.00	Steel 3140-п, 2.00	Steel 3140-п, 4.00	Steel 3140-п, 6.00	Steel 3340-о, 0.95	Steel 3340-о, 2.00	Steel 3340-о, 4.00	Steel 3340-о, 6.00	Steel 3340-у, 0.95	Steel 3340-у, 2.00	Steel 3340-у, 4.00	Steel 3340-у, 6.00
		0.0073	—	—	1800	2448	—	—	—	—	—	1699	—	—	1500	2800	—	—	1600	2200
		0.0100	—	—	2498	3396	—	—	—	—	—	2098	—	—	2250	3245	—	—	2150	3395
		0.0134	—	—	3121	4295	—	—	—	—	—	3047	—	—	2700	4095	—	—	2850	4245
		0.0200	—	2148	4045	6193	—	—	—	—	2748	4145	—	—	3895	5695	—	1750	3945	5695
		0.0272	—	2622	5234	7193	—	—	—	—	3795	5344	—	—	5645	7140	—	2550	5295	6890
		0.0370	—	3470	6713	9638	—	—	—	—	4844	6992	—	—	5895	9190	—	3145	6690	9290
		0.0585	2527	4870	9040	13125	—	1950	2098	4195	6792	9938	2150	4795	8790	12780	2500	4695	9685	13925
		0.0805	3396	6642	11830	16668	2550	—	2897	5444	9588	13475	2995	6195	11730	16415	2850	6295	12580	17510
		0.1100	4445	8090	14920	22645	—	3595	3446	6992	12080	17162	3790	7740	14575	21535	3995	8090	16015	23900
		0.1590	5869	10635	20315	—	4395	—	4595	8688	15717	24000	5295	10335	20195	29620	5395	10735	23165	34450
		0.2210	7369	13674	26280	—	—	5895	6143	11732	20787	—	6990	13330	29085	—	7340	14375	—	—
		0.3000	9940	16566	—	—	7290	—	7640	14622	26845	—	8490	16215	—	—	9190	18105	—	—
		0.4460	11900	22075	—	—	—	10235	10235	19497	—	—	11580	22820	—	—	12230	25080	—	—
		0.6210	14720	28220	—	—	13030	—	13225	25473	—	—	15170	29085	—	—	15520	32550	—	—
		0.8400	18180	—	—	—	—	16115	16265	32905	—	—	18405	—	—	—	19945	—	—	—
		1.2100	24788	—	—	—	21680	—	22620	—	—	—	24540	—	—	—	27135	—	—	—
		1.6850	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 16

Torsional moments, in kilogram-millimeters, during operation of a face milling cutter.
Cutter diameter \(D = 64\); number of teeth \(Z = 10\); milling width \(b = D = 64\) mm.

Workpiece material	\(S_z\), feed per 1 tooth	Steel, 5120-п, 0.50	Steel, 5120-п, 1.00	Steel, 5120-п, 1.03	Steel, 5120-п, 2.00	Steel, 5120-п, 2.00	Steel, 5120-п, 4.00	Steel, 5120-п, 4.10	Steel, 5120-н, 1.0	Steel, 5120-н, 1.0	Steel, 5120-н, 1.0	Steel, 5120-н, 2.0	Steel, 5120-н, 2.0	Steel, 5120-н, 3.9	Steel, 5120-н, 6.0
	0.0073	—	—	—	—	—	1168	—	—	—	—	—	—	1749	2398
	0.0100	—	—	—	—	—	1497	—	—	—	—	—	—	1928	2997
	0.0134	—	—	—	—	—	2295	2296	—	—	—	—	—	2743	3820
	0.0200	—	—	—	—	—	2894	3245	—	—	—	—	—	3520	5370
	0.0272	—	—	—	—	—	4043	4394	—	—	—	—	—	4795	7042
	0.0370	—	—	—	—	—	5342	5843	—	—	—	—	—	5994	8640
	0.0585	1249	—	1847	—	4495	7940	7890	—	2348	2678	4645	4495	8190	13075
	0.0805	1448	—	2846	5445	5893	10283	10735	—	3487	3164	5993	5943	10734	16165
	0.1100	1697	3447	3595	7042	7290	13670	13075	3845	4245	3929	7590	6742	13843	21709
	0.1590	2646	4594	4794	9588	9188	18950	18365	5344	5654	5394	10385	10285	19123	—
	0.2210	3195	6043	6393	12180	12500	25875	24295	6742	6952	6942	12678	13474	25421	—
	0.3000	4294	7891	7990	15070	15540	—	—	8290	8889	8750	15870	16530	—	—
	0.4460	6092	10335	10485	20190	20300	—	—	11234	11282	11282	21285	22412	—	—
	0.6210	7840	13172	13875	25175	25865	—	—	14473	14072	14371	27285	28697	—	—
	0.8400	10235	16605	17063	31990	—	—	—	17858	18403	17817	33970	36800	—	—
	1.2100	13075	—	24098	—	—	—	—	23160	23915	24688	—	—	—	—
	1.6850	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 17

Torques during operation of a face milling cutter. Cutter diameter \(D=64\); number of teeth \(Z=10\); face-milling width \(b=D=64\) mm

Machined material	\(S_z\), feed per tooth	Steel 5140-п, \(t=1.00\) mm	Steel 5140-п, \(t=0.98\) mm	Steel 5140-п, \(t=2.00\) mm	Steel 5140-п, \(t=2.00\) mm	Steel 5140-п, \(t=4.10\) mm	Steel 5140-п, \(t=6.10\) mm	Steel 5140-о, \(t=1.0\) mm	Steel 5140-о, \(t=2.0\) mm	Steel 5140-о, \(t=4.0\) mm	Steel 5140-о, \(t=6.0\) mm	Steel 5140-н, \(t=1.0\) mm	Steel 5140-н, \(t=2.05\) mm	Steel 5140-н, \(t=4.1\) mm	Steel 5140-н, \(t=6.1\) mm	Steel 5145-п, \(t=1.0\) mm	Steel 5145-п, \(t=2.0\) mm	Steel 5145-п, \(t=3.0\) mm	Steel 5145-п, \(t=4.0\) mm	Steel 5145-п, \(t=6.1\) mm
	0.0073	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
	0.0100	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
	0.0134	—	—	—	—	—	2788	—	—	—	—	—	—	—	—	—	—	—	—	—
	0.0200	—	—	—	—	—	4145	—	—	—	—	—	—	—	3663	—	—	—	—	—
	0.0272	—	—	—	—	—	5634	—	—	—	—	—	—	—	5194	—	—	—	—	—
	0.0370	—	—	—	—	—	8889	—	—	—	—	—	—	—	6892	—	—	—	—	—
	0.0585	1758	1699	3397	3905	6743	10685	2298	4215	8270	12278	1631	3971	7466	11531	2100	4250	6000	8450	12275
	0.0805	2218	1998	4470	5195	9189	13874	2930	5726	10874	14894	2678	5354	10236	14949	2750	5670	8530	11250	16260
	0.1100	3072	2847	6293	6593	11630	17708	3896	7192	13693	19346	3496	6892	13075	19207	3750	7160	10950	14588	20633
	0.1590	4375	4070	8623	9413	16354	24640	5274	10037	18750	27763	4869	9578	18552	25474	5070	9920	14847	19440	28418
	0.2210	5953	5619	11349	11805	20329	31295	6433	13175	24933	—	6668	12637	24472	—	6700	13115	18795	24775	—
	0.3000	7092	7392	13973	14470	27137	—	8640	15945	30830	—	8520	16045	31652	—	8450	16240	23795	31390	—
	0.4460	10246	9888	20745	20513	—	—	11292	22619	—	—	11471	22312	—	—	11250	21206	32000	—	—
	0.6210	13185	13415	26843	26550	—	—	14172	29085	—	—	14803	30412	—	—	14470	27825	—	—	—
	0.8400	16423	16389	33858	34295	—	—	18080	—	—	—	17956	—	—	—	17187	32050	—	—	—
	1.2100	23508	22365	—	—	—	—	25049	—	—	—	25827	—	—	—	—	—	—	—	—
	1.6850	31120	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 18

Torques in kilogram-millimeters during operation of an end mill. Cutter diameter \(D = 64\); number of teeth \(Z = 100\), cutter width \(b = D = 64\ \text{mm}\).

Machined material	\(t\), mm	Cast iron No. 1, 0.98	Cast iron No. 1, 2.00	Cast iron No. 1, 4.05	Cast iron No. 1, 6.00	Cast iron No. 2, 1.0	Cast iron No. 2, 2.0	Cast iron No. 2, 4.0	Cast iron No. 2, 6.0	Cast iron No. 3, 1.0	Cast iron No. 3, 2.0	Cast iron No. 3, 4.0	Cast iron No. 3, 6.0
\(S_z\) feed per 1 tooth	0.0073	—	—	—	1424	—	—	—	1900	—	—	—	—
\(S_z\) feed per 1 tooth	0.0100	—	—	—	2048	—	—	—	2550	—	—	—	2200
\(S_z\) feed per 1 tooth	0.0134	—	—	—	3071	—	—	—	3495	—	—	—	3595
\(S_z\) feed per 1 tooth	0.0200	—	—	—	3495	—	—	2700	4645	—	—	2300	3745
\(S_z\) feed per 1 tooth	0.0272	—	—	—	3995	—	—	3595	5645	—	—	3545	4845
\(S_z\) feed per 1 tooth	0.0370	—	—	—	4870	—	—	—	6990	—	—	3895	5595
\(S_z\) feed per 1 tooth	0.0585	1299	2323	4345	6518	—	3695	7190	10035	—	2900	5395	8190
\(S_z\) feed per 1 tooth	0.0805	1299	2696	5493	7930	1800	5045	—	11580	1850	3695	7190	10085
\(S_z\) feed per 1 tooth	0.1100	1778	3445	6800	9867	2250	6195	11035	15320	2300	4695	8690	12880
\(S_z\) feed per 1 tooth	0.1590	2297	4485	8759	12527	2900	7940	—	20240	3145	6495	11930	16415
\(S_z\) feed per 1 tooth	0.2210	3066	5669	11135	16414	3845	10235	18005	26600	—	8190	14920	21930
\(S_z\) feed per 1 tooth	0.3000	3595	7190	14170	20040	5145	12480	23705	—	5195	10335	18355	26650
\(S_z\) feed per 1 tooth	0.4460	4715	8750	19206	27038	6540	16315	—	—	7090	12030	25280	—
\(S_z\) feed per 1 tooth	0.6210	5824	10784	24620	—	8990	20840	—	—	9090	16415	33810	—
\(S_z\) feed per 1 tooth	0.8400	7260	13275	30888	—	11035	25865	—	—	11335	20740	—	—
\(S_z\) feed per 1 tooth	0.2100	10435	18730	—	—	—	—	—	—	14625	25770	—	—
\(S_z\) feed per 1 tooth	0.6850	13450	—	—	—	—	—	—	—	—	—	—	—

Torques in kilogram-millimeters during operation of a face mill. Cutter diameter \(D=64\); number of teeth \(Z=10\), milling width \(b=D=64\) mm. Continuation of Table 18

Work material	\(t\), mm	Cast iron No. 4, 1.0	Cast iron No. 4, 2.0	Cast iron No. 4, 4.0	Cast iron No. 4, 4.0	Cast iron No. 4, 6.0	Cast iron No. 5, 1.0	Cast iron No. 5, 2.0	Cast iron No. 5, 4.0	Cast iron No. 5, 6.0	Bronze, 1.0	Bronze, 2.0	Bronze, 4.0	Bronze, 6.0
\(S_z\), feed per 1 tooth	0.0073	—	—	—	—	1700	—	—	—	1648	—	—	—	—
\(S_z\), feed per 1 tooth	0.0100	—	—	—	—	2500	—	—	—	2347	—	—	—	—
\(S_z\), feed per 1 tooth	0.0134	—	—	—	—	2800	—	—	—	3146	—	—	—	—
\(S_z\), feed per 1 tooth	0.0200	—	—	2450	—	3200	—	—	—	3745	—	—	—	—
\(S_z\), feed per 1 tooth	0.0272	—	—	3150	—	4450	—	—	2797	4694	—	—	—	—
\(S_z\), feed per 1 tooth	0.0370	—	—	3850	—	5450	—	—	3395	5943	—	—	—	—
\(S_z\), feed per 1 tooth	0.0585	1249	2648	5050	—	6650	1648	3396	5493	7840	950	1650	2995	3645
\(S_z\), feed per 1 tooth	0.0805	1698	3296	6000	—	8800	2148	4295	7242	10236	1150	1900	3595	4445
\(S_z\), feed per 1 tooth	0.1100	2097	3845	7450	—	10235	2647	4644	9138	12728	1250	2150	4145	5295
\(S_z\), feed per 1 tooth	0.1590	2946	5294	9400	—	13225	3495	6443	12280	16215	1700	2600	5345	6445
\(S_z\), feed per 1 tooth	0.2210	3645	6592	—	12230	17160	4644	7690	14920	21432	2050	3595	6245	8290
\(S_z\), feed per 1 tooth	0.3000	4145	7390	—	14770	21480	5693	9937	18900	27285	2550	4195	6990	9935
\(S_z\), feed per 1 tooth	0.4460	5993	11233	—	19845	—	7390	14074	26650	—	3145	5545	8990	12930
\(S_z\), feed per 1 tooth	0.6210	7390	13177	—	25865	—	9837	17908	—	—	3850	6640	11085	17110
\(S_z\), feed per 1 tooth	0.8400	9188	16664	—	31410	—	11732	22470	—	—	4850	6540	13475	—
\(S_z\), feed per 1 tooth	1.2100	11582	20540	—	—	—	15773	30543	—	—	6250	10085	—	—
\(S_z\), feed per 1 tooth	1.6850	14870	—	—	—	—	19745	—	—	—	8250	—	—	—

From the tables and graphs, a satisfactory agreement of the results is evident.

On the basis of graphs 70–95, we obtained the values of \(\lambda\), \(K\), and \(l\) for the operation of the face milling cutter (see Table 19).

Table 19

Face milling cutter

No.	Work material	\(\lambda\)	\(K\)	\(l\)
1	Steel 1010-P	200	-0.27	-0.02
2	Steel 1020-P	186	-0.24	-0.04
3	Steel 1020-N	195	-0.26	-0.02
4	Steel 1035-P	187	-0.21	-0.05
5	Steel 1035-N	190	-0.23	-0.09
6	Steel 1040-P	203	-0.2	-0.09
7	Steel 1040-O	198	-0.215	-0.05
8	Steel 1040-N	197	-0.2	-0.05
9	Steel 1040-U	213	-0.205	-0.09
10	Steel 1050-O	190	-0.19	-0.04
11	Steel 5120-P	188	-0.23	-0.09
12	Steel 5120-N	193	-0.25	-0.1
13	Steel 5140-P	190	-0.16	-0.03
14	Steel 5140-O	196	-0.2	-0.06
15	Steel 5140-N	200	-0.175	-0.04
16	Steel 5145-P	193	-0.2	-0.04
17	Steel 3120-N	203	-0.245	-0.1
18	Steel 3140-P	187	-0.2	-0.1
19	Steel 3340-O	210	-0.205	-0.06
20	Steel 3340-U	225	-0.19	-0.03
21	Cast iron No. 1	72	-0.32	-0.07
22	Cast iron No. 2 (pearlitic)	142	-0.26	-0.11
23	Cast iron No. 3 (alloyed)	110	-0.28	-0.1
24	Cast iron No. 4 (alloyed)	90	-0.31	-0.11
25	Cast iron No. 5 (malleable)	123	-0.26	-0.12
26	Bronze	67	-0.34	-0.21

In Table 19, attention is drawn first of all to the fact that the value of \(\lambda\) does not increase in parallel with the increase in hardness of the steel being machined, and it can often be seen that softer steels are characterized by a higher value of \(\lambda\). This was also obtained by us in previous work with face milling cutters\(^{3}\). With regard to cast irons, an increase in \(\lambda\) with an increase in the hardness of the cast iron may be noted. The absence of a regular pattern in the variation of \(\lambda\) for steels is apparently connected with the plastic properties of steels and with the difficult separation of the chip on the tooth of the face milling cutter, where we have a process of non-free cutting with a plan angle of \(90^\circ\).

According to Gipromash data, the constant in the power equation increases with the hardness of the material being machined, which, as we see, is not confirmed by experiment. According to Larin’s work\(^{5}\), just as in our case, the constant does not always increase with an increase in the hardness of the material being machined.

It should further be noted that the value of \(l\) is not equal to zero in all cases. As a result, the depth of cut in the equations of force and power enters with a power less than unity. This is quite natural, since during operation of the face milling cutter we have participation in the cutting process of two cutting edges: the main one, located along the cylinder, and the auxiliary one, located along the face of the cutter. This circumstance is not taken into account by the conclusions of Gipromash, Prof. Reznikov\(^{6}\), and Eng. Ryvkin\(^{7}\).

In Larin’s work, for the depth of milling an exponent greater than unity was obtained, which we consider to be the result of unaccounted harmful resistances, recorded by the dynamometer and increasing with an increase in the depth of milling.

It is of some interest to compare the results obtained from different data. For this purpose we shall use Table 7 of cutting conditions on Gipromash milling machines, where, for the case of milling steel with \(\sigma_{B}=60\text{–}70\ \mathrm{kg/mm^2}\) by a face mill of diameter \(150\ \mathrm{mm}\) with 10 teeth, at a milling width of \(120\ \mathrm{mm}\), depth \(10\ \mathrm{mm}\), feed per tooth \(0.15\ \mathrm{mm}\), and cutter speed 35 rpm, the powers obtained are: according to Gipromash, \(4.3\ \mathrm{hp}\); according to Ryvkin, \(4.2\ \mathrm{hp}\); and according to Reznikov, \(3.9\ \mathrm{hp}\). According to the data obtained by us, for steel 1040-p with \(\sigma_{B}=64.4\ \mathrm{kg/mm^2}\), calculation by formula 9 gives \(N_e=3.48\ \mathrm{hp}\).

A similar comparison with the data of Prof. Reznikov was carried out by us in our previous work\(^{3}\), and in all cases we obtained in experiments smaller torques and powers than those obtained by calculation from Prof. Reznikov’s data.

Power equation 8 can be transformed into the following relation:

\[ N_e = C_N \cdot n \cdot t^{x_p}\cdot S_z^{y_p}\cdot Z \cdot D \tag{23} \]

and since \(N_e=\dfrac{Pv}{4500}\), the equation of the circumferential force on the cutter will be

\[ P = C_p \cdot t^{x_p}\cdot S_z^{y_p}\cdot Z \tag{24} \]

From the last expression one can obtain the power equation

\[ N_e=\frac{C_p\cdot t^{x_p}\cdot S_z^{y_p}\cdot Z\cdot V}{4500} \tag{25} \]

In these relations:

\[ C_N=\frac{2^{-\frac{K}{2}}\cdot \lambda}{1000\cdot 4500}; \]

\[ C_p=\frac{2^{-\frac{K}{2}}\cdot \lambda}{\pi}; \]

\[ x_p=l+1; \]

\[ y_p=K+1. \]

The values of all these quantities are given by us in Table 20.

In Fig. 96 we have plotted the dependence of the quantities \(\lambda\) and \(K\) on Brinell hardness in face milling of cast irons. Here the points with tails are plotted according to the results of our previous work. On the basis of this graph we obtain the following relations for cast irons.

\[ \lambda=0.6\,H_{B} \tag{26} \]

\[ K=-\frac{2}{H_{B}^{0.38}} \tag{27} \]

Table 20

Face Milling Cutter

No.	Workpiece material	$C_N$	$C_p$	$X_p$	$y_p$
1	Steel 1010—P	0.0000489	70	0.98	0.73
2	Steel 1020—P	0.000045	64.5	0.96	0.76
3	Steel 1020—H	0.0000475	68	0.98	0.74
4	Steel 1035—P	0.0000447	64	0.95	0.79
5	Steel 1035—H	0.0000458	65.5	0.91	0.77
6	Steel 1040—P	0.0000482	69	0.91	0.8
7	Steel 1040—O	0.0000475	68	0.95	0.785
8	Steel 1040—H	0.000047	67.3	0.95	0.8
9	Steel 1040—U	0.0000508	72.7	0.91	0.795
10	Steel 1050—O	0.0000451	64.6	0.96	0.81
11	Steel 5120—P	0.0000453	64.9	0.91	0.77
12	Steel 5120—H	0.0000471	67.4	0.90	0.75
13	Steel 5140—P	0.0000447	64	0.97	0.84
14	Steel 5140—O	0.0000468	67	0.94	0.8
15	Steel 5140—H	0.0000472	67.5	0.96	0.825
16	Steel 5145—P	0.0000461	66	0.96	0.8
17	Steel 3120—H	0.0000489	70	0.9	0.755
18	Steel 3140—P	0.0000445	63.6	0.9	0.8
19	Steel 3340—O	0.0000502	71.9	0.94	0.795
20	Steel 3340—U	0.0000535	76.5	0.97	0.81
21	Cast iron No. 1	0.0000179	25.6	0.93	0.68
22	Cast iron No. 2	0.0000346	49.6	0.89	0.74
23	Cast iron No. 3	0.000027	38.6	0.9	0.72
24	Cast iron No. 4	0.0000222	31.8	0.89	0.69
25	Cast iron No. 5	0.00003	42.9	0.88	0.74
26	Bronze	0.0000168	24	0.79	0.66

By the same graphical construction, the following dependences were obtained for steels:

$$ \lambda = 107 \cdot H_B^{0.12} \tag{28} $$

$$ K = -\frac{1.31}{H_B^{0.346}} . \tag{29} $$

These dependences for steels have an accuracy within $\pm 10\%$ and may be regarded only as approximate, just as the analogous dependences for the cylindrical milling cutter.

The power consumption in operation of the face milling cutter may be calculated in the same way as for the cylindrical cutter, from the expression

$$ N_e = A_w \cdot \frac{t \cdot D \cdot S}{1000 \cdot 4500} $$

determining $A_w$ from the corresponding graphs (Figs. 70—95). When $b < D$, the power obtained must be multiplied by the coefficient of equations 9 and 10.

In conclusion, we carried out a series of experiments with a face milling cutter having the same dimensions and geometry as the one described above, but instead of a rounding at the transition from the face blade to the cylindrical blade, this cutter had an angular blade inclined at an angle of 45° to the plane of the face. The angular blade had such dimensions that, at milling depths from 1 to 6 mm used in the experiments, the chip did not engage the cylindrical blade and, over its entire width, was accommodated on the angular blade.

Fig. 96.

Fig. 97.

Fig. 98.

Fig. 99.

These experiments were set up by us in order to determine the influence of the angular cutting edge on power consumption, since milling cutters, and especially milling heads with an angular cutting edge, are widely used in production.

From the literature it is impossible to obtain a clear idea of the influence of the angular cutting edge on the forces and power in milling. According to Ryvkin’s conclusions\(^{7}\), on a tooth with an angular cutting edge the force should be greater than on a tooth without one, but Larin’s experiments gave the opposite results.

According to Schlesinger’s experiments\(^{8}\) with lathe tools, the vertical component of the cutting pressure decreases with an increase in the entering angle; however, this decrease is noticeable only up to an angle of \(45^\circ\). With a further increase of the entering angle, the decrease of the vertical component is extremely insignificant.

Fig. 97 shows a milling cutter with an angular cutting edge. Figs. 98 and 99 present the results of experiments carried out with it. For the experiments we selected steel 1035 and cast iron No. 2 as two metals completely different in their properties.

In Figs. 98 and 99 the straight lines are drawn according to the data of experiments with a cutter without an angular cutting edge for the corresponding metals, and the points are plotted according to the experiments with an angular cutting edge. We see that the points lie well on the straight lines, i.e., that the influence of the angular cutting edge on the force and power is completely absent for the case of the angle \(\varphi = 45^\circ\). The angle of inclination of the chamfer angular cutting edge was taken as \(45^\circ\), as the one most frequently encountered in production.

The obtained absence of any influence of the angular cutting edge on the load of the cutter would seem to contradict logic, since when an angular cutting edge is present the length of the cutting edge is greater than without one, and the cutting pressure would seemingly have to increase. But here it is necessary to take into account the freer separation of the chip; moreover, the rake angle on the angular cutting edge, in the plane normal to its cutting edge, is greater than on the edge of the cylindrical part of the cutter. It is not difficult to calculate its actual value if the tooth of the cutter is identified with a lathe tool having an entering angle of \(45^\circ\) and if, for the calculation, the conclusions of Prof. Semenchenko are used.\(^{9}\)

Let us call \(\gamma_N\) the rake angle in the plane normal to the angular cutting edge.

\(\gamma_1\) — the rake angle in the plane normal to the face of the cutter.

\(\gamma_2\) — the rake angle in the plane perpendicular to the axis of the cutter.

We obtain all these angles if we place the cutter tooth in such a position that the apex of the edge (the point of intersection of the angular and end cutting edges) lies in one horizontal plane with the axis of the cutter.

From direct measurement we have

\[ \gamma_1 = 15^\circ \]

\[ \gamma_2 = 8^\circ \]

Hence \(\gamma_N\) can be found:

\[ \tan \gamma_N = \tan \gamma_1 \cdot \cos \varphi + \tan \gamma_2 \cdot \sin \varphi = 0.289 \]

\[ \gamma_N = 16^\circ . \]

Thus, on the angular cutting edge, for \(\varphi = 45^\circ\), the rake angle exceeds by a factor of two the rake angle on the edge of the cylindrical part of the cutter.

We believe that, since our experiments were carried out both for a sufficiently tough metal (steel 1035) and for a brittle metal (cast iron No. 2), and

for all other metals, we will obtain, perhaps with insignificant deviations, the same result. The values of $A_w$ for the angular cutting edge were calculated from the mean diameter at the given milling depth.

Conclusion.

In all, in the present work we carried out a total of 2,250 tests (counting the tests for reporting the zero readings of the apparatus, but not counting the calibration tests), of which 781 were with a cylindrical milling cutter and 1,479 with a face milling cutter. All tests without exception were taken into account in deriving the corresponding relationships.

All the tests confirmed, for a broader range of materials, our previous conclusions.

On this basis the following may be recommended:

1. For calculating powers and forces during operation of a cylindrical milling cutter, use equations 3, 12, 13 and Tables 10 and 11.

2. For calculating powers and forces during operation of a face milling cutter, use equations 8, 9, 10, 23, 24, 25 and Tables 19 and 20.

3. For an approximate determination of the dynamic characteristics of the material being machined from its hardness, use equations 14–22 and 26–29.

References Cited

1. Rosenberg and Sudnishnikov.—Theory of the operation of a cylindrical milling cutter with a helical tooth. Vestnik metallopromyshlennosti, 1933, No. 4.

2. Rosenberg. Operation of a cylindrical milling cutter. Orga-informatsiya, 1935, No. 12. Izvestiya Tomskogo Industrial’nogo in-ta.

3. Rosenberg—Dynamics of face milling. Izvestiya Tomskogo industrial’nogo in-ta, 1937, vol. 56, issue 1.

Rosenberg.—Dynamics of face milling. Proceedings of the First Leningrad Regional Conference on Metal Cutting. 1937.

Rosenberg.—Operation of a face milling cutter. Orga-informatsiya, 1936, No. 7.

1. Stock—Fräserhandbuch.

2. Larin.—Dynamics of face milling. Orga-informatsiya, 1936, No. 12.

3. Prof. Reznikov.—Theory of metal cutting.

4. Ryvkin.—Problems of cutting in face milling.

5. Schlesinger. V.D.I. 1932, No. 53.

6. Prof. Semenchenko.—Cutting Tool.