Dynamics of Face Milling.

(Theory and Experiment).

The machining of metals with a face (or end) mill occupies a significant place among other methods of machining metals by cutting in mass and large-scale production. Despite this, theoretical and experimental investigations of the operation of face mills are almost entirely lacking.

We can point to only a few attempts to solve the question of the dependence of power consumption on the cutting factors in face milling. Such attempts belong to Prof. Reznikov ¹) and Eng. Rybkin ²); moreover, in the absence of experimental data, the authors have to furnish their equations with coefficients and exponents borrowed from the field of lathe work, which cannot always be regarded as possible.

Of the experimental works in this field, we know only of the work carried out in 1929 at Ohio State University ³). Unfortunately, in carrying out this work on a milling machine, only a wattmeter was used for measuring power, with a preliminary determination of the efficiency of the machine-speed gearbox by means of a Prony brake, and the additional losses in the transmissions during operation were not taken into account at all, nor—what is especially important—the power consumption for the operation of the feed mechanism, which, at large feeds, may amount to as much as 15–20% of the total power expenditure. This circumstance makes it impossible to use the results of the indicated investigation to clarify the dependence of the power expended by a face mill on the operating conditions.

In addition, one can point only to the investigation by Boston and Kraus ⁴), carried out on a pendulum dynamometer with a grooving cutter, the operation of which can rather be identified with the operation of a slotting disk cutter.

The serious objections that the conclusions of this work received from such an authoritative specialist as Prof. Schlesinger compel us to treat these conclusions with caution. In what follows, we examine the question of the power on a face mill and give a brief theory of the operation of a face mill, touching only on the derivation of the power equation; and in the second part we also describe an experimental investigation carried out by us in the metal-cutting laboratory of the Tomsk Industrial Institute, which fully confirmed the relationships proposed by us and made it possible to fill them with coefficients obtained from experiments.

I. Theory of the Operation of a Face Mill.

In the operation of a face mill, unlike a cylindrical one, each tooth cuts the chip with two cutting edges: the main one, located on the cylindrical part of the tool, and an auxiliary one, located...

located on the face. If, in the operation of a cylindrical milling cutter, we had so-called “free cutting,” then here there will be a case of “constrained cutting,” when chip formation takes place along two cutting edges of the tool. The main part of the work falls on the principal cutting edges, but it would be erroneous to think that the additional edges located on the face work only when the cutter is not sufficiently accurately sharpened. Even with absolutely accurate sharpening, the face (additional) edge of each tooth will cut off a chip. According to Fig. 1, let us denote:

\(t\)—the thickness of the removed layer of metal, the depth of milling,
\(b\)—the width of the surface being machined, the width of milling,
\(S_z\)—the feed per tooth,
\(S_e\)—the chip thickness on the tooth at the given instantaneous position of the tooth.

Figure 1 shows the case of operation when \(b = D\), but all these notations are also retained for the case of operation with \(b < D\).

Let us agree to measure the milling depth \(t\) in the direction parallel to the axis of the cutter, and the milling width \(b\) in the direction perpendicular to the direction of feed.

Fig. 1.

Each tooth of the cutter removes a chip of width \(t\) and thickness \(S_e\), and along the “cutting arc” (along the semiperimeter of the section \(t + S_e\)) the chip is cut off in such a way that over the width \(t\) the work of cutting is performed by the main (principal) edge of the tooth, located on the cylindrical surface of the cutter, and over the thickness \(S_e\) by the additional (secondary) edge located on the face surface.

Since the cutting work is performed both over the width and over the thickness, the specific cutting pressure in the operation of a face milling cutter, in the general case, cannot be only a function of the chip thickness, as was the case in the operation of a cylindrical milling cutter, but must be expressed as a function of both the thickness and the width of the chip:

\[ K_s = \lambda \cdot S_e^{k} \cdot t^{l} \tag{1} \]

Here: \(K_s\)—specific cutting pressure in \(\mathrm{kg/mm^2}\),
\(S_e\) and \(t\)—the thickness and width of the chip,
\(\lambda\)—a constant coefficient depending on the kind of material being machined, the cutting angle and the tooth profile, and on the presence or absence of lubrication,
\(k\) and \(l\)—exponents depending mainly on the kind of material being machined; in magnitude they are fractional and negative.

The chip thickness on the tooth at a given moment may be expressed as

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

\(\psi\), the angle determining the given instantaneous position of the tooth, may vary, when \(b = D\), within the limits from \(0^\circ\) to \(180^\circ\).

The chip section on the tooth at a given moment is

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

and there is no point in introducing the influence of the tooth helix, since for normal

of face mills the inclination of the helix to the cutter axis is small and usually does not exceed \(15^\circ\). Therefore, in all subsequent conclusions we shall take the tooth as straight. Moreover, we have proved both theoretically\(^5\) and experimentally\(^6\) that the power consumption of the cutter does not depend on the helix angle.

The total aggregate cross-section of the chip on all \(m\) simultaneously working teeth at the given moment will be:

\[ \sum_{1}^{m} q = t \cdot S_z \sum_{1}^{m} \sin \psi . \]

The circumferential force on a tooth at the given moment is

\[ P = K_s \cdot q = \lambda \cdot S_e^{k+1} \cdot t^{\,l+1}, \]

or

\[ P = \lambda \cdot S_z^{k+1} \cdot t^{\,l+1} \cdot \sin^{k+1}\psi \tag{2} \]

The work of one tooth of the cutter \(A_z\) over one revolution will be

\[ A_z = \lambda \cdot S_z^{k+1} \cdot t^{\,l+1}\frac{D}{2}\int_{0}^{\varphi}\sin^{k+1}\psi\,d\psi \tag{3} \]

Here \(D\) is the diameter of the cutter.

For the case \(b = D\), which we shall analyze below, the contact angle is \(\varphi = 180^\circ\).

The integral in Eq. 3 cannot be evaluated exactly, and two methods of approximate evaluation are known: Salomon’s method\(^7\) and the method based on the property of the integral of the product of two functions\(^8\), which lead to one and the same result:

\[ A_z = \lambda \cdot S_z^{k+1} \cdot t^{\,l+1}\frac{D}{2}\cdot \frac{2^{k+2}}{k+2}\cdot \left[\sin^{k+2}\frac{\psi}{2}\right]_{0}^{\varphi} \tag{4} \]

There is a widespread erroneous opinion that such an approximate evaluation of the integral gives an inaccuracy not exceeding \(1\%\). It must be pointed out here that such a slight inaccuracy can be obtained only for small values of the contact angle \(\varphi\), up to about \(\varphi = 40\text{--}50^\circ\); in the case of face-milling work, where the contact angle may reach \(\varphi = 180^\circ\), this error, increasing with increasing \(\varphi\) and \(k\), may reach \(20\text{--}25\%\), which is entirely inadmissible, but is not taken into account by some authors\(^9\).

In Fig. 2 we have graphically represented the exact and approximate evaluation of the integral, the value of which is determined by the area bounded by one or the other curve. It is clearly seen that (at \(k=-0.5\)) already from \(\varphi=30^\circ\) a difference is noticeable; beginning with \(\varphi=60^\circ\), this difference in areas begins to increase rapidly, which indicates a significant error obtained from Eq. 4. In Fig. 3 the same case is presented in the coordinates \(A_z-\varphi\), and at \(\varphi=180^\circ\) the error of the approximate solution reaches \(27\%\). In Fig. 4 the same as in Fig. 2 is presented for the value \(k=-0.3\). Here the discrepancies are smaller, but the difference in the areas for the case of face-milling work, \(\varphi=180^\circ\), nevertheless reaches inadmissible dimensions of \(16\%\).

Fig. 2.

To determine the inaccuracy of Eq. 4 for various \(\varphi\) and \(k\), we were...

calculated by Simpson’s rule, which gives very high accuracy. The values of \(k\) were taken from \(k=-0.2\) to \(k=-0.5\), i.e., those values that may occur in practice.

Fig. 3.

Fig. 4.

Table 1 summarizes the results of the calculations; it also gives the values of the approximate solution of the integral (according to Eq. 4), and in the last column the coefficient \(\varepsilon\), equal to the ratio of the actual value of the integral (by Simpson’s rule) to the approximate one (according to Eq. 4), which makes it possible to judge the magnitude of the error obtained when using equation 4:

Table 1.

\(\varphi\)	\(\displaystyle \int_0^\varphi \sin^{k+1}\psi\,d\psi\) (by Simpson’s rule), \(k=-0.2\)	\(\displaystyle \int_0^\varphi \sin^{k+1}\psi\,d\psi\) (by Simpson’s rule), \(k=-0.3\)	\(\displaystyle \int_0^\varphi \sin^{k+1}\psi\,d\psi\) (by Simpson’s rule), \(k=-0.4\)	\(\displaystyle \int_0^\varphi \sin^{k+1}\psi\,d\psi\) (by Simpson’s rule), \(k=-0.5\)	\(\displaystyle \frac{2k+2}{k+2}\cdot \sin^{k+2}\frac{\varphi}{2}\), \(k=-0.2\)	\(\displaystyle \frac{2k+2}{k+2}\cdot \sin^{k+2}\frac{\varphi}{2}\), \(k=-0.3\)	\(\displaystyle \frac{2k+2}{k+2}\cdot \sin^{k+2}\frac{\varphi}{2}\), \(k=-0.4\)	\(\displaystyle \frac{2k+2}{k+2}\cdot \sin^{k+2}\frac{\varphi}{2}\), \(k=-0.5\)	\(\varepsilon\), \(k=-0.2\)	\(\varepsilon\), \(k=-0.3\)	\(\varepsilon\), \(k=-0.4\)	\(\varepsilon\), \(k=-0.5\)
\(40^\circ\)	0,2808	0,3111	0,3431	0,3808	0,2805	0,307	0,3381	0,376	1,001	1,013	1,015	1,016
\(60^\circ\)	0,561	0,5998	0,6395	0,6854	0,554	0,586	0,624	0,6655	1,012	1,023	1,025	1,03
\(70^\circ\)	0,772	0,7626	0,8041	0,8517	0,710	0,739	0,777	0,818	1,017	1,032	1,034	1,041
\(90^\circ\)	1,0655	1,1066	1,1521	1,1975	1,036	1,0605	1,09	1,121	1,028	1,044	1,057	1,068
\(110^\circ\)	1,409	1,4506	1,5001	1,5433	1,35	1,3645	1,379	1,399	1,042	1,065	1,088	1,104
\(120^\circ\)	1,570	1,6134	1,6647	1,7097	1,491	1,495	1,505	1,520	1,051	1,079	1,106	1,125
\(140^\circ\)	1,8502	1,9021	1,9611	2,0132	1,73	1,718	1,716	1,716	1,07	1,108	1,143	1,173
\(160^\circ\)	2,0482	2,1158	2,1900	2,2588	1,87	1,85	1,71	1,795	1,09	1,138	1,21	1,258
\(180^\circ\)	2,131	2,2132	2,3042	2,394	1,935	1,908	1,894	1,884	1,1	1,161	1,216	1,271

\[ \varepsilon = \frac{\displaystyle \int_0^\varphi \sin^{k+1}\psi\,d\psi} {\displaystyle \frac{2^{k+2}}{k+2}\cdot\left[\sin^{k+2}\frac{\psi}{2}\right]_0^\varphi} \]

From the magnitude of \(\varepsilon\) it may be concluded that, for expressing the regularities of the operation of a face mill, equation 4 is not suitable if it is extended from \(0^\circ\) to \(180^\circ\), since the errors obtained in this case reach 10–27%.

Still less suitable is this equation for the case when the tooth begins to take a chip from some angle \(\psi \ne 0^\circ\), that is, when \(b < D\). Examples of inaccuracies for certain limits of integration are given in Table 2.

Table 2.

\(\psi_1\)	\(\psi_2\)	\(\dfrac{b}{D}\)	\(\displaystyle \int_{\psi_1}^{\psi_2}\sin^{k+1}\psi\,d\psi\), \(k=-0{,}2\)	\(\displaystyle \int_{\psi_1}^{\psi_2}\sin^{k+1}\psi\,d\psi\), \(k=-0{,}3\)	\(\displaystyle \int_{\psi_1}^{\psi_2}\sin^{k+1}\psi\,d\psi\), \(k=-0{,}4\)	\(\displaystyle \int_{\psi_1}^{\psi_2}\sin^{k+1}\psi\,d\psi\), \(k=-0{,}5\)	\(\displaystyle \frac{2^{k+2}}{k+2}\cdot\left[\sin^{k+2}\frac{\psi}{2}\right]_{\psi_1}^{\psi_2}\), \(k=-0{,}2\)	\(\displaystyle \frac{2^{k+2}}{k+2}\cdot\left[\sin^{k+2}\frac{\psi}{2}\right]_{\psi_1}^{\psi_2}\), \(k=-0{,}3\)	\(\displaystyle \frac{2^{k+2}}{k+2}\cdot\left[\sin^{k+2}\frac{\psi}{2}\right]_{\psi_1}^{\psi_2}\), \(k=-0{,}4\)	\(\displaystyle \frac{2^{k+2}}{k+2}\cdot\left[\sin^{k+2}\frac{\psi}{2}\right]_{\psi_1}^{\psi_2}\), \(k=-0{,}5\)	[[unclear: header not visible]], \(k=-0{,}2\)	[[unclear: header not visible]], \(k=-0{,}3\)	[[unclear: header not visible]], \(k=-0{,}4\)	[[unclear: header not visible]], \(k=-0{,}5\)
\(90^\circ\)	\(110^\circ\)	0,17	0,3435	0,3440	0,348	0,3458	0,314	0,304	0,289	0,278	1,09	1,13	1,205	1,25
\(70^\circ\)	\(110^\circ\)	0,34	0,687	0,688	0,696	0,6916	0,64	0,6255	0,602	0,581	1,07	1,1	1,15	1,19
\(60^\circ\)	\(120^\circ\)	0,5	1,009	1,0136	1,0252	1,0243	0,937	0,909	0,881	0,8545	1,07	1,12	1,16	1,20
\(70^\circ\)	\(140^\circ\)	0,555	1,1282	1,1395	1,1570	1,1615	1,02	0,979	0,939	0,898	1,11	1,16	1,23	1,29
\(90^\circ\)	\(180^\circ\)	0,5	1,0655	1,1066	1,1521	1,1965	0,899	0,8475	0,804	0,763	1,23	1,3	1,43	1,57

On examining Table 2 we see that, even if the last case is disregarded, in which the errors reach 57%, and if we take the case often encountered in the practice of longitudinal milling-machine work, with

\[ \frac{b}{D}=(0{,}5 \div 0{,}55), \]

then even here the error reaches 11–29%. Moreover, it may be concluded that for different ranges of contact angles the errors are different, and this means that for equation 4 one cannot even experimentally determine such values of the constant coefficients \(\lambda\) that would compensate for these errors. Thus equation 4 is suitable for the case of operation of a cylindrical cutter, where the value of \(\varphi\) usually does not exceed \(60^\circ\); for the operation of a face cutter, however, it gives inadmissible errors.

The following method may be proposed for eliminating the errors. During operation, a tooth of a face cutter removes two equal volumes of chip: the volume removed in the passage from \(0^\circ\) to \(90^\circ\) is equal to that in the passage from \(90^\circ\) to \(180^\circ\), and it may be assumed that the works expended in this are also equal, which is expressed by the symmetry of the area representing, in Figs. 2 and 4, the actual work of the tooth with respect to the mean ordinate at \(\varphi=90^\circ\). Thus the total work of one tooth on the path from \(0^\circ\) to \(180^\circ\) is equal to twice its work on the path from \(0^\circ\) to \(90^\circ\), or

\[ A_z=\lambda S_z^{k+1}\cdot t^{l+1}\frac{D}{2}\cdot\int_0^{180^\circ}\sin^{k+1}\psi\,d\psi = \]

\[ =2\lambda S_z^{k+1}\cdot t^{l+1}\cdot\frac{D}{2}\int_0^{90^\circ}\sin^{k+1}\psi\,d\psi; \]

upon substituting

\[ \frac{\psi}{2}=45^\circ \]

we obtain

\[ A_z=2\lambda S_z^{k+1}\cdot t^{l+1}\frac{D}{2}\cdot\frac{2^{k+2}}{k+2}\cdot0{,}707^{k+2}. \tag{5} \]

This expression gives the work of one tooth of the milling cutter at \(b=D\) (for \(b<D\) it cannot be used) and immediately reduces the error from 10% (at \(k=-0.2\)) and 27% (at \(k=-0.5\)) to 2.8–6.8% for the same values of \(k\) (see Table 1, values of \(\varepsilon\) at \(\varphi=90^\circ\)). If expression 5 is multiplied by the corresponding values of \(\varepsilon\) (for \(\varphi=90^\circ\)), then we shall have an entirely exact expression for the work of a tooth of any milling cutter, in any case no less exact than expression 4 for the case of operation of a cylindrical milling cutter. It is not difficult to verify (see Table 3) that for all values of \(k\) for \(\varphi=90^\circ\)

\[ \frac{2^{k+1}}{k+2}\cdot \varepsilon \cong 1 \]

Table 3.

\(k\)	\(\dfrac{2^{k+1}}{k+2}\)	\(\varepsilon\) at \(\varphi=90^\circ\)	\(\dfrac{2^{k+1}}{k+2}\varepsilon\)
\(-0.2\)	0.968	1.034	1.000
\(-0.3\)	0.955	1.044	0.997
\(-0.4\)	0.947	1.056	1.000
\(-0.5\)	0.943	1.068	1.006

Therefore we can give a practically entirely exact expression if we multiply Eq. 5 by \(\varepsilon\) and take

\[ \frac{2^{k+1}}{k+2}\cdot \varepsilon = 1 \]

In this case we shall have

\[ A_z = 2^{-\frac{k}{2}}\cdot \lambda \cdot S_z^{k+1}\cdot t^{l+1}D \ \mathrm{kg/mm} \tag{6} \]

Equation 6 for any milling cutter is simpler than Eq. 4 and, moreover, incomparably more exact. From this it is not difficult to obtain the dependence for the average power consumption in the operation of any milling cutter

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

Substituting Eq. 6 here, we obtain

\[ N_e=\frac{2^{-\frac{k}{2}}}{1000\cdot 4500}\cdot \lambda \cdot n^{-k}\cdot t^{l+1}\cdot S^{k+1}\cdot Z^{-k}\cdot D \tag{7} \]

Here: \(Z\)—number of teeth of the milling cutter,
\(n\)—number of revolutions of the milling cutter per minute,
\(S\)—feed in millimeters per minute.

— Expression 7 relates all variables for the case of operation of any milling cutter at \(b=D\) and can be applied only for this case.

In the case of operation with \(b=D\), the degree of nonuniformity of the resultant circumferential force and of the torque is very close to unity and depends (at

in the absence of beating of the cutter) only on the number of teeth \(Z\), as is evident from Fig. 5, where plots of the resultant circumferential forces are given for various \(Z\). Therefore it makes no sense to determine the sum of forces and the torque for a given instantaneous position of the teeth; rather, one may be satisfied with their mean value, found from the power, especially since it is not possible to take into account the nonuniformity arising from the unavoidable beating of the cutter, which plays a much greater role than the nonuniformity as a function of the cutting geometry,

\[ \left. \begin{aligned} \sum P &= \frac{2^{-\frac{k}{2}}\lambda}{\pi n^{k+1}} \cdot t^{l+1}\cdot S^{k+1}\cdot Z^{-k},\\[6pt] M_{\mathrm{avg}} &= \frac{2^{-\frac{k}{2}}\lambda}{2\pi n^{k+1}} \cdot t^{l+1}\cdot S^{k+1}\cdot Z^{-k}\cdot D \end{aligned} \right\} \tag{8} \]

Fig. 5.

In production there are often cases when a face mill operates with \(b < D\). Let us examine two possible variants in this situation:

1) the cutter operates with \(b \leq \dfrac{D}{2}\) and takes a chip from zero thickness, as in cylindrical milling (see Fig. 6). We shall call this the case of side cutting.

2) the cutter operates with \(b < D\) and is positioned symmetrically with respect to the workpiece (Fig. 7). We shall call this the case of face cutting.

Fig. 6.

Fig. 7.

For case 1 it is not difficult to find the equation for the power \(N'_e\) by a derivation similar to Salomon’s derivation for a cylindrical cutter\(^7\). Retaining our notation and measuring \(t\) parallel to the axis of the cutter and \(b\) perpendicular to the feed, we obtain:

\[ N'_e = \frac{2^{k+1}}{k+2} \cdot \frac{\lambda \cdot n^{-k}}{1000 \cdot 4500} \cdot t^{l+1}\cdot S^{k+1}\cdot b^{\frac{k+2}{2}}\cdot Z^{-k}\cdot D^{-\frac{k}{2}}. \]

Since usually in side cutting by a face mill, in contrast to cy-

cylindrical cutter operates with significant contact angles \(\varphi\), we find it possible here also to introduce the multiplier \(\varepsilon\) and take

\[ \frac{2^{k+1}}{k+2}\cdot \varepsilon = 1 \]

In this case we obtain

\[ N'_e=\frac{\lambda \cdot n^{-k}}{1000\cdot 4500}\cdot u^{k+1}\cdot S^{k+1}\cdot b^{\frac{k+2}{2}}\cdot Z^{-k}\cdot D^{-\frac{k}{2}} \tag{9} \]

Comparing this expression with equation 7, we shall have

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

If in equation 10 we put

\[ b=\frac{D}{2} \]

i.e., \(\varphi=90^\circ\), then we shall have

\[ N'_{e(\varphi=90^\circ)}=\frac{1}{2}N_{e(\varphi=180^\circ)}, \]

which also served as the basic assumption in deriving equation 7.

For case 2 (face cutting) (Fig. 7), the power \(N''_e\) may be obtained if, from the power \(N_e\) (when operating with \(b=D\)), one subtracts twice the power \(N'_e\) required for side cutting with \(b=b_0\)

\[ N''_e=N_e-2N'_e. \]

\[ N''_e=N_e-2N_e\cdot 2^{\frac{k}{2}}\cdot \left(\frac{b_0}{D}\right)^{\frac{k+2}{2}}. \]

According to Fig. 7

\[ b_0=\frac{D-b}{2} \]

\[ N''_e=N_e\cdot \left[1-\left(1-\frac{b}{D}\right)^{\frac{k+2}{2}}\right] \tag{11} \]

The multipliers at \(N_e\) in equations 10 and 11, giving the transition from the case \(b=D\) to the cases of side \(\left(b<\frac{D}{2}\right)\) and face cutting, depend only on the ratio \(\frac{b}{D}\) and on the value \(k\), which is constant for the given machined material. Therefore the multipliers can easily be expressed graphically, and for all three cases of operation of a face mill one can construct a single graph relating the power consumed to the selected cutting conditions, which will also serve for resolving questions concerning utilization of the machine-tool power.

Equations 10 and 11 show that the power consumption does not increase proportionally to \(b\). In addition, from comparison of equations 7 and 11 it may be concluded that, at a constant milling width \(b\), with an increase in the cutter diameter \(D\) (all other conditions being constant), the power consump-

ness decreases. For \(k=-0.3\) (a value encountered in milling work), this decrease in power at \(b=\text{const}\) will be as follows:

\[ \frac{b}{D}=1; \qquad N_e'=1, \]

\[ \frac{b}{D}=0.5; \qquad N_e'=0.89 \]

\[ \frac{b}{D}=0.33; \qquad N_e'=0.87 \]

\[ \frac{b}{D}=0.25; \qquad N_e'=0.85 \]

In this, of course, it is assumed that:

\[ (t,S,z,n)=\text{const}. \]

Thus, from the standpoint of power consumption, it is always more advantageous to mill with a cutter diameter exceeding the milling width \(b\).

It is necessary to take into account that, for different ratios \(\frac{b}{D}\), the degree of nonuniformity of the resultant circumferential force, and also of the torque, may be different; in this respect the most advantageous ratio is:

\[ \frac{b}{D}=\sin\left(C\cdot\frac{360}{2z}\right), \]

where “\(C\)” is any integer. The variation of the resultant circumferential force for several values of \(\frac{b}{D}\) is shown in Fig. 8. In side cutting with \(\frac{b}{D}<\frac{1}{2}\), when the cutter takes up a chip beginning from zero thickness, with an increase in diameter at constant \(b\) the power consumption \(N_e'\) increases. This type of cutting is the least advantageous from the standpoint of power consumption.

In conclusion of the theoretical conclusions it is necessary to point out that, in deriving Eq. 7, we did not take into account a certain difference in work consumption when cutting with a cutter from the thin part of the chip to the thick part, and when cutting from the thick part of the chip to the thin part. This difference in work consumption was noted by Boston and Kraus \(^{4}\), but it has a different character for different materials being machined and, in the theoretical derivation, cannot yet be taken into account.

Fig. 8.
Labels in the figure: \(z=12\); \(b=1.000D\); \(b=0.966D\); \(b=0.866D\); \(b=0.707D\); \(b=0.500D\); \(b=0.250D\); horizontal scale \(0, 30, 60, 90, 120, 150, 180\).

II. Experimental investigation of the operation of a face mill.

The purpose of the experimental part of the work was to verify the correctness of Eqs. 7, 10, and 11 and to find the values of \(\lambda\), \(k\), \(l\) for various materials being machined.

The experiments were carried out on a milling machine of the Tula plant, in whose gearbox a special self-recording hydraulic dynamometer of the author’s design was built in, serving to record the torques on the cutter. The diagram of the dynamometer is shown in Fig. 9; a full description of the diagram is given in our previous work[^6]. Calibration of the dynamometer was performed with a Prony brake; its results are presented in Fig. 10. In addition, the additional losses in the spindle bearings under the influence of the forces acting on the cutter were determined. The additional moments thereby obtained, in kg mm, which distort the readings of the instrument, are presented in Fig. 11. They are extremely small, since the spindle is mounted on tapered roller bearings, and in our experiments amount at most to less than 1.5% of the working moment.

Fig. 9.

The face mills used in the experiments are shown in Fig. 12; their dimensions are given in Table 4.

Table 4.

No.	Diameter \(D\), mm	Cutter material	Number of teeth \(Z\)	Clearance angle on the face	Clearance angle on the cylindrical part
1	39.0	High-speed steel RF 1	6	14°	12°
2	59.6	High-speed steel RF 1	18	12°	9°
3	64.5	High-speed steel RF 1	10	15°	8°
4	109.7	High-speed steel RF 1	10	14°	12°
5	144.5	High-speed steel RF 1	14	14°	8°

Fig. 10.

Vertical axis: \(M\), kg·mm

Left scale: 0, 10,000, 20,000, 30,000, 40,000

Horizontal axis: Height of the ordinate in mm according to the recording instrument.

Lower horizontal axis: Manometer reading in atm.

Legend:

—○—○—○— mechanical calibration of the dynamometer
—×—×—×— calibration by the Prony brake on the machine

Labels on the plotted straight lines: \(M\), kg·mm

Fig. 11.

Vertical axis: \(M_{\mathrm{cr}}\), kg·mm

Horizontal axis: \(P\), kg

Fig. 11.

Milling cutters 1 and 2 had a sharply ground tooth angle, cutter 3—a rounded one, and cutters 4 and 5—a truncated one, as shown in Fig. 13.

For comparison of the values obtained, in parallel with the face mills two cylindrical cutters were used in the work (see Table 5 and Fig. 14).

Fig. 12.

Fig. 13.

Table 5.

No.	Diameter \(D\), mm	Cutter material	Number of teeth \(Z\)	Helix angle, degrees	Rake angle \(\gamma\)
1	74.0	High-speed steel	18	64°	0°
2	59.7	"	8	60°	10°

The materials machined are given in Table 6.

Table 6.

No.	Name	Chemical composition	Brinell hardness \(H_{br}\)	Ultimate tensile strength \(\sigma B\), kg/mm²	Relative elongation \(l\), %
1	Soft cast iron	—	90–100	—	—
2	Cast iron of medium hardness	—	150	—	—
3	Hard cast iron	—	195	—	—
4	Mild steel	Carbon 0.1% Manganese 0.42% Sulfur 0.02% Phosphorus 0.02% Silicon—traces	90	34	39
5	Steel of medium hardness	—	160	61	10
6	Hard steel	Carbon 0.51% Manganese 0.64% Silicon 0.22% Sulfur 0.0% Phosphorus 0.01%	215	76	10
7	Chromium steel 5145	Carbon 0.4–0.48% Silicon 0.2–0.35% Manganese 0.6–0.8% Chromium 0.8–1.1% Nickel \< 0.3%	210	74	16

with such milling cutters. The data obtained for cylindrical cutters fully coincided with the data of our previous work for analogous machined materials and are presented by us here for comparison with the data for face milling cutters.

Fig. 14.

for face milling cutters. The work was carried out either with \(b = D\), or with \(b < D\), according to the type of face cutting.

The depths of milling \(t\) used in the experiments were taken by us for the most part from \(1\ \text{mm}\) to \(8\ \text{mm}\). Sometimes we limited ourselves to a smaller depth; in the case of machining soft cast iron, \(t_{\max} = 12\ \text{mm}\). The feeds per tooth were taken within the limits from \(S_z = 0.014\ \text{mm}\) to \(S_z = 1.69\ \text{mm}\), which fully provides for both finishing and roughing work.

Processing of the Experimental Results

We chose a method for processing the results similar to the method of our previous work with cylindrical cutters \(^{6}\).

In the case \(b = D\), the work of one tooth per one revolution is

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

Let us relate the work of one tooth to the volume of chip removed by it in 1 revolution. In this way we obtain the specific work \(A_w\):

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

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

In deriving equation 7 for \(b = D\), we replaced the work of one tooth within a contact angle of \(180^\circ\) by twice the work within an angle of \(90^\circ\) (thereby reducing the error). Therefore we must conditionally regard equation 7 as having been derived for \(\varphi = 90^\circ\) (despite the fact that \(b = D\)). For \(\varphi = 90^\circ\):

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

that is, \(0.707\,S_z\) will represent the chip thickness at the midpoint of the contact angle, or the mean chip thickness, which we shall denote by \(S_m\):

\[ 0.707\cdot S_z = S_m \tag{12} \]

And hence

\[ A_w=\lambda\cdot S_m^k\cdot t^l \tag{13} \]

Thus the specific expenditure of work in face milling depends, for a given work material, on the mean chip thickness \(S_m\) and the depth of milling; and the greater the latter, the smaller \(A_w\) should be, since in the general case \(l<0\).

Equation 13 can be obtained from equation 7:

\[ A_w=\frac{4500\cdot 1000\cdot N_e}{t\cdot b\cdot S}=\lambda\cdot S_m^k\cdot t^l \]

If equation 13 is expressed graphically in logarithmic coordinates \(S_m\) (abscissa) and \(A_w\) (ordinate), then the tangent of the angle of inclination of the resulting straight line to the abscissa axis will give the value \(k\), while the value of \(A_w\) at \(S_m=1\ \text{mm}\) and \(t=1\ \text{mm}\) will give \(\lambda\). The ratios of \(A_w\) at \(S_m=\text{const}\), but for different \(t\), will make it possible to find \(l\). All tests on a given material at \(t=\text{const}\), but with different \(n, D, S, Z\), must give, in the coordinates \(S_m\)—\(A_w\), a single straight line. This will serve as confirmation of the correctness of our conclusions. If, for some material, tests with different \(t\) give only one straight line, this will show that, for the given material, \(t\) has no influence on the specific work, i.e. \(l=0\).

By recording in the tests the torque on the cutter, we recalculate the value of the torque into the specific work according to the relation:

\[ A_w=\frac{6.28M}{t\cdot D\cdot S_z\cdot Z} \]

we relate it to the correspondingly calculated value \(S_m\), and, expressing \(A_w=f(S_m)\) graphically, we find \(k, l, \lambda\).

When working with \(b<D\), we recalculated the torques obtained into the torques for the case \(b=D\), using the relation between the torques equal to the relation between \(N''_e\) and \(N_e\) according to equation 11. The “reduced” torques thus obtained for \(b=D\) served to find \(A_w=f(S_m)\).

In the case where the values calculated by us lay on the straight lines constructed by us from experiments with \(b = D\), we obtained proof of equation 11.

The transformation we carried out of equation 4 into equation 7, as well as equations 12 and 13 derived by us, make the results of experiments with a face mill comparable with the results of experiments with a cylindrical mill. We can judge the comparative expenditure of work, and the values of \(\lambda\), whereas if we had used equation 4, errors in the theoretical derivation would have entailed distortion of the experimentally calculated values of \(\lambda\) and would have led, in comparison with the operation of a cylindrical mill, to erroneous conclusions.

Results of the experiments.

The results of all experiments are presented by us through the specific work in the coordinates \(S_m\)—\(A_w\), with \(S_m\) plotted on the graphs in microns, and \(A_w\) in kg·mm/mm\(^3\). In Figs. 15–20 the results are presented of experiments on cast iron \(H = 90\)—100 with various depths of milling from \(t = 1\) mm to \(t = 12\) mm, with all face mills. The angle of inclination of the straight lines drawn is one and the same for all values of \(t\).

All straight lines, with the exception of the straight line for \(t = 1\) mm, are located at the same height, which gives \(l = 0\) for \(t > 1\) mm. The straight line for \(t = 1\) mm is located somewhat higher. We give the values of \(\lambda\) separately for \(t = 1\) mm and for \(t > 1\) mm.

Fig. 15.

Fig. 16.

Fig. 17.

Fig. 18.

Fig. 19.

For cast iron \(H=90\text{--}100\) we obtained

\[ K=-0.265;\qquad \lambda=70 \quad (\text{at } t=1\ \text{mm}) \]

\[ \lambda=63 \quad (\text{at } t>2\ \text{mm}) \]

Fig. 20.

The data on the operation of the cylindrical cutter are presented for cast iron \(H=90\text{--}100\) in Fig. 21. Here the results are given of experiments with cutters having a tooth with no undercut and with an undercut of \(10^\circ\).

For the cylindrical cutter we have

\[ K=-0.4;\qquad \lambda=50 \quad (\text{for a tooth without undercut}). \]

An undercut angle of \(10^\circ\) lowers the value of \(\lambda\) by \(12\%\).

Fig. 21.

The data on the operation of the cylindrical cutter agree fully with the results of our previous work, which indicates the reliability of the measuring apparatus.

In Fig. 22 are presented the results of experiments with a face cutter on cast iron \(H=150\).

Here we see the full regularity of the decrease in specific work with increasing milling depth \(t\).

The points for the cutter with a cut-off tooth angle here lie somewhat higher than the others.

For cast iron \(H=150\) we obtained

\[ K=-0.3\div 5;\qquad l=-0.09;\qquad \lambda=85 \]

Fig. 23 contains the results for machining the same cast iron with cylindrical milling cutters.

\[ K=-0.35;\qquad \lambda=66 \]

(for an unrelieved tooth).

Relieving by \(10^\circ\) reduces \(\lambda\) by 8%.

Fig. 22.

In Fig. 24 the results are given for face milling cutters on cast iron \(H=195\). The influence of the depth of milling \(t\) on the specific work is also clearly expressed here.

For cast iron \(H=195\) we have

\[ K=-0.253;\qquad l=-0.06;\qquad \lambda=107 \]

Fig. 25 contains the results of machining the same cast iron with cylindrical milling cutters:

\[ K=-0.33;\quad \lambda=86; \]

backing-off of the tooth by \(10^\circ\) reduces \(\lambda\) by 11%,

Fig. 23.

The latter quantities do not agree with the data of our previous work—and must be corrected there—since they were not the result of our investigation (for lack at that time of the necessary cast iron), but were borrowed from the work of Baleke \(^{10}\), which was carried out without special measuring apparatus; moreover, the additional harmful resistances and the expenditure of work on the feed mechanism were not taken into account.

In Fig. 26 the results of experiments with face mills on steel of medium hardness \(\delta_B=61\ \mathrm{kg/mm^2}\) are presented. There is no influence of the milling depth.

\[ K=-0.22;\quad l=0;\quad \lambda=178. \]

Fig. 27 gives the results of work on the same steel with cylindrical milling cutters:

\[ K=-0.28;\quad \lambda=152\quad \text{(for an unbacked-off tooth).} \]

Backing-off of the tooth by \(10^\circ\) reduces \(\lambda\) by 11%.

In Fig. 28 the results of work with face mills on hard steel are given. Here a certain influence of the milling depth \(t\) on the expenditure of work is noticeable:

\[ K=-0.2;\quad l=-0.04;\quad \lambda=186. \]

Fig. 29 gives, for the same steel, the work of cylindrical milling cutters. The straight line for the unbacked-off tooth was drawn from the materials of the previous work, since the blank subjected to milling was the same.

\[ K=-0.26;\quad \lambda=180\quad \text{(for an unbacked-off tooth).} \]

Backing-off by \(10^\circ\) reduces \(\lambda\) by 12%.

In Fig. 30 the results of work with face mills on chromium steel are given:

\[ K=-0.2;\quad l=0;\quad \lambda=195. \]

Fig. 31 gives, for the same steel, the work of cylindrical milling cutters. The straight line \(\gamma=0^\circ\) was drawn from the materials of the previous work:

\[ K=-0.23;\quad \lambda=205. \]

Fig. 24. Fig. 25.

Undercutting by \(10^\circ\) reduces \(\lambda\) by 13%.

In Fig. 32 the results are given of experiments with face mills on mild steel:

\[ K=-0.29;\qquad l=0;\qquad \lambda=185. \]

Fig. 26.

Fig. 33 gives, for the same steel, the operation of a cylindrical milling cutter. The straight line for \(\gamma=0^\circ\) has been drawn from the data of the previous work.

\[ K=-0.3;\qquad \lambda=142 \quad \text{(for a tooth without undercutting);} \]

\[ K=-0.34;\qquad \lambda=123 \quad \text{(for a tooth undercut by }10^\circ\text{).} \]

Fig. 27.

Top plot annotations:

• \(A_w\)
• \(S\) m.
• Cylindrical cutter
  steel of medium hardness
• \(\varphi = 0^\circ\): \(t = 3\) mm; \(t = 1\) mm
• \(\varphi = 10^\circ\): \(t = 3\) mm; \(t = 1\) mm

Fig. 28.

Plot annotations:

• \(A_w\)
• \(S\) m.
• hard steel
• \(\times\quad D = 109.7\) mm, \(b = 95\) mm
• \(\circ\quad D = 64.5\) mm, \(b = D\)

• \(t = 1\) mm

• hard steel

• \(\times\quad D = 109.7\) mm, \(b = 95\) mm
• \(\circ\quad D = 64.5\) mm, \(b = D\)

• \(t = 2\) mm

• hard steel

• \(D = 64.5\) mm, \(b = D\)

• \(t = 3\) mm

• hard steel

• \(D = 64.5\) mm, \(b = D\)
• \(\circ\quad t = 4\) mm
• \(\rho\quad t = 6\) mm

Fig. 29.

Graph annotations:

• \(R_w\)
• Cylindrical cutter
• Hard steel
  \(D = 59.7\) mm; \(z = 8\); \(\beta = 60^\circ\)
• \(\gamma = 10^\circ\)
• \(\circ\ t = 3\) mm
• \(\times\ t = 1\) mm
• \(\varphi = 0^\circ\)
• \(S_m\)

Fig. 30.

Graph annotations:

• \(R_w\)
• Chromium steel
• \(\circ\ D = 64.5\) mm; \(b = D\)
  \(t = 1\) mm
• \(R_w\)
• Chromium steel
• \(\circ\ D = 64.5\) mm; \(b = D\)
  \(\times\ D = 109.7\) mm; \(b = 92\) mm
  \(t = 2\) mm
• \(R_w\)
• Chromium steel
• \(\circ\ D = 64.5\) mm; \(b = D\)
  \(\times\ D = 109.7\) mm; \(b = 94\) mm
  \(t = 3\) mm
• \(R_w\)
• Chromium steel
• \(D = 64.5\) mm; \(b = D\)
  \(\circ\ t = 4\) mm
  \(\bullet\ t = 5\) mm
• \(S_m\)

Fig. 31.

Fig. 32.

Undercutting by \(10^\circ\) reduces \(\lambda\) by 13%.

When working on all steels, milling was carried out both with \(b = D\) and with \(b < D\). The experimental points for \(b < D\) agreed well with the points for \(b = D\), which indicates the correctness of the relation we derived between \(N_e\) and \(N_e''\) (equation 11). This is also confirmed by Fig. 34, which gives the results of work with various ratios \(\frac{b}{D}\) for cast iron.

Fig. 33.

All straight lines drawn on the graphs were drawn according to the values of \(K\), \(l\), and \(\lambda\) indicated above by us. Thus, from the position of the experimental points relative to the straight lines, one can see how well the given values of \(K\), \(l\), and \(\lambda\) satisfy the arrangement of the experimental points. The values of \(K\), \(l\), and \(\lambda\) are summarized in Table 7. For comparison, values for cylindrical milling cutters are also given here.

From an analysis of Table 7 it follows:

Table 7.

No.	Material machined	Face mill \(\lambda\)	Face mill \(K\)	Face mill \(l\)	Cylindrical mill \(\lambda\) for an undercut tooth	Cylindrical mill \(\lambda\) for a tooth undercut by \(10^\circ\)	Cylindrical mill \(K\)
1	Soft cast iron \(H = 90\text{--}100\) \(t = 1\) mm \(t \ge 2\) mm	70 63	−0.265	0.00	50	44	−0.4
2	Cast iron of medium hardness \(H = 150\)	85	−0.315	−0.09	66	61	−0.35
3	Hard cast iron \(H = 195\)	107	−0.253	−0.06	86	77	−0.33
4	Soft steel \(\sigma_B = 34\)	185	−0.29	0.00	142	123	−0.3 −0.34
5	Steel of medium hardness \(\sigma_B = 61\)	178	−0.22	0.00	152	135	−0.28
6	Hard steel \(\sigma_B = 76\)	186	−0.2	−0.04	180	158	−0.26
7	Chromium steel \(\sigma_B = 74\)	195	−0.2	0.00	205	178	−0.23

The values of $\lambda$ for face mills are higher than for cylindrical mills; this apparently expresses the additional expenditure of work in face cutting and the somewhat more difficult process of chip formation. The values of $K$ for face mills are lower than for cylindrical mills. The values of $K$ for cast iron are higher than for steel. The values of $K$ for face mills come very close to the values for lathe work.

The values of $l$, which determine the degree of influence of chip width on the specific cutting pressure, in most cases are equal to zero when machining steel, whereas when machining cast iron they differ from zero.

If the graphs for face mills are compared with the graphs for cylindrical mills, it may be noted that, owing to the difference in the slope of the straight lines, with a change in chip thickness the difference in energy expenditure in the operation of face and cylindrical mills changes, increasing with an increase in the value of $S_m$. This is apparently explained by the ever-increasing influence of the face cutting edges and by the increasingly difficult formation of the chip.

The opinion of Prof. Reznikov, who considers that the work of the end cutting edges accounts for 3–5% of all the work expended, is erroneous. From the graphs presented it is evident that this percentage is not constant and, at large feeds, may be considerably higher.

Mild steel in our experiments is characterized, during milling, by no smaller expenditure of work than harder steels. This is also noticeable from the data for cylindrical mills; still more is this the case for face mills. In view of the dubiousness of such a result, we repeated the experiments on mild steel a second time—they gave an exact repetition of what had been obtained earlier. Evidently, such an “anomaly” is explained by the considerably greater toughness of mild steel, which can be seen from Table 6, where the relative elongation for mild steel is 39%, while for all the others it is 10–16%.

Fig. 34.

This phenomenon, paradoxical at first glance, expressed in the fact that, when machining a softer steel, we obtained a higher load on the tool than when machining a harder one, has examples in other fields of metal cutting as well. Thus, for example, in drilling the torque is expressed by the dependence

\[ M = k \cdot d^m \cdot S^l, \]

where \(k, m, l\) depend mainly on the quality and strength of the material being machined for a given drill profile.

According to Schlesinger’s data \(^{11}\) (as reworked by Besprozvanny), we have:

Steel \(\sigma_B = 50\ \mathrm{kg/mm^2}\)

\[ M = 0.03\, d^2 S^{0.84}\ \mathrm{kg\,m} \]

Steel \(\sigma_B = 30\ \mathrm{kg/mm^2}\)

\[ M = 0.042\, d^2 S^{0.92} \]

According to Besprozvanny’s simplified formulas: \(^{11}\)

Steel \(\sigma_B = 50\)

\[ M = 0.029\, d^2 S^{0.8} \]

Steel \(\sigma_B = 30\)

\[ M = 0.036\, d^2 S^{0.8} \]

In both cases, the torque in machining the softer steel is considerably (by 24%) higher than in machining the harder steel.

With respect to axial pressure, for the same materials the same picture is obtained, only less sharply expressed.

According to entirely different data from Boston and Oxford \(^{12}\), for drilling we have:

Steel SAE 1020. Brinell hardness \(H = 131\)

\[ M = 0.0726\, S^{0.78} d^{1.8}\ \mathrm{kg\,m} \]

Steel SAE 1035. Brinell hardness \(H = 156\)

\[ M = 0.0611\, S^{0.78} d^{1.8}\ \mathrm{kg\,m} \]

Free-cutting steel. Hardness \(H = 217\)

\[ M = 0.0405\, S^{0.78} d^{1.8} \]

Here we have the same phenomenon, expressed even more distinctly.

The comparatively large expenditure of specific work by face mills in comparison with cylindrical ones (with thick chips) and the above-mentioned “anomaly” for mild steel give rise to an involuntary doubt as to the accuracy of the experiment; this, however, must be categorically rejected on the following grounds.

If we take the graphs \(S_m — A_w\) which we have given for some material for which \(l = 0\), for example, for cast iron \(H = 90—100\) (within the range from \(t = 2\ \mathrm{mm}\) to \(t = 12\ \mathrm{mm}\)) (see Figs. 16—20), and set some constant value of \(S_m\), then, taking \(A_w\) from Fig. 16 for \(t = 2\ \mathrm{mm}\) and \(D = 39\ \mathrm{mm}\), and from Fig. 20 for \(t = 12\ \mathrm{mm}\) and \(D = 64.5\ \mathrm{mm}\), in both cases we shall have the same value of \(A_w\) for one and the same \(S_m\), despite the fact that the torque on the mill in the second case will be 16 times greater than in the first. This is in full agreement with our theoretical conclusions; experimentally, however, with such different torques this can be obtained (and was obtained by us in all experiments) only with an unconditionally accurate and reliable measuring apparatus.

In order once again to verify the accuracy of the results we obtained, we carried out a series of experiments with a face mill \(D = 64.5\ \mathrm{mm}\), making it work at a milling depth \(t = 4\ \mathrm{mm}\) and with various feeds on cast iron \(H_{\mathrm{Br}} = 195\) and steel \(\sigma_B = 61\), in such a way that the teeth located on its face did not participate in the work, and all the work was performed only by the teeth located on the cylindrical surface.

To realize this condition, plates were made from steel \(\sigma_B = 61\) and cast iron \(H_{\mathrm{br}} = 195\), 4 mm thick and of a width exceeding the cutter diameter. These plates were screwed to the sides of a trough-shaped blank, whose distance between the sides was greater than the cutter diameter. By milling such a plate with a face mill, we caused only the teeth located on the cylindrical surface to work, since its end face was located between the milled plate and the bottom of the trough-shaped blank. Such a plate (already partly milled) is shown in Fig. 35.

With this method of milling with a face mill, we should have obtained the same results as when working with a cylindrical cutter, since the influence of the end teeth on the work consumption was eliminated.

Fig. 35.

In Fig. 36, the points represent the results obtained from these experiments. The straight lines in Fig. 36 are drawn through the experiments with the cylindrical cutter from Fig. 25 and Fig. 27. The agreement of the experimental points obtained when working with a face mill “without an end face” with the results of the work of the cylindrical cutter is sufficiently satisfactory, which once again confirms the correctness of all our conclusions and the accuracy of the experiments performed.

It is not without interest to point out that proper sharpening of the cutter has an enormous influence on the work consumption.

We obtained from one of the plants a cutter \(D = 109.7\) mm, in which the tooth angle had been sharpened in such a way that the clearance angle was equal to zero.

In Table 8 we give the results of work with this cutter before correction of the sharpening and after it.

Table 8.

Material Cast iron \(H = 90\text{–}100\)	Material Cast iron \(H = 90\text{–}100\)	Cutter \(D = 109.7\) mm \(Z = 10\)	\(b = 109.7\) \(t = 1\) mm
\(S_n\), feed per 1 revolution	\(S_z\), feed per 1 tooth	Torque with incorrect sharpening	Torque with correct sharpening
0.585	0.0585	5015	1700
1.59	0.159	8540	3650
3.00	0.3	11682	5800
6.21	0.621	15568	10135
8.4	0.84	1807 [[unclear: value as printed]]	12180
12.1	1.21	21682	15666

From the table one sees an increase in work consumption for the incorrectly sharpened cutter, especially significant at small feeds. It should be noted here that in both cases the cutter was sharply sharpened.

In our technical literature on face mills there are only the data of Prof. Reznikov. It is therefore interesting to carry out a comparison of the torques

moments obtained experimentally with those calculated by the formulas and coefficients of Prof. Reznikov. The comparison was carried out by us in Table 9. The calculation of the moments according to Reznikov was performed by us on the basis of formulas

[Figure: graph with vertical axis \(A_w\), horizontal axis \(S_z\). Inscription: “Face mill. 1. Operation without floor? \(D=64.5\ \mathrm{mm}\). \(z=10\).” Legend: “\(\circ\) steel, medium hardness”; “\(\times\) cast iron \(H=195\).”]

Fig. 36.

273 and Tables 42 and 43 of his book (Theory of Metal Cutting); the cutting angles according to his data coincide with the angles of our milling cutters. The calculation was made for a cutter \(D=64.5\ \mathrm{mm};\ Z=10;\ t=4\ \mathrm{mm};\ b=D\).

Table 9.

\(S_z\)	Steel \(\sigma_B=61\): According to Reznikov	Steel \(\sigma_B=61\): Experimental	Medium-hard cast iron: According to Reznikov	Medium-hard cast iron: Experimental	Hard cast iron: According to Reznikov	Hard cast iron: Experimental
Feed per 1 tooth, in mm
0.0585	12200	8050	5550	4860	6850	4625
0.0805	15450	10415	7050	6450	8700	6100
0.11	21400	13645	9700	8300	12000	8150
0.159	25500	19999	11940	10940	14300	11000
0.221	32700	24963	14900	13823	18400	14250
0.3	41200	30443	18800	16762	23200	17700
0.446	—	—	25500	22075	31500	24550
0.631	—	—	32500	28307	40000	29275

It follows from the table that the data of Prof. Reznikov give inadmissible deviations and must be corrected.

Let us note that, using the results of our experiments, expressed as straight lines in the coordinates \(S_z - A_w\), it is very easy to calculate the power required for milling without having a special nomogram for this purpose. To do this, for the selected value of \(S_z\) and \(t\), one must find \(A_w\) on the corresponding graph and substitute it into the following simple expression:

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

We obtain the power in horsepower.

The construction of a nomogram for the graphical solution of equations 7, 10, and 11 likewise presents no difficulty. Such a nomogram, for the case of work on cast iron \(H_{\mathrm{br}}=150\), is given in Fig. 37; moreover, by using it, one can determine the power expenditure for all three cases of operation.

of any face mill, which we analyzed above. The method of using it is explained in Fig. 37.

In concluding the experimental part of the work, we consider it possible to note that the experiments we conducted with such different diameters,

Fig. 37.

such as \(D = 39\ \text{mm}\) and \(D = 144.5\ \text{mm}\), with such different numbers of teeth as \(Z = 6\) and \(Z = 18\), with broad ranges of feeds, depths of milling, and different ratios of milling width to cutter diameter, confirmed the correctness of the derived dependencies of power expenditure on the cutting factors in the operation of any face mill. In addition, it was shown quite convincingly that the mechanical transfer of coefficients and exponents from the field of turning to the field of milling, as was done by Prof. Reznikov, does not provide the necessary accuracy and leads to inadmissible errors in determining the power, the mean circumferential force, and the mean torque.

For calculating the power in face milling, we consider it possible to recommend equations 7, 10, and 11, as verified by a precisely formulated and extensively conducted experiment. The values of the exponents and coefficients are given in Table 7.

III. Nonuniformity in face milling.

In face milling, as a tooth passes under the chip, the circumferential force on each tooth of the cutter undergoes a gradual change according to the relation

\[ P = \lambda \cdot S_z^{k+1} \cdot t^{\,l+1} \cdot \sin^{k+1}\psi, \tag{2} \]

which is the result of the gradual change in chip thickness \(S_e\) and of the specific cutting pressure, which is in turn a function of the variable, in this case the chip thickness.

As a result of this, the total circumferential force and the torque on the cutter undergo a more or less considerable periodic oscillation, passing through all their values over \(\frac{1}{Z}\) part of a revolution of the cutter.

Using the power equations 7, 10, and 11 proposed by us, it is not difficult to determine the circumferential force and the torque on the cutter, since

\[ P_{\mathrm{av}}=\frac{4500N_e}{v}\ \text{kg} \]

and

\[ M_{\mathrm{av}}=716.2\frac{N_e}{n}\ \text{kgm}, \]

but in this way we obtain the average circumferential force and the average torque without taking account of their periodic oscillations.

It was indicated above, and illustrated by Fig. 5, that for the case of milling with \(b=D\), the variation of the total circumferential force, and consequently also of the torque, is very small, and for any instantaneous position of the cutter teeth the total circumferential force and the torque for the given instantaneous position differ only slightly from their average values calculated from the power equation.

In the case, however, where we have face milling with \(b<D\), the instantaneous values of the forces and moments may differ noticeably from their average values. Passing through all its values from minimum to maximum, and then again to minimum, \(z\cdot n\) times per minute, the circumferential force on the cutter, at the moments of its maximum value, may exceed several times the average value calculated from the power equation. As a result, for rational operation of the milling machine it is no less important to know the limits of variation of the total circumferential force and torque than their average values, since otherwise it is impossible to take account of the actual working load on the machine parts. In addition, a considerable periodic oscillation of the forces may cause vibration of the entire machine and thereby set a limit to increasing productivity, even when its power is not being fully utilized.

A special investigation of the phenomenon of nonuniformity in face milling, carried out by us, shows that it is possible to reduce to a minimum the amplitude of oscillation of the instantaneous circumferential force, thereby bringing its maximum and minimum values closer to its average value, making the operation of the machine calmer and, by virtue of this, obtaining the possibility of increasing productivity.

In the further exposition we shall deal only with the maximum instantaneous total circumferential force, leaving its minimum value out of consideration.

The ratio of the maximum total circumferential force \((\Sigma P_{\mathrm{circ}})_{\max}\) to the average circumferential force \(P_{\mathrm{av}}\) will be called the coefficient of nonuniformity \(\psi\) for the given regime:

\[ \psi=\frac{(\Sigma P_{\mathrm{circ}})_{\max}}{P_{\mathrm{av}}}=\frac{M_{\max}}{M_{\mathrm{av}}}. \]

If, for the selected regime, along with \(P_{\mathrm{av}}\), which is easily obtained from the value of \(N_e\), the value \(\psi\) is also known, we can simply determine \((\Sigma P_{\mathrm{circ}})_{\max}\) and \(M_{\max}\), and we shall have everything necessary for determining the actual working loads of the machine parts.

By expression 2

\[ \Sigma P_{\mathrm{circ}}=\lambda \cdot s_z^{k+1}\cdot t^{l+1}\cdot \Sigma \sin^{k+1}\psi . \]

In addition, in face milling with \(b \leq D\)

\[ P_{\mathrm{av}}=\frac{4500N_e}{v} = \left[1-\left(1-\frac{b}{D}\right)^{\frac{k+2}{2}}\right]\cdot \frac{2^{-\frac{k}{2}}\lambda}{\pi\cdot n^{k+1}}\cdot t^{l+1}\cdot s^{k+1}\cdot z^{-k}, \]

or

\[ P_{\mathrm{av}} = \frac{2^{-\frac{k}{2}}\lambda}{\pi}\cdot t^{l+1}\cdot s_z^{k+1}\cdot z\cdot \left[1-\left(1-\frac{b}{D}\right)^{\frac{k+2}{2}}\right]. \]

Hence

\[ \mu= \frac{\lambda\cdot s_z^{k+1}\cdot t^{l+1}\left(\Sigma \sin^{k+1}\psi\right)_{\max}} {\dfrac{2^{-\frac{k}{2}}\lambda}{\pi}\cdot s_z^{k+1}\cdot t^{l+1}\cdot z\cdot \left[1-\left(1-\dfrac{b}{D}\right)^{\frac{k+2}{2}}\right]}. \]

Finally, we obtain the expression for the coefficient of nonuniformity \(\mu\):

\[ \mu= \frac{\pi\left(\Sigma \sin^{k+1}\psi\right)_{\max}} {2^{-\frac{k}{2}}z\cdot \left[1-\left(1-\dfrac{b}{D}\right)^{\frac{k+2}{2}}\right]} \tag{14} \]

Thus the coefficient of nonuniformity \(\mu\) is a function of \(\dfrac{b}{D}\) and \(z\). These two quantities (for \(k=\mathrm{const}\)) completely determine the value of \(\mu\), since \(\left(\Sigma \sin^{k+1}\psi\right)_{\max}\) is also a function of these two quantities.

For any ratio \(\dfrac{b}{D}\), we shall have the maximum sum of sines (and hence the maximum total circumferential force) in the case when, over the given width \(b\), we place under the chip the greatest possible number of teeth, arranging them symmetrically with respect to the axis of the workpiece. Such a position is shown in Fig. 38 for a twelve-tooth cutter (\(z=12\)) with \(b=b_1\). Having calculated for this case

Fig. 38.

\[ \left(\Sigma \sin^{k+1}\psi\right)_{\max} = \sin^{k+1}\psi_1+\sin^{k+1}\psi_2+\sin^{k+1}\psi_3+\sin^{k+1}\psi_4 \]

and substituting all quantities into expression 14, we obtain the coefficient of non-

uniformity coefficient \(\mu\), which will show how many times the maximum total circumferential force will be greater than the mean circumferential force for the given milling case.

If, for this case (\(z=12\)), we begin gradually to reduce the milling width \(b_1\), then the denominator of the fraction in expression 14 will gradually decrease, while the numerator will remain constant until the milling width becomes equal to \(b_2\). In this case, as the width changes from \(b_1\) to \(b_2\), the nonuniformity coefficient \(\mu\) will gradually increase. Under the chip there will always be 4 teeth [at the instant \((\Sigma P)_{\max}\)].

Fig. 39.

If the milling width is reduced still further, then, as soon as it becomes less than \(b_2\), only 3 teeth will be able to fit under the chip; the position corresponding to \((\Sigma \sin^{k+1}\psi)_{\max}\) will correspond to Fig. 39, and in the numerator of the fraction in expression 14 one must substitute

\[ (\Sigma \sin^{k+1}\psi)_{\max} = \sin^{k+1}\psi_{\mathrm I} + \sin^{k+1}\psi_{\mathrm{II}} + \sin^{k+1}\psi_{\mathrm{III}}. \]

At the instant when the milling width (as it is reduced) passes through the value \(b_2\), the numerator of expression 14, which had remained constant, immediately decreases, since

\[ \sin^{k+1}\psi_{\mathrm I} + \sin^{k+1}\psi_{\mathrm{II}} + \sin^{k+1}\psi_{\mathrm{III}} < \sin^{k+1}\psi_1 + \sin^{k+1}\psi_2 + \sin^{k+1}\psi_3 + \sin^{k+1}\psi_4 \]

and the coefficient \(\mu\), which had been gradually increasing, at this instant abruptly lowers its value. With further reduction of the milling width within the limits between \(b_2\) and \(b_3\), we shall again have a gradual decrease of the denominator with a constant numerator, i.e., a gradual increase of \(\mu\). When the milling width reaches the value \(b_3\), we shall again have an abrupt decrease of \(\mu\), and so on.

In Fig. 40 the process of variation of the nonuniformity coefficient \(\mu\) with the change in the ratio \(\dfrac{b}{D}\) is presented for a cutter with \(z=10\). In Fig. 41 the same is presented for a cutter with \(z=30\).

It is not difficult to note that jumps of the nonuniformity coefficient \(\mu\) occur at \(\dfrac{b}{D}\) satisfying the condition:

\[ \frac{b}{D} = \sin\left(\frac{m}{2}\cdot\frac{360^\circ}{z}\right), \]

where \(m\) is the greatest number of cutter teeth that fit under the chip for the given ratio \(\dfrac{b}{D}\);

From Figure 40 one may note that the values of \(\mu\) can be fairly large even at \(z = 10\); thus, at \(\frac{b}{D} = 0.311\), \(\mu = 2\); this means that the maximum total tangential force and the maximum torque exceed the mean values of these quantities by a factor of two. At \(\frac{b}{D} = 0.59\), \(\mu = 1.45\). The fluctuations of the load on the milling cutter, i.e., the ratio of the maximum

Fig. 40.

values of the forces and moments to their minimum values, will be still larger and, approximately, will be equal to twice the value of \(\mu\).

It is also evident from Fig. 40 that if, by changing the cutter diameter at a given milling width, we change the ratio \(\frac{b}{D}\) from \(\frac{b}{D} = 0.311\) to \(\frac{b}{D} = 0.30\), then by this simple measure we reduce the value of \(\mu\) from \(\mu = 2\) to \(\mu = 1.08\), i.e., we practically obtain an entirely uniform load on the cutter, owing to which it is undoubtedly possible to increase productivity by increasing the cutter feed.

Fig. 41.

If in the first case, at \(\frac{b}{D} = 0.311\), the maximum total tangential force on the cutter exceeds the minimum approximately by a factor of 4 \((2\mu)\),

which will undoubtedly cause unsteady operation of the machine at high cutting speeds, then at \(\frac{b}{D}=0.30\) we shall have a practically constant milling-cutter load in magnitude.

In considering Fig. 40 the question arises: to what, in practice, will the coefficient \(\mu\) be equal for values of \(\frac{b}{D}\) associated with a stepwise change in \(\mu\)? Thus, at \(\frac{b}{D}=0.31\), according to Fig. 40 we have two values: the lower \(\mu=1.05\) and the upper value \(\mu=2.02\).

At \(\frac{b}{D}=0.09\), according to Fig. 40, we have two values of \(\mu\): 1.45 and 1.03. Thus, for each ratio \(\frac{b}{D}\) at which a jump in \(\mu\) occurs, we shall have two values of the coefficient of nonuniformity. The question is: which of these two values of \(\mu\) will we actually have during operation?

To answer this question it is necessary to turn to the actual cutting process, with all the phenomena accompanying it.

In Fig. 42 the case \(z=10\), \(\frac{b}{D}=0.59\) is shown.

Fig. 42

In analyzing the question posed, it is not difficult to arrive at the conclusion that the larger value of \(\mu\) will occur in the case in which, at \(\frac{b}{D}=0.59\) (Fig. 42), the chip pressure acts simultaneously on three teeth. According to Fig. 42, this would appear to be the case, since at the moment when tooth 1 is at point \(A\), tooth 3 will already be at point \(B\). In reality, however (and this is confirmed by experimental verification), the chip removed by tooth 1 separates from the workpiece somewhat earlier than the tooth reaches point \(A\); that is, tooth 1 loses its load before tooth 3 reaches point \(B\), enters under the chip, and is loaded by it. That is, the load can be simultaneously on only two teeth, and in practice we shall have the smaller value of the coefficient of nonuniformity \(\mu\). And in all cases in which the ratio \(\frac{b}{D}\) is associated with a jump in \(\mu\), in practice we shall have in operation...

will be the smaller of the two values of \(\mu\), and therefore the ratios \(\frac{b}{D}\) associated with jumps in \(\mu\), i.e., satisfying the condition

\[ \frac{b}{D}=\sin\left(\frac{m}{2}\cdot\frac{360^\circ}{z}\right) \]

will be the most favorable from the point of view of nonuniformity of the cutting process.

From the analysis of equation 14, and also from comparison of Figs. 40 and 41 presented by us, one may conclude that increasing the number of teeth of the milling cutter leads to a decrease in the values of \(\mu\).

From all that has been set forth it should be clear how important it is to know the magnitude of the coefficient of nonuniformity when operating a milling machine, and how necessary it is to have an idea of the laws governing the change in \(\mu\) when selecting or designing a face milling cutter for one operation or another, since by a certain, often insignificant, change in the ratio \(\frac{b}{D}\) or in \(z\), a substantial effect can be achieved. It is therefore necessary to have the possibility of simply finding the value of \(\mu\).

Although equation 14 makes it possible to calculate \(\mu\) exactly, it is inconvenient to use, since each calculation takes much time. The graphs presented in Figs. 40 and 41 are considerably more convenient, but for each number of teeth of the milling cutter it would be necessary to have its own graph.

An accuracy quite sufficient for practice can be obtained from the proposed graph (see Fig. 43), on which, in the coordinates \(\frac{b}{D}\) and \(z\), the lines \(\mu=\mathrm{const}\) are plotted. Here, for any values of \(\frac{b}{D}\) and \(z\), one can simply determine the value of \(\mu\) with an accuracy of up to 0.1–0.2. Some portions of the graph seem unclear at first glance; thus, for example, for \(z=3\) at \(\frac{b}{D}=0.85\) we do not have a value of \(\mu\) on the graph. But we see that for \(z=3\) at \(\frac{b}{D}=0.78\) and at \(\frac{b}{D}=0.9\), \(\mu=1.3\).

Moreover, the lines \(\mu=1.5\), \(\mu=7\), and \(\mu=1.1\) do not reach \(z=3\). From this it may be concluded that for \(z=3\), at \(\frac{b}{D}=0.85\), the value of \(\mu\) will lie between 1.3 and 1.5. Likewise, for \(z=6\) and \(\frac{b}{D}=0.9\), it may be concluded that \(\mu\) will have a value between 1.1 and 1.3.

From the graph it is seen that for small numbers of teeth \(z\), the values of \(\mu\) can be very large, especially at small ratios \(\frac{b}{D}\). In the region of large \(z\) and \(\frac{b}{D}\), on the contrary, we are dealing with coefficients of nonuniformity more or less close to unity.

For \(z \geqslant 20\) and \(\frac{b}{D}\geqslant 0.75\), we shall always have \(\mu<1.1\).

In Fig. 44 a graph is given of the optimal values of \(\frac{b}{D}\), which give, at

for the selected \(z\), the minimal value of \(\mu\), i.e., those satisfying the dependence

\[ \frac{b}{D}=\sin \left(\frac{m}{2}\cdot\frac{360^\circ}{z}\right). \]

From this graph, drawing a horizontal line through the selected \(z\), at the intersections of this horizontal with the curves we obtain the optimal values of \(\dfrac{b}{D}\).

One should not think that in this case we shall have \(\mu=\mathrm{const}\). For \(z=\mathrm{const}\) the coefficient of nonuniformity will be the smaller, the larger the ratio \(\dfrac{b}{D}\). Thus, for \(z=10\), according to Fig. 44, we have 5 optimal values of \(\dfrac{b}{D}\):

0.31; 0.588; 0.811; 0.95; 1.0.

According to Fig. 40, the values of \(\mu\) in this case will be: 1.05; 1.03; 1.02; 1.01; 1.03.

A complete picture of the relation between \(z\), \(\dfrac{b}{D}\), and \(\mu\) is given by Figs. 45 and 46, in which, in spatial coordinates \(z\)—\(\dfrac{b}{D}\)—\(\mu\), the dependence of the coefficient of nonuniformity \(\mu\) on \(z\) and \(\dfrac{b}{D}\) is represented.

For plotting the graphs in Figs. 40, 41, 43, 44, 45, and 46, we carried out all calculations taking, for the quantity \(k\) entering into the exponents of equation 14, the mean value \(k=0.25\).

It follows from Table 7 that, in face milling, the value of this quantity depends on the material being machined and varies within the limits 0.2–0.315.

Thus, for each material being machined, the indicated graphs ought to be constructed according to its own value of \(k\). The error in constructing the graphs according to the mean value \((k=0.25)\) is practically quite imperceptible, but this frees us from the necessity of constructing graphs for different materials being machined.

In the present work we have not touched at all upon the question of nonuniformity in the case of peripheral cutting. This question will be analyzed by us together with the phenomenon of nonuniformity in the operation of a cylindrical cutter, with which it has much in common.

In conclusion to the present work, we give an example showing what effect can be achieved when the coefficient of nonuniformity \(\mu\) is taken into account in face milling.

Let it be given:

spindle power \(4\mathrm{HP}\);

maximum torque on the spindle, according to the strength of the spindle drive,

\(M_{\max}=90\ \text{kgm};\)

machining of cast iron \(H_{\mathrm{br}}=150;\)

\[ b=120\ \text{mm}; \]

\[ t=10\ \text{mm}; \]

\[ D=200\ \text{mm}; \]

\[ z=10; \]

\[ \frac{b}{D}=\frac{120}{200}=0.6 \]

For \(z=10\) and \(\dfrac{b}{D}=0.6\), \(\mu=1.41\).

Fig. 45.

Fig. 46.

If we take \(n = 32\) rpm, in order to obtain a cutting speed \(V \simeq 20\) m/min, then with feed \(S_1 = 110\) mm/min we obtain the average torque on the spindle

\[ M_{\mathrm{av}} = 63.5\ \text{kgm}. \]

Taking into account the coefficient of nonuniformity, we shall have

\[ M_{\max} = \mu \cdot M_{\mathrm{av}} = 1.41 \cdot 63.5 = 90\ \text{kgm}. \]

Thus, at \(n = 32\) rpm and under all the other adopted conditions, in terms of the strength of the spindle drive, which allows \(M_{\max} \leq 90\) kgm, it is impossible to carry out a feed greater than \(S_1 = 110\) mm/min.

If in this same case we take the cutter diameter \(D = 218\) mm and thereby obtain the ratio

\[ \frac{b}{D} = \frac{120}{218} = 0.55, \]

then the coefficient of nonuniformity for this case will be \(\mu = 1.1\).

Applying, with all other conditions constant, the feed \(S_2 = 164\) mm/min, we shall have the average torque

\[ M_{\mathrm{av}} = 82\ \text{kgm}. \]

Taking into account the coefficient of nonuniformity,

\[ M_{\max} = \mu \cdot M_{\mathrm{av}} = 1.1 \cdot 82 = 90.2\ \text{kgm}. \]

In other words, by simply changing the diameter of the cutter, while having on the machine spindle the same maximum torque, we increase productivity by 49%, since

\[ \frac{S_2}{S_1} = \frac{164}{110} = 1.49. \]

Laboratory of Metal Cutting of the Tomsk Industrial Institute

June 1936

LITERATURE

to which references are made in the article:

1. Reznikov.—Theory of Metal Cutting.

2. Ryvkin.—Determination of power in machining planes with face mills.—Orgainformatsiya. 1934, No. 6.

3. ASME Transactions 1934.

4. Boston and Kraus.—The Elements of Milling. ASME Transactions vol. 54, 1932; ASME Transactions vol. 56, 1934.

5. Rozenberg and Sudnishnikov.—Theory of the operation of a cylindrical milling cutter with a spiral tooth. Vestnik metallopromyshlennosti 1933, No. 4.

6. Rozenberg.—Operation of a cylindrical milling cutter. Orgainformatsiya 1935, No. 12.

7. Salomon.—Zur Theorie des Fräsvorganges. Z. d. VDI, 1928.

8. Sludsky.—On the milling integral. Stanki i instrument. 1934, No. 6.

9. Reznikov.—Theory of Metal Cutting, pp. 210 and 249.

10. Bahlecke.—Richtwerte für das Fräsen mit Walzenfräsern. Maschinenbau. 1930. S—437.

11. Orgainformatsiya. 1929, No. 8.

12. Boston and Oxford.—Amer. Mach. Lond. Bd. 71. 1929—30. Maschinenbau. 1933, No. 15/16. S. 395. Schnittkräfte beim Hobeln und Bohren.