A. M. ROZENBERG. Engineer-Docent of the Siberian Mechanical-Machine-Building Institute

and

V. V. SUDNISHNIKOV. Technician of the “Metallist” Plant.

THEORY OF OPERATION OF A CYLINDRICAL MILLING CUTTER WITH A HELICAL TOOTH.

The operation of a milling cutter is a much more complex process than the operation of a lathe or planing tool, or of a drill. This partly explains the fact that, up to the present time, the field of milling is the least developed, both theoretically and experimentally, among the other branches of the theory of metal cutting.

Without dwelling on Fischer’s conclusions,¹ which were based on an incorrect premise (constancy of the specific cutting pressure when the chip cross-sections change), it is necessary to point out in this field the outstanding work of the German engineer Salomon, who gave an impeccable expression for the energy expenditure in the operation of a cylindrical milling cutter with a straight tooth.²

According to his conclusions, the work per one revolution is

\[ A_n=\frac{2^{K+1}}{K+2}\cdot \lambda_1\cdot b\cdot t^{\frac{K+2}{2}}\cdot S_n^{K+1}\cdot z\cdot D^{-K-\frac{K}{2}} \ \text{kg.mm}\ldots;\,* \]

Here (see Fig. 1):

\(D\)—diameter of the milling cutter in mm,

\(\lambda_1\)—a constant quantity depending on the material being machined and on the tooth profile of the milling cutter,

\(b\)—width of milling in mm,

\(t\)—depth of milling in mm,

\(S_n\)—feed in mm per 1 revolution of the milling cutter,

\(Z\)—number of teeth of the milling cutter.

Fig. 1.

*) In Eng. Salomon’s article \(A_n\) is erroneously denoted in kilogram-meters; in reality this is work in kilogram-millimeters.

Passing to the expenditure of power and denoting by \(n\) the number of revolutions of the milling cutter per minute and by \(S\) the feed in mm per minute, we obtain:

\[ N=\frac{A_n}{1000}\cdot\frac{n}{60\cdot75}\ \text{horsepower}; \]

\[ N=\frac{2^{K+1}}{K+2}\cdot\frac{n}{1000\cdot60\cdot75}\cdot \lambda_1\cdot b\cdot t^{\frac{K+2}{2}}\cdot S_n^{K+1}\cdot Z^{-K}\cdot D^{-\frac{K}{2}}; \]

since

\[ S_n=\frac{S}{n}, \]

then:

\[ N=\frac{2^{K+1}}{K+2}\cdot\frac{n^{-K}}{1000\cdot60\cdot75}\cdot \lambda_1\cdot b\cdot t^{\frac{K+2}{2}}\cdot S^{K+1}\cdot Z^{-K}\cdot D^{-\frac{K}{2}} \ldots (1) \]

The derivation is based on the variable specific cutting pressure as a function of the chip thickness at the given moment:

\[ K_S=\lambda_1\cdot S_\varepsilon^K=\lambda_1\cdot S_Z^K\cdot \operatorname{Sin}^K\psi. \]

Here:
\(K_S\)—specific cutting pressure in kg/mm\(^2\),
\(S_\varepsilon\)—chip thickness at the given moment in mm,
\(K\)—an exponent depending on the grade of the material being machined,
\(\lambda_1\)—a constant coefficient depending on the grade of the material being machined and on the cutter profile,
\(S_Z\)—feed per tooth in mm,
\(\psi\)—the angle determining the position of the tooth at the given moment.

Expression (1) gives the average power consumption. Using the reasoning of Eng. Salomon, it would not be difficult to obtain also an expression for the variable circumferential force on a milling cutter with a straight tooth.

Eng. Salomon also gave a nomogram for calculating the power required in milling\(^3\) steel with \(K_Z=50—60\) kg/mm\(^2\).

If from the nomogram we determine the equation on which it is constructed, we obtain

\[ N=\frac{2^{0.72}}{1.72}\cdot\frac{n^{0.28}}{1000\cdot60\cdot75}\cdot170\cdot b\cdot t^{0.86}\cdot S^{0.72}\cdot Z^{0.28}\cdot D^{0.14}\ldots (2) \]

We see that the equation, in its coefficients and exponents, corresponds quite exactly to the theoretically de—

to the expression introduced by him. Hence the specific cutting pressure in milling steel of medium hardness will be

\[ k_s=\frac{170}{S_e^{0.28}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

The nomogram relates all the variable quantities \((S_z, S_n, Z, t, n, D, N)\) and corresponds to an exact expression for the power.

Unfortunately, the conclusions of Eng. Salomon were made for a cutter with a straight tooth, which at the present time has almost completely gone out of use in milling practice; for the operation of a cutter with a helical tooth, however, we still do not have an irreproachable derivation comparable to Eng. Salomon’s derivation for a straight-toothed cutter. To base oneself merely on experiment, because of the complexity of the process, would be incorrect and would give no guarantee against gross errors.

Of the experimental works now available on the helical cutter, the most interesting are the works of Beskh\(^{4}\) and Bahlecke\(^{5}\), which clarify the influence of various factors on the expenditure of energy in milling. As a result of his experiments, Bahlecke gave two slide rules for calculating the required power when milling cast iron of medium hardness and steel of medium hardness.

Having determined the equations on which Bahlecke constructed his rules, we obtain:

\[ \begin{aligned} &\text{For steel:}\\ &N=\frac{n^{0.26}}{32500}\cdot b\cdot t^{0.85}\cdot s^{0.74}\cdot D^{0.26}\ \text{hp.}\\[1em] &\text{For cast iron:}\\ &N=\frac{n^{0.28}}{40500}\cdot b\cdot t^{0.82}\cdot s^{0.72}\cdot D^{0.28}\ \text{hp.} \end{aligned} \qquad\} \ldots \ldots (4) \]

We see an extremely close agreement of the exponents in the equations of Bahlecke (4) and Salomon (2), but in Bahlecke’s equations the influence of the cutter diameter on the power is different than in Salomon’s equation; in addition, the influence of the number of cutter teeth \((Z)\) is not expressed at all, which may be explained either by the specific character of the operation of a cutter with a helical tooth, or, what is more probable, by the fact that Bahlecke worked in conducting his experiments with constant \(Z=8\) and \(D=100\) mm.

It must be pointed out that the experiments of Bahlecke and Beskh could not give completely exact results, since Bahlecke, when calibrating his measuring instruments and determining the forces on the cutter, did not take into account the influence of axial and radial pressures on the harmful resistances in the machine; Beskh based his conclusions on a completely incorrect resolution of the forces acting-

... those acting on the tooth of the cutter, without taking into account the existing vertical pressure of the cutter on the workpiece and considering the resultant force to be always horizontal.

In addition to the experimental works indicated, there are the conclusions of Prof. Savvin\(^6\), which cannot give sufficiently accurate results, on the one hand because they are based on the relative constancy of the specific cutting pressure over the entire path of the cutter tooth

\[ \left\{ K_s = \frac{C}{S_{e\ max}^{\,x}} \right\}, \]

and, on the other hand, because Prof. Savvin gives a completely incorrect expression for determining the chip thickness and the chip section on the cutter tooth at a given moment; this will be explained by us in detail in the subsequent derivations.

Moreover, Savvin’s formulas do not make it possible to calculate the average expenditure of energy and power for the milling process, which is necessary for the design and calculation of milling machines, as well as for the economical use of a milling machine in its work in the shop.

In the following exposition we give the theory of operation of a cylindrical milling cutter with a helical tooth.

THEORY OF OPERATION OF A CYLINDRICAL MILLING CUTTER WITH A HELICAL TOOTH

1. General considerations.

The principal difference between the process of operation of a cutter with a helical tooth and that of one with a straight tooth consists in the fact that, at each given moment, on the cutter tooth located along a helical surface there is a chip section of variable thickness. In addition, during the operation of a helical tooth we do not have an instantaneous exit of the tooth from under the chip, as occurs with a straight tooth; as a result, with a helical tooth the chip section on the tooth, the circumferential force, and the torque, having reached a maximum, decrease gradually, which has a favorable effect on the entire milling process, freeing it from sharp jumps in the load on the cutter and from the vibrations in the machine caused by these jumps.

When a helical cutter is operating, conditions may be set such that the total chip section on all teeth simultaneously engaged in the work will, during rotation of the cutter, remain constant both in magnitude and in shape; the total circumferential force and torque will likewise be constant, and the operation will proceed entirely smoothly, without those load fluctuations and shocks that unfavorably distinguish milling from all other types of metal cutting.

If we denote by

\(\beta\)—the helix lead angle,
\(H\)—the pitch of the helix, or the lead of the tooth line, in mm,
\(b\)—the milling width, in mm,

then

\[ H=\pi \cdot D \cdot \operatorname{tg}\beta . \]

If we take the milling width to be

\[ b=c\cdot \frac{H}{Z} \qquad \ldots\ldots\ldots\ldots\ldots\ldots (5) \]

where “\(c\)” is an integer, then the total cross-section of the chip will be constant throughout the entire operation of the cutter, both in magnitude and in shape; the total circumferential force and the torque will be constant.

Figure 2 shows this condition; the milling surface is shown in developed form, as are two positions of three cutter teeth and the chip cross-section on them*).

In reality, the line of entry of the tooth under the chip, “\(mn\),” and the line of exit from under the chip, “\(pq\),” will not be arranged perpendicular to the direction of feed, but obliquely; however, this inclination, the magnitude of which for each given cutter will depend on the feed per tooth, will have no effect at all on the indicated properties of the process, provided that the relation given by equation (5) is observed, and will be so insignificant that it may be disregarded.

Indeed, from Fig. 3 we see that the segment “\(pr\)” will be traversed in the direction of feed during the time in which the cutter tooth turns through the angle “\(\alpha\)”; in this case:

Fig. 2.

*) The lines bounding the chip thickness are shown conventionally as straight lines in all the drawings.

Fig. 3.

\[ \frac{pr}{S_z \cdot Z}=\frac{\alpha}{2\pi}; \]

\[ \alpha=\frac{2b}{D \cdot \operatorname{tng}\beta}; \]

\[ \operatorname{tng}\gamma=\frac{pr}{b}=\frac{S_z \cdot Z}{\pi D \cdot \operatorname{tng}\beta}. \]

For the customary values of \(S_z\), \(D\), and \(\beta\), the angle \(\gamma\) does not exceed \(1^\circ\), i.e., it may be disregarded without detriment to accuracy.

Thus, with a definite relation between the lead of the tooth spiral \(H\) and the milling width \(b\), the machining process will proceed quite smoothly, which must be taken into account when designing cutters for work in mass production. On the one hand, on the other hand, all milling experiments intended to determine the forces acting on the cutter, the torques, and the energy consumption must be carried out exclusively at a width satisfying the indicated relation, since in this case the influence of inertial forces on the experimental readings is completely eliminated and it becomes possible, with the aid of comparatively simple measuring instruments, to obtain accurate results.

2. Chip thickness on the helical tooth of the cutter.

All subsequent derivations will be made for a cutter with an unrelieved tooth, the front face of which has a radial direction.

Let us denote:

\(S_z\)—feed per tooth,
\(\varphi\)—angle of contact,
\(\psi\)—variable angle determining the position of the given point of the tooth at the given moment,
\(S_e\)—chip thickness at the given moment at the given point of the tooth,
\(\beta\)—angle of rise of the helix (see Fig. 1).

The chip thickness on a straight tooth:

\[ S_e = S_z \cdot \operatorname{Sin}\psi; \]

\[ S_{e\ max}=S_z \cdot \operatorname{Sin}\varphi. \]

The chip thickness on a helical tooth must be measured in a plane perpendicular to the helical (screw) edge of the tooth.

Prof. Savvin gives, for a helical milling cutter:

\[ S_e = S_z \cdot \sin \psi \cdot \sin \beta \ldots \ldots \ldots \ldots (6) \]

In reality, the chip thickness on a helical tooth will be expressed by the same dependence as for a straight tooth; that is, whether we cut the helical milling cutter by a plane perpendicular to its axis (as we do with a straight-tooth cutter), or whether we cut perpendicular to the helical line of the tooth, we shall obtain on the front cutting face of the tooth, situated on the helical surface, one and the same line of intersection, along which we must measure the chip thickness.

Let us prove this.

Let us place the helical surface of the front face of the tooth in a spatial coordinate system so that the axis of the helical surface (the axis of the milling cutter) coincides with the coordinate axis \(OZ\) (Fig. 4). \(AB\) is the development of the helix.

The equation of the helical surface:

\[ \frac{y}{x} = \operatorname{tng}\frac{Z}{R \cdot \operatorname{tng}\beta}; \]

\[ R \cdot \operatorname{tng}\beta = k; \]

\[ \frac{y}{x} = \operatorname{tng}\frac{Z}{k}. \tag{I} \]

The equation of the helical line:

\[ x = R \cdot \cos \psi; \]

\[ y = R \cdot \sin \psi; \]

\[ Z = R \cdot \operatorname{tng}\beta \cdot \psi = k\psi. \]

The equation of the plane normal to the helical line at the point

\[ M(x_1, y_1, z_1): \]

\[ y_1 \cdot x - x_1 \cdot y - k(Z - Z_1) = 0 \ldots \ldots \ldots \ldots (II) \]

Fig. 4.

Let us find the intersection of the helical surface with a plane perpendicular to the \(Z\) axis (the axis of the milling cutter) and passing through the point

\[ M(x_1, y_1, z_1). \]

The equation of such a plane:

\[ Z = Z_1 \ldots \ldots \ldots \ldots \ldots (III) \]

Solving equations (I) and (III) simultaneously, we obtain:

\[ y = x \cdot \operatorname{tng}\frac{Z_1}{k}. \]

Denoting \(\operatorname{tg}\dfrac{z_1}{k}=m=\text{const.}\), we have:

\[ y=mx \ldots \tag{IV} \]

which gives a straight line intersecting the \(Z\)-axis (the axis of the cutter) and perpendicular to it.

Next let us find the intersection of the plane perpendicular to the \(Z\)-axis with the plane normal to the helical line and passing through the point \(M(x_1,y_1,z_1)\). The simultaneous solution of equations (III) and (II) gives

\[ y=\frac{y_1}{x_1}\cdot x \ldots \tag{V} \]

Fig. 5.

but the point \(M(x_1,y_1,z_1)\) lies on the helical surface, and therefore:

\[ \frac{y_1}{x_1}=\operatorname{tg}\frac{z_1}{k}=m \]

and equation (V) assumes the form

\[ y=mx, \]

which coincides with equation (IV). Thus the helical surface, the plane normal to the axis of the cutter, and the plane normal to the helical line of the tooth intersect along one straight line passing through the axis of the cutter and perpendicular to it. In other words, whether we measure the chip thickness in a plane perpendicular to the axis of the cutter or in a plane perpendicular to the helical line of the tooth, we obtain one and the same result:

\[ S_e=S_z\cdot \operatorname{Sin}\psi \ldots \tag{7} \]

Prof. Savvin, in the expression for the chip thickness (6), committed the grossest error (repeated, incidentally, by our authoritative researchers\(^7\)), which completely deprives all subsequent conclusions of his theory of the operation of the spiral cutter of value,

3. Transverse section of the chip on the tooth of a spiral cutter.

Let us denote:

\(q\)—the section of the chip on the tooth of the cutter,
\(l\)—the length of the tooth lying under the chip at the given moment,

$\psi_1$ and $\psi_2$ are the angles determining the instantaneous position of the part of the tooth located under the chip.

The differential of the chip section will be (see Fig. 5):

\[ dq = S_e \cdot dl = S_z \cdot \sin \psi \cdot dl; \]

\[ l = \frac{D}{2} \cdot \frac{\psi_2-\psi_1}{\cos \beta}; \]

\[ dl = \frac{D}{2\cos\beta}\cdot d\psi; \]

\[ dq = \frac{D}{2}\cdot \frac{S_z}{\cos\beta}\cdot \sin\psi \cdot d\psi; \]

\[ q = \frac{D}{2}\cdot \frac{S_z}{\cos\beta}\cdot \int_{\psi_1}^{\psi_2} \sin\psi \cdot d\psi. \]

Finally, we obtain the expression for the chip section at a given instant

\[ q = \frac{D}{2}\cdot \frac{S_z}{\cos\beta}\cdot (\cos\psi_1-\cos\psi_2) \qquad \ldots\ldots\ldots (8) \]

According to Prof. Savvin’s expression we have:

\[ q = \frac{D}{2}\cdot S_z \cdot \tan\beta\, (\cos\psi_1-\cos\psi_2) \qquad \ldots\ldots\ldots (9) \]

which, for angles $\beta < 90^\circ$, gives a value smaller than the actual chip section. The error is the more significant the smaller the angle $\beta$, and at $\beta = 45^\circ$ reaches 30%.

Fig. 6.

In view of the fact that, by our derivations, we give substantive criticism of Prof. Savvin’s generally accepted conclusions, which often appear in our and in foreign literature, we consider it necessary to prove in several ways the correctness of the formula derived for the chip section.

We can obtain the expression for the chip section in the following manner (Fig. 6):

The chip section at a given—

at a given instant, lying on the helical surface of the front face of the tooth within the limits of the angle \((\psi_2-\psi_1)\), is bounded on one side by the helical line of the cutting edge of the tooth, which is projected onto the end surface of the cutter as the arc of the circumference “\(CD\)”; on the other side it is bounded by the curve of intersection of the surface formed in the workpiece by the preceding tooth with the helical surface of the tooth. This curve is projected onto the end surface of the cutter as the curve “\(AB\).” Thus the area \(ABCD\) is the projection of the transverse section of the chip onto the end surface of the cutter, i.e., onto the plane perpendicular to the axis of the cutter.

In other words:

\[ q \cdot \operatorname{Cos}\beta = ABCD; \]

but:

\[ ABCD = OCD - OBA; \]

\[ OCD = \frac{1}{2} R^2(\psi_2-\psi_1). \]

Since, as we have shown earlier, the chip thickness may be measured in the plane perpendicular to the axis of the cutter, for the curve \(AB\) we shall have:

\[ \rho = R - s_z \cdot \operatorname{Sin}\psi; \]

\[ OAB = \frac{1}{2}\cdot \int_{\psi_1}^{\psi_2} \rho^2 \cdot d\psi = \frac{1}{2}\cdot \int_{\psi_1}^{\psi_2} (R - S_z \cdot \operatorname{Sin}\psi)^2\,d\psi; \]

\[ OAB = \frac{1}{2}R^2(\psi_2-\psi_1) - R \cdot s_z(\operatorname{Cos}\psi_1-\operatorname{Cos}\psi_2) + \]

\[ +\frac{1}{4}s_z^2\left(\psi_2-\psi_1-\frac{1}{2}\operatorname{Sin}2\psi_2+\frac{1}{2}\operatorname{Sin}2\psi_1\right). \]

Whence:

\[ q = \frac{R \cdot s_z}{\operatorname{Cos}\beta}(\operatorname{Cos}\psi_1-\operatorname{Cos}\psi_2) - \frac{s_z}{4\operatorname{Cos}\beta} \left(\psi_2-\psi_1-\frac{1}{2}\operatorname{Sin}2\psi_2+ \right. \]

\[ \left. +\frac{1}{2}\operatorname{Sin}2\psi_1 \right) \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (10) \]

In this case we have obtained an expression more accurate than equation (8), since in obtaining equation (8) by integrating over the working length of the tooth, we, as it were, stretched the chip section along its greatest side, inevitably deforming it and thus obtaining an error in its calculation.

[[unclear: beginning of line]] for simplicity and convenience, one may, without appreciable detriment to accuracy, discard the second term in equation (10), since even for the largest chips it amounts to no more than 1% of the first term.

Consequently, we obtain:

\[ q=\frac{R\cdot s_z}{\cos\beta}\,(\cos\psi_1-\cos\psi_2) = \frac{D}{2}\cdot\frac{s_z}{\cos\beta}\,(\cos\psi_1-\cos\psi_2), \]

which is what we had earlier.

Finally, let us take the case in which \(b=\dfrac{H}{z}\). In this case the total chip cross-section is constant throughout the entire operation and may be obtained as the quotient of the minute volume of chips divided by the cutting speed in the direction perpendicular to the front face of the tooth.

From Fig. 7 we have:

at a milling depth \(t_3\), feed \(S\) mm/min, and number of cutter revolutions \(n\) per minute,

\[ q=q_2+q_1=\frac{t_3\cdot S\cdot b}{v\cdot \sin\beta} = \frac{t_3\cdot S\cdot b}{\pi\cdot D\cdot n\cdot \sin\beta} =\text{const.} \]

At a milling depth \(t_2\), on the tooth there will be a chip cross-section

\[ q_2=\frac{t_2\cdot S\cdot b}{\pi\cdot D n\cdot \sin\beta}=\text{const.} \]

Fig. 7.

At a milling depth \(t_1\), on the tooth there will be a chip cross-section

\[ q_1=\frac{t_1\cdot S\cdot b}{\pi\cdot D\cdot n\cdot \sin\beta}=\text{const.} \]

If, on the other hand, for milling we take a width \(b_2<\dfrac{H}{z}\), then [[unclear: beginning of line]] simultaneously and at the given moment there will be

\[ q_2-q_1=\frac{S\cdot b}{\pi\cdot D\cdot n\cdot \sin\beta}\,(t_2-t_1); \]

but:

\[ t_{2}=\frac{D}{2}(1-\cos\psi_{2}); \]

\[ t_{1}=\frac{D}{2}(1-\cos\psi_{1}); \]

\[ b=\frac{H}{z}=\frac{\pi \cdot D \cdot \operatorname{tg}\beta}{z}; \]

\[ \frac{S}{n}=S_{n}=S_{z}\cdot Z; \]

upon substitution we obtain

\[ q_{z}=\frac{D}{2}\cdot\frac{S_{z}}{\cos\beta}(\cos\psi_{1}-\cos\psi_{2}). \]

We have thus obtained, by this method as well, the very same expression for the chip cross-section that we derived earlier.

Thus it may be considered proved that the chip cross-section on the tooth of a helical milling cutter is expressed by the formula

\[ q=\frac{D}{2}\cdot\frac{S_{z}}{\cos\beta}(\cos\psi_{1}-\cos\psi_{2}) \ldots \ldots \ldots (8) \]

The expression of Prof. Savvin, and the derivations of Prof. Krivoukhov and Eng. Besprozvanny\(^{7}\), suffer from the crudest error.

In the case of the simultaneous operation of \(m\) teeth, we shall have the total chip cross-section

\[ q=\frac{D}{2}\cdot\frac{S_{z}}{\cos\beta}\sum_{1}^{m}(\cos\psi_{1}-\cos\psi_{2}) \ldots \ldots \ldots (11). \]

4. Cutting pressure on the tooth of a helical milling cutter.

In the subsequent derivations we shall take as our basis the expression for the specific cutting pressure in milling according to the experiments of Salomon, Oxford and Airey:

\[ K_{s}=\lambda_{1}\cdot s_{e}^{k}. \]

Here: \(\lambda_{1}\)—a constant quantity depending on the type of material being machined and on the profile of the cutter;

\(S_{e}\)—chip thickness,

\(k\)—exponent, constant for the given material being machined.

[[unclear: beginning of line]] for the differential of the force normal to the cutter tooth:

\[ dP_N=dq\cdot k_s=dl\cdot S_e\cdot \lambda_1\cdot S_e^k; \]

\[ dP_N=\lambda_1\cdot S_z^{k+1}\cdot \operatorname{Sin}^{k+1}\psi\cdot \frac{D}{2\cdot \operatorname{Cos}\beta}\cdot d\psi. \]

\[ P_N=\frac{D}{2}\cdot \frac{\lambda_1}{\operatorname{Cos}\beta}\cdot S_z^{k+1} \int_{\psi_1}^{\psi_2}\operatorname{Sin}^{k+1}\psi\cdot d\psi. \]

The tangential force acting in the plane perpendicular to the cutter axis, i.e. in the plane of the torque, will be (see Fig. [[unclear]]):

\[ P=P_N\cdot \operatorname{Sin}\beta; \]

\[ P=\frac{D}{2}\cdot \lambda_1\cdot \operatorname{tng}\beta\cdot S_z^{k+1}\cdot \int_{\psi_1}^{\psi_2}\operatorname{Sin}^{k+1}\psi\cdot d\psi. \]

To evaluate this integral we make the substitution:

\[ \operatorname{Sin}\psi=2\sqrt{x-x^2}; \]

\[ \operatorname{Sin}^2\left(\frac{\psi}{2}\right)=x; \]

\[ d\psi=\frac{dx}{\sqrt{x-x^2}} \]

then:

\[ P=\frac{D}{2}\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \int_{\psi_1}^{\psi_2}(x-x^2)^{\frac{k}{2}}\cdot dx. \]

Expanding the integrand in a Maclaurin series, integrating, and performing the inverse substitution, we obtain:

\[ P=\frac{D}{2}\cdot \lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta\cdot 2^{k+1}\cdot \left[ \frac{2}{k+2}\cdot \operatorname{Sin}^{k+2}\left(\frac{\psi}{2}\right)- \right. \]

\[ \left. -\frac{2}{k+4}\cdot \frac{k}{2}\cdot \operatorname{Sin}^{k+4}\left(\frac{\psi}{2}\right) +\frac{2}{k+6}\cdot \frac{k}{2}\cdot \frac{k-2}{2\cdot 2}\cdot \operatorname{Sin}^{k+6}\left(\frac{\psi}{2}\right)- \right. \]

\[ \left. -\frac{2}{k+8}\cdot \frac{k}{2}\cdot \frac{k-2}{2\cdot 2}\cdot \frac{k-4}{2\cdot 3}\cdot \operatorname{Sin}^{k+8}\left(\frac{\psi}{2}\right)+\ldots \right]_{\psi_1}^{\psi_2}. \]

If, for the sake of simplicity, we discard all terms beginning with the second, then, since the series is alternating, the error obtained thereby will be less than the first discarded term. For the practically used values of the angle \(\psi\) in milling, not exceeding \(60^\circ\), and for \(k=-0.28\) (according to Salomon’s experiments), the relative error in this case will not exceed 1%.

Finally, we obtain the expression for the circumferential force on a cutter tooth

\[ P=D\cdot \lambda_1 \cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}}{k+2}\cdot \left[ \operatorname{Sin}^{k+2}\left(\frac{\psi_2}{2}\right) - \operatorname{Sin}^{k+2}\left(\frac{\psi_1}{2}\right) \right]. \tag{12} \]

Here, as before:

\(S_z\)—feed per 1 tooth in mm,

\(D\)—cutter diameter in mm,

\(\beta\)—helix angle in degrees,

\(\psi_1\) and \(\psi_2\)—angles determining the instantaneous position of the part of the tooth under the chip.

In the case where \(m\) teeth are simultaneously under the chip,

\[ \sum_{1}^{m} P = D\cdot \lambda_1 \cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}}{k+2} \sum_{1}^{m} \left[ \operatorname{Sin}^{k+2}\left(\frac{\psi_2}{2}\right) - \operatorname{Sin}^{k+2}\left(\frac{\psi_1}{2}\right) \right]. \tag{13} \]

5. Torque on the cutter.

The expression for the torque at a given instant is obtained if the force determined by equation (12) is multiplied by the radius of the cutter:

\[ M=P\cdot \frac{D}{2} = D^2 \lambda_1 \cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^k}{k+2} \left[ \operatorname{Sin}^{k+2}\left(\frac{\psi_2}{2}\right) - \operatorname{Sin}^{k+2}\left(\frac{\psi_1}{2}\right) \right]. \tag{14} \]

6. Work and power consumed by a helical cutter.

During the operation of a cutter tooth within the limits of the angle \(\varphi\), two cases may occur:

the first case, when

\[ \frac{D}{2}\varphi \lessgtr \frac{b}{\operatorname{tng}\beta} \quad \text{(see Fig. 8)} \]

Fig. 8.

Fig. 9.

and the second case, when \(\dfrac{D}{2}\varphi < \dfrac{b}{\operatorname{tng}\beta}\) (see Fig. 9).

The first case has the greatest generality, since in the second case, by dividing the entire width “\(b\)” into “\(m\)” parts, we can pass to the first case, in application to which we shall carry out the further derivations*).

Let us find the work of one tooth of the cutter during one revolution, i.e., within the limits of the angle of contact \(\varphi\). For this purpose we shall divide the entire work of the tooth within the limits of the angle \(\varphi\) into three periods.

1. The work of the tooth within the limits of the angle “\(\alpha\)”; here the working length of the tooth increases from zero to “\(l\).” The tooth enters under the chip.

2. The work of the tooth within the limits of the angle \((\varphi-\alpha)\), where the working length of the tooth remains constant, equal to “\(l\).”

3. The work of the tooth within the limits of the angle \((\varphi+\alpha)-\varphi\), where the tooth emerges from under the chip and its working length falls from “\(l\)” to zero (see Fig. 10).

The work of the tooth during one revolution will be equal to the sum of the works over these three periods, separated in Fig. 10 by dotted straight lines.

Fig. 10.

The straight lines 1, 2, 3 denote the instantaneous positions of the tooth, determined by the current angles \(\psi^I,\ \psi^{II},\ \psi^{III}\) in the three corresponding limits of work.

First period:

\[ P=\frac{D}{2}\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta\cdot \int_0^{\psi^I}\operatorname{Sin}^{k+1}\psi\,d\psi; \]

\[ P=\frac{D}{2}\cdot\lambda_1\cdot S_z^{k+1}\operatorname{tng}\beta\cdot \frac{2^{k+1}\cdot 2}{k+2}\cdot \operatorname{Sin}^{k+2}\left(\frac{\psi^I}{2}\right); \]

\[ dA_z^I=P\cdot\frac{D}{2}\cdot d\psi^I. \]

* The limiting case for the two indicated ones, when \(\dfrac{D}{2}\varphi=\dfrac{b}{\operatorname{tng}\beta}\), is also reduced to the first.

The work of the tooth during the first period will be:

\[ A_z^{\mathrm{I}} = \frac{D^2}{4}\cdot \lambda_1 \cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \int_{0}^{\alpha} \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{I}}}{2}\right)\cdot d\psi^{\mathrm{I}}. \]

Second period:

\[ P = \frac{D}{2}\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \int_{\psi^{\mathrm{II}}-\alpha}^{\psi^{\mathrm{II}}} \operatorname{Sin}^{k+1}\psi \cdot d\psi; \]

\[ P = \frac{D}{2}\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \left[ \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{II}}}{2}\right) - \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{II}}-\alpha}{2}\right) \right] \]

\[ A_z^{\mathrm{II}} = \frac{D}{4}\cdot \lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \left[ \int_{\alpha}^{\varphi} \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{II}}}{2}\right)d\psi^{\mathrm{II}} - \int_{\alpha}^{\varphi} \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{II}}-\alpha}{2}\right)d\psi^{\mathrm{II}} \right]. \]

Third period:

\[ P = \frac{D}{2}\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \int_{\psi^{\mathrm{III}}-\alpha}^{\varphi} \operatorname{Sin}^{k+1}\psi \cdot d\psi; \]

\[ P = \frac{D}{2}\cdot \lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \left[ \operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right) - \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{III}}-\alpha}{2}\right) \right]; \]

\[ A_z^{\mathrm{III}} = \frac{D^2}{4}\cdot \lambda_1\cdot S^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \left[ \int_{\varphi}^{\varphi+\alpha} \operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)d\psi^{\mathrm{III}} - \int_{\varphi}^{\varphi+\alpha} \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{III}}-\alpha}{2}\right)\cdot d\psi^{\mathrm{III}} \right]. \]

The work of the tooth during one revolution will be:

\[ A_z = \frac{D^2}{4}\cdot \lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta \cdot \frac{2^{k+1}\cdot 2}{k+2} \cdot \left[ \int_{0}^{\alpha} \operatorname{Sin}^{k+2}\left(\frac{\psi^{\mathrm{I}}}{2}\right)\cdot d\psi^{\mathrm{I}} + \right. \]

\[ +\int\limits_{\alpha}^{\varphi}\operatorname{Sin}^{k+2}\left(\frac{\psi^{II}}{2}\right)\cdot d\psi^{II} -\int\limits_{\alpha}^{\varphi}\operatorname{Sin}^{k+2}\left(\frac{\psi^{II}-\alpha}{2}\right)\cdot d\psi^{II} +\int\limits_{\varphi}^{\varphi+\alpha}\operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)\cdot d\psi^{III} - \]
\[ -\int\limits_{\varphi}^{\varphi+\alpha}\operatorname{Sin}^{k+2}\left(\frac{\psi^{III}-\alpha}{2}\right)\cdot d\psi^{III} \Bigg]. \]

But in this equation:

\[ \left[F\left(\frac{\psi^{I}}{2}\right)\right]_{0}^{\alpha} +\left[F\left(\frac{\psi^{II}}{2}\right)\right]_{\alpha}^{\varphi} -\left[F\left(\frac{\psi^{II}-\alpha}{2}\right)\right]_{\alpha}^{\varphi} -\left[F\left(\frac{\psi^{III}-\alpha}{2}\right)\right]_{\varphi}^{\varphi+\alpha} = \]
\[ =F\left(\frac{\alpha}{2}\right)-F\left(\frac{0}{2}\right) +F\left(\frac{\varphi}{2}\right)-F\left(\frac{\alpha}{2}\right) -F\left(\frac{\varphi-\alpha}{2}\right) + \]
\[ +F\left(\frac{0}{2}\right)-F\left(\frac{\varphi}{2}\right) +F\left(\frac{\varphi-\alpha}{2}\right)=0. \]

And therefore:

\[ A_z=\frac{D^2}{4}\cdot\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta\cdot \frac{2^{k+1}\cdot 2}{k+2}\cdot \int\limits_{\varphi}^{\varphi+\alpha}\operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)\cdot d\psi^{III}. \]

Here \(\operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)\) may be taken out from under the integral sign as a constant quantity (the contact angle under the given operating conditions is a constant quantity), and therefore we shall have:

\[ A_z=\frac{D^2}{4}\cdot\lambda_1\cdot S_z^{k+1}\cdot \operatorname{tng}\beta\, \frac{2^{k+1}\cdot 2}{k+2}\cdot \operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)\cdot \alpha. \]

From Fig. 10 it is seen that:

\[ \frac{D}{2}\cdot \alpha\cdot \operatorname{tng}\beta=b, \]

that is:

\[ A_z=D\cdot\lambda_1\cdot S_z^{k+1}\cdot \frac{2^{k+1}}{k+2}\cdot \operatorname{Sin}^{k+2}\left(\frac{\varphi}{2}\right)\cdot b \tag{15} \]

We can arrive at this same result in a more visual way (see Fig. 11).

To each instantaneous position of the tooth within “\(mnr\)” we can always find the unique instantaneous position of the same tooth within “\(mrsl\),” which will give, as it were, a continuation of the chip section in its form; and in exactly the same way we can find the next position of the tooth within “\(stuy\),” etc., as if further supplementing the chip section on the tooth.

Summing the section in such, as it were, conjugate instantaneous positions of the tooth, we shall obtain the conventional chip section:

\[ q_{\text{conv.}}=\frac{D}{2}\cdot\frac{S_z}{\Cos\beta}\cdot(\Cos 0^\circ-\Cos\varphi); \]

\[ q_{\text{conv.}}=\frac{D}{2}\cdot\frac{S_z}{\Cos\beta}\cdot(1-\Cos\varphi). \]

Fig. 11.

This conventional chip section will also be constant in magnitude and form.

The cutting pressure corresponding to this conventional chip section will likewise be constant and equal to:

\[ P_{\text{conv.}}=\frac{D}{2}\cdot\lambda_1\cdot S_z^{k+1}\cdot\frac{2^{k+2}\cdot 2}{k+2}\cdot \Sin^{k+2}\left(\frac{\varphi}{2}\right)\cdot\tg\beta. \]

It is not difficult to understand that the work of one tooth with the actual variable force over the angle \(\varphi\) (i.e., during 1 revolution) will be equal to the work of the conventional force “\(P_{\text{conv.}}\)” over the angle \(\alpha\), where

\[ \alpha=\frac{b}{\frac{D}{2}\cdot\tg\beta} \]

whence:

\[ A_z=P_{\text{conv.}}\cdot\frac{D}{2}\cdot\alpha = \frac{D}{2}\cdot\lambda_1\cdot S_z^{k+1}\cdot\tg\beta\cdot \frac{2^{k+1}\cdot 2}{k+2}\cdot \Sin^{k+2}\left(\frac{\varphi}{2}\right)\cdot \frac{D}{2}\cdot\alpha. \]

or:

\[ A_z=D\cdot\lambda_1\cdot S_z^{k+1}\cdot \frac{2^{k+1}}{k+2}\cdot \Sin^{k+2}\left(\frac{\varphi}{2}\right)\cdot b, \]

i.e. we obtain equation (15).

since

\[ \sin\left(\frac{\varphi}{2}\right)=\sqrt{\frac{t}{D}}, \]

where \(t\) is the depth of milling in mm,
\(D\) is the diameter of the cutter, ” ”

then:

\[ A_z=\frac{2^{k+1}}{k+2}\cdot \lambda_1\cdot b\cdot t^{\frac{k+2}{2}}\cdot S_z^{k+1}\cdot D^{-\frac{k}{2}}. \]

The work of all the teeth of the cutter in one revolution will be:

\[ A_n=z\cdot A_z. \]

In addition:

\[ S_z=\frac{S_n}{z}. \]

On substitution, we obtain the equation for the work of the cutter in one revolution:

\[ A_n=\frac{2^{k+1}}{k+2}\lambda_1\cdot b\cdot t^{\frac{k+2}{2}}\cdot S_n^{k+1}\cdot z^{-k}\cdot D^{-\frac{k}{2}}. \]

Passing to the expenditure of power, we obtain, in horsepower:

\[ N=\frac{2^{k+1}}{k+2}\cdot \frac{n}{1000\cdot 60\cdot 75}\cdot \lambda_1\cdot b\cdot t^{\frac{k+2}{2}}\cdot S_n^{k+1}\cdot z^{-k}\cdot D^{-\frac{k}{2}} \]

or, since:

\[ S_n=\frac{S}{n} \]

where \(S\) is the feed in millimeters per minute, we shall have:

\[ N=\frac{2^{k+1}}{k+2}\cdot \frac{n^{-k}}{1000\cdot 60\cdot 75}\cdot \lambda_1\cdot b\cdot t^{\frac{k+2}{2}}\cdot S^{k+1}\cdot z^{-k}\cdot D^{-\frac{k}{2}}\ldots \tag{17} \]

Thus, taking into account the process of operation of a cutter with a spiral (helical) tooth, we have obtained an expression for the power expenditure in which the influence of all the factors of the milling operation is reflected:

\(n\)—number of revolutions of the cutter per minute,
\(b\)—width of milling in mm,
\(t\)—depth ” ”
\(S\)—feed in mm per minute,
\(D\)—diameter of the cutter in mm,

\(\lambda_1\)—a quality factor of the material being machined and of the tooth profile.

We see that the power consumed by the cutter is completely independent of the helix angle “\(\beta\).”

The results of Beckh’s experiments indicate an increase in energy expenditure with an increase in the helix angle; the experiments of Orga-Metal\({}^{6}\) assert the exact opposite; Bahlecke’s experiments indicate the absence of any influence of the helix angle on power consumption.

In our opinion, as we have already indicated, finding the dependence for the energy expenditure in milling by a purely experimental method, without the corresponding theoretical conclusions giving the general relationship and the influence of individual operating factors on power consumption, because of the complexity of the process, would not provide a guarantee against gross errors, from which the experimental works mentioned also suffer.

We further propose to undertake a series of experiments, as a result of which it will be possible to determine the values of the coefficients and exponents in equation (17), and also to verify the correctness of the expression for the specific cutting pressure in milling \((k_s=\lambda_1\cdot S_e^k)\), which we have made the basis of our conclusions.

If we compare the equation (17) obtained by us for the power expenditure during operation of a cutter with a helical tooth with equation (1), which Salomon gave for a straight-tooth cutter, then we shall see that we have obtained the same equation, which gives equality of the work of both cutters and indicates the absence of any influence of the helix angle “\(\beta\)” on the power consumption in milling.

Assuming that Salomon’s work gave correct exponents and coefficients in equation (2), one may use this equation and the nomogram constructed from it for the case of operation of a cutter with a helical tooth.

In equation (2) Salomon gives:

\[ k=-0.28 \]

i.e., “\(k\)” is a negative quantity.

Let us denote

\[ -k=x \]

then we shall have a more convenient form of the equation

\[ N=\frac{\lambda_1}{2^{x-1}\cdot(2-x)\cdot1000\cdot4500}\cdot n^x\cdot b\cdot t^{1-\frac{x}{2}}\cdot S^{1-x}\cdot z^x\cdot D^{\frac{x}{2}}. \tag{18} \]

Since “\(x\)” is a positive fractional number, it is not difficult to conclude that, in order to increase productivity, it is advantageous to increase first of all the feed “\(S\),” and then the depth “\(t\).”

At the same time, the power consumption grows more slowly than the productivity; i.e., the specific power consumption decreases.

On the contrary, increasing the number of revolutions of the milling cutter, \(n\), the number of teeth, \(z\), and the diameter, \(D\), without increasing productivity, increases the specific power consumption.

The conclusion we have drawn regarding the equality of the work of a straight-toothed milling cutter and a milling cutter with a helical tooth in no way contradicts the modern tendency to introduce into operation cutters with a small helix angle (40°–50°), owing to the proposition we clarified earlier concerning the steadier operating conditions of a cutter with a helical tooth.

Conclusion.

On the basis of the theory we have developed for the operation of a milling cutter with a helical tooth, the following conclusion may be drawn:

1. The power consumption in the operation of a milling cutter does not depend on the angle of rise of the tooth helix.

2. The results of the experiments of Beckh, Bahlecke, and Orga-Metall suffer from undoubted errors hidden in the experiment.

3. Prof. Savvin’s theory, constructed on an incorrectly derived expression for the chip section, withstands no criticism and cannot be applied in resolving questions connected with the operation of a helical milling cutter.

4. The chip section on the helical tooth of a milling cutter is determined by equation (8).

5. The tangential force on the helical tooth of a milling cutter is determined by equation (12).

6. The power required in the operation of a milling cutter with a helical tooth is determined by equation (18).

7. For a milling cutter with a helical tooth it is possible to set such conditions under which the total, over all teeth simultaneously participating in the work, section and tangential force will remain constant, and the work will take place under conditions entirely and exceptionally favorable for the tool and the machine.

Tomsk.

Laboratory for Metal Cutting of the Siberian
Mechanical-Engineering Institute.

June 1932.

Literature cited in the article:

1. Fischer. Die Werkzeugmaschine. S—16, 1900.
2. Salomon. Zur Theorie des Fräsvorganges. Z. d. V. D. J. 1928. No. 15.
3. Salomon. Loewe-Notizew, Bd. 13 (1928)
   — Khaimovich and Berman. Methods of graphical calculations in mechanical engineering and technology.
4. Beckh. Maschinenbau. 1926 No. 11.
5. Bahleke. Maschinenbau. 1930. No. 13.
6. Sawin. Mechanical Engineering. 1926
   Genie Civil. 1925. Page 127.
7. Sawin. Mechanical Engineering. 1926
   Genie Civil. 1925.
   — Krivoukhov. Metal cutting, 1931, pp. 244—262
   — Besprozvannyi. Orga-Information, 1930, No. 3. Determination of forces and energy expenditure in milling.
8. Orga-Information. 1929, No. 8. The influence of cutter design on energy consumption.