A. N. Eremin.

On the Issue of Chip Thickness on the Tooth of a Cylindrical Milling Cutter

The question of chip thickness on the tooth of a cylindrical helical milling cutter may at present be considered definitively resolved analytically by Docents Rosenberg and Sudnishnikov [1] and confirmed graphico-analytically by the author of the proposed article [2].

In the cited sources it is set forth sufficiently clearly that the chip thickness on the tooth of a cylindrical helical milling cutter has the same mathematical expression as for a cutter with a straight tooth,

\[ \delta_e = S_z \sin \psi \tag{1} \]

and does not depend at all on the magnitude of the helix angle, \(\varepsilon\).

Here: \(S_z\) is the feed per tooth in millimeters,

\(\psi\) is the variable angle determining the position of the given point of the tooth at the given moment.

Expression (1) was the first to refute the incorrect expression for chip thickness on the tooth of a helical cutter (2), which had become established in the technical literature and was proposed by Prof. Savvin [3] in 1925–26:

\[ \delta_e = S_z \sin \psi \cdot \sin \varepsilon . \tag{2} \]

The error made by Savvin was repeated by many authoritative researchers and theorists, among whom we include Prof. Krivoukhov [4], Prof. Besprozvanny [5], Docent Isaev [6], and Prof. Reznikov [7]. They, like Savvin, introduced the quantity \(\sin \varepsilon\) into the expression for chip thickness on the tooth of a helical cutter, evidently without paying serious attention to the correctness of the derivation of \(\delta_e\) (2). One would expect that, after the analytical [1] and graphico-analytical proof [2] of Savvin’s error, there could no longer be any discussion of the influence of the helix angle \(\varepsilon\) on the chip thickness \(\delta_e\). It is enough to have a clear understanding of the process of chip formation on the cutter and to be able to understand the elements of mathematics and drafting in order to be convinced of the incorrectness of expression \(\delta_e\) (2).

We have no doubt that the above-mentioned followers of Savvin—Krivoukhov, Besprozvanny, Isaev, and Reznikov—have already recognized his error and will not delay in abandoning it in the near future.

There is no need to prove how important it is to have the correct expression for the actual chip thickness on the helical tooth of a cutter. From the theory of milling it is known that chip thickness is the initial quantity not only in determining the chip cross-section \(q\), but also in determining the mean cutting force \(P\), the torque \(M\), and the power required at the cutter \(N_e\). Thus, admitting an error in the expression for \(\delta_e\) inevitably leads to erroneous calculations of the indicated quantities \(q\), \(P\), \(M\), and \(N_e\). Comparing expressions (1) and (2) for chip thickness at a given moment, it is easy to see that, according to Savvin, \(\delta_e\) (2) is always obtained as less than the actual value (1), since for cutters with any helix angle \(\varepsilon < 90^\circ\), \(\sin \varepsilon < 1\) always. Consequently, when using expression (2) according to Savvin, we obtain the smaller the value of \(\delta_e\), the smaller \(q\), \(P\), \(M\), and \(N_e\),

the smaller the helix lead angle $\varepsilon$ of the cutter. How great the influence of Savvin’s error is on the indicated quantities can be shown by an example. Suppose we have helical cutters whose lead angle is:

\[ \begin{array}{ll} \varepsilon = 90^\circ; & \sin \varepsilon = 1.000 \\ \varepsilon = 60^\circ; & \sin \varepsilon = 0.865 \\ \varepsilon = 45^\circ; & \sin \varepsilon = 0.707. \end{array} \]

It is evident from this that for a cutter with angle $\varepsilon = 90^\circ$ (a cutter with straight teeth), the reduction of $\delta_e$ (and also of $q, P, M$ and $N_e$) according to Savvin is equal to 0%, while for a cutter with angle $\varepsilon = 45^\circ$ this reduction reaches 30% in comparison with the actual values. Such a magnitude of error, of course, cannot be disregarded.

The example given also reveals to us the reason why, according to Krivoukhov [4], Isaev [6], and other followers of Savvin, the required power at the cutter turns out to be the smaller, the smaller the helix lead angle $\varepsilon$. It is clear that this conclusion is not only false, since it is based on Savvin’s incorrect expression for $\delta_e$, but it also contradicts the conclusions of the scientifically substantiated theory of the operation of a cutter with a helical tooth, developed by Rozenberg and Sudnyashnikov [1] and confirmed experimentally by Rozenberg [8] at the present time.

Despite the absence of any scientific grounds or proofs, in recent years separate works have again appeared in print repeating the false expression for chip thickness (2) on the tooth of a helical cutter. For example, in 1935 an article by Danilenko [9] appeared, in which the author attempts to prove that the expression for $\delta_e$ (2), derived by Isaev [6] (according to Savvin), is correct. Moreover, Danilenko gives in his article a whole series of such definitions (for example, what chip thickness is) and conclusions (for example, on the form of the chip and its thickness along the blade) that characterize the author’s knowledge in the field of milling theory plainly not from a positive side.

In addition to Danilenko’s article, in 1936 a book by A. M. Vulf [10], docent of the Leningrad Industrial Institute, appeared in print. In the milling section of this book, recommended by the author as a textbook for students of higher technical educational institutions, we can also find (on p. 362) an expression for chip thickness on the helical tooth of a cutter corresponding to the above-mentioned expression (2) according to Savvin (instead of $\sin \varepsilon$ of the helix lead angle, Vulf substituted $\cos \varepsilon$ of the angle of inclination). It must be said that Vulf evidently did not attach much importance to how correct the mathematical expression $\delta_e$ (2) is, since, in determining the dependence of the cutting force and the power consumption at the cutter on the magnitude of the helix angle, he gives data opposite to the data of Savvin and his followers (Krivoukhov, Isaev, and others).

The appearance in print of new works not only repeating Savvin’s error but even distorting the very concept of chip thickness (Danilenko [9], p. 46) is the reason that prompted us to give the generally accepted definition of what should be considered chip thickness on the tooth of a cutter.

Let us remind Comrade Danilenko (and other doubters) that chip thickness is customarily taken to be the size of the chip measured along the line of intersection of the front face of the tooth with a plane perpendicular to the cutting edge of the tooth (the blade) (Fig. 1a). This applies when the cutter tooth has no undercut angle; in the case of an undercut cutter tooth, the chip thickness is measured from the cutting edge along the radius line, in a plane perpendicular to the edge of the tooth.

These definitions are sufficiently concrete and can hardly give rise to misunderstanding regarding the direction of measurement of the quantity $\delta_e$, as occurs in the definition proposed by Danilenko [9] (p. 46).

Let us examine one more fundamental error by Danilenko. He, like some followers of Savvin’s error, states: “It should be considered proven that the thickness of the chip removed by one tooth during the operation of a cylindrical milling cutter with a helical tooth can never be equal to the projection of the feed per tooth.”

In confirmation, Danilenko gives the usual expression for the chip thickness on the tooth of a milling cutter with a straight tooth, according to equation (1), for \(\psi=\varphi\)

\[ \delta = S_z \cos(90-\varphi)=S_z \sin\varphi. \]

He then says: “...but since for a helical tooth \(\delta\) cannot be considered in only one plane, we introduce a development (Fig. 2) with the cutting edge represented as a straight line inclined at an angle \(\varepsilon\). The true chip thickness here will be expressed in the form of a perpendicular to the inclined straight line, along the line \(NN\), and will be equal to \(\delta_c\).”

According to this drawing (Fig. 2), incidentally borrowed from Isaev’s book [6] (p. 214), Danilenko obtained the final value of the chip thickness on the tooth of a helical milling cutter from the equation:

\[ \delta_c=\delta\sin\varepsilon=S_z\sin\varphi\sin\varepsilon, \]

that is, the same incorrect equation (2) proposed by Savvin. This coincidence is quite natural, since Isaev’s proof is a repetition of Savvin’s error, as we have said above.

From the cited excerpts and proofs from Danilenko’s article [9], one may judge that he (and, perhaps, also Wolf [10] and others) has no conception of the actual chip thickness on the tooth of a helical milling cutter.

In addition to the proofs available on this question (Rosenberg and Sudnishnikov [1] and the author of the proposed article [2]), which are evidently not sufficiently convincing and clear for Danilenko and others, we give below yet another method of proof, with a more detailed exposition and illustration by axonometric drawings.

...which will considerably facilitate spatial representation for those who are in doubt.

To preserve the logical sequence, we shall begin the proof by considering the chip thickness on a cutter with a straight tooth without an undercut angle. For simplicity, let us assume that the cutter has one tooth. In Fig. 1a, the instant of chip removal by one tooth of the indicated cutter is shown schematically.

Notation:

\(S_z\) — feed per tooth, in mm
\(t\) — depth of milling, in mm
\(D\) — cutter diameter, in mm
\(\varphi\) — contact angle
\(\psi\) — variable angle determining the position of the given point of the tooth at the given instant
\(l\) — length of the contact arc
\(\delta_e\) — chip thickness at the given instant.

From the geometry of the drawing it is evident that the chip thickness—\(\delta_e\)—is a segment measured along the line of intersection of the front face of the tooth with a plane perpendicular to the cutting edge of the tooth. Moreover, it may be noted that the chip thickness \(\delta_e\), as the tooth advances along the contact arc in the direction of rotation of the cutter, is, first, a variable quantity,

\[ \begin{aligned} &\text{at } \psi = 0; \qquad &&\delta_e = 0,\\ &\text{at } \psi = \varphi; \qquad &&\delta_{\max} = S_z \sin \varphi, \end{aligned} \tag{3} \]

and, second, \(\delta_e\) and the front face of the non-undercut tooth are always positioned normally to the tangent line drawn to the contact arc through the point of contact of the tooth edge \(f\). This condition is explained by the fact that the segment \(\delta_e\) (and, consequently, also the front face of the non-undercut tooth) lies on the radius by which the tooth edge describes the contact arc along a curve that is practically taken to be a circle¹.

If now, in accordance with the given instantaneous position of the cutter tooth (Fig. 1a), we unroll the contact arc into a straight horizontal line, we shall see the following picture (Fig. 1b).

In this drawing, along the length of the arc \(l\), the straightened chip depression is schematically shown in the form of the triangle \(nam\)², and the position of the tooth under the chip is indicated. Both the front face of the non-undercut tooth and the chip thickness (the segment \(ff'\)) are arranged perpendicular to the unrolled arc, and this perpendicularity will be preserved throughout the entire time that the tooth moves along the contact arc.

The peculiarity of a straight-tooth cutter consists in the fact that each of its teeth, arranged parallel to the axis of the cutter, enters into operation simultaneously at all points along the length of the edge, and, after removing the chip, simultaneously leaves the chip. As a result, the chip thickness, at each given instantaneous position of the tooth on the contact arc (at a definite \(\psi\)), has the same dimension along the length of the tooth edge.

Consequently, the chip section \(q\) on the straight tooth of a cylindrical cutter at each given instant is, first, the product of the chip thickness \(\delta_e\) and the milling width \(b\),

\[ q = \delta_e \cdot b = b \cdot S_z \sin \psi \tag{4} \]

and, secondly, it will always be located perpendicular to the arc of contact, as is the chip thickness \(\delta_e\).

In the developed view of the arc of contact it will have the following form (Fig. 1в). The line \(nn'\) shows the line of entry of the tooth under the chip \((\psi = 0)\), and \(mm'\) the line of exit of the tooth from under the chip \((\psi = \varphi)\). The plane \(nn'mm'\), bounded by the entry and exit lines, represents the contact surface. The given instantaneous position of the tooth blade, corresponding to the angle \(\psi\) (in Fig. 1a), is indicated in the plane (Fig. 1в) by the line \(KK'\).

Since the plane in which the chip section is located is in reality perpendicular to the arc of contact (Fig. 1б), this plane in the given projection of the drawing (Fig. 1в) is represented normal to the contact surface, and the transverse section of the chip is expressed by one line \(KK'\) (the heavy line). If we wish to obtain the magnitude of the chip thickness at any point along the length of the tooth, for example at point \(f\) (Fig. 1в), we must draw through this point, to the contact surface, a vertical plane \(NN\), perpendicular to the tooth blade \(KK'\), and, consequently, also to the chip section. In this case the segment equal to the chip thickness and lying along the line of intersection of the chip section with the vertical plane \(NN\) is projected onto the drawing plane at point \(f\).

Fig. 3.

Obviously, in order to see the obtained magnitude of the chip thickness, we must conventionally rotate the chip section (to the right or to the left) about the tooth blade until it coincides with the plane of the drawing (Fig. 1г). Here the line \(ff'\) is the chip thickness \(\delta_e\) at the given point of the tooth. Since the cutter is straight-toothed, \(\delta_e = ff'\) is the same over the entire length of the tooth.

The entire course of our reasoning can also be followed from Fig. 3а, б, в, where, in axonometric projection, both the position of the chip section and the procedure for determining its thickness are presented very clearly. The projections are so simple that they require no additional explanation.

Probably many readers, including comrades Danilenko and Vul'f, may ask us why we first cut the vertically positioned chip section by the plane \(NN\) (Fig. 1в) and only then rotated it until it coincided with the plane of the drawing, in order to examine the chip thickness at the given point of the tooth, when it would seem possible to do the opposite.

As a mandatory rule, we must cut the chip section in the position in which it is actually situated with respect to the contact surface, i.e., in the position perpendicular to it, and only after that are we entitled to rotate it conventionally in order to examine \(\delta_e\).

In the present case, when we have a cutter with a straight tooth, when the plane \(NN\), perpendicular to the cutting edge of the tooth, is at the same time also perpendicular to the axis of the cutter, it is of course immaterial whether we first dissect the section of the chip and then rotate it, or conversely, since the line of intersection \(ff'=\delta_c\) remains one and the same in either case.

A different picture is obtained if we wish to determine the actual chip thickness on the helical tooth of a cylindrical cutter. Here it is no longer immaterial in what position we dissect the chip section.

The secret of the error of all Savvin’s successors consists in the fact that, when determining the chip thickness on the helical tooth of a cylindrical cutter, they did not adhere to that obligatory order of operations which we indicated above, using the example of a cutter with a straight tooth. This insignificant “omission” on their part led to their obtaining not the actual value of the chip thickness, expressed mathematically by equation (1), but the false value corresponding to relation (2). In order finally to analyze the error of Savvin’s successors and to show clearly the roots of its origin, let us consider the process of chip formation on the tooth of a cylindrical helical cutter.

By analogy with the straight-toothed cutter, in Fig. 4a we have shown the moment of operation of a cylindrical helical cutter, likewise having only one tooth under the chip, without an undercut angle.

The notation adopted here has the same meanings as in the preceding case.

Fig. 4.

The specific feature of the helical cutter, in contrast to the straight-toothed one, is that its tooth enters under the chip and leaves from under the chip not simultaneously with all its points along the length of the edge, but successively, as the cutter rotates. This circumstance is easy to observe from Fig. 4a, where, in the given instantaneous position of the tooth, the point of its edge \(n\), lying in the plane of the front end of the cutter (facing us), is at the moment of entry under the chip \((\psi=0)\), whereas the point of the tooth edge \(a\), lying in the plane of the rear end of the cutter, is almost at the exit from under the chip (angle \(\psi^1\)).

Since the cutter tooth has no undercut angle, it can be seen from the drawing in Fig. 4a that the intersection of the front face of the tooth with the rear and front end planes of the cutter forms lines running along the radius, and therefore always normal to the tangent drawn to the arc of contact.

through the point of contact of the tooth blade. In Fig. 4a the points of contact are denoted by \(n\) and \(a\).

Since the end planes of the cutter are always perpendicular to its axis, we may say in general that, when the front face of the tooth (with no undercut) is intersected by a plane perpendicular to the cutter axis and passing through any point along the length of the axis, we always obtain a line normal to the tangent to the arc of contact. Thus, when the arc of contact is developed into a straight horizontal line (Fig. 4b), the front face of the tooth (over its entire length) will be perpendicular to this line, and consequently perpendicular also to the surface of contact.

When the surface of contact is represented in plan (Fig. 4c), the front face of the tooth is projected into one line with the blade line \(KK\).

How is the section of the chip situated on the tooth of a helical cutter?

Since the tooth of a helical cutter has no undercut angle, it is obvious that the section of the chip is not only situated along the blade, but also coincides with the plane of the front face of the tooth, as in a cutter with a straight tooth. Consequently, on the surface of contact in plan (Fig. 4c), the chip section \((na)\) will be perpendicular to the surface and will pass along the very same line \(KK\), representing the horizontal projection of the front face of the tooth onto the blade line.

In contrast to a cutter with a straight tooth, the chip thickness along the length of the tooth of a helical cutter is a variable quantity, as may readily be seen from Figs. 4a, b and from the axonometric drawing (Fig. 5a).

Fig. 5.

Let us note that for Comrade Danilenko this circumstance will probably seem a “revelation,” for in his view the chip thickness over the entire length of the tooth of a helical cutter at a given moment has the same magnitude (see his article [9], Fig. 5).

What is the chip thickness on the tooth of a helical cutter at a given point?

If we cut the vertically situated chip section at point \(f\) (Fig. 5c) by the plane \(AA\), perpendicular to the cutter axis, then the line of intersection, equal to the chip thickness at the given place, will be projected onto the plane of the drawing at point \(f\).

The position of point \(f\) on the tooth blade in the vertical projection of the cutter (Figs. 4a and b) corresponds to the angle of rotation \(\psi_1'\), and therefore the chip thickness \(\delta_e\) at this place is expressed by the equation

\[ \delta_e = S_z \sin \psi_1' . \]

i.e., analogous to equation (1) for a cutter with a straight tooth.

Will the chip thickness found on the helical tooth at the point \(f\) be its true value?

According to the definition rule given by us above, chip thickness is measured by a segment along the line of intersection of the front face of the undercut tooth with a plane perpendicular to the tooth edge. In accordance with this rule, through the same point \(f\) of the tooth edge (Fig. 4b) let us draw a vertical plane \(NN\), perpendicular to the cutting edge of the milling-cutter tooth.

As can be seen from the drawing, the line of intersection of the chip section with the plane \(NN\) coincides with the line of intersection of the chip section with the plane \(AA\) and is projected onto the contact surface at one and the same point \(f\). Consequently, the chip thickness at the given point of the tooth, as a result of both cases of intersection, will be one and the same magnitude. Whether we cut by a plane perpendicular to the axis of the cutter or by a plane perpendicular to the tooth edge, the chip thickness on the tooth of a cylindrical helical cutter remains one and the same and, consequently, will have the same expression (1)

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

In other words, the chip thickness on the tooth of a cylindrical helical cutter does not depend on the helix angle \(\varepsilon\), which is what was required to be proved.

If we now turn, conditionally, the vertically located chip section near the tooth edge until it coincides with the contact surface (Fig. 4d), then we shall see both the true segment of chip thickness \((ff'=\delta_e)\) and the area of its section \((naa')\), corresponding to the given instantaneous position of the milling-cutter tooth.

In addition to our proofs, we present Fig. 5: a, b, c, clearly confirming the correctness of the determination of chip thickness, its form, and its position on the helical tooth of the cutter.

Conclusion.

On the basis of the foregoing we can draw the following conclusions:

1. The chip thickness on the tooth of a cylindrical helical cutter does not depend on the helix angle \(\varepsilon\), but has the same mathematical expression as for a cutter with a straight tooth (1)

\[ \delta_e = S_z \sin \varepsilon . \]

1. According to Savvin, the chip thickness on a helical tooth is obtained as less than the true value, since the expression \(\delta_e\) (2) contains \(\sin \varepsilon < 1\). The smaller the helix angle \(\varepsilon\), the greater this reduction.

2. T. Danilenko’s article must be considered not only useless but also harmful to readers, since it contains not a single correctly stated point.

3. The book by Assoc. Prof. A. M. Vul'f, Metal Cutting, recommended by the author as a textbook for students of higher technical educational institutions, because it admits Savvin’s erroneous expression for \(\delta_e\), significantly loses its value from a methodological standpoint.

4. We believe that it is time to stop the further cluttering of the technical literature (journal and textbook) with Savvin’s false expressions for \(\delta_e\).

Tomsk, TII.
Department of Mechanical Assembly Production.
April 1936.

Literature

to which references are made in the article.

1. Rosenberg and Sudnishnikov.—Theory of the operation of a cylindrical helical milling cutter with a helical tooth. Vestnik metallopromyshl., 1936, No. 4.

2. Eremin.—Chip thickness on the tooth of a cylindrical milling cutter. Tekhnormirovanie, 1936, No. 2.

3. Sawin—Mechanical Engineering, 1926; Le Genie Civil, 1925.

4. Krivoukhov.—Metal Cutting. 1931, p. 244.

5. Besprozvannyi.—Determination of force and energy consumption in milling. Orgainformatsiia, 1930, No. 3.

6. Isaev.—Cold Working of Metals. 1932, pp. 213–214.

7. Reznikov.—Theory of Metal Cutting. 1934, p. 222.

8. Rosenberg.—Operation of a cylindrical milling cutter. Jubilee collection of TII, 1936. Orgainformatsiia, 1935, No. 12.

9. Danilenko.—Chip thickness on the tooth of a cylindrical milling cutter. Tekhnormirovanie, 1935, No. 2.

10. Wulff.—Metal Cutting. 1936, p. 362.

1. This is a generally accepted assumption owing to the practically small magnitude of the feed \(S_z\). ↩

2. In reality the line \(na\) is a sinusoidal curve, since the expression for \(\delta_e\) in (3) includes \(\sin \psi\). ↩