Proceedings of the Tomsk Technological Institute of Emperor Nicholas II

Vol. 18. 1910. No. 2.

I

T. I. Tikhonov.

On the Coefficient of Metal Cutting.

(With 2 plates of drawings.)

1–48.

On the Coefficient of Metal Cutting.

T. I. Tikhonov.

Formation of the Chip and Its Form.

The phenomena of cutting were first studied in detail by Professor Time and were set forth by him in 1870 in his work “The Resistance of Metals and Wood to Cutting.” Professor Time considers the following cases:

1) The removal of chips from ductile metals (iron, steel, lead). At first the cutter is pressed into the mass of the metal in the direction of its motion and the mass is compressed. The compression extends, as experiment has shown, not over the entire layer being removed, $chgf$ (Fig. 1, Plate IV), but only over some part of it, bounded by the angle $acm$. But as soon as the pressure on the area of the cutter $cm$ exceeds the resistance to shearing of the metal along $ac$, the chip element $acmf$ is displaced in the direction $ac$, and the cutter begins to compress the next layer (Fig. 2). The element of chip that has formed, owing to the compression of the metal, turns slightly about the point $c$, and thus a curved chip is obtained, smooth on the surface adjacent to the cutter and serrated on the other side of the cutter. Professor Time called such a chip a shearing chip. In the formation of such a chip (Fig. 3), the following characteristic quantities are sharply distinguished: the angle between the direction of motion and the front face of the cutter, $\angle \alpha$—the cutting angle; the angle between the machined surface and the rear face of the cutter, $\angle \gamma$—the angle of inclination; and the angle between the front face of the cutter and the plane of shearing, $\angle \beta$—the shearing angle, or angle of action.

From experiments it was found that the sum of the angles $\alpha+\beta$ for various metals, when $\angle \alpha$ is varied, changes very little; thus, for cast iron,

$\alpha=$	$45^\circ$	$55^\circ$	$65^\circ$	$75^\circ$	$90^\circ$
$\beta=$	$100^\circ$	$90^\circ$	$85^\circ$	$75^\circ$	$65^\circ$
$\alpha+\beta=$	$145^\circ$	$145^\circ$	$150^\circ$	$150^\circ$	$155^\circ$

Hence we see that, when \(\angle \alpha\) is changed from \(45^\circ\) to \(90^\circ\), the position of the shear plane has changed almost not at all.

As for the curvature of the chip, the experiments showed that the radius of curvature is proportional to the thickness of the layer being removed and inversely proportional to the angle \(\alpha\).

If we compare the length of the layer being removed and the length of the chip obtained, it turns out that the chip is considerably shorter; in the formation of the chip, shrinkage of the metal has occurred. Let the length of the layer being removed be denoted by \(L=\sum l\), and the length of the chip by \(L'=\sum l'\); then the coefficient of shrinkage is \(\varphi=\dfrac{\sum l'}{\sum l}\); but from Drawing 3 we have that

\[ \frac{l'}{l}=\frac{\sin \beta'}{\sin \beta}, \]

and therefore

\[ \varphi=\frac{\sin \beta'}{\sin \beta}. \]

If we take the extreme limits for the cutting angle \(\alpha\) encountered in practice, then for

\[ \alpha=45^\circ,\quad \varphi=\frac{\sin(180^\circ-145^\circ)}{\sin 100^\circ}=0.6, \]

\[ \alpha=75^\circ,\quad \varphi=\frac{\sin(180^\circ-150^\circ)}{\sin 75^\circ}=0.5, \]

\[ \alpha=90^\circ,\quad \varphi=\frac{\sin(180^\circ-155^\circ)}{\sin 90^\circ}=0.42. \]

Indeed, direct observations have shown that the coefficient of shrinkage for iron, steel, and bronze fluctuates within these limits; if there is any deviation, it is explained by a change in \(\angle \beta'\).

Turning to Drawing 3 and recalling the process of chip formation, we note that the element of the compressed surface of the chip \(cm\) is in a definite dependence on the element of the sheared surface \(ac\); everything depends on the resistance of the metal to compression and shearing; in general one may admit that \(cm=ac\,\xi\), and consequently the area of the chip is \(\omega=\xi(ac)^2\); but

\[ ac=\frac{\text{thickness of the chip}}{\sin \beta'}=\frac{l}{\sin \beta'} \]

(see Drawing 3), and therefore

\[ \omega=\xi\,\frac{l^2}{\sin^2 \beta'}. \]

As for the appearance of the chip itself, it depends strongly on the cutting angle; at a small cutting angle the displacement of the chip elements

is small, the element will have the form of a trapezoid, and the chip is obtained with a smoother structure (a continuous chip); with a larger cutting angle the displacement of the elements relative to one another is considerable, the bond between the elements is weak, and the appearance of the chip elements themselves becomes triangular. In both cases the surface of the chip adjacent to the cutter has a shiny appearance, and the surface of the workpiece being machined has no especially noticeable roughnesses.

2) Chips of brittle metals.

(Hard bronze, cast iron).

At small cutting angles and with a considerable thickness of the layer being removed, \(l > 3\ \mathrm{m/m}\), the cutter at first penetrates into the metal under the same conditions as when cutting ductile metals, but after some penetration of the cutter the element does not fold over, but breaks off, whereby the chip element receives an entirely different form (Fig. 4). The part of the metal lying below the point of the cutter is torn out, and therefore the machined surface of the workpiece becomes uneven.

Only sometimes, when the metal is very good, do the chip elements not break off completely, but remain connected with one another and form a whole chip (Fig. 5). This latter chip, in contrast to shear chips, has an uneven surface adjacent to the cutter and a smooth concave surface, and is called a fracture chip. Evidently, the fracture chip gives no shrinkage. At cutting angles from \(55^\circ\) to \(65^\circ\), and with a thick layer being removed, only fracture chips are obtained; with thin cutting layers, a mixed chip is sometimes obtained as well. If, however, the cutting angle is changed from \(75^\circ\) to \(90^\circ\), then with thin cutting layers exclusively shear chips are obtained. Consequently, when machining brittle metals, in order to obtain a smooth surface, it is necessary to take a cutter with a large cutting angle, from \(75^\circ\) to \(90^\circ\), and to remove a thin layer. In machining ductile metals, the cutting angle of the cutter has no particular influence on the fineness of the finish of the workpiece. The form of the cutting edge of the cutter has a great influence on the fineness of machining of a metallic surface. Since the thickness of the chip with cutters having a straight cutting edge is constant, and the pressure on the cutter reaches its maximum value periodically, it is evident that the greatest magnitude of pressure on the cutter for the same cross-section of chip will be observed simultaneously, from which the cutter will experience the greatest vibrations. If, however, the cutting edges of the cutter are rounded, then the pressure on the cutter at different places

the chip will fluctuate differently; for one section of the chip along the cutting face of the tool the pressure will be maximal, for another—minimal, i.e., part of the pressure will be mutually compensated, and as a result a tool with a rounded cutting edge will work more quietly, with less vibration, whereby the machined surface obtains a smoother finish.

Resistances in the cutting of metals.

For practice it is very important to determine the magnitude of the resistance which the tool must overcome under various working conditions, i.e., with different material, chip thickness, cutting angle, and speed of motion of the tool itself. Very many have occupied themselves with the elucidation of this question, and one of the first was Cocquil. Cocquil’s experiments were carried out on a lathe; holes were drilled by them, while the object being worked rotated, and the drill itself was held motionless by means of a lever. From the magnitude of the load on the lever they judged the resistance being overcome. Similar experiments on drilling were carried out by the officer of the French artillery, Clarinval, with the only difference that the work was measured by a spring dynamometer. More detailed experiments were carried out by Joessel in 1864 at the French state works in Indret.

The purpose of the investigation was to determine the most advantageous form of the tool. The work expended in removing the chip was determined as the difference between the works measured by means of a dynamometer for the working and idle running of the machine; consequently it was assumed that the loss to harmful resistances in the machine was equal in both cases. There is no doubt that with such a method of determination the magnitude of the work necessary for removing the chip will come out greater than the actual one, for to the latter there is added yet some share of the harmful resistances of the machine. Such a shortcoming is inherent in all experiments where only a driving dynamometer was used. Joessel’s experiments in their time had enormous significance and to this day references to them are encountered, although comparison of them with experiments of later times has shown that they cannot be treated with complete confidence, for facts are encountered that completely contradict the latest experimental data. The experiments of Professor Time *) were carried out in 1868–69 at the Lugansk works with the aim of determining the resistance of metals and wood to cutting. The work was carried out on

) Time. Resistance of metals and wood to cutting*, 1870.

on a planing machine. The necessary force was measured rather simply: on the end of the shaft of the machine, on which the pulleys sit, a freely moving lever was put on, engaging by means of a pawl with a ratchet wheel keyed on the same shaft; during the tests the belt was thrown off the pulley and a weight was hung at the end of the lever; by gradually increasing the weight, the moment was reached when the table began to move and, consequently, the formation of the chip began. Knowing the weight, the length of the lever, and the gear transmission, one can calculate the force required to set the table in motion. The pressure on the tool required to remove the chip will be less than the calculated value, for the harmful resistances of the machine are included here. Professor Time corrected the results obtained by introducing an efficiency coefficient of the machine equal to 0.60; but the efficiency coefficient of a machine varies greatly during operation, and therefore the final results obtained for the measurement of the pressure on the tool cannot be exact. In 1873 Hartig’s book, “Versuche ueber Leitung und Arbeits—Verbrauch der Werkzeugmaschinen,” was published, in which the results of his tests on various machine tools by means of a dynamometer were set forth. The amount of work required to remove a known weight of metal chip was determined. Then, in 1892, very thorough experiments with the same aim were carried out in Kharkov by Professor Zvorykin). Professor Zvorykin took for the experiment an ordinary planing machine with a constant speed of working motion. The tool was fastened to one arm of an elbow lever, the other arm of which rested against the piston of a hydraulic cylinder. An indicator was connected with the cylinder, which gave not only the magnitude of the pressure on the tool, but also the fluctuation of these pressures during the cutting process. The harmful resistances of the machine here could not exert any influence on the results of the measurements; the influence of friction in the dynamometric apparatus itself was also excluded, since the apparatus was previously checked on a special machine for testing materials; for a series of definite pressures on a bar fastened in place of the tool, the corresponding compression of the spring was noted; this made it possible to verify with great accuracy what actual pressures on the tool correspond to the various positions of the pencil on the indicator diagram. In the same year there appeared a report on the experiments of Gausner*), who worked in Austria. In his method of op-

) “Work and force required for the separation of metallic chips.” Moscow, 1893. Technical Collection and Industrial Herald* for 1893.

) Mittheilungen des technologischen Gewerbe-Maschinen in Wien, 1893, S. 117.

the division of the resistance of the chip; the influence of the harmful resistances of the machine was completely eliminated. The planing machine which he used received its motion from a drive, and the cutting speed was not constant. On the table of the machine there was placed a special frame, in whose guides parallel jaws moved, with the piece of material to be worked clamped in them. From the pressure of the tool the jaws moved and compressed a spring placed between the jaws and a projection of the frame. The compression of the spring was recorded by a diagram apparatus. In determining the resistance of the chip it was necessary to introduce a correction only for the friction of the jaws along the guides of the frame. The cross-sections of the chip itself were extremely insignificant (1.57 q mm maximum).

In 1896, in the Zeitschrift des Oesterreichischen Ingenieure und Architecten Vereins *) there appeared a description of the experiments of Sellergren, professor in Stockholm. For his experiments he used a lathe and a simple dynamometric device. An unequal-armed lever, rotating about a horizontal axis, was connected with the tool-rest of the machine; on the short arm of the lever the cutting tool was fixed, the pressure upon which was balanced by a spring acting on the long arm of the lever; from the compression of the spring the force required for removing the chip was calculated. The experiments are especially interesting in that the cutting speed was varied within very wide limits. Later, in 1906–7, very extensive experimental investigations on the question of metal cutting were published, carried out over the course of 25 years at various factories in America by Taylor. Taylor set himself the aim of establishing the dependence of the economy of metal working by cutting on the various factors entering into this working, and thereby making it possible, before beginning any work on a machine, to choose in advance the tool (cutting tool), the amount of feed, the depth of cut, and the cutting speed necessary for the minimum expenditure of energy and time, which directly affects the economy of production.

According to Taylor’s investigation, the final result of work on a machine is influenced by 12 variable quantities, namely the following:

a) the quality of the metal subjected to cutting;
b) the diameter of the object;
c) the depth of cut;
d) the thickness of the chip;
e) the elasticity of the object and of the tool;

*) S. 473.

f) the shape or contour of the cutting edge of the tool, the magnitude of the cutting angle and of the clearance angle;

g) the chemical composition of the steel from which the tool is made, and the method of heat treatment to which it has been subjected;

h) whether the tool is cooled with water;

i) the duration of cutting during which the tool can work without sharpening;

k) the pressure of the chip on the tool;

l) the limits within which the speed and the feed can be varied in machines;

m) the cutting and feeding force of the machine.

He divided the entire investigation into two parts:

A. Determination, by means of a series of experiments, of the principal laws of cutting.

B. Finding a mathematical expression for these laws, one that should be so simple that it could be applied in ordinary practice.

From his investigations Taylor drew the following most important conclusions and deductions:

1) a tool with a rounded point, under given conditions, can run at a much higher cutting speed and, consequently, perform much more work than the old tool with a sharp point;

2) a large feed, accompanied, when necessary, by a slow cutting speed, produces more work than a small feed with a higher cutting speed;

3) a strong jet of water, directed straight at the chip at the place where it separates from the surface being machined by the tool, makes it possible to increase the cutting speed and, consequently, the amount of work performed by 30 to 40%;

4) the thickness of the chip has a far greater influence on the speed of machining than any other of the elements of cutting. The practical result of this finding was that, in the current practice of our factories, tools with long, straight cutting edges were introduced, which allowed us to work with a heavy feed at the same cutting speed as was attained with tools having rounded cutting edges and with a small feed; thus, for a considerable part of the work, the old rule—heavy feed and low cutting speed—was replaced by a new one—heavy feed and high cutting speed.

Some of his conclusions Taylor succeeded in expressing in mathematical formulae. The formula giving the dependence between the cutting speed, the amount of feed, and the depth of cut when working steel with normal cutters having rounded points has the following form:

\[ V= \frac{ c\left(1-\frac{8}{7(32r)}\right) }{ F\left(\frac{2}{5}+\frac{\frac{2+12r}{5}}{32r}\right) \left(\frac{48}{32r}D\right) \left[ \frac{2}{5}+0.06\sqrt{32r} +\frac{0.8(32r)}{6(32r)+48D} \right] }, \]

in which the symbols denote:

\(V\)—the normal cutting speed in feet per minute,
\(F\)—the feed of the cutter in inches,
\(D\)—the depth of cut in inches,
\(r\)—the radius of rounding of the cutter point,
\(c\)—a constant depending on the hardness of the metal being worked and on the quality of the cutter, where \(C\) is the smaller the harder the metal, and the larger the weaker the cutter.

For work on cast iron, the formula derived has a similar structure.

The dependence between the cutting speed and the thickness of the chip is expressed by the formula

\[ V=\frac{1.54}{t^{\frac{2}{3}}}, \]

where \(t\) denotes the thickness of the chip in inches. The dependence between the cutting speed and the width of the chip is as follows:

\[ V=\frac{12.22}{L^{\frac{7}{32}}}, \]

where \(V\) is the cutting speed in ft./min. for a 20-minute duration of cutting, and \(L\) is the width of the chip in inches.

The last two formulae have a form convenient for logarithmic computation; thanks to this, this part of Taylor’s investigations is of the greatest importance, since the result was a slide rule giving an exact mathematical expression of the influence on cutting speed of such factors as the form of the cutter, the thickness and width of the chip, the quality of the metal worked by the cutter, the duration of cutting, and so forth.

This part of Taylor’s investigations makes it possible quickly to establish the normal time within which a given job is to be performed by a worker, and to pay workers bonuses for faster work. The advantage of introducing slide rules is much greater than that of introducing all the other improvements taken together, since it is precisely by means of these slide rules that the chief aim is attained for which Taylor’s investigations were undertaken in 1880: to take the control and management of the course of work out of the hands of a large number of workers and to transfer it entirely into the hands of the shop administration.

Finally, in 1902 and 1903 experiments of the preceding character were carried out in Manchester, England. The experiments were conducted jointly by eight of the most prominent English firms, among which were Armstrong, Whitworth & Co., Vickers and Maxim, John Brown & Co., Thomas Firth and Sons, and others; to them were joined the Manchester Association of Engineers and the Manchester Municipal Technical School. The latter was represented by Dr. J. T. Nicolson, who prepared the final report on these tests under the title “Experiments with a Lathe Tool Dynamometer.”

In 1901 a commission of the Union of German Engineers carried out a series of analogous tests, published in the Zeitschrift des Vereins Deutscher Ingenieure in the issue of September 28, 1901. In Bulletin No. 2 of the University of Illinois, in 1905, there was published the result of tests carried out by Professors Breckenridge and Henry Dirks on the question of metal cutting.

The results obtained by the above-named investigators are not in complete agreement with one another. According to the observations of some, the amount of work required to remove one kilogram of chips does not depend on the thickness of the chip itself, whereas according to the tests of others, this dependence exists; namely, it turns out that less work is expended in removing one kilogram of chips the thicker the layer being removed. In exactly the same way, the majority of observers assert that the resistance to cutting does not depend on the speed of motion of the tool, which is also scarcely probable; a priori one may suppose that the effort must be increased if a considerable deformation beyond the elastic limit is to be produced at an especially considerable speed. Such a divergence of conclusions is, of course, explained by the insufficiency of the experiments and, on the other hand, by the non-uniform conditions under which the experiments were carried out.

On the basis of the material thus obtained, attempts were made to construct a theory of cutting which would reduce the complex phenomenon of cutting to elementary forms of resistance of materials and at the same time would make it possible to establish an analytical connection between the separate processes of cutting. One of the first to appear in this field was Professor Time, who created a theory of cutting on the basis of his own observations. Then, using Jossel’s experimental data, a theory of cutting was proposed by Professor Gadolin. Soon after this, Professor Afanasyev proposed his own theory of cutting, relying almost exclusively on Time’s experiments. Finally, the most recent theory in this field is Professor Zvorykin’s theory, which, in my opinion, agrees most closely with the results of very precise and numerous experiments, both his own and those of others; I therefore consider it necessary to present his theory here, even if in a somewhat abbreviated form.

Theory of cutting. Let us imagine that the cutter, as a wedge, penetrates into the metal and compresses a certain part of the latter over the area \(ab\) (Fig. 6). Let the dimension perpendicular to the plane of the drawing be equal to unity. A force \(P\) acts on the wedge, under the action of which the cutter moves forward, but this motion is opposed by the force of compression of the metal on the area \(ab\); let this latter force be equal to \(Q\) and be directed normally to the surface \(ab\). The force \(Q\), in turn, gives rise to a frictional force \(Qf\), directed parallel to the front face of the wedge. In addition, during operation the cutter is pressed against the machined surface of the work \(ac\), producing a resistance \(R\) and the friction \(Rf\) associated with it. At equilibrium of all the forces, the projections of the latter on the coordinate axes are equal to zero, i.e.

\[ \begin{aligned} P &= Qf\cos\alpha + Q\sin\alpha + Rf,\\ R &= Q\cos\alpha - Qf\sin\alpha,\\ P &= Q\left[2f\cos\alpha + \sin\alpha\,(1-f^{2})\right] \end{aligned} \tag{1} \]

As soon as the force of pressure of the wedge on the surface \(ab\) reaches a certain magnitude, chipping of the chip will immediately occur along some surface. Chipping, of course, will take place along the plane of least resistance of the metal, and therefore the force \(R\) acting on the cutter at that moment must have the least magnitude. On the chip element \(abkd\) act two forces, \(Q\) and \(Qf\). Resolving these forces into forces acting in the plane of chipping \((ak)\) and perpendicular to it \((MN)\), we obtain (Fig. 7)

\[ \begin{aligned} A_{ak}&=Q\cos[90-(\alpha+\beta')]+Qf\sin[90-(\alpha+\beta')]\\ &=Q[\sin(\alpha+\beta')+f\cos(\alpha+\beta')],\\ B_{mn}&=Q[f\sin(\alpha+\beta')-\cos(\alpha+\beta')]. \end{aligned} \]

The force \(A_{ak}\) tends to displace the chip element in the plane \(ak\), which is resisted by the shear stress and the friction of the particles against one another, caused by the action of the force \(B_{mn}\). In equilibrium the following equality must hold:

\[ akT+B_{mn}f'=A_{ak}; \]

but it is known that

\[ ak=\frac{l\ \text{(thickness of the chip)}}{\sin\beta'}, \]

and consequently, substituting their values for the forces \(A_{ak}\) and \(B_{mn}\), we have

\[ \frac{lT}{\sin\beta'}+Q[ff'\sin(\alpha+\beta')-f'\cos(\alpha+\beta')] =Q[\sin(\alpha+\beta')+f\cos(\alpha+\beta')], \]

whence

\[ Q=\frac{lT}{\sin\beta'\{\sin(\alpha+\beta')+f\cos(\alpha+\beta')-ff'\sin(\alpha+\beta')+f'\cos(\alpha+\beta')\}} \]

\[ =\frac{lT}{[(1-ff')\sin(\alpha+\beta')+(f+f')\cos(\alpha+\beta')]\sin\beta'}. \]

Substituting the obtained value \(Q\) into formula (1), we have:

\[ P=\frac{lT\,[2f\cos(\alpha+1-t^2)\sin\alpha]} {[(1-ff')\sin(\alpha+\beta')+(f+f')\cos(\alpha+\beta')]\sin\beta'}. \tag{3} \]

The last formula contains \(\angle\beta'\), and only in the denominator; to determine the value of this angle under the condition that, at the moment of sliding of the chip, the force \(P\) have the least magnitude, it is necessary, as is evident from formula (3), that the denominator for some value of \(\angle\beta'\) have a maximum value, i.e. the first derivative of the denominator with respect to \(\beta'\) must be equal to zero.

\[ \sin\beta'(1-ff')\cos(\alpha+\beta')-(f+f')\sin\beta'\sin(\alpha+\beta')+(1-ff')\cos\beta'\sin(\alpha+\beta') \]

\[ +(f+f')\cos\beta'\cos(\alpha+\beta')=0, \]

\[ (1-ff')\sin(\alpha+2\beta')+(f+f')\cos(\alpha+2\beta')=0, \]

whence

\[ \tang(\alpha+2\beta')=-\frac{f+f'}{1-ff'}. \tag{4} \]

If the value of the coefficient of friction is replaced by the corresponding tangents of the angles of friction \(\varphi\) and \(\varphi'\), then the formula is rewritten as follows:

\[ \tan(\alpha+2\beta')=-\frac{\tan\varphi+\tan\varphi'}{1-\tan\varphi \tan\varphi'} \]

i.e.

\[ \alpha+2\beta'=180^\circ-(\varphi+\varphi'), \qquad \beta'=90^\circ-\frac{\varphi+\varphi'+\alpha}{2}. \tag{5} \]

For this value of \(\beta'\) the denominator does indeed obtain its maximum value, for its second derivative with respect to \(\beta'\), when \(f\) and \(f'\) are less than 1, will be negative. Indeed, \(\beta'=180^\circ-\alpha-\beta\) (see Fig. 7); substituting this value of the angle into the second derivative of the denominator, we obtain:
\[ (1-ff')\cos[-(\alpha+2\beta)]-(f+f')\sin[-(\alpha+2\beta)] =(1-ff')\cos(\alpha+2\beta)+(f+f')\sin(\alpha+2\beta). \]
The latter expression, when \(f\) and \(f'\) are less than unity, is less than zero, since \(\alpha+2\beta>180^\circ\) \((\beta>\beta')\). The values of the quantities \(\varphi\) and \(\varphi'\) are not known, but experimental data permit, for cast iron, taking \(\varphi=23^\circ45'\) and \(\varphi'=20^\circ15'\), which corresponds to 0.44 and 0.34.

Now, varying the angle \(\alpha\), we find from formula (5) different values:

\[ \begin{aligned} \alpha &= 45^\circ \qquad 55^\circ \qquad 65^\circ \qquad 75^\circ \qquad 90^\circ,\\ \alpha+\beta &= 134.5^\circ \qquad 139.5^\circ \qquad 144.5^\circ \qquad 149.5^\circ \qquad 157^\circ; \end{aligned} \]

according to Zvorykin’s observations:

\[ \alpha+\beta=134.5^\circ \qquad ” \qquad ” \qquad ” \qquad 153^\circ, \]

i.e. the value of \(\alpha+\beta\) very closely coincides with that computed from the formula, which indicates a very successful choice of the angles \(\varphi\) and \(\varphi'\). Now, inserting the values of \(\varphi\) and \(\varphi'\) into formula (3), we obtain:

\[ P= \frac{lT\,(0.88\cos\alpha+0.806\sin\alpha)} {\sin\left(68-\frac{\alpha}{2}\right) \left[ 0.85\sin\left(68+\frac{\alpha}{2}\right) +0.78\cos\left(68+\frac{\alpha}{2}\right) \right]}; \tag{6} \]

in general, \(P\) is some \(f(\alpha)\cdot T\), i.e.

From the last formula it is clearly seen that the pressure on the cutter \(P\) depends on the angle of the cutter \((\alpha)\): the larger this angle, the greater the force required for the formation of the chip. Especially convincing in this respect are Zvorykin’s experiments, where the cutting angles varied within wide limits, larger than can be encountered in factory—

practice. The general character of the dependence may be seen from the following table*):

Angle	45°	50°	55°	60°	65°	70°	75°	80°	85°	90°
Resistance	1	1.07	1.16	1.23	1.32	1.42	1.53	1.65	1.78	1.94

From the table it is evident that the resistance increases almost in proportion to the increase of the cutting angle. It may be assumed approximately that an increase of the cutting angle by \(1^\circ\) gives an increase in resistance of \(2\%\).

In formula (6), \(q\) and \(q'\) are so chosen that it gives the magnitude of the least force which must be applied to the cutter in order to remove a chip of thickness \(e^{m}/m\) and width \(1^{m}/m\). Here \(T\) is the coefficient of strength of the material in shearing; for cast iron \(\approx 11.2\), for bronze \(\approx 16.2\), for iron \(\approx 23.2\), and for steel \(\approx 29.2\), per sq. mm. Although from the last formula (6) it follows that it is desirable to use cutters with as small cutting angles as possible, in practice a cutter with an angle less than \(50^\circ\) is rarely taken, since a cutter with a sharper angle digs in, i.e. tends to cut into the depth of the metal, and therefore the machined surface obtained is extremely uneven. In some cases, with especially hard metals, in order to obtain a smooth surface it is necessary to take \(\angle \alpha\) not less than \(90^\circ\), although this is extremely disadvantageous.

Taylor, on the basis of his observations concerning the cutting angle of the cutter, arrived at the following conclusions:

a) For cutting cast iron and hard grades of steel with a carbon content of about \(0.45\%\), the cutter must have a cutting angle of \(68^\circ\) with a clearance angle of the cutter of \(8^\circ\).

b) For cutting soft grades of steel with a carbon content below \(0.45\%\), the cutting angle of the cutter must be \(61^\circ\) with the same magnitude of clearance angle of the cutter \((\angle \gamma)\).

c) For chilled cast iron—\(86\)—\(90^\circ\) with the same magnitude of the clearance angle of the cutter.

d) For steel with a hardness equal to the hardness of tire steel, \(\alpha = 74^\circ\).

e) In shops working chiefly soft grades of steel with a carbon content of \(0.10\)—\(0.15\%\), it is economical to keep \(\angle \alpha\) less than \(61^\circ\).

*) The table was compiled on the basis of the experiments of Prof. Zvorykin, taking the mean values of the magnitude of the resistance for rigid and ductile metals, assuming that the resistance at \(z = 45^\circ\) is equal to 1; see Prof. Gatsuk, “Mechanical Technology,” p. 20.

According to Dr. Nicolson, when choosing the cutting angle for a cutter it is always necessary to settle upon the sharper of them, but one which can nevertheless protect the cutter from chipping.

As is evident from formula (6), the clearance angle of the cutter \((\angle \gamma)\) did not enter at all into the expression \(P\); consequently, from a theoretical point of view, its magnitude has no influence on the cutting process. Nevertheless, this angle is necessary; otherwise the cutter, resting with a broad surface upon the surface being machined, will scratch it badly, and sometimes even seize it, especially if the point of the cutter becomes blunted. In practice the clearance angle usually ranges from \(3^\circ\) to \(15^\circ\). At present-day high cutting speeds, serious attention must be paid to the free removal of the chip itself, so that it is not jammed between the body of the cutter and its cutting face, and also does not bear against the holder; for this purpose the cutter is always set with a certain inclination relative to the axis of the body being machined.

According to Nicolson, the magnitude of this inclination of the cutter should preferably be about \(30^\circ\), in order to reduce the magnitude of the force of the lateral feed of the cutter; thus, according to Nicolson’s investigation, when the cutter is inclined to the axis of the object being machined by \(30^\circ\), the resistance of the lateral feed varies from \(1\) to \(10\%\), whereas with a deviation of \(15^\circ\) the resistance of the lateral feed increases to \(12\)–\(20\%\) of the total pressure on the cutter. Further, according to Taylor’s observations, the indicated inclination of the cutter to the axis of the body being machined is necessary in order to reduce the tendency of the chip to push the cutter away from the body being machined; the greater this pushing away (the trembling of the cutter), the more irregular will be the finishing of the parts. This irregularity is especially noticeable in those cases where the cutter and its holder are not particularly rigid, and where the depth of the chip changes as a result of irregularity in the shape of the object being machined. For ordinary small workshop work, the magnitude of the angle of inclination of the cutter to the object being machined, in Taylor’s opinion, is about \(8^\circ\).

The significance of speed. The question concerning the influence of speed on the magnitude of the chip’s resistance to being cut off long remained insufficiently clarified. The majority of investigators carried out experiments at very insignificant speeds, differing little from zero, and therefore did not notice any special influence of speed on cutting resistance. J. J. Jossel, however, on the basis of his experiments, came to the conclusion that the resistance of the cutter rapidly decreased with increasing speed, reached a minimum at speeds lower than those usually applied in practice, and then again proceeded ...

rapid increase of the resistance. If one takes speed as the axis of abscissas, and as the axis of ordinates the work corresponding to one and the same weight of the chips removed, then the law of variation of the work will be expressed by a curve whose form is, rather roughly, that of a parabola; the late Professor Gadolin, in whose investigations this dependence plays a very important role, also took this curve to be a parabola. Such, of course, will also be the law of variation of the forces of resistance to cutting.

The gap in the information concerning the influence of speed has recently been filled by Sällegren’s experiments, to which great importance may be attached, since his method of observation deserves full confidence, which cannot be said of Hessel’s experiments. The cutting speed in Sällegren’s work varied within considerable limits (from 11.5 m/min to 53.7 m/min). Sällegren found that an increase in speed for brass and cast iron entails a slight lowering of the resistance; within the limits of speeds encountered in practice, this decrease amounts to 3–4%. With ductile metals, for example iron, an increase of the resistance was observed, likewise insignificant—no more than in the first case, within the limits of practice, approximately up to 12%.

In 1908, under my direction, Messrs. students Uftyuzhanin and Yezdakov carried out diploma work in the mechanical workshops of the Tomsk Technological Institute to determine the influence of the speed of cutting metals on the coefficient of cutting.

Let us first consider which elements, besides the one specified, influence the speed of cutting metals, in the order of their relative importance. In order better to characterize the relative importance of these elements, Taylor gives for each of them figures representing the ratio between the lower and upper limits of speed, insofar as the influence of each element is reflected in it; these limits have to be reckoned with in everyday practice.

a) The quality of the metal being machined, i.e. its hardness or other qualities that influence speed. The ratio is equal to 1 in the case of half-hardened steel or chilled cast iron to 100 in the case of soft low-carbon steel.

b) The chemical composition of the steel from which the cutter is made and its heat treatment. The ratio is equal to 1 in the case of cutters made of self-hardening steel to 7 in the case of the best high-speed cutters.

c) The amount of feed. The thickness of the chip is measured at the time when the metal retains its original density, and not the thickness of the actual chip, the metal of which has in part already been worked. The ratio is equal to 1 in the case of a chip thickness of \(3/16''\), to \(3\frac{1}{2}\) in the case of a thickness of \(1/64''\).

d) The outline or contour of the cutting edge of the cutter is the chief factor, because this affects the thickness of the chip. The ratio is equal to 1 in the case of a cutter for cutting screw threads, to 6 in the case of a flat broad cutter.

e) Whether the cutter is abundantly supplied with a jet of water or another cooling liquid. The ratio is equal to 1 in the case of a cutter working dry, to 1.41 in the case of a cutter cooled by water.

f) The depth of cut, or half of that amount by which the diameter of the object being machined is reduced. The ratio is equal to 1 at a depth of cut of \(1/8''\).

g) The duration of cutting, i.e. the time during which the cutter works without regrinding. The ratio is equal to 1 in the case of a cutter sharpened every \(1\frac{1}{2}\) hours, to 1.297 in the case of a cutter sharpened every 20 minutes.

h) The cutting angles and the back angle of the cutter. The ratio is equal to 1 in the case of a cutting angle of \(68^\circ\), to 1.023 in the case of the same angle equal to \(61^\circ\).

i) The yielding of the object being machined and of the cutter as a result of vibration. The ratio is equal to 1 in the case of vibration of the cutter, to 1.15 in the case of using a cutter working without vibration.

In brief, this may be stated as follows:

a) Hardness of the object being machined, 1:100.

b) Chemical composition of the tool steel, 1:7.

c) Thickness of the chip, \(1:3\frac{1}{2}\).

d) Outline of the cutter, 1:6.

e) Cooling of the cutter, 1:1.41.

f) Depth of cut, 1:1.36 (\(1/2''\) and \(1/8''\)).

g) Duration of cutting, 1:1.297 (\(1\frac{1}{2}\) hours and 20 min.).

h) Cutting and clearance angles, 1:1.023 (\(68^\circ\) and \(61^\circ\)).

i) Yielding of the cutter and of the object being machined,
1 (with vibration): 1.12 (absence of vibration).

In beginning the work of determining the dependence of the coefficient of cutting on the cutting speed of metals, we set ourselves the aim of resolving two questions:

a) How does the cutting speed affect the coefficient of cutting when machining various metals?

b) What influence, in this, is exerted by the depth of cut and by different feeds?

In order to answer these questions, a series of experiments was carried out on the machining by cutting of cast iron, iron, and brass. Since in this work the only variable factor was to be the speed of cutting,

cutting, while the remaining quantities were assumed constant, choosing them on the basis of experimental data obtained at different times.

The metals for machining were taken to be more or less homogeneous; for example, the cast-iron blanks were cast specially for this purpose and were distinguished by their density and uniform structure. The iron used was welded iron from a single bar, and it could be considered as homogeneous as the rolling was satisfactory. Comparatively few experiments were made with brass, and they were limited to the machining of one blank from among those used for work in the Institute’s shops.

The work was carried out on a center lathe driven by a separate electric motor and with mechanical feed by the lead screw. The lathe was completely isolated from the others, and observations of the power absorbed were fully assured in their correctness, which was necessary, since, in connection with the main work, it was desirable at the same time to determine the efficiency coefficient of the lathe.

The tools (cutters), in chemical composition, were ordinary ones; the cutting angles for the various metals were the most advantageous. To change the speed, besides the stepped pulley and back gear, an auxiliary Polysius apparatus was available, which made it possible to vary the speed within wide limits, but which later had to be disconnected, since under heavy load it produced rather considerable slipping of the belt between the friction disks and, consequently, uneven operation.

The device for measuring the pressure on the cutter was a spring dynamometer of simple construction, shown in the appended drawing, Fig. 8, Plate IV. The dynamometer is fastened to the upper plate of the carriage and moves together with the carriage from the lead screw. Photograph No. 1 shows the lathe, photographed together with the dynamometer for the indicated tests.

Before proceeding to the experiments, this dynamometer was checked, and a new accurate scale for readings of the load was applied. During the experiments, observations were made of the load at each speed, and the number of revolutions of the workpiece being machined per minute was determined. The diameters of the workpieces being machined, before and after the passage of the cutter, were measured with an accuracy of up to 0.02 m/m. The number of revolutions per minute was measured with the aid of a revolution counter and a stopwatch, and the energy expended with the aid of a wattmeter.

The actual performance of the experiments was conducted as follows: having set the blank accurately and measured its initial diameter, they started

set the machine in motion, and after several readings of the number of revolutions and of the load on the cutter, the speed was changed, gradually passing from lower speeds to higher ones, or conversely, and all the while observing the instruments. The readings thus obtained made it possible to determine the cutting coefficient for the given speed and the cutting speed itself from the formulas:

\[ K=\frac{P}{\omega}, \qquad v=\frac{\pi d n}{60}, \]

where \(\omega\) is the cross-section of the chip being removed in square millimeters,

\[ = \frac{d_0-d_1}{2}\,l, \]

\(l\)—the feed,

\(p\)—the load on the cutter in \(klgr\);

\(n\)—the number of revolutions per minute, and

\(d\)—the mean diameter of the layer being removed.

After a whole series of such experiments and after obtaining various values for the cutting coefficient at the corresponding speeds, curves were constructed from the above formulas; for this purpose the values of all the series of readings were taken that most characteristically expressed the dependence sought.

Values obtained from the experiments, and the curves constructed from them.

A. CAST IRON.

1 Table of readings.

P	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi d n}{60}\)
65,0	90	44,50	42,14	43,32	0,2	0,236	275,4	0,203
67,5	76	—	—	—	—	—	286,0	0,171
68,5	58	—	—	—	—	—	290,3	0,131
69,5	47	—	—	—	—	—	294,5	0,106
70,5	37	—	—	—	—	—	298,7	0,083
71,5	29	—	—	—	—	—	303,0	0,065
72,0	24	—	—	—	—	—	305,1	0,054

2nd curve.

P	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi d n}{60}\)
62,0	90	44,14	39,66	40,90	0,2	124	0 248	250,0
63,5	76	—	—	—	—	—	—	256,0
64,5	58	—	—	—	—	—	—	260,1
65,5	46	—	—	—	—	—	—	264,1
66,5	36	—	—	—	—	—	—	268,1
67,5	29	—	—	—	—	—	—	273,8
68,0	24	—	—	—	—	—	—	275,8

\(P\)	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(l\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi dn}{60}\)
3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.
75,0	91	39,66	36,54	38,10	0,2	1,56	0,312	243,8	0,180
76,5	76	—	—	—	—	—	—	245,2	0,150
77,5	57	—	—	—	—	—	—	248,4	0,130
78,5	46	—	—	—	—	—	—	251,6	0,091
79,0	36	—	—	—	—	—	—	253,2	0,071
79,5	29	—	—	—	—	—	—	244,8	0,057
80,0	24	—	—	—	—	—	—	256,4	0,049

See the diagram of the coefficient of cutting—Table of Drawings IV.

2 Table of Readings

\(P\)	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(l\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi nd}{60}\)
1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.
58,0	88	29,20	27,17	28,18	0,2	1,02	0,204	284,3	0,129
58,5	75	—	—	—	—	—	—	286,8	0,110
59,0	56	—	—	—	—	—	—	289,2	0,082
59,5	46	—	—	—	—	—	—	291,7	0,068
60,0	36	—	—	—	—	—	—	294,1	0,053
61,0	28	—	—	—	—	—	—	299,0	0,041
61,5	24	—	—	—	—	—	—	301,5	0,035
2nd curve	2nd curve	2nd curve	2nd curve	2nd curve	2nd curve	2nd curve	2nd curve	2nd curve	2nd curve
55,6	87	30,54	29,20	29,87	0,4	0,67	0,268	207,1	0,135
56,0	74	—	—	—	—	—	—	209,0	0,115
56,5	55	—	—	—	—	—	—	210,8	0,085
57,0	45	—	—	—	—	—	—	212,7	0,070
57,5	36	—	—	—	—	—	—	214,4	0,056
58,0	28	—	—	—	—	—	—	216,4	0,043
58,5	24	—	—	—	—	—	—	218,3	0,037
3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.
73,5	85	30,00	30,54	31,17	0,4	1,23	0,492	194,4	0,140
74,5	70	—	—	—	—	—	—	151,4	0,116
75,5	52	—	—	—	—	—	—	153,4	0,086
76,5	43	—	—	—	—	—	—	155,5	0,081
77,5	35	—	—	—	—	—	—	157,5	0,058
78,0	28	—	—	—	—	—	—	158,7	0,046
78,5	22,5	—	—	—	—	—	—	159,5	0,037

See the diagram of the coefficient of cutting—Table IV.

B. IRON.

3rd Table of Readings.

P	n	d₀	d₁	d	l	δ	ω	$k=\dfrac{\mathrm{P}}{\omega}$	$r=\dfrac{\pi d n}{60}$
1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.
25,5	24	43,28	42,22	42,75	0,2	0,53	0,106	240,4	0,053
25,2	38	—	—	—	—	—	—	237,7	0,084
25,0	56	—	—	—	—	—	—	235,8	0,124
24,5	77	—	—	—	—	—	—	231,1	0,171
24,7	92	—	—	—	—	—	—	233,0	0,204
21,5	180	—	—	—	—	—	—	212,3	0,340
2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.
25,0	23	47,40	46,30	46,85	0,2	0,55	0,110	227,3	0,056
24,0	37	—	—	—	—	—	—	218,2	0,099
23,0	53	—	—	—	—	—	—	209,1	0,129
23,5	92	—	—	—	—	—	—	213,6	0,220
21,0	180	—	—	—	—	—	—	190,9	0.439
3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.
25,3	24	46,30	45,06	45,68	0,2	0,62	0,124	204,0	0,057
24,3	37	—	—	—	—	—	—	196,0	0,088
23,3	58	—	—	—	—	—	—	188,0	0,128
24,0	78	—	—	—	—	—	—	193,5	0,186
24,5	92	—	—	—	—	—	—	196,6	0,219
21,8	174	—	—	—	—	—	—	175,8	0,414

See diagram—table of drawings IV.

4th Table of Readings.

P	n	d₀	d₁	d	l	δ	ω	$k=\dfrac{\mathrm{P}}{\omega}$	$r=\dfrac{\pi d n}{60}$
1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.	1st curve.
49,5	24	30,82	29,88	30,35	0,4	0,188	0,188	263,3	0,038
47,5	28	—	—	—	—	—	—	252,6	0,047
46,5	36	—	—	—	—	—	—	247,3	0,057
45,5	46	—	—	—	—	—	—	242,0	0,073
45,0	57	—	—	—	—	—	—	239,3	0,090
45,0	71	—	—	—	—	—	—	239,3	0,120
46,0	90	—	—	—	—	—	—	244,7	0,142

P	$n$	$d_o$	$d_1$	$d$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi dn}{60}$
2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.
49,0	24	29,88	28,86	29,37	0,4	0,51	0,204	240,2	0,037
47,5	29	—	—	—	—	—	—	232,4	0,044
45,5	36	—	—	—	—	—	—	223,0	0,055
44,5	46	—	—	—	—	—	—	218,1	0,070
43,5	56	—	—	—	—	—	—	213,2	0,086
43,0	76	—	—	—	—	—	—	210,8	0,116
42,0	90	—	—	—	—	—	—	205,7	0,138
3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.	3rd curve.
46,5	24	25,04	24,00	24,52	0,4	0,52	0,208	223,6	0,031
45,5	29	—	—	—	—	—	—	218,8	0,037
44,0	36	—	—	—	—	—	—	211,5	0,047
42,5	46	—	—	—	—	—	—	204,3	0,059
41,5	56	—	—	—	—	—	—	199,5	0,076
40,5	75	—	—	—	—	—	—	194,7	0,096
39,5	90	—	—	—	—	—	—	190,0	0,115
40,5	17	—	—	—	—	—	—	194,7	0,223

See diagram—table, drawing IV.

5th Table of Reports

P	$n$	$d_o$	$d_1$	$d$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi dn}{60}$
1st curve	1st curve	1st curve	1st curve	1st curve	1st curve	1st curve	1st curve	1st curve	1st curve
28,0	23	20,66	19,60	20,12	0,2	0,52	0,104	269,1	0,024
27,5	29	—	—	—	—	—	—	264,4	0,030
26,5	37	—	—	—	—	—	—	254,8	0,039
26,0	47	—	—	—	—	—	—	250,0	0,049
26,5	57	—	—	—	—	—	—	254,8	0,060
26,0	76	—	—	—	—	—	—	250,0	0,080
26,0	90	—	—	—	—	—	—	250,0	0,095
25,0	174	—	—	—	—	—	—	240,4	0,183
25,0	200	—	—	—	—	—	—	240,4	0,210
2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.	2nd curve.
29,0	24	24,00	22,82	23,41	0,2	0,59	0,118	245,7	0,029
29,0	29	—	—	—	—	—	—	245,7	0,035
29,0	36	—	—	—	—	—	—	275,7	0,044
28,5	47	—	—	—	—	—	—	241,5	0,055
28,0	56	—	—	—	—	—	—	237,2	0,068
27,5	77	—	—	—	—	—	—	233,1	0,094
27,0	89	—	—	—	—	—	—	228,8	0,109
26,0	174	—	—	—	—	—	—	220,3	0,212

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$

3rd curve.

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$
47,0	24	22,82	20,64	21,73	0,2	1,09	0,218	215,6	0,027
46,0	29	—	—	—	—	—	—	211,0	0,033
45,5	37	—	—	—	—	—	—	208,7	0,042
45,0	46	—	—	—	—	—	—	206,4	0,052
44,5	57	—	—	—	—	—	—	204,1	0,064
45,0	75	—	—	—	—	—	—	206,4	0,085
44,5	86	—	—	—	—	—	—	204,1	0,101
45,5	174	—	—	—	—	—	—	208,7	0,197

See the diagram—main drawing IV.

C. BRASS.

6 tables of counts.

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$

1st curve.

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$
13,5	268	29,52	28,58	29,01	0,2	0,51	0,102	332,4	0,403
13,74	176	—	—	—	—	—	—	434,8	0,267
14,0	89	—	—	—	—	—	—	137,2	0,135
14,25	56	—	—	—	—	—	—	139,7	0,085
14,25	37	—	—	—	—	—	—	139,7	0,055
14,50	24	—	—	—	—	—	—	142,3	0,036

2nd curve.

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$
13,5	276	32,58	31,50	32,04	0,2	0,54	0,108	125,0	0,471
13,75	188	—	—	—	—	—	—	127,3	0,313
14,0	172	—	—	—	—	—	—	129,6	0,287
14,5	89	—	—	—	—	—	—	134,3	0,149
14,5	56	—	—	—	—	—	—	134,3	0,094
14,75	37	—	—	—	—	—	—	136,6	0,052
15,0	24	—	—	—	—	—	—	138,9	0,040

3rd curve.

P	$n$	$d$	$d_o$	$d_1$	$l$	$\delta$	$\omega$	$k=\dfrac{P}{\omega}$	$v=\dfrac{\pi d n}{60}$
19,0	268	31,50	29,52	30,31	0,2	0,99	0,198	98,5	0,423
19,75	176	—	—	—	—	—	—	99,7	0,272
20,00	90	—	—	—	—	—	—	101,0	0,141
20,25	57	—	—	—	—	—	—	102,3	0,088
20,5	37	—	—	—	—	—	—	103,5	0,058
20,5	24	—	—	—	—	—	—	103,2	0,038

\(P\)	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(l\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi d n}{60}\)
				4th curve.
20.0	265	28.50	26.44	27.47	0.2	1.03	0.206	97.0	0.379
20.5	176	—	—	—	—	—	—	99.5	0.152
20.75	90	—	—	—	—	—	—	100.7	0.129
21.00	57	—	—	—	—	—	—	102.0	0.082
21.25	37	—	—	—	—	—	—	103.2	0.053
21.25	24	—	—	—	—	—	—	103.2	0.034

See diagram—Table of drawings IV.

7. Table of reports

\(P\)	\(n\)	\(d_0\)	\(d_1\)	\(d\)	\(l\)	\(\delta\)	\(\omega\)	\(k=\dfrac{P}{\omega}\)	\(v=\dfrac{\pi d n}{60}\)
				1st curve.
18.0	174	26.24	25.08	25.66	0.2	0.58	0.116	155.2	0.231
18.5	90	—	—	—	—	—	—	159.5	0.120
19.0	57	—	—	—	—	—	—	163.8	0.075
19.5	37	—	—	—	—	—	—	168.1	0.049
19.5	24	—	—	—	—	—	—	168.1	0.032
				2nd curve.
18.0	174	25.08	23.86	24.47	0.2	0.58	0.122	147.5	0.221
19.0	91	—	—	—	—	—	—	155.7	0.116
19.25	76	—	—	—	—	—	—	157.8	0.097
19.5	58	—	—	—	—	—	—	160.0	0.074
19.75	37	—	—	—	—	—	—	161.8	0.047
19.75	24	—	—	—	—	—	—	161.8	0.030
				3rd curve.
26.0	174	24.86	21.00	22.88	0.2	0.98	0.196	132.7	0.209
26.5	90	—	—	—	—	—	—	135.2	0.107
26.0	76	—	—	—	—	—	—	132.7	0.090
26.0	67	—	—	—	—	—	—	132.7	0.088
26.5	37	—	—	—	—	—	—	135.2	0.044
27.2	24	—	—	—	—	—	—	137.8	0.029

See diagram—Table of drawings IV.

For the tables the following designations are adopted:

\(P\)—pressure on the cutter in \(klgr\); \(n\)—number of revolutions per minute;
\(d_0\)—diameter of the blank before turning in m/m; \(d_1\)—diameter of the blank after turning in m/m; \(d\)—mean diameter of the layer removed; \(l\)—magni-

feed rate in m/m; \(\delta\)—depth of cut in m/m; \(\omega\)—area of the removed chip in \(qmm\), \(K\)—coefficient of cutting in \(klgr/qmm\), and \(v\)—velocity at the circumference of the mean diameter of the layer being removed.

A general view of the curves expressing the dependence of the coefficient of cutting on the cutting speed makes it possible to observe that, in general, the coefficient of cutting of metals decreases with increasing speed. The curves for cast iron and brass express this decrease especially characteristically (Table IV). According to the data of the experiments with cast iron, the fall of the coefficient of cutting with an increase of speed by approximately \(0.100\) metr. is expressed by the following percentage ratio:

\[ \begin{array}{rcl} \text{Curve }1,\ \text{Table IV} & \ldots & 6.60\% \\ \text{”}\quad 2,\ \text{”}\quad \text{IV} & \ldots & 5.56\% \\ \text{”}\quad 3,\ \text{”}\quad \text{IV} & \ldots & 3.48\% \end{array} \left\} \right. \]

\[ \begin{array}{rcl} \text{”}\quad 2,\ \text{”}\quad \text{IV} & \ldots & 5.56\% \\ \text{”}\quad 3,\ \text{”}\quad \text{IV} & \ldots & 6.30\% \end{array} \left\} \right. \]

As is seen from the figures and curves cited, the fall of the coefficient of cutting is affected by the depth of cut and the amount of feed; with an increase in the depth of cut the coefficient of cutting falls more slowly, and with an increase in feed—more rapidly. This circumstance makes it possible to say that it is more advantageous to work with a larger feed than with a considerable depth of cut at one and the same speed.

The curves for iron (Table 4, curves 1, 2, 3) present a certain peculiarity in comparison with the curves of the other metals investigated. Curve 1, Table 4, at a speed of \(0.100\) metr. has a point expressing the minimum value of the coefficient of cutting, and then, with a further increase in speed, rises. Curve 3, Table 4, characterizes the same thing, but at a greater speed and greater depth of cut. In the 2nd curve of Table 4 this point is not clearly present, but there are points of inflection. The curves of Table 3 are similar to one another in general appearance, but differ from the curves of Table 4 in that they have two characteristic points each: the first shows a partial minimum at a speed of about \(0.150\) metr. per sec., which can still be taken in ordinary work, and the second—a partial maximum. From the curves of Table 5 no definite conclusions can be drawn, but in general they are similar in appearance to the curves of Tables 4 and 3.

The curves for brass (Table 6, curves 1, 2, 3, 4, and Table 7, curves 1, 2, 3) are, in character, similar to the curves for cast iron, but the fall of the coefficient of cutting with increasing speed is not so considerable.

Some of the above-cited curves, together with this, give a clear idea of the influence of the depth of cut and the amount of feed on

coefficient of cutting. Comparing the curves of Table 1, we see that, with a small increase in the depth of cut, the 2nd curve runs considerably lower than the 1st, which indicates a decrease in the coefficient of cutting; with a further increase in the depth of cut, the 3rd curve runs still lower, but the distance between the curves is no longer so considerable. Examining the curves, one may observe that, at one and the same speed and depth of cut, but with double the feed, the 3rd curve of Table 2 runs much lower than the 2nd curve of Table 1.

Attempts to find an analytical dependence between the coefficient of cutting and the cutting speed for cast iron and iron, although there were a large number of experiments for this purpose (about 400), have led to no results, and therefore one has to stop at Taylor’s opinion, which completely denies the existence of any definite law.

The complete denial of this law must, it should be supposed, nevertheless be temporary, in view of the fact that an exact accounting of the necessary data for deriving the above-mentioned dependences is not yet possible, since in determining the force on the tool the inconstant cutting ability of the cutter is of great importance. As the experiment proceeds the cutter becomes dull, and the coefficient of cutting increases on account of the decrease in the efficiency of the cutter. In working with brass, the latter phenomenon was especially clearly noticeable. The resistance of brass to cutting is less than that of other metals, and thus the cutting ability of the cutter could be considered more constant; and an attempt to cast the experiments with brass into analytical form led to the following formula:

\[ K \cdot \sqrt[16]{v} = a = const., \]

where \(a\) is the constant value found by us, equal to 213.35; \(K\) is the coefficient of cutting, and \(v\) is the cutting speed.

For comparison of the coefficients of cutting obtained from experiments and from the formula, we may compile the following table (from Table 7):

\(v\), speed	\(K\), obtained from experiment	\(K = \dfrac{a}{\sqrt[16]{v}}\)
49	168.1	168.06
76	163.8	164.12
120	159.5	159.19
231	155.2	152.45

In calculating metal-working machine tools, in order not to place the cutter under severe working conditions, the cutting speed is taken within known limits; during the work of the cutter there is a frictional force of cut-

acts upon the workpiece and the chip being separated; the work of frictional forces increases with the speed of the cutter; consequently, in order to protect the cutter from heating, and above all its edge from rapid dulling, an appropriate speed must be chosen for moving the cutter. The normal cutting speed in practice varies within the following limits:

Material	Cutting speed
Hardened cast iron	35 m/m per second
Hard steel	50 m/m per second
Cast iron	75 m/m per second
Iron	100 m/m per second
Copper alloys	124 m/m per second *)

Depending on the working conditions of the cutter, these speeds vary; for example: for milling cutters, where the cutter is not in contact with the metal all the time, but has time to cool, the cutting speed is taken much higher than the above.

For completeness I shall give Rose’s table**), which may be used when selecting speeds.

Diameter of the object being turned	Steel	Iron	Cast iron	Bronze	Copper
25 m/m and less	100	175	190	600	1750
from 25 m/m to 50 m/m	90	125	175	500	—
from 50 m/m to 75 m/m	"	"	"	"	"
50—100	"	"	150	400	"
50—125	"	"	"	"	1500
75—150	75	"	"	"	"
100—150	"	115	125	350	1100
125—300	"	"	—	—	1100
150—300	—	100	100	30	"
300—500	—	90	100	"	750

Taylor, on the basis of his numerous experiments for his standard cutters made of self-hardening steel of the following composition—vanadium 0.24%, tungsten 18.19%, chromium 5.47%, carbon 0.674%, manganese 0.11%, and silicon 0.043%—gives the following data for cutting speeds; in the tables, the upper number above denotes the smallest upper size of the cutter, and the other dimension is the vertical size of the cutter, 1–1½ times greater than the first; the cutter has a rounded cutting edge.

) See—Mechanical Technology by Professor Ratiu, pp. 34–35.
*) See—Mechanical Technology by Gavrilenko, p. 36 (Theory of Cutting).

Figs. 9, 9a, 9b, Table IV depict the shape of Taylor standard cutters for various machining cases; thus Fig. 9 shows a cutter for cutting cast iron and hard steel; Fig. 9b—for cutting soft and medium-hard steel. If the work of cutting metal is carried out with cooling of the cutter, then the cutting speeds given earlier for simple carbon cutters may be increased by 20% for steel and iron and by 10% for cast iron:

Depth of cut, mm	Feed, mm	Mild steel, cutting speed, m/min	Medium steel, cutting speed, m/min	Hard steel, cutting speed, m/min
Standard cutter 1¼″	Standard cutter 1¼″	Standard cutter 1¼″	Standard cutter 1¼″	Standard cutter 1¼″
2,38	0,4	158	78,8	36
2,38	0,79	112,5	56,8	25,3
2,38	1,58	78,5	39,3	17,8
2,38	2,38	63,7	32	14,5
3,18	0,4	137	68,5	31,1
3,18	0,79	96,5	48,2	21,9
3,18	1,59	68	34,2	15,4
3,18	2,38	55,5	27,7	12,6
3,18	3,18	47,8	23,9	10,9
4,76	0,40	113	56,4	25,6
4,76	0,79	79,2	39,6	18
4,76	1,59	55,7	27,9	12,7
4,76	2,38	45,4	22,7	10,3
4,76	3,18	39,3	19,7	8,9
4,76	4,76	[[unclear: value resembles “113”]]	16	7,3
6,35	0,40	98	49,1	22,3
6,35	0,79	69,2	34,5	15,7
6,35	1,59	48,5	24,3	11
6,35	2,38	39,6	19,8	9
6,35	3,18	34,2	17,1	7,8
6,35	4,79	27,8	13,9	6,3
9,52	0,4	80,5	40,2	18,3
9,52	0,79	56,7	28,4	12,9
9,52	1,59	39,9	20	9,1
9,52	2,38	32,6	16,3	7,3
9,52	3,18	28,1	14,1	6,3
12,7	0,40	70,1	35,1	15,9
12,7	0,79	49,4	24,7	11,2
12,7	1,59	38,8	17,3	7,9
12,7	2,38	28,2	14,1	6,4

Depth of cut, mm	Feed, mm	Mild steel, cutting speed, m/min	Medium steel, cutting speed, m/min	Hard steel, cutting speed, m/min
Standard cutter 1″	Standard cutter 1″	Standard cutter 1″	Standard cutter 1″	Standard cutter 1″
2,38	0,4	149	74,6	33,8
2,38	0,79	103	51,5	23,4
2,38	1,59	71,6	35,7	16,3
2,38	2,38	57,6	28,8	13,1
3,18	0,4	130	65,2	29,6
3,18	0,79	90	45,1	20,5
3,18	1,59	62,5	31,1	14,2
3,18	2,38	50,3	25,3	11,4
3,18	3,18	43,3	21,6	9,8
4,76	0,4	109	54,6	24,8
4,76	0,79	75,3	37,8	17,1
4,76	1,59	52,1	26,1	11,8
4,76	2,38	42,1	21	9,5
4,76	3,17	36	18	8,2
4,76	4,76	29	14,5	6,6
6,35	0,4	96	47,8	21,8
6,35	0,79	66,4	33,2	15,1
6,35	1,59	45,7	22,9	10,4
6,35	2,38	37,9	18,4	8,3
6,35	3,18	31,7	15,8	7,7
9,52	0,4	80	40,2	18,2
9,52	0,79	55,4	27,7	12,6
9,52	1,59	38,4	19,2	8,7
9,52	2,38	30,8	15,4	7
12,7	0,4	70,6	35,4	16,1
12,7	0,79	49	24,5	11,2
12,7	1,59	33,8	17	7,7

Normal cutter \(7/8''\)

Depth of cut, mm	Feed, mm	Mild steel	Medium steel	Hard steel
2,38	0,4	145	72,6	32,9
2,38	0,79	99	49,4	22,5
2,38	1,59	67,7	33,9	15,4
2,38	2,38	54	26,9	12,2
3,18	0,4	128	64,1	29,1
3,18	0,79	87,3	43,7	19,8
3,18	1,59	59,5	29,8	13,5
3,18	2,38	47,6	23,7	10,8
3,18	3,18	40,5	20,2	9,2
4,76	0,4	107	53,7	24,4
4,76	0,79	73,3	36,6	16,6
4,76	1,59	50	25	11,4
4,76	2,38	40	20	9,1
4,76	3,18	34,2	17,1	7,8
6,35	0,4	95,2	47,6	21,6
6,35	0,79	65	32,6	14,7
6,35	1,59	44,3	22,1	10,1
6,35	2,38	35,4	17,7	8,1
9,52	0,4	80,6	40,3	18,3
9,52	0,79	54,9	27,5	12,5
9,52	1,59	37,2	18,6	8,5
12,7	0,4	72,4	36	16,4
12,7	0,79	49,4	24,6	11,2

Normal cutter \(3/4''\)

Depth of cut, mm	Feed, mm	Mild steel	Medium steel	Hard steel
2,38	0,4	147	73,5	33,6
2,38	0,79	98,6	49,2	22,4
2,38	1,59	66,2	33	15
2,38	2,38	52,5	26,2	11,9
3,18	0,4	129	64,7	29,3
3,18	0,79	86,5	43,3	19,7
3,18	1,59	58	29	13,2
3,18	2,38	46,1	23	10,4
3,18	3,18	39,1	19,5	8,9
4,76	0,4	109	54,6	24,8
4,76	0,79	73,3	36,6	16,6
4,76	1,59	49,1	24,5	11,2
4,76	2,38	38,8	19,4	8,8
6,35	0,4	97,6	48,8	22,2
6,35	0,79	65,6	32,7	14,9
6,35	1,59	43,9	21,9	10
9,52	0,4	84,1	42,1	19,1
9,52	0,79	56,5	28,2	12,8

Normal cutter \(5/8''\)

Depth of cut, mm	Feed, mm	Mild steel	Medium steel	Hard steel
1,59	0,4	167	83,6	38,2
1,59	0,79	109	54,6	24,9
1,59	1,59	71,7	35,7	16,2
2,38	0,4	142	71,4	32,3
2,38	0,79	93,4	46,7	21,2
2,38	1,59	61	30,5	13,9
2,38	2,38	47,6	23,8	10,8
3,18	0,4	127	63,8	28,9
3,18	0,79	83,3	41,5	18,9
3,18	1,59	54,6	27,2	12,4
3,18	2,38	42,7	21,3	9,7

Normal cutter \(1/2''\)

Depth of cut, mm	Feed, mm	Mild steel	Medium steel	Hard steel
1,59	0,4	155	77,7	35,4
1,59	0,79	98,1	49,1	22,3
1,59	1,59	62	31,1	14,1
2,38	0,4	136	68	30,8
2,38	0,79	85,6	43	19,5
2,38	1,59	54	27	12,2
2,38	2,38	41,2	20,5	9,4
3,18	0,4	123	61,6	28
3,18	0,79	77,7	39,1	17,6
3,18	1,59	49,1	24,7	11,2

Normal cutter \(5/8''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft steel	Cutting speed, m/min: medium steel	Cutting speed, m/min: hard steel
4,76	0,4	110	55,2	25
4,76	0,79	72	36	16,4
4,76	1,59	47,3	23,6	10,7
6,35	0,4	100	50	22,7
6,35	0,79	65,6	32,6	14,9
14,52	0,4	87,2	43,7	19,8

Normal cutter \(1/2''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft steel	Cutting speed, m/min: medium steel	Cutting speed, m/min: hard steel
4,76	0,4	109	54,6	24,9
4,76	0,79	69	34,5	15,7
6,35	0,4	101	50,4	7,6

Cutting speed in machining cast-iron articles.

Normal cutter \(1\frac{1}{4}''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
2,38	0,4	73	36,5	21,3
2,38	0,79	58,2	29	17
2,38	1,59	43,3	21,6	12,6
2,38	2,38	36	18	10,5
2,38	3,18	31,4	15,7	9,2
2,38	4,76	25,9	13	7,6
3,18	0,4	65,8	33	19,2
3,18	0,79	52,5	26,3	15,3
3,18	1,59	39,1	19,5	11,3
3,18	2,38	32,6	16,3	9,5
3,18	3,18	28,5	14,2	8,3
3,18	4,76	23,4	11,7	6,8
4,76	0,4	57	28,5	16,7
4,76	0,79	45,4	22,8	13,3
4,76	1,59	33,8	16,9	10
4,76	2,38	28,2	14,1	8,2
4,76	3,18	22,3	11,1	6,5
4,76	4,76	20,2	10,1	5,9
6,35	0,4	51,2	25,7	14,9
6,35	0,79	40,8	20,5	11,9
6,35	1,59	30,4	15,2	8,9
6,35	2,38	25,4	12,7	7,4
6,35	3,18	22,3	11,1	6,5
6,35	4,76	18,2	9	5,3

Normal cutter \(1''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
2,38	0,4	68,9	34,5	20,1
2,38	0,79	54	26,9	15,7
2,38	1,59	39,6	19,8	11,5
2,38	2,38	32,6	16,3	9,5
2,38	3,18	28,3	14,1	8,3
2,38	4,76	23	11,5	6,7
3,18	0,4	62,5	31,1	18,2
3,18	0,79	48,8	26	14,3
3,18	1,59	36	17,9	10,5
3,18	2,38	29,6	14,8	7,5
3,18	3,18	25,6	12,8	7,1
3,18	4,76	20,9	10,5	6,1
4,76	0,4	55,2	27,6	16,1
4,76	0,79	43,3	21,6	12,6
4,76	1,59	31,7	15,8	9,2
4,76	2,38	26,2	13,1	7,6
4,76	3,18	22,6	11,3	6,6
4,76	4,76	18,5	9,2	5,4
6,35	0,4	50,3	25,1	14,7
6,35	0,79	39,4	19,6	11,4
6,35	1,59	24,8	14,4	8,4
6,35	2,38	23,7	11,9	9,9
6,35	3,18	20,6	10,3	6
6,35	4,76	16,8	8,4	4,9

Normal cutter \(1\frac{1}{4}"\)

Depth of cut in mm	Feed in mm	Cutting speed in meters per minute: soft cast iron	Cutting speed in meters per minute: medium cast iron	Cutting speed in meters per minute: hard cast iron
9,52	0,4	43,8	21,9	12,8
9,52	0,79	35,1	17,5	10,2
9,52	1,59	25,9	13	7,6
9,52	2,38	22,6	10,8	6,3
9,52	3,18	18,9	9,5	5,5
9,52	4,76	15,5	7,8	4,5
12,7	0,4	40	17	11,7
12,7	0,79	32	16	9,3
12,7	1,59	23,6	11,8	6,9
12,7	2,38	19,7	9,9	5,8
12,7	3,18	17,3	8,6	5
12,7	4,76	14,2	7,1	4,1
19,05	0,4	34,2	17,1	10
19,05	0,79	27,2	13,6	7,9
19,05	1,59	20,2	10,1	5,8
19,05	2,38	16,8	8,4	4,9
19,05	3,18	14,7	7,4	4,3
19,05	4,76	12,1	6	3,5

Normal cutter \(1"\)

Depth of cut in mm	Feed in mm	Cutting speed in meters per minute: soft cast iron	Cutting speed in meters per minute: medium cast iron	Cutting speed in meters per minute: hard cast iron
9,52	0,4	43,6	21,8	12,8
9,52	0,79	34,2	17,1	9,9
9,52	1,59	25	12,5	7,3
9,52	2,38	20,6	10,3	6
9,52	3,18	17,9	8,9	5,2
9,52	4,76	17,5	8,8	5,1
12,7	0,4	40,3	20,2	11,8
12,7	0,79	31,7	15,7	9,2
12,7	1,59	21,1	11,6	6,7
12,7	2,38	19,1	9,5	5,6
12,7	3,18	16,5	8,3	4,8
12,7	4,76	13,3	6,7	3,9

Normal cutter \(\frac{7}{8}"\)

Depth of cut in mm	Feed in mm	Cutting speed in meters per minute: soft cast iron	Cutting speed in meters per minute: medium cast iron	Cutting speed in meters per minute: hard cast iron
2,38	0,4	67,6	33,9	19,6
2,38	0,79	51,6	25,7	15,1
2,38	1,59	37,2	18,2	10,9
2,38	2,38	30,4	15,2	8,9
2,38	3,18	26,3	13,2	7,7
2,38	4,76	21,4	10,7	6,3
3,18	0,4	61,6	30,8	18
3,18	0,79	47,6	23,7	13,8
3,18	1,59	34,2	17,1	10
3,18	2,38	28	14	8,2
3,18	3,18	24,2	12,1	7,1
3,18	4,76	15,6	9,8	5,7
4,76	0,4	54,3	27,1	15,9
4,76	0,79	41,8	20,9	12,2
4,76	1,59	30,3	15,2	8,8
4,76	2,38	24,7	12,3	7,2
4,76	3,18	21,4	10,7	6,2
4,76	4,76	17,3	8,7	5,1
6,35	0,4	49,7	24,8	14,5
6,35	0,79	38,4	19,2	11,2
6,35	1,59	27,7	13,8	8,1
6,35	2,38	22,6	11,3	6,6
6,35	3,18	19,6	9,8	5,7
6,35	4,76	15,9	7,9	4,6

Normal cutter \(\frac{3}{4}"\)

Depth of cut in mm	Feed in mm	Cutting speed in meters per minute: soft cast iron	Cutting speed in meters per minute: medium cast iron	Cutting speed in meters per minute: hard cast iron
2,38	0,4	67,6	33,9	19,8
2,38	0,79	51,6	25,7	15
2,38	1,59	36,6	18,2	10,6
2,38	2,38	29,6	14,8	8,6
2,38	3,18	25,4	12,7	7,4
2,38	4,76	20,2	10,1	5,9
3,18	0,4	61,9	31,1	18,1
3,18	0,79	47,6	23,8	13,9
3,18	1,59	33,5	16,8	9,8
3,18	2,38	27,1	13,5	7,9
3,18	3,18	23,2	11,9	6,8
3,18	4,76	18,6	9,3	5,4
4,76	0,4	55,2	27,6	16,1
4,76	0,79	41,7	20,9	12,2
4,76	1,59	29,8	14,9	8,7
4,76	2,38	23,8	11,8	7
4,76	3,18	20,6	10,3	6
4,76	4,76	16,5	8,3	4,8
6,35	0,4	50,9	25,5	14,9
6,35	0,79	38,4	19,2	11,2
6,35	1,59	27,7	13,8	8
6,35	2,38	22,2	11,1	6,5
6,35	3,18	19,1	9,5	5,6

Normal tool \(7/8''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
9,52	0,4	43,9	21,9	12,8
9,52	0,79	33,8	16,9	9,8
9,52	1,59	24,4	12,2	7,1
9,52	2,38	19,9	9,9	5,8
9,52	3,18	17,2	8,6	5
9,52	4,76	14	7	4,1
12,7	0,4	41,2	20,6	12
12,7	0,79	31,7	15,9	9,3
12,7	1,59	22,9	11,5	6,7
12,7	2,38	18,7	9,4	5,5
12,7	3,18	13,1	6,6	3,8

Normal tool \(3/4''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
9,52	0,4	45,7	22,9	13,4
9,52	0,79	34,5	17,3	10,1
9,52	1,59	24,7	12,3	7,2
9,52	2,38	20	10	5,8

Normal tool \(5/8''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
2,38	0,4	65,8	32,9	19,2
2,38	0,79	48,8	24,4	14,2
2,38	1,59	33,6	16,8	9,8
2,38	2,38	26,9	13,5	7,9
2,38	3,18	23	11,5	6,7
3,18	0,4	61	30,5	17,9
3,18	0,79	45,1	22,6	13,2
3,18	1,59	31,7	15,8	9,2
3,18	2,38	25,2	12,6	7,3
3,18	3,18	21,2	10,6	6,2
4,76	0,4	55,8	27,9	20,7
4,76	0,79	41,2	20,9	12
4,76	1,59	28,7	14,3	8,4
4,76	2,38	23	11,5	6,7
4,76	3,18	19,6	9,8	5,7
6,35	0,4	52,1	26,1	15,3
6,35	0,79	38,4	19,3	11,2
6,35	1,59	26,8	13,4	7,8
6,35	2,38	21,5	10,7	6,3
9,52	0,4	47,6	23,7	13,8
9,52	0,79	35,4	17,6	10,3
9,52	1,59	24,3	12,2	7,1

Normal tool \(1/2''\)

Depth of cut, mm	Feed, mm	Cutting speed, m/min: soft cast iron	Cutting speed, m/min: medium cast iron	Cutting speed, m/min: hard cast iron
2,38	0,4	62,8	31,4	18,3
2,38	0,79	44,8	22,4	13,1
2,38	1,59	29,7	14,9	8,7
2,38	2,38	23,2	11,6	6,8
2,38	3,18	19,5	9,8	5,7
3,18	0,4	59,2	29,6	17,3
3,18	0,79	42,1	21,1	12,3
3,18	1,59	28,4	14,2	8,3
3,18	2,38	22	11	6,5
3,18	3,18	12,7	6,4	3,7
4,76	0,4	55,5	27,7	16,2
4,76	0,79	39	19,5	11,5
4,76	1,59	26,2	13,1	7,7
4,76	2,38	20,6	10,3	6
6,35	0,4	52,7	26,3	15,4
6,35	0,79	37,2	18,6	10,9
6,35	1,59	25	12,5	7,3

Influence of the Dimensions of the Chip

The majority of investigators who have dealt with determining the resistance to cutting have assumed that the force is proportional to the width of the chip. In 1908 N. N. Savin*) confirmed the latter assumption by his experimental investigations in the laboratory of the St. Petersburg Polytechnic Institute.

As regards the influence of the thickness of the chip on the coefficient of cutting, according to the investigations of various observers this dependence is expressed differently.

According to Joessel, the force is proportional to \(L^2\) and \(L\) (\(L\) is the thickness of the chip),

“ Time ” ” \(L\) and \(L^k\), where \(k > 1\), but considerably \(< 2\),

“ Hartig ” ” \(L^x\), where \(x < 1\)

“ Hassner ” ” \(L^x\), where \(x > 1\),

“ Zvorykin ” ” \(L^{\frac{2}{3}}\), with the exponent \(x\) for brittle metals tending to increase to \(\frac{3}{4}\), and for ductile metals decreasing to \(\frac{1}{2}\).

“ Taylor ” ” \(L^{\frac{3}{4}}\) for cast iron, \(L^{\frac{14}{15}}\) for steel.

Savin found that

1) the chips obey the law of similarity,

2) the pressure on the cutter is proportional to the width of the chip and to its thickness, and consequently also to the area of the chip section.

To elucidate the above-mentioned dependence between the coefficient of cutting and the thickness of the chip, in 1909, in the mechanical workshops of the Tomsk Technological Institute, under my supervision, students Mikhail Tauber and Petr Tyulenev carried out experimental investigations on a planing machine of the Chemnitzer Maschinenfabrik works; for measuring the pressures on the cutter, a special dynamometer of the construction described below was used.

*) N. Savin, “On the resistance to the cutting of metals.”

To the carriage of the planing machine (photographs 2 and 3) is screwed, by four bolts, the base \(B\), to which in turn the lever \(C\) and the bar \(D\) are fixed immovably and likewise by bolts. In the latter there are corresponding slots for the passage of the lever \(F\) and for the placement of the ball bearings \(E\). By means of these bearings the lever \(F\), with very slight friction, rotates about the axis \(EE'\). The arrangement of this part of the dynamometer is shown in more detail in the enlarged Fig. 9c, Plate IV. The movable lever \(F\), when deflected from its initial position, is held at a known distance from the fixed lever \(C\) by the spring \(G\), fastened between the upper ends of the aforesaid levers. In the lower part of the lever \(F\) there is a box-shaped enlargement \(S\), into which is inserted the holder \(P\), serving for securing the cutter. The holder \(P\) is freely fitted onto the cotter \(O\), inserted into the cheeks of the box \(S\); as a result, during the idle stroke the cutter together with the holder is thrown back somewhat from the object being planed and slides freely over its surface. The holder and cutter are brought into the working position by their own weight and by the tension of the spring \(T\).

It is evident that during planing the resistance force of the material to cutting will exert pressure on the cutter and thereby will cause the lever \(F\) to deviate from its initial position and will stretch the spring. These deviations of the lever \(F\) and extensions of the spring \(G\) will be proportional to the pressures on the cutter, and therefore may serve for their direct measurement. For this purpose, on the fixed lever \(C\) there is an index \(L\), freely rotating about its axis. The index \(L\) is hinged to the rod \(H\), which can rotate about the screw \(V\), screwed into the movable lever \(F\). Thus the index \(L\) is deflected simultaneously with the lever \(F\), and its lower end indicates the pressures on the cutter corresponding to these deflections, marked on the arc \(M\), which, in turn, is fixed immovably on the lever \(C\).

The introduction of the levers \(I\) and \(K\), visible in the attached photographs 2 and 3, was due to the desire to use a special apparatus for taking diagrams of the change in pressure on the cutter as a function of its travel; but since these forces, when removing a chip of one and the same thickness, changed hardly at all during each observation, there was no need to use this apparatus.

To balance the moving masses of the dynamometer, an oil buffer \(R\) was introduced between the upper ends of the levers \(F\) and \(C\), visible in the attached photographs 2 and 3 and shown schematically in section in Fig. 10, Plate IV. This is an iron turned ...

the cylinder, with a tightly fitted piston, connected by a hinge, by means of an eye on the bottom, to the upper end of lever \(C\). The rod of the piston of this cylinder, passing through the cover and stuffing box screwed to the latter, is hinged to lever \(F\) by means of a cylindrical nut screwed onto the end of the rod and provided with special eyes. The cavities of the cylinder in front of and behind the piston are connected by cocks \(X\) with the common box \(y\) (see phot. 3), into which oil is poured through openings \(Z\), closed by plugs. Thus, during planing, the upper end of lever \(F\), stretching the spring, is deflected and draws after it the piston of the buffer. The piston, meanwhile, compresses the oil situated before it and forces it out through the cocks into the common box. By this means a smooth deflection of lever \(F\) and of pointer \(C\) from their initial position during planing is achieved, and an equally smooth return of them back during the idle stroke of the machine. During the performance of the experiment care was taken that there should be no air in the buffer. When this condition was observed: 1) it was impossible to draw back the upper end of lever \(F\) when the cocks \(X\) were closed; 2) with the cocks open, during the observation itself, the deflection of the pointer took place gradually, without noticeable tremors or jolts.

The graduation of the instrument was carried out after its installation on the planing machine in the following way: to the table of the machine there was fastened a block (drwg. 11, plate IV), over which a steel rope was passed; one end of the rope was connected with the dynamometer by means of an iron plate with cut-outs, fitted onto a prismatic piece of iron (drwg. 12, plate IV) and inserted in the holder of the instrument in place of the cutter, while weights were hung on the other end. At the same time the deflection of pointer \(L\) along arc \(M\) was observed, and on this arc were directly marked the pressures, in kilograms, corresponding to these deflections. The distance \(a\) (drwg. 12) between the mean thickness of the above-mentioned plate and the lower plane of the holder was equal to \(20\ \mathrm{m/m}\), i.e. to the length of the part of the cutter protruding from the holder. This was done in order, as far as possible, to maintain equality of the arms on which act the forces of tension of the rope during graduation and the force of pressure on the cutter during planing of the test bar.

Concerning the dynamometer described, it may be said that it is simple in construction, but, unfortunately, has certain substantial shortcomings: 1) friction is not entirely eliminated in it, which, not being amenable to accounting, affects the result of the experiments and diminishes the sensitivity of the instrument; 2) together with the change in the position of lever \(F\), the position of the cutter relative to the sharp—

of the object being planed, i.e. the cutting angle changes, which, of course, must also affect the final results.

The investigations were carried out with two materials: cast iron and wrought iron. From the cast iron a thick square plate was cast, with dimensions \(618 \times 618\ \mathrm{m/m}\), and for the experiment with iron a bar of three-inch square section, \(618\ \mathrm{m/m}\) long, was taken. In both cases the surface layer had first been removed.

The cutters of the form shown in drawing 13, Plate IV, were made of Rapid steel. Their edge held up very well, and only once during the entire series of experiments was a slight correction required. For cast iron the cutter had a cutting angle of \(55^\circ\), a back rake angle of \(6^\circ\), while the cutting edge, being perpendicular to the direction of motion of the cutter, formed an angle of \(25^\circ\) with the horizontal. The corresponding angles of the cutter for planing iron were \(50^\circ\), \(7^\circ\), and \(30^\circ\). The experiments with cast iron and iron were conducted at an unchanging cutting speed, which was calculated as follows: the length of the bar being planed was \(618\ \mathrm{m/m}\), and the cutter always traversed this distance in 4 seconds; consequently the speed of travel in 1 second was

\[ \frac{0.618}{4} = 0.1545\ \text{m}. \]

Photograph 4.

The calculation of the feed for one stroke of the cutter was based on the following: the lead screw \(a\) (photograph 4), serving to move the carriage along

the horizontal direction, perpendicular to the length of the machine, to have, per 25 m/m of length, 4 threads,

the gear wheel \(b\) to have 18 teeth,

” ” \(c\) ” 50 ” ,

” ” \(e\) ” 50 ” .

Consequently, with one complete revolution of wheel \(b\) the slide-rest moves across the table by \(\frac{25}{4}=6.25\) m/m, and in this time gear \(C\) makes \(\frac{18}{50}\) of its full revolution. When turned by one of its teeth, wheel \(C\) evidently makes only \(\frac{1}{50}\) of its full revolution; in the same time wheel \(b\), and together with it the lead screw \(a\), will turn by \(\frac{1}{18}\) of the whole circumference, and therefore the slide-rest with the cutter is displaced sideways by \(\frac{6.25}{18}=0.347 \simeq 0.35\) m/m from its former position. It is evident that when wheel \(C\) is turned by 2 teeth the feed will also be doubled, i.e. it will be \(=\frac{6.25}{9}=0.694 \simeq 0.7\) m/m. In the experiments with iron only one feed of 0.35 m/m was used, while in the tests with cast iron some observations were made with a feed of 0.35, and others with 0.7 m/m.

The width of the chip removed during each experiment before sharpening was measured directly on the surface of the metal under investigation, and was then calculated quite accurately from the feed and the number of strokes made by the cutter during that time. This number of strokes of the cutter during the entire time of removing a chip of known width was marked by the number of teeth of wheel \(e\) past which the latch \(d\) had passed in doing so. By multiplying the latter number by the above-mentioned feed, the whole width of the chip removed in one observation was determined.

The measurement of the thickness of the chip during the experiments themselves was carried out by means of a simple rule and a strip with divisions marked on it in millimeters, set vertically on the platform of the planing machine. With the aid of these instruments the heights of elevation of the planed surface above the plane of the platform were determined before and after each experiment.

The difference of the readings gave the thickness of the chip being removed.

A more exact value of the chip thickness, used in constructing the diagrams placed below, was calculated from the weight of metal collected during each observation.

To perform each experiment, the surface of the test bar was leveled, the width of chip which it was intended to plane off was marked on it, and the excess was measured by the method mentioned above.

of this surface above the platform of the machine. This latter was set in a position corresponding to the beginning of chip removal. The carriage with the cutter was also brought into the same position by means of screw \(a\) (photographs 2, 3). Then the cutter was lowered by the carriage cutter screw below the surface to be planed by an amount approximately equal to the thickness of the chip that it was intended to remove. At the same time the pawl \(d\) was placed in the working position, and the teeth of wheel \(e\), between which it at that moment stood, were marked with chalk. After all this the machine was set in motion by shifting the belt from the loose pulley to the working one; a chip of the intended width was removed, and the readings of the dynamometer were noted. As soon as the cutter reached the line marked on the metal, the machine was stopped; this stoppage was always made during the idle stroke of the cutter, so that, when stopped, the cutter was outside the test bar. Having stopped the machine, it was noted where the pawl \(d\) then was and how many teeth on wheel \(e\) it had passed since the beginning of the experiment. This number, as was stated earlier, determined the number of strokes of the cutter and served for the exact calculation of the width of the chip removed. Finally, the height by which the planed surface exceeded the table platform was determined, and the chip removed during the experiment was weighed.

All the observed and measured quantities were entered in special tables, placed here at the end. The columns of these tables that were not filled in during the observation itself were completed afterward from the results of the corresponding calculations.

All the observations were divided into groups.

The result of each group of experiments was the construction of diagrams representing the dependence between the total pressure on the cutter and the thickness of the chip. In drawing these diagrams, the various chip thicknesses were plotted along the axis of abscissas, and the corresponding pressures along the axis of ordinates. Comparing curves relating to one and the same material and one and the same feed, and calculating from their ordinates, for different chip thicknesses (at intervals of every \(0.1\ \mathrm{m}/\mathrm{m}\)), the mean ordinates, we draw, from the latter, a curve giving the mean dependence between the observed variables. These latter curves were taken as expressing the sought dependence of the variation of pressure on the cutter in relation to the variable chip thickness.

TABLE I.

Results of experiments with cutting steel at a cutting speed of 0.35 m/min, cutting angle $55^\circ$, and rake angle $6^\circ$.

No. of experiment	Force by dynamometer	Report by divisions: before experiment	Report by divisions: after experiment	Approximate thickness	Number of cutter strokes	Determination of chip thickness by weight: chip weight, kg	Chip width: measured directly	Chip width: calculated by weight	Precise thickness
1	30	101,7	101,1	0,6	57	49,5	20	19,95	0,6
2	30	101,7	101,1	0,6	57	45,8	20	19,95	0,56
3	45	105,6	104,8	0,8	57	71,8	20	19,95	0,87
4	45	105,6	104,8	0,8	58	72,8	20	20,30	0,85
5	60	107,0	105,8	1,2	58	102,0	20	20,30	1,22
6	60	107,0	105,8	1,2	58	99,3	20	20,30	1,18
7	70	104,8	103,4	1,4	58	118,8	20	20,30	1,40
8	72	104,8	103,4	1,4	57	110,0	20	19,95	1,33
9	94	105,6	104,0	1,6	58	156,8	20	20,30	1,85
10	96	105,6	104,0	1,6	56	153,4	20	19,60	1,87
11	146	106,9	104,0	2,9	57	247,0	20	19,95	2,95
12	150	106,9	104,0	2,9	58	243,0	20	20,30	2,89
13	110	105,0	102,8	2,2	57	184,4	20	19,25	2,23
14	115	105,0	102,8	2,2	55	180,3	20	19,95	2,26
15	68	102,6	101,9	0,7	57	78,2	20	19,95	0,95
16	68	102,6	101,9	0,7	58	76,63	19	18,55	1,01
17	48	101,9	101,3	0,6	57	50,1	20	19,95	0,60
18	48	101,9	101,3	0,6	52	45,6	19	18,20	0,60
19	110	107,0	105,3	1,7	60	142,6	20	21,0	1,64
20	114	107,0	105,2	1,8	57	144,2	20	19,95	1,75
21	92	105,2	103,9	1,3	56	106,4	20	19,60	1,31
22	88	105,2	103,9	1,3	57	107,6	20	19,95	1,30
23	146	107,0	104,8	2,2	56	164,0	20	19,60	2,02
24	140	107,0	104,8	2,2	57	170,5	20	19,95	2,07
25	32	103,8	103,5	0,3	50	—	18	17,50	—
26	36	104,0	103,5	0,5	51	—	18	17,85	—
27	172	103,5	101,4	2,1	55	386,5	20	19,25	2,38
28	168	103,5	101,4	2,1	57	386,5	20	19,95	2,38
29	32 | 30	104,0	103,7	0,3	55	65,7	19	19,25	0,41
30	28	104,0	103,7	0,3	55	65,7	19	19,25	0,41
31	58	105,3	104,2	1,1	29	45,5	10	10,15	1,08
32	122	104,2	102,0	2,2	29	93,6	10	10,15	2,23
33	106	104,2	102,2	2,0	30	84,5	10	10,5	1,95
34	78	104,2	102,9	1,3	26	54,2	9	9,10	1,44
35	44	100,5	99,9	0,6	28	25,7	10	9,8	0,63
36	128	99,4	97,2	2,2	28	94,5	10	9,8	2,33

*) In determining chip thickness by weight, a chip length of 6.7 was taken, and for the test bar a diameter of 6⅛ in.

TABLE II.

Results of experiments: cutting speed \(0.7\ \mathrm{m/min}\), cutting angle \(55^\circ\), and inclination angle \(6^\circ\).

Experiment No.	Force by dynamometer	Direct thickness change: reading by marks before experiment	Direct thickness change: reading by marks after experiment	Direct thickness change: approximate thickness	Number of passes	Chip weight, kg	Determination of chip thickness by weight: entire chip	Determination of chip thickness by weight: chip width measured according to divisions	Determination of chip thickness by weight: exact thickness
1	156; 163	103.8	101.5	2.3	29	349.0	21	20.3	2.14
2	170	103.8	101.5	2.3	27	349.0	19	18.9	2.14
3	100	103.8	103.0	0.8	21	61.5	14	14.7	1.01
4	110	103.0	102.0	1.0	22	65.2	15	15.4	1.07
5	130	104.3	103.1	1.2	29	212.2	20	20.3	1.29
6	130	104.3	103.1	1.2	28	212.2	20	19.6	1.29
7	35	101.1	100.9	0.2	56	50.0	20	19.6	0.31
8	36	101.1	100.9	0.2	57	50.0	20	19.6	0.31
9	46	100.9	100.4	0.5	27	70.3	20	18.9	0.45
10	46	100.9	100.4	0.5	27	70.3	20	18.9	0.45
11	100	101.3	100.4	0.9	26	142.0	20	18.2	0.91
12	100	101.3	100.4	0.9	28	142.0	20	19.6	0.91
13	100; 103	101.9	100.6	1.3	28	193.0	20	19.6	1.19
14	106	101.9	100.6	1.3	28	193.0	20	19.6	1.19
15	164; 168	103.2	101.1	2.1	28	326.0	20	19.6	2.01
16	172	103.2	101.1	2.1	28	326.0	20	19.6	2.01
17	140; 141	104.6	103.1	1.5	27	233.7	20	18.9	1.49
18	142	104.6	103.1	1.5	27	233.7	20	18.9	1.49
19	162; 164	103.1	101.5	1.6	29	267.0	20	20.3	1.64
20	166	103.1	101.5	1.6	27	267.0	19	18.9	1.64
21	36; 38	99.9	99.4	0.5	21	53.3	15	14.7	0.42
22	40	99.9	99.4	0.5	22	53.3	15	15.4	0.42
23	70; 71	100.1	99.4	0.7	22	112.3	15	15.4	0.90
24	72	100.1	99.4	0.7	21	112.3	15	14.7	0.90
25	86; 87	100.4	99.4	1.0	22	143.8	15	15.4	1.16
26	88	100.4	99.4	1.0	22	143.8	15	15.4	1.16
27	176	106.8	104.5	2.3	22	142.0	15	15.4	2.22
28	124; 125	106.8	105.0	1.8	21	208.8	15	14.7	1.68
29	126	106.8	105.0	1.8	22	123.5	15	15.4	1.68
30	154	106.8	104.8	2.0	21	80.5	15	14.7	2.00
31	106	106.8	105.5	1.3	22	97.2	15	15.4	1.26
32	118	106.8	105.3	1.5	21	142.8	15	14.7	1.60
33	188	99.4	97.2	2.2	22	119.4	15	15.4	2.23
34	170	99.4	97.7	1.7	22	99.0	15	15.4	1.87
35	150	99.4	98.0	1.4	22	78.8	15	15.4	1.55
36	128	99.4	98.3	1.1	23	64.2	16	16.1	1.24
37	110	106.8	105.8	1.0	22	31.1	15	15.4	1.00
38	94	99.8	99.1	0.7	15	37.3	10	10.5	0.72
39	70	106.6	106.1	0.5	22	128.1	15	15.4	0.59
40	184	104.3	102.2	2.1	22	83.5	15	15.4	2.01
41	140	104.3	103.0	1.3	22	32.7	15	15.4	1.31
42	100	99.4	98.7	0.7	15	36.8	10	10.5	0.75
43	85	106.8	106.3	0.5	22	95.05	15	15.4	0.58
44	212	105.6	103.1	2.5	15	128.5	10	10.5	2.28
45	202	105.3	103.1	2.2	22	76.1	15	15.4	2.00
46	174	105.9	104.2	1.7	15	58.4	10	10.5	1.70
47	140	106.3	104.9	1.4	15	69.5	10	10.5	0.83
48	122	105.9	105.7	0.2	22	52.8	15	15.4	3.55
49	108	105.1	99.4	0.9	22	32.5	15	15.4	5.81
50	78	98.0	97.3	0.7	22	54.7	15	15.4	2.16
51	110	98.3	97.4	0.9	22	54.7	15	15.4	0.10

TABLE III

Results of experiments in cutting cast iron, with a feed of 0.35 mm per revolution, by a cutter with an angle of 50° and a rake angle of 7°.

No. of experiment	Force by dynamometer	Reading before cutting	Reading after cutting	Approximate chip thickness	Number of passes	Chip dimensions by weight: whole chip	Chip width: measured directly	Chip width: calculated from data	Exact chip thickness
1, 142	103,8	102,3	102,3	1,5	33	Chip not weighed	11	11,55	—
2, 140	103,8	102,3	102,3	1,5	—	Chip not weighed	—	—	—
3, 122	103,8	102,5	102,5	1,3	29	Chip not weighed	9	9,15	—
4, 92	103,8	102,8	102,8	1,0	29	Chip not weighed	9	9,15	—
5, 82	103,8	103,0	103,0	0,8	21	Chip not weighed	8	7,35	—
6, 200	101,4	99,4	99,4	2,0	24	Chip not weighed	8	8,30	—
7, 180, 190	102,2	100,2	100,2	2,0	29	Chip not weighed	9	9,15	—
8, 150	102,2	100,7	100,7	1,5	22	Chip not weighed	8	7,70	—
9, 138	102,2	100,9	100,9	1,3	28	Chip not weighed	10	9,80	—
10, 150	102,6	101,1	101,1	1,5	28	Chip not weighed	10	9,80	—
11, 124	102,6	101,4	101,4	1,2	25	Chip not weighed	9	8,75	—
12, 50	100,7	100,3	100,3	0,4	18	Chip not weighed	7	6,30	—
13, 64	100,9	100,3	100,3	0,6	22	Chip not weighed	7	7,70	—
14, 88	101,2	100,3	100,3	0,9	19	Chip not weighed	7	6,65	—
15, 100	118,9	117,0	117,0	1,9	27	69,5	10	9,45	1,85
16, 140	119,4	117,9	117,9	1,5	27	64,0	10	9,45	1,40
17, 120	119,4	118,2	118,2	1,2	28	56,4	10	9,8	1,23
18, 104	119,4	118,5	118,5	0,9	28	46,4	10	9,8	0,98
19, 78	119,4	118,8	118,8	0,6	25	36,84	10	9,1	0,84
20, 58	119,4	118,9	118,9	0,5	18	18,72	6	6,3	0,62
21	29	118,6	118,3	118,3	0,3	27	13,8	10	9,45
22	123	116,8	115,7	115,7	1,1	25	48,92	9	8,75
23	150	117,6	116,1	116,1	1,5	25	63,4	9	8,75
24	106	115,3	117,3	117,3	1,0	28	46,81	10	9,80
25	84	118,3	117,5	117,5	0,8	22	27,75	8	7,70
26	78	118,3	117,7	117,7	0,6	21	25,13	8	7,35
27	50	116,1	115,7	115,7	0,4	22	17,78	8	7,70
28	78	116,5	115,9	115,9	0,6	16	20,34	6	5,60
29	140	117,2	116,0	116,0	1,2	21	46,59	8	7,35
30	156	117,5	116,0	116,0	1,5	25	66,5	9	9,10
31	225	115,3	113,1	113,1	2,2	16	58,75	6	5,60
32	195	115,3	113,3	113,3	2,0	20	65,0	7	7,0
33	170	115,3	113,7	113,7	1,6	24	66,67	8	8,40
34	160	115,3	113,8	113,8	1,5	24	61,41	8	8,40
35	150	117,7	116,3	116,3	1,4	14	33,38	5	4,90
36	156	112,8	111,4	111,4	1,4	12	29,0	5	4,20
37	112	112,8	111,8	111,8	1,0	21	38,5	7	7,35
38	88	113,0	112,1	112,1	0,9	24	35,5	8	8,40
39	78	113,0	112,2	112,2	0,8	18	23,2	7	6,30
40	46	113,5	113,0	113,0	0,5	26	20,8	10	9,10

*) In determining the chip thickness by weight, the specific gravity of cast iron was taken as 7.8 and the length of the test bar as 618 mm.

TABLE IV

for calculating the ordinates of curve \(f\) (diagram A, Table IV) of the sought relation between pressure on the cutter and chip thickness in planing cast iron with a feed of 0.35 m/m.

Chip thickness m/m	Pressure on the cutter, taken from diagram A as ordinates of the curves: a	b	c	d	e	Total sum of pressures for one thickness	Number of terms	Mean pressure on the cutter—ordinate of the sought curve	Chip cross-sectional area	Cutting coefficient (pressure per 1 sq. mm in klgr)
0.3	13	17	23	32	—	85	4	21.25	0.105	200
0.4	19	22	29	37	—	107	4	26.75	0.140	191
0.5	24	27	36	43	—	130	4	32.5	0.175	186
0.6	30	32	42	48	—	152	4	38.0	0.210	181
0.7	35	37	48	54	46	220	5	44.0	0.245	179
0.8	41	42	54	59	49	245	5	49.0	0.280	175
0.9	46	47	61	65	52	271	5	54.2	0.315	172
1.0	50	52	67	71	56	296	5	59.2	0.350	169
1.1	55	56	74	78	59	322	5	64.4	0.385	167
1.2	59	62	81	85	65	352	5	70.4	0.420	168
1.3	64	69	88	91	70	382	5	76.4	0.455	168
1.4	70	75	93	97	76	411	5	82.2	0.490	168
1.5	75	79	99	103	81	437	5	87.4	0.525	166
1.6	80	84	105	108	87	464	5	92.8	0.560	166
1.7	86	88	111	116	92	493	5	98.6	0.595	166
1.8	91	99	118	125	98	525	5	105.0	0.630	167
1.9	96	98	126	134	103	557	5	111.4	0.665	167
2.0	100	102	134	144	109	589	5	117.8	0.700	168
2.1	105	107	143	152	115	622	5	124.4	0.735	169
2.2	109	112	152	159	120	652	5	130.4	0.770	170
2.3	114	117	161	166	126	684	5	136.8	0.805	170
2.4	119	123	170	173	132	717	5	143.4	0.840	171

TABLE V

for calculating the ordinates of curve \(m\) (diagram B) of the sought relation between pressure on the cutter and chip thickness in planing cast iron with a feed of 0.7 m/m.

Chip thickness m/m	Pressure on the cutter, taken from diagram B as ordinates of the curves: g	h	i	k	l	Total sum of pressures for one thickness	Number of terms	Mean pressure on the cutter—ordinate of the sought curve	Chip cross-sectional area	Cutting coefficient (pressure per 1 sq. mm in klgr)
0.3	—	30	36	49	61	176	4	44	0.21	209.5
0.4	—	37	43	57	68	205	4	51.25	0.28	183
0.5	—	43	51	64	76	234	4	58.5	0.35	167.1
0.6	—	50	62	73	86	271	4	67.75	0.42	161.3
0.7	—	57	73	90	95	315	4	78.75	0.49	160.7
0.8	—	64	83	98	105	350	4	87.5	0.56	156.25

Chip thickness, m/m	Pressures on the cutter, taken from diagram B as ordinates of the curves. g	Pressures on the cutter, taken from diagram B as ordinates of the curves. h	Pressures on the cutter, taken from diagram B as ordinates of the curves. i	Pressures on the cutter, taken from diagram B as ordinates of the curves. k	Pressures on the cutter, taken from diagram B as ordinates of the curves. l	Total sum of these pressures for one thickness.	Number of terms	Mean pressure on the cutter—ordinate of the curve sought	Area of the cross-section of the chip.	Coefficient of cutting pressure per 1 sq. mm in klgr.
0,9	—	71	94	104	112	381	4	95,25	0,63	152,8
1,0	—	87	104	108	117	416	4	104	0,70	148,6
1,1	—	104	113	117	122	456	4	114	0,77	148
1,2	94	113	122	125	130	584	5	116,8	0,84	139
1,3	103	123	130	132	138	626	5	105,2	0,91	115
1,4	111	133	135	139	146	624	5	132,4	0,98	135
1,5	114	142	140	146	154	696	5	139,2	1.05	132,6
1,6	118	158	145	153	162	736	5	147,2	1,12	131,4
1,7	127	165	150	159	170	771	5	154,25	1,19	129,6
1,8	137	166	154	166	180	803	5	160,6	1 26	126,7
1,9	145	167	158	173	191	834	5	166,8	1,33	125,4
2,0	154	168	163	183	202	870	5	174.	1,40	124,3
2,1	164	—	168	186	206	724	4	181	1,47	123,1
2,2	175	—	—	188	208	571	3	190,3	1,55	122,8

TABLE VI

for calculating the ordinates of curve t (diagram C) of the required relationship between the pressure on the cutter and the chip thickness when planing iron with a feed of 0,35 m/m.

Chip thickness, in m/m.	Pressures on the cutter, taken from diagram C as ordinates of the curves. n	Pressures on the cutter, taken from diagram C as ordinates of the curves. o	Pressures on the cutter, taken from diagram C as ordinates of the curves. p	Pressures on the cutter, taken from diagram C as ordinates of the curves. r	Pressures on the cutter, taken from diagram C as ordinates of the curves. s	Total sum of these pressures for one thickness.	Number of terms	Mean pressure on the cutter—ordinate of the curve sought	Area of the cross-section of the chip.	Coefficient of cutting pressure per 1 sq. mm in klgr.
0,3	—	29	—	—	—	29	1	29	0,105	276,2
0,4	—	38	—	50	43	131	3	43,6	0,140	311,4
0,5	—	47	49	57	53	206	4	51,25	0,175	292,9
0.6	—	56	60	64	65	245	4	61,25	0,210	291,7
0,7	—	65	72	72	78	287	4	71,75	0,245	292,9
0,8	82	74	82	80	88	386	5	77,2	0,280	275,7
0,9	88	88	88	88	97	449	5	89,8	0.315	285,1
1,0	92	105	100	102	106	505	5	101	0,350	288,6
1,1	103	112	114	112	117	558	5	111,6	0,385	294
1,2	112	118	124	123	127	604	5	120,8	0,420	287,6
1,3	122	128	136	138	140	664	5	132,8	0,455	292,1
1,4	132	140	147	144	148	711	5	142,25	0,490	290,4
1,5	142	152	158	150	156	758	5	151,6	0,525	288,8
1,6	151	163	167	—	—	481	3	160,3	0,560	286,2
1,7	161	173	176	—	—	510	3	170	0.595	286,4
1,8	171	185	187	—	—	543	3	181	0,630	287,8
1,9	180	—	198	—	—	378	2	189	0,665	284,2
2,0	190	—	208	—	—	398	2	199	0,700	284,3
2,1	—	—	218	—	—	219	1	218	0,730	298,6
2,2	—	—	228	—	—	228	1	228	0,770	297,4

As is evident from the diagrams appended below (Plate IV and Plate II), these curves deviate very little from a straight line inclined to the axis of abscissas. Some deviations from this straight line can probably be explained by the imperfection of the measuring instrument and by the insufficient accuracy of the method of investigation.

In any case, considering the above-mentioned diagrams, it may be accepted that, on the average, the pressures on the cutter vary directly in proportion to the thickness of the chip; and since, in carrying out a known group of experiments and in constructing the corresponding curves, the condition of an unchanged feed was taken as a basis, it may therefore be said that the pressures on the cutter vary directly in proportion to the cross-sectional area of the chip removed.

Influence of the Mechanical Properties of the Material

As regards the dependence of the coefficient of cutting on the mechanical properties of the material, then, comparing the resistance of the chip for different metals (under identical conditions), we find that the coefficient of cutting is proportional neither to the tensile strength of the metals nor to the shearing resistance. For example, on the basis of Zworkin’s experiments, Professor Gatschuk compiled the following table, from which it clearly follows that no proportionality exists, as Hart asserts *):

	Cast iron	Bronze	Wrought iron	Cast iron
Tensile strength, kilgr/qmm . . . .	10,2	13,8	31,1	35,9
Shearing resistance, kilgr/qmm . . .	11,2	16,2	23,2	29,2
Coefficient of cutting for a chip thickness = 1 m/m . . . . . . . . . .	56,7	50,5	105,6	159,2
Ratio of the coefficient of cutting to the tensile strength . . . . . . .	5,5	3.6	3,4	4,4
Ratio of the coefficient of cutting to the shearing resistance . . . . . .	5	3,1	4,5	5,4

From Gausner’s experiments it turned out that, in planing three sorts of iron with different carbon contents from \(0,17\%\) to \(0,48\%\),

) See Gatschuk, Mechanical Technology*, p. 42.

therefore, for metals of rather different hardness and, very probably, different tensile strength, the coefficient of cutting came out the same, or rather, its fluctuations were so insignificant that they may be attributed entirely to the errors of experiment.

On the basis of later observations made on the working of metals by cutting, using cutters made of self-hardening grades of steel, one may say with confidence that there is no definite dependence between the coefficients of cutting and the other mechanical coefficients of the metal. A general summary of all these observations is given in the following table:

Place where the experiments were made	Material	Cutters with which the work was done	Depth of cut, mm	Feed, mm	Normal cutting speed, m/min	Chip pressure, kg/sq. cm	Coefficient of cutting, kg/sq. cm	Elongation, %	Total C.	Si.	Mn.
Manchester 1903	Soft forged steel	Various modern English cutters of self-hardening steel containing chromium, tungsten, or molybdenum	4.76	1.59	33.8	18000	4220	29	0.20	0.055	0.605
Manchester 1903	Forged steel of medium hardness	Various modern English cutters of self-hardening steel containing chromium, tungsten, or molybdenum	4.76	1.59	24.4	17000	4500	26	0.28	0.086	0.650
Manchester 1903	Hard forged steel	Various modern English cutters of self-hardening steel containing chromium, tungsten, or molybdenum	4.76	1.59	12.5	23600	7320	14	0.51	0.111	0.792
Wm. Sellers Co. Philadelphia 1902	Annealed forged steel of medium hardness	Similar cutters, but containing chromium and tungsten	4.76	1.59	19.5	19500	4920	30	0.34	0.183	0.600
Wm. Sellers Co. Philadelphia 1902	Soft cast-iron cylinder, 330 mm, with the outer skin removed	Cutters of high-speed steel containing chromium, tungsten, or molybdenum	4.76	1.59	45.2	7310	—	—	—	1.91	—
Wm. Sellers Co. Philadelphia 1902	Same, but the raw outside skin was cut	Cutters of high-speed steel containing chromium, tungsten, or molybdenum	4.76	1.59	23.2	7310	—	—	—	1.91	—
Wm. Sellers Co. Philadelphia 1902	Hard cast iron, of the same size, but containing 15% steel scrap, hollow, with wall 50 mm	Cutters of high-speed steel containing chromium, tungsten, or molybdenum	4.76	1.59	1.37	11230	1690	—	—	1.91	—

	Cutting depth in mm	Feed in m/m	Chip area	Normal cutting speed, m/min	Chip pressure in kg/qcm	Mechanical properties of the metal: tensile strength in kg/qcm	Mechanical properties of the metal: compression coefficient in kg/qcm	Chemical composition of the metal in %: C, total	Chemical composition of the metal in %: bound carbon	Chemical composition of the metal in %: graphite	Chemical composition of the metal in %: Si	Chemical composition of the metal in %: Mn
Manchester, 1900:
Soft cast iron	4.76	1.59	0.08	30.5	7450	890	4220	3.062	0.459	2.603	3.010	1.180
Medium cast iron	4.76	1.59	0.0742	14.9	13200	1760	6880	3.305	0.585	2.720	1.703	0.588
Hard cast iron	4.76	1.59	0.0757	9.8	12930	1250	6600	3.025	1.150	1.875	1.789	0.348

The coefficients of cutting calculated by different authors differ greatly among themselves; the cause of such disagreement must be sought in inaccuracies in the experiments themselves, and also in various secondary circumstances which sometimes have a strong influence on the coefficient of cutting.

For greater clarity, I shall give a comparative table*) of the coefficients of cutting.

Materials	Fischer	Richard	Hart	Time	Zvorykin	Gauscher
Cast iron	70—120	75—90	60—75	52—64	52—62	60—86
Wrought iron	110—170	125—150	90—112	99—107	93—105	—
Cast iron	110—170	125—150	—	—	128—141	109—11
Hard steel	160—240	125—270	—	152—183	—	—
Bronze	—	65—80	—	49—62	50—60	47—61
Red copper	—	—	—	—	—	60

*) The coefficients of cutting are given in kg on sq. m/m. Prof. Fischer’s data are the result of a comparison of the experiments of Time, Smith, Gauscher, and Harting; through Fischer’s error, Time’s numbers are taken without correction for the coefficient of efficiency of the machine; see Zeitschrift des Ver. d. deutsch. Ingenieure, 1897, No. 18, p. 505.

Richard’s data (G. Richard) are taken from the work Traité des machines utiles, p. 1, p. 203.

Time’s data are taken from Foundations of Machine Building, vol. 1, issue 1, p. 408, and corrected for the coefficient of efficiency of the machine, 60%; Hart’s data are calculated on the basis of his examples.

Zvorykin’s data correspond to a chip thickness of 1 m/m and cutting angles usually encountered in practice, but are somewhat rounded.

Gauscher’s figures are the mean of observations relating to ordinary factory practice; see Prof. Gatsuk’s Mechanical Technology, p. 46.

Professor Hartig advises, in determining the pressure on the tool for thick chips, to take: for cast iron \(60 \frac{\mathrm{klgr}}{\mathrm{q\,mm.}}\), for wrought or soft cast iron about \(100 \frac{\mathrm{klg}}{\mathrm{q\,mm.}}\), and for bronze, tombac, and red copper \(60 \frac{\mathrm{klg}}{\mathrm{q\,mm.}}\). Multiplying these quantities by the cross-sectional area of the chip in sq. m/m, we obtain the approximate value of the force in klg required for removing the chip. With a cutting angle of the tool \(\alpha > 70\) and a chip thickness less than \(1\ \mathrm{m/m}\), the coefficient of cutting will have to be somewhat increased.

The coefficient of cutting of metals when machining the latter with cutters of self-hardening steel differs somewhat from the data given earlier and, on the basis of later data cited by Taylor, varies within the following limits (Polyakov, p. 26, table X):

Coefficients of cutting in cutting steel, cast iron, and iron.

Place where the tests were made	Pressure of the chip on the tool, in kg/qcm. Cast iron	Pressure of the chip on the tool, in kg/qcm. Steel	Pressure of the chip on the tool, in kg/qcm. Soft iron	Pressure required for feeding the tool	Pressure*) pressing the tool away in the horizontal direction from the workpiece	Influence of cutting speed on the pressure of the chip
Manchester. Prof. Nicolson.	7450 to 13200	17000 to 23600	—	\(0\%\) to \(20\%\) of the pressure of the chip on the tool.	\(18\%\)—\(78\%\) of the pressure of the chip on the tool.	The pressure on the tool decreases slightly with an increase in cutting speed.
Germany G. Fischer	7000 to 12000	15900 to 23900	11000 to 16900	Equal to the pressure on the tool	—	—
America, Taylor’s investigations.	4900 to 13900	16900 to 20800	—	Equal to the pressure on the tool.	—	No appreciable changes in the pressure on the tool even when the speed is doubled.

Tomsk, August 3, 1909.

*) The pressure pressing the tool away in the horizontal direction from the workpiece.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Key to designations

Material	Designation
Iron	hatched rectangle
Steel	hatched rectangle
Cast iron, copper, and bronze	hatched rectangle

Fig. 8

Fig. 9a

Fig. 9

Fig. 9b

Fig. 9c

Curve constructed from the data of the 1st table

Curve constructed from the data of the 2nd table

Curve plotted from the data of the 5th table.

Curve plotted from the data of the 4th table.

Curves plotted from the data of the 5th table

Curves plotted from the data of the 6th table

Plate I

Fig. 13.

Fig. 11.

Fig. 12.

Fig. 10.

Speed of cutting.

Cutting coefficient.

Curve constructed from the data of [[unclear: name]].

Feed, mm.

Curve labels:

• Kirpichev; \(\mu = 0.3\); \(\delta = 0.5\)
• Kirpichev; \(\mu = 0.3\); \(\delta = 0.1\)
• Artillery Plant; \(\delta = 0.5\)

Axis markings visible on the graph:

• Cutting coefficient: \(1.30,\ 1.40,\ 1.50,\ 1.60,\ 1.70,\ 1.80\)
• Feed, mm: \(0.020,\ 0.040,\ 0.060,\ 0.080,\ 0.100,\ 0.120,\ 0.140,\ 0.160,\ 0.180,\ 0.200,\ 0.220,\ 0.240\)

Diagram A

Note:
a) The ordinates of curves \(a, b, c, d, e\) are plotted on the basis of Table No. [[unclear: table number]], from the results of the diagrams of instrument No. [[unclear: number]] of the tests.
b) The ordinates of curve \(f\) are plotted on the basis of Table No. IV.

[Graph with multiple plotted curves labeled \(a, b, c, d, e, f\), on a rectangular coordinate grid.]

Visible labels on the diagram:

• “Diagram of the dependence of the pressure required for cutting metals on the rate of feed of the tool, at a constant planing depth, for the tool carriage of a planing machine …” [[unclear: remainder of vertical inscription]]
• “Thickness of the chip, in millimeters”
• “Pressure of the cutting edge” [[unclear: exact wording]]
• Numerical scale markings visible along the axes include values from approximately \(0\) to \(190\) on one scale and from approximately \(0.3\) to \(6.4\) on another.

To the article by G. I. Tikhonov: “On the Coefficient of Metal Cutting.”

Diagram B.

Note.
a) The ordinates of curve \(y, m, v, k, l\), are plotted on the basis of table No. 5, for cast-iron working with St. Chamond cutters.
b) The ordinates of curve \(M_1\) are plotted on the basis of table No. 7.

Diagram

Dependence between the resistance to cutting and the product of the cross-section of the chip by the speed.

Longitudinal planing.

For gray cast iron of medium hardness.

\(H = 0.27\ \text{m/m}\).

Vertical axis: Cutting resistance in kilograms.

Horizontal axis: Product of the cross-section of the chip by the cutter speed.

Diagram C. Table II.

Note:
a) The ordinates of the curves M. O. R. P. S. are plotted on the basis of Table No. III, with the serial number of the experiment.
b) The ordinates of curve \(i\) are plotted on the basis of Table No. VI.

Coordinates

Dependence of the pressure on the cutter in kilograms on the thickness of the chip being removed, for the case of cutting iron at \(\varphi = 0.25\ \mathrm{m/m}\).

Horizontal ordinates: pressure in kilograms
Chip thickness in millimeters

To the article by T. I. Tikhonov: “On the Coefficient of Metal Cutting.”