Engineer A. M. Rosenberg

On the Question of Choosing the Most Advantageous Conditions for Metal Cutting

(on lathes)

With 7 drawings

Tomsk
1928

Eng. A. M. Rosenberg.

On the Question of Choosing the Most Advantageous Conditions for Metal Cutting.

(On lathes).

For the article, 7 drawings.

The productivity of work on metal-cutting machine tools depends on a large number of factors, connected with one another in various ways and influencing one another; and, in order to solve the question of the most advantageous machining of metals, it is necessary, in Taylor’s opinion, to solve a complex mathematical problem with many variables¹. This problem, in which the unknowns include the working conditions of cutting, has a large number of possible feasible solutions, each of which is characterized by its own productivity, and only one of all gives a combination of conditions characterized by the maximum possible productivity, i.e. satisfying the requirement of comparative economy.

The criterion of economy of the chip-removal process may be the time spent on producing one and the same operation under different conditions, and that combination of working conditions which is connected with the least expenditure of time will be the most advantageous.

If we denote:

$T$ — time in minutes
$L$ — length of the workpiece being machined, in mm.
$D$ — diameter, in meters.
$n$ — number of revolutions per minute
$s$ — feed in mm/rev. (the path traversed by the tool in 1 revolution parallel to the axis of the workpiece being machined)
$t$ — depth of cut in mm. (half the difference of the diameters before and after removal of the chip)
$v$ — cutting speed in meters/min.
$q$ — cross-section of the chip in mm²,

then the analytical expression for the time required for machining (for operations with rotary motion of the workpiece or of the tool) will be:

$$ T=\frac{L}{ns}\ldots\ldots\ldots\ldots\ldots\ldots (1) $$

Here $L$ is a quantity given in advance, and a change in $T$ can occur only through a change in the quantity $ns$, which represents nothing other than the length of the path traversed by the cutter per unit time parallel to the axis of the workpiece being machined.

On the other hand:

$$ n=\frac{v}{\pi\cdot D}; $$

and

$$ s=q:t; $$

i.e.

$$ T=\frac{L\cdot t\cdot \pi\cdot D}{q\cdot v}\ldots\ldots\ldots\ldots (2) $$

Here the prescribed quantities are \(L\), \(D\), and \(t\) (\(t\) is determined depending on the machining allowance), and the quantity \(T\) depends on the product \(qv\), which is the volume of chips, in \(\mathrm{cm}^3\), removed per minute. The greater the quantities \(ns\) and \(qv\), the smaller the expenditure of time will be, and the greater the productivity; therefore, the quantities of the products \(ns\) or \(qv\) are usually taken as the characteristic of the productivity of the work.

It is necessary to distinguish roughing work from finishing work, since in the first case the objective is the removal of all excess material in as short a time as possible, whereas in the second, in addition, certain additional conditions must be observed, such as the cleanliness of the machined surface, accuracy of dimensions, form, etc.; but even in this case, with the obligatory observance of the additional conditions, the same quantities may serve as an index of productivity.

The work of removing chips cannot be conceived without the presence of three components: the machine tool, the cutting tool (cutter), and the material being machined; the first two are active, and each of them, both the machine tool and the cutter, possesses in each given case a definite power¹) and normal productivity, and only with complete correspondence of their productive capacities and under the condition of their normal use can the maximum effect be attained and the most economical working conditions obtained. But usually there is no complete correspondence between the power of the machine tool and the productive capacity of the cutting tool, and in this case, as Hippler has shown²), in order to increase productivity one must strive first of all to utilize the machine tool by selecting for it an appropriate tool, although theoretically, as we shall show below, with any machine tool and tool one can attain their simultaneous utilization.

The normal productivity of the machine tool and the tool, and the degree of their utilization, can be judged from the relations that we have for the durability of the cutter and the power of the machine tool.

For the machine tool we have:

\[ N_e=\frac{P\cdot v}{4500}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(3) \]

Here:

\(N_e\) — power in HP at the place of chip removal, or the useful power of the machine tool;

\(P\) — cutting pressure in kg;

\(v\) — cutting speed permitted by the power of the machine tool, in m/min.

For the durability of the cutter we have the relation

\[ v_u=\frac{C_1}{s^p\cdot t^r}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(4) \]

Here:

\(v_u\) — cutting speed permitted by the given cutter;

\(s\) — feed per one revolution, in mm.

\(t\) — depth of cut, in mm.

¹) Here, by the power of the machine tool one must understand not only its power in horsepower, but also the stability of its construction, the strength of the transmission mechanisms, etc.

²) Reindl. The Cutting Tool. Hippler’s article: “Economical Chip Removal.”

$C_1$ is a constant quantity depending on the material of the cutter, its dimensions, heat treatment, grinding and setting angles, the form of the cutting edge, the duration of operation without regrinding, the properties of the material being machined, the properties of the cutting fluid used in the work, and the stability of the workpiece and the machine tool.

Thus:

\[ C_1 = K_1 \cdot K_2 \cdot K_3 \cdot K_4 \cdot K_5 \cdot K_6 \cdot K_7 \cdot K_8 \]

$p$ and $r$ are exponents depending on the material being machined.

Equation (4) represents the general form of empirical relationships obtained on the basis of investigations over the last 30 years. All experimenters of this period—Taylor¹, Herbert and Ripper², Friedrich³, Denis⁴, and others—come to the conclusion that the relationship for the cutting speed permissible for a cutter, in other words, for the tool life of the cutter, has the form of a power function, with the exponents $(p$ and $r)$ for feed and depth of cut always less than unity and $p > r$. Prof. Nicolson alone gives a relationship of the form

\[ v = \frac{A}{q + B} + C \ldots \text{ for steel} \]

and

\[ v = A_1 + B_1 \cdot q \ldots \text{ for cast iron,} \]

but for various reasons these data at present have no practical significance⁵.

For the cutting pressure $P$ in formula (3), on the basis of experiments by Taylor⁶, Time⁷, Zvorykin⁸, Nicolson⁹, Savvin¹⁰, Friedrich¹¹, Ripper¹², Tikhonov¹³, Chelyustkin⁹, Klopstock¹⁴, and others, the general form of the relationship may be given by the equation:

\[ P = C_2 \cdot s^l \cdot t^m \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) \]

Here:

$P$ — cutting pressure in kilograms
$s$ — feed per revolution in mm.
$t$ — depth of cut in mm.
$C_2$ — a constant quantity depending on the quality of the material being machined, the grinding angles, the setting angles and the form of the cutting edge of the cutter, the type of cutting fluid, and therefore may be represented in the form:

\[ C_2 = K'_1 \cdot K'_2 \cdot K'_3 \cdot K'_4 \cdot K'_5 . \]

$l$ and $m$ are exponents depending on the material being machined.

Relationship (5), just like relationship (4), is of a purely empirical character, since we still do not have an exact theoretical

substantiation explaining the phenomena occurring when metal is transformed into chips, and attempts to provide such a substantiation have as yet not been crowned with success¹.

Substituting the value for \(P\) from (5) into (3), we obtain:

\[ N_e=\frac{C_2\cdot s^l\cdot t^m\cdot v}{4500}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (6) \]

In most cases the problem of choosing cutting conditions in production is posed as follows: on a given machine, to turn an object of a definite material and a specified diameter; the depth of cut is given or is found from the amount of the machining allowance. It is necessary to find the cutting speed (number of revolutions) and the feed that will make it possible to perform the machining in the shortest possible time, i.e., that are characterized by the value \((ns)_{\max}\) or \((qv)_{\max}\).

The material of the cutter, its form, heat treatment, and working conditions (with cooling or without) are ordinarily constant for the given production operation. To solve the problem posed, one must use the dependences given above for the power of the machine (6) and the endurance of the cutter (4). It would seem that, by solving two equations with two unknowns (the quantities \(N_e, t, p, r, l, m, C_1, C_2\) are constant in each given case), we shall find the quantities \(s\) and \(v\) satisfying the condition of simultaneous utilization of the machine and the tool; but apart from the fact that such a combination by no means always gives the most advantageous machining conditions \((ns_{\max})\)², in the majority of cases we moreover cannot realize the value of the cutting speed \(v\) satisfying the system of equations (4) and (6). Indeed, a machine, possessing an established series of numbers of revolutions, at a given machining diameter can give only quite definite cutting speeds \(v_i=\pi D n_i\), which we are unable to change without special readjustment of the machine. In other words, the simultaneous solution of equations (4) and (6) does not give us an answer to the problem, and it can be solved, as we shall now show, only by a series of trials.

If we take a machine with a stepped pulley, as the type most widespread in our production, then each step of its pulley has its own power and number of revolutions and, at a given diameter, gives a quite definite cutting speed.

Taking any one of its numbers of revolutions \(n_1\) and finding the cutting speed \(v_1=\pi D n_1\), we substitute its value into equalities (4) and (6), together with the power corresponding to this number of revolutions and the other known quantities. Then from equation (4) we can obtain the feed permissible for the cutter:

\[ s_u=\left[\frac{C_1}{t^r\cdot v_u}\right]^{\frac{1}{p}}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (7) \]

and from equation (6), the feed permitted by the power of the machine \((Ne_1)\):

\[ s_{\mathrm{st}}=\left[\frac{Ne_1\cdot 4500}{C_2\cdot t^m\cdot v}\right]^{\frac{1}{l}}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (8) \]

where

\[ s_u \gtrless s_{\mathrm{st}}. \]

Of these two values of the feed we can, provided there is no overloading of the machine tool and the tool, take the smaller, and multiplying it by the number of revolutions \((n_1)\), (or by the depth of cut \(t\) and the cutting speed \(v_1\)), will obtain a characteristic of productivity \(n_1 s_1\), (or \(q v_1\)) for the given number of revolutions. Having made the same computations for other numbers of revolutions—where in each case its own cutting speed will enter into the calculation,

\[ v_i = \pi D \cdot n_i \qquad \ldots\ldots\ldots\ldots\ldots (9) \]

and its own machine power \(Ne_i\), corresponding to the adopted number of revolutions \((n_i)\)—we shall find at what number of revolutions the characteristic of productivity \(n_i s\) (or \(q v_i\)) will be greatest; this will also be the most advantageous combination. In doing this it is necessary to see that the chip section obtained in each case, \(q = ts\), is permissible not only with respect to the durability of the cutter and the power of the machine tool, but also with respect to the strength of the transmission mechanisms (the feed mechanism) and to the condition of stability of the object being machined. Since the usual aid in choosing cutting conditions is a previously prepared technical-rate card (passport) of the machine tool, the maximum chip section permissible by the strength of the feed mechanism is taken from this card, where it should have been entered in due time, obtained by one of the known methods of calculation¹; as for judging compliance with the condition of stability of the object being machined, this can be done on the basis of materials available in the literature, giving the maximum chip section as a function of the kind of material being machined and of the ratio of the dimensions of the workpiece².

Thus, in order to determine the working conditions it is necessary to solve three equations for each number of revolutions of the machine tool. It is quite clear that the method of analytical computation for determining the most advantageous cutting conditions, being rather laborious, taking much time, fatiguing, and undoubtedly not free from accidental errors as a consequence of rather complicated computations, cannot be recognized as suitable for everyday work.

Recently many various instruments have appeared for determining cutting conditions, usually constructed in the form of slide rules, originating from the slide rule first constructed by Taylor and his collaborator Barth. Each of these rules is designed on the basis of the results of those investigations which were either carried out by its author, or are recognized by him as most closely corresponding to reality and deserving attention. In what follows we give the theory and the method of construction of an instrument developed by us, based on the combination of graphs—nomograms—which can easily be carried out, and at the basis of which may be placed all those investigations as a result of which dependences have been obtained for cutting pressure and cutting speed corresponding to the general form of equations (4) and (5). The instrument makes it possible to deter-

to establish the conditions of the most advantageous cutting, and also to resolve certain questions connected with the use of machine tools and the investigation of their kinematic essence.

All further conclusions and constructions will be made on the basis of the fundamental dependences for tool life and machine-tool power:

\[ v_u=\frac{C_1}{s^p\cdot t^r}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (4) \]

\[ N_e=\frac{C_2\cdot s^l\cdot t^m\cdot v}{4500}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (6) \]

Since equations (4) and (6) are power functions, if we plot them in logarithmic coordinates they receive the simplest graphical expression.

Equation (4) is the equation of a plane situated in the spatial system of three coordinates \(\lg v\), \(\lg s\), and \(\lg t\). Indeed, if we take its logarithm, which is necessary for determining the form of the dependence and for constructing it, we obtain

\[ \lg v_u=\lg C_1-p\cdot \lg s-r\cdot \lg t; \]

or

\[ \lg v_u+p\cdot \lg s+r\cdot \lg t-\lg C_1=0\ .\ .\ .\ .\ .\ (10) \]

i.e. an equation of a plane in space of the form

\[ Ax+By+Cz+D=O. \]

It can be brought to the form of the equation of a plane in intercepts on the axes:

\[ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1, \]

and in this case we obtain:

\[ \frac{\lg v_u}{\lg C_1}+ \frac{\lg s}{\frac{\lg C_1}{p}}+ \frac{\lg t}{\frac{\lg C_1}{r}} =1\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (11) \]

This plane will cut off on the coordinate axes the following intercepts: on the \(\lg v\)-axis—the intercept \(\lg C_1\); on the \(\lg s\)-axis—the intercept \(\dfrac{\lg C_1}{p}\); on the \(\lg t\)-axis—the intercept \(\dfrac{\lg C_1}{r}\). The quantities \(C_1\), \(r\), and \(p\) are always positive, with \(C_1>1\), \(r>0\), and \(p>0\), and therefore the intercepts on the axes will all be positive.

Similarly, for equation (6) we shall have:

\[ \lg v+l\cdot \lg s+m\cdot \lg t-\lg\left[\frac{4500\,N_e}{C_2}\right]=0\ .\ .\ .\ (12) \]

Here \(N_e\) is, in each given case, a constant quantity, i.e. dependence (12) is for a machine tool of constant power.

The equation of the plane in intercepts on the axes will be

\[ \frac{\lg v}{\lg\left[\frac{4500\,N_e}{C_2}\right]}+ \frac{\lg s}{\frac{1}{l}\cdot \lg\left[\frac{4500\,N_e}{C_2}\right]}+ \frac{\lg t}{\frac{1}{m}\cdot \lg\left[\frac{4500\,N_e}{C_2}\right]} =1. \tag{13} \]

and the intercepts on the axes \(\lg v\), \(\lg s\), and \(\lg t\), respectively, will be

\[ \lg \left[\frac{4500\,N_e}{C_2}\right],\quad \frac{1}{l}\lg \left[\frac{4500\,N_e}{C_2}\right] \quad \text{and} \quad \frac{1}{m}\lg \left[\frac{4500\,N_e}{C_2}\right]: \]

all of them will be positive, since usually:

\[ 4500\,N_e > C_2;\quad l \le 1;\quad m \le 1. \]

In Fig. 1 the dependence for tool life according to equation (11) is presented. Each point of the plane \(Q\) connects three coordinates satisfying equation (11), i.e., meeting the condition for the use of the given tool; and, knowing the position of the plane \(Q\), one can, for a given value \(v_1=\pi D n_1\), obtain \(s_1\) for a prescribed \(t_1\).

Let us draw a plane satisfying the equality

\[ v_1=\pi D n_1=\mathrm{const}. \]

Being parallel to the coordinate plane \(\lg s,\lg t\), it will intersect the plane \(Q\) along the straight line \(AB\), for which the same equality will hold. The straight line \(AB\), like its projection \(ab\), connects all values \(s\) and \(t\) satisfying the condition for use of the tool at the cutting speed \(v_1=\mathrm{const.}\), and, having specified the value \(t\), one can obtain the value \(s\) associated with it, and conversely. The determination of these quantities from Fig. 1 may proceed in the order indicated by the letters

\[ \begin{aligned} &\lg v_1 \to B \to b \to \\ &\qquad\qquad\qquad\quad k \to \lg s_1;\\ &\lg t_1 \nearrow \end{aligned} \]

from this it follows that, in order to solve equation (11), there is no need to know the position of the plane \(Q\) in space; it is important only to know the position of \(cd\)—the line of intersection of the plane \(Q\) with the coordinate plane \(\lg v,\lg s\)—and the direction of the straight line \(AB\), or of its projection \(ab\), which, as is seen from the drawing, run parallel to \(ed\)—the line of intersection of the plane \(Q\) with the coordinate plane \(\lg s,\lg t\).

On this basis, it is not difficult to replace the spatial system of Fig. 1, as inconvenient for use, by a plane one (Fig. 2), by rotating the coordinate plane \(\lg s,\lg t\), as shown in Fig. 1 by the arrow, through \(90^\circ\) until it coincides with the coordinate plane \(\lg v,\lg s\), so that the axis \(\lg t\) serves as a continuation of the axis \(\lg v\); in doing this we do, it is true, limit the extension of the axes \(v\) and \(t\), but this will not cause any complications, since the quantities \(v\) and \(t\) with which one has to operate in practice have limited ranges.

The resulting plane system is presented in Fig. 2, where the conventional values \(v_0,s_0,t_0\) are taken as the origin of coordinates, as the minimum encountered in practice.\(^{1)}\) The position of the straight line \(cd\) is easily determined from equation (10), if we put \(t=t_0=\mathrm{const.}\) In this case the equation of the plane is transformed into the equation of a straight line

\[ \lg v_u=-p\cdot \lg s+\left[\lg C_1-r\cdot \lg t_0\right]\ldots\ldots\ldots (14) \]

located in the coordinate system \(\lg v,\lg s\) and inclined to the positive direction of the abscissa axis \((\lg s)\) at an angle \(\alpha\), whose tangent is

\[ \operatorname{tg}\alpha=-p\cdot \frac{\mu_v}{\mu_s}\ldots\ldots\ldots (15) \]

\(^{1)}\) In Fig. 2 and subsequently we denote not the logarithms of the quantities, but the quantities themselves, as is usually done for convenience in the use of logarithmic graphs.

where \(\mu_v\) is the scale of the divisions of the \(\lg v\) scale,

\[ \mu_s \qquad \text{”} \qquad \text{”} \qquad \text{”} \qquad \lg s \]

We obtain the equation of the straight line \(ed\) by putting in the same equation (10) \(v_u = v_0 = \text{const.}\)

\[ -\lg t = \frac{p}{r}\cdot \lg s - \frac{1}{r}\cdot \left[\lg v_0 - \lg C_1\right] \ldots \ldots (16) \]

It makes, with the positive direction of the axis of abscissas \((\lg s)\), the angle \(\beta\), and

\[ \operatorname{tg}\beta = \frac{p}{r}\cdot \frac{\mu_t}{\mu_s}\ldots \ldots \ldots \ldots \ldots (17) \]

where \(\mu_t\) is the scale of the divisions of the \(\lg t\) scale.

We shall call the straight line \(cd\) the straight line of tool life of the cutter, and the straight line \(ed\) the direction \(v_u = \text{const.}\), because, moving in this direction, we shall always have different values of \(s\) and \(t\) which satisfy the condition of using the cutter (equation 4), under the condition \(v_u = \text{const.}\)

All the foregoing reasoning presupposes the constancy of the quantities \(C_1\), \(p\), and \(r\), whereas they are only conditionally constant for each given case and may change with a change in the principal factors of the work. A change in the kind of material being machined affects the quantities \(p\) and \(r\); the quantity \(C_1\) is affected by the kind of material being machined, its quality, the quality of the cutter, the presence or absence of cooling, etc., as we have already indicated earlier. A change in the quantities \(p\), \(r\), and \(C_1\) will not change the form of the dependence given by equations (10) and (11), and, by changing in (11) the values of the denominators, it will thereby change the position in space of the plane determined by these equations, which will cause in Fig. 2 a change in the position and direction of the straight lines \(cd\) and \(ed\).

By assigning different values to the quantity \(C_1\), we shall obtain in space (Fig. 1) different planes located parallel to the plane \(Q\), which is due to the proportional change of the intercepts on the axes (the denominators in equation (11)).

In the planar system (Fig. 2) we shall obtain parallel straight lines \(cd\) and \(ed\). A change in the quantities \(p\) and \(r\) will entail a non-proportional change in the intercepts on the axes (the denominators in equation 11), which will give the planes in space different inclinations; in Fig. 2 we shall have straight lines inclined differently.

Further on, in the examination of a concrete example, it will become clear how these changes are reflected in the graph of Fig. 2; for the present, under the condition of constancy of the quantities \(C_1\), \(p\), and \(r\), according to Fig. 2 we can, for each assigned cutting speed \(v\), find the quantities \(s\) and \(t\) which fully utilize the life of the given cutter, as has been done for the particular case \((v_1, s_1, t_1)\) in Fig. 2.

If we were now to carry out, for the equations of the power of the machine tool (12) and (13), reasoning similar to that which was carried out for equations (10) and (11), then the results obtained would be entirely analogous. We would arrive at the conclusion that the spatial system can be replaced by a planar one, similar to Fig. 2, but here, in the coordinates \(\lg v \cdot \lg s\), a straight line of constant power of the machine tool would be plotted; in coordinates

the \(\lg t \cdot \lg s\) plane, or, more precisely, the direction \(P=\mathrm{const.}\)¹) The equation of the first straight line is not difficult to obtain by setting in equation (12) \(t=t_0=\mathrm{const.}\)

\[ \lg v=-l\cdot \lg s+\left[\lg\left(\frac{4500\,N_e}{C_2}\right)-m\cdot \lg t_0\right]\ldots(18) \]

The straight line is inclined to the axis of abscissas at the angle \(\gamma\), where

\[ \tg \gamma=-l\cdot \frac{\mu_v}{\mu_s}\ldots\ldots\ldots\ldots\ldots(19) \]

The equation of the second straight line (\(P=\mathrm{const.}\)) is obtained by setting in (12) \(v=v_0=\mathrm{const.}\)

\[ -\lg t=\frac{l}{m}\cdot \lg s+\frac{1}{m}\left[\lg v_0-\lg\left(\frac{4500\,N_e}{C_2}\right)\right]\ldots(20) \]

The straight line makes with the positive direction of the axis of abscissas the angle “\(\delta\)” and

\[ \tg \delta=\frac{l}{m}\cdot \frac{\mu_t}{\mu_s}\ldots\ldots\ldots\ldots(21) \]

Plotting these straight lines on Drawing 2, we obtain Drawing 3, in which \(fg\) is the straight line \(N_e=\mathrm{const.}\), and \(gh\) is the straight line \(P=\mathrm{const.}\)

Proceeding parallel to the direction \(gh\), we shall obtain values of \(s\) and \(t\) which, at the given speed, make use of a machine of power \(N_e\).

In constructing equations (12) and (13), we adopted as constant quantities \(N_e\), \(C_2\), \(l\), and \(m\). In reality, however, they will be constant only in each given case; changes in them will affect the position of the plane of equation (12) in space, and consequently also the position of the straight lines \(N_e=\mathrm{const.}\) and \(P=\mathrm{const.}\) Changes in the material being machined, its quality, the angles of the cutter and the form of the cutting edge, as well as the presence and type of cooling-lubricating liquid, affect the value of \(C_2\). Without loss of accuracy one may disregard the effect of the presence of cooling, since, according to the experiments of Prof. Savvin²), even with abundant cooling by means ordinarily used in practice, one may expect a change in the value of \(C_2\) within only \(1\%\)—\(3\%\).

A change in the material being machined also entails a change in the quantities \(l\) and \(m\); moreover, for the case of machining iron and steel we shall have one set of values, for cast iron another, for bronze a third, etc.

From equation (13) it may be concluded that, by changing the values of \(C_2\) and \(N_e\), we shall obtain different planes parallel to one another, and in Drawing 3 a series of parallel straight lines \(N_e=\mathrm{const.}\) and \(P=\mathrm{const.}\); but, changing the quantities \(l\) and \(m\), we shall obtain different planes not parallel to one another, and in Drawing 3 non-parallel straight lines.

Thus, taking into account all that has been said above, it may be concluded that Drawing 3 is constructed for machining, with a definite cutter, one particular material being machined, on a machine of a given power, and with the aid of this drawing one can solve questions connected with the use of the data of the cutter and the machine.

¹) We have so called this direction because, proceeding along it, we shall have, for the coordinates of each point, such values of \(s\) and \(t\) as correspond to the use of a machine of the given power \(N_e=\mathrm{const.}\) at \(v=\mathrm{const.}\); in this case, undoubtedly, the condition of constancy of pressure on the cutter is present: \(P=\mathrm{const.}\)

²) Savvin. Cooling of the cutter.

Indeed, if the cutting speed \(v_1\) is specified and it is required to find the conditions (\(s\) and \(t\)) that, at the given cutting speed, simultaneously utilize both the machine tool and the cutting tool, then, on the basis of the preceding, the straight line \(ba\) connects all values \(s\) and \(t\) that utilize the cutter; the straight line \(b'a'\) connects the values \(s\) and \(t\) that, at the same cutting speed, utilize a machine tool of the given power; and their point of intersection, having coordinates \(s_1\) and \(t_1\), satisfies both conditions simultaneously. The straight lines “\(ba\)” and “\(b'a'\),” parallel respectively to the directions \(v_u=\text{const}\) and \(P=\text{const.}\), will intersect at the angle \(x\) (see Fig. 3), where

\[ x=\rho-\delta \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (22) \]

and the angle \(x\) will have a finite value different from zero, since in this latter case (when \(x=0\)) it would be necessary to satisfy the equality

\[ \tg \delta=\tg \rho \]

or, on the basis of the preceding (21) and (17),

\[ \frac{l}{m}=\frac{p}{r}. \]

According to the data of the overwhelming majority of investigations, \(r<p\) and \(l<m\), i.e.:

\[ \frac{l}{m}<1; \]

\[ \frac{p}{r}>1; \]

and

\[ \frac{l}{m}<\frac{p}{r}. \]

Thus, theoretically, in all cases we shall have a point of intersection of the directions \(v_u=\text{const.}\) and \(P=\text{const.}\), i.e., at any cutting speed we can find the values of feed and depth of cut, and consequently also of the chip cross-section, that simultaneously utilize both the machine tool and the tool.

On this basis, in opposition to Engineer Hippler, who asserts that, for a given tool, for each machine-tool power there is only one cutting speed that simultaneously utilizes both the machine and the cutter at one quite definite “characteristic” chip section, it may be stated that for each machine-tool power, at every cutting speed, there exists its own chip section, with quite definite “characteristic” values of \(s\) and \(t\), that simultaneously utilizes both the machine tool and the cutter.

The reason for the disagreement lies in the fact that Hippler, in his conclusions, proceeds from the dependencies of Prof. Friedrich, who, in contrast to all other investigators, assumes in equations (4) and (6)

\[ l=m,\quad \text{and}\quad p=r, \]

which is refuted also by the very latest investigations\(^1\).

\(^1\) Glebov. The Cutting Process as a Single Empirical Formula. 1923, p. 29.
A. Wallisch. Das Gesetz der Schnittgeschwindigkeit beim Drehvorgang. Maschinenbau, 1927, s.—937.
Klopstok. Die Untersuchung der Drehbarkeit. 1926.
Rapatz und Krekeler. Die Prüfung der Bearbeitbarkeit. Stahl und Eisen. 1928, s.—257.

If we were given in advance the depth of cut \(t_2\), then from the condition of tool durability we could, according to Fig. 3, take the feed \(s_2\), and from the condition of using the machine tool, the feed \(s_2'\), at which we should have to stop as the smaller one; in this case the cutter would not be fully utilized.

If in equation (18) we assign various values to \(N_e\), we obtain equations for different machine-tool powers. Plotting these equations in Fig. 3, we obtain a bundle of parallel straight lines, each of which corresponds to a machine tool of a definite power. If, likewise, in equation (14) we assign various values to the quantity \(C_1\), we obtain dependences for the durability of different cutters (of which there should ordinarily be no more than 2 or 3 types), and it will not be difficult to take into account the effect of cooling as well. By plotting these dependences also in Fig. 3, we obtain the graph schematically represented in Fig. 4, which can be constructed for all the most commonly used powers of machine tools and for different cutters. The directions \(P = \mathrm{const}\) and \(v_u = \mathrm{const}\) in all these constructions will remain the same. The graph in Fig. 4 will make it possible to solve questions of the use of machine tools of different powers and of different cutters in machining any one material. If we wished to take another material to be machined, then in the original equations this would change the values \(l, m, p, r, C_1\), and \(C_2\); in Fig. 4, without exception, all the lines would acquire another position and inclination. It does not appear possible to combine all this in one graph, and therefore we are compelled to have a separate graph for each material to be machined, which can be done quite simply, as will be indicated below.

In what follows we shall examine several concrete examples and, in order to give them a simpler form and a more intelligible presentation, we shall take Taylor’s data for the fundamental dependences. By this we do not wish to say that we consider Taylor’s conclusions the only correct ones, but we dwell on them only as the best known and as having the most solid foundation of experimental material.

Taylor divides all grades of steel and cast iron into 40 classes, so that each succeeding class permits, in machining, a typical cutting speed 10% lower than the preceding one. (Thus the subdivision is based on the property which in modern German literature is denoted by the term “Bearbeitbarkeit”—machinability). But in view of the fact that subdivision into 40 classes is extremely cumbersome and practically unrealizable, Taylor himself divided steel and cast iron into 6 classes, which may be considered quite sufficient for practical purposes:

1. Mild steel — \(K_z = 30\text{–}40\ \mathrm{kg}/\mathrm{mm}^2\);
2. Medium-hard steel — \(K_z = 40\text{–}60\ \mathrm{kg}/\mathrm{mm}^2\);
3. Hard steel — \(K_z = 60\text{–}80\ \mathrm{kg}/\mathrm{mm}^2\);
4. Soft cast iron;
5. Medium-hard cast iron;
6. Hard cast iron.

The values of all the quantities in Taylor’s fundamental dependences are given on the assumption of the indicated division of the materials being machined.

When working without cooling with a roughing cutter of medium size \((7/8'')\), made of “the newest high-speed” steel, for a one-and-a-half-hour-

for the duration of cutting, Taylor gives the following values, obtained by him from numerous experiments:

Table 1.

No. in order	Material machined	\(C_1\)	p	r	\(C_2\)	\(l\)	m
1	Mild steel . . . . . . . . . . . .	128	0.56	0.42	180	0.93	1.0
2	“ medium hardness . . . . . . .	64	“	“	200	“	“
3	“ hard . . . . . . . . . . . .	29	“	“	237	“	“
4	Soft cast iron . . . . . . . . . .	58	0.46	0.29	88	0.75	0.93
5	“ medium hardness . . . . . . .	29	“	“	113	“	“
6	“ hard . . . . . . . . . . . .	17	“	“	135	“	“

In this table the quantities \(C_1\) have been calculated by us on the basis of Taylor’s tables of practical cutting speeds\(^{1}\); all the other quantities are available in the technical literature. Thus, according to Taylor’s data, we shall have the basic relations (4) and (6) in the following form:

For cast iron:

\[ v_u=\frac{C_1}{s^{0.46}\cdot t^{0.29}}; \]

\[ N_e=\frac{C_2\cdot v\cdot s^{0.75}\cdot t^{0.93}}{4500}; \]

For steel:

\[ v_u=\frac{C_1}{s^{0.56}\cdot t^{0.42}}; \]

\[ N_e=\frac{C_2\cdot v\cdot s^{0.93}\cdot t}{4500}. \]

Since all our diagrams consist exclusively of straight lines, for their construction it is sufficient to know the angle of inclination of each straight line and the coordinates of one of its points; for the straight lines \(P=\mathrm{const.}\) and \(v_u=\mathrm{const.}\) it is sufficient to know only their direction.

We have taken as initial coordinates the values \(s_0=0.1\) mm/rev., \(v_0=2\) m/min. and \(t_0=2\) mm, as the minimum encountered in practice.

The equation of the tool-life straight line in the coordinates \(\lg v \cdot \lg s\) (the straight line \(cd\) in Fig. 3) will be (see eq. 14):

for cast iron

\[ \lg v_u=-0.46\cdot \lg s+\left[\lg C_1-0.29\cdot \lg 2\right]; \]

for steel:

\[ \lg v_u=-0.56\cdot \lg s+\left[\lg C_1-0.42\cdot \lg 2\right]. \tag{14′} \]

Substituting, in place of \(C_1\), the values from Table 1, and assigning a particular value to \(s\), we obtain the second coordinate “\(v\)” of a point of the tool-life straight line (see Table 2).

\(^{1}\) Taylor. The Art of Cutting Metals, p. 346, 1909.
Tikhonov. On the Cutting Coefficient, pp. 28–32.

Table 2.

\(S\) mm/rev (particular value)	\(v_{u}\), meters per minute — Cast iron, Soft \(C_1 = 58\)	\(v_{u}\), meters per minute — Cast iron, Medium \(C_1 = 29\)	\(v_{u}\), meters per minute — Cast iron, Hard \(C_1 = 17\)	\(v_{u}\), meters per minute — Steel, Soft \(C_1 = 128\)	\(v_{u}\), meters per minute — Steel, Medium \(C_1 = 64\)	\(v_{u}\), meters per minute — Steel, Hard \(C_1 = 29\)
1 mm	47	23,75	13.9	95,5	47,75	21,65

The angle of inclination of the straight line for tool life according to Eq. (15), with equality of the scale divisions \((\mu_v = \mu_s)\), will be:

for cast iron:

\[ \operatorname{tg}\alpha = -0{,}46; \quad \alpha = 155^\circ \]

for steel:

\[ \operatorname{tg}\alpha = -0{,}56; \quad \alpha = 150^\circ \tag{15′} \]

The equations of the straight lines \(N_e = \mathrm{const}\) (the straight line \(fg\) in Fig. 3) will be, according to equation (18):

for cast iron:

\[ \lg v = -0{,}75 \cdot \lg s + \left[\lg\left(\frac{4500\,N_e}{C_2}\right) - 0{,}93 \cdot \lg 2\right] \]

for steel:

\[ \lg v = -0{,}93 \cdot \lg s + \left[\lg\left(\frac{4500\,N_e}{C_2}\right) - \lg 2\right] \tag{18′} \]

Substituting here the values of \(C_2\) from Table 1 and assigning various \(N_e\), we can calculate the coordinates of one of the points of the straight lines (see Table 3).

Table 3.

mm/rev \(S\)	in horse-power \(N_e\)	\(v\), meters per minute — Cast iron, Soft \(C_2 = 88\)	\(v\), meters per minute — Cast iron, Medium \(C_2 = 113\)	\(v\), meters per minute — Cast iron, Hard \(C_2 = 135\)	\(v\), meters per minute — Steel, Soft \(C_2 = 130\)	\(v\), meters per minute — Steel, Medium \(C_2 = 200\)	\(v\), meters per minute — Steel, Hard \(C_2 = 237\)
5	1	8,01	6,23	5,2	3,81	2,51	2,1
5	2	16,02	12,46	10,4	7,62	5,0	4,2
5	3	24,03	18,69	15,6	11,4	7,5	6,3
5	4	32,04	24,92	20,8	15,2	10,0	8,4
5	5	40,05	31,15	26,0	19,0	12,5	10,5
5	6	48,06	37,38	31,2	22,8	15,0	12,6
5	7	56,07	43,61	36,4	26,7	17,5	14,7
5	8	64,08	49,84	41,6	30,4	20,0	16,8
5	9	72,09	56,07	46,8	34,2	22,5	17,9
5	10	80,1	62,3	52,0	38,1	25,1	21,0

The angle of inclination of the straight lines \(N_e=\mathrm{const.}\) is determined from equation (19),

for cast iron:

\[ \operatorname{tg}\gamma=-0.75;\qquad \gamma=143^\circ \]

for steel:

\[ \operatorname{tg}\gamma=-0.93;\qquad \gamma=131^\circ \tag{19'} \]

The straight lines \(v_{\mathrm{c}}=\mathrm{const.}\) will have an angle of inclination according to equation (17), where the ratio of the scales of the axes has been adopted by us, for the construction, as

\[ \frac{\mu_t}{\mu_s}=1.3; \]

for cast iron:

\[ \operatorname{tg}\beta=1.3\,\frac{0.46}{0.29}=2.06;\qquad \beta=64^\circ \]

for steel:

\[ \operatorname{tg}\beta=1.3\,\frac{0.56}{0.42}=1.74;\qquad \beta=60^\circ \tag{17'} \]

The straight lines \(P=\mathrm{const.}\) will have an angle of inclination according to equation (21),

For cast iron:

\[ \operatorname{tg}\delta=1.3\,\frac{0.75}{0.93}=1.06;\qquad \delta=46^\circ \]

For steel

\[ \operatorname{tg}\delta=1.3\cdot 0.93=1.21;\qquad \delta=50^\circ \tag{21'} \]

Using the values obtained in this way, the graph of drawing 5 has been constructed for the case of machining cast iron of medium hardness. Let us give several examples characterizing the applicability and the methods of using the proposed graphs.

Suppose we have a rough-turning lathe with a three-step pulley and back gear. This lathe has the following numbers of revolutions and the corresponding powers:

Pulley step	Spindle revolutions per minute “\(n\)” With back gear	Spindle revolutions per minute “\(n\)” Without back gear	Useful power \(N_e\)
Large . . . . .	\(n_1=13\)	\(n_4=42\)	3.15
Medium . . . . .	\(n_2=18\)	\(n_5=61\)	3.6
Small . . . . .	\(n_3=26\)	\(n_6=86\)	4.25

The maximum chip cross-section permitted by the strength of the feed mechanism when machining cast iron of medium hardness is \(q_{\max}=15\) mm. (All these data are contained in the machine’s specification card.) Diagram²

the numbers of revolutions and cutting speeds, borrowed from the machine-tool specifications compiled according to the AWF system, is presented in Fig. 6. On the lines of the numbers of revolutions the corresponding useful powers are marked; here the slight reduction of the efficiency when the back gear is engaged has not been taken into account, and the power with and without the back gear has been taken as one and the same.

Example 1. Suppose it is necessary to machine a cast-iron cylinder of diameter 250 mm, with a depth of cut \(t = 10\) mm.

It is required to determine at what feed \(s\) and with what number of revolutions \(n\) (with what cutting speed) one must work in order to obtain the maximum productivity \((ns)_{\max}\).

The solution of the problem posed is possible by trial, proceeding from the smallest number of revolutions to the largest, constantly making sure that the chip section does not exceed the limiting value permitted by the strength of the feed mechanism¹.

In our case the limiting chip section is \(15\ \text{mm}^2\), which, for the specified depth \(t = 10\) mm, gives the maximum feed value

\[ S_{\max} = 1.5\ \text{mm/rev}. \]

As was indicated earlier, the starting point in solving the problem posed is the cutting speed, as a quantity fully determined, for a given diameter, for each number of revolutions. Beginning with the smallest number of revolutions \(n_1 = 13\), from Fig. 6 we find that for the specified diameter of 250 mm we shall have a cutting speed \(v = 10\ \text{m/min}\). From Fig. 5, proceeding horizontally from \(v = 10\) on the speed scale to the straight line of tool life (in the direction \(ab'\)), then vertically downward to the feed scale \((b'c')\), then in the direction \(c'd'\), parallel to \(v_u = \text{const}\), to the intersection at point \(d'\) with the horizontal drawn from the specified depth of cut \((t = 10)\), and thence vertically to the feed scale, we obtain the feed allowable for the cutter at the given value of \(v\) and \(t\). This feed will be \(s' = 2.4\ \text{mm/rev}\).

In the same way, from \(v = 10\) on the speed scale, we proceed horizontally to the straight line \(N_e = \text{const}\), corresponding to the machine power at the given number of revolutions. In this case we should reach the straight line \(N_e = 3.15\) HP, but since there is no such line on the diagram, and there are 3 and 4 HP, by interpolating we arrive at point “\(b\),” which approximately corresponds to it. From there we go vertically downward \((bc)\) to the feed scale, then, in the direction \(P = \text{const}\), \((cd)\), to the intersection at point \(d\) with the horizontal \(t = 10\), and thence vertically upward to the feed scale, where we obtain the feed allowable by the machine power, \(s = 1.7\ \text{mm/rev}\). Thus, at this number of revolutions, from the condition of tool life we may take \(s' = 2.4\ \text{mm/rev}\); from the condition of utilizing the machine power, \(s = 1.7\ \text{mm/rev}\). One ought therefore to stop at the latter, and the cutter would then be somewhat underloaded; but in view of the fact that the feed mechanism permits only \(s = 1.5\ \text{mm/rev}\), it will be necessary

this; both the lathe and the cutter will not thereby be used. The quantity \(n_1 \cdot s\), characterizing the productivity of the given combination, will be

\[ n_1 \cdot s = 13 \cdot 1.5 = 19.5; \]

(as a characteristic of productivity one may also take the quantity \(v_1 \cdot s = 10 \cdot 1.5 = 15\)).

For the number of revolutions \(n_2 = 18\), the cutting speed according to the diagram in Fig. 6 will be \(v = 14\) m/min, and from Fig. 5, proceeding along the path \(ef'g'h'\), we find a feed using the cutter \(s'' = 1.1\) mm/rev; proceeding, however, along the path \(efgh\), a feed \(s_2 = 1.3\) mm/rev, using the lathe power of 3.6 HP corresponding to the adopted number of revolutions. Both feed values are less than the limiting one; we can realize \(s'' = 1.1\) mm/rev, while the lathe power remains somewhat unused, and the quantity characterizing the productivity of the given combination will be

\[ n_2 s = 18 \cdot 1.1 = 19.8 \]

i.e. greater (though only very slightly) than in the first case.

Doing the same for the number of revolutions \(n_3 = 26\), we obtain, for using the cutter, a feed \(s''' = 0.53\) mm/rev; for using the lathe, \(s_3 = 1\) mm/rev. We can realize the smaller one, and the productivity characteristic will be

\[ n_3 s = 26 \cdot 0.53 = 13.8 \]

i.e. less than in the second case.

There is no point in doing the same for the remaining numbers of revolutions, since, if productivity decreases in going from the number of revolutions \(n_2 = 18\) to \(n_3 = 26\), then in further attempts to use the higher numbers of revolutions \(n_4\), \(n_5\), and \(n_6\), we shall have an ever decreasing productivity; this is not difficult to verify. Consequently we may conclude that the most advantageous combination will be \(n_2 = 18\) and \(s = 1.1\) mm/rev.

Example 2. If the depth of cut had not been specified and it were required, at the same cutting speeds (10, 14, and 20 m/min), to achieve simultaneous utilization of both the lathe and the tool, then we would seek the points of intersection of the straight lines \(cd\) and \(c'd'\), \(gh\) and \(g'h'\), \(lm\) and \(l'm'\). The coordinates of these points would give us the desired values of \(s\) and \(t\), and the productivity of each combination would be characterized by the quantity \(qv = s.t.v\). One such point is present in Fig. 5 at the intersection of the straight lines \(cd\) and \(c'd'\) (point \(O\)); the others lie on the extensions of the straight lines indicated above. It was shown above that such a point always exists; now we shall merely note that often it will lie outside the limits of the drawing, i.e. its coordinates will be values of \(s\) and \(t\) not used in practice, which, however, does not change the essence of what has been proved. Hence we see that one and the same cutter can be used with lathes of different power, which is what we have in the present case, by selecting the corresponding feed and depth of cut; and conversely, one and the same lathe can be used with different cutters.

In Fig. 7 a diagram is plotted for the case of machining medium-hard steel with various cutters, the upper straight line corresponding to a high-speed-steel cutter, the two lower ones to self-hardening steel. For the self-hardening-steel cutter, two straight lines are plotted, the upper of which corresponds to the case of this cutter working with cooling; for this we used the data given by Glebov.^1

^1 Glebov. The Art of the Most Advantageous Machining of Metals. 1927, p. 68.

When abundant cooling is used, the practically attainable increase in cutting speed will be:

In machining with	Medium and soft steel	Hard steel and cast iron
carbon-steel cutters	15%	5%
self-hardening cutters	20%	7%
high-speed cutters	25%	8%
vanadium cutters	30%	10%

Having such a graph, on which the tool-life relations of several different cutters are plotted, we can already decide the question of choosing one cutter or another for carrying out a given operation on a machine of definite power.

Example 3. Suppose we have a machine with a speed box, i.e., with a constant speed of the driving belt, and therefore also with constant power for all numbers of revolutions (if we neglect the insignificant change in efficiency for the different combinations of transmission gears in the box). For a given machining diameter, each number of revolutions gives a definite speed. Let one of these speeds be \(v = 16\) m/min; the given depth of cut is \(t = 9\) mm; the power of the machine \(N_e = 2.5\) HP.

Proceeding along the path abcde, we find that, when using with cooling a cutter of self-hardening steel, at the given depth of cut it is possible to employ a feed \(s = 0.37\) mm/rev. Proceeding along the path ab′c′de, we find, for using the machine of the indicated power, the very same value \(s = 0.37\) mm/rev. Thus in this case there is simultaneous utilization both of the cutter and of the machine.

Taking a high-speed-steel cutter and proceeding along ab″c″d″e″, we find that this cutter can be used only at a feed \(s'' = 2.3\) mm/rev, which is much greater than the value permitted by the power of the machine. On the other hand, if we were to take a self-hardening cutter without cooling, it could permit a feed \(s_0 = 0.27\) mm/rev, which we would find by proceeding along the path \(ab_0c_0d_0e_0\), and this value would be insufficient for utilizing our machine.

It is not difficult to conclude that, under the given working conditions \((v, t)\), the cutter most suitable for our machine is one of self-hardening steel, provided that it is cooled during operation. But from this it is also evident that the power we have taken is, under the given conditions, the limiting power that the mentioned cutter can still utilize, and that, if we were to take a machine of somewhat greater power, then in order to utilize it we should already have to use a cutter of high-speed steel.

Finally, with the aid of the graphs presented, the question of selecting or designing a machine for a definite operation is solved extremely simply.

Example 4. Suppose it is required to turn steel shafts of diameter 110 mm and length 900 mm, with a depth of cut \(t = 9\) mm. Under this condition, the stability of the workpiece being machined (absence

…burning) the maximum chip cross-section is established as \(q = 9\ \text{mm}^2\),¹⁵ which, at the given depth of cut, gives the limiting value of the feed \(s_{\max} = 1\ \text{mm/rev}\).

It is necessary to find the required power of the machine tool and the number of revolutions corresponding to the simultaneous use of both the machine tool and the tool at the maximum possible values \(t = 9\ \text{mm}\) and \(s = 1\ \text{mm/rev}\). As we shall now see, the results of the solution will differ depending on which cutter we use in the work.

Since we have made it a condition to use the machine tool and the tool simultaneously at \(s = 1\ \text{mm/rev}\) and \(t = 9\ \text{mm}\), it is clear that the directions \(v_u = \text{const.}\) and \(P = \text{const.}\) must intersect at a point having as its coordinates the above-mentioned quantities. In Fig. 7 this is the point \(f\). Proceeding from it to the feed scale in the direction \(v_u = \text{const.}\) to the point \(g\), then vertically to the intersection at the point \(h\) with the durability line of the self-hardening cutter, and then horizontally to the left to the speed scale, we obtain the cutting speed \(v = 7.4\ \text{m/min}\), which, together with the above-mentioned quantities \(s\) and \(t\), fully utilizes the cutter. This same speed must also be utilized by the machine tool, whose power we obtain if, from the same point \(f\), we proceed in the direction \(P = \text{const.}\) to the feed scale to the point \(g'\), and then vertically to its intersection with the obtained speed at the point \(h'\), which corresponds to the machine-tool power \(N_e = 3\ \text{HP}\).

Knowing the cutting speed and the machining diameter, it will not be difficult to find the required number of revolutions.

If we had set ourselves the case of working with a cutter made of high-speed steel, then from the point \(g\) it would be necessary to draw a vertical to the point \(k\), which would determine for us the cutting speed \(v_1 = 25\ \text{m/min}\); the vertical from the point \(g'\), at its intersection with the found speed \(v_1 = 25\) at the point \(k'\), will determine for us the machine-tool power required in the given case: \(N_e = 10\ \text{HP}\).

The speed found, together with the given diameter, determines the required number of revolutions. Here, incidentally, we see confirmation of the long-established proposition that a higher-grade cutter requires, for its use, a more powerful machine tool.

Limiting ourselves to the examples given, in conclusion we shall describe the method of constructing an instrument that uses our graphs. On a sheet of transparent celluloid, logarithmic grids and the scales \(v\), \(s\), and \(t\) are drawn, corresponding to the fields \(ABCD\) and \(EFGH\) in Fig. 7. The celluloid is pasted onto cardboard in such a way that, under the field \(ABCD\) in the direction from \(BC\) to \(AD\), and under the field \(EFGH\) from \(GH\) to \(EF\), sheets of cardboard cut exactly to the width of the fields can be inserted. On the cardboard sheets inserted under the field \(ABCD\), only the straight lines \(N_e = \text{const.}\) and the durability lines of the various cutters are drawn, in such a way that each sheet, when inserted under the logarithmic grid, gives the relationship between \(v\) and \(s\) for one material being machined (as, for example, in Fig. 5 and Fig. 7). Thus the number of sheets inserted under the field \(ABCD\) will be equal to the number of materials being machined—in our case, three for steel and three for cast iron. In order to be able to judge the correctness of insertion, two or three reference lines are drawn on the sheets, corresponding—

...corresponding to some values of \(v\) and \(s\); in addition, it is necessary to indicate to which work material the given sheet corresponds.

Under the field EFGH two sheets of celluloid are inserted at the same time, cut exactly to the size \(EF = GH\). On one of them one or several directions \(v_u = \mathrm{const.}\) are drawn; on the other, \(P = \mathrm{const.}\), and in such a way that, by moving these sheets in the direction \(FE \to GH\), the beginnings of the straight lines \(v_u = \mathrm{const.}\) and \(P = \mathrm{const.}\) can be joined to any point of the feed scale. On the sheet \(P = \mathrm{const.}\) it is useful to write “Machine,” and on the sheet with \(v_u = \mathrm{const.}\), “Cutter.” For lack of celluloid, two sheets of cardboard may be placed under the field EFGH; the edges of one are cut in the direction \(v_u = \mathrm{const.}\), and those of the other in the direction \(P = \mathrm{const.}\). There will be as many such pairs of insert sheets under the field EFGH as there are different directions \(v_u = \mathrm{const.}\) and \(P = \mathrm{const.}\). In our case one pair will be for steel, the other for cast iron, in accordance with which the appropriate markings must be made on them.

The method of use is self-evident. Thus, in the third example (see Fig. 7), under the field ABCD a sheet corresponding to steel of medium hardness is inserted. We proceed mentally from point “a” to “b,” then to \(c\); then, in the field EFGH, we move the insert bearing the inscription “Cutter” until one of the straight lines \(v_u = \mathrm{const.}\) drawn on it joins with its beginning to point \(c\). This straight line will indicate to us the position of point \(d\), and hence also the magnitudes of \(s\) and \(t\); and so on.

The dimensions of the graphs may be made larger than those given by us, which contributes to greater precision in the work. It is useful, next to the field ABCD, to draw a graph of the number of revolutions, similar to Fig. 6, whose axis \(v\) would be adjacent to the axis \(v\) in Fig. 7, with the scales of the velocity scales being the same. If, however, AWF passport cards have to be used, the need for a graph of the number of revolutions disappears, since such a graph is provided on the machine’s own card.

In conclusion, we consider it necessary to express our gratitude to Prof. T. I. Tikhonov for reviewing the present article and for his assistance in its printing.

Tomsk. V—1928.

Laboratory of Mechanical Technology

Metallov S. T. I.

To the article by A. M. Rosenberg

Fig. 1

\[ oc=\lg C_1,\qquad od=\frac{\lg C_1}{p},\qquad oe=\frac{\lg C_1}{2}. \]

Diagram labels: \(\lg v\), \(\lg t\), \(\lg s\), \(A\), \(B\), \(C\), \(d\), \(e\), \(o\), \(K\), \(t\), \(s\), \(\lg v_1\), \(\lg t_1\), \(\lg s_1\), \(\lg s_2\).

Fig. 2

Diagram labels: \(\lg v\), \(\lg t\), \(\lg s\), \(a\), \(b\), \(c\), \(d\), \(e\), \(B\), \(K\), \(S_0\), \(S_1\), \(b\), \(t_0\), \(t_1\), \(\alpha\), \(\beta\), \(\lg v=\mathrm{const.}\).

Fig. 4

Visible inscriptions and labels:

• No. 1
• No. 2
• No. 3
• No. 4
• No. 5
• No. 6
• Cutter 1
• Cutter 4 B
• \(v = \mathrm{const}\)
• \(s = \mathrm{const}\)
• \(P_{qv}\)
• \(P_{qt}\)
• \(P_{qs}\)
• \(t\)
• \(s\)
• \(t_s\)
• \(t_j\)
• \(\alpha\)
• \(\beta\)
• \(\delta\)
• \(\gamma\)

Fig. 3

Visible inscriptions and labels:

• \(N_e = \mathrm{const.}\)
• \(P = \mathrm{const}\)
• \(v = \mathrm{const}\)
• Fig. 3
• \(P_{qv}\)
• \(P_{qt}\)
• \(P_{qs}\)
• \(v\)
• \(v'\)
• \(s_1\)
• \(s_2\)
• \(s'_2\)
• \(t_1\)
• \(t_2\)
• \(d\)
• \(d'\)
• \(C\)
• \(B\)
• \(B'\)
• \(G\)
• \(g\)
• \(n\)
• \(\alpha\)
• \(\beta\)
• \(\gamma\)
• \(\delta\)

Steel of M. Rosenberg.
Cast iron, medium hard.

Fig. 5.

Top diagram:

• Cutting depth, \(t\), in mm.
• Cutting speed, \(v\), in m/min.
• Curve annotations visible:
  \(N_{\max}=1{,}3\); \(N_{\max}=1{,}8\); \(N_{\max}=2{,}6\); \(N_{\max}=4{,}2\); \(N_{\max}=6{,}7\); \(N_{\max}=8{,}8\); \(N_{\max}=12{,}5\); \(N_{\max}=5{,}6\); \(N_{\max}=3{,}15\); \(N_{\max}=1{,}25\); \(N_{\max}=0{,}86\); \(N_{\max}=0{,}1\).
• Marked cutting speeds: \(a\), \(e\), \(i\), with corresponding \(v\) values approximately \(10\), \(14\), and \(20\) m/min.

Middle diagram:

• Cutting speed, \(v\), in m/min.
• Useful power, \(N_e\).
• Feed, \(s\), in mm per revolution.
• Diagonal label: Speed.
• Curve/line annotations visible: \(v=\mathrm{const.}\), \(s=\mathrm{const.}\), \(N_e=\mathrm{const.}\), \(a\), \(e\), \(i\), \(k\), \(f\), \(b\).

Lower diagram:

• Cutting depth, \(t\), in mm.
• Feed, \(s\), in mm per revolution.
• Annotations visible: \(m\), \(n\), \(d\), \(e\), \(i\), \(v=\mathrm{const.}\), \(s=\mathrm{const.}\).

Inscription by A. M. Rozenberg.

Medium-hard steel.

Chart 7.

1. Strictly speaking, the question of the load on the machine-tool feed mechanism, depending on the cutting conditions, cannot be considered sufficiently clarified; from the time of Nicolson’s experiments up to the present day we have had the most contradictory data. Recent investigations, which unquestionably deserve attention (Klopstok. Die Untersuchung der Dreharbeit. 1926), shed an entirely new light on this question, still further extending the possibilities of using the machine tool; but since we still do not have substantiated relationships to which we could give a simple graphical expression, we must assume that the strength of the feed mechanism is characterized by a certain chip section, calculated from one of the sources indicated above, which, for a specified depth of cut, gives the limiting feed value \(S_{\max}\). ↩↩↩↩↩

2. Orentlikher. Technical Rating of Work of Metal-Industry Enterprises, 1926, p. 187.
   Hegner. Manual for Preliminary Cost Calculation, 1926, p. 246.
   Vestnik metalloprom. 1927, No. 11. Orentlikher’s article “The Present State of the Question of Machining Metals.” ↩↩↩

3. Z.V.D.J. 1914, Nos. 10 and 12. Ueber die Wärmevorgänge... ↩

4. R.E.F.A. p. 230; Génie Civil. 1917. ↩

5. R.E.F.A. p. 231. ↩

6. Katkov. Investigation and Use of Machine Tools, 1925; p. 244. ↩

7. Time. Resistance of Metals and Wood to Cutting. 1870. ↩

8. Zvorykin. Work and Forces Necessary for Separating Metal Chips. 1893. ↩

9. Chelyustkin. The Influence of Chip Dimensions on the Force of Metal Cutting. 1925. ↩↩

10. Savvin. On the Resistance of Metals to Cutting. 1908. ↩

11. Friedrich. Ueber den Schnittwiderstand... Z.V.D.J. 1909. ↩

12. Chelyustkin. The Influence of Chip Dimensions on the Cutting Force of Metals. ↩

13. Tikhonov. On the Coefficient of Metal Cutting. Tomsk, 1910. ↩

14. Klopstok. Die Untersuchung der Dreharbeit. 1926; Vestnik Metallopromyshlennosti, 1928, No. 1, p. 71. ↩

15. Hegner, p. 246, Fig. 92 and others. ↩