Associate Professor A. M. ROZENBERG.

ON THE ISSUE OF OPERATING A CONSTANT-POWER DRILLING MACHINE

Questions of the rational operation of metalworking machine tools at the present time, in our machine-building industry, are acquiring especially urgent significance in connection with the leading role of machine building in the second five-year plan. The task before us of mastering modern equipment cannot be conceived merely as the study of equipment from the design-and-kinematic standpoint. The task of mastery is still not completely solved if a machine tool operates and produces output that is satisfactory from the standpoint of technical inspection. This task may be considered fully solved only in the case where we are able not only to start up the machine and remove the finished article from it, but also to use it in the most advantageous way, to obtain from it maximum productivity without detriment to the quality of the product, on the one hand, and without overloading the machine tool and the tool working on it, on the other.

The science of metal cutting proves that even under full load of a metalworking machine tool, the latter can yield different productivity depending on the operating conditions assigned to it. The choice of a rational operating regime for metalworking machine tools is a weak point both of our enterprises and of the technical literature. The authors of works on the theory of metal cutting, while sometimes analyzing the cutting processes in extraordinary detail, say almost nothing, or very little, about the practical application of this science to questions of machine-tool operation, which in the final analysis should be the aim of studying the science of metal cutting.

Questions of machine-tool operation are not so simple that they can be passed over in silence, and the purpose of the present article is, if only to a very small extent, to fill a gap in the technical literature in the section on machine-tool operation. In the article we propose a method for selecting the most advantageous working conditions on constant-power drilling machines, i.e., those having a speed box or an individual motor. This does not exhaust all drilling machines, but in modern production the indicated group is extremely widespread.

General propositions.

The productivity of work on a drilling machine is characterized by the product $ns$,

where $n$ is the number of revolutions of the drill per minute

$s$ is the feed in millimeters per 1 revolution of the drill.

It is not difficult to imagine that the quantity \(ns\) is the depth of the hole drilled in one minute.

The productivity of two drilling processes may also be compared by the quantity \(vs\), where \(v\) is the cutting speed in meters per minute. Since, for a given drilling diameter \(d\) in millimeters,

\[ v=\frac{\pi dn}{1000}, \]

then \(vs\) will be proportional to \(ns\).

When working on a drilling machine, having a definite material to be machined and a specified hole diameter, one must choose \(n\), the number of revolutions of the drill, and \(s\), its feed, so that, under normal use of the machine and the tool, the product \(ns\) or \(vs\) is as large as possible; this will characterize the maximum possible productivity of the process, i.e., the minimum expenditure of time for carrying out the given operation.

The quantity \(n\) has to be chosen on the basis of the cutting speed permissible for the drill when machining the given material, while being guided by the range of spindle speeds available on the machine,

\[ n_1 — n_z. \]

The questions of drill life are at present still an insufficiently studied area of the theory of metal cutting, and, despite a whole series of extensive experimental investigations that have been carried out, we still do not have sufficiently reliable relations connecting the permissible cutting speed \(v\) with the feed \(s\) and the drill diameter \(d\), although such a relation \([v=F(s,d)]\), judging from the nature of the process, must undoubtedly exist. Relations of this kind available in the literature \(1)\) are insufficiently verified, and it cannot be recommended to use them.

Taking this circumstance into account, when choosing \(n\) one has to use tables of recommended cutting speeds \(v\) from handbooks or from prospectuses of factories producing drills. Thus, taking \(v_i\) from the tables, we can determine

\[ n_i=\frac{1000v_i}{\pi d}, \]

above which we cannot go in drilling without detriment to the life of the drill. All numbers of revolutions \(n<n_i\) will be permissible in operation.

The choice of the feed \(s\) is more complicated. Here it is necessary to proceed from the following relations of the theory of metal drilling:

\[ \begin{aligned} M&=k_1\cdot d^{m_1}\cdot s^{l_1}, \ldots\ldots\ldots\ldots 1\\ P&=k_2\cdot d^{m_2}\cdot s^{l_2}, \ldots\ldots\ldots\ldots 3\\ S&=k_3\cdot d^{m_3}, \ldots\ldots\ldots\ldots 3 \end{aligned} \]

These relations have been obtained as a result of all investigations of the last two decades, and their general form may be regarded as quite well established.

Equation 1 is the dependence of the torque on the axis of the drill on the selected feed, the diameter of the drill, its profile, and on the grade of the material being machined.

Here:

\[ s \]

— feed in mm/rev,

\[ d \]

— drill diameter in mm,

\[ M \]

— torque on the drill,

\[ K_1 \]

— factor of the material being machined and of the drill profile (a constant coefficient when machining a constant material with a drill of constant profile, e.g., a twist drill),

\[ m_1,\ l_1 \]

— exponents depending on the grade of the material being machined.

Equation 2 is the dependence of the axial pressure (feed force) on the same factors.

Here:

\[ P \]

— force along the axis of the drill,

\[ K_2 \]

— analogous to \(K_1\) in equation 1,

\[ m_2,\ l_2 \]

— analogous to \(m_1,\ l_1\) in equation 1.

Equation 3 is the dependence of the maximum feed, permitted by the strength of the drill, on the diameter of the drill, the quality of the material being machined, and the quality of the drill.

Here:

\[ K_3 \]

— factor of the material being machined and the material of the drill,

\[ m_3 \]

— fractional exponent.

In these dependences, obtained experimentally in the majority of tests carried out up to the present time:

\[ 2 \geq m_1 > 1; \]

\[ 1 > l_1 > 0; \]

\[ 1 \geq m_2 > 0; \]

\[ 1 > l_2 > 0; \]

\[ 1 > m_3 > 0. \]

Since in what follows we shall speak of the use of a constant-power machine tool, let us introduce an expression for the effective drilling power, which may be obtained in the following way:

\[ N_e=\frac{M_n}{716{,}2}; \]

\[ M=k_1\cdot d^{m_1}S^{l_1}; \]

\[ n=\frac{1000\,v}{\pi d} \]

Whence:

\[ N_e=\frac{k_1\cdot d^{m_1}\cdot S^{l_1}\cdot 1000\,v}{716{,}2\cdot \pi\cdot d} \]

Or finally:

\[ N_e = 0.445 \cdot k_1 \cdot d^{m_1-1} \cdot S^{l_1} \cdot v \qquad \ldots \ldots \ldots \ldots \ldots 4 \]

Equation 4 gives the dependence of the power at the drill on all cutting factors.

When choosing the permissible feed, it is necessary to take into account the four relations given above; moreover, equation 1 will give

\[ S^{\mathrm{I}}=\left[\frac{M}{k_1 \cdot d^{m_1}}\right]^{\frac{1}{l_1}} \qquad \ldots \ldots \ldots \ldots \ldots 5 \]

Equation 5 gives the feed permissible by the maximum torque that can be transmitted by the spindle-rotation drive, i.e., the feed found from the strength condition of the most heavily loaded link of the spindle drive.

Here \(M\) is the magnitude of the torque that we can realize from the strength condition of the spindle-rotation mechanism. Thus, in order to determine \(S^{\mathrm{I}}\), it is necessary to know what maximum torque can be realized on the spindle of the machine selected for the work—not from the condition of the power transmitted to the spindle, but from the strength condition of the spindle-rotation mechanism. In most cases the magnitude \(M\) must be determined from the strength condition of the gear mounted on the spindle.

From equation 2 we obtain:

\[ S^{\mathrm{II}}=\left[\frac{P}{k_2 \cdot d^{m_2}}\right]^{\frac{1}{l_2}} \qquad \ldots \ldots \ldots \ldots \ldots 6 \]

This will be the feed permissible by the strength of the machine’s feed mechanism, and to determine it as applied to the machine selected for the work it is necessary to know the magnitude \(P\), which will be the maximum feed force attainable on the given machine. \(P\) is calculated according to the strength of the rack gear.

From equation 3 we obtain:

\[ S^{\mathrm{III}}=k_3 \cdot d^{m_3} \qquad \ldots \ldots \ldots \ldots \ldots 7 \]

This will be the feed permissible by the strength of the drill.

Finally, from equation 4 we obtain

\[ S^{\mathrm{IV}}=\left[\frac{N_e}{0.445 \cdot k_1 \cdot d^{m_1-1} \cdot v}\right]^{\frac{1}{l_1}} \qquad \ldots \ldots \ldots \ldots \ldots 8 \]

This will be the feed permissible by the power of the machine. To determine it, we need to know the power at the spindle \(N_e\) and the cutting speed \(v\) chosen by us.

Having the values \(S^{\mathrm{I}}, S^{\mathrm{II}}, S^{\mathrm{III}}, S^{\mathrm{IV}}\), we can, from the conditions of normal operation of the machine and the tool, realize only the smallest of them; and the product of it by the chosen speed \(v\) or by the chosen-

…number of revolutions will give the productivity characteristic of the given operation, \(sv\) or \(sn\).

From the dependences presented (5, 6, 7, 8), we see that, for a given workpiece material, drill diameter and profile, and selected machine tool, \(s^{\mathrm{I}}\), \(s^{\mathrm{II}}\), \(s^{\mathrm{III}}\) will be constant quantities; the value \(s^{\mathrm{IV}}\), however, when \(N_e=\mathrm{const.}\), will depend on the selected cutting speed and will be the greater the smaller \(v\) is, since in equation 8

\[ \frac{1}{l_1}>0. \]

Hence we may conclude that if, in selecting the feed, \(s^{\mathrm{IV}}\) turns out to be smaller than \(s^{\mathrm{I}}\), \(s^{\mathrm{II}}\), or \(s^{\mathrm{III}}\), then, by decreasing \(n\) and \(v\), we can thereby increase the selected feed and obtain a different productivity, \(sn\) or \(sv\).

From equation 8 it may be concluded that, from the standpoint of utilizing the power of the machine tool, it is necessary to take a lower speed \(v\) and a larger feed \(s\); in this case we shall obtain a higher productivity. Indeed, in equation 8 the exponent

\[ \frac{1}{l_1}>1, \]

since \(1>l_1>0\). From this equation we obtain:

\[ (s^{\mathrm{IV}}\cdot v)= \left[ \frac{N_e}{0.445\cdot k_1\cdot d^{m_1-1}} \right]^{\frac{1}{l_1}} \cdot \frac{1}{v^{\frac{1}{l_1}-1}}. \]

It is apparent from this that, as \(v\) decreases, the right-hand (and therefore also the left-hand) side of the equation increases, since

\[ \left(\frac{1}{l_1}-1\right)>0. \]

The left-hand side of the equation represents the productivity characteristic, which will increase as the cutting speed decreases, although in the left-hand side of the equation one of the factors, \(v\), will at the same time decrease.

Such an increase in productivity can be used only until \(s^{\mathrm{IV}}\), increasing as \(v\) decreases, becomes equal to \(s^{\mathrm{I}}\), or \(s^{\mathrm{II}}\), or \(s^{\mathrm{III}}\). A further decrease in \(v\) will be inadvisable, since \(s^{\mathrm{I}}\), \(s^{\mathrm{II}}\), and \(s^{\mathrm{III}}\), which in this case limit the feed value, are quantities independent of \(v\).

To summarize, we may say that, in order to select the cutting conditions, we must proceed as follows. Having selected the cutting speed \(v\) from the tables, we determine the value of \(s\) from equations 5, 6, 7, and 8. If \(s^{\mathrm{IV}}\) turns out to be greater than or equal to \(s^{\mathrm{I}}\), \(s^{\mathrm{II}}\), or \(s^{\mathrm{III}}\), then the selected combination will be the most advantageous. If, however, \(s^{\mathrm{IV}}\) turns out to be smaller, then, by reducing \(v\) through switching to a smaller \(n\), we bring, according to equation 8, \(s^{\mathrm{IV}}\) up to the value of one of \(s^{\mathrm{I}}\), \(s^{\mathrm{II}}\), \(s^{\mathrm{III}}\). This combination will be the most advantageous.

We see that, in order to select the cutting conditions, it is necessary each time to solve no fewer than four equations; therefore it is simpler, more convenient

and much more quickly use graphical methods in doing so, employing for this purpose special nomograms or using numerical tables prepared in advance for each machine; this, as we shall show below, makes the work of selecting cutting conditions exceptionally simple and accessible to any merely literate person.

Nomogram for selecting cutting conditions on a constant-power drilling machine.

To construct the nomogram proposed by us, we shall use all those analytical relations which must be used when selecting cutting conditions, i.e., when choosing \(n\) and \(s\). In order that these relations, which have a rather complicated form, should receive the simplest graphical expression, we shall construct the nomogram in logarithmic coordinates. First of all, to solve the equation \(v=\frac{\pi d n}{1000}\), we construct a diagram of speeds; the diagram is found in the machine passports, and we shall not discuss its construction (see Fig. 1).

Then, to the right of this diagram, in coordinate axes \(dv\), we construct the equation of constant machine power:

\[ N_e = 0{,}445\, k_1 \cdot d^{m-1} \cdot S^l \cdot v \ . . . . . . . . . . 4 \]

Let us rewrite Eq. 4 in the following form

\[ v=\frac{N_e}{0{,}445\, k_1 \cdot S^l}\cdot d^{1-m_1} \ . . . . . . . . . . 9 \]

We see that in logarithmic coordinates \(dv\) we shall have a rectilinear expression of this dependence.

If, for each material being machined, characterized in Eq. 9 by the quantity \(K_1\), we construct a separate nomogram, then, assigning to \(s\) the various values available on the machine, we shall have on the nomogram, for each feed, a straight line \(s=\mathrm{const}\), of which there will be as many as the machine has feeds.

All straight lines \(s=\mathrm{const}\) will be inclined to the positive direction of the abscissa axis \((d)\) at the angle \(\alpha_1\), where

\[ \operatorname{tng}\,\alpha_1 = 1 - m_1 \]

To construct these straight lines, knowing the angle \(\alpha_1\), it is necessary and sufficient to know the coordinates of one point of each straight line. For this, assigning in Eq. 9 some value of \(d\) and substituting, in place of \(N_e, K_1, m_1, l_1, s\), the quantities known to us, we obtain the value \(v\), which, together with the chosen \(d\), makes it possible to determine the position of one of the points of the straight line \(s=\mathrm{const}\). Knowing the point and the angle, we have the possibility of constructing the straight line.

The straight lines \(s=\mathrm{const}\) plotted in this way (see Fig. 1), which are plotted only for the values of \(s\) available on the machine, give

the relation between \(v\), \(d\), and \(s\) from the condition of utilizing the power of the machine; i.e., for a selected value of \(v\) and a given \(d\), we are able to determine that value of the feed \(s\) (from those available on the machine) which makes the fullest use of the power of our machine.

From equation 9 it may be said that the lines \(s\) are located on the nomogram the higher, the greater the power of the machine \(N_e\), and the higher, the lower the value of \(K_1\), i.e., the less resistance the material being machined offers to the drill.

On this same nomogram we construct equation 1, knowing for our machine \(M_{max}\)—the maximum torque transmitted by the spindle drive from the condition of strength of the drive

\[ M_{max}=K_1 d^{m_1}s^{\mu}\ldots\ldots\ldots 1 \]

This equation will give on the nomogram the straight line \(M_{max}=\mathrm{const}\). The construction of this straight line can be carried out in two ways. First, by rewriting equation 1 in the form

\[ d=\left[\frac{M_{max}}{k_1.S^\mu}\right]^{\frac{1}{m_1}} \]

substitute into it, instead of \(s\), any two of the values available on the nomogram, determining for each of them the corresponding value \(d\), and, thus having at the intersection of the found \(d\)’s with the corresponding \(s\)’s two points, draw through them the straight line \(M_{max}=\mathrm{const}\).

Secondly, the construction can be made independently of the straight lines \(s=\mathrm{const}\), and even without having them on the nomogram. For this, solving equations 1 and 4 simultaneously, we eliminate \(s\) from them and obtain:

\[ v=\frac{N_e}{0.445\,M_{max}}\cdot d\ldots\ldots\ldots 10 \]

From equation 10 it may be said that \(M_{max}=\mathrm{const}\) will be expressed in the coordinates \(dv\) by a straight line inclined to the abscissa axis \((d)\) at an angle \(\alpha_2\), where

\[ \operatorname{tng}\alpha_2=1, \]

i.e., \(\alpha_2=45^\circ\). In addition, it may be concluded that the position of the straight line \(M_{max}=\mathrm{const}\) depends only on the values of \(N_e\) and \(M_{max}\) and is completely independent of the material being machined, since in equation 10 the characteristics of the material being machined are absent. In other words, on all nomograms constructed for different materials being machined, but for one and the same machine, the straight lines \(M_{max}=\mathrm{const}\) will occupy one and the same position relative to the coordinates \(dv\). The straight lines \(M_{max}=\mathrm{const}\) will be located on the nomogram the lower, the greater the value of \(M_{max}\).

Knowing the angle \(\alpha_2\), it is not difficult to determine the position of the straight line \(M_{max}=\)

\(= const\), by assigning some value to \(d\) and obtaining from equation 10 the corresponding coordinate \(v\).

The straight line \(M_{max} = const\) on our nomogram, at its intersections with different \(d\)’s, will indicate for each diameter the feed \(s\) (from among those available on the machine) permissible from the standpoint of the strength of the machine spindle drive.

In the same way we construct equation 2:

\[ P_{max} = K_2 \cdot d^{m_2} \cdot S^{l_2} \ldots \ldots \ldots 2 \]

This equation can likewise be constructed either from two points, by assigning two values of \(s\) and obtaining from equation 2 two corresponding values of \(d\), or from a point and an angle of inclination, for which from equations 2 and 4 we shall have:

\[ v = \frac{N_e} {0{,}445 \cdot k_1 \left(\dfrac{P^{max}}{k_2}\right)^{\frac{l_1}{l_2}}} \cdot d^{\left(\frac{l_1 \cdot m_2}{l_2} - m_1 + 1\right)} \ldots \ldots \ldots 11 \]

The straight line makes with the abscissa axis \((d)\) the angle \(\alpha_3\)

\[ tng\, \alpha_3 = \left( \frac{l_1 m_2}{l_2} - m_1 + 1 \right). \]

This straight line, at its intersections with different \(d\)’s, will indicate the corresponding values of \(S\) permissible from the standpoint of the strength of the feed mechanism of the machine. The straight line \(P_{max} = const\) will be located on the nomogram the lower, the greater the value of \(P_{max}\) for the given machine.

Finally, in the same way we find the straight line from equation 3:

\[ S = K_3 \cdot d^{m_3} \ldots \ldots \ldots \ldots \ldots 3 \]

again either from two points, or from a point and an angle, having for this from equations 3 and 4 the dependence

\[ v = \frac{N_e}{0{,}445 \cdot k_1 \cdot k_3^{l_1}} \cdot d^{(1 - m_1 - m_3 l_1)} \ldots \ldots \ldots 12 \]

For the angle of inclination \(\alpha_4\) of this straight line to the abscissa axis, we have

\[ tng\, \alpha_4 = (1 - m_1 - m_3 l_1), \]

The straight line \(s = F(d)\) will indicate for each diameter the feed permissible from the standpoint of drill strength.

Thus, on the nomogram (see Fig. 1) we have an expression of all those conditions that must be taken into account when choosing the magnitude of the feed \(S\) for operation on the machine. The broken contour \(ABCD\) bounds the magnitudes of the feeds applicable on the machine.

in machining the given material, and for each drill diameter it indicates the limiting feed from among those available on the machine.

Only the feeds enclosed within the contour may be used (they are drawn more heavily on the nomogram), since by going outside it we thereby fail to satisfy one of the imposed conditions. The straight line \(AB\) limits the increase in feed by the strength of the drill; the straight line \(BC\), by the strength of the feed mechanism; \(CD\), by the strength of the spindle-drive mechanism. For each diameter one may take a feed that intersects the line of the diameter no lower than the contour \(ABCD\).

It is not always possible to use the feed that passes through the point of intersection of the contour \(ABCD\) with the given diameter, since this may involve overloading the machine’s power. One must always strive for this feed, since we shall show below that the greatest productivity for a drill of each diameter will inevitably be associated with the feed passing through the point of intersection of the contour \(ABCD\) with the given diameter, or with the feed, of all those enclosed in the contour, that lies closest to this point (i.e., with the maximum possible feed for the given \(d\)). Therefore \(ABCD\) may be called the contour of maximum productivity.

Let us show by an example how the cutting conditions should be selected using the nomogram, and in what case we shall obtain maximum productivity.

Selection of cutting conditions.

The nomogram shown in Fig. 1 was constructed by us for the following conditions possible in practice:

\[ Ne = 5 \text{ h.p.} \]

\[ P_{\max} = 2200 \text{ kg.} \]

\[ M_{\max} = 50 \text{ kilogram-meters.} \]

The material being machined is steel of medium hardness. The drills are of high-speed steel.

The machine has the following feed values, in mm/rev:

\[ S_1 = 0.1,\quad S_2 = 0.13,\quad S_3 = 0.17,\quad S_4 = 0.22,\quad S_5 = 0.3,\quad S_6 = 0.4,\quad S_7 = 0.52, \]

\[ S_8 = 0.65,\quad S_9 = 0.85. \]

In constructing the nomogram we used Schlesinger’s data as revised by Besprozvannyi\(^2\), according to which, for steel of medium hardness, we shall have:

\[ K_1 = 0.029; \]

\[ m_1 = 2; \]

\[ l_1 = 0.8; \]

\[ K_2 = 84; \]

\[ m_2 = 1; \]

\[ l_2 = 0.75. \]

In other words:

\[ M = 0.029\, d^2 \cdot s^{0.8}\ \text{kgm} \]

\[ P = 84\, d \cdot s^{0.75}\ \text{kg.} \]

The allowable feed according to the strength of the drill was taken from the Gipromash formula*):

\[ S = 0.04\, d^{0.6}, \]

i.e.

\[ K_3 = 0.04 \]

\[ m_3 = 0.6 \]

For the expression for the power expenditure we shall have (eq. 4)

\[ Ne = 0.445 \cdot 0.029 \cdot d \cdot s^{0.8}\, v, \]

i.e.

\[ Ne = 0.0129 \cdot d \cdot s^{0.8}\, v. \]

Let it be specified for machining: steel of medium hardness, and drill diameter \(d = 50\) mm. From the tables we find that, when drilling this material with a high-speed drill, the cutting speed may be taken as \(v = 18\text{--}22\) m/min. With the use of abundant cooling by emulsion, this speed may be increased to \(v = 26\) m/min, to which, judging by the left-hand part of the nomogram (see Fig. 1), there will correspond \(n = 170\) rpm. Thus, going from \(d = 50\) mm to the straight line \(n = 170\) rpm, and then horizontally to the right to the intersection in the right-hand part of the nomogram with the vertical \(d = 50\), we may conclude that, from the condition of utilizing the power of our machine, one may take \(s = 0.22\) mm/rev, which passes through the point of intersection of the horizontal \(v = 26\) and the vertical \(d = 50\). The productivity obtained in this case will be

\[ ns = 170 \cdot 0.22 = 37.4\ \text{mm/min}. \]

In this case neither the strength of the drill, nor the strength of the feed mechanism, nor the strength of the spindle drive will be utilized.

If we take the next lower speed \(n = 132\), then, proceeding in an analogous manner, we arrive at the feed \(s = 0.3\) mm/rev, which again will be obtained from the condition of utilizing the machine power (in this case the power will be slightly overloaded). The productivity for this case will be

\[ ns = 132 \cdot 0.3 = 39.6\ \text{mm/min}, \]

i.e. somewhat higher than the preceding one.

A further transition to \(n = 105\) rpm will make it possible to take the feed \(s = 0.4\) mm/rev, the closest to the contour \(ABCD\). The productivity in this case

\[ ns = 105 \cdot 0.4 = 42\ \text{mm/min}, \]

i.e. still higher.

A further reduction in the number of revolutions to \(n = 85\) rpm will not give an increase in productivity, since at \(d = 50\) mm we cannot take a feed \(s > 0.44\) mm/rev; in doing so we would cross the con-

the contour \(ABCD\), and would overload the feed mechanism of the machine and the strength of the drill.

Thus, we see that it is most advantageous to take \(s = 0.4\), i.e., the feed lying closest of all to the contour of maximum productivity \(ABCD\) for the given diameter, and such a number of revolutions \(n = 105\) that would correspond to the speed which, with the selected feed, would make use of the power of the given machine, provided that this speed is, of course, permissible from the condition of drill life.

Having chosen, in the example analyzed, \(n = 105\) instead of \(n = 170\), we not only raise the productivity from \(37.4\) to \(42\ \mathrm{mm/min}\), i.e., by \(12.3\%\), but also reduce the speed at the circumference of the drill from \(26\ \mathrm{m/min}\) to \(16\ \mathrm{m/min}\), which greatly eases the working conditions of the drill from the standpoint of its life.

If we take several other operating cases (with other \(d\)), nevertheless in the overwhelming majority of cases we shall obtain \((ns)_{\max}\) when choosing a feed lying, for the given diameter, close to the contour \(ABCD\), which therefore may be called the “contour of maximum productivity.” Proceeding from this, it may be recommended that, in operation, one choose such an \(n\) that the resulting speed \(v\) leads to the right-hand part of the nomogram—to a feed close to the contour \(ABCD\) for the given diameter.

In the event that, for some reason, we must confine ourselves to so low a value of \(n\) and \(v\) that we fall in the right-hand part of the nomogram below the contour \(ABCD\), we must proceed as shown by the arrows for \(d = 30\ \mathrm{mm}\) and \(n = 132\) (see Fig. 1). In this case, using the power of the machine, we would have had to take the feed \(s = 0.85\ \mathrm{mm/rev}\), but neither the feed mechanism of the machine nor the strength of the drill permits a feed greater than \(s = 0.3\ \mathrm{mm/rev}\). If, for any reasons, the speed cannot be taken greater than \(v = 12.5\ \mathrm{m/min}\) (i.e., \(n > 132\ \mathrm{rev/min}\)), then the maximum possible productivity will be:

\[ (ns)_{\max} = 132 \cdot 0.3 = 39.6\ \mathrm{mm/min}. \]

The contour of maximum productivity will not necessarily be represented on the nomogram by the intersection of three straight lines. In Fig. 2 a similar nomogram is presented for the case of machining, on the same machine, cast iron of medium hardness, and in Fig. 3—chrome-nickel steel. Here we see that one of the straight lines has fallen out of the contour, lying below it. For the case of machining cast iron, the straight line \(P_{\max} = \mathrm{const}\) did not enter the contour; for the case of machining chrome-nickel steel—the straight line \(M_{\max} = \mathrm{const}\).

With the nomogram we propose, it is possible not only to make a rapid selection of the most advantageous operating conditions, taking into account all the factors affecting them, but also to judge the field of application of the given machine, as well as the rationality of its design.

Thus, from Fig. 1 the following conclusions may be drawn:

1. When machining steel of medium hardness without cooling on the given machine, the power of this machine can be used only when working with drill diameters \(d \geq 40\ \mathrm{mm}\) (high-speed drill).

1. In the case of work with a diameter \(d < 40\) mm, in order to make fuller use of the machine’s power, one should work with abundant cooling.

2. When machining medium-hard steel, feeds of 0.52 mm/rev, 0.65 mm/rev, and 0.85 mm/rev cannot in general be used when drilling in solid material.

3. To make it possible to use these feeds, the machine would have to have stronger feed and spindle-rotation mechanisms (the lines \(P_{max}=\mathrm{const.}\) and \(M_{max}=\mathrm{const.}\) would in that case be located lower). But even then, feeds of 0.65 and 0.85 could be used only with drill diameters \(d > 80\) mm, which is practically inapplicable.

From a comparison of the nomograms in Figs. 1 and 2 it may be concluded that this machine is more suitable for machining steel than cast iron. In machining cast iron, however, the power of the machine, for usable drill diameters, cannot be utilized, since this could be achieved only at speeds inadmissibly high for high-speed drills, or else by using drills with blades made of superhard alloys.

From Fig. 3 it may be said that, in order to use the machine when machining chromium-nickel steel, the machine should have a stronger feed mechanism.

Compilation of Tables of the Most Advantageous Cutting Conditions

Having examined the use of the proposed nomograms, it is not difficult to observe that each drill diameter, when machining a specified material on the given machine, has a quite definite combination of \(s\) and \(n\) that predetermines the maximum productivity.

Thus, for the case of machining medium-hard steel, on the basis of the nomogram (Fig. 1), the following table of the most advantageous working conditions may be compiled.

Machine No. … …

Material—steel \(K_z = 50—60\) kg/mm²

Drill—high-speed steel.

The same applies to cast iron and chromium-nickel steel, taking for cast iron \(V_{max}=15—18\) m/min., and for chromium-nickel steel, with the obligatory use of abundant cooling, \(V_{max}=15\) m/min.

Thus, the compiled tables make the selection of the operating mode an exceptionally simple and rapid operation, since in such tables, compiled for each machine and for the various materials machined on it, the data for \(d\), \(s\), and \(n\) have already been correlated on the basis of maximum productivity, with allowance for all factors affecting that productivity. Even a worker of comparatively low qualification will be able to use such a table, so simple and convenient is it. In addition, it is extremely simple to determine the machine time from it, since the product of the two columns \(n\) and \(s\) gives the depth of the hole drilled in 1 minute.

The certain labor involved in compiling such tables for all drilling machines available in the shop is undoubtedly repaid by the convenience of using them and by the economic effect that will be obtained through the rational utilization of the machines. Let us note that these tables could have been compiled by using not the proposed nomograms, but the analytical relationships given above. But this would involve a great amount of computational work and would not provide the clarity of the graphical method, which can lead to frequent errors.

It would be good if the machine-tool plants of the Soviet Union supplied, with the drilling machines they produce, similar nomograms compiled in accordance with the characteristics of the machine being produced for cases of machining various materials on it, or at least informed the consumer of the necessary data about the machine, such as: the maximum permissible torque on the spindle from the condition of strength of the spindle drive mechanism \((M_{max})\), the maximum permissible feed force \((P_{max})\), the useful

the power \(N_e\), etc. This would greatly simplify for enterprises the mastering of the newest machine-tool equipment, i.e., the solution of one of the principal tasks of the second five-year plan for the construction of socialist industry.

Literature cited in the article:

1) REFA. Handbook on Working Time (p. 260). Krivoukhov.—Machining of Metals by Cutting (p. 232). Isaev.—Cold Working of Metals (p. 165). Tekhnika i proizvodstvo, 1928, No. 6–7. Kondrat’ev.—Drilling of Metals (p. 15).

2) Orga-Informatsiya. 1929, No. 8.

3) Vestnik Metallopromyshlennosti. 1930, No. 2. Krivoukhov (p. 229). Isaev (p. 162). Kondrat’ev (p. 10).

Tomsk, 1934.
Laboratory of Metal Cutting
Siberian Industrial Institute.

A. M. Rosenberg.

A Study of the Working of a Drilling Machine of Constant Power.

A method of selecting the most favourable conditions of cutting with a drilling machine of constant power is examined in this article. The following factors are taken into consideration:

1. Power of the drilling machine.
2. Strength of the spindle gear (maximal admissible twisting moment on the spindle).
3. Strength of the feed mechanism.
4. Strength of the drill-bit and
5. Speed admissible by the condition of durability of the drill.

A nomogram constructed with consideration of all the above-mentioned factors, allowing one to perform a convenient and rapid operation in selecting the most favourable conditions of cutting, is given.

The author proves that the contour limiting the permissible feeds in drilling, resulting from the condition of the maximal twisting moment, from that of the maximal feeding power and from the durability of the drill, is the “contour of the maximum efficiency” of work with this machine-tool.

In conclusion, a method of tabulation for the given machine is offered, these tables warranting a simple and exact selection of its most profitable working.

Fig. 1.

Visible inscriptions in the diagram: \(v\), \(d\), \(n=420\), \(355\), \(265\), \(210\), \(170\), \(132\), \(105\), \(85\), \(67\), \(53\), \(42\), \(n=33\), \(S=f(d)\), \(S=0.1\), \(0.13\), \(0.17\), \(0.22\), \(0.3\), \(0.4\), \(0.52\), \(0.65\), \(0.85\), \(M_{\max}=\mathrm{const}\), \(p_{\max}=\mathrm{const}\), “steel, corresponding,” \(d_1\), \(d_2\), \(d_4\), \(c_3\), \(A\), \(B\), \(C\), \(D\).

Fig. 2.

Visible labels in the figure: \(d\), \(v\), \(A\), \(B\), \(C\), \(P_{\max}=\mathrm{const.}\), \(N_{\max}=\mathrm{const.}\), “Chugun sr. tv.” (cast iron, medium hardness); numerical curve labels: 0.1, 0.13, 0.17, 0.22, 0.3, 0.4, 0.52, 0.65, 0.85.

Fig. 3.