On the Effect

of Supply

on the Coefficient of Cutting of Metals.

T. I. Tikhonov.

[[unclear: library stamp]]

Tomsk.

Printing-Lithography of the Siberian Society of Printing Affairs, corner of Dvoryanskaya St. and Yamskoy Lane, own building.

1912.

On the Influence of Feed on the Coefficient of Cutting of Metals

During the course of producing studies on the coefficient of cutting of metals, published by me in 1910, *) a relationship was observed between feed and the coefficient of cutting of metals. According to the observations obtained in 1910, it followed that it was more advantageous to work with a greater feed at one and the same speed. To verify the indicated relationship, in 1912, in the mechanical workshops of the Tomsk Technological Institute, new observations were organized to determine the dependence of the coefficient of cutting of metals on the feed of the cutter; the observations were carried out on a milling machine, a lathe, and a shaping machine. Four students of the Institute took part in carrying out the observations themselves.

For the first investigations, a cylindrical milling cutter of ordinary construction was taken, and the investigations themselves were carried out on an ordinary universal milling machine driven by a separate electric motor.

The instrument for measuring the pressure on the cutter was an “oil dynamometer” of rather simple construction, designed for this purpose and built in the mechanical workshops of the Tomsk Technological Institute; by means of diagrams of an ordinary indicator, it made it possible to determine the torque of the milling cutter and, from it, to trace the conditions of operation of the cylindrical cutter when the magnitude of the feed was changed, as well as the depth of the layer being removed.

The arrangement of the above-mentioned dynamometer is as follows (Figs. 1 and 2 of the appended plate, 1/5 nat. size): in the hollow spindle of the milling machine there was fastened a mandrel for the cutter. In the spindle of the machine there was fixed the right-hand part of this mandrel, immobile, and it rotated together with the stepped pulley of the machine; this right-hand part of the mandrel ended in a special fork, by means of which the force from the machine belt was transmitted to the pistons of the hydraulic cylinders (1, 2), forming one whole with the cylindrical mandrel on which the cutter under test (3) was fastened.

) See Proceedings of the Tomsk Technological Institute* for 1910.

Oil was poured under the pistons of the hydraulic cylinders, and the air was carefully removed from beneath the pistons of the cylinders (1,2). The milling-cutter mandrel has inside it a channel (5), connected by special tubes with the oil cylinders of the apparatus (1,2), and at the same time the indicated channel (5) is connected with the piston of the indicator. The cylindrical part of the mandrel, on which the cutter sits, can rotate with respect to the right-hand part of the mandrel (the fork), but the joining into one whole with the spindle of the machine (of both halves of the mandrel) takes place only through the pistons of the oil cylinders. Thus the force overcome by the cutter during work produces a definite pressure on the oil under the pistons of the cylinders (1,2), and the oil from the latter, through the channel (5), acts upon the piston of the indicator. The bar (12), from which the chip was removed, was fastened on the table of the machine by means of simple stops. To obtain a diagram on the indicator, beside the cutter on the mandrel of the latter a special drum (4) with a helical groove (11) was fastened. The length of that part (4) of the drum where the groove was made was less than the circumference of the indicator. In the helical recess of the drum (4), when pressed, there could move a pin (10), forming one whole with the oblique flat lever (8). The latter, with its cylindrical guide, can move longitudinally along the mandrel of the cutter in the slots of special grips (7,7), on which the lever (8) is suspended from the oblique knee of the machine. To the end (9) of the cylindrical rod of the lever (8) is attached the end of the indicator cord.

Investigations were carried out both on a tough metal—martensitic iron—and on brittle cast iron and hard bronze.

The speed of rotation of the cutter in each separate case was varied 3 times: \(n_1 = 12\) revolutions per min.; \(n_2 = 20\); \(n_3 = 32\). The feed within the limits of one speed of rotation of the cutter was varied 4 times. The thickness of the layer removed was taken equal to 1 mm., 2 mm., 3 mm., and 4 mm. The total number of all diagrams taken was considerable, but in order to obtain general conclusions on the given question, up to 2 diagrams were taken for calculation, from which the mean indicator pressures were computed for each separate case.

The results of all the computations are placed in the tables below. The scale of the indicator springs was taken as follows: for iron from 1 mm. to 3 mm. inclusive, 20 mm. = 1 kg.; for iron at 4 mm. and for cast iron and bronze—12 mm. = 1 kg.

1. Results of investigations of the operation of a milling cutter on iron with a cutting angle of the cutter teeth of 90°

Table I: mean indicator pressures on each diagram, in kg/qmm.

If now, from the mean indicator pressures obtained for each feed, one subtracts the pressure caused by friction in the apparatus, the absolute pressure is obtained, depending on the cutting resistance of the milling cutter. The pressure caused by friction is taken as equal to 0,13 kg.

Table II of absolute indicator pressures in kg/qmm.

With the aid of the corrected values obtained for the mean indicator pressures, one can calculate the moments acting on the pistons of the oil apparatus.

We have the following equation:

\[ \mathrm{M}=l.S p_i,\quad \text{where} \]

\(\mathrm{M}\)—the moment acting on the pistons;

\(l\)—the arm between the pistons—the distance between the axes of the pistons—\(=200\ \mathrm{mm}\);

\(S\)—the piston area: \(S=\dfrac{\pi d^2}{4}\), where \(d=4.5\ \mathrm{cm}\),

therefore, the piston area

\[ S=\frac{\pi 4.5}{4}=15.806\ \mathrm{cm}^2, \]

\(p_i\)—the mean indicator pressure in \(\mathrm{kg}/\mathrm{cm}^2\); whence:

\[ \mathrm{M}=200.15.86.p_i=3172.p_i \]

\[ \mathrm{M}=3172p_i. \]

The moments calculated by the derived equation are placed in Table III.

Table III of moments on the pistons in kgmm.

With the aid of the moments obtained, it is possible to determine the circumferential forces on the cutter teeth that are in operation at each given moment. For this we have:

\[ \mathrm{M}=\mathrm{W}.r,\quad \text{or}\quad \mathrm{W}=\frac{\mathrm{M}}{r},\quad \text{where} \]

\(\mathrm{W}\)—the circumferential force;

$M$—the moment on the piston of the instrument,

$r$—the outer radius of the milling cutter $=50$ mm, but since

\[ M=3172\,pi,\quad \text{then } W=3172\cdot 0.02\,pi. \]

The results of the computations performed are placed in Table IV.

Table IV of circumferential forces acting on the cutter edge, in kg/qmm.

Next, we shall determine the number of teeth of the milling cutter simultaneously taking part in removing each thickness of layer. We have in the drawing (Fig. 3): $D$—the outer diameter of the cutter, $d$—the thickness of the layer being removed, $\psi$—the angle corresponding to the part of the circumference of the cutter with teeth simultaneously removing a layer of thickness $d$. The angle $\alpha$, corresponding to one tooth of the cutter (with the total number of cutter teeth—22), is determined as follows:

\[ \alpha=\frac{360}{22}=16^\circ 21'49''. \]

The angle $\psi_1$, corresponding to the thickness of the removed layer $d=1$ mm, is determined from the equation

\[ \frac{D}{2}-d=\cos\psi\cdot\frac{D}{2} \quad\text{or}\quad \cos\psi=\frac{\frac{D}{2}-d}{\frac{D}{2}}; \quad \frac{D}{2}=50\ \mathrm{mm}. \quad \text{For } d=1\ \mathrm{mm}; \]

\[ \cos\psi_1=\frac{50-1}{50}=0.98,\quad \text{whence } \psi_1=11^\circ 30'. \]

When comparing $\alpha$ and $\psi_1$, it is seen that, for the thickness of the layer being removed $d = 1$ mm, at each given moment only one tooth of the cutter takes part in the work.

For $d = 2$ mm,

\[ \cos \psi = \frac{50-2}{50} = 0.96; \quad \psi_2 = 16^\circ 15'. \]

At each given moment likewise only one tooth takes part in the work.

For $d = 3$ mm,

\[ \cos \psi_2 = \frac{50-3}{50} = 0.94; \]

\[ \text{whence } \psi_3 = 19^\circ 55'. \]

Here, from a comparison of $\alpha$ and $\psi_3$, it is clear that at each given moment 2 teeth of the cutter take part; the angle corresponding to the second tooth of the cutter is

\[ \psi_3' = \psi_3 - \alpha = 19^\circ 55' - 16^\circ 21'49'' = 3^\circ 33'11'', \]

with corresponding $\cos \psi_3' = 0.9981$. The thickness of the layer $d_3'$, removed by the 2nd tooth, is determined from the equation:

\[ \cos \psi_3' = \frac{\frac{D}{2} - d_3'}{D/2}, \]

or

\[ d_3' = \frac{D}{2} - \frac{D}{2}\cos \psi_3' = 0.095 \text{ mm}. \]

For the thickness of the layer being removed $d = 4$ mm, the angle $\psi_4$ is determined from the equation:

\[ \cos \psi_4 = \frac{D/2 - d_4}{D/2} = \frac{50-4}{50} = 0.92; \quad \text{whence} \]

\[ \psi_4 = 23^\circ 5'. \]

In this case, when comparing $\alpha$ and $\psi_4$, it follows that two teeth of the cutter participate in the work simultaneously. The angle corresponding to the second tooth,

\[ \psi_4' = \psi_4 - \alpha = 23^\circ 5' - 16^\circ 21'49''; \quad \psi_4' = 6^\circ 53'11'', \]

with corresponding

\[ \cos 6^\circ 53'11'' = 0.9928. \]

The thickness of the layer removed by the second tooth is

\[ d_4' = \frac{D}{2} - \frac{D}{2}\cos \psi_4' = 50 - 50 \cdot 0.9928 = 0.36 \text{ mm}. \]

To determine the thickness of the chip removed by one tooth of the cutter, we have drawing (4), in which $bcfiae$ represents the chip removed by one tooth of the cutter, $D/2$ is the radius of the cutter, and $d$ is the thickness of the layer.

For the calculation it is necessary to assume that, during the time when the tooth of the cutter removes the chip $bcfiae$, the cutter itself moves by the amount $mm_1$, equal to the amount of feed per one tooth of the cutter, and that the curve $ebf$ of the outer outline is an arc of radius $D/2$; in this case, the thickness of the chip for a certain position corresponding to the angle $y$ is represented as follows:

\[ X = ac \cdot \sin y, \quad \text{where } ae = mm_1. \]

If \(Z\) is the number of working teeth,

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

\(v\)—the feed rate in \(\mathrm{mm\ sec^{-1}}\), then

\[ ac=\frac{60}{n}\cdot\frac{v}{Z};\quad \text{therefore,}\quad X=\frac{60}{n}\cdot\frac{v}{z}\cdot\sin y. \]

But the greatest chip thickness \(\delta\) will be when \(y=\psi\), i.e.

\[ \delta=\frac{60}{n}\cdot\frac{v}{Z}\sin\psi; \]
for the thickness of the layer being removed \(d\),

\[ \cos\psi=\frac{D/2-d}{D/2},\qquad \sin\psi=\sqrt{1-\cos^2\psi} =\sqrt{4\left(\frac{d}{D}-\frac{D^2}{d^2}\right)} \]

Then the magnitude of the chip thickness removed by one tooth will be expressed as follows:

\[ \delta=\frac{2.60}{n}\cdot\frac{v}{Z}\cdot \sqrt{\frac{d}{D}-\frac{D^2}{d^2}}. \]

This formula may, within the limits of one and the same number of revolutions of the milling cutter \(n\) and one thickness of the layer being removed, be simplified and \(\delta\) represented as a function of the feed rate \(v\).

I. With the thickness of the layer being removed \(d=1\ \mathrm{mm}\), as is already known, at each given moment one tooth of the milling cutter participates in the work. The chip thickness is determined by the formula:

\[ \delta=\frac{2.60}{n.Z}\,v\, \sqrt{\frac{1}{D}-\frac{1}{D^2}} = \frac{2.60}{n}\cdot\frac{v}{Z} \sqrt{\frac{1}{100}-\frac{1}{100^2}}. \]

During our investigations we changed the number of revolutions \(n\) three times: \(n_1=12;\ n_2=20;\ n_3=32\).

a) \(n_1=12;\) instead of \(Z=\dfrac{12}{60}\,2.2=4.4;\)

\[ \delta= \left\{ \frac{2.60}{12}\cdot\frac{1}{4.4}\cdot\frac{1}{100} \sqrt{1(100-1)} \right\}\cdot v \]
or, simplifying this expression,

\[ \delta=0.226\,v. \]

With the aid of so simple a formula we shall readily determine \(\delta\) for each feed \(v\) at \(n_1=12\)

β) \(n_2=20\); instead of \(Z=\dfrac{20.22}{60}=7,4\)

\[ \delta=\left\{\frac{2.60}{20}\cdot \frac{1}{7,4}\cdot \frac{1}{100}\sqrt{1(100-1)}\right\}v=0,081v. \]

γ) \(n_3=32\); instead of \(Z=\dfrac{32.22}{60}=11,7\); then

\[ \delta=\left\{\frac{2.60}{32}\cdot \frac{1}{11,7}\cdot \frac{1}{100}\sqrt{1(100-1)}\right\}v=0,032v. \]

II. With the thickness of the layer being removed \(d=2\) mm, at each given moment one tooth of the cutter participates in the work. The thickness of the chip removed by one tooth will be determined by the formula derived above

\[ \delta_2=\left\{\frac{2.60}{n}\cdot \frac{1}{Z}\sqrt{\frac{d}{D}-\frac{d^2}{D^2}}\right\}v; \]

α) \(n_1=12\);

\[ \delta_2=\left[\frac{2.60}{n}\cdot \frac{1}{4,4}\cdot \frac{1}{100}\sqrt{2(100-2)}\right]v=0,327v. \]

β) \(n_2=20\):

\[ \delta_2=\left[\frac{2.60}{20}\cdot \frac{1}{7,4}\cdot \frac{1}{100}\cdot \sqrt{2(100-2)}\right]v=0,113v; \]

γ) \(n_3=32\)

\[ \delta_2=\left[\frac{2.60}{32}\cdot \frac{1}{11.4}\cdot \frac{1}{100}\sqrt{2(100-2)}\right]v=0,045v. \]

III. Further, when the thickness of the layer being removed is \(d=3\) mm, two teeth of the milling cutter participate in the work at each given moment. The thickness of the chip removed by one tooth of the cutter, for this case, is determined from the relation:

\[ \delta_3'=\left[\frac{2.60}{n}\cdot\frac{1}{Z}\sqrt{\frac{3(D-3)}{D^2}}\right]\cdot v. \]

The thickness of the chip removed by the other tooth of the cutter for the given case is determined as:

\[ \delta_3''=\left[\frac{2.60}{n}\cdot\frac{1}{Z}\sqrt{\frac{0{,}095(D-0{,}095)}{D^2}}\right]\cdot v, \]

since \(d_3'=0{,}095\) mm.

a) \(n_1=12\).

\[ \delta_3'=\left[\frac{2\,60}{12}\cdot\frac{1}{4.4}\cdot\frac{1}{100}\sqrt{3(100-3)}\right]\cdot v=0{,}395\,v, \]

\[ \delta_3''=\left[\frac{2.60}{12}\cdot\frac{1}{4.4}\cdot\frac{1}{100}\sqrt{0{,}095(100-0{,}095)}\right]v=0{,}07\,v. \]

\[ \delta_3=\delta_3'+\delta_3''=(0{,}39+0{,}07)\,v \simeq 0{,}47\,v \]

β) \(n_2=20\). In an analogous manner we obtain

\[ \begin{aligned} \delta_3'&=0{,}14\,v \\ \delta_3''&=0{,}025\,v \end{aligned} \quad\left|\quad \delta_3=\delta_3'+\delta_3''=0{,}165\,v; \right. \]

γ) \(n_3=32\). Also from the general formulas

\[ \begin{aligned} \delta_3'&=0.0843\,v \\ \delta_3''&=0{,}009\,v \end{aligned} \quad\left|\quad \delta_3=\delta_3'+\delta_3''=0{,}0933\,v. \right. \]

IV. Finally, when the thickness of the layer being removed is \(d=4\) mm, two teeth of the milling cutter participate in the work at each given moment, as was stated above. The thickness of the chip removed by the first tooth of the cutter is determined from the relation

\[ \delta_4'=\left[\frac{2.60}{n}\cdot\frac{1}{Z}\sqrt{\frac{4}{D^2}(D-4)}\right]v \]

The thickness of the chip removed by the second tooth of the milling cutter is determined from the relation

\[ \delta_4'=\left[\frac{2.60}{n}\cdot \frac{1}{Z}\sqrt{\frac{0.36}{D^2}(D-0.36)}\right]\cdot v. \]

since \(d_4''=0.36\) mm.

\[ \alpha)\quad n_1=12. \]

\[ \delta_4'=\left[\frac{2.60}{12}\cdot \frac{1}{4.4}\cdot \frac{1}{100}\sqrt{4(100-4)}\right]v=0.445v; \]

\[ \delta_4''=\left[\frac{2.60}{12}\cdot \frac{1}{4.4}\cdot \frac{1}{100}\sqrt{0.36(100-0.36)}\right]v=0.134v; \]

\[ \delta_4=\delta_4'+\delta_4''=(0.445+0.134)V\simeq 0.58v. \]

\[ \beta)\quad n_2=20 \]

\[ \begin{aligned} \delta_4'&=0.159v\\ \delta_4''&=0.048v \end{aligned} \quad\Bigg|\quad \delta_4=0.207v \]

\[ \gamma)\quad n_3=32. \]

\[ \begin{aligned} \delta_4'&=0.0625v\\ \delta_4''&=0.119v \end{aligned} \quad\Bigg|\quad \delta_4=\delta_4'+\delta_4''=0.82v. \]

Determination of the cutting coefficient on the basis of the obtained results is made from the general formula:

\[ W=R\cdot \delta\cdot b,\quad \text{or}\quad R=\frac{W}{\delta\cdot b}, \]

where

\(R\)—the cutting coefficient in kg/qmm.

\(W\)—the circumferential force on the cutter edges working simultaneously.

The values for \(W\) are taken from the table given above, according to each case.

\(\delta\)—chip thickness;

\(b\)—chip width \(=38\) mm.

Substituting into this formula, for each feed, the corresponding values for \(W\) and \(\delta\) (\(b=\) constant), we obtain the values of the cutting coefficient that we require. The results of these substitutions are placed in the following Table V.

Table V of cutting coefficients in kg/mm.

Casting a general glance at the figures given in the table, we clearly see that the decrease of the cutting coefficient is affected by the depth of cut and, chiefly, by the magnitude of the feed; with an increase in the depth of cut, within the limits of one and the same rotational speed of the cutter, the cutting coefficient decreases more slowly, while with an increase in feed it decreases more rapidly. This observation in the work of the milling cutter is in complete agreement with the results of previously published investigations of the work of ordinary cutters, and makes it possible, also with respect to more improved types of cutters, such as milling cutters are, to draw a conclusion of practical importance, previously expressed with respect to ordinary cutters (see T. I. Tikhonov, “On the Cutting Coefficient of Metals”), namely, that it is more advantageous to work with a larger feed than with a considerable depth of cut at one and the same speed.

Entirely analogous experiments were carried out with the same milling cutter and on the same machine with a dynamometer, to determine the dependence of the cutting coefficient of metals on the feed, using cast iron and machine bronze for the investigation. The results of the subsequent observations, to avoid repetition, are briefly given in

in the following, and, in order to avoid misunderstandings, it should be stipulated that the working cutter, both in the first and in the subsequent cases, was regularly sharpened after every 8 hours of work.

Results of investigations of the cutter working on cast iron.

Table VI of average indicator pressures, kg/qmm.

In order to obtain the absolute pressure, dependent on the resistance to cutting by the cutter, as before we subtract from the preceding values the pressure caused by friction in the apparatus—0.13.

Table VIII of corrected indicator pressures in kg/qmm.

By means of the absolute average indicator pressures, the moments acting on the pistons of the oil apparatus are calculated, and

circumferential forces on the cutting edges of the milling cutter that are in operation at any given moment; for this purpose we used the formulas already derived earlier:

\[ \mathrm{M}=l\cdot s\cdot p_i=3172\,p_i, \]

\[ \mathrm{W}=\mathrm{M}:r=3172\,p_i\,0.05. \]

The results of the calculations are given in the following Tables VIII and IX.

Table VIII of moments on the pistons in kgmm.

Table IX of circumferential forces in kg/qmm.

The calculation formulas relating to the determination of the thickness of the chip removed by one tooth of the milling cutter, and also to the determination of the cutting coefficient, remain the same as those we derived earlier for iron.

Chip width \(b=38\) mm.

\[ 1)\quad \delta=\left[\frac{2.60}{n.z.}\cdot \sqrt{\frac{d}{D}-\frac{d^2}{D^2}}\right]v. \]

\[ 2)\quad \mathrm{R}=\frac{\mathrm{W}}{\delta\cdot b}. \]

Therefore, substituting into the 2nd formula, for each value of the feed, the former values of \(\delta\), the calculated values of \(W\), and the indicated value \(b\), we obtain the required values of the cutting coefficient.

The results of the indicated calculations are presented in the following Table X.

Table X; cutting coefficient in kg/qmm.

Results of investigations of machining hard bronze with a milling cutter.

Table XI of mean indicator pressures in kg/qmm.

Table XII of corrected (absolute) mean indicator pressures.

\[ M = 7132 \cdot p_i \]

Table XIII of moments, in kgmm, acting on the pistons of the oil apparatus

\[ W=\frac{M}{R}=7132.p:0.02. \]

Table XIV of circumferential forces in kg/qmm.

\[ R=\frac{W}{\delta b};\quad \delta=\left\{\frac{2.60}{n}\cdot\frac{1}{Z}\sqrt{\frac{d}{D}-\frac{d^{2}}{D^{2}}}\right\}\cdot v;\quad b=32\ \mathrm{mm}. \]

Table XV of cutting coefficients in kg/qmm.

The results of experiments in machining brittle metals—cast iron and hard bronze—with a milling cutter, as was to be expected, fully confirmed the dependence, already observed earlier, of the cutting coefficient on the table feed. Taking a general view of the results obtained in work with a milling cutter (see the table of cutting coefficients), one may note that in places the regularity of the decrease of the cutting coefficient with an increase in feed is violated. But this violation must be explained partly by the heterogeneity of brittle metals, which causes fluctuations in the resistance to the cutting force of the cutter, and partly by not entirely successful

* Speed of rotation of the cutter in mm per second.

...diagrams, owing to the fact that, for cast iron and bronze, it was necessary to use a spring of insufficient sensitivity in the indicator (12 mm = 1 kg).

The experimental investigations on a small planing machine were carried out with the aid of a special dynamometer of the following design. To the carriage of the planing machine (photographs 1 and 2) a pad B is fastened with four bolts; to this pad, in turn, the lever C and the bar D are fixed immovably, also by bolts. In the latter there are corresponding slots for the passage of the lever F and for the placement of the ball bearings E. By means of these bearings the lever F rotates, with very slight friction, about the axis EE′. The construction of this part of the dynamometer is shown in greater detail in fig. 5 of the accompanying plate. The movable lever F, when it is deflected from its initial position, is held at a certain distance from the fixed lever C by the spring G, fastened between the upper ends of the said levers. In the lower part of the lever F there is a box-shaped enlargement S, into which the holder P, serving to secure the cutter, is inserted. The holder P is freely fitted on the pin O, inserted into the cheeks of the box S; consequently, during the idle stroke, the cutter together with the holder is thrown back somewhat from the object being planed and slides freely over its surface. In the working position the holder and the cutter are brought back by their own weight and by the tension of the spring T.

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

To balance the moving masses of the dynamometer, an oil buffer R was introduced between the upper ends of the levers F and C; it is visible in the accompanying photographs 1 and 2 and is shown schematically...

—specifically in the section in Fig. 6 of the appended plate. This is an iron, precisely turned cylinder with a tightly fitted piston, connected by a hinge, by means of an eye on the bottom, to the upper end of lever C. The rod of the piston of this cylinder, passing through a cover and gland screwed to the latter, is attached by a hinge to lever F by means of a cylindrical nut screwed onto the end of the rod and provided with special eyes. Both cavities of the cylinder, in front of and behind the piston, are connected by stopcocks X with a common box y (see photograph 2), into which oil is poured through openings Z closed with plugs. Thus, during planing, the upper end of lever F, stretching the spring, is deflected and pulls with it the piston of the buffer; the piston, in turn, compresses the oil located in front of it and forces it through the stopcocks into the common box. In this way a smooth deflection of lever F and of pointer C from their initial position during planing is achieved, and an equally smooth return of them back during the idle stroke of the machine. During the experiment care was taken that there should be no air in the buffer. If this condition was observed, then, first, it was impossible to pull back the upper end of lever F when the stopcocks X were closed; second, with the stopcocks open, during the observation itself the deflection of the pointer occurred gradually, without noticeable vibrations or jolts.

The graduation of the apparatus was carried out after it had been installed on the planing machine in the following manner: a pulley was fastened to the table of the machine (drawings 7 and 8 of the appended plate), over which a steel cable was passed; one end of the cable was connected to the dynamometer by means of an iron plate with cut-outs, fitted over a prismatic piece of iron (drawing 8), inserted into the holder of the apparatus in place of the cutter, while weights were suspended from the other end. At the same time, the deflection of pointer L along arc M was noted, and directly on it were marked the pressures, in kilograms, corresponding to these deflections. The distance \(a\) (drawing 8) between the middle of the thickness of the aforesaid plate and the lower plane of the holder was equal to 20 mm, i.e. to the length of the part of the cutter projecting from the holder. This was done in order, as far as possible, to maintain equality of the arms on which act the force of tension of the cable during graduation and the force of pressure on the cutter during planing of the test bar.

With regard to the dynamometer described, it may be said that it is simple in construction, but, unfortunately, has certain substantial shortcomings: 1) friction is not fully eliminated in it,

which, not being subject to account, affects the result of the experiments and diminishes the sensitivity of the instrument; 2) together with a change in the position of lever F, the position of the cutter relative to the object being sharpened also changes, i.e., the cutting angle changes, which, of course, must also be reflected in the final results.

Then a major shortcoming of the work on the planing machine was that, owing to the construction of the machine itself, it was impossible to increase significantly the number of feeds, and it was necessary to work with only three feeds: 0.35, 0.70, and 1.05 mm. Then, owing to the narrowness of the problem that could be solved on the planing machine, only cast iron could be taken as the material for investigation. The result of the observations obtained is presented in the following Table XVI.

Table XVI. Cast iron (planing machine).

From an examination of the last table it is seen that, with regard to cast iron, the general assumption concerning a decrease in the coefficient of cutting of metals with an increase in the magnitude of the feed, at one and the same speed within the observed limits, is very close to reality.

Finally, the experimental investigations on the lathe were set up considerably more broadly, since the construction of the lathe itself made it possible to vary the number of different feeds from 8 to 12.

The work was carried out on a lathe driven by a separate electric motor and with mechanical feed by means of a lead screw. The lathe was completely isolated from the others, and observations of the power consumed were fully ensured in their correctness.

The instrument for measuring the pressure on the cutter was a spring dynamometer of simple design, shown in the drawing of Fig. 9 of the appended plate. The dynamometer is attached to the upper board of the carriage and moves together with the carriage from the lead screw. Photograph No. 3 shows the lathe, taken together with the dynamometer for the indicated tests.

To eliminate excessive oscillations, an oil dashpot is inserted between the levers of the dynamometer.

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

The actual conduct of the experiments proceeded as follows: after setting the blank accurately and measuring its initial diameter, the machine was started, and, after several readings of the number of revolutions and of the load on the cutter, the feed was changed, gradually passing from smaller to larger feeds, or conversely, and the instrument was watched throughout. The readings thus obtained made it possible to determine the cutting coefficient for the given speed and feed, and the cutting speed itself, from the formulas:

\[ k=\frac{p}{\omega},\quad v=\frac{\pi d.n}{60}, \]

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

\[ \text{millimeters}=\frac{d_0-d_1}{2}l,\quad l\text{—feed,} \]

\(p\)—load on the cutter in klgr.,

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

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

The cutting speed in the experiments on the lathe was maintained, on average, for iron at 0.1150 meters per second; for cast iron and bronze, at 0.0925 meters per second.

The results of the observations obtained for the last experiments on the lathe are given in the following Tables XVII, where \(g\) is the reading of the dynamometer spring, and \(k\) is the cutting coefficient in kilograms per square millimeter.

Table XVII. IRON.

On the Influence of Feed on the Cutting Coefficient of Metals

CAST IRON.

BRONZE.

If the result of the observations obtained for any material, on the basis of the data presented, is represented in the form of a curve, then we obtain a curve quite reminiscent of Fig. 10 of the appended table, relating to the machining of iron blanks with chips of thickness 0,29 mm. In constructing this curve, the cutting coefficients in kilograms are plotted along the ordinate axis, and the magnitude of the feeds along the abscissa axis.

Thus, summarizing the principal results of the investigations set forth above on the coefficient of cutting of metals, carried out by the above-described method on milling, planing, and turning machines, and taking into account the random deviations of the values of the cutting coefficient caused either by the nonuniformity of the material being machined or by shortcomings in the design of the dynamometers employed, we arrive at the conclusion that the previously expressed assumption concerning the decrease in the coefficient of cutting of metals, at one and the same cutting speed, with an increase in the magnitude of the feed itself, is fully substantiated.

Fig. 2.

Fig. 1.

Fig. 3.

Fig. 5.

Fig. 6.

Fig. 2.

Fig. 10.

Vertical axis: Specific pressure \(P\) in kg per \(1\ \mathrm{mm}^2\) of the cross-sectional area of the chip.

Horizontal axis: Feed \(b\), in hundredths of a millimeter.

I. Tikhonov: “The Effect of Feed on the Coefficient of Cutting of Metals.”

Fig. 5.

Fig. 1.

Fig. 7.

Fig. 8.