Study of the Deformation of Steel Balls Under Impact: With a Table of Drawings and Phototypes
Full Text
PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II.
Vol. 16. 1909. No. 4.
II.
V. N. Pinegin.
A STUDY OF THE DEFORMATION OF STEEL BALLS UNDER THE ACTION OF IMPACT.
(With a plate of drawings and photographs.)
1–41.
Investigation of the Deformations of Steel Balls under the Influence of Impact.
V. N. Pinegin.
In the present article I intend to acquaint readers with the results of my experiments on the investigation of those mutual deformations which are obtained on steel balls placed one upon another and subjected to impact along the line of their centers; these experiments, carried out by me as early as the spring of 1907 in the Charlottenburg laboratory of Prof. Eugen Meyer, are, as it were, a continuation of my investigations on deformations produced by steel balls in various metals under the influence of impact, published in 1908 in the Proceedings of the Tomsk Technological Institute1.
In the earlier investigations the task was to determine the dependence between the live force, the magnitude of the deformation produced by the ball under the influence of this live force on a plate of some metal, and the diameter of this ball; in the present experiments, however, the metal plate was replaced by another ball, on which the deformation from the first ball was obtained, while the latter in turn also received an analogous deformation from the second; these deformations, of course, had to be different, depending on the magnitude of the balls, on whether, moreover, they were of the same size or of different sizes. Thus, the establishment of the dependence of the magnitude of the deformation on the balls upon the magnitude of the diameters of these balls, and also upon the magnitude of the live force, constituted the aim of the present investigations.
The experiments were conducted in the following manner: on the anvil of a drop hammer there was placed a steel plate—\(n\) (see drawings 1 and 2) with a recess corresponding to the given ball; on this plate there was set an iron cylinder, \(1\text{--}1\frac12\ \mathrm{m/m}\) thick, with a slit along
forming a cylinder, parallel to the axis, and with an internal diameter exactly equal to the diameter of the balls being tested (if they were of the same size; if not, then with a diameter equal to the diameter of the larger of the two balls being tested), but on the condition, however, that the balls passed through the cylinder without any friction; the test balls were lowered into this cylinder one upon the other, the height of the cylinder being such that the upper ball protruded from the cylinder by approximately \(1/3\) of its diameter.
Above the ball was placed a bent plate—\(p\), with an opening for the part of the upper ball protruding from the cylinder, resting with its ends on the anvil of the pile-driver (see Fig. 1). On this plate there was placed a steel “underlay”—\(\pi\), with a depression made in it beforehand, exactly corresponding to the dimensions of the upper ball. This depression, like the depression in the steel plate—\(n\), on which the cylinder with the balls was set, as was done in the previous experiments (see the article cited above, p. 4), was obtained by repeated blows of the pile-driver “ram” on the corresponding ball placed on the plate, from a height many times greater than that from which the “ram” had to fall in the experiments themselves; by this means a “settled” state of the depressions in the plates was achieved.
In experiments with balls of different diameters it was necessary to take measures for the proper central positioning of the upper (smaller-diameter) ball, which was usually achieved by putting on the ball, before inserting it into the cylinder, a narrow rubber ring—\(k\) (see Fig. 2), of thickness exactly equal to the difference between the radii of the cylinder and the ball. It is clear that both the slit in the cylinder along a generator parallel to its axis and the rubber ring put on the ball of smaller diameter were intended to give the balls the possibility of expanding freely under impact.
In the first stage of the experiments the blows were delivered by a small weight of \(0.587\ \mathrm{kg}\) in mass. This weight was a piece of iron of the shape shown in Fig. 3. At the end of the thicker cylindrical part it had a round hole—\(o\), through which passed a wire ring—\(k\); to this ring was tied a thread, by which the weight was suspended from the movable crosspiece of the pile-driver, owing to which, on the one hand, this weight could be set at the desired height above the balls being tested, and, on the other hand, the shape of the weight and the manner of suspending it served as a guarantee of the strictly vertical direction of the line passing through the center of gravity of the weight and the direction of the thread on which the weight was suspended,
Investigation of the Deformations of Steel Balls
which in turn served as a guarantee of the correctness of the blow by the load after its release; the release of the load itself was effected by setting fire to the thread with a match.
Since, in general, the living force developed by this load under the conditions of the experiment could not be large, and since, on the other hand, there was no reason to be especially concerned about preserving its end surface, the blows of this load were delivered to the balls without any interposition: the load fell directly upon the balls.
In the second series of experiments the blows were delivered by a load weighing 37.49 kg, released from various heights by means of the usual device used in mines.
The determination of the diameters of the deformation spots obtained on the balls upon impact (see Table 1, Figs. 1, 5) was carried out by means of a Zeiss comparator; moreover, in order to obtain more distinct impressions, the balls were usually coated before the experiment, over an alcohol lamp, with a thin layer of soot1.
In view of the impossibility—because of the errors in the results of the experiments already indicated in my article cited above, errors which I estimated at 2–3%—there was here no need to measure the diameters of the deformation spots more accurately than 1%.
After each blow the balls under test were carefully inspected, and if no cracks were noticed on them, they were again used for a new experiment.
In view of the use, in general, of not particularly strong blows, and in view of the excellent elastic properties of the material of the balls, they usually did not change their spherical shape from the blow; this was constantly checked with a micrometer screw.
The balls were purchased from the Waffen et Munition-Fabrik in Berlin by special order and were of three kinds: 25, 20, and 15 mm in diameter.
In the experiments the height of fall of the load was at first taken to be very small; then it was gradually increased until, finally, one ball or another was destroyed or developed cracks (with the load of 37.48 kg), or simply until the surface of one ball or another was more or less damaged (with the load of 0.587 kg).
The destruction almost always occurred in the form of the splitting of the ball by vertical central planes, in most cases into two, more rarely into several parts; the lower ball sometimes underwent additional destruction from below; but the most characteristic part of the destruction of the balls was invariably the formation of lens-shaped or polyhedral little pyramids at the place of contact of the balls. Some of the most characteristic forms of fracture are presented in the appended drawings1 (see Plate 1, Figs. 2, 3, 4); moreover, as is not difficult to see, there is a great similarity between our forms of fracture and the forms of destruction of balls under the static action of a force indicated by Schwinning2 and Stribeck3. Also characteristic, incidentally, is the phenomenon, so to speak, of the fusing of the balls under large drops of the load; in this case such a large quantity of heat developed at the place of contact of the balls that the adjoining parts of the balls, as well as those lying close to the place of contact, melted, and the balls became welded together, as may be seen in the appended drawing (Plate 1, Fig. 6), where a lens-shaped piece of the destroyed ball is visible, fused with the other ball, which remained intact.
The destruction of the balls occurred either at the very moment of impact, or somewhat later: there were cases when, after the impact, having taken the balls from the hammer anvil and inspected their surface, which proved to be without the slightest cracks, one placed them under the microscope of the comparator for measuring the area of deformation, and suddenly the ball, with a loud crack, flew apart into small pieces, although at least about ten minutes had already passed from the moment of impact to the moment of destruction.
Leaving aside for the time being the question of the living forces destroying the balls, I pass to the consideration of the deformations of the balls under the influence of impact before their destruction.
In the following tables (Nos. 1–7) are given the results of experiments with the smaller load (of 0.587 kg), all the designations being the same as those in my article cited above.
Investigation of the Deformation of Steel Balls
Diameters of the balls \(D_1 = D_2 = 25^{m}/m\).
Table 1.
| Height of fall of the load, in cm. | Live force \(L\), in kg. cm | \(a_1\), in mm. | \(a_2\), in mm. | \(d=\dfrac{a_1+a_2}{2}\), in mm. | Mean diam. \(d_m\) from several tests, in mm. | Area of deformation \(\omega_m\), in cm. | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 2,935 | 1,14 | 1,13 | 1,13 | 1,13 | 0,0100 | 293 | 29300 |
| — | — | 1,13 | 1,13 | 1,13 | 1,13 | 0,0100 | 293 | 29300 |
| — | — | 1,13 | 1,13 | 1,13 | 1,13 | 0,0100 | 293 | 29300 |
| — | — | 1,13 | 1,13 | 1,13 | 1,13 | 0,0100 | 293 | 29300 |
| 10 | 5,87 | 1,33 | 1,35 | 1,34 | 1,33 | 0,0139 | 422 | 30410 |
| — | — | 1,31 | 1,35 | 1,33 | 1,33 | 0,0139 | 422 | 30410 |
| — | — | 1,35 | 1,33 | 1,34 | 1,33 | 0,0139 | 422 | 30410 |
| — | — | 1,30 | 1,33 | 1,31 | 1,33 | 0,0139 | 422 | 30410 |
| 20 | 11,74 | 1,57 | 1,57 | 1,57 | 1,60 | 0,0201 | 584 | 29300 |
| — | — | 1,60 | 1,61 | 1,61 | 1,60 | 0,0201 | 584 | 29300 |
| — | — | 1,61 | 1,61 | 1,61 | 1,60 | 0,0201 | 584 | 29300 |
| — | — | 1,61 | 1,61 | 1,61 | 1,60 | 0,0201 | 584 | 29300 |
| 30 | 17,61 | 1,77 | 1,76 | 1,76 | 1,77 | 0,0246 | 716 | 29110 |
| — | — | 1,77 | 1,75 | 1,76 | 1,77 | 0,0246 | 716 | 29110 |
| — | — | 1,77 | 1,78 | 1,78 | 1,77 | 0,0246 | 716 | 29110 |
| — | — | 1,79 | 1,79 | 1,79 | 1,77 | 0,0246 | 716 | 29110 |
| 40 | 23,48 | 1,87 | 1,87 | 1,87 | 1,88 | 0,0277 | 848 | 30480 |
| — | — | 1,86 | 1,87 | 1,86 | 1,88 | 0,0277 | 848 | 30480 |
| — | — | 1,89 | 1,89 | 1,89 | 1,88 | 0,0277 | 848 | 30480 |
| — | — | 1,89 | 1,89 | 1,89 | 1,88 | 0,0277 | 848 | 30480 |
| 50 | 29,35 | 1,97 | 1,98 | 1,98 | 2,005 | 0,0316 | 929 | 29470 |
| — | — | 2,04 | 2,01 | 2,03 | 2,005 | 0,0316 | 929 | 29470 |
| — | — | 2,03 | 2,02 | 2,02 | 2,005 | 0,0316 | 929 | 29470 |
| — | — | 2,00 | 1,99 | 2,00 | 2,005 | 0,0316 | 929 | 29470 |
| Average | — | — | — | — | 29688 |
As is seen from this table, the greatest difference of the values of
\[ \frac{L}{\omega_m^2} \]
amounts here to
\[ \frac{30480-29110}{29110}\,100 = 4{,}7\%. \]
V. N. Pinegin.
Table 2.
Diameters of the balls \(D_1 = D_2 = 25\ \mathrm{m/m}\). Weight of the falling load \(P = 0{,}587\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diam. \(d_m\) of the impressions | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 2,935 | 1,11 | 1,12 | 1,11 | 1,12 | 0,0098 | 300 | 30730 |
| — | — | 1,12 | 1,12 | 1,12 | 1,12 | 0,0098 | 300 | 30730 |
| 10 | 5,87 | 1,32 | 1,35 | 1,33 | 1,32 | 0,0137 | 429 | 31270 |
| — | — | 1,32 | 1,31 | 1,31 | 1,32 | 0,0137 | 429 | 31270 |
| 20 | 11,74 | 1,58 | 1,59 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| — | — | 1,58 | 1,59 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| — | — | 1,59 | 1,58 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| — | — | 1,59 | 1,59 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| — | — | 1,59 | 1,59 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| — | — | 1,59 | 1,58 | 1,59 | 1,59 | 0,0197 | 596 | 30260 |
| 30 | 17,61 | 1,74 | 1,75 | 1,74 | 1,74 | 0,0238 | 740 | 31110 |
| — | — | 1,74 | 1,75 | 1,74 | 1,74 | 0,0238 | 740 | 31110 |
| 40 | 23,48 | 1,86 | 1,86 | 1,86 | 1,86 | 0,0273 | 856 | 31510 |
| — | — | 1,86 | 1,86 | 1,86 | 1,86 | 0,0273 | 856 | 31510 |
| — | — | 1,86 | 1,87 | 1,87 | 1,86 | 0,0273 | 856 | 31510 |
| — | — | 1,86 | 1,87 | 1,87 | 1,86 | 0,0273 | 856 | 31510 |
| 50 | 29,35 | 2,01 | 2,01 | 2,01 | 1,99 | 0,0309 | 950 | 30630 |
| — | — | 2,00 | 2,01 | 2,01 | 1,99 | 0,0309 | 950 | 30630 |
| — | — | 2,01 | 2,00 | 2,00 | 1,99 | 0,0309 | 950 | 30630 |
| — | — | 2,01 | 2,00 | 2,01 | 1,99 | 0,0309 | 950 | 30630 |
| — | — | 1,95 | 1,95 | 1,95 | 1,99 | 0,0309 | 950 | 30630 |
| — | — | 1,94 | 1,95 | 1,94 | 1,99 | 0,0309 | 950 | 30630 |
| 60 | 35,22 | 2,08 | 2,07 | 2,08 | 2,07 | 0,0338 | 1042 | 30840 |
| — | — | 2,07 | 2,07 | 2,07 | 2,07 | 0,0338 | 1042 | 30840 |
| Mean | — | — | — | — | — | — | — | 30907 |
The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) amounts here to
\[ \frac{31510-30260}{30260}100 = 4{,}2\%. \]
Table 3.
Diameters of the balls \(D_1 = D_2 = 20^{m}/m\). Weight of the falling load \(P = 0.587\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg. cm. | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in sq. cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 2.935 | 1.05 | 1.05 | 1.05 | 1.04 | 0.0085 | 345 | 40590 |
| — | — | 1.04 | 1.05 | 1.05 | 1.04 | 0.0085 | 345 | 40590 |
| — | — | 1.03 | 1.05 | 1.04 | 1.04 | 0.0085 | 345 | 40590 |
| — | — | 1.05 | 1.03 | 1.04 | 1.04 | 0.0085 | 345 | 40590 |
| 10 | 5.87 | 1.25 | 1.24 | 1.25 | 1.24 | 0.0121 | 485 | 40200 |
| — | — | 1.24 | 1.24 | 1.24 | 1.24 | 0.0121 | 485 | 40200 |
| 20 | 11.74 | 1.48 | 1.48 | 1.48 | 1.48 | 0.0172 | 683 | 39800 |
| — | — | 1.48 | 1.48 | 1.48 | 1.48 | 0.0172 | 683 | 39800 |
| — | — | 1.47 | 1.47 | 1.47 | 1.48 | 0.0172 | 683 | 39800 |
| — | — | 1.48 | 1.48 | 1.48 | 1.48 | 0.0172 | 683 | 39800 |
| 30 | 17.61 | 1.64 | 1.63 | 1.64 | 1.63 | 0.0209 | 843 | 40300 |
| — | — | 1.64 | 1.63 | 1.63 | 1.63 | 0.0209 | 843 | 40300 |
| — | — | 1.63 | 1.63 | 1.63 | 1.63 | 0.0209 | 843 | 40300 |
| — | — | 1.63 | 1.63 | 1.63 | 1.63 | 0.0209 | 843 | 40300 |
| 40 | 23.48 | 1.75 | 1.75 | 1.75 | 1.75 | 0.0241 | 974 | 40420 |
| — | — | 1.75 | 1.75 | 1.75 | 1.75 | 0.0241 | 974 | 40420 |
| — | — | 1.75 | 1.74 | 1.75 | 1.75 | 0.0241 | 974 | 40420 |
| — | — | 1.74 | 1.75 | 1.75 | 1.75 | 0.0241 | 974 | 40420 |
| 50 | 29.35 | 1.84 | 1.85 | 1.84 | 1.81 | 0.0266 | 1103 | 41450 |
| — | — | 1.84 | 1.84 | 1.84 | 1.81 | 0.0266 | 1103 | 41450 |
| 60 | 35.22 | 1.93 | 1.94 | 1.93 | 1.94 | 0.0297 | 1186 | 39930 |
| — | — | 1.94 | 1.95 | 1.95 | 1.94 | 0.0297 | 1186 | 39930 |
| — | — | 1.95 | 1.94 | 1.94 | 1.94 | 0.0297 | 1186 | 39930 |
| — | — | 1.94 | 1.95 | 1.95 | 1.94 | 0.0297 | 1186 | 39930 |
| Mean | — | — | — | — | — | — | — | 40415 |
The greatest discrepancy of the values of \(\dfrac{L}{\omega_m^2}\) is
\[ \frac{41450-39800}{39800}100 = 4.1\%. \]
Table 4.
Diameters of the balls \(D_1 = D_2 = 20^{m}/m\). Weight of the falling load \(P = 0.587\) kg.
| Height of fall of the load, in cm | Living force \(L\), in kg·cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 2.935 | 1.04 | 1.05 | 1.04 | 1.04 | 0.00849 | 346 | 40720 |
| — | — | 1.04 | 1.04 | 1.04 | 1.04 | 0.00849 | 346 | 40720 |
| 10 | 5.87 | 1.24 | 1.24 | 1.24 | 1.24 | 0.01208 | 486 | 40210 |
| — | — | 1.24 | 1.25 | 1.24 | 1.24 | 0.01208 | 486 | 40210 |
| — | — | 1.24 | 1.25 | 1.24 | 1.24 | 0.01208 | 486 | 40210 |
| — | — | 1.24 | 1.24 | 1.24 | 1.24 | 0.01208 | 486 | 40210 |
| 20 | 11.74 | 1.48 | 1.48 | 1.48 | 1.48 | 0.0172 | 683 | 39670 |
| — | — | 1.48 | 1.49 | 1.48 | 1.48 | 0.0172 | 683 | 39670 |
| 30 | 17.61 | 1.63 | 1.63 | 1.63 | 1.63 | 0.0209 | 843 | 40300 |
| — | — | 1.63 | 1.62 | 1.63 | 1.63 | 0.0209 | 843 | 40300 |
| 40 | 23.48 | 1.75 | 1.76 | 1.76 | 1.76 | 0.0243 | 966 | 39760 |
| — | — | 1.77 | 1.75 | 1.76 | 1.76 | 0.0243 | 966 | 39760 |
| — | — | 1.76 | 1.76 | 1.76 | 1.76 | 0.0243 | 966 | 39760 |
| — | — | 1.76 | 1.75 | 1.76 | 1.76 | 0.0243 | 966 | 39760 |
| 50 | 29.35 | 1.85 | 1.85 | 1.85 | 1.85 | 0.0269 | 1091 | 40590 |
| — | — | 1.85 | 1.85 | 1.85 | 1.85 | 0.0269 | 1091 | 40590 |
| 60 | 35.22 | 1.93 | 1.94 | 1.93 | 1.93 | 0.0293 | 1203 | 41050 |
| — | — | 1.94 | 1.93 | 1.93 | 1.93 | 0.0293 | 1203 | 41050 |
| Mean | — | — | — | — | — | — | — | 40370 |
The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) is
\[ \frac{41050-39670}{39670}\,100 = 3.5\%. \]
Table 5.
Diameters of the balls \(D_1=D_2=15\ \mathrm{m}/\mathrm{m}\). Weight of the falling load \(P=0.587\ \mathrm{kg}\).
| Height of fall of load, cm | Live force \(L\), kg·cm | Diameter of deformation area \(a_1\), mm | Diameter of deformation area \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 5.87 | 1.17 | 1.17 | 1.17 | 1.18 | 0.0109 | 539 | 49410 |
| — | — | 1.18 | 1.17 | 1.18 | 1.18 | 0.0109 | 539 | 49410 |
| — | — | 1.18 | 1.18 | 1.18 | 1.18 | 0.0109 | 539 | 49410 |
| — | — | 1.18 | 1.18 | 1.18 | 1.18 | 1.18 | 0.0109 | 539 |
| 20 | 11.74 | 1.38 | 1.38 | 1.38 | 1.39 | 0.0152 | 772 | 50820 |
| — | — | 1.39 | 1.39 | 1.39 | 1.39 | 0.0152 | 772 | 50820 |
| — | — | 1.39 | 1.39 | 1.39 | 1.39 | 0.0152 | 772 | 50820 |
| — | — | 1.39 | 1.39 | 1.39 | 1.39 | 0.0152 | 772 | 50820 |
| 30 | 17.61 | 1.55 | 1.55 | 1.55 | 1.55 | 0.0189 | 932 | 49330 |
| — | — | 1.55 | 1.56 | 1.55 | 1.55 | 0.0189 | 932 | 49330 |
| — | — | 1.55 | 1.55 | 1.55 | 1.55 | 0.0189 | 932 | 49330 |
| — | — | 1.56 | 1.55 | 1.55 | 1.55 | 0.0189 | 932 | 49330 |
| 40 | 23.48 | 1.67 | 1.67 | 1.67 | 1.66 | 0.0216 | 1087 | 59280 |
| — | — | 1.67 | 1.66 | 1.66 | 1.66 | 0.0216 | 1087 | 59280 |
| — | — | 1.65 | 1.66 | 1.65 | 1.66 | 0.0216 | 1087 | 59280 |
| — | — | 1.66 | 1.65 | 1.65 | 1.66 | 0.0216 | 1087 | 59280 |
| 50 | 29.35 | 1.75 | 1.74 | 1.75 | 1.75 | 0.0241 | 1218 | 50600 |
| — | — | 1.74 | 1.75 | 1.74 | 1.75 | 0.0241 | 1218 | 50600 |
| — | — | 1.75 | 1.75 | 1.75 | 1.75 | 0.0241 | 1218 | 50600 |
| — | — | 1.74 | 1.75 | 1.75 | 1.75 | 0.0241 | 1218 | 50600 |
| Mean | — | — | — | — | — | — | — | 50088 |
The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) here:
\[ \frac{50820-49330}{49330}\,100=3.0\%. \]
Table 6.
Diameters of the balls \(D_1 = D_2 = 15^{m}/m\). Weight of the falling load \(P = 0.587\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg. cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in sq. cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 2.935 | 0.99 | 0.99 | 0.99 | 0.99 | 0.0077 | 381 | 49490 |
| — | — | 0.99 | 1.00 | 0.99 | 0.99 | 0.0077 | 381 | 49490 |
| 10 | 5.87 | 1.17 | 1.17 | 1.17 | 1.17 | 0.01075 | 546 | 50820 |
| — | — | 1.17 | 1.17 | 1.17 | 1.17 | 0.01075 | 546 | 50820 |
| 20 | 11.74 | 1.39 | 1.40 | 1.40 | 1.40 | 0.0154 | 762 | 49540 |
| — | — | 1.40 | 1.40 | 1.40 | 1.40 | 0.0154 | 762 | 49540 |
| — | — | 1.40 | 1.39 | 1.40 | 1.40 | 0.0154 | 762 | 49540 |
| — | — | 1.40 | 1.40 | 1.40 | 1.40 | 0.0154 | 762 | 49540 |
| 30 | 17.61 | 1.54 | 1.54 | 1.54 | 1.54 | 0.0186 | 947 | 50900 |
| — | — | 1.55 | 1.54 | 1.54 | 1.54 | 0.0186 | 947 | 50900 |
| 40 | 23.48 | 1.66 | 1.66 | 1.66 | 1.66 | 0.0216 | 1087 | 50280 |
| — | — | 1.66 | 1.67 | 1.66 | 1.66 | 0.0216 | 1087 | 50280 |
| Mean | — | — | — | — | — | — | — | 50210 |
The greatest difference of the values \(\dfrac{L}{\omega_m^2}\) here is
\[ \frac{50900-49490}{49490}\,100 = 2.9\%. \]
Investigation of the Deformations of Steel Balls
As is seen from all the tables presented, the ratio of the vis viva to the square of the corresponding area of deformation of the tested balls is a constant quantity for balls of a given diameter: calculations showed that the differences between the individual values
\[ \frac{L}{\omega_m^{2}} \]
for balls of one size fluctuate within the limits—2.9–4.7%, which differences, under the given experimental conditions, cannot be regarded as excessively large.
It is necessary to note that even at such comparatively small vis viva as 2.935 kg. cm. or 5.87 kg. cm., the deformations of the balls proved to be residual, as could easily be verified by examining the surfaces of the balls under a microscope; thus, the results found must be regarded as relating to the properties of the material beyond the elastic limit.
To obtain the dependence between the vis viva, the diameter of the area of deformation of the balls, and the diameters of the latter, from the data of the tables given above there were determined the mean diameters of the areas of deformation of balls of a given size and for a given magnitude of the vis viva; the values of these mean diameters, raised to the fourth power and divided by the value of the diameter of the corresponding ball, gave, for the given magnitude of the vis viva, an almost identical value—let us call it \(\eta\)—for all three kinds of tested balls; finally, it turned out that the ratio of the vis viva to the mean \((\eta_m)\) value of these \(\eta\)’s is a constant quantity for all the magnitudes of the vis viva that occurred; a summary of the results of all the indicated calculations is given in the following Table No. 7.
Table 7.
| Height of fall of the load \(H\) in cm. | Vis viva \(L\), in kg. cm. | Diameter of ball \(D\), in cm. | Diameter of area of deformation \(d_m\), in cm. | \(d_m^4\) | \(\eta=\dfrac{d_m^4}{D}\) | Mean \(\eta_m\) from separate values of \(\eta\) | \(\dfrac{L}{\eta_m}\) |
|---|---|---|---|---|---|---|---|
| 5 | 2.935 | 1.5 | 0.0990 | 0.0000960 | 0.0000640 | 0.0000620 | 47340 |
| — | — | 2.0 | 0.1040 | 0.0001188 | 0.0000583 | 0.0000620 | 47340 |
| — | — | 2.5 | 0.1125 | 0.0001590 | 0.0000636 | 0.0000620 | 47340 |
Table 7.
| Height of fall of the load \(H\), in cm | Live force \(L\), in kg·cm | Diameter of the ball \(D\), in cm | Diameter of the area of deformation \(d_m\), in cm | \(d_m^4\) | \(\eta=\dfrac{d_m^4}{D}\) | Mean \(\eta_m\) from the separate values of \(\eta\) | \(\dfrac{L}{\eta_m}\) |
|---|---|---|---|---|---|---|---|
| 10 | 5,87 | 1,5 | 0,1170 | 0,000188 | 0,000125 | 0,000123 | 47720 |
| — | — | 2,0 | 0,1240 | 0,000237 | 0,000119 | 0,000123 | 47720 |
| — | — | 2,5 | 0,1325 | 0,000313 | 0,000126 | 0,000123 | 47720 |
| 20 | 11,74 | 1,5 | 0,1395 | 0,000376 | 0,000250 | 0,000249 | 47150 |
| — | — | 2,0 | 0,1480 | 0,000480 | 0,000240 | 0,000249 | 47150 |
| — | — | 2,5 | 0,1595 | 0,000645 | 0,000258 | 0,000249 | 47150 |
| 30 | 17,61 | 1,5 | 0,1545 | 0,000569 | 0,000379 | 0,000370 | 47600 |
| — | — | 2,0 | 0,1630 | 0,000708 | 0,000354 | 0,000370 | 47600 |
| — | — | 2,5 | 0,1755 | 0,000945 | 0,000378 | 0,000370 | 47600 |
| 40 | 23,48 | 1,5 | 0,1660 | 0,000759 | 0,000505 | 0,000488 | 48120 |
| — | — | 2,0 | 0,1755 | 0,000945 | 0,000472 | 0,000488 | 48120 |
| — | — | 2,5 | 0,1870 | 0,001225 | 0,000490 | 0,000488 | 48120 |
| 50 | 29,35 | 1,5 | 0,1750 | 0,000936 | 0,000624 | 0,000615 | 47720 |
| — | — | 2,0 | 0,1845 | 0,001170 | 0,000585 | 0,000615 | 47720 |
| — | — | 2,5 | 0,1998 | 0,001592 | 0,000637 | 0,000615 | 47720 |
| 60 | 35,22 | 1,5 | — | — | — | 0,000722 | 48780 |
| — | — | 2,0 | 0,1940 | 0,001414 | 0,000707 | 0,000722 | 48780 |
| — | — | 2,5 | 0,2070 | 0,001840 | 0,000736 | 0,000722 | 48780 |
| Mean | — | — | — | — | — | — | 47766 |
The greatest difference here between the values of \(\dfrac{L}{\eta_m}\) is
\[ \frac{48780-47150}{47150}\,100=3{,}5\%. \]
Thus, with very great accuracy, we have the relation:
$$ \frac{L}{\eta_m}=C $$
or
$$ L=C\eta_m=C\frac{d_m^4}{D}, \tag{1} $$
which is valid for balls of different diameters; here \(C\) is a constant quantity, independent both of the magnitude of the vis viva and of the diameter of the balls, but dependent on the material of the balls.
This completed the experiments of the first stage—with a falling weight of \(0{,}587\ \mathrm{kg}\); the second series of experiments was carried out, as already stated above, with a weight of \(37{,}49\ \mathrm{kg}\).
In experiments with such a falling weight it was already necessary to use a “pad,” in order, on the one hand, not to damage the end surface of the “anvil,” and, on the other hand, not to damage the upper ball with the “anvil,” since it is obvious that, with the “pad,” the latter receives the vis viva over the whole contact surface with the “pad,” and not at one point only, as happens without the “pad.”
The results of the experiments of this series are given in the following Tables 8—21.
Table 8.
Diameters of the balls \(D_1=D_2=25^{m}/m\). Weight of the falling load \(P=37{,}49\ \mathrm{kg}\).
| Height of fall of the load \(H\) in cm | Vis viva \(L\) in kg. cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37,49 | 2,10 | 2,11 | 2,11 | 2,11 | 0,0349 | 1070 | 30780 |
| — | — | 2,13 | 2,11 | 2,12 | 2,11 | 0,0349 | 1070 | 30780 |
| 2 | 74,98 | 2,49 | 2,49 | 2,49 | 2,49 | 0,0487 | 1540 | 31640 |
| — | — | 2,48 | 2,49 | 2,48 | 2,49 | 0,0487 | 1540 | 31640 |
| 3 | 112,47 | 2,76 | 2,77 | 2,77 | 2,77 | 0,0602 | 1870 | 31100 |
| — | — | 2,78 | 2,78 | 2,78 | 2,77 | 0,0602 | 1870 | 31100 |
| 4 | 149,96 | 2,96 | 2,97 | 2,96 | 2,96 | 0,0688 | 2180 | 31680 |
| — | — | 2,96 | 2,96 | 2,96 | 2,96 | 0,0688 | 2180 | 31680 |
| 5 | 187,45 | 3,15 | 3,11 | 3,13 | 3,14 | 0,0774 | 2420 | 31300 |
| — | — | 3,15 | 3,14 | 3,15 | 3,14 | 0,0774 | 2420 | 31300 |
| 10 | 374,9 | 3,73 | 3,72 | 3,72 | 3,72 | 0,1087 | 3450 | 31720 |
| — | — | 3,72 | 3,72 | 3,72 | 3,72 | 0,1087 | 3450 | 31720 |
| Average | — | — | — | — | — | — | — | 31350 |
The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) here is:
$$ \frac{31720-30780}{30780}\,100=3{,}1\%. $$
Table 9.
Diameters of the balls \(D_1 = D_2 = 25\ \mathrm{m/m}\). Height of the falling load \(P = 37.49\ \mathrm{kg}\).
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374,9 | 3,78 | 3,77 | 3,78 | 3,76 | 0,1110 | 3380 | 30450 |
| — | — | 3,78 | 3,78 | 3,78 | 3,76 | 0,1110 | 3380 | 30450 |
| — | — | 3,72 | 3,74 | 3,73 | 3,76 | 0,1110 | 3380 | 30450 |
| — | — | 3,74 | 3,74 | 3,74 | 3,76 | 0,1110 | 3380 | 30450 |
| 20 | 749,8 | 4,46 | 4,46 | 4,46 | 4,46 | 0,1562 | 4800 | 30730 |
| — | — | 4,46 | 4,45 | 4,46 | 4,46 | 0,1562 | 4800 | 30730 |
| 30 | 1124,7 | 4,98 | 4,98 | 4,98 | 4,97 | 0,1940 | 5797 | 29880 |
| — | — | 4,98 | 4,98 | 4,98 | 4,97 | 0,1940 | 5797 | 29880 |
| — | — | 4,97 | 4,96 | 4,96 | 4,97 | 0,1940 | 5797 | 29880 |
| — | — | 4,97 | 4,97 | 4,97 | 4,97 | 0,1940 | 5797 | 29880 |
| 40 | 1499,6 | 5,33 | 5,32 | 5,32 | 5,33 | 0,2231 | 6720 | 30150 |
| — | — | 5,33 | 5,32 | 5,33 | 5,33 | 0,2231 | 6720 | 30150 |
| Mean | — | — | — | — | — | — | — | 30298 |
The difference between the individual values of \(\dfrac{L}{\omega_m^2}\) is
\[ \frac{30730-29880}{29880}\,100 = 2{,}8\%. \]
Table 10.
Diameters of the balls \(D_1 = D_2 = 25^{m/m}\). Weight of the falling load \(P = 37.49\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of deformation areas \(a_1\), in mm | Diameter of deformation areas \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374.9 | 3.84 | 3.84 | 3.84 | 3.84 | 0.1158 | 3240 | 28000 |
| — | — | 3.84 | 3.84 | 3.84 | 3.84 | 0.1158 | 3240 | 28000 |
| 20 | 749.8 | 4.59 | 4.53 | 4.58 | 4.58 | 0.1647 | 4550 | 27650 |
| — | — | 4.58 | 4.59 | 4.58 | 4.58 | 0.1647 | 4550 | 27650 |
| 30 | 1124.7 | 5.07 | 5.06 | 5.06 | 5.06 | 0.2011 | 5590 | 27840 |
| — | — | 5.06 | 5.06 | 5.06 | 5.06 | 0.2011 | 5590 | 27840 |
| 40 | 1499.6 | 5.41 | 5.39 | 5.40 | 5.40 | 0.2290 | 6550 | 28620 |
| — | — | 5.42 | 5.40 | 5.41 | 5.40 | 0.2290 | 6550 | 28620 |
| 50 | 1874.5 | 5.72 | 5.72 | 5.72 | 5.72 | 0.2570 | 7290 | 28380 |
| — | — | 5.73 | 5.72 | 5.72 | 5.72 | 0.2570 | 7290 | 28380 |
| 60 | 2249.4 | 5.98 | 5.98 | 5.98 | 5.98 | 0.2809 | 8010 | 28490 |
| — | — | 5.98 | 5.97 | 5.98 | 5.98 | 0.2809 | 8010 | 28490 |
| Mean | — | — | — | — | — | — | — | 28163 |
Difference between the values of \(\dfrac{L}{\omega_m^2}\):
\[ \frac{28620-27650}{27650}\,100 = 3.5\%. \]
Table 11.
Ball diameters \(D_1 = D_2 = 25\). Weight of the falling load \(P = 37.49\) kg.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of deformation area \(a_1\), in mm | Diameter of deformation area \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37.49 | 2.15 | 2.15 | 2.15 | 2.15 | 0.0363 | 1030 | 28440 |
| — | — | 2.15 | 2.16 | 2.15 | 2.15 | 0.0363 | 1030 | 28440 |
| 3 | 112.47 | 2.81 | 2.82 | 2.82 | 2.82 | 0.0625 | 1800 | 28790 |
| — | — | 2.82 | 2.82 | 2.82 | 2.82 | 0.0625 | 1800 | 28790 |
| 5 | 187.45 | 3.23 | 3.23 | 3.23 | 3.23 | 0.0819 | 2290 | 27940 |
| — | — | 3.22 | 3.23 | 3.23 | 3.23 | 0.0819 | 2290 | 27940 |
| 10 | 374.9 | 3.84 | 3.83 | 3.83 | 3.83 | 0.1152 | 3260 | 28190 |
| — | — | 3.82 | 3.84 | 3.83 | 3.83 | 0.1152 | 3260 | 28190 |
| 20 | 749.8 | 4.57 | 4.58 | 4.58 | 4.57 | 0.1640 | 4570 | 27870 |
| — | — | 4.57 | 4.57 | 4.57 | 4.57 | 0.1640 | 4570 | 27870 |
| Mean | — | — | — | — | — | — | — | 28246 |
The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{28790-27870}{27870}100=3.3\%. \]
INVESTIGATION OF THE DEFORMATION OF STEEL BALLS
From the data of Tables Nos. 8, 9, 10, and 11, the mean results for balls of diameter \(25^{m}/_{m}\), presented in the following table, have been derived.
Table 12.
| Live force in kg. cm. | \(\dfrac{L}{\omega m}\) (8) | \(\dfrac{L}{\omega m}\) (9) | \(\dfrac{L}{\omega m}\) (10) | \(\dfrac{L}{\omega m}\) (11) | Mean | \(\dfrac{L}{\omega m^2}\) (8) | \(\dfrac{L}{\omega m^2}\) (9) | \(\dfrac{L}{\omega m^2}\) (10) | \(\dfrac{L}{\omega m^2}\) (11) | Mean | Note |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 37,4 | 1070 | — | — | 1030 | 1050 | 30780 | — | — | 28440 | 29610 | The figures (8), (9), (10), (11) under \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) denote: from which table the corresponding values of \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) were taken. |
| 112,47 | 1870 | — | — | 1800 | 1835 | 31000 | — | — | 28790 | 29895 | The figures (8), (9), (10), (11) under \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) denote: from which table the corresponding values of \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) were taken. |
| 187,45 | 2420 | — | — | 2290 | 2355 | 31300 | — | — | 27940 | 29620 | The figures (8), (9), (10), (11) under \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) denote: from which table the corresponding values of \(\dfrac{L}{\omega m}\) and \(\dfrac{L}{\omega m^2}\) were taken. |
| 374,9 | 3450 | 3380 | 3240 | 3260 | 3332 | 31720 | 30430 | 28000 | 28190 | 29585 | |
| 749,8 | — | 4800 | 4550 | 4570 | 4640 | — | 30730 | 27650 | 27870 | 28750 | |
| 1124,7 | — | 5797 | 5590 | — | 5693 | — | 29880 | 27840 | — | 28860 | |
| 1499,6 | — | 6720 | 6550 | — | 6635 | — | 30150 | 28620 | — | 29385 | |
| 1874,5 | — | — | 7290 | — | 7290 | — | — | 28380 | — | 28380 | |
| 2249,4 | — | — | 8010 | — | 8010 | — | — | 28490 | — | 28490 | |
| Mean | — | — | — | — | — | — | — | — | — | 29175 |
As is seen from the last table, despite the incompleteness of the data for all magnitudes of live forces, nevertheless the deviation, for all tested balls with a diameter of \(25^{m}/_{m}\), from the dependence \(L = C d^4\) does not exceed
\[ \frac{29895 - 28380}{28380}100 = 5,3\%. \]
Table 13.
Diameters of the balls \(D_1=D_2=20^{m}/m\). Weight of the falling load \(P=37.49\ \mathrm{kg}\).
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), sq. cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^{2}}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374.9 | 3.51 | 3.49 | 3.50 | 3.50 | 0.0962 | 3900 | 40530 |
| — | — | 3.51 | 3.49 | 3.50 | 3.50 | 0.0962 | 3900 | 40530 |
| — | — | 3.49 | 3.51 | 3.50 | 3.50 | 0.0962 | 3900 | 40530 |
| — | — | 3.51 | 3.49 | 3.50 | 3.50 | 0.0962 | 3900 | 40530 |
| 20 | 749.8 | 4.20 | 4.21 | 4.20 | 4.20 | 0.1385 | 5410 | 39090 |
| — | — | 4.20 | 4.21 | 4.21 | 4.20 | 0.1385 | 5410 | 39090 |
| — | — | 4.21 | 4.20 | 4.20 | 4.20 | 0.1385 | 5410 | 39090 |
| — | — | 4.20 | 4.20 | 4.20 | 4.20 | 0.1385 | 5410 | 39090 |
| 30 | 1124.7 | 4.62 | 4.63 | 4.63 | 4.63 | 0.1680 | 6700 | 39880 |
| — | — | 4.64 | 4.62 | 4.63 | 4.63 | 0.1680 | 6700 | 39880 |
| — | — | 4.62 | 4.62 | 4.62 | 4.63 | 0.1680 | 6700 | 39880 |
| — | — | 4.62 | 4.64 | 4.63 | 4.63 | 0.1680 | 6700 | 39880 |
| Mean | — | — | — | — | — | — | — | 39830 |
Here the difference of the values of \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{40530-39090}{39090}\,100=3.7\%. \]
Table 14.
Diameters of the balls \(D_1=D_2=20^{m}/m\). Weight of the falling load \(P=37.49\) kg.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in mm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374.9 | 3.56 | 3.54 | 3.55 | 3.55 | 0.0989 | 3800 | 38260 |
| — | — | 3.56 | 3.54 | 3.55 | 3.55 | 0.0989 | 3800 | 38260 |
| 30 | 1124.7 | 4.65 | 4.64 | 4.65 | 4.66 | 0.1705 | 6600 | 38650 |
| — | — | 4.67 | 4.66 | 4.67 | 4.66 | 0.1705 | 6600 | 38650 |
| 40 | 1499.6 | 4.99 | 4.98 | 4.99 | 4.99 | 0.1956 | 7670 | 39250 |
| — | — | 4.99 | 4.99 | 4.99 | 4.99 | 0.1956 | 7670 | 39250 |
| 50 | 1874.5 | 5.29 | 5.30 | 5.29 | 5.29 | 0.2200 | 8520 | 38730 |
| — | — | 5.30 | 5.28 | 5.29 | 5.29 | 0.2200 | 8520 | 38730 |
| Mean | — | — | — | — | — | — | — | 38725 |
The greatest difference of the values \(\dfrac{L}{\omega_m^2}\):
\[ \frac{39250-38260}{38260}\,100=2.6\%. \]
V. N. Ipatieff
Table 15.
Diameters of the balls \(D_1 = D_2 = 20^{m}/m\). Weight of the falling load \(P = 37.49\) kg.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of deformation areas \(a_1\), in mm | Diameter of deformation areas \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in mm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374.9 | 3.53 | 3.52 | 3.53 | 3.52 | 0.0974 | 3850 | 39520 |
| — | — | 3.52 | 3.52 | 3.52 | 3.52 | 0.0974 | 3850 | 39520 |
| 20 | 749.8 | 4.22 | 4.20 | 4.21 | 4.21 | 0.1392 | 5390 | 38610 |
| — | — | 4.21 | 4.21 | 4.21 | 4.21 | 0.1392 | 5390 | 38610 |
| 30 | 1124.7 | 4.67 | 4.66 | 4.66 | 4.66 | 0.1705 | 6600 | 38770 |
| — | — | 4.65 | 4.65 | 4.65 | 4.66 | 0.1705 | 6600 | 38770 |
| 40 | 1499.6 | 5.01 | 5.00 | 5.00 | 5.01 | 0.1973 | 7600 | 38650 |
| — | — | 5.02 | 5.02 | 5.02 | 5.01 | 0.1973 | 7600 | 38650 |
| 50 | 1874.5 | 5.29 | 5.29 | 5.29 | 5.29 | 0.2198 | 8530 | 38810 |
| — | — | 5.30 | 5.29 | 5.29 | 5.29 | 0.2198 | 8530 | 38810 |
| 60 | 2249.4 | 5.51 | 5.51 | 5.51 | 5.51 | 0.2384 | 9440 | 39740 |
| — | — | 5.50 | 5.51 | 5.51 | 5.51 | 0.2384 | 9440 | 39740 |
| Average | — | — | — | — | — | — | — | 39017 |
The difference of the values of \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{39740 - 38610}{38610} 100 = 2.9\%. \]
Table 16.
Diameters of the balls \(D_1 = D_2 = 20^{m}/m\). Weight of the falling load \(P = 37.49\) kg.
| Height of fall of the load in cm | Live force \(L\) in kg·cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37,49 | 1,98 | 1,98 | 1,98 | 1,98 | 0,03079 | 1220 | 39630 |
| — | — | 1,98 | 1,97 | 1,98 | 1,98 | 0,03079 | 1220 | 39630 |
| 5 | 187,45 | 2,97 | 2,96 | 2,96 | 2,96 | 0,0688 | 2730 | 39590 |
| — | — | 2,95 | 2,96 | 2,96 | 2,96 | 0,0688 | 2730 | 39590 |
| 10 | 374,9 | 3,52 | 3,52 | 3,52 | 3,52 | 0,0974 | 3850 | 39520 |
| — | — | 3,53 | 3,52 | 3,52 | 3,52 | 0,0974 | 3850 | 39520 |
| 20 | 749,8 | 4,22 | 4,23 | 4,22 | 4,22 | 0,1399 | 5360 | 38310 |
| — | — | 4,22 | 4,22 | 4,22 | 4,22 | 0,1399 | 5360 | 38310 |
| Average | — | — | — | — | — | — | — | 39262 |
The greatest difference of the values \(\dfrac{L}{\omega_m^2}\):
\[ \frac{39630-38310}{38310}100 = 3,4\%. \]
Table 17.
Diameters of the balls \(D_1=D_2=20^{m}/m\). Weight of the falling load \(P=37,49\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg. cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374,9 | 3,51 | 3,52 | 3,52 | 3,51 | 0,0968 | 3870 | 4000 |
| — | — | 3,51 | 3,53 | 3,52 | 3,51 | 0,0968 | 3870 | 4000 |
| — | — | 3,50 | 3,50 | 3,50 | 3,51 | 0,0968 | 3870 | 4000 |
| — | — | 3,49 | 3,49 | 3,49 | 3,51 | 0,0968 | 3870 | 4000 |
| 20 | 749,8 | 4,18 | 4,17 | 4,17 | 4,17 | 0,1366 | 5490 | 40180 |
| — | — | 4,17 | 4,17 | 4,17 | 4,17 | 0,1366 | 5490 | 40180 |
| 30 | 1124,7 | 4,67 | 4,67 | 4,67 | 4,66 | 0,1706 | 6590 | 38650 |
| — | — | 4,65 | 4,66 | 4,65 | 4,66 | 0,1706 | 6590 | 38650 |
| 40 | 1499,6 | 5,00 | 5,00 | 5,00 | 5,00 | 0,1964 | 7630 | 39030 |
| — | — | 5,00 | 5,01 | 5,01 | 5,00 | 0,1964 | 7630 | 39030 |
| 50 | 1874,5 | 5,30 | 5,30 | 5,30 | 5,30 | 0,2206 | 8500 | 38570 |
| — | — | 5,31 | 5,30 | 5,30 | 5,30 | 0,2206 | 8500 | 38570 |
| Mean | — | — | — | — | — | — | — | 39286 |
The greatest difference of the values of \(\dfrac{L}{\omega_m^2}\):
\[ \frac{40180-38570}{38570}100=4,2\%. \]
INVESTIGATION OF THE DEFORMATIONS OF STEEL BALLS.
From the data of Tables 13, 14, 15, 16, 17, for balls of \(20^{m}/_{m}\) diameter, the mean results have been derived and are presented in the following table:
Table 18.
| Live force \(L\), in kg·cm | \(\dfrac{L}{\omega_m}\) (13) | \(\dfrac{L}{\omega_m}\) (14) | \(\dfrac{L}{\omega_m}\) (15) | \(\dfrac{L}{\omega_m}\) (16) | \(\dfrac{L}{\omega_m}\) (17) | Mean | \(\dfrac{L}{\omega_m^2}\) (13) | \(\dfrac{L}{\omega_m^2}\) (14) | \(\dfrac{L}{\omega_m^2}\) (15) | \(\dfrac{L}{\omega_m^2}\) (16) | \(\dfrac{L}{\omega_m^2}\) (17) | Mean |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.49 | — | — | — | 1220 | — | 1220 | — | — | — | 39630 | — | 39630 |
| 187.45 | — | — | — | 2730 | — | 2730 | — | — | — | 39590 | — | 39590 |
| 374.9 | 3900 | 3800 | 3850 | 3850 | 3970 | 3874 | 40530 | 38260 | 39520 | 39520 | 40000 | 39566 |
| 749.8 | 5410 | — | 5390 | 5360 | 5490 | 5412 | 39090 | — | 38610 | 38310 | 40180 | 39048 |
| 1124.7 | 6700 | 6600 | 6600 | — | 6590 | 6622 | 39880 | 38650 | 38770 | — | 38650 | 38990 |
| 1499.6 | — | 7670 | 7600 | — | 7630 | 7633 | — | 39250 | 38650 | — | 39030 | 38977 |
| 1874.5 | — | 8520 | 8530 | — | 8500 | 8517 | — | 38730 | 38810 | — | 38570 | 38703 |
| 2249.4 | — | — | 9440 | — | — | 9440 | — | — | 39740 | — | — | 39740 |
| Mean | — | — | — | — | — | — | — | — | — | — | — | 39280 |
The greatest difference here in the values of \(\dfrac{L}{\omega_m^2}\):
\[ \frac{39740 - 38703}{38703}\,100 = 2.7\%. \]
Table 19.
Diameters of the balls \(D_1=D_2=15^{m}/_{m}\). Weight of the falling load \(P=37{,}49\ \mathrm{kg}\).
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187,25 | 2,78 | 2,80 | 2,79 | 2,79 | 0,0611 | 3070 | 49670 |
| — | — | 2,81 | 2,78 | 2,79 | 2,79 | 0,0611 | 3070 | 49670 |
| 10 | 374,9 | 3,32 | 3,34 | 3,33 | 3,33 | 0,0871 | 4300 | 49390 |
| — | — | 3,32 | 3,32 | 2,32 | 3,33 | 0,0871 | 4300 | 49390 |
| — | — | 3,34 | 3,35 | 3,34 | 3,33 | 0,0871 | 4300 | 49390 |
| — | — | 3,34 | 3,34 | 3,34 | 3,33 | 0,0871 | 4300 | 49390 |
| 15 | 561,75 | 3,70 | 3,69 | 3,69 | 3,69 | 0,1069 | 5260 | 49060 |
| — | — | 3,69 | 3,69 | 3,69 | 3,69 | 0,1069 | 5260 | 49060 |
| 20 | 749,8 | 3,95 | 3,96 | 3,95 | 3,96 | 0,1232 | 6090 | 49560 |
| — | — | 3,96 | 3,96 | 3,96 | 3,96 | 0,1232 | 6090 | 49560 |
| 25 | 936,25 | 4,21 | 4,21 | 4,21 | 0,421 | 0,1392 | 6720 | 48510 |
| — | — | 4,21 | 4,21 | 4,21 | 0,421 | 0,1392 | 6720 | 48510 |
| Mean | — | — | — | — | — | — | — | 49238 |
Here the difference in the values of \(\dfrac{L}{\omega_m^2}\):
\[ \frac{49670-48510}{48510}100=2{,}4\%. \]
Table 20.
Diameters of the balls \(D_1=D_2=15\,\mathrm{m/m}\). Weight of the falling load \(P=37.49\) kg.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of deformation area \(a_1\), in mm | Diameter of deformation area \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37.49 | 1.87 | 1.88 | 1.87 | 1.87 | 0.0275 | 1360 | 49590 |
| — | — | 1.87 | 1.87 | 1.87 | 1.87 | 0.0275 | 1360 | 49590 |
| 3 | 112.47 | 2.46 | 2.46 | 2.46 | 2.46 | 0.0475 | 2370 | 49850 |
| — | — | 2.46 | 2.47 | 2.47 | 2.46 | 0.0475 | 2370 | 49850 |
| 5 | 187.25 | 2.81 | 2.81 | 2.81 | 2.81 | 0.0620 | 3020 | 48710 |
| — | — | 2.81 | 2.82 | 2.81 | 2.81 | 0.0620 | 3020 | 48710 |
| 10 | 374.9 | 3.33 | 3.33 | 3.33 | 3.33 | 0.0871 | 4300 | 49390 |
| — | — | 3.32 | 3.34 | 3.33 | 3.33 | 0.0871 | 4300 | 49390 |
| 20 | 749.8 | 3.96 | 3.97 | 3.96 | 3.97 | 0.1238 | 6070 | 49000 |
| — | — | 3.97 | 3.97 | 3.97 | 3.97 | 0.1238 | 6070 | 49000 |
| Mean | — | — | — | — | — | — | — | 49308 |
The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) here is:
\[ \frac{49850-48710}{49710}100=2.3\%. \]
Table 21.
Diameters of the balls \(D_1 = D_2 = 15\ \mathrm{mm}\). Weight of the falling load \(P = 37{,}49\ \mathrm{kg}\).
| Height of fall of the load, cm | Live force \(L\), kg·cm | Diameter of the deformation areas \(a_1\), mm | Diameter of the deformation areas \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Deformation area \(\omega_m\), cm² | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37,49 | 1,85 | 1,86 | 1,86 | 1,86 | 0,0272 | 1380 | 50660 |
| — | — | 1,87 | 1,87 | 1,87 | 1,86 | 0,0272 | 1380 | 50660 |
| 3 | 112,47 | 2,45 | 2,45 | 2,45 | 2,45 | 0,0471 | 2390 | 50710 |
| — | — | 2,45 | 2,46 | 2,46 | 2,45 | 0,0471 | 2390 | 50710 |
| 5 | 187,25 | 2,80 | 2,80 | 2,80 | 2,80 | 0,0616 | 3040 | 49410 |
| — | — | 2,80 | 2,81 | 2,80 | 2,80 | 0,0616 | 3040 | 49410 |
| 10 | 374,9 | 3,32 | 3,33 | 3,32 | 3,32 | 0,0866 | 4330 | 49990 |
| — | — | 3,32 | 3,32 | 3,32 | 3,32 | 0,0866 | 4330 | 49990 |
| 20 | 749,8 | 3,96 | 3,96 | 3,96 | 3,96 | 0,1232 | 6090 | 49560 |
| — | — | 3,97 | 3,96 | 3,97 | 3,96 | 0,1232 | 6090 | 49560 |
| Average | — | — | — | — | — | — | — | 50066 |
The difference of the values of \(\dfrac{L}{\omega_m^2}\) here is:
\[ \frac{50710-49410}{49410}\,100 = 2{,}6\%. \]
Table 22.
Diameters of the balls \(D_1 = D_2 = 15^{m}/_{m}\). Weight of the falling load \(P = 37{,}49\) kg.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 37,49 | 1,88 | 1,87 | 1,87 | 1,87 | 0,0275 | 1360 | 49590 |
| — | — | 1,87 | 1,87 | 1,87 | 1,87 | 0,0275 | 1360 | 49590 |
| 3 | 112,47 | 2,47 | 2,47 | 2,47 | 2,47 | 0,0479 | 2350 | 49030 |
| — | — | 2,48 | 2,46 | 2,47 | 2,47 | 0,0479 | 2350 | 49030 |
| 5 | 187,25 | 2,80 | 2,81 | 2,80 | 2,80 | 0,0616 | 3040 | 49410 |
| — | — | 2,79 | 2,81 | 2,80 | 2,80 | 0,0616 | 3040 | 49410 |
| 10 | 374,9 | 3,34 | 3,33 | 3,34 | 3,33 | 0,0871 | 4300 | 49390 |
| — | — | 3,32 | 3,34 | 3,33 | 3,33 | 0,0871 | 4300 | 49390 |
| Mean | — | — | — | — | — | — | — | 49360 |
The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{49590-49030}{49030}100 = 1{,}1\%. \]
From Tables Nos. 19, 20, 21, and 22 the mean results are again derived in the following table.
Table 23.
| Live force in kg. cm. | $\dfrac{L}{\omega_m}$ (19) | $\dfrac{L}{\omega_m}$ (20) | $\dfrac{L}{\omega_m}$ (21) | $\dfrac{L}{\omega_m}$ (22) | Mean | $\dfrac{L}{\omega_m^{2}}$ (19) | $\dfrac{L}{\omega_m^{2}}$ (20) | $\dfrac{L}{\omega_m^{2}}$ (21) | $\dfrac{L}{\omega_m^{2}}$ (22) | Mean |
|---|---|---|---|---|---|---|---|---|---|---|
| 37,49 | — | 1360 | 1380 | 1360 | 1367 | — | 49590 | 50660 | 49590 | 49950 |
| 112,47 | — | 2370 | 2390 | 2350 | 2370 | — | 49850 | 50710 | 49030 | 49860 |
| 187,25 | 3070 | 3020 | 3040 | 3040 | 3043 | 49670 | 48710 | 49410 | 49410 | 49300 |
| 374,9 | 4300 | 4300 | 4330 | 4300 | 4308 | 49390 | 49390 | 49990 | 49390 | 49540 |
| 561,75 | 5260 | — | — | — | 5260 | 49060 | — | — | — | 49060 |
| 749,8 | 6090 | 6070 | 6090 | — | 6080 | 49560 | 49000 | 49560 | — | 49370 |
| 936,25 | 6720 | — | — | — | 6720 | 48510 | — | — | — | 48510 |
| Mean | — | — | — | — | — | — | — | — | — | 49370 |
The greatest difference here among the mean values of $\dfrac{L}{\omega_m^{2}}$ is:
\[ \frac{49950-48510}{48510}100 = 3\%. \]
Thus, for this series of experiments as well, the dependence found by us above—that the live force is proportional, for the given size of balls, to the fourth power of the diameter of the area of deformation—is confirmed with great accuracy: the deviations from it do not exceed 5.3%.
Likewise, as for the preceding series of experiments, in order to introduce into the dependence of the area of deformation of the balls on the live force also the diameter of the balls, certain calculations have already been made, the results of which are given in the following Table No. 24.
STUDY OF DEFORMATIONS OF STEEL BALLS.
Table 24.
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of the ball \(D\), in cm | Diameter of the area of deformation \(d_m\), in cm | \(d_m^4\) | \(\eta=\dfrac{d_m^4}{D}\) | Mean value \(\eta_m\) from the separate values \(\eta\) | \(\dfrac{L}{\eta_m}\) |
|---|---|---|---|---|---|---|---|
| 1 | 37,49 | 1,5 | 0,187 | 0,001225 | 0,000817 | 0,000803 | 46670 |
| — | — | 2,0 | 0,198 | 0,001537 | 0,000769 | 0,000803 | 46670 |
| — | — | 2,5 | 0,213 | 0,002061 | 0,000824 | 0,000803 | 46670 |
| 3 | 112,47 | 1,5 | 2,46 | 0,003660 | 0,00244 | 0,00245 | 45910 |
| — | — | 2,0 | — | — | — | 0,00245 | 45910 |
| — | — | 2,5 | 2,80 | 0,006147 | 0,00246 | 0,00245 | 45910 |
| 5 | 187,45 | 1,5 | 2,80 | 0,006147 | 0,004098 | 0,004007 | 46780 |
| — | — | 2,0 | 2,96 | 0,007674 | 0,003837 | 0,004007 | 46780 |
| — | — | 2,5 | 3,18 | 0,010221 | 0,004088 | 0,004007 | 46780 |
| 10 | 374,9 | 1,5 | 3,33 | 0,01230 | 0,00820 | 0,00804 | 46640 |
| — | — | 2,0 | 3,52 | 0,01535 | 0,00768 | 0,00804 | 46640 |
| — | — | 2,5 | 3,79 | 0,02061 | 0,00824 | 0,00804 | 46640 |
| 20 | 749,8 | 1,5 | 3,96 | 0,02459 | 0,0164 | 0,0163 | 46000 |
| — | — | 2,0 | 4,20 | 0,03110 | 0,0156 | 0,0163 | 46000 |
| — | — | 2,5 | 4,54 | 0,04244 | 0,0169 | 0,0163 | 46000 |
| 30 | 1124,7 | 1,5 | — | — | — | 0,0244 | 46090 |
| — | — | 2,0 | 4,65 | 0,04666 | 0,02333 | 0,0244 | 46090 |
| — | — | 2,5 | 5,02 | 0,06350 | 0,02540 | 0,0244 | 46090 |
| 40 | 1499,6 | 1,5 | — | — | — | 0,0321 | 46720 |
| — | — | 2,0 | 5,00 | 0,0625 | 0,0313 | 0,0321 | 46720 |
| — | — | 2,5 | 5,36 | 0,0824 | 0,0329 | 0,0321 | 46720 |
| 50 | 1874,5 | 1,5 | — | — | — | 0,0410 | 45720 |
| — | — | 2,0 | 5,29 | 0,0784 | 0,0392 | 0,0410 | 45720 |
| — | — | 2,5 | 5,72 | 0,1069 | 0,0428 | 0,0410 | 45720 |
| 60 | 2249,4 | 1,5 | — | — | — | 0,0486 | 46280 |
| — | — | 2,0 | 5,51 | 0,0922 | 0,0461 | 0,0486 | 46280 |
| — | — | 2,5 | 5,98 | 0,1279 | 0,0512 | 0,0486 | 46280 |
| Mean | — | — | — | — | — | — | 46267 |
The data of this table show that the dependence we found in the first series of experiments between the live force, the diameter of the deformation area of the balls, and the diameter of the latter, expressed by the formula
\[ L=C\frac{d^{4}}{D}, \]
is confirmed here as well: the deviations from this dependence do not exceed:
\[ \frac{46780-45720}{45720}100=2.3\%. \]
And what is still especially remarkable is that the difference between the mean coefficients \((C)\) for the two series of experiments, of which the first was carried out, as is known, without a “shim” and with a falling weight of \(0.587\) kg, while the second was carried out with a steel shim and with a falling weight of \(37.49\) kg, is not particularly large and amounts to only
\[ \frac{47766-46267}{46267}100=3.2\% \]
although, as can be seen, the falling weights differed from one another by 64 times.
Finally, the last series of experiments consisted in the investigation of the mutual deformation upon impact of two balls of different diameters; the results of these experiments are given in the following tables:
Table 25.
Diameters of the balls \(D_{1}=25^{m}/_{m}\), \(D_{2}=20^{m}/_{m}\). Weight of the falling load
\(P=37.49\) kg.
| Height of fall of the load in cm | Live force \(L\), in kg·cm | \(a_{1}\), in mm | \(a_{2}\), in mm | \(d=\frac{a_{1}+a_{2}}{2}\), in mm | Mean diameter \(d_m\), in mm | Deformation area \(\omega_m\), in cm | \(\frac{L}{\omega_m}\) | \(\frac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 10 | 374.9 | 3.65 | 3.66 | 3.66 | 3.65 | 0.1046 | 3580 | 34270 |
| — | — | 3.64 | 3.65 | 3.64 | 3.65 | 0.1046 | 3580 | 34270 |
| 20 | 749.8 | 4.29 | 4.32 | 4.31 | 4.31 | 0.1459 | 5140 | 35200 |
| — | — | 4.31 | 4.31 | 4.31 | 4.31 | 0.1459 | 5140 | 35200 |
| 30 | 1124.7 | 4.80 | 4.81 | 4.81 | 4.81 | 0.1817 | 6190 | 34080 |
| — | — | 4.81 | 4.81 | 4.81 | 4.81 | 0.1817 | 6190 | 34080 |
| 40 | 1499.6 | 5.18 | 5.19 | 5.18 | 5.18 | 0.2107 | 7120 | 33780 |
| — | — | 5.18 | 5.18 | 5.18 | 5.18 | 0.2107 | 7120 | 33780 |
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 50 | 1874,5 | 5,44 | 5,44 | 5,44 | 5,44 | 0,2324 | 8070 | 34840 |
| — | — | 5,44 | 5,44 | 5,44 | 5,44 | 0,2324 | 8070 | 34840 |
| Mean | — | — | — | — | — | — | — | 34434 |
The range of values of \(\dfrac{L}{\omega_m^2}\) is here:
\[ \frac{35200-33780}{33780}100=4,2\%. \]
Table 26.
Diameters of the balls \(D_1=25^{m}/_{m}\), \(D_2=20^{m}/_{m}\). Weight of the falling load
\(P=37,49\ \mathrm{kg}.\)
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187,45 | 3,06 | 3,06 | 3,06 | 3,06 | 0,0735 | 2550 | 34680 |
| — | — | 3,07 | 3,05 | 3,06 | 3,06 | 0,0735 | 2550 | 34680 |
| 10 | 374,9 | 3,63 | 3,64 | 3,64 | 3,64 | 0,1040 | 3610 | 34650 |
| — | — | 3,64 | 3,64 | 3,64 | 3,64 | 0,1040 | 3610 | 34650 |
| 15 | 561,75 | 4,02 | 4,02 | 4,02 | 4,02 | 0,1269 | 4430 | 34890 |
| — | — | 4,01 | 4,02 | 4,02 | 4,02 | 0,1269 | 4430 | 34890 |
| 20 | 749,8 | 4,32 | 4,32 | 4,32 | 4,32 | 0,1466 | 5110 | 34880 |
| — | — | 4,32 | 4,33 | 4,32 | 4,32 | 0,1466 | 5110 | 34880 |
| 30 | 1124,7 | 4,80 | 4,79 | 4,79 | 4,79 | 0,1802 | 6240 | 34710 |
| — | — | 4,79 | 4,79 | 4,79 | 4,79 | 0,1802 | 6240 | 34710 |
| 40 | 1499,6 | 5,16 | 5,15 | 5,15 | 5,15 | 0,2083 | 7200 | 34610 |
| — | — | 5,15 | 5,15 | 5,15 | 5,15 | 0,2083 | 7200 | 34610 |
| 50 | 1874,5 | 5,43 | 5,43 | 5,43 | 5,43 | 0,2316 | 8090 | 34840 |
| — | — | 5,42 | 5,44 | 5,43 | 5,43 | 0,2316 | 8090 | 34840 |
| Mean | — | — | — | — | — | — | — | 34751 |
The difference of the values \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{34890-34610}{34610}100=0{,}81\%. \]
Table 27.
Diameters of the balls \(D_1=25\) mm, \(D_2=15\) mm. Weight of the falling load
\(P=37{,}49\) kg.
| Height of fall of the load, cm | Live force \(L\), kg·cm | \(a_1\), mm | \(a_2\), mm | \(d=\dfrac{a_1+a_2}{2}\), mm | Mean diameter \(d_m\), mm | Area of deformation \(\omega_m\), sq. cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187,45 | 2,97 | 2,97 | 2,97 | 2,96 | 0,0688 | 2730 | 39590 |
| — | — | 2,96 | 2,96 | 2,96 | 2,96 | 0,0688 | 2730 | 39590 |
| 10 | 374,9 | 3,52 | 3,52 | 3,52 | 3,51 | 0,09676 | 3870 | 40010 |
| — | — | 3,51 | 3,51 | 3,51 | 3,51 | 0,09676 | 3870 | 40010 |
| 20 | 749,8 | 4,21 | 4,19 | 4,20 | 4,20 | 0,1385 | 5410 | 39090 |
| — | — | 4,19 | 4,21 | 4,20 | 4,20 | 0,1385 | 5410 | 39090 |
| 25 | 936,25 | 4,44 | 4,42 | 4,43 | 4,43 | 0,1541 | 6080 | 39500 |
| — | — | 4,43 | 4,43 | 4,43 | 4,43 | 0,1541 | 6080 | 39500 |
| Average | — | — | — | — | — | — | — | 39550 |
The greatest difference of the values \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{40010-39500}{39500}100=1{,}3\%. \]
Table 28.
Diameters of the balls \(D_1=25^{m}/_{m}\), \(D_2=15^{m}/_{m}\). Weight of the falling load
\(P=37,49\ \mathrm{kg}\).
| Height of fall of the load, in cm. | Live force \(L\), in kg·cm. | \(a_1\), in mm. | \(a_2\), in mm. | \(d=\dfrac{a_1+a_2}{2}\), in mm. | Mean diameter \(d_m\), in mm. | Area of deformation \(\omega_m\), in cm. | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187,45 | 2,96 | 2,95 | 2,95 | 2,95 | 0,06835 | 2740 | 40220 |
| — | — | 2,95 | 2,95 | 2,95 | 2,95 | 0,06835 | 2740 | 40220 |
| 10 | 374,9 | 3,49 | 3,48 | 3,49 | 3,49 | 0,0957 | 3920 | 40940 |
| — | — | 3,48 | 3,49 | 3,49 | 3,49 | 0,0957 | 3920 | 40940 |
| — | — | 3,50 | 3,48 | 3,49 | 3,49 | 0,0957 | 3920 | 40940 |
| — | — | 3,48 | 3,48 | 3,48 | 3,49 | 0,0957 | 3920 | 40940 |
| 15 | 561,75 | 3,87 | 3,88 | 3,88 | 3,87 | 0,1176 | 4780 | 40710 |
| — | — | 3,86 | 3,86 | 3,86 | 3,87 | 0,1176 | 4780 | 40710 |
| 20 | 749,8 | 4,16 | 4,15 | 4,16 | 4,16 | 0,1359 | 5520 | 40530 |
| — | — | 4,16 | 4,16 | 4,16 | 4,16 | 0,1359 | 5520 | 40530 |
| Average | — | — | — | — | — | — | — | 40600 |
Difference of the values \(\dfrac{L}{\omega_m^2}\):
\[ \frac{40940-40220}{40220}\,100=1,8\%. \]
V. N. Pinyegin
Table 29.
Diameters of the balls \(D_1=20^{m}/m,\ D_2=15^{m}/m\). Weight of the falling load
\(P=37.49\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg. cm | \(a_1\), in mm | \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187,45 | 2,87 | 2,87 | 2,87 | 2,87 | 0,0647 | 2900 | 44760 |
| — | — | 2,87 | 2,87 | 2,87 | 2,87 | 0,0647 | 2900 | 44760 |
| 10 | 374,9 | 3,42 | 3,43 | 3,43 | 3,43 | 0,0924 | 4060 | 43900 |
| — | — | 3,43 | 3,43 | 3,43 | 3,43 | 0,0924 | 4060 | 43900 |
| 15 | 561,75 | 3,79 | 3,79 | 3,79 | 3,79 | 0,1128 | 4980 | 44160 |
| — | — | 3,80 | 3,79 | 3,79 | 3,79 | 0,1128 | 4980 | 44160 |
| 20 | 749,8 | 4,08 | 4,08 | 4,08 | 4,08 | 0,1307 | 5740 | 43850 |
| — | — | 4,09 | 4,08 | 4,09 | 4,08 | 0,1307 | 5740 | 43850 |
| Mean | — | — | — | — | — | — | — | 44168 |
The difference in the values of \(\dfrac{L}{\omega_m^2}\) is:
\[ \frac{44760-43850}{43850}100=2,1\%. \]
Table 30.
Diameter of the balls \(D_1=20^{m}/m,\ D_2=15^{m}/m\). Weight of the falling load
\(P=37.49\ \mathrm{kg}\).
| Height of fall of the load, in cm | Live force \(L\), in kg·cm | Diameter of the deformation area \(a_1\), in mm | Diameter of the deformation area \(a_2\), in mm | \(d=\dfrac{a_1+a_2}{2}\), in mm | Mean diameter \(d_m\), in mm | Area of deformation \(\omega_m\), in sq. cm | \(\dfrac{L}{\omega_m}\) | \(\dfrac{L}{\omega_m^2}\) |
|---|---|---|---|---|---|---|---|---|
| 5 | 187.45 | 2.86 | 2.87 | 2.87 | 2.88 | 0.0651 | 2880 | 44230 |
| — | — | 2.87 | 2.86 | 2.86 | 2.88 | 0.0651 | 2880 | 44230 |
| — | — | 2.90 | 2.89 | 2.89 | 2.88 | 0.0651 | 2880 | 44230 |
| — | — | 2.89 | 2.89 | 2.89 | 2.88 | 0.0651 | 2880 | 44230 |
| 10 | 374.9 | 3.42 | 3.41 | 3.42 | 3.42 | 0.0919 | 4080 | 44390 |
| — | — | 3.42 | 3.43 | 3.43 | 3.42 | 0.0919 | 4080 | 44390 |
| 15 | 561.75 | 3.79 | 3.81 | 3.80 | 3.80 | 0.1134 | 4950 | 43690 |
| — | — | 3.79 | 3.80 | 3.80 | 3.80 | 0.1134 | 4950 | 43690 |
| 20 | 749.8 | 4.08 | 4.07 | 4.08 | 4.07 | 0.1301 | 5760 | 44370 |
| — | — | 4.07 | 4.06 | 4.07 | 4.07 | 0.1301 | 5760 | 44370 |
| 25 | 936.25 | 4.31 | 4.31 | 4.31 | 4.31 | 0.1459 | 6420 | 43950 |
| — | — | 4.31 | 4.32 | 4.31 | 4.31 | 0.1459 | 6420 | 43950 |
| 30 | 1124.7 | 4.50 | 4.50 | 4.50 | 4.50 | 0.1590 | 7070 | 44490 |
| — | — | 4.49 | 4.50 | 4.50 | 4.50 | 0.1590 | 7070 | 44490 |
| Mean | — | — | — | — | — | — | — | 44187 |
The greatest difference in the values of \(\dfrac{L}{\omega_m^2}\) is
\[ \frac{44490-43690}{43690}100=1.8\%. \]
From all these tables it is seen that, on the one hand, the ratio of the living force to the square of the area of the deformation obtained is a quantity constant for each pair of balls being tested, the deviations from this constancy ranging within the limits \(0.81\%\)—\(4.2\%\); and, on the other hand, the coefficient of proportionality for each pair of balls is the arithmetic mean of the coefficients of proportionality obtained in testing balls of each size separately that enter into this pair. The latter proposition follows from the following statements: in testing separately balls of diameter \(25^{m}/m\), \(20^{m}/m\), and \(15^{m}/m\), we obtained, as the mean values of the coefficients under consideration, respectively \(C^{1}_{(25)}=29175\) (Table 12), \(C^{1}_{(20)}=39280\) (Table 18), and \(C^{1}_{(15)}=49370\) (Table 23); the arithmetic mean of the first two will be
\[ \frac{C_{(25)}+C_{(20)}}{2} = \frac{29175+39280}{2} = 34227. \]
The experiments with a pair of balls, of which one is \(25^{m}/m\) and the other \(20^{m}/m\) in diameter, gave (see Tables 25, 26) for the coefficient \(C_{(25,\,20)}\) the value 34592; and thus their difference is
\[ \frac{34592-34227}{34227}100 = 1.06\%. \]
Exactly likewise,
\[ \frac{C_{(25)}+C_{(15)}}{2} = \frac{29175+43370}{2} = 39272, \]
and in testing a pair of balls, of which one had a diameter of \(25^{m}/m\) and the other \(15^{m}/m\), we obtained (Tables 27, 28) as the mean \(C_{(25,\,15)}=40075\), and hence their difference is
\[ \frac{40075-39272}{39272}100 = 2.04\%. \]
Finally,
\[ \frac{C_{(20)}+C_{(15)}}{2} = \frac{39280+49370}{2} = 44325, \]
and the experiments with a pair of balls, of which one was \(20^{m}/m\) and the other \(15^{m}/m\) in diameter, gave (see Tables 29, 30), as the mean value, \(C_{(20,\,15)}=44178\); their difference is
\(^{1}\) Here the numbers (25), (20), and (15) next to \(C\) indicate at what diameter of balls the above-mentioned coefficients were obtained in the test.
Study of the Deformations of Steel Balls
\[ \frac{44325-44178}{44178}100=0{,}33\%. \]
All the differences obtained in this way are extremely small and can quite well be explained by errors of observation and by the conditions of the experiments.
Finally, into the relation found for the mutual deformation of balls of different diameter under a living force, it was desirable to introduce also the diameters of the balls being tested.
For this purpose we carried out a series of calculations, the essence of which is evident from the following table.
Table 31.
| Diameter of the balls tested, in cm. Living force \(L\), in kg. cm. | \(D_1=2{,}5\) cm., \(D_2=2{,}0\) cm. Mean diameter of the deformation area \(d_m\), in cm. | \(D_1=2{,}5\) cm., \(D_2=2{,}0\) cm. \(d_m^4\) | \(D_1=2{,}5\) cm., \(D_2=2{,}0\) cm. \(\eta_1=\dfrac{d_m^4}{D_1+D_2}\) | \(D_1=2{,}5\) cm., \(D_2=1{,}5\) cm. Mean diameter of the deformation area \(d_m\), in cm. | \(D_1=2{,}5\) cm., \(D_2=1{,}5\) cm. \(d_m^4\) | \(D_1=2{,}5\) cm., \(D_2=1{,}5\) cm. \(\eta_2=\dfrac{d_m^4}{D_1+D_2}\) | \(D_1=2{,}0\) cm., \(D_2=1{,}5\) cm. Mean diameter of the deformation area \(d_m\), in cm. | \(D_1=2{,}0\) cm., \(D_2=1{,}5\) cm. \(d_m^4\) | \(D_1=2{,}0\) cm., \(D_2=1{,}5\) cm. \(\eta_3=\dfrac{d_m^4}{D_1+D_2}\) | Mean \(\eta_m\) from \(\eta_1,\eta_2,\eta_3\) | \(\dfrac{L}{\eta_m}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 178,45 | 0,306 | 0,00876 | 0,001947 | 0,296 | 0,00767 | 0,001918 | 0,288 | 0,00687 | 0,001963 | 0,001943 | 96470 |
| 374,9 | 0,364 | 0,01755 | 0,00390 | 0,350 | 0,01500 | 0,00375 | 0,343 | 0,01383 | 0,003952 | 0,003867 | 96950 |
| 561,75 | 0,402 | 0,02611 | 0,005802 | 0,387 | 0,02250 | 0,005625 | 0,380 | 0,02086 | 0,00596 | 0,005796 | 96920 |
| 749,8 | 0,432 | 0,03482 | 0,007738 | 0,418 | 0,0305 | 0,007625 | 0,408 | 0,02772 | 0,00792 | 0,007761 | 96610 |
| 936,25 | — | — | — | 0,443 | 0,0384 | 0,00960 | 0,431 | 0,0345 | 0,00986 | 0,00973 | 96220 |
| 1124,7 | 0,480 | 0,05308 | 0,01180 | — | — | — | 0,450 | 0,0410 | 0,01171 | 0,01176 | 95640 |
| 1499,6 | 0,517 | 0,07129 | 0,01582 | — | — | — | — | — | — | 0,01582 | 94800 |
| 1874,5 | 0,544 | 0,08761 | 0,01947 | — | — | — | — | — | — | 0,01947 | 96280 |
| Mean | — | — | — | — | — | — | — | — | — | — | 96236 |
From this table it is evident that with very great accuracy (the error is
\[ \frac{96950-94800}{94800}\,100=2.3\%), \]
\[ \frac{L}{\eta_m}=C' \]
or
\[ L=C'\eta_m=C'\frac{d_m^4}{D_1+D_2}\ldots\ldots(2) \]
where \(C'\) is an arbitrary constant independent of the diameter of the balls.
It further turns out that this formula (2) contains formula (1) as a special case; indeed, putting in formula (2)
\[ D_1=D_2=D \]
we obtain
\[ L=\frac{C'}{2}\frac{d_m^4}{D}=C''\frac{d_m^4}{D}\ldots\ldots(3) \]
and since \(C''=\frac{C'}{2}=\frac{96236}{2}=48118\), this coefficient \(C''\) differs from the coefficient \(C\) in formula (1) only by (see Table 24)
\[ \frac{48118-46267}{46267}100=4\%, \]
which difference again does not appear large and may quite readily be explained by errors of experiments and observations, and therefore \(C''\) may safely be taken equal to \(C\).
Finally, one more question seemed to us very interesting to resolve: the question of the dependence of the magnitude of the live force destroying a ball on its diameter; but, unfortunately, owing to lack of time it cannot be considered resolved by us.
Thus, for example, balls of diameter \(15^{m}/_{m}\) were destroyed (the balls were tested in pairs, in the same way as in the investigation of their deformation) at live forces of 560–940 kg. cm. (when a weight of 37.49 kg. fell from a height of 15–25 cm.), on average 750 kg. cm., which corresponds to \(424\frac{\mathrm{kg.}\,\mathrm{cm.}}{\mathrm{cm.}^2}\) of the diametral cross-sectional area of the ball.
Study of the Deformations of Steel Balls
Balls with a diameter of \(20^{m}/m\) were destroyed, for the most part, at live forces of \(1100\)—\(2700\) kg. cm. (with corresponding heights of fall of the load of \(30\)—\(70\) cm.), on average \(1900\) kg. cm., which corresponds to \(610 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) of the diametral area of the ball’s cross-section.
The destructive live forces for balls with a diameter of \(25^{m}/m\) fluctuate within still wider limits: from \(1500\) kg. cm. to \(4000\) kg. cm. (corresponding heights of fall of the load \(40\)—\(110\) cm.), on average \(2275\) kg. cm., which corresponds to \(560 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) of the diametral area of the ball’s cross-section.
In the joint testing of balls of different diameters, however, the magnitudes of the destructive live forces were greatly reduced: thus, when testing pairs of balls, of which in each pair one was \(25^{m}/m\), and the other \(20^{m}/m\) in diameter, both the one and the other ball were destroyed at live forces of \(1100\)—\(1900\) kg. cm., on average \(1500\) kg. cm., which corresponds to \(375 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) of the diametral area of cross-section for balls \(20^{m}/m\) in diameter and \(305 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) for balls \(25^{m}/m\) in diameter. When testing pairs of balls \(25^{m}/m\) and \(15^{m}/m\) in diameter in each, destruction occurred already at live forces of \(375\)—\(940\) kg. cm., on average \(660\) kg. cm., which corresponds to \(375 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) of the diametral area of the ball’s cross-section at \(15^{m}/m\) in diameter and \(135 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) for balls \(25^{m}/m\) in diameter.
Finally, when testing pairs of balls with diameters of \(20^{m}/m\) and \(15^{m}/m\), their destruction occurred at live forces of \(375\)—\(1100\) kg. cm., on average \(740\) kg. cm., which corresponds, for balls of \(20^{m}/m\) diameter, to \(226 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\), and for balls of \(15^{m}/m\) diameter, to \(418 \frac{\text{kg. cm.}}{\text{cm.}^{2}}\) of the diametral area of the balls’ cross-section.
Thus, in the joint testing of balls of different diameters, at the time when for the balls of the smallest of the tested sizes (in the present case \(15^{m}/m\) in diameter) the destructive live force falling on \(\text{cm.}^{2}\) of the diametral area
...cross-section of the ball, remained almost unchanged; this live force, for balls of large diameter, has diminished considerably in comparison with the data obtained when testing balls of the same size; and this diminution is the greater, the greater is the difference in the diameters of the balls tested together. Thus, for balls of \(25^{m}/m\) diameter, when tested together with balls of \(15^{m}/m\) diameter, this diminution occurred by a factor of \(4.1\), while for balls of \(20^{m}/m\) diameter it occurred only by a factor of \(2.7\).
Now summarizing all the foregoing, we may formulate the results obtained by us in the form of propositions, valid, of course, only under the conditions and within the limits of the experiments performed:
1) The ratio of the live force to the square of the area of the deformation produced on one of two steel balls of equal size is a constant quantity, depending on the diameter of the balls.
2) The product of the live force, divided by the square of the area of the deformation produced by it on one of two balls of any equal size, and by their diameter, is a constant quantity independent of the diameter of the balls tested.
3) The ratio of the live force to the square of the area of the deformation produced by it on one of two steel balls of different diameter \((D_1\) and \(D_2)\) is a constant quantity, depending on the diameters of the balls tested, and moreover equal to the arithmetic mean of the constants that would have been obtained if an analogous test had been carried out first with two balls of diameter \(D_1\), and then with two balls of diameter \(D_2\).
4) The product of the live force, divided by the square of the area of the deformation produced by it on one of two steel balls of different diameter, and by the sum of the diameters of these balls, is a constant quantity independent of the diameters of the balls tested.
5) The relation between the live force, the diameter of the area of the deformation produced by it on one of two steel balls of equal diameter, and the diameter of the balls, following from the 2nd proposition, can be obtained from the relation between analogous quantities according to the 4th proposition, if only the diameters of the two balls considered here are made equal.
6) Without yet establishing any dependence between the diameter of the balls and the live force that destroys them, one can only say that the magnitude of this live force is greatly reduced for balls of any size if they are destroyed together with balls of another, smaller diameter.
7) Finally, in view of the fact that in the given experiments, with two different weights differing from one another many times over (one being 64 times greater than the other), results were obtained that differ very little from one another, I would permit myself to state the proposition, already indicated by me in my above-cited article, that the observed dependence between the live force, the area of the deformation produced by it on the balls, and the diameter of the balls does not depend also on the velocity of fall of the weight, or, in any case, the influence of this velocity is so insignificant that it does not exceed the observational errors in the given experiments.
City of Tomsk, October 1908.
V. Pinegin.
Plate I.
Fig. 1.
Fig. 2.
Fig. 5.
Fig. 3.
Fig. 6.
Fig. 4.
Photo by P. Pavlov.
To the article by V. N. Pinegin, “Investigation of the Deformations of Steel Balls under the Action of Impact.”
Plate 2
Diagram 1.
Visible labels: \(P\), \(p\), \(\mu\).
Diagram 2.
Visible labels: \(P\), \(p\), \(\mu\).
Diagram 3.
Visible dimensions: \(35\), \(25\), \(90\), \(50\), \(90\).
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I express my deep gratitude to V. V. Mramornov for his kind assistance in photographing the forms of fracture of the balls. ↩↩↩
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Schwinning. Versuche über die zulässige Belastung von Kugeln und Kugellagern. Zeit. d. Ver. d. Ing. 1901, p. 333. ↩
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Stribeck. Prüfverfahren für gehärteten Stahl unter Berücksichtigung der Kugelform. Zeitschrift d. Ver. d. Ing. 1907, p. 1503. ↩