Study of Friction in a Flat Pivot Bearing: With 14 Tables

V. N. Pinegin.

Study of Friction in a Flat Pivot Bearing: With 14 Tables

V. N. Pinegin.

Submitted 1911 | SovietRxiv: ru-191101.07238 | Translated from Russian

Pre-reform orthography. The original is printed in the Russian spelling used before the 1918 reform (ѣ, і, ъ).

Full Text

Proceedings of the Tomsk Technological Institute of Emperor Nicholas II.
Vol. XXIV, 1911, No. 4.

Study of Friction in a Flat Pivot.

V. N. Pinegin.

Already in the article published by me in 1909: “An Experiment in Studying the Distribution of Pressures on a Flat Pivot”¹, I indicated (see p. 11) that, simultaneously with my study of the distribution of pressures on a flat pivot, a study of friction had also been carried out; unfortunately, publication of the results of this latter investigation was delayed for lack of time to process the experimental data, and only now, finally, after almost two years, have I had the opportunity to do so.

In view of the fact that the present article is a natural continuation of the above-mentioned article, I shall allow myself here to refrain entirely from describing the apparatus itself for studying friction in a pivot, which was described in detail and depicted in the drawings in the first article.

The circumstance that this question is a very urgent one prompted me to carry out a study of friction in a flat pivot (made of chilled cast iron), in view of the wide use of such pivots in water turbines, and also the emerging possibility—and even the desirability, expressed by certain engineers—of constructing steam turbines with a vertical axis², in the design of which it would hardly be possible to dispense with the use of flat pivots. The choice of a pivot made of chilled cast iron was made because, in modern turbine construction, flat annular pivots have begun to be used predominantly of chilled cast iron, and only in rare cases are they made of steel and phosphor bronze. The permissible load for such cast-iron pivots is considered, on average, to be within the limits of 30–60 kg/cm², and sometimes rises to

90 kg/cm²¹). The number of revolutions of turbine shafts is also continually being increased, and at the present time 150–200, even 300, revolutions per minute are considered ordinary.

Meanwhile, very few experiments have been carried out to investigate friction in pivots. If one does not count isolated determinations of the coefficient of friction in pivots made by certain experimenters, then, so far as I know, there is only one systematic investigation—that of Woodbury². The latter conducted his experiments with an annular flat pivot having an outer radius of 0.1907 ft. (4.85 cm) and an inner radius of 0.1195 ft. (3.03 cm), with pressure varied within the limits from 1 to 40 pounds per square inch (0.07–2.8 kg/cm²), oil temperature within the limits 40–100° Fahr. (4.44–37.78° C), and with a number of revolutions not exceeding 300 per minute. From this it is clear that, although for the purposes of turbine construction one may still be satisfied with the number of revolutions in Woodbury’s experiments, the limits of loads and temperatures that occurred in his experiments do not at all correspond to the modern working conditions of pivots.

There are, it is true, excellent investigations of friction by Hirn³, Tower⁴, Martens⁵, Thurston⁶, Lasche⁷, Dettmar⁸ and others, and especially by Stribeck⁹, but these investigations concern friction in journals, and it is considered impossible to transfer the results of such experiments to pivots, since “the velocities at the center and at the circumference of a pivot differ very considerably from one another; the lubricant, owing to the influence of centrifugal force and owing to the entirely different character of the plates, is in entirely different conditions as compared with the lubrication of journals”¹⁰.

Finally, there exists, as is known, a hydrodynamic theory of friction, given by our Russian engineer N. Petrov¹, which has found many adherents both among Russian scholars and among foreigners, and has called forth a large literature² on this question; according to this theory, friction in well-lubricated machine parts is reduced, chiefly, to internal friction in the lubricating liquid.

This theory, developed chiefly in application to the friction of a journal in a bearing, extends very simply also to the case of a pivot, as Petrov had already shown³. But if, on the one hand, there is a whole series of experiments confirming his theory, at least in application to the journal, within known limits (the experiments of Petrov himself⁴, Fr. Masi⁵, John Goodman⁶, Collette⁷),

on the other hand, the experiments of Stribeck, Dettmar, Lasche, and others give results that do not agree with this theory; according to Stribeck’s experiments, for example, the law of variation of the coefficient of friction is very complex and is difficult to fit within the framework of any formula.

Bearing all this in mind, I decided, simultaneously with the study of the distribution of pressures on a flat pivot, also to carry out an investigation of friction, in the hope that I might manage to shed at least some light on this difficult and complex question.

Limits of the study of friction. I have already mentioned in the first article that I did not succeed in carrying out investigations of friction and of the distribution of pressures within the limits of loads and speeds that I had initially intended. The reason for this lay in the not entirely successful method of loading the pivot: the lever from which the board for placing the weights was suspended, at higher speeds and loads, entered into a strong oscillatory motion and, from fear, on the one hand, of breaking the machine, and, on the other hand, because it was impossible under such conditions to obtain a steady load on the pivot, it was necessary to reduce the speed and unload the pivot. Thus the maximum possible speed was approximately 280 revolutions of the pivot per minute; the minimum possible speed, in view of the fact that the motor at my disposal did not allow too small a number of revolutions, and it was impossible to arrange an additional transmission, did not go below that which corresponded to 60–50 revolutions of the pivot per minute, and only, as an exception, in a few cases was it possible to obtain the value of the coefficient of friction at a smaller number of revolutions of the pivot (and only at the minimum average load on the pivot of $4.7\ \mathrm{kg}/\mathrm{cm}^{2}$).

The average loads of the pivot, as is already known from the first article, ranged from $4.7\ \mathrm{kg}/\mathrm{cm}^{2}$ to $47.3\ \mathrm{kg}/\mathrm{cm}^{2}$.

Springs for measuring the force of friction and their calibration. The method of measuring the frictional force in the pivot is described in detail in the first article¹. There, among other things, it is stated that in order to measure friction it was necessary first to calibrate the elongation of the springs $s, s$ (see Table II of the drawings in the first article) according to the magnitude of the force stretching them.

In view of the special purpose of the springs in the present case, I ordered them to be made of the best crucible steel on a special

springs—Spiralfedernfabrik von Iohan Bulir (Berlin), and had the following dimensions:

diameter of the spring cross-section—$6\text{ mm}$;

mean diameter of the spring—$30\text{ mm}$;

length of the spring, including the hooks—$200\text{ mm}$;

number of coils of the spring—10;

pitch of the coil—$11.5\text{ mm}$.

The actual graduation of the springs was carried out by me in the following manner.

Decimal scales, with a limiting weighable load of 500 kg, were placed on table $t$ (see schematic drawing 1 of Plate I); the suspended board for placing weights was removed and replaced by a load of corresponding weight, suspended from the lever of the scales by a wire; on the vertical rod $m$, which happened to be in the laboratory and was fastened to the floor, clamp 17 was put on (see Table II of the drawings of the first article) and secured at such a height that spring $s$, being hooked with one of its hooks to the eye of bolt 18 and with the other hook grasping the eye of the scale lever, on which the board for placing weights had previously hung, would, of course, together with the scales, be positioned so that the spring assumed, as far as possible, a vertical position.

On the quadrangular prismatic rods of bolts 18, millimetre and half-millimetre divisions had been marked on a dividing machine; the reading of these divisions, when the bolts moved along the axis, was made according to the position of the quadrangular rods of the bolts relative to the inner flat surface of clamps 17, the surface in which an opening had been made for the passage of the just-mentioned rods of bolts 18. The scale lever was first brought into its uppermost extreme position (corresponding to loading the scales); this position corresponded to the zero division on the prismatic rods of the bolts; then a 50 kg weight was placed on the scale platform; by turning nut 10 (see Table II of the first article and drawing 1 of Plate I), the scale lever was brought into the position of equilibrium, and the number of divisions was noted on the prismatic rod of the bolt by which the latter had moved; this number of divisions gave the magnitude of the extension of the spring under a load of 5 kg (10 times less than the load placed on the platform). Then the platform was loaded with another new weight of 50 kg, the lever again left the position of equilibrium; by turning nut 10 the lever was again brought into the horizontal position, and a new number of divisions was obtained by which it had moved—

the bolt was loosened, and consequently also the magnitude of the stretching of the spring, etc.; the spring was thus loaded up to 45 kg. After this they began gradually to remove weights (50 kg at a time) from the platform and noted, turning the nut 10 in the opposite direction, the shortening of its length corresponding to this decrease in the load on the spring; unloading was carried out down to zero.

Prior to this test, the spring had been stretched several times by a load of 50 kg, in order to bring it into a more or less “settled state.”

The experiment itself, with the testing of each spring (Nos. 1 and 2) and the graduation of its elongation, was repeated twice.

The results of these tests are given in Tables Nos. 1 and 2, and are then represented graphically on Plate I (scale of forces: $1^{m}/m \sim 0.2$ kg; scale of elongations: $1^{m}/m \sim 0.1^{m}/m$ elong.).

Table No. 1.

Load of the springs in kg	Spring No. 1. Total elongation of the spring in m/m	Spring No. 1. Difference of spring elongations at different loads	Spring No. 2. Total elongation of the spring in m/m	Spring No. 2. Difference of spring elongations at different loads
0	0		0
5	4,0	+4,0	3,4	+3,4
10	8,1	+4,1	7,0	+3,6
15	11,6	+3,5	10,8	+3,8
20	15,5	+3,9	14,6	+3,8
25	19,0	+3,5	18,3	+3,7
30	22,6	+3,6	21,9	+3,6
35	26,2	+3,6	25,4	+3,5
40	29,5	+3,3	29,0	+3,6
45	32,8	+3,3	32,5	+3,5
40	29,4	−3,4	29,1	−3,4
35	26,0	−3,4	25,7	−3,4
30	22,8	−3,2	22,3	−3,4
25	19,5	−3,3	19,0	−3,3
20	16,0	−3,5	15,6	−3,4
15	12,1	−3,9	12,3	−3,3
10	8,3	−3,8	9,0	−3,3
5	4,6	−3,7	5,0	−4,0
0	0,5	−4,1	0,8	−4,2

Table No. 2.

Load on the springs in kg	Spring No. 1. Total elongation of the spring in m/m.	Spring No. 1. Difference of elongations of the spring under different loads.	Spring No. 2. Total elongation of the spring in m/m.	Spring No. 2. Difference of elongations of the spring under different loads.
0	0		0
5	4,1	+4,1	4,2	+4,2
10	8,3	+4,2	7,7	+3,5
15	11,6	+3,3	11,5	+3,8
20	15,1	+3,5	15,0	+3,5
25	18,5	+3,4	18,3	+3,3
30	21,9	+3,4	21,8	+3,5
35	25,3	+3,4	25,2	+3,4
40	28,7	+3,4	28,6	+3,4
45	32,1	+3,4	31,9	+3,3
40	28,6	−3,5	28,4	−3,5
35	25,2	−3,4	25,0	−3,4
30	22,1	−3,1	21,5	−3,5
25	18,4	−3,7	18,1	−3,4
20	15,3	−3,1	14,6	−3,5
15	11,6	−3,7	11,1	−3,5
10	8,1	−3,5	7,6	−3,5
5	4,5	−3,6	3,9	−3,7
0	0,2	−4,3	0	−3,9

The elongation curves, both under loading of the springs and under unloading, were plotted from a single point—the origin of coordinates; on the basis of these four curves, for each spring, mean curves were constructed, shown in the drawing by solid lines.

These latter curves were subsequently used to determine, from the elongation of the springs, the corresponding frictional force.

For lubricating the pivot, oil manufactured by the Deutz factory was used, the so-called Gasmotorenöl.

Despite the fact that fresh oil was supplied all the time in a thin stream to the oil bath, with, of course, a corresponding quantity being drawn off from the bath through a special cock, the heating of the oil proceeded rather quickly, and it was very difficult to keep it at a definite temperature.

Measurement of the temperature of the lubricating oil.

Measurement of the temperature of the oil was carried out by means of a thermometer lowered into the oil bath, care being taken that the mercury bulb of the thermometer was always located near the very edge of the rubbing surface of the heel; the oil bath was kept covered at all times with a lid, in which an opening had been made for the passage of the thermometer.

On the basis of Stribeck’s experiments¹ it could be assumed that the difference between the temperature of the oil in the bath, measured in this way, and the temperature of the layer lubricating the rubbing surface of the heel would not be especially significant: as is known, in Stribeck’s experiments it turned out that the temperature of the oil in the bearing reservoir differed from the temperature of the liner near its rubbing surface by $3—8^\circ\text{C}$, and even that at the beginning of the experiment and at 1100 revolutions of the trunnion per minute; at 63 revolutions per minute, however, only by $1—1.5^\circ\text{C}$; subsequently, as the oil warmed up, this difference was reduced almost to nothing. In our case, owing to the very arrangement of the lubricant supply, and with numbers of revolutions below 300 per minute, it could be expected that the difference in the above-mentioned temperatures would in any case be no greater than the corresponding difference in Stribeck’s experiments, and consequently quite small.

Running-in of the heel. In order to give the heel an opportunity to run in, it was made to operate under various loads for a week, several hours a day. Running-in began at the smallest load, $4.7\ \mathrm{kg}/\mathrm{cm}^2$, and only after a more or less steady state of the oil temperature in the bath had been reached at this load did they proceed to the next load, and so on. At the load of $4.7\ \mathrm{kg}/\mathrm{cm}^2$ the steady state of the oil temperature was reached at a temperature of about $48—49^\circ\text{C}$. At higher loads the established temperature lay very high—about $80^\circ\text{C}$, and in the subsequent experiments involving the investigation of the pressure distribution and the friction force such temperatures were not reached.

During the running-in of the heel no measurements of the pressure in the heel or of the friction force were made; the entire task during this time was confined to becoming generally acquainted with the operation of the heel, arranging the manometers conveniently for observations, following the change in temperature of the oil in the bath, observing the operation of the motor at low speeds, and also seeing how successful the fitting of the copper tubes for measuring the pressures at the heel disks had turned out, and whether any

places for overflow of oil, and so forth. At the same time, as I already indicated in the first article, it turned out that a single groove for supplying oil to the rubbing surface of the pivot was insufficient: the pivot heated up very rapidly; it was therefore necessary to make four grooves. It also turned out that, in order to reduce the heating of the pivot, it was necessary to change the circulation of the oil from the main reservoir to the shaft journals and to the pivot itself (see the first article, p. 8).

Only after the pivot had been sufficiently run in was it possible to proceed to measurements of the frictional force and of the distribution of pressures on the pivot.

Conduct of experiments in the study of friction.

Measurements of the magnitude of the frictional force were made not only simultaneously with the determination of the distribution of pressures over the pivot, but in addition, after the completion of the latter experiments, a further series of experiments was carried out specially for measuring the frictional force.

In the latter experiments, particular attention was paid to ensuring that, while maintaining, as far as possible, some definite oil temperature, one could trace, at a chosen load of the pivot, the change in the magnitude of the frictional force as a function of the change in the number of revolutions of the pivot; but this was very difficult to achieve: even at a constant number of revolutions the oil temperature rose with the passage of time, and any increase in the number of revolutions immediately produced a more rapid rise in the oil temperature; this constancy of temperature was better maintained at higher oil temperatures.

Maintaining the number of revolutions while changing the oil temperature also proved difficult, since a change in temperature, and consequently in the frictional force in the pivot, entailed a change in the rotational speed of the motor.

All this was the reason why, for any given load of the pivot, it was necessary to obtain the magnitudes of the frictional force, in most cases, at various temperatures and at various numbers of revolutions of the pivot. Therefore, in order to obtain those magnitudes of the frictional force which corresponded to any one definite temperature and one load of the pivot, it was necessary to select the corresponding values of the frictional force from a whole series of experiments. But, of course, it was impossible to select values of the frictional force that would correspond exactly to a definite degree of temperature; it was necessary to limit oneself to selecting values of the frictional force in small intervals (of $2$–$3^\circ$ C) of oil temperatures.

Thus, we obtained the magnitudes of the frictional force at oil temperatures of $18$–$20^\circ$, $27$–$29^\circ$, $37$–$39^\circ$, $48$–$50^\circ$, and $58$–$60^\circ$ C.

The average temperature in each of these groups of temperatures differs, as is seen, one from another by approximately $10^\circ$ C.

Results of experiments. In the following Tables Nos. 3–46 are given both the actual amounts of extension ($R$) of the springs due to friction in the bearing, and the magnitudes of the forces ($S$), determined from these extensions with the aid of List’s diagram I, which caused the indicated extensions of the springs under different loads and numbers of revolutions ($n$) of the bearing.

Table No. 3.

Load on the bearing $4.7\ \mathrm{kg}/\mathrm{cm}^2$. Oil temperature $18\text{--}20^\circ$ C.

$n$.	60	70	82	104	135	165
$R$ in m/m.	1.4	1.6	1.7	1.8	1.9	2.0
$S$ in kg.	1.7	1.9	2.1	2.2	2.3	2.45
$\mu$.	0.0192	0.0215	0.0236	0.0249	0.0260	0.0274

Table No. 4.

Load on the bearing $10.4\ \mathrm{kg}/\mathrm{cm}^2$. Oil temperature $18\text{--}19^\circ$ C.

$n$.	79	98	135	170
$R$ in m/m.	2.3	2.4	2.5	2.6
$S$ in kg.	2.8	2.95	3.05	3.15
$\mu$.	0.0143	0.0151	0.0156	0.0161

Table No. 5.

Load on the bearing $15.7\ \mathrm{kg}/\mathrm{cm}^2$. Oil temperature $18\text{--}19^\circ$ C.

$n$.	73	95	164
$R$ in m/m.	3.1	3.2	3.3
$S$ in kg.	3.75	3.9	4.0
$\mu$.	0.0128	0.0133	0.0137

Table No. 6.

Pivot load 21.0 kg/cm². Oil temperature 18–20° C.

	55	77	89	135	150
n.	55	77	89	135	150
R in m/m.	3.8	3.8	3.7	3.8	3.9
S in kg.	4.6	4.6	4.5	4.6	4.75
μ.	0.0116	0.0116	0.0114	0.0116	0.0120

Table No. 7.

Pivot load 26.2 kg/cm². Oil temperature 18–20° C.

	61	75	110	150
n.	61	75	110	150
R in m/m.	4.5	4.3	4.2	4.6
S in kg.	5.5	5.25	5.1	5.65
μ.	0.0111	0.0106	0.0103	0.0114

Table No. 8.

Pivot load 31.5 kg/cm². Oil temperature 18–20° C.

	60	74	96	116	124
n.	60	74	96	116	124
R in m/m.	5.5	5.3	5.1	5.0	5.0
S in kg.	6.8	6.5	6.3	6.15	6.15
μ.	0.0114	0.0109	0.0106	0.0103	0.0103

Table No. 9.

Pivot load 36.7 kg/cm². Oil temperature 19–20° C.

	98	130	153
n.	98	130	153
R in m/m.	6.0	(5.7)	5.7
S in kg.	7.4	7.05	7.05
μ.	0.0107	0.0102	0.0102

Table No. 10.

Load on the pivot 42.0 kg/cm². Oil temperature 18–20° C.
n.	100	108	130	155
R in m/m.	6.9	6.8	6.7	6.6
S in kg.	8.5	8.4	8.3	8.2
μ.	0.0108	0.0106	0.0105	0.0104

Table No. 11.

Load on the pivot 47.3 kg/cm². Oil temperature 18–20° C.
n.	55	65	79	106	139	157
R in m/m.	9.5	8.5	8.1	7.8	7.6	7.5
S in kg.	12.1	10.7	10.1	9.7	9.5	9.3
μ.	0.0136	0.0120	0.0113	0.0109	0.0107	0.0104

Table No. 12.

Load on the pivot 4.7 kg/cm². Oil temperature 28–29° C.
n.	17	29	34	70	106	137	217
R in m/m.	0.7	0.8	1.0	1.3	1.5	1.7	1.9
S in kg.	0.825	1.0	1.225	1.6	1.85	2.1	2.3
μ.	0.0093	0.0113	0.0139	0.0181	0.0209	0.0238	0.0260

Table No. 13.

Load on the pivot 10.4 kg/cm². Oil temperature 27–28° C.
n.	73	139	146	169
R in m/m.	1.8	2.3	2.4	2.5
S in kg.	2.2	2.8	2.95	3.05
μ.	0.0112	0.0143	0.0151	0.0156

Table No. 14.

Load on the pivot 15.7 kg./cm². Oil temperature 28–29°.


n.	83	123	187
R in m/m.	2,6	2,9	3,3
S in kg.	3,1	3,5	4,0
μ.	0,0105	0,0118	0,0137

Table No. 15.

Load on the pivot 21.0 kg./cm². Oil temperature 26–28° C.


n.	65	106	131	150	173	185	214
R in m/m.	3,3	3,1	3,3	3,5	3,6	3,8	4,0
S in kg.	4,0	3,75	4,0	4,25	4,4	4,6	4,9
μ.	0,0101	0,0095	0,0101	0,0108	0,0112	0,0116	0,0124

Table No. 16.

Load on the pivot 26.2 kg./cm². Oil temperature 27–28° C.


n.	61	75	98	127	152	198
R in m/m.	4,4	4,2	4,1	3,9	4,0	4,
S in kg.	5,4	5,1	5,0	4,8	4,9	5,0
μ.	0,0109	0,0103	0,0101	0,0096	0,0099	0,0101

Table No. 17.

Load on the pivot 31.5 kg./cm². Oil temperature 27–28° C.


n.	70	93	114	135	160	172	181	197
R in m/m.	5,2	4,9	4,8	4,7	4,6	4,6	4,7	4,7
S in kg.	6,4	6,0	5,9	5,75	5,65	5,65	5,75	5,75
μ.	0,0108	0,0101	0,0099	0,0097	0,0095	0,0095	0,0097	0,0097

Table No. 18.

Load on the heel 36.7 kg/cm². Oil temperature 28–29° C

	79	130	156	167	196
n.	79	130	156	167	196
R in m/m.	5.9	5.5	5.4	5.3	5.2
S in kg.	7.3	6.8	6.7	6.5	6.4
μ.	0.0105	0.0098	0.0097	0.0094	0.0092

Table No. 19.

Load on the heel 42.0 kg/cm². Oil temperature 27–29° C.

	75	85	108	131	140	164	188
n.	75	85	108	131	140	164	188
R in m/m.	7.1	6.9	6.7	6.5	6.3	6.2	6.0
S in kg.	8.95	8.5	8.3	8.1	7.8	7.7	7.5
μ.	0.0112	0.0108	0.0105	0.0103	0.0099	0.0097	0.0095

Table No. 20.

Load on the heel 47.3 kg/cm². Oil temperature 28–29° C.

	58	79	85	92	135	162	220
n.	58	79	85	92	135	162	220
R in m/m.	10.2	7.9	7.8	7.7	7.2	7.0	6.8
S in kg.	13.0	9.9	9.7	9.6	8.95	8.7	8.4
μ.	0.0145	0.0111	0.0109	0.0108	0.0100	0.0098	0.0094

Table No. 21.

Load on the heel 4.7 kg/cm². Oil temperature 37–38° C.

	36	84	130	146	188	199	200	205
n.	36	84	130	146	188	199	200	205
R in m/m.	0.9	1.2	1.5	1.5	1.6	1.7	1.6	1.7
S in kg.	1.1	1.45	1.83	1.83	1.9	2.1	1.9	2.1
μ.	0.0125	0.0164	0.0206	0.0206	0.0215	0.0236	0.0215	0.0236

Table No. 22.

Pivot load 10.4 kg/cm². Oil temperature 38–39° C.

	91	110	129	133	137	160	174	194	205	217
n.	91	110	129	133	137	160	174	194	205	217
R in m/m.	1.9	2.0	2.1	2.1	2.2	2.2	2.3	2.2	2.3	2.3
S in kg.	2.3	2.43	2.55	2.55	2.7	2.7	2.8	2.7	2.8	2.8
μ.	0.0118	0.0124	0.0130	0.0130	0.0138	0.0138	0.0143	0.0138	0.0143	0.0143

Table No. 23.

Pivot load 15.7 kg/cm². Oil temperature 37–39° C.

	83	93	108	127	135	158	177	183	187	198
n.	83	93	108	127	135	158	177	183	187	198
R in m/m.	2.5	2.6	2.7	2.8	2.8	2.9	2.8	2.8	2.9	2.9
S in kg.	3.05	3.15	3.3	3.4	3.4	3.5	3.4	3.4	3.5	3.5
μ.	0.0103	0.0106	0.0111	0.0115	0.0115	0.0118	0.0115	0.0115	0.0118	0.0118

Table No. 24.

Pivot load 21.0 kg/cm². Oil temperature 37–39° C.

	80	96	127	132	154	181	194	214	222	251
n.	80	96	127	132	154	181	194	214	222	251
R in m/m.	3.2	3.3	3.4	3.5	3.6	3.6	3.6	3.7	3.7	3.9
S in kg.	3.9	4.0	4.15	4.3	4.4	4.4	4.4	4.5	4.5	4.7
μ.	0.0098	0.0101	0.0105	0.0109	0.0111	0.0111	0.0111	0.0114	0.0114	0.0119

Table No. 25.

Pivot load 26.2 kg/cm². Oil temperature 38–39° C.

	79	98	129	130	133	152	164	178	208
n.	79	98	129	130	133	152	164	178	208
R in m/m.	4.0	3.9	3.8	3.9	3.9	3.9	3.9	4.0	4.0
S in kg.	4.9	4.75	4.6	4.75	4.75	4.75	4.75	4.9	4.9
μ.	0.0099	0.0096	0.0093	0.0096	0.0096	0.0096	0.0096	0.0099	0.0099

Table No. 26.

Load on the heel 31.5 kg/cm². Oil temperature 37–39° C.

	106	131	141	167	172	191	205
n.	106	131	141	167	172	191	205
R in m/m.	4.6	4.6	4.4	4.3	4.4	4.5	4.6
S in kg.	5.65	5.65	5.35	5.25	5.35	5.5
μ.	0.0095	0.0095	0.0090	0.0088	0.0090	0.0093	0.0095

Table No. 27.

Load on the heel 36.7 kg/cm². Oil temperature 37–38° C.

	79	123	154	183	236	243	260
n.	79	123	154	183	236	243	260
R in m/m.	5.7	5.3	5.0	5.0	5.3	5.4	5.6
S in kg.	7.0	6.5	6.15	6.15	6.5	6.7	6.9
μ.	0.0101	0.0094	0.0089	0.0089	0.0094	0.0097	0.00996

Table No. 28.

Load on the heel 42.0 kg/cm². Oil temperature 37–39° C.

	96	120	141	164	174	191	205
n.	96	120	141	164	174	191	205
R in m/m.	6.6	6.3	6.2	6.0	5.9	5.8	5.9
S in kg.	8.2	7.8	7.7	7.45	7.3	7.15	7.3
μ.	0.0104	0.0099	0.0097	0.0094	0.0092	0.0091	0.0092

Table No. 29.

Load on the heel 47.3 kg/cm². Oil temperature 36–39° C.

	94	98	138	152	162	168	178	189	210
n.	94	98	138	152	162	168	178	189	210
R in m/m.	7.8	7.6	7.2	6.9	6.8	6.7	6.7	6.6	6.5
S in kg.	9.75	9.5	8.95	8.6	8.45	8.3	8.3	8.2	8.1
μ.	0.0109	0.0107	0.0104	0.0096	0.0095	0.0093	0.0093	0.0092	0.0091

Table No. 30.

Heel load 4.7 kg/cm². Oil temperature 48–49° C.

	108	137	163	168	191	200	230	246
n.	108	137	163	168	191	200	230	246
R in m/m.	1.2	1.3	1.4	1.4	1.5	1.5	1.5	1.6
S in kg.	1.45	1.60	1.70	1.70	1.85	1.85	1.85	1.95
μ.	0.0164	0.0181	0.0192	0.0192	0.0209	0.0209	0.0209	0.0220

Table No. 31.

Heel load 10.4 kg/cm². Oil temperature 48–50° C.

	102	131	140	154	164	180	194	208	218	266	272
n.	102	131	140	154	164	180	194	208	218	266	272
R in m/m.	1.8	1.9	1.9	1.95	2.0	2.0	2.1	2.1	2.1	2.2	2.2
S in kg.	2.20	2.30	2.30	2.35	2.45	2.45	2.55	2.55	2.55	2.70	2.70
μ.	0.0112	0.0117	0.0117	0.0120	0.0125	0.0125	0.0130	0.0130	0.0130	0.0138	0.0138

Table No. 32.

Heel load 15.7 kg/cm². Oil temperature 48–50° C.

	84	93	108	148	183	197	205	230	260
n.	84	93	108	148	183	197	205	230	260
R in m/m.	2.4	2.3	2.3	2.4	2.5	2.5	2.5	2.7	2.6
S in kg.	2.9	2.8	2.8	2.9	3.05	3.05	3.05	3.25	3.15
μ.	0.0098	0.0094	0.0094	0.0098	0.0103	0.0103	0.0103	0.0110	0.0107

Table No. 33.

Heel load 21.0 kg/cm². Oil temperature 48–50° C.

	120	146	156	169	189	191	210	230	242
n.	120	146	156	169	189	191	210	230	242
R in m/m.	3.2	3.2	3.1	3.0	3.0	3.0	3.1	3.0	3.1
S in kg.	3.90	3.90	3.75	3.65	3.65	3.65	3.75	3.65	3.75
μ.	0.0098	0.0098	0.0095	0.0092	0.0092	0.0092	0.0095	0.0092	0.0095

Table No. 34.

Heel load 26.2 kg/cm². Oil temperature 49–50° C.

	69	99	132	148	178	185	188	264
n.	69	99	132	148	178	185	188	264
R in m/m.	4.1	4.0	3.95	3.9	3.7	3.6	3.7	4.0
S in kg.	5.0	4.9	4.8	4.75	4.5	4.4	4.5	4.9
μ.	0.0101	0.0099	0.0097	0.0096	0.0091	0.0089	0.0091	0.0099

Table No. 35.

Heel load 31.5 kg/cm². Oil temperature 49–50° C.

	120	140	156	174	183	198	213
n.	120	140	156	174	183	198	213
R in m/m.	4.4	4.2	4.1	4.0	4.0	4.1	4.2
S in kg.	5.4	5.1	5.0	4.9	4.9	5.0	5.1
μ.	0.0091	0.0086	0.0084	0.0082	0.0082	0.0084	0.0086

Table No. 36.

Heel load 36.7 kg/cm². Oil temperature 48–50° C.

	76	96	138	140	176	181	200	250
n.	76	96	138	140	176	181	200	250
R in m/m.	5.7	5.5	5.1	5.2	4.9	4.8	4.8	5.2
S in kg.	7.05	6.8	6.3	6.4	6.0	5.9	5.9	6.4
μ.	0.0102	0.0098	0.0091	0.0092	0.0087	0.0085	0.0085	0.0092

Table No. 37.

Heel load 42.0 kg/cm². Oil temperature 48–49° C.

	110	132	143	161	187	205	220
n.	110	132	143	161	187	205	220
R in m/m.	6.5	6.2	6.1	5.7	5.6	5.5	6.2
S in kg.	8.1	7.7	7.5	7.05	6.9	6.8	7.7
μ.	0.0102	0.0097	0.0095	0.0090	0.0087	0.0086	0.0097

Table No. 38.

Load on the pivot 47.3 kg./cm.² Oil temperature 48—49° C.

	93	104	127	162	170	181	189	208
n.	93	104	127	162	170	181	189	208
R in m/m.	7,7	7,5	7,1	6,9	6,8	6,7	6,5	6,4
S in kg.	9,6	9,3	8,85	8,5	8,45	8,3	8,1	7,9
μ.	0,0108	0,0104	0,0099	0,0095	0,0094	0,0093	0,0091	0,0089

Table No. 39.

Load on the pivot 4.7 kg./cm.² Oil temperature 58—59° C.

	137	180	201	252	282
n.	137	180	201	252	282
R in m/m.	1,3	1,45	1,5	1,6	1,7
S in kg.	1,60	1,75	1,85	1,95	2,10
μ.	0,0181	0,0198	0,0209	0,0220	0,0236

Table No. 40.

Load on the pivot 10.4 kg./cm.² Oil temperature 59—60° C.

	141	170	187	237	265	272
n.	141	170	187	237	265	272
R in m/m.	1,7	1,8	1,9	2,0	2,1	2,1
S in kg.	2,1	2,2	2,3	2,45	2,55	2,55
μ.	0,0107	0,0112	0,0117	0,0125	0,0130	0,0130

Table No. 41.

Load on the pivot 15.7 kg./cm². Oil temperature 59—60° C.

	125	155	195	250	275
n.	125	155	195	250	275
R in m/m.	2,2	2,3	2,5	2,6	2,7
S in kg.	2,70	2,80	3,05	3,15	3,25
μ.	0,0090	0,0093	0,0102	0,0106	0,0110

Table No. 42.

Shoe load 21.0 kg/cm². Oil temperature 59–60° C.


n.	115	137	162	200	270	272
R in m/m.	2.95	2.95	2.9	3.1	3.3	3.3
S in kg.	3.55	3.55	3.5	3.75	4.0	4.0
μ.	0.0090	0.0090	0.0089	0.0095	0.0101	0.0101

Table No. 43.

Shoe load 26.2 kg/cm². Oil temperature 58–60° C.


n.	135	165	202	206	208	262	270
R in m/m.	3.8	3.7	3.5	3.5	3.6	3.8	3.9
S in kg.	4.6	4.5	4.25	4.25	4.4	4.6	4.75
μ.	0.0093	0.0091	0.0086	0.0086	0.0088	0.0093	0.0095

Table No. 44.

Shoe load 36.7 kg/cm². Oil temperature 58–60° C.


n.	96	140	164	185	193	198	202	210	256	260
R in m/m.	5.6	5.2	5.2	4.9	4.9	5.0	4.8	4.8	5.1	5.2
S in kg.	6.9	6.4	6.4	6.0	6.0	6.15	5.9	5.9	6.3	6.4
μ.	0.0099	0.0092	0.0092	0.0087	0.0087	0.0089	0.0085	0.0085	0.0091	0.0092

Table No. 45.

Shoe load 42.0 kg/cm². Oil temperature 58–59° C.


n.	168	187	202	208	226
R in m/m.	6.0	5.8	5.6	5.6	5.7
S in kg.	7.45	7.15	6.90	6.90	7.05
μ.	0.0094	0.0090	0.0087	0.0087	0.0088

Table No. 46.

Load on the pivot 47.3 kg./cm.² Oil temperature 58–60° C.

n.	93	104	148	174	200	220	238	266
R in m/m.	7.4	7.3	7.0	6.8	6.5	6	6.3	6.4
S in kg.	9.25	9.10	8.70	8.45	8.10	7.90	7.80	7.90
μ.	0.0104	0.0102	0.0097	0.0095	0.0090	0.0089	0.0087	0.0089

In the same tables are given, corresponding to the different frictional forces in the pivot, the values of the coefficient of friction $\mu$; the calculation of this coefficient was made on the basis of the assumption of a uniform distribution of pressure over the surface of the pivot and on the assumption that the coefficient of friction is independent of the speed of motion of the rubbing elements of the pivot, i.e. according to the formula

\[ \mu=\frac{S\cdot L}{P\cdot r_m}, \]

where

$S$—the force stretching the spring when the lifting plates (3, 5, 9, see Table II of the drawings of the first article) are restrained from motion, owing to friction on the surface of the pivot;

$L$—the distance of the point of application of the force $S$ to spindle 13 (Table II of the first article) from the center of the pivot, equal to 10 cm.;

$P$—the total load on the pivot in kg.;

$r_m$—the mean radius of the annular surface of the pivot, equal to 2.75 cm.

All the values of the coefficient of friction calculated in this way are represented in the form of curves (see Plates II, III, IV, V and VI; scale for the coefficient of friction: $1^{m}/_{m}\sim 0.0001$, scale for the numbers of revolutions: $1^{m}/_{m}\sim 1$ revolution per minute), showing the dependence of the coefficient of friction (plotted along the ordinate axis), at each definite load on the pivot (per cm.²), on the number of revolutions of the pivot (plotted along the abscissa axis); moreover, where there was no danger of obscuring the drawing, the mean probable curve of variation of the coefficient of friction was drawn with a dotted line.

On the basis of the curves represented on Plates II, III, IV, V and VI, curves were constructed showing the dependence of the coefficient of friction, at definite numbers of revolutions of the pivot, on the change of its mean load; the latter curves are plotted

by broken¹ lines on Plates VIII, IX, X, XI, and XII (scale for the coefficient of friction: $1^{m}/m \sim 0.0001$, scale for loads: $1^{m}/m \sim 0.2\ \mathrm{kg}/\mathrm{cm}^{2}$).

I shall now note that, since for what follows it was sometimes necessary for us to have values of the coefficient of friction beyond the limits of our experiments, the required values of the coefficient were found by interpolation, and the corresponding parts of the curves, or the entire curve, were in that case shown on the diagrams by a dotted line.

Conclusions from the results of the experiments.

Examining the data of the tables and the curves of variation of the coefficient of friction, we first observe a decrease in the coefficient of friction with an increase in the average load of the heel; this decrease, which is especially pronounced at small loads, gradually becomes smaller as the latter increase, and at a certain load of the heel, definite for each given number of revolutions, the coefficient of friction reaches the minimum of its value, and then begins to increase again, although not as strongly as it had previously decreased.

The greater the number of revolutions of the heel, the higher the average load at which the coefficient of friction reaches its minimum. The minimum values of the coefficient of friction differ little from one another at different numbers of revolutions of the heel. Within the limits of the present experiments, approximately up to a heel load of $24\text{--}25\ \mathrm{kg}/\mathrm{cm}^{2}$ (at lower temperatures; at higher ones, up to a somewhat smaller load), the coefficient of friction clearly increases with an increase in the number of revolutions; beginning, approximately, with a load of $28\text{--}30\ \mathrm{kg}/\mathrm{cm}^{2}$, the opposite phenomenon is observed—the coefficient of friction increases as the number of revolutions of the heel decreases. In the interval of loads of approximately $24\text{--}30\ \mathrm{kg}/\mathrm{cm}^{2}$, the curves of variation of the coefficient of friction at different numbers of revolutions of the heel intersect one another.

The influence of temperature is manifested, as is seen from the data of Tables Nos. 47–49, and also from the graphs of Plate XII (the scale for the coefficient of friction here, as before: $1^{m}/m \sim 0.0001$, and the temperature scale: $2^{m}/m$ correspond to $1^\circ \mathrm{C}$), in a decrease of the coefficient of friction with an increase in temperature; this decrease is stronger at smaller loads than at larger ones, and at lower temperatures than at higher ones.

Table No. 47.

Coefficient of friction at 100 revolutions of the pivot per minute.

Pivot load in kg/cm²	Oil temperature in °C: 18–20	Oil temperature in °C: 27–29	Oil temperature in °C: 37–39	Oil temperature in °C: 48–50	Oil temperature in °C: 58–60
5	0.0242	0.0202	0.0171	0.0157	0.0152
10	0.0158	0.0131	0.0127	0.0116	0.0104
15	0.0137	0.0113	0.0111	0.0097	0.0089
20	0.0118	0.0099	0.0103	0.0098	0.0090
25	0.0106	0.0100	0.0099	0.0099	0.0094
30	0.0105	0.0101	0.0096	0.0099	0.0097
35	0.0106	0.0102	0.0097	0.0099	0.0098
40	0.0107	0.0105	0.0101	0.0102	0.0100
45	0.0109	0.0106	0.0105	0.0105	0.0102

Table No. 48.

Coefficient of friction at 150 revolutions of the pivot per minute.

Pivot load in kg/cm²	Oil temperature in °C: 18–20	Oil temperature in °C: 27–29	Oil temperature in °C: 37–39	Oil temperature in °C: 48–50	Oil temperature in °C: 58–60
5	0.0262	0.0250	0.0206	0.0183	0.0178
10	0.0166	0.0158	0.0144	0.0127	0.0114
15	0.0140	0.0130	0.0119	0.0103	0.0095
20	0.0123	0.0112	0.0111	0.0098	0.0090
25	0.0115	0.0101	0.0098	0.0096	0.0091
30	0.0106	0.0097	0.0091	0.0088	0.0092
35	0.0102	0.0097	0.0089	0.0089	0.0093
40	0.0103	0.0098	0.0093	0.0092	0.0094
45	0.0105	0.0099	0.0097	0.0095	0.0096

Table No. 49.

Thrust load, in kg/cm²	Oil temperature in °C	Oil temperature in °C	Oil temperature in °C	Oil temperature in °C
	27–29	37–39	48–50	58–60
5	0.0284	219	0.0203	0.0202
10	0.0183	0.0151	0.0138	0.0125
15	0.0147	0.0122	0.0107	0.0104
20	0.0124	0.0114	0.0096	0.0096
25	0.0106	0.0103	0.0092	0.0089
30	0.0099	0.0096	0.0087	0.0087
35	0.0095	0.0091	0.0085	0.0087
40	0.0094	0.0091	0.0086	0.0087
45	0.0095	0.0092	0.0088	0.0089

At high temperatures (50–60°), under large thrust loads (above 30 kg/cm²), and at 150–200 revolutions of the thrust bearing per minute, even a certain increase in the coefficient of friction with increasing temperature is observed. The minimum values of the coefficient of friction at a given number of revolutions of the thrust bearing also decrease markedly with increasing temperature: thus, at a temperature of 27–29° and at 100 revolutions per minute, the value under consideration of the coefficient of friction is equal to 0.0099 (at a load of 20 kg/cm²), whereas at a temperature of 58–60° it is equal to 0.0089 (at a load of 15 kg/cm²).

Experiments to determine the coefficient of friction “at rest.”

Finally, we also determined the coefficient of friction for the moment at which the thrust bearing began to move from a state of rest (the coefficient of friction “at rest”) under various loads and temperatures in the following manner.

After any one of the above-described experiments with the thrust bearing had been carried out at a definite load, the thrust bearing was brought to rest. The temperature of the oil in the bath at this time was more or less high; the inflow of fresh oil into the bath and the outflow from it were stopped.

Having at this time produced, in an already known manner (by rotating nut 10), a displacement of bearing plate 3¹, we determined, from the extension of the springs, the force required to overcome the frictional force on the surface of the pivot at the moment of this displacement.

Then plate 3 was turned directly by hand to its extreme position, when spindle 14 touches one of the screws $b, b$, and was left in this position for some time at rest; this turning of plate 3 into the indicated position was necessary so that spindle 14, during the next new displacement of the plate for determining the frictional force, would have a place to turn into. After the temperature of the oil in the bath had then fallen by a certain number of degrees, the plate 3 was again displaced by rotating nut 10, and the frictional force was determined, etc.

In this way, at various loads and temperatures, certain values of the force $S$ were obtained; and, assuming in the present case that the pressure on the surface of the pivot was uniformly distributed, it was possible, from the preceding, to calculate the corresponding coefficients of friction.

The values of these coefficients of friction are given in Tables Nos. 50–58 and are represented in the form of curves showing the dependence of the coefficient of friction on the temperature $T$, on Plate VII (scale for the coefficients of friction: $1\ \mathrm{m}/\mathrm{m} \sim 0.0004$, scale for temperatures: $1\ \mathrm{m}/\mathrm{m} \sim 0.2^\circ\mathrm{C}$).

Table No. 50.


Load of the pivot $4.7\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $4.7\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $4.7\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $4.7\ \mathrm{kg}/\mathrm{cm}^2$.
$T^\circ\mathrm{C}$.	23	27	35
$R$ in m/m.	8,2	7.8	5,0
$S$ in kg.	10,3	9,7	6.2
$\mu$.	0.1163	0,1095	0,0700

Table No. 51


Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.	Load of the pivot $10.4\ \mathrm{kg}/\mathrm{cm}^2$.
$T^\circ\mathrm{C}$.	17	23	30	33	49	53
$R$ in m/m.	15,1	14,8	13,5	12,4	12,8	12,9
$S$ in kg.	19,8	19,4	17,6	16,1	16,7	16,8
$\mu$.	0,1010	0,0990	0,0900	0,0822	0,0852	0,0858

Table No. 52.

Pivot load 15.7 kg/cm²

	24	30	35	38
T°C.	24	30	35	38
R in m/m.	24.9	15.7	15.0	13.9
S in kg.	34.1	20.7	19.7	18.2
μ.	0.1150	0.0800	0.0668	0.0615

Table No. 53.

Pivot load 21.0 kg/cm²

	26	35	49
T°C.	26	35	49
R in m/m.	28.2	18.1	13.0
S in kg.	38.95	24.2	16.9
μ.	0.0984	0.0612	0.0439

Table No. 54.

Pivot load 26.2 kg/cm²

	30	38	45	55
T°C.	30	38	45	55
R in m/m.	27.9	22.8	19.2	15.4
S in kg.	38.5	30.9	25.6	20.2
μ.	0.0780	0.0628	0.0520	0.0410

Table No. 55.

Pivot load 31.5 kg/cm²

	40	45	47	50	60
T°C.	40	45	47	50	60
R in m/m	24.9	22.2	21.1	17.1	15.4
S in kg.	34.1	30.1	28.5	22.7	20.2
μ.	0.0573	0.0507	0.0480	0.0382	0.0340

Table No. 56.

Load on the pivot 36.7 kg/cm²

T°C.	36	37	40	41	50	52	60	73
R in m/m.	29.6	27.9	27.5	26.3	23.0	22.5	19.8	15.0
S in kg.	41.1	38.5	37.9	36.2	31.3	30.5	26.6	19.7
μ.	0.0592	0.0553	0.0544	0.0522	0.0449	0.0438	0.0382	0.0284

Table No. 57.

Load on the pivot 41.0 kg/cm²

T°C.	34	38	43	52	64	71
R in m/m.	35.1	33.8	31.9	26.5	21.9	19.6
S in kg.	49.3	47.3	44.4	36.5	29.6	26.4
μ.	0.0622	0.0598	0.0561	0.0461	0.0374	0.0334

Table No. 58.

Load on the pivot 47.3 kg/cm².

T°C.	42	55	60	70
R in m/m.	31.2	28.2	27.8	23.8
S in kg.	43.4	39.0	38.4	32.5
μ.	0.0488	0.0438	0.0432	0.0366

Examining the curves of Plate VII, we observe that they almost coincide with one another at different pressures; only at high temperatures is a certain increase in the coefficient of friction with increasing pressure noticeable, but even this is quite small.

It is true that the curves shown on Plate VII do not fully encompass the dependence of the coefficient of friction, under different loads on the pivot, on temperature over the entire range of its variation in the experiments: at higher temperatures we were unable to obtain values of the coefficient of friction at small loads, and at low ...

at temperatures, we have no values of the coefficient under large loads¹, but nevertheless the course of these curves and their mutual relation indicate, with a high degree of probability, the correctness of the conclusion stated above.

The indicated independence of the coefficient of friction from pressure gives grounds to think that in the present case we are dealing with internal friction in a liquid, which, as is known, according to Newton’s hypothesis does not depend on pressure.

That in the present case we are dealing with internal friction in a liquid is also indicated by the very large fall of the coefficient of friction with temperature.

True, this fall is not as considerable as the lowering with temperature, cited by Stribeck², of the degree of viscosity for the same oil (Gasmotorenöl) of Deutz; but, on the one hand, there is no direct proportionality between the coefficient of friction of a liquid and the degree of viscosity, and, on the other hand, it is possible, of course, that the oil used in our experiments differed in its quality from the oil used by Stribeck; unfortunately, I myself had no possibility of determining the degree of its viscosity.

The latter results of our experiments differ essentially from the results of Stribeck’s experiments with a journal, who found that “the coefficient of friction at rest does not depend on the pressure and temperature of the bearing”³. Stribeck sees the explanation of such a phenomenon in the fact that, at the moments when the rotation of the shaft ceases, oil is no longer supplied to the surface of the journal, but, on the contrary, even that which had been between the journal and the bearing brass flows out into the oil bath of the bearing; in this way, the insignificant quantity of oil remaining between the journal and the bearing brass is not able to protect these surfaces from mutual contact, and at the moment of displacement of the journal friction arises between solid bodies.

In our case, however, it is evident that, in view of the fact that the oil in the bath stands several centimeters above the surface of the heel, the layer of oil

between the rubbing surfaces of the pivot and the thrust-bearing plate there always remains; the thickness of this layer may be insignificant, but sufficient to separate the surfaces of the pivot and the thrust bearing.

Finally, our experiments on the friction of the pivot at the moment of displacement differ from Stribeck’s similar experiments on a journal in a cast-iron bearing in their load limits: in Stribeck’s case these limits are $0.42\text{--}22.6\ \mathrm{kg}/\mathrm{cm}^2$, in ours $4.7\text{--}47.3\ \mathrm{kg}/\mathrm{cm}^2$.

As for the other above-described results of our experiments, they are very close to the results of Stribeck’s experiments (in the latter, the limits of the experiments for loads are $0.43\text{--}32\ \mathrm{kg}/\mathrm{cm}^2$, and for the numbers of revolutions of the journal per minute $5.5\text{--}1100$). The difference consists only in the somewhat higher values, in our case, of the coefficient of friction, and in the slower rise beyond the critical load of the curves of variation of the coefficient of friction with increasing load.

The slower rise of the curves of the coefficient of friction beyond the critical load constitutes, in our opinion, a very substantial advantage of cast-iron pivots as compared with journals in cast-iron bearings, since it makes it possible to increase the load considerably beyond its critical value without fear of excessively increasing the friction in the pivot. Of course, what has just been said is, more or less, valid only within the limits of our experiments.

I now pass to the second stage of the results of our experiments on friction in pivots.

The point is that on the surface of the pivot the velocity of motion of its elements varies greatly depending on their distance from the center of the pivot; moreover, our previous investigations on the distribution of pressure over the surface of the pivot showed that this pressure varies over the pivot.

Therefore, the results we obtained for the mean coefficient of friction, calculated on the assumption of a uniform distribution of pressures over the surface of the pivot and of independence of the coefficient of friction from changes in the sliding velocity of the various elements of the pivot, do not yet give the actual dependence of the true coefficient of friction on velocity and pressure; and, in order to obtain such a dependence, it is necessary to take into account the nonuniformity of the pressure distribution over the pivot and the difference in the sliding velocities of the elements of the pivot surface.

For this purpose the following procedure was adopted. The entire rubbing surface of the pivot was divided on the drawing¹ by two concentric

into three parts (rings) by circles; the radii of these dividing circles were, respectively: 2.03 cm and 3.46 cm, and thus the width of the annular strips of the heel was approximately 1.43 cm. For each such ring of the heel, on the basis of the experimental data on the distribution of pressures over the heel, the mean pressure was computed for each mean load on the heel and the mean velocity.

Next, using the broken curves of the coefficients of friction on Plates VIII, IX, X, XI, and XII, it was possible to compute more exact values of the coefficient of friction in the following manner. Knowing the mean pressure on each ring of the heel at any given mean load on it (the heel), as well as the mean velocity on these rings, we found, from the above-mentioned curves, the values of the coefficients of friction corresponding to these velocities and pressures; taking the coefficient of friction for the middle ring at the corresponding mean velocity as the basis, we expressed the coefficients of friction on the outer rings as fractions of the coefficient of friction of the middle ring; knowing, moreover, the total pressure on each ring, its mean radius, and the force of friction on the heel under the mean load in question, it was easy to find the coefficient of friction at the pressure on the middle ring and at the corresponding mean velocity. Let, for example, the coefficient of friction for the middle ring at the mean pressure and mean velocity under consideration, according to the graphs, be $\mu$, and let the coefficients of friction on the outer rings of the heel at the corresponding mean pressures and velocities, according to the same graphs, be expressed as $\alpha \beta$ and $\beta \mu$; if we denote the total simultaneous pressures on the rings at that time by $P_1, P_2$, and $P_3$, and the mean radii of the rings by $r_1, r_2$, and $r_3$, then we obtain the relation

\[ \mu(P_1 r_1 \alpha + P_2 r_2 + P_3 r_3 \beta) = SL, \]

where $S$ and $L$ are the already indicated designations of the force stretching the spring as a result of friction on the heel, and of the distance of its point of application from the center of the heel. The relation just written makes it possible to determine the coefficient of friction $\mu$, corresponding, obviously, to the given mean pressure on the middle ring of the heel and its mean velocity.

Of course, for greater accuracy of the results, the surface of the heel should have been divided into a larger number of rings, but I considered it inexpedient to do so, in view of the fact that the mean velocities on the outer rings would then turn out to be such as would go beyond the limits of our investigations; thus, for example, at 50 revolutions

of the pivot per minute on the inner ring one would obtain a velocity equal to the mean sliding velocity of the pivot at 18 revolutions per minute; at 75 revolutions of the pivot one would obtain a velocity corresponding to 27 revolutions per minute; on the other hand, on the outer ring, at 175 revolutions of the pivot per minute one would obtain a velocity equal to the mean velocity of the pivot at 288 revolutions of it per minute; at 200 revolutions one would obtain a velocity equal to the mean velocity of the pivot at 330 revolutions of it per minute, etc. Even when the surface of the pivot was divided into three rings, such velocities were obtained on the outer rings that the values of the mean coefficients of friction often had to be found by interpolation.

True, in this respect, the accuracy of the values of the coefficients of friction for the inner outermost ring was not of particular importance, since in general the term $P_3 r_3 \beta$ pertaining to this ring in the above expression for $\mu$ constitutes a very small quantity in comparison with the two other terms; but, on the other hand, the term pertaining to the outer ring, $P_1 r_1 \alpha$, is very large, and the determination of the corresponding value of $\alpha$ had to be treated with extreme caution.

In addition, a comparative computation of the coefficients of friction when dividing the surface of the pivot into 3 and into 5 rings showed that this increase in the number of rings has no very great influence on the values of the coefficient of friction $\mu$.

That this is indeed so is shown by the data of Tables Nos. 59–60, where the values of the coefficient of friction are given in parallel at the velocity $v = 0.286\ \mathrm{mtr./sk.}$ at temperatures of $27\text{–}29^\circ$ and $48\text{–}50^\circ \mathrm{C}$, when computed on the basis of dividing the surface of the pivot into 3 and into 5 rings.

On the basis of the computations indicated above, the values of the coefficients of friction obtained at various pressures, temperatures of the lubricating oil, and sliding velocities of the rubbing surfaces are given in Tables Nos. 61–65 and represented in the form of more or less smooth curves¹ on Plates VIII, IX, X, XI, and XII (scale for coefficients of friction: $1^{\mathrm{m}}/\mathrm{m} \sim 0.0001$, scale for pressures: $1^{\mathrm{m}}/\mathrm{m} \sim 0.2\ \mathrm{kg./cm.^2}$). Of course, these new values of the coefficients of friction cannot be regarded as completely exact values of the coefficient of friction, but only as first approxi-

Table No. 59.

Speed $v = 0.286$ m/sec. Oil temperature $27\text{—}29^\circ$ C.

Pressure between rubbing planes $x$, in kg/cm²	6,4	7,0	13,5	14,0	21,5	22,0	25,0	24,5	29,5	36,0	40,5	40,0	47,0
Coefficient of friction $(\mu_1)$, calculated on the basis of the pressure on 3 boiler sections	0,0129		0,0084		0,0079		0,0075		0,0083	0,0087	0,0090		0,0093
Coefficient of friction $(\mu_2)$, calculated on the basis of the pressure on 5 boiler sections		0,0124		0,0082		0,0076		0,0076	0,0082	0,0087		0,0094	0,0097
Difference in % between the coefficients $\mu_1$ and $\mu_2$, relative to $\mu_2$	+4,04%		+2,44%		+3,95%		−1,32%		+1,22%	00%		−4,26%	−4,13%

Table No. 60.

Speed $v = 0.286$ m/s. Temperature of the oil $48–50^\circ$ C.

	6,4	7,0	13,5	14,0	21,5	22,0	25,0	24,5	29,5	36,0	40,5	40,0	47,0
Pressure between the rubbing surfaces in kg/cm²	6,4	7,0	13,5	14,0	21,5	22,0	25,0	24,5	29,5	36,0	40,5	40,0	47,0
Coefficient of friction $(\mu_1)$, calculated on the basis of an arm length of 3 rings	0,0107		0,0078		0,0075		0,0080		0,0084	0,0084	0,0088		0,0091
Coefficient of friction $(\mu_2)$, calculated on the basis of an arm length of 5 rings		0,0104		0,0077		0,0073		0,0083	0,0088	0,0084		0,0091	0,0096
Difference in %, between the coefficients $\mu_1$ and $\mu_2$ relative to $\mu_2$	+2,88%		+1,30%		+2,74%		−3,61%		−4,55%	0%	−3,30%		−5,21%

Table No. 61.

Coefficient of friction $\mu$ at an oil temperature of 18–20° C.

Sliding speed of the rubbing surfaces, mtr./sk.	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²
	6,4	13,5	21,5	25,0	29,5	36,0	40,5	47,0	55,0
0,143	0,0105	0,0090	0,0087	0,0095	0,0103	0,0104	0,0108	0,0116	0,0148
0,215	0,0134	0,0102	0,0093	0,0088	0,0090	0,0094	0,0096	0,0098	0,0103
0,286	0,0156	0,0110	0,0095	0,0086	0,0084	0,0087	0,0092	0,0094	0,0097
0,358	0,0170	0,0113	0,0093	0,0085	0,0084	0,0084	0,0086	0,0088	0,0092

Table No. 62.

Coefficient of friction $\mu$ at an oil temperature of 27–29° C.

Sliding speed of the rubbing surfaces, mtr./sk.	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²
	6,4	13,5	21,5	25,0	29,5	36,0	40,5	47,0	55,0
0,143	0,0097	0,0073	0,0080	0,0093	0,0104	0,0111	0,0112	0,0131	0,0166
0,215	0,0114	0,0078	0,0076	0,0082	0,0089	0,0093	0,0095	0,0100	0,0110
0,286	0,0129	0,0084	0,0079	0,0075	0,0083	0,0087	0,0090	0,0093	0,0096
0,358	0,0141	0,0088	0,0079	0,0078	0,0077	0,0082	0,0086	0,0089	0,0089
0,430	0,0150	0,0095	0,0083	0,0077	0,0077	0,0079	0,0083	0,0086	0,0087
0,501	0,0158	0,0100	0,0083	0,0079	0,0075	0,0074	0,0076	0,0080	0,0083
0,572	0,0166	0,0104	0,0086	0,0081	0,0075	0,0072	0,0075	0,0077	0,0080

Table No. 63.

Coefficient of friction $\mu$ at an oil temperature of 37–39° C.

Sliding speed of the rubbing surfaces, mtr./sk.	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²	Pressure between the rubbing surfaces, kg./cm²
	6,4	13,5	21,5	25,0	29,5	36,0	40,5	47,0	55,0
0,143	0,0096	0,0075	0,0076	0,0079	0,0087	0,0091	0,0096	0,0107	0,0134
0,215	0,0104	0,0079	0,0073	0,0077	0,0084	0,0086	0,0091	0,0099	0,0107
0,286	0,0118	0,0085	0,0080	0,0078	0,0079	0,0083	0,0088	0,0092	0,0094
0,358	0,0133	0,0089	0,0083	0,0079	0,0073	0,0077	0,0081	0,0088	0,0091
0,430	0,0144	0,0094	0,0086	0,0081	0,0075	0,0074	0,0073	0,0081	0,0085
0,501	0,0149	0,0098	0,0087	0,0083	0,0074	0,0068	0,0069	0,0075	0,0079
0,572	0,0156	0,0101	0,0089	0,0084	0,0074	0,0070	0,0070	0,0072	0,0077

Table No. 64.

Coefficient of friction $\mu$ at an oil temperature of 48–50° C.

Sliding speed of the rubbing surfaces in mtr./sec.	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$
	6,4	13,5	21,5	25,0	29,5	36,0	40,5	47,0	55,0
0,143	0,0092	0,0077	0,0080	0,0084	0,0087	0,0090	0,0096	0,0105	0,0119
0,215	0,0100	0,0074	0,0078	0,0083	0,0086	0,0086	0,0091	0,0096	0,0103
0,286	0,0107	0,0078	0,0075	0,0080	0,0084	0,0084	0,0088	0,0091	0,0094
0,358	0,0114	0,0078	0,0073	0,0080	0,0073	0,0078	0,0082	0,0086	0,0090
0,430	0,0124	0,0082	0,0077	0,0078	0,0075	0,0072	0,0076	0,0080	0,0086
0,501	0,0132	0,0084	0,0075	0,0073	0,0070	0,0066	0,0070	0,0074	0,0083
0,572	0,0139	0,0088	0,0075	0,0073	0,0070	0,0065	0,0066	0,0070	0,0076

Table No. 65.

Coefficient of friction $\mu$ at an oil temperature of 58–63° C.

Sliding speed of the rubbing surfaces in mtr./sec.	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$	Pressure between the rubbing surfaces in kg./cm.$^2$
	6,4	13,5	21,5	25,0	29,5	36,0	40,5	47,0	55,0
0,215	0,0093	0,0069	0,0072	0,0078	0,0084	0,0086	0,0089	0,0093	0,0097
0,286	0,0098	0,0067	0,0068	0,0077	0,0082	0,0085	0,0086	0,0088	0,0092
0,358	0,0104	0,0071	0,0068	0,0074	0,0079	0,0078	0,0083	0,0087	0,0090
0,430	0,0112	0,0074	0,0068	0,0071	0,0073	0,0075	0,0079	0,0083	0,0088
0,501	0,0123	0,0076	0,0072	0,0069	0,0072	0,0073	0,0074	0,0079	0,0083
0,572	0,0133	0,0079	0,0073	0,0072	0,0066	0,0067	0,0066	0,0070	0,0076
0,715	0,0141	0,0083	0,0075	0,0073	0,0071	0,0070	0,0069	0,0069	0,0069

approximation to them. Examining both the data of the tables and the curves, we first of all note a strong decrease in the values of the coefficient of friction in comparison with its initial mean values; thus, the mean values of the coefficient of friction, determined on the basis of the assumption of a uniform distribu-

the distribution of pressures over the surface of the fifth wheel and for the mean sliding speed of its rubbing surface, higher than the corresponding actual values of the coefficient of friction.

As for the character of the curves, it remained the same: to each definite sliding speed of the rubbing surfaces there corresponds a definite pressure (called also by Sommerfeld, in his theoretical investigation¹ of journal friction, the “transition pressure”), at which the coefficient of friction reaches its minimum value; up to this transition pressure the coefficient of friction decreases with increasing pressure, and after the transition pressure the coefficient of friction increases; the transition pressure rises noticeably with increasing speed; the smallest values of the coefficient of friction at different speeds differ little from one another. Within the limits of the experiments, at small pressures (approximately up to a pressure of $20\text{--}24\ \mathrm{kg./cm.^2}$) the coefficient of friction increases with increasing speed, while at larger pressures (approximately from $28\text{--}30\ \mathrm{kg./cm.^2}$) it decreases with increasing speed; to establish, however, a sharp boundary in pressure in this respect does not appear possible. In order to present more clearly the influence of the temperature of the lubricating oil on the coefficient of friction, new tables Nos. 66–68 have been compiled on the basis of the data of Tables Nos. 61–65; in these tables the values of the coefficient of friction are given in parallel at different temperatures and at some one speed. On Plate XIV (the scale for the coefficients of friction: $1^{\mathrm m}/_{\mathrm m} \sim 0.0001$, and for the temperatures: $1^{\mathrm m}/_{\mathrm m} \sim 0.5^\circ\mathrm C$) the dependence of the coefficient of friction on the temperature of the oil is presented in the form of curves. The latter undoubtedly indicate a decrease of the coefficient of friction with increasing temperature of the oil; moreover, it also appears that the greater or lesser decrease of the coefficient with increasing oil temperature depends on the speed and on the pressure: at higher speeds it is greater for smaller pressures, and at low speeds—for larger pressures.

A certain irregularity in the course of the curve of change of the coefficient of friction with increasing temperature at the speed $v = 0.143\ \mathrm{mtr./sk.}$ and at higher pressures may be explained by error in determining the friction force at small numbers of revolutions of the fifth wheel and large loads; thus, for example, I find the coefficient of friction at a temperature of $29^\circ$, with an average load of the fifth wheel of $47.3\ \mathrm{kg./cm.^2}$ and with the number of revolutions $n = 58$, to be highly doubtful, while toward

Table No. 66.

Coefficient of friction at speed $v = 0{,}143$ m/sec.

Pressure between the rubbing surfaces, kg/cm²	Oil temperature, °C: 18–20	Oil temperature, °C: 27–29	Oil temperature, °C: 37–39	Oil temperature, °C: 48–50
6,4	0,0105	0,0097	0,0096	0,0092
13,5	0,0090	0,0073	0,0075	0,0077
21,5	0,0087	0,0080	0,0076	0,0080
25,0	0,0095	0,0093	0,0079	0,0084
29,5	0,0103	0,0104	0,0087	0,0087
36,0	0,0104	0,0111	0,0091	0,0090
40,5	0,0108	0,0112	0,0096	0,0098
47,0	0,0116	0,0131	0,0107	0,0105
55,0	0,0148	0,0166	0,0134	0,0119

Table No. 67.

Coefficient of friction at speed $v = 0{,}286$ m/sec.

Pressure between the rubbing surfaces, kg/cm²	Oil temperature, °C: 18–20	Oil temperature, °C: 27–29	Oil temperature, °C: 37–39	Oil temperature, °C: 48–50	Oil temperature, °C: 58–60
6,4	0,0156	0,0129	0,0118	0,0107	0,0098
13,5	0,0110	0,0084	0,0085	0,0078	0,0067
21,5	0,0095	0,0079	0,0080	0,0075	0,0068
25,0	0,0086	0,0075	0,0078	0,0080	0,0077
29,5	0,0084	0,0083	0,0079	0,0084	0,0082
36,0	0,00[[unclear: digit]]7	0,0087	0,0083	0,0084	0,0085
40,5	0,0092	0,0090	0,0088	0,0088	0,0086
47,0	0,0094	0,0093	0,0092	0,0091	0,0088
55,0	0,0097	0,0096	0,0094	0,0094	0,0092

V. N. Pinegin.

Table No. 68.

Coefficient of friction at speed $v = 0{,}572$ mtr./sec.

Pressure between the rubbing surfaces, in kg./cm²	Oil temperature in C°: 27–29	Oil temperature in C°: 37–39	Oil temperature in C°: 48–50	Oil temperature in C°: 58–60
6,4	0,0166	0,0156	0,0139	0,0133
13,5	0,0104	0,0101	0,0088	0,0079
21,5	0,0086	0,0089	0,0075	0,0073
25,0	0,0081	0,0084	0,0073	0,0072
29,5	0,0075	0,0074	0,0070	0,0066
36,0	0,0072	0,0070	0,0065	0,0067
40,5	0,0075	0,0070	0,0060	0,0066
47,0	0,0077	0,0072	0,0070	0,0070
55,0	0,0080	0,0077	0,007	0,0076

therefore also at smaller loads of 42,0 kg./cm², 36,7 kg./cm², and some others, the magnitude of the friction force at this number of revolutions was not determined; likewise, at temperatures of 18–20° the friction force was not determined for foot loads of 36,7 kg./cm² and 42,0 kg./cm² at numbers of revolutions less than 98–100 per minute.

Relation of the results obtained to N. Petrov’s theory and to the results of Woodbury’s experiments.
In concluding the present article, I cannot pass over in silence the relation of the results obtained by me to the hydrodynamic theory of friction of N. Petrov.

I have already indicated at the beginning of the article that N. Petrov applied his theory also to the friction of a footstep bearing in a thrust bearing, and obtained, as is known, for the coefficient of friction the expression¹), entirely analogous to the expression for the coefficient of friction for a journal:

\[ \mu = \frac{\mu' U} {\left(\varepsilon + \frac{\mu'}{\lambda_1} + \frac{\mu'}{\lambda_2}\right)p} , \]

where:

\[ \mu' \]
— the coefficient of internal friction of the liquid lubricating the footstep;

¹) N. Petrov. Practical results of experiments, etc., p. 73 et seq.

$U$—the velocity, determined by the equation: $U=\omega r$, where, in turn, $\omega$ is the angular velocity of rotation of an element of the heel, and $r$ is the magnitude of the corresponding radius;

$\varepsilon$—the thickness of the layer of liquid lubricating the heel;

$\lambda_1$ and $\lambda_2$—the coefficients of friction of the lubricating liquid against the surface of the heel and of the bearing;

$p$—the pressure falling on a square unit of the surface of the heel.

On the basis of Petrov’s own investigations of friction in journals, and also of his analysis of the experiments of Lamanskii¹ and Woodbury, it may be concluded that, when the pressure $p$ changes, the value of the product

\[ \left(\varepsilon+\frac{\mu'}{\lambda_1}+\frac{\mu'}{\lambda_2}\right)\sqrt{p} \]

remains, with a comparatively high degree of accuracy, constant.² If the validity of this proposition is also assumed for our heel, then it becomes possible to represent the expression given above for the coefficient of friction for it in the form:

\[ \mu=\frac{\mu' U}{k\sqrt{p}}, \]

where

$k$—is a constant quantity when $p$ changes.

If this is so, it follows from this that, at an unchanged velocity $U$, $\mu$ varies inversely as the square root of $p$.

Turning to the results of our experiments, let us take the values of the coefficient of friction up to the transition pressure at $v=0.501\ \mathrm{mtr./sk}$ and at a temperature $t=27\text{–}29^\circ$ (see Sheet IX), and compare the ratios of its values $(\mu_1,\mu_2,\ldots)$ at different pressures $(p_1,p_2,\ldots)$ with the ratios of the corresponding quantities $\dfrac{1}{\sqrt{p_1}},\dfrac{1}{\sqrt{p_2}},\ldots$. In the following Table No. 69 are given the values of the coefficient of friction at the corresponding pressures, and also the magnitudes of the ratios of the coefficients of friction and the ratios of the reciprocal values of the square roots of the corresponding values of $p$.

The data of this table indicate a fairly close agreement of the relations under consideration with one another.

In Tables Nos. 70–71 are given the results of a comparison of the same relations at other velocities and temperatures, and also [[unclear: text continues on next page]]

is observed to be a fairly satisfactory agreement of the data of our experiments with the results of calculations by N. Petrov’s formula, but it is necessary once again to point out that the agreement under consideration has a place only under the assumption of the equality

\[ \left(\varepsilon+\frac{\mu'}{\lambda_1}+\frac{\mu'}{\lambda_2}\right)\sqrt{p}=\mathrm{const}. \]

and within the limits of variation of the coefficient of friction up to the transition pressure.

Table No. 69.

Speed $v = 0.501$ m/sec. Oil temperature $27\text{–}29^\circ$ C.

	7	9	11	13	15	17	19	21	23	25	27	29
$p_1, p_2, \ldots$ in kg/cm$^2$	7	9	11	13	15	17	19	21	23	25	27	29
$\mu_1, \mu_2, \ldots$	0.0148	0.0123	0.0109	0.0101	0.0095	0.0091	0.0087	0.0084	0.0081	0.0079	0.0076	0.0075
$\dfrac{1}{\sqrt{p_1}} : \dfrac{1}{\sqrt{p_2}} = \ldots$	1.135	1.105	1.087	1.072	1.065	1.058	1.052	1.048	1.042	1.038	1.034
$\mu_1 : \mu_2, \ldots$	1.203	1.128	1.080	1.063	1.043	1.046	1.036	1.036	1.026	1.026	1.014

Table No. 70.

$p_1,\ p_2,\ldots$ in kg/cm²	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35
$\dfrac{1}{\sqrt{p_1}} : \dfrac{1}{\sqrt{p_2}},\ldots$	1,135	1,105	1,087	1,072	1,065	1,058	1,052	1,048	1,042	1,038	1,034	1,032	1,030	1,028
Speed $v=0{,}286$ mtr./sk. Temperature of the oil 58–60° C. $\mu_1,\ \mu_2,\ldots$	0,0090	0,0077	0,0071	0,0067	0,0066
Speed $v=0{,}286$ mtr./sk. Temperature of the oil 58–60° C. $\mu_1:\mu_2,\ldots$	1,169	1,084	1,060	1,015
Speed $v=0{,}358$ mtr./sk. Temperature of the oil 18–20° C. $\mu_1,\ \mu_2,\ldots$	0,0157	0,0135	0,0123	0,0115	0,0108	0,0103	0,0098	0,0094	0,0090	0,0088	0,0086	0,0084
Speed $v=0{,}358$ mtr./sk. Temperature of the oil 18–20° C. $\mu_1:\mu_2,\ldots$	1,164	1,100	1,070	1,065	1,050	1,051	1,043	1,044	1,034	1,023	1,023
Speed $v=0{,}572$ mtr./sk. Temperature of the oil 48–50° C. $\mu_1,\ \mu_2,\ldots$	0,0130	0,0109	0,0097	0,0090	0,0084	0,0081	0,0078	0,0075	0,0074	0,0073	0,0072	0,0070	0,0069	0,0067	0,0065
Speed $v=0{,}572$ mtr./sk. Temperature of the oil 48–50° C. $\mu_1:\mu_2,\ldots$	1,193	1,124	1,078	1,071	1,037	1,039	1,040	1,014	1,014	1,014	1,014	1,029	1,015	1,030	1,031

Table No. 71.

Speed $v = 0{,}430$ m/s. Temperature $37—39^\circ$.

$p_1, p_2$, in kg/cm²	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36
$\dfrac{1}{\sqrt{p_1}} : \dfrac{1}{\sqrt{p_2}}$, ....	1,120	1,094	1,080	1,070	1,060	1,055	1,048	1,043	1,040	1,037	1,034	1,032	1,030	1,028
$\mu_1, \mu_2$, ....	0,0118	0,0104	0,0098	0,0093	0,0090	0,0088	0,0087	0,0086	0,0083	0,0079	0,0076	0,0075	0,0074	0,0074	0,0073
$\mu_1 : \mu_2$, ....	1,334	1,061	1,054	1,033	1,023		1,012	1,012	1,036	1,051	1,040	1,013	1,014	1,000	1,014

As for the influence of speed on the coefficient of friction, here it is difficult even to speak of agreement or disagreement between the results of our experiments and the data obtained from N. Petrov’s formula, for the reason alone that an increase in speed must change both the magnitude of the sum

\[ \varepsilon + \frac{\mu'}{\lambda_1} + \frac{\mu'}{\lambda_2} \]

in the denominator of this formula, and the magnitude of the coefficient $(\mu')$ of internal friction of the liquid, which stands in the numerator. How these quantities vary with speed in our case we do not know; even if the above-mentioned sum in the denominator were taken as a constant quantity, as N. Petrov sometimes permits for very rough calculations,¹ and even if we knew the law of variation of $\mu'$ with increasing speed, there is nevertheless, as is known, still a new factor influencing the variation of $\mu$ when the speed changes: this is the thermal conductivity of the rubbing parts of the machine. N. Petrov himself pointed out the enormous influence of thermal conductivity on the character of the dependence of the coefficient of friction on speed: with very high thermal conductivities of the rubbing parts of the machine, the coefficient of friction is almost proportional to the speeds; in machines with small or even ordinary thermal conductivities, however, it may turn out that the coefficient of friction will increase with increasing speed to a lesser degree than the speed itself, or even remain unchanged.² Indeed, in our experiments it turned out that, for example, at a temperature of $18\text{–}20^\circ\mathrm{C}$ and at a pressure of $6.4\ \mathrm{kg}/\mathrm{cm}^2$, the coefficient of friction increases, with a very rough approximation, proportionally to $\sqrt{v}$, while at higher temperatures and under greater loads it increases proportionally to $\sqrt[3]{v}$ (see Table No. 72).

An increase of the coefficient of friction proportionally to $\sqrt{v}$ also follows from the experiments of Stribeck, Dettmar, Tower, and others, only over broader ranges of variation of pressures and speeds.

Finally, turning to Woodbury’s experiments and comparing their results with those of our experiments, one can say only that the values of the coefficient of friction obtained by Woodbury, and the law of its variation with the change of load on the pivot within limits lying below our minimum load, rather confirm³ the results obtained by us than contradict them.

¹ N. Petrov. Practical results of experiments, etc., p. 12.
² N. Petrov. Description and results of experiments, etc., p. 430.
³ Woodbury. Measurements of Friction etc., p. 534, Table II.

Table No. 72.

				0.143	0.215	0.286	0.358	0.430	0.501	0.572
			Speeds $v_1,\ v_2,\ldots$ in m/sec	0.143	0.215	0.286	0.358	0.430	0.501	0.572
			$\dfrac{v_2}{v_1},\ \dfrac{v_3}{v_2},\ldots$		1.50	1.33	1.25	1.20	1.165	1.14
			$\dfrac{\sqrt{v_2}}{\sqrt{v_1}},\ \dfrac{\sqrt{v_3}}{\sqrt{v_2}},\ldots$		1.225	1.153	1.118	1.095	1.077	1.068
			$\dfrac{\sqrt[3]{v_2}}{\sqrt[3]{v_1}},\ \dfrac{\sqrt[3]{v_3}}{\sqrt[3]{v_2}},\ldots$		1.145	1.100	1.080	1.060	1.050	1.045
Oil temperature 18–20° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0105	0.0134	0.0156	0.0170
Oil temperature 18–20° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		1.280	1.164	1.090
Oil temperature 18–20° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0090	0.0102	0.0110	0.0113
Oil temperature 18–20° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		1.133	1.080	1.030
Oil temperature 27–29° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0097	0.0114	0.0129	0.0141	0.0150	0.0158	0.0166
Oil temperature 27–29° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		1.173	1.132	1.093	1.064	1.053	1.051
Oil temperature 27–29° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0073	0.0078	0.0084	0.0088	0.0095	0.0100	0.0104
Oil temperature 27–29° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		1.069	1.077	1.048	1.080	1.053	1.040
Oil temperature 58–60° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0093	0.0098	0.0104	0.0112	0.0123	0.0133	0.0141
Oil temperature 58–60° C.	Pressure between rubbing surfaces	$p=6.4\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		1.054	1.061	1.077	1.098	1.081	1.060
Oil temperature 58–60° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\mu_1,\ \mu_2,\ldots$	0.0069	0.0067	0.0071	0.0074	0.0076	0.0079	0.0083
Oil temperature 58–60° C.	Pressure between rubbing surfaces	$p=13.5\ \mathrm{kg}/\mathrm{cm}^2$	$\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots$		0.971	1.060	1.042	1.027	1.040	1.051

Thus, summarizing all that has been set forth above, I would allow myself to state the following propositions, more or less reliable within the limits of the experiments performed:

1) The so-called mean coefficient of friction during the motion of the pivot, determined on the assumption of a uniform distribution of pressures over the surface of the pivot and of its independence from the sliding velocity of the rubbing elements of the pivot, depends on the mean load of the pivot, its number of revolutions, and the temperature of the lubricating oil.

2) For each definite number of revolutions of the pivot there exists a least value of the coefficient of friction at a quite definite load of the pivot (the so-called transition, or critical, load).

3) The higher the number of revolutions, the greater is the mean load at which the coefficient of friction reaches its minimum value.

4) The least values of the coefficient of friction at different numbers of revolutions are close to one another.

5) Up to the transition load the coefficient of friction decreases with increasing load; after the transition load it increases, but this increase proceeds more slowly than the decrease of the coefficient of friction before the transition load.

6) Any curve showing, for a definite number of revolutions, the dependence of the coefficient of friction on the load, up to its critical value, is situated above the same curve for a smaller number of revolutions; until this latter curve, after passing through its critical load, intersects the first curve and, in turn, lies above this latter.

7) With an increase in the temperature of the oil the coefficient of friction decreases; this decrease is considerable at small loads and lower temperatures (within the range of variation of the number of revolutions: 100–200 per minute); at high temperatures (above 50° C) and at large loads (above 30–35 kg/cm²), a certain increase in the coefficient of friction is observed with increasing temperature (at 150–200 revolutions per minute).

8) The coefficient of friction at the moment of shifting the pivot from a state of rest does not depend on the load of the pivot, but depends on the temperature of the oil: with its increase the coefficient of friction decreases.

9) The above-cited results of experiments on the friction of a pivot in a bearing, with the exception of the last proposition, are in general similar to the results of Stribeck’s experiments on the friction of a journal in a bearing, and thus it appears that the laws of friction in a pivot differ little from the laws of friction in a journal.

Noted Errors

Page	Line	Printed	Should be
10	11 from top	14	1,4
"	13 "	0,0236	0,0237
11	2 from bottom	7,4	7,45
12	7 "	0,0238	0,0237
13	2 "	64	6,4
14	11 from top	8,95	8,85
16	2 from bottom	8,6	8,5
18	1 "	0,0090	0,0089
19	6 from top	0,0095	0,0096
"	11 "	1 60	1,60
20	1 from bottom	0,0088	0,0089
21	6 from top	0,0090	0,0091
22	7 and 8 from bottom	coefficient	coefficient
27	2 from bottom	failed	did not fail
"	8 from top	41,0	42,0
34	5 "	00087	0,0087
35	3 "	in	at
41	3 "	1135	1,135
44	7 "	1090	1,090

The plate of drawings with the curves of the coefficient of friction at $T = 48\text{--}50^\circ\text{C}$ should be designated Plate IV instead of Plate V.

On Plate IX, for the dotted curve at the lowest number of revolutions, the designation $n = 50$ has been omitted.

On page 24, in table No. 49

printed:


5	0,0284	219	0,0203	0,0202
10	0,0183		,0138	0,0125
15	0,0147	0,0122	0,0107	0,0104

should read:


5	0,0284	0,0219	0,0203	0,0202
10	0,0183	0,0151	0,0138	0,0125
15	0,0147	0,0122	0,0107	0,0104

According to the article by V. O. [[unclear: surname]]

Study of friction in a flat pivot

Sheet 1.

Visible labels on the graph:

$ \mathrm{kg} $
$ \mathrm{mm} $
“Coefficient $ \lambda_b $ 1”
“Coefficient $ \lambda_b $ 2”
“Form. 48.”

Visible labels in the inset schematic:

$m$
$a_1$
$b_1$
$c_1$
$s$
$p$

Sheet 5.

To the article by V. D. Pimenov

“Study of friction in a flat pivot.”

Lubricant temperature

$T = 18\text{–}20^\circ$

Visible curve labels:

$p = 4.7$
$p = 10.4$
$p = 16.7$
$p = 24.0$
$p = 26.9$
$p = 31.5$
$p = 38.7$
$p = 42.0$
$p = 47.3$

For the article by V. F. [[unclear: surname]] “Study of Friction in a Flat Pivot.”

Coefficient of friction

Number of revolutions

Patterson and Andrews
$T = 27\text{–}29^\circ$

Stribeck

Visible curve annotations include:
$d = 4{,}3$, $d = 10{,}7$, $d = 21{,}6$, $d = 26{,}2$, $d = 37{,}5$, $d = 65{,}2$, $d = 75{,}6$, $d = 100{,}4$, $d = 157$, $d = 100$, $d = 147{,}30$, $d = 216$, $d = 355{,}2$, $d = 95{,}2$, $d = 45{,}26$, $d = 75$, $d = 15{,}56$, $d = 3{,}7$.

Test VI. Faceplate

Averaging of friction in a flat pivot

Coefficient of friction

Processed data

Revolutions

Lubricant temperature
$T = 37\text{–}39^\circ$

Plate VI

Visible curve annotations include:

$k/d$
$k_0/d$
$k_1/d$
$k_2/d$
$k_3/d$
$k_4/d$
$k_5/d$
$k_6/d$
$k_7/d$
$k_8/d$
$k_9/d$
$k_{10}/d$
$k_{11}/d$
$k_{12}/d$
$k_{13}/d$
$k_{14}/d$
$k_{15}/d$
$k_{16}/d$
$k_{17}/d$
$k_{18}/d$
$k_{19}/d$
$k_{20}/d$
$k_{21}/d$
$k_{22}/d$
$k_{23}/d$
$k_{24}/d$
$k_{25}/d$
$k_{26}/d$
$k_{27}/d$
$k_{28}/d$
$k_{29}/d$
$k_{30}/d$
$k_{31}/d$
$k_{32}/d$

Plate IV.

For the article by V. N. Pimenov, “Study of Friction in a Flat Pivot.”

Lubricant temperature
$T = 48\text{–}50^\circ$

Axes and labels in the diagram:

Vertical axis: Coefficient of friction
Horizontal axis: Angles of rotation
Marked from $0$ to $270$.

Curve annotations visible in the graph:

$p = 1\ \mathrm{kg}$
$p = 2\ \mathrm{kg}$
$p = 4\ \mathrm{kg}$
$p = 5\ \mathrm{kg}$
$p = 6\ \mathrm{kg}$
$p = 7\ \mathrm{kg}$
$p = 8\ \mathrm{kg}$
$p = 10\ \mathrm{kg}$
$p = 16\ \mathrm{kg}$
$p = 20\ \mathrm{kg}$
$p = 21\ \mathrm{kg}$
$p = 24\ \mathrm{kg}$
$p = 28\ \mathrm{kg}$
$p = 32\ \mathrm{kg}$
$p = 40\ \mathrm{kg}$
$p = 48\ \mathrm{kg}$
$p = 56\ \mathrm{kg}$
$p = 64\ \mathrm{kg}$
$p = 72\ \mathrm{kg}$
$p = 80\ \mathrm{kg}$
$p = 104\ \mathrm{kg}$

Vertical scale markings for the coefficient of friction:

$0.0005,\ 0.001,\ 0.002,\ 0.003,\ 0.010,\ 0.011,\ 0.012,\ 0.013,\ 0.014,\ 0.015,\ 0.016,\ 0.017,\ 0.018,\ 0.019,\ 0.020,\ 0.021,\ 0.022,\ 0.023$

Sheet 4.

For steel [[unclear: 58.9 g?]]. Kinematics.

Investigation of friction in a flat pivot.

$T = 58\text{–}60^\circ$

Surface polishing

Coefficient of friction

Number of revolutions

Curve annotations visible on the graph include:

$p = 47$
$p = 42$
$p = 28$
$p = [[unclear]]$
$p = 15.7$
$p = 4 \tfrac{1}{2}$
$p = 3$
$p = [[unclear]]$
$p = 28$
$p = 15.7$
$p = 4 \tfrac{1}{2}$
$p = [[unclear]]$

Graph

On the [[unclear: Bod…–Bunsen]] apparatus.

“Study of friction in a flat pivot.”

Vertical axis: Coefficient of friction.

Horizontal axis: Temperature of the lubricant.

Curve labels visible on the graph:
$p = 5{,}10$; $p = 10{,}1$; $p = 17$; $p = 24$; $p = 41$; $p = 62$; $p = 78$; $p = 120$; $p = 151$; $p = [[unclear]]$.

Sheet VI

From a steel bearing plate

Temperature of the lubricant, $T = 18\text{–}20^\circ$

Coefficient of friction

Load per unit surface $(\mathrm{kg}/\mathrm{cm}^2)$

Speed at unit surface $(\text{revolutions per minute})$

Curve labels visible on the diagram:

$n = 0.143\,\mathrm{m}$
$n = 0.215$
$n = 0.286$
$n = 0.358$

Additional curve labels visible on the lower diagram:

$n = 50$
$n = 75$
$n = 100$
$n = 125$
$n = 150$
$n = 175$

From the article by D. N. Shenevich
“Study of Friction in a Flat Pivot”

Diagram IX

Dependence of the coefficient of friction on the plane.

Temperature of the lubricant
$T = 27\text{–}29^\circ$

Temperature of the journal

Load per unit surface area of the thrust bearing $(\mathrm{kg/cm^2})$

Coefficient of friction

Number of revolutions of the thrust bearing $(\mathrm{rev.}/\mathrm{min.})$

Plate X

For hardened steel specimens.
Friction in the plane of the pivot.

Lubricant temperature
$T = 37\text{–}39^\circ$

Coefficient of friction

Specific pressure per unit surface area
$(\mathrm{kgf}/\mathrm{cm}^2)$

Curve labels visible on the graph include:

$v = 0{,}143$
$v = 0{,}096$
$v = 0{,}058$
$v = 0{,}040$
$v = 0{,}029$
$v = 0{,}023$
$v = 0{,}019$

and

$n = 200$
$n = 155$
$n = 100$
$n = 75$
$n = 52$
$n = 40$
$n = 32$
$n = 24$

Coefficient of friction

Axis label: number of revolutions per minute.

Medium temperature
$T = 48\text{–}50^\circ$

For [[unclear: type/name of steel]], [[unclear: name]],
friction in a flat pivot.

Curve labels include:
$\nu = 0.1245$, $\nu = 0.10$, $\nu = 0.075$, $\nu = 0.055$, $\nu = 0.05$, $\nu = 0.04$, $\nu = 0.035$, $\nu = 0.032$, $\nu = 0.029$

Additional curve labels include:
$m = 50$, $m = 45$, $m = 40$, $m = 33$, $m = [[unclear]]$, $m = 35$, $m = 25$, $m = 21$

Plate XII

Dependence of the coefficient of friction in a flat pivot on the lubricant temperature.

$I = 58\text{–}60^\circ$.

Scale: [[unclear: handwritten note, appears to read “889/85 enlarged”]]

Coefficients of friction

Heating of the bearing surface of the pivot (figures in the photograph)

Curve labels:

$n = 0.215\ \mathrm{m/s}$
$n = 0.286$
$n = 0.258$
$n = 0.230$
$n = 0.201$
$n = 0.172$
$n = 0.145$

Additional curve labels:

$n = 1.25$
$n = 1.00$
$n = 0.25$
$n = 1.175$
$n = 1.15$
$n = 1.00$
$n = 0.780$

[[Chart]] Dependence of [[unclear: initials]] Pinetin. “Study of Friction in a Flat Pivot.”

Plate XII

Vertical scale: Coefficient of friction

Horizontal scale: Time of operation, in minutes

Visible groups on the chart:

Number of revolutions $n = 100$ per minute
Number of revolutions $n = 150$ per minute
Number of revolutions $n = 200$ per minute

Visible curve annotations include loads/pressures marked:

$p = 0.5\ \mathrm{kg/cm^2}$
$p = 1\ \mathrm{kg/cm^2}$
$p = 1.5\ \mathrm{kg/cm^2}$
$p = 2\ \mathrm{kg/cm^2}$
$p = 2.5\ \mathrm{kg/cm^2}$
$p = 3\ \mathrm{kg/cm^2}$
$p = 3.5\ \mathrm{kg/cm^2}$
$p = 4\ \mathrm{kg/cm^2}$
$p = 5\ \mathrm{kg/cm^2}$
$p = 10\ \mathrm{kg/cm^2}$
$p = 15\ \mathrm{kg/cm^2}$

Visible vertical-axis tick labels:

$0.008,\ 0.009,\ 0.010,\ 0.011,\ 0.012,\ 0.013,\ 0.014,\ 0.015,\ 0.016,\ 0.017,\ 0.018,\ 0.019,\ 0.020,\ 0.021,\ 0.022,\ 0.023,\ 0.024,\ 0.025,\ 0.026,\ 0.027,\ 0.028,\ 0.029$

Visible horizontal-axis tick labels, repeated across the separate plotted intervals:

$45,\ 50,\ 55,\ 60,\ 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45,\ 50,\ 55,\ 60,\ 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45,\ 50,\ 55,\ 60,\ 65$

Plate XIV

The dependence of P. F. Steinheil’s coefficient of friction on the bearing pressure.

Coefficient of friction

Steel pivot without flange $C_1$.
Speed $v = 0.14\ \mathrm{m/s}$.

Curve labels visible on the plot:

$p = 5.5$
$p = 55$
$p = 47.5$
$p = 45$
$p = 42.5$
$p = 40$
$p = 35$
$p = 32.5$
$p = 30$
$p = 25$
$p = 21.5$
$p = 14.7$
$p = 50$
$p = 47$
$p = 46$
$p = 35.5$
$p = 30.5$
$p = 25.8$

Steel pivot without flange $C_1$.
Speed $v = 0.26\ \mathrm{m/s}$.

Curve labels visible on the plot:

$p = 54.9$
$p = 65.6$
$p = 55.5$
$p = 47.8$
$p = 41.8$
$p = 36.6$
$p = 30.5$
$p = 25.8$
$p = 66.4$
$p = 55$
$p = 47$
$p = 42.5$
$p = 38.5$
$p = 35.6$
$p = 25$
$p = 24.5$
$p = 14.5$

Steel pivot without flange $C$.
Speed $v = 0.52\ \mathrm{m/s}$.

Curve labels visible on the plot:

$p = 54.9$
$p = 14.5$
$p = 9.5$
$p = 8.5$
$p = 7.5$
$p = 6.5$
$p = 5.5$
$p = 4.5$
$p = 54$
$p = 17.5$
$p = 10.5$
$p = 9.4$
$p = 8.4$
$p = 7.6$
$p = 6.6$
$p = 5.5$
$p = 4.5$

Numerical scale visible on the ordinate:
$0.000,\ 0.002,\ 0.004,\ 0.006,\ 0.008,\ 0.010,\ 0.012,\ 0.014,\ 0.016,\ 0.018,\ 0.020,\ 0.022,\ 0.024,\ 0.026$.

Numerical scale visible on the abscissa:
$0,\ 5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45,\ 50,\ 55,\ 60,\ 65,\ 0,\ 5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45,\ 50,\ 55,\ 60$.

I. Lamanskii. Investigation of lubricating oils, 1884. ↩↩↩↩↩↩↩↩↩↩↩↩
N. Petrov. Description and results of experiments etc., p. 417 ff.; and also: Practical results of experiments etc., p. 12. ↩↩↩↩↩
Ibid, S. 1345—1346. ↩↩↩
N. Petrov. Description and results, etc. ↩↩
Fr. Masi. Le nuove vedute, etc. ↩↩
John Goodman. “Recent researches in friction.” Minutes of Proceedings of the Institution of Civil Engineers. Vol. LXXXV. Session 1885—1886. Part III. ↩↩
Couette. “Études sur le frottement des liquides.” Ann. de Chim. et de Physique. 1890; see also: Petroff. “Sur le frottement,” etc. ↩↩
Dettmar. Neue Versuche über Lagerreibung nebst neuer Berechnungsmethode derselben. Dinglers polyt. Journal, Bd. 315, 1900. ↩
Stribeck. Die wesentlichen Eigenschaften der Gleit-und Rollenlager. Zeit. d. Verein. d. Ingen., 1902, p. 341 ff. ↩
Lasche. Die Reibungsverhältnisse etc., p. 1883. ↩

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[v1] 1911

Study of Friction in a Flat Pivot Bearing: With 14 Tables

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Study of Friction in a Flat Pivot.

Table No. 6.

Table No. 7.

Table No. 8.

Table No. 9.

Table No. 14.

Table No. 15.

Table No. 16.

Table No. 17.

Table No. 22.

Table No. 23.

Table No. 24.

Table No. 25.

Conclusions from the results of the experiments.

Table No. 52.

Table No. 53.

Table No. 54.

Table No. 55.

Table No. 56.

Table No. 57.

Table No. 58.

Table No. 64.

Table No. 65.

Table No. 66.

Table No. 67.

Table No. 68.

Noted Errors

Temperature of the lubricant, \(T = 18\text{–}20^\circ\)

Plate X

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Study of Friction in a Flat Pivot Bearing: With 14 Tables

\(n\).	60	70	82	104	135	165
\(R\) in m/m.	1.4	1.6	1.7	1.8	1.9	2.0
\(S\) in kg.	1.7	1.9	2.1	2.2	2.3	2.45
\(\mu\).	0.0192	0.0215	0.0236	0.0249	0.0260	0.0274


Load of the pivot \(4.7\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(4.7\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(4.7\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(4.7\ \mathrm{kg}/\mathrm{cm}^2\).
\(T^\circ\mathrm{C}\).	23	27	35
\(R\) in m/m.	8,2	7.8	5,0
\(S\) in kg.	10,3	9,7	6.2
\(\mu\).	0.1163	0,1095	0,0700


Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).	Load of the pivot \(10.4\ \mathrm{kg}/\mathrm{cm}^2\).
\(T^\circ\mathrm{C}\).	17	23	30	33	49	53
\(R\) in m/m.	15,1	14,8	13,5	12,4	12,8	12,9
\(S\) in kg.	19,8	19,4	17,6	16,1	16,7	16,8
\(\mu\).	0,1010	0,0990	0,0900	0,0822	0,0852	0,0858

Pressure between rubbing planes \(x\), in kg/cm²	6,4	7,0	13,5	14,0	21,5	22,0	25,0	24,5	29,5	36,0	40,5	40,0	47,0
Coefficient of friction \((\mu_1)\), calculated on the basis of the pressure on 3 boiler sections	0,0129		0,0084		0,0079		0,0075		0,0083	0,0087	0,0090		0,0093
Coefficient of friction \((\mu_2)\), calculated on the basis of the pressure on 5 boiler sections		0,0124		0,0082		0,0076		0,0076	0,0082	0,0087		0,0094	0,0097
Difference in % between the coefficients \(\mu_1\) and \(\mu_2\), relative to \(\mu_2\)	+4,04%		+2,44%		+3,95%		−1,32%		+1,22%	00%		−4,26%	−4,13%

	7	9	11	13	15	17	19	21	23	25	27	29
\(p_1, p_2, \ldots\) in kg/cm\(^2\)	7	9	11	13	15	17	19	21	23	25	27	29
\(\mu_1, \mu_2, \ldots\)	0.0148	0.0123	0.0109	0.0101	0.0095	0.0091	0.0087	0.0084	0.0081	0.0079	0.0076	0.0075
\(\dfrac{1}{\sqrt{p_1}} : \dfrac{1}{\sqrt{p_2}} = \ldots\)	1.135	1.105	1.087	1.072	1.065	1.058	1.052	1.048	1.042	1.038	1.034
\(\mu_1 : \mu_2, \ldots\)	1.203	1.128	1.080	1.063	1.043	1.046	1.036	1.036	1.026	1.026	1.014

				0.143	0.215	0.286	0.358	0.430	0.501	0.572
			Speeds \(v_1,\ v_2,\ldots\) in m/sec	0.143	0.215	0.286	0.358	0.430	0.501	0.572
			\(\dfrac{v_2}{v_1},\ \dfrac{v_3}{v_2},\ldots\)		1.50	1.33	1.25	1.20	1.165	1.14
			\(\dfrac{\sqrt{v_2}}{\sqrt{v_1}},\ \dfrac{\sqrt{v_3}}{\sqrt{v_2}},\ldots\)		1.225	1.153	1.118	1.095	1.077	1.068
			\(\dfrac{\sqrt[3]{v_2}}{\sqrt[3]{v_1}},\ \dfrac{\sqrt[3]{v_3}}{\sqrt[3]{v_2}},\ldots\)		1.145	1.100	1.080	1.060	1.050	1.045
Oil temperature 18–20° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0105	0.0134	0.0156	0.0170
Oil temperature 18–20° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		1.280	1.164	1.090
Oil temperature 18–20° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0090	0.0102	0.0110	0.0113
Oil temperature 18–20° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		1.133	1.080	1.030
Oil temperature 27–29° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0097	0.0114	0.0129	0.0141	0.0150	0.0158	0.0166
Oil temperature 27–29° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		1.173	1.132	1.093	1.064	1.053	1.051
Oil temperature 27–29° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0073	0.0078	0.0084	0.0088	0.0095	0.0100	0.0104
Oil temperature 27–29° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		1.069	1.077	1.048	1.080	1.053	1.040
Oil temperature 58–60° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0093	0.0098	0.0104	0.0112	0.0123	0.0133	0.0141
Oil temperature 58–60° C.	Pressure between rubbing surfaces	\(p=6.4\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		1.054	1.061	1.077	1.098	1.081	1.060
Oil temperature 58–60° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\mu_1,\ \mu_2,\ldots\)	0.0069	0.0067	0.0071	0.0074	0.0076	0.0079	0.0083
Oil temperature 58–60° C.	Pressure between rubbing surfaces	\(p=13.5\ \mathrm{kg}/\mathrm{cm}^2\)	\(\dfrac{\mu_2}{\mu_1},\ \dfrac{\mu_3}{\mu_2},\ldots\)		0.971	1.060	1.042	1.027	1.040	1.051