Proceedings of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 14. 1909. No. 2.

V.

V. N. Pinegin.

Experimental Study of the Distribution of Pressures on a Flat Bearing Surface.

With three plates of drawings.

1–34.

Experimental Study of the Distribution of Pressures on a Flat Bearing Surface

V. N. Pinegin.

The question of the distribution of pressure in a flat bearing surface, if one does not count the assumption of a uniform distribution, adopted for entirely new bearing surfaces, has until now been treated in the technical literature, in most cases, on the basis of the theory of the operation of a bearing surface given by Th. Reye¹ and, almost simultaneously, by Grashof².

As is known, this theory is based, on the one hand, on Reye’s and Grashof’s assumption that the coefficient of friction is independent of pressure and velocity, and, on the other hand, on the assumptions that: a) on all elements \((df)\) of the rubbing surface of the bearing surface, the same part of the total work of friction is expended in the abrasion of the particles of the bearing surface and of the step bearing, and moreover proportional to \(p y\), where \(p\) is the normal pressure on an element of the rubbing surface, and \(y\) is the distance of this element from the axis of rotation; b) the ratio of the normal wear of the bearing surface and of the step bearing at any place is a constant quantity; and c) the resultant wear on all elements of the bearing surface in the direction of the total load \(P\) of the bearing surface must be the same.

As a consequence of these assumptions, we obtain the following obvious equality

\[ \frac{p y}{\operatorname{Cos}\varphi} = Const, \]

where \(\varphi\) is the angle made by the normal to the given element of the bearing-surface surface with the axis of its rotation. The equality written above and the still more obvious relation

\[ P = \int p \operatorname{Cos}\varphi \, df, \]

where the integral extends over the entire rubbing surface of the bearing plate, give the following expression for the pressure on an element of the plate:

\[ p=\frac{P\cos\varphi}{y\int \frac{\cos^2\varphi}{y}\,df}, \]

when applied to a flat plate this expression takes the form, if by \(r_2\) we denote the radius of the outer circumference of the plate:

\[ p=\frac{P}{2\pi r_2 y}, \]

and this shows that the distribution of pressure in the plate, in the direction of its radius, may be represented by an equilateral hyperbola, for which the axis of rotation of the plate serves as an asymptote; at the center of the plate the pressure must be infinitely great. For an annular plate, in an analogous manner we find

\[ p=\frac{P}{2\pi(r_2-r_1)y}, \]

i.e., here also the pressure in the direction of the radius of the plate is distributed according to a hyperbola, increasing toward the edge of the inner annular cut-out.

Experiments of Bach.

Meanwhile, Bach’s experiments¹ in 1885 “on the abrasion of bodies rotating about their own axis and resting on a cast-iron plate” proved that in reality the matter is not quite so: it turned out that, even with good lubrication of the rubbing bodies, the parts lying farther from the axis first begin to wear away; the rubbing surface of the body assumes a convex form. As a consequence of this, the pressure in the central part of the rubbing bodies increases; their wear there becomes greater than in other places, the curvature of the surface of the plate decreases, but as soon as its surface becomes more or less flat, the parts more distant from the axis again begin to wear more, the plate again assumes a convex form, and so on.

Such a change in the abrasion of the parts of the plate gives grounds to suppose that the distribution of pressures over the plate must also change together with the wear: in the initial phase of abrasion of the particles of the plate, when it extends to layers farther removed from the axis, the pressure must likewise increase with distance from the axis. With the passage of time, when the wear of the plate begins from the edges to approach the center, the pressure must increase toward the center; but, as the abrasion

of the central layers, again to diminish here; thus, the maximum pressure should, over time, move from the edges of the bearing surface toward the center and back again.

This is all that is known at the present time about the distribution of pressure in a flat bearing surface. I know of no direct experiments on this question.

Tower, to whose experiments¹ we owe a more or less correct conception of the distribution of pressures in journals, for some reason did not find it necessary to undertake the solution of the analogous question for a bearing surface, although he did determine the coefficient of friction in a bearing surface.

Likewise, other persons² who investigated friction in a bearing surface did not dwell on the distribution of pressure upon it.

Meanwhile, the solution of this question is not only of purely scientific interest, but also has practical significance, in view of the ever-increasing spread, in turbine construction, of flat (chiefly annular) bearing surfaces with ever greater and greater mean pressures; for it is evident that knowledge of the distribution of pressure in bearing surfaces will facilitate their very design with respect to the introduction of lubricant, cooling, and so forth.

In the article offered to the reader are set forth the results of an attempt, made by me as early as 1907 during a foreign assignment, to solve this question by a purely experimental route; at the same time, observations were also made on the magnitude of friction in a bearing surface.

I am, of course, far from thinking that this question has been fully resolved by me; on the contrary, I consider that much work still needs to be done in this direction, and on a broader scale than I was able to carry out.

The fact is that at the very first steps of my work I had to encounter great difficulties, the overcoming of which required above all time; and I had little of it, owing to the expiration of the term of my assignment. Thus, in the apparatus itself for investigating the operation of the bearing surface, shortcomings were encountered that prevented increasing the load of the bearing surface above \(47\ \mathrm{kg}/\mathrm{cm}^{2}\), and the number

to increase the shaft speed above 300 revolutions per minute; nor was it possible to reconstruct the apparatus, for the reason stated above. The experiments themselves required prolonged operation of the apparatus, in order that the gradual course of wear of the bearing might be followed as far as possible; I, however, could devote only two months to the experiments (about 6 months had been spent on designing the apparatus and having it made at the factory), which I consider insufficient.

Therefore I would regard my task as accomplished if I were merely able to show that the question considered here is solvable by the method I have undertaken, and to give at least a few strokes of the picture which the distribution of pressures in a bearing actually presents; the detailed, systematic investigation of this question I leave to another, more favorable time.

To show that the problem is in fact solved by the proposed method is all the more important because this method initially aroused, among those around me, a whole series of objections as to its unsuitability under the given conditions.

The fact is that the method which I used to investigate the distribution of pressures at various points of the bearing is nothing other than Tower’s method, used by him in studying the distribution of pressures in journals; and I mainly had to hear the objection that, in investigating pressures in journals, this method was appropriate and could yield real results, whereas, when applied to a bearing, it could not yield any positive results, since in this case there are not those favorable conditions by virtue of which, in journals, the oil was driven through the tubes into the manometers and made them indicate the pressures at the corresponding places.

If, nevertheless, I was able to carry out the experiments described below, it was only thanks to the kindness of Prof. Eugen Meyer, in whose laboratory the present experiments were performed, and who, despite his own initial doubts as to the possibility of determining pressures by the proposed method, agreed to provide the means for carrying out these experiments.

Therefore, I take this opportunity here to express my sincere gratitude to Prof. Eugen Meyer for the assistance and support he rendered me in this case, most kindly and to the highest degree.

In undertaking the study of the distribution of pressures in the bearing and, incidentally, of friction, I first of all had to take care of a suitable apparatus. Such an apparatus, with the kind assistance of Prof. Eugen Meyer, and also of his assistant Dr. Schmoll von Eisenwerth, was designed by me.

Manufactured at the works of the Berliner Aktiengesellschaft für Eisengisserei und Maschinenfabrikation in Charlottenburg, this apparatus had, in general outline, the following arrangement.

In the cast-iron frame A (Plate I) is placed a vertical shaft 11, carrying a pulley 1 fitted upon it, for transmitting rotation to the shaft by a belt from a motor. Shaft 11, the lower end surface of which also serves as the bearing surface, rests upon a system of plates connected with one another in a known manner (see below), playing the role of a step bearing. The latter rest upon three rather long cylindrical rods 30, having hardened conical ends placed in likewise hardened recesses of plates 28 and 29. This construction was adopted on the advice of Prof. Meyer, with the aim of simplifying the measurement of friction in the step bearing, as will be seen below.

On shaft 11, by means of lever R, one end of which bears against prism 22, fastened by bolts 21 to the frame of the apparatus, while the other end is loaded with weights placed on a board suspended from it by chains, a load is produced which is transmitted to the bearing surface.

Let us now turn to the details of the construction of the apparatus.

As can be seen (see Plate II), in order to achieve greater uniformity in the distribution of pressures over the bearing surface, cylinder 5 rests with its concave spherical surface upon the corresponding convex surface of plate 28, which rests on three rods 30; to prevent cylinder 5 from making excessively large movements over plate 28, there is, as can be seen in the drawing, a pin; on cylinder 5 is placed plate 9, fastened by screws to the supporting step-bearing plate 3, which has at its center a recess O of 12 m/m diameter. Upon this step-bearing plate 3 rests the bearing plate 2, tightly fitted onto the end of shaft 11 and compelled to rotate together with it by means of a special pin. Such an arrangement makes it possible to change the supporting plates 2 and 3, to make them of any desired material, and, consequently, to investigate friction as dependent on the material of the bearing surface. Plate 2 also has at its center a recess of 12 m/m diameter. To plate 3, by means of screws, is attached bath 4, into which oil is poured for lubricating the rubbing surfaces.

In the upper part of cylinder 5 there is made a cross-shaped cut-out k, k, the purpose of which will be clear from what follows.

The method for determining the pressures at various points of the bearing surface was based on the supposition that if, at the considered points of the bearing surface, holes are drilled and connected by tubes with ...

with manometers, then the oil supplied for lubricating the bearing surface and distributed over it in a thin layer will be pressed into the holes and, either through the tubes into the manometers, will cause the latter to indicate the pressures at the corresponding points of the bearing surface.

For this purpose, in the lead plates 3 and 9 joined together, I ordered six holes to be drilled, exactly above one of the cutouts \(k\) in cylinder 5. These holes were arranged along the diameter of the bearing surface, symmetrically with respect to the middle recess \(O\) in plate 3, and, as far as possible, at equal distances from one another (more precisely: the distances of the centers of the holes from the outer edge of the bearing surface were respectively 10, 22, 34 m/m). The holes were drilled with a diameter of 4 m/m. Copper tubes of the same outside diameter and with an inside diameter of 2.5 m/m were tightly inserted into them.

In order to obtain a completely tight and strong connection of the copper tubes with the plates, the procedure was as follows: plates 3 and 9 were separated; the holes in plate 9, on the side adjoining plate 3, were slightly countersunk; then the copper tubes were inserted into the holes of plate 9 and projected from the side where the countersinking was done by an amount slightly greater than the thickness of plate 3; after this, molten solder was poured into the countersunk recesses. After the solder had cooled and its excess on the tubes had been removed, plate 3 was put on with its holes; by light blows of a wooden hammer on plate 3, plates 3 and 9 were made to come tightly together; then they were screwed together, the projecting ends of the tubes were cut off, scraped flush, and their holes were rolled with a thick steel needle. The joint proved so tight that later, when the manometers indicated pressures even up to 48 atmospheres, no seepage of oil around the tubes was observed.

The long ends of the copper tubes emerging from plate 9 were at once bent near the plate at an angle and, after the lead plates 3 and 9 had been placed on cylinder 5, were led out through the cutouts \(k, k\) to the outside.

Initially it was intended to lead these tubes, which were about one meter long, to the distributing board shown in Fig. 1, Plate III, and to fasten them in it, in the corresponding holes, by the method indicated in the drawing; these holes were connected by means of channels with one wider channel ending in a threaded hole for screwing in a manometer. The connecting channels, as may be seen, are provided with cocks and, consequently, at any moment any hole on the surface of the bearing surface can be connected, by opening the corresponding cock, with—

a manometer, which would indicate the pressure above the opening under consideration. But, in view of the rather considerable cost of such a distributing board, it was necessary to abandon it and to make use of the three manometers available in the laboratory for directly measuring the pressures over three openings at once; for this it was only necessary to connect the copper tubes tightly to the manometers. To make such a connection, of course, involved no great difficulty, and it was done in the usual way: round iron plates, of diameter slightly smaller than the outside diameter of the manometer nipples, were fitted and soldered to the ends of the copper tubes; before the plates were soldered to the tubes, nuts with an internal thread exactly corresponding to the external thread of the manometer nipples were fitted onto the latter; by then screwing the nuts onto the manometer nipples, after first placing leather rings under the plates at the edges of the nipples, and tightening them, we obtained a very tight joint.

Since we had at our disposal only three manometers, it was proposed at first to observe the pressures on the plate only on one half of it; after the whole cycle of observations on this half, the manometers were to be moved to the other side and the observations made there.

The loading of the plate was carried out in the following manner: on the vertical shaft 11 there was placed from above a cast-iron washer 25 with an annular hardened recess; in this recess were placed balls 26, on which rested, by means of an annular recess corresponding to the balls, the washer 24. On this washer 24, which had on top a spherical concave surface, there was placed a conical washer 12 having a corresponding convex surface and an inserted hardened prism H. On this last prism rested the lever R, the short arm of which abutted against the crosspiece 22, fitted by its holes onto the bolts 21 screwed into the frame of the apparatus; the crosspiece was fastened on the bolts with nuts. To the long end of the lever there was suspended, by means of a clamp and a hook on chains, a board on which loads of various magnitudes could be placed at will and thus the required load obtained. All the bearing points on the lever were provided with hardened prisms, analogous to prism H, and with corresponding hardened stops.

The dimensions of the lever, according to the original calculation (the ratio of the lever arms was chosen as \(\frac{1890}{90} = 21\)), made it possible to apply to the shaft, and consequently to the plate, a load of 3000 kg.

Transmission of motion to the apparatus.

Owing to the lack of space in the laboratory and the impossibility of arranging a direct transmission from the motor to shaft 11, it was necessary to resort to the following kind of transmission. Opposite our apparatus, at the opposite wall of the room, stood a Martens machine for the investigation of internal friction in oils; this machine, which had a horizontal shaft, was driven by a driving shaft located near the ceiling along the wall and driven, in turn, by a ten-horsepower motor. In view of the complete impossibility of arranging things otherwise, it was necessary to mount an additional pulley on the shaft of the Martens machine and to connect this latter by a crossed belt with pulley 1, fitted on shaft 11 and secured to it by means of a key and a set screw.

Thus, the transmission for imparting rotation to the shaft of our apparatus ran from the motor to the driving shaft, from the latter to the Martens machine, and from this latter already to shaft 11.

The distance between the axis of shaft 11 and the center of the pulley on the Martens machine was equal to 3.75 mtr.

Lubrication of the apparatus.

The lubrication of the moving parts of the apparatus was carried out in the following manner. Onto bolts 21 a tin box, with a capacity of about 2–3 liters, was screwed from above, into which oil was poured; the box was provided with a tap screwed to a copper tube; through this tube the oil entered the channel of the conical cup 12, whence, through a new channel in cup 24, it poured out onto balls 26, lubricated them, and then went further into the fine channels drilled in bearing shell 7 on the surface of its contact with shaft 11; these channels, serving for better lubrication of the shaft, had for the same purpose been drilled along a helical line, but were not carried through to the end of bearing shell 7. The oil, on leaving bearing shell 7, entered the cast-iron bath 32; from here it had originally been intended to pass it through tube 34 for lubricating the neck of the shaft in bearing shell 8, and then directly into bath 4, surrounding the bearing itself; but the very first experiments showed that the oil from bath 32 approaches bearing shell 8 strongly heated, as a result of which the latter, already itself having a tendency to heat up at a large number of revolutions, becomes extremely hot; moreover, the oil leaves bath 32 already fairly contaminated. It is clear that from bearing shell 8 it would leave still more heated and would thereby increase the already existing tendency of the bearing to heat up.

Therefore, after several attempts, it was necessary to abandon the designed circulation of the oil and to stop lubricating the neck (8) and filling bath 4 with oil issuing from bath 32. The first

from these operations was achieved by conducting a special copper tube from the upper reservoir directly to the neck (8); in the same bath 4 it was necessary to arrange a constant inflow of oil from the same reservoir through a third tube, drawing off, of course, the same quantity of oil from it by means of a special cock not shown in the drawing.

In exactly the same way, at first, in order to lubricate the surface of the foot it was intended to make on the surface of the bearing plate only one channel from the outer edge to the central recess O; but in the very first experiment it turned out that, at a load averaging only 4.5 kg/cm² and at a speed of only 190 revolutions per minute, one hour of operation of the foot was sufficient to raise the temperature in the bath from 20° to 61°; it therefore became quite evident that the number of channels had to be increased, which was done at once: four grooves were made in the directions of mutually perpendicular diameters. Their beneficial influence was not slow to show itself in a strong retardation of the rise of temperature in the bath: with the same load, at 200 revolutions per minute, during two hours’ operation of the foot, the temperature in the bath rose from 20° C only to 48° C. The grooves were made 3.5 mm wide and about 2 mm deep.

Measurement of the number of revolutions of the shaft was carried out by means of Dr. Horn’s tachometer, convenient in the respect that it makes it possible to know this number of revolutions at any given moment.

Measurement of the number of revolutions of the shaft.

Owing to friction on the surface of the foot, the plate 3, together with the other plates lying below it, tends always to rotate in the direction of rotation of the shaft 11. To prevent it from making large turns, a spindle 14 (Plate II) is screwed into the cylinder 5; the four-sided end of this spindle is placed between the cheeks of the grip 16; into the cheeks of the latter are screwed the screws b, b, by which, at will, the space for the oscillations of the spindle 14 may be narrowed more or less. Usually we left, for these oscillations, on either side of the spindle a clearance about 1 mm wide.

Measurement of friction in the foot.

In this way, the cylinder 5 and, consequently, the plate 3 can turn only through the very smallest angles about their mean position, corresponding to the oscillations of the spindle 14.

Into the same cylinder 5, at an angle of 90° to the spindle 14, are screwed two spindles 13, having at their ends two small prisms p each; earrings 15 are put on these small prisms; the hooks of spiral springs s catch onto these earrings, while the hooks of the other ends of the springs catch onto the eyes of bolts 18; parts of the rods of the latter have the form of four-sided prisms running in the

in the corresponding holes of the clamps 17, fitted onto the cylindrical rods 19, which are secured in special sockets of the frame by means of the clamping screws 20.

Onto the threaded end of the bolts 18 are screwed the nuts 10; by turning them in one direction or the other, one can move the bolts 18 to one side or the other and, consequently, tighten or loosen the springs \(s, s\). When measuring the friction in the heel, we screw both nuts 10 in so far as to turn the cylinder 5 from the position in which it is placed by the friction on the heel (the spindle 14 being pressed, as a result of this friction, against one or another of the screws \(b, b\)) into the position in which the spindle 14 would stand midway between the screws \(b, b\). In doing this, of course, the springs \(s, s\) will be stretched somewhat, in accordance with the magnitude of the frictional force, and this increase in the length of the springs will serve as the means of measuring the magnitude of the friction on the heel; thus, in the matter itself, it is only necessary first to test the springs and graduate their elongation according to the magnitude of the stretching force, so that afterward, from the elongation of the spring, one may know the force stretching them.

All this, of course, will be correct only in the case where we assume that, when the springs \(s, s\) are stretched, no other force besides the frictional force on the surface of the heel prevents the cylinder 5 and the plate 3 from rotating.

It is true that such forces do exist, but it can be shown that, in comparison with the frictional force on the surface of the heel, they are very small and may be neglected. These forces are the force of friction on the balls surrounding the cylinder 5, and the force of resistance of the three supporting rods 30 to their being turned as a result of the rotation of the plates resting upon them.

As regards the first of these, then, of course, a priori one may say that under the existing conditions it is very small in comparison with the frictional force in the heel; as regards the second, the same can be proved on the basis of the following considerations.

If we denote by \(a\) the distance from the center of the plate 3 to the place where the spindle 14 touches the screw \(b\) (see Plate III, drawing 3), and the clearance between the end of the spindle 14 and the same screw by \(x\), then the angle of rotation of the cylinder 5 together with the plate 3, owing to friction, from the position in which the supporting rods 30 stood vertically, to the greatest possible position for them, will be expressed as \(\alpha = \dfrac{x}{a}\). In this rotation of the plate 3 the upper end of the supporting rod \(M\) is displaced to \(M'\), and \(MM' = r\alpha\), if \(r = OM'\) is the distance of the support—

the rod from the center of the bearing plate. The bearing rod itself is thereby deflected from its vertical position by an angle \(\beta\) (see Table III, drawing 4), where

\[ \beta=\frac{MM'}{l}=\frac{r\alpha}{l}, \]

here \(l\) is the length of the bearing rod. Owing to the latter circumstance, the bearing plate (block 3) is lowered by

\[ h=l(1-\operatorname{Cos}\beta). \]

If we now denote by \(P\) the load on the heel, then, on the basis of the principle of virtual displacements, applying our reasoning at once to the three rods, we obtain:

\[ Pdh=Had\alpha; \]

\[ dh=d[l(1-\operatorname{Cos}\beta)]=l\operatorname{Sin}\beta\,d\beta=\sim l\beta d\beta=l\frac{r^2}{l^2}\alpha d\alpha \]

and consequently

\[ Pl\frac{r^2}{l^2}\alpha d\alpha=Had\alpha, \]

whence

\[ H=P\frac{r^2}{l}\frac{\alpha}{a}. \]

And since \(\alpha=\frac{x}{a}\), then

\[ H=\frac{P}{l}\frac{r^2}{a^2}x. \]

For \(x=0\), i.e., if the bearing rods stand perfectly vertically, we shall have \(H=0\); taking, as in our case, \(x=1\) m/m, \(l=500\) m/m, \(P=3000\) kg., \(r=30\) and \(a=100\) m/m, we obtain

\[ H=\frac{3000}{500}\frac{30.30}{100^2}=0.54\ \text{kg}. \]

Thus, the force necessary in order to rotate cylinder 5 with blocks 3, 9, 28, when the resistance to this rotation is only from the bearing rods, even under the greatest heel load possible for us, is exceedingly small in comparison with the frictional force which it will be necessary to apply to rotate the same parts when friction acts on the surface of the heel; therefore, in the study of friction on the heel, the first force, as we said above, may be neglected.

Leaving aside for the present the question of friction in the heel, which will form the subject of another article, I shall now concern myself exclusively with the distribution of pressures over the heel.

The first point on which we must dwell here is the question of the average pressure on the heel. A more or less exact value

…of the mean pressure is extremely important for us, since with it we must constantly compare our observational results on the pressure at various points of the heel and draw conclusions as to the reliability of these observations. Meanwhile, this question is not a simple one for us, and here is why.

Sectorial bevels on the surface of the heel.

Obviously, nothing would be easier than to determine this mean pressure if the surface of the heel were a plane; true, on this surface there are channels, and in the center there is a recess, to which the full load obviously does not extend, but it would only be necessary to exclude from the total area of the heel the area of these recesses (the area of the central recess, with its diameter of 12 m/m, was 1.13 cm, and the area of all the channels, with the width of two of them 3.8 m/m and of the other two 3.5 m/m, and with a total length of 43.5 m/m, was 3.16 cm, and consequently the sum of these areas was \(f_1 = 4.29\) cm), and we would obtain the actual area of the heel over which the load is distributed.

But in reality the matter was not quite so. When the machine was first put into operation, I was compelled to note the absence of pressure readings on the manometers: under the given conditions the oil in fact was not being driven into the holes drilled in the surface of the heel. In view of this, I decided to make on the thrust-bearing plate 3 sectorial bevels with a gradual inclination toward the oil grooves, as shown in the appended drawing (see Plate III, drawing 2). The lines \(AC\) and \(A'C'\) of the bevels were drawn not quite along the radius, in order somewhat to restrain the oil from radial outflow under the influence of centrifugal force. The inclination of the surfaces \(ABC\) and \(A'B'C'\) toward the grooves was made very slight: the lowering of the surface along the lines \(AB\) and \(A'B'\) did not exceed 1/4 m/m. It was assumed that, with such a beveling of the surface of the heel, when the upper plate 2 rotated in the direction of the arrow \(s\), the oil would be carried from the oil grooves along these bevels to the holes in sufficient quantity and be pressed into them.

The expectations were fully justified: in the very first test with such a thrust-bearing surface, the manometers showed quite definite pressures prevailing over the holes.

Calculation of the mean (per cm.\(^2\)) loads of the heel.

Of course, this was still not enough: it was necessary to be convinced that these readings really represented pressures at the given points of the heel; and for this it was necessary to know the mean load of the heel in the given test. The calculation of this mean load was now complicated by the existence of sectorial bevels on the surface of the heel, over which it was hardly possible to assume the same …

distribution of pressures, as on the other flat parts of it. It was necessary to make some assumption; the simplest and, at the same time, probable assumption was that the pressure, beginning from the line \(AB\) to the line \(AC\), increased along a parabola from zero to that which prevailed over the openings. Assuming this, in determining the mean pressure on the rubbing surface of the bearing, we had to exclude from its area, besides the areas of the central recess and the channels, yet another part equal to two-thirds of the area of one sector \((f_2=4.2\ \mathrm{cm.}^2)\).

Thus, the actual area of the bearing on which the pressure could be assumed uniformly distributed proved to be

\[ F=\frac{\pi D^2}{4}-f_1-f_2=68.5\ \mathrm{cm.}^2. \]

Knowing now the weight of shaft 11 itself with the disks, balls, and pulley 1 mounted on it (55.67 kg), the weight of the lever (29.15 kg), and of the clamp (1.82 kg) for suspending the board from it, and determining the position of the center of gravity of the lever (at a distance of 68.4 cm from its point of support on shaft 11), it was possible to calculate the mean load on the bearing from the weight of shaft 11 itself with the disks and from the pressure of the lever upon it; it proved equal to \(4.7\ \mathrm{kg/cm.}^2\).

According to the original plan, as has already been said above, for a further increase of the load it was intended to suspend a board from the end of the lever and to place specified weights upon it; but in preliminary experiments it turned out that, when the board was suspended in this way, the lever vibrated strongly. Therefore it was necessary to move its suspension to the middle support of the lever—namely, to a distance of \(720\ \mathrm{m/m}\) from it. This suspension of the board alone, with its weight of 43.7 kg, increased the mean load on the bearing to \(10.4\ \mathrm{kg/cm.}^2\). In order not to return once more to this subject, I shall now also indicate the following values of the bearing loads: they were produced by placing each time two new 20-kg weights on the suspended board, which corresponded to an increase of the mean load of the bearing by \(5.26\ \mathrm{kg/cm.}^2\). Thus we obtain the following scale of mean loads of the bearing in our experiments: \(4.7;\ 10.4;\ 15.7;\ 21.0;\ 26.2;\ 31.5;\ 36.7;\ 42.0;\ 47.3\ \mathrm{kg/cm.}^2\).

After these remarks, let us return again to the first readings of the manometers, obtained by us immediately after the bearing surface had been moistened in the manner already described, at its mean load of \(10.4\ \mathrm{kg/cm.}^2\), and let us see how close to this mean load comes the mean pressure on the surface of the bearing as derived from the manometer readings. In the following table No. 1 pri—

the record is given of the first of the present experiments, carried out on August 16, 1907:

Table No. 1.

Load on the bearing surface—10.4 kg/cm²; temperature in the room—18°.

From the table presented we note, first of all, the following: when the apparatus is started, while the oil is still fairly thick, the manometers show a rapid rise of the pressures above the investigated points of the surface of the bearing surface; these pressures very soon reach a certain height, remain at it for some time, and then, as the temperature of the bath rises, begin slowly to fall; this fall in the manometer readings seems indeed to depend on the temperature of the oil and on the properties connected with it, and to depend little on other factors, for example the number of revolutions of the shaft, provided only that it is not changed very greatly. This is evident from the fact that after a sufficiently long operation of the bearing surface, when the temperature of the oil had already risen considerably, I tried, as will be seen from the subsequent records, to stop the apparatus completely, and then again to begin gradually setting it in motion, bringing the number of shaft revolutions up to the previous value: in all these cases the manometer readings were even lower than those at which the apparatus had been stopped; but if the same thing was done at a low temperature

oil, I again obtained the original highest readings of the manometers; I also tried, at a high temperature of the oil, suddenly to increase or decrease the number of revolutions of the shaft, and although such an increase or decrease in the number of revolutions did affect the manometer readings, it did so only very slightly.

Therefore, regarding as the actual pressure readings at the given points of the bearing surface only the highest manometer readings obtained at the lower temperatures of the lubricating oil, and plotting them to a known scale on the drawing (see Plate III, drawing 5), representing the surface of the bearing at full size, in the form of ordinates from the radial straight line MN passing through the centers of the holes; then joining, by a smooth curve, the ends of these ordinates and the end points M, N of the radial straight line, at which points (M, N) the pressures, obviously, must be equal to zero, we obtain the curve of pressure distribution in the radial direction of the bearing surface.

The mean ordinate of this curve must correspond to the mean pressure per unit area of the bearing surface. If this ordinate is calculated for the given experiment, which we did by means of a planimeter, determining the area of the curve and dividing it by the length of the base MN, it proves to be equal to 9.7 mm, which, with our scale of 1 mm = 1 kg/cm², corresponds to a mean pressure of 9.7 kg/cm².

Thus this mean pressure comes out less than that calculated (10.4 kg/cm²) on the basis of the total load on the bearing surface by 0.7 kg/cm². This difference is in any case not especially large and can easily be explained either by the fact that on the other side of the bearing surface the pressure is greater than on the side just investigated, or by the fact that we did not calculate the supporting area of the bearing surface quite correctly.

Let us therefore see what the other experiments tell us at the same load on the bearing surface.

We present the record of the experiment of August 21:

Table No. 2.

Load—10.4 kg/cm²; temperature in the room—19°.

From the data of this table, for the mean ordinate of the pressure curve we find the value \(11,9\ \mathrm{m/m.}\), which corresponds to an average load of the plate of \(11,9\ \mathrm{kg/cm.^2}\), i.e., \(1,5\ \mathrm{kg/cm.^2}\) greater than the calculated average load.

Finally, the record of the experiment of August 24 gives us the following results:

Table No. 3.

Load—\(10,4\ \mathrm{kg./cm^2}\); temperature in the room—\(19.5^\circ\).

The average load according to the data of this table comes out equal to \(10,3\ \mathrm{kg/cm.^2}\), i.e., only \(0,1\ \mathrm{kg/cm.^2}\) less than the calculated one.

If, now, the average load on the bearing surface is determined from the data of all three experiments, it proves to be equal to \(10{,}6\ \mathrm{kg./cm.^2}\), i.e., greater than that calculated by only \(0{,}2\ \mathrm{kg./cm.^2}\)—a result that is truly remarkable.

Let us see, however, what the experiments measuring pressures give us at the same average load of the bearing surface on the other half of the surface of the bearing; here, for example, is the record of the experiment of August 31:

Table No. 4.

Load—\(10{,}4\ \mathrm{kg./cm.^2}\); temperature in the room—\(18^\circ\).

The average load on the bearing surface determined from the data of the last table proves to be equal to \(10{,}0\ \mathrm{kg./cm.^2}\), i.e., \(0{,}4\ \mathrm{kg./cm.^2}\) less than that calculated on the basis of the full load of the bearing surface.

The data of the record of another analogous experiment (see Table 5) give for the average pressure on the bearing surface a value of \(11{,}0\ \mathrm{kg./cm.^2}\), i.e., \(0{,}6\ \mathrm{kg./cm.^2}\) greater than the calculated one.

Table No. 5.

Load—\(10{,}4\ \mathrm{kg./cm.^2}\); temperature in the room—\(16^\circ\).

As the average of these two experiments, we obtain a load on the bearing equal to \(10.5\ \mathrm{kg./cm.^2}\), i.e. \(0.1\ \mathrm{kg./cm.^2}\) greater than the calculated one.

Thus, the mean values of the loads per unit of the supporting surface of the bearing, obtained by us on the basis of experimental data, are so close to the mean load which we calculated on the basis only of the total load of the bearing that this makes it possible to regard the manometer readings with considerable confidence as the actual pressures existing on the given elements of the bearing surface.

All the tables presented, as well as the first, indicate a lowering of the manometer readings with an increase in the oil temperature; a sudden increase in the number of shaft revolutions at a more or less high oil temperature evidently does not play, at least within certain limits of the shaft speed, a significant role in changing the manometer readings. Thus, for example, from Table No. 5 it follows that, at an oil temperature of about \(37^\circ\), an increase in the number of shaft revolutions from 159 to 201, i.e. by approximately \(25\%\), was reflected in an increase in the readings of only two manometers: one by \(0.2\ \mathrm{kg./cm.^2}\), and the other by \(0.1\ \mathrm{kg./cm.^2}\); whereas with the following new increase in the number of revolutions from 201 to 207 the manometer readings even fell.

In exactly the same way, according to Table No. 2, a rapid increase in the number of shaft revolutions from 187 to 237, at an oil temperature of about \(50^\circ\), although it did show some increase in the manometer readings, nevertheless did not bring them up to the initial highest values obtained at a comparatively low oil temperature.

On the other hand, the same tables point to the circumstance that, in order to obtain the higher manometer readings which give the true values of the pressures on the given elements of the bearing, it is apparently necessary, in addition to a low oil temperature, to have a more or less definite number of shaft revolutions: all such manometer readings were obtained with a shaft speed of not less than 130 per minute.

The pressure values obtained on the bearing surface indicate beyond doubt that the load is distributed nonuniformly over

that this distribution also does not obey the law of Reye-Grashof, and that it is very similar to that distribution of pressures along the length of the journal which was found by Tower. Just as there the pressure per unit surface increases gradually from the edges of the journal toward its middle, so here it increases as one approaches the mean circumference drawn on the surface of the bearing at an equal distance from the extreme circumferences bounding the surface of the bearing.

The excess of the greatest pressure over the mean pressure is lower here than in Tower’s experiments for the journal. For the bearing it is expressed as only 40.6%, whereas for the journal it is 90%*).

Let us now see what results we obtain in experiments with other loads on the bearing.

In what follows, first the data of experiments determining the pressures on one side of the bearing are given; and then the results of the experiments for the other half of the bearing will be presented.

Further experiments with the first half of the bearing.

In the experiments with the bearing loaded only up to \(4.7\ \mathrm{kg./cm.^2}\), the results of which are given in the following Tables Nos. 6, 7, and 8:

Table No. 6.

Load—\(4.7\ \mathrm{kg./cm.^2}\); temperature in the room—18°.

) Bach. Maschinen-Elemente*. 1908. p. 490.

Table No. 7.

Load—4,7 kg./cm²; temperature in the room—20°.

Table No. 8.

Load—4,7 kg./cm²; temperature in the room—16°.

The following peculiarity is observed: in the first two experiments, as is seen from Tables No. 6 and No. 7, the load was distributed extremely unevenly and, moreover, increased greatly toward the inner recess O on the surface of the bearing surface; if one calculates, in percent, the ratio of the greatest pressure per unit surface of the bearing surface to the mean, it turns out that the former exceeds the latter by 118%; meanwhile, an experiment carried out with the same load on the bearing surface after prolonged operation of it with loads many times exceeding the one under consideration gives, for the distribution of pressures, a curve (see Table No. 8, or Table III, drawing 5) analogous to that which was obtained when the bearing surface was loaded up to 10,4 kg./cm.\(^2\). And what is still remarkable is that, despite the enormous difference in the law of distribution of the pressures represented by Tables No. 6 and No. 7 on the one hand and Table No. 8 on the other, the mean pressures per unit of bearing surface in both cases are completely the same (5,6 kg./cm.\(^2\)).

It is very probable that such an uneven distribution of pressures as is given by Tables No. 6 and No. 7 can be explained simply by a local unevenness of the surface of the bearing surface, while the normal distribution of pressures is expressed by Table No. 8; this is also indicated by the results of an experiment with the same load for the other half of the bearing surface, which (see below—Table No. 25) are very close to the data of Table No. 8.

In the following Tables No. 9—No. 23 are given the results of experiments for determining the pressures at other, higher loads (15,7—43,7 kg./cm.\(^2\)) of the bearing surface.

Table No. 9.

Load—15,7 kgcm\(^2\); temperature in the hall—16°.

Table No. 10.

Load—21 kg./cm²; temperature in the room—15,5°.

Table No. 11.

Load—26,2 kg./cm²; temperature in the room—18°.

Table No. 12.

Load—26,2 kg./cm²; temperature in the room—16°.

Table No. 13.

Load—26.2 kg./cm²; temperature in the room—16.5°.

Table No. 14.

Load—31.5 kg./cm²; temperature in the room—18°.

Table No. 15.

Load—36,7 kg./cm²; temperature in the room—17°.

Table No. 16.

Load 36,7 kg./cm²; temperature in the room—16,5°.

Table No. 17

Load—36.7 kg/cm²; temperature in the room—16.8°.

Table No. 18.

Load—42 kg/cm²; temperature in the room—15.5°.

Table No. 19.

Load—42 kg/cm²; temperature in the room—16°.

Table No. 20.

Load—47.3 kg./cm²; temperature in the room—16°.

Table No. 21.

Load—47.3 kg./cm²; temperature in the room—16°.

Table No. 22.

Load—47.3 kg/cm²; temperature in the room—17°.

Table No. 23.

Load—47.3 kg/cm²; temperature in the room—16°.

Some of these experiments with specified loads were carried out, like those described above, several times; in drawing the curves (see Table III, drawing 5) of the pressure distribution, the mean values from these experiments were taken; from these same mean values were computed the mean pressure per unit surface of the heel plate, the ratios of the greatest pressures to the mean, etc.; the results of all these computations are given in Table No. 24. Experiments with one and the same load were never carried out consecutively, but were always alternated with experiments at other loads.

Without repeating here those conclusions that have already been drawn on the basis of Tables Nos. 1–8, and which are also readily seen from Tables Nos. 9–23, I shall point out only the circumstance that the highest readings, with a load on the bearing plate of \(26.2\ \mathrm{kg}/\mathrm{cm}^2\), were obtained in one experiment (see Table No. 13) at a shaft speed of only 75 revolutions per minute, and in the experiments with a load of \(47.3\ \mathrm{kg}/\mathrm{cm}^2\), at a shaft speed of 93 (see Table No. 22), i.e. at shaft speeds much smaller than those which occurred in the first experiments. Let us now turn to Table No. 24.

Table No. 24.

From consideration of this table, we may draw the following conclusions: with the exception of the experiments with a mean load on the plate of \(4.7\ \mathrm{kg}/\mathrm{cm}^2\), which gave a discrepancy of \(19.2\%\) between the mean pressure on the surface of the plate obtained from the experiments and the same quantity calculated on the basis of the total load, the given experiments with other plate loads do not give a difference between the quantities just indicated above \(9\%\); in the majority of cases this difference is still smaller, and in some cases even quite negligible.

The fact that all experiments with a plate load of \(15.7\ \mathrm{kg}/\mathrm{cm}\) and higher give a mean pressure per unit surface of the plate lower than that calculated on the basis of the total load of the plate can be explained in part by the existence, though small, of vibrations of the lever, which at the indicated loads there was no possibility of avoiding.

The same Table No. 24 shows that the excess of the greatest observed pressure over the mean pressure is smaller at the higher loads of the heel (17–20%) than at the lower ones (40–118%). Also clearly manifested is a tendency for the greatest pressure to shift with a change in the load on the heel (see Table III, drawing 5): as the load increases, the greatest pressure shifts toward the outer edge of the heel.

As for the constant displacement of the greatest pressure in the radial direction of the heel from its outer edge toward the center and back, observed by Bach, on the basis of all the above-cited results of our experiments it can hardly be established; moreover, this phenomenon could scarcely even have manifested itself, owing to the insufficient duration of our experiments, especially in view of the great hardness of the material of the heel (hardened cast iron): during the experiments the wear of the heel could not manifest itself to a sufficient degree.

Some barely noticeable indications of a displacement of the greatest pressure in the radial direction nevertheless exist: thus, if attention is paid to the data of Tables No. 1—No. 3, then, as is seen both from the tables themselves and from drawing 5 of Table III, the greatest pressure in each of these three experiments shifted slightly toward the outer edge of the heel; it is very interesting that in experiment No. 3, carried out later than experiments No. 20—No. 23, the displacement outward of the greatest pressure is nevertheless noticeable in comparison with similar experiments No. 1 and No. 2.

The same is also indicated by Tables No. 11—No. 13; No. 15—No. 17; No. 20—No. 23; but this displacement is very insignificant, and, in any case, it manifested itself to a lesser degree than the displacement of the greatest pressure when the load on the heel was increased.

Experiments with the other half of the heel. We shall now present the results of the experiments to determine the pressures on the other half of the heel (see Tables No. 25—29), as well as the results of calculations of the mean pressures, the excesses of the greatest pressures over the mean ones, etc. (see Table No. 30).

Table No. 25.

Load \(4.7\ \mathrm{kg}/\mathrm{cm}^2\); temperature in the room—\(16.5^\circ\).

Table No. 26.

Load—21 kg/cm²; temperature in the room—16.0°.

Table No. 27.

Load—26,2 kg/cm²; temperature in the room—16°.

Table No. 28.

Load—31,5 kg/cm²; temperature in the room—17°.

In the interval of time from 10 h. 25 m. to 11 h. the apparatus was stopped.

Table No. 29.

Load—42,0 kg/cm²; temperature in the room—16,5°.

Table No. 30.

All these tables undoubtedly testify to those conclusions which we had already drawn earlier on the basis of experiments with the first half of the shoe: to the great agreement of the mean pressures derived from the results of the experiments and those calculated on the basis of the full load of the shoe; to the decrease, with increasing load, of the excess of the greatest pressures over the mean pressure; and to the displacement of the greatest pressure per unit surface of the shoe toward its outer edge, etc.

Thus, summarizing all that has been set forth above, it may be presented in the form of the following propositions, more or less reliable within the limits of the experiments:

Conclusions.

1) The proposed method of investigating the distribution of pressures over the surface of the bearing plate can give a correct idea of the distribution of this pressure.

2) The experiments carried out indicate a certain analogy, especially up to an average load on the plate of \(21\ \mathrm{kg./cm^2}\), between the law of distribution of pressures along the length of the trunnion, found by Tower, and the law of distribution of pressures on the surface of an annular plate in the radial direction.

3) These experiments, while not fully confirming Bach’s propositions concerning the constant displacement of the greatest pressures in one and the other direction along the radius of the plate—which, however, is quite understandable in view of the insufficient duration of the experiments—nevertheless give some indications of a radial displacement of the greatest pressures with the passage of time.

4) The experiments carried out give grounds for considering the greatest pressures on the surface of the plate to move along the radius in the direction toward its outer edge as the load increases.

5) The excess of the greatest pressure per unit surface of the plate over the average pressure decreases as the load on the plate increases.

V. Pinegin.

Tomsk, May 27, 1909.

Table I.

To V. N. Pinegin’s article “Experimental study of the pressure distribution in a flat heel.”

Plate II.

To the article by V. N. Pinegin
“Experimental study of the pressure distribution in a flat bearing.”

Plate III.

To the article by V. N. Pinegin, “Experience in Studying the Distribution of Pressure in a Flat Bearing.”

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

1. See Proceedings of the Institution of Mechanical Engineers, 1885, and also Engineer, 1884, p. 434, or Zeit. d. Ver. d. Ing., 1885, p. 836. ↩↩↩

2. For example, Woodbury, who carried out experiments with an annular bearing surface under loads \(p = 0.07\text{–}2.8\ \mathrm{kg}/\mathrm{cm}^{2}\) and at speeds not exceeding \(1.5\ \mathrm{m}/\mathrm{s}\); see Zeit. d. Ver. d. Ing., 1885, p. 451, or Thurston, Friction and Lost Work in Machinery and Mill Work. New York, 1898. ↩↩