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above him. He does not, however, take any notice of the comparative range
of those on the top and those at the bottom of the cliff; but wherever he
mentions them he speaks of them as on the cliff, leading me to suppose that
for some reason those at the bottom of the cliff had been abandoned, or
that they were less efficient than those above. If I am right in this
surmise, if the sounds from below did not range so far as those from above,
it is a fact in accordance with refraction, but of which, I think, Prof. Tyndall
has offered no explanation.

[Besides the results of Prof. Tyndall's experiments there are many other
phenomena which are explained by this refraction. Humboldt could hear
the falls of Orinoco three times as loud by night as by day at a distance of
one league; and he states that the same phenomenon has been observed
near every waterfall in Europe. And although Humboldt gave another
explanation*, which was very reasonable when applied to the particular case
at Orinocof, yet it must be admitted that the circumstances were such as
would cause great upward refraction; and hence there can be but little
doubt that refraction had a good deal to do with the diminution of the
sound by day.

In fact if this refraction of sound exists, then, according to Mr Glaisher's
observations, it must be seldom that we can hear distant sounds with any-
thing like their full distinctness, particularly by day ; and any elevation in
the observer or the source of the sound above the intervening ground will
increase this range and distinctness, as will also a gentle wind, which brings
the sound down and so counteracts the effect of refraction. And hence we
have an explanation of the surprising distances to which sounds can some-
times be heard, particularly the explosion of meteors, as well as a reason
for the custom of elevating church-bells and sounds to be heard at great
distances. September 1874.]

* " That the sun acts upon the propagation and intensity of sound by the obstacles met in
currents of air of different density, and by the partial undulations of the atmosphere arising from

unequal heating of different parts of the soil During the day there is a sudden interruption

of density wherever small streamlets of air of a high temperature rise over parts of the soil
unequally heated. The sonorous undulations are divided, as the rays of light are refracted
wherever strata of air of unequal density are contiguous. The propagation of sound is altered
when a stratum of hydrogen gas is made to rise over a stratum of atmospheric air in a tube closed
at one end ; and M. Biot has well explained, by the interposition of bubbles of carbonic acid gas,
why a glass filled with champagne is not sonorous so long as that gas is evolved and passing
through the strata of the liquid." Humboldt's Travels, Bohn's Series, Vol. rr., p. 264.

t The sounds proceeded over a plane covered with rank vegetation interspersed with black
rocks. These latter attained a very considerable elevation of temperature under the effects of the
tropical sun, as much as 48 C., while the air was only 28 ; and hence over each rock there would
be a column of hot air ascending.



17.



ON THE EFFICIENCY OF BELTS OR STRAPS AS
COMMUNICATORS OF WORK.

[From " The Engineer," Nov. 27, 1874.]

IT has often been remarked that it seems to be impossible so to construct
belts that they should drive without slipping. I am not aware that any
reason has ever been given for this ; but, on the other hand, most writers seem
to have assumed that if the belt is made sufficiently tight, so that the tension
on the slack side is from one-half to one-quarter that on the tight side,
according as the strap is in contact with one-half or the whole of the pulleys,
it will not slip. The object of this communication is to show that not only is
a reason to be given for this residual slipping, but that it follows a definite
law, depending on the elasticity of the strap, and independent of its tightness
over and above what is necessary to prevent it slipping bodily round the
wheel.

When a pulley, A, is connected with another pulley B by a belt, so that
A drives B, it is usual to assume that the surfaces of the two pulleys move
with the same velocity, namely, the velocity of the strap ; and that the work
communicated from A to B equals this velocity multiplied by the difference
in the tension on the two sides of the belt. This law would doubtless be true
if the strap were inelastic, and did not stretch at all under the tension to
which it is subjected ; but as all straps are more or less elastic, it can be
shown that this law does not hold rigorously, although with such an elastic
material as leather it is not far from the truth.

Owing to its elasticity, the tight side of the belt will be more stretched
than the slack or slacker side, and will, in consequence, have to move faster.
This is easily seen when we consider that each point on the strap completes
its entire circuit in the same time, so that if at any instant a number of marks
were made on the strap at different points, these marks would all return to



108 ON THE EFFICIENCY OF BELTS OR STRAPS [17

the same points in precisely the same time ; for the velocity at each point
would be equal to the length of strap which passes that point, and on the tight
side this would be the stretched length ; whereas on the other side it would
be the unstretched length, and hence the two sides of the strap would move
with different velocities, according to the degree in which the strap is more
stretched on the one side than on the other.

Now the stretching of a strap will be proportional to the tension, although
the degree will depend on its size and the material of which the strap is
composed. Let XT represent the increase in length per foot in a certain
strap, caused by a tension of r Ib. Then, if TJ and r 2 represent the tensions
on the two sides of the belt respectively, the stretching on these two sides
will be respectively proportional to XT^ and Xr 2 and the difference will be
proportional to X (TJ r 2 ). Therefore the velocities of the two sides will be

in the ratio : = r ^ or 1 + X (T! T 2 ) nearly.



Again, it is easy to see that the velocity of the tight side of the strap must
be equal to that of the surface of the pulley A which drives it ; whereas the
velocity of the pulley B which is driven by the strap, will be the same as that
of the slack side of the strap; and hence the velocities of the two pulleys

differ in the ratio - 1 - . And since the turning effort of the strap on

either pulley is the same, namely, r l r 2 , the difference of its tensions, the work
done by A, which equals its velocity multiplied by this effort, will be greater than

that taken up by B in the ratio 1 . This excess of work will have

been spent in the slipping, or more properly the creeping of the strap round
the pulleys. The manner in which this creeping takes place is easily seen,
as follows: The strap comes on to A tight and stretched, and leaves it
unstretched. It has therefore contracted while on the pulley. This con-
traction takes place gradually from the point at which it comes on to that at
which it leaves, and the result is that the strap is continually slipping over
the pulley to the point at which it first comes on. In the same way with B ;
the strap comes on unstretched and leaves it stretched, and has expanded
while on the wheel, which expansion takes place gradually from the point at
which the strap conies on until it leaves.

The proportion which the slipping bears to the whole distance travelled
by the strap = X (^ r 2 ), which, as previously shown, is the proportion which
the work lost bears to the whole work done by A. From this it appears that
the slipping and work lost are proportional to X, i.e. to the increase which a
tension of 1 Ib would cause in 1 ft. length of the strap ; and hence, the more
inextensible the material is, the better it is suited for belts.



17] AS COMMUNICATORS OF WORK. 109

The actual amount of this slipping may be calculated when we know the
elasticity of the belts. With leather it is very small. One belt, which had
been in use about two years, and was 1'25 in. wide and T 3 g thick the usual
thickness increased in length by sixteen thousandths under a tension of
100 Ib. From this example it appears that, for a leather belt of breadth
b inches,

>- 20 1
100000*6*

Hence the ratio of slipping = '0002 7- (TJ - r 2 ) ; and in practice r x T 2 varies

from 20 Ib. to 60 Ib. per inch width of belt ; therefore the slipping = '008, or
nearly 1 per cent. With new straps it would probably be more. With soft
elastic materials, such as india-rubber, the slipping is very much greater.
In some instances I have been able to make the driving pulley A turn twice
as fast as the pulley B, simply in virtue of this expanding and contracting on
the pulleys. This shows at once how it is that elastic straps, such as can be
made of soft india-rubber, have never come into use, a fact which is otherwise
somewhat astonishing, considering for how many purposes an elastic connection
of this sort would be useful. A similar explanation to the above may also be
given for the friction occurring when elastic tires are used for the wheels of
carriages and engines. The tire is perpetually expanding between the wheel
and the ground. As the wheel rolls on to the tire, it is continually elongating
the part between it and the ground which is in front of the point at which
the pressure is greatest. This elongation can only be accomplished by
sliding the tire over both the surface of the wheel and the ground, against
whatever friction there may be ; and similarly, towards the back of the wheel,
the tire is contracting, also against friction. Even when there is no tire, if
either the wheel or the ground is elastic, a similar action takes place ; and
hence we may probably explain what is usually called rolling friction*, which
has been observed to take place no matter how true or hard the surface of
the wheel and the plane on which it rolls may be.

* See paper 18.



18.



ON ROLLING-FRICTION.

[From the "Philosophical Transactions of the Royal Society of London,"

vol. 166, part 1.]

(Read June 17, 1875.)

Introduction.

ALTHOUGH the motion of wheels and rollers over a smooth plane is attended
with much less resistance or friction than the sliding of one flat surface
over another, however smooth, yet practically it has been found impossible to
get rid of resistance altogether. Coulomb made some experiments on the
resistance which wooden rollers meet with when rolling on a wooden plane,
from which experiments he deduced certain laws connecting this resistance
with the size of the rollers and the force with which they are pressed on to
the plane. These laws have been verified and extended to other materials
by Navier and Morin, and are now set forth in many mechanical treatises as
''the laws of resistance to rolling." It does not appear, however, that any
systematic investigation of this resistance has ever been undertaken, or any
attempts made to explain its nature. When hard surfaces are used it is very
small, and it has doubtless been attributed to the inaccuracies of the surfaces
and to a certain amount of crushing which takes place under the roller.
On closer examination, however, it appears that these causes, although they
doubtless explain a great part of the resistance which occurs in ordinary
practice, are not sufficient to explain the resistance altogether ; and that, if
they could be removed, there would still be a definite resistance depending
on the size and weight of the roller and on the nature of the material of
which it and the plane are composed. If it were not so, a perfectly true
roller when rolling on a perfectly true surface ought to experience no resist-
ance, however soft the roller and the plane might be, provided both were
made of perfectly elastic material so that the one did not permanently crush



18] ON ROLLING-FRICTION. Ill

the other ; and we might expect, although these conditions are not absolutely
fulfilled, that a roller of iron would roll as easily on a surface of india-rubber
as on one of iron, or that an india-rubber roller would experience no more
resistance than one of iron when rolling on a true plane. Such, however, is
not the case. The resistance with india-rubber is very considerable ; my
experiments show it to be ten times as great as with iron. I am not aware that
this fact has been previously recognized; and that it has often been over-,
looked is proved by the numerous attempts which have been made to use
india-rubber tires for wheels, the invariable failure of which may, I think, in
the absence of any other assigned cause, be fairly attributed to the excessive
resistance which attends their use. Another fact which I do not think has
been hitherto noticed, but of which I have had ample evidence, and which
clearly shows the existence of some hitherto unexplained cause of resistance
to rolling, is the tendency which a roller has to oscillate about any position in
which it may be placed on a flat surface.

However true and hard the roller and the surface may be, if the roller is
but slightly disturbed it will not move continuously in one direction until
it gradually comes to rest, but it will oscillate backwards and forwards
through a greater or less angle, depending on the softness of the material.
These oscillations are not due to the roller having settled into a hollow. This
is strongly implied by the fact that the more care is taken to make the sur-
faces true and smooth the more regular and apparent do the oscillations
become. But even if this is not a sufficient proof if it is impossible to
suppose that an iron roller on an iron plane can be made so true that when
the one is resting on the other it will not be able to find some minute
irregularities or hollows in which to settle still we must be convinced when
we find the same phenomenon existing when india-rubber is substituted for
iron, and in such a marked degree that no irregularities there may be in the
surface produce any effect upon it, much less serve to account for it.

These phenomena, with others, have led me to conclude that there is a
definite cause for the resistance to rolling besides the mere crushing of the
surface or accidental irregularities of shape, a cause which is connected with
the softness of the material as well as with the size and weight of the
roller.

Such a force, if its existence be admitted, must either be considered as
exhibiting some hitherto unrecognized action of matter on matter, or must be
supposed to arise in some intelligible manner from the known actions. The
latter is the most natural supposition ; and it is my object in this paper to show
that this force arises from what is ordinarily known as friction. It is to
imply this connexion that I have gone back to the name Rolling-Friction in
place of the more general title resistance to rolling ("resistance au roulement"),



112 ON ROLLING-FRICTION. [18

which Coulomb and subsequent writers have chosen avowedly because they
did not wish to imply such a connexion.

The assumption that this force is due to friction necessarily implies that
there is slipping between the roller and the plane at the point of contact ; and
on the other hand, if it can be shown that there is slipping, it follows as a
natural consequence that there must be friction or resistance to rolling.
Therefore the question as to whether the resistance to rolling is due to
friction, reduces itself into a question as to whether there is any evidence of
slipping between the roller and the surface on which it rolls.

My attention was first called* to the possibility of such slipping while
considering a phenomenon in the action of endless belts when used to transmit
rotary motion from one pulley to another, namely, that it is impossible to
make the belt tight enough entirely to prevent slipping and cause the surfaces
of the two pulleys to move with identically the same velocity. It appears
that this slipping is due to the elasticity of the belt, and, since all material
is more or less elastic, cannot altogether be prevented. This becomes apparent
when we consider that of the two parts of the belt which stretch from pulley
to pulley, the one is tighter and hence more stretched than the other, that is,
when the belt is transmitting power. For that side which is most stretched,
and consequently thinner, will have to move faster than the slacker side in
order to prevent the belt accumulating at one pulley ; and the speed of the
driving-pulley will be equal to that of the tight side of the belt, while the
speed of the following pulley will be equal to that of the slack side. This
difference of speed requires that the belt shall slip over the pulleys ; and this
slipping takes place by the expansion and contraction of the belt on the
pulleys as it passes from the tight side to the slack side, and vice versa. With
leather belts this slipping is very small ; but with soft india-rubber it
becomes so great as practically to bar the use of this material for driving-
belts.

The recognition of this slipping at once suggested to . me that there
must be an analogous slipping when a hard roller rolls on a soft surface,
or when an india-rubber wheel rolls on a hard surface. A single experiment
was sufficient to prove that such was the case an iron roller rolled through
something like three-quarters of an inch less in a yard when rolling on india-
rubber than when rolling on wood or iron.

Having made this discovery, I proceeded to investigate the subject, and
have obtained what I think to be satisfactory evidence that, whatever may be
the material of which the plane and the roller are composed, the deformation
at the point of contact always causes slipping, although, owing to the hard-
ness of the materials, it may be far too small to be measured.

* (See the preceding paper.)



18] ON ROLLING-FRICTION. 113

In the following pages I shall first show that the deformation at the point
of contact caused by the weight of the roller must affect the distance rolled
through, that it must cause slipping, and that this slipping will be attended
with friction. I shall then show that the friction will itself considerably
modify the deformation which would otherwise take place, and endeavour to
trace the exact nature of the actual deformation. The result of my experi-
ments will then be given, together with the description of certain other causes
of rolling-friction which appear under certain circumstances to exist. In
conclusion, I shall indicate the direction in which I hope to continue the
investigation, consider its bearing on the laws discovered by Coulomb, and
discuss certain phenomena connected with the wear of railway-wheels which
have been hitherto unexplained, and which serve to illustrate the importance
of the subject.

The Distance Rolled through.

If a perfectly hard cylinder rolled on a perfectly hard plane and there were
no slipping, then the distance which the cylinder would pass over in one
revolution would be exactly equal to its circumference ; but if, from the weight
of the cylinder or any cause, the length of the surface either of the cylinder
or the plane underwent an alteration near the point of contact, then the
distance traversed in one revolution would not be equal to the natural length
of the circumference. For example, suppose that an iron cylinder is rolling
on a surface of india-rubber across which lines have been drawn at intervals
of '01 of an inch, and suppose that as the cylinder rolls across these lines the
surface of the india-rubber extends so that the intervals become equal to 'Oil
of an inch, closing again after the cylinder is past, then the cylinder will
measure its circumference, so to speak, on the extended plane, and the actual
distance rolled through when measured on the contracted surface will be one-
tenth less than the circumference. In the same way there would be an
alteration in the distance rolled through if the surface of the roller extended
or if either of the surfaces contracted.

In the subsequent remarks I shall call the distance which the roller
would roll through if there were no extension or contraction its geometrical
distance.

Since no material is perfectly hard, when a heavy roller rests on a surface,
the weight of the roller will cause it to indent the surface to a greater or less
extent, according to the softness of the latter ; and in the same way the sur-
face of the cylinder will be flattened at the point of contact in the manner
shown in Fig. 1.

This indentation and flattening will alter the lengths of the surfaces at the
point of contact, and will therefore affect the progress of the roller. When
o. R. 8



114 ON ROLLING-FRICTION. [18

a body of any shape is compressed in one direction it extends in the other
directions ; hence the weight of the roller resting on the plane will, by corn-




Fig. 1.

pressing the material of the plane in a vertical direction, cause it to extend
laterally at the point of contact, and thus the length of the surface which the
cylinder actually rolls over would be greater than the length measured on the
undisturbed plane. From this cause, therefore, the cylinder would roll through
less than its geometrical distance.

On the other hand, the surface of the roller would also be extended
(squeezed out) in a similar manner by the pressure of the plane at the point
of contact; and hence the surface of the roller would be greater than its
natural length, and this would cause the roller to roll through more than its
geometrical distance.

To a certain extent, therefore, the expansion of the surface of the roller
would counteract the expansion of the plane ; and if the two were of the same
material, then the one of these extensions would, if nothing interfered to
prevent it, exactly counteract the other. But if the one was harder than the
other, then the effect on the harder one would be least. Thus an iron cylinder
rolling on an india-rubber plane would roll through less than its geometrical
distance ; whereas, inversely, an india-rubber roller on an iron plane would
roll through more than its geometrical distance.

These things actually take place. But there is, besides softness, another
circumstance, not hitherto mentioned, which affects the lateral extension of
the surface when compressed by the roller, viz. the shape of the surface.

A little consideration will be sufficient to show that a curved indent in a
flat surface will have a greater effect to extend the surface than a flat indent
on a rounded surface. In the case of the rounded surface it will be seen that



18] ON ROLLING-FRICTION. 115

the effect of vertical compression to a certain extent counteracts the effect of
lateral expansion ; whereas in the case of the flat surface these things are
reversed, and the effect of the surrounding material to uphold that which is
depressed will increase the lateral expansion.

From this cause, therefore, even if the cylinder and the plane were made of
the same material, there would still be a difference in the lateral extension of
the surfaces at the point of contact, depending on the smallness of the diameter
of the cylinder, and this difference would still cause the cylinder to roll
through less than its geometrical distance.

If, instead of on a plane, the one cylinder rolled on another parallel cylinder
under a force tending towards the centre, then, if the two cylinders were of
the same material and their diameters were equal, they would roll through
their geometrical distance; but if the one was larger than the other, the
largest would be most retarded.

It appears, therefore, that there are two independent causes which affect
the progress of a roller on a plane the relative softness of the materials and
the diameter of the roller. Of these the curvature of the roller always acts
to retard its progress ; while the other (the relative softness) to retard or to
accelerate, according as the plane is softer than the cylinder, or vice versa,.
These two causes will therefore act in conjunction or in opposition, according
to whether the roller is harder or softer than the plane. In the former case
the roller will be retarded, whereas in the latter it will depend on the relation
between the relative softness and the diameter of the cylinder, whether its
progress is greater than, less than, or equal to its geometrical progress. Thus
an iron roller on an india-rubber plane will make less than its geometrical



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