this arrangement is sufficient. The piston in that case is not forced
upwards by the pressure of steam. During its ascent it is not subject
to the action of any force of steam, and the other end of the beam
falls by the weight of the pump-rods drawing the piston, at the
opposite end A, to the top of the cylinder. Thus the machine is in fact
passive during the ascent of the piston, and exerts its power only
during the descent.
If the machine, however, be applied to purposes in which a constant
action of the moving force is necessary, as is always the case in
manufactures, the force of the piston must drive the beam in its ascent
as well as in its descent. The arrangement just described cannot effect
this; for although a chain is capable of transmitting any force, by
which its extremities are drawn in opposite directions, yet it is, from
its flexibility, incapable of communicating a force which drives one
extremity of it towards the other. In the one case the piston first
_pulls_ down the beam, and then the beam _pulls_ up the piston. The
chain, because it is inextensible, is perfectly capable of both these
actions; and being flexible, it applies itself to the arch-head of the
beam, so as to maintain the direction of its force upon the piston
continually in the same straight line. But when the piston acts upon
the beam in both ways, in pulling it down and pushing it up, the chain
becomes inefficient, being from its flexibility incapable of the latter
action.
The problem might be solved by extending the length of the piston
rod, so that its extremity shall be above the beam, and using two
chains; one connecting the highest point of the rod with the lowest
point of the arch-head, and the other connecting the highest point
of the arch-head with a point on the rod below the point which
meets the arch-head when the piston is at the top of the cylinder,
_fig. 172._
The connection required may also be made by arming the arch-head with
teeth, _fig. 173._, and causing the piston rod to terminate in a
rack. In cases where, as in the steam-engine, smoothness of motion is
essential, this method is objectionable; and under any circumstances
such an apparatus is liable to rapid wear.
The method contrived by Watt, for connecting the motion of the piston
with that of the beam, is one of the most ingenious and elegant
solutions ever proposed for a mechanical problem. He conceived the
motion of two straight rods A B, C D, _fig. 174._,
moving on centres or pivots A and C, so that the extremities B and D
would move in the arcs of circles having their centres at A and C. The
extremities B and D of these rods he conceived to be connected with a
third rod B D united with them by pivots on which it could turn
freely. To the system of rods thus connected let an alternate motion
on the centres A and C be communicated: the points B and D will move
upwards and downwards in the arcs expressed by the dotted lines, but
the middle point P of the connecting rod B D will move upwards and
downwards without any sensible deviation from a straight line.
To prove this demonstratively would require some abstruse mathematical
investigation. It may, however, be rendered in some degree apparent
by reasoning of a looser and more popular nature. As the point B is
raised to E, it is also drawn aside towards the right. At the same
time the other extremity D of the rod B D is raised to E′, and
is drawn aside towards the left. The ends of the rod B D being
thus at the same time drawn equally towards opposite sides, its middle
point P will suffer no lateral derangement, and will move directly
upwards. On the other hand, if B be moved downwards to F, it will be
drawn laterally to the right, while D being moved to F′ will be drawn
to the left. Hence, as before, the middle point P sustains no lateral
derangement, but merely descends. Thus, as the extremities B and D move
upwards and downwards in circles, the middle point P moves upwards and
downwards in a straight line.[6]
[6] In a strictly mathematical sense, the path of the point P is a
curve, and not a straight line; but in the play given to it in its
application to the steam-engine, it moves through a part only of its
entire locus, and this part extending equally on each side of a point
of inflection, the radius of curvature is infinite, so that in practice
the deviation from a straight line, when proper proportions are
observed in the rods, is imperceptible.
The application of this geometrical principle in the steam-engine
evinces much ingenuity. The same arm of the beam usually works two
pistons, that of the cylinder and that of the _air-pump_. The apparatus
is represented on the arm of the beam in _fig. 175._ The beam
moves alternately upwards and downwards on its axis A. Every point of
it, therefore, describes a part of a circle of which A is the centre.
Let B be the point which divides the arm A G into two equal parts
A B and B G; and let C D be a straight rod, equal in
length to G B, and fixed on a centre or pivot C on which it is at
liberty to play. The end D of this rod is connected by a straight bar
with the point B, by pivots on which the rod B D turns freely. If
the beam be now supposed to rise and fall alternately, the points B and
D will move upwards and downwards in circular arcs, and, as already
explained with respect to the points B D, _fig. 174._, the
middle point P of the connecting rod B D will move upwards and
downwards without lateral deflection. To this point one of the piston
rods which are to be worked is attached.
To comprehend the method of working the other piston, conceive a rod
G P′, equal in length to B D, to be attached to the end G of
the beam by a pivot on which it moves freely; and let its extremity
P′ be connected with D by another rod P′ D, equal in length to
G B, and playing on points at P′ and D. The piston rod of the
cylinder is attached to the point P′, and this point has a motion
precisely similar to that of P, without any lateral derangement, but
with a range in the perpendicular direction twice as great. This will
be apparent by conceiving a straight line drawn from the centre A
of the beam to P′, which will also pass through P. Since G P′
is always parallel to B P, it is evident that the triangle
P′ G A is always similar to P B A, and has its
sides and angles similarly placed, but those sides are each twice the
magnitude of the corresponding sides of the other triangle. Hence
the point P′ must be subject to the same changes of position as the
point P, with this difference only, that in the same time it moves
over a space of twice the magnitude. In fact, the line traced by P′
is the same as that traced by P, but on a scale twice as large. This
contrivance is usually called the _parallel motion_, but the same name
is generally applied to all contrivances by which a circular motion is
made to produce a rectilinear one.
CHAP. XIX.
OF FRICTION AND THE RIGIDITY OF CORDAGE.
(320.) With a view to the simplification of the elementary theory of
machines, the consideration of several mechanical effects of great
practical importance has been postponed, and the attention of the
student has been directed exclusively to the way in which the moving
power is modified in being transmitted to the resistance independently
of such effects. A machine has been regarded as an instrument by
which a moving principle, inapplicable in its existing state to the
purpose for which it is required, may be changed either in its velocity
or direction, or in some other character, so as to be adapted to that
purpose. But in accomplishing this, the several parts of the machine
have been considered as possessing in a perfect degree qualities which
they enjoy only in an imperfect degree; and accordingly the conclusions
to which by such reasoning we are conducted are infected with errors,
the amount of which will depend on the degree in which the machinery
falls short of perfection in those qualities which theoretically are
imputed to it.
[Illustration: _C. Varley, del._ _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
Of the several parts of a machine, some are designed to move, while
others are fixed; and of those which move, some have motions differing
in quantity and direction from those of others. The several parts,
whether fixed or movable, are subject to various strains and pressures,
which they are intended to resist. These forces not only vary according
to the load which the machine has to overcome, but also according to
the peculiar form and structure of the machine itself. During the
operation the surfaces of the movable parts move in immediate contact
with the surfaces either of fixed parts or of parts having other
motions. If these surfaces were endued with perfect smoothness or
polish, and the several parts subject to strains possessed perfect
inflexibility and infinite strength, then the effects of machinery
might be practically investigated by the principles already explained.
But the materials of which every machine is formed are endued with
limited strength, and therefore the load which is placed upon it must
be restricted accordingly, else it will be liable to be distorted by
the flexure, or even to be destroyed by the fracture of those parts
which are submitted to an undue strain. The surfaces of the movable
parts, and those surfaces with which they move in contact, cannot in
practice be rendered so smooth but that such roughness and inequality
will remain as sensibly to impede the motion. To overcome such an
impediment requires no inconsiderable part of the moving power. This
part is, therefore, intercepted before its arrival at the working
point, and the resistance to be finally overcome is deprived of it. The
property thus depending on the imperfect smoothness of surfaces, and
impeding the motion of bodies whose surfaces are in immediate contact,
is called _friction_. Before we can form a just estimate of the effects
of machinery, it is necessary to determine the force lost by this
impediment, and the laws which under different circumstances regulate
that loss.
When cordage is engaged in the formation of any part of a machine, it
has hitherto been considered as possessing perfect flexibility. This is
not the case in practice; and the want of perfect flexibility, which
is called _rigidity_, renders a certain quantity of force necessary
to bend a cord or rope over the surface of an axle or the groove of
a wheel. During the motion of the rope a different part of it must
thus be continually bent, and the force which is expended in producing
the necessary flexure must be derived from the moving power, and is
thus intercepted on its way to the working point. In calculating the
effects of cordage, due regard must be had to this waste of power;
and therefore it is necessary to enquire into the laws which govern
the flexure of imperfectly flexible ropes, and the way in which these
affect the machines in which ropes are commonly used.
To complete, therefore, the elementary theory of machinery, we propose
in the present and following chapter to explain the principal laws
which determine the effects of friction, the rigidity of cordage, and
the strength of materials.
(321.) If a horizontal plane surface were perfectly smooth, and free
from the smallest inequalities, and a body having a flat surface also
perfectly smooth were placed upon it, any force applied to the latter
would put it in motion, and that motion would continue undiminished
as long as the body would remain upon the smooth horizontal surface.
But if this surface, instead of being every where perfectly even, had
in particular places small projecting eminences, a certain quantity
of force would be necessary to carry the moving body over these, and
a proportional diminution in its rate of motion would ensue. Thus, if
such eminences were of frequent occurrence, each would deprive the
body of a part of its speed, so that between that and the next it
would move with a less velocity than it had between the same and the
preceding one. This decrease being continued by a sufficient number of
such eminences encountering the body in succession, the velocity would
at last be so much diminished that the body would not have sufficient
force to carry it over the next eminence, and its motion would thus
altogether cease.
Now, instead of the eminences being at a considerable distance asunder,
suppose them to be contiguous, and to be spread in every direction
over the horizontal plane, and also suppose corresponding eminences to
be upon the surface of the moving body; these projections incessantly
encountering one another will continually obstruct the motion of the
body, and will gradually diminish its velocity, until it be reduced to
a state of rest.
Such is the cause of friction. The amount of this resisting force
increases with the magnitude of these asperities, or with the roughness
of the surfaces; but it does not solely depend on this. The surfaces
remaining the same, a little reflection on the method of illustration
just adopted, will show that the amount of friction ought also to
depend upon the force with which the surfaces moving one upon the
other are pressed together. It is evident, that as the weight of
the body supposed to move upon the horizontal plane is increased, a
proportionally greater force will be necessary to carry it over the
obstacles which it encounters, and therefore it will the more speedily
be deprived of its velocity and reduced to a state of rest.
(322.) Thus we might predict with probability, that which accurate
experimental enquiry proves to be true, that the resistance from
friction depends conjointly on the roughness of the surfaces and the
force of the pressure. When the surfaces are the same, a double
pressure will produce a double amount of friction, a treble pressure a
treble amount of friction, and so on.
Experiment also, however, gives a result which, at least at first view,
might not have been anticipated from the mode of illustration we have
adopted. It is found that the resistance arising from friction does not
at all depend on the magnitude of the surface of contact; but provided
the nature of the surfaces and the amount of pressure remain the same,
this resistance will be equal, whether the surfaces which move one upon
the other be great or small. Thus, if the moving body be a flat block
of wood, the face of which is equal to a square foot in magnitude, and
the edge of which does not exceed a square inch, it will be subject to
the same amount of friction, whether it move upon its broad face or
upon its narrow edge. If we consider the effect of the pressure in each
case, we shall be able to perceive why this must be the case. Let us
suppose the weight of the block to be 144 ounces. When it rests upon
its face, a pressure to this amount acts upon a surface of 144 square
inches, so that a pressure of one ounce acts upon each square inch. The
total resistance arising from friction will, therefore, be 144 times
that resistance which would be produced by a surface of one square inch
under a pressure of one ounce. Now, suppose the block placed upon its
edge, there is then a pressure of 144 ounces upon a surface equal to
one square inch. But it has been already shown, that when the surface
is the same, the friction must increase in proportion to the pressure.
Hence we infer that the friction produced in the present case is 144
times the friction which would be produced by a pressure of one ounce
acting on one square inch of surface, which is the same resistance as
that which the body was proved to be subject to when resting on its
face.
These two laws, that friction is independent of the magnitude of the
surface, and is proportional to the pressure when the quality of the
surfaces is the same, are useful in practice, and _generally_ true. In
very extreme cases they are, however, in error. When the pressure is
very intense, in proportion to the surface, the friction is somewhat
_less_ than it would be by these laws; and when it is very small in
proportion to the surface, it is somewhat _greater_.
(323.) There are two methods of establishing by experiment the laws of
friction, which have been just explained.
First. The surfaces between which the friction is to be determined
being rendered perfectly flat, let one be fixed in the horizontal
position on a table T T′, _fig. 176._; and let the other be
attached to the bottom of a box B C, adapted to receive weights,
so as to vary the pressure. Let a silken cord S P, attached to the
box, be carried parallel to the table over a wheel at P, and let a dish
D be suspended from it. If no friction existed between the surfaces,
the smallest weight appended to the cord would draw the box towards
P with a continually increasing speed. But the friction which always
exists interrupts this effect, and a small weight may act upon the
string without moving the box at all. Let weights be put in the dish D,
until a sufficient force is obtained to overcome the friction without
giving the box an accelerated motion. Such a weight is equivalent to
the amount of the friction.
The amount of the weight of the box being previously ascertained, let
this weight be now doubled by placing additional weights in the box.
The pressure will thus be doubled, and it will be found that the weight
of the dish D and its load, which before was able to overcome the
friction, is now altogether inadequate to it. Let additional weights
be placed in the dish until the friction be counteracted as before,
and it will be observed, that the whole weight necessary to produce
this effect is exactly twice the weight which produced it in the former
case. Thus it appears that a double amount of pressure produces a
double amount of friction; and in a similar way it may be proved, that
any proposed increase or decrease of the pressure will be attended with
a proportionate variation in the amount of the friction.
Second. Let one of the surfaces be attached to a flat plane A B,
_fig. 177._, which can be placed at any inclination with an
horizontal plane B C, the other surface being, as before, attached
to the box adapted to receive weights. The box being placed upon the
plane, let the latter be slightly elevated. The tendency of the box
to descend upon A B, will bear the same proportion to its entire
weight as the perpendicular A E bears to the length of the plane
A B (286.). Thus if the length A B be 36 inches, and the
height A E be three inches, that is a twelfth part of the length,
then the tendency of the weight to move down the plane is equal to a
twelfth part of its whole amount. If the weight were twelve ounces, and
the surfaces perfectly smooth, a force of one ounce acting up the plane
would be necessary to prevent the descent of the weight.
In this case also the pressure on the plane will be represented by
the length of the base B E (286.), that is, it will bear the
same proportion to the whole weight as B E bears to B A.
The relative amounts of the weight, the tendency to descend, and the
pressure, will always be exhibited by the relative lengths of A B,
A E, and B E.
This being premised, let the elevation of the plane A B be
gradually increased until the tendency of the weight to descend just
overcomes the friction, but not so much as to allow the box to descend
with accelerated speed. The proportion of the whole weight, which then
acts down the plane, will be found by measuring the height A E,
and the pressure will be determined by measuring the base B E. Now
let the weight in the box be increased, and it will be found that the
same elevation is necessary to overcome the friction; nor will this
elevation suffer any change, however the pressure or the magnitude of
the surfaces which move in contact may be varied.
Since, therefore, in all these cases, the height A E and the base
B E remain the same, it follows that the proportion between the
friction and pressure is undisturbed.
(324.) The law that friction is proportional to the pressure, has been
questioned by the late professor Vince of Cambridge, who deduced from
a series of experiments, that although the friction increases with the
pressure, yet that it increases in a somewhat less ratio; and from
this it would follow, that the variation of the surface of contact
must produce some effect upon the amount of friction. The law, as we
have explained it, however, is sufficiently near the truth for most
practical purposes.
(325.) There are several circumstances regarding the quality of the
surfaces which produce important effects on the quantity of friction,
and which ought to be noticed here.
This resistance is different in the surfaces of different substances.
When the surfaces are those of wood newly planed, it amounts to about
half the pressure, but is different in different kinds of wood. The
friction of metallic surfaces is about one fourth of the pressure.
In general the friction between the surfaces of bodies of different
kinds is less than between those of the same kind. Thus, between wood
and metal the friction is about one fifth of the pressure.
It is evident that the smoother the surfaces are the less will be the
friction. On this account, the friction of surfaces, when first brought
into contact, is often greater than after their attrition has been
continued for a certain time, because that process has a tendency to
remove and rub off those minute asperities and projections on which the
friction depends. But this has a limit, and after a certain quantity
of attrition the friction ceases to decrease. Newly planed surfaces
of wood have at first a degree of friction which is equal to half the
entire pressure, but after they are worn by attrition it is reduced to
a third.
If the surfaces in contact be placed with their grains in the same
direction, the friction will be greater than if the grains cross each
other.
Smearing the surfaces with unctuous matter diminishes the friction,
probably by filling the cavities between the minute projections which
produce the friction.
When the surfaces are first placed in contact, the friction is less
than when they are suffered to rest so for some time; this is proved
by observing the force which in each case is necessary to move the one
upon the other, that force being less if applied at the first moment
of contact than when the contact has continued. This, however, has a
limit. There is a certain time, different in different substances,
within which this resistance attains its greatest amount. In surfaces
of wood this takes place in about two minutes; in metals the time
is imperceptibly short; and when a surface of wood is placed upon a
surface of metal, it continues to increase for several days. The limit
is larger when the surfaces are great, and belong to substances of
different kinds.
The velocity with which the surfaces move upon one another produces but
little effect upon the friction.
(326.) There are several ways in which bodies may move one upon the
other, in which friction will produce different effects. The principal
of these are, first, the case where one body _slides_ over another; the
second, where a body having a round form _rolls_ upon another; and,
_thirdly_, where an axis revolves within a hollow cylinder, or the
hollow cylinder revolves upon the axis.
With the same amount of pressure and a like quality of surface, the
quantity of friction is greatest in the first case and least in the
second. The friction in the second case also depends on the diameter
of the body which rolls, and is small in proportion as that diameter
is great. Thus a carriage with large wheels is less impeded by the
friction of the road than one with small wheels.
In the third case, the leverage of the wheel aids the power in
overcoming the friction. Let _fig. 178._ represent a section of
the wheel and axle; let C be the centre of the axle, and let B E
be the hollow cylinder in the nave of the wheel in which the axle is
inserted. If B be the part on which the axle presses, and the wheel
turn in the direction N D M, the friction will act at B in
the direction B F, and with the leverage B C. The power acts
against this at D in the direction D A, and with the leverage
D C. It is therefore evident, that as D C is greater than
B C, in the same proportion does the power act with mechanical
advantage on the friction.