wheel-work is by the formation of teeth upon their circumferences, so
that these indentures of each wheel fall between the corresponding ones
of that in which it works, and ensure the action so long as the strain
is not so great as to fracture the tooth.
In the formation of teeth very minute attention must be given to their
figure, in order that the motion may be communicated from wheel to
wheel with smoothness and uniformity. This can only be accomplished
by shaping the teeth according to curves of a peculiar kind, which
mathematicians have invented, and assigned rules for drawing. The ill
consequences of neglecting this will be very apparent, by considering
the nature of the action which would be produced if the teeth were
formed of square projecting pins, as in _fig. 103._ When the
tooth A comes into contact with B, it acts obliquely upon it, and,
as it moves, the corner of B slides upon the plane surface of A in
such a manner as to produce much friction, and to grind away the side
of A and the end of B. As they approach the position C D, they
sustain a jolt the moment their surfaces come into full contact; and
after passing the position of C D, the same scraping and grinding
effect is produced in the opposite direction, until by the revolution
of the wheels the teeth become disengaged. These effects are avoided by
giving to the teeth the curved forms represented in _fig. 104._
By such means the surfaces of the teeth roll upon each other with very
inconsiderable friction, and the direction in which the pressure is
excited is always that of a line M N, touching the two wheels, and
at right angles to the radii. Thus the pressure being always the same,
and acting with the same leverage, produces a uniform effect.
(260.) When wheels work together, their teeth must necessarily be of
the same size, and therefore the proportion of their circumferences may
always be estimated by the number of teeth which they carry. Hence it
follows, that in computing the power of compound wheel-work, the number
of teeth may always be used to express the circumferences respectively,
or the diameters which are proportional to these circumferences. When
teeth are raised upon an axle, it is generally called a _pinion_, and
in that case the teeth are called _leaves_. The rule for computing the
train of wheel-work given in (256.) will be expressed as follows: when
the wheel and axle carry teeth, multiply together the number of teeth
in each of the wheels, and next the number of leaves in each of the
pinions; the proportion of the two products will express the power of
the machine. If some of the wheels and axles carry teeth, and others
not, this computation may be made by using for those circumferences
which do not bear teeth the number of teeth which would fill them.
_Fig. 105._ represents a train of three wheels and pinions. The
wheel F which bears the power, and the axle which bears the weight,
have no teeth; but it is easy to find the number of teeth which they
would carry.
(261.) It is evident that each pinion revolves much more frequently in
a given time than the wheel which it drives. Thus, if the pinion C be
furnished with ten teeth, and the wheel E, which it drives, have sixty
teeth, the pinion C must turn six times, in order to turn the wheel
E once round. The velocities of revolution of every wheel and pinion
which work in one another will therefore have the same proportion as
their number of teeth taken in a reverse order, and by this means the
relative velocity of wheels and pinions may be determined according to
any proposed rate.
Wheel-work, like all other machinery, is used to transmit and modify
force in every department of the arts and manufactures; but it is also
used in cases where motion alone, and not force, is the object to be
attained. The most remarkable example of this occurs in watch and
clock-work, where the object is merely to produce uniform motions of
rotation, having certain proportions, and without any regard to the
elevation of weights, or the overcoming of resistances.
(262.) A _crane_ is an example of combination of wheel-work used for
the purpose of raising or lowering great weights. _Fig. 106._
represents a machine of this kind. A B is a strong vertical beam,
resting on a pivot, and secured in its position by beams in the floor.
It is capable, however, of turning on its axis, being confined between
rollers attached to the beams and fixed in the floor. C D is a
projecting arm called a _gib_, formed of beams which are mortised into
A B. The wheel-work is mounted in two cast-iron crosses, bolted on
each side of the beams, one of which appears at E F G H.
The winch at which the power is applied is at I. This carries a pinion
immediately behind H. This pinion works in a wheel K, which carries
another pinion upon its axle. This last pinion works in a larger wheel
L, which carries upon its axis a barrel M, on which a chain or rope
is coiled. The chain passes over a pulley D at the top of the gib. At
the end of the chain a hook O is attached, to support the weight W.
During the elevation of the weight it is convenient that its recoil
should be hindered in case of any occasional suspension of the power.
This is accomplished by a ratchet wheel attached to the barrel M, as
explained in (253.); but when the weight W is to be lowered, the catch
must be removed from this ratchet wheel. In this case the too rapid
descent of the weight is in some cases checked by pressure excited on
some part of the wheel-work, so as to produce sufficient friction to
retard the descent in any required degree, or even to suspend it, if
necessary. The vertical beam at B resting on a pivot, and being fixed
between rollers, allows the gib to be turned round in any direction; so
that a weight raised from one side of the crane may be carried round,
and deposited on another side, at any distance within the range of the
gib. Thus, if a crane be placed upon a wharf near a vessel, weights may
be raised, and when elevated, the gib may be turned round so as to let
them descend into the hold.
The power of this machine may be computed upon the principles already
explained. The magnitude of the circle, in which the power at I moves,
may be determined by the radius of the winch, and therefore the number
of teeth which a wheel of that size would carry may be found. In
like manner we may determine the number of leaves in a pinion whose
magnitude would be equal to the barrel M. Let the first number be
multiplied by the number of teeth in the wheel K, and that product
by the number of teeth in the wheel L. Next let the number of leaves
in the pinion H be multiplied by the number of leaves in the pinion
attached to the axle of the wheel K, and let that product be multiplied
by the number of leaves in a pinion, whose diameter is equal to that of
the barrel M. These two products will express the power of the machine.
(263.) Toothed wheels are of three kinds, distinguished by the position
which the teeth bear with respect to the axis of the wheel. When they
are raised upon the edge of the wheel as in _fig. 105._, they are
called _spur wheels_, or _spur gear_. When they are raised parallel to
the axis, as in _fig. 107._, it is called a _crown wheel_. When
the teeth are raised on a surface inclined to the plane of the wheel,
as in _fig. 108._, they are called _bevelled wheels_.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
If a motion round one axis is to be communicated to another axis
parallel to it, spur gear is generally used. Thus, in _fig. 105._,
the three axes are parallel to each other. If a motion round one
axis is to be communicated to another at right angles to it, a crown
wheel, working in a spur pinion, as in _fig. 107._, will serve.
Or the same object may be obtained by two bevelled wheels, as in
_fig. 108._
If a motion round one axis is required to be communicated to another
inclined to it at any proposed angle, two bevelled wheels can always be
used. In _fig. 109._ let A B and A C be the two axles;
two bevelled wheels, such as D E and E F, on these axles will
transmit the motion or rotation from one to the other, and the relative
velocity may, as usual, be regulated by the proportional magnitude of
the wheels.
(264.) In order to equalise the wear of the teeth of a wheel and
pinion, which work in one another, it is necessary that every leaf
of the pinion should work in succession through every tooth of the
wheel, and not continually act upon the same set of teeth. If the
teeth could be accurately shaped according to mathematical principles,
and the materials of which they are formed be perfectly uniform, this
precaution would be less necessary; but as slight inequalities, both
of material and form, must necessarily exist, the effects of these
should be as far as possible equalised, by distributing them through
every part of the wheel. For this purpose it is usual, especially
in mill-work, where considerable force is used, so to regulate the
proportion of the number of teeth in the wheel and pinion, that the
same leaf of the pinion shall not be engaged twice with any one tooth
of the wheel, until after the action of a number of teeth, expressed
by the product of the number of teeth in the wheel and pinion. Let us
suppose that the pinion contains ten leaves, which we shall denominate
by the numbers 1, 2, 3, &c., and that the wheel contains 60 teeth
similarly denominated. At the commencement of the motion suppose the
leaf 1 of the pinion engages the tooth 1 of the wheel; then after
one revolution the leaf 1 of the pinion will engage the tooth 11 of
the wheel, and after two revolutions the leaf 1 of the pinion will
engage the tooth 21 of the wheel; and in like manner, after 3, 4, and
5 revolutions of the pinion, the leaf 1 will engage successively the
teeth 31, 41, and 51 of the wheel. After the sixth revolution, the
leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus
it is evident, that in the case here supposed the leaf 1 of the pinion
will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51
of the wheel, and no others. The like may be said of every leaf of the
pinion. Thus the leaf 2 of the pinion will be successively engaged with
the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any
accidental inequalities of these teeth will therefore continually act
upon each other, until the circumference of the wheel be divided into
parts of ten teeth each, unequally worn. This effect would be avoided
by giving either the wheel or pinion one tooth more or one tooth less.
Thus, suppose the wheel, instead of having sixty teeth, had sixty-one,
then after six revolutions of the pinion the leaf 1 of the pinion would
be engaged with the tooth 61 of the wheel; and after one revolution of
the wheel, the leaf 2 of the pinion would be engaged with the tooth 1
of the wheel. Thus, during the first revolution of the wheel the leaf
1 of the pinion would be successively engaged with the teeth 1, 11,
21, 31, 41, 51, and 61 of the wheel: at the commencement of the second
revolution of the wheel the leaf 2 of the pinion would be engaged with
the tooth 1 of the wheel; and during the second revolution of the wheel
the leaf 1 of the pinion would be successively engaged with the teeth
10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be
shown, that in the third revolution of the wheel the leaf 1 of the
pinion would be successively engaged with the teeth 9, 19, 29, 39, 49,
and 59 of the wheel: during the fourth revolution of the wheel the
leaf 1 of the pinion would be successively engaged with the teeth 8,
18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it
will appear, that during the tenth revolution of the wheel the leaf
1 of the pinion will be engaged successively with the teeth 2, 12,
22, 32, 42, and 52 of the wheel. At the commencement of the eleventh
revolution of the wheel the leaf 1 of the pinion will be engaged with
the tooth 1 of the wheel, as at the beginning of the motion. It is
evident, therefore, that during the first ten revolutions of the wheel
each leaf of the pinion has been successively engaged with every tooth
of the wheel, and that during these ten revolutions the pinion has
revolved sixty-one times. Thus the leaves of the pinion have acted six
hundred and ten times upon the teeth of the wheel, before two teeth can
have acted twice upon each other.
The odd tooth which produces this effect is called by millwrights the
_hunting cog_.
(265.) The most familiar case in which wheel-work is used to produce
and regulate motion merely, without any reference to weights to be
raised or resistances to be overcome, is that of chronometers. In watch
and clock work the object is to cause a wheel to revolve with a uniform
velocity, and at a certain rate. The motion of this wheel is indicated
by an index or hand placed upon its axis, and carried round with it.
In proportion to the length of the hand the circle over which its
extremity plays is enlarged, and its motion becomes more perceptible.
This circle is divided, so that very small fractions of a revolution
of the hand may be accurately observed. In most chronometers it is
required to give motion to two hands, and sometimes to three. These
motions proceed at different rates, according to the subdivisions of
time generally adopted. One wheel revolves in a minute, bearing a
hand which plays round a circle divided into sixty equal parts; the
motion of the hand over each part indicating one second, and a complete
revolution of the hand being performed in one minute. Another wheel
revolves once, while the former revolves sixty times; consequently the
hand carried by this wheel revolves once in sixty minutes, or one hour.
The circle on which it plays is, like the former, divided into sixty
equal parts, and the motion of the hand over each division is performed
in one minute. This is generally called the _minute hand_, and the
former the _second hand_.
A third wheel revolves once, while that which carries the minute hand
revolves twelve times; consequently this last wheel, which carries
the _hour hand_, revolves at a rate twelve times less than that of
the minute hand, and therefore seven hundred and twenty times less
than the second hand. We shall now endeavour to explain the manner in
which these motions are produced and regulated. Let A, B, C, D, E,
_fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_
represent their pinions, _e_ being a cylinder on the axis of the wheel
E, round which a rope is coiled, sustaining a weight W. Let the effect
of this weight transmitted through the train of wheels be opposed by
a power P acting upon the wheel A, and let this power be supposed
to be of such a nature as to cause the weight W to descend with a
uniform velocity, and at any proposed rate. The wheel E carries on its
circumference eighty-four teeth. The wheel D carries eighty teeth;
the wheel C is also furnished with eighty teeth, and the wheel B with
seventy-five. The pinions _d_ and _c_ are each furnished with twelve
leaves, and the pinions _b_ and _a_ with ten.
If the power at P be so regulated as to allow the wheel A to revolve
once in a minute, with a uniform velocity, a hand attached to the axis
of this wheel will serve as the _second hand_. The pinion _a_ carrying
ten teeth must revolve seven times and a half to produce one revolution
of B, consequently fifteen revolutions of the wheel A will produce two
revolutions of the wheel B; the wheel B, therefore, revolves twice in
fifteen minutes. The pinion _b_ must revolve eight times to produce
one revolution of the wheel C, and therefore the wheel C must revolve
once in four quarters of an hour, or in one hour. If a hand be attached
to the axis of this wheel, it will have the motion necessary for the
minute hand. The pinion _c_ must revolve six times and two thirds to
produce one revolution of the wheel D, and therefore this wheel must
revolve once in six hours and two thirds. The pinion _d_ revolves seven
times for one revolution of the wheel E, and therefore the wheel E will
revolve once in forty-six hours and two thirds.
On the axis of the wheel C a second pinion may be placed, furnished
with seven leaves, which may lead a wheel of eighty-four teeth, so
that this wheel shall turn once during twelve turns of the wheel C. If
a hand be fixed upon the axis, this hand will revolve once for twelve
revolutions of the minute hand fixed upon the axis of the wheel C;
that is, it will revolve once in twelve hours. If it play upon a dial
divided into twelve equal parts, it will move over each part in an
hour, and will serve the purpose of the hour hand of the chronometer.
We have here supposed that the second hand, the minute hand, and the
hour hand move on separate dials. This, however, is not necessary. The
axis of the hour hand is commonly a tube, inclosing within it that of
the minute hand, so that the same dial serves for both. The second
hand, however, is generally furnished with a separate dial.
(266.) We shall now explain the manner in which a power is applied
to the wheel A, so as to regulate and equalise the effect of the
weight W. Suppose the wheel A furnished with thirty teeth, as in
_fig. 111._; if nothing check the motion, the weight W would
descend with an accelerated velocity, and would communicate an
accelerated motion to the wheel A. This effect, however, is interrupted
by the following contrivance:--L M is a pendulum vibrating on the
centre L, and so regulated that the time of its oscillation is one
second. The pallets I and K are connected with the pendulum, so as to
oscillate with it. In the position of the pendulum represented in the
figure, the pallet I stops the motion of the wheel A, and entirely
suspends the action of the weight W, _fig. 110._, so that for
a moment the entire machine is motionless. The weight M, however,
falls by its gravity towards the lowest position, and disengages the
pallet I from the tooth of the wheel. The weight W begins then to take
effect, and the wheel A turns from A towards B. Meanwhile the pendulum
M oscillates to the other side, and the pallet K falls under a tooth
of the wheel A, and checks for a moment its further motion. On the
returning vibration the pallet K becomes again disengaged, and allows
the tooth of the wheel to escape, and by the influence of the weight W
another tooth passes before the motion of the wheel A is again checked
by the interposition of the pallet I.
From this explanation it will appear that, in two vibrations of the
pendulum, one tooth of the wheel A passes the pallet I, and therefore,
if the wheel A be furnished with 30 teeth, it will be allowed to make
one revolution during 60 vibrations of the pendulum. If, therefore, the
pendulum be regulated so as to vibrate seconds, this wheel will revolve
once in a minute. From the action of the pallets in checking the motion
of the wheel A, and allowing its teeth alternately to _escape_, this
has been called the _escapement_ wheel; and the wheel and pallets
together are generally called the _escapement_, or _’scapement_.
We have already explained, that by reason of the friction on the
points of support, and other causes, the swing of the pendulum would
gradually diminish, and its vibration at length cease. This, however,
is prevented by the action of the teeth of the scapement wheel upon the
pallets, which is just sufficient to communicate that quantity of force
to the pendulum which is necessary to counteract the retarding effects,
and to maintain its motion. It thus appears, that although the effect
of the gravity of the weight W in giving motion to the machine is at
intervals suspended, yet this part of the force is not lost, being,
during these intervals, employed in giving to the pendulum all that
motion which it would lose by the resistances to which it is inevitably
exposed.
In stationary clocks, and in other cases in which the bulk of the
machine is not an objection, a descending weight is used as the
moving power. But in watches and portable chronometers, this would be
attended with evident inconvenience. In such cases, a spiral spring,
called the _mainspring_, is the moving power. The manner in which this
spring communicates rotation to an axis, and the ingenious method of
equalising the effect of its variable elasticity by giving to it a
leverage, which increases as the elastic force diminishes, have been
already explained. (255.)
A similar objection lies against the use of a pendulum in portable
chronometers. A spiral spring of a similar kind, but infinitely
more delicate, called a _hair spring_, is substituted in its place.
This spring is connected with a nicely-balanced wheel, called _the
balance wheel_, which plays in pivots. When this wheel is turned to
a certain extent in one direction, the hair spring is coiled up, and
its elasticity causes the wheel to recoil, and return to a position
in which the energy of the spring acts in the opposite direction.
The balance wheel then returns, and continually vibrates in the same
manner. The axis of this wheel is furnished with pallets similar to
those of the pendulum, which are alternately engaged with the teeth of
a crown wheel, which takes the place of the scapement wheel already
described.
A general view of the work of a common watch is represented in
_fig. 111._ _bis._ A is the balance wheel bearing pallets _p_
_p_ upon its axis; C is the crown wheel, whose teeth are suffered to
escape alternately by those pallets in the manner already described
in the scapement of a clock. On the axis of the crown wheel is placed
a pinion _d_, which drives another crown wheel K. On the axis of this
is placed the pinion _c_, which plays in the teeth of the third wheel
L. The pinion _b_ on the axis of L is engaged with the wheel M, called
the centre wheel. The axle of this wheel is carried up through the
centre of the dial. A pinion _a_ is placed upon it, which works in
the great wheel N. On this wheel the mainspring immediately acts.
O P is the mainspring stripped of its barrel. The axis of the
wheel M passing through the centre of the dial is squared at the end
to receive the minute hand. A second pinion Q is placed upon this
axle which drives a wheel T. On the axle of this wheel a pinion _g_
is placed, which drives the hour wheel V. This wheel is placed upon a
tubular axis, which incloses within it the axis of the wheel M. This
tubular axis passing through the centre of the dial, carries the hour
hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels
C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_,
_e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_,
_fig. 111_. From what has already been explained of these wheels,
it will be obvious that the wheel M, _fig. 111._, revolves once
in an hour, causing the minute hand to move round the dial once in
that time. This wheel at the same time turns the pinion Q which leads
the wheel T. This wheel again turns the pinion _g_ which leads the
hour wheel V. The leaves and teeth of these pinions and wheels are
proportioned, as already explained, so that the wheel V revolves once
during twelve revolutions of the wheel M. The hour hand, therefore,
which is carried by the tubular axle of the wheel V, moves once round
the dial in twelve hours.
Our object here has not been to give a detailed account of watch and
clock work, a subject for which we must refer the reader to the proper
department of this work. Such a general account has only been attempted
as may explain how tooth and pinion work may be applied to regulate
motion.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
CHAP. XV.
OF THE PULLEY.