Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

i U
MECHANICS.
Practical 0f the leaves of the pinion (liould be parts of an interior
l1t' Jlnlcs'i epicycloid, generated by a circle of any diameter rolling
upon the concave fuperficies of the pinion, or within the
circle a d h; and the faces a b oi the teeth of the wheel
ihould be portions of an exterior epicycloid formed by
the fame generating circle rolling upon the convex lu-
perficies o dp oi the wheel.
398. But when one circle rolls within another whofc
diameter is double that of the rolling circle, the line ge¬
nerated by any point of the latter is a fraight line, tend¬
ing to the centre of the larger circle. Therefore, if the
generating circle above mentioned Ihould be taken with
its diameter equal to the radius of the pinion, and be
made to roll upon the concave fuperficies a d h of the
pinion, it will generate a ftraight line tending to the
pinion’s centre, which will be the form of the faces of
its leaves ; and the teeth of the wheel will be exterior
epicycloids, formed by a generating circle, whofe dia¬
meter is equal to the radius of the pinion, rolling upon
the convex fuperficies 0 dp of the wheel. Thisre&ili-
Fig. a* neal form of the teeth is exhibited in fig. 2. and is per¬
haps the moft advantageous, as it requires lefs trouble,
and may be executed with greater accuracy, than if the
epieycloidal form had been employed, though the teeth
are evidently weaker than thofe in fig. 1.5 it is recom¬
mended both by De la Hire and Camus as particularly
advantageous in clock and watch work.
Fig. 1. 399. The attentive reader will perceive from fig. 1.
that in order to prevent the teeth of the wheel from act¬
ing upon the leaves of the pinion before they reach the
line of centres AB; and that one tooth of the wheel
may not quit the leaf of the pinion till the fucceeding
tooth begins to adl upon the fucceeding leaf, there mull
be a certain proportion between the number of leaves in
the pinion and the number of teeth in the wheel, or
between the radius of the pinion and the radius of
the wheel, when the dillance of the leaves x^B is
given. But in machinery the number of leaves and
teeth is always known from the velocity which is re¬
quired at the working point of the machine : It be¬
comes a matter therefore of great importance to de¬
termine with accuracy the relative radii of the wheel
and pinion.
Relative 400< For this purpofe, let A, fig. 2. be the pinion hav-
fize of the a&ing faces of its leaves ftraight lines tending to
wheel anti the centre, and B the centre of the wheel. AB will be
pinion. the diftance of their centres. Then as the tooth C is fup-
pofed not to aft upon the leaf Am till it arrives at the
line AB, it ought not to quit Am till the following
tooth F has reached the line AB. But fince the tooth
always afts in the direflion of a line drawn perpendicu¬
lar to the face of the leaf Am from the point of contaft,
the line CH, drawn at right angles to the face of the
leaf Am, will determine the extremity of the tooth CD,
or the laft part of it which Ihould afl upon the leaf
Am, and will alfo mark out CD for the depth of the
tooth. Now, in order to find AH, HB, and CD, put
a for the number of teeth in the wheel, b for the num¬
ber ot leaves in the pinion, c for the diftance of the pi¬
vots A and B, and let x be the radius of the wheel, and
?/ that of the pinion. Then, fince the circumference of
the wheel is to the circumference of the pinion, as the
number of teeth in the one to the number of leaves in
the other, and as the circumferences of circles are pro¬
portional to their radii, we fhall have a : b—x : y, then
5
by compofition (Fuel. v. 18.) a-\-b : b~c : y {c being Practical
equal to a:+y), and confequently the radius of the pinion, f'echanics-i
cb , , . . ,
viz.ym—then by inverting the firft analogy, we
have b ; a=zy : x, and confequently the radius of the'
wheel, viz. x — y being now a known number.
Now, in the triangle AHC, right-angled at C, the
fide AH is known, and likewife ail the angles (HAC
being equal to ^ ; the fide AC, therefore, may be
found by plain trigonometry. Then, in the triangle
ACB, the ^iCxAB, equal to HAC, is known, and
alfo the fides AB, AC, ivhich contain it ; the third
fide, therefore, viz. CB, may be determined \ from
which DB, equal to HB, already found, being fub-
ftradfed, there will remain CD for the depth of the
teeth. When the adlion is carried on after the line
of centres, it often happens that the teeth will not
work in the hollows of the leaves. In order to pre¬
vent this, the CBH muft always be greater than
half the HBP. The ^ HEP is equal to 360
degrees, divided by the number of teeth in the wheel,
and CBH is eafily found by plane trigonometry.
401. If the teeth of wheels and the leaves of pinions be
formed according to the directions already given, they
will act upon each other, not only with uniform force,
but nearly without friction. The one tooth rolls upon
the other, and neither Aides nor rubs to fuch a degree
as to retard the wheels, or wear their teeth. But as
it is impoffible in practice to give that perfect curvature
to the faces of the teeth which theory requires, a quantity
of friction will remain after every precaution has been
taken in the formation of the communicating parts.
402. 2. The fecond mode of action is not fo advantage-gccor^
ous as that which we have been confidering, and Ihould, mociG 0f
if poflible, always be avoided. It is reprefented inadlion.
fig. 3. where A is the centre of the pinion, B that of
the wheel, and AB the line of centres. It is evident ^
from the figure that the tooth C of the wheel acts upon
the leaf D of the pinion before they arrive at the line
B A ; that it quits the leaf when they reach this line,
and have affumed the pofition of E and F; and that
the tooth c works deeper and deeper between the leaves
of the pinion, the nearer it comes to the line of centres.
From this laft circumftance a confiderable quantity of
frieftion arifes, becaufe the tooth C does not, as before,
roll upon the leaf D, but Aides upon it ; and from the
fame caufe the pinion foon becomes foul, as the duft
which lies upon the aefting faces of the leaves is pufiied
into the interjacent hollows. One advantage, how¬
ever, attends this mode of aeftion : It allows us to
make the teeth of the large wheel rectilineal, and thus
renders the labour of the mechanic lefs, and the ac¬
curacy of his work greater, than if they had been of a
curvilineal form. If the teeth C, E, therefore of the
wheel BC are made rectilineal, having their fin-faces
directed to the wheel’s centre, the acting faces of the
leaves D, F, &c. muft be epicycloids formed by a ge¬
nerating circle, whofe diameter is equal to the radius
B 0 of the circle op, rolling upon the circumference
m n of the pinion A. But if the teeth of the wheel
and the leaves of the pinion are made curvilineal as in
the figure, the faces of the teeth of the wheel muft be
portions of an interior epicycloid formed by any gene¬
rating