_Velocity Ratio in Belt Gearing._--Let two pulleys A and B be connected
by a belt, and let their diameters be D_{1} and D_{2}; and let their
speeds, in revolutions per minute, be N_{1} and N_{2} respectively. If
there is no slipping, the speeds of the rims of the pulleys will be the
same as that of the belt, and will therefore be equal. Now the speed of
the rim of A is evidently = D_{1} x 3.1416 x N_{1}; while the speed of
the rim of B is = D_{2} x 3.1416 x N_{2}. Hence D_{1} x 3.1416 x N_{1} =
D_{2} x 3.1416 x N_{2}, and therefore
N_{1} D_{2}
----- = -----.
N_{2} D_{1}
_Pulleys for Flat Bands._--In cross section the rim of a pulley for
carrying a flat band is generally curved as shown in figs. 31 and 32,
but very often the cross section is straight. The curved cross section
of the rim tends to keep the band from coming off as long as the pulley
is rotating. Sometimes the rim of the pulley is provided with flanges
which keep the band from falling off.
Pulleys are generally made entirely of cast iron, but a great many
pulleys are now made in which the centre or nave only is of cast iron,
the arms being of wrought iron cast into the nave, while the rim is of
wrought sheet iron.
The arms of pulleys when made of wrought iron are invariably straight,
but when made of cast iron they are very often curved. In fig. 31, which
shows an arrangement of two cast-iron pulleys, the arms are straight;
while in fig. 32, which shows another cast-iron pulley, the arms are
curved. Through unequal cooling, and therefore unequal contraction of a
cast-iron, pulley in the mould, the arms are generally in a state of
tension or compression; and if the arms are straight they are very
unyielding, so that the result of this initial stress is often the
breaking of an arm, or of the rim where it joins an arm. With the curved
arm, however, its shape permits it to yield, and thus cause a diminution
of the stress due to unequal contraction.
The cross section of the arms of cast-iron pulleys is generally
elliptical.
[Illustration: Fig. 31.]
EXERCISE 31: _Fast and Loose Pulleys_.--Fig. 31 shows an
arrangement of fast and loose pulleys. A is the fast pulley,
secured to the shaft C by a sunk key; B is the loose pulley, which
turns freely upon the shaft. The loose pulley is prevented from
coming off by a collar D, which is secured to the shaft by a
tapered pin as shown. The nave or boss of the loose pulley is here
fitted with a brass liner, which may be renewed when it becomes
too much worn. Draw the elevations shown, completing the left-hand
one. Scale 6 inches to a foot.
By the above arrangement of pulleys a machine may be stopped or
set in motion at pleasure. When the driving band is on the loose
pulley the machine is at rest, and when it is on the fast pulley
the machine is in motion. The driving band is shifted from the one
pulley to the other by pressing on that side of the band which is
advancing towards the pulleys.
[Illustration: FIG. 32.]
EXERCISE 32: _Cast-iron Pulley with Curved Arms and Cone
Keys_.--Draw a complete side elevation and a complete cross
section of the pulley represented in fig. 32 to a scale of 3
inches to a foot. In drawing the side elevation of the arms first
draw the centre lines as shown; next draw three circles for each
arm, one at each end and one in the middle; the centres of these
circles being on the centre line of the arm, and their diameters
equal to the widths of the arm at the ends and at the middle
respectively. Arcs of circles are then drawn to touch these three
circles. The centres and radii of these arcs may be found by
trial. The cone keys for securing the pulley to the shaft were
described on p. 23.
_Pulleys for Ropes_.--Ropes made of hemp are now extensively used for
transmitting power. These ropes vary in diameter from 1 inch to 2
inches, and are run at a speed of about 4,500 feet per minute. The
pulleys for these ropes are made of cast iron, and have their rims
grooved as shown in fig. 33, which is a cross section of the rim of a
pulley carrying three ropes. The angle of the V is usually 45 deg., and the
rope rests on the sides of the groove, and not on the bottom, so that
it is wedged in, and has therefore a good hold of the pulley. The
diameter of the pulley should not be less than 30 times the diameter of
the rope. Two pulleys connected by ropes should not be less than thirty
feet apart from centre to centre, but this distance may be as much as
100 feet.
[Illustration: FIG. 33.]
EXERCISE 33: _Section of Rim of Rope Pulley._--Draw, half size,
the section of the rim of a rope pulley shown in fig. 33.