The arrangement of planetary wheels which has been applied in practice to
the greatest extent and to the most purposes, is probably that in which
the axial motions of the train are derived from a fixed sun wheel.
Numerous examples of such trains are met with in the differential gearing
of hoisting machines, in portable horse-powers, etc. The action of these
mechanisms has already been fully discussed; it may be remarked in
addition that unless the speed be very moderate, it is found advantageous
to balance the weights and divide the pressures by extending the train arm
and placing the planet-wheels in equal pairs diametrically opposite each
other, as, for instance, in Bogardus' horse power, Fig. 31.
[Illustration: PLANETARY WHEEL TRAINS.]
In trains of this description, the velocity ratio is invariable; which for
the above-mentioned objects it should be. But the use of a planetary
combination enables us to cause the motions of two independent trains to
converge, and unite in producing a single resultant rotation. This may be
done in two ways; each of the two independent trains may drive one
sun-wheel, thus determining the motion of the train-arm; or, the train-arm
may be driven by one of them, and the first sun-wheel by the other; then
the motion of the second sun-wheel is the resultant. Under these
circumstances the ratio of the resultant velocity to that of either
independent train is not invariable, since it may be affected by a change
in the velocity of the other one. To illustrate our meaning, we give two
examples of arrangements of this nature. The first is Robinson's
rope-making machine, Fig. 32. The bobbins upon which the strands composing
the rope are wound turn freely in bearings in the frames, G, G, and these
frames turn in bearings in the disk, H, and the three-armed frame or
spider, K, both of which are secured to the central shaft, S. Each
bobbin-frame is provided with a pinion, _a_, and these three pinions
engage with the annular wheel, A. This wheel has no shaft, but is carried
and kept in position by three pairs of rollers, as shown, so that its axis
of rotation is the same as that of the shaft, S; and it is toothed
externally as well as internally. The strands pass through the hollow
axes of the pinions, and thence each to its own opening through the
laying-top, T, fixed upon S, which completes the operation of twisting
them into a rope. The annular wheel, A, it will be perceived, may be
driven by a pinion, E, engaging with its external teeth, at a rate of
speed different from that of the central shaft; and by varying the speed
of that pinion, the velocity of the wheel, A, may be changed without
affecting the velocity of S.
It is true that in making a certain kind of rope, the velocity ratio of A
and S must remain constant, in order that the strands may be equally
twisted throughout; but if for another kind of rope a different degree of
twist is wanted, the velocity of the pinion, E, may be altered by means of
change-wheels, and thus the same machine may be used for manufacturing
many different sorts.
The second combination of this kind was devised by the writer as a
"tell-tale" for showing whether the engines driving a pair of twin
screw-propellers were going at the same rate. In Fig. 33, an index, P, is
carried by the wheel, F: the wheel, A, is loose upon the shaft of the
train-arm, which latter is driven by the wheel, E. The wheels, F and _f_,
are of the same size, but _a_ is twice as large as A; if then A be driven
by one engine, and E by the other, at the same rate but in the opposite
direction, the index will remain stationary, whatever the absolute
velocities. But if either engine go faster than the other, the index will
turn to the right or the left accordingly. The same object may also be
accomplished as shown in Fig. 34, the index being carried by the
train-arm. It makes no difference what the actual value of the ratio A/_a_
may be, but it must be equal to F/_f_: under which condition it is evident
that if A and F be driven contrary ways at equal speeds, small or great,
the train-arm will remain at rest; but any inequality will cause the index
to turn.
In some cases, particularly when annular wheels are used, the train-arm
may become very short, so that it may be impossible to mount the
planet-wheel in the manner thus far represented, upon a pin carried by a
crank. This difficulty may be surmounted as shown in Fig. 35, which
illustrates an arrangement originally forming a part of Nelson's steam
steering gear. The Internal pinions, _a_, _f_, are but little smaller than
the annular wheels, A, F, and are hung upon an eccentric E formed in one
solid piece with the driving shaft, D.
The action of a complete epicyclic train involves virtually and always the
action of two suns and two planets; but it has already been shown that the
two planets may merge into one piece, as in Fig. 10, where the
planet-wheel gears externally with one sun-wheel, and internally with the
other.
But the train may be reduced still further, and yet retain the essential
character of completeness in the same sense, though composed actually of
but two toothed wheels. An instance of this is shown in Fig. 36, the
annular planet being hung upon and carried by the pins of three cranks,
_c_, _c_, _c_, which are all equal and parallel to the virtual train-arm,
T. These cranks turning about fixed axes, communicate to _f_ a motion of
circular translation, which is the resultant of a revolution, _v'_, about
the axis of F in one direction, and a rotation, _v_, at the same rate in
the opposite direction about its own axis, as has been already explained.
The cranks then supply the place of a fixed sun-wheel and a planet of
equal size, with an intermediate idler for reversing the, direction of the
rotation of the planet; and the velocity of F is
V'= v'(1 - f/F).
A modification of this train better suited for practical use is shown in
Fig. 37, in which the sun-wheel, instead of the planet, is annular, and
the latter is carried by the two eccentrics, E, E, whose throw is equal to
the difference between the diameters of the two pitch circles; these
eccentrics must, of course, be driven in the same direction and at equal
speeds, like the cranks in Fig. 36.
[Illustration: PLANETARY WHEEL TRAINS.]
A curious arrangement of pin-gearing is shown in Fig. 38: in this case the
diameter of the pinion is half that of the annular wheel, and the latter
being the driver, the elementary hypocycloidal faces of its teeth are
diameters of its pitch circle; the derived working tooth-outlines for pins
of sensible diameter are parallels to these diameters, of which fact
advantage is taken to make the pins turn in blocks which slide in straight
slots as shown. The formula is the same as that for Fig. 36, viz.:
V' = v'(1 - f/F),
which, since f = 2F, reduces to V' = -v'.
Of the same general nature is the combination known as the "Epicycloidal
Multiplying Gear" of Elihu Galloway, represented in Fig. 39. Upon
examination it will be seen, although we are not aware that attention has
previously been called to the fact, that this differs from the ordinary
forms of "pin gearing" only in this particular, viz., that the elementary
tooth of the driver consists of a complete branch, instead of a
comparatively small part of the hypocycloid traced by rolling the smaller
pitch-circle within the larger. It is self-evident that the hypocycloid
must return into itself at the point of beginning, without crossing: each
branch, then, must subtend an aliquot part of the circumference, and can
be traced also by another and a smaller describing circle, whose diameter
therefore must be an aliquot part of the diameter of the outer
pitch-circle; and since this last must be equal to the sum of the
diameters of the two describing circles, it follows that the radii of the
pitch circles must be to each other in the ratio of two successive
integers; and this is also the ratio of the number of pins to that of the
epicycloidal branches.
Thus in Fig. 39, the diameters of the two pitch circles are to each other
as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used. These pins
must in practice have a sensible diameter, and in order to reduce the
friction this diameter is made large, and the pins themselves are in the
form of rollers. The original hypocycloid is shown in dotted line, the
working curve being at a constant normal distance from it equal to the
radius of the roller; this forms a sort of frame or yoke, which is hung
upon cranks as in Figs. 36 and 38. The expression for the velocity ratio
is the same as in the preceding case: