V¹ = v'(1 - 5/4)= -¼v':
the planet wheel, or epicycloidal yoke, then, has the higher speed, so
that if it be desired to "gear up," and drive the propeller faster than
the engine goes (and this, we believe, was the purpose of the inventor),
the pin-wheel must be made the driver; which is the reverse of
advantageous in respect to the relative amounts of approaching and
receding action.
In Figs. 40 and 41 are given the skeletons of Galloway's device for ratios
of 3:4 and 2:3 respectively, the former having four branches and three
pins, the latter three branches and two pins. Following the analogy, it
would seem that the next step should be to employ two branches with only
one pin; but the rectilinear hypocycloid of Fig. 38 is a complete
diameter, and the second branch is identical with the first; the straight
tooth, then, could theoretically drive the pin half way round, but upon
its reaching the center of the outer wheel, the driving action would
cease: this renders it necessary to employ two pins and two slots, but it
is not essential that the latter should be perpendicular to each other.
In these last arrangements, the forms of the parts are so different from
those of ordinary wheels, that the true nature of the combinations is at
least partially disguised. But it may be still more completely hidden, as
for instance in the common elliptic trammel, Fig. 42. The slotted cross is
here fixed, and the pins, R and P, sliding respectively in the vertical
and horizontal lines, control the motion of the bar which carries the
pencil, S. At first glance there would seem to be nothing here resembling
wheel works. But if we describe a circle upon R P as a diameter, its
circumference will always pass through C, because R C P is a right angle,
and the instantaneous axis of the bar being at the intersection O of a
vertical line through P, with a horizontal line through R, will also lie
upon this circumference. Again, since O is diametrically opposite to C, we
have C O = R P, whence a circle about center C with radius R P will also
pass through O, which therefore is the point of contact of these two
circles. It will now be seen that the motion of the bar is the same as
though carried by the inner circle while rolling within the outer one, the
latter being fixed; the points P and R describing the diameters L M and K
N, the point D a circle, and S an ellipse; C D being the train-arm. The
distance R P being always the diameter of one circle and the radius of the
other, the sizes of the wheels can be in effect varied by altering that
distance.
Thus we see that this combination is virtually the same in its action as
the one shown in Fig. 43, known as Suardi's Geometrical Pen. In this
particular case the diameter of _a_ is half of that of A; these wheels are
connected by the idler, E, which merely reverses the direction without
affecting the velocity of _a's_ rotation. The working train arm is jointed
so as to pivot about the axis of E, and may be clamped at any angle within
its range, thus changing the length of the virtual train arm, C D. The bar
being fixed to _a_, then, moves as though carried by the wheel, _a¹_,
rolling within A¹; the radius of _a¹_ being C D, and that of A¹ twice as
great.
In either instrument, the semi-major axis C X is equal to S R, and the
semi-minor axis to S P.
The _ellipse_, then, is described by these arrangements because it is a
special form of the epitrochoid; and various other epitrochoids may be
traced with Suardi's pen by substituting other wheels, with different
numbers of teeth, for a in Fig. 43.
Another disguised planetary arrangement is found in Oldham's coupling,
Fig. 44. The two sections of shafting, A and B, have each a flange or
collar forged or keyed upon them; and in each flange is planed a
transverse groove. A third piece, C, equal in diameter to the flanges, is
provided on each side with a tongue, fitted to slide in one of the
grooves, and these tongues are at right angles to each other. The axes of
A and B must be parallel, but need not coincide; and the result of this
connection is that the two shafts will turn in the same direction at the
same rate.
The fact that C in this arrangement is in reality a planetary wheel, will
be perceived by the aid of the diagram, Fig. 45. Let C D be two pieces
rotating about fixed parallel axes, each having a groove in which slides
freely one of the arms, A C, A D, which are rigidly secured to each other
at right angles.
The point C of the upper arm can at the instant move only in the direction
C A; and the point D of the lower arm only in the direction A D, at the
same instant; the instantaneous axis is therefore at the intersection, K,
of perpendiculars to A C and A D, at the points C and D. C A D K being
then a rectangle, A K and C D will be two diameters of a circle whose
center, O, bisects C D; and K will also be the point of contact between
this circle and another whose center is A, and radius A K = C D. If then
we extend the arms so as to form the cross, P K, M N, and suppose this to
be carried by the outer circle, _f_, rolling upon the inner one, F, its
motion will be the same as that determined by the pieces, C D; and such a
cross is identical with that formed by the tongues on the coupling-piece,
C, of Fig. 44.
A O is the virtual train-arm; let the center, A, of the cross move to the
position B, then since the angles A O B at the center, and A C B in the
circumference, stand on the same arc, A B, the former is double the
latter, showing that the cross revolves twice round the center O during
each rotation of C; and since A C B = A D B, C and D rotate with equal
velocities, and these rotations and the revolution about O have the same
direction. While revolving, the cross rotates about its traveling center,
A, in the opposite direction, the contact between the two circles being
internal, and at a rate equal to that of the rotations of C and D, because
the velocities of the axial and the orbital motion are to each other as
_f_ is to F, that is to say, as 1 is to 2. Since in the course of the
revolution the points P and K must each coincide with C, and the points M
and N with D, it follows that each tongue in Fig. 44 must slide in its
groove a distance equal to twice that between the axes of the shafts.
Another example of a disguised planetary train is shown in Fig. 46. Let C
be the center about which the train arm, T, revolves, and suppose it
required that the distant shaft, B, carried by T, shall turn once backward
for each forward revolution of the arm. E is a fixed eccentric of any
convenient diameter, in the upper side of which is a pin, D. On the shaft,
B, is keyed a crank, B G, equal in length to C D; and at any convenient
point, H, on B C, or its prolongation, another crank, H F, equal also to C
D, is provided with a bearing in the train-arm. The three crank pins, F,
D, G, are connected by a rod, like the parallel rod of a locomotive; F D,
D G, being respectively equal to H C, C B. Then, as the train-arm
revolves, the three cranks must remain parallel to each other; but C D
being fixed, the cranks, H F and B G, will remain always parallel to their
original positions, thus receiving the required motion of circular
translation.
The result then is the same as though the periphery of E were formed into
a fixed spurwheel, A, and another, _a_, of the same size, secured on a
shaft, B, the two being connected by the three equal wheels, L, M, N. It
need hardly be stated that instead of the eccentric, E, a stationary crank
similar and equal to B G may be used, should it be found better suited to
the circumstances of the case.
It is possible also to apply the planetary principle to mechanism composed
partially of racks; in fact, a rack is merely a wheel of prodigious
size--the limiting case, just as a right line is a circle of infinite
radius. A very neat application of this principle is found in Villa's
Pantograph, of which a full description and illustration was given in
SCIENTIFIC AMERICAN SUPPLEMENT, No. 424; the racks, moving side by side,
are the sun-wheels, and the planet-wheels are the pinions, carried by the
traveling socket, by which the motion of one rack is transmitted to the
other.
Thus far attention has been called only to combinations of circular
wheels. In these the velocity ratios are constant, if we except the cases
in which two independent trains converge, the two sun-wheels, or one of
them and the train-arm, being driven separately--and even in those, a
variable motion of the ultimate follower is obtained only by varying the
speed of one or both drivers. It is not, however, necessary to employ
circular wheels exclusively or even at all; wheels of other forms are
capable of acting together in the relation of sun and planet, and in this
way a varying velocity ratio may be produced even with a fixed sun-wheel
and a single driver. We have not found, in the works of any previous
writer, any intimation that noncircular wheels have ever been thus
combined; and we propose in the following article to illustrate some
curious results which may be thus obtained.
* * * * *