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Page 13
Now, there is no conceivable sense in which the motion of T can be
said to be added to the rotation of F about its axis, and the
expression "absolute revolution," as applied to the motion of the last
wheel in this train, is absolutely meaningless.
Nevertheless, Prof. Goodeve states (Elements of Mechanism, p. 165)
that "We may of course apply the general formula in the case of bevel
wheels just as in that of spur wheels." Let us try the experiment;
when the train-arm is stationary, and A released and turned to the
right, F turns to the left at the same rate, whence:
n
--- = -1; also m' = 0 when A is fixed,
m
and the equation becomes
n' - a
------ = -1, [therefore] n' = 2a:
- a
or in other words F turns _twice_ on its axis during one revolution of
T: a result too palpably absurd to require any comment. We have seen
that this identical result was obtained in the case of Fig. 15, and it
would, of course, be the same were the formula applied to Figs. 5 and
6; whereas it has never, so far as we are aware, been pretended that a
miter or a bevel wheel will make more than one rotation about its axis
in rolling once around an equal fixed one.
Again, if the formula be general, it should apply equally well to a
train of screw wheels: let us take, for example, the single pair shown
in Fig. 8, of which, when T is fixed, the velocity ratio is unity. The
directional relation, however, depends upon the direction in which the
wheels are twisted: so that in applying the formula, we shall have
_n/m_ = +1, if the helices of both wheels are right handed, and
_n_/_m_ = -1, if they are both left handed. Thus the formula leads to
the surprising conclusion, that when A is fixed and T revolves, the
planet-wheel B will revolve about its axis twice as fast as T moves,
in one case, while in the other it will not revolve at all.
[Illustration: PLANETARY WHEEL TRAINS. Fig. 18]
A favorite illustration of the peculiarities of epicyclic mechanism,
introduced both by Prof. Willis and Prof. Goodeve, is found in the
contrivance known as Ferguson's Mechanical Paradox, shown in Fig. 18.
This consists of a fixed sun-wheel A, engaging with a planet-wheel B
of the same diameter. Upon the shaft of B are secured the three thin
wheels E, G, I, each having 20 teeth, and in gear with the three
others F, H, K, which turn freely upon a stud fixed in the train-arm,
and have respectively 19, 20, and 21 teeth. In applying the general
formula, we have the following results:
n 20 n' - a 1
For the wheel F, --- = ---- = ---------, [therefore] n' = - ---- a.
m 19 -a 19
n n' - a
" " " H, --- = 1 = --------, [therefore] n' = 0.
m -a
n 20 n' - a 1
" " " K, --- = ---- = ---------, [therefore] n' = + ---- a.
m 21 -a 21
The paradoxical appearance, then, consists in this, that although the
drivers of the three last wheels each have the same number of teeth,
yet the central one, H, having a motion of circular translation,
remains always parallel to itself, and relatively to it the upper one
seems to turn in the same direction as the train-arm, and the lower in
the contrary direction. And the appearance is accepted, too, as a
reality; being explained, agreeably to the analysis just given, by
saying that H has no absolute rotation about its axis, while the other
wheels have; that of F being positive and that of K negative.
[Illustration: PLANETARY WHEEL TRAINS. Fig. 18]
The Mechanical Paradox, it is clear, may be regarded as composed of
three separate trains, each of which is precisely like that of Fig.
16: and that, again, differs from the one of Fig. 15 only in the
addition of a third wheel. Now, we submit that the train shown in Fig.
17 is mechanically equivalent to that of Fig. 15; the velocity ratio
and the directional relation being the same in both. And if in Fig. 17
we remove the index P, and fix upon its shaft three wheels like E, G,
and I of Fig. 18, we shall have a combination mechanically equivalent
to Ferguson's Paradox, the three last wheels rotating in vertical
planes about horizontal axes. The relative motions of those three
wheels will be the same, obviously, as in Fig. 18; and according to
the formula their absolute motions are the same, and we are invited to
perceive that the central one does not rotate at all about its axis.
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