|
Main
- books.jibble.org
My Books
- IRC Hacks
Misc. Articles
- Meaning of Jibble
- M4 Su Doku
- Computer Scrapbooking
- Setting up Java
- Bootable Java
- Cookies in Java
- Dynamic Graphs
- Social Shakespeare
External Links
- Paul Mutton
- Jibble Photo Gallery
- Jibble Forums
- Google Landmarks
- Jibble Shop
- Free Books
- Intershot Ltd
|
books.jibble.org
Previous Page
| Next Page
Page 12
[Illustration: PLANETARY WHEEL TRAINS. Fig. 15]
To illustrate: Take the simple case of two equal wheels, Fig. 15, of
which the central one A is fixed. Supposing first A for the moment
released and the arm to be fixed, we see that the two wheels will turn
in opposite directions with equal velocities, which gives _n_/_m_ = -1;
but when A is fixed and T revolves, we have _m'_ = 0, whence in the
general formula
n' - a
------ = -1, or n' = 2 a;
-a
which means, being interpreted, that F makes two rotations about its
axis during one revolution of T, and in the same direction. Again, let
A and F be equal in the 3-wheel train, Fig. 16, the former being fixed
as before. In this case we have:
n
--- = 1, m' = 0, which gives
m
n' - a
------- = 1, [therefore] n' = 0;
-a
that is to say, the wheel F, which now evidently has a motion of
circular translation, does not rotate at all about its axis during the
revolution of the train-arm.
[Illustration: PLANETARY WHEEL TRAINS. Fig. 16]
All this is perfectly consistent, clearly, with the hypothesis that
the motion of circular translation is a simple one, and the motion of
revolution about a fixed axis is a compound one.
Whether the hypothesis was made to substantiate the formula, or the
formula constructed to suit the hypothesis, is not a matter of
consequence. In either case, no difficulty will arise so long as the
equation is applied only to cases in which, as in those here
mentioned, that motion of revolution _can_ be resolved into those
components.
When the definition of an epicyclic train is restricted as it is by
Prof. Rankine, the consideration of the hypothesis in question is
entirely eliminated, and whether it be accepted or rejected, the whole
matter is reduced to merely adding the motion of the train-arm to the
rotation of each sun-wheel.
But in attempting to apply this formula in analyzing the action of an
incomplete train, we are required to add this motion of the train-arm,
not only to that of a sun-wheel, but to that of a planet-wheel. This
is evidently possible in the examples shown in Figs. 15 and 16,
because the motions to be added are in all respects similar: the
trains are composed of spur-wheels, and the motions, whether of
revolution, translation, or rotation, _take place in parallel planes
perpendicular to parallel axes_. This condition, which we have
emphasized, be it observed, must hold true with regard to the motions
of the first and last wheels and the train-arm, in order to make this
addition possible. It is not essential that spur-wheels should be used
exclusively or even at all; for instance, in Fig. 16, A and F may be
made bevel or screw-wheels, without affecting the action or the
analysis; but the train-arm in all cases revolves around the central
axis of the system, that is, about the axis of A, and to this the axis
of F _must_ be parallel, in order to render the deduction of the
formula, as made by Prof. Willis, and also by Prof. Goodeve, correct,
or even possible.
[Illustration: PLANETARY WHEEL TRAINS. Fig. 17]
This will be seen by an examination of Fig. 17; in which A and B are
two equal spur-wheels, E and F two equal bevel wheels, B and E being
secured to the same shaft, and A being fixed to the frame H. As the
arm T goes round, B will also turn in its bearings in the same
direction: let this direction be that of the clock, when the apparatus
is viewed from above, then the motion of F will also have the same
direction, when viewed from the central vertical axis, as shown at F':
and let these directions be considered as positive. It is perfectly
clear that F will turn in its bearings, in the direction indicated, at
a rate precisely equal to that of the train-arm. Let P be a pointer
carried by F, and R a dial fixed to T; and let the pointer be vertical
when OO is the plane containing the axes of A, B, and E. Then, when F
has gone through any angle a measured from OO, the pointer will have
turned from its original vertical position through an equal angle, as
shown also at F'.
Previous Page
| Next Page
|
|