CHAPTER XIII.

TRAINS OF ELEMENTARY COMBINATIONS.

316. THE elementary combinations which have been the subject of the preceding chapters consist, for the most part, of two principal pieces only, a driver and a follower; and we have shown how to connect these so as to produce any required constant or varying velocity ratio, or constant directional relation, whatever may be the relative position of the axes of rotation. There are many cases, however, in which, although theoretically possible, it may be practically inconvenient, or even impossible, to effect the required communication of motion by a single combination; in which case a series or train of such combinations must be employed, in which the follower of the first combination of the train is carried by the same axis or sliding piece to which the driver of the second is attached; the follower of the second is similarly connected to the driver of the third, and so on.

317. In all the combinations hitherto considered the principal pieces either revolve or travel in right lines. In a train of revolving pieces, the first follower and second driver being fixed to the same axis, revolve with the same angular velocity; and this is true for the second follower and third driver, and generally for the mth follower and m+1th driver, which will also, if the piece which carries them travel in a right line, move with the same linear velocity. But, for simplicity, let us consider all the pieces in the train to revolve (Art. 36), and let the synchronal rotations of the axes of the train in order be

that is; the ratio of the synchronal rotations of the extreme axes

of

the train is found by multiplying together the separate synchronal ratios of the successive pairs of axes.

angular velocities of the axes, we have

Also if AA...4m be the

318. And since the values of any one of these separate ratios will be unaffected by the substitution of any pair of numbers that are in the same proportion, we may substitute indifferently in any one the numbers of teeth (N), the diameters (D), or radii (R), of rolling wheels, pitch-circles, or pullies, the periods (P) in uniform motion; or express the value of the ratio in any other equivalents that may be most easily obtained from the given machine or train whose motions we wish to calculate, recollecting that

319. Ex. 1.

In a train of wheel-work let the first axis carry a wheel of N1 teeth driving a wheel of n, teeth on the second axis; let the second axis carry also a wheel of N, teeth driving a wheel of n, teeth on the third axis, and so on.

that is, to find the ratio of the synchronal rotations, or angular velocity of the last axis in a given train of wheel-work to those of the first, multiply the numbers of all the drivers for a numerator, and of all the followers for a denominator.

It is scarcely necessary to remark that the number of drivers and of followers in a train of this kind is less by one than the number of axes.

320. Ex. 2. The ratios may each be expressed in a different manner: thus in a train of five axes, let the first revolve once while the second revolves three times;

Let the second carry a wheel of 60 teeth driving a pinion of 20 on the third;

Let the third axis drive the fourth by a belt and pair of pullies of 18 and 6 inches diameter respectively;

S

And let the fourth perform a revolution in ten seconds, and the last in two, when the machinery revolves uniformly;

that is to say, that the first axis will perform one revolution while the last revolves 135 times.

321. In this manner the synchronal rotations of the extreme axes in any given machine may be calculated; their directional relation may also be found, by examining in order the connection. of the axes, and by help of the few remarks which follow.

In a train of wheel-work consisting solely of spur-wheels or pinions with parallel axes, the direction of rotation will be alternately to right and left. If, therefore, the train consist of an even number of axes, the extreme axes will revolve in opposite directions, but if of an odd number of axes, then in the same direction. If an annular wheel be employed, its axis revolves the same way as that of the pinion (Art. 61).

322. If a wheel A (fig. 34, page 45) be placed between two other wheels C and B, it will not affect the velocity ratio of these wheels, which is the same as if the teeth of B were immediately engaged with those of C, but it does affect the directional relation; for if B and C were in contact, they would revolve in opposite directions, but in consequence of the introduction of the intermediate axis of A, B and C will revolve in the same direction. Such an intermediate wheel is termed an idle wheel. 323. When the shafts of two wheels A and B, fig. 256 lie so close together that the wheels cannot be placed in the same plane without making them inconveniently small, they may be fixed as here shown, so as to lie one behind the other, and be connected by an idle wheel C, of rather more than double the thickness of the wheels it connects. Such a thick idle wheel is termed a Marlborough wheel, in some districts. It is employed in the roller frames of spinning machinery.

Fig. 256.

B

324. When the axes in a train are not parallel, the directional relation of the extreme axes can only be ascertained by tracing the separate directional relations of each contiguous pair of axes in order.

Fig. 257.

By intermediate bevil-wheels parallel axes may be made to revolve either in the same or opposite directions, according to the relative positions of the wheels; for example, in fig. 257 the wheel A drives B, upon whose shaft is fixed the wheel E. Now if the wheel C be fixed on the same side of the intermediate axis as A, the parallel axes of A and C will revolve in opposite directions; but if the wheel be fixed as at D, on the opposite

A

D

E

16

side of the intermediate axis, then the axes of A and D will revolve in the same direction, the same number of wheels being employed in both cases.

Endless screws may be represented in calculation by a pinion of one or more leaves, according to the number of their threads (Art. 217), but their effect upon the directional relation of rotation will be different, according as they are right-handed or lefthanded screws. (Art. 211.)

A

Fig. 258.

d

E

325. Two separate wheels or pieces in a train may revolve concentrically about the same axes, as for example, the hands of a clock. Also, in fig. 258, the wheel B is fixed to an axis Cc, and the wheel A to a tube d or cannon, which turns freely upon Cc. If these wheels may revolve in opposite directions, a single bevil-wheel E will serve to connect them, if the three cones have a common apex as in the figure; and since E is an idle wheel (Art. 322), the velocity ratio of B to A will depend solely upon the radii of their own frusta.

B

C

But if the wheels B, A are to revolve in the same direction, they must be made in the form of spur-wheels, and connected by means of two other spur-wheels fixed to an axis parallel to Cc.

326. Millwrights imagine that in a given pair of toothed wheels it is desirable that the individual teeth of one wheel should come into contact with the same teeth of the other wheel as seldom as possible, on the ground that the irregularities of

their figure are more likely to be ground down and removed by continually bringing different pairs of teeth into action.

This is a very old idea, and is stated nearly in the above words by De la Hire. It has also been acted upon up to the present time. Thus Oliver Evans tells us, that great care should be taken in matching or coupling the wheels of a mill, that their number of cogs be not such that the same cogs will often meet; because if two soft ones meet often, they will both wear away faster than the rest, and destroy the regularity of the pitch; whereas if they are continually changing they will wear regular, even if they be at first a little irregular.'*

The clockmakers, on the other hand, think that the wearing down of irregularities will be the best effected by bringing the same pair of teeth into contact as often as possible.†

Let a wheel of M teeth drive a wheel of N teeth, and let M m

= when m and n are the least numbers in that ratio ;

N n

...nM=mN,

and n is the least whole number of circumferences of the wheel M that are equal to a whole number of circumferences of the wheel N.

If, therefore, we begin to reckon the circumferences of each wheel that pass the line of centers, after a given pair of teeth are in contact, it is clear that after n revolutions of M, and m of N, the same two teeth will be again in contact. Neither can they have met before; for as the entire circumference of one wheel applies itself to the entire circumference of the other tooth by tooth, and as the numbers m and n are the least multiples of the respective circumferences that are equal, it follows that it is only after these respective lengths of circumferences have rolled past each other that the beginnings of each can again meet.

If we act on the watchmaker's principle, by which the contacts of the same pair are to take place very often, the numbers of the wheels M and N must be so adjusted that m and n may be the smallest possible, without materially altering the ratio and

M

N' this will be effected by making the least of the two numbers m, n equal to unity, and therefore M a multiple of N.

But if the millwright's principle be adopted, m and ʼn must be

* O. Evans, Young Millwright's Guide, Philadelphia, 1834, p. 193. Vide also Buchanan's Essays, by Rennie, p. 117.

Francœur, Mécaniqne Elémentaire, p. 143.