CHAPTER II.

ON COMBINATIONS FOR PRODUCING AGGRE

GATE VELOCITY.

394. I SHALL in this Chapter proceed to shew the principal methods of obtaining the complex motion of a body in a given path by the simultaneous communication to it of two or more simple motions in that path; arranging the solutions under the same divisions as in the first part of this Work, but taking them in a somewhat different order, for the sake of convenience.

BY LINK-WORK.

395. Let a bar ABC, fig. 203, be bisected in B, and let a small motion Aa perpendicular to the bar be communi

cated to the extremity A, C remaining at rest; then will the

A a

central point B move through a space Bn

=

On the

2

other hand, had A remained at rest, and a small transverse motion Ce been given to the other extremity C, the central

point B would have moved through a space Bm =

Cc

2

If

these two motions are communicated either simultaneously or successively to the two extremities, the center B will be Or, if starting

carried through a space Bb =

Aa + Cc

2

motion mn =

from the position Ac, the two motions had been communicated in the opposite directions, so as to carry the bar into the position a C, then the center of the bar would receive a Aa - Cc The length of the bar being 2 always supposed so great, compared with the motions, that its inclination in the different positions may be neglected, and therefore the lines Cc, Bb, Aa, be all considered perpendicular to AC. Hence two small independent motions being communicated to the extremities of a bar; its center receives half their sum or difference, according as the motions are in the same or in opposite directions.

The bar

If the motions be communicated to A and B, then C will receive the whole motion of A in the opposite direction, and twice the motion of B in the same direction. AC has been divided in half at B for simplicity only, for it is evident that by dividing it in any other ratio we can communicate the component motions in any desired proportions. But in general it is the law of motion which is to be communicated, and the quantity is of less consequence, especially if reduced for both motions in the same proportion.

396. Let FG, fig. 204, be a bar whose center is E, and to whose extremities are

A

B

204

OE

C G D

fixed pins F and G, upon which the centers of other bars AB, CD, turn. Then if four independent motions be communicated to the points A, B, C, D, the motions of A and B will be concentrated upon F, and those of C and D upon G, and the motions of F and G being concentrated in like manner upon E, this point will receive the four motions. By jointing other levers to the extremities of these, and so

on, any number of independent motions may be concentrated upon the point E*.

BY WRAPPING CONNECTORS.

205

397. If a bar Bb, fig. 205, be capable of sliding in the direction of its length and carry a pully A round which is passed a cord DE, then it can be shewn in the same manner, that the bar will receive half the sum of independent motions communicated to the extremities D, E, the bar being supposed to be urged in the direction bB, by a weight or spring. This is a more compendious contrivance than the former, as D the motions may be of considerable extent. the component motions be communicated to one extremity of the string D and to the bar, then will the other extremity E receive the entire motion of D in the reverse direction, and also twice the motion of Bb in the same direction †.

If

398. If a second similar combination be placed at the side of this, with its bar parallel to that of the first, and if a cord whose ends are tied to the upper extremities of each bar be passed over a third intermediate pully, the center of this latter pully will receive the aggregate motion of the cords of the two systems, as shewn for the lever in Art. 396.

399. As an example of the employment of these combinations, let C, fig. 206, be an axis of motion upon which is fixed a small barrel round which the cord e is rolled, and also a disk with an excentric pin c, which by means of a

* Another example of aggregate velocity by Link-work is the well-known reticulated frame termed Lazy tongs, which resembles a row of X's, thus xxxxx. It is too weak from its numerous joints to be of much practical service.

The first application of this principle appears to be the Rouet de Lyon, for winding silk. Vide Enc. Meth. Manufactures, t. II. p. 44.

link cb communicates a reciprocating motion to an arm Aa, whose center of motion is A. The extremity of this arm carries a revolving pully D, and the cord which is coiled round the band is laid over this pully and fixed to a

coiled upon the barrel by which it, as well as E, will receive a slow travelling motion in a constant direction upwards. By what has preceded, therefore, the body E receiving these motions simultaneously, will, as in the example of Art. 387, move vertically with a reciprocating motion, of which the downward trip is shorter than the upward one.

207 B

a

400. Let Aa, fig. 207, be an axis to which are fixed two cylinders B and C, nearly of the same diameter, and let a cord be coiled round B, passed over a pully D, and then brought back and coiled in the opposite direction round C. When a revolves, one end of the cord will be coiled and the other uncoiled, and if R be the radius of B, and r of C, A the angular velocity of the axis, the velocities of the two extremities of the cord will be AR and Ar; and by Art. 397, the center

D

of the pully D will travel with a velocity equal to half the difference of these velocities, since they are in opposite

A (R − r)

directions, or to

This velocity is the same as

2

would be obtained if the center of the pully D were suspended from the axis Aa by a cord wrapped round a single

401. This combination belongs to a class which has received the name of differential motions, their object being to communicate a very slow motion to a body, or rather to produce by a single combination such a velocity ratio between two bodies that under the usual arrangement a considerable train of combinations would be required practically to reduce the velocity, for, theoretically, a simple combination will always answer the same purpose. Thus in the above machine, although theoretically a barrel with a radius

R

2

would do as well as the double barrel, yet its diameter in practice would be so small as to make it useless from weakness. Whereas each barrel of the differential combination may be made as large and as strong as we please.

If a considerable extent of motion however be required, this contrivance becomes very troublesome, on account of the great quantity of rope which must be wound upon the barrels. For by one turn of the differential barrel the space through which the pully is raised = (R − r), but the quantity of rope employed is the sum of that which is coiled upon one barrel, and of that which is uncoiled from the other = 2 π (R + r). Now in the equivalent simple barrel the quantity of rope coiled is exactly equal to the space through which the body is moved, and therefore in this case = π (R − r), so that for a given extent of motion