Imagens das páginas
PDF
ePub
[blocks in formation]

ρη PM

∞ BM × sin BMA varies as the perpendicular upon

AM produced.

But if the curves be an epicycloid turning on the center A, in contact with a radial line which turns round B; then DMB is a right angle,

[blocks in formation]

37. To find the Velocity Ratio in wrapping connexions.

[merged small][merged small][ocr errors][merged small][merged small]

Let A, B be the centers of motion, PQ the wrapping connector touching the curves at P and Q, and let the point P be moved to p very near to its first position, then will Q be drawn to q, and the connector will touch the curves in two new points of contact, which may be r and s respectively. Now, in the action of wrapping or unwrapping, the connector touches the curves in a series of consecutive points between q and S or p and r, and ultimately q coincides with S and p with r. The extremities of the connector may therefore be considered at any given moment as if jointed to the two curves at the points of contact, and turning upon these points in the manner of a link. The relative velocities of the curves are therefore momentarily

the same as if AP, BQ were a pair of rods connected by a link PQ. Hence the angular velocities of the pieces are to each other inversely as the segments into which the connector divides the line of centers.

38. If the line of direction of the link in link-work, of the common normal to the curves in contact motion, and of the connector in wrapping motion, be severally termed the line of action, we can express the separate propositions which relate to the Velocity Ratio, by saying that the angular velocities of the two pieces are to each other inversely as the segments into which the line of action divides the line of centers, or inversely as the perpendiculars from the centers of motion upon the line of action.

I have confined these investigations, for the present, to motions in the same plane. The cases of motions in different planes, are more simply examined as the individual combinations which require them occur.

39. It has been shewn that the principal pieces which constitute a train of mechanism are compelled to move each in a given path. Now it is generally sufficient to consider this path as a circle, for in fact the pieces always either revolve or move in right lines; and the contrivances by which motion is communicated in a rectilinear path, are the same as those by which it is given to a revolving piece, and derived from the latter by that familiar geometrical artifice by which a right line is considered as the arc of a circle whose radius is infinite. It will presently appear that, in this way, much complication of classes will be avoided. Thus, for example, a pinion driving a rack is plainly the same contrivance as a pinion driving a toothed wheel, the rack being considered as a portion of a toothed wheel whose radius is infinite.

It is true, that pieces in a train may be found which describe other paths, such as elliptical, epicycloidal, or sinuous lines; but these are always produced by combining together various circular motions, and therefore the motion of the piece is actually produced by pieces that travel in circular paths. Cases of this kind fall under the head of Aggregate Motions, to which a separate Part of this work has been assigned.

40. The path of a revolving piece may be considered as unlimited in extent in either direction, since the piece may go on performing any number of rotations in the same direction. But a piece that travels in a right line is necessarily limited in its motion either way, to the length of that line.

Again, the method by which motion is communicated from one piece to another, may be of such a nature as to limit the motion of these pieces, although they may be capable of unlimited motion, considered apart from this connexion. For example, if the driver and follower be revolving cylinders, and therefore capable of unlimited motion, the communication of motion may be effected by a string whose ends are fixed one to each cylinder, and coiled round it, so that when the driver revolves it shall communicate motion to the follower by coiling the string round itself and uncoiling it from the follower; in which case the rotations of each cylinder are limited to the number of coils which its circumference contains when the other is empty.

It appears, then, that the motion of a pair of connected pieces may be limited either by the figure of one or both of their paths, or by the nature of their connexion; and a limited connexion may be formed between unlimited paths, or vice versa, but if either the paths or the connexion be limited, the motion of the pieces will be limited.

In classifying the communication of motion, however, the union of unlimited connexions with limited paths, will require but little attention, as the modifications to which they lead are in general sufficiently obvious; but the distinction between limited and unlimited methods of communication is of more importance.

[ocr errors]

CHAPTER II.

ELEMENTARY COMBINATIONS.

CLASS A. {

DIRECTIONAL RELATION CONSTANT.
VELOCITY RATIO CONSTANT.

DIVISION A. COMMUNICATION OF MOTION BY ROLLING
CONTACT.

[ocr errors]

41. IN rolling contact it has been shewn that the point of contact is always in the line of centers; and the angular? J. velocities are inversely as the segments into which the point Aù.. of contact divides that line. Therefore if the velocity ratio is constant, the segments must be constant, and the curves become circles, revolving round their centers, and whose radii are the segments, and no other curves will answer the purpose.

Let R be the radius of the driving circle, and r that of the following circle; L and their synchronal rotations; then as they are (by § 20) in the ratio of the angular velocities:

[merged small][merged small][ocr errors][merged small]

This ratio will be preserved, whatever be the absolute velocity of the driver, but when this is uniform, which is generally the case, let P and p be the respective periods of the driver and follower;

1404

[blocks in formation]

The motions being supposed hitherto to be in the same plane, the axis of rotation of the circles will be parallel.

[ocr errors]
« AnteriorContinuar »