BEARINGS-MOVING PIECES-ELEMENTARY COMBINATION. 423

tion, such as cylinders, spheres, conoids, and flat discs. The bearing of a piece whose motion is helical, must be an exact screw, of a pitch equal to that of the helical motion (Article 382). Those parts of moving pieces which touch the bearings, should have surfaces accurately fitting those of the bearings. They may be distinguished into slides, for pieces which move in straight lines, gudgeons, journals, bushes, and pivots, for those which rotate, and screws for those which move helically.

The accurate formation and fitting of bearing surfaces is of primary importance to the correct and efficient working of machines. Surfaces of revolution are the most easy to form accurately, screws are more difficult, and planes the most difficult of all. The success of Mr. Whitworth in making true planes, is regarded as one of the greatest achievements in the construction of machinery.

429. The Motions of Primary Moving Pieces are limited by the fact, that in order that different portions of a pair of bearing surfaces may accurately fit each other during their relative motion, those surfaces must be either straight, circular, or helical; from which it follows, that the motions in question can be of three kinds only, viz:

I. Straight translation, or shifting, which is necessarily of limited extent, and which, if the motion of the machine is of indefinite duration, must be reciprocating; that is to say, must take place alternately in opposite directions. (See Part III., Chapter II., Section 1.)

II. Simple rotation, or turning about a fixed axis, which motion may be either continuous or reciprocating, being called in the latter case oscillation. (See Part III., Chapter II., Section 2.)

III. Helical or screw-like motion, to which the same remarks apply as to straight translation. (See Part III., Chapter II., Section 3, Article 382.)

430. The Motions of Secondary Moving Pieces relatively to the pieces which carry them, are limited by the same principles which apply to the motions of primary pieces relatively to the frame. But the motions of secondary moving pieces relatively to the frame may be any motions which can be compounded of straight translations and simple rotations according to the principles already explained in Part III., Chapter II., Section 3.

431. An Elementary Combination in mechanism consists of a pair of primary moving pieces, so connected that one transmits motion to the other.

The piece whose motion is the cause is called the driver; that whose motion is the effect, the follower. The connection between the driver and the follower may be

I. By rolling contact of their surfaces, as in toothless wheels.

II. By sliding contact of their surfaces, as in toothed wheels, screws, wedges, cams, and escapements.

III. By bands or wrapping connectors, such as belts, cords, and gearing-chains.

IV. By link-work, such as connecting rods, universal joints, and clicks.

V. By reduplication of cords, as in the case of ropes and pulleys. VI. By an intervening fluid, transmitting motion between two pistons.

The various cases of the transmission of motion from a driver to a follower are further classified, according as the relation between their directions of motion is constant or changeable, and according as the ratio of their velocities is constant or variable. This latter principle of classification is employed by Mr. Willis as the foundation of a primary division of the subject of elementary combinations in mechanism into classes, which are subdivided according to the mode of connection of the pieces. In the present treatise, elementary combinations will be classed primarily according to the mode of connection.

432. Line of Connection.-In every class of elementary combinations, except those in which the connection is made by reduplication of cords, or by an intervening fluid, there is at each instant a certain straight line, called the line of connection, or line of mutual action of the driver and follower. In the case of rolling contact, this is any straight line whatsoever traversing the point of contact of the surfaces of the pieces; in the case of sliding contact, it is a line perpendicular to those surfaces at their point of contact; in the case of wrapping connectors, it is the centre line of that part of the connector by whose tension the motion is transmitted; in the case of link-work, it is the straight line passing through the points of attachment of the link to the driver and follower.

433. Principle of Connection.—The line of connection of the driver and follower at any instant being known, their comparative velocities are determined by the following principle :-The respective linear velocities of a point in the driver, and a point in the fol lower, each situated anywhere in the line of connection, are to each other inversely as the cosines of the respective angles made by the paths of the points with the line of connection. This principle might be otherwise stated as follows:- -The components, along the line of connection, of the velocities of any two points situated in that line, are equal.

434. Adjustments of Speed.-The velocity-ratio of a driver and its follower is sometimes made capable of being changed at will, by means of apparatus for varying the position of their line of connection; as when a pair of rotating cones are embraced by a belt

TRAIN-AGGREGATE COMBINATIONS.

425

which can be shifted so as to connect portions of their surfaces of different diameters.

435. A Train of Mechanism consists of a series of moving pieces, each of which is follower to that which drives it, and driver to that which follows it.

436. Aggregate Combinations in mechanism are those by which compound motions are given to secondary pieces.

CHAPTER IL

ON ELEMENTARY COMBINATIONS AND TRAINS OF MECHANISM.

SECTION 1.-Rolling Contact.

437. Pitch Surfaces are those surfaces of a pair of moving pieces, which touch each other when motion is communicated by rolling contact. The LINE OF CONTACT is that line which at each instant traverses all the pairs of points of the pair of pitch surfaces which are in contact.

438. Smooth Wheels, Rollers, Smooth Racks.—Of a pair of primary moving pieces in rolling contact, both may rotate, or one may rotate and the other have a motion of sliding, or straight translation. A rotating piece, in rolling contact, is called a smooth wheel, and sometimes a roller; a sliding piece may be called a smooth rack.

439. General Conditions of Rolling Contact.—The whole of the principles which regulate the motions of a pair of pieces in rolling contact follow from the single principle, that each pair of points in the pitch surfaces, which are in contact at a given instant, must at that instant be moving in the same direction with the same velocity.

The direction of motion of a point in a rotating body being perpendicular to a plane passing through its axis, the condition, that each pair of points in contact with each other must move in the same direction leads to the following consequences :

I. That when both pieces rotate, their axes, and all their points of contact, lie in the same plane.

II. That when one piece rotates and the other slides, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the sliding piece.

The condition, that the velocities of each pair of points of contact must be equal, leads to the following consequences:—

III. That the angular velocities of a pair of wheels, in rolling contact, must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.

IV. That the linear velocity of a smooth rack in rolling contact with a wheel, is equal to the product of the angular velocity of the wheel by the perpendicular distance from its axis to a pair of points

of contact.

CIRCULAR WHEELS-STRAIGHT RACK.

427

Respecting the line of contact, the above principles III. and IV. lead to the following conclusions :

V. That for a pair of wheels with parallel axes, and for a wheel and rack, the line of contact is straight, and parallel to the axes or axis; and hence that the pitch surfaces are either plane or cylindrical (the term "cylindrical" including all surfaces generated by the motion of a straight line parallel to itself).

VI. That for a pair of wheels, with intersecting axes, the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term "conical" including all surfaces generated by the motion of a straight line which traverses a fixed point).

440. Circular Cylindrical Wheels are employed when an uniform velocity-ratio is to be communicated between parallel axes. Figs. 187, 188, and 189, of Article 388, may be taken to represent pairs of such wheels; C and O, in each figure, being the parallel axes of the wheels, and T a point in their line of contact. In fig. 187, both pitch surfaces are convex, the wheels are said to be in outside gearing, and their directions of rotation are contrary. In figs. 188 and 189, the pitch surface of the larger wheel is concave, and that of the smaller convex; they are said to be in inside gearing, and their directions of rotation are the same.

To represent the comparative motions of such pairs of wheels symbolically, let

CT="2

be their radii: let OC=c be the line of centres, or perpendicular distance between the axes, so that for

outside

inside

} gearing, cr1.......

....(1.)

Let a1, aq be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces; then

(outside } gearing.

the sign±applying to {

441. A Straight Back and Circular Wheel, which are used when an uniform velocity-ratio is to be communicated between a sliding piece and a turning piece, may be represented by fig. 185 of Article 385, C being the axis of the wheel, P T P the plane surface of the rack, and T a point in their line of contact. Letr be the radius of the wheel, a its angular velocity, and v the linear velocity of the rack; then

t=ra.