426

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

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

Let a1, a2, be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces; then

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, PTP the plane surface of the rack, and T a point in their line of contact. Let r be the radius of the wheel, a its angular velocity, and the linear velocity of the rack; then

442. Bevel Wheels, whose pitch surfaces are frustra of regular cones, are used to transmit an uniform angular velocity-ratio between a pair of axes which intersect each other. Fig. 190 of Article 392 will serve to illustrate this case; O A and OC being the pair of axes, intersecting each other in O, O T the line of contact, and the cones described by the revolution of O T about O A and O C respectively being the pitch surfaces, of which narrow zones or frustra are used in practice.

Let a,, a, be the angular velocities about the two axes respectively; and let i1 = A O T, ¿ = ≤COT, be the angles made by those axes respectively with the line of contact; then from the principle III. of Article 439 it follows, that the angular velocityratio is

Which equation serves to find the angular velocity-ratio when the axes and the line of contact are given.

Conversely, let the angle between the axes,

ZAOC=+ i;=j,

be given, and also the ratio 2; then the position of the line of

contact is given by either of the two following equations :

√(a + až + 2 a, a, cos j)

..(2.)

;

Graphically, the same problem is solved as follows:-On the two axes respectively, take lengths to represent the angular velocities of their respective wheels. Complete the parallelogram of which those lengths are the sides, and its diagonal will be the line of contact. As in the case of the rolling cones of Article 393, one of a pair of bevel wheels may be a flat disc, or a concave cone.

T

Fig. 191.

443. Non-Circular Wheels are used to transmit a variable velocity-ratio between a pair of parallel

axes.

In fig. 191, let C1, C2, represent the axes of such a pair of wheels; T1, T2, a pair of points which at a given instant touch each other in the line of contact (which line is parallel to the axes and in the same plane with them); and U1, U2, another pair of points, which touch each other at another instant of the motion; and let the four points, T1,

NON-CIRCULAR WHEELS.

429

T2, U1, U2, be in one plane perpendicular to the two axes, and to the line of contact. Then for every such set of four points, the two following equations must be fulfilled :

C1 U1+ Ñ, U2 = C1 T1 + C2 T2 = C1 C2;

2

arc T, U, arc T, U2;

(1.)

and those equations show the geometrical relations which must exist between a pair of rotating surfaces in order that they may move in rolling contact round fixed axes.

The same conditions are expressed differentially in the following manner :-Let r1, r, be the radii vectores of a pair of points which touch each other; ds1, d s2, a pair of elementary arcs of the cross sections T, U1, T, U,, of the pitch surfaces, and c the line of centres or distance between the axes. Then

If one of the wheels be fixed and the other be rolled upon it, a point in the axis of the rolling wheel describes a circle of the radius e round the axis of the fixed wheel.

The equations 1 and 2 are made applicable to inside gearing by putting instead of and + instead of

The angular velocity-ratio at a given instant has the value

As examples of non-circular wheels, the following may be mentioned ::

I. An ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity-ratio varying from 1 - excentricity 1 + excentricity

to

1 + excentricity 1 - excentricity'

II. A hyperbola rotating about its farther focus, rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocityratio varying between

III. Two logarithmic spirals of equal obliquity rotate in rolling contact with each other through an indefinite angle. (For further examples of non-circular wheels, see Professor Clerk Maxwell's paper on Rolling Curves, Trans. Roy. Soc. Edin., vol. xvi., and Professor Willis's work on Mechanism.)

SECTION 2.-Sliding Contact.

444. Skew-Bevel Wheels are employed to transmit an uniform velocity-ratio between two axes which are neither parallel nor

E

Fig. 192.

H

G

K

Fig. 194.

F

B

Fig. 193.

intersecting. The pitch surface of a skew-bevel wheel is a frustrum or zone of a hyperboloid of revolution. In fig. 192, a pair of large portions of such hyperboloids are shown, rotatfing about axes A B, CD. In fig. 193 are shown a pair of narrow zones of the same figures, such as are employed in practice.

A hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 192, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AB, CD, in opposite directions. The axes will neither be parallel, nor will they intersect each other.

The motion of two such hyperboloids, rotating in contact with each other, has sometimes been classed amongst cases of rolling contact; but that classification is not strictly correct; for although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are neither parallel to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of