and those factors, or if they are too small, multiples of them, used for the numbers of teeth. Should B or C, or both, be at once incon

B

veniently large, and prime, then instead of the exact ratio some C'

ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.

B

Should be greater than 6, the best number of elementary

C

combinations, m - 1, will lie between

log Blog C log B
log 6

and

log C

log 3

(2.)

Then, if possible, B and C themselves are to be resolved each into m 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t, nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity-ratio, found by the method of continued fractions, is

So far as the resultant velocity-ratio is concerned, the order of the drivers N and of the followers n is immaterial; but to secure equable wear of the teeth, as explained in Article 447, Principle V., the wheels ought to be so arranged that for each elementary combination the greatest common divisor of N and n shall be either 1, or as small as possible.

450. Principle of Sliding Contact.-The line of action, or of connection, in the case of sliding contact of two moving pieces, is the common perpendicular to their surfaces at the point where they touch; and the principle of their comparative motion is, that the components, along that perpendicular, of the velocities of any two points traversed by it, are equal.

CASE 1. Two shifting pieces, in sliding contact, have linear velocities proportional to the secants of the angles which their directions of motion make with their line of action.

CASE 2. Two rotating pieces, in sliding contact, have angular velocities inversely proportional to the perpendicular distances from their axes of rotation to their line of action, each multiplied by the sine of the angle which the line of action makes with the particular axis on which the perpendicular is let fall.

In fig. 197, let C1, C2, represent the axes of rotation of the two pieces; A, A,, two portions of their respective surfaces; and T1, T2, a pair of points in those surfaces, which, at the instant under consideration, are in contact with each other. Let P, P, be the common perpendicular of the surfaces at the pair of points T1, T,;

2

PRINCIPLE OF SLIDING CONTACT.

437

that is, the line of action; and let C, P1, C, P., be the common perpendiculars of the line of action and of the two axes respectively. Then at the given instant, the components along the line P, P, of the velocities of the points P1, P2, are equal. Let i, i, be the angles which that line makes with the directions of the axes respectively. Let a, a, be the respective angular velocities of the moving pieces; then

འབ

When the line of action is perpendicular in direction to both axes, then sin i1 = sin i2 = 1; and equation 1 becomes

When the axes are parallel, i

=

=

Let I be the point where the line of action cuts the plane of the two axes; then the triangles P, C, I, P, C, I, are similar; so that equation 1 A is equivalent to the following:

CASE 3. A rotating piece and a shifting piece, in sliding contact, have their comparative motion regulated by the following principle:-Let CP denote the perpendicular distance from the axis of the rotating piece to the line of action; i the angle which the direction of the line of action makes with that axis; a the angular velocity of the rotating piece; v the linear velocity of the sliding piece; j the angle which its direction of motion makes with the line of action; then

When the line of action is perpendicular in direction to the axis of the rotating piece, sin i = 1; and

v=a CP sec ja·IC;..............(2 A.)

where IC denotes the distance from the axis of the rotating piece

to the point where the line of action cuts a perpendicular from that axis on the direction of motion of the shifting piece.

451. Teeth of Spur-Wheels and Racks. General Principle.—The figures of the teeth of wheels are regulated by the principle, that the teeth of a pair of wheels shall give the same velocity-ratio by their sliding contact, which the ideal smooth pitch surfaces would give by their rolling contact. Let B,, B2, in fig. 197, be parts of the pitch lines (that is, of cross sections of the pitch surfaces) of a pair of wheels with parallel axes, and I the pitch point (that is, a section of the line of contact). Then the angular velocities which would be given to the wheels by the rolling contact of those pitch lines are inversely as the segments I C1, I C2, of the line of centres; and this also is the proportion of the angular velocities given by a pair of surfaces in sliding contact whose line of action traverses the point I (Article 450, case 2, equation 1 B). Hence the condition of correct working for the teeth of wheels with parallel axes is, that the line of action of the teeth shall at every instant traverse the line of contact of the pitch surfaces; and the same condition obviously applies to a rack sliding in a direction perpendicular to that of the axis of the wheel with which it works.

452. Teeth Described by Rolling Curves.-From the principle of the preceding Article it follows, that at every instant, the position of the point of contact T, in the cross section of the acting surface of a tooth (such as the line A, T, in fig. 197), and the corresponding position of the pitch point I in the pitch line I B, of the wheel to which that tooth belongs, are so related, that the line IT, which joins them is normal to the outline of the tooth A, T, at the point T. Now this is the relation which exists between the tracingpoint T, and the instantaneous axis or line of contact I, in a rolling curve of such a figure, that being rolled upon the pitch surface B1, its tracing-point T, traces the outline of the tooth. (As to rolling curves, see Articles 386, 387, 389, 390, 393, 396, 397, and Professor Clerk Maxwell's paper there referred to).

In order that a pair of teeth may work correctly together, it is necessary and sufficient that the instantaneous radii vectores from the pitch point to the points of contact of the two teeth should coincide at each instant, as expressed by the equation

and this condition is fulfilled, if the outlines of the two teeth be traced by the motion of the same tracing-point, in rolling the same rolling curve on the same side of the pitch surfaces of the respective wheels.

The flank of a tooth is traced while the rolling curve rolls inside of the pitch line; the face, while it rolls outside. Hence it is

TEETH DESCRIBED-SLIDING OF TEETH.

439

evident that the flanks of the teeth of the driving wheel drive the faces of the teeth of the driven wheel; and that the faces of the teeth of the driving wheel drive the flanks of the teeth of the driven wheel. The former takes place while the point of contact of the teeth is approaching the pitch point, as in fig. 197, supposing the motion to be from P, towards P,; the latter, after the point of contact has passed, and while it is receding from, the pitch point. The pitch point divides the path of the point of contact of the teeth into two parts, called the path of approach and the path of recess; and the lengths of those paths must be so adjusted, that two pairs of teeth at least shall be in action at each instant.

It is evidently necessary that the surfaces of contact of a pair of teeth should either be both convex, or that if one is convex and the other concave, the concave surface should have the flatter curvature.

The equations of Article 390 give the relations which exist between the radius of curvature of a pitch line at the pitch point (r), the radius of curvature of the rolling curve at the same point (r), the radius vector of the tracing-point (r = IT), the angle made by that line with the line of centres of the fixed and rolling curves (8 = TIC), and the radius of curvature of the curve traced by the point T (g), all at a given instant.

୧

୧

When a pair of tooth surfaces are both convex absolutely, that which is a face is concave, and that which is a flank is convex, towards the pitch point; and this is indicated by the values of having contrary signs for the two teeth, being positive for the face and negative for the flank. The face of a tooth is always convex absolutely, and concave towards the pitch point, being positive; so that if it works with a concave flank, the value of e for that flank is positive also, and greater than for the face with which it works. 453. The Sliding of a Pair of Teeth on Each Other, that is, their relative motion in a direction perpendicular to their line of action, is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C, C, to rotate backwards round C, with the angular velocity a1, and the wheel 2 to rotate round C, as before with the angular velocity a relatively to the line of centres C, C, so as to have the same motion as if its pitch surface rolled on the pitch surface of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the motion given by the principles of Article 388; that is, about the instantaneous axis I with the angular velocity Hence the velocity of sliding is that due to this rotation with the radius IT r; that is to say, its value is r (α2 + α);...........

so that it is greater, the farther the point of contact

a

resultant

rotation

is

about I,

...(1.)

from the

line of centres; and at the instant when that point, passing the line of centres, coincides with the pitch point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact. The roots of the teeth slide towards each other during the approach, and from each other during the recess. To find the amount or total distance through which the sliding takes place, let t, be the time occupied by the approach, and t, that occupied by the recess; then the distance of sliding is

8=

ta

'r (a + a,) d t + f "r (a,+ as) d t ;.................(..)

or in another form, if di denote an element of the change of angular position of one wheel relatively to the other, i, the amount of that change during the approach, and i, during the recess, then

454. The Arc of Contact on the Pitch Lines is the length of that portion of the pitch lines which passes the pitch point during the action of one pair of teeth; and in order that two pairs of teeth at least may be in action at each instant, its length should be at least double of the pitch. It is divided into two parts, the arc of approach and the arc of recess. In order that the teeth may be of length sufficient to give the required duration of contact, the distance moved over by the point I upon the pitch line during the rolling of a rolling curve to describe the face and flank of a tooth, must be in all equal to the length of the required arc of contact. It is usual to make the arcs of approach and recess equal.

455. The Length of a Tooth may be divided into two parts, that of the face and that of the flank. For teeth in the driving wheel, the length of the flank depends on the arc of approach,―that of the face, on the arc of recess; for those in the following wheel, the length of the flank depends on the arc of recess,—that of the face, on the arc of approach.

Let be the arc of approach, q, that of recess; 1, the length of the flank, l, the length of the face of a tooth in the driving wheel. Let r, be the radius of curvature of the pitch line, r, that of the rolling curve, r the radius vector of the tracing-point, at any instant. The angular velocity of the rolling curve relatively to the wheel is