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same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT. In like manner, if either the same or any other rolling curve R be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T' shall start from A, it will trace the face AT' of a tooth suitable to work with a flank traced b rolling the same curve R' with the same tracing-point T' insizlk any other pitch-circle.

The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E'I'E', as the case may be) at a fixed point I (or I'). If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, iire traced by rolling the rolling curve on both sides of a straight ll"l€.

The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch surfaces are in rolling contact.

§ 49. Epicycloidal Teeth.-The most convenient rolling curve is which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloid's for wheels, and

both flanks and faces are cycloids for a rack.-For

a pitch-circle of twice the

radius of the rolling or describing the circle. The path of contact

For the proportions approved of by Willis, sin 0=i nearly; r=p fthe pitch) nearly; c=§ p nearly; and, if N be the number of teeth in the wheel, r/ R=6/N nearly; therefore, approximately, -2 L

P °'2' N-12


-2 L

pl"c'2' N-I-12

Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and athe point where a tooth is to cross it. Set off ab=ac=§ p. Draw radii bd,

ce; draw fb, cg, making angles of 75%° with those radii. Make

bf=p'-c, cg=p—c. From f, k

with the radius fa, draw the circular arc ah; from g, with the radius ga, B draw the circular arc ak. Then

ah is the face and ak the flank of b '1the tooth required. A

To facilitate the application of

this rule, Willis published tables of p -c and p'-c, and invented an instrument called the“odontograph."

§ 51. Trundles and Pin-Wheels.-*If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloid's, by rolling the pitch-circle of the pin-wheel or trundle on d C


5 1:


FIG. 105.

the pitch-circle of the driving-wheel, with the centre of a tracing-point, and then drawing curves parallel to, the epicycloid's, at a distance from them equal to the stave. Trundles having only six staves will work wheels.

a stave for

and within

radius of a

with large





~ If


Fig. 104.

that the flanks may be either stra In fig. 104 let BB' be part of

circle (as it is called) the internai epicycloid is a straight line, being, in fact, a diameter of the pitch circle, so that the flanks of the

teeth for such a itch-circle are

planes radiating from the axis.

For a smaller pitch-circle the

fianks would be convex and incurved or under-cut, which would

be inconvenient; therefore the

smallest wheel of a set should

have its pitch-circle of twice the radius of the describing circle, so ight or concave.

the bitch-circle of a wheel with

epicycloid teeth; CIC' the line of centres; I the pitch-point; EIE'. a straight tangent to the pitch-circle at that point; R the internal and R' the equal external describing circles, so placed as to touch the pitch-circle and each other at l. Let DID' be the path of contact, consisting of the arc of approach DI and the arc of recess ID'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID=E'ID; and the mean obliquity is one-half of that angle. It appears from experience that the mean obliquity should not exceed I5°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID' should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six times the pitch.

It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth. § (fo. Nearly Epicycloidal Teeth: Willis's Method.-To facilitate the rawing of epicycloid teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs-one concave, for the flank, and the other convex, for the face-and each having for its radius the mean radius of curvature of the epicycloid arc. Willis's formulae are founded on the following properties of epicycloid's:-

Let R be the radius of the pitch-circle; r that of the describing circle; 6 the angle made by the normal TI to the epicycloid at a given point T, witn a tangent-to the circle at I-that is, the obliquity of the action at T.

Then the radius of curvature of the epicycloid at T is-For an internal epicycloid, p =4r sin; A R+r (28)


Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c =2r sin 6; and therefore

For an external epicycloid, p' =4r sin 0 For the flank, p -c=2r sin 0 rg

R (29)

For the face, p -c=2r sin 0 11727 § 52. Backs of Teeth and Spaces.-Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about kth part wider than the thickness of the tooth on the pitch-circle-that 1S to say, Thickness of tooth ={T pitch;

Width of space =T"I pitch. °

The difference of, lf of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch.

§ 53. Stetpped and Helical T eeth.-R. j. Hooke invented the making of the ronts of teeth in a series of ste s with a view to increase the smoothness of action. A wheel thus fihrmed resembles in shape a series of equal and similar toothed disks placed side by s de, with the teeth of each a little behind those of the preceding (1Sl{. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel work of this kind the contact of each pair of teeth commences a.t the foremost end of the helical front, and terminates at the after most end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next ' pair has commenced.

Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery. § 54. Teeth of Bevel-Wheels.-The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex. The operations of describing the exact figures of the teeth of bevel wheels, whether by in volutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be per ormed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and ° ° Ba

great circles for straight lines. ln consideration of the practical difficulty, especially in the A

case of large wheels, of obtain- / Az in an accurate s herica sur- , f I fage, and of dravxiing upon it 5, E when obtained, the fol owing ”' ""' 'I 1 I approximate method, proposed ' Da originally by T re d g old, is,

generally used:-

Let O(fig.1o6) be the common AI- —- ~ — 1); apex of a pair of bevel-wheels;

OBJ, OBZI their pitch cones; FIG;06 OC1, OC; their axes; OI their

line of contact. Perpendxcular

in Al, Ag; make the outer rims of to OI draw AIIA2, cutting the axes the patterns and of the wheels