there is, therefore, a given relation of the radius of the generating circle of the driving, to that of the driven wheel, which relation being observed in striking the epicycloidal faces and the hypocycloidal flanks of the teeth of two wheels destined to work with one another, those wheels will work with a greater economy of power than they would under any other epicycloidal forms of their teeth. This value of e may be determined by assuming the differential co-efficient of the co-efficient of U, in equation (261) equal to zero, and solving the resulting transcendental equation by the method of approximation.

2

227. THE MODULUS OF THE RACK AND PINION.

If the radius r2 of the pitch circle of the driven wheel be supposed infinite (Art. 213.), that wheel becomes a rack, and the radius r, of the driving wheel remaining of finite dimensions, the two constitute a rack and pinion. To determine the modulus of the rack and pinion in the case of teeth of any form, the number upon the pinion being great, or in the case of involute teeth and epicycloidal teeth of any number and dimensions, we have only to give to r an infinite value in the moduli already determined in respect to these several conditions. But it is to be observed in respect to epicycloidal teeth, that n, becomes infinite with 72, whilst the ratio

12

n2

remains fiuite, and retains its equality

if we represent the ratio by 2e. Making n aud r, infinite

r

in each of the equations (252), (254), and (261), and sub

stituting for in equation (262); we have

e
N2

1. For the modulus of the rack and pinion when the teeth. are very small, whatever may be their forms, provided that they work truly.

2. For the modulus of a rack and pinion, with involute teeth of any dimensions (see fig. 1. p. 280 ),

3. For the modulus of the rack and pinion, with cycloidal and epicycloidal teeth respectively (equation 261),

In each of which cases the value of N is determined by making r infinite in equation (247).

CONICAL OR BEVIL WHeels.

228. These wheels are used to communicate a motion of rotation to any given axis from another, inclined to the first at any angle.

Let AF be an axis to which a motion of rotation is to be

K

D

E

communicated from another axis AE inclined to the first at any angle EAF, by means of bevil wheels.

Divide the angle EAF by the straight line AD, so that DO and DN, perpendiculars from any point D in AD upon AE and AF respectively, may be to one another as the numbers of teeth

which it is required to place upon the two wheels.†

(i+1)=

= because e 2e,

. *. (equation 262) (1+") = 2; ( 1 + ; ;) = {} ( { + 4 ) =

2e

2e

; 1

e

is infinite. The friction of the rack upon its guides is not taken into account in the above equations.

†This division of the angle EAF may be made as follows:-Draw ST and UW from any points S and U in the straight lines AE and AF at

Suppose a cone to be generated by the revolution of the line AD about AE, and another by the revolution of the line AD about AF. Then if these cones were made to revolve in contact about the fixed axes AE and AF, their surfaces would roll upon one another along their whole line of contact DA, so that no part of the surface of one would slide upon that of the other, and thus the whole surface of the one cone, which passes in a given time over the line of contact AD, be equal to the whole surface of the other, which passes over that line in the same time. For it is evident that if n, times the circumference of the circle DP be equal to n, times that of the circle DI and these circles be conceived to revolve in contact carrying the cones with them, whilst the cone DAP makes n, revolutions, the cone DAI will make n revolutions; so that whilst any other circle GH of the one cone makes n, revolutions, the corresponding circle HK of the other cone will make n revolutions: but n, times the circumference of the circle GH is equal to n, times that of the circle HK, for the diameters of these circles, and therefore their circumferences, are to one another (by similar triangles) in the same proportion as the diameters and the circumferences of the circles DP and DI. Since, then, whilst the cones make ni and n2 revolutions respectively, the circles HG and HK are carried through n

A

U

right angles to those lines respectively, and having their lengths in the ratio of the numbers of teeth which it is required to place upon the two wheels; and through the extremities T and W of these lines draw TD and WD parallel to AE and AF respectively, and meeting in D. A straight line drawn from A through D will then make the required division of the angle; for if DO and DN be drawn perpendicular to AE and AF, they will evidently be equal to UW and ST, and therefore in the required proportion of the numbers of the teeth; moreover, any other two lines drawn perpendicular to AE and AF from any other point in AD will manifestly be in the same proportion as DO and DN.

E

and n, revolutions respectively, and that n, times the circumference of HG is equal to n, times that of HK, therefore the circles HG and HK roll in contact through the whole of that space, nowhere sliding upon one another. And the same is true of any other corresponding circles on the cones; whence it follows that their whole surfaces are made to roll upon one another by their mutual contact, no two parts being made to slide upon one another by the rolling of the rest.

The rotation of the one axis might therefore be communicated to the other by the rolling of two such cones in contact, the surface of the one cone carrying with it the surface of the other, along the line of contact AD, by reason of the mutual friction of their surfaces, supposing that they could be so pressed upon one another as to produce a friction equal to the pressure under which the motion is communicated, or the work transferred. In such a case the angular velocities of the two axes would evidently be to one another (equation (227) inversely, as the circumferences of any two corresponding circles DP and DI upon the cones, or inversely as their radii ND and OD, that is (by construction) inversely as the numbers and teeth which it is proposed to cut upon the wheels.

When, however, any considerable pressure accompanies the motion to be communicated, the friction of two such cones becomes insufficient, and it becomes necessary to transfer it by the intervention of bevil teeth. It is the characteristic property of these teeth that they cause the motion to be transferred by their successive contact, precisely as it would by the continued contact of the surfaces of the cones.

229. To describe the teeth of bevil wheels.*

From D let FDE be drawn at right angles to AD, intersecting the axes AE and AF of the two cones in E and F;

* The method here given appears first to have been published by Mr. Tredgold in his edition of Buchanan's Essay on Millwork, 1823, p. 103.

suppose conical surfaces to be generated by the revolution of the lines DE and DF about AE and AF respectively;

X

and let these conical surfaces be truncated by planes LM and XY respectively perpendicular to their axes AE and AF, leaving the distances DL and DY about equal to the depths which it is proposed to assign to the teeth. Let now the conical surface LDPM be conceived to be developed upon a plane perpendicular to AD, and passing through the point D, and let the conical surface XIDY be in like manner developed, and upon the same plane. When thus developed, these conical surfaces will have become the plane surfaces of two segmental annuli MPpm and IXxi*, whose centres are in the points E and F of the axes AE and AF, and which touch one another in the point D of the line of contact AD of the cones.

M

* The lines MP and pm in the development, coincided upon the cone, as also the lines IX and ix; the other letters upon the development in the above figure represent points which are identical with those shown by the same letters in the preceding figure. In that figure the conical surfaces are shown developed, not in a plane perpendicular to AD, but in the plane which contains that line and the lines AE and AF, and which is perpendicular to the last-mentioned plane. It is evidently unnecessary, in the construction of the pattern teeth, actually to develop the conical extremities of the wheels as above described; we have only to determine the lengths of the radii DE and DF by construction, and with them to describe