Pa2++Nr.: . . . . (251).

But the driven or working pressure P, being supposed to remain constant, whilst any two given teeth are in action, Pa, represents the work U, yielded by that pressure whilst those teeth are in contact: also r, represents the space S, described by the circumference of the pitch circle of either wheel whilst this angle is described. Now let be conceived to represent the angle subtended by the pitch of one of the teeth of the driven wheel, these teeth being supposed to act only behind the line of centres, then

senting the number of teeth on the driven

1

=

2

= -, n, repre-
N2
wheel, and

=T +

which relation between the work done at the moving and working points, whilst any two given teeth are in contact, is evidently also the relation between the work similarly done, whilst any given number of teeth are in contact. It is therefore the MODULUS of any system of two toothed wheels, the numbers of whose teeth are considerable.

221. THE MODULUS OF A SYSTEM OF TWO WHEELS WITH INVOLUTE TEETH OF ANY NUMBERS AND DIMENSIONS.

The locus of the points of contact of the teeth has been shown (Art. 201.) to be in this case a straight line DE, which passes through the point of contact A of the pitch circles, and touches the circles (EF and DG) from which the involutes are struck. Let P represent any position of this point of contact, then is AP measured along the given line DE the distance represented by λ in Art. 216., and the angle CAD, which is in this case constant, is that represented by . Since, moreover, the point of contact of the teeth moves precisely as a point P upon a flexible cord DE, unwinding from the circle EF and winding upon DG, would (see note, p. 235.), it is evident

that the distance AP, being that which such a point would traverse whilst the pitch circle AH revolved through a certain angle, measured from the line of centres is precisely equal to the length of string which would wind upon DG whilst this angle is described by it; or to the arc of that circle which subtends the angle. If, therefore, we represent the angle ACD by, so that CD=CA cos. ACD=", cos. 7, then r, cos. . Substituting this value for in equation (249), and observing that 8+9=

T

T

2

from which equation we obtain by the same steps as in Art. 219, observing that is constant,

η

which is the modulus of a system of two wheels having any given numbers of involute teeth.

222. THE INVOLUTE TOOTH OF LEAST RESISTANCE.

It is evident that the value of U, in equation (254), or of the work which must be done upon the driving wheel to cause a given amount U, to be yielded by the driven wheel is dependent for its amount upon the value of the co-efficient of U, in the second member of that equation; and that this co-efficient, again, is dependent for its value (other things being the same) upon the value of representing the angle ACD, or its equal the angle DAI,

19

which the tangent DE to the circles from which the involutes are struck makes with a perpendicular AI to the line of centres. Moreover, that the co-efficient N not involving this factor (equation 247), the variation of the value of U1, so far as this angle is concerned, is wholly involved in the corresponding variation of the co-efficient of U, and becomes a minimum with it; so that the value of which gives to the function of represented by this co-efficient, its minimum value, is the value of it which satisfies the condition of the greatest economy of power, and determines that inclination DAI of the tangent DE to the perpendicular to the line of centres, and those values, therefore, of the radi CD and BE of the circles whence the involutes are struck, which correspond to the tooth of least resistance.

To determine the value of which corresponds to a minimum value of this co-efficient, let the latter be represented by u; then, for the required value of ",

du

:. an

=B sec. (-) tan. (ʼn—¤)— A sin. ❤ { sin. ʼn sec. (~—¤)—

A sin. sec. (—¤){sin. ʼn cos. (~ʼn—❤)—cos. ʼn sin. (~—❤)};

du da

In order, therefore, that may vanish for any value of 7, one of the factors which compose the second member of the above equation must vanish for that value of ; but this can never be the case in respect to the first factor, for the least value of the square of the secant of an arc is the square of the radius. If, therefore, the function u admit of

a minimum value, the second factor of the above equation vanishes when it attains that value; and the corresponding value of is determined by the equation,

B sin. (—)—A sin. 1¤=0 . . . . . (256).

A

.....

or by sin. (—)=sin. p or by "+sin. (sin. ');

or substituting the values of A and B,

Now the function u admits of a minimum to which this value of corresponds, provided that when substituted in d'u

do

this value of gives to that second differential co-efficient of u in respect to ʼn a positive value.

Differentiating equation (255), we have

d'u

dr2

2 sec. '(-) tan. (-){B sin. (—)

A sin. '+B sec. 2(n−q) cos. (n−q)

But the proposed value of n (equation 256) has been shown to be that which, being substituted in the factor {B sin. (-)-A sin. ', will cause it to vanish, and therefore, with it, the whole of the first term of the value of

d'u

dn'

2

: it

corresponds, therefore, to a minimum, if it gives to the second term B sec. (-) cos. (-) a positive value; or, since sec. (-) is essentially positive, and B does not involve, if it gives to cos. (-), a positive value, or if

-1 A

'( sin. ')<

T

A

or if

B

sin. <1; or if

29

LaPa sin. ዋ a2r 2

This condition being satisfied, the value of ", determined

by equation (257), corresponds to a minimum, and deter mines the INVOLUTE TOOTH OF LEAST RESISTANCE. *

223. TO DETERMINE IN WHAT PROPORTION THE ANGLE OF

CONTACT OF EACH TOOTH SHOULD BE DIVIDED BY THE LINE OF CENTRES; OR THROUGH HOW MUCH OF ITS PITCH EACH TOOTH SHOULD DRIVE BEFORE AND BEHIND THE LINE OF CENTRES, THAT THE WORK EXPENDED UPON FRICTION MAY BE THE LEAST POSSIBLE.

Let the proportion in which the angle of contact of each tooth is divided by the line of centres be represented by x,

27

so that x may represent the angular distance from the line

of centres of a line drawn from the centre of the driven wheel to the point of contact of the teeth when they first

2.

come into action before the line of centres, and (1-x) n the corresponding angular distance behind the line of centres when they pass out of contact; and let it be observed that, on this supposition, if U, represent as before the work yielded by the driven wheel during the contact of any two teeth, U, will represent the portion of that work done before, and (1-x)Û, that done behind, the line of centres. Then proceeding in respect to equation (253) by the same method as was used in deducing from that equation the modulus (Equation 254), but integrating first between the limits 0 and, in order to determine the work u, done by

2

the driving pressure before the point of contact passes the line of centres, and then between the limits 0 and (1-x)

2x

n

to determine the work u, done after the point of contact has passed the line of centres; observing moreover, that in the former case is to be substituted in sec. (7) for 7 (Art. 217.), we have

It may easily be shown by eliminating between equations (254) and (256) that the modulus corresponding to this condition of the greatest economy of power, where involute teeth are used, is represented by the formula

U1 = { 1+4A sin. 20+(B2—A2 sin. *9) } 4U2+NS.