ISTOLOTE TEETH.] 



APPLIED MECHANICS. 



889 



engineers generally should determine on some standard 

 form, so that all wheels of equal pitch should work or 

 gear properly together. Unfortunately this is not the 

 case : the wheel made by one machinist is unsuited to 

 that made by another. It often becomes necessary to 

 make costly patterns of wheels to suit some that may 

 have been already made ; these patterns, again, are not 

 suited for other cases ; and thus a great amount of time, 

 labour, and material is misapplied in a matter where a 

 little harmony among the views of machinists might do 

 much to avoid these evils. That such a mutual under- 

 standing is not impossible, may be proved by the fact, 

 that in a similar case, that of screw-threads, among 

 which there once existed quite as great a variety, almost 

 all engineers now adhere to certain forms and proportions; 

 so that the screw which fits one nut, "readily fits any 

 other nut of equal diameter, wherever the screw or the 

 nut may have been manufactured. Perhaps the prin- 

 cipal cause of difference in the forms of teeth, has been 

 the want of knowledge among practical men as to what 

 the true forms should be. Much has been written on 

 this subject, but a great deal is of too abstruse a cha- 

 racter to be generally understood or appreciated ; and 

 the practical mechanic, desiring information, is dis- 

 couraged by the difficulties with which the subject appears 

 to be invested. We will endeavour to point out, as 

 imply as possible, the laws which should govern the 

 forms of teeth, and to lay down a few easy rules for 

 lineating and executing them. 



Fig. 336. 



If we suppose A and B (Fig. 236) to be the bosses or 

 central parts of two wheels having straight teeth or flat 

 blades C and 1) projecting from them respectively, on 

 tracing several positions of these, such as C C 2 and 

 D, D,, we observe that there must be considerable 

 inequalities in their relative movements; or if 0,0, 

 make equal angles with the line of centres A B, D, and 

 l>o do not make equal angles with the same line. Again 

 we observe that the points of each tooth must rub along 

 the flat surface of the other in the course of the motion ; 

 and were this form of tooth practically carried out, the 

 cutting and wear would be considerable. But if, as in 



Fig. 287. 



Fig. 237, the wheel B have a tooth D of some curved 

 outline, we may perhaps find some particular curve for 

 the face of tooth C, such that an equable movement of 

 the one wheel shall produce an equable movement of 

 the other, and that the surfaces of the two teeth shall 

 rather roll along each other than rub with a pushing or 

 sliding movement. Now, although it may be generally 



VOL. I. 



possible to find a proper form for C, whatever be the 

 form of D, yet it is desirable for many reasons that both 

 these curves should be traced according to some fixed 

 type, which shall be constant, whatever the dimensions of 

 the wheels or their numbers of teeth. If we examine 

 somewhat closely the relative motions of two toothed 

 wheels, we may perhaps discover an appropriate form 

 for their teeth. 



INVOLUTE TEETH. If A B (Fig. 238) represent 

 a strap passing round the circumferences of two equal 

 pulleys, and therefore touching both, and if through C, 

 the middle point between their centres, two circular arcs 

 be described, these may represent the pitch circles of two 

 equal toothed wheels, whose relative rotation should be 

 precisely the same from the gearing as it is from the 

 motion of the strap, unwinding from the one pulley and 

 winding on to the other. But farther, if we continue 

 the lines A B and D E, and from some other point F in 

 B E describe a circle touching A B in G, and another 

 circle (dotted) through the point C ; since F C bears 

 the same ratio to C E as F G to E B, it appears that the 

 rotation of the large wheel, if geared with the other at C, 

 should be precisely the same as if it were caused by the 



Fig. 238. 



strap winding on to its pulley, of which F G is the radius. 

 If we assume the angular velocity of the pulley D to be 

 uniform, the rectilineal velocity of the strap A B G must 

 also be uniform, and likewise the angular velocities of 

 the wheels E and F. Whatever, then, be the relative 

 sizes of the geared wheels, the rectilineal motion of the 

 strap, with relation to the rotary motion of any one of 

 them, is always constant ; and if from this relation we 

 can trace out a form of tooth, that form will apply in all 

 cases, whatever be the diameters of the wheels that are 

 geared together. 



Fig. 239. 



In order to avoid complexity, let us first take one 

 wheel, supposing it a circular disc of paper, of which 

 H M K (Fig. 239) is a portion, and let us suppose that 

 the strap A B has a pencil projecting from it at some 

 point P, so that as the disc rotates while the strap 

 travels, the pencil shall trace on the disc a line L P M, 

 compounded of these two motions. If, now, we take the 



6 x 



