SCREW GEABtNO.] 



APPLIED MECHANICS. 



893 



Fig. 249. 



clined to the plane of their wheel, in such a manner 

 as to suit the obliquity 

 of the screw-thread. 

 When the worm is caused 

 to revolve, one portion of 

 the worm pressing against 

 a tooth of the wheel, causes 

 it to advance over a dis- 

 tance equal to the pitch of 

 the thread or of the teeth 

 (which are necessarily 

 alike) ; and by the time 

 one tooth is disengaged 

 from the worm by the re- 

 volution of the wheel, 

 another tooth has come 

 into action, and is caused 

 to advance in like manner. 

 Every revolution of the 



screw round its axis thus causes the wheel to rotate 

 through a portion of a revolution, equivalent to the dis- 

 tance of the centre of one tooth from that of the next ; 

 and if the wheel have, for example, 100 teeth, it thus 

 requires 100 revolutions of the worm to cause one revo- 

 lution of the wheel. It is evident that, whatever be the 

 size of the worm, the same relation of angular velocities 

 subsists, while the number of teeth in the wheel remains 

 constant ; but when we alter the pitch of the screw, that 

 is to gay, the distance from one of its coils to the next, 

 we must alter, to a corresponding extent, the pitch of the 

 toothed wheel ; and, if its diameter be fixed, the number 

 of its teeth must be altered accordingly. 



In deciding on the form of teeth for screw-gearing, we 

 must be guiued by considerations similar to those which 

 we have dealt with in respect of ordinary geaiing. If 

 we suppose the screw-wheel B (Fig. 250) to be a thin 



Fig. 250. 



plate of metal, with teeth of the proper 



involute form cut in its circumference, 



and A the section of the screw to be also 



a thin plate with teeth like those of a 



rack, by drawing the rack along rectili- 



neally in the direction of the arrow, we 



should cause the wheel to rotate in the 



direction of the arrow, and the teeth 



would be of suitable form for their 



relative movements. So when the wheel 



i of some thickness, as shown in section at E, with 



teeth F projecting within the outer circumference 



of the screw, and clearing the solid central portion, vre 

 have only to make the teeth F inclined so as to suit the 

 obliquity of the screw-thread, as indicated by the dotted 

 lines H H on the plan of the screw G ; aud, instead of 

 moving the screw longitudinally like a rack, we may 

 cause it to rotate round its axis ; and while we prevent 

 its longitudinal motion, the teeth of the wheel will be 

 caused to move onwards by the inclined action of the 

 screw-thread. As no part of the screw-thread is a 

 straight line, the teeth of the wheel, indicated by the 

 dotted lines H H, ought theoretically to be curved ; but 

 when the diameter of the screw is large in proportion to 

 its pitch, the portions of the thread, with which the teeth 

 are in contact, approach very nearly to straight lines, 

 and the oblique sides of the teeth may therefore, without 

 much practical error, be made straight. Their proper 

 obliquity may be found thus : Let I be a cylinder of the 

 same diameter with the screw, on which is wound a tri- 

 angular piece of paper, so that its edge shall form the 

 outline of the screw. If at any one of its convolutions 

 K, instead of continuing to wind the paper on the 

 cylinder, we stretch it out straight as K M L, if K L be 

 half the pitch of the screw, or L half-way between K 

 and the point where the next convolution after that at 

 K would cross the axis, then L M must be equal to half 

 the circumference of the cylinder (about 3f times its 

 radius), in order that the point M may be brought round 

 to L when the paper is wound on to the cylinder. By 

 taking K L half the pitch of the screw, and L M half its 

 circumference, and joining K M, we get a line at the 

 proper obliquity to the axis to suit the thread of the 

 screw. By taking a cylinder equal to the inner part of 

 the screw, or of diameter equal to that of the screw at 

 the bottom of its thread, and proceeding in a similar 

 way (as indicated by the dotted lines), we get the line 

 K N of the proper obliquity to suit the bottom of the 

 thread. Now, as the tops or points 

 of the teeth of the wheel gear with 

 the bottoms or hollows of the screw- 

 threads, and the bottoms of the teeth 

 gear with the tops of the screw- 

 threads, we should make the tops of 

 the teeth less inclined to the plane 

 of the wheel than their bottoms, 

 and wo should gradually increase the 

 obliquity from the tops of the teeth 

 downwards. It would be very diffi- 

 cult to effect this in practice ; and, 

 indeed, even if it could be done with 

 ease, it would be positively disadvan- 

 tageous, because some of the teeth 

 are always coming into such a posi- 

 tion with respect to the thread as that 

 marked Q, where the outer portion of 

 the tooth is in contact with the upper 

 part of the thread. Practically, then, 

 it is best to make the obliquity of the 

 teeth like K P a mean between those 

 due to the upper and the lower parts 

 of the screw-thread. In other words, 

 tracing a pitch-line R R for the screw 

 touching the pitch circle of the wheel, 

 and then developing the obliquity K P 

 due to the diameter of the screw mea- 

 sured to that pitch-line, we get an 

 average inclination for the teeth, 

 somewhat in error at points above 

 and below the pitch-line, but correct 

 where the principal contact and com- 

 munication of power takes place. The 

 greater the diameter of the screw, 

 and the smaller its pitch, the greater 

 is the inclination of the thread to 

 its axis, and the less is the error of 

 obliquity in the contact of the teeth 

 and screw-thread above and below 

 the pitch-line. Therefore, when the 

 screw by rotating drives the wheel, the screw should 

 be made as large in diameter as is consistent with 



