HO 



APPLIED MECHANICS. 



KD WHKKLS TIIK INVOHTHL 



oilier wheol (marked by the dotted linos), and suppose 

 it> duo extended to It Q overlapping tho former, the 

 same pencil would truce a corresponding line N P Q on 

 it. Now, nx the ctirves thus produced are traced by the 

 same ]<n>t in the band, aiul under precisely similar con- 

 ditions uf uniform rotation, we miu-ht cut the paper discs 

 t their outlines, and making them rotate in contact, wo 

 I obtain that uniform relative motion which U re- 

 quired. Tho mathematical name of each of tho curve* 

 so described is tho iiirolnleoflke circle, because it u pro- 

 duoed by the point of a thread wound on to (mv 

 a circle, or wound off from a circle. The na- 

 ture of tho curve may be best understood by 

 a reference to Fig. 240. 



1 f a be the centre of a circle or plan of a roller, on 

 which is wound a thread having its end at c, a pen- 

 cil IT u.u-iu; i>"int being fixed at the end of the thread, 



Fig. MO. 



b 



drawing or setting out tooth, then, tho following process 

 will bo found convenient. 



Fig. 211. 



will trace the involute c to o*, as the thread is unwound ; 

 IT h rf being the unwound thread, its point will trace the 

 involute back to c ag it is wound round the circle. If 

 we divide tho circumference into any number of equal 

 parts at d, e, f, y, <tc., and from these points draw 

 tangents or linos touching the circle, at right angles to 

 tho radii ad, at, af, <fcc., respectively, making the lengths 

 of tho tangents equal to tho lengths of circumference 

 nred round from c, the curve joining the extremities 

 of the tangents is the involute the proper outline for the 

 teeth of wheels that we have just described. In applying 

 this theory to the practical formation of teeth, we do not 

 find it necessary to describe tho actual involute form, 

 because the portion of the curve that belongs to any 

 tooth U so small, that we can draw a circular curve so 

 near to the involute as to cause no material error in 

 working Thus, if C (Fig. 241) be the centre of the 

 wheel, ami A, 13 the centres of two adjoining teeth on 

 tin. pitch circle, small circles being described round 

 those centres to fix tho breadth of each tooth D E and 

 F G, and H M N being part of the generating circle of 

 th.. involutes, the portion H K of any involute H K L 

 very nearly corresponds with a circular arc of which M ia 

 tin- ivntre, and tho radius D M is a tangent to the 

 gem-rating circle H M N at M. In practice, therefore, 

 it is only necessary to determine tho circle H M N, and 

 tho length of tho radius D M, for describing the curved 

 lides of the tooth. From what has preceded, it is clear 

 that wo may tike any convenient generating circle 



I H N, bat that whatever bo the one wo may select for 

 any one wheel, that for any other wheel gearing with the 

 formal must bo proportional to it It is found practi- 

 eally convenient to make the radius DM, with which tho 



ide of the tooth is doscril-od, |th of tho pitch radius C A, 



and the generating circle or locus of centres II M N 



be described within tho pitch circle, tho int, , v.,I 



between them A 1' I,, ,,, x Jth of b M or ,',nd of C A. If 



l- adhered to, all wheel* ,,f ,.,,iial pitch will 



gear with one BMttMT, vhatmrtN their diameters. In 



1st From tho centre C, with ratlins OA, de- 

 scribe the pitch circle, and divide it by tho com- 

 passes into equal parts, A B, each equal to the 

 given pitch. The pitch radius may be detenuim .1 

 I'.v the following rule : 



Multiply the number of teeth by 7 times their 

 distance apart, or pitch, and divide by !(. 



Required the pitch radius of a wheel 

 having 28 teeth of f inch pitch. 



28 X $ v 7 

 j^ = 3 '341 inches the pitch radius. 



NOTE. When tho number of tooth is small, and their 

 pitch considerable, the pitch radius must be a little in- 

 creased, as will bo found necessary on trying tho division 

 of the circumference. 



2nd. Hound tho centres A, B describe circles, each of 

 diameter somewhat less than half the pitch : to . \ 

 tho breadth of the teeth. The reason for makin 

 breadth of teeth less than half the pitch, is to give a 

 little room or clearance in the spaces between them, so 

 that in the event of slight irregularities occurring in the 

 workmanship, the teeth of two wheels may not become 

 bound or locked into each other. When the teeth are 

 cut by machinery to their exact form, this clearance is 

 not necessary ; but when they are merely cast and not 

 shaped afterwards, there should bean allowance of about 

 And of an inch for each inch of pitch that is to say, 

 if A B be 1 inch, 1) E should he A an inch, wanting >.. ml 

 of an inch, or jUnds of an inch, while K F is J of" an 

 inch and And of an inch, or iinds (.f an inch. Were 

 A 1 1 2 inches, then DE would be Jgths of an inch, and 

 E F jjths of an inch, and so on in like proportion. 



fed Take AQ - j of AC, and AP - Jth of A Q, 

 and from the contra C describe through P a circle of 

 centres H M K. 



4th. From the point* D, E, F, O, with radius A Q in 

 tho compass, mark oti" tho points II, S, T, U on i 

 II M N, and from these points as centres, with the same 

 radius A Q, describe tho curved sides of the tooth as 

 V II I)K. 



5th. It now only remains to determine the tops of the 

 teeth and the bottoms of the spaces between them. Tim 

 spaces should have somewhat greater depth below tho 

 pitch circle than tho height of the teeth beyond it, to 

 allow the tret h to dear; and it is convenient, both for 

 giving strength of form to the teeth and for providing 

 this clearance of the teeth, especially in coso of dirt 

 getting between tho gearing, to mako the bottom of the 

 lieircuUr. It will be found . i to 



make A \V, the height of the point of the tooth above 



