Woodward — The Efficiency of Gearing under Friction. 97 



Now the friction actually overcome in either case isP sin <p, 

 and since the velocity of sliding is r (a x + # 2 )> ^ ne amount of 

 friction overcome, or the energy lost, during the time dt is 



dU = Psin <p (a 1 + a 2 ) rdl (3). 



These formulae hold at all times and for all kinds of teeth 

 that are of correct outlines. 



3. Epicycloidal leeth. Let the driving moment M 1 be 

 constant, and let the teeth be epicycloidal, described by a 

 rolling circle with radius r . Also let q be the " arc of ap- 

 proach " i. e. the arc of one of the pitch circles which will 

 pass the pitch point while the point of contact T is moving 

 to the pitch point. 



It is evident that during the approach dq \s negative, and 



that 



ajrfit = a 2 r 2 dt = — dq; 



hence 



i a 1 + a 1 )dt = -(±+±Jdq. 



But from the figure, 



r = 2r cos 0, and q = r (7r — 20), 

 hence 



dq — — 2r d0, and since P = M 1 ■+• l x 



we have from (I) and (III) 



M. sin <p I 1 1 \ 

 d U x = -L_? {— + — ) W cos Odd 





7T 



cos 0d# 



sin (0 — <p) + r sin ^ 



<$(£+£)*>{ r"-^~ (4) 



tan0 



(•-*)' 



