Woodward — The Efficiency of Gearing under Friction. 97 



Now the friction actually overcome in either case isP sin ^, 

 and since the velocity of sliding is r (a^ -\- a^), the amount of 

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



(117= Psm<p {a^ + a,)rdi (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^ 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 is negative, and 



that 



a{i\dt = a^r^dt = — dq; 



hence 



But from the figure, 



r = 2rjj cos d, and q = r^ {ir — 2^), 

 hence 



dq = — 2r^dd, and since P = ik/^ -f- 1^ 



we have from (I) and (III) 



,__ M. sin cp / 1 1 \ 



dU, = _L_i'[_ + _j 4r,2cos ddd 



C^. = 4>-o^(^ + ^)^.si°f 



TT 



cos ddd 



?'j sin {6 — ^ ) + )• sin (p 



"'m*^)'4 l. ("^), "> 



tan^ 



