APPLICATION OF GRAPHIC METHODS 337 



angular acceleration of the connecting rod. If A C be the crank, 

 and C B the connecting rod, and if AB be called x, then ^ is the 



FIG. 55. 



j n 



acceleration of the piston ; and if the angle C B A be called 0, is 



ct t 



the angular velocity of the connecting rod, and -- is its angular 



Cf 



acceleration. 



If the angle which the crank makes with A B be called a, is 



d t 



the angular velocity of the crank. 



Then, calling the crank radius r, and the length of the connect- 

 ing rod I, we have 



sin 



dO _ r cos a da 

 d t */ 1~ r 2 sin 2 a ' d t 



rf- 



Difterentiating again, and remembering that 2 = 0, we obtain 



d t 



finally 



d?6 _ r sin a (^ r 2 ) /c?aV 2 



dP (I- r- sin 2 a)? \d tj 



Again 



x = r cos a + 1 cos 0, 



dx . da i * dO 



= r sin a I sin a -=-, 



dt dt dt 



, 

 and -r-^= r cos 



fdaY , nfd6\* , . d~ 2 



a I j I cos 6 [ - } t sin -=-.. 



\dtj \dt) dt* 



Substituting for -=- and -= - their values as determined above, and 

 dt dt- 



putting r sin a for I sin 6, and A /^_ r a s i n 2 a for I cos 6, we obtain 



finally 



r 



r' 2 f" 9/7 J- 7-3 sin *r 



-Ts>= ~ r -r: 1 < COSa + 



-r sn a 

 VOL. II. Z 



