VARIATION OF PRESSURE IN PUMP 607 



- a feet, so that to prevent separation during this stroke we must 

 CI L I g 



have : 



''' a d 9 ( 2 m a d 



Note. Since during the latter half of the stroke the piston is being 

 retarded ( a) is a positive quantity. 



Where the pump is driven from a shaft rotating with uniform angular 

 velocity to radians per second, by means of a crank of radius r, assuming 

 the connecting rod to be of infinite length : 



'a to 2 r cos f.s.s. 

 v = co r sin f.s. 

 where is the crank angle, measured from the inner dead centre, while 

 with a connecting rod of length I these expressions become : 



f 2 f r / 2 cos 2 0+ ; 3 sin 4 ) . 

 i a = or r ; COS u -\ g I i.s.s. 



(I 2 r 2 sin *0f 



r sin 2 ) 



sin -\- -^j 



giving a the maximum value ^ r \ 1 + -V [ according as the crank is on 



the inner or outer dead centres respectively (i.e., according as = or 

 180). 



Neglecting, for simplicity, the effect of the obliquity of the connecting 

 rod, and therefore assuming the piston to have simple harmonic motion, 

 the maximum velocity is given by to r, while the maximum acceleration 

 = to 2 r, and therefore for a given displacement the acceleration is com- 

 paratively reduced by making the stroke of the pump large and the 

 angular velocity of the crank, or the number of strokes per minute, low. 

 This enables a higher piston speed to be adopted, since frictional effects 

 are usually small compared with those due to acceleration. 



Substituting for a and v in terms of the crank angle 0, equation (1) 

 becomes : 



v a h* > ~ \ ** r cos + J ^ \ Sm [ 



a s g { 2 m a s j 



2^ U..* , /rrfnV4" 



, 

 > o> r - - cos 



a s g ( ^ 2 m a s 



(8) 



