s 4 ROYAL SOCIETY OF CANADA 



example is that of a cam operating a tilt hammer, which is raised 

 gradually and allowed to fall suddenly under the action of gravity. 



For many cases motion of the follower is not ])arallel to the shaft 

 carrying the cam, but at right angles to it, and the cam surface must 

 then be traced in polar co-ordinates. The condition for movement to 

 be possible is now that the inclination of the cam curve to the radius 



71 



vector at the point shall be greater than -^ cf, where a =■ tan~^ ^ or 



analytically r-j- >■ tan {- «'. j As in most cases it is easier to fulfil 



the conditions by drawing the displacement curve with rectangular 

 co-ordinates, it will be convenient to enquire what variation is produced 

 in transferring to polar co-ordinates. 



The usual method of transference is to fix upon a minimum radius 

 vector to which corresponds a line in the x, y, plane distant b below the 

 lowest point of the displacement curve. The values of the ordinates cut 

 off between this straight line and the curve of displacements are then 

 taken as polar distances at their appropriate angles, and a smooth curve 

 drawn through their outer ends, and if the curve is such as to satisfy the 

 criterion stated above, the required motion will be obtained. 



It will be sufiicient for our present purpose if we consider the case 

 of the ti-ansformation of a plane cam made up of lines inclined at the 

 maximum angle a = tau~^ ju. 



If ?/' is the corresponding angle which the polar cam makes at any 

 radius r, (p, then 



rd ^/dr = tan ip. 



^j dr dr . 



Now 77, = ^ — -. = tan a. 

 do dç 



. • . r = tan a . tari f. 



= fx tan '/". 



The angle ip is therefore variable, depending upon the value of r, 



7t 



thus when r = o ip ■= o and when r =: œ , ip = -^. 



This shows that for any possible case (excluding finite discontinu- 

 ities) a value of a; can be found, such that the modified criterion can be 

 satisfied, and hence a polar cam curve can be constructed from the Cartesian 

 curve of displacements. 



It is moreover also possible to transform to polar co-ordinates with- 

 out change of angle, but this method of transformation involves an 

 alteration in the displacement given by the cam. 



If we denote a point on the displacement curve by the rectangular 

 co-ordinates x, y, a polar diagram can be drawn, which has, jDoint for 



