Section III., 1903 [ S3 ] Trans. R. S. C. 



VII. — Note on the application of Fourier's Series to the determination 

 of the forms of Cams to fulfil given conditions of displacement, 

 velocity and accleration. 



By E. a CoKER, M.A., (Cantab.), D.Sc. (Edin.) 

 Assistant Professor of Civil Engineering, McGill University, Montreal. 



(Communicated by Dr. H, T. Bovey, and read May 19, 1903.) 



The applications of canas for transmitting and modifying motion are 

 extremely varied on account of the ease with which, any finite displace- 

 ment of one piece with regard to another can be produced, and by suit- 

 ably combining cams a tracing point can be made to occupy successively 

 any point on a curve in a plane, whether non-intersecting or otherwise. 

 This was early shown by Mr. Cowper, who in a lecture before the Eoyal 

 Institution arranged a model in which the tracing point produced a 

 curve forming the letters R. I., and other more complicated forms have 

 been produced since. 



In general cams are designed to produce given displacements only, 

 and their forms are obtained by various artifices. If the required dis- 

 placement of the point considered be marked upon a plane in relation to 

 the angular displacement of the cam from a fixed zero line, then the curve 

 of displacements is evidently a one-valued function of the angle in 

 general, but there may be finite discontinuities corresponding to a sudden 

 rise or fall in the motion of the follower. If, therefore, a predetermined 

 motion is marked out in rectangular co-ordinates, the required cam sur- 

 face is at once produced by wrapping this curve round a right circular 

 cylinder, the periphery of which is equal to the length along the axis of ic 

 corresponding to a complete period of the displacement. This mode of 

 the formation of a cam upon a cylindrical surface has this advantage, that 

 the inclinations of the cam curve to lines parallel to the axis of y, and to 

 the generating lines of the cylinder are equal. For motion to be pot^sible 

 it is essential that the inclination of the cam be less than tan—^ yu, where 

 yw is the co-efficient of friction or analytically dy/dS <^ pi. 



If a curve in the x, y plane be plotted for which at some point of 

 the period o — 2 tt, dy/dd > yu then if every elemental value of dd be 

 altered in a constant ratio of k/\ it is always possible to find k such that 

 dy/kdd < jj. and the required condition can always be fulfilled provided 

 dy/dd is not = d= oo at any place. In that case motion is not possible 

 with the cam alone, but this can be overcome, at the discontinuity, by the 

 application of an extraneous force such as gravity, or the action of a 

 spring arranged to produce motion at the discontinuity. A simple 



