MISCELLANEOUS MECHANISMS 403 
- vertex of the cone is on the axis of the cam, and the acting surface of 
the cam is also conical with its vertex at the vertex of the roller cone. 
For other cases the true form of the roller is not conical, and only when 
_ the acting surface of the cam is a screw surface of constant pitch is it 
possible to give the roller a form which will work correctly on all parts 
of the cam. 
_ A cylindrical cam in which the acting surface is a screw surface 
of pitch p is shown in Fig. 650. R, and R, are the external and internal 
radii of the screw surface respectively, and R is any other radius, 7, and 
r, are the radii of the outer and inner ends of the roller yespectively, 
and r is the radius of the roller CC 
corresponding to the radius R of the 
screw surface. a 
Consider the half of the cam on 
one side of a plane containing the -—-—--s <TR 1-4 
axis of the cam. <A,B,C,, A,B,C,, |-zae TR--+> 
and ABC are the helices which are eRe RST db 
the intersections of the screw surface Fic. 651 
with the surfaces of cylinders of radii an 
R,, R,, and R respectively. ac,, ac,, and ac, the developments of these 
helices, are shown to a reduced scale in Fig. 651. Let the cam make 
half a revolution, then for pure rolling it is evident that 
Ls ee | = bs : = rR? p = rR; Pr 
Fiat aE. Batam a/ PB +E, aye a/R +E, 
2 1 /4?RS 
and ac= wR2+2. Hence ro= - air +e ’ 
4 J 4°R, +p 
4ar*R? + p 
and r= has OME. Whea R=0, r= G5 
p J/4?Ri +p 
and the helix coincides with the axis of the cam, hence the axis of the 
roller must be at a distance from the axis of the cam equal to 
Tp 9, /40?R? + p? 
N4etR? +p 4B? 4p? 
: 4r?R, +p 47? TP 
and rearranging the terms (Sete 72 — (=>) R= 1, and this is 
Tse Pry pry 
ee ¥ 
of the form a a 1, which is the equation to an hyperbola. 
The outline of the roller from R=0 to R=R, is EHLPKF, and 
between R=R, and R=R, the form is EHKF. The true form of the 
roller is an hyperboloid of revolution, but for ordinary cases the part 
EHKF is practically conical. LP is the throat of the hyperboloid. A 
plane section of the hyperboloid by a plane parallel to its axis and 
Squaring both sides of the equation, r= 
3 touching the surface at the throat will be two straight lines, and if this 
ne also contains the axis of the cam, one of the straight lines will 
the line of contact between the roller and the screw surface of the 
eam. In the side elevation to the right in Fig. 650, P’B, is the pro- 
jection on the axis of the cam of the line of contact between the roller 
_ and the screw surface of the cam, and B,P’B, is the true inclination 
