The cam that is designed here is made up of 2 circles with tangent lines between them to form the perimeter as shown below
If we use the equation
It turns out that the angle f is given by the equation:
Therefore, the cam is circular of radius a between angles p/2- f
and 3 p /2+ f, linear between the latter location and (x0+bsin(f),-bcos(f)), circular
of radius b between this latter location and (x0+bsin(f),bcos(f)) and then linear between this latter
location and (asin(f),acos(f)).
Now I illustrate the cam rotated by angle y. The
cam follower is also shown. The center
of the cam follower roller always remains on the x axis so that its coordinates
are (xc,0). A large part of the subsequent
math is involved in computing xc.
and the points of tangency between the cam and follower.
First I show the case where the follower (of radius c) is tangent to the b
circle of the cam. That means that the
two radii of these circles are parallel as shown.
Note that
From this equation I can obtain the angle b:
And from b I can obtain the equation for the center of circle c:
The point of tangency of the c and b circles can also be found:
Finding the point of tangency between the c circle and the 2 straight lines is a bit more involved. Note that the radius of the c circle is going to be perpendicular to this line. Also note that the slope of the originally bottom line will be y+f. To do the computation, for the originally bottom line, I must first compute the distance, r, from the start of the line on circle a to the tangency point..
Then the intersection is:
Similarly, the location of the center of circle c is:
Similar expressions exist for the intersection of the follower with the top line.
For the tangency of circle c with circle a, the y component
is always 0 and the x component is always a.
That means that the center of the c circle is always at xc=a+c.