Page 205 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 193

GEOMETRY OF PLANAR CAM PROFILES 193

a R
T

a

C
r q e

f
y
O l P

FIGURE 7.9. Cam mechanism with oscillating roller follower.

7.4.2.5 Computation of the Local Properties. As shown above, the curvature and the
pressure angle depend on the displacement program and its derivatives. Thus, an adequate
synthesis procedure of the displacement program is required.
The synthesis of the displacement program has been traditionally based on a limited
set of functions giving rise to parabolic, harmonic, cycloidal, trapezoidal, and polynomial
motions. More complex motions, such as those appearing in indexing cam mechanisms,
can be synthesized using spline functions, which are also known as nonparametric
splines.
Nonparametric cubic splines, when applied to the displacement programs s(y) or f(y),
take the simple forms
3
(
(
y y () = A y y ) + ( - 2 C y y )+ D , y £ y £ y (7.37)
B y y ) +
-
-
i i i i i i i i i+1
for i = 1,..., n - 1, and with y representing s in the case of translating followers, f in
the case of their oscillating counterparts. Now, if we define the (n - 1)-dimensional vectors
y, y¢, and y≤ in exactly the same way as defined in Subsec. 7.4.1, then the counterparts of
relations (7.29) and (7.30) take the forms
Ay¢¢ = 6 Cy (7.38)
-
y ¢ = (F GC )y (7.39)
where A, C, F, and G are formally identical to their counterparts in Subsec. 7.4.1. Their
differences are that a i and b i in the entries of these matrices in that subsection change to
Dy i and 1/Dy i, respectively. In this case, obviously,
Dy ∫ y i+1 - y i
i