Page 161 - Cam Design Handbook

THB5 8/15/03 1:53 PM Page 149

CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS 149

FIGURE 5.33. Three-dimensional cam with translating spherical follower.

1 Ï if y £ s < y i 1
i
+
2
s
M () = Ì (5.40)
j,1
2
Ó 0 otherwise
s - ) s () ( y - s M ) s ()
2 (
y M
+
+
-
-
M s () = j j k , 2 1 2 + jk 2 2 j 1 k , 2 1 2 (5.41)
y - y y - y
jk , 2 2
-
+
+
+
jk 2 1 j jk 2 j 1
where x i and y j are used here to denote elements of the knot sequences in each direction.
It should be clear from the earlier discussions that the knot sequences must be nonde-
creasing series of real numbers that satisfy the relations f 2min¢ £ x i £ x i+1 £ f 2max¢ and s 2min¢
£ y i £ y i+1 £ s 2max. Two types of knot sequences must be used. The first, used for the angular
position of the cam, is a uniform knot sequence, for instance [0, 1, 2, 3, 4]. This sequence
will ensure continuity at f 2 = 0 and at f 2 = 2p. A knot sequence having a multiplicity of
values at the ends equal to the order of the B-spline with the internal knot values evenly
spaced is used in the other direction, for example, [0, 0, 1, 2, 3, 3] for B-splines of order
k = 2.
The recursive representation for the derivative of a nonparametric B-spline surface can be
obtained by differentiating Eq. (5.36) to yield (Butterfield, 1976)
f s )
+
∂ ( wv) S ( , n m
()
1 2 2 = ÂÂ N () v ( ) ( s p ) (5.42)
w
f M
,
,
∂f ∂ s v ik 1 2 jk 2 2 ij ,
w
2 2 i=1 j=1