Page 162 - Cam Design Handbook

THB5 8/15/03 1:53 PM Page 150

150 CAM DESIGN HANDBOOK

-
-
f
f
∂ w N () È N ( w 1) () N ( w 1) ()
f
2 ˘
+
-
-
()
w
f
N () ∫ ik , 1 2 = ( k - 1) Í ik , 1 1 2 - i 1 k , 1 1 ˙ (5.43)
∂f x - x x - x
ik , 1 2 w 1
+
+-
+
2 Î ik 1 1 i ik 1 i 1 ˚
()
f
w
f
w
2 ˘
Ê k - 1 ˆ ( È f 2 - xN ) i k , 1 1 () x ( + -f 2 N ) () k , 1 1 ()
2
i 1
i
-
-
+
()
w
N () = Á 1 ˜ ¥ Í + ik 1 ˙ (5.44)
f
ik , 1 2 Ë k - w 1¯ x - x x - x
-
+
+-
1 Î ik 1 1 i ik 1 i 1 ˚
+
-
-
∂ v M s () È M ( v 1) s () M ( v 1) s () ˘
-
+
-
M v () s () ∫ jk , 2 2 = ( k - 1) Í jk , 2 1 2 - j 1 k , 2 1 2 ˙ (5.45)
∂ s 2 Î y jk 2 1 - y i y jk 2 - y j 1 ˚
jk , 2 2 v 2
+
-
+
+
v ()
v ()
y M
2 ( È
Ê k - 1 ˆ s - ) j k , 2 1 s () ( y jk 2 - s M ) j 1 k , 2 1 s () ˘
2
+
-
2
-
+
j
2
v ()
M s () = Á 2 ˜ ¥ Í + ˙ . (5.46)
Ë k -- ¯ Î y jk 2 1 - y i y jk 2 - y j 1 ˚
v 1
jk , 2 2
+
+
-
+
2
The B-spline surface interpolation can also be expressed by matrix as
[ S f , )] = [ N f ()]◊[]◊[ M s ()] (5.47)
(
p
s
1
2
2
2
2
where
1(
1(
1(
s
s
È S f , ) S f , ) L S f , s )˘
2
2 1
2 1
2 2
m
2 1
2 1
Í S f , ) S f , ) L S f , s ) ˙
1(
1(
1(
s
s
[ S f , )] = Í Í 2 2 M 2 1 M 2 2 O M 2 2 2 m ˙ ˙
(
s
2 2
2
1
2
Í S f , ) ) ˙
n 2 (
n 2 (
s
Î 1 n 2 ( s 2 1 S f , ) L S f , s ) ˚ ¥nm
1
2
1
m
2 2
f
f
È N ( ) L N ( )˘
,
1,
2 1
k 1
n k 1
2 1
Í ˙
[ N f ()] = MO M
2 Í ˙
Í Î N ( ) L N ( ) ˚ ˙ nn
f
f
n 2
1,
,
n 2
¥
n k 1
k 1
p È 11 L p 1, m ˘
,
Í ˙
p [] = MO M
Í ˙
˙
Î p Í n 1, K p n m˚ ¥
,
nm
s
È M ( ) L M ( ) ˘
s
2
1,
1,
k 2
2 1
k 2
Í m ˙
[ Ms ()] = M O M .
2 Í ˙
Í M ( ) L s ˙
s
Î mk 2 2 1 M ( ) ˚ ¥
,
2
,
mm
mk 2
m
Follower Motion Synthesis for the Three-Dimensional Cam
Before examining the synthesis process, some basic requirements should be noted:
1. The number of motion constraints is not limited. However, for the nonparametric
forms of the B-spline surface interpolation used here, the pattern of the motion constraints
must be a rectangular grid data form.
2. To guarantee jerk continuity, the synthesized motion function in each parametric
direction must be of at least degree = 4. This means that the order of the B-spline func-
tions must be greater than or equal to 5.
3. Since the profile is a closed surface in the parametric direction of rotation, the
motion function along that parametric direction, i.e., f 2, which is used to form the must