Page 200 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 188

188 CAM DESIGN HANDBOOK

Let (x, y) be the coordinates of one point of the cam profile. The curve describing the
cam profile can be represented in parametric form as
2
3
(
(
(
xp () = A p - ) + B p - ) + C p - )+ D xi (7.26a)
p
p
p
xi
i
xi
i
i
xi
2
3
(
(
yp () = A p - ) + B p - ) + C p - )+ D (7.26b)
(
p
p
p
yi i yi i yi i yi
for p i £ p £ p i+1 and i = 1,..., n - 1, with further definitions:
p = 0, p i 1 = p + D p i (7.27a)
1
i
+
Dp = Dx + Dy 2 (7.27b)
2
i i i
Dx = x i+1 - x , Dy = y i+1 - y . i (7.27c)
i
2
i
Notice that parameter p represents a length measured along the perimeter of the
m
polygon defined by the set of vertices {P i} 1 . In the foregoing description, x i and y i repre-
sent, additionally, the Cartesian coordinates of the ith supporting point (SP) P i of the spline,
while coefficients A ui, B ui, C ui, D ui, for u = x, y, and i = 1,..., n - 1, are determined as
explained presently. Let us define the n¢(∫ n - 1)-dimensional vectors
T
x ∫[x , ..., x ¢ n ] , y ∫[y , ..., y ¢ n ] T (7.28a)
1
1
T
¢,
x ¢ ∫[x ..., x ¢ ] , y ¢ ∫[y ..., y ¢ ] T (7.28b)
¢,
1 ¢ n 1 ¢ n
and
T
T
x ¢¢∫ ¢¢ , ..., x ¢¢] , y ¢¢∫ ¢¢ , ..., y ¢¢] . (7.28c)
[y
[x
¢ n
1
¢ n
1
The relationships between x and x≤ and y and y≤ are linear (Rogers, 2001), namely,
Ax¢¢ = 6 Cx, Ay¢¢ = 6 Cy. (7.29)
Note that the A and C matrices appearing above are themselves functions of the coordi-
nates of the SP. In fact the n¢¥ n¢ matrices A and C are defined as
È 2a 1,n ¢ a 1 0 0 L a ¢ n ˘
Í a 2a a 0 L 0 ˙
Í 1 1 2 , 2 ˙
Í 0 a 2 2a 2 3 , a 3 L 0 ˙
A = Í ˙
Í M M O O O M ˙
Í 0 0 L a n ¢¢¢ 2a n ¢¢¢ ¢¢ a n ¢¢ ˙
,n
Í ˙
Î a ¢ n 0 0 L a n ¢¢ 2a n ¢¢ ¢ ˚
,n
and
- È b b 0 0 L b ˘
1,n ¢ 1 ¢ n
Í b -b b 0 L 0 ˙
Í 1 1 2 , 2 ˙
Í 0 b 2 -b 2 3 , b 3 L 0 ˙
C = Í ˙
Í M M O O O M ˙
Í 0 0 L b -b b ˙
,n
Í n ¢¢¢ n ¢¢¢ ¢¢ n ¢¢ ˙
Î b ¢ n 0 0 L b n ¢¢ -b n ¢¢ ¢ ˚
,n
where