Page 209 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 197

GEOMETRY OF PLANAR CAM PROFILES 197

s
v = i A ( r + B r ) (7.52a)
i ii ii+1
3
where A i and B i are the scalars defined as
1 s 2
◊ +
◊
A ∫ rr + rr i (7.52b)
i i i+1 i i
2 4
1 3
◊ +
◊
2
B ∫ 2rr - rr s . (7.52c)
i i i+1 i i i
2 4
Equations (7.45), (7.48), and (7.49 to 7.52c) are the relations sought.
7.4.3.2.1 Spline Approximation of the Boundary. The formulas introduced in
Angeles et al. (1990) are now applied to regions whose boundary is approximated by peri-
odic parametric cubic splines. Recalling expressions (7.26a and b), the (x, y) coordinates
of one point on the boundary are represented as functions of parameter p, while x i and y i
represent the Cartesian coordinates of the ith supporting point of the spline. Moreover,
integrals (7.42a to c) are approximated as
n-1
A ª Â A i (7.53a)
1
n-1 n-1
Q ª Â Q , Q ª Â Q O (7.53b)
O
O
O
x xi y yi
1 1
n-1 1 n-1 n-1 n-1
I ª Â I , I ª Ê Â I + Â I ˆ , I ª Â I (7.53c)
O
O
O
x xi xy 2 Ë xyi yxi ¯ y yi
1 1 1 1
with
1 p i+1 5
x y dp = Â
A = Ú ( xy¢ + ¢ ) a (D p ) k (7.53d)
i ik i
2 p i k=1
1 p i+1 9
) ¢
Q = Ú ( x + y y dp = Â q (D p ) k (7.53e)
O
2
2
xi xik i
2 p i k=1
1 p i+1 9
Q =- Ú ( x + y x dp = Â q (D p ) k (7.53f)
) ¢
O
2
2
yi yik i
2 p i k=1
where
dx dy
x ¢ ∫ , y ¢ ∫ .
dp dp
The polynomial coefficients a ik , q xik , and q yik appearing in Eqs. (7.53d to 7.53f) are
included in App. C: Polynomial Coefficients. The components I xi, I yi, I xyi, and I yxi of the
inertia matrix are computed as
1 p i+1
¢ )
2
2
I = Ú ( x + y )( xy¢ + 3 x y dp
xi
8 p i
1 p i+1 3 p i+1
) ¢
2
) ¢
2
2
= Ú ( x + y xy dp + Ú ( x + y x ydp (7.54a)
2
8 p i 8 p i