Page 213 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 201

GEOMETRY OF PLANAR CAM PROFILES 201

1 1
(
A = Ú div () = Ú pp p n d ) G (7.73a)
◊
T
P dS
i i i
4 S i 4 i G
1 1
a = Ú div (pp d ) S = Ú p 2 (p ◊ n d ) G (7.73b)
2
i i i i
4 S i 4 i G
1 1
(
b = Ú div ( p pp d ) S = Ú p p p n d ) G (7.73c)
◊
2
2
T
i i i
5 S i 5 i G
where P is a third-rank tensor that is cubic and homogeneous in p. It is omitted here, the
interested reader being referred to Al-Daccak and Angeles (1993) for further details. More-
over, G i denotes the polygonal boundary of S i contained in the plane P i . Note that, just as
the GDT is used in 3-D to reduce volume integrals, it is used in 2-D to reduce surface
integrals to line integrals.
Now let G i,k denote the kth side of the m-sided polygon S i that joins the kth and the
(k + 1)st vertices, numbered counterclockwise when the face of interest is viewed from
outside the polyhedron. Moreover, a sum over subscript k is to be understood, henceforth,
as being modulo m. Furthermore, the position vector p, shown in Fig. 7.12, of any point
of G i,k is defined in the plane P i as
p = m + h , p ŒG 0 £ s £1 (7.74)
,
,
ik , ik i , k ik, ik ,
S
where m i,k and h i,k are the vectors r i,k - r¯ i and r i,k+1 - r i,k, respectively, r i,k being the posi-
tion vector of the kth vertex of the polygon S i. Similar to vector p, vectors m i,k and h i,k lie
solely in the plane P i . Consequently, their representation as 2-D vectors in that plane is
used to apply the GDT to reduce the surface integrals defined over S i to line integrals
defined over G i,k.
Let n i,k be the unit normal vector to G i,k pointing outward of S i and s i,k be the length of
the kth side of G i. Thus, quantities A i, a i, and b i can be evaluated as indicated below,
keeping in mind that s i,k =||h i,k|| and p·n i,k = m i,k ·n i,k, since h i,k ·n i,k = 0. Then
V n i,k
s i,k
h i,k G i,k

p
m i,k
U

FIGURE 7.12. The ith polygon S i contained in the plane P i.