Page 212 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 200

200 CAM DESIGN HANDBOOK

1 È 3 T ˘ 3 1
◊ )
◊
I = Ú r r ( r n 1 rn dS = ( n w 1 ) - w n T (7.68)
-
◊
O
3i Í i i ˙ i i i i i
2 i S 5 Î ˚ 10 2
with w i defined as:
w ∫ ( r r rdS . (7.69)
◊ )
i Ú
S i i
The integral appearing in Eq. (7.69) is evaluated below. To this end, r is expressed as:
r =+ p (7.70)
r
i
where p is a vector lying in the plane of the polygon S i and stemming from the polygon
centroid, as shown in Fig. 7.11. Now, w i becomes
2
+
2
2
w = r D r r ◊ (2 pp + p 1 d ) S + p pdS (7.71)
T
i i i i Ú S i Ú S i
i i
where an exponent k over a vector quantity indicates the kth power of the magnitude of
2
2
the said vector, that is, r ∫ r·r =||r|| .
Three surface integrals over S i are needed in the expression for w i, namely,
A = pp dS (7.72a)
T
i
i Ú S
i
a = p 2 dS (7.72b)
i Ú S i
i
b = pp dS . (7.72c)
2
i Ú S i
i
Since p is a vector lying entirely in the plane P i defined by the polygon S i, it can be
represented uniquely in the 2-D subspace as a 2-D vector of that plane. Consequently, the
GDT can be applied in this 2-D subspace to reduce the surface integrals (7.72a to c) to
line integrals (Al-Daccak and Angeles, 1993), namely,
Z
r i,k
P
r i,k+1
Y

X
FIGURE 7.11. Polygon S i representing one face of a poly-
hedron approximating a closed surface.