Page 214 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 202

202 CAM DESIGN HANDBOOK

1 m 1
A = Â s ( m ◊ n ) pp ds, p ŒG . (7.75a)
T
,
i i k , ik , ik Ú 0 ik ,
4 k=1
where
pp = ( mm )+( mh + hm )+( hh ) s 2 (7.75b)
T
T
T
T
T
ik ,
ik ,
ik ,
ik ,
ik ,
ik ,
ik ,
ik ,
and hence
1 1 1
pp ds = ( mm )+ ( mh + hm )+ ( hh ).
T
T
T
T
T
2 3
Ú 0 ik , ik , ik , ik , ik , ik , ik , ik , (7.75c)
Also
m
1
1
a = Â s (m ◊ n ) p 2 ds, p G .
Œ
,
i i k , i k , i k Ú 0 ik ,
4 k=1
where
2
2
2
p = m + 2 m ◊ h s + h s 2
ik , ik , ik , ik ,
and hence,
1 1
0 Ú p ds = m + m ◊ h + 3 h .
2
2
2
,
ik
ik
ik
,
ik
,
,
Furthermore,
1 m 1
b =  s m ( ◊ n ) p pds, p Œ G

2
i i k , , i k i k Ú 0 ik ,
,
5 k=1
where
2
2
pp = pm + ph s.
2
ik
,
,
ik
Therefore,
1 2 Ê 2 1 2 ˆ
0 Ú ppds = m ik , Ë m + m ◊ h + 3 h ik , ¯
,
,
ik
,
ik
ik
Ê 1 2 1 2 ˆ
+ h m + m ◊ h + h . (7.76)
2
ik , Ë 2 ik , 3 ik , ik , 4 ik , ¯
Once the three surface integrals given in Eqs. (7.72a to 7.72c) have been reduced to
line integrals and evaluated in plane P i, the results obtained in this 2-D subspace should
be mapped to produce the results necessary for calculating w i in the 3-D space, Eq. (7.71).
The scalar quantity a i poses no problem and is readily multiplied by the 3 ¥ 3 identity
matrix 1 in the expression for w i. Furthermore, matrix A i and vector b i in the 2-D sub-
space are transformed to their matrix and vector counterparts, respectively, in the 3-D
space, before being substituted into the expression for w i.
This completes the calculation of w i, in terms of which we can calculate the
matrix second moment of the piecewise-linear approximation of R. See Eqs. (7.68) and
(7.69).