Page 210 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 198

198 CAM DESIGN HANDBOOK

1 p i+1
I =- Ú ( x + y )(3 xy¢ + ¢ )
2
x y dp
2
yi
8 p i
3 p i+1 1 p i+1
) ¢
=- Ú ( x + y xy dp - Ú ( x + y x ydp. (7.54b)
) ¢
2
2
2
2
8 p i 8 p i
In the calculation of the product of inertia appearing in Eq. (7.53c), one summation
O
would suffice to compute the product of inertia I xy, given the symmetry of the inertia
matrix. The two summations are computed here and then averaged to diminish roundoff
or measurement errors that could be introduced if, for instance, the coordinates of the cam
profile—the supporting points—were digitized or picked up with a coordinate measuring
machine. Thus, the terms of both summations are given by
1 p i+1
I xyi = Ú ( x + y xx dp (7.55)
2
) ¢
2
2 p i
1 p i+1
I yxi =- Ú ( x + y yy dp. (7.56)
2
2
) ¢
2 p i
From Eqs. (7.54a to 7.56) it is noted that all components of the inertia matrix are given
as linear combinations of four integrals, namely,
12
p i 1+ k
1 ()
1 ()
2
) ¢
2
I ∫ Ú ( x + y xx dp = Â h (D p ) (7.57a)
i ik i
p i
1
12
p i 1+ k
2 ()
I i 2 () ∫ Ú ( x + y yy dp = Â h (D p ) (7.57b)
2
2
) ¢
ik
i
p i
1
12
p i 1+
3 ()
2
) ¢
2
I i 3 () ∫ Ú ( x + y x ydp = Â h (D p ) k (7.57c)
ik
i
p i
1
12
k
p i 1+
4 ()
I i 4 () ∫ Ú ( x + y xy dp = Â h (D p ) . (7.57d)
2
) ¢
2
ik
i
p i
1
(4)
(1)
The polynomial coefficients h ik ,..., h ik , appearing in Eqs. (7.57a to 7.57d), are also
included in App. C: Polynomial Coefficients.
7.4.3.3 Three-Dimensional Regions. For solid regions, the general relations of
Angeles et al. (1990) reduce to
1
V = Ú ◊ rn dS (7.58)
3
3 S
1 1
(
◊ )
q = ( r r nd = r r n d ) S (7.59)
◊
O
S
3
2 Ú S 4 Ú S
3
T ˘
(
◊
◊
I = S Ú r r È Í Î 10 1 r n) - 1 2 rn dS. (7.60)
O
˙
3
˚
Next, explicit formulas are discussed that are applicable to piecewise-linear approxi-
mations of boundaries of solids of arbitrary shapes.