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194 CAM DESIGN HANDBOOK

The calculation of the presure angle and the curvature of the profile at an arbitrary
n
value of c outside of the set {y i } i calls for values of s¢(y) and s≤(y) or, correspondingly,
f¢(y) and f≤(y) outside of the values arrayed in vectors y¢ and y≤. These derivative values
can be readily calculated by means of Eq. (7.37), namely, as
2
y ¢() = Ay 3 (y -y ) + B2 (y -y )+ C , y £y £y (7.40)
i i i i i i + i 1
y ¢¢() = Ay 6 i (y -y i )+ B2 , y i £y £y . (7.41)
i
+ i 1
With the foregoing interpolated derivative values, the calculation of the pressure angle
and the curvature at arbitrary values of y is straightforward.

7.4.3 Global Properties
The computation of the global geometric properties—area for 2-D contours and volume
for 3-D surfaces, centroid location, and inertia matrix—of cam plates is the subject of this
subsection. Rather than using directly the definitions of these concepts, as given in Sec.
7.3, we resort to the Gauss divergence theorem (GDT) (Marsden and Tromba, 1988) to
transform volume integrals into surface integrals. One second application of the GDT
readily leads to algorithms based on line integrals, which are computationally more eco-
nomical than the direct formulas.

7.4.3.1 Two-Dimensional Regions. In this subsection the general formulas introduced
in Angeles et al. (1990) and recalled below are applied to 2-D regions. Thus, line-inte-
gration formulas will be derived for the computation of the area, the centroid position
vector, and the inertia matrix of planar regions R bounded by a closed contour G.
O
O
Let A, q , and I denote the area, the vector first moment, and the matrix second
O
O
moment of R, q , and I being defined with respect to a given point O in the plane of the
region. The computation of these quantities can be reduced to integration on the bound-
ary G, by application of the GDT, namely,
1
A = ◊ rn dG (7.42a)
2 Ú G
1 1
(
◊ )
◊
O
q = ( r rndG = rr n d ) G (7.42b)
2 Ú G 3 Ú G
È 3 1 T ˘
(
◊
◊
I = r r 1 r n) - rn dG (7.42c)
O
Ú G Í Î8 2 ˙ ˚
where 1 denotes the 2 ¥ 2 identity matrix and n is the unit outward normal of G, the prod-
T
ucts r·n and rn being the inner product and the outer product of vectors r and n, already
introduced in Sec. 7.3. Their products are reproduced below for quick reference, with r i
and n i, for i = 1, 2, denoting the components of vectors r and n, correspondingly, in the
given reference frame, i.e.,
◊
rn = rn + r n
11 2 2
and
r È ˘ rn rn ˘
T
rn = 1 [ nn ] = È 11 1 2 .
Í ˙ 1 2 Í rn rn ˚ ˙
r Î ˚
2 Î 21 2 2