Page 196 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 184

184 CAM DESIGN HANDBOOK

2
2
r + 2 r¢ - rr¢¢
l¢
k = sgn ()
2
( r + r¢ ) 32
2
where the argument q has been dropped for compactness.
7.3 GLOBAL PROPERTIES OF THE CAM PROFILE
The global properties of a planar contour are its area, the position vector of its centroid,
and its moments and products of inertia about a certain point, e.g., its centroid. Likewise,
the global geometric properties of a closed region of the 3-D space are its volume, the
position vector of its centroid, and its 3 ¥ 3 inertia matrix about a certain point such as its
centroid. Below we recall the formal definitions of these items.

7.3.1 Planar Contours
Let R denote the region bounded by a closed contour C in the x-y plane, and p the
position vector of a point P of R in a frame F (O, x, y, z). Moreover, its area is denoted
by A, its first moment by vector q, and its inertia matrix about the given origin O by I O.
Thus,

A = dA (7.23a)
Ú R
q = pdA
Ú R (7.23b)
I = ( p 1 pp )dA
2
-
T
0 Ú R (7.23c)
2
where dA = dx dy, 1 is the 2 ¥ 2 identity matrix, ||p|| denotes the square of the magnitude
of p, that is,
2
p = x 2 + y 2
T
and pp is the 2 ¥ 2 matrix external product recalled below:
x È ˘ x È 2 xy ˘
T
pp = Í ˙ x [ y] = Í ˙ .
2
y Î ˚ Î xy y ˚
Moreover, from the mean-value theorem of integral calculus [1], a vector c exists such
that
q = Ac.
Vector c thus defines the position of the centroid C of R. Henceforth, the two com-
T
ponents of c in F are denoted by c 1 and c 2, i.e., c = [c 1, c 2] .
The moment of inertia of R about C is then defined as
I ∫ [ pc 1 -( pc pc) ] dA.
T
2
-
- )(
-
C Ú
R
Moreover, I C reduces to