Page 197 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 185

GEOMETRY OF PLANAR CAM PROFILES 185

I È I xy˘
I = Í xx ˙
C
I Î xy I yy˚
where
-
I ∫ ( y c ) 2 dx dy
xx Ú R 2
-
I ∫ ( x c ) 2 dx dy
yy Ú R 1
- )
I ∫- ( x c )( y c dx dy.
-
xy
Ú R 1 2
The quantities I xx and I yy are usually referred to as the centroid moments of inertia of
R about the x and y axes, respectively, whereas I xy is termed the product of inertia of R
about its centroid. Moreover, the quantity I zz, defined as
I = I + I yy
xx
zz
is known as the polar moment of inertia of R about its centroid.
The integral appearing in Eq. (7.23b) is called the first moment of R, whereas I C is
referred to as the second moment of R about the centriod C. By extension, the integral of
Eq. (7.23a) is also termed the zeroth moment of R.
Calculating the three moments of R directly as given above calls for a double inte-
gration, one in x and one in y. This is not practical, the foregoing integrals being more
readily calculated if suitable transformation formulas are introduced, as indicated in Sec.
7.4, thereby reducing those calculations to line integrals. Furthermore, if the cam profile
is described in polar coordinates as r = r(q), then the area is given by
1 2p
()
2
A = rq dq
2 Ú 0
which is a simple integral with similar expressions for q and I C.

7.3.2 Solid Regions

Although the global geometric properties of cam disks can be described as those of a planar
contour, the presence of nonsymmetric hubs, keyways, and the like render the cam plate
a 3-D solid of a general shape. By the latter we mean a solid that cannot be generated by
a simple extrusion or sweeping of the planar contour in a direction perpendicular to its
plane. In this case, the global properties are those of a 3-D region R bounded by a closed
surface S, namely, its volume V, its vector first-moment q, and its 3 ¥ 3 inertia matrix
about point O. These are defined
V ∫ dV
Ú R
q ∫ pdV
Ú R
I ∫ ( p 1 pp dV
)
2
-
T
O Ú R
T
where, now, 1 is the 3 ¥ 3 identity matrix, p ∫ [x, y, z] , and hence,
2
p = x 2 + y 2 + z 2
while