Page 198 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 186

186 CAM DESIGN HANDBOOK

x È ˘ x È 2 xy xz ˘
Í ˙ Í ˙
T
pp ∫ y x [ y z] = xy y 2 yz .
Í ˙ Í ˙
z Î Í ˚ ˙ Î Í xz yz z ˚ ˙
2
Again, from the mean-value theorem of integral calculus, a vector c exists such that
q = V c
and vector c, once more, defines the location of the centroid C of R. Correspondingly,
the three components of c in a frame F (O, x, y, z) are given by c i, for i = 1, 2, 3, that is,
T
c = [c 1, c 2, c 3] .
The moment of inertia of R about C is then defined as
I ∫ [ pc 1 -( pc pc) ] dV
T
2
- )(
-
-
C Ú R
which then reduces to
I È xx I xy I xz ˘
Í ˙
I = I I I
C Í xy yy yz ˙
Í I Î I I ˙
xz yz zz ˚
where
I ∫ [ ( y c ) + ( z c ) ] dx dydz
2
2
-
-
xx Ú R 2 3
I ∫ [ ( x c ) + ( z c ) ] dx dydz
2
2
-
-
yy Ú R 1 3
I ∫ [ ( x c ) + ( y c ) ] dx dydz
2
2
-
-
zz Ú R 1 2
- )
-
I ∫- ( x c )( y c dx dydz
xy
Ú R 1 2
- )(
- )
I ∫- ( x c z c dx dydz
xz 1 3
Ú R
- )
-
I ∫- ( y c )( z c dx dydz.
yz
Ú R 2 3
As in the case of planar contours, the foregoing calculations involve multiple integrals,
in this instance, three. Reduced formulas transforming these calculations into surface and
even line integrals are provided later.
7.4 COMPUTATIONAL SCHEMES
For static and dynamic balancing, and also for dynamic analysis (Koloc and Vaclacik,
1993), the determination of the cam global geometric properties is of the utmost impor-
tance. For example, very often the cam axis of rotation and the principal axis of inertia
perpendicular to the cam disk are skew. Therefore, to balance the cam statically and
dynamically, their global volumetric properties must be calculated. The calculation of these
and the local properties requires the introduction of suitable numerical techniques that will
be discussed later.