Page 211 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 199

GEOMETRY OF PLANAR CAM PROFILES 199

One simple approximation of the boundary can be obtained by means of a polyhedron
formed by polygonal faces. The integrals appearing in Eqs. (7.58 to 7.60), therefore, can
be expressed as sums of integrals over the polyhedral faces, the whole boundary S thus
being approximated as

n
S ª U S (7.61)
i
1
where each part S i is a polygonal portion of a plane. The integral formulas thus can be
approximated as
n 1
V ª Â Ú ◊ rn dS (7.62)
3 i i
1 3 i S
1 n 1 n
q ª Â Ú ( r r n dS = Â Ú r r n d ) S (7.63)
(
◊ )
O
◊
3
2 1 i S i i 4 1 i S i i
n 3 1
T ˘
I ª Â Ú r r È 1 r n ) - rn dS . (7.64)
(
◊
◊
O
3 Í i i ˙ i
S i Î 10 2 ˚
1
O
Now, let V 3i, q 3i, and I 3i be the contribution of the ith face S i of the polyhedron to the
O
corresponding integral, D i, r¯ i, and I D i being the area, the position vector of the centroid,
and the inertia matrix of the polygon S i, respectively. (The last two quantities are taken
with respect to O.) Then,
1 1
V = n ◊ d r S = n ◊ rD . (7.65)
i 3 i Ú S i i i i
3 i 3
We calculate the polygon area D i and its centroid r¯ i in a plane defined by the polygon,
using the method applied to 2-D regions, as outlined earlier in this subsection.
By subtracting twice the rightmost-side of Eq. (7.63) from the middle one, we readily
O
O
obtain an expression for q 3 as a summation of terms q 3i having the form
1
◊ )
O
q =- Ú ( [ r r n ) i -( r n r d ] S i
◊
3i
i
2 S i
1
i Ú
=- n ◊ [ ( r r 1 rr d ] S . (7.66)
T
◊ ) -
i
2 S i
O —the second moment of
The second integral of Eq. (7.66) is readily identified as I D i
polygon S i. Hence,
1
q =- n I O (7.67)
◊
O
3i i i D
2
that is, the contribution of S i to the first moment of R is recognized as one-half the pro-
jection onto -n i of the second moment of S i, both moments being taken, of course, with
respect to the same point O. We calculate the centroidal inertia matrix of the polygon at
hand using the formulas derived for 2-D regions in the plane defined by the polygon. Using
O from the calculated cen-
the parallel-axis theorem and a rotation of axes, we then find I D i
troidal inertia matrix.
Additionally, the contribution of the ith polygonal face to the second moment of the
3-D region under study takes the form