Page 207 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 195

GEOMETRY OF PLANAR CAM PROFILES 195

The foregoing formulas, Eqs. (7.42a–c), are applied below to planar regions with
piecewise-linear and cubic-spline approximations of their boundaries, respectively. Note
O
that q can be computed with two alternative formulas, which are given in Eq. (7.42b).
We recall here that the second of these formulas is more suitable for applications involv-
ing piecewise linear approximations, given the simple forms that the r·n term produces
in such cases. Both formulas will prove to be useful in deriving practical simple formu-
las, as shown below.
7.4.3.2 Piecewise-Linear Approximation of the Boundary. If G in Eqs. (7.42a to
7.42c) is approximated by a closed n-sided polygon, then
n
G ª U G . (7.43)
i
1
where G i denotes the ith side of the polygon. The aforementioned formulas, thus, can be
approximated as
1 n
A ª Â Ú ◊ rn dG (7.44a)
2 1 i G i i
1 n 1 n
q ª Â Ú ( r r n dG = Â Ú r r n d ) G . (7.44b)
(
◊ )
O
◊
2 1 i G i i 3 1 i G i i
n È 3 1
I ª Â Ú r r 1 r n ) - rn T ˘ dG . i (7.44c)
(
O
◊
◊
i
i
1 i G Í Î8 2 ˙ ˚
where n i denotes the outward normal unit vector of G i , and hence is a constant along this
side of the approximating polygon.
O
O
Furthermore, let A i, q i , and I i be the contributions of G i, the ith side of the polygon,
to the corresponding integral, s i and r¯ i denoting its length and the position vector of its
centroid, as shown in Fig. 7.10. From this figure, the reduced calculations are readily
derived, namely,
1 1
A = n ◊ d r G i = n i ◊ i i s r . (7.45)
i
2 i Ú G i 2
Using each of the two formulas of Eq. (7.44b), one obtains two alternative expressions
O
for q i, namely,
1
◊
O r rdG i) ◊ n (7.46a)
i i
2 i
q = (Ú G
1
T
O rr dG i) ◊ n . (7.46b)
i i
3 i
q = (Ú G
By subtracting two times both sides of Eq. (7.46a) from three times both sides of Eq.
(7.46b), one obtains
◊ )
-
q =- n ◊ [ ( r r 1 rr d ] G . (7.47)
T
O
i i Ú G i
i
The right-hand side of Eq. (7.47) is readily recognized—see Eq. (7.23c)—as the pro-
jection onto -n i of the matrix second moment of segment G i , with respect to O, repre-
O
sented here as J i . Thus,