Page 201 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 189

GEOMETRY OF PLANAR CAM PROFILES 189

a = D , a = a + a , b =1 a , b = b + b
p
k k ij , i j k k ij , i j
for i, j, k = 1,..., n¢, and
n ¢ =-n 1, n ¢¢ =-n 2, n ¢¢¢ =-n 3.

On the other hand, the relations between x and x¢, and between y and y¢ can be
written as
-
-
x ¢ = (F GC )x, y ¢ = (F GC )y (7.30)
where the n¢¥ n¢ matrices F and G are given by:
- È a a 0 0 L 0 ˘
1 1
Í 0 -a a 0 L 0 ˙
Í 2 2 ˙
Í 0 0 -a 3 a 3 L 0 ˙
F = Í ˙
Í M M O O O M ˙
Í a ¢¢ ˙
Í n ˙
Î a ¢ n -a ¢ n ˚
È 2b 1 b 1 0 0 L 0 ˘
Í 0 2b b 0 L 0 ˙
Í 2 2 ˙
Í 0 0 2b 3 b 3 L 0 ˙
G = Í ˙ .
Í M M O O O M ˙
Í b ¢¢ ˙
Í n ˙
Î b ¢ n 2b ¢ n ˚

For the synthesis of the cam profile, the SPs, and hence, the spline coefficients are
available. In fact, with the SPs known, the coefficients A ui, B ui, C ui, D ui, for u = x, y, and
i = 1,..., n - 1 can be calculated, the coordinates of the interpolated points then follow-
ing from Eqs. (7.26a and b). These points are meant for the actual display of the cam
profile and the generation of the cutter path of the machine tool, if the follower is of the
flat-face type. If the follower is of the roller type, the interpolated points represent the
pitch curve, the cam profile then being obtained as the envelope (Angeles and Lopez-
Cajun, 1991) of the set of positions of the roller. This procedure is illustrated in Sec. 7.6.

7.4.2 Local Properties
In the design phase one has information only about the displacement program, i.e., one
knows s(y) in the case of translational followers, or f (y) in the case of their oscillating
counterparts, but the dimensions of the cam are, as yet, to be determined. Hence, the local
geometric properties of the cam profile are to be computed based on s(y) or, correspond-
ingly, on f(y) and its derivatives with respect to y. Thus, in this subsection we introduce
formulas for calculating the curvature and the pressure angle of the cam profile in terms
of the displacement program and its derivatives.

7.4.2.1 Translating Flat-Face Followers. Apparently, the pressure angle for translat-
ing flat-face followers is constantly equal to zero and needs no further discussion.