Page 190 - Cam Design Handbook

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178 CAM DESIGN HANDBOOK

cutting. This information is also needed in planning the trajectory of the cutter of an NC
machine tool.

7.2.1 Tangent at a Point of a Cam Profile
To systematically obtain a relationship for determining the angle between the position
vector of a point on the cam profile and the tangent at that point, we refer to Fig. 7.1. In
that figure, l measures the arc length from a reference point O l , and e t is a unit vector
parallel to line T, the tangent at point Q of the curve G. Note that e t points in the direc-
tion in which l grows.
It would be desirable to obtain an expression for vector e t as a parametric repre-
sentation of the position vector p of an arbitrary point of the contour, of components x
T
and y, that is, p = [x, y] . However, x and y are usually available as functions of a para-
meter q different from l. Hence, derivatives with respect to l will be obtained using the
chain rule. First, from elementary differential geometry, Marsden and Tromba (1988), we
have
¢()
d p d p dq p q
e = = = (7.1)
t
q
dl dq dl l ¢()
where
¢()
È xq () ˘ È xq ˘
¢()
p q () = and p q = . (7.2)
Í ˙ Í ˙
¢()
Î yq () ˚ Î yq ˚
Next, since e t is a unit vector, we have
l ¢() =± ¢() =±q p q x ¢ ()+ ¢ () q (7.3)
2
2
y
q
where, obviously, the negative sign is used if l¢(q) < 0. Thus,
¢()
¢()
q > 0
Ï p q p q , if l ¢()
e = Ì . (7.4)
t
¢()
p q
q < 0
Ó - ¢() p q , if l ¢()
y
e
t
G g
Q
P
e q l
e r T
q
O l
O x
FIGURE 7.1. Geometric variables of a planar curve.