Page 57 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 45

BASIC CURVES 45

h
From Fig. 2.10, the radius is in which
2p
h h
Displacement y = f - sin f (2.56)
2p 2p
where f = angle of circle rotation, rad.
We see
q f
=
b 2 p
Substituting in Eq. (2.56) yields
Ê q 1 2 pq ˆ
Displacement y = hÁ - sin ˜ (2.57)
Ë b 2 p b ¯
hq
In Fig. 2.10, we see that the first term of the equation, , is the sloping line and the
b
h 2pq
second term, sin , is the subtracted harmonic displacement. Thus
2p b
h Ê 2 pq ˆ
Velocity y ¢ = Á1 - cos ˜ (2.58)
b Ë b ¯
2 hp 2pq
Acceleration y ¢¢ = sin (2.59)
b 2 b
4 hp 2 2pq
Jerk y ¢¢¢ = cos . (2.60)
b 3 b
The above equations are plotted in Fig. 2.11. Note that this curve has no sudden change
in acceleration for the complete cycle and thus has high-speed applications. To simplify

Displacement, y
Dwell
Velocity, y'

y, y', y", h

b
Cam angle q
Dwell
Acceleration, y"
FIGURE 2.11. Cycloidal motion curve—DRD cam.