Page 52 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 40

40 CAM DESIGN HANDBOOK

2.9 CUBIC NO. 2 CURVE

This curve is similar to the constant acceleration and the cubic no. 1 curves. It differs from
these, however, in that there is no discontinuity in acceleration at the transition point and
also in that its acceleration is a continuous curve for the complete rise. Similar to the con-
stant acceleration curve, it has the disadvantages of discontinuity in acceleration at the
beginning and the end of the stroke. The cubic curve has characteristics similar to those
of the simple harmonic motion curve presented next. It is not often employed but has
advantages when used in combination with other curves. No simple construction method
is available.
The characteristic formulas can be found by the same method as that shown for the
parabolic curve.
q 2 Ê q 2 ˆ
Displacement y = h Á3 - ˜ (2.46)
b 2 Ë b ¯
6 hq Ê q ˆ
Velocity y ¢ = Á1 - ˜ (2.47)
b 2 Ë b ¯

6 h Ê 2q ˆ
Acceleration y ¢¢ = Á1 - ˜ (2.48)
b 2 Ë b ¯
12 h
Jerk y ¢¢¢ = (2.49)
b 3
= constant
The procedure used to plot Fig. 2.7 is similar to that for the previous curves.

2.10 SIMPLE HARMONIC MOTION CURVE

This curve (having a cosine acceleration curve) is a very popular choice in combination
with other curves in Chap. 3. In Fig. 2.8, the projection of a radius point P starting at
point O moves vertically at point Q along the diameter h of the circle with simple
harmonic motion.
Let
h
f = angle of rotations with radius
2
The basic harmonic motion displacement function is
h
y = (1 cos f ) (2.50)
-
2
h
The construction of Fig. 2.8 uses a circle of radius . Displacements are taken at angular
2
increments moving through angle f the same increments along the displacement curve.
The relationship between angle f, the generating circle, and the cam angle q is
f p
= (2.51)
q b