Page 48 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 36

36 CAM DESIGN HANDBOOK

The boundary conditions are

q = b y = h
b 2 h
q = y ¢ =
2 b
q = b y ¢ = 0
yield constants
4 h - 2 h
C =- h C =, , C =
0 1 2
b b
Substituting in Eq. (2.20) yields

2 h
- )
displacement y =- (bq 2 (2.21)
h
b 2
4 h Ê q ˆ
velocity y ¢ = Á1 - ˜ (2.22)
b Ë b ¯
4h
acceleration =- (2.23)
b 2
jerk y≤¢ = 0 except at the changes in the acceleration where it equals infinity. Since the
parabolic curve has discontinuities in the acceleration at the dwell ends and the transition
point, it is primarily used for low-speed systems if at all.
Note that the data of this motion could have been accomplished by utilizing from
physics (for constant acceleration action) such that the displacement
1
y = q + aq 2
v
0
2
(2.24)
1
= Vt + At 2
0
2
where
v 0 = initial velocity dimensionless
V 0 = initial velocity
q = cam angle, radians
t = time, sec
a = acceleration, in/rad 2
A = acceleration, in/sec 2
Furthermore, Eq. (2.24) will be used in Chap. 3 where the constant acceleration
curve is blended with other curves to optimize the dynamics of the designed acceleration
curve.
The construction for the constant acceleration curve is shown in Fig. 2.4. Draw line
AC, which is made in odd number increments in this case three (i.e., 1, 3, 5), and draw
line CD to the midpoint of the rise and draw parallel lines, which will be connected to the
three cam increments. This establishes the cam profile since the steps are 1, 4, 9, and so
forth as shown.