Page 50 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 38

38 CAM DESIGN HANDBOOK

Since the velocities are equal at the point q = b 1 differentiating Eqs. (2.25) and (2.26)
yields
Ê b ˆ
C = C Á - 1˜ (2.30)
1 2
Ë b ¯
1
Solving Eq. (2.29) and (2.30) gives
h h
C = , C = (2.31)
1 2
bb (b b
- )b
1 1
Therefore for 0 < q £ b,
h
displacement y = q 2 (2.32)
1
bb
1
2 h
velocity y ¢ = q (2.33)
1
bb
1
2 h
acceleration y 1 ¢¢= (2.34)
bb
1
and for b 1 £ q £ b,
h 2
- )
h
displacement y =- (bq (2.35)
2
- )b
(bb 1
2 h
- )
velocity y ¢ = (bq (2.36)
2
- )b
(bb 1
2 h
acceleration y ¢¢= (2.37)
2
(bb
- )b
1
Note that the jerk values equal zero except where that value is infinite at the points of
acceleration discontinuity.
2.8 CUBIC NO. 1 CURVE
This curve (follower motion) of the polynomial family has a triangular acceleration curve.
It is a modification of the parabolic curve, eliminating the abrupt change in acceleration
at the beginning and the end of the stroke. However it does have an acceleration discon-
tinuity at the midpoint. This curve has limited applications since it also has high veloci-
ties and high acceleration values, giving excessive pressure angle and inertia forces. It is
not very practical except when combined with other curves.
The construction of this curve (Fig. 2.6) is similar to that of the acceleration curve pre-
viously with the increments being 1, 7, 19, and so forth, which is a cubic relationship in
lieu of 1, 3, 5 as with the constant acceleration curve. Thus it will not be shown here.
By methods shown for the parabolic curve, we can solve the cubic curve for
b
0 ££
q
2