Page 88 - Cam Design Handbook

THB3 8/15/03 12:58 PM Page 76

76 CAM DESIGN HANDBOOK

1 q 7
for £ £
8 b 8
È q Ê 4pq p ˆ˘
.
y = h 028005 043990 - 0351506cos Á - ˜
+ .
.
Í Ë 3 b ¯ ˙
Î b 3 ˚
h È Ê 4pq p ˆ˘
y¢ = 043990 +1 31970sin Á + ˜
.
.
b Í Î Ë 3 b 3 ¯ ˙ ˚
h Ê 4pq p ˆ
.
y¢¢ = 5 52796 sin Á + ˜
b 2 Ë 3 b 3 ¯
h Ê Ê 4pq p ˆ
y¢¢¢ = 23 1555 cosÁ + ˜.
.
b 3 Ë 3 b 4 ¯
7 q
for £ £ 1
8 b
È q q ˘
y = h 0 56010 0 43990 - 0 03515006sin 4p ˙
+ .
.
.
Í
Î b b ˚
h È Ê q ˆ˘
.
y¢ = Í 043989 Á1 - cos 4p ˜ ˙
b Î Ë b ¯ ˚
(3.18)
h q
y¢¢ = 5 52796 sin 4p
.
b 2 b
h q
.
y¢¢¢ = 69 4664 cos 4p .
b 3 b
A computer solution is employed to establish the incremental displacement values and the
characteristic curves of the action. The maximum velocity of the modified sine curve is
h h
¢ y = .1 760 , the maximum acceleration is ¢¢ =y . 5 528 , and the maximum jerk is
max max 2
b b
h
y ¢¢¢ = 69 .47 . The nondimensionalized displacement, velocity, and acceleration factors
max
b 3
are given in Table A-4, App. B. Figure 3.10 indicates the comparison (Erdman and Sandor,
1997) of the cycloidal, modified trapezoidal, and modified sine curves. The data shown is
for a 3-inch pitch diameter cam having a 2-inch rise in 6 degrees of cam rotation.
3.8 MODIFIED CYCLOIDAL CURVE
In this section we will reshape the cycloidal curve to improve its acceleration character-
istics. This curve is the modified cycloidal curve that was developed by Wildt (1953).
Figure 3.11 indicates the acceleration comparison between the true cycloidal curve and
the Wildt cycloidal curve. The basic cycloidal curve equation for the displacement
Ê q 1 2 pq ˆ
y = hÁ - sin ˜ (2.58)
Ë b 2 p b ¯
From this equation it is seen that the cycloidal curve is a combination of a sine curve
and a constant velocity line. Figure 3.12a shows the pure cycloidal curve with point A