Page 75 - Cam Design Handbook

THB3 8/15/03 12:58 PM Page 63

MODIFIED CAM CURVES 63

Therefore, the general equations of the curve from B to C are

Ê b ˆ 1 Ê b ˆ 2
y = y + v q - 8 ¯ + 2 a q - 8 ¯
Ë
Ë
0
1
Ê b ˆ
y¢ = v + a q -
0 Ë 8 ¯
y¢¢ = a.
To get displacement, velocity, and acceleration to match at the junction B, it is necessary
that
2 h¢
v =
0
b
8p h¢
a =
b 2
Therefore, the equations from B to C are

Ê 1 1 ˆ 2 h¢ Ê b ˆ 4p h¢ Ê b ˆ 2
y = h¢ - + q - + q - (3.3)
Ë 4 2p ¯ b Ë 8 ¯ b 2 Ë 8 ¯
2 h¢ 8p h¢ Ê b ˆ
y¢ = + q -
b b 2 Ë 8 ¯
8p h¢
y¢¢ = .
b 2
3
When point C is reached, q = b. Substituting in Eq. (3.3), we obtain
8
3 h¢ p h¢
y = h¢ - +
4 2p 4
h¢ p h¢
y y =
\ y = - 1 -
2
2 4
The cycloidal displacement is the sum of a constant velocity displacement and a quarter
sine wave displacement. The displacement equation of the curve from C to D is
3 Ê b
q - b q - ˆ
y = y + y + C + C 2 8 + C sin Á 4p 2 ˜ .
3
1
2
1
b Á Á b ˜ ˜
Ë ¯
Hence,
b
q -
C 4p
y ¢ = 2 + C cos 4p 2
b 3 b b
b
q -
16p 2 2
y ¢¢ =-C 3 sin 4p (3.4)
b 2 b