Page 254 - Cam Design Handbook

THB8 9/19/03 7:25 PM Page 242

242 CAM DESIGN HANDBOOK

Combining Eqs. (8.23) and (8.24) yields
y
Ê ˆ 2 y d
sinq -= + - cos . (8.25)
1
q
Ë ¯ a a
r
Squaring both sides of Eq. (8.25) and rearranging, we get
2
Ê ˆ r 2 Ê ˆ 2 Ê ˆ Ê d ˆ 2 2r 2 Ê d ˆ
r
r
2
sin q - sin q = + - cosq + - cosq . (8.26)
2
Ë ¯ y Ë ¯ Ë ¯ Ë a ¯ ay Ë a ¯
y
a
Substituting for (1/y) from Eq. (8.21) into the above equation gives a quadratic equation
in n
2
Ê ˆ Ï Ê d - cosq ˆ 2 + sin q ¸ n - ) +1 2 r 2 2 Ê d - cosq ˆ ( n - ) +1 Ê ˆ 2 - sin q = 0.
r
r
˝ (
2
2
Ë d ¯ Ì Ó Ë a ¯ ˛ ad Ë a ¯ Ë a ¯
(8.27)
Several observations can be made about the above equation.
1. Since Eq. (8.27) remains the same if we replace q with (2p - q), the two solutions
of n for a given value of q correspond to the angular velocity ratios of the cam and the
roller crank during the initial and reverse traversal of the roller on the cam. That is,
Ê df ˆ
n
n ==
solution1 Ë dq ¯ at 0
Ê df* ˆ
n solution2 = * ˜ . (8.28)
n = Á
Ë dq* ¯
2
at 0* =( p 0
- )
2. When q = 0 or p, Eq. (8.27) becomes independent of r
2 2
d d
d
ˆ
Ê d m 1 ( n 1) + 2 Ê m 1 ( n 1)+ Ê ˆ = 0.
ˆ
2
-
-
Ë a ¯ a Ë a ¯ Ë ¯ (8.29)
a
This can be seen in Fig. 8.20 which shows the solution curves for n for different (r/d)
ratios. It can also be observed in this figure that the region bounded by the two solution
curves for one value of the (r/d) ratio contains similar curves of a larger ratio. As men-
tioned earlier a larger value of (r/d) implies better transmission. Therefore, the region
between two solution curves can be used to specify the angular velocity ratio n so that the
transmission index (r/d) is larger than a minimum specified value. This is a useful feature
in designing the mechanism.
3. Figure 8.21 shows the solutions curves for n for a fixed ratio of (r/d) and varying
ratios of (d/a). It can be seen that n = 0 occurs at the same two points. This is also evident
from Eq. (8.27) if we substitute n = 0 and solve for q
q
0
n =fi= sin -1 ( r d) or p - sin -1 ( r d) (8.30)
which is clearly independent of the ratio (d/a). This feature of the design (see Eq. [8.27])
has an important significance if a long dwell or reversal of motion of the cam is
desired. That is, if we specify a lower bound on the transmission index (r/d), then the
permissible region for n has zero or a negative value over only a limited range of q. For
example, the maximum possible single dwell in the initial traversal of the roller
-1
on the cam is (p - 2sin (r/d)). It should be noted that this is a minimum value of r here,
not r generically. Thus, the ratio is useful not only to achieve a prescribed minimum