Page 217 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 205

GEOMETRY OF PLANAR CAM PROFILES 205

s y s y s y s y s y s y (7.78)
¢¢¢() = ¢¢¢().
¢() = ¢(),
¢¢() = ¢¢(),
On the other hand, the tangent of the pressure angle was derived in Angeles and Lopez-
Cajun (1991) as
s ¢()
y - e
tanay () = . (7.79)
s y ()
Now, substituting Eqs. (7.77) and (7.78) into Eq. (7.79) yields
¢()
sy - e
tanay () = . (7.80)
sy ()+ c
If, moreover, y 1 and y 2 are the values at which the pressure angle attains a maximum
a M and a minimum a m , the corresponding extremality conditions take the form
¢¢() - ¢()tan
sy s y a M = 0 (7.81a)
1
1
sy sy a m = 0 (7.81b)
¢¢() - ¢()tan
2
2
with s i = s(y i), for i = 1, 2, and s¢ i(y i) and s≤ i (y i) defined likewise. The problem thus
reduces to solving two independent nonlinear equations, (7.81a) and (7.81b), for the two
unknowns y 1 and y 2. Furthermore, we have, from Eq. (7.80),
ctana += ¢ - s 1 tana m (7.82)
e s
M
1
ctana += ¢ - s tana . (7.83)
e s
M 2 2 m
Equations (7.82) and (7.83) thus suggest a graphical solution of the optimization
problem at hand, for each of these equations represents a line in the c-e plane.
The optimum values of c and e are found as the coordinates of the intersection of these
two lines, as illustrated in Fig. 7.14, where we have assumed that a m =-a M and a M = 30°.
Moreover, the optimum values c opt and e opt can be obtained by solving for these parame-
ters from Eqs. (7.82) and (7.83), namely,
s
s ¢ -s tan a - ¢ +s tan a
c = 1 1 M 2 2 m (7.84a)
opt
tan a - tan a
M m
(
-tanas ¢ -s tana )+ tana (s ¢ -s tana )
e = m 1 1 M M 2 2 m . (7.84b)
opt
tana - tana
M m
Finally, from the geometry of the translating cam mechanism, the optimum radius of
the base circle of the optimum cam is
2
b = e + c 2 (7.85)
opt opt opt
which can be readily computed using Eqs. (7.84a and b).
Now we apply the foregoing relations to the size-minimization of the cam plate of a
quick-return mechanism. This mechanism is a mechanical transmission that produces a
slow feed motion under a load in one direction, followed by a fast return stroke under no
load in the opposite direction. Quick-return mechanisms are frequently used in manufac-
turing processes, e.g., in pick-and-place operations, metal-cutting, and metal-forming.
Cam mechanisms are well suited for this type of task, because they can readily produce
such a type of motion. The required motion for this case is the dwell-rise-dwell-return dis-