Page 60 - Cam Design Handbook

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

48 CAM DESIGN HANDBOOK

where a e and b e are the semi-major and the semi-minor axes of the ellipse, respectively.
a
If we let n = e and (2.66)
e
b
e
a - b 2 n - 1
2
2
a = e e = e (2.67)
a 2 n 2
e e
It can be shown that
Ê pq ˆ
h Á cos b ˜
Displacement y = Á1 - ˜ (2.68)
2 Á pq ˜
Á 1 -a sin 2 ˜
Ë b ¯
and pq
sin
p h b
Velocity y ¢ = 32 (2.69)
2 b Ê pq ˆ
n 2 Á1 -a sin 2 ˜
e
Ë b ¯
pq
12 2
+ a sin
p 2 h pq b
Acceleration y ¢¢ = cos (2.70)
2 b 2 b Ê pq ˆ 52
n 2 Á1 -a sin 2 ˜
Ë b ¯
e
Figure 2.14 shows the velocity and acceleration curves for n = 1.0 and n = 1.3. The kine-
matic characteristics of the elliptical curve depend on the assumed proportions of the major
and the minor axes. Proper proportions in the ratio between major and minor axes produce
cams of acceptable performance at moderate speeds similar to those of the simple har-
monic motion curve, all having acceleration discontinuity at the beginning and end of the
DRD action.
2.14 COMPARISON OF BASIC CURVES

The basic curves discussed can be applied for low to moderate speeds as a first design
selection. For optimum cam follower performance, combinations of these basic curves and
other mathematical curves for moderate to high speeds are presented in Chap. 3.
Comparing the curves most often used, the trigonometric ones (simple harmonic
motion, cycloidal, and double harmonic curves) give better overall performance than those
in the basic polynomial family (straight-line, parabolic, and cubic curves). The advantages
are smaller cams and reduced translating follower ride thrust. In this chapter, the cycloidal
curve, for most machine requirements, is the first choice. Since it has no abrupt change
in acceleration, it gives the lowest vibrations, wear, stress, noise, and shock. It is easy to
start, requires small springs, and induces low follower side thrust. However, the necessary
accuracy of fabrication is higher than for low-speed curves. It has a slightly larger
maximum acceleration than some others. Figure 2.15 shows the comparison of the basic
curve characteristics of displacement, velocity, and acceleration. The data is that the