Page 115 - Cam Design Handbook

THB4 8/15/03 1:01 PM Page 103

POLYNOMIAL AND FOURIER SERIES CAM CURVES 103

4.8 FOURIER SERIES CURVES—DRD CAM

In Chap. 2 we discussed the basic parabolic, sinusoidal, and cycloidal curves for cam-fol-
lower DRD action. We also showed in Chap. 3 how these curves and others can be mod-
ified and combined to form a variety of cam profiles.
Ignoring the torsional aspect of the cam drive (assuming the cam runs at a constant
speed), the two fundamental dynamic objectives in designing cam profiles are:

1. to have the smallest possible maximum acceleration and thus the minimum dynamic
load on the high-speed follower;
2. To avoid resonant vibratory response in the follower caused by high harmonic content
in the designed cam profile.
These fundamental objectives are not related. Let us compare the basic cam profiles as a
means of describing the harmonic content that is addressed in the later paragraphs.
The parabolic curve has the smallest maximum acceleration but has a high harmonic
content, since the Fourier expansion for a parabolic curve has an infinite number of terms.
Because each term is associated with a certain forcing frequency, it may induce resonance
vibrations in the follower at some of those frequencies at some speed as the system
is brought up to speed. The cycloidal cam, on the other hand, having but a fundamental
frequency in the acceleration curve, has a minimum harmonic content. The cycloidal
profile is the least likely to excite vibration in the follower, but this occurs at the expense
of high acceleration. Furthermore, it has a more than 50 percent higher maximum accel-
eration than does the parabolic profile. In design, it is necessary to find a cam profile
that has the fewest and lowest possible harmonics, though, in the end, nature calls for
a compromised balance of all factors, requiring judicial trade-offs. There is no perfect
solution.
A number of acceptable harmonic curves have been developed in consideration of these
thoughts. Note that for a DRD motion event, the Fourier series used must be an odd har-
monic curve, which has polar symmetry with respect to the midpoint of the curve.

Gutman 1-3 Harmonic (Gutman, 1961)

Èq 15 2 pq 1 6 pq ˘
y = h Í - sin - sin ˙
Î b 32 p b 96 p b ˚
h È 15 2 pq 1 6 pq ˘
y ¢ = Í 1 - cos - cos ˙
b Î 16 b 16 b ˚

hp È 2 pq 6 pq ˘
y ¢¢ = Í 15sin + 3sin ˙
8 b 3 Î b b ˚
hp 2 È 2 pq 6 pq ˘
y ¢¢¢ = Í 15cos + 9cos ˙ . (4.11)
4 b 3 Î b b ˚
Gutman’s 1-3 harmonic curve can be obtained from the Fourier series expansion of the
displacement of the parabolic curve by retaining the first two terms of the series. The
resulting curve has a harmonic content triple the frequency of the cycloidal curve and a
maximum acceleration of about 130 percent of the parabolic curve, or about 80 percent
of the cycloidal curve (Fig. 4.10).