Page 117 - Cam Design Handbook

THB4 8/15/03 1:01 PM Page 105

POLYNOMIAL AND FOURIER SERIES CAM CURVES 105

2p h Ê 2pq 1 6pq 1 10pq ˆ
y ¢¢ = m Ásin + sin + sin ˜
b 2 Ë b 6 b 50 b ¯

4p 2 h Ê 2pq 1 6pq 1 10pq ˆ
y ¢¢¢ = m Ácos + cos + cos ˜
b 3 Ë b 2 b 10 b ¯
This curve has a maximum acceleration of about 125 percent of the acceleration of the
parabolic curve, or about 81 percent of the acceleration of the cycloidal curve.
In this way, other higher-order mulitple harmonic curves can be generated. Baranyi
(1970) has derived and tabulated the Fourier coefficients up to and including the seven-
teenth harmonic of the profile groups for the DRD cam. Unfortunately, these high-order
harmonic curves do not generally produce a satisfactory dynamic response of the follower.
Nevertheless, there is an advantage in using a harmonic series for cam motion design. This
advantage is the direct knowledge of the harmonic content of the forcing function applied
to the cam-and-follower system. With this knowledge, the designer can create a system
that will avoid the resonance at certain critical harmonics.
In the foregoing discussion, we have studied the dwell-to-dwell curves including the
transition between endpoints designed to have finite terminal velocities. Weber (1979) has
presented an approximate method to generate Fourier series of cams with this transition.
His method is based on the superposition principle in which simple curves are combined
to develop a complex curve. In Weber’s work, a curve is considered to be the composite
of two elements: a chord (constant velocity line) connecting the endpoints and a Fourier
sine series having terminal slopes equal and opposite to the chordal slope discontinuities
such that the composite curve is slope-continuous.

REFERENCES

Baranyi, S.M., “Multiple-Harmonic Cam Profiles,” ASME Paper 70-MECH-59, 1970.
Berzak, N., and Freudenstein, F., “Optimization Criteria in Polydyne Cam Design,” Proceedings of
th
5 World Congress on Theory of Machine and Mechanisms, pp. 1303–6, 1979.
Chen, F.Y., Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982.
Dudley, W.M., “A New Approach to Cam Design,” Machine Design (184): 143–8, June 1952.
Freudenstein, F., “On the Dynamics of High-Speed Cam Profiles,” Int J. Mech. Sci. (1): 342–9, 1960.
Gutman, A.S., “To Avoid Vibration—Try This New Cam Profile,” Prod. Eng.: 42–8, December 25,
1961.
th
Matthew, G.K., “The Modified Polynomial Specification for Cams,” Proceedings of 5 World
Congress on Theory of Machine and Mechanisms: 1299–302, 1979.
Stoddart, D.A., “Polydyne Cam Design,” Machine Design 25 (1): 121–35; 25 (2): 146–55; 25 (3):
149–62, 1953.
Thoren, T.R., Engemann, H.H., and Stoddart, D.A., “Cam Design as Related to Valve Train
Dynamics,” SAE Quart. Trans. 1: 1–14, January 6, 1952.
Weber, T., Jr. “Simplifying Complex Cam Design,” Machine Design: 115–20, March 22, 1979.