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y" Cam angle q
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FIGURE 410 Gutman s [1961] 1-3 harmonic acceleration curve
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Similarly, if we retain the rst three terms in the Fourier series expansion of the displacement of the parabolic curve, the equation of Gutman s fth-order harmonic curve can be obtained
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Freudenstein 1-3 Harmonic Curve (Freudenstein, 1960) y= hq h 27 2pq 1 6pq + sin sin b 2p 28 b 84 b h 27 2pq 1 6pq - cos 1 - cos 28 28 b b b 2ph 27 2pq 3 6pq + sin sin 28 b 2 28 b b 4p 2 h 27 2pq 9 6pq + cos cos 28 b 3 28 b b (412)
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In agreement with the foregoing design factors stated, Freudenstein s 1-3 harmonic curve has a maximum acceleration of about 135 percent of the acceleration of the parabolic curve, or 85 percent of the acceleration of the cycloidal curve
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Freudenstein 1-3-5 Harmonic Curve (Freudenstein, 1960) y= m= y = hq hm 2pq 1 6pq 1 10qq + + sin sin sin b 2p b 54 b 1250 b 1125 1192 h b 2pq 1 6pq 1 10pq 1 - m cos b + 18 cos b + 250 cos b (413)
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POLYNOMIAL AND FOURIER SERIES CAM CURVES
y =
2ph 2pq 1 6pq 1 10pq m sin + sin + sin 6 50 b2 b b b 4p 2 h 2pq 1 6pq 1 10pq m cos + cos + cos b3 b b b 2 10
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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 coef cients up to and including the seventeenth harmonic of the pro le 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 nite 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, SM, Multiple-Harmonic Cam Pro les, ASME Paper 70-MECH-59, 1970 Berzak, N, and Freudenstein, F, Optimization Criteria in Polydyne Cam Design, Proceedings of 5th World Congress on Theory of Machine and Mechanisms, pp 1303 6, 1979 Chen, FY, Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982 Dudley, WM, A New Approach to Cam Design, Machine Design (184): 143 8, June 1952 Freudenstein, F, On the Dynamics of High-Speed Cam Pro les, Int J Mech Sci (1): 342 9, 1960 Gutman, AS, To Avoid Vibration Try This New Cam Pro le, Prod Eng: 42 8, December 25, 1961 Matthew, GK, The Modi ed Polynomial Speci cation for Cams, Proceedings of 5th World Congress on Theory of Machine and Mechanisms: 1299 302, 1979 Stoddart, DA, Polydyne Cam Design, Machine Design 25 (1): 121 35; 25 (2): 146 55; 25 (3): 149 62, 1953 Thoren, TR, Engemann, HH, and Stoddart, DA, 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
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