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BRIEF NOTES

Analytically Approximate Solutions for Vibrations of a Long Discrete Chain

[+] Author and Article Information
W. Lee

Department of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 32023, ROC e-mail: wlee@phys.cycu.edu.tw

J. Appl. Mech 70(2), 302-304 (Mar 27, 2003) (3 pages) doi:10.1115/1.1526120 History: Received June 19, 2001; Revised July 26, 2002; Online March 27, 2003
Copyright © 2003 by ASME
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References

Watson, G. N., 1966, Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, pp. 3–5.
Routh, E. J., 1955, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 6th ed., Dover, New York, pp. 403–405.
Spiegel, M. R., 1981, Applied Differential Equations, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, pp. 641–645.
Wang,  C. Y., 1994, “Stability and Large Displacements of a Heavy Rotating Linked Chain With an End Mass,” Acta Mech., 107, pp. 205–214.
Timoshenko, S., Young, D. H., and Weaver, W., 1974, Vibration Problems in Engineering, 4th ed., John Wiley and Sons, New York, pp. 244–246.
McCreech,  J. P., Goodfellow,  T. L., and Seville,  A. H., 1975, “Vibrations of a Hanging Chain of Discrete Links,” Am. J. Phys., 43, pp. 646–648.
Levinson,  D. A., 1977, “Natural Frequencies of a Hanging Chain,” Am. J. Phys., 45, pp. 680–681.
Sujith,  R. I., and Hodges,  D. H., 1995, “Exact Solution for the Free Vibration of a Hanging Cord With a Tip Mass,” J. Sound Vib., 179, pp. 359–361.
Triantafyllou,  M. S., and Howell,  C. T., 1994, “Dynamic Responses of Cables Under Negative Tension: An Ill-Posed Problem,” J. Sound Vib., 173, pp. 433–447.
Weng,  P.-C., and Lee,  W., 1994, “Transverse Vibrations of a Hanging Cable: The Limiting Case of a Hanging Chain,” J. Sound Vib., 171, pp. 574–576.

Figures

Grahic Jump Location
A hanging chain of discrete links. (a) The coordinate system and (b) the free body diagram of the jth link. Counterclockwise displacement angles are taken positive.
Grahic Jump Location
A plot of frequency ratio; i.e., the natural frequency of a chain with rotational inertia divided by that of a hanging cable, versus the radius of gyration, k, for the first three modes of vibration. L=2 m and g=9.8 ms−2 .

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