Closed-Form Solutions for Vibrations of Nearly Vertical Strings and Beams by Means of Asymptotic Methods
Publication: Journal of Engineering Mechanics
Volume 134, Issue 12
Abstract
Top tensioned strings and beams are often used in civil and marine applications. Typically these members have constant cross sections, and a pronounced, usually linear, tension variation, due to the effects of gravity. In this paper simple, approximate formulas for the natural frequency of such strings are derived, based on asymptotic techniques, while for the tensioned beam case approximate closed-form results are developed by the Wentzel–Kramers–Brillouin method. Both derivations are shown in reasonable detail. While similar work is known for a beam with varying axial tension this is believed to be the first time that a single analytic expression is developed for the full length of the beam. A simple example in which the bottom tension is only 9% of the top tension is analyzed for cases with and without bending stiffness, and the solutions have been compared to the exact solution for the string case and to the results from three finite-element programs for the beam case. The accuracy was found to be very good, even in this situation, in which the tension variation is large.
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References
Bender, C. M., and Orszag, S. A. (1978). Advanced mathematical methods for scientists and engineers, McGraw-Hill, Singapore.
Cheng, Y., Vandiver, J. K., and Moe, G. (2002). “The linear vibration analysis of marine risers using the WKB-based dynamic stiffness method.” J. Sound Vib., 251(4), 750–760.
Hildebrand, F. B. (1962). Advanced calculus for applications, Prentice- Hall International, London.
Kim, Y. C. (1983). “Nonlinear vibrations of long, slender beams.” Ph.D. thesis, MIT, Cambridge, Mass.
Kim, Y. C., and Triantafyllou, M. S. (1984). “The nonlinear dynamics of long, slender cylinders.” J. Energy Resour. Technol., 106, 250–256.
Meirovitch, L. (1997). Principles and techniques of vibrations, Prentice-Hall, Upper Saddle River, N.J.
Moe, G., and Chucheepsakul, S. (1988). “The effect of internal flow on marine risers.” Proc., 7th Int. Conf. on Offshore Mechanics and Arctic Engineering, OMAE, Houston, 375–382.
Païdoussis, M. (1998). Fluid-structure interactions, slender structures and axial flow, Vol. 1, Academic, San Diego.
Young, R. D., Fowler, J. R., Fisher, E. A., and Luke, R. R. (1978). “Dynamic analysis as an aid to the design of marine risers.” J. Pressure Vessel Technol., 100(2), 200–205.
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© 2008 ASCE.
History
Received: May 10, 2006
Accepted: Jun 20, 2008
Published online: Dec 1, 2008
Published in print: Dec 2008
Notes
Note. Associate Editor: Andrew W. Smyth
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