Vibration of Tensioned Beams with Intermediate Damper. I: Formulation, Influence of Damper Location
This article is a reply.
VIEW THE ORIGINAL ARTICLEThis article has a reply.
VIEW THE REPLYPublication: Journal of Engineering Mechanics
Volume 133, Issue 4
Abstract
Exact analytical solutions are formulated for free vibrations of tensioned beams with an intermediate viscous damper. The dynamic stiffness method is used in the problem formulation, and characteristic equations are obtained for both clamped and pinned supports. The complex eigenfrequencies form loci in the complex plane that originate at the undamped eigenfrequencies and terminate at the eigenfrequencies of the fully locked system, in which the damper acts as an intermediate pin support. The fully locked eigenfrequencies exhibit “curve veering,” in which adjacent eigenfrequencies approach and then veer apart as the damper passes a node of an undamped mode shape. Consideration of the evolution of the eigenfrequency loci with varying damper location reveals three distinct regimes of behavior, which prevail from the taut-string limit to the case of a beam without tension. The second regime corresponds to damper locations near the first antinode of a given undamped mode shape; in this regime, the loci bend backwards to intersect the imaginary axis, and two distinct nonoscillatory decaying solutions emerge when the damper coefficient exceeds a critical value.
Get full access to this article
View all available purchase options and get full access to this article.
References
Bergman, L. A., and Hyatt, J. E. (1989). “Green functions for transversely vibrating uniform Euler-Bernoulli beams subject to constant axial preload.” J. Sound Vib., 134(1), 175–180.
Bokaian, A. (1990). “Natural frequencies of beams under tensile axial loads.” J. Sound Vib., 142, 481–498.
Cole, J. D. (1968). Perturbation methods in applied mathematics, Blaisdell, Waltham, Mass.
Currie, I. G., and Cleghorn, W. L. (1988). “Free lateral vibrations of a beam under tension with a concentrated mass at the midpoint.” J. Sound Vib., 123(1), 55–61.
Foss, K. A. (1958). “Coordinates which uncouple the equations of motion of damped linear dynamic systems.” J. Appl. Phys., 35, 361–367.
Franklin, P. (1989). “Modal analysis of long systems: A transfer matrix approach.” Master’s thesis, Johns Hopkins Univ., Baltimore.
Gorman, D. F. (1974). “Free lateral vibration analysis of double-span uniform beams.” Int. J. Mech. Sci., 16, 345–351.
Hull, A. J. (1994). “A closed form solution of a longitudinal bar with a viscous boundary condition.” J. Sound Vib., 169(1), 19–28.
Irvine, M. (1981). Cable structures, MIT Press, Cambridge, Mass., reprinted 1992, Dover, New York.
Krenk, S. (2000). “Vibrations of a taut cable with an external damper.” J. Appl. Mech., 67, 772–776.
Krenk, S. (2004). “Complex modes and frequencies in damped structural vibrations.” J. Sound Vib., 270, 981–996.
Krenk, S., and Nielsen, S. R. K. (2002). “Vibrations of a shallow cable with a viscous damper.” Proc. R. Soc. London, Ser. A, 458, 339–357.
Liu, X. Q., Ertekin, R. C., and Riggs, H. R. (1996). “Vibration of a free-free beam under tensile axial loads.” J. Sound Vib., 190(2), 273–282.
Main, J. A. (2002). “Modeling the vibrations of a stay cable with attached damper.” Ph.D. thesis, Johns Hopkins Univ., Baltimore.
Main, J. A., and Jones, N. P. (2001). “Evaluation of viscous dampers for stay-cable vibration mitigation.” J. Bridge Eng., 6(6), 385–397.
Main, J. A., and Jones, N. P. (2002). “Free vibrations of taut cable with attached damper. I: Linear viscous damper.” J. Eng. Mech., 128(10), 1062–1071.
Main, J. A., and Jones, N. P. (2005). “Vibration of tensioned beams with intermediate damper. II: Damper near a support.” J. Eng. Mech., 133(4), 379–388.
McBride, E. J. (1943). “The free lateral vibrations of a cantilever beam with a terminal dashpot.” J. Appl. Mech., 10, A168–A172.
Oliveto, G., Santini, A., and Tripodi, E. (1997). “Complex modal analysis of a flexural vibrating beam with viscous end conditions.” J. Sound Vib., 200, 327–345.
Pacheco, B. M., Fujino, Y., and Sulekh, A. (1993). “Estimation curve for modal damping in stay cables with viscous damper.” J. Struct. Eng., 119(6), 1961–1979.
Perkins, N. C., and Mote, C. D. (1986). “Comments on curve veering in eigenvalue problems.” J. Sound Vib., 106, 451–463.
Preumont, A. (2002). Vibration control of active structures, 2nd Ed., Kluwer, Dordrecht, The Netherlands.
Rasmussen, E. (1997). “Dampers hold sway.” Civ. Eng. Mag., 67(3), 40–43.
Singh, R., Lyons, W. M., and Prater, G. (1989). “Complex eigensolution for longitudinally vibrating bars with a viscously damped boundary.” J. Sound Vib., 133(2), 364–367.
Seebeck, A. (1852). “Über die querschwingungen gespannter und nicht gespannter elastischer stäbe.” Abhandlungen der Mathematisch-Physischen Class der Königlich Sächsischen Gesellschaft der Wissenschaften, 131–168.
Tang, Y. (2003). “Numerical evaluation of uniform beam modes.” J. Eng. Mech., 129(12), 1475–1477.
Wittrick, W. H. (1986). “On the vibration of stretched strings with clamped ends and nonzero flexural rigidity.” J. Sound Vib., 110, 79–85.
Information & Authors
Information
Published In
Copyright
© 2007 ASCE.
History
Received: Oct 28, 2005
Accepted: Sep 12, 2006
Published online: Apr 1, 2007
Published in print: Apr 2007
Notes
Note. Associate Editor: Lambros S. Katafygiotis
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.