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TECHNICAL PAPERS

Nonlinear Dynamics of a Taut String with Material Nonlinearities

[+] Author and Article Information
M. J. Leamy, O. Gottlieb

Faculty of Mechanical Engineering, The Technion-Israel Institute of Technology, Haifa 32000, Israel

J. Vib. Acoust 123(1), 53-60 (Aug 01, 2000) (8 pages) doi:10.1115/1.1325411 History: Received November 01, 1999; Revised August 01, 2000
Copyright © 2001 by ASME
Topics: String , Equations
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References

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Miles,  J. W., 1984, “Resonant Nonplanar Motion of a Stretched String,” J. Acoust. Soc. Am., 75, pp. 1505–1510.
Johnson,  J. M., and Bajaj,  A. K., 1989, “Amplitude Modulated and Chaotic Dynamics In Resonant Motion of Strings,” J. Sound Vib., 128, pp. 87–107.
Molteno,  T. C. A., and Tufillaro,  N. B., 1990, “Torus Doubling and Chaotic String Vibrations: Experimental Results,” J. Sound Vib., 137, pp. 327–330.
O’Reilly,  O., and Holmes,  P. J., 1992, “Non-linear, Non-planar, Non-periodic Vibrations of a String,” J. Sound Vib., 153, pp. 413–435.
Bajaj,  A. K., and Johnson,  J. M., 1992, “On the Amplitude Dynamics and Crisis in Resonant Motions of Stretched Strings,” Philos. Trans. R. Soc. London, Ser. A, 338, pp. 1–41.
O’Reilly,  O., 1993, “Global Bifurcations in the Forced Vibration of a Damped String,” Int. J. Non-Linear Mech., 28, pp. 337–351.
Rubin,  M. B., and Gottlieb,  O., 1996, “Numerical Solutions of Forced Vibration and Whirling of a Non-Linear String Using the Theory of a Cosserat Point,” J. Sound Vib., 197, No. 1, pp. 85–101.
Lee,  C., and Perkins,  N. C., 1992, “Nonlinear Oscillations of Suspended Cables Containing a Two-to-One Internal Resonance,” Nonlinear Dyn., 3, pp. 465–490.
Nayfeh, Samir A., Nayfeh, Ali H., and Mook, Dean T., 1995, “Nonlinear Response of a Taut String to Longitudinal and Transverse End Excitation,” Journal of Vibration and Control, pp. 307–334.
Green, A. E., and Zerna, W., 1968, Theoretical Elasticity, Second Edition, Dover Publications, New York.
Oden, John T., 1972, Finite Elements of Nonlinear Continua, McGraw-Hill, New York.
Meirovitch, Leonard, 1997, Principles and Techniques of Vibrations, Prentice-Hall, London, Section 7.18.
Nayfeh,  A. H., Nayfeh,  J. F., and Mook,  D. T., 1992, “On Methods for Continuous Systems with Quadratic and Cubic Nonlinearities,” Nonlinear Dyn., 1, No. 3, pp. 145–162.
Pakdemirli,  M., Nayfeh,  S. A., and Nayfeh,  A. H., 1995, “Analysis of One-to-One Autoparametric Resonances in Cables—Discretization vs. Direct Treatment,” Nonlinear Dyn., 8, pp. 65–83.
Boyaci,  H., and Pakdemirli,  M., 1997, “A Comparison of Different Versions of The Method of Multiple Scales for Partial Differential Equations,” J. Sound Vib., 204, pp. 595–607.
Luongo, A., and Paolone, A., 1999, “On the Reconstitution Problem in the Multiple Time Scale Method,” Nonlinear Dyn., to appear.
Doedel, E. J., and Wang, X. J., 1995, “AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations,” Technical Report, Center for Research on Parallel Computing, California Institute of Technology, Pasadena, CA 91125. CRPC-95-2.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.

Figures

Grahic Jump Location
Diagram depicting a small element of the string in both the tensioned and the deformed configuration. Material coordinates (x1,x2,x3) identify a point P0 in the tensioned configuration, which is displaced during deformation to point P and is located in space by the inertial coordinates (z1(t),z2(t),z3(t)). After deformation, the material coordinates form a nonorthogonal curvilinear coordinate system (x1,x2,x3) with covariant base vectors (G1,G2,G3).
Grahic Jump Location
Comparison of predicted longitudinal dynamic response for increasing numbers of included modes. Here, a2=−2,a3=6,ωt=1,N=1,r=4,p=0.08,α1=2/2,C=0, and c1=6.25, corresponding to 1 percent critical damping of the first mode. E represents the square root of the modal energy and is given by the L2 norm of all the included modal amplitudes.
Grahic Jump Location
Frequency response curves for different nonlinear material descriptions. In each, ωt=1,N=1,r=4,p=0.08,α1=2/2,a3=6, and μ=6.25, corresponding to 1 percent critical damping.

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