Abstract
In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions.
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References
Dallasta, A. and Leoni, G., ‘Vibrations of beams prestressed by internal frictionless cables’, Journal of Sound and Vibration 222, 1999, 1–18.
Cheng, G. and Zu, J. W., ‘Dynamic analysis of an optical fiber coupler in telecommunications’, Journal of Sound and Vibration 268, 2003, 15–31.
Berkovits, J., ‘On the bifurcation of large amplitude solutions for a system of wave and beam equations’, Nonlinear Analysis 52, 2003, 343–354.
Holubova, G. and Matas, A., ‘Initial-boundary value problem for the nonlinear string-beam system’, Journal of Mathematical Analysis and Applications 288, 2003, 784–802.
Wang, P. H. and Yang, C. G., ‘Parametric studies on cable-stayed bridges’, Computers & Structures 60, 1996, 243–260.
Soyluk, K. and Dumanoglu, A. A., ‘Comparison of asynchronous and stochastic dynamic responses of a cable-stayed bridge’, Engineering Structures 22, 2000, 435–445.
Paolo, C., Francesco, M., Leonardo, L., and Andrea, B., ‘Experimental modal analysis of the Garigliano cable-stayed bridge’, Soil Dynamics and Earthquake Engineering 17, 1998, 485–493.
Ahmed, N. U. and Harbi, H., ‘Mathematical analysis of dynamic models of suspension bridges’, SIAM Journal of Applied Mathematics 58, 1998, 853–874.
Sun, D. K., Xu, Y. L., Ko, J. M., and Lin, J. H., ‘Fully coupled buffeting analysis of long-span cable-supported bridges: Formulation’, Journal of Sound and Vibration 228, 1999, 569–588.
Fung, R. F., Lu, L. Y., and Huang, S. C., ‘Dynamic modeling and vibration analysis of a flexible cable-stayed beam structure’, Journal of Sound and Vibration 254, 2002, 717–726.
Ding, Z. H., ‘Nonlinear periodic oscillations in a suspension bridge system under periodic external aerodynamic forces’, Nonlinear Analysis 49, 2002, 1079–1097.
Ding, Z. H., ‘Multiple periodic oscillations in a nonlinear suspension bridge system’, Journal of Mathematical Analysis and Applications 269, 2002, 726–746.
An, Y. K. and Zhong, C. K., ‘Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load’, Journal of Mathematical Analysis and Applications 279, 2003, 569–579.
Drabek, P. and Necesal, P., ‘Nonlinear scalar model of a suspension bridge: Existence of multiple periodic solutions’, Nonlinearity 16, 2003, 1165–1183.
Royer-Carfagni, G. F., ‘Parametric-resonance-induced cable vibrations in network cable-stayed bridges: A continuum approach’, Journal of Sound and Vibration 262, 2003, 1191–1222.
Gattulli, V. and Lepidi, M., ‘Nonlinear interactions in the planar dynamics of cable-stayed beam’, International Journal of Solids and Structures 40, 2003, 4729–4748.
Gattulli, V., Lepidi, M., Macdonald, J. H. G., and Taylor, C. A., ‘One-to two global-locale interaction in a cable-stayed beam observed through analytical, finite element and experimental models’, International Journal of Non-Linear Mechanics 40, 2005, 571–588.
Cao, D. X. and Zhang, W., ‘Analysis on nonlinear dynamics of a string-beam coupled system’, International Journal of Nonlinear Sciences and Numerical Simulation 6, 2005, 47–54.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Parker, T. S. and Chua, L. O., Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, 1989.
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An erratum to this article is available at http://dx.doi.org/10.1007/s11071-006-9159-0.
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Zhang, W., Cao, D.X. Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom. Nonlinear Dyn 45, 131–147 (2006). https://doi.org/10.1007/s11071-006-2423-5
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DOI: https://doi.org/10.1007/s11071-006-2423-5