Abstract
The paper deals with the subharmonic response of a shallow cable due to time variations of the chord length of the equilibrium suspension, caused by time varying support point motions. Initially, the capability of a simple nonlinear two-degree-of-freedom model for the prediction of chaotic and stochastic subharmonic response is demonstrated upon comparison with a more involved model based on a spatial finite difference discretization of the full nonlinear partial differential equations of the cable. Since the stochastic response quantities are obtained by Monte Carlo simulation, which is extremely time-consuming for the finite difference model, most of the results are next based on the reduced model. Under harmonical varying support point motions the stable subharmonic motion consists of a harmonically varying component in the equilibrium plane and a large subharmonic out-of-plane component, producing a trajectory at the mid-point of shape as an infinity sign. However, when the harmonical variation of the chordwise elongation is replaced by a narrow-banded Gaussian excitation with the same standard deviation and a centre frequency equal to the circular frequency of the harmonic excitation, the slowly varying phase of the excitation implies that the phase difference between the in-plane and out-of-plane displacement components is not locked at a fixed value. In turn this implies that the trajectory of the displacement components is slowly rotating around the chord line. Hence, a large subharmonic response component is also present in the static equilibrium plane. Further, the time variation of the envelope process of the narrow-banded chordwise elongation process tends to enhance chaotic behaviour of the subharmonic response, which is detectable via extreme sensitivity on the initial conditions, or via the sign of a numerical calculated Lyapunov exponent. These effects have been further investigated based on periodic varying chord elongations with the same frequency and standard deviation as the harmonic excitation, for which the amplitude varies in a well-defined way between two levels within each period. Depending on the relative magnitude of the high and low amplitude phase and their relative duration the onset of chaotic vibrations has been verified.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Nielsen, S.R.K., Kirkegaard, P.H.: Super and combinatorial harmonic response of flexible elastic cables with small sag. J. Sound Vib. 251, 79–102 (2002)
Perkins, N.C.: Modal interactions in the nonlinear response of elastic cables under parametric/external excitation. Int. J. Nonlinear Mech. 27, 233–250 (1992)
Pinto da Costa, A., Martins, J.A.C., Branco, F., Lilien, J.L.: Oscillations of bridge stay cables induced by periodic motions of deck and/or tower. ASCE J. Eng. Mech. 122, 613–622 (1996)
El-Attar, M., Ghobarah, A., Aziz, T.S.: Non-linear cable response to multiple support periodic excitation. Eng. Struct. 22, 1301–1312 (2000)
Rega, G: Nonlinear vibrations of suspended cables-Part II: deterministic phenomena. Appl. Mech. Rev. 57, 479–514 (2004)
Larsen, J.W., Nielsen, S.R.K.: Nonlinear stochastic response of a shallow cable. Int. J. Nonlinear Mech. 41, 327–344 (2006)
Irvine, H.M.: Cable Structure, MIT Press, Cambridge, MA, (1981)
Zhou, Q., Nielsen, S.R.K., Qu, W.L.: Semi-active control of three-dimensional vibrations of an inclined sag cable with magnetorheological dampers. J. Sound Vib. 296, 1–22 (2006)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. John Wiley, New York (1995)
Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Wiley , New York (1990)
von Wagner, U., Wedig, W.V.: On the calculation of stationary solutions of multi-dimensional Fokker-Planck equations by orthogonal functions. Nonlinear Dyn. 21, 289–306 (2000)
Clough, R.W., Penzien, J.: Dynamics of structures. McGraw-Hill, New York (1975)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, Q., Larsen, J.W., Nielsen, S.R.K. et al. Nonlinear stochastic analysis of subharmonic response of a shallow cable. Nonlinear Dyn 48, 97–114 (2007). https://doi.org/10.1007/s11071-006-9076-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-006-9076-2