Multiple internal resonances and non-planar dynamics of shallow suspended cables to the harmonic excitations

Abstract In the present study, the nonlinear response of a shallow suspended cable with multiple internal resonances to the primary resonance excitation is investigated. The method of multiple scales is applied directly to the nonlinear equations of motion and associated boundary conditions to obtain the modulation equations and approximate solutions of the cable. Frequency–response curves and force–response curves are used to study the equilibrium solution and its stability. The effects of the excitation amplitude on the frequency–response curves of the cable are also analyzed. Moreover, the chaotic dynamics of the shallow suspended cable is investigated by means of numerical simulations.

[1]  Noel C. Perkins,et al.  Three-dimensional oscillations of suspended cables involving simultaneous internal resonances , 1995, Nonlinear Dynamics.

[2]  Ali H. Nayfeh,et al.  Non-linear responses of suspended cables to primary resonance excitations , 2003 .

[3]  Lianhua Wang,et al.  Non-linear planar dynamics of suspended cables investigated by the continuation technique , 2007 .

[4]  A. H. Nayfeh,et al.  Multiple resonances in suspended cables: direct versus reduced-order models , 1999 .

[5]  Giuseppe Rega,et al.  Nonlinear vibrations of suspended cables—Part I: Modeling and analysis , 2004 .

[6]  Lianhua Wang,et al.  Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances , 2006 .

[7]  A. H. Nayfeh,et al.  Analysis of one-to-one autoparametric resonances in cables—Discretization vs. direct treatment , 1995, Nonlinear Dynamics.

[8]  Nicholas P. Jones,et al.  Free Vibrations of Taut Cable with Attached Damper. I: Linear Viscous Damper , 2002 .

[9]  Ali H. Nayfeh,et al.  Multimode Interactions in Suspended Cables , 2001 .

[10]  Lianhua Wang,et al.  NON-LINEAR DYNAMIC ANALYSIS OF THE TWO-DIMENSIONAL SIMPLIFIED MODEL OF AN ELASTIC CABLE , 2002 .

[11]  R. N. Iyengar,et al.  Internal resonance and non-linear response of a cable under periodic excitation , 1991 .

[12]  Yozo Fujino,et al.  Active Control of Multimodal Cable Vibrations by Axial Support Motion , 1995 .

[13]  G. Rega,et al.  Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation, reduced-order models and nonlinear normal modes , 2007 .

[14]  Noel C. Perkins,et al.  Modal interactions in the non-linear response of elastic cables under parametric/external excitation , 1992 .

[15]  Walter Lacarbonara,et al.  Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems , 2003 .

[16]  R. Alaggio,et al.  Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions , 1995 .

[17]  Yl L. Xu,et al.  Experimental Study of Vibration Mitigation of Bridge Stay Cables , 1999 .

[18]  Lianhua Wang,et al.  On the symmetric modal interaction of the suspended cable: Three-to-one internal resonance , 2006 .

[19]  Fabrizio Vestroni,et al.  Nonlinear oscillations of cables under harmonic loading using analytical and finite element models , 2004 .

[20]  Noel C. Perkins,et al.  Nonlinear oscillations of suspended cables containing a two-to-one internal resonance , 1992, Nonlinear Dynamics.

[21]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[22]  Yozo Fujino,et al.  Semiactive Damping of Stay Cables , 1999 .

[23]  G. Rega,et al.  Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation , 2007 .