Nonlinear Dynamics

, Volume 67, Issue 3, pp 1969–1984 | Cite as

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

Original Paper


The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.


Cable–beam coupled system Internal resonant External resonant Bifurcation Chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wang, L., Ni, Q., Huang, Y.Y.: Bifurcations and chaos in a forced cantilever system with impacts. J. Sound Vib. 296(4–5), 1068–1078 (2006) Google Scholar
  2. 2.
    Younesian, D., Esmailzadeh, E.: Non-linear vibration of variable speed rotating viscoelastic beams. Nonlinear Dyn. 60(1), 193–205 (2010) CrossRefMATHGoogle Scholar
  3. 3.
    Lenci, S., Ruzziconi, L.: Nonlinear phenomenon the single-mode dynamics of a cable-supported beam. Int. J. Bifurc. Chaos Appl. Sci. Eng. 19(3), 923–945 (2009) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Wang, L.H., Zhao, Y.Y.: Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances. Int. J. Solids Struct. 43(25–26), 7800–7819 (2006) CrossRefMATHGoogle Scholar
  5. 5.
    Srinil, N., Rega, G., Chucheepsakul, S.: Large amplitude three-dimensional free vibrations of inclined sagged elastic cables. Nonlinear Dyn. 33(2), 129–154 (2003) CrossRefMATHGoogle Scholar
  6. 6.
    Nayfeh, A.H., Arafat, H.N., Chin, C.M., Lacarbonara, W.: Multi-mode interactions in suspended cables. J. Vib. Control 8(3), 337–387 (2002) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mamandi, A., Kargarnovin, M.H., Farsi, S.: An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions. Int. J. Mech. Sci. 52(12), 1694–1708 (2010) CrossRefGoogle Scholar
  8. 8.
    Kenfack, A.: Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. Chaos Solitons Fractals 15(2), 205–218 (2003) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Musielak, D.E., Musielak, Z.E., Benner, J.W.: Chaos and routes to chaos in coupled Duffing oscillators with multiple degrees of freedom. Chaos Solitons Fractals 24(4), 907–922 (2005) CrossRefMATHGoogle Scholar
  10. 10.
    Zhang, W., Cao, D.X.: Studies on bifurcation and chaos of a string-beam coupled system with two degrees-of-freedom. Nonlinear Dyn. 45(1–2), 131–147 (2006) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Zhang, W., Yao, M.: Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt. Chaos Solitons Fractals 28(1), 42–66 (2006) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Brownjohn, J.M.W., Lee, J., Cheong, B.: Dynamic performance of a curved cable-stayed bridge. Eng. Struct. 21(11), 1015–1027 (1999) CrossRefGoogle Scholar
  13. 13.
    Caetano, E., Cunha, A., Taylor, C.A.: Investigation of dynamic cable–deck interaction in a physical model of a cable-stayed bridge. Part I: modal analysis. Earthquake Eng. Struct. Dyn. 29(4), 481–498 (2000) CrossRefGoogle Scholar
  14. 14.
    Au, F.T.K., Cheng, Y.S., Cheung, Y.K., Zheng, D.Y.: On the determination of natural frequencies and mode shapes of cable-stayed bridges. Appl. Math. Model. 25(12), 1099–1115 (2001) CrossRefMATHGoogle Scholar
  15. 15.
    Zhou, H.B.: The experimental investigation on nonlinear dynamics of cable–beam structure. Ph.D. thesis, Hunan University (2007) (Chinese) Google Scholar
  16. 16.
    Cheng, G., Zu, J.W.: Dynamic analysis of an optical fiber coupler in telecommunications. J. Sound Vib. 268(1), 15–31 (2003) CrossRefGoogle Scholar
  17. 17.
    Gattulli, V., Morandini, M., Paolone, A.: A parametric analytical model for non-linear dynamics in cable-stayed beam. Earthquake Eng. Struct. Dyn. 31(6), 1281–1300 (2002) CrossRefGoogle Scholar
  18. 18.
    Wang, L.H., Zhao, Y.Y.: Multiple internal resonances and non-planar dynamics of shallow suspended cables to the harmonic excitations. J. Sound Vib. 319(1–2), 1–14 (2009) CrossRefGoogle Scholar
  19. 19.
    Kamel, M.M., Hamed, Y.S.: Nonlinear analysis of an elastic cable under harmonic excitation. Acta Mech. 214(3–4), 315–325 (2010) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Shenzhen Graduate SchoolHarbin Institute of TechnologyShenzhenChina

Personalised recommendations