Theoretical and Experimental Nonlinear Vibrations of Sagged Elastic Cables

  • Giuseppe Rega
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 181)


The chapter presents a comprehensive overview of recent advancements in the theoretical and experimental research on modelling, analysis, response, and nonlinear/nonregular phenomena in the finite amplitude, resonant, forced dynamics of sagged, horizontal or inclined, elastic cables. Asymptotic solutions and a rich variety of features of nonlinear multimodal interaction occurring in various resonance conditions are comparatively discussed. Dynamical and mechanical characteristics of some main experimentally observed responses are summarised, along with the relevant robustness, spatio-temporal features, and dimensionality. Challenging issues arising in the characterisation of involved bifurcation scenarios to complex dynamics are addressed, and hints for proper reduced-order modelling in cable nonlinear dynamics are obtained from both asymptotic solutions and experimental investigations, in the perspective of a profitable cross-validation of the observed nonlinear phenomena.


Suspended cable resonant nonlinear dynamics asymptotic solution experimental analysis bifurcation and chaos reduced-order modelling 


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  1. 1.
    Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., Tsimring, L.S.: The analysis of observed chaotic data in physical systems. Rev. Mod. Phys. 65, 1331–1391 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abe, A.: Validity and accuracy of solutions for nonlinear vibration analyses of suspended cables with one-to-one internal resonance. Nonlin Anal: Real World Appl. 11, 2594–2602 (2010)zbMATHCrossRefGoogle Scholar
  3. 3.
    Alaggio, R., Rega, G.: Characterizing bifurcations and classes of motion in the transition to chaos through 3d-tori of a continuous experimental system in solid mechanics. Phys. D 137, 70–93 (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    Alaggio, R., Rega, G.: Exploiting results of experimental nonlinear dynamics for reduced-order modeling of a suspended cable. In: 18th Bienn. Conf. Mechanical vibration and noise DETC01/VIB-21554 CD-Rom. ASME, New York (2001)Google Scholar
  5. 5.
    Alaggio, R., Rega, G.: Low-dimensional model of the experimental bifurcation scenario of a polymeric cable. In: Int. Conf. Nonlinear phenomena in polymer solids and low-dimensional systems, pp. 170–181. Russian Academy of Sciences, Moscow (2008)Google Scholar
  6. 6.
    Alaggio, R., Rega, G.: Unfolding of complex dynamics of sagged cables around a divergence-Hopf bifurcation: theoretical model and experimental results. In: XIX Congr. AIMETA Mecc. Teor Appl. CD-Rom. Ancona, Italy (2009)Google Scholar
  7. 7.
    Anishchenko, V.S., Safonova, M.A., Feudel, U., Kurths, J.: Bifurcations and transitions to chaos through three-dimensional tori. Int. J. Bif. Chaos. 4, 595–607 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Arafat, H.N., Nayfeh, A.H.: Nonlinear responses of suspended cables to primary resonance excitations. J. Sound Vib. 266, 325–354 (2003)CrossRefGoogle Scholar
  9. 9.
    Armbruster, D., Guckenheimer, J., Kim, S.: Chaotic dynamics with square symmetry. Phys. Lett. A 140, 416–420 (1989)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Benedettini, F., Rega, G.: Experimental investigation of the nonlinear response of a hanging cable. Part II: Global analysis. Nonlin. Dyn. 14, 119–138 (1997)Google Scholar
  11. 11.
    Benedettini, F., Rega, G., Alaggio, R.: Nonlinear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. J. Sound Vibr. 182, 775–798 (1995)CrossRefGoogle Scholar
  12. 12.
    Berlioz, A., Lamarque, C.H.: Nonlinear vibrations of an inclined cable. J. Vib. Acoust. 127, 315–323 (2005)CrossRefGoogle Scholar
  13. 13.
    Casciati, F., Ubertini, F.: Nonlinear vibration of shallow cables with semiactive tuned mass damper. Nonlin. Dyn. 53, 89–106 (2008)zbMATHCrossRefGoogle Scholar
  14. 14.
    Chen, H., Zhang, Z., Wang, J., Xu, Q.: Global bifurcation and chaotic dynamics in suspended cables. Int. J. Bif. Chaos. 19, 3753–3776 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Cheng, S.P., Perkins, N.C.: Closed-form vibration analysis of sagged cable/mass suspensions. J. Appl. Mech. 59, 923–928 (1992)zbMATHCrossRefGoogle Scholar
  16. 16.
    Gattulli, V.: Advanced control strategies in cable dynamics. In: Topping, B.H.V. (ed.) Civil Engineering Computations: Tools and Techniques, pp. 243–269. Saxe-Coburg Publications, Stirlingshire (2007)CrossRefGoogle Scholar
  17. 17.
    Gattulli, V., Martinelli, L., Perotti, F., Vestroni, F.: Nonlinear oscillations of cables under harmonic loading using analytical and finite element models. Comput. Meth. Appl. Mech. Eng. 193, 69–85 (2004)zbMATHCrossRefGoogle Scholar
  18. 18.
    Gattulli, V., Alaggio, R., Potenza, F.: Analytical prediction and experimental validation for longitudinal control of cable oscillations. Int. J. Nonlin. Mech. 43, 36–52 (2008)CrossRefGoogle Scholar
  19. 19.
    Goyal, S., Perkins, N.C., Lee, C.L.: Nonlinear dynamics and loop formation in Kirchoff rods with implications to the mechanics of DNA and cables. J. Comput. Phys. 209, 371–389 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Goyal, S., Perkins, N.C., Lee, C.L.: Nonlinear dynamic intertwining of rods with self-contact. Int. J. Nonlin. Mech. 43, 65–73 (2008)zbMATHCrossRefGoogle Scholar
  21. 21.
    Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Phys. D 9, 189–208 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New York (1983)zbMATHGoogle Scholar
  23. 23.
    Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge (1996)zbMATHCrossRefGoogle Scholar
  24. 24.
    Ibrahim, R.A.: Nonlinear vibrations of suspended cables. Part III: random excitation and interaction with fluid flow. Appl. Mech. Rev. 57, 515–549 (2004)Google Scholar
  25. 25.
    Irvine, H.M.: Cable structures. MIT Press, Cambridge (1981)Google Scholar
  26. 26.
    Irvine, H.M., Caughey, T.K.: The linear theory of free vibrations of a suspended cable. Proc. R. Soc. London A 341, 299–315 (1974)CrossRefGoogle Scholar
  27. 27.
    Lacarbonara, W.: Direct treatment and discretizations of non-linear spatially continuous systems. J. Sound Vibr. 221, 849–866 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lacarbonara, W., Pacitti, A.: Nonlinear modeling of cables with flexural stiffness. Math. Probl. Engng., 370–767 (2008)Google Scholar
  29. 29.
    Lacarbonara, W., Rega, G.: Resonant nonlinear normal modes. Part II: Activation/Orthogonality Conditions for Shallow Structural Systems. Int. J. Nonlin. Mech. 38, 873–887 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lacarbonara, W., Arafat, H.N., Nayfeh, A.H.: Non-linear interactions in imperfect beams at veering. Int. J. Nonlin. Mech. 40, 987–1003 (2005)zbMATHCrossRefGoogle Scholar
  31. 31.
    Lacarbonara, W., Paolone, A., Vestroni, F.: Elastodynamics of nonshallow suspended cables: linear modal properties. J. Vib. Acous. 129, 425–433 (2007)CrossRefGoogle Scholar
  32. 32.
    Lacarbonara, W., Paolone, A., Vestroni, F.: Nonlinear modal properties of non-shallow cables. Int. J. Nonlin. Mech. 42, 542–554 (2007)CrossRefGoogle Scholar
  33. 33.
    Lacarbonara, W., Rega, G., Nayfeh, A.H.: Resonant nonlinear normal modes. Part I: Analytical Treatment for One-Dimensional Structural Systems. Int. J. Nonlin. Mech. 38, 851–872 (2003)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lee, C.L., Perkins, N.C.: Nonlinear oscillations of suspended cables containing a two-to-one internal resonance. Nonlin. Dyn. 3, 465–490 (1992)Google Scholar
  35. 35.
    Lee, C.L., Perkins, N.C.: Experimental investigation of isolated and simultaneous internal resonances in suspended cables. J. Vib. Acous. 117, 385–391 (1995)CrossRefGoogle Scholar
  36. 36.
    Lee, C.L., Perkins, N.C.: Three-dimensional oscillations of suspended cables involving simultaneous internal resonances. Nonlin. Dyn. 8, 45–63 (1995)MathSciNetGoogle Scholar
  37. 37.
    Leissa, A.W.: On a curve veering aberration. J. Appl. Math. Phys. 25, 99–112 (1974)zbMATHCrossRefGoogle Scholar
  38. 38.
    Lepidi, M., Gattulli, V., Vestroni, F.: Static and dynamic response of elastic suspended cables with damage. Int. J. Solids Struct. 44, 8194–8212 (2007)zbMATHCrossRefGoogle Scholar
  39. 39.
    Luongo, A., Piccardo, G.: A continuous approach to the aeroelastic stability of suspended cables in 1:2 internal resonance. J. Vib. Control 14, 135–157 (2008)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Luongo, A., Zulli, D., Piccardo, G.: Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables. J. Sound Vib. 315, 375–393 (2008)CrossRefGoogle Scholar
  41. 41.
    Mané, R.: On the dimension of the compact invariant sets of certain nonlinear maps. In: Rand, D.A., Young, L.S. (eds.) Dynamical Systems and Turbulence, vol. 898, pp. 230–242. Springer, Berlin (1981)CrossRefGoogle Scholar
  42. 42.
    Moon, F.C.: Chaotic and fractal dynamics. Wiley, New York (1992)CrossRefGoogle Scholar
  43. 43.
    Nayfeh, A.H.: Introduction to perturbation techniques. Wiley, New York (1981)zbMATHGoogle Scholar
  44. 44.
    Nayfeh, A.H.: Nonlinear interactions. Wiley, New York (2000)zbMATHGoogle Scholar
  45. 45.
    Nayfeh, A.H., Balachandran, B.: Applied nonlinear dynamics. Wiley, New York (1995)zbMATHCrossRefGoogle Scholar
  46. 46.
    Nayfeh, A.H., Arafat, H.N., Chin, C.M., Lacarbonara, W.: Multimode interactions in suspended cables. J. Vib. Control 8, 337–387 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Pakdemirli, M., Nayfeh, S.A., Nayfeh, A.H.: Analysis of one-to-one autoparametric resonances in cables. Discretization vs direct treatment. Nonlin. Dyn. 8, 65–83 (1995)MathSciNetGoogle Scholar
  48. 48.
    Perkins, N.C.: Modal interactions in the non-linear response of elastic cables under parametric/external excitation. Int. J. Nonlinear Mech. 27, 233–250 (1992)zbMATHCrossRefGoogle Scholar
  49. 49.
    Perkins, N.C., Mote Jr., C.D.: Comments on curve veering in eigenvalue problems. J. Sound Vib. 106, 451–463 (1986)CrossRefGoogle Scholar
  50. 50.
    Rega, G.: Nonlinear vibrations of suspended cables. Part I: Modeling and Analysis. Appl. Mech. Rev. 57, 443–478 (2004)Google Scholar
  51. 51.
    Rega, G.: Nonlinear vibrations of suspended cables. Part II: Deterministic Phenomena. Appl. Mech. Rev. 57, 479–514 (2004)Google Scholar
  52. 52.
    Rega, G., Alaggio, R.: Spatio-temporal dimensionality in the overall complex dynamics of an experimental cable/mass system. Int. J. Solids Struct. 38, 2049–2068 (2001)zbMATHCrossRefGoogle Scholar
  53. 53.
    Rega, G., Alaggio, R.: Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation. J. Sound. Vib. 322, 581–611 (2009)CrossRefGoogle Scholar
  54. 54.
    Rega, G., Sorokin, S.: Asymptotic analysis of linear/nonlinear vibrations of suspended cables under heavy fluid loading. In: Kreuzer, E. (ed.) IUTAM Symp. Fluid-Structure Interaction in Ocean Engineering, pp. 217–228. Springer, Berlin (2008)CrossRefGoogle Scholar
  55. 55.
    Rega, G., Srinil, N.: Nonlinear hybrid-mode resonant forced oscillations of sagged inclined cables at avoidances. J. Comput. Nonlin. Dyn. 2, 324–336 (2007)CrossRefGoogle Scholar
  56. 56.
    Rega, G., Alaggio, R., Benedettini, F.: Experimental investigation of the nonlinear response of a hanging cable. Part I: Local Analysis. Nonlin. Dyn. 14, 89–117 (1997)Google Scholar
  57. 57.
    Rega, G., Lacarbonara, W., Nayfeh, A.H., Chin, C.M.: Multiple resonances in suspended cables: direct versus reduced-order models. Int. J. Nonlin. Mech. 34, 901–924 (1999)CrossRefGoogle Scholar
  58. 58.
    Rega, G., Srinil, N., Alaggio, R.: Experimental and numerical studies of inclined cables: free and parametrically-forced vibrations. J. Theor. Appl. Mech. 46, 621–640 (2008)Google Scholar
  59. 59.
    Ricciardi, G., Saitta, F.: A continuous vibration analysis model for cables with sag and bending stiffness. Engng. Struct. 30, 1459–1472 (2008)CrossRefGoogle Scholar
  60. 60.
    Russell, J.C., Lardner, T.J.: Experimental determination of frequencies and tension for elastic cables. J. Engng. Mech. 124, 1067–1072 (1998)CrossRefGoogle Scholar
  61. 61.
    Seydel, R.: Practical bifurcation and stability analysis. Springer, New York (1994)zbMATHGoogle Scholar
  62. 62.
    Sorokin, S., Rega, G.: On modeling and linear vibrations of arbitrarily sagged inclined cables in a quiescent viscous fluid. J. Fluids Structures 23, 1077–1092 (2007)CrossRefGoogle Scholar
  63. 63.
    Srinil, N.: Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities. J. Fluids Structures 26, 1098–1122 (2010)CrossRefGoogle Scholar
  64. 64.
    Srinil, N., Rega, G.: Resonant non-linear dynamic responses of horizontal cables via kinematically non-condensed/condensed modelling. In: Mota Soares, C.A., et al. (eds.) III Eur. Conf. Comput. Mech. Solids, Structures Coupl. Probl. Engng. CD-Rom, Lisbon, Portugal (2006)Google Scholar
  65. 65.
    Srinil, N., Rega, G.: Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal Resonance Activation, Reduced-Order Models and Nonlinear Normal Modes. Nonlin. Dyn. 48, 253–274 (2007)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Srinil, N., Rega, G.: The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables. Int. J. Nonlin. Mech. 42, 180–195 (2007)zbMATHCrossRefGoogle Scholar
  67. 67.
    Srinil, N., Rega, G.: Nonlinear longitudinal/transversal modal interactions in highly-extensible suspended cables. J. Sound Vib. 310, 230–242 (2008)CrossRefGoogle Scholar
  68. 68.
    Srinil, N., Rega, G.: Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables. J Sound Vib. 315, 394–413 (2008)CrossRefGoogle Scholar
  69. 69.
    Srinil, N., Rega, G., Chucheepsakul, S.: Large amplitude three-dimensional free vibrations of inclined sagged elastic cables. Nonlin. Dyn. 33, 129–154 (2003)zbMATHCrossRefGoogle Scholar
  70. 70.
    Srinil, N., Rega, G., Chucheepsakul, S.: Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables. J. Sound Vib. 269, 823–852 (2004)CrossRefGoogle Scholar
  71. 71.
    Srinil, N., Rega, G., Chucheepsakul, S.: Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical Formulation and Model Validation. Nonlin. Dyn. 48, 231–252 (2007)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Takens, F.: Detecting strange attractors in turbulence. In: Rand, D.A., Young, L.S. (eds.) Dynamical systems and turbulence, vol. 898, pp. 366–381. Springer, Berlin (1981)CrossRefGoogle Scholar
  73. 73.
    Triantafyllou, M.S.: The dynamics of taut inclined cables. Q. J. Mech. Appl. Math. 37, 421–440 (1984)zbMATHCrossRefGoogle Scholar
  74. 74.
    Triantafyllou, M.S., Grinfogel, L.: Natural frequencies and modes of inclined cables. J. Struct. Engng. 112, 139–148 (1986)CrossRefGoogle Scholar
  75. 75.
    Wang, L., Rega, G.: Modelling and transient planar dynamics of suspended cables with moving mass. Int. J. Solids Struct. 47, 2733–2744 (2010)zbMATHCrossRefGoogle Scholar
  76. 76.
    Wang, L., Zhao, Y., Rega, G.: Multimode dynamics and out-of-plane drift in suspended cable using the kinematically condensed model. J. Vib. Acous. 131, 061008 (2009)CrossRefGoogle Scholar
  77. 77.
    Wu, Q., Takahashi, K., Nakamura, S.: Formulae for frequencies and modes of in-plane vibrations of small-sag inclined cables. J. Sound Vib. 279, 1155–1169 (2005)CrossRefGoogle Scholar
  78. 78.
    Zhao, Y., Wang, L.: On the symmetric modal interaction of the suspended cable: three-to-one internal resonance. J. Sound Vib. 294, 1073–1093 (2006)CrossRefGoogle Scholar
  79. 79.
    Zhao, Y., Wang, L., Cheng, D., Jiang, L.: Non-linear dynamic analysis of the two-dimensional simplified model of an elastic cable. J. Sound Vib. 255, 43–59 (2002)CrossRefGoogle Scholar
  80. 80.
    Zheng, G., Ko, J.M., Ni, Y.Q.: Super-harmonic and internal resonances of a suspended cable with nearly commensurable natural frequencies. Nonlin. Dyn. 30, 55–70 (2002)zbMATHCrossRefGoogle Scholar
  81. 81.
    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)CrossRefGoogle Scholar
  82. 82.
    Zhou, Q., Larsen, J.W., Nielsen, S.R.K., Qu, W.L.: Nonlinear stochastic analysis of subharmonic response of a shallow cable. Nonlin. Dyn. 48, 97–114 (2007)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Giuseppe Rega
    • 1
  1. 1.Department of Structural and Geotechnical EngineeringSapienza University of RomeRomaItaly

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