Dynamical modeling and non-planar coupled behavior of inclined CFRP cables under simultaneous internal and external resonances

  • Houjun KangEmail author
  • Tieding Guo
  • Weidong Zhu
  • Junyi Su
  • Bingyu Zhao


A dynamic model for an inclined carbon fiber reinforced polymer (CFRP) cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton’s principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments. The steady-state solutions of the nonlinear equations are obtained by the multiple scale method (MSM) and the Newton-Raphson method. The frequency- and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal (between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode, the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young’s modulus, the excitation amplitude, and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.


inclined carbon fiber reinforced polymer (CFRP) cable nonlinear dynamics bifurcation internal resonance external resonance 

Chinese Library Classification


2010 Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Interesting comments by the reviewers are gratefully acknowledged.


  1. [1]
    IRVINE, H. M. Cable Structures, MIT Press, Cambridge (1981)Google Scholar
  2. [2]
    TRIANTAFYLLOU, M. S. Dynamics of cables, towing cables and mooring systems. Shock and Vibration Digest, 23, 3–8 (1991)CrossRefGoogle Scholar
  3. [3]
    STAROSSEK, U. Cable dynamics: a review. Structural Engineering International, 4, 171–176 (1994)CrossRefGoogle Scholar
  4. [4]
    REGA, G. Nonlinear vibrations of suspended cables, part I: modeling and analysis. Applied Mechanics Reviews, 57, 443–478 (2004)CrossRefGoogle Scholar
  5. [5]
    REGA, G. Nonlinear vibrations of suspended cables, part II: deterministic phenomena. Applied Mechanics Reviews, 57, 479–514 (2004)CrossRefGoogle Scholar
  6. [6]
    WEI, M. H., XIAO, Y. Q., and LIU, H. T. Bifurcation and chaos of a cable-beam coupled system under simultaneous internal and external resonances. Nonlinear Dynamics, 67, 1969–1984 (2012)MathSciNetCrossRefGoogle Scholar
  7. [7]
    LUONGO, A. and ZULLI, D. Dynamic instability of inclined cables under combined wind flow and support motion. Nonlinear Dynamics, 67, 71–87 (2012)MathSciNetCrossRefGoogle Scholar
  8. [8]
    GHOLIPOUR, A., FAROKHI, H., and GHAYESH, M. H. In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dynamics, 79, 1771–1785 (2015)CrossRefGoogle Scholar
  9. [9]
    GHAYESH, M. H., FAROKHI, H., and AMABILI, M. In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Composites Part B: Engineering, 60, 423–439 (2014)CrossRefGoogle Scholar
  10. [10]
    GHAYESH, M. H. and FAROKHI, H. Nonlinear dynamics of microplates. International Journal of Engineering Science, 86, 60–73 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    MEI, K., SUN, S., JIN, G., and SUN, Y. Static and dynamic mechanical properties of long-span cable-stayed bridges using CFRP cables. Advances in Civil Engineering, 2017, 1–11 (2017)Google Scholar
  12. [12]
    LIU, Y. Carbon fiber reinforced polymer (CFRP) cables for orthogonally loaded cable structures: advantages and feasibility. Structural Engineering International, 26, 179–181 (2016)CrossRefGoogle Scholar
  13. [13]
    KREMMIDAS, S. C. Improving Bridge Stay Cable Performance under Static and Dynamic Loads, Ph. D. dissertation, University of California, San Diego (2004)Google Scholar
  14. [14]
    KAO, C. S., KOU, C. H., and XIE, X. Static instability analysis of long-span cable-stayed bridges with carbon fiber composite cable under wind load (in Chinese). Tamkang Journal of Science and Engineering, 9, 89–95 (2006)Google Scholar
  15. [15]
    KOU, C. H., XIE, X., GAO, J. S., and HUANG, J. Y. Static behavior of long-span cable-stayed bridges using carbon fiber composite cable (in Chinese). Journal of Zhejiang University, 39, 137–142 (2005)Google Scholar
  16. [16]
    FAN, Z., JIANG, Y., ZHANG, S., and CHEN, X. Experimental research on vibration fatigue of CFRP and its influence factors based on vibration testing. Shock and Vibration, 2017, 1–18 (2017)Google Scholar
  17. [17]
    XIE, X., LI, X., and SHEN, Y. Static and dynamic characteristics of a long-span cable-stayed bridge with CFRP cables. Materials, 7, 4854–4877 (2014)CrossRefGoogle Scholar
  18. [18]
    XIE, X., GAO, J. S., KOU, C. H., and HUANG, J. Y. Structural dynamic behavior of longspan cable-stayed bridges using carbon fiber composite cable (in Chinese). Journal of Zhejiang University, 39, 728–733 (2005)Google Scholar
  19. [19]
    KANG, H. J., ZHU, H. P., ZHAO, Y. Y., and YI, Z. P. In-plane non-linear dynamics of the stay cables. Nonlinear Dynamics, 73, 1385–1398 (2013)MathSciNetCrossRefGoogle Scholar
  20. [20]
    KANG, H. J., ZHAO, Y. Y., and ZHU, H. P. Linear and nonlinear dynamics of suspended cable considering bending stiffness. Journal of Vibration and Control, 21, 1487–1505 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    LEPIDI, M. and GATTULLI, V. Static and dynamic response of elastic suspended cables with thermal effects. Int. Journal of Solids and Structure, 49, 1103–1116 (2012)CrossRefGoogle Scholar
  22. [22]
    LUONGO, A., REGA, G., and VESTRONI, F. Monofrequent oscillations of anon-linear model of a suspended cable. Journal of Sound and Vibration, 82, 247–259 (1982)CrossRefGoogle Scholar
  23. [23]
    RICCIARDI, G. and SAITTA, F. A continuous vibration analysis model for cables with sag and bending stiffness. Engineering Structures, 30, 1459–1472 (2008)CrossRefGoogle Scholar
  24. [24]
    SOUSA, R. A., SOUZA, R. M., FIGUEIREDO, F. P., CAMBIER P., OLIVEIRA, A. C., and SOUZA, R. M. The influence of bending and shear stiffness and rotational inertiain vibrations of cables: an analytical approach. Engineering Structures, 33, 689–695 (2011)CrossRefGoogle Scholar
  25. [25]
    CEBALLOS, M. A. and PRATO, C. A. Determination of the axial force on stay cables accounting for their bending stiffness and rotational end restraints by free vibration tests. Journal of Sound and Vibration, 317, 127–141 (2008)CrossRefGoogle Scholar
  26. [26]
    NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, Wiley, New York (1979)zbMATHGoogle Scholar
  27. [27]
    DING, H., HUANG, L., MAO, X., and CHEN, L. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1–14 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    GUO, T., KANG, H., WANG, L., and ZHAO, Y. Triad mode resonant interactions in suspended cables, Science China: Physics, Mechanics and Astronomy, 59, 1–14 (2016)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Houjun Kang
    • 1
    • 2
    Email author
  • Tieding Guo
    • 1
    • 3
  • Weidong Zhu
    • 2
  • Junyi Su
    • 3
  • Bingyu Zhao
    • 3
  1. 1.Hunan Provincial Key Lab on Damage Diagnosis for Engineering StructuresHunan UniversityChangshaChina
  2. 2.Department of Mechanical EngineeringUniversity of MarylandMarylandUSA
  3. 3.College of Civil EngineeringHunan UniversityChangshaChina

Personalised recommendations