Acta Mechanica

, Volume 108, Issue 1–4, pp 189–206 | Cite as

Elastic equilibria of translating cables

  • O. M. O'Reilly
  • P. Varadi
Original Papers


The equations of motion describing the non-linear behavior of a perfectly flexible travelling cable are derived from first principles. Influences due to changes in the cross-sectional area of the cable and mass conservation are included. A homogenous isotropic non-linearly elastic cable material is assumed and the qualitative nature of a class of its equilibria is analyzed. The dependence of this equilibrium on the constitutive equations and the translational speed is discussed. It is shown that, under gravitational loading, the stretch in this equilibrium is a monotonically increasing function of the translational speed. Furthermore, if this speed is unbounded, so too is the stretch. Related results are proven for the particular cases of a cable composed of a St. Venant-Kirchhoff and an inextensible material.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bajaj, A. K., Johnson, J. M.: On the amplitude dynamics and ‘crisis’ in resonant motion of stretched strings. Phil. Trans. Roy. Soc. London Ser.A338, 1–41 (1992).Google Scholar
  2. [2]
    O'Reilly, O. M.: Global bifurcations in the forced vibrations of damped strings. Int. J. Non-linear Mech.28, 337–351 (1993).Google Scholar
  3. [3]
    Luongo, A., Rega, G., Vestroni, F.: Planar non-linear free vibrations of an elastic cable. Int. J. Non-linear Mech.19, 39–52 (1984).Google Scholar
  4. [4]
    Perkins, N. C., Mote, C. D.: Three dimensional vibration of travelling elastic cables. J. Sound Vibrat.114, 325–340 (1987).Google Scholar
  5. [5]
    Perkins, N. C., Mote, C. D.: Theoretical and experimental stability of two translating cable equilibria. J. Sound Vibrat.128, 397–410 (1989).Google Scholar
  6. [6]
    Ames, W. F., Lee, S. Y., Zaiser, J. N.: Non-linear vibration of a travelling threadline. Int. J. Non-linear Mech.3, 449–469 (1968).Google Scholar
  7. [7]
    Simpson, A.: On the oscillatory motions of travelling elastic cables. J. Sound Vibrat.20, 177–189 (1972).Google Scholar
  8. [8]
    Triantafyllou, M. S.: The dynamics of translating cables. J. Sound Vibrat.103, 171–182 (1985).Google Scholar
  9. [9]
    Irvine, H. M., Caughey, T. K.: The linear theory of free vibrations of a suspended cable. Proc. Roy. Soc. London Ser.A341, 299–315 (1974).Google Scholar
  10. [10]
    Naghdi, P. M.: Finite deformation of elastic rods and shells. In: Proceedings of the IUTAM Symposium on Finite Elasticity Lehigh University 1980. (Carlson. D. E., Shield. R. T., eds.), pp. 47–103. The Hague: Martinus Nijhoff 1982.Google Scholar
  11. [11]
    Truesdell, C., Noll, W.: The non-linear field theories of mechanics, 2nd ed. Berlin New York Tokyo Heidelberg: Springer 1992.Google Scholar
  12. [12]
    Ciarlet, P. G.: Mathematical elasticity. Volume 1: Three-dimensional elasticity. New York: North-Holland 1988.Google Scholar
  13. [13]
    Antman, S. S., Reeken, M.: The drawing and whirling of strings: singular global multiparameter bifurcation problems. SIAM J. Math. Anal.18, 337–365 (1987).Google Scholar
  14. [14]
    Dieudonné, J.: Foundations of modern analysis. London: Academic Press 1960.Google Scholar
  15. [15]
    Serrin, J.: Mathematical principles of classical fluid mechanics. In: Encyclopedia of PhysicsVII/1, (Flügge. S., Truesdell. C., eds.) pp. 125–263. Berlin Heidelberg New York: Springer 1959.Google Scholar
  16. [16]
    Healey, T.J., Papadopoulos, J. N.: Steady axial motions of strings. J. Appl. Mech.57, 785–787 (1990).Google Scholar
  17. [17]
    Antman, S. S.: Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders. Arch. Ration. Mech. Anal.83, 1–52 (1983).Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • O. M. O'Reilly
    • 1
  • P. Varadi
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyU.S.A.
  2. 2.ZürichSwitzerland

Personalised recommendations