Analytical studies of vortex-induced vibrations of structures have often focused on vibrations restricted to either the cross-flow direction or the in-line direction. Nevertheless, experimental evidence has confirmed that coupled cross-flow/in-line responses exist by virtue of the simultaneous excitation in these two directions. In this paper, we explore potential models for these coupled vibrations for elastic cables suspended in uniform cross flows. The coupling of cross-flow and in-line vibrations derives from two principal mechanisms:

  1. structural nonlinearities, and

  2. coupled fluid lift and drag.

Wake-oscillator models for the fluctuating lift and drag forces are combined with the nonlinear partial differential equations of cable motion. Attention focuses on the resonant case when the natural frequencies for cable modes in the cross-flow and in-line directions are in the same 1:2 ratio as the excitation frequencies due to lift and drag. This case is used to investigate three limiting cases:

  • the uncoupled response,

  • the coupled response due to structural nonlinearities, and

  • the coupled response due to coupled lift and drag. For each case, an analytical solution (based on asymptotics) is derived for the predicted periodic motions of the cable/fluid system. The addition of structural nonlinearities (mechanism 1) and coupled lift and drag (mechanism 2) lead to qualitatively different responses (number and stability of periodic motions differ) when compared to the response of the decoupled model.


When structures are exposed to a flowing fluid, the flow may act as an energy source that drives structural vibrations. Such flow induced vibrations occur in diverse applications including heat exchanger tubes, electric power lines, airplane wings, and underwater risers and cables and result from fluid vortex-shedding. Alternate vortex shedding leads to time and space dependent pressure excitation on the structure.

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