Multiple scales perturbation analysis on vortex-induced vibration of a cable with modal internal resonance

Vortex-induced vibration (VIV) of large-span cables has become a significant engineering problem to be solved. As a typical continuous body with quadratic and cubic nonlinearity, the modal resonance of the cable has a great effect on its dynamics, especially with the fluid–structure interaction. The nonlinear dynamic behaviors of multiple modal VIV of a cable are studied in this paper with the combination of theoretical perturbation method and numerical simulation of Computational Fluid Dynamics (CFD). Firstly, motion equations for the cross-flow vibration of the cable under uniform airflow are established, van der Pol wake oscillator is used to simulate the fluid load caused by the vortex shedding. Galerkin’s method is utilized to discretize the partial differential equations into reduced ordinary differential equations. Secondly, modulation equations with 2:1 internal resonance of the cable are derived by the method of multiple scales. Frequency–response curves of the system are plotted and validated by the Runge–Kutta method, where the necessary aerodynamic coefficients for the perturbation analysis are supplied by solving the corresponding two-dimensional (2-D) CFD model based on the Shear Stress Transport (SST) k–ω model. Finally, the parametric influence of structural, aerodynamic, and coupling parameters on the nonlinear dynamic behaviors is explored. Results novelty reveal the bending of the frequency response curve due to structural geometric nonlinearity and nonlinear damping of the wake oscillator is tightly related to the ‘lock-in’ phenomenon. Variation of structural and aerodynamic parameters will significantly change the ‘lock-in’ domain. The low-to-high and high-to-low frequency sweeps have different ‘lock-in’ domains. The jump phenomenon occurs at the endpoints of the ‘lock-in’ domain.