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For the 1 : 1 main parametric resonances problems of cable-bridge coupling vibration, a main parametric resonances model considering cable-beam coupling is developed and dimensionless parametric resonances differential equations are derived. The main parametric resonances characteristics are discussed by means of multiscale approximation solution methods. Using an actual cable of cable-stayed bridge project for research object, numerical simulation analysis under a variety of conditions is illustrated. The results show that when the coupling system causes 1 : 1 parametric resonance, nonlinear main parametric resonances in response are unrelated to initial displacement of the cable, but with the increase of deck beam end vertical initial displacement increases, accompanied with a considerable “beat” vibration. When the vertical initial displacement of deck beam end is 10−6 m order of magnitude or even smaller, “beat” vibration phenomenon of cable and beam appears. Displacement amplitude of the cable is small and considerable amplitude vibration may not occur at this time, only making a slight stable “beat” vibration in the vicinity of the equilibrium position, which is different from 2 : 1 parametric resonance condition of cable-bridge coupling system. Therefore, it is necessary to limit the initial displacement excitation amplitude of beam end and prevent the occurrence of amplitude main parametric excitation resonances.

Theoretical and Numerical Analysis of 1 : 1 Main Parametric Resonance of Stayed Cable Considering Cable-Beam Coupling