Nonlinear Energy Sink (NES), as an innovative passive vibration control technology, has attracted attention for broadband vibration suppression owing to its high efficiency. This paper investigates the parametric resonance and bifurcation behavior of a spatial cable coupled with an NES under parametric excitation. An accurate nonlinear dynamic model of the cable-NES system is first formulated, capturing the essential nonlinear coupling mechanisms and energy transfer pathways. Theoretical analysis reveals complex nonlinear dynamics, including bifurcations induced by system parameters. The dynamic responses under primary parametric resonance are systematically examined using the Homotopy Analysis Method (HAM), which highlights the significant influence of subtle NES parameter variations on the cables vibration. Furthermore, the fourth subharmonic resonance of the cable under parametric excitation is explored. Results demonstrate that optimal placement of the NES at the cable end yields superior vibration mitigation compared to mid-span attachment. Moreover, appropriate tuning of NES damping enables the cable response to settle into a stable steady state. This study elucidates the internal mechanism of parametric resonance in the cable-NES system, providing a theoretical basis and practical guidance for vibration control in bridge engineering.
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