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AbstractThis paper studies the nonlinear vibration of a cable with multiple intra-span elastic supports under primary resonance excitation. The elastic supports are treated as linear springs in the analysis. Lagrange’s equation and modal superposition approach are employed to derive the dynamic equilibrium equations of the cable based on the energy of system. The nonlinear dynamic behaviour of a cable is analysed in this work, and 3:1 internal resonance is investigated, with coupled in-plane and out-of-plane motions discussed as well. A cable with both two edges simply supported subjected to harmonic excitation is studied by using homotopy analysis method. The amplitude-frequency response and the first approximation solutions of the double-modal motion are obtained. Through comparison with four-order Runge-kutta approach, it demonstrates this method is accurate. Finally, numerical analysis is performed to study how the pretensions, the damping coefficients and the linear stiffness coefficients of the elastic supports influence the amplitude-frequency and amplitude-excitation curves. A range of nonlinear dynamic properties of a cable are acquired, especially the internal resonance issue, which requires the cables stay beyond the region of resonance prior to perform structural design.