No abstract

Free vibration of summation and difference resonance of the vertical cable and other coupled structural members were investigated in this article. A model of a vertical cable and two mass-springs was built, with the sling considered to be geometrically nonlinear, and the upper and lower connecting structural members were taken as two mass-springs. Assuming the displacement of the sling, modal superposition method and D'Alembert principle were used to derive the dynamic equilibrium equations of the coupled structure. The nonlinear dynamic equilibrium equations were studied by means of multiple scales method, and the second-order approximation solutions of single-modal motion of the system were obtained. Numerical examples were presented to discuss the amplitude responses as functions of time of free vibration, with and without damping, respectively. Additionally, fourth-order Runge-Kutta method was directly used for the nonlinear dynamic equilibrium equations to complement and verify the analytical solutions. The results show that the coupled system performs strongly nonlinear and coupled characteristics, which is useful for engineering design.

No abstract

An approach is presented to investigate the 1:2 internal resonance of the sling and beam of a suspension sling–beam system. The beam was taken as the geometrically linear Euler beam, and the sling was considered to be geometrically nonlinear. The dynamic equilibrium equation of the structures was derived using the modal superposition method, the D’Alembert principle and the Hamilton principle. The nonlinear dynamic equilibrium equations of free vibration and forced oscillation were solved by the multiple-scales method. We derived the first approximation solutions for the single-modal motion of the system. Numerical examples are provided to verify the correctness of formula derivation and obtain the amplitude–time response of free vibration, the primary resonance response, the amplitude–frequency response, and the amplitude–force response of forced oscillation. According to the analysis, it is evident that the combination system exhibits robust nonlinear coupling properties due to the presence of internal resonance, which are useful for engineering design.