Dynamic Modeling and Theoretical Analysis of Planar Nonlinear Vibrations of an Articulated Hoop Truss Antenna

The dynamic behaviors of large-scale hoop truss antennas (HTAs) are significantly influenced by the nonlinear torque transmission properties of flexible hinges. Due to the nonlinearity of the hinge, the HTA can be easily excited to exhibit internal resonances that result in the energy exchange between the adjacent two or three modes. This paper focuses on the 3:1 internal resonant responses of an articulated HTA induced by the nonlinearity of the hinge. The analytical modes are obtained by using the global mode method (GMM) and are validated by finite element method (FEM). Then the partial differential equations (PDEs) of planar motions are discretized into ordinary differential equations (ODEs) by Galerkin’s technique. The multiple time scale method is employed to obtain the four-dimensional modulation equations of the HTA with primary and 3:1 internal resonance. By using the Newton–Raphson iteration and the pseudo arclength continuation, frequency–response and force–response curves are obtained to investigate the theoretical steady-state vibrations of the HTA. Moreover, the influence of the cubic spring stiffness, damping, and external excitation amplitude on the system’s nonlinear dynamic behaviors are investigated. The numerical simulation reveals that the articulated multi-beam hoop structure exhibits typical nonlinear phenomena such as hardening-spring type characteristic, jumps, bi-stability, and double peaks. The research findings contribute to the reliable structural design and vibration control of the articulated hoop truss antenna.