Nonlinear vibration and stability analysis of mode exchanging induced one-to-one internal resonance in an articulated ring truss antenna

Large-scale ring truss antennas (RTAs) generally adopt flexible hinges to achieve deployable function, whereas introducing significant geometric nonlinearities into the system; especially, due to the geometric and material parameter variations, the low- and dense-frequency properties of RTA are prone to induce mode exchanging, causing various internal resonance conditions and resulting in distinct nonlinear dynamic behaviors. This study focuses on the nonlinear dynamic behaviors of an articulated RTA under different 1:1 internal resonance during mode exchanging. To fully depict the dynamic effect of hinges, the nonlinear torque transmitted by hinge is introduced into the system by balancing the force matching conditions, finding that the variation of hinge linear stiffness can induce complete mode exchanging. The ordinary differential equations (ODEs) of vibrations are derived following the Galerkin method, based on which the four-dimensional modulation equations of the RTA under the primary and various 1:1 internal resonance scenarios are obtained by the multiple scale method. Then the frequency-/force-response curves of two-/single mode solutions are obtained by applying Newton-Raphson iteration combined with arc-length continuation, from which the bifurcations, multi-jumping, and bi-/tri-stability are observed. Simulations indicate that the larger amplitude excitation, larger third-order stiffness and lower damping of hinges will result in a wider resonant region, more significant jumping and more obvious bi-/tri-stability of the RTA under 1:1 internal resonance. Analytical results show that the primary and 1:1 internal resonant regions approach, overlap, superimpose and separate with each other during the process of mode exchanging. The multi-value jumping, bifurcations, and bi-stable and tri-stable characteristics of the RTA are highly modulated by the internal detuning parameter, thus resulting in significant changes of 1:1 internal resonant behaviors of the system during mode exchanging. The findings of this research reveal the intrinsic mechanism and evolution of the changes in the 1:1 internal resonant behaviors of the RTA during mode exchanging.