Internal resonances and their bifurcations of a rigid-flexible space antenna

The nonlinear dynamics of a flexible space antenna supported by a serial of two rigid arms mounted on a heavy satellite is studied under the conditions of 2:1 or 3:1 internal resonances. The nonlinear dynamic equations of the in-plane vibration of the antenna system of three degrees of freedom are derived first via the method of assumed modes. Then, the resonance and non-resonance regions are determined in terms of the dimensionless parameters of a rigid arm and a torsional spring in the antenna system. Afterwards, the approximate analytic solutions of frequency-amplitude response of the antenna system under 2:1 or 3:1 internal resonances are derived by using the method of multiple scales. Furthermore, the topologic description for the bifurcation of each internal resonance with respect to unfolding parameters is studied by means of the theory of singularity. Finally, the accuracy of the approximate analytic solutions is validated via the numerical solutions. The study shows that the internal resonances exist over a wide range of the detuning parameter. In addition, the 3:1 internal resonance exhibits much more complicated bifurcation topology than the 2:1 internal resonance. The approximate analytic solutions of internal resonances and their bifurcation analysis results provide an essential reference for designing the antenna system of concern.