Internal resonance of a flexible beam in a spatial tethered system

As one of important characteristics of the nonlinear system containing the quadratic or cubic nonlinearity, the internal resonance between the system's modes may occur and affect the nonlinear properties of the system. In this paper, the internal resonance conditions of the spatial flexible beam suspended by two springs in a spatial on-orbit tethered system are obtained based on the method of multiple scales firstly. And then, the effects of the internal resonances on the attitude stability and the energy transfer tendency of the tethered system are investigated by the structure-preserving approach in detail. In the numerical simulation, the attitude angle always tends to zero in a finite interval when the internal resonance of the spatial flexible beam occurs even if the mass of the platform is close to that of the beam, which implies that, the occurrence of the internal resonance will improve the attitude stability of the tethered system. In addition, it can be found that, when the internal resonance occurs, the elastic potential energy stored in the springs tends to transfer to the beam even if the initial attitude angle is small, which will shorten the vibration control time of the spatial flexible beam if the damping of the beam is considered. The above findings permit us to optimize the system parameters or select the appropriate initial conditions to improve the attitude stability and to accelerate the total energy dissipation of the tethered system if the damping of the spatial flexible beam is considered.