## 6 A Cube Satellite with Elastic Plate Boundaries and its Dynamic Responses  
In this section, the idea of the elastic boundary is further verified by a 1.5 m\({}^{3}\) cube satellite with elastically supported bottom plate. The settlement of the cube satellite is illustrated in Figure 13. The structural plates are honeycomb sandwich plates, as mentioned above. A uniform mass of 50 kg is applied to each structural plate. The flywheel is mounted on the bottom plate of the satellite. In the finite element model, if the bottom plate is elastically supported, the edges of the bottom plate are connected to the side plates via 60 vertical spring-dashpot elements and 64 transverse spring-dashpot elements, similar to those in Figure 11. The stiffness and damping coefficient of the boundary spring and dashpot are consistent with the previous section. In addition, the responses of the satellite are also predicted for comparison when the bottom plate is rigidly connected to the side plates at its four edges. Figure 12: The force transmissibility curves of the plate-isolator system with elastic boundary: (**a**) compared with the 3-stage simplified theoretical model; (**b**) compared with the plate-isolator system without elastic boundary. In practical applications, multiple flywheels with different mounting angles are often used in a satellite to realize backup and flywheel collaborative speed control. In this paper, the cube satellite with six flywheels is studied: two flywheels mounted at 90 degrees and four flywheels mounted at 45 degrees. The flywheels are mounted on the satellite's bottom plate, and sinusoidal excitations of 1 N are applied simultaneously in the X, Y, and Z directions to the mass centers of each flywheel. The acceleration frequency response curves of the central points of the satellite top plate and side plate are presented in Figure 14.  
It can be seen from Figure 14 that although the flywheels are isolated on the 6-DOF isolators, the responses of the top or the side plates are not as expected when the satellite bottom plate is rigidly connected to the side plates. The acceleration responses at the central  
Figure 14: Frequency response curve of the acceleration amplitude at the central points of the satellite’s top and side plates. Figure 13: Schematic diagram of CubeSat and flywheel installation: (**a**) with no elastic boundary; (**b**) with elastic boundary. points of the top and side plates exhibit multiple peaks due to the local modes. The first peak corresponds to the local mode of the satellite's bottom plate, while the subsequent peaks correspond to the local modes of the side plates and the top plate. In contrast, when the bottom plate is elastically connected to the side plates, although the amplitudes of the first peaks are not obviously affected, the responses of the central points of the satellite's top and side plates are effectively suppressed between 20 Hz and 100 Hz. The average amplitude of acceleration from approximately 1 to 100 Hz has a decrease of 75.4% at the center of the top plate and a decrease of 64.1% at the center of the side plate. In Figure 12, it is seen that with the elastic boundary condition, as the responses corresponding to the local modes of the bottom plate are suppressed, the disturbance of the flywheels is effectively isolated. Thus, the bottom plate-isolator system with elastic plate boundary cannot output enough excitation energy at the frequencies corresponding to the modes of side or top plates, and the response of the cube satellite naturally decreases at the side of the top plates. Note that in this section, the boundary stiffness and damping are determined based on the parameter studies conducted by the simplified 3-stage analytical model in Section 4. Although in the analytical model, the bottom plate is simplified into a spring with one degree of freedom, the obtained optimal boundary stiffness and damping are effective when the bottom plate's multiple degrees of freedom are fully taken into consideration in the finite element model.