## 2 Problem Illustration  
Flywheels are widely utilized in satellites as the attitude controlling device, and the high-frequency vibration of flywheels usually poses significant challenges in achieving stable mounting interfaces for high-precision payloads. Figure 1 illustrates a typical three-axis stabilized satellite with a high-precision optical payload. To attenuate the flywheel-induced vibration, the flywheels are installed on 6-DOF vibration isolation platforms. A 6-DOF vibration isolation platform can be designed to exhibit quite low vibration transmissibility. Thus, theoretically, a well-designed vibration isolator should solve the issue of vibration on a satellite very well. For instance, the 6-DOF vibration isolation platform presented in Figure 1 applies six linear springs with a stiffness \(K_{\text{spring}}\) = 50,000 N/m and the equivalent damping \(C_{\text{spring}}\) = 100 Nm/s. The overall mass of the platform is less than 2 kg, the fundamental frequency of which is about 14 Hz. However, in practice, the isolation platforms are installed on the elastic structural plate of the satellite, which possesses limited stiffness. The elastic structural plate and the isolation platform form an elastic system (plate-isolator system) in essence. Therefore, to effectively suppress the influence of flywheel vibrations, this study will demonstrate that the key is the dynamics of this plate-isolator system rather than merely the dynamics of the isolation platform. To illustrate the influence of the elastic structural plate on the isolation of flywheel vibrations, the dynamics of a plate-isolator (structural plate and isolation platform) system are analyzed. Figure 2 illustrates the plate-isolator system and its finite element model. The flywheel has a mass of 8 kg, while the structural plate is a honeycomb sandwich structure with dimensions of 1500 mm \(\times\) 1500 mm \(\times\) 30 mm. It consists of a carbon fiber skin and an aluminum honeycomb core. Table 1 lists the layering and material properties of the structural plate. The dynamic response of the plate-isolator system is evaluated using the commercial finite element software ABAQUS 6.14. Shell elements (S4R) are employed to model both the flywheel and the structural plate. In addition, the vibration isolation platform is modeled using both spring and dashpot elements, as well as shell elements (S4R). The stiffness and damping coefficient of the spring and damping elements are \(K\) = 50,000 N/m and \(C\) = 100 Nm/s, respectively. The shell element size of the structural plate is 25 mm; it is 5 mm for the platform. Figure 1: Satellite structure with flywheels and flywheel vibration isolators. Figure 2: Finite element model of the plate-isolator (structural plate, isolation platform, and flywheel) system. To analyze the force transmissibility of the vibration isolation platform itself and the plate-isolator system, frequency response analyses are conducted using the \"Steady-state Dynamics, Direct\" analysis step in ABAQUS. Sinusoidal excitations are applied in the X, Y, and Z directions at the mass center of the flywheel. For the isolation platform, fixed constraints are applied to its bottom, and the force transmissibility is calculated by extracting the total reaction force at the bottom of the platform. For the plate-isolator system, fixed constraints are applied to the four edges of the structural plate, and force transmissibility is calculated by extracting the total reaction force from the four edges of the structural plate. Figure 3 compares the force transmissibility curves of the isolation platform and the plate-isolator system in the frequency range of 0-200 Hz, in which obvious differences can be observed. Note that the force transmissibility is defined as 20 \(\times\) Log (\(F_{\text{reaction}}/F_{\text{input}}\)). The force transmissibility of the plate-isolator system, especially in the Y direction, is considerably worse compared to that of the vibration isolation platform.