## 3 Simulation  
### Testing Flywheel Perturbation  
The test of flywheel perturbation is performed with a six-component test bench (Kistler table/HR-FP3402) installed in the air bearing a floating platform, as shown in Figure 3. The flywheel is fixed to the test bench through a rigid fixture [20; 21]. Under push broom imaging conditions during orbit, the flywheel spins at a constant rotation speed. Therefore, the output data are intercepted under steady state in different rotation speeds with the sampling frequency set to 4096 Hz. In light of the misalignment between the centroid of the flywheel and the center of the test bench, the perturbing force/moment of the flywheel can be calculated in the following equation:  
\[\left\{\begin{array}{c}F_{i}=F_{i}^{t}\\ M_{i}=M_{i}^{t}-\sum{(F_{i}^{t}\Delta)}\end{array}\right. \tag{2}\]  
where \(F_{i}/M_{i}\) is the perturbing force/moment of the flywheel in each direction (\(i=1,2,3\)); \(F_{i}^{t}/M_{i}^{t}\) is the measured perturbing force/moment in each direction; and \(\Delta\) represents the distance from the centroid of the flywheel to the center of the test bench in each direction. The disturbing force/moment of the flywheel is attributed to several reasons, including rotor imbalance, bearing irregularity, flexibility of the flywheel structure, and nonlinear stiffness of bearing [22]. To better understand the characteristics of perturbation, the test data (time domain) are transformed to frequency domain by using fast Fourier transform (FFT) [23]. The response curves of perturbation in each DOF are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Image Motion of the Central Image Point on the Focal Plane (mm)**} \\ \cline{2-9}
**Unit Transition/Tilt of Optical** & \multicolumn{2}{c}{**Primary**} & \multicolumn{2}{c}{**Secondary**} & \multicolumn{2}{c}{**Tertiary**} & \multicolumn{2}{c}{**Image**} \\
**Component** & \multicolumn{2}{c}{**Mirror**} & \multicolumn{2}{c}{**Mirror**} & \multicolumn{2}{c}{**Mirror**} & \multicolumn{2}{c}{**Surface**} \\ \cline{2-9}  & **+X** & **+Y** & **+X** & **+Y** & **+X** & **+Y** & **+X** & **+Y** \\ \hline Transition in +\(X\) direction/mm & 0.98 & 0 & \(-\)1.4 & 0 & 1.36 & 0 & \(-\)1 & 0 \\ Tilt about +\(X\) direction/\({}^{\circ}\) & 0 & \(-\)20.66 & 0 & 9.09 & 0 & \(-\)13.46 & 0 & 0 \\ Transition in +\(Y\) direction/mm & 0 & 1 & 0 & \(-\)1.33 & 0 & 1.35 & 0 & \(-\)1 \\ Tilt about +\(Y\) direction/\({}^{\circ}\) & 21.17 & 0 & \(-\)9.2 & 0 & 13.72 & 0 & 0 & 0 \\ Transition in +\(Z\) direction/mm & 0.3 & 0 & \(-\)0.13 & 0 & 0.05 & 0 & \(-\)0.05 & 0 \\ Tilt about +\(Z\) direction/\({}^{\circ}\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The optical amplification factor of each optical component.  
Figure 3: Testing flywheel perturbation.  
As seen from Figure 4, it can be concluded that the perturbation was mainly focused on the first harmonic order because of the rotor imbalance and high frequency (over 300 Hz). The amplitude increased with the rotor speed for the first harmonic order perturbation. Aiming at the lateral perturbation (\(X\)/\(Y\) direction), a V-shape frequency distribution curve was seen around 400-450 Hz because of the rocking whirl mode of flywheel. The perturbing force in the \(Z\) direction at different speeds was concentrated at 330 Hz because of the lateral mode of the flywheel [24; 25].  
Figure 4: Waterfall plot of perturbations in each degree of freedom (DOF).