## 4 Results
### Simulation Results  
In this section, the main results from the simulated attitude maneuvers are introduced, while more details will be discussed in Section 5. First, the system behavior without an active vibration control system is investigated. The time histories of the solar panel tip displacement with different values of damping are shown in Figure 14a for Control A and Figure 15a for Control B. In particular, the black lines indicate the elastic vibrations in the case of low damping (and, therefore, highest vibrations), while the blue and the red lines show the nominal and high damping cases, respectively. It can be noticed how the vibrations persist significantly during and after the attitude maneuvers. In detail, the tip displacements are derived by integrating the mathematical model presented in Equation (28), by pre-multiplying the output modal amplitudes \(\eta\) by the structural modal shapes to obtain displacements in specific nodes of the structure, as described also in [36]. The satellite attitude angles are instead depicted in Figure 14b for Control A and Figure 15b for Control B. It should be noted that, even if the spacecraft is able to reach the target attitude in the desired time period, the panels elastic vibrations are noticeably affecting the final angle, by inducing oscillations which do not dampen out in the first minutes after the re-orientation. In particular, the spacecraft attitude angles are directly obtained by computing the state vector of the system, introduced in Equation (10), according to platform-level sensors with standard accuracy and measurement errors as in [41]. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Maneuver** & **AVC** & \multicolumn{3}{c}{**Damping**} \\ \hline Control A &
\begin{tabular}{l} None \\ Active \\ None \\ Active \\ None \\ Active \\ \end{tabular} & Low: \(\eta=\)0.005 & Nominal: \(\eta=\)0.012 & High: \(\eta=\)0.022 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of simulated maneuvers. In a second phase, different cases of high and low damping are then analyzed in terms of improvement after applying the AVC control system. In particular, the most significant results are illustrated in Figures 16 and 17 for Control A and Figures 18 and 19 for Control B. The passive structure tip displacements are represented in black in Figures 17a, 18a, 19a and 20a, while the tip displacements after applying the AVC strategy is depicted in red in the same pictures. A very good enhancement and vibration reduction can be observed in all the presented displacement time histories. Figure 14: Control A—Structural damping cases: (**a**) tip displacement; (**b**) attitude angle about nadir axis (i.e., satellite \(z\)-axis), and zoom on attitude perturbation due to coupling with elastic dynamics. Figure 15: Control B—Structural damping cases: (**a**) tip displacement; (**b**) attitude angle about nadir axis (i.e., satellite \(z\)-axis), and zoom on attitude perturbation due to coupling with elastic dynamics. Finally, the actuators input voltages are also reported in Figures 20 and 21. These quantities are computed based on Equation (26) for the two different attitude maneuvers. As expected, the voltage peaks have to be provided to the AVC system in correspondence with the thruster's firings, while the structural appendages are more excited by the attitude control. In particular, a symmetric behavior of the actuators input voltage can be noticed, as the attitude maneuver notoriously induces an asymmetric bending of the solar panels in satellite systems. Figure 16: Control A—Low damping—Simulation outcome: (**a**) tip displacement; (**b**) attitude angle about nadir axis (i.e., satellite z-axis), and zoom on attitude perturbation due to coupling with elastic dynamics. Figure 17: Control A—High damping—Simulation outcome: (**a**) tip displacement; (**b**) attitude angle about nadir axis (i.e., satellite z-axis), and zoom on attitude perturbation due to coupling with elastic dynamics.