## 5 Discussion  
It is evident that the proposed vibration control method is able to effectively damp the rigid-motion-induced flexible vibrations on the carbon-reinforced thermoplastic composite. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Case** & & \multicolumn{2}{c}{**Passive Damping**} & \multicolumn{2}{c}{**Active Damping**} \\ \hline Low damping & \(tan(\delta)=\)0.005 & \(\xi_{p}\) & 0.0025 & \(\xi_{a}\) & 0.015 \\ Nominal & \(tan(\delta)=\)0.012 & \(\xi_{p}\) & 0.006 & \(\xi_{a}\) & 0.034 \\ High damping & \(tan(\delta)=\) 0.022 & \(\xi_{p}\) & 0.011 & \(\xi_{a}\) & 0.072 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Damping coefficients: Control A. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Case** & & \multicolumn{2}{c}{**Passive Damping**} & \multicolumn{2}{c}{**Active Damping**} \\ \hline Low damping & \(tan(\delta)=\)0.005 & \(\xi_{p}\) & 0.0025 & \(\xi_{a}\) & 0.019 \\ Nominal & \(tan(\delta)=\)0.012 & \(\xi_{p}\) & 0.006 & \(\xi_{a}\) & 0.043 \\ High damping & \(tan(\delta)=\) 0.022 & \(\xi_{p}\) & 0.011 & \(\xi_{a}\) & 0.081 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Damping coefficients: Control B.  
Figure 21: Control B—Input voltage: (**a**) low damping; (**b**) high damping. Figure 20: Control A—Input voltage: (**a**) low damping; (**b**) high damping. This can be noticed both by observing the time histories of the tip displacement, and the derived damping coefficient in Tables 7 and 8. In particular, only the displacement of one of the two solar arrays is here reported, due to symmetry reasons. The same applies to the actuators driving voltages, which will have the same modulus as the ones on the opposite panel. Both agile maneuvers, indeed, cause relevant elastic oscillations in the order of a few centimeters, to the extent of producing noticeable effects on the attitude of the satellite at the end of the re-orientation. This is shown in the zoomed boxes in Figures 14-19. It can be observed that, in the absence of an active control system, the satellite nadir angle can oscillate around the target value from \(\pm 0.05^{\circ}\) (Control B, high damping case) to \(\pm 0.15^{\circ}\) in the worst case (Control A, low damping case). Such errors can be considered as not in line with expected values for high-performance Earth Observation satellites, which also reach \(\pm 0.005^{\circ}\)[20]. Moreover, another crucial aspect to be considered is the tranquilization time, that is the period needed to reach the desired pointing stability at the end of the maneuver (i.e., attitude remains fixed under a certain error threshold). In case of passive damping, very long tranquilization times are observed to reach a standard \(\pm 0.01^{\circ}\) stability condition: 1 min in case of Control B and high damping, up to 6 min in the worst case of Control A and low damping. The need to wait such long times before starting scientific operations (such as image or data acquisition) is detrimental to performing such fast maneuvers. The AVC system is able to both counteract effectively the attitude error and significantly reduce the tranquilization time, up to 9 s (Control B, high damping) and 30 s (Control A, low damping), respectively. A set of pragmatic factors has also been considered in this work to demonstrate that the method is suitable for practical implementation. To avoid the actuators being fed with a higher input voltage than their physical limit (which leads to suffering a condition of dielectric breakdown), saturation blocks are considered in to the control scheme. In Figures 20a and 21a, the control channel is saturated for a few seconds, but this event does not jeopardize the effective damping capability of the active system.