## 3 Results and Discussion
### Active Control with Decentralized Velocity Feedback Controllers  
In this research, an upper limit of seven channel decentralized feedback controllers is considered. The locations and coordinates of these feedback loops are shown in Figure 6. The control performance of the seven channel feedback loops is validated when the control gain is set as 30, 50, 100, and 200. The kinetic energy and radiated sound power of the ribbed plate before and after control is calculated and plotted in Figures 7 and 8. Note that the kinetic energy of the ribbed plate should be the sum of the kinetic energy of the base plate and the ribs. However, the quality of the base plate used in the simulation (5.4 kg) is much bigger than that of the two ribs (about 0.5 kg, less than one-tenth of the quality of the base plate). Hence, the kinetic energy of the ribbed plate is approximately obtained by calculating the kinetic energy of the base plate using the equation \(E=\frac{\rho ln\omega^{2}\int_{0}^{\omega}\int_{0}^{\omega}w^{2}d\omega y}{2}\), which is expressed in dB with reference \(1\times 10^{-12}\). \(\rho\) and \(h\) are the density and thickness of the base plate, \(w\) is the displacement of the base plate at location \((x,y)\). It can be found that the decentralized feedback controllers can effectively control the vibration and sound radiation of the orthogonally rib stiffened plate. Similar to the case of the unribbed plate--as the gains of the feedback loops are increased--the resonances in the response become more heavily damped. However, if the gains of the feedback loops are increased beyond a certain value, the active control effects will not further be improved remarkably.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline \(m_{p}\) & 0.032 kg \\ \(k_{p}\) & 140 \(N/m\) \\ \(c_{p}\) & 1.39 \(Ns/m\) \\ \(m_{c}\) & 0.050 kg \\ \(k_{c}\) & \(1\times 10^{9}\)\(N/m\) \\ \(c_{c}\) & \(1\times 10^{9}\)\(Ns/m\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The properties of the actuator.  
Figure 6: The locations and coordinates of the feedback loops.  
The decentralized systems with 2, 4, 5, 6, and 7 channel feedback loops are also examined to explore the impact of number of the feedback loops on the control results. The control gain is set as \(h=100\) for these different cases. The kinetic energy and radiated sound power of the ribbed plate in controlled condition for each case with different number of feedback loops are calculated and plotted in Figures 9 and 10. The active control result with two feedback loops (#5 and #6) is not very good, and there is even control spillover on some resonant frequencies. As the number of feedback loops increases, the active noise reduction may also slightly increases. However, too many feedback loops with tiny increment of noise reduction is meaningless for practical application. Hence, the number of the feedback loops should also be chosen appropriately.  
Figure 8: The radiated sound power of the ribbed plate before and after control using seven feedback loops with different control gain.  
Figure 7: The kinetic energy of the ribbed plate before and after control using seven feedback loops with different control gain.