# Automatic Decentralized Vibration Control System Based on Collaborative Stand-Alone Smart Dampers
## 5 Results
### Numerical Results  
At the beginning, the proposed EMC strategy has been tested by means of numerical simulations. In this way, the performance of this control strategy has been verified.  
The starting point for this analysis is the calculation of the final control gains, as shown in Equation (12). The limiting gain between the maximum gain related to the actuator dynamics and saturation gains is the one related to the current saturation, which is assigned to the parameter \(g_{max,i,j}\). Then, the introduced damping and modal displacement weights are calculated according to Equations (10) and (11), considering a given vibration state. Finally, the multiplication between the two weights provide the final weight that modulates \(g_{max,i,j}\).  
The final step regards the tuning of \(\gamma\). As a first trial, \(\gamma\) has been set equal to 1. The saturation degree has been computed for each the three actuators as follows:  
\[S_{j}=\sum_{i=2}^{4}g_{i,j}\frac{\ddot{y}_{i,j}}{\omega_{i}}\frac{1}{F_{a,j, max}} \tag{29}\]  
where \(\dot{y}_{i,j}\) is the modulus of the acceleration related to the \(i\)-th mode measured by the \(j\)-th actuator and \(F_{a,j,max}\) is the maximum electromagnetic force that the \(j\)-th actuator can apply.  
The results showed that the saturation condition was not satisfied for actuator 2, since \(S_{2}>1\). This is due to the final weight \(w_{2,2}\) equal to 1. Since the gain \(g_{max,2,2}\) is already the maximum saturation gain to control only mode 2, if actuator 2 is required to control also mode 3 and 4, the saturation degree \(S_{2}\) overcomes the threshold of 100%. For this reason, the value of \(\gamma\) must be lower than 0.985. Then, the overall combination of gains has been tested considering the control enabled on each actuator. This procedure is necessary to assess the stability of the overall system, otherwise a further reduction of \(\gamma\) is requested.  
The results obtained on the controlled system are compared with the uncontrolled one, as shown in Table 3. In Figure 5 are depicted the FRFs in the sensors positions (Sensor 0, 1 and 2 as reported in Figure 4), both for the uncontrolled and the controlled case. Looking at the results, on the mode 2 (where most of the damping power is focused) the largest reduction is obtained, which is about 90% on the vibration amplitude. About modes 3 and 4, a reduction about 78% and 65% is achieved.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Mode 2** & **Mode 3** & **Mode 4** \\ \hline \(f_{num}\) [Hz] & 91.2 & 202.6 & 331.8 \\ \(f_{exp}\) [Hz] & 90.3 & 204.5 & 335.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between the natural frequencies of the considered modes (numerical and experimental).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mode** & **Sensor 0** & **Sensor 1** & **Sensor 2** \\ \hline \multirow{3}{*}{**Uncontrolled** [m s\({}^{-2}\)]} & **2** & 1.3050 & 0.0559 & 0.9734 \\ \cline{2-5}  & **3** & 0.2424 & 0.4100 & 0.1268 \\ \cline{2-5}  & **4** & 0.0264 & 0.0021 & 0.1251 \\ \hline \multirow{3}{*}{**Controlled** [m s\({}^{-2}\)]} & **2** & 0.1263 & 0.0067 & 0.1396 \\ \cline{2-5}  & **3** & 0.0505 & 0.0800 & 0.0298 \\ \cline{2-5}  & **4** & 0.0093 & 0.0008 & 0.0378 \\ \hline \multirow{3}{*}{**Reduction** [\%]} & **2** & 90.32 & 88.10 & 85.66 \\ \cline{2-5}  & **3** & 79.17 & 80.49 & 76.48 \\ \cline{2-5}  & **4** & 64.79 & 63.04 & 69.82 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Vibration comparison between uncontrolled and controlled case in numerical simulation.