## 3 Overview on Antiresonance Assignment through the Active or Semi-Active Tuned Mass Dampers
### The Active Tuned Mass Damper
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tuning Method Number** & **1** & **2** & **3** & **4** & **5** \\ \hline Reference & [10,23] & [24] & [25,26] & [27] & [24] \\ \hline Tuning rule & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\) & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\sqrt{\frac{2+\mu}{2}}\) & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\sqrt{\frac{2+\mu}{2}}\) & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\) & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{1+\mu}\) & \(\frac{\omega_{f}}{\omega_{0}}=\frac{1}{\sqrt{1+\mu}}\) \\  & \(\xi_{d}=\sqrt{\frac{2+\mu}{2}}\) & \(\xi_{d}=\sqrt{\frac{2+\mu}{2}}\) & \(\xi_{d}=\sqrt{\frac{2+\mu}{2}}\) & \(\xi_{d}=\sqrt{\frac{2+\mu}{2}}\) & \(\xi_{d}=\sqrt{\frac{2+\mu}{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tuning methods for SDOF-TMDs. Figure 2: Sketch of the original system with the ATMD in the numerical example. The results of the simulation of the system forced by a harmonic excitation with amplitude \(F_{0}=10^{-4}\) N at frequency \(\omega_{f}\) and controlled by means of the proposed ATMDs leads to the RMS values reported in Table 2. The RMS value of the control force is reported in percentage with respect to the minimum RMS control force exerted and reveals that the control effort is slightly increased for the 5 th tuning method, which is the one that reduces the most the steady-state vibration of both system masses. Besides evaluating these parameters, stability and robustness margins should be analysed in the case of active controlled systems; values of these parameters that are too low might make infeasible the experimental application of ATMDs in some cases (e.g., low-bandwidth actuators, noisy sensors). The multi-loop robustness margins for the closed-loop system for different ATMDs are summarized in Table 2. It should be noted that tuning 5 [24] leads to the smallest oscillation of the primary system (\(\mathbf{q_{RMS}}\)) with the smallest oscillation of the TMD mass (\(\mathbf{q_{a,RMS}}\)) while ensuring the highest stability margin of the controlled system (the disk-based gain, the values of which should be as high as possible since they quantify the stability of a closed-loop system against gain or phase variations), hence being more robust. On the other hand, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Tuning Method** & \multicolumn{2}{c}{**No Absorber**} & \multicolumn{1}{c}{**1**} & \multicolumn{1}{c}{**2**} & \multicolumn{1}{c}{**3**} & \multicolumn{1}{c}{**4**} & \multicolumn{1}{c}{**5**} \\ \hline qRMS [mm] & 0.72 & 0.39 & 0.28 & 0.60 & 0.45 & 0.24 \\ \hline q\({}_{\text{a,RMS}}\) [mm] & - & 1.49 & 1.13 & 1.49 & 1.49 & 0.76 \\ \hline u RMS [\%] & - & 100.08 & 102.65 & 100 & 100.08 & 104.45 \\ \hline Disk-based gain & \multirow{2}{*}{-} & \multirow{2}{*}{\(\pm 0.79\)} & \multirow{2}{*}{\(\pm 0.72\)} & \multirow{2}{*}{\(\pm 1.16\)} & \multirow{2}{*}{\(\pm 0.89\)} & \multirow{2}{*}{\(\pm 2.49\)} \\ margin [dB] & & & & & & \\ \hline Disk-based phase & \multirow{2}{*}{-} & \multirow{2}{*}{\(\pm 5.2\)} & \multirow{2}{*}{\(\pm 4.7\)} & \multirow{2}{*}{\(\pm 7.6\)} & \multirow{2}{*}{\(\pm 5.9\)} & \multirow{2}{*}{\(\pm 5.2\)} \\ margin [\({}^{\circ}\)] & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the harmonic response of the primary system controlled through different ATMDs and robustness of the closed-loop system. Figure 3: Amplitude of the response of the primary system: comparison of different tunings of the ATMD (normalized by \(k\)). it requires the greatest control effort among the compared techniques, although the increase is smaller than the 5% compared to the less demanding one.