## 5 Effectiveness of ECTMD to Control a Submerged Pipeline  
In general, the mass ratio of a TMD cannot be too large. Therefore, the mass of the designed ECTMD added on the experimental pipeline structure is 32 g, which means the mass ratio \(\mu\) is 4.4%. Considering the mass ratio \(\mu\) = 4.4% and the primary damping ratio of the pipeline structure \(\zeta_{s}\) = 2.4%, the optimal tuning ratio and damping ratio of the ECTMD are obtained from Table 3 to be 0.9508 and 12.63%, respectively. For the implementation of the ECTMD, two tiny cantilever beams are used to provide stiffness for the ECTMD and the eddy current damper are utilized to provide the additional damping for the ECTMD. The natural frequency and damping ratio of the ECTMD can be adjusted  
Figure 8: Optimal frequency tuning ratio \(r_{opt}\) and damping \(\zeta_{opt}\) of a TMD for damped structure: (a) optimal frequency tuning ratio; (b) optimal damping ratio. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mass Ratio** & \multicolumn{2}{c}{\(\zeta_{s}\) = 1\%} & \multicolumn{2}{c}{\(\zeta_{s}\) = 2\%} & \multicolumn{2}{c}{\(\zeta_{s}\) = 3\%} \\ \hline \(\mu\) & \(r_{opt}\) & \(\zeta_{opt}\) & \(r_{opt}\) & \(\zeta_{opt}\) & \(r_{opt}\) & \(\zeta_{opt}\) \\ \hline \(\mu\) = 0.01 & 0.9886 & 0.0625 & 0.9869 & 0.0643 & 0.9851 & 0.0662 \\ \(\mu\) = 0.02 & 0.9783 & 0.0874 & 0.9761 & 0.0886 & 0.9738 & 0.0883 \\ \(\mu\) = 0.03 & 0.9684 & 0.1064 & 0.9658 & 0.1084 & 0.9630 & 0.1100 \\ \(\mu\) = 0.04 & 0.9588 & 0.1213 & 0.9558 & 0.1233 & 0.9526 & 0.1268 \\ \(\mu\) = 0.05 & 0.9493 & 0.1355 & 0.9461 & 0.1371 & 0.9426 & 0.1413 \\ \(\mu\) = 0.06 & 0.9401 & 0.1473 & 0.9367 & 0.1488 & 0.9330 & 0.1518 \\ \(\mu\) = 0.07 & 0.9311 & 0.1584 & 0.9274 & 0.1604 & 0.9237 & 0.1614 \\ \(\mu\) = 0.08 & 0.9222 & 0.1691 & 0.9184 & 0.1702 & 0.9145 & 0.1710 \\ \(\mu\) = 0.09 & 0.9135 & 0.1784 & 0.9095 & 0.1805 & 0.9053 & 0.1817 \\ \(\mu\) = 0.1 & 0.9051 & 0.1869 & 0.9010 & 0.1877 & 0.8967 & 0.1896 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal parameters of a TMD for the damped structure. Figure 7: Mechanical model of damped structure with a tuned mass damper (TMD). through changing the length of the cantilever beams and the gap between the aluminum plates and the magnets. The configuration of the ECTMD and its eddy current damper device for the submerged pipeline are shown in Figure 9. The eddy current damper device consists of two aluminum plates with a thickness of 3 mm and permanent magnets. The ECTMD is installed in the mid-span of the pipeline. To obtain the fundamental natural frequency and damping ratio of the ECTMD, free vibration tests of ECTMD are conducted in the seawater environment. When the length of the cantilever beam is set to 13.5 cm and the gap between magnets and aluminum plates is set to 4.2 mm, the frequency and the damping ratio is quite close to the optimal values. Figure 10 shows the vertical free vibration history of the ECTMD device with and without magnets, and the frequency responses are also shown in Figure 10. It can be seen that the natural frequency and the damping ratio of the ECTMD without magnets are respectively, 3.07 Hz and 6.1%, in the seawater environment. However, when the permanent magnets are added and eddy current damping is activated, the damping ratio of the ECTMD is sharply increased from 6.1% to 13.9%, which helps the ECTMD to achieve the optimal damping. The parameters of the ECTMD in the experiment are summarized in Table 4.  
Figure 10: Vertical free vibration history and its response in frequency domain of the ECTMD with and without eddy current damping in seawater: (**a**) free vibration; (**b**) FFT amplitude. Figure 9: Implementation of the eddy current tuned mass damper (ECTMD) on the pipeline model: (**a**) configuration of ECTMD; (**b**) eddy current damper device. Figure 11 shows the experimental setup for the testing of the submerged pipeline model. A seawater container is employed to simulate the seawater environment.