## 4 Optimal Design of ECTMD for Damped Structure  
If its tuning ratio and damping ratio is off-tuned, the ECTMD will lose its control efficiency. Due to the eddy current damping being linearly proportional to the relative velocity between ECTMD and the primary structure, the optimization procedures of TMD with a linear viscous damper can be employed to design the optimal parameters of the ECTMD. The optimal tuning ratio and damping ratio of a tuned mass damper (TMD) system for an undamped structure are given by [37],  
\[r_{opt}=\frac{1}{1+\mu} \tag{1}\]  
\[\zeta_{opt}=\sqrt{\frac{3\mu}{8(1+\mu)}} \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Result** & **Frequency** & **Damping Ratio** & **Modal Mass** \\ \hline FE computation & 3.26 Hz & - & 0.728 kg \\ Experimental result & 3.20 Hz & 2.4\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Modal parameters of the pipeline structure. Figure 5: Submerged pipeline model and its finite element model: (**a**) polyvinyl chloride (PVC) pipeline in the seawater container; (**b**) finite element (FE) model; (**c**) first vibration mode shape. Figure 6: Vertical response and its Fast Fourier transform (FFT) amplitude of the submerged pipeline in seawater environment: (**a**) free vibration response; (**b**) FFT amplitude. where \(r_{opt}\) and \(\zeta_{opt}\) are the optimal frequency tuning ratio and the optimal damping ratio of the TMD respectively, \(\mu=m_{tmd}/m_{s}\) is the mass ratio of the TMD, \(m_{tmd}\) and \(m_{s}\) are the masses of the primary structure and the TMD, respectively. The damping ratio of the primary structure before a TMD is attached can be calculated by \(\zeta_{s}=c_{s}/(2m_{s}(k_{s}/m_{s})^{1/2})\). For a primary structure with a low damping ratio, such as 0.5% or lower, Equations (1) and (2) can be used to design the optimal parameters of TMD. When the damping ratio of the controlled structure is higher than 2%, such as the tested submerged pipeline in the Section 3, the optimal tuning ratio and damping ratio calculated by Equations (1) and (2) may result in detuning to the ECTMD system, and reduce its optimal control ability [38; 39]. An optimum design of a TMD for a damped controlled structure is proposed by [PERSON] [40] using the frequency focus method. This method for finding the optimal parameter of a TMD is relatively complicated. [PERSON]'s work focused on highly damped structures, but the damping ratios of most civil structures are under 5%. Therefore, the optimal parameters of a TMD system for damped structures based on \(H_{\infty}\) optimization method are studied in this paper. The equation of motion of the damped structure with a linear TMD control, as shown in Figure 7, can be expressed as Equation (3),  
\[\left\{ \begin{array}{c}m_{s}\ddot{x}_{s}+c_{s}\dot{x}_{s}+k_{s}x_{s}+c_{tmd} (\dot{x}_{s}-\dot{x}_{tmd})+k_{tmd}(x_{s}-x_{tmd})=f_{0}\sin(\omega t)\\ m_{tmd}\ddot{x}_{tmd}+c_{tmd}(\dot{x}_{tmd}-\dot{x}_{s})+k_{tmd}(x_{tmd}-x_{s}) =0\end{array} \right. \tag{3}\]  
where \(m_{s}\), \(c_{s}\), \(k_{s}\), \(x_{s}\) are the mass, damping coefficient, stiffness, and displacement of the controlled structure, and \(m_{tmd}\), \(c_{tmd}\), \(k_{tmd}\), \(x_{tmd}\) are the mass, damping coefficient, stiffness, and displacement of theTMD. \(f_{0}\)sin(\(\omega t\)) is the external excitation applied on the controlled structure. The dynamic magnification factor of the primary structure is defined as \(DMF(\omega)=x_{s0}k_{s}/f_{0}\), and it can be expressed as Equations (4),  
\[DMF(\omega)=\frac{x_{s0}}{f_{0}/k_{s}}=\frac{k_{s}}{\left|(k_{s}+i\omega c_{s }-m_{s}\omega^{2})+k_{tmd}+i\omega c_{tmd}-\frac{(k_{tmd}+i\omega c_{tmd})^{2 }}{k_{tmd}+i\omega c_{tmd}-m_{tmd}\omega^{2}}\right|} \tag{4}\]  
where \(i\) is complex unit \((-1)^{1/2}\) and \(x_{s0}\) is the amplitude of steady-state response. The modal mass ratio, tuning ratio, and damping ratio of TMD are defined as \(\mu=m_{tmd}/m_{s}\), \(r=(k_{tmd}/m_{tmd})^{1/2}/(k_{s}/m_{s})^{1/2}\), and \(\zeta=c_{tmd}/(2m_{tmd}(k_{tmd}/m_{tmd})^{1/2})\), respectively.