## 4 Simulation results
### Results  
The damping ratio due to the damper (including the numerical damping) and the according damper efficiency (24, without numerical damping) are given in Tables 1-4 for modes 1 to 4 as function of:  
* Damper types:
* Passive, tuned to mode X (1, 2, 3, 4): linear viscous damper being optimized to the excited mode (19); this is a theoretical benchmark because transverse cable dampers always need to provide the specified damping in several targeted modes; this computation is required to determine the numerical damping. * Passive, m1-m4: linear viscous damper being optimized to the targeted modes 1-4 (12). * Semi-active: clipped VONS including control force constraints due to \(c_{\min}\) and \(c_{\max}\) (21). * Cable model:
* Taut string: cable without flexural rigidity and simply supported ends. * FR and FSC: cable with flexural rigidity and fixed support end conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Damper Type** & \begin{tabular}{c} **Cable** \\ **Model** \\ \end{tabular} _a/L (\%)_ & \(\zeta_{fit}-\zeta_{cable}\) **(\%)** & \(\zeta_{mmn}\) **(\%)** & \(\eta\) **(\%)** \\ \hline \multirow{3}{*}{\begin{tabular}{c} passive, tuned to \\ mode 1 \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & \multirow{3}{*}{\(2.5\%\)} & \multirow{3}{*}{\(1.194\%\)} & \multirow{3}{*}{\(0.034\%\) (\(\gamma\))} & \(78.6\%\) \\ \cline{3-3}  & & & \(0.775\%\) & & & \(59.3\%\) \\ \hline \multirow{3}{*}{semi-active} & \multirow{3}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & \multirow{3}{*}{\(2.363\%\)} & \multirow{3}{*}{\(188.9\%\)} & \multirow{3}{*}{\(100\%\) (\(\gamma\))} \\ \cline{3-3}  & & & \(2.395\%\) & & & \(188.9\%\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} passive, tuned to \\ mode 1 \\ \end{tabular} } & \multirow{3}{*}{\(\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & \multirow{3}{*}{\(1.549\%\)} & \multirow{3}{*}{\(0.00\%\) (\(\gamma\))} & \multirow{3}{*}{\(100\%\) (\(\gamma\))} \\ \cline{3-3}  & & & \(1.471\%\) & & & \(94.8\%\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} passive, tuned to \\ modes 1-4 \\ \end{tabular} } & \multirow{3}{*}{\(3\%\)} & \multirow{3}{*}{\(1.224\%\)} & \multirow{3}{*}{\(0.049\%\) (\(\gamma\))} & \multirow{3}{*}{\(1.5\%\)} & \multirow{3}{*}{\(78.3\%\)} \\ \cline{3-3}  & & & \(0.798\%\) & & & \(78.3\%\) \\ \hline \multirow{3}{*}{semi-active} & \multirow{3}{*}{\
\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & \multirow{3}{*}{\(3\%\)} & \multirow{3}{*}{\(0.978\%\)} & \multirow{3}{*}{\(1.5\%\)} & \multirow{3}{*}{\(10.9\%\)} \\ \cline{3-3}  & & & \(0.978\%\) & & & \(61.9\%\) \\ \hline \multirow{3}{*}{semi-active} & \multirow{3}{*}{
\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & \multirow{3}{*}{\(3.169\%\)} & \multirow{3}{*}{\(0.049\%\) (\(\gamma\))} & \multirow{3}{*}{\(1.5\%\)} & \multirow{3}{*}{\(197.7\%\)} \\ \cline{3-3}  & & & \(3.169\%\) & & & \(208.0\%\) \\ \cline{3-3}  & & & \(3.014\%\) & & & \(197.7\%\) \\ \hline \hline \end{tabular} (*) numerical damping determined from \(\zeta_{fit}\) due to passive viscous damping optimized to excited mode to obtain \(\eta\) = 100\%.  
\end{table} 
Table 1: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
\begin{tabular}{c} taut string \\ FR \& PSC