## 4 Simulation results
### Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Damper Type** & \begin{tabular}{c} **Cable** \\ **Model** \\ \end{tabular} & _a/L (\%)_ & \(\zeta_{fit}-\zeta_{cable}\) (\%)_ & \(\zeta_{mm}\) (\%) & \(\gamma_{2}\) _a/L_ & \(\eta\) (\%) \\ \hline \multirow{2}{*}{\begin{tabular}{c} passive, tuned to \\ mode 3 \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 0.507\% & & & 100\% (\({}^{*}\)) \\ \cline{3-6}  & & & 0.354\% & & & 69.6\% \\ \hline \multirow{2}{*}{\begin{tabular}{c} passive, tuned to \\ modes 1–4 \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 0.354\% & & & 69.4\% \\ \cline{3-6}  & & & 0.402\% & & & 79.0\% \\ \cline{3-6}  & & & 0.497\% & & & 98.0\% \\ \hline \multirow{3}{*}{semi-active} & \multirow{2}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 0.949\% & & & & 188.4\% \\ \cline{3-6}  & & & 0.557\% & & & 110.0\% (\({}^{*}\)) \\ \hline \multirow{2}{*}{\begin{tabular}{c} passive, tuned to \\ mode 4 \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 0.855\% & & & 100\% (\({}^{*}\)) \\ \cline{3-6}  & & & 0.715\% & & & 83.2\% \\ \hline \multirow{2}{*}{\begin{tabular}{c} passive, tuned to \\ modes 1–4 \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 0.675\% & 0.020\% (\({}^{*}\)) & 0.84\% & & 78.4\% \\ \cline{3-6}  & & & 0.850\% & & & 99.4\% \\ \hline \multirow{2}{*}{semi-active} & \multirow{2}{*}{
\begin{tabular}{c} taut string \\ FR \& PSC \\ \end{tabular} } & 1.647\% & & & & 194.9\% \\ \cline{3-6}  & & & 1.290\% & & & 152.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for mode 4 excited. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  
The observation of Tables 1-4 reveals the following main findings:  
* The reduced cable motion at damper position due to cable flexural rigidity and fixed supported ends has strong impact on the cable damping ratio due to passive dampers located at _a/L_ \(\leq\) 2%. This fact is explained by the associated reduced transverse damper motion, see Figure 2b, whereby less energy per cycle can be dissipated. * The damper efficiencies obtained from the simulation case of a cable with flexural rigidity and fixed supported ends, mode 1 excited, typical damper positions around 2% to 2.5% and passive damper being optimized to modes 1-4 match well with the reported damper efficiencies [23; 24; 25; 26; 27; 28; 29; 30; 31]. The according results for mode 2 are greater because a damper being optimized to modes 1-4 generates optimum damping in mode 2 (14), see [19]. In this perspective the according simulation results for modes 3 and 4 of the cable model with bending stiffness and fixed support conditions (FR and FSC) seem to be unrealistically high. This irrational result is interpreted by the fact that the cable model FR and FSC is validated for the modeshape of mode 1 of the test steel wire strand depicted in Figure 2 but not for higher modes. * For some simulation cases, the damping of the cable model with flexural rigidity and fixed supported ends is greater than for the taut string model. This unexpected result is observed for some simulations of modes 3 and 4 and mainly for the passive damper being optimized to modes 1 to 4. An explanation can be that the reduced cable motion at damper position reduces the damper velocity as well whereby the viscous coefficient being optimized for modes 1 to 4 (12) almost generates the optimum viscous force for modes 3 and 4. * Mode 1: The damping is approximately 2.5... 2.6 (taut string) and approximately 2.8... 3.2 (FR and FSC) times greater than the damping due to the passive damper being optimized to modes 1 to 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
* Mode 2: The damping is approximately 2.0... 2.6 (taut string) and approximately 1.6... 2.1 (FR and FSC) times greater than the damping due to the passive damper being optimized to modes 1 to 4.