## 5 Results
In Equation (11), the estimate is shown using a peak factor \(G=3\) (with an estimate provided in this magnitude by [48]) alongside the mean and standard deviation \(\sigma\) of the time series,  
\[Max=Mean(series)+G\;\sigma(series). \tag{11}\]  
A dedicated discussion on the exact choice of the peak factor is out of the scope of the current study, as it serves here as an amplification factor. It is taken as the same throughout the statistical analysis, with its magnitude deemed representative. For special design cases, this term needs a more in-depth investigation. Table 3 provides a summary. Alongside the absolute values, the relative error is also displayed, being evaluated in pairs, between the _TMD-baseline_, _TSD-baseline_ and _TSD-TMD_. Our second scenario represents narrow-banded loading in the cross-wind direction. The dominant force arises in the smooth flow as a result of vortex shedding. Such an excitation case is more typical for the setting that is aimed to be mitigated by AMDs. Passive devices typically work well at particular frequencies. In Figure 11, we include the times series for displacements and accelerations. This is amended by the statistical evaluation in Table 4. The total time used for the evaluation is 200 s, with the initialization being 25 s. This choice resulted from capturing at least \(\approx 25\) periods of the vortex shedding (with \(T=5\) s for the shedding frequency \(f_{sh}=0.2\) Hz matching the weak bending). We can clearly see in the time series that, for the baseline case, around 150 s are necessary to reach the stationary state of damped harmonic oscillations. To put this in context, these very specific flow conditions for a tall structure would need to be stable for 2.5 min to arrive at this state, where it would imply \(\pm 3\) m oscillations at a 180 m height. Both AMDs would be able to reduce this to well under 1 m. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multicolumn{8}{c}{**Along-Wind**} & \multicolumn{8}{c}{**Cross-Wind**} \\  & \multicolumn{2}{c}{\(\mathbf{Disp_{x}}\) [m]} & \multicolumn{2}{c}{\(\mathbf{Acc_{x}}\) [m/s\({}^{2}\)]} & \multicolumn{2}{c}{\(\mathbf{Disp_{y}}\) [m]} & \multicolumn{2}{c}{\(\mathbf{Acc_{y}}\) [m/s\({}^{2}\)]} \\ \cline{2-13}  & _Mean_ & _RMS_ & _Max_ & _Mean_ & _RMS_ & _Max_ & _Mean_ & _RMS_ & _Max_ & _Mean_ & _RMS_ & _Max_ \\ \hline
**Baseline** & 0.40 & 0.41 & 0.56 & 0.00 & 0.24 & 0.72 & 0.00 & 1.62 & 4.88 & 0.00 & 2.58 & 7.75 \\
**With TMD** & 0.39 & 0.39 & 0.45 & 0.00 & 0.10 & 0.30 & 0.00 & 0.13 & 0.39 & 0.00 & 0.21 & 0.62 \\ _Diff. to baseline_ & \([\%]\) & \(-2.5\) & \(-4.9\) & \(-19.6\) & _0.0_ & \(-58.3\) & \(-58.3\) & _0.0_ & \(-92.0\) & \(-92.0\) & _0.0_ & \(-91.9\) & \(-92.9\) \\
**With TSD** & 0.39 & 0.39 & 0.50 & 0.00 & 0.18 & 0.53 & 0.00 & 0.26 & 0.78 & 0.00 & 0.41 & 1.22 \\ _Diff. to baseline_ & \([\%]\) & \(-2.5\) & \(-4.9\) & \(-10.7\) & _0.0_ & \(-25.0\) & \(-26.4\) & _0.0_ & \(-84.0\) & \(-84.0\) & _0.0_ & \(-84.1\) & \(-84.3\) \\ _Diff. to TMD_ & \([\%]\) & _0.0_ & _0.0_ & _11.1_ & _0.0_ & _80.0_ & _76.7_ & _0.0_ & _100.0_ & _100.0_ & _0.0_ & **95.2** & **96.8** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of the kinematics of the top point in smooth flow.