## 4 Numerical Studies on the Effectiveness of SA-TLGDs
### SDOF-Wind Turbine with Single SA-TLGD Attached
The ratio of fluid mass to the modal mass of the SDOF-main system is chosen with \(\mu=1\%\). Inserting this mass ratio into Equation (22) and applying the correction formulas given in Equation (24) to account for the available light structural damping of the main system (\(\zeta_{S}=1.4\%\)) renders the following optimal parameters for the SA-TLGD: \(\omega_{A,opt}=3.39\) rad/s (\(f_{A,opt}=\omega_{A,opt}/2\pi=0.54\) Hz) and \(\zeta_{A,opt}=6.28\%\). To realize the optimal natural frequency \(f_{A,opt}\) in the practical application of the considered wind turbine, the dimensions of the SA-TLGD must be chosen appropriately. The optimal damping ratio \(\zeta_{A,opt}\) is achieved by properly adjusting the diameter of the controllable orifices in the fluid stream. Table 2 lists both the calculated optimal parameters and the chosen dimensions of the SA-TLGD attached to the SDOF-wind turbine. Assuming the time harmonic horizontal excitation force with \(F(t)=F_{0}\ e^{ivt}\), where \(\
u\) is the excitation frequency, inserting the already chosen parameters for the considered wind turbine (Table 1) and for the optimally tuned SA-TLGD (Table 2) into Equation (17), yield the linearized matrix equations of the coupled system with now known mass, stiffness, and damping matrix elements. The steady state solutions for the tower head and fluid mass displacements \(w(\
u)\) and \(u(\
u)\) are determined in the frequency domain after inserting the exponential functions \(w(\
u)=w_{0}\ e^{ivt}\) and \(u(\
u)=u_{0}\ e^{ivt}\) as well as their derivatives into the matrix equation of the coupled system [7]. Figure 5 shows the gained results for the frequency response functions (FRF) of the SDOF-wind turbine with and without activated optimally tuned SA-TLGD. The force amplitude is chosen with \(F_{0}=20,000\) N and the equivalent stiffness parameter of the wind turbine tower is defined by \(k_{S}=m_{S}\ \omega_{S}^{2}=4.09\) MN/m, i.e., the static displacement of the tower head is given by \(w_{st}=F_{0}/k_{S}=0.0049\) m. It is seen that the activated optimally tuned SA-TLGD reduces the maximum horizontal displacement amplitude of the SDOF-wind turbine over the total frequency range of interest more than 70% (from 0.17 m without any vibration absorber to 0.05 m with activated SA-TLGD). The gained reduction in the maximum dynamic wind turbine head displacement corresponds to an effective structural damping ratio of \(\zeta_{S,eff}=4.9\%\) and this equates to an increase in the light structural damping of the wind turbine (\(\zeta_{S}=1.4\%\)) of more than three times. Figure 6 illustrates the steady state response of the fluid mass displacement amplitude \(u\) for the case of the activated SA-TLGD. It is indicated that the maximum fluid mass amplitude results in \(u_{max}=0.37\) m over the total frequency range of interest. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter of SA-TLGD** & **Variable** & **Value** & **Unit** \\ \hline Optimal natural frequency & \(f_{A,opt}\) & 0.54 & Hz \\ Optimal damping ratio & \(\zeta_{A,opt}\) & 6.28 & \% \\ Mass ratio & \(\mu=m_{P}/m_{S}\) & 1.0 & \% \\ Fluid mass & \(m_{P}\) & 3454 & kg \\ Fluid density & \(\rho_{f}\) & 1000 & kg/m\({}^{3}\) \\ Fluid volume & \(V_{f}\) & 3.454 & m\({}^{3}\) \\ Fluid horizontal length & \(L_{f}\) & 4.40 & m \\ Polytropic coefficient & \(n\) & 1.2 & - \\ Initial atmospheric pressure & \(p_{0}\) & \(10^{5}\) & Pa \\ Tube diameter & \(d\) & 1.0 & m \\ Tube cross-sectional area & \(A_{T}\) & 0.79 & m\({}^{2}\) \\ Gas spring length & \(L_{g}\) & 4.75 & m \\ Initial gas volume & \(V_{0}\) & 3.73 & m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and dimensions of the optimized SA-TLGD attached to the SDOF-wind turbine. In the next step it is assumed that the natural frequency of the wind turbine \(f_{S}=0.55\) Hz varies \(\pm 5\%\), i.e., \(f_{S}^{+5\%}=0.58\) Hz and \(f_{S}^{-5\%}=0.52\) Hz, and hence detuning of the SA-TLGD occurs, which obviously influences the achievable damping effect negatively.