## 4 Numerical Studies on the Effectiveness of SA-TLGDs
### MDOF-Shear Frame Structure with Multiple SA-TLGDs Attached
damping absorbers with respect to the expected changes of the fundamental natural frequency during the operating life. Based on the determined optimal tuning parameters the dimensions of the two SA-TLGDs must be chosen with respect to the available space at the 3 rd floor of the MDOF-shear frame structure. The optimal damping ratios \(\xi_{A,iopt}\) are achieved by properly adjusting the diameters of the controllable orifics in the fluid stream. The calculated optimal parameters and chosen dimensions of the installed two SA-TLGDs are listed in Table 3.  
Assuming a time harmonic excitation force \(F_{3}(t)\) with the force amplitude \(F_{03}=1000\) N acting on the 3 rd floor of the MDOF-shear frame structure with the two optimally tuned SA-TLGDs attached (Figure 8) results in the frequency response function (FRF) for the horizontal floor displacement \(w_{3}(t)\) illustrated in Figure 9. The two installed, activated, and optimally tuned SA-TLGDs reduce the maximum floor displacement from 0.143 m to just 0.017 m around the most critical fundamental mode of the MDOF-shear frame structure, i.e., the percentage of vibration reduction is almost 90%. In the case of the activated SA-TLGDs the FRF of the floor displacement indicates three peaks in the vicinity of the fundamental mode and obviously, these peaks arise due to the determined different optimal tuning parameters for both attached SA-TLGDs. Because the two SA-TLGDs are optimally tuned to the fundamental frequency of the MDOF-shear frame structure \(f_{51}=2.38\) Hz, the higher modes \(f_{52}=7.44\) Hz and \(f_{53}=12.29\) Hz are not affected by the installed vibration absorbers. The modal stiffness parameter of the fundamental mode is defined by \(K_{1}=M_{1}\;\omega_{51}^{2}=4.09\;\text{MN/m}\), i.e., the static displacement of the 3 rd floor results in \(w_{3,st}=F_{03}/K_{1}=0.0029\) m. Hence, the gained reduction in the maximum floor displacement \(w_{3}\) corresponds to an effective structural damping ratio of \(\zeta_{51,eff}=8.6\%\) and this equates to an increase in the assumed light structural damping for the fundamental mode (\(\zeta_{51}=1\%\)) of almost nine times. To investigate the detrimental influence of an expected frequency shift of the MDOF-shear frame fundamental frequency \(f_{51}=2.38\) Hz during the operating life on the vibration damping effectiveness gained through the two optimally tuned SA-TLGDs the following assumptions are made: \(f_{51}^{+5\%}=2.50\) Hz and \(f_{51}^{-5\%}=2.26\) Hz. The FRF of the detuned force excited MDOF-shear frame structure with the two sub-optimal tuned SA-TLGDs in parallel connection is illustrated in Figure 10. Note that the FRF of the activated and optimally tuned SA-TLGD is also illustrated in Figure 10. It is seen that due to the assumed frequency shift of the fundamental mode of the MDOF-shear frame structure the maximum floor displacement \(w_{3}\) increases from the minimum value 0.018 m for the two optimally tuned SA-TLGDs, to around 0.023 m. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter of SA-TLGD** & **Variable** & **TLGD 1** & **TLGD 2** & **Unit** \\ \hline Optimal natural frequency & \(f_{A,iopt}\) & 2.18 & 2.49 & Hz \\ Optimal damping ratio & \(\zeta_{A,iopt}\) & 6.08 & 6.72 & \% \\ Total fluid mass & \(m_{f,total}\) & 60 & kg \\ Total mass ratio & \(\mu_{total}=m_{f,total}/M_{1}\) & 4.0 & \% \\ Mass ratio of each SA-TLGD & \(\mu_{i}=m_{p}/M_{1}\) & 2.0 & \% \\ Fluid mass of each SA-TLGD & \(m_{p}\) & 30 & kg \\ Fluid density & \(\rho_{f}\) & 1000 & kg/m\({}^{3}\) \\ Fluid volume & \(V_{f}\) & 0.031 & m\({}^{3}\) \\ Fluid horizontal length & \(L_{f}\) & 0.624 & m \\ Polytropic coefficient & \(n\) & 1.2 & - \\ Initial atmospheric pressure & \(p_{0}\) & \(10^{5}\) & Pa \\ Tube diameter & \(d\) & 0.25 & m \\ Tube cross-sectional area & \(A_{T}\) & 0.049 & m\({}^{2}\) \\ Gas spring length & \(L_{g}\) & 2.05 & 1.57 & m \\ Initial gas volume & \(V_{0}\) & 0.1007 & 0.0772 & m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}