## 5 Calibration of STMD Using Numerical Optimization
### Optimization Results  
An important observation from the optimization is that the optimized STMD reaches a clear efficiency limit around \(K=3\). For \(K>3\), the optimization yields disproportionately large values of \(K_{cor}\) and \(c_{cor}\) without achieving further improvement in the STMD efficiency.  
The optimized correction factors \(K_{cor}\) and \(c_{cor}\) are plotted in Figure 8a,b versus \(K\) from 1 to 3 and for various \(\mu\) from 0.4% to 3%. Within the range of values presented in these figures, both \(K_{cor}\) and \(c_{cor}\) increase with \(K\) in a similar manner for each mass ratio \(\mu\).  
Figure 8c shows the difference between the maximum DAF (10) values calculated for the optimized STMD with mass ratio \(\mu\) and the reference TMD with mass ratio \(K\mu\). This difference serves as a straightforward measure for assessing the results of optimization. The maximum value in Figure 8c is 0.148, obtained for \(K=3\) and \(\mu=0.4\%\). This value implies that the peak response of the STMD exceeds the target \(DAF_{max}=12.95\) by 1.14%. Similarly, when \(K=3\) and \(\mu=3\%\), the maximum difference is 0.125, indicating that the STMD's peak response is 2.6% above the targeted value of 4.82. Given that the maximum percentage error is under 3%, the optimization outcomes are considered acceptable.  
Table 1 lists the optimized values of \(K_{cor}\) and \(c_{cor}\) for selected \(K\) ranging from 1.5 to 3, and for three typical \(\mu\) of 0.5%, 1%, and 2%. It is noticeable that the values for \(K=3\) are exceptionally high compared to the rest of the results. When \(K\) is slightly reduced to 2.98, these values become significantly smaller. This is because, for \(K=3\), the optimized STMD is already at its performance limit.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Correction Factor \(K_{cor}\)**} & \multicolumn{3}{c}{**Correction Factor \(\boldsymbol{c_{cor}}\)**} \\ \cline{2-7} \multicolumn{1}{c}{\(K\)} & \multicolumn{1}{c}{\(\boldsymbol{\mu}\)**=0.5\%**} & \(\boldsymbol{\mu}\)**=1\%** & \(\boldsymbol{\mu}\)**=2\%** & \(\boldsymbol{\mu}\)**=0.5\%** & \(\boldsymbol{\mu}\)**=1\%** & \(\boldsymbol{\mu}\)**=2\%** \\ \hline
1.5 & 1.0585 & 1.0582 & 1.0582 & 1.0285 & 1.0283 & 1.0283 \\
2 & 1.3336 & 1.3331 & 1.3321 & 1.1546 & 1.1539 & 1.1530 \\
2.4 & 2.0131 & 2.0058 & 1.9934 & 1.4201 & 1.4177 & 1.4129 \\
2.6 & 2.9001 & 2.8751 & 2.8307 & 1.7085 & 1.7019 & 1.6891 \\
2.8 & 5.6752 & 5.5289 & 5.2808 & 2.3970 & 2.3679 & 2.3161 \\
2.9 & 11.715 & 10.991 & 9.8577 & 3.4490 & 3.3446 & 3.1709 \\
2.98 & 106.41 & 62.555 & 35.590 & 10.407 & 7.9897 & 6.0348 \\
3 & 1239.4 & 890.90 & 106.90 & 35.534 & 30.165 & 10.463 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimized parameters \(K_{cor}\) and \(c_{cor}\) for \(\mu=0.5\%\), 1%, and 2% and for different \(K\).  
Figure 8: Results of the numerical optimization for different mass ratios \(\mu\) ranging from 0.4% to 3% versus design parameter \(K\): (**a**) Correction factor \(K_{cor}\); (**b**) correction factor \(c_{cor}\); (**c**) resulting difference between maximum dynamic amplification factors for the optimized STMD with mass ratio \(\mu\) and the TMD with mass ratio \(K\mu\).