## 3 Numerical Example 1
### Impact of Primary System Mistuning  
The analysis also covered the impact by the changing mass of the equivalent structure \(\overline{m}_{41,1}\), on the value of the equivalent structure FRF \(\left|\overline{H}_{41,41}\right|\) for a structure with attached MTMDs with optimal parameters, predetermined based on the \(J1\) and \(J2\) objective functions. A 10% reduction and increase in the equivalent structure mass was assumed, which resulted \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**TMD Number** & \(J1\) **Value** & \(J1\) **Change** & \(J2\) **Value** & \(J2\) **Change** \\ \hline  & **[m/(kN*0.5)]** & **[\%]** & **[m/kN]** & **[\%]** \\ \hline SDOF & \(2.4910\times 10^{-2}\) & & \(2.5005\times 10^{-1}\) & \\
1 TMD & \(1.5515\times 10^{-2}\) & \(0.00\%\) & \(7.4579\times 10^{-2}\) & \(0.00\%\) \\
2 TMDs & \(1.5203\times 10^{-2}\) & \(-2.01\%\) & \(6.8248\times 10^{-2}\) & \(-8.49\%\) \\
4 TMDs & \(1.5001\times 10^{-2}\) & \(-3.31\%\) & \(6.4091\times 10^{-2}\) & \(-14.06\%\) \\
8 TMDs & \(1.4875\times 10^{-2}\) & \(-4.12\%\) & \(6.1620\times 10^{-2}\) & \(-17.38\%\) \\
20 TMDs & \(1.4783\times 10^{-2}\) & \(-4.72\%\) & \(6.0202\times 10^{-2}\) & \(-19.28\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The value of the \(J1\) and \(J2\) functions after completed optimization with the SA method. Figure 5: (**a**) Optimal tuning ratios \(\beta_{k}\); (**b**) optimal damping ratios \(\zeta_{k}{}^{t}\) of the individual TMD for \(\mu=0.05\) and \(\zeta_{1}=0\) for SA and GA method. in the change of the natural circular frequencies to a value of \(\omega_{1}{}^{-10\%}=1.05409\) rad/s and \(\omega_{1}{}^{+10\%}=0.95346\) rad/s, respectively. Figure 7a shows a graph of the FRF module for an equivalent system \(\left|\overline{H}_{41,41}\right|\), for a mass reduction of 10% and a different number of TMDs, and with applied J1 objective function, while Figure 7b shows the same for the J2 objective function. Figure 8a,b shows analogous graphs but for a mass increase of 10% (designation +10% and -10% means an increase and reduction of the \(\overline{m}_{41,1}\) equivalent mass, respectively). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**MTMD**} & \multirow{2}{*}{**No.**} & \(\mathbf{m_{k}}^{t}\) & \(\mathbf{k_{k}}^{t}\) & \(\mathbf{\zeta_{k}}^{t}\) & \(\mathbf{\beta_{k}}\) \\ \cline{3-6}  & & [kg] & [kN/m] & [–] & [–] \\ \hline
1 TMD & 1 & 2000 & 1.905820 & 0.089169 & 0.976171 \\ \hline \multirow{2}{*}{2 TMDs} & 1 & 1000 & 0.878697 & 0.056278 & 0.937388 \\  & 2 & 1000 & 1.046257 & 0.062921 & 1.022867 \\ \hline \multirow{4}{*}{4 TMDs} & 1 & 500 & 0.413107 & 0.036608 & 0.908964 \\  & 2 & 500 & 0.456409 & 0.038155 & 0.955415 \\ \cline{1-1} \cline{3-6}  & 3 & 500 & 0.503223 & 0.039621 & 1.003218 \\ \cline{1-1} \cline{3-6}  & 4 & 500 & 0.560626 & 0.041295 & 1.058892 \\ \hline \multirow{4}{*}{8 TMDs} & 1 & 250 & 0.198173 & 0.023339 & 0.890332 \\  & 2 & 250 & 0.210676 & 0.024023 & 0.917988 \\ \cline{1-1} \cline{3-6}  & 3 & 250 & 0.222606 & 0.025988 & 0.943624 \\ \cline{1-1} \cline{3-6}  & 4 & 250 & 0.233932 & 0.028452 & 0.967329 \\ \cline{1-1} \cline{3-6}  & 5 & 250 & 0.245215 & 0.028899 & 0.909383 \\ \cline{1-1} \cline{3-6}  & 6 & 250 & 0.258339 & 0.027081 & 1.016542 \\ \cline{1-1} \cline{3-6}  & 7 & 250 & 0.274253 & 0.026689 & 1.047383 \\ \cline{1-1} \cline{3-6}  & 8 & 250 & 0.293970 & 0.026947 & 1.084380 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal \(k_{k}{}^{t}\), \(\zeta_{k}{}^{t}\), \(\beta_{k}\) values obtained for the J2 objective function.