## 3 Numerical Example 1
### Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \begin{tabular}{c} **Tuning** \\ \(\beta_{opt}\) \\ \end{tabular} & \begin{tabular}{c} **Damping** \\ \(\zeta_{opt}\) \\ \end{tabular} & \begin{tabular}{c} \(J1\) **Value** \\ \(\zeta_{1}=0\) \\ \end{tabular} & \begin{tabular}{c} \(J2\) **Value** \\ \(\zeta_{1}=0\) \\ \end{tabular} & \
\begin{tabular}{c} \(J1\) **Value** \\ \(\zeta_{1}=0.02\) \\ \end{tabular} &
\begin{tabular}{c} \(J2\) **Value** \\ \(\zeta_{1}=0.02\) \\ \end{tabular} \\ \hline \multirow{3}{*}{Den Hartog} & \(\frac{1}{\Gamma+\varphi}\) & \(\sqrt{\frac{3\mu}{8(1+\mu)}}\) & \(1.886\times 10^{-2}\) & \(1.005\times 10^{-1}\) & \(1.559\times 10^{-2}\) & \(7.676\times 10^{-2}\) \\  & \(0.9803922\) & \(0.0870388\) & & & & \\ \hline Warburton & \(\frac{1}{1+\varphi}\sqrt{\frac{2+\mu}{2}}\) & \(\sqrt{\frac{\mu(4+3\mu)}{8(1+\mu)(2+\mu)}}\) & \(1.866\times 10^{-2}\) & \(1.091\times 10^{-1}\) & \(1.550\times 10^{-2}\) & \(8.176\times 10^{-2}\) \\  & \(0.9852819\) & \(0.0701871\) & & & & \\ \hline Warburton & \(\frac{1}{1+\varphi}\sqrt{\frac{2-\mu}{2}}\) & \(\sqrt{\frac{3\mu}{8(1+\mu)(1-\mu/2)}}\) & \(1.891\times 10^{-2}\) & \(1.057\times 10^{-1}\) & \(1.560\times 10^{-2}\) & \(7.497\times 10^{-2}\) \\  & \(0.9754779\) & \(0.0861813\) & & & & \\ \hline Ren & \(\sqrt{\frac{1}{1-\mu}}\) & \(\sqrt{\frac{3\mu}{8(1+\mu/2)}}\) & \(1.941\times 10^{-2}\) & \(1.366\times 10^{-1}\) & - & - \\  & \(1.0101525\) & \(0.0870388\) & & & & \\ \hline Nishihara and & \(\frac{1}{1+\varphi}\Big{(}1-\zeta_{1}\sqrt{\frac{\mu}{1+\varphi-\zeta_{1}^{2}}} \Big{)}\) & \(\frac{\zeta_{1}+\sqrt{\mu}\Big{(}1+\varphi-\zeta_{1}^{2}\Big{)}}{1+\varphi}\) & - & - & \(1.677\times 10^{-2}\) & \(9.865\times 10^{-2}\) \\ [PERSON] & \(0.9776460\) & \(0.1596084\) & & & & \\ \hline Proposed \(J1\) & - & - & \(1.885\times 10^{-2}\) & \(1.148\times 10^{-2}\) & - & - \\ \(\zeta_{1}=0\) & \(0.9890635\) & \(0.0682140\) & \(1.868\times 10^{-2}\) & \(1.04\times 10^{-2}\) & - & - \\ \hline Proposed \(J2\) & - & - & \(1.868\times 10^{-2}\) & \(1.006\times 10^{-2}\) & - & - \\ \(\zeta_{1}=0\) & \(0.9803647\) & \(0.0853719\) & & & & \\ \hline Proposed \(J1\) & - & - & - & \(1.563\times 10^{-2}\) & \(8.521\times 10^{-2}\) \\ \(\zeta_{1}=0.02\) & \(0.9901779\) & \(0.0684623\) & - & & & \\ \hline Proposed \(J2\) & - & - & - & \(1.552\times 10^{-2}\) & \(7.458\times 10^{-2}\) \\ \(\zeta_{1}=0.02\) & \(0.9761710\) & \(0.0891687\) & - & - & \(1.552\times 10^{-2}\) & \(7.458\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimal parameters and \(J1\) and \(J2\) objective function values for \(1\)TMD, based on various authors (values for \(\mu=0.02\)). In [20], [PERSON] and [PERSON] conducted a free analysis of the \(\beta_{k}\), \(\zeta_{k}{}^{t}\) parameters, with a constant value of \(m_{k}{}^{t}\), for 10 and 100 TMDs, with not damping of the primary system \(\zeta_{1}=0\) and a \(\mu=0.05\) mass ratio. They applied optimization based on the \(9t_{2}\) norm and used state space equations for an SDOF system with attached MTMDs. The work also presents tuning increment graphs for individual TMDs in the form of \(\beta_{k}-\beta_{k-1}\). The authors of that article also conducted such simulations for \(n=10\) (number of TMD), \(\zeta_{1}=0\), and \(\mu=0.05\). Figure 10 shows tuning \(\beta_{k}\) and damping ratio \(\zeta_{k}\) distribution based on the number of TMDs, while Figure 11 shows tuning increments and tuning factor for individual TMDs in the form of \(\beta_{k}-\beta_{k-1}\) and \(\zeta_{k}-\zeta_{k-1}\). The obtained \(\beta_{k}\), \(\zeta_{k}{}^{t}\) and \(\beta_{k}-\beta_{k-1}\) parameters exhibited very good conformity with the results of [PERSON] and [PERSON] [20]. Figure 11: Optimal tuning spacings \(\beta_{k}-\beta_{k-1}\) and damping ratios spacings \(\zeta_{k}-\zeta_{k-1}\) of the individual TMD for \(\mu=0.05\) and \(\zeta_{1}=0\) obtained on the basis of \(J1\).