## 3 Numerical Example 1
### Discussion
Figure 10: Optimal tuning ratios \(\beta_{k}\) and damping ratios \(\zeta_{k}{}^{t}\) of the individual TMD for \(\mu=0.05\) and \(\zeta_{1}=0\) obtained on the basis of \(J1\). Figure 9: The module graph for the FRF \(|\overline{H}_{41,41}|\) (value corresponds to the FRF \(|H_{1,1}|\) of a two degree of freedom (TDOF) system based on various studies): **(a)** case \(\zeta_{1}=0\); **(b)** case \(\zeta_{1}=0.02\). In the course of analyzing the results shown in Section 3, which concern the optimization of MTMDs located at the tip of a \(h=160\)-m-high chimney, higher effectiveness of MTMDs relative to a single TMD can be confirmed (see Table 2 and Figure 6). This is particularly clear to optimal parameters determined with the \(J2\) objective function, where the location of the flat section of graph \(\left|\overline{H}_{41,41}\right|\) in the area of \(\omega_{1}\) can be seen increasingly lower (Figure 6b). Another issue that needs to be stressed are the graphs \(\left|\overline{H}_{41,41}\right|\) (Figure 6a) for TMD tuning in the form of white noise \(\
icefrac{{9}}{{6}}\)\({}_{2}\) optimization of the \(J1\) function exhibit a broader damper impact range on the side of higher input frequencies. Furthermore, Figure 12 shows a function variability graph depending on the number of TMDs. Of course, higher MTMD efficiency relative to TMDs can be observed for both objective functions, \(J1\) and \(J2\), applied within the numerical optimization. Furthermore, Figure 12 shows the values of individual objective functions for the optimal parameters obtained using both optimization methods, namely, \(\
icefrac{{9}}{{6}}\)\({}_{2}\) (\(J1\) function) and \(\
icefrac{{9}}{{6}}\)\({}_{\infty}\) (\(J2\) function). It can be seen (see Figure 12a) that in the case of random excitation, which corresponds to the value of the \(J1\) function and TMDs tuned to harmonic excitation (optimal parameters based on the optimization of function \(J2\)), we get a lower MTMD efficiency. Figure 12b indicates an inverse relationship. When analyzing the results concerning the primary system mass change (see Figures 7 and 8), for an approximately \(5\%\) difference in the value of \(\omega_{1}\) (\"structural mismtuning\"), MTMDs are more effective than a single TMD. A lower effectiveness with a higher number of TMDs tuned to harmonic excitations can be observed only in the case of TMDs tuned based on \(J2\) optimization and reducing the primary system mass increase in frequency \(\omega_{1}\)\({}^{-10\%}=1.05409\) rad/s. In the event of such a primary system mistuning, better effectiveness was exhibited by MTMDs tuned based on \(J1\), which resulted from the aforementioned broadband excitation on the side of frequencies higher than \(\omega_{1}\). It can also be concluded that a reduction in the primary system mass had a more adverse influence on the structural response and MTMD operation than mass increase. These relationships are shown in Figure 13. It can also be assumed that with a low difference (below \(5\%\)) between the calculated natural frequencies of the structure and the ones determined for a real structure, it is advisable to use MTMDs instead of a single TMD. The aforementioned analyses confirm the general conclusions included in the works by [PERSON] and [PERSON] [20] as well as [PERSON] and [PERSON] [21] concerning the higher effectiveness of  
Figure 12: Objective function values based on the optimal parameters obtained from \(J1\) and \(J2\), depending on the number of TMDs: (**a**) value of \(J1\); (**b**) value of \(J2\). Additionally, for analysis purposes, it was decided to present the results of optimal MTMD parameters in the form of optimal tuning ratio \(\beta_{k}\) and optimal damping ratio \(\zeta_{k}{}^{t}\) graphs depending on the number of TMDs. The tuning and damping factor increments for individual TMDs in the form of \(\beta_{k}-\beta_{k-1}\) and \(\zeta_{k}{}^{t}-\zeta_{k-1}^{t}\) were also presented.