## 3 Numerical Example 1
### Discussion  
The analyses presented in Section 3 were used as a base to compare the application of the SA optimization method and the \(J1\) and \(J2\) objective functions with known analytical solutions in terms of a single TMD. Table 5 shows known formulas for the determination of the optimal parameters for TMD usually determined from a TDOF system, with stated values of the optimal parameters. The table also presents original results for 1 TMD, taking into account natural damping of the primary system, as well as without it. Figure 8 shows  
Figure 8: Module graph of a FRF \(\left|\overline{H}_{41,41}\right|\) with the mass of the primary system increased by 10% for optimal parameters and a different number of TMDs: (**a**) for the \(J1\) objective function; (**b**) for the \(J2\) objective function. Figure 6: Module graph of a FRF \(\left|\overline{H}_{41,41}\right|\) for optimal parameters and a different number of TMDs: (**a**) for the \(J1\) objective function; (**b**) for the \(J2\) objective function. Figure 7: Module graph of a FRF \(\left|\overline{H}_{41,41}\right|\) with the mass of the primary system reduced by 10% for optimal parameters and a different number of TMDs: (**a**) for the \(J1\) objective function; (**b**) for the \(J2\) objective function. an equivalent system FRF module graph for optimal TMD parameters. When analyzing the obtained solution (see Table 5 and Figure 9a), in the case of no damping of the primary system \(\zeta_{1}=0\) and in comparison with the [PERSON] solution [1], which has been known for years, one can state very good conformity of the optimal tuning parameters \(\beta_{1}\) and the TMD damping factor \({\zeta_{1}}^{t}\), whereas optimal TMD parameters proposed by [PERSON] [2], upon a harmonic excitation, are close to the parameters obtained through calculations of the \(J2\) objective function optimization. The solution proposed by [PERSON] [31] suits a different kind of TMD, where the damping element is not connected with the primary mass but with the substrate (other motion equations); hence, it is impossible to directly apply the optimal parameter formulas proposed in this study as parameters for a standard TDOF system. Furthermore, please note that optimal TMD parameters based on the \(\mathfrak{H}_{2}\) norm (white noise input), i.e., determined from the \(J1\) objective function, provided similar TMD efficiency, as suggested by [PERSON] in [3]. In the case of an analysis covering a structure including natural damping of the primary system, we can observe that FRF module graphs, developed based on optimal Den Hartog and Warburton parameters, are no longer consistent with the graphs based on the parameters obtained through the \(\mathfrak{H}_{2}\) and \(\mathfrak{H}_{\infty}\) methods, that is, the \(J1\) and \(J2\) objective functions. This, of course, stems from the fact that these formulas do not include natural damping \(\zeta_{1}\), whereas, when comparing to the [PERSON] solution [32], where the optimal parameters were obtained based on the stability criterion, taking into account \(\zeta_{1}\), it should be concluded that his solution was optimal neither for harmonic excitations nor for white noise input.