## 3 Numerical Example 1  
### Input Data  
The analysis adopted a cantilever structure in the form of a chimney made of S235 steel, with a height of \(h\)=160 m. The chimney parameters were selected so as to obtain the first circular natural frequency equal to \(\omega_{1}=1\,\mathrm{rad}/\mathrm{s}\). The chimney diameter and its wall thickness were adopted as a constant, along its entire height. These values equal to, respectively, \(D=4010\) mm, \(g=25.5\) mm. The cantilever structure was divided into 40 beam elements with the following parameters: Young modulus \(E=210\) GPa, Poisson coefficient \(\
u=0.3\), and density \(\rho=7850\,\mathrm{kg}/\mathrm{m}^{3}\). First three natural vibration modes and their corresponding eigenvalues were calculated. FE software was used to import a normalized eigenvector for the first mode of the natural vibrations, which was then used to construct motion equations for the structure reduced to SDOF system with attached TMD.  
Figure 4 shows a FEM model with the numberings of the bars and modes, and a presentation of the first three mode shapes and normalized eigenvectors.  
The analysis adopted the issue of tuning TMD and MTMDs to the first natural frequency \(\omega_{1}=1\,\mathrm{rad}/\mathrm{s}\). For this frequency, the adopted structural damping was in the form of a damping ratio \(\zeta_{1}=0.02\). The chimney tip (node 41) was selected as the \(S\) point for reducing the structure to an SDOF equivalent system. The problem involved finding optimal parameters for a single TMD and MTMDs in the form of 2, 4, 8, and 20 TMDs located also at the place of the highest ordinate of the first vibration mode, which was node 41. New \(\overline{\mathbf{M}}\), \(\overline{\mathbf{C}}\), \(\overline{\mathbf{K}}\) matrices for the equivalent system were determined based on the presented modal reduction equations. The equivalent mass determined from the formula \(\overline{m}_{ji}=\widehat{m}_{i}/{a_{ji}}^{2}=1/{a_{ji}}^{2}\) (where \(j\) is the system point of reduction, namely, node 41, whereas \(i=1\) corresponds to the first natural vibration mode) amounted to \(\overline{m}_{41,1}=\widehat{m}_{1}/{a_{41,1}}^{2}=10^{5}\) kg, equivalent stiffness was \(\overline{k}_{41,1}=\widetilde{k}_{1}/{a_{41,1}}^{2}=100\,\mathrm{\,kN/m}\), and equivalent damping was \(\overline{c}_{41,1}=\widetilde{c}_{1}/{a_{41,1}}^{2}=2\zeta_{1}\omega_{1}/{a_ {41,1}}^{2}=4\times 10^{3}\) kg/s. Numerical optimization was conducted for all cases. For each case, the total MTMD mass was equal to the mass of the single TMD, which was adopted as equal to \({m_{1}}^{t}=2\times 10^{3}\) kg. Therefore, the mass factor was \(\mu={m_{1}}^{t}/{\overline{m}_{41,1}}=0.02\). For all other cases, TMD masses were determined from the formula \(m_{k}{}^{t}=\mu\cdot\overline{m}_{41,1}/N_{k}\). The natural circular frequency for a single TMD was determined from the formula \(\omega_{k}{}^{t}=\sqrt{k_{k}{}^{t}/m_{k}{}^{t}}\), whereas the damping factor values for individual TMDs were determined from the formula \(c_{k}{}^{t}=2\zeta_{k}{}^{t}m_{k}{}^{t}\omega_{k}\). Stiffness \(k_{k}{}^{t}\) was used to determine the dimensionless tuning ratio for a single TMD \(\beta_{k}=\omega_{k}{}^{t}/\omega_{1}\). Initial tuning adopted for all dampers within the optimization amounted to \(\beta_{k}=1\). This constituted a base to adopt the starting values \(k_{k}{}^{t}\). Table 1 shows the adopted constant masses of a single TMD and the adopted starting values of the \(k_{k}{}^{t}\) and \(\zeta_{k}{}^{t}\) MTMD parameters (values subject to optimization), depending on the TMD number.  
The calculation process involved using the aforementioned GA and SA optimization methods and the objective functions based on \(\Re_{2}\) and \(\Re_{\infty}\) norms in the form of \(J1\) and \(J2\). The frequency range when determining the objective function was (0.0 rad/s:\(\pi\) rad/s).