## 4 Numerical Example 2  
### Input Data  
The next analysis is an example of the optimization of MTMD mounted on a complex, reinforced concrete structure in the form of a tall building. The analysis included Gray Office \"A\" located in the city of Lublin in Poland. The building is set on a foundation slab with two underground and 13 aboveground stories. The heights of the underground stories are 3.5 m and the overground stories are 3.9 m. The total height of the building is 56.55 m. Figure 17a shows the architectural visualization of the building, while Figure 17b showsthe structural plans of floors 3 and 13. The main load-bearing system is the reinforced concrete core of the building (modified for the present analysis by adding walls in the E-H axes), with wall thicknesses from 20 cm to 65 cm. The remaining load-bearing structure is a reinforced concrete slab ceiling (ceilings thickness from 25 cm to 45 cm), based on reinforced concrete pillars of various dimensions from 60-100 cm. Figure 18 shows the FEM model of the system. The used beam and shell elements with concrete parameters C25/C30 are Young modulus \(E=32\) GPa, Poisson coefficient \(\
u=0.2\), and density \(\rho=2870\) kg/m3 (increasing the density to accommodate the flooring layers). The number of nodes in the model was 11,894, and the total number of equations in the modal analysis was 69,912. The first 11 corresponding natural vibration modes and their eigenvalues were calculated. Table 6 presents the description of particular modes of vibration below the value of \(f<5\) Hz, while Figure 19 shows selected modes' shapes, disregarding the local natural frequencies (\(f_{4}\) to \(f_{8}\)) of the cantilever beams located on the last floor of the building. The frequencies of natural vibrations were separated. The first flexural form of natural vibrations in the \(Y\) direction corresponded to \(f_{1}=0.8431\) Hz, and the next bending form of vibrations occurred at the frequency \(f_{2}=1.3476\) Hz. 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 TMDs. The analysis adopted the issue of tuning MTMD to the first natural frequency \(f_{1}=0.8431\) Hz. The damping value was assumed in the form of a logarithmic damping decrement \(\delta=0.08\). Node 11,110 was accepted was selected as the \(S\) point for reducing the structure to an SDOF equivalent system. The problem involved finding optimal parameters for MTMDs in the form of 2, 4, and 6 TMDs located on the ceiling of the top floor. There were three cases of TMDs attachment: 2 TMDs, at nodes 11,110 and 11,111; 4 TMDs, at nodes 11,243, 10,977, 11,222, and 11,004; 6 TMDs, at nodes 11,243, 11,110, 10,977, 11,222, 11,111, and 11004. The \(Y\) direction was adopted as the operating direction of all TMDs. The designations of the nodes are shown in Figure 20.  
New \(\overline{\textbf{M}}\), \(\overline{\textbf{C}}\), \(\overline{\textbf{K}}\) matrices for the equivalent system were determined based on the presented modal reduction equations. The equivalent mass determined from the formula \(\overline{m}_{ii}=\widetilde{m}_{i}/{a_{ji}}^{2}=1/{a_{ji}}^{2}\) (where \(j\) is the system point of reduction, namely, node 11,110, whereas \(i=1\) corresponds to the first natural vibration mode) amounted to \(\overline{m}_{11110,1}=\widetilde{m}_{1}/{a_{11110,1}^{2}}=6.7449\times 10^{6}\) kg, equivalent stiffness was \(\overline{k}_{11110,1}=\widetilde{k}_{1}/{a_{11110,1}^{2}}=\omega_{1}{}^{2}/{a_ {11110,1}^{2}}=1.8909\times 10^{5}\) kN/m, and equivalent damping was \(\overline{c}_{11110,1}=\widetilde{c}_{1}/{a_{1110,1}^{2}}=2_{\zeta}^{2}\omega_ {1}/{a_{11110,1}^{2}}=9.09\times 10^{2}\) kg/s. Numerical optimization was conducted for all cases. For each case, the total MTMD stiffness was equal to the stiffness of the 1 TMD, which was adopted as equal to \({k_{1}}^{t}=120\) kN/m. For all other cases, TMDs' stiffness was determined