## 2 TMD Optimization Framework
\begin{table}
\begin{tabular}{c c c c c c} \hline  & & \begin{tabular}{c} **Frequency** \\ **Hz** \\ **Dynamic Test** \\ \end{tabular} & \begin{tabular}{c} **Damping** \\ **\%** \\ **D\({}_{\text{F}}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Mass** \\ **\%** \\ **Dynamic Test** \\ \end{tabular} &
\begin{tabular}{c} **Mass** \\ **\%** \\ **Trans.** \\ \end{tabular} \\ \hline
1 & 0.863 & 0.861 & 0.23\% & 4.1 & 81.44 \\
2 & 2.900 & 2.840 & 2.06\% & 1.6 & 14.66 \\ \hline \end{tabular}
\end{table} 
Table 1: Fundamental frequencies of the chimney, with the corresponding participating masses. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Life of the Structure V\({}_{\text{N}}\)** & **Year** & **50** \\ \hline Use category & - & II \\ Coefficient for use category C\({}_{\text{u}}\) & - & 1 \\ Reference life V\({}_{\text{R}}\) & year & 50 \\ Probability of exceedance P\({}_{\text{VR}}\) & \% & 10 \\ Topographic coefficient S\({}_{\text{T}}\) & - & 1 \\ Soil category & - & B \\ PGA & g & 0.094 \\ \hline \end{tabular}
\end{table} 
Table 2: Response spectrum parameters. Figure 7: Finite element model (FEM) of the chimney. \[c_{\rm TMD}=2\cdot\xi_{\rm opt}\cdot(k_{\rm TMD}\cdot m_{\rm TMD})^{0.5}, \tag{3b}\]  
where \(\alpha_{\rm opt}\) is the optimal coefficient of frequencies, defined as (Equation (4)):  
\[\alpha_{\rm opt}=[((1-0.5\mu)^{0.5}/(1+\mu))+(1-2\xi^{2})^{0.5}-1]-[2.375-1.034 \mu^{0.5}-0.426\mu]\cdot\xi\cdot\mu- \tag{4}\]  
\(\mu\) represents the ratio between the TMD mass (m\({}_{\rm TMD}\)) and the participating mass of the structure involved in the first vibration mode (m\({}_{\rm stru,\,I\,mode}\)), as shown in Equation (5):  
\[\mu=m_{\rm TMD}/m_{\rm stru,\,I\,mode}, \tag{5}\]  
\(\xi\) is the damping ratio of the structure considered, in this case, equal to 0.03. The optimal equivalent viscous damping ratio (\(\xi_{\rm opt}\)) is calculated considering Equation (6). \[\xi_{\rm opt}=[(3\mu)^{0.5}/(8\cdot(1+\mu)\cdot(1-0.5\mu))]+(0.151\xi-0.175\xi ^{2})+(0.163\xi+4.98\xi^{2})\cdot\mu, \tag{6}\]  
and \(\omega_{\rm s}\) is defined as the pulsation depending on the frequency of the considered vibration mode (in this case, the frequency of the first mode, f\({}_{\rm I}\)), defined through Equation (7):  
\[\omega_{\rm s}=2\pi\cdot f_{\rm I}, \tag{7}\]  
Changing the \(\mu\) ratio in the range 1-5%, as suggested in [13], a preliminary definition of the TMD parameters is carried out for two different TMD mass positions: (i) at the top of the chimney (40 m height) and (ii) at the top quote of the part of the double skin (about 26.50 m). The optimal \(\mu\) value (depending upon the values of k\({}_{\rm TMD}\) and c\({}_{\rm TMD}\)) significantly reduces the base shear, base moment, and top displacement of the chimney with an acceptable value of the TMD mass [41; 42; 43]. Table 3 shows the main TMD parameters considered in this work for the two different positions. It is important to notice that the two proposed TMD configurations are completely independent and designed to work around the first fundamental frequency that characterizes the chimney's dynamic behavior. The preferable solution, from a construction point of view, is the one where the TMD is installed at quote 26.50 m, using a connection system with the steel structure (Figure 3b). The main advantage of the 26.50 m height positioning of the TMD is the possibility to avoid reinforcement of the upper part of the chimney, where only one skin is present. Figures 8-10 show the optimization curves, expressed in terms of an average value of base shear, bending moment, and top displacement as a function of \(\mu\), obtained by integration of the dynamic equation of the structure under the set of seven spectrum-compatible accelerograms [13; 33] considering the TMD positioned at 26.50 m, while Figures 11-13 show the results of the optimization process for the case of the T