## 2 TMD Optimization Framework  
The main steps of the optimization procedure of the TMD design are summarized in Figure 6. The proposed procedure is based on two main steps. In the first step, the dynamic behavior of the analyzed structure is obtained through the implementation of a simplified  
Figure 5: The historical masonry chimney under study. finite element model to calculate the fundamental characteristics of the TMD after an optimization process performed by the execution of linear time-history analyses. Defined the TMD properties, expressed in terms of mass, equivalent stiffness, and equivalent damping; in the second step, the operating limit of the optimized TMD is evaluated considering a fiber model where the nonlinear behavior of the masonry is considered. Knowing the geometry and the mechanical characteristics of the masonry, a first linear elastic finite element model (FEM_1--CASE A) of the chimney was implemented through Midas GEN software [35], where the structure is schematized only with tapered beam elements (Figure 7) in the typical cantilever configuration with a fully constrained boundary condition at the base. An eigenvalue analysis was executed by updating the stiffness value of the different sections that characterized the chimney under study to reproduce the experimental results obtained from the dynamic test [36] performed by a specialized Italian company in 2015 (Table 1). Four three-axis MEMS accelerometers have (www.microseismic.com, accessed in 5 April 2021) were located along the tower in two different configurations: Config. 1, a distance of 30 m, 38 m, 44 m and 50 m from the base level; Config. 2, a distance of 10.5 m, 20 m, 30 m and 38 m from the base. The reference point of all the measures was the accelerometer located at 38 m of height. The sampling frequency was 128 Hz. After the tests, the well-known frequency domain decomposition (FDD) technique was applied to estimate the fundamental mode of vibration and the associated damping, as shown in Table 1.  
For a better comparison between experimental and numerical results, in Table 1, the \(\mathrm{D_{F}}\) index was reported, which is defined as:  
\[\mathrm{D_{F}}=\left\lvert f_{\mathrm{FEM}}-f_{\mathrm{exp}}\right\rangle/f_{ \mathrm{exp}}\mid\cdot 100 \tag{2}\]  
where \(f_{\mathrm{FEM}}\) is the frequency value obtained from the finite element model, and \(f_{\mathrm{exp}}\) is the one experimentally observed. The lower is \(\mathrm{D_{F}}\), and the better is the agreement between the compared frequencies. Figure 6: Flow diagram of the proposed optimization procedure. The results show that the frequencies, which characterize the first and the second mode, obtained from the FEM, are practically coincident with those evaluated after the dynamic tests performed on the historical masonry chimney. It can be noted that considering the total mass of the chimney equal to m\({}_{\text{T}}\) = 276,465.40 kg, the modal mass involved in mode 1 is m\({}_{\text{stru, 1 mode}}\) = 225,153.29 kg (= 81.4%) [37]. The seismic action is described using a set of seven spectrum-compatible accelerograms [38]. The accelerograms are compatible with the site elastic spectrum prescribed by the code [33] and characterized by the parameters listed in Table 2.  
After evaluating the dynamic behavior of the chimney (FEM_1--CASE A), the TMD is introduced in the FEM (FEM_1--CASE B). The TMD is schematized by a nodal mass linked to the chimney by a linear spring-damper element characterized by an equivalent horizontal stiffness (k\({}_{\text{TMD}}\)) and related equivalent damping (c\({}_{\text{TMD}}\)) calculate according to Equation (3a,b) as a function of the mass of the structure involved in the first vibration mode as reported in [39; 40]:  
\[\text{k}_{\text{TMD}}=\text{m}_{\text{TMD}}\cdot\alpha_{\text{opt}}^{2}\cdot \omega_{\text{s}}^{2}, \tag{3a}\]