*4.1. Tuned Mass Damper (TMD)*

The tuned mass damper (TMD) can be idealized as an SDOF system with mass, stiffness, and damping coefficients. To design a TMD, the mass ratio *γM* is the first parameter that should be determined:

$$
\gamma\_M = \frac{M\_{TMD}}{M\_s} \tag{18}
$$

where *MTMD* and *Ms* are the mass of the TMD and the primary structure, respectively. According to Section 3, the total mass of the primary structure is 8 × 10<sup>5</sup> kg. Four TMDs with various auxiliary mass ratios, i.e., 1%, 2%, 3%, and 4%, are adopted. The corresponding mass of the TMD can easily be calculated using Equation (18).

According to the classical optimizing parameters for TMD proposed by Den Hartog [32], the optimized frequency ratio *γF* and the damping ratio *ξTMD* can be expressed as the function of *γM*:

$$\gamma\_{\mathcal{F}} = \frac{f\_{TMD}}{f\_{\mathcal{s}}} = \frac{1}{1 + \gamma\_M} \tag{19}$$

$$\xi\_{TMD} = \sqrt{\frac{3\gamma\_M}{8(1+\gamma\_M)}}\tag{20}$$

where *fTMD* and *fs* are the frequency of the TMD and the primary structure, respectively. The first-order frequency of the primary structure (*f*1 = 0.98 Hz) can be assigned to *fs*. Other parameters of the TMD, such as stiffness *KTMD* = *MTMD*(<sup>2</sup>*π fTMD*)<sup>2</sup> and damping coefficient *CTMD* = <sup>2</sup>*MTMDξTMD*(<sup>2</sup>*<sup>π</sup> fTMD*), can be determined and are listed in Table 3.


**Table 3.** Parameters of dampers with different mass ratios (mass unit: kg; stiffness unit: N/m; damping coefficient unit: N.s/m).
