*4.2. Tuned Liquid Damper (TLD)*

There are several existing analytical or numerical models for TLD. According to [33], the TLD model developed by Yu et al. [34] has good predictions in both weak and strong wave breaking and in a broad range of frequency ratios. Therefore, Yu et al.'s model is adopted in this study.

Yu et al.'s model is an equivalent nonlinear-stiffness–damping (NSD) model of the TLD. The structural parameters of the TLD model are summarized as follows.


$$f\_w = \frac{\sqrt{\frac{\pi g}{L} \tanh\left(\frac{\pi h}{L}\right)}}{2\pi} \tag{21}$$

where *g* is the gravitational constant. According to Yu et al. [34], the nonlinear stiffness *KTLD* of TLD can be determined by the stiffness hardening ratio of *κ* and *Kw*:

$$\kappa = \frac{K\_{TLD}}{K\_w} = \left(\frac{f\_{TLD}}{f\_w}\right)^2 = \begin{cases} 1.075 \Lambda^{0.007}, \text{for } \Lambda \le 0.03 \text{ weak wave breaking} \\ 2.52 \Lambda^{0.25}, \text{for } \Lambda > 0.03 \text{ strong wave breaking} \end{cases} \tag{22}$$

$$K\_{TLD} = M\_{TLD} \kappa \frac{\pi g}{L} \tanh\left(\frac{\pi h}{L}\right) \tag{23}$$

where Λ = *A*/*L* is the non-dimensional displacement amplitude, and *A* is the displacement amplitude of excitation.

3. Damping: The damping coefficient of the TLD can be obtained as *CTLD* = <sup>2</sup>*MTLDξTLD*(<sup>2</sup>*<sup>π</sup> fTLD*), where the damping ratio *ξTLD* is also a function of Λ:

$$
\xi\_{TLD} = 0.5 \Lambda^{0.35} \tag{24}
$$

$$C\_{TLD} = 2M\_{TLD} \xi\_{TLD} \sqrt{\kappa \frac{\pi \mathcal{g}}{L} \tan \mathbf{h} \left(\frac{\pi h}{L}\right)}\tag{25}$$

To design a TLD, the first parameter is also the mass ratio *γM*, which is the ratio of *MTLD* to *Ms*. In the same manner as for the TMDs, four TLDs with various auxiliary mass ratios, i.e., 1%, 2%, 3%, and 4%, are adopted. The tank length *L* is assigned as a constant: 1 m. To achieve the maximum effectiveness of the TLD, the nonlinear natural frequency *fTLD* of the TLD should be equal to the natural frequency *fs* of the primary structure. Then, the water depth *h* is determined as:

$$h = \frac{L}{\pi} \tan \mathbf{h}^{-1} \left(\frac{4\pi L f\_s^2}{g\kappa}\right) \tag{26}$$

Based on the numerical results presented in Section 3, the displacement amplitude *A* of the fourth story is 0.1149 m; the non-dimensional displacement amplitude Λ and the corresponding stiffness hardening ratio *κ* can be calculated. By substituting the values of *L*, *κ*, and *fs* into Equation (26), the water depth *h* is determined to be 0.3821 m. Therefore, the mass for a certain TLD is a constant, whereas the stiffness and damping are displacement amplitude-related variables and should be updated during time-history analysis. The parameters of the four TLDs with different mass ratios are also listed in Table 3.
