# Adaptive Harmony Search for Tuned Liquid Damper Optimization under Seismic Excitation
## 2 Materials and Methods  
### Design of TLDs and Equations of Motion  
In practice, many TLDs have been included in structures. The British Airways i360 tower is shown as an example (Figure 1). Other examples include the Nagasaki Airport Tower, Tokyo Sofitel Hotel, Yokohoma Marine Tower, Shin Yokohoma Prince Hotel, Sakiatama Bridge, Narita Airport Control Tower, and Emley Moor Tower. While designing tuned liquid dampers, the most important design variables are the geometrical characteristics of the tank, its mass, the damper period, and the damping ratio. The effect of tank geometry on sloshing has been demonstrated by various studies, and a design ratio \(\left(\frac{h}{2R}>0.15\right)\) between tank height and effective liquid length has been proposed [8]. The recommended 5% mass ratio for mass dampers is also used as a suitable ratio for liquid dampers. The selection of the damper period is another factor that should be considered in terms of the performance of the TLD. Since the tank period expresses the circulation time of the liquid in the system, it directly affects the sloshing, and thus, the damping. These parameters, which are interconnected, form the basic motion equations of the system. In this study, a regulated liquid damper with a cylindrical tank was used. In Figure 2a, a 3D scheme of the building plus the TLD is shown. This scheme is also provided as a 2D figure, in Figure 2b, that shows the dimensions and parameters of the TLD. Finally, an idealized figure representing the single-story structure model plus the TLD is provided in Figure 2c. Figure 1: The British Airways i360 tower. Figure 2: Structure plus TLD models: (**a**) 3D; (**b**) 2D; (**c**) Idealised model. The information used in the creation of the equations of motion of the cylindrical TLD device is provided below. The TLD mass ratio (\(\mu\)) is shown in Equation (1). Here, \(m\) denotes the mass of the structure and \(m_{TLD}\) denotes the mass of the damper. \[\mu=\frac{m_{TLD}}{m} \tag{1}\]  
It is known that, in the basic liquid sloshing principle, liquids are not completely shaken, and some liquids remain passive and are not included in the sloshing. The passive liquid that remains dormant in the TLD and the active liquid that is sloshing should be considered separately. Together with the passive liquid TLD tank, this acts as a two-degree-of-freedom system independent of the active liquid tank. Equations (2) and (3) show the calculation of the mass of the sloshing liquid and passive liquid plus the empty tank. Here, \(m_{s}\) is the sloshing liquid, \(m_{st}\) is the total liquid, \(m_{TLD}\) is the TLD mass, and \(m_{d}\) is the mass of the empty tank plus the mass of the passive liquid. The radius and height of the cylindrical tank are indicated by \(h\) and \(R\). \[m_{s}=m_{st}\times R\times\frac{\tanh\left(\frac{1.84h}{R}\right)}{2.2h} \tag{2}\]  
\[m_{d}=m_{m_{TLD}}-m_{s} \tag{3}\]  
The Bessel function is a function used in solving problems related to the wave propagation of cylindrical bodies. The roots of this function form the damping rate parameter (\(\xi\),mn), which specifies the indices of vibration modes in the tangential (m) and radial (n) directions [36]. Damping rate parameter was found to be 1.84 in the first vibration mode in the tangential and radial directions, which accounted for the fundamental mode of the liquid [36; 37]. With the help of this value, the natural frequency of the sloshing liquid can be calculated, as well as the lateral force acting on the walls of the tank, calculated according to the linearized hydrodynamic theory of sloshing [36; 38]. In this way, the sloshing liquid mass (Equation (2)) and sloshing liquid stiffness are derived. Equations (4) and (5) show the TLD stiffness (\(k_{d}\)) and sloshing fluid stiffness (\(k_{s}\)) calculations, while Equations (6) and (7) show the TLD damping coefficient (\(c_{d}\)) and sloshing fluid damping coefficient (\(c_{s}\)) calculations. The damper period is expressed as \(T_{d}\).