## 3 Shaking Table Test
### Conversion Process of Each Method
#### 3.3.1 Method 1
The responses of the displacement \((U_{d})\), velocity \((U_{v})\), and acceleration \((U_{a})\) in the frequency domain were obtained using Equations (24) to (26) by multiplying the complex frequency response function and the load by the frequency in the frequency domain. \[U_{d}\big{(}\omega_{j}\big{)}=H\big{(}\omega_{j}\big{)}\mathrm{F}\big{(} \omega_{j}\big{)} \tag{24}\]  
\[U_{v}\big{(}\omega_{j}\big{)}=i\omega_{j}U_{d}\big{(}\omega_{j}\big{)} \tag{25}\]  
Figure 14: Analysis process for the 10 th-story acceleration in the Kobe earthquake using Method 1: (**a**) discrete Fourier Transform of applied load; (**b**) transfer function; (**c**) acceleration; (**d**) velocity; and (**e**) displacement. \[U_{a}\Big{(}\omega_{j}\Big{)}=-\omega_{j}^{2}U_{d}\Big{(}\omega_{j}\Big{)} \tag{26}\]  
The displacement \((U_{d})\), velocity \((U_{v})\), and acceleration \((U_{a})\) responses in the frequency domain were obtained using the discrete Fourier inverse transform of the response \((u)\), velocity \(\Big{(}\dot{u}\Big{)}\), and acceleration \(\Big{(}\ddot{u}\Big{)}\) in the time domain. The displacement \((u)\) in the time domain was obtained using the discrete Fourier inverse transformation of Equation (27), the displacement \((U_{d})\) in the frequency domain was calculated using Equation (24). The displacement \((u)\) in the time domain of the Manjil earthquake is shown in Figure 12e, that of the Superstition Hills earthquake in Figure 13e, and that of the Kobe earthquake in Figure 14e. Figure 12:0  
The velocity \(\Big{(}\dot{u}\Big{)}\) in the time domain is determined by the discrete Fourier inverse transformation, as in Equation (28), and the velocity \((U_{v})\) by the frequency domain estimated using Equation (25). The displacement \((u)\) in the time domain of the Manjil earthquake is shown in Figure 12d, that of the Superstition Hills earthquake in Figure 13d, and that of the Kobe earthquake in Figure 14d. Figure 12:1  
The acceleration \(\Big{(}\ddot{u}\Big{)}\) in the time domain was obtained using the discrete Fourier inverse transformation in Equation (29) with respect to acceleration \((U_{a})\), and in the frequency domain this was calculated as per Equation (26). The acceleration \(\Big{(}\ddot{u}\Big{)}\) in the time domain of the Manjil earthquake is shown in Figure 12c, that of the Superstition Hills earthquake in Figure 13c, and that of the Kobe earthquake in Figure 14c. Figure 12:2  
Here, as mentioned in the discrete Fourier transform of the load, before the discrete Fourier inverse transformation, the response in the frequency domain needs to be made a symmetrical complex conjugate with respect to \(f_{max}\) on the right and left sides [33]. The damping ratios of high-rise buildings are related to their shape, dimensions, room size, etc., and these were determined to be 3.5-21.7%, based on room size [38]. The average value, 10%, was selected for the Method 1; however, the results of Method 1 were not much affected by the damping ratios. In addition, the baseline drifts were not corrected in Method 1; therefore, the initial value of the displacement graphs did not start at zero point, meaning that zero padding is required.