## 3 Description of the Retrofitted Building Models and Ground Motion Datasets
### Ground Motion Data
#### 3.4.1 Artificial Ground Motions  
In this study, two series of artificial ground motion datasets (the Art-1 and Art-2 series) were generated. As noted in Section 3.2, the equivalent velocity of the total input energy (\(V_{I}\)) was set to 1.41 m/s to determine the properties of the isolation layer. Therefore, the artificial ground motion datasets were generated such that their bidirectional \(V_{I}\) spectrum fit the predetermined target \(V_{I}\) spectrum as follows. The target bidirectional \(V_{I}\) spectrum with a damping ratio of 0.10 was determined such that:  
\[V_{I}(T)=\left\{ \begin{array}{ll}1.41(T/0.75)\ \mathrm{m/s}&:T\leq 0.75\ \mathrm{s}\\ 1.41&:T>0.75\ \mathrm{s}\end{array} \right.. \tag{20}\]  
To generate each horizontal component, it was assumed that the intensity of \(V_{I}\) in both orthogonal components (\(\xi\) and \(\zeta\) components) was identical. The target equivalent velocities of the total input energy of the \(\xi\) and \(\zeta\) components, \(V_{I\xi}(T)\) and \(V_{I\xi}(T)\), respectively, were calculated as:  
\[V_{I\xi}(T)=V_{I\xi}(T)=V_{I}(T)/\sqrt{2}. \tag{21}\]  
The phase angle is given by a uniform random value, and to consider the time-dependent amplitude of the ground motions, a Jenning-type envelope function (\(e(t)\)) was assumed. In this study, two envelope functions were considered. In Art-1-00, the envelope function was set as in Equation (22), while in Art-2-00, the envelope function was set as in Equation (23). \[e(t)=\left\{ \begin{array}{ll}(t/5)^{2}&:0\ s\leq t\leq 5\ s\\ 1&:5\ s\leq t\leq 25\ s\\ \exp\{-0.066(t-25)\}&:25\ s\leq t\leq 60\ s\end{array} \right. \tag{22}\]  
\[e(t)=\left\{ \begin{array}{ll}(t/2.5)^{2}&:0\ s\leq t\leq 2.5\ s\\ 1&:2.5\ s\leq t\leq 12.5\ s\\ \exp\{-0.132(t-12.5)\}&:12.5\ s\leq t\leq 30\ s\end{array} \right. \tag{23}\]  
Figures 13 and 14 show the time-histories of the two components and the orbits of Art-1-00 and Art-2-00, respectively. Because the same target \(V_{I}\) spectrum was applied while the duration of Art-2-00 is the half that of Art-1-00, the peak ground acceleration of Art-2-00 was larger than that of Art-1-00. Note that the artificial ground motions used in this study were generated independently. The correlation coefficients of Art-1-00 and Art-2-00 were close to zero, even though the envelope functions of the two components were the same; therefore, the two components can be considered independently of one another. Next, 11 artificial ground motion datasets were generated from Art-1-00 and Art-2-00 by shifting the phasing angle. The generated artificial ground motion vector \((\mathbf{a_{g}}(t,\Delta\phi_{0}))\) is expressed as:  
\[\mathbf{a_{g}}(t,\Delta\phi_{0})=\left\{ \begin{array}{c}a_{g\xi}(t,\Delta \phi_{0})\\ a_{g\xi}(t,\Delta\phi_{0})\end{array} \right\}=\sum\limits_{n=-N}^{N}\left\{ \begin{array}{c}c_{\xi,n}\\ c_{\xi,n}\end{array} \right\}\exp[i\{\omega_{n}t-\text{sgn}(\omega_{n})\Delta \phi_{0}\}]. \tag{24}\]  
In Equation (24), \(c_{\xi,n}\) and \(c_{\xi,n}\) are the complex Fourier coefficients of the \(n\)th harmonics of \(a_{g\xi}(t)\) and \(a_{g\xi}(t)\), respectively, \(\omega_{n}\) is the circular frequency of the \(n\)th harmonic, and \(\Delta\phi_{0}\) is the constant used to shift the phase angle of all the harmonics. As in previous studies [51,53], the constant \(\Delta\phi_{0}\) was set from \(\pi/12\) to \(11\pi/12\) with an interval of \(\pi/12\). Notably, the phase difference of each ground motion component did not change when shifting the phase angle. In addition, the total input energy spectrum (the \(V_{I}\) spectrum) was independent of the shifting phase angle (\(\Delta\phi_{0}\)). This is because the shifting phase angle does not affect the absolute value of the complex Fourier coefficient, and the total input energy per unit mass can be calculated from the Fourier amplitude spectrum without the phase characteristics of the ground motion, as shown by [PERSON] [61].