## 2 Optimal Design of the TID
### Optimization Problem  
The ground motion acceleration \(\ddot{x}_{g}\) are usually modeled as stationary random processes, making the results independent of the specifics of the ground motion. \(\sigma^{2}\) is expressed as the mean square displacement response of the base-isolated structure under the excitation of random loads. \[\sigma^{2}=\int_{-\infty}^{\infty}\;\rvert\;H_{1}(\omega)\;\rvert^{\;2}S( \omega)d\omega \tag{7}\]  
Figure 1: Base-isolated SDOF structure equipped with a TID. \(S(\omega)\) represents the spectral power density of the ground motion, which characterizes the distribution of power across different frequencies in the ground motion signal. Let ground motion acceleration \(\dot{x}_{g}\) be represented by a stationary white-noise with constant power spectral density \(S_{0}\). This greatly simplifies the analysis of structural dynamic response and becomes a common model for simulating ground motions. Then Equation (7) can be expressed as Equation (8):  
\[\sigma^{2}=\int_{-\infty}^{\infty}|H_{1}(\omega)|^{2}S_{0}d\omega \tag{8}\]  
Substituting the frequency transfer function from Equations (5)-(8), and solving the integrals of form, the \(\sigma^{2}\) of the structure is expressed by  
\[ \begin{array}{l}\sigma^{2}=\frac{\pi S_{0}}{\omega_{b}^{2}}\left[\frac{A_{0} A_{1}\left(-B_{2}^{2}\right)-A_{0}A_{3}\left(B_{1}^{2}-2B_{0}B_{2}\right)+B_{0}^{2 }\left(A_{1}-A_{2}A_{3}\right)}{A_{0}\left(A_{0}A_{3}^{2}+A_{1}^{2}-A_{1}A_{2} A_{3}\right)}\right]\\ A_{0}=f^{2}\\ A_{1}=2f(\xi_{d}+f\xi_{b})\\ A_{2}=1+4f\xi_{b}\xi_{d}+(1+\mu)f^{2}\\ A_{3}=2(\xi_{b}+(1+\mu)f\xi_{d})\end{array}  \tag{9}\]  
It is seen that \(\sigma^{2}\) is a function of \(\omega_{b}\), \(\xi_{b}\), \(\mu\), \(f\), and \(\xi_{d}\). Specifically, \(\omega_{b}\) and \(\xi_{b}\) are modal parameters of the base-isolated structure. Hence, \(\sigma^{2}\) is mainly determined by the TID parameters \(\mu\), \(f\), and \(\xi_{d}\). For convenience, \(\mu\) is usually selected first. Subsequently, \(f\) and \(\xi_{d}\) correspond to various values of \(\mu\) are optimized to minimize \(\sigma^{2}\) for the specified structure. The expression of objective function is as shown in Equation (10):  
\[\left\{ \begin{array}{l}\min_{f,\,\xi_{d}}\sigma^{2}(f,\,\xi_{d})\\ s.t.\,f\geq 0,\,\xi_{d}\geq 0\end{array} \right. \tag{10}\]  
The energy distribution of the white-noise process in the frequency domain is uniform, which cannot reflect the differences caused by the site, distance, etc., in the actual ground motion. Therefore, using the Clough-Penzien spectrum is more suitable for structural seismic performance assessment. The power spectral density function of the Clough-Penzien spectrum is defined by Equation (11):  
\[S(\omega)=\frac{\omega_{g}^{4}+4\xi_{g}^{2}\omega_{g}^{2}\omega^{2}}{\left( \omega_{g}^{2}-\omega^{2}\right)^{2}+4\xi_{g}^{2}\omega_{g}^{2}\omega^{2}} \frac{\omega^{4}}{\left(\omega_{f}^{2}-\omega^{2}\right)^{2}+4\xi_{f}^{2} \omega_{f}^{2}\omega^{2}}S_{w} \tag{11}\]  
in which, \(\omega_{g}\), \(\xi_{g}\), \(\omega_{f}\), and \(\xi_{f}\) are the key parameters of the power spectrum model, which have a strong correlation with the site characteristics. \(S_{W}\) is the spectral density, which is generally believed to have a strong correlation with the amplitude of base acceleration excitation. Ground motion varies significantly with fault distance. To fully account for the impact of seismic loads, it is recommended to select different categories of ground motions for separate analysis. The PEER database provides a method for selecting ground motions at different fault distances. For routine analysis, PEMA P695 provides 20 far-field seismic motions, 12 near-fault non-pulse seismic motions, and 10 near-fault pulse seismic motions [55]. The details of the selected ground motion records are presented in Tables 1-3. Figure 2 shows the pseudo-acceleration response spectra, power spectral density and estimated parameters of the ground motions.